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11 TWO-PHASE FLOW AND HEAT TRANSFER 11.1 Introduction Two-phase flow refers to the interactive flow of two distinct phases – each phase representing a mass or volume of matter – with common interfaces in a channel. Two-phase flow can occur in a single-component or multicomponent system (Table 1.10). Possible phase combinations include: (1) solid-liquid, where solid particles are mostly dispersed in the liquid; (2) solid-gas, where the solid particles are carried by a stream of gas; (3) liquid-vapor (gas), where the volume fraction of one phase relative to the other results in different flow regimes; and (4) a combination of the above. While each of these modes represents a significant area of two-phase flow, liquid-vapor (gas) flow is by far the most common in various industries and thus has been investigated in greater depth. Therefore, the major emphasis of this chapter is on liquid-vapor (gas) flow. Each regime in liquid-vapor (gas) two-phase flow has a characteristic flow behavior that can substantially affect both pressure drop and heat transfer. In the case of a single-component two-phase flow, such as forced convective condensation or evaporation, continuous mass transfer occurs between the vapor and liquid phases. Examples of liquid-gas flow processes include distillation, fractionation, flashing, spray-drying, stripping, and absorption. There are also two-component liquid-gas systems where the gas is noncondensable; these include air/water flow in aeration, deaeration, and humidification or dehumidification processes. A biomedical engineering example is the development of artificial lungs where absorption of oxygen or desorption of CO2 from blood is required. Section 11.2 presents the important parameters for liquid-vapor (gas) twophase flow, and flow patterns in vertical and horizontal tubes. The various twophase flow models, as well as the prediction of pressure drop and void fraction, are presented in Section 11.3. The two-phase flow regimes and heat transfer characteristics for forced convective condensation and boiling are presented in Sections 11.4 and 11.5, respectively. The chapter is closed by a discussion of Chapter 11 Two-Phase Flow and Heat Transfer 853 two-phase flow, condensation, and boiling heat transfer in micro/miniature channels. 11.2 Flow Patterns of Liquid-Vapor (Gas) Two-Phase Flow Among the kinds of two-phase flows summarized in the preceding section, liquid-vapor (gas) flow is the most complex because the interfaces are deformable and the vapor or gas phase is more common. Furthermore, the interfacial configurations in two-phase flow are also very complicated due to heat and mass transfer and can vary over a wide range. The interfacial distribution in the liquid-vapor (gas) flow can be classified into a number of categories known as flow patterns or flow regimes. Flow regimes and flow regime transitions for vertical and horizontal tubes will be discussed. The flow patterns that will be presented in this section apply to two-phase flow in straight tubes only and are not applicable for two-phase flow in bends or coils. In addition, the flow patterns that will be addressed in this section are for cocurrent flow only and are not valid for countercurrent flow. However, one can easily obtain useful two-phase flow information in open literature concerning various configurations in geometry and/or countercurrent flow. Before introducing the flow patterns, it is useful to establish notations and to develop some fundamental concepts and relationships. 11.2.1 Concepts and Notations Since liquid-vapor two-phase flow with phase change will be the focus in this section, one phase is designated liquid and the other as vapor. The vapor or holdup void fraction represents the time-averaged volumetric fraction of vapor in a two-phase mixture, i.e., dV Vv V α= v = (11.1) dV Vv + VA It should be noted that α = ε v . For liquid-vapor two-phase flow in a pipe, the volume of each phase consists of the cross-sectional area of the flow tube covered by that phase times a differential length element. Since the differential length is common in both phases, the void fraction can be considered the time-averaged area fraction: Δz dA Av Av = α= (11.2) Δz dA Av + AA ³ ³ V ³ ³ A 854 Transport Phenomena in Multiphase Systems The density of the two-phase mixture is defined as the average mass per unit volume: ρA dV + ρv dV V Vv ρ= A (11.3) VA + Vv If the density of each phase is constant, the following relationship can be obtained by employing eq. (11.1): ρ = (1 − α ) ρA + αρv (11.4) The phase velocity is the mean velocity of each phase and is defined as the volumetric flow rate of that phase through its cross-sectional area. This areaaveraged velocity should also be considered as time-averaged velocity to eliminate random fluctuations, i.e., Q A wA = A (11.5) AA Q v wv = v (11.6) Av where QA and Qv are the volumetric flow rates of the liquid and vapor phases. The phase velocities defined in eqs. (11.5) and (11.6) are intrinsic-averaged velocity and will be represented by wA and wv in future references for ease of notations. The superficial velocity, or volumetric flux, of each phase is defined as the volumetric flow rate of that phase divided by the total cross-sectional flow area in question, i.e., QA jA = (11.7) AA + Av Qv jv = (11.8) AA + Av The superficial velocity defined in eqs. (11.7) and (11.8) is equal to the velocity that each phase would have if it were to flow alone in the channel at its specified mass flow rate. The relationship between the superficial velocity and the phase velocity of the liquid phase can be obtained by combining eqs. (11.5) and (11.7) and using the definition of the void fraction in eq. (11.2), i.e., jA = wA (1 − α ) (11.9) The relationship between the superficial velocity and the phase velocity of the vapor phase can be obtained in a similar manner: jv = wvα (11.10) Since the void fraction in two-phase flow always ranges between zero and one, it can be seen from eqs. (11.9) and (11.10) that the phase velocities for each phase are greater than the corresponding volumetric flux. The total volumetric flux of the two-phase mixture, j, can be expressed as j = jA + jv (11.11) ³ ³ Chapter 11 Two-Phase Flow and Heat Transfer 855 The volumetric flow fraction, β , is defined as the volumetric flow rate of the vapor divided by the total volumetric flow rate: jv Qv β= = (11.12) jA + jv QA + Qv which is a particularly convenient quantity for the experimentalist since the volumetric flow rates can be readily calculated or measured. It has been observed experimentally that the one-dimensional phase velocity of the vapor is normally greater than the one-dimensional phase velocity of the liquid in flowing two-phase systems. A slip ratio is defined as the ratio of the phase velocity of the vapor to that of the liquid, i.e., w S= v (11.13) wA For homogeneous flow, the velocities for the liquid and vapor phases are identical and therefore the slip ratio S = 1, i.e., wv = wA (11.14) Substituting eqs. (11.9) and (11.10) into eq. (11.14) yields jv j =A (11.15) α 1−α The void fraction for homogeneous flow is then jv α= (11.16) jA + jv which indicates that = for homogeneous flow. The quality is defined as the vapor (gas) content of the two-phase flow. It is a parameter that identifies the dryness or the wetness of the two-phase system. mv x= (11.17) mA + mv where m is the mass flow rate of the liquid or vapor (gas) phase, obtained by (11.18) mA = ρA wA AA mv = ρv wv Av (11.19) The superficial mass flux or mass velocity of liquid and vapor is defined as m (11.20) GA = A = ρA jA = ρA wA (1 − α ) A m Gv = v = ρv jv = ρv wvα (11.21) A The total mass flux in two-phase flow is then (11.22) m′′ = GA + Gv = ρv jv + ρA jA = ρv wvα + ρA wA (1 − α ) The quality, x, can also be defined in terms of mass flux as G x= v (11.23) m′′ 856 Transport Phenomena in Multiphase Systems Substituting eq. (11.23) into eqs. (11.20) and (11.23), one can relate the superficial velocity to the quality as m′′(1 − x) jA = (11.24) ρA jv = m′′x ρv (11.25) 11.2.2 Flow Patterns in Vertical Tubes Two-phase flow in a vertical tube tends to be more symmetric, since gravity acts equally in the circumferential directions. The gravitational force plays a more dominant role in the liquid phase – and therefore in the whole of the two-phase flow – in a vertical channel. Fig. 11.1 shows the flow patterns encountered in vertical upward flow, where the patterns are defined as follows (Hewitt, 1998; Thome, 2004): • • Bubbly flow. Vapor bubbles are dispersed in a continuous liquid phase. The size of the bubbles varies widely but is generally small compared to the diameter of the tube. Slug or plug flow. As the bubble size increases and the bubbles start to consolidate, plugs can form. The plugs are often bullet-shaped in an upward flow and may be separated by regions occupied by liquid with a dispersion of small bubbles. These bullet-shaped plugs are commonly referred to as “Taylor bubbles” after the Taylor instability. Churn flow. As the slug bubbles grow larger, they start to break up, leading to a more random and unstable flow. Although the vapor continuously flows upward, the liquid phase may experience intermittent upward and downward motion, because the shear force from the vapor phase can just balance the imposed pressure gradient and the downward gravitational force. This oscillatory pattern is termed churn flow, and it is an intermediate regime between slug flow and annular flow. When the diameter of the vertical tube is small, the flow pattern can change directly from slug flow to annular flow without going through the churn flow pattern. Annular flow. At relatively high quality, the thin liquid layer flows along the inner wall of the tube and the central core of the flow consists of the vapor (gas) phase. Annular flow results when the interfacial shear of the high velocity gas or vapor on the liquid film becomes dominant over gravity. Liquid is then expelled from the center of the tube to form a film on the tube wall. Since the vapor core velocity is much higher than the liquid velocity, the vapor core may ripple the liquid layer and cause waves on the liquid film. It is also possible that some of the liquid phase may be entrained as small droplets in the gas core, or that some bubbles may be entrained in the liquid film. • • Chapter 11 Two-Phase Flow and Heat Transfer 857 Figure 11.1 Flow regimes in vertical upward cocurrent two-phase flow ( (Hewitt, 1998). liquid vapor) • Wispy annular flow. At a high liquid flow rate, the concentration of the liquid drops in the vapor (gas) core increases. The merging of these liquid droplets can lead to large lumps, breaks, or wisps of liquid in the gas core. In different regimes of two-phase flow, the pressure drop and heat transfer characteristics are significantly different, so it is necessary to identify the conditions corresponding to the flow regimes described above. A number of approaches have been developed to allow two-phase flow regime prediction; these include analytical and numerical models and the use of experimental and operational data that have been collected over the decades by various investigators. The models based on experimental data have resulted in semiempirical correlations that have had some degree of success in predicting twophase flow. The analytical models that are based on physical concepts are often used to derive one-dimensional continuity, momentum, and energy balances for simple boundary conditions and steady state systems. In transient systems and more complex systems where boundary conditions are not exact, numerical models based on physical concepts (Chapter 4) are used to predict two-phase flow. This information and/or experimental details have been transformed into readily available and simple tools that can develop a number of graphic diagrams, referred to as flow maps. These diagrams represent the flow in terms of important and useful parameters providing information about flow rate and operating parameters such as heat transfer coefficient and pressure drop. Based on these data, flow regimes representing different fractions of liquid and gas can be observed. Then, for a given set of operating conditions, the expected flow regime can quickly be determined. The accuracy of each flow map depends on the methods used to develop it. For example, a flow map developed from experimental data for one working 858 Transport Phenomena in Multiphase Systems fluid may not necessarily be representative of another working fluid. On the other hand, when a large amount of data for various fluids and operating conditions are used, a more representative flow map might be possible. One of the limitations of experimental or operational data is the subjectivity in the data as reported by an observer. Furthermore, it is not always easy to distinguish one flow regime from another. Discrepancies in the reporting of flow regimes can translate into errors in a given flow map. Conversely, a flow map based on accurate physical models will provide a better representation of the actual flow regimes. Two-phase flow is generally a complex phenomenon; therefore, to obtain a reasonable analytical prediction, a number of approximations must be made while solving the momentum and energy balances. Generally, the pressure drop across the flow channel and/or heat transfer coefficients are calculated to determine the flow characteristics at a given point and allow prediction of the flow regime. To that end, prediction methods based on heat transfer coefficients and pressure drop correlations are developed. The flow regime map is usually given in terms of superficial velocities or some other generalized flow parameters for liquid and vapor flow. The different regimes are separated by lines that represent the conditions for transition between flow regimes. A flow regime map proposed by Hewitt and Roberts (1969), which represents a fairly wide range of experimental data for upward two-phase flow, is shown in Fig. 11.2. The horizontal and vertical coordinates of the map are the Figure 11.2 Flow regime map obtained by Hewitt and Roberts (1969) for vertical upward twophase flow. Chapter 11 Two-Phase Flow and Heat Transfer 859 superficial momenta of the liquid and vapor phases, respectively. It can be seen that the conditions for different flow regimes and the boundaries that separate them can be expressed as a combination of the superficial momentum fluxes of the liquid and vapor phases. The boundaries between different flow regimes are the criteria that identify points of flow regime transition and are often of interest in practical applications. In addition to the flow map developed by Hewitt and Roberts (1969), there are many flow maps available in the literature. Since the identification of the flow pattern, and even the name of the flow regime, is subjective and may differ from one observer to another, it is very challenging to present every correlation and flow pattern for all of the different flow regimes and configurations. A very detailed review of the boundaries between the different flow regimes, and the corresponding empirical correlations, can be found in Thome (2004). Example 11.1 Determine the flow regime for vertical upward flow of 0.8 Kg/s of R134a in a vertical 0.1 m diameter tube at -20 °C and 25% quality. Solution: The mass flow rate is m = 0.8 kg/s , and the diameter of the tube is D = 0.1m . Therefore, the mass flux of the R11 in the tube is 4 × 0.8 m 4m = = 101.86 kg/m 2 -s m′′ = = A π D π × 0.12 The densities of liquid and vapor R-134 at T= –20 °C can be found from the Table B.34 as ρA = 1358 kg/m3 and ρv = 6.785 kg/m3 . The superficial velocities of the vapor and liquid are obtained from eqs. (11.24) and (11.25), i.e., m′′(1 − x) 101.86 × (1 − 0.25) = = 0.0562 m/s jA = 1358 ρA m′′x 101.86 × 0.25 jv = = = 3.75 m/s 6.785 ρv The superficial momentum of the liquid and vapor phases are, respectively: ρA jA2 = 1358 × 0.05622 = 4.30 kg/s 2 -m ρv jv2 = 6.785 × 3.752 = 95.4 kg/s 2 -m It can be found from Fig. 11.2 that the flow regime is churn flow. The flow regimes discussed above are applicable to vertical upward twophase flow. Although downward two-phase flow finds applications in many areas, especially in condensation, it has received very little attention from investigators, and so there is very little flow pattern data for this case. The regimes for vertical downward two-phase flow are similar to those shown in Fig. 11.1, except that the downward-acting shear force, gravitational force, and imposed pressure gradient eliminate the churn flow regime. The limited research 860 Transport Phenomena in Multiphase Systems jv (m/s) jA (m/s) Figure 11.3 Flow regime map for vertical downward two-phase flow (Golan and Stenning, 1969). shows that the annular flow regime is dominant in vertical downward flow, which is different from upward flow. A flow pattern map based on low-pressure air-water flow data proposed by Golan and Stenning (1969) is shown in Fig. 11.3. 11.2.3 Flow Patterns in Horizontal Tubes Horizontal two-phase flow exhibits flow patterns different from those in vertical two-phase flow, because gravity acts perpendicularly to the flow direction. Twophase flow in horizontal tubes is more complex than vertical two-phase flow, because the flow is usually not axisymmetric, due to the effect of gravity. The flow patterns encountered in horizontal two-phase flow are defined as follows (Hewitt, 1998; Thome, 2004) and shown in Fig. 11.4: • Dispersed bubble flow. The vapor (gas) phase appears as distinct bubbles in a continuous liquid phase. The bubbles tend to rise to the top of the flow due to buoyancy effects. When the liquid velocity is high, the bubbles may be more uniformly distributed in the liquid, as in vertical flow. • Plug flow. An increase of quality results in larger vapor (gas) bubble size and the formation of bullet-shaped plugs, which tend to remain at the top of the flow channel due to buoyancy force. Chapter 11 Two-Phase Flow and Heat Transfer 861 Figure 11.4 Flow regimes in horizontal two-phase flow ( liquid, vapor). • • • • Stratified flow. The liquid and vapor (gas) velocities in this regime are relatively slow, and the quality is relatively high. The liquid flows along the bottom of the tube due to gravity while the vapor (gas) phase flows at the top of the tube. It is also possible that a very thin layer of the liquid forms at the top of the flow channel. Stratified wavy flow. As the vapor (gas) velocity increases in the stratified flows, the shear forces of the vapor (gas) flow over the liquid cause ripples on the top of the liquid phase and result in the formation of waves on the liquid-vapor (gas) interface. The waves climb up the sides of the tube and the liquid layer at the bottom of the channel starts to stretch thin. Slug flow. The amplitude of the waves increases as the liquid flow rate increases. The crests can span the entire tube, and a bridge starts to develop, separating the slugs from one another. However, a substantial liquid phase remains and gravity pulls it to the bottom of the flow channel. The top of the flow channel is still wetted by a relatively thin film of liquid. Annular-dispersed flow. Similar to the flow pattern in vertical two-phase flow, the liquid layer flows near the inner wall of the tube and the vapor (gas) flows in the central core. However, the liquid layer at the bottom is thicker than that on the top of the channel due to the effect of gravitational force. The most widely used flow pattern map for horizontal two-phase flow, proposed by Taitel and Dukler (1976), is shown in Fig. 11.5. This map is based on a semi-theoretical method, and it is computationally more difficult to use than 862 Transport Phenomena in Multiphase Systems other flow maps. The horizontal coordinate of the Taitel and Dukler (1976) map is the Martinelli parameter: ª (dp / dz )A º X =« F (11.26) » ¬ (dpF / dz )v ¼ where (dpF / dz )A and (dpF / dz )v are the pressure gradients for the liquid and vapor phases, flowing along the channel. The vertical coordinates of the Taitel and Dukler (1976) map are defined as ρv jv (11.27) F= ρA − ρv Dg cos θ 1/ 2 ª ρv jv 2 DjA º K =« » ¬ ( ρA − ρv ) Dg cos θ vA ¼ 1/ 2 1/ 2 (11.28) ª (dp / dz )A º T =« (11.29) » ¬ ( ρ A − ρv ) g cos θ ¼ where D is the tube diameter and θ is the angle of inclination of the channel to the horizontal. It can be seen that determination of the flow regime using Taitel and Dukler’s (1976) map requires the pressure gradient for the liquid and vapor phases flowing along the channel, which should be determined using appropriate flow models. Figure 11.5 Flow regime map for horizontal two-phase flow (Taitel and Dukler, 1976). Chapter 11 Two-Phase Flow and Heat Transfer 863 It should be pointed out that the flow maps presented in Figs. 11.2, 11.3, and 11.5 were obtained for adiabatic two-phase flow; however, the transition boundaries between different flow regimes depend on the heat flux. Nevertheless, these flow maps are often used to determine the flow patterns for evaporation and condensation inside tubes, for which external heating or cooling is required. Application of these flow maps to forced convective boiling or condensation inside a tube may not yield reliable results. 11.3 Two-Phase Flow Models A number of models have been proposed to calculate pressure drop and heat transfer in two-phase flow (Chapter 4). These models provide a simple and accurate representation of the flow regimes. The models that have been extensively used by most researchers in two-phase are the homogeneous model and the separated flow model. The latter is a simple version of multi-fluid model (Section 4.5); it allows two phases to have different properties and onedimensional velocities, while the conservation equations are written for the combined flow. The major difference between these two models is the way phase velocity is addressed in each. The homogeneous model lumps both phases together to provide a homogeneous flow, and the behavior of the homogeneous flow is then determined. In the separated flow model, on the other hand, the flow of each phase is determined independently and the effects of the two phases are then summed. The homogeneous flow model provides an easier approach to determining flow properties and behaviors, but it underestimates the pressure drop, particularly in a moderate pressure range. Furthermore, the homogeneous model is less accurate when velocity and flow conditions for both phases are more dispersed. A separated flow model, on the other hand, is somewhat more complex but tends to produce more accurate results. The one-dimensional models for homogeneous flow and separated flow are presented in Sections 11.3.1 and 11.3.2, respectively. 11.3.1 Homogeneous Flow Model The central assumption of the homogeneous flow model is that the two phases travel at equal velocities and mix well; therefore, they can be treated as if there is only one phase. This model works better for two-phase flow near the critical point, where the differences between the properties of the liquid and vapor are insignificant, or when the mass velocity of the two-phase flow is very high so that the flow regime is either bubbly or misty flow. In arriving at the homogeneous model for two-phase flow, area averaging is performed for both phases and the resultant governing equations become one-dimensional in nature. The density of the homogeneous mixture satisfies the relation 864 Transport Phenomena in Multiphase Systems 1 ρ i.e., = x ρv + 1− x ρA (11.30) ρ= ρv ρA ρA x + ρv (1 − x) (11.31) The mass flow rates of the liquid and vapor phases defined in eqs. (11.18) and (11.19) become mA = ρA wAA (11.32) mv = ρv wAv (11.33) Substituting eqs. (11.32) and (11.33) into eq. (11.2) yields mv / ρ v α= (11.34) mv / ρ v + mA / ρA Considering the definition of quality in eq. (11.17), the void fraction becomes x α= (11.35) x + (1 − x) ρv / ρA The total mass flux in the channel becomes m + mv ρA wAA + ρv wAv m′′ = A = = ρw (11.36) A A The governing equations for the homogeneous model include continuity, momentum and energy equations, which are listed below (see Example 4.5): ∂ρ ∂ + (m′′A) = 0 A (11.37) ∂t ∂z ∂m′′ ∂ (m′′2 A / ρ ) ∂ ( pA) A + =− − ρ g cos θ A − τ w P (11.38) ∂t ∂z ∂z ·º 1 ∂ ª § ·º P ∂ª § w2 w2 ∂p ′′ + gz cosθ ¸ » + + gz cos θ ¸ » = qw + q′′′ + «ρ ¨ h + « ρ wA ¨ h + ∂t « © ∂t 2 2 ¹ » A ∂z « © ¹» A ¬ ¼ ¬ ¼ where the angle brackets, “ (11.39) ,” which represent the area average, have been ′′ dropped for ease of notation. P is perimeter, p is pressure, qw is heat flux at the wall, and q′′′ is internal heat generation per unit volume within the fluid. Substituting eq. (11.36) into eq. (11.39), the energy equation becomes ∂ ( ρ h) 1 ∂ + ( m′′Ah) ∂t A ∂z (11.40) ∂ § m′′2 · ∂p P 1 ∂ § m′′3 A · ′′ = qw + q′′′ − − gm′′ cos θ − ¨ + ¨ ¸ ¸ ∂t © 2 ρ ¹ ∂t A A ∂z © 2 ρ 2 ¹ For steady-state two-phase flow in a circular tube with constant crosssectional area, the momentum eq. (11.38) reduces to Chapter 11 Two-Phase Flow and Heat Transfer 865 dp 4τ w ∂ (m′′2 / ρ ) = + + ρ g cos θ (11.41) ∂z dz D The three terms on the right-hand side of eq. (11.41) represent pressure drops due to friction, dpF / dz , acceleration, dpa / dz , and gravity, dpg / dz . Thus, eq. − (11.41) can be rewritten as dpg dp dp dp − =− F − a − (11.42) dz dz dz dz where the pressure drop due to friction, dpF / dz , must be obtained using appropriate correlations. The energy equation (including mechanical and thermal energy) for steadystate two-phase flow in a circular tube with constant cross-sectional area can be obtained by simplifying eq. (11.40), ′′ dh 4qw q′′′ m′′2 d § 1 · (11.43) = + − ¨ ¸ − g cos θ dz m′′D m′′ 2 dz © ρ 2 ¹ The enthalpy of the two-phase mixture can be expressed as h = hA + x(hv − hA ). For a two-phase flow system with condensation or evaporation, where kinetic and potential energy as well as internal heat generation can be neglected, eq. (11.43) reduces to ′′ 4 qw dx = (11.44) dz m′′DhAv 11.3.2 Separated Flow Model Compared with the homogeneous model, the separated flow model has been used more widely, because it provides a better prediction of flow behavior with a manageable level of complexity. The separated flow model assumes that each phase displays different properties and flows at different velocities. It is a simpler version of the two-fluid model discussed in Chapter 4 because it is assumed that only velocities differ between the two phases, while the conservation equations are written only for the combined flow. In addition, the pressure across any given cross-section of a channel carrying a multiphase flow is assumed to be the same for both phases (Hewitt, 1998). The mass flow rates of the liquid and vapor phases are obtained from eqs. (11.18) and (11.19). The cross-sectional area of the channel occupied by liquid and vapor can be obtained by rearranging eqs. (11.18) and (11.19), i.e., AA = mA /( ρA wA ) (11.45) (11.46) Av = mv /( ρv wv ) The void fraction of the two-phase flow can be obtained by substituting eqs. (11.45) and (11.46) into eq. (11.2), i.e., 866 Transport Phenomena in Multiphase Systems mv /( ρv wv ) (11.47) mv /( ρv wv ) + mA /( ρA wA ) Considering the definition of quality in eq. (11.17), the void fraction becomes x α= (11.48) ρw x + (1 − x) v v ρA wA It can be seen that when the vapor velocity, wv , is greater than the liquid velocity, wA (which is often the case for vertical upward and horizontal cocurrent flow) the homogeneous model overpredicts the void fraction. On the other hand, when the vapor velocity, wv , is lower than the liquid velocity, wA , as occurs in vertical downward flow, the homogeneous model underpredicts the void fraction. The governing equations for steady-state two-phase flow in a channel for separated flow model have been given in Example 4.6. More generalized governing equations that are applicable to transient flow are presented here. The continuity equations for the vapor and liquid phases are, respectively ∂ ∂ ′′′ ( ρvα A) + ( ρv wvα A) = mv (11.49) ∂t ∂z ∂ ∂ ′′′ [ ρA (1 − α ) A] + [ ρA wA (1 − α ) A] = mA (11.50) ∂t ∂z ′′′ ′′′ where mv and mA are the mass production rates of vapor and liquid due to phase change in the two-phase flow system. Conservation of mass requires that ′′′ ′′′ the summation of mv and mA equal zero, so the continuity equation for the twophase system can be obtained by adding eqs. (11.49) and (11.50) together, i.e., ∂ ∂ ( ρ A) + [ ρv wvα A + ρA wA (1 − α ) A] = 0 (11.51) ∂t ∂z where ρ is the density of the two-phase mixture from eq. (11.4). Considering the definition of mass flux in eqs. (11.20) – (11.22), the continuity equation can be rewritten as ∂ ∂ ( ρ A) + (m′′A) = 0 (11.52) ∂t ∂z The momentum equations for the vapor and liquid phases are ∂ ∂ ( ρv wvα A) + ρv wv2α A ∂t ∂z (11.53) ∂p = −α A − g ρvα A cos θ − τ w,v Pw,v + FA,v ∂z ∂ ∂ [ ρA wA (1 − α ) A] + ª ρA wA2 (1 − α ) Aº ¼ ∂t ∂z ¬ (11.54) ∂p = −(1 − α ) A − g ρA (1 − α ) A cos θ − τ w,A Pw,A + Fv ,A ∂z α= ( ) Chapter 11 Two-Phase Flow and Heat Transfer 867 where τ w,v and τ w,A are shear stresses at the wall for vapor and liquid, respectively. Pw,v and Pw,A are the portion of the perimeter that is in contact with vapor and liquid, respectively. FA ,v and Fv ,A are the interactive forces between the liquid and vapor phases, which satisfy FA ,v = − Fv ,A as required by Newton’s third law. Assuming constant wall shear stress w around the periphery of the channel, the momentum equation of the two-phase system can be obtained by adding eqs. (11.53) and (11.54), i.e., ∂m′′ ∂ ∂p 2 2 A + ª ρv wv α A + ρA wA (1 − α ) Aº = − A − g ρ A cos θ − τ w P (11.55) ¬ ¼ ∂t ∂z ∂z Substituting eqs. (11.20) and (11.21) into eq. (11.55), the momentum equation becomes GA2 º ½ ∂m′′ ∂ ­ ª Gv2 ∂p ° ° (11.56) A + ®A« + » ¾ = − A − g ρ A cos θ − τ w P ∂t ∂z ° ¬ ρvα ρ A (1 − α ) ¼ ° ∂z ¯ ¿ Considering that the mass flux of the vapor phase can be expressed as Gv = xm′′ [eq. (11.23)] and the mass flux of the liquid is GA = m′′ − Gv = (1 − x)m′′ , the momentum equation in the separated flow model becomes ­ ½ ª x2 τ wP (1 − x) 2 º ° ∂m′′ 1 ∂ ° ∂p (11.57) m′′A « + + ® » ¾ = − − g ρ cos θ − A ∂z ° A ∂t ∂z ¬ ρvα ρA (1 − α ) ¼ ° ¯ ¿ The energy equations for the vapor and liquid phases are 2 § ·º ∂ ª § ·º wv w2 ∂ª + gz cosθ ¸ » + « ρ v wvα A ¨ hv + v + gz cosθ ¸ » « ρvα A ¨ hv + ∂t « 2 2 © ¹ » ∂z « © ¹» ¬ ¼ ¬ ¼ ∂p ′′ ′′′ (11.58) = Pv qw + q′′′α A + α A + qA ,v ∂t 2 § ·º ∂ ª § ·º wA w2 ∂ª + gz cos θ ¸ » + « ρA wA (1 − α ) A ¨ hA + A + gz cos θ ¸ » « ρA (1 − α ) A ¨ hA + 2 2 ∂t « © ¹ » ∂z « © ¹» ¬ ¼ ¬ ¼ ∂p ′′ ′′′ (11.59) = PA qw + q′′′(1 − α ) A + (1 − α ) A + qv ,A ∂t ′′ where the heat flux at the surface of the channel, qw , is assumed to be the same ′′′ for the perimeter of the channel, whether it is in contact with liquid or vapor. qA ,v ′′′ ′′′ ′′′ and qv ,A are the interphase heat transfers that satisfy qA ,v = − qv ,A . The energy equation for the two-phase mixture is then obtained by adding eqs. (11.58) and (11.59), i.e., 868 Transport Phenomena in Multiphase Systems A + § · § ·º w2 w2 ∂ª ρvα ¨ hv + v + gz cosθ ¸ + ρA (1 − α ) ¨ hA + A + gz cos θ ¸ » « 2 2 ∂t « © ¹» © ¹ ¬ ¼ ­ ½ § · § ·º ° w2 w2 ∂° ª A « ρv wvα ¨ hv + v + gz cos θ ¸ + ρA wA (1 − α ) ¨ hA + A + gz cos θ ¸ » ¾ (11.60) ® ∂z ° ¬ 2 2 »¿ © ¹¼ ° © ¹ ¯« ∂p ∂t Substituting eqs. (11.20) and (11.21) into eq. (11.60), the energy equation becomes § · § ·º G2 G2 ∂ª A « ρvα ¨ hv + 2v 2 + gz cos θ ¸ + ρA (1 − α ) ¨ hA + 2 A + gz cos θ ¸ » ∂t « 2 ρv α 2 ρA (1 − α ) 2 © ¹» © ¹ ¬ ¼ ′′ = Pqw + q′′′A + A + ­ ½ · § ·º ° Gv2 GA2 ∂° ª § + gz cos θ ¸ » ¾ ® A «Gv ¨ hv + 2 2 + gz cos θ ¸ + GA ¨ hA + 2 ∂z ° « © 2 ρv α 2 ρA (1 − α ) 2 © ¹» ° ¹ ¼¿ ¯¬ ′′ = Pqw + q′′′A + A ∂p ∂t (11.61) Since Gv = xm′′ and GA = m′′ − Gv = (1 − x)m′′ , eq. (11.61) can be modified as ∂ ∂ A [ ρv hvα + ρA hA (1 − α )] + {m′′A [ xhv + (1 − x)hA ]} ∂t ∂z 3 (1 − x)3 º ½ ∂ ­ m′′ A ª x 3 ° ° ′′ (11.62) = Pqw + q′′′A − ® +2 « » ¾ − m′′Ag cos θ ∂z ° 2 ¬ ρv2α 2 ρA (1 − α ) 2 ¼ ° ¯ ¿ 2 2 2 ­ ½ (1 − x) º ° ∂ ° m′′ ª x ∂p −A ® + « »¾ + A ∂t ° 2 ¬ 2 ρvα 2 ρA (1 − α ) ¼ ° ∂t ¯ ¿ For steady-state two-phase flow in a circular tube with constant crosssectional area (A = constant), the continuity equation (11.52) results in dm′′ / dz = 0 . The momentum equation (11.51) reduces to dp 4τ w d ª x2 (1 − x) 2 º (11.63) − = + m′′ « + » + g ρ cos θ dz D dz ¬ ρvα ρA (1 − α ) ¼ The three terms on the right-hand side of eq. (11.63) represent pressure drops due to friction, dpF / dz , acceleration, dpa / dz , and gravity, dpg / dz . To predict the pressure drop of two-phase flow using the separated flow model, empirical correlations for friction and void fraction are needed, as is evident from eq. (11.63). The energy equation in the separated flow model for steady-state two-phase flow in a circular tube with constant cross-sectional area can be obtained by simplifying eq. (11.62), i.e., Chapter 11 Two-Phase Flow and Heat Transfer 869 4q′′ q′′′ m′′2 d ª x3 d (1 − x)3 º +2 [ xhv + (1 − x)hA ] = ′′ w + ′′ − « » − g cos θ (11.64) dz mD m 2 dz ¬ ρv2α 2 ρA (1 − α ) 2 ¼ For a two-phase flow system with condensation or evaporation, where kinetic and potential energy, as well as internal heat generation, can be neglected, eq. (11.64) also reduces to eq. (11.44). Therefore, the energy equations for the homogeneous model and the separated flow model are the same if kinetic and potential energy, as well as internal heat generation, can be neglected. 11.3.3 Frictional Pressure Drop The pressure drop is very important for the design of two-phase devices, because it dictates the pump power that is required to drive the flow. As shown in eqs. (11.41) and (11.63), the pressure drop includes three parts: frictional, accelerational, and gravitational pressure drops. The accelerational and gravitational pressure drop terms in eqs. (11.41) and (11.63) can be calculated based on physical properties and the void fraction. The frictional pressure drop for steady-state two-phase flow in a circular tube can be calculated by 4τ dp − F= w (11.65) dz D The frictional pressure drop for two-phase flow must be accounted for by using an empirical correlation, which can be based on either the homogeneous or separated flow model. While the former assumes that both liquid and gas move at the same velocity (slip ratio S is 1) and is also referred to as zero-slip model, the latter allows liquid and gas to move at different velocities. Correlations Based on the Homogeneous Model Since the homogeneous model treats the two phases in the mixture as a pseudo-single-phase flow, the frictional pressure gradient can be calculated by (Beattie and Whalley, 1982) 4τ dp 2 f m′′2 − F= w= (11.66) dz D Dρ where f is the two-phase friction factor, which can be determined by empirical correlation for single phase flow. For many two-phase homogeneous flow considerations, particularly in the annular and bubble regimes, the following equation is proposed to determine the homogeneous friction factor, f: ª º 1 2κ 9.35 » = 3.48 − 4log10 « + (11.67) «D f Re f » ¬ ¼ where the term /D represents the surface roughness/diameter ratio. The Reynolds number can be calculated based on a homogeneous flow using ( ) 870 Transport Phenomena in Multiphase Systems Re = m′′D μ μA (11.68) The homogeneous viscosity of the two-phase mixture is obtained by 1 x 1− x = + (11.69) μ μv 11.3.3.2 Correlations Based on the Separated Flow Model The frictional pressure gradient of two-phase flow can be related to that of either the vapor or liquid phase flowing alone in the channel (Lockhart and Martinelli, 1949; Chisholm, 1967). The frictional pressure gradients of the vapor or liquid phase flow in the channel, with their actual flow rate and properties, can be defined as 22 § dp · 2 f m′′ x −¨ F ¸ = v (11.70) D ρv © dz ¹v 2 2 § dp · 2 f m′′ (1 − x) −¨ F ¸ = A (11.71) D ρA © dz ¹A where f v and f A are, respectively, the friction factors for the vapor and liquid phases with their actual mass flux flowing in the channel alone. Similarly, the frictional pressure gradient in the channel – with the same total mass flow rate of the two-phase flow, but with the properties of the vapor or liquid phase – can be defined as 2 f m′′2 § dp · − ¨ F ¸ = v0 (11.72) D ρv © dz ¹vo 2 f m′′ § dp · − ¨ F ¸ = A0 (11.73) D ρA © dz ¹Ao where fv0 is the vapor friction factor if the vapor phase with mass flux, m′′, occupies the entire channel, whereas f A 0 is the liquid fraction factor if the channel is taken by liquid phase with mass flux m′′ alone. Through the standard equations and charts for the single-phase flow, the friction factors defined in eqs. (11.70) – (11.73) can be related to the respective Reynolds numbers: m′′xD Re v = (11.74) 2 μv m′′(1 − x) D ReA = μA m′′D Revo = μv (11.75) (11.76) Chapter 11 Two-Phase Flow and Heat Transfer 871 R e Ao = m′′D μA (11.77) The relationships between the frictional factor and the Reynolds number are different for laminar and turbulent flow. ­ 16 Re < 2000 ° (11.78) f = ® Re °0.079 Re −0.25 Re > 2000 ¯ The frictional pressure gradient of the two-phase flow can be related to those defined in eqs. (11.70) – (11.73) through pressure drop multipliers defined as dpF / dz φv2 = (11.79) ( dpF / dz )v φA2 = 2 φvo = dpF / dz ( dpF / dz )A dpF / dz ( dpF / dz )vo dpF / dz ( dpF / dz )Ao (11.80) (11.81) (11.82) φA2o = Figure 11.6 Lockhart-Martinelli correlations for pressure drop (Hewitt, 1998). 872 Transport Phenomena in Multiphase Systems Two commonly used parameters in two-phase flow investigations are the Martinelli parameter, X, which was defined in eq. (11.26), and the Chisholm parameter, Y, ª (dp / dz )Ao º (11.83) Y =« F » ¬ (dpF / dz )vo ¼ Parameter X, the Martinelli parameter, is a ratio of pressure drops of single-phase flow terms. As can be seen from eqs. (11.79) – (11.82), the pressure drop in twophase flow can be determined if any one of the four multipliers is known. A generalized method to determine the frictional pressure gradient multiplier was proposed by Lockhart and Martinelli (1949), who related the frictional multipliers φv and φA to the Martinelli parameter X as shown in Fig. 11.6. It can be seen that the trends for φv and φA are different because φv increases with increasing X, but φA decreases with increasing X. The multiplier curves also depend on whether the liquid-phase alone flow and the vapor-phase alone flow are laminar or turbulent. There are four curves for φv and φA and each corresponds to the combination of laminar (viscous) and turbulent flow for the vapor- or liquid-phases-alone flows in the channel. For example, φA ,vt represents the multiplier in the liquid alone pressure drop for cases where the liquid-phase flowing alone in the channel is laminar (viscous) but the vapor phase flowing alone in the channel is turbulent. Chisholm (1967) correlated the curves of Lockhart and Martinelli (1949) and recommended the following relationships: C 1 φA2 = 1 + + 2 (11.84) XX φv2 = 1 + CX + X 2 (11.85) Table 11.1 Value of C in eqs. (11.84) and (11.85). Liquid Turbulent Viscous Turbulent Viscous Vapor Turbulent Turbulent Viscous Viscous Subscripts tt vt tv vv C 20 12 10 5 1/ 2 where C is a dimensionless constant that depends on the combination of the natural and the phase-alone flows. The value of the constant C recommended by Chisholm (1967) can be found in Table 11.1. The correlation by Lockhart and Martinelli (1949) can provide a good prediction when μA / μv > 1000 and m′′ < 100 kg/m 2 -s . Alternative correlations should be used when the two-phase flow falls outside these ranges. Chapter 11 Two-Phase Flow and Heat Transfer 873 For cases where μA / μv > 1000 and m′′ > 100 kg/m 2 -s, the following correlation proposed by Chisholm (1973a) should be used: φA20 = 1 + (Y 2 − 1)[ Bx (2− n ) / 2 (1 − x)(2− n ) / 2 + x 2− n ] (11.86) where n is the exponent in the friction factor-Reynolds number relationship ( f Re n = constant). According to eq. (11.78), n equals 1 for laminar flow and 0.25 for turbulent flow. The parameter B is given by ­ 55 0 < Y < 9.5 ° ° m′′ ° 520 B=® 9.5 < Y < 28 (11.87) ° Y m′′ ° 15000 Y > 28 °2 ¯ Y m′′ For cases where μA / μv < 1000, the following correlation developed by Friedel (1979) using a database of 25,000 points can provide a better prediction: 3.24C φA20 = C1 + 0.045 20.035 (11.88) Fr We where § ρ ·§ f · C1 = (1 − x) 2 + X 2 ¨ A ¸¨ v 0 ¸ (11.89) © ρv ¹© f A 0 ¹ C2 = x 0.78 (1 − x) 0.24 § ρA · ¨¸ © ρv ¹ 0.91 § μv · ¨¸ © μA ¹ 0.19 § μv · ¨1 − ¸ © μA ¹ 0.7 (11.90) (11.91) (11.92) Fr = We = m′′2 gD ρ 2 m′′2 D ρσ Bounds on Two-Phase Flow The advantage of the pressure drop correlations based on the separated-flow model is that it is applicable for all flow patterns. This flexibility is accompanied by low accuracy. Awad and Muzychka (2005a) developed rational bounds for two-phase pressure gradients. The lower bound of the friction pressure drop is 0.158m′′ (1 − x) § dp · = ¨¸ D1.25 ρ A © dz ¹ F ,lower 1.75 1.75 μA0.25 ª x· «1 + § ¨ ¸ 1− x ¹ « ¬© 0.7368 § ρA · ¨¸ © ρv ¹ 0.4211 § μv · ¨¸ © μA ¹ º » » ¼ (11.93) 0.1053 2.375 where D is the diameter of the tube. 874 Transport Phenomena in Multiphase Systems 100 100000 1000 100000 frictional pressure gradient (Pa/m) 10000 10000 1000 R 12 D = 14 mm x = 0.3 T0 = 0°C upper (Awad and Muzychka, 2005a) lower (Awad and Muzychka, 2005a) average (Awad and Muzychka, 2005a) Bandel (1973) 1000 100 100 10 10 1 100 1 1000 mass flux (kg/m2.s) Figure 11.7 Pressure gradient versus mass flux (Awad and Muzychka, 2005a) The upper bound of the friction pressure drop is º » » ¼ (11.94) An acceptable prediction of pressure drop can be obtained by averaging the maximum and minimum values, i.e., 0.79m′′1.75 (1 − x)1.75 μA0.25 § dp · = ¨¸ D1.25 ρA © dz ¹ F , ave 0.158m′′ (1 − x) § dp · = ¨¸ D1.25 ρA © dz ¹ F ,upper 1.75 1.75 μA0.25 ª x· «1 + § ¨ ¸ 1− x ¹ « ¬© 0.4375 § ρA · ¨¸ © ρv ¹ 0.25 § μv · ¨¸ © μA ¹ 0.0625 4 4½ 0.4211 0.1053 2.375 ­ 0.7368 º ª § x ·0.4375 § ρ ·0.25 § μ ·0.0625 º ° § ρA · § μv · °ª § x · v A » »¾ ⋅ ®«1 + ¨ + «1 + ¨ ¨¸ ¨¸¨¸ ¨¸ ¸ ¸ « » « ©1− x ¹ » © μA ¹ © ρv ¹ © ρv ¹ © μA ¹ °¬ © 1 − x ¹ ¼ ¬ ¼° ¯ ¿ (11.95) Figure 11.7 shows the lower and upper bounds, and the average fraction pressure gradient versus mass flux. The experimental results by Bandel (1973) for R-12 flow at x = 0.3 and Tsat = 0 °C in a smooth horizontal tube at D = 14 mm are also plotted in Fig. 11.7 for comparison. Equation (11.95) could predict the pressure drop with the root mean square error of 26.4%. Example 11.2 Use the Taitel-Dukler flow map to determine the flow regime for a flow of 8 kg/s of water-steam at 15% quality and 180 °C in a 0.1 m ID horizontal tube. Chapter 11 Two-Phase Flow and Heat Transfer 875 Solution: The thermophysical properties of water at p=10 bar can be found from Table B.48 in Appendix B, i.e., ρA = 887.31 kg/m3 , ρv = 5.1597 kg/m3 , μA = 1493 × 10−7 N-s/m 2 , and μv = 149 × 10−7 N-s/m 2 . The mass flux of the two-phase flow in the horizontal tube is m 4m 4×8 m′′ = = = = 1018.6kg/m 2 -s 2 A πD π × 0.12 The superficial velocities of the vapor and liquid are m′′x 1018.6 × 0.15 = = 29.61m/s jv = 5.1597 ρv m′′(1 − x) 1018.6 × (1 − 0.15) jA = = = 0.976m/s 887.31 ρA The Reynolds numbers of the vapor and liquid phases are obtained from eqs. (11.74) and (11.75), i.e., 1018.6 × 0.15 × 0.1 = 1.025 × 106 μv 149 × 10−7 m′′(1 − x) D 1018.6 × (1 − 0.15) × 0.1 = = 5.799 × 105 ReA = μA 1493 × 10−7 The fraction coefficients for the vapor and liquid phases are determined from eq. (11.78), i.e., f v = 0.079 Re0.25 = 2.51 v Rev = = 0.25 f A = 0.079 ReA = 2.18 The frictional pressure gradients of the vapor or liquid phase flow in the channel, with its actual flow rate and properties, can be found from eqs. (11.70) and (11.71), i.e., 2 f m′′2 x 2 2 × 2.51 × 1018.62 × 0.152 § dp · −¨ F ¸ = v = = 2.271 × 105 Pa/m D ρv 0.1 × 5.1597 © dz ¹v 2 2 § dp · 2 f m′′ (1 − x) −¨ F ¸ = A D ρA © dz ¹A m′′xD 2 × 2.18 × 1018.62 × 0.852 = 3.684 × 104 Pa/m 0.1 × 887.31 Therefore, the Martinelli parameter can be obtained from eq. (11.26), i.e., = ª 3.684 × 104 º ª (dp / dz )A º X =« F =« = 0.403 » 5» ¬ (dpF / dz )v ¼ ¬ 2.271 × 10 ¼ It can be seen from the Taitel and Dukler (1976) flow map that actual flow regime depends on the values of F and K, which can be found from eqs. (11.27) and (11.28), i.e., 1/ 2 1/ 2 876 Transport Phenomena in Multiphase Systems F= ρv ρA − ρv jv Dg = 5.1597 887.31 − 5.1597 1/ 2 29.61 0.1 × 9.8 = 2.29 ª ρv jv 2 ρA DjA º K =« » ¬ ( ρA − ρv ) Dg μA ¼ ª 5.1597 × 29.612 887.31 × 0.1 × 0.976 º =« » = 1742.2 1493 × 10−7 ¬ (887.31 − 5.1597) × 0.1 × 9.8 ¼ The flow regime is annular-dispersed liquid (AD), as indicated by Fig. 11.5. Example 11.3 Determine the pressure gradient due to friction in Example 11.2. Solution: Since the ratio of the viscosity is μA 1493 × 10−7 = = 10 < 1000 μv 149 × 10−7 eq. (11.88) developed by Friedel (1979), should be used. In addition to the properties found in Example 11.2, the surface tension is σ = 42.19 × 10−3 N/m . The Reynolds numbers of the vapor or liquid phases obtained from eqs. (11.76) and (11.77) are m′′D 1018.6 × 0.1 Rev 0 = = = 7.289 × 106 μv 149 × 10−7 m′′D 1018.6 × 0.1 ReA 0 = = = 6.821 × 105 μA 1493 × 10−7 The fraction coefficients for the vapor and liquid phases determined from eq. (11.78) are f v 0 = 0.079 Re0.25 = 4.105 v0 0.25 f A 0 = 0.079 ReA 0 = 2.270 The frictional pressure gradients when the liquid flows in the channel with the total mass flow rate of the two-phase flow, found from eq. (11.73), is 2 f m′′2 (1 − x) 2 § dp · − ¨ F ¸ = A0 D ρA © dz ¹A 0 1/ 2 2 × 2.270 × 1018.62 × 0.852 = 3.836 × 104 Pa/m 0.1 × 887.31 The homogeneous density of the two-phase system is obtained from eq. (11.31): = Chapter 11 Two-Phase Flow and Heat Transfer 877 ρ= ρv ρA 887.31 × 5.1597 = = 33.30kg/m3 ρA x + ρv (1 − x) 887.31 × 0.15 + 5.1597 × 0.85 All the parameters necessary to use eq. (11.88) are obtained from eqs. (11.89) – (11.92): § ρ ·§ f · C1 = (1 − x) 2 + X 2 ¨ A ¸¨ v 0 ¸ © ρv ¹© f A 0 ¹ § 887.31 · § 4.105 · = (1 − 0.15) 2 + 0.4032 × ¨ ¸×¨ ¸ = 51.23 © 5.1597 ¹ © 2.270 ¹ C2 = x = 0.15 0.78 (1 − x) 0.24 § ρA · ¨¸ © ρv ¹ 0.24 0.91 § μv · ¨¸ © μA ¹ 0.19 § μv · ¨1 − ¸ © μA ¹ 0.7 0.78 (1 − 0.15) § 887.3 · ¨ ¸ © 5.1597 ¹ 0.91 § 149 × 10−7 · ¨ −7 ¸ © 1493 × 10 ¹ 0.19 § 149 × 10−7 · ¨1 − −7 ¸ © 1493 × 10 ¹ 0.7 = 14.21 Fr = We = m′′2 1018.62 = = 954.76 gD ρ 2 9.8 × 0.1 × 33.302 = 1018.62 × 0.1 = 7.385 × 104 −3 ρσ 33.30 × 42.19 × 10 Therefore, the pressure multiplier is 3.24C 3.24 × 14.21 φA20 = C1 + 0.045 20.035 = 51.23 + = 74.07 0.045 Fr We 954.76 × (7.385 × 104 )0.035 The pressure gradient due to friction is obtained from eq. (11.82), i.e., dp § dp · − F = −φA2o ¨ F ¸ = 74.04 × 3.836 × 104 = 2.836 × 106 Pa/m dz © dz ¹Ao m′′2 D 11.3.4 Void Fraction Correlations for the frictional pressure drop were discussed in the preceding subsection. To obtain the total pressure drop, accelerational and gravitational pressure drops must also be determined. As can be seen from eqs. (11.41) and (11.63), knowledge of the void fraction is required for determination of the accelerational and gravitational pressure drops for both the homogeneous and separated flow models. In the case of a horizontal circular tube, the gravitational pressure drop term becomes zero but the accelerational pressure drop terms are still present. Therefore, correlations for the void fraction in the two-phase flow will be discussed. 878 Transport Phenomena in Multiphase Systems When the phase velocities differ for the liquid and vapor phases, the slip ratio, defined in eq. (11.13), is frequently used in lieu of the void fraction. Substituting eqs. (11.20) and (11.21) into eq. (11.13), one obtains G ρ (1 − α ) S= v A (11.96) GA ρvα which can be simplified by using eq. (11.23), i.e., ρ x(1 − α ) S= A (11.97) ρv (1 − x)α The slip ratio can also be related to the volumetric flow rate by substituting eqs. (11.5) and (11.6) into eq. (11.13), i.e., QA (11.98) S= v A QA Av Substituting eq. (11.2) into eq. (11.98), one obtains Q (1 − α ) (11.99) S= v QAα The relationships between the void fraction and slip ratio can be obtained by rearranging eqs. (11.97) and (11.99): 1 α= (11.100) 1 − x ρv S 1+ x ρA Qv α= (11.101) SQA + Qv It can be observed from eqs. (11.97) and (11.99), (11.100), and (11.101) that the void fraction may be determined if the slip ratio is known, and vice versa. The void fraction in the homogeneous model obtained by eq. (11.35) is a special case of eq. (11.100) where the slip ratio S = 1. The homogeneous model is the better choice in a two-phase application for bubbly or mist flow, since the liquidvapor interface is very small and the slip ratio is very close to unity. For other flow patterns where the velocity of the vapor phase is significantly higher than the velocity of the liquid phase, the homogeneous model tends to significantly overpredict the void fraction. Another case in which the homogeneous model will provide good results is near critical point ( ρA ≈ ρv ), in which case α ≈ x . To predict the void fraction for cases where the velocities of the two phases differ significantly, correlations based on the separated flow model should be employed. The earliest void fraction correlation is based on the separated flow model proposed by Lockhart and Martinelli (1949), shown in Fig. 11.6. The relationship between the void fraction and the Martinelli parameter, X, is fitted by the following equation: φ −1 α = A ,tt (11.102) φA,tt Chapter 11 Two-Phase Flow and Heat Transfer 879 where φA ,tt is the frictional multiplier for turbulent flow in both vapor and liquid phases, obtained from eq. (11.84) with the appropriate value of C from Table 11.1, i.e., 20 1 φA2,tt = 1 + + 2 (11.103) XX Butterworth (1975) recommended the following simpler correlation to replace eq. (11.102): α = [1 + 0.28 X 0.71 ]−1 (11.104) The void fraction correlation proposed by Lockhart and Martinelli (1949) is the most widely-used correlation. However, it tends to overpredict the void fraction at high mass flow rate. A number of alternative correlations have been proposed to overcome this problem. The correlation proposed by Premoli et al. (1971) is worthwhile to introduce here because it covers a wide range of data. The correlation of Premoli et al. (1971) given in terms of the slip ratio is ª§ y · º2 S = 1 + E1 «¨ ¸ − yE2 » «© 1 + yE2 ¹ » ¬ ¼ where y= 1 (11.105) β 1− β § ρA · ¨¸ © ρv ¹ 0.22 (11.106) (11.107) −0.08 −0.19 E1 = 1.578ReA 0 −0.51 § ρ · E2 = 0.0273We ReA 0 ¨ A ¸ (11.108) © ρv ¹ where β is the volumetric flow fraction of vapor flow defined by eq. (11.12). The Weber number in eqs. (11.107) and (11.108) is defined as m′′2 D We = (11.109) ρAσ The void fraction can be obtained from eq. (11.100) after the slip ratio is obtained from eq. (11.105). Butterworth (1975) compared different void fraction correlations for twophase flow and suggested that the void fractions can be expressed in the following form 1 α= (11.110) s r q § x · § ρ v · § μA · 1+ c¨ ¸¨ ¸¨ ¸ © 1 − x ¹ © ρ A ¹ © μv ¹ 880 Transport Phenomena in Multiphase Systems Table 11.2 Constants and exponents for different void fraction models. Models Homogeneous model Zivi (1964) Turner (1966) Lockhart-Martinelli (1949) Thome (1964) Baroczy (1965) c 1 1 1 0.28 1 1 q 1 1 0.72 0.64 1 0.74 r 1 0.67 0.40 0.36 0.89 0.65 s 0 0 0.08 0.07 0.18 0.13 where the values of the constant and exponents are given in Table 11.2. Chisholm (1973b) presented a simple correlation in terms of slip-ratio ª § ρ ·º S = «1 − x ¨ 1 − A ¸ » (11.111) « © ρv ¹ » ¬ ¼ Awad and Muzychka (2005b) compared these models for two-phase flow of R-22 at Tsat = 50 °C in a smooth tube of D = 10 mm, and the results are shown in Fig. 11.8. It can be seen that no two models can provide the same prediction on the void fraction. 0.5 Figure 11.8 Comparison of void fraction models for two-phase flow (Awad and Muzychka, 2005b). Chapter 11 Two-Phase Flow and Heat Transfer 881 Based on the assumption that both liquid and vapor are turbulent (both ReA and Rev are greater than 2000), Awad and Muzychka (2005b) developed rational bounds for two-phase void fraction. The lower bound of the void fraction is α lower 0.125 16 /19 ½ 0.5 ­ª 0.875 º § ρv · § μA · ° ° §x· » = ®1 + «¨ ¾ ¨ ¸¨ ¸ ¸ » © ρA ¹ © μv ¹ ° «© 1 − x ¹ ° ¼ ¯¬ ¿ −1 (11.112) The upper bound of the void fraction is 0.71 ­ ½ ª§ x ·0.875 § ρ ·0.5 § μ ·0.125 º ° ° v A »¾ (11.113) α upper = ®1 + 0.28 «¨ ¨ ¸¨ ¸ ¸ «© 1 − x ¹ »° © ρ A ¹ © μv ¹ ° ¬ ¼¿ ¯ By averaging the lower and upper bounds, an empirical correlation for void fraction in two-phase flow can be obtained. −1 α ave (11.114) −1 0.125 0.71 ½ 0.5 ­ 0.875 ª§ x · º° § ρ v · § μA · 1° »¾ + ®1 + 0.28 «¨ ¨ ¸¨ ¸ ¸ 2° «© 1 − x ¹ »° © ρ A ¹ © μv ¹ ¬ ¼¿ ¯ Figure 11.9 shows comparisons of the void fraction with Hashizume’s (1983) experimental data for R-12 at Tsat = 50 °C in a smooth horizontal tube at D = 10 mm. The root mean square error is 9.04%. 0.125 16 /19 ½ 0.5 ­ 0.875 º § ρ v · § μA · 1 ° ª§ x · ° » = ®1 + «¨ ¾ ¨ ¸¨ ¸ ¸ 2 ° «© 1 − x ¹ » © ρ A ¹ © μv ¹ ° ¬ ¼ ¯ ¿ −1 Figure 11.9 Void fraction versus quality (Awad and Muzychka, 2005b). 882 Transport Phenomena in Multiphase Systems 11.4 Forced Convective Condensation in Tubes 11.4.1 Two-Phase Flow Regimes Convective condensation in tubes occurs in the condensers of air-conditioning systems and refrigerators. Depending on the tube’s geometric configuration, condensation within can be classified as condensation in vertical or horizontal tubes. Condensation in a vertical tube is relatively simple, because gravity acts parallel to the flow direction and, consequently, an annular liquid film forms on the inner surface of the tube. Therefore, the flow pattern for condensation in a vertical tube is limited to annular flow, and an analytical solution can easily be obtained (see Chapter 8). Condensation in a horizontal tube, on the other hand, is much more complicated, because gravity acts perpendicularly to the flow direction and various flow patterns are possible. When condensation occurs in a horizontal tube, different flow regimes are observed at different positions along its length. Two possible sequences of flow regimes for convective condensation in a horizontal tube are possible, depending on the mass flow rate (Hewitt, 1998). Figure 11.10 (a) shows the sequence of flow regimes that arise during condensation in a horizontal circular tube with higher mass flow rate. Superheated vapor enters the horizontal tube, which has a temperature below the saturation temperature of the vapor. The flow at this point in the tube is singlephase vapor flow, since no condensation occurs near the inlet. After the vapor cools and becomes saturated, condensation starts to occur on the inner wall of the tube. The flow pattern at the beginning of condensation is annular, because the velocity of the vapor is much higher than that of the condensate. The dominant force in the annular flow regime is shear stress at the liquid-vapor interface, with gravity playing a less important role. As condensation continues, the velocity of (a) Higher mass flow rate (b) Lower mass flow rate Figure 11.10 Flow patterns for condensation inside horizontal tubes. Chapter 11 Two-Phase Flow and Heat Transfer 883 the vapor phase decreases and the dominant force shifts from shear force at the interface to gravitational force. Liquid accumulates at the bottom of the tube, while condensation takes place mainly at the top portion of the tube where the liquid film is thin. The void fraction in the tube decreases as the vapor condenses downstream, and the flow patterns change to slug and plug flows. The size of the vapor plugs further decreases downstream, and the two-phase flow enters a brief bubbly flow regime before the completion of condensation. Near the outlet of the horizontal tube, the quality reduces to zero and the flow in the tube becomes single-phase liquid flow. The sequence of flow regimes for convective condensation in a horizontal tube at a lower mass flow rate is shown in Fig. 11.10(b). Superheated vapor enters the tube and the flow is single-phase vapor near the inlet, as in the high mass flow rate case. Another similarity is that the flow regime is annular flow at the commencement of condensation. As condensation continues, the condensate at the top portion of the tube flows to the bottom portion due to gravitational force. Since the mass flow rate is not high enough to produce slug or plug flow, two-phase flow in the horizontal tube enters a stratified flow regime where liquid flows in the bottom portion of the tube and vapor flows in the top portion. Unlike the case of high flow rate, the vapor at lower flow rate never completely condenses. Breber et al. (1980) used an extensive collection of data from the literature to demonstrate that the flow pattern of Taitel and Dukler (1976) is applicable to condensation in horizontal tubes. Experimental data from more than 700 condensation flow regime observations in the literature, including water, R12 and R-113, inner-tube diameters of 4.8 to 50.8 mm, and a wide range of operating parameters, were used for the comparison. With the exception of data from the smallest inner-diameter tube (4.8 mm), agreement was good with the flow regime transition boundaries as predicted by Taitel and Dukler. 2×10 101 6.0 1.0 Spray flow Annular flow Bubbly flow 0.5 Dimensionless vapor velocity j*v 100 -1 10 Stratified flow (wavy) Semiannular flow 0.01 10 -2 Slug flow 10-2 10-1 10-3 -3 10 Liquid component 1− ε 100 101 ε Figure 11.11 Flow pattern map for condensation in a horizontal tube (Tandon et al., 1982). 884 Transport Phenomena in Multiphase Systems A flow map for condensation in a horizontal tube was suggested by Tandon et al. (1982) and is shown in Fig. 11.11. The dimensionless vapor velocity is defined as xm′′ * (11.115) jv = gD ρv ( ρA − ρv ) where D is the diameter of the tube. The abscissa is the relative cross-sectional area of the tube occupied by the liquid and vapor, AA / Av = (1 − ε ) / ε . The flow regimes shown in Fig. 11.11 can be summarized in Table 11.3. The most frequently occurring flow regimes for condensation in a horizontal tube are the annular-dispersed flow and the stratified flow. The dispersed-annular flow can also occur in condensation in a vertical tube if the vapor velocity is high enough to allow the effect of gravity to disregarded. Table 11.3 Flow regimes of convective condensation in a horizontal tube. (1 − ε ) / ε * v * jv Flow regimes Stratified flow j ≤1 (1 − ε ) / ε ≤ 0.5 1≤ j ≤ 6 * v Annular-dispersed flow j >6 * v Spray flow Slug flow Semi-annular flow Bubbly flow j ≤ 0.01 * v (1 − ε ) / ε > 0.5 0.01 ≤ j ≤ 0.5 * v * jv > 0.5 11.4.2 Heat Transfer Predictions Heat transfer characteristics for convective condensation in a tube depend strongly on the flow regimes and the different structures of their liquid-vapor interfaces. Heat transfer in stratified and annular regimes has been extensively studied, and different correlations based on theoretical analysis and experimental data have been proposed. Gravitational effects can be neglected for the high mass flux region, ( m′′ > 400 kg/m 2 -s ), so that annular flow with convective condensation in a horizontal tube can be analyzed with the method applied to the vertical tube in Chapter 8. Furthermore, a detailed analysis of annular flow condensation in a miniature tube presented in Section 11.6.2 under the title “Annular Flow Condensation in a Miniature Tube” is also applicable to conventional single tube. For practical applications, the following empirical correlation, proposed by Shah (1979) from a fit of 474 experimental data points, is recommended: ª 3.8 x 0.76 (1 − x)0.04 º 0.8 Nu = 0.023ReA 0 PrA0.4 «(1 − x)0.8 + (11.116) » ( psat / pc )0.38 ¼ ¬ Chapter 11 Two-Phase Flow and Heat Transfer 885 where ReA 0 is defined by eq. (11.77) and PrA is the liquid Prandtl number. Equation (11.116) was obtained by modifying the Dittus-Boelter equation for single-phase forced convection in a tube. It is applicable to the following conditions: 0.002 ≤ psat / pc ≤ 0.44, 21 o C ≤ Tsat ≤ 310 o C, 3m/s ≤ wv ≤ 300m/s, 0 ≤ x ≤ 1 , 10.8 kg/m 2 -s ≤ m′′ ≤ 1599 kg/m 2 -s, ReA 0 ≥ 350, and PrA > 0.5. Moser et al. (1998) verified eq. (11.116) using 1197 data points for 6 different kinds of refrigerants and concluded that the deviation of eq. (11.116) is 14.37%. At lower vapor velocity, interfacial shear force becomes less important and gravity becomes the dominant force in convective condensation in the tube. As shown in Fig. 11.12, due to gravitational force, the condensate in the top portion of the tube flows into the bottom portion, and the two-phase flow lies in the stratified regime. The stratified regime is very common for low mass flow rates or short tubes. The average heat transfer coefficient for condensation in a tube with stratified flow is θ §θ · (11.117) h = strat hd + ¨1 − strat ¸ hb π π¹ © where hd and hb are heat transfer coefficients of the drainage condensate and the condensate at the bottom. Since condensation occurs primarily at the top portion of the tube where the liquid layer is relatively thin, the condensation occurring on the surface of the stratified liquid can be neglected ( hb  hd ). Thus, eq. (11.117) is simplified as θ h = strat hd (11.118) π Figure 11.12 Cross-section of stratified flow. 886 Transport Phenomena in Multiphase Systems Since condensation occurring in the drainage condensate is film condensation, one can expect that the heat transfer coefficient can be obtained using a correlation similar to the Nusselt solution. ª g ρ ( ρ − ρv )kA3 hAv º ′ hd = F (θ strat ) « A A (11.119) » ¬ μA D(Tsat − Tw ) ¼ where the function F (θ strat ) can be obtained by (Jaster and Kosky, 1976) 1/ 4 sin 2(π − θ strat ) º π ªπ + F (θ strat ) = 0.728 « » θ strat ¬θ strat 2π ¼ 3/ 4 (11.120) The void fraction, , is related to the stratification angle, strat, by sin 2(π − θ strat ) π α= + θ strat 2π Substituting eqs. (11.120) and (11.121) into eq. (11.119) yields (11.121) ª g ρ ( ρ − ρv )kA3 hAv º ′ hd = 0.728 α«AA (11.122) » θ strat μA D ¬ ¼ The average heat transfer coefficient can be obtained by substituting eq. (11.122) into eq. (11.118), i.e., π 1/ 4 3/ 4 ª g ρ ( ρ − ρv )kA3 hAv º ′ (11.123) h = 0.728α « A A » μA D ¬ ¼ The condition for which eq. (11.123) applies is Revo ≤ 3.5 × 104 . The void fraction in eq. (11.123) can be obtained by 3/ 4 1/ 4 ª 1 − x § ρ ·2 / 3 º v (11.124) α = «1 + ¨¸» x © ρA ¹ » « ¬ ¼ In addition to eqs. (11.116) and (11.123), Dobson and Chato (1998) proposed the empirical correlations that took into account the effect of different flow regimes. For annular flow and m′′ ≥ 500 kg/m 2 -s , Dobson and Chato (1998) recommended the following correlation: § hD 2.22 · Nu = = 0.023Re0.8 PrA0.4 ¨1 + 0.89 ¸ (11.125) A kA X tt ¹ © where Xtt is the Martinelli parameter for turbulent flow in both liquid and vapor phases, i.e., § 1 − x · § ρ v · § μA · X tt = ¨ (11.126) ¸¨ ¸¨ ¸ © x ¹ © ρ A ¹ © μv ¹ When the flow pattern is stratified wavy flow, the empirical correlation is 0.9 0.12 0.23Revo ª GaA PrA º hD Nu = = » 0.58 « kA 1 + 1.11X tt ¬ JaA ¼ 0.25 0.5 0.1 −1 §θ + ¨ 1 − strat π © · ¸ Nustrat ¹ (11.127) Chapter 11 Two-Phase Flow and Heat Transfer 887 where the first term on the right-hand side of eq. (11.127) accounts for the contribution of the film condensation on the top portion of the tube. The contribution of the stratified liquid is accounted for by the second term on the right-hand side of eq. (11.127). Ga A is the Galileo number for the liquids defined as gD 3 Ga A = 2 (11.128) νA The angle from the top of the tube to the stratified liquid layer in the bottom θ strat is estimated by θ strat = π − arccos(2α − 1) (11.129) where the void fraction is approximated by eq. (11.100), with the slip ratio 1/ 3 S = ( ρA / ρv ) , i.e., α= x x + (1 − x) ( ρ v / ρA ) 2/3 (11.130) which is referred to as the Zivi void fraction equation (Zivi, 1964). The Nusselt number for forced convective heat transfer of the stratified liquid, Nu strat , in eq. (11.127) is § c· Nu strat = 0.0195Re0.8 PrA0.4 ¨ 1.376 + 1 ¸ A c X tt2 ¹ © where 1/ 2 (11.131) ­4.172 + 5.48FrA − 1.564 FrA2 ° c1 = ® 7.242 ° ¯ and FrA ≤ 0.7 FrA > 0.7 (11.132) ­1.773 − 0.169 FrA FrA ≤ 0.7 c2 = ® (11.133) 1.655 FrA > 0.7 ¯ The transition from annular flow to stratified flow is characterized by the modified Froude transition number, which is defined as 0.039 § 1 + 1.09 X tt · (11.134) Frso = ¨ ¸ X tt © ¹ where c3 = 0.025 and c4 = 1.59 for ReA ≤ 1250 and c3 = 1.26 and c4 = 1.04 for ReA > 1250 . Dobson and Chato (1998) suggested that eq. (11.127) is applicable 1.5 c3 Rec4 A when m′′ < 500 kg/m 2 -s and Frso < 20 . Example 11.4 Steam condenses at 80 °C as it flows inside a horizontal tube with an inner diameter of 5 cm. The mass flow rate is 1.2 kg/s and the quality is x=0.75. Find the heat transfer coefficient. 888 Transport Phenomena in Multiphase Systems Solution: The properties of the water and steam at 80 °C can be found from Table B.48 in Appendix B: psat = 47.359 kPa , hAv = 2308.9 kJ/kg, ρA = 971.82 kg/m3 , ρv = 0.2932 kg/m3 , μA = 3510 × 10−7 N-s/m 2 , μv = 113 × 10−7 N-s/m 2 , kA = 0.669 W/m-K and PrA = μA c pA / kA = 2.2 . The mass flow rate is m = 1.2kg/s , and the diameter of the tube is D = 0.05 m . Therefore, the mass flux of the steam-water mixture in the tube is 4 × 1.2 m 4m = = 611.15kg /(m 2 s ) m′′ = = A π D π × 0.052 Since m′′ > 500 kg/m 2 -s , the heat transfer coefficient can be obtained from eq. (11.125). The Martinelli parameter for turbulent flow in both the liquid and vapor phases, Xtt, is obtained from eq. (11.126): §1− x · X tt = ¨ ¸ ©x¹ 0.9 § ρv · ¨¸ © ρA ¹ 0.5 § μA · ¨¸ © μv ¹ 0.1 0.9 0.5 −7 § 1 − 0.75 · § 0.2932 · § 3510 × 10 · =¨ = 0.0091 ¸¨ ¸¨ −7 ¸ © 0.75 ¹ © 971.82 ¹ © 113 × 10 ¹ The liquid Reynolds numbers are obtained from eq. (11.75), i.e., 0.1 611.15 × (1 − 0.75) × 0.05 = 2.176 × 104 μA 3510 × 10−7 The Nusselt number is obtained from eq. (11.125), i.e., § hD 2.22 · = 0.023Re0.8 PrA0.4 ¨1 + 0.89 ¸ Nu = A kA X tt ¹ © 2.22 · § = 0.023 × (2.176 × 104 )0.8 × 2.20.4 × ¨1 + = 1.363 × 104 0.89 ¸ © 0.0091 ¹ The heat transfer coefficient is then NukA 1.363 ×104 × 0.669 h= = = 1.824 ×105 W/m2 -K 0.05 D ReA = = m′′(1 − x) D 11.5 Forced Convective Boiling in Tubes 11.5.1 Regimes in Horizontal and Vertical Tubes Forced convective boiling in tubes finds application in evaporators for air conditioners and refrigerators, as well as in boilers. The two most common Chapter 11 Two-Phase Flow and Heat Transfer 889 configurations in practical applications are convective boiling in horizontal tubes and upward flow boiling in vertical tubes. The flow regimes and heat transfer correlations for these two configurations are different and will be discussed separately. The typical sequence of flow regimes for upward flow forced convective boiling in a uniformly-heated vertical tube ( q′′ = const ) is shown in Fig. 11.13. Since gravity acts parallel to the flow direction, the upward flow boiling in the Figure 11.13 Regimes for convective boiling in a vertical tube (Collier and Thome, 1994; Reprinted with permission from Oxford University Press). 890 Transport Phenomena in Multiphase Systems tube is axisymmetric. The gravitational force plays a more dominant role while the liquid-vapor interfacial shear force is less important. If the fluid enters the tube as subcooled liquid and leaves the tube as superheated vapor, all of the flow regimes for horizontal tubes discussed in Section 11.2 will be encountered in the interim. The void fraction increases from zero at the inlet of the tube to one at the outlet of the tube. Since the density of the vapor is significantly lower than that of the liquid, the density of the two-phase mixture significantly decreases along the flow direction. To maintain constant mass flux along the flow direction, the mean velocity must increase substantially to match the significant increase in vapor phase velocity. The growing disparity between vapor and liquid velocities along the flow direction will result in changing flow patterns along the flow direction. Subcooled liquid enters the vertical tube and prior to initiation of any bubbles (Zone A), the heat transfer mechanism is single-phase forced convection. When the rising wall temperature exceeds the saturation temperature, vapor bubbles are generated and dispersed in the continuous liquid phase. While the fluid temperature in Zone B (subcooled boiling) is still below saturation temperature, the fluid temperature in Zone C (saturated nucleate boiling) is equal to the saturation temperature. The vapor bubbles in the bubbly flow region rise due to buoyancy force and the external pressure difference. As the vaporization process continues, more liquid is converted to vapor, the void fraction increases, then the flow regime progresses from bubbly to slug, churn, and then to annular (Zones D through F). When the liquid film on the inner surface of the tube is completely evaporated, the flow pattern becomes drop or mist flow (Zone G) and the wall temperature abruptly increases. When the last droplet of the liquid is vaporized, the flow in the tube becomes single-phase (Zone H). Under constant heat flux conditions, both wall and fluid temperatures increase linearly in Zone H. Fig. 11.14 shows the typical sequence of flow regimes for forced convective boiling in a horizontal tube. Two-phase flow with boiling in a horizontal tube is no longer axisymmetric, because the gravitational force acts perpendicularly to the flow direction. When the subcooled liquid enters the horizontal tube, because no bubbles are present, the initial heat transfer mechanism is single-phase forced convection. After vapor is generated, the bubbles are dispersed in the continuous liquid phase and tend to rise to the top portion of the tube due to buoyancy effects. As boiling continues in the horizontal tube, the flow regime changes to plug, annular, and mist flow. As with the vertical flow case, the flow becomes single-phase in the horizontal tube after the last droplet of the liquid is vaporized. Subcooled liquid Bubbly flow Plug flow Annular flow Mist Superheated flow vapor vapor). Figure 11.14 Flow regimes for convective boiling in a horizontal tube ( liquid, Chapter 11 Two-Phase Flow and Heat Transfer 891 Figure 11.15 Kattan-Thome-Favrat flow pattern map for forced convective boiling in horizontal tube (Thome and El Hajal, 2003; Reproduced by permissions of Routledge/Taylor & Francis Group, LLC). Kattan et al. (1998a, b, c) studied forced convective boiling in horizontal tubes, and proposed a new flow map, shown in Fig. 11.15. The boundary to separate different flow regimes in Fig. 11.11 can be found in Thome and El Hajal (2003) or Thome (2004). In contrast to the Taitel-Dukler flow map in Fig. 11.5, which is given in generic form, the Kattan-Thome-Favrat flow map is valid for one fluid (R134a) at a given tube diameter (D = 12 mm). Many techniques have been tried to increase heat transfer coefficients in heat transfer devices. For single-phase applications, these methods concentrate mostly on disruption and destabilization of the thermal boundary layer, where two-phase flow applications rely on heat transfer enhancers or surface modification to increase boiling near the surface. However, these techniques all result in a pressure drop. One way in which heat transfer may be increased and not experience a severe pressure drop is through electrohydrodynamics (EHD). For two-phase flow, the difference in the specific dielectric constant in an electric field can contribute additional body forces, which may lead to an increase in heat transfer. Experimental and analytical studies both show a decrease in the thermal boundary layer, increased convection and interfacial instabilities that result in high heat transfer rates. Existing research on the effects of EHD on convective boiling has confirmed that EHD can significantly increase the heat transfer rates. The optimum enhancement has been attributed to the relative influence of EHD forces and inertial forces on determining the flow regime. At low qualities, EHD effects cause a flow regime redistribution that significantly influences the heat transfer. Its effect diminishes as the inertial forces become the dominant force in determining the flow pattern. Also, some studies show that the liquid extraction 892 Transport Phenomena in Multiphase Systems phenomena may promote early partial dry-out, which negatively affects heat transfer. Modeling of electric field action in two-phase flow is difficult, because fluiddynamics conservation equations are coupled with Maxwell’s and Ohm’s laws and must be solved together. An electric field in two-phase flow adds additional body forces in the fluid as well as at the liquid-vapor interface. The electric force, Fe, applied to the field is given as § ∂ε · º 1 1ª Fe = ρ e E − E 2 ∇ε + ∇ « ρ E 2 ¨ (11.135) ¸» 2 2¬ © ∂ρ ¹T ¼ where e and are the charge density and mass density, respectively, E is the electric field strength and is the permittivity. The three terms on the right-hand side in the above equation are the electrophoretic, dielectophoretic and electrostrictive components of the force, respectively. The electrophoretic force (Coulomb force) results from the net free charge in the fluid or injected form electrodes. Phase interactions are typically associated with this component. The dielectrophoretic force arises from inhomogeneity in the permittivity of the dielectric fluid due to nonuniform electric fields, temperature gradients, and phase differences. The electrostrictive force results from inhomogeneous electric field strength and the variation of the dielectric constant with temperature and density. To examine electroconvective problems, the force, Fe, must be added to the right-hand side of the Navier-Stokes equation as a body force. Some other electrodynamic equations needed in the problem are div ( ε E ) = ρe (11.136) E = −gradϕ (11.137) divJ = 0 (11.138) where e is the charge density, is the electric potential, and J is the current density given by ∂ (ε E ) J = ρe V + σ E E + (11.139) ∂t where charge convection, Ohm’s law and displacement current are accounted for, respectively. Magnetic field effects can be neglected, since the electrical current is low. The most important coupling mechanisms in the flow, temperature, and electrostatic fields are free charge convection, electrical body force Fe, and variations of and with temperature. The presence and displacement of vapor masses can change the electrostatic field distribution. The primary factor causing flow is the variation in the dielectric constant, and therefore the dielectrophoretic effect. A two-phase flow pattern map for annular channels under a DC applied voltage and the application to EHD convective boiling analysis was presented by Cotton et al. (2005). In the Cotton et al. (2005) study, the difference in dielectric constant for liquid was Chapter 11 Two-Phase Flow and Heat Transfer 893 ε A = 9.5 and for vapor ε v = 1.09 at the average saturation temperature of 24 °C for HFC-134a. Given the magnitude for conductivity in typical dielectrics, additional electrical energy consumed by the electric field is considered negligible. A dimensional analysis can be used for the equations of EHD in two-phase flow. The two dimensionless EHD numbers that will result from the analysis of the electric body force are the EHD number or conductive Rayleigh number J o L3 Ehd = (11.140) ρ o v 2 μc A where Io is the outlet current, L is the length, o is the outlet density, v is the kinematic viscosity, c is the ion mobility and A is the area. The Masuda number or dielectric Rayleigh number is given as ε o Eo2To ( ∂ε s / ∂T ) ρ L2 Md = (11.141) 2 ρo v 2 where o is the outlet permittivity, Eo is the outlet electric field strength, To is the outlet temperature, and s is the dielectric constant. The combined effects of electric and forced convection must be considered when Ehd / Re 2  1 and/or M d / Re 2 ~ 1 . If Ehd / Re 2  1 or M d / Re 2  1 electric convection effects can be neglected. If Ehd / Re 2  1 or M d / Re 2  1 forced convection effects may be neglected. This is analogous to buoyancydriven flow, and the EHD numbers can be compared to the Grashof number in the absence of forced convection. This order of magnitude helps determine the range and extent of EHD influence on the flow and needs to be identified to determine the voltage levels required to affect heat transfer through migration of the liquid. Based on dimensionless analysis (Chang and Watson, 1994) it is 2 expected that Ehd / Re1 ~ ≥ 0.1 is sufficient to define the minimum condition above which electric fields significantly influence the flow of liquid. An experimental study by Cotton et al. (2005) was conducted on the tube side boiling heat transfer of HFC-134a in a single-pass countercurrent flow heat exchanger in an electric field. The mechanics of EHD-induced flow and heat transfer have been investigated for various flow conditions by applying 0 – 8 kV to a concentric inner electrode. A theoretical Steiner type two-phase flow pattern map was developed for flow boiling in the annular channel under DC high voltage. Flow regimes (Fig. 11.16) encountered in the convective boiling process were reconstructed experimentally and compared with the EHD map. Results show that when M d  Re 2 , EHD interfacial forces have a strong influence on the flow pattern, which is considered to be the primary mechanism affecting the increase in pressure drop and the augmentation or even suppression of heat transfer. 894 Transport Phenomena in Multiphase Systems Figure 11.16 Proposed reconstructed flow pattern from surface temperature measurements and inlet and outlet flow observations for increasing DC voltage levels: (a) 0 applied voltage, (b) moderate voltage, and (c) high voltage ( m′′ = 100 kg/m2-s, q ′′ = 10 kW/m2, and xin = 0%; Cotton et al., 2005; Reprinted with permission from Elsevier). 11.5.2 Bubble Lift-Off Size in Forced Convective Boiling The bubble lift-off size, which is the bubble size when it detaches from the heating surface, can be different from bubble departure size, which is bubble size when it departs from the nucleate site. The bubble departure size for pool boiling has been discussed in detail in Chapter 10. For forced convective boiling, the bubble departure is usually analyzed by force balance along the flow direction, while the force balance normal to the flow direction plays a crucial role on Chapter 11 Two-Phase Flow and Heat Transfer 895 bubble lift-off size. Situ et al. (2005) analyzed the force balance during the bubble growth to predict the bubble lift-off size; their work will be presented here. Figure 11.17(a) shows an active nucleation site in upward forced convection subcooled boiling. The bubble is nucleated at the nucleation site and grows until it reaches to the departure size, at which point it departs from the nucleate site. The departed bubble slides along the heating surface and vaporization continues at the inner surface of the bubble. Since the liquid is subcooled, condensation may take place at the outer surface of the bubble if the size of the bubble is greater than the superheated liquid layer. Depending on the overall effects of vaporization and condensation, the bubble may either grow or shrink. The bubble eventually lifts-off from the heating surface at some distance from the nucleate site. The force balance of the bubble at the nucleate site is shown in Fig. 11.17(b). The momentum equations along the x- and y-directions are du Fsx + Fdux + Fsl = ρvVb v (11.142) dt dv Fsy + Fduy + Fp + Fg + Fqs = ρ vVb v (11.143) dt where Fs is the surface tension force, Fdu is the unsteady drag force, Fsl is the shear lift force, Fp is the pressure force, Fg is the gravity force, and Fqs is the quasi-steady force, Vb is bubble volume, and uv and vv are bubble velocity in xand y directions, respectively. (a) bubble nucleation phenomena (b) force balance at a nucleation site Figure 11.17 Physical model of bubble lift-off (Situ et al., 2005; Reprinted with permission from Elsevier). 896 Transport Phenomena in Multiphase Systems The surface tension force in the x- and y- directions are (Klausner et al., 1993) (cos θ r − cos θ a ) (11.144) θa − θr π (θ − θ a ) (sin θ a + sin θ r ) Fsy = −1.25 Dwσ 2 r (11.145) π − (θ r − θ a ) 2 where Dw is bubble contact diameter on the heating surface, and θ a and θ r are advancing and receding contact angles [see Fig. 11.17(b)]. The unsteady drag force (also referred to as growth force) is § d 2 H dH dVA · d ( ρAVA ub ) Fdu = = ρA ¨ VA 2 + (11.146) ¸ dt dt dt ¹ dt © where VA is the volume of the virtual added mass (Chen, 2003) 11 3 π Rb (11.147) 12 and H is bubble height measured from the heating wall, ub = dH / dt is the bubble front velocity in the x-direction. For a spherical bubble, H is the diameter of the bubble and the bubble front velocity becomes ub = 2dRb / dt . The unsteady drag force for spherical bubble can be expressed as 11 · 2 § 11 2 Fdu = ρ Aπ Rb ¨ Rb + Rb Rb ¸ (11.148) 6 © 12 ¹ where Rb = dRb / dt and Rb = d 2 Rb / dt 2 . The components of the unsteady drag force in the x- and y-directions are Fdux = Fdu cosθi (11.149) Fduy = Fdu sin θi (11.150) VA = Fsx = − Dwσ π The transient radius of the vapor bubble can be obtained from Zuber’s (1961) bubble growth model (Chapter 10) 2b Rb (t ) = (11.151) Ja α A t where b = 1.73 (Zeng et al., 1993) and the Jakob number is defined as ρ c (T − T ) (11.152) Ja = A A w sat ρ v hAv While eq. (11.151) can provide a good estimate for saturated boiling, the bubble radius will depend on effective Jakob number, Reynolds number, and Prandtl number (Kocamustafaogullari and Ishii, 1983) Rb (t ) = f (Ja e , Re, Pr, t ) (11.153) where ρ c ΔT Ja e = A A e (11.154) ρv hAv π Chapter 11 Two-Phase Flow and Heat Transfer 897 and ΔTe = S (Tw − Tsat ) (11.155) where S is the suppression factor. The shear lift force is (Mei and Klausner, 1994) 1 Fsl = Cl ρ Aπ Rb2 vr2 (11.156) 2 where vr = vA − vv is the relative velocity between the bubble center mass and liquid phase. The shear lift coefficient is (Klausner et al., 1993) 1 − Cl = 3.877Gs / 2 (Reb 2 + 0.014Gs2 )1/ 4 (11.157) where dv R (11.158) Gs = A b dx vr and 2R v (11.159) Reb = b r νA is the bubble Reynolds number. The pressure and gravity forces are Fp = ρA gVb Fg = − ρ v gVb (11.160) (11.161) The quasi-steady drag force is (Klausner et al., 1993) −1.54 0.65 ­ ½ º ° 2 ª§ 12 · ° Fqs = 6πρA vA vr Rb ® + «¨ + 0.862 » (11.162) ¾ ¸ 3 «© Reb ¹ » ° ° ¬ ¼ ¯ ¿ Figure 11.18 shows the force balance in the x-direction, normal to the flow direction, at the moment of bubble lift-off. Since the contact area between the Figure 11.18 Force balance of a vapor bubble at lift-off (Situ et al., 2005; Reprinted with permission from Elsevier). 898 Transport Phenomena in Multiphase Systems bubble and the heating wall becomes zero at lift-off, the effect of surface tension can be neglected. The unsteady drag force (growth force) is also negligible since the inclination angle is zero at lift-off. The force balance in the x-direction is Fdu + Fsl = 0 (11.163) Substituting eqs. (11.148) and (11.156) into eq. (11.163) yields 11 ·1 2 § 11 (11.164) ρAπ Rb ¨ Rb2 + Rb Rb ¸ + Cl ρAπ Rb2 vr2 = 0 6 © 12 ¹2 Substituting eq. (11.151) into eq. (11.164), one obtains α A 3π Cl vr2 = (11.165) tlo 22b 2 Ja 2 where tlo is the time at which a bubble lifts-off, which can be obtained from eq. (11.151), i.e., π r2 tlo = 2 lo2 (11.166) 4b Ja α A Combining eqs. (11.165) and (11.166), a dimensionless lift-diameter is obtained 4 22 / 3b 2 2 −1 * Dlo = (11.167) Ja PrA π where §v D · * Dlo = Cl Reb = Cl ¨ r lo ¸ (11.168) © νA ¹ For forced convective subcooled boiling, the Jakob number in eq. (11.167) should be replaced by the effective Jakob number defined in eq. (11.154). The bubble lift-off size predicted by (11.167) agreed well with the experimental data within an averaged deviation of ±35.2%. 11.5.3 Heat Transfer Predictions The different flow regimes have significant effects on the heat transfer characteristics of convective boiling in a tube. While heat transfer for subcooled liquid and superheated vapor can be easily handled by correlations for singlephase heat transfer, the intermediate heat transfer mechanism is complicated by phase change from liquid to vapor. After boiling is initiated in the tube, vapor bubbles are generated at certain nucleate sites while the rest of the inner surface of the tube remains in contact with the liquid. Under these conditions, the heat transfer mechanism is a combination of two parallel processes: single-phase convection in the liquid and nucleate boiling. The overall heat transfer coefficient for convective boiling in an upward vertical tube can be written as (Chen, 1963) h = FhA + Shb (11.169) Chapter 11 Two-Phase Flow and Heat Transfer 899 where hA and hb are heat transfer coefficients for single-phase convection of the liquid and nucleate boiling, respectively. F and S in eq. (11.169) are dynamic factors that modify the contributions of single-phase liquid convection and nucleate boiling, respectively The single-phase heat transfer coefficient for liquid alone can be obtained by using the Dittus-Boelter/McAdams equation: §k · 0 (11.170) hA = 0.023 ¨ A ¸ ReA .8 PrA0.4 ©D¹ The contribution of nucleate boiling is determined by using the correlation proposed by Forster and Zuber (1955) for pool boiling: ª kA0.79 c 0.45 ρA0.49 º 0.24 0.75 p ,A hb = 0.00122 « 0.5 0.29 0.24 0.24 » [Tw − Tsat ( pA ) ] [ psat (Tw ) − pA ] (11.171) « σ μA h fg ρv » ¬ ¼ The convective boiling factor F can be obtained by a regression of experimental data (Chen, 1963): − ­ 1 X tt 1 ≤ 0.1 ° (11.172) F =® −1 0.736 − X tt 1 > 0.1 ° 2.35(0.213 + X tt ) ¯ where Xtt is the Lockhart-Martinelli parameter obtained by eq. (11.126). The nucleate boiling suppression factor S is 1 S= (11.173) 1 + 2.53 × 10−6 Re1.17 TP where ReTP = ReA F 1.25 (11.174) is the local two-phase Reynolds number. Example 11.5 Calculate the flow boiling two-phase heat transfer coefficient for 300 kg/(sm2) of water at 1atm and 20% quality in an upward vertical tube with a diameter of 2 cm and a wall temperature of 140 °C. Solution: The saturation temperature at pA = 1 atm = 1.013 × 105 Pa is Tsat = 100 o C . The properties of water at this temperature are σ = 58.9 × 10−3 N/m, hAv = 2251.2 kJ/kg, ρA = 958.77 kg/m3 , ρv = 0.5974 kg/m3 , μA = 2790 × 10−7 N-s/m 2 , μv = 121 × 10−7 N-s/m 2 , c pA = 4.216 kJ/kg o C, kA = 0.68W/m- o C, and PrA = μA c pA / kA = 1.73 . The mass flux of m′′ = 300kg/s-m 2 yields a liquid Reynolds number of m′′(1 − x) D 300 × (1 − 0.2) × 0.02 = = 1.720 × 104 ReA = −7 μA 2790 × 10 The single-phase heat transfer coefficient for liquid alone can be obtained from eq. (11.170), i.e., 900 Transport Phenomena in Multiphase Systems § k · 0.8 hA = 0.023 ¨ A ¸ ReA PrA0.4 ©D¹ § 0.68 · 4 0.8 0.4 4 2 = 0.023 × ¨ ¸ × (1.72 × 10 ) × 1.73 = 9.481 × 10 W/m -K © 0.02 ¹ The saturation pressure corresponding to the wall temperature is psat (Tw ) = 3.61 × 105 Pa . The contribution of nucleate boiling is determined from eq. (11.171): ª kA0.79 c 0.45 ρA0.49 º 0.24 0.75 p ,A hb = 0.00122 « 0.5 0.29 0.24 0.24 » [Tw − Tsat ( pA ) ] [ psat (Tw ) − pA ] « σ μA h fg ρv » ¬ ¼ 0.79 ª º 0.68 × (4.216 ×103 )0.45 × 958.770.49 = 0.00122 × « 3 0.24 0.24 » −3 0.5 −7 0.29 ¬ (58.9 ×10 ) × (2790 ×10 ) × (2251.2 ×10 ) × 0.5974 ¼ × [140 − 100] 0.24 × ª3.61×105 − 1.013 ×105 º ¬ ¼ 0.75 = 4.639 ×104 W/m2 -K The Martinelli parameter for turbulent flow in both liquid and vapor phases, Xtt, is obtained from eq. (11.126): §1− x · X tt = ¨ ¸ ©x¹ 0.9 § ρv · ¨¸ © ρA ¹ 0.5 § μA · ¨¸ © μv ¹ 0.1 0.9 0.5 −7 § 1 − 0.2 · § 0.5974 · § 2790 × 10 · =¨ = 0.119 ¸¨ ¸¨ −7 ¸ © 0.2 ¹ © 958.77 ¹ © 121 × 10 ¹ The convective boiling factor F is obtained from eq. (11.172), i.e., − F = 2.35(0.213 + X tt 1 )0.736 = 2.35 × (0.213 + 0.119−1 )0.736 = 11.47 The local two-phase Reynolds number is ReTP = ReA F 1.25 = 1.720 × 104 × 11.471.25 = 3.63 × 105 The nucleate boiling suppression factor S is obtained from eq. (11.173), i.e., 1 1 S= = = 0.11 −6 1.17 −6 1 + 2.53 × 10 ReTP 1 + 2.53 × 10 × (3.63 × 105 )1.17 0.1 Therefore, the heat transfer coefficient is obtained from eq. (11.169) as h = FhA + Shb = 11.47 × 9.481 × 104 + 0.11 × 4.639 × 104 = 1.09 × 106 W/m 2 -K Based on more than 10,000 experimental data points for different fluids, including water, refrigerants, and cryogents, Kandlikar (1990, 1991) proposed the following generalized heat transfer correlation for convective boiling in both vertical and horizontal tubes: Chapter 11 Two-Phase Flow and Heat Transfer 901 h = max ª hNBD , ¬ hCBD º ¼ (11.175) where hNBD and hCBD are the nucleate boiling dominant and convective boiling dominant heat transfer coefficients, and they are obtained by hNBD = ª0.6683Co −0.2 f 2 (FrAo ) + 1058Bo0.7 F f A º (1 − x)0.8 hAo (11.176) ¬ ¼ hCBD = ª1.136Co −0.9 f 2 (FrAo ) + 667.2Bo0.7 F f A º (1 − x)0.8 hAo (11.177) ¬ ¼ where Co is the convective number, Bo is the boiling number, and FrAo is the Froude number. §ρ · Co = ¨ v ¸ © ρA ¹ §1− x · ¨ ¸ ©x¹ ′′ qw Bo = Am′′hAv FrAo = 0.5 0.8 (11.178) (11.179) (11.180) m′′2 ρA2 gD The Froude number multiplier, f 2 (FrAo ) , is 1 vertical or horizontal tubes (FrAo ≥ 0.04) ­ ° f 2 (FrAo ) = ® (11.181) 0.3 horizontal tubes (FrAo <0.04) °(25FrAo ) ¯ For liquids with a Prandtl number between 0.5 and 2000, a range that covers most fluids except liquid metal, the single-phase all-liquid heat transfer coefficient hAo in eqs. (11.176) and (11.177) is ­ kA (ReAo − 1000) PrA ( f / 2) 2300 ≤ ReAo ≤ 104 ° D 1 + 12.7(Pr 2 / 3 − 1)( f / 2)0.5 ° A (11.182) hAo = ® ReAo PrA ( f / 2) 4 6 ° kA 10 ≤ ReAo ≤ 5 × 10 ° D 1.07 + 12.7(PrA2 / 3 − 1)( f / 2)0.5 ¯ where the friction factor f in eq. (11.182) is given by −2 f = (1.58ln ReAo − 3.28 ) (11.183) Table 11.4 F f A for copper tube (Kandlikar, 1990; 1991) Fluid Water R-11 R-12 R-13B1 R-22 R-113 Ff A Fluid R-114 R-134a R-152a Nitrogen Neon Ff A 1.00 1.30 1.50 1.31 2.20 1.30 1.24 1.63 1.10 4.70 3.50 902 Transport Phenomena in Multiphase Systems The fluid-surface parameter F f A depends on the combination of the liquid and tube material used. For stainless steel tubing, F f A is taken as 1 regardless of the type of fluid. For copper tubing, F f A can be obtained from Table 11.4. The above correlation is valid for 0.001 ≤ x ≤ 0.95. Under constant heat flux conditions, the wall temperature will sharply increase when the heat transfer mechanism inside the tube suddenly changes from two-phase to single vapor phase heat transfer. Since the heat transfer inside the tube always begins with the single liquid phase, then changes to convective boiling, and finally to single phase vapor, the critical heat flux (CHF) phenomenon always occurs at a point near the exit of the tube. For this reason, CHF for convective boiling inside the tube is a local phenomenon. There are numerous empirical correlations available in the literature. The most generalized model to predict CHF for forced convective boiling is recommended by Katto and Ohno (1984). h − hin · § ′′ ′′ (11.184) qmax = qco ¨1 + K K A , sat ¸ hAv © ¹ For γ = ρv / ρA < 0.15 ′′ ′′ ′′ qco1 < qco2 ­ qco1 ′′ qco = ® ′′ ′′ ′′ ′′ ¯ min ( qco2 , qco3 ) qco1 > qco2 K K = max ( K K 1 , K K 2 ) ′′ ′′ qco1 < qco5 ′′ ′′ qco1 > qco5 (11.185) (11.186) For γ = ρv / ρA > 0.15 ′′ ­ qco1 ′′ qco = ® ′′ ′′ ¯ min ( qco4 , qco5 ) ­ K K1 KK = ® ¯min ( K K 2 , K K 3 ) (11.187) (11.188) KK1 > K K 2 K K1 ≤ K K 2 −1 where §L· ′′ qco1 = CK m′′hAv We −0.043 ¨ ¸ k ©D¹ ª º 1 ′′ qco2 = 0.10m′′hAvγ 1.33 We −1/ 3 « k » ¬1 + 0.0031( L / D ) ¼ ª ( L / D)0.27 º ′′ qco3 = 0.098m′′hAvγ 1.33 We−0.433 « » k ¬1 + 0.0031( L / D) ¼ ª º 1 ′′ qco4 = 0.0384m′′hAvγ 0.6 We−0.173 « » k −0.233 ( L / D) ¼ ¬1 + 0.280Wek (11.189) (11.190) (11.191) (11.192) Chapter 11 Two-Phase Flow and Heat Transfer 903 ª ( L / D )0.27 º ′′ qco 5 = 0.234m′′hAvγ 0.513 We−0.433 « » k ¬1 + 0.0031( L / D ) ¼ 1.043 K K1 = − 4CK We k 0.043 § 5 · 0.0124 + D / L K K 2 = ¨ ¸ 1.33 −1/ 3 © 6 ¹ γ Wek K K 3 = 1.12 1.52We −0.233 + D / L k 0.6 γ We−0.173 k (11.193) (11.194) (11.195) (11.196) (11.197) (11.198) γ= We k = ρv ρA m′′2 L ρAσ L/D < 50 50 ≤ L/D ≤ 150 L/D > 150 ­ 0.25 ° CK = ®0.25 + 0.009 [ L / D − 50] ° 0.34 ¯ (11.199) The critical heat flux predicted using eq. (11.184) agreed reasonably well with a variety of fluids, including water, ammonia, benzene, ethanol, helium, hydrogen, nitrogen, R-12, R-21, R-22, R-113, and potassium. Heat transfer coefficients for boiling in both vertical and horizontal tubes are often measured from experiments using electrical heating that result in axially and circumferentially uniform heat flux. While this approach can give reasonable boundary conditions for boiling in a vertical tube, Thome (2004) pointed out that electrical heating for boiling in a horizontal tube is not preferred, because circumferential conduction in the tube wall from the hot, dry-wall condition at the top to the colder, wet-wall condition at the bottom yields unknown boundary conditions. Therefore, countercurrent hot water heating that can provide reasonable boundary conditions is preferred. 11.6 Two-Phase Minichannels Flow and Heat Transfer in Micro- and 11.6.1 Two-Phase Flow Patterns The need for reliable, high-performance, price-competitive electronic devices, most notably electronic chips, has created demand for comparably small heat transfer devices capable of removing the required heat load within a limited temperature range. Closed two-phase devices, such as miniature/micro heat pipes and capillary-pumped loops, have been and are being used successfully for this 904 Transport Phenomena in Multiphase Systems application. Typically, these use a spreading strategy and feature a large number of small circular channels arranged in parallel rows in a rectangular body. Whatever configuration is used, the heat energy removed at the chip is transported away and rejected from the system by condensation at a remote location (Begg et al., 1999; Zhang et al., 2001). Therefore, it is necessary to gain a fundamental understanding of two-phase flow and phase change heat transfer in miniature/micro channels. Another area of application is in the refrigeration industry, since miniaturization of the heat exchanger can (a) reduce plant size, (b) lower materials costs, and (c) reduce fluid inventory (Vlasie et al., 2004). Reducing refrigerant charge in the refrigeration system can also minimize possible leakage. The transport phenomena occurring in different scales can vary significantly. In general, the scales of two-phase flow and heat transfer in channels can be classified according to the hydraulic diameter of the channels, although it also depends on the properties of the fluid and flow conditions. One classification is based on the hydraulic diameter of the channels: • • • • • Macro channels: Dh > 6mm Minichannels: 200 m < Dh ≤ 6mm Microchannels: 10 m < Dh ≤ 200 m Transitional channels: 100nm < Dh ≤ 10 m Nanochannels: Dh ≤ 100nm Flow regimes for two-phase flow in both horizontal and vertical tubes have been studied intensively, as outlined in the preceding section. However, most flow regime studies on two-phase flow were performed in conventionally sized (macroscale) tubes with hydraulic diameters greater than 6 mm. Unlike in conventionally-sized passages, in which surface tension effects are limited, surface tension in miniature/micro channels can have significant effects on flow pattern transitions, on overall hydrodynamics, and in particular on the thin films that are believed to be the dominant mechanism controlling the heat transfer characteristics. The flow regime maps for condensation and boiling in conventionally sized tubes may not be relevant to flow in miniature circular tubes, where the surface tension has a significant effect on the hydrodynamics. At present, the data regarding basic flow patterns for two-phase flow, with or without heat transfer in miniature/micro circular tubes, is still very limited. It follows that little is known about the mechanisms associated with transition in these flows. The classifications based on the size of the channel do not necessarily address the threshold where the characteristics of two-phase flow and heat transfer depart from the macroscale descriptions. For example, Barnea et al. (1983) found that the effect of surface tension is important for stratified-slug transition in horizontal flow in small-diameter (4 – 12 mm) tubes. The best Chapter 11 Two-Phase Flow and Heat Transfer 905 threshold criterion for distinguishing the microscale from the macroscale is bubble growth. With its diameter confined by the channel, a bubble can grow only in length rather than diameter. This criterion should be a function of geometry, size and fluid properties and does not exist currently. Kawaji and Chung (2004) presented a thorough review of adiabatic twophase flow in minichannels and microchannels, and suggested that the transition from minichannels to microchannels occurred between 100 and 250 μm. The flow pattern for an air-water system in circular and semi-triangular minichannels with hydraulic diameters between 1.09 and 1.49 mm was investigated by Triplett et al. (1999). They identified five flow patterns for all test sections: bubbly, slug, churn, slug-annular, and annular flow (see Fig. 11.19). Since the gravitational force no longer dominates the two-phase flow in minichannels, the stratified flow and stratified-wavy flow that appeared in the horizontal conventional channels shown in Fig. 11.4 were not observed in minichannels. Other than that, the flow patterns in the minichannels and conventionally-sized channels are similar. Figure 11.19 Flow patterns in horizontal minichannels (Triplett et al., 1999; Reprinted with permission from Elsevier). 906 Transport Phenomena in Multiphase Systems Figure 11.20 Flow patterns in horizontal microchannels (Kawaji and Chung, 2004; Reprinted with permission from Routledge/Taylor & Francis Group, LLC) The studies of flow patterns in microchannels are still quite limited at this time. The flow patterns for an air-water system in 25 and 100 μm microchannels were studied by Feng and Serizawa (1999), Serizawa and Feng (2001), and Serizawa et al. (2000). They identified five flow patterns as shown in Fig. 11.20: dispersed bubbly flow, gas slug flow, liquid-ring flow, liquid lump flow, and liquid droplet flow. Chung and Kawaji (2004) investigated in detail nitrogen gaswater two-phase flow patterns in microchannels with diameters between 50 and 500 μm; they identified slug flow, liquid-ring flow, gas core flow with a serpentine liquid film, and semi-annular flow patterns; dispersed bubbly and droplet flows were not observed. Bubbly flow does not appear in Chung and Kawaji (2004) because it requires bubbles smaller than the channel diameter and would occur at extremely low gas flow rates in adiabatic systems. Another flow pattern that can be observed in microchannels but not minichannels is liquid ring flow, which occurs when the liquid bridge between two consecutive gas slugs becomes unstable at higher flow rates. Chapter 11 Two-Phase Flow and Heat Transfer 907 11.6.2 Flow Condensation Flow Pattern in Two-Phase Condensation in Miniature and Capillary Tubes Condensation in miniature or micro channels finds its applications in electronics cooling, microscale energy, and Micro-Electro-Mechanical Systems (MEMS). Figure 11.21 shows patterns of flow particularly relevant to a capillary tube, i.e., annular, slug, plug, and bubble (Tabatabai and Faghri, 2001). At the beginning of the two-phase flow, an annular layer forms on the inside of the tube. As vapor or gas velocity decreases, it causes ripples to form on the liquid surface, leading to the formation of collars. The collar can also result from condensation of the vapor on the liquid film in the tube. Eventually, the collars grow to form a bridge. The relative size of the gap formed as a result of bridging establishes slug, plug, and bubble regimes. In a large tube, the bridge does not form because of high gravitational pull on the liquid film. The recognition of and accuracy in reporting on various flow regimes and their respective operating conditions is a major consideration in developing a flow map. Some investigators have reported more intermittent (slug and plug) regimes than others. For example, Dobson et al. (1998) reported annular, wavy, wavy-annular and mist-annular regimes for condensation of refrigerants; Soliman (1974) identified annular, semiannular, semiannular-wavy, spray annular, annular wavy, and spray regimes for condensation of refrigerants. In general, there is bound to be some subjectivity in regime reporting that reflects the accuracy of measurement, visualization, and regime identification techniques. This fact becomes even more critical in tubes of smaller diameter where flow regimes may be more difficult to observe. In general, film or interface instability can be used as a criterion for flow transitions. Rabas and Minard (1987) suggested two forms of flow instabilities occurring inside horizontal tubes with complete condensation. The two forms are distinguished by a transition Froude number. It is suggested that the first instability results from the low vapor flow rate associated with a stratified exit Figure 11.21 Flow pattern of two-phase condensation flow in a capillary horizontal tube (Tabatabai and Faghri, 2001). 908 Transport Phenomena in Multiphase Systems condition and from vapor flowing into the tube exit, which causes condensate chugging or water hammer instability. The second instability results from a high vapor flow rate, which produces an inadequate distribution of the vapor and blockage of the tube exit, in turn causing large subcooled condensate temperature variations. Instability can also occur in small-diameter tubes, due mainly to capillary blocking where the liquid film bridges the tube to form a plug. In an integrodifferential approach by Teng et al. (1999), capillary blocking was investigated in a thermosyphon condenser tube with an axisymmetric viscous annular condensate film with a vapor core. It was found that at low relative vapor velocities, surface tension was responsible for film instability in capillary tubes. At high relative vapor velocities, on the other hand, hydrodynamic force was responsible for the instability. Additionally, liquid bridges are maintained by buoyant motion of the vapor bubbles. Tabatabai and Faghri (2001) proposed a detailed flow map that emphasizes the importance of surface tension in twophase flow in horizontal miniature and micro channels. Tabatabai and Faghri’s (2001) flow map for two-phase flow in horizontal tubes plots the ratio of vapor to liquid superficial velocity as defined by j (11.200) SV = v jA versus the ratio of pressure drop due to surface tension and shear: (dp / dz )surface tension (11.201) Δpratio = (dp / dz )A ,shear + (dp / dz )g,shear is shown in Fig. 11.22. A transition boundary based on a force balance that Figure 11.22 Flow map for condensation in miniature/micro channels (Tabatabai and Faghri, 2001). Chapter 11 Two-Phase Flow and Heat Transfer 909 includes shear, buoyancy and surface tension forces is also proposed. The flow map is compared to a number of existing experimental data sets totaling 1589 data points. Comparison of the proposed map and model with previous models shows substantial improvement and accuracy in determining surface tensiondominated regimes. Furthermore, the proposed flow map shows how each regime transition boundary is affected by surface tension. Tabatabai and Faghri (2001) presented a detailed methodology for calculating various pressure drops. It is very difficult to measure the local heat transfer coefficient of flow condensation in a single small-diameter tube, because of complexities involved in controlling the flow conditions, heat flux and vapor quality. Shin and Kim (2005) performed an experimental study of flow condensation of R-134a inside circular channels with diameters of 0.493, 0.691, and 1.067 mm and square channels with hydraulic diameters of 0.494, 0.658, and 0.972 mm. The mass flux varied from 100 to 600 kg/m2-s, and the heat flux varied from 5 to 20 kW/m2. They compared their experimental results with the existing empirical correlations for conventional size, including eq. (11.116), and found that the existing correlations generally underpredict the heat transfer coefficient at lower mass flux regions. As the mass flux increases, the Nusselt number predicted by eq. (11.116) agrees well with the experimental results. Since the heat transfer capacity of a single miniature/micro channel is very low, multiple channels inevitably will be necessary for practical applications. Riehl and Ochterbeck (2002) studied convective condensation heat transfer of methanol in square channels with dimensions of 0.5, 0.75, 1.0 and 1.5 mm; the corresponding channel numbers are 14, 12, 10, and 8, respectively. The experiments were performed at two different saturation temperatures: 45 °C and 55 °C. The experimental results were correlated in the following form: Nu = We − Ja Re PrY (11.202) where ρ w2 L We = A (11.203) Ja = c pA (Tsat − Tw ) hAv m′′Dh σ (11.204) (11.205) Re = μA Re ≤ 65 ­ 1.3 ° (11.206) Y = ® 0.5Dh − 1 Re > 65 ° 2D h ¯ where w is working fluid velocity (m/s), and L is the channel length. Equation (11.202) correlated over 95% of experimental results with a relative error of less than 25%. 910 Transport Phenomena in Multiphase Systems Annular Flow Condensation in a Miniature Tube Analytical methods for condensation in a vertical tube are found to be applicable to the annular flow patterns in both horizontal tube condensation and a reduced-gravity environment. Because of the symmetry in both upward and downward flow, the analysis of condensation in a vertical tube is a problem that has been treated extensively. Many of the studies use the same type of onedimensional analysis as the classic Nusselt approach, but modified to account for pressure drop in the vapor generated by both friction at the liquid/vapor interface, and momentum exchange between the vapor and liquid arising from the condensation mass flux. The objective in this section is to model annular film condensation in miniature circular tubes where capillary phenomena can conceivably result in blocking of the tube cross section with liquid at some distance from the condenser entrance. A physical and mathematical model of annular film condensation in a miniature tube was developed by Begg et al. (1999). The physical description illustrated in Fig. 11.23(a). Vapor condenses inside a tube, forming a stationary liquid-vapor interface. At some point downstream of the inlet, all the incoming vapor is condensed, and only liquid flows in the tube cross section. This can occur due to surface tension effects that are usually neglected in film condensation models for tubes with conventional larger diameters. Overall Condenser Length Condensation Length mv ,in Vapor Tv,in Liquid Condenser Entrance L r o wv Qcond (a) z wA R in (b) Figure 11.23 Description of the physical model for annular film condensation in a miniature tube: (a) structure of two-phase flow for complete condensation, and (b) coordinate system and conventions for film condensation model (Begg et al., 1999). Chapter 11 Two-Phase Flow and Heat Transfer 911 The model condensation in miniature tube differs from traditional models for film condensation in conventionally sized tubes, due to the following features: 1. The disjoining pressure and the interfacial resistance can affect both the liquid and vapor flow for extremely small channels and film thickness. Therefore the terms containing the disjoining pressure and interfacial resistance should be included in the model. 2. The effect of the surface tension on the fluid flow is included in the model and the two principal radii of the liquid-vapor interface curvature are used in the Laplace-Young equation. 3. Liquid subcooling is accounted for in the model, since it can result in a significant variation of the wall temperature under convective cooling conditions. 4. The liquid momentum conservation equation in cylindrical coordinates is utilized and the corresponding expression for the shear stress at the liquid-vapor interface is accounted for. A steady-state mathematical model of condensation leading to complete condensation was developed by Begg et al. (1999). It includes coupled vapor and liquid flows with shear stresses at the liquid free surface resulting from the vapor-liquid frictional interaction and surface tension gradient. The model is based on the following simplifying assumptions: 1. The vapor is saturated and there is no temperature gradient in the vapor in the radial direction. 2. Heat transport in the thin film is due to conduction in the radial direction only. 3. Inertia terms can be neglected for the viscous flow in the liquid films with low Reynolds numbers. 4. Force on liquid due to surface tension is much greater than the gravitational force, thus the gravitational body force is neglected. Therefore the liquid is distributed onto the walls in an axisymmetric film. 5. The solid tube wall is infinitely thin, so that its thermal resistance in the radial direction, as well as the axial heat conduction, can be neglected. The cylindrical coordinate system used is shown in Fig. 11.23(b). Both the vapor and liquid flow along the z-coordinate. The physical situation should be described, taking into consideration the vapor compressibility and the vapor temperature variation along the channel. Also, the second principal radius of curvature of the liquid-vapor interface should be accounted for, although it is usually neglected in modeling of film flows in tubes of larger diameters. Within the assumptions considered above, the mass and energy balances for the liquid film shown in Fig. 11.23(a) yield R Q(z) · 1§ rwA ( r ) dr = (11.207) ¨ mA,in − ¸ R −δ hAv ¹ 2πρA © ³ 912 Transport Phenomena in Multiphase Systems mA ,in is the liquid mass flow rate at the condenser inlet, and Q ( z ) is the rate of heat through a given cross-section, due to phase change for z > 0 , and is defined as follows: Q ( z ) = 2π R ′′ qw ( z ) is the heat flux at the solid-liquid interface due to heat conduction through a cylindrical film with a thickness of balanced with the enthalpy change. Note that Tδ > Tw and therefore Q ( z ) in eq. (11.207) will be a negative quantity. The heat flux at the wall is Tw − Tδ ′′ (11.209) qw = k A R ln ª R / ( R − δ ) º ¬ ¼ where Tδ is the local temperature of the liquid-vapor interface. From eqs. (11.208) and (11.209), we obtain the following equation for the heat rate rejected per unit length of the tube: Tw − Tδ dQ d c p ,A mATA (11.210) = 2π kA − dz º ln ª R / ( R − δ ) ¼ dz ¬ ³ z 0 ′′ qw dz − c p ,A mATA − c p ,A ,in mA ,inTA ,in ( ) (11.208) ( ) where TA is the area-averaged liquid temperature for a given z location. For consideration of subcooling in the condensed liquid, TA is found from an area average given by TA = 2 R −(R −δ ) ³ R R −δ 2 rTA ( r ) dr 2 (11.211) where TA ( r ) is the assumed liquid film temperature profile given by the temperature distribution in a cylindrical wall. T −T r (11.212) TA ( r ) = Tδ + w δ ln R R −δ ln R −δ Substituting eq. (11.212) into eq. (11.211) and integrating results in an expression for TA (r ), the derivate of TA in the axial is approximated by dTA dTδ § dTw dTδ = + − dz dz ¨ dz dz © ·ª R −δ º ¸ « ln R » ¼ ¹¬ −1 2 2 R − (R −δ ) 2 (11.213) 2 ª§ R 1 · R2 ( R − δ ) º × «¨ ln −¸ + » 4 «© R − δ 2 ¹ 2 » ¬ ¼ where the change of film thickness in the axial direction is assumed to be negligibly small. This assumption may need to be reconsidered; however, the reduction in mathematical complexity resulting from its use is significant. Chapter 11 Two-Phase Flow and Heat Transfer 913 The axial momentum conservation for viscous flow in a liquid film in which the inertia terms are assumed to be negligible is 1 ∂ § ∂wA · 1 § dpA · (11.214) r = + ρA g sin ϕ ¸ r ∂r ¨ ∂r ¸ μA ¨ dz © ¹ © ¹ where is the inclination angle. The boundary conditions for the last equation are the nonslip condition at r = R and shear stresses at the liquid-vapor interface due to the frictional liquid-vapor interaction, τ A,v , and the surface tension gradient related to the interfacial temperature gradient along the channel. wA r = R = 0 (11.215) ∂wA dσ dTδ º 1ª (11.216) = « −τ A ,v − dT dz » ≡ E ∂r ( r = R −δ ) μA ¬ ¼ where T is the local liquid-vapor interface temperature, and the term with dσ / dT is due to the Marangoni effect. Taking into account the effect of the condensation process on the shear stress term, an expression from Munoz-Cobol et al. (1996) for annular filmwise condensation in vertical tubes with noncondensable gases is used for τ A ,v . where τ A ,v 0 is defined as the interfacial shear stress in the absence of phase change and is given by where v is the vapor density, wv is the axial vapor velocity and a′ is the ratio of the local condensation mass flow rate to the vapor mass flux rebounding from the interface, as approximated by dQ / ªπ RhAv ρv f A ,v ( wv − wA ,δ ) º (11.219) a′ = − ¼ dz ¬ δ· § f A ,v = f v ¨1 + 360 (11.220) ¸ 2R ¹ © and f A ,v is the interfacial friction factor. From an experimental study for a simulated change of phase the vapor friction factor, fv, is given as follows: f v = 16 ª1.2337 − 0.2337 exp ( −0.0363Re r ) ¼ × ¬ exp (1.2Ma ) ¼ / Re ºª º (11.221) ¬ Rer is found using the vapor suction velocity at the liquid-vapor interface due to condensation, vv ,δ , which is defined through the condensing mass flux. vv ,δ = dQ 1 dz 2π ( R − δ ) hAv ρv (11.222) τ A ,v = τ A ,v 0 a′ / ªexp ( a′ ) − 1º ¬ ¼ (11.217) τ A ,v 0 = 0.5 f A ,v ρv ( wv − wA ,δ ) 2 (11.218) Solving eqs. (11.214) – (11.216), the velocity profile is expressed as follows: 914 Transport Phenomena in Multiphase Systems ª1 ( R − δ )2 ln r º + E R − ln r 2 2 « R −r + » (11.223) ( δ) 2 R» R «4 ¬ ¼ Substituting eq. (11.223) into eq. (11.207), we obtain the following equation for the axial pressure gradient in the liquid: ª 1 §Q º · dpA = ρA g sin ϕ + μA « − mA ,in ¸ + E ( R − δ ) F » ¨ dz « 2πρA © hAv » ¹ ¬ ¼ (11.224) −1 ª R 4 ( R − δ )2 § ·º ( R − δ )2 − R 2 ¸ » ¨F + ׫ + ¨ 2 8 4 ¸» « 16 © ¹¼ ¬ where 1 dpA wA = − μA dz ( ) R 1 · R2 § ln + ¸− (11.225) ¨ 2 © R −δ 2 ¹ 4 The pressure difference between the vapor and liquid phases is due to capillary effects and disjoining pressure, pd, 3 2 −2 ­2ª ½ dδ dδ · º 1 dδ · ° ° § § cos ¨ atan pv − pA = σ ® 2 «1 + ¨ (11.226) »+ ¸ ¸ ¾ − pd R −δ dz ¹ ° © ° dz « © dz ¹ » ¬ ¼ ¯ ¿ The term with cosine on the right-hand side of this equation is due to the second principal radius of interfacial curvature. Introducing an additional variable dδ =Δ (11.227) dz eq. (11.226) can be rewritten as follows: cos ( atan Δ ) · dΔ ª 2 3/ 2 § p − pA + pd = 1+ (Δ) º ¨ v − (11.228) ¸ ¼© dz ¬ R −δ σ ¹ The integral equations of mass conservation for the vapor and liquid flows take the following form: ρv Av wv ( z ) = wv ,in ρv ,in Av ,in + Q ( z ) / hAv (11.229) (R −δ ) F= 2 ρ A AA wA ( z ) = wA,in ρA,in AA ,in − Q ( z ) / hAv 2 (11.230) Av = π ( R − δ ) is the cross-sectional area of the vapor channel, and wv ,in is the average vapor velocity at z = 0 . The compressible quasi-one-dimensional momentum equation for the vapor flow is modified to account for nonuniformity of the vapor cross-sectional area of the liquid-vapor interface, following Faghri (1995). dpv 1 ªd 2 2 = ρv g sin ϕ + « dz − β v ρv wv Av − f v ρv wv π ( R − δ ) dz Av ¬ (11.231) ( ) 2 + 2π ( R − δ ) ρv vv ,δ sin ( atan Δ ) º ¼ Chapter 11 Two-Phase Flow and Heat Transfer 915 with β v = 1.33 for small radial Reynolds numbers. The perfect gas law is employed to account for the compressibility of the vapor flow, p ρv = v (11.232) Rg Tv Therefore, d ρv 1 § dpv 1 pv dTv · = − (11.233) ¨ ¸ dz Rg © dz Tv Tv2 dz ¹ The saturated vapor temperature and pressure are related by the ClausiusClapeyron equation, which can be written in the following form: 2 dTv dpv Rg Tv (11.234) = dz dz pv hAv The seven first-order differential equations, eqs. (11.210), (11.224), (11.227), (11.228), (11.231), (11.233), and (11.234), include the following seven variables: δ , Δ, pA , Q, pv , ρv , and Tv. Therefore, seven boundary conditions are set forth at z = 0 . δ = δ in (11.235) Δ=0 (11.236) 2σ pA = pv ,in − + pd (11.237) R − δ in Q=0 (11.238) pv ≡ pv ,in = pv , sat (Tv ,in ) (11.239) (11.240) (11.241) pv ,in Rg Tv ,in ρv ,in = Tv = Tv ,in In the boundary condition given by eq. (11.235), in is defined from the condition that in the adiabatic zone, just before the entrance of the condenser, the liquid and vapor pressure gradients should be equal. To find in, eqs. (11.224) and (11.231) should be solved for the case of Q = 0 , vv ,δ = 0 , dAv / dz = 0 and dTδ / dz = 0 . The boundary condition, eq. (11.238), directly follows from eq. (11.208). There are also parameters mA ,in and wv ,in , and an additional variable, T , involved in this problem. They will be considered using additional algebraic equations. The parameter mA ,in should be found using a constitutive condition at the entrance of the condenser mA ,in = mt − Qin / hAv (11.242) where Qin is the total heat load into the condenser. Also wv ,in = Qin / ( hAv ρv Av ,in ) . 916 Transport Phenomena in Multiphase Systems The liquid-vapor interface temperature, T , differs from the saturated bulk vapor temperature because of the interfacial resistance and effects of curvature on saturation pressure over liquid films. The interfacial resistance is defined as (Chapter 5) § 2α · hAv ª pv ( psat )δ º ′′ (11.243) qδ = − ¨ − « » ¸ Tδ » © 2 − α ¹ 2π Rg « Tv ¬ ¼ where pv and (psat) are the saturation pressures corresponding to Tv and T , the temperatures associated with the thin liquid film interface, respectively. The following two algebraic equations should be solved to determine T for every point along the z-direction. The relation between the saturation vapor pressure over the thin condensing film, ( psat )δ , affected by the surface tension, and the normal saturation pressure corresponding to T , psat (Tδ ) , is given by the following equation (Chapter 5): ª ( p ) − p (T ) − σ K + pd º (11.244) ( psat )δ = psat (Tδ ) exp « sat δ sat δ » ρA Rg Tδ « » ¬ ¼ where K is the local curvature of the liquid-vapor interface defined by the term in ′′ outer brackets in eq. (11.226). Notice that under steady conditions, qδ is due to heat conduction through the liquid film. It follows form eq. (11.209) and (11.243) hAv ª pv ( psat )δ º R R § 2α · ln (11.245) Tδ = Tw + − « » ¨ ¸× kA ( R − δ ) R − δ © 2 − α ¹ 2π Rg « Tv Tδ » ¬ ¼ Equations (11.244) and (11.245) determine the interfacial temperature, T , and pressure, ( psat )δ , for a given vapor pressure, pv = pv , sat (Tv ) , temperature of the solid-liquid interface, Tw, and the liquid film thickness, . Note that Tw is the local temperature of the wall and can vary along the condenser depending on the cooling conditions. For the case of variable wall temperature, Tw becomes an additional variable. If the convective heat transfer coefficient at the outer tube wall, ho, and the cooling liquid temperature, T∞ , are known, the local wall temperature can be defined using an energy balance 1 dQ ho (Tw − T∞ ) = (11.246) 2π R dz To solve this problem, the first-order ordinary differential equation, eq. (11.246), must be added to the seven previously specified. The additional boundary condition is given by (11.247) z = 0 , Tw = Tw,in Equations (11.210), (11.224), (11.227), (11.228), (11.231), (11.233) and (11.234) with corresponding boundary conditions have been solved using the standard Runge-Kutta procedure by Begg et al. (1999). Algebraic eqs. (11.244) Chapter 11 Two-Phase Flow and Heat Transfer 917 and (11.245), with two unknowns, ( psat )δ and T , have been solved numerically for every point on z using Wegstein’s iteration method. The length of the two-phase zone is given by the numerical solution. At the end of the two-phase zone, d Δ / dz became infinitely large, and the film thickness increased dramatically, causing the solution to break down. Two cases of heat-load-in are presented, one representing complete condensation and the other incomplete condensation. Complete condensation is defined as the condensation of all incoming vapor in a filmwise manner, in which the vapor flow terminates at a well defined location, forming a steady meniscuslike interface. The condition of complete condensation requires the overall energy balance to be satisfied. Thus, the energy convected into the tube, or heatload-in, Qin (W), must be equal to the cumulative heat rate rejected from the condensing vapor in the tube, referred to simply as Q (W). This occurs over a distance from the inlet to the tube known as the condensation length, L (m). Incomplete condensation is said to exist for a heat-load-in greater than Qin (W) for complete condensation. Figure 11.24(a) shows variation of the liquid film thickness along the condenser with a vapor inlet temperature of 363 K and a constant wall temperature of 340 K for two cases of heat-load-in, Qin =10 W and Qin = 12 W. The total mass flow rate of mt = 1.00 × 10−6 (kg/s) is the same for both cases. The film thickness decreases in the downstream direction before converging due to the capillary forces acting at the liquid-vapor interface. Condensation is more intensive in the region where the film thickness is at the minimum. The values of in will be different for each case of heat-load-in even though the total mass flow rate and thermal boundary conditions are the same. At the inlet, the mass flow rates of liquid and vapor adjust themselves to satisfy the energy balance. The vapor pressure drop across the condensation length is insignificant compared to the liquid pressure drop as shown in Figs. 11.24(b) and (c). Figure 11.24(d) shows the cumulative sum of heat removed from the vapor by condensation. For the case of Qin =10 W, representing complete condensation, all the incoming vapor is condensed and a location is calculated for the beginning of single phase (liquid) flow in the tube. Thus all the heat transfer and associated pressure drop occur upstream of this location. Although the incomplete condensation case represents a valid numerical solution for equations modeling axisymmetric film condensation, it must be considered with reference to an overall energy balance to provide a meaningful physical interpretation. For the case of incomplete condensation, Qin =12 W, not all of the heat-load-in to the tube is rejected at the termination of the calculation. At the termination of the calculation the final value of Q at z = Lδ is seen to be less than Qin. Thus, the overall energy balance is not satisfied. During complete condensation, the values of the dependent variables ( in, pA , pv, Q) are the steady field solution of the two-phase flow and are independent of time as calculated and as interpreted physically. The simple physical 918 Transport Phenomena in Multiphase Systems 900 δ (μm) 600 300 0 14 12 10 8 6 4 2 0 -40 Qin = 10 W (complete condensation) Qin = 12 W (incomplete condensation) (a) (pv – pv.in)(Pa) (b) (pl – pv.in)(Pa) -80 -120 -160 -200 0 -2 -4 -6 -8 -10 -12 0 (c) Q (W) (d) 500 100 150 z(μm) 200 250 300 Figure 11.24 Annular film condensation in a circular tube with constant wall temperature boundary condition. R = 1.55mm, Tw = 363 K, m = 0.01 g/s. (a) Film thickness versus position, (b) vapor pressure versus position, (c) liquid pressure versus position, (d) cumulative heat rejected versus position (Begg et al., 1999). interpretation is that the calculated liquid film profile is in fact stationary and goes to the radius of the tube at L . Obviously the vapor mass flow rate becomes zero at this point also. However, during incomplete condensation, the numerical solution is interpreted as representing a time-averaged steady-state two-phase flow field where the vapor mass flow rate does not terminate at the calculated condensation length. For values of Qin greater than but close to the maximum heat-load-in, the incomplete condensation solution predicts that two-phase flow will exist downstream of the point calculated for L . By implication, the flow field will no longer be steady in an instantaneous sense. Observations from a flow visualization experiment of water vapor condensing in a horizontal glass tube, by Begg et al. (1999), confirm the existence and qualitative features of annular film condensation leading to the complete condensation phenomenon in small diameter (d<3.5 mm) circular tubes. Chapter 11 Two-Phase Flow and Heat Transfer 919 11.6.3 Flow Evaporation and Boiling Onset of Nucleate Boiling in Microchannel Flow The first onset of the vapor bubble marks the transition from the single-phase flow in the microchannel to two-phase flow, and for this reason, the prediction of onset of nucleate boiling (ONB) is very important. Liu et al. (2005) thoroughly reviewed the existing works on ONB in subcooled boiling. Hsu (1962) proposed the ONB heat flux correlation based on the minimum superheat criterion for pool boiling k h ρ (T − T ) 2 ′′ qONB = A Av v w sat (11.248) 12.8σ Tsat The effect of contact angle on the ONB heat flux was considered by Davis and Anderson (1966), who proposed the following correlation k h ρ (T − T ) 2 ′′ qONB = A Av v w sat (11.249) 8(1 + cos θ )σ Tsat Both eqs. (11.248) and (11.249) are obtained based on pool boiling, but they are also used to predict ONB in conventional channels. Liu et al. (2005) proposed an analytical model for ONB in microchannels (grooves) as shown in Fig. 11.25, which will be presented here. The physical model of ONB in a microchannel is shown in Fig. 11.26. It is assumed that (1) the bubble shape is a truncated sphere, (2) the liquid temperature is not affected by the bubble nucleus due to its small size, and (3) the vapor and liquid are in equilibrium under saturation. The condition required for the bubble nucleus to grow is that the liquid temperature at y = yb is greater than the superheat requirement [see Fig. 11.26(b)]. wc ww L Hc W q′′ Figure 11.25 Geometric configuration of the microchannels. 920 Transport Phenomena in Multiphase Systems (a) Bubble nucleus at incipience (b) ONB superheat criterion T The superheat equation for the bubble nucleus can be obtained from equilibrium theory 2σ Tb Tb − Tsat = (11.250) hAv ρ v Rb where Tb is the liquid temperature required for the bubble nucleus to grow and Tsat is the saturation temperature corresponding to the liquid pressure pA . Since the bubble shape is a truncated sphere, we have (Fig. 11.26) yb = Rb (1 + cos θ ) (11.251) rc = Rb sin θ (11.252) Substituting eq. (11.251) into eq. (11.250) and solving for Tb , one obtains § 2σ C · Tb = Tsat ¨1 − ¸ © hAv ρ v yb ¹ Figure 11.26 Onset of Nucleate boiling in microchannel (Liu et al., 2005; Reprinted with permission from Elsevier) (11.253) where C = 1 + cos θ . The liquid temperature near the wall of the microchannel can be assumed as ′′ TA ( y ) = Tw − qw y / kA (11.254) ′′ is the effective wall heat flux at the inner wall of the microchannels. where qw If a constant heat flux, q′′ , is applied to the bottom, the liquid temperature at the outlet is related to that at the inlet by q′′WL TA ,out = TA ,in + (11.255) ρ A cA u0 (nwc H c ) where u0 is the liquid inlet velocity, and n is the number of microchannels. If the convective heat transfer coefficient is uniform along the channel surfaces and Chapter 11 Two-Phase Flow and Heat Transfer 921 flow in the channel is fully-developed, the channel wall temperature can be obtained by ′′ qw (11.256) Tw = TA + NukA / Dh where Nu is the Nusselt number for fully developed flow in a three-sided (bottom and side walls) rectangular channel (Shah and London, 1978) Nu = 8.235(1 − 1.883/ α + 3.767 / α 2 − 5.814 / α 3 + 5.361/ α 4 − 2 / α 5 ) (11.257) where α is the aspect ratio of the microchannel. ′′ The effective wall heat flux, qw , is related to the applied heat flux by §α wc + ww · ′′ (11.258) qw = ¨ ¸ q′′ © 1 + 2ηα H c ¹ where is the fin efficiency. The condition for nucleation boiling to occur is TA ( yb ) ≥ Tb . The necessary condition for ONB can be obtained from eqs. (11.253) and (11.254), i.e., § q′′ 2σ C · Tw − w yb = Tsat ¨ 1 − (11.259) ¸ kA © hAv ρv yb ¹ which can be rearranged as an equation of yb, ′′ ′′ qw 2 § 2σ C qw · 2σ C yb − ¨ Tw + Tw = 0 ¸ yb + ρ v hAv kA ¹ kA hAv ρv yb © ′′ ′′ § qw 2σ C 2σ C qw · Tw ≥ 0 ¨ Tw + ¸ −4 ρv hAv kA ¹ kA hAv ρv yb © which can be rearranged to yield the superheat criterion ′′ 2σ C qw Tw − Tsat ≥ ρv hAv kA 2 (11.260) For the roots of eq. (11.260) to be real, the following condition must be satisfied (11.261) (11.262) Equation (11.262) can be used to determine if nucleate boiling will occur in the microchannels under specified parameters. If the liquid inlet conditions are prescribed but the heat flux is allowed to vary, the ONB heat flux can be obtained by §α wc + ww · ′′ ¨ ¸ qONB q′′WL © 1 + 2ηα H c ¹ + + − Tsat ρ A cA u0 (nwc H c ) Nu kA / Dh TA ,in = §α wc + ww · ′′ ¨ ¸ qONB 2σ C © 1 + 2ηα H c ¹ ρv hAv kA (11.263) 922 Transport Phenomena in Multiphase Systems If nucleate boiling is to be avoided or delayed for some application, one can increase the liquid inlet velocity u0, or reduce the liquid inlet temperature, TA ,in . In this case, eq. (11.263) can be used to obtain the minimum liquid velocity or maximum liquid inlet velocity to avoid nucleate boiling. The wall superheat can be obtained from inequality eq. (11.262) as ′′ ′′ 2σ C qw 2σ C qw + 2 Tsat (11.264) Tw − Tsat = ρv hAv kA ρv hAv kA Substituting eq. (11.258) into eq. (11.264), the wall superheat becomes wc + ww · 2σ C § α Tw − Tsat = ¨ ¸ q′′ ρv hAv kA © 1 + 2ηα H c ¹ (11.265) wc + ww · 2σ C § α +2 Tsat ¨ ¸ q′′ ρv hAv kA © 1 + 2ηα H c ¹ Flow Evaporation in Minichannel Heat Sink with Axial Grooves New advanced cooling technologies are needed to meet the demands for dissipating extremely high heat fluxes (over 300 W/cm2) from electronic components. The performance characteristics of a flat high heat flux evaporative mini-channel heat sink with axial capillary grooves on its inner surface, as shown in Fig. 11.27, was modeled by Khrustalev and Faghri (1996). The small axial grooves allow this heat sink to withstand high heat fluxes with a small pressure drop, low thermal resistance, and a comparatively large effective length. To evaluate the advantages of this design compared to existing designs, the maximum heat flux, thermal resistance, and the pressure drop on the element will be estimated for some independent operational parameters, such as the saturatedvapor outlet temperature Tv ,o , the geometry of the grooved surfaces, and the effective lengths of the evaporators under consideration. The cross-sections of the heat sink and the coordinate system are shown in Fig. 11.27. Introduced by a pump, the liquid enters the heat sink and, under the influence of an extremely high heat flux, boils on the wall at a short distance from the entrance. While the vapor phase appears to be moving along the zcoordinate, the vapor and liquid cocurrent flows become separated, with the liquid flowing mostly in the grooves. The separation of the phases in a small, horizontal, flat rectangular channel with smooth walls (19.05×3.18 mm2) has been experimentally investigated by Wambsganss et al. (1991). They have observed the annular flow pattern for a water-air mixture starting with an air velocity of about 10 m/s. Wilmarth and Ishii (1994) examined two test sections with gap widths of 1 and 2 mm with channel widths of 20 and 15 mm, respectively. In their experiments, the annular flow regime was observed with air velocity of about 5 m/s. For the heat sink with a flow channel cross-section of 7×1 mm2, and the heat flux of 100 W/cm2 on both side walls, it can be estimated Chapter 11 Two-Phase Flow and Heat Transfer 923 Figure 11.27 Schematic of the flat miniature evaporator with axial capillary grooves (not to scale; Khrustalev and Faghri, 1996). that the velocity of the vapor reaches 5 m/s at the distance of about 3.3 mm from the entrance (for Tv = 100 °C). This means that for such a heat sink, the flow boiling length preceding the annular flow pattern would be very small, even with smooth walls. Since surface tension tends to pull the liquid into the grooves, the flow boiling length should be even smaller, and the annular regime transforms into the separated cocurrent two-phase flow for the considered heat sink. Surface tension seems to be a stabilizing factor for the liquid flow in a groove; it 924 Transport Phenomena in Multiphase Systems predominates over the destabilizing dynamic vapor-liquid interaction when the Weber number is less than unity. An expression for the Weber number – including the empirically-adjusted characteristic size for a capillary-grooved surface with vapor flow over it – has been given by Vasiliev et al. (1984). Their expression is based on the analysis of the experimental data obtained from the gravity-assisted heat pipe experiments with several fluids, including water. ρ w2 D 2 We ≡ v v h ,A < 1 (11.266) 2πσ W where W is the half-width of the groove. Khrustalev and Faghri (1997) confirmed the importance of the entrainment limitation through experimental results obtained for a miniature heat sink with axial grooves. Therefore, with the inequality given by eq. (11.266) satisfied, the emphasis is on the mode where vapor and liquid flows are separated with the liquid flowing exclusively in the grooves. Since the inlet liquid is supposed to be saturated, and the cross-section of the vapor channel is very small, it is assumed in this analysis that LA = Lb = 0 (Fig. 11.27), which means that only the part of the evaporator with separated vapor and liquid flows is modeled. Another significant feature of this evaporator is that the formation of vapor bubbles in the liquid, which can occur with high heat fluxes, does not obstruct the liquid flow in the grooves (unlike in heat pipes) due to the cocurrency of the liquid and vapor flows. Perturbations in the liquid flow caused by a growing vapor bubble are compensated by the dynamic vapor-liquid interaction during the bubble’s explosion into the vapor space, with the subsequent return of the accelerated liquid droplets to the liquid flow. Moreover, for comparatively moderate heat fluxes, the liquid in the grooves does not boil along most of the effective length, because the liquid superheat is not sufficient for boiling due to the intensive evaporation of the liquid from the surfaces of the thin films attached to the liquid-vapor menisci corners. The analysis also attempts to predict the onset of nucleate boiling using the methodology by Khrustalev and Faghri (1997). Another essential feature of this evaporator is that the liquid flow in the grooves is strongly influenced by the vapor flow. The vapor-liquid frictional interaction can be the most important factor of those affecting the hydraulic resistance of the liquid flow. Another factor of importance to the liquid flow is the axial gradient of the capillary pressure, due to the variation of the main radius of the liquid-vapor meniscus along the groove. This variation takes place not far from the entrance region of the evaporator (see Fig. 11.27), where the grooves are nearly filled with liquid. Farther from the entrance, starting from the point on z where the meniscus main radius Rmen reaches its minimum value Rmen,min, the effective meniscus height Hmen decreases along the z-coordinate, so that at some point closer to the evaporator outlet, the liquid-vapor meniscus interface nearly touches the bottom of the groove. At this interval along z, the liquid flows in a groove and is influenced by the pressure gradient in the vapor and the vaporliquid frictional interaction at the interface. However, the surface tension Chapter 11 Two-Phase Flow and Heat Transfer 925 indirectly controls the liquid distribution by collecting the occasional liquid droplets from the walls of the grooves. With a heat flux close to the maximum, the effective meniscus height, Hmen, can become very small at the outlet of the evaporator; therefore, this region, which can be called the “thin film flow region,” requires a special treatment. Under close examination, it is evident that the bottom of a capillary groove rarely is rectangular, a shape that is essential for modeling the “thin film flow region” (but can be practically insignificant for other regions). This depends upon the manufacturing process used. When the electro-discharge machining process (EDM) is used to manufacture grooves on the inner evaporator surface, the groove bottom is rounded, and the liquid cross-section has a very complicated form. This reality should be kept in mind, even though the present analysis and its model are simplified by use of a rectangular strip to approximate the liquid cross-sectional shape in the “thin film flow region.” Simultaneously, the axial capillary pressure gradient is assumed to be negligible in this region in comparison to the vapor pressure gradient, which is consistent with the description of the previously discussed region with a larger Hmen. With extremely high heat fluxes, dryout of the evaporator can occur for two reasons: (1) hydrodynamic limitations of the fluid transport (the corresponding heat flux is ′′ denoted as the maximum, qmax ), or (2) the beginning of vapor film boiling (the corresponding heat flux is usually referred to as the critical heat flux, CHF). The ′′ present model can predict qmax but CHF should be determined experimentally. At any axial location, the following mass conservation equation must hold over the cross-section of the evaporator modeled under steady-state conditions: mA ( z ) = mA ,o + mv ,o − mv ( z ) (11.267) where mv = wv ρv Av , mA = NwA ρA AA , N is the number of grooves, and the subscript “o” denotes the evaporator outlet. For laminar liquid flow in a groove, the momentum conservation equation is 2 dpA 2 ρ w2 2mA + ρA g sin φ = − f A A A ≡ − f A (11.268) dz Dh ,A Dh ,A ρA AA2 N 2 where φ is the inclination angle. The inertia terms in eq. (11.268) are neglected, since Khrustalev and Faghri (1994) demonstrated that this simplification is true even when the liquid cross-sectional area varies along the flow. For vapor flow in a channel with a small Mach number (Ma  1), the momentum conservation equation is (Faghri, 1995) 2 ρ w2 d 2 ( pv + ρv β v wv ) + ρv g sin φ = − f v v v (11.269) dz Dh ,v The Laplace-Young equation relates the interfacial radius of curvature to the pressure difference between the liquid and vapor. In differential form, the equation is 926 Transport Phenomena in Multiphase Systems · dpv dpA (11.270) − ¸= dz ¹ dz The mass and energy conservation balances relate the vapor mass flow rate with ′′ the heat flux at the inside surface of the vapor channel, qv : ′′( z ) dmv 2qv = (tt − 2tw 2 ) (11.271) dz hAv where ′′ qv = he (Tw − Tv ) he is the local effective heat transfer coefficient describing the evaporative heat transfer from the wall of the evaporator to the saturated vapor, through the liquid films and the fin between the grooves. Tw is the local mean wall temperature. ′′ Note that qv = q′′ for cases in which axial conduction in the wall is insignificant and where q′′ is the heat flux on the outer surfaces of the heat sink. These conservation equations should be solved simultaneously for the inlet interval of z where Rmen varies and tmen = tg (note that Hmen is not constant). However, for the outlet z-interval at the heat sink outlet (see Fig. 11.27), the liquid recedes into the grooves, and tmen varies with z (as does Hmen) while Rmen = Rmen,min. At this “outlet” interval, Hmen is one of the unknown variables, while the pressure gradients in the liquid and vapor are equal: dpA / dz = dpv / dz [see Eq. (11.270)]. For the “outlet” interval, taking the equality of the pressure gradients into consideration, the following transcendental equation for Hmen can be obtained from eqs. (11.267) and (11.268): ª D ρ A2 N 2 § dpv ·º mA ,o + mv ,o − mv ( z ) = « − h ,A A A (11.272) ¨ dz + ρA g sin φ ¸ » 2 fA © ¹» « ¬ ¼ which should be solved for Hmen at every point on z since Dh ,A , AA , f A and 1/ 2 d§ σ ¨ dz © Rmen dpv / dz depend on Hmen. According to eq. (11.271), the vapor mass flow rate at the outlet is Ls dz ′′ mv ,o = 2 qv ( z )(tt − 2tw 2 ) (11.273) 2 hAv The coefficients, f v , f A , Dh ,A , Dh ,v and others appearing in the governing ³ equations, can be found in Khrustalev and Faghri (1996) and will not be repeated here. For the given temperature of the vapor at the outlet, Tv,o, the boundary conditions for eqs. (11.269) and (11.271) at the right-hand end of the evaporator (z = Ls) are pv = psat (Tv ,o ) (11.274) mv = mv ,o (11.275) Chapter 11 Two-Phase Flow and Heat Transfer 927 Taking into consideration that hAv varies along the z-coordinate due to the axial vapor temperature gradient, mv ,o – estimated initially by eq. (11.273) at Tv ,o – should be adjusted so as to satisfy the following condition: mv z =0 = 0 (11.276) When predicting the maximum heat flux, the liquid flow rate at the outlet of the evaporator is infinitesimal: mA ,o = ε mv ,o where ε is a very small number set equal to 10-3 in the numerical experiments. In this case the heat flux, q′′ , should be chosen so as to satisfy the following boundary condition at the beginning of the evaporator (nearly planar liquid-vapor interface): cos θ men ,min 1 = (11.277) Rmen z =0 th / 2 − tw1 − t g which implies that the liquid and vapor flows become separated at z = 0 under the influence of the capillary forces. In cases of a given heat flux below the maximum heat flux, mA ,o should be chosen so as to satisfy eq. (11.277). The numerical procedure begins at z = Ls and moves toward the left-hand end of the vapor zone. At some point on z – which can be denoted as z1 – the condition tmen = tg is reached, and, starting from this point, eqs. (11.268) – Figure 11.28 Performance characteristics of the miniature evaporator along the axial direction: (a) effective height and radius of the liquid-vapor meniscus; (b) vapor and liquid mass flow rates; (c) vapor and liquid pressure variations; and (d) vapor and wall temperature variations, and effective heat transfer coefficient (Khrustalev and Faghri, 1996). 928 Transport Phenomena in Multiphase Systems (11.271) should be solved for pA , pv , Rmen and mv with the following boundary conditions (at z = zI): pA = pv z = z1 − σ Rmen,min z = z1 (11.278) (11.279) (11.280) (11.281) pv = pv mv = mv Rmen = Rmen,min z = z1 Khrustalev and Faghri (1996) solved the above problem numerically, and an example of the characteristics of the copper-water evaporator for the following operating parameters was presented (see Fig. 11.28): W = 0.063 mm , L1 = 0.02 mm , Lt = 80 mm , t g = 0.229 mm , th = 2.58 mm , tt = 7.75 mm , tw1 = 0.5 mm , φ = 0 , θ men,min = 33D , α = 0.05 , Tv ,o = 353 K and q′′ = 65 W/cm 2 . A grooved structure featuring such a thin fin was reported by Plesch et al. (1991). In most numerical experiments, the overall meniscus height at the outlet, tmen,o was larger than tmen,min , and in all cases H men,o was larger than the roughness, Ra. This means that the concept of the “thin film region” did not practically affect the results obtained. Boiling Heat Transfer in a Miniature Axially-Grooved Rectangular Channel Different configurations of miniature heat sinks are needed for the varied applications. The most important characteristics of miniature heat sinks are listed below (Khrustalev and Faghri, 1997): 1. 2. 3. 4. 5. 6. 7. Heated length and external thickness Uniformity of temperature over the surface Effective heat transfer coefficients (or thermal resistance) Maximum attainable heat fluxes on the surface Pressure drop and mass flow rate Stability of operation Operating temperature versus absolute pressure range Obviously, heat sinks with low thermal resistance, and those of sufficient length and capability to withstand high heat fluxes, would be more practical for various technical applications. For this reason, it would be useful to enhance the inner surfaces of the miniature heat sinks. However, the mechanisms of heat transfer and those leading to dryout phenomena in two-phase miniature channels with enhanced surfaces are not completely understood, due to their complexity, especially for the case of discrete, in-line sources addressed by Maddox and Mudawar (1989) and Gersey and Mudawar (1993). While axially-grooved Chapter 11 Two-Phase Flow and Heat Transfer 929 Figure 11.29 Boiling heat transfer in the miniature axially-grooved heat sink (Khrustalev and Faghri, 1997). evaporators have been used in heat pipes (Plesch et al., 1991; Faghri, 1995), Khrustalev and Faghri (1997) attempted to foster a better understanding of the performance of miniature two-phase heat sinks in a minichannel with forced convection. The goal of the experimental investigation presented by Khrustalev and Faghri (1997) was to obtain and analyze the performance characteristics of a flat copper-water miniature two-phase heat sink with small axial grooves on the inner walls, as shown in Fig. 11.29. These capillary grooves provide significant heat transfer enhancement during evaporation and boiling of the liquid, and accelerate rewetting of the inner surface after occasional local dryout. Moreover, since the liquid can flow along the axial grooves under the influence of the capillary pressure and the pressure gradient along the flow channel, the heated length can be larger for the considered configuration than it would be for miniature channels with smooth walls. 930 Transport Phenomena in Multiphase Systems The heat pipe-type (that is, pure evaporative) regime has been modeled by Khrustalev and Faghri (1996; see the preceding subsubsection) for moderate heat loads ( q′′ ≤ 65 W/cm 2 ). Because surface tension tends to pull the liquid into the grooves, the flow boiling length should be even smaller than that for the case with smooth walls, so that the annular regime transforms into separated cocurrent two-phase flow. Surface tension seems to be a stabilizing factor for liquid flow in a groove, and it predominates over the destabilizing dynamic vapor-liquid interaction when the Weber number is small. As mentioned above, the liquid flows in the grooves under the influence of the capillary pressure gradient and the vapor pressure gradient, which can be expressed by the Laplace-Young equation in differential form. The test module shown in Fig. 11.30 was manufactured from a plate of oxygen-free copper using an electro-discharge machining process (EDM). The module consisted of a rectangular cross-section channel with an axially grooved inner surface on the side walls and six thick-film resistors soldered onto the outer surfaces (three resistors per side). The test section had the following dimensions: tt = 7.7 mm , th = 2.6 mm , Lt = 50 mm , and tw1 = tw 2 = 0.5 mm . There were 27 grooves on each of the two inner surfaces of the side with the shape and dimensions shown in the microphotograph of Fig. 11.29(c). The half-width of each groove, W, was about 0.11 mm. The ends of the test module were inserted into copper tubes with crimped ends and soldered so that 44 mm of the section length was available for installation of the resistors. The surface area of a resistor soldered onto the section wall was 6×12 mm2. The gap between the resistors was about 1 mm, so that the maximum effective heated length was about 40 mm. As a Figure 11.30 Test section with six thick film resistors (Khrustalev and Faghri, 1997). Chapter 11 Two-Phase Flow and Heat Transfer 931 result of the manufacturing process, the test section had two longitudinal fins with cross sections of 1×1 mm2 along the centerlines of the 2.6 mm wide lateral walls. Nine copper-constantan thermocouple beads were soldered directly to the test section 5 mm apart, in the corner between the section wall and the fin (Fig. 11.30). This allowed for the observation of local fluctuations in the wall temperature along the heat sink. The thermocouple beads were covered with a high-temperature, low-thermal conductivity cement layer. The test section and the connecting copper tubes were thoroughly insulated to prevent heat losses into the environment. Heat losses from the heat sink were estimated to be less than 1% of the total heat load. The heat flux on the wall of the heat sink was restricted by dryout, which led to a drastic increase in the heated wall temperature. Khrustalev and Faghri (1997) noted that the entrainment of the liquid in the grooves by the high-velocity vapor flow is one of the most important mechanisms for the comparatively long miniature channel. The entrainment of the liquid begins when the dynamic impact of the vapor flow is comparable with the stabilizing effect of the capillary pressure. The latter is usually described by the condition We = 1 , where the Weber number is defined in eq. (11.266). Therefore, for uniform heat flux, and neglecting the effect of subcooling, it follows from eq. (11.266) and the condition We = 1 that (πσ W ρv / 2)1/ 2 ′′ qcrit = hAv [th − 2(tw1 + t g )] (11.282) Leff Dh ,A ′′ Table 11.5 Comparison of the experimental qcrit with the prediction by eq. (11.282). Effective heated length (mm) 40 26 13 13 po, sat (kPa) 101 101 101 20 ′′ qcrit , W/cm2 Experimental 84 108 130 – ′′ qcrit (W/cm2) eq. (11.282) 87 134 269 138 assuming that the grooves are filled with liquid prior to the beginning of entrainment. Table 11.5 compares the critical heat fluxes predicted by eq. (11.282) and those obtained experimentally. For the short length of 13 mm, eq. ′′ (11.282) failed to predict the experimental value of qcrit because in that case, dryout occurred due to the traditionally recognized mechanism whereby the intense vapor bubble formation began on the heated surface before the entrainment could do so. For larger heated lengths, however, the prediction was accurate. The predicted critical heat flux increased with the saturation pressure. To better determine the critical mechanisms inducing dryout, more experimental data with different geometries of the grooved surfaces are needed. For the effective heat transfer coefficients measured, the values were comparatively high (up to 100,000 W/m-K) and of the same order of magnitude 932 Transport Phenomena in Multiphase Systems as those predicted by the evaporative model by Khrustalev and Faghri (1996). The effective heat transfer coefficients did not increase with the heat flux for q′′ ≤ 45 W/cm 2 . This means that with moderate heat flux, the impact of evaporation on heat transfer prevailed over the impact of boiling across the heated length. Actually, the heat sink worked in the mixed boiling-evaporation regime. With small mass flow rates (even with a vapor quality less than 0.3), instabilities and dryout occurred with heat fluxes less than 50 W/cm2. This may have been because the grooves were too shallow to provide the necessary mass flow rate across the entire heated length. With larger mass flow rates, operation was stable due to higher pressure gradients in the channel. The critical heat flux decreased with the heated length. Since the total heat load was proportional to the heated length, it was assumed that dryout was induced mainly by the entrainment of the liquid in the grooves with the vapor flow. However, the mechanisms of dryout in this heat sink should be further investigated. Boiling in Micro/Miniature Channels For two-phase flow in a microchannel, there is no stratified flow; therefore, the orientation of the channel has little effect on the flow pattern. The flow patterns for boiling in microchannels in order of appearance from the inlet are: bubbly flow, elongated bubbles, annular, mist, and flows with partial dryout. Since the diameter of the channel is so small, the bubbly flow regime is very short, as the bubbles grow to the size of the channel very quickly. Once the bubbles grow to the size of the channel, the flow pattern becomes an elongated bubble flow, followed by an annular flow regime. Cyclic dryout is also possible in the elongated bubble flow regime, which is also referred to as the confined bubble regime (Vlasie et al., 2004). Figure 11.31 Three-zone heat transfer model for elongated bubble flow in microchannels (Thome et al., 2004; Reprinted with permission from Elsevier). Chapter 11 Two-Phase Flow and Heat Transfer 933 Thome et al. (2004) developed a three-zone flow boiling model for the evaporation of elongated bubbles in microchannels (Fig. 11.31). The vapor bubble is quickly nucleated and grows to the size of the channel to become an elongated bubble. At any given location in the microchannel that is passed by a liquid slug, an elongated bubble then follows, leaving liquid film behind. Evaporation of the thin film makes the elongated bubble grow longer and move forward. If the thin film dries out before the next liquid slug arrives, a vapor plug (dry zone) passes. This cycle repeats itself so any given point in the microchannel is passed by liquid slugs, elongated bubbles and vapor slugs. Thome et al. (2004) and Dupont and Thome (2004) developed a three-zone heat transfer model based on a homogeneous flow assumption, i.e., the liquid and vapor were assumed to have the same velocity. The model illustrated the importance of strong cyclic variation in the heat transfer coefficient as well as the strong dependency of the heat transfer on bubble frequency, the minimum film thickness at dryout, and the liquid film’s formation thickness. They concluded that the heat transfer coefficient in the elongated film region is several orders of magnitude larger than that in the liquid slug region, while heat transfer in the dry zone is nearly negligible. The time-averaged overall heat transfer coefficient strongly depends on the relative lengths of these three zones. Vlasie et al. (2004) thoroughly reviewed the available empirical correlations for heat transfer in flow boiling in small-diameter channels. Lazarek and Black (1982) studied flow boiling of R-113 in 123 and 246 mm long circular vertical tubes with internal diameters of 3.1 mm. They recommended the following correlation: Nu D = 30 Re0.857 Bo0.714 (11.283) D where q′′ Bo = (11.284) hAv m′′ is the boiling number. Equation (11.283) is obtained by analyzing experimental data performed in the following ranges: 14k W/m 2 < q ′′ < 380 kW/m 2 , 125 kg/m 2 -s < m′′ < 725 kg/m 2 -s , and 130 kPa < p < 140 kPa . In arriving at eq. (11.283), the effect of quality on heat transfer is neglected. Kew and Cornwell (1997) performed experiments of boiling of R-141 in a 500 mm-long, small-diameter tube ( D = 1.39 to 3.69 mm ). While the results for 3.69 and 2.87 mm tubes follow trends similar to those observed in conventionally sized tubes, the results for the 1.39 mm tube depart from these trends significantly. To account for the effect of quality on boiling heat transfer, Kew and Cornwell (1997) proposed a modified Lazerek and Black correlation as follows: Nu D = 30 Re0.857 Bo0.714 (1 − x) −0.143 (11.285) D Despite the fact that there can be three different flow regimes – isolated bubble, confined bubble, and annular flow – in flow boiling in small-diameter 934 Transport Phenomena in Multiphase Systems tubes, nucleate boiling correlations, such as eqs. (11.283) and (11.285), often yield better results. Tran et al. (1996) studied nucleate boiling of R-12 in a circular channel with diameter of 2.46 mm and rectangular channel of 1.70×4.06 mm2 and proposed the following correlation h = 8.4 × 105 (Bo 2 WeA )0.3 ( ρA / ρv ) −0.4 (11.286) where the boiling number, Bo, is defined in eq. (11.284) and the Weber number is m′′2 D WeA = (11.287) ρAσ Equation (11.286) is obtained within the following range: 3.6 kW/m 2 < q′′ < 129 kW/m 2 , 44 kg/m 2 -s < m′′ < 832 kg/m 2 -s , and 510 kPa < p < 820 kPa . Yu et al. (2002) experimentally investigated flow boiling of water in a horizontal tube of 2.98-mm inner diameter at a system pressure of 200 kPa. The inlet temperature is 80 °C and the mass flux ranges from 50 to 200 kg/m2-s. They proposed the following correlations h = 6.4 × 106 (Bo 2 WeA )0.27 ( ρA / ρv ) −0.2 (11.288) To enhance boiling heat transfer on the conventional surface, one can use special surface geometry to promote nucleate boiling by reducing the boiling inception superheat, increasing heat transfer coefficient, and increasing the critical heat flux. The early studies focused on application of rough surface to promote nucleation. However, the effect of roughness will diminish due to aging effects. In order to overcome the aging effect, one can utilize the microchannels with re-entrant cavities as shown in Fig. 11.32 (Koúar et al., 2005). The heat flux Figure 11.32 Microchannels with re-entrant cavities (Koúar et al., 2005; Reprinted with permission from Elsevier). Chapter 11 Two-Phase Flow and Heat Transfer 935 Figure 11.33 Effective heat flux versus average surface temperature (Koúar et al., 2005; Reprinted with permission from Elsevier). versus average surface temperature at different mass flux is shown in Fig. 11.33. It can be seen that there is an abrupt change in slope near 100 °C, where transition from single-phase flow to boiling flow occurs. As the effective heat flux increases to a certain value, the surface temperature significantly increases, which indicates a critical heat flux condition. For nucleate boiling in five parallel rectangular channels with a hydraulic diameter of Dh = 0.233mm , Koúar et al. (2005) proposed the following correlation h = 1.068q′′0.64 (11.289) For convective boiling heat transfer in the same geometric configuration, they suggested: h = 4.068 × 10 4 0.857 ReAo (1 − xe ) 0.8 § 1 − xe · ¨ ¸ © xe ¹ 0.02 (11.290) where xe is the quality at exit. Qu and Mudawar (2004) studied the CHF phenomena in microchannels and found that when there is a significant compressible volume upstream of the heating section, an oscillating flow may lead to CHF. Another phenomenon observed by Qu and Mudawar (2004) that causes CHF is excursive instability caused by backflow of the vapor, as shown in Fig. 11.34. The vapor flows backward from the individual microchannels into the upstream shallow plenum and forms a thick intermittent vapor layer. This layer breaks up into many small vapor bubbles, which propagate further upstream even deeper in the plenum. There, they mix with the incoming liquid. Bergles and Kandlikar (2005) suggested that all CHF data for parallel microchannels are subject to the parallel channel/excursive instability. The incoming flow is simply absorbed by the compressible volume, and the stagnated flow in the headed channel may lead to CHF before the flow through the channel is resumed. Therefore, the upstream compressible volume leads to an oscillating flow in the headed channel, which causes the CHF. 936 Transport Phenomena in Multiphase Systems Figure 11.34 Vapor back-flow near CHF (Qu and Mudawar, 2004; Reprinted with permission from Elsevier). Qu and Mudawar (2004) correlated 42 data points from their experimental results for water in microchannels with Dh = 380 m , as well as data for R-113 in circular tube with D = 510 m and 2.54 mm from their previous study (Bowers and Mudawar, 1994). They recommended the following empirical correlation for critical heat flux in microchannels: ′′ §ρ · qmax = 33.43 ¨ v ¸ ′′hAv m © ρA ¹ where We = m′′L 1.11 §L· We −0.21 ¨ ¸ © Dh ¹ −0.36 (11.291) σρA (11.292) is the Weber number based on the channel heated length L. It should be pointed out that the CHF obtained from eq. (11.291) represents a hydrodynamic CHF, which is caused by excursive instability. It is different from the CHF obtained from a stabilized single channel, in which case the CHF is caused by the usual dryout mechanism. As noted above, boiling heat transfer in micro- and minichannels has attracted significant interest in recent years. However, the complex natures of convective flow boiling and two-phase flow in micro- and minichannels are still not well understood. Additional semi-empirical correlations that are physically based and cover a wide range of conditions in terms of size, fluid properties and flow conditions are needed. Chapter 11 Two-Phase Flow and Heat Transfer 937 References Awad, M.M., and Muzychka, Y.S., 2005a, “Bounds on Two-Phase Flow: Part I Fractional Pressure Gradient in Circular Tubes,” Proceedings of International Mechanical Engineering Congress and Exposition, Orlando, FL (DVD). 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Butterworth, D., 1975, “A Comparison of Some Void-Fraction Relationships for Cocurrent Gas-Liquid Flow,” International Journal of Multiphase Flow, Vol. 1, pp. 845-80. 938 Transport Phenomena in Multiphase Systems Carey, V. P., 1992, Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, Hemisphere Publishing Corp., Washington, D.C. Chang., J.S., Watson, A., 1994, “Electromagnetic Hydrodynamics,” IEEE Transactions on Dielectrics and Electrical Insulation, Vol. 1, pp. 871-895. Chato, J.C., 1962, “Laminar Condensation Inside Horizontal and Inclined Tubes,” ASHRAE Journal, Vol. 4, pp. 52-60. Chen, I.C., 1963, “A Correlation for Boiling Heat Transfer to Saturated Fluids in Convective Flow,” ASME preprint 63-HT-34, Presented at 6th National Heat Transfer Conference, Boston, MA. Chen, Y., Groll, M., Mertz, R. and Kulenovic, R., 2003, “Force Analysis for Isolated Bubbles Growing from Smooth and Evaporator Tubes,” Transaction of the Institute of Fluid-Flow Machinery, Vol. 112, pp. 57–74. Chisholm, D., 1967, “A Theoretical Basis for the Lockhart-Martinelli Correlation for Two-Phase Flow,” International Journal of Heat and Mass Transfer, Vol. 10, pp. 1767- 1778. Chisholm, D., 1973a, “Pressure Gradients Due to Friction During the Flow of Evaporating Two-Phase Mixtures in Smooth Tubes and Channels,” International Journal of Heat and Mass Transfer, Vol. 16, pp. 347-348. Chisholm, D., 1973b, “Void Fraction During Two-Phase Flow,” Journal of Mechanical Engineering Science, Vol. 15, pp. 235-236. Chung, P.M.Y., and Kawaji, M., 2004, “The Effect of Channel Diameter on Adiabatic Two-Phase Flow Characteristics in Microchannels,” International Journal of Multiphase Flow, Vol. 30, pp. 735–761. Collier, J.G., and Thome, J.R., 1994, Convective Boiling and Condensation, 3rd Ed., Oxford University Press, Oxford, UK. Cotton, J., Robinson, A.J., Shoukri, M., and Chang, J.S., 2005, “A Two-Phase Flow Pattern Map for Annular Channels under a DC Applied Voltage and the Application to Electrohydrodynamic Convective Boiling Analysis,” International Journal of Heat and Mass Transfer, Vol. 48, pp. 5563-5579. Crawford, T.J., Weinberger, C.B., and Weismann, J., 1986, “Two-Phase Flow Patterns and Void Fractions in Downward Flows, Part II: Void Fraction and Transient Flow Pattern,” International Journal of Multiphase Flow, Vol. 12, pp. 219-236. Davis, E.J., and Anderson, G.H., 1966, “The Incipience of Nucleate Boiling in Forced Convection Flow,” AIChE Journal, Vol. 12, pp. 774–780. Dobson, M.K., and Chato, J.C., 1998, “Condensation in Smooth Horizontal Tubes,” ASME Journal of Heat Transfer, Vol. 120, pp. 193-213. Chapter 11 Two-Phase Flow and Heat Transfer 939 Dupont, V., and Thome, J.R., 2004, “Evaporation in Microchannels: Influence of the Channel Diameter on Heat Transfer,” Microfluidics and Nanofluidics, Vol. 1, pp. 119-127. Faghri, A., 1995, Heat Pipe Science and Technology, Taylor & Francis, Washington. Feng, Z.P., and Serizawa, A., 1999, “Visualization of Two-Phase Flow Patterns in an Ultra-Small Tube,” Proceedings of the 18th Multiphase Flow Symposium of Japan, pp. 33–36. Forster, H.K., and Zuber, N., 1955, “Dynamics of Vapor Bubbles and Boiling Heat Transfer,” AIChE Journal, Vol. 1, pp. 531-535. Friedel, L., 1979, “Improved Friction Pressure Drop Correlations for Horizontal and Vertical Two-Phase Pipe Flow,” European Two-Phase Flow Group Meeting, Paper E2, Ispra, Italy. Gersey, C.O., and Mudawar, I., 1993, “Orientation Effects on Critical Heat Flux From Discrete, In-Line Heat Sources in a Flow Channel,” ASME Journal of Heat Transfer, Vol. 115, pp. 973-985. Golan, L.P., and Stenning, A.H., 1969, “Two-Phase Vertical Flow Maps,” Proceedings of the Institute of Mechanical Engineers, Vol. 184, No. 3C, pp. 110116. Hashizume, K., 1983, “Flow Pattern, Void Fraction and Pressure Drop of Refrigerant Two-Phase Flow in a Horizontal Pipe, Part I: Experimental Data,” International Journal of Multiphase Flow, Vol. 9, pp. 399-410. Hewitt, G.F., 1998, “Multiphase Fluid Flow and Pressure Drop,” Heat Exchanger Design Handbook, Vol. 2, Begell House, New York, NY. Hewitt, G.F., and Roberts, D.N., 1969, “Studies of Two-Phase Flow Patterns by Simultaneous X-ray and Flash Photography,” AERE-M 2159, HMSO. Hsu, Y.Y., 1962, “On the Size Range of Active Nucleation Cavities on a Heating Surface,” ASME Journal of Heat Transfer, Vol. 84, pp. 207–216. Jaster, H., and Kosky, P.G., 1976, “Condensation Heat Transfer in a Mixed Flow Regime,” International Journal of Heat Transfer, Vol. 19, pp. 95-99. Kandlikar, S.G., 1990, “A General Correlation for Two-Phase Flow Boiling Heat Transfer Coefficient Inside Horizontal and Vertical Tubes,” ASME Journal of Heat Transfer, Vol. 112, pp. 219-228. Kandlikar, S.G., 1991, “Development of a Flow Boiling Map for Subcooled and Saturated Flow Boiling of Different Fluids in Circular Tubes,” ASME Journal of Heat Transfer, Vol. 113, pp. 190-200. 940 Transport Phenomena in Multiphase Systems Kattan, N., Thome, J.R., and Favrat, D., 1998a, “Flow Boiling in Horizontal Tubes: Part 1 - Development of Adiabatic Two-Phase Flow Pattern Map,” ASME Journal of Heat Transfer, Vol. 120, pp. 140-147. Kattan, N., Thome, J.R., and Favrat, D., 1998b, “Flow Boiling in Horizontal Tubes: Part 2 - New Heat Transfer Data for Five Refrigerants,” ASME Journal of Heat Transfer, Vol. 120, pp. 148-155. Kattan, N., Thome, J.R., Favrat, D., 1998c, “Flow Boiling in Horizontal Tubes: Part 3 - Heat Transfer Model Based on Flow Pattern,” ASME Journal of Heat Transfer, Vol. 120, pp. 156-165. Katto, Y., and Ohno, H., 1984, “An Improved Version of the Generalized Correlation of Critical Heat Flux for the Forced Convection Boiling in Uniformly Heated Vertical Tubes,” International Journal of Heat and Mass Transfer, Vol. 27, pp. 1641-1648. Kawaji, M., and Chung, P.M.Y., 2004, “Adiabatic Gas-Liquid Flow in Microchannels,” Microscale Thermophysical Engineering, Vol. 8, pp. 239-257. Kew, P.A, and Cornwell, K., 1997, “Correlations for Prediction of Boiling Heat Transfer in Small Diameter Channels,” Applied Thermal Engineering, Vol. 17, pp. 805-715. Khrustalev, D., and Faghri, A., 1994, “Thermal Analysis of a Micro Heat Pipe,” ASME Journal of Heat Transfer, Vol. 116, pp. 189-198, 1994. Khrustalev, D., and Faghri, A., 1996, “High Flux Evaporative Mini-Channel Heat Sink with Axial Capillary Grooves,” Journal of Enhanced Heat Transfer, Vol. 3, pp. 221-232. Khrustalev, D. and Faghri, A., 1997, “Boiling Heat Transfer in Miniature Axially-Grooved Rectangular Channel with Discrete Heat Sources,” Journal of Enhanced Heat Transfer, Vol. 4, pp. 163-174. Klausner, J.F., Mei, R., Bernhard, D.M., and Zeng, L.Z., 1993, “Vapor Bubble Departure in Forced Convection Boiling,” International Journal of Heat and Mass Transfer, Vol. 36, pp. 651-662 Kocamustafaogullari, G. and Ishii, M., 1983, “Interfacial Area and Nucleation Site Density in Boiling Systems,” International Journal of Heat and Mass Transfer, Vol. 26, pp. 1377-1387. Koúar, A., Kuo, C.J., and Peles, Y., 2005, “Boiling Heat Transfer in Rectangular Microchannels with Reentrant Cavities,” International Journal of Heat and Mass Transfer, Vol. 48, pp. 4867-4886. Lazarek, G.M., and Black, S.H., “Evaporative Heat Transfer, Pressure Drop and Critical Heat Flux in a Small Vertical Tube with R-113,” International Journal of Heat and Mass Transfer, Vol. 25, pp. 945-960. Chapter 11 Two-Phase Flow and Heat Transfer 941 Liu, D., Lee, P.S., and Garimella, S.V., 2005, “Prediction of the Onset of Nucleate Boiling in Microchannel Flow,” International Journal of Heat and Mass Transfer, Vol. 48, pp. 5134-5149. Lockhart, R.W., and Martinelli, R.C., 1949, “Proposed Correlation of Data for Isothermal Two-Phase, Two-Component Flow in Pipes,” Chemical Engineering Progress Symposium Series, Vol. 45, pp. 39-48. Maddox, D.E., and Mudawar, I., 1989, “Single- and Two-Phase Convective Heat Transfer From Smooth and Enhanced Microelectronic Heat Sources in a Rectangular Channel,” ASME Journal of Heat Transfer, Vol. 111, pp. 10451052. Mehendale, S.S., and Jacobi, A.M., 2000, “Evaporation Heat Transfer in Mesoscale Heat Exchanger,” ASHRAE Transactions, Vol. 106, pp. 446-452. Mei, R., and Klausner, J.F., 1994, “Shear Lift Force on Spherical Bubbles,” International Journal of Heat Fluid Flow, Vol. 15, pp. 62–65. Moser, K.W., Webb, R.L., and Na, B., 1998, “A New Equivalent Reynolds Number Model for Condensation in Smooth Tubes,” ASME Journal of Heat Transfer, Vol. 120, pp. 410-417. Munoz-Cobol, J.L., Herranz, L., Sancho, J., Takachenko, I., and Verdu, G., 1996, “Turbulent Vapor Condensation with Noncondensable Gases in Vertical Tubes,” International Journal of Heat and Mass Transfer, Vol. 39, pp. 3249-3260. Qu, W., and Mudawar, I., 2004, “Measurement and Correlation of Critical Heat Flux in Two-Phase Micro-channel Heat Sinks,” International Journal of Heat and Mass Transfer, Vol. 47, pp. 2045-2059. Plesch, D., Bier, W., Seidel, D., and Schubert, K. 1991, “Miniature Heat Pipes for Heat Removal from Microelectronic Circuits,” Micromechanical Sensors, Actuators, and Systems, DSC-Vol. 32, pp. 303-313, ASME, New York. Premoli, A., Francesco, D., Prina, A., 1971, “A Dimensionless Correlation for Determining the Density of Two-Phase Mixtures,” Thermotecnica, Vol. 25, pp. 17-26. Rabas, T.J., and Minard, P.G., 1987, ‘‘Two Types of Flow Instabilities Occurring inside Horizontal Tubes with Complete Condensation,’’ Heat Transfer Engineering, Vol. 8, pp. 40-49. Riehl, R.R., and Ochterbeck, J.M., 2002, “Experimental Investigation of the Convective Condensation Heat Transfer in Microchannel Flows,” The 9th Brazilian Congress of Thermal Engineering and Sciences, Caxambu, MG, Paper CIT02-0495. Serizawa, A. and Feng, Z.P., 2001, “Two-Phase Flow in Micro-Channels,” Proceedings of the 4th International Conference on Multiphase Flow, New Orleans, LA. 942 Transport Phenomena in Multiphase Systems Serizawa, A., Feng, Z., and Kawara, Z, 2000, “Two-Phase Flow in Microchannels,” Experimental Thermal and Fluid Science, Vol. 26, pp. 703-714. Shah, M.M., 1979, “A General Chart Correlation for Heat Transfer during Condensation Inside Pipes,” International Journal of Heat and Mass Transfer, Vol. 22, pp. 547-556. Shah, R.K., and London, A.L., 1978, “Laminar Flow Forced Convection in Ducts,” Advances in Heat Transfer, Suppl. 1, Academic Press, San Diego, CA. Shin, J.S., and Kim, M.H., 2005, “An Experimental Study of Flow Condensation Heat Transfer inside Circular and Rectangular Mini-Channels,” Heat Transfer Engineering, Vol. 26, pp. 36-44. Situ, R., Hibiki, T., Ishii, M., and Mori, M., 2005, “Bubble Lift-off Size in Forced Convective Subcooled Boiling Flow,” International Journal of Heat and Mass Transfer, Vol. 48, pp. 5536-5548. Soliman, H.M., 1974, Analytical and Experimental Studies of Flow Patterns During Condensation Inside Horizontal Tubes, Ph.D. Dissertation, Kansas State University, Manhattan, KS. Taitel, Y., and Dukler, A.E., 1976, “A Model for Predicting Flow Regime Transitions in Horizontal and Near Horizontal Gas-Liquid Flow,” AIChE Journal, Vol. 22, pp. 47- 55. Tabatabai, A., and Faghri, A., 2001 “A New Two-Phase Flow Map and Transition Boundary Accounting for Surface Tension Effects in Horizontal Miniature and Micro Tubes,” ASME Journal of Heat Transfer, Vol. 123, pp. 958-968. Tandon, T.N., Varma, H.K., and Gupta, C.P., 1982, “A New Flow Regime Map for Condensation inside Horizontal Tubes,” ASME Journal of Heat Transfer, Vol. 104, pp. 763-768. Teng, H., Cheng, P., and Zhao, T.S., 1999, ‘‘Instability of Condensate Film and Capillary Blocking in Small-Diameter-Thermosyphon Condensers,” International Journal of Heat and Mass Transfer, Vol. 42, pp. 3071–3083. Thome, J.R.S., 1964, “Prediction of Pressure Drop during Forced Circulation Boiling of Water,” International Journal of Heat and Mass Transfer, Vol. 7, pp. 709-724. Thome, J.R., 2003, “On Recent Advances in Modeling of Two-Phase Flow and Heat Transfer,” Heat Transfer Engineering, Vol. 24, pp. 46-59. Thome, J.R., 2004, Engineering Data Book III, Wolverine Tube, Inc., Huntsville, AL. Thome, J.R., Dupont, V., and Jacobi, A.M., 2004, “Heat Transfer Model for Evaporation in Microchannels. Part I: Presentation of Model,” International Journal of Heat and Mass Transfer, Vol. 47, pp. 3375-3385. Chapter 11 Two-Phase Flow and Heat Transfer 943 Thome, J.R., and El Hajal, J., 2003, “Two-Phase Flow Pattern Map for Evaporation in Horizontal Tubes: Latest Version,” Heat Transfer Engineering, Vol. 24, pp. 3-10. Tran, T.N., Wambsganns, M.W., and France, D.M., 1996, “Small Circular-and Rectangular-Channel Boiling with Two Refrigerants,” International Journal of Multiphase Flow, Vol. 22, pp. 485-498. Triplett, K.A., Ghiaasiaan, S. M., Abdel-Khalik, S. I., and Sadowski, D. L., 1999, “Gas-Liquid Two-Phase Flow in Microchannels – Part I: Two-Phase Flow Pattern,” International Journal of Multiphase Flow, Vol. 25, pp. 377–394. Turner, J.M., 1966, Annular Two-Phase Flow, Ph.D. Dissertation, Dartmouth College, Hanover, NH. Vasiliev, L.L, Grakovich, L.P., and Khrustalev, D.K., 1984, “Gravity-Assisted Heat Pipes for Solar Energy Collectors,” Proceedings of the 5th International Heat Pipe Conference, Tsukuba, Japan, Pre-prints, Vol. 3, pp. 19-24. Vlasie, C., Macchi, H., Guilpart, J., and Agostini, B., 2004, “Flow Boiling in Small Diameter Channels,” International Journal of Refrigeration, Vol. 27, pp. 191-201. Wallis, G.B., 1969, One-Dimensional Two-Phase Flow, McGraw-Hill. Wambsganss, M.W., Jendrzejczyk, J. A. and France, D.M., 1991, “Two-Phase Flow Patterns and Transitions in a Small, Horizontal, Rectangular Channel,” International Journal of Multiphase Flow, Vol. 17, pp. 327-342. Wilmarth, T. and Ishii, M., 1994, “Two-Phase Flow Regimes in Narrow Rectangular Vertical and Horizontal Channels,” International Journal of Heat and Mass Transfer, Vol. 37, pp. 1749-1758. Yu, W., France, D.M., Wambsgans, M.W., and Hull, J.R., 2002, “Two-Phase Pressure Drop, Boiling Heat Transfer, and Critical Heat Flux to Water in a SmallDiameter Horizontal Tube,” International Journal of Multiphase Flow, Vol. 28, pp. 927-941. Zeng, L.Z., Klausner, J.F., Bernhard, D.M., and Mei, R., 1993, “A Unified Model for the Prediction of Bubble Detachment Diameters in Boiling Systems II. Flow Boiling,” International Journal of Heat and Mass Transfer, Vol. 36, pp. 2271-2279. Zhang, Y., Faghri, A., and Shafii, M.B., 2001, “Capillary Blocking in Forced Convective Condensation in Horizontal Miniature Channels,” ASME Journal of Heat Transfer, Vol. 123, pp. 501-511. Zivi, S.M., 1964, “Estimation of Steady-State Void Fraction by Means of the Principle of Minimum Energy Production,” ASME Journal of Heat Transfer, Vol. 86, pp. 247-252. 944 Transport Phenomena in Multiphase Systems Zuber, N., 1961, “The Dynamics of Vapor Bubbles in Nonuniform Temperature Fields,” International Journal of Heat and Mass Transfer, Vol. 2, pp. 83-98. Problems 11.1. 11.2. 11.3. 11.4. Express slip ratio in terms of the total cross-sectional area, total volume flow rate, void fraction, and vapor velocity. Redo Problem 11.1 using the mass flux instead of the total volume flow rate. Express volumetric flow fraction in terms of the total cross-sectional area, total volume flow rate, and gas superficial velocity. Consider liquid-vapor two-phase flow. Find the expression for mass flux as a function of the quality, the individual phase velocities, and the densities. Express total volumetric flux in terms of the individual mass flow rates, the total cross-sectional area, and the phase densities. Determine the flow regime for upward flow of 10 Kg/s of R-11 in a vertical 0.1 m diameter tube at 10 bar and 25% quality. For a horizontal miniature tube of 1 mm inner diameter, determine the flow regime for 1.5×10-3 kg/s of water-steam at 1 atm and 10% quality. The flow of 0.1 kg/s of air and 5 kg/s of water at 20 °C and 1 atm at a given point in a horizontal channel is in the bubbly regime. Describe qualitatively how the two-phase flow should change to achieve stratified flow. Determine the pressure drop due to friction for a 1 kg/s flow of steamwater in a 10 cm inner diameter tube at 2 atm and 40% quality. 11.5. 11.6. 11.7. 11.8. 11.9. 11.10. Calculate the void fraction using the homogeneous model for a twophase water-steam flow with a quality of 35% at 15 bars. 11.11. Specify the governing equations for adiabatic two-phase flow in a circular tube of constant cross-sectional area using (1) the homogeneous model, and (2) the separated-flow model. 11.12. A capillary tube-suction line heat exchanger consists of a capillary tube and a suction line welded together as shown in Fig. P11.1. While evaporation takes place in the capillary tube, the suction line flow is single-phase. Specify the governing equations for fluid flow and heat transfer in the system using the homogeneous model. Chapter 11 Two-Phase Flow and Heat Transfer 945 Figure P11.1 11.13. A mixture of steam-water flows in a 5 cm inner diameter vertical tube at 10 kg/s, 5 atm and 10% quality. What is the void fraction? 11.14. A mixture of steam-water flows in a 5 cm inner diameter vertical tube at 1 atm. The superficial velocities of the liquid and vapor are 1 m/s and 100 m/s, respectively. Determine the void fraction using the correlation proposed by Premoli (1971), eq. (11.105). 11.15. Steam condenses at 80 °C as it flows inside a horizontal tube with an inner diameter of 5 cm. The mass flow rate is 0.5 kg/s and the quality is x = 0.75. Find the heat transfer coefficient. 11.16. Calculate the flow boiling two-phase heat transfer coefficient for a 400 kg/sm2 of R22 at a saturation temperature of -20 °C and 20% quality in an upward vertical tube with an inner diameter of 1 cm and a wall temperature of 0 °C. 11.17. A stainless steel heat exchanger (Fig. P11.2) is being used as a flow boiler. The evaporant flows through 10 tubes with a liquid counter-flow. Find the heat transfer coefficient when the quality, x, is 0.5. The evaporant is ammonia at 200 K and the evaporant passage effective diameter is 1 mm. The wall heat flux is 5,000 W/m2. 11.18. You are given a flow boiler that is fed with saturated liquid ammonia at 200K. 100% saturated vapor exits the boiler. The boiler has 547 passages, each of which is 1.25 mm high and 1.25mm wide. The mass flow rate of the ammonia is 11.45 kg/hr (see Fig. P11.3). Determine the phase velocity at the beginning and the end of the flow boiler. What is the percentage increase in phase velocity from the beginning to the end of the boiler? 946 Transport Phenomena in Multiphase Systems ′′ qw = 5000W/m 2 D=1mm Boiler Tubes x10 Ammonia, 30 kg/hr Figure P11.2 1.25 mm 1.25 mm Figure P11.3 11.19. In an electronic cooling system, a row of parallel tubes with diameters of D are embedded in a horizontal plate with thickness W (Fig. P11.4; Bergles et al., 2003). A mixture of liquid and vapor with void fraction of α flows in the tube and the flow pattern is stratified. The upper surface of the plate is heated at constant heat flux while the bottom of the plate is insulated. The temperature of both liquid and vapor inside the tube is Tsat, but the heat transfer coefficients for liquid and vapor are different. Since the structure is symmetric, it is sufficient to consider only one cell with a width of S. Specify the governing equations and corresponding boundary conditions for heat conduction in the plate. You can use either the Cartesian or the cylindrical coordinate system. Chapter 11 Two-Phase Flow and Heat Transfer 947 Figure P11.4 11.20. The spacing between the tubes in Problem 11.19 is a critical parameter for the performance of the electronic cooling system. While small spacing allows more tubes per unit width, it also increases the conductance resistance from the top of the plate to the bottom of the plate. An optimal spacing will minimize the maximum temperature in the system. Write a computer program to solve the above conduction problem for the following parameters: D = 1 mm, W =2 mm, α = 0.5, q′′ = 106 W/m 2 , hA = 104 W/m 2 K , hv = 100 W/m 2 -K . Perform a parametric study to find the optimal spacing for the system in Problem 11.19. 11.21. A pulsating heat pipe (PHP) is a long miniature tube bent into many turns with the evaporator and condenser sections located at these turns (see Section 1.5.4). Since the diameter of the PHP is very small (less than 5 mm), the working fluid charged in the PHP forms vapor plugs and liquid slugs due to capillary action. Specify the governing equation to describe the oscillatory motion of an arbitrary liquid slug with a length of L in the PHP. The effects of surface tension and friction must be taken into account. 11.22. Evaporation and condensation on the thin film left behind by the liquid slug are the mechanisms that sustain the pressure difference between the two ends of the liquid slug. If the vapor phase satisfies the ideal gas law, obtain the energy equation for the vapor plug. Note that the mass of the vapor plug is not constant due to evaporation and condensation. 948 Transport Phenomena in Multiphase Systems Figure P11.5 11.23. For a miniature passage with a very large length to width ratio, a fullydeveloped flow in both vapor and liquid is very typical. A cross-section of a typical miniature passage with axial rectangular grooves is shown in Fig. P11.5(a). The liquid flows in the capillary microgrooves and the vapor flows in the central part of the passage free of liquid. Fluid flow in the grooved passage can be analyzed by considering a computational domain as shown in Fig. P11.5(b). The vapor and liquid flow along the zcoordinate (cocurrently or countercurrently). It can be assumed that (1) the fluid flow is laminar with fully-developed velocity profiles, (2) the curvature of the liquid-vapor interface is constant, and (3) symmetry conditions are applicable at the three exterior boundaries of the vapor domain. Develop a steady-state mathematical model for the liquid and vapor flows that includes coupled vapor and liquid flows with shear stresses at the liquid free surface due to the vapor-liquid frictional interaction. Chapter 11 Two-Phase Flow and Heat Transfer 949