Flow condensation in micro- and minichannels

From Thermal-FluidsPedia

Condensation in miniature or micro channels finds its applications in electronics cooling, microscale energy, and Micro-Electro-Mechanical Systems (MEMS). The patterns of flow particularly relevant to a capillary tube include 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 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 integro-differential 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 two-phase flow in horizontal miniature and micro channels. Tabatabai and Faghri’s (2001) proposed a flow map for two-phase flow in horizontal tubes plots the ratio of vapor to liquid superficial velocity as defined by

$SV=\frac{{{j}_{v}}}{{{j}_{\ell }}} \qquad \qquad(1)$

versus the ratio of pressure drop due to surface tension and shear:

$\Delta {{p}_{\text{ratio}}}=\frac{{{(dp/dz)}_{\text{surface tension}}}}{{{(dp/dz)}_{\ell \text{,shear}}}+{{(dp/dz)}_{\text{g,shear}}}} \qquad \qquad(2)$

A transition boundary based on a force balance that 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 tension-dominated 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/ $m 2$ . They compared their experimental results with the existing empirical correlations for conventional size, including eq. (1) from Heat Transfer Predictions for Forced Convective Condensation, 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. (1) from Heat Transfer Predictions for Forced Convective Condensation 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:

$\text{Nu}=\text{W}{{\text{e}}^{-\text{Ja}}}\operatorname{Re}{{\Pr }^{Y}} \qquad \qquad(3)$

where

$\text{We}=\frac{{{\rho }_{\ell }}{{w}^{2}}L}{\sigma } \qquad \qquad(4)$ $\text{Ja}=\frac{{{c}_{p\ell }}({{T}_{sat}}-{{T}_{w}})}{{{h}_{\ell v}}} \qquad \qquad(5)$ $\operatorname{Re}=\frac{\dot{{m}''}{{D}_{h}}}{{{\mu }_{\ell }}} \qquad \qquad(6)$ $Y=\left\{ \begin{matrix} 1.3 & \operatorname{Re}\le 65 \\ \frac{0.5{{D}_{h}}-1}{2{{D}_{h}}} & \operatorname{Re}>65 \\ \end{matrix} \right. \qquad \qquad(7)$

where w is working fluid velocity (m/s), and L is the channel length. Equation (3) correlated over 95% of experimental results with a relative error of less than 25%.

Annular Flow Condensation in a Miniature Tube

Figure 1: 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.

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 one-dimensional 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. 1(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.

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. 1(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. 1(a) yield

$\int_{R-\delta }^{R}{r{{w}_{\ell }}\left( r \right)dr}=\frac{1}{2\pi {{\rho }_{\ell }}}\left( {{{\dot{m}}}_{\ell ,in}}-\frac{Q\left( z \right)}{{{h}_{\ell v}}} \right) \qquad \qquad(8)$

${{\dot{m}}_{\ell ,in}}$ is the liquid mass flow rate at the condenser inlet, and $Q\left( z \right)$ is the rate of heat through a given cross-section, due to phase change for $z > 0$ , and is defined as follows:

$Q\left( z \right)=2\pi R\int_{0}^{z}{{{{{q}''}}_{w}}dz}-\left( {{c}_{p,\ell }}{{{\dot{m}}}_{\ell }}{{{\bar{T}}}_{\ell }}-{{c}_{p,\ell ,in}}{{{\dot{m}}}_{\ell ,in}}{{{\bar{T}}}_{\ell ,in}} \right) \qquad \qquad(9)$

${{{q}''}_{w}}\left( z \right)$ is the heat flux at the solid-liquid interface due to heat conduction through a cylindrical film with a thickness of $δ$ balanced with the enthalpy change. Note that $T δ > T w$ and therefore $Q\left( z \right)$ in eq. (11.207) will be a negative quantity. The heat flux at the wall is

${{{q}''}_{w}}={{k}_{\ell }}\frac{{{T}_{w}}-{{T}_{\delta }}}{R\ln \left[ R/\left( R-\delta \right) \right]} \qquad \qquad(10)$

where

$T δ$ is the local temperature of the liquid-vapor interface. From eqs. (9) and (10), we obtain the following equation for the heat rate rejected per unit length of the tube:

$\frac{dQ}{dz}= 2 \pi {k_{\ell }} \frac{T_w}-{T_{\delta }} { ln [ R/( R- \delta) ] } -\frac{d}{dz} ( {c_{p,\ell }} {{\dot{m}}_{\ell}} {{\bar{T}}_{\ell}}) \qquad \qquad(11)$

where

${{\bar{T}}_{\ell }}$ is the area-averaged liquid temperature for a given z location. For consideration of subcooling in the condensed liquid, ${{\bar{T}}_{\ell }}$ is found from an area average given by

${{\bar{T}}_{\ell }}=\frac{2\int_{R-\delta }^{R}{r{{T}_{\ell }}\left( r \right)dr}}{{{R}^{2}}-{{\left( R-\delta \right)}^{2}}} \qquad \qquad(12)$

where ${{T}_{\ell }}\left( r \right)$ is the assumed liquid film temperature profile given by the temperature distribution in a cylindrical wall.

${{T}_{\ell }}\left( r \right)={{T}_{\delta }}+\frac{{{T}_{w}}-{{T}_{\delta }}}{\ln \frac{R}{R-\delta }}\ln \frac{r}{R-\delta } \qquad \qquad(13)$

Substituting eq. (13) into eq. (12) and integrating results in an expression for ${{T}_{\ell }}(r),$ the derivate of ${{\bar{T}}_{\ell }}$ in the axial is approximated by

$\begin{align} & \frac{d{{{\bar{T}}}_{\ell }}}{dz}=\frac{d{{T}_{\delta }}}{dz}+\left( \frac{d{{T}_{w}}}{dz}-\frac{d{{T}_{\delta }}}{dz} \right){{\left[ \ln \frac{R-\delta }{R} \right]}^{-1}}\frac{2}{{{R}^{2}}-{{\left( R-\delta \right)}^{2}}} \\ & \times \left[ \left( \ln \frac{R}{R-\delta }-\frac{1}{2} \right)\frac{{{R}^{2}}}{2}+\frac{{{\left( R-\delta \right)}^{2}}}{4} \right] \\ \end{align} \qquad \qquad(14)$

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.

The axial momentum conservation for viscous flow in a liquid film in which the inertia terms are assumed to be negligible is

$\frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{w}_{\ell }}}{\partial r} \right)=\frac{1}{{{\mu }_{\ell }}}\left( \frac{d{{p}_{\ell }}}{dz}+{{\rho }_{\ell }}g\sin \varphi \right) \qquad \qquad(15)$

where $φ$ 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, ${{\tau }_{\ell ,v}}$ , and the surface tension gradient related to the interfacial temperature gradient along the channel.

${{\left. {{w}_{\ell }} \right|}_{r=R}}=0 \qquad \qquad(16)$

${{\left. \frac{\partial {{w}_{\ell }}}{\partial r} \right|}_{\left( r=R-\delta \right)}}=\frac{1}{{{\mu }_{\ell }}}\left[ -{{\tau }_{\ell ,v}}-\frac{d\sigma }{dT}\frac{d{{T}_{\delta }}}{dz} \right]\equiv E \qquad \qquad(17)$

where $T δ$ is the local liquid-vapor interface temperature, and the term with $d σ / d T$ 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 ${{\tau }_{\ell ,v}}$ .

${{\tau }_{\ell ,v}}={{\tau }_{\ell ,v0}}{a}'/\left[ \exp \left( {{a}'} \right)-1 \right] \qquad \qquad(18)$

where ${{\tau }_{\ell ,v0}}$ is defined as the interfacial shear stress in the absence of phase change and is given by

${{\tau }_{\ell ,v0}}=0.5{{f}_{\ell ,v}}{{\rho }_{v}}{{\left( {{{\bar{w}}}_{v}}-{{w}_{\ell ,\delta }} \right)}^{2}} \qquad \qquad(19)$

where $ρ v$ is the vapor density, $w v$ 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

${a}'=-\frac{dQ}{dz}/\left[ \pi R{{h}_{\ell v}}{{\rho }_{v}}{{f}_{\ell ,v}}\left( {{{\bar{w}}}_{v}}-{{w}_{\ell ,\delta }} \right) \right] \qquad \qquad(20)$

${{f}_{\ell ,v}}={{f}_{v}}\left( 1+360\frac{\delta }{2R} \right) \qquad \qquad(21)$

and ${{f}_{\ell ,v}}$ is the interfacial friction factor. From an experimental study for a simulated change of phase the vapor friction factor, $f v$ , is given as follows:

${f_{\ell, v}} = 16 [ 1.2337-0.2337 exp ( -0.0363{{Re}_r})] \times [ exp( 1.2 Ma)] /Re \qquad \qquad(22)$

R $e r$ is found using the vapor suction velocity at the liquid-vapor interface due to condensation, $v v,δ$ , which is defined through the condensing mass flux.

${{v}_{v,\delta }}=\frac{dQ}{dz}\frac{1}{2\pi \left( R-\delta \right){{h}_{\ell v}}{{\rho }_{v}}} \qquad \qquad(23)$

Solving eqs. (15) – (17), the velocity profile is expressed as follows:

${{w}_{\ell }}=-\frac{1}{{{\mu }_{\ell }}}\frac{d{{p}_{\ell }}}{dz}\left[ \frac{1}{4}\left( {{R}^{2}}-{{r}^{2}} \right)+\frac{{{\left( R-\delta \right)}^{2}}}{2}\ln \frac{r}{R} \right]+E\left( R-\delta \right)\ln \frac{r}{R} \qquad \qquad(24)$

Substituting eq. (24) into eq. (8), we obtain the following equation for the axial pressure gradient in the liquid:

$\begin{align} & \frac{d{{p}_{\ell }}}{dz}={{\rho }_{\ell }}g\sin \varphi +{{\mu }_{\ell }}\left[ \frac{1}{2\pi {{\rho }_{\ell }}}\left( \frac{Q}{{{h}_{\ell v}}}-{{{\dot{m}}}_{\ell ,in}} \right)+E\left( R-\delta \right)F \right] \\ & \text{ }\times {{\left[ \frac{{{R}^{4}}}{16}+\frac{{{\left( R-\delta \right)}^{2}}}{2}\left( F+\frac{{{\left( R-\delta \right)}^{2}}}{8}-\frac{{{R}^{2}}}{4} \right) \right]}^{-1}} \\ \end{align} \qquad \qquad(25)$

where

$F=\frac{{{\left( R-\delta \right)}^{2}}}{2}\left( \ln \frac{R}{R-\delta }+\frac{1}{2} \right)-\frac{{{R}^{2}}}{4} \qquad \qquad(26)$

The pressure difference between the vapor and liquid phases is due to capillary effects and disjoining pressure, $p d$ ,

${{p}_{v}}-{{p}_{\ell }}=\sigma \left\{ \frac{{{d}^{2}}\delta }{d{{z}^{2}}}{{\left[ 1+{{\left( \frac{d\delta }{dz} \right)}^{2}} \right]}^{-{\scriptstyle{}^{3}\!\!\diagup\!\!{}_{2}\;}}}+\frac{1}{R-\delta }\cos \left( \text{atan}\frac{d\delta }{dz} \right) \right\}-{{p}_{d}} \qquad \qquad(27)$

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

$\frac{d\delta }{dz}=\Delta \qquad \qquad(28)$

eq. (27) can be rewritten as follows:

$\frac{d\Delta }{dz}={{\left[ 1+{{\left( \Delta \right)}^{2}} \right]}^{3/2}}\left( \frac{{{p}_{v}}-{{p}_{\ell }}+{{p}_{d}}}{\sigma }-\frac{\cos \left( \text{atan }\Delta \right)}{R-\delta } \right) \qquad \qquad(29)$

The integral equations of mass conservation for the vapor and liquid flows take the following form:

${{\rho }_{v}}{{A}_{v}}{{\bar{w}}_{v}}\left( z \right)={{\bar{w}}_{v,in}}{{\rho }_{v,in}}{{A}_{v,in}}+Q\left( z \right)/{{h}_{\ell v}} \qquad \qquad(30)$

${{\rho }_{\ell}}{{A}_{v}}{{\bar{w}}_{\ell}}\left( z \right)={{\bar{w}}_{\ell,tn}}{{\rho }_{\ell,tn}}{{A}_{\ell,tn}}-Q\left( z \right)/{{h}_{\ell v}} \qquad \qquad(31)$

${{A}_{v}}=\pi {{\left( R-\delta \right)}^{2}}$ is the cross-sectional area of the vapor channel, and ${{\bar{w}}_{v,in}}$ 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).

$\begin{align} & \frac{d{{p}_{v}}}{dz}={{\rho }_{v}}g\sin \varphi +\frac{1}{{{A}_{v}}}\left[ \frac{d}{dz}\left( -{{\beta }_{v}}{{\rho }_{v}}\bar{w}_{v}^{2}{{A}_{v}} \right)-{{f}_{v}}{{\rho }_{v}}\bar{w}_{v}^{2}\pi \left( R-\delta \right) \right. \\ & \left. \text{ }+2\pi \left( R-\delta \right){{\rho }_{v}}v_{v,\delta }^{2}\sin \left( \text{atan }\Delta \right) \right] \\ \end{align} \qquad \qquad(32)$

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,

${{\rho }_{v}}=\frac{{{p}_{v}}}{{{R}_{g}}{{T}_{v}}} \qquad \qquad(33)$

Therefore,

$\frac{d{{\rho }_{v}}}{dz}=\frac{1}{{{R}_{g}}}\left( \frac{d{{p}_{v}}}{dz}\frac{1}{{{T}_{v}}}-\frac{{{p}_{v}}}{T_{v}^{2}}\frac{d{{T}_{v}}}{dz} \right) \qquad \qquad(34)$

The saturated vapor temperature and pressure are related by the Clausius-Clapeyron equation, which can be written in the following form:

$\frac{d{{T}_{v}}}{dz}=\frac{d{{p}_{v}}}{dz}\frac{{{R}_{g}}T_{v}^{2}}{{{p}_{v}}{{h}_{\ell v}}} \qquad \qquad(35)$

The seven first-order differential equations, eqs. (11), (25), (28), (29), (32), (34), and (35), include the following seven variables: $\delta ,\text{ }\Delta ,\text{ }{{p}_{\ell }},\text{ }Q,\text{ }{{p}_{v}},\text{ }{{\rho }_{v}},$ and $T v$ . Therefore, seven boundary conditions are set forth at $z = 0$ .

$\delta ={{\delta }_{in}} \qquad \qquad(36)$

$\Delta =0 \qquad \qquad(37)$

${{p}_{\ell }}={{p}_{v,in}}-\frac{2\sigma }{R-{{\delta }_{in}}}+{{p}_{d}} \qquad \qquad(38)$

$Q=0 \qquad \qquad(39)$

${{p}_{v}}\equiv {{p}_{v,in}}={{p}_{v,sat}}\left( {{T}_{v,in}} \right) \qquad \qquad(40)$

${{\rho }_{v,in}}=\frac{{{p}_{v,in}}}{{{R}_{g}}{{T}_{v,in}}} \qquad \qquad(40)$

${{T}_{v}}={{T}_{v,in}} \qquad \qquad(41)$

In the boundary condition given by eq. (36), $δ i n$ 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 $δ i n$ , eqs. (25) and (32) should be solved for the case of $Q = 0$ , $v v,δ = 0$ , $d A v / d z = 0$ and $d T δ / d z = 0$ . The boundary condition, eq. (39), directly follows from eq. (9). There are also parameters ${{\dot{m}}_{\ell ,in}}$ and ${{\bar{w}}_{v,in}}$ , and an additional variable, $T δ$ , involved in this problem. They will be considered using additional algebraic equations. The parameter ${{\dot{m}}_{\ell ,in}}$ should be found using a constitutive condition at the entrance of the condenser

${{\dot{m}}_{\ell ,in}}= {{\dot{m}}_t} - {Q_{in}} / {h_{\ell v}} \qquad \qquad(42)$

where $Q i n$ is the total heat load into the condenser. Also ${{\bar{w}}_{v,in}}={{Q}_{in}}/\left( {{h}_{\ell v}}{{\rho }_{v}}{{A}_{v,in}} \right)$ .

The liquid-vapor interface temperature, $T δ$ , 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

${{{q}''}_{\delta }}=-\left( \frac{2\alpha }{2-\alpha } \right)\frac{{{h}_{\ell v}}}{\sqrt{2\pi {{R}_{g}}}}\left[ \frac{{{p}_{v}}}{\sqrt{{{T}_{v}}}}-\frac{{{\left( {{p}_{sat}} \right)}_{\delta }}}{\sqrt{{{T}_{\delta }}}} \right] \qquad \qquad(43)$

where $p v$ and ${{({p_{sat}})}_{\delta}}$ are the saturation pressures corresponding to $T v a n d T δ$ , the temperatures associated with the thin liquid film interface, respectively. The following two algebraic equations should be solved to determine $T δ$ for every point along the z-direction. The relation between the saturation vapor pressure over the thin condensing film, ${{\left( {{p}_{sat}} \right)}_{\delta }}$ , affected by the surface tension, and the normal saturation pressure corresponding to $T δ$ , ${{p}_{sat}}\left( {{T}_{\delta }} \right)$ , is given by the following equation (Chapter 5):

${{\left( {{p}_{sat}} \right)}_{\delta }}={{p}_{sat}}\left( {{T}_{\delta }} \right)\exp \left[ \frac{{{\left( {{p}_{sat}} \right)}_{\delta }}-{{p}_{sat}}\left( {{T}_{\delta }} \right)-\sigma K+{{p}_{d}}}{{{\rho }_{\ell }}{{R}_{g}}{{T}_{\delta }}} \right] \qquad \qquad(44)$

where K is the local curvature of the liquid-vapor interface defined by the term in outer brackets in eq. (27). Notice that under steady conditions, $q'' δ$ is due to heat conduction through the liquid film. It follows form eq. (10) and (44)

${{T}_{\delta }}={{T}_{w}}+\frac{R}{{{k}_{\ell }}\left( R-\delta \right)}\ln \frac{R}{R-\delta }\left( \frac{2\alpha }{2-\alpha } \right)\times \frac{{{h}_{\ell v}}}{\sqrt{2\pi {{R}_{g}}}}\left[ \frac{{{p}_{v}}}{\sqrt{{{T}_{v}}}}-\frac{{{\left( {{p}_{sat}} \right)}_{\delta }}}{\sqrt{{{T}_{\delta }}}} \right] \qquad \qquad(45)$

Equations (45) and (46) determine the interfacial temperature, $T δ$ , and pressure, ${{\left( {{p}_{sat}} \right)}_{\delta }}$ , for a given vapor pressure, ${{p}_{v}}={{p}_{v,sat}}\left( {{T}_{v}} \right)$ , temperature of the solid-liquid interface, $T w, a n d t h e l i q u i d f i l m t h i c k n e s s < m a t h > δ$ . Note that $T w$ 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, $T w$ becomes an additional variable. If the convective heat transfer coefficient at the outer tube wall, $h 0$ , and the cooling liquid temperature, ${{T}_{\infty }}$ , are known, the local wall temperature can be defined using an energy balance

${{h}_{o}}\left( {{T}_{w}}-{{T}_{\infty }} \right)=\frac{1}{2\pi R}\frac{dQ}{dz} \qquad \qquad(46)$

To solve this problem, the first-order ordinary differential equation, eq. (47), must be added to the seven previously specified. The additional boundary condition is given by $z = 0$ ,

${{T}_{w}}={{T}_{w,in}} \qquad \qquad(47)$

Equations (11), (25), (28), (29), (32), (34) and (35) with corresponding boundary conditions have been solved using the standard Runge-Kutta procedure by Begg et al. (1999). Algebraic eqs. (45) and (46), with two unknowns, ${{\left( {{p}_{sat}} \right)}_{\delta }}$ and $T δ$ , 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, $d Δ / d z$ 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 meniscus-like interface. The condition of complete condensation requires the overall energy balance to be satisfied. Thus, the energy convected into the tube, or heat-load-in, $Q i n$ (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, $L δ$ (m).

Incomplete condensation is said to exist for a heat-load-in greater than $Q i n$ (W) for complete condensation.

References

Begg, E., Khrustalev, D., and Faghri, A., 1999, “Complete Condensation of Forced Convection Two-Phase Flow in a Miniature Tube,” ASME Journal of Heat Transfer, Vol. 121, pp. 904-915.

Dobson, M.K., and Chato, J.C., 1998, “Condensation in Smooth Horizontal Tubes,” ASME Journal of Heat Transfer, Vol. 120, pp. 193-213.

Faghri, A., 1995, Heat Pipe Science and Technology, Taylor & Francis, Washington.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

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.

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 9^th Brazilian Congress of Thermal Engineering and Sciences, Caxambu, MG, Paper CIT02-0495.

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.

Soliman, H.M., 1974, Analytical and Experimental Studies of Flow Patterns During Condensation Inside Horizontal Tubes, Ph.D. Dissertation, Kansas State University, Manhattan, KS.

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.

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.

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