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C HAPTER 2 The Equations of Convective Heat Transfer 2.1 INTRODUCTION In order to be able to predict convective heat transfer rates there are, in general, three variables whose distributions through the flow field must be determined. These variables (see Fig. 2.1) are: • The pressure, p • The velocity vector, V • The temperatur~ T ~ Once t he distributions o f thes~ quantities are determined, the variation o f any other required quantity, such as the heat transfer rate, can b e obtained. In order to determine the distributions of pressure, velocity, and temperature the principles o f conservation of mass, conservation o f moment~m(Newton's Law) and conservation of energy,Afirst law of Thermodynamics) are applied. These conservation principles represent empirical models o f the behavior o f the physical world. They do not, of course, always apply, e.g., there can b e a conversion o f mass into energy i~ some circumstances, but they are adequate for th~ analysis o f the vast majority o f engineering problems. T hese conservatio~ principles lead to the so-called Continuity, Navier-Stokes and Energy equations respectively. These equations involve, beside the basic variables mentioned above, certain fluid properties, e.g., density, p; viscosity, JL; conductivity, k; and specific h eat,cp • Therefore, to obtain the solution to the equations, the relations between these properties and the pressure and temperature have to b e known. (Non-Newtonian fluids in which JL depends on the velocity field are not considered here.) As discussed in the previous chapter, there are, however, many practical problems i p which the variation o f these properties across the flow field can be ignored, i.e.,\ in which the fluid properties can b e assumed to be constant in obtaining the solution. Such solutions are termed "constant , f / 31 3 2 Introduction to Convective Heat Transfer Analysis vV P ointpV T ,p U FIGURE 2.1 Variables required to determine convective heat transfer rates. fluid property" solutions. They are found to adequately describe many flows i~ which the overall temperature changes are quite large provided that the fluid properties are evaluated at a suitable mean temperature. For this reason, i t will be assumed i n most o f the present chapter and in much o f the remainder o f this book that the fluid properties are consUy;lt. p f course, many o f the methods o f analysis described here can quite easily b e modIfied to account for variations in fluid properties and some discussion o f this will b e given in a later chapter. In the present chapter, it will also be assumed that all body forces are negligible. A discussion o f some flows in which buoyancy forces are important will be given in Chapters 8 and 9. 2.2 C ONTINUITY AND NAVIER-STOKES EQUATIONS T he equations ~ll here be presented in terms o f a cartesian coordinate system, the velocity vector V h aving components u, v, and w in the three coordinate directions x, y, and z, respectively. For the moment, the discussion will also be restricted to steady laminar flow in which the flow variables u, v, W , p , T do not vary with time. 1)..s mentioned above, the continuity equation is obtained b y the application o f the principle o f conservation o f m ass to the fluid flow. A differentially small rectangular control volume with sides parallel to the three coordinate directions is introduced. This is shown in Fig. 2.2. Since steady flow is being considered, the rate at which mass leaves this control volume can be set equal to the rate at which i t enters the control volume. Taking the y ~ x Flow z FIGURE 2.2 Control volume considered . . , CHAPTER 2: The Equations o f Convective Heat Transfer 33 \' F IGURE 2.3 o x Viscous stresses acting on control volume. limiting form o f this result as the size of the control volume tends to zero then gives, because density is being assumed constant [1],[2],[3]: au + av + J w ax ay az = 0 (2.1) This is the continuity equation for constant density flow. As was mentioned above, the Navier-Stokes equations are obtained by the applictltion o f the conservation of momentum principle to the fluid flow. T he same control volume that was introduced above in the discussion o f the continuity equation is contddered and the conservation o f momentum in each o f the three coordinate directions hi separately considered. The net force acting on the control volume in any o f these directions i s then set equal to the difference between the rate at which momentum leaves the control volume in this direction and the rate at which i t enters in this direction. The net force arises from the pressure forces and the shearing forces acting on the faces of t he control volume. The viscous shearing forces for two-dimensional flow (see later) are shown in Fig. 2.3. They are expresse9 in terms o f the velocity field b y assuming the fluid to be Newtonian and are then given b y [4],[5]: Ux = 2 /L- uy = au ax av 2 /Lay (2.2) (2.3) (2.4) T xy ~ JL (~~ + ~:) ~ Tyx (These relations are also required in the derivation o f the energy equation as discussed later in this chapter.) Using the expressions for the shearing stresses in. the momentum balance for the control volume and taking the limit. o f the resulting equations as the size o f the control volume tends to zero then gives the following set o f equations [6],[7], [8],[9]: I 34 Introduction to Convective Heat Transfer Analysis au au au u - +v- +w- = az ax ay av av av u - +v-+w- = ax ay az (2.5) These are the Navier-Stokes equations for steady constant fluid prclperty flow. They are sometimes termed the X -, y-, and z-momentum equations, respectively. Basically these equations state that the net rate o f change o f momentum per unit mass in any direction (the left-hand side) is the sum of the net pressure force and the net viscous force-in MIat direction. It should b e d early noted that since the fluid properties are being assumed constant, the set o f eqs. (2.5) to (2.7) is independent of the temperature field and can, therefore, b e solved independently of the energy equation to give the velocity and pressure distribution. Many practical flows can b e represented with sufficient accuracy by assuming the flow to be-two-dimensional, i.e., by assuming that the axes can be so positioned that one of the velocity components, here always taken as w, is effectively zero. For this reason and because the study o f two-dimensional flows forms the basis of the study o f more complex three-dimensional flows, most of the analyses discussed in later chapters will be for two-dimensional flows. Setting w equal to zero gives the following as the equations governing the velocity and pressure fields in steady twodimensional flow: au + av = 0 ax ay u-.-· + v - (2.8) (2.9) au ax au ay = u - +v- av ax av ay = (2.10) In some problems it is more convenient to have the flow equations expressed in terms of other coordinate schemes. One o f the most important of these, and the only other one that will be used in the present book, is the cylindrical coordinate system which is defined in Fig. 2.4. Here the position of any point being considered is defined i n terms of the coordinates z, r, 1> and the velocity components in these three directions are here designated u, v, and w, respectively. In this coordinate system, the continuity equation becomes: au 1 a 1 aw - + - -(vr) + - - = 0 az r ar r a1> (2.11) CHAPTER 2: The Equations of Convective Heat Transfer 35 z , y-+---------------.--v o Reference Direction F IGURE 2.4 Cylindrical coordinate system. In expressing the Navier-Stokes equations in these coordinates it is convenient to define: - - = u - + v - + - -ra~ D Dt a az a ar wa (2.12) and V2 a2 1a 1 a2 = _ + _ _ + _ _ + a2 _ ar2 r ar r2 a~2 a z2 (2.13) Sq. (2.12) is used because steady flow is being considered. Using this notation, the Navier-Stokes equations become in cylindrical coordi~ nates: Dv _ w2 = Dt r Du Dt = - --- + p az p ar 1 ap vV u 2 (2.14) (2.15) (2.16) _ ! ap + V [V2v _ V ~ _ ~ aw."] r2 r,~!a.d> D w _ vw = _ ~ a p + Dt r pr a~ [V2w + ~ av - r2 a</> r2 w] ExpJ"essions for the flow equations in terms o f general orthogonal coordinates will be found in books on fluid mechanics. 2.3 THE ENERGY EQUATION F OR S TEADY F LOW The present book is concerned with methods o f predicting heat transfer rates. These methods basically utilize the continuity and momentum equations to obtain the velocity field which i s t hen used with the energy equation to obtain the temperature field from which the heat transfer rate can t hen b e deduced. I f t he variation o f fluiq properties with temperature is significant, the continuity and momentum equations 36 Introduction to Convective Heat Transfer Analysis must b e solved simultaneously with the energy equation. As explained above, there are, however, many cases in which this flu~d property variation is negligible. In such cases the velocity field can b e solved for independently o f the energy equation and many such solutions are given i n books on viscous flow theory. These solutions can then be used with the energy equation to derive heat transfer rates. The prediction o f convective heat transfer rates will, however, always involve the solution o f the energy equation. Therefore, because o f its fundamental importance in the present work, a discussion o f the way in which the energy equation is derived will b e given here [2],[3],[5],[7]. For this purpose, attention will be restricted to twodimensional, incompressible flow. Consider flow through a rectangular control volume having sides o f le~gth d x and d y parallel to the x- and y-axes, respectively, and having unit depth, this control volume being shown in Fig. 2.5. (The equation can also be derived by using a closed system, i.e., using an element o f fluid containing a fixed identifiable mass o f fluid.) , F or this control vohlme, the steady flow energy equation requires that: Rate at which enthalpy leaves control volume Rate at which Rate at which Rate at which ki . y + kinetic energy - enthalpy enters - ntetIc enetrgl leaves control volume control volume en ers con ro volume (2.17) Net rate a t which into control volume Net rate at which on control volume = heat is transferred + work is done The left-hand side o f this equation will be considered first. It is convenient to define: I = specific enthalpy + specific kinetic energy = c pT + (u 2 + v2 )/2 (2.18) I is sometimes called the total enthalpy. Consider the x-direction. The rates at which the total enthalpy enters and leaves the control volume in the x-direction are as shown in Fig. 2.6. I f is the mass flow rate through ab, the difference between the rate at which the sum o f enthalpy and kinetic energy leave and enter the control volume in the m y o~----------------------~ x F IGURE 2.5 Control volume considered. CHAPTER 2: T he Equations o f Convective Heat Transfer 37 b r hI a F IGURE 2.6 Enthalpy flows into and out o f control volume. x-direction is, since d x i s assumed to be small, given by: mI + ~ ( ml)dx - mI = ~(mI)dx ax ax (2.19) Since t he control volume has unit width m= Eq. (2.19) can, therefore, be written as: p udy (2.20) [ :x ( p U I)]dXd Y (2.21) Exactly the s ame arguments can b e applied t n t he y-direction and i t follows from this that the difference between the rate at which the sum o f enthalpy and kinetic energy leaves and enters the control volume in the y-direction is given by: [:y (p v I)] d x d y --./- (2.22) Adding Eqs. (2.21) a nd (2.22) and using Eq. (2.18) then shows that the left-hand side of Eq. ( 2.i 7--) is e qual to: i Next, consider t he he~t t ransfer term on the right-hand side o f Eq. (2.17). Heat is transferred into the .eol1trol volume b y conduction through its surface. The x- and y-directions are a gain separately considered. I f Q x i s the rate at which heat is transferredinto t he v olume through the face, ab (see Fig. 2.7), then the difference between the rate a t which h eat is conducted into the control volume i n t he x -direction and the rate at which it i s conducted out i n this direction is given by: (2.24) b ,; ~ J -1 rEI :lL _ _'" "~ ...... F lGURE2.7 . Hdat flows into and out of control volume. 38 Introduction to Convective Heat Transfer Analysis B ut again noting that the control volume\has unit thickness and using Fourier's law gIves: Qx = -kdy , aT ax (2.25) Substituting this into Eq. (2.24) gives the net rate at which heat is transferred i nto, the control volume i n t he x-direction as: (2.26) I f the same procedure is applied i n the y-direction it can b e shown thal the net rate at which heat is transferred into the control volume in this direction is: (2.27) Combining these and noting that k, a fluid property, is being assumed constant then gives the net rate at which heat is transferred into the control volume iri both the x- and y-directions as: (2.28) The last term i n Eq. (2.17) that has to b e considered is the work term. This work is done b y the fluid on the control volume by virtue o f the forces exerted by the flowing fluid on the surface o f the control volume. These forces arise due to the normal pressure and due to the previously discussed shearing stresses on its surface. The way in which this work is done and the way in which the work term is evaluated· are perhaps m o*easily seen by considering the control volume to b e moving through the fluid which is at rest. Considering the stresses as defined in Fig. 2.3 then sho~s that the rate at which work is done on the control volume is given by: { (-UUxdY) + [(UlTx}i Y) + : x(UlTx)dXdY ]} + { ( updy) - [(UPdY) + : x (UP)dXd(,]} + { ( - UTyx d x) + [(UTyX d x) + : y (UTyx ) d x d Y ]} (2.29) + { ( -vu y d x) + [(VlTydX) + : Y (VlTy)dXd Y ]} + {(VPdX) - [(VPdX) + : y(VP)dXdY ]} + { (-VTxydY) + [(VTxydY) + :X(VTXy)d;tdY ]} CHAPTER 2: The Equations o f Convective Heat Transfer 39 I n setting up this equation it has been noted that the pressure acts in the opposite direction to that taken as positive for the normal stresses a x a nd a y. Rearranging Eq. (2.29) then gives the rate at which work is done on the control volume as: a a [-a (uO"x) - -(up) + -a (UTyx ) + -a ( vay)--(vp) + -a (VTxy ) d xdy a a ax ax ay y ay x l.e., 1 aap aTyx aap aaXY)d (uax - u- + u -- + vay - v- + v - - x d y ax ax ay ay ay ax (2.30) - p(~: + ~~ )dXdY + (ux ~: +Tyx ~; + u y ~~ +Txy ; : ) dXdY Now, consider separately the various bracketed terms i n this equation. Expanding the first o f these using the expressions for the stresses given i n Eqs. (2.8) to (2.10) gives: 2 a2u ap a2v) a2v a p a 2 uJ-t- - u- + up.- (a u + - - + 2 vp.-- -. v- + vp.- (au + - v) a x2 ax ay2 d ydx ay2 ay ay ax (2.31) But from the continuity equation it follows that: a2 u a x2 =- a2 v ayax a2u axay and a'lv ay2 1 0 that Eq. =- (2.31) c an be written as: ap ( a 2u a2u [ap ( a 2v a2v u [ - ax + p.- ay2 + a x2 )J + v - ay + p.- ay2 + ax2)J \1 \1 (2.32) I f this equation is compared with the Navier-Stokes equations (2.6) and (2.7) i t will be seen that i t c an be written as: 2au au av 2av pu - + p uv- + p uv- + pv ax ay ax ay (2.33) The second bracketed tet;m i n Eq. (2.30) is zero b y v irtue o f the continuity equation and i t follows, therefore, that the rate at which work is done o n the control volume ~-­ is given b y: (·pu2au ax + puv ay + puv a x + pv ay au . av 2 av) fxdy \ /' (2.34) ( au au \ av a v) +\O"x ax + TyXay +!O"Yay + Txyax d xdy 40 Introduction to Convective Heat Transfer Analysis Substituting Eqs. (2.23), (2.28), and (2.34) into Eq. (2.17) a nd dividing by d x d y gives: (2.35) Since p and cp are fluid properties which are being assumed constant, this equation can be written as: r-- = aT - aT) 1 av au a (u- + v - + -pu2 (au + u v- + v u- + v2 - v) - ax ay c ax ax ay ay +[T + (u 2+ v2\1 (au + av) 2 c p )J ax ay ,av (p~)(a2T + a2T)+ ~pu2(au +uv ax +vu au +v2 ay av) Cp ax2 ay2 C ax ay + (2.36) (p~p )(ux ~: + Tyx ~; + U y ; ; + Txy ~;) T he third term on the left-hand side o f this equation is zero by virtue o f the continuity equation and the second terms on the two sides are identical and therefore can be cancelled. The two-dimensional constant fluid property energy equation, therefore, has the form: (2.37) where <I> is termed the dissipation function and is given by: 1 <I> = IL ( au au av av) a x ax + T yx ay + a y ax + T xy ay (2.38) I f the expressions for the relation between the shearing stresses and the velocity field given in Eqs. (2.2) to (2.4) are used, this dissipation function becomes: C HAPTER 2: T he Equations of Convective Heat Transfer 41 This last term i n Eq. (2.37), the dissipation term, arises, i t will be noted, because o f the work done b y t he viscous stresses. I n low-speed flows this dissipation term will usually be negligible. A discussion of this point will b e g iven later i n this chapter. The above derivation was presented for two-dimensional flow. E xactly the same approach can be used for three-dimensional flow and the resultant energy equation is as follows: · uaT + vaT + waT ax ay az = (~)(a2T + a2T + a21')+ (~)<I> 2 2 p Cp ax ay2 az p Cp (2.40) where, for this three-dimensional flow case, the dissipation function is given by: ~ ~ 2[(~:r +~;r + ~:)} [(~;)+ (::)]' (( (2.41) +[(~;)+ (~:)r +[(~~)+ (~:)r In terms o f t he cylindrical coordinate system defined earlier i n this chapter, the energy equation for steady constant fluid property flow is: uuT + vaT + waT uz ar r acp = (p~)[a2T +!~ (r aTrr2 acp2 + (pCp )+ ~ a2T] ~)<I> Cp az2 r ar a (2.42) where i n this coordinate system the dissipation function is given by: q) = 21(~~)2 +:;)2 +~~; + J] ((~ + [ ! au + aw]2 + [av + au]2 + r acp az az ar [! aacp + aaw _ w]2 v r rr (2.43) I t is perhaps worth pointing out that i f the velocity components are zero the energy equation reduces, o f course, to the heat conduction equation, i.e., i n the case o f Cartesian coordinates, to ~ (2.44) 2.4 SIMILARITY I N C ONVECTIVE HEAT T RANSFER Consideration will now b e given to w hat is basically the same question that was earlier dealt with b y u sing dimensionless 'fmalysis. T he p roblem is, essentially, that o f determining u nder w hat conditions the \fiow a nd temperafure distributions aboht / /' 4 2 Introduction to Convective Heat Transfer Analysis , Vl 1 Yz Xz F IGURE 2 .8 Flow over geometrically I similar bodies. ( Flow Number 1 ) ( Flow Number 2 ) geometrically similar bodies will be " similar", i.e., under what conditions the velocity and temperatUre fields~iabout t he bodies will b e " similar" [7]. To understand what silnilarity means, consider the flow over two g eometriquly s imilar bodies such a s those shown in Fig. 2.8. Assume photographs are taken o f t he flow and temperature fields about the first body and these photographs are then enlarged until the size o f t he body in the photograph is the same as that o f t he sec~ o nd body. I f t he enlarged photographs so obtained show the flow and temperature fields about the s econd body, the two flows are similar. Hence, by similar velocity and temperature fields, i t i s here meant that the velocity components and temperature when expressed in dimensionless form are related to the dimensionless coordinate scheme b y the same functions in all o f t he flows considered. For example, consider two-dimensional flow about two long cylinders. Consider the points a and b in flows 1 and 2. These points are a t geometrically similar positions, i.e., are such that: and YtlDt = Y2 1D2 I f the velocity fields i n t he two flows are similar then the velocity components a t these two points will b e s uch that: (2.45) Attention will first be given to the determination o f conditions under which the velocity fields are similar, the temperature field being dealt with later. The following dimensionless coordinate system is introduced: x = xlL, Y = ylL, Z = z/L (2.46) where L i s some convenient reference dimension. S ince t he series o f bodies being considered are of the same geometrical shape, any convenient dimension can be used for this purpose. I n t he case o f flow over a sphere, for example, the diameter could conveniently be used for L. T he velocity components are also expressed in dimensionless form using the following: CHAPTER 2: The Equations o f Convective Heat Transfer 43 W U = U/UI, = W/UI (2.47) where UI is any convenient characteristic velocity. In the case o f external flows, UI is usually most conveniently taken as the freestream velocity while in internal flows it is usually convenient to take it as the mean velocity. The pressure is nondimensionalized using the dynamic pressure based on U I as follows: P = p /pUr (2.48) p Ut being, o f course, twice the dynamic pressure. In terms o f these dimensionless variables, the equations governing the velocity field, i.e., the continuity and Navier-Stokes equations, become: (2.49) (2.50) (2.51) I t will be seen from this set o f equations that for each value o f the parameter ( pLU 1/ JL) there is a u nique solution, i.e., a unique relation between the dimensionless velocity components and pressure and the position i n terms o f the dimensionless coordinate system. This is equivalent to saying that the flow fields about a series of. geometrically similar bodies will b e d ynamically similar. i f t he Reynolds numbers ( pLUI/ JL) are the same i n all the flows. .. . Consideration will next b e g iven to the determination o f the conditions under which the temperature fields are similar. I n order to define a dimensionless temperature ~ariable, a convenient reference fluid temperature, T I , and some convenient measure o f the wall temperature, Twr , a re introduced. In external flows, Tl is usually most conveniently taken as the freestream temperature while in internal flows it is usually taken as a convenient mean temperature. U sing these the following dimensionless temperature is defined: (2.53) I f the temperature fields i n t he fluid flowing about the series o f geometrically similar bodies are to b e similar, one obvious requirement is that the dimensionless surface temperature distributions must ~e t he same, i.e., at geometrically similar positions on the surfaces, the values o f (T,w - Td/(Twr - Tl) m ust be the same, T w being the local wall temperature a t t he po~nt b eing considered. 44 Introduction to Convective Heat Transfer Analysis I Introducing 0, U, V, W, X, and Z into the energy equation gives: ao ao ao Uax + Vay + Waz = ( )(a¢o a2 a2 ) 0 0 pCpUIL ax2 + ay2 + az2 k + [pCpL(Twr - JLUI TI) ]{ 2 [(aU)2 + (aV)2 + (aW)2] ax ay a z,' + (~~ + ~~r + (~~ + ~~r + (~~ + ~~n \ (2.54) From this i t follows that i f the temperature fields are to b e similar i t is necessary that the following two quantities b e the same in the flows (2.55) I t also, o f course, follows from Eq. (2.54) that i f t he temperature fields are to b e similar the velocity fields must b e similar. The Reynolds number (pUIU JL) has, therefore, already been recognized as a parameter on which the similarity o f the temperature fields depends. The two parameters given i n Eq. (2.55) are, therefore, written as (2.56) From this i t is seen that for the temperature fields to b e similar, i.e., for 0 t9 b e a unique function o f X, y , a nd Z , i t is necessary that, besides the Reynolds number, the following two parameters be the same in the flows (2.57) The first o f these is, o f course, the Prandtl n umber which was previously introduced. The second is termed the Eckert number, i.e.: E ckert number, E c = (T Cp wr 1_ T I ) U2 (2.58) I t should be noted that the Eckert number arises from the dissipation term and that i t can be written as: Ec = (!!L)/(Twr C p TI Tl 1) (2.59) For the case o f gas flows, i f t he gas can be assumed to b e perfect: (1' - 1) (2.60) CHAPTER 2: The Equations o f Convective Heat Transfer 45 where a l is the speed of sound at the temperature, TI. Substituting Eq. (2.60) into Sq. (2.59) then gives: Ec = (1' - I)Mf (~r -1) (2.61) where M I = U 1lal is the Mach number based on U I and T 1. It follows from Eq. (2.61) that the Eckert number and, therefore, the dissipation term will only be significant i f the Mach number is large or i f (TwrIT I) is close ,'to I, i.e., Twr very nearly equal to T 1• I t is significant in the fonner case because 1he velocities are high and in the latter case it is significant because the heat conduction term is very small. In the limiting case where the surface temperature is everywhere equal to T 1, the temperature differences in the fluid are the result of dissipation alone. When Twr is approximately equal to TI the heat transfer rate from the surface will, generally, be small and this case has, therefore, no practical significance in the present work. Thus, for the present purposes, it is concluded that the Eckert number and, therefore, the dissipation term will only be significant i f the Mach number is large. However, constant fluid properties solutions will only be applicable i f the Mach number is low so it is usually consistent with the assumption o f constant fluid properties to ignore the dissipation term. From the above discussion, it can be concluded that the dimensionless temperature field O(X, Y) depends only on the dimensionless numbers R e, Pr, and E c. In order to determine the implications this has for the local heat transfer rate at any point on the surface it is noted that this heat transfer rate is given by Fourier's law as: qw = -k ~~L (2.62) where aTlanl w is the local temperature gradient at the surface measured normal to the surface. Now this derivative can b e written as: aT aT an w = ay I I an I + aT Iw aX~I· ay a ax n w w w (2.63) I t should be noted that the derivatives aylanl w and axlanlw will depend only on the sh~pe o f the surface at the point considered. . Now the right-hand side o f Eq. (2.63) is conveniently rewritten in dimensionless form to give: (2.64) where N =\L n (2.65) Since aXlaNlw and aYlaNlw will d~pend only on the dimensionless shape o f the surface and since the dimensionless temperature gradients aOlaxlw and aOlaYlw 46 Introduction t o Convective Heat Transfer Analysis must depend on the same variables as the dimensionless temperature field, i.e., on Re, Pr, and E c, i t follows that at any position on the surface ao . a N Iw = k ( Twr L Tl) functlon(Re, Pr, E c) Substituting this result into Eq. (2.62) then gives (2.66) qw L (Twr - T 1 )· . (Tw - T1)k = Tw - Tl functlOn(Re, Pr, E c) (2.67) Now, the left-hand side o f this equation is the local Nusselt number, Nu. I t follows, therefore, that for a given surface temperature distribution, i.e., for ~ given distribution o f (Tw - Td/(Twr - T 1): . Nu = function(Re, Pr, E c) (2.68) For constant fluid property flows, the Eckert number will, for the previously discussed reasons, have·a negligible effect and in this case Nu = function(Re, P r) . (2.69) This result was, of course, previously derived by dimensional analysis. 2.5 V ORTICITY A ND T EMPERATURE FIELDS When a viscous fluid flows over a surface, vorticity is generated by the action o f the viscous shearing stresses, the vorticity essentially being proportional to the angular momentum o f the fluid particles. This vorticity is, in general, highest at the surface and decreases with increasing distance from the surface as indicated in Fig. 2.9. The vorticity, i f), which as noted above, is a measure of the angular momentum or rotational motion induced by the action of the viscous stresses acting tangentially around the surface o f the fluid particles, is given in terms o f the velocity components by: (2.70) Distance from Surface Variation o f Magnitude o f Vorticity with Distance from Surface F IGURE 2.9 Vorticity variation in flow. CHAPTER 2: T he Equations of Convective Heat Transfer 47 I f the surface over which the fluid is flowing is held at a different temperature from the bulk of the fluid, there will also be temperature gradients in the fluid near the surface. The difference between the temperature o f the fluid locally and that of the fluid far from the surface is, in general, highest at the surface and decreases with increasing distance from the surface. The purpose of the present section is to show that there is a similarity between these vorticity and temperature fields. For the present purposes, attention will b e restricted to steady, two-dimensional constant fluid property flow and, consistent with this assumption, the dissipation term in the energy equation will be neglected. The result will be derived using the governing equations expressed in Cartesian coordinates although a similar result can, o f course, be obtained in terms o f other coordinate systems. Consider the velocity field first. I t is governed by the continuity and NavierStokes equations which, subject to the assumptions introduced above, are, as previously presented: au + av ax ay uau + vau ax ay uav + vav ax ay = =0 (2.71) (2.72) u -:-.!.. ap + (IL)(a 22 + a2u) pax p ax ay2 _.!.. ap + (IL)(a 22 + a2 v) v pay p ax ay2 = (2.73) T hepressure is eliminated between Eqs. (2.72) and (2.73) by differentiating Eq. (2.72) with respect to y and Eq. (2.73) with respect to x and subtracting the results. This gives: - - - - u - - - - - - v - + - -- +u- + - - +v-2 (2.74) au au ayax a2u ayax avau ayay a2u au av ay2 ax ax a2 avav v ax ax ay a2 v axay Tl1is equation can be arranged as follows: (2.75) But, as noted above, the vorticity, Cd, i s g iven by: \, 48 Introduction to Convective Heat Transfer Analysis so using the continuity equation that Eq. (2.75) can be written as uaw ax + v aw ay = (IL)(a 22 + a2 w) p axw ay2 (2.76) Now when the dissipation term is neglected, the two-dimensional energy equation becomes (2.77) Thus, the vorticity and temperature fields are governed by equations having the same basic form. When P r is equal to 1, the equations have exactly the same form. Even in this case, however, the vorticity and temperature fields will not be identical because the boundary conditions on the two fields at the surface will not in general be identical. However, there will obviously be similarities between the two fields. Vorticity is generated in the flow by the action of viscosity due to the presence of the surface. The temperature differences arise in the flow because the surface is at a temperature which is different f~4?,m the flowing fluid. Thus, Eq. (2.76) essentially describes the rate a t which viscous effects spread into the fluid while Eq. (2.77) describes the rate "at which the effects of the t~mperature changes at the surface spread into the fluid. I t will be seen that the relative rates of. spread depend on the value o f the Prandtl number. . Because they do not contain the pressure as a variable, Eqs. (2.76) and (2.77) have been used quite extensively in solving problems for which the boundary layer equations (see later) cannot be used. For this purpose, instead of solving the NavierStokes and energy, simultaneously with the continuity equation, it is convenient to introduce the stream function, t/!, which is defined such that u= - at/! ay' v= - - at/! ax (2.78) This function, o f course, satisfies the continuity equation, because the left-hand side o f the continuity equation is: au ax + av ay = a2t/! _ a2t/! axay axay = =0 I n terms o f the stream function, Eqs. (2.76) and (2.77) become at/! aw _ at/! aw ayax ax ay at/! aT at/! aT ay ax - ax ay w (IL ) (a 2 + a2 w) p ax2 ay2 (2.79) = Pr ( 1 )(IL )(a22 a2T) p axT + ay2 (2.80) while in terms o f the stream function, the vorticity, as defined by Eq. (2.75), becomes CHAPTER 2: The Equations o f Convective Heat Transfer 49 w= _ (aaX2 + a ay2 2 ", 2 "') (2.81) ,~ These three equations in the three variables " ', W , and T are the set o f equations that must be simultaneously solved subject to the correct boundary conditions to obtain the heat transfer rate. The original set o f equations from which they were deduced, i.e., Eqs. (2.70), (2.71), (2.72), and (2.77), contained four unknowns u, v, p, and T and this reduction in the number o f variables and, therefore, the number o f governing equations i n i tself constitutes a considerable simplification. The governing equations i n t erms o f u, v, p, and T are often said to b e expressed i n "primitive . variable" form. Eqs. (2.79) to (2.81) o r their equivalent i n other coordinate systems have been used i n many numerical solutions o f the two-dimensional Navier-Stokes and energy equations. 2.6 THE EQUATIONS F OR TURBULENT CONVECTIVE HEAT T RANSFEi' In all o f the above discussion i t was assumed that the flow was laminar. The majority of flows encountered i n practice are, however, turbulent. Consideration must, therefore, b e given to t he question o f how t hese equations must b e moaified i n order to apply them to t urbulent flow. I n turbulent flows, even when the mean flow is steady (see below), the flow variables, i.e., velocity, pressure, and temperature, all fluctuate randomly with time due to the superposition o f the turbulent " eddies" on the mean flow. F or example, if the. temperature a t some poiQt i n the. flow is measured by means o f s ome device which has very good time response characteristics and the output o f t his device i s' displayed on a suitable instrument then a signal resembling that shown i n Fig. 2.10 will b e obtained i n turbulent flow. Temperature T T =Instantaneous Value .1' =Mean Value T';"T+T' T' :z:: Fluctuating Component T o~----------~--------------~ T ime,t F IGURE 2.10 Temperature variation with time in turbulent flow. 50 Introduction to Convective H eat Transfer Analysis B ecause the flow variables are fluctuating with time i n this way i t i s necessary t o clearly define what is m eant b y " steady turbulent" flow. To do this, mean values o f t he variables are defined i n t he fo1l6wing way [3],[7],[10],[11],[12]: Ii = 1 t Jr u dt; o 11 = - p = _. t lIt pdt; 0 vdt; w= tot lIt lIt wdt 0 T = 11tf:> Tdt t0 ( 2:82) Here, t i s some interval o f time. T he t erm " steady" t urbulent flow implies that, provided a sufficiently long interval o f time, t, i s taken for this averaging process, constant values o f the mean quantities I i, 11, w, p, a nd T will b e obtained at all points i n t he flow. T hese mean values are. thus often termed time-averaged values. .. F or many purposes, i t i s convenient to express the instantaneous values o f the flow variables i n tenDS o f t hese constant mean values and the fluctuations from these values, i.e., to write: u = Ii + u' ; p= p v = 11 + v'; w = w + w'; + p'; T=T + T' (2.83) w here the primes denote the fluctuating components. I t w ill be obvious that t he m ean (i.e., time averaged) values o f these fluctuating components will, by virtue o f t he definition o f the mean values o f t he variables, i.e., I i,1I, w, p , a nd T , b e e qual to zero, i.e., that u' = ff: u' tit = 0; p' = ff: v' = = ff: 0; v ' d t = 0; T' = p' dt ff: w' = ff: 0 w' d t = 0 (2.84) T' dt = These results can be formally proved i n t he following way, the temperature, T, b eing selected as an example: - lIt T dt lIt (T + T ) dt lIt - dt + -lIt T dt T T =- to =- I to =- I to to B ut T is a constant so T T =- tot It dt + -lIt T dt = T + -lIt, dt -+ T T T -, I - 0 t = 0 therefore, T ' = O. T he m ean value o f the product o f two or more fluctuating components will, however, not i n general b e zero; e.g., u 'u ' ¥: 0; u,2 p ' ¥: u 'T' ¥: 0 u 'v'T' ¥: 0 (2.85) 0; C HAPTER 2: The Equations o f Convective Heat Transfer 51 I n the derivation o f the equations for turbulent flow given below, the following relations, and obvious extensions o f them, concerning the mean and fluctuating values are required. All o f these relations are really obvious but they can be formally proved using the same basic procedure that was used above to prove that T ' = O. If q and r are any two o f the flow variables, i.e., u, v, w, p , T and i f n is any o f the coordinate directions x, v, and z, then qr = aq' an qr; = aan (q') qq' = = 0; aq an qq' = 0; aq an q+r = q+r = (2.86) T he continuity, Navier-Stokes, and energy equations for steady flow were given earlier in this chapter. These equations can b e d erived for unsteady flow using the same basic procedure as that discussed above, the resultant set o f equations being as follows: au + av + aw = 0 ax ay az au + uau + vau + wau = _ ! ap + v (a 2u + a2u + a2u) at ax ay az pax ax2 ay2 az2 av + uav + vav + wav at ax ay az = (2.87) (2.88) _ ! ap + v (a 2v + a2v + a2v) pay ax2 ay2 az2 (2.89) aw + uaw + vaw + waw = _ ! ap + v (a 2w + a2 + a2w) w at ax ay az paz ax2 ay2 az2 aT aT aT aT at + uax + v ay + w az = (a 2T a2T a2T) P r ax2 + ay2 + az 2 v (2.90) (2.91) The dissipation t erm has, for reasons previously given, been neglected in the energy equation. I n the derivation o f these unsteady flow equations, no assumptions regarding the . l1ature of the unsteadiness are made. I n t urbulent flow, therefore, the instantaneous values <:>f the variables will satisfy these equations. Numerical solutions to the above equations are very difficult to obtain and for m ost purposes it is only the mean values of these variables and the mean heat transfer rate that are required. A n attempt is, therefore, usually m ade to express them i n t erms o f t he mean values o f the variables. Attention here will be restricted to the case oftwo-dimensional mean flow, i.e., to flow i n which w = 0, i t being noted that this does not necessarily mean that w' = o. The following values o f t he variables are therefore substituted into the above equa- tions: V u = 11 + u'·, p = p + pi; = !: V + v'; w = W i; T = T + T' (2.92) 52 Introduction to Convective H eat Transfer Analysis Consider first the continuity equation \.(2.85). This becomes a a\ a -x + u') + ---+a ( v + v ') + -(w') (u a y az Taking the time average o f this equation then gives =0 (2.93) !t J(ot [!...-(u + U')]dt + !t Jo [!...-(V + V')]dt ax ay + ! (t [~(w')]dt t r Jo az = 0 I.e., - (u a _ + u ,) + -a - + v ,) + - w' a (v ax ay az au+av ax ay =0 = 0 (2.94) Using the results given in Eq. (2.86) then leads to (2.95) This is the continuity equation for turbulent flow when the mean motion is twodimensional. I t will be noted that this equation has exactly the same form as the continuity equation for two-dimensional steady laminar flow with the mean values o f the velocity components substituted in place o f the steady values that apply in laminar flow. This result can, in fact, be deduced by intuitive reasoning and simply states that i f an elemental control volume through which the fluid flows is considered, then over a sufficiently long period o f time, the fluctuating components contribute nothing to the mass transfer through this control volume. Consider, next, the x-wise Navier-Stokes equation, i.e., Eq. (2.88). Substituting the values o f the variables as given by Eq. (2.92) into this equation and taking the time average o f each o f the terms gives. after multiplying the terms out: - +- au at au' + uau + u u' + u ,Ou + u ,au' + v au -· aat ax ax ax ax ay au' ,au ,au' ,au ,au' + v- + v - + v - + w - + w ay ay ay az az = ------p (2.96) pax 1 ap p ax 1 ap' 2 (a'1i - 'u'+ a2 1i -2- +a21i - - + a- - + a u' - + a u') ax2 ax2 ay2 ay2 az2 az2 Applying rules o f the type given in Eq. (2.86) to this equation then gives on rearrangement since the mean flow is steady, i.e., since au at av = aw = at at 0 (2.97) CHAPTER 2: The Equations o f Convective Heat Transfer 53 the following as the x-wise momentum equation for turbulent flow: _au _au u -+vax ay = 2 2 l ap ,au' ,au') - --+v (a u a- - ~au' -2+ 11) u-+v-+w- (2.98) p ax ax ay2 ax ay az The fluctuating components only occur in the last term o f this equation. This term can be written as follows --a;z +v'- +w'au' au' u 'ax ay az = -- a(u,2) - -;;;; -u'ax ax (2.99) a - ,-,+ - (v u ) - u ,av' + -a "u ) - u ,aw' (w ay ay az az Now i f the continuity equation is multiplied by u and then applied to turbulent flow the following is obtained (11 a a aw' + u ')-(u + u') + (u + u ')-(v + v') + (u + u ')ax ay az 1 1- = 0 (2.100) Time averaging the terms in this equation then gives au av au' av' aw' + u - + u '- + u '- + u 'ax ay ax ay az =0 (2.101) B ut by virtue o f the continuity equation for turbulent flow, i.e., Eq. (2.95), the sum o f the first two terms in this equation is zero so t hat Eq. (2.101) r educes to: ,au' ,av' ,aw' u -+u-+uax ay az = 0 (2.102) Substituting this result into Eq. (2.99) and the result so obtained back into Eq. (2.98) then gives: _au _av u -+vax ay = 2 2 1 ap ul2 , - --+ v (a x11 - y2 - [a.cx ) + -a -, ( + -a (wu - + a 11) vu ') 2a p ax a a ay az I)] (2.103) This is the x-wise momentum equation for constant fluid property turbulent flow when rpe mean flow i s s teady and two-dimensional. Using exactly the same approach will give the following for the y-direction: _av + _av = - -- + v (a 2v + - 2 l ap a u- v-2 ax ay p ay ax ay2 v) (2.104) a- [-(u'v') + _ (v'2) + - (w'v' ) ax ay az o = - (u'w') + :+-:-(v'w') + _(w,2) ax az a- a-] a-- and similarly for the z-direction because, the mean flow is two-dimensional: a. \ \ a-ay (2.105) I 54 Introduction to Convective Heat Transfer Analysis T he t erms involving the d erivativb with respect to -z i n these equations will i n m ost circumstances b e effectively zerQ s ince the m ean flow is two-dimensional. I n this case, the x- and y- momentum equations become: u au 1 ap a-] ua-+v- = - --+v (a 22 a- - [a -u 2 + u) _(u,2)+_(v'u') ax ay pax ax ay2 ax ay a v) -(u'v') + _(v,2) v au 1 ap a-] ua- + v - = - pay + v (axv + - y2 - [ax -- 22 a 2 ax ay a a ay (2.106) / C omparing these equations with the x- and y- Navier-Stokes equatio~s for two':: dimensional laminar flow shows that in turbulent flow extra terms arise Hue to the presence o f t he fluctuating velocity components. These extra terms, which arise because the Navier-Stokes equations contain nonlinear terms, are the result o f the momentum transfer caused by the velocity fluctuating components and are often termed the turbulent o r Reynblds stress terms because o f t heir similarity to the viscous stress terms which a rise d ue to momentum transfer on a molecular scale. This similarity can be clearly seen b y noting that the x-wise momentum equation, for example, for laminar flow c an b e written as: uau + vau = _.!. ap + .!. (au x + aTXY ) ax ay pax p ax ay w here the stress terms are related to the velocity field b y (2.107) ux = 2/L au ay Txy = /L (au + ax ay av) au au 1 ap a 1 12] u - + v-a = - pax + -1 [ -(u x - pu 2) + -a ( fxy - pu) (2.108) -a ax y p ax y where ux a nd T xy a re the time averaged molecular stress terms which are given by: au u x = 2/L ay Txy = /L (ay + T he x -wise momentum equation for turbulent flow c an b e written i n a similar way as follows: au av) ax (2.109) Thus, the extra terms arising from the fluctuating velocity components are such that their effects are the same as an increase in the viscous s hear stress and it is for this reason that they are termed the turbulent or Reynolds stress terms. The presence o f these turbulent stress terms can b e d erived using a more physical approach. To do this, consider, as before, a control volume o f the type shown in Fig. 2.11 through which fluid is flowing, the flow being turbulent. C HAPTER 2: The Equations o f Convective Heat Transfer 55 y o~----------------------~ x F IGURE 2.11 Control volume considered. Consider the flow through face ab o f this control volume and let the area o f this face be d Ax. The rate which x-wise momentum passes into the control volume through this face is given by: Mass flow rate through face xu = p u 2 d Ax Setting u equal to u + u' and time averaging then gives the mean rate at which x-wise momentum crosses the face as: p (u2 + u,2) d Ax Thus, the presence o f the fluctuating turbulent velocity components causes the momentum transfer rate to be different from p x (mean velocity)2·X d A. But in applying the momentum conservation principle to the control volume the presence o f additional momentum transfer is the equivalent o f an additional force on the face o f the control volume in the opposite direction to the momentum transfer. Thus, the additional momentum transfer due to the fluctuating velocity leads to a n equivalent stress, i.e., force per unit area, o f value pu,2 and this is what is termed the turbulent "stress" on the face. "- :; I f the lower face, ad, o f the control volume is considered and i f i t has area d Ay, the mass transfer rate through it will be p VdAy and the rate o f x-wise momentum transfer through it will, therefore, b e p vu d Ay. Time averaging this then shows that on this face there will be an equivalent turbulent "shearing stress" o f value p u'v'. " Using this type, o f argument, the presence o f the turbulent "stresses" i n the other directions can be deduced. Consider, next, the energy equation. Substituting the instantaneous values o f the variables into the unsteady form o f this equation gives: aa/T + T ') + ( ll + u ') a x ( T + T ') + (v + v ') a y ( T + T ') + (w + w ') az ( T + T ') k a = ( p Cp ) [ a x2(T 2 a- a- a- ~2 a2 + T ') + a~2(T + T ') + az2(T + T ') ] (2.110) S6 Introduction to Convective Heat Transfer Analysis Time averaging this equation using the previously discussed rules then gives since the mean temperature distribution is stea4y: _aT +vu - _aT ax ay = (-k)(EPT + - 2T) - pt;,aT' +v ,- +w- (2.111)/ a " aT' , aT') upCp ax2 ay2 ax ay az a a - (u'T') + - (v'T') ax ay Consider the last term in this equation. It can be rewritten as follows: aT' aT' aT' u '- + v '- + w 'ax ay az = (2.112) + ~(w'T') - T rau ' + T,av' + T,aw' az ax ay az B ut i f the continuity equation is multiplied by T ( = T + T') and the terms then time averaged, the following is obtained. au··· T - + TOv+ T,au' T,av' T,aw'- 0 -y -x + -y+ ax a a a az The s um o f t he first two terms in this equation, which can be written as aWay), i s zero b y virtue o f the continuity equation so that: (2.113) T(au/ax + (2.114) T,au' + T,av' + T,aw' ax ay az =0 Substituting this result into Eq. (2.112) and neglecting the term involving the zderivative then gives: u aT + vaT ax ay = (p~)(a2T + a2T)_ [ax ~(U'T') + ~(V'T')l Cp ax2 ay2 ay (2.115) T his is the energy equation for two-dimensional turbulent flow. Comparing it with the equation for laminar flow shows that in turbulent flow extra terms arise because o f t he fluctuating velocity and temperature components. These terms arise because o f t he enthalpy transport caused by these fluctuating terms. Now Eq. (2.115) can be written as u aT + v aT ax ay = (1){ pCp a - ;) ax [ (k ax - (pcpu T, ] aT) (2.116) + :A(k ~~)- (PCPV'T')]} Therefore, since ( - kaT Iax) and ( - kaT lay) are the time-averaged heat conduction rates p er unit area in the x- and y-directions respectively, it will be seen that the effects o f the additional turbulence terms are the same as an increase in the heat transfer rate. For this reason, these extra terms pcpu'T' and pcpv'T' are often termed the turbulent heat transfer terms. Their presence can be demonstrated in a more physical manner using the same line o f reasoning as was adopted in the discussion o f the turbulent stresses. For example, considering the element shown in ~$1; A" $" CHAPTER 2: The Equations of Convective Heat Transfer 57 Fig. 2.11, the rate at which enthalpy enters the face ab parallel to the y-axis is equal to p ucpT d Ax. I f the instantaneous values o f u and T are substituted into this equation and the time-average taken then the mean rate at which enthalpy enters this face per unit area will be found to b e ( pcpuT + p cpu'T'). Thus, because o f the fluctuating components, this rate is greater than ( p X c p X mean velocity X m ean temperature). This additional enthalpy transport is equivalent to additional heat transfer into this face at a rate o f p cpu'T' p er unit area. Similarly, i f the face o f the element ad parallel to the x-axis is considered, the mean rate at which enthalpy enters it per unit area will b e found to be ( pcpvT + p cpv'T'). There is, therefore, again an additional enthalpy transport due to the turbulent fluctuations, this addition being equivalent to an increase in the heat transfer rate per unit area o f p cpv'T'. The four equations governing two-dimensional turbulent flow are, therefore, Eqs. (2.95), (2.106), and (2.116). These contain, beside the four mean flow variables u, V, p , and T, additional terms which depend on the turbulence field. In order to solve this set o f equations, therefore, information concerning these turbulence terms must be available and the difficulties associated with the solution o f turbulent flow problems arise basically from the difficulty o f analytically predicting the values of these terms. The set o f equations effectively contains more variables than the number o f equations. This is termed the turbulence "closure problem". In order to bring about closure o f this set o f equations, extra equations must be generated, these extra equations constituting a "turbulence model". I n the many traditional methods o f calculating turbulent flows, these turbulence terms are empirically defined, i.e., turbulence models that are almost entirely empirical are used. Some success has, however, been achieved by using additional differential equations to h elp i n the description o f these terms. Empiricism is not entirely eliminated, at present, b y t he use o f these extra equations but the empiricism can be introduced in a more systematic and logical manner than is possible i f the turbulence terms in t he m omentum equation are completely empirically described. One o f the most widely used additional equations for this purpose is the turbulence kinetic energy equation and its general derivation will now be discussed. The x-wise momentum equation for two-dimensional mean flow is first multiplied by u' to give: a a a a u '-(u + u') + u '(u + u ')--(u + u') + u '(v + v ')--(u + u') + u 'w'-(u + u') at ax ay az = - u' ~(jJ + p') + (J..t )u' [ a2 (u + u') + a2 (u + u') + aa 2 (u + U')] 2 2 2 p ax p ax ay z (2.117) Time averaging the terms in this equation then gives for steady mean flow . , au' a au ,2au' _ , au' - ,- ,au , , au' , , au' l1u - + u - - + u - + vu - + u v - - + u v - + u w ax ax ax ay ay ay az 2 2u'] 1 , ap' / .L, 2 = - -u-+(-)u [ a u' a u'+ - 2 ay2 a zt +p ax p ax a --=--------------~ 58 Introduction to Convective Heat Transfer Analysis w hich c an b e written as: _1--u + v--u u a ( '2) _1 a ( '2) 2 ax 2 ay '2)du , , au' . , , au') + (u - + -l.v ,)aU ~u - +uv-'+uwu ( - + 2aU.' ax ay ax ay az (2.118) where: a2 a2 a2 V2 = _ +_+_ a x2 ay2 az2 (2.119) Now the bracketed term on the left-hand side o f this equation c an b e written as: 1uav' a - ( au'3 + au,2 v' + au'2 w') - - ,2 (au' + - + - w') 2 a x· ay az 2 ax ay az (2.120) B ut multiplying the continuity equation by u,2 and time averaging shows that: u ,2 ( au'+av' + - ) =0 - aw' ax ay az (2.121) Using this result in Eq. (2.120) and then substituting back into Eq. (2.118) gives: _ 1-- (u u a '2) 2 ax '2)aU 1 , ap' + _1-- (u '2) = - (u - - - (,' - - - u va u v )au 2 ay ax ay p ax _ ~ [a(uxI3 ) + ay ~(uI2v'2) + ~(UI2W'2)] + (11- )(U IV2UI) 2a az p (2.122) Similarly i f the y-momentum equation is multiplied by v' and the result time averaged i t c an b e shown that: _ 1 -- (v u a /2) 2ax _ '2) a 1a + _1-- (v 12) - - -Iv') - .v - (v - v - -v, - p ' va u (a 2 ay ax ay p ay (2.123) - ~ [a x ~(VI2U') + a(v'3) + ~(VI2W/)l + (11- )v IV 2V' ~y 2 az p Lastly, multiplying the z-momentum equation by w', time averaging, and rearranging gives: _ 1 a - /2- _1 a - ,21 , ap' u --(w ) + v --(w ) = - - w 2ax 2 ay p az - ~ [~(WI2UI) + ~(wI2v') + ~(WI3)l 2 ax ay az (2.124) + ( ; )w'V2 w' C HAPTER 2: The Equations of Convective Heat Transfer 59 Adding these three equations, i.e., (2.122), (2.123), and (2.124), together and definmg K = (u,2 + v,2 + w,2)12 (2.125) K thus being the mean kinetic energy associated with the velocity fluctuations, gives: _aK + _aK u- vax ay =- [( u'2)aU + - (' - + - ('- + (v ,)au ,)av '2)av] ax u v ay u v ax ay p ,ap' ,a p'] 1 - - [=¥xa v - + w -' p uax+ ay az a a a - -l [a- + _ (u,2 v ') + _ (u,2 w ') + _ (v,2 u ') _(u'3) 2 ax ay az ax a- a a a a --] + _ (v,3) + _ (v,2 w ') + _ (w,2 u ') + _ (w,2 v ') + _ (w,3) ay az ax ay az (2.126) + (~ )[u'V2 u' + v'v2 v' + w'V2 w'] I t is, o f course, not obvious, at this stage, how this equation can be used in the determination o f the turbulence terms in the momentum equation. I n order to utilize it for this purpose, some o f the terms i n the equation must be related to other flow quantities by approximate relations. A discussion of this will be given in Chapter 5. 2.7 S IMPLIFYING ASSUMPTIONS USED IN THE ANALYSIS O F C ONVECTION The general .equations governing convective heat transfer, i.e., the continuity, Navier-Stokes, and energy equations, together form a very complex set o f simultaneous partial differential equations. Analytical solutions to this set o f equations have only been found for a few, rel~tively very &imple cases. Numerical solutions to these equations can b e obtained using relatively moderately sized computers although the \ computer times involved with three-dimensional flow solutions may be relatively large. There i s still, therefore, considerable interest i n obtaining approximate analytical or simplified numerical solutions to these equations. Now, in many flows of practical importance, some o f the/terms i n t he equations are negligibly small compared to the remaining terms. By dropping these terms, i.e., by assuming that the flow has certain simplifying characteristics, the complexity o f the governing set of equations is often considerably reduced and its solution becomes much more feasible. The following two types of flow are, from a practical viewpoint, probably the most important i n which such simplifying assumptions can be adopted. • I n the case o f flow inside a duct, it c~ b e assumed that the flow is fully develpped. Basically this means that it can be assumed that certain properties o f the flow do not 60 Introduction to Convective H eat T ransfer Analysis change with distance along the duct.l~eal duct flows well away from the entrance or other fittings are very nearly fully developed in many cases. For fully developed duct flows in'which i t can be assumed that the fluid properties are constant, the form o f the velocity and temperature profiles do not change with distance along the duct, i.e., considering the variables as defined in Fig. 2.12" i f the velocity and temperature pro~les are expressed in the form (2.127) then i n fully developed flow, the velocity and temperature profile functions f an<f ' g are independent o f t he distance along the duct. This will mean that the velocity at any distance y from the center line o f the duct will remain constant with distance along the duct and that the temperature at this position will vary in such a way relative to the center line and wall temperatures that ( T - Tc)/(Tw - Tc) remains constant. In fully developed flow because u is not changing, it follows that the velocity components in the radial and tangential coordinate directions will be zero. From 'this discussion it should b e clear that many o f the terms in the general governing equations will b e zero when they are applied to fully developed flow. F or this reason, analytical solutions for fully developed flows can relatively easily be derived. • I n the case o f external flows at high Reynolds numbers, i t c an be assumed that the effects o f viscosity (and, therefore, turbulence) and heat transfer are restricted to a thin region immediately adjacent to the surface over which the fluid is flowing and that outside this region the flow is effectively inviscid and at constant temperature. In this region, termed the boundary layer, in which the effects o f viscosity and heat transfer are important, the velocity and temperature change from the values they have at the wall to the values applicable in the outer inviscid flow. Since the boundary layer is thin in a very large number o f practically important flows, the gradients o f velocity and temperature in the direction normal to the surface are high. As a result it is possible, as will b e shown in the next section, to drop a number o f terms from the governing equations when applying them to the flow in the boundary layer. Further, because the boundary layer is thin, i t is possible to calculate the outer effectively inviscid flow b y neglecting the presence o f the Flow ~ F IGURE 2.12 Velocity and temperature profiles in pipe flow. CHAPTER 2: T he Equations of Convective Heat Transfer 61 boundary layer, i.e., by simply calculating the potential flow over the surface concerned. The velocity at the surface given by this potential flow solution is then used as the boundary condition on velocity at the outer edge of the boundary layer in the boundary layer solution. Because of the simplifications resulting from the introduction of the boundary layer concept, it is possible to obtain analytical solutions to such flows in a number of cases. Numerical solutions to the boundary layer equations are also readily obtained with extremely modest computer resources compared to those needed for the solution of the full equations. 2.8 THE BOUNDARY LAYER EQUATIONS FOR LAMINAR F LOW As explained in the preceding section, it is possible in many external flows in which the Reynolds numbers are large (the reason for this stipulation is discussed later) to consider the flow to b e split into two regions. It is not, of course, possible to define precisely where one region ends and the other begins, the two regions actually merging into each other with no sharp division between them. The two regions are: • A thin layer adjacent to the surface in which the effects of viscosity and heat transfer are important. This region is termed the boundary layer. • An outer region in which the flow is effectively inviscid and at uniform temperature. This region is often termed the freestream. The two regions are shown schematically in Fig. 2.13. The flow in the boundary layer can be either laminar or turbulent. In the present section, attention is being restricted to those cases in which the flow in the boundary layer is laminar. Now, in general, the effects o f viscosity and heat transfer do not extend to the same distance from the surface. For this reason, it is convenient to define both a velocity boundary layer thickness and a thermal or temperature boundary layer thickness as shown in Fig. 2.14. The velocity boundary layer thickness is a measure o f the distance from the surface at which viscous effects cease to be important while the thermal boundary layer thickness is a measure o f the distance from the wall at which heat transfer effects cease to be important. For the flow o f gases, the two boundary layer thicknesses are generally, approximately equal. For the flow o f liquids, however, the two thicknesses are generally F reestream Boundary Layer ~-. ~-------- -~-.-. Freestream ----=--======:::::::= ------------------ F IGURE 2.13 B oundary layer and free stream flows. 62 Introduction to Convective Heat Transfer Analysis Flow _t=~-----E~~&}g;';--;1~;Layer F IGURE 2.14 Velocity and temperature boundary layers. very different although, except in the case o f liquid metals, both are o f the same order o f magnitude. The discussion o f the present section is concerned with the simplifications to the general equations that can be introduced as a result o f the fact that the boundary layer is thin. Due to the action o f viscosity, the velocity o f the fluid in contact with the surface over which it is flowing has the same velocity as this surface. In most cases, this surface velocity is zero in a coordinate system attached to the body. Similarly, the temperature o f the fluid in contact with the surface is the same as that o f the surface. A t the outer "edge->' o f the boundary layer, the fluid velocity and temperature become equal to those in the freestream and so are very different from the values at the wall. Therefore, i f the boundary layer is thin, the gradients o f velocity and temperature across the boundary layer, i.e., normal to the surface, will be much greater than those in the direction parallel to the surface. It is as a result o f this that certain o f the terms in the governing equations are negligibly small compared to the remaining terms and can b e ignored. The equations t hat result when these terms are dropped are called the boundary layer equations. In order to illustrate how the boundary layer equations are derived, [7],[9],[13], [14],[15], consider two-dimensional constant fluid property flow over a plane surface which is set parallel to the x-axis. The following are then defined: = Some measure o f the order o f magnitude o f the velocity in the freestream outside the boundary layer. L = Some characteristic dimension o f the body, e.g., its length. S = Some measure o f the order o f magnitude o f the boundary layer thickness. Ul No distinction is made at this stage between the velocity and temperature boundary layer thicknesses since both are assumed to b e o f the same order o f magnitude. S is, therefore, a measure o f the order o f magnitude o f both boundary layer thicknesses. Now, in the boundary layer, the velocity component U varies from zero at the surface, i.e., at y = 0, to approximately U l at the outer edge o f the layer, where y is o f the same order as S. In the boundary layer, therefore, (U/Ul) has the order o f magnitude o f 1, i.e., 0(1). Similarly, since y varies from to S across the boundary layer, ( y/S) will have 0(1) in the boundary layer. Similarly, ( x/L) has 0(1). In light o f these conclusions regarding the orders o f magnitude o f (U/Ul), (y/S), and ( x/L), all being 0(1), consideration can now be given to the derivation o f the ° CHAPTER 2: The Equations of Convective Heat Transfer 63 boundary layer equations by an order of magnitude analysis of the terms in the general equations. Consider, first, the continuity equation which for the case under consideration is: This can be rewritten as follows a(UIUl) a ( xIL) + a (vlud 1 a (ylo) X ( oiL) = 0 (2.128) From this it follows that O(VIUl) = O(UIUl) o (xIL) X o(yIB) X o (oIL) (2.129) Therefore, since, as discussed above, o (ulud, o (ylo), and q (xIL) are all unity: (2.130) Since the basic assumption o f boundary layer theory is that ( oiL) is small, it follows from this that (VIUl) is also small. Thus, in the boundary layer the lateral velocity component, v, is very much smaller than the longitudinal component, u, which, of course, is what is physically to be expected. Next consider the x-wise momentum equation. For the type o f flow being considered, this has the form u au ax + v au = ay _ "!ap p ax + ( 1-t)[a 2 u + a 2 u] p ax2 ay2 This can be rewritten as ( u ) a(UIUl) Ul a (xIL) + 1 ( v ) a (ulud Ul a (ylo) X ( oiL) (2.131) Now the pressure changes in the flow can at most be o f the same order as the dynamic pressure p ur and because x is o f the same order as L, it follows that the first term on the right-hand side o fEq. (2.131) has, at most, the order o f magnitude o f unity, i.e., 0(1). Therefore, defining (2.132) ReL thus being the Reynolds' number based on Ul a ndL, the orders o f magnitude o f the various terms in Eq. (2.131) are as follows: o ( 1) 0(1) 0(1) + o (oIL) 0(1) ( oiL) = 0(1) 1 • 0(1) + 1 [ 0(1) ReL 0(1) + 0(1) 1 0(1) (0IL)2 1 64 Introduction to Convective Heat Transfer Analysis i.e., 0(1) + 0(1) = 0 (1) +. \[ ( 1) + 0 ReL O (lIReL )] (8IL)2 (2.133) The last term in this equation shows that i f ( 8IL) is to be small, as it must be i f the boundary layer approximations are to be applicable, then ReL must be large, its order o f magnitude being such that: o (lIReL) 0 [(8IL)2] = 0(1) i.e., (2.134) Basically, in deriving the dimensionless equation (2.134), the magnitude o f the viscous term has been compared with the magnitude o f the inertia "force" o f which p ur is a measure. From this comparison, i t then follows that i f these two have the same orders o f magnitude and 81L is small, ReL must be large. Note also from Eq. (2.135) i t follows that 81L = 0 [1/ J ReLl. A s a consequence o f the above result, it is seen that the first term within the square bracket o h the right-hand side o fEq. (2.134), i.e., the term originating from (JLa 2ulax2) will have 0(8IL)2 w hereas the other terms in the equation have 0(1). F or this reason, when dealing with boundary layer flows, this term in the x-wise Navier-Stokes equation is negligible. For such flows, therefore, the x-momentum equation is: (2.135) Next, consider the y-direction Navier-Stokes equation which for the flow situation being considered is: u av ax + v av ay = 2 _.!. ay + ( JL)[a 2v + ay2 ap a v] p p a x2 (2.136) Expressing this in terms o f the same dimensionless variables as were used with the x-direction equation and then multiplying through by (81L) gives: a(VIUI) ( ulu)} a (xIL) ( 8IL) + (VIUI) a (yI8) a(VIUI) (2.137) = _ a (pl pU1 2) a (yI8) + _ 1_ [ a 2(VIU}} (81L) ReL a (xIL)2 + a2(vlul) ] a(yI8)2(8IL) Now it was shown above that (VIUI) was o f the order o f ( 8IL) and that for the boundary layer assumptions to be applicable ReL was o f the order o f 1/(8IL)2. The orders o f magnitude o f the various terms in Eq. (2.137) are then as follows: 0 [(8IL)2] + 0[(81L)2] = - a~~;!) + 0 [(8IL)4] + 0[(81L)2] (2.138) CHAPTER 2: The Equations of Convective Heat Transfer 65 Since terms of the order of magnitude o f (0/L)2 and smaller are being neglected, it follows from this equation that the y-momentum equation for boundary layer flow IS: ap = 0 ay (2.139) Eq. (2.139) indicates that the changes of pressure in the y-direction can be ignored in boundary layer flows. I t follows that the x-direction momentum equation for two-dimensional boundary layer flows, i.e., Eq. (2.135), can be written as: u au ax + v au = ay _~ d p + ( f.L)a 2u p dx p ay2 (2.140) because the pressure depends only on x. The pressure in the boundary layer will, therefore, be essentially the same as that acting along the outer " edge" o f the boundary layer at the same value of x. As previously mentioned, however, because the boundary layer is thin, the potential flow over the surface can be calculated ignoring the presence of the boundary layer and the values of the variables that this solution gives at the surface can be used as the boundary conditions at the outer edge o f the boundary layer. As a result of this, ( d p /dx) becomes a known quantity in the boundary layer solution. Next consider the energy equation which for the type o f flow being considered IS: + (~){2[au]2 + 2 [av]2 + [ au + a v]2} pCp ax ay ay ax (2.141) The following dimensionless temperature variable is introduced: () = ( T - T 1) ( Twr-TI) (2.142) where Twr is some measure o f the order o f magnitude o f the surface temperature and T I is some measure o f the order o f magnitude o f the freestream temperature. In the boundary layer, () will be o f the order o f magnitude o f unity. In terms o f the dimensionless variables, Eq. (2.141) becomes: [ + f.LUI ] { 2 [ a{U/UI) ] pCpL(Twr - TI) a ( ;f/L) 2 (2.143) .. 1. . 1 [aa(V/UI) ]2(0IL)2 + [aa(U/UI) (oiL) + a(vlud ]2} +2 (ylo) (ylo) a (xIL) 66 Introduction to Convective Heat Transfer Analysis . It will b e noted that (Pc'~l) -p-r-~-e-L = and where P r i s the Prandtl number and E c the Eckert number. Therefore, i f tq.ese terms are at most o f the order o f 1 ( E c can o f course b e very small in which case the dissipation term is negligible) the various terms in Eq. (2.143) are as follows: 0(1) + 0 (8/L) _ 2{ 0 (8/L) - o[(8/L)] 0(1) + 0 [(8/L)2] I} + 1 ~[(8/L) ] . 2 . 2} 2 { 0(1) + ~[(8/L)2] q [(8/L)2] + 0[(8/L)2] + 0 [(8/L) ] + 0(1) + 0 [(8/L) ]. (2.144) the last t erm in the square bracket in Eq. (2.143) having been multiplied out. I fterms o f the order (8/L)2 a nd less are again neglected, i t will be seen that th~ energy equation for laminar two-dimensional boundary layer flow becomes': (2.145) To summarize, therefore, the equations for two-dimensional, constant fluid property boundary layer flow over a plane surface are: au + av ax ay uau + vau ax ay = = 0 _.!. dp + (lL)a 2u p dx p ay2 u~~ + v! = (p~. )~; + ( ::. )(~;)2 These equations were derived for flow over a plane surface. They may be applied to flow over a curved surface provided that the boundary layer thickness remains small compared to t he radius o f curvature o f the surface. When applied to flow over ' a curved surface, x i s measured along the surface and y is measured normal to i t al all points as shown i n Fig. 2.15, i.e., body-fitted coordinates are used. In order to solve the above set o f boundary layer equations, the boundary conditions on the variables involved, i.e., u, v, and T, m ust b e known. At the wa1I, the I CHAPTER 2: The Equations o f Convective Heat Transfer 67 y x x x - Tangential to Surface y - Normal to Surface F IGURE 2.15 Coordinate system used for boundary layer flow over a curved surface. no-slip condition requires that the velocity be the same as that o f the wall, i.e., usually zero, so that one such boundary condition is A ty = 0: U =v=O (2.146) I f there is blowing or suction at the surface, the rate o f blowing or sucking will determine the value o f v a t the wall. The boundary conditions on temperature at the wall depend on the thermal conditions specified at the wall. I f the distribution o f the temperature of the wall is specified then, since the fluid in contact with the wall must be at the same temperature as the wall, the boundary condition on temperature is: A ty = 0: T = Tw (2.147) where T w is the temperature o f the wall at the particular value o f x being considered. I f, instead o f the wall temperature, the heat transfer rate at the wall is specified, the boundary condition becomes, using Fourier's law: At y = 0 : -=-- aT qw k ay (2.148) where qw is the specifieg local heat transfer rate from the wall per unit area at the particular value o f x being considered~ T he conditions on velocity at the outer " edge" o f the boundary layer are a little more difficult to define because there is really a n interaction between the boundary layer fl6w and the outer inviscid flow, i.e., because of the reduction in velocity near the surface, the outer flow is somewhat different from that i n truJy inviscid flow over the surface. However, as discussed above, i n m any cases, this effect can be ignored because the boundary l ayer remains thin and i n such cases the inviscid flow over the surface considered is calculated and the value that this solution gives for the velocity on the surface at any position is used as. the boundary condition at the outer "edge" o f the boundary layer, i.e., i f U l ( x) i s the surface velocity distribution given by the solution for inviscid flow over the. ~urface then the boundwy.coriditlon on the boundary layer solution is . For large y : U ~ U l(X) (2.14~) 68 Introduction to Convective Heat Transfer Analysis For example, the inviscid solution for flow over a flat plate is simply that the velocity is constant everywhere and eqQal to the velocity in the undisturbed flow ahead o f the plate, say U l. I n calculating the boundary layer on a flat plate, therefore, the outer boundary condition is that U must tend to U 1 at larg~ y. The term "large y" is meant to imply "outside the boundary layer", the boundary layer thickness, 8 , being by assumption small. There are some cases where this approach fails. O ne such case is that in which significant regions o f separated flow exist. In this case, although the boundary layer equations are adequate to describe the flow upstream o f the separation point, the presence o f the separated region alters the "effective" body shape for the outer inviscid flow and the velocity outside the boundary layer will be different from that given by the inviscid flow solution over the solid surface involved. For example, consider flow over a circular cylinder as shown in Fig. 2.16. Potential theory gives the velocity, U l, on the surface o f the cylinder as: ( UI/U) = 2sin(x/R) (2.150) However, because o f the large wake, the actual velocity distribution outside of the boundary layer is very different from this except·in the vicinity o f the stagnation point. Another case where the boundary layer and th~ freestream interact occurs in the entrance region tp ducts as shown in Fig. 2.17. Separation P oint u FIGURE 2.16 Two-dimensional flow over a circular cylinder. F low ---...... . ~-----------------r------------- F IGURE 2.17 Flow in the entrance region to a duct. CHAPTER 2: T he Equations of Convective Heat Transfer 69 _____..-..-::~:=T;:-::.:::::::'::"_.T __..."....:==_ ·-T-·· ----------__ FIGURE2.1S ====----~------ T~----.-._-~-.--.--............: -. .-.'...-- __. - .-.. . ----------····-···-.--T - ·Freestream- _ - - - - - I--- ----- ----: _ _- - - --- __- -- -. T1_ Temperature is TI everywhere outside the boundary layer. Temperature distribution outside boundary layer. Such flow'can b e treated adequately using the boundary layer assumptions but the freestream velocity gradients exist purely because o f the boundary layer growth and the boundary layer and inviscid core flows must be simultaneously considered. There are, nevertheless, many important practical problems in which such interactions can be ignored. In passing, i t should also b e noted that since the flow outside the boundary is assumed to b e inviscid and since VI is o f a lower order o f magnitude than U I a t the . outer edge o f the boundary layer, the Bernoulli equation gives: - ap ax = - PUl- aUl ax (2.151) This equation then relates the pressure gradient term in the boundary layer equations to the free stream velocity distribution. The last boundary condition that has to b e considered is that on temperature at the outer " edge" o f t he boundary layer. Since,· b y assumption, the effects o f h eat transfer are negligible ou,side t he boundary layer, the fluid temperature outside the boundary layer must everywhere b e t he same and, therefore, equal to the temperature in the undisturbed free stream ahead o f t he surface as shown in Fig. 2.18. ' ijIe outer boundary condition on temperature is, therefore: For large y : (2.152) 2.9 T HE BOUNDARY LAYER EQUATIONS F OR TURBULENT F LOW Compared to the x-wise Navier-Stokes ;equation for two-dimensional laminar flow, the equivalent equation for turbulent flo~ has, when the time averaged values o f the 70 Introduction to Convective He~t Transfer Analysis variables are substituted in place o f t he steady values in laminar flow, the additional term: - - - +-(u'v') [au'x2 aa - ], .a y (2.153) I f the turbulent momentum equation is expressed in nondimensional form i n t he same way as was done in deriving the laminar boundary layer equations then the additional term becomes: - a ut a uI 1 [a(xIL) (u'2) + a(yI8) (U'v') (8IL) ] :(2.154) Now, the rest o f the terms retained in the boundary layer equations have the order o f magnitude o f unity and, therefore, for the boundary layer equations to apply, the dimensionless turbulence terms (u'2IuI) and (u'v'luI), which are assumed to have the same order o f magnitude, will have the order o f magnitude o f (8IL) at most. The first term i n Eq. (2.154) is, therefore, negligible compared to the rest o f the terms in the boundary layer equations. Therefore, the x-wise momentum equation for turbulent boundary layer flow is: _ au _au 1 ap u - +v- = - - ax ay p ax + p- - a2u ay2 -(uv) a - ,-, ay (2.155) Consider next the y-momentum equation. Expressing this in dimensionless form in the same way as was done with the laminar flow equation and neglecting the appropriate mean flow terms, this equation then gives: (2.156) But v'2luI m ust have the same order o f magnitude as the turbulence terms in the x-momentum equation so its order o f magnitude is (8IL). Hence, it follows that i n turbulent flow, as in laminar flow: ap = 0 ay (2.157) The same line o f reasoning can be applied to the energy equation and i f i t is assumed that the turbulence terms (u'T')luI(Twr - TI) and (v'T')luI(Twr - TI) i n the resultant equation have the same order o f magnitude as the turbulence terms in the momentum equation, i.e., ( 8IL), then the energy equation for turbulent boundary layer flow becomes (2.158) III!IIII:JiIlll!lllZIII!IIII.III!IIII• • . ., ii!141iJ£1ilIiIll!llll.l.I!!II'AA!!IIi!;;;;;III!IIII·¥,,;;;;;Q,.................. ' ¥'._""""'III!IIII......;;;;;Q,..i!X_XU;;;;;Q , . _,_ k......" '_'_""'",z- .- - - - - - - - - - - - - - - - - - - . . ...... a ... .• ----- CHAPTER 2: The Equations of Convective Heat Transfer 71 To summarize, the equations for two-dimensional constant fluid property, turbulent boundary layer flow are au + av = 0 ax ay au au 1 ap a2 a - u u- + v - = - - - + p - - -(u'v') ax ay p ax ay2 ay _aT +_aT - (- k )a2T _ - v(T') _ a- ' uax v ay p Cp ay2 ay A s was the case with the full equations, these contain beside the three mean flow variables u, v, and T (the pressure is, o f course, by virtue o f Eq. (2.157) again determined by the external inviscid flow) additional terms arising as a result o f the turbulence. Therefore, as previously discussed, in order to solve this set o f equations, there must be an additional input o f information, i.e., a turbulence model must be used; Many turbulence models are based on the turbulence kinetic energy equation that was previously derived. When the boundary layer assumptions are applied to this equation, it becomes: - -K + _ -K ua va ax ay = - v au 1 v p ') - (u,-, )- - - P iia ay p ay - ! [ay ~(UI2VI) + ~(v'3) + ~(WI3V')l 2 ay ay (2.159) 2.10 T HE BOUNDARY LAYER INTEGRAL EQUATIONS While the boundary layer equations that were derived in the preceding sections are much simpler than the general equations from which they were derived, they still form a complex set o f simultaneous partial differential equations. Analytical solutions to this set o f equations have been obtained in a few important cases. For the majority o f flows, however, a numePical solution procedure must b e adopted. Such solutions are readily obtained today using modest modern computing facilities. This was, however, not always so. F or this reason, approximate solutions to the boundary layer equations have i n the past been quite widely used. While such methods o f solution are less important today, they are still used to some extent. O ne such approach will, therefore, be considered i n t he present text. When only approximate values o f the overall features o f the flow, such as the surface heat transfer rate a nd surface shearing stress, are required, i t is pos,sible to apply the boundary layer assumptions in a different way to obtain, relatively easily, approximate solutions for these quantities. The &~rivation o f the equations required i n this approximate solution procedure will b e discussed in the present section [2],[7],[13). 7 2 Introduction to Convective Heat Transfer Analysis Now the Navier-Stokes and energy equations are derived b y applying the conservation o f m omentum and conservation. o f e nergy principles to a differentially small control volume through which the fluid is flowing a nd then taking the limiting form o f these equations as the dimensions o f this control volume tend to zero. The resultant set o f equations governs the conditions at any point i n t he flow. In the approximate method being discussed here these same conservation principles are applied to a control volume which spans the whole boundary layer as shown in Fig: 2.19. The height, o f t he control volume i s chosen to be gr~ater t han the boundary layer thickness. The limiting forms o f the equations that result from the application o f these conservation princip~his control volume as d x ~ 0 g ive a set o f equations gov~ erning the " average" conditions across the boundary layer. These resulta\nt equations are termed the boundary layer momentum integral and energy integral equations. In order to illustrate how these integral equations are derived, attention will b e given to two-dimensional, constant fluid property flow. First, consider conservation o f momentum. It is. a ssumed that the flow consists of.a boundary layer and an outer inviscid flow and that, because the boundary layer is thin, the pressure is constant across the.boundary layer. The boundary layer is assumed to have a distinct e dge i n' the present analysis. This is shown i n Fig. 2.20. The width o f t he control volume is arbitrary and is taken as unity for convenience. T he w all is assumed to be solid. e, Control Freestream Surface F IGURE 2.19 Control volume used in deriving the boundary layer integral equation. / / ....- -"'=-.---=. ..~"""---- c ----..----------------.... "Edge" o f Boundary Layer .....' F IGURE 2.20 Assumed flow through control _ volume.. C HAPTER 2: The Equations of Convective Heat Transfer 73 Applying the conservation o f m omentum principle to the control volume indicated by a bed i n Fig. 2.20 then gives: Net force Rate at which on control x-momentum volume in leaves through positive face de x-direction Rate at which x-momentum enters through face ab Rate at which x-momentum (2.160) enters through face be The various terms i n this equation will now be separately considered starting with the net force term. Because b e is parallel to the wall and lies in the freestream where, by the basic assumptions, viscous forces are negligible, there can be no force acting o n be. Therefore: Net force on control _ Pressure force on ab in x -direction, Fx Pressure _ Shearing force on de force on a d ~ p e-(p+ ~~dx)e-TWdX ~ -~~dxe-Twdx (2.161) where'Tw is the shearing stress acting on the wall. Next consider the momentum terms i n Eq. (2.160). To find the rate at which momentum enters through ab it is noted that the rate at which mass crosses the strip on a b indicated i n Fig. 2 .20 is equal to ( pu d y), t he control surface having a unit width. T he rate at which momentum enters through this strip is, therefore, equal to ( pu 2 dy). T he rate at which momentum enters through the face ab, w hich will here be t ermed M ab, is, therefore, given b y M ab = p L e u2 dy (2.162) I t should b e noted, and the point will b e discussed further at a later stage, that the equations derived in the present section are applicable to both laminar flow and turbulent flow. I n the case o f turbulent flow, the mean values o f the variables are implied although strictly for turbulent flow, M ab will be given by p I fei 0 dy +p fe0 u,2 d y For reasons discussed i n t he derivation o f t he partial differential equations for turbulent boundary layer flow, the turbulence term is, however, neglected. The rate a t w hich m omentum leaves the control volume through the face dc, which will here b e t ermed Mdc, is, because d x is b y assumption small, related to M ab by: (2.163) Using Eq. (2.162) then gives M dc ~ PJ: u dy + P d : [J: u d Y] dx 2 2 (2.1~) 74 Introduction to Convective H eat Transfer Analysis Lastly, the rate a t w hich momentum 4nters t hrough be, i .e., M bc, must b e obtained. This is done b y noting that since l;Je lies i n t he freestream, any fluid that enters the control volume through i t has a ~elocity U l i n t he x-direction. It follows, therefore, that: ' M bc = M ass flow rate through be X Ul T he mass flow rate through b e i s obtained, i n turn, b y u sing the conservation o f mass principle to give, since the flow is steady: Mass, flow rate Mass flow rate through b e through de a b i s u d y. Therefore, M ass flow rate t hroughab I t was previously noted that the rate at which mass crosses the strip shown on Mass flow rate through a b = p Using this then gives, because d x i s small: Mass flow rate = through de p L e 0 U dy fe ud y + P~ [fe udY]dX dx o Combining these equations then gives: (2.165) Substituting Eqs. (2.162), (2.164), (2.165), and (2.161) into Eq. (2.160) and dividing b y d x t hen gives: - ~~ f - Tw ~ P:x [J.' ud Y] 2 pu, :x [J.' UdY] (2.166) I t is convenient to rearrange this equation by noting that (2.167) Using this result, Eq. (2.166) can b e rewritten as :x [I.'<u, - U)Ud Y] - ~~ [J.' UdY] ~ ~~~f+ ~ (2.168) Because the freestream flow is by assumption inviscid and, as previously discussed, the velocity component normal to the surface, v, i s small compared to that parallel to the surface, u, t he Bernoulli equation applied in the freestream gives as previously noted: dp dx = - pu}- dUl dx (2.169) C HAPTER 2: The Equations o f Convective Heat Transfer 75 Substituting this into Eq. (2.168) and noting that: ! dx't dp p then gives f) = d Ul - Ule d x = - dx dUl Ie uldy 0 (2.170) (2.171) Because U is equal to Ul outside the boundary layer, i.e., because ( u - ud = 0 ,- for y > 8 , t he integrals in the above equation need only be evaluated up to 8 and the equation can, as a result, be written as: :x[I:CU1-U)UdY]+ ~; [J:CU1-U)dY ~ ~ ] (2.172) This is termed the boundary layer momentum integral equation. As previously mentioned, i t is equally applicable to laminar and turbulent flow. In laminar flow, U is the actual steady velocity while i n turbulent flow it is the time averaged value. The way in which the momentum integral equation is applied will b e discussed in detail in the next chapter. Basically, i t involves assuming the form o f the velocity profile, i.e., o f the variation o f U w ith y in the boundary layer. For example, in laminar flow a polynomial variation is often assumed. The unknown coefficients in this assumed form are obtained b y applying the known condition on velocity at the inner and outer edges o f the boundary layer. For example, the velocity must be zero at the wall while at the outer edge o f t he boundary layer i t must become equal to the free stream velocity, Ul. Thus, two conditions that the assumed velocity profile must satisfy are: y= y= 0: 8: U=o U = Ul Other boundary conditions for laminar flow are discussed in the next chapter. I n this way, the velocity profile is expressed i n terms o f U l and 8 . I f t he wall shearing stress Tw is then also related to these quantities, the momentum integral equation (2.173) will allow the variation o f 8 w ith x to b e found for any specified variation o f the free stream velocity, U l. . Consider next the application o f t he conservation o f energy principle to the control volume that was used above in the derivation o f the momentum integral equation. T he height, o f this control volume is taken to be greater than both the velocity and temperature boundary layer thicknesses as shown in Fig. 2.21. As with the velocity boundary layer, the thermal boundary layer is assumed to have a definite thickness, 8T, a nd outside this boundary layer the temperature is assumed to b e constant. Since attention is being restricted to low-speed, two-dimensional constant fluid property flow, dissipation effects o n t he e nergy balance are neglected. T he conservation o f energy principle therefore gives for the flow through the control surface: e, 76 Introduction to Convective Heat Transfer Analysis Layer Layer F IGURE 2.21 " Height" o f control volume considered. ' Rate at which Rate at which enthalpy _ enthalpy leaves enters through dc through a b Rate at which enthalpy enters through bc = Rate at which heat is transferred from the wall into the control volume through a d (2.173) In wrIting this equation, i t has been noted that since b c lies in the freestream where the temperature is constant, there can be no heat transfer into the control volume through it. Longitudinal conduction effects have also been ignored because the boundary layer is assumed to be thin. This is consistent with the neglect of the effects o f longitudinal viscous forces in the derivation o f the momentum integral equation. I f qw is the heat transfer rate per unit area through the wall at the value o f x being considered, i.e., is the local heat transfer rate, then the rate at which heat is transferred into the control volume, which has unit width, through a d is given by (2.174) I f it is again noted that the rate at which mass crosses the strip on ab shown i n Fig. 2.20 will be p udy, it follows that the rate at which enthalpy enters the control volume through this strip is given by p ucpT d y i f c p is assumed to be constant. The rate at which enthalpy enters the control volume through the face ab, which will here be denoted by Hab, is therefore given by Hab = pCp foe u T d y (2.175) The rate at which enthalpy leaves the face d c, termed Hdc is, since d x is small, given by: Hdc Using Eq. (2.175) this gives: (2.177) = d Hab + d x(Hab)dx ( 2.176) CHAPTER 2: The Equations of Convective Heat Transfer 77 In order to determine the rate at which enthalpy crosses b c, i.e., H bc , it is noted that since this face o f the control volume lies in the freestream, any fluid that crosses it will have a temperature o f T l . Hence: H bc = Mass flow rate through b c X C pTl (2.178) Therefore, using the expression for the mass flow rate through b c that was derived when dealing with the momentum integral equation gives: (2.179) Substituting Eqs. (2.174), (2.175), (2.177), and (2.179) into Eq. (2.173) and dividing the result by d x gives: (2.180) Since the freestream temperature rewritten as Tl is independent o f x, this equation can be dx ~ [{t' u (T Jo Tdd Y] = ~ p Cp (2.181) Now outside the thermal boundary layer, i.e., for Y > 8T , T = T l . Eq. (2.181) can, therefore, be written as (2.182) This is termed the boundary layer energy integral equation. The energy integral equation is applied in basically the same way as the momentum integral equation. The form o f the boundary layer temperature profile, i.e., o f the variation o f ( T - T l ) w ith y , is assumed. In the case o f l aminar flow, for example, a polynomial form is again often used. The unknown coefficients in this assumed temperature profile are then determined by applying known boundary conditions on t empeqture at the inner and outer edges o f the boundary layer. For example, the variation o f the wall temperature T w with x may be specified. Therefore, because at the outer edge o f the boundary layer the temperature must become equal to the freestream temperature T 1 , two boundary conditions on the assumed temperature profile in this case are: Y = 0: Y = T: T = Tw T = TI U sing these and other boundary conditions, some o f w hich for laminar flow will b e discussed i n the next chapter, the temperature distribution is expressed as a function o f aT. I f qw is then related to the wall thermal conditions and aT, the energy integral can be solved, using the solution to the momentum integral equation, to give 78 Introduction to Convective Heat Transfer Analysis the variation o f 8T with x. Once this is found, the variation o f the heat transfer rate qw with x can be found. The boundary layer integral equations have been derived above without recourse to the partial differential equations for boundary layer flow. T hey can, however, be detennined directly from these equations. Consider, for example, the laminar momentum equation (2.140). Integrating this equation across the boundary layer to some distance e from the wall, e being greater than the boundary layer thickness, gives because a u/ay is zero outside the boundary layer and because d p/dx is independent o f y : e au I e va u 1 dp au -dy=---e-J-LI u -dy+ o ax 0 ay p dx a y y=o I (2.183) But J-Lau/ayly=o is equal to the wall shearing stress, Also: 'Tw. l and: ea u u -dy o ax = -1 e -u2 y ad 2 0 ax I 1= - d [ I e u2 dy ] 2 dx 0 (2.184) Jo (e v a u d y = (e ~ (vu) d y _ (e u av d y ay ay ay Jo Jo (2.185) = V IUI - Jo (e u av d y ay where VI and UI are the values. o f the velocity components outside the boundary layer. Now the continuity equation gives: av au =ay ax - (2.186) which can be integrated to give VI = - e au I o - x dy a = - -d dx [ Ie u d y ] 0 (2.187) Substituting Eqs. (2.186) and (2.187) into Eq. (2.185) then gives: f.' v~~ dy ~ -u, :x [I: Ud Y] + f.' U ~~ dy (2.188) Now substituting Eqs. (2.184) and (2.188) into Eq. (2.183) and remembering that: - ! d p e = UI dUI e p dx dx CHAPTER 2: The Equations o f Convective Heat Transfer 79 then gives: -d dx [Ie U2y - UI- [Ie udy] = d d] 0 dx 0 dUI Tw U I-C-dx p (2.189) This equation can b e rearranged, as before, to give: :x [L"<Ul - UJUd Y] + : ; [L"<Ul - UJdY] = ~ where i t has again been noted that the upper limit o f t he integration can be taken as ~ 8 since the integrands are zero for y > 8 . T he energy integral equation can be derived i n a s imilar way from the energy equation (2.145). When this equation is integrated across the boundary layer i t gives i f the dissipation term is ignored: i and: o e ua T -dy+ e va T -dy ax 0 i ay =- (- k ) - T a p Cp a y y=o I (2.190) B ut - kaTlayly=o i s t he heat transfer rate a t t he wall qw' Also ie ua T -dy o ax va T -dy o ay = -d' dx [i e 0 uTdy] - Ie T-dy au 0 ax (2.191) ie = ie 0 -a (vT)dy ay Ie T-dy av 0 ay (2.192) = VITI - Jo T a yd y (e av Therefore, substituting Eqs. (2.191) a nd ( 2.192) into Eq. (2.190) and using Eq. (2.187) gives: Because T1 is a constant, i t follows b y u sing the continuity equation that this equation c an b e written as: -d dx [i 8T 0 u(T-Tl)dy] = ~ q p Cp (2.194) The upper limit o f t he integration has b een w ritten as l ir since the integrand is zero for y > 8T • T he quite extensive attention given a bove to t he b oundary layer integral equations must not b e t aken as indicating t hat t hese e quations are, today, widely used. A study of t he derivations does help i n gainiJ;1g a n understanding o f t he meaning o f the equations governing convective heat transfer a nd are, to some extent, the basis 80 Introduction to Convective H eat Transfer Analysis o f some methods o f numerically: solving the general equations governing convective heat transfer. I t is for these reasons that they have been presented i n detail here. 2.11 CONCLUDING REMARKS A flow is completely defined i f t he values o f the velocity vector, the pressure, and the temperature are known at every point i n the flow. The distributIons o f these variables c an b e described by applying the principles o f conservation o f mass, momentum, and " energy, these conservation principles leading to the continuity, t he Na'vier-Stokes, and the energy equations, respectively. I f the fluid properties can be assumed constant, which is very frequently an adequate assumption, the first two o f these equations can b e sImultaneously solved to give the velocity vector and pressure distributions. The energy equation can then be solved to give the temperature distribution. Fourier's law can then b e applied at the surface to g et the heat transfer rates. The flows over a series o f bodies o f the same geometrical shape will be similar, i.e., will differ from each other only in scale, i f the Reynolds and Prandtl numbers are the same in all the flows. From this it follows that the Nusselt number i n forced convection will depend only on the Reynolds and Prandtl numbers. The unsteady governing equations that apply in laminar flow also apply in turbulent flow but essentially cannot b e solved in their full form a t present. In turbulent flow, therefore, the variables are conventionally split into a time averaged value plus a fluctuating component and an attempt is then made to express the governing equations in terms o f the time averaged values alone. However, as a result of the nonlinear terms in the governing equations, the resultant equations contain " extra variables" which depend on the nature o f the turbulence, i.e., there are, effectively, more variables than equations. There is, thus, a closure problem and to bring about closure, extra equations which constitute a "turbulence model" m ust b e introduced. The amount o f effort required to solve the full governing equations can be considerably reduced i f certain assumptions can be introduced that simplify these equations. The most commonly used assumptions are that the flow in a duct is fully developed or that the flow has a boundary layer character. PROBLEMS 2.1. Using the control volume discussed in this chapter, derive the continuity equation for unsteady flow allowing for density variations in the flow. 2.2. Derive the two-dimensional energy equation in cylindrical coordinates using the control volume shown in Fig. P2.1. 2.3. Starting with the two-dimensional energy equation, i.e.: a t ax ay aT + U aT + v aT = (k p Cp )(a22 + a2T) axT ay2 CHAPTER 2: The Equations of Convective Heat Transfer 81 C ontrol Voluffif F IGUREP2.1 and using: U = l i+ u' v = 17 + V' T=T + T' derive the time-averaged energy equation for turbulent flow. 2.4. Write out the continuity, Navier-Stokes, and energy equations in cylindrical coordinates for steady, laminar flow with constant fluid properties. The dissipation term in the energy equation can be ignored. Using this set of equations, investig~te the parameters that determine the conditions under which "similar" velocity and temperature fields will exist when the flow over a series of axisymmetric bodies of the same geometrical shape but w ith different physical sizes is considered. I \ 2.S. Discuss the meaning of the following terms: (i) The turbulence closure problem. (ii) A turbulence model. (iii) A boundary layer. (iv) Similar flow fields. 2.6. Discuss the assumI>~ns used in deriving the boundary layer equations. 2.7. Consider two-dimensional flow over an axisymmetric body. Write the governing equation~ in terms o f a suitably defined stream function and vorticity. 2.8. Two-dimensional flow over a series of geometrically similar bodies having a specified surface temperature was discussed in this chapter. I f the surface heat flux rather than the temperature is specified; a dimensional temperature of the form ( T - T1)/(qwrUk) should be used. Derive the parameters on which the mean surface temperature will depend in this situation. Viscous dissipation effects can be ignored. 2.9. Write the governing equations in Cartesian coordinates given in this chapter in tensor notation, i.e., using Uj = U , U j = V, U l == W , X i = X , X j = y, and X l = (';. Use the Einstein summation convention. 2.10. The turbulence kinetic energy equation for fprced convection 'Vas derived in this chapter. Rederive this turbulence kinetic energy equation by starting with the momentum 82 Introduction to Convective H eat Transfer Analysis. equations that include the buoyancy term f3 g (T - Tr ef) i n the x-momentum equation, T ref b eing a suitable reference temper~ture. 2.11. Consider the derivation, pres.ented i n this chapter, o f the energy integral equation from the governing boundary layer energy equation. Repeat this derivation but include the viscous dis.sipation term in the energy equation. 2.12. In some flows there is a volumetric heat generation due to chemical or other effects. In such situations, the heat generation rate p er unit volume is usually known throughout the flow, its value generally varying with position in the flow. Explain what changes in the steady state energy equation are required to account for such heat gener~tion. ! 2.13. Consider two-dimensional laminar boundary layer flow over a flat isothermal surface. Very close to the surface, the velocity components are very small. I f the pressure changes are assumed to be negligible in the flow being considered, derive an expression for the temperature distribution near the wall. Viscous dissipation effects should be included in the analysis. REFERENCES 1. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, Wiley, New York, 1960. 2. Burmeister, L .c., Convective Heat Transfer, 2 nd ed., Wiley-Interscience, New York, 1993. 3. Eckert, E.R.G. and Drake, R.M., Jr., A nalysis o f H eat a nd M ass Transfer, McGraw-Hill, New York, 1973. 4. Fox, R.W. and McDonald, A.T., Introduction to Fluid Mechanics, 3 rd ed., Wiley, New York,1985. 5. Kakac, S. a nd Yener, Y., Convective Heat Transfer, 2 nd ed., C RC Press, Boca Raton, FL, 1995. 6. Kaviany, M., Principles o f Convective Heat Transfer, Springer-Verlag, New York, 1994. 7. Schlichting, H., B oundary L ayer Theory, 7th ed., McGraw-Hill, New York, 1979. 8. 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