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C HAPTER 4 Internal Laminar Flows 4.1 I NTRODUCTION Attention will be given in this chapter to heat transf~r from the walls of a duct to a fluid flowing through the duct. Attention w ill be restricted to situations in which the flow is laminar. Ducts of various cross-sectional· shape such as those illustrated in Fig. 4.1 w ill be cOJlSidered. Internal flows of the type here being considered occur in heat exchangers,. for example, where the fluid may flow through pipes or between closely spaced plates that effectively form a dUct Although laminar. duct flows do not occur as extensively as turbulent duet flows, they do ·occurin a number of important situations in'which the size of the duct involved is. small or in which the fluid involved has a relatively high viscosity. For·example, fu ail o ilcooler the flow is usually laminar. Conventionally, it is usual te assume that a higher heat transfer rate is achieved with turbulent flow than with laminar flow. However, when the restraints on possible solutions to a particular problem are carefully considered, i t often turns out that a design that involves laminar flow is the most efficient from a heat transfer viewpoint· Near the inlet to a duct or following the disturbance produced by a fitting such as a bepd, there will be a region in which the characteristics of the flow and the heat. transfer rate are changing relatively rapidly with distance along the duct. How~ver, following this "developing flow" region, the flow reaches a "fully .developed"state . i n which the basic characteristics of the flow are not changing with distance. along the duct. This is illustrated in Fig. 4.2. I n this chapter, attenti9n will first be given to heat transfer i n fully developed duct flows. Heat transfer in the developfug region will then b e considered. / 157 1 58 Introduction to Convective Heat Transfer Analysis Pipe Rectangular Duct Plane D uct F IGURE 4.1 Internal laminar flows. Developing Flow Region-Profiles Changing with z Flow Into D uct Fully Developed Flow R egion-Profiles Not Changing with z F IGURE 4.2 Developing duct flow. 4.2 FULLY DEVELOPED LAMINAR PIPE FLOW -Fully developed flow i n a pipe, i.e., a duct with a circular cross-sectional shape, will first be considered [1],[2],[3]. T he analysis is, o f course, carried out using the governing equations written i ll cylindrical coordinates. The z-axis is chosen to lie along the center line o f the pipe and the velocity components are defined i n the same way that they were in Chapter 2, i.e., as shown in Fig. 4.3. F IGURE 4.3 Coordinate system used in analysis o f p ipe . flow. CHAPTER 4: Internal Laminar Flows 159 Assuming the fluid properties c an b e treated as constant [4],[5],[6] a nd that the swirl velocity component w is 0 a nd that the flow is, therefore, symmetrical about the center line, the equations governing t he flow are 1a au - -(vr) + r ar az =0 (4.1) u au+v au az ar = _.!.a p + v 2 u+a2 u+!au) p az 8 z2 8 r2 r ar (8 (4.2) (4.3) (4.4) Attention is being restricted to fully developed flow in the present section which means that the forms o f the velocity and temperature profiles are not changing with distance along the pipe, i.e., that !!:.. = F ucro (!...), (4.5) t he functions F a nd G being independent o f t he distance along the pipe, z. I n t hese expressions, r0 i s the radius o f t he pipe, U c i s t he velocity on the center line, T c i s t he temperature o n t he center line, a nd Tw is t he w all temperature. N ow t he mass flow rate through the p ipe i s given b y m= p . i.e., using Eq. (4.5) i 0 ro u2'1Tr d r (4.6) F / ,m = uc2'1Tp~ Jfo1 . d(!"') ro ( 4.7) B ecause the mass flow rate remaiIis c onstant along the pipe as does, b y a ssumption, t he velocity profile function F, this s hows that: . Uc = constant, I.e., =0 . duc · O· dz = (4.8) . F rom Eq. (4.5) i t then follows that: .au az (4.9) S ubstituting this into Eq. (4.1) t hen g ives on integrating: vr = c onstant (4.10) 160 Introductionto Convective Heat Transfer Analysis B1.Jt the boundary conditions give v = 0 when r = r0 and this equation shows therefore that: v =O (4.11) Using this result in Eq. (4.3) then gives: ap = 0 ar (4.12) and Eq. (4.2) then finally reduces to: v (d 2 U + ! dU)= ! dp d r2 r dr p dz This is conveniently rewritten as: (4.13) !~ (r dU) r dr dr = -.!. d p /L d z (4.14) Integrating this equation subject to the boundary condition d uldr = 0 when r = 0 then gives on rearnmgement: d u = ~dp dr 2/L d z Integrating this equation subject to the boundary condition u then gives on rearrangement: U (4.15) = 0 wh~n r = r0 (4.16) = _ _ dp(~_?) 1 4/L d z 0 This gives t he center line velocity as: Uc = _ _I_dp~ 4/L d z (4.17) 0 DiViding Eq. (4.16) by Eg. (4. 17}then gives: U_ Uc - 1 - (, r ) 2 ro (4.18) This is, of course, the well-known parabolic velocity profile for fully developed laminar pipe flow. Having established the form of the velocity profile, attention must now be turned to the solution o f the energy equation (4.4). Since v waS shown to b e equal to 0 i n fully developed flow, this equation reduces i n this case to u ~~ (;r)[~ :r ~ ~~)+ ~:q = (4.19) '/ ,For the flow o f most fluids under practically significant conditions, the rate at .wJrich heat is conducted down the pipe will be negligible qompared to the rate at CHAPTER 4: Internal Laminar Flows 161 which it is conducted in a radial direction, this being similar to the situation existing in a boundary layer flow. For this reason, the second term on the right-hand side of Eq. (4.19) is ignored compared to the first and the energy equation for fully developed yipe flow is assumed to be: . aT u az = Pr ( v ) 1 a [ a T] a r r ar r (4.20) This is the equation that must be used to give the form of the temperature profile function G given in Eq. (4.5). Now, because G does not depend on z, i t follows that: Le., that: aT = aTw _ [ Tw - T ] [ dTw _ dTe ] az . iJz Tw - :Te dz dz (4.21) To proceed further with the solution, the wall boundary conditions on temperature must be &peeified. Consideration will first be given to the case where the heat flux at the Wall, qw, is uniform and specified. Some discussion of the solution for the case where the wall temperature is kept uniform will be given later. Now using Fourier's law, the heat transfer rate at the wall is given by: qw . a TI = + k- ar r=ro (4.22) The positive sign arises because r is measured from the center toward the wall whereas the heat flux, qw' is U\ken as positive in the inward direction, i.e., in the wall-to-fluid direction. .. ., . . ~. (4.22) can be written using the definition o fthe function G gIven i n &I. (4.5) as: qw =- k(Tw - Te) dG ro d (r/ro) r=:ro I (4.23) Because the case where qw i s uniform is being considered, this equation shows that T w - Te =constant. From this i t follows:that: d Tw dz = d Te dz (4.24) Substituting this result into Eq. (4.21) then shows that when the wall heat flux is uniform: aT d Tw -=-az dz (4.25) Having established this r~u1t from the boundary conditions, attention can now b e returned to the governinKequation (4.20). Substituting Eq. (4.25) into this 162 Introduction to Convective Heat Transfer Analysis equation gives: (4.26) I f the velocity profile as given in Eq. (4.18) is substituted into this equation the following is obtained on rearrangement: This equation can be integrated, subject to the boundary conditions, to give the variation of T with r. The boundary conditions on the solution are: r = 0: a Tlar T =0 = Tw (4.28) The first of these conditions follows, of course, from the reqUirement that the profile be symmetrical about the center line. Integrating Eq. (4.27) once and using the first boundary condition gives: aT a(rlro) = ( pr)u v c ,-1;. d Tw 0 d z · 2 ro [! (~) _ (~)3] ! 4 ro (4.29) Integrating this equation and applying the second boundary condition then gives pr) d Tw [ 3 1 T = Tw - ( V uc~ d z 16 - 4 ro (r )2 + 116 (ro )4] r (4.30) This is, then, the temperature distribution for fully developed laminar pipe flow when the heat flux at the wall is .uniform. I t can be written in tenns of the specified uniform wall heat flux, qw, by noting that when Eq. (4.30) is used to give the value of aTlar/r=ro in Eq. (4.22), the following is obtained: qwro k = (uc~pr)dTw 4 dz (4.31) Substituting this result back into Eq. (4.30) then allows the temperature distri·bution to be written as: (4.32) From this equation it follows that the center line temperature, Tc , is given by: Tw - T. - 3 (qW r0 c ---- ) 4k (4.33) Therefore, the temperature distribution can be written i n the fonn initially introduced for fully developed flow as follows: CHAPrER 4: Internal Laminar Flows 163 1_ ~ (~)2 + ~ (~)4 3 ro 3 ro (4.34) \ The fonns of the velocity and temperature distributions for fully developed flow through a pipe with a unifonn heat flux at the wall as given by Eqs. (4.18) and (4.34) are shown in Fig. 4.4. Now, in fully developed flow it is usually convenient to utilize t he mean fluid temperature, Tm, rather than the center line temperature in defining the Nusselt number. This mean or bulk temperature is given, as explained in Chapter 1, by: . 1 T m = M ass Flow Rate fA p uT d A . (4.35) w here A is the cross-sectional a rea o f the duct. Now: Mass Flow Rate = fA pu d A Hence, for flow i n a pipe where d A is set equal to 27Trdr: Mass Flow Rate = puc~ r(: )21T (:. )d ( :. ) puc~ (4.36) U sing the velocity distribution given i n Eq. (4.18) and integrating then gives: Mass Flow Rate = ; (4.37) This shows, incidentally, that the mean velocity i n f ully developedJaminar pipe flow is half of the center line velocity, a well-known result. CenterLine Wall 1.0 r--'="'""---~--,-------..., 0.5 0 .0 ' --_ _ _ _ _- -I.._ _ _ _ _ _ FIGURE..4.4 ~ 0.0 0.5 rlro 1.0 Velocity and temperature profiles in fully developed flow with a uniform wall heat flux. 164 Introduction to Convective Heat Transfer Analysis Similarly, since: L p uT d A = p u,r. Jol (~ )[Tw - (Tw - T)]2". ( :.) d ( :.) (4.38) it follows using the velocity and temperature profiles given i n Eqs. (4.18) and (4.32) respectively that: ' fA p uTdA = 7T 2 11 2 p uc ro [ Tw - 24 - k- ~ (qwro)ll (4.39) Substituting Eqs. (4.37) and (4.39) into Eq. (4.35) then gives the mean temperature as: T m =T w _!!. (qwR) 24 k = IT ~. 4.~63 . 48. . (4.40) This can b e r earranged to give: N UD .2- (Tw _ Tm)k qwD (4.41) w bere N UD i s the Nusselt nUIilber based o n the difference between the w all temperature and the m ean ·temperature, i.e., ( Tw - T m) and on t he p ipe diameter, D. T his N usselt I)umber is .seen, the~fore, t o b e <?onstant i n fully l aminar flow through a circular pipe with a constant heat transfer rate at the wall. . E XAMPLE 4 .1. I n s ome situations i t is· possible to fiild the heat transfer rate with·ade- quate accuracy b y assuming that the velocity is constant across the duct, i.e., to assume that so-called " slug flow"exists. Find the temperature" distribunonand the Nusselfnumber i n slug flow i n a pipe. when the thennal field is fully developed and when ·there is a unifonn wall heat flux. Solution. T he temperature distribution will still be described b y Eq. (4.26). I n the flow here being considered, U = Um everywh~re so this equation gives: Um d Tw = dz (~)!~ [r a T] P r r ar ar aTtar = 0 T = Tw d Tw ~ dz 2 ( i) T he boundary conditions on the solution are as above: W henr = 0: W henr = ro: Integrating E q.. ( i) o nce and using the first boundary ,condition gives: aT = ar (PT)u · 11 m ( ii) Integrating this equation and applying the s~ond boundary condition then gives: ( iii) CHAPTER 4: Internal Laminar Flows 165 This can be written in t enns o f the specified uniform wall heat flux, qw, by again noting that because qw1k = JTIJr/r=ro ' Eq. ( ii) gives: T (qWro) = (um~v pr) ddzw k 2 Substituting this result into Eq. ( iii) then gives: From this equation it follows that the center line temperature, T e , is given by: T -T = w e r (qWk0) , ' 2 Therefore, the temperature distribution can be written as: The mean or bulk temperature is given: T= m (M ass ~ow R ')f puT dA ate A Le., since, in the case here being considered, u is qonstant a nd equal to Um: Tm = !LTdA Le.: T m:= i.e.: . io.1'T (-rr) d (-ro.)' .2 o r · Integrating then gives: T m = T w ._ (qWro) 4k . qwD ( Tw _ Tm ,k ) This can b e rearranged to give: N UD = =8 where N UD is the Nusselt n umber b ased on the difference between the wall temperature and the mean temperature, i.e., ( Tw - Tm) a nd on the pipe diameter, D . This is higher than the mean Nusselt number that exists with a parabolic velocity profile because o f t he higher velocity near the wall w ith slug flow. ' 166 Introduction to Convective Heat Transfer Analysis . Next, consider fully developed laminar flow through a pipe whose wall temperature is kept constant. For this boundary condition, since d Twldz = 0, Eq. (4.21) gives: a T = (Tw-T)dTc az Tw - Tc d z (4.42) Substituting this result into Eq. (4.20) which has to b e solved to give the temperature distribution and using the velocity distribution given in Eq. (4.18) which is, o f course, independent o f t he temperature distribution and, hence, o f the boundary conditions on temperature, then gives: r wT 1 [1 (r )2]( Tw -- Tc ).ddTzc (Pv) raar (r aaT ) oT r r - UI = (4.43) Since Tw is constant, this equation can be written in terms o f the temperature function, G, defined in Eq. (4.34) as: Prr'loul [ v(Tw - Tl) d Tl]G[l_ ro dz (~)2](!.-) = R _ d d(rlro) [(~) d(rlro) 1 ( 444) dG ro . This equation must be solved to give the variation o f G w ith rlro subject to the following boundary conditions: - = 0: ro r .. (4.45) - = 0: G = 1, ro r - = 1: G ro r =0 No simple closed form solution can b e obtained. However, the variation o f G w ith ( rlr0) can be quite easily obtained to any required degree o f accuracy by using an iterative procedure. This starts with the temperature profile for the constant heat flux case. This profile is used to give a first approximation for the left-hand side o f Eq. (4.44) and this equation c an t hen b e integrated to give a second approximation for G and so on until acceptable convergence is obtained. I n this procedure, Eq. (4.44) is integrated once to give d Gld(rlro) using the boundary condition ( :.) = 0: dG = 0 d (rlro) T his result is then integrated to give the variation o f G with ( rlr0)' T he remaining boundary conditions can b e u sed to determine the constabt integration and to eliminate the unknown quantity: . Prr'louc d Tc v(Tw - Tc) d z . ittEq. (4.44). O nce the temperature profile function q i s obtained, t he Nusselt number can b e determined because, using the results derived i n dealing with the constant wall CHAPTER 4: IntemaI Laminar Flows 167 heat flux case, i t can b e easily shown that: (4.46) The Nusselt number is therefore given using Eq. (4.23) by: N UD = - d (rlro) dG I / f1 [ 4 ( r=ro) 0 G I- (-rr)2] (-ro) d (-ro ) rr o (4.47) The above procedure can b e undertaken using a series-type solution or i t c an b e carried out numerically. A s imple computer program PIPETEM that implements t he procedure is available as discussed i n t he Preface. Alternatively, the procedure c an b e carried' o ut using·a spreadsheet. I f the solution procedure is carried through as outlined above, the following i s obtained for fully deveioped laminar flow through a pipe with constant wall temper. ature: N UD = 3.658 (4.48) This is some 16% lower than t he Nusselt n umber f or flow with a constant wall h eat transfer rate. E XAMPLE 4 .2. A device is cooled by clamping an aIuminum block to i t and passing . cooling air through smaIl diameter holes in this block. The length of the block in the flow direction is fixed by practicaI consideration and the block can be assumed to b e at a uniform temperature of Tw ' The air enters the holes in the block a ta known temperature o f To and the totaI mass flow rate of air through the holes i n the block, AI, is fixed. Investigate the effect o f the number of holes on the heat transfer rate from the block to the air passing through it. Assume that the flow in the cooling holes' is laminar and fully developed. Solution!.. The flow situation being considered is shown in Fig. E4.2. I f n is the number o f holes and D is the diameter o f the holes .then the mass flow rate through each channel is given by: m =n . AI ( i) I f Tm is the mean bulk temperature at any point in a flow channel then the rate o f change o f Tm with respect to distance z aIong ~e:~uct is ~ven b y applying an energy H oles=n FIGURE E4.2 . 168 Introduction to Convective Heat Transfer Analysis balance by: mcp d J: = h7rD(Tw - Tm) ( ii) cp being the specific heat of the air and h the heat transfer coefficient. Because the flow is being assumed to be laminar and fully developed with a uniform wall temperature, Eq. (4.48) gives: N UD = T= hD 3.658 i.e.: k h = 3.658 D Substituting this into Eq. ( ii) and using Eq. ( i) gives on rearrangement: = 3. 658k1Tn Mc p = . which can be written because Tw is constant as: (T. ~ T ) - d (Tw - Tm) w mZ 1 d - 11.81-.M~ kn At the inlet t o the block T m the mean temperature at exit: = To, so Integrating the above equation gives, i f Tme is ( iii) Tw - Tme = e - U .81(knlMCp)L. T w'- To where L is the length of the block. I f the following is defined: L* = ~nL M cp then Eq. (iiz) c an be written as: Tw - Tme = e ... ll.81Lo T w-To (iii) . Now the heat transfer rate to the air as it flows through the block is given by: Q = MCp(Tme - To) = MCp[(Tw - To) - (Tw - Tme)] i.e., using Eq. ( iil): ( iv) . , I f the block was infinitely long the air would b e heated to .the wall temperature, i.e., Tme would equal t w ' Therefore, the maximum possible heat transfer rate is:. Qmax = MCp(Tw - To) Hence: / ~ Qmax = 1 ~ e - ll.811: . CHAPTER 4: Internal Laminar Flows 169 This equation shows that as L* tends to infinity, QIQrnax tends to 1, and that the larger L* the closer Q is to Qmax. Consider the definition of C , i.e.: L* = n Mc p This shows that L * is proportional to the number o f holes, n. Therefore, a design that involves a large number o f holes will produce the highest heat transfer rate. Consider a design that will give: Q = 0.95Qmax I t will be seen from Eq. (iv) t hat this requires: ~L 1 - e- ll .81 L* = 0.95 i.e.: L* = 0.2537 Hence, using the definition o f L*: kL n -.Mc p i.e.: = 0.2537 n - 0.2537 k L _ Mc p 4.3 F ULLY D EVELOPED L AMINAR F LOW I N A P LANE D UCT F ully d eveloped flow in a v ery w ide d uct i s c onsidered i n this section, the flow situation considered being s hown i n Fig. 4.5. B ecause t he d uct i s w ide, changes i n t he flow properties i n t he x-direction a re n egligible, i.e., flow between t wo l arge p arallel p lates i s e ffectively b eing c onsidered. T he a nalysis is, o f course, similar to t hat a dopted i n d ealing with p ipe flow. Flow Essentially Independent of X ; i.e., Flow is two-dimensional F IGURE 4.5 Flow i n a plane duct. 170 Introduction to Convective Heat Transfer Analysis The equations governing the flow are, since there is no velocity component in the x-direction: (4.49) (4.50) (4.51) (4.52) The flow is assumed to b e symmetrical about the center plane so the boundary conditions on the solution are: Y .Y =o: - = 0' v =O, -·= 0 aY aY ·. = w: u = 0, v = 0, T = Tw au . aT . . (4.53) Here, W = W12 i s the h alf width o f the duct. Because attention is being restricted to fully clevelopeq flow i n this section, the forms o f the velocity and temperature profiles do not change with distance along the duct, i.e.: ~ = F(Y)' U W C Tw - T = G (Y) T w-Tc w· . ' . (4.54) the functions F a nd G·being independent o f the.distance along the.duct, z. I n E q. (4.54), uc' is the' veloCity. o n t he center line, Tc i s the temperature'on the center fuie,. . and Tw is~ a s before, t he wall temperature. Now 'the m ass flow rate through the.duct p er u nit Width o f the duct is gj.ven b y: riJ i.e., using Eq. (4.54) = 2p JowUdY (4.55) m=Ucwf>d(~) = constant, I.e., dz = (4.56). B ecause t he m ass flow rate remains constant along the duct as does, b y assumption, the profile function F , Eq. (4.56) shows that: Uc . duc 0 (4.57) From Eq. (4.54) i t then follows that: au az =0 (4.58) CHAPrBR 4: Internal Laminar Flows 171 Substituting this into Eq. (4.49) then gives after integrating: v = constant But the boundary conditions give v shows that: (4.59) = 0 when y = 0 so this equation therefore (4.60) v =O Using this result in Eq. (4.51) then gives: Op = 0 Oy (4.61) a nd Eq. (4.50) then finally reduces to: P-=-- d 2u dy2 1 dp P dz (4.62) ~ Integrating this equation using the boundary condition d uld y g ives o n rearr~gement: du . 1 dp P- = --y dy p dz Integrating this equation using the b oundaiy condition u gives: PU . 0 w hen y =0 (4.63) = 0 whe~ y. = w then (4.64) = _!:. d p [w2 _y2] p dz . 2 2 T his gives: P Uc = ---- 1 d pw2 p.dz 2 (4.65) Dividing the above two equations then gives: ~ = 1-'(YJ2 U W C (4.66) Therefore, as was the c ase w ith fully developed pipe flow, the velocity profile i n f ully developed p lane dUct flow i s parabolic. . '. H aving established t he f onn o f t he velocity profile, attention c an n ow b e t urned to t he solution o f the energy equation, i.e., Eq~ (4.52). Because v was shown to b e e qual to zero i n fully developed flow, this equation reduces i n this case to: .. u~ ~ = (;r)(~:; + ~~) (4.67) As w as the case with p ipe flow, for the flow o f most fluids under practically . ,significant conditions: 1 72 Introduction to Convective Heat Transfer Analysis i.e., the first term on the right-hand side o f Eq. (4.67) can be ignored compared to t he second and the energy equation for fully developed duct flow c an be assumed to be: aT (V) a2 T u az = P r oy2 (4.68) This is the equation that must b e s olved to give the form o f the temperature profile function G given in Eq. (4.54). Now, because G does not depend on z, i t follows that, j ust as with fully developed p ipe flow: ! .-[Tw-T]=O az Tw - Te i.e., that: aT = aTw az az _[ Tw - T ] [dTw Tw - Te dz _ dTe] d z· (4.69) To proceed further with the solution, the wail boundary condition on temperature must, as with pipe flow, b e specified. Consideration will again first be given to the case where the heat transfer rate at the wall is, uniform. Now the heat transfer rate a t t he w all is given using Fourier's L aw by: qw = . +k aTI ay (4.70) y =w T he positive sign arises because y i s m easured from the center line o f t he duct towards the wall. Combiliing this equation with Eq. (4.54) then gives: qw = - k(Tw - Te) ddGI . . ( 431) y y =w B ecause qw i s a constant, this equation shows that Tw - Te = constant, i.e., that: (4.72) dTw dTe dz = dz Substituting this result into Eq. (4.69) t hen shows that when the wall h eat flux i s constant: aT az =:= dTw dz ( 4.73)' . Having established this result from t he boundary condition, attention c an n ow b e r eturned to the governing equation, i.e.·, E q. (4.68). Substituting E q. (4.73) into this e quation gives: U dTw dz = (~)a2T P r oy2 (4.74) CHAPTER 4: .Internal Laminar Flows 173 I f the velocity profile given in Eq. (4.66) is substituted into this equation the following is obtained on rearrangement: ~:; = (7)U[ d~w a rMy = 0 to give: [1- (~ J] (4.75) This equation can be integrated, subject to the boundary condition y = 0, (4.76) gIves: . Integrating this in turn subject to the boundary condition y = w. T , T = Tw - = Tw then [(7)u ~;][ I; -H~J + I~ (~n c (4.77) This equation describes the temperature distribution in fully developed laminar plane duct flow when the wall heat flux. is a constant. I t can be written in terms of the specified wall heat flux, by noting that when Eq. (4.77) is used to give the value of aTlayly=w in Eq. (4.71), the following is obtained: qw, (q~w) = (Zu;:ry:; (4.78) Substituting this result back into Eq. (4.77) then allows the temperature distribution to be written as: (4.79) From this equation i t follows that the center line temperature, T e , is given by: T -T w e = ~(qww) 8k (4.80) The temperature distribution can therefore be written in the form initially introduced as follows: _ Tw - T _ 6 1 G-1---+-. Tw - Te 5w 5w (y)2 (y)4 . (4.81) Now, in duct flows, as previously discussed, i t is usually convenient to utilize the tp.ean fluid temperature, Tm , i n defining the Nusselt number. This mean or bulk temperature is given as explained in Chapter 1 by: Tm = (Mass FIl.w Rate )f puT dA o A (4.82) 174 Introduction to Convective Heat Transfer Analysis where A is the cross-sectional area o f the duct. Now: Mass Flow Rate = L pu d A = 2wpu, fJ~ ) (n d (4.83) Hence, using the velocity distribution given in Eq. (4.66) and integrating gives: Mass Flow Rate = 4 wpuc 3 (4.84) Because the area o f the duct per unit width is 2w and because, by definition, the 'mass flow rate is equal to the (density x area X mean velocity), this shows that , the mean velocity in fully,developed laminar plane duct flow is 6/4 ( = 1.5) times the center line velocity. Similarly because: L p uT d A = 2 pu,w fo' (~ )[Tw - (Tw - T)jd ( ;) (4.85) i t follows using the velocity and temperature profiles given in Eqs. (4.66) and (4.77) respectively and integrating that: . L p uT d A = 2Pu,W[~Tw - Hq~w)l (4.86) Substituting Eqs. ( 4.84) and ( 4.86) into Bq. (4.82) then gives the mean temperature as: T -T :-17(QwW) 35 k mw --- (4.87) This can be rearranged to give: qwW N uw = (Tw - Tm)k = 17 70 = 4 .118 (4.88) where Nuw is the Nusselt humber based on the difference between the wall temperature and the mean temperature, i.e., on ( Tw .- Tm) and on the duct width, W. This Nusselt number is seen, therefore, to be constant for fully developed laminar flow through a plane duct with a con~tant heat transfer rate at the wall. E XAMPLE 4 .3. Consider fully developed flow in a plane duct i n which a uniform heat flux, qw, is applied at one wall and where the other wall is unheated and heavily insulated. Derive expressions for the temperature distribution in the duct and the Nusselt number. Solution. The flow situation being considered is shown i n Fig. E4.3. TheheatfluxattheunheatedwallwillbeO,i.e.,aty = + w,q = O.Here,w = W/2 is the duct half-width and y is measured from the'center line of the duct . Uniform Heat F lux-qw - -- U HHII tttttttt F IGUREE4.3 CHAPTER 4: Internal Laminar Flows 175 T he full duct has to b e considered because the temperature distribution will not be symmetrical about the center line. T he velocity distribution is not dependent on the temperature boundary conditions and is given by Eq. (4.66) as: ~ Uc Uc = 1_ (1'.)2 W being the center line velocity which is equal to 1.5 times the mean velocity, Urn. I t will be assumed that in the fully developed flow the temperature distribution has the form: T wl-T = G(Y) Twi - Tw2 W where w I and Tw2 are the temperatures o f the heated and unheated walls respectively which a re at Y = - wand at Y = +w, respectively. From this equation it follows that: r a [ Twi - T ] _ 0 a z Twi - Tw2 i.e., that: aT az = a Twl _ [ Twi - T ] [dTwl _ dTw2] a z·· Twl - Tw2 dz dz N ow the heat transfer rate at t he w all is given using Fourier's l aw by: qwl = - k aaT I· Y y =-w i.e., using the assumed form o f temperature distribution: qw = k(Twl - T w 2) - d Y dGI y =-w B ecause qw is a constant, this equation shows that Tw I - Tw2 = constant. Consideration o f t he assumed form o f the temperature distribution then shows that T wI - T is a function o f y alone. From these results i t follows that: dT d Twl d Tw2 d z = { fZ = { fZ T he equation governing the temperature distribution is: u aT az = ( p)a 2T P r ay2 w hich c an therefore b e w ritten as: u d !wl = dz [~J a2T P r ay2 I f t he velocity profile is substituted into this equation the following is obtained o n r earrangement: 2 aT = (pr)uc dTwl ay2 p dz [1- (y)2] w 176 Introduction to Convective Heat Transfer Analysis Integrating this equation once gives: where Cl is a constant of integration. Now, the solution must satisfy the boundary condition y = w, aTlay = 0 which gives: CI = - ~ (~r)UI dTwI 3 v dz hence: aT ay = (pr)ucdTwl [(y)_ ! (y)3 _~] v dz W 3 W 3 Integrating this in turn, subject to the boundary condition y = - w, T = Twl, then gives: Twl - T = [(pr)ucdTwl wv dz Because at y = +w, T ][132 _!2 (y)2 + .1!. (y)4 + ~ (1.)~ 1 w 2 3 W w~ = Tw2 the above equation gives: dTw] 4 Tw w2 = [(-pr) Uc-- l-T wv dz 3 Dividing the above two results then gives: ~71_-£ = [:~- H~)\ 1~ (~J + H~)l This equation gives the temperature variation i n terms of the wall temperaiures. Now qw = -kaTtayly=-w so using the equation for aTtay derived above gives: (qkW) = _~(~)ucd~;l . Using this then allows the temperature distribution to be written as: Twl - T = ~ (qwW)[13 _ ! (1.)2 + .!. (y)4 + ~ (l.)~ 4 k 12 2 W 12 W 3. W ~ This. mean or bulk temperature is given by: Tm = But as shown above: (Mass FIlow Rate )J puT.dA A Mass Flow Rate = . 3 4 WPUl Further: .I puTdA .A = pucwJ+l(~)[TWl:- (Twl - T)]d(Y) - I Uc W C HAPTER 4: Internal Laminar Flows 177 so it follows that: Hence: Tm = Twl - 1.346(q~W) This can be rearranged to give: qw W N w = (Tw _ T m)k == 2.692 where N w is the Nusselt number based on the difference between the wall temperature and the mean temperature, i.e., on (Tw - Tm) and on the duct width, W . This Nusselt number is seen, therefore, also to be constant W hen dealing with noncircular ducts i t i s common to assume that the equations . for the heat transfer rate for a circular pipe can b e applied to the noncircular duct provided that the correct " equivalent d iameter" is used for the noncircular duct. Now . for a circular pipe: .. Area 'lTD2/4 Perimeter 'lTD D 4 (4.89) I t i s therefore usual to t ake a s t he " equivalent diameter" the following: D H =4 . A rea Wetted Perimeter (4.90) this being termed the "hydraulic d iameter", the wetted perimeter being the perimeter i n contact with the fluid. F or a p lane duct (see Fig. 4.6), the hydraulic diameter is given·bY: . D . = 4 W X L. H 2 XL = 2W (4.91) where L i s the arbitrary width o f t he d uct considered as s hown i n F ig. 4.6. L .Wetted Perinieter 2L Cross-Sect,ional Area LW = I: = F IGURE 4.6 Hydraulic diameter of a plane duct. 178 Introduction to Convective Heat Transfer Analysis Now for a circular duct, i.e., a pipe with a uniform heat flux at the wall, the analysis discussed in the previous section gave NUD = 4.364. Therefore, for fully developed flow in a plane duct with a uniform heat flux at the wall, this would indicate using the hydraulic diameter concept, that: Nuw = NUD DH = - 2- = 2.182 W 4.364 (4.92) This is very different from the actual value of 4.118 derived above showing that the use of the hydraulic diameter concept can sometimes give very erroneous results for the heat transfer rate. Next consider fully developed laminar flow through a plane duct whose wall temperature is kept constant. As with pipe flow, for tJrls boundary condition, Eq. (4.69) gives: aT ( Tw - T ) dTc a z = Tw - Tc d z (4.93) Substituting this result into Eq. (4.68), which must be solved to give the temperature distribution, and using the velocity distribution given in Eq. (4.66), which is, of course, independent of the temperature distribution, hence, of the boundary conditions on temperature, then gives: UC[I- (YY](Tw -- Tc ) dTc = (P r ay2 T ~)a2T w) Tw dz P .2)][ 42 G [(v(Trw Tc)· ddTz _G 1 - (y)2] = _ d(ylw)2 Uc (4.94) Because Tw is here a: constant, this equation can be written in terms of the temperature profile function, G, defined in Eq. (4.54) as: c w- W (4.95) This equation must be solved to give the variation of G with y lw subject to the following boundary conditions: (~) = 0: dG d(ylw) =0 G =O (;)= 0: G = 1. (~)= 1: (4.96) .No simple closed fonn solution to E q. (4.95) subject to these boundary conditions caD be obtained. However, just as with pipe flo~, the variation of G with (ylw) . can be qUite easily obtained to any required degree.' of accuracy by using an iterative proCedure. This procedure starts with· the temperature profile for the utllform heat flUX case. This profile is used to give a first approximati~n for the left-hand . side ofEq. (4.95) and this equation can then be integrated to give a·second approximation f()r G and so on until acceptable convergence is·obtained. I n this procedure, Eq. (4.95) is integrated once to give d.Gld(ylw) using the boundary condition (ylw)· =? 0·: d Gld(ylw) = o. The resulting equation is then integrated to give the variation o f G with (ylw) using the remaining boundary conditions to detennine the CHAPI'BR 4: Internal Laminar Flows 179 constant of integration and to eliminate the unknown quantity: in Eq. (4.95). Once the temperature profile function G is obtained, the Nusselt number can b e detennined since, using the results derived i n dealing with the constant wall heat flux case, it can b e easily shown that: (Tw - Tm) = ~(Tw 4 Te) r I G(~ H~) The Nusselt number is, therefore, given by: dG 3 d (ylw) Nuw = f>[' ~(~}]d(~) (y/w) = 1 , (4.97) The above procedure can b e undertaken using a series type solution or i t can b e carried out numerically. A simple computer program that implements the procedure is available as discussed i n t he Preface. Alternatively, the procedure can be carried out using a spreadsheet. I f the solution procedure is carried through as outlined above, the following. is obtained for fully developed laminar flow through a plane duct with unifonn wall temperature: . N UD = .,~ 3.771 (4.98) This is some 8% lower than the Nusselt number for flow with a unifonn wall heat transfer rate. 4.4 FULLY DEVELOPED LAMINAR F LOW I N DUCTS· W ITH O THER C ROSS·SECTIONAL SHAPES ~ fully developed flow in a p ipe and i n a p lane duct, as discussed above, the v elocity and, temperature profiles could b e expressed i n t erms o f a single cross stream coordinate, i.e., i n terms o f e ither rlR or y /W. I n m any other situations, however, the cross-sectional shape o f t he duct is s uch that the profiles w ill depend o n two cross stream coordinates, e.g., consider fully developed flows i n the ducts wi1h t he ~ss..sectional shapes shown i n Fig. 4.7. 1'0 illustrate h ow s uch situations can be analyzed, consider fully developed flow i n ~ duct with a rectangular cross-sectional shape, Le., w ith a duct that has the shape shown i n Fig. 4.8 [2],[7],[8],[9]. , Because the flow is fully developed, the velocity components i n t he x and y , coordinate directions will b e zero and the pressure will d epepd only o n z. I f t he / 180 Introduction to Convective Heat Transfer Analysis -f------- j----- ----j-I Rectangular T riangular F IGURE 4.7 Elliptical Circular Sector Ducts in which the fully developed flow is two-dimensional. ! ~ - ------.-----~- ! ,. B ., F IGURE 4.8 Duct with rectangular cross-sectional shape. s ame assumptions as used i n dealing with pipe flow and with plan~ duct flow are . adopted, i.e., i f the diffusion o f h eat in the z-direction is neglected compared to the rates i nx a nd y directions, the governing equations are: o2u 2u) (ox2 + oy2 = ! d p o M dz U (4.99) (4.100) oT oz = ( v )(o2T o2T) P r ox2 + oy2 Because the floW is fully developed: ~ U C = F(~' W w L) ' (4.101) the functions F and G being independent o f the distance along the duct, z. I n these '~expressions U c is the velocity on the center line, Tc i Nhe temperature on the center line, T w is, a s before, the wall temperature, and W is the representative duct cross- CHAPTER 4: Internal Laminar Flows 181 sectional size. Because the flow is fully developed, the center line velocity, uc , ·is not changing. Attention will first be given to the velocity profile. In tenns o f the profile func;tion, F, the momentum equation, i.e., Eq. (4.99), becomes: P F a22 U/L dz a2 (ax2 + ayF) = [W2 d ] c where: ( 4.102) X =W' x y= L W (4.103) Because t he solution will b e symmetrical about the verticui and horizontal center lines, the solution in only a quarter of the duct need be considered, this being shown in Fig. 4.9. The boundary conditions on the solution are then: X = 0: ax = 0, = 0.5A: aF Y = 0: y . aF ay =0 =0 ( 4.104) x F = 0, = 0.5: F Here, A = W IB i s the aspect ratio o f the duct cross-section. I n addition, because o f t he way in which F is defined, the following applies: x = 0, Y =O: F =1 ( 4.105) T here are a number o f analytical and numerical approaches that can b e u sed to solve Eq. ( 4.102) subject to these boundary conditions. A very simple. numerical approach will b e discussed here. Because i t i s a l inear equation in F , the solution is linearly dependent on the value o f the right-hand side o f the equation, i.e., i f Fl is a solution with a particular value o f t he right-hand side, then i f t he value o f t he right-hand side i s C ti,mes as large, the solution will b e F2 = CFl_ Hence, the solution to Eq. ( 4.102) c an b e obtained basically b y setting the right-hand side equal to an arbitrary value, here taken to b e - 1 ( a negative value being u sed because t he pressure gradient is negative so the right-hand side o fEq. ( 4.102) will b e negative) and obtaining t he solution to the equation subject to the boundary conditions given i n Solution Rectangular Duct F IGURE 4.9 Solution domain being considered. 182 Introduction to Convective Heat Transfer Analysis Eq. (4.104), i.e., by solving the following equation subject to the boundary conditions given in Eq. (4.104): a2 F a2F)_ (ax2 + ay2 - -1 (4.106) This solution will be denoted by F l. This solution will not, in general, satisfy the boundary condition given in Eq. (4.105), giving instead a value of F on the center line, say Fe, that is not equal to 1. The correct solution is then given by FllFe and the correct value of the right-hand side that gives a solution that satisfies Eq. (4.105) is given by: - [ - - =U1/J, W2 d P] 1 dz Fe (4.107) which can be written as: W dP ] liFe - [ put d z = Ree (4.108) w hereRee = pueWI/L. The solution to Eq. (4.106) will here be obtained using a finite difference method. The grid system shown in Fig. 4.10 is introduced for this purpose. Using the values of F a t the five nodal points shown in Fig. 4.10, the following finite-difference form o fEq. (4.106) is pbtained using the finite-difference approximations introdq~ed in the previous chapter in the discussion of the numerical solution o f external flows: ( Fi+l,j + F i-l,j IlX.2 2Fi,j)+ (Fi,j+l + F i,j-l . - 2Fi,j) = - 1 a y2 . (4.109) . ~s equation applies to the internal points, i.e., for i = 2 t~ N X - 1 and for j = ,2 to N Y - 1. On the boundary points, the following are .given by the boundary tJ.X j +l tJ. X - -:-- j-l i -I ~ Ii+ 1 ~ tJ. y tJ. y FIG~4.10 i X' Nodal points used i n obtaining fiilite. difference solution. CHAP1ER 4: Internal Laminar Flows 183 conditions: For i = ' 1 to N X: F or j = 1 to N Y: For i For j Fi,1 = F i,2 =0 =0 (4.110) F1,j = F2,j F i,NY F NX,j = 1 to N X: = 1 to N Y: Eq. (4.109) subjectto the b oundary conditions given i nEq. (4.110) can be solved iteratively. The values o f Fi,j a re first approximated. For example, using the plane duct result, the initial values could b e t aken as: Fi.j = (1 - X2)(1 - y2) Eq. (4.109) can be then be applied sequentially to give updated values a t t he internal points and the boundary conditions c an b e u sed to update the boundary point values. I n obtaining the solution, Eq. (4.109) c an b e written as: po . = (Fi+1,j + F i-l,j) (Fi,j+l + Fi,j-1 I ,} I1X2 + l 1y2 + 1)/(~ + ~) I1X2 l1y2 ( 4111) . T his is applied sequentially o ver a nd o ver to all points until convergence is obtained to any specified degree o f accuracy. This gives the values o f F 1. A s d iscussed above~ t he actual solution is t hen g iven b y: F I.,}. = -.!!.L F 1,1 F· . Consideration will next b e g iven to t he solution for the temperature function, G. A s w ith fully developed p ipe a nd p lane d uct flows, the solution depends o n t he n ature o f the thermal boundary conditions a t t he wall. I n t he case o f flow i n a r ectangular d uct there are a variety o f p ossible boundary conditions, some. o f t hese b eing s hown in Fig. 4.11. Here, attention will be restricted to the case where the w all Adiabatic Adiabatic Tw Tw T. I Tw. ITW] Tw Tw Adiabatic I I qw qw I qw Twl I q. qw I I Tw qw I qw Adiabatic Tw'l Twl IT w ' qw F IGURE 4.11 S ome possible wall boundary conditions in· a duct with rectangular cross-sectional shape. 184 Introduction to Convective Heat Transfer Analysis s F IGURE 4.12 Definitions o f n and s i n Eq. (4.112). temperature at any section o f the duct is unifonn but in which the total heat flux from the wall around the section o f the duct is not changing with z. Now, with this boundary condition: JPerimeter an ( k aT ds = qw X length o f perimeter = constant (4.112) where n is t he coordinate nonnal to the surface considered and s is the distance along the surface considered as shown i n Fig. 4.12. I n tenns o f the temperature function G, and defining: N =W' n S =W s (4.113) and recalling that k is being assumed constant, Eq. (4.112) becomes: (Tw - Tl) Jp ~~dS = constant (4.114) P being the actual perimeter divided b y W. From Eq. (4.114) it follows that: (Tw - Tl) = constant (4.115) Hence, using the definition o f G, i t follows that: a T = d Tw = d Tc az dz dz (4.116) From this, i t follows from Eq. (4.102), that the equation governing the temperature . function is: (4.117) T he boundary conditions on the solution are then: X = 0: X = 0.5A: aG ax = 0, Y = O.' a G --..: 0 ay G= 0 (4.118) G = 0, Y = 0.5: Here, again, A = WI B is t he aspect ratio o f t he duct cross-section. I n addition, because o f t he way i n w hich G is defined, the following applies: X = 0, Y = 0: G =1 (4.119) C HAPfER 4: Internal Laminar Flows 185 The solution will, here, b e obtained using the same approach as used in obtaining the solution for the velocity profile function, F. Because, when solving Eq. (4.117), F is a known quantity, Eq. (4.117) is a linear equation in G. Hence, the solution is linearly dependent on the right-hand side o f t he equation, i.e., i f G 1 is a solution with a particular value o f the right-hand side, t hen i f t he value of the right-hand side is C times as large, the solution will b e G2 = C Gl. Therefore, the solution to Eq. (4.117) can b e obtained basically b y setting the right-hand side equal to a n a rbitrary value, here taken to b e F and obtaining the solution to the equation subject to the boundary conditions given in Eq. (4.118), i.e., b y solving the following equation subject to the boundary conditions given i n E q. (4.118): a2 G 2 (aXl- + a G) = aYZ -F (4.120) This solution will b e d enoted by G l. T his solution will not, i n g eneral, satisfy the boundary condition given i n Eq. (4.119) giving instead a value o f G o n t he c enter line, say Gc , t hat is not equal to 1. T he c orrect solution is then g iven b y G l/Gc • T he solution to Eq. (4.120) will h ere b e o btained using a finite difference method. . O nce the solution for G is obtained i n this way, the heat transfer rate c an b e obtained b y noting that, a tany p oint o n t he wall, Fourier's law gives: qw which can b e w ritten as: = + k- aTI w an (4.121) (4.122) The mean h eat transfer r ate i s t hen given by: q ;w ( Tw - Tc)k =( )Perimeter aN w aGI dS (4.123) this c an b e derived from the numerically determined variation o f G . T he N usselt n umbytbased o n ( Tw - T m) c an then b e obtained b y r ecalling that: Tm = (Mass F1~W Rate)L uT dA p -fA pu d A . puc fA F dA (4.124) w here A i s the cross-sectional area o f t he duct. Now: Mass F low R ate = . Similarly, since: (4.125) f p uTdA = p ucf F[Tw - (Tw - T)]dA . A. (4.126) A 186 Introduction to Convective Heat Transfer Analysis Hence: L FGdA Tw - Tm = (Tw - TC> f (4.127) F dA A This equation can be used to determine ( Tw - Tm)/(Tw - Te) by numerical integration of the numerical results. This result can then be combined with that given by Eq. (4.123) to give qw W N u = (Tw - Tm)k (4.128) The numerical solution to Eq. (4.120) is obtained using the same basic procedure as used in obtaining the solution for the velocity profile function, F. The grid system shown in Fig. 4.10 is again used. Using the values of G at the five nodal points indicated in Fig. 4.10, the following finite-difference form of Eq. (4.120) is obtained: ( Gi+ l ,j + G i-l,j flX2 2Gi,j) + ( Gi,j+ I + G i,j-l a y2 - 2Gi,j) = - F.. ',J ( 4 129) • This equation applies to the internal points, i.e., for i = 2 to N X - 1 and for j = 2 to N Y - 1. On the boundary points, the following are given by the boundary conditions: For i = 1 to N X: G"'l = GI'2 . For j = 1 to N Y: F ori = 1 t oNX: For j = 1 t oNY:· = G2,' } Gi,NY = 0 GNK.j = 0 GI , J. (4.130) Eq. (4.129), subject to the boundary conditions given in Eq. (4.130), can be solved iteratively. The values o f Gi,j are first approximated. For example, since,the velocity profile function is determined prior to obtaining the solution for G, the following can be used to start the solution: GI·· = F·· ,} I ,} Eq. (4.129) can then be applied sequentially to give updated values at the internal points and the boundary conditions .can be used to update the boundary point values. I n doing this, Eq. (4.109) can be written as: ., G .. I ,} = (Gi+lljflX2G i-llj ) +. (Gilj+la+ G i,j-l + FI.,}.)/(~ + a }'2 +. . }'2 flX2 2-) .. ( 4131) This i s applied sequentially over and over to all points until convergence is obtained to any specified degree o f accuracy. This gives the values o f Gi. As discussed CHAPTER 4: Internal Laminar Flows 187 above, the actual solution is then given by: G i,j = -.!:L G1,1 G ·· (4.132) A simple computer program, RECDUCT, written in F ORTRAN that implements the procedure outlined above is available as discussed in the Preface. Overrelaxation is used in this program, i.e., i f Fi~j is the new value o f the velocity function given b y Eq. (4.111), the actual new value is taken to be: (4.133) where r is the relaxation factor, here taken to b e greater than 1. Over-relaxation is also used in finding G i,j, i.e., i f F rj is the new value o f t he temperature function given b y Eq. (4.131), the actual new value is taken to be: (4.134) The use o f over-relaxation increases the rate o f convergence. T he variation o f t he m ean N usselt number with aspect ratio as given by this program is shown i n Fig. 4.13. This Nusselt number is based on the duct height, W. Now, for a rectangular duct, the hydraulic diameter is given by: DH =4 4W A rea 4 WB _ W etted Perimeter = 2 B + 2 W 2 +21A (4.135) Using this, the Nusselt n umber b ased on the hydraulic diameter c an b e found and the variation o f this with aspect ratio, A, is also shown in Fig. 4.13. I t will again b e seen 6 4 2 '----'---'--............- '"'"'---"------'-----'"---' 1 2 3 4 5 6 8 7 9 Aspect Ratio F IGURE 4.13 Variation o f Nusselt n umber with aspect ratio for a rectangular duct. 188 Introduction to Convective Heat Transfer Analysis T ABLE 4 .1 Effect o f wall t hermal b oundary c ondition o n t he Nusselt n umber f or fully developed flow i n a r ectangular d uct (Nusselt n umber b ased o n h ydraulic diameter) N u-uniform p eripheral t emperature, uniform axial h eat flux 3.608 4.123 5.331 6.049 6.490 8.235 Nu-uniform p eripheral h eat flux, uniform axial h eat flux 3.091 3.017 2.940 2.930 2.904 8.235 N u-uniform p eripheral and axial t emperature 2.976 3.391 4.439 5.137 5.597 7.541 Aspect ratio 1 2 4 6 8 Plane duct t hat the use o f t he hydraulic diameter as the length scale does not g ive a constant Nusselt number, this again indicating the inadequecy o f this concept for predicting t he h eat transfer rate. The above results were for a rectangular duct with a wall temperature that i s u nifonn around any section o f t he duct b ut i n which the total heat flux from the wall around the section o f t he duct i s n ot changing ·with distance, z, along the duct. A few results for ducts with other thermal boundary conditions [2],[10] are given i n T able4.L Solutions for fully developed flow in ducts with various other cross-sectional shapes have been obtained using techniques similar to those outlined above [2],[10]. T he results given by some o f t hese solutions are presented i n Table 4.2. T ABLE 4 .2 N usselt n umbers f or fully developed flow i n d ucts w ith v arious cross-sectional shapes (NusseJt n umber b ased o n h ydraulic d iameter) N u-uniform p eripheral t emperature, u niform axial h eat flux Cross-sectional s hape ~f duct Equilateral triangle Hexagonal Circular Elliptical-eccentricity = 0.9 Plane duct N u-uniform p eripheral heatfiux,·· u niform a xial hea~fiux N u-uniform p eripheral and a xial t emperature 2.470 3 340 3.657 3.660 7.541 3.111 4.002 4.364 5.099 8.235 1.892 3.862 4.364 4.350 8.235 CHAPfER 4: Internal Laminar Flows 189 4.5 PIPE FLOW WITH A DEVELOPING TEMPERATURE FIELD T he preceding sections w ere c oncerned with fully developed flow in which the forms o f both the velocity profile a nd the temperature profile were not changing with distance along the duct. F ully developed flow is, however, only attained in the flow, well downstream o f t he entrance to the duct or a bend or other fitting in the duct o r o f a region over which there is a change i n t he conditions at the duct wall. T he region in which the flow is developing, i.e., moving towards the fully developed state, is termed the " entrance" r egion as discussed earlier. In general, both the velocity and temperature fields a re s imultaneously developing, i.e., both are changing with distance along the duct. However, i n s ome cases, there is a long section o f d uct i n w hich there is no h eat t ransfer prior to the duct section i n w hich heat transfer takes place. In m any s uch c ases, the velocity profile is then essentially fully developed before the h eat t ransfer occurs and i t is then only the temperature that is developing, i.e., there is o nly a " thermal e ntrance r egion" [1],[2],[11]. This is illustrated i n Fig. 4.14. The situation considered i n this section is traditionally termed the " Graetz" problem. Because there i s no h eat t ransfer i n t he initial portion o f t he d uct flow,· the fluid will have a uniform temperature, Te a t t he point a t w hich h eat transfer starts, Le.: A t Z : - 0: T = Te ( 4.136) Attention will i n this section b e g iven to thermally developing flow i n a pipe. I n this case, the velocity profile, which is not changing with z, i s given, as discussed i n Section 4.2, by: where U c is, as before, t he v elocity o n t he center l ine o f t he pipe. This profile, it m ust b e stressed, exists e verywhere i n t he thermal entrance region. Unheated Wall . . AI .1" Heated Wall •• Temperature ProfIle ~~:'-2~~~:;:-~~~::E-i;-;:Velocity ProfIle (Fully Developed) Temperature ProfIle Velocity ProfIle (Unchanged) F IGURE 4.14 Thennal entrance region. 190 Introduction to Convective Heat Transfer Analysis The temperature profile i s changing with distance z along the duct, the temperature being assumed to b e g overned by: (4.137) the velocity component i n t he radial direction, v, b eing zero because the velocity field is fully developed. T he diffusion o f h eat i n t he axial direction will again be neglected compared t o t hat i n the radial direction, i.e., t he equation governing the temperature is assumed to h ave the form: (4.138) Le.: U aT = (P r ) [r (a (r ajr ~ v1 T)~ az jr ( (4.139) I n order to illustrate t he t ype o f solution obtained i n t he thermal entrance region, attention will b e g iven first to t he c ase where t he w all temperature o f t he pipe is k ept constant i n this thermal e ntrance region, i.e., to t he c ase where: F or z > 0: T = T w a t r = r0 (4.140) I t is then convenient to introduce t he following dimensionJess variables: U = -, Urn U (J = . (Tw - T) (Tw - Te) (4.141) R =D' r Z= z lD R ePr where Urn is' t he m ean v elocity i n t he p ipe a nd Re i s t he Reynolds number b ased o n this m ean v elocity a nd t he d iameter o f t he p ipe, D, w hich i s e qual to 2 ro • T he dimensionless variable Z i s t ermed the Graetz number. I n terms o f t hese v ariables, the equation governing t he developing temperature field is: u!~ = !:R(R!!) . (4.142) I t should b e n oted t hat i f t he second t enn o n t he r ight-hand side o f t he energy equation, Le., a2 Tlail, h ad b een retained, t he r ight-hand s ide o f t he a bove equation would have b een: (4.143) CHAPTER 4: Internal Laminar Flows 191 This shows that the second t enn, i.e., the longitudinal diffusion tenn, is negligible compared to the first tenn, i.e., the radial diffusion tenn, i f the quantity R e P r jp large. The quantity R e P r is termed the Pec1et number, Pe. Generally, i f the Pec1et number is greater than 10, the effects o f longitudinal diffusion are, indeed, negligib~e. Now the velocity profile for fully developed pipe flow discussed above can be written in terms o f the dimensionless variables as: Therefore, E q. (4.142) becomes: 2[1 _(!i.)2] ae = ~ ~ (R aR a ae) 0.5 z R aR· This equation must be solved subject to t he following boundary conditions: (4.144) R = 0: a R = 0, ae R = 0.5: e = 0.0 (4.145) while the initial conditions become i n t enns o f t he dimensionless variables: Z = 0: e= 1 (4.146) The solution to Eq. (4.144) has traditionally been obtained b y using the method o f separation o f variables. T he solution is assumed to have the form: 8 = F(R)G(Z) (4.147) where, as indicated, G i s a function o f Z alone a nd F is a function o f R alone. Substituting this into Eq. ( 4.144) t hen gives: , (R dG 2 [ 1 - 0.5 . F d z i.e.: --= )2] = G d ( d F) R dR R dR 1 dG G dz 1 2 RF 1 - - .[ (R rJ[~~+R~;] 0.5 (4.148) Because G is a function o f Z alone and F i s a function o f R alone, this requires that: (4.149) 192 Introduction to Convective Heat Transfer Analysis and: 1 2RF 1-(-) [ 0.5 R 2J [ dF d 2FJ 2 d R + R dR2 = - A i.e.: d 2F d F 2 3 R dR2 + d R + 2A [R - 4R ]F = 0 (4.150) The solution to the first of these equations, i.e., Eq. (4.149), is: G = Ce->..2z (4.151) where C is a 90nstant o f integration. When the fonn of the second equation is considered, it is found that there are an infinite number of C and A. values and that associated with each of these values is a solution for F. These are denoted by Cn, An, and Fn(R). The values of Fn are given by the second equation, i.e., Eq. (4.150), which can be written as: (4.152) The solution of this equation must be such that when R :whenR = 1 Fn = O. = 0, Fn remains finite and _100 , ...----,----or-------,-------. N UDm 10 1 ' - - - - - ' - - - - - ' -_ _. .,..L.._ _. ..... -. 0.0001 0.001 0 .010.1 z lD RePr 1 FIGURE4.15 Nusselt number variation i n thermally developingftow in a pipe. C HAPrER 4: Internal Laminar Flows 193 T he actual full solution is: () = . L n=O 00 CnRne-A~Z (4.153) T he solution for n = 0 is basically the fully developed flow solution discussed above which applies at large values o f Z . Once () has been determined, the heat flux at the wall and the mean temperature c an be found and the mean Nusselt number c an t hen be found. Exact solutions for values o f n up to 4 can b e r elatively easily obtained and approximate solutions for higher values o f n c an be obtained. The variation o f t he mean Nusselt number with Z g iven by these solutions i s s hown i n Fig. 4.15. InsteaQ o f using the method o f separation o f variables, a numerical solution to Eq. (4.144) can be easily obtained. Here, a numerical finite-difference solution will be discussed. A series o f g rid l ines in the Z a nd R directions are introduced as shown i n Fig. 4.16, a uniform grid spacing, AR, b eing used i n the radial coordinate direction. Consider the four a djacent nodal points shown i n Fig. 4.17. I!.R I!.R I!.R I!.R I!.R ~ ___._-... ... Z F IGURE 4.16 . Nodal system used i n optaining numerical solution. ~ .I j+l ~--~----------~~ -~'-H i -I j i F IGURE 4.17 Nodal ppints Used iD. obtirlniiJ:g finite-dllIereilce approXiniations." "" 194 Introduction to Convective Heat Transfer Analysis Using the values at these points, the following finite-difference fonn of Eq. (4.144) applied at point, i, j , is obtained: .. This can b~ rearranged to give: A /hj + B /hj+l + C/J;,j-l = Dj where: AJ (4.155) .. _ - /l'Z 1 2 [ _ (. Rj+l + R j R.i + R j-l R /O.5· + 2 R j ll.R2 + 'f.R j Il.R2 B . _ _ Rj+1 + R j J2R j ll.R2 C . = R j + R j-l J 2Rjll.R2 )2] (4.156) (4.157) (4.158) (4.159) Dj = :z [1 - (Ri,jlO.S)2 ]6 i - 1,j Since the case o f a constant wall temperature is being considered, i.e., since 6 i,N = 0 and since the symmetry condition on the center line gives, in finitedifference fonn, .6i,1·~: .6t,!l,-the following two results are obtained b y considering . the form.ofEq. (4.155): A l = 1. AN = 1, Cl CN = 0, = 0, Dl DN =0 =0 (4.160) (4.161) Therefore, the finite-difference approximation to the governing equation leads to a set o f N simultaneous linear algebraic equations whose solution requires the inversion o f a tri-diagonal matrix j ust as was the case i n the finite-difference solution . o f the boundary layer equations discussed i n Chapter 3. T he solution is easily obtained using the same method as used in obtaining t he boundary layer solution. This solution gives .the values o f 6 i ,j. T he local h eat transfer .rate a t the wall is then given b y using Fourier's l aw as: qw = + k- . aT or (4.162) r=ro Th~t:Q is 3: ~'+ " i,n, this equation beCause.r i s ineasured radially outward i n t he opposite direction to' that used' i n defining. ·a posi~ve:'value o f qw. C HAPTER 4: Internal Laminar Flows 195 In terms of the dimensionless variables introduced above, Eq. (4.162) becomes: qw D a () k(Tw - Te) = aR I R=O.5 (4.163) T he left-hand side ofthis equation is the local Nusselt number, N UD, i.e., Eq. (4.163) gives: NUD = a () aR I (4.164) R =0.5 The finite difference approximation to this equation is: Nu D - ()i,N - ()i,N-I -AR (4.165) I t should be noted that the Nusselt number being used here is based on the difference between the wall and the inlet temperatures, i.e., on Tw - T e, arid not on the difference between the wall and the mean temperatures. Now the mean temperatqre, i.e., the bulk wean temperature, is given by: Tm = .for uT2'TTr d r f;0 u21Trdr 0 (4.166) i.e., rewriting this equation i n terms o f the dimensionless variables: Tw - Tm = 8 (0.5 U()RdR Tw - Te Jo . (4.167) The Nusselt number based on the difference between the w,all and the mean temperature is then/given by: NUDm = N UD ...........- -:=-- . . .. NuiJ . T w-Tm8fo°.5U()R"dR' Tw - Te -;::-;:----- . (4.168) The value o f the integral c an b e determined b y numerical integration. A computer progr~i DEVPI;PE, written i n FORTRAN based on the above procedure is available as discussed i n the Preface. The variations of N UD, N UDm, and the dimensionless center. line temperatur~, () c , with Z given b y this program are shown i n Fig. 4.18. Next, consider thermally developing flow i n the case where t he wall heat flux, qw, rather than the wall temperature is uniform. I n this case, the folloWing idimensionless temperature is used: (4.169) T he initial conditions a t t he beginning o f the thermally developmg region i ll this case is because o f the way () i s defined: Z = 0: 8 = 0 (4.170) 196 Introduction to Convective Heat Transfer Analysis 100 r-----,----,---.-----, 10 11-------_ 0.1 ' - - _ . . . . . l - _ - - 1 . ._ _L -l..l_ __- -' 0.0001 0.001 0.01 0.1 1 z lD R ePr F IGURE 4.18 Nusselt number and center line temperature variation in dev((loping flow i n a pipe with a uniform wall temperature. and the wall boundary condition is 'obtainoo by recalling that: qw . . aTI = +k~ ur (4.171) r=ro which can be written in terms of the dimensionless temperature as aOI aR R=O.5 8 i.N _1 (4.172) The finite-difference approximation to this equation is: = O i,N-l + Il.R (4.173) this means that in this case: A N = .1, BN = 0, e N' - 1, D N = Il.R (4.174) I n the uniform wall heat. flux case, 1:he definition of the dimensionless tenipera. ture is such that: O. ~ Tw ~ Te~. -1 w. -qwD1k N UD Further since·in this uniform heat flux case: j' T.-: Te , 'O,(9.wP{k); itfonows that; Tm - T i~e!, ,using t:p.e kn~\Vl1 form 0.5 • = e, qw D k fo U ORdR foo5 U RdR . ( 0.5 u '0 (4.175) o f the v~ation o~ U, ~th R: - Tm qw D1k Te = O~ = 8 Jo R· d' R (4.176) CHAPTER 4: Internal Laminar Flows 197 100 . ...----,---..,...--,..-----, 10 1 0.1 '--_~_---' _ _ _J..-'--_--' 0.0001 0.001 0.01 z lD R ePr 0.1 1 F IGURE 4.19 Nusselt number and center line temperature variation in developing flow in a pipe with a uniform wall heat flux. This allows (Jm to be determined at any value o f Z. T he value o f the integral can b e determined by numerical integration. The Nusselt number based on the difference between the wall and,the m ean temperatures is then given b y noting that: , Tw F rom which i t follows that: '.fm = ,(J w - (J m q wDlk NUDm = Ow,- (Jm 1 (4.177) T he computer program DEVPIPE discussed above allows either a unifonn wall heat flux. or a uniform wall temperature to b e considered. The variations of N UD, NUDm, and the dimensionless center line temperature, (J 1 , w ith Z given b y this program for the unifonn wall h eat flux case a re shown i n Fig. 4.19. A solution for the uniform wall heat flux case' c an also b e obtained'using t he separation o f variables approach discussed above. PLANE'DUCT FLOW WITH' A DEVELOPING TEMPERATURE ¥IE~D F low with a developing tel11perature field i n a plane duct is dealt with using the same procedure as used to deal with p ipe flow. T he flow situation is as s howirin Fig. '4.20. In this case, using the s ame assumpti~ns as adopted i n dealing with pipe flow, t he temperature variation w ill b e governed by: , ' 4.6 , , aT ~ g2.r u - - '(- - , az P r ay2 ,v) (4.17~~ 198 Introduction to Convective Heat Transfer Analysis Heated Temp. Profile Velocity P rome (Fully Developed) F IGURE 4.20 Thermally developing plane duct flow. The following dimensionless variables are then introduced: U =-, Urn U o= (Tw - T) (Tw - Te) (4.179) Z = Z/W R ePr where Urn is the mean velocity in the pipe, Re is the Reynolds number based on this mean velocity, and the width o f the d uct is W. In terms o f these variables, the equation governing the developing temperature field is: y= L w' U az ao = aYZ a20 (4.180) Now the velocity profile for fully developed plane duct flow can be written in terms o f the dimensionless variables as: Using this, Eq.(4.180) becomes: , . Y 1.5 [ 1 - ( 0.5 )2] ao = a z . aYZ a2'0' (4.181) This equation must b e solved subject to tl,te foll<?wing boundary conditio~s: y = 0: a y . ao . . .' . = 0, y = 0.5: (} = 0.0 . (4.182) w hile the itliti81 conditions a re i n t~rms o f the dimensiollIess variables: . .. . .. . : z = 0: e = 1 . (4.183) The solution to Eq. (4.181) can either be obtaitied b y u sing the method o f sepaused i n dealing .;With pipe flow. Here, "a numerical'finite-difference solution will b e discussed. The procedure is identical to t hat u sed in dealing with pipe flow. A series o f grid lines i n ~ation o f variables o r n umerically using the same type o f a pproach as CHAPTER 4: Internal Laminar Flows 199 the Z and Y directions are introduced, a uniform grid spacing a Y being used in the Y-coordinate direction. Using the values of the variables at the same nodal points as used in dealing with pipe flow, the following finite-difference form o f Eq. (4.181) ;applied at point i, j is obtained: ~ 1 5[1 _ ( Yj . Q5 )2] (Oi,j -az = Oi,j+l + Oi,j-l Oi-l,j) a~ 20i,j (4.184) This can be rearranged to give, as with pipe flow: A jOi,j + B jOi,j+1 + C jOi,j-1 = D j ~.5 [ - (O.~ 1 Bj (4.185) where in this case: Aj = y.)2] a1z + a y2 2 1 (4.186) (4.187) (4.188) (4.189) = - Ay2 1 C j = - Ay2 D. = 15[1 - (Yi f] Oai-I,j 0.5 J z J.' Because the case o f a constant wall temperature is being considered, i.e., because Oi,N = 0, and since the symmetry condition on the center line gives, i n finitedifference form, Oi,l = Oi,2, t he following two results are obtained by considering . the'form ofEq. (4.185): . , Al AN = 1, CI = 0, CN = 0, (4.190) (4.191) = 1, Therefore the finite-difference approximation to the governing equation again leads to a set of N simultaneous linear algebraic equations whose solution requires the inversion o f a tri-diagonal matrix. The solution is easily obtained using the same method as used in dealing with pipe flow. This solution gives the values o f 0 i ,j' The local heat transfer rate at the wall is then given by using Fourier's law as: . qw = + k aTI y=w ay ao Y=O.5 (4.192) I n terms o f the dimensionless variables ititroduced aboye"Eq. (4.192) becomes: -~- =k (Tw - Te) ay qwW (4.193) The left-hand side o f this equation is the local Nusselt number, Nuw, i.e., Eq. (4.194) , gives: a N uw= - OI a y y =o.S . , (4.194) 200 Introduction to Convective Heat Transfer Analysis The finite-difference approximation to this equation is: AT _ (h,N - (hN-l lVUW - .1.Y (4.195) The Nusselt number being used here is based on the difference between the wall and the inlet temperatures, i.e., on (Tw - Te) and not on t he difference between the wall and the mean temperatures. Now the mean temperature, i.e., the bulk mean temperature, is given by: Tm fow u Tdy = foWudy (4.196) i.e., rewriting this equation i n terms o f the dimensionless variables and using the known form o f t he variation o f U with R gives: Tm - Te = 16 (0.5 U 8dY, Tw - Te 11 Jo The Nusselt n umber b ased on the difference between the wall and the mean temperatures is t hen given by: Nuwm = Nu wTw - Tm - Tw - Te Nuw (16/11) foo. 5 U(J d Y (4.197) The value o f t he i ntegral caD. b e determined b y n umerical integratiop. thus allowing the variation o f N UWm w ith Z to b e found. A computer program, DEVDUCT, b ased on this procedure written i n FORtRAN i s available as discussed i n t he Preface. The variations o f Nuw, Nuwm, a nd the dimensionless center line temperature, ( Jit with Z g iven b y t he a bove program are shown i n F ig. 4.21. 100~--~--~r-~--~----~ 10 1 ,1-------------- FIGURE 4.21 0.1 L .-_ _- ' -_ _ _. .L..-_ _- -'-_..,L.;L-....I 0.0001 0.001 0.01 z lW Re Pr 0.1 1 N usselt number and center line temperature variation i n developing flow in a plane d uct with a uniform wall temperature. / CHAPTER 4: Internal Laminar Flows 201 The modifications to the above analysis to deal with the unifonn wall heat flux case are basically the same as that required for pipe flow and will not be discussed here. 4.7 LAMINAR P IPE F LOW W ITH DEVELOPING V ELOCITY AND TEMPERATURE FIELDS Flow in a pipe when both the velocity and temperature fields are developing is considered in this section, the flow situation thus being as shown in Fig. 4.22 [12],[13],[14],[15]. The ~-axis is again chosen to lie along the center line o f the pipe and the velocity components are defined i n t he same way as in Chapter 2, i.e., as shown in Fig. 4.23. Assuming that the swirl velocity component, w, is zero and that the flow is therefore symmetrical about the center line, the equations governing the flow are: (4.198) (4.199) (4.200) (4.201) 1 I t is assumed that the radius o f the pipe is relatively small compared to the values o f z being considered. I n this case, the same type o f order o f magnitude analysis as used i n deriving the boundary layer equations indicates that v « u and that Velocity Profile Velocity Profile ~--~~~----------~-----------------~~~~ F IGURE 4.22 Temperature Profile Temperature Profile Developing velocity 'and temperature fields. F IGURE 4.23 Coordinate system used. 202 Introduction to Convective Heat Transfer Analysis alaz « alar. As a result, the r-momentum equation gives: ap = 0 ar The governing equations can therefore be written as: 1a au - - (vr) + - = 0 r ar az u au + v au = _.!:. d p + v (a2u + a u) az ar p dz ar2 r ar (4.202) (4.203) (4.204) (4.205) ! aT aT ( v ) ( a 2T 1 a T) u az + v ar = . P r ar2 + ar r These equations. are parabolic in form whereas the full equations are elliptic in form. Eqs. (4.203) to (4.205) are therefore referred to as the parabolized form o f the governing equations. The numerical solution to these equations for the case where both the velocity and temperature profiles are developing and where the wall temperature is uniform will be considered in the present section. The velocity and temperature profiles are assumed to be symmetrical about the center line o f the pipe and the radial velocity component, v, is therefore zero n n the center line. The boundary conditions on the solution on the center line are therefore: At r = 0: - r = 0 ' v a au . = aT 0, -ar = 0 (4.206) At the wall, the velocity components are zero and the wall temperature is constantand specified, i.e., the boundary conditions at the wall are: .. / A t r = ro: u = 0, v = 0, T = Tw (4.207) r0 bei~g the ·radius o f the pipe, i.e., r0 = DI2 where D is tQe diameter o f the pipe. . Althoughit is not anecessary step in obtaining the solution, the governing equations Win be written in t enns of the fOllowmg dimensionless variables: U = ulllm, V = vRePrlum, P = ( p - Po)lpu! (4.208) / Z = zlRePrD, R = rID. (J = { T - To)/(Tw - To) where Um is the mean velocity across t he pipe, Re is the Reynolds number based on Um, and D, i.e., R e= u~Dlv, To-is the temperature on the inlet plane, and Po is s~me reference pressure. The value used for Po i s irrelevant to the solution but,· i f actual values of the pressure are required, i t is often convenient to take Po as the pressure upstream of inlet to the pipe as shownJn Fig. 4.24. Essendally Zero . Velocity . Pressure Po = .~------------------------- '--=--------------------, ---FIGURE 4.24 Reference pressure. Inlet to pipe CHAPTER 4: Internal Laminar Flows 203 I n tenns o f these dimensionless variables, t he governing equations become: (4.209) (4.210) (4.211) I n t enns o f the dimensionless variables, t he houndiuy conditions are: At R and: = 0: aR = 0, au v =o-· =0 ' DR iJ(J (4.212) A tR = 0.5: u = 0, v= Q. (J =1 (4:213) I t will b e noted that the only parameter i n t he above set o f equations i s the Prandtl number~ . I n o rder to obtain a solution to these equations, the conditions on t he i nlet p lane h ave to b e known. I t will b e assumed h ere t hat the velocity and temperature distributions a re u niform on the inlet plane to the p ipe as ll1ustrated i n Fig. 4.25, i.e., that: A t·z =' 0: v = 0, T = To (4.214) .velocity on th~ i nlet plane being equal t o Um because, since the density is b eing a ssumed constan~ t he m ean velocity ~oes n ot change with l with the result t hat i f t he velocity across ~e p ipe s,ection is. uniform, the velocity must b e e qual to U m. Consistent w ith this i s the assumption that v i s zero on the inlet plane. I f t he fluid flows iIito t he pipe through. a bell-shaped inlet section.as s hown i n . . F igure 4.25, t he losses in this inlet section will b e small. I n this case, i f Po i s t aken as the pressure ahead of t he inlet as shown i n F ig. 4.26 a nd i~ the pressure o n t he i nlet plane t hen B ernoulli's equation applied across the inletgives: the Pi Pi + pu~12 = Po (4.215) / FIGURE4.2S Inlet plane conditions. 204 Introduction to Convective Heat Transfer Analysis Pipe I I I p. I F IGURE 4.26 L ow loss inlet section. Le.: (Pi - PO)!pu~ = - 112 (4.216) - In terms of the dimensionless variables introduced above, these inlet conditions are: A tZ ="0, u= 1, v = 0, (J = 0, P = - 0.5 (4.217) A simple finite-difference solution to the above set of equations w ill be discussed here. A series o f nodal lines i n the Z- and R-directions are introduced. Because of the parabolic nature of the assumed form of the governing equations, a forward';marcbing solution i n the Z.,direction can be used. The solution starts with known conditio~oi1 the first R-line. Finite-difference forms o f the governing equations are then used to obtain t4e solution on the next R-line. Once this is obtained, the same pmcedure can be used to "get the solution on the next R-line and so on. This i s the same procedure as was used in Chapter 3 in obtaining a solution to the boundary " layer equations. A n explicit finite-difference procedure will be used here "in dealing with "the momentum and energy equations. Consider the nodal points shown i n Fig. 4.27: I t will be seen that a unifonn grid spacing is used in the R.,.direction. _i F IGURE 4.27 Nodal points use<;! in dealing with momentum and energy equations. CHAPTER 4: Internal Laminar Flows 205 F irst consider the momentum equation. The finite-difference fonn o f this equation is: i Pr ~ V' , (V;,j - Vi-l,j) 1 -1,) LlZ ~ + Pr v. + 1 -1,) , (Vi-l,j+l - Vi-l,j-l) 2LlR _ 1 (Pi - Pi-d - - Pr LlZ ( Vi -I,j+1 + V i -l,j-l) - 2Vi-l,j LlR2 1 ( Vi -l,j+1 - Vi~l,j-l) + Rj which gives: 2LlR (4.218) Ui ,j = ' U,' i -l,j - Pi V i-I,j V i-I,j A + Pi -l - ---=- ( Vi-I,j+1 - Vi-l,j-l) L.l Z U i-l,j V i-l,j 2LlR + Pr [ (Ui-l ' )'+1 + Ui -l ,) '-1) LlR2 2Ui- 1' + - ')'+1 - Ui -l, J' 1 ( Ui-l . Rj 2LlR )'-d] .LlZ U i-l,j (4.219) i.e.: U· ,), = V· - 1,). I 1 P, - U 1 i -I,] + A) · (4.220) where: Aj = P i-l _ V i-I,j ( Ui-l,j+l - Ui-l,j-l ) LlZ V i-l,j Ui -l,j 2LlR + p r[(Ui-l,j+l + Ui -l,j-l LlR2 - 2Ui -I,j) +~ ( Ui -l,j+l - Ui-l,j-I)] LlZ Rj 2LlR V i-l,j (4.221) A j is therefore a known quantity at any stage· o f the calculation because the vall;les o f t he variables on the i - 1 line are known. N ext consider the continuity equation written as: a R(VR) a = -R az au ( 4.222) This is applied to the nodal points shown i n Fig. 4.28. . T he equation is actually applied to the point p , shown i n Fig. 4.28. This lies halfway between nodal points i, j a nd i, j - 1. T he value o f a Ulaz a t point p i s taken as t he average o f the values o f a al a Z a t points i, j and i, j - 1. H ence, the continuity equation gives: R i,jVi,j - A~j-l V i,j-l. = _~ [ ai,j ~~i-l,j + U i,j-l ~~i-l,j-l ] i.e.: AR [ , R i,jVi,j - Ri,j-l V i,j-l = - 2AZ V i,j - Ui-l,j + U i,j-l - Ui-l,j-l ] (4.223) 206 Introduction to Convective Heat Transfer Analysis i -I - .--_...._ _. - - , _.. j I . . i i i i i ~ i ...~ I i : j M l2 I I I i PC' . ontinwty i Equation Applied Here j-l - . - -.._ _. _-_. •- F IGURE 4.28 Nodal points used in dealing with continuity equation. Substituting Eq. (4.220) into this equation then gives: R i,jVi,j - Ri,j-I V i,j-I = - ·2AR [ Pi . AZ - U. L .l , -I,) + Aj - U. Pi . , -1,)-1 + A j _ 1] (4.224) which can b e written as: R i,} V i,j - Ri,j-I V i,j-I = - Pi [ 2!:::..R(l + A Z U i-I,j U i-I,j-I l)~!:::..RZ (A j ~ - 2A + A j -I) (4.225) i.e.: R '·,J· V·,). - Ri, ) - I V',)- I ~= - B·P· - C· . .. l )' J (4.226) where: B. J = !:::..R [ 2 AZ 1 U i-I,j + 1 U i-I,j-I ] (4.227) and: (4.228) I t w ill b e noted that since at any stage o f t he solu~on t he conditions on th~ ( i ':"-1) line are known, the coefficients A bAj~ h 'Bb and C j are known <luantitie~. ' , Now, on the center line, i.e., a t j ' = 1, the quantity R i,jVi.J i s O. Hence, i f Eq. (4.226) is applied sequentially outward from point j = 2 to t he 'point j = N which lies o n the wall; t he following is obtained: Ri,2Vt,2 - 0 = - B2 Pi - C2 Ri,3 Vi,3 - Ri,2 Vi,2 = - B3Pi - C3 R iA Vi,4 - Rt,3 Vi,3 = - B4 P i , - C4 Ri,S Vi,S - Ri,4 Vi,4 = - BsPI- Cs Ri,NVi,N - Ri,N-l V i,N-l = - BNPi·- CN Adding this s et o f equations together then gives: N N Ri,NVi.N = -Pi~Bk k=2 .L: Ck k=2 (4.229) CHAPl'ER 4: Internal Laminar Flows 207 B ut the boundary conditions give V i,N = 0 so Eq. (4.229) gives: (4.230) p. _ 2:f=2 Ck , - 2:f=2 B k This allows the dimensionless pressure, Pi, to be determined. O nce this has been found, Eq. (4.220) can b e u sed to find the Ui,) values and Eq. (4.229) applied sequentially outward from j = 2 as discussed above allows the V i,) values to be found. The boundary conditions give Ui,N = 0 and Ui,l = Ui,2' Lastly, the energy equation, i.e., Eq. (4.211), is used to d etennine the distribution o f Oi,). T he finite-difference f onn o f Eq. (4.211) is obtained using the same nodal points as used in dealing with t he momentum equation. The energy equation then gives: U. , -I,} . (Oi') - Oi-l,)) + v. .( Oi-l,)+1 - Oi-l,)-I) AZ , -I,} 2 AR (Oi-l,)+1 .. = ' (4.231) + O i-l,)-I) - 20i-l,) AR2 + RJ · . 1 (Oi-l,)+1 - Oi-l,)-.) 2 AR w hich gives: O. . = O. .. _ V i-I,) (0;-1,1+ 1 - Oi-l,)-I)tJ.Z I ,) I -I,} U i-l,j 2 AR (Oi-l,)+1 +. OI,N + O i-l,1-1) AR2 (4.232) 20 i- 1,) + 1 (Oi-l,)+1 - Oi-l,)-I) R}· 2 AR This equation allows the 0 u values to b e found. The boundary conditions give = 1 and Oi,1 Oi,2. Once t he 0 values are determined, the heat transfer rate c an b e found. T he focal h eat transfer rate a t the wall at any value o f Z i s given by \ using Fourier's law as; = qw = + k- aTI ar r=ro (4.233) T here is a plus ( +) i n this e quation because r is measured radially outward, i.e., in t he opposite direction to that used i n defining a positive value o f qw. I n tetms o f the dimensionless variables introduced above, Eq. (4.233) becomes: qwD ' k (Tw - To) = aR aOI R=O.S . (4.234) w here as before D = 2 r0 i s the p ipe diameter. T he left-hand side o f t his equation is t he local Nusselt number, N UD, i.e., Eq. (4.234) gives: N UD= - ao R=O.S aR (4.235) T he finite-difference approximation to this equation is: l YUD AT. = --'-~~- OI,N - Oi,N-l AR (4.236) 208 Introduction to Convective Heat Transfer Analysis T he Nusselt number being used here is based on the difference between the wall and the inlet temperatures, i.e., on Tw - To, a nd n ot on the difference between the wall and the mean temperatures. The mean heat transfer rate up to the value o f Z b eing considered can be obtained by integration o f the local wall heat transfer rate or b y noting that the total heat transfer rate u p to the point considered will be equal to the increase in the enthalpy flux between the inlet a nd the z value considered, i.e., that: qw'TTDz = I:o pucpT2'TTrdr ° pumcpTo'TTrt (4.237) i.e., rewriting this equation in terms o f the dimensionless variables: q ;ZRePr = pCpum T T w -.LO i°.5 U(JRdR -(4.238) i.e.: qwD 1 (0.5 k(Tw - To) = z Jo U(JRdR T he l eft-hand side o f this equation is the m ean N usselt n umber u p to dimensionless distance Z f rom t he i nlet considered. H ence,this e quation c an be written as: N UD = z Jo 1 (0.5 ' U(JRdR (4.239) T he value o f t he i ntegral c an b e d etermined b y n umerical integration. T he N usselt numbers introduced above were b ased on t he temperature difference Tw - To.-The N usselt n umbers b ased o n t he t emperature difference Tw - T m w here Tm i s t he m ean t emperature a t a ny v alue o f Z c an be d educed b y noting t hat t he definition o f Tm i s s uch'that: ' qw'TTDz = 'mcp(Tm - To) (4.240) t he l eft-hand side b ejng t he r ate a t w hich h eat i s transferred to t he fluid between the inlet and a point distance z f rom t he inlet. H ence, s ince i s g iven by: m m= i t follows that: pUm'TTD2/4 i.e.: Tm - To = 4-;:r.--Z Tw - To n UD i.e.: / (4.241) CHAPTER 4: Internal Laminar Flows 209 This can be used to find t he local Nusselt number based on T m - To which will be given by: Tw - To NUDm = NUD Tw - Tm = NUD 1 - 4NuD 21Pr (4.242) To summarize, the calculation procedure is as follows: 1. Define the values o f the variables on the inlet plane. As discussed above, these will here be assumed to be: U l.j = 1.0, V 1• j = 0.0, l h,j = 0.0, PI = - 0.5 2. Use Eq. (4.230) to d etermine P2. 3. Use Eq: (4.220) to find the U 2.j values 4. Use Eq. (4.226) applied sequentially outward from j = 2 to give the V 2.j values to b e found . . 5. Use Eq. (4.232) to find the 8 2,j values. 6. Use Eqs. (4.236), (4.239), a nd (4.242) to get the local and mean Nusselt numbers. 7. H aying detennined the values o f the variables on the i = 2 line, the same procedure is used to advance t he solution to the next i line and so o n to the maximum Z value for which a solution i s required. It will be noted that a t any stage o f the solution it is only necessary to know the values o f the variables on two adjacent i lines. ., A computer program, P IPEFLOW, written in FORTRAN, that i s b ased on this proceciure is available as discussed i n the Preface. Because an explicit finite-difference procedure is being used to solve the momentuin and energy equations, the solution can become unstable, i.e., as the solution proceeds i t can diverge increasingly from the actual solution as indicated in Fig. 4.29. A simplified analysis o f t he conditions under which instability occurs is presented i n the next section i n w hich t he numerical solution o f developing flow i n a plane duct is considered. T he r esults obtained there indicate that 'stability exists if: A Z < 0 .5AR2Ui -l. N -l Pr Unstable N umerical Solution (4.243) u Actual Z F IGURE 4.29 Development of the instability. 210 Introduction to Convective Heat Transfer Analysis 10~~--~----~------~ NUDm 1 ' --___- '-____- L...I_ _ _ _ _- -' 0.001 0.01 zlD R ePr 0.1 1 F IGURE 4.30 Nusselt number variations for developing pipe flow for P r = 0.7. will give a good indication o f the maximum value o f Il.Z that can be used without instability developing and this has been incorporated into the program listed above. Typical variations o f N UD and NUDm with Z given b y the above program are shown i n Fig. 4.30. ' E XAMPLE 4 .4. Oil enters a 12-mm diameter pipe at a temperature of 10°C with a unifonn velocity of 0.8 mls. The walls of the pipe are kept at a uniform temperature of 30°C. H the pipe is 60 cm long, plot the variation o f the temperature in the oil at the exit o f the pipe. Assume that for the oil p = 9lOkglm3, p. = 0.008 kg/ms, k = 0.14 W/m-K, and P r = 1100. Solution. The Reynolds number is given by: Re = 'pumD = 910 X 0.8 X 0.012 = 1092 p. 0.008 Hence, i f z is the axial coordinate measured from the inlet to the pipe then at the .exit of the pipe i f eis the length o f the pipe: Z = R e!rD = 1092 X 1~~ X 0.012 = 0.00004163 The above results show that the flow is laminar and that the flow will not be fully developed at the exit of the pipe. The program discussed above gives the variation of dimensionless temperature with dimensionless radius at the pipe exit The program has therefore been run up t oa maximum Z value o f 0.00004163. This gives the dimensionless temperature variation with dimensionless radius at the exit listed in Table E4.4. The actual radius and temperature are then obtained b y ~g that; R = riD, (J = ( T - To)/(Tw - To) from which i t follows that: r = D R = 1 2Dmm a nd that: T = 6 (l'w - To) + To = 6(30 - 10) + 10 = (206 + 10)OC 210 10 Introduction to Convective Heat Transfer Analysis ,,-----.------~------~ 1 L---------'--------.:L.....L..------....J 0.001 0.01 0.1 1 zlD R ePr F IGURE 4.30 Nusselt number variations for developing pipe flow for P r = 0.7. will give a good indication o f the maximum value o f .dZ that can be used without instability developing· and this has been incorporated into the program listed above. Typical variations o f N UD and NUDm with Z given by the above program are shown in Fig. 4.30. E XAMPLE 4 .4. Oil enters a 12-mm diameter pipe at a temperature of 10°C with a uniform velocity o f 0.8 mis. The walls o f the pipe are kept at a uniforrll temperature of 30°C. I f the pipe is 6 0 e m long, plot the variation of the temperature i n die oil at the exit of the pipe. Assume that for the oil p = 910 kg/m3 , J.L = 0.008 kg/ms, k = 0.14 W/m-K, and P r = 1100. Solution. T he Reynolds number is given by: Re = pumD = 910 X 0.8 X 0.012 = 1092 J.L 0 .008· Hence, i f z is the axial coordinate measured from the inlet to the pipe then at the exit o f the pipe i f .e is the length of the pipe: Z = R ePrD .e = 1092 X 0.6 0 6 1100 x 0.012 = 0.00 041 3 The above results show that the flow is laminar and that the flow will not be fully dbveloped at the exit o f the pipe. The program discussed above gives the variation of dimensionless temperature with dimensionless radius at the pipe exit. The program has therefore been run up to a maximum Z value o f 0.00004163. This gives the dimensioue less temperature variation with dimensionless radius at the exit listed in Table E4.4. The actual radius and temperature are then obtained by recalling that: R = rID, 6 = ( T - To)/(Tw - To) from which i t follows that: r = D R = 1 2Dmm and that: T = 6 (Tw - To) + To = 6(30 - 10) + 10 = (206 + 1 0tC C HAPTER 4: Internal Laminar Flows 211 T ABLE E 4.4 R 0.0000 0.3370 0.3587 0.3804 0.4022 0.4239 0.4457 0.4674 0.4783 0.4891 0.5000 () r (nun) 0.0000 0.0000 0.0001 0.0005 0.0028 0.0145 0.0690 0.2813 0.4886 0.7401 1.0000 0.000 4.044 4.304 4.565 4.826 5.087 5.348 5.609 5.740 5.869 6 .000 10.000 10.000 10.001 10.009 10.056 10.291 11.380 15.626 19.772 2 4.802 30.000 Center Line Wall 30r-----~------~------~ ) 4 2 Radius.-mm 6 FIGUREE4.4 The values of the radius, r and T given by these equations are also shown in Table E4.4. The variation ~ftemperature with radius on the exit plane is plotted in Fig. E4.4. )41 In the above discussion, it was assumed that the temperature of the wall of the ~.. pipe wa~ uniform and specified. I f instead of this, the wall heat flux, qw, was uniform I ~. a d spedfied, the dimensionless temperature would be taken as: 0 = ( T - To) .. .. qwDlk (4.244) l!nee: qw = + k- . aTI ar r =ro \ /" th- boundary condition on the temperature" at the wall in this case would be, a R R=O.~ ao I \,= 1 (4.245) 212 Introduction to Convective Heat Transfer Analysis In finite-difference f onn this gives: (Ji,N - (Ji,N-l AR =1 I.e.: (4.246) Thus, the only difference required when the wall heat flux is specified is that instead o f setting (Ji,N = 1, Eq. (4.246) is used to detenn~ne ( Ji,N' T he computer program discussed above allows for either a unifonn wall temperature or a uniform wall heat flux~ 4.8 LAMINAR F LOW I N A PLANE DUCT W ITH D EVELOPING VELOCITY AND T EMPERATURE F IELDS Here, attention will b e g iven to developing flow in a wide duct as shown in Fig. 4.31. Basically, developing two-dimenshfual flow b etween two plates i,s therefore being considered. I f t he Saline "parabolic flow" assumptions as adopted in dealing with pipe flow in the previous section are used, the governing equations for this flow become, i f the coordinate system shown in Fig. 4.32 is used: (4.248) (4.249) Because the flow is assumed to b e symmetrical about the center line, it follows that: , au A t y = 0: a y = 0, v = 0, aT = 0 ay (4.250) Temp. Profile F IGURE 4.31 Flow i n a plane duct. C HAPTER 4: Internal Laminar Flows 213 F IGURE 4.32 Coordinate system used. A t the wall, the velocity components are zero a nd the wall temperature i s u niform and its value i s specified, i.e., the boundary conditions a t t he wall are: A ty = W/2: U = 0, v = 0, T = Tw (4.251) T he g overning equations are written i n t erms o f t he following dimensionless variables: . U ;;;;",u/um , V = v RePr/u m , P = ( p - po)/pu~ Z = zlRePrW, Y = y/W, 0 = (T - To)/(Tw - tTo) , ( (4.252) w here U m i s the m ean velocity across the duct, Re i s the Reynolds n umber b ased on Um a nd W, i .e., is equal to UmW /v, To is t he t emperature o n t he inlet prane, and Po is again some reference pressure. I n terms o f t hese dimensionless variables, the governing equations become: av au ay + az Pr = 0 1 (4.253) 1 (U az au U + v ay = au) = - Pr d Z dP + (da2U) y2 (4.254) (4.255) ao + v ao az ay = 0, a2 + a2 0 0 ay2 ay2 v = 0, I n ,terms o f t hese dimensionless variables, t he b oundary conditions are: I, At and: au y = 0:. ay ao ay 0, =0 (4.256) At Y = 0.5: u= 0, v = 0 =1 (4.257) I n o rder to obtain t he solutions to t hese e quations, the conditions o n t he inlet plane h ave to b e k nown. I t w ill again b e a ssumed t hat t he velocity a nd t emperature d istributions a re u niform on the inl~t p lane a nd t hat t he losses i n t he i nlet a re negligible, i.e., that: ' A tz = 0: U = um, V = 0, T = To, Pi + pu~/2 = Po (4.258) 214 Introduction to Convective Heat Transfer Analysis the uniform velocity on the inlet plane being, as with pipe flow, equal to Urn. In terms of the dimensionless variables introduced above, these inlet conditions are: A tZ = 0: U = 1, v = 0, () = 0, P = - 0.5 (4.259) First consider the momentum equation. Using the same nodal points as those used in dealing with flow in a pipe, the finite-difference form o f this equation is: ~U. . ( Ui,j - Ui-l,j) ~. . ( Ui-l,j+l - Ui-l,j-l) P r I -I.] AZ + P r V I-I.] 2 AY 1 (Pi - Pi-d - Pr AZ ( Ui-l,j+l + + U i-l,j-l - 2Ui- I ,j) A y2 (4.260) I.e.: U-I ,]. = U· - I ,]. 1 p. - U I .• i -l,j + A]· (4.261) where: A. ] = P i- l _ V i-l,j ( Ui-l,j+l - Ui-l,j-l ) AZ U i-l,j Ui-t:.J 2 AY (Ui-l,j+l + Pr [ + U i-l,j-l A y2 - 2Ui -l,j)] ! U. A:Z I -I,] . (4.262) A j is therefore a known quantity at any stage o f the calculatio~ because the values of the variables on the i - I line are known. Next consider the continuity equation which can be written as: - av ay =- au az (4.263) This equation, as with pipe flow, is applied to the nodal points shown in Fig. 4.28. The equation is actually applied to the point p , shown in Fig. 4.28 which lies halfway between nodal points i, j and i, j - 1. T he value o f aUlaz at point p is, as before, taken as the average o f the values o f aUlaz at these two points. Hence, the continuity equation gives: V i,j ~;i,j-I =_ ~ [ Ui,j ~~i-I,j + U i,j-I ~~i-1,j-1 ] i.e.: Vi,j - Vi,j-I = - 2AZ [ Ui,j "'AY - Ui-I,j + U i,j-I - Ui-I,j-I] (4.264) Substituting Eq. (4.261) into this equation then gives: V i,j - Vi,j-I = - 2AAY [ Z L.l. &~i I -I,] . + Aj - U,Pi, 1 -1,]-1 + A j-l] (4.265) CHAP'fER 4: Internal Laminar Flows 21:5 -- which can be written as: I.e.: V· ,). - V· ,)- 1 = - B· p . - C· I 1. ) 1 ) ". where: B. = dY [ 1 + ) 2 d Z U i-l,j and: 1 U i-l,j-l ] (4.268) (4.269) It will again be noted that since at any stage o f the solution the conditions on the ( i - 1) line are known, the coefficients A j , A j-l, B j, a nd C j are known quantities. Now, on the center line, i.e., at j = 1, V i,j is O. Hence, i f Eq. (4.267) is applied sequentially outward from point j = 2 to the point j = N which lies on the wall, the following is obtained: V '3 - V'2 = - B3 P 1, - C3 ~ ~ V '4 - V'3 = - B4 P 1 - C4 , ~ ~ V'5 - V'4 = ' - B5 P 1' - C5 ~ ~ Vi,N - Vj,N-l = - BNPj - CN Adding this set o f equations together then gives: N N Vj,N = " :"PjLB k - LCk k=2 k=2 (4.270) But the boundary conditions give Vj,N = 0 so Eq. (4.270) gives: p. _ I. - 2.:f=2 Ck 2.:f=2 Bk (4.271) This allows the dimensionless pressure, P j, to be determined. Once this has been found, Eq. (4.261) can b e u sed to find the U i,j values and Eq. (4.267) applied sequentially outward from j = 2 as discussed above allows the V j,j values to b e found. The boundary conditions give U j,N = 0 a nd U i,l = U i,2. Lastly, the energy equation, i.e., Eq. (4.255), is used to determine the distribution of f Ji,j' T he finite-difference form o f Eq. (4.255) is obtained using the same nodal 216 Introduction to Convective Heat Transfer Analysis points as used i n dealing with the momentum equation and is: U. . ( (h,j - Oi-l,j) l -l,J 6.Z + V. I -I,J . ( Oi-l,j+l - (Ji-I,j-I) 26.Y ( Oi-l,j+l + O i-l,j-l 6.y2 - 20 i -l,j) (4.272) which gives: O . . = O. .l,J , -l,J Vi-I,j ( (Ji-l,j+I - Oi-l,j-I U i-I,j 2 6.Y )6.Z (4.273) + ( Oi-l,j+l + O i-l,j-l) - 20 i 6.y2 I ,j This equation allows the Oi,j values to be found. T he boundary conditions give = 0 i,2 and either 0 i ,N = 1 o r 0 i ,N = 0 i,N - 1 + 6.T d epending on whether the wall temperature o r t he wall heat flux is specified. Onc~.these values are determined, the heat transfer rate c an b e found. T he local heat transfer rate a t t he wall at any value o f Z is given b y u sing Fourier's law as: (J i, 1 qw:::!· + k- ' k! aTI ay y =WI2 (4.274) In terms o f t he dimensionless variables introduced a bove"Eq. (4.210) becomes: qw k (Tw - To) = a y Y=0.5 w aOI· (4.275) T he left-hand side o f this equation is the local Nusselt number, Nuw; i.e., Eq. (4.275) gives: Nuw = a y Y=0.5 aOI (4.276) T he finite-difference approximation to this equation is: Nuw = O 'N - 0'N-'-1 " 6.y" (4.277) T he m ean h eat t ransfer rate u p to the value o f Z being c onsidered can be obtained by irltegration o f the local wall heat transfer rate or b y noting that the total heat transfer rate u p to the point considered will be equal to the increase in the enthalpy flux between the inlet a nd t he z v alue considered, i.e., that: .. qwz = . Jo ( W/2 W p ucpT d y - pu m cpT 0 2 (4.278) i.e., rewriting t his e quation i n t erms o f t he dimensionless variables, and recallina th~ . . W um 2: f= Jo (WI2 u dy C HAPTER 4: Internal Laminar Flows 217 gIves: qwzRe Pr (0.5 T w-To = pCpumJo U 8dY Le.: qw W k(Tw - To) = 1 (0.5 Z U (}dY Jo (4.279) The left-hand side o f this equation is the mean Nusselt number up to dimensionless distance Z from the inlet considered. Hence, this equation can be written as: 1 rO. 5 N uw =Z Jo U 8dY (4.280) The value o f the integral can be determined by numerical integration. The Nusselt numbers introduced above were based on the temperature difference Tw - To. The Nusselt numbers based on the temperature difference Tw - Tm where Tm is the mean temperature at any value o f Z can be deduced by noting that the definition o f T m is such that considering unit width o f the channel: /~! 2qwz = mCp(Tm - To) the left-hand side being the rate at which heat is transferred to the fluid between the inlet and a point distance, z, from the inlet. Hence, since mis given tiy: m= it follows that: pumW RePr(Tm - To) I.e.: = 2qwzl k Tm-'-To Tw - To = 2 NuwZ Le.: T w-Tm - 1- 2NuwZ Tw - To This c an b e used to find the local Nusselt number based on Tm be given by: - To which will (4.281) . Tw - To Nuw Nuwm = Nuw'Jdw - Tm - 1 - 2NuwZ -The calculation procedure based on t he above equations is the same as that used for pipe flow and a computer program, DUCTSYM, written in FORTRAN, b ased on this procedure is available as discussed i n t he Preface., A s mentioned in the discussion o f p ipe flo,¥, because an explicit finite-difference procedure is being u sed to solve the m omentum and energy equations, the solution can become unstabJe~ i.e., as the solution proceeds i t can diverge increasingly from the actual solution as indicated i n F ig. 4 .29. To determine the conditions under 21S' Introduction to Convective Heat Transfer Analysis which such instability will occur, consider the following simplified fonn o f the momentum equation: Uau pr(a y2 U) az a = 2 (4.282) The finite-:difference f onn o f this equation is: U. . ( Ui. j - Ui-l,j) = I -I.} az p r. ( Ui-I,j+I + U i-l,j-I ay2 - 2Uj _l,j) . which gives: U . . = U. I ,} . . ( Uj-l,j+l I -I.} + + U i-l,j-l - : 2 Uj·- I ,j) a y2 . .. (praz) U i-l.j (4.283) Consider the case where: U t-l,j+l = U i-l,j + a;/ . U i-l,j-l·:;;;; U i-l,j +a I n this case, Eq.(4.283) gives: . U i,j = U i-l,j + . . 2a ( Pr"AZ ). ay2iI_ l.}. . ... I (4.284) From this equation it folIowsthat i f .(. P rllZ ) . Ily2Ui - 1,J > 0.5 then: U i,j (4.285) > U i-l,j +a (4.286) which is physically impossible because i t implies !Qat the presence o f a velocity excess o f II on e ach side o f point i, j incre8$esthe velocity a t point i, j b y IIlO1:e than l l. This is shown iri F i g . 4 . 3 3 . ' " . .. .. Prl::i.Z ot:' 2 > .J: I::i.Y U I-I .} . }+1 . ...- -... j+l.,.-----~ j . ....-~~-+- j--I---+-~ j-l"'--~J-.. j -l ___--~ j i -I , i -lLine iLine F IGURE 4.33 Induc~ ch8nge i n velocity. " CHAPTER 4: Internal Laminar Flows 219 10~----~~----~-------'------' 1 L -_ _- --'-_ _ _. .L.-_ _- --1._ _ _- -' 0.0001 0.001 0.Q1 z /W Re Pr 0.1 1 F IGURE 4.34 Nusselt number variations for developing plane duct flow for P r = 0.7. Hence, for a physically meaningful solution to b e obtained, it is necessary that: !l.Z < 0.5 l ly2Ui _l,j (4.287) Pr I f !l.Z is greater than this, a physi~ally meaningless, unstable solution will b e obtained. Now, the lowest nodal point velocity will b e a t t he nodal point adjacent to the wall, i.e., at j = N - 1, since Eq. (4.287) does not apply at the wall itself. Hence, for stability: (4.288) Because this criterion )Vas obtained using an approximate analysis, it is usual to assume that for stability: IlZ < Klly2Ui-1.N-l Pr (4.289) where K is less than 0.5. This stability criterion is incorporated into the above program. Typical variations o f Nuw a nd lVuwm w ith Z g iven by the program are shown i~ Pig. 4.34. SOLUTIONS T O T HE F ULL NAVIER~STOKES EQUATIONS Approximate solutions to the full governing equations h ave been discussed i n the above sections. T he approximations o n w hich these solutions are based are often not \' I 4.9 220 Introduction to Convective Heat Transfer Analysis applicable. For example, the parabolic forms of the full governing equations which are the basis of the analyses given above do not apply i f there are significant areas of reversed flow or i f the Reynolds number is low. When the parabolic flow assumptions cannot be applied, it is necessary to solve the full governing equations. In almost all cases it is necessary to do this using numerical methods [16],[17],[18],[19]. However, since the full equations are a set of nonlinear, simultaneous partial differential equations, this usually requires a significant computational effort. Commercial computer packages based on finite element methods or on various types o f finite-difference methods are available for obtaining such solutions. These packages usually contain preprocessing and postprocessing modules that make it easy to set up the required nodal system for complex geometrical situations and to display the calculated results in a convenient form. 4.10 CONCLUDING REMARKS This chapter has been concerned with/~e analysis o f l aminar flows in ducts with various cross-sectional shapes. I f the flow is far from the inlet to the'duct or from anything else causing aisturbance i n the flow, a fully developed sJate is reached ~ in many situations, the basic characteristics of the flow in this state not changing with distance along the duct. I f the diffusion of heat down the duct " be neglected, can which is true in most practical situations, it was shown that i n such fully developed flows, the Nusselt number based on the difference between the local wall and bulk mean temperatures is constant. Values o f the Nusselt number for fully developed flow in ducts o f various cross-sectional shape were discussed. In some practical situations, the velocity field becomes fully developed before changes to the temperature field occur. The analysis o f such thermally developing flows was discussed, attention being given to flow i n a circular pipe and between parallel plates, i.e., in a plane duct. I n most real situations, the velocity and temperature fields develop simultaneously. The numerical analysis o f such developing flows was discussed, attention again being restricted to flow i n a pipe and a plane duct. T he analysis was based on the parabolized form of the governing equations. Although many internal flows that are of practical importance can be analyzed to an a dtquate degree o f accuracy using the parabolized form o f the governing equations, there are many situations i n which the full governing equations must be solved. " PROBLEMS 4.1. Consider fully developed laminar flow fluid through a circular pipe with a uniform wa\l heat flux. I f h eat i s generated uniformly\in the fluid, perhaps as the result o f a chemica). reaction, at a rate o f q p er u nit volumJ, determine the value o f the Nusselt number .based on the difference between the wan. temperature and the mean fluid temperatu~~ i n the pipe. ..! . ' C HAPTER 4: Internal Laminar Flows 221 4.2. An organic fluid is to be heated from a n initial temperature o f 10°C to an exit temperature o f SO°C b y passing it through a heated pipe with a diameter of 12 m m a nd a length o f 2 m. The pipe is heated by wrapping an electric resistance heating element uniformly about its surface. The mass flow rate o f the fluid is 0.1 kg/so The properties o f the organic fluid can be treated as constant and the following values can be assumed: P r = 10, p = 800 kg/m 3 , k = 0.12 W / m K, and fL = 0.008 kg/ms. By assuming that the flow is fully developed, find and plot the variations of the pipe surface temperature and the mean fluid temperature with distance along the pipe from the inlet. 4.3. Consider fully developed flow in a circular tube with a uniform heat transfer rate a t the wall. I f the mean fluid velocity in the tube is 1 mis, find the value o f the heat transfer coefficient for the following fluids: (a) air at a mean temperature o f 80°C a nd at standard atmospheric pressure flowing through a 2.S-cmdiameter tube (b) air at a mean temperature o f 80°C a nd at standard atmospheric pressure flowing through a 5.0-cm diameter tube (c) air at a mean temperature o f 80°C a nd at a pressure o f 1 M Pa flowing through a O.S-cm diameter tube (d) hydrogen at a mean temperature o f 8 0°C a nd at a pressure of twice standard atmo, spheric pressure flowing through a 2'~Fm diameter tube (e) water at a mean temperature o f 4 0°C flowing through a 2-rnm diameter tube ( f) liquid sodium at a nTean t emperature o f 80°C flowing through a I -nun d iameter tube 4.4. Water at a mean bulk temperature o f 3 5°C flows through a pipe with' 'a d iameter o f 3 m m whose wall is maintained at a uniform temperature o f SsoC. T he tube length is 2 m. I f the mean velocity o f the water i n t he pipe is O.OS mis, find the heat-transfer rate to the water assuming fully developed flow. 4.5. I n a laminar flow i n a pipe with a diameter o f 2 c m, the velocity and temperature profiles are given by: u = 0.1[1 - (r/0.Ol)2]mls and: for radius, r, i n meters. Determine the b ulk temperature. 4.6. Oil is heated b y passing it through a pipe with an inside diameter o f 1 c m and a length o f 1.5 m. A n electrical heating element is wrapped around the surface o f this pipe and provides a uniform heat transfer rate a t t he p ipe wall. The oil is to be heated from a temperature o f lOoC to a temperature o f 4 0°C a nd the mean velocity o f the o il i n the pipe is 1.3 m ls. F ind t he heat transfer rate required at the wall and, assuming that the flow is laminar and fully developed, i.e., ignoring the entrance region, show how the wall and mean oil temperatures vary along the pipe. Assume that the oil has a density o f 8 90 kg/m3, a specific heat o f 1.9 kJ/kgOC, a coefficient o f viscosity o f 0.1 N-s/m2, a thermal conductivity o f 0.15 W /m K, a nd a P randtl n umber o f 500. 4.'. Oil is heated b y p assing it at a mean velocity o f 1 m ls through a pipe. A uniform heat flux exists at the w all o f this pipe. A t t he e xit e nd o f the pipe the wall has a temperature o f 5 0°C a nd the oil a mean temperature o f 30°C. F ind the lowest temperature 222 Introduction to Convective Heat Transfer Analysis and highest velocity in the oil flow at this exit end o f the pipe. Assume that at the exit, the flow is laminar and fully developed. 4.8. Derive an expression for the Nusselt number in fully developed laminar slug flow through an annulus when the inner and outer surfaces o f the annulus have diameters o f Di and Do respectively and when there is a uniform heat flux applied at the inner surface and when the outer surface is adiabatic. 4.9. In some situations it is possible to find the heat transfer rate with adequate accuracy by assuming that the velocity is constant across the duct, i.e., to assume that so-called "slug flow" exists. Find the temperature distribution and the Nusselt number in slug flow in a plane duct when the thermal field is fully developed and when there is a uniform wall heat flux. 4.10. Air flows at a mean velocity o f 1 mls between flat plates that can be assumed to be at a uniform temperature o f SO°e. The plates have a length o f 15 e m in the direction o f air flow. The gap between the plates is 4 mm and the initial air temperature is 2 0°e. Assuming that the flow between the plates is laminar and fully developed, estimate the mean temperature o f the air leaving the system. 4.11. Water is heated by pa~i!1g i t through an array of parallel, wide heated plates. The plates thus form a series··of parallel plane channels. The distance between the p lates is 1 mm and the mean velocity i n the channels is 1 mls. The plates are electrically heated, the two sides o f e ach plate together transferring heat to the water flow~t a rate of 1600 W1m2 • The water properties can be assumed to be constant and they can b e evaluated at a temperature o f 50°C. I f the flow is assumed to be fully developed, find the rate o f increase o f mean water temperature with distance along the channels, 4.12. Consider fully developed flow in a plane duct in which uniform heat fluxes q wl and qw2 are applied at the two walls, Derive expressions for the temperature distribution in the duct and the Nusselt number, 4.13. The cooling channels in a metal block which is held at a uniform temperature of 50°C have a rectangular cross-sectional shape with a width o f 6 m m and a height o f 2 mm. Air at an initial temperature o f 10°C flows through these channels at a mean velocity o f 1.5 mis, The block has a length o f 22 c m in the flow direction. Find the heat transfer rate to the air in one channel. Assume fully developed laminar flow. 4.14. Water flows in a rectangular 5 m m X 10 m m duct with a mean bulk temperature of 2 0°e. I f the duct wall is k ept at a uniform temperature o f 40°C and i f fully developed laminar flow is assumed to exist'ofind the heat transfer rate per unit length o f the duct. 4.t5. Consider a series o f r ectangular channels o f width W, and height H, all with the same cross-sectional area whose walls are kept at a uniform temperature. The flow in these ducts can be assumed to be fully developed and laminar. (i) I f it is assumed that t he hydraulic diameter concept c an b e used, i.e., that th~ Nusselt number for fully developed flow in a pipe applies provided the hydraulic diameter is used as the length scale, find how the quantity Q /2(H + W )(Tw - T m) varies with WI H for values o f W IH between d5 and 3. Q is the heat transfer rate per unit length o f the duct. CHAPTER 4: Internal Laminar Flows 223 (ii) Determine the variation o f the quantity Q/2(H + W)(Tw - T m) with W IH for values o f WIH between O.S and 3 using the actual Nusselt numbers for flow in a rectangular duct for various values o f WI H. 4.16. Consider laminar fully developed flow through either a circular pipe or a rectangular duct whose cross-section has a length/width ratio of 8. Both o f these ducts have the same cross-sectional area and the average velocity in them is the same. Determine the ratio of the (heat-transfer coefficient X surface area) product for these two flow situations. 4.17. Consider air flow in a plane channel when there is uniform heat flux at one wall and when the other wall is adiabatic. I f the inlet air temperature is known, find how the temperature o f the heated wall at the exit end o f the duct varies with the distance, W, between the two walls. Assume that the mean air velocity is the same in all cases and that the flow is fully developed. 4.18. Find the variation of the local heat transfer rate with axial distance for the thermally developing flow o f oil through a pipe with an inside diameter o f 1 c m and a heated length o f 10 cm. The wall o f the pipe is kept at a temperature of SO°C and the oil enters the heated section o f the pipe at a temperature o f 10°c'.,The mean velocity of the oil in the pipe is 1.3 mls. Assume that the d~ has a density of 890 kg/m3, a specific heat of 1.9 kJlkgOC, a coef~i.ent o f viscosity' o f 0.1 kg/ms, and a thermal conductivity o f 0.15 W /m K. 4.19. Discuss how the computer program for flow in the thermal entrance f rgion to a plane duct must be modified in order to apply to the case where slug flow exists, i.e., for the situation i n which the velocity can be assumed to b e constant across the pipe and equal, o f course, to the mean velocity in the pipe. 4.20. Discuss the modifications required to the computer program given for flow i n the thermal entrance region o f a plane duct to deal with the case where there is a uniform heat flux at one wall o f the duct and where there is a uniform specified temperature at the other wall o f the duct. 4.21. A ir a t 20 kPa and SoC enters a l.5-cm diameter tube at a uniform velocity o f 1.Smls. The tube walls are maintained at a uniform temperature of 4S0C. Estimate the distance from. the entrance at which the flow becomes fully develbped. 4.22. Water \at lSoC enters a plane duct (Le., effectively flows between two large parallel plates} at a uniform velocity o f I.S mls. T he walls o f the duct are separated by a distance of 1 mm and are kept at a uniform temperature o f 25°C. Estimate the distance from the entrance at which the flow can.be assumed to be fully de,yeloped. 4.23. Engine oil enters a S.O-mm diameter tube at a temperature o f 120°C a t a uniform velocity whose value is such that the Reynolds number is 1000. T he tube wall is kept at a uniform temperature o f 60°C, Calculate the mean exit ,oil temperatures for the tube lengths o f l a, 20, and 5 0 cm. '. f.24. . Engine oil at a temperature o f 20°C enters a ~-mm diameter tube at a uniform velocity o f 1.2 mls. T he tube is 1.0 m long. The tube wall is maintained at a uniform temperature o f 60°C. Calculate the mean exit oil temperature. The program for developing flow i n a p lane channel discussed i n this chapter assumed t hat both walls w ere a t the same uniform temperature. Modify the program to allow 425. 224 Introduction to Convective Heat Transfer Analysis for the possibility that the upper and lower walls o f the duct are at different uniform temperatures. Use this modified program to find the variation o f the heat transfer rate with distance along the upper and lower walls for the case of air flowing at a mean velocity o f 0.6 mls through a duct in which the upper and lower surfaces are 1 em apart. The upper surface is kept at a temperature o f 40°C and the lower surface is kept at a temperature o f 30°C. The air enters the channel at a temperature of 15°C. Evaluate the properties o f the air at a temperature o f 20°C. 4.26. Discuss the modifications to the program for developing flow in a pipe that are necessary to allow it to calculate developing flow in an annulus when the inner and outer surfaces o f the annulus have diameters o f Di and Do, respectively, and when both the inner and outer surfaces are kept at the same uniform temperature. 4.27. Oil enters a 10-mm diameter pipe at a temperature o f 20°C with a uniform velocity o f 1 mls. The wall o f the pipe is kept at a uniform temperature of 40°C. I f the pipe is 7 0 cm long, plot the variation of the temperature and velocity in the oil at the exit o f the pipe. Assume t hat for the oil p = 890 kg/m3, JL = 0.0075 kg/ms, k = 0.15 W /m K, a ndPr = 900. 4.28. Air flows at a mean velocity o f 1.5 JR{s between flat plates that can be assumed to be at a uniform temperature o f 40°C. The plates have a length of 20 cm'in the direction o f air flow and are ~de i n the other direction and the gap between th_e plates is 4 mm. The initial air temperature is 20°C and the flow can b e assumed to have a uniform velocity at the inlet to the system. Can the flow between the plat~s be assumed to be fully developed? Evaluate the air properties at a temperature of 3<fC. 4.29. In a heat exchanger, air flows through a pipe o f length, e. The air enters the pipe at a temperature, T;. The heat flux at the wall o f this pipe increases linearly from zero at the inlet o f the pipe to a value o f qw at the end o f the pipe. The velocity in the pipe is such that C/R e P rD = 0.07 where D is the diameter o f the pipe and Re is the Reynolds number. Determine how the Nussel! number based on the local wall heat transfer rate and on the difference between the local wall temperature and the inlet temperature varies with the dimensionless distance, Z , along the pipe. 4.30. Air at a temperature o f 20°C enters a 25-mm diameter pipe with a uniform velocity o f 0.7 mls. The pipe has a length o f30 c m and has a uniform wall temperature o f 50°C. Determine the air outlet temperature (i) assuming fully developed flow throughout the pipe and (ii9 accounting for entrance region effects. 4.31. Water flows through a pipy o f length, e. T he water enters the pipe with a uniform velocity and at a uniform temper~ture o f T i • T he heat flux is uniform around the pipe but , varies with distance z along the pipe, this variation being described by: qw = qwo [1 + A sin(?TVe)] A being a constant.The velocity i n the pipe is such that e /RePrD ::: 0.06 where D i s the diameter o f the pipe and R e is the Reynolds number. Numerically determine how the Nusselt number based on the local wall ~eat transfer rate and on the difference betweelf. the local wall temperature and the inlet ~mperature varies with the dimensionless distance, Z , along the pipe. 4.32. I n a h eat exchanger, air essentially flows through a plane channel of length, e. Thtt air enters the-channel at a temperature, T i • T he temperature o f the walls o f this duct , tAL C HAPTER 4: Internal Laminar Flows 225 increases linearly from Ti to T Rover the first half of the duct and then remains constant at T R • The velocity in the duct is such that f lRePrW = 0.06 where W is the width of the duct and R e is the Reynolds number. Determine how the Nusselt number based on the local wall heat transfer rate and on the difference between the local wall temperature and the inlet temperature varies with the dimensionless distance, Z, along the duct. 4.33. Consider constant-property, fully developed laminar flow between two large parallel plates, i.e., in a wide plane duct. One plate is adiabatic and the other is isothermal and the velocity is high enough for viscous dissipation effects to be significant. Determine the temperature distribution in the flow. 4.34. Consider laminar flow o f a high-viscosity oil in a circular tube with a uniform wall temperature. I f viscous dissipation effects are significant, determine the temperature profile far from the inlet, i.e., in fully developed flow. Assume constant fluid properties. 4.35. A gas flows between large parallel porous plates. The same gas is blown through one wall and exhausted through the other i n such a way that the same normal velocity component, vs , exists at both walls. The plates are k eptat uniform but different temperatures. Assuming fully developed, ffip.stant fluid-property flow, find equations for the velocity and temperature profiles. REFERENCES 1. Schlichting, H., Boundary L ayer Theory, 7th ed., McGraw-Hill, New York, 1979. 2. Shah, R K: and London, A.L., L aminar Forced Convection in Ducts, Academic Press, New York, 1978. ' 3. Siegel, R,Sparrow, E.M., and Hallman, T.M., "Steady L~inar Heat Transfer in a Circular Tube with Prescribed Wall Heat Flux", Appl. Sci. Res., Sect. A, Vol. 7, p. 386, 1958. 4. Worsoe-Schmidt, P.M., "Heat Transfer and Friction for Laminar Flow of Helium and . Carbon Dioxide in a Circular Tube at High Heating Rate", Int. J. Heat-Mass Transfer, V ol.9,pp. 1291-1295, 1966. S. Yang, K.T., "Laminar Forced Convection o f Liquids in T1,lbes with Variable Viscosity", J. H eat Transfer, Vol. 84, pp. 353-362, 1962. 6. Swearingen, T.W. and McEligot, D.M., "Internal Laminar Heat Transfer with Gas Property Vatiation", Trans. ASME, Ser. C, J. H eat Transfer, Vol. 93, pp. 4 32-440, 1971. 7. Clark, S.H. and Kays, W.M., "Laminar Flow Forced Convection in Rectangular Tubes", Trans. ASME, Vol. 75, p. 859, 1953. 8. Shah, R K., "Laminar Flow Friction and Forced Convection Heat Transfer in DUcts o f Arbitrary Geometry", Int. J. H eat Mass Transfer, Vol. 1 8,pp. 849-862, 1975. 9.. Shah, R K. and London, A.L., "Thermal Boundary Conditions and Some Solutions for Laminar Duct Flow Forced Convection", J. H eat Transfer, Vol. 9 6, pp. 159-165,1974. 10. Burmeister, L.C., "Convective Heat Transfer", 2nd ed., Wiley-Interscience, New York, 1993. .' 11. Sellars, 1 .R, Tribus, M., and Klein, 1.S., "Ifeat Transfer to Laminar Flow in a Round Tube or Flat C onduit-The Graetz Problem Extended", Trans. ASME, Vol. 78, p. 441, I 1956. \ Hornbeck, R W., " An All-Numerical Method'.for Heat Transfer in the Inlet o f a Tube", Am. Soc. Mech. Eng., p aper 65-WAIHT-36, 1965. i2. 226 Introduction to Convective Heat Transfer Analysis 13. Kays, W.M., "Numerical Solution for Laminar Flow Heat Transfer in Circular Tubes", Trans. ASME, Vol. 77, pp. 1265-1274, 1955. 14. Shah, R.K., " A Correlation for Laminar Hydrodynamic Entry Length Solutions for Circular and Non-Circular Ducts", J. Fluids Eng. Trans. ASME, Vol. 100, p. 177, 1978. 15. Sparrow, E.M., "Analysis o f L aminar Forced Convection Heat Transfer in the Entrance Region o f Flat Rectangular Ducts", N ACA T N 3331, 1955. 16. Gosman, A D., Pun, W.M., Runchal, A K., Spalding, D.B., and Wolfshtein, M., H eat a nd M ass Transfer in Recirculating Flows, Academic Press, New York, 1969. 17. Patankar, S. V., Numerical Heat Transfer a nd F luid Flow, Hemisphere Publ., Washington, D.C., 1980. 18. Oosthuizen, P.H., "Laminar Forced Convective Heat Trarisfer from Rectangular Blocks Mounted on Opposite Walls o f a Channel", Numerical Methods in Thennal Problems, Vol. VI, Part 1, Proc., 6th Int. Conf., Swansea, U.K., J uly 3 -7, pp. 451-461, 1989. 19. Janssen, L .AM. and Hoogendoom, C.J., " Laminar Convective Heat Transfer in Helically Coiled Tubes", Int. J. H eat Mass Transfer, Vol. 21, pp. 1197-1206,1978. /