effects of viscous dissipation on the heat transfer in a forced pipe flow. part 2: thermally...

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Energy Conversion and Management 46 (2005) 3091–3102

Effects of viscous dissipation on the heat transfer in a forcedpipe flow. Part 2: Thermally developing flow

Orhan Aydin *

Department of Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey

Received 2 May 2004; received in revised form 6 September 2004; accepted 14 March 2005

Available online 29 April 2005

Abstract

In this part of the study, consideration is given to thermally developing laminar forced convection in a

pipe including viscous dissipation. The axial heat conduction in the fluid is neglected. Two different thermal

boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature(CWT). Both the wall heating (the fluid is heated) case and the wall cooling (the fluid is cooled) case are

considered. The distributions for the developing temperature and local Nusselt number in the entrance

region are obtained. Results show that the temperature profiles and local Nusselt number are influenced

by the Brinkman number (Br) and the thermal boundary condition used for the wall. Significant viscous

dissipation effects have been observed for large Br.

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Keywords: Viscous dissipation; Thermally developing flow; Graetz problem; Brinkman number

1. Introduction

The familiar hydrodynamically developed but thermally developing laminar flow problem,which is also called the classical Graetz problem, has traditionally been solved by neglectingthe effect of viscous dissipation. Viscous dissipation changes the temperature distributions by

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Nomenclature

Br Brinkman number, Eq. (6)Brq modified Brinkman number, Eq. (16)cp specific heat at constant pressure (kJ/kgK)D diameter of pipe (m)Nu Nusselt numberk thermal conductivity (W/mK)Pr Prandtl numberqw wall heat flux (W/m2)r radial coordinate (m)r0 radius of pipe (m)R dimensionless radial coordinateRe Reynolds numberT temperature (K)u velocity (m/s)z axial direction (m)Z Graetz number

Greek symbolsa thermal diffusivity (m2/s)l dynamic viscosity (Pas)q density (kg/m3)m kinematic viscosity (m2/s)h dimensionless temperature, Eq. (7)hq dimensionless temperature modified, Eq. (13)

Subscripts

c centerlinem meanvd viscous dissipationw wall

3092 O. Aydin / Energy Conversion and Management 46 (2005) 3091–3102

playing a role like an energy source, which leads to affected heat transfer rates. The merit of theeffect of viscous dissipation depends on whether the pipe is being cooled or heated.

Many studies involving pipe flows in the existing literature have neglected the effect of viscousdissipation. In fact, the shear stresses can induce a considerable power generation. However, inthe existing convective heat transfer literature, this effect is usually regarded as important onlyin two cases: flow in capillary tubes and flow of very viscous fluids. The effects of viscousdissipation in laminar flows have not yet been deeply investigated. For liquids with high viscos-ity and low thermal conductivity, the disregard of viscous dissipation can cause appreciableerrors.

O. Aydin / Energy Conversion and Management 46 (2005) 3091–3102 3093

The work of Brinkman [2] appears to be the first theoretical work dealing with viscous dissipa-tion. The viscous heating effects for single phase, Newtonian fluids in a straight circular tube wereanalyzed. He obtained some temperature distributions in the thermal entrance for the conditionsof a zero wall temperature and an insulated wall. It was found that the effects were pronounced inthe close wall region.

Ou and Cheng [3] employed the separation of variables method to study the Graetz problemwith finite viscous dissipation. They obtained the solution in the form of a series whose eigenvaluesand eigenfunctions satisfy the Sturm–Liouville system. The solution technique follows the sameapproach as that applicable to the classical Graetz problem and, therefore, suffers from the sameweakness of poor convergence behavior near the entrance. They showed that the asymptotic Nus-selt number approaches 48/5, independent of the Brinkman number, and hm approaches �(5/6)Br.

Lin et al. [4] showed that the effect of viscous dissipation was very relevant in the fully devel-oped region if convective boundary conditions were considered. With these boundary conditionsand if viscous dissipation were taken into account, the fully developed value of the Nusselt num-ber was 48/5 for every value of the Biot number and of the other parameters. On the other hand, itis well known that, if a forced convection model with no viscous dissipation is employed, the fullydeveloped value of the Nusselt number for convective boundary conditions depends on the valueof the Biot number.

Basu and Roy [5] analyzed the Graetz problem by taking account of viscous dissipation butneglecting the effect of axial conduction. They showed that the effect of viscous dissipation couldnot be neglected when the wall temperature was uniform.

For the thermal condition, Liou and Wang [6], using uniform wall heat flux, Berardi et al. [7],using convection with an external isothermal fluid and Lawal and Mujumdar [8] and Dang [9],using uniform wall temperature studied the effect of viscous dissipation in the thermal entranceregion in a pipe.

The effect of viscous dissipation in the thermal entrance region of slug flow forced convection ina circular duct was studied by Barletta and Zanchini [10]. The temperature field and the local Nus-selt number were determined analytically for any prescribed axial distribution of wall heat fluxincluding uniform, linearly varying and exponentially varying heat fluxes.

Zanchini [11] studied the asymptotic behavior of the temperature field for laminar and hydro-dynamically developed forced convection of a power law fluid that flows in a circular duct withviscous dissipation taken into account. The asymptotic Nusselt number and the asymptotic tem-perature distribution were evaluated analytically in the cases of either uniform wall temperatureor convection with an external isothermal fluid.

Using a functional analysis method, Lahjomri [12] analytically studied thermally developinglaminar Hartman flow through a parallel plate channel with prescribed transversal uniform mag-netic field, including both viscous dissipation, Joule heating and axial heat conduction with uni-form heat flux. In a recent study, Nield et al. [13] investigated the thermal development of forcedconvection in a parallel plate channel filled by a saturated porous medium with the walls held atuniform temperature and with effects of axial conduction and viscous dissipation included.

This part is a continuation of the first part of this study, Part 1 [1], where the hydrodynamicallyand thermally fully developed flow case has been examined. In this part, an analysis is performedto study the influences of viscous dissipation on convective laminar heat transfer in the thermalentrance region of fully developed, Newtonian pipe flows.


2. Analysis

The flow is considered to be hydrodynamically fully developed but thermally developing. Thisproblem is traditionally termed the ‘‘Graetz’’ problem. Steady, laminar flow having constantproperties (i.e. the thermal conductivity and thermal diffusivity of the fluid are considered to beindependent of temperature) is considered. The axial heat conduction in the fluid and in the wallis assumed to be negligible.

The well known parabolic velocity profile (which is also called Hagen–Poiseuille�s velocity dis-tribution) for fully developed laminar pipe flow is given as follows:

uuc

¼ 1� rr0

� �2

ð1Þ

Since a fully developed velocity profile is assumed for the thermally developing flow, the energyequation including the viscous dissipation effect can be represented by

uoToz

¼ tPr

1

ro

orroTor

� �þ lqcp

ouor

� �2

ð2Þ

where q, cp, k and l are the density, specific heat, thermal conductivity and viscosity of the fluid.The second term on the right hand side is the viscous dissipation term.

Because of the axisymmetry at the center, the thermal boundary condition at r = 0 can be writ-ten as:

oTor

��r¼0

¼ 0 ð3Þ

Two kinds of thermal boundary condition at the wall are considered in this study, namely: con-stant wall heat flux (CHF) and constant wall temperature (CWT). They are treated separately inthe following:

2.1. CWT case

Introducing the following dimensionless variables

U ¼ uum

; h ¼ ðT w � T ÞðT w � T eÞ

; R ¼ rD; Z ¼ z=D

RePrð4Þ

where um is the mean velocity in the pipe and Re is the Reynolds number based on this meanvelocity and the diameter of the pipe, D, which is equal to 2r0. The dimensionless variable Z istermed the Graetz number. Then, Eq. (2) becomes

UohoZ

¼ 1

Ro

oRRohoR

� �� 64BrR2 ð5Þ

where Br is the Brinkman number, which is defined as:

Br ¼ lu2ckðT w � T eÞ

ð6Þ


where uc is the centerline velocity that is

uc ¼ 2um ð7Þ

For the CWT case, the wall temperature is kept isothermal in the entrance region, which ismathematically shown as:

For z > 0: T ¼ T w at r ¼ r0 ð8Þ

In dimensionless form, the thermal boundary conditions that will be applied in the solution ofthe energy equation are given as:

R ¼ 0:ohoR

¼ 0; R ¼ 1: h ¼ 0 ð9Þ

The local Nusselt number is obtained from

NuD ¼ ohoR

��R¼1

ð10Þ

It should be noted that the Nusselt number being used here is based on the difference betweenthe wall and the inlet temperatures, i.e. on Tw � Te and not on the difference between the wall andthe mean temperature. Now, the mean temperature, i.e. the bulk mean temperature is given by[14]

Tm ¼RquT dARqudA

ð11Þ

Rewriting this equation in terms of the dimensionless variables:

T w � Tm

T w � T e

¼ 4

Z 1

0

UhRdR ð12Þ

The Nusselt number based on the difference between the wall and the mean temperature is thengiven by:

NuDm ¼ NuDT w � T e

T w � Tm

¼ NuD

4R 1

0UhRdR

ð13Þ

2.2. CHF case

Now, we will consider the constant heat flux case, qw = c. In this case, the following dimension-less temperature is used:

h ¼ ðT � T eÞðqwD=kÞ

ð14Þ

With this definition, the energy equation can be written in dimensionless form as:

UohoZ

¼ 1

Ro

oRRohoR

� �� 64BrqR2 ð15Þ


where Brq is the modified Brinkman number, which is given as:

Brq ¼lu2cqwD

ð16Þ

The entrance conditions at the beginning of the thermally developing region in this case aredefined as:

Z ¼ 0 : h ¼ 0 ð17Þ
and the wall thermal boundary condition is
koTor

��r¼r0

¼ qw ð18Þ

where qw is positive when its direction is into the fluid (wall heating); otherwise it is negative (wallcooling). In dimensionless form, it can be written as:

ohoR

��R¼1

¼ 1 ð19Þ

For this case, the Nusselt number is given by:

NuD ¼ 1

hwð20Þ

and

NuDm ¼ 1

hw � hmð21Þ

For each case, the energy equation has been solved numerically using the difference method.The details of the solution procedure can be found in Ref. [14].

3. Results and discussion

In the absence of viscous dissipation, the solution is independent of whether there is wallheating or cooling. However, viscous dissipation always contributes to internal heating of thefluid, hence, the solution will differ according to the process taking place. As is well known, theBrinkman number accounts for the relevance of viscous dissipation. As stated earlier, two differ-ent thermal boundary conditions have been considered for the pipe wall: constant heat flux (CHF)and constant wall temperature (CWT). For each boundary condition, both wall heating and wallcooling cases are examined.

In order to understand the heat transfer processes taking place, temperature profiles at certainlocations are needed. For the CHF condition, Figs. 1a and b depict the temperature distributionsfor different Brinkman numbers at different axial locations. Because of the irreversible energy con-version originating from viscous dissipation, with an increase of Brq, the dimensionless tempera-ture distribution is significantly affected. It should be remembered that positive values of Brqcorrespond to the wall heating (heat is being supplied across the walls into the fluid) case

Fig. 1. Dimensionless temperature distributions in terms of Brq for CHF case: (a) wall heating, (b) wall cooling.

Fig. 2. Dimensionless temperature distributions in terms of Br for CWT case: (a) wall heating, (b) wall cooling.



(Tw > Tc), while the opposite is true for negative values of Brq. Fig. 1a reveals the dimensionlesstemperature distribution for the wall heating case. As seen, it is a decreasing function of Br. Forlower values of Brq (Brq = 0.01) the effect of viscous dissipation is found to be negligible. However,increasing values of Brq in the range of Brq > 0.1 severely increases the dimensionless temperature.This increase becomes much more intensified through the flow. For the wall cooling case, thedimensionless wall temperature is a decreasing function of Brq (Fig. 1b). This is due to the internalheating effect of the viscous dissipation on the fluid temperature profile. In fact, wall cooling isapplied to reduce the bulk temperature of the fluid, while the effect of the viscous dissipation isto increase the bulk temperature of the fluid. Therefore, the amount of viscous dissipation maychange the overall heat balance. As will be shown later, when Brq exceeds a certain limiting value,the heat generated internally by the viscous dissipation process will overcome the effect of wallcooling.

Fig. 3. Variation of NuD with Brq for CHF case: (a) wall heating, (b) wall cooling.

Fig. 4. Variation of NuDm with Brq for CHF case: (a) wall heating, (b) wall cooling.


For the CWT condition, Figs. 2a and b illustrate the effect of the Brinkman number on thedeveloping radial dimensionless temperature profiles at various axial positions in the thermalentrance region. The physical mechanisms occurring are very similar to those occurring for theCHF condition. Briefly, there is a dramatic difference between the effect of positive Br and theeffect of negative Br. The case Br > 0 corresponds to the incoming fluid being heated at the walls.The viscous dissipation produces a distribution of positive heat sources, and this reinforces theheating effect as the fluid moves downstream. The case Br < 0 corresponds to incoming fluid beingcooled at the walls, and this cooling at the walls is opposed by the heating due to viscous dissi-pation in the bulk of the fluid.

Of particular practical interest in this study are the effects of viscous dissipation on the heattransfer. For the CHF condition, Figs. 3a and b represent the downstream variation of NuD withBrinkman number for the wall heating and wall cooling cases, respectively. With the increasing


value of Brq, NuD decreases for the wall heating case (Fig. 3a), while it increases for the wall cool-ing case (Fig. 3b). The behavior of NuDm in the downstream with increasing Br suggests similarvariations to that of NuD (Figs. 4a and b). For the CWT case, Figs. 5 and 6 represent the down-stream variation of NuD and NuDm, respectively. For either the wall heating case (Fig. 5a) or thewall cooling case (Fig. 5b), its thermally fully developed value is set by the increasing Brinkmannumber. These values are consistent with those obtained in Part 1 [1]. In the downstream variationof NuDm, very interesting scenarios are observed with the increasing value of Br. For Br = 0, asexpected, NuDm is 3.658. For Br = 0.01, NuDm decreases up to a critical point, where the internallygenerated heat due to viscous dissipation (qvd) balances the heat supplied by the wall (qw). Afterthis critical point qvd suppresses qw. Similar behaviors are observed for Br = 0.1 and 0.5. Withincreasing Br, this critical point is reached at an earlier point from the entrance.

Fig. 5. Variation of NuD with Br for CWT case: (a) wall heating, (b) wall cooling.

Fig. 6. Variation of NuDm with Br for CHF case: (a) wall heating, (b) wall cooling.



4. Conclusions

Hydrodynamically developed, but thermally developing forced convection in a pipe (the Graetzproblem) has been studied by taking the effect of viscous dissipation into account. The axial con-duction in the fluid is neglected. Two types of wall thermal boundary conditions have been con-sidered, namely: constant heat flux (CHF) and constant wall temperature (CWT). Both the wallheating and wall cooling cases are examined. The behavior of the temperature distribution hasbeen compared with that obtained by neglecting the viscous dissipation. It is obtained that theviscous dissipation causes an increase of temperature in the axial direction that does not occurwhen viscous dissipation is neglected. Depending on the heating or cooling situations at the wall,the Brinkman number, Br (for the CWT case) or the modified Brinkman number, Brq (for theCHF case) is shown to play a significant role on the developing Nusselt number.

References

[1] Aydın O. Effects of viscous dissipation on the heat transfer in a forced pipe flow. Part 1. Both hydrodynamically

and thermally fully developed flow. Energy Conversion and Management 2005;46:757–69.

[2] Brinkman HC. Heat effects in capillary flow I. Appl Sci Res 1951;A2:120–4.

[3] Ou WJ, Cheng KC. Viscous dissipation effects in the entrance region: Heat transfer in pipes with uniform heat flux.

Appl Sci Res 1973;28:289–301.

[4] Lin TF, Hawks KH, Leidenfrost W. Analysis of viscous dissipation effect on the thermal entrance heat transfer in

laminar pipe flows with convective boundary conditions. Warme und Stoffubertragung 1983;17:97–105.

[5] Basu T, Roy DN. Laminar heat transfer in a tube with viscous dissipation. Int J Heat Mass Transfer

1985;28:699–701.

[6] Liou CT, Wang FS. Solutions to the extended Graetz problem for a power-law fluid with viscous dissipation and

different entrance boundary conditions. Numer Heat Transfer A 1990;17:91–108.

[7] Berardi PG, Cuccurullo G, Acierno D, Russo P. Viscous dissipation in duct flow of molten polymers. Proceedings

of Eurotherm Seminar 46, Pisa, Italy, July, 1995. p. 39–43.

[8] Lawal A, Mujumdar AS. The effects of viscous dissipation on heat transfer to power law fluids in arbitrary cross-

sectional ducts. Warme Stoffubertragung 1992;27:437–46.

[9] Dang VD. Heat transfer of a power-law fluid at low Peclet number flow. J. Heat Transfer 1983;105:542–9.

[10] Barletta A, Zanchini E. Forced convection in the thermal entrance region of a circular duct with slug flow and

viscous dissipation. Int J Heat Mass Transfer 1997;40(5):1181–90.

[11] Zanchini E. Effect of viscous dissipation on the asymptotic behavior of laminar forced convection in circular tubes.

Int J Heat Mass Transfer 1997;40:169–78.

[12] Lahjomri J, Zniber K, Oubarra A, Alemany A. Heat transfer by laminar Hartmann�s flow in the thermal entrance

region with uniform wall heat flux: The Graetz problem extended. Energy Convers Manage 2003;44:11–34.

[13] Nield DA, Kuznetzov AV, Xiong M. Thermally developing forced convection in porous medium: Parallel plate

channel with walls at uniform temperature, with axial conduction and viscous dissipation effects. Int J Heat Mass

Transfer 2003;46:643–51.

[14] Oosthuizen PH, Naylor D. Introduction to convective heat transfer analysis. New York: McGraw-Hill; 1999.

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