International Communications in Heat and Mass Transfer 37 (2010) 535–539

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International Communications in Heat and Mass Transfer

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On the constant wall temperature boundary condition in internal convection heattransfer studies including viscous dissipation☆

Orhan Aydın ⁎, Mete AvcıDepartment of Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey

☆ Communicated by E. Hahne and K. Spindler.⁎ Corresponding author.

E-mail address: oaydin@ktu.edu.tr (O. Aydın).

0735-1933/$ – see front matter © 2010 Elsevier Ltd. Aldoi:10.1016/j.icheatmasstransfer.2009.11.016

a b s t r a c t

a r t i c l e i n f o

Available online 10 March 2010

Keywords:Viscous dissipationConstant wall temperatureAdiabatic bulk temperatureGraetz problem

In this study, viscous heating effect on convective flow in an unheated adiabatic duct is studied analyticallyand numerically. Two different geometries are considered: circular duct and plane duct between two parallelplates. Two new parameters are defined for internal convection studies: adiabatic wall temperature andadiabatic bulk temperature. Variations of these two parameters with varying intensity of viscous dissipationeffect are determined. In view of the results obtained, usage of the constant wall temperature thermalboundary condition when viscous dissipation is included is discussed and questioned.

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© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Viscous dissipation changes the temperature distributions byplaying a role like an energy source, which affects heat transferrates. Research considering viscous dissipation have increasedrecently because of its importance in microscale flow and heattransfer phenomena as well as flow of fluids with high viscosity andlow thermal conductivity.

For thermally developing flow in internal flows, when the thermalboundary condition of the constant wall temperature is considered atwall, it is shown that an asymptotic value for the thermally fullydeveloped Nusselt number Nu is obtained. When the effect of theviscous dissipation is included (Br≠0) in the analysis for the sameproblem, a different asymptotic value for Nu is obtained. It is disclosedthat no matter what the value of the Brinkman number (adimensionless number characterizing the degree of the viscousdissipation effect) is, i.e. no matter how high the degree of theviscous dissipation, the same asymptotic value for Nu is observed.Interestingly, either for low values (e.g. 10−6) or for very high values(e.g. 106) of the Brinkman number, the same value in the axialdirection is asymptotically reached. This is a common observation inthe open literature (e.g. Krishnan et al. [1], Lin et al. [2], Basu and Roy[3], Valko [4], Aydın [5], Aydın and Avcı [6] for Newtonian fluids;Coelho et al. [7,8] Oliveira et al. [9], Jambal et al. [10,11], Zhang andOuyang [12] for non-Newtonian fluids; Nield et al. [13], Ranjbar-Kaniand Hooman [14], Tada and Ichimaya [15] for porous medium; Chen[16], Jeong and Jeong [17], Aydın and Avcı [18,19], Del Giudice et al.[20], Sun et al. [21] for microscale flows). This case was previously

emphasized by Nield [22] who defined it as worthy of furtherinvestigation.

The purpose of this study is to discuss, question and highlight thisphysically unrealistic situation. In this regard, the viscous heatingeffect on convective flow in an unheated adiabatic duct is studied bothanalytically and numerically at first. Then, the usage of the thermalboundary condition of the constant wall temperature is questionedfrom a thermodynamics viewpoint.

2. Analysis

In this study, for the hydrodynamically fully developed flow,firstly, the thermally fully developed flow case is considered (Case A)and it is extended to the thermally developing case (Case B).

2.1. Case A (Thermally fully developed)

In this case, the flow is considered to be fully developed boththermally and hydrodynamically. Steady, laminar flow havingconstant properties (i.e. the thermal conductivity and the thermaldiffusivity of the fluid are considered to be independent oftemperature) is considered. The axial heat conduction in the fluidand in the wall is assumed to be negligible.

The normalized fully developed velocity profile for the studiedgeometries is given as [23]:

U =uum

=3 + n

21− r

b

� �2� �

ð1Þ

where n=0, 1 for plane duct and circular duct, respectively and b isthe tube radius or half-distance between parallel plates.

Nomenclature

A cross-sectional area (m2)b tube radius or half-distance between parallel plates (m)Br Brinkman numbercp specific heat at constant pressure (kJ/kgK)Dh hydraulic diameter=2b (m)K thermal conductivity (W/mK)n constant to denote plane duct (=0) or circular duct

(=1) geometryPr Prandtl numberr normal coordinate measured from the axial axis of the

ductR dimensionless normal coordinateRe Reynolds numberT temperature (K)u velocity (m/s)U dimensionless velocity=u/umz axial coordinate (m)Z dimensionless axial coordinate

Greek symbolsµ dynamic viscosity (Pa s)ρ density (kg/m3)υ kinematic viscosity (m2/s)θ dimensionless temperature

Subscriptsad,b adiabatic bulkad,w adiabatic wallb bulke fluids enteringm meanvd viscous dissipation

536 O. Aydın, M. Avcı / International Communications in Heat and Mass Transfer 37 (2010) 535–539

The conservation of energy including the effect of the viscousdissipation for both geometries requires

u∂T∂z =

kρcp

1rn

∂∂r rn

∂T∂r

� �+

μρcp

∂u∂r

� �2

ð2Þ

where the second term in the right hand side is the viscous dissipationterm.

For the adiabatic case, the first term in the left-side of Eq. (2) is

∂T∂z =

dTdz

= const: ð3Þ

By introducing the following non-dimensional quantities

R =rDh

; U =uum

; θ =T −Teμu2

m = k; Z =

z =Dh

RePrð4Þ

Eq. (2) can be written as

1Rn

ddR

Rn dθdR

� �=

3 + n2

� �dθdZ

1−4R2� �

− 3 + n2

� �264R2 ð5Þ

For the solution of the dimensionless energy transport equationgiven in Eq. (5), the dimensionless boundary conditions are given asfollows:

∂θ∂R jR = 0:0

= 0 at R = 0

∂θ∂R jR = 0:5

= 0 at R = 0:5

ð6Þ

The solution of Eq. (5) in the fully developed region under thethermal boundary conditions given in Eq. (6) is

dθdZ

= 4 n + 1ð Þ n + 3ð Þ ð7Þ

According to the above equation, the dimensionless axial temper-ature gradient at the wall which termed adiabatic wall temperaturegradient, dθad,w/dZ, is obtained as follows:

For plane duct n = 0ð Þ;dθad;w = dZ = 12 ð8Þ

For circular tube n = 1ð Þ;dθad;w = dZ = 32 ð9Þ

2.2. Case B (Thermally developing)

Taking the assumptions given above into account, thiscase considers hydrodynamically fully developed but thermallydeveloping.

Introducing the same dimensionless variables given in Eq. (4), theenergy transport equation becomes

1Rn

∂∂R Rn ∂θ

∂R

� �=

3 + n2

� �1−4R2

� � ∂θ∂Z−

3 + n2

� �264R2 ð10Þ

The walls of the both ducts are kept adiabatic, which ismathematically shown as:

For z N 0 :∂T∂r = 0; at r = b ð11Þ

In dimensionless form, the thermal boundary conditions that willbe applied in the solution of the energy equations are given as:

R = 0 :∂θ∂R = 0; R = 0:5 :

∂θ∂R = 0 ð12Þ

The mean temperature, i.e., the bulk temperature is given by [18]

Tb =∫ρuTdA∫ρudA

ð13Þ

Rewriting this equation in terms of the dimensionless variables:

θb =∫0:5

0

UθRndR

∫0:5

0

URndR

ð14Þ

3. Results and discussions

In this study, we investigate the sole effect of the viscous heating intwo different adiabatic ducts, i.e. no other heating/cooling effects arepresent. In the absence of the viscous dissipation effect, it is for surethat thermal equilibrium condition will exist.

Fig. 2. Dimensionless temperature distributions at different axial loca

Fig. 1. Downstream variation of the dimensionless adiabatic wall temperature for boththe pipe and plane duct flows.

537O. Aydın, M. Avcı / International Communications in Heat and Mass Transfer 37 (2010) 535–539

Behaving like a heat source, viscous dissipation contributes tointernal heating of the fluid. When this effect is included solely withoutany additional heating/cooling effect, the thermal equilibriumconditionwill be disturbed and, in follows, there will be a heat transfer from theviscous dissipating bulk fluid heated to the adiabatic wall. As a result,both the bulk fluid temperature and thewall temperature will increase.Here, as an original attempt, two new parameters are defined forinternal convection studies: the adiabatic wall temperature and theadiabatic bulk temperature. Note that these two definitions representviscous dissipation affected temperatures for thewall and the bulkfluid,respectively, which are only valid for the case of adiabatic walls.

Fig. 1 shows downstream variations of the dimensionless adiabaticwall temperature for both the pipe and plane duct flows. The wall isheated due to heat transfer from the bulk fluid heated by the viscousdissipation to the wall and, in follows, the wall temperature increasesdownstream. As shown, the dimensionless axial temperature gradi-ents at the adiabatic wall attain their analytically determined-thermally developed values (see Eqs. (8) and (9)).

The dimensionless temperature distributions at different axiallocations are depicted in Fig. 2 for the circular duct (a) and the parallelplane duct (b).

tions: (a) for the circular duct and (b) for the parallel plane duct.

Fig. 3.Downstream variation of the adiabatic wall temperature for the Shell Termia B oilflow.

538 O. Aydın, M. Avcı / International Communications in Heat and Mass Transfer 37 (2010) 535–539

In order to clarify the case much more, dimensional results for apractical cases are illustrated in the following. The flow of the ShellTermia B oil with a Prandtl number of 919 in an adiabatic pipe with adiameter of 5 mm and length of 0.8 m is examined. For an inlettemperature of 20 °C, Figs. 3 and 4 show the downstream variations ofthe adiabatic wall temperature and the adiabatic bulk temperature,respectively. As it can be seen, both the adiabatic wall temperatureand the adiabatic bulk temperature increase noticeably. For a longerpipe, we can predict that this increase will continue linearly in theaxial direction. Note that the axial gradients agree perfectlywith thoseobtained analytically.

3.1. On the constant wall temperature boundary condition

As shown above, the wall is heated due to viscous dissipationeffect. Then one may wonder what happens to this effect when thethermal boundary condition of the constant wall temperature isconsidered. Assumption of this boundary condition assumes thatheat added to the wall by the viscous dissipating fluid is somehowremoved.

Fig. 4. Downstream variation of the adiabatic bulk temperature for the Shell Termia Boil flow.

Let's analyze this case from the thermodynamics viewpoint. Let'sconsider the wall as a thermodynamic system. Applying the First Lawof Thermodynamics to the system, we write

Ein−Eout = ΔEtot

Assuming the wall as a closed system, neglecting the changes inkinetic and potential energies, the above equation reduces to

Qvd−Qout = ΔU

which states heat exchanges of the system with the surroundingwithout any work interactions through the system boundary. Qvd

represents the viscous dissipation heat while Δ∪ representing thechange in the internal energy of the system. Assumption of theconstant temperature thermal boundary condition at wall dictates nochange in the internal energy of the system. Therefore, the aboveequation becomes

Qvd = Qout:

The above equation states that the viscous dissipation heat addedinto the wall has to be removed or counter-balanced somehow byheat transfer to the surrounding. Therefore, by assuming the constanttemperature at wall, we assume the removal of the added heat byviscous dissipation to the surrounding at the same time. From thepractical viewpoint, that doesn't make sense. In fact, while studyingthe effect of the viscous dissipation when the thermal boundarycondition of the constant temperature at wall is considered, wedisregard the effect of the viscous dissipation by forcing the wall to beisothermal. That is why we obtain the same asymptotic value for theNusselt number downstream nomatter how large or small the degreeof the viscous dissipation effect is.

It is disclosed that assuming the constant temperature thermalboundary condition at wall is unrealistic in internal flows when theviscous dissipation is included. Similar disclosure is also validapplying any kinds of volumetric heating in the fluid.

4. Conclusions

Flow of viscous dissipating fluid in an unheated adiabatic duct isstudied analytically and numerically. Two different geometries areconsidered: circular duct and plane duct between two parallel plates.Two new parameters are defined for internal convection studies:adiabatic wall temperature and adiabatic bulk temperature. Both ofthese two parameters are shown to increase downstream in the axialdirection. In the thermally fully developed region, these increases arefound to be linear, which is proved both analytically and numerically.In view of the fact that wall is heated as a result of viscous heating,usage of the constant temperature thermal boundary condition at wallwhen viscous dissipation is included is questioned from thethermodynamics viewpoint. It is disclosed that assuming the constanttemperature thermal boundary condition at wall is unrealistic ininternal flows when the viscous dissipation is included.

Acknowledgment

The first author of this article is indebted to the Turkish Academyof Sciences (TUBA) for the financial support provided under theProgramme to Reward Success Young Scientists (TUBA-GEBIT).

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