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On the constant wall temperature boundary condition in internal convection heat transfer studies including viscous dissipation
International Communications in Heat and Mass Transfer 37 (2010) 535–539
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
On the constant wall temperature boundary condition in internal convection heat transfer studies including viscous dissipation☆ Orhan Aydın ⁎, Mete Avcı Department of Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey
a r t i c l e
i n f o
Available online 10 March 2010 Keywords: Viscous dissipation Constant wall temperature Adiabatic bulk temperature Graetz problem
a b s t r a c t In this study, viscous heating effect on convective flow in an unheated adiabatic duct is studied analytically and numerically. Two different geometries are considered: 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. Variations of these two parameters with varying intensity of viscous dissipation effect are determined. In view of the results obtained, usage of the constant wall temperature thermal boundary condition when viscous dissipation is included is discussed and questioned. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Viscous dissipation changes the temperature distributions by playing a role like an energy source, which affects heat transfer rates. Research considering viscous dissipation have increased recently because of its importance in microscale flow and heat transfer phenomena as well as flow of fluids with high viscosity and low thermal conductivity. For thermally developing flow in internal flows, when the thermal boundary condition of the constant wall temperature is considered at wall, it is shown that an asymptotic value for the thermally fully developed Nusselt number Nu is obtained. When the effect of the viscous dissipation is included (Br ≠ 0) in the analysis for the same problem, a different asymptotic value for Nu is obtained. It is disclosed that no matter what the value of the Brinkman number (a dimensionless number characterizing the degree of the viscous dissipation effect) is, i.e. no matter how high the degree of the viscous 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 axial direction is asymptotically reached. This is a common observation in the 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 and Ouyang [12] for non-Newtonian fluids; Nield et al. [13], Ranjbar-Kani and 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
☆ Communicated by E. Hahne and K. Spindler. ⁎ Corresponding author. E-mail address: [email protected] (O. Aydın). 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2009.11.016
emphasized by Nield [22] who defined it as worthy of further investigation. The purpose of this study is to discuss, question and highlight this physically unrealistic situation. In this regard, the viscous heating effect on convective flow in an unheated adiabatic duct is studied both analytically and numerically at first. Then, the usage of the thermal boundary condition of the constant wall temperature is questioned from 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 both thermally and hydrodynamically. Steady, laminar flow having constant properties (i.e. the thermal conductivity and the thermal diffusivity of the fluid are considered to be independent of temperature) is considered. The axial heat conduction in the fluid and in the wall is assumed to be negligible. The normalized fully developed velocity profile for the studied geometries is given as [23]:
U=
u 3+n = um 2
r 2 1− b
ð1Þ
where n = 0, 1 for plane duct and circular duct, respectively and b is the tube radius or half-distance between parallel plates.
536
O. Aydın, M. Avcı / International Communications in Heat and Mass Transfer 37 (2010) 535–539
Nomenclature cross-sectional area (m2) tube radius or half-distance between parallel plates (m) Brinkman number specific heat at constant pressure (kJ/kgK) hydraulic diameter = 2b (m) thermal conductivity (W/mK) constant to denote plane duct (=0) or circular duct (=1) geometry Prandtl number normal coordinate measured from the axial axis of the duct dimensionless normal coordinate Reynolds number temperature (K) velocity (m/s) dimensionless velocity = u/um axial coordinate (m) dimensionless axial coordinate
A b Br cp Dh K n Pr r R Re T u U z Z
For the solution of the dimensionless energy transport equation given in Eq. (5), the dimensionless boundary conditions are given as follows: ∂θ ∂R ∂θ ∂R
j j
R = 0:0
R = 0:5
= 0 at R = 0 ð6Þ = 0 at R = 0:5
The solution of Eq. (5) in the fully developed region under the thermal boundary conditions given in Eq. (6) is dθ = 4ðn + 1Þðn + 3Þ dZ
ð7Þ
According to the above equation, the dimensionless axial temperature gradient at the wall which termed adiabatic wall temperature gradient, dθad,w/dZ, is obtained as follows: For plane duct ðn = 0Þ; dθad;w = dZ = 12
ð8Þ
For circular tube ðn = 1Þ; dθad;w = dZ = 32
ð9Þ
Greek symbols µ dynamic viscosity (Pa s) ρ density (kg/m3) υ kinematic viscosity (m2/s) θ dimensionless temperature
2.2. Case B (Thermally developing)
Subscripts ad,b adiabatic bulk ad,w adiabatic wall b bulk e fluids entering m mean vd viscous dissipation
1 ∂ 3+n 3+n 2 n ∂θ 2 ∂θ 2 1−4R 64R R = − Rn ∂R 2 2 ∂R ∂Z
Taking the assumptions given above into account, this case considers hydrodynamically fully developed but thermally developing. Introducing the same dimensionless variables given in Eq. (4), the energy transport equation becomes ð10Þ
The walls of the both ducts are kept adiabatic, which is mathematically shown as: ∂T = 0; ∂r
For z N 0 :
The conservation of energy including the effect of the viscous dissipation for both geometries requires
ð2Þ
where the second term in the right hand side is the viscous dissipation term. For the adiabatic case, the first term in the left-side of Eq. (2) is
ð11Þ
In dimensionless form, the thermal boundary conditions that will be applied in the solution of the energy equations are given as: R = 0:
2 ∂T k 1 ∂ μ ∂u n ∂T u = r + n ρcp r ∂r ρ cp ∂r ∂z ∂r
at r = b
∂θ = 0; R = 0:5 : ∂R
∂θ = 0 ∂R
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: ∂T dT = const: = dz ∂z
ð3Þ
∫ UθRn dR 0 θb = 0:5
By introducing the following non-dimensional quantities
R=
r ; Dh
U=
0:5
u ; um
θ=
T − Te ; μu2m = k
Z=
z = Dh RePr
ð14Þ
∫ URn dR 0
ð4Þ 3. Results and discussions
Eq. (2) can be written as 3 + n2 1 d 3 + n dθ n dθ 2 2 R = 1−4R − 64R n R dR dR 2 dZ 2
ð5Þ
In this study, we investigate the sole effect of the viscous heating in two different adiabatic ducts, i.e. no other heating/cooling effects are present. In the absence of the viscous dissipation effect, it is for sure that thermal equilibrium condition will exist.
O. Aydın, M. Avcı / International Communications in Heat and Mass Transfer 37 (2010) 535–539
Fig. 1. Downstream variation of the dimensionless adiabatic wall temperature for both the pipe and plane duct flows.
537
Behaving like a heat source, viscous dissipation contributes to internal heating of the fluid. When this effect is included solely without any additional heating/cooling effect, the thermal equilibrium condition will be disturbed and, in follows, there will be a heat transfer from the viscous dissipating bulk fluid heated to the adiabatic wall. As a result, both the bulk fluid temperature and the wall temperature will increase. Here, as an original attempt, two new parameters are defined for internal convection studies: the adiabatic wall temperature and the adiabatic bulk temperature. Note that these two definitions represent viscous dissipation affected temperatures for the wall and the bulk fluid, respectively, which are only valid for the case of adiabatic walls. Fig. 1 shows downstream variations of the dimensionless adiabatic wall temperature for both the pipe and plane duct flows. The wall is heated due to heat transfer from the bulk fluid heated by the viscous dissipation to the wall and, in follows, the wall temperature increases downstream. As shown, the dimensionless axial temperature gradients at the adiabatic wall attain their analytically determinedthermally developed values (see Eqs. (8) and (9)). The dimensionless temperature distributions at different axial locations are depicted in Fig. 2 for the circular duct (a) and the parallel plane duct (b).
Fig. 2. Dimensionless temperature distributions at different axial locations: (a) for the circular duct and (b) for the parallel plane duct.
538
O. Aydın, M. Avcı / International Communications in Heat and Mass Transfer 37 (2010) 535–539
Let's analyze this case from the thermodynamics viewpoint. Let's consider the wall as a thermodynamic system. Applying the First Law of Thermodynamics to the system, we write Ein −Eout = ΔEtot Assuming the wall as a closed system, neglecting the changes in kinetic and potential energies, the above equation reduces to Qvd −Qout = ΔU which states heat exchanges of the system with the surrounding without any work interactions through the system boundary. Qvd represents the viscous dissipation heat while Δ∪ representing the change in the internal energy of the system. Assumption of the constant temperature thermal boundary condition at wall dictates no change in the internal energy of the system. Therefore, the above equation becomes Qvd = Qout : Fig. 3. Downstream variation of the adiabatic wall temperature for the Shell Termia B oil flow.
In order to clarify the case much more, dimensional results for a practical cases are illustrated in the following. The flow of the Shell Termia B oil with a Prandtl number of 919 in an adiabatic pipe with a diameter of 5 mm and length of 0.8 m is examined. For an inlet temperature of 20 °C, Figs. 3 and 4 show the downstream variations of the adiabatic wall temperature and the adiabatic bulk temperature, respectively. As it can be seen, both the adiabatic wall temperature and the adiabatic bulk temperature increase noticeably. For a longer pipe, we can predict that this increase will continue linearly in the axial direction. Note that the axial gradients agree perfectly with those obtained analytically. 3.1. On the constant wall temperature boundary condition As shown above, the wall is heated due to viscous dissipation effect. Then one may wonder what happens to this effect when the thermal boundary condition of the constant wall temperature is considered. Assumption of this boundary condition assumes that heat added to the wall by the viscous dissipating fluid is somehow removed.
The above equation states that the viscous dissipation heat added into the wall has to be removed or counter-balanced somehow by heat transfer to the surrounding. Therefore, by assuming the constant temperature at wall, we assume the removal of the added heat by viscous dissipation to the surrounding at the same time. From the practical viewpoint, that doesn't make sense. In fact, while studying the effect of the viscous dissipation when the thermal boundary condition of the constant temperature at wall is considered, we disregard the effect of the viscous dissipation by forcing the wall to be isothermal. That is why we obtain the same asymptotic value for the Nusselt number downstream no matter how large or small the degree of the viscous dissipation effect is. It is disclosed that assuming the constant temperature thermal boundary condition at wall is unrealistic in internal flows when the viscous dissipation is included. Similar disclosure is also valid applying any kinds of volumetric heating in the fluid. 4. Conclusions Flow of viscous dissipating fluid in an unheated adiabatic duct is studied analytically and numerically. Two different geometries are considered: 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 of these two parameters are shown to increase downstream in the axial direction. In the thermally fully developed region, these increases are found 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 wall when viscous dissipation is included is questioned from the thermodynamics viewpoint. It is disclosed that assuming the constant temperature thermal boundary condition at wall is unrealistic in internal flows when the viscous dissipation is included. Acknowledgment The first author of this article is indebted to the Turkish Academy of Sciences (TUBA) for the financial support provided under the Programme to Reward Success Young Scientists (TUBA-GEBIT). References
Fig. 4. Downstream variation of the adiabatic bulk temperature for the Shell Termia B oil flow.
[1] K.N. Krishnan, V.M.K. Sastri, Numerical-solution of thermal entry length problem with variable viscosities and viscous dissipation, Warme und Stoffbertragung— Thermo and Fluid Dynamics 11 (2) (1978) 73–79. [2] T.F. Lin, K.H. Hawks, W. Leidenfrost, Analysis of viscous dissipation effect on thermal entrance heat-transfer in laminar pipe flows with convective-boundary
O. Aydın, M. Avcı / International Communications in Heat and Mass Transfer 37 (2010) 535–539
[3] [4]
[5]
[6] [7]
[8]
[9] [10]
[11]
[12]
conditions, Warme und Stoffbertragung—Thermo and Fluid Dynamics 17 (2) (1983) 97–105. T. Basu, D.N. Roy, Laminar heat-transfer in a tube with viscous dissipation, International Journal of Heat and Mass Transfer 28 (3) (1985) 699–701. P.P. Valko, Solution of the Graetz–Brinkman problem with the Laplace transform Galerkin method, International Journal of Heat and Mass Transfer 48 (9) (2005) 1874–1882. O. Aydın, Effects of viscous dissipation on the heat transfer in a forced pipe flow Part 2: Thermally developing flow, Energy Conversion and Management 46 (18–19) (2005) 3091–3102. O. Aydın, M. Avcı, Viscous-dissipation effects on the heat transfer in a Poiseuille flow, Applied Energy 83 (5) (2006) 495–512. P.M. Coelho, F.T. Pinho, P.J. Oliveira, Fully developed forced convection of the Phan–Thien–Tanner fluid in ducts with a constant wall temperature, International Journal of Heat and Mass Transfer 45 (7) (2002) 1413–1423. P.M. Coelho, F.T. Pinho, P.J. Oliveira, Thermal entry flow for a viscoelastic fluid: the Graetz problem for the PTT model, International Journal of Heat and Mass Transfer 46 (20) (2003) 3865–3880. P.J. Oliveira, P.M. Coelho, F.T. Pinho, The Graetz problem with viscous dissipation for FENE-P fluids, Journal of Non-Newtonian Fluid Mechanics 121 (1) (2004) 69–72. O. Jambal, T. Shigechi, G. Davaa, S. Momoki, Effects of viscous dissipation and fluid axial heat conduction on heat transfer for non-Newtonian fluids in ducts with uniform wall temperature—part 1: Parallel plates and circular ducts, International Communication in Heat and Mass Transfer 32 (9) (2005) 1165–1173. O. Jambal, T. Shigechi, G. Davaa, S. Momoki, Effects of viscous dissipation and fluid axial heat conduction on heat transfer for non-Newtonian fluids in ducts with uniform wall temperature, International Communication in Heat and Mass Transfer 32 (9) (2005) 1174–1183. X.H. Zhang, J. Ouyang, Meshless analysis of heat transfer due to viscous dissipation in polymer flow, Engineering Analysis with Boundary Elements 32 (1) (2008) 41–51.
539
[13] D.A. Nield, A.V. Kuznetsov, M. Xiong, Thermally developing forced convection in a porous medium: parallel plate channel with walls at uniform temperature, with axial conduction and viscous dissipation effects, International Journal of Heat and Mass Transfer 46 (4) (2003) 643–651. [14] A.A. Ranjbar-Kani, K. Hooman, Viscous dissipation effects on thermally developing forced convection in a porous medium: circular duct with isothermal wall, International Communication in Heat and Mass Transfer 31 (6) (2004) 897–907. [15] S. Tada, K. Ichimiya, Analysis of laminar dissipative flow and heat transfer in a porous saturated circular tube with constant wall heat flux, International Journal of Heat and Mass Transfer 50 (11–12) (2007) 2406–2413. [16] C.H. Chen, Slip–flow heat transfer in a microchannel with viscous dissipation, Heat and Mass Transfer 42 (9) (2006) 853–860. [17] H.E. Jeong, J.T. Jeong, Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel, International Journal of Heat and Mass Transfer 49 (13–14) (2006) 2151–2157. [18] O. Aydın, M. Avcı, Analysis of micro-Graetz problem in a microtube, Nanoscale and Microscale Thermophysical Engineering 10 (4) (2006) 345–358. [19] O. Aydın, M. Avcı, Thermally developing flow in microchannels, Journal of Thermophysics and Heat Transfer 20 (3) (2006) 628–631. [20] S. Del Giudice, C. Nonino, S. Savino, Effects of viscous dissipation and temperature dependent viscosity in thermally and simultaneously developing laminar flows in microchannels, International Journal of Heat and Fluid Flow 28 (1) (2007) 15–27. [21] W. Sun, S. Kakac S, A.G. Yazicioglu, A numerical study of single-phase convective heat transfer in microtubes for slip flow, International Journal of Thermal Sciences 46 (11) (2007) 1084–1094. [22] D.A. Nield, A note on a Brinkman–Brinkman forced convection problem, Transport in Porous Media 64 (2) (2006) 185–188. [23] R.M. Cotta, M.N. Ozisik, Laminar forced-convection of power-law non-newtonian fluids inside ducts, Wärme und Stoffübertragung—Thermo and Fluid Dynamics 20 (3) (1986) 211–218.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
On the constant wall temperature boundary condition in internal convection heat transfer studies including viscous dissipation☆ Orhan Aydın ⁎, Mete Avcı Department of Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey
a r t i c l e
i n f o
Available online 10 March 2010 Keywords: Viscous dissipation Constant wall temperature Adiabatic bulk temperature Graetz problem
a b s t r a c t In this study, viscous heating effect on convective flow in an unheated adiabatic duct is studied analytically and numerically. Two different geometries are considered: 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. Variations of these two parameters with varying intensity of viscous dissipation effect are determined. In view of the results obtained, usage of the constant wall temperature thermal boundary condition when viscous dissipation is included is discussed and questioned. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Viscous dissipation changes the temperature distributions by playing a role like an energy source, which affects heat transfer rates. Research considering viscous dissipation have increased recently because of its importance in microscale flow and heat transfer phenomena as well as flow of fluids with high viscosity and low thermal conductivity. For thermally developing flow in internal flows, when the thermal boundary condition of the constant wall temperature is considered at wall, it is shown that an asymptotic value for the thermally fully developed Nusselt number Nu is obtained. When the effect of the viscous dissipation is included (Br ≠ 0) in the analysis for the same problem, a different asymptotic value for Nu is obtained. It is disclosed that no matter what the value of the Brinkman number (a dimensionless number characterizing the degree of the viscous dissipation effect) is, i.e. no matter how high the degree of the viscous 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 axial direction is asymptotically reached. This is a common observation in the 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 and Ouyang [12] for non-Newtonian fluids; Nield et al. [13], Ranjbar-Kani and 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
☆ Communicated by E. Hahne and K. Spindler. ⁎ Corresponding author. E-mail address: [email protected] (O. Aydın). 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2009.11.016
emphasized by Nield [22] who defined it as worthy of further investigation. The purpose of this study is to discuss, question and highlight this physically unrealistic situation. In this regard, the viscous heating effect on convective flow in an unheated adiabatic duct is studied both analytically and numerically at first. Then, the usage of the thermal boundary condition of the constant wall temperature is questioned from 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 both thermally and hydrodynamically. Steady, laminar flow having constant properties (i.e. the thermal conductivity and the thermal diffusivity of the fluid are considered to be independent of temperature) is considered. The axial heat conduction in the fluid and in the wall is assumed to be negligible. The normalized fully developed velocity profile for the studied geometries is given as [23]:
U=
u 3+n = um 2
r 2 1− b
ð1Þ
where n = 0, 1 for plane duct and circular duct, respectively and b is the tube radius or half-distance between parallel plates.
536
O. Aydın, M. Avcı / International Communications in Heat and Mass Transfer 37 (2010) 535–539
Nomenclature cross-sectional area (m2) tube radius or half-distance between parallel plates (m) Brinkman number specific heat at constant pressure (kJ/kgK) hydraulic diameter = 2b (m) thermal conductivity (W/mK) constant to denote plane duct (=0) or circular duct (=1) geometry Prandtl number normal coordinate measured from the axial axis of the duct dimensionless normal coordinate Reynolds number temperature (K) velocity (m/s) dimensionless velocity = u/um axial coordinate (m) dimensionless axial coordinate
A b Br cp Dh K n Pr r R Re T u U z Z
For the solution of the dimensionless energy transport equation given in Eq. (5), the dimensionless boundary conditions are given as follows: ∂θ ∂R ∂θ ∂R
j j
R = 0:0
R = 0:5
= 0 at R = 0 ð6Þ = 0 at R = 0:5
The solution of Eq. (5) in the fully developed region under the thermal boundary conditions given in Eq. (6) is dθ = 4ðn + 1Þðn + 3Þ dZ
ð7Þ
According to the above equation, the dimensionless axial temperature gradient at the wall which termed adiabatic wall temperature gradient, dθad,w/dZ, is obtained as follows: For plane duct ðn = 0Þ; dθad;w = dZ = 12
ð8Þ
For circular tube ðn = 1Þ; dθad;w = dZ = 32
ð9Þ
Greek symbols µ dynamic viscosity (Pa s) ρ density (kg/m3) υ kinematic viscosity (m2/s) θ dimensionless temperature
2.2. Case B (Thermally developing)
Subscripts ad,b adiabatic bulk ad,w adiabatic wall b bulk e fluids entering m mean vd viscous dissipation
1 ∂ 3+n 3+n 2 n ∂θ 2 ∂θ 2 1−4R 64R R = − Rn ∂R 2 2 ∂R ∂Z
Taking the assumptions given above into account, this case considers hydrodynamically fully developed but thermally developing. Introducing the same dimensionless variables given in Eq. (4), the energy transport equation becomes ð10Þ
The walls of the both ducts are kept adiabatic, which is mathematically shown as: ∂T = 0; ∂r
For z N 0 :
The conservation of energy including the effect of the viscous dissipation for both geometries requires
ð2Þ
where the second term in the right hand side is the viscous dissipation term. For the adiabatic case, the first term in the left-side of Eq. (2) is
ð11Þ
In dimensionless form, the thermal boundary conditions that will be applied in the solution of the energy equations are given as: R = 0:
2 ∂T k 1 ∂ μ ∂u n ∂T u = r + n ρcp r ∂r ρ cp ∂r ∂z ∂r
at r = b
∂θ = 0; R = 0:5 : ∂R
∂θ = 0 ∂R
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: ∂T dT = const: = dz ∂z
ð3Þ
∫ UθRn dR 0 θb = 0:5
By introducing the following non-dimensional quantities
R=
r ; Dh
U=
0:5
u ; um
θ=
T − Te ; μu2m = k
Z=
z = Dh RePr
ð14Þ
∫ URn dR 0
ð4Þ 3. Results and discussions
Eq. (2) can be written as 3 + n2 1 d 3 + n dθ n dθ 2 2 R = 1−4R − 64R n R dR dR 2 dZ 2
ð5Þ
In this study, we investigate the sole effect of the viscous heating in two different adiabatic ducts, i.e. no other heating/cooling effects are present. In the absence of the viscous dissipation effect, it is for sure that thermal equilibrium condition will exist.
O. Aydın, M. Avcı / International Communications in Heat and Mass Transfer 37 (2010) 535–539
Fig. 1. Downstream variation of the dimensionless adiabatic wall temperature for both the pipe and plane duct flows.
537
Behaving like a heat source, viscous dissipation contributes to internal heating of the fluid. When this effect is included solely without any additional heating/cooling effect, the thermal equilibrium condition will be disturbed and, in follows, there will be a heat transfer from the viscous dissipating bulk fluid heated to the adiabatic wall. As a result, both the bulk fluid temperature and the wall temperature will increase. Here, as an original attempt, two new parameters are defined for internal convection studies: the adiabatic wall temperature and the adiabatic bulk temperature. Note that these two definitions represent viscous dissipation affected temperatures for the wall and the bulk fluid, respectively, which are only valid for the case of adiabatic walls. Fig. 1 shows downstream variations of the dimensionless adiabatic wall temperature for both the pipe and plane duct flows. The wall is heated due to heat transfer from the bulk fluid heated by the viscous dissipation to the wall and, in follows, the wall temperature increases downstream. As shown, the dimensionless axial temperature gradients at the adiabatic wall attain their analytically determinedthermally developed values (see Eqs. (8) and (9)). The dimensionless temperature distributions at different axial locations are depicted in Fig. 2 for the circular duct (a) and the parallel plane duct (b).
Fig. 2. Dimensionless temperature distributions at different axial locations: (a) for the circular duct and (b) for the parallel plane duct.
538
O. Aydın, M. Avcı / International Communications in Heat and Mass Transfer 37 (2010) 535–539
Let's analyze this case from the thermodynamics viewpoint. Let's consider the wall as a thermodynamic system. Applying the First Law of Thermodynamics to the system, we write Ein −Eout = ΔEtot Assuming the wall as a closed system, neglecting the changes in kinetic and potential energies, the above equation reduces to Qvd −Qout = ΔU which states heat exchanges of the system with the surrounding without any work interactions through the system boundary. Qvd represents the viscous dissipation heat while Δ∪ representing the change in the internal energy of the system. Assumption of the constant temperature thermal boundary condition at wall dictates no change in the internal energy of the system. Therefore, the above equation becomes Qvd = Qout : Fig. 3. Downstream variation of the adiabatic wall temperature for the Shell Termia B oil flow.
In order to clarify the case much more, dimensional results for a practical cases are illustrated in the following. The flow of the Shell Termia B oil with a Prandtl number of 919 in an adiabatic pipe with a diameter of 5 mm and length of 0.8 m is examined. For an inlet temperature of 20 °C, Figs. 3 and 4 show the downstream variations of the adiabatic wall temperature and the adiabatic bulk temperature, respectively. As it can be seen, both the adiabatic wall temperature and the adiabatic bulk temperature increase noticeably. For a longer pipe, we can predict that this increase will continue linearly in the axial direction. Note that the axial gradients agree perfectly with those obtained analytically. 3.1. On the constant wall temperature boundary condition As shown above, the wall is heated due to viscous dissipation effect. Then one may wonder what happens to this effect when the thermal boundary condition of the constant wall temperature is considered. Assumption of this boundary condition assumes that heat added to the wall by the viscous dissipating fluid is somehow removed.
The above equation states that the viscous dissipation heat added into the wall has to be removed or counter-balanced somehow by heat transfer to the surrounding. Therefore, by assuming the constant temperature at wall, we assume the removal of the added heat by viscous dissipation to the surrounding at the same time. From the practical viewpoint, that doesn't make sense. In fact, while studying the effect of the viscous dissipation when the thermal boundary condition of the constant temperature at wall is considered, we disregard the effect of the viscous dissipation by forcing the wall to be isothermal. That is why we obtain the same asymptotic value for the Nusselt number downstream no matter how large or small the degree of the viscous dissipation effect is. It is disclosed that assuming the constant temperature thermal boundary condition at wall is unrealistic in internal flows when the viscous dissipation is included. Similar disclosure is also valid applying any kinds of volumetric heating in the fluid. 4. Conclusions Flow of viscous dissipating fluid in an unheated adiabatic duct is studied analytically and numerically. Two different geometries are considered: 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 of these two parameters are shown to increase downstream in the axial direction. In the thermally fully developed region, these increases are found 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 wall when viscous dissipation is included is questioned from the thermodynamics viewpoint. It is disclosed that assuming the constant temperature thermal boundary condition at wall is unrealistic in internal flows when the viscous dissipation is included. Acknowledgment The first author of this article is indebted to the Turkish Academy of Sciences (TUBA) for the financial support provided under the Programme to Reward Success Young Scientists (TUBA-GEBIT). References
Fig. 4. Downstream variation of the adiabatic bulk temperature for the Shell Termia B oil flow.
[1] K.N. Krishnan, V.M.K. Sastri, Numerical-solution of thermal entry length problem with variable viscosities and viscous dissipation, Warme und Stoffbertragung— Thermo and Fluid Dynamics 11 (2) (1978) 73–79. [2] T.F. Lin, K.H. Hawks, W. Leidenfrost, Analysis of viscous dissipation effect on thermal entrance heat-transfer in laminar pipe flows with convective-boundary
O. Aydın, M. Avcı / International Communications in Heat and Mass Transfer 37 (2010) 535–539
[3] [4]
[5]
[6] [7]
[8]
[9] [10]
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