Transactions on Engineering Sciences vol 5, 1994 WIT Press, ISSN

Transcription

1 Heat transfer from thermally developing flow of non-newtonian fluids in rectangular ducts B.T.F. Chung & Z.J. Zhang Department of Mechanical Engineering, The University of Akron, Akron, ABSTRACT The purpose of this work is to determine numerically the Nusselt number of hydrodynamically fully developed and thermally developing flow of power law non-newtonian fluids in a rectangular duct. Three boundary conditions, namely, T, HI and H2 are taken into account. The effects of the power law index and duct aspect ratio on Nusselt number and the thermal entrance length are investigated. Under certain limiting conditions, the present solutions compare very well with some numerical results available in the literature. NOMENCLATURE a 1/2 duct width b 1/2 duct height Dh duct hydraulic diameter, characteristic length f fanning friction factor, (-dp/dx)(dj/4)/(pujv2) h heat transfer coefficient k fluid thermal conductivity K parameter in power law stress strain relationship L*& thermal entrance length n power law index Nu Nusselt number, hd^/k Pe Peclet number, u JV«q' surface heat flux per unit length imposed on one quadrant of the duct q" surface heat flux per unit area Re+ generalized Reynolds number, pu^'tvvk T dimensionless temperature T = (t-tj/(vtj, (t-t;)/(q'/k), (t-t;)/q"d,/k)for the T, HI and H2 boundary conditions respectively U dimensionless fluid axial velocity, u/u^ x, y, z dimensional rectangular coordinates X dimensionless axial coordinate, x/(e\pe)

2 184 Heat Transfer Y, Z dimensionless transverse coordinates, Y=y/D^, Z=z/D^ Greek symbols: a fluid thermal diffusivity a* duct aspect ratio, b/a %* dimensionless complex velocity gradient f integration variable along duct wall Subscripts: HI HI boundary condition H2 H2 boundary condition i duct inlet m mean s integration path along duct wall T T boundary condition w wall INTRODUCTION The fluid dynamics and heat transfer behavior of laminar flow through rectangular ducts are of special interest because of the wide application of such geometries in electronic cooling systems and compact heat exchangers. As a result, extensive heat transfer studies have been carried out on such geometries. Non-Newtonian fluids are found wide applications in many industries such as the chemical, pharmaceutical, biological and food industries. Therefore, it is important to develop an understanding of the hydrodynamics and heat transfer behavior of such fluids in rectangular duct. In this study, consideration is given to purely viscous fluids. The power law model is employed to describe the shear stress and shear rate relation of such non-newtonian fluids and all thermal boundary conditions described below will be considered. The thermal boundary conditions for convective heat transfer in rectangular ducts are in general represented by the following three types. 1) constant wall temperature both peripherally and axially, known as T boundary condition. 2) constant heat input per unit axial distance and constant peripheral wall temperature at each axial position with wall temperature varying axially only; generally is referred to as HI boundary condition. 3) constant heat input per unit axial distance as well as per unit peripheral distance; this is denoted by H2 boundary condition. Shah and London [1] summarized the fluid flow and heat transfer of Newtonian fluid in rectangular duct flow. An comprehensive review of fluid dynamics and heat transfer aspects of Newtonian and non-newtonian fluids in rectangular duct flow was given by Hartnett and Kostic [2]. Extensive theoretical and experimental studies have been carried out for heat transfer of Newtonian fluids in laminar flow through rectangular ducts. However, for power law non-newtonian fluids, most studies have been restricted to the plane parallel plate geometry, e.g. References [3-10]. Using a finite difference method, Chandrupatla [11] investigated the developing Nusselt number for the T,H1 and H2 boundary conditions with n=0 and n^o.5 for the duct aspect ratio a* = 0.5 only.

3 Heat Transfer 185 Our literature survey reveals that the Nusselt numbers of power law fluid in rectangular duct are not available for non-square duct geometries (a*^ 1.0). Even for a square duct geometry, the corresponding Nusselt numbers are not available for n less than 0.5. ANALYSIS Consideration is given to the system shown in Figure 1 which depicts a flow of non-newtonian fluid in a horizontal rectangular duct. The origin of the rectangular coordinates is set at the center of the duct inlet cross section. The analysis is restricted to the quadrant of the rectangular duct due to the symmetry in the geometry as well as the flow and heat transfer boundary conditions. The present analysis is based on the following assumptions: 1) Hydrodynamically fully developed laminar flow; 2) Steady state, incompressible flow; 3) Constant thermophysical properties of the fluid; 4) Negligibly small axial conduction and viscous dissipation; 5) Non-Newtonian fluids obeying the power law. Governing Equations a) Momentum Equation Under the aforementioned assumptions, the momentum equation of a power law fluid in rectangular duct is given as the following dimensionless form by Chung and Zhang [12] ~~. - (i) where U=u/Um, Y = y/d,, Z = z/d,, f=(-dp/dx)(d,/4)/(pu^/2), Re+=puJ and,* = Equation (1) is subjected to the nonslip boundary condition at the wall. The following constraining condition relates the friction factor fre+ to the parameters of%* and n. b) Energy Equation 16«* f< r <-, u(y,z)dydz = 1 (3) *y J<> jo The associated boundary conditions are Case 1) T boundary condition uh = PL + *n. (4) ax ay= az= T = Y = (l + a*)/(4a*) (5a) T = = (l + cx*)/4 (5b) dt/dy = Y = 0 (5c) at/az = z = o (5d) T = X = 0 (5e)

4 186 Heat Transfer Case 2) HI boundary condition, l+«* /nrpv 1 * «* /nrpv f ~*~ *! dz + f "*=*" Jo \ay/y=^* 4a* Jo \az/z=i*«* <T~ dy = 1 (6a) 8T/8Y = Y = 0 (6b) at/az = Z = 0 (6c) T = X = 0 (6d) Case 3) H2 boundary condition dt/dy Y = (l + a*)/(4a*) (7a) at/az = z = (i+«*)/4 (?b) ax/ay = Y = o (7c) at/az = z = o (?d) T = X = 0 (7e) Numerical Scheme The momentum equation of power law fluids in rectangular duct has been solved numerically for various duct aspect ratios and power law indices by Chung and Zhang [12] and hence will not be repeated here. The Successive Overrelaxation method is employed to solve the energy equation. The two-point forward difference and three-point central difference representations are employed for the first and second order derivatives of temperature respectively. The values at the preceding axial position are used as an initial estimate for the temperature. Iteration is repeated until the difference in temperature between two successive iterations agrees to within the required accuracy of 10 *. The convergence of the iteration is enhanced by using the overrelaxation factor, 0 equal to 1.6. To conserve space, details of numerical analysis will not be presented; the interested reader may refer to the report of Zhang [13]. RESULTS AND DISCUSSION The temperature distribution and heat transfer in rectangular ducts are determined for a wide range of duct aspect ratios and power law indices. In the present computation, the gride size in Y and Z directions ranges from to depending on #* and is kept at 2.5 x 10"* along the axial direction. Limiting Solutions-Thermally Fully Developed Flow The accuracy of the present numerical approach is examined under the limiting condition of n=1.0 and X -» oo. The numerical solutions of limiting Nusselt numbers NU?, Nu^, Nu%2 for Newtonian fluids (n= 1) under the fully developed condition (X -> oo) are found to agree excellently with those of Shah and London [1], Further comparisons of the present solution of developed Nusselt numbers of power law fluids in a square duct geometry with the previous solutions under the T,H1 and H2 boundary conditions are also made. It is found that the maximum difference between the present solutions and those of Chandrupatla [11] is about 0.5%.

5 Heat Transfer 187 Thermally Developing Solutions The thermally developing Nusselt numbers of power law fluids flowing through rectangular duct for the T boundary condition are shown in Figures 2 and 3 for the aspect ratio of 0.5 and 0.2 respectively. Each figure includes five different geometries with n ranging from 0 to 1.0. As shown in Figure 2, the present solutions and the solutions by Wibulswas [14] for Newtonian flow agree very well. For a fixed aspect ratio, the developing Nusselt number increases as the power law index decreases. In Figures 4 and 5 Nu^ is plotted against axial position, X with a* equal to 0.5 and 0.2 respectively. Similar to the case of N%, N%i increases as the power-law index decreases. The local Nusselt numbers of the power law fluids as a function of axial distance for the H2 boundary condition are shown in Figures 6 and 7. A quite different behavior of the developing Nusselt number with respect to the power law index is observed in Figure 7 which shows that all curves intersect and the thermally developed Nusselt number with n=0.5 serves as a lower bound for all curves. Thermal Entrance Length Another point of interest in this study is to investigate the thermal entrance length, L V It is defined as the duct length at which the local Nusselt number has reached within 5% of its fully developed values. The thermal entrance length as a function of the power law index with the aspect ratio as a parameter is shown in Figures 8-10 for the T,H1 and H2 boundary conditions respectively. In Figure 8, the thermal entrance lengths are plotted for the T boundary condition. For a*= 1.0, L\ increases almost linearly from a value of at n=0 to a value of at n=1.0. Good agreement is observed between the present solution and the solution of Chandrupatla [11] depicted by the diamond symbol. For e* = 0.5, L\ increases form at n = 0 to at n = 0.85 and then decreases slightly to 0.05 at n=1.0. Similar pattern is observed for other value of a*. It is of interest to note that there exists a maximum value of L** for each aspect ratio and this maximum value shifts toward the left hand side (smaller n) as the aspect ratio decreases from 1.0 to 0 or changes from a square duct to infinite parallel plates. The change of thermal entrance lengths as a function of the power law index for the HI boundary condition is demonstrated in Figure 9. Unlike the case of HI condition, there is no significant change in thermal entrance length with respect to n for 0.3<n< 1. The solutions of Chandrupatla [11] for 0<n<1.0 and Shah and London [1] for n=1.0 are also included for comparison. The numerical values for n= 1.0 from the above two authors and the present solution are 0.068, and respectively. As n decreases, the discrepancy between the present solution and that of Chandrupatla [11] becomes smaller. The thermal entrance lengths for the H2 boundary condition are illustrated in Figure 10 for various values of the aspect ratio. There is no major change in L** with respect to n. Therefore, for this purpose the Newtonian result can be used for all values of n. It is noted that L** increases drastically as the aspect ratio decreases. For the square duct geometry, the present solution of L**

6 188 Heat Transfer agrees well with that of Chandrupatla [11] shown by the diamond symbol in this figure. REFERENCES 1. Shah, R.K., and London, A.L., Laminar Flow Forced Convection in Ducts, Advances in Heat Transfer, supplement 1, Academic Press, Inc., New York, Hartnett, J.P., and Kostic, M., Heat Transfer to Newtonian and Non- Newtonian Fluids in Rectangular Ducts, Advances in Heat Transfer, pp , Academic Press, Inc., Harcourt Brace Jovanovich, Publishers, Cotta, R.M., and Ozisik, M.N., Laminar Forced Convection of Power-Law Non-Newtonian Fluids Inside Ducts, Wdrme Stqfftibertrag, Vol. 20, p. 211, Javeri, V., Magnetohydrodynamic Channel Flow Heat Transfer For Temperature Boundary Conditions of The Third Kind, Int. J. Heat Mass Transfer, Vol. 20, pp , Kwant, P.B., and Van Ravenstein, Th.N.M., Non-Isothermal Laminar Channel Flow, Chem. Eng. ScL, Vol. 28, pp , Lin, T., and Shah, V.L., Numerical Solution of Heat Transfer to Yield Power Law Fluids Flowing in the Entrance Region, Int. Heat Transfer Conf. 6th, Toronto 5, p. 317, Shah, R.K., and Bhatti, M.S., Laminar Convective Heat Transfer in Ducts, in Handbook of Single Phase Convective Heat Transfer, p. 3, Wiley, New York, Skelland, A.H.P., Non-Newtonian Flow and Heat Transfer, Wiley, New York, Tien, C., Laminar Heat Transfer of Power-Law Non-Newtonian Fluid- Extension of Graetz-Nusselt Problem, Can. J. Chem. Eng., pp , Vlachopoulos, J., and Keung, C.K.J., Heat Transfer to a Power-Law Fluid Flowing Between Parallel Plates, AIChE Journal, Vol. 18, pp , Chandrupatla, A.R., Analytical and Experimental Studies of Flow and Heat Transfer Characteristics of a Non-Newtonian Fluid in A Square Duct, Ph.D. thesis, Indian Institute of Technology, Madras, India, Chung, B.T.F., and Zhang, Z.J., Velocity and Friction Factor of Fully Developed Flow of Non-Newtonian Power Law Fluids in Rectangular Ducts, to present at ASME Winter Annual Meeting, Zhang, Z.J., Laminar Forced Convection of Non-Newtonian Fluids in the Entrance Region of Rectangular Ducts, Master's Thesis, Department of Mechanical Engineering, University of Akron, Akron, Ohio, Wibulswas, P., Laminar-Flow Heat-Transfer in Non-Circular Ducts, Ph.D. dissertation, Department of Mechanical Engineering, University of London, 1966.

7 Heat Transfer 189 U,>- Figure 1. Flow in a rectangular duct Nu n=1.0 n = 0.2 n = 0^3 Wibulswas for n= Figure 2. Thermally developing Nusselt numbers Figure 3. Thermally developing Nusselt numbers under T boundary condition, a* = 0.5. under T boundary condition, a* = ^ Nu, 9 "HI n=1.0 n = 0.5 Wibulswas for n = 1 n Nu Figure 4. Thermally developing Nusselt numbers Figure 5. Thermally developing Nusselt numbers under H1 boundary condition, a* = 0.5. "rider H1 boundary condtion, a" =

8 190 Heat Transfer Figure 6.. Thermally under H2 boundary developing Nusselt numbers Figure 7. Thermally developing Nusselt numbers condition, a* =0.5. under H2 boundary condition, a* = "T 1 1 O a* = 1.0 V a* = 0.25 a* = 0.5 V a* = 0.2 O Chandrupatla, a* =1.0 _. ^ n Figure 8. Thermal entrance lengths for T boundary conditon. v * ' I i i i ~r - # a*=1.0 V a*=0.25 ' V a*=0.5 D a'=0.2 + Shah and London, a*=1.0 _ O Chandrupatla, a"=1.0 " ^ % tdzj ~~* !_. 1 1, 1, 1, n Figure 9. Thermal entrance lengths for H1 boundary condition O «'=1.0 V a'=0.25 # a"=0.5 T a'=0. O Chandrupatla, a'= n 0.6 Figure 10. Thermal entrance lengths for H2 boundary condition.