TWO DIFFERENT APPROACHES FOR ANALYSING HEAT TRANSFER IN A POWER-LAW FLUID FLOW BETWEEN PARALLEL PLATES

In this paper, Nusselt numbers for a power-law fluid in a fully developed laminar flow between parallel plates with constant, and different, wall heat fluxes in the presence of dissipation effects are presented. The Nusselt numbers values were obtained following two different approaches. One is the “classical” approach, based on a single bulk temperature, and this approach is used in this work to obtain for the first time generic analytical expressions for Nusselt numbers. In the new approach, different bulk temperatures are used for each Nu′ determination, one bulk temperature for each side of the location of the temperature profile where the derivative ∂T/∂y=0.


INTRODUCTION
An analytic study regarding heat transfer in a fully developed laminar flow between parallel plates of a power-law fluid with constant, but different, wall heat fluxes, in the presence of viscous dissipation, is presented in this work. As it is explained in detail below, the same subject has been partially addressed in the literature previously but not in the precise manner undertaken here which, as we will show, may be advantageous.
In the review article of Hartnett and Kostic (1989), restricted to the hydrodynamics and heat transfer aspects of non-Newtonian flow in rectangular duct geometries, the analytical solution to this problem presented by the authors was due to Skelland (1967), which is only valid for constant and equal heat fluxes at the walls in the absence of viscous dissipation. In the review work of Lawal and Mujumdar (1987) where an overview of laminar duct flow and heat transfer regarding purely viscous non-Newtonian fluids taking into account viscous dissipation and the effect of variables properties is presented, again just the particular case of constant and symmetric wall heat fluxes is discussed. Etemad and Majumdar (1994) carried out a numerical study regarding the simultaneously developing laminar flow and heat transfer of a power-law fluid flowing between two parallel plates. Several different thermal boundary conditions were examined. They showed that the Nusselt number is significantly affected by the variation of the fluid viscosity with temperature, viscous dissipation, the power-law index value as well as the fluid Prandtl number and thermal boundary conditions.
Only recently was the asymmetric wall heat fluxes case addressed analytically in the literature by Tso et al. (2010). In this work the authors present analytical solutions for the temperature profiles and Nusselt numbers, Nu, but, because they use only one coordinate system in the mathematical development, the general result is, as admitted to by the authors themselves, too complex. Simpler expressions for four specific values of the power-law index, n, namely 0.25, 0.5, 1 and 2, are then presented by the authors in order to revel the heat transfer characteristics, but the only verification done was via results from the literature for n=1. For the particular case of equal heat fluxes at the walls in the absence of viscous dissipation, the Nu values given by those simpler expressions, based on the hydraulic diameter, for n=0.25, n=0.5 and n=2, are 8.0, 6.9 and 6.3, respectively. Those values are different from the corresponding ones in the literature, Skelland (1967) and Baptista et al. (2013), that for the same power-law index values are of 9.5, 8.8 and 7.9, respectively, which may indicate a problem in this analytical solution. Also, by not using the generalized Brinkman number definition, Br * , Coelho and Pinho (2009), the Nusselt number values rapidly decrease with an increase of the Brinkman number, something that, as shown in Coelho and Faria (2011), hinders the graphical representations of Nu.
Considering all of the studies discussed above, it is fair to state there is currently no simple and generic analytical solution for the Nusselt numbers, Nu, i.e., valid for any values of the power index, n, wall heat flux ratio, w,1 w,2 q q Φ =   , and generalized Brinkman number. Such an analytical solution and the underlying mathematical approach, is one of the main contributions of the present work. The current study starts by showing the calculation procedure that leads to the analytical expressions of the Nusselt numbers at the duct walls. Figure 1 shows schematically the plane walls 1 and 2 of the duct, spaced apart by a distance of 2H, with the wall heat fluxes applied, w,1 q  and w,2 q  , respectively. The coordinate systems used and an asymmetric temperature profile are also shown. By using two coordinate systems, I y and II y , cf. Fig. 1, the y variable in the velocity profile remains always positive, allowing an easy integration of the differential equations for any value of n and giving rise to simpler mathematical expressions. As far as the authors are aware, the use of two coordinate systems in this type of problems is also new to the literature and here we demonstrate its utility to such problems. The wall temperatures, w,1 T and w,2 T , represented in Fig. 1, are naturally a function of the longitudinal coordinate x.

Frontiers in Heat and Mass Transfer
Available at www.ThermalFluidsCentral.org Fig. 1 Schematic representation of the parallel plates duct, the boundary conditions and of the coordinate systems. A temperature profile is also shown, which is divided by the point where ∂T/∂y=0.
For the analytical expressions of the Nusselt numbers at the walls 1 and 2, Nu1 and Nu2 respectively, the "classical" approach was used, i.e., a single bulk temperature was considered for the entire duct crosssection. When the temperature profile is asymmetric, this approach may lead to negative Nusselt number values and discontinuities in the Nu curves and, as will be seen in section 3, even to a case where Nu=4 regardless the values of n or the heat flux ratio, In order to obtain Nusselt number values free of the above mention anomalies, what we will term "Nu' ", and therefore comparable with the existing values in the literature for cases where a symmetric temperature profile exists, e.g. pipe flow, it is necessary to use two different bulk temperatures, 1 T and 2 T , cf. Fig. 1 , effectively divides the duct cross-section in two independent zones, each one with a temperature profile, named Profile 1 and Profile 2 in Fig. 1, since there is no heat transfer between the two zones. The use of this new approach in the flow between parallel plates is an additional contribution of the present work.
In a flow with a symmetrical temperature profile, the proposed approach reproduces Nu′ values equal to the ones obtained using the "classical" approach. Therefore, it can be stated that in situations with asymmetric temperature profiles, the Nusselt numbers obtained using the new approach are also comparable with the Nu values of the literature cases where the temperature profile is symmetric.
Generally, this new approach can be used whenever the temperature profile is asymmetric, e.g. in annular flow, where the flow between parallel plates is a limiting case, as explained in the work of Coelho and Poole (2017).

CALCULATION PROCEDURE
In a fully developed laminar flow between parallel plates of a powerlaw fluid, the dimensionless velocity profile is given by the following equation, 1 * * 2 1 1 1 n n u n u y n U where u is the local velocity, U is the bulk velocity and * y y H = is the transversal coordinate in dimensionless form, Fig. 1.
The differential form of the energy conservation equation in Cartesian coordinates, for a fully developed flow between parallel plates in the presence of viscous dissipation, is shown in Eq. (2), where T is the temperature, x the longitudinal coordinate, Fig. 1, ρ, c and k are density, specific heat and conductivity of the fluid, respectively, and yx τ is the absolute value of the local shear stress.
Using a similar method to the one shown in Çengel and Turner (2005) for a constant wall heat flux in a pipe, but considering the presence of viscous dissipation, it can also be shown that, where T is the bulk temperature, w q  is the average wall heat flux, , and τw is the wall shear stress.
Replacing T x ∂ ∂ by dT dx in equation (2)  ( ) Since * y is raised to the power ( ) , it must be always positive in order to make the integration of Eq. (6) possible. To assure this, two coordinate systems are used. The x-axis is the same while the y-axis are distinct, yI and yII axis, one for each region I and II, separated by the duct symmetry plane as shown in Fig. 1. Equation (6) is then integrated in both regions I and II, subject to the following boundary conditions, The boundary conditions given by Eqs. (7) and (8), on one hand, and the boundary conditions given by Eqs. (9) and (10) on the other, allow the temperature profile equations in the regions I and II, respectively, to be obtained. The boundary condition (11) was used to validate the resulting expressions for * * dT dy .

RESULTS
The integration of Eq. (6) in the regions I and II, cf. Fig. 1, leads to the following expressions for the temperature profiles, (13), the following expression for the wall 2 temperature, Although * w,1 T is unknown and a function of x * , which requires the use of a temperature difference in a graphical representation of the temperature profiles, for example * * w,1 T T − , this does not affect the Nusselt number calculations, since these are based on a temperature difference that is always independent of the wall temperatures, as will be demosntrated below. The derivative of the temperature profile in the duct axis is given by the following expression, as it can be observed, the expression is independent of n and Br * , since these two variables have a symmetrical effect on the temperature profile. The temperature at the duct axis is generally a function of the variables n, Br * and Φ, except for the case where the heat supplied at the duct walls equals the heat generated by viscous dissipation, i.e., Br * =0.125. In this case this temperature is given by the following expression, being in this particular case also independent of n. Figure 2 shows three temperature profiles for three representative where hi is the convection coefficient, the expressions (17) and (18) were used, respectively. They arise via rendering dimensionless the equation w, w, The bulk temperature, T , used in the Nusselt numbers calculation was obtained through the following integral, where b is the duct spanwise length. The analytical expression for this bulk temperature is given by Eq. (20).
The mathematical expressions for the Nusselt numbers at both walls, Nu1 and Nu2, are given by Eqs. (21) and (22) The validation of Eqs. (21) and (22) Equation (23) shows that regardless of the value of n, for Br * >0.25 the wall temperature * w,1 T is always higher than the bulk temperature, * T , regardless of the positive value of Φ.
The variation of Nu1 and Nu2 with Br * for different values of Φ and n=0.5 is shown in Fig. 5. In general, it can be seen, once again, that an increase in the Brinkman number values leads to a reduction in the Nusselt numbers and for wall heat flux ratios between 0<Φ<0.5, according to Eq. (23), the Nu1 values may become infinity and negative. An interesting fact noted when analyzing Fig. 3, 4, and, 5, is that for Br * =0.25, i.e., visc dissip w 2 q q =   , both Nusselt numbers are equal, Nu1=Nu2 regardless of the values of n or Φ. In fact, when replacing Br * by 0.25 in Eqs. (21) or (22) the result is Nu1=Nu2=4. The use of a single bulk temperature for the entire duct cross-section is responsible for this outcome and again shows that the heat transfer coefficients thus obtained are far from the real value. Essentially, if correct, this would mean the "real" convection coefficient is simultaneously independent of velocity and temperature profiles as varied when n and Φ vary between zero to infinity, i.e., a velocity profile varying between plug flow and almost triangular. and different values of the wall heat flux ratio, Φ. Data obtained using the "classical" approach for the bulk temperature calculation.
The results presented in Figs 3-5 show that the "classical" approach, i.e., the use of a single bulk temperature for the whole duct crosssection, while useful from a practical standpoint, since this temperature is easily determined, experimentally and mathematically, sometimes yields Nusselt numbers that deviate from the values and behaviours of the expected heat transfer coefficient. By correctly relating the wall temperature with the bulk temperature, which, obviously, is the most relevant in practice, the "classical" approach is an important tool and it will continue to be used. The next section presents a different approach for the Nusselt numbers calculation that eliminates the singularities referred to above, while allowing a direct comparison between the Nu′ values thus obtained, regardless of the temperature profile shape, with those in the literature for the cases where the temperature profile is symmetrical.
The new Nusselt numbers are calculated using Eqs. (17) and (18) The relationship between both convection coefficients, given by the "classical" and the new approach, is shown in Equation (27), also valid for the Nusselt numbers. This equation was deduced knowing that the wall heat flux, w,i q  , is the same regardless of the approach used.
Equation (27) shows that both approaches allow correct calculations of the wall temperature, w,i T , given the corresponding Nusselt number and bulk temperature, i T or T . The advantage of the new approach, as previously mentioned, is that it allows more realistic heat transfer coefficients to be obtained, comparable with the existing values in the literature for flows inside ducts when the temperature profile is symmetrical. Although the new bulk temperatures have these advantages, their calculation is not straightforward potentially hindering its use. Figures 6, 7 and 10, show the results already presented in figures 3 to 5 but now obtained through the use of the new approach, i.e., using different bulk temperatures for calculating the Nusselt number on each wall. Figure 6 shows the variation of the Nusselt numbers,  For Φ<0.2, when the coordinate * * * I, 0 dT dy y = approaches wall 1, the temperature profile 1 is also closer to that wall and to the zone where the heat generated by viscous dissipation occurs, cf. Fig. 1. Because of that, the Nusselt number, 1 Nu′ , also starts to depend strongly on the Brinkman number, Br * , as shown in Fig. 7a). In the "classical" approach, this dependence goes unnoticed, cf. Fig. 4a).
For As expected, it is clear that the heat transfer coefficient decreases when the Brinkman number increases, since the heat generated by viscous dissipation hinders the heat transfer.
Once again, it can be seen in Fig. 8a), by contrast with Fig. 8b),   and different values of the wall heat flux ratio, Φ. Data obtained using two bulk temperatures, the new approach.
In section 3 it was found that for Br * =0.25, both Nusselt numbers were constant, Nu1=Nu2=4, and simultaneously independent of Φ and n. This singular case is inherent to the "classical" approach and does not occur in this new and more realistic approach.

CONCLUSIONS
In this work, analytical expressions for the Nusselt number in a laminar flow of a power-law fluid between parallel plates were obtained. These results are valid for a fully developed flow, with constant and different heat fluxes at the walls in the presence of viscous dissipation. In these analytical solutions the "classical" approach was used, i.e., both Nusselt numbers are, as usual, based in the same bulk temperature calculated for the entire duct cross-section. This approach occasionally leads to negative values and discontinuities in the Nusselt number plots, and for the particular case of the Brinkman number, Br * , equal to 0.25 ( visc dissip w 2 q q =   ) a singular result is obtained, i.e., Nu1=Nu2=4 regardless of the wall heat flux ratios, Φ, and the power-law index, n, values. It was found that the temperature profile derivative at the duct axis only depends of Φ, being independent of the values of n or Br * since the heat transfer across the duct cross-section is symmetrically affected by these two variables. In the particular case in which the heat supplied at the duct walls is equal the heat generated by viscous dissipation, i.e., Br * =0.125, the temperature value, T * , at the duct axis is independent of the power law index, n Generally, for low values of Br * , the decrease in the value of n leads the velocity profile to become closer to the plug profile, i.e., greater velocities near the walls, and the Nusselt number increases. For higher values of Br * , the heat generated by viscous dissipation, that approaches the walls when n decreases, surpasses the effect of the increased velocity near the walls and Nu decreases.
The Nusselt numbers obtained using the traditional approach do not always reflect the anticipated behaviour of the real heat transfer coefficient. In order to obtain Nu′ values that are closer to the real ones, and comparable with the literature values for the many other cases where the temperature profile is symmetric, an new approach for the Nusselt numbers determination was also undertaken. The calculation is made using two different bulk temperatures, 1 T and 2 T , one for each side of the temperature profile where ∂T/∂y=0, with the transverse coordinate 0 T y y y ∂ ∂ = = , avoiding the anomalies discussed above.
Both approaches allow the correct calculation of the wall temperature, once known the corresponding Nusselt number and bulk temperature of each approach. The advantage of this new approach is that it also allows the calculation of a more realistic heat transfer coefficient, comparable with the current values in the literature for flows inside ducts having symmetrical temperature profiles, although at the expense of a bulk temperature calculation that is not as straightforward which may hinder its practical implementation. The Nusselt number plots, obtained using the new approach, do not show discontinuities or negative values. Also from a didactic point of view, it is relevant to show that negative values of Nusselt numbers and discontinuations in the curves that reflect their behavior are not inevitable and that there are ways to avoid this apparently unrealistic behavior.
Another thing that this new approach revealed was the strong variation of the Nusselt numbers with the Brinkman number, at the wall with lower heat flux, for Φ values smaller than about 0.2. Finally the 8 two different approaches produce Nu values in a given wall much more similar to each other the further away from that wall the coordinate * * * 0 dT dy y = is.

ACKNOWLEDGMENTS
The help of Prof. Rob J. Poole (Univ. Liverpool) for proofreading the original manuscript and the helpful comments of the anonymous reviewers are gratefully acknowledged.