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A Reduced-Boundary-Function Method for Convective Heat Transfer with Axial Heat Conduction and Viscous Dissipation

We introduce a new method of solution for the convective heat transfer under forced laminar flow that is confined by two parallel plates with a distance of 2a or by a circular tube with a radius of a. The advection-conduction equation is first mapped onto the boundary. The original problem of solving the unknown field T(x,r,t) is reduced to seek the solutions of T at the boundary (r=a or r=0, r is the distance from the center line shown in Fig. 1), i.e. the boundary functions. In this manner, the original problem is significantly simplified by reducing the problem dimensions from 3 to 2. The unknown field T(x,r,t) can be eventually solved in terms of these boundary functions. The method is applied to the convective heat transfer with uniform wall temperature boundary condition and with heat exchange between flowing fluids and its surroundings that is relevant to the geothermal applications. Analytical solutions are presented and validated for the steady state problem using the proposed method.


Introduction
Convective heat transfer in circular tube and flat plates is of fundamental interest and practical importance and attracts considerable attention in the past. The analysis of convective heat transfer in tube was originally studied by Graetz [1,2] with assumptions of steady and incompressible flow with fully developed velocity profile, negligible viscous dissipation, axial conduction, and temperature independent fluid properties. These assumptions were made to simplify the complexity while solving the heat transport equation.
A steady-state flow of fluid with a constant temperature Ti passing through a channel is shown in Fig.1. The wall boundary is prescribed with a uniform temperature Tw in the original work of Graetz [1,2] and Nusselt [3], and the model has been referred to as the Graetz problem that represents a class of simplified convective heat transfer problems.
A number of studies have been carried out on various extensions of the original Graetz problem for different applications. For example, the so-called "extended Graetz problem" refers to the including of axial conduction in the original Graetz problem and has been studied extensively [4][5][6][7][8]. The Peclet number (the dimensionless ratio of convective to conductive heat transfer) in the flow could be sufficiently large in many practical applications so that the axial heat conduction can be neglected. However, it is not the case for the heat transfer with small Peclet numbers, for example, in compact heat exchangers where liquid metals are used as the working fluids and axial conduction can play an important role [9].
The thermal-fluid system becomes increasingly smaller with the rapid growth of semiconductor technology. The technological applications in such applications as microelectromechanical systems (MEMS) require more efficient convective heat transfer in a relatively smaller space at the micro-scale level. As the size of flow channel is reduced, the convective heat transfer does not follow the standard results based on the continuum assumption, which leads to the extension of the original Graetz problem to include the slip flow boundary and to allow the viscous dissipation and temperature jump at the wall [10][11][12].
The specific objective of this paper is to present a method of solution to the convective heat transfer with applications to two different boundary conditions. The example problems consider the effects of both axial conduction and viscous dissipation. Previous attempts to solve Graetz problem and its extensions are mostly taking the way of plugging a solution of series expansion into the advection-conduction equation. The method of solution presented in this paper uses a reduced-boundary-function approach, which significantly simplifies the original problem and reduces the problem dimensionality without losing numerical accuracy.
Moving boundary problems are important and numerically difficulty to solve since boundary or interface location is not known. Lagrangian particle methods have been developed to solve such problems without explicit interface tracking/capturing [13][14][15]. The method of solution presented in this paper can be applied to these problems in order to reduce the problem dimensionality. Examples are the solute precipitation and dissolution due to the chemical reaction [16,17] and crystal growth. It was also applied to the thermal oxidation problem [18], where an oxidized layer is constantly growing due to the oxidation reaction.
In the problem of convective heat transfer, the advection-conduction equation is mapped onto the boundary, and the original problem to find temperature solution ( ) , , T x r t is reduced to find the temperature solutions at the boundary The original problem is significantly simplified with this method because the dimensionality of the problem is reduced from 3 to 2. In this paper, we introduce the formulation of the reduced boundary function method in Section 2, followed by the analytical, numerical solutions, and comparison for two boundary conditions in Sections 3 and 4.

The Formulation of the Reduced-Boundary-Function Method
The rigorous mathematical formulation is based on the advection-conduction equation for the convective heat transfer. The problem considered here is that a fluid is flowing with a steady Poiseuille flow through a channel confined by two flat plates or a circular tube. The fluid enters the channel at a prescribed temperature Ti. Two types of boundary conditions are considered. First one is a uniform wall temperature Tw applied on the flow boundary. The second one is a mixed Dirichlet and Neumann boundary to mimic the heat exchange between the flowing fluid and the surrounding environment. The physical parameters are assumed to be independent of temperature. The geometry of the problem is defined in Fig. 1. The dynamics of temperature field T is governed by the advection-conduction equation in the cylindrical coordinates, where ( ) Obviously, the velocity v(r) also satisfy where 0 u is the maximum velocity at r=0 and the mean velocity u is written as An additional relation can be derived from the l'Hôpital's rule in calculus and will be used later, a T x t T x r a t ≡ = represents the temperature along the wall (r=a) and Since Eq. (1) must be satisfied everywhere including the boundary r=a and r=0, the following relationship can be obtained from Eqs. (1)-(6) for boundary functions a T and 0 T , ( ) The same set of equations can be rewritten as ( ) by introducing short notations to represent derivatives at r=0 and r=a, In principle, the temperature T can be expressed in terms of these derivatives through a Taylor expansion, where n starts from 2 because ( ) The following relation can be easily verified with expansion (12), By taking derivative of the governing equation (1) with respect to r, we have ( ) Again, Eq. (14) can be mapped onto the boundary r=a and r=0 and the following relationships can be established for boundary functions 1,a T and 3,0 T with the help of Eq.(13), More generally, it was found that due to the symmetry of T with respect to r. Up to now, we do not use any temperature boundary conditions at wall boundary. Therefore, Eqs. (9)-(10) and Eqs. (15)- (17) are valid for any arbitrary temperature boundary conditions. Additionally, we do not make any assumptions, such as negligible viscous dissipation, small axial conduction, which are common in order for an analytical solution.

First Type of Boundary --The Uniform Wall Temperature Tw
First, we consider the standard Graetz problem, where the wall of the channel or the tube is maintained at a uniform temperature Tw so that Eq. (9) relating the boundary functions 1,a T and 2,a T is simplified to We first use expansion (12) with terms up to the fourth order to approximate the temperature field T, namely The following relationships from the expansion can be easily found, 3 1, This can be rewritten in the matrix form, By inversing the matrix B and using Eq. (20), we can solve for 2,0 T and 4,0 T , The uniform wall temperature boundary should be satisfied, where It is shown from Eq. (29) that 0 T follows an advection-diffusion-reaction equation. The steady-state solution can be easily found for 0 T , where it is assumed that 1, Instead of using expansion with terms up to the fourth order, we can try the solution with terms up to the sixth order to see the difference of solutions, Equation (15) was not used before and must be used here because we have an additional unknown boundary function 6 T  T  d  d  A  D  T  T  T  t x a a a The matrix form of the relationship between , The characteristic equation (fourth order algebraic equation) for this inhomogeneous fourth order ODE is ( ) If inlet boundary has a uniform temperature i T , With the substitution of solution of centerline temperature 0 T into the steady state form of Eq. (40), the solution for boundary function 1,a T can be found, where ( ) (30)) gives good results for 0 T .

Second Type of Boundary -Heat Exchange with Surroundings
In this section, we consider heat exchange between confined flow and its surroundings.
where o q is the heat flux at external surface and h is the heat transfer coefficient between external surface and its surrounding and is assumed to be independent of the temperature difference. At the internal surface, the heat flux i q is expressed as, Heat equilibrium between external and internal surface requires and we obtained the boundary condition for the convective heat transfer, We still use the expansion (Eq. (26)) with terms up to the fourth order for the temperature field T. The matrix form of the relationship between , where ( )   interplay between e p and α determines the steady state temperature distribution.

Conclusion
The convective heat transfer was studied in great detail for a fully developed viscous flow