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Forced Convection

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

[+] Author and Article Information
Zhijie Xu1

Energy Resource Recovery and Management, Idaho National Laboratory, Idaho Falls, ID 83415zhijie.xu@pnnl.gov

1

Present address: Computational Mathematics Group, Fundamental and Computational Sciences Directorate, Pacific Northwest National Laboratory, Richland, WA 99352.

J. Heat Transfer 134(7), 071705 (May 22, 2012) (7 pages) doi:10.1115/1.4006112 History: Received July 21, 2011; Revised December 09, 2011; Published May 22, 2012; Online May 22, 2012

## Abstract

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 centerline shown in Fig. 1), i.e., the boundary functions $Ta(x,t)≡T(x,r=a,t)$ and/or $T0(x,t)≡T(x,r=0,t)$. In this manner, the original problem is significantly simplified by reducing the problem dimensionality 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.

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## Figures

Figure 5

A schematic representation of the convective heat transfer with heat exchange with surroundings

Figure 6

Dependence of dimensionless parameter β1 on the Peclet number pe and parameter α for the flow confined by two flat plates (d = 0) and by a circular tube (d = 1)

Figure 4

Comparison of dependence of dimensionless centerline temperature θ on the axial position ξ1 for Eqs. 30, 46, and 58

Figure 3

Dependence of dimensionless parameter β2 (solved from Eq. 42) on the Peclet number pe for the flow confined by two flat plates (d = 0) and by a circular tube (d = 1) for expansion with terms up to the sixth order

Figure 2

Dependence of the dimensionless parameter β1 on the Peclet number pe for the flow confined by two flat plates (d = 0) and by a circular tube (d = 1) for expansions with terms up to the fourth order (Eq. 31) and up to the sixth order (solved from Eq. 42)

Figure 1

A schematic representation of the convective heat transfer that is confined by two flat plates (d = 0, left) and by a circular tube (d = 1, right)

## Errata

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