## Document Type

Thesis - Open Access

## Award Date

1968

## Degree Name

Master of Science (MS)

## Department / School

Mechanical Engineering

## Abstract

It has been recognized that many convection heat transfer problems involve internal heat generation. Nuclear fuel elements, chemically reactive liquids and transparent shell solar energy collectors are a few examples. Theoretically, this type of problem has not been studied extensively. Lumsdaine solved the internal heat generation problem with constant temperature boundary conditions in spherical coordinates. The purpose of this thesis is to follow Lumsdaine's analysis to study those problems where internal heat generation is a function of radius. The governing partial differential equation in spherical coordinates is of the second order, non-homogeneous and has variable coefficients. The non-homogeneity arises from the internal heat generation function in the energy equation. The boundary conditions are also non-homogeneous. In order to solve this problem, steady state, an incompressible inviscid fluid, a constant fluid entrance temperature, and constant ambient temperatures are assumed. In solving the problem, the method of superposition is first used to form a non-homogeneous ordinary differential equation and a homogeneous partial differential equation. By doing this, the boundary conditions are also simplified. These two equations are then solved separately. The ordinary differential equation can be integrated directly. The partial differential equation is solved by using separation of variables. The solution is given as an infinite series of Euler functions. The temperature field is finally obtained by superimposing the two solutions. The expression for the average exit temperature is derived from the temperature field. The average exit temperature is non-dimensionalized in terms of Graetz number, Nusselt number, radius ratio and other dimensionless groups. Computer programs are developed to obtain numerical results. In the first part of the thesis, the inner shell is assumed to be at a constant temperature. The problem is first solved in general without specifying the internal heat generation function; therefore the solution holds for any type of internal heat generation. In the sample problem the internal heat generation function is assumed to consist _of a constant term and an exponential term to account for the decay from the outer shell. In the second part of the thesis, the solution emphasizes its -application to the solar heat exchanger. By assuming the outer shell transparent, the absorption of solar energy by the water can be interpreted as internal heat generation. The inner shell is assumed to be a black body which absorbs all the energy not absorbed by the water; this energy is then conducted back to the fluid. The efficiency of this type of solar heat exchanger should be higher than in exchangers with an opaque outer shell since the latter have higher temperatures at the outer shell and thus lose more heat to the surroundings.

## Library of Congress Subject Headings

Heat exchanges -- Solar energy

## Format

application/pdf

## Publisher

South Dakota State University

## Recommended Citation

Tsou, John Lin, "Mathematical Model of a Spherical Shell Heat Exchanger with Exponential Internal Heat Generation" (1968). *Electronic Theses and Dissertations*. 3507.

https://openprairie.sdstate.edu/etd/3507