Doctoral Dissertations

Date of Award

5-1991

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mechanical Engineering

Major Professor

Robert J. Krane

Committee Members

J. W. Hodgson, J. R. Parsons, M. Parang, J. J. Perona

Abstract

An analytical study was performed to determine optimal heat flux distributions along one vertical wall of a two-dimensional rectangular enclosure cooled by natural convection. Two different optimization problems were investigated. The primary problem of interest, referred to as "optimization problem number 1", was to determine the local heat flux distribution along the heated vertical wall of the rectangular enclosure such that either the average or peak temperature of the heated wall is minimized with the constraint that the total heating rate along the wall is a specified constant. The alternative problem, referred to as "optimization problem number 2," was to determine the local heat flux distribution along the heated vertical wall of the rectangular enclosure such that the total heating rate is maximized subject to the constraints that the average temperature of the heated wall is a specified constant and the peak temperature of the heated wall is less than a specified constant. To perform these studies a numerical model of the steady state, laminar, natural convection process that occurs in the enclosure was constructed using the finite volume approach as embodied in the SIMPLER algorithm. The natural convection model was coupled with a numerical optimization code which is based on the generalized reduced gradient (GRG) method. Preliminary studies were performed for a reference set of parameters to define appropriate grid sizes and to fix the methods to be used for a parametric study. The reference parameters included a modified Rayleigh number (based on average heat flux on the heated vertical wall) of 105, a Prandtl number and enclosure aspect ratio of unity, and boundary conditions with adiabatic top and bottom walls and an isothermal vertical wall opposite the heated vertical wall. Results of grid refinement studies show that a 13 x 13 nonuniformly spaced grid is adequate to accurately predict the behavior of the natural convection process and is compatible with performing optimization studies which require many repeated solutions of the natural convection problem. Solutions to optimization problems number 1 and number 2 are identical for corresponding constraints. However, because optimization problem number 2 has constraints that are nonlinear functions of the heat flux, considerably more computing time was required for its solution. Two alternative methods for representing the local nonuniform heat flux distribution on the heated vertical wall were also evaluated. Approximating the continuous heat flux distribution as the set of heat fluxes at each discrete grid point along the heated wall was shown to achieve more optimal results and required only slightly more computing time than using the coefficients of polynomial heat flux distributions as the independent variables. Parametric studies were performed to determine the sensitivity of the optimal solutions to the nondimensional parameters and type of thermal boundary conditions. Based on the results of the preliminary studies, these parametric studies were performed using a 13 x 13 nonuniform grid with the heat flux at the grid points treated as the independent variables and the total heating rate along the heated vertical wall fixed (optimization problem number 1). Rayleigh numbers from 10 to 106, Prandtl numbers from 0.1 to 100, enclosure aspect ratios from 0.5 to 2, and three sets of thermal boundary conditions on the remaining walls of the enclosure were considered in the parametric sudies. Optimization study results demonstrated that compared to a uniform heat flux distribution significant reductions in either average or peak temperature of the heated wall are possible for optimal heat flux distributions. For example, at reference parameters, the minimal average nondimensional temperature of the heated vertical wall is 17% lower than the average nondimensional temperature achieved for a uniform heat flux distribution with the same total heating rate.

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