SOME HEAT TRANSFER PROBLEMS

Abstract

This thesis is devoted to studying some multi dimensional, transient boundary-value problems in Heat Conduction, primarily with aview to applying them for predicting the thernal behaviour 8* building elements and building enclosures. The results may, however, find applications in wider fields like geophysics, highway engineering and machine design. The thesis has been divided into two parts and an introductory chapter. The introductory chapter outlines the objectives of the investigations and provides the necessary perspective. The first part deals with the basic problems and consists of six chapters. Applications to buildings have been considered in the second part. The first chapter is introductory in character and reviews the mechanism of heat transfer in buildings. It briefly explains the heat exchange at the outside surface, heat flow through the fabric and the heat exchange at the inside surface. It further considers the various types of problems encountered and locates the areas of investigation. With t;is background, the motiv ation for the choice of basic problems, dealt with in subsequent chapters in the first part of the thesis, can be easily comprehended. The second chapter deals with the heating of a half space, on the surface of which, different rates of heating are applied to a strip and the region outside it. The heat inputs are arbitrarily time-dependent. Expressions for the time-varying temperature distribution are obtained in an integral form for the general case as well as constant inputs, uniform heating and periodic heat inputs. The integrals are evaluated numerically on a digital computer for the periodic case and values are tabul ated for different strip widths and frequencies. The numerical procedure for quadrature, in the case of oscillatory integrals with large range, has been specially devised with the aim of reducing computat ional effort for any desired accuracy and is given in an appendix. The temperature distribution in a homogeneous and isotropic semi-infinite solid, to which heat is supplied at arbitrarily time-dependent rates but with different magnitudes on the insides and outsides of the rectangular region on the surface, has been considered in chapter three. On the application of Fourier trans forms to the three dimensional heat conduction equation and the boundary conditions, the problem is reduced to a pair of dual integral equations. By solving these, the temperature is obtained in a closed form in terms of heat input functions and the Error functions. For the case of periodic inputs, the results obtained have been compared with those obtained in chapter two. In the fourth chapter, the problem studied is regarding the temperature distribution in an infinitely long prism with rectangular cross-section and having uniform and fixed thermal properties, when it is subjected to time-dependent boundary conditions. Convective boundary conditions with different values of surface coefficients of heat transfer have been imposed on one set of coordinate surfaces and heat flux boundary conditions on the other set. A method due to Olcer, which combines the finite integral transforms with separation of variables technique, has been used to derive the solution in the form of a double series. In the case of periodic inputs, complex representation has been used to express the thermal transmission func tions for all the inputs. These have been computed on a digital computer for various points of three different walls and at three different frequencies. The fifth chapter deals with a homogeneous rectangular parallelepiped subjected to heat flux and convective boundary conditions on its various lit surfaces. The thermal inputs are general timevarying functions and surface coefficients of heat transfer have fixed values, which may be different on different surfaces. Olcer's method is applied to the three dimensional heat conduction equation and the boundary conditions to obtain a triple series solution for the temperature distr ibution. Six complex thermal transmission functions corresponding to the six periodic thermal inputs have been computed for three frequencies and three types of walls and compared to those obtained in chapter four. In the sixth chapter, variational methods, due to Biot and Prigogine et.al. have been used to obtain the penetration depth for a homogeneous semi-infinite solid, subjected to transient direct and convective heating on its surface. To keep the resulting first order nonlinear differential equation in a tractable form, quadratic temperature profile and Chu's approximation for the convective heat transfer have been assumed. For power law variation of the stream temperature and surface coefficient, an iterative procedure has been empl oyed and the solution obtained in a closed form. For exponential variation, Runge-Kutta method has been used to integrate the initial value problem. IV Penetration depth versus time curves and transient temperatare-time history have been plotted for various values of thermal and time variation para meters. In the seventh chapter, transient temperature distribution for a slab of finite thickness has been determined when it is subjected to convective heating on one face and it is losing heat to a medium at zero temperature on the other face.Lagrangian methods due to Biot along with Chu's approximation are used for time-dependent streaa temperature an3 surface coeffi cients of heat transfer. To specify the initial value, transit time is computed from the sixth chapter. Using a quadratic temperature profile, the Lagrangian equi valent of me dimensional heat conduction equation comes out to be a first order linear differential equation with time dependent coefficients. This is integrated to yield the temperature time history for various values of thermal and time-variation parameters. In the second part of the thesis comprising the applications aspect of the heat transfer problems studied in part one, computational methods have been developed to predict the thermal behaviour of building enclosures which may be on the ground or a part of higher storeys and which may have one or more unusually thick building elements exposed at edges. In chapter eight, three types of buildings have been considered. For the simplest case of building enclosures forming apart of the upper storeys and having normally thick building elements, the matrix method, based on one dimensional heat flow considerations, has been coupled to an indoor convective radiative exchange network so as to take into account the interaction between the different elements. In the case of the building enclosures having it's floor laid directly on the ground, the modified matrix method is extended to include the integrated effect of the floor and the ground by using the expressions derived in chapters two and three. In the case of enclosures built on raised plinths on the ground, as Is the normal Indian practice, the floor is coupled to the ground through the plinth by using the expressions for thermal transmission functions derived in chapters four and five. In the case of long buildings, two dimensional expressions are used and in the case of end rooms of long buildings or isolated rooms of normal dimensions, three dimensional expressions are used. The conditions for the methods to be applicable to unconditioned, conditioned and partially conditioned buildings and the terms to take into account the Internal masses, ventilation and internal loads due to lighting, appliances and VI occupancy are explicitly stated. In chapter nine, an experimental set-up has been described for measuring the thermal data for experimental verification of the procedures evolved. A comparison has been made between the experimentally measured temperatures and computed values for an actual building unit laid on ground on a raised plinth and the results are found to agree within less than five percent. The analyses and the results embodied in the thesis are the candidate's original work except for the major part of introductory chapter one, which has been included to make the account complete.