Table Of ContentANALYTICAL
HEAT DIFFUSION THEORY
A.V. LUIKOV
HEAT AND MASS TRANSFER INSTITUTE
MINSK, BSSR, USSR
Edited by James P. Hartnett
DEPARTMENT OF ENERGY ENGINEERING
UNIVERSITY OF ILLINOIS, CHICAGO, ILLINOIS
1968
ACADEMIC PRESS
A Subsidiary of Harcourt Brace Jovanovich, Publishers
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First published in the Russian language under the title
TEORIYA TEPLOPROVODNOSTI
2nd Edition, Izd. "Vysshaya Shkola," Moscow, 1967
COPYRIGHT © 1968, BY ACADEMIC PRESS, INC.
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EDITOR'S PREFACE
The editor is pleased to have played a role in introducing this textbook
by Academician A. V. Luikov, member of the BSSR Academy of Sciences
and Director of the Heat and Mass Transfer Laboratory in Minsk, BSSR,
USSR.
This work is a revised edition of an earlier book by Academician Luikov
which was widely used throughout the Soviet Union and the surrounding
socialist countries. The presentation is unique in that it not only treats
heat conduction problems by the classical methods such as separation of
variables, but, in addition, it emphasizes the advantages of the transform
method, particularly in obtaining short time solutions of many transient
problems. In such cases, the long time solution may be obtained from the
classical approach, and by interpolation, a very good estimate is obtained
for intermediate times. The text is also noteworthy in that it covers a wide
variety of geometrical shapes and treats boundary conditions of constant
surface temperature, and constant surface heat flux, as well as the technically
important case of a convective boundary condition.
The level of the book is advanced undergraduate or graduate. In addition
to its value as a textbook, the availability of many technically important
results in the form of tables and curves should make the book a valuable
asset to the practicing engineer. The editor is convinced that the work will
be well received by the English reading audience.
JAMES P. HARTNETT
October, 1968
INTRODUCTION
The study of heat transfer is one of the most important fields of modern
science and is of great practical significance in industrial power engineering,
chemical technology, civil engineering, light industry, and other branches
of technology. For example, design of thermal apparatus, design of walls of
buildings undergoing changing thermal effects (heat insulation of buildings,
furnaces, pipe lines), heating of machines, thermal stresses in bridges, and
many other problems involve the solution of unsteady heat conduction
problems. Investigation of the kinetics of sorption, drying, combustion, and
other processes of chemical technology depends on the solution of diffusion
problems, similar to those of transient heat conduction. Some aspects of
unsteady heat transfer acquire special importance in rocket and propulsion
engineering, where thermal apparatus operate under transient conditions.
Thus the analytical heat conduction theory is widely applicable to the solu-
tion of various engineering problems.
The present work differs considerably from earlier monographs of the
author, the last being published in 1952. While in the book " Heat Conduc-
tion and Diffusion " [68] all the problems were solved by contour integra-
tion, in the books published in 1948 and 1952 [69, 71] operational calculus
and the Laplace transformation were used as the main solution methods.
In the present monograph, numerous problems of the heat conduction
theory are solved by different methods: the method of separation of variables
(a classical method for the solution of such problems), the integral Laplace,
Fourier, and Hankel transformations, as well as operational calculus. For
xiii
xîv INTRODUCTION
asymptotic solutions, the contour integration should be applied. Thus, the
present monograph is a summary of all those previously published by the
present author.
The book is a text for students and it considers in detail the solution of
unsteady-state heat conduction problems of basic bodies (semi-infinite body,
infinite plate, solid cylinder, sphere, hollow cylinder) by various methods
(separation of variables, the operational Hankel and Fourier integral trans-
formations).
Thus, the reader who becomes acquainted with peculiarities of every
approach can choose for his work the most realistic method of solution
with regard to conservation of time, labor, and accuracy for application to
design problems.
The solutions are presented in generalized variables and obtained by the
methods of the similarity theory. Much numerical information in the form
of tables and text figures drawn by the author and excellent nomograms
taken from Schneider's work [102] is given, allowing rapid engineering
calculation which will prompt wider application of the solution to engineer-
ing practice. In addition, the solution of the most important problems are
presented in two forms, one of which is convenient for calculation with
small Fourier numbers, and the other with large ones.
The experience of the author in teaching heat conduction theory at various
institutions has shown the necessity of presenting the whole process of
solution in detail with main manipulations and calculations, and to consider
the problems according to their difficulty.
In Chapters 4-6, detailed solutions are therefore illustrated by a large
number of numerical examples and the problems are classified according to
the mode of interaction of a body with the surroundings but not according
to the geometry of bodies. This approach has been more effective from a
pedagogical viewpoint.
Great attention is paid to the solution of problems with boundary con-
ditions of the fourth kind. Such solutions are necessary for realistic studies
in the field of unsteady convective heat transfer. A special chapter (Chapter
13) is devoted to the solution of problems with variable thermal coefficients.
In Chapter 14, a short description is given of application of the Laplace,
Fourier, and Hankel transforms to the solution of unsteady-state heat
conduction problems.
Readers who are interested in more profound problems of the heat con-
duction theory (asymptotic approximations, etc.) are referred to Chapter 15,
where a brief description of the theory of analytical functions and their
application to the solution of heat conduction problems is presented.
INTRODUCTION xv
Reference material in the form of formulas and tables is given in the
Appendixes.
The book is designed mainly for students and engineers who have the
usual background in mathematics. The author aims to educate them in the
solution of problems of unsteady-state heat transfer encountered in engineer-
ing design.
The author expresses his deep gratitude to Professor J. P. Hartnett for
the interest shown in this manuscript and the work done by him in editing
the English translation. He also wishes to thank Mrs. T. Kortneva, G.
Malyavskaya, and E. Bogacheva from the Heat and Mass Transfer Institute
of the BSSR Academy of Sciences for the English translation of this book.
CHAPTER
1
PHYSICAL FUNDAMENTALS OF HEAT TRANSFER
In the present chapter the main principles of the phenomenological heat
conduction theory are given.
When heat is. transferred from one part of a body to another, or from
one body to another which is in contact with it, the process is usually referred
to as heat conduction. In the phenomenological heat conduction theory,
the molecular structure of a substance is neglected; the substance is con-
sidered as a continuous medium (continuum), but not as a combination of
separate discrete particles. Such a model of the substance may be adopted
when solving problems on heat propagation, provided that differential
volumes are large compared with molecule sizes and distances between them.
In all the following calculations and examples, the body is assumed isotropic
and uniform.
1.1 Temperature Field
Any physical phenomenon, including the heat transfer process, occurs
in time and space. Analytical investigation into heat conduction reduces,
therefore, to the study of space-time variations of the main physical quantity
(temperature) peculiar to a certain process, i.e., to the solution of the equation
t=Äx,y>z*r), 0-1-1)
where x, y, z are Cartesian space coordinates and τ is time.
The instantaneous values of temperature at all points of the space of
interest is called a temperature field. Since temperature is a scalar quantity,
then so is a temperature field.
1
2 1. PHYSICAL FUNDAMENTALS OF HEAT TRANSFER
A distinction is made between a steady and transient temperature field.
A transient temperature field is one in which the temperature varies not
only in space but also with time; in other words, "temperature is a function
of space and time" (the unsteady state). Equation (1.1.1) is a mathematical
representation of a transient temperature field.
A steady temperature field is one in which the temperature at any point
never varies with time, i.e., it is a function of the space coordinates solely
(the steady state):
t = φ(χ, y z\ dt/θτ = 0. (1.1.2)
9
In some problems, a transient temperature field becomes asymptotically
steady when τ -> oo.
A temperature field governed by Eq. (1.1.1) or (1.1.2) is spatial (three-
dimensional) since Ms a function of three coordinates. When temperature
is a function of two coordinates, the field is then two-dimensional:
/ = F(x, y, τ), dt /dz = 0.
If temperature is a function of one space coordinate alone, then the field
is one-dimensional:
t = <p(x, τ), dt/dy = dt/dz = 0.
The field of an infinite plate (i.e., the width and the length are infinitely
large compared with its thickness) affords an example of a one-dimensional
temperature field, the heat flow being normal to the plate surface.
If the points of the field with equal temperatures are connected, an iso-
thermal surface results. Intersection of the isothermal surface with a plane
Fig. 1.1 Isotherms of temperature field.
(Letters with arrows correspond to bold-
face type in the text.)
12 The Fundamental Fourier Heat Conduction Law 3
yields a family of isotherms on the plane surface (lines corresponding to
equal temperature). Isothermal surfaces and isothermal lines never inter-
sect inside the field when it is continuous. In Fig. 1.1, several isothermal
lines are shown, each differing from its neighbor by the amount At.
By definition, the temperature does not change along the isotherm, but
in any other direction it may vary. Temperature change per unit length
is a maximum along the normal to the isothermal surface.
The temperature increase along the normal to the isothermal surface is
characterized by a temperature gradient (grad /). A temperature gradient
is a vector along the normal to the isothermal surface in the direction of
the increasing temperature, i.e.,
grad t = n (dt/dn), (1.1.3)
0
where n denotes a unit vector,1 along the normal in the direction of the tem-
0
perature change (see Fig. 1.1) and dt/dn is the temperature derivative along
the normal (n) to the isothermal surface. A temperature gradient is there-
fore the first temperature derivative along the normal to the isothermal
surface. The gradient is also denoted by V. Gradient components along
the Cartesian coordinates are identical to the appropriate partial derivaties,
so that
*-, . dt , . dt , _ dt , ..
ίΛ
= r = - - - + *— (1.1.4)
gndt t 11 + J g
where i, j, k are mutual orthogonal vectors of a unit length along the coor-
dinate axes. This relation is possible because any vector may be represented
as a vectorial sum of three components along the coordinate axes.
The concept of a temperature-field intensity may be introduced:
E= -grad*. (1.1.5)
The vector E is referred to as a vector of the temperature field intensity.
1.2 The Fundamental Fourier Heat Conduction Law
The necessary condition for heat conduction is the existence of a tem-
perature gradient. Experience shows that heat is transferred by conduction
in the direction normal to the isothermal surface from a higher tempera-
ture level to a lower one.
1 Vectors are shown by boldface type and their scalar values by italic type.
4 1. PHYSICAL FUNDAMENTALS OF HEAT TRANSFER
The quantity of heat transferred per unit time per unit area of the isothermal
surface is referred to as a heat flux ; the appropriate vector is obtained by
the relation
q (1 2 1}
-<-*>w^· · ·
where dQ/dr is quantity of heat transferred per unit time, or the heat-flow
rate, S is the isothermal surface area, and (—n) is a unit vector along the
0
normal to the area S in the direction of the decreasing temperature.
The vector q is therefore called a heat flux vector, the direction of which is
opposite to that of the temperature gradient (both vectors follow the nor-
mal to the isothermal surface, but their directions are opposite to each other).
The projection of the vector q on any arbitrary direction / is also the vec-
tor q, the scalar quantity of which is q cos(«, /).
t
The lines which coincide with the direction of the vector q are referred
to as heat-flow lines. These are perpendicular to the isothermal surfaces
at the intersection points. A tangent to the heat-flow lines taken in the
opposite direction yields the temperature gradient direction (Fig. 1.1).
The fundamental heat-conduction law may be formulated as follows: the
heat flux is proportional to the temperature field intensity, or the heat flux
is proportional to the temperature gradient, i.e.,
q = XE = - λ grad t= -XVt= - λη (βί/θη), (1.2.2)
0
where λ is the proportionality factor called thermal conductivity.
To reveal the physical significance of the thermal conductivity, we shall
write the basic relation (1.2.2) for a steady one-dimensional temperature
field for the situation where the temperature depends only on one coordinate
which is normal to the isothermal surfaces. The scalar quantity of the heat-
flux vector is
: dt (dt dt dt n\ n o ^
If the temperature gradient is a constant value (dt/dx = const), which
means the temperature variation with x follows the linear law, then it may
be written
A A ^A = const. (1.2.4)
=
ax X2 X\
Hence the heat-flow rate dQ/dr is also a constant value
