PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Vol. 89　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1957

STUDY OF NONSTATIONARY FIELDS OF TEMPERATURE AND MOISTURE IN AN INFINITE PLATE WITHOUT MOISTURE EVAPORATION

S. P. KUZNETSOV

Fourier’s law lies at the foundation of the mathematical theory of heat conduction; it establishes the dependence between the density of the heat flux \(q\) and the temperature gradient \(\nabla t\), i.e.

\[ q=-\lambda \nabla t. \]

In this formulation Fourier’s law is valid for isotropic bodies in which the transfer of heat is due to molecular transport.

In the case of solid bodies, Fourier’s law holds for isotropic materials that do not absorb moisture or absorb little of it. Many thermal insulators, most building materials, and also materials of the food industry absorb moisture from the surrounding medium and thus are moist materials. In moist materials the transfer of heat is complicated by the phenomenon of moisture transfer, as a result of which the fundamental Fourier law is modified: the density of the heat flux depends not only on the temperature gradient, but also on the density of the moisture flux, i.e.

\[ q=-\lambda \nabla t+Ji, \]

where \(J\) is the enthalpy of moisture in kcal/kg.

Consequently, in moist materials the transfer of heat is inseparable from the transfer of moisture, and conversely.

In the earliest works, heating of moist materials was equated with heating of bodies with a negative heat source. Among the first works in this field were those of Prof. A. V. Lykov [1]. Thus, for example, he gave the solution of the problem of heating an infinite plate when, between one surface of the plate and the surrounding medium with constant temperature, heat exchange occurs according to Newton’s law with some additional heat source, while on the other surface a constant temperature \(t_n\) is maintained. He also gave the solution of a more complicated problem, when evaporation takes place on its opposite surfaces, with different intensities.

The heating of moist soils was studied by A. V. Lykov and A. A. Pomerantsev [3]. They investigated the motion of the drying front in moist soil and the distribution of temperatures in it during heating. The question of moisture transfer in the soil was not touched upon in the work: the simplest scheme of moisture distribution in the soil was used—the concentration of moisture in the dried part of the soil was taken to be equal to zero, and in the moist part, to the initial value. Existing solutions for the heating of moist materials have substantial shortcomings:

1) it is assumed that evaporation of moisture inside the material is absent, i.e. evaporation occurs only at the surface of the material;

2) the influence of moisture transfer on heat transfer is not taken into account, i.e., heat exchange inside a moist material is considered separately from moisture exchange.

This state of affairs in the theory of heating moist materials is explained by the fact that until recently the laws of moisture transfer inside a material had not been well studied.

The basic laws of moisture transfer, both in the liquid and in the gaseous state, are: the law of diffusion (molecular transfer) and the law of capillary motion (molar transfer). Under isothermal conditions, these two laws are combined into a single law—the law of moisture conductivity: the moisture flux density \((i)\) is directly proportional to the concentration gradient \((\nabla \omega)\) or to the moisture-content gradient \((\nabla u)\) in the direction normal to an isoconcentration surface, i.e.,

\[ i=-k\nabla\omega=-k\gamma_0\nabla u, \]

where: \(k\) is the coefficient of moisture conductivity, in \(m^2/\text{h}\),
\(i\) is the moisture flux density, in \(\text{kg}/m^2\text{h}\).

The mechanism of moisture transfer depends on the form in which the moisture is bound to the material. According to the classification of Academician P. A. Rebinder, the following are distinguished:

1) chemical bonding,

2) physicochemical bonding,

3) mechanical bonding.

The moisture-conductivity coefficient \(k\) depends not only on the moisture content of the material, but also on the temperature. As the temperature increases, the moisture-conductivity coefficient increases.

If there is a temperature difference inside a moist body, then, under the influence of the temperature gradient, moisture is transferred in the direction of the heat flux. It may be transferred both in the form of liquid and in the form of vapor.

One of the causes of such moisture migration is the molecular motion of liquid and vapor—the phenomenon of thermal diffusion.

A second cause is the decrease of the capillary potential with increasing temperature. In places with a higher temperature the capillary potential is lower, and therefore moisture moves toward lower temperatures.

A third cause may be the action of “entrapped” air.

The totality of these factors, which cause moisture transfer in a capillary-porous body, is called thermal moisture conductivity. The phenomenon of thermal moisture conductivity plays an important role in soil science, structural mechanics, drying theory, etc.

Thus, if inside a moist material there is not only a moisture-content gradient but also a temperature gradient, then the law of moisture transfer will have the form:

\[ i=-k\gamma_0(\nabla u+\delta\nabla t), \]

where \(\gamma_0\) is the specific weight of the dry substance, in \(\text{kg}/m^3\),
\(\delta\) is the coefficient of thermal moisture conductivity, in \(\%/\text{degree}\).

Until now, in studying the moisture-content field, the effect of thermal moisture conductivity has been neglected. Under this assumption, the differential equation of moisture conductivity becomes the differential equation of heat conduction, in which \(t\) and \(\lambda\) are replaced respectively by \(u\) and \(k\), i.e.,

\[ \frac{\partial u}{\partial \tau}=k\nabla^2 u. \]

Using ready-made solutions for the heat-conduction equation under the corresponding boundary conditions, solutions were obtained for various problems of determining the moisture field. Such a treatment of the question may be admissible when the temperature field has little effect on the moisture field of the material.

In connection with the intensification of hygrothermal processes by means of temperature fields with large gradients, the existing solutions for the moisture field and the temperature field become unsuitable, since at high intensities the temperature field significantly affects the moisture field, and vice versa. The development of a new theory taking into account the interrelation of the moisture and temperature fields has become an urgent necessity.

The study of nonstationary thermo-hygrothermal fields is associated with the solution of a system of differential equations:

\[ \frac{\partial u}{\partial \tau} = \kappa\left(\nabla^2 u+\delta \nabla^2 t\right) \]

\[ -\frac{\partial t}{\partial \tau} = a\nabla^2 t+\frac{\varepsilon r}{C}\frac{\partial u}{\partial \tau}, \tag{1} \]

where \(a\) is the coefficient of thermal diffusivity,

\[ \nabla^2=\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2} \quad \text{is the Laplace operator,} \]

\(r\) is the heat of evaporation, in \(\mathrm{kcal/kg}\),

\(C\) is the reduced specific heat capacity of the body, in \(\mathrm{kcal/kg\ deg}\),

\(\varepsilon\) is the evaporation criterion, which characterizes the ratio between the amount of moisture moving in the form of vapor and in the form of liquid \((0<\varepsilon<1)\).

If \(\varepsilon=0\), then all the moisture moves in the form of liquid (there is no evaporation inside); if \(\varepsilon=1\), then all the moisture moves in the form of vapor, and the change in moisture inside the body occurs only on account of evaporation.

For the uniqueness of the solutions of this system in any particular cases, boundary conditions must be appended to them. Under prescribed boundary conditions, the temperature and moisture content are uniquely determined by system (1).

Nonstationary moisture and temperature fields without moisture evaporation (boundary condition of the first kind)

At comparatively low temperatures \((t \leqslant 50^\circ)\), evaporation of moisture inside a moist body almost does not occur; consequently, the internal evaporation criterion \(\varepsilon \to 0\). In this case the system of differential equations takes the form:

\[ -\frac{\partial t}{\partial \tau}=a\nabla^2 t \]

\[ \frac{\partial u}{\partial \tau} = \kappa\nabla^2 u+\kappa\delta\nabla^2 t. \tag{2} \]

The temperature field of the moist body will be described by the same differential equation as for a dry body, since internal negative sources of heat (moisture evaporation) are absent. Such a situation will exist when moisture moves in the form of liquid. The latter

and does occur; this is attested by experimental studies at low temperatures.

For the simplest boundary conditions the problem can be formulated as follows.

We have a moist body of simple geometric shape at temperature \(t_0\) and moisture content \(u_0\). At the initial instant of time the surface of the body instantaneously assumes a temperature equal to \(t_c\) (in the special case \(t_c=0\)), which is then maintained constant throughout the entire cooling process \((Bi \to \infty)\). The surface of the body has moisture insulation that conducts heat well (a metal shell), as a result of which the mean moisture content of the body remains unchanged and equal to the initial moisture content throughout the entire cooling process. It is required to find the moisture field and the temperature field for any instant of time.

The author has given a solution of this problem for those classical forms which, in the theory of heat conduction, are called the fundamental bodies (an infinite plate, a sphere, and an infinite cylinder). For bodies of complex configuration, the solution of the given problem can be obtained from the solutions for the fundamental bodies by means of the method of A. I. Veinik [2]. The essence of this method is based on certain features of nonstationary temperature fields. If regions near the surface of the body are excluded from consideration, then the temperature field of a body of complex configuration can be replaced by the temperature field of a body of fundamental form, introducing the approximate similarity criterion \(E\) as a factor multiplying the coefficient of thermal diffusivity (the first variant of Veinik’s method) or the heat-transfer coefficient (the second variant). The approximate similarity criterion \(E\) is equal to the ratio of the surface area \((s)\) of the body of complex configuration to the surface area \((s_{\text{fund.}})\), equal in volume, of the fundamental body, i.e.

\[ E=\frac{s}{s_{\text{fund.}}}. \]

Thus, solutions of problems for bodies of fundamental form acquire still greater practical significance, since they can be used to calculate the nonstationary fields of bodies of complex configuration.

Solution for an Infinite Plate

The problem is formulated mathematically by the system:

\[ \frac{\partial t(x,\tau)}{\partial \tau} = a\,\frac{\partial^2 t(x,\tau)}{\partial x^2} \]

\[ \frac{\partial u(x,\tau)}{\partial \tau} = \kappa\,\frac{\partial^2 u(x,\tau)}{\partial x^2} + \kappa\delta\,\frac{\partial^2 t(x,\tau)}{\partial x^2} \tag{3} \]

under the boundary conditions:

\[ t(x,0)=t_0=\text{const} \]

\[ u(x,0)=u_0=\text{const} \]

\[ \frac{\partial u(0,\tau)}{\partial x}=0. \]

\[ \frac{\partial t(0,\tau)}{\partial x}=0. \]

\[ t(R,\tau)=0, \]

\[ u_0=\frac{1}{R}\int_{0}^{R}u(x,\tau)\,dx, \]

where \(R\) is half the thickness of the plate; the remaining notation is as before.

We shall integrate system (3), under the indicated boundary conditions, by means of the Laplace transform. From the first equation of system (3) it follows that

\[ T''(x,s)-\frac{s}{a}T(x,s)+\frac{t_0}{a}=0, \tag{4} \]

where

\[ T(x,s)=\int_{0}^{\infty}t(x,\tau)e^{-s\tau}\,d\tau=L[t(x,\tau)]. \]

The general solution of the differential equation (4) is

\[ T(x,s)-\frac{t_0}{s} = A\operatorname{ch}\sqrt{\frac{s}{a}}\,x + B\operatorname{sh}\sqrt{\frac{s}{a}}\,x, \tag{5} \]

where \(A\) and \(B\) are constants independent of \(x\), but containing \(s\); they can be found from the boundary conditions, which for the transform take the form

\[ T'(0,s)=0, \]

\[ T(R,s)=0, \]

from which it follows that the coefficient \(B=0\),

\[ A=-\frac{t_0}{s\operatorname{ch}\sqrt{\frac{s}{a}}\,R}. \]

Thus, the solution for the transform (5) can be represented in the form

\[ T(x,s)=\frac{t_0}{s} - \frac{t_0\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {s\operatorname{ch}\sqrt{\frac{s}{a}}\,R}. \]

Applying the inverse Laplace transform to the last equality, we find the original—the temperature distribution function in the plate. The inverse transform is determined by the integral relation

\[ t(x,\tau)=L^{-1}[T(x,s)] = \frac{1}{2\pi i} \int_{\sigma-\infty i}^{\sigma+\infty i} e^{s\tau}\cdot T(x,s)\,ds, \]

which for the present problem takes the form

\[ t(x,\tau)-t_0 = -\frac{t_0}{2\pi i} \int_{\sigma-\infty i}^{\sigma+\infty i} \frac{ e^{s\tau}\operatorname{ch}\sqrt{\frac{s}{a}}\,x }{ s\operatorname{ch}\sqrt{\frac{s}{a}}\,R } \,ds. \tag{6} \]

The integrand is a single-valued function with respect to \(s\), whose singular points will be simple poles: \(s=0\) and the numbers \(s_n\), determined by the equality:

\[ i \sqrt{\frac{s_n}{a}}\, R = (2n-1)\frac{\pi}{2} = \psi_n, \]

i.e.

\[ s_n = -\psi_n^2 \cdot \frac{a}{R^2} \qquad \text{for } n=1,\,2,\,3\ldots \]

It is easy to see that one pole is located at the origin, while all the others are located on the negative direction of the real axis.

To compute the integral (6), we shall apply Cauchy’s residue theorem. Consider the closed contour \(MNPM\), where \(MN\) is parallel to the imaginary axis and lies at a distance \(\sigma\) from it.

If the radius of the circle \(R_1\) is denoted by

\[ an^2\frac{\pi^2}{R^2}, \]

then the circle will not pass through any of the poles of the integrand. In mathematical analysis it is proved that in this case the integral over the arc \(MPN\) tends to 0 as \(n \to \infty\). Consequently, the integral:

\[ \frac{1}{2\pi i} \int_{\sigma-\infty i}^{\sigma+\infty i} \frac{e^{s\tau}\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {s\,\operatorname{ch}\sqrt{\frac{s}{a}}\,R}\,ds \]

may be replaced by the limit of the integral over the closed contour \(MNPM\) as \(n \to \infty\).

Fig. 1

On the basis of Cauchy’s theorem it is equal to the product of the sum of the residues of the integrand by \(2\pi i\), i.e.

\[ \frac{t(x,\tau)-t_0}{t_0} = -\left\{ \operatorname{res.} \left[ \frac{e^{s\tau}\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {s\,\operatorname{ch}\sqrt{\frac{s}{a}}\,R} \right] + \sum_{n=1}^{\infty} \operatorname{res.} \left[ \frac{e^{s\tau}\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {s\,\operatorname{ch}\sqrt{\frac{s}{a}}\,R} \right] \right\}. \]

The first residue for the pole \(s=0\) is equal to 1. The sum of the residues for the remaining poles, located on the negative direction of the axis of abscissas, is equal to:

\[ -\sum_{n=1}^{\infty} (-1)^{n+1} \frac{2\cos\psi_n \dfrac{x}{R}}{\psi_n} e^{-\psi_n^2 \dfrac{a\tau}{R^2}}. \]

Thus, finally,

\[ \frac{t(x,\tau)-t_0}{t_0} = -1 + 2\sum_{n=1}^{\infty} (-1)^{n+1} \frac{\cos\psi_n \dfrac{x}{R}}{\psi_n} e^{-\psi_n^2 \dfrac{a\tau}{R^2}}, \]

or

\[ \Theta(x,\tau)=\frac{t(x,\tau)}{t_0} =2\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\cos \mu_n \dfrac{x}{R}}{\mu_n}\cdot e^{-\mu_n^2\dfrac{a\tau}{R^2}}, \tag{7} \]

where \(\mu_n=(2n-1)\dfrac{\pi}{2}\),

\[ F_0=\frac{a\tau}{R^2} \]
is the thermal Fourier criterion,

\[ \frac{x}{R} \]
is the dimensionless coordinate,

\(\Theta(x,\tau)\) is the relative temperature of the plate at any point.

To determine the function of moisture distribution in the plate, we apply the Laplace transform to the second equation of system (3). We obtain

\[ \kappa F''(x,s)-sF(x,s)+\kappa\delta T''(x,s)+u_0=0, \]

where

\[ F(x,s)=\int_{0}^{\infty}u(x,\tau)e^{-s\tau}\,d\tau, \]

which can be represented as

\[ F''(x,s)-\frac{s}{\kappa}\left[F(x,s)-\frac{u_0}{s}\right] -\frac{\delta t_0}{a}\, \frac{\operatorname{ch}\sqrt{\dfrac{s}{a}}\,x} {\operatorname{ch}\sqrt{\dfrac{s}{a}}\,R} =0. \]

Applying the Laplace transform once more with respect to \(x\), after some operations we obtain

\[ F(x,s)-\frac{u_0}{s} =A\,\operatorname{ch}\sqrt{\frac{s}{\kappa}}\,x +B\,\operatorname{ch}\sqrt{\frac{s}{\kappa}}\,x+ \]

\[ +\frac{\delta t_0\left[\operatorname{ch}\sqrt{\dfrac{s}{a}}\,x-\operatorname{ch}\sqrt{\dfrac{s}{\kappa}}\,x\right]} {s\left(1-\dfrac{a}{\kappa}\right)\cdot \operatorname{ch}\sqrt{\dfrac{s}{a}}\,R}. \]

The coefficients \(A\) and \(B\) are determined from the boundary conditions:

\[ F'(0,s)=0 \]

\[ \frac{1}{R}\int_{0}^{R}F(x,s)\,dx=\frac{u_0}{s}, \]

and we obtain

\[ A=-\frac{\delta t_0\left[\sqrt{\dfrac{a}{\kappa}}\,\operatorname{sh}\sqrt{\dfrac{s}{a}}\,R-\operatorname{sh}\sqrt{\dfrac{s}{\kappa}}\,R\right]} {\left(1-\dfrac{a}{\kappa}\right)\cdot s\,\operatorname{ch}\sqrt{\dfrac{s}{a}}\,R\cdot \operatorname{sh}\sqrt{\dfrac{s}{\kappa}}\,R} \]

and

\[ B=0. \]

Consequently, the image of the moisture-distribution function has the form:

\[ F(x,s)-\frac{u_0}{s}= \]

\[ = \frac{ \delta t_0\left[ \sqrt{\frac{a}{\kappa}}\,\operatorname{sh}\sqrt{\frac{s}{a}}\,R\, \operatorname{ch}\sqrt{\frac{s}{\kappa}}\,x - \operatorname{ch}\sqrt{\frac{s}{a}}\,x\, \operatorname{sh}\sqrt{\frac{s}{\kappa}}\,R \right] }{ \left(\frac{a}{\kappa}-1\right)s\, \operatorname{sh}\sqrt{\frac{s}{\kappa}}\,R\cdot \operatorname{ch}\sqrt{\frac{s}{a}}\,R } = \]

\[ = \frac{\delta t_0}{\left(\dfrac{a}{\kappa}-1\right)} \cdot \frac{\Phi(s)}{\psi(s)}. \tag{8} \]

The ratio \(\dfrac{\Phi(s)}{\psi(s)}\) can be represented as the ratio of two generalized polynomials in \(s\), with the polynomial in the denominator having degree one higher than the degree of the numerator, which makes it possible, in passing to the original, to use the expansion theorem. Applying the expansion theorem, we obtain

\[ u(x,\tau)-u_0= \frac{\delta t_0}{\left(\dfrac{a}{\kappa}-1\right)} \cdot \left[ \lim_{s\to 0}\frac{\Phi(s)}{\psi'(s)} + \sum_{n=1}^{\infty} \frac{\Phi(s_n)}{\psi'(s_n)}e^{s_n\tau} + \right. \]

\[ \left. + \sum_{m=1}^{\infty} \frac{\Phi(s_m)}{\psi'(s_m)}e^{s_m\tau} \right], \tag{9} \]

where

\[ \psi'(s)= \operatorname{sh}\sqrt{\frac{s}{\kappa}}\,R\cdot \operatorname{ch}\sqrt{\frac{s}{a}}\,R + \frac{1}{2}\sqrt{\frac{s}{\kappa}}\,R\cdot \operatorname{ch}\sqrt{\frac{s}{a}}\,R + \]

\[ + \frac{1}{2}\sqrt{\frac{s}{a}}\,R\, \operatorname{sh}\sqrt{\frac{s}{a}}\,R\cdot \operatorname{sh}\sqrt{\frac{s}{\kappa}}\,R, \]

\[ s=0,\qquad s_n=-\mu_n^2\frac{a}{R^2},\qquad s_m=-\mu_m^2\frac{\kappa}{R^2} \]

and

\[ \mu_n=i\sqrt{\frac{s}{a}}\,R=(2n-1)\frac{\pi}{2}, \qquad \mu_m=i\sqrt{\frac{s}{\kappa}}\,R=m\pi. \]

After evaluating each term of equality (9), the moisture-distribution function in the plate takes the form

\[ \vartheta(x,\tau)=\frac{u(x,\tau)-u_0}{u_0}= \]

\[ = \frac{\delta t_0}{u_0\left(\dfrac{a}{\kappa}-1\right)} \left[ 2\sum_{n=1}^{\infty} \frac{ \cos(2n-1)\dfrac{\pi x}{2R}\cdot \sin\sqrt{\dfrac{a}{\kappa}}\,(2n-1)\dfrac{\pi}{2} }{ (-1)^{n+1}(2n-1)\dfrac{\pi}{2}\cdot \sin\sqrt{\dfrac{a}{\kappa}}\,(2n-1)\dfrac{\pi}{2} } \right. \]

\[ -\frac{(-1)^{n+1}\sqrt{\dfrac{a}{\kappa}}\cos\left[\sqrt{\dfrac{a}{\kappa}}\,(2n-1)\,\dfrac{\pi x}{2R}\right]} {(-1)^{n+1}(2n-1)\dfrac{\pi}{2}\cdot \sin\left[\sqrt{\dfrac{a}{\kappa}}\,(2n-1)\,\dfrac{\pi}{2}\right]} e^{-\mu_n^2\dfrac{a\tau}{R^2}} + \]

\[ +2\sum_{m=1}^{\infty} \frac{\sqrt{\dfrac{a}{\kappa}}\sin\left[\sqrt{\dfrac{\kappa}{a}}\,m\pi\right]\cdot \cos m\pi\,\dfrac{x}{R}} {(-1)^m m\pi\cos\left[\sqrt{\dfrac{\kappa}{a}}\,m\pi\right]} e^{-\mu_m^2\dfrac{\kappa\tau}{R^2}} \tag{10} \]

where \(\vartheta(x,\tau)\) is the relative humidity,

\[ F_0=\frac{a\tau}{R^2} \]
is the thermal Fourier criterion,

\[ F'_0=\frac{\kappa\tau}{R^2} \]
is the hygrometric Fourier criterion,

\[ i_n=\frac{\delta t_0}{u_0} \]
is the Posnov criterion.

Analysis of the solutions obtained.
From solution (7) it is seen that the relative temperature is a function of the Fourier criterion \(\dfrac{a\tau}{R^2}\) and of the dimensionless coordinate.

It is convenient to use equality (7) for large Fourier criteria \((F_0>0.1)\), since series (7) converges rapidly as the Fourier criterion and \(\mu_n\) increase. For small Fourier criteria \((F_0<0.1)\), solution (7) can be represented in the form

\[ \Theta(x,\tau)=\Phi\left(\frac{x}{R},F_0\right)=1- \]

\[ -\left[ \sum_{n=1}^{\infty}(-1)^{n+1}\operatorname{erfc} \frac{(2n-1)-\dfrac{x}{R}}{2\sqrt{F_0}} + \right. \]

\[ \left. +\sum_{n=1}^{\infty}(-1)^{n+1}\operatorname{erfc} \frac{(2n-1)+\dfrac{x}{R}}{2\sqrt{F_0}} \right], \]

where

\[ \operatorname{erfc}z=1-\frac{2}{\pi}\int_0^z e^{-z^2}\,dz . \]

Solution (10) is convenient for calculations at large values of the Fourier criterion, when one may restrict oneself to the first term alone of the entire sum. For small values of \(F_0\), solution (10) can be given a more convenient form if equality (8) is first represented in the following form:

\[ F(x,s)-\frac{u_0}{s} = \frac{\delta t_0\sqrt{\dfrac{a}{\kappa}}} {\left(\dfrac{a}{\kappa}-1\right)} \sum_{n=1}^{\infty}(-1)^{2n}\frac{1}{\sqrt{s}}\, e^{-\dfrac{(2n-1)R\mp x}{\sqrt{\kappa}}\sqrt{s}} . \]

\[ \cdot \sum_{n=1}^{\infty}(-1)^{n-1}\left[ e^{-\frac{(2n-3)R}{\sqrt{a}}\sqrt{s}} -\frac{1}{\sqrt{s}}e^{-\frac{(2n+1)R}{\sqrt{a}}\sqrt{s}} \right] - \]

\[ -\frac{\delta t_0}{\left(\dfrac{a}{\kappa}-1\right)} \sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{\sqrt{s}}\, e^{-\frac{(2n-1)\mp x}{\sqrt{a}}\sqrt{s}}, \]

and then pass from the image to the original by means of the relations

\[ L^{-1}\left[\frac{1}{\sqrt{s}}e^{-\lambda\sqrt{s}}\right] = \frac{1}{\sqrt{\pi\tau}}e^{-\frac{\lambda^2}{4\tau}} \]

\[ L^{-1}\left[\frac{1}{s}e^{-\lambda\sqrt{s}}\right] = \operatorname{erfc}\left(\frac{\lambda}{2\sqrt{\tau}}\right). \]

The temperature field of a moist body is analogous to the temperature field of a dry body, since the differential equation of heat conduction for the conditions of our problem is the same.

In the stage of the regular regime \((F_0>F'_0)\), when all terms of the series except the first may be neglected, since the inequalities hold

\[ \mu_1<\mu_2<\cdots<\mu_n, \]

the relation between \(\Theta\) and \(F\) may be written as follows:

\[ \Theta=A_1\Phi\left(\mu_1\frac{x}{R}\right)e^{-\mu_1^2 F_0} \quad \text{for } F_0>F'_0, \tag{11} \]

where \(A_1\) is the initial thermal amplitude, equal to \(\dfrac{4}{\pi}\),

\[ \Phi\left(\mu_1\frac{x}{R}\right) \]

is the function describing the temperature distribution along the characteristic coordinate \(x\),

\[ \mu_1=\frac{\pi}{2} \]

is the first root of the characteristic equation

\[ \operatorname{ch}\sqrt{\frac{s}{a}}\,R=0, \]

\[ F_0=\frac{a\tau}{R^2} \]

is the thermal Fourier criterion.

From relation (11), relying on the theory of the regular regime developed by Prof. Kondrat'ev, one can determine the coefficient of thermal diffusivity \(a\).

Indeed, if one plots the graph

\[ \ln \Theta=f(\tau), \]

then in the stage of the regular regime it will have the form of a straight line with angular coefficient

\[ \operatorname{tg}\psi=\mu_1^2\frac{a}{\tau}. \]

and the computational formula for the coefficient of thermal diffusivity will have the form

\[ a=\frac{R^2}{\mu_1^2}\tg \psi=\frac{R^2}{\mu_1^2}\,2.3\,\frac{\lg \Theta_1-\lg \Theta_2}{\tau_2-\tau_1}. \tag{12} \]

The moisture field is described not only by the criteria \(F_0\) and \(\dfrac{x}{R}\), but also by the new criteria \(\kappa_1\) and \(P_n\), specially introduced into the theory of heat and moisture exchange,

\[ \vartheta=f\left(\frac{x}{R},\,F_0,\,\kappa_1,\,P_n\right), \]

where \(\vartheta\) is the relative moisture content of the body,

\[ \kappa_1=\frac{k}{a} \]

is a criterion equal to the ratio of the coefficient of moisture diffusivity to the coefficient of thermal diffusivity, characterizing the relative inertia of the moisture field as compared with the temperature field,

\[ P_n=\frac{\delta t_0}{u_0} \]

is the Posnov criterion, characterizing the nonuniformity of the moisture field caused by thermomoisture conductivity.

The curve of the change in moisture content with time for any point of the body has an extremum. This conclusion follows directly from the physical nature of the problem. The moisture distribution before cooling and after cooling is uniform, i.e., the moisture content at any point of the body is the same and is equal to \(u_0\).

During the cooling process, the moisture field is disturbed by the nonstationary temperature field, and the moisture content at any point of the body either increases, reaching a maximum, then decreases and again becomes equal to \(u_0\), or else the moisture content first decreases, reaching a minimum, and then increases and becomes equal to the initial value.

The curve of moisture distribution with respect to the coordinate \(\dfrac{x}{R}\) also has such an extremal value for small values of \(F_0\). For large values of \(F_0\), the curve of moisture distribution has no extremum, but intersects the straight line (the initial moisture distribution) at a certain point

\[ \left(\frac{x}{R}\right)_m . \]

This character of the moisture distribution allows us to propose a new method for determining the coefficient of moisture diffusivity \(k\) and the coefficient of thermomoisture conductivity \(\delta\).

LITERATURE

1. Lykov A. V. Theory of Heat Conduction. GITTL, 1952.
2. Veinik A. I. — ZhTF 20, 11, 1950.
3. Lykov A. V. and Pomerantsev A. A. — Calculation of the temperature field under conditions of underground gasification of coals — Underground Gasification of Coals, 9, 1935.