Application of the Theory of Nonstationary Temperature and Humidity Fields to the Determination of Thermal and Moisture Conductivity Coefficients
Abstract Generated abstract
This paper develops a nonstationary method for determining moisture conductivity and thermal moisture conductivity coefficients in moist spherical bodies, intended to avoid the long duration and apparatus complexity of stationary methods. It formulates coupled heat and moisture exchange equations for a sphere with constant surface temperature and moisture insulation, solves them using Laplace transforms, and derives series expressions for the temperature and moisture fields. The resulting criteria relate measurable moisture distributions during cooling to the coefficients \(k\) and \(\delta\), allowing both to be estimated from a single experiment. An illustrative procedure with moistened quartz sand in an aluminum spherical mold shows moisture migration toward the cooled surface under a temperature gradient and indicates how experimental profiles can be processed to obtain the desired coefficients.
Full Text
Application of the Theory of Nonstationary Fields of Temperature and Humidity to the Determination of the Coefficients of Thermal and Moisture Conductivity
S. P. Kuznetsov
Existing stationary methods for determining the coefficients of thermal-moisture conductivity require complex apparatus and a long duration of the experiment. Methods of nonstationary mass exchange for determining the moisture-conductivity coefficient \(k\) from curves of the distribution of moisture within a material during drying are suitable under isothermal conditions, when the body temperature is constant and the temperature gradient is absent. The method we propose for determining the coefficients of thermal-moisture conductivity \(k\) and \(\delta\) is free of these substantial shortcomings and is based on measuring the moisture field under a nonstationary temperature field. The theory of the method is based on the solution of the differential equations of heat and moisture exchange of moist bodies of spherical shape in a medium with constant temperature. The problem is posed as follows.
We have a body of spherical shape at temperature \(t_0\) and moisture content \(u_0\). At the initial moment 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. The surface of the body has moisture insulation that conducts heat well (a metallic shell), as a result of which the average moisture content of the body remains unchanged and equal to the initial moisture content. It is required to determine the moisture field and the temperature field for any moment of time, under the condition that the transfer of moisture under the influence of the temperature gradient occurs in the form of liquid. Mathematically the problem is formulated by the following system of partial differential equations:
\[ \frac{\partial [r t(r,\tau)]}{\partial(\tau)} = a \frac{\partial^2 [r t(r,\tau)]}{\partial r^2} \tag{1} \]
\[ \frac{\partial [r u(r,\tau)]}{\partial \tau} = k \frac{\partial^2 [r u(r,\tau)]}{\partial r^2} + k\delta \frac{\partial^2 [r t(r,\tau)]}{\partial r^2} \]
with boundary conditions:
\[ t(r,0)=t_0 \]
\[ u(r,0)=u_0 \]
\[ \frac{\partial t(0\tau)}{\partial r} = \frac{\partial u(0\tau)}{\partial r} = 0 \]
\[ t(R\tau)=0 \]
\[ \frac{3}{R^{3}}\int_{0}^{R} r^{2}u(r,\tau)\,dr=u_{0}, \]
where \(R\) is the radius of the sphere,
\(t, u\) are, respectively, the temperature and moisture content at any point of the sphere,
\(k\) is the moisture-conductivity coefficient,
\(\delta\) is the thermal-moisture-conductivity coefficient.
Applying the Laplace transform to the first equation of system (1),
\[ L[t(r,\tau)]=\int_{0}^{\infty} t(r,\tau)e^{-s\tau}\,d\tau=T(r,s), \]
after well-known operations we obtain the temperature-distribution function for the sphere
\[ \Theta(r,\tau)=\frac{t(r,\tau)-t_c}{t_0-t_c} =\sum_{n=1}^{\infty}2(-1)^{n+1} \frac{R\,\operatorname{sh}\mu_n \dfrac{r}{R}}{r\mu_n} e^{-\mu_n^{2}F_o}, \tag{2} \]
where \(\Theta(r,\tau)\) is the relative temperature,
\[ F_o=\frac{a\tau}{R^{2}} \]
is the thermal Fourier criterion,
\[ \mu_n=i\sqrt{\frac{s}{a}}\,R=n\pi,\qquad (n=1,2,3\ldots) \]
is the set of roots of the equation
\[ \operatorname{sh}\sqrt{\frac{s}{a}}\,R=0. \]
When using equality (2) for small Fourier criteria \((F<0.1)\), it is necessary to take several first terms of the expansion; for significant Fourier criteria \((F_o>0.1)\), the series converges very rapidly and only the first term of the expansion may be used for calculation. The error of the approximate solution can be determined easily.
Applying the Laplace transform to the second equation of system (1),
\[ L[u(r,\tau)]=\int_{0}^{\infty}u(r,\tau)e^{-s\tau}\,d\tau=F(r,s) \]
\[ L[ru(r,\tau)]=U(r,s)=rF(r,s) \]
\[ L[rt(r,\tau)]=V(r,s)=rT(r,s), \]
we shall have
\[ kU''(r,s)+k\delta V''(r,s)=sU(r,s)-ru_0. \]
Knowing that
\[ V(r,s)= -\frac{(t_c-t_0)R\,\operatorname{sh}\sqrt{\dfrac{s}{a}}\,r} {s\,\operatorname{sh}\sqrt{\dfrac{s}{a}}\,R} +\frac{t_0r}{s}, \]
we can write
\[ k U''(r,s)-s\left[U(r,s)-\frac{r u_0}{s}\right] +\frac{(t_c-t_0)R\,\operatorname{sh}\sqrt{\frac{s}{a}\,r\cdot k\delta}} {a\,\operatorname{sh}\sqrt{\frac{s}{a}\,R}}=0. \]
Using once again the Laplace transform, but now with respect to \(r\),
\[ L_r[U(r,s)]=\int_0^\infty e^{-Sr}U(r,s)\,dr=\Theta(S,s), \]
after some transformations we obtain
\[ \Theta(S,s)=U(0,s)\frac{S}{S^2-\frac{s}{k}} +\frac{U'(0,s)}{S^2-\frac{s}{k}} -\frac{u_0}{kS^2\left(S^2-\frac{s}{k}\right)} +\frac{D}{\left(S^2-\frac{s}{a}\right)\left(S^2-\frac{s}{k}\right)}, \tag{3} \]
where
\[ D=\frac{\delta(t_0-t_c)R\sqrt{\frac{s}{a}}} {a\,\operatorname{sh}\sqrt{\frac{s}{a}\,R}}. \]
Applying the inverse Laplace transform to equality (3), we find the transform of the function of moisture distribution in the sphere:
\[ F(r,s)=\frac{A}{r}\operatorname{ch}\sqrt{\frac{s}{k}}\,r +\frac{B}{r}\operatorname{sh}\sqrt{\frac{s}{k}}\,r +\frac{u_0}{s} + \]
\[ +\frac{\delta(t_0-t_c)R\left[ \operatorname{sh}\sqrt{\frac{s}{a}}\,r -\sqrt{\frac{k}{a}}\operatorname{sh}\sqrt{\frac{s}{k}}\,r \right]} {rs\left(1-\frac{a}{k}\right)\operatorname{sh}\sqrt{\frac{s}{a}}\,R}. \]
To determine the coefficients \(A\) and \(B\) we use the boundary conditions, which for the transform will have the form:
\[ F'(0,s)=0 \]
and
\[ \frac{3}{R^3}\int_0^\infty r^2F(r,s)\,dr=\frac{u_0}{s}. \]
Having determined the coefficients \(A\) and \(B\), we write the transform of the moisture distribution function
\[ F(r,s)-\frac{u_0}{s}= \]
\[ =\frac{\delta(t_0-t_c)R}{r(k-a)} \left[ \frac{ k\,\operatorname{sh}\sqrt{\frac{s}{a}}\,r \left(\sqrt{\frac{s}{k}}\,R\,\operatorname{ch}\sqrt{\frac{s}{k}}\,R -\operatorname{sh}\sqrt{\frac{s}{k}}\,R\right) }{ s\,\operatorname{sh}\sqrt{\frac{s}{a}}\,R \left(\sqrt{\frac{s}{k}}\,R\,\operatorname{ch}\sqrt{\frac{s}{k}}\,R -\operatorname{sh}\sqrt{\frac{s}{k}}\,R\right) } -\right. \]
\[ -\frac{ a\left(\sqrt{\frac{s}{a}}R\operatorname{ch}\sqrt{\frac{s}{a}}R-\operatorname{sh}\sqrt{\frac{s}{a}}R\right)\cdot \operatorname{sh}\sqrt{\frac{s}{k}}r }{ s\cdot \operatorname{sh}\sqrt{\frac{s}{a}}R \left(\sqrt{\frac{s}{k}}R\operatorname{ch}\sqrt{\frac{s}{k}}R-\operatorname{sh}\sqrt{\frac{s}{k}}R\right) } \right]. \tag{4} \]
To determine the original \(U(r,\tau)\) from the image (4), we shall use the expansion theorem.
Let us denote by
\[ \Phi(s)= k\operatorname{sh}\sqrt{\frac{s}{a}}\,r \left(\sqrt{\frac{s}{k}}R\operatorname{ch}\sqrt{\frac{s}{k}}R -\operatorname{sh}\sqrt{\frac{s}{k}}R\right) - \]
\[ -a\left(\sqrt{\frac{s}{a}}R\operatorname{ch}\sqrt{\frac{s}{a}}R -\operatorname{sh}\sqrt{\frac{s}{a}}R\right) \operatorname{sh}\sqrt{\frac{s}{k}}\,r \]
and
\[ \psi(s)= s\operatorname{sh}\sqrt{\frac{s}{a}}R \left(\sqrt{\frac{s}{k}}R\operatorname{ch}\sqrt{\frac{s}{k}}R -\operatorname{sh}\sqrt{\frac{s}{k}}R\right). \]
The roots of the polynomial \(\psi(s)\) will be:
-
\[ s=0 \]
-
The roots of the equation
\[ \sin i\sqrt{\frac{s}{a}}R=0, \qquad \text{i.e.}\qquad s_n=-\frac{\mu_n^2 a}{R^2}, \]
where
\[ \mu_n=i\sqrt{\frac{s}{a}}R=n\pi,\qquad n=1,2,3,\ldots \]
- The roots of the equation
\[ \sqrt{\frac{s}{k}}R\operatorname{ch}\sqrt{\frac{s}{k}}R -\operatorname{sh}\sqrt{\frac{s}{k}}R=0, \]
i.e.
\[ s_m=-\frac{\mu_m^2 k}{R^2}, \qquad \text{where}\qquad \mu_m=i\sqrt{\frac{s}{k}}R. \]
The constants \(\mu_m\) are determined from the characteristic equation
\[ \operatorname{tg}\mu_m=\mu_m. \]
This equation has an infinite set of real roots, as is easily concluded from the infinite set of points of intersection of the tangent curve \(y=\operatorname{tg}\mu\) with the straight line \(y=\mu\).
The first four roots of this equation will be:
\[ \mu_1=0.0000;\qquad \mu_2=4.4934;\qquad \mu_3=7.7252 \quad \text{and}\quad \mu_4=10.9041. \]
Passing from the operational solution (4) to the original, we obtain
\[ u(r,\tau)-u_0= \]
\[ = \frac{\xi(t_0-t_c)R}{r(k-a)} \left[ \lim_{s\to 0}\frac{\Phi(s)}{\psi'(s)} + \sum_{n=1}^{\infty}\frac{\Phi(s_n)}{\psi'(s_n)}e^{s_n\tau} + \sum_{m=1}^{\infty}\frac{\Phi(s_m)}{\psi'(s_m)}e^{s_m} \right], \tag{5} \]
where
\[ \psi'(s)=\operatorname{sh}\sqrt{\frac{s}{a}}\,R \left( \sqrt{\frac{s}{k}}\,R\,\operatorname{ch}\sqrt{\frac{s}{k}}\,R -\operatorname{sh}\sqrt{\frac{s}{k}}\,R \right) + \]
\[ +\frac{1}{2}\sqrt{\frac{s}{a}}\,R\,\operatorname{ch}\sqrt{\frac{s}{a}}\,R \left( \sqrt{\frac{s}{k}}\,R\,\operatorname{ch}\sqrt{\frac{s}{k}}\,R -\operatorname{sh}\sqrt{\frac{s}{k}}\,R \right) + \]
\[ +\frac{1}{2}\sqrt{\frac{s}{k}}\,R\cdot \sqrt{\frac{s}{k}}\,R\cdot \operatorname{sh}\sqrt{\frac{s}{k}}\,R\cdot \operatorname{sh}\sqrt{\frac{s}{a}}\,R . \]
After some transformations we find that
\[ \lim_{s\to 0}\frac{\Phi(s)}{\psi'(s)}=0 . \]
Replacing in equality (5) each term by the corresponding quantity, we obtain the function of moisture distribution in the sphere under the given boundary conditions
\[ \vartheta(r,\tau)=\frac{u(r,\tau)-u_0}{u_0}= \]
\[ = \frac{\delta(t_0-t_c)R}{r u_0\left(1-\dfrac{a}{k}\right)} \left[ 2\sum_{n=1}^{\infty} \frac{ \left( \sin\sqrt{\dfrac{a}{k}}\,\mu_n -\sqrt{\dfrac{a}{k}}\,\mu_n\cos\sqrt{\dfrac{a}{k}}\,\mu_n \right) \sin\mu_n\dfrac{r}{R} }{ (-1)^n\mu_n \left( \sin\sqrt{\dfrac{a}{k}}\,\mu_n -\sqrt{\dfrac{a}{k}}\,\mu_n\cos\sqrt{\dfrac{a}{k}}\,\mu_n \right) } \right. \]
\[ \left. + \frac{ \mu_n(-1)^n\dfrac{a}{k}\, \sin\sqrt{\dfrac{a}{k}}\,\mu_n\dfrac{r}{R} }{ (-1)^n\mu_n \left( \sin\sqrt{\dfrac{a}{k}}\,\mu_n -\sqrt{\dfrac{a}{k}}\,\mu_n\cos\sqrt{\dfrac{a}{k}}\,\mu_n \right) } \cdot e^{-\mu_n^2 F_0} \right. \]
\[ \left. -2\sum_{m=1}^{\infty} \frac{ \left( \sin\sqrt{\dfrac{k}{a}}\,\mu_m -\sqrt{\dfrac{k}{a}}\,\mu_m\cos\sqrt{\dfrac{k}{a}}\,\mu_m \right) \dfrac{a}{k}\, \sin\mu_m\dfrac{r}{R} }{ \mu_m^2\sin\mu_m\cdot \sin\sqrt{\dfrac{k}{a}}\,\mu_m } \cdot e^{-\mu_m^2\cdot\dfrac{k}{a}F_0} \right], \tag{6} \]
where \(\mu_n=n\pi,\quad n=1,2,3\ldots\)
\(\mu_m\) are the roots of the characteristic equation \(\operatorname{tg}\mu=\mu\).
The relative temperature \(\Theta(r,\tau)\) is a function of the relative coordinate \(X/R\) and of the thermal Fourier criterion.
In the stage of the regular regime \((F_0>F_0')\), the relation between \(\Theta\) and \(F_0\) can be written in the form
\[ \Theta=A_1\Phi\left(\mu_1\frac{X}{R}\right)e^{-\mu_1^2F_0}, \]
where \(A_1=\dfrac{2}{\pi}\) is the initial thermal amplitude,
\(\Phi\left(\mu_1 \dfrac{X}{R}\right)\)—a function describing the temperature distribution with respect to the characteristic coordinate,
\(\mu_1=\pi\)—the first root of the characteristic equation,
\[ F_0=\frac{a\tau}{R^2} \]
—the thermal Fourier criterion.
The moisture field is described by the criteria \(F_0,\ \dfrac{X}{R},\ k_1\) and \(P_n\), i.e.
\[ \vartheta(r,\tau)=f\left(F_0,\frac{X}{R},k_1,P_n\right), \]
where
\(\vartheta(r,\tau)\)—the relative moisture content of the body,
\[ k_1 \]
—the criterion equal to
\[ \frac{k}{a}, \]
\[ P_n \]
—the Posnov criterion, equal to
\[ \frac{\delta(t_0-t_c)}{u_0}, \]
characterizing the nonuniformity of the moisture field caused by the presence of thermomoisture conductivity.
The obtained solutions (2), (6) of the system of differential equations of heat and moisture exchange (1) make it possible to determine the thermomoisture coefficients directly from a single experiment.
Quartz sand, previously moistened to the required moisture content, is poured into a spherical mold, \(R=10\) cm, made of aluminum. Five thermocouples are introduced into the bulb: \(\dfrac{r}{R}=1\) (surface of the bulb), \(\dfrac{r}{R}=0.8,\ \dfrac{r}{R}=0.6,\ \dfrac{r}{R}=0.4,\ \dfrac{r}{R}=0\) (center of the bulb).
At first the bulb with sand is heated in an air thermostat at a constant temperature (from 40 to \(60^\circ\)C) until a uniform distribution of the temperature and moisture content of the sand is obtained. After heating is completed, it is quickly removed from the thermostat and placed in a water thermostat with a constant temperature for cooling.
In order to obtain curves of the distribution of the sand moisture content at different moments of cooling time, sampling is carried out. After a specified cooling interval has elapsed, by means of a sampling tube an analysis of the sand moisture content is made along the radius of the bulb. The moisture content of the sand is determined by the ordinary gravimetric method—drying in a thermostat to constant weight at \(105^\circ\)C.
The experimental data obtained are processed in the form of graphs of the distribution of temperature and moisture content.
Figure 1 presents the curve of moisture distribution corresponding to the radial distribution after 25 minutes of cooling, in a medium with temperature \(t_c=9.8^\circ\)C, of sand previously heated to \(60^\circ\)C.
The initial moisture content, equal to \(10.9\%\), is indicated as a straight line parallel to the abscissa axis. At the initial moment of time the moisture content of the sand inside the bulb is the same; later, at the surface it will be \(13.3\%\), and at the center \(10.1\%\), i.e. under the influence of the temperature gradient the moisture has moved from the center to the surface.
The relative moisture drop at the surface is
\[ \vartheta_n=\frac{u(R,\tau)-u_o}{u_o}=\frac{13.3-10.9}{10.9}=0.22 . \]
In the experiment shown in Fig. 1, the temperature-conductivity coefficient of the sand, determined by Professor G. M. Kondrat'ev’s method, was equal to \(2.3\cdot 10^{-3}\ \text{m}^2/\text{hour}\).
Fig. 1
If, in solution (6), which represents the nonstationary moisture field for a body in the form of a sphere, the Fourier criterion is assigned values \(F_o \geq 0.09\) and it is taken into account that \(k_1=\dfrac{k}{a}\) and varies in the interval from 0 to 1, then one may obtain the dependence
\[ \varphi(k_1,\xi_m)=0 \tag{7} \]
between the criterion \(k_1\) and the relative coordinate \(\xi_m\).
From the value of \(\xi_m\) found experimentally (Fig. 1), the corresponding value of the criterion \(k_1\) can be found from the graph of dependence (7), after which, from the known quantities \(k_1\) and \(a\), and from the relation
\[ k_1=\frac{k}{a} \]
one can readily determine the value of the moisture-conductivity coefficient \(k\).
The method considered by us makes it possible to determine Posnov’s criterion \(P_n\), and consequently also the coefficient of thermal-moisture conductivity \(\delta\).
Solution (6), for \(\dfrac{r}{R}=1\), gives the relative moisture content at the surface of the sphere, from which, for values of \(k_1\) lying in the interval from 0 to 1, one may obtain the dependence
\[ f\left(\frac{\vartheta_n}{P_n}, F_o\right)=0 \tag{8} \]
between \(\dfrac{\vartheta_n}{P_n}\) and the criterion \(F_o\), for \(F_o>0.09\).
From the indicated value of the Fourier criterion \(F_o\), from the graph of dependence (8), one can find the value of \(\dfrac{\vartheta_n}{P_n}\). Knowing the relative moisture difference at the surface
\[ \vartheta_n=\frac{u(R,\tau)-u_o}{u_o} \]
and the value \(\dfrac{\vartheta_n}{P_n}\), one can find the Posnov criterion \(P_n\), and then, from the relation
\[ P_n=\frac{\delta(t_o-t_c)}{u_o} \]
determine also the coefficient of thermal-moisture conductivity.