Nonstationary Temperature and Humidity Fields with Moisture Evaporation Inside a Body under a Boundary Condition of the First Kind

S. P. Kuznetsov

(Presented by Prof. P. M. Alabuzhev, Doctor of Technical Sciences)

Statement of the problem. If the moisture content of a body is small, then the moisture in it is bound mainly by adsorption forces, and its motion during heating of the body occurs both in the form of liquid and in the form of vapor. The system of differential equations describing heat and mass transfer inside a moist capillary-porous body may be written as follows:

\[ \frac{\partial t}{\partial \tau} = a \nabla^{2} t + \frac{\varepsilon \rho}{c}\, \frac{\partial u}{\partial \tau} \]

\[ \frac{\partial u}{\partial \tau} = k \nabla^{2} u + k \delta \nabla^{2} t, \]

where \(a\) is the coefficient of thermal diffusivity, \(k\) is the coefficient of moisture diffusivity, \(u\) is the moisture content, \(\rho\) is the specific heat of evaporation, \(c\) is the reduced specific heat,

\[ (c=c_{0}+c_{в}u). \]

The criterion of internal evaporation \(\varepsilon\) in this case is not equal to zero; its value lies between zero and unity. In the presence of ideal moisture insulation of the body, which must at the same time be a good conductor of heat, the problem may be formulated as follows. At the time \(\tau=0\), a moist body with initial temperature \(t_{0}\) and initial moisture content \(u_{0}\) is immersed in a medium with temperature equal to zero, which is maintained at this level throughout the entire process of cooling of the body. It is necessary to investigate the temperature and humidity fields at any moment of time, assuming that the total amount of moisture remains unchanged and that moisture transport occurs in the form of liquid and vapor.

Investigation of fields for an infinite plate. Mathematically, the problem is formulated by the system of differential equations

\[ \frac{\partial t(x,\tau)}{\partial \tau} = a \frac{\partial^{2} t(x,\tau)}{\partial x^{2}} + \frac{\varepsilon \rho}{c} \frac{\partial u(x,\tau)}{\partial \tau}, \]

\[ \frac{\partial u(x,\tau)}{\partial \tau} = \kappa \frac{\partial^2 u(x,\tau)}{\partial x^2} + \kappa' \delta \frac{\partial^2 t(x,\tau)}{\partial x^2} \tag{1} \]

under the following boundary conditions:

\[ t(x,0)=t_0, \qquad u(x,0)=u_0, \]

\[ \frac{\partial t(0,\tau)}{\partial x} = \frac{\partial u(0,\tau)}{\partial x} = 0, \tag{2} \]

\[ t(R,\tau)=0, \qquad \frac{1}{R}\int_0^R u(x,\tau)\,dx=u_0. \]

To solve the systems (1)—(2), we shall use the Laplace integral transform. The Laplace transforms of the temperature and humidity functions, respectively, will be denoted by

\[ L[t(x,\tau)] = \int_0^\infty e^{-s\tau} t(x,\tau)\,d\tau = T(x,s), \]

\[ L[u(x,\tau)] = \int_0^\infty e^{-s\tau} u(x,\tau)\,d\tau = F(x,s). \]

Here \(s\) belongs to the region \(\operatorname{Re}(s)>\sigma_1\), in which the indicated improper integrals exist. Applying the Laplace transform to the first equation of system (1) and to the boundary conditions (2), we obtain an ordinary nonhomogeneous differential equation of the fourth order with constant coefficients, whose general solution, after obvious transformations, will be

\[ T(x,s)=A_1 \operatorname{ch} r_1 x + A_2 \operatorname{ch} r_2 x + \frac{t_0}{s}, \tag{3} \]

where

\[ r_1=\sqrt{\frac{s}{a}\,N_1}, \qquad r_2=\sqrt{\frac{s}{a}\,N_2}, \]

\[ N_{1,2} = \left[ \left( \frac{1}{2} + \frac{\varepsilon \rho \delta}{2c} + \frac{a}{2\kappa} \right) \pm \sqrt{ \left( \frac{1}{2} + \frac{\varepsilon \rho \delta}{2c} + \frac{a}{2\kappa} \right)^2 - \frac{a}{\kappa} } \right]^{1/2}. \]

The coefficients \(A_1\) and \(A_2\) can be found by means of the transformed boundary conditions. Finally, the image of the temperature-distribution function takes the form

\[ T(x,s)-\frac{t_0}{s} = -t_0 \frac{ \sigma \operatorname{sh} r_2 R \operatorname{ch} r_1 x - \operatorname{sh} r_1 R \operatorname{ch} r_2 x }{ s\left[ \sigma \operatorname{sh} r_2 R \operatorname{ch} r_1 R - \operatorname{ch} r_2 R \operatorname{sh} r_1 R \right] } = -t_0 \frac{\Phi(s)}{\psi(s)}, \tag{4} \]

where

\[ \sigma = \frac{N_1}{N_2} \left( \frac{1-N_2^2}{1-N_1^2} \right). \]

To determine the original \(t(x,\tau)\), we shall use the expansion theorem, since expression (4) can be represented in the form of a ra-

relations of two generalized polynomials with respect to \(s\), and the degree of the polynomial in the denominator is greater by one than the degree of the polynomial in the numerator. As a result we obtain

\[ \Theta(x,\tau)=\frac{t(x,\tau)}{t_0}= \]

\[ =2\sum_{m=1}^{\infty} \frac{ \sigma \sin b\mu_m \cos \mu_m \dfrac{x}{R} - \sin \mu_m \cos b\mu_m \dfrac{x}{R} }{ \mu_m\left[(1-\sigma b)\cos \mu_m \cos b\mu_m+(\sigma-b)\sin \mu_m \sin b\mu_m\right] } \cdot e^{-\frac{\mu_m^{2}a\tau}{N_1^{2}R^{2}}}, \tag{5} \]

where

\[ \mu_m=i\sqrt{\frac{s}{a}}\,N_1R \]

are the roots of the equation

\[ \operatorname{ctg} b\mu-\sigma\operatorname{ctg}\mu=0, \qquad b=\frac{N_2}{N_1}. \]

\(\Theta(x,\tau)\) is the dimensionless temperature. Applying the Laplace transform to the second equation of system (1) and taking into account the value for the image of the temperature \(T(x,s)\), we first find the image \(F(x,s)\) for the moisture-distribution function, and then, using the expansion theorem, we also find the original \(u(x,\tau)\).

Finally we have:

\[ \vartheta(x,\tau)=\frac{u(x,\tau)-u_0}{u_0} = \frac{2\kappa t_0\delta}{u_0(\kappa N_1^2-a)(\kappa N_2^2-a)} \times \]

\[ \times \sum_{n=1}^{\infty} \frac{ \sqrt{\dfrac{a}{\kappa}}(M-P)\sin\mu_n\sin b\mu_n \cos\frac{1}{N_1}\mu_n\frac{x}{R} + \left(PN_1\sin b\mu_n\cos\mu_n\frac{x}{R} - M N_2\sin\mu_n\cos b\mu_n\frac{x}{R}\right) \sin\frac{1}{N_1}\sqrt{\frac{a}{\kappa}}\mu_n }{ \mu_n\left\{\dfrac{1}{N_1}(N_1^2-N_2^2) \left(N_1N_2\cos b\mu_n\cos\mu_n-\sin b\mu_n\sin\mu_n\right)\right\} \sin\frac{1}{N_1}\sqrt{\frac{a}{\kappa}}\mu_n } \,e^{-\frac{\mu_n^2 a\tau}{N_1^2R^2}} + \]

\[ + \sum_{p=1}^{\infty} \frac{ \left[(M-P)\sqrt{\dfrac{a}{\kappa}}\, \sin\sqrt{\dfrac{\kappa}{a}}\,N_1\mu_p\, \sin\sqrt{\dfrac{\kappa}{a}}\,N_2\mu_p\, \cos\mu_p\frac{x}{R}\right] \cdot e^{-\frac{\mu_p^2\kappa\tau}{R^2}} }{ (-1)^p\mu_p\left\{ N_1(1-N_2^2)\sin\sqrt{\dfrac{\kappa}{a}}\,N_2\mu_p \cos\sqrt{\dfrac{\kappa}{a}}\,N_1\mu_p - N_2(1-N_1^2) \right. } \]

\[ \left. {}\times \sin\sqrt{\dfrac{\kappa}{a}}\,N_1\mu_p \cos\sqrt{\dfrac{\kappa}{a}}\,N_2\mu_p \right\} } \tag{6} \]

where

\[ M=N_2^2(1-N_1^2)(\kappa N_1^2-a), \qquad P=N_1^2(1-N_2^2)(\kappa N_2^2-a), \]

\[ s_n \text{ are the roots of the equation }\quad N_1(1-N_2^2)\operatorname{ctg} b\mu - N_2(1-N_1^2)\operatorname{ctg}\mu =0, \]

\[ s_p \text{ are the roots of the equation }\quad \sin i\sqrt{\frac{s}{\kappa}}\,R=0, \qquad \mu=i\sqrt{\frac{s}{a}}\,N_1R, \qquad b=\frac{N_2}{N_1}. \]

Analysis of the solution. In considering the solutions (5), (6)—the temperature and moisture content of the body—it may be concluded that, along with the known

criteria (the thermal Fourier criterion, the hygrothermal Fourier criterion), new criteria also enter here:

1. The Posnov criterion

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

which characterizes the nonuniformity of the moisture field.

1. The Kossovich criterion

\[ K_0=\frac{\rho u_0}{c t_0}, \]

which characterizes the ratio of the heat expended on evaporation of moisture to the heat required for heating a moist body; in this case the heat is calculated per unit mass of dry substance. In addition to these new criteria, the solutions include the dimensionless quantities

\[ N_1=\sqrt{\frac{1}{2}\left[\left(1+\frac{\varepsilon\rho\delta}{c}+\frac{a}{\kappa}\right)+ \sqrt{\left(1+\frac{\varepsilon\rho\delta}{c}+\frac{a}{\kappa}\right)^2-\frac{4a}{\kappa}}\right]}, \]

\[ N_2=\sqrt{\frac{1}{2}\left[\left(1+\frac{\varepsilon\rho\delta}{c}+\frac{a}{\kappa}\right)- \sqrt{\left(1+\frac{\varepsilon\rho\delta}{c}+\frac{a}{\kappa}\right)^2-\frac{4a}{\kappa}}\right]}. \]

The quantities \(N_i,\ i=1,2\), are functions of the Lykov criterion

\[ Lu=\frac{\kappa}{a}, \]

which characterizes the relative inertia of the moisture field, and of the new criterion

\[ K_\varepsilon=\frac{\varepsilon\rho\delta}{c}. \]

This criterion has a definite physical meaning. It is equal to the ratio of the heat required for the internal evaporation of moisture retained by thermal-moisture conductivity to the heat required to heat a moist body by one degree; here the calculation is made per unit mass of the body. The criterion \(K_\varepsilon\) is equal to the product of three previously considered criteria \(\varepsilon\), \(P_n\), \(K_0\), namely:

\[ \varepsilon P_n K_0=\varepsilon\cdot\frac{\delta t_0}{u_0}\cdot\frac{\rho u_0}{c t_0} =\frac{\varepsilon\rho\delta}{c}=K_\varepsilon . \]

It may be written that

\[ N_{1,2}=\sqrt{\frac{1}{2}\left[\left(1+K_\varepsilon+\frac{1}{Lu}\right)\pm \sqrt{\left(1+K_\varepsilon+\frac{1}{Lu}\right)^2-\frac{4}{Lu}}\right]}, \]

that is,

\[ N_i=f(Lu,\varepsilon,P_n,K_\varepsilon)\quad \text{for } i=1,2 \]

or

\[ N_i=\varphi(Lu,K_\varepsilon). \tag{7} \]

By formula (7) one can compile a table for determining the criteria \(N_i\) as functions of the criteria \(Lu\) and \(K_\varepsilon\). From the data of such a table, for different values of the criterion \(K_\varepsilon\) from 0 to 1, one can construct the graphs

\[ N_i=f_1(Lu), \]

which make it possible to find easily the values \(N_1\) and \(N_2\) for specified values of the Lykov criterion and the criterion \(K_\varepsilon\).

The obtained solutions (5), (6) make it possible to carry out the calculation of the temperature and moisture fields through the thickness of the plate as a function of the Fourier criterion.

The considerations set forth may be used in calculating thermophysical coefficients.

References

1. A. V. Lykov. Transport Phenomena in Capillary-Porous Bodies. Moscow, Publishing House, 1954.