On the Theory of Heat and Mass Transfer in an Infinite Cylinder
Abstract Generated abstract
This paper develops an analytical treatment of coupled heat and moisture transfer in a homogeneous isotropic moist body, based on Lykov’s differential equations. For the special case of an infinite cylinder with constant initial temperature and moisture content, radial symmetry, a prescribed surface temperature, and conservation of mean moisture, the author applies Laplace transforms and Bessel function expansions to solve the governing equations. The result is a series expression for the relative temperature field and a corresponding expression for the relative moisture distribution, accounting for thermomoisture coupling when moisture migrates in liquid form. These formulas extend classical heat conduction theory to moist materials under the stated simplifying assumptions.
Full Text
ON THE THEORY OF HEAT AND MASS TRANSFER IN AN INFINITE CYLINDER
S. N. Kuznetsov
The study of the phenomena of heat and moisture transfer in moist materials is based on a system of differential equations of heat and mass exchange, first proposed by A. V. Lykov [1].
Consider a homogeneous isotropic moist body, inside which there is both a temperature gradient and a moisture gradient. In this case the moisture flux \(q_{\mathrm{в}}\) is equal to:
\[ q_{\mathrm{в}}=-k\gamma_0\frac{\partial u}{\partial n} -k\gamma_0\delta\frac{\partial t}{\partial n}, \tag{1} \]
where \(u\) is the moisture content of the material, kg/kg of dry substance;
\(k\) is the moisture-conductivity coefficient, \(\dfrac{m^2}{\text{hour}}\);
\(\delta\) is the thermomoisture-conductivity coefficient, \(\dfrac{1}{\text{deg}}\);
\(\gamma_0\) is the density of the absolutely dry body, \(\dfrac{kg}{m^3}\);
\(n\) is the normal to the surface.
If in the body under consideration we isolate some volume \(v\), bounded by a smooth surface \(s\), then the increase of moisture in it will occur due to the inflow of moisture from outside, as a result of thermal and moisture conductivity. We denote by \(Q_1\) the amount of moisture flowing through the surface \(s\) per unit time. If it is assumed that the transfer of moisture occurs only in the direction of the \(ox\) axis, then, using Ostrogradsky’s formula, one may write
\[ Q_1=\iiint\limits_{(v)} k\gamma_0 \left( \frac{\partial^2 u}{\partial x^2} +\delta\frac{\partial^2 t}{\partial x^2} \right)\,dv. \tag{2} \]
Assuming that at time \(\tau\) the moisture content in the volume under consideration is
\[ \iiint\limits_{(v)}\gamma_0 u(\tau)\,dv \]
and the increase of moisture referred to unit time is
\[ Q_2=\iiint\limits_{(v)} \gamma_0\,\frac{\partial u}{\partial \tau}\,dv, \tag{3} \]
then the equality will hold:
\[ \iiint\limits_{(v)} k\gamma_0\left(\frac{\partial^2 u}{\partial x^2}+\delta\,\frac{\partial^2 t}{\partial x^2}\right)dv = \iiint\limits_{(v)} \gamma_0\,\frac{\partial u}{\partial \tau}\,dv. \]
After minor transformations, under the condition that the thermophysical coefficients \((k,\delta)\) are constant, we obtain the basic differential equation of moisture conductivity
\[ \frac{\partial u}{\partial \tau} = k\,\frac{\partial^2 u}{\partial x^2} + k\delta\,\frac{\partial^2 t}{\partial x^2}. \tag{4} \]
In a moist body, heat is transferred not only under the action of a temperature difference, but is also carried by the migrating moisture; as a result, Fourier’s hypothesis takes the form
\[ q=-(\lambda+\gamma_0 k\delta I)\,\frac{\partial t}{\partial n} - k\gamma_0\,\frac{\partial u}{\partial n}. \tag{5} \]
Denoting the amount of heat passing per unit time through the surface of the body along the \(ox\) axis by \(Q_1\), on the basis of Ostrogradsky’s formula one may write
\[ Q_1=\iiint\limits_{(v)} \left[ (\lambda+k\gamma_0\delta I)\,\frac{\partial^2 t}{\partial x^2} + k\gamma_0 I\,\frac{\partial^2 u}{\partial x^2} \right]dv, \]
considering the enthalpy \(I\) to be a constant quantity, since its change in the body under consideration is very small. If evaporation occurs inside the body, it is necessary to take into account the amount of heat absorbed by the internal negative sources, determined by the formula
\[ Q_{\mathrm{II}}= \iiint\limits_{(v)} r\gamma_0\,\frac{\partial u'}{\partial \tau}\,dv, \tag{6} \]
where \(\dfrac{\partial u'}{\partial \tau}\) is the amount of evaporated moisture per unit density of dry matter per unit time;
\[ r \]
is the heat of vaporization, \(\dfrac{\text{kcal}}{\text{kg}}\).
If at the time \(\tau\) the selected volume contained the amount of heat
\[ Q_{\mathrm{III}}= \iiint\limits_{(v)} (c_0+c_{\mathrm{в}}u)\gamma_0 t\,dv, \tag{7} \]
where \(c_0\) and \(c_{\mathrm{в}}\) are, respectively, the specific heat capacity of the absolutely dry substance and of moisture, \(\dfrac{\text{kcal}}{\text{kg}\,\text{deg}}\) \((c_{\mathrm{в}}\simeq 1)\), then the change in heat content within the volume \(v\) per unit time is equal to
\[ Q_{\mathrm{IV}}=\iiint\limits_{(v)} \left[ (c_0+c_v u)\gamma_0\frac{\partial t}{\partial \tau} + c_v t\gamma_0\frac{\partial u}{\partial \tau} \right]\,d\tau, \tag{8} \]
We note that the enthalpy \(I\) of the moving moisture \(\left(\dfrac{\mathrm{kcal}}{\mathrm{kg}}\right)\) is basically accounted for by the moisture in the liquid state, since the mass of the migrating vapor is small in comparison with the mass of the moving liquid; therefore one may take \(c_v t \approx I\).
Setting up the heat balance in the volume \(v\) under consideration and taking into account the differential equation of moisture conductivity (4), we obtain
\[ c\gamma_0\frac{\partial t}{\partial \tau} = \lambda\frac{\partial^2 t}{\partial x^2} + r\gamma_0\frac{\partial u'}{\partial \tau}, \tag{9} \]
where \(c=c_0+c_v u\).
If evaporation or condensation of moisture does not occur inside the body, then the obtained differential equation reduces to the ordinary heat-conduction equation
\[ \frac{\partial t}{\partial \tau} = a\frac{\partial^2 t}{\partial x^2}, \quad \text{where } a=\frac{\lambda}{c\gamma_0}. \]
In drying theory it is proved that, with some approximation, a direct proportionality may be assumed between the rate of evaporation and the rate of change of moisture content \(\dfrac{\partial u}{\partial \tau}\):
\[ \frac{\partial u'}{\partial \tau} = \varepsilon\frac{\partial u}{\partial \tau}, \]
and then differential equation (9) takes the form
\[ \frac{\partial t}{\partial \tau} = a\frac{\partial^2 t}{\partial x^2} + \frac{\varepsilon r}{c}\cdot\frac{\partial u}{\partial \tau}, \tag{10} \]
where \(a\) is the coefficient of thermal diffusivity, \(\dfrac{m^2}{\text{hour}}\), and \(\varepsilon\) is the criterion of internal evaporation.
Thus, the study of the processes of heating and cooling of moist bodies is associated with the solution of the system of differential equations (4), (10) of heat and moisture exchange under the corresponding initial and boundary conditions.
Let us consider the solution of this system for an infinite cylinder in the case of unchanged moisture content and under the condition that it moves in the form of a liquid.
The problem is mathematically formulated by the equations:
\[ \frac{\partial t(r,\tau)}{\partial \tau} = a\left( \frac{\partial^2 t(r,\tau)}{\partial r^2} + \frac{1}{r}\frac{\partial t(r,\tau)}{\partial r} \right), \tag{11} \]
\[ \frac{\partial u(r,\tau)}{\partial \tau} = k\left( \frac{\partial^2 u(r,\tau)}{\partial r^2} + \frac{1}{r}\frac{\partial u(r,\tau)}{\partial r} \right) + k\delta\left( \frac{\partial^2 t(r,\tau)}{\partial r^2} + \frac{1}{r}\frac{\partial t(r,\tau)}{\partial r} \right); \]
under the initial and boundary conditions:
\[ t(r,0)=t_0=\mathrm{const}, \]
\[ u(r,0)=u_0=\mathrm{const}; \tag{12} \]
\[ \frac{\partial t(0,\tau)}{\partial r} = \frac{\partial u(0,\tau)}{\partial r} =0, \]
\[ t(R,\tau)=0, \tag{13} \]
\[ u_0=\frac{2}{R^2}\int_0^R u(r,\tau)r\,dr. \]
Applying the Laplace transform
\[ L[t(r,\tau)]=\int_0^\infty e^{-s\tau}t(r,\tau)d\tau=T(r,s) \]
to the first equation of system (11), we obtain the general solution of the transformed equation for the temperature
\[ T(r,s)=AI_0\left(\sqrt{\frac{s}{a}}\,r\right) +BK_0\left(\sqrt{\frac{s}{a}}\,r\right) +\frac{t_0}{s}, \tag{14} \]
where \(I_0\left(\sqrt{\frac{s}{a}}\,r\right)=J_0\left(i\sqrt{\frac{s}{a}}\,r\right)\) is the modified Bessel function of zero order, of the first kind; \(K_0\left(\sqrt{\frac{s}{a}}\,r\right)=\)
\[ -I_0\left(\sqrt{\frac{s}{a}}\,r\right) \left(\ln \frac{1}{2}\sqrt{\frac{s}{a}}\,r+C\right) +\left(\frac{1}{2}\sqrt{\frac{s}{a}}\,r\right)^2+ \]
\[ +\left(1+\frac{1}{2}\right) \frac{\left(\frac{1}{2}\sqrt{\frac{s}{a}}\,r\right)^4}{(2!)^2} +\ldots \]
is the modified Bessel function of zero order, of the second kind, \(C=0.5772\ldots\) is Euler’s constant, and \(A\) and \(B\) are coefficients independent of \(r\), but dependent on \(s\).
From the symmetry condition it follows that \(B=0\). We determine the coefficient \(A\) from the boundary condition for the transform \(T(R,s)=0\); we obtain
\[ A=-\frac{t_0}{sI_0\left(\sqrt{\frac{s}{a}}\,R\right)}. \]
The transform of the temperature takes the form
\[ T(r,s)=\frac{t_0}{s} -\frac{I_0\left(\sqrt{\frac{s}{a}}\,r\right)} {sI_0\left(\sqrt{\frac{s}{a}}\,R\right)}. \tag{15} \]
For passing from the transform to the original \(t(r,\tau)\) we use the integral relation
\[ t(r,\tau)-t_1 = -\frac{t_0}{2\pi i} \int_{\gamma-i\infty}^{\gamma+i\infty} \frac{ e^{s\tau} I_0\!\left(\sqrt{\frac{s}{a}}\,r\right) }{ s I_0\!\left(\sqrt{\frac{s}{a}}\,R\right) }\,ds . \]
The integrand has simple poles, one of which is at the origin \(s=0\), while the remaining ones,
\[ s_n=-\mu_n^2\frac{a}{R^2}, \]
are situated on the negative direction of the real axis.
Passing to the contour integral and using the residue theorem, we obtain
\[ \frac{t(r,\tau)}{t_0} = -\left\{ \operatorname{res} \left[ \frac{ e^{s\tau} I_0\!\left(\sqrt{\frac{s}{a}}\,r\right) }{ s I_0\!\left(\sqrt{\frac{s}{a}}\,R\right) } \right]_0 + \sum_{n=1}^{\infty} \operatorname{res} \left[ \frac{ e^{s\tau} I_0\!\left(\sqrt{\frac{s}{a}}\,r\right) }{ s I_0\!\left(\sqrt{\frac{s}{a}}\,R\right) } \right]_{s_n} \right\}. \]
The residue with respect to the pole \(s=0\) is equal to unity. The sum of the residues with respect to the remaining poles is equal to
\[ \sum_{n=1}^{\infty} \frac{ 2J_0\!\left(\mu_n\frac{r}{R}\right) }{ \mu_n J_1(\mu_n) } \cdot e^{-\mu_n^2\frac{a\tau}{R^2}}, \]
since
\[ I_1\!\left(\sqrt{\frac{s}{a}}\,R\right) = i^{-1}J_1\!\left(i\sqrt{\frac{s}{a}}\,R\right), \qquad \mu_n = i\sqrt{\frac{s_n}{a}}\,R . \]
Thus, the relative temperature at any point of the infinite cylinder at any instant of time is determined by the equality
\[ \Theta(r,\tau) = \frac{t_0-t(r,\tau)}{t_0} = 1 - 2\sum_{n=1}^{\infty} \frac{ J_0\!\left(\mu_n\frac{r}{R}\right) }{ \mu_n J_1(\mu_n) } \cdot e^{-\mu_n^2\frac{a\tau}{R^2}} . \tag{16} \]
To determine the function of moisture distribution in the infinite cylinder, we apply the Laplace transform to the second equation of system (11).
Denoting
\[ L[u(r,\tau)] = \int_0^\infty e^{-s\tau}u(r,\tau)\,d\tau = F(r,s) \]
and using the relations
\[ T(r,s) = -\frac{t_0}{ s I_0\!\left(\sqrt{\frac{s}{a}}\,R\right) } \cdot I_0\!\left(\sqrt{\frac{s}{a}}\,r\right) + \frac{t_0}{s}, \]
\[ Z I'_n(Z)+n I_n(Z)=Z I_{n-1}(Z), \]
we obtain
\[ F''(r,s)+\frac{1}{r}F'(r,s)-\frac{s}{k}F(r,s)+\frac{u_0}{k} -\frac{t_0\delta\left(\dfrac{s}{a}\right) I_0\left(\sqrt{\dfrac{s}{a}}\,r\right)} {s I_0\left(\sqrt{\dfrac{s}{a}}\,R\right)} =0. \tag{17} \]
Applying the Laplace transform to the last equation,
\[ L[F(r,s)]=Q(\lambda,s), \]
after some transformations we obtain
\[ Q(\lambda,s)=\frac{u_0}{\lambda s} + \frac{t_0\delta} {a I_0\left(\sqrt{\dfrac{s}{a}}\,R\right) \left(\dfrac{s}{a}-\dfrac{s}{k}\right) \left(\lambda^2-\dfrac{s}{a}\right)^{\nu_2}} + \frac{c}{\left(\lambda^2-\dfrac{s}{k}\right)^{\nu_2}}, \tag{18} \]
where \(c\) is an arbitrary constant.
By means of the inverse Laplace transform of equation (18), we pass to the image of the moisture-distribution function
\[ F(r,s)-\frac{u_0}{s} = \frac{k t_0\delta}{(k-a)} \cdot \frac{ I_1\left(\sqrt{\dfrac{s}{k}}\,R\right) I_0\left(\sqrt{\dfrac{s}{a}}\,r\right) - \sqrt{\dfrac{a}{k}}\, I_1\left(\sqrt{\dfrac{s}{a}}\,R\right) I_0\left(\sqrt{\dfrac{s}{k}}\,r\right) } { s I_0\left(\sqrt{\dfrac{s}{a}}\,R\right) I_1\left(\sqrt{\dfrac{s}{k}}\,R\right) }. \tag{19} \]
The arbitrary constant \(c\) is determined by means of boundary condition (13), taking into account the relation
\[ \int_0^R x I_0(ax)\,dx=\frac{R}{a}I_1(aR). \]
To pass from the image (19) to the original \(u(r,s)\), we introduce the notation:
\[ \Phi(s)=I_1\left(\sqrt{\dfrac{s}{k}}\,R\right)\cdot I_0\left(\sqrt{\dfrac{s}{a}}\,r\right) - \]
\[ -\sqrt{\dfrac{a}{k}}\, I_1\left(\sqrt{\dfrac{s}{a}}\,R\right)\cdot I_0\left(\sqrt{\dfrac{s}{k}}\,r\right), \]
\[ \Psi'(s)=s I_0\left(\sqrt{\frac{s}{a}}\,R\right)\cdot I_1\left(\sqrt{\frac{s}{k}}\,R\right). \]
The roots of the polynomial \(\Psi(s)\) will be:
\[ \begin{aligned} &1.\quad s=0;\\[4pt] &2.\quad s_n=-\frac{\mu_n^2 a}{R^2},\quad \text{where } \mu_n=i\sqrt{\frac{s}{a}}\,R;\\[4pt] &3.\quad s_m=-\frac{\mu_m^2 k}{R^2},\quad \text{where } \mu_m=i\sqrt{\frac{s}{k}}\,R, \end{aligned} \]
which, for
\[ n=1,\,2,\,3,\ldots \]
\[ m=1,\,2,\,3,\ldots \]
are located on the negative direction of the real axis.
Applying the expansion theorem to image (19), we obtain the function of the relative moisture distribution in an infinite cylinder under the initial and boundary conditions under consideration:
\[ V(r,\tau)=\frac{u(r,\tau)-u_0}{u_0} \]
\[ = -\frac{t_0\delta k}{u_0(k-a)} \left[ 2\sum_{n=1}^{\infty} \frac{ \sqrt{\frac{a}{k}}\,J_1(\mu_n)J_0\!\left(\sqrt{\frac{a}{k}}\,\mu_n\frac{r}{R}\right) - J_1\!\left(\sqrt{\frac{a}{k}}\,\mu_n\right)J_0\!\left(\mu_n\frac{r}{R}\right) }{ \mu_n J_1(\mu_n)\cdot J_1\!\left(\sqrt{\frac{a}{k}}\,\mu_n\right) } \cdot e^{-\mu_n^2\frac{a\tau}{R^2}} \right. \]
\[ \left. -2\sum_{m=1}^{\infty} \frac{ \sqrt{\frac{a}{k}}\,J_1\!\left(\sqrt{\frac{k}{a}}\,\mu_m\right)\cdot J_0\!\left(\mu_m\frac{r}{R}\right) }{ \mu_m J_0\!\left(\sqrt{\frac{k}{a}}\,\mu_m\right)\cdot J_1(\mu_m) } \cdot e^{-\mu_m^2\frac{k\tau}{R^2}} \right]. \tag{20} \]
The results obtained, (16) and (20), are a further development of the analytical theory of heat conduction in moist materials.
LITERATURE
- Lykov A. V. Theory of Heat Conduction, published 1952.
- Lavrent'ev M. A. and Shabat B. V. Theory of Functions of a Complex Variable, published 1951.