Differential Equations of the Dynamics and Statics of Furnaces for Drying Wire Insulation under Constant Heat Generation in the Wall
Abstract Generated abstract
This paper derives dynamic and static equations for vertical cylindrical furnaces used to dry wire insulation in cable production, where automatic temperature control requires knowledge of furnace characteristics. Modeling the furnace as an open vertical cylindrical cavity with constant wall heat generation, the authors formulate heat balances that include convection, wall heat accumulation, external radiation, radiation through the open ends, and radiant exchange along the inner surface. The result is a nonlinear integro-differential system coupling wall temperature and heat-carrier temperature, with parameters corresponding to heat-transfer coefficients, gains, and time constants. The formulation also yields limiting cases for black surfaces, absent radiation, zero heat generation, and steady-state operation, and is presented as a general structure applicable to related heat exchangers with ideal jacket mixing.
Full Text
DIFFERENTIAL EQUATIONS OF THE DYNAMICS AND STATICS OF FURNACES FOR DRYING WIRE INSULATION UNDER CONSTANT HEAT GENERATION IN THE WALL
Yu. M. Trifonov, Yu. A. Trifonova, P. A. Andriyanov
(Presented by the Department of Machines and Apparatuses of Chemical Production)
Widely used objects of temperature control in the cable industry are vertical furnaces for drying the insulation of winding wires and enamel wires. Temperature control in these furnaces is carried out automatically. To select the tuning parameters of the controller, it is necessary to know the static and dynamic characteristics of the furnace. These characteristics can be determined if the equations describing the dynamic and static properties of the controlled object are known. In the literature there are no equations of dynamics and statics that take into account the complex heat exchange in furnaces of cable production. Below is given a derivation of such equations, taking account of heat exchange by radiation and convection.
The structural design of the furnaces under consideration is such that they may be represented as a vertical cylindrical chamber, open at the ends. In furnaces of this type, heat is released by electric heaters arranged vertically or horizontally along the height of the furnace. The temperature of the furnace walls $T_2(\tau, h)$ in the steady state is a function of the coordinate $h$. The mean value of $T_2(\tau, h)$ lies within the limits $773^\circ\mathrm{K}$—$873^\circ\mathrm{K}$. The heat carrier in the furnace is air. The difference in densities of the air heated in the furnace and the cold ambient air is the cause of the motion of the heat carrier.
In deriving the differential equations of dynamics and statics, we take into account heat exchange by radiation inside the furnace. The derivation of the equations is carried out under the following assumptions: 1) we replace the cylindrical chamber of arbitrary profile by a round, vertical, cylindrical cavity (tube) open at the ends and of height $H$ (Fig. 1), the lateral surfaces of which are gray and optically homogeneous; 2) the heat generation in the wall $Q$ is taken as constant over the height of the tube; 3) the temperatures of the tube and of the heat carrier in the section under consideration $h$ are constant and respectively equal to $T_2(\tau, h)$ and $T_1(\tau, h)$, while the change in temperatures occurs only with motion along the axis $h$ (the point of entry of the heat carrier into the tube is taken as the origin); 4) we neglect thermal conductivity in the heat carrier and in the tube wall, since the Peclet criterion $Pe$ is greater than 100; 5) the heat carrier is taken as diathermic; 6) in deriving the equations we use mean values of the heat capacities, densities, and heat-transfer coefficients over the entire length of the cylindrical cavity; 7) the heat consumed for heating the wire is not taken into account [1, 2, 3]; 8) the material of the tube from the end surfaces—
does not participate in heat exchange; 9) back radiation of the surrounding medium is not taken into account.
For an elementary cylindrical section of length \(\Delta h\), located at a distance \(h\) from the origin of coordinates, the following heat-balance equation is valid:
\[ \Delta Q_1+\Delta Q_2+\Delta Q_3+\Delta Q_4+\Delta Q_5=\Delta Q+\Delta Q_6, \tag{1} \]
where \(\Delta Q\) is the heat generation of an element of the cylindrical wall of length \(\Delta h\) during the time \(\Delta\tau\);
Fig. 1. Vertical cylindrical cavity, open at the ends
\[
\Delta Q_1
\]
— the quantity of heat transferred to the heat carrier from the wall by convection;
\[
\Delta Q_2
\]
— the quantity of heat given off by the outer surface of the pipe to the surrounding medium by convection;
\[
\Delta Q_3
\]
— the quantity of heat accumulated by the pipe wall;
\[
\Delta Q_4
\]
— the quantity of heat given off by radiation from the outer surface of the cylindrical ring of height \(\Delta h\);
\(\Delta Q_5\) — the amount of heat given off by radiation by the elementary cylindrical annular element through the bases of the cylinder;
\(\Delta Q_6\) — the resultant radiant flux between the inner surface of the annular element and the entire inner surface of the cylinder.
The terms of equation (1), respectively, are equal to:
\[ \Delta Q=\frac{Q}{H}\cdot \Delta h\cdot \Delta\tau, \tag{2} \]
\[ \Delta Q_1=\alpha_{21}\,[T_2(\tau,h)-T_1(\tau,h)]\cdot \pi\cdot d_1\cdot \Delta h\cdot \Delta\tau, \tag{3} \]
\[ \Delta Q_2=\alpha_{23}\,[T_2(\tau,h)-T_3]\cdot \pi\cdot d_2\cdot \Delta h\cdot \Delta\tau, \tag{4} \]
\[ \Delta Q_3=c_2\cdot \rho_2\cdot F_2\, \frac{\partial T_2(\tau,h)}{\partial \tau}\cdot \Delta h\cdot \Delta\tau, \tag{5} \]
\[ \Delta Q_4=c_0\cdot \varepsilon\cdot 10^{-8}\cdot T_2^4(\tau,h)\cdot \pi\cdot d_2\cdot \Delta h\cdot \Delta\tau, \tag{6} \]
\[ \Delta Q_5=c_0\cdot \varepsilon\cdot 10^{-8}\cdot [\psi(h)+\psi(H-h)]\cdot T_2^4(\tau,h)\cdot \pi\cdot d_1\cdot \Delta h\cdot \Delta\tau, \tag{7} \]
where
\[
T_1(\tau,h)
\]
— the temperature of the heat carrier flowing inside the cylindrical cavity;
\[
T_2(\tau,h)
\]
— the wall temperature;
\[
T_3
\]
— the ambient temperature;
\[
\alpha_{21}, \alpha_{23}
\]
— the heat-transfer coefficients, respectively from the inner and outer surfaces of the cylindrical cavity;
\[
d_2 \text{ and } d_1
\]
— respectively the outer and inner diameters of the tube;
\[
c_2
\]
— the mean heat capacity of the tube material;
\[
\rho_2
\]
— the density of the tube material;
\[
\varepsilon
\]
— the degree of blackness of the tube surfaces;
\[
c_0
\]
— the radiation coefficient of an absolutely black body;
\[
\psi(h),\ \psi(H-h)
\]
— irradiation coefficients from an elementary cylindrical ring to the bases of the cylinder.
To determine the term \(\Delta Q_6\), divide the cylinder into \((n+m)\) elementary cylindrical rings of height \(\delta h=\Delta h_1\). Above the coordinate \(h\) the number of rings is \(n\), and below it is \(m\). Introduce the coordinate \(h_1\). It characterizes the distance from the cylindrical ring under consideration, with coordinate \(h\), to the moving ring with coordinate \(h+h_1\) or \(h-h_1\). Taking into account that the function \(T_2(\tau,h)\), determined without allowance for radiation [4, 5], represents the limiting temperature distribution, then the resultant radiant flux between annular elements with coordinates \(h,\ h-h_1\) is approximately equal to
\[ \Delta^2 E_{h,h-h_1} = \varepsilon_{\mathrm{p}}\cdot c_0\cdot 10^{-8}\cdot \Delta\varphi(h_1)\cdot [T_2^4(\tau,h)-T_2^4(\tau,h-h_1)]\cdot \pi\cdot d_1\cdot \Delta h\cdot \Delta\tau, \tag{8} \]
where
\[
\Delta\varphi(h_1)
\]
— the irradiation coefficient between two cylindrical rings located at a distance \(h_1\) from one another;
\[
\varepsilon_{\mathrm{p}}
\]
— the reduced degree of blackness of the system of two cylindrical elementary rings.
The degree of blackness of the tube surface is \(\varepsilon \ge 0.8\). The reduced degree of blackness in expression (8) can be determined from the formula for two arbitrarily arranged surfaces [6] \(\varepsilon_{\mathrm{p}}=\varepsilon_1\cdot \varepsilon_2\); in our case \(\varepsilon_{\mathrm{p}}=\varepsilon^2\).
The resulting radiant flux between rings with coordinates \(h,\ h+h_1\) is equal to:
\[ \Delta^2 E_{h,h+h_1} = c_0\cdot \varepsilon_{\mathrm p}\cdot 10^{-8}\cdot \Delta\varphi(h_1)\cdot \left[T_2^4(\tau,h+h_1)-T_2^4(\tau,h)\right]\cdot \pi\cdot d_1\cdot \Delta h\cdot \Delta\tau . \tag{9} \]
The expression for the irradiation coefficient between two elementary cylindrical rings [7] in increments can be represented as
\[ \Delta\varphi(h_1)=f(h_1)\cdot \Delta h_1 . \]
Summing expressions of the type (8) and (9), after certain transformations we have
\[ \Delta Q_6 = c_0\cdot \varepsilon_{\mathrm p}\cdot 10^{-8}\cdot \left\{ \int_0^{H-h} f(h_1)T_2^4(\tau,h+h_1)\,dh_1 + \int_0^h f(h_1)\cdot T_2^4(\tau,h-h_1)\,dh_1 - \left[ \int_0^{H-h} f(h_1)\,dh_1 + \int_0^h f(h_1)\cdot dh_1 \right]\cdot T_2^4(\tau,h) \right\}\cdot \pi\cdot d_1\cdot \Delta h\cdot \Delta\tau . \tag{11} \]
Considering jointly equations (1) through (7) and (11), we obtain
\[ T_{21}\frac{\partial T_2(\tau,h)}{\partial \tau} + (1+a)T_2(\tau,h) = \kappa Q + T_1(\tau,h) + aT_3 + \kappa_2 \left[ \int_0^{H-h} f(h_1)T_2^4(\tau,h+h_1)\,dh_1 + \int_0^h f(h_1)T_2^4(\tau,h-h_1)\,dh_1 \right] - \]
\[ - \kappa_1\cdot T_2^4(\tau,h) - \kappa_2 \left\{ \left(\frac{1}{\varepsilon}-1\right) \left[\psi(h)+\psi(H-h)\right] + 1 \right\} \cdot T_2^4(\tau,h), \tag{12} \]
where the following notation has been introduced:
\[ a=\frac{\alpha_{23}\cdot d_2}{\alpha_{21}\cdot d_1}, \qquad \kappa=\frac{1}{\pi\cdot d_1\cdot H\cdot \alpha_{21}}, \qquad \kappa_1=\frac{\varepsilon\cdot c_0\cdot 10^{-8}\cdot d_2}{\alpha_{21}\cdot d_1}, \]
\[ \kappa_2=\frac{\varepsilon^2\cdot c_0\cdot 10^{-8}}{\alpha_{21}}, \qquad T_{21}=\frac{c_2\cdot \rho_2\cdot \delta}{\alpha_{21}}. \]
The physical meaning of the quantities denoted is as follows: \(a\) is a dimensionless heat-transfer coefficient; \(\kappa\) is a gain coefficient, \({}^{\circ}K/\mathrm{W}\); \(\kappa_1,\ \kappa_2\) are gain coefficients, \(1/{}^{\circ}K^3\); \(T_{21}\) is a time constant, sec.
Finally, the dynamics of the furnace model under consideration, taking into account the equation of variation of the heat-carrier temperature [5], [7], has the form:
\[ \left\{ \begin{aligned} &T_{21}\frac{\partial T_2(\tau,h)}{\partial \tau} + (1+\alpha)T_2(\tau,h) = \kappa Q + T_1(\tau,h) + \alpha T_3 +\\ &\quad + \kappa_2 \left[ \int_0^{H-h} f(h_1)T_2^4(\tau,h+h_1)\,dh_1 + \int_0^h f(h_1)T_2^4(\tau,h-h_1)\,dh_1 \right] -\\ &\quad - \kappa_1\cdot T_2^4(\tau,h) - \kappa_2\cdot \left\{ \left(\frac{1}{\varepsilon}-1\right)\cdot \left[\psi(h)+\psi(H-h)\right] + 1 \right\} \cdot T_2^4(\tau,h). \end{aligned} \right. \tag{13} \]
\[ \tag{13-a} \]
\[ v\cdot T_{12}\frac{\partial T_1(\tau,h)}{\partial h} + T_{12}\frac{\partial T_1(\tau,h)}{\partial \tau} = T_2(\tau,h)-T_1(\tau,h), \tag{13-б} \]
where
\[ v \cdot T_{12}=\frac{v \cdot F_1 \cdot \rho_1 \cdot c_1}{a_{21}\cdot \pi \cdot d_1}. \]
Here the following parameters of the heat-transfer medium occur: \(v\)—velocity of motion; \(T_{12}\)—time constant; \(\rho_1\)—density; \(c_1\)—heat capacity.
Relations (13-a), (13-b) constitute a nonlinear integro-differential system of equations, for which it is at present difficult to give an exact analytical solution owing to the lack of developed mathematical methods for solving such systems.
It appears possible to obtain solutions of the system of equations (13) by the following methods: 1) by calculation on electronic digital machines; 2) by approximate methods using approximations of the nonlinear terms of expression (13-a); 3) after processing experimental data by methods of similarity theory.
From the system of equations (13) one can obtain the following frequently encountered cases: 1) for \(\varepsilon=1\) we have the equations of the dynamics of a furnace model with absolutely black surfaces; 2) for \(\varepsilon=0\) the equations describe the dynamics of a heat exchanger with ideal mixing in the jacket in the presence of heat generation in the wall; 3) for \(Q=0\) the system of equations corresponds to the dynamics of a furnace if the temperature of the heating heat-transfer medium is constant; 4) by setting the time derivatives equal to zero, in the indicated cases we obtain equations of static characteristics. Some combinations of the enumerated cases are possible.
Thus, the system of equations (13) has been obtained with allowance for heat transfer by radiation from the more heated inner surface of the cylindrical cavity to its less heated part. In the structure of its terms it is general for heat exchangers with ideal mixing in the jacket.
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