Quasi-Steady Heating of a Plate in the Presence of Additional Walls
Abstract Generated abstract
This paper examines quasi-steady heating of a plate when electric heaters act through thin additional walls, a configuration needed especially for metal specimens and relevant to experimental determination of thermophysical properties. The authors formulate the one-dimensional heat conduction problem for the main plate and the added wall, analyze the heat flux at their interface, and show that after an initial interval the system reaches a quasi-stationary regime in which temperature differences remain constant while local temperatures vary linearly with time. Using this regime, they derive a correction formula for the effective constant heat flux entering the specimen, accounting for heat absorbed by the additional wall, and illustrate the effect with experimental and calculated heating curves.
Full Text
QUASI-STEADY HEATING OF A PLATE IN THE PRESENCE OF ADDITIONAL WALLS
G. P. BOIKOV, N. N. NEGODYAEVA, G. V. NIKOLAEVA
Presented by Professor G. I. Fuks
At the present time, an experimental method for determining the thermophysical characteristics of a substance, based on the quasi-steady regime of heating of bodies, is finding ever wider application. The experimental setup is based on the idea of heating an infinite plate made of the material under investigation under the action of a heat flux that is constant in time (Fig. 1).
The transfer of a heat flux constant in time into the plate is practically carried out by means of electric heaters of constant power, pressed against the lateral planes of the plate. If an electric heater is applied directly to a plate made of an insulator, then to a metal plate it can be applied only by means of an electrically conductive additional thin wall (Fig. 2). In the indicated scheme, the propagation of heat is described by the system of equations (1)—(6).
Fig. 1
Fig. 2
\[ \frac{\partial t'(x,\tau)}{\partial \tau} = a_R \frac{\partial^2 t'(x,\tau)}{\partial x^2} \tag{1} \]
\[ \frac{\partial t''(x,\tau)}{\partial \tau} = a_\delta \frac{\partial^2 t''(x,\tau)}{\partial x^2} \tag{2} \]
\[ a \leq x \leq R \]
\[ R \leq x \leq R+\delta \]
\[ t'(x,0)=t''(x,0)=t_0 \tag{3} \]
\[ t'(R,\tau)=t''(R,\tau) \tag{4} \]
\[ \frac{\partial t'(0,\tau)}{\partial x}=0 \tag{5} \]
\[ \lambda \cdot \frac{\partial t''(R+\delta;\tau)}{\partial x}=g_c \tag{6} \]
Introducing a new variable \(g(x,\tau)\) by the formula
\[ g(x,\tau)=-\lambda \frac{\partial t}{\partial x}, \]
the system of equations (1)—(6) can be rewritten as follows:
\[ \frac{\partial g'(x,\tau)}{\partial \tau} = a_R \cdot \frac{\partial^2 g'(x,\tau)}{\partial x^2} \tag{1a} \]
\[ \frac{\partial g''(x,\tau)}{\partial \tau} = a_\delta \frac{\partial^2 g''(x,\tau)}{\partial x^2} \tag{2a} \]
\[ g'(x,0)=g''(x,0)=0 \tag{3a} \]
\[ g'(R,\tau)=g''(R,\tau)=g_R \tag{4a} \]
\[ g'(0,\tau'')=0 \tag{5a} \]
\[ g''(R+\delta;\tau)=g_c \tag{6a} \]
Analysis of the system (1a)—(6a) makes it possible to assert that the magnitude of the flux \(g\), in the general form, is determined by the relation:
\[ g_R=N-Me^{-a_\delta \cdot k_\delta^2 \cdot \tau} \tag{7} \]
where \(N\), \(M\), \(k\) are constant quantities.
At some \(\tau>\tau_*\), the magnitude of the flux \(g_R\) will become a constant quantity.
If it is assumed that the additional wall of thickness \(\delta\) consists of the same substance as the homogeneous wall, then expression (7) takes the form:
\[ g_R=A-B \cdot e^{-a_R \cdot k_R^2 \cdot \tau}, \tag{8} \]
which in its form coincides with the relation for determining the heat flux given in the analytical theory of heat conduction [1].
Thus, in the system described by conditions (1)—(6), after the elapse of a certain moment of time \(\tau_*\) from the beginning of heating, a quasi-stationary heating regime sets in. The temperature difference at two points of the system will remain one and the same, and the temperature of each point will change according to a rectilinear law. The latter is also confirmed by experimental data (Fig. 3).
In order that the heated plate more closely fall under the concept of an unbounded one, its thickness \(2R\) is taken sufficiently small in size. This contributes to the circumstance that the quasi-stationary heating regime sets in immediately from the first moments of time.
According to the data of A. V. Lykov [1], for large values of the Fourier criterion the temperature field in a plate of thickness \(2R\) is described by the dependence:
\[ t'(x,\tau)=t_0+\frac{g_R}{\lambda_R} \left( \frac{a_R \cdot \tau}{R} - \frac{R^2-3x^2}{6R} \right), \tag{9} \]
where \(t_0\) is the initial temperature of the body,
Fig. 3
- Change in the temperature difference between the center and the surface of the plate when it is heated through additional walls.
- Change in the temperature of the center of the plate when it is heated by the heater without additional walls.
- The same, in the presence of additional walls.
- Main plate made of vinyl plastic, of thickness \(2R = 36\) mm.
- Additional plates made of cardboard, of thickness \(\delta = 2\) mm.
- Electric heaters of constant power.
7 and 8. Locations where the thermocouple junctions are installed.
I. Main set of the installation.
II. Auxiliary (protective) sets of the installation, mounted analogously to the main one.
\(g_R\) is the constant heat flux at the surface of the plate under investigation, determined by the relation:
\[ g_R = g_c - \delta \cdot (c \cdot \gamma)_\delta \cdot \frac{dt_{\mathrm{cp}}}{d\tau} \]
Here \(t_{\mathrm{cp}}\) is the mean temperature of the additional wall of thickness \(\delta\).
One of the characteristics of the quasi-steady heating regime is the identical change with time of the temperature of all points in space. Therefore
\[ \frac{dt_{\mathrm{cp}}}{d\tau} = \frac{dt}{d\tau} \]
According to formula (9):
\[ \frac{dt}{d\tau} = \frac{g_R \cdot a_R}{\lambda_R \cdot R} = \frac{g_R}{R \cdot (C \cdot \gamma)_R} \]
Substituting the value of the derivative into relation (10), we shall have:
\[ g_R = g_c - g_R \cdot \frac{\delta}{R} \cdot \frac{(C \cdot \gamma)_\delta}{(C \cdot \gamma)_R} \]
whence we find
\[ g_R=\frac{g_c}{1+\dfrac{\delta}{R}\cdot \dfrac{(C\gamma)_\delta}{(C\gamma)_R}} \tag{11} \]
Using expression (11), one can determine the value of the constant heat flux acting on the surface of a plate made of the material under investigation.
Figure 4 shows the heating of a steel plate of thickness \(2R=20\) mm at a heat flux from the heater \(g_c=10000\ \mathrm{kcal}/\mathrm{m}^2\mathrm{h}\).
Line 1 shows the change in temperature at the center of the steel plate, obtained by calculation when it is heated without additional walls.
Line 2 shows the change in the temperature of the center according to formula (9), in the presence of asbestos insulating gaskets of thickness \(\delta=3\) mm.
Fig. 4
Calculation of the heating of a steel plate of thickness \(2R=20\) mm at a heat flux from the heater \(g_c=10000\ \mathrm{kcal}/\mathrm{m}^2\mathrm{h}\).
1 — change in temperature at the center of the steel plate when it is heated without additional walls.
2 — change in the temperature of the center, according to formula (9), in the presence of asbestos insulating gaskets of thickness \(=3\) mm.
CONCLUSION
The method presented can be used in experimental work to determine the thermophysical quantities of materials on the basis of quasi-steady heating of bodies. The reasoning given makes it possible, to a certain extent, also to take into account the heat going to heat the heater itself and the device on which the heater is mounted.
LITERATURE
- A. V. Lykov — Theory of Heat Conduction, GITTL, Moscow, 1952.