On the Required Accuracy of Two-Position Temperature Control Taking into Account the Attenuation of Temperature Oscillations Inside Bodies

A. A. Gurchenok

(Submitted by the Department of Automation of Thermal-Power Processes of Industrial Enterprises)

For automatic temperature control in furnaces and other apparatus, a two-position “on—off” control system is often used (Fig. 1, a).

Because of a certain dead zone of the controller, switching the regulating action on and off occurs at different values of the controlled variable. The control process is then an oscillatory process with a constant, undamped amplitude. The magnitude of the oscillation amplitude of the controlled parameter for an object without lag will be minimal and equal to one-half of the controller dead zone (Fig. 1, b).

Fig. 1. a — thermal action, b — temperature, \(\vartheta-\vartheta\) — dead zone of the controller.

With a periodic character of variation of the temperature of the heat sources, it should be expected that both at the surface and in deeper...

layers of the heated object, the temperature will also vary periodically.

The desired temperature function must satisfy the condition

\[ \frac{\partial \vartheta}{\partial t}=a\frac{\partial^2\vartheta}{\partial x^2} \tag{1} \]

(the case of a one-dimensional temperature field).

The boundary conditions may be

either

\[ \vartheta_0=\vartheta_{0ц}\cos\left(2\frac{\pi}{t_0}t\right), \tag{1-а} \]

or

\[ \vartheta_0=\vartheta_{0ц}\sin\left(2\frac{\pi}{t_0}t\right). \tag{1-б} \]

Conditions (1-а) and (1-б) are equivalent and differ only by a shift of the zero point of the time axis by \(\pi/2\). If the temperature changes not according to the laws of harmonic oscillation, but in the form of a discontinuous or some other type of periodically repeating curve, then this function should be expanded into its constituent harmonics. Then, for each harmonic, the corresponding surface temperature must be determined and, finally, all these temperatures must be summed.

The solution of equation (1), after determining the arbitrary constants of integration from condition (1-а) at

\[ t=0,\qquad x=0, \]

\[ \vartheta_0=\vartheta_{0ц} \tag{1-в} \]

takes the form

\[ \vartheta=\vartheta_{0ц}e^{-x\sqrt{\frac{\pi}{at_0}}} \cos\left(x\sqrt{\frac{\pi}{at_0}}-2\frac{\pi}{t_0}t\right). \tag{2} \]
[1]

Here \(t\) is the time, counted from an arbitrary initial instant, hours;

\(\vartheta,\vartheta_0,\vartheta_{0ц}\) are, respectively, the temperature oscillations at a distance \(x\) from the surface, at the surface, and the maximum amplitude of the oscillations, \({}^{\circ}\mathrm{C}\);

\(a\) is the thermal diffusivity coefficient of the body, \(\mathrm{m}^2/\mathrm{hour}\).

Analysis of the equation obtained makes it possible to find an instantaneous picture of the distribution of temperature oscillations. At \(t=0\), depending on \(x\), the quantity \(\vartheta\) will vary according to the law

\[ \vartheta_x=\vartheta_{0ц}e^{-x\sqrt{\frac{\pi}{at_0}}} \cos x\sqrt{\frac{\pi}{at_0}}. \tag{3} \]

This dependence is the equation of a wave (cosinusoid) with a decaying amplitude. The maximum value of the amplitude will vary as a function of \(x\) according to the law

\[ \vartheta_{xц}=\vartheta_{0ц}e^{-x\sqrt{\frac{\pi}{at_0}}}. \tag{4} \]

From (4) the problem is solved: at what depth \(x\) will the temperature oscillations decrease to the \(n\)-th part of their value at the surface,

\[ x=\sqrt{\frac{at_0}{\pi}}\ln n. \tag{5} \]

The depth of penetration of temperature waves is the greater, the greater the thermal diffusivity and the more slowly the oscillations occur.

Expression (4) makes it possible to determine the magnitude of the permissible dead zone of the regulator, provided that the specified magnitude of temperature oscillations inside the controlled object is maintained.

Example: In controlling the temperature of the charge in mica-synthesis furnaces, it is required to maintain the temperature with an accuracy of \(\pm 0.5^\circ\mathrm{C}\).

Determine the permissibility of setting a temperature regulator with a dead zone of \(\pm 10^\circ\mathrm{C}\), if the rate of heating of the heaters when the controlled power is switched on in a step is 20 seconds (the full period of oscillation is 40 sec—0.011 hour). The crucible material is fireclay; the thermal diffusivity coefficient is \(a = 6 \cdot 10^{-3}\ \mathrm{m^2/hour}\); the wall thickness is \(0.02\ \mathrm{m}\). The crucible with the filled mass is to be regarded as a semi-infinite body.

Let us take the amplitude of the temperature oscillations on the surface of the crucible to be equal to the amplitude of the heater oscillations, \(\vartheta_{0\mu} = \pm 10^\circ\mathrm{C}\).

We consider the periodic function of temperature variation to be: a) a sinusoid, b) a stepwise function.

For case (a), from (4) we have

\[ \vartheta_{x\mu} = \pm 10 \cdot e^{-0.02 \sqrt{\frac{3.14}{6 \cdot 10^{-3} \cdot 0.011}}}, \]

\[ \vartheta_{x\mu} = \pm 0.12^\circ\mathrm{C}. \]

Case (b). A stepwise function (Fig. 1, \(a\)) can be represented by the following series [2]:

\[ \vartheta_o = \frac{4}{\pi}\,\vartheta_{0\mu} \left[ \cos\left(2\frac{\pi}{t_o}t\right) + \frac{1}{3}\cos 3\left(2\frac{\pi}{t_o}t\right) + \frac{1}{5}\cos 5\left(2\frac{\pi}{t_o}t\right) \right]. \tag{6} \]

For the boundary conditions (6), equation (4) will have the form

\[ \vartheta_{x\mu} = 1.27\,\vartheta_{0\mu} \left[ e^{-x\sqrt{\frac{\pi}{a t_o}}} + \frac{1}{3}e^{-x\sqrt{\frac{3\pi}{a t_o}}} + \frac{1}{5}e^{-x\sqrt{\frac{5\pi}{a t_o}}} \right]. \tag{7} \]

After substituting the conditions of the problem into (7), we obtain

\[ \vartheta_{x\mu} = 1.27(\pm 10)[0.012 + 0.002 + 0.0001] \simeq \pm 0.18^\circ\mathrm{C}. \]

Conclusion: The specification of the accuracy of two-position temperature control should be made taking into account the attenuation of temperature oscillations inside bodies.

REFERENCES

1. Gröber H., Erk S., Grigull U. Fundamentals of the Theory of Heat Exchange. 1958, IL.
2. Romanovsky P. I. Fourier Series. Field Theory. GIDML, 1958.