Determination of the Steady-State Temperature from Partial Heating Experiment Data
Abstract Generated abstract
This paper addresses the problem of estimating the final steady-state temperature of a heated body when an experiment cannot be continued until equilibrium is reached. Starting from the standard exponential heating law, the author derives an analytical relation between temperature excess and its rate of change, replacing graphical extrapolation methods with a simple formula based on two equal time intervals in a partial heating record. An example illustrates calculation of the maximum temperature excess from early measurements, and the method is noted to be most accurate when the compared temperature increments differ substantially. The approach is limited to cases in which the heat input per unit time remains constant.
Full Text
M. F. Filippov.
DETERMINATION OF THE STEADY-STATE TEMPERATURE FROM THE DATA OF A PARTIAL HEATING EXPERIMENT
If, in heating some body, it is necessary to know the final heating temperature corresponding to the steady state, but the heating experiment cannot be carried through to the end, indirect methods are used to determine the maximum temperature.
Most indirect methods (Loppe, Tonnelle, Union of German Electrical Engineers, etc.) require graphical constructions, which, at the moment of the experiment, when it is required to predetermine the maximum heating temperature, cannot be carried out sufficiently quickly and accurately. In this case preference should be given to methods that do not require graphical constructions, such as, for example, the methods of Lissajous and Buro. Below the author proposes another analytical method, somewhat simpler than the methods of Lissajous and Buro.
As is known, the heating curve is expressed by the equation
\[ \tau = \tau_m \left[1 - e^{-\frac{t}{T}}\right], \tag{1} \]
where: \(\tau\) is the excess of the temperature of the body over that of the surroundings after a time interval \(t\) from the beginning of the heating experiment,
\(\tau_m\) is the maximum excess of the temperature of the body over that of the surroundings, corresponding to the steady state,
\(T\) is the time constant of heating.
From equation (1) we may obtain two equations,
\[ \tau_m - \tau = \tau_m e^{-\frac{t}{T}}, \tag{2} \]
\[ \frac{d\tau}{dt} = \frac{\tau_m}{T} e^{-\frac{t}{T}}, \tag{3} \]
solving the last two equations jointly, we obtain:
\[ \tau = \tau_m - T \frac{d\tau}{dt}, \tag{4} \]
i.e., \(\tau = f\!\left(\frac{d\tau}{dt}\right)\) represents a straight line which cuts off on the ordinate axis the segment \(\tau_m\), the desired maximum temperature excess.
This property is used in the method of the Union of German Electrical Engineers. But this method, as was indicated earlier, requires graphical constructions. In the method proposed by the author, graphical constructions are eliminated.
Equation (4), as the equation of a straight line, can be expressed through the intercepts on the coordinate axes (Fig. 1)
\[ \frac{\Delta \tau}{\Delta t} \Big/ \frac{\Delta \tau_0}{\Delta t} + \frac{\tau}{\tau_m} = 1 . \]
whence
\[ \tau_m=\frac{\tau\cdot \dfrac{\Delta\tau_0}{\Delta t}} {\dfrac{\Delta\tau_0}{\Delta t}-\dfrac{\Delta\tau}{\Delta t}}, \tag{5} \]
where \(\Delta\tau_0\) is the increase in temperature over the time interval \(\Delta t\) from the beginning of heating, and \(\Delta\tau\) is the increase in temperature over the same time interval \(\Delta t\) beyond an arbitrary intermediate value of the excess \(\tau\).
Fig. 1.
Fig. 2.
In Fig. 2 the heating curve is shown, and the segments corresponding to the elements of equation (5) are indicated.
Since the interval \(\Delta t\) must be the same in determining \(\Delta t_0\) and \(\Delta t\), equation (5) assumes the form
\[ \tau_m=\frac{\tau\cdot\Delta\tau_0}{\Delta\tau_0-\Delta t}. \tag{6} \]
The resulting equation can be used to determine the maximum temperature excess from data of a partial heating experiment.
Example: From the heating experiment the following data were obtained:
| Time \(t\), min. | 0 | 2 | 6 | 8 | 12 |
|---|---|---|---|---|---|
| Temp. excess, \(\tau^\circ\mathrm{C}\) | 0 | 18 | 45 | 55 | 70 |
we compute the maximum temperature excess
\[ \tau_m=\frac{45\cdot18}{18-10}=101^\circ\mathrm{C}. \]
The greater the difference between \(\Delta\tau_0\) and \(\Delta t\), the more accurate the result, despite possible experimental inaccuracies.
The proposed method, like all the methods indicated above, applies to the case in which the amount of heat expended on heating per unit time is a constant quantity.