The Celsius and Fahrenheit Scales — Question 8

Question 8

Converting a thermostat preserves its physical hysteresis

Use numerical readings $F=\frac95C+32$ and $T/\mathrm K=C+273.15$; these conversions are exact. A heating controller switches on below $18.0^\circ\mathrm C$ and off above $22.0^\circ\mathrm C$. In the ideal cycle, temperature rises at $1.0$ Celsius degree per minute when on and falls at $.50$ Celsius degree per minute when off; switching delays are negligible.

1. Convert both thresholds and both signed rates to Fahrenheit units.

2. Calculate the heating time, cooling time and on-time fraction in either scale. Show why including 32 in a rate conversion would be invalid.

Question 8 - Solution

Converting a thermostat preserves its physical hysteresis

(a) Method and calculation

$$F_{\mathrm{on}}=64.4,\qquad F_{\mathrm{off}}=71.6,\qquad\Delta F=7.2.$$$$\dot F_{\mathrm{heat}}=+1.8\,{}^\circ\mathrm F/\mathrm{min},\quad\dot F_{\mathrm{cool}}=-.90\,{}^\circ\mathrm F/\mathrm{min}.$$

(b) Interpretation and further calculation

$$t_h=4/1=7.2/1.8=4\,\mathrm{min},\quad t_c=4/.50=7.2/.90=8\,\mathrm{min}.$$$$D=\frac{t_h}{t_h+t_c}=\frac13,\qquad\frac{dF}{dt}=1.8\frac{dC}{dt}.$$The derivative of the constant 32 is zero; the cycle duration and duty fraction are scale-independent.

Physical check

The faster heating stage occupies less time even though it crosses the same physical temperature interval.