Temperature — Question 8

Question 8

A stationary profile is not global thermal equilibrium

A rod is maintained between reservoirs at $300$ and $340\,\mathrm K$. Its steady field is $T(x,t)=300\,\mathrm K+(100\,\mathrm{K/m})x$, for $0\le x\le.40\,\mathrm m$. The supplied transport model is $P=-kA\,\partial T/\partial x$, with $kA=.020\,\mathrm{W\,m/K}$; positive $P$ points toward increasing $x$.

1. Find the temporal derivative, spatial derivative and energy-transfer rate.

2. Does an unchanging local thermometer reading establish global equilibrium? State the conditions that fail here and the direction of transfer.

Question 8 - Solution

A stationary profile is not global thermal equilibrium

(a) Method and calculation

$$\frac{\partial T}{\partial t}=0,\qquad\frac{\partial T}{\partial x}=100\,\mathrm{K/m}.$$$$P=-.020(100)=-2.0\,\mathrm W.$$Every local temperature is constant in time, but the field remains nonuniform.

(b) Interpretation and further calculation

$$T(.40)-T(0)=40\,\mathrm K\ne0,\qquad P\ne0.$$The negative sign means transfer from the right, hotter end toward the left. Maintained boundary reservoirs sustain this steady state. Global thermal equilibrium would require no such temperature-driven net transfer.

Physical check

Zero time derivative and zero spatial gradient are different statements; steady and equilibrium cannot be interchanged.