The fundamental of heat conduction is dQ/dt = -k middot A middot dT/ds dQ/dt is

Question

The fundamental of heat conduction is dQ/dt = -k middot A middot dT/ds dQ/dt is the time rate of heat transfer across area A; dT/ds is the temperature gradient perpendicular to area A, and k is a constant of proportionality called the thermal conductivity. Consider a rod of cross-sectional area A and length L between two objects held at fixed temperatures of T_1 and T_2. (This might represent a silver spoon with one end in a hot cup of coffee and the other end in your hand.) What is the steady state temperature profile T(x) between the two objects? Your result should show the dependence of temperature on x, k, A, L, T_1, and T_2. In steady state, the rate of heat transfer is constant. For additional bonus points, determine the temperature profile between two concentric spheres of radius r_1 and r_2 held at temperatures T_1 and T_2. (This might represent the earth's inner core and its outer crust.) Assume that r_1

Explanation / Answer

we know that heat flowing through rod is

dQ/dt = kA (T2-T1)/L

consdera a cross section at a distance x from on end (T2 )

since heat flow is same

we have

dQ/dt = kA(T2- Tx)/x

from both equations we have

T2-Tx = x (T2-T1)/L

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