- #1

CAF123

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## Homework Statement

2)A certain heat source (a heat reservoir or thermal reservoir), is always at temperature ##T_0##. Heat is transferred from it through a slab of thickness ##L## to an object which is initially at a temperature ##T_1 < T_0##. The object, which is otherwise thermally insulated from its surroundings, has a mass m = 0.5 kg and a specific heat capacity ##c = 4 \times 10^{3} J kg^{-1}K^{-1}##. Heat is conducted through the slab at a rate (in J s^{-1}) specified by the

formula ##KA((T_0 - T)/L),## where K is the thermal conductivity of the slab, A is the area of the slab through which heat is transferred and T is the instantaneous temperature of the object.

a)Show that provided certain assumptions are made, ##KA((T_0 - T)/L)\Delta t = mc \Delta{T}##

b)Show that for small t, and hence small ΔT, the above equation can be rearranged and

integrated to give $$T_2 - T_1 = (T_0 - T_1)\left(1 - \exp\left(-\frac{KA(t_2 - t_1)}{Lmc}\right)\right)$$

## The Attempt at a Solution

2) a)Assuming all energy gained by the slab is then gained by the object ( and there is then no losses because of insulation), $$\frac{KA(T_o - T)}{L} \Delta t = cm (T + \Delta T - T)$$ and result follows.

b)Integrate $$\frac{KA}{L} \int_{t_1}^{t_2} dt = cm \int_{T_1}^{T_2} \frac{dT}{T_o - T}$$

which gives $$\frac{KA}{Lmc}(t_2 - t_1) = \ln\left(\frac{T_o - T_1}{T_o -T_2}\right).$$ It is given that ##T_o > T_1## so the numerator of the log is definitely postive. However, I am not so sure about the denominator. Since there are no heat losses, I think there may exist a temperature ##T_2## where ##T_2 >T_o##.

Thank you.