Question

For your cabin in the wilderness, you decide to build a primitive refrigerator out of Styrofoam, planning to keep the interior cool with a block of ice that has an initial mass of 27.0 kg . The box has dimensions of 0.500 m ×0.850 m ×0.500 m . Water from melting ice collects in the bottom of the box. Suppose the ice block is at 0.00 ∘C and the outside temperature is 17.0 ∘C .
If the top of the empty box is never opened and you want the interior of the box to remain at 7.00 ∘C for exactly one week, until all the ice melts, what must be the thickness of the Styrofoam?

Answer 1

Answer:

$x = 0.0485 m$

Explanation:

Total surface area of the box is given as

$A = 2(L\times H + H \times W + W\times L)$

so we have

$A = 2(0.500\times 0.850 + 0.850 \times 0.500 + 0.500 \times 0.500)$

so we have

$A = 2.2 m^2$

now rate of heat flow in conduction mode is given as

$\frac{dQ}{dt} = kA\frac{dT}{dx}$

here we know

$k = 0.033$

$dT = 17 - 7 = 10^o C$

$\frac{dQ}{dt} = \frac{mL}{T}$

$\frac{dQ}{dt} = \frac{27 \times 335,000}{7\times 24 \times 3600}$

$\frac{dQ}{dt} = 14.95 watt$

Now we have

$14.95 = 0.033\times 2.2 \times \frac{10}{x}$

$x = 0.0485 m$