Part A In the winter a pond has a layer of ice on top that i

Part A

In the winter, a pond has a layer of ice on top that is becoming thicker. The air above the layer is at a constant -11 C and the water below is at a constant 0 C. Starting from zero thickness, how much time is needed for the layer of ice to reach a thickness of 20 cm ? (Hint: Call the thickness through which the heat conducts x, then express dQ in the heat conduction equation in terms of dx, then separate t and x and integrate both sides.) Express your answer using two significant figures.

t = ? s

Part B

Assuming that the depth of the pond is 40 m at all points, calculate how much time would be needed for the pond to freeze entirely.

Express your answer using two significant figures.

t = ? s

Solution

Let the pond already have an ice upto depth x at time t
After time dt, another dq energy is lost from the water and it forms a new layer of thickness dx, volume dV, mass dm
Temperature outside = To, Temperture of water = T

so dq/dt = k*A*(T - To)/x
where k is thermal conductivity of ice, A is area of pond

this dq is the latent heat of fusion of that amound of water
so dq = L*dm (L = latent heat of fusion of water)
dq = L * (rho) * A * dx ( rho -> density of water)

so (rho) L A dx = k A (T - To) dt / x
=> (rho) L xdx = k (T - To) dt
=> t = [ rho * L x ^2 ] / [2 k (T - To)]

1. x = 0.2m , rho = 1000 kg/m^3 , L = 334000 J/kg , k = 2.2 , T = 0 , To = -11

t = 76.675 Hours

2. x = 40m , rho = 1000 kg/m^3 , L = 334000 J/kg , k = 2.2 , T = 0 , To = -11
t = 127,793.0823 days = 350.11 years (365 days / year)