A large container is separated into two halves by a 2.0 cm thick glass divider. The area of the divider is 0.25 m^2 and the thermal conductivity of glass is 0.90W/m•K. Each side is filled with 30 kg of water, specific heat 4180 J/kg•K, at different temperatures. Initially, the temperature difference is 20 K. How long will it take before the temperature difference is 1.0 K?
Assume the water on each side of the divider is at uniform but changing temperature, due to convection or stirring.
Let the two temperatures on opposite sides be T1 and T2
The rate of heat conduction through the divider is
dQ/dt = (T1- T2)*k*A/(thickness)
= (T1 - T2)(0.9)(0.25)/0.02
= (T1 - T2)*11.25 J/(K*s)
The energy conservation equation is
dT1/dt = -dT2/dt = -(1/MC)dQ/dt
d/dt(T1 - T2) = -[2/(M*C])*dQ/dt
= -1.595*10^-5 (K/J) dQ/dt
Let X be the temperature difference, T1 - T2
dX/dt = -1.595*10^-5 dQ/dt
= -1.79*10^-4 X
dX/X = -1.794*10^-4 *t
X = Xo*exp(-t/5573 s)
Where Xo = 20K is the inityial tyemperature difference.
Solve for t when X = 1 C
Thanks a lot.
To find out how long it will take before the temperature difference is 1.0 K, we need to calculate the rate at which heat flows through the glass divider. Once we know the rate of heat flow, we can calculate the time it takes to transfer the required amount of heat.
First, let's calculate the rate of heat flow through the glass divider using Fourier's Law of Heat Conduction:
Q = (k * A * ΔT) / d
where:
Q is the heat flow rate,
k is the thermal conductivity of glass,
A is the area of the glass divider,
ΔT is the temperature difference across the divider,
and d is the thickness of the glass divider.
Given:
k = 0.90 W/m•K,
A = 0.25 m^2,
ΔT = 20 K,
and d = 0.02 m.
Plugging in the values:
Q = (0.90 * 0.25 * 20) / 0.02
Q = 9.0 W
The heat flow rate through the glass divider is 9.0 watts.
Next, let's calculate the amount of heat required to reduce the temperature difference from 20 K to 1.0 K. The heat required can be calculated using the equation:
Q = m * C * ΔT
where:
Q is the heat required,
m is the mass of the water,
C is the specific heat of water,
ΔT is the desired temperature difference.
Given:
m = 30 kg,
C = 4180 J/kg•K,
and ΔT = 20 K - 1.0 K = 19 K.
Plugging in the values:
Q = 30 * 4180 * 19
Q = 238,7400 J
The heat required to reduce the temperature difference from 20 K to 1.0 K is 238,7400 J.
Now, let's calculate the time it takes to transfer this amount of heat using the formula:
Q = P * t
where:
Q is the heat required,
P is the power or rate of heat flow,
and t is the time.
Rearranging the formula:
t = Q / P
Plugging in the values:
t = 238,7400 J / 9.0 W
t = 265,266.67 s
The time it will take to reduce the temperature difference from 20 K to 1.0 K is approximately 265,266.67 seconds.