A black thin-walled cylindrical can, 10.0 cm in diameter and 20.0 cm tall is filled with water at 100°C and suspended in a 20°C room. Determine the initial rate of heat transfer due to radiation, and the initial rate of change of temperature of the water. (Neglect any calorimetric effects of the can)
Note that they only ask for the heat transfer due to radiation. Use the Stefan-Boltzmann law.
The can's temperatuire will be 373 K, initially.
The room temperature is 293 K.
Q = M*C*dT/dt
You want the cooldown rate dT/dt
Q = A*sigma*(373^4 - 293^4)
C = 4184 J/kg*K is the specific heat of water
A is the can's outside arwea in m^2
sigma is the Stefan-Boltzmann constant. Assume can and room wall emissivity = 1.
M is the water mass
To determine the initial rate of heat transfer due to radiation and the initial rate of change of temperature of the water, we can use the Stefan-Boltzmann law and the equation for thermal conduction.
1. Initial rate of heat transfer due to radiation:
The heat transfer due to radiation can be calculated using the Stefan-Boltzmann law, which states that the rate of heat transfer by radiation is proportional to the difference in temperatures raised to the fourth power. The equation for the rate of heat transfer by radiation is:
Q_rad = σ * A * (T_can^4 - T_room^4)
Where:
Q_rad is the rate of heat transfer by radiation,
σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/(m^2·K^4)),
A is the surface area of the can,
T_can is the initial temperature of the can (100°C + 273.15), converted to Kelvin,
T_room is the temperature of the room (20°C + 273.15), converted to Kelvin.
First, calculate the surface area of the can:
A = π * r^2 + 2 * π * r * h
Where:
r is the radius of the can (10.0 cm / 2),
h is the height of the can (20.0 cm).
Then, substitute the values into the equation for the rate of heat transfer by radiation and calculate Q_rad.
2. Initial rate of change of temperature of the water:
The initial rate of change of temperature of the water can be determined by considering thermal conduction. The equation for the rate of heat transfer by conduction is:
Q_conduction = k * A * (dT/dx)
Where:
Q_conduction is the rate of heat transfer by conduction,
k is the thermal conductivity of water (approximately 0.6 W/(m·K)),
A is the cross-sectional area of the water (π * r^2),
dT/dx is the temperature gradient between the can and the water.
Assuming that the can and water are in thermal equilibrium initially, dT/dx can be calculated as (T_can - T_room) / (h / 2), where h is the height of the can.
Finally, substitute the values into the equation for the rate of heat transfer by conduction and calculate Q_conduction.
Please note that this calculation neglects the calorimetric effects of the can, which means that the heat capacity of the can is not considered.