Question:
A metal rod with a length of 1.5 meters and a cross-sectional area of 0.02 square meters is initially at a temperature of 20°C. The rod is heated at one end to a temperature of 200°C. The heat flows through the rod from the hot end to the cold end. It is found that the time it takes for the temperature at the cold end of the rod to increase from 20°C to 120°C is 300 seconds.
a) Calculate the thermal conductivity of the metal rod. b) If the rod is made of aluminum with a specific heat capacity of 900 Joules per kilogram per Kelvin, calculate the amount of heat energy transferred during the time interval specified.
Assume one-dimensional heat flow and neglect any heat loss to the surrounding.
Answer:
a) The rate of heat conduction through a material is given by Fourier's law:
Q = -kA (ΔT/Δx)
Where Q is the rate of heat conduction, k is the thermal conductivity of the material, A is the cross-sectional area, ΔT is the change in temperature, and Δx is the length of the material.
Given: Length of rod (Δx) = 1.5 m Cross-sectional area (A) = 0.02 m² Temperature change (ΔT) = 120°C - 20°C = 100°C
Substituting the given values into the formula:
Q = -kA (ΔT/Δx)
Solving for k:
k = -Q(Δx) / (AΔT)
Since the question mentions that the time it takes for the temperature at the cold end of the rod to increase from 20°C to 120°C is 300 seconds, we can determine the heat conduction rate using the formula:
Q = mcΔT
Where m is the mass of the rod, c is the specific heat capacity, and ΔT is the change in temperature.
b) Given: Specific heat capacity (c) = 900 J/kg⋅K Temperature change (ΔT) = 120°C - 20°C = 100°C Time interval (Δt) = 300 s
To calculate the amount of heat energy transferred during the time interval, we need to determine the mass of the rod. Using the formula:
Q = mcΔT
Rearranging the formula to solve for mass:
m = Q / (cΔT)
Substituting the given values into the formula:
m = Q / (900 J/kg⋅K) * (100°C)
Once the mass is determined, the amount of heat energy can be calculated using the formula:
Q = mcΔT
Substituting the values obtained:
Q = (mass) * (900 J/kg⋅K) * (100°C)
Provide your final answer for part a) and part b) in appropriate units.