Two rods of equal length and temperature at end of the rods is T1 and T2. What is the condition for equal flow of heat through them
The condition for equal flow of heat through two rods of equal length is when the rate of heat transfer, also known as heat flux, is the same for both rods. Heat flux measures the amount of heat transferred per unit of time per unit of area.
To determine this condition, we need to consider two factors: the thermal conductivity of the rods and the temperature difference between the ends of each rod.
The heat flux, Q, for each rod can be calculated using Fourier's Law of Heat Conduction:
Q = -k * A * (dT/dx)
Where:
- Q is the heat flux
- k is the thermal conductivity of the rod
- A is the cross-sectional area of the rod
- (dT/dx) is the temperature gradient along the length of the rod
For equal flow of heat through the rods, we can set the heat flux equations of both rods equal to each other:
Q1 = Q2
-k1 * A1 * (dT1/dx) = -k2 * A2 * (dT2/dx)
Assuming the cross-sectional area and length of both rods are the same, we can simplify the equation:
k1 * (dT1/dx) = k2 * (dT2/dx)
This equation indicates that for equal flow of heat, the temperature gradients along the length of the rods must be in inverse proportion to their respective thermal conductivities.
In other words, the temperature difference per unit length (dT/dx) should be larger for the rod with lower thermal conductivity and smaller for the rod with higher thermal conductivity, in order to balance the heat transfer rates.
Therefore, the condition for equal flow of heat through two rods of equal length is that the temperature gradient along each rod is inversely proportional to its thermal conductivity.