To tackle the problem of finding the temperature at a distance \( x \) from the cold end of a uniform rod with temperature-dependent thermal conductivity, we need to apply Fourier's law of heat conduction along with the given relationship for thermal conductivity. Let's break this down step by step.
Understanding the Problem
We have a rod of length \( l \) with one end at temperature \( t_1 \) (the cold end) and the other at \( t_2 \) (the hot end), where \( t_2 > t_1 \). The thermal conductivity \( k \) of the rod varies inversely with temperature, expressed as:
k = A/T
Here, \( A \) is a constant, and \( T \) is the temperature at any point along the rod. We need to find the temperature at a distance \( x \) from the cold end and the thermal current per unit area.
Setting Up the Equation
Using Fourier's law, the heat current \( Q \) through a cross-sectional area \( A \) can be expressed as:
Q = -k \cdot A \cdot \frac{dT}{dx}
Since \( k \) varies with temperature, we can substitute \( k \) in the equation:
Q = -\frac{A}{T} \cdot A \cdot \frac{dT}{dx}
Rearranging gives us:
Q = -\frac{A}{T} \cdot \frac{dT}{dx}
Integrating to Find Temperature
To find the temperature distribution along the rod, we can express \( dT \) in terms of \( dx \) and integrate. Rearranging the equation gives:
Q \cdot T = -A \cdot dT/dx
We can separate variables:
Q \cdot T \, dx = -A \, dT
Now, integrating both sides from \( t_1 \) to \( T(x) \) and from \( 0 \) to \( x \):
∫(T dx) = -\frac{A}{Q} ∫dT
This results in:
Q \cdot x = -A \ln(T) + C
Here, \( C \) is a constant of integration. To find \( C \), we can use the boundary condition at \( x = 0 \) where \( T = t_1 \):
Q \cdot 0 = -A \ln(t_1) + C
Thus, \( C = A \ln(t_1) \). Substituting back gives:
Q \cdot x = -A \ln(T) + A \ln(t_1)
Rearranging leads to:
Q \cdot x = A \ln\left(\frac{t_1}{T}\right)
Exponentiating both sides gives:
T = \frac{t_1}{e^{\frac{Q \cdot x}{A}}}
Finding Thermal Current per Unit Area
The thermal current per unit area \( J \) can be expressed as:
J = \frac{Q}{A}
Substituting \( k \) into Fourier's law gives:
J = -\frac{A}{T} \cdot \frac{dT}{dx}
Using the previously derived expression for \( T \), we can find \( J \) as a function of \( x \) and the temperatures at the ends of the rod.
Final Thoughts
In summary, we derived the temperature at a distance \( x \) from the cold end of the rod and expressed the thermal current per unit area. This approach illustrates the interplay between temperature, thermal conductivity, and heat flow in a material with temperature-dependent properties. If you have any further questions or need clarification on any steps, feel free to ask!
