Question

Find the temperature distribution in a substance placed between two parallel plates kept at temperatures $T_1$ and $T_2$. The plate separation is equal to $l$, the heat conductivity coefficient of the substance $x \propto \sqrt{T}$.

Answer 1

The temperature distribution in the substance is given by: $T(x) = \left[ \left(1 - \frac{x}{l}\right) T_1^{3/2} + \left(\frac{x}{l}\right) T_2^{3/2} \right]^{2/3}$

Answer 2

The problem asks for the temperature distribution $T(x)$ in a substance between two parallel plates at temperatures $T_1$ and $T_2$, separated by a distance $l$. The thermal conductivity $k$ of the substance varies with temperature as $k \propto \sqrt{T}$, which can be written as $k = C\sqrt{T}$ where $C$ is a constant.

1. Fourier's Law of Heat Conduction: In one dimension and steady state, heat flux $J$ is given by $J = -k \frac{dT}{dx}$. Since heat flows from higher to lower temperature, $J$ is constant throughout the substance.
2. Incorporate Variable Conductivity: Substitute $k = C\sqrt{T}$ into Fourier's Law: $J = -C\sqrt{T} \frac{dT}{dx}$
3. Separate Variables: Rearrange the equation to separate temperature and position variables: $J dx = -C\sqrt{T} dT$
4. Integrate over the entire length: To find the constant heat flux $J$, integrate from one plate ($x=0$, $T=T_1$) to the other ($x=l$, $T=T_2$): $\int_{0}^{l} J dx = \int_{T_1}^{T_2} -C\sqrt{T} dT$ $J l = -C \left[ \frac{T^{3/2}}{3/2} \right]_{T_1}^{T_2} = -\frac{2C}{3} (T_2^{3/2} - T_1^{3/2})$ $J = \frac{2C}{3l} (T_1^{3/2} - T_2^{3/2})$
5. Integrate for Temperature Distribution: Now, integrate from the first plate ($x=0$, $T=T_1$) to an arbitrary position $x$ with temperature $T(x)$: $\int_{0}^{x} J dx' = \int_{T_1}^{T(x)} -C\sqrt{T'} dT'$ $J x = -C \left[ \frac{T'^{3/2}}{3/2} \right]_{T_1}^{T(x)} = -\frac{2C}{3} (T(x)^{3/2} - T_1^{3/2})$ $J x = \frac{2C}{3} (T_1^{3/2} - T(x)^{3/2})$
6. Solve for T(x): Substitute the expression for $J$ from step 4 into the equation from step 5: $\frac{2C}{3l} (T_1^{3/2} - T_2^{3/2}) x = \frac{2C}{3} (T_1^{3/2} - T(x)^{3/2})$ Cancel $\frac{2C}{3}$ and rearrange: $\frac{x}{l} (T_1^{3/2} - T_2^{3/2}) = T_1^{3/2} - T(x)^{3/2}$ $T(x)^{3/2} = T_1^{3/2} - \frac{x}{l} (T_1^{3/2} - T_2^{3/2})$ $T(x)^{3/2} = \left(1 - \frac{x}{l}\right) T_1^{3/2} + \frac{x}{l} T_2^{3/2}$ Raise both sides to the power of $2/3$ to get $T(x)$: $T(x) = \left[ \left(1 - \frac{x}{l}\right) T_1^{3/2} + \frac{x}{l} T_2^{3/2} \right]^{2/3}$