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The problem given provides the temperature distribution across a wall and asks to determine three things:

### Temperature distribution:
\[ T(x) = a + bx + cx^2 \]
Where:
- \( a = 900^\circ C \),
- \( b = -300 \, \frac{^\circ C}{m} \),
- \( c = -50 \, \frac{^\circ C}{m^2} \),
- Wall thickness = \( 1 \, m \),
- Heat generation rate \( \dot{g} = 1000 \, W/m^3 \),
- Wall area = \( 10 \, m^2 \),
- Material properties: \( \rho = 1600 \, kg/m^3 \), \( k = 40 \, W/m.K \), and \( C_p = 4 \, kJ/kg.K \).

Now, let’s break the tasks down:

### (i) Rate of heat transfer:
We are asked to determine the heat transfer rate at \( x = 0 \, m \) (entering) and \( x = 1 \, m \) (leaving the wall).

Heat flux is calculated using Fourier's law of conduction:
\[
q_x = -k \frac{dT(x)}{dx}
\]
First, we compute \( \frac{dT(x)}{dx} \).

### (ii) Rate of change of energy storage in the wall:
The energy stored in the wall depends on the heat generation and the heat storage capacity of the material. We can compute the rate of change of energy storage using:
\[
\frac{dE}{dt} = \rho C_p A \int_0^1 \frac{\partial T(x)}{\partial t} dx
\]
We need to find the change in internal energy of the system based on the temperature change over time.

### (iii) Time rate of temperature change at \( x = 0, 0.25, 0.5 \, m \):
The time rate of temperature change is given as \( \frac{\partial T}{\partial t} \) at the specific points.

Let’s start solving by first calculating the heat transfer at \( x = 0 \, m \) and \( x = 1 \, m \) by finding the derivative of the temperature distribution.

The derivative of the temperature distribution \( T(x) \), which gives the temperature gradient, is:

\[
\frac{dT(x)}{dx} = -100x - 300
\]

Now, evaluating this at the specified points:
- At \( x = 0 \, m \), \( \frac{dT(x)}{dx} = -300 \, \frac{^\circ C}{m} \).
- At \( x = 1 \, m \), \( \frac{dT(x)}{dx} = -400 \, \frac{^\circ C}{m} \).

We can now calculate the rate of heat transfer entering and leaving the wall using Fourier's law:

\[
q_x = -k \frac{dT(x)}{dx}
\]
where \( k = 40 \, W/m.K \).

Let’s compute the heat flux at both \( x = 0 \) and \( x = 1 \).

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Here are 5 related questions:
1. How do we determine the heat flux in a material using Fourier's law?
2. What is the significance of thermal conductivity in heat transfer problems?
3. How do material properties like \( \rho \) and \( C_p \) affect energy storage in a system?
4. Can we use the temperature distribution function to calculate temperature at any point \( x \)?
5. How does heat generation within a wall affect its temperature profile over time?

**Tip:** When working with heat transfer problems, always check the units carefully to ensure consistent calculations, especially when working with temperature gradients.

The temperature distribution across the wall (1 m thick) at a certain instant of time is given as T(x) = a + bx + cx^2 where T is in °C and x is in meters, while a = 900°C, b = -300°C/m and c = -50°C/m^2. A uniform heat generation of 1000 W/m³ is present in the wall of area 10 m² having the properties ρ = 1600 kg/m³, k = 40 W/m.K, and Cp = 4 kJ/kg.K. Determine (i) The rate of heat transfer entering the wall (x = 0) and leaving the wall (x = 1 m). (ii) The rate of change in energy storage in the wall. (iii) The time rate of temperature change at x = 0, 0.25, and 0.5 m.

This detailed solution explores the temperature distribution across a 1m thick wall, analyzing heat transfer, energy storage, and time rate of temperature change at various points. Learn how to apply Fourier's Law of Conduction and thermodynamics principles to this problem.

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