Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures 2T and 3T respectively. The temperature of the middle (i.e. second) plate under steady state condition is ?

To determine the temperature of the middle plate in a system with three large, parallel plates, we can apply the principles of heat transfer and thermal equilibrium. Given that the first plate is at a temperature of 2T, the third plate is at 3T, and all plates are ideal black surfaces with high thermal conductivity, we can analyze the thermal equilibrium conditions.

Understanding Heat Transfer Between Plates

In steady-state conditions, the rate of heat transfer between the plates reaches a constant value. The heat transfer occurs through radiation, and given that the plates are ideal black bodies, we can assume that they absorb and emit radiation perfectly. The temperature of the middle plate will be influenced by the temperatures of the other two plates.

Set Up the Problem

Let's denote the temperatures of the plates as follows:

• Temperature of Plate 1 (T1) = 2T
• Temperature of Plate 2 (T2) = ? (this is what we want to find)
• Temperature of Plate 3 (T3) = 3T

In steady-state, the heat absorbed by the middle plate from the first plate must equal the heat emitted by the second plate to the third plate:

Applying the Stefan-Boltzmann Law

The power radiated per unit area by a black body is given by the Stefan-Boltzmann law:

P = σT^4

Where σ is the Stefan-Boltzmann constant. For the middle plate, we set up the following equation:

Heat Absorbed from Plate 1

The heat absorbed by the second plate from the first plate is:

P_1→2 = σ(T1^4 - T2^4) = σ((2T)^4 - T2^4) = σ(16T^4 - T2^4)

Heat Emitted to Plate 3

The heat emitted by the second plate to the third plate is:

P_2→3 = σ(T2^4 - T3^4) = σ(T2^4 - (3T)^4) = σ(T2^4 - 81T^4)

Setting Up the Equation

At steady-state, these two rates of heat transfer must be equal:

σ(16T^4 - T2^4) = σ(T2^4 - 81T^4)

We can cancel out σ since it is common on both sides:

16T^4 - T2^4 = T2^4 - 81T^4

Simplifying the Equation

Rearranging the terms gives us:

16T^4 + 81T^4 = 2T2^4

97T^4 = 2T2^4

Solving for T2

Now we can solve for T2:

T2^4 = (97/2)T^4

T2 = (97/2)^(1/4)T

Final Temperature Calculation

Calculating this gives us the temperature of the middle plate. Approximating the value:

T2 ≈ 3.16T

Thus, the temperature of the middle plate under steady-state conditions is approximately 3.16T. This result shows how the middle plate's temperature is influenced by the temperatures of the adjacent plates, emphasizing the balance of heat transfer in thermal systems.