The plane wall of an oven is of thickness L=0.05 m and is exposed to the surrounding at 300 K. The outter surface of the oven is exposed to the surrounding is at temperature 320 K in steady state. The energy radiated (in W/m²) by the outter surface is proportional to the temperature difference between the surface and the surrounding, with proportionality constant 20 W/m²K . What is the temperature of the inner surface of the oven in steady state if the wall has a thermal conductivity of K = 0.5 W/mK.

(a) 400K (b) 340 K (c) 380 K (d) 360 K

To determine the temperature of the inner surface of the oven in steady state, we can apply the principles of heat transfer through conduction and radiation. The problem provides us with the necessary parameters, including the thickness of the wall, the thermal conductivity, and the temperatures of the surrounding environment and the outer surface of the oven. Let's break this down step by step.

Understanding the Heat Transfer Mechanisms

In this scenario, we have two main modes of heat transfer at play: conduction through the oven wall and radiation from the outer surface. The outer surface of the oven radiates energy to the surroundings based on the temperature difference, while the heat conducted through the wall will reach the inner surface.

Given Data

• Thickness of the wall (L): 0.05 m
• Outer surface temperature (T_outer): 320 K
• Surrounding temperature (T_surround): 300 K
• Proportionality constant for radiation (h): 20 W/m²K
• Thermal conductivity (K): 0.5 W/mK

Calculating the Heat Loss by Radiation

The energy radiated by the outer surface can be calculated using the formula:

Q_rad = h * (T_outer - T_surround)

Substituting the values:

Q_rad = 20 W/m²K * (320 K - 300 K) = 20 W/m²K * 20 K = 400 W/m²

Applying Fourier's Law of Heat Conduction

Next, we need to find the temperature of the inner surface (T_inner) using Fourier's law, which states:

Q_cond = K * (T_inner - T_outer) / L

Since we are in steady state, the heat conducted through the wall must equal the heat radiated from the outer surface:

Q_cond = Q_rad

Thus, we can set up the equation:

K * (T_inner - T_outer) / L = Q_rad

Substituting the known values:

0.5 W/mK * (T_inner - 320 K) / 0.05 m = 400 W/m²

Solving for T_inner

Now, we can rearrange the equation to solve for T_inner:

0.5 * (T_inner - 320) = 400 * 0.05

0.5 * (T_inner - 320) = 20

T_inner - 320 = 20 / 0.5

T_inner - 320 = 40

T_inner = 360 K

Final Answer

Therefore, the temperature of the inner surface of the oven in steady state is 360 K. This corresponds to option (d).

In summary, we utilized the principles of heat transfer to analyze the system, applying both radiation and conduction equations to arrive at the inner surface temperature. Understanding these concepts is crucial for thermal management in engineering applications.