Two long parallel surfaces, each of emissivity 0.7 are at different temperatures and accordingly have radiation exchange between them. It is desired to reduce 75% of this radiant heat transfer by inserting thin parallel shields of equal emissivity 0.7 on both sides. What should be the number of shields?a)2b)4c)1d)3Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Introduction:
The problem is about reducing radiant heat transfer between two long parallel surfaces by inserting thin parallel shields. The emissivity of both the surfaces and the shields is given as 0.7. The goal is to reduce 75% of the radiant heat transfer.

Solution:

Step 1: Calculate the initial radiant heat transfer:
The initial radiant heat transfer between the surfaces can be calculated using the Stefan-Boltzmann law:

Q_initial = ε1 * ε2 * σ * A * (T1^4 - T2^4)

where Q_initial is the initial radiant heat transfer, ε1 and ε2 are the emissivities of the two surfaces, σ is the Stefan-Boltzmann constant, A is the area of the surfaces, T1 is the temperature of the first surface, and T2 is the temperature of the second surface.

Step 2: Calculate the desired reduction in radiant heat transfer:
The desired reduction in radiant heat transfer is 75% of the initial radiant heat transfer:

Q_reduction = 0.75 * Q_initial

Step 3: Calculate the number of shields:
To reduce the radiant heat transfer, thin parallel shields are inserted between the surfaces. Each shield has an emissivity of 0.7. Let's assume the number of shields to be n.

The radiant heat transfer through the shields can be calculated using the Stefan-Boltzmann law:

Q_shields = ε_shields * σ * A * (T1^4 - T2^4)

where Q_shields is the radiant heat transfer through the shields, ε_shields is the emissivity of the shields, σ is the Stefan-Boltzmann constant, A is the area of the surfaces, T1 is the temperature of the first surface, and T2 is the temperature of the second surface.

The total radiant heat transfer after inserting the shields can be calculated as:

Q_total = Q_initial - Q_shields

Since the goal is to reduce 75% of the initial radiant heat transfer, we can write the equation:

Q_total = Q_initial - Q_shields = 0.25 * Q_initial

Substituting the values and rearranging the equation, we get:

0.25 * Q_initial = ε1 * ε_shields^n * σ * A * (T1^4 - T2^4)

Taking the logarithm of both sides, we get:

log(0.25) + log(Q_initial) = log(ε1) + n * log(ε_shields) + log(σ) + log(A) + 4 * log(T1) - 4 * log(T2)

Simplifying the equation, we get:

log(Q_initial) - log(0.25) = log(ε1) + n * log(ε_shields) + log(σ) + log(A) + 4 * log(T1) - 4 * log(T2)

The number of shields can be calculated using the above equation.

Step 4: Calculate the number of shields:
By substituting the values in the equation mentioned in Step 3, we can calculate the number of shields required to reduce 75% of the radiant heat transfer.

Conclusion:
The correct answer is option 'D', which states that three shields should be inserted

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