Question

Heat is lost through a brick wall $(k=0.72 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\(, which is \)4 \mathrm{~m}\( long, \)3 \mathrm{~m}\( wide, and \)25 \mathrm{~cm}\( thick at a rate of \)500 \mathrm{~W}$. If the inner surface of the wall is at \(22^{\circ} \mathrm{C}\), the temperature at the midplane of the wall is (a) \(0^{\circ} \mathrm{C}\) (b) \(7.5^{\circ} \mathrm{C}\) (c) \(11.0^{\circ} \mathrm{C}\) (d) \(14.8^{\circ} \mathrm{C}\) (e) \(22^{\circ} \mathrm{C}\)

Short answer

Answer: (d) 14.8°C

Step-by-step solution

Note given values

Write down the given values: Thermal conductivity (k) = 0.72 W/m.K Length of the wall (L) = 4 m Width of the wall (W) = 3 m Thickness of the wall (t) = 25 cm = 0.25 m Heat loss rate (Q) = 500 W Temperature of the inner surface (T_inner) = 22°C Our goal is to find the temperature at the midplane of the wall (T_midplane).

Calculate the area of the wall

Find the total area (A) of the wall by multiplying the length and the width: A = L × W = 4 m × 3 m = 12 m²

Apply the heat conduction equation

The heat conduction equation is given by: Q = -k × A × (dT/dx) where dT is the temperature difference across the wall and dx is the thickness of the wall. The negative sign indicates the heat flow from the hot side to the cold side. Since we are looking for the midplane temperature, we need to find dT/2, which represents half of the temperature difference across the entire wall.

Rearrange the equation to find dT/2

Rearrange the equation from Step 3 to solve for dT/2: \(dT/2 = -(Q/2) / (k × A × dx)\) Now plug in the values for Q, k, A, and dx: \(dT/2 = -(500 W / 2) / (0.72 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K} × 12 \mathrm{~m}^{2} × 0.25 \mathrm{~m})\)

Calculate dT/2

Calculate the temperature difference across half the thickness of the wall: \(dT/2 = - (250 W) / (2.16 \mathrm{~W} / (\mathrm{K} ⋅ \mathrm{m}^{2}) × 0.25 \mathrm{~m}) = -7.16 \mathrm{K}\)

Calculate the midplane temperature

Since dT/2 represents half the temperature difference across the wall, and the temperature difference is negative, we will add it to the inner surface temperature to find the midplane temperature: T_midplane = T_inner + dT/2 = 22°C + (-7.16°C) = 14.84°C Looking at the options provided, the closest answer is: (d) 14.8°C