Calculate Pressure Difference at Chimney Cap

To find the pressure difference $\Delta P$ between the top and bottom surfaces of the cap when the chimney is closed, we proceed step-by-step:

1. Density of the Hot Air ($\rho$)

Assuming air behaves as an ideal gas, the pressure, density, and temperature are related by $P = \frac{\rho R T}{M}$.

Since the air enters the furnace at pressure $P_a$ and stays at constant pressure inside the furnace: $\rho T = \rho_a T_a$

Substituting the given values ($\rho_a = 1.2 \text{ kg m}^{-3}$, $T_a = 300 \text{ K}$, and $T = 360 \text{ K}$): $\rho = \rho_a \left(\frac{T_a}{T}\right) = 1.2 \times \frac{300}{360} = 1.0 \text{ kg m}^{-3}$

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2. Pressure Distribution

Let $z = 0$ correspond to the bottom of the furnace, where atmospheric air enters and the pressure both inside and outside is $P_a$.

The total height of the hot air column from the bottom of the furnace to the top of the chimney is: $H_{\text{total}} = H + h = 1 \text{ m} + 9 \text{ m} = 10 \text{ m}$

• Outside Pressure at the top of the cap ($P_{\text{top}}$): Due to the ambient air column of density $\rho_a$: $P_{\text{top}} = P_a - \rho_a g (H + h)$

• Inside Pressure at the bottom of the cap ($P_{\text{bottom}}$): Due to the hot air column of density $\rho$: $P_{\text{bottom}} = P_a - \rho g (H + h)$

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3. Calculation of Pressure Difference ($\Delta P$)

The pressure difference across the cap is: $\Delta P = P_{\text{bottom}} - P_{\text{top}} = (\rho_a - \rho) g (H + h)$

Substituting the values ($g = 10 \text{ m s}^{-2}$): $\Delta P = (1.2 - 1.0) \times 10 \times 10 = 0.2 \times 100 = 20 \text{ N m}^{-2}$

Final Answer: The value of $\Delta P$ is 20.