Natural draught design: For maximum discharge through a chimney, should the flue gas temperature in the chimney be only slightly greater than atmospheric temperature? Judge the correctness of this statement.

Introduction:
Chimneys create natural draught due to density differences between hot flue gases and cooler ambient air. The achievable mass flow (discharge) depends on both the buoyancy-induced pressure difference and the gas density. Understanding the trade-off between temperature (density) and pressure head is key to optimizing discharge.

Given Data / Assumptions:

• Chimney of height H with uniform cross-section.
• Ambient air at temperature T_a; flue gas inside at average temperature T_g.
• Ideal gas behavior and steady flow.

Concept / Approach:
The pressure head (draught) is proportional to (ρ_a − ρ_g) * g * H. Increasing T_g reduces ρ_g, increasing the pressure difference. However, the mass flow rate is ρ_g * velocity * area; extremely high T_g lowers ρ_g and hence mass flow even if draught rises. The optimum for maximum discharge occurs at a temperature significantly higher than ambient, not just “slightly more”. Therefore, the statement claiming “slightly more than atmospheric” for maximum discharge is incorrect.

Step-by-Step Solution:
1) Draught ∝ (ρ_a − ρ_g) * H: higher T_g (lower ρ_g) improves pressure difference.2) Mass flow m_dot = ρ_g * A * √(2 * Δp / ρ_g) = A * √(2 * Δp * ρ_g).3) Substitute Δp ∝ (ρ_a − ρ_g) * H to see m_dot depends on both ρ_g and (ρ_a − ρ_g).4) The optimal T_g is well above T_a; hence “slightly more” is not the maximizing condition.

Verification / Alternative check:
Differentiating the idealized expression for m_dot with respect to ρ_g shows an internal extremum at a ρ_g value below ρ_a, not near-equal densities (i.e., not “slightly above” atmospheric temperature).

Why Other Options Are Wrong:
Conditional variants (very tall chimneys, air preheaters, low barometric pressure) do not change the fundamental physics; maximum discharge still requires a sizable temperature difference.

Common Pitfalls:
Assuming larger draught always means larger mass flow; ignoring the counteracting effect of reduced gas density.

Final Answer:
Incorrect