In an underground coal mine, the air quantities \( Q_A \) and \( Q_B \) pass through ventilation districts A and B, respectively, as shown, with a pressure drop of \( \Delta P \) across them. The air quantity in district B is to be increased to \( Q_B' \) by installing a booster fan, as shown, without affecting the air quantity in ventilation district A. The capacity of the booster fan \( (P_B) \) can be expressed as
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Write the pressure balance for the two parallel airways and see what extra head the fan must supply once B's flow goes up.
\( P_B = \Delta P \left( \dfrac{Q_B}{Q_B'} \right)^2 - \Delta P \)
\( P_B = \Delta P \left( \dfrac{Q_B'}{Q_B} \right)^2 - \Delta P \)
\( P_B = \Delta P - \Delta P \left( \dfrac{Q_B'}{Q_B} \right)^2 \)
\( P_B = \Delta P + \Delta P \left( \dfrac{Q_B'}{Q_B} \right)^2 \)
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The Correct Option isB
Solution and Explanation
Step 1: Set up the pressure balance for the two parallel airways.
Ventilation districts A and B run in parallel between the same two junctions, so the pressure drop across each one must be the same value \( \Delta P \), even though the air quantities \( Q_A \) and \( Q_B \) can differ if the resistances differ.
Using the mine ventilation square law, pressure drop equals resistance times the square of the airflow:
\[ \Delta P = R_A Q_A^2 = R_B Q_B^2 \]
Step 2: Get the resistance of district B before the booster fan is added.
From \( \Delta P = R_B Q_B^2 \):
\[ R_B = \frac{\Delta P}{Q_B^2} \]
This resistance \( R_B \) is a property of the airway itself, its size, shape and roughness, so it does not change when the booster fan is switched on.
Step 3: Work out the new frictional drop once the flow rises to Q_B'.
The airway still obeys the same square law, just at the new higher flow:
\[ R_B (Q_B')^2 = \frac{\Delta P}{Q_B^2}(Q_B')^2 = \Delta P \left( \frac{Q_B'}{Q_B} \right)^2 \]
Since the problem states that \( Q_A \), and therefore the pressure available between the two junctions, must stay at \( \Delta P \), only \( \Delta P \) of pressure is available naturally to push air through district B. Anything beyond that has to come from the booster fan.
Step 4: Find the fan capacity.
The fan must supply the extra pressure not covered by the available \( \Delta P \):
\[ P_B = \Delta P \left( \frac{Q_B'}{Q_B} \right)^2 - \Delta P \]
Option (A) inverts the ratio of \( Q_B \) and \( Q_B' \), which would mean the fan capacity shrinks as the flow rises, the opposite of what a booster fan does. Options (C) and (D) get the sign of the subtraction wrong, either flipping which term is subtracted from which, or adding instead of subtracting.
Final Answer:
The booster fan capacity is \( P_B = \Delta P \left( \dfrac{Q_B'}{Q_B} \right)^2 - \Delta P \), which is option (B).
\[ \boxed{P_B = \Delta P \left( \dfrac{Q_B'}{Q_B} \right)^2 - \Delta P} \]
