Question

Three road ways A, B and C are connected in parallel and are having same cross-sectional area and surface characteristics. But length of three roadways are 100 m, 200m and 300 m respectively. The ratio of Quantity of air flowing through the roadways Q\(_A\): Q\(_B\): Q\(_C\) is

• A. 1 : 2 : 3
• B. 1 : \( \sqrt{2} \) : \( \sqrt{3} \)
• C. 1 : 1/2 : 1/3
• D. 1 : \( 1/\sqrt{2} \) : \( 1/\sqrt{3} \)

Hint:

Remember the key rules for parallel and series circuits in ventilation:
- Series: Quantities are equal (\(Q_T = Q_A = Q_B\)), Resistances add up (\(R_T = R_A + R_B\)).
- Parallel: Pressures are equal (\(H_A = H_B\)), Quantities add up (\(Q_T = Q_A + Q_B\)).
For parallel flow, air will always prefer the path of least resistance, so the shortest airway gets the most air.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to find the ratio of air quantities flowing through three parallel airways that are identical except for their lengths.

Step 2: Key Formula or Approach:
1. For airways in

parallel, the pressure drop (H) across each branch is the same.
2. The fundamental ventilation equation is \( H = R Q^2 \), where H is pressure, R is resistance, and Q is quantity.
3. From the previous question, we know that for these airways, resistance is directly proportional to length (\( R \propto L \)).

Step 3: Detailed Explanation:
Since the pressure drop is the same for all three roadways: \[ H = R_A Q_A^2 = R_B Q_B^2 = R_C Q_C^2 \] This means that \( Q^2 \) is inversely proportional to R: \[ Q^2 \propto \frac{1}{R} \] Taking the square root, we find that the quantity Q is inversely proportional to the square root of the resistance: \[ Q \propto \frac{1}{\sqrt{R}} \] Since resistance is proportional to length (\( R \propto L \)), we can say: \[ Q \propto \frac{1}{\sqrt{L}} \] Now we can write the ratio of the quantities: \[ Q_A : Q_B : Q_C = \frac{1}{\sqrt{L_A}} : \frac{1}{\sqrt{L_B}} : \frac{1}{\sqrt{L_C}} \] Substitute the given lengths \( L_A=100 \), \( L_B=200 \), \( L_C=300 \): \[ Q_A : Q_B : Q_C = \frac{1}{\sqrt{100}} : \frac{1}{\sqrt{200}} : \frac{1}{\sqrt{300}} \] \[ Q_A : Q_B : Q_C = \frac{1}{\sqrt{100 \times 1}} : \frac{1}{\sqrt{100 \times 2}} : \frac{1}{\sqrt{100 \times 3}} \] \[ Q_A : Q_B : Q_C = \frac{1}{10\sqrt{1}} : \frac{1}{10\sqrt{2}} : \frac{1}{10\sqrt{3}} \] To simplify the ratio, we can multiply all parts by 10: \[ Q_A : Q_B : Q_C = 1 : \frac{1}{\sqrt{2}} : \frac{1}{\sqrt{3}} \]

Step 4: Final Answer:
The ratio of the quantities is 1 : \( 1/\sqrt{2} \) : \( 1/\sqrt{3} \).
This corresponds to option (D).