Question

A control valve with hyperbolic characteristics has a turndown ratio (ratio of maximum flow to minimum controllable flow) of 50. The flow rate through the valve at 70% open is 5 \(\text{m}^3\,\text{s}^{-1}\). Assuming constant fluid density and pressure drop across the valve, which one of the following is the flow rate (in \(\text{m}^3\,\text{s}^{-1}\)) through the valve at 30% open?

• A. 1.1
• B. 2.2
• C. 3.3
• D. 4.4

Hint:

Use $f(l)=1/[1+R(1-l)]$ for the hyperbolic valve; find $Q_{max}$ from the 70% open data point, then apply it at 30% open.

Answer 1

The Correct Option is B

Step 1: Hyperbolic valve characteristic.

\[ f(l) = \frac{Q(l)}{Q_{max}} = \frac{1}{1 + R(1-l)} \]

Step 2: Evaluate at 70% open.

\[ f(0.7) = \frac{1}{1+50(0.3)} = \frac{1}{16} = 0.0625 \]

Step 3: Find Q_max.

\[ Q_{max} = 5/0.0625 = 80\ \text{m}^3\,\text{s}^{-1} \]

Step 4: Evaluate at 30% open.

\[ f(0.3) = \frac{1}{1+50(0.7)} = \frac{1}{36} = 0.02778 \]

Step 5: Compute the flow rate.

\[ Q(0.3) = 0.02778 \times 80 = 2.222 \approx 2.2\ \text{m}^3\,\text{s}^{-1} \]

\[ \boxed{Q(30\%) \approx 2.2\ \text{m}^3\,\text{s}^{-1}} \]