Question

Water (density = 1000 kg/m$^3$) flows steadily with a flow rate of 0.05 m$^3$/s through a venturimeter having throat diameter of 100 mm. If the pipe diameter is 200 mm and losses are negligible, the pressure drop (in kPa, rounded off to one decimal place) between an upstream location in the pipe and the throat (both at the same elevation) is $________________$.

Hint:

For venturimeter problems, compute velocities using continuity, then apply Bernoulli to find pressure difference.

Answer 1

Correct Answer: 18.5

Step 1: Compute areas.
Pipe diameter = 0.2 m $\Rightarrow$ \[ A_1 = \frac{\pi}{4}(0.2)^2 = 0.03142 \ \text{m}^2 \] Throat diameter = 0.1 m $\Rightarrow$ \[ A_2 = \frac{\pi}{4}(0.1)^2 = 0.007854 \ \text{m}^2 \]
Step 2: Compute velocities.
\[ V_1 = \frac{Q}{A_1} = \frac{0.05}{0.03142} = 1.59 \ \text{m/s} \] \[ V_2 = \frac{Q}{A_2} = \frac{0.05}{0.007854} = 6.37 \ \text{m/s} \]
Step 3: Apply Bernoulli’s equation.
\[ \Delta p = \frac{1}{2} \rho (V_2^2 - V_1^2) \] \[ \Delta p = 0.5 \times 1000 \times (6.37^2 - 1.59^2) \] \[ = 500 \times (40.6 - 2.53) = 500 \times 38.07 = 19035 \ \text{Pa} \]
Step 4: Convert to kPa.
\[ \Delta p = 19.0 \ \text{kPa} \] Final Answer: \[ \boxed{19.0 \ \text{kPa}} \]