Question

Find pressure difference between A and B.

• A. \(75\,MPa\)
• B. \(85\,MPa\)
• C. \(95\,MPa\)
• D. \(65\,MPa\)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For an ideal fluid flowing through a horizontal pipe with varying cross-sections, we use the equation of continuity and Bernoulli's theorem.
Step 2: Key Formula or Approach:
Continuity Equation: $A_1 v_1 = A_2 v_2$.
Bernoulli's Equation for horizontal flow: $P_A + \frac{1}{2}\rho v_A^2 = P_B + \frac{1}{2}\rho v_B^2$.
Step 3: Detailed Explanation:
Given data at point A: $A_1 = 10 \text{ cm}^2 = 10 \times 10^{-4} \text{ m}^2$, $v_A = 10 \text{ m/s}$.
Given data at point B: $A_2 = 20 \text{ mm}^2 = 20 \times 10^{-6} \text{ m}^2$.
Fluid density $\rho = 600 \text{ kg/m}^3$.
First, find velocity at B using continuity equation:
\[ (10 \times 10^{-4}) \times 10 = (20 \times 10^{-6}) \times v_B \]
\[ 10^{-2} = 20 \times 10^{-6} \times v_B \implies v_B = \frac{10^{-2}}{2 \times 10^{-5}} = \frac{1000}{2} = 500 \text{ m/s} \]
Now, use Bernoulli's equation to find the pressure difference:
\[ P_A - P_B = \frac{1}{2}\rho (v_B^2 - v_A^2) \]
\[ P_A - P_B = \frac{1}{2} \times 600 \times (500^2 - 10^2) \]
\[ P_A - P_B = 300 \times (250000 - 100) = 300 \times 249900 \text{ Pa} \]
\[ P_A - P_B = 74970000 \text{ Pa} = 74.97 \times 10^6 \text{ Pa} \]
Approximating to the nearest whole number in options:
\[ \Delta P \approx 75 \text{ MPa} \]
Step 4: Final Answer:
The pressure difference between A and B is 75 MPa.