Question

For Inlet diameter $>$ Outlet diameter the Velocity and pressure along the flow will be as

• A. Velocity decreases, pressure increases
• B. Velocity increases, pressure decreases
• C. Velocity constant, pressure decreases
• D. Velocity increases, pressure constant

Hint:

This configuration behaves exactly like a convergent nozzle.
In convergent nozzles:
Area $\downarrow$ $\implies$ Velocity $\uparrow$ $\implies$ Pressure $\downarrow$.
This is a standard application of Venturi principles.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the changes in fluid velocity and pressure along a horizontal pipe where the inlet diameter is greater than the outlet diameter.

Step 2: Key Formula or Approach:
This problem is solved using the Continuity Equation and Bernoulli's Principle.
The Continuity Equation for incompressible flow is:
\[ A_{\text{in}} V_{\text{in}} = A_{\text{out}} V_{\text{out}} \]
Bernoulli's Equation for horizontal flow (where elevation change is negligible) is:
\[ P_{\text{in}} + \frac{1}{2}\rho V_{\text{in}}^2 = P_{\text{out}} + \frac{1}{2}\rho V_{\text{out}}^2 \]

Step 3: Detailed Explanation:

• Since the inlet diameter ($D_{\text{in}}$) is greater than the outlet diameter ($D_{\text{out}}$), the cross-sectional area of the inlet ($A_{\text{in}}$) is larger than that of the outlet ($A_{\text{out}}$).

• From the continuity equation:
\[ V_{\text{out}} = V_{\text{in}} \left(\frac{A_{\text{in}}}{A_{\text{out}}}\right) \] Since $A_{\text{in}} > A_{\text{out}}$, the fluid velocity must increase at the outlet ($V_{\text{out}} > V_{\text{in}}$).

• According to Bernoulli's equation, as the velocity of the fluid increases, its kinetic energy increases.

• Because the total mechanical energy of the fluid must remain constant along a streamline, an increase in velocity (kinetic energy) must be balanced by a corresponding decrease in static pressure ($P_{\text{out}} < P_{\text{in}}$).

Step 4: Final Answer:
As the fluid flows from the larger inlet to the smaller outlet, the velocity increases and the pressure decreases.