Question

A liquid flows under steady and incompressible flow conditions from station 1 to station 4 through pipe sections P, Q, R, and S as shown in the figure. Consider, \( d \), \( V \), and \( h \) represent the diameter, velocity, and head loss, respectively, in each pipe section with subscripts ‘P’, ‘Q’, ‘R’, and ‘S’. \( \Delta h \) represents the head difference between the inlet (station 1) and outlet (station 4). All the pipe sections are placed on the same horizontal plane for which the figure shows the top view. 
(Insert diagram here, if possible)

• A. \( \Delta h = h_p + h_q + h_r + h_s \) and \( V_p d_p^2 = V_q d_q^2 = V_r d_r^2 = V_s d_s^2 \)
• B. \( \Delta h = h_p + h_q + h_r \) and \( V_p d_p^2 = V_q d_q^2 = V_r d_r^2 = V_s d_s^2 \)
• C. \( \Delta h = h_p + h_q + h_r \) and \( V_p d_p^2 = V_q d_q^2 = V_r d_r^2 = V_s d_s^2 \)
• D. \( \Delta h = h_p + h_q + h_r + h_s \) and \( V_p d_p^2 = V_q d_q^2 = V_r d_r^2 = V_s d_s^2 \)

Hint:

In fluid mechanics, for steady incompressible flow, the flow rate remains constant, which leads to a relationship between the velocities and diameters of the pipe sections.

Answer 1

The Correct Option is C

In steady incompressible flow, the head loss across each section is a result of factors like friction, changes in velocity, and other physical conditions. According to the energy equation for each section, we can write: 
- Head Loss Equation: The total head difference between station 1 and station 4 is the sum of the head losses across each section. Therefore, the total head loss \( \Delta h \) is the sum of \( h_p \), \( h_q \), and \( h_r \), and it can be expressed as: \[ \Delta h = h_p + h_q + h_r \] - Velocity and Diameter Relationship: Since the flow is steady, the flow rate \( Q \) is constant throughout the system. For each section, we can express the flow rate as: \[ Q = V \cdot A = V \cdot \left( \frac{\pi d^2}{4} \right) \] Therefore, for steady incompressible flow, the relationship between the velocity and diameter for each section is: \[ V_p d_p^2 = V_q d_q^2 = V_r d_r^2 = V_s d_s^2 \] This ensures that the volumetric flow rate is conserved at each section of the pipe. Thus, the correct option is: \[ \Delta h = h_p + h_q + h_r {and} V_p d_p^2 = V_q d_q^2 = V_r d_r^2 = V_s d_s^2 \]