Velocity distribution at entry to the pump intake is inversely propor...
The velocity distribution is prescribed by the relation V = C / r 2
At r = 1 m, u = 0.75 m⁄s and hence C = 0.75 Acceleration,
a = ∂V / ∂t + V ∂V / ∂r
or steady state flow ∂V / ∂t = 0
a = V ∂V / ∂r = C / r2 × ∂/∂r (C / r2 )
= −2C /r5
Substituting C = 0.75
a = −2 × (0.75)2 /r 5 = −1.125 / r5
i) When r = 0.5 m
a = −1.125 / (0.5)5= −36 m⁄s2
(ii) When r = 1.5 m
a = −1.125 /(1.5)5 = −0.148 m/s2
The minus sign indicates that acceleration is directed towards the intake.
Velocity distribution at entry to the pump intake is inversely propor...
Given information:
- Velocity distribution at entry to the pump intake is inversely proportional to the square of the radial distance from the inlet to the suction pipe.
- Velocity at a radial distance of 1 m from the pipe inlet is 0.75 m/s.
To find:
- Acceleration of flow at 0.5 m and 1.5 m from the inlet.
Let's solve this step by step:
1. Velocity distribution equation:
- According to the given information, the velocity distribution equation can be written as:
V = k / r²
where V is the velocity, r is the radial distance, and k is a constant.
2. Find the value of constant k:
- We know that at a radial distance of 1 m from the inlet, the velocity is 0.75 m/s.
- Substituting these values in the velocity distribution equation, we get:
0.75 = k / 1²
k = 0.75
3. Acceleration of flow at 0.5 m:
- To find the acceleration at 0.5 m, we need to differentiate the velocity equation with respect to time, since acceleration is the rate of change of velocity.
- Differentiating V = 0.75 / r² with respect to time, we get:
dV/dt = 0.75 * (-2) / r³
dV/dt = -1.5 / r³
- Substituting the value of r as 0.5 m, we get:
dV/dt = -1.5 / (0.5)³
dV/dt = -1.5 / 0.125
dV/dt = -12 m/s²
4. Acceleration of flow at 1.5 m:
- Similarly, substituting the value of r as 1.5 m in the velocity equation, we get:
V = 0.75 / (1.5)²
V = 0.75 / 2.25
V = 1/3 m/s
- Again, differentiating V = 1/3 m/s with respect to time, we get:
dV/dt = 0
Since the velocity is constant, the acceleration is zero.
Therefore, the calculated accelerations are:
- At 0.5 m from the inlet: -12 m/s²
- At 1.5 m from the inlet: 0 m/s²
The correct answer is option A: -36 m/s²; -0.148 m/s² respectively.