Answer:
To calculate the volumetric flow rate of water at the exit pipe, we can use the principle of conservation of mass. According to this principle, the mass flow rate of water at the inlet is equal to the mass flow rate of water at the exit.
First, we need to calculate the mass flow rate at the inlet. The mass flow rate (m_dot) is given by the equation:
m_dot = density * volumetric flow rate
The density of water is given as 1000 kg/m^3. Let's assume the volumetric flow rate at the inlet is Q. Therefore, the mass flow rate at the inlet is:
m_dot_in = 1000 * Q
Next, we need to calculate the pressure drop across the pipe. The pressure drop (ΔP) can be calculated using the Darcy-Weisbach equation:
ΔP = f * (L/D) * (ρ/2) * (V^2)
Where:
- f is the Darcy friction factor (which depends on the flow regime and pipe roughness)
- L is the length of the straight pipe (25 m in this case)
- D is the inside diameter of the pipe (0.09 m in this case)
- ρ is the density of water (1000 kg/m^3)
- V is the average velocity of water in the pipe
To calculate the average velocity (V), we need to know the cross-sectional area of the pipe (A). The area can be calculated using the equation:
A = π * (D/2)^2
Once we have the area, we can calculate the average velocity as:
V = Q / A
Now, we can substitute the values into the Darcy-Weisbach equation and solve for the pressure drop (ΔP).
Next, we need to calculate the work done by the pump (W). The work done is given by the equation:
W = m_dot_in * ΔP
Finally, we can calculate the volumetric flow rate at the exit pipe (Q_exit) using the equation:
Q_exit = W / (density * 44.33)
Substitute the known values and solve for Q_exit to find the volumetric flow rate at the exit pipe.
Please note that this is a simplified explanation and there might be additional factors or considerations depending on the specific problem and assumptions made.