Answer:
To compute the power requirements of the pump, we need to determine the head loss and the pump's efficiency. The head loss in the pipeline is given by the Darcy-Weisbach equation:
hL = f (L / D) (V^2 / 2g)
where hL is the head loss, f is the friction factor, L is the length of the pipe, D is the inside diameter of the pipe, V is the average fluid velocity, and g is the acceleration due to gravity.
First, we need to calculate the fluid velocity:
Q = A * V
where Q is the flow rate, A is the cross-sectional area of the pipe, and V is the fluid velocity.
The cross-sectional area of the pipe is:
A = π/4 * D^2
A = π/4 * (0.025 m)^2
A = 4.91 x 10^-4 m^2
So, the fluid velocity is:
V = Q / A
V = 10 m^3/h / (3600 s/h) / (4.91 x 10^-4 m^2)
V = 5.04 m/s
Next, we need to calculate the Reynolds number to determine the friction factor:
Re = (ρVD) / μ
where ρ is the fluid density and μ is the fluid viscosity.
Re = (975 kg/m^3)(5.04 m/s)(0.025 m) / (4.3 x 10^-4 Pa s)
Re = 5.73 x 10^5
Using the Moody chart or a Colebrook equation solver, we can determine the friction factor for the given Reynolds number and roughness of the steel pipe. For simplicity, we will assume a friction factor of 0.02.
The head loss due to friction in the pipe is:
hL = f (L / D) (V^2 / 2g)
hL = 0.02 (30 m / 0.025 m) (5.04 m/s)^2 / (2 x 9.81 m/s^2)
hL = 24.4 m
The head loss due to the two elbows is:
hL = K (V^2 / 2g)
where K is the equivalent length of the elbow in diameters and is equal to 20 diameters each. From a piping handbook, K for a long radius 90° elbow is approximately 30 diameters.
hL = 30 (5.04 m/s)^2 / (2 x 9.81 m/s^2)
hL = 7.82 m
The total head loss is:
hL_total = hL_friction + hL_elbows
hL_total = 24.4 m + 7.82 m
hL_total = 32.2 m
The power required by the pump is:
P = ρQhL_total / η
where η is the pump efficiency.
We will assume a pump efficiency of 75%.
P = (975 kg/m^3)(10 m^3/h)(3600 s/h)(32.2 m)/(0.75)
P = 1.13 x 10^6 W or 1.13 MW
Therefore, the power requirements of the pump are 1.13 MW.