University is all fun and games!

1. The problem statement, all variables and given/known data

A Farmer builds a water storage tank 12m above ground, as shown below:



The water is fed via a 200m long, 125mm diameter pipe to a field for irrigation purposes. The pipe has a friction factor of 0.008. Using the K data below, show that the water flow rate (kg/s) in the pipe is approximately 26kg/s

Data:

K Factors: Tank Exit 0.5, Bends 0.9 (x2), Valve 1.0 (Total: 3.3)
Water Density: 1000kg/m3

2. Relevant equations

Bernoulli's Equation: (P_1/ρ*g) + (V_1^/2*g) + z1 = (P_2/ρ*g) + (V_2^/2*g) + z2
Darcy Formula: H_f = 4*f*(L/D)*(V^/2*g)
Moody Diagram

3. The attempt at a solution

Right then, I've been asked to find the water flow rate, which I'm presuming means the mass flow rate (kg/s) - so in order to find that I need the volume flow rate, and in order to find that I need the velocity

I haven't been given the velocity in the original problem, however I think I need to use the Bernoulli Equation in order to find the velocity.

I've made a few assumptions with the Bernoulli Equation, which are that the pressure at point 1 (the water tank) is atmospheric (101325pa) - because on the picture the tank looks to be open to atmosphere, I've also assumed that the pressure at point 2 (the end of the pipe) is also atmospheric (101325pa) because the pipe is leading to a field.

As well as this I've assumed that the velocity at point 1 is approximately 0m/s (because the water is going from a large tank into a small pipe)

So at the moment, my Bernoulli equation looks like:

(101325 / (1000 x 9.81)) + (0^ / (2 x 9.81)) + 12 = (101325 / (1000 x 9.81)) + (V_2^ / (2 x 9.81)) + 0

However I understand that this equation isn't full, as I also have to take into account the friction (head loss), which would change the Bernoulli equation into:

(P_1/ρ*g) + (V_1^/2*g) + z1 = (P_2/ρ*g) + (V_2^/2*g) + z2 + h_f

So in order to accurately find the velocity using the Bernoulli equation, I have to also take into account the Head loss?

But the Darcy Formula is: H_f = 4*f*(L/D)*(V^/2*g)

So in order to use the Darcy Formula I have to know the velocity - but I don't know the velocity (as that's what I'm trying to find)? - Unless there's a way of finding the head loss without knowing the velocity?

Which somewhat puts me at a dead end.

Another route I was thinking of using to find the velocity was to use the Moody Diagram to try to find the Reynolds number, then using Re=(ρ*V*D)/μ to find the velocity (as I know all the other values.)

However the only other Moody diagram value I know is the Friction Factor of 0.008, and I'm unable to find the Relative roughness (k/D) as the problem doesn't state what sort of pipe is being used (so I can't find the roughness.)