Determining the power required by pump using the Bernoulli's theorem

A pump is lifting water (density equal to 62.4 lb/ft^3) by 50 ft and increasing the pressure of the water by 30 psig. The water flows through a 3-inch inside diameter tube at 40 lb/second. Starting with the general energy balance. Determine the power that is required by the pump in hp?

We have to use Bernoulli's:theorem here

Let  A = Area      ρ=density;   v = velocity of flow;     g =acceleration  due to gravity

        Z =Δheight;                  p = pressure;             P = power   and  Q =f low rate

Q =   mass/ density = (40lb/s) / 62.4 lb/ft³ = 0.641 ft³/s

Considering density and flow rate to be based on lbm,

Q=A·v = m / ρ ==> v = m / (Aρ) = (40lb/s) /(¼π·¼² · 62.4 lb/ft³) = 13.06 ft/s

Q = (40lb/s) / 62.4 lb/ft³ = 0.641 ft³/s

Total energy    =     K.E + P.E +  pressure

P/Q      =   ½ρv² + ρgz + p   

P          = [(½v² + gz)·ρ + p]·Q

Considering that   ρ = 62.4/32.2

P =   [(½{13.06}² + 32.2·50ft)·62.4/32.2 + 30*144]·[0.641] ft³/s

P =   [1695.267 · 62.4/32.2 + 30*144]·[0.641] ft³/s

P = [3285.24 + 4320]·0.641 ft³/s = 4875.2 ft·lb/sec

Power of the pump  = 8.86 HP                  ( since  HP/(550 ft·lb/sec)

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