Bernoulli's Equation — Question 7

Question 7

A pump adds energy while losses remove it

Water at volume rate $Q=0.0020\,\mathrm{m^3/s}$ passes through a pump and pipe system. From inlet to outlet, pressure rises by $20\,\mathrm{kPa}$, speed rises from $1.0$ to $3.0\,\mathrm{m/s}$, and elevation rises by $5.0\,\mathrm{m}$. Irreversible losses equal $2.0\,\mathrm{m}$ of water head. Use $\rho=1000\,\mathrm{kg/m^3}$ and $g=10\,\mathrm{m/s^2}$.

1. Find the mechanical energy per unit mass that the pump must add, including the specified loss.

2. Find hydraulic power and required shaft power for pump efficiency $0.80$. Explain why the unmodified Bernoulli equation cannot connect these endpoints.

Question 7 - Solution

A pump adds energy while losses remove it

(a) Derivation and calculation

Per unit mass, the extended mechanical-energy balance gives$$w_p=\frac{p_2-p_1}{\rho}+\frac{v_2^2-v_1^2}{2}+g(z_2-z_1)+gh_L.$$$$\boxed{w_p=\frac{20000}{1000}+\frac{9-1}{2}+10(5)+10(2)=94.0\,\mathrm{J/kg}}.$$

(b) Result and interpretation

$$\dot m=\rho Q=(1000)(.00200)=2.00\,\mathrm{kg/s}.$$$$\boxed{P_{\mathrm{hydraulic}}=\dot mw_p=188\,\mathrm W},\qquad\boxed{P_{\mathrm{shaft}}=P_{\mathrm{hydraulic}}/.80=235\,\mathrm W}.$$Unmodified Bernoulli omits both the pump addition and dissipation; it cannot connect these endpoints.

Physical check

The pump supplies 74 J/kg of endpoint energy increase plus 20 J/kg of pipe loss.