Conservation of Energy Explained

The energy equation requires that the energy between two points may change form but the total is always conserved:

EnergyInFluid_1 + EnergyAdded_{1 \to 2} = EnergyInFluid_2 + EnergyLost_{1 \to 2}
or in more technical terms:

Complete Energy Equation:

\frac{P_1}{\gamma} + \frac{v^2_1}{2g} + Z_1 + I_1 + e_h + e_m = \frac{P_2}{\gamma} + \frac{v^2_2}{2g} + Z_2 + I_2 + h_L

The energy in the fluid at point 1 and 2 is the total of pressure, kinetic, potential, and internal energy. The energy added from point 1 to 2 is heat and mechanical energy. The energy lost from point 1 to 2 is called "head loss" (hL). Head loss is the source of many additional fluid equations including the: Darcy Weisbach Equation, Hazen Williams Equation, and Manning's Equation

The energy equation is simplified in the case of fluids where the temperature doesn't change (I1 = I2) , no heat is added (eh=0), and the only mechanical energy added is by a pump (em is replaced with hp):

\frac{P_1}{\gamma} + \frac{v^2_1}{2g} + Z_1 +h_P = \frac{P_2}{\gamma} + \frac{v^2_2}{2g} + Z_2 + h_L

where:

  • P/γ = pressue head (ft)
  • v2/2g = kinetic head (ft)
  • Z = potential head (ft)
  • hp = head of the pump (ft)
  • hL = head loss (ft)

Note: It is almost always simplest to treat all the heads in feet even though pressure heads, pump heads, and head losses are sometimes be given in psi.

Simplified to Bernoulli's Equation:

The energy equation is greater simplified in the case of fluids where the temperature doesn't change (I1 = I2), no heat is added (eh = 0), no pump is involved (em = 0), and losses are small (hL = 0). In such cases, the simplified energy equation is know as Bernoulli's Theorem which says that between any two points, the total of the pressure, kinetic, and potential energy is equal:

\frac{P_1}{\gamma} + \frac{v^2_1}{2g} + Z_1 = \frac{P_2}{\gamma} + \frac{v^2_2}{2g} + Z_2

and more generally

\frac{P_1}{\gamma} + \frac{v^2_1}{2g} + Z_1 = constant

References:


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