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Fluid Dynamics
  1. Continuity Equation
  2. Energy Equation

Continuity Equation

The continuity equation states that the mass flow rate is constant at steady state:

\dot{m} = \rho_1A_1 V_1 = \rho_2 A_2 V_2 = constant


ρ = mass density (slugs/ft3)
A = cross-sectional area of the flow(ft2)
V = velocity of the flow (ft/sec)

Because liquids can be assumed to have constant density over fairly large vertical distances, the continuity equation simplifies to:

Q = A_1V_1 = A_2V_2 = AV


Q = discharge (ft3/sec)
A = cross-sectional area of the flow (ft2)
V = velocity of the flow (ft/sec)

Energy Equation

The energy equation requires that the energy between two points may change form but the total is always conserved (See Bernoulli's Equation for more information):

EnergyInFluild_1 + EnergyAdded_{1\rightarrow2} = EnergyInFluid_2 + EnergyLost_{1\rightarrow2}


\frac{P_1}{\gamma} + \frac{v_1^2}{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).

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_1^2}{2g} + Z_1 + h_P = \frac{P_2}{\gamma} + \frac{v_2^2}{2g} + Z_2 + h_L


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.

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_1^2}{2g} + Z_1 = \frac{P_2}{\gamma} + \frac{v_2^2}{2g} + Z_2

and more generally

\frac{P}{\gamma} + \frac{v^2}{2g} + Z = constant



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