Fluid Mechanics

Bernoulli's Equation

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Contents:

  1. Bernoulli’s Equation
  2. Head Losses

Bernoulli’s Equation

Bernoulli’s equation arises from the classic thermodynamics energy equation for incompressible flow:

Q˙−W˙=m˙[u2+p2ρ+gz2+v222−(u1+p1ρ+gz1+v122)]\dot{Q}-\dot{W}=\dot{m}\left[u_2+\dfrac{p_2}{\rho}+gz_2+\dfrac{v_2^2}{2}-\left(u_1+\dfrac{p_1}{\rho}+gz_1+\dfrac{v_1^2}{2}\right)\right]

By taking heat rate, work, and internal energy change as zero we arrive at Bernoulli’s equation for fluid flow:

0=p2ρ+gz2+v222−(p1ρ+gz1+v122)0=\dfrac{p_2}{\rho}+gz_2+\dfrac{v_2^2}{2}-\left(\dfrac{p_1}{\rho}+gz_1+\dfrac{v_1^2}{2}\right)

We can rewrite this equation into the classic form of Bernoulli’s equation:

p1+ρgz1+12ρv12=p2+ρgz2+12ρv22p_1+\rho gz_1+\dfrac{1}{2}\rho v_1^2=p_2+\rho gz_2+\dfrac{1}{2}\rho v_2^2

There are three important terms in Bernoulli’s Equation:

  • pp: Static Pressure, associated with the pressure of the fluid at rest.
  • ρgz\rho gz: Hydrostatic Pressure, associated with the weight of fluid at a given depth.
  • 12ρv2\dfrac{1}{2}\rho v^2: Dynamic Pressure, associated with the motion of the fluid.

Head Losses

Bernoulli’s equation clearly models an idealized form of fluid flow, since it does not account for heat loss through friction. This energy loss through friction is termed head loss, hlh_l.

p1+ρgz1+12ρv12=p2+ρgz2+12ρv22+hlp_1+\rho gz_1+\dfrac{1}{2}\rho v_1^2=p_2+\rho gz_2+\dfrac{1}{2}\rho v_2^2+h_l

Due to the complexity of friction, head loss is typically measured empirically and put in tables. Head loss can be broken into two components:

  • Major Losses: Friction loss due to flow through a long, straight pipe.
  • Minor Losses: Friction loss due to abrupt changes in geometry, like bends, nozzles, or valves.

Major and minor losses typically have empirical or semi-analytical solutions which can be found in any standard fluid mechanics textbook.