13.21:
Energy Conservation and Bernoulli’s Equation
Assume an incompressible, laminar fluid at a steady state passing through a non-uniform cross-sectional tube.
As the fluid is incompressible, the volume of the fluid element and its mass remain constant.
The speed of the fluid element entering and leaving the tube is different, resulting in net kinetic energy change.
The force acting on the fluid element is due to the fluid's surrounding pressure.
So, the net work done on the fluid element by the surrounding fluid during displacement is the difference between the work done while entering and leaving the tube.
When the fluid element traverses in the elevated region, it gains gravitational potential energy and the change in gravitational potential energy is determined.
According to the work-kinetic energy theorem, the net work done on the fluid element equals the change in kinetic energy.
Dividing throughout by a common value, an equation is obtained.
Along a streamline, the total energy per unit volume of a fluid flowing is a constant, termed the Bernoulli's equation.
13.21:
Energy Conservation and Bernoulli’s Equation
Applying the conservation of energy principle or the work-energy theorem to an incompressible, inviscid fluid in laminar, steady, irrotational flow leads to Bernoulli's equation. It states that the sum of the fluid pressure, potential, and kinetic energy per unit volume is constant along a streamline.
All the terms in the equation have the dimension of energy per unit volume. The kinetic energy per unit volume is called the kinetic energy density, and the potential energy per unit volume is called the potential energy density.
It is important to note that the liquid's density should not change through the flow; that is, it should be incompressible. The flow should also be laminar and not turbulent. Bernoulli's equation is applicable for gases that have negligible compressibility effects. For such gases, the density is assumed to be constant and is treated as an incompressible fluid. Since gases are generally compressible, the equation does not apply to them.
Although a simple restatement of the energy conservation principle with a few critical assumptions, the equation makes it easy to calculate pressure at different points if speeds are known.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson: section 12.5; pages 385–386.
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/1-introduction: section 14.6; pages 716–718.