How is Bernoulli’s equation expressed? What is energy density?

Bernoulli’s principle: What exactly does this mean?

Bernoulli's principle is not an energy conservation equation with an open end. For this Bernoulli equation to apply, the following assumptions must be met: [1]?the flow must be steady, that is, the flow parameters (velocity, density, etc.) at any?point cannot change with time; [2]?the flow must be incompressible,?even if pressure varies, the density must remain constant along a streamline, and [3]?friction by viscous forces must be negligible.

How does one look at Bernoulli’s principle in relation to the first law of thermodynamics?

Look at these constraints; they mean no heat generation [due to no change in density] into the fluid, no work done by the fluid [due to the fluid being non-compressible], and no entropy [due to constant density] Essentially, it means that the total energy [enthalpy h1=h2] is constant at two points 1 and 2. This further means, because there is no thermodynamic work [PdV =0], the internal energy at two points U1=U2 in a moving liquid is constant. When a liquid flows through a pipe, its total energy is its internal energy, and Bernoulli's principle states that the internal energy of a moving liquid is constant at all points. The internal energy per unit volume of the liquid is called energy density. This is precisely what Bernoulli’s principle is about.

Further explanation

Energy density

Fundamentally, Bernoulli’s principle state that in flowing fluids, once a steady state has been reached, all locations in the connected fluid system must also have the same total energy density. Total energy density = Work energy / Volume + Kinetic energy/volume + Potential energy/volume. The term 'P ' in Bernoulli's equation is never zero even when the liquid is at rest and gauze pressure is showing zero pressure since molecules are hitting each other and generating momentum. Only its magnitude may be different. When a liquid is at rest P = Atmospheric pressure. When a liquid is at rest there is no consumption of kinetic energy and hence the potential energy of the liquid is constant.

The sum of three energy densities, work energy, kinetic energy, and potential energy is constant at any point connecting two points in a fluid flow. Energy Density = Energy × [Volume]-1. = [M1 L2 T-2] × [M0 L3 T0]-1 = [M1 L-1 T-2]. Therefore, the energy density is dimensionally represented as [M1 L-1 T-2].?

How is energy/volume arriving at

There are two most general formats in which Bernoulli’s equation is written.

[1] P + 1/2 x v2 x rho + rho x g x h = Constant

[2] P/rho + 1/2 v2 + g x h = Constant

P is pressure, V is velocity, rho is density and h is the elevation

Let us take equation [1]

P +1/2 x v2 x rho + rho x g x h = constant

Step by step

The first term of the equation: ?P

P = F / A, multiply numerator and denominator by l [ l is length, F is force, A is cross-section]

You get, P = F X l/ A x l, F x l = work, [ force x distance = work] A x l = Volume [cross-section x length = volume]

P = Work energy / Volume

Therefore P, the first term of Bernoulli’s equation is work energy per unit volume.

Explanation

F is the force exerted by the fluid at a certain point, A is the cross-sectional area of the flowing fluid at that particular point.?l refers to the distance the fluid at that point moves and V represents the volume of fluid in that distance l. Fl is the work done by the fluid in moving the fluid ahead of it by a distance l so P ??then becomes the work per unit volume of fluid moved in moving V amount of fluid by a distance of l

The second term of the equation: The second term of equation [1] is ? x rho x v2

We can write, ? x rho x v2= [1/2mv2] / V [ V = Volume, v = velocity, m is mass and rho density. Density =m/V

?We get, the second term of the equation, ? x rho x v2= [1/2mv2] / V = Kinetic energy / Volume

Therefore, the second term of Bernoulli’s equation [1] kinetic energy per unit volume

The third term of the equation: The third term of equation [1] is rho x g x h

We can write, rho x g x h = m/V x g x h = m x g x h /V, mgh = Potential energy [For the gravitational force the formula for potential energy is mgh]

Therefore, the third term of equation [1] potential energy per unit volume

Therefore, Bernoulli’s equation, P + 1/2 x v2 x rho + rho x g x h = Sum of [ Work energy/volume + Kinetic energy / volume + Potential energy / Volume] = Constant

[1] P + 1/2 x v2 x rho + rho x g x h = Constant

This is energy density per unit volume

The second expression of Bernoulli’s equation P/rho + 1/2 v2 + g x h = Constant is specific energy per unit mass.