Compressible Fluids EXCEPT

Compressible Fluids

•most fluids are somewhat compressible •in chemical-engineering processes, compressibility is unimportant at most operating pressures •even gases may be modeled as incompressible if ∆ p

Compressible Fluids

How is pressure information transmitted in liquids and gases?

Fa Fb The Hydraulic Lift operates a b on Pascal’s principle 2ra 2rb Pressure exerted on an enclosed liquid is transmitted equally to fluid of density ρ every part of the liquid and to Fb the walls of the container. lb a b Fc c d la

1 Compressible Fluids

For static incompressible liquids,

Compressible Fluids

For static compressible fluids (gases), pressure causes volume change.

500g

air

N N N p p p + ∆p V V V − ∆V

2 Compressible Fluids

For moving incompressible liquids and gases,

The presence of the obstacle is felt by the upstream fluid (pressure) and that information is transmitted very rapidly throughout the fluid.

flow

The streamlines adjust according to momentum conservation.

Compressible Fluids

For compressible fluids moving near sonic speeds, information (pressure) and the gas itself are moving at comparable speeds.

Shock wave Pressure piles up at the shock wave

3 Compressible Fluids

Velocity of a pressure wave = constant = speed of sound Velocity of a fluid = variable = supersonic, sonic, subsonic

A shock forms where the pressure waves from the obstacle stack up, and the speed of the pressure wave traveling upstream equals the speed of the fluid traveling downstream. © Faith A. Morrison, Michigan Tech U.

Compressible Fluids Relief Valves (Safety Valves)

The rapid flows in relief valves can become sonic. For sonic flow, the flow rate is constant tank no matter what the pressure drop is.

(pressure waves pile up) Choked Flow Choked flow can be understood from basic equations of compressible fluid mechanics. © Faith A. Morrison, Michigan Tech U.

4 Compressible Fluids Momentum and Energy in Compressible Fluids incompressible Microscopic momentum balance: τ = −µ(∇v + ()∇v T )  ∂v  ρ + v ⋅∇v = −∇p − ∇⋅τ + ρ g  ∂t 

T  2  τ = −µ()∇v + ()∇v +  µ −κ ∇ ⋅v 3 compressible   κ = bulk viscosity

Mechanical energy balance: incompressible ∆p ∆V 2 W + + g∆z + F = s,on ρ 2α m& compressible? © Faith A. Morrison, Michigan Tech U.

Compressible Fluids

Mechanical energy balance (compressible) Back up one step and reintegrate without constant ρ assumption. dp dW +VdV + gdz + dF = s,on ρ m& Assume: •constant cross section •constant mass flow ρAV=GA G ≡ ρV = mass velocity •neglect gravity •no shaft work

5 Compressible Fluids Mechanical energy balance (compressible)

Ideal Gas Law pV = NRT For isothermal flow: V RT = = p1V1 NRT = N p p2V2 NRT V RT = p1 = V2 MN pM p2 V1 1 RT = ρ M ρ pM Also, av = pav RT ρ 2 av = M + p1 p2 RT

Compressible Fluids

Mechanical energy balance (compressible) G ≡ ρV = mass velocity G 2 p 2 fG 2 ()p − p + ln 1 + ()L − L = 0 2 1 ρ ρ 2 1 av p2 av D

(matches equation 3.187 in handout)

The compressible MEB predicts that there is a maximum velocity at (see handout) p RT V = 2 = = isothermal speed of sound max ρ 2 M

6 Compressible Fluids

A better assumption than isothermal flow is adiabatic flow (no heat transferred). For this case,

γ p γ RT V = 2 = = adiabatic speed of sound max ρ 2 M C γ = p Cv (see handout)