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For more analyses of compressible flow through nozzles and diffusers, click the topic “Gas Dynamics” please: Gas Dynamics
A nozzle is a device which accelerates fluid. During this process, velocity of fluid increases with decreasing pressure.
A diffuser is a device which slows down fluid. That means, velocity of fluid decreases with increasing pressure.
The 1^{st} law of thermodynamics:
No work is involved in nozzles and diffusers:
ΣW_{j}=0
The change of potential energy of fluid flowing into and out of nozzles and diffusers is negligible because of almost no height change.
(e_{pot})_{out }– (e_{pot})_{in}≈0 → g∙(z_{out }– z_{in})≈0
Nozzles and diffusers are also regarded as steadyflow engineering device, so the term at the righthand side equals zero:
Furthermore, m_{out}=m_{in} because of conservation of mass.
So now we obtain a simplified expression for nozzles and diffusers:
q + (h_{in }+o.5∙c^{2}_{in}) – (h_{out}+o.5∙c^{2}_{out}) =0
q + (h_{in }–h_{out}) + 0.5∙ (c^{2}_{in} – c^{2}_{out}) =0
where
 q=heat transferred per unit mass
 h_{in}= specific enthalpy of inlet fluid
 h_{out}= specific enthalpy of outlet fluid
 c_{in}= velocity of inlet fluid
 c_{out}= velocity of outlet fluid
We notice that velocity appears in the equation of energy balance, so the conservation of mass is usually taken into consideration in order to solve the problems:
ρ_{in}∙c_{in}∙A_{in}= ρ_{out}∙c_{out}∙A_{out}
where
 A= section area
 ρ=density
Another necessary equation is the law of ideal gas:
p= ρ∙R∙T
where
 p=pressure
 R=specific gas constant
 T=temperature
So usually we need three equations to solve the problems related to nozzles and diffusers:

The conservation of energy:
q + (h_{in }– h_{out}) + 0.5∙ (c^{2}_{in} – c^{2}_{out}) =0

The conservation of mass:
ρ_{in}∙c_{in}∙A_{in}= ρ_{out}∙c_{out}∙A_{out}

The law of ideal gas: