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Explain Bernoulli's principle equation for adiabatic process.
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Solution:

In case of an adiabatic process,

pvγ= constant or pργ= constant =c2 (say) ργ=pc2 or ρ=(pc2)1/γ

Hence, dpρ=dp(p/c2)1/γ==(c2)1/γ1p1/γdp=(c2)1/γp1/γdp

=(c2)1/γ[p1γ+1(1γ+1)]=(c2)1/γ(p)(γ1γ)(γ1γ)=γγ1(c2)1/γ(p)(γ1γ)=(γγ1)(pργ)1/γ(p)(γ1γ)(c2=pργ)=(γγ1)(p1/γγ×1γ)(p)(γ1γ)=(γγ1)(p)(1γ+γ1γ)ρ=(γγ1)pρ

Substituting the value of, dpp , we get

(γγ1)pρ+V22+gz= constant 

Dividing both sides by g, we get, (γγ1)pρg+V22g+z= constant  

The Bernoulli's equation for compressible flow undergoing adiabatic process.

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