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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/γ∫1p1/γdp=(c2)1/γ∫p−1/γdp
=(c2)1/γ[p−1γ+1(−1γ+1)]=(c2)1/γ(p)(γ−1γ)(γ−1γ)=γγ−1(c2)1/γ(p)(γ−1γ)=(γγ−1)(pργ)1/γ(p)(γ−1γ)(∵c2=pργ)=(γγ−1)(p1/γγ×1γ)(p)(γ−1γ)=(γγ−1)(p)(1γ+γ−1γ)ρ=(γγ−1)pρ
Substituting the value of, ∫dpp , we get
(γγ−1)pρ+V22+gz= constant
Dividing both sides by g, we get, (γγ−1)pρg+V22g+z= constant
The Bernoulli's equation for compressible flow undergoing adiabatic process.