Air initially at rest (M1 = 0) with an initial pressure P1 = 1 atm and temperatu

Question

Air initially at rest (M1 = 0) with an initial pressure P1 = 1 atm and temperature T1= 300k flows through a nozzle into a partial vacuum with P2 = 0.2 atm. There is no elevation change.

What is the air temperature at the end of the nozzle (T2)?

a detailed response would be appreciated.

Explanation / Answer

d be the density PV = mRT =>P = dRT P + 0.5*d*v^2 = constant [bernoulli's principle] P1 = .5*(P2/RT2)*v^2 + P2 =>2(P1-P2)*(RT2/P2) = v^2 energy is conserved. mCv(T1-T2) = 0.5*m*v^2 [since no heat is added and no external work is done] =>(T1-T2) = 0.5v^2/Cv = (P1-P2)*(RT2/P2*Cv) Cv = 1.0035 J/g-K = 1003.5 J/kg-K =>T1 = T2 * (1 + [(P1-P2)*R/P2*Cv] ) = T2 * { 1 + (0.8/.2)* (287.04/1003.5) } =>300 = T2 * { 1 + 4*.286} =>T2 = 139.925K

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