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418 ENGINEERING THERMODYNAMICS

dharm

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Again,

m

M

= n and Mcv = Cv

∴ U = nCvT ...(9.23)

Similarly, H = nCpT ...(9.24)

By the Gibbs-Dalton law,

U = Σ Ui and H = Σ Hi

∴ nCvT = Σ niCvi T and nCpT = Σ niCpiT

i.e., Cv =
n
n

i

∑ Cvi ...(9.25)

and Cp = n
n

i

∑ Cpi. ...(9.26)

9.6. Adiabatic Mixing of Perfect Gases

— Fig. 9.3 shows two gases A and B separated from each other in a closed vessel by a thin

diaphragm. If the diaphragm is removed or punctured then the gases mix and each

then occupies the total volume, behaving as if the other gas were not present. This

process is equivalent to a free expansion of each gas, and is irreversible. The process can

be simplified by the assumption that it is adiabatic ; this means that the vessel is

perfectly thermally insulated and there will therefore be an increase in entropy of the

system.

Diaphragm

Closed vessel

Gas A Gas B

mA nA

pA

TA VA

mB nB

pB

TB VB

Mixture of

gas A and gas B

m = m + mAB

p, T

V = V + VAB

n = n + nAB

Fig. 9.3
In a free expansion process, the internal energy initially is equal to the internal energy finally.
In this case, from eqn. (9.23),

U 1 = nACvA TA + nBCvBTB

and U 2 = (nACvA + nBCvB)T

If this result is extended to any number of gases, we have

U 1 = Σ niCviTi and U 2 = T Σ niCvi

Then U 1 = U 2

i.e., Σ niCviTi = T Σ niCvi

i.e., T =

Σ

Σ

nC T

nC

ivii

ivi

...(9.27)

— When two streams of fluid meet to form a common stream in steady flow, they give an-

other form of mixing (Fig. 9.4).