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Modern Engineering Thermodynamics

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280 CHAPTER 9: Second Law Open System Applications<br />

and<br />

Z<br />

ðS T Þ m = sdm<br />

This equation can also be written as<br />

Z Z<br />

ðS T Þ m = sdm=<br />

<br />

s dm Z<br />

dt =<br />

dt<br />

_ms dt<br />

where _m is the mass flow rate crossing the system boundary. Differentiation of this equation with respect to<br />

time yields the mass transport rate term as<br />

_S T<br />

m = _ms<br />

Unlike heat and work transports of entropy, it is customary in mass flow transport to write a more explicit formula<br />

for the net mass flow entropy transport rate as<br />

Entropy transport rate due to mass flow<br />

h i<br />

_S T m<br />

= ∑ _ms −∑ _ms (9.1)<br />

net<br />

in out<br />

where the summations are over all inlet and outlet flow streams (the flow streams normally are pipes or ducts<br />

that convey mass into or out of the system).<br />

9.3 MASS FLOW PRODUCTION OF ENTROPY<br />

Mass flow entropy production is due to mass flow that occurs inside the system. No entropy production is due<br />

solely to mass crossing the system boundary, because it is an imaginary boundary of zero thickness. The two<br />

main sources for this type of internal mass flow entropy production are viscous dissipation and diffusion of<br />

dissimilar chemical species.<br />

Viscous dissipation is really a work mode entropy production mechanism and has already been treated in<br />

Eqs. (7.71) and (7.72). Diffusion of dissimilar chemical species inside the system is an advanced topic not treated<br />

in this text. Consequently, if we neglect diffusion, there is no mass flow production of entropy inside a<br />

system, and<br />

ð _S P Þ m = 0<br />

9.4 OPEN SYSTEM ENTROPY BALANCE EQUATIONS<br />

The general entropy rate balance given by Eq. (7.3) equates the net rate of gain of total entropy by a system _S G<br />

to the sum of the net entropy transport rate into the system _S T plus the net entropy production rate within the<br />

system _S P ,or<br />

_S T + _S P = _S G (7.3)<br />

where<br />

dS<br />

_S G = _S system =<br />

dt system<br />

and the transport and production rates of entropy have been identified as caused by heat transfer, work, and<br />

mass flow, or<br />

and<br />

_S T = ð _S T Þ Q<br />

+ ð _S T Þ W<br />

+ ð _S T Þ m<br />

(9.2)<br />

_S P = ð _S P Þ Q<br />

+ ð _S P Þ W<br />

+ ð _S P Þ m<br />

(9.3)

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