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# Introduction to Compressible Flow (Part - 2) Civil Engineering (CE) Notes | EduRev

## Civil Engineering (CE) : Introduction to Compressible Flow (Part - 2) Civil Engineering (CE) Notes | EduRev

The document Introduction to Compressible Flow (Part - 2) Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Fluid Mechanics - Notes, Videos, MCQs & PPTs.
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Entropy and Second Law of Thermodynamics

• Equation (38.24) does not tell us about the direction (i.e., a hot body with respect to its surrounding will gain temperature or cool down) of the process. To determine the proper direction of a process, we define a new state variable, entropy, which is

(38.25)

where s is the entropy of the system,  Î´qrev is the heat added reversibly to the system and T is the temperature of the system. It may be mentioned that Eqn. (38.25) is valid if both external and internal irreversibilities are maintained during the process of heat addition

• Entropy is a state variable and it can be associated with any type of process, reversible or irreversible. An effective value of Î´qrev can always be assigned to relate initial and end points of an irreversible process, where the actual amount of heat added is Î´q . One can write

(38.26)

It states that the change in entropy during a process is equal to actual heat added divided by the temperature plus a contribution from the irreversible dissipative phenomena. It may be mentioned that dsirrev implies internal irreversibilities if T is the temperature at the system boundary. If T is the temperature of the surrounding dsirrev implies both external and internal irreversibilities. The irreversible phenomena always increases the entropy, hence

(38.27)

• Significance of greater than sign is understandable. The equal sign represents a reversible process. On combining Eqs (38.26) and (38.27) we get

(38.28)

If the process is adiabatic, , Eq. (38.28) yields

(38.29)

• Equations (38.28) and (38.29) are the expressions for the second law of thermodynamics. The second law tells us in what direction the process will take place. The direction of a process is such that the change in entropy of the system plus surrounding is always positive or zero (for a reversible adiabatic process). In conclusion, it can be said that the second law governs the direction of a natural process.

• For a reversible process, it can be said that  Î´W = - pdv where dv is change in volume and from the first law of thermodynamics it can be written as

Î´q - pdv = de                                                 (38.30)

• If the process is reversible, we use the definition of entropy in the form  Î´qrev = Tds the Eq. (38.30) then becomes,
Î´q - pdv = de
Tds = de + pdv                                         (38.31)

• Another form can be obtained in terms of enthalpy. For example, by definition

h = e +pv

Differentiating, we obtain

dh = de + pdv + vdp                                  ( 38.32)

Combining Eqs (38.31) and (38.32), we have

Tds = dh + vdp                                  ( 38.33)

• Equations (38.31) and (38.33) are termed as first Tds equation and second Tds equation, respectively.

• For a thermally perfect gas, we have dh= cpdt  (from Eq. 38.20) , substitute this in Eq. (38.33) to obtain

( 38.34)

Further substitution of pv = RT into Eq. (38.34) yields

( 38.35)

Integrating Eq. (38.35) between states 1 and 2,

( 38.36)

If   cp  is a variable, we shall require gas tables; but for constant  cp  we obtain the analytic expression

( 38.37)

In a similar way, starting with Eq. (38.31) and making use of the relation  the change in entropy can also be written as

( 38.38)

Isentropic Relation

An isentropic process is a reversible-adiabatic process. For an adiabatic process  Î´q = 0 and for a reversible process, dsirrev = 0 From Eq. (38.26), for an isentropic process, ds = 0 However, in Eq. (38.37), substitution of isentropic condition yields

(38.39)

Using    , we have

(38.40)

Considering Eq. (38.38), in a similar way, yields

(38.41)

Using            we get

(38.42)

• Using  we can write

(38.43)

• Combining Eq. (38.40) with Eq. (38.43), we find,

(38.44)

This is a key relation to be remembered throughout the chapter.

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