Introduction to Compressible Flow - 2 | Fluid Mechanics for Mechanical Engineering PDF Download

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,  δq_rev 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 δq_rev can always be assigned to relate initial and end points of an irreversible process, where the actual amount of heat added is δq . 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 ds_irrev implies internal irreversibilities if T is the temperature at the system boundary. If T is the temperature of the surrounding ds_irrev 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  δW = - pdv where dv is change in volume and from the first law of thermodynamics it can be written as
  
  δq - pdv = de                                                 (38.30)                          
  
                            
• If the process is reversible, we use the definition of entropy in the form  δq_{rev =} Tds the Eq. (38.30) then becomes,
  δq - 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= c_pdt  (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   c_p  is a variable, we shall require gas tables; but for constant  c_p  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  δq = 0 and for a reversible process, ds_irrev = 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.