7.5 Polytropic processes

 

The name polytropic is given to reversible thermodynamic processes that satisfy the equation

 

                                                                                                                         (7.78)

 

with an arbitrary value of n constant for the given polytropic process.

The quantity n is referred to as the polytropic exponent, which varies for different polytropic processes from  to .

The curve representing a polytropic process on the diagram of state is called a polytrope.

The concept of polytropic processes was introduced in thermodynamics by an analogy with the concept of the adiabatic processes.

The equation of a polytropic process (7.78) is similar to that of an adiabat (7.51) in external appearance, the only essential difference between the two equations being that while the isentropic (adiabatic) exponent k is variable in the general case, the very concept of the polytropic process is based on the assumption that the polytropic exponent n is constant. In a polytropic process heat may be added to or removed from a system.

The concept of polytropic processes finds wide application in studying the processes of compression and expansion in gas engines; polytropic processes often prove to be helpful in approximating the real gas processes in engines. The real processes of compression in gas engines are usually neither adiabatic nor isothermal but occupy an intermediate position between the two. That is why the values of the polytropic exponent n most frequently encountered in practical problems range from 1 to k.[1]

For any point lying on a polytrope Eq. (7.78) reduces to

 

                                                              

 

If a polytropic process occurs in an ideal gas, it is easy to obtain from Eq. (7.78) and Clapeyron's equation that

 

                                                                                                                         (7.79)

 

and

 

                                                                                                                      (7.80)

 

 

These equations, just as Eq. (7.78), relate p, v and T at any two points on a polytrope. Equation (7.78) is true both for real and ideal gases, whereas Eqs. (7.79) and (7.80) are applicable only to ideal gases.

In addition to the practical expediency mentioned above, the introduction of the concept of the polytropic process is of great importance from the methodological viewpoint. The concept of the polytropic process correlates all other known thermodynamic processes; it can be easily proved that the isochoric, isobaric and adiabatic processes are particular cases of polytropic processes.

In fact, from the equation of the polytropic process (7.78) it follows that a polytropic process with an exponent n = 0 is merely a conventional isobaric process p = const.

From Eq. (7.78), presented in the form

 

                                                             

 

it is clear that a polytropic process with an exponent  is an isochoric process, v = const.

The equation of a polytrope with an exponent n = k turns into Poisson's equation for an adiabat.

Finally, the equation of a polytrope with an exponent n = 1 is the equation for an ideal gas

 

                                                               

 

(It will be emphasized that this equation is only true for an isothermal process in an ideal gas; it is impossible to derive from the equation of a polytrope (7.78) an equation for the isotherm of a real gas. The process n = 1 can, of course, take place in a real gas, but this process is not isothermal.)

Figure 7.8 shows a p-v diagram on which the curves of various polytropic processes are plotted.

 

 

Fig. 7.8

 

The work of expansion done by a system undergoing a polytropic process between points 1 and 2 is determined with the aid of Eq. (3.1),

 

                                                               

 

Since, by Eq. (7.78), for a polytrope

 

                                                                                                                             (7.81)

 

integrating the expression for the work of expansion l1-2 (note that  is constant), we obtain for the work of expansion in a polytropic process:

 

                                                                                                       (7.82)

 

or, which is the same

 

                                                                                                     (7.83)

 

 

For an ideal gas Eqs. (7.82) and (7.83) can be rearranged in the following way:

 

                                                                                                       (7.84)

 

                                                                                                     (7.85)

 

                                                                                                              (7.86)

 

                                                                                                             (7.87)

 

                                                                                                       (7.88)

 

Equations (7.82) and (7.83) are true for polytropic processes both in real and ideal gases, and Eqs. (7.84) to (7.88) are applicable only to ideal gases.

The amount of heat added to or removed from a system undergoing a polytropic process is determined in the following way. The amount of heat q can be found with the aid of the mathematical statement of the first law of thermodynamics:

 

                                                                                                                     

 

calculating the work of expansion done by the system [with the aid of Eqs. (7.82) or (7.83)] and the change in the internal energy of the system undergoing the polytropic process between points 1 and 2.

The difference in the internal energies at points 1 and 2 is determined in the usual way. Integrating relationship (2.29),

 

                                                                                                          

 

between the limits represented by the points 1 and 2 on the polytrope, and taking into account the fact that, by Eqs. (2.31) and (4.25),

 

                                                                                                                                    

 

and

 

                                                                                                                  

 

we obtain:

 

                                                             (7.89)

 

 

To use this equation for calculations, the parameters of the two points on the polytrope must be known: T1 and v1, T2 and v2.

If we are dealing with a polytropic process in an ideal gas (it will be noted that in calculating a number of polytropic processes in gas engines and compressors, the ideal-gas approximation proves to be quite sufficient for engineering calculations), the equations for calculating q2-1 can be reduced to a simpler form. Indeed, inasmuch as for an ideal gas

 

                                                                                                               (7.90)

 

If the dependence of the constant-volume heat capacity cv on temperature can be ignored for an ideal gas, it follows that

 

                                                                                                   (7.91)

 

 

Now calculate q2-1 with the aid of Eq. (2.18). Substituting into the latter the value of u2 – u1 from Eq. (7.91) and the value of l1-2 from equation (7.86), we obtain:

 

                                                                                                   (7.92)

 

Since, by Mayer's relation,

 

                                                                                                                                   

 

we find from Eq. (7.92) that

 

                                                                                                       (7.93)

 

 

As was shown in Sec. 7.4, for an ideal gas the ratio of the heat capacities cp/cv is the exponent of the isentrop of an ideal gas (7.55),

 

                                                                                                                                     

 

Taking the above ratio into account, Eq. (7.93) becomes

 

                                                                                                         (7.94)

 

 

Equation (7.94) determines the amount of heat added to an ideal gas (or removed from it) in a polytropic process.

Since, in accordance with the general definition of heat capacity

 

                                                                                                                                      

 

the heat capacity of a polytropic process can be determined as

 

                                                                                                                             (7.95)

 

it is clear that

 

                                                                                                                             (7.96)

 

and

 

                                                                                                                        (7.97)

 

If within the temperature interval considered the heat capacity cn remains constant between points 1 and 2, Eq. (7.97) permits us to derive the following equation for calculating the amount of heat added to a system in a polytropic process:

 

                                                                                                                   (7.98)

 

It is interesting to note that inasmuch as for an ideal gas whose heat capacity does not change with temperature

 

                                                                                                                       

 

[see Eq. (7.91)], the above relationship and Eq. (7.98) give:

 

                                                                                                                         (7.99)

 

 

In this case, the heat capacity of an ideal gas does not change with temperature and the ratio cv/cn remains constant throughout the given polytropic process; consequently, by Eq. (7.99), in this process

 

                                                                                                                       (7.100)

 

 

Comparing Eqs. (7.94) and (7.98), we conclude that the expression cv (n - k)/(n - 1) in the right-hand side of Eq. (7.94) is nothing but the heat capacity of an ideal gas undergoing a polytropic process:

 

                                                                                                                      (7.101)

 

Let us analyze this relationship for the polytropic heat capacity of an ideal gas.

At n = 0, Eq. (7.101) reduces to

 

                                                                                                                                        

 

or, taking into account Eq. (7.55),

 

                                                                                                                                          

 

This result is understandable since, as it was shown above, a polytrope with an exponent n = 0 is an isobar.

With  rearranging Eq. (7.101) as

 

                                                                                                                                

 

we obtain:

 

                                                                                                                                          

 

which is also clear, since a polytrope with an exponent  is an isochor.

At  a zero appears in the denominator of the right-hand side of Eq. (7.101) and, consequently,

 

                                                                                                                                        

 

it will be recalled that the heat capacity of an isothermal process is infinitely large.

Finally, at

 

                                                                                                                                           

 

which is the heat capacity of an adiabatic process.

It is interesting to note that, as Eq. (7.101) shows, for 1 < n < k the heat capacity cn is negative. In polytropic processes an expanding gas produces work, considerably exceeding the amount of heat added to the gas undergoing expansion. Work is then performed not only at the expense of the heat added to the gas but also due to the expenditure of some amount of the internal energy of the gas. All the heat added to the gas undergoing expansion converts into work, and the decrease in the internal energy leads to a drop in gas temperature. Thus, we are concerned here with a rather peculiar process: heat is added to the system, but the system temperature decreases. According to the general definition of heat capacity,

 

                                                                                                                                      

 

we conclude that the heat capacity of such a polytropic process is negative. In Fig. 7.9 the heat capacity of a polytropic process cn is plotted as a function of the polytropic exponent n.

 

 

Fig. 7.9

 

The change in system entropy in a polytropic process, i.e. the difference in the entropies corresponding to the points 1 and 2 on the polytrope, is determined from the relationship

 

                                                                                                           (7.102)

Since it is clear that

 

                                                                                                                       (7.103)

 

Eq. (7.102) acquires the following form:

 

                                                                                                                 (7.104)

 

 

If the heat capacity of a polytropic process cn remains constant in a given interval of properties of state, Eq. (7.104) becomes

 

                                                                                                                 (7.105)

 

 

Thus, the dependence of entropy on temperature in a polytropic process is logarithmic.[2]

In order to determine the polytropic exponent for a particular gas process, experimental data on this process should be available; it is most convenient to make use of the curve of the process plotted on a p-v diagram. Taking the logarithm of the equation of a polytrope,

 

                                                                                                                                  

 

we obtain:

 

                                                                                                             (7.106)

 

 

 

 

Fig. 7.10

 

Plotting the p-v diagram of the process on log paper, on which the curve becomes a line (Fig. 7.10), it is easy to find the exponent n for the given curve of the process; as can be seen from Eq. (7.106), the exponent n is the slope of this line on the logarithmic p-v diagram. If the curve of the polytropic process fails to be completely straight on the logarithmic p-v diagram and has a certain curvature, this line should be divided into several straight sections, the value of the exponent n for each of these sections should be found, and then the mean value of the exponent n for the entire line should be calculated. Having determined the exponent n for the given real process, we can use the previously derived equations to calculate the work of expansion, l1-2, the amount of heat participating in the process, q2-1, the temperature at any point of the process, etc.

 

 

 


[1] It follows from Eq. (7.78) that at n = 1 this equation coincides with Boyle's law for an isothermal process in an ideal gas (pv = const).

[2] The method of derivation and the obtained equation (7.105) are not applicable to the process with n = 1 (isothermal process); with n = 1 an indeterminate form  appears in the right-hand side of Eq. (7.105).