7.4 The adiabatic process

 

The name adiabatic is given to thermodynamic processes in the course of which heat is neither added nor removed from a system, in other words, no heat transfer occurs during the process, i.e.

 

                                                                                                                                   (7.36)

 

A thermodynamic system in which an adiabatic process proceeds can be visualized as a certain volume confined by boundaries fitted with ideal heat insulation through which no heat transfer can take place; boundaries of this kind are said to be adiabatic.

In real conditions, a process is said to be adiabatic when the system is outfitted with proper heat insulation or when the expansion (or compression) of a gas proceeds so rapidly that no noticeable heat transfer between the gas and the surroundings has time to take place.

Inasmuch as for a reversible process, in accordance with Eq. (3.121),

 

                                                                

 

with allowance for Eq. (7.36), it follows that in a reversible adiabatic process

 

                                                                                                                                  (7.37)

 

i.e. the entropy of a system remains constant. In other words, a reversible adiabatic process is an isentropic process at the same time.

It is not by mere chance emphasized here that the process dealt with above is a reversible adiabatic process, since irreversible adiabatic processes are also possible. For example, consider the flow of a real gas in a rough tube covered with an ideal heat insulation preventing any heat transfer through the walls of the tube. The gas will then be in adiabatic flow since no heat is added to it from the outside and no heat is rejected from the gas to the surroundings. But inasmuch as the flow of a real gas in a rough tube is always accompanied by friction, leading to dissipation of the energy of flow, this process is irreversible: as in any irreversible adiabatic process, it proceeds with an increase in the system's entropy. Irreversible processes are characterized by the following inequality

 

                                                                

 

In this irreversible adiabatic process dq = 0, but ds > 0, hence, an irreversible adiabatic process is not isentropic.

Thus, any isentropic process developing in an isolated system can be said to be adiabatic, but not any adiabatic process is isentropic (only reversible-adiabatic processes are isentropic).

Here, we shall deal only with reversible adiabatic processes (ds = 0).

Let us now turn to the relation between the parameters of various states in a reversible adiabatic process.

To elucidate this problem, it is necessary to formulate the differential equation of an isentropic process.

For an isentropic process (ds = 0) the combined mathematical statement of the first and second laws of thermodynamics,

 

                                                            

 

and

 

                                                            

 

acquire the following form:

 

                                                                                                                           (7.38)

 

                                                                                                                             (7.39)

 

whence,

 

                                                                                                                         (7.40)

 

                                                                                                                           (7.41)

 

These relationships can be used to obtain

 

                                                                                                             (7.42)

 

Relationship (7.41) is the differential equation of an isentropic process, showing how the change in the caloric properties of a system (h and u) is related with the change in its thermal properties (p and v) in an isentropic process.

We shall introduce the following notation:

 

                                                                                                                           (7.43)

 

The quantity k will be referred to as the exponent of an isentropic process (or the isentropic exponent). Taking into account this new notation, relationship (7.42) becomes:

 

                                                                                                                     (7.44)

 

The product present in the left-hand side of Eq. (7.44) can be transformed as follows:

 

                                                    

 

allowing for the above relationship, Eq. (7.44) can be presented in the form:

 

                                                                                                                  (7.45)

 

Differential equation (7.45) relates pressure p and volume v in an isentropic process.

Integrating this relationship between the limits 1 and 2 on an isentrop, we obtain:

 

                                                                                                                (7.46)

 

If in the interval of the change of state considered (from point 1 to point 2) the isentropic exponent remains constant, k can be taken out of the integral sign, and Eq. (7.46) reduces to

 

                                                                                                                   (7.47)

 

or, which is the same,

 

                                                                                                                  (7.47a)

 

This relationship, in turn, can be presented in the form

 

                                                                                                                       (7.48)

 

Taking antilogarithms for Eq. (7.48), we obtain:

 

                                                                                                                             (7.49)

 

therefore,

 

                                                                                                                         (7.50)

 

By analogy, taking this system into any third state along the isentrop with parameters p3 and v3, we can show that

 

                                                 

 

Thus, for any state of a system undergoing an isentropic process (provided the isentropic exponent k remains constant),

 

                                                                                                                         (7.51)

 

This relationship is called the Poisson equation of an adiabat.

If the isentropic exponent changes with a change in the state of a system and the dependence of k: on the isentrop is known, to calculate p2 from p1, v1, v2 known, we must calculate the integral present in the right-hand side of Eq. (7.46), applying numerical methods with the values of k known.

We can use the mean (in the interval of states considered) isentropic exponent[1] km. Applying then a method similar to that described above, from Eq. (7.46) we get:

 

                                                                                                                     (7.51a)

 

It will be noted that the equations of the isentropic process (7.51) or (7.51a) are true for a gas, liquid and solid (while deriving the equations, no assumptions were made regarding the nature of the system undergoing an isentropic process).

The isentropic exponent k (whose value by itself can be used as a property of state) happens to differ significantly for substances in different states of aggregation. For solids and liquids k is rather large, with the magnitude of k varying noticeably with temperature. Consequently, for water at a temperature t = 0 °C, k = 3602000; at t = 50 °C, k = 187000; and at t = 100 °C, k = 22300. For gases and vapors the value of k changes (diminishes) rather slightly with temperature, for most gases being equal from 1.3 to 1.7. The values of A: vary noticeably only near the boundary curve. The values of the isentropic exponent k for water vapor are plotted in Fig. 7.6.

 

 

Fig. 7.6

 

The isentropic exponent for an ideal gas is determined in the following way. Equation (7.44),

 

                                                            

 

taking account of relationship (4.59)

 

                                                       

 

 

can also be presented in the form

 

                                                                                                               (7.53)

 

Since for an ideal gas, as follows from Clapeyron's equation,

 

                                                                                                                       (7.54)

 

for an ideal gas Eq. (7.53) reduces to

 

                                                                                                                             (7.55)

 

Since, in accordance with Eq. (2.54), for an ideal gas

 

                                                               

 

it follows from equation (7.55) that

 

                                                                                                                        (7.56)

 

Since the heat capacities of an ideal gas vary only slightly with temperature, the quantity kideal can be considered with a high degree of accuracy to be independent of temperature. The molar isochoric (constant-volume) heat capacity μcv of an ideal gas is known to be equal to about 13 kJ/(kmol·K) ≈ 3 kcal/(kmol·K) for a monatomic ideal gas, 21 kJ/(kmol·K) ≈ 5 kcal/(kmol·K) for a diatomic one, and 29 kJ/(kmol·K) ≈ 7 kcal/(kmol-K) for triatomic and multiatomic gases. Inasmuch as μR ≈ 8.3 kJ/(kmol·K) ≈ 2 kcal/(kmol-K), Eq. (7.56) can be used to obtain the following approximate isentropic exponents for an ideal gas, kideal:

 

Monatomic gas                     1.67

diatomic gas                         1.40

tri- and multiatomic gases    1.29

 

For air in an ideal-gas state the isentropic exponent is about 1.35. It was stated above that for any two points laying on an isentrop the quantities p and v are related by Eq. (7.50),

 

                                                              

 

where k is the isentropic exponent, constant in the interval of states between points 1 and 2; but if k is variable in this interval of parameters of state, then Eq. (7.50) must contain the quantity km, which is the mean exponent for this interval of state parameters.

The relation between the temperatures T1 and T2 at the two points on the isentrop can be found as follows.

From relationship

 

                                                                                         (7.57)

 

remembering that

 

                                                                                                     (7.58)

 

and also allowing for Eqs. (4.22) and (4.45), we obtain:

 

                                                                                      (7.59)

 

To calculate T2 from a known T1 with the aid of Eq. (7.59), the values of  and cp must be available. The calculations are also hindered by the fact that the temperature T is under the integral sign. To facilitate the problem of finding the various parameters for an isentropic process, isotherms, isochors, lines of constant entropy, isentrops, as well as isobars, are often plotted on the diagrams of state. In Fig. 7.7 isentrops are plotted on the p-v, p-T and T-v diagrams.

 

 

Fig. 7.7

For an ideal gas undergoing an isentropic process, Eq. (7.50) can be used to obtain formulas relating T and v and also T and p, on an isentrop. In fact, allowing for an ideal gas that

 

                                                                 

 

we find from Eq. (7.50) that

 

                                                                                                                       (7.60)

 

i.e.

 

                                                                                                                       (7.60a)

 

Further, substituting into equation (7.50)

 

                                                                 

 

we obtain:

 

                                                                                                                        (7.61)

 

It follows from Eq. (7.61) that a decrease in pressure on an isentrop leads to a decrease in the temperature of the ideal gas. For a real gas, it is clear from Eq. (7.59) that inasmuch as cp is always greater than zero, cp > 0, and usually[2]  is also greater than zero, the integral in the right-hand side of this equation is positive. Consequently, if p1 < p2, then T1 < T2. In this manner a substance undergoing isentropic (reversible adiabatic) expansion cools. As will be seen subsequently, the process of reversible adiabatic expansion is used effectively for cooling gases.

The work of expansion done by a system in an isentropic process is determined in the following manner. It is clear from Eq. (2.23)

 

                                                             

 

that for any adiabatic process (dq = 0), including the reversible adiabatic (i.e. isentropic) process,

 

                                                                                                                           (7.62)

 

and, consequently, in accordance with Eq. (3.1)

 

                                                                                                                           (3.1a)

 

for the work of expansion performed by a system undergoing an adiabatic process we obtain:

 

                                                                                                                           (3.1a)

 

Thus, in an adiabatic process the work of expansion is performed by a system at the expense of its internal energy. This is understandable - indeed no heat is added to the system undergoing an adiabatic process, and the only source of energy which would permit work to be performed is the internal energy of the system itself.

Equation (7.63) is valid not only for an isentropic, i.e. reversible adiabatic process, but also for an irreversible adiabatic process. The equations set forth below, however, are true only for isentropic processes, since in deriving these equations the concept of the isentropic exponent is being used, which pertains only to isentropic processes.

The formula for work performed in an isentropic process, l1-2, can be presented in another form. Inasmuch as, in accordance with Eq. (7.50), for an isentrop

 

                                                                                                                             (7.64)

 

substituting this expression for p in Eq. (3.1a),

 

                                                                                                                     (7.65)

 

and integrating, we obtain (taking into account that  is const):

 

                                                                                                       (7.66)

 

or, allowing for Eq. (7.50),

 

                                                                                                       (7.67)

 

 

It should be emphasized that Eqs. (7.66) and (7.67) are suitable to calculate l1-2, provided the isentropic exponent k remains constant between the states 1 and 2. But if k varies and Eqs. (7.66) and (7.67) are used, calculations should be based on the mean value of k, i.e. km, for the given interval.

For an ideal gas the formulas for the work of expansion can be represented in a different form. Since for an ideal gas

 

                                                                

 

Eqs. (7.66) and (7.67) acquire the following form:

 

                                                                                                       (7.68)

 

and

 

                                                                                                       (7.69)

 

For an ideal gas undergoing an isentropic process, in accordance with Eq. (7.60a),

 

                                                             

 

 

and from Eq. (7.68) we obtain:

 

                                                                                                               (7.70)

 

From Eq. (7.66)

 

                                                                                                               (7.71)

 

Thus

 

                                                                                                       (7.72)

 

(inasmuch as ).

 

Since the internal energy of an ideal gas depends only on temperature and is independent of volume, in accordance with Eq. (2.37),

 

                                                                

 

and, consequently,

 

                                                                                                   (7.73)

 

Thus, by Eq. (7.63),

 

                                                                                                                         (7.74)

 

If the dependence of the heat capacity cv of an ideal gas on temperature is ignored or use is made of the concept of the mean heat capacity for a given temperature interval, then Eq. (7.74) takes the following form:

 

                                                                                                                   (7.75)

 

It will be emphasized once more that Eqs. (7.63), (7.66) and (7.67) are applicable both to real substances and ideal gases, whereas Eqs. (7.68) to (7.75) are valid only for ideal gases.

The amount of heat added to a system undergoing an isentropic process is equal to zero:

 

                                                           or                                                      (7.76)

 

The change of entropy in an isentropic process is also zero. Naturally, in an isentropic process the heat capacity is zero:

 

                                                                                                                       (7.77)

 

This conclusion stands to reason, for, if a system undergoes an isentropic process, the system's temperature changes, although no heat is added to the system.

 

 

 


[1] The mean value of k is determined from the relationship

 

                                                                                                                                                               (7.52)

 

 

[2] It will be recalled that on rare occasions the quantity  becomes negative (for instance, for water in the region of density anomaly at low temperatures). An increase in pressure p on an isentrop then leads to a drop in T, as can be seen from Eq. (7.59).