CALCULATION OF ADIABATIC AND POLYTROPIC PROCESSES USING AVERAGE HEAT CAPACITY

G. I. Fuks

For calculations of adiabatic and polytropic processes with variable heat capacity, the use of heat diagrams is usually recommended. This is indeed expedient, especially when a series of calculations must be carried out. But in individual cases, especially for single calculations, it is desirable to have the possibility of analytical calculations, using the usual formula relationships for heat capacities or tabulated values. As indicated below, these calculations can be carried out with a sufficient degree of accuracy by an appropriate selection of the value of the average heat capacities.

1. Adiabatic Processes

The relation between \(p\) and \(T\) for adiabatic processes is

\[ C_p \frac{dT}{T} - AR \frac{dP}{P} = 0. \tag{1} \]

Adopting a linear dependence of heat capacity on temperature,

\[ C_p = a_p + bT, \tag{2} \]

after integration we obtain

\[ a_p \ln \frac{T_2}{T_1} + b(T_2 - T_1) - AR \ln \frac{P_2}{P_1} = 0. \tag{3} \]

If integration of (1) is carried out while considering the heat capacity constant and equal to the average value \(c_p\), then we obtain

\[ c_p \ln \frac{T_2'}{T_1} - AR \ln \frac{P_2}{P_1} = 0. \tag{4} \]

For a linear dependence of heat capacity on temperature,

\[ c_p = a_p + b \frac{T_1 + T_2'}{2}, \tag{5} \]

and, taking (4) into account,

\[ \left[a_p + \frac{b}{2}(T_1 + T_2')\right]\ln \frac{T_2'}{T_1} - AR \ln \frac{P_2}{P_1} = 0. \tag{6} \]

From (3) and (6) it follows that

\[ \ln \frac{T_2}{T_1} + \frac{b}{a_p} T_2 = \ln \frac{T_2'}{T_1} + \frac{b}{a_p} \left[ \frac{T_1 + T_2'}{2}\ln \frac{T_2'}{T_1} + T_1 \right]. \tag{7} \]

Obviously, in general,

\[ T_2 \ne T_2', \]

i.e., using the value of the heat capacity from (5) to calculate the final temperature of an adiabatic process gives an incorrect result. But the average heat capacity over a given temperature interval is equal to the true heat capacity at some average temperature (in particular, to the arithmetic mean temperature when the heat capacity depends linearly on temperature). One may try to choose such an average temperature that the heat capacity determined from it will give the correct value of the final temperature of the adiabatic process. When the heat capacity depends linearly on temperature, for this purpose it is sufficient in relation (7) to put

\[ \frac{T_1+T_2'}{2}=\frac{T_2-T_1}{\ln \dfrac{T_2}{T_1}} . \tag{8} \]

Indeed, in this case relation (7) becomes

\[ \ln \frac{T_2}{T_1}+\frac{b}{a_p}T_2 = \ln \frac{T_2'}{T_1} + \frac{b}{a_p} \left[ \frac{T_2-T_1}{\ln \dfrac{T_2}{T_1}} \ln \frac{T_2}{T_1} +T_1 \right], \]

which, for

\[ T_2=T_2' \]

turns into an identity. Consequently, if, when integrating the adiabatic equation (1), the heat capacity is taken to be that which it has at the logarithmic mean between the initial and final temperatures of the process, then the integral equation will give the correct final result. Since in reality the dependence of heat capacity on temperature is not linear, but of a more complex character, the indicated rule is not exact but approximate. Nevertheless, it can quite well be adopted, since the error of the calculations then proves to be negligible, as is shown below in several examples.

In actual calculations, of course, there is no need to carry out integration, etc. The matter reduces to using the ordinary relations for an adiabatic process with constant heat capacity:

\[ \frac{T}{p^{\frac{k_1-1}{k_1}}}=\text{const} \tag{9} \]

\[ Tv^{k_1-1}=\text{const}, \tag{10} \]

\[ pv^{k_1}=\text{const}, \tag{11} \]

with the distinction that the adiabatic exponent \(k_1\) must be selected as the ratio of the heat capacities at the logarithmic mean temperature between the beginning and the end of the process. If the initial and final temperatures of the process are given, then the quantity \(k_1\) is found at once, which also makes it possible at once to find the final pressure or the final volume of the process. If the final temperature of the process is not given, then its determination from (9) or (10) must be carried out by trial: first the value of \(k_1\) is assigned by guess, the value of the final temperature is calculated, from it the value of \(k_1\) is refined, and so on until sufficient agreement is obtained. This is obtained on the second or third calculation (not counting the preliminary one). The details of the calculation will depend on what is used: formulas for heat capacities, tabli-

quantities, and which ones in particular. Let us examine in more detail the case of using the table for enthalpies given in the appendix.

To determine \(\dfrac{k_1-1}{k_1}\), we have

\[ \frac{k_1-1}{k_1} = \frac{\dfrac{C_p}{C_v}-1}{\dfrac{C_p}{C_v}} = \frac{(c_p-c_v)(t_2-t_1)}{c_p(t_2-t_1)} = \frac{AR\Delta t}{\Delta i}, \tag{12} \]

and, when calculating per \(\text{nm}^3\),

\[ \frac{k_1-1}{k_1}=\frac{0.0886\Delta t}{\Delta i}. \tag{13} \]

For a preliminary calculation one may in all cases take \(t_2=t_2'=0\), i.e.

\[ \frac{k_1-1}{k_1}=\frac{0.0886t_1}{i_1}. \tag{13a} \]

Using formulas (9) and (10) (depending on the nature of the problem), the final temperature \(T_2\) is determined preliminarily; from it the value of \(k_1\) must be refined. For this purpose the conditional temperature is determined from the relation

\[ T_{2k}=2\,\frac{T_2-T_1}{\ln\dfrac{T_2}{T_1}}-T_1, \tag{14} \]

or

\[ t_{2k}=2\,\frac{T_2-T_1}{\ln\dfrac{T_2}{T_1}}-T_1-273^\circ\text{C}. \tag{14a} \]

From the table, for this temperature, we determine the enthalpy value \(i_{2k}\), then

\[ \Delta i=i_{2k}-i_1, \]

\[ \Delta t=t_{2k}-t_1 \]

and then, by (13), the new value of \(\dfrac{k_1-1}{k_1}\).

Example No. 1: adiabatic compression of air from \(p_1=1\ \text{ata}\), \(t_1=300^\circ\text{C}\) to \(p_2=100\ \text{ata}\).

Solution: from the table, \(i_1=94.4\ \dfrac{\text{kcal}}{\text{nm}^3}\). By (13a),

\[ \frac{k_1-1}{k_1}=\frac{0.0886\cdot 300}{94.4}=0.281. \]

From (9)

\[ \lg\frac{T_2}{T_1}=0.281\lg\frac{p_2}{p_1}=0.562, \]

\[ T_2=1995^\circ\text{K}. \]

From (14a)

\[ t_{2k}=2\,\frac{1995-573}{\ln\dfrac{1995}{573}}-573-273=1351^\circ\text{C}. \]

The remaining calculations are given in Table 1

Table 1

Note: 1. In the intermediate calculations, the values of $t_2k$ were rounded in order to simplify calculation from the table.
2. An exact calculation of the final temperature gives the value $T_2 = 1812^\circ\mathrm{K}$.

Example No. 2: Compression of $CO_2$ under the same conditions. The calculations are given in Table 2.

Table 2

Note: The refined calculation gives $T_2 = 1194^\circ\mathrm{K}$.

Example No. 3: Compression of $CH_4$ under the same conditions. The calculation is given in Table 3.

Table 3

Note: The exact calculation gives $T_2 = 1055^\circ\mathrm{K}$.

For calculating the work of an adiabatic process, the general relation may serve:

\[ Aw_a=u_1-u_2. \tag{15} \]

If the calculation is carried out from an enthalpy table, the more convenient expression is

\[ Aw_a=i_1-i_2-AR(t_1-t_2), \tag{16} \]

or, in a calculation per \(1\ \mathrm{nm}^3\),

\[ Aw_a=i_1-i_2-0.0886(t_1-t_2)\ \frac{\mathrm{kcal}}{\mathrm{nm}^3}. \tag{16a} \]

Taking into account the dependence of internal energy on temperature, one may also write:

\[ Aw_a=c_v(T_1-T_2), \tag{17} \]

where \(c_v\) is the mean heat capacity in the temperature interval \(T_1(t_1)—T_2(t_2)\). One may also obtain the relation

\[ w_a=\frac{R}{k-1}(T_1-T_2), \tag{18} \]

where \(k\) is the ratio of the mean heat capacities in the temperature interval \(T_1(t_1)—T_2(t_2)\). Taking (9) and (10) into account, it is likewise easy to obtain

\[ w_a=\frac{p_1v_1}{k-1} \left[ 1-\left(\frac{v_1}{v_2}\right)^{k_1-1} \right], \tag{19} \]

\[ w_a=\frac{p_1v_1}{k-1} \left[ 1-\left(\frac{p_2}{p_1}\right)^{\frac{k_1-1}{k_1}} \right], \tag{20} \]

where \(k_1\) is the ratio of the mean heat capacities in the temperature interval \(T_1(t_1)—T_{2k}(t_{2k})\), see (14) and (14a).

Comparison of the values of the final temperatures \(T_2\), obtained by the indicated method, with the true values given in the notes to the tables, shows that the work of the adiabatic process can be calculated with entirely satisfactory accuracy.

The calculation for given volumes \(v_1\) and \(v_2\) has no essential differences. For the computation we have:

\[ k_1-1=\frac{AR\Delta t}{\Delta i-AR\Delta t}, \tag{21} \]

where

\[ \Delta t=t_{2k}-t_1, \]

\[ \Delta i=i_{2k}-i_1. \]

Similarly,

\[ k_1=\frac{\Delta i}{\Delta i-AR\Delta t}. \tag{22} \]

2. Polytropic Processes

Calling polytropic those processes which obey the relation

\[ pv^n=\mathrm{const}, \tag{23} \]

where \(n\) is any constant number, we obtain the generally known relations between the parameters and the expressions for the work of a polytropic process, also valid for variable heat capacities \(C_p\) and \(C_v\). Taking into account that

\[ \Delta u=c_v(T_2-T_1), \tag{17} \]

it is easy to obtain for the heat of a polytropic process

\[ q=c_v\,\frac{n-k}{n-1}\,(T_2-T_1), \tag{24} \]

where \(c_v\) and \(k\) are the mean heat capacity and the ratio of the mean heat capacities \(\dfrac{c_p}{c_v}\) in the temperature interval \(T_1—T_2\). The quantity

\[ c_n=c_v\,\frac{n-k}{n-1}, \tag{25} \]

should be regarded as the mean heat capacity of the polytropic process in the temperature interval of the process. Assigning to the polytropic exponent \(n\) the values \(0,1\), and \(\infty\), we obtain the corresponding relations between the parameters, as well as expressions for work and heat (through the mean heat capacities) for isobaric, isothermal, and isochoric processes. Thus, a polytropic process, even with variable heat capacities, may be regarded as a generalization of certain basic processes.

The only exception is the adiabatic process. For \(n=k\), from (25) one obtains \(c_n=0\), i.e., a polytropic process with total heat exchange equal to zero. But this does not exclude the possibility of heat supply and removal in equal amounts in separate parts of this process, which is how it differs from an adiabatic one.

For calculations using an enthalpy table, the following relations are most convenient:

\[ Aw_n=\frac{AR}{n-1}\,(t_1-t_2), \tag{26} \]

and

\[ q=i_2-i_1-\frac{n}{n-1}\,AR(t_2-t_1). \tag{27} \]

In particular, per \(1\ \text{nm}^3\),

\[ Aw_n=\frac{0.0886}{n-1}\,(t_1-t_2), \tag{26a} \]

\[ q=i_2-i_1+0.0886\,\frac{n}{n-1}\,(t_2-t_1). \tag{27a} \]

Table 4

Enthalpy of ideal gases per nm³ (0°C, 760 mm Hg), in kcal/nm³