On the Calculation of the Adiabatic Process of an Ideal Gas with Variable Heat Capacities
Abstract Generated abstract
The paper addresses tabular calculation of adiabatic processes for ideal gases with temperature dependent heat capacities when the final gas volume, rather than final pressure or temperature, is specified. It reformulates the entropy relation by introducing a tabulated auxiliary quantity based on the standard entropy function and a logarithmic temperature term, allowing entropy to be expressed as this quantity plus the gas constant times the logarithm of volume. The resulting relations provide a practical procedure for determining the final state from initial pressure and temperature and a specified final volume, and the same construction is noted to apply to gas mixtures.
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On the Calculation of the Adiabatic Process of an Ideal Gas with Variable Heat Capacities
G. I. Fuks
A method is proposed for the tabular calculation of an adiabatic process when the final volume of the gas is specified.
As shown in [1], the calculation of the adiabatic process of an ideal gas with variable \(C_p\) and \(C_v\) is conveniently carried out using tables of thermodynamic properties, for example [2]. The relation used is
\[ S = S_0 - R \ln P, \tag{1} \]
where
\[ S_0 = \int_{273.15}^{T} C_p \frac{dT}{T}. \]
For calculating the adiabatic process in this case, in addition to knowing the initial parameters \(P_1\) and \(t_1\), it is necessary to specify at the end of the process the pressure \(P_2\) (or the temperature \(t_2\)). But if at the end of the process the volume \(V_2\) is specified, then calculation by (1) is difficult.
Let us write relation (1) in expanded form:
\[ S = \int_{273.15}^{T} C_p \frac{dT}{T} - R \ln \frac{P}{P_0}, \]
where \(P_0\) and \(t = 0\) are the pressure and temperature at which the entropy value is taken to be zero. The entropy value can also be calculated from the relation
\[ S = \int_{273.15}^{T} C_v \frac{dT}{T} + R \ln \frac{v}{v_0}, \]
where \(v_0\) is the gas volume at \(t = 0\) and \(P = P_0\).
Since \(C_v = C_p - R\), then
\[ S = \int_{273.15}^{T} C_p \frac{dT}{T} - R \ln \frac{T}{T_0} - R \ln \frac{v}{v_0} \]
or
\[ S = S_0 - R \ln T + R \ln \frac{T_0}{v_0} + R \ln v. \]
From the equation of state \(\dfrac{T_0}{V_0}=\dfrac{P_0}{R}\), for example, for a kilomole,
\[ R\ln \frac{P_0}{R}=19.599\ \text{kJ}/(\text{kmol}\cdot\text{deg}). \]
Thus,
\[ S=\psi+R\ln v, \tag{2} \]
where
\[ \psi=S_0+19.599-R\ln T. \tag{3} \]
The values of \(\psi\) at different temperatures can be entered in a table. Then the value of the entropy can be obtained by specifying \(t\) and \(v\). The calculation of the adiabatic process is carried out according to the relations
\[ \psi_1+R\ln V_1=\psi_2+R\ln V_2 \]
or
\[ S_{01}-R\ln P_1=\psi_2+R\ln V_2. \]
For gas mixtures, the quantity \(\psi\) is calculated in the same way as the quantity \(S_0\).
REFERENCES
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G. I. Fuks. Calculation of the adiabatic process of an ideal gas from tables. Izvestiya vuzov. Energetika, No. 6, Minsk, 1962.
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M. P. Vukalovich, V. A. Kirillin, S. A. Remizov, V. S. Siletskii, V. N. Timofeev. Thermodynamic Properties of Gases. Mashgiz, Moscow, 1963.