GENERAL METHOD FOR CALCULATING INTERNALLY IRREVERSIBLE PROCESSES

G. I. Fuks

1

A reversible expansion process from the initial state \((p_1, T_1)\) is represented in the \(T\)-\(S\) diagram by the line \(1—2_0\) (Fig. 1). The supplied heat \(q\), the external work \(l_0\), and the change in the parameters of state are related by

\[ q=i_{20}-i_1+l_0. \tag{1} \]

The presence of internal irreversibility (friction, nonequilibrium of the working body, and so on), with the same heat supply \(q\), will give a work output \(l_{\mathrm{n}}<l_0\). In this case

\[ q=i_{2\mathrm{n}}-i_1+l_{\mathrm{n}}, \tag{2} \]

where

\[ i_{2\mathrm{n}}>i_{20} \]

is the enthalpy of the working body at the end of the irreversible process.

Term-by-term subtraction of (2) from (1) gives the loss of work due to irreversibility

\[ \Delta' l_{\mathrm{n}}=l_0-l_{\mathrm{n}}=i_{2\mathrm{n}}-i_{20}. \tag{3} \]

Fig. 1.

If, on the basis of experiment or by calculation, the value \(\Delta' l_{\mathrm{n}}\) has been found, then from this relation one finds the work of the real process \(l_{\mathrm{n}}\) and the values of the state parameters at the end of the process. Let us now make the assumption that the irreversible process proceeds so slowly that it may be regarded as an equilibrium process. For this process the heat \(q'\) supplied to the working body and the external work \(l'\) will be related by

\[ q'=i_{2\mathrm{n}}-i_1+l'. \tag{4} \]

Term-by-term subtraction of (2) from (4) gives the amount of heat released by irreversibility,

\[ \Delta q_{\mathrm{n}}=q'-q=l'-l_{\mathrm{n}}. \tag{5} \]

If the quantity \(\Delta q_{\mathrm{n}}\) is known from experience or has been calculated, then from this relation the values of \(q\) and \(l_{\mathrm{n}}\) are computed. The quantities \(\Delta' l_{\mathrm{n}}\) and \(\Delta q_{\mathrm{n}}\) are not equal to each other. As can be seen from Fig. 1, the quantity \(\Delta q_{\mathrm{n}}\) is measured by the sum of the areas \(\alpha \sim\) area \(1—2_{\mathrm{n}}—2_0—1\) and \(\beta = \sim\) area \(2_0—2_{\mathrm{n}}—b—d—2_0\). The quantity \(\Delta' l_{\mathrm{n}}\), in accordance with (3), is the area \(\beta\).

If the dependence of the work loss due to irreversibility on the pressure of the working body is given, \(\Delta' l_{\mathrm{ir}} = f(p)\), then for any intermediate pressure \(p'\) one can find the value \(\Delta' l_{\mathrm{ir}}\). Then for this pressure \(i'_{2\mathrm{ir}} = i'_{20} + \Delta' l_{\mathrm{ir}}\). This makes it possible to find the value \(i'_{2\mathrm{ir}}\). Thus one can find a number of points of the line \(1 - 2_{\mathrm{ir}}\), and then the values \(\alpha\) and \(\beta\). In this way the heat release due to the irreversibility of the real process is determined:

\[ \Delta q_{\mathrm{ir}} = \alpha + \beta . \]

Instead of absolute values, relative quantities may be used for the calculation:

\[ \frac{\Delta' l_{\mathrm{ir}}}{l_0},\quad 1-\frac{\Delta' l_{\mathrm{ir}}}{l_0},\quad \frac{\Delta q_{\mathrm{ir}}}{q'},\quad 1-\frac{\Delta q_{\mathrm{ir}}}{q'} . \]

The calculations are especially simple to carry out on an \(i-s\) diagram, in which the values of \(\beta\) will be measured by segments along the ordinate axis.

2

The reversible process of compression of the working body from pressure \(p_1\) to \(p_2 > p_1\) is shown in Fig. 2 in the \(T-s\) diagram. The heat \(q\) supplied to the working body and the work \(l_0\) are related by relation (1). But in this case work is expended, \(l_0 < 0\). It is more convenient to use the relation

\[ (-l_0)=i_{20}-i_1-q, \tag{6} \]

where \((-l_0)>0\), i.e., the expended work is given as a positive number. In the presence of internal irreversibility, the expended work \((-l_{\mathrm{ir}}) > (-l_0)\). In this case

\[ (-l_{\mathrm{ir}})=i_{2\mathrm{ir}}-i_1-q . \tag{7} \]

Term-by-term subtraction of (6) from (7) gives the additional value of the work expended due to irreversibility:

\[ \Delta(l_{\mathrm{ir}})=(-l_{\mathrm{ir}})-(-l_0)=i_{2\mathrm{ir}}-i_{20}. \tag{8} \]

Fig. 2.

If the real irreversible process proceeds sufficiently slowly, it may be regarded as equilibrium. The supplied heat \(q'\) will be measured by the area \(1-2_{\mathrm{ir}}-b-a-1\). In the reversible process \(1-2_0\), the supplied heat \(q\) is measured by the area \(1-2_0-c-d-1\). Their difference, measured by the area of the triangle \(d-2_0-2_{\mathrm{ir}}=\alpha\), gives the excess heat supplied in the irreversible compression process:

\[ \Delta q_{\mathrm{ir}}=q'-q . \tag{9} \]

For the process \(1-2_{\mathrm{ir}}\),

\[ (-l')=i_{2\mathrm{ir}}-i_1-q' . \tag{10} \]

Term-by-term subtraction of (10) from (7), taking (9) into account, gives

\[ (-l_{\mathrm{ir}})-(-l')=\Delta q_{\mathrm{ir}} . \tag{11} \]

If, in addition to the initial process \(1-2_0\), the work loss due to irreversibility \(\Delta'(l_{\mathrm{ir}})\) is given, then from (8) the quantities \((-l_{\mathrm{ir}})\) and \(i_{2\mathrm{ir}}\) are found. If

If the irreversible compression process is considered equilibrium, then this is equivalent to specifying the line \(1—2_{\mathrm{n}}\). Taking, in addition, as a known quantity the heat release from irreversibility \(\Delta q_{\mathrm{n}}\), from (9) the heat supplied from external sources is determined as

\[ q=q' - \Delta q_{\mathrm{n}}, \tag{12} \]

and from (11) the work expenditure in the irreversible process is

\[ (-l_{\mathrm{n}})=(-l')+\Delta q_{\mathrm{n}}. \tag{13} \]

Comparison of expressions (3) and (8) indicates that the influence of irreversibility in irreversible expansion and compression processes is different. In expansion processes, because of irreversibility, less work is obtained than in the reversible process. In compression, the irreversibility of the real process leads to an increase in the work expended. A limiting case of an irreversible expansion process is possible in which its work will be equal to zero, which corresponds to the condition

\[ i_{2\mathrm{n}}-i_{2o}=l_o. \]

In irreversible compression processes, an increase in irreversibility may proceed without bound, accompanied by a corresponding increase in the work expenditure. This is explained by the fact that in all irreversible processes there is a one-way conversion of work into heat.

In conclusion, let us note that the calculation methods indicated above can also be used to calculate the work of irreversible processes \(W_{\mathrm{n}}\) (without accounting for the work of filling and displacement). For this, in all the calculation relations written above, the enthalpy quantities \(i\) should be replaced by the corresponding values of the internal energies of the working body \(u\), and, in calculations by diagrams, lines of constant volumes should be used.

3

In [1] a method is given for calculating irreversible processes of an ideal gas, considering the process to be equilibrium. For the calculation, the ratio of the work obtained during gas expansion to the sum of this same quantity with the work expended in overcoming friction forces is used. At the same time, a second coefficient is introduced, equal to the ratio of the heat supplied to the sum of this heat with the heat released by friction. With the help of these two indicators, the equilibrium real process of a gas with friction is reduced to a certain conditional polytropic process. This device cannot be considered general. Indeed, subsequently, when considering irreversible processes of vapors, this device is not used.

Some particular cases of calculating irreversible processes are available in [2]. In both of the indicated works, the consideration of an irreversible isochoric process raises doubts. Internal irreversibility can occur only with mutual conversion of heat and work. In an isochoric process, heat is expended only on changing the internal energy of the working body. This is especially clearly seen from [2], where the process of expansion into a vacuum is considered as an irreversible isochoric process. Consideration of irreversible polytropic processes of an ideal gas is also given in [3] and [4].

REFERENCES

1. A. I. Andryushchenko. Fundamentals of the Technical Thermodynamics of Real Gases. “Vysshaya shkola,” 1967.

2. M. P. Vukalovich, I. I. Novikov. Technical Thermodynamics, “Energiya,” 1968.

3. V. Traupel. Thermal Turbomachines, Vol. 1, GEI, 1961.

4. A. V. Lapshin. Calculations of irreversible polytropic processes. Izv. vuzov, Energetika, 1966, No. 4.