ON THE MATHEMATICAL FORMULATION OF THE FIRST LAW OF ENGINEERING THERMODYNAMICS

G. I. Fuks

The first law of thermodynamics is an application of the universal law of conservation and transformation of energy to the phenomena of interconversion of thermal and other forms of energy. Its general mathematical formulation in engineering thermodynamics is the expression:

\[ Q = U_2 - U_1 + AW . \tag{1} \]

For a reversible process of this regularity, with heat \(Q\) supplied to the working body, work \(W_o\) is obtained and the internal energy of the working body increases by the amount \(U_{2o} - U_1\):

\[ Q = U_{2o} - U_1 + AW_o . \tag{2} \]

For an elementary reversible process:

\[ dQ = dU_o + A dW_o . \tag{3} \]

If an irreversible process is performed (whether equilibrium or nonequilibrium), then, with the same heat supply \(Q\) and the same general character of the process (for example, with the same change in the volume of the working body), work \(W_n < W_o\) will be obtained. Part of the work is “lost,” i.e., passes into heat. The work of the nonequilibrium process will be

\[ W_n = W_o - W_r , \tag{4} \]

(where \(W_r\) is the work converted into heat as a result of the irreversibility of the process, for example, friction, etc.). From (2) and (4) we have:

\[ Q = U_{2o} - U_1 + AW_n + AW_r . \tag{5} \]

The work converted into heat as a result of the irreversibility of the process causes a corresponding increase in the internal energy of the working body. The value of the internal energy at the end of the irreversible process will be:

\[ U_{2n} = U_{2o} + AW_r . \tag{6} \]

Therefore

\[ Q = U_{2n} - U_1 + AW_n . \tag{7} \]

For an elementary irreversible equilibrium process:

\[ dQ = dU_n + A dW_n . \tag{8} \]

The mathematical expression of the first law for a moving working body with a reversible change of its state has the form:

\[ q = i_{2o} - i_1 + A \frac{c_{2o}^{2} - c_1^{2}}{2g} + A l_T , \tag{9} \]

where \(i_{20}\) is the enthalpy of the working body at the end of the process,
\(c_{20}\) is the final velocity of the working body.

For an irreversible process of change of state of the working body this relation must be written as:

\[ q=i_{20}-i_1+A\frac{c_{20}^{2}-c_1^2}{2g}+Al_t, \tag{10} \]

where \(i_{20}\) and \(c_{20}\) are, respectively, the enthalpy and velocity of the working body at the end of the actual process.

In a number of books ([1], [2], [3] and [4]) there is given a not quite correct or, in any case, imprecise formulation of the first law for nonequilibrium (or irreversible equilibrium) processes. Thus, M. P. Vukalovich and I. I. Novikov, having written on page 81 of [1] equation (10) in the general form

\[ E_1+p_1V_1=E_2+p_2V_2+L'-Q \tag{2—67} \]

and, in particular, for \(1\ \mathrm{kg}\):

\[ i_1+\frac{W^2}{2g}=i_2+\frac{W_2^2}{2g}+l'-q, \tag{2—68} \]

then add: “The heat of friction \(Q_{tr}\) is not included in the quantity \(Q\). Indeed, in the presence of friction, work \(L_{tr}\) must be expended to overcome the forces of friction; since the work against frictional forces is completely converted into heat, within the given quantity of flowing gas an amount of heat \(Q_{tr}\), equivalent to \(L_{tr}\), will be released. To take into account the influence of friction on the flow of gas (or liquid), in the right-hand side of equation (2—68) one must substitute the values \(L_{tr}\) and \(Q_{tr}\), which are component parts of the external work \(L\) and of the total amount of heat \(Q\). Owing to the equivalence of the work of friction \(L_{tr}\) and the heat of friction \(Q_{tr}\), both these quantities mutually cancel and thus drop out of equation (2—67).”

These arguments and, in particular, the last conclusion cannot be recognized as correct. The heat equivalent to the work against frictional forces gives a corresponding increase in the internal energy (and, consequently, also in the enthalpy) of the working body, i.e., it does not drop out of the equation expressing the first law of thermodynamics.

In the book by V. S. Zhukovsky [2] this question is interpreted analogously. First the phenomenon is considered without taking friction into account. The energy-conservation equation for a flow is written as

\[ dq=di+Ad\frac{W^2}{2g}+Adl_t \tag{VII, 2} \]

and the generalized Bernoulli equation:

\[ v\,dp+d\frac{W^2}{2g}+dl_t=0. \tag{VII, 3} \]

Term-by-term subtraction of them gives:

\[ dq=du+Ap\,dv=di-Av\,dp. \tag{VII, 4} \]

Further, on p. 353 we read: “Since the internal work of friction was transferred into the category of external actions on the working body, in the energy-conservation equation it is necessary, to the amount of heat supplied from outside \(dq\)

one must also add the heat of friction \(dq_{TP}\), as though the latter were transmitted to the element through its surface. On the basis of what has been said,

\[ dq+dq_{TP}=di+Ad\frac{W^2}{2g}+Adl_T+Adl_{TP}. \]

“... The quantities \(dq_{TP}\) and \(Adl_{TP}\), entering into the last formula, are mutually equal and are subject to cancellation. As a result we arrive at the energy-conservation equation (VII, 2), which was established earlier as applied to a nonviscous flow.”

Next the Bernoulli equation is written:

\[ vdp+d\frac{W^2}{2g}+dl_T+dl_{TP}=0 \tag{VII, 3a} \]

and, subtracting it from (VII, 2), we obtain:

\[ dq=di-Aidp-Adl_{TP}=du+Apdv-Adl_{TP}. \tag{VII, 4a} \]

“The independence of formula (VII, 2) from whether friction in the flow manifests itself or does not manifest itself is explained as follows.

Formula (VII, 2) gives the total effect of the energy interaction of an element, separated from the flow, with the surrounding medium. It is obvious that the increment of the energy of the system for a given amount of heat received from outside and for a given magnitude of external work is determined in a completely unambiguous way. In this connection it is entirely immaterial what particular internal processes occurred in the element on the path between its initial and final positions. Hence it follows that the change in the energy of the element

\[ \Delta i + A\Delta \frac{W^2}{2g} \]

does not in any measure depend on whether friction does or does not manifest itself during the flow of a liquid or gas in a pipe. However, in the most direct way, friction affects the distribution of the increment of energy among its individual components—enthalpy and kinetic energy.”

The author is, of course, right in asserting that, for a given supply of heat, the sum

\[ \Delta i + A\Delta \frac{W^2}{2g}\left(+Al_T\right) \]

remains a constant quantity, regardless of whether friction takes place or not. This follows directly from the law of conservation and transformation of energy. But the practical meaning of writing the mathematical expression of the first law consists in explicitly bringing out the connection between the heat supplied from outside and the amount of useful work obtained. In irreversible processes with friction, for which equation (VII, 3a) has been written, the change of the state parameters \(v\) and \(p\) and of the velocity \(w\) will be different from that in reversible processes, for which equation (VII, 2) has been written. Therefore subtracting (VII, 3a) from (VII, 2) is an unlawful operation.

The formulas (VII, 2), (VII, 3), and (VII, 4), referring to reversible processes, should be written as follows:

\[ dq=di_o+Ad\frac{w_o^2}{2g}+Adl_T \tag{VII, 2)'} \]

\[ v_o\,dp_o+d\frac{w_o^2}{2g}+Adl_T=0 \tag{VII, 3)'} \]

\[ dq=du_o+Ap_o\,dv_o=di_o-Av_o\,dp_o, \tag{VII, 4} \]

where the index “\(o\)” emphasizes that all quantities in these equations pertain to a reversible process.

For an equilibrium irreversible process the fundamental relation of the first law of thermodynamics has the form:

\[ dq=di_n+A\,d\frac{w_n^2}{2g}+A\,dl_T . \tag{a} \]

Bernoulli’s equation should be written as follows:

\[ v_n\,dp_n+d\frac{w_n^2}{2g}+dl_T+dl_{TR}=0. \tag{VII,3a)' \]

Subtracting (VII, 3a) from (a), we obtain:

\[ dq=di_n-Av_n\,dp_n-A\,dl_{TR}=du_n+Ap_n\,dv_n-A\,dl_{TR}. \tag{VII,4a} \]

The index “\(n\)” in these relations emphasizes that in an irreversible process the change in the state parameters of the working body follows a different law than in reversible processes. Otherwise the internal energy of the working body will change, and the work of displacement will also be different.

N. I. Belokon’ [4], considering equilibrium irreversible processes, introduces the concept of thermodynamic (reduced) heat exchange in the form of the sum

\[ \delta Q=\delta Q^*+\delta Q^{**}; \]

where \(\delta Q^*\) and \(\delta Q^{**}\) are, respectively, the heat supplied from outside and the heat of internal heat exchange (from friction), and the concept of thermodynamic work

\[ \delta L=\delta L^*+\delta L^{**}, \]

where \(\delta L^*\) and \(\delta L^{**}\) are, respectively, the effective external work and the irreversible loss of work.

The fundamental equation is written by him in the form (p. 64):

\[ \delta Q=du+A\delta L. \tag{ж} \]

If one takes into account that the heat of friction is equivalent to the work expended in overcoming frictional forces,

\[ \delta Q^{**}=A\delta L^{**}, \]

then the reduced relation becomes

\[ \delta Q^*=du+A\,dL^*; \]

i.e., the same result is obtained as for a reversible process. In reality, however, for one and the same supply of heat from outside, an irreversible process will give a smaller amount of external work than a reversible one.

The expediency of introducing the concepts of thermodynamic heat exchange and thermodynamic work raises serious doubts. The quantities of external heat and external work are of practical significance. The introduction of new concepts complicates and needlessly confuses the accepted terminology. Thus, for example, an adiabatic process in N. I. Belokon’s understanding is obtained in the case when the thermodynamic heat exchange \(\delta Q=0\). In the usual understanding of this term (cf. [5], term 84), an adiabatic process occurs in the absence of a supply of external heat. A throttling process, according to N. I. Belokon’, will be not an adiabatic process, but an “externally adiabatic” process. But if the throttling process is accompanied by removal of heat—

then it can become adiabatic. It is not clear what benefit such a change in generally accepted terminology can bring.

The loss of work in a nonequilibrium (irreversible) process, as compared with a reversible one, is equal to the work expended in overcoming frictional forces. By this same amount, in irreversible processes the final value of the internal energy of the working body is increased. If the irreversible and reversible processes are compared with one another for the same heat supply and the same change in the volume of the working body, then this is evident from relations (4), (6), and (7).

In a moving working body, the heat equivalent to the work spent in overcoming frictional forces also causes a corresponding increase in the internal energy of the working body and a decrease in the external work. At the same time, in an irreversible process the state parameters of the working body change according to a different law than in a reversible process. This causes a change in the magnitude of the work of pushing the gas through. We are usually interested in the loss of work in the form of kinetic energy and external work \((l_T)\).

Let us compare the results of reversible and irreversible processes in a moving working body for one and the same heat supply, identical initial states, and identical final pressures. Comparing, under these conditions, \((a)\) and \((\mathrm{VII}, 2)'\), we obtain:

\[ \frac{w_{20}^{2}}{2g}-\frac{w_{2H}^{2}}{2g}=i_{2H}-i_{20}. \]

But

\[ i_{2H}=u_{2H}+A p_2 v_{2H} \]

\[ i_{20}=u_{20}+A p_2 v_{20}. \]

Further,

\[ u_{2H}=u_{20}+A l_{TP}. \tag{6'} \]

Therefore:

\[ \frac{w_{20}^{2}}{2g}-\frac{w_{2n}^{2}}{2g} = A l_{TP}+A p_2 (v_{2H}-v_{20}). \tag{11} \]

Thus, under the stated conditions, the loss of kinetic energy in irreversible processes is greater than the expenditure of work in overcoming frictional forces by the amount by which the work of pushing the gas through is increased in the irreversible process.

A. Litvin [3], considering the phenomenon of gas outflow with resistances, comes to the conclusion (p. 233) that the kinetic energy of actual outflow may be obtained as the kinetic energy in ideal outflow minus the heat equivalent of the work of friction, plus the energy of outflow that has appeared as a result of supplying to the working body the heat of friction. Thus it follows that the loss of kinetic energy in outflow with resistances is less than the work of the frictional forces. This incorrect result was obtained because, without any justification, it was assumed that the heat of friction is expressed by the area under the conditional curve of the actual process of gas expansion in the \(S\)–\(T\) diagram. At the same time it is pointed out that the form of this expansion curve plays no role, since the corresponding area cancels out in the final summation. Meanwhile it is obvious that, for a given heat supply and given initial and final states of the working body, the expenditure of work in overcoming frictional forces is a definite quantity.

Conclusions

1. In irreversible processes, the loss due to irreversibility causes a corresponding increase in the internal energy of the working body. The work of expansion over the same interval of volumes as in the reversible process is reduced by the same amount.

2. In a moving working body, the presence of resistances leads to an increase in the work of pushing the gas through. Therefore, the loss of kinetic energy is greater than the work expended in overcoming frictional forces.

References

1. Bukalovich M. P., Novikov I. I. — Engineering Thermodynamics. Gosenergoizdat, 1952.

2. Zhukovsky V. S. — Engineering Thermodynamics. State Publishing House for Technical-Theoretical Literature, 1952.

3. Latvin A. M. — Engineering Thermodynamics. Gosenergoizdat, 1947.

4. Belokon N. I. — Thermodynamics. Gosenergoizdat, 1954.

5. Committee on Technical Terminology of the Academy of Sciences of the USSR. Terminology of Thermodynamics. Academy of Sciences of the USSR, 1952.