Mean Thermodynamic Temperature

G. I. Fuks

Professor, Doctor of Technical Sciences

Calculation of the Mean Thermodynamic Temperature

The mean thermodynamic temperature¹ of a process is the temperature of an isothermal process that gives the same thermal effect as the process under consideration. The introduction of this concept simplifies the consideration of questions of technical thermodynamics, and in some cases gives the proofs an obviousness.

Let \(AB\) be the representation of the process in \(TS\) coordinates, and let \(\Delta S\) be the increase in entropy of the working body (Fig. 1). The heat imparted to the working body is represented by the area \(ABba\). To find the m.-t. temperature of the process, a rectangle equal in area to this area is constructed with base \(\Delta S\). The height of this rectangle will be \(T_c\), the m.-t. temperature of the process. If the heat of the process \(AB\) is \(q\), then

\[ T_c=\frac{q}{\Delta S}. \tag{1} \]

Let the heat capacity in the process \(AB\) be \(c\). Then

\[ q=\int_A^B c\,dT \tag{2} \]

and

\[ \Delta S=\int_A^B c\,\frac{dT}{T}. \tag{3} \]

Consequently,

\[ T_c= \frac{\displaystyle\int_A^B c\,dT} {\displaystyle\int_A^B c\,\frac{dT}{T}}. \tag{4} \]

Fig. 1

For constant heat capacity of the process \(AB\), expression (4) may be written as follows:

\[ T_c=\frac{T_2-T_1}{\ln \dfrac{T_2}{T_1}}. \tag{5} \]

It is clear that relation (5) can be applied to any process under the condition \(C=\mathrm{Const}\), in particular to the isochoric, isobaric, and polytropic processes of a gas (with \(C_v=\mathrm{Const}\)).

With an unchanged heat capacity of the process, its m.t.t. is calculated as the logarithmic mean of the final and initial absolute temperatures of the process.

If the transfer of heat takes place by several processes, then one may pose the question of the m.t.t. of heat transfer to the working body. Let (Fig. 2) heat be transferred in the processes \(AB\) and \(BC\) in the amounts \(q_1\) and \(q_2\), with the m.t.t. of each process \(T_c'\) and \(T_c''\). The m.t.t. of the processes \(ABC\) will be

\[ T_c=\frac{q_1+q_2}{\Delta S_1+\Delta S_2}. \tag{6} \]

Fig. 2

According to the definition,

\[ T_c'=\frac{q_1}{\Delta S_1}, \tag{1'} \]

\[ T_c''=\frac{q_2}{\Delta S_2}. \tag{1''} \]

Consequently,

\[ T_c=\frac{q_1+q_2}{\dfrac{q_1}{T_c'}+\dfrac{q_2}{T_c''}}, \tag{7} \]

whence

\[ \frac{T_c}{T_c'}=\frac{1+m}{1+m\dfrac{T_c'}{T_c''}}, \tag{9} \]

where

\[ m=\frac{q_2}{q_1}. \tag{9} \]

From the last relation it follows that the m.t.t. of heat transfer to the working body in two processes is a mean between the m.t.t. of the first and the second process. In particular, if a second process of heat transfer is added to a given process, proceeding at a higher m.t.t. than the first, then the m.t.t. of heat transfer to the working body always increases. These conclusions will be used below.

It is also not difficult to write a relation analogous to (8) for the transfer of heat to the working body in 3 and more processes.

Efficiency

Let an arbitrary cycle of reversible processes be given (Fig. 3). Let us plot on the diagram the m.t.t. of heat transfer to the working body \(T_c'\) and of heat removal \(T_c''\). Then the supplied and removed heat will be, respectively,

\[ q_1=T_c'\Delta S \tag{10} \]

and

\[ q_2=T_c''\Delta S. \tag{11} \]

Consequently, the efficiency is

\[ \eta_t=\frac{T'_c-T''_c}{T'_c}=1-\frac{T''_c}{T'_c}. \tag{12} \]

On the basis of this relation, a visually clear method is obtained for comparing cycles by efficiency. Thus, for the Otto and Diesel cycles at identical maximum pressures, we have (see Fig. 4) the same mean thermodynamic temperature of heat rejection from the working body and different mean thermodynamic temperatures of heat addition, \(T'_{co}\) and \(T'_{cd}\). Since in the present case

\[ T'_{cd}>T'_{co}, \]

then, obviously,

\[ \eta_{td}>\eta_{to}. \]

By this method one may compare cycles by efficiency in more complicated cases, when the usual method gives no solution. Thus, the question of comparing by efficiency the cycles 123 and 141, shown in Fig. 5, is resolved as follows. The mean thermodynamic temperature of heat addition of both cycles is, obviously, the same. According to (5), the mean thermodynamic temperature of the processes 31 and 41 is also the same \((C_v=\mathrm{Const})\). Consequently, the efficiencies of the compared cycles are identical.

Fig. 3

The introduction of superheat increases the efficiency of the steam cycle, since with an unchanged mean thermodynamic temperature of heat rejection from the working body we have, on the basis of (8), a higher mean thermodynamic temperature of heat addition to the steam.

The introduction of intermediate superheating of steam may be regarded as the addition to the working body of an additional quantity of heat at a mean thermodynamic temperature \(T''_c\), distinct from the mean thermodynamic temperature of the main heating process, at one and the same mean thermodynamic temperature of heat rejection from the working body.

Fig. 4

Fig. 5

The condition for increasing the efficiency with intermediate superheating, according to (8), will be the existence of the inequality

\[ 1+m>1+m\frac{T'_c}{T''_c}, \]

whence

\[ T''_c>T'_c . \]

Consequently, intermediate superheating of steam increases the efficiency of the cycle in the case when the mean thermodynamic temperature of the intermediate superheating is higher than the mean thermodynamic temperature of heat supply in the main cycle.

But, on the other hand, an increase in efficiency achieved by introducing intermediate superheating of steam does not always mean an increase in the economy of the installation, i.e., a reduction in fuel consumption. Thus, in the case of steam intermediate superheating, since the mean thermodynamic temperature of the heating steam is always higher than the mean thermodynamic temperature of the heated steam, an increase in the efficiency of the cycle cannot yield a saving in fuel.

In some cases relation (12) makes it possible to write at once an expression for \(\eta_t\) of the cycle without intermediate calculations. In Figs. 6a and 6b there is plotted the cycle of an ideal gas turbine with combustion at \(p=\mathrm{Const}\). (2—3 and 5—1 are processes in the regenerator).

Fig. 6a

Fig. 6b

Heat removal from the working body occurs in process 1—2 at the constant temperature \(T_1\). Heat supply occurs at \(p=\mathrm{Const}\) (process 3—4). With constant heat capacity, the mean thermodynamic temperature of heat supply by (5) is

\[ T=\frac{T_4-T_3}{\ln \frac{T_4}{T_3}} . \tag{5'} \]

The thermal efficiency of the cycle by (12) is

\[ \eta_t=1-\frac{T_1}{\dfrac{T_4-T_3}{\ln \dfrac{T_4}{T_3}}}. \tag{13} \]

If there are irreversible processes in the cycle (for example, the cycle of the steam-power plant in Fig. 7, where the expansion 12 is plotted on the diagram conditionally), then, obviously,

\[ q_1=T'_c \Delta S_1, \tag{14} \]

\[ q_2=T''_c \Delta S_2. \tag{15} \]

The efficiency of this cycle will be

\[ \eta=1-\frac{T''_c \Delta S_2}{T'_c \Delta S_1}. \tag{16} \]

This expression is convenient to use in the course of certain proofs. For calculations it may be given another form.

The irreversibility of the expansion process leads to the fact that the change in entropy in the process of heat removal from the working body, \(\Delta S_2\), is greater than its change in the process of heat addition, \(\Delta S_1\). Let us call

\[ \delta S=\Delta S_2-\Delta S_1. \tag{17} \]

Then from (16)

\[ \eta=1-\frac{T''_c}{T'_c}-\frac{T''_c\delta S}{T'_c\Delta S_1}. \tag{18} \]

But

\[ T''_c\delta S=\delta q, \tag{19} \]

where \(\delta q\) is the additional rejection of heat to the condenser due to irreversibility in the operation of the machine. From (14), (16), and (19) we have

\[ \eta=1-\frac{T''_c}{T'_c}-\frac{\delta q}{q_1}. \tag{20} \]

Fig. 7

The difference between the real cycle and the theoretical one is evaluated by the relative efficiency of the cycle \(\eta_{io}\). Obviously,

\[ \eta_{io}=\frac{q_1-q'_2-\delta q}{q_1-q'_2} =1-\frac{\delta q}{q_1-q'_2}, \tag{21} \]

where \(q'_2\) is the removal of heat from the working body in the theoretical cycle.

For the theoretical cycle

\[ \eta_t=\frac{q_1-q'_2}{q_1}=1-\frac{T''_c}{T'_c}. \tag{22} \]

From (20), (21), and (22)

\[ \eta=\eta_{io}\left(1-\frac{T''_c}{T'_c}\right). \tag{23} \]

If the quantity of substance participating in the cycle changes (for example, extraction of steam for heating feed water), but the plant delivers exclusively mechanical work, then the expression for the efficiency will be as follows.

Let heat be supplied in the boiler to \(g_1\) kg of working fluid in the amount \(q_1\) kcal/kg. The total amount of heat supplied to the working fluid will be

\[ Q_1 = g_1 q_1 = g_1 T'_c \Delta S_1, \tag{24} \]

where \(T'_c\) and \(\Delta S_1\) are the mean thermodynamic temperature and the increase in entropy of the working fluid per kg when heat is supplied to it.

Similarly, the heat removed is

\[ Q_2 = g_k q_2 = g_k T''_c \Delta S_2. \tag{25} \]

The efficiency of the cycle is

\[ \eta = 1 - \frac{g_k}{g_1}\frac{T''_c \Delta S_2}{T'_c \Delta S_1}. \tag{26} \]

This expression may be regarded as the general expression for the efficiency of a purely power installation, from which the particular formulas (12) and (16) are easily obtained. It will be used below to study the regenerative cycle of a steam-power installation.

A more general case will be that in which the quantity of working substance in the engine changes not only because of internal heat exchange in the installation, but also because of extraction of working substance for district-heating purposes. In this case the efficiency of the cycle is

\[ \eta = 1 - \frac{g_k}{g_1}\frac{T''_c \Delta S_2}{T'_c \Delta S_1} - \frac{\sum g T_c \Delta S}{g_1 T'_c \Delta S_1}. \tag{27} \]

Here \(g\), \(T_c\), and \(\Delta S\) are the amount of district-heating extraction, its mean thermodynamic temperature, and the change in its entropy per 1 kg. Relation (27), of course, gives the thermal efficiency of the power part, and not of the entire installation as a whole.

Fig. 8

From the external form of expression (27) it is impossible to determine the difference in the influence on the efficiency of extractions for internal and external (district-heating) heating. This difference is manifested in the fact that extraction for internal heating affects the magnitude of the mean thermodynamic temperature of heat supply \((T'_c)\), raising it. District-heating extractions have no direct influence on \(T'_c\).

Regenerative process of a steam-power installation

Let there be a regenerative cycle with one point of steam extraction (see Fig. 8) at point 10. Let \(g_1\) kg of steam enter the engine, per 1 kg of fuel, and let \(g_k\) kg of steam enter the condenser. According to (28), the efficiency of this cycle will be

\[ \eta_p = 1 - \frac{g_k}{g_1}\frac{T''_c \Delta S_2}{T'_c \Delta S_1}. \tag{26} \]

The quantities \(g_1\) and \(g_k\) are determined from the heat balance of the heater. For mixing heaters, the latter is written as follows:

\[ g_k \Delta q = (g_1 - g_k) T_p \Delta S_p, \tag{28} \]

whence

\[ g_k = g_1 \frac{T_p \Delta S_p}{\Delta q + T_p \Delta S_p}. \tag{29} \]

Here \(\Delta q\) is the amount of heat used to heat 1 kg of water in the heater. From (26) and (29) we have

\[ \eta_p = 1-\frac{T_c''\Delta S_2}{T_c'\Delta S_1}\frac{T_p\Delta S_p}{\Delta q+T_p\Delta S_p}. \tag{30} \]

The last fraction on the right-hand side of (30) is a proper fraction. Let us add to the numerator and denominator of this fraction the same number

\[ i_1-i_{01}. \]

Then the absolute value of this fraction will increase, i.e.

\[ \eta_p > 1-\frac{T_c''\Delta S_2}{T_c'\Delta S_1}\frac{T_p\Delta S_p+(i_1-i_{01})}{\Delta q+T_p\Delta S_p+(i_1-i_{01})}. \tag{31} \]

But \(T_p\Delta S_p+(i_1-i_{01})\) is equal to the area \(10'q1pn10'\), i.e. it represents the heat expended per 1 kg of steam in the boiler in the cycle with regeneration, or

\[ T_p\Delta S_p+(i_1-i_{01})=T_c'\Delta S_1. \tag{32} \]

Similarly, the quantity

\[ \Delta q+T_p\Delta S_p+(i_1-i_{01}), \]

measured by the area \(mk'01'q1pm\), gives the heat expenditure per kg of steam in the cycle without regeneration, i.e.

\[ \Delta q+T_p\Delta S_p+(i_1-i_{01})=T_R'\Delta S_R', \tag{33} \]

where the subscript \(R\) refers to the Rankine cycle. From (31), (32), and (33) we have

\[ \eta_p > 1-\frac{T_c''\Delta S_2}{T_R'\Delta S_R'}, \tag{34} \]

which, taking (16) into account, gives

\[ \eta_p>\eta_R. \tag{35} \]

Thus, extraction of steam for heating the feed water gives an increase in the efficiency of the installation. But the absolute magnitude of this increase depends on the extraction pressure (temperature). Thus, if extraction is carried out at boiler pressure, then

\[ \begin{aligned} T_p&=T_1',\\ \Delta S_p&=\Delta S_1',\\ \Delta q&=q-\text{area }mk'qrm. \end{aligned} \tag{a} \]

In this case (30) becomes

\[ \eta_p=1-\frac{T_c''\Delta S_2}{T_1'\Delta S_1}\frac{T_1'\Delta S_1'}{q+T_1'\Delta S_1'}. \tag{36} \]

But

\[ q+T_1'\Delta S_1'=T_R'\Delta S_R', \tag{37} \]

whence, taking (16) into account, we obtain

\[ \eta_p=\eta_R, \]

i.e. extraction of live steam for heating the feed water is useless.

In the same way, extraction of steam for heating the feed water at condenser pressure makes no sense. In this case

\[ \Delta q=0. \tag{b} \]

From the balance equation of the heater (29) it follows that

\[ g_k=g_1; \tag{c} \]

Further,

\[ T'_c = T'_R, \tag{d} \]

\[ \Delta S_1 = \Delta S'_R. \]

Consequently, from (26) and (16)

\[ \eta_p = \eta_R. \]

At all intermediate bleed pressures

\[ \eta_p > \eta_R. \]

The expression most convenient for direct use for the efficiency of a regenerative cycle with one bleed point can be obtained from (30). Taking into account that (see Fig. 8)

\[ \begin{aligned} T''_c \Delta S'' &= i_k - i'_k,\\ T_p \Delta S_p &= i_{01} - i'_{01},\\ T'_c \Delta S_1 &= i_1 - i'_{01},\\ \Delta q + T_p \Delta S_p &= i_{01} - i'_k, \end{aligned} \tag{38} \]

we have

\[ \eta_p = 1 - \frac{i_{01}-i'_{01}}{i_1-i'_{01}}\, \frac{i_k-i'_k}{i_{10}-i'_k}. \tag{39} \]

From this relation it is once again confirmed that bleeding steam at boiler and condenser pressures gives no benefit. Indeed, in the first case

\[ i_1 = i_{01}, \]

and in the second

\[ i_{01} = i_k \]

and

\[ i'_{01} = i'_k, \]

which in both cases gives

\[ \eta_p = \eta_R. \]

Fig. 9

Passing to a plant with several steam bleed points, let us first show that the addition of a second bleed point (after the first) leads to an increase in the efficiency of the plant. Let, in a plant with two bleed points, whose process is shown in Fig. 9, in the first part, per 1 kg of fuel, \(g_1\) kg of steam operate, in the second—\(g_2\), and in the third—\(g_k\) kg. The efficiency of this plant according to (26) will be

\[ \eta_p = 1 - \frac{g_k}{g_1}\, \frac{T''_c \Delta S_2}{T'_c \Delta S_1}. \tag{40} \]

Let us write the heat balances of the heaters:

\[ g_k \Delta q'' = (g_2 - g_k) T''_p \Delta S''_p . \tag{41} \]

\[ g_2 \Delta q' = (g_1 - g_2) T'_p \Delta S'_p . \tag{42} \]

Here

\[ \Delta q'' \sim \text{area } k' \, 02' \, n m k', \]

\[ \Delta q' \sim \text{area } n \, 02' \, 01' \, г n. \]

From (40), taking (41) and (42) into account, we have

\[ \eta_p=1-\frac{T_c''\Delta S_2}{T_c'\Delta S_1}\frac{T_p'\Delta S_p'}{\Delta q'+T_p'\Delta S_p'}\frac{T_p''\Delta S_p''}{\Delta q''+T_p''\Delta S_p''}. \tag{43} \]

Both of the last fractions on the right-hand side are, obviously, proper. If to the numerator and denominator of the last fraction one adds the same number

\[ i_{01}-i_{02}, \]

then the absolute value of this fraction will increase, i.e.,

\[ \eta_p>1-\frac{T_c''\Delta S_2}{T_c'\Delta S_1} \frac{T_p'\Delta S_p'}{\Delta q'+T_p'\Delta S_p'} \frac{T_p''\Delta S_p''+(i_{01}-i_{02})}{\Delta q''+T_p''\Delta S_p''+(i_{01}-i_{02})}. \tag{44} \]

But

\[ T_p''\Delta S_p''+(i_{01}-i_{02})\sim \text{area } n\,02'\,01'\,O/sn\sim \Delta q'+T_p'\Delta S_p', \tag{45} \]

\[ \Delta q''+T_p''\Delta S_p''+(i_{01}-i_{02})\sim \text{area } mnk'\,02'\,01'\,01\,sm\sim \]

\[ \sim \Delta q'+\Delta q''+T_p'\Delta S_p'=\Delta q+T_p'\Delta S_p'. \tag{46} \]

Thus,

\[ \eta_p>1-\frac{T_c''\Delta S_2}{T_c'\Delta S_1} \frac{T_p'\Delta S_p'}{\Delta q+T_p'\Delta S_p'}. \tag{47} \]

According to (30), the right-hand side of the last relation is equal to the efficiency of the cycle with one extraction point. Consequently, the addition of a second extraction point at \(c\)-tt \(T_p''\)

\[ T_p'>T_p''>T_k \tag{48} \]

gives an increase in the efficiency of the installation. It can be shown by an analogous device that the addition of a third extraction point gives a further increase in efficiency.

However, this increase has a limit, which is not difficult to calculate. With an infinitely large number of steam extraction points, the efficiency according to (26) will be

\[ \eta_p=1-\frac{g_k}{g_1}\frac{T_c''\Delta S_2}{T_c'\Delta S_1}. \tag{26} \]

With an infinitely large number of extraction points, heat exchange in each heater occurs with an infinitely small temperature difference, i.e., quasi-statically, without an increase in entropy. If the expansion also proceeds quasi-statically, then the entropy of all bodies of the “system” must not change, i.e.,

\[ g_k\Delta S_2=g_1\Delta S_1. \tag{49} \]

From (26) and (49) it follows that

\[ \eta_p=1-\frac{T_c''}{T_c'}\quad (\text{for } n=\infty). \tag{50} \]

Thus, if the supply and removal of heat proceed isothermally, then with an infinitely large number of extraction points the efficiency of the cycle will be the same as for the Carnot cycle.

To calculate the efficiency of the regenerative cycle with two extraction points we have

\[ \eta_p=1-\frac{T_p'\Delta S_p'}{T_c'\Delta S_1} \frac{T_p''\Delta S_p''}{\Delta q'+T_p'\Delta S_p'} \frac{T_c''\Delta S_2}{\Delta q'+T_p''\Delta S_p''}. \tag{43} \]

According to Fig. 9,

\[ \begin{aligned} T_p'\Delta S_p'&=i_0'-i_{01}',\\ T_p''\Delta S_p''&=i_{02}'-i_{02}'',\\ T_c''\Delta S_2&=i_k-i_k',\\ T_c'\Delta S_1&=i_1-i_{01}'. \end{aligned} \tag{51} \]

\[ \Delta q' + T'_p \Delta S'_p = i_{01} - i'_{01}, \]

\[ \Delta q'' + T''_p \Delta S''_p = i_{02} - i'_k . \]

Consequently,

\[ \eta_p = 1 - \frac{i_{01} - i'_{01}}{i_1 - i'_{01}}\, \frac{i_{02} - i'_{02}}{i_{01} - i'_{02}}\, \frac{i_k - i'_k}{i_{02} - i'_k}. \tag{52} \]

Using the indicated method, one can write a general expression for the efficiency of a regenerative cycle with any number of extraction points:

\[ \eta_p = 1 - \frac{i_{01} - i'_{01}}{i_1 - i'_{01}}\, \frac{i_{02} - i'_{02}}{i_{01} - i'_{02}}\, \cdots\, \frac{i_{0n} - i'_{0n}}{i_{0(n-1)} - i'_{0n}}\, \frac{i_k - i'_k}{i_{0n} - i'_k}. \tag{53} \]

Expression (53) can be written in another way. Thus, for the cycle \(1\ O1\ O1'\ 1\) (Fig. 9) we have:

\[ \eta_1 = \frac{i_1 - i_{01}}{i_1 - i'_{01}}, \tag{54} \]

whence

\[ 1 - \eta_1 = \frac{i_{01} - i'_{01}}{i_1 - i'_{01}}. \tag{55} \]

Similarly, for the cycle \(O1\ O2\ O2'\ O1'\ O1\):

\[ 1 - \eta_2 = \frac{i_{02} - i'_{02}}{i_{01} - i'_{02}}. \tag{56} \]

and so on. From (53), (55), (56), etc., we obtain

\[ 1 - \eta = (1 - \eta_1)(1 - \eta_2)\cdots(1 - \eta_n)(1 - \eta_{n+1}), \tag{57} \]

where \(\eta_1, \eta_2, \ldots, \eta_n, \eta_{n+1}\) are the efficiencies of the cycles from the first to the \((n+1)\)-th, into which the main cycle is divided by the extraction isobars.

NOTED MISPRINTS

1. Hereinafter denoted in abbreviated form as m.-t. ↩