SOME POSSIBILITIES FOR INCREASING THE OUTPUT OF THERMAL INSTALLATIONS

G. I. Fuks

1

Of all the proposals that have been made in recent years with the aim of increasing the output of thermal installations, the Velox installation deserves special attention. Analysis of the operation of a steam-power installation [3] shows that the most vulnerable point in the chain of energy conversion is the steam boiler, despite its high efficiency. The Velox boiler, apart from the sharp reduction in the dimensions of the installation and other well-known advantages, is of interest as the first successful attempt to influence the combustion process in the furnace. The issue is not the high efficiency of the Velox boiler installation. If this efficiency is obtained at the expense of the additional power supplied for compressing the air, then this will produce not an increase but a decrease in the overall output of the installation [4]. But when the compressor–turbine group begins to deliver work, an increase in the economic efficiency of the installation is obtained at ordinary values of the efficiency of the boiler section. Modern achievements in the field of gas turbines make this path sufficiently realistic.

However, Velox installations require the use of a special and, generally speaking, scarce fuel. We know of not a single installation of this type designed for pulverized fuel, and still less for the combustion of fuel in a stratified process. Operation of the boiler with supercharging in the gas section is also not always desirable; therefore, such a modification of the Velox installation seems expedient that would make it possible to use its thermal advantages in combination with ordinary steam-power installations.

Fig. 1

Fig. 2

The diagram of such an installation is given in Fig. 1, and the process in thermal \((TS)\) coordinates in Fig. 2. The compressor compresses the air to pressure \(p_2\), process \(1\text{--}2\). The compressed air passes through an air heater, operating on the heat of the exhaust gases of the boiler installation, heating—

is brought to state 4, process 2–4, and enters the turbine. After the turbine the air enters the boiler furnace (state 5). On the boiler heating surfaces the combustion products are cooled from K to \(4'\); in the air heater the gases are cooled in process \(4'—6\), and then are directed to the water economizer, where process 6—7 takes place.

Here the task is to assess the relative advantage of the scheme under consideration; therefore the calculations have been carried out with certain simplifications. Thus, the influence of the change in chemical composition during combustion on the enthalpy per \(nm^3\), air inleakage in the boiler installation, etc., are neglected as quantities of minor importance. Under these conditions the work obtained in the given installation from \(nm^3\) of air will be

\[ AL=AL_m+\eta_{te}^{n}\left[(i_k-i'_4)+(i_6-i_7)-\Sigma\pi_k\right]. \tag{1} \]

The notation adopted is:

\(AL_m\)—the excess work of the compressor–turbine group, equal to the difference between the effective works of the turbine and the compressor per \(nm^3\) of air

\[ AL_m=AL_{me}-AL_{ke}, \tag{2} \]

\(\eta_{te}^{n}\)—the effective thermal efficiency of the steam part.

The expression in square brackets is the heat transferred to the steam and water (per \(nm^3\) of air), \(i_k\)—the theoretical value of the enthalpy of the air (gas) after combustion, \(\Sigma \pi_k\)—the sum of the losses of the boiler part to the external environment.

But according to Fig. 2

\[ (i_k-i'_4)+(i_6-i_7)=(i_k-i_7)-(i'_4-i_6). \tag{a} \]

If the heat release per \(nm^3\) of air is \(q\ \dfrac{\text{kcal}}{nm^3}\) and the efficiency of the furnace is \(\eta_m\), then

\[ i_k=i_5+\eta_m q. \tag{b} \]

The heat balance of the air heater is

\[ i'_4-i_6=i_4-i_2, \tag{c} \]

therefore,

\[ (i_k-i'_4)+(i_6-i_7)-\Sigma\pi_k =\eta_m q-\Sigma\pi_k-(i_7-i_1)-(i_4-i_2)+(i_5-i_1). \tag{3} \]

The quantity

\[ \eta_m q-\Sigma\pi_k-(i_7-i_1) \]

is the utilized heat of the boiler installation,

\[ \eta_m q-\Sigma\pi_k-(i_7-i_1)=\eta_{ky}q. \tag{4} \]

The quantity

\[ (i_4-i_2)-(i_5-i_1) \]

according to Fig. 2 is the difference between the heat input and heat removal in the compressor–turbine group. It is equal to the difference between the internal works of the turbine and the compressor

\[ (i_4-i_2)-(i_5-i_1)=AL_{mi}-AL_{ki}. \tag{d} \]

But

\[ AL_{me}=AL_{mi}\eta_{mm}, \tag{e} \]

\[ AL_{ke}=\frac{AL_{ki}}{\eta_{mk}}, \tag{f} \]

where \(\eta_{mm}\) and \(\eta_{mk}\) are the mechanical efficiencies of the turbine and compressor, respectively. It is also convenient to introduce the ratio of the works (effective powers) of the compressor and turbine

\[ w=\frac{AL_{ke}}{AL_{me}}. \tag{g} \]

Then

\[ (i_4-i_2)-(i_5-i_1)=\frac{AL_m}{1-w}\left(\frac{1}{\eta_{mm}}-w\eta_{mk}\right). \tag{5} \]

From (1), taking into account (3), (4), and (5), we obtain

\[ AL=\eta_{ky}\eta_{te}^{n}q+AL_m\left[1-\frac{\eta_{te}^{n}\left(\frac{1}{\eta_{mm}}-w\eta_{mk}\right)}{1-w}\right]. \tag{6} \]

In an ordinary steam-power installation the magnitude of the work per nm\(^3\) of air will be

\[ AL'=\eta_{te}^{n}q=\eta_{ky}\eta_{te}^{n}q . \tag{4} \]

The introduction of a compressor-turbine group will be advantageous in the case when the second term on the right-hand side of (6) is positive. For this it is necessary that two conditions be satisfied:

\[ AL_m>0, \tag{7} \]

\[ 1-\frac{\eta_{te}^{n}\left(\frac{1}{\eta_{mm}}-w\eta_{mk}\right)}{1-w}>0 . \tag{8} \]

From relations (2) and (9) we have

\[ AL_m=AL_{me}(1-w) \tag{9} \]

The turbine always delivers work,

\[ AL_{me}>0, \]

therefore condition (7) gives

\[ 1>w>0. \tag{i} \]

From condition (8), after simple transformations, one may obtain

\[ w<\frac{1-\frac{\eta_{te}^{n}}{\eta_{mm}}}{1-\eta_{te}^{n}\eta_{mk}} . \tag{10} \]

Since \(\eta_{te}^{n}\) and \(\eta_{mm}\) always represent proper fractions, the quantity \(w\) will also be a proper fraction; therefore condition (i) is superfluous. There remains only condition (10), the observance of which is necessary and sufficient for the addition of a compressor-turbine group to bring about an increase in the economic efficiency of the steam-power installation.

In Fig. 3 are given the results of calculations of the limiting values of the ratios of the effective powers of the compressor and the turbine for two values of the mechanical efficiency. It is evident from the figure that even with a mechanical efficiency low for rotating machines,

\[ \eta_{mm}=\eta_{mk}=0.95 \]

Fig. 3

and with high values of \(\eta_{te}\), the work of compression must be less than 90% of the turbine work. It is not difficult to carry out an installation with a relative expenditure of work for compression of 75–82% [5]. On the basis of (6) and (10), this means that the addition of a compressor-turbine group should practically lead to an increase in the economic efficiency of the installation.

Before proceeding to an estimate of the benefit obtained from introducing a compressor-turbine group into the installation, let us consider the operating conditions of the latter. From relation (6) it is clear that the increment in work obtained from introducing the compressor-turbine group, for specified operating conditions of the steam part, depends not only on the work \(AL_m\), but also on the ratio of the compressor and turbine works \(w\), the mechanical degrees of perfection \(\eta_{km}\) and \(\eta_{mk}\), and the absolute value of the thermal efficiency \(\eta_{te}^{n}\). The quantities \(AL_m\) and \(w\) are determined by the conditions of temperature and pressure under which the compressor-turbine group operates. Indeed, the effective work per \(nm^3\) in the turbine and compressor may be calculated as follows:

\[ AL_{em}=c_pT_4\left[1-\left(\frac{p_1}{p_2}\right)^{\frac{k-1}{k}}\right]\eta_{oe}^{m}, \tag{11} \]

\[ AL_{ek}=\frac{c_pT_1}{\eta_{ad}}\left[\left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}}-1\right]. \tag{12} \]

Here the notation is:

\(c_p\) — heat capacity of air, in \(\dfrac{\mathrm{kcal}}{nm^3\,\mathrm{deg}}\),

\(\eta_{oe}^{m}\) — relative effective efficiency of the turbine,

\(\eta_{ad}\) — adiabatic efficiency of the compressor,

\(k\) — adiabatic exponent. In the calculation \(k=1.40\) is adopted.

Introducing the notation

\[ \left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}}=x, \tag{a} \]

we obtain

\[ AL_{em}=c_pT_4\frac{x-1}{x}\,\eta_{oe}^{m}, \tag{13} \]

\[ AL_{ek}=\frac{c_pT_1}{\eta_{ad}}(x-1), \tag{14} \]

\[ w=\frac{AL_{ek}}{AL_{em}}=\frac{T_1x}{T_4\eta_{oe}^{m}\eta_{ad}}. \tag{15} \]

On this basis, taking (2) into account, from (6) the function \(U\) is formed, having its maximum value at the same \(x\) as the value \(AL\):

\[ U=(x-1)\left(\frac{T_4\eta_{oe}^{m}}{x}-\frac{T_1}{\eta_{ad}}\right) \left( 1-\eta_{te}^{n} \frac{ \dfrac{1}{\eta_{km}}-\dfrac{T_1x}{T_4\eta_{oe}^{m}\eta_{ad}\eta_{mk}} }{ 1-\dfrac{T_1x}{T_4\eta_{oe}^{m}\eta_{ad}} } \right). \tag{16} \]

The condition for the maximum of this function,

\[ \frac{du}{dx}=0 \]

gives

\[ x=\sqrt{ \frac{T_4}{T_1}\eta_{oe}^{m}\eta_{ad}\, \frac{ 1-\dfrac{\eta_{te}^{n}}{\eta_{km}} }{ 1-\eta_{te}^{n}\eta_{mk} } }. \tag{17} \]

Figures 4 and 5 show the results of calculations of the most advantageous pressure ratios \(\dfrac{p_2}{p_1}\) according to (17) and \((a)\), and of the corresponding values of \(w\) according to (15), for different \(\eta_{oe}^{m}\) and with different degrees of perfection of the compressor-turbine group. In all cases the most advantageous pressure ratio lies within the range 1.8—3.0, which is not difficult to obtain in an axial compressor and is readily utilized in turbines with a small number of stages.

Fig. 4

Fig. 5

Thus, Prof. Stodola [5] gives test data for a gas turbine operating with gases at \(t = 490—550^\circ\mathrm{C}\), the values of the relative efficiencies being

\[ \eta_{oe}^{m}=0.884, \]

\[ \eta_{ad}=0.846—0.849, \]

the power ratio

\[ w=0.74—0.78. \]

The turbine and compressor had a small number of stages. With the low figures adopted for the comparative calculation in Fig. 5, the value of \(w\) does not exceed 78% even at the smallest values of \(\eta_{ke}^{n}\).

3

A concrete idea of the advantage of an installation according to the scheme presented, and of the conditions of its operation, can be obtained by calculating, from (6), \(AL\) and the efficiency of the installation from the general relation

\[ \eta=\frac{AL}{q}. \tag{a} \]

In each case the most advantageous pressure ratio of the compressor-turbine group is selected for the given case. The work of the latter, \(AL_m\), according to the relation (2) given above, will be

\[ AL_m = AL_{me} - AL_{ke}. \tag{2} \]

For the effective work of the turbine we have

\[ AL_{me} = (i_4 - i_{5a})\eta_{oe}^{m}, \tag{b} \]

where \(i_{5a}\) is the enthalpy of air per \(1\,\mathrm{nm}^3\) at the end of the adiabatic expansion \(4—5a\) (Fig. 2).

But

\[ i_4-i_{5a}=i_4\left(1-\frac{i_{5a}}{i_4}\right)\simeq i_4\left(1-\frac{t_{5a}}{t_4}\right) =\frac{i_4}{t_4}(t_4-t_{5a})= \]

\[ =\frac{i_4}{t_4}(T_4-T_{5a})=\frac{i_4}{t_4}T_4\left(1-\frac{T_{5a}}{T_4}\right). \]

From the adiabatic equation

\[ \frac{T_{5a}}{T_4}=\left(\frac{p_1}{p_2}\right)^{\frac{k-1}{k}}=\frac{1}{x}, \]

therefore,

\[ AL_{me}=\frac{i_4}{t_4}T_4\frac{x-1}{x}\eta_{oe}^{m}. \]

Similarly, for the compressor work

\[ AL_{ek}=\frac{i_1}{t_1}\frac{T_1}{\eta_{ad}}(x-1) \tag{19} \]

and, from (2),

\[ AL_m=\left(\frac{i_4}{t_4}\frac{T_4}{x}\eta_{oe}^{m}-\frac{i_1}{t_1}\frac{T}{\eta_{ad}}\right)(x-1). \tag{20} \]

The results of the corresponding calculations are given in Figs. 6, 7, 8, and 9. Fig. 6 summarizes the data for good, but attainable, efficiency of the compressor-turbine group and for the temperature before the turbine \(t_4=500^\circ\mathrm{C}\). For comparison, two heat releases per \(nm^3\) are taken—\(600\ \frac{\mathrm{kcal}}{nm^3}\) and \(475\ \frac{\mathrm{kcal}}{nm^3}\), which covers a sufficient range of fuels and excess air. The absolute values of the economic efficiency of the installation \(\eta\), the saving in fuel consumption relative to

\[ E=\frac{\eta-\eta_{te}^{n}}{\eta}, \]

as well as the percentage of work delivered by the compressor-turbine group, \(AL\%\), and the turbine work as a percentage of the total, \(AL_{me}\%\), were calculated.

Fig. 6

It is immediately evident that the addition of a compressor-turbine group to an ordinary steam-power installation increases the overall efficiency of the installation. This increase is greater at a small heat release per \(nm^3\), when the share of the work delivered by the compressor-turbine installation increases.

Usually steam-power installations operate with an economic efficiency on the order of 20–35%, which in the calculations corresponds to \(\eta_{te}^{n}=25\text{–}40\%\). In this case

the saving in fuel consumption is 9–4%. The compressor-turbine group delivers 13–7% of the total work of the installation. The work of the turbine amounts to 51–26% of the total work of the installation group. In this case the flow-through part of the turbine will have sufficient dimensions, which will ensure for it a sufficiently high internal relative efficiency.

Fig. 7 gives a summary of the results of calculations at a higher temperature before the turbine

\[ t_4 = 600^\circ \mathrm{C}. \]

The performance indicators of the installation are clearly improved. The fuel saving reaches 13–6%. But it is clear that this is obtained at the expense of the durability of the installation. Thus, the walls of the air heater will in this case have, at the hot end, temperatures of the order of 650°C.

For comparison, Figs. 8 and 9 give the results of analogous calculations with considerably worse performance indicators of the compressor-turbine group (the data are written on the figure). The saving in fuel consumption in this case is reduced to 5–2% at a temperature of 500°C before the turbine and to 9–4% at 600°C. Thus, the degree of perfection of the compressor-turbine group is of great importance. However, a reduction in the perfection of the compressor-turbine group affects the output of the entire installation not so strongly as in the case of a gas turbine. The reason for this is obvious: the losses of the compressor-turbine installation, with the exception of mechanical losses, pass into heat and are partially utilized in the steam part.

Fig. 7

Fig. 8

Fig. 9

On the whole it may be considered established that a steam-power installation with a compressor-turbine group according to the scheme of Fig. 1, without changing the operation of the steam part, can give a fuel saving of the order of 13–5%. The same saving will be obtained in the Velox installation, which differs in principle from the scheme of Fig. 1 in that not air but combustion products pass through the turbine. The Velox installation will have the advantage that it is easier in it to realize higher temperatures before the turbine. Therefore, where the fuel and other conditions permit, it is better to carry out the installation in this way. But the installation according to the scheme of Fig. 1 has the advantage that it permits the use in it of any steam part and any fuel. On the other hand, in the indicated scheme there is inevitably a high air temperature at the furnace inlet (of the order of 400–460°C), which may sometimes create difficulties in designing the furnace.

Raising the air temperature before the turbine from \(500^\circ\mathrm{C}\) to \(600^\circ\mathrm{C}\) increases the fuel-consumption saving from 9–4% to 13–6%. At the same time the air heater is placed under more severe conditions, since the metal temperature at its hot end must reach \(650^\circ\mathrm{C}\). But a sufficiently high temperature before the turbine can be obtained in another way—by burning a certain amount of fuel in the compressed air. For this purpose there must be available a small amount of liquid or gaseous fuel, which can be burned in the combustion chamber before the gas turbine. The diagram of such an installation is given in Fig. 10, and the process in thermal \((TS)\) coordinates in Fig. 11. After heating of the air compressed in the compressor (process \(2—8\)), the gas temperature rises to \(t_4\) through combustion of the fuel.

Fig. 10

Fig. 11

For this case, as for the preceding one,

\[ AL=AL_m+\eta_{te}^{n}\left[(i_k-i'_4)+(i_6-i_7)-\sum \pi_k\right]. \tag{1} \]

But

\[ (i_k-i'_4)+(i_6-i_7)=(i_k-i_7)-(i'_4-i_6), \tag{a} \]

\[ i_k=i_5+\eta_m q_1, \tag{b} \]

where \(q_1\) is the heat release in the furnace per \(\mathrm{Nm}^3\) of air. From the heat balance of the air heater,

\[ i'_4-i_6=i_8-i_2=(i_4-i_2)-(i_4-i_8). \tag{c} \]

The rise in temperature from \(t_8\) to \(t_4\) is obtained through fuel combustion in the combustion chamber, i.e., heat release at \(p=\mathrm{idem}\),

\[ i_4-i_8=q_2, \tag{d} \]

where \(q_2\) is the heat release in the combustion chamber per \(\mathrm{Nm}^3\) of air drawn in by the compressor. Taking into account, in addition, that

\[ (i_4-i_2)-(i_5-i_1)= \frac{AL_m}{1-w} \left( \frac{1}{\eta_{mi}}-w\eta_{mm} \right), \tag{e} \]

we obtain

\[ AL=\eta_{te}^{n}(\eta_{ky}q_1+q_2)+AL_m \left[ 1- \frac{ \eta_{te}^{n} \left( \frac{1}{\eta_{mi}}-w\eta_{mm} \right) }{1-w} \right]. \tag{21} \]

To determine the efficiency of the installation we have

\[ \eta_i=\frac{AL}{q_1+q_2}. \tag{22} \]

The first bracket on the right-hand side of (21) represents the heat released in the installation per \(nm^3\) of air. The effort to reduce losses with the exhaust gases makes it necessary to achieve, as far as possible, the use of all the oxygen of the air. If part of the oxygen is consumed in the combustion chamber before the turbine, then the heat release in the furnace will correspondingly decrease. In practice this will lead to the fact that the sum

\[ q_1 + q_2 \]

will change little when its terms are varied. Taking this into account, we obtain the former formula for the condition of the maximum efficiency:

\[ x = \sqrt{ \frac{T_4}{T_1}\, \eta_{oe}^{m}\eta_{ad}\, \frac{1-\dfrac{\eta_{ie}^{m}}{\eta_{um}}} {1-\eta_{ie}^{m}\eta_{mk}} }, \tag{17} \]

where

\[ x=\left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}} . \tag{1} \]

Taking all this into account, we arrive at the conclusion that calculations according to relations (21) and (22) should give results that do not differ from those obtained without burning fuel in the combustion chamber. Thus, if the gas temperature before the turbine is raised by combustion of fuel in the combustion chamber to \(600^\circ\mathrm{C}\), then, according to the calculations whose results are given in Figs. 7 and 9, the saving in fuel consumption will be \(13\text{--}6\%\) with good performance of the compressor-turbine group \((\eta_{mt}=\eta_{mk}=0.98;\ \eta_{oe}^{m}=0.85,\ \eta_{ad}=0.80)\), down to \(9\text{--}4\%\) with poor performance \((\eta_{mt}=\eta_{mk}=0.95;\ \eta_{oe}^{m}=0.80,\ \eta_{ad}=0.75)\). In this case the fuel consumption for the combustion chamber, if it is assumed that heating the air in the air heater gave \(t_3=500^\circ\mathrm{C}\), will amount to \(5\text{--}8\%\) of the total consumption, counted by heat release or in units of standard fuel.

5

A considerable increase in the output of a heat engine can be achieved by a rational combination of an internal-combustion engine and a steam installation. The advantage in such combined operation is obtained because different temperature intervals are used in these engines. An installation can be obtained whose output is higher than that of each component part. Taking into account, on the other hand, that rotary engines should be used in a modern installation, the scheme of the installation shown in Fig. 12 appears rational. A two-stroke piston internal-combustion engine with free pistons works directly on the compressor. The engine is scavenged with compressed air, for which a portion of it is diverted. The main part of the compressed air after the compressor is heated in the air heater by the heat of the gases leaving the boiler. After this air is mixed with the scavenging air and the combustion products of the engine, it enters the gas turbine and, after expansion, the boiler furnace. Ordinary fuel is burned in the furnace.

The process of the installation on the gas side is plotted in thermal \((TS)\) coordinates in Fig. 13. \(1\text{--}2\) is the line of air compression in the compressor. Further compression \(2\text{--}a\) takes place inside the engine cylinder.

The working process of the engine is determined by the type of fuel and other usual circumstances. The state of the products of combustion after expansion in the engine and mixing with the scavenging air is shown by point 3. Heating of the compressed air after the compressor takes place in process 2--8; the gases are cooled in the air heater from \(4'\) to 6. The state of the gas before the turbine (a mixture of heated air and combustion products with scavenging air) is shown by point 4. Point 5 represents the state of the gas (air) when it is supplied to the boiler furnace. Point \(K\) represents the conditional state of the products of combustion in the boiler furnace (with adiabatic combustion of the fuel). Heat is transferred from the gases in the boiler in process \(K--4'\), and in the water economizer in process \(6--7\). The condensate from the condenser of the steam turbine is passed through the engine jacket.

Fig. 12

Fig. 13

b

The calculation of the operation of the installation is carried out per \(nm^3\) of air working in the engine. The effect of the change in enthalpy due to the change in chemical composition during combustion is neglected. Then

\[ Al = AL_{me} + \eta_{te}^{u}\left\{\beta\left[(i_k - i'_4) + (i_6 - i_7) - \sum \pi_k\right] + q_{oxl}\right\}. \tag{23} \]

Here, in addition to the notation introduced earlier,

\(\beta\) is the compressor capacity, in \(nm^3\) per \(nm^3\) of air inside the engine,

\(q_{oxl}\) is the heat of engine cooling.

If the calculation is made for an uncooled compressor and the losses of the engine and compressor to the external surroundings are neglected, then the energy balance of the engine--compressor will be

\[ AL_d = \beta (i_2 - i_1), \tag{a} \]

where \(AL_d\) is the effective work of the engine per \(nm^3\) of air. Consequently,

\[ \beta = \frac{AL_d}{i_2 - i_1}. \tag{24} \]

On the other hand,

\[ AL_d = (\xi - r)\eta_{oi}^{d}\eta_{m}^{d}\eta_{t}^{d} q_z, \tag{25} \]

where \(\xi\) is the coefficient of heat release of the engine, allowing for losses from chemical and mechanical incomplete combustion,

\(r\) is the heat transferred to the cooling water in the process of combustion and expansion, as a percentage of the theoretical heat release,

\(\eta_t^{d}\) is the thermal efficiency of the engine cycle.

It may be assumed that

\[ \eta_t^{d}=1-\frac{B}{x_0}, \tag{8} \]

where, depending on the nature of the cycle,

\[ B \geq 1 \tag{c} \]

and

\[ x_0=\left(\frac{p_\delta}{p_z}\right)^{\frac{n-1}{n}}, \tag{d} \]

\(n\) is the exponent of the adiabatic compression (and expansion) in the theoretical cycle of the engine.

For an uncooled reciprocating compressor it may be assumed that all the work of friction is converted into heat, which is transferred to the gas being compressed, i.e.,

\[ i_2-i_1=\frac{i_{2a}-i_1}{\eta_{mk}} =\frac{i_1}{\eta_{mk}}\left(\frac{i_{2a}}{i_1}-1\right). \tag{e} \]

Within the range of those temperature changes that will occur in the compressor during compression in one stage, it may be taken that

\[ \frac{i_{2a}}{i_1}=\frac{t_{2a}}{t_1}. \tag{f} \]

Consequently,

\[ i_2-i_1=\frac{i_1}{\eta_{mk}}\left(\frac{t_{2a}}{t_1}-1\right) =\frac{i_1}{t_1}\frac{T_1}{\eta_{mk}}(x-1), \tag{26} \]

where

\[ x=\left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}} =\frac{T_{2a}}{T_1}. \tag{g} \]

From (24), taking into account (26) and (g), we have

\[ \beta=\frac{AL_0\,t_1\,\eta_{mk}}{i_1\cdot T_1\cdot (x-1)}. \tag{27} \]

The work of the turbine is

\[ AL_{me}=\beta (i_4-i_5)\eta_{mt}. \tag{28} \]

An analogous transformation gives

\[ AL_{me}= \frac{i_1}{t_4}\frac{t_i}{i_1}\frac{T_4}{T_1}\, \frac{\eta_{mt}\eta_{oe}^{m}}{x}\,AL_0 . \tag{29} \]

The quantity in square brackets in (23) is the heat transferred from the gases in the boiler per \(nm^3\) of air,

\[ (i_k-i_4')+(i_6-i_1)-\sum \tau_k =(i_k-i_1)-(i_4'-i_i)-\sum \tau_k. \tag{h} \]

Let the heat release per \(nm^3\) in the boiler furnace be \(q_1\ \dfrac{\mathrm{kcal}}{nm^3}\). Of the \(\beta\,nm^3\) of air supplied by the compressor, \(1\,nm^3\) goes to the engine, from which

for combustion, \(a\,\text{Nm}^3\) is used. The heat release in the furnace per \(\text{Nm}^3\) of air supplied by the compressor will be

\[ q_k=\frac{\beta-a}{\beta}\,q_1, \tag{i} \]

therefore,

\[ i_k=i'_k+\frac{\beta-a}{\beta}\,q_1\eta_m, \tag{k} \]

and the heat transferred from the gases in the boiler unit is

\[ \beta\left[(i_k-i'_k)+(i_6-i_1)-\sum \pi_k\right]= \]

\[ =\beta\left[\frac{\beta-a}{\beta}\eta_m q_1-(i_i-i_1)-\sum \pi_k-(i_4-i_6)+(i_5-i_1)\right]. \tag{l} \]

The first three terms on the right-hand side represent the heat obtained from fuel combustion in the furnace

\[ \frac{\beta-a}{\beta}\eta_m q_1-(i_i-i_1)-\sum \pi_k = \eta_{ky}\frac{\beta-a}{\beta}q_1. \tag{m} \]

For scavenging the engine, \((1+b)\) of air must be supplied per \(\text{Nm}^3\) in state 2 (Fig. 13). The remainder of the compressed air is heated at the expense of the heat of the boiler exhaust gases, i.e.

\[ [\beta-(1+b)](i_3-i_2)=\beta(i'_4-i_6). \tag{30} \]

After this, mixing occurs of \([\beta-(1+b)]\,\text{Nm}^3\) with enthalpy \(i_4\) and \((1+b)\,\text{Nm}^3\) with enthalpy \(i_3\). This gives \(\beta\,\text{Nm}^3\) of gas with enthalpy

\[ i_4=\frac{[\beta-(1+b)]i_4+(1+b)i_3}{\beta}, \tag{31} \]

which, taking (30) into account, gives

\[ \beta(i'_4-i_6)-\beta(i_4-i_1)=\beta(i_4-i_2)-\beta(i_5-i_1)-(1+b)(i_3-i_2). \tag{n} \]

The first two terms on the right-hand side represent (Fig. 13) the difference between the internal works of the turbine and the compressor. Taking (24) into account, one may write

\[ \beta(i_4-i_2)-\beta(i_5-i_1)=\frac{AL_{me}}{\eta_{mm}}-AL_0. \tag{o} \]

The last term on the right-hand side of (n) gives the heat that can be released from the engine exhaust gases if they are cooled to \(t_2\), i.e.

\[ (1+b)(i_3-i_2)=q'. \tag{32} \]

From (23), taking into account (l), (m), (n), and (o), we have

\[ AL=AL_{me}+\eta_{te}^{\,n}\left[\eta_{ky}(\beta-a)q_1-\frac{AL_{me}}{\eta_{mm}}+AL_0+q'+q_{ox.l}\right]. \tag{p} \]

But from the heat balance of the engine

\[ \xi q_2=AL_0+q'+q_{ox.l}, \tag{33} \]

therefore,

\[ AL=AL_{me}+\eta_{te}^{\,n}\left[\eta_{ky}(\beta-a)q_1-\frac{AL_{me}}{\eta_{mm}}+\xi q_2\right]. \tag{34} \]

The efficiency of the installation is calculated from the relation

\[ \eta=\frac{AL}{(\beta-a)q_1+q_2}. \tag{35} \]

Of the quantities that must be calculated beforehand in order to carry out the calculations by relations (34) and (35), we should dwell on determining the state of the gas before the turbine (\(t_4\) or \(i_4\)). This quantity is necessary for calculating the work of the turbine \(AL_{mв}\) by (28) or (29).

If we specify the temperature of the gases leaving the boiler and directed to the air heater (\(t'_4\) or \(i'_4\)), then the temperature of the air heating (\(t_8\) or \(i_8\)) can be assigned, taking into account the practically necessary temperature difference at the hot end of the air heater,

\[ t_8 = t'_4 - \Delta t. \tag{36} \]

On the other hand, from (31) and (32) we have:

\[ i_4 = \frac{[\beta - (1+b)]i_8 + q' + (1+b)i_2}{\beta}, \tag{37} \]

where \(i_2\) is determined from (26), and \(q'\) from (33).

Having determined the enthalpy \(i_4\), and from it the temperature \(t_4\), it is necessary to assess whether this temperature is permissible for the turbine blading. If it proves to be impermissibly high, then one must specify the maximum permissible temperature \(t_4\), and then from (37) calculate the corresponding temperature or enthalpy of air heating.

To carry out all the indicated calculations, it is necessary to specify the pressure of the compressed air \(p_2\), which will make it possible to calculate

\[ x=\left(\frac{p_2}{p_1}\right)^{\frac{\kappa-1}{\kappa}}. \tag{a} \]

At first glance the choice of \(p_2\) seems arbitrary. In reality, however, there are so many limiting factors here that there will hardly be any arbitrariness. First of all, obviously,

\[ 1 < p_2 < p_0, \]

where \(p_0\) is the final compression pressure in the engine cylinder. As \(p_2\) increases, \(\beta\) will decrease; consequently, relatively more heat will be released in the engine and less in the boiler furnace. At the same time the overall efficiency of the installation will increase, but the fuel consumption of the engine will also increase. Usually this fuel is more expensive and scarcer than the fuel burned in the boiler furnace. In most cases, therefore, the maximum compression pressure will be determined by the availability of fuel for the engine. The greater this availability, the higher the air compression pressure can be taken, and the greater the overall efficiency of the installation will be.

8

A concrete idea of the results of the installation’s operation can be obtained by comparing the results of a series of calculations. The results of all these computations are given in Figs. 14–17.

In Figs. 14 and 15 the results of calculations are given for an installation with a diesel-compressor. In the calculation it was assumed:

the overall compression ratio in the engine \(\varepsilon = 20\);

the coefficient of heat release in the engine \(\xi = 0.95\);

the coefficient of heat loss to the cooling water [6] \(r = 0.15\);

the adiabatic exponent in the engine \(n = 1.38\);

relative internal efficiency of the engine \(\eta_{oi}=0.80\);

mechanical efficiency of the engine (there is no connecting-rod-and-crank mechanism) \(\eta_{m}^{d}=0.90\);

mechanical efficiency of the compressor \(\eta_{mk}=0.90\);

temperature of the gases before the turbine not above \(t_4=600^\circ\mathrm{C}\);

effective relative efficiency of the turbine \(\eta_{oe}^{m}=0.85\);

mechanical efficiency of the turbine \(\eta_{mt}=0.98\);

efficiency of the boiler installation \(\eta_{ku}=0.85\).

Heat release per \(nm^3\):

a) in the engine

\[ q_2=488\,\frac{\mathrm{kcal}}{nm^3}, \]

which corresponds to excess air in the engine \(\sim 1.8\),

b) in the furnace

\[ q_1=600\,\frac{\mathrm{kcal}}{nm^3}. \]

The straight lines in Fig. 14 give the values of the efficiency of the installation at various degrees of compression in the compressor \(\dfrac{p_2}{p_1}\) and correspondingly different values of the relative heat release in the engine \(m_1\), inscribed on the straight lines, as a function of the effective thermal degree of perfection of the steam part of the installation \(\eta_{te}^{n}\). The magnitude of the output of the installation increases with the growth of \(\eta_{te}^{n}\) and with an increase in the air-compression pressure in the compressor. For comparison, the straight line labeled \(\eta_{te}^{n}\) gives the effective output of an ordinary steam-power installation.

Fig. 14

Fig. 15

The results of calculating the saving in the specific heat consumption per unit of power, in comparison with an ordinary steam-power installation, \(E_1\%\), are very indicative. Within the usual limits of output of a steam-power installation, \(20\text{--}35\%\), this saving reaches:

With air compression to \(2\) ata and \(m_1=11.2\%\), \(E_1=20\text{--}11\%\),

» » » » \(3\) ata and \(m_1=28.2\%\), \(E_1=27\text{--}16\%\).

The curves labeled \(E_2\) give the saving in the specific heat consumption in the installation under consideration in comparison with an internal-combustion engine. Thus, at \(\eta_{w}^{n}=0.35\) and \(\dfrac{p_2}{p_1}=2\), the given installation consumes per unit of power \(5\%\) less heat than a Diesel engine, and at

\[ \frac{p_2}{p_1}=3 \]

by 10%, although the heat consumption in the liquid fuel amounts to only 11.2–28.2% of the total heat consumption in the fuel.

In Fig. 15, for these same conditions, the results of calculations are plotted for the power of the gas turbine as fractions of the total power of the installation \((AL_{me}^{0}\%)\) and of the power generated in the engine \((AL_{0}^{0}\%)\). The corresponding values will be

\[ \text{for } \frac{p_2}{p_1}=2 \ldots AL_{me}^{0}\%=25\text{–}16\%,\quad AL_{0}^{0}\%=15\text{–}9\%, \]

\[ \text{for } \frac{p_2}{p_1}=3 \ldots AL_{me}^{0}\%=34\text{–}24\%,\quad AL_{0}^{0}\%=22\text{–}14\%. \]

A direct-acting diesel compressor, owing to the absence of a connecting-rod-and-crank mechanism, is a relatively simple machine that can be made sufficiently high-speed. Judging from the values of \(AL_{0}\) given above, no special difficulties should be expected in achieving considerable installation powers.

Fig. 16

Fig. 17

Still more interesting are the results of calculations for an installation with a direct-acting gas compressor, presented in Figs. 16 and 17. In the engine it is assumed that blast-furnace gas is burned with a heat release of

\[ 475\ \frac{\text{kcal}}{\text{nm}^{3}}, \]

which corresponds to an excess-air coefficient of \(\sim 2\). The overall compression ratio in the engine is taken as \(\varepsilon = 8.0\), and the internal efficiency of the engine as \(\eta_{oi}^{0}=0.75\). The remaining quantities are taken the same as in the preceding example.

From Fig. 16 it is evident that the total saving in the specific heat consumption, at economic efficiencies of the steam part of 25–35%, is:

\[ \text{for } \frac{p_2}{p_1}=2,\quad m_1=14.4\%,\quad Э_1=18\text{–}8\%, \]

at

\[ \frac{p_2}{p_1}=3,\qquad m_1=27\%,\qquad \mathcal{E}_1=28-16\%. \]

It is characteristic that in this case there is a substantial saving in the specific heat consumption as compared with a gas engine \((3_2\%)\), reaching 20% at \(\frac{p_2}{p_1}=2\) and 27% at \(\frac{p_2}{p_1}=3\).

The data of Fig. 17 show that the capacity of the gas turbine and engine lies within moderate limits that are feasible in practice.

9

A comparison of the calculations presented above, summarized in Table 1, makes it possible to draw certain general conclusions.

Table 1

In many modern large metallurgical and chemical combines there is a possibility of consuming in the boiler plant a small quantity of gas—the waste gas of the main production process. In these cases it is possible not only to replace with gas some part of the boiler-plant fuel, but also to obtain a very appreciable saving in the total fuel consumption, or a corresponding increase in the capacity generated by the combine’s station. Thus, at one metallurgical combine the average fuel consumption per month (according to data for January—June) was as follows:

Table 2

In calculating the fuel consumption in percent by standard fuel, mazut was excluded from the fuel balance, since it was used for melting and in emergency cases. The quantity of coke-oven gas is very small and its supply is irregular. It is therefore difficult to imagine any other use for it than burning directly in the furnace. If one assumes the installation of a direct-acting gas compressor, then, with air compression \(p_2=2\) ata, the heat release in the engine should amount to 14.4% of the total (Table 1).

and Fig. 16). At the thermal power station of the combine under consideration, steam was generated at a pressure \(p = 30\) ata. With a purely condensing installation, the economic efficiency of the station would hardly exceed 25%. Taking into account the production of district-heating steam, we shall take the economic degree of perfection [3] as \(\sim 30\%\). According to Fig. 16, the saving in the specific heat consumption will then be 10%. If it is assumed that the consumption of gas and coke fines remains unchanged, then the consumption of hard coal will decrease by \(6070 \ \frac{\text{tons}}{\text{month}}\), or by 15%.

It is interesting to note that, according to the data of Table 1, a noticeable saving in fuel consumption is obtained with a relatively small consumption of gaseous fuel. This places the question of the possibility of using gas generators at central power stations and thermal power stations in a quite different position than is usually the case.

On the other hand, certain considerations must also be taken into account, the consideration of which may somewhat reduce the expected effect in comparison with the figures calculated above. Thus, in our calculation, as a preliminary one, the influence of the change in the chemical composition of the gas during combustion on its enthalpy was not taken into account; the influence of air leaks in the boiler installation, etc., was not taken into account entirely accurately. At very high degrees of perfection of the steam-power part \((\eta_{\omega}^{n} = 35—40\%)\), obtaining the saving in heat consumption calculated above may prove completely impossible, since the methods described somewhat limit the possibility of applying regeneration in the steam part. The questions of regulating the installation also require detailed consideration, although in principle this question raises no doubts [1].

CITED LITERATURE

1. Vaisman, Varshavskii—Paths for the modern use of internal-combustion engines as generators of the working medium. Zhurnal tekhnicheskoi fiziki, issues 23—24, 1940.

2. Ramzin—Ramzin’s once-through boiler and prospects for the use of high-pressure steam in the USSR. Izvestiya Akademii Nauk SSSR. Otdel. tekh. nauk, Nos. 1—2, 1944.

3. Fuks—Thermodynamic evaluation of the degree of perfection of a heat-power installation. Dissertation, Tomsk, 1941.

4. Fuks, Novikov—Thermal calculation of the Velox boiler. Izvestiya Tomskogo ordena Trudovogo Krasnogo Znameni politekhnicheskogo in-ta, vol. 59, issue 2.

5. Stodola—Load Tests of a Combustion Turbine. Power, 1940.

6. Pescara—La combustion dans les chambres a volume variable. Chaleur et Industrie, 1939, Janvier.