The “Quantity—Quality” Diagram and Its Use for Illustrating Ideas in Energy Studies

V. T. Yurinskii

In explaining a number of propositions of modern heat engineering, for example, in elucidating the question of the advantages provided by district heating, as a rule the \(T—S\) diagram of water vapor is used in teaching.

In the practice of teaching special courses, the author has for a number of years been applying another illustrative method. For this purpose the “quantity—quality” diagram is used, constructed in the coordinates \(i—D\). This diagram is much more visual than the \(T—S\) diagram; it shows wherein lie the advantages of the combined process of producing electric and thermal energy. With the aid of this diagram it is easy to judge how various methods of covering thermal and electrical loads, load fluctuations, fluctuations of the outdoor-air temperature, etc., are reflected in the economy of operation of an individual station or of an energy system.

Fig. 1

Let us consider the following concrete example of solving an energy-supply problem.

A thermal electric power station on which a turbine with steam extraction is installed (Fig. 1) supplies the energy needs of a district. At the moment under consideration the outdoor-air temperature is low, and therefore the network water is heated in the main and peak heaters. The peak heater receives reduced steam from the boilers of the thermal power station; the main heater is supplied with steam from the controlled extraction of the turbine. Let the heat content of the live steam be \(i_0\); the heat content of the steam entering the

condenser, \(i_k\); the heat content of the steam going to extraction, \(i_{no}\). The quantity of steam going to the condenser is \(D_k\); extracted from the turbine, \(D_{no}\); taken through the reducing valve from the boiler, \(D_{nn}\) kg/hr.

Let the heat contents of the condensate formed from the indicated three steam flows be, respectively, \(i'_{2k}\), \(i'_{2no}\), \(i'_{2nn}\).

The internal power developed by the steam flow working down to the pressure in the condenser will be

\[ N_{ik}=\frac{D_k(i_o-i_k)}{860}\ \text{kW}. \]

The internal power developed by the steam flow going to extraction will be

\[ N_{ino}=\frac{D_{no}(i_o-i_{no})}{860}\ \text{kW}. \]

The total steam consumption from the boilers is

\[ D=D_k+D_{no}+D_{nn}\ \text{kg/hr}. \]

This total consumption, as well as its components, can be plotted along the abscissa axis of the diagram (Fig. 2).

By plotting the corresponding heat contents along the ordinate axis, we obtain rectangular areas showing how all the energy contained in the steam taken from the boilers has been distributed: to the production of mechanical energy, to the production of heat energy, and to covering losses. The internal power of the steam flow working down to the pressure in the condenser is represented in the diagram by the area \(azdb\), since \(860N_{ik}=D_k(i_o-i_k)\). Similarly, the area \(gezde\) represents the value of the internal power of the steam flow working down to the pressure at the extraction point, since \(860N_{ino}=D_{no}(i_o-i_{no})\).

Fig. 2

As shown in Fig. 2, the steam flow delivered by the boilers is divided into three flows. The flow \(D_k\) produces only electric energy, the flow \(D_{no}\) produces electric and heat energy, and the flow \(D_{nn}\) produces only heat energy. The flow \(D_{no}\) is the most desirable. To show this, it is placed in the center of the diagram.

The task of district heating is to displace, as completely as possible, the condensation kilowatt-hours, i.e., to develop the central flow. For the moment under consideration, heat consumption is measured by the vertically hatched areas. Since heat consumption is not determined by the station operating mode, but depends on external conditions, that is, on consumers, it is evident that the steam flow \(D_{nn}\) is a forced flow. The necessity of the flow \(D_{nn}\) is determined only by the fact that, at the moment under consideration, it is necessary to heat the water in the supply main of the network to a temperature higher than that attainable from the extraction parameters (or from the quantitative limitations of extraction).

If, for a given heat consumption, i.e., for a given magnitude of the area marked on the diagram by vertical hatching, it were possible to increase the steam consumption from extraction, then in this case we would have:

1) a reduction in the consumption of reduced steam from the boilers for the peak heater,

2) an increase in the generation of electric energy from heat consumption (with \(i_o\), \(i_{no}\) unchanged, the central flow \(D_{no}\) will have a greater width on the diagram),

3) a reduction in the generation of electric energy by the condensation method.

We see that the central steam flow, as it develops, displaces both the steam flow providing condensation generation of electric energy and the steam flow taken directly from the boilers to satisfy heat consumption.

It is thought that the “quantity--quality” diagram is a visual means that makes it possible, in a most accessible form, to illustrate the basic ideas of district heating.

But the illustrative merits of the proposed diagram are not exhausted by this. It is also easy to depict in it the electromechanical losses of the turbine generator. In fact, these losses represent mechanical energy obtained in the turbine but not delivered to electrical consumers, instead being expended to cover friction in the bearings, the action of the turbine control system, heating of the generator windings, etc. The electric power is related to the internal power by the obvious relation

\[ N_e = \eta_g \eta_m \cdot N_i . \]

Here

\[ \eta_g \text{ is the efficiency of the generator,} \]

\[ \eta_m \text{ is the mechanical efficiency of the turbine.} \]

Since the internal power, determined by the steam flow working down to the condenser pressure, will be expressed by the area of the rectangle \(a g d b a\) (\(\text{area } a g d b a = 860\,N_{ik}\)), the electromechanical losses of this flow can be represented by the area of the rectangle \(a_1 g b_1 a_1\)

\[ \text{area } a_1 g b_1 a_1 = (1 - \eta_g \eta_m)\,\text{area } a g d b a \]

\[ 860\,N_{ik}(1 - \eta_g \eta_m) = 860\,N_{e\text{-}mk}, \]

where \(860\,N_{e\text{-}mk} = 860\,(N_{ik} - N_{ek})\) represents the power expended to cover the electromechanical losses of the flow.

Similarly, the area of the rectangle \(z_1 e zh d_1 z_1\) measures the electromechanical losses of the steam flow extracted at an intermediate point of the turbine,

\[ \text{area } z_1 e zh d_1 z_1 = 860\,N_{e.mno} = (1 - \eta_g \eta_m)\,N_{ino}. \]

Let the electric load be \(N_e\) kW and the heat load \(Q\ \dfrac{\text{Mcal}}{\text{h}}\).

In this case

\[ Q = Q_{op} + Q_{pp}, \]

\(Q_{op}\) is the heat load that can be covered through the use of extraction (under quantitative or qualitative extraction constraints),

\(Q_{pp}\) is the heat load that must be covered by consuming reduced steam from the boilers.

The electrical capacity of the station is

\[ N_e = N_{ek} + N_{epo}. \]

The quantity of steam going to extraction will be determined as

\[ D_{po} = \frac{Q_{op}}{i_{po} - i'_{2po}}, \]

and the electrical capacity provided by the steam extracted from the turbine is

\[ N_{epo} = \frac{D_{po}(i_o - i_{po})}{860}\,\eta_m \eta_2. \]

Accordingly, we find

\[ N_{ek} = N_e - N_{epo}, \]

whence the steam consumption for the condensing flow is

\[ D_k = \frac{860\,N_{ek}}{(i_o - i_k)\eta_m\eta_2} = \frac{860}{(i_o - i_k)\eta_m\eta_2}(N_e - N_{epo}). \]

The required value \(D_{pp}\), that is, the steam consumption for the peak heater, is determined just as easily:

\[ D_{pp} = \frac{Q_{pp}}{i_o - i'_{2pp}} = \frac{Q - Q_{op}}{i_o - i'_{2pp}}. \]

If the electrical load remains unchanged, but the heat consumption covered by extraction steam increases, then the flow \(D_{po}\) develops, displacing the condensing flow; that is, the condensing generation of electric energy is reduced, which is also reflected very clearly on the diagram.

When the heat load is redistributed between peak and main heaters, the steam consumptions \(D_{po}\) and \(D_{pp}\) are redistributed; this is again clearly reflected in the diagram and again leads to a change in the ratio between electric-power generation by condensing and by district-heating processes.

The \(i\)--\(D\) diagram can also be used successfully to illustrate the question of the distribution of losses and the economy of operation of a complex electric-power system comprising a CHP plant, a condensing power plant, and a hydroelectric power plant. Here we must represent energy generation under the following headings:

1) generation of electric energy and coverage of heat consumption by the district-heating method;

2) generation of electric energy by the condensing method;

3) generation of heat energy to cover heat consumption by means of reduced steam from boilers;

4) generation of electric energy at hydroelectric power plants.

Let the system consist of 1 CHP plant and 1 hydroelectric power plant.

Let the electrical load supplied by the system be

\[ N_e = N_{ek} + N_{epo} + N_e^{\mathrm{HPP}}. \]

Here \(N_{\mathrm{e}}^{\mathrm{HPP}}\) is the load covered by the hydroelectric power station.

The electric power of the HPP is

\[ N_{\mathrm{e}}^{\mathrm{HPP}}= \frac{1000\,W\cdot H}{3600\cdot 102}\, \eta_{\mathrm{hT}}\cdot \eta_{\mathrm{m}}^{\mathrm{hT}}\cdot \eta_{\mathrm{g}}^{\mathrm{e}}, \]

\(W\) — water flow through the turbines, \(m^3/\text{hour}\),

\(H\) — the available head, \(m\),

\(\eta_{\mathrm{hT}}\) — efficiency of the hydraulic turbine,

\(\eta_{\mathrm{m}}^{\mathrm{hT}}, \eta_{\mathrm{g}}^{\mathrm{e}}\) — mechanical efficiency of the hydraulic turbine and efficiency of the hydrogenerator.

Expressing the power of the HPP in electrical calories, one may write:

\[ 860\,N_{\mathrm{e}}^{\mathrm{HPP}}= \frac{D_{\mathrm{HPP}}\cdot H}{427}\, \eta_{\mathrm{hT}}\eta_{\mathrm{m}}^{\mathrm{hT}}\cdot \eta_{\mathrm{g}}^{\mathrm{e}}, \]

\(D_{\mathrm{HPP}}\) — water flow in \(kg/\text{hour}\),

\[ \frac{H}{427}\eta_{\mathrm{hT}} \]
— the conventional “heat-content difference” of the HPP, expressed in \(kcal/kg\).

When constructing a rectangle whose area expresses \(860\,N_{\mathrm{e}}^{\mathrm{HPP}}\) in the coordinates \(i—D\), it is more convenient to increase the scale of the ordinates (that is, \(\eta_{\mathrm{hT}}\dfrac{H}{427}=(\Delta i)_{\mathrm{conv}}\)) by 427, 4270, … times, and to reduce the scale of the abscissas for the quantity \(D_{\mathrm{HPP}}\) by 427, 4270, … times, so that the rectangle obtained is not too elongated along the abscissa axis. It is clear that the area of the rectangle will not change thereby. The scales may also be changed by another identical number of times, depending on the magnitudes of the water flow and the head.

Then the diagram characterizing the energy production by the various units of the system will have the form shown in Fig. 3. In constructing the diagram, the “conventional heat-content difference” of the hydraulic turbine is laid off from the abscissa axis.

Such a diagram is extremely graphic. Nothing prevents us, in rectangles 8 and 9, from singling out rectangles \((1-\eta_{\mathrm{TS}})\cdot \text{pl. }8\), \((1-\eta_{\mathrm{TS}})\cdot \text{pl. }9\), in order to represent the losses of the heat network. Similarly, in rectangles 1, 2, 3, rectangles \((1-\eta_{\mathrm{e}}^{\mathrm{SN}})\cdot \text{pl. }1\), etc., may be singled out, showing the expenditure of electric energy for station auxiliary needs. Losses in electric networks may also be singled out. The number of separate flows in the diagram is not limited. Therefore, taking account of the specific situation, \(i—D\) diagrams may be constructed side by side for all stations included in the system.

In the case of joint operation of a CHPP and an HPP, we distinguish 4 flows.

1. Condensing flow. Its existence leads to large losses. Here there are both electromechanical and thermal losses. Development of this flow is not advantageous.

2. District-heating flow. Here there are only electromechanical losses and losses of the heat and electric networks (including the expenditure of electric energy for auxiliary needs). This flow must and is advantageous to develop, since it is necessary to supply both heat and electric demand under the specific conditions of the energy region.

3. HPP flow. Electric energy is generated without fuel expenditure, that is, there are no thermal losses, and only electromechanical losses and network losses exist, including the expenditure of electric energy for auxiliary needs. This flow is advantageous. Theoretically speaking, it should be developed in order to displace condensing kilowatt-hours. But this flow can be developed only insofar as this is necessary for displacing condensing kilowatt-hours. Displacement of district-heating

kilowatt-hours now leads to the development of a fourth flow—the flow of reduced steam—and may lead to overconsumption of fuel.

Thus, warmer weather under heating heat consumption should stimulate the transfer of load to hydroelectric power stations.

1. The flow of peak steam. It should be developed only to the extent necessary to satisfy heat demand. This flow is a forced flow.

As we see, the \(i-D\) diagram splendidly illustrates the basic ideas of modern power engineering. With its aid it is also easy to depict the generation of energy at hydroelectric power stations, clearly showing how the generation of electric power without fuel consumption through the use of “white coal” is reflected in heat losses.

Fig. 3

\[ \begin{aligned} 1&\;-\;860\,N_{\mathrm{ek}}\\ 2&\;-\;860\,N_{\mathrm{epo}}\\ 3&\;-\;860\,N_{\mathrm{GES}}' \end{aligned} \qquad \left\} \begin{array}{l} \text{electric-power generation} \end{array} \right. \]

\[ \begin{aligned} 4&\;-\;(1-\eta_m\eta_2)\,860\,N_{\mathrm{ek}}\\ 5&\;-\;(1-\eta_m\eta_2)\,860\,N_{\mathrm{epo}}\\ 6&\;-\;(1-\eta_m^{T}\eta_2^{T})\,860\,N_{\mathrm{e}}^{\mathrm{GES}} \end{aligned} \qquad \left\} \begin{array}{l} \text{electromechanical losses} \end{array} \right. \]

7 — Losses with the cooling water in the condenser.
8 — Heat generation through extraction of steam.
9 — Heat generation through reduced steam.

We shall use the same diagram to illustrate the benefits of regenerative heating of water at steam power stations.

Let us compare two turbine installations. One of them is of an obsolete type, without regenerative heating of feedwater. The second has regenerative heating by using three points of intermediate steam extraction. The heating of the water is carried out in mixing heaters according to the scheme shown in Fig. 4.

Both installations are identical in capacity and have identical values of the quantities \(\eta_m\), \(\eta_2\), \(i_0\), \(i_k\). In order to show how flexible the method of illustration by means of the \(i-D\) diagram is, we shall construct the diagram for \(1\ \mathrm{kWh}\) of the turbine’s internal power; that is, we shall construct the diagram in the coordinates \(i-d\), where \(d\) is the specific steam consumption per kilowatt-hour.

For the condensing installation without regeneration, the diagram will be very simple (Fig. 5a).

Fig. 4

In subfigure a):

• vertical axis: \(i,\ \dfrac{\mathrm{kcal}}{\mathrm{kg}}\);
• horizontal axis: \(d_i,\ \dfrac{\mathrm{kg}}{\mathrm{kWh}}\);
• top level: \(i_0\);
• intermediate level: \(i_k\);
• lower level: \(i_2'\);
• width: \(d_k\);
• shaded upper rectangle: \(860\ \dfrac{\mathrm{kcal}}{\mathrm{kWh}}\);
• main area: \(Q_{2k}\).

To the right of subfigure a):

\[ d' < d_k; \]

\[ Q_{2p} < Q_{2k}. \]

In subfigure b):

• shaded upper rectangle: \(860\ \dfrac{\mathrm{kcal}}{\mathrm{kWh}}\);
• levels: \(i_0\), \(i_{10}\), \(i_{20}\), \(i_{30}\), \(i_k\), \(i_2'\);
• main area: \(Q_{2p}\);
• widths: \(d'\), \(d_p\);
• side labels: \(\alpha_{10} d_p\), \(\alpha_{20} d_p\), \(\alpha_{30} d_p\);
• horizontal axis: \(d_i,\ \dfrac{\mathrm{kg}}{\mathrm{kWh}}\).

In subfigure c):

• vertical axis: \(i,\ \dfrac{\mathrm{kcal}}{\mathrm{kg}}\);
• shaded upper rectangle: \(860\ \dfrac{\mathrm{kcal}}{\mathrm{kWh}}\);
• levels: \(i_0\), \(i_{10}\), \(i_{20}\), \(i_{30}\), \(i_k\), \(i_2'\), \(i_{230}\);
• main area: \(Q_{2p}'\);
• widths: \(d'\), \(d_p\);
• side labels: \(\alpha_{10} d_p\), \(\alpha_{20} d_p\), \(\alpha_{30} d_p\);
• lower marked quantity: \(\Delta Q_g\);
• horizontal axis: \(d_i,\ \dfrac{\mathrm{kg}}{\mathrm{kWh}}\).

Fig. 5

Converted into mechanical work

\[ d_k(i_0-i_k)=860\ \text{kcal/kWh}. \tag{1} \]

Lost with the cooling water

\[ d_k(i_k-i_2')\ \text{kcal/kWh}. \]

Returned with the feedwater

\[ d_k i_2'\ \text{kcal/kWh}. \]

For an installation with regeneration, the equation of the energy balance, in calculation per kilowatt-hour of internal power, takes the form

\[ d_p(1-\alpha_{10}-\alpha_{20}-\alpha_{30})(i_0-i_k) +d_p\alpha_{10}(i_0-i_{10}) +d_p\alpha_{20}(i_0-i_{20})+ \]

\[ +d_p\alpha_{30}(i_0-i_{30})=860\ \frac{\text{kcal}}{\text{kWh}}. \]

After simple transformations,

\[ 860=d_p(i_0-i_k)\left(1-\alpha_{10}\frac{i_{10}-i_k}{i_0-i_k} -\alpha_{20}\frac{i_{20}-i_k}{i_0-i_k} -\alpha_{30}\frac{i_{30}-i_k}{i_0-i_k}\right); \]

\[ 860=d_p(i_0-i_k)\left(1-\sum \alpha_{m0}\frac{i_{m0}-i_k}{i_0-i_k}\right). \tag{2} \]

From comparison of (1) and (2) we obtain

\[ d_k(i_0-i_k)=d_p(i_0-i_k)\left(1-\sum \alpha_{m0}\frac{i_{m0}-i_k}{i_0-i_k}\right); \]

\[ d_p=\frac{d_k}{1-\sum \alpha_{m0}\dfrac{i_{m0}-i_k}{i_0-i_k}}. \]

Here \(i_{10}, i_{20}\) are the heat contents of the steam at the extraction points, kcal/kg; \(\alpha_{10}, \alpha_{20}\) are the fractional coefficients of the extractions.

We see that \(d_p>d_k\), which is understandable, since part of the steam works in the turbine with regeneration not down to the final parameters. But the heat losses with the cooling water in the turbine without regeneration will be greater. These losses are:

a) for the turbine without regeneration

\[ Q_{2k}=d_k(i_k-i_2')\ldots\frac{\text{kcal}}{\text{kWh}}, \]

b) for the turbine with regeneration

\[ Q_{2p}=d_p\left(1-\sum \alpha_{m0}\right)(i_k-i_2') =d_k(i_k-i_2')\frac{1-\sum \alpha_{m0}}{1-\sum \alpha_{m0}\dfrac{i_{m0}-i_k}{i_0-i_k}} \]

(see Fig. 5b)

\[ \frac{Q_{2p}}{Q_{2k}} = \frac{1-\sum \alpha_{m0}} {1-\sum \alpha_{m0}\cdot\dfrac{i_{m0}-i_k}{i_0-i_k}} <1. \]

In the \(i—d\) diagrams of the two turbines being compared, this will be reflected in the fact that the area of the triangle representing the losses of the turbine with regeneration will be smaller than the area representing the losses of the turbine without regeneration (with the same height of both of these rectangles, their widths will be different). This is also evident without the insets. In fact, the quantity of heat converted into mechanical energy is the same in both cases: \(860\ \text{kcal/kWh}\); that is, the areas marked by oblique hatching inclined to the left must be equal in area. Hence it follows inevitably that, for the turbine with regeneration, the area of the rectangle representing the losses will be smaller.

Let us note that the benefit of regeneration could also have been shown in the \(i—D\) diagram, and not in the \(i—d\) diagram, as we have done. With the aid of the \(i—d\) diagram it is easy, for example, to show what role is played by the arrangement for connecting the heaters in regenerative feedwater heating. Let the fractional extraction coefficients \(\alpha_{10}, \alpha_{20}\ldots\) remain unchanged. Let \(i_{10}, i_{20}\ldots\) also remain unchanged.

Fig. 6

But regenerative feedwater heating is carried out not according to the scheme shown in Fig. 4, but in surface heaters with the quite usual cascade removal of drains from the heaters (Fig. 6), that is, with removal of all drains from the low-pressure heater to the condenser. As shown in Fig. 5c, in this case the amount of heat additionally lost per each internal kilowatt-hour will be

\[ \Delta Q_{e}=d_{p}\sum \alpha_{m0}\left(i'_{230}-i'_{2k}\right)\frac{\text{kcal}}{\text{kWh}}, \]

which is easy to represent also in the \(i—d\) diagram (Fig. 5c).

With the aid of the \(i—D\) diagram one can vividly clarify certain points that do not lend themselves to consideration when ordinary methods of illustration are used. For example, in considering the question of the heat carrier in heat networks, we easily arrive at the conclusion that the electric power needed to circulate the water in the networks is produced by the condensing process. Indeed, when using water as the heat carrier, in addition to covering the district load (including here also the station’s own electrical auxiliaries), we must generate electric power for the network

pumps. The heat consumption of the district does not depend on what heat carrier is in the network. Therefore, for the district’s heat consumption, the district-heating method can produce only a limited quantity of electric power, determined by the magnitude \(D_{po}\), that is, determined by the heat consumption through the use of extraction. The additional kilowatt-hours for the network pumps will be generated by the condensing process (see

Fig. 7

Fig. 7). In the diagram this will be expressed by an increase in the width of the flow \(D_k\), since it is necessary to provide the additional condensing generation \(860N^{ch}_{ek}\), where \(N^{ch}_{ek}\) is the power consumed by the network pumps.

The conclusions that, it seems to us, should be drawn are as follows. Visual clarity of illustrations, simplicity of proofs, the use of a minimum of concepts while preserving the necessary clarity, and the consideration of phenomena and processes in their development should be the basis of teaching methodology in Soviet higher education. We have had occasion to hear objections to the use of the \(i-D\) diagram. The basis of these objections was that the \(T-S\) diagram was proposed for illustrations and calculations by Clausius’s contemporaries (Stodola believes that the \(T-S\) diagram was first used by Belpaire in 1873), and that therefore abandoning the illustrative use of the \(T-S\) diagram is an impermissible violation of the customary canons of technical thermodynamics. Since no one uses the \(T-S\) diagram for calculations of processes in steam-power plants, and its use is confined purely to illustrative purposes, we believe that the \(i-D\) diagram, as an illustration, has unquestionable advantages over the \(T-S\) diagram, since it makes it possible to illustrate both the qualitative and the quantitative aspects of the processes occurring in power installations, and, moreover, very simply shows the interrelation of the various methods of energy production in its development and as a function of the existing external conditions, determined by the ratio of electric and heat consumption.

Noted Misprints