ON THE USE OF HEAT FROM FLUE GASES OF INDUSTRIAL FURNACES

V. N. Smirenskii

General Considerations

The very sharp difference in the efficiency of fuel-consuming installations used purely for power generation (steam generators) and of various industrial furnaces cannot fail to attract the attention of heat engineers. Whereas modern, well-designed boiler installations operate with an efficiency on the order of 85–90%, in industrial furnaces the efficiency, as a rule, nowhere exceeds 20–30%, falling in individual cases to 5–10%.

Analysis of the operation of various industrial furnaces shows that in all cases the principal share of losses is due to the loss of heat with the outgoing gases, which have very high temperatures. The lowest values of the temperature of flue gases usually lie within the range 400–500°C, while the maximum values reach 1000–1200°C and higher. Taking into account the relative share of the various furnace units in their total number, it may be considered that, on average, this loss amounts to about 45%, i.e., a very substantial figure.

It should also be noted that in modern industry the predominant fuel consumers are mainly large installations, which yield large quantities of flue gases amounting to several tens of thousands of normal cubic meters per hour. This creates favorable conditions for their utilization.

The heat of the flue gases of various industrial installations contains enormous unused reserves. Modest theoretical calculations, made as applied to 1951, show that with the proper use of the heat of flue gases from only relatively large installations, an annual fuel saving for the national economy of the USSR of more than 2 million tons of standard fuel can be obtained. There is no doubt that these reserves must and will be used. In the USSR there are all the conditions necessary for this. First of all, we are obliged to do so by our fuel policy, which demands a careful attitude toward fuel resources, of a completely different kind from that in capitalist countries. Secondly, under a planned socialist economy there are favorable conditions for the broad and rapid introduction of every useful measure.

However, it cannot be said that in our industry utilization installations have already found their proper application. On the contrary, at many existing industrial furnaces, where the use of the waste heat of gases is entirely possible and profitable, utilization installations are encountered rather rarely. This is by no means always provided for in the design of new furnace installations or even of new shops. Several principal reasons for this state of affairs may be enumerated.

First, there is sometimes an underestimation of the significance that utilization installations may have for individual enterprises and for the entire national economy as a whole.

Secondly, there is also the conservatism of some production personnel, an unwillingness to complicate the production installation, and fear of interference with production and of unnecessary trouble.

Thirdly, the circumstance is also of great importance that, to this day, there are still not sufficiently good and fully established designs and types of heat-recovery installations that operate reliably under different specific production conditions.

Fourthly, there has been insufficient scientific and technical development of the basic questions involved in the use of secondary resources in industry.

Among the various heat-transfer media obtained by utilizing the heat of exhaust gases from industrial furnaces, the most expedient is the production, in heat-recovery installations, of steam—and, above all, steam of sufficiently high parameters with the possibility of using it for power purposes. Other possible heat-transfer media—water and air—cannot be regarded as the principal ones for the complete utilization of all the thermal possibilities of waste heat from industrial furnaces at the overwhelming majority of installations, chiefly large ones. Hot water and air obtained in heat-recovery installations at large enterprises (metallurgical, cement, glassmaking, and other industries) will often be unable to find nearby a sufficient number of consumers. Therefore such heat-recovery installations should be regarded as certain special cases of heat use at enterprises where the appropriate conditions for this exist. The production of steam of sufficiently high parameters in waste-heat boilers is the most universal solution to the problem of heat utilization, since such steam will always find application either in power installations for generating electric power, or for production purposes.

Many of the attempts made earlier to utilize the heat of exhaust gases must be acknowledged as insufficiently successful, since sometimes random types of boilers were employed, often without any consideration of the specific operating conditions of the heat-recovery unit and the local conditions of its installation. Horizontal water-tube boilers, vertical water-tube boilers, and boilers with fire tubes of normal and special types were installed. There are isolated cases of installations of waste-heat boilers with forced circulation as well.

For a heat-recovery installation it is impossible to specify a universal type of waste-heat boiler suitable for all possible cases of using the heat of gases, since the operating conditions may be of the most varied kind. Here it is necessary to keep in mind the temperatures of the heating gases, the quantity and quality of the entrainment in the gases—chiefly with respect to its melting temperature—the presence in the gases of products of incomplete combustion and of sulfur compounds, the data on local conditions necessary for the layout of the heat-recovery unit, and so forth. But if one disregards some of the specificity of the operation of individual heat-recovery units, they must all satisfy the following basic requirements:

1. A high degree of compactness, because, as a rule, heat-recovery installations will be placed within strict, limited overall dimensions.

2. Simplicity of manufacture for different capacities and parameters with low metal consumption.

3. Reliability of operation and simplicity of servicing.

From the first basic requirement it follows, first, that the heating surfaces must operate highly efficiently and, second, that, from the standpoint of design, the installation must be readily adaptable to various local dimensional conditions, without requiring major and complicated design changes.

In those cases where the temperatures of the flue gases from industrial furnaces are sufficiently high, waste-heat boilers will operate with the use, to one degree or another, of radiant heat exchange. But in the overwhelming majority of cases, judging by the temperature conditions of the heating gases, convective heat exchange will take place in the waste-heat installations. Thus, in contrast to ordinary boiler installations, in waste-heat boilers the convective heating surface will be the principal, and often the only, one. Therefore, in the design and structural arrangement of waste-heat boilers, these operating features must be taken into account first of all.

Some Data and Conclusions from Theoretical Studies

The high values of the heat-transfer coefficients needed to make waste-heat boilers compact can be obtained by reducing the tube diameter and increasing the gas velocities. In order to study the influence of these factors on the heat-transfer coefficient, the heating surface of the installation, gas resistances, dimensions, the quantity of net steam delivered by the boiler, metal consumption, and payback periods of the installation, special theoretical studies were carried out. It should be noted that, in its general form, this problem cannot be solved because of the presence of a number of extraneous factors, also of substantial importance, which cannot be taken into account. Therefore several series of comparative calculations were performed. The resulting data for some of them are given below. The relationships shown in the graphs presented refer to a waste-heat boiler with a superheater and economizer, the calculations for which were based on the following initial data.

Initial temperature of the flue gases . . . 600°C
Hourly quantity of gases . . . 75000 nm³/h
Capacity of the installation (gross) . . . 13500 kg/h
Steam pressure . . . 35 ata
Superheat temperature . . . 420°C
Feed-water temperature . . . 50°C
Temperature of the exiting gases . . . 188°C

Since the steam pressure was taken as elevated, the case of gas flow inside the tubes (the type of boiler with fire tubes) was excluded from the comparative calculations. To obtain regularities over broader limits, the calculations were carried out up to values of tube diameters and gas velocities of 80 mm and 40 m/sec, respectively. Figures 1, 2, and 3 show the change in the total heating surface of the boiler, superheater, and economizer (solid lines) and in the gas resistances of the heating surfaces of the waste-heat installation (dashed lines) as a function of changes in the outside tube diameter and gas velocities. The calculations were made by ordinary methods. The break in the curves at tube-diameter values of 40 mm is explained by the fact that, up to 40 mm, the tube diameter of all elements of the installation was changed simultaneously. At tube-diameter values above 40 mm, only the dimensions of the boiler and economizer tubes were changed, while the diameters of the superheater tubes remained constant and equal to 40 mm. In all three cases of tube arrangement and gas-flow direction, the ratio of pitch to diameter was taken as \(s/d = 2.5\). These remarks, however, also apply to all the other graphs. From Figs. 1–3 it is evident over what a wide range the total heating surface of the waste-heat installation and the gas resistances change with changes in gas velocities, tube diameters, and their arrangement

and the direction of the gas flow as applied to the case under consideration. But on the basis of these regularities it is still impossible to give preference to one variant or another, since a number of conditions, very important for the waste-heat installation, remain unclear: the volume necessary for placing the installation, the net steam output of the boiler, the cost of steam, the metal consumption, and the payback period of the installation.

Figures 4–6 show the variation in the volume occupied by the installation, calculated from the internal dimensions of the gas ducts (solid lines),

Fig. 1

and of the specific metal consumption for the tubes (dashed lines), as a function of the change in tube diameter and gas velocities.

Figures 7–9 give an idea of the variation in the net output of the installation \(D_k\) (solid lines), the cost of a ton of steam \(Ц\) (dashed lines), and the payback period of the installation \(X\) (dash-dot lines), as a function of the change in tube diameter and gas velocity. Here, in calculating the cost \(m\) of steam and the payback period of the installation, the following data were adopted. The average assumed cost of the installation, per 1 sq. m of heating surface, including installation, foundation, frame, steam pipelines, and auxiliary—

heating equipment—800 rubles. Deductions for depreciation of the equipment, repairs, and materials—10%.

The cost of chemically purified water was taken conditionally on the basis of 1 ruble/m³, allowing for 50% return.

The annual operating time of the installation is 7000 hours.

Fig. 3

In accounting for the costs of operating personnel, the least favorable case was adopted—1 stoker per boiler. With simultaneous operation of several boilers, the adopted norm may be reduced.

The cost of 1 t of steam obtained in a normal boiler installation, needed for calculating the payback periods of the installation, was taken conditionally as 15 rubles.

Fig. 2

The cost of the electric power required to drive the induced-draft fan, feed pumps, etc., was not calculated directly in monetary terms. This component was accounted for automatically, since from the total steam output of the gross boiler the corresponding steam consumption was deducted.

for auxiliary service. The remainder constituted the net steam available for delivery by the boiler.

The steam consumption for driving the induced-draft fans was determined from the expression:

\[ D_{\partial}=k'\,\frac{V_{\text{sec}}\Sigma h}{102\,\eta_{\partial}}\,d\ \text{kg/hour}, \]

where
\(V_{\text{sec}}\ \text{m}^3/\text{sec}\) — the per-second capacity of the induced-draft fan,
\(\Sigma h\) — mm water column — the calculated head,
\(\eta_{\partial}\) — efficiency of the induced-draft-fan installation,
\(d\ \text{kg}/\text{kWh}\) — specific steam consumption of the induced-draft fan,
\(k'\) — overall reserve coefficient.

For average values of \(\eta_{\partial}\), \(d\), and \(k'\), we obtain

\[ D_{\partial}=0.15V_{\text{sec}}\Sigma h\ \text{kg/hour}. \]

If the steam consumption for pumps and other purposes is taken as 5% of the boiler output \(D_{\mathrm{br}}\), then the total steam consumption for the plant’s own needs is expressed by the quantity

\[ D_{\mathrm{br}} - D_{\mathrm{n}} = 0.15 V_{\mathrm{sec}}\Sigma h + 0.05 D_{\mathrm{br}}, \]

Fig. 7.

whence the quantity of net steam delivered by the boiler is

\[ D_{\mathrm{n}} = 0.95 D_{\mathrm{br}} - 0.15 V_{\mathrm{sec}}\Sigma h \ \text{kg/hour}. \]

In this way the quantity of steam delivered by the boiler was determined.

Fig. 6.

The relationships shown in Figs. 4–9 already give a more complete picture for the various arrangements of the waste-heat boiler. If all specific data applicable to local conditions are available, it is possible to outline the most correct solution to the question of arrangement.

The curves in Figs. 10–12 give an idea of the change in the net steam delivered by the boiler (solid lines), the cubage required for the installation

(dashed lines), and the payback periods (dash-dotted lines) as functions of the gas velocities at different pipe diameters.

From these graphs, as well as from the curves shown in the preceding figures, the effect of the pipe diameter is perfectly clear.

With a decrease in the pipe diameter, in all cases the overall dimensions of the installation are sharply reduced, which is of very substantial importance for waste-heat boilers. The metal consumption and the payback periods of the installation are also reduced. At the same time, the quantity of steam delivered by the boiler increases somewhat.

As regards the effect of the gas velocity, as is evident from the graphs, the quantity of net steam delivered by the boiler decreases with increasing velocity in all cases. For longitudinal gas flow this decrease is less pronounced, and the sharpest reduction in the quantity of steam delivered is observed with a staggered arrangement of the tubes and transverse washing of them by the gases.

Fig. 10

Fig. 11

The regularity in the change of the volume required for placing the installation is the same for all cases; but at the same gas velocities the bulkiest installations are obtained with longitudinal gas flow, and the most compact with a staggered arrangement of the tubes and transverse gas flow. This does not mean, however, that if it is necessary to have a compact installation, preference should be given to transverse gas flow, since in that case it is necessary to take into account the expenditure of energy for internal needs, or the net steam delivery. The decision may prove more correct when choosing longitudinal flow with increased gas velocities, especially if the latter contain a considerable amount of solid entrainment.

It is interesting to trace the influence of a change in gas velocity on the payback periods of the installation for different arrangements of the heating surfaces. For longitudinal flow, the higher the gas velocity, the shorter the payback period of the installation. In this case, as gas velocities increase to 15–20 m/sec, this reduction in payback periods is very pronounced,

Fig. 12

whereas at higher velocities the reduction is less noticeable. The floor space required for placing the installation changes with the gas velocity in approximately the same way, while the amount of steam delivered by the boiler gradually decreases. For transverse gas flow, on the contrary, the payback periods, both with staggered and with in-line arrangement of the tubes, increase as the gas velocity increases; and the smaller the tube diameter, the less noticeable this increase is. If, in addition, attention is paid to the sharp decrease in the amount of steam delivered by the boiler net, it becomes quite obvious that the payback period of the installation alone cannot serve as the principal criterion in choosing the type of installation.

Above, certain data from theoretical studies were presented as applied to an initial temperature of the heating gases of 600°C, i.e., close to the case most often encountered in practice for waste-heat installations when their heating surfaces operate as convective ones. In the course of the calculations, a whole series of quantities was chosen that may be subject to corrections to one degree or another; however, changes in these quantities obviously cannot affect the character of the regularities and the conclusions that can be drawn from an analysis of these regularities. Taking into account the requirements that a waste-heat installation must satisfy, the following principal conclusions may be drawn on the basis of the material presented.

1. Waste-heat boilers should, in the main, be made of tubes of small diameter. For convective heating surfaces, tubes with an internal diameter on the order of 20–30 mm should be regarded as the most acceptable.

Small tube diameters will be of greater importance at lower temperatures of the gases being utilized.

1. Gas velocities should be chosen with allowance for local conditions and should be coordinated with the dimensions available for the waste-heat installation.

The most expedient values of gas velocities for longitudinal flow will evidently be on the order of 15–20 m/sec. For transverse washing of the tubes, they should not exceed 8–10 m/sec; moreover, the lower velocity values should be adopted for a staggered arrangement of the tubes.

At the indicated gas velocities, the total resistances of the installation will not exceed 100–120 mm water column.

1. With respect to metal consumption and overall dimensions of the installation, at the indicated values of gas velocity, the most advantageous is the use of a staggered arrangement of tubes with transverse gas flow; next comes an in-line arrangement of tubes and, finally, a layout with longitudinal gas flow.

2. The expenditure of energy for complete self-servicing of the installation, expressed in terms of steam, will amount to about 10% of the gross boiler output.

3. If the above conditions are fulfilled, the payback periods of the installation will usually be within the range of 1–2 years.

Note. 1. In choosing the layout of the heating surfaces, other data not mentioned here must also be taken into account: clogging of the gas by entrainment and the possibility of wear of the tubes or of its deposition in the gas ducts, the quality of the entrainment and the possibility of slagging of the heating surfaces, high initial temperatures of the heating gases, etc.

1. A number of the conclusions presented are also applicable to the convective heating surfaces of ordinary boiler units (for example, water economizers).

Thus, if we set aside certain special cases of utilizing the heat of gases with very high initial temperatures, then, in the main, waste-heat installations will operate chiefly through convective heat exchange. As was shown above, this will require the use of small-diameter tubes in order to reduce metal consumption and obtain compact installations. The use of small-diameter tubes for heating surfaces and the creation of compact installations will cause large hydrodynamic resistances in the steam-water path, and this, in turn, will inevitably require the use of forced circulation. As is known, boilers with forced circulation have a number of substantial advantages, especially valuable for waste-heat installations. They make it possible to vary widely the arrangement of heating surfaces when designing the heat exchanger and make it possible to obtain a very compact installation. At the same time, reliable circulation is ensured under considerable fluctuations of the heat load, which will be a frequent occurrence for waste-heat boilers. But, along with this, boilers with forced circulation also have weak points. Of lesser importance here is the question of the additional expenditure of energy for driving the circulation pump. Undoubtedly of greater importance for a waste-heat installation, from which simplicity of design and reliability of operation are required, will be the presence of the circulation pump itself, operating under severe conditions at high pressure on water close to the boiling state. This complicates both the installation itself and its operation, makes it more expensive, and raises concern over reliability of operation.

In order to eliminate this complex and undesirable component from the scheme of a boiler with forced circulation, a variant of a boiler with forced circulation without a circulating electric pump has been proposed. The functions of the latter are to be replaced by a steam-jet water pump installed ahead of the lower header or drum on the downcomer branch of the circulation circuit. A steam-jet pump has substantial advantages in comparison with a circulating electric pump: absence of moving parts, elementary simplicity of its design, manufacture, and servicing, compactness, low cost, and reliability of operation.

A steam-jet pump can operate on steam taken from an external source, for example, the plant boiler-house, power station, or production facility; from a specially installed natural-circulation boiler when there is a large number of waste-heat boilers; or, finally, a combined self-sustaining waste-heat boiler may be built. At the same time it is necessary to point out the absence of a theory of steam-jet water pumps and methods for calculating them. Below are presented some theoretical considerations and an approximate calculation method for a steam-jet water pump operating in the scheme of a boiler with forced circulation.

Elementary theory of a steam-jet water pump operating in the scheme of a boiler with forced circulation

Jet pumps are, in their construction, elementarily simple. They find application in the most diverse branches of industry and technology: in mining, in the chemical industry, in the production of large earthworks, for pumping and transporting various liquids, gaseous and solid granular bodies, for creating a vacuum, pulverizing, etc., etc. As the working medium in them, various substances are also used: gas, liquid, steam. However, despite the simplicity of design of these devices, the phenomena occurring in them during

operation remain, to this day, completely unclear and unstudied. This applies especially to jet pumps operating with different driving and driven media, in which, during impact and mixing inside the pump itself, a change in the aggregate state of the substances occurs and does not go to completion. It is precisely to this type of apparatus that the steam-jet water pump belongs, intended to operate as a circulating pump for a boiler with forced circulation. The pump must transport water at a temperature close to the boiling temperature corresponding to the given pressure. In this case the following phenomena take place in the pump: conversion of the potential energy of the steam into kinetic energy, suction of the liquid, impact of two jets of steam and liquid and the associated loss of kinetic energy, mixing with partial change in the aggregate state of the steam and water, friction against the walls and mutual friction at the boundary of the two phases of the moving jets, local losses, and, finally, conversion of the kinetic energy of the total mixture flow into pressure energy. Some of these phenomena are so complex that at present it does not seem possible to account for and evaluate them in any way. These include such questions as interaction at the boundary of the two phases—liquid and steam—when they move at different velocities and under different parameters; evaluation of the energy loss upon impact of two jets; questions of steam discharge into an aqueous medium under different parameters of the steam and water; the rate of the processes during mixing of steam and water jets under different parameters and velocities; and a number of other questions. All this leads to the fact that, at present, it does not yet seem possible to approach the problem of calculating such a pump in a strictly theoretical manner and to evaluate quantitatively each phenomenon occurring in it. Even in simpler cases, such as, for example, a water-water jet pump, where there is no change in the aggregate state of the bodies—which greatly simplifies the solution of the problem—there still remain many uncertainties and difficulties in the calculation.

The present work does not aim to provide a solution to the questions listed above. These questions have been posed as current problems for solution at the Department of Heat-Power Installations of TPI. Here a possible method is presented for calculating a steam-jet water pump for a boiler with forced circulation, developed on the basis of the usual concepts of hydraulics, mechanics, and thermodynamics, taking into account a certain amount of available experimental material. In the proposed calculation method, a quantitative evaluation of the separate losses associated with the phenomena occurring in the pump is bypassed, and the total internal losses of the pump may be evaluated at the end of the calculation as a summed quantity expressed in fractions of the kinetic energy of the working steam jet.

The usually very low mechanical efficiency of jet pumps, in this case, when it is used as a circulating pump for a boiler, is not of essential significance, since the losses associated with the imperfection of its operation are returned to the mixture in the form of heat and are utilized by the boiler. Thus, the thermal efficiency of the steam-jet pump is very high and approaches 100%.

Basic Calculation Formulas

For the following discussion we shall adopt the following notation.

For the steam entering the nozzle:

\[ \begin{aligned} \text{pressure} \ . . . . .\ & p_1 \ \text{ata},\\ \text{heat content} \ . . .\ & i_1 \ \text{kcal/kg},\\ \text{specific weight} \ . . .\ & \gamma_1 \ \text{kg/m}^3,\\ \text{flow rate} \ . . . . .\ & G_1 \ \text{kg/sec},\\ \text{velocity at the outlet}\\ \text{from the nozzle} \ . . .\ & c_1 \ \text{m/sec}. \end{aligned} \]

Specific weight of the steam at the nozzle outlet . . . . . . . . . . . $\gamma_1^{x}\ \mathrm{kg/m^3}$,
Heat content of the steam at the outlet from the nozzle . . . . . . . . . . . $i_1^{x}\ \mathrm{kcal/kg}$.

For water at the inlet to the mixing chamber of the pump:

flow rate . . . . . . . . . . . . . . . . $G_2\ \mathrm{kg/sec}$,
heat content . . . . . . . . . . . . . . $i_2\ \mathrm{kcal/kg}$,
specific weight . . . . . . . . . . . . $\gamma_2\ \mathrm{kg/m^3}$,
mean velocity . . . . . . . . . . . . . $c_2\ \mathrm{m/sec}$.

For the mixture in the throat of the diffuser:

flow rate . . . . . . . . . . . . . . . . $G\ \mathrm{kg/sec}$,
heat content of the mixture . . . . . . . $i\ \mathrm{kcal/kg}$,
specific weight ” . . . . . . . . . . . . $\gamma\ \mathrm{kg/m^3}$,
velocity . . . . . . . . . . . . . . . . $c\ \mathrm{m/sec}$,
specific weight of water . . . . . . . . $\gamma_b\ \mathrm{kg/m^3}$,
specific weight of steam . . . . . . . . $\gamma_n\ \mathrm{kg/m^3}$,
steam content . . . . . . . . . . . . . $x$,
heat of vaporization . . . . . . . . . . $r\ \mathrm{kcal/kg}$,
heat content of water . . . . . . . . . $i'\ \mathrm{kcal/kg}$.

Steam pressure in the boiler . . . . . . $p_k\ \mathrm{ata}$.
Heat content of the water in the boiler . . . $i'_k\ \mathrm{kcal/kg}$.
Specific weight of steam at the pressure in the boiler . . . . . . . . . . $\gamma_n^k\ \mathrm{kg/m^3}$.
Specific weight of water at the pressure in the boiler . . . . . . . . . . $\gamma_b^k\ \mathrm{kg/m^3}$.
Heat content of steam in the boiler . . . $i''_k\ \mathrm{kcal/kg}$.
Heat content of water ” . . . . . . . . . $i'_k\ \mathrm{kcal/kg}$.
Heat of vaporization ” . . . . . . . . . $r_k\ \mathrm{kcal/kg}$.

Distance from the pump axis to the water level in the upper drum, $h\ \mathrm{m}$.

Multiplicity of pump delivery, or injection coefficient,

$$ k=\frac{G_2}{G_1}\ \mathrm{kg/kg}. $$

Circulation ratio with respect to the quantity

of mixture leaving the pump . . . . . . . . . . . . . .

$$ K=\frac{G}{G_n}\ \mathrm{kg/kg}. $$

Quantity of steam formed in the evaporating part of the circuit owing to the heat of the gases . . . . . . . . . . . $G_n\ \mathrm{kg/sec}$.

Resistances of the rising section of the circulation circuit from the outlet of the diffuser to the upper drum . . . . . . . . . . . . . . . $\Delta p\ \mathrm{kg/m^2}$.

Resistances of the descending section from the upper drum to the receiving chamber . . . . . . . . . . . . . . . $\Delta p_{on}\ \mathrm{kg/m^2}$.

Mean specific weight in the heated (rising) circuit . . . . . . . . . . . . . . $\gamma_{\mathrm{cp}}\ \mathrm{kg/m^3}$.

Specific weight of the steam–water mixture at its outlet into the drum . . . . . . . . . . . . . . $\gamma'\ \mathrm{kg/m^3}$.

Pressure in the mixing chamber . . . . . . . . . . $p_0\ \mathrm{ata}$.

Coefficient of diffuser resistance . . . . . . . . . $\zeta$.

Coefficient of resistance of the annular section for the passage of water into the mixing chamber . . . . . . . . $\zeta_2$.

Velocity coefficient of the steam nozzle . . . . . . $\varphi_1$.

Conventional mechanical efficiency of the pump . . . $\eta_m$.

Basing ourselves on the theorem of conservation of the quantities of motion of two colliding bodies before and after impact, we may write:

\[ \varphi_k\left(\frac{G_1}{g}c_1+\frac{G_2}{g}c_2\right)=\frac{G}{g}c, \]

where \(\varphi_k\) is the velocity coefficient of the mixing chamber. On the basis of the laws of hydrodynamics it may be assumed that, in the present case, the velocity coefficient for the mixing chamber will differ little from unity. Therefore, in what follows we consider it possible to take \(\varphi_k=1\).

After division by \(G_1\), we obtain:

\[ c_1+kc_2=(k+1)c. \]

Hence we shall have expressions for \(k, c_1\), and \(c_2\):

\[ k=\frac{c_1-c}{c-c_2}\ \text{kg/kg} \tag{1} \]

\[ c_1=(k+1)c-kc_2\ \text{m/sec} \tag{2} \]

\[ c_2=\frac{(k+1)c-c_1}{k}\ \text{m/sec}. \tag{3} \]

Expression (1) gives the value of the maximum feed ratio for a pump operating under known conditions, provided certain velocity relations are satisfied.

Let us now find the relation of the feed ratio to certain parameters characterizing the state of the working steam, the water, and the mixture. For this purpose we shall write the heat-balance equation for the working steam and water entering the pump and for the mixture leaving it, taking into account that the loss of kinetic energy occurring in the impact of the two jets is returned to the mixture in the form of heat. We write the balance for 1 kg of working steam.

For a steam heat content at the nozzle outlet \(i_{1x}\) and velocity \(c_1\), the total quantity of heat in the steam flowing from the nozzle is equal to

\[ i_{1}^{x}+\frac{Ac_1^2}{2g}\ \text{kcal/kg}. \]

If the heat content of the water entering the pump is \(i_2 \simeq i'_k\), and the feed ratio is \(k\), then the total quantity of heat of the water moving with velocity \(c_2\) into the mixing chamber, per 1 kg of working steam, will be

\[ ki_2+k\frac{Ac_2^2}{2g}\ \text{kcal/kg}, \]

Fig. 13

Let us assume that the process of heat exchange during the mixing of two streams of steam and water takes place sufficiently rapidly, ending essentially in the mixing chamber, although this may not be entirely in accord with reality. The solution of this question requires special investigations.

During mixing and impact of the jets, the steam, losing part of its kinetic energy, gives it up in the form of heat to the water being entrained. The heat content of this water was lower than the heat content of water at boiling, corresponding to the pressure in the mixing chamber. In this case it may turn out that all the water will be heated to boiling, and part of the steam will remain in excess, i.e., we shall have a mixture in the form of “wet steam” (mainly water). With the heat content of the mixture equal to \(i\), the quantity of heat in the mixture per \(1\ \mathrm{kg}\) of working steam will be equal to

\[ (k+1)i\ \text{kcal/kg}. \]

In addition, the mixture in the throat of the diffuser possesses kinetic energy, which, when converted into heat, will amount to

\[ (k+1)\frac{Ac^{2}}{2g}\ \text{kcal/kg}. \]

The heat balance is written as follows:

\[ i_1^x+\frac{Ac_1^2}{2g}+k\left(i_2+\frac{Ac_2^2}{2g}\right) =(k+1)\left(i+\frac{Ac^2}{2g}\right). \]

Bearing in mind that

\[ i_1^x+\frac{Ac_1^2}{2g}=i_1, \]

from the balance equation the following expressions may be obtained:

\[ i=\frac{i_1+k\left(i_2+\frac{Ac_2^2}{2g}\right)}{k+1} -\frac{Ac^2}{2g}\ \text{kcal/kg}, \tag{4} \]

\[ k=\frac{i_1-i-\frac{Ac^2}{2g}} {i-i_2+\frac{A}{2g}(c^2-c_2^2)}\ \text{kg/kg}. \tag{5} \]

Let us note that in these expressions the influence of the terms containing the velocities is very small. Thus, if for \(c\) one takes the value \(30 \div 40\ \mathrm{m/sec}\), then the value of the term

\[ \frac{Ac^2}{2g} \]

will be of the order of \(0.1 \div 0.2\ \text{kcal/kg}\).

As for the term

\[ \frac{Ac_2^2}{2g}, \]

in all cases it will not exceed thousandths of a calorie. Thus, when using formulas (4) and (5), in a large number of calculation cases the indicated terms may be neglected, all the more so in preliminary calculations, for which formulas (4) and (5) may be represented in the following form:

\[ i \simeq \frac{i_1+ki_2}{k+1}\ \text{kcal/kg}, \tag{4′} \]

\[ k \simeq \frac{i_1-i}{i-i_2}\ \text{kg/kg}. \tag{5′} \]

Expression (5) can be represented in another form. Since \(i=i' + xr\), then

\[ k = \frac{ i_1-i' - \dfrac{Ac^2}{2g} - xr }{ i' - i_2 + \dfrac{A}{2g}(c^2-c_2^2) + xr } \ \text{kg/kg}. \tag{6} \]

From this equation it is evident that the lower the steam content in the mixture, the greater the delivery ratio will be.

Let us derive several additional thermo- and hydrodynamic relations that will be needed subsequently in calculating the steam-jet water pump.

From the expression for the specific volume of wet steam

\[ v=xv_n+(1-x)v_b \]

or

\[ \frac{1}{\gamma}=\frac{x}{\gamma_n}+\frac{1-x}{\gamma_b}, \]

we obtain

\[ \gamma=\frac{\gamma_b\gamma_n}{x\gamma_b+(1-x)\gamma_n}. \tag{7} \]

In addition, for \(x\) one may write the expression:

\[ x=\frac{i-i'}{r}. \]

Substituting the value of \(x\) into the expression for \(\gamma\) (7), we obtain:

\[ \gamma = \frac{\gamma_b\gamma_n}{ \dfrac{i-i'}{r}(\gamma_b-\gamma_n)+\gamma_n } \ \text{kg/m}^3. \tag{8} \]

By expression (8), the specific weight of the steam-water mixture in the throat of the diffuser can be determined. If the working steam is completely condensed at the low temperature of the incoming water and the final heat content of the mixture is below the heat content of the liquid at the pressure in the mixing chamber, then the specific weight of the mixture \(\gamma\) may be taken from tables according to the actual heat content of the mixture.

During operation of the boiler, the steam-jet pump, at a certain (normal) operating regime, must overcome the known resistances of the circulation circuit. The pump overcomes only part of the total resistances of the circuit, since the other part is overcome by the natural driving head of circulation, created by the difference in the specific weights of water in the downcomer section of the circuit and of the mixture of steam with water in the riser section at a known height \(h\). The specific weight of this mixture will depend both on the thermal stress of the heated surfaces and on the specific weight of the mixture leaving the pump. The resistances of the circuit must be determined for the selected (desired) circulation ratio, which, for waste-heat boilers with forced circulation operating in the region of low gas temperatures, may be chosen somewhat lower than for non-waste-heat boilers. In calculating the resistances it should be borne in mind that the amount of water passing through the downcomer section \(G_2\) will be less than the per-second amount for the riser section \(G\) by the amount of the per-second steam consumption \(G_1\).

Let us find an expression for determining the pressure in the mixing chamber immediately beyond section I—I. It can be found as the difference between the sum of the pressures acting in the downcomer section from the water level of the upper drum to the mixing chamber, and the resistances in the same section, i.e.

\[ 10000\,p_0=10000p_k+h\gamma_2-\Delta p_{on}-\zeta_2\frac{c_2^2}{2g}\gamma_2-\frac{c_2^2}{2g}\gamma_2\ \text{kg}/\text{m}^2. \]

or

\[ p_0=p_k+\frac{h\gamma_2-\Delta p_{on}-(1+\zeta_2)\frac{c_2^2}{2g}\gamma_2}{10000}\ \text{ata} \tag{9} \]

and

\[ c_2=\sqrt{\,2g\,\frac{10000(p_k-p_0)+h\gamma_2-\Delta p_{on}}{\gamma_2(1+\zeta_2)}\,}\ \text{m/sec}. \tag{9'} \]

In the last two formulas the quantities \(p_0\) and \(c_2\) are mutually related, and if one of them were known it would be possible to find the other. Unfortunately, owing to the extreme complexity of the phenomena occurring in the mixing chamber when the two streams entering it meet, it is not possible to find the value of the pressure at the beginning of the mixing chamber \(p_0\), and consequently the velocity of water entry \(c_2\). From the practice of injector operation it is known that in individual cases they develop a vacuum of up to \(0.6\div 0.7\) atm. This vacuum depends on a number of factors: the outflow velocity of the working steam (or \(p_1\)), the ratio of the cross sections for the passage of water and steam, and the temperature of the water being sucked in. However, the action of injectors differs sharply from the operation of a jet steam-water pump. If in the former the vacuum in the mixing chamber is created both by the action on the surface of the jets and by the condensation of the working steam, then in the latter the condensation of steam will manifest itself much more weakly. Therefore, evidently, the relative vacuums (with respect to the pressure in the receiving chamber) obtained in the mixing chamber of steam-jet water pumps serving boilers with forced circulation will be lower than in injectors. If in injectors the velocities of water entry are usually about \(4—6\ \text{m/sec}\), then in steam-jet water pumps operating on hot water they will more often be within the range up to \(4\ \text{m/sec}\): in the experiments carried out, with a small difference between the temperatures of the steam and of the inlet water, the velocity \(c_2\) varied in different cases from 1 to \(3\ \text{m/sec}\). The magnitude of the inlet velocity will depend to a considerable extent on the difference between the temperature of the working steam and the temperature of the water sucked into the chamber. The greater this difference, the more advantageous in all respects the operation of the pump. Therefore colder feedwater should be supplied not into the boiler drum, but directly into the downcomer pipes. This is all the more important when the resistance of the circuit is large and the boiler height \((h)\) is small. It should be noted here that if feedwater is supplied into the downcomer pipes through a jet water-water device as the working fluid, then the operation of the steam-jet device will be considerably facilitated and improved.

Let us write the analogous equation for the riser (heated) section of the circuit from section II—II (the throat of the diffuser) to the upper drum:

\[ 10000p_0+\frac{c^2}{2g}\gamma=10000p_k+h\gamma_{\text{cp}}+\Delta p_n+\zeta\frac{c^2}{2g}\gamma, \]

whence the velocity in the throat of the diffuser will be equal to

\[ c=\sqrt{\,2g\,\frac{\Delta p_n+h\gamma_{\text{cp}}+10000(p_k-p_0)}{\gamma(1-\zeta)}\,}\ \text{m/sec}, \tag{10} \]

or, using expression (9) for \(p_0\),

\[ c=\sqrt{\,2g\frac{\Delta p+\left(1+\zeta_2\right)\dfrac{c_2^2}{2g}\gamma_2-h(\gamma_2-\gamma_{\mathrm{cp}})}{\gamma(1-\zeta)}\,}\ \text{m/sec}, \tag{10'} \]

where

\[ \Delta p=\Delta p_n+\Delta p_{\mathrm{op}}. \]

In the last expressions, \(\gamma_{\mathrm{cp}}\) represents the value of the specific weight, averaged over the height \(h\), in the heated part of the circuit. Taking into account the steam leaving the pump and self-evaporation as the mixture rises to the upper drum, this quantity can be found as follows. Let \(K\) denote the circulation ratio with respect to the quantity of mixture leaving the pump, i.e.,

\[ K=\frac{G}{G_n}, \]

where \(G_n\), kg/sec, is the quantity of steam obtained in the evaporating part of the circuit as a result of its heating. Then the heat content of the steam-water mixture as it leaves the riser tubes into the drum will be equal to

\[ i_{\mathrm{out}}\simeq \frac{(K-1)i+i_k''}{K}\ \text{kcal/kg}, \tag{11} \]

where \(i_k''\) is the heat content of steam at the pressure in the upper drum of the boiler, and \(i\) is the heat content of the mixture at the diffuser throat.

By an equation analogous to (8), the specific weight of the steam-water mixture as it leaves the tubes into the boiler drum may be determined as

\[ \gamma'=\frac{\gamma_b^k\gamma_n^k}{\dfrac{i_{\mathrm{out}}-i_k'}{r_k}(\gamma_b-\gamma_n)+\gamma_n}\ \text{kg/m}^3, \tag{12} \]

where \(\gamma_b^k\), \(\gamma_n^k\), \(i_k'\), and \(r_k\) must be taken at the steam pressure in the upper drum of the boiler.

With the specific weight of the mixture at the entrance to the riser tubes equal to \(\gamma\), the mean specific weight of the steam-water mixture over the height \(h\) of the rising section of the circuit will be

\[ \gamma^{\mathrm{cp}}\simeq \frac{\gamma+\gamma'}{2}. \tag{13} \]

During operation of the steam-jet pump, a loss of kinetic energy will occur, arising mainly as a result of the impact of two jets—steam and water—moving at very different velocities. Because of this loss, the mechanical efficiency of such a pump is very low. At the same time, however, it should be noted that, with subsequent utilization of the heat into which this loss is transformed, its thermal efficiency becomes very high. This loss of kinetic energy may be written in the following form:

\[ G_1\frac{c_1^2}{2g}+G_2\frac{c_2^2}{2g}-G\frac{c^2}{2g}\ \frac{\text{kgm}}{\text{sec}} \]

or, per \(1\) kg of working steam,

\[ \frac{c_1^2}{2g}+\frac{kc_2^2}{2g}-\frac{(k+1)c^2}{2g}\ \frac{\text{kgm}}{\text{sec}}. \]

Expressing this loss in heat units, we obtain:

\[ A\left[\frac{c_1^2+kc_2^2-(k+1)c^2}{2g}\right]\frac{\text{kcal}}{\text{kg}}. \tag{14} \]

Relating the loss to the energy of the jet of working steam and expressing it as a part of this energy, equal to \((1-\eta_m)\dfrac{c_1^2}{2g}\), we may write that

\[ \eta_m\frac{c_1^2}{2g}+k\frac{c_2^2}{2g}=\frac{c^2}{2g}(k+1), \]

whence the mechanical efficiency of the pump is obtained as

\[ \eta_m=\frac{(k+1)c^2-kc_2^2}{c_1^2}. \tag{15} \]

By \(\eta_m\) here one should understand not the efficiency of an isolated pump, but of a pump operating in the system of the boiler circulation circuit.

Method for Calculating a Steam-Jet Water Pump

For calculating a steam-jet pump intended for operation with a boiler with artificial circulation, it is necessary to have an accurate diagram of the boiler with all structural dimensions. It is also necessary to have data from a preliminary thermal calculation of the boiler pertaining to the heated section of the circuit. The location for installing the pump is selected before the lower header, on the downcomer branch of the circuit. In accordance with the operating conditions of the boiler and the quality of the feedwater, the circulation ratio \(K\) is selected. Thus, with the circuit (boiler) capacity \(G_n\), taken from the thermal calculation of the boiler, known, the total flow rate of the mixture \(G\), kg/sec, leaving the pump and delivered to the heated part of the circuit can now be determined. Here the preliminary thermal calculation of the boiler is meant, because the final capacity of the boiler may be greater than that taken from the preliminary calculation owing to the working steam of the pump. For boilers without a superheater this increase in capacity will not require a repeated thermal calculation. If, however, the boiler has a superheater, then, in connection with the increase in the quantity of steam delivered by the boiler, the thermal calculation must necessarily be refined.

In the proposed method, the calculation of the steam-jet water pump is not carried out in isolation, but in connection with the quite definite operating conditions of the assigned circulation circuit. In the scheme of this circuit the pump must perform a definite task: to overcome the known circuit resistances falling to its share, including the resistances of the pump itself. On the basis of these conditions, the corresponding velocities of the water and of the mixture, and the required exit velocity from the nozzle of the working steam, must be determined. Therefore, by the mechanical efficiency determined at the end of the calculation one should understand the efficiency not only of the pump itself, but of the pump together with the circulation circuit operating under a definite regime.

Bearing in mind the operation of the pump on saturated or wet steam, it is recommended that the calculation be carried out in the following order:

1. Several values are assigned for the delivery ratio of the pump and for the velocity of water entering the mixing chamber \(c_2\), taking into account the indications given earlier. For these selected quantities, parallel po-

the following preliminary calculations, as a result of which the dependence \(k=f(p_1)\) should be obtained. After such a graph has been constructed, it is possible to outline the final variant, for which a refined calculation is carried out, if such is required, and then the structural dimensions of the pump are determined. The choice of variant may depend, for example, on the steam parameters of the existing steam boiler or boiler house, the steam from which is to be used as the working steam for the pump. In the absence of a ready source for supplying the pump with steam, one with the appropriate parameters may be selected on the basis of various considerations in analyzing the graphs obtained.

1. The heat content of the mixture in the throat of the diffuser is determined from expression (4). In doing so, the heat content \(i_1^{pr}\) is first selected in accordance with the intended pressure of the working steam at the inlet to the nozzle and the desired limits of the pump delivery ratio. The minimum value of the working steam pressure before the nozzle can be estimated approximately by the following expression:

\[ p_1 \geq p_k+\frac{h\gamma_2+\Delta p}{10000}\ \text{ata}. \]

When operating on wet steam, the value \(i_1^{pr}\) is taken in accordance with the moisture content. The value \(\Delta p=\Delta p_n+\Delta p_{op}\) is determined in accordance with the design scheme and the operating regime of the boiler.

Let us note that the heat content of saturated steam changes little with a small change in pressure. Thus, at pressures of \(13\text{–}16\ \text{ata}\), with a change in pressure by \(1\ \text{at}\), the heat content changes only by \(0.5\div0.6\ \text{kcal/kg}\), and in the pressure interval \(25\div35\ \text{ata}\), by only \(0.1\ \text{kcal/kg}\). Consequently, the error in determining the value \(i\) will be insignificant. The value \(i_2\) is taken according to the heat content of the water in the upper drum of the boiler, taking into account its cooling on the way to the pump. When feed water is supplied into the downcomer pipes, the heat content of the water \(i_2\) must be determined separately.

1. From expression (11), the heat content of the mixture at the outlet into the upper drum—\(i_{out}\)—is determined.

2. From formula (12), the specific weight of the mixture at the outlet into the upper drum—\(\gamma'\)—is found.

3. The pressure in the mixing chamber \(p_0\) is determined from equation (9). In this expression the value \(\gamma_2\) is taken according to the actual temperature of the water supplied to the pump.

For \(\zeta_2\) one may take the value \(\simeq 0.2\div0.3\).

As for the value \(\Delta p_{op}\), it is calculated according to the actual flow rate of water in the downcomer bundle \(G_2\), which can be determined in the following way.

Since

\[ K=\frac{G}{G_n}, \qquad \text{and} \qquad G=\frac{G_2(k+1)}{k}, \]

then, substituting the value \(G\) into the expression for \(K\) and determining \(G_2\) from it, we obtain:

\[ G_2=\frac{G_n kK}{(k+1)}. \tag{16} \]

The resistances \(\Delta p_{op}\) will have values identical for all calculations carried out in parallel.

1. The specific weight of the mixture \(\gamma\) is found from expression (8), or it is taken in accordance with the indications made for that formula.

1. From expression (13), the average specific gravity of the mixture in the heated section of the circuit is found—\(\gamma_{\mathrm{av}}\).

2. The velocity of the mixture in the throat of the diffuser, \(c\) m/sec, is determined from equation (10) or (10′). For this purpose, in each parallel calculation the sum of the resistances of the heated section from the pump to the upper drum, \(\Delta p\), must be determined. To speed up preliminary calculations, all resistances may be reduced to friction resistances or local resistances, and recalculations made in accordance with the change in the average velocity of the mixture. Lacking the structural dimensions and the actual velocity, for \(\zeta\) one may, in preliminary calculations, take the value equal to 0.2.

3. From formula (2), the value of the velocity at the outlet from the steam nozzle, \(c_1\) m/sec, is found.

4. From the formula for the outflow of steam

\[ c_1=\varphi 91.5\sqrt{i_1-i_1^x}=\varphi 91.5\sqrt{\Delta i} \]

the quantity is determined

\[ \Delta i=i_1-i_1^x=\frac{c_1^2}{8375\varphi^2}\ \frac{\mathrm{kcal}}{\mathrm{kg}}. \tag{17} \]

For \(\varphi\) one may take the value \(=0.95\).

1. Considering the process of steam outflow to be adiabatic, we find from the \(J\)—\(S\) diagram the required initial steam pressure \(p_1\) and its heat content \(i_1\). The position of the segment \(\Delta i\) in the \(J\)—\(S\) diagram must be chosen, when operating with dry steam, between the saturation curve \(x=1\) and the pressure curve corresponding to the value \(p_0\), and when operating with wet steam, respectively between the curves \(x=\mathrm{const}\) and \(p_0\) (see Fig. 14).

Fig. 14

1. From the data obtained from the parallel calculations, we construct the curves \(k=f(p_1)\).

2. From formula (15), the efficiencies of the pump are determined; their values may also be plotted on the graph.

From these or other considerations, an acceptable variant is selected on the constructed curve \(k=f(p_1)\). If, in the accepted variant, the heat content \(i_1\) differs only slightly from that adopted at the beginning of the calculation, then the calculation is not repeated.

The further calculation reduces to determining the principal structural dimensions of the jet pump. First of all, the steam and water flow rates \(G_1\) and \(G_2\) are determined from the previously adopted total flow rate \(G\) and the pump delivery ratio \(k\):

\[ G_1=\frac{G}{k+1}\ \mathrm{kg/sec}, \]

\[ G_2=\frac{G}{k+1}\,k=G_1k\ \mathrm{kg/sec}. \]

Further, for known values of the velocities \(c_1, c_2\) and \(c\) and the corresponding specific weights \(\gamma_1^x, \gamma_2\) and \(\gamma\), the areas of the flow sections are found as

\[ f_1=\frac{G_1}{\gamma_1^x c_1}; \qquad f_2=\frac{G_2}{\gamma_2 c_2} \quad \text{and} \quad f=\frac{G}{\gamma c}\ \text{m}^2 \]

and from them the diameters

\[ d_1=\sqrt{1.27 f_1}; \qquad d_2=\sqrt{1.27 f_2+(d_1)^2_n}; \qquad d_3=\sqrt{1.27 f_2}\ \text{m}, \]

where \((d_1)_n\) is the outside diameter of the steam nozzle at the mouth. The remaining dimensions of the pump, the shape of the steam nozzle, the taper angle of the diffuser, and the linear dimensions may be tentatively specified in accordance with the data available in the literature on jet apparatus and, in particular, on injectors.

Data from a Calculation Example

Table 1 below contains data from the calculation of a steam-jet water pump, carried out in accordance with the method presented. The initial data for the calculation were as follows:

\[ \begin{aligned} \text{boiler capacity} \qquad & D=10\ \text{t/h},\\ \text{steam pressure in the boiler} \qquad & p_k=14\ \text{ata},\\ \text{height} \qquad & h=4\ \text{m},\\ \text{resistances} \qquad & \Delta p_n=9750\ \text{kg/m}^2,\\ & \Delta p_{op}=250\ \text{kg/m}^2,\\ \text{circulation ratio} \qquad & K=3. \end{aligned} \]

In columns 5–7 of Table 1 are given the calculation data for the case of feeding the boiler into the drum and supplying water to the jet pump directly from the drum. Columns 1–4 refer to the case of supplying feed water into the downcomer pipes of the boiler; moreover, in each case the maximum temperature of the feed water for the boiler \((i^k_{\text{pit}})\) was taken to be that at which complete condensation of the working steam was theoretically possible. On the basis of these calculated data the graph shown in Fig. 15 was constructed.

Table 1

This graph shows the variation of the initial pressure of the working steam \(p_1\) and the pump efficiency \(\eta_n\) as a function of the pump delivery ratio. The dashed curves refer to the graphs of Tables 5–7, and the solid curves to 1–4. From the curves it is clearly seen that, as \(k\) increases, an increase in the pressure of the working steam is required, but at the same time the efficiency falls very noticeably. A comparison

Fig. 15

of the dashed and solid curves gives an idea of the sharp improvement in pump operation in the case of feeding colder boiler feed water into the boiler downcomer pipes. And the lower the temperature of the water in the downcomer pipes and the greater \(\gamma_2\), the more advantageous this is for pump operation. The curves of Fig. 15 facilitate the selection of the most acceptable variant, after which further calculations may be carried out to determine the pump’s structural dimensions.

Noticed Misprints