DETERMINATION OF THE COMPRESSION RATIO IN THE STAGES OF A STEAM EJECTOR

E. N. SHADRIN

In calculating a steam ejector, the pressure \(p_k\) of the steam-air mixture at the inlet to the mixing chamber of the first stage and the pressure \(p_v\) of the mixture of steam and air at the outlet from the last stage are specified. The pressure \(p_k\) is usually taken to be equal to the pressure of the steam-air mixture at the suction point of the last condenser, while \(p_v\) is taken somewhat higher than atmospheric, taking into account the resistance of the exhaust pipe. The intermediate pressures in the stages are selected from the condition of an equal compression ratio in the individual stages [1; 2; 3]. Thus, for a two-stage ejector, the pressure of the steam-air mixture in the mixing chamber of the second stage is

\[ p_{np}=p_k\cdot \sqrt{\frac{p_v}{p_k}}, \]

for a three-stage ejector—the pressure in the second stage

\[ p_{np2}=p_k\sqrt[3]{\frac{p_v}{p_k}}, \]

and in the third—

\[ p_{np3}=p_{np2}\cdot \sqrt[3]{\frac{p_v}{p_k}}. \]

This method of determining the pressures in the intermediate stages of an ejector is borrowed from the method of calculating and designing multistage air compressors, in which the intermediate coolers serve only to cool the air, which reduces the work expended on compressing this air. It is easy to show that, in this case, when the air is cooled in the intermediate cooler to the initial temperature, the minimum work expended on compressing the air will occur when the pressure differences in the stages are equal.

The work of adiabatic compression of one kilogram of air in a two-stage compressor may be written as

\[ l=\frac{k}{k-1}p_1v_1\left[\left(\frac{p_{np}}{p_k}\right)^{\frac{k-1}{k}}-1\right] +\frac{k}{k-1}p_{np}v_{np}\left[\left(\frac{p_v}{p_{np}}\right)^{\frac{k-1}{k}}-1\right]. \tag{1} \]

If we take into account that, when the air is cooled in the intermediate cooler to the initial temperature, \(p_1v_1=p_{np}v_{np}\), then, after equating to zero the first derivative,

\[ \frac{dl}{dp_{np}}=0, \]

we obtain

\[ p_{np}=p_k\sqrt{\frac{p_v}{p_k}}. \]

In steam ejectors, in the intercoolers there occurs not only cooling of the steam-air mixture, but also condensation of part of the steam of the mixture. As a result, the quantity of mixture passing through the individual stages will be different. If, under these conditions, equal compression ratios are adopted in all stages, then the greatest load will fall on the first stage, through which the maximum quantity of steam-air mixture passes. This means that the minimum work expended on removing the steam-air mixture will not be ensured with equal compression ratios in the individual stages of the ejector.

In determining the optimal pressure distribution one should proceed from the condition of minimum steam consumption. But, as investigations have shown, this considerably complicates the problem. Therefore we shall proceed from the condition of the minimum work required to create a specified rarefaction. We shall solve the question as applied to a two-stage ejector.

Fig. 1.

The work expended in a two-stage ejector in the adiabatic compression of one kilogram of steam-air mixture will be expressed by the formula

\[ l= \frac{k'_{sm}}{k'_{sm}-1}\cdot p_k\, \frac{R_v T'_{sm}}{p_k-p'_n} \left[ \left(\frac{p_{pr}}{p_k}\right)^{\frac{k'_{sm}-1}{k'_{sm}}} \right] + \]

\[ + \frac{k''_{sm}}{k''_{sm}-1}\cdot p_{pr}\, \frac{R_v T''_{sm}}{p_{pr}-p''_n} \left[ \left(\frac{p_z}{p_{pr}}\right)^{\frac{k''_{sm}-1}{k''_{sm}}} -1 \right]. \tag{2} \]

Here, in contrast to formula (1),

\[ \frac{R_v T'_{sm}}{p_k-p'_n} \quad \text{and} \quad \frac{R_v T''_{sm}}{p_{pr}-p''_n} \]

take into account the presence of steam in the air; \(p'_n\) and \(p''_n\) are the partial pressures of steam in the mixture before the first and second stages. The quantities \(p'_n\) and \(p''_n\) depend only on the temperatures \(T'_{sm}\) and \(T''_{sm}\) of the mixture before the stages. In what follows we shall assume \(T'_{sm}=T''_{sm}\).

Under actual conditions \(T''_{\text{mix}}\) is often tens of degrees greater than \(T'_{\text{mix}}\). Since absolute temperatures enter into the formula, a difference of \(10^\circ\text{C}\) will not introduce a large error.

To determine \(l_{\min}\), take the first derivative \(\dfrac{dl}{dp_{\text{pr}}}\) and set it equal to zero. Then, after the simplest transformations, we obtain, putting \(p_c=1\) ata,

\[ (k''_{\text{mix}}-1)\frac{p_k^{1/k'_{\text{mix}}}}{p_k-p'_n}\cdot \frac{p_{\text{pr}}-p''_n}{p_{\text{pr}}^{1/k'_{\text{mix}}}} + \frac{p_{\text{pr}}^{1/k''_{\text{mix}}}}{p_{\text{pr}}} - \]

\[ - k''_{\text{mix}}\cdot \frac{p_{\text{pr}}^{1/k''_{\text{mix}}}}{p_{\text{pr}}-p''_n} - k''_{\text{mix}} + k''_{\text{mix}}\frac{p_{\text{pr}}}{p_{\text{pr}}-p''_n} =0. \tag{3} \]

Equation (3) is expediently solved graphically. To reveal the influence of individual factors on \(p_{\text{pr}}\), all terms of equation (3) should be grouped so that in one part of the equality there are terms containing \(p'_n\), and in the other, \(p''_n\). Then one may write

\[ y_1=(k''_{\text{mix}}-1)\frac{p_k^{1/k'_{\text{mix}}}}{p_k-p'_n}\cdot \frac{p_{\text{pr}}-p''_n}{p_{\text{pr}}^{1/k'_{\text{mix}}}}, \tag{4} \]

\[ y_2=k''_{\text{mix}}\frac{p_{\text{pr}}}{p_{\text{pr}}-p''_n} \left(1-p_{\text{pr}}^{1/k''_{\text{mix}}-1}\right) + p_{\text{pr}}^{1/k''_{\text{mix}}-1} - k''_{\text{mix}}. \tag{5} \]

Having plotted, from equations (4) and (5), the dependences \(y_1=f_1(p_{\text{pr}})\) and \(y_2=f_2(p_{\text{pr}})\), we easily determine the value of \(p_{\text{pr}}\) at the intersection of the curves \(y_1=f_1(p_{\text{pr}})\) and \(y_2=f_2(p_{\text{pr}})\).

In Fig. 1 the curves \(y_1\) and \(y_2\) are plotted as functions of \(p_{\text{pr}}\) for a particular example, when \(p_k=0.04\) ata, \(p_c=1\) ata, \(t'_{\text{mix}}=25^\circ\text{C}\), \(t''_{\text{mix}}=35^\circ\). It is seen from Fig. 1 that the optimum value \(p_{\text{pr}}=0.08\) ata is considerably

Fig. 2.

different from \(p_{\text{pr}}=0.2\) ata, determined from the condition of an equal degree of compression. The physical meaning of the conclusion obtained is quite clear if one takes into account the presence of vapor in the mixture, which it is expedient to remove at a lower pressure.

From equation (4) it is seen that with an increase in \(p'_n\), which will occur at a higher temperature \(t'_{\text{mix}}\), \(y_1\) increases and

the curve will become steeper. As a result, the point of intersection of the curves will shift to the left, toward smaller values of \(p_{pr}\). With a decrease in \(t'_{sm}\), on the contrary, \(p_{pr}\) will increase, approaching \(p_{pr}=0.2\). This is explained by the fact that, as \(t'_{sm}\) decreases, the amount of steam in the mixture will be smaller. At \(p'_n=p''_n=0\), which will occur when only pure air enters the ejector, the intermediate pressure is \(p_{pr}=0.2\) ata; that is, in this case we arrive at the already known uniform distribution of pressure drops over the individual stages.

Figure 2 shows the curves \(y_1\) as functions of \(p_{pr}\) for different mixture temperatures \(t'_{sm}\). Figure 1 confirms the considerations just expressed regarding the influence of \(t'_{sm}\) on the value of \(p_{pr}\).

If we compare the amount of work required to compress the steam-air mixture at \(p_{pr}=0.2\) ata and \(p_{pr}=0.08\) ata, substituting these values successively into formula (2), we obtain a reduction in work at \(p_{pr}=0.08\) ata, compared with \(p_{pr}=0.2\) ata, of approximately 40–45%.

Conclusions

1. A uniform distribution of pressure drops over the stages of a steam ejector does not ensure the minimum work required to remove the steam-air mixture.

2. To ensure the minimum work for removing the steam-air mixture, the pressure drops must increase from the first stage to the next.

Literature

1. Makeev V. A. Condensing Devices of Steam-Power Installations. ONTI, 1933.
2. Kazanskii A. M. Condensing Devices. GONTI, 1939.
3. Blyudov V. P. Condensing Devices of Steam Turbines. Gosenergoizdat, 1951.