On the Assessment of the Energy Efficiency of Regenerative Schemes with Cascade Drainage Discharge into the Condenser
Abstract Generated abstract
The paper examines how to assess the energy efficiency of regenerative feedwater heating schemes in turbine units when extraction drains are discharged partly or fully into the condenser. It argues that the commonly used efficiency expression based only on the regenerative energy coefficient is insufficient because it neglects additional heat losses carried by the drains, and derives a generalized formula incorporating a drain heat loss coefficient. Comparative calculations for several cascade drainage schemes show that a higher energy coefficient does not necessarily imply higher economy, since increases in drain losses can offset the benefit of regenerative extraction. The proposed formulation clarifies how turbine efficiency depends jointly on the base-unit efficiency, the regenerative energy coefficient, and the drain loss coefficient.
Full Text
On the Assessment of the Energy Efficiency of Regenerative Schemes with Cascade Drainage Discharge into the Condenser
V. A. Bragin
(Presented by the scientific seminar of the Department of TPP)
In [1, 2], for assessing the energy efficiency of regenerative water heating in turbine-unit schemes, the following expression is proposed:
\[ \eta_r = \eta_0 \frac{1 + A_r}{1 + A_r \cdot \eta_0}, \tag{1} \]
which makes it possible to establish the advantage of regeneration simply and clearly. Here \(\eta_0\) denotes the efficiency of the initial plant without regeneration, and
\[ A_r = \frac{\sum \alpha_r \cdot h_r}{\alpha_k \cdot H_k} \]
is the energy coefficient. The greater the energy coefficient, other conditions being equal, the higher the efficiency \(\eta_r\) of the turbine unit.
However, in this form expression (1) can be used only for analyzing the energy efficiency of those regenerative schemes of plants in which the heat of extraction drainage is completely used for heating the water. If the drainage of the heating steam is discharged, even partially, into the condenser and its heat is lost with the cooling water, then the presence of an additional heat loss is not taken into account by expression (1).
Fig. 1
In deriving it, it was assumed that only the heat of the exhaust steam entering the condenser is lost. In addition, it is easy to show that an increase in the energy coefficient does not always lead to an increase in the efficiency of the turbine unit. For example, the energy coefficient for the plant with the regenerative scheme shown in Fig. 1 will be smaller than for the scheme in Fig. 2, a, but the efficiency of the plant operating according to the scheme in Fig. 1 will be higher. This is explained by the fact that the total heat loss in the condenser of the plant for the scheme in Fig. 1 will be smaller than for the scheme shown in Fig. 2, a. The additional heat loss in the case of a plant with the scheme shown in Fig. 2, a, plays a more significant role than the increase—
increase in power generation on the basis of regenerative extractions. These points sometimes mislead students when analyzing the energy efficiency of the regenerative schemes shown in Fig. 2. It therefore seems expedient to derive a more general expression for the efficiency of turbine installations, which could be used both for analyzing the economy of installations with schemes without discharge of drains into the condenser and with various methods of discharging them into the condenser.
Fig. 2
In the case of an installation with the regenerative schemes shown in Fig. 2, the total heat loss in the condenser will be equal to
\[ Q_k=\alpha_k q_k+\sum \alpha_r \cdot q'_k . \tag{2} \]
Here, as in (1), \(\alpha_k\) is the fraction of steam admitted to the condenser, kg/kg;
\[ q_k=i_k-t_k \]
is the heat loss in the condenser with the exhaust steam, kJ/kg;
\[ \sum \alpha_r \]
is the total fraction of regenerative extractions, kg/kg.
By
\[ q'_k=t'_r-t'_k \]
is denoted the heat of the drains of the regenerative extractions, transferred to the cooling water in the condenser, kJ/kg. The heat consumption for the turbine installation may be represented as
\[ Q_0=\alpha_k q_k+\alpha_k H_k+\sum \alpha_r h_r+\sum \alpha_r q'_k, \tag{3} \]
where \(H_k=i_0-i_k\) is the heat drop of the steam going to the condenser, kJ/kg,
and \(h_r = i_0 - i_r\) is the heat drop of the steam going to the regenerative extraction, kJ/kg.
Then the efficiency of the unit, in accordance with the generally accepted formula, will be
\[ \eta_r = 1- \frac{\alpha_k \cdot q_k' + \sum \alpha_r \cdot q_k'} {\alpha_k \cdot q_k + \alpha_k \cdot H_k + \sum \alpha_r \cdot h_r + \sum \alpha_r \cdot q_k} \]
or, after simple transformations, this expression can be reduced to the form
\[ \eta_r = \eta_0 \frac{1 + A_r}{1 + A_r \cdot \eta_0 (1 + k_r)} . \tag{4} \]
Here
\[ k_r = \frac{\sum \alpha_r \cdot q_k'}{\sum \alpha_r \cdot h_r}, \]
is the coefficient of additional heat loss with the drains, showing what quantity of heat of the extraction drains is lost in the condenser in relation to the work performed by the steam going to the regenerative extractions. If discharge of the drains into the condenser is absent and there is no additional heat loss, then \(\sum \alpha_r \cdot q_k'\) and \(k_r\) are equal to zero and formula (4) takes the form (1). Thus, formula (4) can be used for analyzing any regenerative schemes.
It follows from (4) that the efficiency of a unit with regeneration, in the general case, depends on the efficiency of the simple unit without regeneration \(\eta_0\), the energy coefficient \(A_r\), and the coefficient of heat loss with the drains \(k_r\).
The relative increase in efficiency compared with a unit without regeneration will be equal to:
\[ \Delta \eta_r = \frac{\eta_r - \eta_0}{\eta_0} = \frac{1 - \eta_0 (1 + k_r)} {\dfrac{1}{A_r} + \eta_0 (1 + k_r)} . \tag{5} \]
It follows from (5) that the increase in efficiency when regeneration is introduced into the unit scheme will be greater the lower the efficiency of the simple unit \(\eta_0\), the greater the energy coefficient \(A_r\), and the smaller the coefficient of heat loss with the drains \(k_r\). The value of the coefficient \(k_r\) may be less than or greater than unity. Everything depends on the value of the enthalpy \(\bar{i}_r\), with which the drains are discharged into the condenser, and on the quantity of discharged drains. The smaller \(\bar{i}_r\) and the quantity of discharged drain, all other conditions being equal, the smaller the value of the coefficient \(k_r\), and the greater the relative role played by the energy coefficient in increasing \(\Delta \eta_r\). The energy coefficient has the greatest influence on the increase in efficiency in the case when the coefficient \(k_r\) is equal to zero.
Therefore, the transition from the scheme in Fig. 2, \(a\), to the scheme in Fig. 1, although it is associated with some decrease in \(A_r\), nevertheless, because for the scheme in Fig. 1 the coefficient \(k_r = 0\), the efficiency of the unit becomes higher. The values of the coefficients \(A_r\) and \(k_r\) are closely interrelated, and a change in one coefficient leads to a change in the other. Most often, in the schemes shown in Fig. 2, an increase in the energy coefficient is associated with an increase in the coefficient of heat loss with the drains. However, for example, the transition from the scheme in Fig. 2, \(a\), to the scheme in Fig. 2, \(b\), is associated with a decrease in the energy coefficient and an increase in the coefficient of heat loss with the drains. This is explained by the fact that, in this transition, the additional heat loss with the drains in the condenser decreases to a lesser degree than the total amount of steam extractions for regeneration is reduced. Therefore the coefficient \(k_r\) increases, while economy decreases.
In the table below, for a number of representative schemes with cascade drainage discharge into the condenser (Figs. 1, 2), values are shown for the energy coefficients and the coefficients of heat loss with drains, as well as their comparative economy \((P_0 = 29.4\ \text{bar},\ t_0 = 420^\circ\text{C},\ P_k = 0.039\ \text{bar},\ P_1 = 5.75\ \text{bar},\ P_2 = 2.48\ \text{bar},\ P_3 = 0.892\ \text{bar},\ P_4 = 0.268\ \text{bar},\ \eta_0 = 0.284)\).
Table 1
| Scheme | \(A_r\) | \(\kappa_r\) | \(\kappa_r/A_r\) | \(\Delta\eta_r\) |
|---|---|---|---|---|
| Fig. 1 | 0,1260 | 0 | 0 | 0,088 |
| Fig. 2, a | 0,1385 | 0,318 | 2,29 | 0,082 |
| Fig. 2, b | 0,1354 | 0,324 | 2,39 | 0,080 |
| Fig. 2, c | 0,1570 | 0,690 | 4,40 | 0,074 |
| Fig. 2, d | 0,1700 | 1,090 | 6,42 | 0,067 |
The data in the table clearly show what influence, in addition to the energy coefficient, the coefficient of heat loss with drains has on the relative increase in efficiency. Any change in these coefficients that leads to an increase in the ratio \(\kappa_r/A_r\) causes a decrease in the economy of the scheme. Thus, for the scheme shown in Fig. 2, d (a combination of the “reverse” cascade with drainage discharge into the condenser), the coefficient \(\kappa_r\), as well as the ratio \(\kappa_r/A_r\), has the largest value among the other schemes; therefore, despite the largest value of the energy coefficient, the relative increase in efficiency proves to be the smallest. If, in the plant scheme shown in Fig. 2, a, a drain cooler is provided on the path of their discharge from heater P4 to the condenser, and it is assumed that the drain flow lowers its temperature in it by approximately \(5^\circ\text{C}\), then the coefficient of heat loss with drains will decrease from 0,318 to 0,276 (\(\kappa_r/A_r\) will decrease from 2,29 to 2,02), and the efficiency of the scheme will increase by approximately 0,4%.
When the number of regenerative heaters in the scheme in Fig. 2, a is reduced to one while maintaining the feedwater temperature, the ratio \(\kappa_r/A_r\) increases to approximately 21,5 (the specific share of the additional heat loss with the drain increases sharply), and therefore the relative economy of the plant falls by almost 6%.
REFERENCES
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