The direct air-cooled unit operates in a high back pressure heating mode, and the turbine exhaust steam flow rate is D _C . A portion of the exhaust steam flow D _HC is used as the heat source for the heating condenser, which can fully utilize the waste heat. The remaining D _AC is cooled by the air-cooled island. As shown in Figure 1, after the D _HC heats the return water of the heat network, the heat supply extraction steam is used for secondary heating to meet the water supply temperature requirements of the heat network.

This article takes a certain 620 MW direct air-cooled cogeneration unit as an example, and the main parameters are shown in Table 1. After the high back pressure transformation, the rated back pressure is 35 kPa. The parameters of the heating condenser are shown in Table 2. The temperature of the circulating water in the heat network is 40 °C, and the rated flow rate is 16,000 t/h. The unit retains the heating extraction steam and combines high back pressure heating to achieve energy stepwise utilization, with a maximum extraction steam flow rate of approximately 250 t/h.

The purpose of the unit’s off-design condition calculation is to determine the steam parameters at each extraction port and exhaust end of the steam turbine, as well as the corresponding parameters of the regenerative system. The calculation method for off-design conditions of steam turbine thermal system can be described as follows:

First, conduct a heat balance calculation for the unit under rated or base conditions. The efficiency ηᵢ_(r) of each stage group r of the steam turbine is calculated using Equation 1. The main steam flow rate and stage group efficiency vary under different operating conditions and have a specific functional relationship. After fitting the main steam flow rates and stage group efficiencies, the efficiency of each stage group under off-design conditions is determined (Zhang et al., 2021). η i r = h 1 r − h 2 r H s r (1) where h₁(r) is the inlet steam enthalpy of stage group r, kJ/kg; h₂(r) is the outlet steam enthalpy of stage group r, kJ/kg; H_s(r) is the isentropic expansion enthalpy drop of steam in stage group r within the steam turbine, kJ/kg.

During the off-design condition calculation, the new main steam flow rate is an unknown quantity and needs to be obtained through iterative calculation after assuming an initial value. To improve the efficiency of iterative calculation, the assumed value can be roughly determined based on the ratio to the electrical load, i.e., the ratio of main steam flow rate to electrical load under off-design conditions is equal to that under base conditions (Equation 2): D = N d N d 0 D 0 (2) where D is the main steam flow rate under off-design conditions/kg·h^-1;D₀ is the main steam flow rate under base conditions/kg·h^-1; N_n is the unit electrical load under off-design conditions/kW; N_n0 is the unit electrical load under base conditions/kW.

The off-design conditions of the unit and thermal system are mainly reflected in the change of the steam inlet flow rate of the turbine or the steam flow rate passing through the turbine. The change in steam flow rate inside the turbine will cause changes in the pressure before and after the stage group, and their relationship is determined by the Flügel formula (Equation 3) (Wang et al., 2019): D 1 D 10 = P 1 2 − P 2 2 P 10 2 − P 20 2 T 10 T 1 (3)

The work done per unit of new steam (H₀) can be obtained from the extraction share of each stage group, and then the new steam flow rate is determined by the Equation 4: D 01 = 3600 N d η m η g H 0 (4) where P _r is the extraction pressure at the current extraction port under off-design conditions/MPa; P _r+1 is the extraction pressure at the next extraction port under off-design conditions/MPa; D _r is the extraction flow rate under off-design conditions/kg·h^-1; D _r0 is the extraction flow rate under rated conditions/kg·h^-1; P _r0 is the extraction pressure at the current extraction port under rated conditions/MPa; P _(r+1)0 is the Extraction pressure at the next extraction port under rated conditions/MPa.

This study focuses on a direct air-cooled (dry cooling) system, distinct from conventional evaporative cooling towers (which need a latent heat term for water evaporation). This dry cooling system only uses sensible heat exchange (affected by dry-bulb temperature and frontal wind speed).

The direct air-cooled unit relies on air-cooled fan to cool the exhaust steam of the turbine. Heat exchange occurs between the steam and the air through the outer surface of the heat exchanger tube bundle. The efficiency - number of heat transfer units method is often used to analyze the off-design condition characteristics of the air-cooled system (Zhou and Li, 2011).

Heat release from the condensation of the steam D _AC entering the air-cooled island is calculated using Equation 5: Q A C = D A C h C − h A C (5)

The heat absorbed by the air outside the heat exchanger tubes is equal to the heat released by the condensation of the D _AC : Q A C = G a C a t a 2 − t a 1 = F F v N F ρ C a t a 2 − t a 1 (6) where G a = F F v N F ρ is the air flow rate, kg/s; v _NF is the frontal wind speed of the air-cooled heat exchanger, m/s; ρ is the local air density, kg/m³; C _a is the specific heat of air, J/(kg·°C).

In the heat exchanger, condensation on the steam side releases heat, and the efficiency ε of the heat exchanger is Equation 7: ε = 1 − e − N T U = t a 2 − t a 1 t C − t a 1 (7) where the number of heat transfer units (NTU) is Equation 8: N T U = K a A a F F v N F ρ C a (8)

The saturation temperature t _C of the turbine exhaust steam is: t C = D A C h C − h A C F F v N F ρ C a · 1 1 − e − N T U + t a 1 (9) where D _AC is the turbine exhaust steam flow ratev entering the air-cooled island, kg/s; h _C is the specific enthalpy of exhaust steam, kJ/kg; h _AC is the specific enthalpy of condensate water in the air-cooled island, kJ/kg; F _F is the windward area of the air-cooled heat exchanger, m²; K _a is the heat transfer coefficient of the air-cooled island, W/(m²·°C); A _a is the heat dissipation area of the air-cooled island heat exchange tube bundle, m²; t _a1 and t _a2 are respectively the temperatures of the air before and after entering the air-cooled condenser, °C.

This heat balance Equation 6 is specific to the direct air-cooled system studied herein. Unlike conventional evaporative cooling towers (which require an additional latent heat term for water evaporation), the direct air-cooled system only involves sensible heat exchange—hence (Equation 6) excludes latent heat-related parameters. Notably, Equation 9 is also applicable solely to the direct air-cooled system, as it is derived based on Equation 9 and dry cooling-specific heat transfer mechanisms.

For a well-designed direct air-cooled system, the total heat dissipation area A _a and the total windward area F _F have been determined. While the local air physical parameters ρ and C _a are affected by the ambient temperature t _a1, and the heat transfer coefficient is affected by the frontal wind speed v _NF and the ambient temperature t _a1. Thus, the influencing factors of the turbine exhaust steam pressure P _C are obtained (Equation 10): P C = f t C = f D A C , v N F , t a 1 (10)

The turbine exhaust steam and the return water of the heat network undergo heat exchange in the heating condenser to recover the waste heat of the exhaust steam and improve the overall thermal economy of the unit. The heat transfer analysis was carried out by using the efficiency - number of heat transfer units method. The efficiency ε represents the ratio of the actual heat transfer effect of the heat exchanger to the maximum possible heat transfer effect, and the number of heat transfer units characterizes the comparative relationship between the heat transfer performance and thermal convection performance of the heat exchanger.

In the heating condenser, the cold and hot fluids exchange heat in counter-flow, and the steam undergoes phase change heat transfer. The efficiency formula is Equation 11 (Bejan, 1988): ε = 1 − e − N T U = t w 2 − t w 1 t C − t w 1 (11) where t _w1 and t _w2 are respectively the inlet and outlet temperatures of the return water in the heat network, °C; t _C is the saturation temperature of the turbine exhaust steam, °C.

NTU represents the number of heat transfer units, and its calculation formula is Equation 12: N T U = K w A w C w G w (12) where K _w is the heat transfer coefficient of the heating condenser, W/(m²·°C); A _w is the heat exchange area of the heating condenser, m²; C _w is the specific heat of the return water in the heat network, J/(kg·°C); G _w is the circulating water flow rate of the heat network through the condenser, in kg/s.

The saturation temperature t _C of the turbine exhaust steam is also the saturation temperature corresponding to the pressure of the heat supply condenser. According to the above formula, it can be obtained (Equation 13) (Zhou and Li, 2011): t C = t w 1 + D H C h C − h H C C w G w 1 − exp − K w A w C w G w (13) where D _HC is the turbine exhaust steam flow rate entering the condenser, kg/s; h _C is the specific enthalpy of exhaust steam, kJ/kg; h _HC is the specific enthalpy of condensate water in the heating condenser, kJ/kg.

The turbine exhaust steam pressure P _C corresponding to t _C can be obtained through the steam parameter table. For the designed heating condenser, the heat exchange area A _w has been determined. The heat exchange coefficient is affected by the circulating water flow rate G _w of the heat network and the inlet temperature t _w1 of the return water of the heat network. During the heat exchange process of the return water of the heat network, the specific heat C _w does not change much and is mainly affected by the inlet temperature t _w1. Thus, the influencing factors of the turbine exhaust steam pressure P _C are obtained (Equation 14): P C = f t C = f D H C , G w , t w 1 (14)

The saturation temperature corresponding to the condenser pressure can also be calculated from the inlet and outlet temperatures of the return water in the heat network and the heat transfer end difference, that is Equations 15–17: t C = t w 1 + Δ t w + δ t (15) Δ t w = t w 2 − t w 1 = D H C h C − h H C C w G w (16) δ t = t C − t w 2 = Δ t w e K w A w C w G w − 1 (17)

Where Δt _w is the temperature rise of the return water of the heat network in the heating condenser, °C; δt is the heat transfer end difference, °C.

Combined with the efficiency calculation formula, the relationship between efficiency and the return water temperature rise Δt _w of the heat network, as well as the heat transfer end difference δt, can be derived (Equation 18): ε = 1 δ t Δ t w + 1 (18)

The number of heat transfer units NTU reflects the combined influence of the heat transfer coefficient K _w , the circulating water flow rate G _w of the heat network, and the inlet temperature t _w1 of the return water of the heat network. The influencing factors of the turbine exhaust steam pressure P _C are simplified as being characterized by the more measurable return water temperature rise Δt _w of the heat network and the heat transfer end difference δt (Equation 19): P C = f D H C , N T U = f D H C , δ t , Δ t w (19)

The common calculation methods for off-design conditions of condensers include the fixed back pressure method, the logarithmic mean temperature difference method, and the efficiency - number of heat transfer units method. In the actual operation of the unit, it is difficult to maintain a stable back pressure, so the fixed back pressure method does not conform to the actual situation. The logarithmic mean temperature difference method is based on the fixed end difference and is also difficult to truly reflect the actual operation of the heating condenser. The number of heat transfer units can comprehensively reflect the influence of the condenser heat transfer coefficient and cold end parameters on the heat transfer effect, and the efficiency changes little in actual operation.

As shown in Figure 2, it is the back pressure, end difference and efficiency change trend of the extraction - high back pressure unit on a certain day. The average value and standard deviation of the data within a day are calculated respectively, as shown in Table 3. The standard deviation of the efficiency is the smallest, and the change range is very small. It can basically be regarded as remaining unchanged. Therefore, the fixed efficiency method can be adopted to conduct off-design condition analysis on the heating condenser.

The energy utilization rate is defined as Equation 20: η t p = 3.6 P e + Q h Q t p (20) where Pe is the power generation output in MW; Qh is the total heat load of the plant, GJ/h; Qtp is the total heat consumption of the plant, GJ/h.

The coal consumption rate coal consumption rate is calculated by Equation 21: b e = Q t p P e Q L H V · 10 − 6 (21) where Q_LHV is the low heat value of coal, kJ/kg.

When establishing the thermodynamic model of the direct air-cooled unit, the design parameters of the unit under the Boiler Maximum Continuous Rating (BMCR) condition, Turbine Heat Acceptance (THA) condition, and 30% of the rated main steam flow rate condition, provided by the combined heat and power (CHP) plant, were adopted. Off-design condition simulation calculations were conducted for the above three operating conditions, and the results were compared with the parameters from the heat balance diagram provided by the steam turbine manufacturer. As shown in Table 4, the simulation calculation errors of the main steam flow rate and heat rate under the three operating conditions are all less than 1%. This indicates that the established model has high accuracy and can be used for the thermal system simulation of the unit under off-design conditions.

The back pressure of the direct air-cooled unit decreases with the increase of the head-on wind speed and increases with the increase of the ambient temperature (Zhao et al., 2014). This study focuses on discussing the operation characteristics of the air-cooled extraction - high back pressure unit. The head-on wind speed and ambient temperature can be regarded as remaining constant, and the exhaust steam flow rate entering the air-cooled island affects the heating situation of the unit.

When the power load N _d and the heating steam D _HS remain constant, the influence of the change in the exhaust steam D _AC entering the air-cooled island on the operating parameters of the unit is shown in Figures 3, 4, and the influence on the energy consumption characteristics of the unit is shown in Figure 5.

With the increase of the exhaust steam flow entering the air-cooled island, the main steam pressure and the extraction steam pressure at each stage of the regenerative system decrease uniformly, and the extraction steam share also decreases uniformly, resulting in a reduction in the main steam flow. The exhaust steam used for heating is more affected by the exhaust steam entering the air-cooled island, with a significant reduction in flow rate. Correspondingly, the heat load of the exhaust steam decreases, the back pressure also drops, and the energy consumption for heating decreases. Moreover, the exhaust steam entering the air-cooled island, as a cold source loss, keeps increasing, thus the coal consumption rate rises.

The high back pressure unit relies on the heating condenser to provide the heat load. The heat load of exhaust steam is affected by the heat exchange capacity of heating condenser, the exhaust steam parameters, and the return water parameters of the heat network.

When the power load N _d and the heating steam D _HS remain constant, and the exhaust steam flow D _AC entering the air-cooled island is kept at 150 t/h, the influence of the return water flow of the heat network on the operating parameters of the unit is shown in Figures 6, 7. With the increase of the heat network return water flow, the main steam pressure of the unit and the extraction steam pressure at each stage of the regenerative system decrease. The greater the heat network return water flow, the smaller the decrease, and the extraction steam share also decreases accordingly, resulting in a reduction in the main steam flow.

The impact on the energy consumption characteristics of the unit is shown in Figure 8. As the heat network return water flow increases, the proportion of heating exhaust steam decreases, and the heat absorption capacity of the return water increases. Since the efficiency of the heating condenser remains unchanged, the saturation temperature of the heating condenser decreases, the back pressure drops, and the heat load of the exhaust steam also decreases accordingly. If the D _AC remains unchanged, the cold source loss remains basically the same, the heat absorption capacity of the return water increases, the coal consumption of the unit decreases as the main steam flow decreases, the overall energy consumption drops, and the coal consumption rate decreases.

When the power load N _d and the heating steam D _HS remain constant, and the exhaust steam flow D _AC entering the air-cooled island is kept at 150 t/h, the influence of the return water inlet temperature of the heat network on the operating parameters of the unit is shown in Figures 9, 10. As the return water inlet temperature increases, the main steam pressure of the unit and the extraction steam pressure at each stage of the regenerative system rise uniformly, and the extraction steam share also rises uniformly, resulting in an increase in the main steam flow rate.

The impact on the energy consumption characteristics of the unit is shown in Figure 11. As the return water inlet temperature increases, the proportion of heating exhaust steam increases, the saturation temperature of the heating condenser and back pressure rise, the heat load of the exhaust steam increases, and the coal consumption of the unit increases with the increase of the main steam flow. The overall energy consumption rises, and the coal consumption rate increases.

After the unit is modified to have a high back pressure, the rated back pressure is 35kPa, and it is not less than 25 kPa in actual operation. After adding extraction steam for heating, up to 250 t/h of extraction steam can be provided for heating, with a maximum heating capacity of approximately 195 MW. The operation of the unit is restricted by the maximum main steam flow rate, the minimum back pressure, the minimum extraction steam flow rate for heating (without extraction steam), and the maximum extraction steam flow rate for heating. When the exhaust steam flow rate D _AC entering the air-cooled island is 150 t/h, the operation domain is shown in Figure 12.

When the exhaust steam flow of the air-cooled island changes, the unit operation domain also changes. When the exhaust steam flow (D_AC) of the air-cooled island is 56.8 t/h, 150 t/h, and 300 t/h respectively, the unit operation domains are shown in Figures 13, 14. As D_AC increases, the area of the electric-heat load operation domain gradually decreases, and the flexibility of the unit decreases.

When the D_AC is 56.8 t/h, the back pressure operation range of the unit is 24.5–54.5 kPa. However, when the D_AC increases to 300 t/h, the back pressure operation range of the unit is reduced to 24.5–38 kPa, a reduction of 55%. At the same time, the larger the exhaust steam flow of the air-cooled island, the more stable the unit’s back pressure. However, when the D_AC increases from 56.8 t/h to 300 t/h, the coal consumption rate of the unit increases by 50 g/kWh.

The heat load of the extraction - high back pressure unit is composed of the extraction heat load and the exhaust heat load together. When the external heating load and the supply water temperature remain unchanged, changing the cold end parameters will affect the proportional relationship between the heat load of the extraction and the heat load of the exhaust, and thereby affect the overall energy consumption of the cogeneration unit. As shown in Figures 15–17, they respectively represent the influence of the changes in the exhaust steam flow rate entering air-cooled island, the heat network return water flow rate and the return water inlet temperature on the sharing ratio of heat load of extraction steam and exhaust steam of the unit and energy consumption.

With the increase of the exhaust steam flow rate entering air-cooled island, the heat load of exhaust steam decreases, the proportion of the heat load of extraction steam increases, the cold source loss of the unit increases, the coal consumption rate rises, and the energy utilization efficiency decreases. When the exhaust steam flow rate entering air-cooled island remains unchanged, with the increase of the return water flow rate, the heat exchange capacity of the heating condenser is enhanced, the back pressure is reduced, the proportion of the heat load of the extraction steam increases, but the waste heat of the exhaust steam is fully utilized, the coal consumption rate decreases, and the energy utilization efficiency improves. As the return water temperature rises, the heat exchange capacity of the heating condenser weakens, the back pressure increases, the proportion of the heat load of the extraction steam decreases, the coal consumption rate increases, and the energy utilization efficiency drops, which is not conducive to the effective utilization of waste heat.

The maximum extraction heating capacity can reach 250 t/h, with a heating capacity of approximately 195 MW. When the total heat load is 700 MW, the proportion of the extraction heat load is approximately 27.9% at its maximum. By adjusting the exhaust steam flow rate entering air-cooling island, the return water flow rate and temperature of the heat network, the proportion of the heat load of the extraction steam can be changed to maximize the heat extraction steam supply. The adjustment results are shown in Table 5. Compared with no extraction steam heat load, the adjustment of the air-cooled island will increase the coal consumption rate by approximately 3.78 g/(kW·h) and reduce the energy utilization efficiency by 0.44%. The regulation effect of the return water flow and temperature of the heat network is consistent, which can reduce the coal consumption rate by approximately 0.1 g/(kW·h) and increase the energy utilization efficiency by 0.43%.