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Simulation of steam superheater operation under conditions of pressure decrease
Accepted Manuscript Simulation of steam superheater operation under conditions of pressure decrease
Wiesław Zima PII:
S0360-5442(19)30148-3
DOI:
10.1016/j.energy.2019.01.132
Reference:
EGY 14605
To appear in:
Energy
Received Date:
19 September 2018
Accepted Date:
25 January 2019
Please cite this article as: Wiesław Zima, Simulation of steam superheater operation under conditions of pressure decrease, Energy (2019), doi: 10.1016/j.energy.2019.01.132
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ACCEPTED MANUSCRIPT Simulation of steam superheater operation under conditions of pressure decrease Wiesław Zima (corresponding author) The Cracow University of Technology, Institute of Thermal Power Engineering, al. Jana Pawła II 37, 31-864 Kraków, Poland Phone number: 0048 12 6283653, e-mail: [email protected] Abstract A decrease in the live steam pressure at the turbine inlet results in a pressure decrease in the boiler and enables an immediate increase in the steam mass flow rate at the boiler outlet. Assuming that the rate of the pressure-decrease process is known, the correct value of the target pressure decrease can only be determined computationally. This value should ensure the required total increment in the steam mass flow rate at the boiler outlet which is necessary to achieve the requested increase in the power-unit load. The total increment is the sum of increments which occur in live steam superheaters and in the evaporator. Therefore, a robust mathematical model of the steam superheater is essential for the correct computation of the increase in the steam mass flow rate at the boiler outlet which occurs as the pressure decreases. In this paper, an efficient in-house model of a superheater is proposed. It is a distributed-parameter model based on the solution of the equations which describe the principles of mass, momentum, and energy conservation. The model enables the determination of the value by which the pressure must be decreased to ensure the required increment in the steam mass flow rate. Simulation computations related to the increase in the steam mass flow rate at the superheater outlet were performed. Different rates of the pressuredecrease process and different values of the target decrease in the pressure were considered. The obtained results were compared with the results of the simplified calculations for steady states. The developed model is intended for the computation of the live-steam mass flow rate increments at the boiler outlet when rapid increases in loads are required within a short time. The computations are necessary for the generation of the modified sliding curves of the power unit. Keywords: power boiler, steam superheater, steam pressure decreasing, rapid increase in the steam mass flow rate, mathematical model, simulation computations Nomenclature A – surface area, m2 c – specific heat, J/(kg·K) Cd – correction factor taken into account in flows through circular channels (where heating occurs from the outside or inside only), Cl – correction factor taking into account the ratio between the surface length and the equivalent diameter, Cs – correction factor taking into account the tube bank system geometry, Ct – correction factor depending on the fluid and wall temperature, Cz – correction factor taking into account the number of tube rows on the flue gas path, d – diameter, m
ACCEPTED MANUSCRIPT f g h i k L m
M p Pr q r t T fg
– friction factor, – gravitational acceleration, m/s2 – heat transfer coefficient, W/(m2·K) – enthalpy, kJ/kg – thermal conductivity, W/(m·K) – length, m – mass flow rate, kg/s – the number of cross-sections, – pressure, bar – Prandtl number, – heat flux, W/m2 – radius, m – temperature, °C
U w z
– mean flue gas temperature in the control area, K – circumference, m – velocity, m/s – space coordinate, m
fg o o σ z
– thickness, m – fouling factor, (m2·K)/W – flue gas emissivity, – tube outer surface emissivity, – wall mean temperature, °C – tube outer surface temperature, K – density, kg/m3 – Stefan-Boltzmann constant, W/(m2·K4) – time, s – tube inclination angle, ° – time step, s – control volume length, m
Subscripts c – convection eq – equivalent fg – flue gases in – inner j – the number of control volumes along the tube m – mean o – outer r – radiation w – wall 1. Introduction Considering the increasing role of renewable energy sources in electricity generation, power units are now required to display considerable operating flexibility. A rapid increase in the power-unit load is achieved mainly by decreasing the boiler operating pressure. This process is realised by opening the turbine throttle valve. Meanwhile,
ACCEPTED MANUSCRIPT the mass flow rate of fuel fired in the boiler combustion chamber is increased. The pressure decrease process is carried out to obtain an immediate temporary increase in the live-steam mass flow rate at the boiler outlet, which is needed in order to achieve a required increment in the power-unit load within a set time. It is impossible to achieve this increase only by firing more fuel mass flow rate (it results from the mill characteristics and the high thermal inertia of the boiler evaporator). The process of the steam pressure decrease at the turbine inlet causes an immediate pressure decrease in the entire pressure part of the boiler, which results in an increase in the live-steam mass flow rate at the boiler outlet. The increase is due to the decreasing saturation pressure in the evaporator (within subcritical pressure values) and the increase in the specific volume of the steam in the “water volume” of live-steam superheaters. Therefore, the computation of the rapid increase in the mass flow rate of steam at the boiler outlet requires an accurate mathematical model of the superheater. The literature review presented below indicate that issues related to simulations of superheater operation are still relevant today. The mathematical models presented in [1–2] concerned a steam superheater characterised by a complex tube configuration. A detailed analysis was performed in [1] regarding the impact of the outer and inner deposit layers on the temperature of the steam, wall, and flue gases at the steady state. It was demonstrated that deposits on the inner surface of the superheater tube caused a great increase in the tube wall temperature despite the decrease in the heat flux transferred from the flue gases to the steam. Moreover, the proposed model enabled the consideration of the impact of temperature nonuniformity of the flue gas which existed in the cross-section of the flue-gas duct on the wall temperature of the superheater tubes. In [2], relations were proposed which enabled the calculation of the tube wall temperature, as well as the steam pressure and temperature along the steam flow path under steady-state conditions. A control volume-based finite element method was used to determine the temperature distribution in a superheater tube. The advantage of the method was that the results were obtained very quickly. The time needed to obtain the results was several times less compared with computational fluid dynamics (CFD) computations. The authors of [3] analysed the heat transfer in the tubes of a supercritical steam superheater. The developed method was a combination of three-dimensional CFD simulations (thermal processes of external flue gases) and a 1D analytical model (the steam temperature distribution). Approaching the problem in this manner, it was possible to determine the boundary conditions on both surfaces of the tube. The heat transfer was analysed along the entire length of the superheater tubes, which enabled appropriate selection of the tube material to prevent overheating. In [4], the authors analysed a similar problem. They developed a mathematical model enabling the estimation of the tube-wall temperature distribution of heating surfaces operating in high temperatures. The model enabled the consideration of the inner surface fouling which may cause overheating of the tubes. A detailed analysis was performed on the superheater and reheater tubes. The proposed model was based on the finite volume method and was complemented with steam temperature measurements at the inlet and outlet of the superheater/reheater stage. In the model, it was assumed that the pressure of the steam flowing through the superheater was constant. The superheater and reheater tubes were investigated in [5] as well, where a new method was presented to determine the heat fluxes transferred in the tubes of these heating surfaces of the boilers. The proposed procedure was iterative and was developed using empirical relations. The finite element method was employed to find the temperature distribution in the tube. The forced convection on the inner surface of the tube and the cross flow of the flue gases were considered as well. In [6], attention was drawn to the fact that considerable deviations of the steam temperature in the boiler superheater and reheater could substantially affect the safe and economic operation of the facility. The deviations constituted some of the main causes of tube
ACCEPTED MANUSCRIPT failure, which account for approximately 40% of forced breaks in the power plant operation. To predict potential tube failure, the authors of [6] proposed a model of the thermal load deviation for the superheater and the reheater. The model was based on in situ thermodynamic parameters of the power plant and the deviation theory of the flow rate. The results which were obtained using the model were in good agreement with experimental results. A three-dimensional model was presented in [7] to simulate the thermal and flow phenomena occurring in the reheater of the power boiler. The equations formulated in the process were solved using a CFD code. The model was complemented with empirical relations which were used for the calculation of the heat transfer coefficients on the outer surface of the tube. Moreover, the proposed model enabled the use of the results of the velocity, temperature, and gas composition measurements as boundary conditions. Using the model, it was possible to analyse the impact of the boiler operating conditions on the temperature of the tubes and to predict which tubes were exposed to the highest temperatures. A detailed, three-dimensional CFD model for the superheater region of heat-recovery boilers was proposed in [8]. The model included the superheater region geometry and it was supplemented with sub-models to describe the flow field and the heat transfer. The results obtained using the model were compared with measurements and presented good agreement. The developed model can be used to optimise the efficiency of heat recovery boilers. In [9], attention was drawn to the fact that the control of the temperature of superheated steam at the superheater/reheater outlet is an issue of great importance. A new method was proposed for the modelling and control of steam temperature, and was based on a stable fuzzy model predictive controller (SFMPC). An SFMPC was developed for the output regulation for the entire range of the power plant operation. By using the controller, the input-state stability could be guaranteed. In the literature, steam superheater/reheater models incorporated into models of entire power plants have been presented as well, e.g. [10, 11]. In [10], the authors developed original mathematical models of integrated power boiler units (economiser, drum, and superheater). The models were based on equations describing the principles of mass and energy conservation. The modelling enabled the determination of time transients for characteristic load changes. The results were verified experimentally. The models were proposed as prediction tools for the optimisation of the control design. Detailed modelling results of the entire power plant operation are presented in [11]. The results were obtained using the Advanced Process Simulation Environment (APROS) – a commercial simulation programme developed in Finland [12]. To show the influence of control system changes on the power plant dynamic behaviour, the authors developed the simulation model of a 550 MWel hard coal-fired power plant. The results obtained from the model were verified through a comparison with the results of the measurements, which had been collected during the steadyand transient-state operations of the power plant. The agreement between the results was very good (the maximum deviations were less than 5%). The results presented in [11] prove that the APROS software is a highly effective tool for the analysis of the operation flexibility of the power unit. Several other commercial programmes have been developed to perform such simulations (e.g. ENBIPRO, ASPEN Plus DYNAMICS, SIMULINK, MODELICA, and DYMOLA), which were reviewed and described in detail in [13]. The authors presented programmes for simulating the steady-state operation and dynamic processes of thermal power plants. The complexity of the balance equations used in the codes depends, among other factors, on whether the analysed flow problem is a steady-state, a quasi-steady-state or a dynamic one. The dimension of the flow problem is a significant factor as well (the problem can be zero-dimensional, one-dimensional, two-dimensional, or three-dimensional). Programmes simulating transient-state processes enable the analysis of increments in the power-unit load, which are achieved by increasing the steam mass flow rate at the boiler
ACCEPTED MANUSCRIPT outlet. However, these increments are simulated mainly as a result of firing more fuel mass flow rate or condensate stop. Moreover, the details of the results are limited and are determined by the fact that, for example, the source codes of the programmes cannot be modified by the user or are unknown. Another common problem is that the programmes are very costly. Building an in-house program intended for solving a specific problem offers insight to the generated results at any stage of the computations. The equations used in such a program can be analysed at any level of detail. The developed in-house mathematical models can be applied to the on-line mode in a real facility. The flexibility of the operation of power units is also essential in terms of forecasting the demand for electricity. The demand is predicted using different algorithms, such as those presented in [14–16]. These research works deal with long-term forecasting. In [14], an analysis was conducted on the electricity consumption in Iran. The results which were obtained using an artificial cooperative search algorithm were verified based on a comparison with real data from 1992–2013. Moreover, they were compared with the results produced by other optimisation methods. The electricity demand in Iran until 2030 was estimated as well. The optimised gene expression programming (GEP) was used in [15] to identify the relationship between historical data and electrical energy consumption. The data included the electricity consumption of the Association of Southeast Asian Nations (five countries). Similar to [14], the results obtained in [15] were compared with the results obtained from other methods and were validated using data from 1971–2011. The GEP was also used in [16] as an expression-driven approach. The method was applied to estimate the electricity demand in Iran based on economic criteria. The results obtained using the methods presented in [14– 16] appear to be more accurate compared with the results generated by other optimisation methods. The literature review indicates that the phenomenon of the increase in the steam mass flow rate at the superheater outlet caused by the decrease in the pressure at its inlet has not yet been simulated computationally in detail. The decrease in pressure is a temporary process which causes an immediate increase in the steam mass flow rate. It is realised with a set rate and has a duration ranging from a few dozen seconds to approximately one minute, depending on the power-unit initial load. After the pressure-decrease process stops, there is an immediate drop in the steam mass flow rate which is almost equal to the initial increase. In the real facility, the pressure decreases by opening the turbine throttle valve, and the process is carried out in parallel with an increase in the mass flow rate of the fuel fired in the boiler combustion chamber. The total increase in the steam mass flow rate at the boiler outlet is the effect of the two processes, and it should ensure the required increase in the power-unit load within a set time. The required total increments in the live-steam mass flow rate depend on the power-unit initial loads and they are known (they are specified by the turbine manufacturer). After the pressure decrease stops, the required mass flow rate of steam should be maintained only by firing more fuel. The duration of the simultaneous processes of the decrease in pressure and the increase in the fuel mass flow rate, as well as the value by which the pressure should decrease, can only be determined using simulation computations. For this purpose, verified mathematical models are required to perform simulations of the operation of steam superheaters, of the feed water heaters (economisers), and of the boiler evaporator in transient conditions. The models are important, particularly in the case of power units (the supercritical ones as well), which are now under construction. These new facilities are required to operate with great flexibility. Owing to this requirement, power boiler manufacturers have to develop what are known as modified sliding curves which are used as input data for the automatic control system of the power-unit operation. The curves include the changes in the steam pressure at the turbine inlet.
ACCEPTED MANUSCRIPT The total increment in the steam mass flow rate at the boiler outlet is the sum of increments in the rate that they occur in live-steam superheaters and in the boiler evaporator. The only manner in which the increments can be identified is through the development of accurate mathematical models of the heating surfaces and through simulation computations. In this paper, a proposal is presented for an in-house mathematical model of the steam superheater. The model enables a detailed analysis of the increase in the steam mass flow rate at the superheater outlet as the boiler pressure is decreased. It is a one-dimensional model with distributed parameters, and it is based on solving the equations describing the principles of mass, momentum, and energy conservation. The increments in the steam mass flow rate at the superheater outlet which are analysed in this paper are mainly the effect of the pressuredecrease process in the superheater. The simultaneous increase in the steam mass flow rate at the superheater inlet is taken into consideration as well. 2. Mathematical model of the steam superheater The simulations of the increase in the mass flow rate of the steam at the superheater outlet were performed using the mathematical model described below. It is a one-dimensional model and is based on solving the following equations, which describe the principles of mass, momentum, and energy conservation, respectively [17–19]: 1 m , A z
(1)
2 p p f m 1 m A g sin , A z z z 1
i 1 p m 1 p i 1 p f 1 i A z z z
(2)
U 1 p m q . A A z
(3)
The equations are solved using an implicit-difference scheme with the use of a nonstaggered mesh. A characteristic feature of the scheme is that the conservation equations are integrated along the j, j-1 region. Time derivatives are replaced in the equations with forward difference quotients:
j j ,
m m j m j ,
i i j i j ,
(4)
whereas space derivatives are replaced with backward difference quotients: m j 1 m m , j z z
p p j 1 p , j z z
i i j i j 1 . z z
(5)
After the derivatives in Equations (1)–(3) are substituted by the difference schemes, namely Equations (4)–(5), the following relations are obtained after relevant transformations, which enable the determination of the time- and space-dependent distributions of the steam:
ACCEPTED MANUSCRIPT –
mass flow rate: j m j 1 Az m
pressure:
pj
p j 1
z A
m
j
m j
j j
;
j=2, ..., M
2 m j 1 1 m j A2 j j 1
z p 2
f
z
(6)
z j g sin
j
j=2, ..., M
and enthalpy:
ij
1 p i j 1 j i
j
1
4h j j t j d in
m 1 p p ij ij 1 1 p f j j 1 j A j j z z j z
j
1 p A j
m
j
j
j 1 m ; z
j = 2, ..., M.
(7)
j
(8)
p and p in Equation (8) are also approximated using difference quotients. i The pressure loss is expressed using the Darcy–Weisbach formula:
Derivatives
p f z
j
f j m j m j 2d in A2 j
.
(9)
The distributions of the steam density and temperature can then be found as a function of pressure and enthalpy. For this purpose, functions and sub-programmes were developed which enable the precise determination of the physical properties of the fluid in the entire range. The functions were developed based on [20–22]. To obtain a stable solution for the analysed one-dimensional problems, the following Courant–Friedrichs–Lewy stability condition over time step should be satisfied [23]: 1,
z , w
(10)
w is the Courant number. z The temperature of the steam superheater tube wall can be calculated from the following transient heat-conduction one-dimensional equation:
where
c w w
1 rk w . r r r
(11)
In [24–25], the equation was solved for two cases, where the wall was divided into two control volumes and one control volume (on the wall thickness), respectively. The division into two control volumes enables the calculation of the history of changes in the temperature of both sides of the wall—the flue gas side fg and the steam side st (Fig. 1). Such an
ACCEPTED MANUSCRIPT approach is right if great differences in temperature occur on the wall thickness. In the case of steam superheaters, where the temperature drop along the thickness totals only a few degrees (34 K), it is justified that Equation (11) be solved by taking into account the control volume. The result is then a relation describing the history of the mean temperature, . The physical properties of the wall material (cw, w, and kw) are always calculated on-line as a function of temperature.
Fig. 1. Division of the steam superheater tube wall into one or two control volumes [24–25]
In this case, after simple transformations, Equation (11) is written as:
c w w
r
2 o
rin2 rk w rk w . 2 r r ro r r rin
(12)
Considering boundary conditions
kw
h r rin t h t , r r rin
kw
h fg ,eq t fg r r h fg ,eq t fg , o r r ro
(13)
(14)
the following differential equation is obtained: d B t fg C t . d
(15)
After replacing the time derivative with a forward-difference scheme, an equation is obtained which enables the computation of the history of the wall mean temperature: j
1 B C j t fg , j t j ; D D D
j = 1, ..., M.
(16)
In the above equations, namely Equations (15)–(16), B
h fg , eq d o cw w wd m
, C
hdin cw w wd m
, D
1
B C and d m
d o din 2
.
ACCEPTED MANUSCRIPT In the computations, it is assumed that the flow of the steam through all tubes of the superheater and the flow of the flue gases in the entire cross-section of the flue gas duct are uniform. Considering the above, the computations are performed for one tube of the steam superheater after appropriate calculations of the free-flow cross-section of the flue gas per tube (Afg). In the mathematical model, the superheater tube length corresponds to the actual value, as do the values of the outer diameter and the wall thickness. Equations (6)–(8) and (16) are complemented with a relation enabling the calculation of the flue-gas temperature history. The relation can be derived from the following energy balance equation (Fig. 1):
zA fg c fg fg
t fg
m fg i fg m fg i fg z
h fg ,eq d o z t fg . z z
(17)
After relevant rearrangements and assuming that 0 and z0, Formula (17) results in the following differential equation: t fg
F t fg G
t fg z
.
(18)
Approximating the derivatives in the above equation using an implicit-difference scheme and by performing appropriate transformations, a relation is obtained which enables the computation of the flue gas history: t fg,j
1 F G t fg , j j t fg , j 1 . P P Pz
(19)
The “+” sign in Equations (17)–(19) refers to the counter-flow at j = 1, ..., M-1, and the “–” sign to the parallel flow at j = 2, ..., M. The physical properties of the flue gas (cfg and fg) are calculated on-line as a function of temperature. In Equations (18) and (19)
F
h fg , eqd o A fg c fg fg
, G
m fg A fg fg
and P
1
F
G
z
.
The equivalent heat-transfer coefficient on the flue-gas side is found using the following relation: h fg ,eq
1 1 h fg
,
(20)
where h fg h fg ,c h fg ,r .
(21)
For the cross-flow arrangement, the convection heat-transfer coefficient is calculated as [26]:
ACCEPTED MANUSCRIPT
h fg ,c
k wd 0.2C z Cs o do
0.65
Pr 0.33 .
(22)
In the case of the single-phase fluid flowing longitudinally, the Dittus–Boelter formula, complemented with correction factors which have been determined experimentally, is used [26]: 0.8
k wd h 0.023 in Pr 0.4 Ct Cd Cl . d in
(23)
The radiation heat-transfer coefficient for dust-laden flue gases is found from the following relation [27]: h fg ,r
T 4 Θ o4 T 4 Θ o4 (1 o ) (1 0.8) . fg fg 5.67 10 8 fg fg 2 T fg Θ o 2 T fg Θ o
(24)
The flue gas emissivity in Formula (24) is calculated as
fg 1 e as .
(25)
The blackness degree of the flue-gas jet, a, and the thickness of the radiating layer, s, are determined according to [26]. In the proposed model, the boundary conditions (steam temperature, mass flow rate, and pressure, as well as the flue-gas mass flow rate and temperature) can be time-dependent. The conditions can be determined from measurements or from simulation of the preceding superheater or economiser stage operation. Equations (6)–(8) can also be used to determine the history of the fluid mass flow rate, pressure, and enthalpy in the boiler evaporator. For this purpose, the homogeneous flow relations in the two-phase flow region should be assumed [28–29]. The flow can be described using relations derived for the single-phase flow; however, in such case, the liquid- and the gaseous-phase properties should be averaged appropriately. 3. Results and discussion In this section, the results of the simulation computations related mainly to the decrease in the steam pressure in the superheater will be presented. Two methods which enable the computation of the increase in the steam mass flow rate will be introduced. In the first case, the decrease in the pressure was realised at a set rate and the increase in the mass flow rate was determined using the mathematical model of the steam superheater which was described in Section 2. In the second case, simplified calculations will be presented. The results obtained from the two methods will be compared with one another. A short analysis of the simultaneous processes of the decreasing pressure and the increasing steam mass flow rate at the superheater inlet will be presented as well. The simulation computations were performed for the steam superheater stage (Fig. 2). The assumed geometrical data of the superheater and the input data are listed in Tables 1 and 2, respectively.
ACCEPTED MANUSCRIPT
Fig. 2. Diagram of the analysed steam superheater (one panel) Table 1. Assumed geometrical data of the superheater
Parameter Number of tubes
Value 900
Unit -
Number of panels
25
-
36 43 6.3 (from 0 to 15 m in length) 43 7.1 (from 16 m to 30 m in length) 43 8 (from 31 m to 45 m in length) 43 8.8 (from 46 m to 60 m in length) 60
-
Number of tubes in a single panel Outer diameter tube wall thickness
Single tube total length
mm mm mm mm m
Longitudinal pitch
80
mm
Transverse pitch
900
mm
33.16
m3
Superheater “water volume” Material of tubes
X10CrNiCuNb18-9-3 Table 2. Assumed input data of the superheater
Parameter
Value
Unit
Steam temperature
500
°C
Steam mass flow rate
400
kg/s
Steam pressure
250
bar
Flue gas temperature
900
°C
Flue gas mass flow rate
500
kg/s
First, the steam superheating process was simulated until the operating conditions became steady. The proper computations, consisting of simulations of decreasing steam pressure at different rates, were then carried out from the level of the same steady operating conditions. Furthermore, an analysis of the different values by which the pressure was reduced was carried out. This is of paramount importance because the value by which the pressure would be reduced at the same pressure decrease rate determines the process duration and,
ACCEPTED MANUSCRIPT consequently, the time at which the increased steam mass flow rate would be obtained at the superheater outlet. Steady-state conditions of the steam superheater operation were achieved after approximately 160 s, as presented in Figs. 3 and 4, which present the computed histories of the temperature of steam and of the flue gases at the superheater outlet, respectively.
Fig. 3. Curve illustrating changes in the steam temperature at the superheater outlet
Fig. 4. Curve illustrating changes in the fluegas temperature at the superheater outlet
In the real facility, the pressure-decrease process of the boiler is carried out to achieve an immediate increase in the power-unit load of up to a few percentiles compared with the nominal value. The effect of the process is an increment in the steam mass flow rate at the boiler outlet, which means higher power in the power unit. Because the required increase in the power-unit load is related to the nominal power of the facility, it is a constant value independent of the initial load (the level from which the increase is to be realised). For example, if the nominal power of the power unit totals 900 MW and the required increment constitutes 5% of the value, the pressure-decrease process should ensure 45 MW of extra power capacity above the initial load level. Therefore, assuming that the rate of the process is known, it is necessary to find the appropriate value by which the pressure should be decreased. This value should ensure the required increment in the steam mass flow rate (necessary to achieve the required increase in the power-unit load) within a set time. The required increments in the steam mass flow rates, which depend on the initial loads of the power unit, are known (they are specified by the turbine manufacturer). Appropriate values of the decrease in pressure can only be determined through simulation computations. The results of such computations, which were performed for a single steam superheater stage, are presented below. An analysis was carried out regarding the increased steam mass flow rates which were obtained at the superheater outlet owing to the decrease in the steam pressure of - 5 bar and 10 bar at the rate of 10 bar/min, - 10 bar and 15 bar at the rate of 15 bar/min, - 15 bar and 20 bar at the rate of 20 bar/min. 3.1. Simulation of the increase in the steam mass flow rate performed using the developed model of the steam superheater The process of decreasing the pressure by 5 bar at the rate of 10 bar/min (which means that the process lasts for 30 s) is presented in Fig 5. The curve illustrating changes in the steam
ACCEPTED MANUSCRIPT mass flow rate at the superheater outlet which corresponds to this process is presented in Fig 6. Once the pressure-decrease process starts, a jump increase occurs in the steam mass flow rate at the superheater outlet of approximately 2.2 kg/s (Fig. 6). Because the process rate is constant, the increased mass flow rate of the steam is maintained at an almost constant level as the process continues (for 30 s). After the decrease in the pressure stops, there is an immediate decrease in the steam mass flow rate which is almost equal to the initial increase. After a few dozen seconds, the conditions become stabilised and the value of the steam mass flow rate reaches the value it had before the pressure decrease process (400 kg/s).
Fig. 5. Pressure-decrease process—by 5 bar at the rate of 10 bar/min
Fig. 6. Curve illustrating changes in the steam mass flow rate at the superheater outlet
Fig. 7. Curves illustrating changes in the steam temperature and density at the superheater outlet
Figure 7 presents the computed histories of the steam temperature and density at the superheater outlet. The temperature values were found as a function of enthalpy and pressure, which are known from the solution of the equations describing the principles of energy and momentum conservation. As can be seen, the change in the steam temperature related to the pressure-decrease process is only slight, and the value increases by approximately 0.9 K. Owing to the thermal inertia of the superheater, the temperature becomes stabilised after approximately 350 s. The steam temperature stabilisation causes the stabilisation of the steam density, being a function of pressure and temperature. As the pressure decreases (Fig. 5), the steam density decreases as well (Fig. 7). The result is an increase in the specific volume and
ACCEPTED MANUSCRIPT the mass flow rate values of the steam (Fig. 6). If the pressure-decrease process stops, a jump decrease occurs in the steam mass flow rate to a value close to the initial one (i.e. the value before the process). The following figures depict the results of subsequent simulations; they illustrate only the pressure-decrease processes and the changes in the steam mass flow rates at the superheater outlet. The changes in the steam temperature or density values were not analysed. Figures 8 and 9 present the process of the decrease of 10 bar in the pressure at the aforementioned rate of 10 bar/min and the curve illustrating the changes in the steam mass flow rate at the superheater outlet corresponding to the process, respectively. Because the rate of the pressure-decrease process has not changed, when the decrease begins, there is a jump increase in the steam mass flow rate of approximately 2.2 kg/s (Fig. 9) as well. The effect of the decrease of 10 bar in the pressure is that the process lasted for 60 s, i.e. it lasted twice as longer compared with the previous case.
Fig. 8. Pressure-decrease process—by 10 bar at the rate of 10 bar/min
Fig. 9. Curve illustrating changes in the steam mass flow rate at the superheater outlet
The process the decrease of 10 bar in the pressure at the rate of 15 bar/min (which means the process lasted for 40 s) and the curve illustrating the changes in the steam mass flow rate at the superheater outlet are presented in Figs. 10 and 11, respectively.
Fig. 10. Pressure decrease process—by 10 bar at the rate of 15 bar/min
Fig. 11. Curve illustrating changes in the steam mass flow rate at the superheater outlet
ACCEPTED MANUSCRIPT If the pressure-decrease process is carried out faster, there is a greater increase in the steam mass flow rate. It follows from the analysis of the Fig. 11 that the increase totals approximately 3.3 kg/s and it is maintained for the entire duration of the pressure-decrease process (40 s). Figures 12 and 13 present the process of the decrease of 15 bar in the pressure at the aforementioned rate of 15 bar/min and the curve illustrating the changes in the computed steam mass flow rate at the superheater outlet, respectively. In this case, the increment in the steam mass flow rate is approximately 3.3 kg/s as well (the rate of the pressure-decrease process has not been changed), and it lasts for 60 s (Fig. 13).
Fig. 12. Pressure decrease process—by 15 bar at the rate of 15 bar/min
Fig. 13. Curve illustrating changes in the steam mass flow rate at the superheater outlet
The process of the decrease of 15 bar in the pressure at the rate of 20 bar/min (which means that the process lasted for 45 s) and the curve illustrating the changes in the steam mass flow rate at the superheater outlet are presented in Figs. 14 and 15, respectively. Increasing the pressure-decrease rate to 20 bar/min causes an increase in the steam mass flow rate by approximately 4.4 kg/s (Figs. 15 and 17).
Fig. 14. Pressure decrease process—by 15 bar at the rate of 20 bar/min
Fig. 15. Curve illustrating changes in the steam mass flow rate at the superheater outlet
ACCEPTED MANUSCRIPT Figures 16 through 17 present the process of the decrease of 20 bar in the pressure at the aforementioned rate of 20 bar/min and the curve illustrating the changes in the steam mass flow rate at the superheater outlet corresponding to the process, respectively. In a similar manner to Figs. 9 and 13, the increase which can be observed in Fig. 17 lasts for 60 s.
Fig. 16. Pressure decrease process—by 20 bar at the rate of 20 bar/min
Fig. 17. Curve illustrating changes in the steam mass flow rate at the superheater outlet
After analysing Figures 5 through 17, it may be stated that starting the steam pressuredecrease process causes an immediate jump increase in the steam mass flow rate at the superheater outlet. Once the pressure decrease stops, there is an immediate jump decrease to a value close to the initial one. The pressure-decrease processes realised at a constant rate results in jump increments in the steam mass flow rate with constant values (Figs. 6 and 9, Figs. 11 and 13, and Figs. 15 and 17). However, owing to the different values to which the pressure values can decrease to, the pressure-decreasing periods and, consequently, the periods when increased mass flow rates are achieved, are different as well. The pressure-decrease duration is a function of the value by which pressure is to be decreased (at the same rate of the process). The above simulations were performed assuming that the steam mass flow rate at the superheater inlet is a constant value. In the actual facility, the total increase in the steam mass flow rate at the boiler outlet is the effect of a decrease in the boiler pressure which occurs simultaneously with an increase in the quantity of the fired fuel. The pressure-decrease process cannot last too long. The appropriate process duration should be computed in order for the required steam mass flow rate to be maintained only by a higher mass flow rate of the fired fuel (after the pressure-decrease process has stopped). Considering the energy losses, the steam pressure should only be decreased during the required period; then, the pressure value should be “reconstructed”. The only manner to identify the correct value of the decrease in the pressure at a set rate (and, consequently, the process duration) is to develop a mathematical model of the entire boiler and perform simulation computations.
ACCEPTED MANUSCRIPT
Fig. 18. Variation in the steam mass flow rate at the superheater outlet as a result of the pressure decrease by 15 bar and an increase in the quantity of fired fuel
Fig. 19. Variation in the steam mass flow rate at the superheater outlet as a result of the pressure decrease by 20 bar and an increase in the quantity of fired fuel
Figures 18 and 19 present the simulation results obtained during a simultaneous pressure decrease in the superheater and an increase in the steam mass flow rate at the superheater inlet. It was assumed that the pressure decreased by 15 bar and 20 bar at the rate of 20 bar/min (Figs. 14 and Fig. 16, respectively). In both cases, it was also assumed that the steam mass flow rate at the superheater inlet increases linearly from 400 kg/s to 410 kg/s (as a result of increase in the evaporator output). The increase continues for 1 minute and is caused by firing more fuel (in the real facility the rate of the increase results from the characteristic of the mills). Once the pressure-decrease process starts, a jump increase occurs in the steam mass flow rate. At the same time, a linear increase in the mass flow rate of steam at the superheater inlet appears. The total increment in the mass flow rate at the superheater outlet is the effect of both these processes. The different values of the target decrease in pressure (15 bar and 20 bar) resulted in different periods during which the increased mass flow rate of the steam related to the decrease would be generated (45 s and 60 s, respectively). The duration of the process of the increase in the mass flow rate of the fired fuel (and the higher mass flow rate of the steam related to this process) was the same in both these cases and it was 60 s. Figures 18 and 19 indicate that there is a rapid decrease in the steam mass flow rate when the pressure-decrease process stops. In Fig. 18, the decrease can be observed after 45 s and it causes a temporary decrease in the steam mass flow rate to the minimum value of approximately 408 kg/s. Then, for the next 15 s, there is an increase in the steam mass flow rate, which is only due to the firing of a larger quantity of fuel. A decrease in the steam mass flow rate to 410 kg/s is visible in Fig. 19 (after 60 s). Therefore, different values of the target decrease in pressure enable the identification of different minimum values of the steam mass flow rate at the same rate of the decrease process. As previously mentioned, the required values of the increase in the steam mass flow rate at the boiler outlet are specified by the turbine manufacturer. A part of the increase, which is attributed to the characteristic of the mills, is characterised by a constant rate and a constant duration within which it is achieved. In this case, the unknown is the necessary value of the decrease in pressure. Different values of the pressure decrease combined with a constant rate of the increase in the load of the mills make it possible to achieve different minimum values
ACCEPTED MANUSCRIPT of the steam mass flow rate after the pressure-decrease process in the boiler stops (as shown in Figs. 18 and 19 for a single stage of the superheater). To find a value of the minimum mass flow rate which is higher than the required value (after the pressure-decrease process stops), it is necessary to perform simulation computations for different values of the target pressure decrease. This is of fundamental practical importance because the computation results are used to generate the power-unit modified sliding curves, which determine the power-unit operation under conditions of rapid changes in loads. The curves are generated by the boiler manufacturer. 3.2. Simplification of the computations The increment in the steam mass flow rate at a superheater outlet which is achieved by only decreasing the steam pressure can also be estimated in a simplified manner. This simplified manner comprises the calculation of the mean values of steam density (e.g. as a function of pressure and temperature) along the superheater for its two states of operation, i.e. for steady-state conditions before the pressure decreases and for the new steady-state conditions after the pressure decrease. The latter differ from the former only by the pressure value. The steam temperature remains almost unchanged. The simulation computation results indicate that after the pressure decreases within the range of 5–20 bar, the temperature value increases by approximately 1–3 K only; this increase can be neglected (small negligible impact on the steam density). The obtained results also show that it is sufficient to calculate the steam density value as the arithmetic mean at the superheater inlet and outlet for a given state of operation. By multiplying the difference in the mean values of steam density (for two states of operation) by the superheater “water volume”, the total increment in the steam mass is obtained. The increments obtained in this manner are listed in Table 3 for the same pressure-decrease conditions as those assumed for the simulation computations described in Subsection 3.1. As a result of this computation, the increment in the steam mass flow rate lasting for a specific period is achieved. Table 3. Comparison between increments in the steam mass obtained by the decrease in pressure in the superheater (results of simulation computations and simplified calculations)
Parameters of the pressure-decrease process Pressuredecrease rate, bar/min
10 15 20
5
65.8
Total increment in the steam mass calculated using the simplified method, kg 67
10
132.1
133.3
10
131.4
133.3
15
197.1
198.9
15
196.4
198.9
20
261.4
263.8
Value by which pressure was reduced, bar
Total increment in the steam mass computed using the mathematical model of the steam superheater, kg
To enable a comparison between the results of the simulations and those of the simplified calculations, the obtained mass flow rate increment should be multiplied by the known duration of the pressure-decrease process. The results of the multiplication are listed in Table 3 together with those of the simplified calculations. After analysing the results, a good
ACCEPTED MANUSCRIPT agreement is observed. In every case, the increment value obtained using the mathematical model is slightly lower. If this model is applied, once the pressure-decrease process stops, a jump decrease is obtained in the steam mass flow rate to a value which is minimally higher compared with the initial rate (Figs. 6, 9, 11, 13, 15, and 17). This occurs because of the lower pressure in the superheater (after a few dozen seconds, the conditions become stabilised and the steam mass flow rate reaches the initial value of 400 kg/s). The results of the simplified calculations should be treated only as informative. Nevertheless, they enable the assessment of the correctness of the results produced by the developed mathematical model of a steam superheater operating under the conditions of decreasing pressure only. If the live-steam pressure is reduced and the fired-fuel mass flow rate simultaneously increases (a higher mass flow rate of steam is generated in the evaporator), simplified calculations cannot be performed. In this case, the robust and efficient mathematical model of the superheater presented in Section 2 should be applied. 4. Conclusions One of the main requirements that newly designed and constructed power units must satisfy is the high flexibility of operation. It is the effect of the increasing role of renewable energy sources in electricity generation. Because of the characteristics of mills and the considerable thermal inertia of the boiler evaporator, the required increase in the power-unit load (for example, increase by 5% of the nominal power capacity within 30 s) cannot be achieved by merely firing more fuel. One of the methods of ensuring a rapid increase in the power-unit load is to decrease the boiler operation pressure simultaneously with an increase in the mass flow rate of the fuel which is fed into the boiler. The decrease in the pressure causes an immediate increment in the steam mass flow rate at the boiler outlet. The increment is the effect of the decrease in the saturation pressure value in the boiler evaporator and the increase in the specific volume of the steam in the high-pressure part of the boiler which is occupied by steam. Once the required increment in the live-steam mass flow rate is achieved by only firing more fuel, the pressure-decrease process should stop. To obtain the required increase in the power-unit load in the entire range of the load variability, it is necessary to know the appropriate increments in the live-steam mass flow rates at the boiler outlet. These increments are specified by the turbine manufacturer. The only manner in which the correct values of the pressure decrease parameters can be found is through simulation computations; this means that a mathematical model of the entire boiler must be created. The model should consider all the heating surfaces of the boiler, with a special focus on the steam superheater and the boiler evaporator. The mathematical model of the superheater should enable computations of the increment in the steam mass flow rate at the superheater outlet (and, consequently, at the outlet of the boiler) owing to a decrease in the pressure. This is a new approach to modelling the operation of steam superheaters under transient conditions. The total increment in the steam mass flow rate at the boiler outlet, which is the sum of increments at individual stages of the live-steam superheater and in the boiler evaporator, should be computed using verified mathematical models. In this study, an in-house one-dimensional mathematical model of the steam superheater was proposed based on the relations describing the principles of mass, momentum, and energy conservation, which meets the aforementioned requirements. The developed model enables the computation of the increase in the steam mass flow rate at the superheater outlet as an effect of a decrease in the pressure. The variability of steam parameters over time at the superheater inlet can be taken into consideration (in the proposed model, the boundary conditions can be time-dependent). This variability may result from the fact that the evaporator output varies owing to changes in the quantity of the fired fuel.
ACCEPTED MANUSCRIPT The fact that this is an original model developed in-house is of great importance. The model enables the analysis of all parameters at any instant of the pressure-decrease process and the increase in the steam mass flow rate at the superheater outlet. It is also possible to quickly create mathematical models of steam superheaters with any tube configuration. The analysis carried out in this paper demonstrated the effectiveness of the developed model of the steam superheater. The model enables analysis using any pressure-decrease process rate value and any target decrease in pressure. The results obtained are the different increments in the steam mass flow rate and different durations of these increments. This is of fundamental practical importance for the generation of the modified sliding curves of the power unit and constitutes the main contribution of the paper in comparison with previous works. The curves must be generated for new power units (including the supercritical ones), as well as for the existing facilities, which are intended for high-flexibility operation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Trojan M, Taler D. Thermal simulation of superheaters taking into account the processes occurring on the side of the steam and flue gas. Fuel 2015; 150: 75–87. Madejski P, Taler D, Taler J. Numerical model of a steam superheater with a complex shape of the tube cross section using Control Volume based Finite Element Method. Energy Conversion and Management 2016; 118: 179–192. Qi J, Zhou K, Huang J, Si X. Numerical simulation of the heat transfer of superheater tubes in power plants considering oxide scale. International Journal of Heat and Mass Transfer 2018; 122: 929–938. Hong X, Bo D, Dongfang J, Yongzhong N, Naiqiang Z. The finite volume method for evaluating the wall temperature profiles of the superheater and reheater tubes in power plant. Applied Thermal Engineering 2017; 112: 362–370. Purbolaksono J, Khinani A, Rashid A.Z, Ali A.A, Ahmad J, Nordin N.F. A new method for estimating heat flux in superheater and reheater tubes. Nuclear Engineering and Design 2009; 239: 1879–1884. Lijun X, Jamil A.K, Zhihang Ch. Thermal load deviation model for superheater and reheater of a utility boiler. Applied Thermal Engineering 2000; 20: 545–558. Prieto M.M, Suárez I, F.J. Fernández F.J, Sánchez H, Mateos M. Theoretical development of a thermal model for the reheater of a power plant boiler. Applied Thermal Engineering 2007; 27: 619–626. Maakala V, Järvinen M, Vuorinen V. Computational fluid dynamics modeling and experimental validation of heat transfer and fluid flow in the recovery boiler superheater region. Applied Thermal Engineering 2018; 139: 222–238. Xiao W, Jiong S, Yiguo L, Kwang Y.L. Fuzzy modelling and predictive control of superheater steam temperature for power plant. ISA Transactions 2015; 56: 241–251. Sreepradha C, Panda R.C, Bhuvaneswari N.S. Mathematical model for integrated coalfired thermal boiler using physical laws. Energy 2017; 118: 985–998. Hentschel J, Zindler H, Spliethoff H. Modelling and transient simulation of a supercritical coal-fired power plant: Dynamic response to extended secondary control power output. Energy 2017; 137: 927–940. APROS Advanced process simulation software. See also: www.apros.fi/en/. Alobaid F, Mertens N, Starkloff R, Lanz T, Heinze Ch, Epple B. Progress in dynamic simulation of thermal power plants. Progress in Energy and Combustion Science 2016; 59: 79–162. Kaboli S. Hr. Aghay, Selvaraj J, Rahim N.A. Long-term electric energy consumption forecasting via artificial cooperative search algorithm. Energy 2016; 115: 857–871.
ACCEPTED MANUSCRIPT [15] Kaboli S. Hr. Aghay, Fallahpour A, Selvaraj J, Rahim N.A. Long-term electrical energy consumption formulating and forecasting via optimized gene expression programming. Energy 2017; 126: 144–164. [16] Kaboli S. Hr. Aghay, Fallahpour A, Kazemi N, Selvaraj J, Rahim N.A. An ExpressionDriven Approach for Long-Term Electric Power Consumption Forecasting. American Journal of Data Mining and Knowledge Discovery 2016; 1(1): 16–28. [17] Zima W. Simulation of dynamics of a boiler steam superheater with an attemperator. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 2006; 220 (7): 793-801. [18] Zima W, Nowak-Ocłoń M, Ocłoń P. Simulation of fluid heating in combustion chamber waterwalls of boilers for supercritical steam parameters. Energy 2015; 92: 117-127. [19] Grądziel S, Majewski K. Simulation of Thermal and Flow Phenomena in Smooth and Internally Rifled Tubes. Journal Heat Transfer Engineering 2018; 39: 1243-1250. [20] Haar L, Gallagher J, Knell G. NBS/NRC Steam Tables, Thermodynamic and Transport Properties and Computer Programs for Vapour and Liquid States of Water in SI Units, McGraw-Hill International, Toronto 1984. [21] Thermodynamic and Transport of Steam, ASME Steam Tables, Sixth Edition, New York 1993. [22] Wagner W, Kretzschmar H.J. International Steam Tables – Properties of Water and Steam Based on the Industrial Formulation IAPWS-IF97, Springer-Verlag, Berlin 2008. [23] Gerald C.F, Wheatley P.O. Applied Numerical Analysis, Addison-Wesley Publishing Company, New York 1994. [24] Zima W. Mathematical Modelling of Dynamics of Boiler Surfaces Heated Convectively, Heat Transfer – Engineering Applications, Editor – Vyacheslav S. Vikhrenko, InTech, Rijeka 2011. [25] Zima W, Grądziel S. Simulation of transient processes in heating surfaces of power boilers, LAP LAMBERT Academic Publishing; Saarbrücken 2013. [26] Kuznetsov N.W, Мitor W.W, Dubovski I.Е, Karasina E.S. Teplovoi raschet kotelnyh agregatov, ENERGIA, Мoskva 1973 (in Russian). [27] Hottel H.C, Sarofim A.F. Radiative Transfer, McGraw-Hill, New York 1967. [28] Hewitt G.F, Shires G.L, Bott T.R. Process Heat Transfer, CRC Press, Begell House, Boca Raton 1994. [29] Thome J.R. Engineering Data Book III, Wolverine Tube, Inc., 2006.
ACCEPTED MANUSCRIPT Highlights An in-house model of superheater operation under an unsteady-state is proposed The steam pressure-decrease process causes a jump increase in the mass flow rate A rapid increase in the steam mass flow rate at the superheater outlet is computed The change in the steam temperature related to the pressure decrease process is slight The faster the pressure-decrease process, the greater increase in the steam mass flow
Wiesław Zima PII:
S0360-5442(19)30148-3
DOI:
10.1016/j.energy.2019.01.132
Reference:
EGY 14605
To appear in:
Energy
Received Date:
19 September 2018
Accepted Date:
25 January 2019
Please cite this article as: Wiesław Zima, Simulation of steam superheater operation under conditions of pressure decrease, Energy (2019), doi: 10.1016/j.energy.2019.01.132
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ACCEPTED MANUSCRIPT Simulation of steam superheater operation under conditions of pressure decrease Wiesław Zima (corresponding author) The Cracow University of Technology, Institute of Thermal Power Engineering, al. Jana Pawła II 37, 31-864 Kraków, Poland Phone number: 0048 12 6283653, e-mail: [email protected] Abstract A decrease in the live steam pressure at the turbine inlet results in a pressure decrease in the boiler and enables an immediate increase in the steam mass flow rate at the boiler outlet. Assuming that the rate of the pressure-decrease process is known, the correct value of the target pressure decrease can only be determined computationally. This value should ensure the required total increment in the steam mass flow rate at the boiler outlet which is necessary to achieve the requested increase in the power-unit load. The total increment is the sum of increments which occur in live steam superheaters and in the evaporator. Therefore, a robust mathematical model of the steam superheater is essential for the correct computation of the increase in the steam mass flow rate at the boiler outlet which occurs as the pressure decreases. In this paper, an efficient in-house model of a superheater is proposed. It is a distributed-parameter model based on the solution of the equations which describe the principles of mass, momentum, and energy conservation. The model enables the determination of the value by which the pressure must be decreased to ensure the required increment in the steam mass flow rate. Simulation computations related to the increase in the steam mass flow rate at the superheater outlet were performed. Different rates of the pressuredecrease process and different values of the target decrease in the pressure were considered. The obtained results were compared with the results of the simplified calculations for steady states. The developed model is intended for the computation of the live-steam mass flow rate increments at the boiler outlet when rapid increases in loads are required within a short time. The computations are necessary for the generation of the modified sliding curves of the power unit. Keywords: power boiler, steam superheater, steam pressure decreasing, rapid increase in the steam mass flow rate, mathematical model, simulation computations Nomenclature A – surface area, m2 c – specific heat, J/(kg·K) Cd – correction factor taken into account in flows through circular channels (where heating occurs from the outside or inside only), Cl – correction factor taking into account the ratio between the surface length and the equivalent diameter, Cs – correction factor taking into account the tube bank system geometry, Ct – correction factor depending on the fluid and wall temperature, Cz – correction factor taking into account the number of tube rows on the flue gas path, d – diameter, m
ACCEPTED MANUSCRIPT f g h i k L m
M p Pr q r t T fg
– friction factor, – gravitational acceleration, m/s2 – heat transfer coefficient, W/(m2·K) – enthalpy, kJ/kg – thermal conductivity, W/(m·K) – length, m – mass flow rate, kg/s – the number of cross-sections, – pressure, bar – Prandtl number, – heat flux, W/m2 – radius, m – temperature, °C
U w z
– mean flue gas temperature in the control area, K – circumference, m – velocity, m/s – space coordinate, m
fg o o σ z
– thickness, m – fouling factor, (m2·K)/W – flue gas emissivity, – tube outer surface emissivity, – wall mean temperature, °C – tube outer surface temperature, K – density, kg/m3 – Stefan-Boltzmann constant, W/(m2·K4) – time, s – tube inclination angle, ° – time step, s – control volume length, m
Subscripts c – convection eq – equivalent fg – flue gases in – inner j – the number of control volumes along the tube m – mean o – outer r – radiation w – wall 1. Introduction Considering the increasing role of renewable energy sources in electricity generation, power units are now required to display considerable operating flexibility. A rapid increase in the power-unit load is achieved mainly by decreasing the boiler operating pressure. This process is realised by opening the turbine throttle valve. Meanwhile,
ACCEPTED MANUSCRIPT the mass flow rate of fuel fired in the boiler combustion chamber is increased. The pressure decrease process is carried out to obtain an immediate temporary increase in the live-steam mass flow rate at the boiler outlet, which is needed in order to achieve a required increment in the power-unit load within a set time. It is impossible to achieve this increase only by firing more fuel mass flow rate (it results from the mill characteristics and the high thermal inertia of the boiler evaporator). The process of the steam pressure decrease at the turbine inlet causes an immediate pressure decrease in the entire pressure part of the boiler, which results in an increase in the live-steam mass flow rate at the boiler outlet. The increase is due to the decreasing saturation pressure in the evaporator (within subcritical pressure values) and the increase in the specific volume of the steam in the “water volume” of live-steam superheaters. Therefore, the computation of the rapid increase in the mass flow rate of steam at the boiler outlet requires an accurate mathematical model of the superheater. The literature review presented below indicate that issues related to simulations of superheater operation are still relevant today. The mathematical models presented in [1–2] concerned a steam superheater characterised by a complex tube configuration. A detailed analysis was performed in [1] regarding the impact of the outer and inner deposit layers on the temperature of the steam, wall, and flue gases at the steady state. It was demonstrated that deposits on the inner surface of the superheater tube caused a great increase in the tube wall temperature despite the decrease in the heat flux transferred from the flue gases to the steam. Moreover, the proposed model enabled the consideration of the impact of temperature nonuniformity of the flue gas which existed in the cross-section of the flue-gas duct on the wall temperature of the superheater tubes. In [2], relations were proposed which enabled the calculation of the tube wall temperature, as well as the steam pressure and temperature along the steam flow path under steady-state conditions. A control volume-based finite element method was used to determine the temperature distribution in a superheater tube. The advantage of the method was that the results were obtained very quickly. The time needed to obtain the results was several times less compared with computational fluid dynamics (CFD) computations. The authors of [3] analysed the heat transfer in the tubes of a supercritical steam superheater. The developed method was a combination of three-dimensional CFD simulations (thermal processes of external flue gases) and a 1D analytical model (the steam temperature distribution). Approaching the problem in this manner, it was possible to determine the boundary conditions on both surfaces of the tube. The heat transfer was analysed along the entire length of the superheater tubes, which enabled appropriate selection of the tube material to prevent overheating. In [4], the authors analysed a similar problem. They developed a mathematical model enabling the estimation of the tube-wall temperature distribution of heating surfaces operating in high temperatures. The model enabled the consideration of the inner surface fouling which may cause overheating of the tubes. A detailed analysis was performed on the superheater and reheater tubes. The proposed model was based on the finite volume method and was complemented with steam temperature measurements at the inlet and outlet of the superheater/reheater stage. In the model, it was assumed that the pressure of the steam flowing through the superheater was constant. The superheater and reheater tubes were investigated in [5] as well, where a new method was presented to determine the heat fluxes transferred in the tubes of these heating surfaces of the boilers. The proposed procedure was iterative and was developed using empirical relations. The finite element method was employed to find the temperature distribution in the tube. The forced convection on the inner surface of the tube and the cross flow of the flue gases were considered as well. In [6], attention was drawn to the fact that considerable deviations of the steam temperature in the boiler superheater and reheater could substantially affect the safe and economic operation of the facility. The deviations constituted some of the main causes of tube
ACCEPTED MANUSCRIPT failure, which account for approximately 40% of forced breaks in the power plant operation. To predict potential tube failure, the authors of [6] proposed a model of the thermal load deviation for the superheater and the reheater. The model was based on in situ thermodynamic parameters of the power plant and the deviation theory of the flow rate. The results which were obtained using the model were in good agreement with experimental results. A three-dimensional model was presented in [7] to simulate the thermal and flow phenomena occurring in the reheater of the power boiler. The equations formulated in the process were solved using a CFD code. The model was complemented with empirical relations which were used for the calculation of the heat transfer coefficients on the outer surface of the tube. Moreover, the proposed model enabled the use of the results of the velocity, temperature, and gas composition measurements as boundary conditions. Using the model, it was possible to analyse the impact of the boiler operating conditions on the temperature of the tubes and to predict which tubes were exposed to the highest temperatures. A detailed, three-dimensional CFD model for the superheater region of heat-recovery boilers was proposed in [8]. The model included the superheater region geometry and it was supplemented with sub-models to describe the flow field and the heat transfer. The results obtained using the model were compared with measurements and presented good agreement. The developed model can be used to optimise the efficiency of heat recovery boilers. In [9], attention was drawn to the fact that the control of the temperature of superheated steam at the superheater/reheater outlet is an issue of great importance. A new method was proposed for the modelling and control of steam temperature, and was based on a stable fuzzy model predictive controller (SFMPC). An SFMPC was developed for the output regulation for the entire range of the power plant operation. By using the controller, the input-state stability could be guaranteed. In the literature, steam superheater/reheater models incorporated into models of entire power plants have been presented as well, e.g. [10, 11]. In [10], the authors developed original mathematical models of integrated power boiler units (economiser, drum, and superheater). The models were based on equations describing the principles of mass and energy conservation. The modelling enabled the determination of time transients for characteristic load changes. The results were verified experimentally. The models were proposed as prediction tools for the optimisation of the control design. Detailed modelling results of the entire power plant operation are presented in [11]. The results were obtained using the Advanced Process Simulation Environment (APROS) – a commercial simulation programme developed in Finland [12]. To show the influence of control system changes on the power plant dynamic behaviour, the authors developed the simulation model of a 550 MWel hard coal-fired power plant. The results obtained from the model were verified through a comparison with the results of the measurements, which had been collected during the steadyand transient-state operations of the power plant. The agreement between the results was very good (the maximum deviations were less than 5%). The results presented in [11] prove that the APROS software is a highly effective tool for the analysis of the operation flexibility of the power unit. Several other commercial programmes have been developed to perform such simulations (e.g. ENBIPRO, ASPEN Plus DYNAMICS, SIMULINK, MODELICA, and DYMOLA), which were reviewed and described in detail in [13]. The authors presented programmes for simulating the steady-state operation and dynamic processes of thermal power plants. The complexity of the balance equations used in the codes depends, among other factors, on whether the analysed flow problem is a steady-state, a quasi-steady-state or a dynamic one. The dimension of the flow problem is a significant factor as well (the problem can be zero-dimensional, one-dimensional, two-dimensional, or three-dimensional). Programmes simulating transient-state processes enable the analysis of increments in the power-unit load, which are achieved by increasing the steam mass flow rate at the boiler
ACCEPTED MANUSCRIPT outlet. However, these increments are simulated mainly as a result of firing more fuel mass flow rate or condensate stop. Moreover, the details of the results are limited and are determined by the fact that, for example, the source codes of the programmes cannot be modified by the user or are unknown. Another common problem is that the programmes are very costly. Building an in-house program intended for solving a specific problem offers insight to the generated results at any stage of the computations. The equations used in such a program can be analysed at any level of detail. The developed in-house mathematical models can be applied to the on-line mode in a real facility. The flexibility of the operation of power units is also essential in terms of forecasting the demand for electricity. The demand is predicted using different algorithms, such as those presented in [14–16]. These research works deal with long-term forecasting. In [14], an analysis was conducted on the electricity consumption in Iran. The results which were obtained using an artificial cooperative search algorithm were verified based on a comparison with real data from 1992–2013. Moreover, they were compared with the results produced by other optimisation methods. The electricity demand in Iran until 2030 was estimated as well. The optimised gene expression programming (GEP) was used in [15] to identify the relationship between historical data and electrical energy consumption. The data included the electricity consumption of the Association of Southeast Asian Nations (five countries). Similar to [14], the results obtained in [15] were compared with the results obtained from other methods and were validated using data from 1971–2011. The GEP was also used in [16] as an expression-driven approach. The method was applied to estimate the electricity demand in Iran based on economic criteria. The results obtained using the methods presented in [14– 16] appear to be more accurate compared with the results generated by other optimisation methods. The literature review indicates that the phenomenon of the increase in the steam mass flow rate at the superheater outlet caused by the decrease in the pressure at its inlet has not yet been simulated computationally in detail. The decrease in pressure is a temporary process which causes an immediate increase in the steam mass flow rate. It is realised with a set rate and has a duration ranging from a few dozen seconds to approximately one minute, depending on the power-unit initial load. After the pressure-decrease process stops, there is an immediate drop in the steam mass flow rate which is almost equal to the initial increase. In the real facility, the pressure decreases by opening the turbine throttle valve, and the process is carried out in parallel with an increase in the mass flow rate of the fuel fired in the boiler combustion chamber. The total increase in the steam mass flow rate at the boiler outlet is the effect of the two processes, and it should ensure the required increase in the power-unit load within a set time. The required total increments in the live-steam mass flow rate depend on the power-unit initial loads and they are known (they are specified by the turbine manufacturer). After the pressure decrease stops, the required mass flow rate of steam should be maintained only by firing more fuel. The duration of the simultaneous processes of the decrease in pressure and the increase in the fuel mass flow rate, as well as the value by which the pressure should decrease, can only be determined using simulation computations. For this purpose, verified mathematical models are required to perform simulations of the operation of steam superheaters, of the feed water heaters (economisers), and of the boiler evaporator in transient conditions. The models are important, particularly in the case of power units (the supercritical ones as well), which are now under construction. These new facilities are required to operate with great flexibility. Owing to this requirement, power boiler manufacturers have to develop what are known as modified sliding curves which are used as input data for the automatic control system of the power-unit operation. The curves include the changes in the steam pressure at the turbine inlet.
ACCEPTED MANUSCRIPT The total increment in the steam mass flow rate at the boiler outlet is the sum of increments in the rate that they occur in live-steam superheaters and in the boiler evaporator. The only manner in which the increments can be identified is through the development of accurate mathematical models of the heating surfaces and through simulation computations. In this paper, a proposal is presented for an in-house mathematical model of the steam superheater. The model enables a detailed analysis of the increase in the steam mass flow rate at the superheater outlet as the boiler pressure is decreased. It is a one-dimensional model with distributed parameters, and it is based on solving the equations describing the principles of mass, momentum, and energy conservation. The increments in the steam mass flow rate at the superheater outlet which are analysed in this paper are mainly the effect of the pressuredecrease process in the superheater. The simultaneous increase in the steam mass flow rate at the superheater inlet is taken into consideration as well. 2. Mathematical model of the steam superheater The simulations of the increase in the mass flow rate of the steam at the superheater outlet were performed using the mathematical model described below. It is a one-dimensional model and is based on solving the following equations, which describe the principles of mass, momentum, and energy conservation, respectively [17–19]: 1 m , A z
(1)
2 p p f m 1 m A g sin , A z z z 1
i 1 p m 1 p i 1 p f 1 i A z z z
(2)
U 1 p m q . A A z
(3)
The equations are solved using an implicit-difference scheme with the use of a nonstaggered mesh. A characteristic feature of the scheme is that the conservation equations are integrated along the j, j-1 region. Time derivatives are replaced in the equations with forward difference quotients:
j j ,
m m j m j ,
i i j i j ,
(4)
whereas space derivatives are replaced with backward difference quotients: m j 1 m m , j z z
p p j 1 p , j z z
i i j i j 1 . z z
(5)
After the derivatives in Equations (1)–(3) are substituted by the difference schemes, namely Equations (4)–(5), the following relations are obtained after relevant transformations, which enable the determination of the time- and space-dependent distributions of the steam:
ACCEPTED MANUSCRIPT –
mass flow rate: j m j 1 Az m
pressure:
pj
p j 1
z A
m
j
m j
j j
;
j=2, ..., M
2 m j 1 1 m j A2 j j 1
z p 2
f
z
(6)
z j g sin
j
j=2, ..., M
and enthalpy:
ij
1 p i j 1 j i
j
1
4h j j t j d in
m 1 p p ij ij 1 1 p f j j 1 j A j j z z j z
j
1 p A j
m
j
j
j 1 m ; z
j = 2, ..., M.
(7)
j
(8)
p and p in Equation (8) are also approximated using difference quotients. i The pressure loss is expressed using the Darcy–Weisbach formula:
Derivatives
p f z
j
f j m j m j 2d in A2 j
.
(9)
The distributions of the steam density and temperature can then be found as a function of pressure and enthalpy. For this purpose, functions and sub-programmes were developed which enable the precise determination of the physical properties of the fluid in the entire range. The functions were developed based on [20–22]. To obtain a stable solution for the analysed one-dimensional problems, the following Courant–Friedrichs–Lewy stability condition over time step should be satisfied [23]: 1,
z , w
(10)
w is the Courant number. z The temperature of the steam superheater tube wall can be calculated from the following transient heat-conduction one-dimensional equation:
where
c w w
1 rk w . r r r
(11)
In [24–25], the equation was solved for two cases, where the wall was divided into two control volumes and one control volume (on the wall thickness), respectively. The division into two control volumes enables the calculation of the history of changes in the temperature of both sides of the wall—the flue gas side fg and the steam side st (Fig. 1). Such an
ACCEPTED MANUSCRIPT approach is right if great differences in temperature occur on the wall thickness. In the case of steam superheaters, where the temperature drop along the thickness totals only a few degrees (34 K), it is justified that Equation (11) be solved by taking into account the control volume. The result is then a relation describing the history of the mean temperature, . The physical properties of the wall material (cw, w, and kw) are always calculated on-line as a function of temperature.
Fig. 1. Division of the steam superheater tube wall into one or two control volumes [24–25]
In this case, after simple transformations, Equation (11) is written as:
c w w
r
2 o
rin2 rk w rk w . 2 r r ro r r rin
(12)
Considering boundary conditions
kw
h r rin t h t , r r rin
kw
h fg ,eq t fg r r h fg ,eq t fg , o r r ro
(13)
(14)
the following differential equation is obtained: d B t fg C t . d
(15)
After replacing the time derivative with a forward-difference scheme, an equation is obtained which enables the computation of the history of the wall mean temperature: j
1 B C j t fg , j t j ; D D D
j = 1, ..., M.
(16)
In the above equations, namely Equations (15)–(16), B
h fg , eq d o cw w wd m
, C
hdin cw w wd m
, D
1
B C and d m
d o din 2
.
ACCEPTED MANUSCRIPT In the computations, it is assumed that the flow of the steam through all tubes of the superheater and the flow of the flue gases in the entire cross-section of the flue gas duct are uniform. Considering the above, the computations are performed for one tube of the steam superheater after appropriate calculations of the free-flow cross-section of the flue gas per tube (Afg). In the mathematical model, the superheater tube length corresponds to the actual value, as do the values of the outer diameter and the wall thickness. Equations (6)–(8) and (16) are complemented with a relation enabling the calculation of the flue-gas temperature history. The relation can be derived from the following energy balance equation (Fig. 1):
zA fg c fg fg
t fg
m fg i fg m fg i fg z
h fg ,eq d o z t fg . z z
(17)
After relevant rearrangements and assuming that 0 and z0, Formula (17) results in the following differential equation: t fg
F t fg G
t fg z
.
(18)
Approximating the derivatives in the above equation using an implicit-difference scheme and by performing appropriate transformations, a relation is obtained which enables the computation of the flue gas history: t fg,j
1 F G t fg , j j t fg , j 1 . P P Pz
(19)
The “+” sign in Equations (17)–(19) refers to the counter-flow at j = 1, ..., M-1, and the “–” sign to the parallel flow at j = 2, ..., M. The physical properties of the flue gas (cfg and fg) are calculated on-line as a function of temperature. In Equations (18) and (19)
F
h fg , eqd o A fg c fg fg
, G
m fg A fg fg
and P
1
F
G
z
.
The equivalent heat-transfer coefficient on the flue-gas side is found using the following relation: h fg ,eq
1 1 h fg
,
(20)
where h fg h fg ,c h fg ,r .
(21)
For the cross-flow arrangement, the convection heat-transfer coefficient is calculated as [26]:
ACCEPTED MANUSCRIPT
h fg ,c
k wd 0.2C z Cs o do
0.65
Pr 0.33 .
(22)
In the case of the single-phase fluid flowing longitudinally, the Dittus–Boelter formula, complemented with correction factors which have been determined experimentally, is used [26]: 0.8
k wd h 0.023 in Pr 0.4 Ct Cd Cl . d in
(23)
The radiation heat-transfer coefficient for dust-laden flue gases is found from the following relation [27]: h fg ,r
T 4 Θ o4 T 4 Θ o4 (1 o ) (1 0.8) . fg fg 5.67 10 8 fg fg 2 T fg Θ o 2 T fg Θ o
(24)
The flue gas emissivity in Formula (24) is calculated as
fg 1 e as .
(25)
The blackness degree of the flue-gas jet, a, and the thickness of the radiating layer, s, are determined according to [26]. In the proposed model, the boundary conditions (steam temperature, mass flow rate, and pressure, as well as the flue-gas mass flow rate and temperature) can be time-dependent. The conditions can be determined from measurements or from simulation of the preceding superheater or economiser stage operation. Equations (6)–(8) can also be used to determine the history of the fluid mass flow rate, pressure, and enthalpy in the boiler evaporator. For this purpose, the homogeneous flow relations in the two-phase flow region should be assumed [28–29]. The flow can be described using relations derived for the single-phase flow; however, in such case, the liquid- and the gaseous-phase properties should be averaged appropriately. 3. Results and discussion In this section, the results of the simulation computations related mainly to the decrease in the steam pressure in the superheater will be presented. Two methods which enable the computation of the increase in the steam mass flow rate will be introduced. In the first case, the decrease in the pressure was realised at a set rate and the increase in the mass flow rate was determined using the mathematical model of the steam superheater which was described in Section 2. In the second case, simplified calculations will be presented. The results obtained from the two methods will be compared with one another. A short analysis of the simultaneous processes of the decreasing pressure and the increasing steam mass flow rate at the superheater inlet will be presented as well. The simulation computations were performed for the steam superheater stage (Fig. 2). The assumed geometrical data of the superheater and the input data are listed in Tables 1 and 2, respectively.
ACCEPTED MANUSCRIPT
Fig. 2. Diagram of the analysed steam superheater (one panel) Table 1. Assumed geometrical data of the superheater
Parameter Number of tubes
Value 900
Unit -
Number of panels
25
-
36 43 6.3 (from 0 to 15 m in length) 43 7.1 (from 16 m to 30 m in length) 43 8 (from 31 m to 45 m in length) 43 8.8 (from 46 m to 60 m in length) 60
-
Number of tubes in a single panel Outer diameter tube wall thickness
Single tube total length
mm mm mm mm m
Longitudinal pitch
80
mm
Transverse pitch
900
mm
33.16
m3
Superheater “water volume” Material of tubes
X10CrNiCuNb18-9-3 Table 2. Assumed input data of the superheater
Parameter
Value
Unit
Steam temperature
500
°C
Steam mass flow rate
400
kg/s
Steam pressure
250
bar
Flue gas temperature
900
°C
Flue gas mass flow rate
500
kg/s
First, the steam superheating process was simulated until the operating conditions became steady. The proper computations, consisting of simulations of decreasing steam pressure at different rates, were then carried out from the level of the same steady operating conditions. Furthermore, an analysis of the different values by which the pressure was reduced was carried out. This is of paramount importance because the value by which the pressure would be reduced at the same pressure decrease rate determines the process duration and,
ACCEPTED MANUSCRIPT consequently, the time at which the increased steam mass flow rate would be obtained at the superheater outlet. Steady-state conditions of the steam superheater operation were achieved after approximately 160 s, as presented in Figs. 3 and 4, which present the computed histories of the temperature of steam and of the flue gases at the superheater outlet, respectively.
Fig. 3. Curve illustrating changes in the steam temperature at the superheater outlet
Fig. 4. Curve illustrating changes in the fluegas temperature at the superheater outlet
In the real facility, the pressure-decrease process of the boiler is carried out to achieve an immediate increase in the power-unit load of up to a few percentiles compared with the nominal value. The effect of the process is an increment in the steam mass flow rate at the boiler outlet, which means higher power in the power unit. Because the required increase in the power-unit load is related to the nominal power of the facility, it is a constant value independent of the initial load (the level from which the increase is to be realised). For example, if the nominal power of the power unit totals 900 MW and the required increment constitutes 5% of the value, the pressure-decrease process should ensure 45 MW of extra power capacity above the initial load level. Therefore, assuming that the rate of the process is known, it is necessary to find the appropriate value by which the pressure should be decreased. This value should ensure the required increment in the steam mass flow rate (necessary to achieve the required increase in the power-unit load) within a set time. The required increments in the steam mass flow rates, which depend on the initial loads of the power unit, are known (they are specified by the turbine manufacturer). Appropriate values of the decrease in pressure can only be determined through simulation computations. The results of such computations, which were performed for a single steam superheater stage, are presented below. An analysis was carried out regarding the increased steam mass flow rates which were obtained at the superheater outlet owing to the decrease in the steam pressure of - 5 bar and 10 bar at the rate of 10 bar/min, - 10 bar and 15 bar at the rate of 15 bar/min, - 15 bar and 20 bar at the rate of 20 bar/min. 3.1. Simulation of the increase in the steam mass flow rate performed using the developed model of the steam superheater The process of decreasing the pressure by 5 bar at the rate of 10 bar/min (which means that the process lasts for 30 s) is presented in Fig 5. The curve illustrating changes in the steam
ACCEPTED MANUSCRIPT mass flow rate at the superheater outlet which corresponds to this process is presented in Fig 6. Once the pressure-decrease process starts, a jump increase occurs in the steam mass flow rate at the superheater outlet of approximately 2.2 kg/s (Fig. 6). Because the process rate is constant, the increased mass flow rate of the steam is maintained at an almost constant level as the process continues (for 30 s). After the decrease in the pressure stops, there is an immediate decrease in the steam mass flow rate which is almost equal to the initial increase. After a few dozen seconds, the conditions become stabilised and the value of the steam mass flow rate reaches the value it had before the pressure decrease process (400 kg/s).
Fig. 5. Pressure-decrease process—by 5 bar at the rate of 10 bar/min
Fig. 6. Curve illustrating changes in the steam mass flow rate at the superheater outlet
Fig. 7. Curves illustrating changes in the steam temperature and density at the superheater outlet
Figure 7 presents the computed histories of the steam temperature and density at the superheater outlet. The temperature values were found as a function of enthalpy and pressure, which are known from the solution of the equations describing the principles of energy and momentum conservation. As can be seen, the change in the steam temperature related to the pressure-decrease process is only slight, and the value increases by approximately 0.9 K. Owing to the thermal inertia of the superheater, the temperature becomes stabilised after approximately 350 s. The steam temperature stabilisation causes the stabilisation of the steam density, being a function of pressure and temperature. As the pressure decreases (Fig. 5), the steam density decreases as well (Fig. 7). The result is an increase in the specific volume and
ACCEPTED MANUSCRIPT the mass flow rate values of the steam (Fig. 6). If the pressure-decrease process stops, a jump decrease occurs in the steam mass flow rate to a value close to the initial one (i.e. the value before the process). The following figures depict the results of subsequent simulations; they illustrate only the pressure-decrease processes and the changes in the steam mass flow rates at the superheater outlet. The changes in the steam temperature or density values were not analysed. Figures 8 and 9 present the process of the decrease of 10 bar in the pressure at the aforementioned rate of 10 bar/min and the curve illustrating the changes in the steam mass flow rate at the superheater outlet corresponding to the process, respectively. Because the rate of the pressure-decrease process has not changed, when the decrease begins, there is a jump increase in the steam mass flow rate of approximately 2.2 kg/s (Fig. 9) as well. The effect of the decrease of 10 bar in the pressure is that the process lasted for 60 s, i.e. it lasted twice as longer compared with the previous case.
Fig. 8. Pressure-decrease process—by 10 bar at the rate of 10 bar/min
Fig. 9. Curve illustrating changes in the steam mass flow rate at the superheater outlet
The process the decrease of 10 bar in the pressure at the rate of 15 bar/min (which means the process lasted for 40 s) and the curve illustrating the changes in the steam mass flow rate at the superheater outlet are presented in Figs. 10 and 11, respectively.
Fig. 10. Pressure decrease process—by 10 bar at the rate of 15 bar/min
Fig. 11. Curve illustrating changes in the steam mass flow rate at the superheater outlet
ACCEPTED MANUSCRIPT If the pressure-decrease process is carried out faster, there is a greater increase in the steam mass flow rate. It follows from the analysis of the Fig. 11 that the increase totals approximately 3.3 kg/s and it is maintained for the entire duration of the pressure-decrease process (40 s). Figures 12 and 13 present the process of the decrease of 15 bar in the pressure at the aforementioned rate of 15 bar/min and the curve illustrating the changes in the computed steam mass flow rate at the superheater outlet, respectively. In this case, the increment in the steam mass flow rate is approximately 3.3 kg/s as well (the rate of the pressure-decrease process has not been changed), and it lasts for 60 s (Fig. 13).
Fig. 12. Pressure decrease process—by 15 bar at the rate of 15 bar/min
Fig. 13. Curve illustrating changes in the steam mass flow rate at the superheater outlet
The process of the decrease of 15 bar in the pressure at the rate of 20 bar/min (which means that the process lasted for 45 s) and the curve illustrating the changes in the steam mass flow rate at the superheater outlet are presented in Figs. 14 and 15, respectively. Increasing the pressure-decrease rate to 20 bar/min causes an increase in the steam mass flow rate by approximately 4.4 kg/s (Figs. 15 and 17).
Fig. 14. Pressure decrease process—by 15 bar at the rate of 20 bar/min
Fig. 15. Curve illustrating changes in the steam mass flow rate at the superheater outlet
ACCEPTED MANUSCRIPT Figures 16 through 17 present the process of the decrease of 20 bar in the pressure at the aforementioned rate of 20 bar/min and the curve illustrating the changes in the steam mass flow rate at the superheater outlet corresponding to the process, respectively. In a similar manner to Figs. 9 and 13, the increase which can be observed in Fig. 17 lasts for 60 s.
Fig. 16. Pressure decrease process—by 20 bar at the rate of 20 bar/min
Fig. 17. Curve illustrating changes in the steam mass flow rate at the superheater outlet
After analysing Figures 5 through 17, it may be stated that starting the steam pressuredecrease process causes an immediate jump increase in the steam mass flow rate at the superheater outlet. Once the pressure decrease stops, there is an immediate jump decrease to a value close to the initial one. The pressure-decrease processes realised at a constant rate results in jump increments in the steam mass flow rate with constant values (Figs. 6 and 9, Figs. 11 and 13, and Figs. 15 and 17). However, owing to the different values to which the pressure values can decrease to, the pressure-decreasing periods and, consequently, the periods when increased mass flow rates are achieved, are different as well. The pressure-decrease duration is a function of the value by which pressure is to be decreased (at the same rate of the process). The above simulations were performed assuming that the steam mass flow rate at the superheater inlet is a constant value. In the actual facility, the total increase in the steam mass flow rate at the boiler outlet is the effect of a decrease in the boiler pressure which occurs simultaneously with an increase in the quantity of the fired fuel. The pressure-decrease process cannot last too long. The appropriate process duration should be computed in order for the required steam mass flow rate to be maintained only by a higher mass flow rate of the fired fuel (after the pressure-decrease process has stopped). Considering the energy losses, the steam pressure should only be decreased during the required period; then, the pressure value should be “reconstructed”. The only manner to identify the correct value of the decrease in the pressure at a set rate (and, consequently, the process duration) is to develop a mathematical model of the entire boiler and perform simulation computations.
ACCEPTED MANUSCRIPT
Fig. 18. Variation in the steam mass flow rate at the superheater outlet as a result of the pressure decrease by 15 bar and an increase in the quantity of fired fuel
Fig. 19. Variation in the steam mass flow rate at the superheater outlet as a result of the pressure decrease by 20 bar and an increase in the quantity of fired fuel
Figures 18 and 19 present the simulation results obtained during a simultaneous pressure decrease in the superheater and an increase in the steam mass flow rate at the superheater inlet. It was assumed that the pressure decreased by 15 bar and 20 bar at the rate of 20 bar/min (Figs. 14 and Fig. 16, respectively). In both cases, it was also assumed that the steam mass flow rate at the superheater inlet increases linearly from 400 kg/s to 410 kg/s (as a result of increase in the evaporator output). The increase continues for 1 minute and is caused by firing more fuel (in the real facility the rate of the increase results from the characteristic of the mills). Once the pressure-decrease process starts, a jump increase occurs in the steam mass flow rate. At the same time, a linear increase in the mass flow rate of steam at the superheater inlet appears. The total increment in the mass flow rate at the superheater outlet is the effect of both these processes. The different values of the target decrease in pressure (15 bar and 20 bar) resulted in different periods during which the increased mass flow rate of the steam related to the decrease would be generated (45 s and 60 s, respectively). The duration of the process of the increase in the mass flow rate of the fired fuel (and the higher mass flow rate of the steam related to this process) was the same in both these cases and it was 60 s. Figures 18 and 19 indicate that there is a rapid decrease in the steam mass flow rate when the pressure-decrease process stops. In Fig. 18, the decrease can be observed after 45 s and it causes a temporary decrease in the steam mass flow rate to the minimum value of approximately 408 kg/s. Then, for the next 15 s, there is an increase in the steam mass flow rate, which is only due to the firing of a larger quantity of fuel. A decrease in the steam mass flow rate to 410 kg/s is visible in Fig. 19 (after 60 s). Therefore, different values of the target decrease in pressure enable the identification of different minimum values of the steam mass flow rate at the same rate of the decrease process. As previously mentioned, the required values of the increase in the steam mass flow rate at the boiler outlet are specified by the turbine manufacturer. A part of the increase, which is attributed to the characteristic of the mills, is characterised by a constant rate and a constant duration within which it is achieved. In this case, the unknown is the necessary value of the decrease in pressure. Different values of the pressure decrease combined with a constant rate of the increase in the load of the mills make it possible to achieve different minimum values
ACCEPTED MANUSCRIPT of the steam mass flow rate after the pressure-decrease process in the boiler stops (as shown in Figs. 18 and 19 for a single stage of the superheater). To find a value of the minimum mass flow rate which is higher than the required value (after the pressure-decrease process stops), it is necessary to perform simulation computations for different values of the target pressure decrease. This is of fundamental practical importance because the computation results are used to generate the power-unit modified sliding curves, which determine the power-unit operation under conditions of rapid changes in loads. The curves are generated by the boiler manufacturer. 3.2. Simplification of the computations The increment in the steam mass flow rate at a superheater outlet which is achieved by only decreasing the steam pressure can also be estimated in a simplified manner. This simplified manner comprises the calculation of the mean values of steam density (e.g. as a function of pressure and temperature) along the superheater for its two states of operation, i.e. for steady-state conditions before the pressure decreases and for the new steady-state conditions after the pressure decrease. The latter differ from the former only by the pressure value. The steam temperature remains almost unchanged. The simulation computation results indicate that after the pressure decreases within the range of 5–20 bar, the temperature value increases by approximately 1–3 K only; this increase can be neglected (small negligible impact on the steam density). The obtained results also show that it is sufficient to calculate the steam density value as the arithmetic mean at the superheater inlet and outlet for a given state of operation. By multiplying the difference in the mean values of steam density (for two states of operation) by the superheater “water volume”, the total increment in the steam mass is obtained. The increments obtained in this manner are listed in Table 3 for the same pressure-decrease conditions as those assumed for the simulation computations described in Subsection 3.1. As a result of this computation, the increment in the steam mass flow rate lasting for a specific period is achieved. Table 3. Comparison between increments in the steam mass obtained by the decrease in pressure in the superheater (results of simulation computations and simplified calculations)
Parameters of the pressure-decrease process Pressuredecrease rate, bar/min
10 15 20
5
65.8
Total increment in the steam mass calculated using the simplified method, kg 67
10
132.1
133.3
10
131.4
133.3
15
197.1
198.9
15
196.4
198.9
20
261.4
263.8
Value by which pressure was reduced, bar
Total increment in the steam mass computed using the mathematical model of the steam superheater, kg
To enable a comparison between the results of the simulations and those of the simplified calculations, the obtained mass flow rate increment should be multiplied by the known duration of the pressure-decrease process. The results of the multiplication are listed in Table 3 together with those of the simplified calculations. After analysing the results, a good
ACCEPTED MANUSCRIPT agreement is observed. In every case, the increment value obtained using the mathematical model is slightly lower. If this model is applied, once the pressure-decrease process stops, a jump decrease is obtained in the steam mass flow rate to a value which is minimally higher compared with the initial rate (Figs. 6, 9, 11, 13, 15, and 17). This occurs because of the lower pressure in the superheater (after a few dozen seconds, the conditions become stabilised and the steam mass flow rate reaches the initial value of 400 kg/s). The results of the simplified calculations should be treated only as informative. Nevertheless, they enable the assessment of the correctness of the results produced by the developed mathematical model of a steam superheater operating under the conditions of decreasing pressure only. If the live-steam pressure is reduced and the fired-fuel mass flow rate simultaneously increases (a higher mass flow rate of steam is generated in the evaporator), simplified calculations cannot be performed. In this case, the robust and efficient mathematical model of the superheater presented in Section 2 should be applied. 4. Conclusions One of the main requirements that newly designed and constructed power units must satisfy is the high flexibility of operation. It is the effect of the increasing role of renewable energy sources in electricity generation. Because of the characteristics of mills and the considerable thermal inertia of the boiler evaporator, the required increase in the power-unit load (for example, increase by 5% of the nominal power capacity within 30 s) cannot be achieved by merely firing more fuel. One of the methods of ensuring a rapid increase in the power-unit load is to decrease the boiler operation pressure simultaneously with an increase in the mass flow rate of the fuel which is fed into the boiler. The decrease in the pressure causes an immediate increment in the steam mass flow rate at the boiler outlet. The increment is the effect of the decrease in the saturation pressure value in the boiler evaporator and the increase in the specific volume of the steam in the high-pressure part of the boiler which is occupied by steam. Once the required increment in the live-steam mass flow rate is achieved by only firing more fuel, the pressure-decrease process should stop. To obtain the required increase in the power-unit load in the entire range of the load variability, it is necessary to know the appropriate increments in the live-steam mass flow rates at the boiler outlet. These increments are specified by the turbine manufacturer. The only manner in which the correct values of the pressure decrease parameters can be found is through simulation computations; this means that a mathematical model of the entire boiler must be created. The model should consider all the heating surfaces of the boiler, with a special focus on the steam superheater and the boiler evaporator. The mathematical model of the superheater should enable computations of the increment in the steam mass flow rate at the superheater outlet (and, consequently, at the outlet of the boiler) owing to a decrease in the pressure. This is a new approach to modelling the operation of steam superheaters under transient conditions. The total increment in the steam mass flow rate at the boiler outlet, which is the sum of increments at individual stages of the live-steam superheater and in the boiler evaporator, should be computed using verified mathematical models. In this study, an in-house one-dimensional mathematical model of the steam superheater was proposed based on the relations describing the principles of mass, momentum, and energy conservation, which meets the aforementioned requirements. The developed model enables the computation of the increase in the steam mass flow rate at the superheater outlet as an effect of a decrease in the pressure. The variability of steam parameters over time at the superheater inlet can be taken into consideration (in the proposed model, the boundary conditions can be time-dependent). This variability may result from the fact that the evaporator output varies owing to changes in the quantity of the fired fuel.
ACCEPTED MANUSCRIPT The fact that this is an original model developed in-house is of great importance. The model enables the analysis of all parameters at any instant of the pressure-decrease process and the increase in the steam mass flow rate at the superheater outlet. It is also possible to quickly create mathematical models of steam superheaters with any tube configuration. The analysis carried out in this paper demonstrated the effectiveness of the developed model of the steam superheater. The model enables analysis using any pressure-decrease process rate value and any target decrease in pressure. The results obtained are the different increments in the steam mass flow rate and different durations of these increments. This is of fundamental practical importance for the generation of the modified sliding curves of the power unit and constitutes the main contribution of the paper in comparison with previous works. The curves must be generated for new power units (including the supercritical ones), as well as for the existing facilities, which are intended for high-flexibility operation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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ACCEPTED MANUSCRIPT Highlights An in-house model of superheater operation under an unsteady-state is proposed The steam pressure-decrease process causes a jump increase in the mass flow rate A rapid increase in the steam mass flow rate at the superheater outlet is computed The change in the steam temperature related to the pressure decrease process is slight The faster the pressure-decrease process, the greater increase in the steam mass flow