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ISA Transactions 47 (2008) 374–385www.elsevier.com/locate/isatrans
A PDE model of a waterwalls steam generation process
Miguel A. Delgadilloa,∗, Dionisio A. Suareza,1, Jaime A. Morenob,2
a Instituto de Investigaciones Electricas, Calle Reforma 113 Col. Palmira, 62490 Cuernavaca, Morelos, Mexicob Instituto de ingenierıa, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, Edificio 12, Circuito Exterior, Coyoacan 04510, Mexico DF, Mexico
Received 22 January 2007; received in revised form 10 April 2008; accepted 2 July 2008Available online 9 August 2008
Abstract
This paper describes a model of a forced circulation waterwalls steam generator, derived from first principles. The distributed parameter criteriawere applied to the heat transfer process and to the steam production inside the waterwalls. The model is capable of representing swell and shrinkeffects as well as the condensation–vaporization phenomena that take place inside the waterwall tubes, when large drum steam pressure variationsare introduced. The swell and shrink effects are responsible for water displacement from the waterwalls to the drum and from the drum to thewaterwalls. Open loop simulated test were produced with the steam pressure disturbance. Closed loop tests, including the models of the drumlevel and the combustion system and their control systems are presented.c© 2008 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Steam generator; Dynamic modeling; Simulation; Swell and shrink
1. Introduction
The “Termoelectrica Francisco Perez Rıos” is a CombustionEngineering (Canada) boiler, wall-fired type, radiant andpressurized furnace; forced water circulation, oil and gas fueledwith tangential burners. This power plant was designed togenerate 300 MW. A frequent source of power plant instabilitiesis the so-called swell and shrink effects in the waterwalls ofsteam generators. Therefore, these effects are studied in here.
1.1. Historical background
In the literature surveyed, the steam generators dynamicshas been studied for almost 50 years. Some of thesestudies, where simplified models were obtained, are: Chienet al. [18]; Speedy–Goodwin [19], McDonald–Kwantny [20];Herget–Park [27] and Astrom–Eklund [5]. Other works forcoal fired units are Kwan and Anderson [26] and Armor
∗ Corresponding author. Tel.: +52 777 3623811x7725; fax: +52 7773623811x2100.
E-mail addresses: [email protected] (M.A. Delgadillo),[email protected] (D.A. Suarez), [email protected] (J.A. Moreno).
1 Tel.: +52 777 3623811x7286; fax: +52 777 3623811x2110.2 Tel.: +52 55 5623 3600x8811; fax: +52 55 5623 3681.
et al. [28]; the first one applies the distributed properties alongthe waterwall tubes in a linearized model solution and thesecond presents a module modeled system but no equation wasshown. Maffezzoni [22] proposes that the two-phase flow inthe evaporator of a steam generator may be treated as a singlehomogeneous phase with equivalent average fluid properties. Inthe study of Ferrarini–Maffezzoni [23] the partial differentialequations (PDE) of heat exchangers, is proposed. However,none of the studies included the swell and shrink effects.
Once-through boiler models are presented in Laustereret al. [29] where the two-phase flow were studied; however,in this type of boilers there is not a drum level to be studied.In conventional power plants (with wall fired furnace) Flynn-O’Malley [8] developed a lumped model for risers (waterwalls)with a linear distribution of steam quality; however, they stateda general overview analysis without distinction of the differentparts of the processes and equipment, that is, global mass andenergy balances were applied. Maffezzoni, references [9,10],stated a model from global mass and energy balances; andLu [21] gives an example of a dynamic simulation for a coaland gas fired power plant. Therefore, in these jobs, a distributedparameter analysis was avoided.
Some other researchers have avoided stating models fromfirst principles since they consider such analysis to be complex.Habbi et al. [11] derived a fuzzy logic application of power
0019-0578/$ - see front matter c© 2008 ISA. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.isatra.2008.07.002
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M.A. Delgadillo et al. / ISA Transactions 47 (2008) 374–385 375
plant control. Kallappa [12] developed a supervisory controllerwith rule-based expert knowledge. These models are valid onlyin the short ranges that were stated.
1.2. Background for the swell and shrink phenomena
Some papers that study the swell and shrink effects onnuclear plants like: Zhao et al. [13] who proposed a first orderlag expression to represent the swell and shrink effect, andKothare et al. [14] where a non-minimum phase term wasintroduced in a first order lag. Also, for nuclear power plantsDai–Thompson [15] developed a neural network to model thedynamic behavior of level. Habibiyan et al. [16] proposed awater level controller for a “U” tubes steam generator usingneural networks. Lin–Lin [17] developed a neural networkapplication on a nuclear reactor steam generator using amultilayer perceptron feedforward neural network.
Some works that studied the swell and shrink ef-fects in fossil-fueled power plants are: Astrom–Bell [1]analytically treated the swell and shrink effect, and acondensation–boiling equation was obtained. In the work ofAleksandrova–Davydov [7] a simplified boiler model, applyingglobal energy and mass balances, is developed. Astrom–Bell [2]presents a comparison with plant experiments data, of modelsimulation results. Bell–Astrom [4] presents results of valida-tion of a forth order model. Finally, in Astrom–Bell [3] anempirical equation is added to the model stated before in ref-erence [2], that accounts for “the steam flow through the liquidsurface in the drum”. However, all these models were statedwith the same considerations of global mass and energy bal-ances and linear distribution of water–steam mass ratio alongthe risers.
1.3. The paper approach
In spite of so many papers that deal with swell and shrinkeffects, none of them, to our knowledge, applies a distributedparameter analysis with the effect of water displacement fromthe waterwalls to the drum and vice versa. Moreover, in thesestudies there is no clear distinction between the swell and shrinkeffects, as a consequence of the water–steam mixture heating,and boiling–condensation effect due to steam pressure changes.In this paper, a nonlinear model based on physical principles isobtained. The distributed parameter analysis is applied to studythe heat transfer in the furnace and, through the waterwalls, tothe water–steam mixture. Also, a detailed analysis of the waterdisplacement due to the additional bubble formation duringa heating change, or because of a drum pressure change, isapplied.
2. Swell, shrink, boiling and condensation effects
Fig. 1 is a schema of the real plant showing the maincomponents of the steam generator. The feedwater enters thedrum and the circulation pump sucks the water from the drum.After that, the water is distributed in the waterwall tubes in thefurnace where the fuel combustion takes place heating the water
Fig. 1. Heat transfer and boiling effect in the waterwalls of a steam generator.
entering the waterwalls, at about 50 K below the saturationtemperature. When a sudden generator load rejection takesplace and the control system reacts closing the steam controlvalve, a large increase of the drum pressure occurs. Then, theinstantaneous condensation of the steam inside the waterwalltubes takes place. An increase in the drum pressure makes thecontrol system react closing the fuel valve, and then producinga reduction of the drum pressure; therefore, boiling of waterin the waterwall tubes takes place. The sudden condensationor boiling produces water displacement from the drum to thewaterwalls, or from the waterwalls to the drum respectively.Therefore, the condensation–boiling phenomena may producelarge drum level variation.
Besides, waterwalls heating produces a fixed water–steamratio; therefore, if heating is increased, the steam proportionin the water–steam mixture increases also. This is called swelleffect. Moreover, additional bubbling inside the waterwallstubes is produced and these bubbles push water from thewaterwalls into the drum. On the contrary, a reduction ofthe heating rate results in a reduction of the steam inthe water–steam mixture which means a reduction in steamvolume. This phenomena is called shrink effect, which suckswater from the drum to the waterwalls.
On the other hand, if the heating flow rate remains constantbut a drum depressurization occurs changing the equilibriumpoint, then the new water enthalpy value is below its formervalue, before the pressure change. Then, the exceeding heat isset free by evaporating the liquid water needed to reach the newenthalpy value. At the same time, the depressurization processalso produces steam condensation due to steam expansion.As an issue of swell–shrink and boiling–condensation effects,when the steam flow to the turbine is increased, the drum steampressure decreases causing the level to initially rise instead offalling as should be expected from the mass balance in thedrum. A decrease in the steam flow rate leads to an oppositeeffect of collapsing the steam bubbles as a consequence ofincreasing the pressure in the drum.
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376 M.A. Delgadillo et al. / ISA Transactions 47 (2008) 374–385
Fig. 2. Lamber’s law applied to flame radiation in the furnace.
3. The process model
3.1. Assumptions
The main assumptions for developing the model are: (a)the same pressure along the waterwall tubes is assumed sincethe pressure difference between the top and the bottom ofthese tubes is approximately 38,000 Pa; (b) perfect mixing ofwater–steam mixture is reasonable since there is a turbulentflow inside the waterwall tubes; (c) The slip between phasescomplicates the analysis due to the difference in the steam andwater velocities. Refs. [22,18,8]; state that the slip betweenphases, in power plant boilers, “is negligible in practice”. Thesame consideration is made hereby; (d) the phase changes fromliquid to steam and vice versa, are so fast compared to thedominant lags of heat transfer; that thermodynamic equilibriummay be stated; (e) The gases pressure change and the gasesflow in the furnace are not the main aims in the presentwork. Therefore, the inertia of hot gases is neglected; (f) thesteam generator is a tube wall furnace, therefore equidistantwaterwalls may be assumed; (g) the steam generator is a forcedcirculation type with a constant velocity pump, then it followsthat a constant flow in the waterwalls input is reasonable.
3.2. Steam generation analysis
3.2.1. Energy balance in the furnaceIn accord with Lambert’s cosine law, see reference [24], the
flame radiation is stated as 1Q = σ cos θ(
T 4f − T 4
pw
)1A,
where σ is the Stefan–Boltzman constant, T f is the flametemperature, Tpw the waterwall tubes temperature, the radiationarea is A = L1Z , and changing the cos θ (see Fig. 2), by itsequivalent, it follows that:
1Q =k√
r2h + Z2
(T 4
f − T 4pw
)1Z
where k = Lσ . The heat transfer is carried out mainlyby flame and hot gas radiation in the furnace. Gases radiationof oxygen and nitrogen are negligible, see reference [24],therefore the carbon dioxide emissivity, εCO2 and water vapor
Fig. 3. Control volume of combustion gases in the furnace.
emissivity, εH2O, are the only considered combustion gasesradiation. Then, it follows that:
1QCO2H2O
= τ(εCO2 + εH2O
) [(Tgh
100
)4
−
(Tpw
100
)4]
ALtc1Z .
A heat balance in a control volume in the furnace, Fig. 3,gives us:
Heat flow input ofhot gases to the con-trol volume
−
Heat flow output ofhot gases from thecontrol volume
−
Heat rate transferredby flame and gasesradiation to the wa-terwalls
=
Accumulation rate ofthermal energy in thecontrol volume
.
The thermal energy drop through the waterwall tubes isCpgGg1Tgh , therefore:
−CpgGg∂Tgh
∂ Z−
[(Tgh
100
)4
−
(Tpw
100
)4]
K(εCO2 + εH2O
)−
[(T f h
100
)4
−
(Tpw
100
)4]
K√r2
h + Z2
= AT hCpg
[ρg
∂Tgh
∂ t+ Tgh
∂ρg
∂ t
](1)
where AT h is the cross sectional area of the furnace (see Fig. 3).
3.2.2. Waterwall tubes temperature
A heat balance from the hot combustion gases to the wall ofthe waterwalls gives us:
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M.A. Delgadillo et al. / ISA Transactions 47 (2008) 374–385 377
∂Tpw
∂t=
[(T f100
)4−
(Tpw
100
)4]
K√r2
h +Z2+
[(Tgh100
)4−
(TPw
100
)4]
K(εCO2 + εH2O
)− h pw ALtc
(Tpw − TL
)AanLnt Cppwρp
Box I.
Fig. 4. Control volume for the thermal energy transferred to the water–steammixture.
Rate of accumulatedthermal energy in thewaterwall tubes inthe control volume
=
Input of thermal en-ergy flow of flameand gases radiation tothe control volume
−
Heat flow transferredthrough the water-wall tubes in the con-trol volume
.
Since the heat transferred through the waterwall tubes ish pw ALtc(Tpw − TL), it follows the equation in Box I.
3.2.3. Energy balance inside the waterwallsFig. 4 shows the thermal energy transferred by the wall of
the waterwall tubes, then:Heat flow transferredto the water–steammixture through thewaterwall tubes
+
Heat flow thatreceives the water–steam mixture
−
Heat flow with thewater–steam mixturethat leaves the con-trol volume
=
Heat rate accumula-tion in the water–steam mixture in thecontrol volume
.
h pw Atc(Tpw − TL
)1Z − 1 (Gm Hm)
=AT t nt 1Z 1 (ρm Hm)
1t, hence:
h pw Atc(Tpw − TL
)− Gm
∂ Hm
∂ Z− Hm
∂Gm
∂ Z
= AT t nt
[ρm
∂ Hm
∂t+ Hm
∂ρm
∂t
]. (2)
The steam mass fraction may be calculated as follows
Xv =HmLv − HL
Hv − HL. (3)
Fig. 5. Control volume for mass balance of the water–steam mixture in thewaterwalls.
3.2.4. Mass balance inside the waterwallsA mass balance in the water–steam mixture in the control
volume, see Fig. 5, give us:
Accumulating rate ofwater–steam mixture
=
Water–steam mixtureflow that inputs thecontrol volume
−
Water–steam mixtureflow that leaves thecontrol volume
−Water flow displacedby the steam
−
Water flow sockedby bubble collaps-ing due condensingsteam
.
1Mmac
1t
∣∣∣∣CtrlVol
= −1Gm |CtrlVol − 1G ldc|CtrlVol
−1G ldei|CtrlVol + 1Gvcdo|CtrlVol. (4)
The liquid mass displaced by heating, Mldc, depends onthe water density, ρL , and on the steam volume generatedby heating, Vvc, that is, Mldc = ρL Vvc. Since Vvc may besubstituted by its equivalent Mvc/ρv; then, by differentiatingthe resulting equation with respect to time, assuming ρLconstant, we obtain:
G ldc =ρL
ρv
Gvc − ρLMvc
ρ2v
∂ρv
∂t. (5)
In the same way, the water flow generated due to waterboiling when depressurization takes place, G ldei, is calculatedconsidering that the mass of the displaced liquid, Mldei, dependson the displaced volume, Vvei, which is calculated as Mvei/ρv ,then we obtain:
G ldei =ρL
ρv
Gvei − ρLMvei
ρ2v
∂ρv
∂t. (6)
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378 M.A. Delgadillo et al. / ISA Transactions 47 (2008) 374–385
dρm
dt=
[ρv + Xv (ρL − ρv)] ρLdρv
dt − ρvρL
[(1 − Xv)
dρv
dt + (ρL − ρv)dXv
dt
][ρv + Xv (ρL − ρv)]2
Box II.
The mass balance of the steam inside the waterwalls resultsas follows:
∂ Mvac
∂t= Gvc + Gvei − Gvcdo. (7)
Here, the accumulated steam mass is Mvac = Mvc + Mvei −
Mvcdo, then, by substituting Eqs. (5) and (6) in (4) and usingEq. (7), we obtain:
∂ Mmac
∂t= ∂Gm −
ρL
ρv
∂ Mvac
∂t+ Gvcdo
(1 −
ρL
ρv
)+
1
ρ2v
∂ρv
∂t(Mvac + Mcdo) . (8)
But
∂ Mmac
∂t= AT t nt∂ Z
∂ρm
∂t. (9)
And since the accumulated steam mass is Mvac =
1Z AT t ntρm Xv , and applying the derivative with respect totime to this equation:
∂ Mvac
∂t= ∂ Z AT t nt
(Xv
∂ρm
∂t+ ρm
∂ Xv
∂t
). (10)
But since the water–steam mixture flow is Gm = Glr +
Gvei − Gvcdo, and since circulation flow; Glr , is constant, thenthe derivative of this equation is:
∂Gm
∂ Z=
∂Gvei
∂ Z−
∂Gvcdo
∂ Z. (11)
Then by substituting equations (9) and (10) into (8), andusing (11), then we obtain:
∂ρm
∂t
(1 +
ρL
ρv
Xv
)+
ρmρL
ρv
∂ Xv
∂t
=1
AT t nt
[∂Gvei
∂ Z−
ρL
ρv
∂Gvcdo
∂ Z
]+
XvρmρL
ρ2v
∂ρv
∂t(12)
where (∂ Mvcdo/∂ Z),is considered negligible. The mixturedensity, ρm , is as follows:
ρm =ρvρL
ρv + Xv (ρL − ρv). (13)
By differentiating this equation with respect to time, weobtain Box II.
3.2.5. Instant boiling and condensation
The bulk water boiling, during depressurization, may beevaluated as follows:
Saturated water en-thalpy at time “t” be-fore depressurization
−
Saturated water en-thalpy at time “t”just after depressur-ization at time t +1t
=
Thermal energy usedfor mass water boil-ing due to depressur-ization
.
−1 (ML HL) = Gvei (HV − HL) 1t |CtrolVol.
(14)
But the mass of water is ML = 1Z AT t ntρm(1 − Xv); thenEq. (14) give us:
1AT t nt
∂Gvei
∂ Z
=ρm HL
∂ Xv
∂t − ρm (1 − XV )∂ HL∂t − HL (1 − XV )
∂ρm∂t
Hv − HL. (15)
Steam condensation also takes place when depressurizationoccurs and it may be evaluated in the same way as the boilingeffect. Therefore, the thermal energy balance is:
−1 (Mvs Hvs) = Gvcdo (HV − HL) 1t | VolCtrol
. (16)
But, the steam mass is Mvs = 1Z AT t ntρm Xv; hence, bysubstitution of Eq. (16), we obtain:
1AT t nt
∂Gvcdo
∂ Z
=−ρm Hvs
∂ Xv
∂t − ρm XV∂ Hvs∂t − Hvs Xv
∂ρm∂t
Hvs − HL. (17)
3.3. Drum level and steam pressure
Drum level. The drum is a horizontal cylindrical vessel with el-liptical covers. The volumes for the horizontal cylindrical, Vcy ,and that of the covers, Vell , are level (ND) functions as follows:
Vcy = L D
[(ND − rD)
√2NDrD N 2
D + r2D cos−1
(1 −
NDrD
)]and the covers Vcll = Ccll N 2
D
(32 DD − ND
).
A mass balance, accounting for the total water volume(Vcy, +Vell), see Fig. 1, gives:
dND
dt=
Gm (1 − Xv) − Glr + G f w
ρL
(2L D
√DD ND − N 2
D + C1 ND (DD − ND)
) .
(18)
Drum steam pressure. We assume that a variation of pressuredoes not generate steam condensation or water vaporization in
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M.A. Delgadillo et al. / ISA Transactions 47 (2008) 374–385 379
the drum (see Fig. 1).
d (EvacD MvacD)
dt= G mLv Xvs Hvs − GvoD Hvs . (19)
The saturated steam enthalpy Hvs , and specific volume Svs ,are drum pressure functions, and since the internal energyEvacD , may be expressed as EvacD = Hvs − Pvs Svs , therefore:
dPvs
dt=
Pvs SvsdMvacD
dt
Vvac(K1 + K2 Pvs + K3 P2
vs
) (20)
where the K ’s constants are K1 = C12 − C21, K2 =
2(C13−2C22) and K3 = −3C23. The steam mass balance gives:
dMvacD
dt= Gm Xvs − GovD. (21)
3.4. Combustion reaction and flame temperature
Since natural gas composition (mainly methane and ethane)is known, this fuel was used. The NOx production appearstypically in coal reaction and in accordance to reference [30]NOx reactions do not have a worthy effect in the flametemperature. Moreover, air pollution, including the productionof CO is out of the scope of the present work. Then thecombustion reaction is as follows:
CH4 + C2H6 + 11/2O2 = 3CO2 + 5H2O.
Taking a volume of gases with a fixed mass, inside thefurnace, the energy balance gives:
dT f h
dt
=Gca J + GairCpeair (Teair − T0) η f − Cpg f Ggc
(T f h − T0
)Cpg f Macgh
(22)
where the natural gas as fuel flow, Gc, is a function of valveopening, Yc, and pressure difference:
Gc = C f cYc
√P2
ic − P2gh . (23)
The mass ratio of actual fuel to the stoichiometric fuel massa is evaluated as
a =Gceq
Gc. (24)
If a is equal to 1, this means that fuel is just the quantityrequired to react with the oxygen in the air. If a is less than 1,then fuel is in excess. On the contrary, if a is bigger than 1,then the air is in excess. The stoichiometric mass flow Gceq, iscalculated as:
Gceq = GO2qc/O2 . (25)
Here qc/O2 is the mass fuel ratio actually reacted with thestoichiometric oxygen, and the actual oxygen flow GO2 , isproportional to the mass fraction of oxygen in the air (b):
GO2 = Gairb. (26)
Fig. 6. Three element feedwater control system.
Fig. 7. Combustion control system.
4. Drum level and combustion control systems
The three element typical level control system, see Fig. 6,was used to perform closed loop tests. This figure shows thedrum level, the feedwater flow and the steam flow signalsas inputs; here the controllers are in a standard cascadearrangement of master and slave form; therefore, the mastercontroller output signal enters as the setpoint signal of the slavecontroller.
The combustion control system (see Fig. 7) has the steampressure signal as the controlled variable. The air flow signalis characterized by a function F(x) that consider the squaredmeasurement of the air and the nonlinearity of the forced draftvane position with respect to the air delivered by the forced draftfan. The fuel–air flow ratio is used as input in another controllerand the output of this controller is added as a feedforward signal
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Table 1Parameters at base load
Parameter Value Parameter Value
AT h 314 m2 Glr 594.542 kg/sAltc 75.06 m2/m J 33,782,026 J/kgAT t 0.00114 m2 K 531.13 J/(ms K)b 0.01105 Adim. L t 18.745 mCcll 0.402 m rh 8.86 mCpg 1287.2146 J/(kg K) nt 1254h pw 393.23 J/(s K) εCO2 0.150 1/mGgc 293.2339 kg/s εH2O 0.275 1/m
with output of the master controller, the resulting signal sets theair forced draft vanes position in accordance with the fuel–airflow ratio; then, the air follows the fuel requirements.
Other side control systems, such as the steam temperatureand deareator level, are outside the scope of the present papersince they do not contribute under a direct effect to the drumpressure and level.
No formulation of the control systems is presented since theyare based on the well known standard form of PID. Also most ofthe control valves and other parts of the process were omittedfor the sake of simplicity, although they were included in thesimulation tests. Some effort has been made to apply PDE incontrol strategies. As examples of this are the references [31,32]. Then, it may be useful to look for applications of the PDEmodel, here presented, in control strategies of drum level andcombustion of steam generators.
5. Numerical solution
Some researchers, like Maffezzoni [22] and Ferrarini–Maffezzoni, have found parasitic oscillations (Gibbs effect)in solving PDE in the solution of heat transfer models statedas PDE, when lumped parameter techniques are applied. TheExplicit Finite Difference (EFD) is a suitable method for a PDEsystem having the same partial derivative in several equations.Then, by applying the EFD (see reference [25]) method itresults in an algebraic linear differential equation system thatmay be solved by a simultaneous solution. This procedurehas good stability if a proper integration step size is chosen;therefore, the EFD method was applied on the steam generationmodel solution. The integration step, for running the steamgeneration module, was found by trial and error, beginningwith a reasonable step size and reducing it in each test untilthe stability solution was obtained. Fig. 8 shows the steps forsolution.
Most coefficients of the system equations were obtainedfrom construction data of the “Central Termoelectrica FransiscoPerez Rıos” plant, located in Tula Mexico. However, someconstants were evaluated from steady state values of variablesat base load (100% of load), as the simulation starting point, seeTable 1 for the calculated parameters and Table 2 for the initialconditions.
Table 2Initial conditions at base load
Variable Value Variable Value
Gm 594.542 kg/s Tpw 1049.34255 KGvei 0 kg/s Xv 0.43376954a
Gvcdo 0 kg/s ρgc 1.418 kg/m3
HL 1,657,279.6425 J/kg. ρm 213.74482568 kg/m3a
HmLv 2,052,754.7993 J/kga ρL 592.8951 kg/m3
Hv 2,568,996.84 J/kg ρv 116.496692 kg/m3
PvD 16,551,765 Pa ∂Tgc/∂ Z −33.23 K/mTgc 1,473.2 Ka ∂Gvcdo/∂ Z 0 kg/(s m)TL 623.26 K ∂Gvei/∂ Z 0 kg/(s m)
a Values distributed along variable Z are fixed for a initial linear distributionfor ρm and Xv .
6. Simulation results
6.1. Open loop tests
The open loop tests applied to the steam generation model,were performed as shown in Fig. 9. The drum pressure wasdisturbed with the Agnesi function, Pvs = a3/(t2
+ a2).The effect of the pressure disturbance in the internal
variables is shown in graphs of Figs. 10–14. Each graph has16 lines and each line corresponds to each piece in which thewhole waterwall tubes length was divided, and going from thebottom to the top, the first line in the bottom corresponds tothe first piece of the tube. Fig. 10 shows the steam qualityand has a little descent when the pressure is going up, evennear the pressure top. Therefore, the effect in the steam qualityfor ascending pressure is larger than in the case of descendingpressure. In Fig. 11 the boiling water flow, Gvei, is shown.In this case, this flow shows a descending trend for pressureascent, this means, that this flow must be subtracted from theactual water–steam mixture flow Gm . Fig. 12 shows the steamcondensate flow, Gvcdo, with the contrary behavior to Gveidue to the fact that an increase in the drum pressure resultsin an increase in condensation. Fig. 13 shows a reasonabledecrease in the water–steam mixture enthalpy when the drumpressure increases; however, when the drum pressure decreasesa larger increase in the water–steam mixture enthalpy takesplace. In Fig. 14 the water–steam mixture flow shows a largedecrease (almost 150 kg/s) when the drum pressure increasesand a reasonable increase in this variable (less than 100 kg/s)when the drum pressure decreases. Large movements of theevaporated steam flow (Fig. 11) and in the mixture low Gm(Fig. 14) show the expected behavior in accordance withoperators reported performance when large pressure changestake place in the drum.
6.2. Closed loop tests
Closed loop tests were applied to the complete model,including the control strategy for the drum level and forthe steam pressure, as is shown in Fig. 15. Steam pressureperturbations were introduced by first closing the steam throttlevalve (from 80 to 20%) and once the stability in all variableswas recovered, a sudden opening of the throttle valve returns
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M.A. Delgadillo et al. / ISA Transactions 47 (2008) 374–385 381
Fig. 8. Flow chart for the steam generation equations system solution.
it to its original position (80%). Figure 16 shows the drumpressure and the steam valve movements. It is clear that closingthe steam valve results in a more oscillatory behavior than thatshown when the valve position is back to the original position(80%). This means that at low load the process stability is moredifficult than at high loads as was also found in papers [3,4,12].Fig. 17 shows the drum level behavior when the steam valve isclosed to 20% position. Some reasonable oscillations may beappreciated. When the steam valve goes back to its initial 80%position, in the first part of the graph, the level shows a clearly
good stability. However, as the drum pressure is recoveringslowly, the drum level goes down due to the compressing effect.Until the drum pressure goes down to its setpoint, the levelsignal shows a very steep ascent due to the water vaporization;then afterwards, almost instantaneously, the level falls to itssetpoint value.
The sudden drop in the drum level, about the time 1300 s(see Fig. 17), is due to the continuous decrease in the waterflow that enters to the drum with the water–steam mixture thisflow is calculated as Gm(1 − Xv) in Eq. (18), as it is shown in
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Fig. 9. Open loop test arrangement of the steam generation process model.
Fig. 10. Steam mass fraction (Xv) behavior.
Fig. 11. Evaporated steam flow (Gvei) behavior.
Fig. 18. Note that at the same time (1300 s) the minimum valueof this flow is reached. The level drop takes place in spite ofthe continuous increase of the feedwater flow. When the dropof the drum pressure is overcome (see Fig. 16), the drum levelis steeply recovering as it is shown on Fig. 17.
6.3. A plant run
In order to show the plant behavior when the drum pressurechanges, the Fig. 19 presents a plant run carried out onNovember 3rd of 2006 in our Tula–Mexico plant (unit 1);here, drum level and the combustion controls were in automaticmode; therefore, this graph shows closed loop results where thelevel control system reacts just after the drum level goes upsteeply. However it is clear that a drum pressure descent makes
Fig. 12. Condensate steam flow (Gvcdo) trajectory.
Fig. 13. Water–steam mixture enthalpy (Hm ) behavior.
the level go up, in accordance with the simulated response(Fig. 14) that shows an increase of the water–steam mixtureflow that enters in the drum. A vertical line is added in Fig. 19in order to show the time when the pressure descent starts. Notethat this pressure descent causes a drum feedwater flow increasethat contributes additionally to the drum level increase.
It has to be noted that the level trajectory in the graph agreeswith the expected behavior according to the mathematicalmodel; therefore, it may be considered as a qualitativevalidation of the model here developed.
7. Conclusions
A detailed mathematical model for the steam generationprocess in a power plant has been presented. A separateanalysis is applied to study each of the shrink, swell, condensateand boiling effects that take place inside the waterwalls of asteam generator of a power plant. The analysis accounts fordistribution of the water and steam mass and volume along thewaterwall tubes (distributed parameter criteria); that is, globalstatements (lumped parameter) were avoided. Other studies are
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Fig. 14. Water–steam mixture flow (Gm ) trajectory.
featured by global analysis to represent the swell and shrinkphenomena, see references [1–4,6–10].
An open loop simulation test, with steam pressuredisturbance, was applied to the steam generation processmodule (represented by partial derivatives). The results showcongruence with the expected behavior of internal and outputvariables (Figs. 10–14). Closed loop simulation tests wereapplied to the steam generation module in conjunction withthe drum level, the combustion systems, as well as theconventional control systems of the drum level and pressure(combustion system). These tests show the impact of the swell,shrink, condensing and boiling effects which appear with thedrum pressure changes and with the fuel combustion energyvariation. It may be noted that the drum level variation wasrepresented with a strict non-linear differential equation (seeEq. (18)). The graphs of Figs. 16 and 17 show the difficulties incontrolling two of the fundamental variables for power plantscontrol: the steam pressure and the drum level. The model
Fig. 16. Drum steam pressure (Pvs ) behavior in closed loop test.
developed here may be used to evaluate control strategies oflevel and combustion control systems. Moreover, Eqs. (15) and(17) are responsible for the sudden condensation and boilingeffects that appear with a pressure change. Therefore, thesetwo equations may be introduced in Eq. (12) and the resultingequation may be used in a model-based control strategy whichmay account for the drum level and for the drum steam pressurevariation.
The affected variables for a steam pressure variation showthe expected trend in the simulated closed loop results. Theplant run here presented (Fig. 19) also shows the expectedbehavior, i.e. a drum pressure decrease results in a drum levelincrease; therefore, it may be said that there is a qualitativevalidation of the mathematical model developed in this paper.
Acknowledgements
The authors wish to thank the reviewers of this paper fortheir worthy comments. They would also like to express theirgratitude to Pablo Villasenor Dıaz and Fany Mendez Vergara
Fig. 15. Process model and control systems in closed loop arrangement.
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Fig. 17. Drum level (ND) behavior in closed loop test.
Fig. 18. Water flow (G f w) behavior in closed loop test.
Fig. 19. Drum level and steam pressure variation in a plant run.
for helping them in parameters calculations and for running thetests.
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