Electric Power Systems Research, 18 (1990) 175 - 189 175

D e v e l o p m e n t o f a S ta t i c -S ta te E s t i m a t o r for a P o w e r S ta t ion Boi ler Part I. M a t h e m a t i c a l Model

K. L. LO and Z. M. SONG

Power Systems Research Group, Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow G1 1XW (U.K.)

E. MARCHAND and A. PINKERTON

South of Scotland Electricity Board, Cathcart House, Glasgow (U.K.)

(Received October 23, 1989)

ABSTRACT

This paper forms Part I of a two-part series which describes the overall construction of the computer simulation and the estimation al- gorithm of a static-state estimator for a power station boiler. A mathematical model, estab- lished by an analytical approach, for the steady-state estimation is presented in this pa- per. The boiler system is divided into nine subsystems according to its physical construc- tion, and the energy conservation principle is then applied to each to form the subsystem models; finally, all of the subsystem models are coupled together to form a complete system model for state estimation. A set of suitable empirical correlations has been chosen to deter- mine the system parameters. The validity of the model and some necessary assumptions made in modelling have been tested by running a pro- gram which simulates the working behaviour of the boiler under several different loading condi- tions. The effects of the variation of certain parameters on the performance of the model have also been tested.

1. INTRODUCTION

This paper presents a mathematical model for the steady-state estimation of a coal-fired boiler in a conventional power station. The boiler referred to is that of a 300 MW drum-type natural circulation unit. Its con- t inuous operation rate for steam is 900 tonne/ h at a temperature and pressure of 568 °C and 169 bar, respectively, and for reheating steam 703tonne/h at 568°C and 39.7bar, respectively.

There are many mathematical models of power station boilers suggested in the litera- ture [1-7] which are used for different pur- poses. Some sophisticated models, expressed by high order partial and/or ordinary differen- tial non-linear equations, can represent boiler dynamic processes over a relatively wide working range. These models can also be used to represent state variations during start-up and shut-down. At the other end of the scale, models deduced completely by identification scheme methods may only illustrate some of the fundamental characteristics and are mainly used to represent the boiler as one of the elements in the power system. For the purpose of steady-state on-line estimation, a complicated model should be avoided in order to minimize computation time; too simple a model, which would only give the terminal relationships, is not, however, appropriate for estimation purposes either. Therefore, the model developed in this paper is between these two extremes. The proposed model should be sufficiently accurate to represent the behaviour of the boiler and also suffi- ciently simple for direct application. In this context the model is established with refer- ence to the work of Kwan and Anderson [1] and includes some of the features suggested b y Chien et al. [2] and Adams et al. [3].

Summarizing the work done by the authors of refs. 1-7, there are at least four aspects of the problem which give rise to difficulties in describing the working procedure of the boiler mathematically. Firstly, the distribution of the fluid means that the model has to be repre- sented in terms of three-dimensional partial differential equations in the case of dynamic

0378-7796/90/$3.50 © Elsevier Sequoia/Printed in The Netherlands

176

analysis; secondly, the inclusion of high non- linearity in the fundamental thermodynamic variables causes the computing procedure to be time consuming; thirdly, the uncertainty of some of the physical and operational parame- ters of the boiler makes modelling difficult; fourthly, some unmeasurable quantities with- in the boiler require estimation techniques to obtain their values. A comprehensive consid- eration of these four problems, particularly of the first and second, led the authors to the following approach in establishing the mathe- matical model. The boiler system is divided into nine sections and each section is treated as a lumped-parameter subsystem. The mea- surement equations of each subsystem can be established individually by using the thermo- dynamic balance relationships with some nec- essary assumptions. The subsystem models are then coupled together to obtain the whole system model.

Most of the physical data in the model were obtained from the drawings of the 300 MW coal-fired boiler, and the rest from engineering estimations and assumptions. Heat transfer coefficients are determined by using a set of appropriate empirical correlations, and the specific heat is obtained by a set of equations which fit the Steam Table [8].

2. SYSTEM DESCRIPTION AND GENERAL ASSUMPTIONS

A typical coal-fired drum-type boiler system is shown schematically in Fig. 1.

When coal is burned in the combustion chamber, heat energy is generated in the form of flames. This generated energy heats up its surroundings and the air. When the heated air (usually called hot gas) is driven along the gas path, it heats up sections progressively by transferring heat energy to the fluid flowing inside metal tubes.

Within the combustion chamber, the proce- dure of burning is very complex and its de- tailed study is beyond the scope of this paper. According to the Stefan-Boltzmann law, the rate of heat transfer in the radiative mode at a constant emissivity varies approximately as the fourth power of the absolute temperature, and from the fact that the rate of heat transfer by convection varies directly as the tempera- ture difference, it is reasonable to assume that

the heat transfer in the combustion chamber is dominated by radiation since the flame tem- perature is much higher than that of the hot gas. This assumption simplifies part of the calculation problem, even though there are still difficulties in calculating the heat trans- fer in the combustion chamber. For example, changes in the combustion temperature and flame configuration will have a relatively great effect on the radiation because the in- tensity of radiation received by a surface varies inversely as the square of its distance from the source. Local conditions at the tube wall also have a marked effect in many ways: the pressure of a deposit layer will lower the rate of heat transfer; variations in the direc- tion and velocity of the flowing gas will form a s tagnant gas film of high thermal resistance close to the tubes. When the hot gas flows through the rest of the area along the gas path, heat transfer is mainly by convection. The amount of heat transferred by convection depends on the construction of the section, the velocity of the gas and the heat transfer co- efficient. As the combustion temperature, the emissivities of the flame and gas, the gas flow velocity, and the convective heat transfer co- efficient are difficult to determine, the energy conservation law will be used to determine the heat transfer at the gas side of the tubes.

Water is fed into the boiler by the feed- water pumps. In the economizer, water ab- sorbs heat from the departing gas and its temperature increases to a certain value which is usually below the saturation temper- ature corresponding to the drum pressure. From the economizer, water flows into the vaporization circulator which is constituted of the drum, the downcomer and the waterwalls. The circulator is a critical part of the boiler; in it, the water acquires a great amount of heat energy so that boiling takes place. In a natural circulation boiler, the fluid motion around this circle is set up by buoyancy effects resulting from the density difference caused by the temperature difference in the fluid. As the water from the bottom of the waterwalls flows upward, the process of heat transfer takes place in four stages: subcooled convection, subcooled boiling, saturated nu- cleate boiling and two-phase heat transfer through the liquid film. At the top of the waterwalls a certain percentage of the water may be vaporized. When this water-steam

8 Re~e~er FLed. B~,a~k

? Re l~er

3 Pr'~.m~" B~l~erkeJZer

1 Eco~mi~er

1

I l I

J ge¢o~'3l g~per~er geooa~. B~k

f f

j Aivl~ter 11

Dow'~omer 12

Comb~tio~ clu~ber 10

P .F.B1Jr~r

177

I

Fig. 1. Schematic construction diagram of a typical coal-fired boiler.

mixture flows back into the drum, condensa- tion and vaporization take place simulta- neously. For the sake of simplicity, it is assumed here that all of the water component separates from the water-s team mixture and mixes with the water at the bottom of the drum. This water has a temperature slightly below the saturat ion temperature and will flow into the downcomer with water from the economizer for the next circulation. The sepa- rated steam, at the saturat ion temperature, leaves the drum and enters the superheater section.

In the superheater region, the steam passes through various sections; some are mounted in the horizontal position and some are verti- cal. Although the construction of these super- heater sections may differ, the heat transfer is dominated by convection. The properties of the steam in this region do not change as much as those during vaporization. There is a spray at temperator between the outlet of the platen superheater and the inlet of the first bank of the secondary superheater. The attem-

perator is used only for controlling the tem- perature of the steam at the outlet of the boiler.

The first and second banks of the reheater consti tute another part of the boiler through which the exhausted steam from the high pres- sure turbine is reheated and used to drive the intermediate pressure turbine.

From the preceding description, the whole working procedure can be divided roughly by temperature into three regions: the heat re- covery region, the combustion chamber, and the high temperature region. Their character- istics, for the sake of analysis, can be summa- rized as follows. (1) In the heat recovery region, like in the economizer, the working fluid is in the liquid phase, and the flux of the transferred heat energy is not so large that the properties of the fluid change remarkably. Therefore, many relationships can be approxi- mated linearly. (2) In the combustion chamber the large exchange of heat energy causes a change in the properties of the fluid. Although the temperature of the fluid increases a little

178

around the saturat ion temperature, the water absorbs a lot of latent heat so that the en- thalpy of the water-s team mixture at the out- let of the waterwalls is much greater than that of the water at the inlet of the water- walls. There is a fundamental change in the properties of the fluid. (3) In the high tempera- ture region, which includes various super- heater and reheater sections, the change in the fluid properties is much smoother than that in the combustion chamber, but there are strong non-linear characterist ic relationships between the variables. In this region the working temperature and pressure are desig- nated to reach values appropriate to the mate- rial. An excessive increase in temperature may exceed the tolerance of the metal tubes and damage them, therefore the temperature at different locations in this region should be calculated accurately.

According to the preceding analysis and physical division of the boiler, the system is divided into the following nine sections: (1) economizer, (2) waterwalls, (3) drum, (4) pri- mary superheater, (5) platen superheater, (6) first secondary superheater bank, (7) second secondary superheater bank, (8) first reheater bank, (9) second reheater bank.

The block diagram of the divided system is shown in Fig. 2, where the solid line denotes the water and steam paths and the broken line denotes the gas path.

In order to facilitate a mathematical formu- lation while still retaining the essential fea- tures of the boiler, some assumptions have to be made. General assumptions used for the whole system are given in the following list, whereas part icular ones will be mentioned in the appropriate places.

(1) The properties of the fluid flow for each

To chimney

Drum

Pd~

Air

tT~o

EconomJser ~ _ _

_ * t T ~ i * T ~ i c - .

feedvater

Wa~t~nfll

fuel

!

I Superhea~r

Fig. 2. Block d i a g r a m of t he boi ler sys tem.

t *r 1 J

t't' ii T~,I I t I .

Trli Rehea~er [Trlo

P,I" t

I

i PIA~n

r I

% • Rehea~er

final bank

J

superhea~r I P.'.

M"

Key: ........ fluid path . . . . gas ~th

~ r a e n t metal tube tempts ~ .

lumped-parameter section only vary in the ax- ial direction and linearly in space.

(2) The mass flow rate of the water and steam in each section is constant.

(3) The boundary condition for fluid flow between two adjacent sections is that the mass flow rate at the inlet of a section is equal to that at the outlet of the previous one.

(4) Each section is an ideal heat exchanger in which the heat energy absorbed by the metal is zero, and 100% of the heat energy given up by the hot gas is transferred to water and steam.

(5) The kinetic and potential energies are negligible in the energy equations.

(6) Enthalpy is a function of temperature and pressure which are dependent variables and their relationship is given in the Steam Table. Since enthalpy depends much more on temperature than on pressure in certain parts of the boiler, the enthalpy is only considered as a temperature-dependent variable at the inlet and outlet of each section with a corre- sponding specific heat.

(7) Heat transfer in all sections, except for the waterwalls, is dominated by the convec- tive mode.

3. NON-LINEAR MEASUREMENT EQUATIONS

The derivation of the measurement equa- tions depends to a great extent on the loca- tions of the meters. The measurement system in a power station usually consists of various types of measuring equipment and the princi- pal quantities measured are temperature, pres- sure, and fluid and gas mass flow rates. A good measurement system can provide an operation engineer with a complete knowledge of the working state of the boiler. Owing to the con- struction and operational conditions of the boiler, some quantities cannot be measured easily. Each measurement in each section can be expressed in the form of an equation. The equation will give the value of the measure- ment in terms of the state variables which are governed by physical laws and engineering judgments.

In this paper it is assumed that the follow- ing quantities are measurable: the mass flow rates of the fluid and gas, the temperatures and pressures of the fluid at the inlet and outlet of each section, the temperature and

179

pressure in the drum, and the gas tempera- tures at the inlet and outlet of the economizer stage. The gas temperatures within the com- bustion chamber and the superheater and re- heater sections are not measured.

3.1. The e c o n o m i z e r The economizer consists of a number of

metal tubes bound together by a Set of metal plates. The plates are welded to the outside of the tubes to improve the heat transfer. The economizer is composed of two geometrically symmetrical parts, one being mounted below the primary superheater and other below the first bank of the reheater. Each part consists of the top and the bottom bank, but, for the purposes of state estimation, all of these parts are treated as a lumped section.

The heat absorbed by the water in the econ- omizer is the difference in energy flux at the inlet and outlet of the tubes and can be ex- pressed mathematically as

Qe -- MHeo - MHei (1)

If the enthalpy is expressed in terms of tem- perature, eqn. (1)becomes

Qe = MCeo Teo - MCei Hei (2)

where M is the mass flow rate in the econo- mizer, T~i and Teo are the water temperatures at the economizer inlet and outlet respec- tively, and Cei and C~o the corresponding spe- cific heats of the water.

The heat transfer at the water side of the tube is by forced convection. According to Nusselt's law [9], the mass flow rate is the most important variable affecting the perfor- mance of heat transfer. When the effects of the rest of the variables on the heat transfer are summarized and represented by a heat transfer coefficient K.1, the heat flux from the tubes to the water is

Q~m = KeIM°'S(Tem - T,o) (3)

where Tern is the average temperature of the metal tubes, and K,~ is the heat transfer co- efficient at the water side of the tubes.

The value of Ke~ depends on the type of flow (either laminar or turbulent), the geome- try of the economizer, the flow passage area, the physical properties of the fluid, and the average temperature of the water. Its accurate determination is complicated, and will be dis- cussed in ~4.

180

The heat flux transferred from the gas to the tubes is

Q~,e = Ke2M~°'6(Tgeo - Tme) (4)

where K¢2 is the heat transfer coefficient at the gas side of the tubes, Mg the mass flow rate of the gas, and Tg~o the gas temperature at the economizer stage outlet.

The heat energy given up by the gas is

Qge = Cge Tg~, - Cge Tgeo (5)

where Cg¢ is the specific heat of the gas, which depends on the temperature; Tgei and Tg~o are the gas temperatures at the economizer stage inlet and outlet, respectively.

Tei, Teo, Tern, Tgeo, Tgei, M and M~ are measured variables. We can express them in terms of the state variables by using the en- ergy conservation relationship. Under steady- state conditions, Qe = Qem = Qge -- Q'ge. From the relation Qe = Q.m, we have

CeoM \ Tern- gelC~iMM°s T~i + 1 + ~ ) T e o (6)

By putting Q.m = Qg~, and combining eqns. (3) and (4), we have

1

1 1

(7)

and on the gas side of the relation Qge = Q~e gives

r g e i - - - - C e i M ~- Ke2 aAr/-~-~ Tei \ ge g

M 0.8

1 &)] + K~2MO~ + Teo (8)

The above three equations plus the direct measurements of Tei and T.o constitute the temperature measurement equations. The derivation in the following sections also fol- lows the same philosophy.

3.2. The waterwalls In the waterwalls, fundamental changes in

the properties of the fluid will take place dur- ing vaporization. The value of the enthalpy

will increase suddenly and the density of the water will fall for a slight increase in tempera- ture around the saturation value. As more and more water vaporizes into steam the main heat transfer mode changes from convection to radiation. All of these extreme variations will cause several non-linearities in the equa- tions. On the other hand, all the saturation properties of the fluid depend uniquely on the drum pressure Pdr- Once Pdr is specified, the saturat ion temperature of the steam, Tdr, the density of the saturated steam, Pdrs, and the density of the saturated water, Pd~, can be determined from the Steam Table [8]. The equations to follow represent the saturation temperature, Tdr, the enthalpy of the satu- rated steam, Hdrs, the enthalpy of the satu- rated water, Hdr~, the density of the saturated steam, p . . . . and the density of the saturated water, p . . . . in terms of the drum pressure Pdr. They are valid within the range 160 bar ~ Pdr ~<188 bar, which covers the working range under full load conditions. Any value outside this range under full load conditions is consid- ered to be unacceptable.

Tdr ----- -0.00176 Pdr 2 + 0.88196 Pdr

+ 236.301346 (9)

Hdr ~ = 0.006622 Pdr 2 + 1.959846 Pdr

+ 1167.302612 (10)

Hd~,,, = --0.02366 Pdr 2 -- 4.480249 Par

+ 2473.766113 (11)

Pwo~ = 6.4 x 10 ~ Pdr 2 -- 0.015095 Par

+ 2.486906 (12)

Pwow = 3.03 x 10- ' Par 2 - 0.193746 Par

+ 32.553112 (13)

From experimental study [10] the heat flux transferred from the tubes to the water and steam mixture is

Qwm = Kwi(Twm - Tdr) a (14)

where Mw is the mass flow rate in the water- walls, Kwl the film boiling heat transfer co- efficient, and T,, m the tube temperature of the waterwalls.

Because there is no net increase of heat energy in the drum, the heat absorbed in the waterwalls is equal to the energy difference between the outlet of the drum to the primary

181

superheater and the inlet of the drum con- nected with the economizer. This can be ex- pressed as

Q,, = MHdr,- MHeo (15)

Combining eqn. (14) with (15) by putting Qw = Qwm, we have

gwlr(Twm - Tdr) 3 = M(Hdr s - Heo ) (16)

It is assumed that Twin can be measured, so eqn. (16) is one of the measurement equations for the waterwalls section. The drum tempera- ture Td~ is also a direct measurement. The water level in the drum has been assumed to be an unmeasured quantity, but it can readily be expressed in terms of Pd~ when the volume of the drum is known.

3.3. The primary superheater This is a section mounted on the top of the

economizer and its tubes are horizontal, paral- lel with the first bank of the reheater. The primary superheater consists of the top, inter- mediate and bottom three banks. The percent- age of hot gas flowing through this section depends on the design and construction. From the experimental results when testing the boiler model, the value was taken to be 66% in this model. Similar to the economizer section, assuming that the heat energy absorbed by the fluid is equal to the heat flux transferred from the tubes to the stearn, the temperature of the metal tubes can be expressed in terms of the state variables Tpi and Tpo as follows:

CpoM \ Tpm-- Kp 1CpiMM0.8 Tpi + (1 -~ Kp---~.8)Tpo (17) where Tpi and Tpo are the steam temperatures at the primary superheater inlet and outlet respectively, Cpi and Cpo are the correspond- ing specific heat of the steam, and Kp~ is the heat transfer coefficient at the steam side of the tubes.

Equation (17) plus the direct measurements of Tpi and Tpo comprise the temperature mea- surements for the primary superheater.

3.4. The platen superheater This is a vertical section mounted at the top

of the combustion chamber. Although some parts of the tubes receive direct radiation from the flame, the whole effect of the heat transfer is assumed to be in the convective mode. The temperature of the metal tubes is

CpLoM \ TpLm = KpmCpLiM °sM Tpai + 1 + KpL-~--Ss) TpL o

(18)

where Tpi i and Tpi o are the steam tempera- tures at the platen superheater inlet and out- let respectively, CpLi and CpLo are the corresponding specific heats of the steam, and KpL 1 is the heat transfer coefficient at the steam side of the tubes.

Equation (18) plus the direct measurements of TpLi and TpLo comprise the temperature measurements of the platen superheater.

3.5. The first bank of the secondary superheater This section lies next to the platen super-

heater. The construction is the same as that of the platen superheater, being suspended on the top enclosure of the boiler. The tempera- ture of the metal tubes is

C~loM T.~ m - K~11CsliMMos T~I i -[- 1 h ~ 8 ) T s l o (19)

where T,1 i and T,1 o are the steam tempera- tures at the first bank of the secondary super- heater inlet and outlet respectively, C.ai and C.~o are the corresponding specific heats of the steam, and K~11 is the heat transfer coefficient at the steam side of the tubes.

Equation (19) plus the direct measurements of T.li and T,1 o comprise the temperature mea- surements of the first bank of the secondary superheater.

3.6. The second bank of the secondary superheater

Its physical dimensions are similar to those of the first bank of the secondary superheater, but the temperature of the fluid within this section is the highest in the boiler. The tem- perature of the metal tubes is

C,2o M "~ T~2 m -- C~2iM Ts2 i + 1 + K,21MOS K,2-~-~.8)T,2o (20)

where T.2i and Ts2 o are the steam tempera- tures at the first bank of the secondary super- heater inlet and outlet respectively, C.2 i and C~2o are the corresponding specific heats of the steam, and K.21 is the heat transfer coefficient at the steam side of the tubes.

Equation (20), together with the direct mea- surements of TB2 i and T.2o comprise the tem- perature measurements of the first bank of the secondary superheater.

182

3. 7. The first bank of the reheater Similar to the primary superheater, this is

also a horizontal section and mounted above the economizer. The reheating steam is fed into this section. The total gas flowing along the gas path is separated into two components, one flowing into the gas path for the primary super- heater and the other into the path for this section. The temperature of the metal tubes is

Crlo M r "~ Tr,m-- grl Mr ° C i`Mr Trl + g )Trio

(21)

where T~ i and Tri o are the steam temperatures at the first bank of the reheater inlet and outlet respectively, Crli and Crlo are the correspond- ing specific heats of the steam, and Krli is the heat transfer coefficient at the steam side of the tubes.

Equation (21) together with the direct mea- surements of Trl i and Trl o comprise the temper- ature measurements of the first bank of the reheater.

3.8. The second bank of the reheater This lies behind the second bank of the

secondary superheater following the gas path, and it is connected to the outlet of the first bank of the reheater. The temperature of the metal tubes is

Tr2 m - Cr2i M r ( Kr21M o.s T~2i + \1

Cr2oMr "~ +

(22)

where Tr2i and Tr2o are the steam temperatures at the second bank of the reheater inlet and outlet respectively, Cr2 i and Cr2 o a r e the corre- sponding specific heats of the steam, and Kr21 is the heat transfer coefficient at the steam side of the tubes.

Equation (22) together with the direct mea- surements of Tr2 i and Tr2 o comprise the temper- ature measurements of the second bank of the reheater.

3.9. The pressure measurements Pressure loss in the boiler is mainly caused

by friction, section entrance loss and gravita- tional loss when the fluid is flowing through a section. These factors are related to the density of the fluid, the geometry of the section, and the polish of the tubes. A set of empirical equations obtained from a power station test

[11] has been adopted here as the measurement equations for the pressures:

Poo=O.1Po~ +O.9 Pdr

Ppo = 0.77/)dr + 0.23 P82o

PPLo = 0.56 Pdr + 0.44 P~2o

Bsli ---- 0.48 Pdr ÷ 0.52 Ps2 o

Ps~o = 0.30 Pdr + 0.70 Ps2o

Pr2i = 0.66 P~li + 0.34 Prli

(23)

(24)

(25)

(26)

(27)

(28)

All the variables on the left-hand side of the above equations are measured quantities. The subscript denotes the location of the respective measurement. The pressure at the economizer inlet, Pei, and at the outlet of the second bank of the secondary superheater, P82o, the drum pressure, Pdr, the pressure at the inlet of the first bank of the reheater, Prli, and at the outlet of the second bank of the reheater, Prli, are direct measurements as well as the state vari- ables in the model.

In the steady-state condition, the mass flow rate along all the sections is assumed to be a constant. That is, the mass flow rate in the economizer is equal to that through all the superheater sections and equal to the mass flow rate at the boiler outlet, M; the mass flow rate through all the reheater sections is Mr, and the gas flow rate through all the sections except to the primary superheater and the first bank of the reheater is Mg. Three of these are direct measurements in the model.

Between adjacent sections the following boundary conditions apply: Tpo = TpLi, TpLo-- T~li, Ts~o = Ts2i, and Trio = Tr2i. Arranging all the above-defined measurements into a vector Z, we can get a system model in terms of the state variable vector X; this is represented in a general compact form by

Z = g(X) (29)

where g(X) indicates the 40 measurement equa- tions.

The elements of the measurement vector Z are

Z = [Tei , reo , Tern , T,,o, T, ei, Td~, Tan,, Tw=,

Tpi, Tpo, Tpm, TpLi, TpLo, TpL:, T.li, T.io,

T,I=, T,2i, T82o, T.2m, Trli, Trio, Trim, Tr2i,

Tr2o, Tr2m, P.~, Peo, P~, P~o, PP~o, P.i~, P.io, P.~o, Pr~i, Prlo, Pr2o, M, Mr, M~I T

The elements of the state vector X are

X = [Tel , Teo, Tpi, Tpo, TpLo, T~lo, T,2o, Trli,

Trio, Tr2o, Pei, Pdr, P. o, Prii, Pr2o, M, Mr, MgV

4. PARAMETER DETERMINATION AND BOILER

SIMULATION

In the previous sections, enthalpy is repre- sented in terms of temperature by using the specific heat which is treated as a constant. This specific heat possesses a non-linear rela- tionship with temperature and pressure, but its value can be viewed as a constant within small variations of temperature and pressure. However, when a big change takes place in the working conditions of the boiler, for exam- ple, when the boiler output changes from one level to another, this assumption will not be true and the value of the specific heat should be modified. Any modification would require reference to the relationships presented in the Steam Table.

4.1. The Steam Table fitting The aim of the Steam Table fitting is to find

a set of simple equations which represents the relationship among given variables. At a given pressure the relationship between en- thalpy and temperature can be established by determining the coefficients of a preliminary equation by minimizing the quadratic errors between the fitted value and the Table value. This preliminary equation is selected by 'watching' the corresponding plot of enthalpy versus temperature. In this study a set of discrete algebraic equations is selected be- cause the enthalpy is extremely non-linear at the saturat ion point. The relationships in the subcooled and superheated ranges are each fitted by separate quadratic equations, while the saturat ion range is represented by a linear function with a sharp slope, as shown in the following equations:

200 ~< T ~< 350

H = 0.009091 T 2 + 0.1868 T + 4.661 (30a)

350 ~< T ~< 360

H = -90.93 T - 30165.7 (30b)

360 ~< T ~< 580

H = -0.010552 T2+ 13.7342 T - 942.84 (30c)

183

TABLE 1

Comparison between the Steam Table value and the fitted value for enthalpy

Temp. Table value Fitted value Error (°C) (kJ kg -l °C -I) (kJ kg -I °C-I) (%)

200 695.5 867.11 -0.890 250 1086.5 1081.01 0.505 300 1335.7 1340.36 -0.349 350 1659.8 1645.60 0.856 400 2890.3 2862.57 0.959 450 3104.0 3100.83 0.102 500 3269.6 3286.33 -0.512 550 3416.1 3419.80 -0.108

An error of less than 1.0% holds for the fitted values, as demonstrated in Table 1.

Since the Steam Table only offers enthalpy values at some standard pressure points at certain given intervals (e.g at 170bar and 175 bar), the values between these standard pressures are calculated by linear interpola- tion. The same method is used in fitting the density with respect to temperature and pres- sure.

A set of algebraic equations has also been established which satisfactorily covers the en- thalpy and density for temperatures ranging from 200 to 600 °C and pressures from 170 to 190 bar, respectively.

The specific heat of the gas is obtained by fitting to the Gas Table at atmospheric pres- sure in a manner similar to that given in the Appendix of ref. 8.

4.2. Determination of the heat transfer coefficient

There are many empirical equations avail- able for calculating heat transmission, but each of them is obtained under certain experi- mental conditions and therefore can only be applied in the corresponding cases. For the case of forced convection in the boiler system, Petukhov's equation [9] is adopted for the determination of the heat transfer coefficient at the inner side of the tubes:

K= -s\ J \b) (31)

where

fl = 1.07 + 12.7(Pr 2/a - 1)(f/8) 1/2

f = (1.82 log Re - 1.64)-2

184

is the f r ic t ion factor , the Reynolds number Re = uDfi,, u and ~ denote the ve loc i ty and viscosity, respect ively , of the fluid, D is the inside di- amete r of the tube, k denotes the the rmoconduc- t iv i ty and Pr is the P rand t l number .

#b//~. denotes the ra t io of the viscosi ty of the fluid at the mean t e m p e r a t u r e to tha t at the wall t e mpe ra tu r e and is approx imate ly equal to un i ty in the boiler.

All of these var iables will affect the hea t t r ans fe r when the t empe ra tu r e and pressure are varied. The combined effect is best illus- t r a t ed by a d iagram shown in Fig. 3. The d iagram is ob ta ined from eqn. (31) using a mass flow ra te of 12 m s 1 and an inside d iamete r of 3.2 cm. Region I cor responds to the va lue of hea t t r ans fe r in the hea t r ecove ry area, region II to the s a tu r a t i on area, and region III to the supe rhea t e r area. I t is obvious t ha t at a given pressure the va lue in region I is much g rea te r t han tha t in region III. The va lue of the co- efficient is g rea tes t a t the s a tu ra t i on tempera- tu re then decreases sharp ly with increas ing t em pe ra tu r e in region II. The va r i a t ion of pres- sure has l i t t le inf luence on the va lue of the coefficient in general , bu t its effect on s team is la rger t ha n tha t on water .

The use of eqn. (31) can be demons t ra t ed by the fol lowing example.

A hea t exchanger consists of N = 144 tubes in paral le l ca r ry ing 250 kg s -1 of steam. The tubes are bent t h rough 180 ° every metre, the length L of each tube being approximate ly equal to 9 m. The inside diameter , outside di- amete r and th ickness of each tube are 3.55 cm, 5.08 cm and 0.7632 cm respect ively. The s team t empera tu re and pressure at the inlet and out- let of the tubes are 356.8 °C, 179.1 bar and 433.6 °C, 176.8 bar respect ively. The fol lowing ca lcu la t ion is self-explanatory.

Cross sec t ional a rea A of tube = n(0.0355/2) 2

= 9.9 × 10 -4 m 2

Mean s team t empera tu re = 0.5(356.5 + 433.6)

= 395.05 °C

Mean s team pressure = 0.5(179.1 + 176.8)

= 177.95 bar

At mean t em p e ra tu r e and pressure,

? =0.3195 x 10-6m2 s-1 p = 8 6 . 4 9 k g m -a

k = 0.09725 W m -1 °C -1 Pr = 1.59

I - - Z

H 0 i,-,.-,I bL LL W 0

W U-

Z

¢ v I,--

I,.-

W "1-

X105 2 . 0 0

1 . 8 0

I . 6 0 .

1 . 4 0

I .20.

I . 0 0 .

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0.60.

0.40.

0 . 2 0 .

0 . 0 0 , 2 0 0

I

I §6o

III

~4o ~aO ~20 400 44o 4eo ~2o ~eo 600 TEMPERATURE

Fig. 3. H e a t t r a n s f e r coefficient as a func t ion of temperature and pressu re u s i n g P e t u k h o v ' s e q u a t i o n (31) for a mass flow r a t e of 12 m s - I and an ins ide d i a m e t e r of 3.2 cm: E3, P = 170 bar; V, P = 180 bar; Z&, P = 190 bar.

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186

the mass flow rate is

M / N u -

Ap

250/144 - 20.28m s ~

= 9.898 × 10 -4 × 86.49

and

R e - uD

?

20.28 x 0.0355 - - 2.254 x I0 s

0.3195 x 10 -6

Therefore,

f = (1.82 log Rc - 1.64)-e = 0.01016

fl = 1.07 + 12.7(Pr 2/3 - 1)(]:/8) I/2 = 1.234

K R e P r f k = l . 0 1 x l 0 4 W m _ 2 o c _ , 8D

The total equivalent heat transfer coeffi- cient is

NK(L~D) = 1.460 x 103 kW °C-i

The dimensions of all the sections of the boiler are given in Table 2. The heat transfer coefficient is determined, as shown in the above example, with a computer program sub- routine.

Leuy's empirical equation [9] is employed for calculating KwI where

Qwlr 283.2 Awl Pdr4/3Tdr 3 Kwl = T~m - Tdr---- (Twin __ Tdr) 3 (32)

where Qwl represents the heat flux transferred from the tube waterwalls to the fluid flowing in the tubes.

The heat transfer coefficient Kw2 can be calculated according to the Stefan-Boltzmann law [9]:

Kw2 - Qg~ (33) 4 - T . m 4

where Qgw is the heat flux released by fuel gas in the combustion chamber, and Tg~ is the gas temperature in the combustion chamber deter- mined by calculating the combustion pro- cesses.

The values of the specific heat at the inlet and outlet of each section under full load conditions are given in Table 3.

The values of the heat transfer coefficient at the internal and external sides of the tubes are given in Table 4.

TABLE 3

The specific heat (kJ kg -1 °C -1) at the inlet and outlet of each section under full load conditions

Cei 4.336 C~o 4.450 Cpi 7.136 Cpo 7.156 CpLo 6.711 C~1 o 6.342 C~2 o 6.121 Crl i 8.587 Crl o 7.111 Cr2 o 6.356

TABLE 4

Heat transfer coefficients

Ks1 572.0343 Ke2 196.3408 Kwl , 420.70 x 102 Kw2 28.37 × 10 -3 Kpl 325.8137 gp2 34.2807 KpL 1 123.8363 KpL 2 3.2056 Ks21 148.8930 Ks22 1.5993 Krl, 217.0571 Krl 2 84.5185 Kr21 180.2339 Kr~ 3.9689

4.3. Boiler simulation As a prelude to the state estimation a com-

puter simulation of the boiler has been estab- lished. If the state estimation is viewed as a process to determine the state of the boiler from a set of measurements, the simulator here will work as a boiler when a set of input variables is input to the established system. The simulation represents the working be- haviour of the boiler in terms of temperature and pressure. The performance of the simula- tion can be used to verify the stated assump- tions, the empirical equations used and the validity of the model. The associated output from the simulator can be used as measure- ments to test the estimator off-line.

Both physical and operation data in the simulator are identical with the model for estimation. The values of M, Tei, Psi, Trli, Prli, Ps2o, and the fuel rate Mf are the input control variables. The values of the temperature and pressure of each separate section are the out- put variables.

In the simulation model the energy conser- vation law is a dominant principle when the mathematical model is established. Apart from considering the pressure drop and heat trans- fer from the tubes to the fluid which have been processed in section 3, the heat transfer from the gas to the tubes in each section is calcu- lated in the same manner as that stated by

eqn. (6) for the economizer. The total heat flux transfer from the gas to the tubes in a section is equal to the difference in the gas enthalpies at the inlet and outlet of the stage.

Calculations in the gas path are carried out in the following order: the combustion cham- ber, the platen superheater, the first bank of the secondary superheater, the second bank of the secondary superheater, the second bank of the reheater, the primary superheater and the first bank of the reheater in parallel, and the economizer, as indicated by the broken line in Fig. 2. As the hot gas flows through these sections, heat is absorbed by the metal tubes and the gas temperature drops progressively. The following general equation is used:

Qg ~- MgCg(Tg i - Tg o) (34)

where Qs is the heat released by the gas, M s is the mass flow rate of the gas, Cg the specific heat of the gas, and Tgi and Tg o are the tem- peratures of the gas at the inlet and outlet, respectively, of the stage.

The temperature in the combustion cham- ber is determined by

c f ( 1 - Mash)Mf + CsM Tair = + C wM Tg

(35)

where

M s = (1 - M~h)Mf + Ra Mf

Cf is the calorific value of the fuel, assumed to be a constant, M~, h is the boiler loss, which is 10% of the fuel, Ra is the air/fuel ratio, as- sumed to be 10, and Tai ~ is the temperature of the gas leaving the economizer, assumed to be 280 °C.

Equation (35) indicates that the total heat input to the boiler, given by the left-hand side of the equation, is the sum of the heat in the coal and that in the air in the combustion chamber, and must balance with the heat ab- sorbed by the waterwalls plus that leaving the combustion chamber.

In the waterwalls, if Z per cent of water has been vaporized into steam after one circula- tion, the mass flow rate in the waterwalls becomes M,--M/Z in the steady-state condi- tion, where M and Mw are the mass flow rates in the economizer and the waterwalls respec- tively, and the value of X is determined by

1 1 1 - X + - - P,,,o P~,o~ Pwow

187

P,o is the average density of the water-s team mixture at the outlet of the waterwalls, and P,o8 and Pwo~ are calculated by eqns. (12) and (13) respectively.

The spray mass flow rate is assumed to be 6% M. The position of the orifice plate in the gas path, which controls how much gas flows through the primary superheater, is obtained by a test method in order to achieve a boiler output temperature of 568 °C.

The simulation results under full load con- ditions are listed in Table 5.

Several numerical experiments have been performed and the results show that the model works well over a relatively wide range. The aim of these tests was to observe the perfor- mance of the model while adjusting some main control variables and to analyse the sensitiv- ity of the model with respect to the variation of the parameters. The control variables were varied as follows: (1) the feedwater flow rate Mei from 100% of the rated value to 50%; (2) the pressure at the throttle, P~2o, from 100% of the rated value to 50%; (3) the fuel flow rate Mf from 100% of the rated value to 50%. The model performed satisfactorily in all the ex- periments. The analysis of the sensitivity of the model to changes in the parameters was carried out by investigating the performance of the model when the value of a parameter was changed within ___ 10% of the base value. This involved an extensive number of tests be- cause there is a large number of parameters in the model and each parameter has to be tested individually. These parameters are the specific heat and the heat transfer coefficient in every section. The tests show that the model per- forms well under parameter changes. More- over, it was found that the model is more sensitive to changes in the specific heat than to changes in the heat transfer coefficient. Different values of the specific heat of the gas also have a noticeable influence on the energy distribution within the boiler. The simulation results have been checked by an operation engineer satisfactorily.

5. CONCLUSION

A mathematical model for the state estima- tion of a power station boiler has been estab- lished in this paper. The use of the model for state estimation and related algorithms will

188

TABLE 5

Bo i l e r s i m u l a t i o n r e su l t s

Input control variables Feedwater flow rate Steam flow rate in reheater Spray mass flow rate Temperature at economizer inlet Temperature at first bank of reheater inlet Feedwater pressure Throttle pressure Fuel mass flow rate Calorific value of fuel Gas flowing through primary superheater/total gas

Simulation results Temperature at economizer inlet Temperature at economizer outlet Temperature of economizer tubes Temperature of gas at economizer stage inlet Temperature of gas at economizer stage outlet

Temperature of water in drum Temperature of steam in drum Temperature of waterwal] tubes Temperature of gas in combustion chamber

Temperature at primary superheater inlet Temperature at primary superheater outlet Temperature of primary superheater tubes Temperature of gas at primary superheater stage inlet Temperature of gas at primary superheater stage outlet

Temperature at platen superheater inlet Temperature at platen superheater outlet Temperature of platen superheater tubes Temperature of gas at platen superheater stage inlet Temperature of gas at platen superheater stage outlet

Temperature at secondary superheater 1st bank inlet Temperature at secondary superheater 1st bank outlet Temperature of secondary superheater 1st bank tubes Temperature of gas at secondary superheater let bank stage inlet Temperature of gas at secondary superheater 1st bank stage outlet

Temperature at secondary superheater 2nd bank inlet Temperature at secondary superheater 2nd bank outlet Temperature of secondary superheater 2nd bank tubes Temperature of gas at secondary superheater 2nd bank stage inlet Temperature of gas at secondary superheater 2nd bank stage outlet

Temperature at reheater 1st bank inlet Temperature at reheater 1st bank outlet Temperature of reheater 1st bank tubes Temperature of gas at reheater 1st bank stage inlet Temperature of gas at reheater let bank stage outlet

Temperature at reheater 2nd bank inlet Temperature at reheater 2nd bank outlet Temperature of reheater 2nd bank tubes Temperature of gas at reheater 2nd bank stage inlet Temperature of gas at reheater 2nd bank stage outlet

Pressure at feedwater pump outlet Pressure at economizer inlet Pressure at economizer outlet Pressure in drum Pressure at primary superheater inlet Pressure at primary superheater outlet Pressure at platen superheater inlet Pressure at platen superheater outlet Pressure at secondary superheater 1st bank inlet Pressure at secondary superheater 1st bank outlet Pressure at secondary superheater 2nd bank inlet Pressure at secondary superheater 2nd bank outlet Pressure at reheater 1st bank inlet Pressure at reheater 1st bank outlet Pressure at reheater 2nd bank inlet Pressure at reheater 2nd bank outlet

250.00 kg s -1 195.30 kg s 1 15.00 kg s

242.00 °C 364.00 °C 185.00 bar 169.54 bar 38.65 kg s '- i

24000 kJ kg - I 0.745

242.00 °C 302.37 °C 303.11 °C 514.14 °C 339.61 °C

342.30 °C 357.41 °C 359.33 °C

1751.12 °C

344.18 °C 435.60 °C 441.56 °C 902.63 °C 512.86 °C

435.60 °C 521.15 °C 527.71 °C

1284.11 °C 1142.54 °C

476.25 °C 537.34 °C 542.75 °C

1142.54 °C 1047.24 °C

537.34 °C 567.96 °C 571.38 °C

1047.24 °C 997.60 °C

364.00 °C 464.51 °C 469.77 °C 902.63 °C 516.61 °C

464.51 "C 567.64 °C 577.35 °C 997.60 °C 902.63 °C

189.95 bar 185.00 bar 180.40 bar 180.00 bar 179.62 bar 178.36 bar 178.38 bar 175.96 bar 175.04 bar 172.98 bar 172.98 bar 169.54 bar 41.85 bar 41.12 bar 41.12 bar 39.71 bar

be presented in Par t H of this series [12]. In the model the temperature and pressure are the two fundamental variables. Many proper- ties are determined by these two variables. In the proposed model the pressure can affect the temperature only in the drum section. An al- ternative approach is to represent enthalpy as a function of both temperature and pressure at a given point and the pressure will affect the energy balance equation directly in every section. However, the mathematical complex- ity of the model will increase considerably and may not be practical. The combustion chamber is a critical part in the model, its representation affecting the overall perfor- mance. The effects of the downcomer and air heater are neglected in the present study. However, their inclusion introduces no major difficulty and should not change the solution under steady-state conditions. The test perfor- mance of the model for various loading condi- tions is satisfactory.

NOMENCLATURE

A area, m 2

C specific heat, k J kg - 1 °C- 1 H enthalpy, kJ kg -~ K heat transfer coefficient M mass rate, k g s - ~ P pressure, bar Q energy transfer rate, k J s - ~ T temperature, °C

p density, k g m -3

Subscripts

dr drum drs steam in drum drw water in drum e economizer f fuel g gas gw gas in combustion chamber h enthalpy i at inlet m metal tube o at outlet PL platen superheater PLo outlet of platen superheater

p primary superheater r radiation r l first bank of reheater r2 second bank of reheater sl first bank of secondary superheater s2 second bank of secondary superheater si inlet of secondary superheater sp spray w waterwalls wm metal tube of waterwalls wo outlet of waterwalls

189

REFERENCES

1 H.W. Kwan and J. H. Anderson, A mathematical model of a 200 MW boiler, Int. J. Control, 12 (1970) 977 - 998.

2 K. L. Chien, E. I. Ergin, C. Ling, A. Lee and A. Calif, Dynamic analysis of a boiler, Trans. ASME, 80 (1958) 1809 - 1819.

3 J. Adams, D. R. Clark, J. R. Louis and J. P. Spanbauer, Mathematical modeling of once-through boiler dynam- ics, IEEE Trans., PAS.84 (1965) 146- 156.

4 J. P. McDonald, H. G. Kwatny and J. H. Spare, A nonlinear model for a reheater boiler turbine~genera- tor system, Part I. General description and evaluation, Proc. 12th Joint Automatic Control Conf., 1971, Am. Soc. Mech. Eng., New York, pp. 219- 226.

5 H. G. Kwatny, J. P. McDonald and J. H. Spare, A nonlinear model for a reheater boiler turbine-genera- tor system, Part II. Development, Proc. 12th Joint Automatic Control Conf., 1971, Am. Soc. Mech. Eng., New York, pp. 227- 236.

6 E. B. Swidenbank and B. W. Hogg, On-line identifica- tion of large turbogenerators, Proc. 6th Europ. Conf. on Electrotechnics: Computers in Communication and Con- trol, 1984, pp. 74-78.

7 T. Thumm and E. Welforden, Modular steam-generator and turbine model for on-line process control and power system simulation, Modelling and Control of Electric Power Plant, IFAC, 1983.

8 United Kingdom Committee on the Properties of Steam, UK Steam Table in SI Units, 1970, Edward Arnold, London, 1970.

9 M. N. Ozisik, Heat Transfer: A Basic Approach, McGraw-Hill, London 1985.

10 W. F. Davidson, P. H. Hardie, C. G. R. Humphreys, A. A. Markson, A. R. Mumford and T. Ravese, Studies of heat transmission through boiler tubing at pressures from 500 to 3300 pounds, Trans. ASME, 65 (1943) 553- 591.

11 D. R. Booth and E. Marchand, Cockenzie boiler inves-~ tigation, Internal Rep. No. Rd 5/69, South of Scotland Electricity Board, Cathcart House, Glasgow.

12 K. L. Lo, Z. M. Song, E. Marchand and A. Pinkerton, Development of a static-state estimator for a power station boiler, Part H. Estimation algorithm and bad data processing, Electr. Power Syst. Res., 18 (1990) 191- 203.