NONLINEARITY OF DYNAMIC PROCESSES IN A DRUM BOILER CIRCULATION LOOP

Copyright © " ' A C Modelling & C< ol KU'uric Power Pl.mis

NONLINEARITY OF DYNAMIC PROCESSES IN A DRUM BOILER CIRCULATION LOOP V. Kecman Faculty

<>ι''Mechanical luigineering

1

ana Xax'al

Architecture.

l'ui\>ers'it\ of Zagreb. )'u!>;o.\la\'ia

Abstract. The paper deals with mathematical modelling and numerical si­ mulation of the dynamics of flow and heat processes inside the circula­ tion loo;> of a power steam generator. The processes in the evaporator, drum and downcomers are modelled separately. The evaporator is treated as a system with distributed parameters and is described by three nonli­ near, partial differential equations for the mass flow, entropy and pressure. An eigenvalue analysis of evaporator system matrix A has been made. Linear and nonlinear models of circulation loop describe the phe­ nomena of shrink and swell of drum water level. The comparative analysis of nonlinear model responses to the inputs of the same magnitude obta­ ined at tbre different load levels, clearly presents the nonlinearity of the circulation loop processes. Keywords. toilers; modelling; nonlinear simulation; linear analysis; partial differential equations; discretisation INTRODUCTION

part of the riser is the basic condition for the validity of the model of entire circulation path. Therefore, a special concern is given to the model ling of this part of the circulation loop. In contrast to most papers dealing \*rith drum boilers the processes in the evaporator, boiler's drum and downcomers are modelled separatly. The riser is treated as a system with distributed parameters and the processes of the flow and heat exchange in it are described by partial differential equa­ tions (PDE) in terms of the mass flow (m), entropy (s) and pressure (P). The analy­ sis of the eigenvalues of the discretised and linearised set of equations obtained from these IDE presents a good insight into dynamic characteristics of the eva­ porator. The transients of pressure and water level in the drum are described with two nonlinear, ordinary differential equ­ ations (ODE). The do\imcomers' dynamics is neglected.

Successful application of modern control methods depends mostly upon validity of mathematical models which form the base for a control algorithmic synthesis. The­ se models, besides tiiey should be able to describe well the process dynamics, ought to be in the form of linear dynamic equa­ tions of state space (i.e. in the form of matrix differential equations), because the modern control theory uses just this mathematical tool . There is always the sa­ me question about the linearisation - what is the range of change of state variables inside which the obtained linear model can still reproduce a process dynamics ve­ ry well. It is the question about the nonlinearity of modelled processes. In this paper, the analysis of nonlinear character of the processes which take part inside the circulation path of power plant boiler with forced circulation is given. A necessary tool for fulfilling this goal is an appropriate mathematical model able to give a true representation of the pro­ cess dynamics on the steam - water side. The precise description of the evaporating

The mathematical model was applied to the circulation path of the No. Γ? unit of Cromby station, Philadelphia, PSA. All necessary geometric data were found at C lei land (.1976). 21

V. Ke cman

22

i.vric.

JL·

I'A.>I..':.)JL

FPMMMENTAL ^ , , Ρ Λ Τ Ι Ο Ν Ι :

I'i^ure ί illustrates shematicaly a circu­ lation loop for which a proper mathemati­ cal model should he established. The w h o ­ le loop is divided into thre parts - eva­ porator (riser), drum, downcomers, and each part has been modelled separatly. The model is developed on the basis of a first principle analysis, that i s , g o ­ verning equations are equations of mass, momentum and energy balance. T h e develop­ ment of these equations will be not given (it can be found in any good book on flu­ id dynamics) and just their final forms will be presented here. Nevertheless, a curious reader could find out that it lias been tryed to be in accordance with Kwatny, McDonald, Spare ( 1 9 7 1 ) , Konopacki (1979), Kecman (1980,198°), Cermak, Peterka, Zavorka ( 1 9 7 ° ) .

dm n_ _ P -Γn-1, _ Pmn vn mn -mn-1, rnn Λ n dt ~ "' Δζ A Δζ ~A v n -vn-1 . ds n "dt

(4)

Λ(ΔΡα+ΔΡκ)η

Δζ

( 1 Oq

n

n

- " νη(ΛΓ ο^ " IT ~T1.

n-1

n n A ,, x

+

TT- A , W

n

n (5)

dP

Λ

n-1 "HI tf,

, 1

_

J

Dq

v* , m -m , n-1 n n-1 , A Δζ O ··, III m111 i, (3s —-s m n-1

n

n-1

n = ?,

Now,

ν

Bn-l n■· , vV i

n-1

n-lAn Δ1

Δ Ζ — + ΤΓ-.;

^ΤΓ—· öt - — n-3

γ ö

NE + 1

\

Ί·η-] > (6)

n-1

this system of equations can be used

for the numerical simulation of dynamic behaviour of the evaporator. Mathematicaly speaking, from the thre PDE, we derive a system of 3·ΝΕ ODE in time, where NE is the number of elements.

Evaporator

Before simulation, the boundary conditions

The riser section lias been treated as the

should be stated. In the riser, which is

system with distributed parameters and

the closed circulation loop, the physical

the model of dynamic processes in it has

boundary conditions are inlet flow m, inlet

the following form,

enthalpy h ,

dm Dt

.DP

ds _ V /J__ 5 ϋ Dt " \ \ T Dz

1

^mv Dm Λ Dz

JB ^ s

A "Sz

cVV

Λ(ΔΓ·„+Δ!' )

DZ

vj

Γ

(1) NIV

ΤΛ A

\

ΛΤ Δι ν>

(?)

m Ds , mv A M xi dq IT Dz ~ A Dz -Ds Dt

convenient for computer simulation and it should be previously rewriten in a proper form. The most common means for solving this system of equations is the discreti­ sation method. Essentially, this means dividing the evaporating part of the riser into a number of elements and any obtained section spatial derivative replace with back-ward difference. (Clearly, the grea­ ter the number of sections, the better the approximation, but the larger the model the more expensive is the simulation). Applying this back-ward method to the riser's nonlinear model the following set of equations is obtained:

dt

¥

dh, dF

Oi - h flc ) - Ap'-h' (7)

(3)

This set of three nonlinear PDE is not

= P,

-r—L -rr c>q - m , v( l l, - h , / ) - TT-L An' v(ττ— Dh' dp ac ? s Dp a t dz s dc

dL

DP Dt

and outlet pressure P

- dc * nn d with the heat flux dq/dz as the driving force over the heated length. In this p a ­ per, internal variable is the moving subcooling length L which can be calculated from the following equations,

(At Konopacki (1979) and Kecman (1980, 198?) L is c o n s t a n t ) . s In order to get better insight into dyna­ mic characteristic of the evaporator part, a linearisation of the nonlinear model has been made and the analysis of the eigenva­ lue of the system matrix A bas been done. Discretised and linearised set of equati­ ons for flow rate, entropy and pressure is given below, ^Διη n

Dt

= (+ (-

4m _ ΑΔζ V n nffr An ΑΔΖ

m-tf rBn>

(- -TÄ

ΑΔζ

, )ΔΠΙ + ( -

V

n-1

+ -T-); Δ Ρ , Δζ n-1

■) A S

ΑΔζ '

77—

, + (-

n-1

nP# An ΛΔζ

m' tfOn T

v

Δ z

Pmv

<8>

n

ΑΔζ A A

Δζ

>Δν

n-1 )Δ1 } η

( 8 )

'on linearity

ÖÄS

s - s _,

n

v

of Dynanic

Γ ι

^(v-vjC" D! 'dt 7ΓΓ

v. -TK _,.(--^-|2)

τκ = 3Δ;

η-1

=

^Τ— -

Δ

;

(Οϊ M V

ϊ-^-ϊ

r,

I01

- ^7- Ι '· ^^Ι ' )As ΛΔζϊ[, η _/) -Δ ^η +^ ίΤ- ρ™

3*

(lrt)

(^Μΐ1

η +

3Τ

( ν■

Λ τη_,

23

v rn

Δζ

■(^^

Processes

η

ΪΛη-1

- <ρ-/>·

N° =

'Dp' ,, ,'

c}h' ,, ■

öp"

■(v-Vh"+i£
~'

Cl v~ T

*Λη-1

, n 1

' ~

Δζ

*Hn-l ' *An~l

* / ί η- ΐ

"Λ

Δ ζ

V

n-^

S

n"Sn-!. Δζ

-)AS

,+ (10)

f- ^ ^"~' . n~l x 7>An

Input variables are: steam flow m , feedwater flow m , circulation flow m, e ' riser outlet flow and enthalpy (entropy) m , h (s ) . Output variables are pressu­ re ί* and water volume V (or level H ) . y d w Linear analysis gives that the drum, ana­

^n-1

lysed separatly, shows integral

character

for water level H and proportional charac­ Mg:. ° illustrates the state variable

ter (T»oO seconds) for drum pressure i^.

form for the case NE=4 (evaporating part

Downcomers

is divided into four elements), rig. o indicates the influence of the number of elements (NE), n t MX, > of load, on the width of the eigenvalue spectrum. Clearly, the rrreater the NE, the wider the range of eigenvalues reproduced. It should be noted that real eigenvalues move toward the origin and the right hand plane, i.e. towards the area of instability, when NE is decreasing. After a number of calcula­ tions it has been established that the model will be unstable if the chancre of entropy (As) per element (and that is the only criterion) is greater than C ^ I P

The downcomers' dynamics is neglected and merely the mixing of two streams is given (for the calculation of the enthalpy h, ) dc bei ow, rn h + (m - m /)h e e Q h (13) dc Algebraic equation shows that the «cumula­ tion of mass or energy inside the downco­ mers is not taken into the account. But this equation is important because the subcooling length L is determined bv h , , s dc' and the dynamics of the riser depends upon

kJ/kg K. This criterion gives the smallest

L . s

NE which ensures the stable model. For

Finally, the nonlinear and linear model

Cromby No °. Unit at 100 ;■!> of load the

of the entire circulation loop is obtained

smallest NE equals °.

by coupling the appropriate equations of all three subsystems.

A dynamics of the processes in the drum is described with the following two ODE,

SIMULATION RESULTS The division of the riser has a double impact on the size of the model and cost

dP dt

d

(me+mr-m-mu)

#

of the simulation of the boiler's dynamics.

(Pn'~j"h^

N (mrhr-(m-me)h'-mdtf)
(11)

Firstly, the smaller the N E , the smaller the number of differential equations des­ cribing the evaporation processes. Se­ condly, the smaller the N E , the smaller

V. Kecman

24 the real

COKCLi'S] UNS

and imaginary parts of the

eigenvalues (Fig. 5) i.e. tlie larger the time constants describing; the process, and because tlie step of integration

At

is proportional to tlie time constant, the larger the step of integration At, consenquently, the cheaper the simulation. Table 1 shows the integration steps At which, for given NE, still ensure the stable simulation runs. (For NE = ° and At = G.üü°5 s first G°0 seconds of dyna­ mics take °rV9' of U M VAC i 100 computer time using °7 K of memory).

A drum boiler circulation

loop has nonli­

near dynamic characteristics. Though, for the whole upper half of load changes, li­ near model obtained at 75

;

of load can

give acceptable responses. If the evapo­ rating part of the loop is treated as a distributed process, simulation runs are pretty costly because of very small inte­ gration step. The most recommended

further

continuation of this work would he to apply a more appropriate (faster) integra­ tion procesure as well as to analyse po­

Fig. 4 indicates the influence of NE on

ssible reductions of the evaporator model,

transients of water level I! and pressure

neglecting momentum equations. The divi­

P at Π0 > of load in the case of sudden

sion of riser has not an impact on pressu­

increase of heat rate for 10

re dynamics and the phenomena of shrink

J. The swell

phenomenom is more pronounced in tlie case

and swell are more pronounced in the case

of rougher discretisation, but even more

of rough discretisation. There is a limit

interesting is that the pressure dynamics

in the smallest number of elements obta­

does not depend upon division of the eva­

ined by the division, and this limit is

porating part.

determined by the entropy change along the

To get the answer to the question about

discrete riser element.

the nonlinearity of processes which take place inside the circulation loop, simu­ lation runs have been made with the dis­ turbances of tlie same magnitude at three different load levels. Fig;. 5 and 6 illus­ trate the dependence of dynamic behaviour of circulation loon in the case of incre­ ase of heat rate for G.?85 kJ/ms (Fig.5) and in the case of increase of feedwater rate for 9 kg/s (Fig.6). Both figures in­ dicate the difference in dynamic behavi­ our at different levels and show that the linear model obtained at one load level cannot describe the dynamics in the whole area of load changes. At tlie same time simulation runs show that the linear mo­ del obtained at 75 % of load could repre­ sent the dynamics in area from 50 yo to full load, satisfactorily.

REFERENCES Clelland, P.J. (1976). Power Plant Data Base, PECO Cromby No. ° Unit, Volume 1. Kwatny, II. G. , McDona Id, J . P . , Spare , J. Γ;. (1971). A nonlinear model for reheat boiler-turbinesenerator

systems,

Part II Development, JACC of tlie AACC, Paper No. 5-D5. Konopacki, W.A.(1979). The Dynamics and Stability of Evaporation in Steam Ge­ nerators, Ph.D. Thesis, Drexel University, Philadelphia. Kecman, V. (1980). Simulation Analysis of the Dynamics of a Drum Boiler Circula­ tion Loop, Research Report No. 80-845-01, Drexel university, V)\i Fa del phia. Kecman, V. (198°). Modeliranje i simula-

In this paper, the results presented have

ciona analiza dinamickih procesa u par-

been obtained by simulating just the dy­

nom kotlu (in Serbocroatian), Ph. D.

namics of circulation loop, and it is rather hard to compare them with values

Thesis, University of Zagreb. Cermak, 1., Peterka, V., Zavorka, J. (197°)

measured on the whole plant. Such experi­

Dinamika reguliruemih sistem v teplo-

mentally obtained values for Cromby No.

energetike i himii (in Russian, trans­

2 Unit are presented in Thompson (1967), and Fig. 7 shows the comparison of results

lated from Czech) Mir, Moscow. Thompson, F.T. (1967). A Dynamic Model of

at 90 ;o of load and for the change of he­

a Drum-Type Boiler System, IEEE Trans.

ating of 5 ;υ. Tlie short dashed curve is

on PAS, Vol. PAS - 86, No. 5,

the Cromby Plant response, while the solid

S. 6°5-635.

curve gives the model response.

Üonlinearity of Dynamic Processes

n

25

d . rd

x x x x

X X (Λ

U

o ε o ' υ

X

(Λ •H

X X

h

2

X X

■ aq/az

3

P2 X

m

X X

X

h

X X

P 3 m5

X

X X X X

^

4

X X X

X X

öq/öz

s

X X X

X

o n

'ΊΚ- 1. Tlie b o i l e r c i r c u l a t i o n loop.

m3

x x

X X

§

X

s

Χ Χ Χ X X

u Φ

X X

x x x x x x x x x x x x x x

s

5

KJ

X

Fig. ?. S t a t e v a r i a b l e form of l i n e a r r i s e r model.

1-400

30 id

a 2

>-0-c££ O

real f 2igenvaluqs_

,

- £i

■WNA/Vfr- • NE>8

NE«2

Fig. 3. Influence of NE on the eigenvalues spectrum - 100% load.

TABLE 1 Dependence of the Greatest Integration Step At upon NE NE

AU[«!

1

0.02

2

0,004

3

0.001

5

0,00075

7

0.0005

10

0.0001

26 ΛΗ(ΓΓ}

ΔΡ

[bar]

NE - 2,3,5

150

200

250

300

'-'i»;. 4 . I n f l u e n c e of r i s e r d i v i s i o n (NE) on l o o p ' s 50-\> l o a d , Ιϋ,ο i n c r e a s e in h e a t r a t e . Δ

t [s]

transients,

Η [m]

Fig. 5. Dependence of transients upon load level, increase of heat rate for 0.?85 kJ/ms.

27

Nonlinearity of Dynamic Processes

ΛΗ :m3

·/·

of

load

. loo 0,2

^<^^'

10

^**&<·**

0,1

__50^^.<-*" , i00

150

200

250

300

200

250

300

t[s]

Λ? [bar]

% —

of

t[s],

load

100 50

— 10

F i g . 6. Dependence of l o o p ' s t r a n s i e n t s upon load l e v e l , increase of feedwater r a t e for 9 k g / s .

Δ Η prj

Pip;. 7. Model and plant responses, 90',-o load, 3(;ό decrease in heat rate.