lime modelling to control lime rotary kilns

7/27/2019 Lime modelling to control lime rotary kilns

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Lime : Mathematical model for the simulation and

control of rotary lime kilns

Mathematical model for the simulation and control of rotary lime kilns Open-loop simulations agree

with the literature

TEXT SIZE

By: J.J. Castro, F.J. Doyle III and T. Kendi

2001-07-02

Some of the earliest work modeling industrial rotary kilns was developed for cement kilns in the 1960's.

Lyons et al. [1] developed a model to simulate the steady-state behaviour of a cement kiln. This model was

the basis for a lime kiln model developed by Koivo and Chase [2], which included a methodology forparameter identification. Another significant model was developed by Kochar [3] to study the effects of

alternative fuels and dams for operation of lime kilns. That work also included a complete set of parameters

for an industrial lime kiln and detailed steady-state data temperature profiles. As computer processing power

increased, it was possible to study the dynamic behaviour of rotary kilns. Spang [4] extended Lyons work

and developed a dynamic model of a cement kiln. Similarly, Smith [5] extended the work of Kochar to

develop a dynamic model for simulation and control of lime kilns. Le Blanc et al. [6] have recently

presented a new dynamic mathematical model for rotary lime kilns using the "cinematic" [7] technique to

solve the partial differential equations (PDEs) that describe the process.

In this work, a fundamental dynamic model of the lime kiln for simulation and control studies is presented.

The model is an extension of the work by Kochar [3] and Smith [5]. The model, constructed from firstprinciples mass and energy balance equations, is described. Simulation results, including steady-state

temperature profiles, solids compositions and dynamic responses to potential manipulated variables for

control studies, are included.

PROCESS DESCRIPTION

The lime kiln is one of the significant pieces of processing equipment within the recovery section of a pulp

mill, Fig. 1. It is a counter-current rotary furnace that recovers spent lime from "lime mud," a mixture of

approximately 30% water and 70% solids (mostly calcium carbonate). The calcination reaction is

accomplished by heating the lime mud to a sufficiently high temperature (approximately 1400K) where the

calcium carbonate (CaCO3) undergoes a decomposition reaction to produce lime (CaO) and carbon dioxide(CO2).

The operational objective of a kiln is to produce "good" lime, typically determined by the residual

concentration of calcium carbonate, at a constant rate to maintain lime inventory. Secondary objectives

include minimization of the energy utilization (fuel), keeping the front-end temperature (temperature of the

hot lime) within specified bounds, and controlling the gas stack oxygen composition. The measured outputs

are: the front end temperature (FET), the back end temperature (BET) and the stack gas oxygen

concentration. The residual carbonate concentration is measured every two hours. The manipulated variables

are the fuel flow-rate and the air flow-rate. The most important process disturbances are the lime mud

composition and flow-rate.

LIME KILN MODEL


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A fundamental model for the lime kiln was developed dividing the kiln into three distinct zones (which

parallel the development of Koivo and Chase [2]): the drying zone, heating zone and the combustion zone.

The lime kiln is a distributed parameter system and is approximated as the interconnection of continuous

stirred tank reactors (CSTRs) in series. The dynamics of the gas are two orders of magnitude faster than the

mud. Therefore, the gas phase is modeled by steady-state equations [8]. The solid phase (lime mud) and the

wall are modelled using dynamic equations. The modelling strategy is depicted schematically in Fig. 2.

The lime kiln is modeled using 66 CSTRs in series with 13 ordinary differential equations (ODEs) and 8

algebraic equations per CSTR approximation. The solids are described by one energy balance (solids

temperature) and four mass balances (CaCO3, CaO, H2O, and inerts). The kiln wall is discretized into 10

nodes (eight inner nodes, and the internal and external walls of the kiln). The inner nodes are described by

an energy balance and the remaining nodes are solved using boundary conditions. The gas phase is modelled

using steady-state equations with one energy balance (gas temperature) and five mass balances (CH4, O2,

CO2, H2O and N2).

The movement of solids within the kiln is dictated by the kiln dimensions, inclination slope, and rotational

speed according to the following equation [3]:

wmDi

vs = (1)

0.308(q + 24.0)

A change in the volumetric flowrate of the mud entering the kiln changes the volume occupied by the solids,

rather than the solids residence time. To incorporate such an effect, a novel formulation is proposed to

model the solids holdup within a CSTR section of the lime kiln:

Fin if j = 0

Fj = 5vsVsj (2)

if j > 0

Lj

dVsj

= Fj-1 - Fj (3)

dt

where Fj is the mud flow-rate leaving section j and Vsj is the volume of section j. The mass and energy

balances are described by the following equations:

dCij 1 Cij dVsj rij

= (Fj-1Cij-1 - FjCij) - + (4)

dt Vsj Vsj dt Vsj

dTsj -H(Tsj) - Qrxn + Qconv + Qrad

=


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dt N s

dt Vsj ^ CijCpi(Tsj)

i=1

N s

H(Tsj) = Fj ^ Cij #TTssjj-1Cpi(x)dx (5)

i=1

Qrxn = DHcal(Tsj)rjCaCO3 + DHvap(Tsj)rjH2O

Qconv = f2A2(Tgj - Tsj) + f3A3(Twj _ Tsj)

Qrad = sA41(Tgj)4 - (Tsj)42 + sA61(Twj)4 - (Tsj)42

Several authors have modelled the rate of calcination as an elementary first-order reaction for CaCO3 [2, 9],and the rate of water evaporation as a zero-order reaction at high moisture content and first-order at low

moisture. These are described in the following equations:

Ecal

rjCaCO3 = -kcal exp3- 4VsjCjCaCO3

RTsj

rjCaO = -(MWCaO/MWCaCO3)rjCaCO3 (6)

rjCO2 = -(MWCO2/MWCaCO3)rjCaCO3

-kevapexp(-Eevap/RTsj)Vsj if xH2O * 0.1

rjH2O = 5

-kevapexp(-Eevap/RTsj)VsjXjH2O if xH2O * 0.1

The wall energy balance was solved by discretizing the radial coordinate into 10 nodes and applying the two

boundary conditions inside and outside of the kiln wall.

dTjw kw d2Tjw 1 dTjw

= ( + )

dt rwCpw dr2 r dr

dTjw

0 = -pDiLj + Q1 + Q2, r = Di/2 B.C.1

dr

Q1 = f1A1(Tjw - Tjg) + f3A3(Tjw - Tjs) (7)


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Q2 = sA51(Tjw)4 - (Tjg)42 + sA61(Tjw)4 - (Tjs)42

dTjw

0 = -k - f4(Tjw - Tamb), r = Do/2 B.C.2

dr

The gas mass and energy balances are described by the following equations:

^Fij = ^Fij+1 + Gij + ^rij

GjCO2 = rjCO2 (8)

GjH2O = rjH2O

0 = -^H(Tgj) + ^Qrxn + ^Qconv + ^Qrad

N g Tgj N s Tgj

^H(Tgj) = ^ ^Fij+1 #Cpi(x)*x + ^ Gij #Cpi(x)*x

i=1 Tjg+1 i=1 Tjs+1

^Qrxn = DHcomb(Tsj)rjCH4 (9)

^Qconv = -f2A2(Tgj - Tsj) + -f1A1(Tgj - Tjw)

^Qrad = -sA41(Tgj)4 - (Tsj)42 - sA51(Tgj)4 - (Tjw)42

Under typical operating conditions, combustion is usually complete within the first 4 to 10 m. Since the heat

transfer dominates the calcination reaction, it is reasonable to assume a combustion profile to simplify

dynamics and avoid a coupled set of equations for the mass balances. Therefore the gas rates of reaction can

be expressed as:

^rjCH4 = -Fj C+H14/a

^rjO2 = 2.0(MWO2/MWCH4)^rjCH4 (10)

^rjCO2 = -(MWCO2/MWCH4)^rjCH4

^rjH2O = -2.0(MWH2O/MWCH4)^rjCH4

where a is an adjustable parameter that determines the speed of the combustion reaction and can be used to

change the length of the combustion zone (flame).

RESULTS

The steady-state profiles for the temperatures, and solid concentrations are shown in Figs. 3 and 4. The

steady-state temperature profiles are in good agreement with data published in literature [3, 8]. The solids

concentration profiles also exhibit the expected trends. The water is completely evaporated within the firstthird of the kiln and the calcination reaction begins two-thirds of the distance along the kiln.

Figures 5 and 6 show the dynamic response of the front end temperature (FET), and the back end

temperature (BET) to step changes in the manipulated variables (fuel flow-rate, induced air flow rate). The


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responses have the same qualitative nature as that presented by Osmond et al. [10] and Charos et al. [11].

Both outputs have a positive gain with respect to the fuel flow-rate and opposite gains for changes in the air

flow-rate. The FET response is very sensitive to changes in the fuel flow-rate and it is highly asymmetric for

positive vs. negative step changes. This is due to the fact that at low CaCO3 concentrations, the temperature

of the lime mud is very sensitive to positive changes in the fuel flow due to the first-order dependency of the

rate of calcination. The BET has a very fast initial response followed by a slow ramp and it is symmetric

with respect to the fuel flow. The FET response to a step change in the air flow-rate behaves like a second-

order system with a first-order lead term. Again, the BET shows a very fast initial response followed by a

very slow ramp

SUMMARY

The lime kiln is a complex unit operation and a challenging modeling problem. There are three phases

present: gas, wall and mud. Since the gas balances are not equimolar (because of the contribution of the mud

phase), the volumetric flow of the gas changes through the kiln. In addition, heat transfer occurs by

conduction, convection and radiation, which complicate the energy balances and can create stiffness in the

numerical solution.

The gas phase was modelled using steady-state mass and energy balances. This was required forcomputational efficiency in the dynamic simulation. When dynamic balances for the gas phase are

introduced, the problem becomes stiff and the convergence of the model takes several hours. The current

model can simulate 50 hr in only 4 min with an AMD K6-2 500-MHz PC with 128-MB SDRAM. A new

feature introduced in this model is the handling of the volume occupied by the mud within a CSTR section,

which can vary through the kiln and it is dependent on the incoming flow-rate of lime mud to the kiln and

the rotational speed. Dynamic simulations show that the model reproduces the steady-state results for

temperatures as well as compositions. The model also exhibits the expected qualitative dynamic behaviour

when compared to other results presented in the literature.

Our next step is to include the effects of dams, dust losses and variable heat transfer coefficients as part of

the mathematical model. The mathematical model described above will be used as a "virtual kiln" forclosed-loop simulations using decentralized PI control and model predictive control (MPC). This lime kiln

model will be part of a general model for the simulation and control of pulp mill processes to be used for

testing different plant-wide control strategies including heuristics approaches for decentralized control as

well as model-based methods.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the support of the National Science Foundation under Grant CTS

9729782, the University of Delaware Process Control and Monitoring Consortium, and the University of

Delaware Presidential Fellowship.

LITERATURE

1. LYONS, J.W., H.S. MIN, P.E. PARISOT, and J.F. PAUL. Experimentation With a Wet-Process Rotary

Cement Kiln Via the Analog Computer. Ind. Eng. Chem., Process Des. Dev. 1:29-33 (1962).

2. KOIVO, A., and J. CHASE. Determination of a Mathematical Model for a Lime Kiln. In IFAC/IFIP

Conference on Digital Computer Applications to Process Control, Part 2, Paper XII-3, Helsinki (1971).

3. KOCHAR, P.S. Simulation of Rotary Lime Kilns: Impact of Alternative Fuels and Dams. PhD Thesis.

University of Idaho, Department of Chemical Engineering, Moscow (1983).

4. SPANG, H. A Dynamic Model of a Cement Kiln. Automatica 8:309-323 (1972).


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5. SMITH, D.B. Dynamic Mathematical Model of a Rotary Lime Kiln. PhD Thesis. University of Idaho,

Chemical Engineering, Moscow, Idaho (1991).

6. LE BLANC, G.D., D.E. SEBORG, K.L. HOLMAN, N.G. SWEERUS. Physically-Based Dynamic Model

of a Lime Kiln. In AIChE Annual Meeting, Los Angeles, CA, (1997).

7. LAKSHMANAN, C.C., and O.E. POTTER. Cinematic Modelling of Dynamics of Countercurrent

Systems. Comp. Chem. Eng. 14(9):945-956 (1990).

8. MUMFORD,W., D. SMITH, L. EDWARDS, and A. VEGEGA. Reducing Lime Kilns Fuel Usage:

Retrofit Results From Actual Mill Installations. Tappi J. 72(1):28 (1989).

9. URONEN, P., and K. LEIVISKA. Modelling and Control of a Rotary Lime Kiln in the Sulphate Pulp

Mill, PLAIC report 116. Purdue Laboratory for Applied Industrial Control, Schools of Engineering, Purdue

University, West Lafayette, IN (1979).

10. OSMOND, D.R., F.J.C. TESIER, and M. SAVOIE. Control of an Industrial Lime Kiln Operating Close

to Maximum Capacity. Tappi J. 77(2):187-194 (1994).

11. CHAROS, G.N., and Y. ARKUN. Model Predictive Control of an Industrial Lime Kiln. Tappi J.

74(2):203-211 (1991).

Abstract: A dynamic mathematical model was developed for the simulation and control of rotary lime kilns.

The lime kiln is modelled using 66 continuous stirred tank reactors (CSTRs) in series with 13 ordinary

differential equations (ODEs) and eight algebraic equations per CSTR approximation. Open-loop

simulations show that the steady-state and dynamic behaviour of the model are in agreement with the results

presented in the literature. The model is suitable for simulation and control studies as well as for

optimization and design.

Reference: CASTRO, J.J., DOYLE III, F.J., KENDI, T. Mathematical model for the simulation and controlof rotary lime kilns. Pulp Paper Can. 102(7): T210-213 (July 2001). Paper presented at the 2000 Control

Systems Conference of the Pulp and Paper Technical Association of Canada in Victoria, BC, on May 2 to 4,

2000. Not to be reproduced without permission of PAPTAC. Manuscript received February 22, 2000.

Revised manuscript approved for publication by the Review Panel September 29, 2000.

Keywords: MATHEMATICAL MODELS, SIMULATION, PROCESS CONTROL, LIME KILNS,

ROTARY MACHINES, OPTIMIZATION, MACHINE DESIGN.

Rsum: Un modle mathmatique dynamique a t dvelopp pour la simulation et le contrle des fours

chaux rotatifs. Le four chaux est modlis l'aide de 66 racteurs agits en srie avec 13 quations

diffrentielles ordinaires et 8 quations algbriques par approximation de racteur agit. Des simulations deboucles ouvertes indiquent que le comportement dynamique et en rgime continu du modle est en accord

avec les rsultats prsents dans la documentation. Le modle convient des tudes de contrle et de

simulation ainsi qu' l'optimisation et la conception.

Table I. Nomenclature

Symbol Description

A1-A3 areas of heat exchange (convection), m2

A4-A6 areas of heat exchange (radiation), m2

C solid concentration, kg/m3

pCp heat capacity, J/(kg K)

Di kiln internal diameter, m

Do kiln external diameter, m


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DHcal heat of calcination, J/(kg CaCO3)

DHcomb heat of combustion, J/(kg CH4)

DHvap heat of evaporation, J/(kg H2O)

Ecal activation energy of calcination, J/(mol K)

Evap activation energy of evaporation, J/(mol K)

F solids volumetric flow-rate, m3/hr

^F gas mass flow-rate, kg/hrf1-f4 heat transfer coefficients, J/(hr m2)

G gas mass flow-rate (from solids), kg/hr

K conduction constant, J/(m hr K)

kcal calcination rate constant, kg CaCO3/(hr m3)

kvap calcination rate constant, kg H2O/(hr m3)

L length of a CSTR section, m

MW molecular weight, kg/mol

M slope of kiln, degrees

Ns number of solid componentsNg number of gas components

R ideal gas constant, J/(mol K)

r solid phase reaction rate, kg/hr

^r gas reaction rate, kg/hr

T temperature, K

Tamb external temperature (outside the kiln), K

Vs solids volume of CSTR section, m3

vs solids velocity, m/hr

s Plank's constant, J/(hr m2 K4)

q solids angle of repose, degrees

w kiln rotational speed, rpm

r density, kg/m3

i subscript, denotes species i

g subscript, denotes gas phase

s subscript, denotes solid phase (lime mud)

w subscript, denotes the kiln wall

j superscript, denotes CSTR section j

lime modelling to control lime rotary kilns

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