A dynamic model of a cement kiln

Automatica, Vol. 8, pp. 309-323. Pergamon Press, 1972. Printed in Great Britain.

A Dynamic Model of a Cement Kiln* Un module dynamique d'un four h ciment Ein dynamisches Modell eines Zementofens ~[HHaMHqecKa~ MoRe.rib ueMeHTHO~ neqn H. A. SPANG, I I I t

A cement kiln is a highly non-linear distributedprocess. A partial-differential equation model incorporating an improved model of the flame gives a dynamic response which appears to match the qualitative behavior of an actual kiln. Sununary--A cement kiln is a counter-flow type of process in which a complex series of exothermic and endothermic reactions take place. To understand its behavior and to provide a means of improving computer control, a dynamic partial differential model of the kiln has been developed. In this paper, the basic reactions and assumptions of the model will be discussed. A model of a coal]oil flame which takes into account the surrounding temperatures is developed. The dynamic behavior of the kiln as exhibited by this model is then indicated. In particular, it is shown that the interaction between the two exothermic reactions causes the kiln to be basically unstable with respect to regulating the position of the burning zone.

studied by GILBERT [3] and LYONS et al. [4]. A transient solution is indicated by LYONS et al. [5], but a limited amount of response data is presented. In this paper, a dynamic model of a kiln will be developed. In order to reduce the computational problems, the equations are averaged with respect to the radius of the kiln. The resulting equations are similar to those of LYONS et al. [5] and the author has used many of their values for the coefficients of the model. An important addition is a dynamic model of the flame. Instead of Lyon's model of the flame which assumes a rate of reaction dependent only on flame length and not the surrounding temperature, a more detailed model is developed taking into account the effect of temperature on the rate of reaction. The interaction between the flame and the solid causes an unstable behavior with respect to burning zone position which is not evident with non-interactive flame models. The response of the model to gas velocity and kiln speed is examined. It is further shown that, like the actual process, cement can be made for a period of time after the flame has been turned off. In general, the behavior of the model appears reasonable, especially in the burning zone/flame dynamics. However, due to a lack of adequate sensors, quantitative confirmation has not been possible.

NOTATION In order to simplify the writing, we use the following standard symbols for the cement components: C=CaO, A=A12Oa, S=SiO2, F=Fe203, = CaCO a and ~b= CO2. Thus 2 C a O . SiO2 = C2S, 4 C a O . A1203 • F e 2 O a = C 4 A F , etc. A list of all symbols is given in Appendix B at the end of the paper. All components are normalized with respect to the pounds of CaO in the burning zone. We use, in the material balance equations, the following abbreviations: A = # A 1 2 0 3 / # CaO, F = # Fe203/ #CaO, S= #SiO2/#CaO, C= #CaO/#CaO, 0~= # CaS / 7~CaO, fl = ~ C2S / ~ CaO, ~ = # CaA / # CaO, 6 = # C 4 A F / # CaO, 09= # n20 / ~ CaO, = # C O 2 / # CaO and ~k= # C a C O 3 / # CaO. INTRODUCTION A CEMENT kiln is a distributed parameter system which has a highly complex dynamic behavior due to the chemical reactions. The chemistry of making cement has been studied in some detail [1, 2]. The steady state temperature behavior has also been

QUALITATIVE DESCRIPTION OF A CEMENT KILN A highly simplified diagram of a cement kiln is shown in Fig. 1. It basically consists of a drum approximately 400-600 ft long and 10-20 ft in diameter. The length of the kiln mainly depends on how long it takes to heat up the raw materials to the "clinkering" temperature, which is approximately 2500°F. In some plants, a cyclone preheater is used to raise the initial temperature of the solids in which case the length of the kiln is greatly shortened.

* Received 19 April 1971; revised 29 November 1971. The original version of this paper was accepted for presentation at the IFAC Symposium on The Control of Distributed Parameter Systems which was held in Banff, Canada during June 1971. It was recommended for publication in revised form by Associate Editor H. Kwakernaak. t GE Research and Development Center, Schenectady, N.Y. 12309. 309

H . A . SPANG,III

310

SOLIDS

Z:O

Z=

FIG. 1. Simplified geometry of a cement kiln.

The raw materials fed into the kiln contain calcium carbonate ( C a C O 3 ) , silica (SiO2), shale (A1203), and iron ore (Fe203). These are ground to a very fine powder and mixed according to the type of cement being made. Upon heating by the hot gases, various reactions occur, as shown in Table 1. A typical temperature curve for the kiln is shown in Fig. 2. There are three important zones TABLE 1 Temp. (°F) 212° 1620° and above 2200° 2300° above 2300°

Reaction

Heat change

Evaporation of free water Endothermic Evolution of carbon dioxide Endothermic from calcium carbonate CaCO3-~ C.aO+ CO2 Start of combination of lime Exothermic and silica 2CaO+SiO2~C2S Liquid formation Endothermic 4CaO +A1203 +Fe2Oa--* C4AF 3CaO +A1203-~C3A Completion of cement compounds 2CaO +SiO2-~C2S Exothermic C2S +CaO-oC3S Endothermic

which determine the behavior of the kiln: the preheat zone, the calcining zone, and the burning zone. The position of the boundaries of the zones depends on the temperature and chemical reactions taking place in the solid. Hence, they shift in time and length depending on the amount of thermal energy obtained from the gas and wall. In the preheat zone, the solid material is heated to the point where the calcining reactions can begin. The initial rise in temperature is retarded by evaporative cooling as water is removed. At 1620°F (2000°R) the highly endothermic calcining zone defined by the reaction of CaCO3 converting into CaO begins. This is entirely a heat limited reaction with the solid temperature increasing very slowly. The speed of the reaction is entirely determined by the amount of heat that can be supplied by the gas and wall. The burning zone occurs when the calcining is completed. The solid temperature rises about 1500°R due to the exothermic formation of C2S. Three slightly endothermic reactions also take place. The first two are the formation of a molten liquid consisting of C4AF and C3A. Thi~ liquid causes the mix to cohere into small balls known as

3000

_9 o

~20~

10(~

16o B

260

DISTANCE ALONGLENGTHOF KiLN (f))

FIG. 2. A typical temperature profile.

38o

A dynamic model of a cement kiln

311

"clinkers". Although the required reactions could not occur without this liquid, its effect on the dynamics of the kiln is felt to be second order and was only included in the model implicitly by using lower values of activation energy in the C2 S, C3 S reactions. The third endothermic reaction occurs when the C2S combines with the remaining CaO to form CaS. This reaction has minimal effect on the temperature profile. The position in the kiln of the burning zone appears to depend almost entirely on the length of the preheat and calcining region. In all of the simulation runs, the calcining reaction is completed before the C2S reaction beings.

in Appendix A. The thermodynamic equations for the temperature of the solid, gas and kiln wall have been simplified by averaging out any variation in the angular and radial directions. Thus, these temperatures will depend only on the position along the length of the kiln and time. The resulting temperatures will be fairly representative of the actual temperatures in the kiln, especially for the solid and the gas since these are well mixed. It is a somewhat cruder approximation for the wall temperature due to the inherent thermal mass. The equation for heat conduction for any cylinder of cross-section, F, is

SUMMARY OF ASSUMPTIONS

O2T, ~2T' kVZT'=k[ l f~l'rOT~' +-~r Or\ ] -~1 ~02 +-~z2 1

In developing the model, every effort is made to make it as simple as possible while maintaining the essential dynamics that have been observed qualitatively. Some of these assumptions could be relaxed without increasing the computational complexity. However, until more detailed and accurate measurements are made, one has little justification for changing the model. The assumptions which are made are as follows:

=

Cv P -~T' _ -~- + ~

where k is the thermal conductivity of the cylinder, C~ is the specific heat, p is the density, and Q is the net amount of heat generated within and/or moving into a unit volume other than by conduction. We define the average temperature per unit area as

(1) The inside and outside diameters of the kiln are constant.

T=lffrdrdOrT'

(2) The solids transported in the gas are not explicitly included in the model. (3) The clinking rate expressions do not take into account the effect of trace compound concentrations. (4) Reaction rates are determined by Arrhenius' law. (5) Specific heats, latent heats, and heats of reactions are independent of temperature and position. (6) Coefficients of convection and emissivity are independent of temperature and position. (7) Conduction in the solid and gas is negligible. The conduction in the axial direction of the wall is also considered negligible. (8) The velocity and mass of the solid and gas is constant throughout the length of the kiln. (9) The heat transfer due to the release of water vapor and CO2 into the gas can be treated as convective heat transfer. THE MODEL In this section, due to space limitations, only the general form and assumptions will be indicated for the thermodynamic and material balance equations used in the model. The specific equations are given

O)

(2)

where A is the cross-sectional area, A =ffr

rdrdO.

Multiplying equation (1) by r, integrating, and using the divergence theorem, we obtain t~2T

~ 8T

t3T

kA.-£2-f_2 + k I ~ @ = CvpA-- + AQ Oz Jr, On Ot

(3)

where the second term on the left-hand side is the derivative of the temperature with respect to the outward normal integrated along the boundary of the cross section, 1-"t. This term represents the heat transfer across the boundary due to convection and radiation whose effects will be assumed to be additive. It will be assumed that the heat transfer by convection will obey Newton's law of cooling which states that the heat transfer is proportional to the linear diffelence in temperature. We shall further assume that radiation is governed by the fourth power law of black-body radiation. Under these assumptions, the heat transferred across the boundary for the solid, for example, can be written in the quasi-linear form: fl2(Ta- T~) + fl3(Tw- T~)

(4)

H . A . SPANG,III

312 where /,'x f12 = 2 r l sln(2)[f2 + 1"73 × 10- 9eges(7"2 2

+ Ts )(T o+ 7 w)] f13 = rx(21r - P)[.[3 + 1.73 x 10- 9hewe~(T2,

+ r~)(vw + rs)]. h = i + 2ho sin(p/2) 2z~-p and p is defined by Fig. 1. The other symbols are given in Appendix B. The terms multiplying the square brackets represent the length of the boundary between the solid and the gas and between the solid and the wall. The first term, f, is Newton's proportionality constant and the remaining terms account for the heat transfer due to radiation. The solid receives heat by radiation not only from the area of contact but also some through the gas from the rest of the inner wail. The parameter ho is the fraction of radiation not absorbed by the gas and h is the effective increase in the radiational boundary. The heat transfer into the gas and wall are handled in a similar manner. The heat loss between the wall and the outside air must be handled slightly differently since the outside wall temperature will be at an intermediate value between the inside wall and the ambient air temperature. This outside temperature is determined by the conduction through the wall, and the radiation and convection heat losses. Using the approximation, given in Ref. [6], that the convection loss is proportional to the 5/4 power of the difference between the outside wall temperature and the ambient, the total heat loss per unit area is: q=

k (Tw- To)= 1.73 × lO-%w,[T~-T~]

F2 --/"1

+0"27(To-T,) s/4 •

k~"T=O2oz

and

~T Q=Cppv-~z- q

where q is the heat generated by the chemical reactions in the solid and by the flame in the gas. Since the gas will reach steady state very rapidly, as compared with the solid and wall, we shall assume that

OTg/~t = 0 . Due to the thermal constants of the brick of the kiln wall, very little conduction occurs in the z direction which again means that k~2Z____~w= 0.

~z z

(5)

where ew, is emissivity of the outside wall. The first term is the conduction through the wall which must be equal to the sum of the radiation and convection losses. Solving for T o for various inside wall and ambient temperatures, it can be shown that a reasonable approximation is: 2~r2]4(T,- Tw)

solid should be decreased to account for this loss. However, there are indications that the total mass of the solid may actually increase especially in the burning zone. The reason for this is due to changes in the physical properties which cause the transit time across a unit volume to vary quite widely [7] depending on temperature and chemical composition. The modeling of this effect is quite complex and could not be included in this study. Therefore, the greatly simplifying assumption is made that the material flow in both the solid and the gas is constant independent of material balance considerations. The heat transfer between the solid and the gas due to this material release can be shown to be proportional to the difference in the solid and gas temperatures. It can therefore be modeled by the convection term of the previous heat transfer equations. The gas and solid both are well mixed with the movement of heat into a region primarily caused by the flow of the solid and gas and only slightly by conduction. This is equivalent to:

(6)

where 1~. varies only from 0.6 to 0.75 which is a 25 per cent increase for a 5 fold increase in Tw from 600 ° to 3500°R. During the heating of the solid, water vapour and CO2 is released into the gas stream. From strictly material balance considerations, the mass of the

Applying the above assumptions to equation (1) yields the three thermodynamic equations shown in Appendix A. It should be observed that for the coordinates shown in Fig. 1, the velocity of the solid, v~, is negative. The solid to solid and liquid to solid reactions which take place in the kiln are very difficult to describe analytically because they depend upon many factors such as the amount of surface provided by the grains, the conditions of contact, the phase of the particles, and the temperature. A reasonable approximation to the rate constant for the reactions can be expressed by the normal gas to gas reaction equation: ki = Ai exp( - Ei/R Ts)

(7)

A dynamic model of a cement kiln where A i is a constant dependent on the texture of the material, but relatively independent of temperature; R is the gas constant; Ts is the temperature of the solid, Ei is the activation energy which represents the energy necessary to bring a particle to the state of reactivity. We shall assume that it is independent of temperature. The change with time in the concentration of any compound in a given cross-section must be equal to the amount of that compound moving into the region plus the amount generated in that region. Thus, the material balance equations are of the form:

313

The heat generated by the chemical reactions is proportional to the rate of reaction. Using the convention that the coefficient is positive for endothermic reactions, the heat generated in the solids is: P~

qc=(l + Ai+ Fi+ S~)

[_AHckl~-AH,oR,o

- AH#k#S(C) 2 - AH~k~Cp].

(10)

The constant multiplying the brackets is the weight of CaO for a total solid mass of A,ps. The heat generated by the flame is

q: = ( - AH:)AgM~R dt

-

vs

oz

+R(K)

(8)

where K is' the concentration of a given compound and R(K) is the amount generated. The detailed equations are given in Appendix A. Since CO2 is released into the gas, its concentration depends on velocity of the gas rather than the velocity of the solid. LYONS et al. [5] used a simplified model of the flame which assumed that the reaction rate of the flame is inversely proportional to the distance from the end of the kiln. While this greatly simplifies the model, it effects the dynamic response of the model by ignoring the interaction of the rate of reaction of the flame with the surrounding solid and gas temperatures. A slightly more complex model which treats the flame as an exothermic chemical reaction is developed in Appendix C. In this model, the flame is treated as an exothermic chemical reaction which transfers its heat directly to the gas. This is, of course, an over-simplification which is reasonable for coal or oil flames. In that case the flame is luminous over its entire length due to the burning of small particles or droplets of the fuel. The rate of reaction is determined by the rate of diffusion of the oxygen to the particles. Assuming a first order reaction equation, we show in Appendix C that the reaction rate is

1 V~bMC,(M,P) 2-] -I - IkedoCe

pgLP#~Io~(RTg)2J

(9)

where do is the diffusion coefficient. This model takes into account that the reaction rate will be slower at high temperatures since the density of the oxygen is inversely proportional to temperature. A more detailed discussion of the behavior of the flame can be found in Ref. [7]. Again, it is reasonable to assume that the flame is always in steadystate.

--G y ( - AH :)[ ~MC:( MaP)~lk~IoC~ . (11) pgV, LPrMo,(RTg)~J Computational approach It can be seen that in order to solve the equations in Appendix A one must solve a two point boundary value problem since the initial temperature and chemical composition of the solid are known at one end of the kiln while the temperature of the gas and the amount of fuel are known at the other. The approach taken was to linearize the energy balance equations about the previous values of temperature. The two point boundary value problem can then be uncoupled by using the Riccati transformation technique which sets

Tg(t, z)=v(t, y)+k(t, z)T~(t, Z ~1 • By substituting the above equation into the linearized material balance equations for v(t, z) and k(t, z) can be obtained which have boundary conditions at only one end of the kiln. The solution can then be iterated at each time increment to obtain the true solution to the non-linear problem. During each iteration the true non-linear material balance equations are used. The approach was implemented on the computer using a first order finite difference scheme which can be shown to be consistent and stable in the sense of RICHTMYER [9] for all integration step sizes in z and t. For accuracy, a variable step size was used in the z direction while a step size of 0.07 hr was used tbr the time increment. On a G E 635 or IBM 360/50 computer, each time increment takes approximately 0.18 sec with the cases shown in Figs. 6-8 taking approximately 36 sec each. RESPONSE OF THE MODEL The first test of the model was made without the flame model. With the input gas temperature set at 3000°R and with twice the velocity of the gas, sufficient Btu was supplied by the gas to obtain a

314

H . A . SPANG, III

stable steady-state profile. When the input gas temperature was increased to 3200 °, the burning zone moved down the kiln at a rate of about 7 ft/hr, as shown in Fig. 3. This is considerably slower than observed in actual kilns and is caused by the small temperature difference between the gas and solid under these input conditions. This test illustrates that the gas profile can have a substantial effect on the dynamic behavior of the kiln. 4000 350O

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120 160 200 240 6EE{3~ 36O 400 LENGTH(FEET)

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practice, a similar approach is used to start a kiln, but the depth and velocity are initially low and slowly increased. This tends to minimize the unproductive raw material waste and the total amount of fuel consumption. It was found necessary to reduce the radiation coefficients, ~, eg and e , from those used without the flame. The original values allowed a large heat transfer between the burning zone and the flame. The inherent instability due to the positive feedback caused extremely high temperatures in the burning zone with little heat reaching the preheat zone. Reducing the radiation coefficient reduces this positive feedback without markedly effecting the heat transfer characteristics in the preheat zone due to its lower temperatures. To improve the heat transfer from the gas to the solid, iron chains are often hung in the preheat zone. In order to match more realistically such a kiln, the effect of chains in the preheat zone was simulated by multiplying by a constant the heat transfer coefficient fl~ during the first 50 ft of the kiln. The effect of various multiplying constants on the solid temperature is shown in Fig. 4. These curves are temperature profiles at T = 1.40 hr after the effects of the chains were simulated starting with the same initial profile. 4000

....

:

-

;

-

T

~ - - -

F --- ;

---.

i. . . .

3500" LENGTH(FEET)

Fzo. 3. Transient effect of a 200° increase in initial gas temperature. When the equations for the flame model were included into the cement kiln model serious numerical difficulties were encountered in establishing an initial temperature profile. These difficulties were caused by the two exothermic reactions of the burning flame and of the CGS reaction in burning zone of the solid. These reactions cause an essentially positive feedback situation which allow high temperatures to build up in the calcining and burning zones of the kiln. In fact, no truly steady-state solution was obtained though the final profile, given in Fig. 5, is close. The true implications of this with respect to an actual kiln is unknown since it is impossible to get accurate enough measurements to determine its true dynamic state. However, it would seem to indicate that possibly the actual kiln is unstable with respect to holding a constant position for the burning zone. The procedure used to initialize the complete model was to start with the solid, gas, and wall at room temperature and to heat it to the desired operating condition. During this process, as a matter of convenience, the bed depth and the velocity of solid were held constant. In actual

:3000 -- 25002OOO

GF = 5500 #/HR T = 1"40 HR

I0 XCHAIN~ =.-.,..7.-/ :

/.,~•../ ~ ' i 5

I000 - / / j ~ / / ~ 500 f--

x CHAIN

NO CHAIN

] V

0

40 80 120 160 200 240 280 320 360 400 LENGTH (FEET) FIG. 4. Effect of chains in first 50 ft on the temperature of the solid• The higher the multiplying factor the faster the initial heating of the solid. However, the rate of increase is less after the chain section. This results in the position of the calcining and burning zones remaining fixed. The higher multiplying factor causes the gas to give up more heat in the chain section but less in the remainder of the preheat zone because there is less temperature differential between the gas and the solid. One would expect that, since the initial calcining temperature is arrived at earlier, the position of the burning zone would start to be affected if these runs had been continued for a longer time.

A dynamic model of a cement kiln The final profile obtained with the flame is shown in Fig. 5. Although not completely steady-state, especially the wall temperature, this profile is used as the starting point for the transient studies. Comparing with the no-flame case, in Fig. 3, the preheat zone is longer due to the lower gas temperatures in that region. However, the hotter burning zone temperatures allow a much higher gas temperature in the calcining zone. This results in a much shorter calcining zone. 4500

1

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l

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4000.

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1 I I I I I J/i '0 120 160 200 240 280 320 360 400 LENGTH (FEET}

FIG. 5. Final (nonsteady-state) profile used as start for rest of studies: (a) Temperature.

31 5

We will now examine the step function response to changes in gas velocity, kiln speed, and the turning off of the flame. Each of these runs were started with the profile given in Fig. 5. The step change in all cases is large in order to show the effect more easily. A smaller change would to a lesser degree provide the same effect. In Fig. 6, the gas velocity has been reduced from 40,000 to 30,000 ft/hr. This instantaneously shortens the flame which concentrates the burning to a shorter area and creates a higher flame temperature. The flow of heat out of any cross section is also reduced. The net effect is to increase greatly the temperature of the gas prior to and over the peak of the burning zone, but to decrease the gas temperature within the calcining and preheat zones. Since the effective thermal time constant of the solid and wall at the burning zone temperature is short, the temperature of the solid builds up rapidly in the flame region. The increase of the peak solid temperature in the burning zone further increases the peak gas temperature causing a continual increase in the gas, wall and solid temperatures in the last 40 ft of the kiln. The flattening of the peak gas temperatures at T=0.91 and 1.47 hr is due to off-scale plotting and not the simulation.

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~,,~y.;..J ..,U:............ "

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LEMfTH (FEET/

Comparing the two cases it can be shown that the total input/output efficiency of the kiln with the flame is twice as good as that without the flame. The reason for this is the more efficient use of the thermal energy in the calcining and burning zones. This increased efficiency is obtained even though more Btu are used in the preheat and calcining regions since the wall temperatures are lower.

i,.o~ ~,,,~

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......

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LEM6TH ~FEET)

FIG. 6. Effect of a 25 per cent reduction in gas velocity: (a) Temperatures plotted as a function of distance, Even though the gas, wall and solid temperatures are increasing, the burning zone is moving toward the flame end of the kiln at a rate of 14 ft/hr. This apparent contradiction is caused by the reduced amount of heat being transported down the kiln.

316

H.A.

Since less thermal energy is available, both the preheat and calcining zones become longer moving the burning zone down the kiln. In actual practice, measurements of the solid and gas temperatures cannot be made as a function of length, but only at a few fixed points along the kiln. As illustrated in Fig. 6(b), this type of measurement can be very misleading. If one were looking at the solid temperatures at 380 or 400 ft, one might feel that the burning zone moves back toward the flame increasing the solid temperature and then forward with a resulting decrease. Actually, as Fig. 6(a) shows, the burning zone moves in only one direction. The change in temperature is due to the narrowing of the zone as more heat is applied. The reading at 350 ft gives the "true" picture of the zone movement, though it can only be interpreted properly because we know the state of the entire kiln. i

4500

:

450012-"---~G-...., ----

4000.

4000

II I

SPANG,

at the rate of 14 ft/hr due to less heating of the solid in a given length. The lengthening of the zone occurs slowly because of the higher time constants at that lower temperature. The calcining zone also is lengthened due to less energy transfer from the gas, causing the burning zone to move out at a rate of 28 ft/hr. 45OOi

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FIG. 7. Effect of a 26.5 per cent increase in the flow rate of the solid:

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(a) Temperatures plotted as a function of distance.

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.....

~L

! ,o §

._

o ~

-T-G - "

Z ~400 FT

05

4500

1~1

3500

207

TIME (HRI

FIG. 6. Effect of a 25 per cent reduction in gas velocity: Co) Temperatures plotted as a function of time.

Increasing the rotary kiln speed is modeled as an increase in the linear velocity of the solid from 150 to 190 ft/hr, and this has an entirely different effect on the temperature profiles as shown in Fig. 7. The increase in velocity means that the solid material resides in the kiln for a shorter period of time which causes less heating of the material. Any hot material that is initially there moves out of the kiln faster than before. The net effect is an immediate decrease in the burning zone peak temperature. The peak gas temperature is also reduced due to less thermal feedback from the solid. Though the temperature of the gas remains essentially constant throughout the preheat zone, the length of that zone increases

~

Z,IOO FT

45¢0

:

4500

:

] ! ]

4OOO-

4 0 0 0 L ~

I

3500 ~_

35OO

t

~

T

"

2. 3oco ~: 2500

7 TIUE

I

4000 •

2000

. . . . . T_G .... r- T*

1500-...,

i

Ii

dI

.,..TS ...........

~000

-.0"5

i 007 (;'57 ~

4000

Z,ESO rf

Z ,400 ET

3~00; ~

"!

. . . . Lug L . . . . .

I ~

~ TiME

I

TW

;

~7

TIME (HRI

;

i =

5OO

...... -%L__ 007 ff57 I'07 1.5: 2'OT

! 1"57 207

TIME (ItRi

~soo

moo

~1'0~

500

35OO~ 3000~

Z ,~50 FT

I000-

500

:

4000

~ ~500v

I000[

~0! J _ 01 O',37 057 1-07 1"57 2"07 TIME {HRI 4500-

Z =500 FT

1500 L

~57 ~ {Hill

--

I~ 2soo : ~ L~TW

....... ..TS

•

~ zoool ~

!

05

IOOO!

!

90, ~T

i

= ] t

i

0"5

i 057 107

F57 2,07

TIME (HRt

FIG. 7. Effect of a 26.5 per cent increase in the flow rate (b) Temperatures plotted as a function of time.

A dynamic model of a cement kiln For this increase in kiln speed, the measurements of the temperature at fixed points give an accurate picture of the behavior of the kiln as shown in Fig. 7(b). However, there is an immediate slight increase in the exit solid temperature as more of the initially hot solids move out of the kiln. It is well known that a kiln will make cement long after the flame has been turned off due to the storage of heat in the walls. This effect is shown in Fig. 8 with the velocity of gas reduced to 10,000 ft/hr. The temperature of the gas in the burning zone reduces immediately to the wall temperature. Since the gas is still being heated in the burning zone, it continues to supply heat to the preheat and calcining zone. This internal feedback greatly extends the time cement is made. From the plot of the chemical composition at the flame end of the kiln, shown in Fig. 8, we see that good cement can be made for at least ½ hr, if not longer. Of course, the temperature of the initial profile will influence this time. The time is also inversely proportional to the velocity of gas and the velocity of solid since these determine how fast the heat moves out of the kiln.

3500

3~F

•

T'C4)T HR

"~

i;f

I~¢ /

45O('[

:

317 4500

T. . . . .

7~--

-- [

--~,--T-

4 o o 0 i-

3500 -

~

Z=lO0 FT

3500 -

Z * 500 FT

~

3500

TW

3000 2500[

... 2000 . . . .

~oooI.

~

TG

TS

Q

i

....... Ts

-~

"........ T~....

I

500~

OOT 057 1'07 1"57 2'OT TME (HR

TIME (HRI

TIME (HRI

] 2000~

"

TS

=

-

"

,ooo_

~!

1

~or-

'

TiME (HRI

TIME (HRI

FIG. 8. Transient response after turning off the flame: (b) Temperatures plotted as a function of time.

1'8

i

~

I

1"6

ic~r . /

1'4

500~

k~P,t

LENGTH(FEETI

(~.rrl

1"2 T.O'~ HR

I'0

Z =400 FT

0"8

C3S

500 ...... ,'o

( ~

Lr.~'mFrzn

. C°C03

0"6

450¢

40GOF-

4C¢~ , 3500

T,O'gl HR

~OOh"

T'1"47 HR

0'4

,..\~

0

0"2 I-- C3A /\I ',, 0

@0¢

Lt k ~ ' 1 1 / 1 ~

o, ,~ ~, ~o ,~0~o(FEET] ',,o~,o'~o",oo

0"07

0"57

1'07

TIME (HR)

1'57

2"07

FIG. 8. Transient response after turning off the flame: FIG. 8. Transient response after turning off the flame: (a) Temperatures plotted as a function of distance.

(c) Chemical composition plotted as a function of time.

I

318

H . A . SPANG, I11 CONCLUSIONS

The model given in this paper appears to m a t c h qualitatively the behavior o f actual kilns fairly well. This is especially true at the burning zone end o f the kiln with the all important flame-solid interactions. There are, however, two areas which require refinement. The first is indicated by the peak temperatures in the burning zone which seems somewhat high. This would indicate too high emissivity coefficients in the burning zone requiring the use o f coefficients for each zone. Since there are large physical changes in the material in each zone, changes in emissivity are physically reasonable. The second refinement is needed because temperature o f the gas in the preheat zone o f our model is less sensitive to changes in the burning zone than has been observed. This is primarily caused by the wall dynamics in the preheat zone indicating a better approximation to the wall should be developed. As in all models, it is highly desirable to obtain a quantitative comparison with the actual kiln. This has not been possible primarily due to the lack o f adequate sensors. There are usually only two points o f gas temperature measurement and one highly questionable measurement o f the burning zone solid temperature. A specially instrumented kiln appears to be necessary.

APPENDIX A

Thermodynamic energy balance equations Gas: ~T

,

AaCmPaVo2-~ = ,[J1( "Fw- To) + fl2(T~ - To) + q f T o(L, t) = Tg i Solid :

AsCpspsVs~z s = f l 2 ( T g - 7'~)+ fl3( T w - Ts) C v , p ~ A ~ 2 + Asq , T~(O, T)= Ts~ Wall: ~?Tw

,

A wCpwpw-'~- = fi, (rg - T,,,) + fl a(T, - Tw)

+/~,(T. - T~). Material balance equations Water: &o

&o

fko,~o

R

~ot = - R , o - v , - ~ z

~' = / k , ,

co~0-1

o9>0"1

~o(o, 0 =~, CO 2 :

Acknowledgements---The author acknowledges the many helpful exchanges and discussions that he has had with John Friedly during the development of this model. The author is indebted to the people in the GE Industrial Sales Division and Process Computer Business Department for their help in understanding the physical behavior of a kiln.

at

Ag p~ Mc

O(L, t)=o

Oz

CaCO 3:

~(o, t)=~ REFERENCES

[1] R. H. BOOUE: The Chemistry of Portland Cement, 2nd ed. Reinhold, New York (1955). [2] F. M. LEA: The Chemistry of Cement and Concrete, rev. exl. Edward Arnold, London (1956). [3] W. GmaERT: Heat transmission in rotary kilns: Cement and Cement Manufacture, Part I, 5, 1932; Part II-VI, 6; 1933; Part VII-X, 7, 1934, Part XI, 8, 1935; Part XIIXIII, 9, 1936. [4] J. W. LYONS,H. S. MIN, P. E. PAmSOTand J. F. PAUL: Experimentation with a wet-process rotary cement kiln via analog computer. Monsanto Chemical, St. Louis (unpublished). [5] H. S. MrN, P. E. PAmsor, J. F. PAULand J. W. LYONS: Computer-simulation of a wet-process cement kiln operation. Preprint No. 202-LA61, Instrument Society of America, Fall Instrument-Automation Conf., Los Angeles, California, September 1961. [6] J. H. PERRY (Editor): Chemical Engineers' Handbook, 3rd ed. McGraw-Hill, New York (1956). 17] Join RtrlXS: Investigationof Material Transport in WetProcess Rotary Kilns by Radio Isotopes, pp. 120-133, Pit and Quarry, July, 1955. [8] H. A. SPANO, III: A Model of a Coal/Oil Flame for Rotary Kilns, to be published. [9] R. D. RlCnTMYIm: Difference Methods for Initial Value Problems. Interscience, New York, 1957.

C3S:

~(o, 0 = 0

~t -~k~( C)~- ~'Tz C2S:

Ofl- Mp,kp(C)2s_Mpk~(C)fl ~, t

2M ,

"

M ,.

v~Off -

-~z

fl(0, t) = 0 C3A:

d y _ M~,krCaA_ v,f7 ~t 3M c ~z

r(o, 0 = o

C4AF:

&5_ M a k a C * A F - v~~--~ at 4Me ~z

a(o, 0 = o

A dynamic model of a cement kiln

319

available fuel:

Fe20 3 : dF My 4 OF . -=-= _ k~C AF-vs=--6t 4M~ Oz

F(O, t)=F,

do = 1

kr (Do3/r 2) + kr

Diffusion constants:

A1203:

Do=aoTa 3/2

8___AA=_ M a k ~ C ' A F - Ma k , , C 3 A - v f A dt 4Mc 3M~ " Oz

Reaction rate coefficients:

X(O, t)=A i SiO 2 :

~S at =

k,=A, e x p ( - E i ' ~ \ RT,,] M s k#S(C)2 -ZM~

dS Vs~z

i=1, a, fl, y, 6, ~o

S(0, t)= S~ k r = 3 A r e x p ( - Ev ~ re \ RTJ

CaO: Coefficients of heat transfer:

0C t?t - k t ~ - k~Cf l - kpS( C) 2 - k.~C3A

- k~C*AF- vs~z

fl, = r l v [ f 1 + 1.73 x

10-9(1 -

ho)eoew(T2 + T 2)

(T o+ Tw)]

C(O, t) =0

f12 = 2 q sin/2)[f2 + 1.73 x 10-%oss(T 2

Fuel:

+ T~)(Tg + T,)] ~z

f13= r1(2~ - P)[f3 + 1"73 x 10- 9hewss(T2

pavoLPvMo~(RTa) J

+ T2)(T~, + rs)]

Ce(L, t) = CFi

fl,=2nf4r2

Heat of reaction:

%=

h = 1 d 2ho sin(p/2) 27t-p

Ps

. _ [-AHek~q-AH,oRo, (1 + Ai+ l"~+ S~)

Area coefficients:

- anpkaS(C) 2 - AB~k,C~]

/.2

Ao= 2 ( p - sin p) Heat from flame:

/.2

As = 2 (2re - p + sin p)

qj

A..)I. Po%

e Mo)qk.doC . A w = 2 n ( r 2 - r 2)

LPFMo,(RTo)~I APPENDIX B Nomenclature

Notation a n

A Aa

At Av As Aw AI A~ Ap Ar

Proportionality constant for D # Aluminium oxide (A12Oa) / # CaO Area of gas at given cross section Initial value of # A 1 2 O a / # C a O at input Pre-exponential factor--fuel Area of solid at given cross section Area of wall at given cross section Pre-exponential factor--CaCOa Pre-exponential factor--CaS Pre-exponential factor--C2S Pre-exponential factor--CaA

Value used

Units

0.0538 ft 2

0.0469 5-6 x 101°

1.64 x 1035 4.8 x 108 1.48 x 109 3.0 x 10 l°

# / # CaO 1/hr ft 2 ft 2 1/hr 1]hr 1/hr 1/hr

320

H.A. SPAN6, IlI Notation

Ao, C

%

c,w C2S CaA C4AF

do Do

Pre-exponential factor--C4AF Pre-exponential factor--water Calcium oxide (CaO) Normalize amount of available fuel Specific heat of gas Specific heat of solid Specific heat of wall 2CaO • SiO 2 3CaO • A12Oa 4CaO • A120 3 • Fe2Oa available oxygen Diffusion coefficient for oxygen Activation constant--fuel

Value used

Units

3.0 x 1012 7.08 x 107

l/hr 1/hr

0.28 0.26 0.26

Btu/# °F Btu/# °F Btu/# °F

2.45 x 104

Btu/lb mole Btu lb mole Btu lb mole Btu lb mole Btu lb mole Btu lb mole Btu lb mole Btu hr (ft)2°F Btu hr (ft)2°F Btu hr (ft)2°F Btu hr (ft)2°F

E1

Activation constant--CaCOa

3.46 x 105

E,

Activation constant--CaS

1.10x 105

Activation constant--CzS

8.3 x 10"

Activation constant--CaA

8.33 x 104

Activation constant--C4AF

7.95 x 104

E,o

Activation constant--H20

1.807 x 10 4

A

Coefficient of conduction--gas to wall

4

A

Coefficient of conduction--solid to gas

4

A

Coefficient of conduction--wall to solid

4

f,

Coefficient of conduction--wall to outside air

0.7

F Fi Gr ho

Iron oxide # Fe203/# CaO Initial value of # Fe2Oa/# CaO Amount of fuel per hour Fraction of radiation

0.0469 5300 0.0758

k

Thermal conductivity of wall

0.9

Btu hr (ft)°F

k~ kl

Reaction rate--fuel Reaction rate--CaCO2 Reaction rate--CaS Reaction rate--C2S Reaction rate--C3A Reaction rate--C4AF Reaction rate--water Total length of kiln Depending on the subscript molecular weight of the chemicals [see BOGOE(1) for values]

400

ft

E~

ks kr k ,~ L M

P

Angle subtended by surface of solid (Fig. 1)

qc

Heat generated by chemical reactions

2

#/#CaO #/hr

radians Btu ft3hr

A dynamic model of a cement kiln Value used

Notation Heat generated by fuel

Q

Heat generated or moving into a region Particle size of fuel Inside radius of kiln Outside radius of kiln Ratio of heat transfer in chain section Reaction rate of water

]'2 r3

S St

7.

rg 7g, 7~ 7~ 7w 1)g I)s z

8

10 -2 5 6.0 5

AH~ AH,

AH, AH~, 'go ~s ~w

P9 P~ Pw

cm ft ft

# Si02/# CaO 0.3398 561.7

Initial value of # SiO2/# CaO Temperature outside kiln Temperature of gas Input temperature of gas Temperature of solid Input temperature of solid Temperature of wall Velocity of gas Velocity of solid Distance along kiln # C3S/ # CaO # C2S/# CaO

#/#CaO oR

oR 1700

oR oR

562

oR oR

40,000 - 150

ft/hr ft/hr ft

Btu hr °F

81, 82, 83, 84 Heat transfer coefficients 5

Units Btu ft hr

qF

rF rl

321

# C3A/# CaO # C4AF/# CaO Heat of reaction--CaCO3 Heat of reaction fuel Heat of reaction--CaS Heat of reaction--C2S Heat of reaction--water Radiation coefficient--gas Radiation coefficient--solid Radiation coefficient--wall # CaCO3/# CaO Initial value of # CaCO3 / # CaO at input Density of gas Density of solid Density of wall

1275 - 14,000 11 -381 970 0.273 0.500 0.751

Btu/#CaCO 3 Btu/# Btu/#CsS Btu/#C2 S Btu/# water

1.784 0.05 56 112

#/#CaO #/fP #/ft 3 #/ft 3

0.0649

#/#CaO

80,000 0.447 1.0 1.0

ft/hr

# C02/# CaO # Water/# CaO Initial value of # Water/# CaO Changes in constants for no flame case vo Velocity of gas % Radiation coefficient--gas ~ Radiation coefficient--solid 8w Radiation coefficient--wall 03 031

APPENDIX C Development of the flame model The combination of fuel particles with the oxygen is a diffusion process. In this appendix, we shall first obtain a simple diffusion model and then determine the total reaction rate for the flame. Let x be the oxygen reacting with a sphere of the fuel which has radius r o. In spherical coordinates the steady-state diffusion equation is

V2x_°2x l_2 ax_0. -7,2 re,.

(Cl)

We shall assume that far from the sphere x is equal to its initial concentration, Xo. At the surface of the sphere, we shall assume that both a first- and second-order reaction takes place. Therefore, the boundary conditions for equation ((21) are

H . A . SPANG, I11

322 r~ ro

X=X 0 D S d x _ kpx o "dr-

r = ro

(C2) (C3)

where D is the diffusion constant, S, is the ratio of the surface area to the volume of the sphere and k~, k2 are reaction rate coefficients. The solution of equation (C1) can be shown to be of the form

We define do as the percentage of x available for the reaction at the surface of the sphere.

(C4)

Assuming that the particles are nearly spherical, S~ = 3/r r and 3Do do-

r~kF 3Do

P

Co~= ~ o

M,

Mo~ "

Similarly, the density of the gas will be

P Pa = MORT a •

rpkexo X=Xo= kr+(DoS,/ro}"

d°=~oot,o

where Gv is the total amount of fuel burned and CF is the percent of fuel left at any given point. The concentration of oxygen is given by the equation of state for a gas.

(c5)

r2kv + 1

In the previous equations, P is the pressure, ~ the percent oxygen in the air and M,, Mo2 the molecular weight of air and oxygen. We have assumed in deriving the gas energy balance equation that the total mass flow of gas was independent of temperature. Thus, the total rate per unit volume is

R = GF r~b(PMa)2Mc1.]k#loCr. F,AgLPFMo~(RTg)~ ]

(c10)

Assuming that the fuel particles move at the same velocity as the gas, the concentration of fuel is K v = FF .

(CI1)

vgAg where r v is the particle size, D O is the diffusion constant for oxygen given by

D o= aoTa 3/2

(C6)

and k r is the rate constant for the reaction of the fuel.

kr=3A%xg-Evl. rv "LRTa.]

(C7)

We have written k r in a slightly different form in order to reflect its dependence on ratio of surface area to volume of the particle. The reaction rate per particle of fuel will be

R' = krdoCo2

(C8)

where Cos is the total concentration of oxygen. The total rate of reaction in any given volume of air will depend on the fuel-to-air ratio and is given by

R v - F r p ~ C t krdoCo2 . FaPF

(C9)

mC 1 is the molecular weight of carbon. F r, PF, Fa and Pa are the pounds per hour and the density of the fuel and gas, respectively. To put this rate in a more usable form, let

Fr=GeCr

It is again reasonable to assume that the reaction of the fuel will always be in a steady-state. Therefore, we obtain the material balance equation for the fuel given in Appendix A.

R&amar----Unfour ~tciment est un type de prochd6 ~tcontrecourant dam lequel a lieu une srfie complexe de rractions exothermiques et endothermiques. Un modrle dynarnique, ~tdrrivOespartielles, du four a 6t6 6tablie g fin de comprendre son comportement et fournir tm moyen d'amrliorer la commande par calculateur. Dam le prrsent article, les rractions et les hypotheses fondamentales du modrle seront discutres. Un modrle d'une flamme charbon/mazout tenant compte des temp&atures ambiantes est 6tabli. Le comportement dynanuque du four illustr6 par ce modrle est ensuite indiqur. I1 est montr~, en particulier, que l'interaction entre les deux rractions exothermiques conduit le four ~t~tre imtable en ce qui concerne le rrglage de la position de la zrne de combustion. Zusammenfassnng--Ein Zementofen stellt einen ProzeB vom GegenfluBtyp dar, in dem eine komplexe Folge von exothermen und endothermen Reaktionen stattfindet. Um sein Verhalten zu verstehen, und Mittel zur Verbesserung der Steuerung dutch Rechner zu finden, wurde mater Benutzung partieller Differentialgleichungen ein dynamjsches Modell des Ofens entwiekelt. Die Hauptreaktion und -Annahme des Modells werden er6rtert. Ein Modell einer Kohle-OlFlamme, die die Umgebungstemperaturen in Rechnung zieht, wird entwickelt. Das dynamische Verhalten des Ofens, wie es durch dieses Modell dargelegt ist, wird dann angegeben. Speziell wird gezeigt, dab die Wechselwirkung zwischen den zwei exothermen Reaktionen verursacht, dab tier Ofen im Hinblick auf die Regelung der Lage der Brennzone grtmdsitzlich imtabli ist.

A dynamic model of a cement kiln Pe3mMe---I~eMeHTHaS n e ~ s npe~cTaa~seT c060fl iIpOTHBOTO~ n p o u e c c B KOTOpOM IIpOHCXO~IiT p ~ 3K3OTepMHqeCKHX H 3H~OTepMHqeCKIiX p e a x H l ~ . Pazpa6oTaHa RUHaMH~a~ napLtKa.~Has ~ Y [ ~ e p c m m a ~ H a ~ MO~CYH, C ~IcYn~o ~TO6bl HOH~ITh ee HOBeHeHHC H y ~ T b yHpaBHCHHe IIPOHeCCa BbI~HCHHTCYIbHO~ MamHHO]]. B CTaTbC paCCMOTpeHs~ OCHOBH/~Ie pCaKHHH H rIpC~rlOHO~KCHII~ Mo~eHH.

323

Pa3pa6oTaHa MO~Ca~ yroYmHO-MaayTHOtt II3IaMCHH, npHHHMa~ nO VHHMmme o x p y z a ~ m H e TeMnepaTyp~. 3aTeM y x a 3 a u o ~HaMU~ecxO¢ noBe~eHHe Helm x a x iloKa3ano 3TOi~ MO~ICJ~O. ]3 LlaCTHOCTH lIolta3aHo ~TO B ~ O ~ I C ~ C T B H e MCX.B,y ~ByMH 3K3OTCpMH~eL'KHMH peammsMH BM3MBaeT HeCTa~HJIbHOCTI~ HCLIHOTHOCHTeHbHO peryJIHpOBaHH~[ MeCTOHOJIO~KeHH~I 3OHLI ropeHH~.