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Modellingofaclinkerrotarykilnusingoperatingfunctionsconcept

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MODELLING OF A CLINKER ROTARY KILN USINGOPERATING FUNCTIONS CONCEPT

Kyarash Shahriari1*,† and Stanislaw Tarasiewicz2‡

1. Centre de Recherche Industrielle du Quebec (CRIQ), Quebec City, QC, Canada

2. Laboratory of Complex Automation and Mechatronics, Mechanical Engineering Department, Laval University, Quebec City,QC, Canada

Modelling and parameter identification of complex dynamic systems/processes is one of the main challenging problems in control engineering.An example of such a process is clinker rotary kiln (CRK) in cement industry. In the prevailing models independently of which structure is usedto describe the kiln’s dynamics and the identification algorithm, parameters are assumed to be centralised and constant while the CRK is wellknown as a distributed parameter system with a strongly varying dynamic through time. In this work, the kiln’s dynamic is described in theform of a state-space representation with three state variables using a system of partial differential equations (PDE). The structure is chosen sothat it can easily be embedded in classical state-space control algorithms. The parameters of the PDE system are called operating functionssince their numerical values vary with respect to different operating conditions of the kiln, to their position in the kiln, and through time. Aphenomenological approach is also proposed in this paper to identify the operating functions for a given steady-state operation of the kiln. Themodel is then used to perform a semi-dynamic simulation of the process through manipulating main process variables.

La modelisation et l’identification des parametres de systemes/procedes dynamiques complexes constituent l’un des principaux problemes engenie des controles automatiques. Un exemple d’un tel procede est le four rotatif clinker dans l’industrie du ciment. Dans les modeles actuelsindependamment de la structure utilisee pour decrire la dynamique du four et l’algorithme d’identification, on presume que les parametres sontcentralises et constants, alors que le FRC est bien connu comme systeme de parametres distribue avec une dynamique qui varie beaucoup avecle temps. Dans ce travail, la dynamique du four est decrite sous forme de representation d’espace d’etats avec trois variables d’etats faisant usaged’un systeme d’equations differentielles partielles. La structure est choisie pour etre facilement integree en algorithmes de commande d’espaced’etats traditionnels. Les parametres du systeme d’EDP sont nommes fonctions d’operation, puisque leurs valeurs numeriques varient concernantles differentes conditions de fonctionnement du four, leur position dans le four et dans le temps. Une approche phenomenologique est egalementproposee dans le present document pour determiner les fonctions d’operation pour un fonctionnement en etat stationnaire donne du four. Lemodele est ensuite utilise pour realiser une simulation semi-dynamique du procede par la manipulation des principales variables du procede.

Keywords: phenomenological modelling, systems identification, cement industry, clinker rotary kiln, operating function

INTRODUCTION

In a cement production plant, clinker rotary kiln (CRK) isthe most important element of the manufacturing chain whosecapacity generally determines the overall capacity of the

plant. It is also highly energy-consuming and greenhouse-gas-emitting. Rough estimations show that the cement industry iscurrently the responsible of 5–7% CO2 emission and that thisvolume will reach up to 25% by the year 2050 considering thecurrent trend of increasing demand for cement and the avail-able production technology (Portland Cement Association—PCA,https://www.cement.org; Cement Association of Canada—CAC,https://www.cement.ca). Consequently, improving the efficiencyof the CRK and reducing CO2 emission are among the main

concerns in cement manufacturing technology. The latter isalso strongly supported by World Business Council for Sustain-able Development under the theme of CO2 emission reductionand responsible use of fuel and raw materials (World Business

†R&D Researcher.‡Head of the Laboratory of Complex Automation and Mechatronics.∗Author to whom correspondence may be addressed.E-mail address: [email protected]. J. Chem. Eng. 89:345–359, 2011© 2010 Canadian Society for Chemical EngineeringDOI 10.1002/cjce.20398Published online 9 November 2010 in Wiley Online Library(wileyonlinelibrary.com).

| VOLUME 89, APRIL 2011 | | THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING | 345 |

Council for Sustainable Development, Cement SustainabilityInitiative—WBCSD-CSI, www.wbcsdcement.org). In this regard,an intermediate milestone is to develop an easy-to-handle fromthe engineering point of view and at the same time accuratedescriptive mathematical model of the process to underpin amodel-based control system and to explore potential technicalenhancements.

Whilst the CRK is in operation in cement industry since early1900s, its efficient modelling has been delayed for decades. Oneof the first descriptive models of a CRK was proposed by Spang(1972) that consisted of a complete description of chemicalreactions occurring within the kiln (Bogue, 1947), bed motion(Saeman, 1951), heat and mass exchanges between gas and solidphases, and a dynamic model of the flame. The model is derivedfrom mass and energy balance (MEB) equations for gas and solidphases in a moving control volume along the kiln’s length that ledto a system of partial differential equations with known boundaryconditions. The two-point boundary-value problem is then solvedusing appropriate numerical techniques to obtain the tempera-ture profiles and species mass fractions for both bed materialsand freeboard gas along the horizontal axe. In Guruz and Bac(1981); and Mujumdar and Ranade (2006), the same methodol-ogy has been employed to predict the concentration of materialsas well as bed and freeboard gas temperature profiles along thelength of the kiln. In the above MEB models, it is assumed thatthe model parameters are independent of time, temperature, andposition in the kiln or in other words, centralised real-valuedconstants.

The inconvenience of a 1D model for the gas flow is that in akiln, it takes 3D complex patterns due to burners, the geometryof the kiln as well as physical and chemical phenomena tak-ing place within the kiln. These patterns affect considerably heatexchange mechanisms between bed materials and freeboard gaswhilst a 1D model cannot capture them. To overcome this draw-back, computational fluid dynamics (CFD) codes for freeboardgas are coupled with 1D models for bed materials. Lockwood etal. (1995) used CFD codes to describe flow field and coal com-bustion in a CRK. Another attempt to enhance CRK modellingusing CFD codes was carried out by Mastorakos in which a 3DCFD code for the freeboard gas was coupled with a 1D modeldescription of the bed materials (Mastorakos et al., 1999). Thismodel incorporated three modules; the first module comprised amultidimensional CFD code for coal combustion and freeboardgas based on the results presented in Lockwood et al. (1995),the second a 3D heat conduction equation to describe tempera-tures in the walls developed in Boateng and Barr (1996), and thethird a 1D axial model presented in Spang (1972) for materialsfractional composition. Three modules have been solved simulta-neously to determine the gas and materials temperatures as wellas bed composition. In a recent work, the CFD code has been usedto develop a comprehensive model of the CRK and the respectivesimulator to explore energy efficiency of the kiln (Mujumdar etal., 2006). As for freeboard gas temperature profile, an approxi-mation obtained by a CFD model has been used. Heat exchangeequations have been adapted from Gorog et al. (1981, 1983); andBarr et al. (1989) while chemical reactions have been consideredas shrinking core process by numerical parameters stated in Irfanand Dogu (2001). This work has been followed in Mujumdar etal. (2007) in which pre-heater and cooler have also been incor-porated into the model to provide more complete mathematicaldescription of the process and enhancing simulation results. Inthe stated works, the model parameters are yet again real-valuedconstants.

In the above presented models, the bed in the transverse planeis assumed to be well mixed while evidences on real processesdemonstrate that it should not be the case; that is, non-uniformproduction of the CRK and dead-burning zone of large particleswhile fine particles are fully calcinated suggest the existence ofa temperature gradient within the bed in transverse plane. Thisis especially due to the bed motion and bed mixing/segregationmechanisms. A quasi 3D model has been developed and proposedin Boateng and Barr (1996) to consider this issue based on theearlier work of Barr et al. (1989). In this work a 1D axial modelalong the kiln’s length for freeboard gas has been incorporatedwith a 2D transversal model of bed materials. Using it, activelayer and plug flow zones have been distinguished in transversalplane. However, the key assumption is that physical properties ofparticles do not change along the length of the kiln.

The application of rotary kilns (RK) is not limited to cementplants and they are also vastly used in other industries, for exam-ple, rotary dryers or incinerators whose modelling has massivelybeen investigated. There are three main issues to be scrutinisedin any RK that are solid flow and bed form/motion in transver-sal plane, particle mixing and segregation, and heat exchangebetween freeboard phase and bed materials as well as heat trans-fer within the bed. For any of these issues, several works canbe stated. For instance, solid flow and bed motion, bed depth,and axial velocity of materials have been investigated in Sae-man (1951); Sriram and Sai (1999); Spurling (2000), bed form intransversal plane in Mellmann (2001), and material’s residencetime in Sai et al. (1990), particle mixing/segregation in Pollardand Henein (1989); Boateng and Barr (1996); McCarthy et al.(1996); Khakhar et al. (1997), and heat transfer in Cook andCundy (1995); Dhanjal et al. (2004); Chaudhuri et al. (2006);Shi et al. (2008).

Although the kiln is well known as a strongly time-variant dis-tributed parameter system along which materials’ physical andchemical properties as well as heat and mass exchange parame-ters are subject to radical changes, this issue is taken into accountin few of the prevailing CRK models. Furthermore, even thoughnumerical models are vastly used for simulation and processoptimisation/control, a comprehensive and easy-to-understandmathematical description of the process is still essential and apowerful source of knowledge for process engineers to attaindesign/production objectives and optimise resource management.The objective of this work is therefore to move away from purelynumerical models and to study the feasibility of an analyticaldescription of the CRK being in the same time simple and accurate.In this regard, the tempo-spatial state-space representation origi-nally developed in Tarasiewicz and Shahriari (2008b) is proposedwith minor modifications in the Model Structure Section. Thisrepresentation consists of three state variables that are freeboardgas temperature, bed materials temperature, and bed materialsmass distribution profiles along the kiln. The evolution of anystate variable through time is determined by the phenomena tak-ing place in the kiln; any phenomenon is described individuallywith the aid of a process variable and an operating function as itscoefficient. The identification procedure of the operating functionsfor a steady-state operation of the kiln is explained in the Oper-ating Functions Identification Section. The model is then used inthe fourth Section to perform a semi-dynamic simulation of theprocess which is also considered as a validation procedure. Con-clusion is then followed in the fifth section in which the possibilityof embedding this model into classical model-based control andoptimisation algorithms is also provided as the perspective of thecurrent work.

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MODEL STRUCTUREInvestigations on the CRK reveals strongly coupled physical andchemical phenomena within the kiln (Bogue, 1947; Tarasiewiczand Shahriari, 2008a). Applying classical modelling proceduresleads to a system of complex mathematical equations with dif-ferent time scales for freeboard and bed materials dynamics. Toreduce model structure complexity while persisting on modelaccuracy, we adopt in this work the phenomenological modellingapproach; main phenomena are identified and described individ-ually with the aid of an appropriate process variable and therespective operating functions as its coefficient. Three key pro-cess variables are chosen as state variables that are bed materials’temperature TC(x,t), freeboard gas temperature TG(x,t), and bedmaterials mass distribution MC(x,t). The evolution of the statevariables in the tempo-spatial state-space representation is thenexpressed using a system of partial differential equations derivedfrom phenomena affecting any state variable and MEB equationsin a mobile control volume.

Operating Function ConceptThe concept of operating function and its utilisation in modellingcomplex processes has originally been developed in Tarasiewiczet al. (1994). This concept has then been employed in Kada andTarasiewicz (2004) for wood drying systems modelling and sim-ulation. The idea of using operating functions comes from thefact that in mathematical models of complex processes, parameteridentification requires a deep study of the process and determiningseveral physical/chemical coefficients which are generally chal-lenging due to technical constraints and the lack of knowledgeand observations/experiences on the process. For instance, sup-pose that we wish to calculate thermal energy supplied to theCRK. This consists of the heat recycled by primary and secondaryair from the cooler and satellites as well as the heat generated bycombustion of fuel. For the former, we have:

Qair(x, t) = fG(x, t)∂TG(x, t)

∂x

where:

fG(x, t) = mpa(x, t)cpa(Tpa(x, t)) + msa(x, t)csa(Tsa(x, t))

+ mf(x, t)cf(Tf(x, t))

In the above expression, m(·)(x, t) represents the distributedmass and c(·)(T(·)(x, t)) the specific heat of primary air, secondaryair, and fuel at temperature T(·)(x, t). Moreover, the heat generatedby fuel combustion is calculated from:

Qfuel(x, t) = hf(x, t)∂Mf(x, t)

∂x= hf(x, t)

∂Mf(x, t)∂TG(x, t)

× ∂TG(x, t)∂x

where hf(x,t) is the heat released per fuel mass unit and Mf(x,t)is the injected fuel mass rate. Therefore, total heat transferred tothe kiln is expressed as:

Qtotal(x, t) =(

fG(x, t) + hf(x, t)∂Mf(x, t)∂TG(x, t)

)∂TG(x, t)

∂x(1)

Calculating Qtotal(x,t) from Equation (1) requires accurate mea-surement of process variables m(·)(x, t), Mf(x,t), and T(·)(x, t) andidentification of coefficients c(·)(T(·)(x, t)) which can hardly be

performed due to technical constraints. On the other hand, Equa-tion (1) can be reformulated as:

Qtotal(x, t) = FQ(x, t)∂TG(x, t)

∂x

where the so-called operating function FQ(x,t) embeds the men-tioned process variables and coefficients. Calculating FQ(x,t)directly through an identification procedure will bypass themeasuring and identifying of m(·)(x, t), Mf(x,t), T(·)(x, t), andc(·)(T(·)(x, t)) to compute Qtotal(x,t).

Generally speaking, operating functions replace complex math-ematical descriptions of phenomena in a process. Any operatingfunction includes a number of inaccessible process variables aswell as unknown physical/chemical coefficients. To employ oper-ating functions in modelling a process, we decompose it in thefirst place into its basis phenomena. The key process variable isthen determined for any phenomenon, and the evolution of thephenomenon is described in terms of its respective key processvariable with the aid of an operating function as coefficient. Theresulting expression should be in such a way that the operatingfunction can directly be identified using available observations onthe process. As for the example of the energy supplied to the CRK,transferred heat Qtotal(x,t) is expressed in terms of the evolution offreeboard gas temperature along the kiln’s length and operatingfunction FQ(x,t) as its coefficient. Using this approach, modellingand identification procedures are simplified by introducing a phe-nomenological description of the process that brings the problemto a higher and more user-friendly level than detailed mathemati-cal equations. Moreover, down-to-up accumulation of parametersuncertainty in identification procedure from physical/chemical tophenomenological representation is excluded which results in amore accurate model of the process under study.

Operating function approach is not the only method to take intoaccount tempo-spatial variation of model parameters. An alter-native solution is to consider the data as separate time seriescorrelated in space or extending traditional spatial methods suchas kriging to the spatio-temporal domain by adding an extradimension (Sampson and Guttorp, 1992; Oehlert, 1993). Someresearchers have also proposed to use Kalman filters. In Wikle andCressie (1999) a statistical model that is temporally dynamic andspatially descriptive has been applied to a data set of near surfacewinds in atmospheric science. However, the main drawback ofthese models is that the parameters are derived statistically anda reach data set is required to mitigate measuring uncertaintieswhile this set is generally not available for a CRK due to mea-suring technical problems. Moreover, a priori knowledge cannotexplicitly be integrated in statistical models that are a major sourceof information for CRK modelling.

Phenomenological Decomposition of the ProcessWe identify in this early work six main phenomena taking placewithin the CRK that are as follows:

(a) Heat exchange between freeboard gas, bed materials, refrac-tory, shell, and surroundings. This is expressed in theform of a linear relation of the absolute temperature orthe temperature potential between two points accompany-ing with an operating function, that is, Fi,j(x, t)T(·)(x, t)or Fi,j(x, t)(T(·)(x, t)−T(·)(x, t)). Examples are refractory-bedand gas-bed heat exchanges expressed as F0,1(x, t)TC(x, t)and F1,5(x, t)(TG(x, t)−TC(x, t)), respectively. Thermal coef-ficients are lodged in the operating functions.

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(b) Diffusion of the materials into the gas and vice versa.It is described as the absolute temperature times by therespective operating function, that is, F1,1(x, t)TC(x, t) andF2,1(x, t)TG(x, t) for diffusion of materials into gas and gasinto materials, respectively.

(c) The freeboard gas and bed materials displacement alongthe length of the kiln. It is defined with the aid of thederivative of the temperature with respect to x and theoperating function resulting in F1,2(x, t)(∂TG(x, t)/∂x) andF2,2(x, t)(∂TC(x, t)/∂x) for freeboard gas and bed materialsdisplacements. It is worth to state that velocities vG(x,t) andvC(x,t) are embedded into the operating functions.

(d) Heat transferred from combustion of fuel to freeboard gas.It is described as F1,3(x, t)Mf(x, t) by the fuel feed mass rateMf(x,t). Thermal properties of the fuel are considered by theoperating function.

(e) Gas and materials heat exchange due to physical/chemicalreactions. Any reaction results in mass evolution andhence, the latter is expressed as F1,4(x, t)(∂MC(x, t)/∂x) andF2,3(x, t)(∂MG(x, t)/∂x) for bed materials and freeboard gas,respectively.

(f) Gas and materials mass exchanges due to physi-cal/chemical reactions. Identical to the previous case,F3,1(x, t)(∂MC(x, t)/∂x) and F3,2(x, t)(∂MG(x, t)/∂x) describemass exchanges between freeboard and solid phases. For twolast cases, reaction coefficients and rates are included in theoperating functions.

It should be mentioned that the identified phenomena and theirrespective mathematical descriptions are subject to further mod-ifications if additional accuracy would be required or a prioriknowledge and observations on the kiln would be available.

Distributed Parameter ZoneDue to the nature of the CRK, the model consists of two parts.The first part describes the process in distributed and the secondin lumped parameter zone. The former corresponds to the inte-rior of the kiln whereas the latter describes its two ends knownas boundaries. We first start with two expressions—that are notexplicitly a part of the model but will be used to identify the operat-ing functions in a later stage—describing heat exchanges betweenfreeboard gas, bed materials, kiln’s body, and surroundings:

TB(x, t) = f0,1(x, t) + f0,2(x, t) + f0,3(x, t) (2a)

TS(x, t) = f0,4(x, t) + f0,5(x, t) (2b)

Table 1. Heat exchange operating functions

Function Description Proposed structure

f0,1(x,t) Refractory and bed heat exchange F0,1(x,t)TC(x,t)f0,2(x,t) Refractory and gas heat exchange F0,2(x,t)TG(x,t)f0,3(x,t) Refractory and shell heat exchange F0,3(x,t)TS(x,t)f0,4(x,t) Shell and refractory heat exchange F0,4(x,t)TB(x,t)f0,5(x,t) Shell and surroundings heat exchange F0,5(x,t)TSur(x,t)

Detailed description of any fi,j(x,t) in Equation (2) is given inTable 1 in which TB(x,t), TS(x,t), and TSur(x,t) are, respectively,firebrick, shell, and surroundings temperatures and Fi,j(x,t) areoperating functions.

The evolution of any state variable through time is determinedusing MEB equations in a moving control volume along the lengthof the kiln (see Figure 1). This volume is a thin slice of the CRK atx with thickness �x perpendicular to the horizontal axe. It shouldbe mentioned that contrary to the kiln’s cross-section, the movingcontrol volume is elliptical due to inclination angle �. Consideringthe main phenomena taking place inside the kiln (bed materialsand freeboard gas displacements along the kiln, diffusion of bedmaterials into freeboard gas and vice versa, fuel combustion, heatexchanges between solid and gas phases, and physical/chemicalreactions) resulting MEB equations in the control volume (seeFigure 2) (Tarasiewicz and Shahriari, 2008a), we have:

∂TG(x, t)∂x

= f1,1(x, t) + f1,2(x, t) + f1,3(x, t) + f1,4(x, t) + f1,5(x, t)

+ f1,6(x, t) (3)

for freeboard gas temperature,

∂TC(x, t)∂x

= f2,1(x, t) + f2,2(x, t) + f2,3(x, t) + f2,4(x, t) + f2,5(x, t)

(4)

for bed materials temperature, and

∂MC(x, t)∂x

= f3,1(x, t) + f3,2(x, t) + f3,3(x, t) + f3,4(x, t) + f3,5(x, t)

(5)

for bed materials mass distribution. Detailed description offunctions fi,j(x,t) in Equations (3)–(5) representing phenomenawithin the CRK is given in Table 2.

Figure 1. Kiln’s geometry and moving control volume.

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Figure 2. Mass and energy balance in moving control volume.

Lumped Parameter ZoneHaving developed model structure in distributed parameters zone,we now focus on the boundaries of the CRK. Boundary conditionsare mathematically expressed for freeboard gas as:

∂TG(0, t)∂x

= f4,1(0, t) + f4,2(0, t) (6a)

∂TG(L, t)∂x

= 0 (6b)

for bed materials temperature as:

∂TC(0, t)∂x

= 0 (7a)

∂TC(L, t)∂x

= f5,1(L, t) + f5,2(L, t) (7b)

and for clinker mass distribution as:

∂MC(0, t)∂x

= 0 (8a)

∂MC(L, t)∂x

= f6,1(L, t) (8b)

where x = 0 is the lower end and x = L is the upper end of the kiln.For freeboard gas, x = 0 is the fixed and x = L is the free boundarywhilst for bed temperature and mass distribution x = L is the fixedand x = 0 is the free boundary. Detailed description of functionsfi,j(x,t) in Equations (6)–(8) is given in Table 3. As soon as themodel structure is established, we tackle with the identificationof the operating functions.

OPERATING FUNCTIONS IDENTIFICATIONThe first step to calculate the operating functions is preparing anidentification data set. A series of observations and measurementshas been performed on a wet kiln in its nominal steady-state oper-ation. Flame and clinker temperatures have been measured by aspectrometer at the lower end of the kiln while thermocoupleshave been used to observe TG(x) and TC(x) at the upper end. Also,shell temperature has been observed at several points by installedthermocouples on the shell through which TG(x) and TC(x) havealso been calculated by the I&C system (see Figure 3). To mit-igate noise effect and perturbation, mid-value of the measuredtemperatures over several minutes has been considered. Usingthe mentioned measuring values, the temperature profiles for fullrange of x have been reproduced for TG(x), TC(x), and TS(x) usingpiecewise cubic Hermite interpolation.

When the kiln has been operating in the respective steady state,the flame has been shut down and the raw materials feedinghas been stopped for a mechanical maintenance. The kiln con-tinued to rotate while the output mass flow rate and the materialsdensity have continuously been measured until the kiln is com-pletely empty. Respective measured density for five points andthe reconstructed bed materials mass distribution profile fromthe measurements are illustrated in Figure 4. Although not ideal,but this procedure has been identified as the best feasible way ofmeasuring mass distribution profile within the kiln.

Table 2. Distributed parameter zone operating functions

Function Description Proposed structure

f1,1(x,t) Diffusion of materials into gas −F1,1(x, t)TC(x, t)f1,2(x,t) Displacement of gas along the kiln F1,2(x, t)(∂TG(x, t)/∂x)f1,3(x,t) Heat transferred from combustion of fuel to gas F1,3(x, t)Mf (x)f1,4(x,t) Gas and bed heat exchange due to physical/chemical reactions F1,4(x, t)(∂MC(x, t)/∂x)f1,5(x,t) Gas and bed heat exchange due to radiation and conduction −F1,5(x, t)(TG(x, t)−TC(x, t))f1,6(x,t) Gas and refractory heat exchange due to radiation and conduction −F1,6(x, t)(TG(x, t)−TB(x, t))f2,1(x,t) Diffusion of gas into materials F2,1(x, t)TG(x, t)f2,2(x,t) Displacement of bed along the length of the kiln F2,2(x, t)(∂TC(x, t)/∂x)f2,3(x,t) Bed and gas heat exchange due to physical/chemical reactions −F2,3(x, t)(∂MG(x, t)/∂x)f2,4(x,t) Bed and gas heat exchange due to radiation and conduction F2,4(x, t)(TG(x, t)−TC(x, t))f2,5(x,t) Bed and refractory heat exchange due to radiation and conduction F2,5(x, t)(TC(x, t)−TB(x, t))f3,1(x,t) Bed and gas mass exchange due to physical/chemical reactions −F3,1(x, t)(∂MC(x, t)/∂x)f3,2(x,t) Gas and bed mass exchange due to physical/chemical reactions F3,2(x, t)(∂MG(x, t)/∂x)f3,3(x,t) Diffusion of materials into gas −F3,3(x, t)TC(x, t)f3,4(x,t) Diffusion of gas into materials F3,4(x, t)TG(x, t)

f3,5(x,t) Displacement of bed along the length of the kiln F3,5(x, t)(∂•

MC

(x, t)/∂x)

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Table 3. Lumped parameter zone operating functions

Function Description Proposed structure

f4,1(x,t) Gas and bed heat exchange F4,1(0,t)(TG(0,t) − TC(0,t))f4,2(x,t) Gas and refractory heat exchange F4,2(0,t)(TG(0,t) − TB(0,t))f5,1(x,t) Bed and gas heat exchange F5,1(L,t)(TG(L,t) − TC(L,t))f5,2(x,t) Bed and refractory heat exchange F5,2(L,t)(TB(L,t) − TC(L,t))f6,1(x,t) Mass positioning in the upper end of the kiln F6,1(L,t)MC(L,t)

Figure 3. Measured and reconstructed temperature profiles.

The observed data including TG(x), TC(x), TS(x), and MC(x)together with the design parameters given in Table 4 and ther-mal properties of the kiln are considered as the identificationdata set. The identification procedure of the operating functions isexplained in which follows. The first step is to determine the valid-ity domain of any operating function through phenomenologicaldecomposition of the process and then to use complementarymathematical equations to calculate it.

Figure 4. Measured and reconstructed bed density and massdistribution profiles.

Heat Exchange EquationsConsidering the kiln as a long horizontal cylinder, its thermalequivalent circuit is obtained as shown in Figure 5. Respectiveheat resistances to any heat exchange mechanism is calculatedfrom Rcond = ln(r2/r1)/2k��x for conduction, Rconv = 1/2�hr�x

for convention, and Rrad = 1/ε�(TS−TSur)(T2S −T2

Sur) for radiation,where k represents the conductivity and h the convection coeffi-cients (Incropera et al., 2007).

The heat exchange between TB(x) and TS(x) is uniquely theresult of conduction. Assuming that the gas and materials temper-atures in the control volume for �x = 0.1 m are constant, and thatrefractory and shell thermal conductivities are kref = 4.0 W/mKand kshell = 46.0 W/mK, we have:

R4 = ln(r2/r1)2kref��x

= 0.0462 K/W, R5 = ln(r3/r2)2kshell��x

= 0.0005 K/W

Heat exchange rate q6(x) is the consequence of radiationfrom the kiln’s surface to surroundings. Using shell temper-ature profile in Figure 3, R6(x) is calculated for Tsur = 300 K,Stefan–Boltzmann constant � = 5.67 × 10−8 W/m2K4, and emis-sion coefficient ε = 0.9 m2. On the other hand, q7(x) representsheat rate due to convection. The respective thermal resistance forconvection coefficient h = 5 K/Wm2 is:

R7 = 12�hr3�x

= 0.1721 K/W

Knowing R6(x) and R7 as well as the thermal potentialTsur(x) − TS(x), heat rate lost qloss(x) is calculated from:

qloss(x) = TSur−TS(x)Req(x)

(9)

where Req = R6(x)||R7. As soon as qloss(x) is known, we have:

TSin (x) = TS(x) + R5qloss(x)

TB(x) = TS(x) + (R4 + R5)qloss(x)

For simplicity in this early work, we ignore materials coatingto the refractory layer and consequently, R3 is void. Therefore,

Table 4. Design parameters

Parameter Description Value (m)

r1 Inner radius with refractory 1.625r2 Inner radius without refractory 1.825r3 Outer radius 1.85L Kiln length 120

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Figure 5. Thermal equivalent circuit of the kiln.

q1(x) = TG(x)−TBin (x)R1(x)

(10a)

q2(x) = TC(x)−TBin (x)R2(x)

(10b)

q(x) = q1(x) + q2(x) (10c)

To calculate R1(x) and R2(x), one more equation is required. Weassume that R1(x) and R2(x) are proportional to the gas-refractoryand materials-refractory contact surfaces mathematically statedas:

(�−˛(x))R1(x) = ˛(x)ˇ(x)R2(x), 0<ˇ(x)≤1 (11)

where ˛(x) is the filling angle and ˇ(x) is a corrective coefficient.Combining Equations (9)–(11) we have:

R1(x) = 1qloss(x)

(TG(x)−TB(x)) + ˛(x)ˇ(x)(�−˛(x))qloss(x)

(TC(x)−TB(x))

(12a)

R2(x) = �−˛(x)˛(x)ˇ(x)qloss(x)

(TG(x)−TB(x)) + 1qloss(x)

(TC(x)−TB(x))

(12b)

Again for simplicity, we chose ˇ(x) = 1.0 in this work. Ther-mal resistances R1(x) and R2(x) are then calculated by solvingthe system of Equations (10) and (12). The respective operatingfunctions are consequently computed as follows:

F0,1(x) = R1(R4 + R5)R1R2 + (R1 + R2)(R4 + R5)

F0,2(x) = R2(R4 + R5)R1R2 + (R1 + R2)(R4 + R5)

F0,3(x) = R1R2

R1R2 + (R1 + R2)(R4 + R5)

F0,4(x) = Req

Req + R4 + R5

F0,5(x) = R4 + R5

Req + R4 + R5

which are illustrated in Figure 6. Considering Equations (9)–(12),the effect of thermal properties of the kiln on heat loss qloss(x),

and temperature profiles TB(x) and TS(x) are determined helpingto assess the effect of any further modifications; for example, theeffect of convection and radiation from the shell on the total heatloss using Equation (9), or the thermal residence of the refractorylayer R4 on the temperature profiles as well as the heat loss of thekiln using operating functions F0,i(x) and Equation (2).

Freeboard Gas Temperature EquationsAs the first step to identify the operating functions in Equation (3),their validity domains are determined through decomposition ofthe process into four operating zones that are drying zone whereTC ∈ [0, 500], preheating TC ∈ [600, 1050], calcination TC ∈ [1100,1400], and clinkerization TC ∈ [1500, 1800] (Bogue, 1947; Spang,1972; Guruz and Bac, 1981). Considering temperature profilesin Figure 3, these ranges correspond to x ∈ [90, 120], x ∈ [65,80], x ∈ [40, 60], and x ∈ [15, 35], respectively. Mainly due tothe lack of adequate information and precise measurements,heat generated from exothermic reactions and the effect of thediffused materials into freeboard gas on TG(x) are not taken intoaccount in this work. Therefore, F1,1(x) and F1,4(x) are void. Onthe other hand, considering the proposed zone decompositionand the definition of TG(x) operating functions given in Table 2,F1,2(x,t), F1,3(x,t), F1,5(x,t), and F1,6(x,t) are valid in x ∈ [10, 30],x ∈ [0, 120], x ∈ [5, 120], and x ∈ [0, 120], respectively. F1,6(x)represents heat loss in the kiln. Total kiln’s heat loss rate qloss(x)has already been calculated in the Heat Exchange EquationsSection. The evolution of freeboard gas temperature due to heat

Figure 6. Operating functions of heat exchange equations.

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loss in the mobile control volume is characterised by:

�TG(x)�t

= − 1Cp,G(x)MG(x)

qloss(x) (13)

where Cp,G(x) is the specific heat and MG(x) is the mass distribu-tion of freeboard gas. As for the latter, following data are available:

vG(x)|x=0∈[14.76, 36.3] m/s

mG(x) = 4.600 kg/s

TG(x)|x=0 = 557 K

Since no direct measurement is available on the kiln, thegas velocity is approximated by hyperbolic function vG(x) =2.5 tanh(−0.03(x−60)) + 27.5 m/s from which freeboard massdistribution profile is calculated as follows:

MG(x) = mG(x)vG(x)

Also, from Equations (3) and (13) we have:

F1,6(x)(TG(x)−TB(x)) = 1Cp,G(x)MG(x)

qloss(x) (14)

from which F1,6(x) is identified and illustrated in Figure 7 forCp,G(x) = 1200 J/kg K. The operating function acquires highernumerical values symbolising more heat loss for the same temper-ature potential TG(x) − TB(x) where the absolute gas and materialstemperatures are higher, that is, around the flame position whichalso agrees with evidences and observations on the kiln.

F1,5(x) represents freeboard gas and bed materials heatexchange. The evolution of freeboard gas temperature due to gasand bed heat exchange rate qC(x) is expressed in a similar way as

Figure 7. Operating functions of freeboard gas temperature equation.

in Equation (13) and F1,5(x) is identified from a similar expres-sion as in Equation (14). The only unknown term is qC(x) whichconsists of two parts. The first part is associated with the dryingzone to evaporate moisture from raw materials and the secondpart elsewhere in the kiln to heat up bed materials. Dried massdistribution profile including the mass coated in the chains anddiffused in the gas has already been defined as MC(x). The ini-tial concentration of water in raw materials is known to be 30%.Considering the operating zone limits, moisture concentration isapproximated by the empirical hyperbolic function Cmoisture(x) =0.15 tanh(0.1(x−90)) + 0.15 in drying zone and therefore, mois-ture mass distribution profile is:

Mmoisture(x) = Cmoisture(x)MC(x)

Water evaporation rate is therefore:

Rate(x) = −∂(Mmoisture(x)vC(x))∂x

where vC(x) is the bed materials’ velocity. Bed velocity hasan inverse relation with bed mass distribution. Mathematicallystated:

vC(x)∝ 1MC(x)

Using mean value vC = 0.015 m/s provided in Spang (1972) fora similar kiln, bed velocity profile is computed along the lengthof the kiln. Required heat rate for drying is then calculated from:

qC1(x) = Rate(x)CW

where CW = 2.270 × 106 J/kg is the latent heat of water evapora-tion. qC(x) also raises the bed materials temperature. Thermalenergy E(x) accumulated in the bed along the kiln’s length is:

E(x) = Cp,C(x)MC(x)TC(x)

where Cp,C(x) is the specific heat of bed materials. Consequently,

qC,2(x) = −∂(E(x)vC(x))∂x

In this work, we fix Cp,C = 1200 J/kg K (Lide, 2008). It is alsoobserved in the kiln in operation that about 30% of materials arepartially melt in above TC > 1600 K that in this case takes place inx ∈ [10, 35]. Consequently, the required heat rate for melting is:

qC,3(x) = −∂(HCCm(x)MC(x)vC(x))∂x

where Cm(x) is the melting percentage of the materials along thekiln. The latent heat is assumed to be HC = 2 × 106 J/kg fromthe value given for CaO in Lide (2008). As soon as qC(x) =qC,1(x) + qC,2(x) + qC,3(x) is calculated, F1,5(x) is identified froman expression similar to Equation (14) which is also illustrated inFigure 7. Two facts are pointed out; firstly the operating functionis negative at the lower end of the kiln before the flame positionrevealing an inverse heat gradient from bed to gas. This is in accor-dance with heat recovery from the hot clinker leaving the kiln byinlet air. Secondly, a peak value in the flame position is observed

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Figure 8. Weights of the phenomena on the evolution of gastemperature.

which is due to the fact that radiation from the flame increasesconsiderably the heat exchange rate from gas to bed materials.

Thermal energy required to heat up the kiln is provided bycombustion of fuel. Considering energy balance equations in themobile control volume we have:

F1,3(x)Mf(x) = qF(x)1

Cp,G(x)MG(x)

where qF(x) is the heat rate produced by combustion. The oper-ating function F1,3(x) is then calculated and illustrated in Figure7 for qF(x) = 16.75 × 106 J/s. The function is non-zero only in theflame position where the fuel is injected into the kiln and zeroelsewhere.

As mentioned before, the observations have been performedin steady-state operation of the kiln leading to ∂TG(x)/∂t = 0 inEquation (3). Therefore, operating function F1,2(x) is computedas the residual from:

F1,2(x)∂TG(x)

∂x= −F1,3(x) + F1,5(x)(TG(x)−TC(x))

+ F1,6(x)(TG(x)−TB(x))

One more informative diagram presented here is the effect andthe weight of any individual phenomenon on the evolution of thegas temperature profile through time by multiplying the operatingfunctions and the respective process variable. We note in Figure8 that the gas displacement represented by F1,2(x,t) (upper left) isthe dominant factor in the equilibrium before the flame positionwhilst it is the fuel who takes the main role where the flame ispositioned (upper right). Heat recovered by the inlet air (lowerleft) before the flame cannot be neglected either in this regardwhich is then used to heat up the materials in the upper partsof the kiln. However, heat exchange between gas and refractorylayer (lower right) follows a smooth pattern, it is higher in theflame position and reduces as it approaches the upper end of thekiln.

The discontinuity of F1,2(x) in the neighbouring of x = 20 m isdue to the fact that the derivative of gas temperature profile is zeroresulting a division-by-zero while computing the function. How-ever, it causes no inconvenience since the term F1,2(x)(∂TG(x)/∂x)is employed together as seen in Figure 8 rather than F1,2(x) and∂TG(x)/∂x individually. This is shown in the fourth section whileperforming semi-dynamic simulation of the process.

Bed Materials Temperature EquationsAs stated before and due to the lack of information, heat gener-ated from exothermic reactions and the effect of the diffused gasinto materials on TC(x) are ignored. Therefore, F2,1(x) and F2,3(x)representing these two phenomena are void. Also, consideringthe kiln decomposition into operating zones, operating functionsF2,2(x,t), F2,4(x,t), and F2,5(x,t) are valid in x ∈ [0, 120].

Operational function F2,4(x) represents the effect of the free-board gas temperature on the evolution of bed temperaturethrough time. Heat exchange rate qC(x) has already been calcu-lated in Freeboard Gas Temperature Equations Section. Knowingthe value of qC(x) and considering energy balance in the mobilecontrol volume, we have:

�TC(x)�t

= − 1Cp,C(x)MC(x)

qC(x) (15)

which implies:

F2,4(x)(TG(x)−TC(x)) = 1Cp,C(x)MC(x)

qC(x) (16)

from which F2,4(x) is identified.Heat exchange rate q2(x) between the bed materials and the

kiln’s body which is mostly a result of conduction and radia-tion has also been calculated in the Heat Exchange EquationsSection. The evolution of the bed materials temperature in the con-trol volume in terms of q2(x) is described by a similar expressionas Equation (15) and operating function F2,5(x) is consequentlycomputed from a similar expression as Equation (16).

Being in steady-state operation implies that ∂TC(x)/∂t = 0 inEquation (4). Therefore,

F2,2(x)∂TC(x)

∂x= −F2,4(x)(TG(x)−TC(x))−F2,5(x)(TC(x)−TB)

from which F2,2(x) is calculated. The operating functions of thebed materials temperature equation are illustrated in Figure 9.We observe in this figure that the results follow the same patternsand are consistent with the ones from Figure 7; likewise F1,5(x)and F2,4(x) have a negative peak before the flame position demon-strating heat gradient from bed to gas which corresponds to theheat recovered by the inlet air from the hot outlet clinker. It is thenfollowed by a positive peak in the flame position showing the max-imum heat exchange rate from the gas to bed to provide requiredclinkerization heat as well as melting latent energy. At the end,it takes a smooth pattern for almost a constant exchange rate.The same similarity is also observed for F2,5(x), the counterpartof F1,6(x) representing heat transfer from bed to refractory layer.As for F2,2(x), it represents the effect of bed materials displace-ment on its temperature evolution through time. Since ∂TC(x)/∂x

is zero in the neighbouring of x = 20 m, a discontinuity is observedin the respective profile. However, as stated before, it is not prob-lematic since F2,2(x)(∂TC(x)/∂x) rather than F2,2(x) and ∂TC(x)/∂x

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Figure 9. Operating functions of bed materials temperature equation.

is used in Semi-Dynamic Simulation of the Process Section (seealso Figure 10).

As before, the weight of any individual phenomenon on bedtemperature evolution is calculated by multiplying the operatingfunctions and its respective process variable. As seen in Figure9, the most influential factor is bed displacement (upper left)demonstrating that any increase in bed motion will increase bedtemperature after and decrease it before the flame position whichwill result in the moving of the peak temperature toward the lowerend. An important amount of heat required for the process isrecovered from the outlet clinker by the inlet air (upper right)which reduces considerably the consumption of the energy in thekiln. The discontinuity due to division by zero for F2,2(x) is alsodisappeared. Bed-gas heat transfer with the least effect (lower left)follows the same pattern identified in Figure 7.

Figure 10. Weights of the phenomena on the evolution of bedtemperature.

Table 5. Initial species mass fraction in raw materials

CaCO3 66%SiO2 22%Al2O3 5%Fe2O3 4%CaOMgO 2%CaSO4 1%

Mass Distribution EquationsConsidering the operating zones and the definitions provided inTable 2, the validity of operating functions F3,1(x,t), F3,2(x,t),F3,3(x,t), and F3,5(x,t) are determined as x ∈ [60, 120], x ∈ [60,120], x ∈ [40, 100], and x ∈ [0, 120], respectively. Again due tolack of precise information and measurements on the amount ofdiffused bed materials into freeboard gas, we assume that the dif-fusion does not change the bed mass distribution which impliesF3,4(x) = 0.

Operating functions F3,1(x) and F3,2(x) represent massexchanges between gas and solid phases due to physical andchemical reactions that are briefly recalled in Appendix A. Toidentify F3,1(x), the initial concentration of water in raw materialis known to be 30%. Moreover, considering raw materials initialcomposition provided in Table 5 and the clinkerization reactions,we estimate that 5% of raw materials’ mass enters into the free-board gas in zones 1 and 2 in the form of CO2 and SO2 and that nomore mass exchange due to physical or chemical reactions takesplace in zones 3 and 4. Therefore, the distribution of mass releasedfrom the solid into gas phase is expressed by:

Mreleased(x) = MC(x)Creleased(x)

The released mass concentration is approximated by the empir-ical hyperbolic function Creleased(x) = 0.15 tanh(0.068(x−80)) +0.15 in the first and the second operating zones. Operational func-tion F3,1(x) is then identified from:

F3,1(x)∂MC(x)

∂x= ∂Mreleased(x)

∂x

On the other hand and due to mass balance between solid andgas phases in the control volume, we have:

F3,2(x)∂MG(x)

∂x= F3,1(x)

∂MC(x)∂x

from which F3,2(x) is calculated.As for operating function F3,5(x), it represents the bed materials

displacement along the kiln’s length. We therefore have

F3,5(x)∂MC(x)

∂x= ∂(vC(x)MC(x))

∂x

which implies F3,5(x) = 1 knowing that MC(x) = vC(x)MC(x).Since ∂MC(x)/∂x = 0 in the steady-state operation, F3,3(x) is

calculated from Equation (6) as the residual as follows:

F3,3(x)TC(x) = −F3,1(x)∂MC(x)

∂x+ F3,2(x)

∂MG(x)∂x

+ F3,5(x)∂MC(x)

∂x

The respective operating functions of the bed materials massdistribution are illustrated in Figure 11.

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Figure 11. Operating functions of bed materials mass distributionequation.

The main factors affecting the evolution the mass distributionprofile through time is its displacement represented by F3,5(x).Mass exchanges between gas and bed represented by F3,1(x) andF3,2(x) cancel each other while diffusion has a minor part.

Boundary Condition EquationsComputing the operating functions in Equations (6)–(8) is the laststage in the identification procedure. The evolution of freeboardgas temperature in the mobile control volume with respect to heatexchange rate q1(x) has already been calculated and expressed asfollows:

�TG(x)�x

= q1(x)Cp,GvG(x)MG(x)

which implies:

F4,2(x)(TG(x)−TB(x)) = q1(x)Cp,GvG(x)MG(x)

from which F4,2(x) is identified for x ≥ 0. It is void for x < 0 sincethe gas is not in contact with the refractory layer in this region.Accordingly, operating function F4,1(x) is computed for x ≥ 0 from:

F4,1(x)(TG(x)−TC(x)) = ∂TG(x)∂x

−F4,2(x)(TG(x)−TB(x))

Being acquainted with the kiln’s feeding installation is manda-tory to identify F4,1(x) for x < 0.

Likewise for freeboard gas temperature, we have:

�TC(x)�x

= q2(x)Cp,CvC(x)MC(x)

and consequently:

F5,2(x)(TB(x)−TC(x)) = q2(x)Cp,CvC(x)MC(x)

Figure 12. Operating functions of boundary condition equations.

from which F5,2(x) is identified. As soon as F5,2(x) is determined,F5,1(x) is obtained from:

F5,1(x)(TG(x)−TC(x)) = ∂TC(x)∂x

−F5,2(x)(TB(x)−TC(x))

In Figure 12, the negative sign of operational function F5,1(x)is due to the counter-current flow of materials with respect to thehorizontal axis. It is also observed that operating function F5,1(x)is negligible compared to that of F5,2(x). This demonstrates thatheat at right-hand side of the kiln is mostly provided from therefractory rather than the freeboard gas to the bed materials.

To identify F6,1(x), we assume that raw materials are fed in aconstant rate M0 = 30 kg/s and the feeding installation providesa homogeneous distribution in the entrance of the kiln fromx = L − l to x = L. Here, l represents the transition zone betweenthe distributed and lumped parameter regions at the upper endof the kiln. We also suppose that no mass exchange takes placebetween gas and solid phases and the velocity of the materialsin L − l ≤ x < L is constant value vC. Kiln’s geometry implies thatMC(x)|x≥L = 0. Mass distribution is then expressed as:

MC(x)|x≥L = 0

MC(x)|L−l≤x<L = 1vC

L−l∫L

M0

ldx

MC(x)|x=L−l = −M0

vC

which implies:

∂(MC(x)vC(x))∂x

|x≥L = 0

∂(MC(x)vC(x))∂x

|L−l≤x<L = M0

l

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∂(MC(x)vC(x))∂x

|x=L−l = 0

Operating function F6,1(x) is then computed from Equation (8)and the above system of equations as follows:

F6,1(x)|x≥L = 0

F6,1(x)|L−l≤x<L = M0

lMC(x)

F6,1(x)|x=L−l = 0

The operating functions of lumped parameters zone are illus-trated in Figure 12. Although they are not employed in this workfor process simulation, their identification paves the path to imple-ment the proposed model in process control algorithms in futureworks.

SEMI-DYNAMIC SIMULATION OF THE PROCESSThe objective of this section is to perform a semi-dynamic sim-ulation of the process using the proposed model structure andidentified operating functions. This section can also be seen asa model validation procedure. However, contrary to many otherprocesses and systems and due to technical constraints and mea-suring difficulties, few observations and measurements of thekiln’s process variables in transient periods are available. There-fore, we limit ourselves in this work to qualitative rather thanquantitative reasoning to validate the simulation results.

The simulation is called semi-dynamics since the couplingbetween Equations (3), (4), and (5) for simplicity reasons is notconsidered. This simplification is justified considering the differ-ent response times of the state variables (long response time oforder of several 10 min for materials temperature and mass distri-bution profiles while short response time of order of some secondsfor gas temperature profile) and the simulation period. We firststart with the freeboard gas temperature profile. The manipulatedvariables are the primary/secondary airflow rate and the amountof the injected fuel while other variables and parameters are keptconstant. As for the first case, we reduce 50% of the airflow ratein the kiln. vG(x) is consequently reduced by 50%. This is sim-ulated by multiplying operating function F1,2(x)—representinggas displacement along the kiln—by 0.5. Equation (3) is thensolved through time using an appropriate solver. The gas tem-perature profile is calculated after 24 s and is shown in Figure13. We observe for the first simulation case that two events takeplace; the peak temperature and equivalently the flame positionmove toward the lower end of the kiln while its absolute valueincreases considerably. As for the second case, we decrease theairflow rate by 50% while the amount of injected fuel is alsodiminished by 80%. Similar to the previous case, the airflowrate reduction is applied by multiplying operating function F1,2(x)by 0.5 and injected fuel rate by multiplying operating functionF1,3(x)—representing heat transferred from the combustion—by0.2. After 24 s, we observe that the peak temperature moves againtoward the lower end of the kiln due to vG(x) reduction while theabsolute temperature also decreases since the amount of injectedfuel is diminished. It should be mentioned that the gas temper-ature does not decrease considerably since a significant amountof heat is always recovered from the hot clinker at the lower end

Figure 13. Evolution of freeboard gas temperature profile insemi-dynamic simulation of the kiln.

Figure 14. Evolution of mass distribution profile in semi-dynamicsimulation of the kiln.

of the kiln. Evidences and experiences on the kiln justify qualita-tively the simulation results while quantitative observations areunfortunately not available at this stage to validate them.

The second semi-dynamic simulation is performed on massdistribution profile while inlet raw materials mass rate is con-sidered as the manipulated variables. Other process variables arekept invariant. The simulation is performed for 90 min applyingM0 = MC(x = 120) shown in Figure 14. The evolution of the massdistribution profile through time for t = 400 and 1800 s is alsoillustrated in the same figure. The mass profile follows the rawmaterials entry pattern which is in accord with the qualitativeevidences on the kiln. However, likewise the previous case, quan-titative observations due to technical constraints are not availablein this stage to validate the results. Meanwhile, a preliminary con-clusion is that the effect of manipulating the raw materials entrywill be seen approximately after 30 min at the other end of thekiln which is quite reasonable for the kiln under study.

CONCLUSION AND PERSPECTIVEIn this work, a new model has been proposed to describe the com-plex dynamic of a CRK. The model structure has been in the form

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of a state-space representation of the process having three statevariables whose evolution through time has been described bya system of partial differential equations. The model parameterscalled operating functions have then been identified both for dis-tributed and lumped parameter zones for a steady-state operationof the kiln through a phenomenological decomposition of the pro-cess into four operating zones. The model has then been used toperform a semi-dynamic simulation of the process to predict theevolution of the freeboard gas temperature and bed materials massdistribution profiles while manipulating some process variables.

The operating functions in this work have been identified for agiven steady-state operation of the kiln and the model would bevalid in the neighbouring of this point. Considering the fact thatCRK is designed to work in a specific operating point—which is inthis case the steady-state point—and is generally operated aroundthis point, model validity can be justified for a large portion ofthe kiln operation time. However, continuous real-time identifica-tion is suggested to re-tune operating functions to follow minorchanges of the kiln’s dynamics.

The ultimate objective of this project is to use the proposedmodel to underpin a kiln’s simulator and a model-based opti-mal/predictive control system of the process. As for the former, asemi-dynamic simulation has been performed to test the feasibil-ity and to validate the results qualitatively. Though the numericalvalues of operating functions are only valid in the neighbouring ofthe current operating state and therefore, simulation may divergefrom the real profiles for long prediction horizons. Determiningthe evolution of the operating functions with respect to the kiln’sdynamics is then the first problem to be tackled in future works.As for the control of the process, the model structure has beendeveloped in the form of a state-space representation providingfacilities to be easily embedded into classical algorithms. How-ever, this should also be scrutinised in more details in furtherstudies.

NOMENCLATUREc(·)(T(·)(x, t)) specific heat of primary air, secondary air, and

fuel at temperature T(·)(x, t)Cm(x) melting percentage of the materials along the kilnCmoisture(x) moisture concentration in raw materialsCp,C(x) specific heat of bed materialsCp,G(x) specific heat of freeboard gasCrelease(x) release mass concentration from the bed to free-

board phaseCW latent heat of water evaporationGf gas flow rateh convection coefficienthf(x,t) heat released per fuel mass unitHC latent heat of materials meltingk(·) thermal conductivity of shell and refractory lay-

ersl transition zone between the distributed and

lumped parameter regionsL Kiln’s lengthm(·)(x, t) primary and secondary air distributed massmG(x) entering freeboard gas mass rateMC(x,t) bed materials mass distributionMf(x,t) injected fuel mass rateMG(x,t) freeboard gas mass distributionq(·)(x) thermal heat exchange rateqF(x) heat rate produced by combustionQair(x,t) heat recycled by primary and secondary air

Qfuel(x,t) heat supplied by the combustion of fuelQtotal(x,t) total heat transferred to the kilnRi(x) thermal resistanceTB(x,t) firebrick temperatureTC(x,t) bed materials temperatureTG(x,t) freeboard gas temperatureTS(x,) shell temperatureTSur(x,t) surroundings temperaturevC(x) bed materials’ velocityvG(x) freeboard gas’ velocity

Greek Symbols˛(x) filling angleˇ(x) a corrective coefficient� inclination angle� = 5.67 × 10−6 W/m2 K4 Stefan–Boltzmann constant

END NOTESAs soon as the missing information would be available, the model

can be updated, respectively. The fact that a phenomenon can beeliminated/embedded easily from/into the model—considering theavailability of the corresponding information—should be regarded asan advantage of this structure that adapt itself to available measuringsystem on the kiln.

APPENDIX A: PHYSICAL AND CHEMICALREACTIONS IN A CLINKER ROTARY KILNA typical initial species mass fraction in raw materials is givenin Table 5. The physical and chemical reactions in the kiln arebriefly mentioned in four operating zones along the kiln’s length.

Drying ZoneIn a wet process, fed raw materials contain 25–40% of moisture.This water is evaporated while the following chemical reactionstake place:

CaO + H2O→Ca(OH)2 + E

CaMgO + H2O→Ca(OH)2 + MgO + E

Preheating ZoneThis zone begins after the drying zone and ends before the temper-ature of clinker reaches 875 K. In this zone, materials are heatedup and the calcination begins. The following reactions take placein this zone:

CaCO3MgCO3 + E→CaOMgO + 2CO2

CaCO3 + E→CaO + CO2

CaO + SO2 + 12

O2→CaSO4

CaSO42H2O + E→CaSO4 + 2H2O

Calcination ZoneThis zone begins where the bed temperature reaches 975 K andends as soon as the temperature achieves 1325 K. Calcination iscompleted in this zone. Other reactions in this zone are as follows:

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Ca(OH)2 + CO2→CaCO3 + H2O

CaCO3 + E→CaO + CO2

Clinkering ZoneIn this zone, the bed materials are partially melted and the fol-lowing reactions take place:

2CaO + SiO2→C2Si + E

C2Si + CaO→C3Si + E

4CaO + Al2O3 + Fe2O3→(CaO)4Al2O3Fe2O3

3CaO + Al2O3→(CaO)3Al2O3

C + O2→CO2 + E

2H2 + O2 + N2→2H2O + N2 + E

2H2 + 2O2 + S→2H2O + SO2

S + O2→SO2

ACKNOWLEDGEMENTSThe present work is made possible partly by grants from theNatural Sciences and Engineering Research Council of Canada(NSERC) and the Research Center on Concrete Infrastructures(CRIB).

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Manuscript received April 23, 2009; revised manuscriptreceived April 20, 2010; accepted for publication April 20, 2010.

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