An Inverse Problem Method for Gas Temperature Estimation ...scientiairanica.sharif.edu/article_2914_095891944dff0ae...Mathematically, inverse heat transfer problems belong to a class

Scientia Iranica, Vol. 15, No. 5, pp 584{595c Sharif University of Technology, October 2008

An Inverse Problem Method for Gas TemperatureEstimation in Partially Filled Rotating Cylinders

M.M. Heydari1 and B. Farhanieh1;�

The objective of this article is to study gas temperature estimation in a partially �lled rotatingcylinder. From the measured temperatures on the shell, an inverse analysis is presented forestimating the gas temperature in an arbitrary cross-section of the aforementioned system. A�nite-volume method is employed to solve the direct problem. By minimizing the objectivefunction, a hybrid e�ective algorithm, which contains a local optimization algorithm, is adoptedto estimate the unknown parameter. The measured data are simulated by adding random errors tothe exact solution. The e�ects of measurement errors on the accuracy of the inverse analysis areinvestigated. Two optimization algorithms are used in determination of the gas temperature. Theconjugate gradient method is found to be better than the Levenberg-Marquardt method, sincethe former produces more accurate results for the same measurement errors. A good agreementbetween the exact value and the estimated result has been observed for both algorithms.

INTRODUCTION

In recent years, much e�ort has been devoted tostudy inverse analysis in many design and manufac-turing problems, especially when direct measurementof surface conditions is not possible. The analysisof Inverse Heat Conduction Problems (IHCP) hasnumerous applications in various branches of scienceand engineering.

Inverse heat transfer problems, in contrast to di-rect heat transfer problems, belong to a set of problemsthat become unstable under small changes in the initialdata. In fact, well and ill-posed problems of mathemat-ical physics are distinguished and di�culties associatedwith the solution of IHTP should also be recognized.Mathematically, inverse heat transfer problems belongto a class called ill-posed [1-7], whereas standard heattransfer problems are well-posed.

Conventionally, the inverse problem includes twophases: The process of analysis and the process ofoptimization. In the analysis process, the unknown pa-rameters or functions are assumed and the results of theproblem are solved directly through either analyticalmethods [8], such as Laplace transform [9], or numeri-

1. Center of Excellence in Energy Conversion, Departmentof Mechanical Engineering, Sharif University of Technol-ogy, P.O. Box 11155-9567, Tehran, Iran.

*. To whom correspondence should be addressed. E-mail:[email protected]

cal methods, such as �nite di�erence and �nite elementmethods [10-14] and dynamic programming [15]. Inthe optimization process, an optimizer, such as in theLevenberge-Marquardt method, conjugated gradientmethod and steepest descent method [16,17], is usedto guide the exploring points systematically to searchfor a new set of guess parameters. These new datacould be substituted for unknown parameters in theanalysis process.

Several numerical and experimental studies havebeen performed for the estimation of unknown pointheat sources and boundary conditions in cylindricalcoordinate systems using IHCPs [18-22]. The investiga-tion of thermal behavior in hollow cylinders employingan inverse methodology has also been reported [23-25].

A literature survey of the available work, pointsout the lack of investigation of thermal behavior in arotating cylindrical container using an inverse method-ology.

A partially �lled rotating cylinder is a funda-mental physical concept that can be used in mathe-matical modeling of many industrial processes, suchas drying, heating (cooling), calcining, separating,mixing, reducing, roasting, sterilizing, food making andmetallurgical processes and in the sintering of a varietyof materials [26-36].

In this study, a parameter estimation of quasi-steady conduction-convection heat problems has beenundertaken using a numerical inverse analysis solverbased on �nite volume formulation in cylindrical coor-

Gas Temperature Estimation 585

dinates. The gas temperature in an arbitrary cross-section of a rotating cylinder has been determinedusing an inverse algorithm. The e�ectiveness of theinverse algorithm was assessed using a sensitivity study.For assurance as to the e�ectiveness of the developedalgorithm, di�erent experimental data were examined.To discuss the correlation between measuring locationsand the accuracy of the results, two types of measuringlocation were considered. Two inverse heat techniqueswere used for hot gas estimation in an arbitrary cross-section of an in�nite long rotating cylinder. Fordamping instabilities in the inverse analysis procedure,iterative parameter estimation techniques were consid-ered.

PROBLEM FORMULATION

The considered partially �lled rotating cylinder ispresented, schematically, in Figure 1. The geometricalproperties are 4.5 m for the internal radius, 6 m forthe external radius and 2.25 m for the bed depth. Therotational speed of the hollow cylinder is 1.5 rad/s andthe used material properties in the direct model aregiven in Table 1.

GOVERNING EQUATION

In the cylinder, the bed (plug ow) region rotates as arigid body about the hollow cylinder axis. A cylindricalcoordinate system is used in this region, as well as forthe wall itself. Energy conservation for any cylindricalcontrol volume in the plug ow region and the wall(refractory lining), as shown in Figure 2, requires the

Figure 1. 3-D schematic presentation of partially �lledrotating cylinder.

Table 1. Relevant physical properties.

Property Charge Wall

k (W/m/�K) 0.692 0.4� (kg/m3) 1680 1334

Cp (J/kg/�K) 1298 1100-1300

Figure 2. Two-dimensional cross-section model.

following:1r@@r

�kir

@T@r

�+

1r2

@@�

�ki@T@�

�+

@@z

�ki@T@z

�= �iCpiu�

@Tr@�

; i = w; p; (1)

where;

u� = r!:

In order to predict temperature distribution in theconsidered long rotating cylinder, the following as-sumptions were made:

1. Due to high length-to-diameter ratio e�ects inthe system, the conduction heat transfer in thelongitudinal direction is negligible, i.e. @Ti@z = 0:0,

2. The charge behavior at the cross-section is as inrolling bed mode,

3. The hydrodynamic behavior of the charge at thebed is as plug ow,

4. Both the hollow cylinder wall and the bed are takento be relatively gray [37-40],

5. The owing gas (freeboard gas) is taken to berelatively gray,

6. There are no radial temperature gradients in thefreeboard gas, so that either phase could be char-acterized by a unique temperature at a given cross-section of the hollow cylinder.

Applying appropriate simpli�cations to the phys-ical model, the governing partial di�erential equation(Equation 1) reduces as follows [28]:

1r@@r

�kir

@T@r

�+

1r2

@@�

�ki@T@�

�= �iCpi!

@T@�

; i = w; p: (2)

586 M.M. Heydari and B. Farhanieh

According to Figure 2, the following boundary condi-tions could be used:

a. At the exposed inner wall for �0 � � � 2� � �0:r = Ri;

�0 � � � 2� � �0 :

�k@T@r

= (hCg!ew + hRg;p!ew)(Tg � T ew): (3)b. At the charge surface (exposed solid surface) for

2� � �0 � � � �0;r = R(�);

2� � �0 � � � �0 :

�k@T@r

= (hCg!p + hRg;ew!p)(Tg � T p): (4)The value of Tg in Equations 3 and 4 should beobtained by inverse analysis.

c. Initial conditions for �;

Ri�Tic�r�Ro :T (r; � = 0) = T (r; � = 2�)

@T (r;�=0)@r =

@T (r;�=2�)@r :

(5)

The radiative heat transfer coe�cients, i.e.hRg;p!ew ; hRg;w!p , can be calculated from a multi-zone radiation model [40,41].

The convective heat transfer coe�cient be-tween the owing hot gas and the exposed wallcan be determined from the following correla-tion [32,40,42,43]:

hCg!ew = 0:036kgD

Re0:8Pr0:33�DL

�0:055: (6)

The convective heat transfer coe�cient from the

owing hot gas to the bed (plug ow) can be de-termined from the following correlation [28,32,40]:

hCg!p = 0:4G0:62g : (7)

d. At the outer hollow cylinder surface (shell) for 0 �� 2�;r = Ro;

0 � � � 2� :

�k@T@r

= (hCsh!1 + hRsh!1)(T sh � T1): (8)

The radiative heat transfer coe�cient at the outershell, hRsh!1 , could be evaluated using:

hRsh!1 =�"sh(T

4sh � T 41)

(T sh � T1) : (9)

The natural convection, (Re2D < GrD), heat trans-fer between the shell surface of the cylinder and thesurroundings could be evaluated by [44]:

Nu = hshDk1

= 0:6

+0:386Ra1=6

[1+(0:559=Pr)9=16]8=27;

10�4


the gradient of S(P), with respect to the vector ofparameter P to zero, that is:

rS(P) = 2"�@T c

T(P )

@P

#[T e � T c(P )] = 0: (13)

Sensitivity Analysis

The solution of inverse problems and the accuracy ofthe estimated parameters and functions are directlyrelated to the sensitivity of the measurements to theerrors in input data, due to the location of the sensorsand frequency of oscillations. The sensitivity matrix isdetermined by di�erentiating Equation 12, with respectto the vector of the unknown parameters or functionsas follows:

J(P ) =@T c

T(P )

@P: (14)

The elements of the sensitivity matrix are called sen-sitivity coe�cients. A sensitivity coe�cient is veryimportant because it indicates the magnitude of achange in the temperature, due to perturbations in thevalue of parameters or functions.

Using the de�nition of sensitivity from Equa-tion 14, the sensitivity coe�cient distribution can bedetermined by di�erentiating governing Equation 2,with respect to the vector of the unknown parametersor functions, P , as follows:

1r@@r

�kir

@J@r

�+

1r2

@@�

�ki@J@�

�= �iCpi!

@J@�; i = w; p: (15)

The initial and boundary conditions read as follows:

r = Rin;

�0 � � � 2� � �0 :

�k@J@r

= (hCg!ew + hRg;p!ew); (16)

r = R(�);

2� � �0 � � � �0 :

�k@J@r

= (hCg!p + hRg;ew!p); (17)

Ri� Tic � r � Ro :J(r; � = 0) = J(r; 2�);

@J(r; � = 0)@r

=@J(r; � = 2�)

@r: (18)

Inverse Analysis Methods

In inverse analysis methods, unknown quantities canbe estimated by minimizing the least squares, Equa-tion 12. These methods, based on estimation of anunknown quantity, can be divided into two groups;parameter estimation and function estimation. Thereare several parameter estimation methods available forinverse analysis to treat the ill-posed nature of inverseproblems, e.g., the least squares method modi�ed bythe addition of a regularization term, and the use ofa conjugate gradient method, where the regularizationis inherently built into the iterative procedure [1,2,5,6].There are some powerful techniques for solving inverseheat transfer problems. These techniques are asfollows [24]:1. Levenberg-Marquard method for parameter estima-

tion,2. Conjugate Gradient method for parameter estima-

tion,3. Conjugate Gradient method with an adjoint prob-

lem for parameter estimation,4. Conjugate Gradient method with an adjoint prob-

lem for function estimation.The parameter estimation approaches are, in general,nonlinear and the number of the unknown parametersis strongly limited [7]. However, in the parameter esti-mation methods, there is a priori information regardingthe functional forms of the unknown parameters andtheir locations. It is also generally assumed thatthe number of these parameters/functions and theircoe�cients are known.

However, for cases with no priori informationof the functional forms, one must use the functionestimation methods of the inverse analysis, such asthe adjoint conjugate gradient method and the func-tion speci�cation method. The solution of functionestimation problems is highly sensitive to noise in themeasured data. Even if data were measured exactly,the inverse function estimation problems would bedi�cult. However, for function estimation, if someinformation is available on the functional form of theunknown function, the inverse problem of estimatingthis function is reduced to estimating a �nite numberof parameters.

The Levenberg-Marquardt and the conjugate gra-dient methods, based on minimization of the objectivefunction, are the two iterative algorithms used in thisstudy for parameter estimations. These inverse heatconduction methods for estimating unknown parame-ters and functions are conducted for certain sequentialitems, i.e. solution of the direct problem, solution ofthe inverse problem and a convergence criterion. Thesemethods can be suitably arranged in iterative proce-dures. In this research, both Levenberg-Marquardt


and Conjugate Gradient methods are used for gastemperature estimation.

Levenberg-Marquardt MethodIn this parameter estimation method, for the minimiza-tion of Equation 12, its gradient, with respect to thevector of the unknown parameters, must be equated tozero.

Using the de�nition of the sensitivity matrix,Equation 13 could be written as:

�2JT (P )[T e � T c(P )] = 0: (19)The solution of Equation 19 for nonlinear estimationproblems requires an iterative procedure, which isobtained by linearizing the vector of the estimatedtemperatures, T ci (P), with a Taylor series expansionaround the current solution, Pk, at iteration k. Such alinearization is given by:

T c(P ) = T c(P k) + Jk(P � P k): (20)Equation 20 is substituted into Equation 19 and theresulting expression is rearranged to yield the followingiterative procedure, in order to obtain the vector ofthe unknown parameters, P. Inverse problems aregenerally very ill conditioned, especially near the initialguess used for the unknown parameters. Levenberg-Marquardt alleviates such di�culties by utilizing aniterative procedure in the following form [24,25]:

P k+1 =P k+[(Jk)TJk+�kk]�1(Jk)T [T e�T c(P k)];(21)

where k is a diagonal matrix used for alleviatingthe ill-conditioned problem and �k is a positive scalarcalled the damping factor.

Conjugate Gradient MethodThe conjugate gradient method is a powerful itera-tive parameter or function-estimation algorithm forthe solution of linear and nonlinear inverse problems.In this method, the following principle relations areconsidered [24,25].

In the conjugate gradient method, the minimiza-tion procedure is performed by using the followingiteration:

P k+1 = P k � �kdk: (22)At each iteration step, a suitable step size, �k, is takenalong the direction of descent in order to minimize theobjective function.

The direction of the descent is obtained as a linearcombination of the negative gradient direction at thecurrent iteration, with the direction of descent of theprevious iteration, which is as follows:

dk = rS(P k) + kdk�1; (23)

where:

k =

NPj=1f[rS(P k)]j [rS(P k)�rS(P k�1)]jg

NPj=1

[rS(P k)]2j;

0 = 0: (24)

The gradient direction vector is determined as:

rS(P k) = �2(Jk)T [T e � T c(P k)]: (25)Temperature vector, T (P k � �kdk), can be linearizedwith a Taylor series expansion and, then, the mini-mization, with respect to �k, is performed to yield thefollowing expression for the search step size:

�k =

NPj=1

[Jkdk]T [T c(P k)� T e]NPj=1f[Jk]T dkg2

: (26)

Convergence CriterionThe above-mentioned iterative algorithms need a crite-rion to stop the iterative procedure of the solution. Thefollowing condition is used for terminating the iterativeprocess:

S(P k+1) < "1;

(Jk)T [Y � T (P k)]

< "2;

P k+1 � P k

< "3: (27)The solution of inverse problems using an iterativealgorithm may become well posed if the convergencecriterion is used to stop the iterative procedure.

NUMERICAL ANALYSIS

Direct Method Algorithm

The �nite volume-method is employed to solve the bed-wall governing equation for the cross-section of thepartially �lled rotating cylinder. Rectangular gridswith cylindrical control volumes are used in both theplug ow (charge) and the wall regions. The gridindependency study, as shown in Figure 3, on the meshin the solution domain, indicates that the 131 � 45grids in � and r directions, respectively, are su�cientto produce grid independent results. The generatedmesh for the computational domain is presented inFigure 4.


Figure 3. The e�ect of di�erent grid sizes on radialtemperature distribution.

Figure 4. Computational grid for partially �lled rotatingcylinder (cross section).

Inverse Method Algorithm

A quasi-steady-state heat conduction-convection in-verse problem was developed for the partially �lledrotating cylinder. As shown in Figure 5, the gastem perature in the arbitrary cross-section, Tgas, couldbe estimated using the inverse problem method. Forestimation of the unknown parameter, the objectivefunction, S(Pk), should be optimized. This objectivefunction is produced, based on the measured and com-putational temperature di�erences at the shell surfaceof the rotating hollow cylinder. Due to employmentof the L.M.M. algorithm, Equations 20 or 21 shouldbe computed iteratively. To perform the iteration,according to C.G.M, the direction of the descent vector,dk, in Equation 23 and the step size, �k, in Equation 26

Figure 5. Layout of cross-section model withthermocouple arrangement shown.

require computation. For both optimization algo-rithms, the iteration producer is continued until oneof the criteria introduced in Equation 27 is satis�ed.

RESULTS

The properly developed inverse algorithm should leadto the correct value of the parameter in question. Thee�ectiveness of the inverse algorithm can be assessedusing the sensitivity study. To assess the e�ectivenessof the developed algorithm, di�erent experimental datawith 0%, 1% and 5% error were examined. To discussthe correlation between the measuring locations andthe accuracy of the results, two types of measuringlocation were considered. The �rst location is theentire perimeter of the shell surface of the rotatingcylinder's cross-section and the second location is thebedside zone of the shell. For each location, threesets of thermocouple arrangement, as shown in Fig-ure 5, were considered. Both Levenberg-Marquardtand Conjugate-Gradient optimization algorithms wereapplied for each thermocouple arrangement.

Sensitivity Analysis

Sensitivity coe�cients are, in some sense, the key to thesuccess of the estimation procedures of the parameterand verify the inuence of the measured parameters(measured temperatures). The sensitivity variationsfor gas temperature estimation are computed in fourdi�erent layers of the rotating cylinder wall and arepresented in Figure 6. As seen from this result, thesensitivity coe�cient for gas temperature estimation atthe inner wall is better, with respect to the outer wall(shell), for all thermocouple arrangements. In addition,the sensitivity coe�cient in the bedside zone of thecross-section (at inner or outer walls) is considerable


Figure 6. Sensitivity study for gas temperatureestimation in di�erent wall layers of cross section.

with respect to other zones. Therefore, it is expectedthat computational analysis based on covered wallactual temperature data, converges faster comparedwith computational analysis based on other isoclinesexperimental data.

Experimental Data

In order to simulate the measured temperature,Tmeasure, (by imaginary thermocouples) realistic noisesare introduced to contaminate the theoretical tem-perature, Texact, computed from the solution of thedirect problem. The results involving the random mea-surement errors, the normally distributed uncorrelatederrors with non-zero mean and constant standard devi-ation were assumed. Thus, the measured temperature,Tmeasure, could be expressed as follows:

Tmeasure = Texact(1 + ��); (28)

where Texact is the solution of the direct problem withthe exact gas temperature, 1600�K, � is the standarddeviation of the measurement and � is the randomvariable, which is within �2:576 � � � 2:576 for99.43% con�dence bound [16,22-25]. The producedmeasuring data for shell and bedside zones of a cylinderwith 0%, 1% and 5% errors are shown in Figures 7and 8, respectively.

Numerical Test Case 1

As shown in Figure 5, 27, 33 and 44 thermocouples,in turn, are arranged at the perimeter shell surfaceat an arbitrary cross-section of the rotating cylinder.The results of this inverse study for C.G.M and L.M.Malgorithms are shown in Figures 9 and 10, respectively.The options (a), (b) and (c) are related to the 26,

Figure 7. Experimental temperature with di�erent errorfor (a) 27, (b) 33 and (c) 44 thermocouples arrangementon the shell.

33 and 44 thermocouples, respectively. These �guresillustrate that the estimated results have excellentapproximations with the exact value of the gas tem-perature when the measurement error is not consideredto be � = 0:0. In addition, the estimated results


Figure 8. Experimental temperature with di�erent errorfor (a) 18, (b) 26 and (c) 51 thermocouples arrangementon the bedside zone.

are in good agreement with the aforementioned exactvalue, even when the measurement error, � = 0:01,is considered. These results show that the accuracyof the gas temperature increases as the number ofthermocouples are increased to an optimum value.

Figure 9. Gas temperature estimation for (a) 27, (b) 33and (c) 44 thermocouples on the shell based on C.G.M.algorithm.

For example, as shown in Figures 9 and 10, it isobvious that an optimum value for the thermocouplesarrangement is 33. Also, the results obtained by usingC.G.M are better than the results obtained by usingL.M.M. regarding accuracy and CPU time.


Figure 10. Gas temperature estimation for (a) 27, (b) 33and (c) 44 thermocouples on the shell based on L.M.M.algorithm.

Numerical Test Case 2

18, 26 and 51 thermocouples, in turn, are arrangedat an arbitrary cross-section on the bedside zone ofthe shell of the rotating cylinder. The results of

Figure 11. Gas temperature estimation for (a) 18, (b) 26and (c) 51 thermocouples on the bedside zone based onC.G.M. algorithm.

this test case are presented in Figures 11 and 12.These �gures illustrate that the estimated results haveexcellent approximations with the exact value of thegas temperature when the measurement error is notconsidered to be � = 0:0. Also, the estimated results


Figure 12. Gas temperature estimation for (a) 18, (b) 26and (c) 51 thermocouples on the bedside zone based onL.M.M. algorithm.

are in good agreement with the aforementioned exactvalue, even when the measurement error, � = 0:01,is considered. In addition, these results show thatthe accuracy of the gas temperature increases as thenumber of thermocouples is increased to an optimum

value. As shown in Figures 11 and 12, it is obvious thatan optimum value for thermocouple arrangement is 10.Also, the results obtained using C.G.M are better thanthe results obtained using L.M.M. regarding accuracyand CPU time.

CONCLUSION

In this paper, a method for estimating the hot gastemperature in a partially �lled rotating cylinder isdeveloped. This parameter is estimated by an ex-perimental temperature via an inverse method. Twoexamples were used to show the robustness of theproposed method. The Levenberg-Marquardt methodand the conjugate gradient method with a �nite vol-ume approach were successfully applied for solutionof the inverse problem, in order to determine the gastemperature, by utilizing temperature readings at theshell of the rotating cylinder. The results show thatthe C.G.M does not require an accurate initial guessof unknown quantities, needs very short CPU time andrequires fewer sensors, when compared with the L-MM,in performing the inverse calculations.

From the results, it appears that, by using theproposed method, without measurement error, theexact solution (exact value for gas temperature) can befound, even with only few measurement points. Whenmeasurement errors are included, in order to enhancestability and accuracy, temperature data requires moremeasuring points on the shell surface. Furthermore,fewer measurement points on the bedside of the shellsurface, with respect to the whole surface of the shell,may be used to obtain accurate results.

Consequently, the results con�rm that the pro-posed inverse method is e�ective and e�cient for gastemperature estimation in a partially �lled rotatingcylinder.

NOMENCLATURE

A areaCp speci�c heatD hollow cylinder diameterd direction of descent vectorE emissive powerG mass ow rate of gash convection coe�cientJ sensitivity matrixJ radiosityk conductivityM number of measuring TemperatureN number of unknownsNu Nusselt number


P estimated parameterPr Prandtl numberr polar coordinateRi inner radius of hollow cylinderRo outer radius of hollow cylinderRe Reynolds numberRa Rayleigh numberS square root of errorsT temperatureT average temperatureTic bed depth� absorptivity� search step size (descent parameter)

conjugate coe�cient! hollow cylinder angular velocity" emissivity"1; "2; "3 tolerance for convergence of the

iterative inverse methods�r reectivity� angular coordinate� density� measurement error�s Stephan-Boltzman constant� the probability or random value� dynamic viscosity�0 �ll percent� dimensionless temperature�S gradient of S

Superscript

c computationale experimentalk iteration stepT transpose� average valueSubscript

a;1 ambientp plug ow (bed)cw covered wallew exposed wallexact exact valueg gasi measured parameter indexj unknown parameter indexmeasure measurement valueover overallsh shellw wall

REFERENCES

1. Hadamard, J., Lectures on: Cauchy's Problem inLinear Di�erential Equations, Yale University Press,New Haven, CT (1923).

2. Tikhonov, A.N. and Arsenin, V.Y., Solution of IllPosed Problems, Winston & Sons, Washington, DC(1977).

3. Backus, G. and Gilbert, P. \Numerical applicationsof a formalism for geophysical inverse problems",Geophys J. Roy. Astron, 13, pp 247-276 (1967).

4. Beck, J.V. and Arnold, K.J., Parameter Estimation:in Engineering Estimation, John Wiley & Sons Pub.(1977).

5. Beck, J.V. and Blackwell, B., Inverse Heat Conduc-tion: Ill-Posed Problems, Wiley, New York (1985).

6. Beck, J.V., Blackwell, B. and St. Clair, C.R., InverseHeat Conduction-Ill Posed Problem, Wiley, New York(1985).

7. Mozorov, V.A. and Stressin, M. \RegularizationMethod for Ill-posed problem", CRC Press Boca Ra-ton, FL (1993).

8. Burggraf, O.R. \An exact solution of the inverseproblem in heat conduction theory and applications",ASME Journal of Heat Transfer, 84, pp 373-382(1964).

9. Arledge R.G. and Haji-Sheykh, A. \An iterative ap-proach to the solution of inverse conduction heatproblem", Numerical Heat Transfer, 1, pp 365-376(1978).

10. Fidelle, T.P. and Zinsemeister, G.E. \A semi-discreteapproximate solution of the inverse problem of tran-sient heat conduction", A.S.M.E. Paper No. 68-WA/HT-26 (1968).

11. Beck, J.V. \Nonlinear estimation applied to the non-linear inverse heat conduction problem", InternationalJournal of Heat and Mass Transfer, 13, pp 703-716(1970).

12. D'Souza, N. \Numerical solution of one-dimensionalinverse transient heat conduction by �nite di�erencemethod", A.S.M.E. Paper No. 75-WA/HT-81 (1975).

13. Krutz, G.W., Schoenhals, R.J. and Hore, P.S. \Appli-cation of the �nite element method to the inverse heatconduction problem", Numerical Heat Transfer, 1, pp489-498 (1978).

14. Beck, J.V., Litkouhi, B. and St Clair, Jr, C.R.,\E�cient sequential solution of the nonlinear heatconduction problem", Numerical Heat Transfer, 5, pp272-286 (1982).

15. Beck, J.V. and Litkouhi, B. \E�cient sequentialsolution of the nonlinear heat conduction problem",Numerical Heat Transfer B, 27, pp 291-307 (1995).

16. Necati Ozisik, M., Inverse Heat Transfer, Taylor &Francis 29 West 35th Street, New York, NY (2000).

17. Liu, G.R. and Han, X., Computational Inverse Tech-niques in Nondestructive Evaluation, CRC PRESSLLC (2003).


18. Su, J. and Silva Neto, A.J. \Two-dimensional inverseheat conduction problem of source strength estimationin cylindrical rods", Appl. Math. Modeling, 25, pp 861-872 (2001).

19. Raynaud, M. \Some comments on the sensitivity tosensor location of inverse heat conduction problemsusing Beck's method", Int. J. Heat Mass Transfer,29(5), pp 815-817 (1986).

20. Bartsch, G., Schroeder-Richter, D. and Huang, X.C.\Quenching experiments with a circular test section ofmedium thermal capacity under forced convection ofwater", Int. J. Heat Mass Transfer, 37(5), pp 803-818(1994).

21. Nguyen, K.T. and Bendada, A. \An inverse approachfor the prediction of the temperature evolution duringinduction heating of a semi-solid casting billet", Mod-elling Simul. Mater. Sci. Eng., 8, pp 857-870 (2000).

22. Lin, J.H., Chen, C.K. and Yang, Y.T. \Inverse esti-mation of the thermal boundary behavior of a heatedcylinder normal to a laminar air stream", Int. J. HeatMass Transfer, 43(21), pp 3991-4001 (2000).

23. Lin, J.H., Chen, C.K. and Yang, Y.T. \An inversemethod for simultaneous estimation of the center andsurface thermal behavior of a heated cylinder normalto a turbulent air stream", Int. J. Heat Mass Transfer,124, pp 601-608 (2002).

24. Yang, Y.T., Hsu, P.T. and Chen, C.K. \A three-dimensional inverse heat conduction problem approachfor estimating the heat ux and surface temperatureof a hollow cylinder", Journal of Physics D: AppliedPhysics, 30, pp 1326-1333 (1997).

25. Hsu, P.T., Yang, Y.T. and Chen, C.K. \Simultane-ously estimating the initial and boundary conditionsin a two-dimensional hollow cylinder", InternationalJournal of Heat and Mass Transfer, 41(1), pp 219-227(1998).

26. Brimacombe, J.K. and Watkinson, A.P. \Heat transferin direct-�red rotary Kiln, I- Pilot plant and experi-mentation, II- Heat ow results and their interpreta-tion", Met. Trans. B., l9B, pp 201-219 (1978).

27. Barr, P.V., Brimacombe, J.K. and Watkinson, A.P.\A heat transfer model for the rotary kiln, Part I.Pilot hollow cylinder trials, Part II. Development ofcross-section mode", Met. Trans. B., 20B, pp 391-419(1989).

28. Boateng, A.A. \Rotary kiln transport phenomena;study of the bed motion and heat transfer", PhD.Dissertation, The University of British Columbia, Van-couver (1993).

29. Marcio, A. and Learndo, S. \Modeling and simulationof petroleum coke calcinations in rotary kiln", FUEL,80, pp 1-15 (Jan. 2001).

30. Baker, C.G.J. \Cascading rotary dryers", Advances inDrying, 2, pp 1-51 (1983).

31. Maciej, S. and Mohamed, M. \Modeling of rotatingdrum dryer for sugar", AVH Association - 11th Sym-posium - Reims (March 2004).

32. Kadir, B.B. \Flow phenomena in horizontal axiallyrotated partially �lled cylinder", Thesis for Degree ofDoctor of Philosophy, Dalhousie University (1998).

33. Carslow, H.S. and Jager, J.C., Conduction of Heat inSolid, Clarendon Press, Oxford, GBR (1973).

34. Cannon, J.N. \Heat transfer to a uid owing insidea pipe rotating about its longitudinal axis", J. HeatTransfer, 91, pp 135-139 (1969).

35. Gavish, J., Chadwick, R.S. and Gut�nger, C. \Viscous

ow in a partially �lled rotating horizontal cylinder",Israel J. Techno, 1(16), pp 264-272 (1978).

36. Heydari, M.M. and Farhanieh, B. \Bed depth esti-mation in rotary kiln by inverse problem method",International Conference on Numerical Analysis andApplied Mathematics, WILEY (2005).

37. Gorog, J.P., Brimacombe, J.K. and Adams, T.N.\Radiative heat transfer in rotary kiln", MetallurgicalTransaction B, 12B, pp 55-70 (March 1981).

38. Gorog, J.P., Brimacombe, J.K. and Adams, T.N. \Re-generative heat transfer in rotary kiln", MetallurgicalTransaction B, 13B, pp 153-163 (June 1982).

39. Gorog, J.P., Brimacombe, J.K. and Adams, T.N.\Heat transfer from ames in rotary kiln", Metallurgi-cal Transaction B, 14B, pp 411-424 (Sept. 1982).

40. Ghoshdastidar, P.S. and Anandan Unni, V.K. \Heattransfer in the non-reacting zone of a cement rotarykiln", ASME Journal of Engineering for Industry, 118,pp 169-172 (1996).

41. Siegel, R. and Howell, J.R., Thermal Radiation HeatTransfer, McGraw-Hill, New York (1981).

42. Tscheng, S.C. and Watkinson, A.P. \Convective heattransfer in a rotary kiln", The Canadian J. of Chem.Eng., 57 (1979).

43. Kreith, F. and Black, W.Z., Basic Heat Transfer,Harper & Row, New York, NY (1980).

44. Ozisik M.N., Heat Transfer, 2nd Ed Montr�eal, Mc-Graw Hill (1985).

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