ESTIMATION OF THE HEAT TRANSFER COEFFICIENT IN THE SPRAY COOLING OF CONTINUOUSLY CAST SLABS

Transcription

1 ESTIMATION OF THE HEAT TRANSFER COEFFICIENT IN THE SPRAY COOLING OF CONTINUOUSLY CAST SLABS Helcio R. B. Orlande and Marcelo J. Colaço Federal Univerity of Rio de Janeiro, UFRJ Department of Mechanical Engineering, EE/COPPE Cx. Potal 6853, Rio de Janeiro, RJ, 95-97, Brail FAX: , Alexandre A. Malta COSIPA, Steel Company of São Paulo Cubatão, Brail ABSTRACT An invere problem involving the etimation of the heat tranfer coefficient at the urface of a plate, with no information regarding the functional form of the unnown, i olved by applying the conjugate gradient method with adjoint equation. Thi paper i part of an experimental and numerical imulation of the actual pray cooling proce in continuou cating machine. Reult obtained with imulated meaurement, for the etimation of the time and patial variation of the unnown pray heat tranfer coefficient are ummarized. The conjugate gradient method i found to provide accurate etimate for the unnown, even for function containing harp corner and dicontinuitie, which are the mot difficult to be recovered by invere analyi. The effect of number and location of enor on the invere problem olution are alo addreed on the paper. NOMENCLATURE A dimenionle length of the plate B dimenionle width of the plate Bi dimenionle heat tranfer coefficient (θ) dimenionle temperature-dependent pecific heat d direction of decent given by equation (.b) e RMS RMS error defined by equation (3) J functional defined by Eq. () J gradient of the functional given by Eq. (3) K(θ) dimenionle temperature-dependent thermal conductivity S number of enor X, Y, Z dimenionle coordinate GREEKS β earch tep ize given by Eq. (5) θ enitivity function; olution of the enitivity problem given by Eq. (7) γ conjugation coefficient given by Eq. (.c) λ Lagrange multiplier; olution of the adjoint problem given by Eq. () µ dimenionle meaured temperature θ dimenionle temperature; olution of the direct problem given by Eq. () σ tandard deviation of the meaurement dimenionle time dimenionle final time f SUBSCRIPTS refer to the enor number o reference value ε perturbed quantity SUPERESCRIPTS * dimenional property number of iteration INTRODUCTION The pray cooling technique utilized in continuou cating have direct influence on the temperature ditribution in the lab, a well a on the local thicne of the olidifying hell. Thermal tree arie in continuouly cat lab due to irregular urface cooling, while mechanical tree come into picture becaue of the preure exerted

2 by the machine roll. Hence, the ability to accurately control and predict the behavior of uch a cooling ytem can reduce defect caued by thermal and mechanical tree in continuouly cat product (Hibbin and Brimacombe, 98, Kohno et al, 98). Different tudie can be found in the literature on the etimation of the heat tranfer coefficient of pray cooling ytem, and how it i affected by different parameter, including average water flux, lab urface temperature, ditance of the pray to the lab, etc. (Brimacombe et al, 98, Hibbin and Brimacombe, 98, Miziar, 97). However, uch tudie were baed on average value for the heat tranfer coefficient, or the analyi did not involve the olution of an invere problem, but a comparion of meaured and etimated temperature by trial-and-error. In thi paper we olve the invere problem of etimating the heat tranfer coefficient of an air-mit pray, by uing a function etimation approach baed on the conjugate gradient method with adjoint equation. The heat tranfer coefficient i allowed to vary with the poition over the urface of the lab, a well a with time. The conjugate gradient method of function etimation i a powerful iterative technique, which ha been uccefully applied to the olution of linear and non-linear invere problem (Jarny et al, 99, Orlande and Ozii, 993, 99, Alifanov, 99, Huang et al, 995, Danta and Orlande, 996, Machado and Orlande, 997). We ue here imulated experimental data in order to ae the accuracy of uch a method, a applied to the etimation of the heat tranfer coefficient. Different functional form were teted in order to generate the imulated data, including thoe containing harp corner and dicontinuitie, which are the mot difficult to be recovered by invere analyi. The ue of imulated data alo permitted the planning of an experimental apparatu, currently under contruction, with repect to the number and location of enor required to obtain accurate etimate for the unnown function. PHYSICAL PROBLEM The phyical problem conidered here involve a laboratory imulation of the actual cooling proce of continuouly cat lab. A teel plate i heated up to temperature of the order of o C. It top urface i then cooled by an air-mit pray, and tranient temperature recording are taen at everal location inide the plate. The plate lateral urface are ept inulated, while the tranient temperature of the bottom urface i meaured with an infrared radiation pyrometer. The etimation of the heat tranfer coefficient of the air-mit pray, by uing the conjugate gradient method with adjoint equation, conit of the following baic tep: direct problem, invere problem, enitivity problem, adjoint problem, gradient equation, conjugate gradient method of minimization, topping criterion and computational algorithm. We preent below the detail of the each of thee ditinct tep. DIRECT PROBLEM The direct problem i concerned with the determination of the temperature field in the plate, when the pray heat tranfer coefficient, a well a the phyical propertie, initial condition and Z -A/ -B/ B/ A/ X Figure - Geometry and Dimenionle Coordinate other parameter appearing in boundary condition are nown. The mathematical formulation of thi heat conduction problem i given in dimenionle form by Y θ K + Y K Y + K Y -A/<X<A/, -B/<Y<B/, <Z<, for > at X-A/, XA/, for > at Y-B/, YB/, for > in (.a) (.b,c) (.d,e) θ ϕ ( X, Y, ) at Z, for > (.f) θ K + Biθ at Z, for > (.g) θ φ ( X, Y, Z ) for, in -A/<X<A/, -B/<Y<B/, <Z< (.h) Figure illutrate the geometry and coordinate, while variou dimenionle group are defined a a b x y z A, B, X, Y, Z (.a-e) c c c c c * Ko t hc T T, Bi, θ * ρ * Ko T o T Co c * (.f-h) where a, b and c are the length, width and thicne of the plate, repectively, while T i the cooling fluid temperature and T o i a reference temperature for the plate. In order to write the direct problem in dimenionle form, we aumed the temperature dependence of thermal conductivity and pecific heat to be in the form:

3 * K ( T ) * K o K ( θ ) * * ( T ) Co ( θ ) (3.a) (3.b) where K * and C * are reference value for thermal conductivity and pecific heat, repectively, while K(θ) and (θ) are dimenionle function of θ. The lab denity ρ* wa aumed contant. The upercript * above refer to dimenional phyical propertie. INVERSE PROBLEM For the invere problem, the dimenionle heat tranfer coefficient of the air-mit pray, Bi(X,Y, ), i regarded a unnown. Such a function i to be etimated by uing the tranient reading of S temperature enor located inide the plate, at poition (X, Y, Z ),,,S, during the time interval f. An etimate for Bi(X, Y, ) i obtained o that the following functional i minimized: J S X Y Z () f [ Bi ( X, Y, )] [ θ (,,, ; Bi) µ ( ) ] d where θ(x, Y, Z, ;Bi) and µ () are the etimated and meaured temperature at the meaurement location, repectively. The etimated temperature are obtained from the olution of the direct problem by uing an etimate for Bi(X, Y, ). In order to apply the conjugate gradient method for minimizing the functional given by Eq. (), we need to develop and olve two auxiliary problem, nown a the enitivity and adjoint problem, a decribed next. SENSITIVITY PROBLEM The enitivity problem i developed by auming that the temperature θ(x, Y, Z, ) i perturbed by an amount ε θ(x, Y, Z, ), when the Biot number Bi(X, Y, ) i perturbed by ε Bi(X, Y, ), where ε i a real number. Due to the non-linear character of the problem, a perturbation on temperature caue perturbation on the temperature dependent propertie, a well. Thu, we can write the following perturbed quantitie: Biε ( X, Y, ) Bi ( X, Y, ) + ε Bi ( X, Y, ) (5.a) θ ε ( X, Y, Z, ) θ ( X, Y, Z, ) + ε θ ( X, Y, Z, ) (5.b) d ( θ ε ) ( θ ) + ε θ (5.c) dθ dk K ( θ ε ) K ( θ ) + ε θ (5.d) dθ The enitivity problem i obtained by applying the following limiting proce: L ( ) ( ) lim ε Biε L Bi ε ε where L ε (Bi ε ) and L(Bi) are the operator form of the direct problem, written for the perturbed and unperturbed quantitie, repectively. The following problem reult for the enitivity function θ(x, Y, Z, ): ( K θ ) ( K θ ) ( K θ ) ( θ ) + + Y ( K θ ) ( K θ ) Y in -A/<X<A/, -B/<Y<B/, <Z<, for > at X-A/, XA/, for > at Y-B/, YB/, for > (6) (7.a) (7.b,c) (7.d,e) θ at Z, for > (7.f) ( K θ ) + Bi θ Biθ at Z, for > (7.g) θ for in -A/<X<A/, -B/<Y<B/, <Z< (7.h) ADJOINT PROBLEM An adjoint problem for a Lagrange Multiplier come into picture, becaue the temperature θ(x, Y, Z, ; Bi) appearing in the functional () need to atify a contraint, given by the olution of the direct problem. In order to develop the adjoint problem, we multiply the differential equation (.a) of the direct problem by the Lagrange Multiplier λ(x, Y, Z, ), integrate over the time and pace domain, and add the reulting expreion to the functional given by Eq. (). The following extended functional i obtained: J S A B / / f [ Bi] [ θ ( ) µ ( ) ] X A/ Y B/ Z δ ( r r ) d A/ B/ f X A/ Y B/ Z θ K K Y Y dz dy dx + θ θ K λ( X, Y, Z, ) d dz dy dx (8)

4 where δ( ) i the Dirac delta function and r i the vector with the poition of enor, i.e., r (X, Y, Z ). An expreion for the directional derivative of J[Bi] in the direction of the perturbation Bi(X, Y, ) i obtained by applying the following limiting proce: D J Bi [ Bi] lim ε J [ Biε ] J [ Bi ] where J[Bi ε ] i the functional (8) written for the perturbed quantitie given by Eq. (5). After performing ome integration by part on the reulting expreion for D Bi J[Bi] and applying the boundary and initial condition of the enitivity problem, we let the term containing θ(x, Y, Z, ) to go to zero. The following adjoint problem i then obtained for the Lagrange Multiplier λ(x, Y, Z, ): + S λ λ λ λ K K K ( θ ) δ ( r r ) µ Y in -A/<X<A/, -B/<Y<B/, <Z<, for > (.a) λ λ Y at X-A/, XA/, for > ε at Y-B/, YB/, for > (9) (.b,c) (.d,e) λ at Z, for > (.f) λ K + Biλ at Z, for > (.g) λ for f in -A/<X<A/, -B/<Y<B/, <Z< (.h) GRADIENT EQUATION In the proce of obtaining the adjoint problem, the following integral term i left: D A/ f J[ Bi] Bi Y X A / Bi( X, Y, ) dy B/ B / dx d J ' ( X, Y, ) () Therefore, by comparing Eq. () and (), we obtain the gradient equation for the functional a J ' ( X, Y, ) λ ( X, Y,, ) θ ( X, Y,, ) (3) CONJUGATE GRADIENT METHOD OF MINIMIZATION The iterative procedure of the conjugate gradient method (Jarny et al, 99, Alifanov, 99), a applied to the etimation of the unnown heat tranfer coefficient, i given by Bi + ( X, Y, ) Bi ( X, Y, ) β d ( X, Y, ) (.a) where the upercript denote the number of iteration. The direction of decent i a conjugation of the gradient direction and of the previou direction of decent, given in the form d ( X, Y, ) J' ( X, Y, ) + γ d ( X, Y, ) (.b) The conjugation coefficient utilized here wa obtained from the Fletcher-Reeve expreion (Alifanov, 99): γ A/ B/ f X A / Y B / A/ B/ f X A/ Y B / ' J ( X, Y, ) ' J ( X, Y, ) with γ d dy dx d dy dx for,, (.c) The earch tep ize β i obtained by minimizing the functional J[Bi + ] given by Eq. () with repect to β. The following expreion reult: D A/ f J[ Bi] Bi X A/ Y B/ B / θ ( X, Y,, ) Bi( X, Y, ) dy λ( X, Y,, ) dx d () By auming that Bi(X, Y, ) belong to the Hilbert pace of quare integrable function in the domain (, f ) x (-A/, A/) x (-B/, B/), we can write β f S f [ θ ( ) µ ( )] S θ ( d ) θ ( d ) d d (5) where θ (d ) i the olution of the enitivity problem given by Eq. (7), obtained by etting Bi(X, Y, ) d (X, Y, ).

5 STOPPING CRITERION We top the iterative procedure of the conjugate gradient method when the functional given by Eq. () become ufficiently mall, that i, + J Bi ( X, Y, ) < ε (6) If the meaurement are aumed to be free of experimental error, we can pecify ε a a relatively mall number. However, actual meaured data contain experimental error, which will introduce ocillation in the invere problem olution, a the etimated temperature approach thoe meaured. Such difficulty can be alleviated by utilizing the Dicrepancy Principle (Alifanov, 99) to top the iterative proce. Hence, we aume that the invere problem olution i ufficiently accurate when the difference between etimated and meaured temperature i le than the tandard deviation, σ, of the meaurement. Thu, the value of the tolerance ε i obtained from Eq. () a ε Sσ (7) f COMPUTATIONAL ALGORITHM Suppoe an etimate Bi (X, Y, ) i available for the unnown heat tranfer coefficient Bi(X, Y, ) at iteration. Thu: STEP : Solve the direct problem given by Eq. () to obtain the etimated temperature θ(x, Y, Z, ); STEP : Chec the topping criterion given by Eq. (6) Continue if not atified; STEP 3: Solve the adjoint problem given by Eq. () to obtain the Lagrange Multiplier λ(x, Y, Z, ); STEP : Compute the gradient of the functional J (X, Y, ) from Eq. (3); STEP 5: Compute the conjugation coefficient γ from Eq. (.c) and then the direction of decent d (X, Y, ) from Eq. (.b); STEP 6: Solve the enitivity problem given by Eq. (7) to obtain θ(x, Y, Z, ) by etting Bi(X, Y, ) d (X, Y, ); STEP 7: Compute the earch tep ize β from Eq. (5); STEP 8: Compute the new etimate Bi + (X, Y, ) from Eq. (.a) and return to tep. RESULTS AND DISCUSSION A a tet-problem, we conider here the cooling of a teel plate with dimenion a.3m, bm and c.m, initially at a uniform temperature T o C. For time t>, the top urface of the plate i cooled by an air-mit pray, and the temperature of the cooling fluid i uppoed to be T 5 o C. The bottom urface of the plate i uppoed to be maintained at the contant temperature of 5 o C. The phyical propertie of the plate are aumed contant and their value are taen a (Ozii, 993): ρ*7753 Kg/m 3, K *36 W/m o C and C *.86 KJ/Kg o C. The final experimental time i taen a t f.7 min. The dimenionle variable aociated with the value above are: A3, B, f., ϕ and φ. We ue imulated experimental data in order to ae the accuracy of the conjugate gradient method with adjoint equation, a applied to the etimation of Bi(X, Y, ). For the olution of the preent invere problem, tranient meaurement of multiple enor are required, in order to recover the patial and time dependencie of the unnown function. The imulated experimental data were obtained from the olution of the direct problem for an a priori aumed functional form for Bi(X, Y, ). The olution of the direct problem provide the exact meaurement, µ exa. Meaurement containing error are imulated by adding a random error term to µ exa in the form µ µ exa + ωσ (8) where ω i a random variable with normal ditribution, zero mean and unitary tandard-deviation. It i obtained with the ubroutine DRRNOR of the IMSL (996). The value of σ i the tandard deviation of the meaurement error, which i aumed to be contant. In order to generate the imulated meaurement, the function Bi(X, Y, ) wa written in the following form: Bi 6 f ( ) f X ( X ) f Y ( Y ) (9) We note that the contant 6 appearing in Eq. (9) give a heat tranfer coefficient of approximately W/m o C for f () f X (X) f Y (Y). Value for the heat tranfer coefficient were reported in the range to 6 W/m o C, depending on the pray operating condition and plate urface temperature (Miziar, 97, Hibbin and Brimacombe, 98, Lally et al, 99). Different functional form were teted for f (), f X (X) and f Y (Y), including: f ( ) (.a) for.333or.667 f ( ) (.b) for.333 < <.667 f ( ) for.333 or.667 for.333 <.5 for.5 < <.667 (.c)

6 e X f X ( X ) () f Y ( Y ) (.a) e Y f Y ( Y ) (.b) Equation (.b,c) were choen becaue they repreent function containing dicontinuitie and harp corner in time, repectively. Such ind of function are the mot difficult to be recovered by invere analyi. Equation (.a) wa ued to tet the method for a function with no time dependence. Equation (, ) were choen becaue an exponential decay for Bi(X, Y, ), with the ditance from the point where the pray nozzle i located, ha been reported (Lally et al, 99). Therefore, a patial variation for Bi(X, Y, ) obtained from Eq. () and (.a) would correpond to a row of pray nozzle located above the Y axi. Similarly, the combination of Eq. () and (.b) would correpond to a ingle pray nozzle located above XY.

7 Table I - RMS error for different number of enor Number of Senor Bi(X, Y, ) σ e X X Y 6 e e Table II - X and Y poition for enor location Number of X Y Senor.;.75;.5.;.5; 3.5; 5. 6.;.5;.5;.5.;.5; 3.5; 5..;.5;.5;.5.;.;.5;.; 5. Due to the ymmetry with repect to the X and Y axe of the patial variation teted, Eq. (, ), we olved the direct, enitivity and adjoint problem in the reduced domain X A/, Y B/ and Z. Thee problem were olved by finite difference by dicretizing the patial domain with xx point in the X x Y x Z direction, repectively, and with 6 time-tep. Such number of point were choen by comparing the finite-difference olution for the direct problem with a nown analytical olution involving a contant Bi (Ozii, 993). The agreement between the two olution wa better than.%. We have alo performed a grid convergence analyi for cae involving Bi(X, Y, ) obtained with different combination of Eq. (-). The reult obtained with the dicretization above were within a maximum difference of.66%, with repect to thoe obtained by doubling the number of point in each patial direction or in the time domain. We examine here two different tandard-deviation for the meaurement error: σ (errorle meaurement) and σ.. The tandard deviation σ. i characteritic of the meaurement ytem to be ued in the actual experiment and yield error of the ame order of thoe reported by Hibbin and Brimacombe (98). Table I preent the RMS error obtained with, 6 and enor, for two patial variation of Bi(X, Y, ). The function wa uppoed to be contant in time and 6 tranient meaurement per enor were ued in the invere analyi. The enor were located at Z.95, which correpond to 5 mm below the pray cooled urface, and in a grid formed by the X and Y poition hown in Table II. The RMS error i defined here a: where I i the total number of meaurement. The ubcript et and ex refer to the etimated and exact dimenionle heat tranfer coefficient, repectively. Table I how that there i a large reduction on the RMS error when 6 enor are ued intead of. However, the reduction on the RMS error i not ignificant when the number of enor i increaed to. Hence, an examination of Table I reveal that the tranient reading of a minimum of 6 enor hould be ued in order to recover the patial variation of the heat tranfer coefficient. Table III preent a comparion of the invere problem olution, obtained with 6 tranient meaurement of 6 enor, located at Z.95 and at Z.975, correponding to 5 mm and.5 mm below the urface, repectively. A tep variation in time for Bi(X, Y, ), Eq. (.b), wa ued for thi comparion. A expected, we note in Table III a reduction on the RMS error by locating the enor cloer to the urface with the unnown heat tranfer coefficient, ince le information i lot due to the diffuive character of the problem. Figure and 3 preent the exact and etimated function for the ame time and patial variation conidered in Table III. The reult preented in thee figure were obtained with meaurement containing random error (σ.) of 6 enor located at Z.975. An examination of Eq. (.h) and (3) reveal that the gradient equation i null at the final time. Hence, the initial gue ued for Bi (X, Y, f ) i not changed by the iterative procedure of the conjugate gradient method, and ocillation on the etimated function are oberved in the neighborhood of f. In order to avoid uch difficultie, the initial gue wa taen a the exact olution for the final time, which wa aumed nown a priori. We note in Fig. that accurate reult are obtained for the patial variation given by 6 e X. The method i able to correctly predict the exponential decay in the X direction, while the olution i independent of Y. Figure how that the etimated function follow the exact tep variation in time quite cloely, even for X, where the larget dicontinuity tae place. Note alo that the method i able to recover mall variation, uch a for X.5, reaonably well. Figure 3 how the reult obtained for a function involving a patial variation with X and Y in the form 6 e X Y e, given by Eq. () and (.b). We note that the exponential decay in X and Y are correctly predicted. The etimation of the tep variation in time i alo quite good, except at the poition X, Y, where the larget dicontinuity tae place. e RMS I I i [ (,, ) (,, )] Biet X i Y i i Biex X i Yi i (3) 6 6 e Table III - RMS error for different enor poition Senor Poition Bi(X, Y, ) σ Z.95 Z.975 eq. (.b) e X e X Y eq. (.b)

8 Biot (X, Y, ) Biot (X, Y, ) 8 8 σ. X. X.5 X. X.5 X.5 X Dimenionle Time, Figure - Solution for a tep variation in time and exponential decay in X σ. Y. X. Y. Y.5 Y3.5 Y5. Etimated Biot (X, Y, ) Biot (X, Y, ) 8 3 σ. Y. Y.5 Y3.5 Y5. X Dimenionle Time, Figure 3.b - Solution for a tep variation in time and exponential decay in X and Y for X.5 σ. Y. Y.5 Y3.5 Y5. Etimated X.5 Etimated Y.5 Y3.5 Y Dimenionle Time, Figure 3.a - Solution for a tep variation in time and exponential decay in X and Y for X Dimenionle Time, Figure 3.c - Solution for a tep variation in time and exponential decay in X and Y for X.5 Figure how the reult obtained for condition imilar to thoe of Fig., but for a triangular variation in time. We note that the general character of the time and patial variation of Bi (X, Y, ) are correctly recovered, although ome moothne i noticed in the pea at.5, for X. A function with exponential decay in X and Y, Eq. () and (.b), wa alo teted with a triangular variation in time. The reult obtained were imilar to thoe hown in Fig. 3 and, therefore, are omitted here for the ae of brevity.

9 Biot (X, Y, ) 8 σ. X. X.5 X.5 X Dimenionle Time, Figure - Solution for a triangular variation in time and exponential decay in X Y. Y.5 Y3.5 Y5. CONCLUSIONS In thi paper, we olved the invere problem of etimating the heat tranfer coefficient at the urface of a plate. Such heat tranfer coefficient wa uppoed to vary in time and patially over the plate urface. The conjugate gradient method with adjoint equation wa applied a a function etimation approach. Reult obtained with imulated meaurement, for function involving exponential decay in the patial domain, with tep and triangular variation in the time domain, revealed that the method i table with repect to meaurement error and capable of providing accurate etimate for the unnown heat tranfer coefficient. Any dependence of the heat tranfer coefficient with temperature can alo be recovered with the preent approach, ince the local urface temperature of the plate are etimated a part of the olution of the invere problem. The ue of imulated meaured data alo allowed for the deign of an experimental apparatu, with repect to the number and location of the temperature enor. Such experimental apparatu i currently under contruction and will be applied to the etimation of the heat tranfer coefficient of air-mit pray, utilized in the cooling of continuouly cat lab. ACKNOWLEDGEMENT Thi wor wa partially upported by COSIPA under the contract number ET-. The upport provided by CNPq, an agency of the Brazilian government, i alo greatly appreciated. REFERENCES Alifanov, O.M., 99, Invere Heat Tranfer Problem, Springer- Verlag, Berlin. Brimacombe, J., Agarmal, P., Hibbin, S., Prabhaer, B. and Baptita, L., 98, Spray Cooling of the Continuou Cating of Steel, in Continuou Cating Vol. II, Brimacombe, J., Samaraeera, I. and Lait, J. (editor), Iron and Steel Society of AIME. Danta, L.B. and Orlande, H.R.B., 996, A Function Etimation Approach for Determining Temperature-Dependent Thermophyical Propertie, Invere Problem in Engineering, vol. 3., pp Hibbin, S.G. and Brimacombe, J.K., 98, Characterization of Heat Tranfer in the Secondary Cooling Sytem of a Continuouly Slab Cater, in Continuou Cating Vol. II, Brimacombe, J., Samaraeera, I. and Lait, J. (editor), Iron and Steel Society of AIME. Huang, C.H., Ju, T.M. and Teng, A.A., 995, The Etimation of Surface Thermal Behavior of the Woring Roll in Hot Rolling Proce, Int. J. Heat and Ma Tranfer, vol. 38, pp IMSL Librarie, 996, Uer Manual, Houton, TX. Jarny, Y., Ozii, M.N. and Bardon, J.P., 99, A General Optimization Method uing Adjoint Equation for Solving Multidimenional Invere Heat Conduction, Int. J. Heat Ma Tranfer, vol. 3, pp Kohno, T., Shima, T., Kumabara, T. Yamamoto, T., Wae, M. and Tuneoa, A., 98, Improvement of Surface Crac by Air-Water Mit Cooling in Straud Cating, in Continuou Cating Vol. II, Brimacombe, J., Samaraeera, I. and Lait, J. (editor), Iron and Steel Society of AIME. Lally, B., Biegler, L. and Henein, H., 99, Finite Difference Heat Tranfer Modeling for Continuou Cating, Metallurgical Tranaction B, vol. 8, pp Machado, H.A. and Orlande, H.R.B., 997, Invere Analyi of Etimating the Timewie and Spacewie Variation of of the Wall Heat Flux in a Parallel Plate Channel, Int. J. Num. Met. Heat of Fluid Flow, (in pre). Miziar, E.A., 97, Sp ray Cooling Invetigation for Continuou Cating Billet and Bloom, Iron and Steel Engineer, june, pp Orlande, H.R.B. and Ozii, M.N., 993, Invere Problem of Etimating Interface Conductance Between Periodically Contacting Surface, AIAA J. Thermophyic and Heat Tranfer, vol. 7, pp Orlande, H.R.B. and Ozii, M.N., 99, Determination of the Reaction Function in a Reaction-Diffuion Parabolic Problem, ASME J. Heat Tranfer, vol. 6, pp. -. Ozii, M.N., 993, Heat Conduction, John Wiley, New Yor.