Finite Element Methods to Optimize by Factorial Design the Solidification of Cu-5wt%Zn Alloy in a Sand Mold 379 1.1 Mathematical solidification heat transfer model The mathematical formulation of heat transfer to predict the temperature distribution during solidification is based on the general equation of heat conduction in the unsteady state, which is given in two-dimensional heat flux form for the analysis of the present study (Ferreira et al., 2005; Santos et al., 2005; Shi & Guo, 2004; Dassau et al., 2006).   (  ) =       +       +  (1) where ρ is density [kgm -3 ]; c is specific heat [J kg -1 K -1 ]; k is thermal conductivity [Wm -1 K -1 ]; ∂T/∂t is cooling rate [K s -1 ], T is temperature [K], t is time [s], x and y are space coordinates [m] and   represents the term associated to the latent heat release due to the phase change. In this equation, it was assumed that the thermal conductivity, density, and specific heat vary with temperature. In the current system, no external heat source was applied and the only heat generation was due to the latent heat of solidification, L (J/kg) or ΔH (J/kg).   is proportional to the changing rate of the solidified fraction, f s , as follow (Ferreira et al, 2005; Santos et al, 2005; Shi & Guo, 2004).   =∆     =     =       (2) Therefore, Eq. (2) is actually dependent on two factors: temperature and solid fraction. The solid fraction can be a function of a number of solidification variables. But in many systems, especially when undercooling is small, the solid fraction may be assumed as being dependent on temperature only. Different forms have been proposed to the relationship between the solid fraction and the temperature. One of the simple forms is a linear relationship (Shi & Guo, 2004; Pericleous et al., 2006):   = 0   (   )/(    )     1   (3) where   and   are, respectively, the liquid and solid temperature (K). Another relation is the widely used Scheil relationship, which assumes uniform solute concentration in the liquid but no diffusion in the solid (Shi & Guo, 2004):   =1            (4) where k o the equilibrium partition coefficient of the alloy. Eq. (1) defines the heat flux (Radovic & Lalovic, 2005), which is released during liquid cooling, solidification and solid cooling in classical models. The heat evolved after solidification was assumed to be equal zero, i.e. for   ,  =0. However, experimental investigations have showed that lattice defects and vacancy are condensed in the already solidified part of the crystal and the enthalpy of the solid increases and thus the latent heat will decrease (Radovic & Lalovic, 2005). Due to this fact, another way to represent the change of the solid fraction during solidification can be written as (Radovic & Lalovic, 2005):   (    ) +   (     ) (1    (  ) (    )  (     ) (1  2/) (5) Convection and Conduction Heat Transfer 380 Considering c´, as pseudo specific heat, as   =    and combining Eqs. (1) and (2), one obtains (Shi & Guo, 2004 ; Radovic & Lalovic, 2005): (  )  =() (6) The boundary condition applied on the outside of the mold is:    =(  ) (7) Here h is the heat transfer coefficient for air convection and T o is the external temperature. 1.2 The factorial design technique The factorial design technique is a collection of statistical and mathematical methods that are useful for modeling and analyzing engineering problems. In this technique, the main objective is to optimize the response surface that is influenced by various process parameters. Response surface methodology also quantifies the relationship between the controllable input parameters and the obtained response surfaces (Kwak, 2005). The design procedure of response surface methodology is as follows (Gunaraj & Murugan, 1999): i. Designing of a series of experiments for adequate and reliable measurement of the response of interest. ii. Developing a mathematical model of the second-order response surface with the best fittings. iii. Finding the optimal set of experimental parameters that produce a maximum or minimum value of response. iv. Representing the direct and interactive effects of process parameters through two and three-dimensional plots. If all variables are assumed to be measurable, the response surface can be expressed as follows (Aslan, 2007; Yetilmezsoy et al., 2009; Pierlot et al., 2008; Dyshlovenko et al. 2006): y= f( x 1 , x 2 , x 3 , … x k ) (8) where y is the answer of the system, and x i the variables of action called variables (or factors). The goal is to optimize the response variable y. It is assumed that the independent variables are continuous and controllable by experiments with negligible errors. It is required to find a suitable approximation for the true functional relationship between independent variables (or factors) and the response surface. Usually a second-order model is utilized in response surface methodology: =  +    +     +         +       (9) where x 1 , x 2 ,…,x k are the input factors which influence the response y; β o , β ii (i=1, 2,…,m), β ij (i=1, 2,…,m; j=1,2,…,m) are unknown parameters and ε is a random error. The β coefficients, which should be determined in the second-order model, are obtained by the least square method. Finite Element Methods to Optimize by Factorial Design the Solidification of Cu-5wt%Zn Alloy in a Sand Mold 381 The model based on Eq. (9), if m=3 (three variables) this equation is of the following form: =  +    +    +    +     +     +     +      +      +      + (10) where y is the predicted response, β o model constant; x 1 , x 2 and x 3 independent variables; β 1 , β 2 and β 3 are linear coefficients; β 12 , β 13 and β 23 are cross product coefficients and β 11 , β 22 and β 33 are the quadratic coefficients (Kwak, 2005). In general Eq. (9) can be written in matrix form (Aslan, 2007). Y = b X + ε (11) where Y is defined to be a matrix of measured values, X to be a matrix of independent variables. The matrixes b and ε consist of coefficients and errors, respectively. The solution of Eq. (11) can be obtained by the matrix approach (Kwak, 2005; Gunaraj & Murugan, 1999). b = (X‘X ) - 1 X‘ Y (12) where X’ is the transpose of the matrix X and (X’X) -1 is the inverse of the matrix X’X. The objective of this work was to study the solidification process of the alloy Cu-5 wt %Zn during 1.5 h of cooling. It was optimized through the factorial design in three levels, where the considered parameters were: temperature of the mold, the convection in the external mold and the generation of heat during the phase change. The temperature of the mold was initially fixed in 298, 343 and 423 K, as well as the loss of heat by convection on the external mold was fixed in 5, 70 and 150 W/m 2 .K. For the generation of heat, three models of the solid fraction were considered: the linear relationship, Scheil´s equation and the equation proposed by Radovic and Lalovic (Radovic & Lalovic, 2005). As result, the transfer of heat, thermal gradient, flow of heat in the system and the cooling curves in different points of the system were simulated. Also, a mathematical model of optimization was proposed and finally an analysis by the factorial design of the considered parameters was made. 2. Methodology of the numerical simulation The finite elements method was used in this study (Su, 2001; Shi & Guo, 2004; Janik & Dyja, 2004; Grozdanic, 2002). Software program Ansys version 11 (Handbook Ansys, 2010) was used to simulate the solidification of alloy Cu-5 wt %Zn in green-sand mold. Effects due to fluid motion and contraction are not considered in the present work. The geometry of the cast metal and the greensand mold is illustrated in Figure 1(a), which is represented in three-dimensions. However, in this work the analysis was accomplish in 2-D, which is illustrated in Figure 1(b). Some material properties of Cu-5 wt %Zn alloy were taken from the reference Miettinen (Miettinen, 2001), the other properties were taken from Thermo-calc software (Thermo-calc software, 2010), and in Figure 2 the enthalpy and the phase diagram of alloy Cu-Zn are presented (Thermo-calc software, 2010). Three pseudo specific heat (c´) obtained from the equations (3), (4) and (5) were used and these equations were denoted respectively by models A, B and C, and the sand thermo-physical properties was given by Midea and Shah (Midea and Shah, 2002). In this study, the Box–Behnken factorial design in three levels (Aslan, 2007; Paterakis et al., 2002; Montgomery, 1999) was chosen to find out the relationship between the response 38 Fi g Fi g 5 w fu n th e of so l w a in t 2 g . 1. The cast par t g . 2. (a) Enthalp y w t%Zn allo y (The r n ctions. Indepen e mold temperat u the latent heat l idification. The f a s adopted, whe r t ermediate state b (a) t and mold in (a) and phase dia g r a r mo-calc softwar dents variables ( u re (x 1 ), the conv e release, (Z) re p f actorial desi g n i s r e for the inferi o by (0) and for the three dimension a a m of Cu-5 wt % Z e, 2010) ( factors) and the i e ction phenome n p resents the res u s shown in Tabl e o r state of the v superior state b y Convection an d (b) a l and (b) bi dim e Z n allo y and (b) p i r coded/actual l n on (x 2 ) and the m u lt of the temp e 1. For this desi g v ariable it was d e y (+1). d Conduction Heat T e nsional p hase dia g ram o f l evels considere d m athematical mo d erature after 1. 5 g n t y pe a nomen c enoted b y (-1), f T ransfe r f Cu- d were d el (x 3 ) 5 h of c lature f or the Finite Element Methods to Optimize by Factorial Design the Solidification of Cu-5wt%Zn Alloy in a Sand Mold 383 x 1 Mold Temperature x 2 Convection phenomenon (h f ) x 3 Mathematic model Z - Temperature after 1.5 h of solidification (K) -1 298 K 5 W/m 2 K A 0 343 K 70 W/m 2 K B +1 423 K 150 W/m 2 K C 1 -1 -1 -1 806.799 2 -1 -1 0 775.945 3 -1 -1 +1 862.902 4 -1 0 -1 800.301 5 -1 0 0 769.408 6 -1 0 +1 855.752 7 -1 +1 -1 798.197 8 -1 +1 0 767.562 9 -1 +1 +1 854.967 10 0 -1 -1 840.174 11 0 -1 0 809.199 12 0 -1 +1 897.176 13 0 0 -1 833.699 14 0 0 0 802.835 15 0 0 +1 890.279 16 0 +1 -1 832.200 17 0 +1 0 801.430 18 0 +1 +1 890.110 19 +1 -1 -1 899.860 20 +1 -1 0 868.171 21 +1 -1 +1 958.587 22 +1 0 -1 893.996 23 +1 0 0 862.277 24 +1 0 +1 953.026 25 +1 +1 -1 893.136 26 +1 +1 0 861.015 27 +1 +1 +1 952.674 Table 1. Factorial design of the solidification process parameters Convection and Conduction Heat Transfer 384 The initial and boundary conditions were applied to geometry of Figure 1 according to Table 1. The boundary condition was the convection phenomenon and this phenomenon was applied to the outside walls of the sand mold, as shown in Table 1. The convection transfer coefficient at the mold wall was considered constant in this work, due to lack of experimental data. The effects of the refractory paint and of the gassaging process were not taken into consideration either. The final step consisted in solving the problem of heat transfer of the mold/cast metal system using equation (6), in applied boundary condition and in controlling the convergence condition. Heat transfer is analyzed in 2-D form, as well as the heat flux, the thermal gradient, and in addition, the thermal history for some points in the cast metal and in the mold is discussed. 3. Result and discussion The result for solidification was discussed for some particular cases, at condition given in lines 7, 8 and 9 from Table 1, which correspond respectively to the lowest temperatures for each mathematical model of latent heat release. Each one of the lines corresponds to the temperature of the mold for the lower state (-) and for convection phenomenon for the higher state (+). (a) (b) Fig. 3. Temperature distribution in (a) sand mold system, (b) cast metal (line 9 of Table 1) The condition mentioned on line 9 of Table 1 was chosen to present heat transfer results, where the temperature field is shown in Figure 3(a) in all the system mold and in the cast metal (Figure 3(b)). This last case can be visualized in more detail in part (b), where an almost uniform temperature is observed. In the geometric structure of the mold there is a core constituted of sand that is represented by a white circle in Figure 3(b), which can be verified also in Figure 1(a). In Figure 4 the results of the thermal gradient and the thermal flux are shown, where the thermal gradient goes from the cold zone to the hot zone. On the other hand, the thermal flux goes from the hot zone to the cold zone. Also the convergence of the solution was studied; this point is discussed in more detail by Houzeaux and Codina (Houzeaux & Codina, 2004). Finite Element Methods to Optimize by Factorial Design the Solidification of Cu-5wt%Zn Alloy in a Sand Mold 385 (a) (b) Fig. 4. (a) Thermal gradient (K/m) in vector form and (b) Heat flux (W/m 2 ) in vector form (line 9 of Table 1) In order to simulate the cooling curves, two points were considered, as shown in Figure 5: one located in the core (point 2) and the other in the metal (point 1). The three forms of latent heat release were applied into the mathematical model and the resulting thermal profiles were compared. Fig. 5. Reference points for the mold/metal system The cooling curves were studied for condition of line 7, 8 and 9 from Table 1 as shown in Figure 6. Figure 6 (a) shows a comparison of temperature evolution at point (2) for the three formulations of latent heat release: linear (model A), Scheil (model (B) and Radovic and Lalovic (model C). It can be observed that the highest temperature profile corresponds to model A, followed by model C and last by model B, mainly after the solidification range. Although not presented, a similar behavior has occurred at other positions in the casting. Chen and Tsai (Chen and Tsai, 1990) analyzed theoretically four different modes of latent heat release for two of alloys solidified in sand molds: Al-4,5wt%Cu (wide mushy region, 136K ) and a 1wt% Cr steel alloy (narrow mushy region, 33.3K). In their work, they conclude that no significant differences can be observed in the casting temperature for different modes of latent heat release, when the alloy mushy zone is narrow. The alloy used in the present work, Cu-5wt%Zn, as shown in Figure 2(b), has a narrow mushy zone (less than 10K). Figure 6(a) shows that there is a significant temperature profile difference due to the three different latent heat release modes. In addition, it is important to remark that the latent heat release form has strongly influenced the local solidification time. 2 1 38 S u se c so l co r ar m et Fi g fo r pr o is r T h li n w h fa c Fi g 1. ( A Y e an Fi g le v ad th e pa re l In an in d an 6 u ch solidificatio n c ondar y dendriti l idification (t SL ) r relatin g ultimat e m spacin g s have al., 2000). g ure 6 (b) sho w r mulations of lat o file correspond s r epeated for the o h e si g nificant var n ear re g ression a h ich is a respo n c torial desi g n. g . 6. Thermal pro f ( a) Inside of the c three level Box - e tilmezso y et al., d x 3, based in T a g ure 7. In this fi g v el (p) of 95%, sh o opted for this a n e main effect of rameter 2 (conv e l ease form). Fi g ure 7, two si g d x 3 (mathemat i d ependent varia b al y sis, other t y p e (a) n parameter affe c arm spacin g s. are well know n e tensile stren g th shown that (  ) w s a compariso n ent heat release. s to model A, foll o o ther points in th iables indicated nd anal y sis of v n se surface met h f iles for the mol d c ast – point 1, (b) I - Behnken desi g ( 2009) was used a ble 1. The resu l g ure the estimate d o win g the varia b n al y sis was, “L” m the first factor e ction phenome n g nificant influen c i cal model) wit h b les and interac t e of standard g r a cts the microst r Correlations bet w n in the literat u (  ) and second a increases with d e n of temperatur e It can be obser v o wed b y model C e mold. b y the Pareto c h v ariance) were o p h odolo gy , based d /metal s y stem c o I nside of the mol d ( Aslan, 2007; Pa t to determine th e l t of this anal y si s d valor of the re s b les with and wit h m eans linear, “Q ” and “2L b y 3 Q n on) with the qu a c es were found: x h linear and qu a t ions are ne g li g i b a ph was accomp l Convection an d r ucture characte r w een dendritic s u re (Rosa et al. a r y (SDAS) or pr e creasin g (SDAS ) e evolution at p v ed a g ain, that t h C and last b y mo d h art (which was o p timized usin g a on a hi g hl y fr a o ncernin g condit i d – point 2 t erakis et al., 20 0 e responses of t h s is shown in th s ult Z is present e h out si g nificant i n ” means quadrat i Q ” means the li n a dratic effect of p x 1 (mold temper a a dratic effects. T b le in this fi g ur e l ished, and it is s (b) d Conduction Heat T r ized b y prima r s pacin g s and loc a , 2008). Investi g imar y (PDAS) d e ) or (PDAS) (Qu a p oint (1) for th e h e hi g hest temp e d el B, and this be o btained after m a Box-Behnken d a ctionalized thre i on 7, 8 and 9 of T 0 2; Mont g omer y, h e three variable s h e Pareto’s dia g r a e d with the si g ni f n fluences. The n o i c. For example, “ n ear interaction p arameter 3 (late n a ture) with linea r T he other effect e . To clarif y mo r hown in Fi g ure 8 T ransfe r ry and a l time g ations e ndrite a resma e three e rature havior ultiple d esi g n, e-level T able , 1999; s x 1 , x 2 a m of f icance o tation “ (1)” is of the n t heat r effect of the r e this 8 . This Fi n D e Fi g fi g ( M in f p o ob in i pa ne ar o g i v In co e co e in fo l A cr i T h eq u n ite Element Metho d e sign the Solidificati g . 7. Pareto chart g ure of the facto r M ont g omer y , 199 9 f luence can be o o ints that belon g t served in Fi g ure i tial temperature rameter 3 (laten t g ative influence o und zero, as pr e v en Z the followi n =802.7889 + this equation t h e fficients are ne g e fficients belon g Fi g ures 7 and 8 l lowin g equation : =802.7889 + quadratic equati i tical points of t h h en, the derivati o u ations, bein g e q d s to Optimize by F on of Cu-5wt%Zn A of standardized e r ial desi g n was b 9 ). In this fi g ure bserved b y thos e t o the concentrat e 8 that the bi gg e s ) with linear be t heat release fo r on the factorial e sented in Fi g ure ng equation: + 46.4582  3. 9 + 59.2232   + 1.27212   h e linear and q u g li g ible (the y ar e to the variables w 8 . Then accordi n : + 46.4582  3. 9 + 59.2232   on that correlat e h is equation ca n o n of this equati o q uations 15, 16 a n F actorial A lloy in a Sand Mol d e ffects for the ful l b uilt based on t h the main effect s e dispersed poi n e d re g ion points s t positive influe n havior, followe d r m). Parameter x desi g n and the o 8. For this anal ys 9 348   28.363 8 + 0.4273    + 0    + 0.2196     + u adratic coeffici e e considered as r w hich stron g l y i n ng to this consi d 9 348   28.363 8 + 0.4273    + 0 e s the variables a n be estimated t h o n in relation to n d 17: d l factorial desi gn h e Student’s pro b s and their inter a n ts (around of t h are of the ne g li gi n ce is due to the d b y the linear a 2 with linear be h o ther effects ha d s is a mathematic a 8    13.0766   0 .7476    +0.1 9 + 0.6686      + 0 e nts are most i m r esidue). Precise l n fluence the resu l d eration, equati o 8    13.0766   0 .7476    +0.1 9 a nd the response h rou g h this mat h (x 1 ), (x 2 ) and ( x b abilit y distribu t a ctions with si gn h e strai g ht line). ible influence. It main effect of x 1 a nd quadratic ef f h avior presents a d a ne g li g ible be h a l model was pro  2.5486   9 88    0 .3273     m portant and th e ly the most si gn l t, as it can be ob s o n (13) reduces  2.5486   9 88    was obtained, a n h ematical relati o x 3 ) results in thr e 387 t ion (t) n ificant Those can be (mold f ect of a small h avior, posed, (13) e other n ificant s erved to the (14) n d the o nship. e e new 38 Fi g T h th e re c va m o A n ac c ap ar e eq u fo r he T h te m m i fo r m o fi n 8         g . 8. Curve of sta n h e critical point i n e condition of   c ommendations o lues for the criti c o del. n al y zin g this res u c ordin g to the c r proach to these v e a bit different u ation (13). The n r m, where x 1 =- 1 at release, 150 W / h e values of x 1 =- 1 m perature after 1 i nimum tempera t r the optimizatio n o delin g represen t n ite elements, see =46.4582+2 6     =3.9348 + 0 =28.3638+0. 7 n dardized effect s n the surface re s Z  =0, Z   =0 an d o f the authors M a c al point are: x 1 = u lt, we know tha t r iteria adopted i n v alues adopted, b of these values n accordin g to t h 1 .7850 ≈-1 (mold / m 2 .K) and x 3 =- 0 1 , x 2 = + 1 and x 3 1 .5h of solidificat i t ure of factorial d n process of the c t s a proof trivial . Fi g ure 6. 6 .1532x  + 0.42 7 0 .4273  + 5.097 2 7 476  + 0.1988  s of the factorial d s ponse are foun d d Z   =0. This c a rtendal et al. ( M = -1.7850 K, x 2 = 0. t the variables x 1 , n factorial desi gn b ut it can be obse r adopted, this is h ese consideratio n temperature, in 0 .2298 ≈ 0 (math e 3 = 0 in Table 1 c i on that is 767.56 2 d esi g n. This solu t c astin g b y factor i . Also this result Convection an d 7 3x  +0.7476x  2   + 0.1988    +118.4464  d esi gn d b y solvin g thes riterion of solut M artendal el al., 2 9306 W/m 2 .K, x 3 x 2 and x 3 must t a n . Because the c a r ved that some s o possible b y si m n s, the approxi m environment), x 2 e matic model, Sc h orresponds to th 2 K, j ustl y this v a t ion proved the v i al desi g n, despi t is to a g ree with d Conduction Heat T e equation s y ste m t ion was based o 007) and the cal c 3 =-0.2298 mathe m a ke values -1 or 0 a lculated values s o lutions of x 1 , x 2 m plif y in g consid e m ations must be i n 2 = 0.9306 ≈ + 1 h eil relationship) . e line 8, that me a a lue correspond s v alidit y of the m o t e that this prove the result obtai n T ransfe r (15) (16) (17) m s for o n the c ulated m atical 0 or +1, s hould and x 3 e red in n such (latent . a ns Z - s to the o delin g of the n ed b y [...]... the Solidification of Cu-5wt%Zn Alloy in a Sand Mold 389 (a) (b) (c) Fig 9 Response surface plots showing the effect of the (a) x2 and x3 factors, x1 was held at zero level, (b) x1 e x3 factors, x2 was held at zero level and (c) x1 and x2 factors, x3 was held at zero level 390 Convection and Conduction Heat Transfer While a quantitative analysis of equation (14) was made, will soon be made a qualitative... 1359-6454 Chen, J.H & Tsai, H.L (1990) Comparison on different modes of latent heat release for modeling casting solidification AFS Transactions, (1990), pp 539-546 ISSN 00658375 392 Convection and Conduction Heat Transfer Dassau, E., Grosman, B & Lewin, D.L (2006) Modeling and temperature control of rapid thermal processing Computers and Chemical Engineering, Vol 30, No 4, (February 2006), pp 686–697, ISSN... Department of Mechanical, Industrial and Nuclear Engineering Cincinnati, 2001 Thermo-Calc Software, Stockholm, Sweden, 2010 Xiao, Z & Vien, A (2004) Experimental designs for precise parameter estimation for nonlinear models Minerals Engineering, Vol 17, No 3, (March 2004), pp 431–436, ISSN 0892-6875 394 Convection and Conduction Heat Transfer Yetilmezsoy, K., Demirel, S & Vanderbei, R J (2009) Response surface... Modelling and Simulaton in Materials Science and Engineering, Vol 13, (September 2005), pp 1071–1087, ISSN 0965-0393 Shi, Z & Guo, Z X (2004) Numerical heat transfer modeling for wire casting Materials Science and Engineering, Vol A 365, No 1-2, (January 2004), pp 311–317, ISSN 09215093 Su, X “Computer aided optimization of an investment bi-metal casting process” Ph.D Thesis, University of Cincinnati, Department... transition and microstructure evolution during transient directional solidification of pb-sb alloys Metallurgical and Materials Transactions, Vol A 39, (September 2008), pp 2161-2174 ISSN 1073-5623 Santos, C A., Fortaleza, E L., Ferreira, C R F., Spim, J A & Garcia A (2005) A solidification heat transfer model and a neural network based algorithm applied to the continuous casting of steel billets and blooms... tool to optimize and predict results in quantitative and qualitative way This tool can be used in research, optimization and prediction of industrial processes, saving manpower, material and time in order to improve cost and quality of the product Finite Element Methods to Optimize by Factorial Design the Solidification of Cu-5wt%Zn Alloy in a Sand Mold 391 This work has done the numerical simulation... qualitative form and combining with a response surface methodology was employed for modeling and optimizing three operations parameters of the casting process According to this study, it was observed when the parameters of the solidification process are in the following state, such as, mold temperature in the environment, convection phenomenon in its fullest expression and the latent heat release to... process and thus, a special care needs to be taken during the project elaboration of the casting Also this optimization tool can be used in other research areas, optimizing and predicting industrial processes, saving manpower, material and time in order to improve cost and quality of the product 5 Acknowledgment The authors acknowledge financial support provided by CNPq (The Brazilian Research Council) and. .. variation is to project Z on the x2 and x3 plane, in terms of color band The region limited by the white points on this curve represents the Z confidence interval For other levels of x1, the surface graph behavior has the same characteristic as previously mentioned Figure 9(b) shows the effect of mold temperature (x1) and latent heat release form (x3) at zero level of convection phenomenon (x2) In this... x3 and this fact can be observed by the points x3(L) and x3(Q) mentioned at the graph of Figure 8 Note that, for a given value of x1, as the x3 increases and the Z increases reaching the highest value of 953.026, as can be seen in Table 1 Figure 9(c) shows the effect of mold temperature (x1) and convection phenomenon (x2) at zero level of latent heat release (x3) Note that, as the x1 factor increases, .  (  ) (    )  (     ) (1  2/) (5) Convection and Conduction Heat Transfer 380 Considering c´, as pseudo specific heat, as   =    and combining Eqs. (1) and (2), one obtains (Shi &. parameters Convection and Conduction Heat Transfer 384 The initial and boundary conditions were applied to geometry of Figure 1 according to Table 1. The boundary condition was the convection. level and (c) x 1 and x 2 factors, x 3 was held at zero level (a) (b) (c) Convection and Conduction Heat Transfer 390 While a quantitative analysis of equation (14) was made, will