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Discussion of “A theoretical study of Gasarite eutectic growth”
Scripta Materialia 52 (2005) 799–801 www.actamat-journals.com
Discussion of ‘‘A theoretical study of Gasarite eutectic growth’’ a,*
Ludmil Drenchev
, Jerzy Sobczak b, Savko Malinov c, Wei Sha d, Adrian Long
d
a
Institute of Metal Science, 67 Shipchenski Prohod Street, Sofia 1574, Bulgaria Foundry Research Institute, 73 Zakopianska Street, 30-418 Krakow, Poland School of Mechanical and Manufacturing Engineering, The Queens University of Belfast, Belfast BT7 1NN, UK d School of Civil Engineering, The Queens University of Belfast, Belfast BT7 1NN, UK b
c
Abstract An attempt to develop a mathematical model for the simultaneous growth of the gas pore and the solid phase in gasar technology was offered in the paper ‘‘A theoretical study of Gasarite eutectic growth’’. Unfortunately the paper incorporates mistakes which have led to erroneous results and incorrect conclusions. In order to rectify this, the most important ones are analyzed by the authors. 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction The paper by Liu and Li [1], proposes a mathematical description of microstructure formation in gasar technology which was developed by Shapovalov [2]. Although gasar has been in existence for more than 10 years, a comprehensive mathematical description of the basic physical phenomena involved in the formation process does not exist or has never been published in the literature. Because of this, an attempt to develop a mathematical model of the entire process or part of it is welcome. The theoretical study presented in Ref. [1] focuses on the relationships between the porosity, the inner-pore spacing and the processing parameters. Mistakes have been found in this paper—some are basic errors but others are more fundamental. Only those which are important for developing an understanding of the process are discussed. 2. Diffusion problem The solution of the diffusion problem, Eqs. (1)–(4) in Ref. [1], plays an essential role in the theoretical analy*
Corresponding author. Tel.: +359 2 870 40 33; fax: +359 2 870 32
07. E-mail address: [email protected] (L. Drenchev).
sis. Our comments on the definition of the diffusion problem are as follows: (a) Eq. (2) is valid only when the solidification velocity m is constant and this should have been stated in Ref. [1] explicitly. The assumption that m = const is reasonable but users need to know when the solution is applicable. From a mathematical viewpoint, a partial differential equation possesses a unique and continuous solution only when the equation is considered in a strictly determined geometrical area and, on the basis of this area, a full set of boundary conditions is defined. An initial condition is necessary if time dependent functions are considered. In Ref. [1], the geometrical area for equation definition and a full set of boundary conditions are not specified. For example, the conditions on the radial axis are not defined. (b) Additionally, even after these boundary conditions are added to the Eqs. (1)–(4), the problem of finding the gas distribution in the melt is not defined correctly. The reason for this is that expressions (3) and (4) define a function at z = 0 which is not continuous. A continuous function is a precondition for the existence of a unique solution of Eq. (2). The lack of continuity means that it is impossible to find an analytical solution for the problem
1359-6462/$ - see front matter 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2004.12.023
800
L. Drenchev et al. / Scripta Materialia 52 (2005) 799–801
discussed. Although Liu and Li cited such a solution, in Eqs. (5)–(9), it is unclear what problem is being solved as boundary condition (3) is not satisfied. It should be noted that concentration field problems of this nature can be solved approximately by numerical methods.
3. Porosity calculation The determination of gasar ingot porosity as a function of the main processing parameters (solidification velocity, partial gas pressure, initial melt and mold temperatures) is the eventual goal of a mathematical model relevant to gasar technology. The paper also considers this matter but again in a manner that lacks rigor. In Section 3.1 of Ref. [1], instead of Eq. (1) a different one (10) was used for the determination of the porosity obtained during gasar structure formation in a copper– hydrogen system. This expression includes hydrogen concentrations in liquid and solid at the solidification front. To determine these concentrations the temperature dependent Sieverts law was applied (11 and 12). Obviously, these formulae give constant gas distribution in liquid and constant gas distribution in solid and can be applied only in cases where there is no solidification or when the solidification velocity is negligible (m 0) but this is not valid for the gasar process. It is difficult to understand why Liu and Li attempted to obtain a non-uniform gas distribution in the previous section of their paper if it was not intended for use. The gas pressure, P, in Eq. (12) has three components: total gas pressure in the furnace atmosphere, Pg capillary pressure, Pc and hydrostatic pressure, Ph where P = Pg + Pc + Ph. It follows from Ref. [1] that the gas distribution in the solid is determined by the pressure P but this is not the real situation in the process considered. If it were true, the gas concentration in the solid would have reached very high values (even many times higher than the concentration in the melt for small pore size), which is not valid for the gasar process. Each formula, which reflects a physical phenomenon, is an approximation of reality and is normally only suitable for application under certain conditions. To use the formula one has to keep in mind these conditions and to state them explicitly when necessary so as to define the limits of application of the results. These were not taken into account in Eqs. (10)–(12) by Liu and Li as their equations describe a situation where • the temperature of the melt is constant and equal to the melting point of copper, Tm; • the temperature of the solid is constant and equal to Tm;
• all quantity of gas rejected on the solid/liquid interface is trapped (sucked) into gas pores and floating bubbles are not formed; • gas diffusion in the melt and the solid can be neglected, i.e. the velocity of solidification is approximately zero, m 0. These equations are not appropriate to gasar structure formation and m 0 is not typical. The first two conditions never occur during the gasar structure formation process. In particular, the gas diffusion and the temperature gradient (although it is a relatively small one) play an essential role in the process [3]. Eqs. (10)–(12) cannot be used to obtain the influence of melt temperature on the gasar porosity for the following reasons: (i) the temperature in these equations is constant and equal to the melting temperature, Tm of Cu, and (ii) Eq. (12) does not exist for temperatures higher than Tm (there is no solid phase for temperatures above melting point). For these reasons, Fig. 4 is misleading. According to Liu and Li, expressions (5–9) were obtained as a solution of Eqs. (2)–(4) and the porosity, e, given by Eq. (1). In the paragraph under the section title ‘‘3.2. Solute distribution’’ the authors analyzed the hydrogen concentration in the melt at the solid–liquid interface. The conclusions were made on the basis of Eqs. (5)–(10), which gives porosity, e. The substitution of expression in Eq. (10) for e is incorrect because (a) the solution (5)–(9) contains e obtained by (1); (b) if one substitutes expression (10) into (7) it means that the solution (5), i.e. gas distribution in liquid, does not depend on the radius of the gas pore, rg. This is in contradiction with the boundary condition in Eq. (3). Another incorrect conclusion in the same part of the paper is that ‘‘the inner-pore spacing decreases with rising solidification rate’’ because of ‘‘the diffusion of the hydrogen atom in the radial direction’’. No reason is given in the text that supports this assertion. One basic result reported in Ref. [1] is the correlation between the inner-pore spacing, L = 2rs and the solidification velocity, m. Undoubtedly, it is useful to have a formula like (16) but only if the authors have derived it correctly. This result should be discarded for two reasons. First, in obtaining the explicit expression for undercooling, DT ¼ f ðrs ; C ls . . .Þ, a formula, stated three lines above Eq. (13), was used which does not have any physical justification. Second, anyone attempting to repeat the procedure described in the paper will obtain m Æ (L C2) = A, which is not what is presented in Eq. (16). The last group of figures, Figs. 9(a)–(c), shows the influence ‘‘of the solidification rate, the gas pressure
L. Drenchev et al. / Scripta Materialia 52 (2005) 799–801
and the melt temperature on the inner-pore spacing (L) and the pore size (2rg)’’. However, no formula in Ref. [1] gives the dependence of L and 2rg on the above three parameters and the authors did not explain how they obtained the curves in the figures. Only a scrupulous reader can detect this, as it is impossible to find the basis for the correlation between L, rg and melt temperature in Fig. 9(c). All conclusions related to melt temperature are incorrect because in this work the thermal problem was not considered. As commented above, the temperature is present only in expressions (11) and (12) and this is the melting temperature of copper but not the temperature in the melt; the latter is of course always higher than the former temperature. Finally, we would like to comment on Fig. 9(a) which gives the dependence of L and 2rg on the solidification velocity. We found that the curve ‘‘L’’ could be reproduced by applying formula (16) with A = 17 · 1013, i.e. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 17 1013 L¼ ; m 2 ð0; 1000Þ m pffiffiffiffiffiffiffiffiffi Curve ‘‘2rg ’’ could be reproduced using 2rg ¼ 0:25L, which can be easily derived from Eq. (1) with e = 0.25 and L as above. The curves were obtained without any reference to partial pressures, temperatures and others parameters. The purpose of specifying on Fig. 9(a)
801
values of partial argon pressure, partial hydrogen pressure and temperature is therefore unclear.
4. Conclusion The paper [1] contains many errors and unsubstantiated statements which could lead to wrong or speculative results and incorrect conclusions. Because of this, a reader wishing to use the results must exercise extreme caution and preferably refer to other sources of information on this topic [4,5].
References [1] Liu Y, Li Y. Scripta Mater 2003;49:379–86. [2] Shapovalov VI. U.S. Patent No. 5,181,549, 26 January, 1993. [3] Drenchev L, Sobczak J, Asthana R, Malinov S. J Comput Aided Mater Des 2004;10(1):35–54. [4] Drenchev L, Sobczak J, Sha W, Malinov S. In: Sabotinov N, Piperov N, editors. IV International Congress ‘‘Mechanical Engineering Technologies 04’’, 23–25 September, 2004, Varna, Bulgaria, vol. 3, p. 55–8. [5] Drenchev L, Sobczak J, Malinov S, Sha W. Mathematical model for simultaneous growth of gas and solid phases in gas-eutectic reaction. In: Fourth international conference on high temperature capillarity (HTC-2004), 31 March–3 April, 2004, Sanremo, Italy.
Discussion of ‘‘A theoretical study of Gasarite eutectic growth’’ a,*
Ludmil Drenchev
, Jerzy Sobczak b, Savko Malinov c, Wei Sha d, Adrian Long
d
a
Institute of Metal Science, 67 Shipchenski Prohod Street, Sofia 1574, Bulgaria Foundry Research Institute, 73 Zakopianska Street, 30-418 Krakow, Poland School of Mechanical and Manufacturing Engineering, The Queens University of Belfast, Belfast BT7 1NN, UK d School of Civil Engineering, The Queens University of Belfast, Belfast BT7 1NN, UK b
c
Abstract An attempt to develop a mathematical model for the simultaneous growth of the gas pore and the solid phase in gasar technology was offered in the paper ‘‘A theoretical study of Gasarite eutectic growth’’. Unfortunately the paper incorporates mistakes which have led to erroneous results and incorrect conclusions. In order to rectify this, the most important ones are analyzed by the authors. 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction The paper by Liu and Li [1], proposes a mathematical description of microstructure formation in gasar technology which was developed by Shapovalov [2]. Although gasar has been in existence for more than 10 years, a comprehensive mathematical description of the basic physical phenomena involved in the formation process does not exist or has never been published in the literature. Because of this, an attempt to develop a mathematical model of the entire process or part of it is welcome. The theoretical study presented in Ref. [1] focuses on the relationships between the porosity, the inner-pore spacing and the processing parameters. Mistakes have been found in this paper—some are basic errors but others are more fundamental. Only those which are important for developing an understanding of the process are discussed. 2. Diffusion problem The solution of the diffusion problem, Eqs. (1)–(4) in Ref. [1], plays an essential role in the theoretical analy*
Corresponding author. Tel.: +359 2 870 40 33; fax: +359 2 870 32
07. E-mail address: [email protected] (L. Drenchev).
sis. Our comments on the definition of the diffusion problem are as follows: (a) Eq. (2) is valid only when the solidification velocity m is constant and this should have been stated in Ref. [1] explicitly. The assumption that m = const is reasonable but users need to know when the solution is applicable. From a mathematical viewpoint, a partial differential equation possesses a unique and continuous solution only when the equation is considered in a strictly determined geometrical area and, on the basis of this area, a full set of boundary conditions is defined. An initial condition is necessary if time dependent functions are considered. In Ref. [1], the geometrical area for equation definition and a full set of boundary conditions are not specified. For example, the conditions on the radial axis are not defined. (b) Additionally, even after these boundary conditions are added to the Eqs. (1)–(4), the problem of finding the gas distribution in the melt is not defined correctly. The reason for this is that expressions (3) and (4) define a function at z = 0 which is not continuous. A continuous function is a precondition for the existence of a unique solution of Eq. (2). The lack of continuity means that it is impossible to find an analytical solution for the problem
1359-6462/$ - see front matter 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2004.12.023
800
L. Drenchev et al. / Scripta Materialia 52 (2005) 799–801
discussed. Although Liu and Li cited such a solution, in Eqs. (5)–(9), it is unclear what problem is being solved as boundary condition (3) is not satisfied. It should be noted that concentration field problems of this nature can be solved approximately by numerical methods.
3. Porosity calculation The determination of gasar ingot porosity as a function of the main processing parameters (solidification velocity, partial gas pressure, initial melt and mold temperatures) is the eventual goal of a mathematical model relevant to gasar technology. The paper also considers this matter but again in a manner that lacks rigor. In Section 3.1 of Ref. [1], instead of Eq. (1) a different one (10) was used for the determination of the porosity obtained during gasar structure formation in a copper– hydrogen system. This expression includes hydrogen concentrations in liquid and solid at the solidification front. To determine these concentrations the temperature dependent Sieverts law was applied (11 and 12). Obviously, these formulae give constant gas distribution in liquid and constant gas distribution in solid and can be applied only in cases where there is no solidification or when the solidification velocity is negligible (m 0) but this is not valid for the gasar process. It is difficult to understand why Liu and Li attempted to obtain a non-uniform gas distribution in the previous section of their paper if it was not intended for use. The gas pressure, P, in Eq. (12) has three components: total gas pressure in the furnace atmosphere, Pg capillary pressure, Pc and hydrostatic pressure, Ph where P = Pg + Pc + Ph. It follows from Ref. [1] that the gas distribution in the solid is determined by the pressure P but this is not the real situation in the process considered. If it were true, the gas concentration in the solid would have reached very high values (even many times higher than the concentration in the melt for small pore size), which is not valid for the gasar process. Each formula, which reflects a physical phenomenon, is an approximation of reality and is normally only suitable for application under certain conditions. To use the formula one has to keep in mind these conditions and to state them explicitly when necessary so as to define the limits of application of the results. These were not taken into account in Eqs. (10)–(12) by Liu and Li as their equations describe a situation where • the temperature of the melt is constant and equal to the melting point of copper, Tm; • the temperature of the solid is constant and equal to Tm;
• all quantity of gas rejected on the solid/liquid interface is trapped (sucked) into gas pores and floating bubbles are not formed; • gas diffusion in the melt and the solid can be neglected, i.e. the velocity of solidification is approximately zero, m 0. These equations are not appropriate to gasar structure formation and m 0 is not typical. The first two conditions never occur during the gasar structure formation process. In particular, the gas diffusion and the temperature gradient (although it is a relatively small one) play an essential role in the process [3]. Eqs. (10)–(12) cannot be used to obtain the influence of melt temperature on the gasar porosity for the following reasons: (i) the temperature in these equations is constant and equal to the melting temperature, Tm of Cu, and (ii) Eq. (12) does not exist for temperatures higher than Tm (there is no solid phase for temperatures above melting point). For these reasons, Fig. 4 is misleading. According to Liu and Li, expressions (5–9) were obtained as a solution of Eqs. (2)–(4) and the porosity, e, given by Eq. (1). In the paragraph under the section title ‘‘3.2. Solute distribution’’ the authors analyzed the hydrogen concentration in the melt at the solid–liquid interface. The conclusions were made on the basis of Eqs. (5)–(10), which gives porosity, e. The substitution of expression in Eq. (10) for e is incorrect because (a) the solution (5)–(9) contains e obtained by (1); (b) if one substitutes expression (10) into (7) it means that the solution (5), i.e. gas distribution in liquid, does not depend on the radius of the gas pore, rg. This is in contradiction with the boundary condition in Eq. (3). Another incorrect conclusion in the same part of the paper is that ‘‘the inner-pore spacing decreases with rising solidification rate’’ because of ‘‘the diffusion of the hydrogen atom in the radial direction’’. No reason is given in the text that supports this assertion. One basic result reported in Ref. [1] is the correlation between the inner-pore spacing, L = 2rs and the solidification velocity, m. Undoubtedly, it is useful to have a formula like (16) but only if the authors have derived it correctly. This result should be discarded for two reasons. First, in obtaining the explicit expression for undercooling, DT ¼ f ðrs ; C ls . . .Þ, a formula, stated three lines above Eq. (13), was used which does not have any physical justification. Second, anyone attempting to repeat the procedure described in the paper will obtain m Æ (L C2) = A, which is not what is presented in Eq. (16). The last group of figures, Figs. 9(a)–(c), shows the influence ‘‘of the solidification rate, the gas pressure
L. Drenchev et al. / Scripta Materialia 52 (2005) 799–801
and the melt temperature on the inner-pore spacing (L) and the pore size (2rg)’’. However, no formula in Ref. [1] gives the dependence of L and 2rg on the above three parameters and the authors did not explain how they obtained the curves in the figures. Only a scrupulous reader can detect this, as it is impossible to find the basis for the correlation between L, rg and melt temperature in Fig. 9(c). All conclusions related to melt temperature are incorrect because in this work the thermal problem was not considered. As commented above, the temperature is present only in expressions (11) and (12) and this is the melting temperature of copper but not the temperature in the melt; the latter is of course always higher than the former temperature. Finally, we would like to comment on Fig. 9(a) which gives the dependence of L and 2rg on the solidification velocity. We found that the curve ‘‘L’’ could be reproduced by applying formula (16) with A = 17 · 1013, i.e. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 17 1013 L¼ ; m 2 ð0; 1000Þ m pffiffiffiffiffiffiffiffiffi Curve ‘‘2rg ’’ could be reproduced using 2rg ¼ 0:25L, which can be easily derived from Eq. (1) with e = 0.25 and L as above. The curves were obtained without any reference to partial pressures, temperatures and others parameters. The purpose of specifying on Fig. 9(a)
801
values of partial argon pressure, partial hydrogen pressure and temperature is therefore unclear.
4. Conclusion The paper [1] contains many errors and unsubstantiated statements which could lead to wrong or speculative results and incorrect conclusions. Because of this, a reader wishing to use the results must exercise extreme caution and preferably refer to other sources of information on this topic [4,5].
References [1] Liu Y, Li Y. Scripta Mater 2003;49:379–86. [2] Shapovalov VI. U.S. Patent No. 5,181,549, 26 January, 1993. [3] Drenchev L, Sobczak J, Asthana R, Malinov S. J Comput Aided Mater Des 2004;10(1):35–54. [4] Drenchev L, Sobczak J, Sha W, Malinov S. In: Sabotinov N, Piperov N, editors. IV International Congress ‘‘Mechanical Engineering Technologies 04’’, 23–25 September, 2004, Varna, Bulgaria, vol. 3, p. 55–8. [5] Drenchev L, Sobczak J, Malinov S, Sha W. Mathematical model for simultaneous growth of gas and solid phases in gas-eutectic reaction. In: Fourth international conference on high temperature capillarity (HTC-2004), 31 March–3 April, 2004, Sanremo, Italy.