Introduction to the special issue Calphad Tools for Calphad Tools for the Metallurgy of Solidification
Abstract
This collection of papers is mainly aimed at proposing a common simplified approach for different classes of materials and processes related to the solidification of metallic alloys.
This introduction has the aim of giving a common framework for the papers presented here, which could be exploited by any Calphad software provided a relevant thermodynamic database would be utilized.
It has also the aim of providing a quick "cookbook" approach in order to exploit the wide availability of software and databases, proposing for a specific software - Pandat by Computherm - the relevant steps to extract from the Scheil solidification curve - which is normally presented in Calphad software as the relationship between the solid fraction fs with the solidification temperature T and numerically calculate the derivative of the this function. This numerical derivative is able to depict in a simple way the main features of solidification processes; while this approach has been presented in the past only for the growth restriction factor on solidification,?some simple mathematics demonstrate that is possible to express two other important approached customarily used in semi-solid processing and in evaluating hot tearing.
Specifically, the derivative function of T(fs) could be exploited:
for fs-> 0 for the growth restriction factor Q ( comment this paper and /or recent paper on High Chromium Cast Iron?and or the papers from Maria Balart in Materials Metallurgical Transactions)
for 0.3<fs<0.7 for semi-solid processing ( i.e. rheocasting and thixoforming) ( comment one paper from Ernesto Zoqui)
for fs->1 for evaluating hot tearing (comment?two papers from Sindo Kou)
Implicitly, this approach has been already worked out?in most of the papers presented in this special issue. Here?we will propose the relevant calculations and - where available - the open thermodynamic databases in order to reproduce results with other Calphad software using just one function.
Possible outline
1.?????Calphad method
2.?????Scheil solidification Curve and its numerical derivative
3.?????Three different approaches for the metallurgy of solidification
4.?????Perspectives of the use of this?methodology for the development of new metallic materials and processes
1.?????Calphad method
There are excellent books, an association which organizes and annual conference and a specific scientific journal, on the Calphad – Calculation of Phase Diagram - method. Since the introduction of a fortran code to the appendix of Kaufman and Nesor book in 1970, Calphad method has been useful in calculation of multiple components phase diagrams and derived values. While Larry Kaufman introduced the concept of “lattice stability” i.e. the thermodynamic properties of metastable allotropes with a brilliant and visionary perspective , the current approach of Calphad commonly exploited is based on a paper by Dinsdale published in 1991. Most of the commercial softwares include this thermodynamic database as the first brick of an "inverted pyramid" ; notwithstanding some ongoing work on the development of a new database for elements including ab-initio data (i.e. those obtained at 0 K) the Dinsdale database approach, available for free at https://lnkd.in/eV_wGZdy it is the one universally accepted.
In the figure, the binary "ideal" phase diagram calculated with Dinsdale/SGTE unary database. The binary - in such a way "ideal" database - for Titanium and Coppers is reported here. Similiar binary database using Dinsdale/SGTE data may really easily be constructed such as this
Copper Titanium binary system ( in red) along with one calculated (in blue) using currently available database at NIMS https://lnkd.in/edXP-xYr ( file available for free upon registration). This database has been extracted from thermodynamic optimisation of the Cu-Ti system, K.C. Hari Kumar, I.Ansara, P.Wollants, L.Delaey, Z.Metallkd., 87 (1996) 666-672.
领英推荐
There are indeed more recent approaches to this system which have just been published as a review https://lnkd.in/epSrhrFF . Calphad programs, such as Computherm Pandat, allow easily to calculated contour maps of liquid phase fractions, as in the phase diagram calculated using the mentioned Cu-Ti binary database https://lnkd.in/dyuuEpVq Specified a chemical composition, it is the possible to calculate the solid fraction fs using the lever rule
2.?????Scheil solidification Curve and its numerical derivative
With the same database https://lnkd.in/dyuuEpVq it is then easy to calculate the solid fraction fs in the solidification building a temperature T solid fraction fs relationship and graph, for both the lever and Scheil rules
It is then possible to calculate the numerical derivative of the above T(fs) relationship
This may be done also by extracting/ exporting data from the Tables of the calculation results; other functional relationships are also relatively?easy to be obtained.
T fl fs phase_name Q H_Latent H_tot f_tot(@LIQUID) f_tot(@BCC) w(CU) w(TI) f(@LIQUID) f(@BCC) G x(CU) x(TI) Label #NOME? T_1 -fs//
C mole/mole mole/mole J/mole J/mole J/mole % % mole/mole mole/mole J/mole % %
1553,11 1 0 LIQUID+BCC 0 0 63200,73 1 0 10 90 1 0 -114021 7,72515 92,2749 BCC 0 1826,26 0,016884
1553,093 0,999727 0,000273 LIQUID+BCC -4,32483 -3,43281 63196,41 0,999727 0,000273 10 90 0,999727 0,000273 -114019 7,725147 92,27485 BCC -0,00027 1826,243 0,014821
1553,073 0,999455 0,000545 LIQUID+BCC -8,64813 -6,86414 63192,08 0,999455 0,000545 10 90 0,999455 0,000545 -114017 7,725147 92,27485 BCC -0,00054 1826,223 0,013619
1553,033 0,99891 0,00109 LIQUID+BCC -17,2909 -13,7232 63183,44 0,99891 0,00109 10 90 0,99891 0,00109 -114013 7,725147 92,27485 BCC -0,00109 1826,183 0,01361
1552,953 0,997822 0,002178 LIQUID+BCC -34,5606 -27,4264 63166,17 0,997822 0,002178 10 90 0,997822 0,002178 -114006 7,725147 92,27485 BCC -0,00218 1826,103 0,013592
1552,793 0,995651 0,004349 LIQUID+BCC -69,0373 -54,7735 63131,69 0,995651 0,004349 10 90 0,995651 0,004349 -113990 7,725147 92,27485 BCC -0,00435 1825,943 0,013555
1552,473 0,991325 0,008675 LIQUID+BCC -137,74 -109,231 63062,99 0,991325 0,008675 10 90 0,991325 0,008675 -113959 7,725147 92,27485 BCC -0,00868 1825,623 0,013483
1551,833 0,982742 0,017258 LIQUID+BCC -274,153 -217,209 62926,58 0,982742 0,017258 10 90 0,982742 0,017258 -113897 7,725147 92,27485 BCC -0,01726 1824,983 0,01334
1550,553 0,965848 0,034152 LIQUID+BCC -543,086 -429,488 62657,65 0,965848 0,034152 10 90 0,965848 0,034152 -113773 7,725147 92,27485 BCC -0,03415 1823,703 0,013062
1547,993 0,933108 0,066892 LIQUID+BCC -1065,94 -839,888 62134,79 0,933108 0,066892 10 90 0,933108 0,066892 -113526 7,725147 92,27485 BCC -0,06689 1821,143 0,012531
1545,433 0,901692 0,098308 LIQUID+BCC -1569,81 -1232,12 61630,92 0,901692 0,098308 10 90 0,901692 0,098308 -113279 7,725147 92,27485 BCC -0,09831 1818,583 0,012029
1542,873 0,871519 0,128481 LIQUID+BCC -2055,84 -1607,3 61144,89 0,871519 0,128481 10 90 0,871519 0,128481 -113033 7,725147 92,27485 BCC -0,12848 1816,023 0,011558
1540,313 0,842516 0,157484 LIQUID+BCC -2525,08 -1966,45 60675,65 0,842516 0,157484 10 90 0,842516 0,157484 -112788 7,725147 92,27485 BCC -0,15748 1813,463 0,011114
1537,753 0,814615 0,185385 LIQUID+BCC -2978,49 -2310,51 60222,24 0,814615 0,185385 10 90 0,814615 0,185385 -112544 7,725147 92,27485 BCC -0,18539 1810,903 0,010696
1535,193 0,787752 0,212248 LIQUID+BCC -3416,98 -2640,37 59783,75 0,787752 0,212248 10 90 0,787752 0,212248 -112300 7,725147 92,27485 BCC -0,21225 1808,343 0,010302
1532,633 0,761871 0,238129 LIQUID+BCC -3841,36 -2956,81 59359,37 0,761871 0,238129 10 90 0,761871 0,238129 -112056 7,725147 92,27485 BCC -0,23813 1805,783 0,009929
1530,073 0,736915 0,263085 LIQUID+BCC -4252,41 -3260,59 58948,32 0,736915 0,263085 10 90 0,736915 0,263085 -111814 7,725147 92,27485 BCC -0,26308 1803,223 0,009577
1527,513 0,712837 0,287163 LIQUID+BCC -4650,84 -3552,39 58549,89 0,712837 0,287163 10 90 0,712837 0,287163 -111572 7,725147 92,27485 BCC -0,28716 1800,663 0,009243
1524,953 0,689589 0,310411 LIQUID+BCC -5037,3 -3832,86 58163,43 0,689589 0,310411 10 90 0,689589 0,310411 -111330 7,725147 92,27485 BCC -0,31041 1798,103 0,008927
1522,393 0,667129 0,332871 LIQUID+BCC -5412,41 -4102,6 57788,32 0,667129 0,332871 10 90 0,667129 0,332871 -111089 7,725147 92,27485 BCC -0,33287 1795,543 0,008628
1519,833 0,645415 0,354585 LIQUID+BCC -5776,76 -4362,16 57423,98 0,645415 0,354585 10 90 0,645415 0,354585 -110848 7,725147 92,27485 BCC -0,35458 1792,983 0,008343
1517,273 0,624412 0,375588 LIQUID+BCC -6130,86 -4612,06 57069,88 0,624412 0,375588 10 90 0,624412 0,375588 -110608 7,725147 92,27485 BCC -0,37559 1790,423 0,008073
1514,713 0,604083 0,395917 LIQUID+BCC -6475,21 -4852,78 56725,52 0,604083 0,395917 10 90 0,604083 0,395917 -110369 7,725147 92,27485 BCC -0,39592 1787,863 0,007815
1512,153 0,584397 0,415603 LIQUID+BCC -6810,3 -5084,77 56390,43 0,584397 0,415603 10 90 0,584397 0,415603 -110130 7,725147 92,27485 BCC -0,4156 1785,303 0,007571
1509,593 0,565322 0,434678 LIQUID+BCC -7136,54 -5308,46 56064,19 0,565322 0,434678 10 90 0,565322 0,434678 -109891 7,725147 92,27485 BCC -0,43468 1782,743 0,007337
1507,033 0,54683 0,45317 LIQUID+BCC -7454,36 -5524,24 55746,37 0,54683 0,45317 10 90 0,54683 0,45317 -109653 7,725147 92,27485 BCC -0,45317 1780,183 0,007115
1504,473 0,528894 0,471106 LIQUID+BCC -7764,13 -5732,48 55436,6 0,528894 0,471106 10 90 0,528894 0,471106 -109416 7,725147 92,27485 BCC -0,47111 1777,623 0,006902
1501,913 0,511489 0,488511 LIQUID+BCC -8066,23 -5933,53 55134,51 0,511489 0,488511 10 90 0,511489 0,488511 -109178 7,725147 92,27485 BCC -0,48851 1775,063 0,0067
1499,353 0,494591 0,505409 LIQUID+BCC -8360,97 -6127,72 54839,76 0,494591 0,505409 10 90 0,494591 0,505409 -108942 7,725147 92,27485 BCC -0,50541 1772,503 0,006506
1496,793 0,478179 0,521821 LIQUID+BCC -8648,7 -6315,36 54552,03 0,478179 0,521821 10 90 0,478179 0,521821 -108705 7,725147 92,27485 BCC -0,52182 1769,943 0,006321
1494,233 0,46223 0,53777 LIQUID+BCC -8929,7 -6496,73 54271,03 0,46223 0,53777 10 90 0,46223 0,53777 -108469 7,725147 92,27485 BCC -0,53777 1767,383 0,006143
1491,673 0,446725 0,553275 LIQUID+BCC -9204,26 -6672,11 53996,47 0,446725 0,553275 10 90 0,446725 0,553275 -108234 7,725147 92,27485 BCC -0,55328 1764,823 0,005973
1489,113 0,431646 0,568354 LIQUID+BCC -9472,65 -6841,76 53728,08 0,431646 0,568354 10 90 0,431646 0,568354 -107999 7,725147 92,27485 BCC -0,56835 1762,263 0,005811
1486,553 0,416974 0,583026 LIQUID+BCC -9735,12 -7005,92 53465,61 0,416974 0,583026 10 90 0,416974 0,583026 -107764 7,725147 92,27485 BCC -0,58303 1759,703 0,005655
1483,993 0,402694 0,597306 LIQUID+BCC -9991,92 -7164,81 53208,81 0,402694 0,597306 10 90 0,402694 0,597306 -107530 7,725147 92,27485 BCC -0,59731 1757,143 0,005505
1481,433 0,388789 0,611211 LIQUID+BCC -10243,3 -7318,67 52957,47 0,388789 0,611211 10 90 0,388789 0,611211 -107296 7,725147 92,27485 BCC -0,61121 1754,583 0,005361
1478,873 0,375245 0,624755 LIQUID+BCC -10489,4 -7467,69 52711,36 0,375245 0,624755 10 90 0,375245 0,624755 -107062 7,725147 92,27485 BCC -0,62475 1752,023 0,005223
1476,313 0,362047 0,637953 LIQUID+BCC -10730,4 -7612,06 52470,29 0,362047 0,637953 10 90 0,362047 0,637953 -106829 7,725147 92,27485 BCC -0,63795 1749,463 0,00509
1473,753 0,349183 0,650817 LIQUID+BCC -10966,7 -7751,98 52234,06 0,349183 0,650817 10 90 0,349183 0,650817 -106596 7,725147 92,27485 BCC -0,65082 1746,903 0,004963
1471,193 0,336638 0,663362 LIQUID+BCC -11198,2 -7887,61 52002,48 0,336638 0,663362 10 90 0,336638 0,663362 -106363 7,725147 92,27485 BCC -0,66336 1744,343 0,00484
1468,633 0,324402 0,675598 LIQUID+BCC -11425,3 -8019,13 51775,39 0,324402 0,675598 10 90 0,324402 0,675598 -106131 7,725147 92,27485 BCC -0,6756 1741,783 0,004722
1466,073 0,312463 0,687537 LIQUID+BCC -11648,1 -8146,69 51552,61 0,312463 0,687537 10 90 0,312463 0,687537 -105899 7,725147 92,27485 BCC -0,68754 1739,223 0,004608
1463,513 0,30081 0,69919 LIQUID+BCC -11866,7 -8270,44 51334 0,30081 0,69919 10 90 0,30081 0,69919 -105667 7,725147 92,27485 BCC -0,69919 1736,663 0,004498
1460,953 0,289432 0,710568 LIQUID+BCC -12081,3 -8390,53 51119,39 0,289432 0,710568 10 90 0,289432 0,710568 -105436 7,725147 92,27485 BCC -0,71057 1734,103 0,004392
1458,393 0,278321 0,721679 LIQUID+BCC -12292,1 -8507,08 50908,65 0,278321 0,721679 10 90 0,278321 0,721679 -105205 7,725147 92,27485 BCC -0,72168 1731,543 0,004291
1455,833 0,267465 0,732535 LIQUID+BCC -12499,1 -8620,24 50701,65 0,267465 0,732535 10 90 0,267465 0,732535 -104975 7,725147 92,27485 BCC -0,73254 1728,983 0,004192
1453,273 0,256857 0,743143 LIQUID+BCC -12702,5 -8730,12 50498,24 0,256857 0,743143 10 90 0,256857 0,743143 -104744 7,725147 92,27485 BCC -0,74314 1726,423 0,004097
1450,713 0,246487 0,753513 LIQUID+BCC -12902,4 -8836,85 50298,31 0,246487 0,753513 10 90 0,246487 0,753513 -104514 7,725147 92,27485 BCC -0,75351 1723,863 0,004005
1448,153 0,236349 0,763651 LIQUID+BCC -13099 -8940,53 50101,74 0,236349 0,763651 10 90 0,236349 0,763651 -104284 7,725147 92,27485 BCC -0,76365 1721,303 0,003917
1445,593 0,226433 0,773567 LIQUID+BCC -13292,3 -9041,27 49908,41 0,226433 0,773567 10 90 0,226433 0,773567 -104055 7,725147 92,27485 BCC -0,77357 1718,743 0,003831
1443,033 0,216732 0,783268 LIQUID+BCC -13482,5 -9139,17 49718,22 0,216732 0,783268 10 90 0,216732 0,783268 -103826 7,725147 92,27485 BCC -0,78327 1716,183 0,003749
1440,473 0,20724 0,79276 LIQUID+BCC -13669,7 -9234,34 49531,06 0,20724 0,79276 10 90 0,20724 0,79276 -103597 7,725147 92,27485 BCC -0,79276 1713,623 0,003669
1437,913 0,197949 0,802051 LIQUID+BCC -13853,9 -9326,86 49346,84 0,197949 0,802051 10 90 0,197949 0,802051 -103368 7,725147 92,27485 BCC -0,80205 1711,063 0,003591
1435,353 0,188854 0,811146 LIQUID+BCC -14035,3 -9416,83 49165,46 0,188854 0,811146 10 90 0,188854 0,811146 -103140 7,725147 92,27485 BCC -0,81115 1708,503 0,003516
1432,793 0,179947 0,820053 LIQUID+BCC -14213,9 -9504,33 48986,82 0,179947 0,820053 10 90 0,179947 0,820053 -102912 7,725147 92,27485 BCC -0,82005 1705,943 0,003444
1430,233 0,171223 0,828777 LIQUID+BCC -14389,9 -9589,44 48810,85 0,171223 0,828777 10 90 0,171223 0,828777 -102684 7,725147 92,27485 BCC -0,82878 1703,383 0,003373
1427,673 0,162676 0,837324 LIQUID+BCC -14563,3 -9672,24 48637,45 0,162676 0,837324 10 90 0,162676 0,837324 -102456 7,725147 92,27485 BCC -0,83732 1700,823 0,003305
1425,113 0,1543 0,8457 LIQUID+BCC -14734,2 -9752,82 48466,55 0,1543 0,8457 10 90 0,1543 0,8457 -102229 7,725147 92,27485 BCC -0,8457 1698,263 0,003239
1422,553 0,146091 0,853909 LIQUID+BCC -14902,7 -9831,23 48298,07 0,146091 0,853909 10 90 0,146091 0,853909 -102002 7,725147 92,27485 BCC -0,85391 1695,703 0,003175
1419,993 0,138042 0,861958 LIQUID+BCC -15068,8 -9907,55 48131,94 0,138042 0,861958 10 90 0,138042 0,861958 -101775 7,725147 92,27485 BCC -0,86196 1693,143 0,003113
1417,433 0,13015 0,86985 LIQUID+BCC -15232,6 -9981,86 47968,08 0,13015 0,86985 10 90 0,13015 0,86985 -101549 7,725147 92,27485 BCC -0,86985 1690,583 0,003053
1414,873 0,12241 0,87759 LIQUID+BCC -15394,3 -10054,2 47806,43 0,12241 0,87759 10 90 0,12241 0,87759 -101322 7,725147 92,27485 BCC -0,87759 1688,023 0,002995
1412,313 0,114817 0,885183 LIQUID+BCC -15553,8 -10124,6 47646,92 0,114817 0,885183 10 90 0,114817 0,885183 -101096 7,725147 92,27485 BCC -0,88518 1685,463 0,002938
1409,753 0,107367 0,892633 LIQUID+BCC -15711,2 -10193,3 47489,48 0,107367 0,892633 10 90 0,107367 0,892633 -100871 7,725147 92,27485 BCC -0,89263 1682,903 0,002883
1407,193 0,100055 0,899945 LIQUID+BCC -15866,7 -10260,1 47334,07 0,100055 0,899945 10 90 0,100055 0,899945 -100645 7,725147 92,27485 BCC -0,89994 1680,343 0,00283
1404,633 0,092879 0,907121 LIQUID+BCC -16020,1 -10325,2 47180,6 0,092879 0,907121 10 90 0,092879 0,907121 -100420 7,725147 92,27485 BCC -0,90712 1677,783 0,002778
1402,073 0,085833 0,914167 LIQUID+BCC -16171,7 -10388,6 47029,04 0,085833 0,914167 10 90 0,085833 0,914167 -100195 7,725147 92,27485 BCC -0,91417 1675,223 0,002728
1399,513 0,078914 0,921086 LIQUID+BCC -16321,4 -10450,4 46879,33 0,078914 0,921086 10 90 0,078914 0,921086 -99969,8 7,725147 92,27485 BCC -0,92109 1672,663 0,002679
1396,953 0,072118 0,927882 LIQUID+BCC -16469,3 -10510,6 46731,41 0,072118 0,927882 10 90 0,072118 0,927882 -99745,1 7,725147 92,27485 BCC -0,92788 1670,103 0,002631
1394,393 0,065443 0,934557 LIQUID+BCC -16615,5 -10569,3 46585,24 0,065443 0,934557 10 90 0,065443 0,934557 -99520,7 7,725147 92,27485 BCC -0,93456 1667,543 0,002585
1391,833 0,058885 0,941115 LIQUID+BCC -16760 -10626,6 46440,76 0,058885 0,941115 10 90 0,058885 0,941115 -99296,5 7,725147 92,27485 BCC -0,94112 1664,983 0,00254
1389,273 0,05244 0,94756 LIQUID+BCC -16902,8 -10682,4 46297,92 0,05244 0,94756 10 90 0,05244 0,94756 -99072,6 7,725147 92,27485 BCC -0,94756 1662,423 0,002496
1386,713 0,046105 0,953895 LIQUID+BCC -17044 -10736,7 46156,69 0,046105 0,953895 10 90 0,046105 0,953895 -98848,8 7,725147 92,27485 BCC -0,95389 1659,863 0,002453
1384,153 0,039878 0,960122 LIQUID+BCC -17183,7 -10789,8 46017,01 0,039878 0,960122 10 90 0,039878 0,960122 -98625,3 7,725147 92,27485 BCC -0,96012 1657,303 0,002412
1381,593 0,033756 0,966244 LIQUID+BCC -17321,9 -10841,5 45878,86 0,033756 0,966244 10 90 0,033756 0,966244 -98402 7,725147 92,27485 BCC -0,96624 1654,743 0,002372
1379,033 0,027735 0,972265 LIQUID+BCC -17458,6 -10891,9 45742,17 0,027735 0,972265 10 90 0,027735 0,972265 -98178,9 7,725147 92,27485 BCC -0,97227 1652,183 0,002332
1376,473 0,021813 0,978187 LIQUID+BCC -17593,8 -10941,2 45606,92 0,021813 0,978187 10 90 0,021813 0,978187 -97956 7,725147 92,27485 BCC -0,97819 1649,623 0,002294
1373,913 0,015988 0,984012 LIQUID+BCC -17727,7 -10989,2 45473,07 0,015988 0,984012 10 90 0,015988 0,984012 -97733,3 7,725147 92,27485 BCC -0,98401 1647,063 0,002257
1371,353 0,010257 0,989743 LIQUID+BCC -17860,2 -11036 45340,57 0,010257 0,989743 10 90 0,010257 0,989743 -97510,8 7,725147 92,27485 BCC -0,98974 1644,503 0,002221
1368,793 0,004617 0,995383 LIQUID+BCC -17991,3 -11081,7 45209,4 0,004617 0,995383 10 90 0,004617 0,995383 -97288,5 7,725147 92,27485 BCC -0,99538 1641,943 0,002185
1367,513 0,001831 0,998169 LIQUID+BCC -18056,4 -11104,1 45144,3 0,001831 0,998169 10 90 0,001831 0,998169 -97177,5 7,725147 92,27485 BCC -0,99817 1640,663 0,002168
1366,873 0,000446 0,999554 LIQUID+BCC -18088,9 -11115,2 45111,87 0,000446 0,999554 10 90 0,000446 0,999554 -97122 7,725147 92,27485 BCC -0,99955 1640,023 0,00216
1366,713 0,000101 0,999899 LIQUID+BCC -18097 -11117,9 45103,77 0,000101 0,999899 10 90 0,000101 0,999899 -97108,1 7,725147 92,27485 BCC -0,9999 1639,863 0,002157
1366,673 1,47E-05 0,999985 LIQUID+BCC -18099 -11118,6 45101,75 1,47E-05 0,999985 10 90 1,47E-05 0,999985 -97104,6 7,725147 92,27485 BCC -0,99999 1639,823 0,00121
1366,653 0 1 BCC -18099,8 -11118,6 45100,96 1 10 90 1 -97102,9 7,725147 92,27485 BCC -1 1639,803 0,000264T
3.?????Three different approaches for the metallurgy of solidification?
For reproducing results of some of the papers presented here we need to be able to calculate:
Limit(diff(T(fs),fs),fs = 0)
This value in fact, according to an approach originally proposed in a series of papers by Kozlov and Schmidt-Fetzer is the so called growth restriction factor Q, which is customarily expressed by Q=mc0 (k-1) so here
Another approach, also proposed in two papers of this special issue, is relating the hot cracking to a similiar value
?Here we work some simple mathematics and obtain:
?
Finally criteria for semi-solid, as originally proposed by and re-elaborated by Zoqui also in a paper in this special issue set up bound for the calculation of semi-solid window, in terms of the so called liquid fraction sensitivity:
Where fL=1-fs is the solid fraction