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Analyzing Batch Crystal Population Balance and Cooling Rate Effects on Crystallization Dynamics

This study addresses the batch crystal population balance equation using the nucleation and growth rate relationships established by Cashman (1993) and a cooling rate expression from Jaeger (1956). It focuses on the dynamics of crystallization in a half-sheet magma system, presenting detailed analytical solutions under defined initial and boundary conditions. Key parameters, including cooling rates and growth rates, are calculated from synthetic crystal size distributions (CSDs) derived from numerical cooling results. The findings provide insights into the crystallization process in magmatic systems.

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Analyzing Batch Crystal Population Balance and Cooling Rate Effects on Crystallization Dynamics

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  1. Abstract

  2. AT-67 14.0 12.0 10.0 ln(n) (no./cm4) 8.0 6.0 4.0 2.0 0.0 0.0 0.5 1.0 1.5 2.0 L (mm) A Typical CSD

  3. (of the liquid) The Problem Solve the batch crystal population balance equation given by: using the crystal nucleation rate (I) and growth rate (G) relations of Cashman (1993): with a cooling rate expression from Jaeger (1956) for an infinite half-sheet of magma: Symbols and values are given in the Symbol Table, below.

  4. ...This gives the following: Equation Initial Conditions Boundary Conditions Solution (for t, L > 0)

  5. 22 Analytical 20 ln(n) (no./cm4) 18 16 log(I’) log(G’) m p 0.47 -8.050 1.37 1.00 14 0.00 0.02 0.04 0.06 0.08 L (mm)

  6. 20.0 y = -12.02x + 18.04 18.0 16.0 14.0 ln(n) (no./cm4) 12.0 2 R = 0.992 10.0 8.0 6.0 log(I) = 0.47 + 1.37*log(T/t) 4.0 0.00 0.25 0.50 0.75 1.00 1.25 L (mm)

  7. 22 Analytical Numerical 20 ln(n) (no./cm4) 18 log(I’) log(G’) m p 0.47 -8.050 1.37 1.00 16 14 0.0 0.02 0.04 0.06 0.08 L (mm)

  8. 22 p = 0.88 p = 1.00 20 ln(n) (no./cm4) 18 16 log(I’) log(G’) m 0.47 -8.050 1.37 14 0.00 0.05 0.10 0.15 L (mm)

  9. log(I’) log(G’) m p 0.72 -7.810 1.35 0.88 20 18 16 14 12 log(I’) log(G’) m p -2.47 -7.150 1.79 0.64 ln(n) (no./cm4) 10 8 6 4 2 0.0 0.5 1.0 1.5 2.0 2.5 L (mm)

  10. * * * *

  11. ln(n) where: Cooling Rate = t = (time at complete solidification) - (time of crystallization of first solids) L ln(n°) ln(n) ln(n) Slope = -1/Gt I = n°G L L (mm) ln(n) log(growth rate) log(nucleation rate) L log(cooling rate) log(cooling rate) Numerical Sill Solidification Model Synthetic CSDs Calculated From Numerical Cooling Results t, cooling rate, G, and I Calculated from Synthetic CSDs and Numerical Cooling Results log(I) = 0.47 + 1.37*log(T/t) and mass-distribution growth rate mechanism. Wallrock Sill Log(cooling rate), log(G), and log(I) are plotted Wallrock

  12. Show sill CSDs

  13. log(I) = 0.47 + 1.37*log(cooling rate) -5 -6 -7 y = 0.87x - 7.97 -8 log(Growth Rate), cm/sec -9 2 y = 0.88x - 8.22 R = 1 -10 -11 -12 -3 -2 -1 0 1 2 log(Cooling Rate), °C/hr

  14. log(I) = 0.47 + 1.37*log(cooling rate) 3 2 y = 1.36x + 0.45 2 R = 0.9997 1 log(Nucleation Rate), no/cm3/sec 0 y = 1.37x + 0.47 -1 -2 -3 -3 -2 -1 0 1 2 log(Cooling Rate), °C/hr

  15. Table 1 Meters Slope Intercept Res.Time T/t log(T/t) G (from CSDs) log(G) log(I) (from contact) cm-1 no./cm4 cm/sec cm/sec no./cm3.sec hours °C/hour °C/hour 1 -180.04 19.68 482 0.498 -0.303 3.20 x 10-09 -8.49 0.05 2 -148.30 18.88 2186 0.110 -0.959 8.57 x 10-10 -9.07 -0.87 3 -133.76 18.47 5120 0.047 -1.329 4.06 x 10-10 -9.39 -1.37 4 -125.92 18.23 9288 0.026 -1.588 2.38 x 10-10 -9.62 -1.71 5 -120.21 18.04 14690 0.016 -1.787 1.57 x 10-10 -9.80 -1.97

  16. Symbol Table

  17. References Cited

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