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Abstract

Abstract. AT-67. 14.0. 12.0. 10.0. ln(n) (no./cm 4 ). 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. (of the liquid). The Problem. Solve the batch crystal population balance equation given by:.

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Abstract

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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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