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Modeling Crystal Populations: Moment-Transformed BPBE Simulation

Derive and compare crystal size distributions using numerical simulations and moment-transformation methods. Invert moments to generate 'typical' CSD. Solve non-linear ODEs. Linearly invert moments for CSD plot. Achieve agreement with Resmini's model.

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Modeling Crystal Populations: Moment-Transformed BPBE Simulation

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  1. Model Populations with the Moment- Transformed BPBE Describe Numerical Simulation of CSDs Derive Moment- Transformed BPBE Obtains Moments of the CSD as a Function of Time Compare CSDs with Those Obtained from the Inverted Moments Invert Moments to Obtain the ‘Typical’ CSD ln(n) vs. L Generate Numerical CSDs Summary and Conclusions Symbol Table Outline 1 2 3 4 5 6 7 8

  2. The Batch Population Balance Equation (BPBE) is: Here, nucleation is included as a source term and the derivation of Hulburt and Katz (1964) is followed. The moments of a crystal size distribution (CSD) are calculated as follows (see also, e.g., Randolph and Larson, 1988): This transformation is applied to the BPBE:

  3. For the third term, assume: This gives: The integration is applied to each term of the BPBE. First term: For the second term, assume G  G(L). Integration by parts gives:

  4. ...but, since mj = mj(t): j = 1, 2, 3,... Assembling terms yields the moment-transformed BPBE: Thus, a single partial differential equation (the BPBE) may now be represented by a set of ordinary differential equations (ODEs). Now, let crystal growth rate, G, be given by: The crystal nucleation rate, I, is given by Cashman (1993):

  5. (cooling rate of the liquid ...with latent heat) Now: where L0 = 0.5 x 10-6 m The cooling rate is from Jaeger (1957) for an infinite half-sheet of magma: This combination of G and I mechanisms has been demonstrated to yield CSDs typical of those in natural rocks (Resmini, 2001; 2002). This now yields the following set of ODEs (for j = 0 to 7; see below):

  6. (Implicit in derivation of cooling rate expression.) This is a set of nonlinear ordinary differential equations (ODEs) solvable numerically using a fourth-order Runge-Kutta method. Though the set of ODEs is closed after j = 2, eight equations are utilized to facilitate an inversion of the moments. The solution to the set of ODEs is a table of eight moment values as a function of time (from first nucleation event to complete solidification).

  7. Moments … … … … … … … … … For a position located one meter from the contact within the infinite half-sheet described by the Jaeger (1957) cooling-rate expression, the moments are as follows: Complete solidification at x’ = 1 meter after 457 hours

  8. The set of eight moment values may be inverted to yield the more familiar crystal size distribution plot; i.e., ln(n) vs. L. This is done using constrained linear inversion according to the following equation: where: n(L) is the crystal population density as a function of crystal size; K is a matrix of quadrature coefficients (aka kernel function); g is a Lagrange multiplier; H is a smoothing matrix of 2nd order finite-difference coefficients; and mj* is a vector of the first eight moments (calculated above) K is calculated with a weighting function to smooth n(L) vs. L Constrained linear inversion as applied here is described in, e.g., Twomey (1977), Steele and Turco (1997) and King (1982).

  9. and is a smoothing matrix. n(L) is obtained by multiplying by h(L) after inversion: For brevity, the terms for the inversion are shown, below, for four moments: The matrix on the left is K and is derived from a quadrature approximation to the integral equation that defines the moments of the crystal population. h(L) is a weighting function to smooth n(L). La - Ld represent four crystal sizes. The right-most vector is mj*. The ‘usual’ CSD is generated by computing ln(n(L)) and plotting vs. L.

  10. Comparison with the Numerical Simulation of Resmini (2001)

  11. Moment 25 20 From Resmini (2001) 15 ln(n), no./cm4 10 5 From the inversion 0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 L (cm) The first four moments are compared to those obtained from the model of Resmini (2001) for 100% solidification. Agreement is excellent. Both CSDs are typical of those observed in natural rocks. The CSD is compared to one obtained from the model of Resmini (2001).

  12. Summary and Conclusions • A moment transformation is applied to the batch population balance equation(BPBE), a number continuity equation describing the evolution of crystalpopulations in closed magmatic systems such as sills. • The more typical CSD, usually presented as a plot of the natural log ofcrystal population density (ln(n)) vs. crystal size (L), is obtained by inversionof the moments. • The CSDs recovered by constrained linear inversion of the momentsare similar to those observed in natural rocks. • The moments of the crystal population are, to within ~0.4%, identical to thosegenerated by the numerical simulation described in Resmini (2001). • These results demonstrate an equivalence between equation-based modelingand a numerical simulation of evolving crystal populations. • Modeling crystal populations with the moment-transformed BPBE is faster thanthe iterative, computationally-intensive simulation of Resmini (2001).

  13. Acknowledgements Partial funding for this work provided by The Boeing Company. References Cited

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