Model Populationswith the Moment-

TransformedBPBE

Obtains Momentsof the CSD as a

Function ofTime

DeriveMoment-

TransformedBPBE

Invert Momentsto Obtain the‘Typical’ CSD

ln(n) vs. L

Compare CSDswith Those

Obtained from theInverted Moments

DescribeNumerical

Simulation ofCSDs

GenerateNumerical

CSDs

1 2 3

4 65

7Summary

andConclusions

8

Outline

Symbol Table

LIL

nG

t

n

0

j dLLIL

nG

t

nL

0

jj dLt,LnLt,L

The Batch Population Balance Equation (BPBE) is:

Here, nucleation is included as a source term and thederivation of Hulburt and Katz (1964) is followed.

The moments of a crystal size distribution (CSD) arecalculated as follows (see also, e.g., Randolph and Larson, 1988):

This transformation is applied to the BPBE:

tdL

t

nL j

0

j

t,LjGdLL

nLGdL

L

nGL 1j

0

j

0

j

The integration is applied to each term of the BPBE. First term:

For the second term, assume G G(L). Integration by parts gives:

j0

0

j LIdLLIL

For the third term, assume: 0LLL This gives:

2TT

isandsolidsofareasurfacetotalAwhereA

1

t

m1G

m''

t

TIIgivesthis

t

Tm)Ilog()Ilog(

The crystal nucleation rate, I, is given by Cashman (1993):

Assembling terms yields the moment-transformed BPBE:

Now, let crystal growth rate, G, be given by:

Thus, a single partial differential equation (the BPBE) may now berepresented by a set of ordinary differential equations (ODEs).

j01j

j ILjGt

j = 1, 2, 3,...j01j

j ILjGdt

d

...but, since j = j(t):

m'0

t

TI

dt

d

This now yields the following set of ODEs (for j = 0 to 7; see below):

)))t(2

'x(exp(t

)(

'x

)b(erf1

TT

2

1

t

T 2

22

12

21

23

21

(cooling rate of the liquid...with latent heat)

The cooling rate is from Jaeger (1957) for an infinite half-sheet of magma:

TA

1

t

T

T

m1Ggiveswhich

t

T

T

m

t

mthatso

t

m

t

mLet

Now:

This combination of G and I mechanisms has been demonstrated toyield CSDs typical of those in natural rocks (Resmini, 2001; 2002).

0

m'

2a

01 Lt

TI

kt

Tk

dt

d

where L0 = 0.5 x 10-6 m

30

m'

a

3L

t

TI

t

T

k

k3

dt

d

40

m'

2 a

3 4L

t

TI

k t

Tk4

dt

d

50

m'

2 a

4 5L

t

TI

k t

Tk5

dt

d

60

m'

2 a

5 6L

t

TI

k t

Tk6

dt

d

70

m'

2 a

6 7L

t

TI

k t

Tk7

dt

d

20

m'

2 a

1 2L

t

TI

k t

Tk2

dt

d

.constT

m1k

(Implicit in derivation of cooling rate expression.)

This is a set of nonlinear ordinary differential equations (ODEs) solvablenumerically using a fourth-order Runge-Kutta method.

Though the set of ODEs is closed after j = 2, eight equationsare utilized to facilitate an inversion of the moments.

The solution to the set of ODEs is a table of eight momentvalues as a function of time (from first nucleation event to

complete solidification).

For a position located one meter from the contact within the infinitehalf-sheet described by the Jaeger (1957) cooling-rate expression,

the moments are as follows:

Complete solidification at x’ = 1 meter after 457 hours

Time (hrs) 0th 1st 2nd 3rd 4th 5th 6th 7th10 1776.012 77.809 0.214 4.450E-04 3.485E-05 2.285E-07 9.113E-10 2.833E-1211 2517.336 78.287 0.246 5.870E-04 3.528E-05 2.647E-07 1.216E-09 4.364E-1212 3458.459 78.974 0.280 7.570E-04 3.585E-05 3.028E-07 1.582E-09 6.462E-1213 4623.948 79.926 0.315 9.550E-04 3.661E-05 3.430E-07 2.012E-09 9.247E-1214 6036.531 81.209 0.352 1.183E-03 3.758E-05 3.853E-07 2.510E-09 1.285E-1115 7716.693 82.889 0.390 1.442E-03 3.880E-05 4.297E-07 3.078E-09 1.739E-1116 9682.406 85.037 0.430 1.732E-03 4.030E-05 4.763E-07 3.720E-09 2.299E-1117 11948.970 87.725 0.471 2.055E-03 4.210E-05 5.254E-07 4.436E-09 2.979E-1118 14528.950 91.023 0.514 2.409E-03 4.424E-05 5.771E-07 5.228E-09 3.789E-1119 17432.220 94.998 0.558 2.794E-03 4.674E-05 6.317E-07 6.098E-09 4.739E-1120 20666.030 99.713 0.605 3.212E-03 4.961E-05 6.893E-07 7.046E-09 5.839E-11

450 2126360.000 6027.355 31.057 2.374E-01 2.461E-03 3.263E-05 5.269E-07 9.795E-09451 2127810.000 6032.731 31.087 2.377E-01 2.463E-03 3.266E-05 5.274E-07 9.804E-09452 2129254.000 6038.090 31.117 2.379E-01 2.466E-03 3.269E-05 5.278E-07 9.813E-09453 2130693.000 6043.434 31.147 2.381E-01 2.468E-03 3.272E-05 5.283E-07 9.822E-09454 2132126.000 6048.761 31.177 2.384E-01 2.470E-03 3.275E-05 5.288E-07 9.831E-09455 2133554.000 6054.073 31.207 2.386E-01 2.473E-03 3.278E-05 5.293E-07 9.840E-09456 2134977.000 6059.368 31.236 2.388E-01 2.475E-03 3.281E-05 5.298E-07 9.850E-09457 2136394.000 6064.648 31.266 2.391E-01 2.477E-03 3.284E-05 5.303E-07 9.859E-09

Moments

… … … … … … … … …

The set of eight moment values may be inverted to yield the morefamiliar crystal size distribution plot; i.e., ln(n) vs. L. This is done

using constrained linear inversion according to the following equation:

Constrained linear inversion as applied here is described in,e.g., Twomey (1977), Steele and Turco (1997) and King (1982).

*j

T1T KHKKLn

where:n(L) is the crystal population density as a function of crystal size;K is a matrix of quadrature coefficients (aka kernel function); is a Lagrange multiplier;H is a smoothing matrix of 2nd order finite-difference coefficients; and

j* is a vector of the first eight moments (calculated above)

K is calculated with a weighting function to smooth n(L) vs. L

3

2

1

0

d

c

b

a

3dd

3cc

3bb

3aa

2dd

2cc

2bb

2aa

1dd

1cc

1bb

1aa

0dd

0cc

0bb

0aa

n

n

n

n

LL)L(h5.0LL)L(hLL)L(hLL)L(h5.0

LL)L(h5.0LL)L(hLL)L(hLL)L(h5.0

LL)L(h5.0LL)L(hLL)L(hLL)L(h5.0

LL)L(h5.0LL)L(hLL)L(hLL)L(h5.0

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 j*.

2100

1210

0121

0012

H and is a smoothing matrix.

)L(n

)L(h

)L(h

)L(h

)L(h

n

n

n

n

d

c

b

a

d

c

b

a

n(L) is obtained by multiplying by h(L) after inversion:

The ‘usual’ CSD is generated by computing ln(n(L)) and plotting vs. L.

Comparison with the Numerical Simulation of Resmini (2001)

The CSD obtained from inversion of the moments is compared to that obtained

by the numerical simulation of Resmini (2001). The numerical simulation of

Resmini (2001) tracks batches of crystals as they nucleate and grow within the

solidification interval and is not based on a finite-difference or other numerical

approximation of the BPBE. In Resmini (2001) the nucleation mechanism is

identical to that described above; crystal growth, however, proceeds by applying

the solids formed during cooling to each pre-existing nucleus and crystal. The

amount of solids formed is easily calculated with the linear fraction of solids (f)

vs. temperature relationship within the solidification interval used in the modified

heat capacity method employed by Jaeger (1957). Thus, a single growth rate is

calculated such that if every pre-existing crystal, crystallite, or nucleus in the

system grows at that rate, all newly formed solid is consumed in crystal growth.

All particles grow at the same rate at each time-step; growth rate dispersion or

size-dependent growth rate mechanisms are not employed.

The CSD is compared to one obtained from the model of Resmini (2001).

The first four moments are compared to those obtained from themodel of Resmini (2001) for 100% solidification. Agreement is excellent.

Numerical Simulation

Moment Transformed

BPBE

Relative Error (%)

0th 2136394 2136394 0.0000

1st 6077.71 6064.65 0.2153

2nd 31.38 31.27 0.3768

3rd 0.24 0.24 -0.1337M

om

ent

Both CSDs are typicalof those observed innatural rocks.

0

5

10

15

20

25

0.00 0.01 0.02 0.03 0.04 0.05 0.06

L (cm)

ln(n

), n

o./c

m4

From Resmini (2001)

From the inversion

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).

References Cited

AcknowledgementsPartial funding for this work provided by The Boeing Company.

Cashman, K.V., (1993). Relationship between plagioclase crystallization and cooling rate in basaltic melts. Contributionsto Mineralogy and Petrology, v. 113, pp. 126-142.

Hulburt, H.M., and Katz, S., (1964). Some problems in particle technology: a statistical mechanical formulation. ChemicalEngineering Sciences, v. 19, pp. 555-574.

King, M.D., (1982). Sensitivity of constrained linear inversions to the selection of the Lagrange multiplier. Journal of theAtmospheric Sciences, v. 39, pp. 1356-1369.

Randolph, A.D., and Larson, M.A., (1988). Theory of Particulate Processes: Analysis and Techniques of ContinuousCrystallization, 2nd ed. Academic Press, San Diego, 369 p.

Resmini, R.G., 2001. The crystal size distribution (CSD) intercept vs. slope relationship: a numerical simulation. EOS,Trans., A.G.U., v. 82, no. 20, p. S432.

Resmini, R.G., 2000. Numerical simulation of crystal size distributions (CSDs) in sills. EOS, Trans., A.G.U., v. 81, no. 19,p. S435.

Steele, H.M., and Turco, R.P., (1997). Retrieval of aerosol size distribution from satellite extinction spectra usingconstrained linear inversion. Journal of Geophysical Research, v. 102, no. D14, 27 July, pp. 16737-16747.

Twomey, S., (1977). Introduction to the Mathematics of Inversion and Indirect Measurements. Development inGeomathematics, no. 3, Elsevier Scientific Publishing, Amsterdam, (republished by Dover Publ., 1996), 243 p.