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Towards an Analytical Expression for the Formation of Crystal Size

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Towards an Analytical Expression for the Formation of Crystal Size

Distributions (CSDs) in Closed Magmatic Systems

Ronald G. Resmini

The Boeing Company, Chantilly, Virginia 20151

v: (703) 735-3899, f: (703) 735-3305

e: ronald.g.resmini@boeing.com

Introduction

- Resmini (1993, 2000, 2001, and 2002) has shown that the batchpopulation balance equation and associated expressions (see slide 3)generate CSDs and properties of CSDs observed in natural rocks.
- The equations on slide 3 must be solved numerically.
- However, it can be shown that the 2nd and 3rd CSD moments varylinearly in time during the formation of a crystal population.
- This is important because the 2nd CSD moment may be cast by aterm that’s easily, analytically calculated: the 3rd CSD moment isrelated to the amount of solids present in the crystallizing system.
- It is then possible to analytically calculate a CSD for a closed system.
- All of the calculations are based on the analytical expression ofJaeger (1957) for the cooling and solidification of an infinite half-sheetof magma with latent heat.

initial condition

boundary condition, where:

...from Cashman (1993)

(cooling rate of the liquid

...with latent heat)

For an infinite half-sheet of magma; from Jaeger (1957)

Batch Population

Balance Equation

(BPBE)

Integro-Partial

Differential Equation

BTW...This is

Analytically

Intractable!

(See slide 15 for definition of all symbols.)

The integro-partial differential equation (batch population balance equation)on slide 2 is intractable.

The BPBE may be cast as a set of nonlinear ordinary differential equations

and solved numerically for moments of the CSD vs. time (Resmini, 2002).

This yields the important result that the 2nd and 3rd CSD moments vary linearly.This is shown on slide 5 and is from Resmini (2002).

This is important because the 2nd CSD moment (intractable term in the integro-partial differential equation) may be cast by a term that’s easily, analytically calculated:

the 3rd CSD moment—which is related to amount of solids presentin the crystallizing system.

Key Relationship: The 2nd CSD Moment Varies Linearly with the 3rd Moment

2nd CSD Moment, cm2/cm3

( to total amount of surface area)

Increasing Time

3rd CSD Moment, cm3/cm3

( to amount of mass present)

Amount of mass present is easily calculated from the expression of Jaeger (1957).

where G is now given as

(and it’s not an integral):

Can Now Recast Integro-Partial Differential Equation as Follows:

The initial condition is still:

...but the new boundary condition, n(0,t) = I/G, becomes:

(of the liquid)

A Bit More About Recasting G...

Recasting G is based on the Jaeger (1957) expression for temperature vs. time

in a cooling and solidifying half-sheet of magma—with latent heat:

This expression is based on a linear relationship between the amount of solids

precipitated vs. temperature (within the solidification interval).

Thus, temperature may be easily used to estimate amount of solids present within

the solidification interval—which in turn is linearly related to the 2nd CSD moment.

Note the expression for TLiquid, above, in the equation for G on slide 6—

with some terms for the linear relationship between the 2nd and 3rd CSD moments.

e

{

{

{

{

{

{

{

{

{

{

a

{

q

h

c

d/t

f

j

c

g

sqrt(d)/sqrt(t)

...and the values of the other variables are:

The Solution:

The Analytical CSD

ln(n), no./cm4

L (mm)

CSD, after complete solidification, for a position 1 meter

from the Jaeger (1957) half-sheet/wallrock contact

All values used to generate this CSD are the same as those used togenerate the CSD from Resmini (2002) and shown next...

ln(n), no./cm4

L (mm)

Numerical CSD

CSD, after complete solidification, for a position 1 meter

from the Jaeger (1957) half-sheet/wallrock contact.

Discussion

- The analytical CSD resembles those observed in natural rocks.
- It is not, however, equivalent to that generated by the numerical model.
- This is because of the initial condition of n(0,L) = 0.
- Note on slide 12 that the linear relationship between the 2nd and 3rdmoments does not hold at the very beginning of solidification.
- The linear relationship is established very early in the solidification interval.
- The problem that should be solved is that given on slide 13; notethe initial condition that should be employed; f(L) = n(L) early inthe solidification interval.

...it’s not linear at the very beginningof the solidification interval...

...quickly becomes linear...

2nd CSD Moment

2nd CSD Moment

3rd CSD Moment

3rd CSD Moment

Note Different Initial Condition

...but same B.C., n(0,t) = I/G :

(of the liquid)

Should really solve:

Summary and Conclusions

- An analytical model for the formation of crystal size distributions (CSDs) inclosed magmatic systems such as sills has been presented.
- The model is based on noting (from previous modeling efforts) that the 2ndand 3rd CSD moments vary linearly in time during the formation of acrystal population.
- This important relationship facilitates the conversion of an intractable integro-partial differential equation (the batch population balance equation)to a tractable form.
- The model produces CSDs similar to those observed in natural rocks.
- However, the model must incorporate non-zero initial conditions (i.e.,n(0,L) = f(L)); this is a direction of future work.
- An analytical expression for the generation of CSDs can facilitate closercoupling of quantitative petrography with geochemical thermodynamicmodels of igneous petrogenesis.

Symbol Table

References Cited