Simple Explanation of the CompressiblePacking Model

Cameron Kellough

February 20, 2008

1 Introduction

The compressible packing model was developed by Francois de Larrard todescribe the packing density achieved by a granular mixture. 1 This modelhas been described in other places besides the footnoted book including in apaper by the United States National Institute of Standards and Technology2.The major tenet of the model is that all size classes in the mixture interactwith all other sizes classes in the mixture affecting the overall density. It isa semi-empirical model.

The compressible packing model models the actual packing density whichcan be achieved using a mixture of actual aggregates. The prediction of actualdensities differentiates this model from similar models and statistical exper-iments which predict the theoretical density that could be achived placingaggregate grain by grain. This latter concept is known as the virtual packingdensity.

2 Virtual Packing Density

Of specific interest in the model is the virtual packing density for a class whenit is dominant. It should be noted here that the virtual packing density is a

1See Concrete Mixture Proportioning, a Scientific Approach by Francois de Larrard,c©1999, E & FN SPON, 11 New Fetterer Lane, London EC4P 4EE.

2http://fire.nist.gov/bfrlpubs/build98/PDF/b98023.pdf

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theoretical concept that is only a component of the achievable density of anactual mixture.

A class of large grains is dominant when it fills all space available such thatadding a few smaller grains would merely fill interstitials without forcing thelarge particles apart. A class of small grains is dominant when it consumesmore space than the interstitials available between the large particles.

The virtual packing density of a single component mixture is denoted βi

and is about 0.72 for spheres. 2

The overall virtual packing density for a mixture of any number of par-ticle size classes with independent beta values is defined by the followingequation where the value yi represents the volume fraction of the ith sizeclass when each of the i size classes to be mixed are measured in beakersbefore combination.

The book equation for γi is the following:

γi =βi

1−∑ i−1j=1[1− βi + bijβi(1− 1

βj)]yj −

∑ nj=i+1[1− aij

βi

βj]yj

(1)

After thinking about it however and trying to get results to match thebook’s graphs, I believe the following to be correct:

γi =βi

1−∑ i−1j=1[1− βi + bijβi(1− 1

βj)]yj −

∑ i+1j=n[1− aij

βi

βj]yj

(2)

3 The Loosening Effect

The loosening effect describes an effect whereby the introduction of smallparticles forces apart larger particles. The theoretical basis for this was acurve fit of an analysis of several researchers data over the course of morethan 50 years.

aij =

√1− (1− dj

di

)1.02 (3)

The value di represents is the average particle diameter.

2http://www.niss.org/technicalreports/tr104.pdf

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4 The Wall Effect

The wall effect describes an effect whereby larger particles cause interstitialsin the mixture which are too small to be filled by other particle classes. Thetheoretical basis for this was a curve fit of an analysis of several researchersdata over the course of more than 50 years.

bij = 1− (1− di

dj

)1.50 (4)

5 The Compressible Packing Model Equation

The final equation for the compressible packing model involves a bunch ofderivation which is not of great interest to those implementing the model.As a result, I will state without proof that the final equation for the modelis given below where Φ represents the percentage solids of the mixture andthe value of K, a constant related to the manner in which the mixture iscompacted (corresponding to vibration under 15psi of pressure) is equal tothe value 9.0.

The situation in the book is confused by the notation in which Ki isdescribed. The original equation in the book is the following:

9.0 = K =∑

ni=1Ki =

∑ni=1

yi

βi

1Φ− 1

γi

(5)

What was really meant is this:

Ki =

yi

βi

1Φ− 1

γi

(6)

9.0 = K =∑

ni=1Ki (7)

To get a mixture of maximal density, one must optimize the above equa-tion for Φ.

A theoretical result for a simplified model elsewhere in de Larrard’s bookindicates that an optimal mixture with size classes of the form dn = λdn

max

with n = 0 for the largest and n = i for the ith size class should have equalvalues of Ki when the density is maximized.

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6 The Segregation Potential

Once again, avoiding the derivation, we reproduce an equation for a factorcalled the segregation potential. This value represents the tendency for amixture to segregate.

S = 1− min1<=i<=n

Ki

1 + Ki

(8)

7 Solving for Packing Density

To find the optimal mixture, one must solve the compressible packing modelequation given above such that Φ is maximized and S is minimized. It can beobserved from the segregation potential equation that the minimum value forKi is achieved when all Ki are equal since 9.0 = K =

∑ nj=1Ki. This condition

minimizes both the segregation potential and maximizes the density. Onecan optimize for Φ without optimizing for S but the mixture will have alarger tendency to segregate which is not suitable for Epoxy Granite. Thefinal solution then is to optimize the compressible packing model equationfor Ki = Ki+1 = ....

8 Measuring βi

βi is measured by placing a mass MD of aggregate in a cylinder width diam-eter D which is more than 5 times the maximum size of the aggregate. Thecylinder is closed with a piston which exerts a force of 10kPa and vibratedat an acceleration of 4G for 2 minutes. This corresponds to the conditionsfor K = 9. After the vibration, the height h of the packed aggegate is mea-sured and the actual packing density calculated according to the followingequation:

Φ =4MD

πD2hρD

(9)

where ρD is the intrinsic density of the dry aggregate.Assuming that the aggregate is from a single sieve size where the ratio

between sieve sizes is less than 1.26:1, the uncorrected mean virtual packingdensity can be expressed as follows:

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β = (1 +1

K)Φ (10)

The corrected virtual packing density is then expressed as follows:

β =β

1− (1− kw)[1− (1− d/D)2(1− d/h)](11)

where kw is the wall effect coefficient, d is the average diameter of theaggregate and h is the height in the cylinder. The value of kw was measuredby de Larrard for round and crushed grains in 30 widely varying cases. Forround grains, kw = 0.88. For crushed grains, kw = 0.73

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