Lami's theorem

2014-01-24T15:54:31Z

117.198.127.145:

{{for|the settlement in the North West province of South Africa|Slurry, North West}}
[[Image:High angle.jpg|right|thumb|A slurry composed of glass beads in [[silicone oil]] flowing down an inclined plane.]]
A '''slurry''' is a thin sloppy mud or cement or, in extended use, any fluid mixture of a pulverized solid with a liquid (usually water), often used as a convenient way of handling solids in bulk.<ref>Oxford English Dictionary 2nd ed.: Slurry</ref> Slurries behave in some ways like thick [[fluid]]s, flowing under gravity but are also capable of being [[pump]]ed if not too thick.

==Examples==
Examples of slurries include:
* Cement slurry, a mixture of [[cement]], water, and assorted dry and liquid additives used in the [[Petroleum industry|petroleum]] and other industries<ref>[http://www.glossary.oilfield.slb.com/Display.cfm?Term=cement Shlumberger: Oilfield glossary]</ref><ref>[http://www.rheonova.fr/en/chemistry Rheonova : Measuring rheological propertis of settling slurries]</ref>
* Soil/cement slurry, also called Controlled Low-Strength Material (CLSM), flowable fill, controlled density fill, flowable mortar, plastic soil-cement, K-Krete, and other names<ref>[http://www.cement.org/basics/concreteproducts_clsm.asp Portland Cement Association: Controlled Low-Strength Material]</ref>
* A mixture of [[thickening agent#weapon use|thickening agent]], oxidizers, and water used to form a [[Water gel explosive|gel explosive]]{{Citation needed|date=May 2012|reason=is this called a slurry?}}
* A mixture of [[pyroclastic]] material, rocky debris, and water produced in a [[volcanic eruption]] and known as a [[lahar]]
* A mixture of [[bentonite]] and water used to make [[slurry wall]]s
* [[Coal slurry]], a mixture of coal waste and water, or crushed coal and water<ref>[http://www.redvalve.com/rv/index.php?option=com_content&task=view&id=137&Itemid=56 Red Valve Company: Coal Slurry Pipeline]</ref>
* A mixture of wood pulp and water used to make [[paper]]
* A mixture of animal waste, organic matter, and sometimes water known simply as "slurry" in [[Agriculture|agricultural]] use, used as [[fertilizer]] after ageing in a [[slurry pit]]
* [[Meat slurry]], a mixture of finely ground meat and water, centrifugally dewatered and used as food
* An abrasive substance used in [[chemical-mechanical polishing]]
* [[Slurry ice]], a mixture of ice crystals, freezing point depressant, and water
* A mixture of raw materials and water involved in the [[rawmill]] manufacture of [[Portland cement]]
* A mixture of minerals, water, and additives used in the manufacture of [[ceramic]]s
* A [[Bolus (digestion)|bolus]] of chewed food mixed with saliva<ref>[http://www.rheonova.fr/en/fodd-industry Rheonova : Measuring food bolus properties]</ref>

==Calculations==

===Determining solids fraction===
To determine the percent solids (or solids fraction) of a slurry from the density of the slurry, solids and liquid<ref>Wills, B.A. and Napier-Munn, T.J, ''Wills' Mineral Processing Technology: an introduction to the practical aspects of ore treatment and mineral recovery, ISBN 978-0-7506-4450-1'', Seventh Edition (2006), Elsevier, Great Britain</ref>
:<math>\phi_{sl}=\frac{\rho_{s}(\rho_{sl} - \rho_{l})}{\rho_{sl}(\rho_{s} - \rho_{l})}</math>
where
:<math>\phi_{sl}</math> is the solids fraction of the slurry (state by volume)
:<math>\rho_{s}</math> is the solids density
:<math>\rho_{sl}</math> is the slurry density
:<math>\rho_{l}</math> is the liquid density

In aqueous slurries, as is common in mineral processing, the specific gravity of the species is typically used, and since <math>SG_{water}</math> is taken to be 1, this relation is typically written:
:<math>\phi_{sl}=\frac{\rho_{s}(\rho_{sl} - 1)}{\rho_{sl}(\rho_{s} - 1)}</math>
even though specific gravity with units tons/m^3 is used instead of the SI density unit, kg/m^3.

===Liquid mass from mass fraction of solids===
To determine the mass of liquid in a sample given the mass of solids and the mass fraction:
By definition
:<math>\phi_{sl}=\frac{M_{s}}{M_{sl}}</math>*100
therefore
:<math>M_{sl}=\frac{M_{s}}{\phi_{sl}}</math>
and
:<math>M_{s}+M_{l}=\frac{M_{s}}{\phi_{sl}}</math>
then
:<math>M_{l}=\frac{M_{s}}{\phi_{sl}}-M_{s}</math>
and therefore
:<math>M_{l}=\frac{1-\phi_{sl}}{\phi_{sl}}M_{s}</math>
where
:<math>\phi_{sl}</math> is the solids fraction of the slurry
:<math>M_{s}</math> is the mass or mass flow of solids in the sample or stream
:<math>M_{sl}</math> is the mass or mass flow of slurry in the sample or stream
:<math>M_{l}</math> is the mass or mass flow of liquid in the sample or stream

===Volumetric fraction from mass fraction===
:<math>\phi_{sl,m}=\frac{M_{s}}{M_{sl}}</math>
Equivalently
:<math>\phi_{sl,v}=\frac{V_{s}}{V_{sl}}</math>
and in a minerals processing context where the specific gravity of the liquid (water) is taken to be one:
:<math>\phi_{sl,v}=\frac{\frac{M_{s}}{SG_{s}}}{\frac{M_{s}}{SG_{s}}+\frac{M_{l}}{1}}</math>
So
:<math>\phi_{sl,v}=\frac{M_{s}}{M_{s}+M_{l}SG_{s}}</math>
and
:<math>\phi_{sl,v}=\frac{1}{1+\frac{M_{l}SG_{s}}{M_{s}}}</math>
Then combining with the first equation:
:<math>\phi_{sl,v}=\frac{1}{1+\frac{M_{l}SG_{s}}{\phi_{sl,m}M_{s}}\frac{M_{s}}{M_{s}+M_{l}}}</math>
So
:<math>\phi_{sl,v}=\frac{1}{1+\frac{SG_{s}}{\phi_{sl,m}}\frac{M_{l}}{M_{s}+M_{l}}}</math>
Then since
:<math>\phi_{sl,m}=\frac{M_{s}}{M_{s}+M_{l}}=1-\frac{M_{l}}{M_{s}+M_{l}}</math>
we conclude that
:<math>\phi_{sl,v}=\frac{1}{1+SG_{s}(\frac{1}{\phi_{sl,m}}-1)}</math>

where
:<math>\phi_{sl,v}</math> is the solids fraction of the slurry on a ''volumetric'' basis
:<math>\phi_{sl,m}</math> is the solids fraction of the slurry on a ''mass'' basis
:<math>M_{s}</math> is the mass or mass flow of solids in the sample or stream
:<math>M_{sl}</math> is the mass or mass flow of slurry in the sample or stream
:<math>M_{l}</math> is the mass or mass flow of liquid in the sample or stream
:<math>SG_{s}</math> is the bulk specific gravity of the solids

==See also==
{{commons category|Slurry}}
* [[Slurry pipeline]]
* [[Slurry transport]]

==References==
{{reflist}}

==External links==
{{wiktionary|slurry}}
* Antonio Bonapace - "[http://solids-hydraulic-transport.com/introduction.html A General Theory of the Hydraulic Transport of Solids in Full Suspension]"
* [http://www.sciencedirect.com/science/article/pii/S089417771200310X Ravelet, F., Bakir, F., Khelladi, S., Rey, R. (2012). Experimental study of hydraulic transport of large particles in horizontal pipes. Experimental thermal and fluid science.]
* [https://www.westerndredging.org/index.php/information/proceedings-presentations/category/62-session-4b-dredging-research?download=248:2-ming-hydraulic-transport-of-coarse-gravela-laboratory-investigation-into-flow-resistancepdf Ming, G., Ruixiang, L., Fusheng, N., Liqun, X. (2007). Hydraulic Transport of Coarse Gravel-A Laboratory Investigation Into Flow Resistance.]

[[Category:Materials]]

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Lami's theorem