Editing Microfiltration (section)

===Fundamental design equations===

As separation is achieved by sieving, the principal mechanism of transfer for microfiltration through micro porous membranes is bulk flow.<ref>Seadler, J & Henley, E 2006, ''Separation Process Principles'', 2nd Edn,  John Wiley & Sons Inc. New Jersey p.540-542</ref>

Generally, due to the small diameter of the pores the flow within the process is laminar ([[Reynolds Number]]  < 2100) The flow velocity of the fluid moving through the pores can thus be determined (by [[Hagen–Poiseuille equation|Hagen-Poiseuille]]'s equation), the simplest of which assuming a [[parabola|parabolic]] [[boundary layer|velocity profile]].

:<math> v = \frac{D^2*\Delta P}{32*\mu *L} </math>

'''Transmembrane Pressure''' (TMP)<ref>Cheryan, M 1998, ''Fouling and Cleaning. in Ultrafiltration and Microfiltration Handbook'' 2nd edn., CRC Press, Florida, 645.</ref>

The transmembrane pressure (TMP) is defined as the mean of the applied pressure from the feed to the concentrate side of the membrane subtracted by the pressure of the permeate. This is applied to dead-end filtration mainly and is indicative of whether a system is fouled sufficiently to warrant replacement.

:<math> v = \frac{P_F + P_C}{2} - P_P </math>

Where

* <math>P_f</math> is the pressure on the Feed Side
* <math>P_c</math> is the pressure of the Concentrate
* <math>P_p</math> is the pressure of the Permeate

'''Permeate Flux'''<ref>Ghosh, R, 2006, ''Principles of Bioseparations Engineering'', Word Scientific Publishing Co.Pte.Ltd, Toh Tuck Link, p.233</ref>

The permeate flux in microfiltration is given by the following relation, based on [[Darcy's Law]]

:<math> J_v = \frac{1}{A_M}*\frac{dV}{dt} = \frac{\Delta P}{\mu *(R_u + R_c)}</math>

Where
* <math>R_u</math> = Permeate membrane flow resistance (<math>m-1</math>)
* <math>R_c</math> = Permeate cake resistance (<math>m-1</math>)
* μ = Permeate viscosity (kg m-1 s-1)
* ∆P = Pressure Drop between the cake and membrane

The cake resistance is given by:

:<math> R_c= r*\frac{V_S}{A_m} </math>

Where

* r = Specific cake resistance (m-2)
* Vs = Volume of cake (m3)
* AM = Area of membrane (m2)

For micron sized particles the Specific Cake Resistance is roughly.<ref>Ghosh, R, 2006,''Principles of Bioseparations Engineering'', Word Scientific Publishing Co.Pte.Ltd, Toh Tuck Link, p.234</ref>

:<math> r= \frac{180*(1-\epsilon)}{\epsilon^3*d_s^2 } </math>

Where
* ε = Porosity of cake (unitless)
* d_s = Mean particle diameter (m)

'''Rigorous design equations'''<ref>Polyakov, Yu,  Maksimov, D & Polyakov, V, 1998 'On the Design of Microfilters' ''Theoretical Foundations of Chemical Engineering'', Vol. 33, No. 1, 1999, pp. 64–71.</ref>

To give a better indication regarding the exact determination of the extent of the cake formation, one-dimensional quantitative models have been formulated to determine factors such as

* Complete Blocking (Pores with an initial radius less than the radius of the pore)
* Standard Blocking
* Sublayer Formation
* Cake Formation

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