u
=
e
(1.29)
T he im portance o f this equation resides in the fact that it relates the perm eability to the porosity o f the m edium ,
K -
(1.30)
180(l-e)2
A nalysis o f a great deal o f data has led to a change in the value o f the constant from 180 to 150, resulting in the B lake-K ozeny equation (M cC abe and Sm ith, 1976).
T hese equations w ere developed for the flow of fluids through rigid porous m edia, how ever, extensions have been m ade in order to m ake possible the use o f these equations with beds of com pressible particles.
1.4.3.3 M odels for com pressible beds
F o r non-com pressible beds the pressure drop at low R eynolds num bers < 10, is a linear function o f flow rate; and the flow resistance is constant.
In the case o f beds packed w ith non-rigid particles the bed void fraction and therefore the resistance to flow vary with the degree o f particle deform ation. T he deform ation in turn is a function o f the pressure drop, the colum n length and the colum n diam eter. W ith this type o f particle the pressure drop through a bed is a non-linear function o f linear flow rate. M odels fo r com pressible beds, in w hich these variables have been related, are presented in this section. Joustra et al (1967) m odified D arcy ’s equation to fit the properties o f S ephadex gels by assum ing an exponential relation between the perm eability K, and the pressure drop AP. T his assum ption led to the follow ing equation:
= -
w here a and K„ are constants. represents the perm eability w hen no pressure is being applied on the bed, this is the perm eability at zero pressure drop (no flow), a is a constant fo r a particular gel, and is a function o f the colum n dim ensions and the particle rigidity.
A ccording to this equation there is a flow rate m axim um corresponding to the optim al pressure drop (i.e. the largest pressure drop ju st before clogging occurs), w hich according to hydrodynam ic considerations is determ ined by the state o f com pression o f the bottom section o f the colum n (this section being subjected to the largest com pression force). In their analysis, they considered the force exerted by the w eight o f the gel to be sm all w hen com pared to the pressure drop, and since their m odel did not take into account friction forces against the colum n walls, they assum ed the supporting force o f the colum n wall to be negligible for beds o f diam eters larger than 15 cm. The optim al pressure drop is, therefore alm ost independent o f bed height, and depending on the colum n diam eter it m ay also be alm ost independent o f colum n diam eter. The assum ptions m ade in the developm ent o f this m odel should result in poor predictions o f experim ental data in particular when wall friction is important.
B uchholz and G odelm ann (1978) considered that the flow through a com pressible bed m ust take into account varying hydraulic radii in the How path, caused by particle deform ation. The pressure drop then becomes a function o f bed height h, u^, and R^. B ased on this consideration, they developed a m odel based on elastic deform able spheres w ith a m odulus A defined by
— = A AP
(1.32)
w here Rp is the particle radius, and AR, represents the particle deform ation. T hey calculated AP=f(h) by approxim ation and com pared the results w ith AP = f ( u j , w hich was experim entally determ ined. T he approxim ation was achieved by calculating the pressure drop at each lay er o f spheres form ing the bed (in a dense colum n packing) by m eans o f the H agen-PoiseuiU e equation, and using this inform ation to evaluate the particle deform ation and the reduction o f void volum e at each o f these layers. T he sum o f all these losses in pressure and void fraction gave the total loss along the colum n. By using this m odel it was found th at the m ain pressure drop and occlusion occurred at the bottom o f the fixed bed.
In a study related to glucose isom erase reactors V erhoff and Furjanic (1983) developed a m odel w hich uses the K ozeny-C arm an equation to describe the fluid dynam ics. T hey analyzed the forces acting on a bed o f deform able particles and derived a m odel that predicts the solids pressure at each position along a column. The force analysis included the w eight o f the particles as w ell as the drag force on the particles due to the flow o f a fluid, and unlike all the previous m odels, it considered the support from the walls o f the container resulting from friction forces arising at this location. T hese researchers were unable to develop a satisfactory theoretical description o f the change in void volum e as a function o f applied pressure, and used the follow ing em pirical relationship to com plete their m odel, (full derivation o f this m odel is presented in section 2.2),
e = —
(1. 33)
( 1 + a P ^ )
The m odel was used to investigate the influence o f the wall friction coefficient, the colum n dim ensions, the particle size, and the m atrix com pressibility on colum n hydrodynam ics. V erhoff and F u ijan ic (1983) also suggested experim ents for testing various deform able m aterials for use in packed beds and em ployed the m odel to predict the perform ance o f plant-scale equipm ent. D avies and BeUhouse (1989) in a study o f the perm eability o f beds o f agarose-based particles, developed a m odel based on the following force balance on a cross-sectional elem ent o f a colum n,