56 SHORT COMMUNICATIOPii

A mte on the equations for compresii~e cake obtained from a reduction of voidage in the cake. fiMration

The theory of the filtration process has been devel- oped by a number of workers and Heertjes’ has reviewed the present state of knowledge_ It is the purpose of this note to reconsider the fundamental partial differential equations governing cake filtra- tion_ These, together with consolidation and per- meability data and the boundary conditions ap propriate to the mode of operation enable filters to be designed in a logical manner.

In the following analysis it is assumed that the solid particles and the liquid are both incom- pressible, ie_ cake compression is due to particle distortion and rearrangement with negligible den- sity change_ It is assumed that the filter cake is saturated with liquid at all times, and that the system is isothermal. The small effects due to gravity are not considered_

P

4 r

X

L P a &

P

e

liquid pressure, solid effective stress liquid superficial velocity solid superticial velocity u.&mce from cake base cake depth applied filtration pressure specific resistance of cake void fraction liquid viscosity time

Subscripts I liquid 0 outlet condition s solid 72 space velocity

Theory

At low liquid velocities, the one-dimensional flow through filter cakes may be analysed in terms of the modified Darcy equation

where q(O) is the superIicial liquid velocity at time 8 in a stationary cake in which the liquid pressure

Ruth’ and Grace3*” have

shown that this equati& provides a sound basis for industrial analysis of incompressible and compress- ible cakes Tiller - lo has extended this work and has shown that significant effects may occur especially in the filtration of concentrated slurries owing to a variation of the liquid superficial velocity with position in the cake. In such instances the superficial velocity can vary by a factor as much as two’O; the balance of the outlet liquid is

Filter cake

__1______ -- 1 I --- x=0

q_ (61

Fig. 1. Diagrammatic representation of the compressibie cake !ilvation pr-.

Consider unit cross section of the cake shown diagrammatically in Fig. 1. In order to conform to the normal conventions of fluid mechanics the flows are negative in the direction shown on the figure. From continuity, at some time, @, the outlet liquid superficial velocity mllst equal the sum of the superficial liquid and solid velocities at some height x (Otxt L)_ Thus

qde)=q(~~)+~(~,~). (3

Neglecting the small effects due to the change of momentum of the liquid and solid with position in the filter cake, the applied filtration pressure must equal the sum of the liquid pressure, pt. and the effective stress, pP on the solids. p, is the consolidat- ing force on the solid per unit area of solid phis liquid.

p(e) = Pi (x, 0) f P&G 0) - (3)

ps determines the equilibrium voidage” and it is assumed that this voidage is attained instantane- ously. In many cases this is believed to be a good approximation3*4*‘2.

Powdm TeclmoL, 1(1%7) 54-57

The space velocity of the solids in the cake is given by

(4)

Darcyk law is required. Now the permeability data is determined from a stationary experiment General equations have been developed which but the solid is in motion. Thus the super&Sal allow for the m0vemer.t of both liquids and solids velocity of the liquid with respect to the space in filter cakes. The analysis given will be most velocity of the solids must be inserted into the important when slxries of high solid content -which expression for the law_ Hence give a compressible cake are fihered ic items of

equipment in which the deposition time is in- herently low, as for example in rotating horizontal drum filters.

It is at this point the equations differ s@ificantly from those discussed previously_ The second term on the left hand side may not be neglected. Combin- ing eqns (2), (3) and (5) yields

4(x=@ - (40w-qke) ( 1 -+,e) ) =-g-e);_) Pembroke Street,

Cambridge (Gt. Britain)

I P. M. HEERTJa Trmr_ INI. C&m hgrs.. 42 (19M) X56.

By mass balance on an element of cake, the 2 B. F_ RLEH. hi. Eng_ Chem. 27 (1935) 708.

Terzaghi equation may be obtained 3 H. P. GRAPE, C&m Eng. Progr., 49 (1953) 303. 4 H. P_ Grurz Chon. Eng. Progr., 49 (1953) 367.

(cgq = _ (y); 5 F. M. Tm CSan. .Eng_ Progr_. 49 (1953) 462 6 F. hf. TILLER. Chem. fig. Pmgr_. 53 (1955) 282. 7 F_ M. Txu.n_ A_tCXE. J_, 4 (1958) 170.

SHORT COXMUNl CATIONS 57

then be integrated numerically with appropriate boundary conditions using the consolidation and permeability data.

Discss~-on

J. BRIDGWATER

Department of Chemical Engineering. University of Cambridge,

Although expressions such as the Kozeny- 8 E M. TILLER AXD H. R COOPE& A_ZLX..E J_. 6 (1960) 595. 9 F. M. TILLER AND H. COOPER, A.Z_Ch.E J., 8 (1962) 445.

Carman equation13 may be used to develop eqns. 10 F. M. Tm AhD M. SHIRATO, A.I.Ch.E_ J.. IO (1961) 61.

(6) and (7), when it is deemed that the effects llkl%S ~HEIDEGGER, The Phs-s&v of ~70~. rhmqh Porous

discussed here are significant in an industrial ~fcdti, University of Toroaro Press, 1957. P_ 20.

situation it would be essential to determine the 12 F_A.Ko- AhD D-R BOYLAX A.I.C.X.E. J-.4(19%)

175.

consolidation (p, zx E) and permeability (pS us. a) 13 P_CC~~!~~~,~ranshsf. Chem. Engrs_.16(1938)168_

characteristics of the cake in the manner discussed by Grace and Tiller3*5*5. The eqns. (6) and (7) can Received August 4, 1966