FLUID MECHANICS. TUTORIAL No.4 FLOW THROUGH POROUS PASSAGES


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1 FLUID MECHANICS TUTORIAL No.4 FLOW THROUGH POROUS PASSAGES In thi tutorial you will continue the work on laminar flow and develop Poieuille' equation to the form known a the Carman  Kozeny equation. Thi equation i ued to predict the flow rate through porou paage uch a filter, filter bed and fluidied bed in combution chamber. On completion of thi tutorial you hould be able to do the following. Derive the Carman  Kozeny equation. Solve problem involving the flow of fluid through a porou material. In order to do thi tutorial you mut be familiar with Poieuille' equation for laminar flow and thi i covered in tutorial 1.
2 FLOW THROUGH POROUS PASSAGES The following are example where porou flow occur. A filter element made of thick intered particle. Thi might be a cylinder with radial flow. A and bed filter for cleaning water. The water percolate down through the filter though long tortuou paage. The depth of water on top of the filter govern the rate at which the water i forced through. A layer of rock through which water, ga or oil might eep. Thi i imilar to a radial flow filter but on a much larger cale.
3 When a fluid pae through a porou material, it flow through long thin tortuou paage of varying cro ection. The problem i how to calculate the flow rate baed on nominal thickne of the layer. Thi wa tackled by Kozeny and later by Carman. The reult i a formula, which give a mean velocity of flow in the direction at right angle to the layer plane in term of it thickne and other parameter. The paage between the particle i o mall that the velocity in them i mall and the flow i well and truly laminar. Poieuille' Equation for laminar flow tate p/l  µu'/d. Kozeny modelled the layer a many mall capillary tube of diameter D making up a layer of cro ectional area A. The actual cro ectional area for the flow path i A'. The difference i the area of the olid material. The ratio i ε A'/A and thi i known a the poroity of the material. The volume flow rate through the layer i Q. Figure 1 Kozeny ued the notion that Q Au where u i the mean velocity at right angle to the layer. The volume flow rate i alo Q A'u' where u' i the mean velocity in the tube. Equating u' ua/a' ε void fraction A'/A. u' u/ε. Carman modified thi formula when he realied that the actual velocity inide the tube mut be proportionally larger becaue the actual length i greater than the layer ul thickne. It follow that u εl where l i the layer thickne and l' the mean length of the paage. Subtituting thi in Poieuille' Equation give : p µ ul l εld p µ ul rearranging l lεd Thi i uually expreed a a preure gradient in the direction of the mean flow (ay x) and it become : dp µ ul...(1) dx lεd
4 A ca of the olid A' ca of the tube A ca of the layer A' + A. A A ε A A + A Multiply top and bottom by thye length l and the area become volume o ε + Q where Q' i the volume of the tube and Q i the volume of the olid. ε + Q Q Q ( ) Q Q Q Q εq Q Q Q εq Q Q + Q S Surface Area of the tube. Divide both ide by S εq S S nπd l Q' i made up of tube diameter D and length l' o 4 and S nπdl' where n i the number of tube which cancel when thee are 4εQ ubtituted into the formula. Thi reult in : D S 1 ε ( ) Next we conider the olid a made up of pherical particle of mean diameter d. S urface area of tube but alo the urface area of the olid particle. πd Q Hence Q and S πd and 6 S εd It follow that D 1 ε ( ) d 6 Subtitute thi into equation (1) and : Reearch ha hown that (l'/l) i about.5 hence : dp dx dp dx 7µ ul ld ε d 180µ u ε Thi i the Carman Kozeny.
5 WORKED EXAMPLE No.1 Water i filtered through a and bed 150 mm thick. The depth of water on top of the bed i 10 mm. The poroity ε i 0.4 and the mean particle diameter i 0.5 mm. The dynamic vicoity i 0.89 cp and the denity i 998 kg/m. Calculate the flow rate per quare metre of area. SOLUTION The preure difference acro the and bed i aumed to be the head of water ince atmopheric preure act on top of the water and at the bottom of the bed. Firt convert the head into preure difference. p ρg(h h 1 ) {0 (998 x 9.81 x 0.1)} Pa. The length of the bed (L) i 10 mm. Aume that dp/dx p/l The dynamic vicoity i 0.89 x 10  N/m. Uing the Carman Kozeny equation where the preure gradient i aumed to be linear. dp dx 180µ u 1 d ε u 0.54x10 Q 0.54x10 ( ε)  dp x 0.89 x 10 u (10.4)  dx x10 x u m / m / per quare meter of area.
6 WORKED EXAMPLE No. Calculate the flow rate through a filter 70 mm outide diameter and 40 mm inide diameter and 100 mm long given that the preure on the outide i 0 kpa greater than on the inide. The mean particle diameter d i 0.04 mm and the void fraction i 0.. The dynamic vicoity i 0.06 N /m. SOLUTION The flow i radial o dp/dx dp/dr ince radiu increae in the oppoite ene to x in the derivation. The equation may be written a : dp dx r i the radiu. Putting in value: d 180µ u ε 0.) dp 180x0.06u 9 1.5x10 u dr x0. Conider an elementary cylinder through which the oil flow. The velocity normal to the urface i Q Q 1 u 1.591Qr πlr πx0.1xr hence: dp 1.5 x 109 x 1.591Q r 1 dr Q r 1 dr Integrating between the outide and inide radiu yield: Ro p Q ln R i 5 p Q ln 0 Q 18.x10 9 m / 18.mm /
7 SELF ASSESSMENT EXCERCISE No.1 Q.1 Outline briefly the derivation of the CarmanKozeny equation. dp 180µ u dl ε d dp/dl i the preure gradient, µ i the fluid vicoity, u i the uperficial velocity, d i the particle diameter and ε i the void fraction. A cartridge filter conit of an annular piece of material of length 150 mm and internal diameter and external diameter 10 mm and 0 mm. Water at 5oC flow radially inward under the influence of a preure difference of 0.1 bar. Determine the volumetric flow rate. (1.5 cm/) For the filter material take d 0.05 mm and ε 0.5. µ0.89 x 10 N /m and ρ 997 kg/m. Q. (a) Dicu the aumption leading to the equation of horizontal vicou flow dp 180µ u through a packed bed dl ε where p i the preure drop acro a bed of depth L, void fraction ε and effective particle diameter d. u i the approach velocity and µ i the vicoity of the fluid. (b) Water percolate downward through a and filter of thickne 15 mm, coniting of and grain of effective diameter 0. mm and void fraction The depth of the effectively tagnant clear water above the filter i 0 mm and the preure at the bae of the filter i atmopheric. Calculate the volumetric flow rate per m of filter. (. dm/) (Note the denity and vicoity of water are given in the intruction on all exam paper) µ0.89 x 10 N /m and ρ 997 kg/m d
8 Q. Oil i extracted from a horizontal oilbearing tratum of thickne 15 m into a vertical bore hole of radiu 0.18 m. Find the rate of extraction of the oil if the preure in the borehole i 50 bar and the preure 00 m from the bore hole i 50 bar. Take d 0.05 mm, ε 0.0 and µ 5.0 x 10  N /m.
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