# FLUID FRICTION IN POROUS MEDIA

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FLUID FRICTION IN POROUS MEDIA
FLOW IN PACKED BEDS FLUID FRICTION IN POROUS MEDIA PACKED TOWERS Packed towers are finding applications in adsorption, absorption, ion-exchange, distillation, humidification, catalytic reactions, regenerative heaters etc., Packing is to provide a good contact between the contacting phases. Based on the method of packing, Packings are classified as (a) Random packings (b) Stacked packings

The packings are made with clay, porcelain, plastics or metals.

Principal requirements of a tower packing
It must be chemically inert to the fluids in the tower. It must be strong without excessive weight. It must contain adequate passages for the contacting streams without excessive pressure drop. It must provide good contact between the contacting phases. It should be reasonable in cost.

FLUID FRICTION IN POROUS MEDIA
In this approach, the packed column is regarded as a bundle of crooked tubes of varying cross sectional area The theory developed for single st. tubes is used to develop the results of bundle of crooked tubes….. Laminar flow Turbulent flow Transition flow

Laminar flow in packed beds
Porosity (void fraction) is given by ε = (volume of voids in the bed / (total volume of bed ) Superficial velocity ‘vs’ = (Q / Apipe) Interstitial velocity ‘vI’ = (Q / ε Apipe) EMPTY TOWER VELOCITY Velocity based on the area actually open to the flowing fluid

i.e., AREA AVAILABLE FOR FLOW = A ε
In a packed bed consider a set of crooked tubes (non-circular CSA) rH = (cross sectional area of channel) / (wetted perimeter of channel) Multiply and divide by LENGTH of bed =rH = (A ε ) L / (wetted perimeter) L =rH = ε (volume of bed) / (Total wetted surface area of solids)

To find wetted surface area……. Total wetted surface area of solids
= (no. of spherical particles) x (surface area of one particle) and we know…. No. of particles = (volume of bed) (1- ε) / (volume of one particle) volume fraction

ERGUN defined NRe,p without the constant term (4/6) for PACKED BED

Only if NRe,p < 10 (LAMINAR)
By several experiments it has been found that the constant value should be 150 KOZNEY-CARMANN EQN. Only if NRe,p < 10 (LAMINAR)

Turbulent flow in packed beds
By several expts it has been found that for turbulent flow, the ‘ 3f ’ should be replaced by a value 1.75 BURK- PLUMMER EQN. Only if NRe,p > 1000 (TURBULENT)

FOR TRANSITION REGION…
ERGUN EQN. if NRe,p between 10 and (TRANSITION)

PROB….. Calculate the pressure drop of air flowing at 30ºC and 1 atm pressure through a bed of 1.25 cm diameter spheres, at a rate of 60 kg/min. The bed is 125 cm diameter and 250 cm height. The porosity of the bed is The viscosity of air is cP and the density is gm/cc.

Data: Mass flow rate of Air = 60 kg/min = 1 kg/sec Density of Air (r) = gm/cc = kg/m3 Viscosity of Air (m) = cP = x 10-3 kg/(m.sec) Bed porosity (e) = 0.38 Diameter of bed (D)= 125 cm = 1.25 m Length of bed (L) = 250 cm = 2.5 m Dia of particles (Dp)= 1.25 cm = m

Volumetric flow rate = mass flow rate / density = 1 / 1. 156 = 0
Volumetric flow rate = mass flow rate / density = 1 / = m3/sec Superficial velocity Vo = / ( (p/4) D2 ) = / ( (p/4) ) = m/sec NRe,P = x x / ( x 10-3 x ( ) ) = 903…….Transition region We shall use Ergun equation to find the pressure drop. Dp = N/m2

Pressure Drop in Regenerative Heater
A regenerative heater is packed with a bed of 6 mm spheres. The cubes are poured into the cylindrical shell of the regenerator to a depth of 3.5 m such that the bed porosity was If air flows through this bed entering at 25ºC and 7 atm abs and leaving at 200ºC, calculate the pressure drop across the bed when the flow rate is 500 kg/hr per square meter of empty bed cross section. Assume average viscosity as cP and density as 6.8 kg/m3.

Mass flow rate of Air / unit area = 500 kg/(hr.m2) = 0.139 kg/(sec.m2)
Density of Air (r) = 6.8 kg/m3 Viscosity of Air (m) = cP = x 10-3 kg/(m.sec) Bed porosity (e) = 0.44 Length of bed (L) = 3.5 m Dia of particles (Dp)= 6 mm = m

Superficial velocity Vs = mass flow rate per unit area / density = 0
Superficial velocity Vs = mass flow rate per unit area / density = / 6.8 = m/sec NRe,p = x x 6.8 / (0.025 x 10-3 x ( ) ) = 59.45 We shall use Ergun equation to find the pressure drop. ∆P = N/m2

Design of Packed Tower with Berl Saddle packing
7000 kg/hr of air, at a pressure of 7 atm abs and a temperature of 127oC is to be passed through a cylindrical tower packed with 2.5 cm Berl saddles. The height of the bed is 6 m. What minimum tower diameter is required, if the pressure drop through the bed is not to exceed 500 mm of mercury? For Berl saddles, Dp = (1.65 x 105 Z Vs1.82 r 1.85 ) / Dp1.4 where Dp is the pressure drop in kgf/cm2, Z is the bed height in meter, r is the density in g/cc, Dp is nominal diameter of Berl saddles in cm, Vs is the superficial linear velocity in m/sec.

Data: Mass flow rate = 7000 kg/hr = kg/sec Height of bed (Z) = 6 m Dp = 2.5 cm 760 mm Hg = 1 kgf/cm2 = 1 atm Dp = 500 mm Hg = (500/760) x 1 kgf/cm2 = 0.65 kgf/cm2 Formula: Ideal gas law: PV = nRT Formula given, Dp = (1.65 x 105 Z Vs1.82 r 1.85 ) / Dp1.4

Calculations: r = M(n/V) = M(P/RT) = 29 x 7 x x 105 / (8314 x ( ) ) = kg/m3 = x 10-3 g/cc Dp = (1.65 x 105 Z Vs1.82 r 1.85 ) / Dp1.4 0.65 = (1.65 x 105 x 6 x Vs1.82 x (6.185 x 10-3 )1.85 ) / Vs1.82 = Vs = m/sec Volumetric flow rate = mass flow rate/density = 1.944/6.185 = m3/sec Required Minimum Diameter (D) = m.

Air flows thro a packed bed of powdery material of 1cm depth at a superficial gas velocity of 1cm/s. A manometer connected to the unit registers a pressure drop of 1cm of water. The bed has a porosity of 0.4. Assuming that Kozney-Carmann equation is valid for the range of study, estimate the particle size of the powder? Density of air = 1.23kg/m3 viscosity of air = 1.8x10-5 kg/m-s Dp=1.24x10-4m

Flow Rate of Water through Ion-Exchange Column
Figure shows a water softener in which water trickles by gravity over a bed of spherical ion-exchange resin particles, each 0.05 inch in diameter. The bed has a porosity of Calculate the volumetric flow rate of water. Assume laminar flow.

g(∆z) = hf hf = ∆p/ρ=3.7376 J/kg
Applying Bernoulli's equation from the top surface of the fluid to the outlet of the packed bed and ignoring the kinetic-energy term and the pressure drop through the support screen, which are both small, we find ……… Since Laminar flow, apply Kozney-Carmann equation vs = m/sec = Q = 21cm3/sec g(∆z) = hf hf = ∆p/ρ= J/kg

Water trickles by gravity over a bed of particles each 1mm dia in a bed of 6cm and height 2m. The water is fed from a reservoir whose dia is much larger than that of packed bed, with water maintained at a height of 0.1m above the top of the bed. The bed has a porosity of calculate the volumetric flow rate of water if its viscosity is 1cP

Shape factor-Sphericity factor
For non-spherical particles instead of diameter an equivalent diameter is defined. Sphericity Φs is defined as the surface-volume ratio for a sphere of dia Dp divided by the surface-volume ratio for the particle whose nominal size is Dp. Φs = (6/Dp) / (sp/vp) Therefore, actual dia to be used in Ergun eqn is = Φs Dp

For a non-spherical particle, Ergun eqn is given by………