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Conventional Filtration CE 547

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Filtration is a unit operation of separating solids from liquids. Types of Filters (to create pressure differential to force the water through the filter): Gravity filters Gravity filters Pressure filters Pressure filters Vacuum filters Vacuum filters In terms of media Perforated plates Perforated plates Septum of woven materials Septum of woven materials Granular materials (such as sand) Granular materials (such as sand) Sand Filters Slow sand filters (1.0 – 10 m 3 /m 2.d) Slow sand filters (1.0 – 10 m 3 /m 2.d) Rapid sand filter (100 – 200 m 3 /m 2.d) Rapid sand filter (100 – 200 m 3 /m 2.d)

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Medium Specification for Granular Filters 1. Medium is the most important component of granular filters Small grain sizes tend to have higher head losses Small grain sizes tend to have higher head losses Large grain sizes may not be effective in filtering Large grain sizes may not be effective in filtering The most effective grain sizes are found from previous experience The most effective grain sizes are found from previous experience 2. Effective size of medium is specified in terms of : Effective size Effective size Uniformity coefficient Uniformity coefficient

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What is Effective Size? It is the size of sieve opening that passes the 10% finer of the medium sample (the 10 th percentile size, P 10 ) What is Uniformity Coefficient? It is the ratio of the size of the sieve opening that passes the 60% finer of the medium sample (P 60 ) to the size of the sieve opening that passes the 10% finer of the medium sample (P 10 ). It is P 60 to P 10. For Slow Sand Filters P 10 = 0.25 – 0.35 mm P 10 = 0.25 – 0.35 mm P 60 /P 10 = 2 – 3 P 60 /P 10 = 2 – 3 For Rapid Sand Filters P 10 = 0.45 mm and higher P 10 = 0.45 mm and higher P 60 /P 10 = 1.5 and higher P 60 /P 10 = 1.5 and higher

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Example 7.1

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Linear Momentum Equation Applied to Filters (Fig 7.10) Movement of water through a filter bed is similar to the moment of water in parallel pipes, except that the motion is not straight but tortuous. If momentum equation applied on water flow in the downward direction of the element, then:

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F 2 = unbalanced force in downward (z) direction (inertia force) p = hydrostatic pressure A = cross-sectional area of the cylindrical element of the fluid F g = weight of water in the element F sh = shear force acting on the fluid along the surface areas of the grains d V = volume of element dl = differential length of the element A s = surface area of the grains k = factor which converts As into an area such that ( kA s dl = d V ) = porosity of the bed a 2 = acceleration of fluid element in downward (z) direction = fluid mass density = fluid element velocity in the (z) direction = fluid element velocity in the (z) direction t = time

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Since K i = proportionality constant V = average water velocity

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Since P s = drop in pressure due to shear force = liquid viscosity l = length of pipe D = diameter of pipe

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K s = proportionality constant r H = hydraulic radius = (area of flow / wetted perimeter) then Since F g is constant, it can be included in K i and K s and removed from the equation. Then: This is a good linear momentum equation which can be applied to any filter

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If particles are spherical d = diameter of particle

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Thus r H = (area of flow / wetted perimeter) = (volume of flow / wetted area) if N = number of grains v p = volume of each grain Thus Since V can be expressed as = (v s / ) v s = superficial velocity

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Head Loss in Grain Filters There are two categories head loss in clean filters head loss in clean filters head loss due to the deposited materials head loss due to the deposited materials A. Clean-filter Head Loss S p = surface area of a particle N = number of grains in the bed Volume of bed grains = S 0 l(1- ) S 0 = empty bed or surficial area of the bed = porosity of bed l = length of bed

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if v p is volume of a grain, then:

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Now we have

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Substitute (2), (3), and (4) in (1)

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f p is a form of friction factor. Since:

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Since particles are not spherical and = shape factor d p = sieve diameter After backwash, the grain particles are allowed to settle. That means, the particles will deposit layer by layer, and hence particles will be of different sizes.

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The bed is said to be stratified, and the head loss will be the sum of head losses of each layer. Then: If x i is the fraction of the d i particles in the i th layer, then Assume is the same throughout the bed, then

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Example 7.2

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Head Losses Due to Deposited Materials If q = deposited materials per unit volume of the bed, then: If h L0 = clean-bed head loss, then C = concentration of solids introduced into bed l = length of the filter bed

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How to determine (a) and (b) in: It is clear that the equation represents a straight line if ln (h d ) was plotted against ln (q).

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This means that only two data points are needed in order to determine (a) and (b). If we have:

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Examples

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Backwashing Head Loss in Granular Filters During backwashing, the filter bed expands. For this to happen, a force must be applied: l e = expanded depth of the bed l e = expanded depth of the bed l’ = the difference between the level at the trough and the limit of bed expansion l’ = the difference between the level at the trough and the limit of bed expansion h Lb = backwashing head loss h Lb = backwashing head loss w = specific weight of water w = specific weight of water

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weight of suspended solids is: e = expanded porosity of bed p = specific weight of particles weight of column water

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Weights of suspended grains and column water are acting downward against the backwashing force. Thus: Solving for backwashing head loss:

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If v b = backwashing velocity v b = backwashing velocity v = settling velocity of grains v = settling velocity of grains Then, and according to the following empirical equation:

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for stratified bed if xi = fraction of particles in layer I, then: So,

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Assuming expanded mass = unexpanded mass; then: Then, the fraction bed expression is:

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Study Example 7.6

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Cake Filtration This is happening when solids precipitate on the surface of the filter medium. In cake filtration, the general momentum equation can be applied:

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Because solids are tightly packed in cake filtration, the first term is equal to zero due to the fact that the inertia force must disappear upon entrance of flow into the cake and the flow is not accelerated. So:

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because A = S 0 ; and 36K s = 150; then:

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if p is the density of solids, the mass (dm) in the differential thickness of cake (dl) is: = specific cake resistance

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integrating from P 2 to P 1 m c = total mass of solids collected on the filter bed If the resistance of the filter medium to be included, then: R m = resistance of medium

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Determination of V = volume of filtrate collected at any time, t if m c = cV c = mass of cake collected per unit volume of the filtrate, then = average specific cake resistance (since changes with time)

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the equation represents a straight line with slope m of : by determining the slope of the line from the experimental data, can be determined as:

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Where n1 and n2 are the number of elements in the respective group

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Example 7.7

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Design of Cake Filtration Equation Practically, the resistance of the filter medium is negligible. Then: Introducing a new parameter called the filter yield, Lf, defined as: Lf = the amount of cake formed per unit area of the filter per unit time

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If t was called the formation time (t f ) that is when cake starts to form. Also some filters operate on a cycle. Call cycling time as t c. In such cases t f may be expressed as a fraction of t c. So: Then In vacuum filtration, f is equal to the fraction of submergence of the drum.

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For compressible cakes, is not constant and the above equation must be modified. In such cases: s = measure of cake compressibility If s = zero, then the cake is incompressible and = 0. For compressible cakes:

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Determination of Cake Filtration Parameters What design parameters? L f L f - P - P f t c t c c , s , s

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This is a straight line equation with slope = s

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Examples 7.8 and 7.9

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