Standard Methods for Examination of Water & Wastewater_6 pot

On the other hand, pressure and vacuum filters are filtersthat rely on applying some mechanical means to create the pressure differentialnecessary to force the water through the filter..

Conventional Filtration

Filtration is a unit operation of separating solids from fluids Screening is defined

as a unit operation that separates materials into different sizes Filtration also rates materials into different “sizes,” so it is a form of screening, but filtration strictlypertains to the separation of solids or particles and fluids such as in water Themicrostrainer discussed in Chapter 5 is a filter In addition to the microstrainer, otherexamples of this unit operation of filtration used in practice include the filtration ofwater to produce drinking water in municipal and industrial water treatment plants,filtration of secondary treated water to meet more stringent discharge requirements

sepa-in wastewater treatment plants, and dewatersepa-ing of sludges to reduce their volume

filtration Chapter 8 uses membranes as the medium for filtration; thus, it is titledadvanced filtration

Mathematical treatments involving the application of linear momentum to tration are discussed Generally, these treatments center on two types of filters calledgranular and cake-forming filters These filters are explained in this chapter

fil-7.1 TYPES OF FILTERS

Figures 7.1 to 7.8 show examples of the various types of filters used in practice.Filters may be classified as gravity, pressure, or vacuum filters Gravity filters arefilters that rely on the pull of gravity to create a pressure differential to force thewater through the filter On the other hand, pressure and vacuum filters are filtersthat rely on applying some mechanical means to create the pressure differentialnecessary to force the water through the filter

The filtration medium may be made of perforated plates, septum of wovenmaterials, or of granular materials such as sand Thus, according to the mediumused, filters may also be classified as perforated plate, woven septum, or granular filters The filtration medium of the microstrainer mentioned above is of perforatedplate The filter media used in plate-and-frame presses and vacuum filters are ofwoven materials These units are discussed later

Figures 7.1 and 7.2 show examples of gravity filters The media for these filtersare granular In both figures, the influents are introduced at the top, thereby utilizinggravity to pull the water through the filter Figure 7.1a is composed of two granularfilter media anthrafilt and silica sand; thus, it is called a dual-media gravity filter

Figure 7.1a is a triple-media gravity filter, because it is composed of three media:anthrafilt, silica sand, and garnet sand

Generally, two types of granular gravity filters are used: slow-sand and sand filters In the main, these filters are differentiated by their rates of filtration

rapid-7

Slow-sand filters normally operate at a rate of 1.0 to 10 m3/d.m2, while rapid-sand filters normally operate at a rate of 100 to 200 m3/d.m2 A section of a typical gravityfilter is shown in Figure 7.2

The operation of a gravity filter is as follows Referring to Figure 7.2, drainvalves C and E are closed and influent value A and effluent valve B are opened.This allows the influent water to pass through valve A, into the filter and out of thefilter through valve B, after passing though the filter bed

For effective operation of the filter, the voids between filter grains should serve

as tiny sedimentation basins Thus, the water is not just allowed to swiftly passthrough the filter For this to happen, the effluent valve is slightly closed so that thelevel of water in the filter rises to the point indicated, enabling the formation of tinysedimentation basins in the pores of the filter As this level is reached, influent andeffluent flows are balanced It is also this level that causes a pressure differentialpushing the water through the bed The filter operates at this pressure differentialuntil it is clogged and ready to be backwashed (in the case of the rapid-sand filter).Backwashing will be discussed later in this chapter In the case of the slow-sand filter,

FIGURE 7.1 (a) Dual-media filter; (b) triple-media filter.

FIGURE 7.2 A typical gravity filter.

Influent

Influent

Underdrain chamber

Underdrain chamber

Anthrafilt

Silica sand

Anthrafilt

Silica sand

Garnet sand

Water level during filtering

Water level during backwashing

Wash-water trough

Bed expansion limit

Sand Influent

Drain Effluent

A

C

B Drain Underdrain system Wash water

500 mm

650 mm

600 mm freeboard

500 mm

10 m

Wash-water tank

hLb

l2

E Cornroller

it is not backwashed once it is clogged Instead, the layer of dirt that collects on top

of the filter (called smutzdecke) is scraped for cleaning

As shown, the construction of the bed is such that the layers are supported by anunderdrain mechanism This support may simply be a perforated plate or septum.The perforations allow the filtered water to pass through The support may also bemade of blocks equipped with holes The condition of the bed is such that the coarserheavy grains are at the bottom Thus, the size of these holes and the size of theperforations of the septum must not allow the largest grains of the bed to pass through

Figure 7.3 shows a cutaway view of a pressure filter The construction of thisfilter is very similar to that of the gravity filter Take note of the underdrain construc-tion in that the filtered water is passed through perforated pipes into the filtered wateroutlet As opposed to that of the gravity filter above, the filtered water does not fallthrough a bottom and into the underdrain, because it had already been collected bythe perforated pipes The coarse sand and graded gravel rest on the concrete subfill Using a pump or any means of increasing pressure, the raw water is introduced

to the unit through the raw water inlet It passes through the bed and out into theoutlet The unit is operated under pressure, so the filter media must be enclosed in

a shell As the filter becomes clogged, it is cleaned by backwashing

Thickened and digested sludges may be further reduced in volume by ing Various dewatering operations are used including vacuum filtration, centrifuga-tion, pressure filtration, belt filters, and bed drying In all these units, cakes areformed We therefore call these types of filtration cake-forming filtration or simply

dewater-cake filtration Figure 7.4a shows a sectional drawing of a plate-and-frame press In

pressure filtration, which operates in a cycle, the sludge is pumped through the unit,forcing its way into filter plates These plates are wrapped in filter cloths With thefilter cloths wrapped over them, the plates are held in place by filter frames inalternate plates-then-frames arrangement This arrangement creates a cavity in theframe between two adjacent plates

FIGURE 7.3 Cutaway view of a pressure sand filter (Courtesy of Permutit Co.)

Raw water inlet

Filtered water outlet

Weir Drain Sump

Manhole Inlet baffle

Fine sand Coarse sand Graded gravel Concrete subfill Header lateral strainer system with expansible strainer heads Adjustable jack legs

FIGURE 7.4 (a) Sectional drawing of a plate-and-frame press (from T Shriver and Co.); (b) an installation of a plate-and-frame

press (courtesy of Xingyuan Filtration Products, China).

(a)

Side rails Filter cloths

Movable head Fixed headSolids collect in framesPlate Frame

© 2003 by A P Sincero and G A Sincero

Two channels are provided at the bottom and top of the assembly The bottomchannel serves as a conduit for the introduction of the sludge into the press, whilethe top channel serves as the conduit for collecting the filtrate The bottom channelhas connections to the cavity formed between adjacent plates in the frame The topchannel also has connections to small drainage paths provided in each of the plates.These paths are where the filtrate passing through cloth are collected

As the sludge is forced through the unit at the bottom part of the assembly (at

a pressure of 270 to 1,000 kPa), the filtrate passes through the filter cloth into thedrainage paths, leaving the solids on the cloth to accumulate in the cavities of theframes As determined by the cycle, the press is opened to remove the accumulatedand dewatered sludge Figure 7.4b shows an installation of a plate-and-frame press unit

Figures 7.5 to Figure 7.7 pertain to the use of rotary vacuum filters in vacuumfiltration In vacuum filtration, a drum wrapped in filter cloth rotates slowly whilethe lower portion is submerged in a sludge tank (Figure 7.7a) A vacuum applied inthe underside of the drum sucks the sludge onto the filter cloth, separating the filtrateand, thus, dewatering the sludge

A rotary vacuum filter is actually a drum over which the filtration medium iswrapped This medium is made of a woven material such as canvas This medium isalso called a filter cloth The drum is made of an outer shell and an inner shell.These two shells form an annulus The annulus is then divided into segments, whichare normally 30 cm in width and length extending across the entire length of thedrum Figure 7.7a shows that there are twelve segments in this vacuum filter Theouter shell has perforations or slots in it, as shown in the cutaway view of Figure7.6 Thus, each segment has a direct connection to the filter cloth The purpose ofthe segments is to provide the means for sucking the sludge through the cloth while

it is still submerged in the tank

Each of the segments are connected to the rotary valve through individualpipings As shown in Figure 7.7a, segments 1 to 5 are immersed in the sludge, while

FIGURE 7.5 A rotary vacuum filter in operation (Courtesy of Oliver United Filters.)

Water Wash liquor Filtrate

Air Blowback Drum Scraper or“doctor knife”

Cake being removed

Drive

segments 6 to 12 are not Pipes V1 and V2 of the rotary valve are connected to anexternal vacuum pump, as indicated in Figure 7.7b The design of the rotary valve

is such that when segments are submerged in the sludge such as segments 1 to 5,

segments are not submerged such as segments 6 to 12, the design is also such that

sludge into the filter cloth over the segment when it is submerged (V1) and drying

of the sludge when the segment is not submerged (V2)

We can finalize the description of the operation of the vacuum filter this way

As the segments that had been sucking sludge while they were still submerged in

FIGURE 7.6 Cutaway view of a rotary vacuum filter (Courtesy of Swenson Evaporator Co.)

FIGURE 7.7 (a) Cross section of a rotary vacuum filter; (b) flow sheet for continuous vacuum filtration.

Stationary valve plate

Rotating wear plate Port Segments Drum Wash spray nozzles

Filter medium Agitator arm Scraper

Repulper

Tank

To separator and vacuum pump

Discharge head

Valve

Filter drum Roll

Rotating valve Scraper

Air connection Continuous rotary filter Cake

9.1 m

Vacuum receivers Filtrate Filtrate Pump (b)

Air out

Dry vacuum pump

Barometric seal

1 12 11 10 9 8 7 6 5 4

the tank emerge from the surface, their connections are immediately switched from

V1 to V2 The connection V2 completes the removal of removable water from thesludges, whereupon the suction switches to sucking air into the segments promotingthe drying of the sludges The “dry” sludge then goes to the scraper (also called

doctor blade) and the sludge removed for further processing or disposal

Figure 7.7b shows the fate of the filtrate as it is sucked from the filter cloth.Two tanks called vacuum receivers are provided for the two types of filtrates: thefiltrate removed while the segments are still submerged in the tank and the residualfiltrate removed when the segments are already out of the tank Vacuum receiversare provided to trap the filtrate so that the filtrate will not flood the vacuum pump.Also note the barometric seal As shown, this is in parallel connection with thesuction vacuum of the filter The vacuum pressure is normally set up to a value of

66 cm Hg below atmospheric Any vacuum set for the filter will correspondinglyexert an equal vacuum to the barometric seal, on account of the parallel connection.Hence, the length of this seal should be set equivalent to the maximum vacuumexpected to be utilized in the operation of the filter If, for example, the filter is to

be operated at 51 cm,

where 13.6 is the mass density of mercury in gm/cc, and 1 is the density of wateralso in gm/cc Thus, from this result, the length of the barometric seal should be6.94 m if the operational vacuum is 51 cm Hg The design in Figure 7.7b shows thelength as 9.1 m

Figure 7.8a shows another type of filter that operates similar to a rotary vacuumfilter in that it uses a vacuum pressure to suck sludge into the filter medium Thistype of filter is called a leaf filter A leaf filter is a filter that operates by immersing

a component called a leaf into a bath of sludge or slurry and using a vacuum to suckthe sludge onto the leaf An example of a leaf filter is shown in Figure 7.8b Asindicated, it consists of two perforated plates parallel to each other, with a separatorscreen providing the spacing between them A filter is wrapped over the plate assem-bly, just like in the plate-and-frame press Each of the leaves are then attached into

a hub through a clamping ring The hub has a drainage space that connects into thecentral pipe through a small opening As indicated in the cutaway view on the right

of Figure 7.8a, several of these leaves are attached to the central pipe Each of theleaves then has a connection to the central pipe through the small opening from thedrainage space The central pipe collects all the filtrates coming from each of the leaves

In operation, a vacuum pressure is applied to each of the leaves The feed sludge

is then introduced at the feed inlet as indicated in the drawing The sludge creates

a slurry pool inside the unit immersing the leaves Through the action of the vacuum,the sludge is sucked into the filter cloth As the name implies, this is a rotary leaffilter The leaves are actually in the form of a disk The disks are rotated, immersingpart of it in the slurry, just as part of the drum is immersed in the case of the rotaryvacuum filter As the immersed part of the disks emerge from the slurry pool intothe air, the filtrate are continuously sucked by the vacuum resulting in a dry cake

51 13.6( ) = ( )∆h1 H2O; ∆hH2O = 6.94 m

Hollow shaft

Filtrate outlet Central pipe

Filter leaf hub

Filter leaf hub Central pipe

(b)

Drainage space

Drainage space

© 2003 by A P Sincero and G A Sincero

is operated as a batch Thus, filters may also be classified as continuous and tinuous Only the plate-and-frame press is discussed in this chapter as a representation

discon-of the discontinuous type, but others are used, such as the shell-and-leaf filters andthe cartridge filters The first operates in a mode that a leaf assembly is inserted into

a shell while operating and retracted out from the shell when it is time to remove thecake The second looks like a “cartridge” in outward appearance with the filter mediuminside it The medium could be thin circular plates or disks stacked on top of eachother The clearance between disks serves to filter out the solids

7.2 MEDIUM SPECIFICATION FOR

GRANULAR FILTERS

The most important component of a granular filter is the medium This mediummust be of the appropriate size Small grain sizes tend to have higher head losses,while large grain sizes, although producing comparatively smaller head losses, arenot as effective in filtering The actual grain sizes are determined from what expe-rience has found to be most effective The actual medium is never uniform, so thegrain sizes are specified in terms of effective size and uniformity coefficient Effective size is defined as the size of sieve opening that passes the 10% finer of the mediumsample The effective size is said to be the 10th percentile size P10 The uniformity coefficient is defined as the ratio of the size of the sieve opening that passes the 60%finer of the medium sample (P60) to the size of the sieve opening that passes the10% finer of the medium sample In other words, the uniformity coefficient is theratio of the P60 to the P10 For slow-sand filters, the effective size ranges from 0.25

mm to 0.35 mm with uniformity coefficient ranging from 2 to 3 For rapid-sandfilters, the effective size ranges from 0.45 mm and higher with uniformity coefficientranging from 1.5 and lower

Plot of a sieve analysis of a sample of run-of-bank sand is shown in Figure 7.9

by the segmented line labeled “stock sand ….” This sample may or may not meetthe required effective size and uniformity coefficient specifications In order totransform this sand into a usable sand, it must be given some treatment The figureshows the cumulative percentages (represented by the “normal probability scale” onthe ordinate) as a function of the increasing size of the sand (represented by the

“size of separation” on the abscissa)

Let p1 be the percentage of the sample stock sand that is smaller than or equal

to the desired P10 of the final filter sand, and p2 be the percentage of the samplestock sand that is smaller than or equal to the desired P60 of the final filter sand.Since the percentage difference of the P60 and P10 represents half of the final filtersand, p2−p1 must represent half of the stock sand that is transformed into the final

filter sand Letting p3 be the percentage of the stock sand that is transformed into

the final filter sand,

Of this p3, by definition, 10% must be the P10 of the final sand Therefore, if p4

is the percentage of the stock sand that is too fine to be usable,

The plot in the figure shows an increasing percentage as the size of separation

increases, so the sum of p4 and p3 must represent the percentage of the sample stock

sand above which the sand is too coarse to be usable Letting p5 be this percentage,

Now, to convert a run-of-bank stock sand into a usable sand, an experimental

curve such as Figure 7.9 is entered to determine the size of separation corresponding

to p4 and p5 Having determined these sizes, the stock sand is washed in a sand

washer that rejects the unwanted sand The washer is essentially an upflow settling

FIGURE 7.9 Sieve analysis of run-of-bank sand.

% 1.05

1.49 2.10 2.97 4.2 5.9 8.4 11.9 16.8 23.8 33.6

0.2 0.9 4.0 9.9 21.8 39.4 59.8 74.4 93.3 96.8 100.0

tank By varying the upflow velocity of the water in the washer, the sand particlesintroduced into the tank are separated by virtue of the difference of their settlingvelocities The lighter ones are carried into the effluent while the heavier ones remain.The straight line in the figure represents the size distribution in the final filter sand

when the p4 and fractions greater than p5 have been removed

Example 7.1 If the effective size and uniformity coefficient of a proposed filter

is to be 5(10−2) cm and 1.5, respectively, perform a sieve analysis to transform therun-of-bank sand of Figure 7.9 into a usable sand

Solution: From Figure 7.9, for a size of separation of 5(10−2) cm, the percent

p1 is 30 Also, the P60 size is 5(10−2)(1.5) = 7.5(10−2

) cm From the figure, the percent

p2 corresponding to the P60 size of the final sand is 53

Therefore, the sand washer must be operated so that the p4 sizes of 4.5(10−2) cm and

smaller and the p5 sizes of 1.1(10−1) cm and greater are rejected Ans

7.3 LINEAR MOMENTUM EQUATION

APPLIED TO FILTERS

The motion of water through a filter bed is just like the motion of water throughparallel pipes While the motion through the pipes is straightforward, however, themotion in the filter bed is tortuous Figure 7.10 shows a cylinder or pipe of fluidand bed material Inside this pipe is an element composed of fluid and bed material

being isolated with length dl and interstitial area A and subjected to forces as shown.

(We use the term interstitial area here because the bed is actually composed of grains.The fluid is in the interstitial spaces between grains.) The equation of linear momen-tum may be applied on the water flow in the downward direction of this element, thus,

(7.4)

where ∑F z is the net unbalanced force in the downward z direction; p is the static pressure; A is the interstitial cross-sectional area of the cylindrical element of fluid; F g is the weight of the fluid in the element; F sh is the shear force acting onthe fluid along the surface areas of the grains; is the volume of element of space;

hydro-dl is the differential length of the element, l being any distance from some origin;

A s is the surface area of all the grains; k is a factor that converts A s into an area such

that kA s dl = ηd ; η is the porosity of the bed; az is the acceleration of the fluid

element in the downward z direction; ρ is the fluid mass density; v is the component fluid element velocity in the z direction; and t is the time Since the fluid is in the interstitial spaces, d needs to be multiplied by the porosity to get the fluid volume.

The law of inertia states that a body at rest will remain at rest and a body inuniform motion will remain in this uniform motion unless acted upon by an unbal-anced force ∑F z= ρkA s dl(dv /dt) is this unbalanced force that breaks this inertia; thus, it is called the inertia force By the chain rule, (dv /dt) = (dv/dl)(dl/dt) = v(dv/dl).

Thus, ρkA s dl(dv /dt) = ρkA s dlv(dv /dl) = ρkA s vdv Let be some characteristic

aver-age pipe velocity

The velocity through the pipe could vary from the entrance to the exit representsthe average of these varying values; hence, it is a constant Note that all the velocitiesreferred to here are interstitial velocities, the true velocities of the fluid as it travelsthrough the pores

Now, let v∗= v/ Hence, dv∗= dv/ , and ρkA s vdv = ρkAs( v∗) dv∗= ρkAs

To address F sh, let us recall the Hagen–Poiseuille equation from fluid mechanics

FIGURE 7.10 Free-body diagram of a cylinder (pipe) of fluid and bed material.

Cylinder of fluid and bed

This is written as

(7.7)

−∆p s is the pressure drop due to shear forces; µ is the absolute viscosity of the fluid;

l is the length of pipe; and D is the diameter of pipe In a bed of grains, the

cross-sectional area of flow is so small that the boundary layer created as the flow passesaround one grain overlaps with the boundary layer formed in a neighboring grain.Because boundary layer flow is, by nature, laminar, flows through beds of grain islaminar and Equation (7.7) applies

F sh is a shear force acting on the fluid along the surface areas of the grains Theshear stress is this −∆p s ; thus, F sh = −∆p s A s From this relationship and the equationfor ∆p s , we glean that the F sh is directly proportional to µ, , and l and inversely

proportional to the square of D The granular filter is not really a pipe, so we replace

D by the hydraulic radius r H With D expressed in terms of r H, the equation becomes

general In other words, it can be used for any shape, because r H is simply defined

as the area of flow divided by the wetted perimeter The proportionality relation is

denominator Calling the overall proportionality constant as K s , F sh = K s(µ A s /r H);and Equation (7.6) becomes

(7.8)

To address F g, it must be noted that for a given filter installation it is a constant

It is a constant, so its effect when the variables are varied is also a constant This

effect will be subsumed into the values of K i and K s We can therefore safely remove

F g from the equation and write simply

(7.9)This is the general linear momentum equation as applied to any filter

7.4 HEAD LOSSES IN GRAIN FILTERS

Head losses in granular filters may be divided into two categories: head loss in cleanfilters and head loss due to the deposited materials We will now discuss the firstcategory

7.4.1 C LEAN -F ILTER H EAD L OSS

To derive the clean-filter head loss, we continue with Equation (7.9) by expressing

A s , r H, and in terms of their equivalent expressions Let s p be the surface area of

a particle and let N be the number of grains in the bed Thus, A s = Ns p Let S o bethe empty bed or superficial area of the bed With the porosity η and length l, the

V

volume of the bed grains is S o l(1 − η) Letting vp represent the volume of a grain,

N is also calculated as Thus, A s is

volume of the filter is Nv p/(1 − η) Therefore, the volume of flow is η[Nvp/(1 − η)]and the hydraulic radius becomes

(7.13)

The velocity is the interstitial velocity of the fluid through the pores of the

porosity η (The cross-sectional area of flow is much constricted, hence, the velocity

(7.14)Substituting Eqs (7.11), (7.13), and (7.14) in Equation (7.9) and simplifyingproduces

(7.15)

(7.16)

where

A is equal to S oη Substituting in Equation (7.17), the pressure drop across thefilter is finally given as

(7.19)

γ is the specific weight of the fluid In terms of the equivalent height of fluid, weuse the relationship −∆p = γ h L , where h L is the head loss across the filter Thus,

(7.20)

The equations treated above all refer to the diameter of spherical particles; yet

in practice, not all particles are spherical To use the above equations for thesesituations, the sieve diameter must be converted to its equivalent spherical diameter

In Chapter 5, the relationship was given as d = (6/π )1/3

β1/3

d p, where β is the shape

factor and d p is the sieve diameter

Equation (7.20) has been derived for a bed of uniform grain sizes In some types

of filtration plants such as those using rapid sand filters, however, the bed is washed every so often This means that after backwashing, the grain particles areallowed to settle; the grain deposits on the bed, layer by layer, are of different sievediameters The bed is said to be stratified To find the head loss across a stratifiedmedium, Equation (7.20) is applied layer by layer, each layer being converted toone single, average diameter The head losses across each layer are then summed

back-to produce the head loss across the bed as shown next

(7.21)

where the index i refers to the ith layer If x i is the fraction of the d i particles in the

ith layer, then l i equals x i l Assuming the porosity η is the same throughout the bed,the equation becomes

(7.22)

Note, again, that d i is for a spherical particle For nonspherical particles, the sieve

diameter d p must be converted into its equivalent spherical particle by the equationmentioned in a previous paragraph

Example 7.2 A sharp filter sand has the sieve analysis shown below The ity of the unstratified bed is 0.39, and that of the stratified bed is 0.42 The lowesttemperature anticipated of the water to be filtered is 4°C Find the head loss if the

poros-sand is to be used in (a) a slow-poros-sand filter 76 cm deep operated at 9.33 m3/m2⋅ d and

(b) a rapid-sand filter 76 cm deep operated at 117 m3/m2⋅ d

=9.33m3/m2⋅d=1.08 10( 4) m/s

Re

+

f p 1.75 150(1–0.39)

0.0324 -

=3.056 10( 5)5.24 10( 4) - 0.058 m Ans

7.4.2 H EAD L OSSES D UE TO D EPOSITED M ATERIALS

The head loss expressions derived above pertain to head losses of clean filter beds

In actual operations, however, head loss is also a function of the amount of materials

deposited in the pores of the filter Letting q represent the deposited materials per

materials may be modeled as

h d = a(q) b

(7.23)

where a and b are constants Taking logarithms,

This equation shows that plotting lnh d against lnq will produce a straight line By

performing experiments, Tchobanoglous and Eliassen (1970) showed this statement

=1.24 0.77( )0.333

d pi(0.00135) 1000( )

15 10( 4) - 1022.98d pi

=

Letting h Lo represent the clean-bed head loss, the total head loss of the filter bed

h L is then

where h d is the head loss over the several layers of grains in the bed due to thedeposited materials

Now, let us determine the expression for q This expression can be readily derived

from a material balance using the Reynolds transport theorem This theorem isderived in any good book on fluid mechanics and will not be derived here Thederivation is, however, discussed in the chapter titled “Background Chemistry andFluid Mechanics.” It is important that the reader acquire a good grasp of this theorem

as it is very fundamental in understanding the differential form of the materialbalance equation This theorem states that the total derivative of a dependent variable

is equal to the partial derivative of the variable plus its convective derivative In

terms of the deposition of the material q onto the filter bed, the total derivative is

(7.26)

is the volume of the control volume The partial derivative (also called local ative) is

deriv-(7.27)and the convective derivative is

(7.28)

The symbol A means that the integration is to be done around the surface area ofthe volume The vector is a unit vector on the surface and normal to it and thevector is the velocity vector through the surface for the flow into the filter Now,substituting these equations into the statement of the Reynolds theorem, we obtain

(7.29)

In the previous equation, the total derivative is also called Lagrangian derivative,

material derivative, substantive derivative, or comoving derivative The combination

of the partial derivative and the convective derivative is also called the Eulerian

derivative Again, it is very important that this equation be thoroughly understood.

It is to be noted that in the environmental engineering literature, many authorsconfuse the difference between the total derivative and the partial derivative Someauthors use the partial derivative instead of the total derivative and vise versa Asshown by the previous equation, there is a big difference between the total derivativeand the partial derivative If this difference is not carefully observed, any equationwritten that uses one derivative instead of the other is conceptually wrong; this

dt

- q V d V

- q V d V

confusion can be seen very often in the environmental engineering literature Thus,caution in reading the literarture should be exercised.

Over a differential length dl and cross-sectional area A of bed,

(7.30)

results from the fact that at the inlet the unit normal vector is in opposite direction

to that of the velocity vector ) Therefore,

(7.31)

V s is the superficial velocity of flow in the bed

Also, over the same differential length dl and cross-sectional area S o of bed,

(7.32)

d in the second term has been taken out of the parentheses, because it is arbitrary

and therefore independent of t.

Substituting Eqs (7.31) and (7.32) in Equation (7.29),

(7.33)

(Since the solids are conservative substances, the total derivative is equal to zero.)

Dividing out S o dl and rearranging,

(7.34)

The numerical counterpart of Equation (7.34), using n as the index for time and

m as the index for distance, is

(7.35)

for the first time-step Solving for q n+1,

(7.36)The equation for the second time-step is

V

∂q

∂t -S o dl V s S o ∂c

∂l -dl

-=

q n +1,m q n,m ∆tV s

∆l - c( n,m−1–c n,m)

∆c n

∆l

–

Over a length ∆l, the gradients ∆c/∆l varies negligibly from time step to time

step Thus, ∆c n /∆l = ∆c n+1 /∆l = ∆c n+2 /∆l and so on Equation (7.37) may then be

of the drinking water treatment influent For these reasons, in order to use theprevious equations for determining head losses, a pilot plant study should be con-ducted for a given type of influent

Determination of constants a and b As noted before, h d plots a straight line

in logarithmic form with q This means that only two data points are needed in order

to determine the constants a and b Let the data points be (h d1 , q1) and (h d2 , q2) The

two equations for h that can be used to solve for the constants are then

-=

q n+k,m q n,m k ∆tV s ∆c

∆l

–

-=

q n +k,m q n,m tV s ∆c

∆l

–

-=

q tV s ∆c

∆l

–

-=

These equations may be solved simultaneously for a and b producing

(7.44)

(7.45)

If the number of data points is more than two, the data may be grouped to form

two data points Using i as the index for the first group and j as the index for the second,

(7.46)

(7.47)

s i and s j refer to the total number of member elements in the respective groups Theprevious equations are actually calculating the means of the two groups

Example 7.3 In order to determine the values of a and b of Equation (7.23),

experiments were performed on uniform sands and anthracite media yielding the

fol-lowing results below Calculate the a’s and b’s corresponding to the respective diameters.

- 

  q2

q1

ln

Solution: Uniform sand, diameter = 0.5 mm:

Similar procedures are applied to the rest of the data The following is the finaltabulation of the answers:

From the results of this example, the equations below with the values of b averaged

for uniform anthracite, 2.0 mm diameter

The h’s and q’s in the previous equations are in meters and mg/cm3, respectively.Also, the head losses for diameters not included in the equations may be interpolated

or extrapolated from the results obtained from the equations

Example 7.4 The amount of suspended solids removed in a uniform anthracitemedium, 1.8 mm in diameter is 40 mg/cm3 Determine the head loss due to suspendedsolids.

Solution:

Interpolating, let x be the head loss corresponding to the 1.8-mm-diameter media.

Example 7.5 According to a modeling evaluation of the Pine Hill Run sewagedischarge permit, a secondary-treated effluent of 20 mg/L can no longer be allowedinto Pine Hill Run, an estuary tributary to the Cheseapeake Bay To meet the new,more stringent discharge requirement, the town decided to investigate filtering theeffluent A pilot study was conducted using a dual-media filter composed of anthracite

as the upper 30-cm part and sand as the next lower 30-cm part of the filter The results

are shown in the following table, where c o is the concentration of solids at the influent

and c is the concentration of solids in the water in the pores of the filter.

If the respective average sizes of the anthracite and sand layers are 1.6 mm and0.5 mm, what is the length of the filter run to a terminal head loss of 3 m at afiltration rate of 200 L/m2⋅ min? Assume the clean water head loss is 0.793 m Note:Terminal head loss is the loss when the filter is about to be cleaned

1.32 m

1.59–x

1.59–1.32 - 1.6–1.8

1.6–2.0 -; x 1.46 m Ans