Part 2 Change in concentration emerging from any layer

Prior to backwashing the axial concentration distribution within the bed can be determined fi"om the filtration model using equation (55).

Depending on the number of layers used in the model an average concentration for each layer can be calculated. This then forms the initial concentration of the filtrate suspension in each layer, prior to backwashing.

During the expansion phase of the backwash each layer experiences an increase in concentration fi-om dislodged deposit and the incoming concentration fi*om preceding layers. The exception is the bottom most layer which only experiences an increase in concentration fi-om dislodged deposit assuming that the incoming backwash water at

this point is clean. Following the expansion phase each layer will experience further changes in concentration due to the incoming flow of backwash water and from the volume of deposit sheared from media grains into suspension. Again the bottom most layer will only experience changes in concentration from deposit sheared off into suspension during this phase.

At the beginning of the steady state phase, suspended deposit is removed by the backwash water at rate CQ, where C is the concentration of the backwash water and

Q is the backwash volumetric flow rate. For any given layer the concentration is dependent on the in-flowing concentration and the concentration generated within the layer from dislodged deposits.

L a y e r ( n + 1)

In itia l

c o n ce n tra tio n

C o n c e n tr a tio n rem oved from la y e r (C )

L a y e r ( n )

F lu id v o lu m e V f

In ter n a lly g en er a ted c o n c e n tr a tio n (Vcm (t) ) In flo w in g c o n ce n tra tio n ( Cb)

L a y e r ( n - I)

Backwash volumetric flowrate (Q)

Figure 3-5. Backwash model - Layer mass balance principle.

A mass balance equation can therefore be formed for any layer during the steady state phase. Considering any layer, the rate of change of concentration is equal to the concentration generated in the layer, plus the incoming concentration, minus the outgoing concentration.

Mathematically the mass balance can be written as

V^(C(t + At)-C(t)) = { v ^ it + A t ) - V „ ( t ) ) - C Q A t + C . Q A t (81)

where Van(t) is the instantaneous volume of deposit dislodged into suspension for any layer and can be determined using equation (79). Divide throughout by At and writing in diSerential form we obtain

Substituting for Von(t) from Part 1, equation (79), the differential equation (82) becomes

^ = rexp(-K,){D„,„ + BIexp(-K,)Y ~ ^ { C - C , ) (83)

ALA [l-8,)7rBlK,

where y = —

For layer Nr. 1 we can assume that the incoming concentration is zero i.e. the incoming backwash is assumed to be clean water. Setting Cb equal to zero and rearranging equation (83) into standard linear form to get

^ ^ ( - K .t) Y (84)

which we can solve in the usual way to give

exp(-K,t) 2 y D ^ Blaq?(-2K,t) y B l aq)(-3K,t) ^

^ ^ ---

where

(9 = Q_ Vr

and Ai is di constant of integration which can be determined from the initial conditions.

For layer Nr. 1 at the start of the steady state phase the concentration in the layer will be equal to some initial concentration dependent on the volume of deposit dislodged into suspension during layer expansion. Let the initial concentration for layer Nr. 1 be termed Coi and the steady state phase start at time t=0. We can now solve for the constant of integration^/ in equation (85), which gives

A\ = C ^ , - y A

e - K , e - i K , 0-2>K, (86)

The mass balance equation can be solved for subsequent layers using the more generalised form of equation (83) as follows

^ = r„ exp(-K,)(D„^ +B1, exp(-K,t)f - ^ ( C „ (87)

where n is the layer number and Cb^„./j is the incoming concentration from the

previous layer. For example to solve equation (87) for layer Nr.2, substitute equation (85) for CB(n-i)y adjusted to allow for the layer expansion time and rearrange into standard linear form and solve in the normal way. The solutions for layer Nr.2 onward have a fixed pattern which can be written as a general form

-exp(-A:,0 r=n-\ ar 1+ X -^exp(-r/:,de/) r = l r=n-l nr 1+ V Q x \ > { - r 2 K d e l ) B\^ r „ ^ e x p ( - 3 / : , 0 r=«-l nr

1+ V d e l ) _{n terms involving constants}

where del is the time taken to expand a single layer and

p \ ^ e - K , p i = e-2K^ P2> = 0 - ^ K ,

The initial concentration Con in any layer at the start of the steady state phase is dependent on three factors:

(i) the concentration in the layer immediately after the filtration run (C,„);

(ii) the concentration increase caused by layer expansion (Ce„);

(iii) in-flowing concentration fi-om the preceding expanded layer (Q^y;).

Such that Con for any layer is given by:

^on ~ ^sn + (89)

for layer Nr. 1 the third term of equation (89) is assumed to be zero.

Csn can be determined fi-om the filtration model, equation (55). The concentration increase due to layer expansion Cen can be determined by making use of equation (61) and dividing by the expanded layer fluid volume. The in-flowing concentration is determined from the previous layer concentration fimction C(„.j/n.del), again where

del is the time required to expand one layer and is dependent on the magnitude of the backwash velocity.

3.3 Model set-up

The complete model is implemented using a mathematical software package, Mathcad Plus 6.0 professional edition, running on a desktop personal computer. An example of a Mathcad work sheet for the model set-up is shown in appendix II. The

theoretical filter bed is modelled on one of the experimental filter columns described in more detail in chapter 4 and has the following basic parameters:

Diameter = 100 mm

Cross sectional area = 7853.98 mm^ Initial porosity = 0.43

Bed depth = 450 mm