Surface Water Treatment by Roughing Filters  A Design, Construction and Operation Manual (SANDEC  SKAT, 1996, 180 p.) 
Annexes 
Filtration is more an art than a science. This saying also applies to roughing filtration. Numerous researchers have tried to describe the filtration mechanisms in mathematical models applying either the phenomenological or the trajectory approach. The first one uses simple but important variables such as filtration rate, filter size, depth and porosity to describe filter efficiency. The second approach focuses more on transport mechanisms of the individual particle and its behaviour in the single collector. The phenomenological and trajectory approach will be used in this short summary on filtration to provide some more theoretical information on the mechanisms of roughing filtration.
Transport Mechanisms
The trajectory approach, describing the route of a clay particle through a roughing filter, has been vividly depict in Chapter 9.2. Additional analytical considerations regarding this mechanism are given hereafter.
Screening, as shown in Fig 4/1, is not relevant in roughing filters since the pore sizes are considerably larger than the particles generally encountered in suspensions. The ratio between a clay particle of 4 mm in diameter d_{p} and different pore sizes d_{o} is illustrated in the following table.
gravel size d_{g} 
[mm] 
16 
8 
4 
pore size d_{o} 
[mm] 
2.5 
1.25 
0.63 
ratio d_{o}/d_{p} 
[] 
625 
313 
156 
Sedimentation is the next possible process for solid matter separation. Under the conditions described in Fig. 4/2 and shown in the following table, the ratio between the settling distance d_{s} travelled by the clay particle during its flow through the pore and required total settling height h_{s} is very important.
settling velocity 
vs 
0.01 mm/s 
for a 4 mm particle 
pore length 
I_{p} 
4 mm 
for 16 mm gravel 
filtration rate 
v_{F} 
0.5 m/h  
flow velocity 
v_{eff} 
0.4 mm/s 
for 35 % porosity 
flow time 
t_{f} 
10 s 
(I_{p} / v_{eff}) 
settling distance 
d_{s} 
0.1 mm 
(v_{s }´_{ }t_{f}) 
settling height 
h_{s} 
1.25 mm 
(h_{s} = 0 5 do) 
ratio 
h_{s}/d_{s} 
12.5 

Fig. 4/1
Fig. 4/2
Fig. 4/3
Fig. 4/4
Interception decreases porosity and settling height hs and enhances solid matter removal by sedimentation. However, as illustrated in Fig. 4/3, solids accumulation in roughing filters does not significantly improve solid matter separation. This is also presented in the following table.
initial porosity 
p_{o} 
35 %  
filter load 
s 
5 9/1 
(accumulated solid per filter volume) 
taken up volume 
m_{a} 
2.5 % 
for a 0.2 g/cm³ density 
actual porosity 
p_{a} 
32.5 % 
(PO  ma) 
Hydrodynamic forces are capable of carrying the particles in still water zones as illustrated in Fig. 4/4. In such prevailing conditions, the clay particle can settle on the gravel surface as calculated in the table below.
settling velocity 
v_{s} 
0.01 mm/s 
for a 4 mm particle 
settling distance 
d_{s} 2 mm 
 
settling time 
t_{s} 
200 s 
(I_{s}/v_{s}) 
The "1/32/3 Filter Theory''
The following very simplified model elucidates the filter removal kinetics and is based on the considerations described on page IX4 of Chapter 9.
gravel layer 
separated particles 
remaining particles 
removal [%] 

300 mg/l 
(removal in % per layer)  
1 
100 
200 
33 
2 
67 
133  
3 
44 
89  
4 
30 
59  
5 
20 
39  
 

90 
6 
13 
26 
(16.5% layer) 
7 
9 
17  
8 
6 
11  
9 
4 
7  
10 
2 
5  
11 
1.5 
3.5  
  
99 
12 
1.2 
2.3 
(1.5% layer) 

2.3 mg/l 

This simple arithmetic exercise clearly proves that solid matter separation by filtration can be described by an exponential equation as subsequently exemplified by equation (1). However, filter efficiency does not only depend on particle concentration but also on size and settling characteristics. Furthermore, filter variables such as filtration rate and size of filter medium strongly influence filter performance. Finally, the accumulated volume of separated solids per unit of filter bed volume, known as filter load, also determines the actual filter efficiency.
Extensive parameter tests were conducted to determine the influence of different design parameters on the performance of horizontalflow roughing filters. The tests were conducted in the laboratory with filter cells of 10  30 cm and 20  40 cm length for differently sized filter material and different filtration rates varying between 0.5 and 2 m/h. A kaolin stock suspension was used to simulate a suspended solids concentration of about 200 mg/l. Particle size counts were performed with a Coulter Counter TA II. These laboratory tests are described in [10] and the data obtained were evaluated by a multiple linear regression analysis to develop a filtration model for horizontalflow roughing filtration of which the following is an excerpt.
According to the established filter theory, the filter efficiency can be expressed by the filter coefficient _{} [cm^{1}] (described by Iwasaki's equation) or by some other collector efficiency factors
_{} (1)
with c as solids concentration and x filter depth. The filter coefficient _{} is a function of the interstitial flow pattern (depending on filtration rate and pore size distribution), of the grain surface area (depending on size and shape of the filter medium) and of Stoke's law parameters of the water and the suspended particles (particle size, density). Straining mechanisms are neglected and surface chemical conditions are assumed to be constant. The volume of retained solids increases with progressive filtration time and hence, augments the filter surface area available for deposition but decreases at the same time the filter porosity. The degree of filter clogging can be expressed by the volume filter load _{} which is the volume of deposited material per unit filter bed volume. _{} varies with position x in the filter as well as with filtration time t. _{} is therefore not a first order removal rate constant, but varies with time and position in the filter. A more appropriate model parameter is considered to be the particle specific filter coefficient _{} which for a short time interval is constant throughout a homogeneous filter layer. The removal of a particle fraction of the size dpi can thus be formulated by
_{} (2)
with _{} as concentration of particles of size d_{pi}. Assuming the total filter length as a multistore reactor consisting of a series of small filter cells, the performance of a HRF can be calculated on the base of the filter cell test results. For each of the cell tests _{} may be approximated by
_{} (3)
resulting in different relations of _{} as function of filtration velocity, grain size, particle size and the time dependent filter load c, according to experimental conditions.
Knowing _{} as function of the different design variables and of the filter load _{}, it is possible to calculate at a certain time t in steps of layer thickness _{} (close to the length of the experimental filter cells) the effluent of each particle fraction by
_{} (4)
and the total suspended solids concentration after an element _{}
_{} (5)
The volume filter load _{} may be calculated in short time intervals _{} from the particle volume balance equation for a small filter element _{}
_{} (6)
with _{} as filter velocity, _{} as removed particle volume of size d_{pi} and _{} as k^{th}, time interval from the beginning.
All the dependencies of _{} from the various filtration variables could be derived from the small filter cell parameter tests by empirical analysis of the test data.
The influence of the particle capture volume _{} on the filter coefficient was formulated according to Ives^{5} and transformed to the particle specific filter coefficient _{} Starting with an initial filter coefficient _{}, the filter coefficient _{} becomes
_{} (7)
where _{} considers the increased surface area available for deposition (k = constant) and the third term accounts for the porosity decrease and the resulting increase of the interstitial velocity. _{} is the initial porosity and _{} is a constant describing the influence of the gradually constricting pores. Exhaustion of the filter is attained when the suspended particles of a certain size are no longer retained (_{} = 0) and the quantity of deposits in the pores attains its ultimate value _{}. It can be noticed that _{} is the volume deposit of all particles together, but _{} varies with particle size d_{pi}.
From the experimental results in Fig. 4/5, it may be concluded that _{} does not substantially increase with progressive filter load _{}. Apparently, the effect of surface area increase for additional deposition plays a minor role in HRF and straining effects may be completely neglected. A conservative assumption is made by setting
k = 0 (8)
Thus, equation (7) is simplified considerably. At _{} = 0, _{} may be expressed as function of _{} and _{} to
_{} (9)
The resulting equation for _{} therefore becomes
_{} (10)
The initial filter coefficient _{} and the ultimate filter load _{} are determined on the basis of the parameter test results summarised in Fig. 4/5. The general considerations of Boller^{4} for the determination of the filter constants were adapted and applied accordingly.
The value of the initial filter coefficient _{} depends on the process variables _{} (filtration rate), d_{g} (filter grain size) and varies with particle size d_{pi}. A matrix comprising the measured initial filter coefficients for different values of the process variables and sizes of suspended solids was transformed by a multiple linear regression analysis to the following general equation
_{} (11)
The values for
_{} = 0.02
[cm^{1}]
_{} = 0.88
_{} = 0.85
_{} = 1.0
were determined from 36 data points with a correlation coefficient of 0.96.
The ultimate filter load _{} is similar to the initial filter coefficient a function of the different process variables. The volumetric filter load _{}[ml/l] was determined by the calculated and measured mass filter load _{} [g/l] applying a specific wet sludge density of 1.15 g/ml. The transformation of a similar matrix by multiple linear regression analysis resulted in the equation
_{} (12)
with the following values
b_{o} = 10 [ml/l]
_{} =
0.80
_{} = 0.18
_{} = 0.35
The 20 data points used showed a correlation coefficient of 0.97.
With the established equations for _{} and _{}, it is possible to calculate in time steps _{} and filter layer elements _{} the resulting particle size distribution in function of time and space. Changes in grain size, filter velocity and particle size distribution may be adjusted by adapting _{}. Hence, the filter performance of a full scale HRF can be simulated by the arrangement of a number of short filter layer elements each specified with its own _{}. The increment of filter load within each element is calculated over a time step _{} and its influence on _{} is considered in the next time interval.
The above studies have only focused on the physical removal mechanisms. Roughing filters may, however, also develop biological activities which enhance particle removal. Such investigations were carried out with suspensions containing clay (kaolin), algae (Scenedesmus) or a combination as described in [11]. The laboratory tests were also evaluated by multilinear regression models. The following equations were obtained for steady state conditions.
for kaolin: 
C_{e}/C_{o} = 0.188 + 0.0231 media +0.136 flow  0.101 depth 
for Scenedesmus algae: 
C_{e}/C_{o} =  0.170 + 0.253 flow + 0.142 media  0.021 depth  0.0128 media^{2} 
for kaolin + algae: 
C_{e}/C_{o} = 0.0280 + 0.0902 flow + 0.0181 media  0.0558 depth 

where 

 C_{e} is the effluent concentration in [mg/l] 
 C_{o} is the inlet concentration in [mg/l] 
 "media" is the gravel size in [mm] 
 "flow" is the filtration rate in [m/h] 
 "depth" is the filter length in [cm] 
Fig. 4/5 Filter Coefficient in
Relation to Filtration Rate, Grain Size and Filter Load
This research has also revealed that filter efficiency is dependent on design variables such as filtration rate, gravel size and filter length. However, as outlined in other investigations [36, 47], flow direction is of minor importance for filter performance. These laboratory tests have shown that kaolin removal is enhanced by the addition of algae which destabilise the clay into aggregates that are more efficiently removed by roughing filtration. However, hydraulic filter cleaning is more difficult when the clay is coated with organic matter. Hence, the presence of biomass in a roughing filter probably does enhance solid matter separation but may also hinder hydraulic filter cleaning.
The chemical properties of the suspension; i.e., the suspension stability is, however, not taken into consideration in these filter models. Filter models are not universally applicable to all types of raw water as filter efficiency is strongly influenced by the raw water quality. Such semiempirical models may therefore be used to investigate the overall influence of specific design parameters or to optimise treatment plant design on the basis of a comprehensive pilot plant field test programme.