CHAPTER 9

Sand Filtration Models

Introduction

This chapter describes the deep bed granular filtration models available in GPS-X. The continuous and massbalance models are based on empirical removal efficiencies. The simple1d model is a mechanistic model based on continuity and kinetic equations that describe the removal of suspended particles by deep bed granular filters.

Continuous Model

The basis for the continuous model is the direct specification of the filter performance through two parameters: the backwash flow fraction and the backwash solids mass fraction (Figure 9‑1). The backwashoutconnection (bottom left of object) is not used in this model.

The backwash flow fraction (frqbw) is the fraction of the incoming flow to the filter (input connection - top left of object) that is used for backwash. Using this parameter, GPS-X will calculate a continuous backwash flow (Q_b) associated with the backwashout stream (top right of the object):

Equation 9‑1

The continuous output flow (bottom right of the object), is then calculated from the difference between the input flow and the backwash flow:

Equation 9‑2

The backwash solids mass fraction (frxbw) is the fraction of incoming solids that is captured by the filter, ending up in the backwash stream.  Using this parameter, GPS-X calculates the solids concentration in the backwash stream (X_b):

Equation 9‑3

The solids concentration in the output stream (X_o) is calculated from:

Equation 9‑4

Figure 9‑1 – Operational Parameters Form – Continuous Model

Mass Balance Model

The massbalance model is based on empirical removal efficiency for suspended solids, BOD and TKN parameters. A `best' value is set for removal efficiency immediately after backwash. The removal efficiency decreases during the filter cycle time as the filter is fouled until the next backwash. The decrease in the removal efficiency is modelled with an exponential function:

Equation 9‑5

where:

out                   = filter effluent component concentration (mg/L)

in                     = filter influent component concentration (mg/L)

bestefficiency   = best removal efficiency after backwash (%)

foulingcoeff      = fouling coefficient

cycletime          = time elapsed since last backwash (h)

A minimum efficiency value for suspended solids removal is also defined as an operational parameter. This parameter does not affect removal computed by the model but warns the user that, once removal of solids gets below this minimum value, the filter is undergoing high headloss.

Figure 9‑2 – Operational Parameters Form – Massbalance Model

The massbalance model accounts for the mass accumulated in the filter during the filter cycle time. The accumulated mass is removed during backwash. Therefore, stoichiometric parameters for the filter effluent and the backwash may be specified. The operational parameters for the massbalance model are shown in Figure 9‑2.

One-Dimensional Model

The basis for the simple1d model is the combination of the continuity (mass balance) and the kinetic partial differential equations by Horner et al. (1986), which describe the removal of suspended particles by a granular filter:

Equation 9‑6

where:

s          = volume of deposited solids per unit bed volume

l         = filtration coefficient

C          = concentration of suspended particles at depth L and time t

u          = approach velocity (velocity of the fluid above the filter bed)

e_d         = porosity of deposited solids

t           = time

When combined with defining equations for the deposited (attached) solids (X= s×d) and the unattached solids (X = C ×dd) in the filter, the following equation is derived for the simple 1d model:

Equation 9‑7

where:

X          = unattached solids

X_d = attached (deposited) solids

δd        = density

The filter bed is divided into layers and it is assumed that the specific deposit is uniform across each layer.  During the backwash cycle, the average deposit through all layers of the filter is used.  The average deposit after backwash is used as the initial condition for the subsequent filter run.

Model Parameters

This section discusses the various model parameters and inputs that the user encounters when using the simple1d model.

Physical

These menu items are found under the Parameters sub-menu item Physical and contain both real physical dimensions to describe the actual filter modelled, and model dimensions which allow the user to specify the number of layers and an effective particle diameter. The real dimensions of the filter are the bed surface area and the bed depth.

Operational

Operational items, found under the Parameters sub-menu, include the basis for the duration of the filtration run. The duration of the filtration run may be based on a user-specified time, headloss or effluent suspended solids concentration. The filtration run may also be set manually. Other operational parameters include the influent flow and specification of the backwash parameters (duration, rate).

Stoichiometric

The stoichiometric fractions available for the model are the ratios of particulate COD to VSS, VSS to TSS and BOD₅ to BOD_ultimate.

Filter

The filtration constants required for the model are shown in Figure 9‑3. The filtration constants include the clean bed filtration coefficient (l_o), the initial porosity of the filter bed (e_o) and the porosity of deposited solids (e_d), the ultimate bulk specific deposit (s_u), and the density and dry material content of the solids. The packing factor is used in the defining equation for the variation of the filtration coefficient with the bulk specific deposit as given by Ojha and Graham (1992).

Figure 9‑3 - Filter Parameters

Output Variables

The flow and characteristics of the filter flow and backwash can be displayed on output graphs. There are various filter variables available in the sub-menus under Output Variables. The output variables include dilution, headloss and rates, the unattached and attached solids in the filter and filter conditions (i.e. bulk deposit, bed porosity).

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