Review of Stormwater Quality, Quantity and Treatment Methods Part 2: Stormwater: Quality Modelling
 Author: Aryal Rupak, Kandasamy J., Vigneswaran S., Naidu R., Lee S. H.
 Organization: Aryal Rupak; Kandasamy J.; Vigneswaran S.; Naidu R.; Lee S. H.
 Publish: Environmental Engineering Research Volume 14, Issue3, p143~149, 30 Sep 2009

ABSTRACT
In this paper, review of stormwater quality and quantity in the urban environment is presented. The review is presented in three parts. This second part reviews the mathematical techiques used in the stromwater quality modelling and has been undertaken by examining a number of models that are in current use. The important features of models are discussed.

KEYWORD
Stormwater , Quality and quantity , Mathematical models , Analytical technique

1. Introduction
A review of stormwater quality and quantity in the urban environment is presented. The review is presented in three parts. The first part reviewed the mathematical methods used in stromwater quantity modelling. This second part reviews the mathematical techiques used in stromwater quality modelling and has been undertaken by examining a number of models that are in current use.
2. Urban Runoff Quantity Problems and Models
2.1. Pollutant Buildup and Washoff Model
2.1.1. Regression Model
Tasker and Driver (1988) developed simple regression model on the basis of long term urban runoff data and made it applicable for the unmonitored watershed based on some physical (drainage area, impervious percentage, percentage residential or/and industrial) and climatological data (total rainfall, storm duration, mean annual rainfall).1) The model uses the following generalized regression formula for calculating loads:
where,
L = pollutant load,X _{n} = physical, land use or climatic characteristics, β_{n }? = regression coefficients, andBCF = Bias Correction FactorThe model parameters are estimated by a generalizedleastsquare regression method that accounts for cross correlation and differences in reliability of sample estimates between sites. The regression models account for 20 to 65 percent of the total variation in observed loads.
2.1.2. Simple Empirical Model
Schueler (1987) introduced an easy empirical equation based model known as Simplified Urban Nutrient Output Model (SUNOM) for urban pollutants load prediction based on five years data collected by United States Environmental Protection Agency (USEPA). The method uses the flowweighted mean concentration. ^{2)} The generalized equation is as follows:
where
L_{p} = pollutant load,H_{r} = total rainfall (mm),P_{j} = percent of rainfall contributes to runoff (equals to 1 for individual storm events), Rv = runoff coefficients estimated as 0.05+0.009^{*} (impervious percentage),C = flowweighted pollutant mean concentration (mg/L),A = area (ha)According to Schueler, the simple method does not consider base flow runoff and associated pollutant loads, and is better used at small watersheds. The model is rarely appered in the literature. Recently, the model was applied by Flint (2004) to estimate water quality an ultra urban area in Maryland, US.^{3)}
2.1.3. Sartor and Boyd Model
James Sartor and Gail Boyd first introduced this model in 1972 (Sartor and Boyd, 1972).^{4)} This model provides the knowledge of pollutants transport and their quantification. The model shows the dislodging of the particles during a rainfall event is dependent on the street characteristics, rainfall intensity and the particle sizes where the wash off can be described by the following equation:
where,
P(t) is the amount of the pollutants washed off in time t,P _{o} is the initial loading,k = washoff coefficient, andI = rainfall intensity and t is the time andQ = rainfall volumeMany models such as PSRMQUAL are based on equation 26 (PSRMQUAL Users Manual, 1996) and kinematic wave equations.^{5)} Once the particle is dislodged the shear forces generated by the runoff cause its movement when the runoff is above the critical velocity (velocity at which drage force and resistance forces are equal). Critical velocity is given by
where
V _{cr} is the critical velocity,C _{d} is drag coefficient,C _{s} is static coefficient of friction, g is gravitational acceleration constant,r is the average sediment radius, and SG is sediment specific gravity. United State Environmental Protection Agency (USEPA, 1979) estimated the pollutant load using the following equation.^{6)}where,
M (t) is the pollutant washoff for time periodt (kg),L (t  Δt ) is the pollutant accumulation per unit area at the previous time periodt  Δt (kg ha^{1}), A is the drainage area (ha), and K_{w} is the watershed washoff coefficient (mm^{1}) which is a function of imperviousness of the watershed and the type of simulation, i.e., single event or continuous.Haiping and Yamada (1996) applied Sartor and Boyd equation with refinement by adding some constants such as i) maximum amount of constituents on impervious areas (
k _{1}) and ii) removal due to wind and vehicles as well as biological and chemical decay (k _{2}) besides washoff coefficient.^{7)} The amount of pollutant accumulation on impervious surface is given bywhere,
P _{oR} is the residual amount of constituents on impervious surface after street sweeping or storm runoff in grams. Residual amount of constituents on impervious areas (P_{R} ) after washoff in storm is given by;where,
k_{3} is the washoff coefficient in mm1. Q is the total runoff volume (mm). The equation reflects both effects of accumulation in dry weather (Q=0) and washoff in wet weathers (Q>;0). This can be used as a tool for continuous simulation of urban non point source pollution for a long term prediction.Furumai et al., (2003) applied Sartor and Boyd model with some modification in urban catchment in Japan. They assumed that the runoff from road and roof are different so that the washoff behavior as follows.^{8)} They provided two different runoff coefficients for road and roof.
where,
P (t ) is the amount of the pollutants washed off in timet ,k = washoff coefficient, andI = rainfall intensity at timet ,i = 1 and 2 the roof and road respectively.The above model (Furumai et al., 2003) was applied by Murakami et al., (2004) to predict the washoff behavior of particlebound PAHs from road and roof and stated that the model could explain suspended solids and particlebound PAHs runoff well except during and after heavy rainfall (>;10mm/hr).
Aryal (2003) applied Sartor and Boyd model to predict the pollutants washoff behavior in highway runoff. As this equation states that the quantity of the constituents available for washoff decreases exponentially with runoff volume during the event, the model could not be applied to the events where the two or more pollutants loading pattern were observed due to the change in rainfall intensity (intermittent rainfall) during wet weather period.^{9)} The Sartor and Boyd model found not suitable to the events where two or more SS loading patterns observed. This indicated difficulty in applying the model in those runoff events where the rapid fluctuation of concentration occurred. The following equation establishes the relationship between concentration and the Sartor and Boyd model.
where,
C (t ) = concentration, quantity/volumeF (t ) =A ^{*}R (t ) = Flow rate (L/sec)A = Area (ha)R (t ) = Runoff rate (mm/hr)This equation (11) shows that the primary difficulty of the Sartor and Boyd equation is that it always produces decreasing concentrations as a function of time regardless of the time distribution of runoff. This is counterintuitive, since it is expected that high runoff rates during the middle of the storm might produce higher concentration than those proceeding. Aryal (2003) descretized the storm event and applied the model which he finally summed up to calculate the pollutant load.
Egodawatta et al., (2007) also applied the modified version of Sartor and Boyd model by introducing the capacity factor (CF). They reported that a storm event has the capacity to washoff only a fraction of pollutants available and this fraction varies primarily with rainfall intensity, kinetic energy of rainfall and characteristics of the pollutants. They then modified the Sartor and Boyd equation in order to incorporate the washoff capacity of rainfall by introducing the ‘capacity factor’ CF. According to them, the fraction washoff can be written as
where, C_{F} is the value ranging from 0 to 1 depending on the rainfall intensity. Other factors such as road surface condition, characteristics of the available pollutants and slope of the road may also have influence on C_{F}.
Chen and Adams (2007) also applied the Sartor and Boyd washoff with refinement by introducing the pollutants accumulation rate based on OsuchPajdzinksa and Zawilski (1998) that can accommodate the dry weather period also.^{10,11)} The rate of pollutant accumulation is:
where
M_{b} is amount of pollutant per unit area on catchment surface,h is the fraction of the impervious area of the catchment,m_{d} is a constant rate of pollutant deposition (dust fall),m_{w} is the quantity of street sweeping effectiveness parameter, η is the street sweeping effectiveness parameter,k_{b} is a constant pollutant removal rate, b is the time elapsed since the last rainfall, β_{1} is the conversion of the mass of particulate matter into a parameter of a given type of pollutant and β_{2} describe the conversion of mass of sweeping into a parameter of a given type of pollutant. Integrating the above equation, Mb is:where, M_{o} is residual pollutant mass not washed off by the previous runoff event.
In their study they assumed that the rate of pollutant washoff from the catchment surface is proportional to the amount of pollutant buildup on the catchment surface and is directly related to the volume of runoff.
where _{r} v is the average runoff rate in mm/hr,
k_{w} is the decay or washoff coefficient, in mm^{1}. Performing integrated yields2.2. Advective Diffusion Model (Mass Transport Equation)
It is the one dimensional conservative advectivediffusion equation, that incorporates the advection and diffusion process is to describe the behaviour of a pollutant in stream;
where,
C is the thermal energy or constituent concentration,t the time,x si the distance,u is the advection velocity, Ax the crosssectional area,D_{x} is the diffusion coefficient andS (C, x, t ) are all sources and sinks.This equation includes the advection of pollutants by the flowing water, diffusion of pollutants in the stream, constituent reactions, interactions and sources and sinks. Assuming that
A _{x} andD _{x} are constants and using the flow continuity equation then:Then
which is the form of the advectivediffusion equation used in model like HEC5Q and WQRRS.
Shaw et al., (2006) proposed a new stochastic physical model that is primarily focused on the rain flow transportation.^{12)} The model was mainly based on Hairsine and Rose (1991) which states that the flow does not exceed the threshold for particles entrainment, mass conservation of suspended particles in the water layer:
where,
e is rate of particles enter the shallow flow by raindropinduced ejection,h is the rate of particle settleout of the shallow,M _{s} is the suspended particle mass (g cm^{2}), x is the down slope distance, and v is the fluid velocity (cm min^{1})Particles mass on the surface,
M _{g} (g cm^{2}), at a distinct spatial position is given by:The value e was defined by
e =aPM _{g}where,
a (cm^{1}) is an experimentally determined “detachability” constant that accounts for mass loss per drop and P is the precipitation rate (cm min^{1}). Similarly particle settling rate is given by (Equation Ommited) where α adjusts bulk concentration to account for variations near the surface, v_{set} is the particle settling velocity (cm s^{1}), and D is the depth (cm).They also applied water balance at a pint x by using the equation
where, P is the rain intensity per unit width (mL min^{1} cm^{1}), q is the flow reate per unit width (mL min^{1} cm^{1}) and qo (mL min^{1} cm^{1}) is the constant upslope inflow per unit width.
2.3. Kinematic Wave Equation Model
It is another governing onedimensional equation for pollutant transport on a unit width basis, where solute is injected instantaneously, can be written as
where,
C is solute concentration defined as mass of solute per unit volume of water, w is the mass of solute (M ) per unit area A of the plane (M/A ), w is the mass of pollutant per unit surface area and δ(t )is the instantaneous unit flux of the solute (I/T ). The pollutant discharge Q_{s} is defined asQs = CQ
2.4. Other Stormwater Quality Models
2.4.1. SWMM
SWMM is one of the most successful model produced by United States Environmental Protection Agency (USEPA). This model is not exclusively designed for urban drainage and singleevent or long term (continuous) simulation. The earlier SWMM model used the linear build up formulation. The model provides three options for pollutant build up as follows:
Among the above equation, exponential and MichaelisMenton functions clearly define asymptotes or upper limits. Upper limits for linear or power function buildup may be imposed if desired. The washoff equation as follows: using an exponential washoff equation as follows:
where,
P (t ) is the washoff load rate at timet ,K _{w} is the washoff coefficient,R is the runoff rate (mm/hr),n is the power of runoff rate, andP is the amount of pollutant remaining on the catchment.As mentioned above earlier, primary difficulty in this equation is always producing decreasing concentrations as a function of time regardless of the time distribution of runoff (Aryal et al., 2003). This problem is overcome in SWMM by making washoff at each time step,
P (t ), proportional to runoff rate to a power,K _{w} :where,
P (t ) is constituent load washed off at time, t, quantity/sec,P _{o} (t + Δt ) is quantity of constituent available for washoff at time,t , (e.g., mg),K _{w} is washoff coefficient andR (_{t}) is runoff rate.It may be seen that if equation is divided by runoff rate to obtain concentration, then concentration is now proportional to ?r^{Kw1}. Hence, if the increase in runoff rate is sufficient, concentrations can increase during the middle of a storm even if PSHED is diminished.
From the basic equation (48), the washoff parameters, washoff coefficient and exponent are determined from a finite difference approximation (Nix, 1994) which produces:^{13)}
where,
P _{o} (t + Δt ) is the amount of pollutant washoff during simulation time step ((t + Δt ), P_{o} (t )is the amount of pollutant on land surface during a time step (t ), Kw is the washoff or decay coefficient, Δt is the time step, 0.5[R(t)^{n} + R( t + Δt )^{n}] is average runoff rate over a time step and n is the power function of runoff rate.2.4.2. HydroWorks/Infoworks
HydroWorks/InfoWorks calculates the surface pollutant build up for each subcatchment, during a build up (or dry weather) period, before a rainfall event. The basic hypothesis is one of a timelinear accumulation of pollutant on the ground, which depends on the type of activities present on the catchment/subcatchment or in the vicinity. The buildup equation is based on hypothesis that on a clean surface the rate of pollutants accumulation is linear but as the surface mass increases the accumulation rate decays exponentially. The buildup equation is written as:
where, M is mass of the deposit per surface unit (kg/ha), P is buildup factor (kg/ha/day),
k _{d} decay factor (1/day)The software carries out the following process to determine the build up of pollution for each subcatchment: i. Determine the decay factor ii. Determine the buildup factor iii. Determine the mass of deposit at the end of the buildup period:
where,
M _{o} is the mass of sediment at the end of the buildup period (kg/ha)M _{d} is the initial mass of deposit in kg/ha (from catchment sediment data (.CSD) file).,k _{d} is the decay factor (day^{1}),N _{j} is the duration of the dry weather period (days), and Ps is the buildup factor (kgha^{1}day^{1}).The surface washoff model is based, as the runoff module, on the single linear reservoir model. The model consists of sediment erosion and its washoff. First the amount of sediment eroded from the surface and held in suspension in the storm water is calculated. Then the amount of sediment washed into the drainage system is calculated using a single linear reservoir routing method. The amount of sediment washed into the drainage drainage system is calculated as
where,
M _{e} is mass of sediments dissolved or in suspension per unit active surface (kg/ha),M (_{t}) is mass of surface deposit pollutants (kg/ha),K _{a} is erosion/dissolution coefficient (1/s) and is calculated aswhere,
i (t ) is effective rainfall and C_{1}, C_{2} and C_{3} are coefficients. Sediment washoff is given bywhere,
k is linear reservoir coefficientf (t ) is sediment flow Then the sediment flow per subcatchment (kg/s) is calculated (sediment inflow to each node)where,
C is portion of subcatchment,A is subcatchment area.2.4.3. MOUSE Trap
The MOUSE TRAP model provides several submodules for the simulation of sediment transport and water quality for both urban catchments surfaces and sewer systems. Since pollutants are carried by sediment, the model tries its best to correlate sediment transport process and water quality in sewer systems. Mouse Trap can also model the first flush phenomenon based on temporal and spatial distribution of sediment on the catchment surface and sewer system. Surface Runoff Quality (SRQ) computes the pollutant buildup and transport on catchment surfaces. Two major processes that are involved in SRQ are:
1. Buildup and washoff of sediment particles on the catchment.
2. Surface transport of pollutants attached to the sediment particles.
2.4.4. MUSIC
MUSIC is one of the most popular stormwater model used in Australia developed by Cooperative Research Centre for Catchment Hydrology (CRCCH) Australia (CRCCH 2005).^{14)} The model uses simple first order kinetics for the pollutant wash off from the surface. According to the model the pollutant concentrations in the parcel tend to move by an exponential decay process towards an equilibrium value for that site at that time.
where, C_{out} is the output concentration, C* is the equilibrium value or background concentration, C_{in} is the input concentration,
k is the exponential rate constant andq is the hydraulic loading (flow rate per surface area) of the treatment measure.2.4.5. ASTROM
ASTROM model uses the following pollutant build up equation.
where,
M _{b} is the amount of pollutant per unit area on the catchment surface (kg/m^{2}),k _{o} is constant rate of pollutant deposition (kg/m^{2}.h),k _{b} is constant pollutant removal rate (h^{1}), andb is interevent time.Integrating above equation yields
where,
M _{m} (=k _{o}/k _{b}) represents the maximum amount of pollutant buildup (kg/m^{2}) andM _{o} is residual amount of pollutant after the previous runoff or street sweeping event (kg/m^{2}).The pollutant washoff model is defined as
where,
l is mass of pollutant washed off per unit area per rainfall event (kg/m^{2}),v _{r} is runoff event volume (mm) andk _{w} is pollutant washoff coefficient.The model assumes that rainfall event pollutant washoff load is proportional to, or dependent upon, the accumulated pollutant mass on the catchment surface before the runoff event, and the pollutant washoff load is a direct function of runoff volume.
Besides, there are several literatures appeared to describe the washoff behaviour of pollutants during wet weather period. Here, few are described.
Kim et al., (2005) introduced new washoff model for highway stormwater runoff that incorporates many parameters such as antecedent dry weather periods, rainfall intensity and runoff coefficient.^{15)} The equation can be initially expressed as
where,
C (t ) is pollutant concentration;Q (t ) is runoff flow rate discharged at timet , α is washoff rate coefficient,C (t ) is pollutant concentration at timet ; V_{TRu} is total runoff volume (Equation Ommited) which they solved and rearranged in the formwhere,
M (t ) is the pollutant mass emission rate at timet , V_{nRu} (t ) is the normalized cumulative volume, 0≤V_{nRu} (t )≤1.0where, δ is an initial concentration related to antecedent dry weather period. The parameters α and γ* are related total runoff. The β* is related to rainfall, runoff coefficient, and storm duration.
This model has two different functions. The first is linear, γ *
V _{nRu} (t)+δ , and the second takes the form of a gamma type function, β *V _{nRu} (t ).Exp [?α .V_{nRu} (t )]. To use this model it is necessary to predict the total runoff volume, which must be based upon weather forecast or other information.Kanso et al., (2006) applied simple classical pollutant accumulation followed by the washoff model to describe the water quality.^{16) }He described two accumulation behaviours. The first equation calculates the accumulation of pollutants assumed to follow an asymptotic behaviour that depends on two parameters: an accumulation rate
D _{a} (kg/ha/day) and a dry erosion rateD _{e} (day^{1}).where,
M _{a}(t ) (kg) is the available pollutant’s mass at timet and S_{i} (ha) is the impervious area. The model depends on two parameters: an accumulation coefficientK _{a} and maximum accumulated massM _{1}. It is supposes that the accumulation is proportional to the mass still to be accumulated before reaching the maximumM _{1}, which is equivalent to the (D _{a} /D _{e}).He described the evolution of the available pollutant mass during the stormwater period by applying the following equation.
(mg/l) is the SS concentration produced by erosion,
q (t ) is the discharge (m^{3}/s) at the outlet of the watershed at timet , andI (t ) is the rainfall intensity (mm/hr).3. Conclusion
This paper reviews mathematical methods used in stomwater quality modelling and has been undertaken by examining a number of models that are in current use. The analytical techniques are presented in this paper. The important feature of models is discussed.