AQUATIC WATER QUALITY MODELLING August 8, 2007 Research Professor Tom Frisk Pirkanmaa Regional Environment Centre P.O.Box 297, FIN-33101 Tampere, Finland E-mail tom.frisk@ymparisto.fi Phone +358 500 739 991

3. RIVER MODELS   3.1 Streeter-Phelps model Streeter and Phelsps presented in 1925 a model for predicting dissolved oxygen concentration in rivers The model is based on the assumption that dissolved oxygen concentration is dependent on two independent processes: decomposition of organic matter and reaeration. Both processes are described as first order reactions. The model is based on plug flow hydraulics.

The basic equations of the model can be written as follows: dL ---- = -K1L (3.1) dτ dC ---- = -K1L + K2(Cs - C) (3.2) dτ where: L = concentration of organic matter measured as ultimate BOD (M L-3), τ = travel time (T), K1 = decomposition rate coefficient of organic matter (T-1), C = dissolved oxygen concentration (M L-3), K2 = reaeration coefficient (T-1), Cs = saturation concentration of dissolved oxygen (M L-3)

The solution of Eq. (3.1) wit the initial condition L=L0 when τ=0 is the following: -K1τ L = L0 e (3.3) When Eq. (3.3) is taken into account the following solution is gained to Eq. (3.2) with the initial condition C=C0 when τ=0: K1 L0 - K1τ - K2τ - K2τ C = Cs - ---------- (e - e ) - (Cs -C0)e (3.4) K2 - K1 Reaction rate coefficients K1, K2 and saturation concentration of dissolved oxygen Cs are dependent on temperature.

If the values of the reaction rate coefficients K1 and K2 are equal Eq. (3.4) cannot be used because of division by zero. The following equation must be used instead: - K1τ - K1τ C = Cs - K1τ L0 e - (Cs -C0)e (3.5) In the model, organic matter is expressed as ultimate BOD. Ultimate BOD can be estimated on the basis of shorter-term BOD measurements (in Nordic countries BOD7 , in other countries most often BOD5) applying first order kinetics. BOD7 is oxygen consumption during seven days due to decomposition of organic matter.

Fig. 3.1 (Kylä- Harakka 1979) C = Oxygen sag curve A = contribution of decomposition B = contribution of reaeration

Ultimate BOD (L) is a measure of organic matter indicating the amount of oxygen (mg l-1) which is needed in total decomposition. The concentration of organic matter in the beginning of the BOD test is = L0 and after seven days = L(7). If we assume first order kinetics we can write: -K1* ∙7d L(7) = L0 e (3.6) where: K1* = decomposition rate coefficient in lab (T-1) and

-K1* ∙7d BOD7 = L0 - L(7) = L0(1 – e ) (3.7) Ultimate BOD can now be calculated on the basis of BOD7: BOD7 L0 = ------------------ (3.8) -K1* ∙7d 1 - e Eq. (3.8) can also be written as follows: L0 = f · BOD7 (3.9) where: f = the stoichiometric coefficient between ultimate BOD and BOD7.

Fig. 3.2. BOD curve. L0 = ultimate BOD, L(7) = BOD which is left after seven days (Kylä-Harakka 1979)

In the following table the dependence of f on decomposition rate coefficient (in lab) is presented: K1* (d-1) f 0.05 3.39 0.1 1.9 0.15 1.4 0.2 1.33 0.3 1.14 0.4 1.06

Derivation of Eqs. (3.6) - (3.8) is based on first order kinetics with no lag phase in the beginning of decomposition. In reality, in the BOD test only readily decomposing organic compounds affect. Thus ultimate BOD can be defined as the total concentration of readily decomposing organic matter. Decomposition rate coefficient in the lab is not the same as in the river. This is because there are different temperatures and, in general, the conditions in the test bottles are different than in natural waters.

The saturation concentration of oxygen is dependent on temperature. E.g. the following third degree polynom can be used Cs = 14.6221 - 0.407699 T + 0.00812934 T2 - 0.0000794448 T3 (3.10) where: T = temperature (°C)

The dependence of reaction rates is according to Streeter and Phelps (1925) the following: T - Ts K(T) = K(Ts) Θ (3.11) where: K(T) = reaction rate coefficient at temperature T Ts = standard temperature (can be selected, most often 20°C) Θ = an empirical constant, characteristic of each reaction rate coefficient For decomposition rate coefficient K1 the value of Θ = 1.047 ja for the reaeration coefficient K2 the value of Θ = 1,024 is often used.

The model is based on PFR hydraulics and travel time can be calculated as the ratio of distance (x) and velocity of flow (u): τ = x/u (3.12) Example 3.1. The discharge of the river is = 30 m3 s-1 and the loading of organic matter as BOD7 = 15 t d-1. The average cross- sectional area is = 2000 m2. Calculate dissolved oxygen concentration at the distance of 10 km from the waste- water pipe at temperature 14°C using the following values for the coefficients at standard temperature (20°C): K1 = 0.15 d-1 and K2 = 0.08 d-1.

Dissolved oxygen concentration above the wastewater pipe is = 9.2 mg l-1. It is assumed that BOD7 in the river water above the wastewater pipe is = 0. The ratio Ultimate BOD/BOD7 is assumed to be = 1.5. The mixing concentration of ultimate BOD is first calculated: L0 = 1.5 *15 t d-1*106 g t-1/(30 m3 s-1* 86400 s d-1)  = 8.68 mg l-1 According to Eq. (3.10) at temperature 14°C   Cs = 10.29 mg l-1

Reaction rate coefficients at temperature 14°C : 14-20 K1(14°C ) = 0.15 d-1 *1.047 = 0.1139 d-1 K2(14°C ) = 0.08 d-1 *1.024 = 0.0694 d-1 Travel time to the distance of 10 km is τ = 10· 103 m/(30 m3 s-1· 86400 s d-1/2000 m2) = 7.716 d

Introduction to Eq. (3.4) gives C= 10.29 mg l-1 - (0.1139 d-1 * 8.68 mg l-1/( 0.0694 d-1 – 0.1139d-1 ) * -7.716 d*0.1139 d-1 -7.716 d * 0.0694 d-1 (e - e ) - -7.716 d * 0.0694 d-1 (10.29 mg l-1 – 9.2 mg l-1)e = 10.29 + (-22.217)*(0.4152 – 0.5854) – 1.09 * 0.5854 = 10.29 – 3.781 – 0.6381 = 5.871 mg l-1 = 5.9 mg l-1

3.2 Additions to the River Model There are many additions made to the Streeter-Phelps model, and there are several modifications of the river models one of the most famous of which is called QUAL II, existing as many different versions. The most important additions made to the model are the following: Consideration of additional discharge and input (Eqs. 1.24 and 2.17) Consideration of benthic oxygen demand Consideration of sedimentation of BOD

- Consideration of nitrification - Consideration of photosynthesis and respiration - New state variables - Consideration of dispersion (Eqs. 1.27 and 1.28) - Inclusion of non-steady hydraulics. Description of chemical and biological processes will be presented in Chapters 4 and 5.