1. 1 INTRODUCTION - Q2KFortran2_11
QUAL2K:
A Modeling Framework for Simulating River and Stream Water Quality
(Version 2.11)
A river and stream water quality model that is intended to represent
a modernized version of the QUAL2E model
Chapra, Greg Pelletier and Hua Tao
August, 11th, 2015
Dongil Seo

2. “If the physics isn’t correct, you don’t have a prayer of getting
QUAL2K PHYSICS
“If the physics isn’t correct, you
don’t have a prayer of getting
the water quality right!”
Segmentation
Flow Balance
Depth, Velocity and Travel Time
Dispersion
Temperature Modeling
- Slide by S. Chapra

3. QUAL2K Segmentation Scheme
The model represents a river as a series of reaches
These represent stretches of river that have constant hydraulic characteristics
(e.g., slope, bottom width, etc.).
The reaches are numbered
in ascending order starting
from the headwater of the river’s main stem.

4. REACHES AND ELEMENTS n = 4 Elements Reach
The model’s fundamental computational unit which consists of an equal length subdivision of a reach. done by specifying the number of elements that are desired
A length of river with constant hydraulic characteristics
If desired, any model reach can be
further subdivided into a series of n equal-length elements

5. SEGMENTATION Headwater boundary 1 2 3 4 5 6 8 7 Point source
Point abstraction
Point source
Non -
point
abstraction
Non -
point
source
Point source
Downstream boundary
- Slide by S. Chapra

6. QUAL2K segmentation scheme and headwater and tributary numbering schemes
For systems with tributaries the reaches are numbered in ascending order starting at reach 1 at the headwater of the main stem.

7. 3.1 Flow Balance
A steady-state flow balance is implemented for each model reach
Qi = Qi-1 + Qin,i – Qab,I
Total Inflow
Total Outflow

8. STEADY-STATE FLOW BALANCEG QA QC QF QE QD QB QG A
Upstream
Lake
- Slide by S. Chapra

9. WHAT HAPPENS WHEN YOU ADD FLOW TO A SLOPING CHANNEL?
The water velocity and depth increase
Given Qoutflow, compute depth (H) and velocity (U):
Weir
Rating Curves
Continuity and Manning equation
- Slide by S. Chapra

10. 3.2 Hydraulic Characteristics
Once the outflow for each reach is computed,
the depth and velocity are calculated
1) If a weir height is entered,
the weir option is implemented.
2) If the weir height and width are zero
and rating curve coefficients
are entered (a and a),
the rating curve option is implemented.
3) If neither of the previous conditions are met,
Q2K uses the Manning equation
WEIR
RATING CURVES
MANNING EQUATION

11. WEIRS Height of water in reach upstream of weir
(a) Side
(b) Cross-section Bw Hh Hh Hd Hi Hi Hw Hw
Hi+1
elev2i
elev2i
elev1i+1
elev1i+1
Height of water in reach upstream of weir
Drop of water between reaches
- Slide by S. Chapra

12. where Qi is the outflow from the segment upstream
For a sharp-crested weir where Hh/Hw < 0.4, flow is related to head by (Finnemore and Franzini 2002)
where Qi is the outflow from the segment upstream
of the weir in m3/s, and Bi and Hh are in m.
Equation (4) can be solved for
(4) B H w i h

13. DROP OF WATER BETWEEN REACHES SEPARATED BY A WEIR (For reaeration)
This result can then be used to compute the depth of reach i, and the drop over the weir
and Hd Hi
elev2i
elev1i+1
Hi+1
- Slide by S. Chapra

14. Rating Curves
Power equations can be used to relate mean velocity and depth to flow,
where a, b, α and β are empirical coefficients
that are determined from
velocity-discharge and
stage-discharge rating curves, respectively.

15. Rating Curves U = 0.0279 Q H = 1.6487 Q 0.5652 0.1622 H (m) U (m/s) Q10 1 U
(m/s) Q (m 3
/s) Q (m 3
/s) 1
0.1
100
500
100
500
- Slide by S. Chapra

16. Ratng Curve ComputationQi
U = aQib
H = aQib U Qi Ac = H Ac B =
- Slide by S. Chapra

17. CONSTANT DEPTH AND VELOCITY
U = aQb
H = aQb
a = U; b = 0
a = H; b = 0 1
U = UQ0
H = HQ0
- Slide by S. Chapra

18. The values of velocity and depth can then be employed to determine the cross-sectional area and width by
The exponents b and β typically take on values listed in Table 1.
Note that the sum of b and β must be less than or equal to 1.
If their sum equals 1, the channel is rectangular.

19. Table 1 Typical values for the exponents of rating curves used to determine velocity and depth from flow (Barnwell et al. 1989).

20. 3.2.3 Manning Equation
Each reach is idealized as a trapezoidal channel
Given Q, uses Manning Equation to compute H, and U
where Q = flow [m3/s],
S0 = bottom slope [m/m],
n = the Manning roughness coefficient,
Ac =the cross-sectional area [m2], and
P = the wetted perimeter [m].

21. The cross-sectional area of a trapezoidal channel is computed as
where B0 = bottom width [m],
ss1 and ss2 = the two side slopes [m/m], and H = reach depth [m].
The wetted perimeter is computed as
Notice that time is measured in seconds in this and other formulas used to characterize hydraulics within Q2K.
However, once the hydraulic characteristics are determined they are converted to day units to be compatible with other computations.

22. Hk+1 can be iteratively solved as
where k = 0, 1, 2, …, n, where n = the number of iterations.
The method is terminated when the estimated error falls below a specified value of 0.001%.

23. [ ] [ ] Manning Approach ) ( . + ø ö ç è æ = H B S Qn ) ( 5 . + = H B2 1 10 / 3 5 ) ( . + ø ö ç è æ = s H B S Qn [ ] 2 1 ) ( 5 . + = s c H B A c A Q U =
- Slide by S. Chapra

24. The estimated error is calculated as
Velocity can be determined from the continuity equation with the cross-sectional area
The average reach width, B [m], can be computed as

25. Manning roughness coefficient for various open channel surfaces
(from Chow et al. 1988)
Manning’s n typically varies with flow and depth
As the depth decreases at low flow,
The relative roughness increases.

26. Typical published values of Manning’s n,
range from about for smooth channels
to about 0.15 for rough natural channels,
when the flow is at the bankfull capacity
Critical conditions of depth for evaluating water quality are
generally much less than bankfull depth, and
the relative roughness may be much higher.

27. RIVER HYDROGEOMETRY (STEADY)Q
Manning
Rating Curves
U, H
Weir
Ac = Q/U
B = Ac /H
As = Dx B
V = B H Dx
tw = Dx / U = V/Q
- Slide by S. Chapra

28. 3.2.4 Waterfalls The drop of water over a weir
to compute the enhanced reaeration
that occurs in such cases.
In addition to weirs,
such drops can also occur at waterfalls
QUAL2K computes such drops for cases where
the elevation above sea level drops abruptly
at the boundary between two reaches.

29. For both the rating curve and the Manning equation options
Hd= elev2i + Hi - elev1i+1 – Hi+1
The drop is only calculated for
elev2i > elev1i+1.
Hi+1 Hi
elev2i Hd
elev1i+1

30. 3.3 Travel Time The residence time of each reach is computed as
where tk = the residence time of the kth reach [d]
Vk = the volume of the kth reach [m3] = Ack∆xk
and ∆xk = the length of the kth reach [m]
The travel time from the headwater to the downstream end of reach i,
where tt,i = the travel time [d]

31. 3.4 Longitudinal Dispersion
1) the user can simply enter estimated values.
2) if the user does not enter values,
internally compute dispersion based on the channel’s hydraulics (Fischer et al., 1979)
Ep,i = the longitudinal dispersion between reaches i and i +1 [m2/s],
Ui = velocity [m/s], Bi = width [m],
Hi = mean depth [m], and Ui* = shear velocity [m/s],
g = acceleration due to gravity [= 9.81 m/s2] and
S = channel slope [dimensionless].

32. DISPERSION
E = 0
E = moderate
E = high
- Slide by S. Chapra

33. IF YOU DO NOT ENTER A SLOPE
If the user has entered a roughness coefficient, Manning’s equation is used to estimate the slope
If the user has neither entered a slope nor a roughness coefficient, 2 3 / ) ( ú û ù ê ë é = P A nU S c
E = 0
- Slide by S. Chapra

34. The bulk dispersion coefficient
The bulk dispersion coefficient is computed as
Note that a zero dispersion condition is applied at the downstream boundary.
The net heat load from sources is computed as
Tps,i,j = the jth point source temperature for reach i [oC]
Tnps,i,j = the jth non-point source temperature for reach i [oC].

35. 4 TEMPERATURE MODEL A heat balance can be written for reach i as
Ti = temperature in reach i [oC], t = time [d],
E’i = the bulk dispersion coefficient
between reaches i and i + 1 [m3/d],
Wh,i = the net heat load from point and
non-point sources into reach I [cal/d],
ρw = the density of water [g/cm3],
Cpw = the specific heat of water [cal/(g oC)],
Jh,i = the air-water heat flux [cal/(cm2 d)], and
Js,i = the sediment-water heat flux [cal/(cm2 d)].

36. Q2K HEAT BALANCE i atmospheric transfer heat load heat abstraction
inflow
outflow i
dispersion
dispersion
sediment-water
transfer
sediment
- Slide by S. Chapra

37. TEMPERATURE MODELING ANALOGY BETWEEN MASS AND HEAT
extensive
intensive
relationship
MASS BALANCE
mass
concentration
c = M V
sweetness
sugar
COFFEE
lumps
cup
HEAT BALANCE
heat
temperature r
T = V H Cp
- Slide by S. Chapra

38. The Atmospheric Window

39. Shortwave radiation (visible light)
Shortwave radiation is basically radiant energy (light).
contains a lot of energy)
ultraviolet (UV) rays,
visible light
or Infra Red
Some of shortwave radiation
which enters the Earth
is reflected to space
due to reflection,
with the others being absorbed
by the surface or other things

40. longwave radiation (infrared light)
Longwave radiation is energy leaving the Earth as infrared energy.
Earth emits a large amount of longwave energy, only around 5% is directly lost to space,
with the majority being absorbed by greenhouse gases and being converted to heat energy which is then emitted as atmospheric longwave radiation, resulting in the greenhouse effect.
The atmospheric longwave radiation emission is split into 2 categories with 40% being lost to space, and the other 60% being transmitted to the Earth’s surface where it is absorbed and becomes heat energy.

41. For cloudy conditions the atmospheric emissivity may increase
as a result of increased water vapor content.
High cirrus clouds (권운, 卷雲)
may have a negligible effect
Lower stratus clouds (층운, 層雲) and
Cumulus clouds (적운, 積雲)
may have a significant effect.

42. 4.1 Surface Heat Flux
Surface heat exchange is modeled as a combination of five processes:
I(0) = net solar shortwave radiation at the water surface,
Jan = net atmospheric longwave radiation,
Jbr = longwave back radiation from the water,
Jc = conduction, and Je = evaporation [cal/cm2/d]

43. DAILY SOLAR RADIATION Summer Solstice (하지) Winter Solstice (동지) 500
500
1000
1500
2000 6 12 18 24
Winter
Solstice
(동지)
Summer
Solstice (하지)
Equinox
(춘분, 추분)
- Slide by S. Chapra

44. Cloud Attenuation
Attenuation of solar radiation due to cloud cover is computed with
where CL = fraction of the sky covered with clouds.

45. Reflectivity A and B are coefficients related to cloud cover
Coefficients used to calculate reflectivity based on cloud cover

46. Shade
the fraction of potential solar radiation that is blocked by topography and vegetation
- Slide by S. Chapra

47. 4.1.1 Solar Radiation
The model computes the amount of solar radiation entering the water at a particular latitude (Lat) and longitude (Llm) on the earth’s surface.
A function of the radiation at the top of the earth’s atmosphere
attenuated by atmospheric transmission, cloud cover, shade, and reflection
I(0) = Io at ac (1-Rs) (1 – Sf)
where I(0) = solar radiation at the water surface [cal/cm2/d],
I0 = extraterrestrial radiation (i.e., at the top of the earth’s atmosphere)
at = atmospheric attenuation,
CL = fraction of sky covered with clouds,
Rs = albedo (fraction reflected; 반사계수),
Sf = effective shade (fraction blocked by vegetation and topography).
Extraterrestrial
radiation
Atmospheric
Attenuation
Cloud
Attenuation
Reflection
Shading

48. WHAT YOU NEED??? (METEOROLOGICAL DATA)
Air
Temperature
Dew Point
Temperature
evaporation
conduction
Water longwave radiation
Atmospheric longwave radiation
Solar
Shortwave
radiation ( ) )( 15
273 1
031 .
air 4 e U f T c R A J s w L sn - + = es
Solar
(Cloud Cover)
Wind
Speed
- Slide by S. Chapra

49. 4.2 Sediment-Water Heat Transfer
A heat balance for bottom sediment underlying a water reach i can be written as
Ts,i = the temperature of the bottom sediment below reach i [℃]
ρs= the density of the sediments [g/cm3]
Cps = the specific heat of the sediments [cal/(g ℃)]
Jh,i = the air-water heat flux [cal/(cm2 d)]
Js,i = the sediment-water heat flux [cal/(cm2 d)]
Hsed,i = the effective thickness of the sediment layer [cm]
Ts,i Ti

50. The flux from the sediments to the water can be computed as
as = the sediment thermal diffusivity [cm2/s]
Presence of solid material in stream sediments leads
to a higher coefficient of thermal diffusivity
than that for water or porous lake sediments.
In Q2K, a value of cm2/s used

51. 5. CONSTITUENTS OF MODEL s mi o cs cf no na nn po pi ap mo X Alk cT ab
Variable
Symbol
Units*
Conductivity s
mhos
Inorganic suspended solids mi
mgD/L
Dissolved oxygen o
mgO2/L
Slowly reacting CBOD cs
Fast reacting CBOD cf
Organic nitrogen no
gN/L
Ammonia nitrogen na
Nitrate nitrogen nn
Organic phosphorus po
gP/L
Inorganic phosphorus pi
Phytoplankton ap
gA/L
Detritus mo
Pathogen X
cfu/100 mL
Alkalinity
Alk
mgCaCO3/L
Total inorganic carbon cT
mole/L
Bottom algae biomass ab
mgA/m2
Bottom algae nitrogen
INb
mgN/m2
Bottom algae phosphorus
IPb
mgP/m2

52. General Mass Balance
For all but the bottom algae, a general mass balance for a constituent in a reach is written as i
inflow
outflow
dispersion
mass load
mass abstraction
atmospheric
transfer
sediments
bottom algae
Wi = the external loading of the constituent to
reach i [mg/d],
Si = sources and sinks of
the constituent due to reactions and mass transfer mechanisms
[mg/m3/d]

53. The external load is computed as
where cps,i,j is the jth point source concentration for reach i [mg/L or mg/L], and
cnps,i,j is the jth non-point source concentration for reach i [mg/L or mg/L].

54. the transport and loading terms are omitted,
For bottom algae,
the transport and loading terms are omitted,
where Sb,i = sources and sinks of bottom algae due to reactions [gD/m2/d].

55. CE-QUAL-W2 Water Quality Kinetics

56. Interaction of WASP7 EUTRO Module
Phytoplankton
NH3
Dis.
Org. P
Org. N
CBOD1
CBOD2
CBOD3
PO4
Settling
Photosynthesis
Denitrification
Nitrification
atmosphere DO
NO3
Oxidation
Mineralization
Reaeration N2 N P C
Detritus
sediment
Death
(Detritus Only)
Growth
Respiration
(Dissolved organic and
inorganic release) fo
(1-fo)
SSinorg
Adsorption

57. Interaction of WASP7 MPM (Multiple Algal Model)
Phytoplankton
NH3
Dis.
Org. P
Org. N
CBOD1
CBOD2
CBOD3
PO4
SSinorg
Settling
Photosynthesis
Denitrification
Nitrification
atmosphere DO
NO3
Adsorption
Oxidation
Mineralization
Reaeration N2 C
Detritus
Periphyton
Sediment 1 2 3 Si N
Org. Si
Death
(Detritus)
Growth
(Inorganic
Uptake)
Respiration
(Dissolved organic
& Inorganic) qP qN

58. EFDC Water Quality Interactions
RPOC
LPOC
DOC
RPON
LPON
DON
NH4
NO23
RPOP
LPOP
DOP
PO4t
PO4d
PO4p
SAd
SAp SU SA
Denitrification
Growth
Predation BM DO
COD
Photosynthesis
TAM
TSS
light
respiration or
reaeration
FCB Bc
Bvg Bd

59. EFDC-HEM3D and WASP7_EUTRO
WASP7_MPM
WASP7_EUTRO
1) cyanobacteria
2) diatom algae
3) green algae
4) refractory POC
5) labile particulate organic C
6) dissolved organic carbon
7) refractory particulate organic P
8) labile particulate P
9) dissolved organic
10) total phosphate
11) Refractory PON
12) labile particulate organic N
13) dissolved organic nitrogen
14) ammonia nitrogen
15) nitrate nitrogen
16) particulate biogenic silica
17) dissolved available silica
18) chemical oxygen demand
19) dissolved oxygen
20) total active metal
21) fecal coliform bacteria
22) macroalgae
Phytoplankton 1
Detrital P
Detrital N
Detrital C
Dissolved Organic P
Ortho Phosphate
Dissolved Organic N
Ammonia
Nitrate
Dissolved Oxygen
CBOD1
CBOD2
CBOD3
Ammonia Nitrogen
Nitrate Nitrogen
Dissolved Organic N
Inorganic Phosphate
Dissolved Organic P
Inorganic Silica
Dissolved Organic Silica
CBOD1 (ultimate)
CBOD2 (ultimate)
CBOD3 (ultimate)
Dissolved Oxygen
Detrital Carbon
Detrital Nitrogen
Detrital Phosphorus
Detrital Silica
Total Detritus
Salinity
Benthic Algae
Periphyton Cell Quota N
Periphyton Cell Quota P
Inorganic Solids 1
Inorganic Solids 2
Inorganic Solids 3
Phytoplankton 1
Phytoplankton 2
Phytoplankton 3

60. CAP Water Quality Model (CMF And PF Water Quality Model)
For PFR
For CMF

61. Important Processes in Q2K
Kinetic processes :
dissolution (ds)
hydrolysis (h)
oxidation (x)
nitrification (n)
denitrification (dn)
photosynthesis (p)
death (d)
respiration (r)
Mass transfer processes :
reaeration (re)
settling (s)
sediment oxygen demand (SOD)
sediment inorganic carbon flux (cf)

62. QUAL2K Kinetics: Sources and Sinks for the state variables
Kinetic processes: dissolution (ds), hydrolysis (h), oxidation (ox), nitrification (n), denitrification (dn), photosynthesis (p), death (d), and respiration/excretion (r).
Mass transfer processes: reaeration (re), settling (s), sediment oxygen demand (SOD), sediment exchange (se), and sediment inorganic carbon flux (cf).

63. cT o rdx rod mo cf cs cT o rnd rpd mi Alk x na no nn rpx rnx rcn ap ab
sod cf re cT o
rdx d
rod mo ds cf h cs x cT o
rnd
rpd se s s mi
Alk x h na no n nn dn
rpx
rnx
rcn s ap ab r p cT o pi po h
rcp s
phosphorus
nitrogen
organic carbon
plants
non-living solids
oxygen/inorg C
other

64. 5.2.2 Stoichiometry of Organic Matter
The model requires that the stoichiometry of organic matter (plants and detritus) be specified by the user.
Suggested representation as a first approximation
100 gD: 40 gC: 7.2 gN: 1.0 g P: 1.0 gA
where gX = mass of element X [g]
mgY = mass of element Y [mg]
The terms D, C, N, P and A refer to dry weight, carbon, nitrogen, phosphorus, and chlorophyll a, respectively.
It should be noted that chlorophyll a is the most variable of these values with a range of approximately gA

65. These values are then combined to determine stoichiometric ratios as in
For example, the amount of nitrogen that is released when 1 gD of detritus is dissolved can be computed as

66. 5.2.2.1 Oxygen Generation and Consumption
If ammonia is the substrate,
oxygen generated for each gram of plant matter that is produced through photosynthesis.

67. If nitrate is the substrate, the following ratio applies
For nitrification, the following ratio is used

68. 5.2.2.2 CBOD Utilization Due to Denitrification
CBOD is utilized during denitrification,

69. 2.4.2.7 Effect of Denitrification on Dissolved Organic Carbon
(2.43)
(2.44)
KRORDO = denitrification half-sat constant for DO (g O m-3)
KHDNN = denitrification half-saturation constant for nitrate (g N m-3)
AANOX = ratio of denitrification rate to oxic dissolved organic carbon respiration rate.

70. 5.2.3 Temperature Effects on Reactions
The temperature effect for all first-order reactions used in the model is represented by
k(T) = the reaction rate [/d] at temperature T [oC]
q = the temperature coefficient for the reaction.

71. 5.3 Composite Variables

72. 5.4 Relationship of Model Variables and Data
For all but slow and fast CBOD (cf and cs), there exists a relatively straightforward relationship between the model state variables and standard water-quality measurements.

73. 5.4.1 Non-CBOD Variables and Data
TKN = total Kjeldahl Nn (mgN/L) or TN = total N (mgN/L)
NH4 = ammonium N (mgN/L), NO2 = nitrite N (mgN/L)
NO3 = nitrate N (mgN/L)
CHLA = chlorophyll a (mgA/L)
TP = total P (mgP/L), SRP = soluble reactive P (mgP/L)
TSS = total SS (mgD/L), VSS = volatile SS (mgD/L)
TOC = total organic C (mgC/L),
DOC = dissolved organic C (mgC/L)
DO = dissolved oxygen (mgO2/L), PH = pH
ALK = alkalinity (mgCaCO3/L), TEMP = temperature (oC)
COND = specific conductance (mmhos)

74. 5.4.1 Non-CBOD Variables and Data
s = COND
mi = TSS – VSS or TSS – rdc (TOC – DOC)
o = DO
no = TKN – NH4 – rna CHLA or
no = TN – NO2 – NO3 – NH4 – rna CHLA
na = NH4
nn = NO2 + NO3
po = TP – SRP – rpa CHLA
pi = SRP
ap = CHLA
mo = VSS – rda CHLA or rdc (TOC – DOC) – rda CHLA
pH = PH
Alk = ALK

75. 5.4.2 Carbonaceous BOD
The interpretation of BOD measurements in natural waters is complicated by three primary factors:
Filtered versus unfiltered.
If the sample is unfiltered, the BOD will reflect oxidation of both dissolved and particulate organic carbon.
Since Q2K distinguishes between dissolved (cs and cf) and particulate (mo and ap) organics,
an unfiltered measurement alone does not provide an adequate basis for distinguishing these individual forms.

76. 5.4.2 Carbonaceous BOD Nitrogenous BOD.
Along with the oxidation of organic carbon (CBOD), nitrification also contributes to oxygen depletion (NBOD).
Thus, if the sample (a) contains reduced nitrogen and (b) nitrification is not inhibited, the measurement includes both types of BOD.

77. 5.4.2 Carbonaceous BOD cf + cs = CBODFNU
where CBODFNU = the ultimate dissolved carbonaceous BOD [mgO2/L],
CBODFN5 = the 5-day dissolved carbonaceous BOD [mgO2/L], and
k1 = the CBOD decomposition rate in the bottle [/d]
cf + cs = CBODFNU

78. Slow versus Fast CBOD
As in Figure 19, such rates suggest that much of the readily oxidizable CBOD will be exerted in about 20 to 30 days.

79. 5.5 Constituent Reactions
5.5.1 Conservative Substance (s)
By definition, conservative substances are not subject to reactions:
Ss = 0
5.5.2 Phytoplankton (ap)
Phytoplankton increase due to
photosynthesis
They are lost via
respiration, death, and settling
Sap = PhytoPhoto–PhytoSettl-PhytoDeath-PhytoResp

80. 5.5.2.1 Photosynthesis Phytoplankton photosynthesis is a function of
temperature, nutrients, and light
mp = phytoplankton photosynthesis rate [/d]
kgp(T) = the maximum photosynthesis rate at temperature T [/d],
fNp = phytoplankton nutrient attenuation factor
fLp = the phytoplankton light attenuation coefficient

81. Nutrient Limitation
Michaelis-Menten equations are used to represent growth limitation for inorganic nitrogen and phosphorus
The minimum value is then used to compute the nutrient attenuation factor,
where ksNp = nitrogen half-saturation constant [mgN/L] and
ksPp = phosphorus half-saturation constant [mgP/L].

82. Light Limitation Beer-Lambert Law,
where PAR(z) = photosynthetically available radiation (PAR) at depth z below the water surface [ly/d]2, and
ke = the light extinction coefficient [m−1].
ly/d = langley per day (A langley = a calorie per cm2)
Note that a ly/d is related to the µE/m2/d by the following approximation: 1 µE/m2/s ≅ 0.45 Langley/day
The PAR at the water surface is assumed to be a fixed fraction of the solar radiation

83. 5.5.2.2 Losses Respiration. Death Settling.
krp(T) = temp-dependent phytoplankton respiration rate [/d].
Death
kdp(T) = temperature-dependent phytoplankton death rate [/d].
Settling.
va = phytoplankton settling velocity [m/d].

84. 5.5.3 Bottom algae (ab) Bottom algae increase due to photosynthesis.
They are lost via respiration and death.
Sap = BotAlgPhoto-BotAlgResp-BotAlgDeath

85. Photosynthesis
The representation of bottom algae photosynthesis is a simplification of a model developed by Rutherford et al. (1999).
Photosynthesis is based on
a temperature-corrected zero-order rate
attenuated by nutrient and light limitation,
Cgb(T) = the temperature-dependent maximum photosynthesis rate [gD/(m2 d)],
φNb = bottom algae nutrient attenuation factor [dimensionless number between 0 and 1], and
φLb = the bottom algae light attenuation coefficient

86. Temperature Effect
As for the first-order rates, an Arrhenius model is employed to quantify the effect of temperature on bottom algae photosynthesis,

87. Nutrient Limitation.
The photosynthesis rate is dependent on intracellular nutrients using a formulation developed by Droop (1974),
to compute the nutrient attenuation coefficient,
qN and qP = the cell quotas of N [mgN/mgA] and P [mgP/mgA]
q0N and q0P = the minimum cell quotas of N [mgN/mgA] and P [mgP/mgA]
ksCb = inorganic C half-saturation constant for the bottom algae [mole/L].
The minimum cell quotas are the levels of intracellular nutrient at which growth ceases.

88. Light Limitation
In contrast to the phytoplankton, light limitation at any time is determined by the amount of Photosynthetically Active Radiation (PAR) reaching the bottom of the water column.
This quantity is computed with the Beer-Lambert law evaluated at the bottom of the river,

89. Losses
Respiration
represented as a first-order rate that is attenuated at low oxygen concentration,
krb(T) = temperature-dependent bottom algae respiration rate [/d]
Death.
represented as a first-order rate,
kdb(T) = temperature-dependent bottom algae death rate [/d]

90. 5.5.4 Bottom Algae Internal Nitrogen (INb)
The change in intracellular nitrogen in bottom algal cells is calculated from
where
BotAlgUpN = the uptake rate of nitrogen by bottom algae (mgN/m2/d),
BotAlgDeath = bottom algae death (mgA/m2/d), and
BotAlgExN = the bottom algae excretion of nitrogen (mgN/m2/d),
which is computed as
keb(T) = the temperature-dependent bottom algae excretion rate [/d].

91. 5.5.4 Bottom Algae Internal Nitrogen (INb)
The N uptake rate depends on both external and intracellular nutrients as in (Rhee 1973),
where
rmN = the maximum uptake rate for N [mgN/mgA/d],
ksNb = half-saturation constant for external N [mgN/L]
KqN = half-saturation constant for intracellular N [mgN/mgA].

92. 5.5.5 Bottom Algae Internal Phosphorus (IPb)
The change in intracellular phosphorus in bottom algal cells is calculated from
BotAlgUpP = the uptake rate of phosphorus by bottom algae (mgP/m2/d),
BotAlgDeath = bottom algae death (mgA/m2/d)
BotAlgExP = the bottom algae excretion of phosphorus (mgP/m2/d), which is computed as
keb(T) = the temp-dependent bottom algae excretion rate [/d].

93. 5.5.5 Bottom Algae Internal Phosphorus (IPb)
The P uptake rate depends on both external and intracellular nutrients as in (Rhee 1973),
rmP = the maximum uptake rate for P [mgP/mgA/d]
ksPb = half-saturation constant for external P [mgP/L]
KqP = half-saturation constant for intracellular P [mgP/mgA].

94. 5.5.6 Detritus (mo)
Detritus or particulate organic matter (POM) increases due to plant death.
It is lost via dissolution and settling and settling
Smo=rdaPhytoDeath+BotAlgDeath-DetrDiss-DetrSettl
where kdt(T) = the temperature-dependent detritus
where vdt = detritus settling velocity [m/d].

95. 5.5.7 Slowly Reacting CBOD (cs)
increases due to detritus dissolution.
It is lost via hydrolysis.
Ssc=rodDetrDiss-SlowCHydr
khc(T) = temperature-dependent slow CBOD hydrolysis rate [/d]

96. 5.5.6 Fast Reacting CBOD (cf)
Fast reacting CBOD is gained
via the hydrolysis of slowly-reacting CBOD.
It is lost via oxidation and denitrification.
Ssf=SlowCHydr-FastCOxid-rondnDenitr
kdc(T) = the temperature-dependent fast CBOD oxidation rate [/d]
Foxcf = attenuation due to low oxygen [dimensionless].
The parameter rondn is the ratio of oxygen equivalents lost per nitrate nitrogen that is denitrified.
The term Denitr is the rate of denitrification [µgN/L/d].

97. Foxcf : Oxygen Attenuation
Half Saturation
Exponential
Second Order Half Saturation
where Ksocf = half-saturation constant for second-order effect of oxygen on fast CBOD oxidation [mgO22/L2].

98. 5.5.7 Dissolved Organic Nitrogen (no)
increases due to detritus dissolution.
It is lost via hydrolysis.
Sno=rndDetrDiss-DONHydr
khn(T) = temperature-dependent organic N hydrolysis rate [/d]

99. 5.5.8 Ammonia Nitrogen (na) Ammonia nitrogen increases due to
dissolved organic nitrogen hydrolysis and plant respiration.
It is lost via
nitrification and plant photosynthesis:
Sno=DONHydr+rnoPhytoResp+rndBotAlgResp-
NH4Nitrif-rnaPapPhytoPhoto-rndPabPBotAlgPhoto

100. Nitrification Rate Preference Factor
khnxp = preference coef. of phytoplankton for ammonium
khnxb = preference coef. of bottom algae for ammonium [mgN/m3].

101. 5.5.9 Unionized Ammonia (NH3)
The model simulates total ammonia.
In water, the total ammonia
consists of two forms:
NH4+, and NH3
ammonium ion and
unionized ammonia
At normal pH (6 to 8),
most of the total ammonia will be in the ionic form.
However at high pH,
unionized ammonia predominates.

102. Amount of unionized ammonia can be computed as
nau = the concentration of unionized ammonia [µgN/L], and
Fu = the fraction of the total ammonia that is in unionized form,
where Ka = the equilibrium coefficient for the ammonia dissociation reaction, which is related to temperature by
where Ta is absolute temperature [K] and pKa = −log10(Ka).

103. 5.5.10 Nitrate Nitrogen (nn) Nitrate nitrogen increases
due to nitrification of ammonia.
lost via denitrification and plant photosynthesis:
Sni=NH4Nitrif-Denitr-rna(1-Pap)Phytophoto
-rnd(1-Pab)BotAlgPhoto

104. The denitrification rate is computed as
kdn(T) = temp-dependent denitrification rate of NO3 [/d]
Foxdn =effect of low oxygen on denitrification [dimensionless] as modeled by Eq(127) to (129) with the oxygen dependency represented by the parameter Ksodn

105. 5.5.11 Dissolved Organic Phosphorus (po)
increases due to dissolution of detritus.
It is lost via hydrolysis.
Spo=rpdDetrDiss-DOPHydr
khp(T) = the temperature-dependent organic phosphorus hydrolysis rate [/d].

106. 5.5.12 Inorganic Phosphorus (pi)
Inorganic phosphorus increases due to
dissolved organic phosphorus hydrolysis and
plant respiration.
It is lost via plant photosynthesis:
Spi = DOPHydr+rpaPhytoResp+rpdBotAlgResp
-rpaPhytoPhoto-rpdBotAlgPhoto

107. 5.5.13 Inorganic Suspended Solids (mi)
Inorganic suspended solids are lost via settling,
Smi = -InorgSettl
where vi = inorganic suspended solids settling velocity [m/d].

108. 5.5.14 Dissolved Oxygen (o) Dissolved oxygen increases
due to plant photosynthesis.
It is lost via
fast CBOD oxidation, nitrification and plant respiration
gained or lost via reaeration,
So = roaPhytoGrowth+rodBotAlgGrowth-rocFastCOxid
-ronNH4Nitr-roaPhytoResp-rodBotAlgResp+OxRear
where ka(T) = the temp.-dependent oxygen reaeration coef. [/d],
os(T, elev) = the saturation concentration of oxygen [mgO2/L] at temperature, T, and elevation above sea level, elev.

109. The effect of elevation is accounted for by
The following equation is used to represent the dependence of oxygen saturation on temperature (APHA 1992)
where os(T, 0) = the saturation concentration of dissolved oxygen in freshwater at 1 atm [mgO2/L]
and Ta = absolute temperature [K] where Ta = T
The effect of elevation is accounted for by
where elev = the elevation above sea level [m].

110. 5.5.14.2 Reaeration Formulas O’Connor-Dobbins Owens-Gibbs Churchill
If H < 0.61 m
use the Owens-Gibbs formula
If H > 0.61 m and H > 3.45U2.5
use the O’Connor-Dobbins formula
Otherwise
use the Churchill formula
O’Connor-Dobbins
Owens-Gibbs
Churchill

111. Reaeration can also be internally calculated based on the following scheme patterned after a plot developed by Covar (1976) (Figure 18):

112. 5.5.14.3 Effect of Control Structures: Oxygen
Butts and Evans (1983) suggested the following formula,
where rd = the ratio of the deficit above and below the dam, Hd = the difference in water elevation[m] as calculated with Equation (7), T = water temperature (0C) and ad and bd are coefficients that correct for water-quality and dam-type.

113. Coefficient values used to predict the effect of dams on stream reaeration.
(a) Water-quality coefficient
Polluted state ad
Gross
0.65
Moderate
1.0
Slight
1.6
Clean
1.8
(b) Dam-type coefficient
Dam type bd
Flat broad-crested regular step
0.70
Flat broad-crested irregular step
0.80
Flat broad-crested vertical face
0.60
Flat broad-crested straight-slope face
0.75
Flat broad-crested curved face
0.45
Round broad-crested curved face
Sharp-crested straight-slope face
1.00
Sharp-crested vertical face
Sluice gates
0.05
o’i−1 = the oxygen concentration
entering the reach [mgO2/L],

114. 5.5.15 Pathogen (x) Pathogens are subject to death and settling,
Sx = -PathDeath-PathSettl

115. Death
Pathogen death is due to natural die-off and light (Chapra 1997).
The death of pathogens in the absence of light is modeled as a first-order temperature-dependent decay and the death rate due to light is based on the Beer-Lambert law,
where kdx(T) = temperature-dependent pathogen die-off rate [/d].

116. 5.5.15.2 Settling Pathogen settling is represented as
where vx = pathogen settling velocity [m/d].

117. Alkalinity Total Alkalinity.
the equivalent sum of the bases that are titratable with strong acid (Stumm and Morgan, 1981)."
"mHCO3- + 2mCO3-2 + mB(OH)4- + mH3(SiO)4- + mHS- + morganic anions + mOH- - mH+".
AT = [HCO3−]T + 2[CO3−2]T + [B(OH)4−]T + [OH−]T + 2[PO4−3]T + [HPO4−2]T + [SiO(OH)3−]T − [H+]sws − [HSO4−]
In most natural waters, all ions except HCO3- and CO3−2 have low concentrations.
Thus carbonate alkalinity, which is equal to mHCO3- + 2mCO3-2 is also approximately equal to the total alkalinity.

118. pH
The following equilibrium, mass balance and electroneutrality equations define a freshwater dominated by inorganic carbon (Stumm and Morgan 1996),
where K1, K2 and Kw are acidity constants,
Alk = alkalinity [eq L−1],
H2CO3* = CO2 + H2CO3
HCO3− = bicarbonate ion, CO32− = carbonate ion,
H+ = hydronium ion, OH− = hydroxyl ion, and
Tc = total inorganic carbon concentration [mole L−1].
The brackets [ ] designate molar concentrations.

119. Note that the alkalinity is expressed in units of eq/L for the internal calculations.
For input and output, it is expressed as mgCaCO3/L. The two units are related by
Alk(mgCaCO3/L) = 50,000 x Alk(eq/L)

120. Harned and Hamer (1933):
Plummer and Busenberg (1982):

121. The nonlinear system of five simultaneous equations can be solved numerically for the five unknowns: [H2CO3*], [HCO3−], [CO32− ], [OH−], and [H+].
an efficient solution method can be derived by combining Equations to define the quantities (Stumm and Morgan 1996).
where α0, α1, and α2
= the fraction of
total inorganic carbon in
carbon dioxide, bicarbonate,
and carbonate, respectively

122. Equations can then be combined to yield,
Thus, solving for pH reduces to determining the root, [H+], of
where pH is then calculated with

123. 5.5.17 Total Inorganic Carbon (cT)
Total inorganic carbon concentration increases due to fast carbon oxidation and plant respiration.
It is lost via plant photosynthesis. Depending on whether the water is undersaturated or oversaturated with CO2, it is gained or lost via reaeration,
where kac(T) = the temperature-dependent carbon dioxide reaeration coefficient [/d], and
[CO2]s =the saturation concentration of carbon dioxide [mole/L].

124. The stoichiometric coefficients are derived from Equation

125. 5.5.17.1 Carbon Dioxide Saturation
The CO2 saturation is computed with Henry’s law,
where KH = Henry's constant [mole (L atm)−1] and
pCO2 = the partial pressure of carbon dioxide in the atmosphere [atm].
Note that users input the partial pressure to Q2K in ppm.
The program internally converts ppm to atm using the conversion: 10−6 atm/ppm.

126. The value of KH can be computed as a function of temperature by (Edmond and Gieskes 1970)

127. The partial pressure of CO2 in the atmosphere has been increasing, largely due to the combustion of fossil fuels (Figure 20).
Values in 2003 are approximately atm (= 372 ppm).

128. Mauna Loa Carbon Dioxide

129. CO2 Gas Transfer
The CO2 reaeration coefficient can be computed from the oxygen reaeration rate by

130. 5.5.17.3 Effect of Control Structures: CO2
CO2 gas transfer in streams can be influenced by the presence of control structures.
Q2K assumes that carbon dioxide behaves similarly to dissolved oxygen (recall Sec ).
Thus, the inorganic carbon mass balance for the reach immediately downstream of the structure is written as
where c'T,i−1 = the concentration of inorganic carbon entering the reach [mgO2/L],

131. Alkalinity (Alk)
The present model accounts for changes in alkalinity due to plant photosynthesis and respiration, nitrification, and denitrification.
where the r’s are ratios that translate the processes into the corresponding amount of alkalinity.

132. The stoichiometric coefficients
Phytoplankton Photosynthesis (Ammonia as Substrate) and Respiration:
Phytoplankton Photosynthesis (Nitrate as Substrate):
Bottom Algae Nitrogen Uptake and Nitrogen Hydrolysis:
Bottom Algae Phosphorus Uptake and Phosphorus Hydrolysis:
Nitrification:
Denitrification

133. 5.6 SOD/Nutrient Flux Model
Sediment nutrient fluxes and sediment oxygen demand (SOD) are based on
a model developed by Di Toro (Di Toro et al. 1991, Di Toro and Fitzpatrick. 1993, Di Toro 2001).
The present version also benefited from James Martin’s (Mississippi State University, personal communication) efforts to incorporate the Di Toro approach into EPA’s WASP modeling framework.
A schematic of the model is depicted in Figure 21.
As can be seen, the approach allows oxygen and nutrient sediment-water fluxes to be computed based on the downward flux of particulate organic matter from the overlying water

134. Schematic of SOD-nutrient flux model of the sediments

135. The sediments are divided into 2 layers:
a thin (≅ 1 mm) surface aerobic layer underlain by
a thicker (10 cm) lower anaerobic layer.
Organic carbon, nitrogen and phosphorus are delivered to the anaerobic sediments
via the settling of particulate organic matter (i.e., phytoplankton and detritus).
There they are transformed by mineralization reactions into dissolved methane, ammonium and inorganic phosphorus.

136. These constituents are then transported to the aerobic layer where some of the methane and ammonium are oxidized.
The flux of oxygen from the water required for these oxidations is the sediment oxygen demand.
The following sections provide details on how the model computes this SOD along with the sediment-water fluxes of carbon, nitrogen and phosphorus that are also generated in the process.

137. 5.6.1 Diagenesis Conversion of Particulate organic matter (POM)
into soluble reactive forms
in the anaerobic sediments.
The first step in the computation
How much of the downward flux of POM is converted

138. 1) Total downward flux is computed as the sum of
the fluxes of phytoplankton and
detritus settling from the water column
where JPOM = the downward flux of POM [gD m−2 d−1],
rda = the ratio of dry weight to chlorophyll a [gD/mgA],
va = phytoplankton settling velocity [m/d],
ap = phytoplankton concentration [mgA/m3],
vdt = detritus settling velocity [m/d], and
mo = detritus concentration [gD/m3].

139. Representation of how settling particulate organic particles (phytoplankton and
detritus) are transformed into fluxes of dissolved carbon (JC) , nitrogen (JN) and
phosphorus (JP) in the anaerobic sediments.

140. 2) Stoichiometric ratios are then used to divide the
POM flux into carbon, nitrogen and phosphorus.
POC
as oxygen equivalents using the stoichiometric coefficient roc.
3) Each of the nutrient fluxes is further broken down into three reactive fractions:
labile (G1),
slowly reacting (G2) and
non-reacting(G3).

141. 4) These fluxes are then entered into mass balances
to compute the concentration of each fraction in the anaerobic layer.
For example, for labile POC, a mass balance is written as
where H2 = the thickness of the anaerobic layer [m], POC2,G1 = the concentration of the labile fraction of POC in the anaerobic layer [gO2/m3], JPOC,G1 = the flux of labile POC delivered to the anaerobic layer [gO2/m2/d], kPOC,G1 = the mineralization rate of labile POC [d−1], qPOC,G1 = temperature correction factor for labile POC mineralization [dimensionless], and w2 = the burial velocity [m/d].

142. At steady state, above equation can be solved for
The flux of labile dissolved carbon, JC,G1 [gO2/m2/d], can then be computed as

143. In a similar fashion, a mass balance can be written and solved for
the slowly reacting dissolved organic carbon.
This result is then added to arrive at the total flux of dissolved carbon generated in the anaerobic sediments.
Similar equations are developed to compute the diagenesis fluxes of nitrogen, JN [gN/m2/d], and phosphorus JP [gP/m2/d].

144. 5.6.2 Ammonium
mass balances for total ammonium in the aerobic layer and the anaerobic layers,
where H1 = the thickness of the aerobic layer [m], NH4,1 and NH4,2 = the concentration of total ammonium in the aerobic layer and the anaerobic layers, respectively [gN/m3], na = the ammonium concentration in the overlying water [mgN/m3], κNH4,1 = the reaction velocity for nitrification in the aerobic sediments [m/d], θNH4 = temperature correction factor for nitrification [dimensionless], KNH4 = ammonium half-saturation constant [gN/m3], o = the dissolved oxygen concentration in the overlying water [gO2/m3], and KNH4,O2 = oxygen half-saturation constant [mgO2/L], and JN = the diagenesis flux of ammonium [gN/m2/d].

145. The fraction of ammonium in dissolved (fdai) and particulate (fpai) form are computed as
where mi = the solids concentration in layer i [gD/m3], and
pai = the partition coefficient for ammonium in layer i [m3/gD].

146. The mass transfer coefficient for particle mixing due to bioturbation between the layers, w12 [m/d], is computed as
where
Dp = diffusion coefficient for bioturbation [m2/d],
qDp = temperature coefficient [dimensionless],
POCR = reference G1 concentration for bioturbation [gC/m3] and
KM,Dp = oxygen half-saturation constant for bioturbation [gO2/m3].

147. The mass transfer coefficient for pore water diffusion between the layers, KL12 [m/d], is computed as,
where Dd = pore water diffusion coefficient [m2/d], and
qDd = temperature coefficient [dimensionless].

148. The mass transfer coefficient between the water and the aerobic sediments, s [m/d], is computed as
where SOD = the sediment oxygen demand [gO2/m2/d].

149. At steady state, equations for ammonium are two simultaneous nonlinear algebraic equations.
The equations can be linearized by assuming that the NH4,1 term in the Monod term for nitrification is constant.
The simultaneous linear equations can then be solved for NH4,1 and NH4,2.
The flux of ammonium to the overlying water can then be computed as

150. 5.6.3 Nitrate
Mass balances for nitrate can be written for the aerobic and anaerobic layers as
where NO3,1 and NO3,2 = the concentration of nitrate in the aerobic layer and the anaerobic layers [gN/m3], nn = the nitrate concentration in the overlying water [mgN/m3], kNO3,1 and kNO3,2 = the reaction velocities for denitrification in the aerobic and anaerobic sediments [m/d], and qNO3 = temperature correction factor for denitrification [dimensionless].

151. In the same fashion as for Eqs. 170 and 171, Eqs
In the same fashion as for Eqs. 170 and 171, Eqs. 178 and 179 can be linearized and solved for NO3,1 and NO3,2.
The flux of nitrate to the overlying water can then be computed as

152. Denitrification requires a carbon source as represented by the following chemical equation
The carbon requirement (expressed in oxygen equivalents per nitrogen) can therefore be computed as

153. Therefore, the oxygen equivalents consumed during denitrification, JO2,dn [gO2/m2/d], can be computed as

154. 5.6.4 Methane
The dissolved carbon generated by diagenesis is converted to methane in the anaerobic sediments.
Because methane is relatively insoluble, its saturation can be exceeded and methane gas produced.
As a consequence,
rather than write a mass balance for methane in the anaerobic layer,
an analytical model developed by Di Toro et al. (1990) is used
to determine the steady-state flux of dissolved methane corrected for gas loss delivered to the aerobic sediments.

155. First, the carbon diagenesis flux is corrected for the oxygen equivalents consumed during denitrification,
where JCH4,T = the carbon diagenesis flux corrected for denitrification [gO2/m2/d].
In other words, this is the total anaerobic methane production flux expressed in oxygen equivalents.

156. If JCH4,T is sufficiently large (≥ 2KL12Cs), methane gas will form.
In such cases, the flux can be corrected for the gas loss,
where JCH4,d = the flux of dissolved methane (expressed in oxygen equivalents) that is generated in the anaerobic sediments and delivered to the aerobic sediments [gO2/m2/d], Cs = the saturation concentration of methane expressed in oxygen equivalents [mgO2/L].
If JCH4,T < 2KL12Cs, then no gas forms and

157. The methane saturation concentration is computed as
where H = water depth [m] and T = water temperature [oC].
A methane mass balance can then be written for the aerobic layer as
where CH4,1 = methane concentration in the aerobic layer [gO2/m3], cf = fast CBOD in the overlying water [gO2/m3], kCH4,1 = the reaction velocity for methane oxidation in the aerobic sediments [m/d], and qCH4 = temperature correction factor [dimensionless].

158. At steady, state, this balance can be solved for
The flux of methane to the overlying water, JCH4 [gO2/m2/d], can then be computed as

159. 5.6.5 SOD
The SOD is equal to the sum of the oxygen consumed in methane oxidation and nitrification,
where CSOD = the amount of oxygen demand generated by methane oxidation [gO2/m2/d] and NSOD = the amount of oxygen demand generated by nitrification [gO2/m2/d].
These are computed as
where ron = the ratio of oxygen to nitrogen consumed during nitrification [= 4.57 gO2/gN].

160. 5.6.6 Inorganic Phosphorus
Mass balances can be written total inorganic phosphorus in the aerobic layer and the anaerobic layers as
where PO4,1 and PO4,2 = the concentration of total inorganic phosphorus in the aerobic layer and the anaerobic layers, respectively [gP/m3], pi = the inorganic phosphorus in the overlying water [mgP/m3], and JP = the diagenesis flux of phosphorus [gP/m2/d].

161. The fraction of phosphorus in dissolved (fdpi) and particulate (fppi) form are computed as
where ppi = the partition coefficient for inorganic phosphorus in layer i [m3/gD].

162. The partition coefficient in the anaerobic layer is set to an input value.
For the aerobic layer, if the oxygen concentration in the overlying water column exceeds a critical concentration, ocrit[gO2/m3], then the partition coefficient is increased to represent the sorption of phosphorus onto iron oxyhydroxides as in
where DpPO4,1 is a factor that increases the aerobic layer partition coefficient relative to the anaerobic coefficient.

163. If the oxygen concentration falls below ocrit then the partition coefficient is decreased smoothly until it reaches the anaerobic value at zero oxygen,
Equations 194 and 195 can be solved for PO4,1 and PO4,2. The flux of phosphorus to the overlying water can then be computed as

164. 5.6.7 Solution Scheme
Although the foregoing sequence of equations can be solved, a single computation will not yield a correct result because of the interdependence of the equations.
For example, the surface mass transfer coefficient s depends on SOD.
The SOD in turn depends on the ammonium and methane concentrations which themselves are computed via mass balances that depend on s.
Hence, an iterative technique must be used.

165. The procedure used in QUAL2K is
1. Determine the diagenesis fluxes: JC, JN and JP.
2. Start with an initial estimate of SOD,
where ron’ = the ratio of oxygen to nitrogen consumed for total conversion of ammonium to nitrogen gas via nitrification/denitrification [= gO2/gN].
This ratio accounts for the carbon utilized for denitrification.
3. Compute s using

166. 4. Solve for ammonium, nitrate and methane, and compute the CSOD and NSOD.
5. Make a revised estimate of SOD using the following weighted average
6. Check convergence by calculating an approximate relative error

167. 7. If ea is greater than a prespecified stopping criterion es then set SODinit = SOD and return to step 2.
8. If convergence is adequate (ea ≤ es), then compute the inorganic phosphorus concentrations.
9. Compute the ammonium, nitrate, methane and phosphate fluxes.

168. 5.6.8 Supplementary Fluxes
Because of the presence of organic matter deposited prior to the summer steady-state period (e.g., during spring runoff), it is possible that the downward flux of particulate organic matter is insufficient to generate the observed SOD.
In such cases, a supplementary SOD can be prescribed,
SODt = the total sediment oxygen demand [gO2/m2/d], and
SODs = the supplementary SOD [gO2/m2/d].
In addition, prescribed ammonia and methane fluxes can be used to supplement the computed fluxes.