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Equations for nitrogen supersaturation are of two types. One type constitutes empirical equations with no underlying theory but which provide a general fit to observed supersaturation data as a function of spill. The other type constitutes mechanistic equations which define nitrogen levels in terms of physical processes producing spill. CRiSP.1 contains two empirical models and two mechanistic models. CRiSP.1 is calibrated to all submodels. In general, we recommend using the model called Gas Spill 2. Relevant parameters in the submodels are illustrated in Fig. 45.
Exponential Saturation Equation
An empirical nitrogen supersaturation equation based on an exponential relationship between spill flow and supersaturation in the spilled water can be expressed
(112)
where
- Ns = percent supersaturation above 100%
- Fs = spillway flow volume in kcfs
- a, b and k = coefficients specific to each dam derived from nitrogen rating curves provided by the Corps of Engineers.
The exponential equation was developed first and was used in CRiSP.1 version 3. It is retained in version 4 for backward compatibility of models. The hyperbolic model fits the data better than the exponential model.
Hyperbolic Saturation Equation
The nitrogen supersaturation equation data can also be fit with a hyperbolic relationship between spill flow and supersaturation. The relationship is
(113)
where
- Ns = percent supersaturation above 100%
- Fs = spillway flow volume in kcfs
- a, b and h = coefficients specific to each dam and can be derived from nitrogen rating curves available from the Corps of Engineers.
Although this submodel can produce a degree of supersaturation at zero spill flow (when h = 0), this does not contribute to supersaturation in the tailrace water since the contribution of spill water to the tailrace is zero with zero spill as is defined in eq (123). This model is the preferred empirical model and should be used in place of the exponential model if an empirical model is selected.
Mechanistic Equation
The theory for nitrogen supersaturation from the physical process of spilling water and dissolving excess nitrogen in the tailrace water is developed below.1 The mechanistic model begins with an equation for nitrogen concentration as
(114)
where
- F = total flow in kcfs
- Fs = spill in kcfs
- Ns = nitrogen concentration in tailrace in mg/l
- Nfb = nitrogen concentration in the forebay in mg/l
- Neq = nitrogen equilibrium concentration as a function of temperature in degrees C at one atmospheric pressure
- This is approximated by:
(115)
- L = length of the stilling basin in feet
- P = average hydrostatic pressure in the main flow of the stilling basin in atmospheres
- This is defined
(116)
- P0 = barometric pressure in atmospheres (assume P0 is 1)
= density of water (0.0295 atm/ft) - 0 = specific gravity of the roller at the base of the spill
- This depends on the degree of aeration of the roller.
- W = spillway width
- D = water depth at the end of the stilling basin
- Y0 = thickness of the spill at the stilling basin entrance, where
(117)
- H = hydraulic head expressing the forebay elevation minus the elevation of the spilling basin floor (H is in ft and gravity constant, g, is 32 ft s-2)
= differential pressure factor defined
(118)
- Ke = bubble entrainment coefficient with units of ft s-1atm-1/3 and is defined
(119)
- T = temperature in degrees C
- K20 = temperature compensated entrainment coefficient.
- The coefficient are estimated using different relationships depending upon the dam. These are known as Gas Spill 1 and Gas Spill 2 and are detailed as follows.
Gas Spill 1
Gas Spill 1 is a three-parameter multiplicative model, used by the Corps of Engineers at Bonneville Dam only. The equation is
(120)
Gas Spill 2
Gas Spill 2 is a three-parameter additive model used at all other dams. It is defined
(121)
where
- E = energy loss rate expressed as total headloss divided by residence time of water in the stilling basin
(122)
- P = forebay percent saturation
- a, b, and c = dam dependent empirical coefficients.
The original WRE model used a two-parameter multiplicative model, which was identical to Gas Spill 1 with c = 0.
Nitrogen in the Tailrace
Nitrogen supersaturation in the tailrace results from mixing spill water with water passing through turbines (Fig. 45). The equation is
(123)
where
- F = total flow through the dam in kcfs
- Fs = spill flow in kcfs
- N = tailwater nitrogen supersaturation (in percent)
- Nfb = forebay nitrogen supersaturation (in percent)
- Ns = spill water nitrogen in percent saturation as defined by an empirical or mechanistic saturation equation.
Nitrogen at a Confluence
The nitrogen at a confluence is determined by the addition of two flows with different nitrogen levels. The equation is
(124)
where
- F i = flow in kcfs in segment i
- N i = nitrogen in percent supersaturation in segment i of the confluences.
Nitrogen Dissipation
Nitrogen levels above the saturation level are lost from the river as a first order process. This is defined (WRE) by a total flux equation for a segment as
(125)
where
= flux of nitrogen across the air water interface - N = nitrogen supersaturation concentration in the segment
- Neq = nitrogen equilibrium concentration
- A = surface area of the segment
- Kd = transfer coefficient defined
(126)
where
- Dm = molecular diffusion coefficient of nitrogen
- U = hydraulic stream velocity
- D = depth of the segment
To express the loss in terms of concentration we divided eq (125) by AD to give
(127)
To put the calculation in units of miles and days, note that one mile = 16.0934 x 104 cm = 5280 ft, and one day = 8.64 x104 seconds. Expressing U in miles/day and D in feet and Dm in cm2/s, the diffusion coefficient per unit square mile of river is
(128)
where the coefficient k is expressed
(129)
assuming:
- Dm = order2 of 2 x 10-5 cm2s-1
- U = order of (20 miles/day) 1.2 ft/s. Note this changes on a daily basis and for each reach in the model
- D = order of 30 ft. Note this changes on a reach specific basis and is dependent on reservoir elevation.
- The constant 700.75 gives the coefficient k in unit of day-1
Nitrogen loss rate due to degassing can be expressed as a function of time since the water entered the tailrace. The equation describing the nitrogen level as a function of residence time in a river segment is
(130)
where
- Neq = nitrogen equilibrium concentration defined by eq (115)
- N(0) = tailrace concentration defined by eq (123)
- k = dissipation coefficient defined by eq (129)
- t = time in a river segment
Noting that in the models N is in terms of percent above supersaturation we then set Neq = 0.
Adjustments of k
The nitrogen dissipation coefficient depends on the average depth as defined in eq (129). The average depth is variable according to the geometry of the reservoir and the pool elevation. This depth is defined as
(131)
where
- Volume = pool volume at a specific elevation
- W = average pool width at full pool
- L = length of pool
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1This model was developed by Water Resources Engineers (WRE), Inc. (1971) for the Corps of Engineers and reviewed by Boyer (1974).
2F.A. Richards 1965.
Columbia River Salmon Passage Model CRiSP.1.5 Theory, Calibration & Validation Manual
Copyright © 1996, Columbia Basin Research. All rights reserved.
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