Qual2K: a modeling Framework for Simulating River and Stream Water Quality (Version 11)

Flow Balance : Continuity Equation

As described in the last section, Q2K’s most fundamental unit is the element. A steady-state flow balance is implemented for each model element as (
)

()
where Q_i = outflow from element i into the downstream element i + 1 [m³/d], Q_i–1 = inflow from the upstream element i – 1 [m³/d], Q_in,i is the total inflow into the element from point and nonpoint sources [m³/d], and Q_out,i is the total outflow from the element due to point and nonpoint withdrawals [m³/d]. Thus, the downstream outflow is simply the difference between inflow and source gains minus withdrawal losses.

Figure Element flow balance.
The total inflow from sources is computed as
()
where Q_ps,i,j is the jth point source inflow to element i [m3/d], psi = the total number of point sources to element i, Q_nps,i,j is the jth non-point source inflow to element i [m³/d], and npsi = the total number of non-point source inflows to element i.
The total outflow from withdrawals is computed as
()
where Q_pa,i,j is the jth point withdrawal outflow from element i [m³/d], pai = the total number of point withdrawals from element i, Q_npa,i,j is the jth non-point withdrawal outflow from element i [m³/d], and npai = the total number of non-point withdrawal flows from element i.
The non-point sources and withdrawals are modeled as line sources. As in Figure , the non-point source or withdrawal is demarcated by its starting and ending kilometer points. Its flow is then distributed to or from each element in a length-weighted fashion.

Figure The manner in which non-point source flow is distributed to an element.

2. Hydraulic Characteristics (Problem Of Final Exam)

Once the outflow for each element is computed, the depth and velocity are calculated in one of three ways: weirs, rating curves, and Manning equations. The program decides among these options in the following manner:

• 
  If weir height and width are entered, the weir option is implemented.
  
• 
  If the weir height and width are zero and rating curve coefficients are entered (a and ?), the rating curve option is implemented.
  
• 
  If neither of the previous conditions is met, Q2K uses the Manning equation.
  

1. Weirs

Figure shows how weirs are represented in Q2K. Note that a weir can only occur at the end of a reach consisting of a single element. The symbols shown in Figure are defined as: H_i = the depth of the element upstream of the weir [m], H_i+1 = the depth of the element downstream of the weir [m], elev2_i = the elevation above sea level of the tail end of the upstream element [m], elev1_i+1 = the elevation above sea level of the head end of the downstream element [m], H_w = the height of the weir above elev2_i [m], H_d = the drop between the elevation above sea level of the surface of element i and element i+1 [m], H_h = the head above the weir [m], B_w = the width of the weir [m]. Note that the width of the weir can differ from the width of the element, B_i.

Figure A sharp-crested weir occurring at the boundary between two reaches.
H : Height (??), D : Depth(??), elev : Elevation (??, ??? ??)

Hi : ??????(??), Hi+1 : ??????, Hd : ????? ??????

Hw : ??? ??, Hh : ????-????
For a sharp-crested weir where H_h/H_w < 0.4, flow is related to head by (Finnemore and Franzini 2002)
()
where Q_i is the outflow from the element upstream of the weir in m³/s, and B_w and H_h are in m. Equation (4) can be solved for
()
This result can then be used to compute the depth of element i,
()
and the drop over the weir
()
Note that this drop is used to compute oxygen and carbon dioxide gas transfer due to the weir (see pages 63 and 69).
The cross-sectional area, velocity, surface area and volume of element i can then be computed as
()
()


where B_i = the width of element i, ?x_i = the length of element i. Note that for reaches with weirs, the reach width must be entered. This value is entered in the column AA (labeled "Bottom Width") of the Reach Worksheet.

2. Rating Curves

Power equations (sometimes called Leopold-Maddox relationships) can be used to relate mean velocity and depth to flow for the elements in a reach,

()
()
where a, b, ? and ? are empirical coefficients that are determined from velocity-discharge and stage-discharge rating curves, respectively. The values of velocity and depth can then be employed to determine the cross-sectional area and width by
()
()
The surface area and volume of the element can then be computed as


The exponents b and ? typically take on values listed in Table . Note that the sum of b and ? must be less than or equal to 1. If this is not the case, the width will decrease with increasing flow. If their sum equals 1, the channel is rectangular.
Table Typical values for the exponents of rating curves used to determine velocity and depth from flow (Barnwell et al. 1989).

Equation	Exponent	Typical value	Range
		b	0.43	0.4?0.6
		b	0.45	0.3?0.5

In some applications, you might want to specify constant values of depth and velocity that do not vary with flow. This can be done by setting the exponents b and ? to zero and setting a equal to the desired velocity and ? equal to the desired depth.

3. Manning Equation

Each element in a particular reach can be idealized as a trapezoidal channel (Figure ). Under conditions of steady flow, the Manning equation can be used to express the relationship between flow and depth as

()
where Q = flow [m³/s]¹, S₀ = bottom slope [m/m], n = the Manning roughness coefficient, A_c = the cross-sectional area [m²], and P = the wetted perimeter [m].

Figure Trapezoidal channel.
The cross-sectional area of a trapezoidal channel is computed as
()
where B₀ = bottom width [m], s_s1 and s_s2 = the two side slopes as shown in Figure [m/m], and H = element depth [m].
The wetted perimeter is computed as
()
After substituting Eqs. (15) and (16), Eq. (14) can be solved iteratively for depth (Chapra and Canale 2006),
()
where k = 1, 2, …, n, where n = the number of iterations. An initial guess of H₀ = 0 is employed. The method is terminated when the estimated error falls below a specified value of 0.001%. The estimated error is calculated as
()
The cross-sectional area is determined with Eq. (15) and the velocity can then be determined from the continuity equation,
()
The average element width, B [m], is computed as
()

The top width, B₁ [m], is computed as

The surface area and volume of the element can then be computed as


Suggested values for the Manning coefficient are listed in Table . Manning’s n typically varies with flow and depth (Gordon et al. 1992). As the depth decreases at low flow, the relative roughness usually increases. Typical published values of Manning’s n, which range from about 0.015 for smooth channels to about 0.15 for rough natural channels, are representative of conditions when the flow is at the bankfull capacity (Rosgen, 1996). Critical conditions of depth for evaluating water quality are generally much less than bankfull depth, and the relative roughness may be much higher.
Table The Manning roughness coefficient for various open channel surfaces (from Chow et al. 1988).

MATERIAL	n
Man-made channels
Concrete		0.012
Gravel bottom with sides:
Concrete		0.020
mortared stone		0.023
Riprap		0.033
Natural stream channels
Clean, straight		0.025-0.04
Clean, winding and some weeds		0.03-0.05
Weeds and pools, winding		0.05
Mountain streams with boulders		0.04-0.10
Heavy brush, timber		0.05-0.20

4. Waterfalls

In Section 3.2.1, the drop of water over a weir was computed. This value is needed in order to compute the enhanced reaeration that occurs in such cases. In addition to weirs, such drops can also occur at waterfalls (Figure ). Note that waterfalls can only occur at the end of a reach.

Figure A waterfall occurring at the boundary between two reaches.
QUAL2K computes such drops for cases where the elevation above sea level drops abruptly at the boundary between two reaches. Equation (7) is used to compute the drop. It should be noted that the drop is only calculated when the elevation above sea level at the downstream end of a reach is greater than at the beginning of the next downstream reach; that is, elev2_i > elev1_i+1.

3.