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

Sediment-Water Heat Transfer

A heat balance for bottom sediment underlying a water element i can be written as

()
where T_s,i = the temperature of the bottom sediment below element i [^oC], J_s,i = the sediment-water heat flux [cal/(cm² d)], ?_s = the density of the sediments [g/cm³], C_ps = the specific heat of the sediments [cal/(g ^oC)], and H_sed,i = the effective thickness of the sediment layer [cm].
The flux from the sediments to the water can be computed as
()
where ?_s = the sediment thermal diffusivity [cm²/s].
The thermal properties of some natural sediments along with its components are summarized in Table . Note that soft, gelatinous sediments found in the deposition zones of lakes are very porous and approach the values for water. Some very slow, impounded rivers may approach such a state. However, rivers tend to have coarser sediments with significant fractions of sands, gravels and stones. Upland streams can have bottoms that are dominated by boulders and rock substrates.
Table Thermal properties for natural sediments and the materials that comprise natural sediments.


Inspection of the component properties of Table suggests that the 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, we suggest a default value of 0.005 cm²/s for this quantity.
In addition, specific heat tends to decrease with density. Thus, the product of these two quantities tends to be more constant than the multiplicands. Nevertheless, it appears that the presence of solid material in stream sediments leads to a lower product than that for water or gelatinous lake sediments. In Q2K, we suggest default values of ?_s = 1.6 g/cm³ and C_ps = 0.4 cal/(g ^oC). This corresponds to a product of 0.64 cal/(cm^{3 o}C) for this quantity. Finally, as derived in Appendix C, the sediment thickness is set by default to 10 cm in order to capture the effect of the sediments on the diel heat budget for the water.

5. CONSTITUENT MODEL

1. Constituents and General Mass Balance

The model constituents are listed in Table .

Table Model state variables

Variable	Symbol	Units*
Conductivity		s	?mhos
Inorganic suspended solids		mi	mgD/L
Dissolved oxygen		o	mgO2/L
Slowly reacting CBOD		cs	mgO2/L
Fast reacting CBOD		cf	mgO2/L
Organic nitrogen		no	?gN/L
Ammonia nitrogen		na	?gN/L
Nitrate nitrogen		nn	?gN/L
Organic phosphorus		po	?gP/L
Inorganic phosphorus		pi	?gP/L
Phytoplankton		ap	?gA/L
Phytoplankton nitrogen		INp	?gN/L
Phytoplankton phosphorus		IPp	?gP/L
Detritus		mo	mgD/L
Pathogen		X	cfu/100 mL
Alkalinity		Alk	mgCaCO3/L
Total inorganic carbon		cT	mole/L
Bottom algae biomass		ab	mgA/m^2
Bottom algae nitrogen		INb	mgN/m^2
Bottom algae phosphorus		IPb	mgP/m^2
Constituent i
Constituent ii
Constituent iii

* mg/L ? g/m³; In addition, the terms D, C, N, P, and A refer to dry weight, carbon, nitrogen, phosphorus, and chlorophyll a, respectively. The term cfu stands for colony forming unit which is a measure of viable bacterial numbers.
For all but the bottom algae variables, a general mass balance for a constituent in an element is written as (Figure )
()
where W_i = the external loading of the constituent to element i [g/d or mg/d], and S_i = sources and sinks of the constituent due to reactions and mass transfer mechanisms [g/m³/d or mg/m³/d].

Figure Mass balance.
The external load is computed as (recall Eq. 2),
()
where c_ps,i,j is the jth point source concentration for element i [mg/L or ?g/L], and c_nps,i,j is the jth non-point source concentration for element i [mg/L or ?g/L].
For bottom algae, the transport and loading terms are omitted,
()
()
()
where S_b,i = sources and sinks of bottom algae biomass due to reactions [mgA/m²/d], S_bN,i = sources and sinks of bottom algae nitrogen due to reactions [mgN/m²/d], and S_bP,i = sources and sinks of bottom algae phosphorus due to reactions [mgP/m²/d].
The sources and sinks for the state variables are depicted in Figure (note that the internal levels of nitrogen and phosphorus in the bottom algae are not depicted). The mathematical representations of these processes are presented in the following sections.

Figure Model kinetics and mass transfer processes. The state variables are defined in Table . Kinetic processes are dissolution (ds), hydrolysis (h), oxidation (ox), nitrification (n), denitrification (dn), photosynthesis (p), respiration (r), excretion (e), death (d), respiration/excretion (rx). Mass transfer processes are reaeration (re), settling (s), sediment oxygen demand (SOD), sediment exchange (se), and sediment inorganic carbon flux (cf).

2.