APPENDIX C: SEDIMENT-WATER HEAT EXCHANGE

10 APPENDIX C: SEDIMENT-WATER HEAT EXCHANGE

Although the omission of sediment-water heat exchange is usually justified for deeper systems, it can have a significant impact on the heat balance for shallower streams. Consequently, sediment- water heat exchange is included in QUAL2K.

A major impediment to its inclusion is that incorporating sediment heat transfer often carries

a heavy computational burden. This is because the sediments are usually represented as a vertically segmented distributed system. Thus, inclusion of the mechanism results in the addition of numerous sediment segments for each overlying water reach.

In the present appendix, I derive a computationally-efficient lumped approach that yields comparable results to the distributed methods.

The conduction equation is typically used to simulate the vertical temperature distribution in

a distributed sediment (Figure 56a)

This model can be subjected to the following boundary conditions:

T ( 0 , t ) = T + T a cos [ ω ( t − φ ) ]

2 − where T = sediment temperature [ 1 C], t = time [s], α = sediment thermal diffusivity [m s ], and z = depth into the sediments [m], where z = 0 at the sediment-water interface and z increases

downward, o T = mean temperature of overlying water [ C], T

a = amplitude of temperature of

overlying water [ 1 C], ω = frequency [s − ]=2 π /T p ,T p = period [s], and φ = phase lag [s]. The first boundary condition specifies a sinusoidal Dirichlet boundary condition at the sediment-water interface. The second specifies a constant temperature at infinite depth. Note that the mean of the surface sinusoid and the lower fixed temperature are identical.

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(a) distributed

(b) Lumped

Figure 56. Alternate representations of sediments: (a) distributed and (b) lumped.

Applying these boundary conditions, Equation (177) can be solved for (Carslaw and Jaeger 1959)

a e cos [ ω ( t − φ ) − ω z ' ] (208)

− where 1 ω ’ [m ] is defined as

The heat flux at the sediment water interface can then be determined by substituting the derivative of Equation (178) into Fourier’s law and evaluating the result at the sediment-water interface (z = 0) to yield

J ( 0 , t ) = ρ C p ωα T a cos [ ω ( t − φ ) + π / 4 ] (210)

where J(0, t) = flux [W/m 2 ].

An alternative approach can be developed using a first-order lumped model (Figure 56b),

dT s α s ρ s C ps

[ T + T a cos [ ω ( t − φ ) ] − T s ]

H s ρ s C ps =

dt

where H 3 sed = the thickness of the sediment layer [m], ρ s = sediment density [kg/m ], and C ps =

− sediment heat capacity [joule (kg 1 C)] ]. Collecting terms gives,

dT +

k h T = k h T + k h T a cos [ ω ( t − φ ) ]

dt

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2 H sed

After initial transient have died out, this solution to this equation is

T a cos [ ω ( t − φ ) − tan ( ω / k h ) ] 2 (211)

which can be used to determine the flux as

cos [ ( t − φ )

C T  cos ( t − = 1 ρ

− tan ( ω / k ) 

(212)

H sed   

It can be shown that Equations (180) and (182) yield identical results if the depth of the single layer is set at

1 H sed =

ω (213) '

Water quality models typically consider annual, weekly and diurnal variations. Using α = 0.0035 cm 2 /s (Hutchinson 1957), the single-layer depth that would capture these frequencies can be

calculated as 2.2 m, 30 cm and 12 cm, respectively.

Because QUAL2K resolves diel variations, a value on the order of 12 cm should be selected for the sediment thickness. We have chosen of value of 10 cm as being an adequate first estimate because of the uncertainties of the river sediment thermal properties (Table 4).

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