Time rate of change of custom rate laws

[TBush]

Hi,

We have been working with specifying custom rate laws for microbial respiration in React, and have been following the example specified on Page 67 of the reaction modelling guide.

 

This integrates the differential equation for the time rate of change of the biomass so that the biomass is then updated according to:

rpave = (1.0-Theta)*rprime0 + Theta*rprime

biom = biomass0*exp((growth_yield*rpave-decay_con)*Deltat)

We were wondering about the numerical justification for this. It seems like it assumes that the chemical concentrations change slower over time than the biomass does, because the concentration appears in "rpave" (actually in rprime). Basically this integral seems to have ignored fact that the chemical concentration is a function of time, which would be fine if the chemical concentration were varying in time much slower than the biomass, but we were just wondering about the

exact numerical justification for this.

Does anyone know about the numerical justification for this integral?

Also we are a bit confused by the line rpave = (1.0-Theta)*rprime0 + Theta*rprime and what the variable Theta actually does. So if someone could shed some light on this question as well then that would be great.

Thanks very much.

Regards,

Tim Bush

[Brian Farrell]

Hi Tim,

As you say, the rate of reaction depends on the chemical concentration, which changes with time. React can account for this by figuring a weighted average (rpave) of the reaction rate at the beginning (rprime0) and end (rprime) of the time step. Theta, the time-weighting variable, determines how the average is calculated. A value of 0 would cause React to use the rate from the beginning of the step, a value of 1 would cause React to use the rate at the end of the step, and a value of 0.5 would equally weight the rates. The default used by the program is 0.6, but you can set theta to any number between 0 and 1. For more information, please see:

6.104 in the GWB Reference Manual

4.1 (Setting kinetic reactions) in the GWB Reaction Modeling Guide

16.3 (Numerical solution) in the Geochemical and Biogeochemical Reaction Modeling text

Hope this helps,

Brian Farrell

Aqueous Solutions