McKay, Donald. "Front matter" Multimedia Environmental Models ...

8.1.2 Model-Building Strategies
The general model-building strategy is first to evaluate the system being simulated,
then to decide how many compartments and thus mass balances are required.
There is a compelling incentive to start with a simple model then build up complexity
only when justified. The volumes and bulk Z values are deduced for each compartment.
All inputs and outputs are identified, preferably as arrows on a mass balance
diagram. Equations are written for each flux, either as an emission or a Df product.
For a steady-state model, the inputs are equated to outputs for each of the n compartments,
leading to n equations with n unknown fugacities. These equations are
solved, either algebraically or using a matrix method.
For a dynamic system, the differential equations for each medium are written in
the form
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d(VZf)/dt = inputs – outputs mol/h
These equations are then solved, either analytically or numerically, for a defined
initial condition and defined inputs. The integration time step can be selected as 5%
of the shortest half-time for transport or transformation and the stability of the result
checked by decreasing the time step systematically. Integration is best done using
a Runge–Kutta method, but the simple Euler’s method may be adequate.
Results should be checked for a mass balance. For steady-state models, this is
simply a comparison of inputs and outputs for each compartment. For dynamic
models, the initial mass plus the cumulative inputs should equal the final mass and
the cumulative outputs. To gain a pictorial appreciation of the results, a mass balance
diagram should be drawn listing all inputs and outputs beside the appropriate arrows.
The dominant processes then become apparent.
It is often useful to play “sensitivity games” with the model to gain an appreciation
of how variation in an input quantity such as an emission rate, Z value, or D
value propagates through the calculation and affects the results. An input quantity
can be increased by 1% and the effect on the desired output quantity determined.
The best way to quantify this sensitivity S is to deduce
S = (Doutput/output)/( Dinput/input)
For example, if the input quantity of 100 is increased to 101 and the output
quantity changes from 1000 to 1005, then S is (5/1000)/(1/100) or 0.5. This is
actually an estimate of the partial derivative of log output with respect to log input,
and it is dimensionless. For linear systems of the types treated here, all values of S
should be less than 1.0. It may be useful to list the input parameters, deduce S for
each, then rank them in order of decreasing S. The most sensitive parameters, for
which the most accurate data are necessary, are then identified. Often, the sensitivity
of a parameter is surprisingly small, and only a rough estimate is needed. It may be
desirable to revisit and improve the accuracy of the estimates of most sensitive
parameters.