ISCST3 Tech Guide
6.2.3 The Short-Term Area Source Model
The ISC Short Term area source model is based on a numerical integration over the area in the upwind and crosswind directions of the Gaussian point source plume formula given in Equation (1-1). Individual area sources may be represented as rectangles with aspect ratios (length/width) of up to 10 to 1. In addition, the rectangles may be rotated relative to a north-south and east-west orientation. As shown by Figure 1-9, the effects of an irregularly shaped area can be simulated by dividing the area source into multiple areas. Note that the size and shape of the individual area sources in Figure 1-9 varies; the only requirement is that each area source must be a rectangle. As a result, an irregular area source can be represented by a smaller number of area sources than if each area had to be a square shape. Because of the flexibility in specifying elongated area sources with the Short Term model, up to an aspect ratio of about 10 to 1, the ISCST area source algorithm may also be useful for modeling certain types of line sources.
The ground-levelconcentration at a receptor located downwind of all or a portion of the source area is given by a double integral in the upwind (x) and crosswind (y) directions as:
QA = area source emission
rate (mass per unit area per unit time)
K = units scaling coefficient (Equation (1-1))
V = vertical term (see Section 1.1.6)
D = decay term as a function of x (see Section 1.1.7)
The Vertical Term is given by Equation (1-50) or Equation (1-54) with the effective emission height, he, being the physical release height assigned by the user. In general, he should be set equal to the physical height of the source of emissions above local terrain height. For example, the emission height he of a slag dump is the physical height of the slag dump.
Since the ISCST algorithm estimates the integral over the area upwind of the receptor location, receptors may be located within the area itself, downwind of the area, or adjacent to the area. However, since Fz goes to 0 as the downwind distance goes to 0 (see Section 126.96.36.199), the plume function is infinite for a downwind receptor distance of 0. To avoid this singularity in evaluating the plume function, the model arbitrarily sets the plume function to 0 when the receptor distance is less than 1 meter. As a result, the area source algorithm will not provide reliable results for receptors located within or adjacent to very small areas, with dimensions on the order of a few meters across. In these cases, the receptor should be placed at least 1 meter outside of the area.
In Equation (1-65), the integral in the lateral (i.e., crosswind or y) direction is solved analytically as follows:
where erfc is the complementary error function.
In Equation (1-65), the integral in the longitudinal (i.e., upwind or x) direction is approximated using numerical methods based on Press, et al (1986). Specifically, the ISCST model estimates the value of the integral, I, as a weighted average of previous estimates, using a scaled down extrapolation as follows:
where the integral term refers to the integral of the plume function in the upwind direction, and IN and I2N refer to successive estimates of the integral using a trapezoidal approximation with N intervals and 2N intervals. The number of intervals is doubled on successive trapezoidal estimates of the integral. The ISCST model also performs a Romberg integration by treating the sequence Ik as a polynomial in k. The Romberg integration technique is described in detail in Section 4.3 of Press, et al (1986). The ISCST model uses a set of three criteria to determine whether the process of integrating in the upwind direction has "converged." The calculation process will be considered to have converged, and the most recent estimate of the integral used, if any of the following conditions is true:
1) if the number of "halving intervals" (N) in the trapezoidal approximation of the integral has reached 10, where the number of individual elements in the approximation is given by 1 + 2N-1 = 513 for N of 10;
2) if the extrapolated estimate of the real integral (Romberg approximation) has converged to within a tolerance of 0.0001 (i.e., 0.01 percent), and at least 4 halving intervals have been completed; or
3) if the extrapolated estimate of the real integral is less than 1.0E-10, and at least 4 halving intervals have been completed.
The first condition essentially puts a time limit on the integration process, the second condition checks for the accuracy of the estimate of the integral, and the third condition places a lower threshold limit on the value of the integral. The result of these numerical methods is an estimate of the full integral that is essentially equivalent to, but much more efficient than, the method of estimating the integral as a series of line sources, such as the method used by the PAL 2.0 model (Petersen and Rumsey, 1987).