Page 474 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

15.2 Source Sampling 455

Fig. 15.9 Equal area method for a circular duct (Left equal area determination. Right sampling
point determination)

Sampling points are distributed over equal areas of the cross section of the stack or
duct.
A typical stack is circular. Consider a circular stack with an inner diameter of D,
2
its cross section area can be easily determined as A ¼ pD =4. If samples are to be
taken, for whatever reasons, from two equal areas as shown in Fig. 15.9, the area of
2
2
the inner circle (dashed line) is A=2. In this case, pD =4 ¼ A=2 ¼ pD =8 and
1
p ffiffiffi
D ¼ D= 2. However, the sampling points are NOT to be located on the thick
1
dashed circle defined by D 1 . Rather there will be eight sampling points evenly
distributed on two other circles indicated by the dotted lines defined by D s1 and D s2
in Fig. 15.9. These dotted lines divide the inner circle and the O-ring into equal
areas again. Then four evenly distributed sampling points on each dotted line are
determined finally.
A reader may have noticed that the total stack area was actually divided into four
equal ones by the dashed circle and two dotted circles. The ultimate goal of this
practice is to determine the sampling points, which is defined by the imaginary
sampling circles (dotted ones). A fast approach to this can bypass the step for
determining the diameters of the circles (dashed lines). For the above practice, the
stack is first divided into four instead of two equal areas, then the sampling points
are located every other circles starting from the innermost.
Despite the tedious work, one can determine the corresponding sampling points
for any stack sampling following the same principles introduce above. Consider a
general case where the stack with an inner diameter of D and an area of A is to be
divided into n equal annular areas (with diameters of D i ) and the corresponding
4n sampling points are determined as follows. In order to determine the sampling
points without calculating the n equal areas, treat it as if the total area were to be
divided into 2 n equal annular areas, then the diameter ith circle (D si) counting
from the inner-most outward is defined by

i
2 2
D ¼ D ð i ¼ 1; 3; ... 2n 1Þ ð15:14Þ
si
2n
On each sampling circle, there are four sampling points evenly distributed.