322
Chapter 8
y / { X ) x
= J
m f (m , X m ) d m
(8.8)
0
Thus, flow cytometry measurements may also be used to obtain the distribution function based on
mass fraction.
The population balance in Eq. (8.2) must satisfy an initial condition describing the state of the
population, i.e.,
J(
y,0) should be known. Generally, this is sufficient since the model normally
satisfies consistency criteria, i.e., that the flux is zero at the boundaries of the property space. On
some occasions, boundary conditions do enter the analysis, but this depends on how the problem is
formulated. The solution to the dynamic balance Eq. (8.2) can be found by one of several weighted
residual methods,
as has been illustrated for a one-dimensional distribution function by
Subramanian and Ramkrishna (1971) [see also Ramkrishna (1985) for a discussion of various
solution methods for the dynamic population balance].
In many situations it is, however, sufficient to obtain qualities pertaining to the average
composition and the standard deviation for the population. These can be calculated from the
moments of the distribution functions, where the nth moment of a one-dimensional distribution
function is defined by
The zeroth moment is equal to the total number of cells per unit volume [see Eq. (8.1)], and from
the first moment the average cell composition can be calculated:
we find a relationship between the variance and the second moment of the distribution function:
(
8
.
9
)
(8.10)
From the definition of the variance
° 2
of the distribution function,
(
8
.
11
)
« (0
n { t)
(8.12)
The definitions of the moments can easily be extended to the case of a multidimensional
distribution function. Thus, for a two-dimensional property space (see also Example 8.4),