# Linear Growth

McGraw (1997) has shown that a homogeneous growth problem with linear growth rate can be shown as

and in transformed form

For any initial distribution function * f* (<* 0), there is an exact analytical solution given by

Substituting Eq. (4.27) in Eq. (4.11) and combining with Eq. (4.8) gives the following set of ordinary differential moment equations that are closed regardless of the solution method (e.g., method of moments):

with an initial size distribution function in the finite domain [?_{min}(t), ?max(0L

**a**

And by scaling the distribution function with respect to - the dimensionless

2^{-+-}

form of the function becomes

Then, a set of equations resulting from Eq. (4.29) was implemented in ANSYS Fluent in the form of Eq. (4.24). The value of K used in this simulation was * 0.05* s

^{-1}, while - = 2 and - = 8. It is essential to multiply the source term by the flow density

*because of the format of the UDS equations in the solver:*

**p**_{p}

The equations were discretized using a second-order time discretization scheme with a fully implicit integration formula. The advantage of the fully implicit scheme is that it is unconditionally stable with respect to time-step size.

The simulation results are shown in Figs. 4.2 and 4.3. Figure 4.2 shows the initial dimensionless particle size distribution along with the comparison between the numerical simulation, obtained using the first eight moments * (i = 0-7),* and the exact solution at

*It shows that the numerical solution closely represents*

**t = 10 s.**Fig. 4.2 Particle size distribution at t = 10 s computed using eight moments (This figure was originally published in *Chemical Engineering Science* 102, 2013 and has been reused with permission)

**Fig. 4.3 **Particle size distribution at t *= 10 s* computed using ten moments (This figure was originally published in *Chemical Engineering Science* 102, 2013 and has been reused with permission)

the exact solution; however, it poses some negative values in the tails of the size distribution curve that are not physically possible. Figure 4.3 indicates that the final size distribution can be accurately predicted by increasing the number of moments from 8 to 10. The same conclusion has been made by Strumendo and Arastoopour (2008) as they showed the convergence of FCMOM for various particle growth processes.