# Solution

In this section, the CFD model of Case 2 is coupled with a FCMOM-based population balance equation (PBE) through the two-way coupling algorithm presented in Chap. 4 to account for the temporal and spatial evolution of the sorbent density distribution inside the reactor. In formulation of the PBE, for an inhomogeneous and univariate particulate system, with the particle density being the only internal coordinate, Eq. (4.1) becomes

In the above equation, it is assumed that the system is nondiffusive (i.e., no diffusion term).

Since, in the FCMOM approach, the solution of the PBE is sought in the finite interval between the minimum and maximum values [?_{min}, * ?_{max}]* of the particle density,

*and*

**?**_{min}*can be considered as the density of the fresh sorbent and the density of the sorbent with complete conversion, respectively.*

**?**_{max}The final form of the moments transport equations (Eq. 4.9) for this specific case becomes

The moments evolution equation, Eq. (5.19), must be coupled with the moving boundary conditions providing the governing equations for ?_{min}(f, x). There is also a source term due to the reaction, and ?_{min}(f, x) is convected because of the convective particle movement in the presence of spatially inhomogeneous conditions. The definition of the moving boundary conditions for ?_{min}(f, x) is based on the evaluation of (a) the spatial derivatives of ?_{min}(f, x) and (b) the velocity v_{p},_{min}(t, x) of the particles whose densities are equal to ?_{min}(f, x).

The moving boundary condition for ?_{min}(f, x) is then given by

where G is the growth rate of the particle density due to the reaction defined as (Abbasi et al. 2015)

<*2 is the density of the product layer (which is higher density), and * ^_{1}* is the density of the inner core fresh sorbent (which is lower density) as shown in Fig. 5.4. The partial derivative -§4 is provided by the heterogeneous reaction kinetic model between CO

_{2}and the sorbent based on a two-zone variable diffusivity shrinking core model, as described by Eq. (5.16).

The mean density value was calculated based on the ratio of the second moment of the distribution function (Fig. 5.8) to the first moment of the distribution function

**(И***1***/И***0***).**

**Fig. 5.9 **Contours of the time-averaged minimum and mean density of the solid phase at t *=* 20 s (This figure was originally published in *Powder Technol* 286, 2015 and has been reused with permission)

Figure 5.9 shows contours of the time-averaged minimum and mean solid density in the reactor. The minimum density contours of the solid provide valuable information about the location of the fresh sorbent front. Based on Eq. (5.20), the minimum density boundary (fresh sorbent front) moves with the reaction rate, and, because the reaction rate is significantly higher in the lower part of the reactor, the minimum density values rise faster in this region. However, the mean density values, calculated based on the moments of the distribution, are increasing along the height of the reactor.

Although in this specific case the change in the density of the sorbent is not significant and has a very small effect on the hydrodynamics of the system, it can become important in the cases where the sorbent has a significantly higher reaction rate or a higher residence time in a different reactor design.

In addition, the coupled CFD-PBE model is a valuable tool for design and optimization of the reactor and can be useful in the calculation of the regeneration rate and the rate of sorbent make-up to the system. Another application of such a model is studying the effect of particle size distribution on the reactor performance and in the cases in which particle attrition and particle breakage exist.