Reacting Channel Coupling

The Reacting Channel Coupling feature couples a 3D Simcenter STAR-CCM+ simulation with the 1D Plug Flow Reactor which uses the CVODE solver.

Reacting Channel Plug Flow Reactor Formulation

The flow in the reacting channel is simulated using the one-dimensional Plug Flow Reactor (PFR). At any axial point within the reacting channel, the flow variables are radially uniform. For a given reaction $k$ , the axial diffusion of heat and species are consistent and are therefore ignored. The evolution of species and temperatures in the PFR are represented as follows:

Species:
Figure 1. EQUATION_DISPLAY

$\frac{d Y_{k}}{d t} = \frac{{\dot{ω}}_{k}}{ρ}$
(3810)
where $Y_{k}$ is the mass fraction of all species in reaction $k$ and $ρ$ is the density.
Temperature:
Figure 2. EQUATION_DISPLAY

$\frac{d T}{d t} = \frac{\sum_{k} {\dot{ω} H}_{k} + {\dot{Q}}_{C}}{ρ C_{p}}$
(3811)

where $H_{k}$ is the total enthalpy of reaction $k$ and $C_{p}$ is the specific heat. In the reacting channel, the convective heat transfer source, ${\dot{Q}}_{C}$ , is determined using:

Figure 3. EQUATION_DISPLAY

{\dot{Q}}_{C} = h_{f} (T_{W} - T)

(3812)

$T_{W}$ , the reacting channel wall temperature, is averaged from the 3D outer flow temperature field on the resolved reacting channel wall and $T$ is the reacting channel bulk temperature.

The convective heat transfer coefficient,

h_{f}

, of the reacting channel is determined depending on the type of heat transfer correlation that is specified:

For open pipes which use the Gnielinski heat transfer correlation:
Figure 4. EQUATION_DISPLAY

$h_{f} = \frac{N u K_{C}}{D_{C}}$
(3813)

$N u$ is the Nusselt number which is determined according to fully developed pipe flow empirical correlations:
- For laminar flow where Re < 3000:
  Figure 5. EQUATION_DISPLAY
  
  $N u = 3.66$
  (3814)
- For turbulent flow where Re > 3000, $N u$ is determined using Gnielinski’s correlation for turbulent flow:
  Figure 6. EQUATION_DISPLAY
  
  $N u = \frac{(f / 2) (Re - 1000) \Pr}{1 + 12.7 \sqrt{(f / 2)} (\Pr^{2 / 3} - 1)}$
  (3815)
Where $\Pr$ is the Prandtl number which is determined by:

Figure 7. EQUATION_DISPLAY

$\Pr = \frac{μ C_{p}}{K_{c}}$
(3816)
and $f$ is the Fanning friction factor.
For packed bed pipes that use the Leva / Grummer heat transfer correlation:
Figure 8. EQUATION_DISPLAY

$h_{f} = f_{h t g} 0.813 \frac{K_{C}}{2 r_{i n}} \exp (- 3 \frac{D_{p}}{r_{i n}}) {Re}^{0.9}$
(3817)
where $D_{p}$ is the equivalent packed bed particle diameter, $f_{h t g}$ is the heat transfer factor that is defined, and
Figure 9. EQUATION_DISPLAY

$Re = \frac{D_{p} G_{s}}{μ}$
(3818)
For packed bed pipes that use the Beek heat transfer correlation:
Figure 10. EQUATION_DISPLAY

$h_{f} = f_{h t g} \frac{K_{C}}{D_{p}} (2.58 {Re}^{1 / 3} \Pr^{1 / 3} + 0.094 {Re}^{4 / 5} \Pr^{0.4})$
(3819)

Where $f_{h t g}$ is the heat transfer factor that is defined and the Prandtl number, $\Pr$ , is determined by Eqn. (3816).
For packed bed pipes that use the De Wasch / Froment heat transfer correlation:
Figure 11. EQUATION_DISPLAY

$h_{f} = \frac{K_{C}}{D_{p}} (h_{f}^{0} + 0.033 \Pr Re)$
(3820)

For smooth, cylindrical, reacting channels, the pipe friction factor

f

, is determined depending on the type of friction factor correlation that is specified:

For open pipes that use the Blasius friction factor correlation:
Figure 12. EQUATION_DISPLAY

$f = 0.0791 {Re}^{- 0.25}$
(3821)
For open pipes that use the Filonenko friction factor correlation:
Figure 13. EQUATION_DISPLAY

$f = \frac{0.25}{{(0.79 \log (Re) - 1.64)}^{2}}$
(3822)
For packed bed pipes that use the Hicks friction factor correlation:
Figure 14. EQUATION_DISPLAY

$f = 6.8 \frac{{(1 - χ)}^{1.2}}{χ^{3}} {Re}^{- 0.2}$
(3823)
For packed bed pipes that use the Ergun friction factor correlation:
Figure 15. EQUATION_DISPLAY

$f = \frac{(1 - χ)}{χ} [1.75 + \frac{150 (1 - χ)}{Re}]$
(3824)

The equations in the PFR are solved using the stiff CVODE Ordinary Differential Equation (ODE) solver with time-steps that are based on the grid size of the reacting channel and the local reacting channel velocity.

The pressure drop is calculated as:

Figure 16. EQUATION_DISPLAY

\frac{\partial (ρ v^{2} + p)}{\partial t} = - ρ v^{3} \frac{f}{D}

(3825)

where

f

is the friction factor and

D

is the characteristic length of the system. The characteristic length of the system is either:

for open pipes, the hydraulic radius $\frac{D_{H}}{2}$
for packed beds, the packed bed representative particle diameter

Outer Flow STAR-CCM+ Formulation

The outer flow, which you can define as reacting or non-reacting, is simulated using three-dimensional Simcenter STAR-CCM+. You can mesh the three-dimensional outer flow geometry as long as the geometrical channels within the geometry that represent the reacting channel remain unmeshed.

The heat flux boundary condition at the reacting channel wall, ${\dot{Q}}_{3 D}$ , is calculated as an under-relaxation of the heat that the outer flow gains from the reacting channel. This heat flux is determined using:

Figure 17. EQUATION_DISPLAY

{\dot{Q}}_{3 D} = - a {\dot{Q}}_{C} - (1 - a) {\dot{Q}}_{C}^{0}

(3826)

When the solution for the simulation converges, the heat flux from the reacting channel is equal to the flux gain from the outer flow.