 ## Fluids

• julia.hartig
Subscriber

I'm struggling to understand the meaning behind the "algebraic" solution to the granular energy equation (i.e. neglecting convection and diffusion in the granular temperature PDE). The terms in the equation from the Fluent theory manual (top of attachment) don't seem to match the algebraic form of the granular temperature equation it references in the MFiX documentation (bottom of attachment). I tried plugging in and rearranging the forms of the collisional dissipation and diffusive granular energy terms from the Fluent manual into the simplified granular PDE, but I still can't come up with the K1m/K2m/etc. coefficients in the MFiX documentation.

Even if you could simply point me to the 2005 paper the MFiX documentation refers to, that would be helpful. I've tried a literature search for the derivation of this MFiX algebraic granular temperature and haven't found anything. I've also tried talking to several PhD students and postdocs who've worked with MFiX and they weren't familiar with the algebraic form of granular temperature either.

• DrAmine
Ansys Employee
No transport processes is the backbone of algebraic.
• julia.hartig
Subscriber

I guess I wasn't confused about the concept of algebraic granular temperature, but about how the coefficients from the MFiX documentation relate to the equation from the Fluent theory manual. I finally figured it out - I hadn't realized that you can rearrange and solve the (nonlinear) algebraic granular energy equation using u-substitution and the quadratic formula.

For anyone else who is curious/confused, this is what those MFiX coefficients actually mean:

K1m = coefficient for granular pressure (basically P_s/theta_s). The MFiX documentation ignores the kinetic term of granular pressure and just includes the collisional granular pressure coefficient.

K2m = coefficient for viscous energy generation from velocity divergence "Dmii" (i.e. (lambda_s - 2/3*mu_s)/sqrt(theta_s))
K3m = coefficient for viscous energy generation from velocity gradient "Dmij" (i.e. mu_s/sqrt(theta_s))
K4m = coefficient for collisional dissipation (i.e. gamma_theta/(theta_s)^(3/2))

So basically, K1m through K3m come from the "generation of energy by the solids stress tensor" and K4m comes from "the collisional dissipation of energy" in the Fluent theory manual. I wrote out a derivation but it's rather messy, if anyone reading this post wants some more details feel free to let me know and I can post them! 