adding very short write up for QB case biseg

Bidisperse Segregation 

====================== 

wdf TODO ASAP

In this simple test conducted at NETL, a uniform, random mixture 

of nylon 

(:math:`d_p = 3.19` mm, :math:`\rho_p = 1130` kg/m\ :sup:`3` \)

and ceramic 

(:math:`d_p = 4.25` mm, :math:`\rho_p = 2580` kg/m\ :sup:`3` \)

particles. When fluidized, the smaller, lighter nylon particles 

(:math:`U_{mf} ~ 1.1` m/s) segregate out of the mixture to the top, 

with the larger, heavier ceramic particles  (:math:`U_{mf} ~ 1.8` m/s)

remaining at the bottom. The batch segregation test was conducted in 

a small bed with a square cross-section of side length 60.325 mm. 

Similar to previous observations of fluidized segregation of 

bidisperse mixtures [GLMK03]_, the optimal separation was found to 

occur just above the larger of the two  minimum fluidization velocities. 

Below the larger :math:`U_{mf}`, lack of fluidization inhibits particle 

movement and too much above :math:`U_{mf}` vigorous bubbling promotes 

mixture in the bed. 

The optimal batch segregation experiment is simulated with MFiX-Exa 19.08

discretized onto a :math:`8 \times 24 \times 8` mesh. No-slip walls are 

set at vertical domain extents with a mass inflow and pressure outflow at

bottom and top, respectively. The :cpp:`Gidaspow` drag law is applied. 

A defluidization curve of all ceramic particles was traced to find 

:math:`U_{mf} \approx 1.9` m/s, close to the experimental value. The inflow 

velocity is set to `2.0` m/s. The image below shows the simulated segregation

(inset) occurs much more rapidly than observed experimentally. In this case, 

as the nylon particles begin to leave the mixture, the ceramic particles 

defluidize. In the final state, the ceramic particles are essentially static 

with a fluidized layer of nylon particles floating on top, in contrast to the 

experiment, in which the full bed showed signs of fluidization. Investigation 

into the over-segregation in this case is on-going. 

.. figure:: figs/netl_biseg_1908_small.png

:math:`d_p = 140` microns (:math:`dx^* \approx 1.8`) and density of 

:math:`\rho_p = 1050` kg/m\ :sup:`3` \. The resitution and sliding friction 

coefficients are set to 0.9 and 0.25, which are believed to be representative 

of the polystyrene material. The Wen and Yu drag law [WY66]_ is applied. 

1.12M particles are initially randomly distributed throughout the domain. 

The gravity force on the particles is modified by :math:`\tanh 100(t - 2)` 

so that the particles initially settle upwards, and then the body force is 

quickly inverted around :math:`t = 2` s, causing the particles to fall 

downward into the quiescent air. 

of the polystyrene material. The :cpp:`WenYu` drag law of Wen and Yu drag 

[WY66]_ is applied. 1.12M particles are initially randomly distributed 

throughout the domain. The gravity force on the particles is modified by 

:math:`\tanh 100(t - 2)` so that the particles initially settle upwards, 

and then the body force is quickly inverted around :math:`t = 2` s, 

causing the particles to fall downward into the quiescent air. 

.. figure:: figs/granRT_1908_small.png

larger system size: :math:`L^*_x = L^*_y = 256`. The system is thin in the 

depth dimension, :math:`L^*_z = 8` in order to highlight the clustering 

phenomena. Therefore, :math:`N_p = 50000`. Because the system is hypothetical, 

the ideal ``BVK2`` DNS drag law is applied, see [BvK07]_, [TPKKv15]_.

the ideal :cpp:`BVK2` DNS drag law is applied, see [BvK07]_, [TPKKv15]_.

.. figure:: figs/hcs_ke_1908.png

   Kuipers. Drag force of intermediate Reynolds number flow past mono- 

   and bidisperse arrays of spheres. *AIChE Journal*, **53**, 489--501 (2007).

.. [GLMK03] M.J.V. Goldschmidt, J.M. Link, S. Mellema, and J.A.M. Kuipers. 

   Digital image analysis measurements of bed expansion and segregation dynamics 

   in dense gas-fluidised beds. *Powder Technology*, **138** (2-3), 135--159 (2003).

.. [BPLPM19] C.M. Boyce, A. Penn, M. Lehnert, K.P. Pruessmann, and C.R. Müller. 

   Magnetic resonance imaging of single bubbles injected into incipiently 

   fluidized beds. *Chemical Engineering Science*, **200**, 147--166 (2019).

A separate defluidization simulation was first carried out first to determine

:math:`U_{mf} \approx 0.66` m/s using the `WenYu` [WY66]_ drag law, slightly 

:math:`U_{mf} \approx 0.66` m/s using the :cpp:`WenYu` [WY66]_ drag law, slightly 

below the experimentally measured value of :math:`U_{mf} = 0.7` m/s. The bed 

was prepared with two initialization simulations. First, the particle initial

condition is fluidized above :math:`U_{mf}` at :math:`0.8` m/s for one second