To do that we need to define dimensions for the different moments (defined on the pureGrowth tutorial as: [0 -3 0 0 0 0 0], [0 -2 0 0 0 0 0],[0 -1 0 0 0 0 0] ... My question is what is the criteria for it?
Also, a similar vector is used in different instances like the growthModel properties:
minAbscissa minAbscissa [0 -2 0 0 0 0 0] 0.0;
maxAbscissa maxAbscissa [0 -2 0 0 0 0 0] 1.0;
Cg Cg [0 3 -1 0 0 0 0 ] 1.0;
What is the reason for using those particular values? Where can I find an explanation for that?
Post by Alberto Passalacqua on Aug 9, 2017 9:36:52 GMT -6
The unit of the moments become more clear if you consider the unit of the internal coordinate of your population balance. For example, if you use length as internal coordinate, then the unit should be meters, which would lead to [0 1 0 0 0 0 0] for the abscissa. Moments are calculated based on the quadrature as:
m_k = sum_i (weight_i * abscissa_i^k)
where k is the order of the moment, and weight_i is the number of particles per cubic meter (or some other way to express the number density: the code will work as long as units are consistent).
For the constant growth model example, the choice made in the tutorial is to consider Cg as the volumetric growth rate, which led to the units you see.
However, it is not clear to me how the different moments are set up. Is there a physical interpretation for them? How to decide how many moments to use?
Yes, moments have a physical meaning. Remember they are statistics of a number density function (or equivalent distribution). For example, if your number density is based on the particle length: n(L), then:
m0 = integral of the number density with respect to size = number of particles m1 = integral of (L * n(L)) dL = mean length ...
The number of moments you need to have accurate results depends on what information you need. For population balance problems, typically 3 to 4 primary quadrature nodes is the recommended setting, which corresponds to 6 - 8 moments for QMOM and 7 - 9 moments for EQMOM. Note that with EQMOM you are likely to need less moments than with QMOM. You may need more nodes if your number density has a particularly complex shape, or if you are interested in higher-order statistics. For example, if you need to know d43, you will need at least m4. If you use QMOM ("basic" quadrature in OpenQBMM), then m4 will be exactly preserved with 2 nodes.
To avoid an extended answer, a reference to where I can find these answers is also welcomed.
A good introduction and a lot of details of the methods we implement in OpenQBMM are in: D.L. Marchisio, R.O. Fox, Computational models for polydisperse particulate and multiphase systems, Cambridge University Press, Cambridge, 2013.
The implementation is however done differently (See references in header files).
Thanks again for the clear explanation regarding the moment interpretation.
Regarding the choice of how many moments to use for each method, is it really as you stated(?):
Note that with EQMOM you are likely to need less moments than with QMOM
It seems from the book chapter that it is rather # EQMOM moments > # QMOM moments:
6 - 8 moments for QMOM and 7 - 9 moments for EQMOM
More than that, I am having troubles testing the available tutorials (pbeFoam/aggregationBreakup). I focused on that in order to, later on, set-up simple a 1-D phase PBE-EQMOM using length as internal coordinate, perhaps using more than 4 nodes.
At this point, I can run the tutorial, but I do not find how to change the initial conditions (number of particles and respective size, especially when a population with different sizes is present) If I understood correctly, the output is saved on postProcessing folder for each moment and time point.
Again, to avoid an extensive answer, pointing to a tutorial explanation (if exists) will be equally welcomed.
Post by Alberto Passalacqua on Sept 9, 2017 19:33:37 GMT -6
You are comparing QMOM and EQMOM with a given number of primary quadrature nodes. Since EQMOM preserves more moments by definition, you may be good with a lower number of moments (= less primary nodes). It all depends on how many moments you really need to preserve for your problem.
In order to change the initial conditions, you need to set the moments of the number density function at time zero. The number of particles will be your m0, the mean size your m1/m0, and so on. This is all explained in the theory of moment methods, which is summarized in the first chapters of Marchisio and Fox (2013).
P.S. Sorry for the late reply. For some reason the forum did not notify me of your post.