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Discretization of mass conservation equation in a 2D staggered grid 

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April 15, 2016, 15:34 
Discretization of mass conservation equation in a 2D staggered grid

#1 
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kiwi
Join Date: Apr 2016
Posts: 3
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Hi everybody,
As mentioned in the title I would like to solve the mass conservation to simulate the boiling phenomenon into a tube bundle. I am facing some issues to discretize these equations into a staggered grid after to have applied an integration into a finite volume. Accordingly to the classical staggered grid: the flow properties are content in the control volume whereas the velocity information is on the bondaries of this control volume. Here is a representation of the staggered grid I use (for the notations) My governing equation is: Where the subcript k is for the phase type (1 for liquid; 2 for gas) and are the evaporation and condensation rate defined as: Where H" is the staturated enthalpy of the gas, H' is the saturated enthalpy of the liquid and is the relaxation time. First question: does someone have a definition to calculate that relaxation time? After having integrated the mass conservation equation for the control volume I;J, I get: By making the assumption that the velocity field is known, I can resolve this equation and find the void fraction and the density. My issue arrives now: I don't know how to approximate the terms of void fraction and density which are at the bondaries of the control volume because the information is not stocked there (ex: ) If someone can help me on this I will be very grateful. In advance, thank you. Kiwi 

April 15, 2016, 17:35 

#2 
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Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,292
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your grid arrangement is not totally clear to me from the picture...
I immagine that rho(i,j), u(i,j), v(i,j) are staggered each other, right? For example, if rho(i,j) is at the position x(i),y(j), then u(i,j) is at x(i)+dx/2,y(j) and v(i,j) is at x(i),y(j)+dy/2? If it is true, the integration for rho is for the CV centred at x(i),y(j). 

April 16, 2016, 07:26 

#3 
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kiwi
Join Date: Apr 2016
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Hi FMdenaro and thank you for your answer.
Yes you are right; if we use the notation (i,j) (i+1/2) and (j+1/2) the equation can consequenlty be written as: Is it more clear for you with that writing? So you mean that for the terms I can make the approximation ? So the equation would become: Is that correct ? I read also that we can do this approximation with the UPWIND scheme but I am not confident to use this scheme because it is based on the value of the velocity; for example: if OR if But I have also make the test for and so the fill in of my matrix coefficients become very tough ... What is the best option? 

April 16, 2016, 14:32 

#4  
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Filippo Maria Denaro
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Quote:
not at all ... you have a convective flux at one section it can not be discretized as first derivative! You can use zerothorder polynomial in a first order upwind reconstruction or a linear polynomial for a second order reconstruction 

April 16, 2016, 15:57 

#5 
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kiwi
Join Date: Apr 2016
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Understood I can't use a first derivative solution.
But I am not sure to fully understand what you mean by "linear polynomial for a second order reconstruction". Could you give me an example ? Thank you. 

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