- 23,701
- 5,915
I would call it the net rate at which mass leavescasualguitar said:Got it
To describe the above equations I would say the following, where every term is per unit volume:
1) The rate of increase of mass plus the rate at which mass leaves in the ##x## direction ##= 0##
I would call it "rate of increase of enthalpy per unit volume" and "the net rate at which enthalpy per unit volume leaves the control volume in the x direction"casualguitar said:2) The rate of increase of energy plus the rate at which energy leaves in the ##x## direction = the rate of axial dispersion in the x direction plus the heat transfer between the fluid and solid
Yes.casualguitar said:(I notice we have arrived at the same equation I started the post with (!), just using enthalpy instead of temperature. Nice)Understood
Understood. I think I have seen some correlations for ##D## the axial dispersion coefficient but I guess this is of secondary importance for now.
The mass balance is straightforward, however I do have questions on the energy balance to clear up.
Points of confusion:
1) Are you using the term 'heat balance' and 'energy balance' interchangeably?
As we said before, it represents the net advection of enthalpy per unit volume out of the control volume.casualguitar said:2) Is this an advection term: ##\frac{\partial (\phi h)}{\partial x}##? I'm not familiar with advection, other than its definition as the transfer of heat through fluid flow, so I guess this is it
Like you said, it is a catch all for the net of all these effects. It's probably not going to be worthwhile trying to quantify it to a more detailed extent, and, in my judgment, it will be preferable to treat the dispersion length l as an adjustable parameter that the experimental results will be used to calibrate.casualguitar said:3) So the axial dispersion coefficient is a 'catch all' term for a number of energy transport processes as far as I know (diffusion and conduction within the fluid, convection by the fluid in the axial direction, axial and transverse mixing of the fluid, etc, as far as I know). What terms are included in this term in our case?
No. The heat transfer coefficient will depend on the fluid parameters, velocities, and bed geometry, so, in reality, it can be a function of location and time. I too will be difficult to quantify, and I recommend, at least initially, treating it as a constant and, like the dispersibity, calibrating it to the experimental results. (This is how we treated it in the lumped model) Later, if necessary, we can "sharpen our pencils."casualguitar said:4) The final term seems to be the convection between the fluid and the solid term. Why does this term not have a time or space derivative, because it will surely depend on space and time? Is it because 'time/space dependence' is built in, in the sense that ##T## and ##T_S## will change with time and space, meaning that the convection term will also change with time and space indirectly?
No. The solid energy balance is still used. We have just had the add the mass balance because the density is a function of x and t (which needs to be accounted for, since the density varies by a factor of 1000 from liquid to vapor).casualguitar said:Got it
So now instead of solving a fluid and solid energy balance, we will be solving the mass balance for the fluid, and the energy balance for the fluid/solid instead?