How Can Unreacting Flow Dynamics Be Modeled in a Packed Bed Reactor?

[niketas]

Hello all!

This is a great site, congratulations!

I think I need help for proceeding with my reaction engineering project.

Now the problem is this: I have a packed bed reactor whose dynamic modelling I am supposed to complete. At t = 0, there is still air in the reactor with no net flow (Initial condition). When t > 0, I let air flow into the reactor with some initial velocity, which will, at certain intervals, will be increased (Time-dependent boundary condition; will be fixed after a certain while though). The unreacting flow in the reactor will reach steady state. Then I will let the reactive mixture into the reactor. This second stage I have no problem dealing with, i.e., reactive flow.

I am concerned with the initial unreacting flow. Since my fluid dynamics background is not strong, I needed your consultance. When the forced flow meets the still air residing in the reactor, what kind of modelling can I use? Can it be classified as a Riemann problem that moves downstream the reactor until there happens to be a moving chunk fluid at the lower end (downstream)?

Any other modelling difficulties, singularities that can arise when the moving fluid "bounces" into the stationary fluid? Shocks, instabilities?

Normally when I work with reactive flow, I do not take into account gravitational forces, that is, whether the reactor is horizontal or vertical. I suppose gravity comes into play in the unreacting flow case.

I would be grateful upon your suggestion of what model I should use, numerical method that needs to employed and any kind of tips.

If this is not the forum the thread should have been started, please feel free to move it to the appropriate forum.

Thank you for taking your time.

Best regards...

Niketas Konstantinidis
National Technical University
School of Chemical Engineering
Athens, Greece

 

[niketas]

Follow-up post

I forgot to mention...

This is supposed to be a plug flow reactor.

In view of the information I gave in the previous post, is it in any way possible that the steady-state flow will be of the plug-flow type?

And I do not, at this stage, consider diffusion. Only convection with unreacting flow, and convection + reaction with reacting flow.

I take the flowing and stationary fluid temperatures equal for the sake of simplicity. But I guess an energy balance still has to be included for predicting the spatial and temporal evolution of temperature, right?

 

[Hawknc]

I'll admit to knowing pretty much nothing about reactors, but it looks to be a reasonably standard scenario for air moving in and out of a cavity. What happens in the reactor in terms of outflow? Is there any? If not, then the air flowing into the reactor will simply increase the pressure and temperature (ideal gas law), but be warned that this WON'T reach a steady state - it's going to keep increasing. If you have air leaving the reactor (which I guess is the more likely case), then provided that your mass flow in = mass flow out, you'll reach steady state conditions.

As the air enters the reactor, you'll probably get turbulence (unless it's reaaaally slow) and thus good mixture. Shocks are highly unlikely unless it's a very fast flow (in the order of 340 m/s). Don't think of fluids as a bunch of particles, bouncing off each other; fluids mix, they swirl, they tumble, they have cohesion and viscosity.

Now I await the PF oldbies to come in and rip my post to pieces for my numerous inaccuracies.

PS. I don't know anything about plug-flow reactors aside from what I just read on Wikipedia, but the assumptions that both make are largely the same. What I posted doesn't take into account chemical reactions, but for the most part the fluid mechanics would probably be the same.

 

[niketas]

I'll admit to knowing pretty much nothing about reactors, but it looks to be a reasonably standard scenario for air moving in and out of a cavity. What happens in the reactor in terms of outflow? Is there any? If not, then the air flowing into the reactor will simply increase the pressure and temperature (ideal gas law), but be warned that this WON'T reach a steady state - it's going to keep increasing. If you have air leaving the reactor (which I guess is the more likely case), then provided that your mass flow in = mass flow out, you'll reach steady state conditions.

As the air enters the reactor, you'll probably get turbulence (unless it's reaaaally slow) and thus good mixture. Shocks are highly unlikely unless it's a very fast flow (in the order of 340 m/s). Don't think of fluids as a bunch of particles, bouncing off each other; fluids mix, they swirl, they tumble, they have cohesion and viscosity.

Efcharisto (Thank you!) Hawknc :)

In the reactor, what flows in - in terms of total mass - flows out of the reactor, but with different composition. Yes, there is outflow.

I am interested in how I can model the evolution of air flow to steady state. Standard scenario. :)

Turbulence is likely because the flow is fast but not as fast so to cause a shock (I got your point). I still wonder if the phenomenon taking place at the step change (v = 0 ---> v > 0; IC to BC) can be modeled as a discontinuity.

When I try to use the continuity equation (partial_C_wrt_time + velocity x partial_C_wrt_z = 0)*, the solution blows up upon inserting the boundary condition since the initial velocity is taken to be zero. Everything is considered to continuous. Is this an erroneous assumption or is it the numerical scheme I am using (Lax-Friedrichs/Wendroff) that is insufficient?

Could I make myself clear?

Thank you again. Have a nice day...

*C: Concentration
v: Velocity of moving fluid (gas)

 

[Hawknc]

Hmmm...modelling fluid flow isn't easy. The Navier-Stokes equations are what are usually used, but for any reasonable degree of accuracy they're sufficiently complex that it's easier to get a computer to model it for you. Transient flow (which is what I'm guessing you're interested in) isn't something I know a huge amount about, unfortunately, but most work I've seen in the area uses computational fluids packages to work out pressure, temperature etc. at specific locations within the fluid flow. It's certainly a complex problem to solve if you want to do it analytically.

 

[niketas]

Hmmm...modelling fluid flow isn't easy. The Navier-Stokes equations are what are usually used, but for any reasonable degree of accuracy they're sufficiently complex that it's easier to get a computer to model it for you. Transient flow (which is what I'm guessing you're interested in) isn't something I know a huge amount about, unfortunately, but most work I've seen in the area uses computational fluids packages to work out pressure, temperature etc. at specific locations within the fluid flow. It's certainly a complex problem to solve if you want to do it analytically.

Transient flow modeling is what I want. Exactly.

But I am definitely not pursuing an analytical solution.

Have a good day. Thanks...

 

[Clausius2]

An oldbie:

Are you using a 2 dimensional model or one dimensional? I mean, how is the geometry of your problem? Are you modelling with small variation of cross area?. Anyway, imagining the most simple geometry and assuming unidirectional flow towards the exit inside the reactor, your model should be well posed with the unsteady Euler equations in compressible flow:

$$\frac{\partial \rho}{\partial t}+\frac{\partial (\rho u)}{\partial x}=0$$ (CONTINUITY)

$$\frac{\partial \rho u}{\partial t}+\frac{\partial \rho u^2}{\partial x}=-\frac{\partial P}{\partial x}$$ (MOMENTUM)

$$P/\rho^{\gamma}=const$$ (ENERGY) assuming isentropic flow.

where the initial condition is given by $$u=0$$ everywhere and the boundary condition modelling the sudden inflow is $$u(0,t)=H(t)$$ where H(t) is the step function. This system of equation is elliptic in space and parabolic in time (so you have to pose a right boundary condition), does not give you any difussion (which I particularly think should be important in the transmission of the motion to the still air if the incoming velocity scale is not very big compared with the viscous velocity scale $$\nu/a$$), and of course no turbulence effect is allowed (it makes no sense to talk about one dimensional turbulence) as the flow is considered inviscid. In order to start to integrate the problem, specify the initial velocity and pressure field, make spatial derivatives, and then advance in time. Of course your code may blow up, and the reason is the step gradient you pose in the left boundary condition. Your scheme may be sensible and gets unstable because of such huge gradient. In order to stabilize your code add a little amount of artificial viscosity in order to flatten the gradient of velocity of the incoming flow.

 

[niketas]

Thank you so much Clausius!

This is exactly what I was looking for. I have cylindrical geometry but am using a 1-D model which is governed by the equations you have just presented.

I can now see to it which combination of numerical methods I should use.

Farewell

Niketas

 

[Clausius2]

No problemo.

I'm pretty sure that your problem has an exact analytical solution. If you think about it it has very much to do with a propagation of a characteristic line. Moreover, if you neglect the convective term assuming that the change on velocities is not very large, your equations belong to the theory of Linear Acoustics. Furthermore, if you manipulate both equations you will arrive to a Wave Equation for the density and another one for the velocity, saying that both of them are propagated by characteristic lines. In order to solve the problem, you have to take care of the reflections of the characteristics on each extreme of the domain. Now I look carefully to the problem, I think the standing air is set into motion in times of order L/c, being c the speed of sound. The convection plays little role in the initial stage, and the compressibility is negligible if the velocity imparted is small compared with the speed of sound ($$M<<1$$.