Chamber Wall Loss Scoping Study

[Gorkowski]

Development of the particle wall-loss terms for a steel smog chamber.

Stainless Steel Box Length: 739.14 mm Width: 607.64 mm Height: 1663.7 mm

[ngmahfouz]

@Gorkowski, sorry I wasn't attentive to getting this done. Here's a quick summary. I will write some generic utilities to do these calculations soon (or likely whenever you push me to do so 😉)

Let's assume we only care about diffusive (both turbulent and random) and gravitational sedimentation for now. Letting length, width, and height be $L$, $W$, and $H$, then the wall loss coefficient, $\beta$, in this rectangular shape is generally of the form

$$ \beta_0 = (LWH)^{-1} (4H(L+W) \sqrt{k_t D}/\pi + v_g LW \coth{[(\pi v_g)/(4\sqrt{k_t D}})]) $$

where $k_t$ is some turbulent mixing parameter (corresponding to something like a fan speed), $D$ is the particle diffusion coefficient, $v_g$ is the terminal settling velocity, and $\coth$ is the the hyperbolic cotangent (which actually comes from contrasting the top and bottom surfaces --- the $v_g$ action).

This can be derived step by step if needed. It leaves us with only one uncertain parameter ($k_t$) which we will take to account for all flow stuff.

If we are interested in adding electric forces, which we could do relatively easily, we can manipulate the expression above by simply adding an electric terminal velocity, $v_e$. If we assume some uniformity (an electric field just near the wall of some sort, i.e., not an external one), we can just write it like so

$$ \beta_1 = \beta_0 + (LWH)^{-1} (2(LW+HL+HW) v_g (\exp{[(\pi v_g)/2\sqrt{k_t D}]})^{-1}) $$

The terminal velocities can be calculated as:

• The gravity terminal velocity can be calculated using Stokes law or something like that (simple)
• The electric terminal velocity is less common, but there's essential an equivalent of that Stokes law for electricity (replacing regular mobility with electric one, etc.)

In either case, there's really only one unknown parameter ($k_t$), but there are a few tunable parameters throughout (e.g., the assume averaged electric field, etc.).

(Note: for an external electric field, we can just treat it like we treat gravity --- I mean, it is essentially the same exact thing, meaning, we need to know something about the geometry/direction of the field. For gravity, we assume it is going down towards the center of earth, i.e., negative at the top and positive at the bottom surfaces)

[ngmahfouz]

I should add, the above $\beta$-like simplification is really mostly meant for a Eulerian representation (i.e., calculating how many particles we are expected to lose to wall loss at a time step). Of course, you cannot apply that for the Lagrangian stuff, and I think you don't need this for that (but correct me if I am wrong --- if you do, we can think about representing wall loss for Lagrangian processes as well)

[Gorkowski]

Right, I'm thinking of the Lagrangian processes as separate. Eulerian is better suited for the data interpretation or fitting done here.