VPM Overview

[Lakshman]

Hello Robert,

On Wed, Mar 12, 2014 at 11:18 AM, Robert Helling <helling at atdotde.de> wrote:
>
> On 11.03.2014, at 15:17, Robert Helling <helling at atdotde.de> wrote:
>
>> I have more to say about the later parts of that web page (from “applying VPM to diving”). But have to rush now.
>
>
> Here is a continuation:
>
> Regardingt the derivation of eqs. (19)/(20): I have tried very hard but only got something somewhat close but distinctly different. I have the strong suspicion that there is typo in (18a), the cube just looks so wrong (not only unit wise). With a slightly modified (18) I believe i could obtain (19).
>
Equation (18) clearly is not a Laplace equation with the term p_f. By
removing p_f (currently shown as the coefficient to the surface
tension part) and using this along with Eqs. (11), (15), (17) and
another equation, i.e. instead of (14), for the case of p_m <= p^*, we
get a similar equation that looks like (11), that defines relation
between p^* and p_m, I could obtain Eq. (19). However I had to use p_f
= p_{in} - p_{amb}, I was so lost in the conventions they used after
going through their description, I will get back to the validity of
this (in case this is valid?).

In the same lines, Eq. (20) can be obtained using Eqs. (11), (14),
(15), (17) and corrected (18a).

This point must have been clear to us, however for the sake of
completeness I am adding the following point. As mentioned about p_m
<= p^*, p*v is no more constant, since "moles" of gas is no more
constant due to permeable nature, hence modified Eq. (14) is used to
get Eq. (19).

> I haven’t gotten around completely understanding the derivation the quadratic equation that leads to (40). But maybe I once again I haven’t tried hard enough.
>
Except Eq. (39), which I didn't spend time to derive, everything else,
they described from Eqs. (43) - (54) was clear in the derivation.
(Personally I didn't pen down equations myself for these steps).

> I also have to admit all the talking around the bubble regeneration with tau_R sounds fishy to me: given it the half-time is 20160min which happens to be two weeks. This is much longer than any timescale encountered in recreational (and recreation tech) diving.
>
I will get back to this point.

> Thus it could equally well taken to be infinite. In particular things like exp(-t/tau_r) where t is any diving relevant time, I am sure can be safely approximated by 1. Such a long timescale should not influence anything in diving. Making this simplification makes a lot of things shorter.
>
By making the approximation as you said, we are validating Eq. (15),
i.e. instantaneous growth from r_m to r_s. This brings us back to Eqs.
(19), (20), which were derived based on single inert gas
approximation. I need to make myself more clear with part again.

> Regarding “more intert gases”: what is said around (42) sounds to me to speak against the theory of different bubbles for different gases.
>
Please correct me if I am wrong. But I haven't seen this assumption in
this model, as far as I could recall, this theory was developed with
the considering one inert gas in the solution.


> Best
> Robert
>
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> Robert C. Helling     Elite Master Course Theoretical and Mathematical Physics
>                       Scientific Coordinator
>                       Ludwig Maximilians Universitaet Muenchen, Dept. Physik
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Thank you,
Lakshman