relationship between Visibility and Optical Depth

[CK Lam]

Hi,

I would like to know the relationship between visibility and optical
depth.

According to FDS User's guide, Visibility (S) can be found from
dividing C (the non-dimensional constant characteristic of the type of
the object viewed through the smoke = 3 for light-reflecting sign) by
K (the light extinction coefficient).

Therefore, S = C / K.

For optical depth, OD = K / 2.3 according to FDS User's guide.

(1) In "OD = K / 2.3", I would like to know where the value of "2.3"
comes from.
(2) As S = C / K and OD = K / 2.3, is that OD = (3 / S) (1 / 2.3)?
(3) Should optical depth (OD) have the same physical meaning as the
light extinct coefficent (K)? However, it seems that they are not the
same as "OD = K / 2.3". Could anyone explain this to me?

Thanks.

[TimoK]

Well, there are many constant numbers in the world.
2.3 seems to be quite close to ln(10). And the manual
has:

The gas phase output quantity ’EXTINCTION COEFFICIENT’ is K. A similar
quantity is the ’OPTICAL
DENSITY’, K/2.3, the result of using log10 in the definition
D 􀀀
1
L
log10

I
I0

= K log10 e (14.10)


Timo

[Kevin]

Yes, for some reason optical depth is based on log_10 rather than ln.
The SFPE Handbook defines optical density, K/2.3

[CK Lam]

Thanks for all your clarifications.

[TimoK]

The log_10 is also in the logarithmic intensity level
unit decibel. The optical density might be defined,
because engineers like log_10, number ten is
much easier to understand than the neper, which
is the other dimensionless logarithmic unit for
ratios. It has a nasty number e, "Euler's number".
Engineers are rational and they can not understand
irrational numbers, especially when it is also
a transcendental one ;-)

So, if D*L = 1.0 then intensity is ten times
less, if D*L = 2.0 then hundred times less,
just like in the acoustics. Might also be related
to the culture, some cultures like to use
number 10 in their system of units and some others
not ;-)

And similar thing is the "half life" in the
nuclear physics, it is much easier to understand
than the decay time constant lambda, where one
once again have the nasty number e playing a role.

But I like much more logarithms with the base e
than log_10. Well, especially I like the
exponential function, the only function that
I can differentiate even when I'm in sleep.

TimoK

[DavidShep]

Have some perspective and a sense of history. Smoke measurements were
performed for many decades before computers were available to
engineers because of tests like the Steiner Tunnel and the ASTM smoke
chamber. Log 10 is used for optical density because engineer’s slide
rulers used log base 10. (FYI.. I am not old enough to have used a
slide rule, but I remember my dad having one that he treated like
gold.)

Extinction coefficient using natural log is the more scientific smoke
obscuration measurement. I assume that if it were purely a scientific
decision that log based 10 optical density would have faded into
history by this time. Unfortunately, it is not a scientific
decision. Product acceptance requirements have been written into laws
and regulations that refer to optical density. Therefore, optical
density will exist forever.