Plume Calculations
==================
Scott D. Brown
:stem: latexmath

== Mass Rate Conversion
This calculation outlines the conversion from the concentration rate in
ppm/s input to the Blackadar model to a mass rate in gm/s.

=== Derivation
The Blackadar is bouyancy driven, and the Briggs buoyancy flux
parameter (latexmath:[F_b]) can be computed in various ways.  The
most practical is from the following equation:

[latex]
\begin{equation}
  F_b = g \frac{V_s}{\pi} \frac{T_s - T_a}{T_s}
      = g v_s r^2 \frac{T_s - T_a}{T_s}
\end{equation}

where latexmath:[g] is the gravity constant, latexmath:[V_s] is the
stack volume flow rate [m^3^/s], latexmath:[v_s] is the stack gas
exit velocity [m/s], latexmath:[r] is the stack radius [m],
latexmath:[T_s] is the stack gas temperature [K] and latexmath:[T_a]
is the ambient air temperature [K].

Since latexmath:[F_b], latexmath:[r], latexmath:[T_s] and latexmath:[T_a]
are provided was inputs, we can compute the volume flow rate
latexmath:[V_s] as:
[latex]
\begin{equation}
  V_s = \frac{F_b \pi}{g} \frac{T_s}{T_s - T_a}
\end{equation}

The release mass rate is provided as a concentration rate [ppm/s].
If we want to compute the mass flow rate, then we need to do some
calculations to compute the relative volume of the gas.

The release concentration rate (latexmath:[\dot{C}]) is ratio of the
volume rate of the gas, latexmath:[V_g], and the volume rate of the
air, latexmath:[V_a]:
[latex]
\begin{equation}
  \dot{C} = \frac{V_g [m^3/s]}{V_a [m^3/s] \cdot (1 \times 10^6)}
\end{equation}

If we assume the gas is a small fraction of the overall volume, we can
assume latexmath:[V_a] equals latexmath:[V_s] and solve for the volume
rate of the gas latexmath:[V_g] as:
[latex]
\begin{equation}
  V_g = \frac{\dot{C}}{V_s \cdot (1 \times 10^6)}
\end{equation}

The Ideal Gas Law (latexmath:[PV = nRT]) allows us to relate the volume of
gas to the number of moles (latexmath:[n]) of the gas:
[latex]
\begin{equation}
  n = \frac{PV}{RT}
\end{equation}

where, latexmath:[P] is the pressure in Pa, latexmath:[R] is the
universal gas constant (8.31446261815324 joules/K/mole) and
latexmath:[T] is the temperature in K.  Since the pressure is not
explicitly provided as an input, we will assume standard pressure
P~std~ = 1 x 10^5^ Pa (100 kPa) or 1 x 10^5^ N/m^2^.  It is assumed
that temperature of the released air/gas mixture is the release
temperature, latexmath:[T = T_s].  Finally, since we are ultimately
solving for the mass rate latexmath:[\dot{m}] [gm/s], we will modify
this equation to compute the molar rate latexmath:[\dot{n}] [moles/s]
by using the gas volume rate (latexmath:[V_g]) rather than just a
volume:
[latex]
\begin{equation}
  \dot{n}~[moles/s] = \frac{P_{std} V_g}{R T_s} =
    \frac{P_{std} \dot{C}}{R T_s V_s (1 \times 10^6)} = 
    \frac{P_{std} \dot{C} g (T_s - T_a)}{R T_s^2 F_b \pi (1 \times 10^6)}
\end{equation}

Finally, the mass rate latexmath:[\dot{m}] can be computed by folding in
the molecular weight (latexmath:[m]) in grams per mole.  Note that the
molecular weight is _not_ included as an input to the model and must be
externally obtained:
[latex]
\begin{equation}
  \dot{m}~[gm/s] =
      \frac{P_{std} \dot{C} m g (T_s - T_a)}{R T_s^2 F_b \pi (1 \times 10^6)}
\end{equation}

Conversely, if we want to compute the concentration rate,
latexmath:[\dot{C}], from a mass rate, latexmath:[\dot{m}]:
[latex]
\begin{equation}
  \dot{C}~[ppm/s] =
      \frac{\dot{m} R T_s^2 F_b \pi (1 \times 10^6)}{P_{std} m g (T_s - T_a)}
\end{equation}

=== Example #1
Computing the mass release rate, latexmath:[\dot{m}], of a
methane (CH~4~) plume with
a release concentration rate of latexmath:[\dot{C}] = 100 ppm/s,
a molecular mass latexmath:[m] of 16.04 gm/mole,
a Briggs Bouyancy Flux of latexmath:[F_b] = 10.0 m^4^/s^3^,
a release temperature of latexmath:[T_s] = 350 K and 
an air temperature of latexmath:[T_a] = 300 K:
[latex]
\begin{equation}
  \dot{m}~[gm/s] =
      \frac{(1 \times 10^5)(100)(16.04)(9.8)(350 - 300)}
      {(8.3144)(350^2)(10) \pi (1 \times 10^6)}
      = 2.45 \times 10^{-3} ~[gm/s]
      = 8.84 ~[gm/hr]
\end{equation}

=== Example #2
Computing the concentration release rate, latexmath:[\dot{C}], of a
methane (CH~4~) plume with 
a mass release rate of latexmath:[\dot{m}] = 100 kg/hr (22.777 gm/s),
a molecular mass latexmath:[m] of 16.04 gm/mole,
a Briggs Bouyancy Flux of latexmath:[F_b] = 10.0 m^4^/s^3^,
a release temperature of latexmath:[T_s] = 350 K and 
an air temperature of latexmath:[T_a] = 300 K:
[latex]
\begin{equation}
  \dot{C}~[ppm/s] =
      \frac{(22.77)(8.3144)(350^2)(10) \pi (1 \times 10^6)}
      {(1 \times 10^5)(16.04)(9.8)(350 - 300)}
      = 1793.4 ~[ppm/s]
\end{equation}
