The compressible fluid flow coefficient formula is just the incompressible formula with different units and with the ideal gas law applied under a particular convention.
I can find references to the compressible fluid flow formula going back to Brockett in 1952, but I cannot find a published derivation online. After reverse engineering the formula, however, it turns out to be relatively simple, but with confusing units and one less-than-obvious convention. I have since found a paper copy of an article that is consistent with my derivation: "Development of a universal gas sizing equation for control valves" (Buresh, James F., and Charles B. Schuder. ISA Transactions 1964 Vol. 3 Iss. 4, Pages 322-328).
Incompressible Flow
Let's first review the inspiration for the incompressible flow coefficient formula.
The volume flow rate through a valve of area $A$ is just
$$
Q_v=A v_o
$$
where $v_o$ and $A$ are the flow velocity and area of valve outlet. For a horizontal valve, from Bernoulli's principle we expect
$$
\frac{1}{2}\rho v_i^2+P_i = \frac{1}{2}\rho v_o^2+P_o
\tag{Eq. A.1}$$
For zero inlet velocity, $v_i$, we have
$$
v_o^2 = 2\frac{P_i-P_o}{\rho} \tag{Eq. A.3}
$$
so the volumetric flow through a valve is
$$
Q_v=A\sqrt{\frac{2 \Delta P}{\rho}} = C_v \sqrt{\frac{\Delta P}{S_l}} \tag{Eq. A.3}
$$
where $\Delta P \equiv P_i-P_o$, $C_v=A\sqrt{2/\rho_L}$ is the naive volumetric flow coefficient, and
$$
S_l=\rho/\rho_L \tag{Eq. A.4}
$$
is the specific gravity relative to a reference liquid density $\rho_L$.
Compressible Flow
There is much discussion in the literature about the best way to parameterize $C_v$.
For example, Turnquist reported that the experimental volume flow in 32 different valves was better described with
$$Q_v =C_v \left(1399 - 636 \frac{\Delta P}{P} \right) \sqrt{\frac{P\Delta P}{S_g T}} \tag{Eq. A.5}$$
For large pressure differences, the flow rate for a compressible fluid has a more complex relationship to pressure that depends on factors such as the heat capacity ratio $\gamma$, e.g.
$$Q_v =C A \sqrt{2\frac{P_i}{\rho_i}\left(\frac{\gamma}{\gamma-1}\right)\left[(P_o/P_i)^{2/\gamma}-(P_o/P_i)^{(\gamma+1)/\gamma}\right]} \tag{Eq. A.6}$$
For small pressure differences, however, the gas can be treated as an incompressible fluid, but by convention with different units and and with the flow coefficient defined for a standard density.
Because mass, but not gas volume, is conserved, I found it conceptually easier to start with the mass flow per unit time
$$
Q_m =\rho Q_v = \rho C_v \sqrt{\frac{\Delta P}{S_l}} = C_v \sqrt{\rho \rho_L \Delta P} \tag{Eq. A.7}
$$
The simplest assumption about how to modify the incompressible formula for compressible fluids is to assume the ideal gas law, $PV=nRT$, so the fluid density will be
$$
\rho =n/V = P/RT = \rho_0 \frac{T_0}{P_0}\frac{P}{T} \tag{Eq. A.8}
$$
where the reference gas density ($\rho_0$) is defined at some reference temperature ($T_0$) and pressure ($P_0$).
The mass flow (Eq. A.7) can now be rewritten as
$$
Q_m = C_v \sqrt{\rho_L \rho_0 \frac{T_0}{P_0}\frac{P\Delta P}{T} } \tag{Eq. A.9}
$$
By a standard convention (see note below), the volumetric flow coefficient is defined at the reference density:
$$
Q_v = \frac{Q_m}{\rho_0} = C_v \sqrt{\frac{\rho_L}{\rho_0} \frac{T_0}{P_0} \frac{P\Delta P}{T} } \tag{Eq. A.10}
$$
and the flow coefficient is :
$$
C_v \equiv Q_v \sqrt{\frac{\rho_0}{\rho_L} \frac{P_0}{T_0} \frac{T}{P\Delta P} } \tag{Eq. A.11}
$$
This is a general form, but Equations 1 & 2 in the question are defined under the following conventions:
- Flow rates are given in SCFH (Standard Cubic Feet per Hour) for gases, but in GPM (US Gallons Per Minute) for liquids. The conversion factor is 1 GPM = 8.02 SCFH.
- The standard reference gas temperature and pressure is 60°F (519.7°R) and 1 atmosphere (14.696 PSI).
- The specific gravity for compressible gas flow is defined relative to air, not water as is the case for incompressible liquid flow. Since at the standard pressure and temperature, the water density is $\rho_{water} = 999.0\,\mathrm{kg/m^3}$ and the air density is $\rho_{air} = 1.222\,\mathrm{kg/m^3}$, the specific gravity for gases is $S_g=817.5S_l$, where $S_g\equiv \rho_0/\rho_{air}$ and $S_l\equiv \rho_0/\rho_{water}$.
- All temperatures in (°R) and and pressures in (PSI).
- The average pressure is $P=(P_1+P_2)/2$.
So Equation A.11 can be rewritten as
$$
C_v = \frac{Q_v(\mathrm{GPM})}{Q_v(\mathrm{SCFH})}
\sqrt{ \frac{\rho_{air}}{\rho_{water}}}
\sqrt{ \frac{P_0}{T_0}}
\,Q_v(\mathrm{SCFH}) \sqrt{\frac{\rho_0}{\rho_{air}}
\frac{T}{P\Delta P}} \tag{Eq. A.12}
$$
and inserting the numerical values gives
$$
C_v = \frac{1}{8.02} \sqrt{\frac{1}{817.5}}
\sqrt{\frac{14.696\,\mathrm{PSI}}{519.7\mathrm{°R}}}
\,Q_v(\mathrm{SCFH}) \sqrt{ \frac{S_gT}{P\Delta P} } \tag{Eq. A.13}
$$
This reduces to
$$
C_v =
\frac{Q_v(\mathrm{SCFH})}{1364\,\mathrm{PSI}^{-1/2}\mathrm{°R^{1/2}}} \sqrt{ \frac{S_g T}{P\Delta P} } \tag{Eq. A.14}
$$
or if using $P_1+P_2$ instead of $P=(P_1+P_2)/2$
$$
C_v =
\frac{Q_v(\mathrm{SCFH})}{964\,\mathrm{PSI}^{-1/2}\mathrm{°R^{1/2}}} \sqrt{ \frac{S_g T}{(P_1+P_2)\Delta P} } \tag{Eq. A.15}
$$
If you round "964" to "960", this is just a rearranged version of the question's Equation 2, with it clear that "960" is not a dimensionless constant but has units.
Notes:
There are various versions of the formula with slightly different coefficients, e.g. this 1961 article on "Comparing Gas flow Formulas for Gas Valve Sizing" has "1360", "1364", and "1390". Aside from possibly using slightly different values for water and air densities, the coefficient also depends on whether the input, output, or average pressure is used in the definition of $C_v$.
According to Buresh & Schuder, the above formula (Eq. A.15)is accurate for $\Delta P /P < 0.02$, but looking at some of the figures in that paper and Brockett, it seems to often work reasonably well up to $\Delta P /P \sim 0.3$.
Getting the correct $T$ and $P$ dependence is confusing. It helped me to remember $C_v$ is the volume of fluid that flows through the valve per unit time for a given pressure. According to Bernoulli's principle we expect this to be inversely proportional to $\sqrt{S}$, i.e. the denser the fluid the lower the flow. Since a fixed value for the specific gravity is used in the flow formula, but actual gas density is proportional to $P/T$, we must have $C_v \propto \sqrt{T/P}$ to compensate.
Using this fixed value is consistent with the ISA-75.01.01 Standard on "Flow Equations for Sizing Control Valves" that states that "Volumetric flow rates for compressible fluids … refer to normal or standard conditions". Accordingly, equation A.14/15 is a definition of $C_v$ that corrects the observed volume flow $Q_v$ at some temperature and pressure to give the flow coefficient that would be observed under standard conditions (1013.25 mbar / 14.7 PSI and 288.6 K / 60°F).
The original question had a mistake in the the units for the incompressible flow coefficient, which are actually $\frac{GPM}{\sqrt{PSI}}$.
Give me SI units any day.
