As you can see, both equations use the ideal gas low with the universal gas constant in J/kg.k Both equations are used in published articles as follows:
"Application of dynamic programming to the optimal management of a hybrid power plant with wind turbines, photovoltaic panels and compressed air energy storage" and "Design and thermodynamic analysis of a multi-level underwater compressed air energy storage system"
Should the air specific heat ratio "K" be included in the ideal gas law or not ?
The (specific) heat capacity ratio never appears in the ideal gas law
$$PV=nRT;$$ $$PV=mR_gT,$$
with pressure $P$, volume $V$, amount (in moles) $n$, universal gas constant $R$, temperature $T$, mass $m$, and mass-specific gas constant $R_g$.
Despite what another answer implies, it also doesn't appear in the corresponding relation for real gases
$$PV=ZnRT,$$
with compressibility factor $Z$; I believe the other answer confuses the compressibility factor with the specific heat ratio.
I believe you're asking why the parameter $k$, defined in the quoted paper as the specific heat ratio, appears in the unnumbered relation
$$\frac{dp}{dt}=\frac{kR}{V}\left(\dot m_{in}T_{in}-\dot m_{out}T_{cv}\right),$$
which may appear to be a time-derivative version of the ideal gas law, or perhaps a rate dependence on a difference between ideal-gas-law terms. This is not accurate; the resemblance is superficial. The relation is instead obtained from an energy balance of a control volume. Specifically, the rate of change in internal energy $U$ of a control volume is given by the enthalpy difference of the incoming and outgoing material in terms of their entrance and exit rates:
$$\frac{dU}{dt}= \dot m_{in}h_{in}-\dot m_{out}h_{out},$$
where $h$ is the specific enthalpy.
(The reason the enthalpy is used is that the incoming material is pushed into the volume $V$ with input pressure $P_{in}$ and the outgoing material must push its way out against output pressure $P_{out}$, which involves work done in addition to internal energy entrance and exit. Enthalpy $H\equiv U+PV$ is defined to incorporate this work.)
Using the equations of state for the ideal gas
$$U=mc_VT;$$ $$H=mc_PT\mathrm{,~or~}h=c_PT,$$
with constant-volume and constant-pressure specific heat capacities $c_V$ and $c_P$, respectively, and applying the ideal gas equation, we have
$$\frac{d(mc_VT)}{dt}= \dot m_{in}c_PT_{in}-\dot m_{out}c_PT_{out};$$
$$\frac{d(PV/R_g)}{dt}= \frac{c_P}{c_V}\left(\dot m_{in}T_{in}-\dot m_{out}T_{out}\right);$$
$$\frac{dP}{dt}= \frac{kR_g}{V}\left(\dot m_{in}T_{in}-\dot m_{out}T_{out}\right),$$
which exactly matches the relation of interest with specific heat capacity ratio $k\equiv\frac{c_P}{c_V}$ and if $T_{out}$ is set to $T_{cv}$, as discussed in the text (which uses $p$ for $P$ and $R$ for $R_g$). Again, this is not the ideal gas law but the result of an energy balance.
The specific heat ratio constant comes from the equation for a polytropic process:
$$PV^k = \rm{constant} $$
$$ \implies {P_1V_1^k=P_2V_2^k}$$
This means an expansion or compression process, Where $k$ is different depending on the type of expansion or compression.
For a constant pressure process, $k=0$. For an isothermal process, $k=1$ (which agrees with the ideal gas law). For an isentropic (adiabatic) process, $k$ is the specific heat ratio, usually 1.4 or 1.67.
The polytropic equation is a relationship that is separate from the ideal gas law and used alongside it. The IGL comes from the inherent properties of the gas in different states (i.e. $P,T,V,m$) while the polytropic equation comes from analysis of how work and energy are transferred into and out of the gas.
Whether it is added into the equation depends on how accurate you want your theoretical model to be compared to real world observations. The ideal gas law is not very good in describing real gases, so extra considerations are needed to improve its description. Looking at your snippet of the papers, these may be engineering problems, in which case I do not think the model is descriptive enough, even with the specific heat ratio taken into account.
