In fact, $\delta Q/T$ is the entropy of $\delta Q$, and if $\delta Q$ is considered as the heat energy in transfer, it follows that $\delta Q/T$ is the entropy in transfer. For the inequality \begin{align}\oint dS \ge \oint \frac{\delta Q}{T}.\end{align} Whether reversible or not does not need to be considered for $\delta Q/T$ in that the entropy production in heat transport process is that \begin{align} d_iS=\Delta \left(\frac{1}{T}\right){\delta Q}.\end{align} From non-equilibrium thermodynamics, we have \begin{align} dS =d_eS+d_iS.\end{align} Where $d_eS $ denotes the entropy flux, and $d_iS $ denotes the entropy production. 749
A well known fact is that $\delta Q/T$ is $d_eS$, such that the real meaning of <PERSON> inequality is that \begin{align} dS =d_eS+d_iS \geq d_eS=\frac{\delta Q}{T}.\end{align} Only when $d_iS=0$, we can get that \begin{align} dS = d_eS=\frac{\delta Q}{T}.\end{align} Why don't we define the entropy not by $dS$ for an arbitrary path but by $d_eS=\delta Q/T$?
“this creates often confusion since in irreversible processes there are other sources of heat, for instance that due to the dissipation of turbulences or friction, which in fact should not be counted in the δQ.”
If we consider that the heat energy is a non-conserved quantity, the changes in the heat energy within a system come from the two sources, one comes from the heat transfer $\delta Q$, another one comes from the heat conversion. Using $d_eq$ denotes the heat transfer $\delta Q$, and $d_iq$ denotes the heat production, a new function can be defined by \begin{align}dq=d_eq+d_iq\end{align} Where, $q$ is the heat energy within the system, then the fist law can be expressed as \begin{align}dU=dq+Ydx+\sum_j \mu_jdN_j\end{align} The entropy $ S$ for the system under consideration \begin{align}dS=\frac{dU}{T}-\frac{Ydx}{T}-\sum_j\frac{\mu_jdN_j}{T}+\frac{pdV}{T}=\frac{dq}{T}+\frac{p}{T}dV.\end{align} In thermodynamics, we have defined the heat in transfer $Q$, but failed to take into account a function to describe the energy of thermal motion within the system. It means that there is a state function, the internal heat energy that was lost. 749