In an arbitrarily unstructured mesh, shown in the figure below, in the context of finite volume method, I want to obtain an approximation of $\phi_f$, where $N$ and $P$ are cell centers of adjacent cells and $f$ is the face center- $N$,$P$ and $f$ are not aligned. CFD textbooks say: $$ \phi_f=g\phi_P+(1-g) \phi_N$$ where $g$ is a weight factor. They also say that it is a second-order accurate approximation. But how?
Let's use Taylor expansion around $f$: $$ \phi_P=\phi_f+(\nabla\phi)_f.(r_P-r_f)+O(2)$$ $$\phi_N=\phi_f+(\nabla\phi)_f.(r_N-r_f)+O(2)$$ Question:
How to eliminate the terms containing $\nabla$ and obtain the early mentioned relation?