ratio between heat transfer coefficient and thermal conductivity
Most likely, you can't assume that $h/k$ might be constant. Reason is that those parameters are characteristics of two very separate systems.

Thermal conductivity shows how fast heat can be propagated through material. That is, it answers question: if block of metal has $T_0$ temperature on its surface A, and you apply temperature $T_1$ (higher) to opposite surface B, how long it will take for whole block to get to temperature $T_1$.

Heat transfer coefficient is a characteristic of a system. For example, when you have coolant on one side of metal plate, and heated air on the other. Question is how efficiently heat will be transferred from one side to the other. For more you can check this link. As you can see, transfer coefficient $U$ is a function of conductivity of the metal plate $k$. Hence, generally, relationship is non-linear.