(a) Monolith pressure loss characterization.

The classical approach that links pressure drop and velocity in flows through porous media is the Forchheimer equation (Equation (1)), that can be derived from the Navier–Stokes equation for one-dimensional, incompressible and steady laminar flow of a Newtonian fluid in a rigid porous medium [29]:

$$-\frac{dp}{dx} = \frac{1}{\kappa}(\mu \cdot v) + \beta \left(\rho \cdot v^2\right) \tag{1}$$

The second term in the right can be interpreted as a second-order correction to account for the contribution of inertial forces, but, at sufficiently low velocities, this effect is negligible and Equation (1) can be reduced to Darcy's law (Equation (2)) [30]:

$$-\frac{dp}{d\mathbf{x}} = \frac{1}{\kappa}(\mu \cdot \mathbf{v})\tag{2}$$

Darcy's law can be rewritten in the form of Equation (3), relating the average fluid velocity *v* through the pores with pressure drop Δ*p* along a segmen<sup>t</sup> of length *L*.

$$-\frac{\Delta p}{\mu L} = \frac{1}{\kappa} \,\mathrm{v}$$

If the equation from the linear fit of the scatter plot of −Δ*p μL*vs. *v* is (Equation (4)):

$$
\mathcal{Y} = B\mathcal{X} \tag{4}
$$

then Darcy's constant *κ* can be derived as follows (Equation (5)):

$$\mathbf{x} = \frac{1}{B} \tag{5}$$
