Ergun Equation

When designing packed bed reactors understanding the pressure drop across the reactor is very important as this can affect the flow in the bed and the reaction pressure. A commonly used equation for this is the Ergun-equation and combines both the laminar and turbulent components of the pressure loss across a packed bed, $$-\frac{\Delta P}{\Delta L}=150\left(\frac{\mu q}{d_p^2}\right)\frac{(1-\varepsilon)^2}{\varepsilon^3}+1.75\left(\frac{\rho q^2}{d_p}\right)\frac{(1-\varepsilon)}{\varepsilon^3}$$ The Ergun equation tells us a number of things. It tells us the pressure drop along the length of the packed bed given some fluid velocity. It also tells us that the pressure drop depends on the packing size, $d_p$, length of bed, $L$, fluid viscosity, $\mu$, and fluid density, $\rho$. The first term basically represents laminar flow conditions and is essentially the Carman–Kozeny equation. In the laminar region ($\text{Re}_b < 10$) the pressure drop through the packed bed is independent of fluid density and has a linear relationship with superficial velocity.

The second term basically represents turbulent flow conditions ($\text{Re}_b > 1000$). In the turbulent region the pressure drop increases with the square of the superficial velocity and has a linear dependence on the density of the fluid passing through the bed.

The graph below shows the pressure drop over the bed length and shows the key parameters affecting the pressure drop of fluid flow through a packed bed. Investigate these parameters effecting the pressure drop over the length of a 2 meter packed bed. The inlet pressure is 100 kPa at $L = 0$ m.