Equation Ergun

The Ergun equation relates the pressure drop in a packed bed to the flow rate and the properties of particle and gas. However, the application of this equation has been extended beyond the limits of fixed bed systems since it was first formulated in 1951. Thus, a detailed account of the origin of this equation is necessary. 

The first term represents the pressure loss due to viscous drag (this is essentially the Carman-Kozeny equation) whilst the second term represents kinetic energy losses, which are significant at higher velocities (kinetic energy being proportional to velocity squared). Equation 1.43 is valid in the range 1 Re 2000 where the Reynolds number is defined by 

Writing Ergun s equation for minimum fluidizing conditions, and assuming the particles to be spherical ( =1), gives 

This is a quadratic equation in Re f where the Galileo number Ga is defined by 

Knowledge of the size and density of the particles to be fluidized, of the density and viscosity of the fluidizing gas (at the relevant 

Minimum fluidizing velocity as a function of terminal falling velocity 

---

The overall pressure drop is expressed as the sum of a laminar term proportional to FT/DT and a turbulent term proportional to FP/DI to yield the Ergun equation (1) ... 

Figure 3.4.3 illustrated that the pressure drop is independent of the catalyst quantity charged at any one RPM. This must be so, as will appear later on the modified Ergun equation. Since RPM is constant, so is AP on the RHS of the equation. Therefore, on the LHS, if bed depth (L/dp) is increasing, u must drop to maintain equality. Results over 5, 10, and 15 cm catalyst, and pumping air, all correlate well with the simple equation ... 

Pressure drop in the tube can be expressed with the use of the friction factor from the Ergun equation ... 

Axial pressure drop was calculated using the Ergun equation ... 

The convective wave cycle was described in 5.2.4 but its heat transfer properties not quantified. Critoph and Thorpe [22] and Thorpe [23] have measured the convective heat transfer coefficient between flowing gas and the grains within the bed. Preliminary results imply that the pressure drop through the bed can be expressed by a modified Ergun equation ... 

AP equation arising from simultaneous turbulent kinetie and viseous energy losses that is applieable to all flow types. Ergun s equation relates the pressure drop per unit depth of paeked bed to eharaeteristies sueh as veloeity, fluid density, viseosity, size, shape, surfaee of the granular solids, and void fraetion. The original Ergun equation is ... 

For paeked beds in either laminar or turbulent flow the Ergun equation is... 

The next step is to eharaeterize the resistanees offered by the porous eatalyst bed and support sereens. Several eorrelations relating the pressure drop through porous beds and veloeity and bed eharaeteristies are available. We seleet an Ergun equation to represent the resistanee of eatalyst bed ... 

The pressure drop is given by the Ergun equation (Ergun, 1952), expressed by... 

For packed beds in either turbulent or laminar flow, the Ergun equation is often satisfactory ... 

The pressure is also given by an ODE. The Ergun equation. Equation (3.17), applies to a packed bed ... 

Geometrically Similar Scaleups for Packed Beds. As was the case for scaling packed beds in series, the way they scale with geometric similarity depends on the particle Reynolds number. The results are somewhat different than those for empty tubes because the bed radius does not appear in the Ergun equation. The asymptotic behavior for the incompressible case is... 

Note that Sr appears here even though it is missing from the Ergun equation. It arises because throughput is proportional to F u. 

These asymptotic forms may be useful for conceptual studies, but the real design calculations must be based on the full Ergun equation. Turning to the case of compressible fluids, scaleup using geometric similarity with Sr = Sl = S is generally infeasible. Simply stated, the reactors are just too long and have too much inventory. 

The void fraction should be the total void fraction including the pore volume. We now distinguish Stotai from the superficial void fraction used in the Ergun equation and in the packed-bed correlations of Chapter 9. The pore volume is accessible to gas molecules and can constitute a substantial fraction of the gas-phase volume. It is included in reaction rate calculations through the use of the total void fraction. The superficial void fraction ignores the pore volume. It is the appropriate parameter for the hydrodynamic calculations because fluid velocities go to zero at the external surface of the catalyst particles. The pore volume is accessible by diffusion, not bulk flow. 

Several correlating equations for the friction factor have been proposed for both the laminar and turbulent flow regimes, and plots of fM (or functions thereof) versus Reynolds number are frequently presented in standard fluid flow or chemical engineering handbooks (e.g., 96, 97). Perhaps the most useful of the correlations is that represented by the Ergun equation (98)... 

The pressure drop through the packed bed reactor is governed by the Ergun equation discussed earlier as equation 12.7.4. 

This equation with a value of 150 instead of 180 is called the Ergun equation and is simply the sum of Eqs (13-15) and (13-16). (The more recent references favor the value of 180, which is also more conservative.) Obviously, for ARcRM < 10 the first term is small relative to the second, and the Ergun equation reduces to the Blake-Kozeny equation. Likewise, for A Re PM > 1000 the first term is much larger than the second, and the equation reduces to the Burke-Plummer equation. 

If the definitions of /PM and ARcRM are inserted into the Ergun equation, the resulting expression for the frictional energy loss (dissipation) per unit mass of fluid in the medium is... 

Consider the limit of high particle Reynolds numbers where the inertial term in the Ergun equations dominates. 

Since the same simplified set of dimensionless parameters holds exactly at both high and low Reynolds numbers, it is reasonable to expect that it will hold, at least approximately, over the entire range of conditions for which the drag coefficient can be determined by the Ergun equation or an equation of similar form. 

Downcomer Pressure Drop. When the downcomer is less than minimally fluidized, the pressure drop can be estimated with a modified Ergun equation substituting gas-solid slip velocities for gas velocities (Yoon and Kunii, 1970), as shown in Eq. (9). 

There are few reported comparisons to experimental pressure drop data taken by the same workers. An exception is Calis et al. (2001) who compared CFD, the Ergun correlation and experimental data for N — 1-2. They found 10% error between CFD and experimental friction factors, but the Ergun equation... 

The validation of CFD codes using pressure drop is most reliable when actual experimental data are taken in equipment identical to the situation that is being simulated. Existing literature correlations such as the Ergun equation are known to have shortcomings with respect to wall effects, particle shape effects, application to ordered beds and validity at high Re. The applicability of literature correlations to typical CFD simulation geometries needs to be examined critically before fruitful comparisons can be made. 

Contributions to pressure drop have also been studied by lattice Boltzmann simulations. Zeiser et al. (2002) postulated that dissipation of energy was due to shear forces and deformational strain. The latter mechanism is usually missed by capillary-based models of pressure drop, such as the Ergun equation, but may be significant in packed beds at low Re. For a bed of spheres with N — 3, they found that the dissipation caused by deformation was about 50% of that... 

---