Modified Ergun equation

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 ... 

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 ... 

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). 

The gas pressure gradient set up by gas-particle friction can be described by the modified Ergun equation (Ergun, 1952) and Kwauk equation (Kwauk, 1963), shown respectively as follows ... 

The small particle diameters may lead to a relatively high pressure drop in the MSR, which can be estimated with the modified Ergun equation in SI units [35] (Figure 3) ... 

Equation 108 is a wall modified one-dimensional Ergun equation, and equation 109 can be considered as the wall modified Ergun equation in a multidimensional format. 

Figures 18 and 19 show the experimental data of Liu et al. (32) for the packed bed of small glass beads as compared with the theoretical predictions. Here, D = 4.47 mm, ds = 1.917 mm, and e = 0.4529. We observe that the 2-dimensional model of Liu et al. (32) agrees fairly well with the experimental results. The modified Ergun equation predicts the experimental data well in the low Rem range as is shown in Figure 16 and underpredicts the experimental data as Rem is increased (Figure 18).

Table IV gives a summary of the packed beds that we made use of in this section. The term Cm< reflects the two-dimensional effects for Rem = 0. A value of zero, for 100(Cm / — 1), would indicate no two-dimensional effects. We can observe that the wall effects on the viscous term, Cw2, range from about 6% for the experimental data of Fand et al. (110) to 274% for the packed bed of Liu et al. (32) with a ds/D = 0.7123 as used here. The wall effects on the inertial term, Cwh range from around 1 to 19%. The two-dimensional effects are also significant for the packed beds of large particle to tube diameter ratios. The 2-dimensional model of Liu et al. (32) predicts quite well over a wide range of wall effects. In contrast, the wall modified Ergun equation significantly underpredicts at low porosity (Figures 15 and 19) and overpredicts at high porosity (Figures 16 and 17) the experimental data.

Comparison between the models given by the modified Ergun equation 108 or 109 and by LAM equation 106 or 107 with available experimental data would indicate a preference to using LAM-type equations. 

The pressure drop in ceramic and metallic foams can thus be estimated based on a modified Ergun equation [39]. Recently, Dietrich [40] evaluated the pressure drop measurements of nearly 100 different foams reported by about 25 authors. He could describe the experiments with an error of 40% in a broad range of Re numbers (10 < Re < 10 ) with the following correlation ... 

For fixed beds consisting of mono-sized spherical particles, a modified Ergun equation is proposed [44] (and demonstrated in Example 6.1) ... 

From what has been mentioned above, the modified Ergun Equation for CAP bed can be described by ... 

When fluid phase flowing through the catalyst, the frictional resistance is created. The coefficient of the frictional force can be calculated by the modified Ergun equation [6] ... 

For the case of the trapezoidal spout, the Ergun equation can be modified as follows ... 

Although Ergun equation is widely accepted in predicting the pressure drop for flow-through porous media, it is a known fact that the Ergun equation or its modified forms overpredict the pressure drop by as much as 100% at high porosity and underpredict the pressure drop by as much as 300% for low porosity medium such as sandstones (31). A more accurate equation has been developed by Liu et al. (32) based on a revised Kozeny-Carman theory. 

To the degree of uncertainties and variety of the beds studied, these investigators were able to incorporate the wall effects into the Ergun equation by replacing the particle diameter ds with a wall modified particle diameter dp,... 

Figures 14 and 15 show the normalized pressure drop factor for a densely packed bed of monosized spherical particles. For Rem < 7,fv is fairly independent of Rern, and at high Rem values, it increases fairly linearly with Rem. The data points are the experimental results taken from Fand et al. (110), where the bed diameter is D = 86.6 mm and the particle diameter is ds = 3.072 mm. One can observe that the 2-dimen-sional model of Liu et al. (32), referred to as equation 107, agrees with the experimental data fairly well in the whole range of the modified Reynolds number. From Figure 14, one observes a smooth transition from the Darcy s flow to Forchheirner flow regime. The one-dimensional model of Liu et al. (32) (i.e., equation 106) showed only slightly smaller fv value. Hence, the no-slip effect or two-dimensional effect for this bed is small. As shown in Figures 14 and 15, the Ergun equation consistently underpredicts the pressure drop. The deviation becomes larger when flow rate is increased.

Ergun then used published data to confirm the validity of this equation. Note that Eq. (4.100) is for spheres. For structured solids other than spheres, we modify the equation by including a sphericity factor P, which is defined as... 

Gibilaro et al. (1985) modified the Ergun equation and proposed an alternative pressure drop equation on the basis of theoretical considerations ... 

Leva et al (1949) and Ergun (1952) developed similar useful equations. Later, these were refined and modified by Handley and Heggs (1968 ) and by MacDonald et al (1979). All these equations try to handle the transient regime between laminar and turbulent flow somewhat differently but are based on the same principles. 

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