Ergun relationship

Pressure drop calculations are included, which use the Ergun relationship. This relationship predicts the pressure drop through packed beds, such as the primary reformer catalyst tubes and the secondary reformer (auto-thermal) bed. Measurements provide feedback that is used to update appropriate parameters in this relationship. 

Pressure drop calculations in the primary and secondary reformer models are based on the Ergun relationship, as presented in Reference 25. Both the laminar flow and turbulent flow terms are included. The natural parameter that arises from the Ergun equation, which can be updated with measured pressure drop information, is the TURBULENT DP COEF term in the models of both the primary and secondary reformers. This term affects only the pressure drop, whereas another term, the bed void fraction, which might also have been used as the parameter to update with measure pressure drop, also affects all the reaction rates. The bed void fraction affects the amount of catalyst in a fixed volume reformer tube, and is not an appropriate parameter to use in this case. The void fractions of typical packed beds are shown in Figure 5.70 of Reference (26). Void fractions of 0.4 to 0.6 are typical, and can be determined for specific catalysts sizes and shapes from vendor specification sheets, by measurement, or, with more difficulty, by calculation. 

Wen and Yu(6) have examined the relationship between voidage at the minimum fluidising velocity, emf, and particle shape, (ps, which is defined as the ratio of the diameter of the sphere of the same specific as the particle d, as used in the Ergun equation to the diameter of the sphere with the same volume as the particle dp. 

The Richardson-Zaki equation has been found to agree with experimental data over a wide range of condifions. Equally, if is possible fo use a pressure drop-velocity relationship such as Ergun to determine minimum fluidization velocity, just as for gas-solid fluidizafion. An alternative expression, which has the merit of simplicify, is fhaf of Riba ef al. (1978)... 

In particulate fluidization, for us > u[m and Rep <10, the relationship between the fluidized bed voidage and velocity can be derived from the Ergun equation (McCabe et al., 1983) ... 

Hydrodynamics The particle size can be evaluated from the superficial velocity at incipient fluidization fm and the Ergun equation by trial and error (eq. (3.451)). For this calculation, we need the bed porosity at incipient fluidization for the assumed particle size, which can be evaluated by using the relationship of Broadhurt and Becker (eq. (3.466)). Note, that the resulting value cannot be lower than the fixed-bed porosity. Since we assume spherical particles, a reasonable value of bed porosity is 0.41. This procedure results in a particle size of 0.077 mm and sfm = 0.47. 

As an alternative, Thrower and Nagle (19) have adapted a suggestion by Ergun (20) and used the relationship... 

The Eulerian continuum approach, based on a continuum assumption of phases, provides a field description of the dynamics of each phase. The Lagrangian trajectory approach, from the study of motions of individual particles, is able to yield historical trajectories of the particles. The kinetic theory modeling for interparticle collisions, extended from the kinetic theory of gases, can be applied to dense suspension systems where the transport in the particle phase is dominated by interparticle collisions. The Ergun equation provides important flow relationships, which are useful not only for packed bed systems, but also for some situations in fluidized bed systems. 

The review published by Ergun (E2) provides a definitive description of pressure drop in packed tubes when the ratio of particle diameter to tube diameter is sufficiently low. In addition, although the complicated relationship between the diameter ratio, the fraction void and the friction factor can not be accurately represented without some explicit dependence of the friction factor on the diameter ratio, Ergun showed that his correlation does work for a wide variety of experimental conditions. The friction factor is calculated from the expression... 

In the preceding analysis, the velocity-hold-up relationship was expressed in terms of the Richardson-Zaki equation. Alternatively, the relationship can be derived from the Ergun (1952) equation. This latter approach was used by Molerus (1993). 

Using the relationship between z and W [Equation (4-26)] we can change our variables to express the Ergun equation in terms of catalyst weight ... 

For a quick first approximation in the case of trickle flow, pressure loss may be expressed by an Ergun-type relationship in which the porosity c is replaced by c(l - /3), the total liquid extraparticle holdup /3 being calculated either by the correlation proposed by Sato et al. for spheres (S7) of diameter 2 mm or larger ... 

The state of fluidization begins at the point of minimum or incipient fluidization. At the minimum fluidization point, the pressure drop for a fixed bed and that for a fluidized bed are equivalent. This relationship is used as the basis for the formation of the predictive equation for the minimum fluidization velocity. The pressure drop in the fixed bed can be described by the Ergun equation. Under the minimum fluidization condition, the Ergun equation can be expressed as... 

A hydrocracker is a three-phase operation. The gas phase supplies hydrogen, the liquid phase supplies the heavy hydrocarbons, and the catalyst is the solid phase. This unit can be operated as a trickle bed reactor, with gas and liquid phases fed in at the top. Products, removed from the bottom, are in both the gas and the liquid phases. The key steps and analysis are similar to the packed bed reactor above. Pressure drop and holdup can be determined from the Ergun equation and gas and liquid phase Reynolds numbers. Relationships for transitions to pulsating and other flows can also be developed. ... 

Consequently, the reactor model is constituted by a system of N+1 equations, where N is the number of chemical species present in the system (NO, NO2, N2 and O2, neglecting the presence of N2O N = 4) and another unknown variable is pressure. The equations are one momentum balance (in the form of simplified Ergun Law), and four mass balance relationships. The presence of NO2 among the reaction products has been related to the catalytic activity of Cu-ZSM5 towards the oxidation of NO to NO2, as revealed by our previous investigation in similar experimental conditions [7], as well as by the present results (Fig. 1). It has been hypothesised that reaction (2) proceeds in parallel to NO decomposition, having not assumed that NO2 formation is responsible for copper reduction from Cu (inactive in decomposing NO) to Cu (the active site), as also proposed by some author [20-21,23]. 

Mcked beds of adsoibent panicles is usually piedicied by relationships such as those of Ergun and of Leva. Goienilly. the pressure drop per unit loigth of packed bed is inversely proportional to the panicle size to a power not 1 then unity. Thus, pressure drop can be reduced by selecting the laiger particle size. 

First, if we ignore the filter medium and consider only the cake itself, the pressure drop versus liquid flow relationship is described by the Ergun equation [Equation (6.15)]. The particle size and range of liquid flow and properties commonly used in industry give rise to laminar flow and so the second (turbulent) term vanishes. For a given slurry (particle properties fixed) the resulting cake resistance is defined as ... 

If the relative upward velocity of the gas Uf- Up) is less than the relative velocity at incipient fluidization (Llf - Up), then packed bed flow results and the relationship between gas velocity and pressure gradient is in general determined by the Ergun equation [see Chapter 6, Equation (6.11)]. 

The Ergun equation is usually expressed in terms of the superficial gas velocity through the packed bed. However, for the purposes of standpipe calculations it is useful to write the Ergun equation in terms of the magnitude of the velocity of the gas relative to the velocity of the solids Lfrei (= Uf — Up ). (Refer to Section 8.1.4 for clarification of relationships between superficial and actual velocities.)... 

The pressure gradient in packed bed flow is generated by the upward flow of gas relative to the solids in the standpipe. The Ergun equation [Equation (8.25)] provides the relationship between gas flow and pressure gradient in a packed bed. 

ILLUSTRATIVE EXAMPLE 18.16 Comment on the relationship between the Ergun equation and the Burke-Plummer and Blake-Kozeny equations. 

By equating the rate of change in fluid momentum caused by the exchange flow with the inertial pressure drop term in the Ergun equation a relationship between F and S is established ... 

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