Effects of Pressure Drop

In the preceding sections, we discussed the operation of plug-flow reactors with gas-phase reactions under the assumption that the pressure does not vary along the reactor. However, in some applications, the pressure significantly changes and, therefore, affects the reaction rates. In this section, we incorporate the variation in pressure into the design equations. For convenience, we divide the discussion into two parts tubular tube with uniform diameter and packed-bed reactors. 

We consider first a cylindrical reactor of uniform diameter D. To derive an expression for the pressure drop, we write the steady-state momentum balance equation for a reactor element of length dL and cross-sectional area A  

The first term on the right indicates the pressure drop due to friction, and the second indicates the pressure drop due to change in velocity (kinetic energy). In many apphcations, the seeond term in Eq. 7.5.2 is small in comparison to flie first, and noting that u — v/A, the momentum balance equation reduces to 

Equation 7.5.4 provides an approximate relation for the ehanges in pressure along a plug-flow reactor, expressed in terms of dimensionless extents and temperature. It is applicable when the velocity does not exceed 80-90% of the sound velocity. For these situations, we solve Eq. 7.5.4 simultaneously with the design equation 

When the gas velocity approaches the sound velocity, the kinetic energy and viscous work terms in the energy balance equation are not negligible (as assumed in Chapter 5). For these cases, we write the general energy balance equation for a differential plug-flow reactor with length dL (see Eq. 5.2.44), 

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Effect of Pressure Drop and Nozzle Size For a nozzle with a developed pattern, the average drop size can be estimated to fall with rising AP (pressure drop) by Eq. (14-196) ... 

Figure 9-37. Comparison effect of pressure drop across support plate and bed of VA in. Intalox saddles. Used by permission of Leva, M., Lucas, J. M., and Frahme, H. H., Ind, Eng Chem. V. 46, No. 6 (1954) all rights reserved.

Vertical Thermosiphon Reboiler Comparison Effect of Pressure Drop and Pipe Size on Selection... 

Figure 13. Effect of pressure drop and number of reactors in series on theoretical horsepower for hot gas recycle compressor H CO = 3 1, standard ff/day = 250,000,000...

To evaluate the effect of pressure drop on performance, differential equations for the pressure drop (15.2-11), material balance (15.2-4), and energy balance (15.2-10) must be integrated simultaneously to solve for P,fA, and T as functions of axial position, x ... 

In Equation (e) an average value for the heat transfer coefficient U is assumed, ignoring the effect of pressure drop. U depends on the working fluid and the operating temperature.) Let the cost per unit area of the exchanger be CA and the annualization factor for capital investment be denoted by r. Then the annualized capital cost for the boiler is... 

Note that while the fluid density may be a function of the pressure in the bed in a compressible flow, the superficial mass velocity is constant. The Ergun equation in the form given in eq. (3.450) is more convenient when analyzing the effects of pressure drop in the fluid density. 

Reactor model for a first-order reaction To illustrate the effect of pressure drop, consider an isothermal two-phase fixed-bed operation (gas-solid system). In terms of a reactant, the intrinsic reaction rate is... 

Fig. 12. Effect of pressure drop on the solid fraction of soot cake deposits as a function of soot aggregate size.

Figure 5.1. Effect of pressure drop on column flow at various points along the column length.

Effects of change in temperature on heat capacity and heats of vaporization are negligible. Heat losses from the column are negligible. Effects of pressure drop over the column may be neglected. 

The preceding equations are the basis for the nomograph presented in Fig. 14-2, and this figure can be used for estimating the optimum diameter of steel pipe under ordinary plant conditions. Equations (15) and (16) should not be applied when the flowing fluid is steam, because the derivation makes no allowance for the effects of pressure drop on the value of the flowing material. Equation (15) is limited to conditions in which the viscosity of the fluid is between 0.02 and 20 centipoises. 

We now must determine the ratio F/Fq a function of volume V or the catalyst weight, W to account for pressure drop. We then can combine the concentration, rate law, and design equation. However, whenever accounting for the effects of pressure drop, the differential form of the mole balance (design equation) must be used. 

In liquid-phase reactions, the concentration of reactants is insignificantly affected by even relatively large changes in the total pressure. Consequently, we can totally ignore the effect of pressure drop on the rate of reaction when sizing liquid-phase chemical reactors. However, in gas-phase reactions, the concentration of the reacting species is proportional to the total pressure and consequently, proper accounting for the effects of pressure drop on the reaction system can, in many instances, be a key factor in the success or failure of the reactor operation. 

The model of Tayakout et al. [117,118] in addition accounts for the possibility of axial dispersion effects in the tubeside and shellside. The inclusion of axial dispersion effects in regions (1) and (4) necessitates a different set of initial conditions at Z = 0 and a companion set of conditions at Z = L. The effect of pressure drop through the catalytic bed could be included in this type of model using Ergun s equation. 

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