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4.1. On theother hand, if the solids loading is given, the pressure dropincreases with increasing superficial gas velocity and theextent of the increase seems to increase with solids loading.Physically, two main factors contribute to the pressuredrop of a single phase flow through a given packed bedaccording to Ergun equation (or modified Ergun equation),viscous contribution proportional to the superficial gasvelocity, and inertia contribution proportional to thesquare of the superficial gas velocity, with the proportion-ality factors given, respectively, as ½ð1 emÞ2e3mlm and ½1 eme3mqm ,where qm, lm and em denote, respectively, density, viscosityand voidage of the two-phase mixture. The two-phase mix-ture voidage can be expressed as em = eg bst with eg andbst being the bed voidage at zero solids loading and staticsolids hold-up. Given the superficial gas velocity, the rela-tionship between the pressure drop and the solids loading isa function of the two proportionality factors. As the static hold-up is very small, it is expected that its influence on thetwo factors is small. As a consequence, the observed depen-dence of pressure drop on the solids loading is likely to bedue to the effect of the solids loading on the viscosity anddensity of the gas–solid two-phase mixture.Fig. 10(b) compares the pressure drop data at the hightemperature with those at 19 C (the ambient temperature)for 112.5 lm glass beads. The following observations canbe made:(a) Given the superficial gas velocity, the nonlineardependence of the pressure drop on the solids loadingoccurs at both ambient and elevated temperatures.(b) Given the solids loading, the pressure drop at thehigher temperature is higher than that at the ambienttemperature particularly at high solids loadings andhigh superficial gas velocities.(c) At very low solids loading, the effect of temperaturetends to be diminishing, suggesting that the large dif-ference in the pressure drops at the two temperaturesis mainly due to the presence of suspended particles.As mentioned in Section 4.1,
the use of Euler number isunable to unify the experimental results. Effort was thenmade to develop a relatively simple mode for predicting the pressure drop. The starting point for the effort was theErgun equation [13,14]. It is found that, if the mixtureparameters are used, the Ergun equation fits the experi-mental data well up to a pressure drop of 15000 Pa/m(Fig. 11a). At pressure drop higher than 15000 Pa/m,the model overpredicts measurements and the highest devi-ation is found to be 25% (Fig. 11b).Fig. 12 illustrates the total solids hold-up, defined as thesummation of the dynamic and static solids hold-ups, as afunction of solids mass flux at different superficial gasvelocities. Given the superficial gas velocity, the total solidshold-up is seen to increase rapidly with increasing solidsloading. This is in a broad agreement with the cold stateexperiments (Fig. 6), possibly due to narrow range of solidsloading used in the cold state experiments. An attempt ismade to relate the total solid hold-up to other parameters.It was found, by trail and error, that btotal/(1 ep) relatesto (Us/Ug)(ds/dp)0.5well by the following expression:btotal=ð1 epÞ¼ 5:83½ðUs=UgÞðds=dpÞ0:5 0:32ð1Þwhere Us is the solids velocity, ep is the volume fraction ofthe packed particles, and ds and dp are packed and sus-pended particle diameters, respectively. Eq. (1) agrees withthe experimental data within 30%.The total solids hold-up at 100 C is compared with thatat 19 Cin Fig. 13. The lower temperature is seen to gener-ate a higher solids hold-up, particularly at low superficialgas velocities. Physically, this is associated with the temper-ature dependence of the gas viscosity; the higher the temper-ature, the higher the gas viscosity, hence a lower gas–solidslip velocity and the lower solids hold-up. The effect ofthe gas velocity can be explained by using the Ergun equa-tion – viscous term is dominating the flow hydrodynamicsin packed beds at low superficial gas velocities.4.4.