Scaleup with geometric similarity

For laminar flow of a Newtonian fluid, scaleup at constant pressure drop is identical to scaleup with geometric similarity. For constant pressure drop in turbulent flow, the reactor diameter increases somewhat faster, than for scaleup with... 

Increase the tube diameter, either to maintain a constant pressure drop or to scale with geometric similarity. Geometric similarity for a tube means keeping the same length-to-diameter ratio L/dt upon scaleup. Scaling with a constant pressure drop will lower the length-to-diameter ratio if the flow is turbulent. 

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... 

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. 

Constant-Pressure Scaleups for Laminar Flows in Tubes. As shown in the previous section, scaling with geometric similarity, Sr = Sr = 5 /, gives... 

Solution Now, Ar=107°C. Scaling with geometric similarity would force the temperature driving force to increase by S = 1.9, as before, but the scaled-up value is now 201°C. The coolant temperature would drop to —39°C, which is technically feasible but undesirable. Scaling with constant pressure forces an even lower coolant temperature. A scaleup with constant heat transfer becomes attractive. 

The reactor volume scales as S, and the aspect ratio of the tube decreases upon scaleup. The external surface area scales as SrSl = >S, 6 /27 compared with A2/3 for the case with geometric similarity. The Reynolds number also scales as S16/27. It increases upon scaleup in both cases, but less rapidly when the pressure drop is held constant than for geometric similarity. 

As shown in the previous section, scaling with geometric similarity, Sr = Sl = gives constant pressure drop when the flow is laminar and remains laminar upon scaleup. This is true for both liquids and gases. The Reynolds number and the external area increase as. Piston flow is a poor assumption for laminar flow in anyfhing but small tubes. Conversion and selectivity of the reaction is likely to worsen upon scaleup unless the pilot reactor is already so large that molecular and thermal diffusion are negligible on the pilot scale. Ways to avoid unpleasant surprises are discussed in Chapter 8... 

When scaling with geometric similarity, all linear dimensions—for example, the impeller diameter and blade width, the tank diameter, the distance that the impeller is off the bottom, the height of the liquid in the reactor, and the width of the baffles—scale as The scaleup relations are comparatively simple when scaling with geometric similarity and when the small-scale vessel is fully turbulent. The Reynolds number for a mechanically agitated vessel is defined as... 

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