Scaling with Geometric Similarity

The case of a compressible fluid is more complicated since it is the inventory and not the volume that scales with A. The case of laminar flow is the simplest and is one where scaling with geometric similarity can make sense. 

Geometrically Similar Scaleups for Laminar Flows in Tubes. The pressure drop for this method of scaleup is found using the integrated form of the Poiseuille equation  

The same result is obtained when the fluid is compressible, as may be seen by substituting Sr = Si = S into Equations (3.40) and (3.41). Thus, using geometric similarity to scale isothermal, laminar flows gives constant pressure drop provided the flow remains laminar upon scaleup. The large and small reactors will have the same inlet pressure if they are operated at the same outlet pressure. The inventory and volume both scale as S. 

Geometrically Similar Scaleups for Turbulent Flows in Tubes. Integrating Equation (3.15) for the case of constant density and viscosity gives 

In laminar flow, the pressure drop is constant when scaleup is carried out by geometric similarity. In turbulent flow, it increases as the square root of throughput. There is extra pumping energy per unit volume of throughput, which gives 

Geometrically Similar Scaieups for Laminar Flow in a Tube 

The external area scales as 5 so that this design has the usual problem of surface area rising more slowly than heat generation. Thus, large scaieups using 

Geometrically Similar Scaleups for Turbulent Flow in a Tube 

Turning to the case where the working fluid is an ideal gas, substituting Sr = 

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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... 

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. 

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... 

Scaling in parallel or series is preferred when heat transfer is a dominant consideration. The third method, scaling with geometric similarity, is cheaper for reactions that permit adiabatic operation. 

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