Scale-Up Rules

Herbst et al. [International J. Mineral Proce.ssing, 22, 273-296 (1988)] describe the software modules in an optimum controller for a grinding circuit. The process model can be an empirical model as some authors have used. A phenomenological model can give more accurate predictions, and can be extrapolated, for example from pilot-to full-scale apphcation, if scale-up rules are known. Normally the model is a variant of the popiilation balance equations given in the previous section. 

Samant and Ng [28] compared various scale-up rules for agitated reactors. They suggested that a scale-up rule of power per unit volume and constant average residence time (where the power per unit volume and average residence time cannot be increased) is the most suited in many operations. However, this still may not improve or preserve the performance of the systems. Therefore, adequate consideration must be given to a tradeoff between performance and operating constraints. 

Follow the example of Reference [32], using scale-up rules. A pilot plant test run has been conducted using a laboratory equipped test vessel. Design equivalent process results for a 10,000 gallon tank are ... 

Scale-up rules have been established for liquid-liquid, liquid-solid, and liquid-gas systems [199]. 

Each dimensionless group represents a rule for scale-up. Frequently these individual scale-up rules conflict. For example, scale-up on dynamic similarity should depend chiefly upon a single dimensionless group that represents the ratio of the applied to the opposing forces. The Reynolds, Froude and Weber numbers are the ratios of the applied to the resisting viscous, gravitational and surfaces forces, respectively. 

Mutually conflicting scale-up rules are given by equations 5.32, 5.33 and 5.34. Other possible ways of scaling up are a constant tip speed uT, and a constant ratio of circulating capacity to head Qa a, and a constant power per unit volume P /F. Since PA is proportional to NiD and V is proportional to D, the ratio PA/T is proportional to N3D%. 

Potente, H., Existing Scale-Up Rules for Single-Screw Plasticating Extruders, Int. Polym. Process., 6, 267 (1991)... 

This statement is obviously useless as a scale-up rule because meat is bought according to weight and not to surface. We can remedy this simply. 

This is the scale-up rule for baking or cooking time in cases involving meat of the same kind a, p = idem). It states that when the mass of meat is doubled, the cooking time will increase by 2 = 1.58. 

Until rigorous scale-up rules are determined, these cautionary rules are the state of the art for now. We offer a first step toward rigorous scaling rules by scaling particle surface velocities but caution that this work is only preliminary in nature. The best advice is to be cautious—understand the physics behind the problem and that statistics of the data collected. Remember that a fundamental understanding of the issues is still limited and luck is unlikely to be on your side, hence frustrating trial-and-error is still likely (and unfortunately) necessary to be employed. 

Potente H. Existing scale-up rules for single-screw plasticating extruders. Int Polym Process 1991 4 267-278. 

Bigio D, Wang K. Scale-up rules for mixing in a non-intermeshing twin-screw extruder. Polym Eng Sci 1996 36(23) 2832-2839. 

Practical Scale-up Rules for Certain Reactor Types... 

PRACTICAL SCALE-UP RULES FOR CERTAIN REACTOR TYPES 6.2.1 Fixed beds... 

Using the above criteria, the other variables that determine reactor performance are affected. For example, using the scale-up rule of Eqn. 5.3-25, the process-side heat-transfer coefficient in the large reactor may be lower by 25% or so. Since the reaction may be strongly exothermic, the new heat transfer area should be scaled up, with more area per unit volume installed in the large reactor than in the pilot plant unit. 

As a conclusion, there is no one scale-up rule that applies to many different kinds of mixing operations. Theoretically we can scale up based on geometrical and dynamic similarities, but it has been shown that it is possible for only a few limited cases. However, some principles for the scale-up are as follows (Oldshue, 1985) ... 

The process conditions should remain the same, e.g., reactant concentrations, flow rate ratio, mixer type, relative feed position. If the friction factor, mixer voidage, and turbulence-generating efficiency do not vary significantly with scale, then the following scale-up rules can be applied (where k = 2new/ 2oid> the mixer diameter should be rescaled from dold to dnew). 

For the sake of completeness, two different scale-up rules are briefly considered here, and their different results are shown. First, let us define the stirring power P as... 

Information entropy offers a good possibility of becoming a consistent viewpoint to treat phenomena that must be expressed by the probability terms. By using information entropy, it will become possible to define the evaluation indices for mixing and separation operations/equipment, to estimate turbulent flow structure in a chemical equipment, to establish scale-up rules based on the turbulent flow structure, to present a general particle size probability density distribution, and to define the amount of anxiety/expectation. 

Challenge 4.3. Reliability of traditional scale-up rule of a stirred vessel n 13-18 19... 

Many scale-up rules have been used in industries. However, the reliability of these rules has not been discussed. In order to scale up equipment without causing any problem, it is indispensable to verify the reliability of these rules. Additionally, in this section, a new scale-up rule is proposed based on the viewpoint of information entropy. 

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