“Scale-up is the task of producing an identical, if possible, process result at a larger production rate than previously accomplished,” says Leon Levine of the engineering consulting firm Leon Levine & Associates, Albuquerque, N.M. (phone 505-856-5914). His chapters 10 and 11 in the book *Food Processing Operations and Scale-up *by K.J. Valentas, L. Levine, and J.P. Clark (Marcel Dekker, 1991) are among the few resources on this topic with a focus on food processing.

Levine states that scale-up usually, but not always, requires experimentation at two scales, though not necessarily at the scale that is targeted. The key objective is to identify a criterion that remains constant regardless of scale. An example is the ratio t/m^{2/3}, where t is the time required to reach a given internal temperature and m is mass. This ratio remains constant in cooking objects of different mass but essentially the same geometric shape, such as a turkey or roast.

This result can be derived from fundamental principles of heat transfer and geometry, but often experimental data must be manipulated to derive the appropriate criterion. If the underlying physics and chemistry of a process are understood, the scale-up criteria may be known or a form suggested. In the fields of heat transfer, mass transfer, and fluid flow, there are well-established criteria known as dimensionless numbers with such names as Prandtl, Nusselt, Sherwood, and Reynolds. These typically combine important parameters in such a way that the units or dimensions cancel and thus are independent of scale.

One of the best-known dimensionless numbers is the Reynolds number for fluid flow, which is defined as LVρ/μ where L is some characteristic dimension, such as pipe diameter, V is velocity, ρ is density, and μ is viscosity. For flow through a pipe, if the Reynolds number is below about 4,000, the flow is laminar, and above that value the flow is turbulent. This has a significant effect on residence time distribution, as in hold tubes for pasteurizers or aseptic processes. The residence time of the fastest particle in laminar flow is about half that of the average particle, while in turbulent flow, all of the fluid particles have about the same residence time. One can imagine developing a process in small-diameter tubing and then wishing to scale up to a larger flow rate and larger tubing. It is important to maintain the same Reynolds number, if possible, to assure the same process result.

Another instance where Reynolds number is important is in cleaning of process piping. For good cleaning-in-place (CIP), it is important that the flow be turbulent. In practice, in the United States, the minimum velocity in the cleaning cycle is specified to be 5 ft/sec regardless of the pipe diameter. This means that the Reynolds number is not constant at different pipe sizes, but it is more convenient and easier to enforce a constant velocity. European practice is to adjust velocity to maintain a turbulent Reynolds number but not to expend pumping power needlessly by keeping velocity constant.

Often there are secondary criteria which prevent exact scale-up. For example, in fluid flow, there may be pressure drop considerations which prevent exact duplication. In other cases, heat transfer may not scale up in the same way that the primary process criterion does. This situation often arises when scaling a kettle cooking or cooling process because the surface-to-volume ratio of a vessel goes down as the vessel gets larger while retaining geometrical similarity. This means that heating or cooling area will not remain in proportion to the mass of contents. One solution is to provide a supplemental heat-transfer surface either external to the kettle or by adding coils inside the vessel. Coils may change the mixing pattern in the vessel and can be hard to clean. An external heat exchanger requires additional investment, addition of a pump, and extra piping.

The lesson is that scale-up can be complex in even the simplest-appearing situations. What about seemingly very complex processes, such as extrusion? Levine points out that screw speed is one of the most important parameters in extrusion and that, combined with the number of die openings (assuming constant die shape and constant mix formula), it should be a primary experimental variable. Often, specific mechanical energy input or total energy input (mechanical and thermal) correlates well with product properties. Experimentation is still required at several scales (e.g., screw diameters in extrusion) to permit confident extrapolation to larger machines.

Solids mixing is a common operation where experimentation can occur at one scale and then be scaled up reliably. It is known that the time to mix powders to some desired degree in a given mixer is a function of the Froude number Fr, defined as N^{2}D/g, where N is rate of rotation, D is size of the mixer, and g is acceleration of gravity, all in consistent units. The product Nt is a function of Fr, and is determined by mixing at various values of Fr, usually by varying speed. (It is assumed that degree of fill is maintained constant, usually below 50%.)

Once the appropriate Fr is determined and fixed, then the speed at a larger scale is inversely proportional to the square root of the size. That is, as a mixer gets larger, the speed to maintain the constant Fr decreases. As a result also, the time to mix to a desired degree increases because Nt is constant. It was long believed that mix times for a given mixture remained constant upon scale-up, but Levine’s analysis shows that this is not true. There are obvious consequences in process design.

**Dimensionless Analysis**

When theory does not suggest the appropriate correlation or scale-up criteria, dimensionless analysis may help. “Dimensionless analysis is the method of reducing the equations that describe a process into a form containing no reference to units of measurement, that is, a dimensionless form, “ says Levine in his chapter on this topic. Furthermore, he says, “Unfortunately, . . . dimensionless analysis cannot be applied [to] the effect of process variables on most product qualities.” The problem with product qualities is that there is no absolute scale for subjective properties, such as taste or texture. However, the technique is very useful for such properties as temperature or degree of mixing.

It is not always necessary to develop new dimensionless numbers or groups—many have been found by researchers to be useful. Levine provides several tables of common dimensionless numbers, and similar lists can be found in other textbooks. For a given problem, one selects the appropriate dimensionless numbers, knowing the expected phenomena of interest (heat transfer, mass transfer, pressure drop, etc.) and conducts experiments in which the selected numbers are the parameters. If a correlation is found, then it can be used for scale-up.

An example of a useful result from such studies is the observation that power per volume should remain constant in fluid mixing scale-up, if possible. Secondary issues, as previously mentioned, may prevent exact application, perhaps because additional heat-transfer surface distorts the geometrical similarity or some other consideration applies. In fluid mixing, the shape of the agitator, shape of the vessel, degree of fill, and the baffling must all be similar for small-scale experimentation to apply on a larger scale.

Levine analyzes dough mixing, which can be very complex, and concludes that development time can be identical in a small and large mixer operated at the same speed if the two mixers are geometrically identical except for scale. However, he notes, that is rarely the case.

In another analysis, he concludes that the dimensionless power number P/(ρN^{3}D) is inversely proportional to Fr (at low values of Fr) for mixers of similar geometry but different scale and the same degree of fill. This permits measurement of power consumption at one scale and confident extrapolation to a larger scale. The result applies to such devices as pan coaters as well as typical mixers. Energy per unit mass increases at constant Fr, which may mean more damage to products at larger scale.

It is important to actually operate processes at their design conditions and to resist the temptation to overfill mixers, for instance. Most people know that clothes dry faster when a clothes dryer is not too full. Operations such as packaging usually perform better when run at something less than their theoretical maximum, because there are fewer interruptions when running at a more “comfortable” speed. These factors, while not expressed quantitatively, can be experienced and observed in most plants.

**A Challenging Task Not Often Taught**

Scale-up is one of the more challenging tasks facing the food engineer or food technologist. Often the underlying physics or chemistry is not well known, though when it is, great insight may be gained with careful experimentation. The common technique of experimental design must be carefully applied using the correct combinations of variables. Scale-up criteria and dimensionless groups convert the basic variables of temperature, time, size, power, mass, and pressure drop into parameters that have no dimensions and thus apply to any scale. If an experimental design is developed with the basic variables, such as power or time, it may become distorted and less useful when the variables are combined into dimensionless groups. Many experimental designs are orthogonal, meaning they are symmetrical in the experimental parameters. After variables are converted, this convenient relationship may be distorted. It is better to develop the orthogonal design with dimensionless parameters if possible.

It takes some degree of intuition and physical insight to simplify seemingly intractable scale-up problems. Much of this wisdom comes with experience, but studying pertinent examples can help in acquiring these skills. Sadly, the topic is not often taught in engineering, let alone in food science, so must be acquired on the job.

**by J. PETER CLARK**

Contributing Editor

Consultant to the Process Industries

Oak Park, Ill.

[email protected]

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