Which scale-up equations or approaches have proven effective for vertical powder mixers?
For vertical powder mixers, there is no universally reliable scale-up equation as exists for many liquid agitators. Geometric similarity, constant circumferential speed, the number of tool revolutions, torque and power demand, and carefully conducted trials with the original product are the most important foundations. The Froude number, by contrast, plays a role mainly for powder mixers with horizontally rotating mixing tools; for vertical mixers it is not a general or primary scale-up criterion.
Why powders are more difficult
Liquids can be described comparatively well via density, viscosity, surface tension and other measurable material properties. For vertical liquid mixers, numerous design and scale-up approaches therefore exist, each taking the impeller geometry used into account. With known geometry, flow regime, power input, suspension behaviour, heat and mass transfer, and mixing time can often be calculated with usable accuracy, or simulated with CFD support.
For powders and bulk materials, this is substantially more complex. Their behaviour depends not only on particle size and bulk density, but additionally on particle shape, size distribution, moisture, cohesion, adhesion, electrostatic charging, abrasiveness, breakage strength, surface structure, air content, fill level, order of addition, and wall friction. Effects such as percolation, segregation, bridging, agglomeration, compaction, and particle breakage add to this. Even small variations in a raw-material lot can significantly change the flow and mixing behaviour of a recipe.
A vertical mixer can therefore show very different mixing kinetics for two outwardly similar powders. Furthermore, it is not enough to assess homogeneity solely within the mixing chamber. During discharge, conveying, screening, intermediate storage, or filling, a previously good mixture can segregate again. The effects of apparatus geometry, mode of operation, and product properties on the outcome are, for solids mixers, often not fully calculable; optimal conditions are therefore in many cases determined empirically through trials.
Suitable orientation criteria
Despite these limitations, several criteria can be used to prepare a scale-up in a technically sound manner. They do not replace a trial, but help to align the laboratory, pilot and production plant with one another in a sensible way.
Geometric similarity is the most important starting basis. Vessel proportions, cone angle, the ratio of mixing-tool to vessel diameter, helix pitch, clearance to the vessel wall, discharge geometry, feed positions, and fill level should remain as comparable as possible. Only then can similar product circuits and similar movement patterns develop. Even small changes in the tool-to-wall clearance, in the discharge area, in the vessel shape, or in internals can affect mixing quality, residual discharge and segregation tendency.
Constant circumferential speed is particularly relevant where product protection, particle breakage, attrition, or fines formation are to be limited. It keeps the maximum speed at the mixing tool within a comparable range. As tool diameter increases, the rotational speed must be reduced accordingly. This approach is helpful above all for sensitive granules, crystals, coated particles, fibres, sensitive food components, and recipes with limited mechanical load capacity. For convective vertical mixers too, circumferential speed can be a sensible starting point for transferring product stress.
Specific power input can serve as additional information. For powder mixtures, however, it is considerably less universal than for liquids. Power demand can change during a mixing process, for example through moisture uptake, liquid addition, compaction, deaeration, the breaking up of agglomerates, or the transition to a tough, plastic mass. A comparable power value therefore does not automatically mean that the same homogeneity, the same product protection, or the same discharge quality will be achieved at large scale.
The torque profile is particularly informative for many powder mixers. It shows how the product's resistance changes during charging, mixing, liquid addition, agglomeration, compaction, or drying. For scale-up, sufficient torque reserve must be available to reliably master unfavourable but permissible product states as well. This is particularly relevant where powdery mixes temporarily become cohesive, sticky, moist, or tough and plastic during the process.
The number of tool revolutions needed to reach the target homogeneity is a further practical figure. In geometrically similar vertical mixers, it can provide good initial orientation. It must, however, be confirmed through a mixing-kinetics study with the original product. Several mixing times are examined for this, and mixing quality is assessed by means of representative sampling. The aim is not the shortest mixing time under ideal conditions, but a robust process window that functions reliably under permissible variations in fill level, raw-material quality, and dosing.
The Froude number describes the ratio of inertial to gravitational influences. It is important above all for powder mixers with horizontally rotating mixing tools, when assessing throw pattern, centrifugal effects, mechanical fluidisation, or the movement behaviour of a rotating product bed. For vertical mixers with slow-running, wall-travelling helix or screw-band tools, by contrast, controlled convective product circulation is the primary consideration. The Froude number is therefore not a universal or primary scale-up criterion for this design.
Calculation and simulation
CFD simulations are often informative for liquids, because liquid flows can be described continuously. For powder mixtures, this is considerably more difficult. Discrete element methods, or DEM for short, can make the movement of individual particles, contact forces, local shear zones, and fundamental mixing mechanisms visible in model form. They can help compare variants of tool geometry, rotational speed, fill level, or internals with one another.
A DEM simulation, however, is only robust if it is carefully calibrated with real material data. Industrial powders contain an extremely large number of particles, often micrometre-sized and non-spherical. Fully modelling every particle would be computationally very demanding. In practice, particles are therefore simplified, enlarged, or modelled as spheres. In addition, friction, cohesion, rolling resistance, wall contact, and further parameters must be adjusted. Validated studies show that while DEM can represent the temporal development of mixing well, it can nevertheless significantly mispredict the actual mixing quality or RSD.
Simulations are therefore a useful supplement, but not a replacement for product trials. They can shorten development time, reveal critical variants, and support the selection of possible tool geometries. For the binding design of mixing quality, particle breakage, segregation tendency, residual discharge, discharge stability, and cleanability, trials with the original product remain necessary.
The reliable route
For powder mixtures, it is often more cost-effective, faster and more reliable to carry out meaningful trials in suitable mixers than to develop a purely theoretical model to a supposed level of certainty. In practical mixing trials, exactly the phenomena become visible that are hard to capture in simplified equations and simulations: agglomerate formation, wall build-up, percolation of fines, segregation during discharge, the influence of order of addition, liquid distribution, particle breakage, local over-wetting, and changing flow properties.
Trial planning begins with material characterisation. At minimum, particle size distribution, particle shape, bulk density, moisture, flow function, cohesiveness, wall friction, electrostatic properties, and, where applicable, abrasiveness should be recorded. For cohesive products, shear tests and wall-friction investigations can additionally be sensible.
This is followed by mixing kinetics with the original components. Different fill levels, real orders of addition, intended rotational speeds, several mixing times, and, where applicable, liquid additions are examined here. In addition to homogeneity within the mixing chamber, stability after discharge should also be checked. Samples from different zones of the mixing chamber are supplemented by discharge samples staggered over time, in particular from the beginning, the middle, and the end of the discharge.
The evaluation covers not only the mean and RSD or CoV of a marker component. It should also include conformance to the setpoint, minimum and maximum, particle breakage, proportion of agglomerates, residual discharge, dust generation, product temperature, and stability after conveying, intermediate storage, and filling. Only from this does it become clear whether a process can be operated reliably at industrial scale.
A professional scale-up therefore combines constructional similarity with measured data from trials. Geometry, fill-level window, tool concept, circumferential speed, torque reserve, and mixing time serve as the starting point. The final design is defined on the basis of the measured mixing kinetics, product movement, energy consumption, torque profiles, discharge, and actual discharge quality. Studies on cohesive granules do show that, under certain conditions, simplified scaling relationships can occur. Such results, however, are always material-, geometry- and process-specific, and cannot be generally transferred to other powders, mixing tools, or sizes.
Which scaling rules amixon® applies — and where trials replace them
For vertical precision mixers, product circulation can be approximately described using a conveying equation derived from screw-conveying technology. It explains why a vertical mixing helix can circulate the mix three-dimensionally: the product is conveyed upward near the wall, sinks back down in the centre under the influence of gravity, and then re-enters the outer mixing zone. These recurring product circuits continuously distribute all constituents throughout the entire mixing chamber and form the basis for a high, reproducible mixing quality.
The volumetric product flow I_V of a vertical mixing helix can be approximately described via the effective annular conveying area and the axial conveying speed. In plain-text formatting, the formula reads:
I_V = A · v_ax
Here, A is the effective cross-sectional area of the product stream moved by the mixing helix, and v_ax is the axial conveying speed of the mix.
For a vertical mixing helix, this results in the following simplified relationship:
I_V = (π / 4) · (D² − d²) · φ · S · n · ζ
Here, I_V denotes the conveying capacity or volume flow of the mixing helix. D is the outer diameter of the mixing helix, while d describes the inner diameter. The fill level φ indicates what proportion of the mixing chamber is filled with product. S denotes the pitch of the mixing helix, n the rotational frequency of the mixing tool, and ζ the velocity coefficient.
The velocity coefficient ζ accounts for the fact that real powders do not behave like an ideal conveying medium. Depending on moisture, cohesion, particle size, wall friction, particle shape, and fill level, they can slip, flow back, adhere, compact, or participate in the conveying motion to varying degrees. It is precisely this factor that makes clear that the conveying equation is a well-founded technical approximation, but not a complete description of the actual mixing kinetics of every bulk material.
The equation shows that convective circulation capacity increases with the effective helix area, the fill level, the helix pitch, and the rotational frequency. In geometrically similar mixers, the essential ratios of vessel, mixing helix, and outlet remain comparable. If, in addition, a comparable fill-level window and a product-gentle circumferential speed are maintained, the specific product circulation can be designed in a similar way across different sizes. The conveying equation is therefore a helpful starting point for sizing the mixing tool, rotational speed, and unit size.
It describes, however, only the main convective motion. A high theoretical conveying capacity alone does not yet guarantee an ideal mixture. For the actual mixing quality, the central downward flow, the cross-mixing, the order of addition, the homogeneity of dosing, the flow behaviour, the particle size distribution, and the stability of the mixture during discharge are likewise decisive. Especially with cohesive, moist, strongly differing-density, or very fine components, agglomeration, percolation, wall build-up, electrostatic charging, or particle breakage can alter the mixing kinetics. The velocity coefficient ζ is therefore to be understood as a product-dependent quantity that must be established through trials with the original product.
For amixon® precision mixers, the time course of mixing quality can additionally often be approximated with an empirical exponential function. This function describes the approach of an initially inhomogeneous product state toward an ideal, or practically achievable, limiting state. In plain-text formatting, the formula reads:
f(t) = C_ideal + (C_0 − C_ideal) · e^(−k · t)
Time t is plotted on the abscissa and corresponds to the mixing duration. The value f(t) on the ordinate describes the mixing quality, or a selected measurable property of the mixture, at the respective point in time. Depending on the evaluation, this can be, for example, the concentration of a marker component at a particular sample position, a dispersion figure, the coefficient of variation, the difference between the measured value and the setpoint, or another analytically determinable quality parameter.
C_0 is the initial value at time t=0. It describes the starting state before the mixing process begins, for example the high concentration scatter of a not-yet-homogenised mixture. C_ideal is the horizontal asymptote of the function. It describes the ideal or practically achievable limit value that the mixture approaches with increasing mixing time. If mixing quality is expressed as the coefficient of variation, C_ideal can, in idealised terms, tend toward zero. In industrial practice, however, a residual scatter often remains, arising from product characteristics, sampling, analytics, and the physically achievable mixed state. If the function describes a concentration, C_ideal instead corresponds to the setpoint of the target concentration.
The number e is Euler's number, with a value of approximately 2.718. It describes the exponential shape of the curve. The decisive parameter is the mixing-rate constant k. It determines how quickly mixing quality approaches the limit value. A high k value means rapid approach to the homogeneous state. A low k value points to a slower mixing process, for example due to unfavourable flow properties, high cohesion, limited product movement, unfavourable dosing locations, or an unsuitable mixing-tool and vessel design.
The empirical function is particularly valuable because it does not merely assess a single point in the mixing time, but describes the entire mixing kinetics. In a mixing trial, several mixing times are examined for this purpose. Representative samples are taken for each time and evaluated analytically. From the measured values, a mixing curve can be constructed and fitted to the exponential function. This allows the mixing-rate constant k, the achievable limit value C_ideal, and the time after which a predefined target mixing quality is reliably reached to be determined.
Such a mixing-kinetics study prevents the mixing time being chosen arbitrarily too short or unnecessarily long. With too short a mixing time, the desired homogeneity may not yet have been reached. An unnecessarily long mixing time, by contrast, increases energy input, particle stress, and process time, without further improving product quality. In some cases, mixing for too long can even increase the tendency to segregate, where differing particle sizes, bulk densities, or particle shapes are present. The practical aim is therefore a robust mixing-time window, not merely the highest theoretical homogeneity value at a single point in time.
The combination of the conveying-capacity estimate and empirical mixing kinetics provides a technically sound approach for the design and scale-up of vertical mixing processes. The conveying equation describes how quickly the mixing helix convectively circulates the product through the mixing chamber. The exponential function describes how the resulting mixing quality approaches the achievable limiting state over time. For geometrically similar amixon® precision mixers, these relationships can be an important basis for pre-planning unit size, fill level, rotational speed, mixing time, and the required mixing intensity.
Nevertheless, neither of the two equations replaces the product trial. The conveying equation idealises the product movement, and the exponential mixing curve is an empirical description of the observed mixing progress. Both must be verified with the original product, realistic fill levels, the planned order of addition, and representative sampling. Only then can the product-specific velocity coefficient ζ, the mixing-rate constant k, the actually achievable mixing quality, and a robust mixing-time window be established.