E32 Mixed

Topic

Kitchen mixing encompasses a range of operations—from stirring a sauce with a spoon to whipping cream with an electric mixer—all aimed at homogenizing concentration or temperature, or incorporating air into the fluid being mixed; the fluid's flow regime—whether laminar or turbulent—determines which of these mechanisms dominates the process. If the fluid reaches a turbulent regime rather than remaining in a laminar one, the mixing process consequently relies on a different dominant mechanism to homogenize the fluid.

The mixing process's Reynolds number—the same dimensionless quantity used in another model in this set—compares the inertial forces generated by the agitator's movement within the fluid against the viscous forces the fluid exerts in opposition to that movement. When the Reynolds number falls below a low threshold, the flow remains laminar and mixing proceeds slowly, dominated by the stretching of fluid layers over one another; conversely, when the Reynolds number exceeds a much higher threshold, the flow becomes turbulent and turbulent diffusion takes over, proving far more efficient than the layer-stretching characteristic of the laminar regime. Consequently, a higher Reynolds number results in more efficient mixing.

The power consumed by the agitator scales with its rotational speed and diameter in a manner that is far more than proportional to either parameter: doubling the rotational speed requires significantly more than double the power, while doubling the diameter requires even more additional power than doubling the speed does. The power number specific to a given agitator depends not only on the agitator's geometry but also on the flow regime—laminar or turbulent—in which the fluid exists.

In a turbulent flow regime, the fluid's Kolmogorov scales characterize its smallest eddies—the sites where mixing energy is ultimately dissipated. For a low-viscosity fluid (such as water or a liquid sauce), these Kolmogorov scales are very small, resulting in highly efficient mixing down to the molecular level; conversely, for a viscous fluid (such as dough or purée), turbulence fails to fully develop within the fluid, and mixing remains slow. Thus, higher fluid viscosity hinders the development of turbulence, thereby slowing the mixing process.

Consequently, based on the mixing Reynolds number and Kolmogorov scales introduced earlier, the agitator's mixing number—defined as the product of rotational speed and required mixing time—remains approximately constant for a given agitator geometry. This constancy implies that mixing time is inversely proportional to rotational speed; doubling the speed therefore halves the required mixing time. Spinning the agitator faster thus reduces the required mixing time—consistent with the constant mixing number—though this comes at the cost of the significantly higher power input necessitated by the increased rotational speed.

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