Does convection suffice to clarify a broth?
COOKBOOKS MAKE MANY CURIOUS CLAIMS. One by the late Bernard Loiseau asserts that adding ice cubes to a cloudy broth “stuns” the suspended particles, causing them to fall to the floor of the stock pot. One may quibble with this way of stating the matter, but does it contain an element of truth?
The Modeling of Broth
Let’s begin by selecting particles close in size to those that actually cloud beef broth. Ground coffee is a good candidate because it consists of particles of various sizes. But because, unadulterated, it would excessively tinge the color of the broth, let’s dilute it by running water through the grounds until the coloring agents are rinsed out. The result is a black powder of mixed granularity.
Let’s now divide this powder into two equal parts and put them in identical glasses containing the same quantity of water. After we heat the contents of the two glasses in a microwave oven, the particles suspended in both liquids reveal the presence of energetic currents that cease after a few seconds. Now, very carefully, put ice cubes in one glass and in the other a mass of hot water equal to that of the ice cubes. Nothing happens in the latter case, but in the glass with the ice cubes the suspended powder shows signs of intense agitation.
The observed motion is not surprising: The ice cubes cool the water in the upper part of the glass while melting and releasing cold—that is, dense—water. This dense water falls, and the hot water at the bottom rises and cools on contact with the ice cubes, which are warmed in turn, and so on until the ice cubes have melted.
What happens to the particles? Have they all been “floored” by a knockout blow from the cold water? Not really. A strange segregation appears: Although both the large and small particles in suspension are carried downward by the convective current and seem to be deposited at the bottom of the glass, in reality the upward current carries the smallest particles back with it. Why don’t the large particles rise again as well? Probably because particles that fall in a fluid have a maximum speed.
If the liquid were immobile, the particles would be subject to two forces: the weight of the particles themselves, pulling them toward the bottom, and the upward thrust—or buoyancy—described by Archimedes’ principle (equal to the weight of the fluid displaced by the particles). Ultimately the particles wind up forming a deposit because they are denser than the water and because the resultant of these forces pushes them toward the bottom.
Yet the falling particles are subject to another force that slows them down. The intensity of this drag depends on the viscosity of the liquid and the radius and velocity of the particles. To simplify matters, let’s begin by considering this force in isolation, acting independently of the others. During free fall, the force is initially zero (because the rate of fall is zero), and the particles are accelerated by the downward force. Gradually, however, the upward drag asserts itself, offsetting the resultant of the weight and the Archimedean thrust so that the particles end up falling at a constant speed, which is their maximum velocity.
The Segregation of Particles
When the liquid rises, after having fallen to the bottom of the glass, it tends to carry both the small and the large particles with it. However, these particles have different maximum rates of fall and therefore react differently. Because their radius is small, the small particles fall to the bottom at a speed that is less than that of the fluid’s upward motion. By contrast, the large particles, with their greater maximum speed, fall too fast for the ascending fluid to be able to carry them back up again; they remain at the bottom of the glass.
How can we test this hypothesis? Would it be possible to reproduce the experiment in a more viscous fluid and in this way modify the maximum velocity of the particles? Marc Fermigier of the École Supérieure de Physique et de Chimie in Paris, citing Darcy’s law for flow through porous media, observes that the increase in viscosity may slow the speed of convection and thus alter the phenomena being studied: In pure water the current pushes the particles back up along the curve of its path, below the convection cell, because it is able to penetrate the granular medium of these particles; the more viscous the fluid, however, the harder it is for the current to pass between them.
Fermigier’s colleague Eduardo Weisfred adds that the large particles, being more inert than the small ones, tend to shift to another line of current below the convection cell, so that they are carried toward areas where the current is less swift and where sedimentation is possible; the small particles, on the other hand, follow the fluid throughout its complete motion.
What is the practical lesson of all this? That only the large particles form a significant deposit, and the small ones will continue to cloud the bouillon—just as the lamb, the innocent polluter in La Fontaine’s fable, is said to have fouled the wolf’s drinking water.