# Catsup conundrum solved?

November 16, 1998

In the 9/7/98 issue of Design News, we challenged readers to tell us why the following problem cannot be solved: "How long will it take catsup to flow across Canada?"Several readers responded, and the editors chose Mike Holdoway's as the most thorough. A retired research chemist from the U.K., (where, he is careful to point out, the correct term is "ketchup") he writes that while difficult, it is by no means an insolvable problem--provided one makes a few assumptions. His humorous analysis is as follows:

By "flow," one assumes that the whole of Canada has to be covered and for the purposes of this calculation, I have assumed a depth of 1 mm. This depth was chosen to provide the minimum disruption to the Canadian economy and to minimize any inconvenience to the local population.

Given that the area of Canada is 3,849,675 square miles (9,970,610 sq km) and neglecting the 7.6% that is inland water and assuming that it is flat, I calculate that we need 9,970,610,000 cubic meters of catsup. Call it 10 billion liters to allow for theft and evaporation.

I have assumed that the flow would emanate from a single pile of catsup. Obviously, the best place to deliver the catsup would be on top of the Rockies, but logistics would prevent that and would violate my assumption that the country is flat. Similarly, economic factors would suggest that the catsup should emanate from the middle of the country but, for this exercise, I have assumed Quebec, since this might give the Separatists something else to worry about. Naturally, since Quebec is on the edge of the country, we will need more catsup since much of it will spread into the sea, Massachusetts, New York, and other neighboring land masses.

So to be on the safe side, we should order 20 billion liters of catsup--so make sure you have bought some shares in Heinz first!

If we assume that the catsup can be delivered in one pile, we may assume that it will be in the approximate form of a circular cone. To simplify matters, I have assumed that the height of the cone is equal to the diameter of the base. Naturally, this approximation can be improved upon by experiment.

Thus: 1/12p 3 diameter^{2} 3 height = 2 X 10^{10} l

So the height of the catsup pile would be the cube root of 24 3 10^{10}p or approximately 4 kilometers, and the base would be 4 kilometers from side to side. More than enough for the Separatists, don't you think?

Now, the rate of flow depends upon the height of the catsup, its density (taken as unity and thus ignored), and viscosity. Thus: rate = constant 3 (height/viscosity). The constant was found to be 42, by experiment on the kitchen table. Naturally, the rate will diminish with time as the height of the pile is reduced as the catsup spreads.

Unfortunately catsup is a non-Newtonian fluid and exhibits thixotropy, which means that it has a high viscosity which falls dramatically if shaken. Consequently, I have to assume there are no earthquakes during the experiment so that a viscosity of 500 centipoise may be assumed as an average value. It should be noted that the annual Canadian snowfall will dilute the catsup, with a resultant decrease in viscosity. The hot Canadian summers, on the other hand, will cause evaporation, with a consequent rise in viscosity. For the purpose of this calculation, I have assumed that these two seasons will balance each other.

Thus, armed with this information, it is only necessary to set up a computer simulation calculating the height, and thus, the flow at various distances from the epicenter until the whole country is covered.

Unfortunately, my computer reported an "overflow error" (presumably in Quebec), but I think that we can all agree it will take quite a long time.

For his illuminating analysis of the catsup conundrum, Mike will receive a nifty Design News Mini MagLite(R) flashlight. Any complaints can be sent to Mike at [email protected] or e-mail your comments to us at [email protected]

## Condiment flow study available on web

Reader K. Foote, of General Dynamics, produced an intriguing and humorous four-page report on the catsup flow problem. It defines four flow equations but identifies at least 29 separate variables, including the brand of catsup, flow restriction due to migratory animals, splash pattern, and the correolis effect, that affect catsup flow across Canada. Foote believes that the study could be used by the Canadian Defense Department, if it ever investigates catsup as a means of repelling a foreign nation's amphibious assault on Quebec. For the full report, check out Design News' website at www.designnews.com/webx/catsup.pdf. Foote's shining insight earns him a Design News Mini MagLite(R) flashlight.

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