# Coffee Break Mistake

April 28, 2014
Here's a little puzzle to ruminate on during a coffee break. Maybe you'll win a gift card! Give it your best shot in the comment section below. Enter as many times as you want. A \$25 gift card will go to the first right answer. Another \$25 winner will be selected from the other right answers. We need your email to send the gift card so make sure you're registered (and you have verified your registration via email) when you leave your comment in the box below.

The Solution: What a bunch of smart readers we have! Here’s a solution, by no means covering all the bases.

Craig should have added the cream and sugar at the coffee shop for several reasons. The biggest being that the cream and sugar would have lowered the coffee temperature. The cold (or room temperature) obviously lowers the coffee temperature. But so does the sugar, not only because it adds a colder mass to the solution but because it also requires energy (absorbs heat) to break down the sugar crystal lattice structure so it can “dissolve”. Granted the latter effect is small but there were 3 packets of sugar so it could be significant (I’ll leave it to chemistry geniuses among our readers to figure it out if they have the neurotic inclination).

Other reasons for adding the cream and sugar are that both the coffee’s mass and its density (and viscosity) are increased. The increased mass decreases the rate of temperature change due to any heat loss mechanism. The increase in viscosity reduces the rate of the cup of coffee’s internal heat transfer between the warmer middle and the cooler cup surface where it loses heat due to various mechanisms.

Heat conduction between the cup and the interior of the bag is the biggest cause of cooling, and it’s close to proportional to the temperature difference. So pre-cooling by adding the cream and sugar would reduce this loss. But, as at least one astute reader picked up, there is the heat loss due to radiation (Stefan–Boltzmann law) which is approximately proportional to the temperature raised to the fourth power (again, readers who thrive on black body radiation calculations can have a ball).

Of course, besides adding the cream and sugar at the shop. Craig should have gotten double cups or wrapped the cup in napkins – all sorts of possibilities. However, the puzzle doesn't tell us that he didn't so those are off the table.

That’s my take. Let me know if I’m wrong or missed something.

The Puzzle: Jeff, a mechanical engineer and Craig, an electrical engineer, both work at Hinterland Electric. One cold, windy January day they find themselves at an unheated urban substation. Jeff, the senior guy, asks Craig to walk down and get both of them some coffee from the coffee shop a few blocks away. “We haven’t had lunch so I’ll take extra cream and 3 sugars. And I’m freezing so keep the coffee as hot as possible”, said Jeff, “Don’t let it get cold.”

Craig comes back with the two lidded take out coffees in a bag along with the creamers, sugar packs, napkins and stir sticks.

Jeff just shakes his head as he looks into the bag. “I know you electrical types don’t have many thermodynamic classes,” he said, “but I thought you’d do better than this!”

Why did Jeff say that? This is a tricky one. The most complete answer wins a \$50 gift card. Make sure you register AND verify your registration through email.