Practical chemical engineering.

It's day two of my training course, and as I contemplate my mug of decaf, I am suddenly flashing back to a question that was rumored to be part of the chemical engineering qualifying exam in my chemistry graduate program. As it's an intriguing problem, I thought I'd share it here:

In the dead of winter, a professor sends his grad student out into the cold to fetch him a hot beverage from the cafe. "Coffee with two creams, and make sure it's HOT when it gets to me!" the professor barks.

Shivering from fear as much as cold, the grad student procures a 12-ounce styrofoam cup of hot coffee and two little containers (maybe 20 mL each) of half and half at the cafe. To maximize the temperature of the coffee when it is delivered to the prof, should he add the half and half to the coffee before he walks it through the cold or after?

Feel free to work together on this problem, and please show your work in the comments.

21 responses so far

  • Nat says:

    nice try, but I stopped reading at "...mug of decaf..."


  • Ryan says:

    The coffee with the half and half in it has a lower surface area to lose heat than the coffee with the half and half out, doesn't it?

  • NFQ says:

    Ryan, I was thinking the same thing about the surface area. How to weigh this against the fact that adding the room-temperature half-and-half will lower the coffee's temperature a bit right away?

  • melissasbench says:

    If I recall, rate of heat transfer scales with temperature difference. The steaming hot coffee will lose more heat to the cold than slightly cooler coffee. So add the half-and-half right away.

  • --bill says:

    12 oz coffee and 20 mL cream? trick question--inconsistent units.

  • Dan Hicks says:

    Part of the trick is that the rate of change of temperature is proportional to the difference between the temperature of the cup of coffee and the ambient temperature: coffee-without-half-and-half starts at a higher temperature but cools faster and coffee-with-half-and-half starts at a lower temperature but cools slower. Since I don't remember just what the heat equation looks like, though, I don't remember just how those interactions play out.

    And let me complain about how the problem's underspecified. If the cafe is sufficiently far away, the coffee will cool to within an epsilon of the ambient temperature whether the half-and-half is added or not.

  • xander says:

    It doesn't matter. As soon as the grad student gets back to the lab, she should toss the "coffee"* into a convenient microwave, and reheat it. Alternatively, assuming that no microwave is convenient, empty the "coffee"/cream mixture into an Erlenmeyer flask, find a Bunsen burner, and heat the "coffee."

    * Decaf is not coffee.

  • As I recall, the underspecification of the problem was intentional. The students answering it were supposed to identify how particular issues like the outdoor temperature and travel distance would bear on the right thing to do.

  • Peter B says:

    Add the cream first.

    The Science: Conductive heat loss is linear with temperature difference. Radiative heat loss varies with the difference of the forth power of the two temperature.

    We then make several reasonable assumptions.

    1) The two creams start out colder than the coffee
    2) It’s colder outside than the coffee
    3) Unopened cream does not cool enough on the way back to make any difference.

    Now a little way back of the envelope math -
    let x = coffee heat loss if cream is added after returning
    let y = coffee heat loss if cream is added first

    Clearly x>y. Heat loss is the key to returning with the hotter coffee.

    Better to almost boil the coffee in a microwave. (Decaf is already spoiled anyway.) Add the cream and return. Warn the professor that the coffee is hot.

    FWIW, I had P Chem, a math class with a focus on physics and a little chemistry, at SJS 1963. Dr. Acravos(sp?) lectured from one 3x5 card filling whiteboards around the room twice in an hour. Her husband, also Ph.D., taught at Stanford.

  • becca says:

    Can the student carry the cream in their inside pocket (i.e. close to a 37 degree heat source)? If so, it's definitely better to add the cream after the walk. My mind went straight to Newtons law of cooling- which would kind of imply it would be better to add the cream immediately. I hadn't even considered surface area : volume.

    More importantly is this "the dead of winter" in say, Seattle? Or winter in Duluth? Cause I'm thinking that there are times where the *grad student* would freeze before getting back, let alone the coffee. I maintain that Duluth renders the question moot for that reason.

  • Nick says:

    "Your grad student will freeze before she reaches the first marker!"


    Although "The Empire Strikes Back" does bring to mind another way to keep the coffee warm...

  • bill says:

    1. the coffee will be hotter if you add the milk immediately, as per the temperature differential explanations above

    1a. unless the outside temperature is very cold, and/or the distance between cafe and prof is very long so that the transit time is long relative to the cooling rate -- then it will be cold on arrival no matter what

    2. why not buy two cups of coffee and four containers of milk and do the experiment?

    2a. if you did that, I'd be willing to bet that even if a thermometer could tell the difference in the prof's office, the prof won't be able to -- the effect you are measuring is below the limit of detection of your crucial readout

    3. wtf is wrong with the professor that he can't get his own fucking coffee? I'd spit in it.

    3a. that won't change the temperature much either

  • bill says:

    Hmm. If you put the milk under your armpit on the walk back, you might warm it up enough to overcome the temperature-differential effect. You could also hope that it would make the coffee taste like armpit, see 3 above.

  • Bill says:

    Hit the lab on the way back. Pour the coffee into a clean Pyrex. Apply flame until scalding hot. Replace coffee in styrofoam cup. Add half and half. Serve to professor.

  • Dennis says:

    Mix the cream in at the coffee shop, that way no one you know will notice you spitting in the coffee.

  • Tualha says:

    Doesn't matter. Grad student is doomed anyway, because the prof said "cream" and the student got half-and-half. Dooooooooooooomed, I say!

  • Autistic Lurker says:

    no clue about the half & half thing but thinking outside the box lead me to suggest taking two styrofoam container and to slip the one container having coffee in the second one to help insulation.


  • Argon says:

    The right answer: The student should replace the cream with urine.

  • Eric Juve says:

    I have been asked a similar question in two separate job interviews, the answer I gave was add the half and half before.

  • Doug Spoonwood says:

    I wonder how the student taking such an exam would fare if they proposed the following solution:

    Thermometers don't cost much, so we'll come up with similar conditions and hypothesize about why which solution works better after we do the following experiment. We'll first heat up a cup of coffee for x seconds and measure its temperature. Then we'll place it outside (perhaps right next to the building's door or maybe on a window sill) for 5 minutes (or whatever amount of time seems similar to the time it would take the student to walk from where s/he bought the coffee). Then we'll take a second cup of coffee in a similar cup and heat it up for x seconds. Next we'll put in the half and half, and then let this second cup stand outside for 5 minutes. We'll then measure its temperature also.

    Next we'll just compare the temperatures to determine which cup came out hotter. We'll also compare the difference of temperatures which allows us to determine (or at least hypothesize) whether adding the half and half before or after changes the temperature enough to get noticed by someone or make a real difference in heating a professor who weighs y pounds.

    I notice Bill already proposed this, I've just spelled out details. Even though I took an only took two semesters of physics and an introductory astronomy course at my public university back in the late 90s/early 2000s and I don't really know, I have trouble believing that a thinking student who proposed doing an experiment would get given credit in such a case... or on any other problem. How often do science students have exams where they get asked "describe an experiment or observation by which this hypothesis can get more fully explored." Having majored in philosophy I find it difficult to believe that shallow answers like "before" or writing what amounts to "after" in a "complete sentence" comes as preferable here (other than *merely* for quick grading purposes, which perhaps IS the problem), than a short-essay response above which gets *away* from the a priori hypothesizing/authority trusting "reasoning" of what the problem suggests as acceptable.

    It comes as one thing to not do experiments or observations because they of potential dangers or health concerns, or that you don't understand the techniques required and you need to develop them first, or because their cost gets prohibitive, or not everyone can do them do to time constraints, resources, etc. Make no mistake, there DO exist real reasons to infer theoretically instead of doing experiments. But, if there don't exist serious objections to doing an experiment, on what basis should one infer theoretically and propose that we infer theoretically like Aristotle? And why not *prefer* describing how realizable experiments could get done instead of inferring theoretically? How could real scientists prefer theoretical information in such a case to experimental information? Preferring theoretical inference over experimental information seems like preferring a synopsis of Kant's Critique of Pure of Reason to Kant's own book.