SPU27: Problem Set 6 Due on Canvas by 11 PM on Saturday, October 25th Please type or write your answers within this document or on a separate sheet of paper. Then save your work as either a Microsoft word document (.doc or .docx) or as a PDF file (.pdf) and upload to Canvas. If you write your answers by hand, you may scan your work and paste the images into either of these file types, but submissions that are NOT either of these file types will not upload properly. Your work must be organized and legible – if your TF can’t understand what you wrote, they won’t give you credit. Show your work for derivations and calculations. YOU WILL NOT RECEIVE FULL CREDIT WITHOUT SHOWING YOUR WORK. Be sure to calculate all results fully (don’t leave numbers in fraction form, or in terms of pi, etc) and to provide answers in the requested units, if applicable. Equations of the Week Concept Description Units L Distance of diffusion m D Diffusion coefficient m2/s t Time to diffuse s Problem 1: Lab Follow-up (28 points) In last week’s lab, you made molten chocolate cakes. The protocol called for you to measure the temperature at different distances from the center as it either steamed or baked. Because of how difficult it can be to collect data on your own using the original recipe for molten chocolate cake, we didn’t ask you to make it in an oven without a water bath. However, you can collect this data if you have enough people to help out. Below is a table with some sample data for molten chocolate cakes baked using the original recipe. Time (min) Width of crust (cm) Diffusion coefficient (cm2/s) 0 0 N/A 4 1.8 8 2.5 10 2.8 12 3.1 13 3.3 14 3.5 15 3.5 a. Use the equation of the week to estimate how long it will take for heat to diffuse to the center of the cake? L = radius cake (3.5cm) + width of ramekin (0.5cm). See diagram below. Assume for now that the heat diffusion coefficient of cake batter is approximately 1.4·10-3cm2/s, since the cake batter is largely made of water. (3 points) b. Is this calculation consistent with how long you cooked the cake in lab? Why or why not? (2 points) c. Calculate the effective heat diffusion coefficient for each of the time points in the table and write them in the table (or recreate the table for this part of the question). You don’t need to show your work for each calculation, but write the equation you use to find D in general terms. Ignore the ramekin width in your calculations (i.e. use only the value in the table for L) (9 points) d. What is the average value for D? How does this average compare to the average of the D values you calculated in your lab?.(6 points) e. Here is a rough sketch of the temperature of the cake at different distances. The three lines show the heat distribution for the cake at three different time points (it’s a bit hard to see, but the solid line is the 0 time point, the dotted is the middle, and the dashed line is the last time point). Compare the temperature distribution within the cake to the thickness of the crust at these time points. What is the apparent transition temperature at which the cake batter solidifies? (4 points) f. Which ingredient in the cake batter is likely responsible for its solidification? (2 points) g. How does this ingredient’s transition temperature compare with the apparent transition temperature for cake batter from the graph? (2 points) Problem 2: Why is it hard to cook steak? (20 points) Please watch the “Why it’s hard to cook a steak” video online (http://cm.dce.harvard.edu/cs50player/youtube.html?title=&youtube_id=a_yTHsPypSk& srt_url=srts/a_yTHsPypSk.srt) as well as this New York Times video on how to cook a steak (http://www.nytimes.com/video/dining/100000001945249/how-to-cook-a- steak.html) a. What are the four reasons it is difficult to cook a steak? Name one method for solving each of these problems. (4 points) b. Why is it important to leave the fat on the steak? (2points) c. In the video, Jake Dickson says that letting the steak sit will correct many of your cooking mistakes. What kinds of mistakes could be fixed by letting the steak sit for 5-7 minutes after cooking? What is happening in the steak during this time? (2 points) Many properties of the perfect steak are also properties of delicious burgers. Please watch Nathan Myhrvold’s Cryoburger video: (http://cm.dce.harvard.edu/cs50player/youtube.html?title=&youtube_id=w3PvY7E5pgY& srt_url=srts/w3PvY7E5pgY.srt) e. What are the three steps to making a cryoburger? How does this recipe solve the difficulties presented in part a? (6 points) f. Heat can diffuse into or out of a material. For simplicity, we can consider diffusion moving OUT of a food to work as though “cold” was moving IN to a food by calculating the loss of heat as the movement of a “cold front” inward. In the case of the cryoburger, the “cold” from the nitrogen will move into the patty with a diffusion coefficient identical to that of heat in water (as we used in Problem 1). How far does the cold diffuse into the burger if it is frozen in liquid nitrogen for 30 seconds? (2 points) g. After being frozen, the burger should immediately go into the deep fryer. Why is this important? (2 points) h. How long does the burger need to stay in the deep fryer for the heat to penetrate as far as the cold had when you removed the burger from the nitrogen? (2 points) Problem 3: Fried Ice Cream (22 points) Fried Ice Cream is a dish that would seem impossible at first - how can you fry something that will melt? However, it’s a perfect illustration of heat transfer - the fact that heat takes time to diffuse through the ice cream ball means that short frying times can allow for a cooked breading exterior without melting all of the ice cream on the interior. The following is a recipe for fried ice cream from Allrecipes.com Fried Ice Cream 1 quart vanilla ice cream 3 cups crushed cornflakes cereal 1 teaspoon ground cinnamon 3 egg whites 2 quarts oil for frying 1) Scoop ice cream into ½ cup sized balls (8 in total). Place on a baking sheet and freeze until firm, about 1 hour 2) In a shallow dish, combine cornflakes and cinnamon. In another dish, beat egg whites until foamy. Roll ice cream balls in egg whites, then in cornflakes, covering ice cream completely. Repeat if necessary. Freeze again until firm, 3 hours. 3) In deep fryer or large, heavy saucepan, heat oil to 190℃ 4) Using a basket or slotted spoon, fry ice cream balls 1 or 2 at a time for 10 to 15 seconds (until golden). Drain quickly on paper towels and serve immediately. a. You decide that you want a thicker, delicious breading layer for each ice cream ball and that ½ cup is too large for one ball using this amount of batter. Instead, you make balls that are ⅓ cup in size. What is the radius of this smaller ice cream ball? Assume that 1 cup is equal to 237mL. (4 points) b. What is the surface area of each ⅓ cup ball? (2 points) c. The quart of ice cream in this recipe should give you 12 balls in total. Given this, what is the thickness of the egg white layer on each assuming that 1 egg white has a volume of 21 cm3? Assume that all of the egg whites get distributed equally on the balls. (4 points) d. How long will it take for the heat from the frying oil to diffuse through the entire layer of egg white? Assume that the heat diffusion coefficient of egg white is 1.4·10-3 cm2/s. (2 points) e. Assume that the cornflake layer around the egg white is 0.3cm thick. Do you think your ⅓ cup ice cream balls will melt using the frying times suggested by the original recipe? Why or why not? What other information do you need to know to know for sure? (6 points) f. You found out that your guests are going to be late at the last minute, but you have already started frying your fried ice cream balls. You want to see if you can freeze them in liquid nitrogen to prevent the heat from melting the interior after frying similar to how shocking vegetables works (see 5f below). Assuming you’re using the same ice cream balls as before, how long will you have to immerse them in liquid nitrogen to stop the melting of the ice cream? (2 points) g. What do you think would happen if you froze the ice cream in liquid nitrogen before frying instead of after? (2 points) Problem 4: Chocolate (10 points) a. What temperature does Enric Rovira cool his chocolate to in order to achieve a proper temper? (2 point) b. You pour a layer of chocolate that is 5mm thick onto a room temperature (23˚C) slab of marble. You want to know how long it will take for the heat to diffuse out of the chocolate.