Solution. outside Styrofoam

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Stanley Hutchinson
6 years ago
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1 Ice chest You are in charge o keeping the drinks cold or a picnic. You have a Styrooam box that is illed with cola, water and you plan to put some 0 o ice in it. Your task is to buy enough ice to put in the box at am so that the temperature stays at 0 o C until the picnic starts at pm. You don t want to buy too much ice because that means that you ll have less money to spend on ood and other picnic items. You plan on keeping cold six 2-liter bottles o cola. How much ice will you need? Today will be beautiully sunny: a perect day or a picnic. The temperature will start out at 10 o C at am and then the temperature will steadily increase, reaching 35 o C at pm. It will be partially cloudy with a mild breeze rom the west. 1

2 Solution Strategy Our ice chest is a box o thickness and total area A. The chest will be let or 12 hours, and some heat will enter rom the. On the other hand, the cola, whose initial temperature is above 0C, will also release some heat cola onto the ice. The goal is to make sure that total heat absorbed by the ice is, at most, enough to melt all o it, so that the temperature never goes above 0C. We have thus two constraints or the low o heat in the system. The irst one is about the rate at which heat lows into the chest rom air. The second one is about how the heat exchange between all the parts o the system (, ice, cola) are related among them, and thus, ultimately, to the mass o ice, which is what we are looking or. Step 1 The rate o heat low between the inside and the o the chest depends on the thermal conductivity o Styrooam and on the temperature dierence between inside and : d dt k Styrooam A T out T in Since most o the inner surace o the chest is in contact with the ice (or icy water), the temperature T in in the equation above is 0C. (Also, the cola cools down to 0C pretty quickly, since it is not thermally insulated rom the ice. So the entire contents o the chest will be at 0C most o time.) We are also told that T out is expected to increase linearly with time. Thereore, through integration o the equation above, we can calculate how much total heat will enter the chest through the Styrooam in 12 hours. Note that we are not given the exact dimensions o the chest, i.e., the values o A and. These quantities are clearly needed, so we will simply make an estimation o the size o a typical Styrooam cooler. 2

3 Step 2 The total heat exchanged between all the parts o the system is dictated by the low diagram below: Air Ice melt Water Cola cola I the worst-case scenario is that all the ice just melts at pm, then 1 Note that melt depends on the mass o ice. cola melt Step 3 I we put together the expressions rom steps 1 and 2, we will be able to solve or the mass o ice. Detailed setup and calculation Step 1 As mentioned above, the rate o heat low between the inside and the o the chest is: d dt k Styrooam A T out T in with T in = 0C and T out increases linearly with time: Tout at b 1 Another way o writing this relation is 0, but then you must make sure that the terms have cola melt the appropriate sign: <0 or heat released by the ice and >0 or heat absorbed by the ice. 3

4 The constants a and b can be determined with the inormation given about the temperatures at am and pm (this will be done urther down, since we do not need the numerical values yet). Putting all this together, d dt A kstyrooam at b Through integration, we obtain the total heat that enters the cooler rom the between t = 0 ( am) and t = t ( pm) t A A1 2 kstyrooam at bdt kstyrooam at bt 0 2 [Equation 1] Step 2 As explained in the strategy section, the ice absorbed heat rom the and rom the cola: cola melt melt or a mass m o ice at 0C is: melt m with ice = 334 kj/kg ice The heat released by the cola as it is cooled down rom its initial temperature T 0 to 0C is: 0 m c 0 T with c cola = c water = 418 J/(kgC) cola cola cola Note that this is a negative heat because it is calculated rom the point o view o the cola (released heat). For our energy balance, we are using only positive quantities: m c T cola cola cola 0 We are not told what the initial temperature o the cola is, but we can make an estimation o what that temperature would be based on the typical temperature in a supermarket. T 0 = 20C sounds like a reasonable value. Putting all this together gives: m m c T [Equation 2] ice cola cola 0 4

5 Step 3 Equations 1 and 2 together yield: And thus the minimum required amount o ice is: A1 m k at bt m c T 2 2 ice Styrooam cola cola 0 A1 2 kstyrooam at bt mcolac colat0 2 m [Equation 3] ice Numerical evaluation et us start by inding the values o the parameters a and b involved in the time-dependence o the temperature: Tout at b At am (which we shall take to be t = 0), the temperature is 10C. At pm, t = 12 h = s and the temperature is 35C: b 10 b 10C a b 35 a C/s A typical Styrooam chest has sides o about 0.4 m and thickness o about 2 cm. A cube has sides 2. Thereore, 0.02 m 2 2 A 0.4 m 0.9 m As mentioned beore, we also need to do an estimation o the initial temperature o the cola beore it was placed in the container. We shall take T 0 = 20C as a reasonable estimation o the temperature inside the supermarket. 2 One o the sides is probably not in contact with air but with the ground, whose temperature will be dierent than that o air. We are neglecting the special treatment o this surace as a irst approximation. 5

6 The rest o the quantities in equation 3: t 12 h s m 12 kg (12 liters o cola, or water, whose density is 1 kg / liter) cola And, rom the tables 3 : k Styrooam W m C J ccola cwat er 418 kg C 5 J ice kg With all these numerical values, we can evaluate the mass o ice in equation 3, as well as the magnitudes o the heat exchanged in the process: m.7 kg cola melt J J J Conclusions and checks The minimum amount o ice required under the assumed conditions is.7 kg. One easy, quick check is that this sounds like a reasonable number or a situation that we all have a daily-lie intuition on. We can also check on the limits o dierent quantities in equation 3: As expect by intuition, equation 3 does predict that more ice will be required i any o the ollowing happens: a) The cooler is larger (larger A) b) The cooler walls are thinner (smaller ) c) The cooler is made o a material that is a worse insulator (larger k) d) We want to increase the amount o drinks (large m cola) e) The cola is at a larger initial temperature T 0 3 In most tables, the units or these quantities are given in Kelvins rather than degrees Celsius. Since both heat and heat conductivity are proportional to a temperature increment, it does not make any dierence! In this problem, using Celsius seems more convenient because the target temperature is 0 C and that simpliies the calculations.

7 Also, in the (strange) limit where the cola is initially at 0C and the temperature is always 0C, too, we should require no ice at all. Indeed, i T = 0C, this means a = b = 0. I we also have T 0 = 0C, equation 3 gives m = 0. O course, the inal result will vary a little depending on your estimations or T 0, A and. It is worth noticing that, with our estimations, about hal o the heat absorbed by the ice comes rom the cooling o the cola to 0C. This should not be surprising: Styrooam is a good insulator, and cola (i.e., water) has a large speciic heat. This means that i we started with cola that is already cold (T 0 ~ 0C), then cola ~ 0 and the only heat we have to worry about is, whose value ( J) is independent o the amount o cola. In this case, equation 3 becomes: m 3.9 kg ice So you can save on ice by placing your cola in the ridge overnight assuming that you are not in charge o the electricity bill, otherwise you are just moving the cost rom one place to another. The ridge is probably more eicient that a Styrooam cooler, but that is really beyond the scope o this problem! In this calculation, we only took into account heat exchanges due to conduction. Convection and radiation are also present. Radiation eects can be minimized by placing the cooler in the shade. 7

Chapter 10 Test Form B

Chapter 10 Test Form B Chapter 10 Test Form A 1. B 2. A 3. A 4. B 5. D 6. B 7. B 8. A 9. A 10. A 11. B 12. D 13. A 14. C 15. No, heat and cold do not flow between objects. Energy transferred between objects changes the temperature