A chip that is of length L 5 mm on a side and thickness t 1 mm is encased in a ceramic substrate, and its exposed surface is convectively cooled by a dielectric liquid for which h 150 W/m2 K and T 20 C. In the off-mode the chip is in thermal equilibrium with the coolant (Ti T). When the chip is energized, however, its temperature increases until a new steady state is established. For purposes of analysis, the energized chip is characterized by uniform volumetric heating with . Assuming an infinite contact resistance between the chip and substrate and negligible conduction resistance within the chip, determine the steady-state chip temperature Tf . Following activation of the chip, how long does it take to come within 1 C of q this temperature? The chip density and specific heat are 2000 kg/m3 and c 700 J/kg K, respectively.

Koshar Amy Brogan March 14 & 16, 2016 Week 10 Apportionment Part 2 Terms Review: Quota: o The value of how many items are going to be distributed before adjusted apportionment o Found by dividing the population by the standard devisor Employees / employees per manager = mangers going to each factory Standard Devisor: o The value that determines the rate at which an item is distributed o Found by dividing the total population by the number of items being distributed Employees of all factories combined / mangers available = employees per manager Practice Example 1: 12 iPads being sent among 3 high schools by population of students A: 3,700 B: 2,200 C: 6,100 Total: 12,000 Standard Devisor: 12,000 / 12 iPads = 1,000 students per iPad Quota: population / standard devisor Ex. A: 3,700 / 1,000 = 3.7 iPads Hamilton Method Quota Round down Highest fraction End result School A 3.7 3 .7 +1 4 School B 2.2 2 2 School C 6.1 6 6 Totals: 12,000 11 out of 12 12 Jefferson Method: After rounding down, adjust the standard devisor so that when it is used in the quota it rounds down to the total number of items available for apportioning. Jefferson Method Quota Round down Adjusted SD 1: 900 Round down End Result School A 3.7 3 4.11 4 4 School B 2.2 2 2.44 2 2 School C 6.1 6 6.77 6 6 Totals: 12,000 11 / 12 12 The outcomes just happened to come out the same in both processes. Do not rely on coincidence. Example 2: New country with 4 states creating a house with 20 representatives based on population State A: 25,000 State B: 18,000 State C: 29,000 State D: 31,000 Total: 103,000 Standard Devisor: 103,000 / 20 reps = 5,150 Hamilton Method Quota Round Down Highest fraction End result State A 4.854 4 .854 = +1 5 State B 3.495 3 3 State C 5.631 5 .631 = +1 6 State D 6.019 6 6 Total: 103,000 18 / 20 20 Jefferson Method Quota Round Down Q2: 5000 Q3: 4800 End Result State A 4.854 4 5 5.208 5 State B 3.495 3 3.6 3.75 3 State C 5.631 5 5.8 6.042 6 State D 6.019 6 6.2 6.458 6 Total: 103,000 18 / 20 19 / 20 20 20 Create: Try coming up with an example where the Hamilton and Jefferson outcomes will be different. Use the idea of 10 graphing calculators need to be distributed among 3 classes. How many students are in each class Identify Are each of these situations an example of apportionment 30 managers being sent out to 6 factories based on number of employees 500 reams of paper going to 10 office divisions based on number of employees 5 trucks being sent out to 18 warehouses by their daily output Why or why not Because the managers must be distributed as a whole, this is an example of apportionment Reams of paper can be portioned out by number of sheets as needed, so this is not an example The trucks can’t be split, so this is an example of apportionment even though there are more warehouses than trucks What are the units of the Standard Devisor and the Quota for each working example Standard Devisor: Population / item of apportionment Employees / manager = employees per manager Output per truck Quota Population / standard devisor o Quota = the number of items being apportioned Employees / employees per manager employees cancel out = managers Output / output per truck outputs cancel out = trucks Hamilton Drawbacks This process was used in the US, but in 1881 a study was made of its productability. At the time there were 299 seats in the House and Alabama had 8 representatives. The next year the number of seats went up, but Alabama lost a rep. They eventually made a law to give Alabama its rep back, but the issue of losing a rep even when the number of possible seats goes up is called the Alabama Paradox. Webster After the quota is reached, instead of rounding down as with Hamilton and Jefferson, we round up as we usually do in math, and then adjust from there as in Jefferson, but adjusting in either direction until the distribution is reached. If the outcome is too low find a smaller devisor If the outcome is too high find a larger devisor Example 3: 30 managers to be sent out to 3 factories based on the population of employees Factory 1: 120 Factory 2: 91 Factory 3: 200 Total: 411 Standard Devisor: 411 employees / 30 managers = 13.7 employees per manager Webster Method Quota Round Up Adjust devisor: 14 Round Up End result Factory 1 8.759 9 8.571 9 9 Factory 2 6.642 7 6.5 7 7 Factory 3 14.599 15 14.286 14 14 Total: 411 31 / 30 30 / 30