Your company has developed a micro-end milling process that allows you to easily fabricate an array of very small fins in order to make heat sinks for various types of electrical equipment. the end milling process removes material in order to generate the array of fins. your initial design is the array of pin fins shown in the figure below. you have been asked to optimize the design of the fin array for a particular application where the base temperature is tbase = 120°c and the air temperature is tair = 20°c. the heat sink is square; the size of the heat sink is w = 10 cm. the conductivity of the material is k = 70 w/m-k. the distance between the edges of two adjacent fins is a, the diameter of a fin is d, and the length of each fin is l.
d
l
w=10cm
arrayoffins k=70w/m-k
120 cbaset 
20 c, air th 
a
air is forced to flow through the heat sink by a fan. the heat transfer coefficient between the air and the surface of the fins as well as the unfinned region of the base, h , has been measured for the particular fan that you plan to use and can be calculated according to:   0.4 0.3 2 w40 m k 0.005 m 0.01 m ad h                  mass is not a concern for this heat sink; you are only interested in maximizing the heat transfer rate from the heat sink to the air given the operating temperatures. therefore, you will want to make the fins as long as possible. however, in order to use the micro-end milling process you cannot allow the fins to be longer than 10x the distance between two adjacent fins. that is, the length of the fins may be computed according to: 10la  . you must choose the most optimal value of a and d for this application. a.) prepare a model using ees that can predict the heat transfer rate for a given value of a and d. use this model to predict the heat transfer rate from the heat sink for a = 0.5 cm and d = 0.75 cm. note that the number of fins should be computed based on how many can fit onto the heat sink base (rather than counting the number of fins in the figure). b.) prepare a plot that shows the heat transfer rate from the heat sink as a function of the distance between adjacent fins, a, for a fixed value of d = 0.75 cm. be sure that the fin length is calculated using 10la  . also be sure that the number of fins installed on the heat sink varies as a is varied. your plot should exhibit a maximum value, indicating that there is an optimal value of a. c.) prepare a plot that shows the heat transfer rate from the heat sink as a function of the diameter of the fins, d, for a fixed value of a = 0.5 cm. be sure that the fin length is
calculated using 10la  and the number of fins is computed. your plot should exhibit a maximum value, indicating that there is an optimal value of d. d.) determine the optimal value of a and d using ees' built-in optimization capability