1. You are tasked with designing a heat sink to cool a flat surface that operates...

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1. You are tasked with designing a heat sink to cool a flat surface that operates at a temperature T₁ = 90°C. The dimension of this surface is a 5 cm x 5 cm square. Your plan is to cover this surface with rectangular fins. Each fin is as wide as surface (w = 5 cm), has a thickness t = 1 mm, and has a length L = 6 cm. All fins will be parallel to one another. The surface and the fins are made of aluminum, which has a thermal conductivity k = 225 W/(m*K). Ambient air will be blown over these fins with a temperature To = 25°C, resulting in a heat transfer coefficient h = 150 W/(m²*K) at the aluminum-air interface. a. First derive a symbolic expression for the total cooling of the surface, Q, at steady-state as a function of w, t, L, h, k, Th, To, N (the number of fins), and Qr (the rate of heat transfer associated with a single fin). You may use the form of Qr derived in class. Assume the spacing of the fins does not matter and that none of the fins are touching each other. b. Your initial design consists of 10 fins. What is the rate of cooling? How does this compare to the case where no fins are present (i.e., N=0)? c. Someone on your team overestimated the amount of clearance space available for the fins and you've now learned that 6 cm is too long of a length. You must reduce the length of your fins to 1.5 cm. How many fins are required to match or exceed the same level of performance as the original design? d. It should be clear from part a that the amount of cooling increases as the number of fins increases. Are there any practical limitations to using too many fins? Explain your answer. 1. You are tasked with designing a heat sink to cool a flat surface that operates at a temperature T₁ = 90°C. The dimension of this surface is a 5 cm x 5 cm square. Your plan is to cover this surface with rectangular fins. Each fin is as wide as surface (w = 5 cm), has a thickness t = 1 mm, and has a length L = 6 cm. All fins will be parallel to one another. The surface and the fins are made of aluminum, which has a thermal conductivity k = 225 W/(m*K). Ambient air will be blown over these fins with a temperature To = 25°C, resulting in a heat transfer coefficient h = 150 W/(m²*K) at the aluminum-air interface. a. First derive a symbolic expression for the total cooling of the surface, Q, at steady-state as a function of w, t, L, h, k, Th, To, N (the number of fins), and Qr (the rate of heat transfer associated with a single fin). You may use the form of Qr derived in class. Assume the spacing of the fins does not matter and that none of the fins are touching each other. b. Your initial design consists of 10 fins. What is the rate of cooling? How does this compare to the case where no fins are present (i.e., N=0)? c. Someone on your team overestimated the amount of clearance space available for the fins and you've now learned that 6 cm is too long of a length. You must reduce the length of your fins to 1.5 cm. How many fins are required to match or exceed the same level of performance as the original design? d. It should be clear from part a that the amount of cooling increases as the number of fins increases. Are there any practical limitations to using too many fins? Explain your answer.

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A The rate of heat transfer associated with a single fin Q can be calculated using the formula Q hwt Th Tair where Th is the temperature of the hot surface 90C in this case Tair is the temperature of ... View the full answer