Explanation Given Information: ρ = 995 k g / m 3 , q = ε C min ( T b − T c , i ) P e l e c = ε C min ( T b − T c , i ) 1800 = 2.047 × 10 − 3 × 29931.192 ( T b − 17 ) T b − 17 = 29.37 T b = 29.37 + 17 T b = 46.37 C C p = 4178 J / k g . K , Thermal conductivity of the aluminium heat sink , k o l = 180 w / m . K Number of fins, N=6 Thickness of each fin, t= 10 mm Pitch of the fin, S= 18mm Length of the fin, L f =50 mm Thickness of the base of heat sink, L b = 10 mm Dimensions of heat sink are : w 1 = 100 mm ; w 2 = 100 mm Velocity of the water flow, u 8 = 3m/s Power dissipation of the chips, P e l c = 1800 W Initial temperature of the water = T c , i = 17°C Let the base temperature of heat sink be T b Properties of water are given below: k = 0.62 W / m . K ; ρ = 995 k g / m 3 ; Pr = 5.2 , c p , c = 4178 J / k g ⋅ K ; v = 7.73 × 10 − 7 m 2 / s Since the base temperature of channels is almost uniform, the heat sink behaves as a single stream heat exchanger. Heat capacity of water is given as , C c = m ⋅ c c p , c = N ρ u ∞ L f ( S − t ) C p , c = 6 × 995 × 3 × 50 × 10 − 3 × ( 18 × 10 − 3 ) × 4178 = 29931.192 W / K Since the heat sink behaves as a single steam heat exchanger, C min = C c = 29931.192 W / K And heat capacity ratio tends to zero i.e ., C r → 0 Reynolds number for external flow of water is given as, Re x = u ∞ w 2 v = 3 × 100 × 10 − 3 7.73 × 10 − 7 = 388098.3 Average Nusselt number for fully developed laminar flow over flat plat is give by, N u ¯ = 0.664 Re x 1 / 2 Pr 1 / 3 = 0.664 × ( 388098.3 ) 1 / 2 × ( 5.2 ) 1 / 3 = 716.64 h ¯ w 2 k = 716.64 h ¯ = 716.64 100 × 10 − 3 = 4443.16 W / m 2 ⋅ K The thermal resistance of the base is given by, R t , b a s e = L b K a t w 1 w 2 = 10 × 10 − 3 180 × 100 × 10 − 3 × 100 × 10 − 3 = 5.55 × 10 − 3 K / W Corrected length of the fin is given by : L c = L f + ( t / 2 ) = 50 × 10 − 3 + ( 10 × 10 − 3 / 2 ) = 50 × 10 − 3 + 5 × 10 − 3 = 55 × 10 − 3 m Surface area of all fins is given : A f = 2 L c w 2 N = 2 × 55 × 10 − 3 × 100 × 10 − 3 × 6 = 0

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ISBN: 9780470501979

Author: Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine

Publisher: Wiley, John & Sons, Incorporated

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Chapter 11, Problem 11.90P

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The temperature of the heat sink.

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The temperature of the heat sink.

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