Explanation Given: Surface area of chip is 10 mm × 10 mm . Length of micro channel is 10 mm . Width of micro channel is 50 μ m . Height of channel is 200 μ m . Inlet temperature of water is 290 K . Flow rate of water is 10 − 4 kg/s . Temperature of chip and cap is 350 K . The number of micro-channels is 50 . Concept used: Refer to table A-4 “Thermo physical properties of air at atmospheric pressure” to obtain the density for air as 997 kg/m 3 , Specific heat as 4.197 × 10 3 J/kg ⋅ K , dynamic viscosity as 885 × 10 − 6 N ⋅ s/m 2 , thermal conductivity as 613 × 10 − 3 W/m ⋅ K and Prandtl number as 5.83 corresponding to 300 K temperature. Write the expression for Hydraulic diameter for micro channel. D h = 4 A c P Here, D h is the hydraulic diameter, A c is the area of cross-section, P is the perimeter of channel. Substitute H W for A c and 2 ( H + W ) for P in above expression. D h = 4 H W 2 ( H + W ) ..............(1) Here, H is the height of channel and W is the width of channel. Write the expression for mean velocity of air. u m = m ˙ ρ A c Here, u m is mean velocity of air, m ˙ is the mass flow rate of air and ρ is the density of air. Substitute H W for A c in above expression. u m = m ˙ ρ H W ..............(2) Write the expression for Reynolds number. Re D = ρ u m D h μ ..............(3) Here, Re D is the Reynolds number and μ is the dynamic viscosity. Write the expression for Nusselt number. Nu D = h D h k ..............(4) Here, Nu D is the Nusselt number, h is the convective heat transfer coefficient and k is thermal conductivity. Write the expression for overall heat convection coefficient. T m , o − T s T m , i − T s = exp ( − P L h ¯ m ˙ c p ) Here, T m , o is the temperature at outlet, T m , i is the temperature of fluid at inlet, T s is the surface temperature tube, L is the length of channel, m ˙ is the mass flow rate of air and c p is the specific heat of air. Substitute 2 ( H + W ) for P in above expression. T m , o − T s T m , i − T s = exp ( − 2 ( H + W ) L h ¯ m ˙ c p ) ..............(5) Write the expression for total heat dissipation by each micro-channel. q = m ˙ c p ( T m , o − T m , i ) N ..............(6) Here, q is the total heat dissipation by each micro-channel and N is the number of micro-channels. Calculation: Substitute 200 μ m for H and 50 μ m for W in equation (1). D h = 4 ( 200 μ m ) ( 1 m 10 6 μ m ) ( 50 μ m ) ( 1 m 10 6 μ m ) 2 [ ( 200 μ m ) ( 1 m 10 6 μ m ) + ( 50 μ m ) ( 1 m 10 6 μ m ) ] = 8 × 10 − 5 m Substitute 10 − 4 kg/s for m ˙ , 997 kg/m 3 for ρ , 200 μ m for H and 50 μ m for W in equation (2)

7th Edition

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 8, Problem 8.107P

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The water outlet temperature and the chip power dissipation.

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Chapter 8, Problem 8.105P

Chapter 8, Problem 8.108P

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The water outlet temperature and the chip power dissipation.

Solution Summary: The author explains the water outlet temperature and the chip power dissipation in the table A-4 "Thermo physical properties of air at atmospheric pressure".

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