problem4-77 - PROBLEM 4.77 KNOWN Heat sink for cooling...

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PROBLEM 4.77 KNOWN : Heat sink for cooling computer chips fabricated from copper with microchannels passing fluid with prescribed temperature and convection coefficient. FIND : (a) Using a square nodal network with 100 μ m spatial increment, determine the temperature distribution and the heat rate to the coolant per unit channel length for maximum allowable chip temperature T c,max = 75 ° C; estimate the thermal resistance betweeen the chip surface and the fluid, (m K/W); maximum allowable heat dissipation for a chip that measures 10 x 10 mm on a side; (b) The effect of grid spacing by considering spatial increments of 50 and 25 μ m; and (c) Consistent with the requirement that a + b = 400 μ m, explore altering the sink dimensions to decrease the thermal esistance. t,c f R r SCHEMATIC : ASSUMPTIONS : (1) Steady-state, two-dimensional conduction, (2) Constant properties, and (3) Convection coefficient is uniform over the microchannel surface and independent of the channel dimensions and shape. ANALYSIS : (a) The square nodal network with Δ x = Δ y = 100 μ m is shown below. Considering symmetry, the nodes 1, 2, 3, 4, 7, and 9 can be treated as interior nodes and their finite-difference equations representing nodal energy balances can be written by inspection. Using the, IHT Finite - Difference Equations Tool , appropriate FDEs for the nodes experiencing surface convection can be obtained. The IHT code along with results is included in the Comments. Having the temperature
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