2. [30] A computer chip is supplied with 35 W of electric power. The temperature of...

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2. [30] A computer chip is supplied with 35 W of electric power. The temperature of the chip is spatially uniform, but changing with time. The exterior of the chip is cooled by convective heat transfer to the room air, with parameters given in the table below. *Assume that the internal energy of the chip is given by the following equation: U=mcT, where the temperature is in K, and other values are given in the table. Use the MechE sign convention. Convective heat transfer coefficient h 100 0.008 Chip surface area exposed to the air. This number is larger than you might expect because the chip has fins to promote heat loss Temperature of the cooling air Mass of the chip m (appears in equation for U) Specific heat of the chip c (in equation for U) Chip temperature before the chip is turned on 300 0.030 750 300 W/(m2 K) m K kg J/(kg K) K a) What is the temperature of the chip when it reaches steady state? b) Now we will find how the chip temperature changes with time, after it is turned on. Write down an instantaneous rate version of the energy equation for the chip. Then substitute in expressions for du/dt and Q in terms of the chip temperature T. W (which is constant) should stay in the equation. This is a differential equation for T. W + hA hA We-at. Here, t c) Since some of you have not learned how to solve differential equations, you have a simpler job: to check the solution and find an expression for one of the parameters that appears in it. The solution to the equation is the following: T = Tair represents the time after the chip was turned on. Note that W is a negative number (following the MechE sign convention). Substitute this expression into the differential equation to find an expression and value for a, with units. d) Plot the chip temperature vs t (the time after the chip was turned on). Our expectation is well formatted computer generated plot. A well formatted and quantitative hand plot is also acceptable. - e) Find the values of T at t=0 and as t approaches infinity. Explain why you think these values make sense (or not). f) At t=0, what fractions of electrical power go into heat loss vs energy storage in the chip? As tinfinity, what fractions of electrical power go into heat loss vs energy storage in the chip? 2. [30] A computer chip is supplied with 35 W of electric power. The temperature of the chip is spatially uniform, but changing with time. The exterior of the chip is cooled by convective heat transfer to the room air, with parameters given in the table below. *Assume that the internal energy of the chip is given by the following equation: U=mcT, where the temperature is in K, and other values are given in the table. Use the MechE sign convention. Convective heat transfer coefficient h 100 0.008 Chip surface area exposed to the air. This number is larger than you might expect because the chip has fins to promote heat loss Temperature of the cooling air Mass of the chip m (appears in equation for U) Specific heat of the chip c (in equation for U) Chip temperature before the chip is turned on 300 0.030 750 300 W/(m2 K) m K kg J/(kg K) K a) What is the temperature of the chip when it reaches steady state? b) Now we will find how the chip temperature changes with time, after it is turned on. Write down an instantaneous rate version of the energy equation for the chip. Then substitute in expressions for du/dt and Q in terms of the chip temperature T. W (which is constant) should stay in the equation. This is a differential equation for T. W + hA hA We-at. Here, t c) Since some of you have not learned how to solve differential equations, you have a simpler job: to check the solution and find an expression for one of the parameters that appears in it. The solution to the equation is the following: T = Tair represents the time after the chip was turned on. Note that W is a negative number (following the MechE sign convention). Substitute this expression into the differential equation to find an expression and value for a, with units. d) Plot the chip temperature vs t (the time after the chip was turned on). Our expectation is well formatted computer generated plot. A well formatted and quantitative hand plot is also acceptable. - e) Find the values of T at t=0 and as t approaches infinity. Explain why you think these values make sense (or not). f) At t=0, what fractions of electrical power go into heat loss vs energy storage in the chip? As tinfinity, what fractions of electrical power go into heat loss vs energy storage in the chip?