CIRCLE YOUR DIVISION: Div. 1 (9:30 am) Div. 2 (11:30 am) Div. 3 (2:30 pm) Prof. Ruan Prof. Naik Mr. Singh
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1 CICLE YOU DIVISION: Div. (9:30 am) Div. (:30 am) Div. 3 (:30 pm) Prof. uan Prof. Naik Mr. Singh School of Mechanical Engineering Purdue University ME35 Heat and Mass ransfer Exam # ednesday, September, 00 Instructions: rite your name on each page Closed-book exam a list of equations is given Please write legibly and show all work for your own benefit. rite on one side of the page only. eep all pages in order You are asked to write your assumptions and answers to sub-problems in designated areas. Only the work in its designated area will be graded. Performance otal 00
2 Problem [30 pts] Consider a plane wall of thickness 00 mm and thermal conductivity /m- shown below. One side of the wall is perfectly insulated. he other side of the wall is exposed to surrounding air at temperature 5C and convective heat transfer coefficient of 0 /m -. he radiative heat loss from the same side to surrounding air is 400 /m. here is uniform volumetric heat generation of 0,000 /m 3 in the wall. q = 0,000 m 3 Assume that conduction through the wall is one-dimensional and at steady-state. (a) Calculate the temperature (C) of the wall surface exposed to the surrounding air. (b) rite the differential equation and the necessary boundary conditions to obtain temperature distribution (x) through the wall. Start with the generalized heat diffusion equation in rectangular coordinate system. Solve the differential equation to obtain (x). List your assumptions here [3 pts]: Steady-state One-dimensional conduction in the wall Constant Properties Uniform convective heat transfer coefficient on the surface Start your answer to part (a) here [0 pts]: Consider energy balance at the surface: E in E out E gen E st qv qconv qrad " 00 q LAhAs qrada0, 000 m 0 3 s 5 C 400 m 000 m - C m 85 C s
3 Problem - cont. Start your answer to part (b) here [7 pts]: Consider heat diffusion equation in rectangular coordinates: d q kx ky kz q Cp x x y y z z t dx k 0 d Boundary Conditions: () 0 at x = 0 (insulated); () dx s at x = 00 mm d qx Integrating: c dx k d qx Using (), we get: c = 0 dx k qx Integrating again: x c k 00 0,000 3 m m 000 Using (), we get: 85 C s c c 0 C m- qx emperature distribution: x 0 C k
![Problem [35 pts] An electronic chip has a rectangular cross section (width w = 80 mm, and height t = 0 mm) and negligible thickness, as shown in the figure. It dissipates heat at the rate of q = 0.](/docs-images/72/67310160/images/4-0.jpg "o assist heat removal to the ambient, a long fin made of aluminum with the same cross section and with a thermal conductivity k = 37 /m- is mounted to the left side of the chip.")
4 Problem [35 pts] An electronic chip has a rectangular cross section (width w = 80 mm, and height t = 0 mm) and negligible thickness, as shown in the figure. It dissipates heat at the rate of q = 0. o assist heat removal to the ambient, a long fin made of aluminum with the same cross section and with a thermal conductivity k = 37 /m- is mounted to the left side of the chip. On the right side, two long fins made of copper, with a square cross section 0 mm 0 mm are mounted, and the thermal conductivity is k = 400 /m-. he system is exposed to the ambient air with a convective heat transfer coefficient h = 00 /m - and far-field temperature = 4 C. (a) Find the steady-state temperature (C) of the chip c. (b) Using words, define the concept fin effectiveness. hen evaluate the effectiveness of the aluminum fin starting from your definition. List your assumptions here [3 pts]: Steady-state One-dimensional conduction in fins Constant Properties Negligible contact resistance Convective heat transfer coefficient same for fin and unfinned surface Start you answer to part (a) here [ pts]: Consider energy balance for the chip: q qfin, qfin, Cu qunfinned, Cu q M hp k A ; P wtand Ac, wt fin, c, b q M hp k A ; P 4tand fin, Cu Cu c, Cu b q ha ; Aunfinned wt t unfinned, Cu unfinned b Ac, m37 m c t m m m400 m c m m- 000 m m c c 3 C
5 Problem - cont. Start your answer to part (b) here [0 pts]: Fin effectiveness is the ratio of the rate of heat transfer with the fin to the rate of heat transfer that would exist without the fin from the cross-sectional area of the fin at the base m q fin, M hpk Ac, b kp m fin, ha 80 0 cb hacb hac, b hac, 00 m m fin, 7.
6 3. Problem 3 [35 pts] A cylindrical aluminum pipe with inner radius r = 0.5 m and outer radius r = 0.6 m as shown below carries water. he inner surface of the pipe is at a temperature i = 60 o C. Square insulation of width m experiences uniform convection heat transfer at h = 0 /m - and = 0 o C. Assume the following: k aluminum = 30 /m-, k insulation = /m-, and unit length of the pipe. (a) Assuming that the heat conduction through the cylindrical aluminum pipe is one-dimensional in the radial direction, draw the thermal resistance network from i to. Clearly label all the temperatures and calculate all the resistances involved (using appropriate D resistances and D shape factors). (b) Calculate the heat transfer rate () from the inner surface to the ambient. h=0 /(m.) =0 o C Insulation 0.5m 0.6m uminum i =60 o C m List your assumptions here [3 pts]: Stead- state Constant properties One-dimensional conduction through the aluminum pipe; two-dimensional conduction through the insulation Uniform convective heat transfer coefficient on the insulation surface Start you answer to part (a) here [5 pts]: hermal resistance network is shown below. cond, D r ln r Lk conv ha cond, D Sk insulation
7 Problem 3 - cont. D conduction resistance of the cylindrical aluminum pipe: r 0.6 m ln ln r 0.5 m cond,d cond,d Lk m37 m- D conduction resistance of the insulation: L m cond,d ; Using able 4.: S 0.7 m Skinsulation.08w.08 m ln ln D r. m cond,d cond,d m m- Convection resistance on the insulation surface: l conv ha h 4wL 0 4m m- Start your answer to question (b) here [7 pts]: otal resistance: total cond, D cond,d conv conv Heat transfer rate from the inner surface to the ambient: q q i total
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