Examination Heat Transfer

Transcription

1 Examination Heat Transfer code: 4B680 date: June 13, 2008 time: Note: There are 4 questions in total. The first one consists of independent subquestions. If possible and necessary, guide numbers are given for the answers. Good luck

2 Question 1 (all sub-questions can be solved independently) a) Consider a long cylinder submerged in a flowing fluid with a constant heat transfer coefficient h c (no variations in length and circumferential direction). At time t = 0 the cylinder temperature equals T = Ti and an internal heat source q is switched on. 1) Give the describing differential equation and accompanying boundary and initial conditions for the temperature in the cylinder as function of time t and position r. 2) Which dimensionless numbers determine the solution. 3) Give an expression for the temperature at t. hc b) If the heat transfer of a flow over a flat plate can be described by calculate then Nu L and express Nu L in Nu L (Nusselt at position x = L). Nu = 0.4 Re Pr, x 0,6 0,4 x c) Water at the rate of 68 kg/min is heated from 35 to 75 0 C by an oil having a specific heat of 1.9 kj/kg. 0 C. The oil enters the exchanger at C and leaves at 75 0 C. The overall heat transfer coefficient is 320 W/m 2.K. The fluids are used in a shell-and-tube heat exchanger with the oil making one shell pass and the water making two tube passes. Calculate the heat exchanger area using the figure below.

3 Question 2 (all sub-questions can be solved independently) A small ball of steel (diameter 4 cm) falls into a constant temperature bath. The initial temperature of the ball is C and the material properties of steel can be taken as: k =10 W/m K, ρ = 8000 kg/m 3 and c = 500 J/kg K The cooling medium is oil or water, the bath temperature is 20 0 C and the accompanying heat transfer coefficients are: Oil Water h = 100 W/m 2 K h = W/m 2 K a) Determine for both situations the Biot-number. Take the radius as characteristic length scale. Based on the value of the Biot-number determine whether a lumped capacity analysis is valid or not. Using the lumped capacity analysis determine for both situations: b) the contact time needed till the ball reaches a temperature of C; c) the amount of heat that is transferred to the environment when reaching this temperature of C. Using the full analysis (using the figures on the next page) determine for both situations: d) the contact time needed till the maximum temperature in the ball is dropped to C; e) the maximum occurring temperature difference in the ball at that time level. Based on the value of this maximum temperature difference determine again whether a lumped capacity analysis is valid or not.

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5 Question 3 (all sub-questions can be solved independently) A system for heating water from an inlet temperature of T m,i = 20 0 C to an outlet temperature of T m,o = 60 0 C involves passing the water through a thick-walled bronze tube (k = 50 W/m.K) having inner and outer diameters of 20 and 40 mm, respectively. The outer surface of the tube is well insulated, and electrical heating within the wall provides for a uniform generation rate of q = 10 6 W/m 3. It is assumed that this uniform heat source results in a constant heating rate q from the wall to the water. The water flow rate equals to m = 0,1 kg/s. a) How long must the tube be to achieve the desired temperature? (L = 20 m) b) Determine whether the flow is hydrodynamically and thermally fully developed at tube outlet. c) What is the local convection heat transfer coefficient at tube outlet? (h x=l = 1500 W/m 2 K) d) What is the inner surface temperature at tube outlet? e) What is the outer surface temperature at tube outlet?

6 Question 4 (all sub-questions can be solved independently) Two concentric cylinders with diameters D 1 = 10 cm and D 2 = 20 cm are situated in a large room with a temperature T 3 = 300 K. The cylinders are long L = 20 cm. Cylinder 1 is massive and has a temperature T 1 = 1000 K. Cylinder 2 is thin-walled. Furthermore the emissivities are equal to ε 1 = 0.8 and ε 2 = 0.2. The whole system is in thermal equilibrium.. Cylinder 1 Cylinder 2 D 1 = 10 cm L= 20 cm T 3 = 300K D 2 = 20 cm Room 3 a) Calculate the view factors F 12, F 13, F 23,i en F 23,o, with F 23,i the view factor from the inner side of cylinder 2 to the surrounding room 3 and F 23,o from the outer side of cylinder 2 to the surrounding room 3. Make use of the figures on the next page. (F 12 =0.8, F 13 =0,15, F 23,i = 0,23 en F 23,o = 0.9) b) Below the thermal network of the problem is presented. It concerns a 4-surface enclosure because cylinder 2 is radiating in 2 directions, indicated in the network with the radiosities J 2i for the inner surface and J 2o for the outer surface. Fill out the missing thermal resistances (first reproduce the thermal network) and determine all the values of the thermal resistances in the network. (R 13 = 150; R 12 = 20; R 23i = 30; R 23o = 8; r 1 = 5; r 2 = 35; all values in 1/m 2 ) = r 2 = r 2 = c) Set up the equations with which the radiosities J 1, J 2i, J 2o en J 3 can be calculated. Take for the values of the radiosities: J 1 = W/m 2, J 2i = W/m 2, J 2o = W/m 2, J 3 = 500 W/m 2. d) Determine the temperature T 2 of cyinder 2. e) Calculate the total radiation rate of cylinder 1 to cylinder 2 and the environment.

7 Zichtsfactoren voor twee concentrische cilinders van eindige lengte. Boven: buiten cilinder naar binnen cilinder; Beneden: buiten cilinder naar zichzelf.