Examination Heat Transfer code: 4B680 date: June 13, 2008 time: 14.00-17.00 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
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 110 0 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.
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 400 0 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 = 1 000 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 100 0 C; c) the amount of heat that is transferred to the environment when reaching this temperature of 100 0 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 100 0 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.
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?
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 = 50 000 W/m 2, J 2i = 25 000 W/m 2, J 2o = 3 000 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.
Zichtsfactoren voor twee concentrische cilinders van eindige lengte. Boven: buiten cilinder naar binnen cilinder; Beneden: buiten cilinder naar zichzelf.