Examination Heat Transfer

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1 Examination Heat Transfer code: 4B680 date: 17 january 2006 time: hours NOTE: There are 4 questions in total. The first one consists of independent sub-questions. If necessary, guide numbers are given for the answers. Good luck!

2 Question 1 (all sub-questions can be solved independently) a) The heat transfer coefficient can be expressed as: h c = k T f ( Tw T ) y + y= 0 with y the normal wall coordinate, see figure. Derive this expression and make plausible (no derivation) that h c only depends on the Reynolds and Prandtl number for a forced convection flow. b) Consider a long cylinder with internal heat generation, which is exposed to a convection environment with a heat transfer coefficient h c (see figure). The situation is stationary. 1) Give the describing differential equation and the boundary conditions. 2) Solve for the temperature distribution T(r). c) Suppose that the following relation holds for a free convection problem along a flat plate 1/ 4 Nu x = C Gr x. C is a constant. Based on this, calculate the mean Nusselt number defined as Nu L h L L k and express the mean Nusselt number in the local Nusselt number Nu L at position L. L is the length of the plate. All material properties and the wall temperature can be taken constant.

3 Question 2 (all sub-questions can be solved independently) A spherical, thin-walled metallic vessel is used to store liquid nitrogen at 77 K. The vessel contains a saturated liquid-vapour mixture, which is kept at a constant pressure using a vent. The vessel has an inner diameter of 0.5 m and is covered with an evacuated, reflective insulation composed of silica powder with a thermal conductivity of W/m.K. The insulation is 25 mm thick, and its outer surface is exposed to ambient air at 300 K. The convection coefficient is known to be 20 W/m 2.K. The latent heat of vaporization and the density of liquid nitrogen are J/Kg and 804 kg/m 3, respectively. To start with it can be assumed that radiation from the insulation towards the environment can be neglected. a) Sketch the thermal circuit of the vessel and indicate the thermal resistances involved. To this end use the thermal conduction resistance as derived for spherical coordinates. b) Suppose now that the convection resistance from the nitrogen to the vessel and the conduction resistance of the vessel wall are negligible. Determine then the rate of heat transfer to the nitrogen. (q = 15 W) c) Determine the loss volume per day of liquid nitrogen and compare this with the total volume of the vessel. d) Assume now that the reflection coefficient of the insulation is equal to 0.9. Indicate if it is still allowed to neglect radiation to the environment. ṁh fg Vent Air Thin-walled spherical container, r 1 = 0.25 m 2 T, 2 = 300 K, h = 20W / m K Insulation outer surface, r 2 = m q Liquid nitrogen, 3 5 T = 77 K, ρ = 804kg / m, h fg = 2 10 J / kg, 1

4 Question 3 (all sub-questions can be solved independently) A counter-flow double-pipe heat exchanger is used to heat water from 20 0 C to 80 0 C at a rate of 1.2 kg/s. The heating is to be accomplished by geothermal water available at C and at a mass flow rate of 0.2 kg/s. Due to practical reasons the cold water flows through the inner tube, which is thin-walled and has a diameter of 1.5 cm. The inner diameter of the outer tube equals 4 cm. Assume the heat losses to the environment to be negligible. Take all water properties at C. a) Estimate the overall heat transfer coefficient. Assume both flows to be thermally and hydrodynamically fully developed. (U = 900 W/m 2.K) b) Determine the heat rate needed to heat the cold water from 20 0 C to 80 0 C and calculate the Logarithmic Mean Temperature Difference LMTD. (q = 20 kw; LMTD = 85 0 C) c) Using this LMTD, calculate the total length needed to heat the cold water from 20 0 C to 80 0 C. d) Determine the effectiveness of this heat exchanger, which is used in the ε- NTU method. 0.2 kg/s 0.1 kg/s

5 Question 4 (all sub-questions can be solved independently) Consider an air heating system, which consists of a half-round pipe and a flat bottom. The flat bottom is kept at a temperature of 1000 K, while the other surface is ideally insulated (no heat losses to the environment). The required heating rate per meter length of the furnace to keep the flat bottom at a temperature of 1000 K is denoted in the figure as q 1,ext (W/m). The pipe radius is 20 mm and both surfaces have an emissivity of 0.8. Air is flowing through the pipe with a mass flow rate of 0.01 kg/s and a bulk temperature of 400 K. So, convective heat transfer takes place between the walls and the air, in the figure denoted as q 1,conv en q 2,conv (W/m). a) Give the thermal network for radiation including the resistances and the driving potentials. Denote in this network also the convective and externally applied heat transfer rates. b) Determine the convective heat transfer coefficient h between the furnace walls and the air. (h = 25 W/m 2.K) c) Determine the temperature of the isolated surface. This temperature can be determined by setting up a balance between the convective heat transfer rate and the net radiation exchange between both surfaces. (T 2 = 600 K) d) Determine the heat rate per meter length needed to keep the flat surface at a constant temperature of 1000 K.

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