A HEAT TRANSFER TEXTBOOK - THIRD EDITION Episode 3 Part 5 pot

Problems 589 radiation but that it offers very little resistance to conduction. Thus, the window temperature is almost uniform.) 10.28 A very effective low-temperature insulation is made by evacu- ating the space between parallel metal sheets. Convection is eliminated, conduction occurs only at spacers, and radiation is responsible for what little heat transfer occurs. Calculate q between 150 K and 100 K for three cases: (a) two sheets of highly polished aluminum, (b) three sheets of highly polished aluminum, and (c) three sheets of rolled sheet steel. 10.29 Three parallel black walls, 1 m wide, form an equilateral trian- gle. One wall is held at 400 K, one is at 300 K, and the third is insulated. Find Q W/m and the temperature of the third wall. 10.30 Two 1 cm diameter rods run parallel, with centers 4 cm apart. One is at 1500 K and black. The other is unheated, and ε = 0.66. They are both encircled by a cylindrical black radiation shield at 400 K. Evaluate Q W/m and the temperature of the unheated rod. 10.31 A small-diameter heater is centered in a large cylindrical radi- ation shield. Discuss the relative importance of the emittance of the shield during specular and diffuse radiation. 10.32 Two 1 m wide commercial aluminum sheets are joined at a 120 ◦ angle along one edge. The back (or 240 ◦ angle) side is insulated. The plates are both held at 120 ◦ C. The 20 ◦ C sur- roundings are distant. What is the net radiant heat transfer from the left-hand plate: to the right-hand side, and to the surroundings? 10.33 Two parallel discs of 0.5 m diameter are separated by an infi- nite parallel plate, midway between them, with a 0.2 m diame- ter hole in it. The discs are centered on the hole. What is the view factor between the two discs if they are 0.6 m apart? 10.34 An evacuated spherical cavity, 0.3 m in diameter in a zero- gravity environment, is kept at 300 ◦ C. Saturated steam at 1 atm is then placed in the cavity. (a) What is the initial flux of radiant heat transfer to the steam? (b) Determine how long it will take for q conduction to become less than q radiation . (Correct for the rising steam temperature if it is necessary to do so.) 590 Chapter 10: Radiative heat transfer 10.35 Verify cases (1), (2), and (3) in Table 10.2 using the string method described in Problem 10.14. 10.36 Two long parallel heaters consist of 120 ◦ segments of 10 cm di- ameter parallel cylinders whose centers are 20 cm apart. The segments are those nearest each other, symmetrically placed on the plane connecting their centers. Find F 1–2 using the string method described in Problem 10.14.) 10.37 Two long parallel strips of rolled sheet steel lie along sides of an imaginary 1 m equilateral triangular cylinder. One piece is 1 m wide and kept at 20 ◦ C. The other is 1 2 m wide, centered in an adjacent leg, and kept at 400 ◦ C. The surroundings are distant and they are insulated. Find Q net . (You will need a shape factor; it can be found using the method described in Problem 10.14.) 10.38 Find the shape factor from the hot to the cold strip in Prob- lem 10.37 using Table 10.2, not the string method. If your instructor asks you to do so, complete Problem 10.37 when you have F 1–2 . 10.39 Prove that, as the figure becomes very long, the view factor for the second case in Table 10.3 reduces to that given for the third case in Table 10.2. 10.40 Show that F 1–2 for the first case in Table 10.3 reduces to the expected result when plates 1 and 2 are extended to infinity. 10.41 In Problem 2.26 you were asked to neglect radiation in showing that q was equal to 8227 W/m 2 as the result of conduction alone. Discuss the validity of the assumption quantitatively. 10.42 A 100 ◦ C sphere with ε = 0.86 is centered within a second sphere at 300 ◦ C with ε = 0.47. The outer diameter is 0.3 m and the inner diameter is 0.1 m. What is the radiant heat flux? 10.43 Verify F 1–2 for case 4 in Table 10.2.(Hint: This can be done without integration.) 10.44 Consider the approximation made in eqn. (10.30) for a small gray object in a large isothermal enclosure. How small must A 1 /A 2 be in order to introduce less than 10% error in F 1–2 if Problems 591 the small object has an emittance of ε 1 = 0.5 and the enclo- sure is: a) commerical aluminum sheet; b) rolled sheet steel; c) rough red brick; d) oxidized cast iron; or e) polished elec- trolytic copper. Assume both the object and its environment have temperatures of 40 to 90 ◦ C. 10.45 Derive eqn. (10.42), starting with eqns. (10.36–10.38). 10.46 (a) Derive eqn. (10.31), which is for a single radiation shield between two bodies. Include a sketch of the radiation net- work. (b) Repeat the calculation in the case when two radia- tion shields lie between body (1) and body (2), with the second shield just outside the first. 10.47 Use eqn. (10.32) to find the net heat transfer from between two specularly reflecting bodies that are separated by a specularly reflecting radiation shield. Compare the result to eqn. (10.31). Does specular reflection reduce the heat transfer? 10.48 Some values of the monochromatic absorption coefficient for liquid water, as ρκ λ (cm −1 ), are listed below [10.4]. For each wavelength, find the thickness of a layer of water for which the transmittance is 10%. On this basis, discuss the colors one might see underwater and water’s infrared emittance. λ (µm) ρκ λ (cm −1 ) Color 0.30.0067 0.40.00058 violet 0.50.00025 green 0.60.0023 orange 0.80.0196 1.00.363 2.069.1 2.6–10.0 > 100. 10.49 The sun has a diameter of 1.391 × 10 6 km. The earth has a diameter of 12,740 km and lies at a mean distance of 1.496 × 10 8 km from the center of the sun. (a) If the earth is treated as a flat disk normal to the radius from sun to earth, determine the view factor F sun–earth . (b) Use this view factor and the measured solar irradiation of 1367 W/m 2 to show that the effective black body temperature of the sun is 5777 K. 592 Chapter 10: Radiative heat transfer References [10.1] E. M. Sparrow and R. D. Cess. Radiation Heat Transfer. Hemi- sphere Publishing Corp./McGraw-Hill Book Company, Washing- ton, D.C., 1978. [10.2] M. F. Modest. Radiative Heat Transfer. McGraw-Hill, New York, 1993. [10.3] D. K. Edwards. Radiation Heat Transfer Notes. Hemisphere Pub- lishing Corp., Washington, D.C., 1981. [10.4] R. Siegel and J. R. Howell. Thermal Radiation Heat Transfer. Tay- lor and Francis-Hemisphere, Washington, D.C., 4th edition, 2001. [10.5] J. R. Howell. A Catalog of Radiation Heat Transfer Configuration Factors. University of Texas, Austin, 2nd edition, 2001. Available online at http://www.me.utexas.edu/∼howell/. [10.6] A. K. Oppenheim. Radiation analysis by the network method. Trans. ASME, 78:725–735, 1956. [10.7] W J. Yang, H. Taniguchi, and K. Kudo. Radiative heat transfer by the Monte Carlo method. In T.F. Irvine, Jr., J. P. Hartnett, Y. I. Cho, and G. A. Greene, editors, Advances in Heat Transfer, volume 27. Academic Press, Inc., San Diego, 1995. [10.8] H. C. van de Hulst. Light Scattering by Small Particles. Dover Publications Inc., New York, 1981. [10.9] P. W. Atkins. Physical Chemistry. W. H. Freeman and Co., New York, 3rd edition, 1986. [10.10] G. Herzberg. Molecular Spectra and Molecular Structure. Kreiger Publishing, Malabar, Florida, 1989. In three volumes. [10.11] D. K. Edwards. Molecular gas band radiation. In T. F. Irvine, Jr. and J. P. Hartnett, editors, Advances in Heat Transfer, volume 12, pages 119–193. Academic Press, Inc., New York, 1976. [10.12] H. C. Hottel and A. F. Sarofim. Radiative Transfer. McGraw-Hill Book Company, New York, 1967. References 593 [10.13] D. K. Edwards and R. Matavosian. Scaling rules for total absorp- tivity and emissivity of gases. J. Heat Transfer, 106(4):684–689, 1984. [10.14] M. Iqbal. An Introduction to Solar Radiation. Academic Press, Inc., New York, 1983. [10.15] J. A. Duffie and W. A. Beckman. Solar Engineering of Thermal Processes. John Wiley & Sons, Inc., New York, 2nd edition, 1991. [10.16] H. G. Houghton. Physical Meteorology. MIT Press, Cambridge, MA, 1985. [10.17] P. Berdahl and R. Fromberg. The thermal radiance of clear skies. Solar Energy, 29:299–314, 1982. [10.18] A. Skartveit, J. A. Olseth, G. Czeplak, and M. Rommel. On the estimation of atmospheric radiation from surface meteorological data. Solar Energy, 56:349–359, 1996. [10.19] P. Berdahl and M. Martin. The emissivity of clear skies. Solar Energy, 32:663–664, 1984. [10.20] J. A. Fay and D. S. Gollub. Energy and Environment. Oxford Uni- versity Press, New York, 2002. [10.21] J. Hansen, R. Ruedy, M. Sato, M. Imhoff, W. Lawrence, D. Easter- ling, T. Peterson, and T. Karl. A closer look at United States and global surface temperature change. J. Geophysical Research, 106: 23947, 2001. [10.22] R. T. Watson, editor. Climate Change 2001: Synthesis Report. Third assessment report of the Intergovernmental Panel on Cli- mate Change. Cambridge University Press, New York, 2002. Also available at http://www.ipcc.ch. [10.23] P. A. Stott, S. F. B. Tett, G. S. Jones, M. R. Allen, J. F. B. Mitchell, and G. J. Jenkins. External control of 20th century temperature by natural and anthropogenic forcings. Science, 290:2133–2137, 2000. [10.24] F. Kreith and J. F. Kreider. Principles of Solar Engineering. Hemi- sphere Publishing Corp./McGraw-Hill Book Company, Washing- ton, D.C., 1978. 594 Chapter 10: Radiative heat transfer [10.25] U.S. Department of Commerce. Solar Heating and Cooling of Res- idential Buildings, volume 1 and 2. Washington, D.C., October 1977. Part V Mass Transfer 595 11. An introduction to mass transfer The edge of a colossal jungle, so dark-green as to be almost black, fringed with white surf, ran straight, like a ruled line, far, far away along a blue sea whose glitter was blurred by a creeping mist. The sun was fierce, the land seemed to glisten and drip with steam. Heart of Darkness, Joseph Conrad, 1902 11.1 Introduction We have, so far, dealt with heat transfer by convection, radiation, and diffusion (which we have been calling conduction). We have dealt only with situations in which heat passes through, or is carried by, a single medium. Many heat transfer processes, however, occur in mixtures of more than one substance. A wall exposed to a hot air stream may be cooled evaporatively by bleeding water through its surface. Water vapor may condense out of damp air onto cool surfaces. Heat will flow through an air-water mixture in these situations, but water vapor will diffuse or convect through air as well. This sort of transport of one substance relative to another is called mass transfer; it did not occur in the single-component processes of the preceding chapters. In this chapter, we study mass transfer phenomena with an eye toward predicting heat and mass transfer rates in situations like those just mentioned. During mass transfer processes, an individual chemical species trav- els from regions where it has a high concentration to regions where it has a low concentration. When liquid water is exposed to a dry air stream, its vapor pressure may produce a comparatively high concentration of wa- ter vapor in the air near the water surface. The concentration difference between the water vapor near the surface and that in the air stream will drive the diffusion of vapor into the air stream. We call this evaporation. 597 598 An introduction to mass transfer §11.1 Figure 11.1 Schematic diagram of a natural-draft cooling tower at the Rancho Seco nuclear power plant. (From [11.1], courtesy of W. C. Reynolds.) In this and other respects, mass transfer is analogous to heat trans- fer. Just as thermal energy diffuses from regions of high temperature to regions of low temperature (following the temperature gradient), the mass of one species diffuses from regions high concentration to regions of low concentration (following its concentration gradient.) Just as the diffusional (or conductive) heat flux is directly proportional to a temper- ature gradient, so the diffusional mass flux of a species is often directly proportional to its concentration gradient; this is called Fick’s law of dif- fusion. Just as conservation of energy and Fourier’s law lead to equations for the convection and diffusion of heat, conservation of mass and Fick’s law lead to equations for the convection and diffusion of species in a mixture. The great similarity of the equations of heat convection and diffusion to those of mass convection and diffusion extends to the use of con- vective mass transfer coefficients, which, like heat transfer coefficients, relate convective fluxes to concentration differences. In fact, with sim- ple modifications, the heat transfer coefficients of previous chapters may often be applied to mass transfer calculations. [...]... Gas mixture air-carbon dioxide air-ethanol air-helium air-napthalene air-water T (K) 276 31 3 276 30 3 31 3 D12 (m2/s) 1.42×10 5 1. 45 6.24 0.86 2.88 argon-helium 2 95 628 1068 8 .3 32.1 81.0 (dilute solute, 1 )-( liquid solvent, 2) T (K) D12 (m2/s) ethanol-benzene benzene-ethanol water-ethanol carbon dioxide-water ethanol-water 288 298 298 298 288 2. 25 10−9 1.81 1.24 2.00 1.00 methane-water 2 75 33 3 0. 85 3. 55 ... molecules travel a distance close to one mean free path, —call it a , where a is a number on the order of one The molecular flux travelling rightward across x0 , from its plane of origin at x0 − a , then has a fraction of molecules of A equal to the value of NA /N at x0 − a The leftward flux, from x0 + a , has a fraction equal to the value of NA /N at x0 + a Since the mass of a molecule of A is MA /NA (where... 0. 85 3. 55 pyridene-water 288 0 .58 where kT is called the thermal diffusion ratio and is generally quite small Thermal diffusion is occasionally used in chemical separation processes Pressure gradients and body forces acting unequally on the different species can also cause diffusion Again, these effects are normally small A related phenomenon is the generation of a heat flux by a concentration gradient (as distinct... of aluminum foil as radiation shielding, the first measurements of velocity and temperature fields in a natural convection boundary layer, and a once widely-used graphical procedure for solving unsteady heat conduction problems He was among the first to develop the analogy between heat and mass transfer 4 Warren K Lewis (1882–19 75) was a professor of chemical engineering at M.I.T from 1910 to 19 75 and... (where NA is Avogadro’s number), the net mass flux in the x-direction is then jA x0 = η NC MA NA NA N x0 a − NA N x0 +a (11 .30 ) where η is a constant of proportionality Since NA /N changes little in a distance of two mean free paths (in most real situations), we can expand the right side of eqn (11 .30 ) in a two-term Taylor series expansion about Diffusion fluxes and Fick’s law §11 .3 611 Figure 11 .5 One-dimensional... away, with a net flow of carbon through the interface If the system is at steady state and, if a separate analysis shows that carbon is consumed at the rate of 0.00241 kg/m2 ·s, find the mass and mole fluxes through an imaginary surface, s, that stays close to the gas side of the interface For this case, concentrations at the s-surface turn out to be mO2 ,s = 0.20, mCO2 ,s = 0. 052 , and ρs = 0.29 kg/m3... pO2 R◦ T = (2.1 23 × 104 Pa) (30 0 K)( 831 4 .5 J/kmol·K) cO2 = = 0.00 851 0 kmol/m3 Finally, eqn (11.4) gives the partial density ρO2 = cO2 MO2 = (0.00 851 0 kmol/m3 ) (32 .00 kg/kmol) = 0.27 23 kg/m3 6 03 604 An introduction to mass transfer §11.2 Velocities and fluxes Each species in a mixture undergoing a mass transfer process will have an species-average velocity, vi , which can be different for each species in... evaporation, falls to the bottom, where it is collected and recirculated The temperature of the air rises as it absorbs the warm vapor and, in 59 9 An introduction to mass transfer 600 §11.2 the natural-draft form of cooling tower shown, the upper portion of the tower acts as an enormous chimney through which the warm, moist air buoys, pulling in cool air at the base In a mechanical-draft cooling tower, fans are... §11 .3 611 Figure 11 .5 One-dimensional diffusion x0 and obtain Fick’s law: jA x0 MA NA = η NC − 2a dmA = −2 a( C )ρ dx d(NA /N ) dx x0 (11 .31 ) x0 (for details, see Problem 11.6) Thus, we identify DAA = (2 a) C (11 .32 ) and Fick’s law takes the form jA = −ρDAA dmA dx (11 .33 ) The constant, a, in eqn (11 .32 ) can be fixed only with the help of a more detailed kinetic theory calculation [11.2], the result of which... to pull air through the packing Mechanical-draft towers are much shorter and can sometimes be seen on the roofs of buildings (Fig 11.2) The working mass transfer process in a cooling tower is the evaporation of water into air The rate of evaporation depends on the temperature and humidity of the incoming air, the feed-water temperature, and the air-flow characteristics of the tower and the packing When . sur- roundings are distant. What is the net radiant heat transfer from the left-hand plate: to the right-hand side, and to the surroundings? 10 .33 Two parallel discs of 0 .5 m diameter are separated by an. apart? 10 .34 An evacuated spherical cavity, 0 .3 m in diameter in a zero- gravity environment, is kept at 30 0 ◦ C. Saturated steam at 1 atm is then placed in the cavity. (a) What is the initial flux of radiant heat. Corp./McGraw-Hill Book Company, Washing- ton, D.C., 1978. [10.2] M. F. Modest. Radiative Heat Transfer. McGraw-Hill, New York, 19 93. [10 .3] D. K. Edwards. Radiation Heat Transfer Notes. Hemisphere Pub- lishing