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978-0123869449 Chapter 1 Lecture Note
RADIATIVE HEAT TRANSFER Third Edition Michael F. Modest University of California at Merced SOLUTION MANUAL Academic Press New York San Francisco London PREFACE Most solutions have been checked, and some have been rechecked. Still, only a major miracle could have […]
978-0123869449 Chapter 1 Solution Manual Part 1
PREFACE This manual/web page contains the solutions to many (but not all) of the problems that are given at the end of each chapter, in particular for problems on topics that are commonly covered in a first (or, at least, […]
978-0123869449 Chapter 1 Solution Manual Part 2
1.6 A rocket in space may be approximated as a black cylinder of length L=20 m and diameter D=2 m. It flies past the sun at a distance of 140 million km such that the cylinder axis is perpendicular to […]
978-0123869449 Chapter 10 Part 1
CHAPTER 10 10.1 A semi-infinite medium 0 ≤z<∞consists of a gray, absorbing-emitting gas that does not scatter, bounded by vacuum at the interface z=0. The gas is isothermal at 1000 K, and the absorption coefficient is κ=1 m−1. The interface […]
978-0123869449 Chapter 10 Part 2
278 RADIATIVE HEAT TRANSFER 10.8 Repeat Problem 10.6, but assume that the temperature is uniform at 2000 K. Also, there is no heat production, meaning that the sphere cools down. How long will it take for the sphere to cool […]
978-0123869449 Chapter 11 Part 1
CHAPTER 11 11.1 Estimate the eigenfrequency for vibration, νe, for a CO molecule. Solution According to equation (11.25) the energy associated with a vibrational transition is, for a harmonic oscillator, ∆ǫij =ǫj−ǫi=hνe(vj−vi)=hνe∆v, where ∆vis the change in vibrational quantum number. […]
978-0123869449 Chapter 11 Part 2
CHAPTER 11 309 11.20 Consider the spectral absorption coefficient for a narrow band range of ∆ηas given by the sketch. Carefully sketch the corresponding k-distribution. Determine the mean narrow band emissivity of a layer of thickness Lfrom this k-distribution. Solution […]
978-0123869449 Chapter 11 Part 3
324 RADIATIVE HEAT TRANSFER 11.32 In a combustor the air-fuel ratio is controlled by measuring the total band absorptance of the fuel (methane) for its 3.3µm band. The mixtures inlet conditions are 1 atm total pressure, temperature is 400 K, […]
978-0123869449 Chapter 12 Part 1
CHAPTER 12 12.1 A mass of m(kg) of coal is ground into particles of equal size a(µm), which may be assumed to be “large” and black. Determine the optical thickness of the resulting spherical particle cloud, assuming that the particles […]
978-0123869449 Chapter 12 Part 2
350 RADIATIVE HEAT TRANSFER 12.13 A semi-infinite space is filled with black spheres. At any given distance, z, away from the plate the particle number density is identical, namely NT=6.3662 ×108m−3. However, the radius of the suspended spheres diminishes monotonically […]
978-0123869449 Chapter 14
CHAPTER 14 14.1 The gap between two parallel black plates at T1and T2, respectively, is filled with a particle-laden gas. Radiative equilibrium prevails, the particle loading is a fixed volume fraction, with particles manufactured from two different materials (one a […]
978-0123869449 Chapter 15 Part 1
CHAPTER 15 15.1 Derive the jump boundary condition for the diffusion approximation, equation (15.26), for the case of concen- tric cylinders. Assume the heat transfer to be one-dimensional (only radial, no azimuthal or axial dependence). Hint: Introduce a local Cartesian […]
978-0123869449 Chapter 15 Part 2
CHAPTER 15 391 Tmax =2172 K, Q′′′ qwall =|Ψ(0)|˙ κ=−5 2×106W/m3 5 m−1=5×105W/m2. 392 RADIATIVE HEAT TRANSFER 15.12 Do Problem 15.10 using the Milne-Eddington approximation. Solution For radiative equilibrium we have ∇·q=dq dz =˙ Q′′′,or dq dτ=˙ Q κ, and, […]
978-0123869449 Chapter 16 Part 1
CHAPTER 16 16.1 Consider a gray medium at radiative equilibrium contained within a long black cylinder with a surface temperature of T(r=R,z)=Tw(z). Find the relevant boundary conditions for the P1-approximation directly from equation (16.23), i.e., in a manner similar to […]
978-0123869449 Chapter 16 Part 2
CHAPTER 16 417 16.9 Two infinitely long concentric cylinders of radii R1and R2with emittances ǫ1and ǫ2both have the same constant surface temperature Tw. The medium between the cylinders has a constant absorption coefficient κ and does not scatter; uniform heat […]
978-0123869449 Chapter 17
CHAPTER 17 17.1 Consider a gray, isothermal and isotropically-scattering medium contained between large, isothermal, gray plates at temperatures T1and T2, and emittances ǫ1and ǫ2, respectively. Determine the radiative flux between the plates using the S2-approximation. dq dτ=(1 −ω)(4πIb−G), dG dτ=−1 […]
978-0123869449 Chapter 19
CHAPTER 19 19.1 A semi-infinite, gray, isotropically scattering medium, originally at zero temperature, is subjected to colli- mated irradiation with a constant flux qonormal to its nonreflecting surface. Set up the integral relationships governing steady-state temperature and radiative heat flux […]
978-0123869449 Chapter 2
CHAPTER 2 2.1 Show that for an electromagnetic wave traveling through a dielectric (m1=n1), impinging on the interface with another, optically less dense dielectric (n2<n1), light of any polarization is totally reflected for incidence angles larger than θc=sin−1(n2/n1). Hint: Use […]
978-0123869449 Chapter 20 Part 1
CHAPTER 20 20.1 A long, cylindrical furnace bounded by a cold, black wall of 1 m radius contains pure CO2that is isothermal at 1700 K and at a pressure of patm. Using the mean-beam-length method, determine the nondimensional wall heat […]
978-0123869449 Chapter 20 Part 2
466 RADIATIVE HEAT TRANSFER 20.8 The new planet in an adjacent solar system recently found by Penn State (and other) researchers has been determined to have an atmosphere consisting of nitrogen with 1% by volume NO. The planet’s surface has […]
978-0123869449 Chapter 21
CHAPTER 21 21.1 Consider the (highly artificial) absorption coefficient of Problem 11.22. Find narrow band averages for the absorption coefficient and the transmissivity using Monte Carlo integration (use mcint.f90 or write your own code). Compare with answers from Problem 11.22. […]
978-0123869449 Chapter 3 Part 1
CHAPTER 3 3.1 A diffusely emitting surface at 500 K has a spectral, directional emittance that can be approximated by 0.5 in the range 0 < λ < 5µm and 0.3 for λ > 5µm. What is the total, hemispherical […]
978-0123869449 Chapter 3 Part 2
CHAPTER 3 47 3.20 Determine the total, normal emittance of copper, silver and gold for a temperature of 1500 K. Check your results by comparing with Fig. 3-8. Gold 4.10 ×1050.820 ×1051.8293 ×10−20.1353 0.075 These ǫncoincide with the dashed line […]
978-0123869449 Chapter 4 Part 1
CHAPTER 4 4.1 For Configuration 11, Appendix D, find Fd1−2by (a) area integration, and (b) contour integration. Compare the effort involved. Solution (a) From equation (4.23) Scos θ1=ˆn1·s12 =ˆ k·s12 =z, Scos θ2=−ˆn2·s12 =−ˆı ·s12 =c, and, from equation (4.21) […]
978-0123869449 Chapter 4 Part 2
CHAPTER 4 83 4.16 2d 2d A2 A1 2d d/2 d/2 Consider the configuration shown; determine the view factor F1−2assuming the configuration a) to be axisymmetric (1 conical, 2 a disk with hole), or b) to be two-dimensional Cartesian (1 […]
978-0123869449 Chapter 5 Part 1
CHAPTER 5 5.1 Tsky fire ground h 4 4 4 3 2 1 A fire fighter (approximated by a two-sided black surface at 310 K 180 cm long and 40 cm wide) is facing a large fire at a dis- […]
978-0123869449 Chapter 5 Part 2
120 RADIATIVE HEAT TRANSFER 5.15 TC ss ∋ ∋ ss, ho ∋ ∋ vacuum helium ss, hi TC, hTC Tamb A thermocouple used to measure the temperature of cold, low pressure helium flowing through a long duct shows a […]
978-0123869449 Chapter 5 Part 3
140 RADIATIVE HEAT TRANSFER (b) With a silver foil the radiative resistance becomes R23 =1 A21 ǫ2−1+1 A2 +2 As1 ǫs−1+1 As +1 A31 ǫ3−1 =1 +1 A2ǫ2 As2 ǫs−1+1 A31 ǫ3−1, 2πLR23 =1 0.100×0.2+1 0.1045 2 0.02 −1+1 0.109 […]
978-0123869449 Chapter 5 Part 4
154 RADIATIVE HEAT TRANSFER 5.33 Consider Configuration 33 in Appendix D with h=w. The bottom wall is at constant temperature T1and has emittance ǫ1; the side wall is at T2=const and ǫ2. Find the exact expression for q1(x) if ǫ2=1. […]
978-0123869449 Chapter 6 Part 1
CHAPTER 6 6.1 An infinitely long, diffusely reflecting cylinder is opposite a large, infinitely long plate of semiinfinite width (in plane of paper) as shown in the adjacent sketch. The plate is specularly reflecting with ρs 2=0.5. As the center […]
978-0123869449 Chapter 6 Part 2
188 RADIATIVE HEAT TRANSFER 6.15 Repeat problem 5.7 for the case that the flat part of the rod (A1)is a purely specular reflector. Solution Including specular reflections the governing equations become: Ebi −X j1−ρs jFs i−jEbj =qi ǫi−X j ρd […]
978-0123869449 Chapter 6 Part 3
CHAPTER 6 201 0.5−1 0.5−1×0.5199 Q′ 1=q1b=−119.0 W/cm length of plate. q1=(1−0.5199)5.670 ×10−12 ×3004−15.84−0.2×0.6770 ×13.12 1 =−11.90 W/cm2, 202 RADIATIVE HEAT TRANSFER 6.26 w w w w T, ∋ S´ S´ (2) A2 A1 An infinitely long corner of […]
978-0123869449 Chapter 7 Part 1
CHAPTER 7 7.1 φ qsun d d A1 A2 Two identical circular disks of diameter d=1 m are connected at one point of their periphery by a hinge. The configuration is then opened by an angle φ. Surface 1 is […]
978-0123869449 Chapter 7 Part 2
228 RADIATIVE HEAT TRANSFER Solv- ing the problem via GRAYDIFFXCH requires the following input, with results given below the code: PROGRAM GRAYDIFFXCH IMPLICIT NONE INTEGER,PARAMETER :: N=2 DOUBLE PRECISION :: A(N),F(N,N),EPS(N),HO(N),PIN(N),POUT(N),rr,qsun rr=0.1d0 qsun=1.d3 ! Surface 1 (concentrator) A(1)=12*rr HO(1)=(2*pi-2)*rr/A(1)*qsun ! […]
978-0123869449 Chapter 8 Part 1
CHAPTER 8 8.1 Prepare a little Monte Carlo code that integrates I(z)=Rb af(z,x)dx. Apply your code to a few simple integrals, plus si(z)=−Zπ/2 0 e−zcos xcos(zsin x)dx =Si(z)−π 2. Note: Si(1) =0.94608. A=0. B=1.570796 WRITE(*,*) ’INPUT # OF BUNDLES’ READ(*,*) […]
978-0123869449 Chapter 8 Part 2
CHAPTER 8 253 WRITE(LU,*) ’ZONE’ ! ! BREAK INTO GROUPS OF 100S, INITIALIZE VIEW FACTORS PSI=2.*PI*RAN1(ISEED) ! azimuthal angle of emission COSPS=COS(PSI) SINPS=SIN(PSI) SINSQTH=RAN1(ISEED) ! sinˆ2 of polar angle of emission SINTH=SQRT(SINSQTH) COSTH=SQRT(1.-SINSQTH) SX=SINTH*COSPS ! unit direction vector of emission, […]
978-0123869449 Chapter 9
CHAPTER 9 9.1 ∋ ∋ ad s R qsol A satellite shaped like a sphere (R=1 m) has a gray-diffuse surface coating with ǫs=0.3 and is fitted with a long, thin, cylindrical antenna, as shown in the adjacent sketch. The […]