Chapter 13: Radiation Heat Transfer

1. Chapter 13: Radiation Heat Transfer Yoav Peles Department of Mechanical, Aerospace and Nuclear Engineering Rensselaer Polytechnic Institute

2. Objectives When you finish studying this chapter, you should be able to: • Define view factor, and understand its importance in radiation heat transfer calculations, • Develop view factor relations, and calculate the unknown view factors in an enclosure by using these relations, • Calculate radiation heat transfer between black surfaces, • Determine radiation heat transfer between diffuse and gray surfaces in an enclosure using the concept of radiosity, • Obtain relations for net rate of radiation heat transfer between the surfaces of a two-zone enclosure, including two large parallel plates, two long concentric cylinders, and two concentric spheres, • Quantify the effect of radiation shields on the reduction of radiation heat transfer between two surfaces, and become aware of the importance of radiation effect in temperature measurements.

3. The View Factor • Radiation heat transfer between surfaces depends on the orientationof the surfaces relative to each other as well as their radiation properties and temperatures. • View factor is defined to account for the effects of orientation on radiation heat transfer between two surfaces. • View factoris a purely geometric quantity and is independent of the surface properties and temperature. • Diffuse view factor ─ view factor based on the assumption that the surfaces are diffuse emitters and diffuse reflectors. • Specular view factor ─ view factor based on the assumption that the surfaces are specular reflectors. • Here we consider radiation exchange between diffuse surfaces only, and thus the term view factorsimply means diffuse view factor.

4. The view factor from a surface ito a surface jis denoted by Fi→jor just Fij, and is defined as • Fij=the fraction of the radiation leaving surface i that strikes surface j directly. • Consider two differential surfaces dA1 and dA2 on two arbitrarily oriented surfaces A1 and A2, respectively. • The rate at which radiation leaves dA1 in the direction of q1 is: I1cos q1dA1 • Noting that dw21=dA2cos q2/r2, • the portion of this radiation that strikes dA2 is (13-1)

5. The total rate at which radiation leaves dA1 (via emission and reflection) in all directions is the radiosity (J1=pI1) times the surface area: • Then the differential view factor dFdA1→dA2 (the fraction of radiation leaving dA1 that strikes dA2) • The view factor from a differential area dA1 to a finite area A2 is: (13-2) (13-3) (13-4)

6. (13-5) • The total rate at which radiation leaves the entire A1 in all directions is • considering the radiation that leaves dA1 and strikes dA2, and integrating it over A1, • Integration of this relation over A2 gives the radiation that strikes the entire A2, (13-6) (13-7)

7. Dividing this by the total radiation leaving A1(from Eq. 13–5) gives the fraction of radiation leaving A1 that strikes A2, which is the view factor F12, • The view factor F21 is readily determined from Eq. 13–8 by interchanging the subscripts 1 and 2, • Combining Eqs. 13–8 and 13–9 after multiplying the former by A1 and the latter by A2 gives the reciprocity relation (13-8) (13-9) (13-10)

8. When j=i: Fii=the fraction of radiation leaving surfaceithat strikes itself directly. • Fii=0: for plane or convex surfaces and • Fii≠0: for concave surfaces • The value of the view factor ranges between zeroand one. • Fij=0 ─ the two surfaces do not have a direct view of each other, • Fij=1─ surface jcompletely surrounds surface.

9. View FactorsTables for Selected Geometries (analytical form)

10. View FactorsFigures for Selected Geometries (graphical form)

11. View Factor Relations • Radiation analysis on an enclosure consisting of Nsurfaces requires the evaluation of N2 view factors. • Fundamental relations for view factors: • the reciprocity relation, • the summation rule, • the superposition rule, • the symmetry rule.

12. The Reciprocity Relation • We have shown earlier that the pair of view factors Fij and Fji are related to each other by • This relation is referred to as the reciprocity relationor the reciprocity rule. • Note that: (13-11)

13. The Summation Rule • The conservation of energy principle requires that the entire radiation leaving any surface iof an enclosure be intercepted by the surfaces of the enclosure. • Summation rule ─ the sum of the view factors from surface i of an enclosure to all surfaces of the enclosure, including to itself, must equal unity. (13-12)

14. The summation rule can be applied to each surface of an enclosure by varying ifrom 1 to N. • The summation rule applied to each of the Nsurfaces of an enclosure gives Nrelations for the determination of the view factors. • The reciprocity rule gives 1/2N(N-1) additional relations. • The total number of view factors that need to be evaluated directly for an N-surface enclosure becomes

18. The Superposition Rule • Sometimes the view factor associated with a given geometry is not available in standard tables and charts. • Superposition rule ─ the view factor from a surface i to a surface j is equal to the sum of the view factors from surface i to the parts of surface j. • Consider the geometry shown in the figure below. • The view factor from surface 1 to the combined surfaces of 2 and 3 is • From the chart in Table 13–2: • F12and F1(2,3) and then from Eq. 13-13: • F13 (13-13)

21. The Symmetry Rule • Symmetry rule ─ two (or more) surfaces that possess symmetry about a third surface will have identical view factors from that surface. • If the surfaces j and k are symmetric about the surface i then • Using the reciprocity rule, it can be shown that

28. View Factors between Infinitely Long Surfaces: The Crossed-Strings Method • The view factor between two-dimensional surfaces can be determined by the simple crossed-strings method developed by H. C. Hottel in the 1950s. • Consider the geometry shown in the figure. • Hottel has shown that the view factor F1 2 can be expressed in terms of the lengths of the stretched strings as (13-16)

33. Radiation leaving the entire surface 1 that strikes surface 2 Radiation leaving the entire surface 2 that strikes surface 1 - = Radiation Heat Transfer: Black Surfaces • Consider two black surfaces of arbitrary shape maintained at uniform temperatures T1 and T2. • The net rate of radiation heat transfer from surface 1 to surface 2 can be expressed as • Applying the reciprocity relation A1F12=A2F21yields • For enclosure consisting of N black surfaces (13-18) (13-19) (13-20)

39. Radiation Heat Transfer: Diffuse, Gray Surfaces • To make a simple radiation analysis possible, it is common to assume the surfaces of an enclosure are: • opaque (nontransparent), • diffuse (diffuse emitters and diffuse reflectors), • gray (independent of wavelength), • isothermal, and • both the incoming and outgoing radiation are uniform over each surface.

40. For a surface i that is grayand opaque(ei=aiand ai+ri=1), the radiosity can be expressed as where: • For a surface that can be approximated as a blackbody (ei=1), the radiosity relation reduces to: (13-21) (13-22)

41. Radiation leaving entire surface i Radiation incident on entire surface i - = Net Radiation Heat Transfer to or from a Surface • The net rate of radiation heat transfer from a surface iof surface area Aiis expressed as • Solving for Gifrom Eq. 13–21 and substituting into Eq. 13–23 yields (13-23) (13-24)

42. In an electrical analogy to Ohm’s law, Eq.13-24 can be rearranged as where surface resistance to radiation is • For a blackbody Ri=0 and the net rate of radiation heat transfer in this case is determined directly from Eq. 13–23. (13-25) (13-26)

43. Reradiating surface ─ an adiabatic surface: • when convection effects is negligible, • under steady-state conditions. • Reradiating surface must lose as much radiation energy as it gains, thus: • The temperature of a reradiating surface is independent of its emissivity. Eq. 13-25 (13-27)

44. Radiation leaving the entire surface j that strikes surface i Radiation leaving the entire surface i that strikes surface j - = Net Radiation Heat Transferbetween Any Two Surfaces • Consider two diffuse, gray, and opaque surfaces of arbitrary shape maintained at uniform temperatures. • The net rate of radiation heat transfer from surface ito surface jcan be expressed as (13-28)

45. Applying the reciprocity relation AiFij= AjFji yields • In analogy to Ohm’s law where space resistance to radiation is (13-29) (13-30) (13-31)

46. In an N-surface enclosure, the conservation of energy principle requires • Combining Eqs. 13–25 and 13–32 gives (13-32) (13-33)

47. Methods of Solving Radiation Problems • In the radiation analysis of an enclosure, either • the temperature or • the net rate of heat transfer must be given for each of the surfaces. • Two methods commonly used: • surfaces with specified net heat transfer rate (Eq. 13-32) • surfaces with specified temperature (Eq. 13-33) (13-34) (13-35)

48. The equations above give Nlinear algebraic equations for the determination of the Nunknown radiosities for an N-surface enclosure. • Once the radiosities J1, J2, . . . , JNare available, the unknown heat transfer rates can be determined from Eq. 13–34. • The unknown surface temperatures can be determined from Eq. 13–35. • Two methods to solve the system of N equations: • direct method • very suitable for use with today’s popular equation solvers • network method • not practical for enclosures with more than three or four surfaces • simple and emphasis on the physics of the problem.

49. Radiation Heat Transfer in Two-Surface Enclosures • Consider an enclosure consisting of twoopaque surfaces at specified temperatures. • Need to determine the net rate of radiation heat transfer. • Known: T1, T2, e1, e2, A1, A2, F12. • Surface resistances: • two surface resistances, • one space resistance. • The net rate of radiation transfer is expressed as (13-36)

50. Simplified forms of Eq. 13–36 for some familiar arrangements