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Selected Topics in Transport Phenomena

Selected Topics in Transport Phenomena. Energy Transport. By Radiation (2). §16.4 Direct Radiation between Black Bodies in Vacuum (1). 1. Energy transport by radiation between two black bodies.

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Selected Topics in Transport Phenomena

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  1. Selected Topics in Transport Phenomena Energy Transport By Radiation (2)

  2. §16.4 Direct Radiation between Black Bodies in Vacuum (1) 1. Energy transport by radiation between two black bodies We choose the solid angle dw1=sin1d1d1 in such a way that dA2 is included in the cone formed by dw1 . On Lambert’s cosine law, the radiation of dA1 in dw1 is equal to (16.4-1)

  3. §16.4 Direct Radiation between Black Bodies in Vacuum (2) The project of dA2 on the plane normal to r12 is cos2dA2 . The transection area of the cone formed by dw1 on A2 is r122sin1d1 d1 Then the ratio Is the fraction of dQ1 falling onto dA2 . Because dA2 is a black body surface, this part of energy is absorbed completely by dA2 .

  4. §16.4 Direct Radiation between Black Bodies in Vacuum (3) Therefore, (16.4-5) Similarly, (16.4-6) The net energy transport from dA1 to dA1 is (16.4-7)

  5. §16.4 Direct Radiation between Black Bodies in Vacuum (4) The integration of Eq.(16.4-7) over A1 and A2givesthe totalenergy net transport fromA1 to A2 , (16.4-8) In the above derivation, we have introduced an assumption implicitly that the laws in §16.3derived at equilibrium state be valid in this non-equilibrium process.

  6. §16.4 Direct Radiation between Black Bodies in Vacuum (5) 2. The view factor By Eq.(16.4-8), Q12 is the product of two parts. The first part is the difference of total radiation intensities of two bodies The second part is the fraction of the totaldifference absorbed by A2 , which is determined only by the geometric relations.

  7. §16.4 Direct Radiation between Black Bodies in Vacuum (6) • Definition • It is convenient to rewrite Eq.(16.4-8) in the following form (16.4-9) in which F12, known as the view factor, is defined as (16.4-10) and stands for the fraction of total energy radiation of A1 being transported to A2.

  8. §16.4 Direct Radiation between Black Bodies in Vacuum (7) 2) Properties The view factor has the following properties (1) It can be proved immediately from the definition of the view factor. (2) IfA1is enclosed by nsurfaces Ai, then (16.4-10) It is easy to understand from the physical meanings of view factors because all radiation of A1 must fall onto the enclosure including itself.

  9. §16.4 Direct Radiation between Black Bodies in Vacuum (8) (3) Proof

  10. §16.4 Direct Radiation between Black Bodies in Vacuum (9) (4) Proof

  11. §16.4 Direct Radiation between Black Bodies in Vacuum (10) 3) View factor algebra By use of the basic properties of view factor and the combination of geometries, the view factors of many geometries can be derived from those known. For example, the view factors of two classes of geometries have been given in the textbook, adjacent rectangles in perpendicular planes and some opposed identical shapes in parallel planes. Then,

  12. §16.4 Direct Radiation between Black Bodies in Vacuum (11) (1) (2)

  13. §16.4 Direct Radiation between Black Bodies in Vacuum (12) (3) Since Hence

  14. Application Examples of View factor algebra (1) Body 1 is an isosceles right triangle with it vertices at (0, 0, 0), (0, a, 0) and (0, a, a) . Body 2 is a sphere of diameter b < a with its center at (a, 0, 0) . Please try to find the view factor F12 .

  15. Application Examples of View factor algebra (2) Body 1 is extended in a way shown in the right picture to form a cubic enclosure, denoted as body 3, consisting of 48 isosceles right triangles with body 2 locating in its center. Then

  16. Application Examples of View factor algebra (3) Because the radiation from the sphere are received equivalently by all the 48 triangles, the view factor form the sphere to any triangle should be The area of body 2 is The area of body 1 is Therefore,

  17. §16.4 Direct Radiation between Black Bodies in Vacuum (13) 4)Contour integration method Let A1 and A2 be two arbitrary surfaces enclosed by curves C1 and C2 respectively. The view factor from A1 to A2 is The shapes of A1 and A2 may be very complicated that the integration may be too difficult to be executed analytically.

  18. §16.4 Direct Radiation between Black Bodies in Vacuum (14) Because r1 is constant on A2 , the following relation is true

  19. §16.4 Direct Radiation between Black Bodies in Vacuum (15) And

  20. §16.4 Direct Radiation between Black Bodies in Vacuum (16) That is

  21. §16.4 Direct Radiation between Black Bodies in Vacuum (17) Therefore, by use of Stokes’ curl theorem, the surface integrals on A2 may be transformed into a contour integrals of A2.

  22. §16.4 Direct Radiation between Black Bodies in Vacuum (18) Exchanging the order of integrations on A1 and A2 , we have The integrand in the square bracket can be transformed into the curl of an isotropic tensor.

  23. §16.4 Direct Radiation between Black Bodies in Vacuum (19)

  24. §16.4 Direct Radiation between Black Bodies in Vacuum (20) By use of Stokes’ curl theorem, the surface integrals on A1 may also be transformed into a contour integrals of A1.

  25. §16.4 Direct Radiation between Black Bodies in Vacuum (21) Upon the above result, we have If the contours could be taken as the coordinate curves, the line integrals would be much simpler to execute than the original surface integral.

  26. Application Examples of Contour integration method(1) Adjacent Rectangles in perpendicular planes

  27. Application Examples of Contour integration method(2) Inserting the expressions of r12 intoit

  28. Application Examples of Contour integration method(3) The integration results

  29. §16.4 Direct Radiation between Black Bodies in Vacuum (22) 3. Energy transport by radiation in an enclosure of black bodies Consider an enclosure consisting of n pieces of black body surfaces, A1 , A2 , … , An . • The temperatures of n surfaces are kept at T1 , T2 , … , Tn, respectively. • The amount of energy needed for keeping the temperature of the ith surface is equal to the net heat transport of the surface to all other surfaces.

  30. §16.4 Direct Radiation between Black Bodies in Vacuum (23) On Eq.(16.4-9), the net radiation from Ai to Aj is Then, Since Hence (16.4-13)

  31. §16.4 Direct Radiation between Black Bodies in Vacuum (24) 2) All surfaces but the A1 and A2 are adiabatic , A1 and A2 are kept at T1 and T2. It is evident that the temperatures of all adiabatic surfaces must be between T1 and T2 . So, besides the direct radiation Qd,12, there must be indirect heat transport in following way,

  32. §16.4 Direct Radiation between Black Bodies in Vacuum (25) On Eq.(16.4-13), we have (16.4-a) (16.4-b) (16.4-c) Denoting , Eq.(16.4-c) may be rewritten as (16.4-d)

  33. §16.4 Direct Radiation between Black Bodies in Vacuum (26) It is a set of (n-2) linear equations. We might as well assume that the equations be linearly independent of each other. If not, for example, the ith and jth equations are linearly dependent, there must be Ti=Tj . Then, these two surfaces can be considered as a combined surface and n reduces by 1 so that we obtain a linearly independent set of (n-3) equations. Eq.(16.4-d) may be written in a form of matrix as (16.4-e)

  34. §16.4 Direct Radiation between Black Bodies in Vacuum (27) The coefficient matrix [Gij] is a non-singular symmetric matrix of (n-2) by (n-2) . Let [Gij-1] be the inverse matrix of [Gij] . The solution to Eq.(16.4-e) is (16.4-f) Substitution of Eq.(16.4-f) into Eq.(16.4-a,b) gives

  35. §16.4 Direct Radiation between Black Bodies in Vacuum (28) Since Q1+Q2=0 , hence Because T1 and T2 are arbitrary, F’12=F’21 and G-1ij=G-1ji , there must be

  36. §16.4 Direct Radiation between Black Bodies in Vacuum (29) and (16.4-g) Defining an overall view factor as (16.4-h) We have (16.4-15) The difference characterizes the indirect radiation from A1to A2.

  37. §16.5 Radiation between Non-Black Bodies (1) Because of the partially reflection of radiation on the surfaces of non-black bodies, the absorption and reflection of a given radiation is a repeating process of infinite times. In general, the dependence of reflection on angle and frequency is not well known, so an accurate calculation of radiating heat transport for real bodies is difficult. The diffuse reflection and gray body are the simplifications commonly adopted in the radiation calculation of non-black bodies. Even with these simplifications, only some simple cases can be solved analytically.

  38. §16.5 Radiation between Non-Black Bodies (2) 1. A small convex surface in a large isothermal surrounding This situation is a satisfactory approximation for many practical applications. In this situation, the surrounding is considered as a cavity, so that the situation simplifies to the radiation between a non-black body ‘ 1’ and a black enclosure ‘ 2’. (16.5-1) (16.5-2)

  39. §16.5 Radiation between NonBlack Bodies (3) Since e1 and a1 may depend on the frequency distribution of radiation, we have e1= e1 (T1) in Eq.(16.5-1) and a1= a1 (T2) in Eq.(16.5-2) . The net radiation from ‘ 1’ to ‘ 2’ is (16.5-3) If ‘ 1’ can be treated as a gray body, we have (16.5-3’)

  40. Example 16.5-3The maximum air temperature for frosting In late autumn, we often see frost on the surface of grass or roof in the morning, though the lowest temperature in the night, according to the weatherman, was higher than ice point. Was the weatherman wrong? Physical picture For a poorly heat-conducting surface placed in outdoor surroundings, heat is transported mainly in two ways: convective transport between air and the surface, radiation transport between the surface and the sky. In an autumn night, the air is very clear and transparent so the sky means the outer space.

  41. Example 16.5-3The maximum air temperature for frosting • 1. Physical model • Steady state. • Heat conduction between the surface and supporting object is negligible. • The heat effect of evaporation or condensation of water on the surface is negligible. • Heat transport between air and the surface is by free convection. • Back radiation from the surroundings and atmosphere is negligible. • The temperature in a dew drop may be taken as uniform.

  42. Example 16.5-3The maximum air temperature for frosting • 2. Mathematical model • On P.M.1), 2), 3) and 6), heat balance equation is 2) On P.M.4), heat transport flux by free convection on a horizontal plate can be calculated by 3) On P.M.5), the heat radiation flux from the dew drop to the sky can be calculated by

  43. Example 16.5-3The maximum air temperature for frosting 3. Solution of M.M. The combination of M.M.1), 2) and 3) gives The temperature of outer space is 3 K, the frosting temperature of dew drop is 273.15 K, and the emissivity of water is 0.95. The substitution of these into the above equation gives

  44. Example 16.5-3The maximum air temperature for frosting 4. Analysis This result does say that frosting can happen under an air temperature higher than the ice point. But the predicted maximum temperature is much higher than that we observed practically. What is the main source of error ? The back radiation from the surroundings is not negligible and the air is not completely transparent. So usually Tfmax is about several degrees above the ice point.

  45. §16.5 Radiation between NonBlack Bodies (4) 2. Equi-view geometries By the term ‘equi-view’ we mean that the surrounding looks the same at any point on A1 and so does on A2 . The instances include two infinite parallel plates, two coaxial cylinders infinite in lengths and two concentric spheres. Let qO,j be the output radiation flux of Aj including emission and reflection, and qI,j the incident radiation flux on Aj . (*1)

  46. §16.5 Radiation between NonBlack Bodies (5) 1) Since Hence and inserting it into Eq.(*1) gives (*2)

  47. §16.5 Radiation between NonBlack Bodies (6) 2) On the other hand, since hence and (*3)

  48. §16.5 Radiation between NonBlack Bodies (7) 3) From Eqs.(*2,*3) , or (*4) Similarly, (*5)

  49. §16.5 Radiation between NonBlack Bodies (8) The combination of Eqs.(*4,*5) gives (*6) 4) Inserting it into Eq*3) , we obtain (*7)

  50. §16.5 Radiation between NonBlack Bodies (9) The apparent view factor is defined as (*8) Then, (*9) in which T3 and T4 are between T1 and T2 .

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