Thermodynamics and Propulsion

19.4 Radiation Heat Transfer Between Black Surfaces of Arbitrary Geometry

In general, for any two objects in space, a given object 1 radiates to object 2, and to other places as well, as shown in Figure 19.10.

Figure 19.10: Radiation between two bodies

Figure 19.11: Radiation between two arbitrary surfaces

We want a general expression for energy interchange between two surfaces at different temperatures. This is given by the radiation shape factor or view factor, $F_{i - j}$ . For the situation in Figure 19.11,

$F_{1-2}$ =	fraction of energy leaving 1 which reaches 2
$F_{2-1}$ =	fraction of energy leaving 2 which reaches 1
$F_{1-2}$ , $F_{2-1}$	are functions of geometry only

For body 1, we know that $E_b$ is the emissive power of a black body, so the energy leaving body 1 is $E_{b1} A_1$ . The energy leaving body 1 and arriving (and being absorbed) at body 2 is $E_{b1} A_1 F_{1-2}$ . The energy leaving body 2 and being absorbed at body 1 is $E_{b2} A_2 F_{2-1}$ . The net energy interchange from body 1 to body 2 is

$$E_{b1} A_1 F_{1-2} - E_{b2} A_2 F_{2-1} = \dot{Q}_{1-2}.$$ (19..4)

Suppose both surfaces are at the same temperature so there is no net heat exchange. If so,

$$E_{b1} A_1 F_{1-2} - E_{b2} A_2 F_{2-1} = 0,$$

but also $E_{b1} = E_{b2}$ . Thus

$$A_1 F_{1-2} = A_2 F_{2-1}.$$

Equation (19.4) is the shape factor reciprocity relation. The net heat exchange between the two surfaces is

$$\dot{Q}_{1-2} = A_1 F_{1-2} (E_{b1} - E_{b2})\qquad [\textrm{or}\quad A_2 F_{2-1}(E_{b1} - E_{b2})].$$

19.4.1 Example: Concentric cylinders or concentric spheres

Figure 19.12: Radiation heat transfer for concentric cylinders or spheres

The net heat transfer from surface 1 to surface 2 of Figure 19.12 is

$$\dot{Q}_{1-2} = A_1 F_{1-2}(E_{b1}-E_{b2}).$$

We know that $F_{1-2} = 1$ , i.e., that all of the energy emitted by 1 gets to 2. Thus

$$\dot{Q}_{1-2} = A_1(E_{b1} - E_{b2}).$$

This can be used to find the net heat transfer from 2 to 1.

$$\dot{Q}_{2-1} = A_2 F_{2-1}(E_{b2} - E_{b1})= A_1 F_{1-2} (E_{b2} - E_{b1}) = A_1 (E_{b2} - E_{b1}).$$

View factors for other configurations can be found analytically or numerically. Shape factors are given in textbooks and reports (they are tabulated somewhat like Laplace transforms), and examples of the analytical forms and numerical values of shape factors for some basic engineering configurations are given in Figures 19.13 through 19.16, taken from the book by Incropera and DeWitt.

Figure: Total emittances for different surfaces [from: A Heat Transfer Textbook, J. Lienhard]

Figure 19.13: View Factors for Three-Dimensional Geometries [from: Fundamentals of Heat Transfer, F.P. Incropera and D.P. DeWitt, John Wiley and Sons]

Figure 19.14: View factor for aligned parallel rectangles [from: Fundamentals of Heat Transfer, F.P. Incropera and D.P. DeWitt, John Wiley and Sons]

Figure 19.15: View factor for coaxial parallel disks [from: Fundamentals of Heat Transfer, F.P. Incropera and D.P. DeWitt, John Wiley and Sons]

Figure 19.16: View factor for perpendicular rectangles with a common edge [from: Fundamentals of Heat Transfer, F.P. Incropera and D.P. DeWitt, John Wiley and Sons]