Fundamentals of Heat and Mass Transfer - CH013

Fundamentals of Heat and Mass Transfer - CH013

(Parte 1 de 10)


KNOWN: Various geometric shapes involving two areas A1 and A2.

FIND: Shape factors, F12 and F21, for each configuration. ASSUMPTIONS: Surfaces are diffuse.

ANALYSIS: The analysis is not to make use of tables or charts. The approach involves use of the reciprocity relation, Eq. 13.3, and summation rule, Eq. 13.4. Recognize that reciprocity applies to two surfaces; summation applies to an enclosure. Certain shape factors will be identified by inspection. Note L is the length normal to page.

But F12 = F13 by symmetry, hence F12 = 0.50<

(c) Long duct (L):

But F12 = F13 by symmetry, hence F12 = 0.50<

By reciprocity,

(e) Sphere lying on infinite plane Summation rule,F11 + F12 + F13 = 1

But F12 = F13 by symmetry, hence F12 = 0.5<

Continued …..

PROBLEM 13.1 (Cont.)

By inspection, F12 = 1.0<


Note that by inspection you can deduce F22 = 0.5 (g) Long open channel (L):

COMMENTS: (1) Note that the summation rule is applied to an enclosure. To complete the enclosure, it was necessary in several cases to define a third surface which was shown by dashed lines.

(2) Recognize that the solutions follow a systematic procedure; in many instances it is possible to deduce a shape factor by inspection.


KNOWN: Geometry of semi-circular, rectangular and V grooves.

FIND: (a) View factors of grooves with respect to surroundings, (b) View factor for sides of V groove, (c) View factor for sides of rectangular groove.

ASSUMPTIONS: (1) Diffuse surfaces, (2) Negligible end effects, “long grooves”.

ANALYSIS: (a) Consider a unit length of each groove and represent the surroundings by a hypothetical surface (dashed line).

Semi-Circular Groove:

sin sinθθ

Hence, ()12 12W1F1 or F1 sin . W/ 2 /sin 2 θθ

COMMENTS: (1) Note that for the V groove, F13 = F23 = F(1,2)3 = sinθ, (2) In part (c), Fig. 13.4 could also be used with Y/L = 2 and X/L = ∞. However, obtaining the limit of Fij as X/L → ∞ from the figure is somewhat uncertain.


KNOWN: Two arrangements (a) circular disk and coaxial, ring shaped disk, and (b) circular disk and coaxial, right-circular cone.

FIND: Derive expressions for the view factor F12 for the arrangements (a) and (b) in terms of the areas A1 and A2, and any appropriate hypothetical surface area, as well as the view factor for coaxial parallel disks (Table 13.2, Figure 13.5). For the disk-cone arrangement, sketch the variation of F12 with θ for 0 ≤ θ ≤ π/2, and explain the key features.

ASSUMPTIONS: Diffuse surfaces with uniform radiosities.

ANALYSIS: (a) Define the hypothetical surface A3, a co-planar disk inside the ring of A1. Using the additive view factor relation, Eq. 13.5,

F A A FA F121 where the parenthesis denote a composite surface. All the Fij on the right-hand side can be evaluated using Fig. 13.5.

(b) Define the hypothetical surface A3, the disk at the bottom of the cone. The radiant power leaving

That is, the same power also intercepts the disk at the bottom of the cone, A3. From reciprocity,

The variation of F12 as a function of θ is shown below for the disk-cone arrangement. In the limit when θ → π/2, the cone approaches a disk of area A3. That is, F /2F1213θπ→=


KNOWN: Right circular cone and right-circular cylinder of same diameter D and length L positioned coaxially a distance Lo from the circular disk A1; hypothetical area corresponding to the openings identified as A3.

FIND: (a) Show that F21 = (A1/A2) F13 and F22 = 1 - (A3/A2), where F13 is the view factor between two, coaxial parallel disks (Table 13.2), for both arrangements, (b) Calculate F21 and F22 for L = Lo = 50 m and D1 = D3 = 50 m; compare magnitudes and explain similarities and differences, and (c)

Magnitudes of F21 and F22 as L increases and all other parameters remain the same; sketch and explain key features of their variation with L.

ASSUMPTIONS: (1) Diffuse surfaces with uniform radiosities, and (2) Inner base and lateral surfaces of the cylinder treated as a single surface, A2. ANALYSIS: (a) For both configurations, since the radiant power leaving A1 that is intercepted by A3 is likewise intercepted by A2. Applying reciprocity between A1 and A2,

Substituting from Eq. (1), into Eq. (2), solving for F21, find

Apply reciprocity between A2 and A3, solve Eq. (3) to find and since F32 = 1, find

PROBLEM 13.4 (Cont.)

(b) For the specified values of L, Lo, D1 and D2, the view factors are calculated and tabulated below. Relations for the areas are:

The view factor F13 is evaluated from Table 13.2, coaxial parallel disks (Fig. 13.5); find F13 = 0.1716.

It follows that F21 is greater for the disk-cone (a) than for the cylinder-cone (b). That is, for (a), surface A2 sees more of A1 and less of itself than for (b). Notice that F22 is greater for (b) than (a); this is a consequence of A2,b > A2,a.

(c) Using the foregoing equations in the IHT workspace, the variation of the view factors F21 and F22 with L were calculated and are graphed below.

Right-circular cone and disk


F21 F22

(Parte 1 de 10)