In this article we will discuss about the mechanism of heat exchange between two or more surfaces.
Heat Exchange between Black Bodies:
Consider heat exchange between elementary areas dA1 and dA2 of two black radiating bodies having areas A1 and A2 respectively. The elementary areas are at a distance r apart and the normal to these areas make angles 01 and 02 with the line joining them. The surface dA1 is at temperature and the surface dA2 is at temperature T2.
If the surface dA2 subtends a solid angle dco1 at the centre of the surface dA1, then radiant energy emitted by dA1 and impinging on (and absorbed by) the surface dA2 is-
The solution to this equation is simplified by introducing a term called radiation shape factor, geometrical factor, configuration factor or view factor. The configuration factor depends only on the specific geometry of the emitter and collection surfaces, and is defined as-
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“The fraction of the radiative energy that is diffused from one surface element and strikes the other surface directly with no intervening reflections.”
The radiation shape factor is represented by the symbol Fij which means the shape factor from a surface, Ai to another surface, Aj. Thus the radiation shape factor F12 of surface A1 to surface A2 is-
The above result is known as a reciprocity theorem. It indicates that the net radiant interchange may be evaluated by computing one way configuration factor from either surface to the other. Thus the net heat exchange between surfaces A1 and A2 is-
(Q12)net = A1F12 σb (T14 – T24 ) = A2F21 σb (T14 – T24 ) … (8.7)
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Equation 8.7 applies only to black surfaces and must not be used for surfaces having emissivities very different from unity.
The evaluation of the integral of equation 8.3 for determining the geometrical factor is rather complex and cumbersome. Accordingly results have been obtained and presented in graphical form for the geometries normally encountered in engineering practice. Geometrical factors for parallel planes (disks and rectangles) directly opposed and those for radiation between perpendicular rectangles with a common edge are depicted in Figs. 8.2 to 8.4.
Shape Factor Algebra and Sailent Features of the Shape Factor:
The shape factors for complex geometries (for which shape factor charts or equations are not available) can be derived in terms of known shape factors for other geometries. For that the complex shape is divided into sections for which the shape factor is either known or can be readily evaluated.
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The unknown configuration factor is worked out by adding and subtracting known factors of related geometries. The method is based on the definition of shape factor, the reciprocity principle and the energy conservation law.
The inter-relation between various shape factors is called shape factor algebra.
The following facts and properties will be useful for the calculation of shape factors of specific geometries and for the analysis of radiant heat exchange between surfaces:
(i) The value of shape factor depends only on the geometry and orientation of surfaces with respect to each other. Once the shape factor between two surfaces is known, it can be used for calculating the radiant heat exchange between the surfaces at any temperature,
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(ii) The net heat exchange between surfaces A1 A2 is-
(Q12) net = A1 F12 σ1 T14 – A2 F21 σ2 T24
When the surfaces are thought to be black (σ1 = σ2 = σb) and are maintained at the same temperature (T1 = T2 = T), there is no heat exchange and as such –
0 = (A1F12 – A2F21) σbT14
Since σb and T are both non-zero entities,
A1 F12 = A2 F21
This reciprocal relation is particularly useful when one of the shape factors is unity.
(iii) All the radiation streaming out from a convex surface 1 is intercepted by the enclosing surface 2. As such the shape factor of convex surface with respect to the enclosure F12 is unity. Then in conformity with reciprocity theorem, the other shape factor F21 is merely the ratio of areas.
(iv) The radiant energy emitted by one part of concave surface is intercepted by another part of the same surface. Accordingly a concave surface has a shape factor with respect to itself. The shape factor of a surface with respect to itself is denoted by F11.
For a flat or convex surface, the shape factor with respect to itself is zero. This aspect stems from the fact that for any part of flat or convex surface, one cannot see any other part of the same surface.
(v) If one of the two surfaces (say Ai) is divided into sub areas Ai1 Ai2, …. Ain, then-
Ai Fij = ∑Ain Fin j
Thus with respect to Fig. 8.5 (a), wherein the radiating surface A1 has been split up into areas A3 and A4,
A1 F12 = A3 F32 + A4 F42
Obviously F12 ≠ F32 + F42
Thus if the transmitting surface is sub divided, the shape factor for that surface with respect to the receiving surface is not equal to the sum of the individual shape factors.
If the receiving surface A2 [Fig. 8.5b] is divided into subareas A3 and A4, we have-
A1 F12 = A1 F13 + A1 F14 or F12 = F13 + F14
Apparently the shape factor from a radiating surface to a subdivided receiving surface is simply the sum of the individual shape factors.
(vi) Any radiating surface will have finite area and therefore will be enclosed by many surfaces. The total unit radiation being emitted by the radiating surface will be received and absorbed by each of the confining surfaces. Since a shape factor is the fraction of total radiation leaving the radiating surface and falling upon a particular receiving surface, the energy balance would give-
The interior surface of a complete enclosed space has been subdivided in n-parts-each part having a finite area A1, A2, An. Thus-
Accordingly when a radiating surface exchanges heat with a number of black surfaces comprising the enclosure, the net heat transfer from the radiating surface will be-
Heat Exchange between Non-Black Bodies:
Heat Exchange between Infinite Parallel Planes:
The analysis of radiant heat exchange between two non-black parallel surfaces shall be based on the following assumptions:
(i) The surfaces are arranged at small distance from each other and are of equal area so that practically all radiation emitted by one surface falls on the other. The configuration factor of either surface is therefore unity.
(ii) The surfaces are diffuse and uniform in temperature, and that the reflective and emissive properties are constant over all the surface.
(iii) The surfaces are separated by a non-absorbing medium such as air.
The surface 1 emits radiant energy E1 which strikes the surface 2. From it a part α2 E1 is absorbed by the surface 2 and the remainder (1 – α2) E1 is reflected back to surface 1. On reaching surface 1, a part α1 (1 – α2) E1 is absorbed and the remainder (1 – α1) (1 – α2) is reflected and so on.
The amount of radiant energy which left surface 1 per unit time is:
Since P is less than unity, the series 1 + P + P2 + ……………………, when extended to infinity gives 1/(1 – P)
Similarly the surface 2 emits radiation of emissive power E2. From it a part α1 E2 is absorbed by surface 1 and the remainder (1 – α1) E2 is reflected back to it. On reaching surface 2, a part α2 (1 – α1) E2 is absorbed and the rest (1 – α1) (1 – α2) E2 is reflected and so on. Proceeding exactly in the same way, we can determine the amount of heat which leaves surface 2 per unit time.
Heat Exchange between Infinite Long Concentric Cylinders:
Consider two large concentric cylinders of areas A1 and A2, emissivities ϵ1 and ϵ2 and their surfaces maintained at temperatures T1 and T2 respectively
From reciprocity theorem: A1 F12 = A2 F21.
The inner cylinder is completely enclosed by the outer cylinder and as such the entire heat radiations emitted by the inner cylinder are intercepted by the outer cylinder i.e., F12 = 1 and therefore,
F21 = A1/A2
Consider the energy emitted per unit area by the inner cylinder. All this emitted energy at any instant will eventually come to rest either back to inner cylinder or in the outer cylinder.
The process involves the following sequence of absorption and reflection:
Continuation of this process would show that total energy lost by inner cylinder per unit area is-
Similarly the heat energy lost by the outer cylinder per unit area would workout –
is the interchange factor or equivalent emissivity for radiant heat exchange between infinite long concentric cylinders.
Equation 8.15 is equally applicable to concentric spheres except that for concentric cylinders of equal length –
Heat Exchange between Small Gray Bodies:
Consider two small gray bodies having emissivities e1 and e2, and absorptivities α1 and α2 respectively. The small size of the bodies does signify that their size is very small compared to the distance between them. The radiant energy emitted by surface 1 would be partly absorbed by surface 2, and the unabsorbed reflected portion would be lost in space. It will not be reflected back to surface 2 because of its small size and large distance between the two surfaces.
Where, f12 = ϵ1 ϵ2 represents the equivalent emissivity or interchange factor for radiant heat exchange between two small gray bodies.
Heat Exchange between Small Body in a Large Enclosure:
The large gray enclosure acts like a black body; it absorbs practically all the radiation incident upon it and reflects negligibly small energy back to the small gray body. Further, the entire radiations emitted by the small body would be intercepted by the outer large enclosure and as such F12 = 1.
where f12 = ϵ1 represents the equivalent emissivity or interchange factor for radiation heat exchange between a small body and a large enclosure.
While calculating the radiant interchange between two gray surfaces, both the interchange factor f12 and the geometric factor F12 are considered and net heat interchange is computed from the relation-
Qnet = f12 F12 σb A1 (T14 – T24) ………(8.18)