Geometrical Concepts in Radiation Heat Transfer P M V Subbarao Professor Mechanical Engineering Department IIT Delhi A New Algebra …
The elements dAi and dAj are isothermal at temperatures Ti and Tj respectively. • The normals of these elements are at angles qi and qj respectively to their common normal. • The total energy per unit time leaving dAi and incident upon dAj is: dwi is the solid angle subtended by dAj when viewed from dAi.
Radiative Heat Exchange between Two Differential Area Elements Configuration Factor for rate of heat Exchange from dAi to dAj Configuration Factor for Energy Exchange from dAj to dAi
Reciprocity of Differential-elemental Configuration Factors Consider the products of :
Net Rate of Heat Exchange between Two differential Black Elements The net energy per unit time transferred from black element dAi to dAj along emissive path r is then the difference of i to j and j to i.
Ib of a black element = Finally the net rate of heat transfer from dAi to dAj is:
Configuration Factor between a Differential Element and a Finite Area dAi Aj, Tj qj qj qi dAi, Ti
Configuration Factor for Two Finite Areas dAi Aj, Tj qj qi Ai, Ti
Shape factors for other simple geometries can be calculated using basic theory of geometry. • For more complicated geometries, the following two rules must be applied to find shape factors based on simple geometries. • The first is the summation rule. • This rule says that the shape factor from a surface (1) to another (2) can be expressed as a sum of the shape factors from (1) to (2a), and (1) to (2b). • The second rule is the reciprocity rule, which relates the shape factors from (1) to (2) and that from (2) to (1) as follows:
Thus, if the shape factor from (1) to (2) is known, then the shape factor from (2) to (1) can be found by: If surface (2) totally encloses the surface 1:
Radiation Exchange between Two Finite Areas The net rate of radiative heat exchange between Ai and Aj
T1,A1 T2,A1 TN,AN J1 JN J2 . . . . . Ji . . . . . Ti,Ai Configuration Factor Relation for An Enclosure Radiosity of a black surface i For each surface, i The summation rule !
T1,A1 T2,A1 TN,AN J1 JN J2 . . . . . Ji . . . . . Ti,Ai • The summation rule follows from the conservation requirement that al radiation leaving the surface I must be intercepted by the enclosures surfaces. • The term Fii appearing in this summation represents the fraction of the radiation that leaves surface i and is directly intercept by i. • If the surface is concave, it sees itself and Fii is non zero. • If the surface is convex or plane, Fii = 0. • To calculate radiation exchange in an enclosure of N surfaces, a total of N2 view factors is needed.
Real Opaque Surfaces Kichoff’s Law: substances that are poor emitters are also poor absorbers for any given wavelength At thermal equilibrium • Emissivity of surface (e) = Absorptivity(a) • Transmissivity of solid surfaces = 0 • Emissivity is the only significant parameter • Emissivities vary from 0.1 (polished surfaces) to 0.95 (blackboard)
Complication • In practice, we cannot just consider the emissivity or absorptivity of surfaces in isolation • Radiation bounces backwards and forwards between surfaces • Use concept of “radiosity” (J) = emissive power for real surface, allowing for emissivity, reflected radiation, etc
Radiosity of Real Opaque Surface • Consider an opaque surface. • If the incident energy flux is G, a part of it is absorbed and the rest of it is reflected. • The surface also emits an energy flux of E. Rate of Energy leaving a surface: J A Rate of Energy incident on this surface: GA Net rate of energy leaving the surface: A(J-G) Rate of heat transfer from a surface by radiation: Q = A(J-G)
Enclosure of Real Surfaces T1,A1 T2,A1 TN,AN J1 JN J2 . . . . riGi Ei Gi . Ji . . . . . Ti,Ai For Every ith surface The net rate of heat transfer by radiation:
For any real surface: For an opaque surface: If the entire enclosure is at Thermal Equilibrium, From Kirchoff’s law: Substituting all above:
Ji riGi Ei Gi Qi Surface Resistance of A Real Surface Real Surface Resistance Ebi Black body Ji Actual Surface qi . Ebi–Ji : Driving Potential :surface radiative resistance .
Radiation Exchange between Real Surfaces • To solve net rate of Radiation from a surface, the radiosity Ji must be known. • It is necessary to consider radiation exchange between the surfaces of encclosure. • The irradiation of surface i can be evaluated from the radiosities of all the other surfaces in the enclosure. • From the definition of view factor : The total rate at which radiation reaches surface i from all surfaces including i, is: From reciprocity relation
This result equates the net rate of radiation transfer from surface i, Qi to the sum of components Qij related to radiative exchange with the other surfaces. Each component may be represented by a network element for which (Ji-Jj) is driving potential and (AiFij)-1 is a space or geometrical resistance.
Relevance? • “Heat-transfer coefficients”: • view factors (can surfaces see each other? Radiation is “line of sight” ) • Emissivities (can surface radiate easily? Shiny surfaces cannot)
Basic Concepts of Network Analysis Analogies with electrical circuit analysis • Blackbody emissive power = voltage • Resistance (Real +Geometric) = resistance • Heat-transfer rate = current
T1,A1 Qi1 T2,A1 TN,AN J1 J1 JN J2 . . . Qi2 . J2 riGi Ei Gi . Ji . . Ebi Ji Qi3 . J3 . . Ti,Ai JN-1 QiN-1 JN QiN Resistance Network for ith surface interaction in an Enclosure Qi