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## Geometrical Concepts in Radiation Heat Transfer

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**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