Radiation heat transfer between surfaces is all about view factors. These numbers show how much radiation from one surface hits another. They depend on the surfaces' shapes, sizes, and positions.

Calculating view factors can get tricky, but there are tricks to make it easier. We'll learn about reciprocity and summation rules, which help us figure out view factors faster. We'll also explore how to calculate radiation exchange between surfaces.

View Factors in Radiation Heat Transfer

Definition and Significance

• View factor (F12) represents the fraction of radiation leaving surface 1 that is intercepted by surface 2
• View factors are essential in determining radiation exchange between surfaces by quantifying the fraction of emitted radiation reaching another surface
• View factor values depend on size, shape, and relative orientation of the surfaces involved in radiation exchange
• View factors are used in the radiosity method to calculate net radiation exchange between diffuse, gray surfaces

Factors Influencing View Factors

• Geometry of the surfaces plays a crucial role in determining view factors
• Relative orientation of surfaces affects the amount of radiation intercepted by each surface
• Distance between surfaces influences the view factor, with closer surfaces having larger view factors
• Obstructions or intervening surfaces can block radiation and reduce view factors between the original surfaces

Calculating View Factors for Simple Geometries

Analytical Methods

• Analytical methods involve evaluating the double integral of the view factor expression over the surfaces involved in radiation exchange
• View factor expression is derived from the definition of view factor and involves cosine of angles between surface normals and line connecting differential area elements, as well as distance between elements
• For simple geometries (parallel plates, perpendicular plates, concentric cylinders or spheres), the double integral can be simplified and evaluated analytically to obtain closed-form expressions for view factors

Simplification Techniques

• In cases where surfaces have symmetry or can be divided into smaller, simpler shapes, view factors for smaller shapes can be calculated and combined using summation rule to obtain overall view factor
• Tabulated view factors for common geometries can be used to simplify calculations
• Graphical methods (view factor charts, nomograms) can be employed to estimate view factors for specific geometric configurations

Reciprocity and Summation Rules for View Factors

Reciprocity Theorem

• Reciprocity theorem states that the product of area and view factor for one surface is equal to the product of area and view factor for the other surface: A1 × F12 = A2 × F21
• Reciprocity theorem allows calculation of one view factor from knowledge of the other view factor and areas of surfaces involved
• Reciprocity theorem is useful in reducing the number of view factors that need to be calculated in an enclosure

Summation Rule

• Summation rule states that the sum of all view factors from a given surface to all other surfaces in an enclosure, including itself, is equal to unity: ΣF1i = 1
• Summation rule is useful in determining unknown view factors when some view factors are known or can be calculated using analytical methods or reciprocity theorem
• Summation rule can be used to check consistency and accuracy of calculated view factors in an enclosure
• Summation rule applies to both convex and concave surfaces in an enclosure

Radiation Exchange Between Diffuse, Gray Surfaces

Radiosity Method

• Radiosity method is used to calculate net radiation exchange between diffuse, gray surfaces in an enclosure, considering both emission and reflection of radiation
• Radiosity (J) is the total radiation energy leaving a surface per unit area, equal to the sum of emitted and reflected radiation: J = ε × σ × T^4 + (1 - ε) × G, where ε is emissivity, σ is Stefan-Boltzmann constant, T is surface temperature, and G is irradiation (incoming radiation) on the surface
• Net radiation exchange between surfaces is determined by setting up a system of linear equations relating radiosity of each surface to radiosities of all other surfaces in the enclosure, using view factors and surface properties (emissivity and temperature)

Solving Radiosity Equations

• System of linear equations can be solved using matrix methods or iterative techniques to obtain unknown radiosities and net radiation heat transfer rates between surfaces
• Matrix methods involve forming a coefficient matrix based on view factors and emissivities, and solving for radiosities using matrix inversion or Gaussian elimination
• Iterative techniques (Gauss-Seidel, Jacobi) involve starting with initial guesses for radiosities and updating them iteratively until convergence is achieved
• Net radiation heat transfer rates can be calculated from the difference between radiosity and irradiation for each surface, multiplied by the surface area

Assumptions and Limitations

• Radiosity method assumes all surfaces are diffuse (emit and reflect radiation uniformly in all directions) and gray (emissivity and absorptivity are independent of wavelength)
• Enclosure is assumed to be at a steady state, with no transient effects
• Radiosity method does not account for specular (mirror-like) reflection or transmission of radiation through surfaces
• Presence of participating media (gases, particles) between surfaces can affect radiation exchange and requires additional considerations beyond the basic radiosity method