The Apparent Absorptivity of the Infinite V-groove Rydge B. Mulford, Nathan R. Collins, Matthew R. Jones, Brian D. Iverson Brigham Young University

The Cavity Effect - Definition Reflections inside a cavity result in increased absorption

The Cavity Effect - Definition A cavity concentrates emission towards the cavity opening, causing the opening to emit more than a flat surface

The Cavity Effect - Variables The apparent radiative properties of a cavity are a function of: Collimated Irradiation Diffuse Irradiation Specular Reflection Specular Reflection Diffuse Reflection Isothermal Diffuse Reflection

The Cavity Effect - Equivalence The apparent radiative properties of the cavity are identical when: Irradiation is diffuse for apparent absorptivity Emission is diffuse and cavity is isothermal for apparent emissivity Valid for specular and diffuse reflection Diffuse Irradiation Diffuse Emission But what are the applications of the cavity effect? 1988, Ohwada, Y., J. Opt. Soc. Am.

The Cavity Effect - Application A radiator with variable radiative properties would give better temperature control Origami provides the mechanism by which the cavity effect might be controlled, giving a variable radiator High Emissivity Reject as much heat as possible Low Emissivity Hold in as much heat as possible Need apparent radiative properties of the V-groove

V-groove – Existing Works Models have been developed for the apparent radiative properties of the V-groove Organized by irradiation type and reflection type Collimated Irradiation Diffuse Irradiation Irradiation Diffuse Collimated Reflection Specular Modest Sparrow Specular Reflection Diffuse Reflection 2013, Modest, M. F., Radiative Heat Transfer, pp. 202-213. 1962, Sparrow, E. M., and Lin, S. H., IJHMT

V-groove – Existing Works Irradiation Diffuse Collimated Reflection Specular Modest Sparrow Irradiation Diffuse Collimated Reflection Specular Modest Sparrow Modest’s model is easy to implement and the derivation is simple to follow No apparent experimental or numerical validation exists 2013, Modest, M. F., Radiative Heat Transfer, pp. 202-213.

V-groove – Existing Works Irradiation Diffuse Collimated Reflection Specular Modest Sparrow Partial Illumination No Published Model Exists Full Illumination 1962, Sparrow, E. M., and Lin, S. H., IJHMT

V-groove – Existing Works Irradiation Diffuse Collimated Reflection Specular Modest Sparrow Sparrow’s diffuse-reflector models are simultaneous integral equations These models are exact but can be difficult to implement More accessible models would be useful But why do we need new models? 1962, Sparrow, E. M., and Lin, S. H., IJHMT

V-groove – Motivation A motorized accordion fold is exposed to varying heat inputs The fold actuates to maintain a given surface temperature Reliable apparent emissivity / absorptivity models that are amenable to a control system are required Thermal Camera Thermocouple Motorized Sample Heater Vacuum Chamber Motor

Research Objectives We use Monte Carlo Ray Tracing to: Irradiation Validate the specular, diffuse model Develop and validate a collimated, specular model for partial illumination Develop accessible correlations for diffuse reflection Irradiation Diffuse Collimated Reflection Specular Modest Sparrow 2013, Modest, M. F., Radiative Heat Transfer, pp. 202-213. 1962, Sparrow, E. M., and Lin, S. H., IJHMT

Monte Carlo Ray Tracing Rays are emitted from a cavity surface or irradiated into the cavity. Number of rays emitted (Nemit) and absorbed (Nabsorb) are counted. Apparent Emissivity Apparent Absorptivity 1990, Steinfeld, A., IJHMT

Objective 1: Validation Method Irradiation Diffuse Collimated Reflection Specular Modest Sparrow Apparent emissivity was calculated using Ray Tracing for all cavity angles and intrinsic emissivities Ray tracing results were compared to the results of Modest’s model

Objective1: Results Irradiation Diffuse Collimated Reflection Specular Modest Sparrow Ray tracing results and Modest’s model agree very well with an average error of 0.018% Modest’s model is accurate Modest 2013, Modest, M. F., Radiative Heat Transfer, pp. 202-213

Objective 2: Full Illumination Model Irradiation Diffuse Collimated Reflection Specular Modest Sparrow Total fraction of incident ray energy that is absorbed by the cavity Fraction of rays incident on the left-hand side that reflect n times Total fraction of energy incident on the left surface that is absorbed, adjusted for an uneven number of reflections Left Side Right Side Total Projected Area

Objective 2: Full Illumination Model Irradiation Diffuse Collimated Reflection Specular Modest Sparrow The “ray plane angle” ( ) increases with every reflection The ray is assumed to exit the cavity once the ray plane angle exceeds 180° # of Reflections = n - 1 # of Reflections = n Rounded Down

Objective 2: Partial Illumination Model Irradiation Diffuse Collimated Reflection Specular Modest Sparrow The right-side term is eliminated from the apparent absorptivity equation The projected area of the left side is now the same as the total projected area X’ must be adjusted to allow for shading

Objective 2: Partial Illumination Model Irradiation Diffuse Collimated Reflection Specular Modest Sparrow In the case of partial illumination, Sparrow’s n-counting model does not apply Ray tracing was used to count number of reflections for all cavity angles and collimation angles

Objective 2: Number of Reflections Results Irradiation Diffuse Collimated Reflection Specular Modest Sparrow We found a pattern that describes the number of reflections for partial illumination With the number of reflections known, the apparent absorptivity may now be calculated Full Illumination (Sparrow) Partial Illumination (this work) rounded down to nearest integer

Objective 2: Apparent Absorptivity Irradiation Diffuse Collimated Reflection Specular Modest Sparrow The ray tracing data and model results agree for both the full and partial illumination regions The average error for the full data set is 0.025% Model Model = Partial Illumination = Full Illumination

Objective 3: Correlation Method Irradiation Diffuse Collimated Reflection Specular Modest Sparrow We developed a basic model for each condition utilizing the assumption that the panel radiosity is uniform A correction function was introduced to remedy the error introduced by the uniform radiosity assumption This correction function was calculated through ray tracing and correlated through successive curve fits Diffuse Irradiation ` Collimated Irradiation Full Illumination ` ` Collimated Irradiation Partial Illumination

Objective 3: Final Expressions Irradiation Diffuse Collimated Reflection Specular Modest Sparrow The completed expressions remove the need to solve simultaneous integral equations The infinite summation was carried out to 100 terms Diffuse Irradiation Collimated Irradiation – Full Illumination

Objective 3: Final Expressions Irradiation Diffuse Collimated Reflection Specular Modest Sparrow Collimated Irradiation – Partial Illumination

Objective 3: Apparent Properties Irradiation Diffuse Collimated Reflection Specular Modest Sparrow The average error between ray tracing data and correlation is 1% with a maximum error of about 4% Benchmarking correlation vs. Sparrow’s results gives an average error of 0.5% and a maximum error of 2% Collimated Irradiation Diffuse Irradiation = Partial Illumination = Full Illumination

Conclusions Objective 1 Objective 2 Objective 3 Modest’s diffuse irradiation, specular reflection model is accurate. Objective 2 Sparrow’s collimated irradiation, specular reflection model for full illumination is accurate. A new model has been developed for partial illumination that shows excellent agreement with ray tracing results. Objective 3 Three new expressions have been developed that give the apparent absorptivity of an infinite V-groove for diffuse reflection and diffuse or collimated irradiation. Two of the three expressions show excellent agreement with Sparrow’s results. One expression cannot be validated at this time.

Acknowledgements This research was made possible by a NASA Space Technology Research Fellowship (Grant Number NNX15AP494) The authors would like to acknowledge the initial work of Michael Farnsworth This work is currently being prepared for publication Works Cited: Steinfeld, A., 1990, "Apparent absorptance for diffusely and specularly reflecting spherical cavities," International Journal of Heat and Mass Transfer, 34(7), pp. 1895-1897. Modest, M. F., 2013, Radiative Heat Transfer, Academic Press, London, pp. 202-213. Sparrow, E. M., and Lin, S. H., 1962, "Absorption of thermal radiation in a v-groove cavity," International Journal of Heat and Mass Transfer, 5, pp. 1111-1115. Contact information: Brian D. Iverson bdiverson@byu.edu

Ray Tracing – Emissivity Thermal Model From the definition of Qemit Plug into apparent emissivity equation Using the original energy balance and equating thermal model to ray tracing results gives: Rearrange to give apparent emissivity

Ray Tracing – Absorptivity Thermal Model Assuming opaque Final Expression From definition of apparent reflectivity From Ohwada[1] we learn that apparent absorptivity and apparent emissivity are equivalent for an isothermal cavity [1] - Ohwada, Y. 1988 “Mathematical proof of an extended Kirchoff Law for a cavity having direction-dependent characteristics” Journal of the Optical Society of America 5(1). 141-145.

Objective 2: Partial Illumination Model Irradiation Diffuse Collimated Reflection Specular Modest Sparrow The fraction of rays that are reflected n times (X’) must also be adjusted to account for shading The variable X’’ is the fraction of surface that is shaded

Objective 3: Apparent Properties Irradiation Diffuse Collimated Reflection Specular Modest Sparrow Correlation results match Sparrow’s results with an average error of 0.5% and a maximum error of 2% The collimated irradiation correlations are less accurate than the diffuse irradiation correlation Collimated Irradiation - Full Illumination Diffuse Irradiation