Apparent Emissivity Research Articles

Non-contact monitoring of soda-lime glass surface changes by its LWIR apparent emissivity: Possibilities and limitations

Soda-lime glass is widely used in many areas of daily life because of it is inexpensive, chemically stable, reasonably hard, extremely workable, and has excellent optical properties.In use, it is subject to weathering degradation. This causes surface changes that reduce its optical properties. After carrying out a spectral characterization of their effects, we highlight that the changes produced have a specific signature in the LWIR band due to the IR absorbing properties of the glass. Therefore, LWIR cameras used as radiometers are perfectly suited for their quantification by a non-contact measurement of their LWIR apparent emissivity. The LWIR apparent emissivity is provided in a quantitative thermographic approach. Based on the imaging capabilities of the thermal camera, it is possible to generate a 2D mapping of the surface changes in the form of an emissogram.The metrological capabilities of LWIR bolometric thermal cameras are then tested under laboratory conditions (strict control of influence temperatures) and under outdoor conditions. Under laboratory conditions, LWIR apparent emissivity is capable of quantifying specral anomalies as small as A=5%μm. The indicator is suitable for managing the effects of erosion and hydration of glass, respectively from 8 % of the eroded surface and from 100h of humid heat (2 years of natural exposure in a temperate climate). In outdoor conditions, the reflected temperature is an advantage because it improves the Type B accuracy of LWIR apparent emissivity measurements. Assuming a 3 °C non-uniformity of surface temperature, as it is the case for PV systems with active cooling, we conclude that erosion effects should be quantifiable from 48 % of the eroded surface, allowing optical transmittance losses of glass to be detected from 10 %. The effects of glass hydration remain difficult to manage due to the weakness of the spectral anomalies they produce.

Simultaneous measurement of normal spectral emissivity and temperature of materials with low thermal conductivity

This study develops a measurement system and a method for measuring the normal spectral emissivity and temperature of low thermal conductivity materials at high temperatures. A self-design heating device using a synergistic heating treatment including a high-power laser and a radiation heating cavity is introduced, and a Fourier transform infrared spectrometer is utilized to measure the radiation signal of sample. Based on the temperature hypothesis model, which considers normal spectral emissivity keeps constant within small temperature difference less than 10.0 K, a method is proposed for realizing the inversion of normal spectral emissivity and temperature with the multispectral radiation signal. A genetic algorithm-based technique with big mutations is used to estimate the normal spectral emissivity and temperature of sample at high temperatures. The variation of blackbody radiation signals with the blackbody radiance is analyzed to verify the linearity for the detector of FTIR spectrometer. The apparent normal spectral emissivity of an alumina ceramic fiber is determined in the spectral range of 2.5–25.0 μm at 673–1273 K. Furthermore, the uniformity of temperature distribution on the sample surface and inside the sample is numerically evaluated. The standard uncertainty in the experiment at 1273 K is less than 4.0 % in the investigated bands.

Thermal‐Infrared Emission Spectroscopy of Graybody Minerals (Sulfide): Implications for Extraterrestrial Exploration for Magmatic Ore Deposits

AbstractGraybody materials exhibit systematically low emissivity across their spectrum. This characteristic violates the key assumption of unit emissivity at some wavenumber in the spectrum used to calibrate thermal‐infrared emission data. This assumption makes graybody materials “appear colder” than their actual physical temperature and imparts a slope in emission spectra that is non‐physical in nature, both of which affect interpretations of planetary surfaces. Pyrrhotite derived from the Stillwater Complex's J‐M Reef in Montana, USA exhibits systemic graybody behavior across its mid‐infrared spectrum and thus has a steep negative spectral slope from high to low wavenumbers when calibrated using conventional methods. A new measurement technique is introduced for deriving the absolute emissivity of graybody materials using reference samples with known Christiansen Frequencies during calibration. The reference temperature method significantly reduces the spectral slope of and provides a more accurate estimation of the absolute emissivity of graybody materials. After correcting the temperature of pyrrhotite using results from a series of reference experiments, we conclude that the emission spectrum of pyrrhotite is spectrally featureless and has a maximum emissivity of ∼0.7. If sulfide mineral deposits are exposed on Mars, they will not be identified using spectral features found in the mid‐infrared (5–40 μm). However, they could be located by identifying basaltic terrain that appears colder than their surroundings and with apparent emissivity spectra that exhibit negative spectral slopes from high to low wavenumbers and are spectrally neutral.

Apparent normal spectral emissivity of TBC(ZrO2)-coated TC4 alloy

Apparent normal spectral emissivity of TBC(ZrO2)-coated TC4 alloy

Measured spectral, directional radiative behavior of corrugated surfaces

Spacecraft thermal control is entirely reliant upon radiative heat transfer with its surroundings for temperature regulation. Current methods are often static in nature and do not provide dynamic control of radiative heat transfer. As a result, modern spacecraft thermal control systems are typically ‘cold-biased’ with radiators that are larger than necessary for many operating conditions. Deploying a variable radiator as a thermal control technique in which the projected surface area can be adjusted to provide the appropriate heat loss for a given condition can reduce unnecessary heat rejection and reduce power requirements. However, the radiative behavior of the apparent surface representing the expanding/collapsing radiator changes in addition to the projected surface area size. This work experimentally quantifies the spectral, directional emissivity of an apparent surface comprised of a series of V-grooves (e.g. corrugated surface), as a function of angle and highlights its emission characteristics that trend toward black behavior. The experimental setup for quantifying this apparent radiative surface behavior is described and utilized to show the influence of surface geometry, direction and wavelength. Experiments on test samples were performed at 573 K. The experimental design is validated and demonstrated using fully oxidized, nearly diffuse, copper, corrugated test samples. The results presented in this work demonstrate, for the corrugated oxidized copper surfaces tested, that (1) higher emissivity values correspond to higher wavelengths in the spectral range of 2.5 to 15.4 µm (2) apparent emissivity values increase with decreasing V-groove angle resulting in less spectral variation in emissivity and greater blackbody like behavior, (3) azimuth dependence can be relatively small despite the obvious pattern associated with a corrugated surface, and (4) as the V-groove angle decreases, higher emissivity values are associated with θ→0∘ and ϕ→0∘. Results provide a foundation for future radiator design, improved spacecraft thermal control methods, and improved emissivity testing methods for patterned or angular surfaces.

Ground based thermographic screening of bagworm (<em>Metisa plana</em> Walker) infestation in oil palm and identification of their growth stages

Bagworms (Lepidoptera: Psychidae) are one of the main species of vicious leaf-eating insect pests that affects oil palm plantations in Malaysia. A moderate bagworm attack of 10-50% leaf damage may cause 43% yield loss. In 2020, the economic loss due to the bagworm attack in Malaysia is estimated as RM 180 million. Hence, it is necessary to monitor closely the bagworm outbreak in infested areas. However, precise data collection with accuracy is a challenging task. In this context, we explored the possibility of using thermal infrared (TIR) cameras (T 440) for detection of bagworms and identification of bagworm growth stages at the infested areas for effective monitoring. The reflector method was applied to detect the reflected apparent temperature and emissivity of the bagworms using thermographic measurement techniques. The results have revealed that the bagworms from the 1st to 7th larval instar and pupal stages exhibit emissivity values ranging from 0.88 to 0.89. Two rounds of our observation have shown that the bagworms can be detected during evening and afternoon vision as compared to night, midnight and morning vision, with consideration of emissivity, solar radiation, and snapshot distance at a percentage accuracy of 74 and 85%, respectively. The classification of the larval and pupal stages was carried out by grouping the larval and pupal stages based on their real size; Group I: larval stage 1-3, Group II: larval stage 4-7; and Group III: pupal stage. Identification of the bagworm stages according to the classified group and the object led to differentiation between the larval stages from Gr. I, II, and III, with the per cent detection of 60, 84 and 89%, respectively. The results indicate that the thermographic imaging approach can be applied to detect bagworms, provided all conditions and time observation are taken into consideration. This thermographic technique deserves thorough testing and optimization for detection of bagworms in oil palm plantations.

Modulation of apparent optical properties using arrayed mesoscale structures

In this study, a method for using arrays of mesoscale structures to modify the apparent optical properties of an opaque composite surface has been theoretically demonstrated to both raise and lower the apparent emissivity as compared to the intrinsic properties of the constitutive materials. For design problems where thermomechanical and optical material properties are both of importance, mesoscale surface structuring can greatly expand the design space. Analysis via the net radiosity method herein illustrates the ability to achieve a wide range of spectral apparent optical properties. Notably, a hexagonal array of spheres on a planar surface can raise the apparent emissivity of a planar surface by 50%. Conversely, a hexagonal enclosure of reradiating surfaces, realized by thin adiabatic walls, can reduce the apparent emissivity of a blackbody by half. As this method of modifying apparent optical properties utilizes structures much larger than the wavelengths of interest, the relationship between intrinsic planar emissivity, geometry, and apparent emissivity can be computed semi-analytically at low computational expense. Passive solar cooling, thermophotovoltaic cells, aerodynamic surfaces exposed to intense heating, and solar absorbers are presented as case studies that could benefit from the use of mesoscale structures on opaque surfaces to modify the apparent optical properties.

The thermal emission of Saturn’s icy moons

Context.The effects of space weathering and other alteration processes on the upper surface of Saturn’s icy moons are yet to be explored.Aims.We present a thermophysical model parametrised by way of regolith properties such as porosity, grain size, and composition, as well as the local topography. The modelled surface temperature and apparent emissivity are intended to be compared to measurements taken by Cassini’s Composite Infrared Spectrometer (CIRS), using its focal plane FP1. We study how they are impacted by the topographic model and the regolith properties.Methods.As an example, we coupled the topography of the Dione moon with our model. Simulations provide the thermal history of the surface elements of the shape model included in the FP1 footprints at the viewing geometries along one CIRS observation. The heat transfer in the regolith may occur through conduction or radiation. Its bolometric albedo,A, and hemispherical emissivity,εh, are expressed as a function of grain properties.Results.The model roughly reproduces the observed variations of surface temperature,TF, and apparent emissivity,εF, in the chosen example, while assuming uniform regolith properties. The dispersion of temperatures within the footprints due to the difference in local time of the surface elements explains most of the directionality of the apparent emissivity,εF(Em), at emission angles of Em ≥ 30°. Adding topography at the 8-km scale amplifies this effect by a few percent. Refining the scale to 1 km increases it again by a single percent but at a high computational cost. This particular anisotropy ofεF(Em) cannot be explained by the directional emissivity,εd, of the regolith. The temperatureTFis less affected by this dispersion or by the topographic resolution. Adding regional variations of grain size significantly improves the agreement between the model and observations.Conclusions.This model demonstrated its good performance and, thus, it is ready for testing current hypotheses on regolith processing by space weathering on Saturn’s icy moons, such as regional changes in grain size.

Study on the combustion behavior and soot formation of single coal particle using hyperspectral imaging technique

This paper proposes a hyperspectral imaging technique for capturing radiative images at a high temporal–spatial resolution, for the purpose of spectrally visualizing the instantaneous combustion behaviors of single coal particles. We initially use a hyperspectral imaging device to record the combustion processes of two types of single pulverized coal particles on a McKenna flat-flame burner. From the snapshot images captured by the device, we observe the combustion processes, ignition characteristics, flame morphologies, and sizes of the two single coal particles. The observations indicate that the two single coal particles undergo volatile release/combustion and char ignition/oxidation during the luminous combustion process; moreover, both particles exhibit homogeneous ignition characteristics under the experimental conditions. Furthermore, based on the spectral radiative intensity images obtained via the device, the soot temperature and volume fraction images of the two single coal particles in the envelope flame are obtained at different burning times. Thereafter, the influences of the combustion behaviors and soot formation mechanisms on the distribution rules of the soot temperature and concentration at different combustion periods are discussed. Finally, the temperature and apparent spectral emissivity values of the two single char particles during the char oxidation process are also calculated to analyze their combustion behaviors. The results show that the temperature changes within a relatively narrow range with respect to the burning time during the oxidation stage, whereas the spectral emissivity continues to increase with the oxidation of chars and subsequently decreases.

Spot size, distance and emissivity errors in field applications of infrared thermography

Abstract Infrared thermography is increasingly emerging as an analytical approach within the thermal ecology research community, providing unique and rapid temperature information crucial to understanding how plants and animals exchange heat with their environment. What is difficult to appreciate are the numerous ways in which thermography may still yield inaccurate (i.e. deviation from the ‘correct’ value) information if certain tenets are not followed. In this paper, we examine, demonstrate and discuss these tenets with an aim to provide methodological advice to ecologists interested in employing thermography. We found that spot size and distance strongly influenced the surface temperature estimates of known, calibrated temperature sources, with similar results observed in maximum eye temperature measurements in wild birds. We also report on how the angle of incidence affects the apparent emissivity of various biological surfaces (fur, feather, skin, leaves), another source of uncertainty in thermography. The variation in temperature caused by variation in distance and uncertainty in emissivity are large enough to raise flags for field applications of thermography where accuracy is necessary but control over study subjects is limited. Since accurate emissivity and distance parameters are crucial to thermography calculations, our results should serve as a framework to assist ecologists in better experimental design with respect to the use of thermography.

Experimental Determination of the Emissivity of Powder Layers and Bulk Material in Laser Powder Bed Fusion Using Infrared Thermography and Thermocouples

Recording the temperature distribution of the layer under construction during laser powder bed fusion (L-PBF) is of utmost interest for a deep process understanding as well as for quality assurance and in situ monitoring means. While having a notable number of thermal monitoring approaches in additive manufacturing (AM), attempts at temperature calibration and emissivity determination are relatively rare. This study aims for the experimental temperature adjustment of an off-axis infrared (IR) thermography setup used for in situ thermal data acquisition in L-PBF processes. The temperature adjustment was conducted by means of the so-called contact method using thermocouples at two different surface conditions and two different materials: AISI 316L L-PBF bulk surface, AISI 316L powder surface, and IN718 powder surface. The apparent emissivity values for the particular setup were determined. For the first time, also corrected, closer to real emissivity values of the bulk or powder surface condition are published. In the temperature region from approximately 150 °C to 580 °C, the corrected emissivity was determined in a range from 0.2 to 0.25 for a 316L L-PBF bulk surface, in a range from 0.37 to 0.45 for 316L powder layer, and in a range from 0.37 to 0.4 for IN718 powder layer.

Apparent emissivity measurement of semi-transparent materials part 2: Theoretical concept

Semi-transparent materials are increasingly applied in various industrial and scientific applications. Because there is a strong coupling between optical and thermal properties of these materials, they pose unique challenges for their modelling and experimental characterization. At PTB, the facilities for spectral emissivity measurement of opaque materials have been expanded for the characterization of semi-transparent materials in the near-, mid- and far-IR region and for temperatures between 20 °C and 90 °C by using a new sample holder. The experimental realization and validation of the new facility for emissivity measurement of semi-transparent materials is presented in part 1 of this publication. Here the evaluation procedure of the measurement data obtained with the new sample-holder is discussed in detail. A model of radiative transport within semi-transparent samples considering emission, absorption, scattering and multiple reflections was formulated and solved over a wide spectral range. The model was inverted using a least squares approach and an extensive sensitivity analysis was performed.

Apparent emissivity measurement of semi-transparent materials part 1: Experimental realization

Semi-transparent materials are increasingly applied in various industrial and scientific applications such as non-destructive testing of fiber-reinforced polymers by active or passive thermography, reflecting and transmitting components in high precision infrared optics and novel thermal insulation systems. Because there is a strong coupling between optical and thermal properties of these materials, they pose unique challenges for their modeling and experimental characterization. At PTB, the facilities for spectral emissivity measurement of opaque materials have been expanded for the characterization of semi-transparent materials in the near-, mid- and far-infrared region and for temperatures from 20 °C to 90 °C by using a new sample holder. The sample holder was modelled, realized and thermometrically characterized. For a first validation of the new setup, measurements of natural PTFE foil were performed at temperatures from 27 °C to 52 °C and in the wavelength range from 4.5 µm to 20 µm yielding the spectral directional emittance, spectral optical thickness, and spectral reflectance simultaneously.

An Accurate Device for Apparent Emissivity Characterization in Controlled Atmospheric Conditions Up To 1423 K

Emissivity is a material property that must be measured before an accurate noncontact temperature measurement can be made. We have developed a novel instrument for measuring apparent emissivity under a controlled atmosphere, providing data for applications in radiation thermometry. Our instrument employs a split furnace, a sample-blackbody component, two custom-designed radiometers, and a controlled atmospheric system. We measure across the temperature range from 973 to 1423 K and the spectral range from 0.85 to 1.1 μm; this range is matched to the majority of high-temperature radiation thermometers. The sample and reference approximate blackbody are heated and maintained in the thermal equilibrium, with a temperature difference of better than 1 K at 1423 K. The combined standard uncertainty of the system is lower than 0.0590 (at k = 2) over the whole temperature range. Apparent emissivity of type 304 stainless steel (SS304) was studied under different oxidizing procedures. Nitrogen and compressed air were input into the system to control the oxidization process. We elucidated the relationship between the apparent emissivity variations and the surface composition changes of SS304 during oxidization. This article aims toward accurate and traceable apparent emissivity data, with well-investigated uncertainty, for use in radiation thermometry.

Study of temperature, apparent spectral emissivity, and soot loading of a single burning coal particle using hyper-spectral imaging technique

A hyper-spectral imaging device that can simultaneously capture three-dimensional data information of spectra, space, and time is employed to experimentally investigate the combustion behavior of a single coal particle. Three types of single pulverized coal particles are injected and burned on a Hencken flat-flame burner, and their flames are captured by a high-speed camera and the hyper-spectral imaging device. The camera observations demonstrate that the three single coal particles undergo particle heat-up, volatile release and combustion, and volatile and char oxidation during the combustion process. Furthermore, the image captured by the hyper-spectral imaging device records the spectral and spatial history of the single burning coal particles throughout the combustion process. Then, the time–space temperature and apparent spectral emissivity of the single burning char particles are calculated from the time–space spectra radiative intensity obtained by the hyper-spectral imaging device. The gray characteristic exhibited by the spectral emissivity of the three single burning char particles in the longer wavelength range demonstrates the feasibility of calculating the temperature using the two-color method. Finally, the soot temperature and volume fraction distribution in the volatile flames are obtained to investigate the formation of soot during the combustion of a single coal particle. The results show that an elongated tail-like soot cloud is formed around the two single bituminous coal particles during the combustion process, but a nearly spherical soot cloud is formed around the single lignite coal particle. Moreover, the range of soot volume fractions in the three envelope volatile flames is comparable to the order of magnitude of the values measured by other researchers.

Long-Term Arctic Snow/Ice Interface Temperature from Special Sensor for Microwave Imager Measurements

The Arctic sea ice region is the most visible area experiencing global warming-induced climate change. However, long-term measurements of climate-related variables have been limited to a small number of variables such as the sea ice concentration, extent, and area. In this study, we attempt to produce a long-term temperature record for the Arctic sea ice region using Special Sensor for Microwave Imager (SSM/I) Fundamental Climate Data Record (FCDR) data. For that, we developed an algorithm to retrieve the wintertime snow/ice interface temperature (SIIT) over the Arctic Ocean by counting the effect of the snow/ice volume scattering and ice surface roughness on the apparent emissivity (the total effect is referred to as the correction factor). A regression equation was devised to predict the correction factor from SSM/I brightness temperatures (TBs) only and then applied to SSM/I 19.4 GHz TB to estimate the SIIT. The obtained temperatures were validated against collocated Cold Regions Research and Engineering Laboratory (CRREL) ice mass balance (IMB) drifting buoy-measured temperatures at zero ice depth. It is shown that the SSM/I retrievals are in good agreement with the drifting buoy measurements, with a correlation coefficient of 0.95, bias of 0.1 K, and root-mean-square error of 1.48 K on a daily time scale. By applying the algorithm to 24-year (1988–2011) SSM/I FCDR data, we were able to produce the winter-time temperature at the sea ice surface for the 24-year period.

Thermal modeling and efficiency of a dual-stage sodium heat engine

A thermal design of an axisymmetric dual-stage sodium thermal electrochemical converter (Na-TEC) is presented, where a reduced-order finite-element model is used in conjunction with a Na-TEC thermodynamic model to determine the total parasitic heat loss of this dual-stage design. A number of simplifications are applied in the reduced-order model to decrease the computational time while maintaining acceptable accuracy. These include the use of effective thermal conductivities to account for complex corrugated geometries and apparent surface emissivities to accommodate the effect of radiation shields. Furthermore, a simplified analytical model is developed to account for conduction bypass through the Na-TEC liquid-return path. An iterative procedure between the finite-element and thermodynamic models is described in detail, and the thermal performance of the design proposed herein is analyzed. According to this analysis, a maximum efficiency of 29% and a maximum power output of 125 W can be achieved.

Impact of Ice Surface and Volume Scatterings on the Microwave Sea Ice Apparent Emissivity

AbstractEmissivity retrieval for sea ice from passive microwave measurements has been an important problem in climate/environmental research because of its link to various snow/ice variables. However, so far, it has been a difficult task because of the influences of surface and snow/ice induced volume scatterings. Here we examine the influences of scatterings on the ice emissivity from 10.65, 18.7, 23.8, and 36.5 GHz Advanced Microwave Scanning Radiometer (AMSR)‐E brightness temperatures over the Arctic Ocean. In doing so, we use a two‐dimensional roughness parameterization, modified with surface facet orientations with an assumption that the facet emission follows the Fresnel relationship. Emitting layer temperature and refractive index retrieved from AMSR‐E 6.9 GHz brightness temperature measurements were used in this study and applied to other channels of interest. We demonstrated that the obtained roughness index has a strong linear relationship with the root‐mean‐square height measured by Atmospheric Terrain Mapper on the National Aeronautics and Space Administration (NASA) P‐3 aircraft. The obtained roughness index showed that surface scattering on the emissivity is generally insignificant except for some first‐year ice regions in particular at higher frequencies. This fact implies that Fresnel relations can be applicable for most of sea ice at the low‐frequency microwave spectrum. By contrast, volume scattering is found to be significant in emissivity retrieval in case of multiyear ice. Nonetheless, volume scattering influence over first‐year ice appears to be minor. We suggest that Fresnel‐type emissivity can be estimated once a correction factor is used for removing surface scattering and volume scattering contributions from the apparent emissivity.

Mapping Surface Quartz Content in Sand Dunes Covered by Biological Soil Crusts Using Airborne Hyperspectral Images in the Longwave Infrared Region

Biological soil crusts (BSCs), composed of cyanobacteria, algae, mosses, lichens, and fungi, are important ecosystem engineers that stabilize the quartz-rich dunes in the Nitzana study area near the Israel–Egypt border. The longwave infrared (LWIR) region of the electromagnetic spectrum is very useful for quartz identification since quartz reflectance in the visible, near infrared, and shortwave infrared (VIS-NIR-SWIR, 0.4–2.5 µm) spectral regions lacks identifying features, whereas in the LWIR region, the quartz emissivity spectrum presents a strong doublet feature. This emissivity feature can be used as a diagnostic tool for BSCs development in desert environments, because BSCs attenuate the quartz feature as a function of their successional development. A pair of day and night airborne hyperspectral images were acquired using the Specim AisaOWL LWIR sensor (7.7–12 µm) and processed using an innovative algorithm to reduce the atmospheric interference in this spectral domain. The resulting day and night apparent emissivity products were used to produce a surface quartz content map of the study area. The significant reduction in atmospheric interference resulted in a high correlation (R2 = 0.88) between quartz content in field samples determined by X-ray powder diffraction analysis and emissivity estimations from the airborne images. This, in turn, served as the ground truth to our quartz content map of the surface, and by proxy to the BSC.

Total hemispherical apparent radiative properties of the infinite V-groove with specular reflection

Multiple reflections in a cavity geometry augment the emission and absorption of the cavity opening relative to a flat surface in a phenomenon known as the cavity effect. The extent of the cavity effect is quantified using apparent absorptivity and apparent emissivity. Analysis of complicated thermal systems is simplified through application of apparent radiative properties to cavity geometries. The apparent radiative properties of a specularly-reflecting, gray, isothermal V-groove have been derived analytically, but these results have not been validated experimentally or numerically. Additionally, the model for apparent absorptivity of an infinite V-groove subjected to partial illumination in the presence of collimated irradiation is not available. In this work, the following existing models for a specularly-reflecting V-groove are collected into a single source: (1) the apparent absorptivity of a diffusely irradiated V-groove, (2) the apparent emissivity of an isothermal V-groove and (3) the apparent absorptivity of a V-groove subject to collimated irradiation with full-illumination. Further, a new analytical model is developed to predict the apparent absorptivity of an infinite V-groove subject to collimated irradiation with partial-illumination. A custom, Monte Carlo ray tracing solver is used to predict the apparent radiative properties for all cases as a means of numerical verification by comparing the ray tracing data with the results from the new model in this work and the previously existing models. For diffuse irradiation, the analytical model and ray tracing data show excellent agreement with an average discrepancy of 4.4 × 10−4, verifying the diffuse-irradiation analytical model. Similar agreement is found for collimated irradiation, where the full and partial illumination models indicate average discrepancies of 4.9 × 10−4 and 4.6 × 10−4 when compared with ray tracing data.