A Comparison of Industrial-Scale Radiometer Heat Flux Measurements Between Pulverized-Coal and Coal/Biomass Co-Firing Combustion

Biomass is an abundant resource that can be used to produce renewable energy. In 2020, it provided roughly 5% of the energy consumed in the United States [1]. There is ever-strengthening concern to reduce net CO₂ emissions in the energy industry. Dreizler et al. [2] discuss the need to utilize carbon-neutral, renewable energy sources in order to implement climate neutrality into the world wide energy industry. Advancements in the implementation of biomass combustion are necessary to achieve this goal. Currently, the cost (both monetarily and in terms of CO₂ expenditure) to harvest and transport biomass is higher—often significantly—than when using fossil fuels [3,4]. However, there is also motivation to manage the forestlands in the United States. Co-firing biomass in power plants near these forests could prove symbiotically beneficial to both wildfire management and lowering CO₂ emissions [5].

While there are myriad ways to utilize biomass in energy production, biomass co-firing is commonly seen as the optimal solution [6]. It is relatively easy to deploy, requires few furnace modifications, can reduce other pollutant emissions when strategically implemented, and can readily switch between running purely on fossil fuels when biomass fuel is not available [7].

For biomass co-firing to be feasible, any changes to the heat transfer in the boiler must be minimal. Many co-firing tests have been run at full-scale utility boilers to evaluate this impact [7–11]. However, these tests typically rely solely on the operational data routinely collected at the plants, which can be limited in scope and do not include incident heat flux measurements.

More commonly, data to analyze the potential effects of biomass co-firing are taken at smaller scales [12–20], which can then be extrapolated to the industrial-scale through modeling. Pilot and particularly laboratory scales allow for much more control, repeatability, accessibility, and flexibility in choice of measurement technique than at the scale of interest. However, the relative ease of experimenting at smaller scales must be tempered with caution in extrapolating the conclusions to the industrial scale. Bäckström et al. [21] reported unexpected differences in heat transfer trends in a 400 kW test furnace when firing two very similar coals. The paper emphasized the need to measure radiative heat transfer from flames “under conditions representing the applications of interest.” The computer simulation of biomass co-firing in industrial boilers is of great import because of the large diversity in biomass sources and the extreme cost and difficulty of performing experiments at this scale. Milićević et al. [22] numerically examine the effect of co-firing pulverized-coal with three types of agricultural residues at three different boiler loads. Qin et al. [23] utilize boiler operational data, two-color pyrometry, and NO_x measurements to create an online deep learning model that can predict NO_x emissions in oxy-biomass combustion. Providing experimental data at the scale of interest enables better verification and validation of these complex models.

Optical methods are often appropriate for industrial applications due to their non-invasive nature. Radiometers are particularly apt due to their relatively simple setup and operation, along with the ability to function well in harsh environments. As such, the use of radiometers is extremely common in measuring radiative heat transfer in many combustion situations [21,24–32]. Presently, there is a limited number of studies presenting radiative heat flux data at industrial-scale. As radiation is the dominant heat transfer mechanism in industrial boilers [33], experimental heat flux measurements are very important to facilitate the accurate modeling of these boilers. Costa et al. [34] took incident wall heat flux measurements with a radiometer as well as local gas species concentrations and temperatures in a 300 MW_E utility boiler. Li et al. [35] measured gas species, temperatures, char burnout, and wall heat flux in a 200 MW_E utility boiler. They utilized an optical pyrometer to present the heat flux profile of the boiler in four elevations at three different loads. Butler and Webb [36] reported local temperature and wall heat flux measurements in an 80 MW_E coal-fired boiler. The heat flux measurements were taken with an ellipsoidal radiometer (ER) at six boiler elevations. These measurements were repeated in a subsequent paper with the addition of particle temperature measurements and results were compared when firing two different coals [37]. With the exception of Costa et al. [34], these papers do not present any replicates of the heat flux measurements. None of the papers indicate the level of temporal variability seen during each measurement. Neither do any present heat flux data with taken with multiple instruments. One of Butler and Webb papers examines [37] two types of coal, but none of the above papers examine the effect of biomass co-firing on radiative heat flux.

The objective of this work is to present radiative heat transfer data in a 471 MW_E utility boiler. Furthermore, a co mparison of heat transfer profiles between standard pulverized-coal operation and co-firing a coal/biomass blend is presented to identify the effects of co-firing on the system. These data, while somewhat preliminary, are valuable as they provide the heat flux profile of the boiler at both conditions, provide insight regarding the feasibility of biomass co-firing, and yield insights on taking additional data in such challenging conditions in the future.

This work was done in collaboration with multiple teams of experimentalists who performed the following:

1. Incident wall radiation measurements (University of Utah, Chalmers University).

2. Deposit and entrained aerosol ash sample measurements (Brigham Young University, University of Utah).

3. Nitric oxide (NO) measurements (Chalmers University).

4. Various plant impact measurements (PacifiCorp, Brigham Young University).

Item (1) is the sole subject of this study. Li et al. [38] have published their findings from Item (2). They concluded that fly ash and aerosol size distributions and compositions are broadly unchanged when co-firing. They also saw a reduction in total deposition amount when co-firing with only slight differences in deposit composition. Allgurén et al. [39] have published their findings from Item (3), which show that co-firing reduced NO formation beyond what could be expected solely from the reduction in fuel nitrogen. Not all measurements taken during Item (4) have yet been published.

The experimental work discussed in this paper was conducted at the Rocky Mountain Power Hunter Power Plant, which is located in central Utah. The experimental work presented here was conducted on Unit 3, which provides the electrical grid with 471-MW_E through the combustion of pulverized-coal. The total inner height of the boiler is roughly 51 m with a maximum cross section of 15.3 × 15.8 m. Measurements presented in this paper were conducted through ports in the furnace wall located at Floors 10, 12, and 14 (Figs. 1 and 2). These ports are located above the burners. During the experimental work, the unit was operated at 90% load with normal burner and staging operating conditions.

In the experimental test campaign, two different fuel conditions were studied. The baseline case utilized the Utah-sourced bituminous coal primarily used to fuel the unit. The co-firing case utilized the same coal (85% by weight) and a torrefied, pelletized, woody biomass (15% by weight) sourced from local Utah forests and produced by Amaron Energy in Salt Lake City. The biomass was collected from the Wasatch-Cache National Forest roughly 150 miles from the Hunter Power Plant. This biomass is very similar in composition to that of Manti-La Sal National Forest, the border of which is only 20 miles from the Hunter Power Plant. Minimizing the distance from the biomass source to the plant is important to make biomass utilization feasible. The ultimate analysis for the coal, the torrefied biomass, and a calculated coal/biomass co-firing blend analysis are shown in Table 1. For the co-firing case, the coal and biomass pellets were milled together at the plant and then directly fed to the burners. Each fuel condition was tested for a 24 h period, which required roughly 720 tons of biomass during the co-firing case. It took approximately 8 h of operation with the biomass blend before the biomass percentage in the fuel flow stabilized.

Researchers from several universities collaborated to take data that included wall radiation, gas composition, deposit, and aerosol ash sampling, as well as the standard operational (PI) data measured by the plant. However, the work presented in this paper focuses solely on the radiative heat transfer data, which were obtained using the following three instruments:

• UofU NAR: The University of Utah (UofU) used a narrow-angle radiometer (NAR) to measure the narrow-angle radiative intensity.

• CTH NAR: Chalmers University of Technology (CTH) used a separate NAR to measure the narrow-angle radiative intensity.

• CTH ER: Chalmers University of Technology used an ER to measure the wide-angle radiative heat flux.

Radiation data from all three instruments were taken at three elevations in the boiler (Floors 10, 12, and 14).

The narrow-angle radiometer used by the University of Utah (UofU NAR) was initially developed for use in a high pressure oxy-fuel combustor, the construction of which is detailed in Dobó [40]. Briefly, a thermopile detector is installed at the end of small diameter (ID of 3.5 mm) view tube 305 mm in length (see Fig. 3). The thermopile absorbs incoming radiation onto a small detector (0.49 mm²) which then outputs a corresponding voltage. The inside of the view tube is coated with pulverized-coal to eliminate scattering onto the detector from rays not in the line-of-sight. This, coupled with the relatively long view tube, results in a narrow field of view (FOV) (6.70 × 10⁻⁵ sr). Thus, this probe measures radiative intensity. A thermistor located on the non-irradiated side of the thermopile provides an ambient temperature measurement, which is necessary to account for any changes in ambient conditions between calibration and operation. The radiometer was placed in a water-cooled water jacket during data collection, with a nitrogen purge flowing over it to prevent any interfering media in the line-of-sight path. During the data collection, the radiometer was centered in each measurement port facing directly into the boiler.

The narrow-angle radiometer instrument used by Chalmers University (CTH NAR) has been used in several previous studies to measure the radiative heat transfer in various flame conditions [21,26–28]. To ensure a narrow view angle, the probe has a total length of 2.3 m with an inner tube diameter of 1 cm. A thermopile detector is installed at the back end of the probe and a lens is used to concentrate the light to the detector (Fig. 4). With its narrow field of view (1.23 × 10⁻⁵ sr), the instrument can be considered to measure exclusively the radiative intensity in its line-of-sight. The inner tip of the probe is threaded to remove reflections in the inner tube wall. A small flow of nitrogen is used to prevent any gases or particles entering the probe in order to prevent signal distortion or probe damage. To prevent the probe from bending in the high temperature environment, it was constructed of titanium and is water-cooled during operation.

To maintain a constant sensor temperature, a separate cooling system is typically applied to the probe and the sensor housing. However, in the rugged conditions of the power plant, this refrigeration system was not feasible to use. A high flowrate of cooling water was used in its place to keep the sensor temperature as cool and stable as possible. The measured voltage signal from the thermopile corresponds to the radiative intensity incident on the detector. This signal is dependent on the temperature of the sensor as well. Neither the cooling water nor sensor temperature was directly measured, so—given the high temperature at the measurement locations (roughly 55 °C)—the sensor temperature was assumed to be 45 °C during these experiments. The narrow line-of-sight of the probe allowed for separate measurements with the probe directed in various directions within the boiler (e.g., boiler walls, center of the flame, etc.). At each of three elevations, measurements in multiple directions were conducted in order to provide comparisons with both the UofU-NAR data (pointed toward the center of the boiler) and the ellipsoidal radiometer data (using different view angles to combine measurements in various directions to approximate total heat flux).

The incident total radiative heat flux to the furnace wall was measured using an ellipsoidal radiometer from Chalmers University (CTH ER). Radiation enters through a small orifice at the tip of a water-cooled probe into a highly-reflective, gold-plated cavity. The radiation is then focused via a lens onto a thermopile detector (Fig. 5). In contrast with the NAR probes, the view angle for this detector is close to hemispherical and the detector’s voltage signal corresponds to all incident radiation reaching the measurement position, thus approaching the total radiative heat flux. During the measurements, the probe was centered in each measurement port with the probe tip directed straight into the center of the boiler.

Blackbody generators were used to provide known radiative intensity values to calibrate both NARs at different ambient temperatures. The UofU NAR was calibrated with blackbody temperatures from 600 °Cto1500 °C in 100 °C increments and at three ambient temperatures (⁠$18,32,and46∘C$⁠). The CTH NAR was calibrated with a separate blackbody unit between 400 °Cand1500 °C in 100 °C increments at two ambient temperatures (⁠$20and45∘C$⁠). The matlab nonlinear “fit” function was used with each radiometer’s calibration data to find the best-fit values for the coefficients K₁–K₄ and the associated 95% confidence intervals, which are summarized in Table 2. As shown in Table 2, each of the four calibration parameters are on a similar scale between the two radiometers. The differences in the coefficient magnitudes between the two radiometers are due to the differences in the radiometer design (e.g., different thermopiles, different dimensions, different view angles, etc.).

Port 10 was positioned below the nose of the furnace (right wall in Fig. 2), while Port 12 was positioned at the opposite wall (left wall in Fig. 2) at the height of the lower end of the superheater (SH). Port 14 was positioned between the superheater and the convection path tubes (wall facing the reader in Fig. 2). The roughly approximated view factors from the left wall (L), right wall (R), directly opposite (O) wall, SH, as well as the top (T) and bottom (B) of the furnace are presented in Table 3 for Floors 10, 12, and 14. These view factors are used in conjunction with the CTH NAR measurements in the associated direction to approximate a total incident heat flux at each measurement elevation.

As shown in subsequent results, this assumption of diffuse emission is not perfectly valid, as we measure smaller intensities at the periphery versus the center of the flame. However, the comparison is still deemed useful. Thus, the results from all three probes are presented in units of heat flux.

The measured radiative heat fluxes are presented in Fig. 6 for both fuel conditions at Floors 10, 12, and 14 as a function of time. The UofU measurements lasted roughly 30 min per replicate with a sampling frequency of roughly one measurement per second. The width of each data point’s rectangular marker corresponds to the time duration of the measurement. The measurement times for the two CTH probes were much shorter (typically 30–60 s), and the width of the corresponding data markers is not correlated to the measurement duration. Both CTH probes utilized a sampling frequency of approximately 10 Hz. The UofU NAR probe was used twice for each case and floor and was directed toward the center of the furnace. The data presented in Fig. 6 from the CTH NAR were also conducted with the probe directed toward the center of the furnace, although measurements with this probe pointed in additional directions are presented later. The average heat flux for each instrument is plotted with error bars representing two standard deviations of the temporal variation in the signal at each replicate. The horizontal regions on the plots indicate the total variation in heat flux measurements between the three instruments at a given fuel condition and height in order to aid comparison between fuels. During the experimental campaign, soot blowing was periodically conducted from the blowers located along the walls of the boiler in to clear deposit accumulation. Fifty-five soot blowers operated in the elevations of interest. Times of active soot blowing at any position in the elevation range of interest are represented in Fig. 6 by the semi-transparent vertical bands.

Overall, it may be observed from Fig. 6 that the heat flux substantially decreases with ascending boiler height, presumably due to heat siphoning from the flame into the steam tubes. When examining the narrow-angle radiometer data, the heat flux measurements overlap between replicates and fuel conditions. These data have much higher standard deviations than those of the ellipsoidal radiometer due to their narrow FOVs. This large variability shows the range of heat flux values from the different turbulent structures of the flame. The ellipsoidal radiometer collects rays from a much larger spatial area, averaging the highly luminous parts of the flame along with any cooler eddies together and, thus, has much smaller error bars. This difference in collection area also explains the consistently lower ellipsoidal radiometer averages when compared to the two NARs. Diffuse radiation was assumed to convert the NAR measured intensity to heat flux (Eq. (8)). All radiometers were directed toward the center of the radiant center of the boiler, but the ellipsoidal radiometer captured radiation from comparatively cooler areas on the periphery as well. This disparity between measurement techniques demonstrates that assuming diffuse reflection to convert the NAR intensity data to heat flux was not a completely valid assumption, despite the large size of the boiler. Even so, in general, the ellipsoidal data remain within the error bounds of the NARs.

Thus, generally, all three data sets overlap between fuel conditions (compare shaded areas at each floor level), implying the addition of biomass does not significantly change the heat flux profile of the boiler. The biomass measurement at Floor 10 shows a larger spread, but this is due to a single NAR measurement, which seems atypically high. If this measurement is considered an outlier, the similarity between both fuel conditions increases. The variation between all measurements at a given condition and floor is presumably due to bias error between the different instruments (such as the averaging over different FOVs) and the timing of soot blowing and other transient changes in the furnace conditions. Costa et al. [34] measured heat flux at 39 points in the flame on two different runs. The average normalized difference between the two runs was 18%, despite extensive efforts to keep the operating conditions stable and avoid soot blowing. Encouragingly, the average normalized difference between the two UofU NAR replicates was quite similar at 15%.

The standard deviations of both NAR probes decrease with ascending height in the boiler. This decrease is presumably due to more flame uniformity with less turbulent fluctuations higher in the boiler, which matches visual observations made by the experimentalists at the time.

Care was taken to procure measurements during times with no active soot blowing; however, soot blowing did overlap with two of the UofU NAR measurements. These points are marked in Fig. 6 as solid black rectangles rather than the standard hollow markers, and these measurements occurred during replicates on Floors 10 and 12 during coal-only firing. In the case of Floor 10, when active soot blowing was noticed, the probe was removed, and a new data point was taken after the soot blowing was completed. For Floor 12, the probe remained in place before, during, and after the ∼5 min soot blowing, as conditions proved hazardous to remove it.

Typically, one would expect to see flame heat flux increase with duration of time after soot blowing, since growing deposits inhibit heat absorption by the steam tubes. In this data set, this behavior is most clearly seen in the UofU NAR data during coal firing on Floor 12 (Fig. 6). However, this behavior is not as clearly seen in the case of coal firing on Floor 10, in which the first UofU replicate (first hollow rectangular data point) has a higher average heat flux than the second replicate (solid rectangular data point). However, during this second replicate, soot blowing began occurring somewhere in the boiler before producing a noticeable effect—which is when the replicate was aborted (note the vertical red line overlapping the second replicate before it was terminated in Fig. 6). This soot blowing may have affected the heat flux of the flame. It is also possible the heat flux profile of the boiler rhythmically shifts throughout the day due to soot blowing, ambient weather conditions (data were taken in August, so days were quite hot with cooler nights) and other unknown transient effects. Thus, perhaps there is not a constant heat flux value at a given boiler elevation but rather a range of typical values.

Table 4 summarizes the heat flux data at each boiler elevation and fuel condition averaged between the three radiometers. A student’s t-test was performed on the data in order to ascertain the effect of replacing part of the coal with biomass. While there are only four data points per sample group, de Winter [42] concludes that t-tests can be safely applied for very small sample groups (N ≤ 5) when the effects of the intervention are large. For this experiment, the null hypothesis was that the intervention of changing fuels did not change the heat flux profile of the boiler. P-values were calculated using the two-sample t-test and are presented in the final column of Table 4. P-values under a threshold value of 0.05 lead to a rejection of the null hypothesis, while values higher than 0.05 lead to the conclusion that the addition of biomass did not affect the heat flux profile to any large degree. The parenthetical values for Floor 10 were calculated without including the outlier replicate from the CTH NAR for that floor. Encouragingly, even when including the outlier at Floor 10, the lowest P-value is significantly larger than 0.05, which still leads to the conclusion that there is no significant change due to biomass co-firing. This result is promising for the implementation of co-firing regularly in the future.

Subplots B and D in Fig. 7 present the calculated total heat flux data from the CTH NAR probe (circular data points labeled “CTH NAR 3D”), which were calculated using the view factors presented in Table 3 and the measured radiative intensities at three different directions into the boiler. CTH NAR data at a single direction pointing into the center of the flame (triangular points labeled “NAR 1D”) from Fig. 6 are presented again in this figure (subplots A and C) for ease of comparison along with the total heat flux measurements from the ellipsoidal radiometer (“CTH ER”). The green icons refer to the biomass co-firing case while the black icons refer to the coal-only case.

In similarity to the observations made in Fig. 6, apart from the calculated heat flux value at Floor 10 for the biomass co-firing case, the multi-directional NAR and the ellipsoidal radiometer data sets overlap fairly well, although the CTH NAR data are always higher in the coal-only case. When examining the difference between the single direction NAR measurements versus the multi-directional measurements, we see that for both fuel cases, the multi-directional measurements are closer to those of the ellipsoidal radiometer. This difference is due to the directional dependency of the flame radiation. It is predicted that increasing the number of measurement directions with the CTH NAR probe would decrease the difference seen between the two instruments.

Heat flux data from three radiometers were taken at two different fuel cases (85% coal/15% torrefied, woody biomass co-firing versus firing only coal) and three different elevations in an industrial power plant. Intensity data from the two narrow-angle radiometers were converted to total heat flux assuming diffuse radiation in order to compare results between the three instruments. Multi-directional data from the Chalmers narrow-angle radiometer were also used to calculate a more accurate total heat flux to compare with the ellipsoidal radiometer data. Notable conclusions from the data are as follows:

• Heat flux substantially decreased with boiler height as heat was absorbed from the flame. Variation in heat flux over the collection time for a single data point also decreased with boiler height as the flame sheet became more uniform.

• The narrow-angle radiometers exhibited higher heat flux values and much larger standard deviations than the ellipsoidal radiometer data due to differences in each instrument’s field of view. The narrow-angle instruments measured radiation from the hottest part of the flame and averaged radiation over a much smaller spatial area, resulting in higher radiation measurements and greater sensitivity to fluctuations in the flame.

• Calculated total heat flux values from the multi-directional Chalmers radiometer data were closer to the ellipsoidal results than the single direction measurements, although still overpredicted total heat flux. Adding more measurement directions presumably would increase that agreement even more. This disparity due to radiometer view angle indicates that the incident radiative intensity at the boiler ports is highly directional, despite the large size of the boiler. The effect of the cooler boiler periphery should be considered when analyzing and modeling radiative data in boilers.

• No obvious trends were observed in the heat flux measurements as a function of time since soot blowing, despite the fact that deposit growth inhibits heat transfer from the flame. This lack of pattern potentially indicates that the heat flux profile in the boiler is not constant and perhaps cycles periodically with time.

• Large variations between replicates at any given condition were observed. Presumably, this variation is due to a combination of factors, including instrument uncertainty, bias error between instruments, and changes in the nature of the flame as a function of time—one cause of which was certainly soot blowing. Despite this high variation in replicates, a student’s t-test analysis on the data indicates that the biomass co-firing did not have any substantial effect on the heat flux profile as compared to coal-only firing.

Regarding future work of a similar nature taken in a similar setting, there are multiple improvements and ideas that could be implemented:

• Increase the number of replicates at each floor and fuel condition.

  Increasing the number of replicates would improve confidence in the statistical comparison between fuel conditions. In this campaign, the number of replicates was limited by duration of the fuel and the time it took to move equipment between floors. The use of multiple identical radiometers is recommended for future campaigns, as the equipment could be setup once and left in place. This approach would also provide the ability to take data at different elevations simultaneously, which would provide a better comparison between floors and a clearer temporal picture of the heat flux profile of the boiler.

• Increase the number of narrow-angle radiometer measurements in various directions into the boiler.

  This increase in directional measurements would enable a more direct comparison to any ellipsoidal radiometer results and would also yield valuable insight for modeling the boiler.

• Increase repeatability in the operation of the radiometers.

  This recommendation specifically applies to radiometer alignment in each port and the ambient temperature of the radiometers. Regarding alignment, custom rigs that allow for precise adjustments of the radiometer position and alignment is recommended. Ports must be kept closed except for short windows of active measurement, so the ability to reinsert the radiometers precisely at the same position for every replicate would be ideal. This ability would require custom designs for each floor, as each port has different access constraints (see Fig. 1). Greater repeatability with instrument positioning could reduce differences between replicates, as the results showed that radiation from the cooler periphery can have a large impact on the measurements when changing radiometer direction. Regarding ambient temperature, thermopiles tend to be extremely sensitive to temperature gradients across the detector. The temperatures at the measurement locations were quite high (∼55 °C). Cooling water jackets were used on all instruments while taking measurements, but with limited access to cooling water and the need to move between floors, the use of cooling jackets between measurements was intermittent. Ideally, all radiometers would remain in cooling jackets at all times to ensure a steady-state constant temperature profile across the radiometer thermopiles.

• Increase replicates at a single location to gain better understanding of transient boiler effects.

  It would be of great interest to systematically take replicates at a single port over a time period that included multiple soot blowing events. These replicates could provide insight into what the actual “steady-state” heat flux of the boiler is at a particular elevation. The data from this campaign imply that the heat flux at a given floor is not a constant value. Understanding the bounds of this variation would greatly aid in comparing different fuel conditions and in accurate modeling of the boiler.

In conclusion, the challenging nature of taking data in an industrial setting and the associated constraints make it difficult to procure copious data sets. However, it appears the heat flux profile between biomass co-firing and coal-only firing is unchanged—or at least that the change is below the level of measurement resolution. These results bode well for the ability of the plant to implement biomass co-firing at the levels investigated as a reasonable option to utilize biomass from forest management and to reduce net greenhouse gas emissions.

The authors gratefully acknowledge PacifiCorp/Rocky Mountain Power and the State of Utah for the financial support provided through the Utah Sustainable Transportation and Energy Plan (STEP). We greatly appreciate the assistance from Alex Prlina, Stan Harding, Thomas Allgurén, and Dan Gall in collecting these measurements in challenging conditions. We also greatly appreciate the collaboration with the staff at the Hunter Power Plant, whose cooperation and support was invaluable.

• PacifiCorp (Contract No. 3300000705).

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

A =

area (m²)

E =

total radiative heat flux (kW/m²)

F =

view factor

I =

radiative intensity (kW/m²/sr)

T =

temperature (K)

V =

voltage (mV)

$R$ =

electrical responsivity (mV/kW)

K₁ =

instrument model parameter 1 (kW/m²/mV/sr)

K₂ =

instrument model parameter 2 (kW/m²/K⁴/sr)

K₃ =

instrument model parameter 3 (kW/m²/K⁴/sr)

K₄ =

instrument model parameter 4 (K/mV)

R_t =

thermal resistance (K/kW)

θ =

polar angle from probe tip to surface (rad)

ϕ =

solid angle from narrow wall sections to probe tip

Φ =

radiative power (kW)

Ω =

solid angle from surface to probe tip

a =

ambient environment of radiometer thermopile

d =

thermopile detector

e =

emitted

i =

view factor radiating surface

j =

view factor intercepting surface

s =

radiation source

w =

radiometer view tube walls

B =

view factor surface, bottom of boiler

E =

electrical output of power plant

L =

view factor surface, left wall of boiler

O =

view factor surface, opposite wall of boiler

R =

view factor surface, right wall of boiler

T =

view factor surface, top of boiler

SH =

view factor surface, superheater section of boiler