Radiative Heat Flux Measurement

RADIATIVE HEAT FLUX MEASUREMENT

Spectral Transmissivity Correction for Nextherm Sensing Heat Flux Sensors

Reference : NXT-NT-RAD-001

Version : 1.0

1. Purpose

Nextherm Sensing heat flux sensors—specifically the CHF (cooled) and GHF (uncooled) series—can be used to measure radiative heat flux originating from sources such as flames, hot bodies, furnaces, plasmas, laser beams, or concentrated solar radiation.

In many applications, the sensor’s sensing element must be isolated from the source by an optical window to eliminate the contribution of convective heat transfer.

However, this window acts as an optical element positioned directly in the path of the radiation. A portion of the incident flux is reflected or absorbed by the window, meaning only the transmitted fraction reaches the sensor’s sensing element.

Since the window’s transmissivity is wavelength-dependent, the correction must be tailored to the specific spectrum of the source.

This note outlines a simple method for correcting measurements taken with a Nextherm Sensing radiative heat flux sensor in the following scenarios:

• the sensor was calibrated at a specific wavelength;

• the measured radiation is monochromatic or quasi-monochromatic;

• or the radiation has a broad spectrum, such as that of a flame or concentrated solar radiation.

2. Measurement Principle

The heat flux sensor measures the radiative flux actually received by its sensing surface.

When an optical window is present, the incident flux (before the window) is denoted as Φ_inc, and the flux transmitted to the sensor is:

Φ_mes = τ_eff × Φ_inc

where:

• Φ_mes: the heat flux density measured by the sensor prior to correction;

• τ_eff: effective transmittivity of the window for the spectrum under consideration;

• Φ_inc: incident heat flux density.

Thus, in the ideal case:

Φ_inc = Φ_mes/τ_eff

The difficulty arises from the fact that τ_eff is generally not equal to the window's transmittivity at an arbitrary wavelength.

3. Specifics of Nextherm Sensing calibration

Nextherm Sensing sensors are systematically calibrated on a laser test bench at a wavelength of λ0 = 1070 nm. We define τ0 = τ(λ0). The window's transmittivity at 1070 nm therefore serves as the spectral reference for the calibration.

It is essential to distinguish between two approaches:

Case A — Calibration referenced to the flux received by the sensor

If the flux indicated by the sensor corresponds to the flux "actually" received after passing through the window, then:

Φ_corr = Φ_mes/τ_eff

Case B — Calibration referenced to the flux before the window

If the chosen approach considers the flux applied before the window, then the effect of the transmittivity at 1070 nm is already factored into the calibration. For a measurement at a different wavelength or with a different spectrum, the correction is as follows:

Φ_corr = Φ_mes × τ0/τ_eff = Φ_mes × Cr

Where:

• τ0: transmittivity of the window at the reference (calibration) wavelength;

• Cr: spectral correction factor.

This formulation—the most practical for the user—is the one adopted by Nextherm Sensing.

4. Case of monochromatic radiation

For a monochromatic source with a wavelength of λ, the correction is particularly simple. The effective transmittivity is simply:

τ_eff = τ(λ)

In the case of a calibration at 1070 nm referenced before the window:

Φ_corr = Φ_meas × τ(1070)/τ(λ)

Example: Suppose τ(1070) = 0.85, and consider a measurement at λ = 1550 nm, for which the window has a transmittivity of τ(1550) = 0.90. If the sensor indicates Φ_meas = 100 kW/m², then:

Φ_corr = 100 × 0.85/0.90 = 94.4 kW/m²

In this instance, the raw measurement slightly overestimated the incident flux, as the window transmits more radiation at 1550 nm than it does at 1070 nm.

For a monochromatic source (such as a laser), the correction can be applied by using a discrete value for the correction factor, Cr, based on our reference window (1 mm thick sapphire, with two polished, non-anti-reflective coated surfaces):

Note that for monochromatic radiation, the correction remains relatively minor up to 5000 nm.

5. Why is an average transmittivity necessary for a broadband source?

A flame, a hot body, or the Sun does not emit at a single wavelength; these are known as polychromatic sources. Consequently, the correction requires calculating an average value over the spectral band under consideration.

The incident spectral flux can be represented as E(λ). The total incident radiative power is therefore:

Φ_inc = ∫E(λ) dλ

After passing through the window:

Φ_mes = ∫E(λ) × τ(λ) dλ

The effective transmittivity of the window is thus:

τ_eff = ∫E(λ) × τ(λ) dλ / ∫E(λ) dλ

This equation constitutes the general recommended method for correcting the effect of the window's transmittivity in a general polychromatic case.

It simply means that the effective transmittivity is the average of the window's transmittivity, weighted by the energy actually emitted.

6. Practical method using discrete spectral data

It is not necessary to perform an analytical integration. If the user has discrete values ​​for the spectrum:

then one can calculate:

τ_eff = ∑(E(λi) x τ(λi) x Δλi) / ∑(E(λi) x Δλi)

For regular intervals:

τ_eff = ∑(E(λi) x τ(λi)) / ∑E(λi)

This formula is particularly simple and can be adapted for use in a spreadsheet, for example.

7. How to obtain the source spectrum?

When a radiative source is difficult to characterize spectrally, its spectrum can nonetheless be roughly estimated based on its temperature. The source can, in fact, be approximated as a black body or a gray body at a temperature Ts. Planck's law can then be used to estimate the spectral distribution:

Eλ(λ, Tₛ) = (2πhc² / λ⁵) × 1 / [exp(hc / (λ.kB.Tₛ)) − 1]

The calculation of τ_eff also becomes a function of Ts:

τ_eff(Tₛ) = ∫(Eλ(λ,Tₛ) × τ(λ)).dλ / ∫Eλ(λ,Tₛ).dλ

The following table provides, for reference, the value of the correction for a blackbody at various equivalent temperatures (blackbody equivalent temperature, or BBET):

This approach provides an estimate rather than a spectral characterization of the source. Naturally, it is up to the user to determine the best method for calculating the transmissivity correction.

8. Uncertainty related to spectral estimation

The uncertainty in τ_eff can become the primary contributor to the overall uncertainty when the source spectrum is not well known. An initial estimate can be obtained via:

u(Φ_corr) / Φ_corr = √[(u(Φ_inc) / Φ_inc)² + (u(τ_eff) / τ_eff)² + (u(τ₀) / τ₀)²]

Knowledge of the spectrum is therefore particularly important when τ(λ) varies significantly across the band in question.

Conversely, if the optical window has a nearly constant transmittance over the entire useful band, spectral uncertainty becomes much less critical.

9. Influence of the angle of incidence

Transmittance values ​​are generally provided for near-normal incidence. In a real-world configuration, the angle of incidence can alter:

• Fresnel reflection;

• the optical path length;

• the response of any anti-reflective (AR) coating;

• polarization.

The proposed method should therefore, whenever possible, utilize data corresponding to the actual geometry.

For an optical window positioned perpendicular to the flux with a small angle of incidence, this correction can generally be neglected as a first-order approximation.

Conclusion

The presence of an optical window on fluxmeters necessitates a distinction between the flux received by the sensor and the incident flux prior to the window.

The simplest and most robust correction involves comparing the effective transmittance of the window for the source spectrum against the transmittance corresponding to the calibration wavelength.

This approach allows for a consistent treatment of applications involving:

• lasers;

• flames;

• furnaces;

• thermal radiation;

• concentrated solar radiation;

• broadband radiative sources.