Measuring two-dimensional heat flux using a plate sensor and infrared thermography

Abstract

The thin-skin calorimeter sensor is used to measure heat flux. This device is limited to measurements at a single physical location. A new method, denoted HFITS, has been developed to obtain heat flux measurements over a two-dimensional surface. A large flat plate sensor is positioned near a heat source at the location where field heat flux measurements are desired; exemplar plate sensor designs and property data are provided in this work to support method application. An infrared camera is used to measure the temperature of the plate sensor over time. Thermogram sequences are translated to a rectilinear grid, and a two-dimensional inverse heat transfer model is used to derive measurements of net heat flux to the plate sensor. Incident heat flux (consisting of both radiative and convective components) is calculated by applying numerical approximations, optimizations, and engineering correlations. A validation experiment was performed, and the HFITS method was shown to produce measurements consistent with Schmidt-Boelter heat flux gauges, having an RMSE of 0.5 kW m^-2 for an incident radiative heat flux of 18 kW m^-2 (< 3 %). Limitations of the method, such as response time, sensor survivability, noise, and measurement uncertainty, are detailed.

• •The HFITS method provides measurements of two-dimensional heat flux fields to a target location.
• •The plate sensor is simple to fabricate, inexpensive, and is practical to deploy in various thermal engineering applications.
• •Exemplar plate sensor designs, property data, and infrared camera considerations are provided to support method application.

Graphical abstract

Specifications table

Subject area:	Engineering
More specific subject area:	Heat transfer
Name of your method:	HFITS: A method for measuring two-dimensional Heat Flux fields using Infrared Thermography and a plate Sensor
Name and reference of original method:	ASTM E459–22 Standard Test Method for Measuring Heat Transfer Rate Using a Thin-Skin Calorimeter, ASTM International, West Conshohocken, PA (2022). doi: 10.1520/E0459–22.
Resource availability:	https://github.com/ulfsri/HFITS

Background

This methodology, hereinafter denoted the “HFITS” method, involves the use of a plate sensor (also referred to as a thin-skin calorimeter), and infrared (IR) thermography to obtain measurements of heat flux over a two-dimensional surface exposed to an arbitrary heat source. This method builds upon ASTM E459–22 [1], which is intended for measurements at a single location, by providing spatially-resolved measurements. The primary application for which the HFITS method was developed is fire safety testing.

The HFITS method involves the use of a metallic plate sensor and an infrared camera to derive measurements of heat flux over a two-dimensional surface. The plate sensor is positioned near a heat source, an infrared camera is positioned on the opposite side, and heat flux to the sensor is derived from the measured thermogram sequence. Fig. 1 depicts the typical application and workflow for the HFITS method. Additional details of the instrument design and analysis procedure are provided in the Method Details section.

Fig. 1.

Overview of the HFITS methodology: (A) apparatus configuration for the intended application of HFITS; and (B) workflow for measurement and visualization of temporally and spatially varying surface heat flux.

The original method, ASTM E459–22 [1], describes a thin-skin calorimeter as a thin metallic plate with a thermocouple welded to one side of the plate. The side without the thermocouple is exposed to a heat source, and the temperature measurements are supplied to a zero-dimensional (thermally thin) inverse heat transfer model to calculate the net heat flux to the sensor. Given knowledge of convective heat transfer conditions, the incident radiative heat flux to the surface may be deduced. Inherent in this sensor design is the neglect of lateral heat transfer over the plate surface, making this sensor suitable for measurements of heat flux to a small area; for example, a sensor length scale of 10 cm or smaller is typical in fire testing applications [[2], [3], [4]].

The HFITS method improves upon the original method by expanding the capability from a single measurement location to a two-dimensional field measurement, and by replacing thermocouple measurements with non-contact infrared thermography. The HFITS method requires that: (1) the sensor area be discretized into a number of smaller elements; (2) independent measurements of the temperature of each discrete element be acquired; (3) lateral conduction heat transfer between elements be accounted for; and (4) convection heat transfer must account for the variation in film temperature over the surface of the sensor. The sensor size is limited only by practical fabrication constraints; to date, the method has been applied to sensor areas as large as 4 m² [[5], [6], [7], [8], [9], [10]].

Details of the HFITS methodology, including sensor design considerations and the analysis procedure, are provided in the Method Details section. The Validation section details a validation experiment that was performed to demonstrate the utility of the method. Limitations of the method, including sensor survivability and sensitivity, are provided in the Limitations section.

Method details

Operating principle

The purpose of the HFITS method is to measure the thermal exposure from a heat source, such as a fire or hot object, to a target location. While heat flux sensors typically used in the fire science field, such as Gardon [11] and Schmidt-Boelter [12] gauges, provide measurements at a single location, the HFITS method provides measurements over a two-dimensional field. The operating principle of the HFITS method (and the thin-skin calorimeter method on which it is based) is to infer heat flux to the sensor from measurements of the sensor’s temperature using the inverse formulation of an energy conservation model.

In the original method for a small plate sensor [1], net heat flux to the sensor may be expressed from conservation of energy on the plate sensor volume as:

$${\dot{q}}_{net,f}^{″}+{\dot{q}}_{net,b}^{″}=\rho c_p\delta \frac{\mathrm{d}T}{\mathrm{d}t}$$ (1)

where ${\dot{q}}_{net}^{″}$ is the net heat flux into the front ($f$) and back ($b$) sides of the plate (W m^-2), $\rho$ is the density of the plate material (kg m^-3), $c_p$ is the specific heat capacity of the plate material (J kg^-1 K^-1), $\delta$ is the thickness of the plate (m), $T$ is the temperature of the plate (K), and $t$ is time (s). The plate temperature is assumed to be uniform both in-depth ($z$) and over the plate surface ($x$ and $y$), and equal to the measurement provided by a thermocouple welded to the back face. The in-depth temperature gradient (and any error associated with neglecting this) may be minimized by using a thermally conductive, thin plate material. Common choices include copper, carbon steel, stainless steel, and Inconel 600 (a nickel-chromium-iron alloy). On the other hand, lateral heat transfer (and any error associated with neglecting this) may be minimized by using a material with a lower thermal conductivity; stainless steel and Inconel 600 are suggested [1]. Finally, there is some error associated with the accuracy of the temperature measurement, both in terms of the thermoelectric response and heat losses by conduction into the thermocouple wires [13].

While the plate temperature is measured over time, the required quantity in Eq. (1) is the rate of temperature change over time. This can be estimated from temperature measurements using a numerical approximation of the temporal derivative, but this has the undesired consequence of magnifying small perturbations in the temperature measurements (e.g., noise) to large fluctuations in the derived heat flux. This can be partially mitigated by applying signal processing filters to the temperature measurements, decimating, or applying a curve-fit (e.g., smoothing spline) and calculating the analytical derivative of the fit. However, such measures may risk smoothing over real phenomena (e.g., oscillations associated with flame radiation). Furthermore, Eq. (1) requires that the density and specific heat capacity of the plate material, which may vary with temperature, be known.

The net heat flux boundary condition on each side of the plate sensor consists of radiative and convective components, and may be expressed as:

$${\dot{q}}_{net}^{″}=\alpha {\dot{q}}_i^{″}-\epsilon \sigma T^4-{\dot{q}}_c^{″}$$ (2)

where ${\dot{q}}_i^{″}$ is the incident radiative heat flux (W m^-2), ${\dot{q}}_c^{″}$ is the convective heat flux out of the surface (W m^-2), $T$ is the surface temperature (K), $\alpha$ is the absorptivity of the surface, $\epsilon$ is the emissivity of the surface, and $\sigma$ is the Stefan-Boltzmann constant (5.6704 × 10^–8 W m^-2 K^-4). Assuming that the surface is diffuse and gray, the radiative properties may be considered to be spectrally and directionally independent ($\epsilon$ = $\alpha$).

Convective heat flux depends on the temperature of the plate surface and the nearby gas, the gas velocity, the intrinsic properties of the gas, and the surface characteristics of the plate. It can be represented in terms of a convective heat transfer coefficient:

$${\dot{q}}_c^{″}=h_c\left(T-T_{\infty }\right)$$ (3)

where $h_c$ is the convective heat transfer coefficient (W m^-2 K^-1) and $T_{\infty }$ is the free-stream gas temperature (K).

In many applications the quantity of interest is the incident radiative heat flux to the front of the sensor, which is a characteristic of the heat source and its configuration relative to the sensor. If it can be assumed that the surroundings are large and at a uniform temperature of $T_s$ (K), and that there is no heat source on the back side (${\dot{q}}_{i,b}^{″}=\sigma T_s^4$), then Eqs. (1) to (3) can be solved for the incident radiative heat flux to the front side of the sensor:

$${\dot{q}}_{i,f}^{″}=\frac{\rho c_p\delta }{\epsilon }\frac{\mathrm{d}T}{\mathrm{d}t}-\sigma T_s^4+2\sigma T^4+\frac{h_{c,f}}{\epsilon }\left(T-T_{\infty ,f}\right)+\frac{h_{c,b}}{\epsilon }\left(T-T_{\infty ,b}\right)$$ (4)

Eq. (4) requires knowledge of the temperature of the surroundings and the free-stream gas temperatures. In a controlled lab environment, it may be appropriate to assume a value of $T_s$ equal to the ambient temperature far from the heat source. Additional measurements of the local gas temperature near the sensor may be used to represent free-stream temperatures, though depending on the relative position of the heat source it may be appropriate to also approximate the free-stream temperatures as equal to the ambient temperature. Convection heat transfer coefficients are also required, and may be estimated using engineering correlations or characterized experimentally. Finally, the total hemispherical emissivity of the plate material must be known.

In the modified method (HFITS), a large plate sensor area is used, and the temperature is no longer assumed to be uniform over the plate surface ($x$ and $y$). Fig. 2 depicts a differential element of a plate sensor having an area of $\mathrm{d}x$ by $\mathrm{d}y$, and a thickness of $\delta$ [14]. The element thickness is assigned a discrete value owing to the thermally thin assumption. Heat conduction (${\dot{q}}_x$ and ${\dot{q}}_y$) and exposure (${\dot{q}}_{net,f}$ and ${\dot{q}}_{net,b}$) heat transfer terms are shown.

Fig. 2.

Net heat flux into a differential element of the plate sensor.

Applying conservation of energy to this differential element, and assuming that the thermal conductivity ($\lambda$) is not spatially dependent, yields a modified form of the Fourier field equation (FFE) [14]:

$${\dot{q}}_{net,f}^{″}+{\dot{q}}_{net,b}^{″}+\lambda \delta \left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)=\rho c_p\delta \frac{\partial T}{\partial t}$$ (5)

In this equation temperature is the measured quantity, but the first temporal derivative and second spatial derivatives of temperature are needed to calculate the net heat flux. This inverse heat transfer (IHT) problem is inherently ill-posed, and requires the application of numerical procedures to approximate the derivative terms [15]. As depicted in Fig. 3, the plate sensor area is discretized into elements of finite size. By convention of the energy balance method [14], conduction heat transfer is directed into the element. Energy conservation yields:

$${\dot{q}}_{net,f}^{″}+{\dot{q}}_{net,b}^{″}+\frac{\delta }{\Delta x}\left({\dot{q}}_{x-}^{″}+{\dot{q}}_{x+}^{″}\right)+\frac{\delta }{\Delta y}\left({\dot{q}}_{y-}^{″}+{\dot{q}}_{y+}^{″}\right)=\rho c_p\delta \frac{\partial T}{\partial t}$$ (6)

where ∆x and ∆y are the width and height of the element in the $x$ and $y$ ordinates, respectively.

Fig. 3.

Net heat flux into a discrete element of the plate sensor.

Under the same assumptions as the zero-dimensional case, the expression for incident radiative heat flux to the front side of the plate sensor becomes:

$${\dot{q}}_{i,f}^{″}=\frac{\rho c_p\delta }{\epsilon }\frac{\mathrm{d}T}{\mathrm{d}t}-\sigma T_s^4+2\sigma T^4+\frac{h_{c,f}}{\epsilon }\left(T-T_{\infty ,f}\right)+\frac{h_{c,b}}{\epsilon }\left(T-T_{\infty ,b}\right)-\frac{\delta }{\epsilon \Delta x}\left({\dot{q}}_{x-}^{″}+{\dot{q}}_{x+}^{″}\right)-\frac{\delta }{\epsilon \Delta y}\left({\dot{q}}_{y-}^{″}+{\dot{q}}_{y+}^{″}\right)$$ (7)

The lateral conduction heat flux in the $x-$ direction is given by Fourier's law:

$${\dot{q}}_{x-}^{″}=-\lambda {\frac{\partial T}{\partial t}|}_{x-}\approx \frac{\lambda }{\Delta x}\left(T_{x-}-T\right)$$ (8)

where $\lambda$ is the thermal conductivity of the plate material (W m^-1 K^-1). Similar equations are developed for the $x+$, $y-$, and $y+$ directions. As shown in the equation, the spatial temperature derivative can be expressed using a first-order finite difference approximation, where the conduction path length is equal to the element size in the $x$ and $y$ directions; gradient-based approaches or higher order finite difference approximations may also be used to approximate the derivative terms. First-order spatial approximations yield the following equation for net heat flux to an internal element:

$${\dot{q}}_{net,f}^{″}+{\dot{q}}_{net,b}^{″}+\frac{\delta \lambda }{\Delta x^2}\left(T_{x-}-2T+T_{x+}\right)+\frac{\delta \lambda }{\Delta y^2}\left(T_{y-}-2T+T_{\mathrm{y}+}\right)=\rho c_p\delta \frac{\partial T}{\partial t}$$ (9)

For edge and corner elements, the conduction terms must be represented using the appropriate boundary conditions. Examples include adiabatic boundary conditions for sensors utilizing an insulated frame (Eq. (10)) and fixed-temperature boundary conditions for sensors utilizing a fixed-temperature frame (Eq. (11)).

$${\dot{q}}_{x-,\text{insulated}}^{″}=-\lambda {\frac{\partial T}{\partial t}|}_{x-}=0$$ (10)

$${\dot{q}}_{x-,\text{fixed}}^{″}=-\lambda {\frac{\partial T}{\partial t}|}_{x-}\approx \frac{\lambda }{\Delta x/2}\left(T_{\text{fixed}}-T\right)$$ (11)

Method development

The HFITS method has been developed based on insights from several decades of research. A brief history of recent developments leading to the recommended sensor design (detailed in the follow section) is provided.

Dillon [5] used a two-dimensional plate sensor to measure heat flux to a surface in a compartment fire scenario. The carbon steel plate sensor measured 1.2 m wide by 0.6 m tall by 4.7 mm (7 ga) thick. Plate temperatures were measured using an array of thermocouples affixed to the back side of the sensor with steel strips and screws. The plate area was spatially discretized such that each thermocouple location corresponded to the center of an element. Insulation was installed on the back of the sensor, creating a pseudo-adiabatic boundary condition (${\dot{q}}_{net,b}^{″}\approx 0$). A black paint coating was applied to the plate, but the optical properties of that surface were not measured. A limitation of this technique was that the thick plate material, while robust in this fire application, had the effect of damping the thermal response and decreasing measurement sensitivity. Furthermore, the spatial resolution was fairly coarse, owing to practical limitations in using thermocouples to measure the surface temperature distribution.

DiDomizio [6] built upon Dillon’s approach for measuring compartment fire exposures with a refined sensor design and calculation method. A carbon steel plate sensor was used, measuring 1.9 m wide by 1.9 m tall by 1.3 mm (18 ga) thick, with an array of thermocouples welded to the back side of the sensor. The back side of the plate sensor was uninsulated, and the temperature of gases and the surroundings on the unexposed side were measured to account for ${\dot{q}}_{net,b}^{″}$ in the calculation of ${\dot{q}}_{i,f}^{″}$ (Eq. (7)). The sensing surface was coated in a high-temperature high-emissivity (HTHE) flat black paint, and the optical properties of that surface were independently measured. Additionally, the Bayesian uncertainty in heat flux measurements was calculated. This approach improved measurement sensitivity and reduced uncertainty, but was limited by the coarse spatial resolution provided by the thermocouple array.

Rippe and Lattimer [7] further improved upon this methodology by using infrared thermography to measure the surface temperature of a plate sensor exposed to fire. They performed experiments using a 304 stainless steel plate sensor measuring 0.6 m wide by 0.6 m tall by 1.0 mm (20 ga) thick. The plate area was discretized such that each measurement location (based on the projected pixel resolution of the camera) corresponded to the center of an element in the subsequent IHT calculations. This approach was shown to produce heat flux measurements with greatly improved spatial resolution. The authors reported that filtering of temperature measurements was necessary due to the amplification of noise in the derivative terms (first-order finite difference numerical approximations were used).

Hodges et al. [8] and DiDomizio et al. [9] would further improve the procedure, ultimately reporting that a plate sensor constructed of 0.79 mm (22 ga) 304 stainless steel is optimal for measuring exposures in fire testing applications up to the thermal and structural limitations of the sensing surface. One of the limiting factors of this surface is the integrity and optical properties of the steel coating, which is required to ensure that radiative heat transfer is properly accounted for in the IHT model. Bellamy et al. [16] investigated the thermal stability and optical properties of seven different HTHE paints for coating the sensing surface, and provide property data and general guidance on their use for this application.

DiDomizio and Butta [10] recently deployed a 1.2 m wide by 2.4 m tall by 0.79 mm (22 ga) 304 stainless steel plate sensor to measure heat flux distributions from a variety of fire sources (gas burner, liquid pools, wood cribs, and upholstered furniture) to a freestanding wall. Grids of bare-bead thermocouples were installed on both sides of the plate sensor to measure gas temperatures near the surface. Schmidt-Boelter heat flux gauges were installed at five positions to validate the plate sensor measurements. Although previous studies accounted for convective heat transfer using empirical correlations, in this study an optimization routine and interpolated measurements of the free-stream gas temperatures were used. Furthermore, for the first time, a water-cooled frame was employed to impose a constant temperature boundary condition on the four edges of the plate. It is unclear whether these modifications significantly improved the accuracy of heat flux measurements, but this work demonstrated that alternative treatments of convective heat transfer and conduction boundary conditions can produce measurements that are consistent with traditional gauges.

Building upon the insights of these past works, a refined methodology has been developed for the measurement of two-dimensional heat flux using a plate sensor and infrared thermography.

Instrument design

Plate sensor

The plate sensor should consist of a sensing surface (over which heat flux measurements will be obtained), a mounting frame (within which the sensing surface will be mounted), and a supporting system (to support the plate sensor in the desired location).

The sensing surface should be constructed of a thin metallic material coated in HTHE paint. The calculation procedure adopts a thermally thin assumption; that is, in-depth heat conduction is instantaneous. Although some difference between the temperatures on the front and back surfaces is expected, in practice this difference can be minimized by selecting a thin material for the sensing surface. Thinner materials will increase measurement sensitivity, which will improve the ability to resolve small changes in heat flux, but will also manifest increased noise in the derived heat flux. Another consideration is that thinner materials may be more susceptible to deformation under severe heating, which may compromise the integrity of the high-emissivity coating at high temperatures. Based on the findings of previous work, the recommend sensing surface for this application is 0.79 mm thick (22 ga) 304 stainless steel with a 35 ± 5 µm thick coating of Rust-Oleum Specialty High Heat (flat black) paint applied to both sides. This surface has been demonstrated to be suitable up to temperatures of 550 °C before failure of the coating [10,16].

The thermophysical and optical properties of the recommended sensing surface have been characterized in previous works [10]. Temperature-dependent specific heat capacity and thermal conductivity were measured using the laser flash method [17], and are provided in Fig. 4. Density was measured to be 7590 kg m^-3. Spectral near-normal emissivity of the surface, shown in Fig. 5, was measured using a Bruker Invenio R Fourier Transform Infrared Spectrometer (FTIR) equipped with a Bruker A562-G gold-coated integrating sphere and a liquid nitrogen-cooled Mercury Cadmium Telluride (MCT) detector. The spectral measurement range of the apparatus was 1.33 µm to 16.66 µm, which accounts for the majority of thermal radiation that is relevant in the intended applications [10]. Using a weighting function for an effective blackbody with source temperatures ranging from 20 °C to 1000 °C, the average emissivity over this band was calculated to be 0.94, which is analogous to the total emissivity for this application. As detailed below, the microbolometer IR camera that is recommended for these measurements has a spectral range of 7.5 µm to 14 µm. Using the same methodology, the total emissivity was calculated to be 0.94 over this band as well.

Fig. 4.

Thermophysical properties of the recommended sensing surface: (A) specific heat capacity; and (B) thermal conductivity [10].

Fig. 5.

Spectral emissivity of the recommended sensing surface [10].

The sensing surface should be mounted in a rigid frame that secures the surface while also accepting connections for additional supporting members. The frame should be as thin as practically possible in the $z$ ordinate to eliminate any obstruction of the view factor between the heat source and the sensing surface, or between the IR camera and the sensing surface. To simplify sensor fabrication and repair, a framing system is recommended over welded stock. For large sensors (side lengths greater than 60 cm), 4.1 cm (1–5/8 in.) steel strut channel has been found to be suitable. The sensing surface and edge insulation may be mounted within the channel, allowing room for thermal expansion. For smaller frames, 2.5 cm (1 in.) aluminum T-slot extrusions have been found to be suitable. The sensing surface and insulation may be mounted in aluminum U-channel that has been bolted to the aluminum framing members.

The sensing surface should be mounted in the frame such that its edges are either insulated or maintained at a fixed temperature. Exemplar CAD models of square plate sensors with side lengths of 1.2 m (4 ft) are shown in Fig. 6 (insulated edges) and Fig. 7 (fixed-temperature edges). In each case, the frames were designed using steel strut channels and fittings. Fittings are shown in green color, for clarity. U fittings are provided on the top and bottom strut channels to be used as mounting points.

Fig. 6.

Exemplar design of a plate heat flux sensor with insulated edges.

Fig. 7.

Exemplar design of a plate heat flux sensor with fixed temperature (water-cooled) edges.

For the insulated edge case, the strut channels are packed with either ceramic fiber board or batt insulation, and the sensing surface is compressed into the insulation. For the fixed-temperature case, the sensing surface is fit into copper support clips that contain a 1.2 cm (0.5 in.) deep slot on the inside, and a U-channel on the outside. This U-channel is sized to tightly fit 1.0 cm (3/8 in.) diameter copper tubing. Once the clips are installed around the sensor, copper tubing and fittings are soldered into the channel. A tee fitting is soldered into the cooling loop at top and bottom of the sensor for the water supply and drain, respectively. Strut channels filled with insulation are then attached around the water-cooled sensor assembly.

An advantage of using a framing system over welded stock is the ease of attachment of additional fittings to support the plate sensor at the measurement location. The support system must prevent any movement of the plate sensor relative to the IR camera, since the data analysis procedure assumes a fixed geometric reference. Examples of support systems include suspending the plate sensor from metal cables or chains, or mounting it to a rigid structure. Care should be taken to ensure that the support system is thermally protected, to prevent the plate sensor from displacing as a result of thermal expansion.

The exemplar plate sensor designs provided here are recommendations based on the most recent research and engineering considerations in applying the HFITS method. Alternative framing, edge treatments, and sensing surface designs may be suitable for use with the HFITS method if appropriate property determination and model validation accompany those designs.

Infrared thermography

An infrared thermal imaging camera, or “IR camera”, measures radiance - the radiant power emitted from a source per unit of projected area and solid angle. Thermal radiation is transmitted through the camera's optics and focused onto a spectrally selective sensor. The digital count produced by the sensor is converted to radiance values using a calibration function unique to the sensor and camera model. Radiance measurements are obtained over the camera's field of view (FOV) and projected onto a 2D grid analogous to pixels in an optical camera. The temperature at each pixel location is derived from the radiance values by applying corrections for atmospheric attenuation, reflected radiation from the surroundings, and transmission effects from any attached optics [18]. The computed temperature field data at a single time instance is referred to as a thermogram.

In the HFITS application, the IR camera is used to produce thermogram sequences within which the plate sensor area is encapsulated. These thermograms define the $T\left(x,y,t\right)$ values of discrete elements in Eq. (9) (see also Fig. 3). The IR camera must be selected to achieve the desired spectral selectivity, temperature range, resolution, field of view, and recording frame rate for the intended experimental configuration.

The spectral range of an IR camera with an uncooled microbolometer sensor (typically 7.5 µm to 14 µm) is appropriate for measurements of the recommended sensing surface, for which the band-averaged emissivity is equal to the total emissivity over the relevant portion of the infrared spectrum. The temperature range should accommodate the expected range of the plate sensor temperature; for the recommended sensing surface, the temperature should not exceed 550 °C at any location to ensure the survivability of the coating.

The IR camera should be sufficiently separated from the plate sensor to avoid thermal damage. The maximum internal temperature of an IR camera for safe operation is typically reported to be between 50 °C and 60 °C. Thermal damage can be mitigated by cooling the external housing and protecting the optics with windows that are partially transparent to infrared radiation. If an external window is used (e.g., Germanium), it is crucial that the impact of this window on temperature measurements be accounted for in the IR corrections. Spectrally selective windows may require recalculating the band-averaged emissivity.

The spacing between the IR camera and the plate sensor (separation distance) should be selected to achieve the greatest spatial resolution while maintaining thermal safety of the IR camera. Spatial resolution is dictated by the separation distance and the IR camera’s resolution and FOV. The IR camera's FOV is usually specified in terms of horizontal ($AFO{V}_x$) and vertical ($AFO{V}_y$) angular fields of view. The projected fields of view at a separation distance $d$ can be calculated from:

$$FO{V}_i=2d\tan \left(\frac{AFO{V}_i}{2}\right)$$ (12)

where $i$ denotes the horizontal ($x$) or vertical ($y$) ordinate, respectively. The thermogram pixel size is:

$${\Delta }_i=\frac{FO{V}_i}{r_i}$$ (13)

where $r_i$ is the IR camera's resolution (px) in the horizontal ($x$) or vertical ($y$) ordinate, respectively. Many IR cameras use sensors that project measurements to square pixels, in which case ∆_x = ∆_y = ∆.

For reference, Table 1 shows the resolution ($r_i$), angular fields of view ($AFO{V}_i$), projected fields of view ($FO{V}_i$), and thermogram pixel size ($\Delta$) at a 3 m separation distance for several IR camera models. Note that while many IR camera models are available with multiple lens options, only those with an $AFO{V}_x$ between 50° and 60° were selected for this comparison. Additionally, Fig. 8 shows the dependence of $\Delta$ and $FO{V}_x$ on the separation distance for a common sensor resolution of 640 px by 480 px and a range of $AFO{V}_x$ values. This table and figure can assist in selecting a suitable camera model based on the safe separation distance and sensor area for a given application.

Table 1.

Projected field of view and thermogram pixel size at 3 m separation distance for several IR camera models.

Camera Model	$r_x$	$r_y$	$AFO{V}_x$	$AFO{V}_y$	$FO{V}_x$	$FO{V}_y$	$\Delta$
	(px)	(px)	(°)	(°)	(m)	(m)	(mm)
InfraTec VarioCAM HD head 900	1024	768	60.3	47.1	3.5	2.6	3.4
FLIR A70	640	480	50.2	38.7	2.8	2.1	4.4
FLIR A50	464	348	51.4	39.7	2.9	2.2	6.2
Seek G300	320	240	56.0	42.0	3.2	2.3	9.8
ICI FMX 640 P	640	512	50.0	37.5	2.8	2.0	4.2
Optris PI 640i	640	480	60.0	45.0	3.5	2.5	5.3
Fotric 628CH	640	480	50.0	38.0	2.8	2.1	4.3
Mikron MCL640	640	480	57.0	44.3	3.3	2.4	5.1

Fig. 8.

Thermogram pixel size and horizontal field of view for an IR camera with 640 px by 480 px resolution.

Finally, the IR camera should be selected with a recording frame rate appropriate for the time scale of heat transfer events in the given application. Many uncooled microbolometer IR cameras support a maximum frame rate of 25 Hz to 30 Hz at full resolution, while some models offer higher frequency recording at reduced resolutions. Recording at a lower frequency can greatly reduce the size of thermogram sequences and processing times, at the risk of potentially obfuscating higher frequency phenomena. If computational resources are not practical limitations, the highest possible recording frequency should be used. If necessary (and if appropriate for the phenomena of interest) IHT calculations may be performed on a decimated dataset to reduce the required computational load.

Additional measurements

For fixed-temperature plate sensor designs, the temperature of the edge elements must be known to define the conduction boundary conditions on the four edges of the sensing surface (Eq. (11)). If a cooling loop is used as depicted in Fig. 7, it is recommended to measure the water temperature using mineral-insulated metal-sheathed (MIMS) thermocouples installed in-line with compression fittings. Thermocouples could be installed around the perimeter of the sensing surface, or at the inlet and outlet. An ideal fixed-temperature boundary condition would manifest no difference between inlet and outlet water temperatures, but in practice the outlet temperature is expected to be slightly greater than the inlet due to the water absorbing heat from the perimeter of the sensing surface. The flow rate of water should be high enough to minimize this temperature difference, and thus minimize the error in the boundary condition model.

The HFITS method, and the thin-skin calorimeter method upon which it is based, rely on an estimation of the convective heat flux on both sides of the plate to calculate the incident radiative heat flux (Eq. (2)). One common approach for representing convective heat transfer is to use an empirical model to estimate local values of the convective heat transfer coefficient over the surface; as shown in Eq. (3), this requires knowledge of the free-stream gas temperature over the surface of the plate sensor. If the heat source is far from the plate sensor and local gas velocities are low, a natural convection model may be appropriate. In this case the free-stream temperature can be approximated as a global sink temperature (i.e., ambient). On the other hand, if the heat source is close to the sensor, or reactive media (e.g., flames) are close to its surface, then such correlations will no longer apply. A grid of discrete gas temperature measurements near the plate sensor could be used to represent the local sink temperatures, and convective heat transfer coefficients could be calculated experimentally. For example, DiDomizio and Butta [10] successfully employed a grid of fine wire bare bead thermocouples (36 ga) positioned 2.5 cm from the surface of a plate sensor to characterize the local free-stream temperatures for convective heat transfer estimates.

Analysis procedure

The HFITS method involves three analysis steps: thermogram processing, heat transfer analysis, and post-processing. The authors have developed an open-source software to automate these analysis procedures [19]. This software is by no means a requirement for application of the HFITS method, but it serves as a useful reference implementation for practitioners of the method.

Thermogram processing

Before being supplied to the IHT model, spatial corrections must be applied to raw thermograms to account for radial and perspective distortion and cropping. This procedure, referred to as “rectification” by some authors [[7], [8], [9], [10]], is depicted in Fig. 9.

Fig. 9.

Depiction of the application of spatial corrections to raw thermograms.

All lenses impart some amount of radial distortion in thermograms. The IHT analysis relies on a Cartesian discretization of the Fourier field equation; therefore, radial distortion correction must be employed to ensure that the plate sensor area coincides with a rectilinear grid. Radial distortion correction models, such as the division model [20], can be applied to thermograms in a manner similar to that of digital photographs.

Perspective distortion may also be present if the IR camera is not aligned normal to the plate sensor, making the rectangular sensor area appear trapezoidal. Since the plate area must coincide with a rectilinear grid, projective geometry transformations may be required to correct this form of distortion. It is recommended to align the camera as close to normal to the plate sensor as possible, to minimize the perspective distortion corrections required. It should also be noted that the IHT model assumes that surface properties are not directionally dependent (i.e., the emissivity, measured at a near-normal angle of incidence, is representative of the hemispherical emissivity). This assumption may break down at large angles of incidence (e.g., >45°).

Finally, the distortion-corrected thermograms should be cropped to only include only the region of interest of the plate sensor (the rectilinear region within which each thermographic pixel corresponds to a discrete element in the IHT model, bounded by edge elements which adjoin either insulation or a fixed-temperature element).

Some spatial corrections may be applied using commercial software packages developed by an IR camera's manufacturer, such as IRBIS 3.1 for InfraTec IR cameras. It is also possible to perform these corrections using established algorithms such as warpPerspective from the OpenCV [21] Python package.

Inverse heat transfer analysis

Numerical methods must be employed to derive estimates of the incident radiative heat flux over the plate sensor from surface temperature measurements using the underlying physical model (Eq. (5)).

The discretized form of the two-dimensional Fourier field equation is given by Eq. (7). An explicit finite difference solution to this equation may be implemented, using finite difference approximations for the spatial and temporal derivative terms. Since the temperature of all elements is known at the current time step, posterior values may be calculated using thermophysical properties and convective heat transfer coefficients calculated at the same time step; prior values need not be considered when using this approach. Even so, small perturbations in temperature measurements can propagate to larger apparent noise in the derived heat flux - this is further explored in the Limitations section. Alternatively, gradient-based optimization approaches [15] may be employed to solve the FFE. Such approaches may consider both spatial and temporal gradients, effectively smoothing the calculated field data. While such approaches may yield heat flux measurements with reduced noise, the impact on measurement uncertainty has not been investigated to date.

Although it is possible to compute the net heat flux to the plate sensor (${\dot{q}}_{net,f}^{″}+{\dot{q}}_{net,b}^{″}$) considering only energy storage and lateral conduction heat transfer terms, ultimately the objective of this method is to measure the thermal exposure to the front side of the plate sensor from an arbitrary heat source. This is best expressed in terms of incident radiative heat flux (${\dot{q}}_{i,f}^{″}$), which requires that convective heat flux on both sides be known (or estimated). For this, the quantities $h_{c,f}$ and $h_{c,b}$ must be calculated for each finite element on both the exposed and unexposed sides of the plate (using the corresponding film temperatures on each side). Uncertainty associated with convective heat transfer estimates has been reported to be one of the greatest contributors to the expanded measurement uncertainty with IHT-based heat flux measurement methods [7]. In light of this, some authors [[7], [8], [9]] have opted to use constant values of convective heat transfer coefficient, propagating this uncertainty to the derived ${\dot{q}}_{i,f}^{″}$. Others [10] have employed numerical optimization methods using replicate experiments to deduce convective heat transfer coefficients that are appropriate for a given experimental configuration, but the uncertainty associated with such optimizations is difficult to quantify. Additional study is needed to understand the impact of the convective heat transfer model on the accuracy and expanded uncertainty of measurements made with the HFITS method.

Post-Processing

The main quantity of interest derived from the heat transfer analysis is the incident radiative heat flux, which varies both spatially (over the surface of the plate sensor) and temporally. The recommended output of the HFITS method is contour plots at specified time instances, with $x$ and $y$ corresponding to the abscissa and ordinate axes, and ${\dot{q}}_{i,f}^{″}$ corresponding to the depth axis. Contour sequences may be generated to display the temporal progression of the measured heat flux field; sequences may also be rendered into video files. Additionally, line plots may be generated of heat flux at specified positions on the plate sensor over time on the ordinate and abscissa axes, respectively. Examples of these are provided in the Method Validation section.

Method validation

An experiment was carried out to validate the HFITS method. In this experiment a vertically-oriented square plate sensor was centrally aligned with a propane-fired radiant panel heater. The sensing surface of the plate sensor was 0.79 mm (22 ga) thick 304 stainless steel coated in Rust-Oleum Specialty High Heat (flat black) paint, with side dimensions of 60 cm. The sensing surface was mounted in an insulated aluminum frame and was positioned 40 cm from the emitting surface of the radiant panel, which measured 45.7 cm tall by 30.5 cm wide. An infrared camera (Infratec VarioCAM HD head 980, having a microbolometer sensor with a spectral range of 7.5 µm to 14 µm and resolution of 1024 px by 768 px) was positioned on the unexposed side of the sensor. After distortion corrections (see below), the thermogram pixel size was 1.7 mm.

Experiments were also conducted with Medtherm 64-series water-cooled Schmidt-Boelter total heat flux (THF) gauges in place of the plate sensor. Schmidt-Boelter THF gauges are widely adopted in the fire science field for single-point heat flux measurements, and are suitable as a validation reference. THF gauges were mounted in calcium silicate board (2.5 cm thick and having the same width and height dimensions as the plate sensor) at three locations. THF gauge experiments were conducted in triplicate.

Both the thermogram sequence and the heat flux gauge measurements were recorded at 1 Hz (higher frequency phenomena were not relevant in this case). The experimental procedure involved first igniting and stabilizing the burner for 1 min with a radiation shield (a 120 cm by 120 cm panel consisting of 2.5 cm thick ceramic fiber insulation affixed to 1.9 cm thick plywood) positioned between the burner and the target. Once stable, the shield was removed, and the target was exposed for 9 min. Fig. 10 shows a photograph of the experiment setup.

Fig. 10.

Setup and apparatus used for HFITS validation experiments.

Thermograms were rectified and IHT calculations were performed using the reference HFITS code [19]. Fig. 11 shows a processed thermogram and the derived incident radiative heat flux corresponding to the end of the exposure duration (9 min). Convective heat transfer coefficients of 20 W m^-2 K^-1 (see below) and free-stream temperatures of 22.6 °C (the ambient temperature) were assumed on both the front and back sides. The region of greatest temperature and heat flux coincided with the center of the plate sensor, which is consistent with the radiant panel and plate sensor being centrally aligned. Both the temperature and heat flux fields were horizontally symmetric, and slightly favored the upper region of the sensor. This is owing to radiation from the hot surface immediately above the panel, which was heated by combustion gases expelled from the radiant panel.

Fig. 11.

Field measurements from the validation experiment after 9 min of exposure: (A) plate temperature; and (B) incident radiative heat flux.

The locations of the three THF gauges used in those experiments are also indicated in Fig. 11. Heat flux measured with the THF gauges was compared to that measured with the HFITS method at each corresponding location; results are shown in Fig. 12. In the figure, THF gauge measurements for each replicate experiment are depicted as small black circles, while the timeseries average of all three experiments is depicted as a black line. A detailed investigation of convective heat transfer to the plate sensor is not within the scope of the present paper; rather, the convective heat transfer coefficient was considered to be an independent variable in this validation exercise. IHT analysis was carried out using $h_c$ values ranging from 15 W m^-2 K^-1 to 25 W m^-2 K^-1. Measurements with the HFITS method are shown in Fig. 12 for three values of convective heat transfer coefficient. Using $h_c$ = 20 W m^-2 K^-1 produced an excellent match between HFITS and THF gauge measurements at all three locations, with root mean squared differences of 0.54 kW m^-2, 0.27 kW m^-2, and 0.15 kW m^-2 at locations 1 through 3, respectively.

Fig. 12.

Comparison of heat flux measurements from the validation experiment using THF gauges and the HFTIS method at: (A) location 1 (center); (B) location 2 (above center); and (C) location 3 (bottom corner).

Measurements with the HFITS method exhibited an initial spike corresponding to the removal of the radiation shield. This is a consequence of the numerical approximation of the temporal derivative (a first-order finite difference approximation was used), and is an expected response to the sudden change in exposure. This behavior is further addressed in the Limitations section.

This validation exercise has demonstrated that the HFITS method is capable of producing measurements of incident radiative heat flux that are in agreement with the measurements of Schmidt-Boelter heat flux gauges. Care must be taken with the treatment of convective heat transfer in this procedure. Future developments of the HFITS method will address convective heat transfer sensitivity and optimization procedures. Additional limitations of the method are addressed in the following section.

Limitations

Response time

Both the original method and the modified method rely on the assumption that the sensing surface is thermally thin. This assumption can be verified by calculating an effective Biot number (including radiative and convective components). A Biot number <0.1 is considered an acceptable criterion for the thermally thin assumption (Bi = 0.1 corresponds to an error of <3 % in a zero-dimensional heat transfer model) [14]. An effective Biot number of 0.0035 was calculated using a forward heat transfer calculation on a finite element having $\delta$ = 0.79 mm, ${\dot{q}}_{i,f}^{″}=50$ kW m^-2, $h_c$ = 20 W m^-2 K^-1, and $T_s$ = $T_{\infty }$ = 20 °C. This demonstrates that the resistance to in-depth conductive heat transfer is significantly less than the heat transfer into the surface, making in-depth temperature gradients negligible and justifying the thermally thin assumption.

Although the thermally thin assumption is justified for this method, there is expected to be a brief initial transient period during which the temperature of the back (unexposed) surface of the plate sensor will lag behind that of the front (exposed) surface. As was demonstrated in the validation experiment, this can manifest an apparent spike in the heat flux derived by the HFITS method in response to a sudden change in exposure. As specified in the original method, the response time of the plate sensor is defined as the time for the rate of temperature change of the unexposed surface to reach 99 % of the rate of temperature change of the exposed surface [1]. As shown in Fig. 13, the response time of the recommended sensing surface is approximately 80 ms for a plate temperature of 20 °C.

Fig. 13.

Response time of the plate sensor as a function of plate temperature.

Sensor survivability

The main limitation on the survivability of the sensor is delamination or decomposition of the HTHE coating. For the recommended sensing surface, 550 °C is considered the maximum operating temperature for survivability. This does not necessarily correspond to a specific threshold value of heat flux over which the plate sensor will not survive — the time-history of exposure is relevant (Eq. (5)). However, to aid in experimental design, it is useful to consider steady-state exposure limits.

Fig. 14(A) shows the steady-state temperature that will be reached when the recommended sensing surface is exposed to a constant incident radiative heat flux of the specified intensity. Fig. 14(B) shows the time to reach a steady-state surface temperature of 550 °C when the plate sensor is exposed to a constant incident radiative heat flux. In calculating these values, lateral heat transfer was neglected (reasonable when the temperature of a particular element is close to that of adjoining elements), convective losses were set to zero, and $T_s$ = 20 °C. From this analysis, the plate sensor is expected to survive exposures less than approximately 50 kW m^-2 indefinitely. For exposures greater than 50 kW m^-2, the sensor's survivability may be measured on the order of minutes or even seconds.

Fig. 14.

Survivability of a plate sensor exposed to constant irradiance: (A) steady-state temperature; and (B) time to reach the critical temperature for failure.

Sensor orientation

The exemplar sensor designs (Fig. 6, Fig. 7) are intended for a vertical (perpendicular to grade) or near-vertical orientation. In a horizontal orientation (parallel with grade), such designs may cause the large metallic sensing surface to deform under its own weight, requiring alternative framing solutions. Other works [7,9] have employed large plate sensors in a horizontal orientation by laying the sensing surface flat on thin insulation strips or small metal pins. Either of these approaches will affect heat transfer at those locations and must be accounted for in the IHT model. Regardless of orientation, care must be taken to ensure that the sensing surface is flat. Contact with other surfaces should be minimized, and must be limited to the edges of the sensing surface. The goal is to minimize the thermal impact of the framing design on heat transfer in the sensing surface.

Numerical approximations and noise

Numerical approximations of the derivative terms in the IHT model can cause small perturbations in the measured temperature of the plate sensor to manifest greater apparent noise in the derived heat flux. This is an inherent limitation of the method, and should be considered in its application and data treatment. The effect of temporal discretization on noise is explored here.

A temporally varying incident radiative heat flux curve was defined for a 5 min exposure, and a forward heat transfer model was used to calculate the plate temperature corresponding to this input exposure. Lateral heat transfer was neglected (reasonable when the temperature of a particular element is close to that of adjoining elements), convective losses were set to zero, and $T_s$ = 20 °C. Fig. 15(A) shows the prescribed incident radiative heat flux and resulting plate temperature. The forward model was run using time steps of 0.1 s, 0.2 s, and 0.5 s. For each case, random white noise of 0.2 K was added to the temperature results to represent measurement noise. The synthetic temperature data was then supplied to an inverse model to deduce the incident radiative heat flux to the plate sensor. A first order finite difference approximation of the temporal derivative was used. As shown in Fig. 15(B), the small amount of noise added to the temperature data was significantly magnified in the heat flux computed by the inverse model. Furthermore, the noise in the derived heat flux decreased as the time step used for the IHT calculation increased.

Fig. 15.

Effect of temporal discretization on noise: (A) temperature calculated with a forward model supplied with a prescribed heat flux; and (B) heat flux calculated using an inverse model supplied with synthetic temperature data.

While it is recommended to record thermogram sequences at a high frame rate to ensure that heat transfer phenomena are captured effectively (e.g., 30 Hz), this example demonstrates that processing thermogram sequences with the IHT model at a lower frame rate (e.g., 1 Hz) will reduce the noise in the derived heat flux measurements. This can readily be achieved by decimation (i.e., process every n^th frame of the thermogram sequence), though care should be taken to ensure that the processing time scale is less than the time scale of the relevant heat transfer phenomena. Alternatively, multidimensional interpolation and smoothing algorithms might be employed to reduce the noise in derived heat flux measurements. Care must be taken in the use of such algorithms, and the impact on model accuracy and uncertainty must be considered. Finally, noise may be reduced by employing higher order numerical approximations of the finite difference terms, with a corresponding computational expense.

Measurement uncertainty

Uncertainty in measurements of incident radiative heat flux using the HFITS method is derived both from uncertainty in thermogram measurements and uncertainty in the assumptions inherent in the thermal model. Uncertainty in inverse problems is notoriously difficult to quantify due to the reliance on numerical approximations and optimization techniques. Rippe and Lattimer [7] investigated the uncertainty in inverse measurements of heat flux using plate sensors through the propagation of errors technique [22], reporting that uncertainties associated with the temperature measurements (IR camera) and convective heat transfer coefficients were the greatest contributors to the expanded uncertainty in the derived heat flux measurements.

Temperature measurement uncertainty is dependent on the thickness and thermophysical properties of the sensing surface and the spatial and temporal discretization used in the IHT model. The temperature measurement uncertainty coefficient, $S_T$, may be expressed as [7]:

$$S_T=\left(h_{c,f}+h_{c,b}\right)+8\left({\epsilon }_f+{\epsilon }_b\right)\sigma T^3+\frac{4\lambda \delta }{{\Delta }^2}+\frac{\rho c_p\delta }{\Delta \mathrm{t}}$$ (14)

Although both emissivity and convective heat transfer coefficients are included in this term, these do not significantly affect the temperature measurement uncertainty. Rather, the last two terms in the equation (associated with thermal diffusion and energy storage, respectively) drive this uncertainty. Fig. 16 shows the effect of element size and time step on the temperature measurement uncertainty for the recommended sensing surface (Eq. (14)). From this, it is recommended that an element size between 1 mm and 10 mm be used to minimize temperature measurement uncertainty. The processing time step should be as large as possible while still resolving the relevant heat transfer phenomena for a given experimental configuration.

Fig. 16.

Effects of spatial and temporal discretization on temperature measurement uncertainty.

Measurement uncertainty can also be quantified using Bayesian methods. For example, DiDomizio [6] implemented an inverse model for heat flux estimation using a Kalman Filter framework. Each model parameter was treated as a continuous random variable and an algorithm was implemented to calculate posterior values based on prior information (previous time steps). Uncertainties associated with each parameter were integrated into the process model. With such an approach, additional (relevant) information may be integrated into the model to reduce the Bayesian uncertainty in derived heat flux measurements. For example, additional temperature or heat flux gauge measurements near the plate sensor, or radiation estimates derived from view factor relationships. The approach developed by DiDomizio [6] was unique for that particular application, and a generalized approach suitable for application to the HFITS method has not been developed to date.

Ethics statements

None

CRediT authorship contribution statement

Matthew J. DiDomizio: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization. Parham Dehghani: Software, Investigation, Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.