Thermal Emissivity and Heat Capacity of Composite Metal Foam

Abstract

Composite metal foam (CMF) is a new class of material based on a mixture of metal matrix composites and metal foams. While the mechanical properties of CMF are well studied, its thermal properties, particularly at extreme temperatures, are yet to be evaluated and established. This study investigates the specific heat capacity of stainless-steel composite metal foam at temperatures up to 1200 °C while comparing data obtained using the laser flash method and a differential scanning calorimetry method (DSC). Moreover, it outlines a detailed procedure for investigating the surface emissivity of composite metal foam (CMF) as a function of the emissivity of separate components (spheres and matrix). It uses experimental and analytical procedures to show how emissivity is directly affected by surface roughness, temperature, sphere curvature and viewing angles. The CMF used in this study consists of 316L stainless steel matrix and stainless-steel hollow spheres with varying sphere sizes.

1. Introduction

Composite metal foam (CMF) is a composite material made by tightly packed air-filled metallic hollow spheres embedded in the matrix. As such, when the material is cut, its surface profile consists of two major surface morphologies, the exposed inner surface of cut hollow spheres and the matrix surrounding the spheres, that exhibit different emissive properties. Characterization procedures are necessary to better understand the intrinsic and extrinsic properties of this novel material [1,2]. Such properties in the engineering space typically include thermal, mechanical, electrical and physical properties. While many of the mechanical and physical properties of CMF have been well studied and reported, its thermal properties are yet to be understood. It is notable that although lab-scale experimental tests are more manageable, large-scale experimental studies, particularly those that need to be carried out at high temperatures (such as torch fire and jet fire tests), are cost-intensive and require numerical simulations to predict material performance [3,4,5]. Some important thermal properties needed for developing a finite element model (FE) of material under extreme heat are their thermal conductivity, specific heat capacity, surface emissivity and coefficient of thermal expansion. Thermal conductivity, diffusivity and coefficient of thermal expansion of composite metal foam at a variety of temperatures have been investigated before [6,7]. In this study, specific heat capacity (Cp) measurements are conducted using Laser Flash and Differential Scanning Calorimetry (DSC), along with studying the surface emissivity of CMF by considering the effect of various components of material on its global surface emissivity. The methodology established in this study to predict the global surface emissivity for CMF can be applicable to other composites as well.

It is worth noting that there are direct and indirect methods for measuring the surface emissivity of monolithic materials using ASTM E1933-14 (2018) [8] and ASTM C1371-04 [9], but challenges in characterizing the surface emissivity of composite materials have motivated this part of the research.

2. Materials and Processing

Steel–Steel Composite metal foam (S-S CMF) samples were manufactured using stainless steel hollow spheres embedded in a 316L stainless steel matrix [10,11,12]. Hollow spheres were manufactured using the lost core technique with average outer diameters of 2 mm, 4 mm and 6 mm and approximate wall thicknesses of 100, 200 and 300 µm, respectively. They were manufactured in Dresden, Germany, by Hollomet GmbH. The hollow steel spheres were shaken into a random-loose packing arrangement within a steel mold surrounded by 316L stainless steel powder from North American Höganas, with an average particle size of 44 µm. The mold was heated in a vacuum hot press and allowed to cool under a high vacuum to room temperature. Samples were then extracted from the molds, cut and machined to various sizes for testing. The elemental composition of the components (hollow steel spheres and matrix) can be found in

Table 1. Chemical composition of S-S CMF components by weight percent.

	Chemical Composition (Weight Percent)
Material	Fe	C	Mn	Si	Cr	Ni	Mo
Stainless Steel Hollow Spheres	Balance	0.68	0.13	0.82	16.11	11.53	2.34
316L Steel Matrix	Balance	0.03	2.00	1.00	16.00–18.00	10.00–14.00	2.00–3.00

.

3. Experimental Procedures

3.1. Specific Heat Capacity Measurements

The specific heat capacity of composite metal foam was conducted using two known techniques: differential scanning calorimetry (DSC) and laser flash. The following section discusses the experimental procedures adhered to for each method.

3.1.1. DSC Measurements

The first method for determining the heat capacity of CMF is based on the use of Differential Scanning Calorimetry. DSC is a thermal analysis method that measures enthalpy changes in samples due to changes in physical or chemical properties as a function of time and temperature [13]. It measures the amount of heat required to raise the temperature of a unit mass of a given material and compares it to that of a material with known specific heat capacity at the same test conditions. To find the specific heat capacity of a material using DSC, the following test procedures are conducted: (1) A baseline test, where empty crucibles are heated to yield an inherent signal bias in the system; (2) A reference test where a standard sample with known heat capacity is tested; and (3) The actual experimental heat capacity test for the unknown sample. The baseline test is important to eliminate system bias from the measurements and ensure it is running consistently, whilst the reference test is used as a comparative analysis for the calculation of heat capacity for the unknown sample [14].

In this study, three batches of composite metal foam shavings were machined from larger CMF samples using a Buehler ISOMET 4000 linear precision saw (Buehler, Lake Bluff, IL, USA). Three sets of shavings were weighed to meet the weight requirements of the DSC test equipment [10–20 mg], as shown in

Table 2. CMF sample mass is used to measure the specific heat capacity (Cp) via the DSC method.

	Sample Mass for Cp Measurement via DSC Technique
Sample number	1	2	3
Mass (mg)	10.1	18.7	16.0

. These were thoroughly cleaned using water in an ultrasonic cleaner and dried on a hot plate to ensure that no impurities were present in the samples, as the DSC test is sensitive to any potential contamination affecting data.

A Mettler Toledo DSC3 system with a maximum heat capacity of 400 °C was utilized to evaluate the specific heat capacity of CMF, as shown in Figure 1.

As noted above, a baseline test was conducted to ensure the system heat flow was consistent. This was done by heating the system with empty sample pans from 25 °C to 400 °C at a rate of 10 °C/min. The system was purged with nitrogen gas as it is non-reactive. This was repeated three times to ensure the heat flow curve was consistent, thus showing the stability of the system. For the reference test, a sapphire 4.8 mm calibration standard provided by Mettler Toledo was used as a means of comparison to calculate the specific heat capacity of CMF samples. The reference sample was first placed in a testing pan, covered with a lid and sealed using a standard sample pan crimper press. The crimped sample was then placed carefully in the sample holder and heated at the same rate of 10 °C/min up to 400 °C in a nitrogen atmosphere. The obtained heat capacity curve was saved in the system and applied in the calculation for Cp of CMF [15]. After the reference test, the CMF samples were prepared in a similar fashion by crimping in the sample pan. These were conducted individually at the same temperatures and heating rates in nitrogen as the baseline and reference test. The results were calculated against the reference sapphire standard to obtain the specific heat capacity of the CMF. The results were then extrapolated up to 1200 °C.

3.1.2. Laser Flash Test

Since the DSC machine did not have the capacity to measure CMF heat capacity up to 1200 °C, an alternative methodology was approached to further explore the heat capacity of CMF at higher temperatures. The laser flash test is a method by which the diffusivity of a material is found by a laser impulse and used to determine the thermal conductivity and specific heat capacity of a material [16,17]. Diffusivity (α) is a temperature-dependent thermal property that defines the speed of heat propagation by conduction through a material. Diffusivity (α) is defined as a function of the density (ρ), specific heat capacity (Cp) and thermal conductivity (k) by the following formula:

α = k/(ρ × Cp)

(1)

In the laser flash technique, a brief laser pulse is shot through the sample at one side within 300 to 400 µs at a pulse energy up to 17 J. Temperature at the other side of the sample is monitored with a high-speed infrared detector operating at a 16-bit resolution. This phenomenon is depicted in a schematic in Figure 2. Diffusivity is then determined using Parker’s relationship [18] as shown in Figure 2b where α represents the thermal diffusivity (mm²/s), T the measuring temperature in °C, d the thickness of the sample and t1/2 the amount of time in seconds, required for the sample’s opposite end to heat to half its maximum temperature after being heated with the laser pulse.

Three cylindrical composite metal foam samples were machined from larger CMF samples using the Buehler ISOMET 4000 linear precision diamond saw. To ensure evenness across the sample cross-sections, the samples were carefully polished using a Buehler AUTOMET workstation. This process utilized a SiC (silicon carbide) sandpaper with increasing grit sizes of 180, 240, 320, 600 and 800. These were polished on the wheel at a speed of 50 RPM, and samples were thoroughly cleaned in water using an ultrasonic cleaner in between different grit sizes to avoid cross-contamination. After polishing, the samples were cleaned further with water and acetone in an ultrasonic cleaner and were thoroughly dried on a hot plate to ensure that there were no traces of residue or moisture in the samples. The samples’ diameter and height were then carefully measured with a laboratory caliper with a ±0.01 mm precision and weighed using an Adventurer OHAUS (M/N AR3130) scale at a ±1 mg accuracy. These measurements were necessary to calculate the densities of these samples, which are summarized in

Table 3. CMF sample dimensions for Cp measurements using laser flash technique.

Sample Number	Sample Dimensions for Laser Flash Technique
	Diameter (mm)	Thickness (mm)	Density (g/cm3)
1	25.3	5.96	2.63
2	25	5.72	2.56
3	25	5.91	2.68

. To ensure easy absorptivity and heat transfer from the laser to the samples with minimal reflections, both sides of the samples were sprayed with colloidal graphite.

Experimental tests were carried out using the Discovery Laser Flash DLF-1200 testing device in accordance with the ASTM E1461 standard [19]. To find the Cp, the DLF-1200 system computes the diffusivity and thermal conductivity of the samples. To begin measurements, a standard thermo-graphite reference sample was placed in one of the two sample holders, and the CMF sample was placed in the other holder. The samples were heated from room temperature to 1200 °C in a nitrogen gas atmosphere at a rate of 10 °C/min, similar to DSC measurements and allowed to reach equilibrium at the desired temperatures. Once equilibrium was reached, the laser pulses were released and temperatures were monitored by the IR detector on the opposite end of the samples. The Clark and Taylor model [20] is utilized by the TA instruments’ software 5.1.1 to account for energy losses and produce diffusivity values. This single flash procedure was repeated three times per temperature point to strike an average. Based on the known diffusivity, thermal conductivity and densities, the specific heat capacity is calculated using Equation (1) for the various samples.

3.2. Surface Emissivity Measurements

The emissivity test setup was developed following the ASTM E1933-14 “Standard Practice for Measuring and Compensating for Emissivity using Infrared Imaging radiometers” [8]. In this standard, there are two methods of measuring emissivity: the contact thermometer method and the non-contact thermometer method. For this study, the contact thermometer method was used.

3.2.1. Sample Preparation

Cylindrical composite metal foam samples with various sphere sizes of 2 mm, 4 mm and 6 mm were machined from larger cylindrical CMF samples using the Buehler ISOMET 4000 linear precision diamond saw. The sample dimensions were carefully measured with a laboratory caliper with ±0.01 mm precision, as seen in

Table 4. Dimensions of cylindrical S-S CMF samples used for emissivity tests.

Sample	Sample Diameter (mm)	Sample Height (mm)
2 mm—sphered S-S CMF	25.59	25.26
4 mm—sphered S-S CMF	38.24	25.28
6 mm—sphered S-S CMF	37.61	25.21

. The samples were ground using a Buehler AUTOMET workstation. This process utilized a SiC (silicon carbide) sandpaper with increasing grit sizes of 180, 600 and 1200 at a speed of 50 RPM. The samples were thoroughly cleaned in water using an ultrasonic cleaner in between different grit sizes to avoid cross-contamination. In the final stage, the samples were cleaned further with acetone in an ultrasonic cleaner and thoroughly dried on a hot plate to ensure that there was no moisture in the samples.

3.2.2. Test Setup for Emissivity Measurements

A Thermo Scientific Thermolyne HPA2235MQ (Thermo Fisher Scientific, Waltham, MA, USA) analog hot plate with a maximum heating capacity of 371 °C was used as the heat source for conducting the tests. A ceramic insulation block was used to cover the entire surface of the hotplate except for the area where the CMF sample is located to prevent any heat loss during the test. A circular shape is cut from the center of the ceramic insulation block to match the cylindrical samples’ diameter. The CMF samples were directly placed on the hot plate, with its bottom being in direct contact with the hotplate, the top exposed to the atmosphere for emissivity measurements and all perimeters surrounded by the ceramic insulating blocks. Thirty AWG type-R thermocouples were installed on the top and bottom surfaces of each CMF sample to measure the incoming and outgoing temperatures, whilst a third 30 AWG type-R thermocouple was installed on the hotplate to measure the input control temperature of the samples. A high-temperature tape was used to secure each thermocouple on the ceramic insulating blocks, ensuring each thermocouple bead touched the CMF sample. The thermocouples were connected to a computer running LabVIEW software NXG 5.1 via a National Instruments cDAQ 9171 (National Instruments, Austin, TX, USA) data acquisition unit. An enclosure with a black interior was built to house the hotplate and sample to prevent external reflections that could affect the accuracy of the measurements. A FLIR E40 infrared radiometer camera with a thermal sensitivity of <0.07 °C and spectral range of 7.5 to 13 µm was used in emissivity measurements. Figure 3 shows the test setup used for the emissivity measurements.

3.2.3. Calibration Test

To ensure the test setup produced reliable data, a set of three calibration tests were conducted on a bulk stainless-steel sample. The sample surface was ground with sandpaper to achieve three distinctly different surface finishes: (1) ground on 600 grit {13–16 µm roughness} sandpaper, (2) ground on 800 grit {9.8–12.3 µm roughness} sandpaper, and (3) ground on 1200 grit {4.5–6.5 µm roughness} sandpaper. These were chosen to confirm the IR camera sensitivity in detecting emissivity for various surface roughness. The samples were heated from room temperature to 250 °C, and the data obtained from these runs were compared to the literature data available on the emissivity of 316L stainless steel within those temperature ranges and with similar surface finishes.

3.2.4. CMF Emissivity Measurement Procedures

In order to deconvolute the emissivity measurement of a complex surface like CMF, the measurement was broken down into two segments: the measurement of the CMF matrix emissivity and the measurement of its hollow sphere emissivity.

3.2.5. Measurement Procedure for CMF Matrix

To measure the emissivity of the CMF matrix, each sample was heated separately on the hot plate gradually from room temperature to 200 °C. At every heating stage, the IR camera was held normal to the surface of the matrix using a stable stand while the laser was triggered. The IR camera emissivity readout was then adjusted until the temperature readings from the camera corresponded to that of the temperature from the thermocouples on the sample surface. This adjusted emissivity value becomes the emissivity of the CMF matrix at that temperature. The procedure was repeated three times at each temperature to ensure IR camera readings were consistent. A visual representation of data obtained from the IR camera for samples ground at 180 grit for temperatures 30 °C, 50 °C and 100 °C is shown in Figure 4. This measurement was repeated for CMF sample surface roughness created by grinding at 180 (63–88 µm roughness), 600 (13–16 µm roughness) and 1200 grit-sized (4.5–6.5 µm roughness) sandpapers to investigate the effect of surface roughness on emissivity.

3.2.6. Sphere Emissivity Determination Procedure

The measurement procedures to determine the emissivity of spheres within the CMF samples are divided into a direct measurement of the sphere emissivities and an indirect numerical evaluation that considers the effect of sphere curvature on emissivity.

3.2.7. Direct Measurement of Sphere Emissivity

The procedure for measuring sphere emissivity followed a similar protocol to that of the matrix. Various samples were heated on the hot plate from room temperature to 200 °C, with two type-R 30 AWG thermocouples placed in two cut-open spheres on the sample’s top and bottom surfaces and were secured by high-temperature tape. The IR camera laser was pointed into an adjacent, cut-open sphere with a diameter larger than that of the camera’s laser beam diameter to ensure accurate readings. A sphere of diameter smaller than the diameter of the laser would record the matrix readouts, affecting data accuracy. This phenomenon is described in Figure 5.

Once a suitable sphere was identified, the emissivity value on the IR camera was adjusted until the temperature on the IR camera matched that of the thermocouple temperature. This value becomes the emissivity of the interior surface of the sphere. However, due to the variation in the size of exposed spheres at the sample surface, there is a limit on the potential spheres that can be used for such measurement. This challenge necessitates the need for indulging a combined experimental and numerical approach that considers the effect of roughness and curved topography of the interior surface of spheres on their emissivity.

3.2.8. Indirect Method of Sphere Emissivity Measurements

For the measurement of sphere emissivity, the surface roughness of the sphere interior surface and the effect of the sphere curvature on the emissivity are being investigated in three steps: (i) measuring the surface roughness of the interior walls of the spheres and duplicating this surface roughness on a solid 316L stainless steel flat sample through sandblasting; (ii) measuring the emissivity of the 316L flat sample; and (iii) use the data from emissivity of flat sample to develop an analytical model to predict the emissivity of the interior surface of spheres considering the effect of sphere curvature. The following sections discuss all three steps in detail.

3.2.9. Roughness Measurements for Spheres

A Keyence VKx1100 optical profilometer (Keyence, Itasca, IL, USA) was used to measure roughness, as shown in Figure 6. This device scans the sample surface with a laser, which in turn detects the height (z) information across the x and y planes of the sample surface and recreates a three-dimensional surface profile [21]. An example of a profile of a sphere generated in our measurements is seen in Figure 7. The VK_Analyzer software 3.5.0.40 produces a measurement of roughness (Ra) value for the surfaces scanned that is applicable to a flat surface. For each sample, three spheres were scanned at multiple points in the sphere to give an average Ra value for the surface roughness of the sphere’s inner surfaces.

3.2.10. Roughness and Emissivity Measurement for Flat Stainless-Steel Samples

To overcome the size limitation mismatch between the IR camera beam radius and the curvature/diameter of hollow spheres, flat stainless-steel samples were prepared with similar surface roughness as those of the spheres’ inner surfaces. To match the roughness of the spheres’ interior surfaces, a solid stainless-steel sample was sandblasted with aluminum oxide sand using a BB850/1050LED-BVR-PR sandblaster (BADBAOY Blasters, Inc., Canton, OH, USA). The sand mesh is selected carefully to provide the desired roughness on the steel surface. After sandblasting, the samples were cleaned with acetone in a sonic cleaner and dried before using a Keyence VKx1100 optical profilometer to measure the surface roughness to confirm its agreement with the sphere’s inner roughness. The sandblasting continued until the Ra value of flat stainless steel became close enough to the Ra range of the sphere roughness for 2 mm, 4 mm and 6 mm spheres (within a ±2.5% tolerance). At this point, the emissivity measurements were conducted on the flat, sandblasted sample between room temperature and 200 °C in the exact same way the CMF matrix emissivity was measured. The results would be considered representative of the hollow spheres’ (2 mm, 4 mm and 6 mm) emissivity, excluding the effect of their curvatures.

3.2.11. Analytical Determination of CMF Emissivity Considering the Sphere Morphology

Aside from the internal surface roughness of the spheres, their curvature can act as another major factor that affects the emissivity as depicted in Figure 8.

To include the effects of sphere curvature on emissivity, a numerical formulation was derived. This was achieved by generating an equation based on a curve fit of some literature data that shows the effects of surface curvature on the emissivity of 316L stainless steel [22,23,24]. Based on the results from these data sources combined with emissivity data from the sandblasted flat 316L sample, a predictive equation for the emissivity of the sphere was derived as follows:

$${\epsilon }_{sphere}=1e^{-11}{\varnothing }^6-2e^{-9}{\varnothing }^5+2e{\varnothing }^4-1e^{-5}{\varnothing }^3+0.0002{\varnothing }^2-0.0027\varnothing +0.1697$$

(2)

where ε_sphere = emissivity of sphere and $\varnothing$ = measurement angle varied from 0° to 180°.

4. Global Emissivity of CMF Using Rule of Mixtures

A lower-bound and upper-bound rule of mixture was applied to find the global emissivity of CMF using the emissivity of its components (hollow sphere and the matrix) using the following equations:

$$Upper Bound: {\epsilon }_{cmf}=A_{spheres}·{\epsilon }_{spheres}+A_{matrix}·{\epsilon }_{matrix}$$

(3)

$${Lower Bound: \epsilon }_{cmf}=\frac{1}{\left[{(A}_{matrix}/{\epsilon }_{matrix})+(A_{sphere}/{\epsilon }_{spheres}\right]}$$

(4)

where ε_cmf = emissivity of CMF, ε_spheres and ε_matrix = emissivity of spheres and matrix, respectively, and A_spheres and A_matrix = % area of spheres and the matrix within the CMF surface, respectively.

To find the average surface area of the spheres to the matrix for each sample, digital images of various sample surfaces were first taken using a digital camera. ImageJ version 2.14.0/1.54f software was then used to evaluate the average percentage area of spheres and the matrix on each sample, as seen in Figure 9. With the percentage areas found, the rule of mixtures is applied to various samples of CMF with different sphere sizes at temperatures up to 200 °C. The obtained data was then extrapolated to show emissivity predictions for each sample at higher temperatures up to 1200 °C.

5. Results and Discussions

5.1. Heat Capacity Results and Discussions

Table 5. Specific heat capacity results for steel–steel CMF at various temperatures using the DSC technique. Note that the actual data are highlighted in Bold Italic font, and the remaining data are a result of extrapolation.

Nominal Temperature (°C)	Specific Heat Capacity Using DSC (kJ/Kg·K)
	Sample 1	Sample 2	Sample 3	Average
30	0.49	0.37	0.52	0.46
100	0.53	0.42	0.55	0.50
200	0.60	0.49	0.59	0.56
300	0.67	0.56	0.63	0.62
400	0.74	0.63	0.67	0.68
500	0.81	0.70	0.71	0.74
600	0.88	0.77	0.75	0.80
700	0.95	0.84	0.79	0.86
800	1.02	0.91	0.83	0.92
900	1.09	0.98	0.87	0.98
1000	1.16	1.05	0.91	1.04
1100	1.23	1.12	0.95	1.10
1200	1.30	1.19	0.99	1.16

and

Table 6. Specific heat capacity results for steel–steel CMF at various temperatures using laser flash technique.

Nominal Temp (°C)	Specific Heat Capacity Using Laser Flash Technique (kJ/Kg.K)
	Sample 1	Sample 2	Sample 3	Average
26	0.491	0.556	0.541	0.529
200	0.59	0.617	0.646	0.618
400	0.661	0.703	0.742	0.702
600	0.707	0.762	0.782	0.75
700	0.743	0.799	0.81	0.784
823	0.778	0.812	0.81	0.8
900	0.829	0.818	0.84	0.829
1000	0.865	0.919	0.893	0.892
1100	0.972	1.015	0.944	0.977
1200	1.02	1.101	1.098	1.073

show tabulated data obtained for the laser flash technique and DSC technique, respectively. As mentioned earlier, data for the DSC measurements were extrapolated due to the system’s temperature limit. Figure 10 gives a graphical representation of the results, where Cp obtained for both processes show an approximate ±5% deviation between each other, which is a good level of agreement. It can be observed that Cp data for the two methods are close for temperatures up to about 600 °C and begin to deviate at higher temperatures. This can be due to extrapolation and possible differences in the sensitivity of the two methods at higher temperatures. Figure 10 also compares the Cp of the composite metal foam to that of stainless steel. It is seen that the Cp of stainless steel is relatively lower than that of the composite metal foam, especially at high temperatures. This can be attributed to the higher carbon content in the steel hollow spheres used in the structure of composite metal foams (0.68 weight % of carbon in steel hollow spheres compared to 0.03 weight % in stainless steel). These major differences in Cp values of the composite metal foam to those of stainless steel are significant, as they can affect the accuracy of data obtained when conducting numerical simulations involving composite metal foams.

5.2. Emissivity Calibration Test Results

Figure 11 shows the emissivity measurement results for 316L stainless steel calibration samples. As can be seen, the results indicate a variation in emissivity data for each grit size, confirming the IR camera’s sensitivity in detecting changes in sample surface roughness. Also, the data obtained for the 600 grit 316L emissivity is comparable to the literature data for a laser-powder-bed-fused 316L sample with similar surface roughness in the ranges of 16–4.5 µm [25], indicating a margin of about 5%. The success of the calibration test proves the accuracy of the procedure and apparatus in providing acceptable data.

5.3. CMF Matrix Surface Emissivity Results

Figure 12 shows the emissivity values of the CMF matrix that range from 0.60 to 0.22 between room temperature and 200 °C. Since these measurements are only focused on the CMF matrix emissivity, they are not impacted by the sphere sizes. However, due to the minute varying porosities in the matrix resulting from the packing of matrix powders around different size spheres, the surface profile is impacted on a micron scale, and as such, there are marginal differences (1–15%) between emissivity values of 2 mm, 4 mm and 6 mm sphere CMF matrixes. It can also be observed that the emissivity is affected by testing temperature, as data shows a decrease in emissivity values with an increase in temperature. This phenomenon is seen in a study where direct measurements of emissivity of laser-powder-bed-fused 316L stainless steel are found to decrease from approximately 0.26 to 0.19 when the temperature increased from 60 °C to 350 °C [25]. The same study shows a similar phenomenon with IN718, where the emissivity value decreases with increasing temperature. From Figure 12a–c, data obtained also confirm that the surface emissivity values are directly impacted by the surface roughness, as all sample emissivity values show a considerable decrease with a lower surface roughness.

5.4. S-S CMF Hollow Sphere Surface Emissivity Measurements

5.4.1. Direct Sphere Measurements Results

Figure 13 shows the data obtained for the direct measurements of the surface emissivity of various spheres in the CMF samples. Emissivity values obtained for the spheres ranged from 0.71 to 0.46, which confirms that the inner surfaces of spheres are relatively rougher than the ground surfaces of the CMF matrix. Also, it is observed that as the temperature increases, emissivity values decrease. Data also shows an inconsistent emissivity pattern of the various sphere sizes because of the varying depths as seen from the IR camera at the sectioned surface, hence the need to include an analytical formulation to better predict the emissivity of the spheres’ inner surfaces. The analytical formulation will give a more holistic prediction of emissivity and help in overcoming the ambiguity of the direct measurement of surface emissivity of hollow spheres’ inner surfaces.

5.4.2. Roughness and Emissivity Measurement Results for CMF Spheres and Flat Stainless-Steel Sample

Average roughness measurements made for various CMF spheres and flat stainless-steel samples are shown in

Table 7. Measured sphere surface roughness (Ra) values for various steel CMF samples.

Sample	2 mm-Sphered	4 mm-Sphered	6 mm-Sphered	Flat 316L Steel
Ra (μm)	5.58	5.82	5.62	5.69
Standard Deviation	1.54	0.68	0.54	0.26

, along with their respective standard deviations after multiple scans across the sample surfaces. Roughness values within the sphere surfaces show that spheres of different diameters have similar roughness, and this is successfully replicated on the flat 316L stainless steel sample. Direct emissivity measurements made on the flat 316L stainless steel plate are shown in

Table 8. Emissivity measurements obtained for a flat 316L stainless steel sample mimicking the inner surface of spheres.

Temperature (°C)	Emissivity
50	0.8
100	0.8
150	0.75
200	0.63
250	0.58

.

5.4.3. Effects of Sphere Curvature on Emissivity Using the Numerical Method

Figure 14 shows the effect of curvature on the emissivity of the CMF spheres at varying temperatures based on the analytical formulation. At each curvature angle, the emissivity values differ based on the depth of reflection, which is a consistent phenomenon as seen in prior studies [22,23,24]. This method is important as it captures these minute details that impact the overall emissivity of the spheres. These emissivity values are averaged across the sphere curvature shown in

Table 9. Average calculated emissivity values of 2 mm, 4 mm and 6 mm hollow steel spheres in the CMF samples based on the sphere’s inner curvature at various temperatures.

	Emissivity
Temperature (°C)	2 mm	4 mm	6 mm	Average	Deviation
50	0.765	0.808	0.761	0.778	±0.026
101	0.745	0.807	0.782	0.778	±0.031
152	0.702	0.756	0.726	0.728	±0.027
203	0.601	0.612	0.611	0.608	±0.006
248	0.531	0.563	0.58	0.558	±0.025

. The averaged data ranges from 0.78 to 0.56 within the temperature range of 50–250 °C. This is marginally higher than that of the data obtained from direct measurements, confirming the effects of surface curvatures.

5.5. Global Emissivity of CMF Samples

After the emissivity values were found for the components (hollow spheres and the matrix), they were combined using the rule of mixtures. Two sets of data were generated for the global emissivity using matrix emissivity ground at 1200 grit and (A) emissivity of spheres using the analytical formulation of the effects of sphere curvature and (B) emissivity of spheres using the direct measurement method. Figure 15, Figure 16 and Figure 17 show the CMF emissivity ranges for lower and upper bound limits between room temperature and 200 °C, which is then extrapolated to 1200 °C. Comparing methods (A) and (B), it is seen that using method (A) generates higher emissivity values than (B) due to the inclusion of the sphere curvature on the emissivity of CMF. However, method (B) leaves room for error as the data were based on a limited number of spheres with varying geometries. These errors are corrected when emissivity is considered in minute detail, as in the case of method (A).

To further investigate the accuracy of this method, data for the 2 mm CMF was compared to experimental measurements made in a previous study [26], where the emissivity of a 2 mm CMF was found using the non-contact thermometer method. This sample was ground to a mirror-like finish to be comparable to the mirror-like finish of the 1200 grit-ground samples used in the previous study. Figure 18 shows that the emissivity from that study falls within the upper–lower limits of the emissivity values obtained using the global emissivity method in this study, which further confirms the accuracy of this experimental-analytical approach.

Comparing the various samples based on the spheres’ outer diameters (2 mm, 4 mm and 6 mm), it can be deduced from the data that emissivity is not directly impacted by the individual sphere sizes but by the ratio of sphere surface areas to the matrix surface area, as seen in the ImageJ analysis in Figure 9. Since the spheres have a rougher surface than the matrix, fewer spheres in a specific area translates to a lower global emissivity. Generally, global emissivity may vary based on the distribution of spheres in a local region. However, based on the CMF manufacturing, there will always be an approximate 65 to 35% ratio of the matrix-to-sphere distribution, thus providing a relatively consistent global emissivity.

Though this study uses CMF as a case study, the developed method can be applied to different types of composite materials that have components with different geometries and a non-uniform surface profile. In the case of homogenous and monolithic materials, it is easier to use direct measurements as any area captured by IR camera laser beams is representative of the entire material, whereas in composite materials, there are additional levels of complexity added to the equation. The proposed mix of experimental and analytical approaches makes it easier to capture the emissivity of the various components of composite materials and put them together to accurately predict their collective emissivity.

6. Uncertainty Assessment

For future thermal numerical simulations where these emissivity values can be applied, it is imperative to know the derived level of uncertainty in the data, as a small change in emissivity can impact the resulting data. The following sections give an uncertainty assessment for the calculated global surface emissivity of CMF samples. This is based on the ISO/IEC Guide 98-3 [27], also referred to as the GUM (Guide on Uncertainty of Measurements), which describes the concepts and calculation methods to assess the uncertainty of the measurements. Uncertainty measurements were made based on experimentally acquired global emissivity values for 2 mm, 4 mm and 6 mm sphered CMF samples.

Calculated Uncertainties of Global Emissivity of CMF

The global emissivity uncertainties of the various sphere-sized CMF samples were defined as functions of the following measurable input quantities, as categorized below:

• 2 mm sphered CMF:

  o

  X1 = Matrix Emissivity, ε_m

  o

  X2 = Sphere Emissivity, ε_s

  o

  X3 = % Area of matrix, A_m

  o

  X4 = % Area of sphere, A_s

• 4 mm sphered CMF:

  o

  X5 = Matrix Emissivity, ε_m

  o

  X6 = Sphere Emissivity, ε_s

  o

  X7 = % Area of matrix, A_m

  o

  X8 = % Area of sphere, A_s

• 6 mm sphered CMF:

  o

  X1 = Matrix Emissivity, ε_m

  o

  X2 = Sphere Emissivity, ε_s

  o

  X3 = % Area of matrix, A_m

  o

  X4 = % Area of sphere, A_s

Table 10. Standard uncertainties or ranges of the input quantities.

Input Quantity	Symbol	Base Value	Uncertainty or Range
X1	εm	Table 11	±2%
X2	εs	Table 11	±2%
X3	Am	0.368	±0.025
X4	AS	0.632	±0.025
X5	εm	Table 11	±2%
X6	εs	Table 11	±2%
X7	Am	0.327	±0.025
X8	AS	0.673	±0.025
X9	εm	Table 11	±2%
X10	εs	Table 11	±2%
X11	Am	0.343	±0.025
X12	AS	0.657	±0.025

provides the estimates of the uncertainties and ranges of the input quantities used to define the global emissivity of CMF. Column three denotes the input quantity values obtained as an average from tests that are used in calculating the upper bound emissivity of samples.

Table 11. Average emissivity values for various sphere-sized CMF samples.

	2 mm-CMF		4 mm-CMF		6 mm-CMF
Temperature (°C)	Sphere	Matrix	Sphere	Matrix	Sphere	Matrix
	εs	εm	εs	εm	εs	εm
50	0.71	0.58	0.71	0.53	0.66	0.3
75	0.7	0.34	0.69	0.34	0.64	0.26
100	0.68	0.26	0.67	0.24	0.57	0.24
150	0.59	0.24	0.64	0.23	0.49	0.23
200	0.57	0.22	0.63	0.22	0.46	0.23

shows the sphere and matrix emissivity base values used in the uncertainty analysis. Owing to the nature of the rule of mixtures, uncertainties obtained for the upper bound will be repeatable for the lower bound, hence only the upper bound was considered for the purposes of this uncertainty measurement. The modes of determining the uncertainties for the input quantities are briefly discussed below.

-

The emissivity of the matrix, ε_m, and emissivity of the spheres, ε_s, were estimated based on the measurement uncertainty of the FLIR E40 infrared radiometer camera as obtained in its manual.

-

The uncertainties for A_m and A_s were found by conducting multiple % area measurements in ImageJ version 2.14.0/1.54f software for each sphere-sized sample and calculating the deviations obtained from these measurements.

In estimating the uncertainty of the CMF emissivity, two methods were used. In the first method, the uncertainty of CMF global emissivity was calculated directly using the statistical summation of the distribution of standard errors from the various input parameters. This follows a numerical calculation as seen in the equation below.

$$u_c\left(y\right)=\sqrt{{\sum }_{i=1}^N{\left[\frac{\delta f}{{\delta X}_i}{|}_{x_i}\right]}^2{u}^2(x_I)}\equiv \sqrt{{\sum }_{i=1}^N{\left[c_i·u\left(x_i\right)\right]}^2}$$

(5)

where u = standard uncertainty, u_c = combined standard uncertainty and c_i = sensitivity coefficients.

The sensitivity coefficients derived using this method were obtained by numerical estimation based on the change of the calculated global emissivity for a small change of each of the input quantities.

The second method of estimating uncertainty of the CMF emissivity applied a Monte Carlo simulation consisting of 1000 realizations. With each realization, the input quantities were varied randomly according to a normal and uniform distribution for the component emissivity (sphere and matrix) and their percentage areas, respectively. Based on 1000 realizations, this method increases the accuracy of the obtained uncertainties.

The results after applying these two approaches to all samples are shown in

Table 12. Calculated combined standard uncertainties for global CMF emissivity.

	2 mm Sample			4 mm Sample			6 mm Sample
Temperature (°C)	Emissivity	Equation (5)	MC a	Emissivity	Equation (5)	MC a	Emissivity	Equation (5)	MC a
RT	0.63	±0.0017	±0.0013	0.59	±0.0003	±0.0012	0.42	±0.0006	±0.0009
75	0.47	±0.0014	±0.0009	0.45	±0.0006	±0.0009	0.39	±0.0006	±0.0008
100	0.41	±0.0138	±0.0009	0.38	±0.0007	±0.0008	0.35	±0.0005	±0.0007
150	0.37	±0.0006	±0.0007	0.36	±0.0007	±0.0007	0.32	±0.0003	±0.0006
200	0.35	±0.0006	±0.0007	0.35	±0.0007	±0.0007	0.31	±0.0002	±0.0006

^a MC—Results for the Monte Carlo simulation.

.

These uncertainty values are based on a 95% confidence level. Since emissivity is a dimensionless number with high sensitivity and ranges between 0 and 1, it is seen that uncertainty values obtained are up to four decimal places, which confirms the accuracy of data obtained experimentally, as uncertainty values as high as one or two decimal places will dramatically elevate emissivity values. Based on the results, the uncertainty values show an acceptable difference of no more than 3% with data obtained from the Monte Carlo simulation. Confirming the assertion made by the measurement standard for the emissivity evaluation, it is observed that uncertainty reduces with increasing temperature, showing that there is a higher margin of error when measuring emissivity at lower temperatures. Based on the uncertainty analysis, there is a 95% level of confidence that CMF emissivity will fall within the same ranges if measurements are reproduced under similar conditions.

7. Conclusions

This study successfully determined the specific heat capacity of the steel–steel composite metal foam, comparing two different methods and investigating their effect on the heat capacity values. It shows that the Cp of CMF is higher than that of its parent stainless steel material due to some variation in the composition of the hollow spheres used in the structure of CMF. The study also developed a novel approach to measure the surface emissivity of CMF. An analytical method was developed to predict the emissivity of composite materials using a rule of mixture. It is shown that the surface emissivity is directly impacted by spheres in the matrix ratio in CMF. It also confirms that the surface emissivity of a CMF is a function of its surface roughness, but sphere diameter has little to no effect on the global emissivity of CMF if the volume fraction or areal fraction of spheres and matrix stays the same in samples with various sphere sizes. The uncertainty analysis indicated a 95% confidence in the CMF emissivity measurements, showing near certainty that the test would be reproduced in a different laboratory.