Experimental Thermal Conductivity Measurement of Hollow-Structured Polypropylene Material by DTC-25 and Hot Box Test

Abstract

In this study, experimental measurements of porous polypropylene (PP) using the DTC-25 TA laboratory equipment and a hot box test are compared. The thermal conductivity of building materials indicates their insulation capability. Excellent building materials will have a lower thermal conductivity value as well as other building insulator performance metrics. While results show that increasing the volume fraction of fluid in porous PP has an inverse association with the thermal conductivity of the material, as predicted by porous media theories, there is a marked difference in the measured values of thermal conductivity using the two methods. The thermal conductivity values of porous PP from the DTC-25 and the hot box test were 0.21 and 0.0033 W/mK, respectively. The difference in the thermal conductivity values was due to the misapplication of the Fourier’s guarded heat flow model in the DTC-25 device to a convective fluid porous medium.

1. Introduction

In the last decade, attention toward energy and the environment has grown extensively and many international and national policies have been developed to achieve a more sustainable future for the environment while also focusing on reducing the cost of energy. Across the world, buildings seem to consume the most energy. In Europe, 40% of the total energy utilization is attributed to the building sector [1]. The United States Energy Information Administration (EIA) estimated that about 39% (or about 38 quadrillion British thermal units) of the total US energy consumption was consumed by the residential and commercial building sector in 2017 [2]. Residential buildings have more than a 50 percent share of the total building energy utilization in the United States.

One way to reduce this vast amount of energy utilization in buildings, which will lower the cost of energy utilization and reduce the greenhouse effect [3,4,5], is to engage in energy-saving practices. Energy savings are implemented through energy efficiency practices and through the judicious use of the energy available to our residential building envelope. Since a considerable amount of energy consumption in residential buildings is expended in HVAC systems [6,7], windows and door insulation [8,9], thermal bridge losses [10] and external walls [11,12], any design or material improvement in terms of energy utilization would lead to more energy-efficient housing.

The development of external walls has greatly contributed to significant energy savings and presents research opportunities in the development of new and advanced insulation materials. In future buildings, external wall insulation should guarantee optimum performance throughout the entire life cycle of the building. Other non-thermal considerations when choosing insulation materials are the environmental safety of the material [13], fire resistivity [14], sound shielding [15,16], mechanical impact tolerance and cost effectiveness [17].

While inroads have been made in newer innovative building insulation such as vacuum insulation panels (VIPs), gas-filled panels (GFPs), phase change materials (PCMs), dynamic insulators, etc., and in meeting the goal of saving more energy than the conventional insulating materials such as glass wool, cellulose, polystyrene etc., conventional insulation is still very popular in old and recent residential buildings due to the high cost and advanced manufacturing processing of innovative insulating materials [18].

This work explored alternative material designs that would increase energy savings. We have selected the material polypropylene because it potentially provides a mix of good insulation, impact resistance and recyclability. Polypropylene exhibits good building insulator characteristics, as outlined by Al-Homoud [19]. Polypropylene board insulation has the edge over conventional insulations like blanket and batt, blown and spray in its eco-friendliness, and it poses no harm to humans and animals. Its high impact resistance [20,21] has the potential to enhance the hail storm resistance of buildings.

A multi-hollowed polypropylene plastic is manufactured through 3D printing to create a foam-like macrostructure. The efficient thermal insulation of polar bears’ hair [22,23,24,25] is, in part, explained by their hollow macrostructure. Hollow hair reduces thermal conductivity in severe cold weather, thereby enhancing thermal insulation. Dohrn et al. [26] note that the insulation capacity of the polyurethane (PUR) form is primarily due to the gases contained in the foam, which significantly improve the thermal insulation associated with vacuum insulations.

Effective thermal conductivity represents the mean thermal conductivity value for a composite material or porous medium. Effective thermal conductivity is an important heat transfer property of materials. Kulkani and Doraiswamy [27] found that the effective thermal conductivity of a porous material is influenced by the thermal conductivities of the solid, gas, porosity, emissivity of the particles’ surface and temperature. Similarly, other research studies [28,29,30] have established that effective thermal conductivity is determined by factors such as the properties of the particles and matrix, and the microstructure of a composite polymer. Effective thermal conductivity can be calculated analytically [31,32], assessed through numerical simulation [33,34] or measured experimentally [35,36]. In this paper, an experimental method was used to measure the effective thermal conductivity of multi-hollowed polypropylene board using the DTC-25 and hot box testing. The thermal conductivity measurements of samples were obtained by means of the DTC-25 conductivity meter from TA Instruments, which uses the guarded heat flow method to obtain thermal conductivity measurements [37]. The hot box test follows the ASTM C1046 [38] standard practice for in situ measurement of heat flux and temperature in building envelope components. This method establishes the heat flux measurements through localized materials and between the hot box’s internal and external environmental temperatures using heat flux sensors and temperature transducers. While the hot box test method can provide accurate measurements of the thermal conductivity of building materials in real-world usage [39], it requires an understanding of a complex data-logging tool for post-test data processing.

2. Theory

Wang et al. [40] define a hollow structure as a solid structure with voids. The existence of voids breaks the continuity of heat transport pathways in hollow-structured materials [41]. Figure 1 shows the morphology of a hollow-structured material with three immiscible phases. Phase I is a solid for continuous heat transport pathway. Phase II is the void. Phase III is an interfacial bridge which could be a solid component. The multi-hollow polypropylene polymer was modeled as a pore structure. Pore defects in polymer composites are capable of lowering the effective thermal conductivities of polymers [28] and improving their insulation properties.

Many researchers have developed models to explain the effective thermal conductivity of porous materials [36,42,43,44,45,46]. The multi-hollow polypropylene polymer was modeled as a pore structure. Pore defects in polymer composites are capable of lowering the effective thermal conductivities of polymers [28] and improving their insulation properties. Wang and Pan [47] reviewed the effective thermal conductivity models that are applied to multiphase materials according to the size and structure of the pores. The basic structural model includes the series, parallel, Maxwell–Eucken and effective medium theory (EMT) models shown in Equations (1)–(4), respectively. The series and parallel models assume the simplest structure with heat flow. In the series model, heat flows across the pore, while the parallel model assumes heat from along the porous media. The Maxwell–Eucken model proposes a spherical dispersed phase in a continuous solid phase with no contact. The EMT model supposes that a void or filler is surrounded by a homogeneous, effective medium.

$$k_e=\frac{1}{(1-{\mathsf{\theta }}_2)/k_I+{\mathsf{\theta }}_2/{k_2}_{}}$$

(1)

$$k_e=k_1\left(1-{\mathsf{\theta }}_2\right)+k_2{\mathsf{\theta }}_2$$

(2)

$$k_e=k_1\frac{2k_1+k_2-2\left(k_1-k_2\right){\mathsf{\theta }}_2}{2k_1+k_2+\left(k_1-k_2\right){\mathsf{\theta }}_2}$$

(3)

$$\left(1-{\mathsf{\theta }}_2\right)\frac{k_1-k_e}{k_1+2k_e}+{\mathsf{\theta }}_2\frac{k_2-k_e}{k_2+2k_e}=0$$

(4)

where $k_e$ is the effective thermal conductivity of the porous medium, $k_1$ and ${\mathsf{\theta }}_1$ are the thermal conductivity and phase volume of the solid medium, respectively, and $k_2$ and ${\mathsf{\theta }}_2$ are, respectively, the thermal conductivity and volume fraction of the fluid medium.

Utilizing the local thermal equilibrium hypothesis, which supposes that the temperature (T) is the same for both the solid and the fluid phase, the heat transfer equation is derived from the solid and fluid energy mixture. The energy quantity present in the fluid and solid phases is given by Equations (5) and (6).

$${\left(\mathsf{\rho }{C}_P\right)}_e\frac{\partial \mathrm{T}}{\partial \mathrm{t}}+{\mathsf{\rho }}_f{C}_{P.f}\mathrm{u}\cdot \nabla \mathrm{T}+\nabla .\mathrm{q}=\mathrm{Q}$$

(5)

$$q={-\mathrm{k}}_e\nabla \mathrm{T}$$

(6)

where ${\left(\mathsf{\rho }{C}_P\right)}_e={\mathsf{\theta }}_{\mathrm{s}}{\mathsf{\rho }}_{\mathrm{s}}{C_{p,s}}_{}+{{\epsilon }_p\mathsf{\rho }}_{\mathrm{f}}{C}_{p.f}$

$${\mathsf{\rho }}_f=\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d} \mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}$$

$$C_{P.f}=\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d} \mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t} \mathrm{c}\mathrm{a}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y} \mathrm{a}\mathrm{t} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t} \mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}$$

$${\left(\mathsf{\rho }{C}_P\right)}_e=\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c} \mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t} \mathrm{c}\mathrm{a}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y} \mathrm{a}\mathrm{t} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t} \mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}$$

$${\mathsf{\theta }}_{\mathrm{s}}=\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d} \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x} \mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e} \mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$

$${\mathsf{\rho }}_{\mathrm{s}}=\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d} \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x} \mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}$$

$$C_{p,s}=\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d} \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x} \mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t} \mathrm{c}\mathrm{a}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y}$$

$${\mathrm{k}}_e=\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}$$

$$\mathrm{q}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t} \mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}$$

$$\mathrm{u}=\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y} \mathrm{f}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{d}, \mathrm{interpreted}\mathrm{as}\mathrm{Darcy}\mathrm{velocity}$$

$$\mathrm{Q}=\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t} \mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e} \mathrm{o}\mathrm{r} \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{k}$$

The term ${\left(\mathsf{\rho }{C}_P\right)}_e\frac{\partial \mathrm{T}}{\partial \mathrm{t}}$ vanishes for a steady state problem. Since the heat flows in series across the porous medium, with all the heat flux passing through both solid and fluid, the effective thermal conductivity refers to the weighted harmonic mean of conductivities $k_f$ and $k_s$.

$$\frac{1}{k_e}=\frac{{\mathsf{\theta }}_{\mathrm{s}}}{k_s} + \frac{{\epsilon }_p}{k_f}$$

(7)

Equation (7) provides a lower bound for the effective thermal conductivity of porous media. Equation (7) is a modification of Equation (1), which considers the impact of porosity and convection in the fluid moving through porous media.

The experiment performed by Liang [36] to measure the thermal conductivity of polypropylene–hollow glass bead composites underscore the fact that the effective thermal conductivity decreases linearly with the increase in the volume fraction of hollow glass beads (HGBs). Other materials also display similar behavior. For example, Nait-Ali et al. [48] determined the thermal conductivity of highly porous zirconia ceramic with nanometric-sized grains and found that its effective thermal conductivity decreases as the pore volume fraction increases. The finding of Liang’s experiment demonstrates that creating a void in polymer composites could decrease the thermal conductivity of the polymer concerned. Figure 2 shows the correlation between effective thermal conductivity ($K_{\mathrm{e}\mathrm{f}\mathrm{f}}$) and the volume fraction (${\mathrm{\varphi }}_{\mathrm{f}}$) of a polypropylene–hollow glass bead (HGB) composite (PP/TK35) supplied by the Molus company in Germany [36]. From the plot, it can be observed that the effective thermal conductivity $K_{\mathrm{e}\mathrm{f}\mathrm{f}}$ and volume fraction ${\mathrm{\varphi }}_{\mathrm{f}}$ of HGB show an inverse relationship.

3. Methodology

3.1. Experimental Methods

We measured the effective thermal conductivity of multi-hollow polypropylene board by employing the discovery thermal conductivity tester (DTC-25) manufactured by TA Instruments and the hot box built in Dr. Sanjeev’s laboratory at the University of Missouri. DTC-25 is suited for quality control and accurately determining the thermal conductivity of polymers, metals, ceramics, composites, glass, rubber and graphite products in the laboratory.

The hot box test follows the ASTM C1046 standard [38] practice for in situ measurement of heat flux and temperature in building envelope components. This hot box is a more robust measurement of the thermal conductivity of building materials as it closely reflects reality in application. The method involves measuring heat flux through localized materials and the hot box’s internal/external environmental temperatures with heat flux transducers and temperature transducers, respectively. Heat flux is the heat energy transfer rate through a given material. It is defined as the energy flow per unit area per unit of time. The SI unit is given by watts per square meter (W/m²). The heat flux determination through the PP insulation board using the hot box test provides a basis for the calculation of the thermal conductivity of the porous PP board at steady state, via the formula of Fourier’s law of heat transfer given in Equation (8).

$$q={-k}_e\nabla \mathrm{T}$$

(8)

$$q={-k}_e\frac{\mathsf{\Delta }\mathrm{T}}{\mathsf{\Delta }x}$$

$$q=\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t} \mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}$$

$$k_e=\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}$$

$$\nabla \mathrm{T}=\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r} \mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l} \mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r} \mathrm{o}\mathrm{n} \mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}$$

$$\mathsf{\Delta }\mathrm{T}=\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e} \mathrm{i}\mathrm{n} \mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}$$

$$\mathsf{\Delta }x=\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{k}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}$$

3.2. Method for DTC-25 Test

The three-step procedure in the DTC-25 test includes sample preparation, sample loading and an experimental test. We prepared a multi-hollow polypropylene sample measuring 50 mm in diameter and 10 mm in thickness for the experimental measurement of thermal conductivity. The sample (Figure 3) was cut away from the 3D-printed, hollow-structured PP block shown in Figure 4. Three different porous PP samples with hole sizes of 4 mm, 2 mm and 1 mm were produced for this experimental test. It is necessary that test samples are smooth and flat on both surfaces. Before beginning the experimental run, a heat sink compound or thermal paste should be applied to both surfaces to maximize heat transfer and dissipation between the test stack carrying heat and the polypropylene board. Silicone paste was used as a thermal paste for this test. The sample is loaded into the test stack with the aid of a free/move switch on the electronic enclosure, which allows for the up/down movement of the stack rod. The test pressure should be kept between 40 and 60 psi, ensuring that the sample is in line with the top and bottom sample plate before it is locked for testing.

The temperature controller setpoint should be maintained at 45 °C and should be adjusted whenever it deviates from the required temperature. Experimental reading takes 30 min, which allows the voltages on the LCD display to reach equilibrium. The thermistor output voltages taken for experimental analysis included reference voltage, upper voltage, lower voltage and heat sink voltage. The voltages were converted to temperature in degrees Celsius with help from computer software.

3.3. Method for Hot Box Test

Thermal conductivity is known to be affected by thermal bridging, changes in material property with temperature, airflow in and through the building envelope, internal convection, phase changes and the hygric storage effect [49], especially in contemporary insulation materials and how they control heat flow. To obtain a more accurate predictive thermal conductivity of a material, a physical facility is needed to replicate field usage conditions. A hot box was deployed for the experimental measurement of the hollow-structured PP’s thermal conductivity. A hot box apparatus is generally large and expensive [49,50]. A few of them can be found in the US National Laboratories. The hot box utilized was constructed in Professor Sanjeev Khanna’s laboratory at the University of Missouri by a group of students in 2015. Figure 5 shows the constructed hot box for measuring the heat flux through building materials. It measures 50 in length by 50 in width by 75 in height and is completely insulated except for two metering windows, which allow for the measurement of heat flux passing through materials. Two incandescent light sources provide the heating for the enclosure. A datalogger, CR1000, manufactured by Campbell Scientific, provides a connection for measuring the heat flux and the hot box’s internal and external environmental temperature. A datalogger software, PC400, accompanies the RC1000 for post-processing recorded measurements.

The hot box is well insulated with two windows measuring 14 in by 14 in on both sides of the building envelope enclosure. An extruded polystyrene (XPS) insulator with a known thermal conductivity is placed on one window while the other window is covered with porous PP material. We are able to establish the PP’s thermal conductivity if the sensors and data-logging system correctly determined the known thermal conductivity value of the XPS insulator. Cai et al. [51] reviewed the thermal conductivities of XPS to be 0.026–0.040 W/mK. These thermal conductivity values were confirmed in a previous technical report on the comparison of expanded and extruded foam polystyrene insulation in roadway and airport embankments [52].

One HFP01 sensor is attached to the outer surface of the XPS insulator. Similarly, the other HFP01 sensor is attached to the outer surface of the PP’s insulation. The Campbell Scientific datalogger system with connected heat flux and temperature sensors was used to log the data for 5 h.

Circling back to Equation (8), XPS and hollow-structured PP thermal conductivity can be calculated when the heat flux is measured using the sensors. Figure 6 shows a schematic of the experimental setup. Figure 7 illustrates the physical test showing the Campbell Scientific datalogger system.

4. Results and Discussion

For DTC-25, the thermistor output voltages are converted to temperature using a DTC-25 computer software with an algorithm to calculate the thermal conductivity of porous PP. The results are tabulated in

Table 1. Measurement values from DTC-25.

PP hole size (mm)	1.00	2.50	4.00
Upper temperature (°C)	41.10	41.44	41.10
Lower temperature (°C)	5.94	5.62	5.94
Heat sink temperature (°C)	3.68	3.57	3.68
Mean sample temperature (°C)	23.52	23.53	23.52
Thermal conductivity (W/mK)	0.28	0.25	0.21

. These values compare to the porous media models described by Wang and Pan [47], showing that an increase in the volume fraction of the material lowers its thermal conductivity. A graph of thermal conductivity of porous PP versus the size of the holes in the material is shown in Figure 8.

To use the hot box, processing of the sensors’ measurements in the datalogger is carried out in the Campbell Scientific PC400 software. The datalogger logs a measurement every 5 s. A snapshot of the sensors’ measurements is shown in Figure 9 after 5 h.

From the results, the heat flux through the PP board is −48.72 W/m²K while the heat flux through the extruded polystyrene is −20.15 W/m²K. The negative values of the heat flux indicate that heat flows from a high to low temperature. The hot box’s interior temperature reading is 142.18 F (61.21 °C), and the average outer temperature reading is 73.35 F (22.97 °C). The PP’s board thickness is 50 mm with a pore size of 4 mm. Using Fourier’s equation of conduction heat transfer in Equation (8), the thermal conductivity of the XPS specimen can be deduced. Figure 10 illustrates the heat flow through the XPS insulation board. In this case, the measured heat transfer is the heat conduction through both the solid and fluid phases. As noted in various publications on heat transfer through metal and insulating foams [53,54,55,56,57], convection and radiation heat transfer are considered negligible to the extent that conduction is the only heat transfer mechanism.

Given:

$$k_e=\frac{q\mathsf{\Delta }x}{\mathsf{\Delta }T}$$

Thermal conductivity $k$ of XPS =$\frac{20.15 \times 0.05}{38.24}$= 0.0263 W/mK.

The thermal conductivity value of the extruded polystyrene (XPS) in the hot box test corresponds to the reviewed values in Cai et al. [51] and Connor [52].

With some innovative porous media, the effect of convective heat transfer plays a crucial role in the overall heat transfer through the media, i.e., the total heat transfer is a summation of conductive heat transfer through the solid and convective heat transfer through the pores. Such materials must surpass a critical Rayleigh number for convective heat transfer to show any influence. The Rayleigh number is a dimensionless number that assesses how the effects of buoyancy forces and viscosity forces compare to conductive heat transfer. Harold Jeffreys [58] first calculated the critical Rayleigh number for a layer of fluid to be 1708 in the year 1926 in his widely published original article titled “The stability of a layer of fluid heated below”. Determination of the Rayleigh number will indicate whether the heat transfer mode in the pore of the porous PP is conduction or convection. Equation (9) is a depiction of the Rayleigh number $Ra$ problem highlighted in Nygard and Tyvand [59,60,61]. A graph of pore size against the Rayleigh number is shown in Figure 11.

$$Ra=\frac{g\mathsf{\beta }\mathsf{{\rm K}}\mathsf{\Delta }\mathsf{{\rm T}}\mathrm{h}}{\upsilon \kappa }$$

(9)

$$Ra=\mathrm{R}\mathrm{a}\mathrm{y}\mathrm{l}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h} \mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}$$

$$g=\mathrm{a}\mathrm{c}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{d}\mathrm{u}\mathrm{e} \mathrm{t}\mathrm{o} \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}$$

$$\mathsf{\beta }=\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{o}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}$$

$$\mathsf{{\rm K}}=\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$$

$$\mathsf{\Delta }\mathsf{{\rm T}}=\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e} \mathrm{i}\mathrm{n} \mathrm{p}\mathrm{o}\mathrm{r}\mathrm{e} \mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}$$

$$h=\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l} \mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{o}\mathrm{f} \mathrm{p}\mathrm{o}\mathrm{r}\mathrm{e}$$

$$v=\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l} \mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y} \mathrm{o}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}$$

$$\mathrm{k}=\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c} \mathrm{v}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y} \mathrm{o}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}$$

$Ra=1.5167\times {10}^6$ implies that the critical Rayleigh number is exceeded, and convection must take precedence in the porous PP. The analysis infers that a two-dimensional heat transfer mode, conduction and convection, is present in the PP porous media if we consider the effect of radiative heat transfer negligible. Following this, Fourier’s conduction heat transfer equation is insufficient to model the total heat transfer through the porous PP media to determine the thermal conductivity of the hollow-structured material. The guarded heat flow technique utilizes the Fourier steady heat transfer model to measure the thermal conductivity of materials in many laboratory apparatuses such as DTC-25. With porous media, a critical Rayleigh number for the fluid media must be determined to understand if the heat flow in the pores is conduction or convection. There is a potential measurement error in determining the thermal conductivity of porous media when the total heat flow through the material is a combination of conduction and convection. A more accurate model for such porous media was put forward by Adrian Bejan [62,63]. The Bejan model demonstrates how the heat flux $\mathrm{q}$ relates to the Rayleigh number $Ra$ in the porous medium shown in Equation (10).

$$\frac{\mathrm{q}}{k_e\mathsf{\Delta }T}=0.319{Ra}^{1/2}$$

(10)

where:

$$\mathrm{q}=48.72 \mathrm{W}/{\mathrm{m}}^2\mathrm{K}$$

$$\mathsf{\Delta }T=38.24 \mathrm{K}$$

$$Ra=1.5167\times {10}^6$$

The effective thermal conductivity $k_e$ of the porous PP can be deduced from the above equation and determined to be $0.00325 \mathrm{W}/\mathrm{m}\mathrm{K}$. This effective thermal conductivity value of porous PP is about eight times less than the thermal conductivity of conventional building insulators. Figure 12 compares experimental thermal conductivity tests using DTC-25 and the hot box.

There is a marked difference in the effective thermal conductivity measurement of porous PP between the DTC-25 and the hot box test. The hot box thermal conductivity measurement is 70 times lower than DTC-25. This difference is due to the exclusion of convective heat transfer in the porous PP, when it is in fact present when using the DTC-25. The guarded heat flow method that considers only conductive heat flow through the solid and fluid phase is the operational principle of the DTC-25, whereas the hot box allows us to consider the influence of convective heat flow through the porous PP medium, as demonstrated in our approach. Selected building insulators’ thermal conductivities are compared with the thermal conductivity of our porous PP in

Table 2. Thermal conductivity values of building insulators.

Building Insulator	Thermal Conductivity (W/mK)
Rock wool	0.04 [64]
Glass wool	0.034 [65]
Wood wool	0.04 [66]
EPS	0.045 [67]
XPS	0.028 [67]
PUR	0.024 [68]
Cellulose	0.040 [69]
VIPs	0.004 [70]
Aerogel	0.012 [71]
Porous PP	0.0033 (current study)

. In Figure 13, we created a bar chart of the table for better data visualization.

5. Conclusions

Experimental measurements of porous PP using the DTC-25 laboratory equipment from TA Instruments and a hot box test have been compared. The thermal conductivity of building materials indicates their insulation capability. Excellent building materials will have a lower thermal conductivity value as well as other building insulator performance metrics. While results show that increasing the volume fraction of fluid in the porous PP has an inverse association with the thermal conductivity of the material, as predicted by porous media theories [47], there is a marked difference in the measured values of thermal conductivity using the two methods. The DTC-25 test utilizes Fourier’s guarded heat flow model, which assumes conductive heat transfer throughout the multiphase material. The assessment of the Rayleigh number of the fluid phase confirmed that a critical Rayleigh number has been exceeded. When the Rayleigh number is below a critical value for a fluid, there is no flow of heat and heat is transferred purely by conduction, which is sufficient for the operation of the DTC-25 device. A deviation from the pure conductive heat model is attained beyond a critical Rayleigh number, which triggers heat transfer by natural convection. Our porous PP medium is an example of a material that has a high Rayleigh number. An accurate model developed by the acclaimed authority on thermodynamics and heat transfer, Adrian Bejan, to predict a combined heat transfer by conduction and convection in porous PP was employed in the studies. The combined heat flow model was applied in the hot box test. The thermal conductivity value of the porous PP derived from the heat flux measurement in the hot box experiment was 0.0033 W/mK. This value exceeds the performance of conventional building insulators compared in Figure 13. The insulating metric exceeds the performance of traditional building insulators by a multiple of eight and compares favorably with the most innovative insulators, vacuum insulation panels (VIPs). Porous PP materials exhibit superior thermal insulation attributes due to the presence of a convective porous structure that inhibits the direct flow of heat across the porous PP board. The results demonstrate that porous PP has an efficient thermal insulation capability. There is the potential to use hollow-structured polypropylene as insulation for buildings’ external walls or exterior cladding to enhance building insulation, especially for renovated homes. This could save more energy cost than conventional insulators and is a more environmentally friendly option.

Future work should focus on a scaled production of 3D-printed, multi-hollow-structured polypropylene board, or manufacturing techniques for the production of multi-hollow polypropylene. The thermal conductivities of other insulators with multi-hollow structures should be investigated using the combined DTC-25 and hot box method to determine the impact of the Rayleigh number on their thermal conductivities.