Thermal Insulation Performance and Reliability of the “Structure-Insulation” Integrated Wall Panel (SIW) for Storage Grain Bungalows

Abstract

The traditional brick bungalow is not conducive to long-term grain storage because of its poor thermal insulation. In this paper, a new type of wall element for grain bungalows with both load-carrying and thermal insulation functions, called a “Structure-Insulation” integrated wall panel (SIW), is proposed for improving the grain storage environment. To study the thermal insulation reliability of SIW under multivariable randomness and the availability of different grain storage zones, a finite element model was established based on the test. Then, the failure criterion was established with the heat transfer coefficient as the key point and 1,000,000 sampling calculations were carried out by the Monte Carlo method. The reliability was discussed and sensitivity of random parameters was quantified. The thermal performance test shows that the heat transfer coefficients of the two designed SIW wall panels compared with the traditional brick bungalow wall are reduced by 45.81% and 56.13%, respectively. The thickness of the insulation panel is sensitive mostly to the thermal insulation performance, with a correlation coefficient of 0.877. When the thickness of the insulation panel is 80, 94, and 107 mm, the wall panel can meet the limit requirements of the heat transfer coefficient of the granary enclosure structure of 0.59, 0.53, and 0.46 $\left[\mathrm{W}/\left({\mathrm{m}}^2·\mathrm{K}\right)\right]$, with reliability indexes of 3.08, 1.82, and 1.75, respectively. The research results provide an important reference for the design, optimization, and application of SIW wall panels in thermal insulation.

1. Introduction

The granary is a special structure, which is different from traditional civil buildings. The environment in which grain is stored is critical. The growth of microorganisms can be inhibited and the quality of the grain itself can be maintained by improving the thermal insulation performance of the granary enclosure structure. There are bungalow warehouses, silos, underground warehouses, building warehouses, etc., at present in China, as shown in Figure 1 [1]. However, the grain bungalow is used the most. Because of the earlier construction time and China’s bad early economic conditions, grain bungalows were mostly built with brick walls. Many of them are still in use today. However, they have poor airtightness and thermal insulation performance, which is unfavorable to grain storage [2]. As China is a big country of grain storage and transportation, traditional sintered clay brick walls can no longer meet China’s current requirements for thermal insulation performance of grain bungalows enclosure structures [3]. Therefore, there is an urgent need for a wall with excellent thermal insulation properties specially used for granaries.

In civil buildings, a layer of thermal insulation to the outer wall, sandwich composite wall panels, and double-layer energy-saving walls have been used for thermal insulation [4,5,6,7]. There are many kinds of wall insulation materials, including polyurethane foam (PU), expanded polystyrene (EPS), and extruded polystyrene (XPS) [8,9]. Among them, PU has the advantages of thermal insulation and airtightness. However, it has poor weather resistance, flammability, and relatively high cost [10]. The EPS and XPS are polystyrene resins that have good thermal insulation effects, easy processing, low cost and other advantages. However, they are flammable, have poor anti-aging ability, and have other disadvantages. XPS has a perfect honeycomb structure compared with EPS. The closed-cell ratio of XPS is above 99% and the heat insulation performance is slightly better than EPS [11]. Therefore, adding XPS material to the granary as a composite wall can effectively improve its thermal insulation. Another key for composite wall panel structures is that connectors are required to connect different layers of wall materials into a whole. Some scholars have carried out mechanical and thermal performance tests of composite wall panels by using the steel-GFRP composite connector (SGCC), basalt-fiber-reinforced polymer (BFRP), carbon-fiber-reinforced polymer (CFRP), glass-fiber-reinforced polymer (GFRP), and other materials as connectors [12,13,14,15,16]. It is found that BFRP can reach 85% of the latter bearing capacity and has better ductility compared with CFRP. However, the thermal conductivity of GFRP is lower than theirs in terms of thermal performance. For SGCC, although it has a higher bearing capacity and lower thermal conductivity, the material cost is higher. Therefore, it is a comprehensive choice to choose GFRP as the connector of the sandwich wall of the granary. However, the granary wall should not only have thermal insulation properties but also undertake the huge lateral pressure from the grain. Therefore, XPS material, two-layer reinforced concrete, and GFRP material can be selected to make a new type of composite wall for the granary. They are used as the middle thermal insulation layer, two layers of stress-bearing layers, and connecting parts of the sandwich structure, respectively. The key problem of poor thermal insulation of traditional brick walls is solved. Because of the application of wrapped XPS, the flammable shortcoming is also overcome. At present, the existing thermal insulation technology includes adding new materials to make composite materials, such as bio-composite materials and organic fibers. However, most of them have high cost and their flexural bearing capacity is related to the proportion of added materials, which is needed for further research. The wall of the grain bungalow is tall, resulting in a large lateral pressure generated by the grain. Taking into account the composite wall panel structure using reinforced concrete and XPS thermal insulation panel, it can meet both the requirements of stress and thermal insulation. In addition, the material and production cost is more economical.

To study the thermal insulation performance of the wall panels, a thermal performance test should be carried out. In previous studies, the thermal path method and the calibrated hot-box method were used to study the thermal resistance and the heat transfer coefficient of wall panels, respectively [17,18,19,20]. The heat transfer coefficient was also measured by the co-heating method [21]. By the thermal path method, the concrete is divided into separate units for each layer and they were mixed with insulation. Then, the heat transfer is proportional to the length of the thermal path to calculate the thermal resistance. Although this method is relatively novel, it delays the time of heat transfer and consumes more time. The heat flow and surface temperature of the wall panel can be measured by the heat flow meter. Therefore, based on the principle of one-dimensional steady heat transfer, the air temperature, the surface temperature, and the heat flux of the specimen can be measured by the temperature control box-heat flow meter method. More importantly, it is convenient and accurate to calculate the heat transfer coefficient. Although the co-heating method can also study the heat transfer coefficient, its reliability remains to be proven. Since the key parameter of the granary enclosure structure is the heat transfer coefficient, which corresponds to the key research of the temperature control box-heat flow meter method. Further, this method is more mature, which is more suitable for this study.

However, most of the previous studies were deterministic analyses. However, any kind of structure is actually in an environment full of uncertainty, such as uncertain load and geometric structure [22,23]. In particular, the new wall panel, as a complex composite structure, is greatly affected by randomness. There is an urgent need to study the uncertain parameters in the thermal insulation process. Meanwhile, some studies have found that the reliability analysis is beneficial for saving costs and obtaining a better design of structure [24,25]. According to engineering practice statistics, most of the physical parameters in reliability analysis are random variables and obey a certain probability distribution, which includes extreme value distribution, lognormal distribution, and Gaussian distribution. It has been found that most of the physical quantities tend to conform to the Gaussian distribution in the state of static load [26]. In the field of reliability, many methods can be used for research, such as the first-order second-moment method, the Monte Carlo method, and the response surface method [27,28,29]. Among them, the first-order second-moment method is not accurate enough for the calculation of nonlinear functional functions. The response surface method also needs a large number of accurate sample values to fit the response function. Otherwise, the calculation result is still inaccurate. Since the thermal insulation performance of granaries is related to the safety of grain storage, the national economy, and people’s livelihood, the Monte Carlo method with the most accurate results is more suitable for the research content of this paper [30,31,32]. In previous studies, the stability reliability analysis of bored-pile walls considering parameter uncertainty has been studied and the seismic reliability evaluation of steel–timber hybrid shear wall systems has also been analyzed [33,34,35,36]. However, most of them are the analysis of mechanical properties. The influence of uncertain parameters on random thermal states of structures has been analyzed in terms of thermal performance [37,38,39,40,41]. The thermal response of the structure under a steady random temperature field has also been discussed [42,43]. However, the reliability of the heat transfer coefficient of the granary structure has not been studied yet. For the thermal performance study, if the structure is designed with the maximum possible temperature and the minimum possible thermal insulation parameters to guarantee the absolute thermal insulation performance of the structure, it is uneconomical and unreasonable. Therefore, it is necessary to analyze the reliability of the thermal performance of granary wall panels. The sensitivity of random parameters is then determined by reliability analysis [44].

This paper proposes a new type of grain bungalow wall element named “Structure-Insulation” integrated wall panel (SIW), which is composed of inner, outer leaf reinforced concrete, and XPS insulation panels through GFRP connectors, as shown in Figure 2. The thermal performance test of the SIW wall panel is carried out by the temperature control box-heat flow meter method to analyze the thermal insulation performance of the wall panel focusing on the heat transfer coefficient. However, uncertainty is ubiquitous. At present, the randomness of wall panels is generally not considered under existing solutions. The real thermal insulation performance of wall panels cannot be accurately reflected. Therefore, based on the reliability theory, the Monte Carlo method is used to randomize the deterministic analysis in this paper. The thermal insulation performance of the SIW wall panel is represented by probability results. The thermal insulation performance indexes and sensitivity parameters are also studied. Additionally, the applicabilities of SIW wall panels of different thicknesses in different grain storage ecological zones are explored. Finally, a theoretical basis for reliability for the design, optimization, and application of SIW wall panels is proposed in practical engineering.

2. Method and Materials

2.1. Design of the SIW Wall Panel Specimen

The specimens were cast on site and the wall panels were 1700 mm in length and width. The wall panel reinforcement is shown in Figure 3. According to the SIW wall panel mechanical performance test [45], the inner and outer leaf walls are independently stressed. The ratio of the loads on the inner and outer leaf walls is equal to the ratio of their stiffness, which is directly related to their thickness. The inner leaf wall is in direct contact with the grain and undertakes the main grain lateral pressure. The thicker the inner leaf wall, the less likely the connector will be twisted, which can avoid stress concentration and prevent concrete cracking. Therefore, the thickness of the inner leaf wall is much larger than that of the outer leaf wall. Two layers of steel bars are also arranged to improve the bearing capacity of the inner leaf wall. In the original mechanical test of SIW wall panels, three sizes of wall panels were designed with the prototype 1/3 model. The thickness of the inner leaf wall is 85, 100, and 115 mm, respectively [46]. The test results show that the requirements of mechanical properties can be met by the three specimens. Therefore, the thinnest 85 mm wall panel of the inner leaf wall was selected to carry out a full-scale thermal test by expanding it three times. For the convenience of production, the thickness of the inner leaf wall is selected to be approximately 250 mm. Correspondingly, the outer leaf wall is relatively thin, playing mainly the role of the enclosure. The inner leaf wall was concrete with a thickness of 250 mm, equipped with two layers of two-way steel bars with a diameter of 8 mm. The difference was that the thickness of the outer leaf wall was 60 mm, equipped with a single layer of two-way steel bars with a diameter of 6 mm. The standard value of compressive strength of a concrete cube was 30 MPa. The rebar was the hot-rolled ribbed bar with a standard value of yield strength of 400 MPa. The XPS insulation panel was set with two thicknesses of 50/80 mm, named S-50XPS and S-80XPS specimens, respectively. The thermal performance test was carried out after the wall panel curing. Meanwhile, another thermal performance test was carried out on the traditional 490 mm thick brick wall named B-490 to compare with the SIW wall panel.

2.2. The Thermal Performance Test and Measuring Point Arrangement

The instrument used in this thermal test was the DTKS-JWCX-1 type building envelope heat transfer coefficient on-site detector. There were also the supporting temperature control box, heat flow meter, thermocouple, and data processing and acquisition system. The temperature control box, heat flow meter, thermocouple, and data processing and acquisition system were, respectively, responsible for inputting heat, obtaining heat flux, obtaining wall panel surface temperature, and recording data. The temperature control box-heat flow meter method was selected for this test, maintaining the temperature difference at 10–15 °C on both sides. Under this temperature condition, the growth of microorganisms can be inhibited and the use of chemical drugs can be reduced to the greatest extent. The schematic diagram of the temperature control box-heat flow meter test is shown in Figure 4. The side of the outer leaf wall was a natural environment temperature of approximately 26 °C. Correspondingly, a temperature control box was arranged on one side of the inner leaf wall. The temperature of the temperature control box was set to a stable 39 °C. The tests were performed on S-50/80XPS and B-490, respectively, as shown in Figure 5.

The thermocouples and heat flow meters were set on both sides of the wall panel to obtain the inner, outer surface temperature, and heat flux values. As shown in Figure 6 and Figure 7, three measuring points were arranged at different positions on the inner and outer sides. In Figure 7, the heat flow meters and thermocouples are represented by rectangles and circles, respectively. The data were automatically collected by the data acquisition instrument once every minute in the test. Because of the long-term storage of grain, only the part of heat transfer in steady-state at the later stage of the test was studied in this paper.

3. Results

3.1. Analysis of Test Results

The test data of the three specimens are shown in Figure 8, Figure 9 and Figure 10. It can be seen that the data of S-50XPS and S-80XPS tend to be stable after approximately 7200 min. This shows that the SIW wall panel is less affected by the environmental temperature and has good thermal insulation performance. However, the data of B-409 are not stable. The heat flux and outer surface temperature of B-490 are floating data with time. It shows that the thermal insulation performance of the traditional brick wall is poor, which is greatly affected by weather, sunshine, and environmental temperature.

The average value of the three measuring points after data stabilization is used to reflect the overall heat insulation condition of the SIW wall panels. Similarly, the average value of data fluctuation is used to reflect B-490, except for invalid data from the initial measurement. As shown in Figure 11, there are average inner surface temperature, average outer surface temperature, and average heat flux from the test results.

The arithmetic average method is used to analyze the test data and calculate the thermal resistance and the heat transfer coefficient. The calculation equations are as follows.

$$R=\frac{W_{\mathrm{i}}-W_{\mathrm{o}}}{q}$$

(1)

where R is the thermal resistance (${\mathrm{m}}^2·\mathrm{K}/\mathrm{W}$); W_i is the inner surface temperature (°C); W_o is the outer surface temperature (°C)(); q is the heat flux ($\mathrm{W}/{\mathrm{m}}^2$).

$$K=\frac{1}{R_{\mathrm{e}}+R+R_{\mathrm{i}}}$$

(2)

$$R_{\mathrm{i}}=\frac{1}{{\alpha }_{\mathrm{i}}}$$

(3)

$$R_{\mathrm{e}}=\frac{1}{{\alpha }_{\mathrm{e}}}$$

(4)

where K is the heat transfer coefficient $\left(\mathrm{W}/({\mathrm{m}}^2·\mathrm{K}\right))$; R_i is the inner surface heat transfer resistance (${\mathrm{m}}^2·\mathrm{K}/\mathrm{W}$); a_i is the inner surface heat convection coefficient $\left(\mathrm{W}/({\mathrm{m}}^2·\mathrm{K}\right))$; R_e is the outer surface heat transfer resistance (${\mathrm{m}}^2·\mathrm{K}/\mathrm{W}$); a_e is the outer surface heat convection coefficient $\left(\mathrm{W}/({\mathrm{m}}^2·\mathrm{K}\right))$. According to the Chinese standard GB50176-2016 Code for thermal design of civil building [47], 8.7 and 23 $\left(\mathrm{W}/({\mathrm{m}}^2·\mathrm{K}\right))$ are taken for a_i and a_e, respectively.

The tests data were substituted into Equations (1)–(4) to calculate the thermal resistance and heat transfer coefficient, respectively, as shown in

Table 1. Thermal resistance and heat transfer coefficient.

Specimens	$\mathbf{Thermal} \mathbf{Resistance} \left({\mathbf{m}}^2\mathbf{·}\mathbf{K}/\mathbf{W}\right)$	$\mathbf{Heat} \mathbf{Transfer} \mathbf{Coefficient} \left[\mathbf{W}/\left({\mathbf{m}}^2\mathbf{·}\mathbf{K}\right)\right]$
S-50XPS	1.03	0.84
S-80XPS	1.31	0.68
B-490	0.49	1.55

.

From Table 1, it can be concluded that the thermal insulation performance of S-80XPS is the best, S-50XPS is second, and B-490 is the worst. Compared with the traditional brick bungalow wall, the heat transfer coefficient of the SIW wall panel is reduced by 45.81% and 56.13%, respectively. It shows that the thermal insulation performance of the SIW wall panel has been greatly improved compared with the traditional brick wall. Through increasing the thickness of insulation panel from 50 to 80 mm, the thermal resistance of the SIW wall panel increases by 27.18% and the heat transfer coefficient decreases by 19.05%. Therefore, to explore its applicability, the following chapters are only used to analyze the thermal performance reliability of the SIW wall panel with better thermal insulation performance.

3.2. Theoretical Calculation

Meanwhile, theoretical calculations were carried out to verify the experimental data and the subsequent finite element model. According to the Chinese standard DG/TJ08-2158-2017 Technical specification for precast concrete sandwich wall panel [48], the thermal conductivity of thermal insulation panels connected with FRP connectors should be revised. Meanwhile, it stipulates that the calculated correction coefficient of the XPS insulation panel is 1.5. Similarly, because of the non-standard construction process and damp exposure of the insulation panel, the correction coefficient of exterior wall heat transfer coefficient is 1.2 according to the Chinese standard GB50189-2015 Design standard for energy efficiency of public buildings [49]. The heat transfer coefficient of the wall panel is calculated as shown in Equation (5).

$$K=\frac{1}{R_0}=\frac{1}{R_{\mathrm{i}}+\sum \frac{c_{\mathrm{j}}}{{\sigma }_{\mathrm{j}}}+R_{\mathrm{e}}}$$

(5)

where K is the heat transfer coefficient $\left(\mathrm{W}/({\mathrm{m}}^2·\mathrm{K}\right))$; R₀ is the heat transfer resistance (${\mathrm{m}}^2·\mathrm{K}/\mathrm{W}$); $\sum \frac{c_j}{{\sigma }_j}$ is the sum of the ratio of thickness to the thermal conductivity of each layer, j = 1,2,3.

By the test requirements, the corresponding parameters are substituted into Equation (5). Then, K₅₀ = 0.89$\left(\mathrm{W}/({\mathrm{m}}^2·\mathrm{K}\right))$ and K₈₀ = 0.61$\left(\mathrm{W}/({\mathrm{m}}^2·\mathrm{K}\right))$ are calculated. The theoretically calculated values are compared with the experimentally measured values, with differences of 5% and 10%, respectively. When the thickness of the insulation panel is slightly larger, the concrete slurry will flow into the joint in the pouring process because of the incompleteness of the insulation panel in the test. This makes the heat insulation effect worse, resulting in a slightly large heat transfer coefficient in the test.

4. Discussion

4.1. Finite Element Simulation Analysis

4.1.1. Modeling

In this study, finite element software was used for modeling [50]. Hexahedral element of eight nodes was used for the inner, outer leaf concrete, and the insulation panel. The rebar and connector were simulated with a two-dimensional heat transfer element, which has only two nodes, by defining its cross-section dimensions to simulate the actual structure. The link element can only transfer heat in one direction, which is suitable for two-dimensional steady-state thermal analysis. Each node has only one temperature freedom. The geometric sizes were corresponding to the test specimens. Then, binding constraints were set for them. Meshing is critical for finite element models. The number of mesh and the element shape are key factors affecting the simulation results. The uniform, regular, and square element shape can make the results accurate. The appropriate number of elements can improve computational efficiency. According to these two principles, the length and width of the SIW wall panel element were both 50 mm, the thickness of the inner leaf wall element was 30 mm, and the thickness of the insulation panel and the outer leaf wall element were 20 mm. Connector and rebar elements only had a length dimension, so their elements had the same length as concrete and insulation panel elements that shared the same node. The finite element models after meshing were shown in Figure 12.

The properties of each material were set as shown in

Table 2. Material properties.

Materials	$\mathbf{Density} (\mathbf{k}\mathbf{g}/{\mathbf{m}}^3)$	$\mathbf{Specific} \mathbf{Heat} \left[\mathbf{J}/\left(\mathbf{k}\mathbf{g}\mathbf{·}\mathbf{K}\right)\right]$	$\mathbf{Thermal} \mathbf{Conductivity} \left[\mathbf{W}/\left(\mathbf{m}\mathbf{·}\mathbf{K}\right)\right]$
Concrete	2300	920	1.74
Rebar	7850	480	58.2
XPS insulation panel	25	1380	0.033
GFRP connectors	2000	1100	0.43

, based on the test materials and relevant specifications.

To ensure that the model is accurate, the environment of the model was set to the same as the test. According to the tests, the temperature load of the nodes on the inner surface of the model was set at 39 °C and the heat convection coefficient of the inner surface was set at 8.7 $\left(\mathrm{W}/({\mathrm{m}}^2·\mathrm{K}\right))$. Similarly, the boundary condition of the outer leaf wall surface nodes was set at 26 °C and the outer surface heat transfer coefficient was set at 23$\left(\mathrm{W}/({\mathrm{m}}^2·\mathrm{K}\right))$. After the heat transfer is stabilized, the surface temperature and heat flux on both sides of the wall panel could be obtained.

4.1.2. Analysis of Simulation Results

The steady-state thermal analysis was carried out on two SIW wall panels. The temperature contour and heat flux diagram were obtained. The S-50XPS temperature contour, the S-80XPS temperature contour, the S-50XPS heat flux diagram, and the S-80XPS heat flux diagram are shown in Figure 13a,b and Figure 14a,b, respectively. Meanwhile, S-80XPS is taken as an example to output the heat flux vector diagram, as shown in Figure 15. It can be seen from the heat flux diagram that more heat will be transferred at the connectors, appearing as the “cold-hot bridge phenomenon”. This causes more heat to be passed through the wall from the connectors, which is certainly harming the thermal insulation performance. The greater the thickness, the more obvious this situation. However, it can be seen that the difference in heat flux in the position with or without connectors is small. As in Figure 14b, the maximum value is approximately 7.89 and the minimum value is approximately 7.78. This is because GFRP materials have relatively low thermal conductivity and relatively good thermal insulation.

If GFRP is replaced by a large thermal conductivity of the material as a connector, the “cold-hot bridge phenomenon” will be obvious. Then, the overall insulation performance of the wall panel will be harmed. For example, the same analysis is carried out for rebar instead of GFRP as connectors. Call this fictitious specimen S-Rebar. The same steady-state thermal analysis was performed for S-Rebar. Then, the S-Rebar heat flux vector diagram is shown in Figure 16. It can be seen from the figure that the maximum heat flux density at the connector can reach 59 $\left(\mathrm{W}/{\mathrm{m}}^2\right)$ and the minimum heat flux density at the non-connector is only 6 $\left(\mathrm{W}/{\mathrm{m}}^2\right)$. This shows that a lot of heat is being transferred from the connector, resulting in poor insulation performance. Meanwhile, the S-Rebar temperature contour of the section of the wall panel at the connector is shown in Figure 17. The “cold-hot bridge phenomenon” is obvious. However, as shown in Figure 18, this phenomenon is extremely small in the S-80XPS connector. The good thermal insulation performance of the SIW wall panel with GFRP as the connector is proved.

According to Figure 13 and Figure 14, the inner and outer surface temperature and heat flux values are recorded in

Table 3. Finite element simulation results.

Specimens	Inner Surface Temperature °C	Outer Surface Temperature °C	$\mathbf{Inner} \mathbf{Surface} \mathbf{Heat} \mathbf{Flux} \left(\mathbf{W}/{\mathbf{m}}^2\right)$	$\mathbf{Outer} \mathbf{Surface} \mathbf{Heat} \mathbf{Flux} \left(\mathbf{W}/{\mathbf{m}}^2\right)$
S-50XPS	27.45	37.82	10.25	10.39
S-80XPS	26.34	38.10	7.78	7.89

. Here, the heat flux is taken as the maximum value in favor of safety considerations.

The data in Table 3 are substituted into Equations (1)–(4), respectively. The thermal resistance $R=1.01 \left({\mathrm{m}}^2·\mathrm{K}\right)/\mathrm{W}$ and the heat transfer coefficient $K=0.86 \left(\mathrm{W}/({\mathrm{m}}^2·\mathrm{K}\right))$, by calculating and analyzing the data obtained from S-50XPS. Similarly, the S-80XPS thermal resistance $R=1.49 \left({\mathrm{m}}^2·\mathrm{K}\right)/\mathrm{W}$ and the heat transfer coefficient $K=0.61 \left(\mathrm{W}/({\mathrm{m}}^2·\mathrm{K}\right))$.

4.1.3. Comparison of Results

The simulation values of S-50XPS and S-80XPS are compared with the theoretical values and test values, as shown in Figure 19. It can be seen from the figure that the simulation value of the heat transfer coefficient of the two specimens is close to the theoretical value and the test value, with a difference of 3%, 2%, 0%, and 10%, respectively. When the thickness of the insulation panel is larger, the concrete slurry flows into the joint because of the imperfect integrity of the insulation panel. It is responsible for the slightly larger heat transfer coefficient of the test. The results show that the model is reasonable and can be used to study the reliability of the heat transfer coefficient under uncertain conditions.

4.2. Reliability Analysis

4.2.1. Monte Carlo Method

The Monte Carlo method is a mathematical statistics method, which is famous for the exact solution of the result [44,51]. In engineering applications, the parameters of various types of structures are regarded as input variables with mathematical-statistical characteristics, including average, variance, and the coefficient of variation. The output variables are also defined. The functional function Z that needs to complete the predetermined function conditions is established. After a sufficient number of loop calculations, enough solutions of functional function are obtained. They are summarized and statistically analyzed to obtain the failure probability and reliability probability [52]. According to the results, whether the structure can meet the requirements of predetermined conditions is studied. Assuming that the random sampling value is N and the number of times $Z_{\mathrm{i}}<0$ is n_f, the calculation formula of structural failure probability is Equation (6).

$$P_{\mathrm{f}}=\raisebox{1ex}{$n_{\mathrm{f}}$}\!\left/ \!\raisebox{-1ex}{$N$}\right.$$

(6)

4.2.2. Establish Functional Function

According to weather, humidity, temperature, and other climatic conditions, China is divided into seven grain-storage ecological zones by the Chinese standard GB/T29890-2013 Technical criterion for grain and oil-seeds storage [53]. Meanwhile, the ecological characteristics of each zone and the limit of heat transfer coefficient are shown in

Table 4. Limit values of the heat transfer coefficient of each grain storage ecological zone enclosure structure.

Regional Division	Ecological Characteristics	$$\begin{gathered} \mathbf{Limit} \mathbf{Value} \mathbf{of} \mathbf{Heat} \mathbf{Transfer} \\ \mathbf{Coefficient} \left[\mathbf{W}/\left({\mathbf{m}}^2\mathbf{·}\mathbf{K}\right)\right] \end{gathered}$$
First Zone	High cold and dry grain storage zone	0.59–0.70
Second Zone	Low temperature and dry grain storage zone	0.59–0.70
Third Zone	Low temperature and high humidity grain storage zone	0.59–0.70
Fourth Zone	Medium temperature and dry grain storage zone	0.53–0.58
Fifth Zone	Medium temperature and high humidity grain storage zone	0.46–0.52
Sixth Zone	Medium temperature and low humidity grain storage zone	0.53–0.58
Seventh Zone	High temperature and high humidity grain storage zone	0.46–0.52

. It can be seen from Table 4 that the maximum value of the heat transfer coefficient in the first, second and third zones is 0.7. The maximum value of the heat transfer coefficient in the fourth and sixth zones is 0.58. The maximum value of the heat transfer coefficient in the fifth and seventh zones is 0.52.

Therefore, in this paper, the maximum value of the heat transfer coefficient in each zone is taken as the predetermined requirement condition. The functional function Z under the condition of thermal performance control is then established as Equation (7).

$$Z_1=K_{\lim }-K$$

(7)

where K_lim is the limit of the heat transfer coefficient in Table 4; K is the heat transfer coefficient obtained for each simulation.

According to the results of the previous chapters and Table 4, S-50XPS does not meet the requirements of heat transfer coefficients in any zones. However, S-80XPS can meet the requirements of heat transfer coefficients in the first, second, and third zones. Therefore, the reliability of the insulation capability of the S-80XPS in the first, second, and third grain storage ecological zones will be investigated next. Here, the maximum limit value K_lim of the heat transfer coefficient in the first, second, and third zones is 0.7 $\left(\mathrm{W}/\left({\mathrm{m}}^2·\mathrm{K}\right)\right)$.

4.2.3. Random Variables

The SIW wall panel is a complex structure consisting of inner and outer reinforced concrete walls, the XPS insulation panel, and connectors. Combined with the reliability theory, there are uncertain conditions for the geometric parameters of materials, thermal performance parameters, and thermal loads. In this paper, the thermal conductivity, density, and specific heat capacity of materials are selected as random thermal performance parameters. The material thickness and width are random geometric dimensions. The thermal temperature is a random load. According to the law of large numbers, the geometrical dimensions, material properties, and dead loads of the structure all obey the Gaussian distribution in the steady state (static load) [26,54]. The value of 5% coefficient of variation is comprehensively selected according to the practical experience and References [32,54,55]. Combined with the finite element model, as many random parameters as possible should be selected, as detailed shown in

Table 5. Distribution characteristics of random variables.

Random Variables	Distribution Characteristics	Average Value μ	Coefficient of Variation δ	Code Name
$\mathrm{Thermal} \mathrm{conductivity} \mathrm{of} \mathrm{concrete} \left[\mathrm{W}/\left(\mathrm{m}·\mathrm{K}\right)\right]$	GUSS	1.74	0.05	HD
$\mathrm{Thermal} \mathrm{conductivity} \mathrm{of} \mathrm{XPS} \left[\mathrm{W}/\left(\mathrm{m}·\mathrm{K}\right)\right]$	GUSS	0.033	0.05	XD
$\mathrm{Thermal} \mathrm{conductivity} \mathrm{of} \mathrm{rebar} \left[\mathrm{W}/\left(\mathrm{m}·\mathrm{K}\right)\right]$	GUSS	58.2	0.05	GD
$\mathrm{Thermal} \mathrm{conductivity} \mathrm{of} \mathrm{connection} \left[\mathrm{W}/\left(\mathrm{m}·\mathrm{K}\right)\right]$	GUSS	0.43	0.05	LD
$\mathrm{Density} \mathrm{of} \mathrm{concrete} \left(\mathrm{kg}/{\mathrm{m}}^3\right)$	GUSS	2300	0.05	HDENS
$\mathrm{Density} \mathrm{of} \mathrm{XPS} \left(\mathrm{kg}/{\mathrm{m}}^3\right)$	GUSS	25	0.05	XDENS
$\mathrm{Density} \mathrm{of} \mathrm{rebar} \left(\mathrm{kg}/{\mathrm{m}}^3\right)$	GUSS	7850	0.05	GDENS
$\mathrm{Density} \mathrm{of} \mathrm{connection} \left(\mathrm{kg}/{\mathrm{m}}^3\right)$	GUSS	2000	0.05	LGENS
$\mathrm{Specific} \mathrm{heat} \mathrm{of} \mathrm{concrete} \left[\mathrm{J}/\left(\mathrm{kg}·\mathrm{K}\right)\right]$	GUSS	920	0.05	HC
$\mathrm{Specific} \mathrm{heat} \mathrm{of} \mathrm{XPS} \left[\mathrm{J}/\left(\mathrm{kg}·\mathrm{K}\right)\right]$	GUSS	1380	0.05	XC
$\mathrm{Specific} \mathrm{heat} \mathrm{of} \mathrm{rebar} \left[\mathrm{J}/\left(\mathrm{kg}·\mathrm{K}\right)\right]$	GUSS	480	0.05	GC
$\mathrm{Specific} \mathrm{heat} \mathrm{of} \mathrm{connection} \left[\mathrm{J}/\left(\mathrm{kg}·\mathrm{K}\right)\right]$	GUSS	1100	0.05	LC
Thickness of outer leaf wall (m)	GUSS	0.06	0.05	WH
Thickness of XPS (m)	GUSS	0.08	0.05	XH
Thickness of inner leaf wall (m)	GUSS	0.25	0.05	NH
Length and width of wall panel (m)	GUSS	1.7	0.05	L
Temperature load (°C)	GUSS	39	0.05	T

.

4.2.4. Sampling Calculation

The Monte Carlo Latin Hypercube Sampling was used for random input parameter variables 10⁶ times [56,57]. The data from Table 5 were then substituted into Equations (6) and (7) one by one to calculate the failure probability. Taking XDENS as an example, the random sampling is shown in Figure 20 and the distribution histogram is shown in Figure 21. As can be seen from the figures, the maximum value of XDENS sampling is 31.485, the minimum value is 18.934, and the average value is 25. The maximum value of the abscissa of Figure 20 is 10⁶ times and the distribution shape of Figure 21 is the GUSS function, which are conformed to the definition of random sampling distribution in Table 5. These indicate that enough data have been extracted to support this study of reliability.

After 10⁶ times sampling calculations, the cumulative distribution curve is shown in Figure 22. It can be seen that the cumulative distribution curve is smooth without mutation, indicating that the cumulative distribution function has good convergence and the result is accurate. The slope of the curve is the highest in the range of 0.05–0.15, proving that the probability of Z₁ is the highest in this range. Meanwhile, the minimum value of Z₁ is −0.0614, the maximum value is 0.204, and the average value is 0.09275. The failure probability $P_{\mathrm{f}}=1.01379\times {10}^{-3}$ and the reliability $P_{\mathrm{r}}=0.99897$. The results show that the reliability of the SIW wall panel with 80 mm thickness of insulation panel is 99.897% under the random condition of a 5% variation coefficient. This translates to a reliability index of 3.08. Therefore, it has good thermal reliability performance and can well meet the requirements of the first to third grain storage ecological zones on the heat transfer coefficient of the enclosure structure.

Then, the limit value of heat transfer coefficient K_lim is adjusted appropriately to analyze the change of reliability probability, as shown in Figure 23. The results show that when the limit of the heat transfer coefficient is between 0.68 and 0.72, all of the reliability is above 99%. Therefore, the SIW wall panel can be considered to own a relatively stable thermal reliability. It can be seen that the reliability probability is increased with the increase of the limit value of the heat transfer coefficient, approaching 1 infinitely. However, the increased amplitude is getting smaller and smaller.

4.2.5. Sensitivity Analysis

The calculation results were statistically analyzed to study the influence of random parameters on structural response. The input parameters that affect the significance level of Z₁ results exceeding 2.5% are taken for analysis, as shown in Figure 24. Other parameters that do not exceed 2.5% indicate that the influence ability is small. The results show that the relatively large parameters affecting the reliability of thermal performance of SIW wall panels are XH, XD, HD, NH, WH, and LD. Among them, the sensitivities of XH, NH, and WH are beneficial for the reliability of thermal performance. On the contrary, the sensitivities of XD, HD, and LD are harmful to the reliability of thermal performance. It can be seen that XH is the parameter of greatest influence on the reliability of the thermal performance of the SIW wall panel. Therefore, the influence of the thickness of the XPS insulation panel on the structure should be mainly considered in engineering applications.

Similarly, the correlation between input variables and Z₁ is shown in

Table 6. Correlation coefficients.

Variables	Correlation Coefficient	Variables	Correlation Coefficient	Variables	Correlation Coefficient
HD	−0.114	XD	−0.418	GD	<0.001>
LD	−0.013	HDENS	<−0.002>	XDENS	<0.001>
GDENS	<0.001>	LDENS	<−0.001>	HC	<0.001>
XC	<0.001>	GC	<−0.001>	LC	<−0.001>
L	<0.001>	XH	0.877	WH	0.042
NH	0.092	T	<0.001>

. The influence ability of less than 2.5% is represented by <>. The results show that the correlation coefficients of XH, XD, HD, NH, WH, and LD are 0.877, −0.418, −0.114, 0.092, 0.042, and −0.013, respectively. Meanwhile, the correlation coefficient of XH is the largest, which is consistent with the sensitivity analysis result.

4.2.6. Numerical Analysis

To study the applicabilities of SIW wall panels in different grain storage ecological zones, the thickness of the insulation panel will be taken as the key parameter for analysis, because of its maximum influence ability. By increasing the thickness of the insulation panel, the Monte Carlo method is still used to analyze the reliability of the SIW wall panel under the uncertain condition of a 5% variation coefficient.

The finite element simulation is carried out for every 2 mm increase, with the insulation panel thickness of 80 mm as the benchmark. The simulation results are shown in Figure 25. It can be seen that the thickness of the insulation panel is linear with the heat transfer coefficient. Then, the scatter diagram is fitted to obtain the linear equation of XPS insulation panel thickness-heat transfer coefficient, as shown in Equation (8).

$$K=0.95467-0.00447T$$

(8)

where K is the heat transfer coefficient ($\mathrm{W}/({\mathrm{m}}^2·\mathrm{K}$)); T is the thickness of XPS with the scope of 82–112 ($\mathrm{mm}$).

To explore the applicabilities of SIW wall panels in various grain storage ecological zones, the heat transfer coefficient limit $K_{\lim }=0.58 \left(\mathrm{W}/\left({\mathrm{m}}^2·\mathrm{K}\right)\right)$ of the fourth and sixth grain storage ecological zones is selected. The functional function is still expressed by Equation (7). Then, random variables with the same variability and dispersion are used to conduct 10⁶ times Monte Carlo sampling simulations for SIW wall panels with different thicknesses. Finally, they are substituted into the functional function to calculate the failure probability. The calculation results are shown in Figure 26a. In this study, a 5% failure probability is taken as the threshold value, corresponding to the variability. Therefore, the failure probability lower than 5% is regarded as having good reliability.

When the thickness of the XPS insulation panel is 93 mm, the failure probability is calculated of $P_{\mathrm{f}}=0.05059>5\%$. When the thickness of the XPS insulation panel is 94 mm, the failure probability is calculated of $P_{\mathrm{f}}=0.03428<5\%$. This translates to a reliability index of 1.82. Therefore, when the thickness of the XPS insulation panel is increased to 94 mm, the SIW wall panel can also meet the requirements of heat transfer coefficient under uncertain conditions in the fourth and sixth grain storage ecological zones, except for the first, second, and third zones.

Then, the XPS thickness–failure probability curve is fitted as an exponential function, as shown in Figure 26b and Equation (9).

$$P_{\mathrm{f}}=3.11602\times {10}^8\times e^{\frac{-T}{4.20381}}-0.01935$$

(9)

where P_f is the failure probability ($P_{\mathrm{f}}\ge 0$); T is the thickness of XPS with the scope of 86–100 ($\mathrm{mm}$).

Similarly, the limit value of the heat transfer coefficient $K_{\lim }=0.52\left(\mathrm{W}/\left({\mathrm{m}}^2·\mathrm{K}\right)\right)$ in the fifth and seventh grain storage ecological zones is selected. The functional function is still expressed by Equation (7). Then, random variables with the same variability and dispersion are also used to conduct 10⁶ times Monte Carlo sampling simulations for SIW wall panels with different thicknesses. Finally, they are also substituted into the functional function to calculate the failure probability. The calculation results are shown in Figure 26c.

When the thickness of the XPS insulation panel is 106 mm, the failure probability is calculated of $P_{\mathrm{f}}=0.05569>5\%$. When the thickness of the XPS insulation panel is 107 mm, the failure probability $\mathrm{is} \mathrm{calculated} \mathrm{of} P_{\mathrm{f}}=0.039942<5\%$. This translates to a reliability index of 1.75. Therefore, when the thickness of the XPS insulation panel is increased to 107 mm, the SIW wall panel can meet the requirements of the heat transfer coefficient of all grain storage ecological zones under uncertain conditions.

Then, the XPS thickness–failure probability curve is fitted as an exponential function, as shown in Figure 26d and Equation (10).

$$P_{\mathrm{f}}=8.67113\times {10}^7\times e^{\frac{-T}{5.13026}}-0.02979$$

(10)

where P_f is the failure probability ($P_{\mathrm{f}}\ge 0$); T is the thickness of XPS with the scope of 98–112 ($\mathrm{mm}$).

In general, the requirements of the heat transfer coefficient limit of 0.59 $\left(\mathrm{W}/\left({\mathrm{m}}^2·\mathrm{K}\right)\right)$ for the enclosure structure in grain storage ecological zone can be met by S-80XPS with a reliability index of 3.08. However, to meet the requirements of the heat transfer coefficient limit of 0.53 $\left(\mathrm{W}/\left({\mathrm{m}}^2·\mathrm{K}\right)\right)$, the thickness of the insulation panel needs to be increased to 94 mm with a reliability index of 1.82. Similarly, to meet the heat transfer coefficient limit of 0.46 $\left(\mathrm{W}/\left({\mathrm{m}}^2·\mathrm{K}\right)\right)$, it needs to be increased to 107 mm with a reliability index of 1.75.

5. Conclusions

In this paper, a new type of SIW wall panel for the wall element of the grain bungalow is proposed. To study its thermal insulation performance, thermal performance tests were carried out. Then, the thermal insulation reliability of the SIW wall panel under multivariable random conditions was studied by the Monte Carlo method. Meanwhile, the influence of each parameter on the thermal insulation performance of the panel was quantified. Finally, the availabilities of wall panels to different grain storage ecoregions were discussed. The main findings were drawn as follows:

(1)

The heat transfer coefficients of S-50XPS and S-80XPS are reduced by 45.81% and 56.13%, respectively, compared with the traditional brick bungalow B-490 wall. SIW wall panels have significantly better thermal insulation performance than traditional brick walls.

(2)

The SIW wall panel with an insulation panel of 80 mm thickness can meet the requirements of the heat transfer coefficient limit of 0.59 $\left(\mathrm{W}/\left({\mathrm{m}}^2·\mathrm{K}\right)\right)$ for the granary enclosure structure in the grain storage ecological zone.

(3)

The thickness of the insulation panel is sensitive mostly to the thermal insulation performance, with a correlation coefficient of 0.877.

(4)

The reliability index of S-80XPS meeting the heat transfer coefficient limit for the granary enclosure structure of 0.59 $\left(\mathrm{W}/\left({\mathrm{m}}^2·\mathrm{K}\right)\right)$ is 3.08. When the thickness of the insulation panel is increased to 94 mm and 107 mm, the wall panel can meet the requirements of the heat transfer coefficient limits of 0.53 $\left(\mathrm{W}/\left({\mathrm{m}}^2·\mathrm{K}\right)\right)$ and 0.46 $\left(\mathrm{W}/\left({\mathrm{m}}^2·\mathrm{K}\right)\right)$ with the reliability index of 1.82 and 1.75, respectively.

The research results provide an important reference for the design, optimization, and application of SIW wall panels in thermal insulation.