Impact of Insulation Type and Thickness on the Dynamic Thermal Characteristics of an External Wall Structure

sustainability

Article Impact of Insulation Type and Thickness on the Dynamic Thermal Characteristics of an External Wall Structure

Jihui Yuan ID

Department of Architectural Engineering, Graduate School of Engineering, Osaka University, Suita 5650871, Japan; [email protected]; Tel.: +81-80-4768-3098

Received: 13 July 2018; Accepted: 7 August 2018; Published: 9 August 2018

Abstract: The dynamic thermal characteristics of external wall structures are directly related to indoor thermal comfort and energy savings in buildings; they are also complicated and worth investigating. Thermal insulation in external wall structures has become a popular topic of investigation in the domain of building energy efficiency. This study aims to find the impact of insulation type and thickness on the dynamic thermal characteristics of external wall structures using a homogeneous multi-layer building external wall structure and three types of insulation materials that are widely used in Japan. The impact of insulation type and thickness on seven thermal characteristics of external walls, including thermal transmittance, decrement factor or amplitude attenuation, time lag, thermal admittance, time lead for thermal admittance, surface factor, and thermal capacity, was evaluated by numerical methods in this study. It was shown that insulation type and thickness would have a significant effect on thermal transmittance, decrement factor and time lag, but yield no significant change in thermal admittance, time lead for thermal admittance, surface factor, and the thermal capacity of external wall structures.

Keywords: external wall structure; thermal characteristics; insulation type; insulation thickness; numerical analysis

1. Introduction The appropriate design of the building envelopes is widely known as one important factor in the energy-efficiency strategies in buildings [1]. The building envelope is the physical separator between the interior and exterior of a building. The amplitude of the sinusoidal heat transfer wave driven by the daily temperature cycle shrinks as it penetrates through a building envelope. This shrinkage of the heat wave from the outdoor side to the indoor side of external wall is due to the thermal mass of the building envelope materials, and is known as the “decrement factor” or “amplitude attenuation”, which is defined as a ratio of the amplitude of the wave in the inner and the outer surfaces of the wall. The time taken for the heat sine wave to penetrate through the building envelope is known as the decrement delay [2]. Building envelope materials should have a low decrement factor to maintain comfortable indoor conditions [3]. Therefore, much research has been implemented to study the impact of the insulation type, thickness, and position of the external wall structure on the time lag and decrement factor [4,5]. Optimum insulation position was investigated with consideration of maximum time lag and minimum decrement factor by using the Crank–Nicolson method under periodic convection boundary conditions [4]. A total of six different configurations were selected as the initial state, and swept across the wall cross-section where time lags and decrement factors were calculated. The result showed that to place half of the insulation in the mid-center plane of the wall structure and half of it in the outer surface of the wall structure yields a high time lag and low decrement factor, which is

Sustainability 2018, 10, 2835; doi:10.3390/su10082835 www.mdpi.com/journal/sustainability Sustainability 10 Sustainability2018 2018,, 10,, 2835 x FOR PEER REVIEW 22 ofof 1413

yields a high time lag and low decrement factor, which is considered to be the optimum. A study has considered to be the optimum. A study has examined the effect of insulation thickness and position on examined the effect of insulation thickness and position on the time lag and decrement factor by the the time lag and decrement factor by the Crank–Nicolson method under periodic convection boundary Crank–Nicolson method under periodic convection boundary conditions [5]. The result showed that conditions [5]. The result showed that the insulation thickness and position have a very profound the insulation thickness and position have a very profound effect on the time lag and decrement effect on the time lag and decrement factor. An admittance procedure was used to explore the surface factor. An admittance procedure was used to explore the surface factor of building envelopes [6]. factor of building envelopes [6]. The calculation of the surface factor was proposed to investigate the The calculation of the surface factor was proposed to investigate the sensitivity of common sensitivity of common single-layered walls to the thickness and the thermophysical properties of the single-layered walls to the thickness and the thermophysical properties of the slab. The effect of wall slab. The effect of wall orientation and exterior wall surface solar absorptivity on the time lag and orientation and exterior wall surface solar absorptivity on the time lag and decrement factor for decrement factor for specific climatic conditions was carried out using the implicit finite difference specific climatic conditions was carried out using the implicit finite difference method (FDM) [7]. method (FDM) [7]. A simplified calculation method to calculate the thermal and solar properties of a A simplified calculation method to calculate the thermal and solar properties of a double-glazing double-glazing façade along with its with comfort performance was developed, based on standards façade along with its with comfort performance was developed, based on standards EN410 and EN673, EN410 and EN673, by taking the thermal mass of the glazing into account [8]. Furthermore, the by taking the thermal mass of the glazing into account [8]. Furthermore, the performance calculated in performance calculated in terms of internal surface temperature was verified with experimental data terms of internal surface temperature was verified with experimental data collected from a full-scale collected from a full-scale façade element test facility at Aalborg University in Denmark. The internal façade element test facility at Aalborg University in Denmark. The internal surface temperature surface temperature of the glazing calculated using the simplified method, window information of the glazing calculated using the simplified method, window information system (WIS) software, system (WIS) software, and measurements taken across a whole week was compared. It showed that and measurements taken across a whole week was compared. It showed that the R2 between the the R2 between the measurement and the simplified method for the whole week was about 0.83, and measurement and the simplified method for the whole week was about 0.83, and the R2 between the the R2 between the measurement and the WIS software for the whole week was about 0.88. measurement and the WIS software for the whole week was about 0.88. Much research on the time lag and decrement factor of wall has been carried out. However, Much research on the time lag and decrement factor of wall has been carried out. However, there there has been little study on the other thermal characteristics of external walls as related to has been little study on the other thermal characteristics of external walls as related to decrement decrement factor and time lag. Similar to the international standard EN ISO 13786 [9], this study aims to factor and time lag. Similar to the international standard EN ISO 13786 [9], this study aims to use a use a simplified numerical method that is based on Japanese Industrial Standards (JISs) [10–12] to simplified numerical method that is based on Japanese Industrial Standards (JISs) [10–12] to evaluate evaluate thermal characteristics for different multi-layer wall structures worldwide, and find the thermal characteristics for different multi-layer wall structures worldwide, and find the impact of the impact of the insulation type and thickness of the commonly used external wall structure on the insulation type and thickness of the commonly used external wall structure on the dynamic thermal dynamic thermal characteristics, including (i) thermal transmittance; (ii) decrement factor or characteristics, including (i) thermal transmittance; (ii) decrement factor or amplitude attenuation; amplitude attenuation; (iii) time lag; (iv) thermal admittance; (v) time lead for admittance; (vi) (iii) time lag; (iv) thermal admittance; (v) time lead for admittance; (vi) surface factor; and (vii) thermal surface factor; and (vii) thermal capacity, using the simplified numerical method. capacity, using the simplified numerical method.

2.2. MaterialsMaterials andand MethodsMethods

2.1.2.1. ExternalExternal WallWall StructureStructure ThisThis studystudy considered considered a homogeneousa homogeneous multi-layer multi-layer building building external external wall structurewall structure that is widelythat is usedwidely in used Japan in [13 Japan], and [13], is illustrated and is illustrated in Figure in1 Figure. The multi-layer 1. The multi-layer structure structure is composed is composed of, in order of, in fromorder indoor from indoor to outdoor: to outdoor: 15 mm 15 of mm wood, of wood, L mm L of mm insulation of insulation material material (expanded (expanded polystyrene polystyrene (EPS) or(EPS) glass or wool glass or wool wood or cementwood cement board), board), a 20-mm a 20-mm air layer, air 120 layer, mm 120 of concrete,mm of concrete, and 30 mmand of30 plaster.mm of Theplaster. thermophysical The thermophysical properties properties of each materialof each mate in therial external in the external wall structure wall structure are shown are in shown Table 1in. TheTable thermophysical 1. The thermophysical properties properties of the materials of the materials are those are at a those temperature at a temperature of 20 ◦C. of 20 °C.

FigureFigure 1.1. Cross-sectionalCross-sectional viewview ofof thethe externalexternal wallwall structure.structure. Sustainability 2018, 10, 2835 3 of 14

Table 1. Thermophysical properties of building materials at a temperature of 20 ◦C.

Thermal Conductivity Speciﬁc Heat (C ) Building Material Density (ρ) [kg/m3] p (k) [W/mK] [J/kgK] Wood 0.179 530 2300 Expanded polystyrene 0.037 16 1340 Glass wool 0.051 10 837 Wood cement board 0.16 680 1675 Air layer 0.026 1.2 1006 Concrete 0.9 2000 879 Plaster 0.43 1250 1050

2.2. External Wall Structure For a homogeneous wall structure, the heat transfer through the thickness of wall structure can be expressed by Davies [14] based on Fourier’s law:

2 σ T(x, t) ρ · Cp σT(x, t) = · (1) σx2 k σt

3 where ρ is the density of the material, [kg/m ]; Cp is the speciﬁc heat of the material, [J/kgK]; k is the thermal conductivity of the material, [W/mK]; T(x,t) is the temperature, [◦C]; x is the distance coordinate that starts at zero from the external surface toward the internal surface, [m]; and t is the time, [s]. The temperature at the x-position of the wall and at time t, T(x,t) can be obtained using the method outlined by Davies [15]:

T(x, t) = [Q · sinh(γ · x + j · γ · x) + R · cosh(γ · x + j · γ · x)] · exp(j · 2π · t/P) (2) q γ = π · ρ · Cp/k · P (3) √ j = −1 (4) where Q is the heat ﬂux from a unit external wall surface, [W/m2]; R is the thermal resistance of the wall, [m2K/W]; P is the time period, [s]; and j is the imaginary number, [-]. The temperature at the external wall surface (x = 0) and the heat ﬂux at the external wall surface (x = 0) can be calculated using the following equations according to Urano and Uezono [16–18]: " # " #" # T cosh(µ + j · µ)(sinh(µ + j · µ))/a T ext = int (5) qext sinh(µ + j · µ) · a cosh(µ + j · µ) qint

s 2 π · ρ · Cp · l µ = (6) k · P r j · 2π · k · ρ · Cp a = (7) P ◦ where Text is the temperature at the external wall surface x = 0, [ C]; qext is the heat ﬂux at the external 2 wall surface x = 0, [W/m ]; l is the thickness of the wall material, [m]; Tint is the temperature at the ◦ 2 internal wall surface x = l,[ C]; and qint is the heat ﬂux at the internal wall surface x = l, [W/m ]. The above Equations (5)–(7) can be written as follows: " # " #" # T A + j · A )(A + j · A )/a T ext = 1 2 3 4 int (8) qext (−A4 + j · A3) · a A1 + j · A2) qint Sustainability 2018, 10, 2835 4 of 14

A1 = cosh(µ) · cos(µ) (9)

A2 = sinh(µ) · sin(µ) (10) √ A3 = [cosh(µ) · sin(µ) + sinh(µ) · cos(µ)]/ 2 (11) √ A4 = [cosh(µ) · sin(µ) − sinh(µ) · cos(µ)]/ 2 (12)

For a homogeneous multi-layer wall structure, Tint and qint can be represented by Equation (13): " # " #" #" # " #" #" # T 1 −R a b a b a b 1 −R T int = int i i i+1 i+1 ... n n ext ext (13) qint 0 1 ci di ci+1 di+1 cn dn 0 1 qext where Rint and Rext are the internal wall surface resistance and external wall surface resistance, 2 respectively, [m K/W]; they are taken as 1/9 and 1/23, respectively in this study. Meanwhile, ai, bi, ci, and di are the elements of the wall layer-i matrix, and n is the number of wall layers. The above Equation (13) can be represented by the following Equations (14) and (15): " # " #" # T E E T int = 11 12 ext (14) qint E21 E22 qext

" # " #" # " # E E 1 −R A + j · A (A + j · A )/a 1 −R 11 12 = int 1 2 3 4 ..... ext (15) E21 E22 0 1 (−A4 + j · A3) · a A1 + j · A2 0 1 f where E11, E12, E21, and E22 are the elements of the multi-layer wall transmission matrix. The thermal characteristics of external wall structures are dynamic and unsteady. This periodic change can be characterized by quantities such as thermal admittance and amplitude attenuation. These quantities were ﬁrst introduced by Loudon based on a matrix analysis of periodic heat ﬂow in solids [19]. This method is popularly known as the admittance method or the Chartered Institution of Building Services Engineering (CIBSE) method. The CIBSE (2006) method is used for estimating the unsteady heat transfer parameters of building envelopes. The dynamic thermal characteristics of various external wall enclosures can be computed by the following relationships shown in Equations (16)–(22).

2.2.1. Thermal Transmittance (U) The thermal transmittance of the wall structure and its thermal resistance (R) are a reciprocal relationship. The lower the thermal transmittance, the higher the thermal insulation of the external wall structure. The thermal transmittance can be calculated using Equation (16):

1 U = (16) xi xi+1 xn Rext + ( ) + ( ) + ... + ( ) + Rair + Rint ki ki+1 kn

2 where U is the thermal transmittance of the wall structure, [W/m K]; xi is the thickness of wall layer-i, [m]; ki is the thermal conductivity of wall layer-i, [W/mK]; and Rair is the thermal resistance of air 2 layer [m K/W], which is used instead of the corresponding xi/ki term.

2.2.2. Thermal Admittance (Y) The thermal admittance of the wall structure is a measure of the thermal mass of the external surface material, and represents a surface’s ability to absorb heat from the atmosphere and release it back to the same over a given time period. It can be calculated using Equation (17):

qint Y = ( ) = |−E11/E12| (17) Tint Text=0 Sustainability 2018, 10, 2835 5 of 14

2 where Y is the thermal admittance of the wall structure, [W/m K]; E11 and E12 are calculated using the above Equation (15).

2.2.3. Time LAG (ϕ) The time lag is the time that it takes for a heat ﬂow from the outer surface of the wall to reach the inner surface. It is also called “decrement delay”. It can be calculated with Equation (18):

1 12 Im(− U·E ) ϕ = · arctan( 12 ) (18) π Re(− 1 ) U·E12 where ϕ is the time lag, [h]. For a thin wall structure with lower thermal capacity, its ϕ is close to zero.

2.2.4. Time Lead for Thermal Admittance (ω) The time lead for thermal admittance is a measure of the time difference between the time of peak heat ﬂow at the internal wall surface and the peak temperature of the internal wall surface. It can be derived by Equation (19): −E11 12 Im( E ) ω = · arctan( 12 ) (19) π Re( −E11 ) E12 where ω is the time lead for thermal admittance, [h].

2.2.5. Decrement Factor or Amplitude Attenuation (δ) The decrement factor of the wall structure is the difference between the outside and inside wall surface temperature swings; it is also called “amplitude attenuation”. It is known that the higher the thermal mass, the lower δ becomes. The decrement factor can be calculated using Equation (20):

1 δ = − (20) U · E12 where δ is the decrement factor or amplitude attenuation of the wall structure, [-].

2.2.6. Surface Factor (F) The surface factor is the ratio of shortwave radiant heat ﬂow re-admitted to the atmosphere from the external wall surface to the heat ﬂow striking upon the external wall surface. It can be calculated using Equation (21):

−E11 F = 1 − Rint · (21) E12 where F is the surface factor, [-].

2.2.7. Thermal Capacity (η) The thermal capacity is the amount of heat stored within the wall surface per unit area of building element per unit degree of atmospheric temperature swing. This heat stored in the wall surface is released to the atmosphere when the outside temperature falls during the nighttime. It can be calculated using Equation (22):

t E22 − 1 η = (22) 2π E12 where η is the thermal capacity, [J/m2K]; and t is the time period, [s]. Sustainability 2018, 10, x FOR PEER REVIEW 6 of 13

t E −1 η = 22 π (22) 2 E12

where η is the thermal capacity, [J/m2K]; and t is the time period, [s].

3. Results and Discussion In order to evaluate the impact of insulation type and thickness on the dynamic thermal Sustainabilitycharacteristics2018, 10of, 2835the external wall structure, seven thermal characteristics, including thermal6 of 14 transmittance (U), decrement factor (δ), time lag (φ), thermal admittance (Y), time lead for admittance (ω), surface factor (F), and thermal capacity (η) were evaluated by the above numerical 3. Results and Discussion methods, while using one of three types of insulation material (EPS, glass wool, and wood cement board)In and order varying to evaluate the insulation the impact thickness of insulation of the external type and wall thickness structure onfrom the 0 dynamicm to 0.2 m thermal with a characteristicsthickness interval of of the 0.01 external m, under wall the structure, MATrix LABoratory seven thermal (MATLAB) characteristics, environment. including The MATLAB thermal transmittancecodes for calculating (U), decrement the seven factor thermal (δ), timecharac lagteristics (ϕ), thermal are shown admittance in Appendixes (Y), time A lead and for B. admittance (ω), surface factor (F), and thermal capacity (η) were evaluated by the above numerical methods, while using3.1. Impact one of on three Thermal types Transmittance of insulation (U) material (EPS, glass wool, and wood cement board) and varying the insulation thickness of the external wall structure from 0 m to 0.2 m with a thickness interval of Figure 2 shows the impact of insulation type and thickness on the thermal transmittance (U) of 0.01 m, under the MATrix LABoratory (MATLAB) environment. The MATLAB codes for calculating the external wall structure. It was found that the thermal transmittance decreased for all three types the seven thermal characteristics are shown in AppendixsA andB. of insulation material, as insulation thickness was increased from 0 m to 0.2 m. If EPS was used as 3.1.the insulation Impact on Thermal material Transmittance in the wall structure, (U) the thermal transmittance of the external wall structure decreased by 0.67 W/m2K (81.7%); it also decreased by 0.63 W/m2K (76.4%) if glass wool was used, and Figuredecreased2 shows by the0.42 impact W/m2K of (50.8%) insulation if typewood and cement thickness board on thewas thermal used, while transmittance increasing ( U )the of theinsulation external thickness wall structure. from 0 Itm was to 0.2 found m. As that the the insu thermallation thickness transmittance increases decreased from for0 m all to three0.08 m, types the ofthermal insulation transmittance material, of as the insulation external thickness wall struct wasure increased decreased from more 0 sharply m to 0.2 when m. If the EPS EPS was and used glass as thewool insulation were used material as insula intion the wallmaterials, structure, compared the thermal with the transmittance condition where of the the external wood wallcement structure board 2 2 decreasedwas used as by insulation 0.67 W/m material.K (81.7%); As itthe also insulation decreased thickness by 0.63 increases W/m K (76.4%)from 0.08 if glassm to 0.2 wool m, wasthere used, was 2 anda slight decreased decrease by 0.42of the W/m thermalK (50.8%) transmittance if wood cement for the board three was types used, of while insulation increasing materials, the insulation which thicknessshowed a fromdecrease 0 m of to 0.15 0.2 m.W/m As2K the (49.0%) insulation for EPS, thickness a decrease increases of 0.16 from W/m 02 mK (45.9%) to 0.08 m,for the glass thermal wool, transmittanceand a decrease of of the 0.18 external W/m2K wall(30.5%) structure for wood decreased cement moreboard. sharply when the EPS and glass wool wereThe used result as insulation indicates materials, that the impact compared of insulation with the conditionthickness whereon the thethermal wood transmittance cement board of was an usedexternal as insulationwall structure material. is different As the insulation owing to thickness the different increases types from of the 0.08 insulation m to 0.2 m, materials. there was In a slightaddition, decrease it was of found the thermal that there transmittance is a critical for thickness the three for types the thermal of insulation insulation materials, of the which external showed wall 2 2 astructure, decrease and of 0.15 there W/m is a Ktrend (49.0%) of diminishing for EPS, a decrease returns ofas 0.16more W/m insulationK (45.9%) is used for above glass wool, this critical and a 2 decreasethickness. of This 0.18 has W/m beenK verified (30.5%) forin previous wood cement research board. [20,21].

FigureFigure 2. ImpactImpact of of insulation insulation type type and and thickness thickness on onthe thethermal thermal transmittance transmittance of the of external the external wall wallstructure. structure.

The result indicates that the impact of insulation thickness on the thermal transmittance of an external wall structure is different owing to the different types of the insulation materials. In addition, it was found that there is a critical thickness for the thermal insulation of the external wall structure, and there is a trend of diminishing returns as more insulation is used above this critical thickness. This has been veriﬁed in previous research [20,21].

3.2. Impact on Decrement Factor or Amplitude Attenuation (δ) Figure3 shows the impact of insulation type and thickness on the decrement factor or amplitude attenuation (δ) of the external wall structure. It was found that the decrement factor of the external wall structure decreased for all of the three types of insulation materials, as insulation thickness increased Sustainability 2018, 10, x FOR PEER REVIEW 7 of 13

3.2. Impact on Decrement Factor or Amplitude Attenuation (δ) Figure 3 shows the impact of insulation type and thickness on the decrement factor or amplitude attenuation (δ) of the external wall structure. It was found that the decrement factor of the external wall structure decreased for all of the three types of insulation materials, as insulation thickness

Sustainabilityincreased from2018, 10 0, xm FOR to PEER0.2 m. REVIEW If EPS was used as an insulation material in the wall structure,7 of the 13 decrement factor of the external wall structure decreased by 4.45 × 10−4 (93.8%); it decreased by 3.14 × 10−4 3.2.(66.2%) Impact if glasson Decrement wool was Fact used,or or and Amplitude it decreased Attenuation by 4.74 (δ )× 10−4 (about 100%) if wood cement board Sustainability 2018, 10, 2835 7 of 14 was used, while increasing the insulation thickness from 0 m to 0.2 m. Among the three types of Figure 3 shows the impact of insulation type and thickness on the decrement factor or amplitude insulation materials, the wood cement board had the sharpest decrease in decrement factor (as the attenuation (δ) of the external wall structure. It was found that the decrement factor of the external frominsulation 0 m to thickness 0.2 m. If EPSincreased was used to 0. as06 anm, insulation the decrement material factor in thefell wall to nearly structure, 0), followed the decrement by EPS. factor The wall structure decreased for all of the three types of insulation materials, as insulation thickness ofglass the wool external had wall the smallest structure decrease decreased in bydecrement 4.45 × 10 factor.−4 (93.8%); it decreased by 3.14 × 10−4 (66.2%) if increased from 0 m to 0.2 m. If EPS was used as an insulation material in the wall structure, the glassThe wool result was indicates used, and that it decreased the decrement by 4.74 factor× 10 of−4 the(about wall 100%)structure if wood could cement possibly board be reduced was used, by decrement factor of the external wall structure decreased by 4.45 × 10−4 (93.8%); it decreased by 3.14 × 10−4 whileincreasing increasing the insulation the insulation thickness thickness of the wall. from The 0 m degree to 0.2 of m. decrease Among in the the three decrement types offactor insulation varied (66.2%) if glass wool was used, and it decreased by 4.74 × 10−4 (about 100%) if wood cement board materials,depending the on woodthe different cement types board of had insulation the sharpest materi decreaseal. However, in decrement whether factorit is positive (as the or insulation negative was used, while increasing the insulation thickness from 0 m to 0.2 m. Among the three types of thicknesseffect depends increased on the to insulation 0.06 m, the material decrement type and factor the fell setting to nearly position 0), followedof insulation by EPS. in the The wall glass structure wool insulation materials, the wood cement board had the sharpest decrease in decrement factor (as the had(i.e., theinternal smallest thermal decrease insulation, in decrement external factor.thermal insulation, middle thermal insulation) [5]. insulation thickness increased to 0.06 m, the decrement factor fell to nearly 0), followed by EPS. The glass wool had the smallest decrease in decrement factor. The result indicates that the decrement factor of the wall structure could possibly be reduced by increasing the insulation thickness of the wall. The degree of decrease in the decrement factor varied depending on the different types of insulation material. However, whether it is positive or negative effect depends on the insulation material type and the setting position of insulation in the wall structure (i.e., internal thermal insulation, external thermal insulation, middle thermal insulation) [5].

Figure 3. Impact of insulation type and thickness on the decrement factor or amplitude attenuation Figure 3. Impact of insulation type and thickness on the decrement factor or amplitude attenuation of of the external wall structure. the external wall structure. 3.3. Impact on Time Lag (φ) The result indicates that the decrement factor of the wall structure could possibly be reduced by increasingFigure 4 shows the insulation the impact thickness of insulation of the type wall. and The thickness degree of on decrease the time in lag the (φ decrement) of external factor wall variedstructures. depending It was onfound the differentthat the typeschange of trend insulation of the material. time lag However, for external whether wall structures it is positive with or negativedifferentFigure effecttypes 3. Impact depends of insulation of insulation on the materials insulation type and were material thickness differ type enton thewhile and decrement the increasing setting factor position insulation or amplitude of insulation thickness attenuation in from the wall0 m structureto 0.2of m. the The (i.e., external change internal wall period thermalstructure. of insulation,the time lag external with increasing thermal insulation,insulation thickness middle thermal for the insulation) wood cement [5]. board was the shortest (0.04 m per change period), followed by EPS (0.15 m per change period). The 3.3.glass Impact wool onhad Time the Laglongest ((ϕφ) change period of time lag (0.16 m per half-change period). Figure4 4 shows shows thethe impactimpact ofof insulationinsulation typetype andand thicknessthickness onon thethe timetime lag lag ( (ϕφ) ofof externalexternal wallwall structures. ItIt was was found found that that the the change change trend trend of the of time the lagtime for lag external for external wall structures wall structures with different with typesdifferent of insulationtypes of insulation materials materials were different were differ whileent increasing while increasing insulation insulation thickness thickness from 0 m from to 0.2 0 m.m Theto 0.2 change m. The period change of period the time of lag the with time increasing lag with increasing insulation insulation thicknessfor thickness the wood forcement the wood board cement was theboard shortest was the (0.04 shortest m per (0.04 change m per period), change followed period), by followed EPS (0.15 by m EPS per (0.15 change m per period). change The period). glass wool The hadglass the wool longest had the change longest period change of time period lag (0.16of time m perlag (0.16 half-change m per half-change period). period).

Figure 4. Impact of insulation type and thickness on the time lag of the external wall structure.

Figure 4. Impact of insulation type and thickness on the time lag of the external wall structure. Figure 4. Impact of insulation type and thickness on the time lag of the external wall structure. Sustainability 2018, 10, x FOR PEER REVIEW 8 of 13 Sustainability 2018, 10, 2835 8 of 14 The result shows that there is a change period of time lag with as the insulation thickness increases, and it differs for different types of insulation materials. In addition, it is found that the insulationThe result types shows and thatthickness there isof a the change external period wall of timestructure lag with strongly as the affect insulation the time thickness lag of increases, the wall Sustainability 2018, 10, x FOR PEER REVIEW 8 of 13 andstructure. it differs Thus, for differentan appropriate types of selection insulation of materials.insulation In types addition, and thickness it is found becomes that the insulationmore important types and thickness of the external wall structure strongly affect the time lag of the wall structure. Thus, an in termsThe ofresult energy shows saving that in therebuildings. is a change period of time lag with as the insulation thickness appropriate selection of insulation types and thickness becomes more important in terms of energy increases, and it differs for different types of insulation materials. In addition, it is found that the saving3.4. Impact in buildings. on Thermal Admittance (Y) insulation types and thickness of the external wall structure strongly affect the time lag of the wall 3.4.structure. ImpactFigure Thus, on 5 Thermalshows an appropriate the Admittance impact ofselection (Y) insulation of insu typelation and thicknesstypes and on thickness the thermal becomes admittance more important (Y) of the inexternal terms ofwall energy structure. saving It in was buildings. found that there was almost no significant change in the thermal Y admittanceFigure5 for shows any of the the impact three oftypes insulation of insulation type and materials, thickness holding on the at thermalaround admittance18.4 W/m2K, ( while) of the3.4.increasing externalImpact oninsulation wall Thermal structure. thicknessAdmittance It was from (Y) found 0 m thatto 0.2 there m. was almost no signiﬁcant change in the thermal 2 admittanceThus, it for can any be ofconsidered the three that types the of insulation insulation type materials, and thickness holding will at around yield no 18.4 significant W/m K, change while increasingFigure insulation 5 shows the thickness impact fromof insulation 0 m to 0.2 ty m.pe and thickness on the thermal admittance (Y) of the externalin the thermal wall structure.admittance It towas the found external that wall there structure. was almost no significant change in the thermal admittance for any of the three types of insulation materials, holding at around 18.4 W/m2K, while increasing insulation thickness from 0 m to 0.2 m. Thus, it can be considered that the insulation type and thickness will yield no significant change in the thermal admittance to the external wall structure.

FigureFigure 5. 5. ImpactImpact of insulation of insulation type and type thickness and thickness on the th onermal the admittance thermal admittanceof the external of wall the structure. external wall structure. 3.5. Impact on Time Lead for Admittance (ω)

Thus,Figure it 6 can shows be considered the impact that of insulation the insulation type typeand andthickness thickness on the will time yield lead no for signiﬁcant admittance change (ω) inof thetheFigure thermalexternal 5. Impact admittance wall of structure. insulation to the typeIt externalwas and found thickness wall that structure. on th theere th ermalwas almost admittance no ofsignificant the external change wall structure. in the time lead for admittance for all three types of insulation materials, holding at around 2.45 h, while 3.5.3.5.increasing Impact oninsulation Time LeadLead thickness forfor AdmittanceAdmittance from 0 ( (ωmω)) to 0.2 m. FigureFigure6 6 shows shows the the impact impact of of insulation insulation type type and and thickness thickness on on the the time time lead lead for for admittance admittance ( ω )(ω of) theof the external external wall wall structure. structure. It was It was found found that that there th wasere was almost almost no signiﬁcant no significant change change in the in time the leadtime forlead admittance for admittance for all for three all types three of types insulation of insulati materials,on materials, holding atholding around at 2.45 around h, while 2.45 increasing h, while insulationincreasing thicknessinsulation from thickness 0 m to from 0.2 m. 0 m to 0.2 m.

Figure 6. Impact of insulation type and thickness on the time lead for thermal admittance of the external wall structure.

Thus, it is considered that the insulation types and thickness will yield no significant change in the time lead for admittance to the external wall structure. Figure 6. Impact of insulation type and thickness on the time lead for thermal admittance of the Figure 6. Impact of insulation type and thickness on the time lead for thermal admittance of the external wall structure. external wall structure.

Thus, it is considered that the insulation types and thickness will yield no significant change in the time lead for admittance to the external wall structure.

Sustainability 2018, 10, 2835 9 of 14

Thus, it is considered that the insulation types and thickness will yield no signiﬁcant change in theSustainability time lead 2018 for, 10 admittance, x FOR PEER toREVIEW the external wall structure. 9 of 13 Sustainability 2018, 10, x FOR PEER REVIEW 9 of 13 3.6.3.6. Impact on Surface Factor (F) 3.6. Impact on Surface Factor (F) FigureFigure7 7shows shows the the impact impact of insulationof insulation type ty andpe thicknessand thickness on the on surface the surface factor ( Ffactor) of the (F external) of the wallexternal structure.Figure wall 7 structure.shows It was foundthe It impactwas that found there of insulationthat was there almost wasty nope almost signiﬁcantand thickness no significant change on inthe change the surface surface in thefactor factor surface ( (FF)) of forfactor the all threeexternal(F) for types all wall three of structure. insulation types of It materials, wasinsulation found holding thatmaterials, there at around washolding almost 1.38, at whilenoaround significant increasing 1.38, whilechange insulation increasing in the thickness surface insulation factor from 0(thicknessF m) for to 0.2all fromm.three 0 typesm to 0.2of m.insulation materials, holding at around 1.38, while increasing insulation thickness from 0 m to 0.2 m.

Figure 7. Impact of insulation type and thickness on the surface factor of the external wall structure. Figure 7. Impact of insulation type and thickness on the surface factor of the external wall structure. Figure 7. Impact of insulation type and thickness on the surface factor of the external wall structure. Thus, it is considered that the insulation type and thickness will yield no significant change in the surfaceThus, it itfactor isis consideredconsidered to the external that that the the wall insulation insulation structure. type type and and thickness thickness will will yield yield no no significant significant change change in the in surfacethe surface factor factor to the to externalthe external wall wall structure. structure. 3.7. Impact on Thermal Capacity (η) 3.7. Impact on Thermal Capacity (η) Figure 8 shows the impact of insulation type and thickness on the thermal capacity (η) of the Figure8 shows the impact of insulation type and thickness on the thermal capacity ( η) of the externalFigure wall 8 showsstructure. the Itimpact was found of insulation that there ty pewa ands almost thickness no significant on the thermal difference capacity in the (η thermal) of the external wall structure. It was found that there was almost no significant difference in the thermal externalcapacity wallfor EPSstructure. and glass It was wool, found holding that there at around was almost 4.5 × no10 4significant J/m2K, while difference increasing in the insulation thermal capacity for EPS and glass wool, holding at around 4.5 × 104 J/m2K, while increasing insulation capacitythickness forfrom EPS 0 m and to 0.2 glass m. As wool, the insulation holding thicknessat around increased 4.5 × 10 from4 J/m 02K, m towhile 0.2 m, increasing the thermal insulation capacity thickness from 0 m to 0.2 m. As the insulation thickness increased from 0 m to 0.2 m, the thermal thicknessfor wood fromcement 0 m board to 0.2 also m. As had the almost insulation no si gnificantthickness change,increased holding from 0 at m around to 0.2 m, 4.6 the × 10thermal4 J/m2K, capacity which capacity for wood cement board also had almost no significant change, holding at around 4.6 × 104 foris higher wood thancement that board of EPS also and had glass almost wool. no siThisgnificant is because change, both holding the thermal at around conductivity 4.6 × 104 J/m and2K, density which J/m2K, which is higher than that of EPS and glass wool. This is because both the thermal conductivity isof higherthe wood than cement that of board EPS and were glass larger wool. than This for isEPS because and glass both wool. the thermal conductivity and density andof the density wood ofcement the wood board cement were larger board werethan for larger EPS than and forglass EPS wool. and glass wool.

Figure 8. Impact of insulation type and thickness on the thermal capacity of the external wall structure. Figure 8. Impact of insulation type and thickness on the thermal capacity of the external wall structure. Figure 8. Impact of insulation type and thickness on the thermal capacity of the external wall structure. Thus, it is indicated that the insulation type and thickness will yield no significant change in the thermalThus, capacity it is indicated to the thatexternal the insulationwall structure, type an sinced thickness the thermophysical will yield no significantproperties change(i.e., thermal in the Thus, it is indicated that the insulation type and thickness will yield no signiﬁcant change in thermalconductivity capacity and todensity) the external are far wall different structure, from sincethat ofthe the thermophysical other materials properties (i.e., concrete, (i.e., thermalplaster, the thermal capacity to the external wall structure, since the thermophysical properties (i.e., thermal conductivitysteel) in the external and density) wall structure. are far different from that of the other materials (i.e., concrete, plaster, steel) in the external wall structure. 4. Conclusions 4. Conclusions In order to find the impact of insulation type and thickness on the dynamic thermal characteristicsIn order toof thefind external the impact wall ofstructure, insulation a homogeneous type and thickness multi-layer on buildingthe dynamic external thermal wall characteristics of the external wall structure, a homogeneous multi-layer building external wall Sustainability 2018, 10, 2835 10 of 14 conductivity and density) are far different from that of the other materials (i.e., concrete, plaster, steel) in the external wall structure.

4. Conclusions In order to ﬁnd the impact of insulation type and thickness on the dynamic thermal characteristics of the external wall structure, a homogeneous multi-layer building external wall structure widely used in Japan was selected, and seven thermal characteristics of the external wall, including U, δ, ϕ, Y, ω, F, and η, were investigated by changing the insulation thickness (from 0 m to 0.2 m) and varying the type of insulation material (EPS, glass wool, and wood cement board). The knowledge obtained is as follows:

• The U of the external wall structure varies depending on the different insulation type and thickness of the external wall structure. It was found that there is a critical thickness for the thermal insulation of the external wall structure, and the effect of thermal insulation will not be obvious when the thickness exceeds the critical value. • The δ of the wall structure could possibly be reduced by increasing the insulation thickness of the wall. The degree of decrease in the decrement factor varies depending on the different type of insulation material. • There is a change period of ϕ with the increasing insulation thickness, and it differs for different types of insulation material. The insulation type and thickness in the external wall structure strongly affects the ϕ of the wall structure. • The insulation type and thickness will yield no signiﬁcant change in the Y, ω, F, and η thermal characteristics of the external wall structure.

Future research will be focused on evaluating the impact of the dynamic thermal characteristics of more insulation types and different positions in more external wall structures.

Funding: This research received no external funding. Acknowledgments: The author is very grateful to Emura and Farnham of Osaka City University for their supports. Conﬂicts of Interest: The author declares no conﬂict of interest.

Appendix A The ﬁrst step for solving the matrix X = [E11 E12; E21 E22].

clear; P=3600; % Time periods [s] pi=3.1415926; drt=5.67*10-8; % Stefan-Boltzmann constant j=sqrt(-1); % Imaginary number

Fe=0.9; % Emissivity factor of wall surface which is equal to K*e [K=1; e=0.9] Tsi=15+273.15; % Internal surface absolute temperature [K] Tso=15+273.15; % External surface absolute temperature [K]

hi=9.; % For internal wall surface convective heat transfer coefﬁcient [W/m2K] ho=23.; % For external wall surface convective heat transfer coefﬁcient [W/m2K]

Rsint=1/hi; % Internal surface resistance [m2K/W] Rsext=1/ho; % External surface resistance [m2K/W] Sustainability 2018, 10, 2835 11 of 14

% Themophysical properties of building wall enclosure materials (from indoor to outdoor) kw=0.179; % Thermal conductivity of wood [W/mK] Dw=530; % Density of wood [kg/m3] Cw=2300; % Speciﬁc heat of wood [J/kgK] xw=0.015; % Thickness of wood [m]

ke=0.16; % Thermal conductivity of Wood cement board [W/mK] De=680; % Density of Wood cement board [kg/m3] Ce=1675; % Speciﬁc heat of Wood cement board [J/kgK] xe=0.06; % Thickness of Wood cement board [m]

ka=0.026; % Thermal conductivity of air layer [W/mK] Da=1.2; % Density of air layer [kg/m3] Ca=1006; % Speciﬁc heat of air layer [J/kgK] xa=0.02; % Thickness of air layer [m]

kc=0.9; % Thermal conductivity of concrete [W/mK] Dc=2000; % Density of concrete [kg/m3] Cc=879; % Speciﬁc heat of concrete [J/kgK] xc=0.12; % Thickness of concrete [m]

kp=0.43; % Thermal conductivity of plaster [W/mK] Dp=1250; % Density of plaster [kg/m3] Cp=1050; % Speciﬁc heat of plaster [J/kgK] xp=0.03; % Thickness of plaster [m]

% The homogeneous wall enclosure transmission matrix coefﬁcients % For wood material uw=sqrt(pi*Dw*Cw*xw*xw/(kw*P)); aw=sqrt(j*2*pi*kw*Dw*Cw/P);

Awo=cosh(uw)*cos(uw); Awt=sinh(uw)*sin(uw); Aws=(cosh(uw)*sin(uw)+sinh(uw)*cos(uw))/sqrt(2); Awf=(cosh(uw)*sin(uw)-sinh(uw)*cos(uw))/sqrt(2);

% For Wood cement board material ue=sqrt(pi*De*Ce*xe*xe/(ke*P)); ae=sqrt(j*2*pi*ke*De*Ce/P);

Aeo=cosh(ue)*cos(ue); Aet=sinh(ue)*sin(ue); Aes=(cosh(ue)*sin(ue)+sinh(ue)*cos(ue))/sqrt(2); Aef=(cosh(ue)*sin(ue)-sinh(ue)*cos(ue))/sqrt(2);

% For air layer ua=sqrt(pi*Da*Ca*xa*xa/(ka*P)); aa=sqrt(j*2*pi*ka*Da*Ca/P);

Aao=cosh(ua)*cos(ua); Aat=sinh(ua)*sin(ua); Aas=(cosh(ua)*sin(ua)+sinh(ua)*cos(ua))/sqrt(2); Aaf=(cosh(ua)*sin(ua)-sinh(ua)*cos(ua))/sqrt(2); Sustainability 2018, 10, 2835 12 of 14

% For concrete material uc=sqrt(pi*Dc*Cc*xc*xc/(kc*P)); ac=sqrt(j*2*pi*kc*Dc*Cc/P);

Aco=cosh(uc)*cos(uc); Act=sinh(uc)*sin(uc); Acs=(cosh(uc)*sin(uc)+sinh(uc)*cos(uc))/sqrt(2); Acf=(cosh(uc)*sin(uc)-sinh(uc)*cos(uc))/sqrt(2);

% For plaster material up=sqrt(pi*Dp*Cp*xp*xp/(kp*P)); ap=sqrt(j*2*pi*kp*Dp*Cp/P);

Apo=cosh(up)*cos(up); Apt=sinh(up)*sin(up); Aps=(cosh(up)*sin(up)+sinh(up)*cos(up))/sqrt(2); Apf=(cosh(up)*sin(up)-sinh(up)*cos(up))/sqrt(2);

% Solve matrix X=[E11 E12; E21 E22] A=[1 -Rsint; 0 1]; % For internal layer B=[Awo+Awt*j (Aws+Awf*j)/aw; (-Awf+Aws*j)*aw Awo+Awt*j]; % For 1st wood layer C=[Aeo+Aet*j (Aes+Aef*j)/ae; (-Aef+Aes*j)*ae Aeo+Aet*j]; % For 2nd Wood cement board layer D=[Aao+Aat*j (Aas+Aaf*j)/aa; (-Aaf+Aas*j)*aa Aao+Aat*j]; % For 3th air layer E=[Aco+Act*j (Acs+Acf*j)/ac; (-Acf+Acs*j)*ac Aco+Act*j]; % For 4rd concrete layer F=[Apo+Apt*j (Aps+Apf*j)/ap; (-Apf+Aps*j)*ap Apo+Apt*j]; % For 5th plaster layer G=[1 -Rsext; 0 1]; % For external layer

X=A*B*C*D*E*F*G ——

Appendix B The second step for calculating different thermal characteristics of external wall structure using the matrix value obtained in AppendixA.

clear; drt=5.67*10-8; % Stefan-Boltzmann constant pi=3.1415926; i=sqrt(-1);

hi=9.; % For internal wall surface convective heat transfer coefﬁcient [W/m2K] ho=23.; % For external wall surface convective heat transfer coefﬁcient [W/m2K]

Rsint=1/hi; % Internal surface resistance [m2K/W] Rsext=1/ho; % External surface resistance [m2K/W]

ka=0.026; % Thermal conductivity of air layer [W/mK] Da=1.2; % Density of air layer [kg/m3] Ca=1006; % Speciﬁc heat of air layer [J/kgK] xa=0.02; % Thickness of air layer [m]

kc=0.9; % Thermal conductivity of concrete [W/mK] Dc=2000; % Density of concrete [kg/m3] Cc=879; % Speciﬁc heat of concrete [J/kgK] xc=0.12; % Thickness of concrete [m]

kp=0.43; % Thermal conductivity of plaster [W/mK] Dp=1250; % Density of plaster [kg/m3] Cp=1050; % Speciﬁc heat of plaster [J/kgK] xp=0.03; % Thickness of plaster [m]

E11= 1.0e+07 *(0.3913 − 0.2623i); E12= 1.0e+07 *(-0.0255 − 0.0014i); E21= 1.0e+07 *(-2.0146 + 3.2741i); E22= 1.0e+07 *(0.1943 − 0.0766i);

% Thermal transmittance (U) [W/m2K] U=1/(Rsext+(xw/kw)+(xe/ke)+(xa/ka)+(xc/kc)+(xp/kp)+Rsint)

% Decrement factor or attenuation factor or amplitude attenuation (d) [-] d=abs(-1/(U*E12))

% Decrement delay or time lag or phase shift (tlag) [h] tlag=(12/pi)*atan(imag(-1/(U*E12))/real(-1/(U*E12)))

% Thermal admittance (Y) [W/m2K] Y=abs(-E11/E12)

% Time lead for thermal admittance (w) [h] w=(12/pi)*atan(imag(-E11/E12)/real(-E11/E12))

% Surface factor (F) [-] F=abs(1-Rsint*(-E11/E12))

% Associated time delay of surface factor (stlag) [h] stlag=(12/pi)*atan(imag(1-Rsint*(-E11/E12))/real(1-Rsint*(-E11/E12)))

% Thermal heat capcity (Cap) [J/m2K] T=3600; % Times [s] Cap=(T/2*pi)*abs((E22-1)/E12)

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