# Semianalytical Solution of Transient Heat Transfer for Laminated Structures under Time-Varying Boundary Conditions

Authors: Xi-Xia Li; Li Wang; Chang-Yu Li; Yi-Yu Wan; Yu-Chen Qian
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In order to solve the transient heat transfer problem of the laminated structure, a semianalytical method based on calculus is adopted. First, the time domain is divided into tiny time segments; the analytical solution of transient heat transfer of laminated structures in the segments is derived by using the method of separation of variables. Then, the semianalytical solution of transient heat transfer in the whole time domain is obtained by circulation. The transient heat transfer of the three-layer structure is analyzed by the semianalytical solution. Three time-varying boundary conditions(a: square wave, b: triangular wave, and c: sinusoidal wave) are applied to the surface of the laminated structure. The influence of some key parameters on the temperature field of the laminated structure is analyzed. It is found that the surface temperature of the laminated structure increases fastest when heated by square wave, and the maximum temperature can reach at 377°C, the temperature rises the most slowly when heated by the triangular wave, and the maximum temperature is 347°C. The novelty of this work is that the analytical method is used to analyze the nonlinear heat transfer problem, which is different from the general numerical method, and this method can be applied to solve the heat transfer problem of general laminated structures.


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## Body

## 1. Introduction

In practical engineering applications, there are many problems of transient heat transfer in laminated structures [1, 2]. For example, transient heat transfer of human skin during hyperthermia, transient heat transfer of multilayer insulation materials, transient heat transfer of thermal barrier coating, or new thermoelectric materials [3–7]. More and more attention has been paid to transient heat transfer in laminated structures. The relevant research studies can be summarized as follows.Xu et al. established a three-layer heat transfer model aiming at the skin tissue heat transfer problem by means of the finite difference method [8]. Liu et al. solved the temperature field of the three-layer heat transfer structure using the inverse Fourier transform method [9]. Campo and Arıcı exploited a radically different approach applying the Method of Lines (MOL) forthwith to the one-dimensional, unsteady, and heat conduction equation with prescribed uniform heat flux [10]. Li and Lin used the method of separation of variables to obtain the semianalytical solution of heat transfer in a laminated structure [11]. Yu et al. studied the heat transfer of multilayer structures by using the smooth particle manifold (SPH) method [12]. Salehi et al. investigated flow and heat transfer of magnetohydrodynamic squeezing nanofluid flow between two infinity parallel plates by Akbari-Ganji’s method [13]. Fu and Cheng solved the semianalytical solution of temperature field analysis of the three-dimensional structure of the laminated cylinder by using the fine integration method [14]. Ramadan solved the transient heat transfer problem of the multilayer structure by the semianalytical method and analyzed the non-Fourier characteristics of the heat transfer process [15]. Li et al. studied transient heat transfer in laminated structures by using a backward differential method [16]. Liu and Shi used the lattice Boltzmann method to solve the numerical solution of heat transfer in multilayer structures [17]. Shan et al. studied the heat transfer of a multilayer structure using a modified scale-boundary finite element method [18]. Guo et al. simulated the transient heat transfer problem of the three-layer structure by using the improved time-domain discontinuous Galerkin method and improved the numerical fluctuation problem of the traditional method [19].From the literature analysis, it can be seen that, for the heat transfer of laminated structures, numerical methods are mostly used, and the boundary conditions generally do not change with time. The transient heat transfer of multilayer materials is a nonlinear problem because the thermophysical parameters of each layer are different; in addition, the boundary conditions change with time, and it is difficult to obtain the analytical solution of the problem directly [20–23]. In this paper, the boundary conditions are differentiated in the time domain, and a special method of separating variables is used to obtain the analytical solution of the temperature field of the laminated structure in a small time period, and then, the cycle is carried out to obtain the analytical solution of the temperature field in the whole time domain. Then, the temperature field of the three-layer structure heated by time-varying heat flow is analyzed by using the method presented in this paper. The temperature field of the structure under three heating conditions (square wave, triangular wave, and sine wave) has been obtained, and the influence of surface convective heat transfer coefficient, heating frequency, and other parameters on the temperature field distribution was studied.


## 2. Mathematical Model

As shown in Figure1, the number of layers of the laminated structure is n, and the heat flow of f1(t) is heated on the left side of the laminated structure. At the same time of heating, the surface of the laminated structure is cooled by fluid. The temperature on the right side of the laminated structure is fn(t). The thermophysical parameters of each layer of the laminated structure are consistent, but they vary from layer to layer.Figure 1
Schematic diagram of the model.A mathematical and physical model can be established based on the problem studied, and the corresponding governing equation can be expressed as [24](1)ρici∂Tix,t∂t−ki∂2Tix,t∂x2+hiTix,t−Tbas−Qit=0,where Qit is the heat generation and hiTix,t−Tbas is the interacts with the environment by convective heat transfer from upper and lower surfaces. In view of some special conditions, there is a heat transfer phenomenon in the laminated material through a certain medium. Here, hi is defined as the internal heat transfer coefficient of ith layer.The boundary conditions are as follows.Whenx = 0,(2)k1∂T10,t∂x−h0T10,t=−h0T∞−f1t.Whenx=ln,(3)Tnln,t=fnt.The initial condition is(4)Tinax,0=Tbas.After dimensionless, the governing equation, boundary condition, and initial conditions can be expressed as follows:(5)rρirci∂θi∂τ−rki∂2θi∂ξ2+Nuiθi−Q¯it=0.The boundary condition of dimensionless is as follows.Whenξ=0,(6)rk1∂θ10,τ∂τ−Nu0θ10,τ=−h¯0θ∞−f^1τ=f¯1τ.Whenξ=1, in other words, at the left end of the laminated structure, the temperature changes according to f¯nτ, where f¯nτ=fnt−Tbas/Tbas:(7)θn1,τ=f¯nτ.The initial conditions for dimensionless is(8)θi,inaξ,0=0.


## 3. Solution Method

### 3.1. Differential in Time Domain

In general, for the heat transfer problem of the laminated structure under time-varying boundary conditions, the analytical solution cannot be obtained directly by the separation of variables method because the thermophysical parameters of each layer of the structure are not consistent and the boundary conditions change with time. In order to solve the transient heat transfer problem of the laminated structure with time-varying boundary conditions, it is necessary to carry out differentiation in the time domain. Suppose there is a sufficiently small time intervalΔτ, according to the idea of calculus, in the time domain of [0 Δτ], the following hypothesis is true.Whenξ=0,(9)f¯1τ≈f¯1Δτ−f¯10Δττ=f¯1,1τ.Whenξ=1,(10)f¯nΔτ≈f¯nΔτ−f¯n0Δτ=f¯n,1τ.Whenξ=li/ln=ξi (i = 1,2,3, …, n),(11)θi,i+1τ=Ki,i+1⋅τ,where f¯1,1τ is the differential approximation function of the left boundary condition of the laminated structure in [0 Δτ]: θi,i+1τ is the function of temperature change at the junction of layer i and layer i + 1 of the laminated structure and Km,m+1 is the slope of the function θm,m+1τ, which is an unknown constant. At this point, boundary conditions on both sides of each layer of the inner layered structure at time [0 Δτ] are known. Therefore, the method of separation of variables can be used to solve the nonlinear problem.


### 3.2. The First Analytic Solution in Time [0Δτ]

For the first layer in the first [0Δτ] time domain, the governing equation is as follows:(12)rρ1rc1∂θ1∂τ−rk1∂2θ1∂ξ2+Nu1θ1−Q¯1τ=0.The corresponding boundary conditions are as follows.Whenξ=0,(13)rk1∂θ10,τ∂τ−Nu0θ10,τ=f¯1,1τ.Whenξ=l1/ln=ξ1,(14)θ1,2ξ1,τ=K1,2⋅τ.The initial condition is(15)θ1,inaξ,0=0.Through analysis, it can be known that the boundary conditions of governing equation are inhomogeneous. In order to homogeneous the boundary conditions, equations (16) and (17) are assumed [25]:(16)θ1ξ,τ=ϕ1ξ,τ+ψ1ξ,τ,(17)ψ1ξ,τ=aξ+bf¯1,1τ+cξ+dK1,2⋅τ.Under homogeneous boundary conditions,ψ1ξ,τ needs to satisfy equations (18) and (19):(18)rk1∂ψ10,τ∂ξ−Nu0ψ10,τ=f¯1,1τ,(19)ψ1ξ1,τ=K1,2⋅τ.By substituting equation (17) into equations (18) and (19), then a, b, c, and d in equation (17) can be derived:(20)a=1rk1+Nu0ξ1,b=−ξ1rk1+Nu0ξ1,c=Nu0rk1+Nu0ξ1,d=rk1rk1+Nu0ξ1.By substituting equation (16) into equation (12), equation (21) can be obtained:(21)rρ1rc1∂ϕ1∂τ=rk1∂2ϕ1∂ξ2+Nu1ϕ1+rk1∂2φ1∂ξ2−rρ1rc1∂φ1∂τ+Nu1φ1−Q¯1τ.The characteristic equation of equation (21) is(22)rρ1rc1∂ϕ1∂τ=rk1∂2ϕ1∂ξ2.In order to separate variables, let(23)ϕξ,τ=Y1ξG1τ.By substituting (23) into (22), equations (24) and (25) can be obtained:(24)d2Y1dξ2+λG1=0,(25)dG1dτ+λrk1rρ1rc1Y1=0.It can be seen from equations (13), (14), (18), and (19) that(26)rk1∂ϕ10,τ∂τ−Nu0ϕ10,τ=0,(27)φ1,2ξ1,τ=0.Substituting equation (23) into equations (26) and (27), we can obtain(28)rk1dY10dξ−Nu0Y10=0,(29)Y1ξ1=0.By combining equations (24), (28), and (29), which is a second-order homogeneous ordinary differential equation, its characteristic function is(30)Ymξ=cosλmξ1ξ+Nu0rk1λmsinλmξ1ξ,where λm is the positive solution of(31)rk1λNu0+tanλξ1=0.Equation (32) is established by calculation:(32)∫0ξ1Y1,mξY1,kξdξ=0,form≠k,δ1,m,form=k.Equation (32) expresses that Ymξ satisfies the orthogonality property. In order to separate variables, let(33)ϕ1ξ,τ=∑m=1∞Y1,mξB1,mτ.By substituting equation (33) into equation (21) and integrating within [0, ξ1], equation (34) can be obtained:(34)dB1,mdτ+rk1λm+Nu1rρ1rc1B1,m=1rρ1rc11δ1,m∫01Y1,mrk1∂2φ1∂ξ2−rρ1rc1∂φ1∂τ+Nu1φ1−Q¯1τdξ.Equation (34) is a first-order ordinary differential equation with respect to B1,mξ, and its solution is [26](35)B1,mτ=e−α1τB1,m0+e−α1τ∫0τeα1τqts1dτ,where(36)α1=rk1λ1,m+Nu1rρ1rc1,(37)qts1=1rρ1rc11δ1,m∫01Y1,mrk1∂2φ1∂ξ2−rρ1rc1∂φ1∂τ+Nu1φ1−Q¯1τdξ,(38)B1,m0=−1δ1,m∫01φ1ξ,0Y1,mξdξ.Combined with the above derivation process, it can be seen that the solution of the first layer of the laminated structure in the first differential period can be expressed by(39)θ1ξ,τ=∑m=1∞B1,mτY1,mξ+φ1ξ,τ.


### 3.3. Theith Layer Analytic Solution in Time [0 Δτ]

The difference between the mathematical and physical models of the second toith layers and the first layer is that the left side of the first layer is the time-varying heat flow boundary condition and the right side is the time-varying temperature boundary condition. The left and right sides of layers 2 to i are time-varying temperature boundary conditions. The analytical solution of the ith (i ≥ 2) layer in the first differential period can be expressed as equation (40) by the same method as that of the first layer:(40)θiξ,τ=∑m=1∞Bi,mτYi,mξ+φiξ,τ.The parameters in equation (40) are shown as follows:(41)φnξ,τ=−1ξi−1−1ξ+ξi−1ξi−1−1f¯iτ+−1−ξi−1+1ξ+1−ξi−1+1f¯i−1,iτ,(42)Yi,mξ=cosλmξsinλmξi−1−cosλmξi−1sinλiξ,where λm is the positive root of(43)cosλsinλξi−1−cosλξi−1sinλ=0,(44)Bi,mτ=e−αiτBi,m0+e−αiτ∫0τeαiτqtsi,mdτ,(45)qtsi=1rρirci1δi,m∫ξi−1ξiYi,mrki∂2φi∂ξ2−rρirci∂φi∂τ+Nuiφi−Q¯iτdξ,(46)δi,m=∫ξi−1ξiYi,mYi,kdξ=0,form≠k,14−2ξi−1+2ξi+sin2ξi−1−ξnλi,mλi,m,form=k,(47)Bi,m0=−1δi,m∫ξi−1ξiφiξ,0Yi,mξdξ.


### 3.4. The Solution of the Temperature Field in the Whole Time Domain

According to Sections3.1–3.3, the analytic solutions of each layer structure in the first [0 Δτ] time domain are derived. However, there is an unknown coefficient which included in the analytic solutions. According to the equal temperature at the junction and the equal heat flow out and in at the junction, equation (48) can be obtained:(48)∂θ10,τ∂ξ∂θ20,τ∂ξ∂θ20,τ∂ξ∂θ30,τ∂ξ⋯⋯∂θn−10,τ∂ξ∂θn0,τ∂ξrk1rk2⋯rkn=00⋯0.Ifτ=Δτ is substituted into equation (48), equation (49) can be obtained:(49)a1,2K1,2b1,2K1,2a2,3K2,3b2,3K2,3⋯⋯an−1,nKn−1,nbn−1,nKn−1,nrk1rk2⋯rkn=00⋯0.So,(50)ai−1,i=∂θi−10,Δτ∂ξ,(51)bi−1,n=∂θi0,Δτ∂ξ.Ki−1,i can be solved by equation (49). At this time, the analytic solution of the first [0 Δτ] time domain can be completely solved, and θiξ,Δτ can be solved. With Δτ at this time as the initial value of the next differential period, the semianalytic solution of each layer of the layered structure in the entire time domain can be solved by executing the loop.


## 3.1. Differential in Time Domain

In general, for the heat transfer problem of the laminated structure under time-varying boundary conditions, the analytical solution cannot be obtained directly by the separation of variables method because the thermophysical parameters of each layer of the structure are not consistent and the boundary conditions change with time. In order to solve the transient heat transfer problem of the laminated structure with time-varying boundary conditions, it is necessary to carry out differentiation in the time domain. Suppose there is a sufficiently small time intervalΔτ, according to the idea of calculus, in the time domain of [0 Δτ], the following hypothesis is true.Whenξ=0,(9)f¯1τ≈f¯1Δτ−f¯10Δττ=f¯1,1τ.Whenξ=1,(10)f¯nΔτ≈f¯nΔτ−f¯n0Δτ=f¯n,1τ.Whenξ=li/ln=ξi (i = 1,2,3, …, n),(11)θi,i+1τ=Ki,i+1⋅τ,where f¯1,1τ is the differential approximation function of the left boundary condition of the laminated structure in [0 Δτ]: θi,i+1τ is the function of temperature change at the junction of layer i and layer i + 1 of the laminated structure and Km,m+1 is the slope of the function θm,m+1τ, which is an unknown constant. At this point, boundary conditions on both sides of each layer of the inner layered structure at time [0 Δτ] are known. Therefore, the method of separation of variables can be used to solve the nonlinear problem.


## 3.2. The First Analytic Solution in Time [0Δτ]

For the first layer in the first [0Δτ] time domain, the governing equation is as follows:(12)rρ1rc1∂θ1∂τ−rk1∂2θ1∂ξ2+Nu1θ1−Q¯1τ=0.The corresponding boundary conditions are as follows.Whenξ=0,(13)rk1∂θ10,τ∂τ−Nu0θ10,τ=f¯1,1τ.Whenξ=l1/ln=ξ1,(14)θ1,2ξ1,τ=K1,2⋅τ.The initial condition is(15)θ1,inaξ,0=0.Through analysis, it can be known that the boundary conditions of governing equation are inhomogeneous. In order to homogeneous the boundary conditions, equations (16) and (17) are assumed [25]:(16)θ1ξ,τ=ϕ1ξ,τ+ψ1ξ,τ,(17)ψ1ξ,τ=aξ+bf¯1,1τ+cξ+dK1,2⋅τ.Under homogeneous boundary conditions,ψ1ξ,τ needs to satisfy equations (18) and (19):(18)rk1∂ψ10,τ∂ξ−Nu0ψ10,τ=f¯1,1τ,(19)ψ1ξ1,τ=K1,2⋅τ.By substituting equation (17) into equations (18) and (19), then a, b, c, and d in equation (17) can be derived:(20)a=1rk1+Nu0ξ1,b=−ξ1rk1+Nu0ξ1,c=Nu0rk1+Nu0ξ1,d=rk1rk1+Nu0ξ1.By substituting equation (16) into equation (12), equation (21) can be obtained:(21)rρ1rc1∂ϕ1∂τ=rk1∂2ϕ1∂ξ2+Nu1ϕ1+rk1∂2φ1∂ξ2−rρ1rc1∂φ1∂τ+Nu1φ1−Q¯1τ.The characteristic equation of equation (21) is(22)rρ1rc1∂ϕ1∂τ=rk1∂2ϕ1∂ξ2.In order to separate variables, let(23)ϕξ,τ=Y1ξG1τ.By substituting (23) into (22), equations (24) and (25) can be obtained:(24)d2Y1dξ2+λG1=0,(25)dG1dτ+λrk1rρ1rc1Y1=0.It can be seen from equations (13), (14), (18), and (19) that(26)rk1∂ϕ10,τ∂τ−Nu0ϕ10,τ=0,(27)φ1,2ξ1,τ=0.Substituting equation (23) into equations (26) and (27), we can obtain(28)rk1dY10dξ−Nu0Y10=0,(29)Y1ξ1=0.By combining equations (24), (28), and (29), which is a second-order homogeneous ordinary differential equation, its characteristic function is(30)Ymξ=cosλmξ1ξ+Nu0rk1λmsinλmξ1ξ,where λm is the positive solution of(31)rk1λNu0+tanλξ1=0.Equation (32) is established by calculation:(32)∫0ξ1Y1,mξY1,kξdξ=0,form≠k,δ1,m,form=k.Equation (32) expresses that Ymξ satisfies the orthogonality property. In order to separate variables, let(33)ϕ1ξ,τ=∑m=1∞Y1,mξB1,mτ.By substituting equation (33) into equation (21) and integrating within [0, ξ1], equation (34) can be obtained:(34)dB1,mdτ+rk1λm+Nu1rρ1rc1B1,m=1rρ1rc11δ1,m∫01Y1,mrk1∂2φ1∂ξ2−rρ1rc1∂φ1∂τ+Nu1φ1−Q¯1τdξ.Equation (34) is a first-order ordinary differential equation with respect to B1,mξ, and its solution is [26](35)B1,mτ=e−α1τB1,m0+e−α1τ∫0τeα1τqts1dτ,where(36)α1=rk1λ1,m+Nu1rρ1rc1,(37)qts1=1rρ1rc11δ1,m∫01Y1,mrk1∂2φ1∂ξ2−rρ1rc1∂φ1∂τ+Nu1φ1−Q¯1τdξ,(38)B1,m0=−1δ1,m∫01φ1ξ,0Y1,mξdξ.Combined with the above derivation process, it can be seen that the solution of the first layer of the laminated structure in the first differential period can be expressed by(39)θ1ξ,τ=∑m=1∞B1,mτY1,mξ+φ1ξ,τ.


## 3.3. Theith Layer Analytic Solution in Time [0 Δτ]

The difference between the mathematical and physical models of the second toith layers and the first layer is that the left side of the first layer is the time-varying heat flow boundary condition and the right side is the time-varying temperature boundary condition. The left and right sides of layers 2 to i are time-varying temperature boundary conditions. The analytical solution of the ith (i ≥ 2) layer in the first differential period can be expressed as equation (40) by the same method as that of the first layer:(40)θiξ,τ=∑m=1∞Bi,mτYi,mξ+φiξ,τ.The parameters in equation (40) are shown as follows:(41)φnξ,τ=−1ξi−1−1ξ+ξi−1ξi−1−1f¯iτ+−1−ξi−1+1ξ+1−ξi−1+1f¯i−1,iτ,(42)Yi,mξ=cosλmξsinλmξi−1−cosλmξi−1sinλiξ,where λm is the positive root of(43)cosλsinλξi−1−cosλξi−1sinλ=0,(44)Bi,mτ=e−αiτBi,m0+e−αiτ∫0τeαiτqtsi,mdτ,(45)qtsi=1rρirci1δi,m∫ξi−1ξiYi,mrki∂2φi∂ξ2−rρirci∂φi∂τ+Nuiφi−Q¯iτdξ,(46)δi,m=∫ξi−1ξiYi,mYi,kdξ=0,form≠k,14−2ξi−1+2ξi+sin2ξi−1−ξnλi,mλi,m,form=k,(47)Bi,m0=−1δi,m∫ξi−1ξiφiξ,0Yi,mξdξ.


## 3.4. The Solution of the Temperature Field in the Whole Time Domain

According to Sections3.1–3.3, the analytic solutions of each layer structure in the first [0 Δτ] time domain are derived. However, there is an unknown coefficient which included in the analytic solutions. According to the equal temperature at the junction and the equal heat flow out and in at the junction, equation (48) can be obtained:(48)∂θ10,τ∂ξ∂θ20,τ∂ξ∂θ20,τ∂ξ∂θ30,τ∂ξ⋯⋯∂θn−10,τ∂ξ∂θn0,τ∂ξrk1rk2⋯rkn=00⋯0.Ifτ=Δτ is substituted into equation (48), equation (49) can be obtained:(49)a1,2K1,2b1,2K1,2a2,3K2,3b2,3K2,3⋯⋯an−1,nKn−1,nbn−1,nKn−1,nrk1rk2⋯rkn=00⋯0.So,(50)ai−1,i=∂θi−10,Δτ∂ξ,(51)bi−1,n=∂θi0,Δτ∂ξ.Ki−1,i can be solved by equation (49). At this time, the analytic solution of the first [0 Δτ] time domain can be completely solved, and θiξ,Δτ can be solved. With Δτ at this time as the initial value of the next differential period, the semianalytic solution of each layer of the layered structure in the entire time domain can be solved by executing the loop.


## 4. Case Study and Discussion

### 4.1. Description of the Problem

In order to verify the correctness of the semianalytical solution derived in this paper, the semianalytical method has been used to analyze the temperature field variation of a three-layer structure under time-varying boundary conditions. The dimensions of the three-layer structure and thermophysical parameters of each layer are shown in Table1 [10]. Time-varying boundary conditions are applied to the left side of the laminated structure. The change of heating heat flow with time is shown in Figure 2. Three heating conditions are considered in this study (a: square wave heating, b: triangular wave, and c: sinusoidal wave). The period and amplitude of the three heating cases are the same. The former period of time is heating and cooling at the same time. In the later period, only cooling is carried out. When heating, the heat flow amplitude is 20 W/cm2. When cooling, the fluid temperature is set as 0°C. The convective heat transfer coefficient of the fluid and the left boundary is set as h0 = 10 W/m2K or h0 = 300 W/m2K, the initial temperature of the structure is Tina = 37°C, and the right boundary condition of the structure is f3(t) = 37°C.Table 1
Parameter value.
Parameteri = 1i = 2i = 3Thickness (mm)0.11.54.4Density (kg/m3)11901116971Coefficient of heat conduction (W/m·K)0.2350.4450.185Specific heatc (J/kg·K)360033002700Heat production W/m3368.1368.1368.3Internal heat transfer coefficienth (W/m2·K)010Figure 2
Diagram of time-varying boundary conditions.


### 4.2. Analysis and Discussion

Figure3 shows the change of surface temperature of the laminated structure with time. The data corresponding to line O are the result of the method of presented in this paper, and it is found that the results of this paper are completely consistent with those of Liu et al. [9]. Lines A, B, and C correspond to three different boundary conditions (a: square wave, b: triangular wave, and c: sinusoidal wave). When h0 = 10 W/m2K, as can be seen from Figure 3 in the top 10 s, temperature rise by fluctuations on the surface of the laminated structure. When the square wave is heated, the amplitude of the wave is the largest, the sine wave is the second, and the triangle wave is the smallest; that is, it is caused by different effective heating areas. After 10 s heating, the surface temperature of the structure showed a downward trend due to the continued fluid cooling. Lines D, E, and F correspond to three different boundary conditions (a: square wave, b: triangular wave, and c: sinusoidal wave). When h0 = 300 W/m2K, the surface temperature of the laminated structure also fluctuates. However, compared with h0 = 10 W/m2K, it increases more slowly. The reason is that the convective heat transfer coefficient is larger and the cooling effect is better. Line G and line A are the results calculated by the finite element method and the semianalytical method in this paper under exactly the same conditions. It can be seen from Figure 3 that they are completely coincident, which further verifies the correctness of the method in this paper.Figure 3
Surface temperature of the laminated structure varies with time (x = 0).Figure4 shows the temperature change over time at the junction of the first and second layers of the laminated structure, where lines A, B, and C, respectively, represent the temperature change with time under the conditions of square wave heating, triangular wave heating, and sine wave heating, when h = 10 W/m2 K, and lines D, E, and F, respectively, represent the temperature change with time under the conditions of square wave heating, triangular wave heating, and sine wave heating when h = 300 W/m2 K. Its variation trend is consistent with the surface trend of the laminated structure. The first 10 s heating process also has temperature fluctuation, but it is relatively gentle.Figure 4
Temperature variation with time at the junction of the first and second layers of the laminated structure (x = 0.1 mm).Figure5 shows the temperature change over time at the junction of the second and third layers of the laminated structure. Here, the data of A, B, and C are calculated with h0 = 10 W/m2K and the data of D, E, and F are calculated with h0 = 300 W/m2K. It can be seen from Figure 5 that the temperature change is not fluctuating at this position. It is inferred that this location is far away from the heat source, and the fluctuation of heat flux of heat source cannot cause the fluctuation of temperature at this position. It can also be seen from Figure 5 that the larger the convective heat transfer coefficient, the slower the temperature rise. The temperature rise in this position is much lower than the temperature rise on the surface of the structure.Figure 5
Temperature variation with time at the junction of the second and third layers of the laminated structure (x = 1.6 mm).Figure6 shows the temperature distribution along the length direction of the laminated structure at different times. The blue line is the result of the method presented in this paper, which is consistent with that of Liu et al. [9]. Thus, the correctness of the method presented can be verified. It can be seen from Figure 6 that, at 9.5 s, the temperature from the left to the right decreases successively because the fluctuating heating on the surface of the laminated structure has just finished. When h0 = 300 W/m2K, the heat dissipation is better and the temperature rises rapidly, so the overall temperature of the structure is higher than that of h0 = 10 W/m2K at the end of heating. It can also be seen that the temperature is highest when square wave heating is applied and lowest when triangle wave heating is applied. When t = 15 s, the heating ended and a period of cooling was carried out. It can be seen that, from the left to right side of the laminated structure, the temperature first increased and then decreased because the cooling effect was better near the surface of the structure on the left side.Figure 6
Temperature distribution alongx direction at different time periods.Figure7 shows the effect of heating period on the surface temperature change at the surface of the laminated structure. In Figures 7(a)–7(c), respectively, represent the temperature changes with period Tc = 0.5 s, Tc = 1 s, and Tc = 2 s. It can be seen from Figure 7 that the larger the heating period is, the larger the temperature fluctuation amplitude is. It can also be seen from Figure 7 that the temperature change is the most obvious when the square wave is heated, and the temperature change curve when the triangle wave is heated is relatively smooth. It is inferred that the larger the period is, the larger the limited heating area is, so the faster the temperature rises.Figure 7
Temperature variation of the structure surface with time at different heating frequencies (x = 0).
(a)(b)(c)Figure8 shows the effect of heating period on the surface temperature change at the junction of the first and second floors of the laminated structure. Figures 8(a)–8(c), respectively, represent the temperature change with period Tc = 0.5 s, Tc = 1 s, and Tc = 2 s. It can be seen from Figure 8 that the temperature change trend at the junction is consistent with the surface temperature change trend. However, because it is far away from the surface heating source, the range of temperature change is smaller than the surface. At the same time, it can be seen that the longer the period is, the smaller the frequency is and the larger the range of temperature fluctuation is.Figure 8
Temperature variation with time at the junction of the first and second layers at different heating frequencies (x = 0.1 mm).
(a)(b)(c)


### 4.3. Summary of Results

Through the above analysis and discussion, the relevant results can be summarized, as shown in Table2.Table 2
Summary of relevant results.
Influence factorResultTriangular waveThe temperature rise is the slowestSine waveThe temperature rises slowlySquare waveThe temperature rise is the fastestHeating frequencyThe higher the frequency, the slower the temperature rise


## 4.1. Description of the Problem

In order to verify the correctness of the semianalytical solution derived in this paper, the semianalytical method has been used to analyze the temperature field variation of a three-layer structure under time-varying boundary conditions. The dimensions of the three-layer structure and thermophysical parameters of each layer are shown in Table1 [10]. Time-varying boundary conditions are applied to the left side of the laminated structure. The change of heating heat flow with time is shown in Figure 2. Three heating conditions are considered in this study (a: square wave heating, b: triangular wave, and c: sinusoidal wave). The period and amplitude of the three heating cases are the same. The former period of time is heating and cooling at the same time. In the later period, only cooling is carried out. When heating, the heat flow amplitude is 20 W/cm2. When cooling, the fluid temperature is set as 0°C. The convective heat transfer coefficient of the fluid and the left boundary is set as h0 = 10 W/m2K or h0 = 300 W/m2K, the initial temperature of the structure is Tina = 37°C, and the right boundary condition of the structure is f3(t) = 37°C.Table 1
Parameter value.
Parameteri = 1i = 2i = 3Thickness (mm)0.11.54.4Density (kg/m3)11901116971Coefficient of heat conduction (W/m·K)0.2350.4450.185Specific heatc (J/kg·K)360033002700Heat production W/m3368.1368.1368.3Internal heat transfer coefficienth (W/m2·K)010Figure 2
Diagram of time-varying boundary conditions.


## 4.2. Analysis and Discussion

Figure3 shows the change of surface temperature of the laminated structure with time. The data corresponding to line O are the result of the method of presented in this paper, and it is found that the results of this paper are completely consistent with those of Liu et al. [9]. Lines A, B, and C correspond to three different boundary conditions (a: square wave, b: triangular wave, and c: sinusoidal wave). When h0 = 10 W/m2K, as can be seen from Figure 3 in the top 10 s, temperature rise by fluctuations on the surface of the laminated structure. When the square wave is heated, the amplitude of the wave is the largest, the sine wave is the second, and the triangle wave is the smallest; that is, it is caused by different effective heating areas. After 10 s heating, the surface temperature of the structure showed a downward trend due to the continued fluid cooling. Lines D, E, and F correspond to three different boundary conditions (a: square wave, b: triangular wave, and c: sinusoidal wave). When h0 = 300 W/m2K, the surface temperature of the laminated structure also fluctuates. However, compared with h0 = 10 W/m2K, it increases more slowly. The reason is that the convective heat transfer coefficient is larger and the cooling effect is better. Line G and line A are the results calculated by the finite element method and the semianalytical method in this paper under exactly the same conditions. It can be seen from Figure 3 that they are completely coincident, which further verifies the correctness of the method in this paper.Figure 3
Surface temperature of the laminated structure varies with time (x = 0).Figure4 shows the temperature change over time at the junction of the first and second layers of the laminated structure, where lines A, B, and C, respectively, represent the temperature change with time under the conditions of square wave heating, triangular wave heating, and sine wave heating, when h = 10 W/m2 K, and lines D, E, and F, respectively, represent the temperature change with time under the conditions of square wave heating, triangular wave heating, and sine wave heating when h = 300 W/m2 K. Its variation trend is consistent with the surface trend of the laminated structure. The first 10 s heating process also has temperature fluctuation, but it is relatively gentle.Figure 4
Temperature variation with time at the junction of the first and second layers of the laminated structure (x = 0.1 mm).Figure5 shows the temperature change over time at the junction of the second and third layers of the laminated structure. Here, the data of A, B, and C are calculated with h0 = 10 W/m2K and the data of D, E, and F are calculated with h0 = 300 W/m2K. It can be seen from Figure 5 that the temperature change is not fluctuating at this position. It is inferred that this location is far away from the heat source, and the fluctuation of heat flux of heat source cannot cause the fluctuation of temperature at this position. It can also be seen from Figure 5 that the larger the convective heat transfer coefficient, the slower the temperature rise. The temperature rise in this position is much lower than the temperature rise on the surface of the structure.Figure 5
Temperature variation with time at the junction of the second and third layers of the laminated structure (x = 1.6 mm).Figure6 shows the temperature distribution along the length direction of the laminated structure at different times. The blue line is the result of the method presented in this paper, which is consistent with that of Liu et al. [9]. Thus, the correctness of the method presented can be verified. It can be seen from Figure 6 that, at 9.5 s, the temperature from the left to the right decreases successively because the fluctuating heating on the surface of the laminated structure has just finished. When h0 = 300 W/m2K, the heat dissipation is better and the temperature rises rapidly, so the overall temperature of the structure is higher than that of h0 = 10 W/m2K at the end of heating. It can also be seen that the temperature is highest when square wave heating is applied and lowest when triangle wave heating is applied. When t = 15 s, the heating ended and a period of cooling was carried out. It can be seen that, from the left to right side of the laminated structure, the temperature first increased and then decreased because the cooling effect was better near the surface of the structure on the left side.Figure 6
Temperature distribution alongx direction at different time periods.Figure7 shows the effect of heating period on the surface temperature change at the surface of the laminated structure. In Figures 7(a)–7(c), respectively, represent the temperature changes with period Tc = 0.5 s, Tc = 1 s, and Tc = 2 s. It can be seen from Figure 7 that the larger the heating period is, the larger the temperature fluctuation amplitude is. It can also be seen from Figure 7 that the temperature change is the most obvious when the square wave is heated, and the temperature change curve when the triangle wave is heated is relatively smooth. It is inferred that the larger the period is, the larger the limited heating area is, so the faster the temperature rises.Figure 7
Temperature variation of the structure surface with time at different heating frequencies (x = 0).
(a)(b)(c)Figure8 shows the effect of heating period on the surface temperature change at the junction of the first and second floors of the laminated structure. Figures 8(a)–8(c), respectively, represent the temperature change with period Tc = 0.5 s, Tc = 1 s, and Tc = 2 s. It can be seen from Figure 8 that the temperature change trend at the junction is consistent with the surface temperature change trend. However, because it is far away from the surface heating source, the range of temperature change is smaller than the surface. At the same time, it can be seen that the longer the period is, the smaller the frequency is and the larger the range of temperature fluctuation is.Figure 8
Temperature variation with time at the junction of the first and second layers at different heating frequencies (x = 0.1 mm).
(a)(b)(c)


## 4.3. Summary of Results

Through the above analysis and discussion, the relevant results can be summarized, as shown in Table2.Table 2
Summary of relevant results.
Influence factorResultTriangular waveThe temperature rise is the slowestSine waveThe temperature rises slowlySquare waveThe temperature rise is the fastestHeating frequencyThe higher the frequency, the slower the temperature rise


## 5. Conclusion

In this paper, a general semianalytical solution for transient heat transfer of the laminated structure under time-varying boundary conditions is derived, and the transient heat transfer of a three-layered structure under time-varying heating is studied by using the semianalytical solution. The specific work and the conclusion can be summarized as follows:(1)
The time-varying boundary conditions and temperature change functions at the boundary are assumed to be differential and linear.(2)
In a small time period, the analytic solutions of each layer of the laminated structure are deduced by using the special separation of variable method.(3)
The analytic solutions of each layer in each minute period are obtained by a loop, and then, the semianalytic solutions in the whole time domain are obtained.(4)
It is found that when the boundary of the laminated structure is heated by time-varying heat flow, the difference of effective heating area will lead to different amplitude of temperature change. Square wave heating temperature variation amplitude is the largest, sine wave is the second, and triangle wave is the smallest.(5)
The larger the heating frequency of the time-varying heat flux is, the smaller the amplitude of temperature fluctuation is.



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*Source: 5545566-2021-10-06.xml*