Thermal Step Response of N-Layer Composite Walls—Accurate Approximative Formulas

Hagentoft, Carl-Eric; Pallin, Simon

doi:10.1115/1.4045642

Abstract

For many industrial applications, heat flow through composites relates directly to energy usage and thus is of highest interest. For multilayer composites, the heat flow is a result of multiple variables, such as the temperature gradient over the surface boundaries and each material's thermal conductivity, specific heat, and thickness. In addition, the transient heat flux also depends on how the materials are aligned together. The heat flow through composites can be estimated using advanced computer simulations for applied heat transfer. Although these tools are powerful, they are also time consuming. Therefore, approximations that allow the estimation of heat flow through composites can be very useful. This paper presents approximations to solve transient heat transfer in multilayer composites, with and without an interior surface resistance. Since the energy use for various applications relates to the heat transferred at the surface boundary, the main focus of this paper is to define approximate solutions for interior heat flow. In other words, these approximations are found by applying a unit step change in temperature on one side of a composite and then in real-time emulating the surface heat flux on the opposite side from which the step change occurs. The approximations are presented based on lumped analyses and Laplace network solutions and are validated against analytical and numerical solutions.

1 Introduction

Quantifying transient heat flow through composite materials is relevant for various industries [1]. For many applications, the surface temperatures around a component/system vary over time, causing heat flows at the surface boundaries to fluctuate. In addition to varying temperature gradients, heat flow depends on many other variables, such as thermal conductivity, specific heat, and material thickness. For composites, the transient heat flux also depends on how the materials are aligned together [2,3].

Under steady-state conditions, the heat flow through composites is typically a straightforward approach. Unless a material represented in the composite is significantly anisotropic [4], an effective thermal conductivity can usually be estimated [5,6]. For transient heat transfer though, the specific heat comes into play [7]; and unfortunately, an effective specific heat for the composite cannot easily be defined [8].

Zedan and Mujahid [9] presented a solution for heat transfer in composite walls using the Laplace transform. Toutain et al. [10] also showed that the Laplace transform can be useful to determine thermal characteristics of composites. For homogeneous materials, approximations of a depth-dependent heat flow have been presented [11], including an external surface resistance [12]. Depending on how the surface temperatures of the composite vary (pulsed or periodic), different approaches to estimating the heat flow may apply [13]. The time scale also impacts the accuracy of the approximation method. For heat flows at relatively smaller time scales, the Laplace transform can be used effectively [14], whereas Fourier series are more suitable for larger time scales [15].

For two-layer composites, the heat flows at the surface boundaries mainly depend on the thicknesses and thermal properties of the two materials. In addition, external and internal heat surface resistance transfer coefficients may have a significant impact on the overall heat flow [16]. Fortunately, in terms of calculation efforts, for many applications, one of the surface transfer coefficients can be eliminated because of a known, measured, or estimated surface temperature. Such applications include, but are not limited to, heat transfer through pipes, ducts, and any conduits that channel fluids. Additionally, multilayer composites apply to walls, roofs, or any construction component for which the thermal performance is of relevance.

For many applications, heat flow relates directly to energy usage. Advanced computer simulations for applied heat transfer are powerful tools but typically are time consuming. Therefore, an approximation that allows estimation of the heat flow through composites can be very useful. This paper presents approximations for solving transient heat transfer in multilayer composites, with and without interior surface resistance (Fig. 1). The main focus of this paper is to approximate the surface heat flux of a composite on the opposite side from which a defined step change in surface temperature occurs. The approximations are presented based on lumped analyses and Laplace network solutions and are validated against analytical and numerical solutions.

Fig. 1

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An N-layer composite wall with a surface resistance on the right-hand side of material N

2 Problem Formulation—Multilayer Composite

In this study, the unit step change of an external surface temperature is investigated. For a general case, the magnitude of the generated heat flux on the interior side is directly proportional to the actual temperature change. Using superposition techniques and/or convolution integrals, any changes in the external surface temperature can be handled.

The partial differential equation for the linear heat transfer problem reads (conditions of continuity for heat flow and temperature must be fulfilled at all interfaces)

\begin{matrix} \frac{\partial T}{\partial x} (λ \frac{\partial T}{\partial x}) = ρ c \frac{\partial T}{\partial t} 0 < x < L_{tot} \\ L_{tot} = \sum_{i = 1}^{N} L_{i} \end{matrix}

(1)

Now, introducing a unit step-change in temperature and the initial condition reads

T = 0 t = 0 0 < x < L_{tot}

(2)

With no surface resistances at

x = 0

⁠, the boundary conditions are

{\begin{cases} T = 1 x = 0 \\ - \frac{\partial T}{\partial x} = \frac{T}{d} x = L_{tot}, \frac{d}{λ_{N}} = R_{s} \end{cases}

(3)

An equivalent depth $d$ (m) is introduced. This depth corresponds to a conductive thickness of the innermost material layer N, which by Eq. (3) has the same thermal resistance as the surface resistance, $R_{s}$ ⁠.

The main target of this study is to approximate the interior heat flux

q

(W/m²) of a composite/slab after a left-hand side (exterior) temperature change is introduced

q = - λ_{N} \frac{\partial T}{\partial x} = \frac{T}{R_{s}} x = L_{tot}

(4)

For steady-state conditions, there is a straightforward solution

q_{s} = \frac{1}{\sum_{i = 1}^{N} \frac{L_{i}}{λ_{i}} + R_{s}} = U

(5)

The thermal problem can also be described for an N-layer composite with constant thermal properties in which all layers must fulfill the heat conduction equation.

The Laplace transform reads

\begin{matrix} \frac{\partial^{2} \bar{T}}{\partial x^{2}} = \frac{s}{a_{n}} \bar{T} \sum_{i = 1}^{n - 1} L_{i} < x \leq \sum_{i = 1}^{n} L_{i} \\ a_{n} = \frac{λ_{n}}{ρ_{n} \cdot c_{n}} \end{matrix}

(6)

The Laplace transform $\bar{T}$ must also be continuous at the interfaces; that rule also must apply to the corresponding heat flows. The left-hand boundary temperature equals $s^{- 1}$ ⁠.

3 Solution Techniques

The approximation presented in this paper is based on solution techniques from lumped analysis and Laplace formulations. Both techniques were found useful to calculate surface heat flux.

3.1 Lumped Analysis.

For a lumped analysis, which is basically equivalent to a finite difference solution, the following equation system must be solved:

\frac{d T}{dt} = A T + b

(7)

Here, $T$ is a vector with element j = 1…N, representing the center temperature of each layer j. $A$ is a tridiagonal matrix.

As seen in Fig. 1, the thermal conductance between two neighbor layers n − 1 and n reads as follows:

\begin{matrix} K_{n - 1, n} = \frac{1}{L_{n - 1} / (2 λ_{n - 1}) + L_{n} / (2 λ_{n})} \\ 2 \leq n \leq N \end{matrix}

(8)

At the left- and right-hand boundaries, the following is the thermal conductance between the boundary layer and the boundary temperature:

\begin{matrix} K_{left, 1} = \frac{1}{L_{1} / (2 λ_{1})} \\ K_{N, right} = \frac{1}{L_{N} / (2 λ_{N}) + R_{s}} \end{matrix}

(9)

The matrix

A

components then become

\begin{matrix} A_{1, 1} = - \frac{1}{L_{1} ρ_{1} c_{1}} (K_{left, 1} + K_{1, 2}) \\ A_{1, 2} = \frac{K_{1, 2}}{L_{1} ρ_{1} c_{1}} \\ A_{N, N} = - \frac{1}{L_{N} ρ_{N} c_{N}} (K_{N - 1, N} + K_{N, right}) \\ A_{N, N - 1} = \frac{K_{N - 1, N}}{L_{N} ρ_{N} c_{N}} \end{matrix}

(10)

{\begin{cases} A_{i, i - 1} = \frac{K_{i - 1. i}}{L_{i} ρ_{i} c_{i}} \\ A_{i, i + 1} = \frac{K_{i . i + 1}}{L_{i} ρ_{i} c_{i}} \\ A_{i, i} = - (A_{i, i - 1} + A_{i, i + 1}) \end{cases} i = 2 \dots N - 1

(11)

The column vector

b

is

{\begin{cases} b_{1} = \frac{K_{left, 1}}{L_{1} ρ_{1} c_{1}} \\ b_{i} = 0 2 \leq i \leq N \end{cases}

(12)

The initial condition is zero for all the

T

vector elements. A temperature step change equal to one is introduced on the left-hand boundary side, while the temperature change remains zero on the right-hand side. With the temperature vector known at each time, the heat flux to the interior becomes

q (t) = λ_{N} \frac{T_{N} (t)}{\frac{L_{N}}{2} + d}

(13)

Equation (7) represents a system of ordinary differential equations. It can be solved by various techniques [17]. One way of solving Eq. (7) is outlined in Eq. (30).

3.2 Periodic and Laplace Networks.

There are systematic ways of setting up periodic thermal problems by using matrix solutions [18] or by using network solutions [11].

There are also systematic reduction rules that can be used to simplify the networks, as seen in Fig. 2.

Fig. 2

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Network reduction rules

A technique for the systematic solution of steady periodic problems uses a complex formulation of the temperature

T (x, t) = ℜ {\overset{⌢}{T} (x) \cdot e^{\frac{2 π i t}{t_{p}}}}

(14)

Here,

\overset{⌢}{T} (x)

is a complex-value amplitude and t_p is the time period. For a layer with constant material properties, we then have

\frac{\partial^{2} \overset{⌢}{T}}{\partial x^{2}} = \frac{2 π i}{a t_{p}} \overset{⌢}{T}

(15)

Using a Laplace formulation, the heat conduction equation reads

\frac{\partial^{2} \bar{T}}{\partial x^{2}} = \frac{s}{a} \bar{T}

(16)

Figure 3 shows the Laplace network for one homogeneous layer. This is directly found by the following substitution of the periodic network components in Ref. [11]:

\sqrt{2 i π / (a t_{p})} \to \sqrt{s / a}

(17)

Fig. 3

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Thermal conductance network for a one-layer material with a right-hand side surface resistance

For the case depicted in Fig. 3, the transmittive and admittive conductances are

{\bar{K}}_{t} = λ \frac{1}{\sinh ((L \sqrt{s / a}))} \sqrt{\frac{s}{a}}

(18)

{\bar{K}}_{a} = λ \frac{\cosh (L \sqrt{s / a}) - 1}{\sinh ((L \sqrt{s / a}))} \sqrt{\frac{s}{a}}

(19)

4 Two and N-Layer Composite Including Surface Resistance—Small Time Period

For any specified N-layer composite, reciprocal rules show that the heat flux on the reverse side of a composite is identical regardless of the side on which the step change in temperature occurs [19]. The reciprocal theorem simplifies the approximative Laplace network solution provided in this section and allows such a solution to be valid irrespective of the direction of the heat flow.

4.1 Two-Layer Laplace Network Solution.

Figure 4 shows the thermal conductance network for two layers without outer surface resistance. To simplify the solutions below, an internal step change in temperature was chosen instead of an external one. Owing to the linearity of the configuration, i.e., reciprocity [19], the heat flow at the opposite side of where the step change occurs is always the same, regardless of heat flow direction.

Fig. 4

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Thermal conductance network for two layers without an internal surface resistance

The temperature at the interface between the two layers is

{\bar{T}}_{surface} (s) = \frac{1}{R_{s}} {\bar{K}}_{1} \frac{1}{s} \frac{\frac{1}{1 / R_{s} + {\bar{K}}_{a 2}}}{{\bar{K}}_{1} + {\bar{K}}_{a 1} + {\bar{K}}_{a 2} + {\bar{K}}_{t 1}}

(20)

The transmittive and admittive conductances are

\begin{matrix} {\bar{K}}_{t 1} = \frac{1}{\sinh (β_{1} \sqrt{s})} \frac{λ_{1} β_{1}}{L_{1}} \sqrt{s}, β_{1} = \frac{L_{1}}{\sqrt{a_{1}}} \\ {\bar{K}}_{t 2} = \frac{1}{\sinh (β_{2} \sqrt{s})} \frac{λ_{2} β_{2}}{L_{2}} \sqrt{s}, β_{2} = \frac{L_{2}}{\sqrt{a_{2}}} \\ {\bar{K}}_{a 1} = \frac{\cosh (β_{1} \sqrt{s}) - 1}{\sinh (β_{1} \sqrt{s})} \frac{λ_{1} β_{1}}{L_{1}} \sqrt{s} \\ {\bar{K}}_{a 2} = \frac{\cosh (β_{2} \sqrt{s}) - 1}{\sinh (β_{2} \sqrt{s})} \frac{λ_{2} β_{2}}{L_{2}} \sqrt{s} \end{matrix}

(21)

For Eq.

\frac{1}{{\bar{K}}_{1}} = \frac{1}{{\bar{K}}_{t 2}} + \frac{1}{1 / R_{s} + {\bar{K}}_{a 2}}

(22)

The heat flux at the surface is given by

q (t) = L^{- 1} {\bar{q}} = L^{- 1} {{\bar{K}}_{t 1} \cdot {\bar{T}}_{surface}}

(23)

The asymptotic behavior for relatively small time periods t is of interest. It can be found using Eqs. (20)–(23) and allowing the Laplace variable to tend to infinity, followed by implementing the inverse Laplace transform.

The asymptotic behavior of the used conductance becomes

\begin{matrix} {\bar{K}}_{t} \to 2 λ \frac{β}{L} \sqrt{s} \cdot e^{- β \sqrt{s}} \\ {\bar{K}}_{a} \to λ \frac{β}{L} \sqrt{s} \\ β = \frac{L}{\sqrt{a}} \end{matrix}

(24)

The details of the derivation are given in Ref. [20]. There is an explicit, and remarkably simple, solution for the inverse Laplace transform of this expression. The time-dependent solution for short times becomes

\begin{matrix} q (t) = \frac{λ_{2}}{d} \frac{b_{1}}{b_{1} + b_{2}} 4 e^{\frac{a_{2} t}{d^{2}} + \frac{\sqrt{a_{2}}}{d} (\frac{L_{1}}{\sqrt{a_{1}}} + \frac{L_{2}}{\sqrt{a_{2}}})} \\ \cdot erfc (\frac{\sqrt{a_{2} t}}{d} + \frac{L_{1}}{\sqrt{4 a_{1} t}} + \frac{L_{2}}{\sqrt{4 a_{2} t}}) \end{matrix}

(25)

where the effusivities are

b_{1} = \sqrt{λ_{1} \cdot ρ_{1} c_{1}}

(26)

b_{2} = \sqrt{λ_{2} \cdot ρ_{2} c_{2}}

(27)

d = λ_{2} \cdot R_{s}

(28)

Regardless of the thermal characteristics of the two arbitrary material layers, Eq. (25) approximates the interior heat flux very well. For larger time periods, an approximation is presented in Sec. 5.

4.2 N-Layer Laplace Network Solution.

In accordance with the interior heat flux approximated by Eq. , an analogous explicit inverse Laplace transform can be used for an N-layer composite. A successive reduction of the network, in line with what was done in Sec. 4.1, was performed. The details can be found in Ref. [20]. The time-dependent interior heat flux for small time periods becomes

\begin{matrix} q (t) = \frac{λ_{N}}{d} \frac{b_{1}}{b_{N}} 2^{N} {\prod_{i = 2}^{N} \frac{b_{i}}{b_{i - 1} + b_{i}}} e^{\frac{a_{N} t}{d^{2}} + \frac{\sqrt{a_{N}}}{d} \sum_{i = 1}^{N} \frac{L_{i}}{\sqrt{a_{i}}}} \\ \cdot erfc (\frac{\sqrt{a_{N} t}}{d} + \sum_{i = 1}^{N} \frac{L_{i}}{\sqrt{4 a_{i} t}}) \end{matrix}

(29)

As expected, solving Eq. (29) for N = 2 generates identical approximations as found for two-layer composites; see Eq. (25).

5 N-Layer Lumped Analysis—Large Time Period

For relatively larger time periods, the approximate solution can be found from the lumped analysis. Matrix

A

⁠, defined by Eqs. –, gives the basis for the approximation in which each element corresponds to the layers of the composite. The general solution for the lumped case has the following form [21]:

T = T_{s} - V exp (D t) V^{- 1} T_{s}

(30)

The elements in the

T_{s}

vector represent the steady-state temperature of the problem in Eq. . The matrix

V

of A has the eigenvectors of column vectors. The matrix

D

has the corresponding eigenvalues as diagonal elements. The principal solution of Eq. will be of the following type:

T_{i} = T_{s, i} + \sum_{j = 1}^{N} c_{i j} \cdot e^{- t / t_{c j}} t_{c j} = - \frac{1}{D_{j j}}

(31)

Here, we have introduced the characteristic time,

t_{c j}

⁠, as the inverse of the eigenvalue

D_{j j}

with changed sign. The largest characteristic times will determine the behavior at large times. Under this statement, an approximate expression for the interior heat flux at large times uses the two largest characteristic times

{\begin{cases} \frac{q (t)}{U} = 1 - C_{1} \cdot e^{- t / t_{c 1}} - C_{2} \cdot e^{- t / t_{c 2}} \\ t_{c 1}, t_{c 2} = \max (t_{c j}) \end{cases}

(32)

The chosen breakpoint,

γ

⁠, for which Eq. is used over Eq. can be determined under the following conditions:

\frac{q (t_{γ})}{U} = γ = 1 - C_{1} \cdot e^{- t_{γ} / t_{c 1}} - C_{2} \cdot e^{- t_{γ} / t_{c 2}}

(33)

Here,

q (t_{γ})

represents the interior heat flux found in Eq. at time

t_{γ}

⁠. The breaking point is chosen as the time when the heat flux has reached a certain level of full implementation (steady-state). After this breakpoint, the approximation for short times becomes less accurate with increasing time. Consequently, this breakpoint also represents a time for which the temperature step change has propagated to a certain degree through the composite. The following conditions of continuous slope of the curve can be formulated:

\frac{1}{U} \frac{d q (t_{γ})}{d t} = \frac{C_{1}}{t_{c 1}} \cdot e^{- t_{γ} / t_{c 1}} + \frac{C_{2}}{t_{c 2}} \cdot e^{- t_{γ} / t_{c 2}}

(34)

The two conditions in Eqs. (33) and (34) determine the constants C₁ and C₂. Investigations have shown that the value of $γ$ should be in the range of 0.1 to 0.3. Preferably, $γ$ is optimized depending on the thermal characteristics of the material layers.

Figure 5 illustrates the accuracy of the two approximations in Eqs. (29) and (32) for a three-layer composite.

Fig. 5

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((a) and (b)) Interior heat flux of a three-layer composite including interior surface resistance. Comparisons are made from results of Eqs. (23), (29), and (32). The criterion is given in header and reads 0.1 (a) and 1.0 (b).

The most suitable breakpoints, $γ$ ⁠, for switching from the solution for smaller times, Eq. (29), to the one best suited for larger times, Eq. (32), are marked by circles in Fig. 6.

Fig. 6

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Breakpoint for which Eq. (32) is used over Eq. (29). The solid curves represent the exact Laplace solution and the dashed lines represent the two approximations. The circles depict where the breakpoints occur. P represents the ratio defined in the header.

The largest absolute error from using the approximations is 0.007 under the conditions presented in Fig. 6. The maximum error will depend on the value of $γ$ and should be in the range of 0.1 to 0.3. According to Fig. 6, $γ$ seems to decrease with increasing value of P.

6 Discussion and Conclusion

This paper presents approximations for solving the interior heat flux of a composite when an exterior unit step change in temperature is introduced. The approximations are valid for one, two, and N-layer composites, with and without interior surface resistance. These solutions are found by using lumped analyses and Laplace network solutions and are validated against analytical and numerical solutions.

As seen in Eq. (29), a solution exists for estimating the interior heat flux of composites at smaller time periods with great accuracy. For N-layer composites, the approximation for larger time periods is found using the two largest characteristic time scales by using a lumped analysis approach, Eq. (32). The largest absolute error from using these approximations is 0.007, for a unit step change-induced internal heat flow of a three-layer composite.

Acknowledgment

This paper has been authored [or co-authored] by UT-Battelle LLC under contract DE-AC05-00OR22725 with the U.S. Department of Energy (DOE).

Research conducted for this paper at Chalmers Technical University was realized through Swedish governmental core funding.

Funding Data

Chalmers Technical University (Swedish governmental core funding) (Funder ID: 10.13039/501100002835).

Nomenclature

$a$ =: thermal diffusivity (m²/s)
$A$ =: tridiagonal matrix (s⁻¹)
$b$ =: thermal effusivity ((W √s)/(m² K))
$c$ =: specific heat capacity (J/(kg K))
$d$ =: depth (m)
$K$ =: thermal conductance (W/(m² K))
$L$ =: thickness (m)
$M, N$ =: variable
$q$ =: heat flux (W/m²)
$R$ =: thermal resistance ((m² K)/W)
$s$ =: Laplace variable (s⁻¹)
$t$ =: time (s)
$T$ =: temperature (°C, K)
$U$ =: thermal conductance (W/(m² K))
$λ$ =: thermal conductivity (W/(m K))
$ρ$ =: material density (kg/m³)

$e$ =: external
$i$ =: internal
$p$ =: time period
$s$ =: surface or steady-state

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Thermal Step Response of N-Layer Composite Walls—Accurate Approximative Formulas

Abstract

1 Introduction

2 Problem Formulation—Multilayer Composite

3 Solution Techniques

3.1 Lumped Analysis.

3.2 Periodic and Laplace Networks.

4 Two and N-Layer Composite Including Surface Resistance—Small Time Period

4.1 Two-Layer Laplace Network Solution.

4.2 N-Layer Laplace Network Solution.

5 N-Layer Lumped Analysis—Large Time Period

6 Discussion and Conclusion

Acknowledgment

Funding Data

Nomenclature

References

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Thermal Step Response of N-Layer Composite Walls—Accurate Approximative Formulas

Abstract

1 Introduction

2 Problem Formulation—Multilayer Composite

3 Solution Techniques

3.1 Lumped Analysis.

3.2 Periodic and Laplace Networks.

4 Two and N-Layer Composite Including Surface Resistance—Small Time Period

4.1 Two-Layer Laplace Network Solution.

4.2 N-Layer Laplace Network Solution.

5 N-Layer Lumped Analysis—Large Time Period

6 Discussion and Conclusion

Acknowledgment

Funding Data

Nomenclature

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