Heat Transfer in Composite Cylinders Under Harmonically Oscillating Ambient Conditions

Abstract

An analytical solution is presented for transient heat conduction in a two-layer composite cylinder subjected to outer-surface convection with a general time-dependent ambient temperature. Using Duhamel’s principle, closed-form series expressions are derived and then specialized to harmonic ambient fluctuations, recovering the classical constant-ambient solution in the zero-frequency limit. A parametric study shows that the ratio of the inner layer conductivity to the conductivity of the outer layer strongly shapes interfacial gradients and mean-temperature evolution, with sensitivity concentrated at small ratios and diminishing when the ratio is larger than 0.1. Increasing Biot number accelerates the heat transfer and approaches the isothermal-surface limit as it becomes extremely large. The geometric aspect ratio is most influential when the inner layer is resistive, and becomes weak for large conductivity ratio, supporting thin-coating approximations. Under harmonic ambient fluctuations, the response rapidly reaches a periodic steady state; higher frequency decreases amplitude and increases phase lag, while larger Biot numbers amplify oscillations and reduce delay. The coupled effects of the aspect ratio and the conductivity ratio govern penetration and phase behavior.

1. Introduction

Concentric, multi-layer cylinders arise whenever a cylindrical component must simultaneously carry load and manage heat, for example, a metallic pipe protected by an insulating jacket, an electrical conductor surrounded by polymer insulation and an outer sheath, or a cladded fuel rod. In these systems, the dominant pathway is radial heat conduction across dissimilar layers, so the interface conditions (continuity of temperature and heat flux) and property contrasts determine both the net thermal resistance and the local gradients that drive thermal stress and material degradation. Because many such components operate under start-up, shut-down, and cyclic environments, peak temperatures and thermal lag are set by transient, not steady, conduction. Standard heat-conduction references treat composite cylindrical media as canonical examples in which discontinuous material properties and boundary conditions shape the solution structure [1,2,3,4].

Composite-cylinder conduction models are used as the basis of practical rating calculations. For high-voltage power cables, the allowable continuous current (ampacity) is set by the maximum permissible conductor and insulation temperature. Standards such as IEC 60287 compute this by treating the cable as concentric cylindrical layers and modeling heat flow outward using a thermal-resistance network: radial conduction resistances of each layer plus the external heat-transfer resistance to the surroundings [5]. In nuclear fuel rods, radial conduction through fuel and cladding, together with gap and surface heat transfer, sets peak fuel and cladding temperatures and therefore margin to safety limits. Standard reactor-thermal texts idealize the rod as a layered cylinder and emphasize the role of material properties and boundary conditions [6,7]. Layered conduction is also central in thermal protection and thermal management: thermal-barrier coatings are designed to reduce metal temperature in hot components [8], and battery thermal-management studies highlight how temperature nonuniformity and transient loading affect performance and safety [9].

Closed-form or semi-analytical solutions are valuable for layered cylinders because they clarify how conductivity ratios, Biot numbers, and layer thicknesses shape the response. These solutions also enable rapid parametric studies without remeshing and provide benchmarks for verification. Classical heat-conduction references develop eigenfunction expansions in cylindrical coordinates and use superposition and transform ideas to treat transient boundary-value problems [1,2,3,4]. More recent syntheses continue to organize and extend analytical solutions for multilayer conduction in composite structures across geometries and boundary conditions [10,11,12,13].

Practically, the external thermal environment is rarely steady. Ambient temperature and convection conditions vary with operating cycles, weather changes, or process transients, and such forcing can be represented as periodic or general time-dependent boundary conditions. For solids subjected to periodic ambient variations, prior analyses show attenuation and phase lag between the boundary signal and internal temperatures [14]. In composite cylinders, the same dynamics couple with interfacial continuity and property contrast, so the amplitude and phase of the response can differ substantially from the homogeneous-cylinder case.

Analytical and semi-analytical studies of heat conduction in composite media have been widely reported in the literature. Classical treatments focus primarily on steady-state or transient conduction in homogeneous or multilayer cylinders under constant or simplified boundary conditions [1,2,3,4,10,11,12,13]. Several studies have extended these formulations to composite geometries, emphasizing eigenfunction expansions and interface continuity conditions. More recent works have addressed transient behavior in multilayer cylindrical systems under more complex boundary conditions. For example, transient solutions for multilayer composite cylinders and time-varying convective boundary conditions have been addressed in [15,16,17]. Other studies have explored hybrid analytical formulations [18] and semi-analytical approaches for multilayer systems [19], highlighting the increasing complexity of such problems. Additional extensions include the incorporation of porous media effects [20] and spatially varying convective conditions [21]. In parallel, numerical and computational approaches have been developed to address multilayer heat conduction problems under general conditions [22,23], providing flexible alternatives when analytical solutions are difficult to obtain.

Despite these advances, most available formulations are restricted to specific forms of boundary forcing or rely on numerical and semi-analytical techniques rather than fully explicit closed-form solutions. In particular, analytical solutions for composite cylinders subjected to general time-dependent ambient conditions with convection remain relatively limited, with only a few studies addressing specific forms of time-dependent boundary conditions [12,15]. While periodic boundary conditions have been studied for homogeneous solids [14], their extension to layered cylindrical systems introduces additional coupling through interfacial conditions, significantly affecting amplitude attenuation and phase behavior. A comprehensive analytical framework capturing these combined effects is still lacking, as also highlighted in recent reviews [10].

Typical thermophysical properties of engineering materials and metals, including conductivity, density, and heat capacity, are reported in standard reference compilations [24,25,26]. In addition, related composite analyses have employed simplified relationships between material properties for parametric studies [27].

Accordingly, this paper derives an analytical solution for transient radial conduction in a two-layer composite cylinder with convection at the outer surface and a prescribed, time-dependent ambient temperature. The temperature field in each layer is represented by Bessel-function expansions, and Duhamel’s principle is used to express the response to arbitrary ambient histories through a convolution in time. The general formulation is then specialized to harmonic ambient fluctuations to quantify amplitude attenuation and phase lag and to recover the constant-ambient limit as the forcing frequency tends to zero. The paper could be useful for applications such as those in [28].

The main contributions of this work are summarized as follows:

• A unified analytical treatment of a two-layer composite cylinder with general time-dependent ambient forcing via Duhamel’s principle;
• Explicit harmonic specialization, including systematic analysis of amplitude attenuation and phase lag;
• A combined parametric study of conductivity ratio, Biot number, geometric aspect ratio, and frequency within an exact analytical framework; and
• The derivation and organization of several nontrivial limiting regimes within a single formulation.

2. The Model

Consider a two-layer composite cylinder consisting of an inner solid core ($0\le r\le a$) and an outer annulus $(a\le r\le b)$, initially at a uniform temperature $T_0$, Figure 1.

The two layers may be composed of different materials; subscripts 1 and 2 denote the inner and outer layers, respectively. Heat is exchanged with the surrounding medium by convection at the external surface. The temperature field $T(r,t)$ within the composite cylinder (with constant properties and no internal heat generation) satisfies the transient energy equation:

$${\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\displaystyle \frac{\partial T}{\partial r}}\right)={\displaystyle \frac{1}{\alpha }}{\displaystyle \frac{\partial T}{\partial t}}$$

(1)

where $\alpha$ is the thermal diffusivity, $r$ denotes the radial distance, and t is time. The initial temperature distribution through the cylinder is given by

$$T_1(r,0)=T_2(r,0)=T_0$$

(2)

The temperature at the center of the cylinder must remain finite, that is

$$T_1(0,t)<\infty$$

(3)

At the interface, both the temperature and the flux are continuous.

$$T_1(a,t)=T_2(a,t)$$

(4)

$$k_1{{\displaystyle \frac{\partial T_1}{\partial r}}|}_{r=a}=k_2{{\displaystyle \frac{\partial T_2}{\partial r}}|}_{r=a}$$

(5)

where $k$ is the thermal conductivity. At the outer surface, convective heat exchange with the surrounding fluid is described by the boundary condition

$$-k_2{{\displaystyle \frac{\partial T_2}{\partial r}}|}_{r=b}=h\left(T_2(b,t)-T_{\infty }\left(t\right)\right)$$

(6)

where $h$ is the convective heat transfer coefficient and $T_{\infty }\left(t\right)$ (possibly time-dependent) is the ambient fluid temperature.

We now introduce the following dimensionless variables

$$\xi ={\displaystyle \frac{r}{b}},\hspace{1em}\epsilon ={\displaystyle \frac{a}{b}},\hspace{1em}\tau ={\displaystyle \frac{{\alpha }_2}{b^2}}t,\hspace{1em}u={\displaystyle \frac{T-T_0}{T_m}},\hspace{1em}Bi=hb/k_2$$

(7)

where $T_m$ is a reference temperature scale and $Bi$ is the Biot number. With these definitions, the system (1)–(6) can be rewritten in dimensionless form as

$${\displaystyle \frac{{\partial }^2{u}_1}{\partial {\xi }^2}}+{\displaystyle \frac{1}{\xi }}{\displaystyle \frac{\partial u_1}{\partial \xi }}={\displaystyle \frac{{\alpha }_2}{{\alpha }_1}}{\displaystyle \frac{\partial u_1}{\partial \tau }}\hspace{1em}\mathrm{and} {\displaystyle \frac{{\partial }^2{u}_2}{\partial {\xi }^2}}+{\displaystyle \frac{1}{\xi }}{\displaystyle \frac{\partial u_2}{\partial \xi }}={\displaystyle \frac{\partial u_2}{\partial \tau }}$$

(8)

$$u_1(\xi ,0)=u_2(\xi ,0)=0$$

(9)

$$u_1(0,\tau )<\infty$$

(10)

$$u_1(\epsilon ,\tau )=u_2(\epsilon ,\tau )$$

(11)

$$k_1{{\displaystyle \frac{\partial u_1}{\partial \xi }}|}_{\xi =\epsilon }=k_2{{\displaystyle \frac{\partial u_2}{\partial \xi }}|}_{\xi =\epsilon }$$

(12)

$${\left[{\displaystyle \frac{\partial u_2}{\partial \xi }}+Bi{ u}_2(\xi ,\tau )\right]}_{\xi =1}=Bi{ u}_{\infty }(\tau )$$

(13)

To construct a solution, we first consider an auxiliary problem in which $\phi \left(\xi ,\tau \right)$ satisfies the system (8)–(13), except that the forcing term $Bi{ u}_{\infty }\left(\tau \right)$ on the right-hand side of (13) is replaced by unity.

Equation (13), which now reads ${\left[{\displaystyle \frac{\partial {\phi }_2}{\partial \xi }}+Bi{ \phi }_2(\xi ,\tau )\right]}_{\xi =1}=1$, can be made homogeneous by defining $w\left(\xi ,\tau \right)=\phi \left(\xi ,\tau \right)-{\displaystyle \frac{1}{Bi}}$. The system (8)–(13) can be recast in terms of $w\left(\xi ,\tau \right)$ as follows:

$${\displaystyle \frac{{\partial }^2{w}_1}{\partial {\xi }^2}}+{\displaystyle \frac{1}{\xi }}{\displaystyle \frac{\partial w_1}{\partial \xi }}={\displaystyle \frac{{\alpha }_2}{{\alpha }_1}}{\displaystyle \frac{\partial w_1}{\partial \tau }}\hspace{1em}\mathrm{and}\hspace{1em}{\displaystyle \frac{{\partial }^2{w}_2}{\partial {\xi }^2}}+{\displaystyle \frac{1}{\xi }}{\displaystyle \frac{\partial w_2}{\partial \xi }}={\displaystyle \frac{\partial w_2}{\partial \tau }}$$

(14)

$$w_1\left(\xi ,0\right)=w_2\left(\xi ,0\right)=-{\displaystyle \frac{1}{Bi}}$$

(15)

$$w_1\left(0,\tau \right)<\infty$$

(16)

$$w_1(\epsilon ,\tau )=w_2(\epsilon ,\tau )$$

(17)

$$k_1{{\displaystyle \frac{\partial w_1}{\partial \xi }}|}_{\xi =\epsilon }=k_2{{\displaystyle \frac{\partial w_2}{\partial \xi }}|}_{\xi =\epsilon }$$

(18)

$${\left[{\displaystyle \frac{\partial w_2}{\partial \xi }}+Bi w_2(\xi ,\tau )\right]}_{\xi =1}=0$$

(19)

Equation (14) admits a separable solution, with radial eigenfunctions expressed in terms of the order-zero Bessel functions of the first and second kinds, $J_0$ and $Y_0$. For example, by assuming $w_2\left(r,t\right)=R_2\left(r\right) Z\left(t\right)$, the equation ${\displaystyle \frac{{\partial }^2{w}_2}{\partial {\xi }^2}}+{\displaystyle \frac{1}{\xi }}{\displaystyle \frac{\partial w_2}{\partial \xi }}={\displaystyle \frac{\partial w_2}{\partial \tau }}$ separates into ${\displaystyle \frac{R_2^{\prime \prime }+\frac{1}{\xi } R_2^{\prime }}{R_2}}={\displaystyle \frac{Z^{\prime }}{Z}}=-{\lambda }^2$. The radial equation $R_2^{\prime \prime }+{\displaystyle \frac{1}{\xi }} R_2^{\prime }+{\lambda }^2 R_2=0$ is Bessel’s differential equation with solutions ${ J}_0$ and $Y_0$.

Enforcing conditions (16) and (19) and then applying (17) to the superposition yields:

$$w_1(\xi ,\tau )={\sum }_{n=1}^{\infty }{A}_n J_0\left({\lambda }_n\sqrt{{\displaystyle \frac{{\alpha }_2}{{\alpha }_1}}} \xi \right)e^{- {\lambda }_n^2 \tau }$$

(20)

$$w_2(\xi ,\tau )={\sum }_{n=1}^{\infty }{A}_n{ \Omega }_1 \left({ J}_0\left({\lambda }_n \xi \right)+{\Omega }_0 Y_0\left({\lambda }_n \xi \right)\right){ e}^{-{\lambda }_n^2 \tau }$$

(21)

where ${\Omega }_0={\displaystyle \frac{ {\lambda }_{n }{J}_1\left({\lambda }_n\right) - Bi{ J}_0\left({\lambda }_n\right)}{ -{ \lambda }_n Y_1\left({\lambda }_n\right) + Bi{ Y}_0\left({\lambda }_n\right)}}$ and ${\Omega }_1={\displaystyle \frac{J_0\left({\lambda }_n \sqrt{\frac{{\alpha }_2}{{\alpha }_1}} \epsilon \right)}{J_0\left({\lambda }_n \epsilon \right)+ {\Omega }_0 Y_0\left({\lambda }_n \epsilon \right)}}$.

Equation (18) is used to determine the eigenvalues ${\lambda }_n$,

$${\displaystyle \frac{k_1}{k_2}} \sqrt{{\displaystyle \frac{{\alpha }_2}{{\alpha }_1}}}{ J}_1\left({\lambda }_{n }\sqrt{{\displaystyle \frac{{\alpha }_2}{{\alpha }_1}}} \epsilon \right)-{\Omega }_1 \left(J_1\left({\lambda }_n \epsilon \right)+{\Omega }_0 Y_1\left({\lambda }_n \epsilon \right)\right)=0$$

(22)

where $J_1$ and $Y_1$ denote the first- and second-kind Bessel functions of order one, respectively. The eigenvalues are determined numerically by a root-finding routine. In accordance with the standard Sturm–Liouville theorem, these eigenvalues are real, positive, and can be arranged in an increasing order (${\lambda }_1<{\lambda }_2<{\lambda }_3<\dots \to \infty$).

The coefficients in (20) and (21) are determined by enforcing the initial condition (15). It can be shown that

$$A_n=-{\displaystyle \frac{1}{Bi}} {\displaystyle \frac{\frac{k_1}{{\alpha }_1}{\int }_0^{\epsilon }\xi { R}_{1n} d\xi +\frac{k_2}{{\alpha }_2} {\int }_{\epsilon }^1\xi { R}_{2n} d\xi }{\frac{k_1}{{\alpha }_1} {\int }_0^{\epsilon }\xi { R}_{1n}^2 d\xi +\frac{k_2}{{\alpha }_2}{\int }_{\epsilon }^1\xi R_{2n}^2 d\xi }}$$

(23)

where $R_{1n}(\xi )= J_0\left({\lambda }_n\sqrt{{\displaystyle \frac{{\alpha }_2}{{\alpha }_1}}} \xi \right)$ and $R_{2n}\left(\xi \right)={\Omega }_1 \left({ J}_0\left({\lambda }_n \xi \right)+{\Omega }_0 Y_0\left({\lambda }_n \xi \right)\right)$.

The coefficients $A_n$ can be evaluated analytically in closed form; however, the resulting expression is lengthy, so we do not reproduce it here. Instead, an exact Mathematica command for computing $A_n$ and the lengthy expression are provided in Appendix A. It was made sure, by watching the tail of the series, that enough terms in the series were taken to ensure that the error does not exceed ${10}^{-6}$.

In deriving (23), we used the orthogonality condition:

$${\displaystyle \frac{k_1}{{\alpha }_1}} {\int }_0^{\epsilon }\xi { R}_{1m}{ R}_{1n} d\xi +{\displaystyle \frac{k_2}{{\alpha }_2}} {\int }_{\epsilon }^1\xi R_{2m} R_{2n} d\xi =0\mathrm{if} n\ne m$$

(24)

The proof of (24) is detailed in Appendix A.

Now, applying Duhamel’s principle [2], the time-dependent solution can be expressed as

$$u\left(\xi ,\tau \right)={\int }_{\eta =0}^{\tau }Bi{ u}_{\infty }\left(\eta \right){\displaystyle \frac{\partial \phi \left(\xi ,\tau -\eta \right)}{\partial \tau }} d\eta =-Bi {\int }_{\eta =0}^{\tau } u_{\infty }\left(\eta \right){\displaystyle \frac{\partial w(\xi ,\tau -\eta )}{\partial \eta }}d\eta$$

(25)

This representation is valid for an arbitrary time-dependent ambient condition. The time dependence enters only through the convolution integral, which couples the exponential kernel of the eigenfunction expansion with the prescribed ambient temperature. In the special case of a constant ambient temperature ($u_{\mathrm{\infty }}=\mathrm{const}.$), the integral simplifies and the solution reduces to $Bi{ u}_{\infty } \phi \left(\xi ,\tau \right)$.

3. Discussion

We consider an ambient temperature that varies harmonically about the composite cylinder’s initial temperature:

$$T_{\infty }\left(t\right)=T_0+T_m \cos ({\nu }_0 t)$$

(26)

where ${\nu }_0$ is the frequency.

The corresponding dimensionless ambient temperature is

$$u_{\infty }(\tau )={\displaystyle \frac{T_{\infty }\left(\tau \right)-T_0}{T_m}}=\cos ({\omega }_0 \tau )$$

(27)

where ${\omega }_0={\displaystyle \frac{b^2 {\nu }_0}{{\alpha }_2}}$ is the dimensionless frequency. Applying Duhamel’s principle to solutions (20) and (21) then yields expressions of the form

$$u_1\left(\xi ,\tau \right)=-Bi {\sum }_{n=1}^{\infty }{A}_n {\lambda }_n^2 J_0\left(\sqrt{{\displaystyle \frac{{\alpha }_2}{{\alpha }_1}}}{\lambda }_n \xi \right)\left({\displaystyle \frac{-e^{-{\lambda }_n^2 \tau } {\lambda }_n^2+{\lambda }_n^2 \cos \left({\omega }_0 \tau \right)+{\omega }_0\sin ({\omega }_0 \tau )}{{\lambda }_n^4+{\omega }_0^2}} \right)$$

(28)

$$u_2\left(\xi ,\tau \right)=-Bi {\sum }_{n=1}^{\infty }{A}_n {\lambda }_n^2 {\displaystyle \frac{-e^{-{\lambda }_n^2 \tau } {\lambda }_n^2+{\lambda }_n^2 \cos \left({\omega }_0 \tau \right)+{\omega }_0\sin \left({\omega }_0 \tau \right)}{{\lambda }_n^4+{\omega }_0^2}} {\Omega }_1 \left({ J}_0\left({\lambda }_n \xi \right)+{\Omega }_0 Y_0\left({\lambda }_n \xi \right)\right)$$

(29)

3.1. Constant Ambient Temperature

When the frequency vanishes (${\omega }_0=0$), the problem reduces to a composite cylinder exposed to a constant ambient temperature ($u_{\mathrm{\infty }}=1$). The corresponding solution is obtained directly from (28) and (29) by setting ${\omega }_0=0$.

We examine the influence of the conductivity ratio $k_1/k_2$ on the temporal evolution of the dimensionless temperature in the inner and outer layers. As noted by Alassar and Al-Attas [27], the ratios $k_1/k_2$ and ${\alpha }_1/{\alpha }_2$ are often comparable for commonly used metals; we adopt the same assumption here. Figure 2 presents radial profiles of the dimensionless temperature at selected dimensionless times for $Bi=1$ and $\epsilon =1/2$. For small values of $k_1/k_2$ (Figure 2a), the inner layer offers a relatively large thermal resistance, so its temperature changes only weakly, even at later times. A pronounced “kink” appears at the interface, reflecting an abrupt change in the slope required to accommodate the discontinuity in thermal conductivity while maintaining continuity of heat flux. This kink disappears when $k_1/k_2=1$ (Figure 2c), corresponding to a homogeneous cylinder. For large conductivity ratios (Figure 2d), the interfacial slope change reverses sign, indicating a redistribution of the temperature gradient consistent with the higher conductivity of the inner layer.

The average dimensionless temperature of the composite cylinder is calculated by

$$\overline{u}\left(\tau \right)=2\left({\int }_0^{\epsilon }\xi { u}_1 d\xi +{\int }_{\epsilon }^1\xi { u}_2 d\xi \right)$$

(30)

Figure 3a indicates that, for small conductivity ratios, the transient evolution of the average dimensionless temperature is sensitive to $k_1/k_2$. However, once $k_1/k_2$ exceeds roughly $0.1$, the response becomes only weakly dependent on further increases in conductivity ratio. In this regime, the thermal behavior is governed primarily by the outer layer (the annulus), and the curves nearly collapse for $k_1/k_2\ge 1$.

The limiting case $k_1/k_2\to \mathrm{\infty }$ (also shown in Figure 3a) yields the fastest heat-transfer rate. At the other extreme, $k_1/k_2\to 0$ represents a lower bound in which the interface effectively behaves as an insulated boundary for the annulus. This may be modeled by replacing condition (12) with ${{\displaystyle \frac{\partial u_2}{\partial \xi }}|}_{\xi =\epsilon }=0$. A closed-form solution for this special case is provided in Appendix A.

Figure 3b illustrates the influence of the Biot number on heat transfer. As $Bi$ increases, the heat-transfer rate increases but approaches an upper limit. This limiting behavior, corresponding to $Bi\to \infty$, is also derived in Appendix A. Physically, $Bi\to \infty$ implies an isothermal outer surface, i.e., a Dirichlet boundary condition obtained by replacing (13) with $u_2(1,\tau )=u_{\infty }(\tau$).

Figure 4 illustrates the influence of the aspect ratio $\epsilon$ on the temporal evolution of $\overline{u}\left(\tau \right)$ for several values of $k_1/k_2$. In Figure 4a, $\epsilon$ is chosen as $\sqrt[3]{{\displaystyle \frac{1}{10}}}, \sqrt[3]{{\displaystyle \frac{2}{10}}}, \dots , \sqrt[3]{{\displaystyle \frac{9}{10}}}$ so that the two regions partition the (normalized) domain into equal-volume increments.

For small $k_1/k_2$, the inner layer offers a large thermal resistance, and decreasing $\epsilon$ (i.e., reducing the inner-layer thickness) leads to a faster overall heat-transfer rate. As $k_1/k_2$ approaches unity, the sensitivity to $\epsilon$, as expected, diminishes. When $k_1=k_2$, the cylinder behaves as a single homogeneous material and the interface location becomes immaterial. When $k_1/k_2>1$, the trend reverses (though the effect is comparatively quite modest): smaller $\epsilon$ corresponds to a slightly slower heat-transfer rate.

A particularly relevant limiting case is $\epsilon \to 1$, which models an extremely thin outer coating. Spray-on coatings are typically thinner than $1 \mathrm{mm}$ and often have thermal conductivities in the range $0.04-0.08 \mathrm{W}/\mathrm{m}.\mathrm{K}$, whereas common metals such as gold, copper, steel, and aluminum have conductivities on the order of 40–500 $\mathrm{W}/\mathrm{m}.\mathrm{K}$ [24,25,26]. The resulting conductivity ratio $k_1/k_2$ therefore ranges from $\mathcal{O}\left({10}^2\right)$ to above $\mathcal{O}\left({10}^4\right)$. Since we have shown that, for $k_1/k_2>1$, the evolution of $\overline{u}\left(\tau \right)$ becomes largely insensitive to further increases in $k_1/k_2$, it is reasonable over this range to assume that $k_1/k_2\to \mathrm{\infty }$.

Figure 5 clearly illustrates the transient evolution of the cross-sectional average temperature, $\overline{u}\left(\tau \right)$, in the limiting case $\epsilon \to 1$ and $k_1/k_2\to \mathrm{\infty }$, highlighting the influence of the Biot number on the thermal response. These curves quantify the transient response of a cylinder coated with an extremely thin, low-conductivity layer.

3.2. Harmonically Fluctuating Ambient Temperature

The dimensionless temperature field under a harmonically fluctuating ambient condition can be evaluated directly from solutions (28) and (29).

In the periodic steady-state regime, the thermal response can be characterized in terms of amplitude attenuation and phase lag relative to the ambient forcing. For a harmonic input of the form $u_{\mathrm{\infty }}\left(\tau \right)=\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_0\tau \right)$, the temperature response at a given radial location may be expressed in the form $u(\xi ,\tau )\approx A\left(\xi \right)\mathrm{c}\mathrm{o}\mathrm{s}({\omega }_0\tau -\varphi (\xi \left)\right),$ where $A\left(\xi \right)$ represents the amplitude (gain) and $\varphi \left(\xi \right)$ denotes the phase lag. Inspection of the analytical solutions (28) and (29) shows that each mode contributes a harmonic response with amplitude proportional to ${\displaystyle \frac{{\lambda }_n^2}{\sqrt{{\lambda }_n^4+{\omega }_0^2}}},$ and phase lag given by ${\varphi }_n={\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}\left({\displaystyle \frac{{\omega }_0}{{\lambda }_n^2}}\right).$ These expressions indicate that increasing the forcing frequency reduces the amplitude and increases the phase lag, consistent with the attenuation and delay observed in the numerical results. The overall response reflects the superposition of modal contributions, with lower modes dominating the long-time periodic behavior.

Figure 6 shows the radial distribution of the dimensionless temperature $u(\xi ,\tau )$ at selected times over one cycle for $k_1/k_2=0.1$, $Bi=1$, and $\epsilon =3/4$: (a) ${\omega }_0=\pi /2$, (b) ${\omega }_0=2\pi$. The curves correspond to different time instants within the cycle. The results show that the start-up transients decay after only a few cycles; in the example shown, a quasi-steady state is essentially established by the fifth cycle. At lower frequency, the thermal oscillations penetrate more deeply into the composite cylinder and exhibit a larger amplitude.

To facilitate comparison among different frequencies, we normalize the cycle length by introducing the variable ${\tau }^{*}={\displaystyle \frac{\tau }{2\pi /{\omega }_0}}$ so that one full cycle has a unit period length. We focus on the quasi-steady (periodic) regime, i.e., after the initial transients have decayed over a few cycles. Figure 7 shows the evolution of the average dimensionless temperature $\overline{u}$ over the fifth cycle for $k_1/k_2=10$, $Bi=1$, and $\epsilon =1/2$. As the frequency increases, the normalized phase lag becomes more pronounced and the oscillation amplitude decreases. The influence of Biot number on both the amplitude and phase lag is illustrated in Figure 8 for $k_1/k_2=1$, ${\omega }_0=\pi /4$, and $\epsilon =1/2$. The limiting solution for very large $Bi$ is provided in Appendix A. As $Bi$ increases, the amplitude of $\overline{u}$ increases, as expected, and the thermal response becomes more immediate, i.e., the phase lag relative to the ambient oscillations is reduced.

Figure 9 compares the average dimensionless temperature response for different aspect ratios $\epsilon$. For large conductivity ratios ($k_1/k_2\gg 1$), variations in $\epsilon$ produce no noticeable change across the Biot numbers considered.

In this regime, increasing $Bi$ primarily amplifies the response and reduces the phase lag (e.g., compare Figure 9b with Figure 9d, or Figure 9a with Figure 9c). In contrast, when $k_1/k_2\ll 1$, the aspect ratio influences both the amplitude and the phase of $\overline{u}$. As shown in Figure 9a, increasing $\epsilon$ increases the relative influence of the low-conductivity inner layer, which slows the thermal response and results in larger delays and more pronounced phase shifts relative to the ambient fluctuations.

4. Conclusions

We presented an analytical solution for transient heat conduction in a two-layer composite cylinder subject to convection at the outer surface with a time-dependent ambient temperature. Using expansions in terms of Bessel functions and invoking Duhamel’s principle, we derived closed-form expressions for the dimensionless temperatures in the inner and outer layers. These expressions are valid for a general time-dependent boundary condition. We specialized the general solution to the case where the ambient temperature varies harmonically about the initial temperature. The expressions reduce, in the limiting case as the frequency ${\omega }_0\to 0$, to the classical constant-ambient problem ($u_{\mathrm{\infty }}=1$).

A parametric investigation clarified how material and boundary parameters control the transient thermal responses. For constant ambient conditions, the conductivity ratio $k_1/k_2$ strongly affects the radial temperature gradients and produces a distinct interfacial “kink” when the conductivities differ, consistent with continuity of heat flux. The average dimensionless temperature is sensitive to $k_1/k_2$ only for small ratios; once $k_1/k_2\gtrsim 0.1$, the evolution becomes largely governed by the outer annulus and changes little for $k_1/k_2\ge 1$. The limiting bounds $k_1/k_2\to 0$ (effectively insulated interface for the annulus) and $k_1/k_2\to \mathrm{\infty }$ (fastest transfer) were identified, with supporting closed-form developments provided in Appendix A. Similarly, increasing the Biot number accelerates heat transfer but approaches an upper bound as $Bi\to \mathrm{\infty }$, corresponding to an isothermal outer surface (Dirichlet condition).

The aspect ratio $\epsilon =a/b$ plays an important role when the inner layer is resistive ($k_1/k_2\ll 1$), where reducing $\epsilon$ generally enhances the heat-transfer rate. Its influence diminishes as $k_1/k_2\to 1$ and becomes quite weak for $k_1/k_2\gg 1$. A practically important regime is when $\epsilon \to 1$, representing extremely thin coatings. For typical low-conductivity coatings on highly conductive metals, $k_1/k_2$ is very large, and our results justify the approximation $k_1/k_2\to \mathrm{\infty }$; the corresponding $\overline{u}\left(\tau \right)$ response under different Biot numbers quantifies the thermal “shielding” effect of thin insulating layers.

Under harmonic fluctuations of the ambient conditions, the solution transitions rapidly to a periodic steady state after only a few cycles. Frequency primarily controls thermal penetration, amplitude, and phase lag: lower ${\omega }_0$ produces deeper penetration and larger oscillation amplitudes, whereas increasing ${\omega }_0$ attenuates the response and increases the phase lag. The Biot number amplifies the oscillation amplitude and reduces the phase delay, yielding a more immediate tracking of ambient fluctuations. Finally, the coupling between $\epsilon$ and $k_1/k_2$ governs the dynamic response: when $k_1/k_2\gg 1$, $\epsilon$ has little effect, but when $k_1/k_2\ll 1$, increasing $\epsilon$ strengthens the influence of the low-conductivity inner layer and produces substantial delays and larger phase shifts.