Heat Transfer in Composite Cylinders Under Harmonically Oscillating Ambient Conditions

Rajai S. Alassar; Mohammed Abushoshah; Husain Al-Attas; Said Al Algarni

doi:10.20944/preprints202603.1665.v1

Submitted:

19 March 2026

Posted:

23 March 2026

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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.

Keywords:

composite cylinder

;

conduction

;

Biot

;

oscillation

;

Duhamel’s principle

Subject:

Engineering - Mechanical Engineering

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 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.

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 reduction and phase shift and to recover the constant-ambient limit as the forcing frequency tends to zero. A parametric study highlights how the conductivity ratio, Biot number, and geometric aspect ratio govern interfacial gradients and mean-temperature evolution.

The Model

Consider a two-layer composite cylinder with inner radius

a

and outer radius

b (b> a)

, initially at a uniform temperature

T_{0}

.

Figure 1. The Problem Configuration.

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 satisfies the transient energy equation:

\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial T}{\partial r}) = \frac{1}{α} \frac{\partial T}{\partial t}

(1)

where

α

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} {\frac{\partial T_{1}}{\partial r}|}_{r = a} = k_{2} {\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} {\frac{\partial T_{2}}{\partial r}|}_{r = b} = h (T_{2} (b, t) - T_{\infty} (t))

(6)

where

h

is the convective heat transfer coefficient and

T_{\infty} (t)

(possibly time-dependent) is the ambient fluid temperature.

We now introduce the following dimensionless variables

ξ = \frac{r}{b}, ε = \frac{a}{b}, τ = \frac{α_{2}}{b^{2}} t, u = \frac{T - T_{0}}{T_{m}}, B i = h b / k_{2}

(7)

where

T_{m}

is a reference temperature scale and

B i

is the Biot number. With these definitions, the system (1)–(6) can be rewritten in dimensionless form as

\frac{\partial^{2} u_{1}}{\partial ξ^{2}} + \frac{1}{ξ} \frac{\partial u_{1}}{\partial ξ} = \frac{α_{2}}{α_{1}} \frac{\partial u_{1}}{\partial τ} and \frac{\partial^{2} u_{2}}{\partial ξ^{2}} + \frac{1}{ξ} \frac{\partial u_{2}}{\partial ξ} = \frac{\partial u_{2}}{\partial τ}

(8)

u_{1} (ξ, 0) = u_{2} (ξ, 0) = 0

(9)

u_{1} (0, τ) < \infty

(10)

u_{1} (ε, τ) = u_{2} (ε, τ)

(11)

k_{1} {\frac{\partial u_{1}}{\partial ξ}|}_{ξ = ε} = k_{2} {\frac{\partial u_{2}}{\partial ξ}|}_{ξ = ε}

(12)

{[\frac{\partial u_{2}}{\partial ξ} + B i u_{2} (ξ, τ)]}_{ξ = 1} = B i u_{\infty} (τ)

(13)

To construct a solution, we first consider an auxiliary problem in which

φ (ξ, τ)

satisfies the system (8)–(13), except that the forcing term

B i u_{\infty} (τ)

on the right-hand side of (13) is replaced by unity.

Defining

w (ξ, τ) = φ (ξ, τ) - \frac{1}{B i}

, the system (8)–(13) can be recast in terms of

w (ξ, τ)

as follows:

\frac{\partial^{2} w_{1}}{\partial ξ^{2}} + \frac{1}{ξ} \frac{\partial w_{1}}{\partial ξ} = \frac{α_{2}}{α_{1}} \frac{\partial w_{1}}{\partial τ}

and

\frac{\partial^{2} w_{2}}{\partial ξ^{2}} + \frac{1}{ξ} \frac{\partial w_{2}}{\partial ξ} = \frac{\partial w_{2}}{\partial τ}

(14)

w_{1} (ξ, 0) = w_{2} (ξ, 0) = - \frac{1}{B i}

(15)

w_{1} (0, τ) < \infty

(16)

w_{1} (ε, τ) = w_{2} (ε, τ)

(17)

k_{1} {\frac{\partial w_{1}}{\partial ξ}|}_{ξ = ε} = k_{2} {\frac{\partial w_{2}}{\partial ξ}|}_{ξ = ε}

(18)

{[\frac{\partial w_{2}}{\partial ξ} + B i w_{2} (ξ, τ)]}_{ξ = 1} = 0

(19)

Equations (14) admit 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}

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

w_{1} (ξ, τ) = \sum_{n = 1}^{\infty} A_{n} J_{0} (λ_{n} \sqrt{\frac{α_{2}}{α_{1}}} ξ) e^{- λ_{n}^{2} τ}

(20)

w_{2} (ξ, τ) = \sum_{n = 1}^{\infty} A_{n} Ω_{1} (J_{0} (λ_{n} ξ) + Ω_{0} Y_{0} (λ_{n} ξ)) e^{- λ_{n}^{2} τ}

(21)

where

Ω_{0} = \frac{λ_{n} J_{1} (λ_{n}) - B i J_{0} (λ_{n})}{- λ_{n} Y_{1} (λ_{n}) + B i Y_{0} (λ_{n})}

and

Ω_{1} = \frac{J_{0} (λ_{n} \sqrt{\frac{α_{2}}{α_{1}}} ε)}{J_{0} (λ_{n} ε) + Ω_{0} Y_{0} (λ_{n} ε)}

.

Equation (18) is used to determine the eigenvalues

λ_{n}

,

\frac{k_{1}}{k_{2}} \sqrt{\frac{α_{2}}{α_{1}}} J_{1} (λ_{n} \sqrt{\frac{α_{2}}{α_{1}}} ε) - Ω_{1} (J_{1} (λ_{n} ε) + Ω_{0} Y_{1} (λ_{n} ε)) = 0

(22)

where

J_{1}

and

Y_{1}

denote the first- and second-kind Bessel functions of order one, respectively. The coefficients in (20) and (21) are determined by enforcing the initial condition (15). It can be shown that

A_{n} = - \frac{1}{B i} \frac{\frac{k_{1}}{α_{1}} \int_{0}^{ε} ξ R_{1 n} d ξ + \frac{k_{2}}{α_{2}} \int_{ε}^{1} ξ R_{2 n} d ξ}{\frac{k_{1}}{α_{1}} \int_{0}^{ε} ξ R_{1 n}^{2} d ξ + \frac{k_{2}}{α_{2}} \int_{ε}^{1} ξ R_{2 n}^{2} d ξ}

(23)

where

R_{1 n} (ξ) = J_{0} (λ_{n} \sqrt{\frac{α_{2}}{α_{1}}} ξ)

and

R_{2 n} (ξ) = Ω_{1} (J_{0} (λ_{n} ξ) + Ω_{0} Y_{0} (λ_{n} ξ))

.

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}

is provided in the Appendix.

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

\frac{k_{1}}{α_{1}} \int_{0}^{ε} ξ R_{1 m} R_{1 n} d ξ + \frac{k_{2}}{α_{2}} \int_{ε}^{1} ξ R_{2 m} R_{2 n} d ξ = 0 if n \neq m

(24)

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

u (ξ, τ) = \int_{η = 0}^{τ} B i u_{\infty} (η) \frac{\partial φ (ξ, τ - η)}{\partial τ} d η = - B i \int_{η = 0}^{τ} u_{\infty} (η) \frac{\partial w (ξ, τ - η)}{\partial η} d η

(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_{\infty} = const .

), the integral simplifies and the solution reduces to

B i u_{\infty} φ (ξ, τ)

.

Discussion

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

T_{\infty} (t) = T_{0} + T_{m} \cos (ν_{0} t)

(26)

where

ν_{0}

is the frequency.

The corresponding dimensionless ambient temperature is

u_{\infty} (τ) = \frac{T_{\infty} (τ) - T_{0}}{T_{m}} = \cos (ω_{0} τ)

(27)

where

ω_{0} = \frac{b^{2} ν_{0}}{α_{2}}

is the dimensionless frequency. Applying Duhamel’s principle to solutions (20) and (21) then yields expressions of the form

u_{1} (ξ, τ) = - B i \sum_{n = 1}^{\infty} A_{n} λ_{n}^{2} J_{0} (\sqrt{\frac{α_{2}}{α_{1}}} λ_{n} ξ) (\frac{- ⅇ^{- λ_{n}^{2} τ} λ_{n}^{2} + λ_{n}^{2} \cos (ω_{0} τ) + ω_{0} \sin (ω_{0} τ)}{λ_{n}^{4} + ω_{0}^{2}})

(28)

u_{2} (ξ, τ) = - B i \sum_{n = 1}^{\infty} A_{n} λ_{n}^{2} \frac{- ⅇ^{- λ_{n}^{2} τ} λ_{n}^{2} + λ_{n}^{2} \cos (ω_{0} τ) + ω_{0} \sin (ω_{0} τ)}{λ_{n}^{4} + ω_{0}^{2}} Ω_{1} (J_{0} (λ_{n} ξ) + Ω_{0} Y_{0} (λ_{n} ξ))

(29)

Constant Ambient Temperature:

When the frequency vanishes (

ω_{0} = 0

), the problem reduces to a composite cylinder exposed to a constant ambient temperature (

u_{\infty} = 1

). The corresponding solution is obtained directly from (28) and (29) by setting

ω_{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 [15], the ratios

k_{1} / k_{2}

and

α_{1} / α_{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

B i = 1

and

ε = 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.

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} \geq 1

.

The limiting case

k_{1} / k_{2} \to \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

{\frac{\partial u_{2}}{\partial ξ}|}_{ξ = ε} = 0

. A closed-form solution for this special case is provided in the Appendix.

Figure 3b illustrates the influence of the Biot number on heat transfer. As

B i

increases, the heat-transfer rate increases but approaches an upper limit. This limiting behavior, corresponding to

B i \to \infty

, is also derived in the Appendix. Physically,

B i \to \infty

implies an isothermal outer surface, i.e., a Dirichlet boundary condition obtained by replacing (13) with

u_{2} (1, τ) = u_{\infty} (τ

).

The average dimensionless temperature of the composite cylinder is calculated by

\bar{u} (τ) = 2 (\int_{0}^{ε} ξ u_{1} d ξ + \int_{ε}^{1} ξ u_{2} d ξ)

(30)

Figure 4 illustrates the influence of the aspect ratio

ε

on the temporal evolution of

\bar{u} (τ)

for several values of

k_{1} / k_{2}

. In Figure 4a,

ε

is chosen as

\sqrt[3]{\frac{1}{10}}, \sqrt[3]{\frac{2}{10}}, \dots, \sqrt[3]{\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

ε

(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

ε

, 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

ε

corresponds to a slightly slower heat-transfer rate.

A particularly relevant limiting case is

ε \to 1

, which models an extremely thin outer coating. Spray-on coatings are typically thinner than

1 mm

and often have thermal conductivities in the range

0.04 - 0.08 W / m . K

, whereas common metals such as gold, copper, steel, and aluminum have conductivities on the order of 40-500

W / m . K

[16,17,18]. The resulting conductivity ratio

k_{1} / k_{2}

therefore ranges from

O (10^{2})

to above

O (10^{4})

. Since we have shown that, for

k_{1} / k_{2} > 1

, the evolution of

\bar{u} (τ)

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 \infty

.

Figure 5 presents

\bar{u} (τ)

for the limiting case

ε \to 1

and

k_{1} / k_{2} \to \infty

under different Biot numbers. These curves quantify the transient response of a cylinder coated with an extremely thin, low-conductivity layer.

Harmonically Fluctuating Ambient Temperature:

The dimensionless temperature field under a harmonically fluctuating ambient condition can be evaluated directly from solutions (28) and (29). Figure 6 presents radial profiles of the dimensionless temperature at several instants over a single period. 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. The case considered corresponds to

k_{1} / k_{2} = 0.1

,

B i = 1

, and

ε = 3 / 4

, with frequencies

ω_{0} = π / 2

and

ω_{0} = 2 π

. At the 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

τ^{*} = \frac{τ}{2 π / ω_{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

\bar{u}

over the fifth cycle for

k_{1} / k_{2} = 10

,

B i = 1

, and

ε = 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

,

ω_{0} = π / 4

, and

ε = 1 / 2

. The limiting solution for very large

B i

is provided in the Appendix. As

B i

increases, the amplitude of

\bar{u}

increases, as expected, and the thermal response becomes more immediate, i.e., the phase lag relative to the ambient oscillations is reduced.

,

B i = 1

,

ε = 1 / 2 .

Figure 9 compares the average dimensionless temperature response for different aspect ratios

ε

. For large conductivity ratios (

k_{1} / k_{2} ≫ 1

), variations in

ε

produce no noticeable change across the Biot numbers considered. In this regime, increasing

B i

primarily amplifies the response and reduces the phase lag (e.g., compare Figure 9b with 9d, or Figure 9a with 9c). In contrast, when

k_{1} / k_{2} ≪ 1

, the aspect ratio influences both the amplitude and the phase of

\bar{u}

. As shown in Figure 9a, increasing

ε

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.

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

ω_{0} \to 0

, to the classical constant-ambient problem (

u_{\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} ≳ 0.1

, the evolution becomes largely governed by the outer annulus and changes little for

k_{1} / k_{2} \geq 1

. The limiting bounds

k_{1} / k_{2} \to 0

(effectively insulated interface for the annulus) and

k_{1} / k_{2} \to \infty

(fastest transfer) were identified, with supporting closed-form developments provided in the Appendix. Similarly, increasing the Biot number accelerates heat transfer but approaches an upper bound as

B i \to \infty

, corresponding to an isothermal outer surface (Dirichlet condition).

The aspect ratio

ε = a / b

plays an important role when the inner layer is resistive (

k_{1} / k_{2} ≪ 1

), where reducing

ε

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} ≫ 1

. A practically important regime is when

ε \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 \infty

; the corresponding

\bar{u} (τ)

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

ω_{0}

produces deeper penetration and larger oscillation amplitudes, whereas increasing

ω_{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

ε

and

k_{1} / k_{2}

governs the dynamic response: when

k_{1} / k_{2} ≫ 1

,

ε

has little effect, but when

k_{1} / k_{2} ≪ 1

, increasing

ε

strengthens the influence of the low-conductivity inner layer and produces substantial delays and larger phase shifts.

Acknowledgments

The author would like to acknowledge King Fahd University of Petroleum & Minerals (KFUPM) for supporting this research.

Appendix A

The general problem under consideration in dimensionless form is that described by equations (8-13). We consider the following limiting/special cases.

A) The Case

B i \to \infty

The only change is in equation (13) in the general system. The system may be rewritten as

\frac{\partial^{2} u_{1}}{\partial ξ^{2}} + \frac{1}{ξ} \frac{\partial u_{1}}{\partial ξ} = \frac{α_{2}}{α_{1}} \frac{\partial u_{1}}{\partial τ}

(A1)

\frac{\partial^{2} u_{2}}{\partial ξ^{2}} + \frac{1}{ξ} \frac{\partial u_{2}}{\partial ξ} = \frac{\partial u_{2}}{\partial τ}

(A2)

u_{1} (ξ, 0) = u_{2} (ξ, 0) = 0

(A3)

u_{1} (0, τ) < \infty

(A4)

u_{1} (ε, τ) = u_{2} (ε, τ)

(A5)

k_{1} {\frac{\partial u_{1}}{\partial ξ}|}_{ξ = ε} = k_{2} {\frac{\partial u_{2}}{\partial ξ}|}_{ξ = ε}

(A6)

u_{2} (1, τ) = u_{\infty} (τ)

(A7)

Let

φ (ξ, τ)

be the solution for the system with the right-hand side of equation (A7) set to 1. Following the same procedure, one can show that

φ_{1} (ξ, τ) = 1 + \sum_{n = 1}^{\infty} A_{n} J_{0} (\sqrt{\frac{α_{2}}{α_{1}}} λ_{n} ξ) e^{- λ_{n}^{2} τ}

(A8)

φ_{2} (ξ, τ) = 1 + \sum_{n = 1}^{\infty} A_{n} ψ_{1} [J_{0} (λ_{n} ξ) + ψ_{0} Y_{0} (λ_{n} ξ)] e^{- λ_{n}^{2} τ}

(A9)

where

ψ_{0} = - \frac{J_{0} (λ_{n})}{Y_{0} (λ_{n})}

and

ψ_{1} = \frac{J_{0} (\sqrt{\frac{α_{2}}{α_{1}}} λ_{n} ε)}{J_{0} (λ_{n} ε) + Ω_{0} Y_{0} (λ_{n} ε)}

. The coefficients

A_{n}

are given by

A_{n} = - \frac{\frac{k_{1}}{α_{1}} \int_{0}^{ε} ξ R_{1 n} d ξ + \frac{k_{2}}{α_{2}} \int_{ε}^{1} ξ R_{2 n} d ξ}{\frac{k_{1}}{α_{1}} \int_{0}^{ε} ξ R_{1 n}^{2} d ξ + \frac{k_{2}}{α_{2}} \int_{ε}^{1} ξ R_{2 n}^{2} d ξ}

(A10)

where

R_{1 n} (ξ) = J_{0} (λ_{n} \sqrt{\frac{α_{2}}{α_{1}}} ξ)

and

R_{2 n} (ξ) = ψ_{1} [J_{0} (λ_{n} ξ) + ψ_{0} Y_{0} (λ_{n} ξ)]

. The expression for

A_{n}

is easily calculated in closed form. We use (A6) to determine the eigenvalues

λ_{n}

,

\frac{k_{1}}{k_{2}} \sqrt{\frac{α_{2}}{α_{1}}} J_{1} (λ_{n} \sqrt{\frac{α_{2}}{α_{1}}} ε) - ψ_{1} (J_{1} (λ_{n} ε) + ψ_{0} Y_{1} (λ_{n} ε)) = 0

(A11)

Using Duhamel’s principle, the solution is

u (ξ, τ) = \int_{η = 0}^{τ} u_{\infty} (η) \frac{\partial φ (ξ, τ - η)}{\partial τ} d η

(A12)

B) The Case

ε = 0

In this case, the general system reduces to

\frac{\partial^{2} u_{2}}{\partial ξ^{2}} + \frac{1}{ξ} \frac{\partial u_{2}}{\partial ξ} = \frac{\partial u_{2}}{\partial τ}

(B1)

u_{2} (ξ, 0) = 0

(B2)

u_{2} (0, τ) < \infty

(B3)

{[\frac{\partial u_{2}}{\partial ξ} + B i u_{2} (ξ, τ)]}_{ξ = 1} = B i u_{\infty} (τ)

(B4)

Let

φ_{2} (ξ, τ)

be the solution for the system with the right-hand side of equation (B4) set to 1. Following the same procedure, one can show that

φ_{2} (ξ, τ) = \frac{1}{B i} + \sum_{n = 1}^{\infty} A_{n} J_{0} (λ_{n} ξ) e^{- λ_{n}^{2} τ}

(B5)

A_{n} = \frac{- 2 J_{1} (λ_{n})}{B i λ_{n} (J_{0}^{2} (λ_{n}) + J_{1}^{2} (λ_{n}))}

(B6)

The eigenvalues are determined by

- λ_{n} J_{1} (λ_{n}) + B i J_{0} (λ_{n}) = 0

(B7)

C) The Case

ε = 1

In this case, the general system reduces to

\frac{\partial^{2} u_{1}}{\partial ξ^{2}} + \frac{1}{ξ} \frac{\partial u_{1}}{\partial ξ} = \frac{α_{2}}{α_{1}} \frac{\partial u_{1}}{\partial τ}

(C1)

u_{1} (ξ, 0) = 0

(C2)

u_{1} (0, τ) < \infty

(C3)

{[\frac{\partial u_{1}}{\partial ξ} + \frac{k_{2}}{k_{1}} B i u_{1} (ξ, τ)]}_{ξ = 1} = \frac{k_{2}}{k_{1}} B i u_{\infty} (τ)

(C4)

This equivalent to Case B except for the scaling by

α_{2}

in (C1).

Let

φ_{1} (ξ, τ)

be the solution for the system with the right-hand side of equation (C4) set to 1. Following the same procedure, one can show that

φ_{1} (ξ, τ) = \frac{1}{\frac{k_{2}}{k_{1}} B i} + \sum_{n = 1}^{\infty} A_{n} J_{0} ({\sqrt{α_{2} / α_{1}} λ}_{n} ξ) e^{- λ_{n}^{2} τ}

(C5)

A_{n} = \frac{- 2 J_{1} (\sqrt{α_{2} / α_{1}} λ_{n})}{\frac{k_{2}}{k_{1}} B i λ_{n} \sqrt{α_{2} / α_{1}} (J_{0}^{2} (\sqrt{α_{2} / α_{1}} λ_{n}) + J_{1}^{2} (\sqrt{α_{2} / α_{1}} λ_{n}))}

(C6)

The eigenvalues are determined by

- λ_{n} \sqrt{\frac{α_{2}}{α_{1}}} J_{1} (\sqrt{\frac{α_{2}}{α_{1}}} λ_{n}) + \frac{k_{2}}{k_{1}} B i J_{0} (\sqrt{α_{2} / α_{1}} λ_{n}) = 0

(C7)

Using Duhamel’s principle, the solution is

u (ξ, τ) = \int_{η = 0}^{τ} \frac{k_{2}}{k_{1}} B i u_{\infty} (η) \frac{\partial φ (ξ, τ - η)}{\partial τ} d η

(C8)

D) The Case

\frac{k_{1}}{k_{2}} \to 0

In this case, the interface works as an insulator and the general system reduces to

\frac{\partial^{2} u_{2}}{\partial ξ^{2}} + \frac{1}{ξ} \frac{\partial u_{2}}{\partial ξ} = \frac{\partial u_{2}}{\partial τ}

(D1)

u_{2} (ξ, 0) = 0

(D2)

{\frac{\partial u_{2}}{\partial ξ}|}_{ξ = ε} = 0

(D3)

{[\frac{\partial u_{2}}{\partial ξ} + B i u_{2} (ξ, τ)]}_{ξ = 1} = B i u_{\infty} (τ)

(D4)

Let

φ_{2} (ξ, τ)

be the solution for the system with the right-hand side of equation (D4) set to 1. Following the same procedure, one can show that

φ_{2} (ξ, τ) = \frac{1}{B i} + \sum_{n = 1}^{\infty} A_{n} R_{n} (ξ) e^{- λ_{n}^{2} τ}

(D5)

where

R_{n} (ξ) = J_{0} (λ_{n} ξ) + \frac{- B i J_{0} (λ_{n}) + λ_{n} J_{1} (λ_{n})}{B i Y_{0} (λ_{n}) - λ_{n} Y_{1} (λ_{n})} Y_{0} (λ_{n} ξ)

, and

A_{n}

are found by

A_{n} = \frac{- 1}{B i} \frac{\int_{ε}^{1} ξ R_{n} (ξ) d ξ}{\int_{ε}^{1} ξ R_{n}^{2} (ξ) d ξ}

(D6)

The eigenvalues are determined by

J_{1} (λ_{n} ε) + \frac{- B i J_{0} (λ_{n}) + λ_{n} J_{1} (λ_{n})}{B i Y_{0} (λ_{n}) - λ_{n} Y_{1} (λ_{n})} Y_{1} (λ_{n} ε) = 0

(D7)

Using Duhamel’s principle, the solution is

u (ξ, τ) = \int_{η = 0}^{τ} B i u_{\infty} (η) \frac{\partial φ (ξ, τ - η)}{\partial τ} d η

(D8)

E) Mathematica commands to evaluate the coefficients

Ω0 = (δ*BesselJ [1, δ] − Bi*BesselJ[0, δ])/ ((-δ)*BesselY[1, δ] + Bi*BesselY[0, δ]);

Ω1 = BesselJ[0, Sqrt[α2/α1]*δ*ϵ]/(BesselJ[0, δ*ϵ] + Ω0*BesselY[0, δ*ϵ]);

R1 = BesselJ[0, Sqrt[α2/α1]*δ*ξ];

R2 = Ω1*(BesselJ[0, δ*ξ] + Ω0*BesselY[0, δ*ξ]);

(Integrate[ξ*R1, {ξ, 0, ϵ}] + (k2/(k1*(α2/α1)))*Integrate[ξ*R2, {ξ, ϵ, 1}])/(Integrate[ξ*R1^2, {ξ, 0, ϵ}] + (k2/(k1*(α2/α1)))*Integrate[ξ*R2^2, {ξ, ϵ, 1}])

References

Carslaw, H.S.; Jaeger, J.C.; Feshbach, H. Conduction of Heat in Solids. Phys. Today 1962, 15, 74–76. [CrossRef]
M. N. Ozisik, Heat Conduction, 2nd ed. New York, NY, USA: John Wiley & Sons, 1993.
J. Crank, The Mathematics of Diffusion, 2nd ed. Oxford, UK: Oxford University Press, 1975.
F. P. Incropera, D. P. DeWitt, T. L. Bergman, and A. S. Lavine, Fundamentals of Heat and Mass Transfer, 7th ed. Hoboken, NJ, USA: Wiley, 2011.
International Electrotechnical Commission, 2023, Electric cables—Calculation of the current rating—Part 1-1: Current rating equations (100% load factor) and calculation of losses—General, IEC 60287-1-1:2023, Ed. 3.0, May 22, 2023.
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Figure 2. Dimensionless temperature along the radius for the case

B i = 1

,

ε = 1 / 2

,

ω_{0} = 0 .

(a)

k_{1} / k_{2} = 0.001

(b)

k_{1} / k_{2} = 0.1

(c)

k_{1} / k_{2} = 1

(d)

k_{1} / k_{2} = 10 .

Figure 2. Dimensionless temperature along the radius for the case

B i = 1

,

ε = 1 / 2

,

ω_{0} = 0 .

(a)

k_{1} / k_{2} = 0.001

(b)

k_{1} / k_{2} = 0.1

(c)

k_{1} / k_{2} = 1

(d)

k_{1} / k_{2} = 10 .

Figure 3. Development of the average temperature for

ε = \frac{1}{2}

, (a)

B i = 1

, (b)

k_{1} / k_{2} = 0.1 .

Figure 3. Development of the average temperature for

ε = \frac{1}{2}

, (a)

B i = 1

, (b)

k_{1} / k_{2} = 0.1 .

Figure 4. Impact of aspect ratio on the average dimensionless temperature,

B i = 1

. (a)

k_{1} / k_{2} = 0.01

(b)

k_{1} / k_{2} = 0.2

(c)

k_{1} / k_{2} = 1

(d)

k_{1} / k_{2} = 100 .

Figure 4. Impact of aspect ratio on the average dimensionless temperature,

B i = 1

. (a)

k_{1} / k_{2} = 0.01

(b)

k_{1} / k_{2} = 0.2

(c)

k_{1} / k_{2} = 1

(d)

k_{1} / k_{2} = 100 .

Figure 5. Average dimensionless temperature,

ε \to 1

and

k_{1} / k_{2} \to \infty .

Figure 5. Average dimensionless temperature,

ε \to 1

and

k_{1} / k_{2} \to \infty .

Figure 6. Development of the dimensionless temperature

k_{1} / k_{2} = 0.1

,

B i = 1

,

ε = 3 / 4 .

(a)

ω_{0} = π / 2

, (b)

ω_{0} = 2 π .

Figure 6. Development of the dimensionless temperature

k_{1} / k_{2} = 0.1

,

B i = 1

,

ε = 3 / 4 .

(a)

ω_{0} = π / 2

, (b)

ω_{0} = 2 π .

Figure 7. Average dimensionless temperature over a cycle,

k_{1} / k_{2} = 10

Figure 7. Average dimensionless temperature over a cycle,

k_{1} / k_{2} = 10

Figure 8. Average dimensionless temperature over a cycle,

k_{1} / k_{2} = 1

,

ω_{0} = π / 4

,

ε = 1 / 2 .

Figure 8. Average dimensionless temperature over a cycle,

k_{1} / k_{2} = 1

,

ω_{0} = π / 4

,

ε = 1 / 2 .

Figure 9. Average dimensionless temperature over a cycle,

ω_{0} = π / 16

. (a)

B i = 10

,

\frac{k_{1}}{k_{2}} = 0.05

, (b)

B i = 10

,

\frac{k_{1}}{k_{2}} = 20

, (c)

B i = 0.1

,

\frac{k_{1}}{k_{2}} = 0.05

, (d)

B i = 0.1

,

\frac{k_{1}}{k_{2}} = 20 .

Figure 9. Average dimensionless temperature over a cycle,

ω_{0} = π / 16

. (a)

B i = 10

,

\frac{k_{1}}{k_{2}} = 0.05

, (b)

B i = 10

,

\frac{k_{1}}{k_{2}} = 20

, (c)

B i = 0.1

,

\frac{k_{1}}{k_{2}} = 0.05

, (d)

B i = 0.1

,

\frac{k_{1}}{k_{2}} = 20 .

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Heat Transfer in Composite Cylinders Under Harmonically Oscillating Ambient Conditions

Abstract

Keywords:

Subject:

Introduction

The Model

Discussion

Conclusions

Acknowledgments

Appendix A

References

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