Analytical Solution of Heat Conduction for Hollow Cylinders with Time-Dependent Boundary Condition and Time-Dependent Heat Transfer Coefficient

Abstract

An analytical solution for the heat transfer in hollow cylinders with time-dependent boundary condition and time-dependent heat transfer coefficient at different surfaces is developed for the first time. The methodology is an extension of the shifting function method. By dividing the Biot function into a constant plus a function and introducing two specially chosen shifting functions, the system is transformed into a partial differential equation with homogenous boundary conditions only. The transformed system is thus solved by series expansion theorem. Limiting cases of the solution are studied and numerical results are compared with those in the literature. The convergence rate of the present solution is fast and the analytical solution is simple and accurate. Also, the influence of physical parameters on the temperature distribution of a hollow cylinder along the radial direction is investigated.

Te-Wen Tu 1 and Sen-Yung Lee 2

Academic Editor:Assunta Andreozzi

1, Department of Mechanical Engineering, Air Force Institute of Technology, No. 198 Jieshou W. Road, Gangshan Township, Kaohsiung 820, Taiwan
2, Department of Mechanical Engineering, National Cheng Kung University, No. 1 University Road, Tainan 701, Taiwan

Received 5 May 2015; Accepted 21 July 2015; 1 October 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The problems of transient heat flow in hollow cylinders are important in many engineering applications. Heat exchanger tubes, solidification of metal tube casting, cannon barrels, time variation heating on walls of circular structure, and heat treatment on hollow cylinders are typical examples. It is well known that if the temperature and/or the heat flux are prescribed at the boundary surface, then the heat transfer system includes heat conduction coefficient only; on the other hand, if the boundary surface dissipates heat by convection on the basis of Newton's law of cooling, the heat transfer coefficient will be included in the boundary term.

For the problem of heat conduction in hollow cylinders with time-dependent boundary conditions of any kinds at inner and outer surfaces, the associated governing differential equation is a second-order Bessel differential equation with constant coefficients. After conducting a Hankel transformation, the analytical solutions can be obtained and found in the textbook by Özisik [1].

For the heat transfer in hollow cylinders with mixed type boundary condition and time-dependent heat transfer coefficient simultaneously, the problem cannot be solved by any analytical methods, such as the method of separation of variable, Laplace transform, and Hankel transform. Few studies in Cartesian coordinate system can be found and various approximated and numerical methods were proposed. By introducing a new variable, Ivanov and Salomatov [2, 3] together with Postol'nik [4] transformed the linear governing equation into a nonlinear form. After ignoring the nonlinear term, they developed an approximated solution, which was claimed to be valid for the system with Biot number being less than 0.25. Moreover, Kozlov [5] used Laplace transformation to study the problems with Biot function in a rational combination of sines, cosines, polynomials, and exponentials. Even though it is possible to obtain the exact series solution of a specified transformed system, the problem is the computation of the inverse Laplace transformation, which generally requires integration in the complex plane. Becker et al. [6] took finite difference method and Laplace transformation method to study the heating of the rock adjacent to water flowing through a crevice. Recently, Chen and his colleagues [7] proposed an analytical solution by using the shifting function method for the heat conduction in a slab with time-dependent heat transfer coefficient at one end. Yatskiv et al. [8] studied the thermostressed state of cylinder with thin near-surface layer having time-dependent thermophysical properties. They reduced the problem to an integrodifferential equation with variable coefficients and solved it by the spline approximation.

In addition, different approximation methods such as the iterative perturbation method [9], the time-varying eigenfunction expansion method with finite integral transforms [10, 11], generalized integral transforms [12], and the Lie point symmetry analysis [13] were used to study this kind of problems. Various inverse schemes for determining the time-dependent heat transfer coefficient were developed by some researchers [14-20].

According to the literature, because of the complexity and difficulty of the methodology, none of any analytical solutions for the heat conduction in a hollow cylinder with time-dependent boundary condition and time-dependent heat transfer coefficient existed. This work extends the methodology of shifting function method [7, 21, 22] to develop an analytical solution with closed form for the heat transfer in hollow cylinders with time-dependent boundary condition and time-dependent heat transfer coefficient simultaneously. By setting the Biot function in a particular form and introducing two specially chosen shifting functions, the system is transformed into a partial differential equation with homogenous boundary conditions and can be solved by series expansion theorem. Examples are given to demonstrate the methodology and numerical results are compared with those in the literature. And last but not least, the influence of physical parameters on the temperature profile is studied.

2. Mathematical Modeling

Consider the transient heat conduction in heat exchanger tubes as shown in Figure 1. A fluid with time-varying temperature is running inside the hollow cylinder and the heat is dissipated by the time-dependent convection at the outer surface into an environment of zero temperature. The governing differential equation of the system is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the temperature, [figure omitted; refer to PDF] is the space variable, [figure omitted; refer to PDF] is the thermal diffusivity, [figure omitted; refer to PDF] is the time variable, and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote inner and outer radii, respectively. The boundary and initial conditions of the boundary value problem are [figure omitted; refer to PDF] Here, [figure omitted; refer to PDF] is a time-dependent temperature function at the inner surface, [figure omitted; refer to PDF] is the thermal conductivity, [figure omitted; refer to PDF] is a time-dependent heat transfer coefficient function, and [figure omitted; refer to PDF] is an initial temperature function. For consistence in initial temperature field, [figure omitted; refer to PDF] must be equal to [figure omitted; refer to PDF] . The above problem can be normalized by defining [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a constant reference temperature, and the dimensionless boundary value problem will then become [figure omitted; refer to PDF]

Figure 1: Hollow cylinders with time-dependent temperature and time-dependent heat transfer coefficient at inner and outer surfaces.

[figure omitted; refer to PDF]

To keep the boundary condition of the third kind at outer surface in the following analysis, one sets the Biot function [figure omitted; refer to PDF] in the form of [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are defined as [figure omitted; refer to PDF]

It is obvious that [figure omitted; refer to PDF] , and the boundary condition at [figure omitted; refer to PDF] can be rewritten as [figure omitted; refer to PDF]

3. The Shifting Function Method

3.1. Change of Variable

To find the solution for the second-order differential equation with time-dependent boundary condition and time-dependent heat transfer coefficient at different surfaces, the shifting function method [7, 21, 22] was extended by taking [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is called transformed function, [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) are two shifting functions to be specified, and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) are the auxiliary time functions defined as [figure omitted; refer to PDF]

Substituting (11) into (4), (5), (10), and (7), one has the following equation: [figure omitted; refer to PDF] and the associated boundary and initial conditions now are [figure omitted; refer to PDF]

Something worthy to mention is that (13) contains three functions, that is, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), and hence it cannot be solved directly.

3.2. The Shifting Functions

For convenience in the analysis, the two shifting functions are specifically chosen in order to satisfy the following conditions: [figure omitted; refer to PDF] Consequently, the shifting functions can be easily determined as [figure omitted; refer to PDF] Substituting these shifting functions and auxiliary time functions into (11) yields [figure omitted; refer to PDF] When setting [figure omitted; refer to PDF] in the equation above, one has the relation [figure omitted; refer to PDF] Therefore, two functions in governing differential equation (13) are integrated to one. With (16) and (18), (13) can be rewritten in terms of the function [figure omitted; refer to PDF] as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are defined as [figure omitted; refer to PDF] Meanwhile, the associated boundary conditions of the transformed function turn to homogeneous ones as follows: [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , hence, the associated initial condition can be simplified as [figure omitted; refer to PDF]

3.3. Series Expansion

To find the solution for the boundary value problem of heat conduction, that is, (19)-(22), one uses the method of series expansion with try functions: [figure omitted; refer to PDF] satisfying the boundary conditions (21). Here the characteristic values [figure omitted; refer to PDF] are the roots of the transcendental equation [figure omitted; refer to PDF] The try functions have the following orthogonal property: [figure omitted; refer to PDF] where the norms [figure omitted; refer to PDF] are [figure omitted; refer to PDF]

Now, one can assume that the solution of the physical problem takes the form of [figure omitted; refer to PDF] where [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) are time-dependent generalized coordinates. Substituting solution from (27) into differential equation (19) leads to [figure omitted; refer to PDF] Expanding [figure omitted; refer to PDF] and [figure omitted; refer to PDF] on the right hand side of (28) in series forms we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) are given as [figure omitted; refer to PDF] From (29), one can let [figure omitted; refer to PDF] After taking the inner product with try function [figure omitted; refer to PDF] and integrating from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , the resulting differential equation now is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is [figure omitted; refer to PDF] The associated initial condition is [figure omitted; refer to PDF] As a result, the complete solution of the ordinary differential equation (33) subject to the initial condition (36) is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is [figure omitted; refer to PDF]

After substituting (16), (18), (23), and (27) back to (11), one obtains the analytical solution of the physical problem [figure omitted; refer to PDF] where the summation is taken over all eigenvalues [figure omitted; refer to PDF] of the problem.

3.4. Constant Heat Transfer Coefficient at [figure omitted; refer to PDF]

When the heat transfer coefficient [figure omitted; refer to PDF] at [figure omitted; refer to PDF] is time-independent, the Biot function is a constant [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The infinite series solution, (39), is reduced to [figure omitted; refer to PDF] where the generalized coordinates [figure omitted; refer to PDF] are [figure omitted; refer to PDF] The [figure omitted; refer to PDF] 's for the problem under consideration are [figure omitted; refer to PDF] Introducing (42) in (41) and performing integration by parts, we can get [figure omitted; refer to PDF] Substituting (43) into (40) yields the temperature distribution: [figure omitted; refer to PDF] This solution is the same as that obtained via the integral transform method by Özisik [1].

4. Verification and Example

To illustrate the previous analysis and the accuracy of the three-term approximation solution, one examines the following case.

The time-dependent boundary condition [figure omitted; refer to PDF] considered at [figure omitted; refer to PDF] is taken as [figure omitted; refer to PDF] and differentiating it with respect to [figure omitted; refer to PDF] leads to [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are two arbitrary constants and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are two parameters.

The Biot function considered at boundary [figure omitted; refer to PDF] is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are two arbitrary constants and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are two parameters. According to (8)-(9), we obtain [figure omitted; refer to PDF]

Consequently, the temperature distribution in the hollow cylinder is [figure omitted; refer to PDF] where the [figure omitted; refer to PDF] 's are defined in (37). The associated [figure omitted; refer to PDF] now is [figure omitted; refer to PDF]

To avoid numerical instability that occurred in computing [figure omitted; refer to PDF] , (37) is rewritten as [figure omitted; refer to PDF]

Since the initial conditions cannot have effect on the steady-state response, we consider only the heat conduction in a hollow cylinder with constant initial temperature [figure omitted; refer to PDF] as prescribed in the previous sections. The [figure omitted; refer to PDF] 's are now computed as [figure omitted; refer to PDF] For consistence in the temperature field, the constant [figure omitted; refer to PDF] is taken as zero in the following examples.

In comparison with the literature, the example of constant Biot function is studied first. [figure omitted; refer to PDF] and time-dependent temperature function, [figure omitted; refer to PDF] , are chosen in the case. In Table 1, we find that the convergence of the present solution is faster than that of Özisik [1]. The error of three-term approximation in present study is less than [figure omitted; refer to PDF] ; on the contrary, at least twenty-term approximation is required to get the same accuracy in Özisik's [1] cases.

Table 1: Temperatures of the hollow cylinder at outer surface and at various times [ [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ].

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
2 terms	3 terms	20 terms
A	B	A	B	A	B
0.1	0.0263	0.0155	0.0261	0.0337	0.0262	0.0250
0.5	0.2287	0.1841	0.2296	0.2610	0.2297	0.2248
1	0.3981	0.3263	0.3998	0.4502	0.3998	0.3919
5	0.6541	0.5416	0.6591	0.7364	0.6598	0.6449
10	0.6578	0.5456	0.6675	0.7417	0.6690	0.6495

A: present solution, (39); B: Özisik [1], (44).

In the case of time-dependent boundary condition and time-dependent heat transfer coefficient at both surfaces, we consider the time-dependent temperature function, [figure omitted; refer to PDF] , and the Biot function, [figure omitted; refer to PDF] . From Table 2, one can find that the error of three-term approximation is less than [figure omitted; refer to PDF] . Because of large values of [figure omitted; refer to PDF] , the internal conductance of the hollow cylinder is small, whereas the heat transfer coefficient at the surface is large. In turn, the fact implies that the temperature distribution within the hollow cylinder is nonuniform. Therefore, we find that the larger the Biot function, that is, when [figure omitted; refer to PDF] approaches to [figure omitted; refer to PDF] in Table 2, the more the iteration numbers.

Table 2: Temperatures of the hollow cylinder at outer surface and at various times [ [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ].

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
2 terms	3 terms	10 terms	20 terms
0.1	0.0267	0.0265	0.0266	0.0266
0.5	0.2398	0.2408	0.2408	0.2408
1	0.4163	0.4185	0.4184	0.4185
5	0.4900	0.4951	0.4951	0.4958
10	0.4889	0.4987	0.4989	0.5001

Figure 2 depicts the temperature profiles along the radial of the hollow cylinder at different times, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . We find that the temperature at [figure omitted; refer to PDF] is higher than the temperature at [figure omitted; refer to PDF] and the temperature profile decreases at the negative slope for every case. It is clear since the heat source comes to the hollow cylinder from inner surface [figure omitted; refer to PDF] , and the heat dissipates from [figure omitted; refer to PDF] to the surrounding environment.

Figure 2: Temperature distribution and variation along the radial of the hollow cylinder with different parameter [figure omitted; refer to PDF] of temperature function [figure omitted; refer to PDF] , [ [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ].

[figure omitted; refer to PDF]

Variable heat source versus variable Biot function is drawn to show the temperature variation of the hollow cylinder at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with respect to [figure omitted; refer to PDF] in Figures 3(a) and 3(b), respectively. Two cases of Biot function [figure omitted; refer to PDF] (solid lines) and [figure omitted; refer to PDF] (dash lines) are considered. Due to the fact that the function [figure omitted; refer to PDF] severely decays as time goes, therefore, in the same temperature function [figure omitted; refer to PDF] the temperature in [figure omitted; refer to PDF] is less than that in [figure omitted; refer to PDF] as [figure omitted; refer to PDF] proceeds. That is to say, more heat will be dissipated into the surrounding environment for [figure omitted; refer to PDF] as [figure omitted; refer to PDF] goes.

Figure 3: Influence of function parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] on the temperatures of the hollow cylinder at middle and right surfaces, [ [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ].

(a) At [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

(b) At [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

Figure 4 depicts the effect of the parameter [figure omitted; refer to PDF] of temperature function [figure omitted; refer to PDF] upon the temperature variation of the hollow cylinder. It is found that, in the same temperature function [figure omitted; refer to PDF] , the temperature for [figure omitted; refer to PDF] is less than that for [figure omitted; refer to PDF] . Besides, as [figure omitted; refer to PDF] increases from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , the difference between temperatures at [figure omitted; refer to PDF] and at [figure omitted; refer to PDF] becomes significant.

Figure 4: Temperature distribution and variation along the radial of the hollow cylinder with different parameter [figure omitted; refer to PDF] of temperature function [figure omitted; refer to PDF] , [ [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ].

[figure omitted; refer to PDF]

Periodical heat source versus time-varying Biot function is drawn to show the temperature variation of the hollow cylinder at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with respect to [figure omitted; refer to PDF] in Figures 5(a) and 5(b), respectively. Two cases of heat source [figure omitted; refer to PDF] (solid lines) and [figure omitted; refer to PDF] (dash lines) are considered. At the same [figure omitted; refer to PDF] , the temperature of [figure omitted; refer to PDF] is less than that of [figure omitted; refer to PDF] for constant [figure omitted; refer to PDF] . The reason is that more heat has been dissipated into the surrounding environment at the case of [figure omitted; refer to PDF] . It can be observed that as [figure omitted; refer to PDF] proceeds, in the beginning, the temperatures are nonsensitive with [figure omitted; refer to PDF] parameters, as shown in Figure 5.

Figure 5: Influence of [figure omitted; refer to PDF] parameter and [figure omitted; refer to PDF] parameter on the temperature variation of the hollow cylinder at middle and right surfaces, [ [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ].

(a) At [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

(b) At [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

5. Conclusion

An analytical solution for the heat conduction in a hollow cylinder with time-dependent boundary conditions of different kinds at both surfaces was developed for the first time. The surface is subject to a time-dependent temperature field at inner surface, whereas the heat is dissipated by time-dependent convection from outer surface into a surrounding environment at zero temperature. The methodology is an extension of the shifting function method and the present results are identical to those in the literature when constant Biot function is considered. Since the methodology does not use integral transform, it has a proven result. The proposed method can also be easily extended to various heat conduction problems of hollow cylinders with time-dependent boundary conditions of different kinds at both surfaces.

Nomenclatures

[figure omitted; refer to PDF] :

Inner and outer radii (m)

[figure omitted; refer to PDF] :

Arbitrary constants used to express temperature and Biot functions

[figure omitted; refer to PDF] :

Biot function

[figure omitted; refer to PDF] :

Auxiliary functions

[figure omitted; refer to PDF] :

Auxiliary time functions

[figure omitted; refer to PDF] :

Biot function minus a constant

[figure omitted; refer to PDF] :

Shifting functions

[figure omitted; refer to PDF] :

Time-dependent heat transfer coefficient at outer surface (W·m^-2 ·K^-1 )

[figure omitted; refer to PDF] :

Variable temperature function at inner surface (K)

[figure omitted; refer to PDF] :

Bessel function of order zero of the first kind

[figure omitted; refer to PDF] :

Thermal conductivity (W·m^-1 ·K^-1 )

[figure omitted; refer to PDF] :

Norm of try functions

[figure omitted; refer to PDF] :

Time-dependent generalized coordinates

[figure omitted; refer to PDF] :

Space variable (m)

[figure omitted; refer to PDF] :

Ratio of inner radius over outer radius

[figure omitted; refer to PDF] :

Dimensionless radius

[figure omitted; refer to PDF] :

Parameters used to express temperature and Biot functions

[figure omitted; refer to PDF] :

Time variable (sec)

[figure omitted; refer to PDF] :

Temperature (K)

[figure omitted; refer to PDF] :

Constant reference temperature (K)

[figure omitted; refer to PDF] :

Initial temperature (K)

[figure omitted; refer to PDF] :

Auxiliary function

[figure omitted; refer to PDF] :

Bessel function of order zero of the second kind.

Greek Symbols

[figure omitted; refer to PDF] :

Thermal diffusivity (m² ·s^-1 )

[figure omitted; refer to PDF] :

Auxiliary functions

[figure omitted; refer to PDF] :

Initial value of Biot function

[figure omitted; refer to PDF] :

Eigenfunctions

[figure omitted; refer to PDF] :

Auxiliary integration variable

[figure omitted; refer to PDF] :

Auxiliary functions

[figure omitted; refer to PDF] :

Eigenvalues

[figure omitted; refer to PDF] :

Dimensionless temperature

[figure omitted; refer to PDF] :

Dimensionless initial temperature

[figure omitted; refer to PDF] :

Dimensionless time variable

[figure omitted; refer to PDF] :

Parameters for temperature and Biot functions

[figure omitted; refer to PDF] :

Auxiliary function

[figure omitted; refer to PDF] :

Auxiliary functions to express integration terms of Bessel functions

[figure omitted; refer to PDF] :

Dimensionless time-dependent temperature function

[figure omitted; refer to PDF] :

Auxiliary integration variable.

Subscripts

[figure omitted; refer to PDF] :

Indices.

Acknowledgment

It is gratefully acknowledged that this work was supported by the National Science Council of Taiwan, under Grants NSC 103-2221-E-006-048 and NSC 95-2221-E-344-001.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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