#### 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.

#### 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 iswhere is the temperature, is the space variable, is the thermal diffusivity, is the time variable, and and denote inner and outer radii, respectively. The boundary and initial conditions of the boundary value problem areHere, is a time-dependent temperature function at the inner surface, is the thermal conductivity, is a time-dependent heat transfer coefficient function, and is an initial temperature function. For consistence in initial temperature field, must be equal to . The above problem can be normalized by definingwhere is a constant reference temperature, and the dimensionless boundary value problem will then become

To keep the boundary condition of the third kind at outer surface in the following analysis, one sets the Biot function in the form ofwhere and are defined as

It is obvious that , and the boundary condition at can be rewritten as

#### 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 takingwhere is called transformed function, () are two shifting functions to be specified, and () are the auxiliary time functions defined as

Substituting (11) into (4), (5), (10), and (7), one has the following equation:and the associated boundary and initial conditions now are

Something worthy to mention is that (13) contains three functions, that is, and (), 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:Consequently, the shifting functions can be easily determined asSubstituting these shifting functions and auxiliary time functions into (11) yieldsWhen setting in the equation above, one has the relationTherefore, two functions in governing differential equation (13) are integrated to one. With (16) and (18), (13) can be rewritten in terms of the function aswhere , are defined asMeanwhile, the associated boundary conditions of the transformed function turn to homogeneous ones as follows:Since and , hence, the associated initial condition can be simplified as

##### 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:satisfying the boundary conditions (21). Here the characteristic values are the roots of the transcendental equationThe try functions have the following orthogonal property:where the norms are

Now, one can assume that the solution of the physical problem takes the form ofwhere () are time-dependent generalized coordinates. Substituting solution from (27) into differential equation (19) leads toExpanding and on the right hand side of (28) in series forms we obtainwhere and arein which () are given asFrom (29), one can letAfter taking the inner product with try function and integrating from to , the resulting differential equation now iswhere and areand isThe associated initial condition isAs a result, the complete solution of the ordinary differential equation (33) subject to the initial condition (36) iswhere is

After substituting (16), (18), (23), and (27) back to (11), one obtains the analytical solution of the physical problemwhere the summation is taken over all eigenvalues of the problem.

##### 3.4. Constant Heat Transfer Coefficient at

When the heat transfer coefficient at is time-independent, the Biot function is a constant and . The infinite series solution, (39), is reduced towhere the generalized coordinates areThe ’s for the problem under consideration areIntroducing (42) in (41) and performing integration by parts, we can getSubstituting (43) into (40) yields the temperature distribution: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 considered at is taken asand differentiating it with respect to leads towhere and are two arbitrary constants and and are two parameters.

The Biot function considered at boundary iswhere and are two arbitrary constants and and are two parameters. According to (8)-(9), we obtain

Consequently, the temperature distribution in the hollow cylinder iswhere the ’s are defined in (37). The associated now is

To avoid numerical instability that occurred in computing , (37) is rewritten as

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 as prescribed in the previous sections. The ’s are now computed asFor consistence in the temperature field, the constant is taken as zero in the following examples.

In comparison with the literature, the example of constant Biot function is studied first. and time-dependent temperature function, , 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 ; on the contrary, at least twenty-term approximation is required to get the same accuracy in Özisik’s [1] cases.

In the case of time-dependent boundary condition and time-dependent heat transfer coefficient at both surfaces, we consider the time-dependent temperature function, , and the Biot function, . From Table 2, one can find that the error of three-term approximation is less than . Because of large values of , 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 approaches to in Table 2, the more the iteration numbers.

Figure 2 depicts the temperature profiles along the radial of the hollow cylinder at different times, and . We find that the temperature at is higher than the temperature at 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 , and the heat dissipates from to the surrounding environment.

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

**(a)**At

**(b)**AtFigure 4 depicts the effect of the parameter of temperature function upon the temperature variation of the hollow cylinder. It is found that, in the same temperature function , the temperature for is less than that for . Besides, as increases from to , the difference between temperatures at and at becomes significant.

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

**(a)**At

**(b)**At#### 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

: | Inner and outer radii (m) |

: | Arbitrary constants used to express temperature and Biot functions |

: | Biot function |

: | Auxiliary functions |

: | Auxiliary time functions |

: | Biot function minus a constant |

: | Shifting functions |

: | Time-dependent heat transfer coefficient at outer surface (W·m^{−2}·K^{−1}) |

: | Variable temperature function at inner surface (K) |

: | Bessel function of order zero of the first kind |

: | Thermal conductivity (W·m^{−1}·K^{−1}) |

: | Norm of try functions |

: | Time-dependent generalized coordinates |

: | Space variable (m) |

: | Ratio of inner radius over outer radius |

: | Dimensionless radius |

: | Parameters used to express temperature and Biot functions |

: | Time variable (sec) |

: | Temperature (K) |

: | Constant reference temperature (K) |

: | Initial temperature (K) |

: | Auxiliary function |

: | Bessel function of order zero of the second kind. |

*Greek Symbols*

: | Thermal diffusivity (m^{2}·s^{−1}) |

: | Auxiliary functions |

: | Initial value of Biot function |

: | Eigenfunctions |

: | Auxiliary integration variable |

: | Auxiliary functions |

: | Eigenvalues |

: | Dimensionless temperature |

: | Dimensionless initial temperature |

: | Dimensionless time variable |

: | Parameters for temperature and Biot functions |

: | Auxiliary function |

: | Auxiliary functions to express integration terms of Bessel functions |

: | Dimensionless time-dependent temperature function |

: | Auxiliary integration variable. |

*Subscripts*

: | Indices. |

#### Conflict of Interests

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

#### 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.