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

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.


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.
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.

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 where  is the temperature,  is the space variable,  is the thermal diffusivity,  is the time variable, and  and  denote inner and outer radii, respectively.The Here, () is a time-dependent temperature function at the inner surface,  is the thermal conductivity, ℎ() is a timedependent heat transfer coefficient function, and  0 () is an initial temperature function.For consistence in initial temperature field, (0) must be equal to  0 ().The above problem can be normalized by defining where   is a constant reference temperature, and the dimensionless boundary value problem will then become (, 0) =  0 () , when  = 0.
To keep the boundary condition of the third kind at outer surface in the following analysis, one sets the Biot function Bi() in the form of Bi () =  +  () , (8) where  and () are defined as It is obvious that (0) = 0, and the boundary condition at  = 1 can be rewritten as

Change of Variable.
To find the solution for the secondorder 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 where are two shifting functions to be specified, and   () ( = 1, 2) 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, V(, ) and   () ( = 1, 2), and hence it cannot be solved directly.

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 as Substituting these shifting functions and auxiliary time functions into (11) yields When setting  = 1 in the equation above, one has the relation 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 V(, ) as where   (), ( = 1, 2) are defined as Meanwhile, the associated boundary conditions of the transformed function turn to homogeneous ones as follows: Since  2 (0) = −(0)(1, 0) and (0) = 0, hence, the associated initial condition can be simplified as

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 equation The 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 of where   () ( = 1, 2, 3, . ..) are time-dependent generalized coordinates.Substituting solution from (27) into differential equation ( 19) leads to Expanding  1 () and  1 () on the right hand side of (28) in series forms we obtain where   and   are in which   ( = 0, 1, 2, 3;  = , ) are given as From (29), one can let After taking the inner product with try function   () and integrating from  to 1, the resulting differential equation now is where   and   are and   () is The associated initial condition is As a result, the complete solution of the ordinary differential equation (33) subject to the initial condition (36) is where   () is After substituting ( 16), ( 18), (23), and (27) back to (11), one obtains the analytical solution of the physical problem where the summation is taken over all eigenvalues   of the problem.

Constant Heat Transfer
Coefficient at  = 1.When the heat transfer coefficient ℎ at  = 1 is time-independent, the Biot function is a constant  and () = 0.The infinite series solution, (39), is reduced to where the generalized coordinates   () are The   (0)'s for the problem under consideration are Introducing ( 42) in (41) and performing integration by parts, we can get Substituting ( 43) into (40) yields the temperature distribution: This solution is the same as that obtained via the integral transform method by Özisik [1].

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 as and differentiating it with respect to  leads to where  1 and  1 are two arbitrary constants and  1 and  1 are two parameters.The Biot function considered at boundary where  2 and  2 are two arbitrary constants and  2 and  2 are two parameters.According to ( 8)-( 9), we obtain Consequently, the temperature distribution in the hollow cylinder is where 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  0 () =  0 as prescribed in the previous sections.The   (0)'s are now computed as For consistence in the temperature field, the constant  0 is taken as zero in the following examples.
In comparison with the literature, the example of constant Biot function is studied first.Bi() = 1 and time-dependent temperature function, () = 1 −  − , 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 threeterm approximation in present study is less than 0.4%; 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, () = 1 −  − cos , and the Biot function, Bi() = 2 −  − .From Table 2, one can find that the error of three-term approximation is less than 0.4%.Because of large values of Bi(), 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 10 in Table 2, the more the iteration numbers.
Figure 2 depicts the temperature profiles along the radial of the hollow cylinder at different times,  = 0.1 and  = 1.We find that the temperature at  = 0.6 is higher than the temperature at  = 1 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  = 0.6, and the heat dissipates from  = 1 to the surrounding environment.
Variable heat source versus variable Biot function is drawn to show the temperature variation of the hollow cylinder at  = 0.8 and  = 1.0 with respect to  in Figures 3(a  function () the temperature in Bi() = 2 −  −2 is less than that in Bi() = 2 −  − as  proceeds.That is to say, more heat will be dissipated into the surrounding environment for Bi() = 2 −  −2 as  goes.Figure 4 depicts the effect of the parameter  1 of temperature function () upon the temperature variation of the hollow cylinder.It is found that, in the same temperature function (), the temperature for  = 0.4 is less than that for  = 1.0.Besides, as  increases from 0.4 to 1.0, the difference between temperatures at  1 = 1 and at  1 = 2 becomes significant.
Periodical heat source versus time-varying Biot function is drawn to show the temperature variation of the hollow cylinder at  = 0.8 and  = 1.0 with respect to  in Figures 5(a) and 5(b), respectively.Two cases of heat source () = 1 − cos  (solid lines) and () = 1 − cos 2 (dash lines) are considered.At the same , the temperature of Bi() = 2− − is less than that of Bi() = 2− −0.1 for constant ().The reason is that more heat has been dissipated into the surrounding environment at the case of Bi() = 2 −  − .It can be observed that as  proceeds, in the beginning, the temperatures are nonsensitive with  1 parameters, as shown in Figure 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 timedependent 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 timedependent boundary conditions of different kinds at both surfaces.

Figure 1 :
Figure 1: Hollow cylinders with time-dependent temperature and time-dependent heat transfer coefficient at inner and outer surfaces.
) and 3(b), respectively.Two cases of Biot function Bi() = 2 −  − (solid lines) and Bi() = 2 −  −2 (dash lines) are considered.Due to the fact that the function  −2 severely decays as time goes, therefore, in the same temperature At  = 1