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

Authors: Te-Wen Tu; Sen-Yung Lee
Journal: Journal of Applied Mathematics
 (2015)

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## 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.


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## Body

## 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 Figure1. 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(1)∂2T∂r2+1r∂T∂r=1α∂T∂t,a<r<b,t>0,where T is the temperature, r is the space variable, α is the thermal diffusivity, t is the time variable, and a and b denote inner and outer radii, respectively. The boundary and initial conditions of the boundary value problem are(2)T=Ht,atr=a,-k∂T∂r=htT,atr=b,Tr=T0r,whent=0.Here, H(t) is a time-dependent temperature function at the inner surface, k is the thermal conductivity, h(t) is a time-dependent heat transfer coefficient function, and T0(r) is an initial temperature function. For consistence in initial temperature field, H(0) must be equal to T0(a). The above problem can be normalized by defining(3)θ=TTr,R=rb,τ=αtb2,r-=ab,ψτ=HtTr,Biτ=htbk,θ0R=T0rTr,where Tr is a constant reference temperature, and the dimensionless boundary value problem will then become(4)∂2θ∂R2+1R∂θ∂R=∂θ∂τ,r-<R<1,τ>0,(5)θ=ψτ,atR=r-,(6)∂θ∂R+Biτθ=0,atR=1,(7)θR,0=θ0R,whenτ=0.Figure 1
Hollow cylinders with time-dependent temperature and time-dependent heat transfer coefficient at inner and outer surfaces.To keep the boundary condition of the third kind at outer surface in the following analysis, one sets the Biot functionBi(τ) in the form of(8)Biτ=δ+Fτ,where δ and F(τ) are defined as(9)δ=Bi0,Fτ=Biτ-Bi0.It is obvious thatF(0)=0, and the boundary condition at R=1 can be rewritten as(10)∂θ∂R+δθ=-Fτθ,atR=1.


## 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(11)θR,τ=vR,τ+g1Rf1τ+g2Rf2τ,where v(R,τ) is called transformed function, gi(R) (i=1,2) are two shifting functions to be specified, and fi(τ) (i=1,2) are the auxiliary time functions defined as(12)f1τ=ψτ,f2τ=-Fτθ1,τ.Substituting (11) into (4), (5), (10), and (7), one has the following equation:(13)∂2vR,τ∂R2+1R∂vR,τ∂R=∂vR,τ∂τ+g1Rdf1τdτ-d2g1RdR2+1Rdg1RdRf1τ+g2Rdf2τdτ-d2g2RdR2+1Rdg2RdRf2τ,and the associated boundary and initial conditions now are(14)vr-,τ+g1r-f1τ+g2r-f2τ=f1τ,atR=r-,∂v1,τ∂R+δv1,τ+dg11dR+δg11f1τ+dg21dR+δg21f2τ=f2τ,atR=1,vR,0+g1Rf10+g2Rf20=θ0R.Something worthy to mention is that (13) contains three functions, that is, v(R,τ) and gi(R) (i=1,2), 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:(15)g1r-=1,g2r-=0,atR=r-g11=0,g21=0,dg11dR=0,dg21dR=1,atR=1.Consequently, the shifting functions can be easily determined as(16)g1R=1-R1-r-2,g2R=R-1R-r-1-r-.Substituting these shifting functions and auxiliary time functions into (11) yields(17)θR,τ=vR,τ+1-R1-r-2ψτ-R-1R-r-1-r-Fτθ1,τ.When setting R=1 in the equation above, one has the relation(18)θ1,τ=v1,τ.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(R,τ) as(19)∂vR,τ∂τ-∂2vR,τ∂R2-1R∂vR,τ∂R-g2RddτFτv1,τ+d2RFτv1,τ=-g1Rψ˙τ+d1Rψτ,where di(R), (i=1,2) are defined as(20)d1R=4R-2R1-r-2,d2R=4R-1+r-R1-r-.Meanwhile, the associated boundary conditions of the transformed function turn to homogeneous ones as follows:(21)vr-,τ=0,∂v1,τ∂R+δv1,τ=0.Since f2(0)=-F(0)θ(1,0) and F(0)=0, hence, the associated initial condition can be simplified as(22)vR,0=θ0R-g1Rψ0=v0R.


### 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:(23)ϕnR=J0λnR-J0λnr-Y0λnr-Y0λnR,n=1,2,3,…,satisfying the boundary conditions (21). Here the characteristic values λn are the roots of the transcendental equation(24)J0λnr-Y1λn-J1λnY0λnr-J0λnr-Y0λn-J0λnY0λnr-=δλn.The try functions have the following orthogonal property:(25)∫r-1RϕmRϕnRdR=0,m≠nNn,m=n,where the norms Nn are(26)Nn=12λn2λn2+δ2ϕn21-4π2Y02λnr-.Now, one can assume that the solution of the physical problem takes the form of(27)vR,τ=∑n=1∞ϕnRqnτ,n=1,2,3,…,where qn(τ) (n=1,2,3,…) are time-dependent generalized coordinates. Substituting solution from (27) into differential equation (19) leads to(28)∑n=1∞ϕnRq˙nτ-ϕn′′Rqnτ-1Rϕn′Rqnτ-g2Rϕn1F˙τqnτ+Fτq˙nτ+d2Rϕn1Fτqnτ=-g1Rψ˙τ+d1Rψτ.Expanding g1(R) and d1(R) on the right hand side of (28) in series forms we obtain(29)∑n=1∞ϕnRq˙nτ-ϕn′′R+1Rϕn′Rqnτ-g2Rϕn1Fτq˙nτ+F˙τqnτ+d2Rϕn1Fτqnτ=∑n=1∞-γnϕnRψ˙τ+γ-nϕnRψτ,where γn and γ-n are(30)γn=∫r-1RϕnRg1RdRNn=11-r-2ξJ1-2ξJ2+ξJ3-J0λnr-Y0λnr-ξY1-2ξY2+ξY3,γ-n=∫r-1RϕnRd1RdRNn=21-r-22ξJ1-ξJ0-J0λnr-Y0λnr-2ξY1-ξY0,in which ξWi (i=0,1,2,3;W=J,Y) are given as(31)ξW0=∫r-1W0λnRdRNn,ξW2=∫r-1R2W0λnRdRNn,ξW1=∫r-1RW0λnRdRNn=1λnNnW1λn-r-W1λnr-,ξW3=∫r-1R3W0λnRdRNn=1λn3Nn2λnW0λn+λn2-4W1λn-2λnr-2W0λnr--r-λn2r-2-4W1λnr-.From (29), one can let(32)ϕnRq˙nτ-ϕn′′R+1Rϕn′Rqnτ-g2Rϕn1Fτq˙nτ+F˙τqnτ+d2Rϕn1Fτqnτ=-γnϕnRψ˙τ+γ-nϕnRψτ.After taking the inner product with try function Rϕn(R) and integrating from r- to 1, the resulting differential equation now is(33)q˙nτ+λn2-βnF˙τ+β-nFτ1-βnFτqnτ=ξnτ,where βn and β-n are(34)βn=ϕn1∫r-1RϕnRg2RdRNn=ϕn11-r-ξJ3-1+r-ξJ2+r-ξJ1-J0λnr-Y0λnr-ξY3-1+r-ξY2+r-ξY1,β-n=ϕn1∫r-1RϕnRd2RdRNn=ϕn11-r-4ξJ1-1+r-ξJ0-J0λnr-Y0λnr-4ξY1-1+r-ξY0,and ξn(τ) is(35)ξnτ=-γnψ˙τ+γ-nψτ1-βnFτ.The associated initial condition is(36)qn0=∫r-1RϕnRv0RdRNn.As a result, the complete solution of the ordinary differential equation (33) subject to the initial condition (36) is(37)qnτ=Qnτqn0+∫0τξnφQnφdφ,where Qn(τ) is(38)Qnτ=e-∫0τλn2-βnF˙ζ+β-nFζ/1-βnFζdζ.After substituting (16), (18), (23), and (27) back to (11), one obtains the analytical solution of the physical problem(39)θR,τ=∑n=1∞ϕnR-ϕn1g2RFτqnτ+g1Rψτ,where the summation is taken over all eigenvalues λn of the problem.


### 3.4. Constant Heat Transfer Coefficient atR=1

When the heat transfer coefficienth at R=1 is time-independent, the Biot function is a constant δ and F(τ)=0. The infinite series solution, (39), is reduced to(40)θR,τ=∑n=1∞ϕnRqnτ+γnϕnRψτ,where the generalized coordinates qn(τ) are(41)qnτ=e-λn2τqn0+∫0τ-γnψ˙φ+γ-nψφ·eλn2φdφ.The qn(0)’s for the problem under consideration are(42)qn0=∫r-1RϕnRθ0R-ψ0g1RdRNn.Introducing (42) in (41) and performing integration by parts, we can get(43)qnτ=e-λn2τNn∫r-1RϕnRθ0RdR-2πY0λnr-∫0τeλn2φψφdφ-γnψτ.Substituting (43) into (40) yields the temperature distribution:(44)θR,τ=∑n=1∞e-λn2τNnϕnR∫r-1RϕnRθ0RdR-2πY0λnr-∫0τeλn2φψφdφ.This solution is the same as that obtained via the integral transform method by Özisik [1].


## 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(11)θR,τ=vR,τ+g1Rf1τ+g2Rf2τ,where v(R,τ) is called transformed function, gi(R) (i=1,2) are two shifting functions to be specified, and fi(τ) (i=1,2) are the auxiliary time functions defined as(12)f1τ=ψτ,f2τ=-Fτθ1,τ.Substituting (11) into (4), (5), (10), and (7), one has the following equation:(13)∂2vR,τ∂R2+1R∂vR,τ∂R=∂vR,τ∂τ+g1Rdf1τdτ-d2g1RdR2+1Rdg1RdRf1τ+g2Rdf2τdτ-d2g2RdR2+1Rdg2RdRf2τ,and the associated boundary and initial conditions now are(14)vr-,τ+g1r-f1τ+g2r-f2τ=f1τ,atR=r-,∂v1,τ∂R+δv1,τ+dg11dR+δg11f1τ+dg21dR+δg21f2τ=f2τ,atR=1,vR,0+g1Rf10+g2Rf20=θ0R.Something worthy to mention is that (13) contains three functions, that is, v(R,τ) and gi(R) (i=1,2), 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:(15)g1r-=1,g2r-=0,atR=r-g11=0,g21=0,dg11dR=0,dg21dR=1,atR=1.Consequently, the shifting functions can be easily determined as(16)g1R=1-R1-r-2,g2R=R-1R-r-1-r-.Substituting these shifting functions and auxiliary time functions into (11) yields(17)θR,τ=vR,τ+1-R1-r-2ψτ-R-1R-r-1-r-Fτθ1,τ.When setting R=1 in the equation above, one has the relation(18)θ1,τ=v1,τ.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(R,τ) as(19)∂vR,τ∂τ-∂2vR,τ∂R2-1R∂vR,τ∂R-g2RddτFτv1,τ+d2RFτv1,τ=-g1Rψ˙τ+d1Rψτ,where di(R), (i=1,2) are defined as(20)d1R=4R-2R1-r-2,d2R=4R-1+r-R1-r-.Meanwhile, the associated boundary conditions of the transformed function turn to homogeneous ones as follows:(21)vr-,τ=0,∂v1,τ∂R+δv1,τ=0.Since f2(0)=-F(0)θ(1,0) and F(0)=0, hence, the associated initial condition can be simplified as(22)vR,0=θ0R-g1Rψ0=v0R.


## 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:(23)ϕnR=J0λnR-J0λnr-Y0λnr-Y0λnR,n=1,2,3,…,satisfying the boundary conditions (21). Here the characteristic values λn are the roots of the transcendental equation(24)J0λnr-Y1λn-J1λnY0λnr-J0λnr-Y0λn-J0λnY0λnr-=δλn.The try functions have the following orthogonal property:(25)∫r-1RϕmRϕnRdR=0,m≠nNn,m=n,where the norms Nn are(26)Nn=12λn2λn2+δ2ϕn21-4π2Y02λnr-.Now, one can assume that the solution of the physical problem takes the form of(27)vR,τ=∑n=1∞ϕnRqnτ,n=1,2,3,…,where qn(τ) (n=1,2,3,…) are time-dependent generalized coordinates. Substituting solution from (27) into differential equation (19) leads to(28)∑n=1∞ϕnRq˙nτ-ϕn′′Rqnτ-1Rϕn′Rqnτ-g2Rϕn1F˙τqnτ+Fτq˙nτ+d2Rϕn1Fτqnτ=-g1Rψ˙τ+d1Rψτ.Expanding g1(R) and d1(R) on the right hand side of (28) in series forms we obtain(29)∑n=1∞ϕnRq˙nτ-ϕn′′R+1Rϕn′Rqnτ-g2Rϕn1Fτq˙nτ+F˙τqnτ+d2Rϕn1Fτqnτ=∑n=1∞-γnϕnRψ˙τ+γ-nϕnRψτ,where γn and γ-n are(30)γn=∫r-1RϕnRg1RdRNn=11-r-2ξJ1-2ξJ2+ξJ3-J0λnr-Y0λnr-ξY1-2ξY2+ξY3,γ-n=∫r-1RϕnRd1RdRNn=21-r-22ξJ1-ξJ0-J0λnr-Y0λnr-2ξY1-ξY0,in which ξWi (i=0,1,2,3;W=J,Y) are given as(31)ξW0=∫r-1W0λnRdRNn,ξW2=∫r-1R2W0λnRdRNn,ξW1=∫r-1RW0λnRdRNn=1λnNnW1λn-r-W1λnr-,ξW3=∫r-1R3W0λnRdRNn=1λn3Nn2λnW0λn+λn2-4W1λn-2λnr-2W0λnr--r-λn2r-2-4W1λnr-.From (29), one can let(32)ϕnRq˙nτ-ϕn′′R+1Rϕn′Rqnτ-g2Rϕn1Fτq˙nτ+F˙τqnτ+d2Rϕn1Fτqnτ=-γnϕnRψ˙τ+γ-nϕnRψτ.After taking the inner product with try function Rϕn(R) and integrating from r- to 1, the resulting differential equation now is(33)q˙nτ+λn2-βnF˙τ+β-nFτ1-βnFτqnτ=ξnτ,where βn and β-n are(34)βn=ϕn1∫r-1RϕnRg2RdRNn=ϕn11-r-ξJ3-1+r-ξJ2+r-ξJ1-J0λnr-Y0λnr-ξY3-1+r-ξY2+r-ξY1,β-n=ϕn1∫r-1RϕnRd2RdRNn=ϕn11-r-4ξJ1-1+r-ξJ0-J0λnr-Y0λnr-4ξY1-1+r-ξY0,and ξn(τ) is(35)ξnτ=-γnψ˙τ+γ-nψτ1-βnFτ.The associated initial condition is(36)qn0=∫r-1RϕnRv0RdRNn.As a result, the complete solution of the ordinary differential equation (33) subject to the initial condition (36) is(37)qnτ=Qnτqn0+∫0τξnφQnφdφ,where Qn(τ) is(38)Qnτ=e-∫0τλn2-βnF˙ζ+β-nFζ/1-βnFζdζ.After substituting (16), (18), (23), and (27) back to (11), one obtains the analytical solution of the physical problem(39)θR,τ=∑n=1∞ϕnR-ϕn1g2RFτqnτ+g1Rψτ,where the summation is taken over all eigenvalues λn of the problem.


## 3.4. Constant Heat Transfer Coefficient atR=1

When the heat transfer coefficienth at R=1 is time-independent, the Biot function is a constant δ and F(τ)=0. The infinite series solution, (39), is reduced to(40)θR,τ=∑n=1∞ϕnRqnτ+γnϕnRψτ,where the generalized coordinates qn(τ) are(41)qnτ=e-λn2τqn0+∫0τ-γnψ˙φ+γ-nψφ·eλn2φdφ.The qn(0)’s for the problem under consideration are(42)qn0=∫r-1RϕnRθ0R-ψ0g1RdRNn.Introducing (42) in (41) and performing integration by parts, we can get(43)qnτ=e-λn2τNn∫r-1RϕnRθ0RdR-2πY0λnr-∫0τeλn2φψφdφ-γnψτ.Substituting (43) into (40) yields the temperature distribution:(44)θR,τ=∑n=1∞e-λn2τNnϕnR∫r-1RϕnRθ0RdR-2πY0λnr-∫0τeλn2φψφdφ.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 conditionψ(τ) considered at R=r- is taken as(45)ψτ=a1-b1e-s1τcos⁡ω1τ,and differentiating it with respect to τ leads to(46)ψ˙τ=b1e-s1τs1cos⁡ω1τ+ω1sin⁡ω1τ,where a1 and b1 are two arbitrary constants and s1 and ω1 are two parameters.The Biot function considered at boundaryR=1 is(47)Biτ=a2-b2e-s2τcos⁡ω2τ,where a2 and b2 are two arbitrary constants and s2 and ω2 are two parameters. According to (8)-(9), we obtain(48)δ=a2-b2,Fτ=b21-e-s2τcos⁡ω2τ,F˙τ=b2e-s2τs2cos⁡ω2τ+ω2sin⁡ω2τ.Consequently, the temperature distribution in the hollow cylinder is(49)θR,τ=∑n=1∞qnτϕnR-R-1R-r-1-r-ϕn1b21-e-s2τcos⁡ω2τ+1-R1-r-2a1-b1e-s1τcos⁡ω1τ,where the qn(τ)’s are defined in (37). The associated ξn(τ) now is(50)ξnτ=-γnψ˙τ+γ-nψτ1-βnb21-e-s2τcos⁡ω2τ.To avoid numerical instability that occurred in computingqn(τ), (37) is rewritten as(51)qnτ=Qnτqn0+∫0τξnφ·QnτQnφdφ.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 temperatureθ0(X)=θ-0 as prescribed in the previous sections. The qn(0)’s are now computed as(52)qn0=ξJ1-J0λnr-Y0λnr-θ-0-γnψ0.For consistence in the temperature field, the constant θ-0 is taken as zero in the following examples.In comparison with the literature, the example of constant Biot function is studied first.Bi(τ)=1 and time-dependent temperature function, ψ(τ)=1-e-τ, 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 0.4%; 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 [ψ(τ)=1-e-τ, Bi(τ)=1, θ-0=0].
τ
θ
(
1
,
τ
)
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,ψτ=1-e-τcos⁡τ, and the Biot function, Bi(τ)=2-e-τ. From Table 2, one can find that the error of three-term approximation is less than 0.4%. Because of large values of Bi(τ), 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 τ approaches to 10 in Table 2, the more the iteration numbers.Table 2
Temperatures of the hollow cylinder at outer surface and at various times [ψ(τ)=1-e-τcos⁡τ, Bi(τ)=2-e-τ, θ-0=0].
τ
θ
(
1
,
τ
)
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
0.5001Figure2 depicts the temperature profiles along the radial of the hollow cylinder at different times, τ=0.1 and τ=1. We find that the temperature at R=0.6 is higher than the temperature at R=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 R=0.6, and the heat dissipates from R=1 to the surrounding environment.Figure 2
Temperature distribution and variation along the radial of the hollow cylinder with different parameters1 of temperature function ψ(τ), [ψ(τ)=1-e-s1τ, Bi(τ)=2-e-τ, θ-0=0].Variable heat source versus variable Biot function is drawn to show the temperature variation of the hollow cylinder atR=0.8 and R=1.0 with respect to τ in Figures 3(a) and 3(b), respectively. Two cases of Biot function Bi(τ)=2-e-τ (solid lines) and Bi(τ)=2-e-2τ (dash lines) are considered. Due to the fact that the function e-2τ severely decays as time goes, therefore, in the same temperature function ψ(τ) the temperature in Bi(τ)=2-e-2τ is less than that in Bi(τ)=2-e-τ as τ proceeds. That is to say, more heat will be dissipated into the surrounding environment for Bi(τ)=2-e-2τ as τ goes.Figure 3
Influence of function parameterss1 and s2 on the temperatures of the hollow cylinder at middle and right surfaces, [ψ(τ)=1-e-s1τ, Bi(τ)=2-e-s2τ, θ-0=0].
(a)
AtR=0.8
(b)
AtR=1Figure4 depicts the effect of the parameter ω1 of temperature function ψ(τ) upon the temperature variation of the hollow cylinder. It is found that, in the same temperature function ψ(τ), the temperature for τ=0.4 is less than that for τ=1.0. Besides, as τ increases from 0.4 to 1.0, the difference between temperatures at ω1=1 and at ω1=2 becomes significant.Figure 4
Temperature distribution and variation along the radial of the hollow cylinder with different parameterω1 of temperature function ψ(τ), [ψτ=1-cos⁡ω1τ, Bi(τ)=2-e-τ, θ-0=0].Periodical heat source versus time-varying Biot function is drawn to show the temperature variation of the hollow cylinder atR=0.8 and R=1.0 with respect to τ in Figures 5(a) and 5(b), respectively. Two cases of heat source ψτ=1-cos⁡τ (solid lines) and ψτ=1-cos⁡2τ (dash lines) are considered. At the same τ, the temperature of Bi(τ)=2-e-τ is less than that of Bi(τ)=2-e-0.1τ for constant ψ(τ). The reason is that more heat has been dissipated into the surrounding environment at the case of Bi(τ)=2-e-τ. It can be observed that as τ proceeds, in the beginning, the temperatures are nonsensitive with ω1 parameters, as shown in Figure 5.Figure 5
Influence ofω1 parameter and s2 parameter on the temperature variation of the hollow cylinder at middle and right surfaces, [ψ(τ)=1-cos⁡ω1τ, Bi(τ)=2-e-s2τ, θ-0=0].
(a)
AtR=0.8
(b)
AtR=1


## 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.



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*Source: 203404-2015-10-01.xml*