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Nuclear Engineering and Design 241 (2011) 144–154

Contents lists available at ScienceDirect

Nuclear Engineering and Design

journa l homepage: www.e lsev ier .com/ locate /nucengdes

inite integral transform method to solve asymmetric heat conduction in aultilayer annulus with time-dependent boundary conditions

uneet Singha, Prashant K. Jainb,∗, Rizwan-uddinc

Energy Science and Engineering, Indian Institute of Technology (IIT) Bombay, Mumbai, IndiaReactor and Nuclear Systems Division, Oak Ridge National Laboratory, 1 Bethel Valley Road, Oak Ridge, TN, USANuclear, Plasma and Radiological Engineering, University of Illinois at Urbana–Champaign, USA

r t i c l e i n f o

rticle history:eceived 21 October 2008eceived in revised form 16 August 2010ccepted 18 October 2010

a b s t r a c t

The separation of variables (SOV) method has recently been applied to solve time-dependent heat conduc-tion problem in a multilayer annulus. It is observed that the transverse (radial) eigenvalues for the solutionin polar (r-�) coordinate system are always real numbers (unlike in the case of similar multidimensionalCartesian problems where they may be imaginary) allowing one to obtain the solution analytically. How-ever, the SOV method cannot be applied when the boundary conditions and/or the source terms aretime-dependent, for example, in a nuclear fuel rod subjected to time-dependent boundaries or heatsources. In this paper, we present an alternative approach using the finite integral transform method tosolve the asymmetric heat conduction problem in a multilayer annulus with time-dependent boundaryconditions and/or heat sources. An eigenfunction expansion approach satisfying periodic boundary con-dition in the angular direction is used. After integral transformation and subsequent weighted summationover the radial layers, partial derivative with respect to r-variable is eliminated and, first order ordinarydifferential equations (ODEs) are formed for the transformed temperatures. Solutions of ODEs are theninverted to obtain the temperature distribution in each layer. Since the proposed solution requires thesame eigenfunctions as those in the similar problem with time-independent sources and/or boundaryconditions—a problem solved using the SOV method—it is also “free” from imaginary eigenvalues.

© 2010 Elsevier B.V. All rights reserved.

. Introduction

In modern engineering applications, multilayer components are extensively used due to the added advantage of combining physical,echanical and thermal properties of different materials. Many of these applications (for example, in various automotive, space, chemical,

ivil and nuclear industries) require a detailed knowledge of transient temperature and heat-flux distribution within the component layers.ime-dependent temperature distribution in such components, with the presence of sources (with all three types of boundary conditions)ay either be obtained using analytical or numerical methods. Nonetheless, numerical solutions are preferred and prevalent in practice,

ue to either unavailability or higher computational complexity of the corresponding analytical (or exact) solutions. Exact solutions, whenvailable, are however very useful in analyzing governing physics of the problem and thus, are more insightful compared to the numericalolutions. Nowadays, such solutions also find their applications in validating and comparing various numerical algorithms, which mayelp improve computational efficiency of computer codes that currently rely on numerical techniques.

Analytical solutions for 1D time-dependent multilayer heat conduction problems were developed several decades ago. Mathematicalheory for such problems in more than one dimension was also developed during the same time (Ozisik, 1993; Salt, 1983a,b; Mikhailovnd Ozisik, 1986; Yener and Ozisik, 1974). Several of these approaches were based on the separation of variables (SOV) and the finite integralransform (FIT) methods. However, application of these approaches in producing exact solutions was hindered due to the difficulty of solvinghe corresponding eigenvalue problems (which are an essential part of such solution methodologies). Furthermore, in 2D and 3D Cartesian

oordinates, these eigenvalues can be imaginary, rendering the solutions even more difficult. It has recently been shown that similarroblems in 2D cylindrical (r,�), 2D spherical (r,�) and 3D spherical (r, �, �) coordinate systems do not possess imaginary eigenvalues (Singht al., 2008; Jain et al., 2009, 2010) and therefore, exact solutions can be obtained using the softwares capable of symbolic manipulationsfor example, MapleTM (Maple, 2010), Mathematica (Mathematica, 2010) etc.).

∗ Corresponding author. Tel.: +1 865 574 6272; fax: +1 865 574 2032.E-mail address: jainpk@ornl.gov (P.K. Jain).

029-5493/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.nucengdes.2010.10.010

S. Singh et al. / Nuclear Engineering and Design 241 (2011) 144–154 145

Nomenclature

aimp, bimp coefficients of Bessel functions in transverse (radial) eigenfunction (see Eq. (29))Ain, Bin, Cin coefficients in inner surface boundary condition (see Eqs. (2), (18) and (25))Aout, Bout, Cout coefficients in outer surface boundary condition (see Eqs. (3), (19) and (26))fi(r, �) initial temperature distribution in the ith layer at t = 0gi(r, �, t) time-dependent volumetric heat source distribution in the ith layerJm Bessel function of the first kind of order mki thermal conductivity of the ith layerM number of angular eigenfunction used in the solutionNmp norm for r-direction (see Eq. (30))P number of radial eigenfunction used in the solution corresponding to each angular eigenvalueq′ ′ heat-fluxq0 magnitude of heat flux in the illustrative example (see Eq. (48))r radial coordinateri outer radius for the ith layerRimp(�impr) transverse eigenfunctions for the ith layert timeTi(r, �, t) temperature distribution for the ith layerYm Bessel function of the second kind of order m

Greek symbols˛i thermal diffusivity of the ith layer� angular coordinate�imp transverse (radial) eigenvalues� angle subtended by the multilayers

Abbreviations1D, 2D, 3D one-dimensional, two-dimensional, three-dimensionalFIT finite integral transform methodODEs ordinary differential equationsRHS right hand sideSOV separation of variables method

Subscripts and superscripts

eaAofcs

saacacp

ctmtssmb(

i layer or interface numbern outermost layer number

Recent advances in computational resources for symbolic manipulations have created renewed interest among researchers (Singht al., 2008; Jain et al., 2009, 2010; Lu and Viljanen, 2006; de Monte, 2006, 2000, 2003, 2002; Lu et al., 2006a,b, 2005; Haji-Sheikhnd Beck, 2002) in developing exact analytical solutions of problems for which numerical solutions are currently more prevalent.lthough multilayer heat conduction problems have been studied in great detail and various solution methods have been devel-ped in the past (for review, see Ozisik, 1993), there is a continued need to explore and develop novel methods to solve problemsor which exact solutions are still in infancy. One such problem is to determine the unsteady temperature distribution in cylindri-al polar coordinates (r,�) with multiple layers in the radial direction exposed to time-dependent boundary conditions and/or heatources.

Salt (1983a,b) solved the time-dependent heat conduction problem by applying an orthogonal expansion technique in a 2D compositelab with no internal heat source, and homogenous boundary conditions. Mikhailov and Ozisik (1986) solved an analogous 3D problem inCartesian nonhomogenous finite medium. Recently, Lu et al. (Lu and Viljanen, 2006; Lu et al., 2006a, 2005) combined a Laplace Transformpproach and the separation of variables method for a rectangular and cylindrical multilayer slab with time-dependent periodic boundaryonditions. Treatment in the cylindrical coordinates is however restricted to the r − z coordinates only. Haji-Sheikh and Beck (2002) appliedGreen’s function approach to solve a 3D Cartesian two-layer orthotropic heat conduction problem. They also accounted for the effects of

ontact resistance in their solution. Eigenfunction expansion method is applied by de Monte (2003) to solve the unsteady heat conductionroblem in a 2D two-layer isotropic slab with homogenous boundary conditions.

An exact solution based on the separation of variables (SOV) method is recently developed by Singh et al. (2008) for multilayer heatonduction in polar coordinates. However, that exact solution is applicable only to the domains with pie slice geometry (� < 2�, where � ishe azimuthal angle subtended by the layers) and time-independent boundary conditions. Recently, the same approach, based on the SOV

ethod, is extended to determine the temperature distribution in complete disc-type (i.e., � = 2�) layers by Jain et al. (2009). However, dueo the limitation of the SOV method, it cannot be conveniently extended to include the effects of time-dependent boundary conditions and/or

ources. One typical example of such a problem is a nuclear fuel rod, which consists of concentric layers of different materials and oftenubjected to asymmetric time-dependent boundary conditions. Moreover, numerous other applications including multilayer insulationaterials, several cryogenic systems, tall buildings and other cylindrical structures may be subjected to time-dependent sources and/or

oundary conditions, and may benefit from having an exact solution. This paper presents an approach based on the finite integral transformFIT) method to solve for such temperature distributions. Solution is valid for any combinations of time-dependent, inhomogeneous

146 S. Singh et al. / Nuclear Engineering and Design 241 (2011) 144–154

bt

dgpm2

2

altb

•

•

•

•

Fig. 1. Schematic representation of an n-layer annulus.

oundary conditions at inner and outer radii of the domain. Results for an illustrative problem involving a three-layer annulus subjectedo asymmetric, time-dependent heat-flux are also presented.

It should be noted that the generalized integral-transform method for solving heat-conduction in arbitrary finite composite regions iseveloped by Yener and Ozisik (Ozisik, 1993; Yener and Ozisik, 1974). However, reducing the scheme to closed-form solutions for specificeometry and coordinate system is not trivial which is particularly true in the context of cylindrical polar coordinates. However, recentrogress in computer based symbolic manipulations (Maple, 2010; Mathematica, 2010) and increase in computational power have nowade it possible to override such computational difficulties, and yield closed-form analytical solutions (Singh et al., 2008; Jain et al., 2009,

010).

. Mathematical formulation

Consider an n-layer annulus (r0 ≤ r ≤ rn) as shown schematically in Fig. 1. All the layers are assumed to be isotropic in thermal propertiesnd are in perfect thermal contact. Let ki and ˛i be the temperature independent thermal conductivity and thermal diffusivity of the ithayer. At t = 0, each ith layer is at a specified temperature fi(r, �) and time-dependent heat sources gi(r, �, t) are switched on for t > 0. Both,he inner (i = 1, r = r0) as well as the outer (i = n, r = rn) surfaces of the annulus may be subjected to any combination of time-dependentoundary conditions of the first, second or the third kind.

Under these assumptions, the governing heat conduction equation along with the boundary and initial conditions, are as follows:Governing equation:

1˛i

∂Ti

∂t(r, �, t) = 1

r

∂

∂r

(r

∂Ti

∂r(r, �, t)

)+ 1

r2

∂2Ti

∂�2(r, �, t) + gi(r, �, t)

ki, ri−1 ≤ r ≤ ri, 1 ≤ i ≤ n (1)

Boundary conditions:

Inner surface of the 1st layer (i = 1)

Ain∂T1

∂r(r0, �, t) + BinT1(r0, �, t) = Cin(�, t) (2)

Outer surface of the nth layer (i = n)

Aout∂Tn

∂r(rn, �, t) + BoutTn(rn, �, t) = Cout(�, t) (3)

Periodic boundary conditions (i = 1, 2, ..., n)

Ti(r, � = 0, t) = Ti(r, � = 2�, t) (4)

∂Ti

∂�(r, � = 0, t) = ∂Ti

∂�(r, � = 2�, t) (5)

Interface of the ith layer (i = 2, ..., n)

Ti(ri−1, �, t) = Ti−1(ri−1, �, t) (6)

ki∂Ti

∂r(ri−1, �, t) = ki−1

∂Ti−1

∂r(ri−1, �, t) (7)

BEbw

3

c

Se

C

Sigm

•

•

•

S. Singh et al. / Nuclear Engineering and Design 241 (2011) 144–154 147

Initial condition:

Ti(r, �, t = 0) = fi(r, �). (8)

oundary conditions either of the first, second or third kind may be imposed at r = r0 and r = rn by choosing the appropriate coefficients inqs. (2) and (3). However, the case in which Bin and Bout are simultaneously zero is not considered. In addition, asymmetric, time-dependentoundary conditions can be applied by choosing �- and t-dependent Cin and Cout. Furthermore, temperature distribution in multiple layersith zero inner radius (r0 = 0) can be solved by assigning zero values to constants Bin and Cin in Eq. (2).

. Eigenfunction expansion in the �-direction

Due to periodic boundary conditions in the �-direction, Ti(r, �, t) can be expanded into angular eigenfunctions (cos(m�), sin(m�) and aonstant) as follows:

Ti(r, �, t) = T�i0(r, t) +

∞∑m=1

T�imc(r, t) cos(m�) +

∞∑m=1

T�ims(r, t) sin(m�). (9)

imilarly, specified expressions for the sources, boundary conditions, and initial condition, gi(r, �, t), Cin(�, t), Cout(�, t) and fi(r, �), are alsoxpanded as:

gi(r, �, t) = g�i0(r, t) +

∞∑m=1

g�imc(r, t) cos(m�) +

∞∑m=1

g�ims(r, t) sin(m�) (10)

Cin(�, t) = C�in,0(t) +

∞∑m=1

C�in,mc(t) cos(m�) +

∞∑m=1

C�in,ms(t) sin(m�) (11)

Cout(�, t) = C�out,0(t) +

∞∑m=1

C�out,mc(t) cos(m�) +

∞∑m=1

C�out,ms(t) sin(m�) (12)

fi(r, �) = f �i0(r) +

∞∑m=1

f �imc(r) cos(m�) +

∞∑m=1

f �ims(r) sin(m�). (13)

oefficients in Eqs. (10)–(13) can be evaluated by applying the orthogonality conditions in the �-direction. For example, in Eq. (10):

g�i0(r, t) = 1

2�

∫ 2�

0

gi(r, �, t)d� (14)

g�imc(r, t) = 1

�

∫ 2�

0

gi(r, �, t) cos(m�)d� (15)

g�ims(r, t) = 1

�

∫ 2�

0

gi(r, �, t) sin(m�)d�. (16)

ubstituting Eqs. (9)–(13) in Eqs. (1)–(8) yields the following set of equations for coefficients in Eq. (9) for each positive integer m [Note thatn the formulation that follows, subscripts 0, c or s are omitted for clarity because coefficients with these different subscripts are essentiallyoverned by the same set of equations as shown below (see Eqs. (17)–(22)). Moreover, in the equations for coefficient with subscript 0,= 0 is taken.]:

1˛i

∂T�im

∂t(r, t) = 1

r

∂

∂r

(r

∂T�im

∂r(r, t)

)− m2

r2T�

im(r, t) + g�im

(r, t)

ki, ri−1 ≤ r ≤ ri, 1 ≤ i ≤ n (17)

Boundary conditions:

Inner surface of the 1st layer (i = 1)

Ain∂T�

1m

∂r(r0, t) + BinT�

1m(r0, t) = C�in,m(t) (18)

Outer surface of the nth layer (i = n)

Aout∂T�

nm

∂r(rn, t) + BoutT

�nm(rn, t) = C�

out,m(t) (19)

Interface between ith and (i − 1)st layer (i = 2, ..., n)

T�im(ri−1, t) = T�

i−1,m(ri−1, t) (20)

ki

∂T�im

∂r(ri−1, t) = ki−1

∂T�i−1,m

∂r(ri−1, t) (21)

1

•

4

A

4

a

•

•

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N

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tn

4

48 S. Singh et al. / Nuclear Engineering and Design 241 (2011) 144–154

Initial condition:

T�im(r, t = 0) = f �

im(r). (22)

. Finite integral transform in the r-direction

Operating Eq. (17) by∫ ri

ri−1rRim(r)dr and using integration of parts twice on the first term on the right hand side (RHS), one gets (see

ppendix A for details):∫ ri

ri−1

(1˛i

∂T�im

∂t(r, t)

)rRim(r)dr =

∫ ri

ri−1

(1r

d

dr

(r

dRim

dr(r)

)− m2

r2Rim(r)

)rT�

im(r, t)dr

+[

rRim(r)∂T�

im

∂r(r, t) − rT�

im(r)∂Rim

∂r(r)

]ri

ri−1

+∫ ri

ri−1

rRim(r)g�

im(r, t)

kidr. (23)

.1. Eigenvalue problem in the r-direction

Rim(r) in Eq. (23) is chosen so as to satisfy,

1r

d

dr

(r

dRim

dr(r)

)+

(−m2

r2+ �2

im

)Rim(r) = 0 (24)

nd boundary conditions:

Inner surface of the 1st layer (i = 1)

AindR1m

dr(r = r0) + BinR1m(r = r0) = 0 (25)

Outer surface of the nth layer (i = n)

AoutdRnm

dr(r = rn) + BoutRnm(r = rn) = 0 (26)

Interface between ith and (i − 1)st layer (i = 2, ..., n)

Rim(ri−1) = Ri−1,m(ri−1) (27)

ki∂Rim

∂r(ri−1) = ki−1

∂Ri−1,m

∂r(ri−1). (28)

olutions of the above equations are eigenfunctions Rimp(r) corresponding to the eigenvalues �imp, and given by:

Rimp(�impr) = aimpJm(�impr) + bimpYm(�impr). (29)

he eigenfunctions �imp satisfy the following orthogonality condition:

n∑i=1

ki

˛i

∫ ri

ri−1

rRimp(�impr)Rimq(�imqr)dr =[

0 if p /= qNmp if p = q

. (30)

ote that the above condition is satisfied if and only if (Singh et al., 2008):

˛i�2imp = ˛1�2

1mp. (31)

pplication of the boundary conditions (Eqs. (25) and (26)) and interface conditions (Eqs. (27) and (28)) to the transverse eigenfunctionimp(�impr) yields a (2n × 2n) matrix for each integer value of m. Eigencondition for the transverse direction is obtained by setting theeterminant of this matrix equal to zero. Roots of which, in turn, yield the infinite number of eigenvalues �1mp corresponding to the first

ayer for each integer value of m (similar to Singh et al., 2008).It has been shown earlier by Singh et al. (2008) and Jain et al. (2009) that in polar coordinates, dependence of the eigenvalues in the

ransverse direction on those in the other direction is not explicit. Absence of explicit dependence leads to a complete solution which doesot have imaginary transverse eigenvalues, and thus �imp are real.

.2. Formulation of the time-dependent ODEs and their solution

In view of Eq. (24), Eq. (23) can be written as,∫ ri

ri−1

(1˛i

∂T�im

∂t(r, t) + �2

impT�im(r, t)

)rRimp(r)dr =

[rRimp(r)

∂T�im

∂r(r, t) − rT�

im(r)∂Rimp

∂r(r)

]ri

ri−1

+∫ ri

ri−1

rRimp(r)g�

im(r, t)

kidr. (32)

N

M

N

a

E

N

D

w

a

S

w

S. Singh et al. / Nuclear Engineering and Design 241 (2011) 144–154 149

ow using Eq. (31), one gets:

1˛i

∫ ri

ri−1

(∂T�

im

∂t(r, t) + ˛1�2

1mpT�im(r, t)

)rRimp(r)dr =

[rRimp(r)

∂T�im

∂r(r, t) − rT�

im(r)∂Rimp

∂r(r)

]ri

ri−1

+∫ ri

ri−1

rRimp(r)g�

im(r, t)

kidr. (33)

ultiplying the above equation by ki and summing over all the n layers, one gets,

n∑i=1

ki

˛i

∫ ri

ri−1

(∂T�

im

∂t(r, t) + ˛1�2

1mpT�im(r, t)

)rRimp(r)dr =

n∑i=1

ki

[rRimp(r)

∂T�im

∂r(r, t) − rT�

im(r)∂Rimp

∂r(r)

]ri

ri−1

+n∑

i=1

∫ ri

ri−1

rRimp(r)g�im(r, t)dr. (34)

ow defining,

Tr�mp(t) ≡

n∑i=1

ki

˛i

∫ ri

ri−1

rRimp(r)T�im(r, t)dr (35)

nd,

gr�mp(t) ≡

n∑i=1

∫ ri

ri−1

rRimp(r)g�im(r, t)dr (36)

q. (34) reduces to:

dTr�mp

dt(t) + ˛1�2

1mpTr�mp(t) =

n∑i=1

ki

[rRimp(r)

∂T�im

∂r(r, t) − rT�

im(r, t)∂Rimp

∂r(r)

]ri

ri−1

+ gr�mp(t). (37)

ow application of the interface conditions, Eqs. (20) and (21) and (27) and (28) yields (see Appendix B for details):

dTr�mp

dt(t) + ˛1�2

1mpTr�mp(t) = knrn

Cr�out,mp(t)︷ ︸︸ ︷[

Rnmp(rn)∂T�

nm

∂r(rn, t) − T�

nm(rn)dRnmp

dr(rn)

]

− k1r0

[R1mp(r0)

∂T�1m

∂r(r0, t) − T�

1m(r0)dR1mp

dr(r0)

]︸ ︷︷ ︸

Cr�in,mp

(t)

+ gr�mp(t) (38)

efining Cr�out,mp(t) and Cr�

in,mp(t) as the first and second square bracket on the RHS of the above equation, one gets:

dTr�mp

dt(t) + ˛1�2

1mpTr�mp(t) = knrnCr�

out,mp(t) − k1r0Cr�in,mp(t) + gr�

mp(t), (39)

here Cr�in,mp

(t) can be evaluated from Eqs. (18) and (25) as,

Cr�in,mp(t) =

⎡⎢⎣

C�in,m

(t)R1mp(r0)

Ain, if Ain /= 0

−C�

in,m(t)((dR1mp)/(dr))(r0)

Bin, if Ain = 0

(40)

nd similarly, Cr�out,mp(t) can be evaluated from Eqs. (19) and (26) as,

Cr�out,mp(t) =

⎡⎢⎣

C�out,m(t)Rnmp(rn)

Aout, if Aout /= 0

−C�

out,m(t)((dRnmp)/(dr))(rn)

Bout, if Aout = 0

. (41)

olution of Eq. (39) is:

Tr�mp(t) = Tr�

mp(0)e−˛1�2

1mpt + e

−˛1�21mp

t∫ t

0

(knrnCr�out,mp(�) − k1r0Cr�

in,mp(�) + gr�mp(�))e

˛1�21mp

�d�, (42)

here Tr�mp(0) is evaluated using the initial condition (Eq. (22)) and the definition in Eq. (35) as:

Tr�mp(t = 0) =

n∑i=1

ki

˛i

∫ ri

ri−1

rRimp(r)f �im(r)dr. (43)

150 S. Singh et al. / Nuclear Engineering and Design 241 (2011) 144–154

F(

5

T

N

Aa

6

t

igb

ig. 2. Asymmetric heat conduction in a three-layer annulus. Each layer has a different thermal conductivity (ki) and thermal diffusivity (˛i). The lower-half of the annulus� ≤ � ≤ 2�) is kept insulated, while the upper-half (0 ≤ � ≤ �) is subjected to a (�, t)-dependent incoming heat-flux.

. Inversion formula

T�im

(r, t) can be expanded in the generalized Fourier series as follows:

T�im(r, t) =

∞∑p=1

cmp(t)Rimp(r). (44)

he time-dependent coefficients are found by applying the orthogonality condition (Eq. (30)),

cmp(t) =

n∑i=1

ki˛i

∫ ri

ri−1rRimp(r)T�

im(r, t)dr

Nmp. (45)

ow using the definition in Eq. (35),

cmp(t) = Tr�mp(t)

Nmp. (46)

s stated before, the above formulation and solution is valid for T�i0(r, t), T�

imc(r, t) and T�

ims(r, t) in Eq. (9). Substituting Eq. (44) with

ppropriate subscripts (0, c or s) in Eq. (9), one gets:

Ti(r, �, t) =∞∑

p=1

Tr�0p(t)

N0pRi0p(r) +

∞∑m=1

∞∑p=1

Tr�mpc(t)

NmpRimp(r) cos(m�) +

∞∑m=1

∞∑p=1

Tr�mps(t)

NmpRimp(r) sin(m�). (47)

. Illustrative example and results

A three-layer annulus (r0 ≤ r ≤ r3, 0 ≤ � ≤ 2�; see Fig. 2) is initially at a uniform zero temperature. For time t > 0, following �- and-dependent heat-flux,

q′′(r = r3, �, t) =[

q0�2(� − �)2f (t), 0 ≤ � ≤ �0, � ≤ � ≤ 2�

(48)

s applied at the outer surface (r = r3) while the inner surface (r = r0) is maintained isothermal at zero temperature. The details of f(t) areiven later in this section. This leads to the coefficients Ain = 0, Bin = 1, Aout = k3, Bout = 0, Cin(�, t) = 0 and Cout(�, t) = q′′(r3, �, t) in the respectiveoundary condition equations. There is no volumetric heat generation in any of the layers, i.e., gi(r, �, t) = 0.

Parameter values used in this problem are, k2/k1 = 2, k3/k1 = 4; ˛2/˛1 = 4, ˛3/˛1 = 9; r1/r0 = 2, r2/r0 = 4, r3/r0 = 6.It should be noted that, in the results that follow, r, t, and Ti(r, �, t) are in the units of r0, r2

0 /˛1 and, q0r0/k1, respectively.

S. Singh et al. / Nuclear Engineering and Design 241 (2011) 144–154 151

Fig. 3. A comparison of temporal and steady state radial temperature distributions at different angular positions: (a) � = 0, (b) � = �/4, (c) � = �/2, (d) � = 3�/2, (e) � = 7�/4,with the results reported in Jain et al. (2009). Lines represent data from Jain et al. (2009) and circles are from data obtained in current work.

152 S. Singh et al. / Nuclear Engineering and Design 241 (2011) 144–154

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7

dbliea

Fig. 4. Temporal variation of temperature at � = �/2 and mid-point of different layers. Non-oscillating plots are for f(t) = 1 and oscillating ones are for f(t) = sin(t).

For this particular problem, the infinite series solution for the temperature Ti(r, �, t) is truncated at p = P and m = M, leading to,

Ti(r, �, t) =P∑

p=1

Tr�0p(t)

N0pRi0p(r) +

M∑m=1

P∑p=1

Tr�mpc(t)

NmpRimp(r) cos(m�) +

M∑m=1

P∑p=1

Tr�mps(t)

NmpRimp(r) sin(m�) + εi(r, �, t; M, P) (49)

here εi(r, �, t ; M, P) is the truncation error. Both, M and P are chosen to be 15 for the results shown in this paper. [The change in thealues of temperature for (M = 10 and P = 10) and those obtained with (M = 15 and P = 15) is of the order of 10−4. Since, series is sufficientlyonverged; we have used 15 terms for both M and P.]

The analysis is carried out for the following two cases:

Case 1: f(t) = 1Case 2: f(t) = sin(t).

In the Case 1, boundary condition is time-independent. However, solution of the problem is obviously time-dependent as temperaturencreases with time from its initial value to finally attain a steady state temperature distribution. This problem can be solved by the separationf variables (SOV) method as well as the finite integral transform (FIT) method discussed in this paper. Therefore, solution obtained fromhese two methods are compared to show the accuracy and correctness of the method developed in the current work. Radial temperatureistributions at different angular positions and for different values of t obtained from applying the SOV as well as the FIT method are shown

n Fig. 3. Results obtained using the FIT method based time-dependent formulation are in excellent agreement with those obtained earliersing the SOV method for the time-independent boundary conditions. It should be noted that correctness and accuracy of the SOV methodave already been established by Jain et al. (2009) for the same problem.

Solution for the Case 2 is used to show the application of the methodology for time dependent boundary condition using the FIT method.ote that it is not possible to solve this problem using the SOV method. Temporal variations of temperature for the Case 2, at � = �/2 andid-point of different layers (r = 1.5, 3.0 and 5.0) are shown in Fig. 4. Solution for the case 1 (at the same points) is also plotted in the same

raph. As expected, the solution for f(t) = sin(t) oscillates about the solution for f(t) = 1. Frequency of oscillations at these three differentoints are the same, however, there is a phase difference due to diffusion lag.

. Conclusions

In this paper, an analytical solution is presented for the asymmetric transient heat conduction in a multilayer annulus subjected to time-ependent boundary conditions. Each layer can have spatially varying as well as time-dependent volumetric heat sources. Inhomogeneous

oundary condition of the first, second or third kind can be applied in the radial direction. Proposed solution is also applicable to the

ayered-structures with zero inner radius (r0 = 0). In polar coordinates, dependence of the eigenvalues in the transverse direction on thosen the other direction is not explicit. Absence of explicit dependence leads to a complete solution which does not have imaginary transverseigenvalues. Numerical evaluation of the series solution shows that a reasonable number of terms are sufficient to obtain results withcceptable errors for engineering applications.

A

U

N

A

E

S

A

N

∑

F

S. Singh et al. / Nuclear Engineering and Design 241 (2011) 144–154 153

ppendix A.

Operating Eq. (17) by∫ ri

ri−1rRim(r)dr leads to:∫ ri

ri−1

(1˛i

∂T�im

∂t(r, t)

)rRim(r)dr =

∫ ri

ri−1

d

dr

(r

dT�im

dr(r)

)Rim(r, t)dr︸ ︷︷ ︸

Term A

−∫ ri

ri−1

(m2

r2Rim(r)

)rT�

im(r, t)dr +∫ ri

ri−1

rRim(r)g�

im(r, t)

kidr. (A1)

sing integration by parts on Term A in the above equation gives:

Term A =[

rdT�

im

dr(r, t)Rim(r)

]ri

ri−1

−∫ ri

ri−1

rdT�

im

dr(r, t)

dRim(r)dr

dr︸ ︷︷ ︸Term B

(A2)

ow, using integration by parts on Term B, we get:

Term A =[

rdT�

im

dr(r, t)Rim(r)

]ri

ri−1

−[

rT�im(r)

dRim

dr(r)

]ri

ri−1

+∫ ri

ri−1

d

dr

(r

dRim

dr(r)

)T�

im(r, t)dr . (A3)

Substitution of Eq. (A3) into Eq. (A1) yields Eq. (23).

ppendix B.

Consider the first term on the RHS of Eq. (37), which is as follows:

n∑i=1

ki

[rRimp(r)

∂T�im

∂r(r, t) − rT�

im(r, t)∂Rimp

∂r(r)

]ri

ri−1

. (B1)

valuating the above expression at ri and ri−1, one gets:

n∑i=1

ki

[riRimp(ri)

∂T�im

∂r(ri, t) − riT

�im(ri)

∂Rimp

∂r(ri)

]−

n∑i=1

ki

[ri−1Rimp(ri−1)

∂T�im

∂r(ri−1, t) − ri−1T�

im(ri−1)∂Rimp

∂r(ri−1)

]. (B2)

eparating out the last term in the first summation and first term in the second summation yields:

n−1∑i=1

ki

[riRimp(ri)

∂T�im

∂r(ri, t) − riT

�im(ri)

∂Rimp

∂r(ri)

]+ knrn

[Rnmp(rn)

∂T�nm

∂r(rn, t) − T�

nm(rn)dRnmp

dr(rn)

]

− k1r0

[R1mp(r0)

∂T�1m

∂r(r0, t) − T�

1m(r0)dR1mp

dr(r0)

]−

n∑i=2

ki

[ri−1Rimp(ri−1)

∂T�im

∂r(ri−1, t) − ri−1T�

im(ri−1)∂Rimp

∂r(ri−1)

]. (B3)

ppropriately changing the indexing and range of first summation in the above expression, leads to:

n∑i=2

ki−1

[ri−1Ri−1,mp(ri−1)

∂T�i−1,m

∂r(ri−1, t) − ri−1T�

i−1,m(ri−1,)∂Ri−1,mp

∂r(ri−1)

]+ knrn

[Rnmp(rn)

∂T�nm

∂r(rn, t) − T�

nm(rn)dRnmp

dr(rn)

]

− k1r0

[R1mp(r0)

∂T�1m

∂r(r0, t) − T�

1m(r0)dR1mp

dr(r0)

]−

n∑i=2

ki

[ri−1Rimp(ri−1)

∂T�im

∂r(ri−1, t) − ri−1T�

im(ri−1)∂Rimp

∂r(ri−1)

]. (B4)

otice the same range of both the summations in the above expression. Combining the two summations together, one gets:

n

i=2

[ki−1

{ri−1Ri−1,mp(ri−1)

∂T�i−1,m

∂r(ri−1, t) − ri−1T�

i−1,m(ri−1,)∂Ri−1,mp

∂r(ri−1)

}−ki

{ri−1Rimp(ri−1)

∂T�im

∂r(ri−1, t) − ri−1T�

im(ri−1)∂Rimp

∂r(ri−1)

}]

+ knrn

[Rnmp(rn)

∂T�nm

∂r(rn, t) − T�

nm(rn)dRnmp

dr(rn)

]− k1r0

[R1mp(r0)

∂T�1m

∂r(r0, t) − T�

1m(r0)dR1mp

dr(r0)

]. (B5)

rom the interface conditions (Eqs. (20), (21), (27) and (28)), it can be immediately seen that following holds for i = 2, ..., n:

ki−1

{ri−1Ri−1,mp(ri−1)

∂T�i−1,m

∂r(ri−1, t) − ri−1T�

i−1,m(ri−1,)∂Ri−1,mp

∂r(ri−1)

}= ki

{ri−1Rimp(ri−1)

∂T�im

∂r(ri−1, t) − ri−1T�

im(ri−1)∂Rimp

∂r(ri−1)

}.

(B6)

1

T

T

R

ddd

dHJJ

LLLLh2MOS

SS

Y

54 S. Singh et al. / Nuclear Engineering and Design 241 (2011) 144–154

herefore, each term in the summation in (B5) is zero. Hence, expression in (B5) reduces to:

knrn

[Rnmp(rn)

∂T�nm

∂r(rn, t) − T�

nm(rn)dRnmp

dr(rn)

]− k1r0

[R1mp(r0)

∂T�1m

∂r(r0, t) − T�

1m(r0)dR1mp

dr(r0)

](B7)

he above discussion shows that the expression in (B1) and (B7) are equal. Hence, one can get Eq. (38) from Eq. (37).

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