generalized functions in thermal conductivity problems for multilayered constructions

ISSN 0018�151X, High Temperature, 2013, Vol. 51, No. 6, pp. 830–840. © Pleiades Publishing, Ltd., 2013.Original Russian Text © I.V. Kudinov, V.A. Kudinov, E.V. Kotova, A.E. Kuznetsova, 2013, published in Teplofizika Vysokikh Temperatur, 2013, Vol. 51, No. 6, pp. 912–922.

830

INTRODUCTION

It is very difficult to obtain exact analytical solu�tions of nonstationary thermal conductivity problemsfor multilayered constructions because of the necessityin the fulfillment of conjugacy conditions assigned inthe form of equality of temperatures and heat fluxes inlayer contact points. The fulfillment of these condi�tions is connected with solving the characteristic sys�tem in the form of the chained transcendent equationrelative to proper numbers of the boundary problemwhose exact solution is possible only by numericalmethods. Solving such problems is very effective bymethods for reduction of thermal conductivity prob�lems for multilayered constructions to single�layerconstructions with the discontinuous (piecewise uni�form) physical properties of the medium. One suchmethod is connected with the use of the asymmetryunit function (Heaviside function). This paper usesthis method for obtaining the solution in the first stepof the nonstationary process. The basis of the secondmethod is the orthogonal Kantorovich method inwhich the reduction of the multilayer construction toa single�layer construction is realized by the use of theglobal set of unknown time functions. This method isused for solving the second step of the nonstationaryprocess.

APPLICATION OF THE MATHEMATICAL APPARATUS OF GENERALIZED FUNCTIONS

IN THE INTEGRAL METHOD FOR THERMAL BALANCE

The main advantage of methods in which multilay�ered constructions are reduced to single�layer is thesignificant simplification of the fulfillment of conju�gacy conditions. For example, in the case of using theasymmetry unit function, these conditions are fulfilledexactly due to the special construction of the basic dif�ferential equation obtained here [1–5]. By the Kan�

torovich method, one can construct a set of coordinatefunctions such that in any approximation they exactlymeet the boundary conditions and conjugacy condi�tions. Analytical solutions obtained by these twomethods allow one practically with the required accu�racy to carry out investigations of the temperaturestate of multilayered constructions in all time range ofthe nonstationary process, including its small andminute values.

The methods allowing one to neglect the difficul�ties for solving at small values of the time variable areintegral thermal balance methods [1, 5–9]. The gen�eral problem for their use is insufficient accuracy ofthe results obtained. In [1, 5, 8, 9] they applied addi�tional boundary conditions allowing for obtaining theanalytical solutions practically with the required accu�racy. Let us consider the application of this method tosolving the nonstationary thermal conductivity prob�lem for a multilayered plate in the following mathe�matical formulation (Fig. 1):

(1)

(2)

(3)

(4)

where Θ = is the relative excess tem�perature, x is the coordinate, Tw is the wall tempera�ture at T0 is the initial temperature, τ is the time,n is the layer number, δi is the thickness of the ith layer,

is the heat conductivity, C(x) = is thevolume heat capacity, is the specific heat, and

is the density.

Thermophysical characteristics and of amultilayered body represented in the form of single

( )( )

( )( )

( )

, ,

,n

x xC x x

x x> x x

∂Θ τ ∂Θ τ⎡ ⎤∂= λ⎢ ⎥∂τ ∂ ∂⎣ ⎦

τ < <0;0

( ),0 0,xΘ =

(0, ) 1,Θ τ =

( , ) 0,nx x∂Θ τ ∂ =

− −w0 0( ) ( )T T T T

0,x =

( )xλ ( ) ( )c x xρ

( )c x( )xρ

( )xλ ( )C x

Generalized Functions in Thermal Conductivity Problemsfor Multilayered Constructions

I. V. Kudinov, V. A. Kudinov, E. V. Kotova, and A. E. KuznetsovaSamara State Technical University, Samara, Russia

Received August 31, 2012

Abstract—Based on the mathematical apparatus of the generalized function theory and the Kantorovichorthogonal method, approximate analytical solutions of the thermal conductivity problem for a multilayeredplate are obtained that allow for evaluating the thermal state of a construction practically in the entire timerange of nonstationary process, including its small and minute values.

DOI: 10.1134/S0018151X13050131

HEAT AND MASS TRANSFER AND PHYSICAL GASDYNAMICS

HIGH TEMPERATURE Vol. 51 No. 6 2013

GENERALIZED FUNCTIONS IN THERMAL CONDUCTIVITY PROBLEMS 831

layer are described by means of the asymmetry unitfunction by the following relationships:

(5)

(6)

where is the asymmetry unit function deter�mined by the formula

(7)

For simplifying Eq. (1), let us introduce the newindependent variable

(8)

where by analogy with (5) is determined in the

following way:

(9)

Substituting (9) into (8), we obtain

(10)

Determining the integrals in (10), we find

(11)

From relationship (11) it follows that

(12)

From (12) we have

(13)

From this one can conclude that the new indepen�dent variable in the physical meaning presents thethermal resistance of the corresponding wallpart.

In order to describe problem (1)–(4) in the newindependent variable, relationship (8) is differentiatedwith respect to the variable x

( ) ( ) ( )1

1 1

1

,n

i i i

i

x H x x−

+

=

λ = λ + λ − λ −∑

( ) ( ) ( )1

1 1

1

,n

i i i

i

C x C C C H x x−

+

=

= + − −∑( )− iH x x

( )<⎧

− = ⎨≥⎩

at ;

at

0

1 .i

ii

x xH x x

x x

( )z z x=

( )( )

( )1 0, ,

i

x

x

z x dx i nx

= =λ∫

( )1xλ

( )( )

1

1 11

1 1 1 1 .n

ii ii

H x xx

−

+=

⎛ ⎞= + − −⎜ ⎟λ λ λ λ⎝ ⎠

∑

( ) ( )

0

1

1 11

1 1 .

i

x x n

ii iix x

dxz x H x x dx−

+=

⎛ ⎞= + − −⎜ ⎟λ λ λ⎝ ⎠

∑∫ ∫

( ) ( ) ( )1

0

1 11

1 1 .n

i ii ii

x xz x x x H x x

−

+=

⎛ ⎞−= + − − −⎜ ⎟λ λ λ⎝ ⎠

∑

( )−

+=

−

= =

⎛ ⎞−= + − −⎜ ⎟λ λ λ⎝ ⎠

− δ= =

λ λ

∑

∑ ∑

1

0

1 11

1

1 1

1 1

.

n

nn n i

i ii

n n

i i i

i ii i

x xz x x

x x

1 0 11

1 1

2

1 0 2 12

1 2 1

,

,....i

ii

x xz

x x x xz

=

− δ= =

λ λ

− δ−= + =

λ λ λ∑

R = δ λ

(14)

Because the variable z is the function of x, the fol�lowing relationships are true:

(15)

From (15) it follows that the derivative of tempera�ture with respect to the variable z is the measure of heatflux

(16)

With allowance made for (14) and (16), the differ�ential operator of the right side of Eq. (1) is

(17)

From this the mathematical formulation of prob�lem (1)–(4) in the variable z takes the form

(18)

(19)

(20)

(21)

( )

( )1 .

dz x

dx x=λ

( )( )

( )( )

( )( )

( )

( )

∂Θ τ ∂Θ τλ = λ

∂ ∂

∂Θ τ ∂Θ τ= λ =

∂ λ ∂

, ,

, ,1 .

x z dzx xx z dx

z zx

z x z

( )( )

( ), ,.

z xx

z x

∂Θ τ ∂Θ τ= λ

∂ ∂

( )( )

( ) ( )

( )

∂Θ τ⎡ ⎤∂ λ⎢ ⎥∂ ∂⎣ ⎦

∂Θ τ ∂ Θ τ⎡ ⎤∂= =⎢ ⎥∂ ∂ λ⎣ ⎦ ∂

2

2

,

, , 1 .

xx

x x

z zdzz z dx zz

( )( )

( )

( )2

2

, ,1 ( 0;0 ),nz z

C z z zz z

∂Θ τ ∂ Θ τ= τ > < <

∂τ λ ∂

( ),0 0,zΘ =

(0, ) 1,Θ τ =

( , ) 0.nz z∂Θ τ ∂ =

0 x1 x2 xn–1 x xn

Θ

λ1 λ2 λn

a1 a2 an

δ1 δ2 δn

δ

q(τ)

Fig. 1. Calculational scheme of heat exchange.

832


KUDINOV et al.

It is evident that Eq. (18) is simpler for integrationin comparison with (1).

Problem (18)–(21) is solved by the integral heatbalance method with the attraction of additionalboundary conditions [1, 5, 8, 9]. Following themethod, the heating process is separated into two timesteps: and For that we introducea boundary (front of temperature disturbance) movingin time and separating the initial area intotwo subareas and where

is the function determining the motion of theboundary in time (Fig. 1). At the same time, the areabehind the temperature disturbance front remains theinitial temperature. The first step of the process ends atthe attainment by the moving boundary of the coordi�nate i.e., then During the second step,the temperature change occurs over the entire bodyvolume (step of the regular mode of heatexchange). At this process step, the concept of thetemperature disturbance front loses its meaning, andthe problem is solved by the orthogonal Kantorovichmethod (see below).

The mathematical problem formulation for thefirst process step includes Eq. (18) with boundary con�dition (20) and the following boundary conditions ful�filled on the temperature disturbance front:

(22)

(23)where relationships (22) and (23) present conditionsof mating the heated and unheated zones. Relation�ship (22) determines the equality of the body temper�ature at the point to its initial temperature.According to condition (23), the heat flux is not prop�agated behind the temperature disturbance front (con�dition of the adiabatic wall). The mathematical proofof the necessity to introduce boundary conditions oftypes (22) and (23) at the temperature disturbancefront is given in [10].

Let us pay attention to the evident fact that prob�lem (18), (20), (22), and (23) at the first process step isnot generally determined behind the temperature dis�turbance front, i.e., in the range There�fore, there is no need to fulfill the initial condition ofform (19) over the entire plate thickness. In this case itis quite enough to fulfill boundary condition (22),according to which for all the body tempera�ture is equal to the initial temperature. In addition,also the boundary condition of form (21) is lacking inthe given problem because it has no effect upon heatexchange in the first step of the process.

Let us note that problem (18), (20), (22), and (23)is not related to the class of problems in which the endvelocity of the heat wave is taken into account. Theirsolution is reduced to integration of the hyperbolic(wave) heat conductivity equation [1, 11]. The tem�perature disturbance front introduced in problem

≤ τ ≤τ10 τ ≤ τ< ∞1 .

nz z≤ ≤0( )z q≤ ≤ τ0 ( ) ,nq z zτ ≤ ≤

( )q τ

,nz z= 1.τ = τ

nz z≤ ≤0

( ), 0,qΘ τ =

( )∂Θ τ ∂ = ≤ ≤ τ0 ( ), 0 ( ),q z z q

( )z q= τ

τ ≤ ≤( ) .nq z z

( )z q= τ

(18), (20), (22), and (23) is an analog of the movingisotherm (but no heat wave) by the physical meaning.Because the initial temperature is main�tained at the temperature disturbance front during itsmotion over the coordinate z, consequently, it is theanalog of the zero isotherm (isotherm of the initialcondition). In papers [1, 5] it is shown that as theapproximation number increases, the time τ1 of themotion of the temperature disturbance front inthe range decreases and in the limit at

there is . In other words, the tempera�ture disturbance front velocity is directed towards theinfinite value with an increase in the approximationnumber. The given result correlates completely withthe hypothesis of the infinite velocity of the heat distur�bance, which is laid on the basis of the derivation of theparabolic thermal conductivity equation of form (1).Let us note that with increasing the approximationnumber, the accuracy of the solution obtained risessubstantially. Investigations in [1] showed that thetemperature in the center of the infinite plate in thesecond, fourth, seventh, and fourteenth approxima�tions differs from the exact solution by 0.31, 0.0028,0.26 × 10–5, and 0.28 × 10–12%, respectively.

The solution of problem (18), (20), (22), and (23)is found in the form of the following polynomial:

(24)

where are unknown coefficients determinedfrom boundary conditions (20), (22), and (23). Afterdetermining them, relationship (24) takes the form

(25)

To determine unknown function , we form theresidual of Eq. (18) and determine the integral of itwithin the depth of the thermal layer (which is equiva�lent to the construction of the heat balance integral)

(26)

Determining the integral in the right part of rela�tionship (26) with allowance made for (23) and (25),we find

(27)

The product in the left part of relationship(26) by the analogy with (5) and (6) can be written as

(28)

( , ) 0qΘ τ =

( )q τ

0 n nz z≤ ≤

→ ∞n τ →1 0

0

( , ) ( ) ,n

kk

k

z b q z=

Θ τ =∑

kb q( )

( ) ( )2, 1 .z z qΘ τ = −

q τ( )

( ) ( )( ) ( )

2

2

0 0

, ,.

q qz z

z C z dz dzz

∂Θ τ ∂ Θ τλ =

∂τ ∂∫ ∫

2

2

0

( , ) ( , ) (0, ) 2.

q

q qdz

z z qz

∂ Θ τ ∂Θ τ ∂Θ τ= − =

∂ ∂∂∫

z C zλ( ) ( )

( ) ( )1

1 1 1 1

1

( ) ( ) .n

i i i i i

i

z C z C C C H z z−

+ +

=

λ = λ + λ − λ −∑



Substituting (25), (27), and (28) into (26), weobtain

In determining the integrals in the last relationship,there is

(29)

Integrating Eq. (29) at the initial condition, we find

(30)

for example, for the double�layer body we have

(31)

Assuming from (30) we determine the timeof the termination of the first step of the process as

Relationship (31) relative to presents the non�linear equation whose solution can be found only bynumerical methods. The lack of a possibility for theexplicit expression of forms defined inconve�niences both for calculations of the motion of the tem�perature disturbance front and for the temperaturestate of the construction. To obtain the approximateanalytical solution of Eq. (29), let us use the methodfor solving the nonlinear equations that is given in [1]according to which we find

(32)

The results of calculations of by relationships(31) and (32) are given in Fig. 2. Their analysis allowsone to conclude that they are practically congruent at

. Consequently, formula (32) with sufficientaccuracy can be used as the analytical solution ofEq. (29).

Relationships (25) and (32) represent the solutionof problem (18)–(21) in the first approximation.

( ) ( )−

+ +

=

⎡ ⎛ ⎞ ⎛ ⎞∂ ⎢ − + λ + − +⎜ ⎟ ⎜ ⎟∂τ ⎢ ⎝ ⎠ ⎝ ⎠⎣

⎤× λ − λ − =⎥

⎥⎦

∫ ∫

∑

2 2

1 12 2

0

1

1 1

1

2 21 1

2.

i

q q

z

n

i i i i i

i

z zz C dz zq qq q

C C H z z dzq

( )

( )

−

+ +

=

⎡λ + λ − λ⎢

τ ⎢⎣

⎤⎛ ⎞× − + − =⎥⎜ ⎟⎝ ⎠ ⎦

∑1

1 1 1 1

1

2 33 26.

n

i i i i

i

i ii

dqq C C C

d

z zq H q z

q q

( )0 0q =

( )

( )

−

+ +

=

⎛λ + λ − λ −⎜

⎝

⎞− + − = τ⎟

⎠

∑ ln1

2 2 21 1 1 1

1

32

6

4 3 12 ;

n

i i i i iii

ii i

qq C C C q z

z

zz H q z

q

( )

( )

⎛λ + λ − λ −⎜

⎝

⎞− + − = τ⎟

⎠

ln2 2 21 1 2 2 1 1 1

1

3211 1

6

4 3 12 .

qq C C C q z

z

zz H q z

q

nq z=

1.τ = τ

( )q τ

( )q τ

( )0.70.00048 .q τ = τ

( )q τ

τ ≥ 0.05

Let us find the solution of the concrete problem fora double�layer plate (steel–titanium) at the followinginput data [7]:

(33)

Relationship (30) at (sin�gle�layer plate) is reduced to the following quadraticequation relative to function

Extracting the positive root (the value of the tem�perature disturbance front by physical meaning can�not be negative) from the solution of the last equation,we find

(34)

The termination time of the first step of the pro�cess, according to relationship (13), is observed at

(35)

Substituting (35) into (34), we have

(36)

Taking into account that from (36) we find

(37)

−

−

δ = δ = = ×

= × λ =

λ = =

= ρ =

ρ =

2

2

3

3

m, m, m s,

m s, W (m К),

W/(m К), J (kg К),

J (kg К); kg m ,

kg m .

61 2 1

62 1

2 1

2 1

2

0.002 0.004 12.5 10

6 10 45.24

16.24 462

527 7900

5136

a

a

C

C

1 2 ,λ = λ = λ 1 2C C C= =

( ):q τ

12 .qλ τ = τ2C ( )

( ) ( )12 .q Cτ = τ λ

( ) ( ) 1 21 2

1 2

.q q zδ δ δ

τ = τ = = + =λ λ λ

112 ( ).Cδ λ = τ λ

,C c= ρ

21 (12 ).cτ = δ ρ λ

0.1000

0.5

1.0

1.5

2.0

3.0

2.5

0.200 0.300 0.400 0.484

13

24

q(τ) q(

τ)

τ1 τ1

τ, s

z 104

m2/K( )/W,×

Fig. 2. Motion of the temperature disturbance front q(τ)over the coordinate z in time: (1) by formula (31), (2) (54),(3) (32), and (4) (55).

834


KUDINOV et al.

Paper [1], as applied to the single�layer plate, for

gives the formula where Therefore,

(38)

The termination time of the first process step,according to [1, 5, 8, 9] is observed at ξ = x/δ = 1, i.e.,at where ξ is the dimensionless spatial vari�able. Relationship (38) at takes the form

(39)

Therefore,

(40)

( )q Fo

( ) 12 ,q =Fo Fo 2 ;a= τ δFo( ).a c= λ ρ

( ) ( )212 .q cτ = λτ δ ρ

( )1 1,q τ =

( )1 1q τ =

( )2112 1.cλτ δ ρ =

21 (12 ).cτ = δ ρ λ

Relationships (37) and (40) are congruent. Conse�quently, relationship (30) for the single�layer plate isreduced to the known solution obtained in the firstapproximation by means of the use of the integralthermal balance method. Taking τ1 = Fo1δ

2/a in for�mula (40), we obtain Fo1 = 1/12 = 0.08333. This valueof Fo1 is congruent with its value found in [1, 5, 12] forthe single�layer plate.

The calculation results for the motion of the temper�ature disturbance front over coordinates z and x, whichis obtained for a double�layer plate by formulas (31) and(32), are given in Figs. 2 and 3. The dimensionlesstemperatures found by formula (25) in variables z andx are represented in Figs. 4 and 5, where τ* is the timeof the attainment by the temperature disturbance frontof the coordinate z = z1.

Analysis of obtained results allows one to concludethe following. The sharp bend on curves and Θ(z, τ)in Figs. 2 and 4 is absent, whereas the sharp bend ispresent for the same functions with the variable x. Thisis connected with the fact that the relationship withthe variable z for the equality of heat fluxes for the dou�ble layer body, according to (15), has the form

(41)

Taking into account that the variable z is the functionof x, relationship (41) with allowance made for (14) canbe rewritten in the form

(42)

The value 1/(λ(x)) in the right and left parts of rela�tionship (42) is related to different layers and, conse�quently, according to relationship (9), forthe left part and for the right part. There�fore, (42) with the variable z takes the form

(43)

The obtained relationship verifies the lack of thesharp bend in curves and Θ(z, τ) determined withthe variable z.

For enhancement of the accuracy of the solution,there is a need to increase the degree of approximatingpolynomial (24). The appearing unknown coefficientsbk are found from basic (20), (22), (23) and additionalboundary conditions. For their determination equa�tion (18) and boundary conditions (20), (22), and (23)are used. In particular, from differentiation of condi�tions (20) with respect to the variable τ and compari�son of the obtained relationship with Eq. (18), asapplied to the point z = 0, we find the first additionalboundary condition [8, 9, 13]

(44)

Let us differentiate relationship (22) with respect tothe variable τ. Because there is a need to find Θ(z, τ)from (22) at the point z = q(τ), z is a function of τ and,

( )q τ

( ) ( ) ( ) ( )1 1 1 2 1 11 2

, ,.

z dz x z dz x

z dx z dx


∂ ∂

1 1 2 11 2

( , ) ( , )1 1 .( ) ( )

z z

z x z x


∂ λ ∂ λ

11 ( ) 1xλ = λ

21 ( ) 1xλ = λ

( ) ( )1 1 2 1, , .z z z z∂Θ τ ∂ = ∂Θ τ ∂

( )q τ

( )2 20, 0.z∂ Θ τ ∂ =

0.4840 0.080

1

x1

3

4

5

6

0.160 0.240 0.320τ*

q(τ) q(τ)1

2

τ, s

τ1

x ×103, m

Fig. 3. Motion of the temperature disturbance front q(τ)over the coordinate z in time: (1) by formula (32), (2) (55).

0.2

300 3 6 9 12 15 18 21 24

0.4

0.6

0.8

1.0Θ

τ * = 0.027

0.002

0.109

0.283

τ1 = 0.484

z × 105

Fig. 4. Temperature distribution in the double�layer platein the variable z by (25) (first approximation).



consequently, Θ(z, τ) is a complex function. Thenthere is

(45)

With allowance made for (23), relationship (45) is(46)

Comparing (46) with Eq. (18), as applied to thepoint z = q(τ), the second additional boundary condi�tion is found:

(47)Let us differentiate boundary condition (23) with

respect to the variable τ with allowance made for thefact that the variable z is a function of τ:

(48)

Relationship (48) with allowance made for (47) isreduced to the form

(49)Let us differentiate Eq. (18) with respect to the

variable z and apply the obtained relationship to thepoint :

(50)

Comparing (49) and (50) with allowance made for(47), the third additional boundary condition is found:

(51)The physical meaning of the additional boundary

conditions is in the fulfillment of Eq. (18) and its vari�ables of various order at the point z = 0 and at the tem�

( )[ ]=

==

∂Θ τ ∂Θ τ∂ ⎡ ⎤ ⎡ ⎤Θ τ = +⎢ ⎥ ⎢ ⎥⎣ ⎦∂τ ∂ τ ∂τ⎣ ⎦

( , ) ( , ), .

z qz qz q

z zdzzz d

( ), 0.q∂Θ τ ∂τ =

( )2 2, 0.q z∂ Θ τ ∂ =

== =

⎡ ⎤ ⎡ ⎤∂Θ τ ∂ Θ τ ∂ Θ τ∂ ⎡ ⎤ = +⎢ ⎥ ⎢ ⎥⎢ ⎥∂τ ∂ τ ∂ ∂τ⎣ ⎦ ∂⎣ ⎦ ⎣ ⎦

2 2

2

( , ) ( , ) ( , ).

z q z q z q

z z zdzz d zz

( ) ( )2 , 0.q z∂ Θ τ ∂ ∂τ =

( )z q= τ

( ) ( )

( ) ( )

=

=

⎡ ⎤∂ Θ τ ∂ Θ τ∂= ⎢ ⎥∂ ∂τ ∂ λ ∂⎣ ⎦

⎡ ⎤ ∂ Θ τ+ ⎢ ⎥λ ∂⎣ ⎦

2 2

2

3

3

( , ) ( , )1

( , )1 .

z q

z q

q qz z C z z z

qC z z z

( )3 3, 0.q z∂ Θ τ ∂ =

perature disturbance front Because the range

of the motion of the temperature distur�bance front includes the entire region of thechange of the spatial variable, consequently, thegreater the number of additional boundary conditionsused, the better the fulfillment of Eq. (18) within theregion under consideration.

The basic (20), (22), (23) and additional (44), (47),(51) boundary conditions allow one to find already sixcoefficients of series (24). Substituting (24) into theabove�mentioned boundary conditions, we obtain rel�

ative to unknown coefficients a set of

six algebraic linear equations. The determination ofcoefficients and their substitution into relation�ship (24) give

(52)

Substituting (52) into thermal balance integral (27)relative to the unknown function , we obtain thefollowing ordinary differential equation:

(53)

Integrating the last equation at the initial condition, we find

( ) .z q= τ

0 ( ) nq z≤ τ ≤

0 nz z≤ ≤

( ) ( )0,5kb q k =

( )kb q

4

( , ) 1 1.5 1 .z zzq q

⎛ ⎞⎛ ⎞Θ τ = + −⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

( )q τ

( )

( )

−

+ +

=

⎡ ⎛λ + λ − λ −⎢ ⎜

τ ⎢ ⎝⎣

⎞⎤+ − + − − =⎟⎥

⎠⎦

∑1

2 2 31 1 1 12

1

4 5 6

2 3 4

1 54

515 16 5 0.2

n

i i i i i

i

i i ii

dqq C C C q z

d q

z z zH q z

qq q q

( )0 0q =

0 0.4 0.8 1.2 1.6 2.0 2.5 3.0 3.5 4.0 4.5 6.0

0.2

0.4

0.6

0.8

1.0

τ =

0.0010.0050.013

τ =

τ * = 0.027

0.078

0.191

τ = τ

1 = 0.484

x × 103, m

Θ

Fig. 5. Temperature distribution in the double�layer plate in the variable x: curves by (25), points by (52).

836


KUDINOV et al.

For example, the last relationship for the double�layer plate takes the form

(54)

Assuming from (54) one can determined the

termination time of the first step of the process.

Using methods [1, 9], we find the approximateanalytical solution of Eq. (53)

(55)

Calculation results for by relationship (54) andformula (55) are given in Figs. 2 and 3 from which itfollows that, as the approximation number increases,

the time τ1 of the attainment of the coordinate

( ) by the temperature disturbance front

decreases and in the limit at [1]. Thecalculation results for the temperature by formula (52)in comparison with the first approximation (calcula�tion by formula (25)) are given in Fig. 5.

( )

( )

−

− +

=

⎡λ + λ − λ −⎢

⎣

⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + − + ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎥⎦

× − = τ

∑ ln1

2 2 21 1 1 1

1

2 3 42 2 2 2

10

32 5 35153 2 6

20 .

n

i i i i iii

i i ii i i i

i

qq C C C q z

z

z z zz z z z

q q q

H q z

( )

( )

⎡ ⎛ ⎞λ + λ − λ − −⎢ ⎜ ⎟

⎝ ⎠⎢⎣

⎤⎛ ⎞ ⎛ ⎞+ − + − = τ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎥⎦

ln2

2 2 2 2 11 1 2 2 1 1 1 1

1

3 42 2 21 11 1 1 1

10 15

32 5 35 20 .3 2 6

zqq C C C q z z

z q

z zz z z H q z

q q

2,q z=

1τ = τ

( ) 0.70.00061 .q τ = τ

( )q τ

nx x=

nz z=

1 0τ → n → ∞

APPLICATION OF THE ORTHOGONAL KANTOROVICH METHOD

The mathematical problem formulation for thesecond process step has the form (double�layer construction is considered, see Fig. 6)

(56)

(57)

(58)

(59)

(60)

(61)

where is the temperature of the ith layer; x isthe coordinate; τ is the time; δ1, δ2 are thicknesses oflayers; is the total thickness of the double�layer system; λi, ai are the thermal conductivity andthe thermal diffusivity of the ith layer; T0 is the initialtemperature; and Tw is the wall temperature at x = δ.

Let us introduce the following dimensionless vari�ables:

(62)With allowance made for notation (62), problem

(56)–(61) takes the form

(63)

(64)

(65)

(66)

(67)

(68)where a is the least of thermal diffusivity ai (i = 1, 2).

The solution of problem (63)–(68), according tothe orthogonal Kantorovich method, is

(69)

where are unknown functions and arecoordinate functions.

As the coordinate functions, the following aretaken [14]:

(70)

Relationship (69) with the use of relationships ofform (70) as the coordinate functions in any approxi�mation exactly fulfills boundary conditions (65), (66)and conjugacy conditions (67), (68).

To obtain solution of problem (63)–(68) in the firstapproximation with the limitation by one term of

( )1τ ≤ τ < ∞

2 2

1 2 0 2

( , ) ( , )

( 0; ; 1,2; 0; ),i i iT x a T x x

x x x i x x

∂ τ ∂τ = ∂ τ ∂

τ > ≤ ≤ = = = δ

0( ,0) ,iT x T=

1(0, ) 0,T x∂ τ ∂ =

δ τ = w2( , ) ,T T

1 1 2 1( , ) ( , ),T x T xτ = τ

1 1 1 2 2 1( , ) ( , ) ,T x x T x xλ ∂ τ ∂ = λ ∂ τ ∂

( 1,2)iT i =

1 2δ = δ + δ

Θ = − − ξ = δ = τ δw w Fo 20( ) ( ), , .i i iT T T T x a

2

2

1 2 0 2

( , ) ( , )

( 0; ; 1,2; 0; 1),

i i ia

a

i

∂Θ ξ ∂ Θ ξ=

∂ ∂ξ

> ξ ≤ ξ ≤ ξ = ξ = ξ =

Fo Fo

Fo

Fo

( ,0) 1,iΘ ξ =

1(0, ) 0,∂Θ ∂ξ =Fo

2(1, ) 0,Θ =Fo

1 1 2 1( , ) ( , );Θ ξ = Θ ξFo Fo

1 1 1 2 2 1( , ) ( , ) ,λ ∂Θ ξ ∂ξ = λ ∂Θ ξ ∂ξFo Fo

1

( , ) ( ) ( ) ( 1,2),n

i k ki

k

f i=

Θ ξ = ϕ ξ =∑Fo Fo

( )kf Fo ( )kiϕ ξ

2 22 21 1

1 1

22

( ) 1 1 ,

( ) 1 ( 1, ).

k kk

kk k n

⎛ ⎞λ λϕ ξ = − − ξ − ξ⎜ ⎟λ λ⎝ ⎠

ϕ ξ = − ξ =

λ1 λ2

a1 a2

δ1 δ2

x1

x2

x0

T

Fig. 6. Calculational scheme of the double�layer construc�tion.



series (69), the residual of differential equation (63) isformed and the residual orthogonality to the firstapproximation coordinate functions and is required (it is taken into account that, according toinput data (33), thermal diffusivity a2 is the least):

(71)

Substituting (69) into (71), we find

where is the first variable of the function of with respect to the variable Fo; and are

the second variables of functions withrespect to the variable ξ.

The last relationship relative to the unknown func�tion presents the following ordinary differentialequation:

(72)

where + and

+

Equation (72) for input data (33) takes the form

(73)Integrating Eq. (73), we obtain

(74)

where С is the integration constant.Substituting (74) into (69), we find

(75)For determining the integration constant, the

residual of initial condition (64) is formed and theresidual orthogonality to coordinate functions and is required:

(76)


(77)

Determining integrals in (77) relative to the integra�tion constant C, we obtain the following algebraic equa�tion: 1.40005С – 1.0194 = 0. Its solution is С = 0.7281.

11( )ϕ ξ 12( )ϕ ξ

( )1

2 211

1 1 1221

( )( )( ) ( ) ( ) 0

1,2 .

i

i

i ii i

i

aff d

a

i−

ξ

= ξ

⎡ ⎤∂ ϕ ξ∂ ϕ ξ − ϕ ξ ξ =⎢ ⎥∂ ∂ξ⎣ ⎦

=

∑ ∫Fo

FoFo

ξ

ξ

ξ

ξ

⎛ ⎞⎜ ⎟ϕ ϕ ξ + ϕ ϕ ξ⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟− ϕ ϕ ξ + ϕ ϕ ξ =⎜ ⎟⎝ ⎠

∫ ∫

∫ ∫

Fo

Fo

1

1

1

1

1

1 11 11 12 12

0

1

11 11 11 12 12

20

'( )

'' ''( ) 0,

f d d

af d d

a

1'( )f Fo

1( )f Fo 11'' ,ϕ 12''ϕ

( )11 ,ϕ ξ ( )12ϕ ξ

1( )f Fo

1 1( ) ( ) 0,df d f− ν =Fo Fo Fo

2 1 ,ν = μ μ

ξ

μ = ϕ ϕ ξ∫1

1 11 110

dξϕ ϕ ξ∫

1

1

12 12 ,d

2µξ

= ϕ ϕ ξ∫11

11 110

2

''ad

a ξϕ ϕ ξ∫

1

1

11 12'' .d

1 1'( ) ( ) 0.f f− =Fo 3.957553 Fo

( )1( ) exp( ) 1,2 ,f C i= ν =Fo Fo

1( , ) exp( ) ( ).i iCΘ ξ = ν ϕ ξFo Fo

11( )ϕ ξ

12( )ϕ ξ

[ ]

1

2

1

1

( ,0) 1 ( ) 0.i

i

i

i

d

−

ξ

= ξ

Θ ξ − ϕ ξ ξ =∑ ∫

1

1

1

11 11 12 12

0

( ( ) 1) ( ) ( ( ) 1) ( ) 0.C d C d

ξ

ξ

ϕ ξ − ϕ ξ ξ + ϕ ξ − ϕ ξ ξ =∫ ∫

With allowance made for the found value of theintegration constant, the solution of problem (63)–(68) in the first approximation takes the form

(78)Relationship (78) as applied to the single�layer

plate ( ) is reduced to the form

(79)Formula (79) coincides with the solution in the first

approximation for the single�layer plate that isobtained in [5, 14, 15].

Let us solve problem (63)–(68) in the secondapproximation. Forming the residual of Eq. (63) andrequiring the orthogonality of the residual to coordi�nate functions and we obtain

(80)

Relationship (80) is reduced to the following set oftwo ordinary differential equations relative tounknown functions and

Determining integrals, we find

(81)

where b1 = 1.238258, b2 = 1.400050, b3 = 5.540774,b4 = 4.991920, l1 = 1.238258, l2 = 1.138087, l3 =5.280966, l4 = 6.189359.

The particular solutions of the set of Eq. (81) aredetermined in the form

(82)where A, D, and r are certain constants.

1 1 1( , ) exp( ) ( ) ( 1,2,3).i iC iΘ ξ = ν ϕ ξ =Fo Fo

1 2 ,λ = λ = λ 1 2a a a= =

2( , ) 1.25exp( 2.5 )(1 ).Θ ξ = − − ξFo Fo

1 ( )iϕ ξ 2 ( )iϕ ξ =( 1, 2),i

1

2

1 1 2 2 1 1 2 221

' ' '' ''( ) 0

( 1,2).

i

i

ii i i i ji

i

af f f f d

a

j−

ξ

= ξ

⎡ ⎤ϕ + ϕ − ϕ + ϕ ϕ ξ =⎢ ⎥⎣ ⎦

=

∑ ∫

1( )f Fo 2( ):f Foξ

ξ

ξ

ξ

ϕ + ϕ ϕ ξ + ϕ + ϕ ϕ ξ

− ϕ + ϕ ϕ ξ

− ϕ + ϕ ϕ ξ =

∫ ∫

∫

∫

1

1

1

2

1

1 11 2 21 11 1 12 2 22 12

0

11 11 2 21 11

20

1

1 12 2 22 12

' ' ' '( ) ( )

'' ''( )

'' ''( ) 0,

f f d f f d

af f d

a

f f d

ξ

ξ

ξ

ξ

ϕ + ϕ ϕ ξ + ϕ + ϕ ϕ ξ

− ϕ + ϕ ϕ ξ

− ϕ + ϕ ϕ ξ =

∫ ∫

∫

∫

1

1

1

2

1

1 11 1 21 21 1 12 1 22 22

0

11 11 2 21 21

20

1

1 12 2 22 22

' ' ' '( ) ( )

'' ''( )

'' ''( ) 0.

f f d f f d

af f d

a

f f d

1 1 2 2 3 1 4 2

1 1 2 2 3 1 4 2

' ' 0,

' ' 0,

b f b f b f b f

l f l f l f l f

⎫+ + + = ⎪⎬⎪+ + + = ⎭

1 2( ) exp( ), ( ) exp( ),f A r f D r= =Fo Fo Fo Fo

838


KUDINOV et al.


(83)

The set of uniform algebraic equations (83) has anontrivial solution in the case when its determinant iszero:

(84)

Expanding the determinant relative to propernumbers , we obtain the quadratic equation.Its solution is

Substituting the value of the root r1 in the set ofequations (83), we obtain

(85)

For uniform set (85) one can take A1 = const =1.Then D1 = –0.237004.

Similarly, substituting r2 into (83), we find A2 = 1,D2 = –1.133491.

With allowance made for found values of constants , relationships (82) take the form

(86)

In order to find the general solution of the set ofequations (81), let us multiply the particular solutionincluding the root r1 by the arbitrary constant C1 andsolution including the root r2 by the constant C2. Substi�tuting the obtained general solutions into (69) (at n = 2),we find

(87)

To determine constants C1 and C2, we form theresidual of initial condition (64), and the orthogonal�ity of the residual to coordinate functions and

(i = 1, 2) is required:

(88)

Relationship (88) relative to unknown constants C1and C2 is reduced to the following set of two algebraiclinear equations

+ ⎫⎪+ + = ⎪⎬

+ ⎪⎪+ + = ⎭

1.400050 5.540774

1.238258 4.991920

1.238258 5.280966

1.138087 6.189359

( )

( ) 0,

( )

( ) 0.

A r

D r

A r

D r

+ +

=

+ +

1.400050 5.540774 1.238258 4.991920

1.238258 5.280966 1.138087 6.1893590.

r r

r r

( 1,2)ir i =

1 ,r = −3.937965 2r = −33.516547.

1 1

1 1

0.115700 0,

0.

A D

A D

+ = ⎫⎬

+ = ⎭

0.027424

0.404746 1.70761 1

, ,i i iA D r ( 1,2)i =

1 2

1 2

1 1 2

2 1 2

( ) ,

( ) .

r r

r r

f A e A e

f D e D e

= +

= +

Fo Fo

Fo Fo

Fo

Fo

Θ ξ = + ϕ

+ + ϕ =

Fo Fo

Fo Fo

Fo 1 2

1 2

1 1 2 2 1

1 1 2 2 2

( , ) ( )

( ) ( 1,2).

r ri i

r ri

C A e C A e

C D e C D e i

1 ( )iϕ ξ

2 ( )iϕ ξ

[

]−

ξ

= ξ

+ ϕ

+ + ϕ − ϕ ξ = =

∑ ∫1

2

1 1 2 2 1

1

1 1 2 2 2

( )

( ) 1 0 ( 1,2).

i

i

i

i

i ji

C A C A

C D C D d j

(89)

where E1 = C1A1 + C2A2, E2 = C1D1 + C2D2

Determining the integrals, we reduce set of equa�tions (89) to the form

(90)

From the solution of the set of equations (90), weobtain C1 = 0.914979, C2 = 1.969779.

After determining the constants C1 and C2, the solu�tion of problem (63)–(68) in the second approxima�tion has been found from (87). If we assume

, relationship (87) is reducedto the form

(91)

Relationship (91) coincides with the solution of thesimilar problem in the second approximation, which isgiven in [13, 15].

Similarly, the solution of problem (63)–(68) can beobtained in the following approximation. Methods forthe construction of sets of coordinate functions for thegreater number of contacting body are given in [12, 14].For example, coordinate functions for a three�layerplate at any number of approximations for the first, sec�ond, and third layers, respectively, are written as

where = =

1; δ1, δ2, δ3 are layer thicknesses; and n is thenumber of approximations.

Calculation results by formulas (52) (secondapproximation) and (69) (eighth approximation) aregiven in Fig. 7. Their analysis allows one to concludethat the maximum divergence of results found by for�mulas (52) and (69) does not exceed 2.5% in the rangeof where τ1 = 0.342 s is the time ofattainment by the temperature disturbance front of thecoordinate ξ = 1. As the time decreases, the diver�

ξ

ξ

ξ

ξ

⎫⎪ϕ + ϕ − ϕ ξ⎪⎪⎪

+ ϕ + ϕ − ϕ ξ = ⎪⎪⎪⎬⎪

ϕ + ϕ − ϕ ξ ⎪⎪⎪⎪

+ ϕ + ϕ − ϕ ξ = ⎪⎪⎭

∫

∫

∫

∫

1

1

1

1

1 11 2 21 11

0

1

1 21 2 22 12

1 11 2 21 21

0

1

1 21 2 22 22

( 1)

( 1) 0,

( 1)

( 1) 0,

E E d

E E d

E E d

E E d

1 2

1 2

1.106577 0.003505 1.019400 0,

0.968526 0.051753 0.988124 0.

C C

C C

− − = ⎫⎬

− − = ⎭

1 2 ,λ = λ = λ 1 2a a a= =

− −

− −

Θ ξ = + − ξ

− + − ξ

Fo Fo

Fo Fo

Fo 2.47 25.52 2

2.47 25.52 42

( , ) (1.55 3.3 )(1 )

(0.28 2.9048 )(1 ).

e e

e A e

2 2 23 3 3 31 1 2

2 1 2 1

( ) 1 1 ,k k kk

⎛ ⎞ ⎛ ⎞λ λ λ λϕ ξ = − − ξ − − ξ − ξ⎜ ⎟ ⎜ ⎟λ λ λ λ⎝ ⎠ ⎝ ⎠

2 2 23 32 2 3

2 2

( ) 1 1 , ( ) 1 ,k k kk k

⎛ ⎞λ λϕ ξ = − − ξ − ξ ϕ ξ = − ξ⎜ ⎟λ λ⎝ ⎠1 1 ,ξ =δ δ ( )2 1 2 ,ξ = δ +δ δ 3ξ 1 2 3( )δ + δ + δ δ

1, ,k n=

10.001 ,≤ τ ≤ τ



gence rises that is connected with the lower accuracyfor the solution obtained by the Kantorovich method.The increase in the accuracy by this method dependson an increase in the number of approximations; how�ever, in this case relative to proper numbers, we havethe characteristic equation with high degrees and greatsets of algebraic linear equations at the fulfillment ofinitial conditions of the boundary problem. The matri�ces of coefficients of such systems, being as filled qua�dratic matrices with the great coefficient scatteringwith respect to the absolute value, are ill�conditioned.At the same time, let us note that the limitation on thetime in the range of its small values are practicallylacking in the case of the use of the integral thermalbalance method [1, 12].

Thus, combining two methods, the integral ther�mal balance method with the use of the asymmetryunit function and the orthogonal Kantorovich methodwith the use of systems of coordinate functions exactlymeeting the boundary conditions and conjugacy con�ditions, one can obtain sufficiently simple approxi�mate analytical solutions of the thermal conductivityproblems for multilayered constructions in the entiretime range of the nonstationary process (including itssmall and minute values) practically with the requiredaccuracy.

CONCLUSIONS

(i) Theoretical propositions of the method forobtaining approximate analytical solutions of bound�

ary problems of thermal conductivity for multilayeredconstructions at the step of the inertial (irregular)mode of the nonstationary process are developed onthe basis of the application of the generalized functiontheory by means of determining the temperature dis�turbance front and additional boundary conditions.Due to using the additional boundary conditions,there is the possibility for obtaining an approximateanalytical solution practically with the required accu�racy in the entire time range of the irregular mode ofheat exchange, including its small and minute values.

(ii) The theoretical propositions of the method forconstruction of the approximate analytical solutionsof boundary problems of thermal conductivity formultilayered constructions are given on the basis of theuse of the orthogonal Kantorovich method by meansof reduction of them to single�layer with the variable(piecewise�uniform) medium properties. Realizationof such a method of obtaining the solutions is found tobe possible due to the use of the global system of coor�dinate functions that exactly meet the boundary con�ditions and conjugacy conditions in any approxima�tion.

REFERENCES

1. Kudinov, V.A. and Kudinov, I.V., Metody resheniya par�abolicheskikh i giperbolicheskikh uravnenii teploprovod�nosti (Methods for Solution of Parabolic and Hyper�bolic Heat Conductivity Equations), Moscow:Librokom, 2011.

1.00.20

0.2

0.4

0.6

0.8

1.0

0.4 0.6 0.8

0.001

0.005

0.0130.027

0.078

0.191

0.484

0.95

1.6

2.6

τ = 5 s

Θ

ξ

Fig. 7. Temperature distribution in the double�layer: points by (52), curves by (69) (eighth approximation).

840


KUDINOV et al.

2. Kolyano, Yu.M., in Matematicheskie metody i fiziko�mekhanicheskie polya (Mathematical Methods andPhysico�Mechanical Fields), Kiev: Naukova Dumka,1978, issue 7, p. 7.

3. Kolyano, Yu.M. and Popovich, V.S., Fiz. Khim. Obrab.Mater., 1975, no. 5, p. 16.

4. Kecz, W. and Teodorescu, P., Introducere in teoria dis�tributiilor cu aplicati in tehnica, Bucharest: Editura Teh�nica, 1975.

5. Kudinov, V.A., Averin, B.V., Stefanyuk, E.V., and Naz�arenko, S.A., Analiticheskie metody teploprovodnosti(Analytical Methods for Estimating Thermal Conduc�tivity), Samara: Samara State Technical University,2004.

6. Gudmen, T., in Problemy teploobmena. Sbornik nauch�nykh trudov (Problems of Heat Exchange: A Collectionof Scientific Works), Moscow: Atomizdat, 1967, p. 41.

7. Belyaev, N.M. and Ryadno, A.A., Metody nestatsionar�noi teploprovodnosti (Methods for Estimating Non�Steady�State Thermal Conductivity), Moscow:Vysshaya Shkola, 1978.

8. Kudinov, V.A. and Stefanyuk, E.V., Izv. Akad. Nauk,Energ., 2008, no. 5, p. 141.

9. Stefanyuk, E.V. and Kudinov, V.A., High Temp., 2009,vol. 47, no. 2, p. 250.

10. Kartashov, E.M., Analiticheskie metody v teorii teplopro�vodnosti tverdykh tel (Analytical Methods in the Theoryof Thermal Conductivity of Solids), Moscow: VysshayaShkola, 1985.

11. Kudinov, V.A. and Kudinov, I.V., Izv. Akad. Nauk,Energ., 2012, no. 1, p. 119.

12. Kudinov, V.A., Kartashov, E.M., and Stefanyuk, E.V.,Tekhnicheskaya termodinamika i teploperedacha (Tech�nical Thermodynamics and Heat Transfer), Moscow:Yurait, 2012.

13. Stefanyuk, E.V. and Kudinov, V.A., High Temp., 2010,vol. 48, no. 2, p. 272.

14. Kudinov, V.A., Kartashov, E.M., and Kalashnikov, V.V.,Analiticheskie resheniya zadach teplomassoperenosa itermouprugosti dlya mnogosloinykh konstruktsii (Analyt�ical Solutions of the Problems on Heat and Mass Trans�fer and Thermoelasticity for Multilayered Structures),Moscow: Vysshaya Shkola, 2005.

15. Tsoi, P.V., Metody rascheta zadach teplomassoperenosa(Methods for Calculation of Heat and Mass TransferProblems), Moscow: Energoatomizdat, 1984.

Translated by S. Ordzhonikidze

generalized functions in thermal conductivity problems for multilayered constructions

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