ERROR ANALYSIS OF MIXED LUMPED-DIFFERENTIAL FORMULATIONS IN DIFFUSION PROBLEMS

Error Analysis

of the

Improved Lumped - Differential Formulations

L. S. de B. Alves, L. A. Sphaier and R. M. Cotta.

Laboratory of Heat Transmission and Technology – LTTC

Mechanical Engineering Department – E.E. – COPPE / UFRJ

Universidade Federal do Rio de Janeiro

Cx Postal 68503 – R.J., R.J., CEP : 21945 – 970, Brasil

ABSTRACT

The aims of these formulations are the task of modeling problems, prior to the

choice of solutions strategies, trying to reduce, as much as possible, and within

prescribed accuracy requirements, the number of dimensions of the problem.

This paper show how appropriate integration strategies can be employed to

deduce mathematical formulations of comparable simplicity and improved accuracy

in comparison with the classical well – established lumping procedures, and further

on, how approximate final error expressions, based on the boundary and / or initial

conditions, of the solutions can be found.

This is the so – called C.I.E.A. (Coupled Integral Equations Approach) and

reduction of the computational effort, keeping the accuracy needed, is the ultimate

goal of this work.

INTRODUCTION

Is of great interest for engineering practice to propose simpler formulations of

partial differential systems, through a reduction of the number of independent

variables by integrating the correspondent equations where the potential variations

with these variables can be negligible, but retaining some information in the direction

integrated out by approximating adequately the boundary potentials with the average

ones.

2

Different Levels of approximations can be used starting from the classical

lumped system analysis toward improved formulations obtained through Hermite type

integral approximations [1-3].

Hermite [1] developed a way of approximate an integral, based on the values

of the integrand and on its derivatives at the integration limits. A detailed derivation

was first presented by Mennig [2]. This is called Hα,β approximation and is given by:

βα

β

ν

νννα

ν

νν αββα ,0

)(1

01

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

Erroxyhcxyhcdxxy iiviiv

x

x

i

i

+−+= ∑∑∫=

+

=−

+

−

(1.a)

where,

1−−→ iii xxh (1.b)

)!2()!()!1()!1()!1(),(

++−++++−+

→βανανβανα

βανc (1.c)

and, y(x) and its derivatives y(ν)(x) are defined for all x ⊙ (xi-1;xi) and its assumed that

y(ν)(xi-1) = y(

ν)i-1 for ν = 0, 1, 2, …, α and y(

ν)(xi) = y(

ν)i for ν = 0, 1, 2, …, β.

The error formulae of this approximation is:

∫ ++++++

−++

−=

hdxhxxxyErro

0

11)2(2

, )()()!2(

)1( αββαβα

βα βα (1.d)

In the following sections we consider four different approximations, given by:

))()0((21)(

00,0 hyyhdxxyHh

+≅→ ∫ (2.a)

)(61))(2)0((

31)( )1(2

01,0 hyhhyyhdxxyHh

−+≅→ ∫ (2.b)

)0(61))()0(2(

32)( )1(2

00,1 yhhyyhdxxyHh

++≅→ ∫ (2.c)

))()0((121))()0((

21)( )1()1(2

01,1 hyyhhyyhdxxyHh

−++≅→ ∫ (2.d)

The former and the latter ones correspond to the well known trapezoidal and

corrected trapezoidal integration rules.

The respective error expressions are written as:

),0(; )(121 )2(3

0,0 hyhErro ∈−→ ηη (3.a)

),0(; )(721 )3(4

1,0 hyhErro ∈+→ γγ (3.b)

3

),0(; )(721 )3(4

0,1 hyhErro ∈−→ δδ (3.c)

),0(; )(7201 )4(5

1,1 hyhErro ∈→ ζζ (3.d)

which will be employed to compose an approximate final error expression based on

boundary and / or initial conditions of each problem studied.

The entire process of obtaining this error expression will be demonstrated in

two different cases shown bellow: An Extended Surface (Fin), where we work with a

two dimensional potential and A Diffusion One Dimensional Transient Problem

(Slab).

LONGITUDINAL FINS

We consider a longitudinal fin of rectangular profile, also analyzed, subjected

to the boundary conditions and variable ranges shown below:

10 10 ; 0),(),(2

22

2

2

≤≤∴≤≤=∂

∂+

∂∂ XY

YYXK

XYX θθ

(4.a)

1),( 0 =→XYXθ ),1( ),( *

1

YBiKXYX

X

θθ

−=#$

%&'

(∂

∂

→

(4.b, c)

0),(

0

=!"

#$%

&∂

∂

→YYYXθ (X,1) ),(

1

θθ BiYYX

Y

−=#$

%&'

(∂

∂

→

(4.d, e)

with the dimensionless groups,

2 ;

2 Bi;

2

Bi; 2

; ; ),(),( *

wL

khw

khw

wy

XLx

XTTTYXT

YX xy

b

=Κ====−

−=

∞

θ (5.a, f)

• Exact Solution:

This problem has been reviewed and solutions have been provided within an

unified frame in [4]. It was also solved by separation of variables [5] to yield,

*

*

1

)Sinh(K )Cosh(K ) X)-(1Sinh(K ) X)-(1Cosh(K

)( )(

) ( )(2),(

BiBi

CosSinYCosSin

YX

mmm

mmm

m mmm

mm

λλλλλλ

λλλλλ

θ

+

+

⋅+

= ∑∞

= (6.a)

where λm`s are the positive roots of the transcendental equation,

BiTan mm =)( λλ (6.b)

4

The average dimensionless temperature at any cross section of the fin is

obtained from,

∫=1

0),()( dYYXXav θθ (7.a)

giving,

*

*

1

2

)Sinh(K )Cosh(K ) X)-(1Sinh(K ) X)-(1Cosh(K

))( )(( )(

2)(

BiBi

CosSinSin

X

mmm

mmm

m mmmm

mav

λλλλλλ

λλλλλ

θ

+

+

⋅+

= ∑∞

= (7.b)

• Classical Lumped System Analysis:

In many practical situations the solution obtained above might be considered

too involved and costly in terms of computational efforts, so we seek for a simplified

formulation by integrating out the variable Y in the partial differential equation (4.a),

which results in an ordinary differential problem for the spatially averaged

temperature [6].

Invoking the θav definition and the boundary conditions of our problem we

obtain,

1X0 ; 0)1,( )( 22

2

≤≤=−∂

∂ XKBiXXav θ

θ (8.a)

1)( 0 =→Xav Xθ )1( )( *

1av

X

av BiKXX

θθ

−=#$

%&'

(∂

∂

→

(8.b, c)

The classical lumped approximation assume negligible the temperature

gradient along the transversal coordinate Y, in other words, that the boundary

potential can be reasonably well approximated by the average potential as:

)()1,( XX avθθ = (9.a)

so, equation (8.a) becomes:

0)( )( 22

2

=−∂

∂ XKBiXX

avav θθ (9.b)

which combined with the boundary conditions (8.b, c) results in,

) () ( ))1( ())1( ( )(

*

*

KBiSinhBiKBiCoshBiXKBiSinhBiXKBiCoshBiXav

+

−+−=θ (9.c)

5

It is quite evident that as Biot number Bi increases the transversal gradients

became less negligible which impose severe limitations on the use of the classical

lumped system analysis.

• Improved Lumped – Differential Equations:

The main goal of these formulations is to find a relation between the boundary

and average potentials that carries more information about the problem, if compared

to the classical one.

New relations, with different degrees of approximation, were developed by the

author et al. in this paper from Hermite – type approximations of integrals,

maintaining the same level of mathematical simplification and also increasing overall

accuracy, depending of the Hermite integration involved, as now more closely

discussed.

1. H0,0 / H0,0 Approximation.

The integrals that define the average temperature and heat flux are:

)(),(1

0XdYYX avθθ =∫ (10.a)

)0,()1,(),(1

0XXdY

YYX

θθθ

−=∂

∂∫ (10.b)

and are approximated both through the trapezoidal rule H0,0 (2.a) to yield,

))1,()0,((21)( XXXav θθθ +≅ (11.a)

!!"

#$$%

&!"

#$%

&∂

∂+!

"

#$%

&∂

∂≅−

→→ 10

),(),(21)0,()1,(

YY YYX

YYXXX θθ

θθ (11.b)

By employing the boundary conditions and with little effort we obtain the new

desired relation,

)(4

4)1,( XBi

X avθθ+

= (12)

which provides us, with the boundary conditions (4.b, c), a new ordinary differential

system,

0)( )( 22

2

=−∂

∂ + XKBiXX

avav θθ (13.a)

were the modified Biot number is given by,

6

4 4+

=+

BiBiBi (13.b)

It is clear that the classical and the lumped formulations approach each other

as Bi -> 0, and we can see that improved relations provide some corrections but

retaining the same simple mathematical formulations as offered in classical analysis.

Including the error expression (3.a) in the H0,0 / H0,0 approximations (11.a, b)

we find the exact form of equation (13.a), that is given by:

1X0 ; )()( )( 22

2

≤≤=−∂

∂ + XErroXKBiXX

EQNavav θθ (14.a)

where,

10 10

),(2),(24

)( 2

2

3

32

≤≤∴≤≤

##

$

%

&&

'

(##$

%&&'

(

∂∂

−##$

%&&'

(

∂∂

=→→

+

ηζ

θθ

ηζ YYEQN Y

YXYYXKBi

XErro (14.b)

Now we try to provide a priori error estimate on the average potential by

employing the boundary information of the problem. This analysis results in four

improved combined error expressions shown below:

1 1 ; )()4(32)(Bi8)( 23

2

→∴→+

+= ηζθ XK

BiBi

XErro avEQN (15.a)

0 0 ; )()4(32)(Bi8)( 23

2

→∴→+

+= ηζθ XK

BiBi

XErro avEQN (15.b)

0 1 ; )()4(3

1)(Bi16)( 23

2

→∴→+

+= ηζθ XK

BiBi

XErro avEQN (15.c)

1 0 ; )()4(3

16)( 23

2

→∴→+

= ηζθ XKBiBi

XErro avEQN (15.d)

The source term that appear in equation (14.a), changes the value of the

constants of the original problem (13.a), so finding an explicit final error formulae

that would give us the absolute value of our average potential error turns out to be

impossible.

But equation (14.a) can still be solved if we rewrite it as is shown bellow,

( ) 1X0 ; 0)( )( 22

2

≤≤=−−∂

∂ + XCoefKBiXX

avav θθ (16)

where Coef can be found in expressions (15.a-d) and avθ is the improved

dimensionless average temperature, final error formulae would be,

7

)()()( XXXErro avavav θθθ −= (17)


The integrals (10.a, b) that define the average temperature and heat flux are

approximated by (2.b, a) rules to yield,

1

),(61))1,(2)0,((

31)(

→

"#

$%&

'∂

∂++=

Yav Y

YXXXX θθθθ (17.a)

!!"

#$$%

&!"

#$%

&∂

∂+!

"

#$%

&∂

∂=−

→→ 10

),(),(21)0,()1,(

YY YYX

YYXXX θθ

θθ (17.b)


desired relation,

)(3

3)1,( XBi

X avθθ+

= (18.a)

that provides us a new modified Biot number, given by,

3 3+

=+

BiBiBi (18.b)

Including the error expression (3.b, a) in the H0,1/H0,0 approximation (17.a, b)

we find a new value for the source term from equation (14.a), now given by:

10 10

),(),(272

)( 3

3

3

32

≤≤∴≤≤

##

$

%

&&

'

(##$

%&&'

(

∂∂

+##$

%&&'

(

∂∂

=→→

+

ηζ

θθ

ηζ YYEQN Y

YXYYXKBi

XErro (19)




1 1 ; )()3(8

9)( 23

3

→∴→+

= ηζθ XKBiBi

XErro avEQN (20.a)

0 0 ; 0)( →∴→= ηζXErroEQN (20.b)

0 1 ; )()3(4

3)( 23

3

→∴→+

= ηζθ XKBiBi

XErro avEQN (20.c)

1 0 ; )()3(8

3)( 23

3

→∴→+

= ηζθ XKBiBi

XErro avEQN (20.d)

8

If we reformulate our problem as we did in (16), we can find new values of

Coef from (20.a-d) that would make possible to us use expression (17) to find a final

error expression of this approximation.


The difference of this approximation from the latter one is the integral (17.a)

that define the average temperature that is approximated by rule (2.c),

0

),(61))1,()0,(2(

31)(

→

"#

$%&

'∂

∂++=

Yav Y

YXXXX θθθθ (21)

By employing the boundary conditions we find the same result of H0,1/H0,0

approximation,

)(3

3)1,( XBi

X avθθ+

= (22.a)

that provides us the same modified Biot number,

3 3+

=+

BiBiBi (22.b)

Including the error expression (3.c, a) in the H1,0/H0,0 approximation (21, 17.b)

we find the exact form of equation (14.a), but Erroeqn(X) is now given by,

10 10

),(),(472

)( 3

3

3

32

≤≤∴≤≤

##

$

%

&&

'

(##$

%&&'

(

∂∂

−##$

%&&'

(

∂∂

=→→

+

ηζ

θθ

ηζ YYEQN Y

YXYYXKBi

XErro (23)




1 1 ; )()3(8

9)( 23

3

→∴→+

= ηζθ XKBiBi

XErro avEQN (24.a)

0 0 ; 0)( →∴→= ηζXErroEQN (24.b)

0 1 ; )()3(2

3)( 23

3

→∴→+

= ηζθ XKBiBi

XErro avEQN (24.c)

1 0 ; )()3(8

3)( 23

3

→∴→+

−= ηζθ XKBiBi

XErro avEQN (24.d)

By applying the same procedure of the other approximations but now using

expressions (24.a-d) we find four different improved final error expressions for the

average potential.

9


The difference of this approximation from the other ones is the integral (17.a)

that define the average temperature that is now approximated by rule (2.d),

!!"

#$$%

&!"

#$%

&∂

∂−!

"

#$%

&∂

∂++=

→→ 10

),(),(121))1,()0,((

21)(

YYav Y

YXYYXXXX θθ

θθθ (25)

By employing the boundary conditions we find the same result of H0,1/H0,0 and

H1,0/H0,0 Approximations,

)(3

3)1,( XBi

X avθθ+

= (26.a)

that provides us the same modified Biot number,

3 3+

=+

BiBiBi (26.b)

Including the error expression (3.d, a) in the H1,1/H0,0 approximation (25, 17.b)

we find the exact form of equation (14.a), but Erroeqn(X) is now given by,

10 10

),(301),(

24)( 4

4

3

32

≤≤∴≤≤

##

$

%

&&

'

(##$

%&&'

(

∂∂

+##$

%&&'

(

∂∂

=→→

+

ηζ

θθ

ηζ YYEQN Y

YXYYXKBi

XErro (27)




( ) 1 1 ; )()3(8031109)( 24

3

→∴→+

+= ηζθ XK

BiBiBi

XErro avEQN (28.a)

( ) 0 0 ; )()3(16029)( 2

4

3

→∴→+

+= ηζθ XK

BiBiBi

XErro avEQN (28.b)

( ) 0 1 ; )()3(16062219)( 24

3

→∴→+

+= ηζθ XK

BiBiBi

XErro avEQN (28.c)

1 0 ; )()3(80

9)( 23

3

→∴→+

= ηζθ XKBiBi

XErro avEQN (28.d)

By applying the same procedure of the other approximations but now using

expressions (28.a-d) we find four different improved final error expressions for the

average potential.

10

ONE - DIMENSIONAL SLAB

We consider heat conduction in a one – dimension slab, subjected to the

boundary conditions and variable ranges shown below:

0 10 ; ),(),(2

2

≥∴≤≤+∂

∂=

∂∂

ττθ

ττθ XG

XXX

(29.a)

1),( 0 =→ττθ X (29.b)

0),(

0

=!"

#$%

&∂

∂

→XXX τθ )(1, ),(

1

τθτθ Bi

XX

X

−=$%

&'(

)∂

∂

→

(29.c, d)

with the dimensionless groups,

( )∞∞ −====

−−

=TTkLg

kLh

Lt

LxX

TTTYXTYX

0

20

20

G ; Bi; ; ; ),(),( α

τθ (30.a, e)’

• Exact Solution:

This problem is solved with the Classical Integral Transform Technique [5] by

the authors to yield,

!!"

#$$%

&+!!"

#$$%

&−

⋅⋅!!"

#$$%

&

+

++=

−

∞

=∑

GSin

GSinSin

CosBiBiBi

X

m

m

m

m

m

m

mm m

m

m33

122

22

)()()(E

X) (21),(

2

λλ

λλ

λλ

λλ

λτθ

τλ

(31.a)

where λm`s are the positive roots of the transcendental equation,

BiTan mm =)( λλ (31.b)

The slab average dimensionless temperature as a function of the dimensionless

time only is obtained from,

∫=1

0),()( dXXav τθτθ (32.a)

giving,

!!"

#$$%

&+!!"

#$$%

&−

⋅⋅!!"

#$$%

&

+

++=

−

∞

=∑

GSinGSinSin

SinBiBiBi

m

m

m

m

m

m

m

m

m m

mav

m33

122

22

)()()(E

)(21)(

2

λλ

λλ

λλ

λλ

λλ

τθ

τλ

(32.b)

11

• Classical Lumped System Analysis:

By integrating out the variable X in equations (29.a, b) and invoking the

average temperature definition (32.a) and the boundary conditions of our problem

(29.c, d) we find an integral – differential representation of system (29) that show us

the approximation path we must follow. This new system is shown below,

0 ; ),1()(≥=+

∂

∂ττθ

ττθ GBiav (33.a)

1)( 0 =→ττθav (33.b)

As we saw in the other example, the longitudinal fin, we can rewrite equations

(33.a, b), using the modified Biot number, to find,

0 ; )()(≥=+

∂

∂ + ττθττθ GBi av

av (34.a)

1)( 0 =→ττθav (34.b)

The solution of the ordinary differential system (34) is easily obtained to yield,

ττθ+−

++⋅%&

'()

* −+= Biav E

BiG

BiG 1)( (34.c)

The Classical Lumped System Analysis (C.L.S.A.) assume that the

temperature gradients are smooth enough to allow the boundary potential to be

reasonably approximated by the averaged one, as:

)(),1( τθτθ av= (35.a)

or

BiBi =+ (35.b)

Using relation (35.b) in solution (34.c) we find the C.L.S.A. approximated

solution of the average potential. This solution has very strict applicability limits, and

in general is restricted to problems with Bi < 0.1.

• Improved Lumped – Differential Equations:

As said before, the main goal of these formulations is to find a relation

between the boundary and average potentials that carries more information about the

problem, if compared to the classical one, maintaining the same level of mathematical

simplification and increasing overall accuracy [7], in other words,

[ ])(),1( τθτθ avf≅ or [ ]BifBi ≅+ (36.a, b)

12


The integrals that define the average temperature and heat flux are:

)(),(1

0τθτθ avdXX =∫ (37.a)

),0(),1(),(1

0τθτθ

τθ−=

∂∂∫ dX

XX (37.b)

and are approximated both through the trapezoidal rule H0,0 (2.a) to yield,

)),1(),0((21)( τθτθτθ +≅av (38.a)

!!"

#$$%

&!"

#$%

&∂

∂+!

"

#$%

&∂

∂≅−

→→ 10

),(),(21),0(),1(

XX XX

XX τθτθ

τθτθ (38.b)


desired relation,

)(4

4),1( τθτθ avBi += (39)

which provides us a new modified Biot number,

4 4+

=+

BiBiBi (40)

Using relation (40) in solution (34.c) we find the H0,0 / H0,0 approximated

solution of the average potential. It is clear that this approximation provides some

correction to the classical formulation.

Including the error expression (3.a) along the derivations, we find the exact

form of equation (34.a), that is given by:

0 ; )()( )(≥+=+

∂

∂ + τττθττθ

EQNavav ErroGBi (41.a)

where,

10 10

),(),(21

12)( 2

2

3

3

≤≤∴≤≤

##

$

%

&&

'

(##$

%&&'

(

∂∂

−##$

%&&'

(

∂∂

=→→

+

ηζ

τθτθτ

ηζ XXEQN X

XXXBi

Erro (41.b)


employing all the information about the problem. This analysis results in four


1 1 ; )4(3

)()4(32)(Bi8)( 2

2

3

2

→∴→+

−+

+= ηζτθτ G

BiBi

BiBi

Erro avEQN (42.a)

13

0 0 ; )4(3

)()4(32)(Bi8)( 2

2

3

2

→∴→+

−+

+= ηζτθτ G

BiBi

BiBi

Erro avEQN (42.b)

0 1 ; )4(3

3 )()4(3

1)(Bi16)( 2

2

3

2

→∴→+

−+

+= ηζτθτ G

BiBi

BiBi

Erro avEQN (42.c)

1 0 ; )4(3

)43( )()4(3

16)( 23

2

→∴→+

−−

+= ηζτθτ G

BiBiBi

BiBi

Erro avEQN (42.d)

The solution of system (41, 34.b) is equation (34.c), that is the solution of

system (34), plus an extra term that is a function of the time dependent source term

that appear in equation (41.a). This extra term is the final exact error of the average

potential, and is given by:

( )∫ −− +

=τ ττ

θ τττ0

1 1)1()( dErroEErro EQNBi

av (43)

We could find an explicit formula for the average potential error (43) because

we are working on an initial value problem (41, 34.b), this was not possible in system

(14), the longitudinal fin problem, because it was a boundary value problem.

By introducing equations (42.a-d) into formula (43) we find four different

approximate final error expressions of the average temperature, that will be later on

compared.


The integrals that define the heat flux and average temperature are

approximated through rules (2.a, b) respectively to yield,

1

),(61)),1(2),0((

31)(

>−

"#

$%&

'∂

∂−+≅

Xav X

X τθτθτθτθ (44.a)

!!"

#$$%

&!"

#$%

&∂

∂+!

"

#$%

&∂

∂≅−

→→ 10

),(),(21),0(),1(

XX XX

XX τθτθ

τθτθ (44.b)


desired relation,

)(3

3),1( τθτθ avBi += (45)

which provides us a new modified Biot number,

3 3+

=+

BiBiBi (46)

14

Using relation (46) in solution (34.c) we find the H0,1 / H0,0 approximated

solution of the average potential.

Including the error expressions (3.a, b) along the derivations, we find another

definition for the time dependent source term of equation (41.a), that is:

10 10

),(),(272

)( 3

3

3

3

≤≤∴≤≤

##

$

%

&&

'

(##$

%&&'

(

∂∂

+##$

%&&'

(

∂∂

=→→

+

ηζ

τθτθτ

ηζ XXEQN X

XXXBi

Erro (47)

By employing all the information about the problem we find four improved

combined error expressions shown below:

1 1 ; )3(8

3 )()3(8

9)( 2

2

3

3

→∴→+

−+

= ηζτθτ GBiBi

BiBi

Erro avEQN (48.a)

0 0 ; 0)( →∴→= ηζτEQNErro (48.b)

0 1 ; )3(4

)()3(4

3)( 2

2

3

3

→∴→+

−+

= ηζτθτ GBiBi

BiBi

Erro avEQN (48.c)

1 0 ; )3(8

)()3(8

3)( 2

2

3

3

→∴→+

−+

= ηζτθτ GBiBi

BiBi

Erro avEQN (48.d)



compared.


The integral that defines the average temperature is approximated through rule

(2.c) to yield,

0

),(61)),1(),0(2(

31)(

>−

"#

$%&

'∂

∂++≅

Xav X

X τθτθτθτθ (49)

The integral that defines the average heat flux is approximated through rule

(2.a) to yield equation (38.b).

By employing the boundary conditions and with little effort we obtain the

same relation of approximation H0,1 / H0,0 (45), so the modified Biot number will be

the same too,

3 3+

=+

BiBiBi (50)

15

So we have the same solution of the last approximation for the average

temperature.

Including the error expressions (3.a, c) along the derivations, we find another


10 10

),(),(472

)( 3

3

3

3

≤≤∴≤≤

##

$

%

&&

'

(##$

%&&'

(

∂∂

−##$

%&&'

(

∂∂

=→→

+

ηζ

τθτθτ

ηζ XXEQN X

XXXBi

Erro (51)



1 1 ; )3(8

3 )()3(8

9)( 2

2

3

3

→∴→+

−+

= ηζτθτ GBiBi

BiBi

Erro avEQN (52.a)

0 0 ; 0)( →∴→= ηζτEQNErro (52.b)

0 1 ; )3(2

)()3(2

3)( 2

2

3

3

→∴→+

−+

= ηζτθτ GBiBi

BiBi

Erro avEQN (52.c)

1 0 ; )3(8

)()3(8

3)( 2

2

3

3

→∴→+

++

−= ηζτθτ GBiBi

BiBi

Erro avEQN (52.d)


approximate final error expressions of the average temperature, that will be, again,

later on compared.


The integral that defines the average temperature is now approximated through

corrected trapezoidal rule (2.d) to yield,

!!"

#$$%

&!"

#$%

&∂

∂−!

"

#$%

&∂

∂++≅

>−>− 10

),(),(121)),1(),0((

21)(

XXav X

XXX τθτθ

τθτθτθ (53)

The integral that defines the average heat flux is approximated through rule

(2.a) to yield equation (38.b), as all other approximations.

By employing the boundary conditions and with little effort we obtain the

same relation of approximation H0,1 / H0,0 (45), so we will have the same modified

Biot number (46) too.

The solution for the average temperature will be same of the last two

approximations.

16

Including the error expressions (3.a, d) along the derivations, we find another


10 10

),(301),(

24)( 4

4

3

3

≤≤∴≤≤

##

$

%

&&

'

(##$

%&&'

(

∂∂

+##$

%&&'

(

∂∂

=→→

+

ηζ

τθτθτ

ηζ XXEQN X

XXXBi

Erro (54)



1 1 ; )3(80

)3110( 3 )()3(80

)3110(9)( 3

2

4

3

→∴→+

+−

+

+= ηζτθτ G

BiBiBi

BiBiBi

Erro avEQN (55.a)

0 0 ; )3(160

)2( 3 )()3(160

)2(9)( 3

2

4

3

→∴→+

+−

+

+= ηζτθτ G

BiBiBi

BiBiBi

Erro avEQN (55.b)

0 1 ; )3(160

)6221( 3 )()3(160

)6221(9)( 3

2

4

3

→∴→+

+−

+

+= ηζτθτ G

BiBiBi

BiBiBi

Erro avEQN (55.c)

1 0 ; )3(80

3 )()3(80

9)( 3

2

4

3

→∴→+

−+

= ηζτθτ GBiBi

BiBi

Erro avEQN (55.d)



compared.

RESULTS

All the analysis done until now and the numerical and graphical data present

from this point on have been made making use of the software Mathematica [8].

Numerical and graphical results were obtained for both test cases used in this

work, through classical, Hermite – type and fully exact differential formulations.

Values of Biot and Aspect Ratio, for the first case, Biot and Dimensionless Heat

Generation, for the second case were chosen so as to show the extension range of the

new approximations and the limits of applicability of the classical one.

• Test – Case One

Figures (1.a-c) present the dimensionless temperature evolution along its

longitudinal profile for three different Biot numbers (Bi = 0.1, Bi = 1, Bi = 10). The

first figure shows good agreement between all solutions because the Biot used is the

accuracy limit for the classical solution.

17

Figure 1.a – Comparison of the dimensionless average temperature between the

formulations (Slab - Bi = 0.1 & K = 1)

Figure 1.b – Same as above, for Bi = 1.0

Figure 1.c – Same as above, for Bi = 10

18

As Biot is increased the improvement offered by the proposed formulations

becomes evident as we can see in figure (1.b), and markedly more evident as we can

see in figure (1.c).

In table 1 results for the dimensionless average heat transfer rate at the base of

the fin are presented with different aspect ratios and, K, and Biot numbers, Bi. An

aspect ratio value of two represents a square. H.A.1 represent the H0,0 / H0,0

approximation and all the other Hermite approximations are represented by H.A.2,

once they have the same results.

Table 1 – Comparison of the dimensionless average heat transfer rate (X = 0)

between all formulations

Bi K C.L.S.A. H.A.1 H.A.2 Sep. Var.

0.1 0.5 0.0713 0.0707 0.0705 0.0708

0.1 1 0.1794 0.1773 0.1767 0.1773

0.1 2 0.4707 0.4639 0.4618 0.4631

1 0.5 0.5000 0.4681 0.4600 0.4784

1 1 1.0000 0.91125 0.8883 0.9259

1 2 2.0000 1.7944 1.7399 1.8138

10 0.5 1.6522 1.1002 1.0514 1.2983

10 1 3.1682 1.7741 1.6302 2.1579

10 2 6.3246 3.3862 3.0485 4.1032 C.L.S.A. – represents the dimensionless average temperature given by classical solution

H.A.1 – represents the dimensionless average temperature given by H0,0 / H0,0 solution

H.A.2 – represents the dimensionless average temperature given by other Hermite solutions

Sep. Var. – represents the dimensionless average temperature given by the Exact solution

Figures (2.a-c) show the variation of the final approximate error expression of

the dimensionless average temperature with the dimensionless longitudinal distance

of one of the improved formulations, with different Biot numbers and aspect ratios.

The exact error of the classical and improved formulations is provided too.

In these figures there are only three final approximate error curves. This

happen because in H0,0 / H0,0 approximation, the error relative to the boundaries

η = 1 & ζ = 1 and η = 0 & ζ = 0 are equal.

19

Figure 2.a – Comparison between the classical, H0,0 / H0,0 and exact final error

solutions of the dimensionless average temperature (Fin – Bi = 0.1 & K = 0.5)

Figure 2.b – Same as above, for Bi = 1.0


20

By analyzing equation (14.b) and the boundary conditions (4.b, c) we can

previously say that the boundary η = 0 & ζ = 1 will provide us the higher final

approximate error expression as we can see in figures (2.a-c) and confirm in table 2.

In table 2, percentage results for the dimensionless average heat transfer rate

errors at the base of the fin are presented with different aspect ratios and, K, and Biot

numbers, Bi. Exact error values of the classical and H0,0 / H0,0 approximation are

present for comparison. The error using the second boundary combination is not

shown for the same reason it didn’t appear in figures (2.a-c).

Table 2 – Relative percentage results for the dimensionless average heat transfer rate

errors at the base of the fin of all formulations.

Bi K C.L.S.A. c1 H.A.1 c3 H.A.1 c4 H.A.1 Exact

0.1 0.5 0.0426 0.0191 0.0200 0.0182 0.0146

0.1 1 0.2083 0.0699 0.0732 0.0666 0.0014

0.1 2 0.7554 0.2243 0.2350 0.2136 0.0832

1 0.5 2.1517 1.0290 1.3705 0.6868 1.0416

1 1 7.4125 2.8911 3.8439 1.9329 1.4628

1 2 18.6184 6.7899 9.0069 4.5502 1.9382

10 0.5 35.3936 10.0067 18.0080 1.7002 19.8072

10 1 101.0310 28.0670 49.3594 4.8873 38.3759

10 2 222.1370 62.7211 108.4520 11.2063 71.7025 C.L.S.A. - represents the exact error of the classical analysis

c1 H.A.1 - represents the error using the boundary η = 1 & ζ = 1 in H0,0 / H0,0 approximation



Exact - represents the exact error of H0,0 / H0,0 approximation

The final error approximations of all other three Hermite formulations don’t

give us so good results. We could not find approximate error values higher than the

exact ones for all Biot number and aspect ratios used.

• Test – Case Two

Figures (3.a-c) show the behavior of the dimensionless average temperature

with a dimensionless time and with G = 1, for Bi = 0.1, 1 and 10 respectively.

21

Figure 3.a – Comparison between all formulations of the dimensionless average

temperature (Slab – Bi = 0.1 & G = 1)

Figure 3.b – Same as above, for Bi = 1


22

In figure (3.a), we can notice that for low Biot number all formulations have a

good agreement with the exact solution, as we saw in figure 1.a, test case – one.

In this figures we can’t see approximations H0,1 / H0,0 and H1,0 / H0,0 results

because they are equal to approximation H1,1 / H0,0. In figures (3.b, c) it is easier to see

that these approximations have a better approach than H0,0 / H0,0 in steady state

solution.

We can see clearly in these figures that the classical approach deviates from

the exact one, and this became severe when we have greater Biot numbers, as we can

see in figures (3.b, c).

Table 3 – Comparison between all formulations of the dimensionless average

temperature with dimensionless heat generation (G = 0.1)

Bi τ C.L.S.A. H.A.1 H.A.2 I.T.T.

0,1 0.0 1.0000 1.0000 1.0000 1.0000

0,1 5.0 1.0000 1.0095 1.0128 1.0127

0,1 7.5 1.0000 1.0130 1.0172 1.0171

0,1 10.0 1.0000 1.0156 1.0207 1.0206

0,1 15.0 1.0000 1.0192 1.0255 1.0255

1 0.0 1.0000 1.0000 1.0000 1.0000

1 1.0 0.4311 0.5182 0.5427 0.5402

1 2.0 0.2218 0.3017 0.3267 0.3274

1 4.0 0.1165 0.1607 0.1765 0.1775

1 10.0 0.1000 0.1253 0.1338 0.1338

10 0.0 1.0000 1.0000 1.0000 0.9999

10 0.5 0.0167 0.2662 0.3451 0.3429

10 1.0 0.0100 0.0904 0.1385 0.1513

10 2.0 0.0100 0.0382 0.0528 0.0573

10 4.0 0.0100 0.0350 0.0434 0.0434 C.L.S.A. – represents the dimensionless average temperature given by Classical solution



I.T.T. – represents the dimensionless average temperature given by the Exact solution

23

In table 3 data for G = 1 and Bi = 0.1, 1 and 10 is presented. For values of

dimensionless time higher than the last ones that appear in this table, for each Biot

number, the dimensionless average temperature, relative to H0,1 / H0,0, H1,0 / H0,0 and

H1,1 / H0,0 approximations, present a four - digit convergence.

Table 4, that appear bellow, present data for G = 10 and Bi = 0.1, 1 and 10.

This is an extreme case and shows the quality of Hermite – type approximations

regardless the classical one. A criteria similar to the one used in table 3, but for

three – digit convergence, have been used here.

Table 4 – Comparison between all formulations of the dimensionless average

temperature with dimensionless heat generation (G = 10)

Bi τ C.L.S.A. H.A.1 H.A.2 I.T.T.

0,1 0.0 1.000 1.000 1.000 1.000

0,1 5.0 39.954 40.182 40.256 40.249

0,1 20.0 86.602 88.077 88.561 88.555

0,1 40.0 98.187 100.451 101.201 101.199

0,1 70.0 99.901 102.390 103.216 103.216

1 0.0 1.000 1.000 1.000 1.000

1 1.0 6.689 7.333 7.507 7.449

1 2.0 8.782 10.178 10.581 10.526

1 4.0 9.835 12.031 12.719 12.695

1 10.0 10.000 12.496 13.326 13.326

10 0.0 1.000 1.000 1.000 1.000

10 0.5 1.000 2.901 3.282 3.105

10 1.0 1.000 3.356 4.002 3.891

10 1.5 1.000 3.466 4.229 4.174

10 4.5 1.000 3.500 4.333 4.333 C.L.S.A. – represents the dimensionless average temperature given by Classical solution



I.T.T. – represents the dimensionless average temperature given by the Exact solution

In figures (4.a-c) we show the final approximate error expressions of the

dimensionless average temperature for all boundaries of H1,1 / H0,0 approximation

with the exact error of the classical and H1,1 / H0,0 approximation for comparison.

24

Figure 4.a – Comparison between the classical, H1,1 / H0,0 and exact final error

solutions of the dimensionless average temperature (Slab – Bi = 0.1 & G = 10)

Figure 4.b – Same as above, for Bi = 1


25

In table 5 we present some very important data. We found the higher exact

absolute error of H0,0 / H0,0 approximation, them got the values of θ(τ), classical exact

absolute error and H0,0 / H0,0 higher approximated absolute error for the same

dimensionless time τ.

Data is shown for twelve different combinations of dimensionless heat

generation (G = 0.1, 1 and 10) and Biot numbers (Bi = 0.1, 1 and 10).

We can see in this table that our higher estimated error is always superior than

the exact one, and even better, lower than the classical one.

Table 5 – Absolute dimensionless average temperature errors of the Classical, Exact

H0,0 / H0,0 and solutions for twelve combinations of Bi and G values.

G Bi C.L.S.A. H.1 b. H.1 e. τ θ(τ)

0 0.1 0.0121 0.0032 0.0030 10.50 0.3620

0 1 0.0998 0.0380 0.0237 1.60 0.3017

0 10 0.2544 0.2310 0.0767 0.60 0.2568

0.1 0.1 0.0331 0.0083 0.0083 50.00 1.0331

0.1 1 0.1113 0.0388 0.0261 1.70 0.3757

0.1 10 0.2751 0.2298 0.0788 0.60 0.2876

1 0.1 0,3331 0.0833 0.0833 95.00 10.3324

1 1 0.3333 0.8333 0.0833 15.00 1.3333

1 10 0.4043 0.1726 0.1047 0.90 0.5044

10 0.1 3.3314 0.8333 0.8333 100.00 103.327

10 1 3.3332 0.8332 0.8333 15.00 13.3331

10 10 3.3333 0.8333 0.8333 5.25 4.3333 C.L.S.A. – Exact error of the classical solution

H.1 b. – Higher H0,0 / H0,0 improved error

H.1 e. – Exact error of H0,0 / H0,0 solution

The same problem we had in test - case one appear here, but in a very lower

amplitude. Approximations H0,1 / H0,0, H1,0 / H0,0 and H1,1 / H0,0 don’t present so good

results for lower values of Biot number, specially when combined with lower

dimensionless heat generation values.

In the other hand H0,0 / H0,0, approximation has an excellent error estimate in

all cases even loosing precision in comparison to H1,1 / H0,0, approximation.

Table 6 – Integral Hermite Approximation and Equation Errors for both Trapezoidal Approximations

H0,0 / H0,0

No B. C. at X = 0 B. C. at X = 1 ∫ ∂

∂10 2

2 ),( dXXX τθ ErroEQN(τ)

1 1A=θ 22 C=+∂

∂θ

θ BX

( ) ( ){ })(1B2C2BA2 2221 τθ av+−++ ( )!!

"

#

$$

%

&!!"

#$$%

&

∂

∂+!!

"

#$$%

&

∂

∂+

−

>−>− ζη

τθτθ

XX XX

XXB 3

3

2

2

2),(),(12

61

2 11 C=+∂

∂θ

θ BX

2A=θ ( ) ( ){ })(1B2C2BA2- 1112 τθ av−−+− ( )!!

"

#

$$

%

&!!"

#$$%

&

∂

∂+!!

"

#$$%

&

∂

∂−

>−>− ζη

τθτθ

XX XX

XXB 3

3

2

2

1),(),(12

61

3 11 C=+∂

∂θ

θ BX

22 C=+∂

∂θ

θ BX

( ) ( )

( ) )}(BBBB2

C2BC2B{4BB

2

2121

211221

τθ av−+−

−++−−

( )( )

( ) )),(

),(12(4BB

61

3

3

21

2

2

21121

ζ

η

τθ

τθ

>−

>−

&&'

())*

+

∂

∂++

&&'

())*

+

∂

∂−+

−−

−

X

X

XXBB

XXBBB

H1,1 / H0,0

No B. C. at X = 0 B. C. at X = 1 ∫ ∂

∂10 2

2 ),( dXXX τθ ErroEQN(τ)

4 1A=θ 22 C=+∂

∂θ

θ BX

( ) ( ){ })(1B2C2BA4B

62221

2

τθ av+−+++

( )

( )!!

"

#

$$

%

&!!"

#$$%

&

∂

∂−!!

"

#$$%

&

∂

∂+

+>−>− ζη

τθτθ

XX XX

XXB

B 3

3

4

4

22

),(30),(14

601

5 11 C=+∂

∂θ

θ BX

2A=θ ( ) ( ){ })(1B2C2BA4B

61112

1

τθ av−−+−−

( )

( )!!

"

#

$$

%

&!!"

#$$%

&

∂

∂−!!

"

#$$%

&

∂

∂−

−>−>− ζη

τθτθ

XX XX

XXB

B 3

3

4

4

11

),(30),(14

601

6 11 C=+∂

∂θ

θ BX

22 C=+∂

∂θ

θ BX

( ) ( )( )

( ) ( ) )}(BBBB2 C2B

C2B{3B4B4B

2

212121

12212

τθ av−+−−

+++−+

( ) ( )( )( )

( ) )),(30 ),(

1(3B4B4B

601

3

3

214

4

211212

ζη

τθτθ

>−>−&&'

())*

+

∂

∂+−&&

'

())*

+

∂

∂

⋅−++−+

XX XXBB

XX

BBB

B. C. at X = 0 : Boundary Condition at X = 0 ; B. C. at X = 1 : Boundary Condition at X = 1

NOMECLATURE

Bi, Bi* - Biot numbers

Bi+ - modified Biot number

Cν - coeficient in Hermite integration

Dν - coeficient in Hermite integration

ErroEQN - Hermite equation error

Erroα,β - Hermite integration error

Erroθav - Hermite dimensionless average temperature error

g0 - constant volumetric heat generation

G - dimensionless heat generation term

h* - heat transfer coefficient

hι - integration interval

Hα,β - Hermite approximation orders α and β

k - thermal conductivity

K - aspect ratio

L - length

m - order of eigenquantity

t - time

T - temperature

T0 - initial temperature distribution

Tb - fin base temperature distribution

T∞ - ambient temperature

x - space coordinate

xi - integration limit

X - dimensionless coordinate

y - integrand

α - order of Hermite integration

α - thermal diffusivity

β - order of Hermite integration

η - unitary parameter

λm - eigenvalue

ν - summation index

θ - dimensionless temperature distribution

28

θav - dimensionless average temperature

avθ - improved dimensionless average temperature

τ - dimensionless time

ζ - unitary parameter

REFERENCES

1. M.Ch. Hemite, “Sur la Formule d’Interpolation de Lagrange”, J. Crelle, V. 84

(1878).

2. J. Mennig, T. Auerbach, and W. Hälg, “Two Point Hermite Approximation for the

Solution of Linear Initial Value and Boundary Value Problems”, Comp. Meth.

Appl. Mech. Eng., V. 39, 199-224 (1983).

3. R. M. Cotta and M. D. Mikhailov, “Heat Conduction:- Lumped Analysis,

Integral Transforms, Symbolic Computation”, John Wiley, Chichester (1997).

4. M. D. Mikhailov and M. N. Özisik, “Unified Analysis and Solutions of Heat

and Mass Diffusion”, John Wiley, New York (1984); also, Dover Publications

(1994).

5. M. N. Özisik, “Heat Conduction”, John Wiley, New York (1980).

6. J. B. Aparecido and R. M. Cotta, “Improved One - Dimensional Fin Solutions”,

Heat Transf. Eng., V. 11, no. 1, 49-59 (1989).

7. R. M. Cotta, “Improved Lumped – Differential Formulations in Heat Transfer”,

Invited Chapter, Volume 3 – Modelling of Engineering Heat Transfer

Phenomena, Eds. B. Sundem & M. Faghri, Heat Transfer Series, Computational

Mechanics Publications, UK. (1997).

8. S. Wolfram, “Mathematica: A System for Doing Mathematics by Computer”,

Addison – Wesley (1991).

ERROR ANALYSIS OF MIXED LUMPED-DIFFERENTIAL FORMULATIONS IN DIFFUSION PROBLEMS

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