Influence of Convective and Radiative Cooling on Heat Transfer for

a Thin Wire with Temperature-Dependent Thermal Conductivity

OKEY OSELOKA ONYEJEKWE

Robnello Unit for Continuum Mechanics and Nonlinear Dynamics, Umuagu Oshimili South,

Asaba Delta State, NIGERIA

Abstract In this study, a numerical prediction of temperature profiles in a thin wire exposed to

convective, radiative and temperature-dependent thermal conductivity is carried out using a finite-

difference linearization approach. The procedure involves a numerical solution of a one-dimensional

nonlinear unsteady heat transfer equation with specified boundary and initial conditions. The resulting

system of nonlinear equations is solved with the Newton-Raphson’s technique. However unlike the

traditional approach involving an initial discretization in space then in time, a different numerical

paradigm involving an Euler scheme temporal discretization is applied followed by a spatial

discretization. Appropriate numerical technique involving partial derivatives are devised to handle a

squared gradient nonlinear term which plays a key role in the formulation of the Jacobian matrix. Tests on

the numerical results obtained herein confirm the validity of the formulation.

Key-words: temperature-dependent, thermal conductivity, finite difference, nonlinearity, unsteady heat

transfer, Newton-Raphson technique, Jacobian matrix, Euler scheme,

Received: May 15, 2021. Revised: October 22, 2021. Accepted: December 16, 2021. Published: January 2, 2022.

1 Introduction

Non-linearity in heat transfer problems occurs

when thermo-physical parameters are

temperature –dependent or when boundary

conditions are nonlinear. Examples of this occur

frequently in practice for example in

groundwater flow, heat exchangers,

environmental pollution, fin design, biological

systems etc. Although the method of separation

of variables has wide applicability, it is limited

to linear problems. The resulting nonlinear

differential equations describing such systems

are usually computed iteratively until a certain

error tolerance value is satisfied. However for

strongly nonlinear problems, the iterative

process can diverge and cause numerical

instability. Many researchers have been able to

deal with this problem satisfactorily by

manipulating the Jacobian matrix encountered in

the Newton-Raphson (NR) method [1-7]. This

approach has been facilitated by the advent of

the latest generation of high speed computers.

The thin wire under consideration is assumed to

be of uniform cross-section and is long enough

so that temperature variation is only relevant in

the axial direction. Hence heat transfer process

is one-dimensional. The thermal conductivity is

temperature-dependent and can be modelled by

power law or linear dependency on temperature

[8-11].

In this work, we shall consider the effects

radiation as well as convection and nonlinear

conduction in the overall heat transfer process.

Radiation for example is a huge contributor to

nonlinearity and its impact on the temperature is

quite considerable especially in the performance

of heat exchangers at high temperatures [12].

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2. Numerical Formulation

Consider a heat transfer problem of a 1D

conductive-convective-radiative thin metal with

a fin profile area

p

A

, length L, perimeter P.

The rod has a fixed base temperature

b

T

and

extends into the surrounding fluid of

temperature

a

T

. At the fin surface, heat loss

occurs by convection and radiation. The energy

balance for the longitudinal hotwire is given by:

 

44

0, 0 1

va

rs

TT

c kA hP T T

t X X

P T T X L





  



  



  



    

where k is thermal conductivity, h is the heat

convection transfer coefficient,

,v

c



are the

density and volumetric heat capacity,



is the

Stefan-Boltzmann constant, X and t are the

spatial and temporal variables, and T is the

temperature. It is assumed that the metal is

homogeneous and isotropic, convection heat

coefficient between the thin metal and the

environment is constant and uniform over the

entire surface of the solid. Heat dissipated from

the surface obeys the Stefan-Boltzmann law.

The thermal conductivity k is dependent on the

local temperature. Other thermo-physical

parameters such as the heat transfer coefficient,

h and surface emissivity

r



are assumed to be

constant. The boundary and initial conditions

can be specified as:

     

,0 , , 2

bL

T t T T t L T a

Initially the metal is kept at the ambient

temperature:

   

0, 2

a

T X T b

To facilitate computation, the following

dimensionless parameters are introduced:

   

2

23

22

,,

,

a

v

a

b a a

b

r

a

kt

X

xL c L

kT

TT D

T T k

PL T hP

ML

k A k A



















where

2

M

is the thermo-geometric parameter,

a

k

is the thermal conductivity of the rod at

ambient temperature.

Equation (1) together with the boundary

conditions are given as :

 

24

, 0 1 3

r

D

xx

M x a







  

  





  



   

where

     

0exp 3Db

  



The boundary and initial conditions can also be

rewritten as:

   

 

0, ) 0, 1, 1, ,0 0 (4xa

    

  

and

   

2

24

2

15

r

DM

x D u x

  

  





   



    





   







Equation (5) is a two-point nonlinear boundary

value problem (TPBVP) and is solved iteratively

by the NR method. For the numerical

implementation we adopt the method described

in [13] with some modifications. Unlike the

traditional approach where the governing

differential equation is first discretized in space,

to yield an initial-value (Cauchy) problem

involving a system of first order ordinary

differential equations (ODEs); the equation is

first discretized in time to obtain a sequence of

TPBVPs. In order to facilitate stability, an

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implicit scheme is utilized for the temporal

discretization. The finite difference method is

employed for the spatial discretization.

Complete discretization of the temporal and

spatial components of equation (5) yields an

approximate discrete equation, which is applied

to each node of the problem ID domain to yield

a system of algebraic equation. This allows the

application a modified NR iteration technique.

Equation (5) is simply written as :

   

2

1

2, , 6

n n n

fv

x









where

   

1

,,

, , 7

n n n

n

v

fv D

  

 





hence

 

1

2

24 1

,,

8

n n n

nn

r

v

D

x

M

  







   





























In accordance to the time discretization, the

values of

   

1

, , ,

nn

xx

   



are given as::

 

11

,

nn

x







The spatial gradient is

nn

vx



  

The NR iteration scheme for the computation of

a nonlinear system of equation is defined as:

     

 

   

 

1

19

k k k k

n n n n

  



 n

JG

Equation (9) is a matrix equation and its

manipulation can be made clearer if it is put in

the form:

 

   

 

1

110

k k k

n n n







   n

JG

where

n

G

can ‘loosely’ be described as the

right hand side (RHS) column vector of known

quantities and houses the boundary and initial

condition ( first guess). It is defined in [12 ] as:

 

,1 ,2 ,

, .................. 11

T

n n n N

G G G a





n

G

And

 

,

2

, 1 , , 1 ,

2 11

ni

n i n i n i n i

G

h f b

  





  

where

 

, , , 1,

,,

n i n i n i n i

f f v







The first and last entries should reflect the

boundary conditions at both ends of the problem

domain

 

,1 ,1 ,

n n n

G

  



and

 

,, ,

n N n N n

G

  



The Jacobian matrix is defined as:

   

 

12

kk

n

a









n

nn

G

J

It is expressed at the grid points as:

 

   

 

   

 

   

 

,

1

2

,

1

1 1 2 ,

2,

1 1 2 12

kk

ni

kk

ni

kk

ni

J hp

J h q

J hp b







  



where

       

 

     

 

, , , 1,

, , ,

, , 12

k k k k

n i n i n i n i

k k k

n i n i n i

q q v

p p v c









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   

11

,

,,

2 12

vi

ii

qf

pf

v h d









  



The Jacobian matrix is easily computed and is

found to be tridiagonal because each

 

i



J

only depends on the grids at

1, , 1i i i

.

Hence the nonzero elements of the Jacobian

matrix for rows 2,3…N-1(the first and the last

rows are apriorily determined by the boundary

conditions) are computed as:

 

,1

1

2

11

1 13

i

ii

i

ii

G

L

f f v

ha

vv

















   





   



 

,

2

1 13

i

ii

i

ii

G

L

f f v

hb

vv













   





   



 

,1

1

2

11

1 13

i

ii

i

ii

G

L

f f v

hc

vv

















   





   



1,1 , 1

NN

LL

for Dirichlet boundary

condition specifications

The various components of the partial

derivatives are defined as:

     

114

nn

n n n n

D

f

q f a

D





   







  



  



   

114

nn

n

nn

f

pb

v D v









 





   

2

114

n

nn

n

Dvc

x













   

2 14

nn

Dvd

v













3. Results and Discussion

In the foregoing work, we have obtained

physically realistic results to confirm the

efficacy of our numerical technique. Equation

(3) is solved for N = 51 mesh-points, M = 41

temporal grids,

0

0.5, 0.1





, with

integration range

0 20





, for time, and

13x

for space, time step

0.25





, and

spatial step h = 0.05.

0.5, 0.5

r

M





Iteration is carried out until

the difference between current and previous

results satisfy a predetermined error tolerance;

1kk

nn

  



where for this study

.0001





.

The results for the nonlinear diffusion heat

transfer case are shown in Figs. 1a, 1b and 1c.

Fig. 1a is consistent with the physics of heat

transfer.

Fig. 1a : Temperature field : Nonlinear Diffusion

It can be seen that higher temperatures move

from the higher to the cooler end and tend to

‘smother’ the effects of the lower temperature

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imposed on the right hand side (RHS) boundary.

The temperature bar at the right end of the figure

confirms the extent to which this happens. The

domain of the cooler temperatures is confined to

the right end of the rod. The 3D plot of the

temperature profile shown in Fig. 1b confirms

the observation in Fig. 1a. A close look reveals

Fig.1b:3D Temperature field : Nonlinear

Diffusion

that a sudden profile change (a profile separation

almost) happens around x = 1.5 where the

temperature profiles from the cooler end return

to the x axis instead of progressing to the hotter

end . Further progress to the hotter end would

have been a contradiction to the law of

conservation of energy. Fig. 1c shows that the

temperature profiles become more linear as time

increases.

Fig. 1c: Transient Temperature field: Nonlinear

Diffusion

This is not surprising because the whole system

tends to steady state.

Figs. 2a, 2b and 2c show the influence of

convection and nonlinear diffusion in the heat

transfer process. Starting from Fig. 2a ,

Temperature field: Nonlinear Diffusion and

Convection

it is interesting to observe how the influence of

the colder end on the temperature profiles seems

to have changed especially in the vicinity of the

RHS boundary. This is as a result of cooling by

convective heat transfer. In comparison with the

previous case, colder temperature profiles are

observable near the right boundary. Most of this

change happens within the

1.6 2.8x

as

confirmed by Fig.2b.

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Fig. 2b : 3D Temperature profile for Nonlinear

Diffusion and Convection

Fig. 2c shows that unlike Fig. 1c, there is little

Fig.2c: Transient Temperature Profiles for

Nonlinear Diffusion and Convection

change in dimensionless temperature with time

especially for

5, 10





. In addition, the

temperatures are much less than previously as

heat is convected away from the surface.

The effects of Nonlinear diffusion and radiation

heat loss can be observed in Figs. 3a, 3b and 3c.

Fig. 3a Temperature field for Nonlinear

Diffusion and Radiation

More heat transfer activities can now be seen to

be happening at the left and right end

boundaries. As the influence of the colder

dimensionless temperature profiles are felt more

at the RHS boundary, higher temperatures

profiles are confined to the left boundary in

accordance with conservation of energy. This

balance is very mush shown in figure 3b where

Fig. 3b: 3D Temperature Profile for Nonlinear

Diffusion and Radiation

it is further demonstrated that the region of fast

transitions is positioned within

1.8 2.7x

.

This is in total agreement with Fig.3a.

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Fig. 3c: Transient Temperature field for

Nonlinear Diffusion and Radiation

Fig. 3c illustrates a significant influence of

radiation cooling in the overall heat transfer

process. As can be observed, there is an overall

cooling effect on the surface of the rod despite

the time change. The cooling effect is

significantly lower than in the previous cases

considered. Figs. 4a, 4b and 4c illustrate

nonlinear conduction, radiation and convection

effects on the overall heat transfer activity. Fig.

4a demonstrates the importance of considering

both convection and radiation in the energy

equation.

Fig. 4a: Temperature field for Nonlinear

Diffusion, Convection and Radiation

The hotter temperature profiles can be seen to

be moving more uniformly towards the cooler

end. There are no profile singularities closer to

the cooler end as was the case in previous

considerations. This is confirmed in Fig. 4b

below.

Fig. 4b: Scalar Profile for Diffusion,

Convection and Radiation.

Fig. 4c shows that a combination of heat transfer

rates resulting from nonlinear conduction,

radiation and convection produce lower surface

temperatures than in the previous cases.

Fig.4c: Transient Temperature field for

Nonlinear Diffusion, Convection and Radiation

4 Conclusion

In the work reported herein, heat transfer

computations involving temperature-dependent

thermal conductivity as well as convection and

nonlinear radiation effects have been carried out

using an FD modified NR approach. The results

are physically plausible and amply demonstrate

the impact of convection and radiation on the

overall heat transfer process. Of these two

(convection and radiation), radiation effects are

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more significant. It has previously been

demonstrated that for temperature distribution in

a metal cooled by convection and radiation, the

heat loss due to radiation, contributes to 15-20

percent of the total loss [14,15]. Hence it plays a

significant role in improving the thermal

performance of heat loss components such as

fins and more importantly in devices with low

heat transfer coefficients. We hasten to comment

that for this particular problem the non-

dimensionless temperature profiles in the region

1.3 2.7x

needs further study in terms of

relating it to the influence of radiation

parameter

r



thermogeometric parameter M,

conduction parameters

0,



. A look at the

profiles within this region, suggests that certain

combinations of these parameters may lead to

physically unrealistic results and numerical

instability which may significantly affect design.

References

[1] U.M. Ascher, R.M.M. Mattjeij, R.D. Russel,

Numerical solution of boundary value problem

for ordinary differential equations in; Claasics in

Applied Mathematics, Vol. 13, 1995, SIAM,

Philadelphia

[2] A. Marasco, A. Romano, Scientific

computing with Mathematica: mathematical

problems for ordinary differential equations in:

Modelling and Simulation in Science,

Engineering and Technology,2001, Birkhauser

Boston, MA

[3] R.E. White, Computational mathematics:

models, methods, and analysis with MATLAB

and MPI, 2004, Chapman and Hall/CRC, Boca

Raton, London, NY Washington.

[4] A.U. Keskin, Boundary value problems for

engineers with MATLAB solutions, Springer

Nature Switzerland , ISBN 978-3-030-21079-3

2019

[5] I. Farago Numerical methods for ordinary

differential equations, TypoTech, Budapest,

http://faragois.web.elte.hu/ODE_angol.pdf

2014

[6] J. Lambers, Finite difference methods for

two=point boundary value problems. MAT

461/561, lecture 26 notes , department of

mathematics, university of southern Mississippi

http://www.math.usm.edu/lambersmat461/spr10

/lecture26.pdf 2010

[7] O.O. Onyejekwe, Localized boundary-

domain integro-partial differential formulations

for transient scalar transport problems,

International journal of applied and

computational mathematics Vol. 2,No.3, 2016

DOI 10.1007/s40819-016-0235-y

[8] G. Grarslan, Numerical modelling of linear

and nonlinear differential equations y compact

finite difference method, Applied Mathematics

and Computation, Vol. 216, 2010 pp. 2472-2478

[9] R.J.Moitsheki, T. Hayat, M. Y. Malik, Some

exact solution of the fin problem with power

temperature-dependent thermal conductivity,

Nonlinear Analysis, Real World Appl. Vol.11,

No.5 2010, pp. 3287-3294

[10] J. Hristov, Integral solutions to transient

nonlinear heat (mass) diffusion with power law

diffusivity: a semi-infinite medium with fixed

boundary conditions, Heat Mass Transfer Vol.

52 No.3 2016, pp. 635-655

[11] A.S.Vatsala, T.G. Melton, Generalized

quasilinearization method for higher order of

convergence for second-order boundary-value

problems, Bound. Value. Problems, 2006, pp. 1-

15

[12] Y. Huang, X-F. Liang Exact and

approximate solution of convective –radiative

fins with temperature dependent thermal

conductivity using integral equation method,

International Journal of Heat and Mass

Transfer, Vol. 150, 2020, pp. 1-8

[13] S.M. Filipov, I.D. Gospodinov and I.

Farago, Replacing the finite difference methods

for nonlinear two-point boundary value

problems by successive application of the linear

shooting method, Journal of Computational and

Applied Mathematics, Vol. 358, 2019, pp.46-60

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[14] A.S. Dogonchi, M. Waqas S.M. Seyyedi,

M. Hashemi-Tilehnoee, D.D. Ganji, CVFEM

analysis for

3 4 2

Fe O H O

nanofluid in an

annulus subject to thermal radiation, Int. Jnl.

Heat Mass Transf Vol 132, 2019, pp. 473-483

[15] A.S. Dogonchi, S. Hashim, Heat transfer by

natural convection of

3 4 2

Fe O H O

nanofluid

in annulus between a wavy-circular circular

cylinder and a rhombus, Int. Jnl. Heat Mass

Transf. Vol. 130, 2019, pp. 320-332

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DOI: 10.37394/232012.2022.17.1

Okey Oseloka Onyejekwe

E-ISSN: 2224-3461

9

Volume 17, 2022