effect of axial wall conduction on heat transfer parameters for

Fundamental J. Thermal Science and Engineering, Vol. 2, Issue 2, 2012, Pages 63-87 Published online at http://www.frdint.com/

:esphras and Keywords axial wall conduction, conjugate heat transfer, laminar flow, step

change, numerical study, P-1 approximation, participating medium.

*Corresponding author

Received December 14, 2012

© 2012 Fundamental Research and Development International

EFFECT OF AXIAL WALL CONDUCTION ON HEAT

TRANSFER PARAMETERS FOR PARALLEL-PLATE

CHANNEL HAVING RADIATION PARTICIPATING MEDIUM

AND STEP CHANGE IN BOUNDARY CONDITIONS

M. A. HASSABa, M. KHAMIS MANSOUR

b,* and M. SHAWKY ISMAIL

a

aDepartment of Mechanical Engineering

Faculty of Engineering

Alexandria University

Alexandria, Egypt

bDepartment of Mechanical Engineering

Faculty of Engineering

Beirut Arab University

Beirut, Lebanon

e-mail: [email protected]

Abstract

This paper presents a theoretical model to investigate the effect of axial

wall conduction on the heat transfer characteristics for laminar flow

through a thick-walled channel, taking into account the radiation effect in

gases as a participating medium in the radiation exchange. A partition

separates the channel of interest into two regions (upstream and

downstream region) in which two different coolant fluids bound the

channel in those regions. A numerical model has been developed in this

study using a finite-difference technique to solve the conjugate heat

transfer problem by creating a self-made computer code. The modified

P-1 approximation is employed to solve the radiation part of the problem.

HASSAB, MANSOUR and ISMAIL

64

The calculated results from the computer program have been compared

with a commercial code “ANSYS FLUENT 12” and a good agreement has

been obtained. The effects of the optical thickness ,oτ the conduction-to

radiation parameter N, single scattering Albedo coefficient ω and wall

emissivity ζ on the heat transfer characteristics are presented in this

study. The heat transfer characteristics are represented by local Nusslet

number Nu, fluid temperature distribution, and solid temperature

distribution.

The study shows that the significant change in those parameters could lead

to an enhancement of 24% for the average Nusselt number (Nu) at certain

conditions.

List of Symbols

iB Biot number, w

i K

thB ∞=

pC Specific heat at constant pressure, ( )K.kgJ °

1∞h Outside heat transfer coefficient for the upstream fluid,

( )K.mW 2 °

2∞h Outside heat transfer coefficient for the downstream fluid,

( )K.mW 2 °

H Vertical distance from center line to the inner surface of the solid

domain, m

fK Fluid domain thermal conductivity, ( )K.mW °

wK Solid domain thermal conductivity, ( )K.mW °

rK Thermal conductivity ratio, fw KK

pK Planck mean coefficient

RK Rosseland mean coefficient

L Domain length parallel to the fluid flow, m

N Conduction to radiation parameter, 34

*

i

f

T

KN

σ

β=

EFFECT OF AXIAL WALL CONDUCTION ON HEAT …

65

Nu Local Nusselt number, k

Hhi=Nu

P pressure, kPa

P* Dimensionless pressure, 2inletu

PP*

ρ=

Pe Peclet number, Pr*RePe =

Pr Prendtl number, f

p

k

cPr

µ=

inconvQ Dimensionless convective heat transfer from inner fluid, *

*

yx

∂

θ∂∆

outconvQ Dimensionless convective heat transfer to the outer fluid,

( )∞θ−θ∆ *xKB ri

axialQ Dimensionless axial heat transfer through the solid domain,

*xtK rr

∂

θ∂

radQ Radiative heat flux, 2mW

*rQ Dimensionless radiative heat flux, ( )

−σ ∞1

3rad 4 TTTQ ii

Re Reynolds number, µ

ρ=

UHRe

t Thickness of the solid domain, m

rt Thickness ratio of the solid domain, Httr =

T Temperature, K°

iT Inlet fluid temperature, K°

meanT Local bulk temperature of the fluid domain, K°

surfaceT Interface temperature between the solid and fluid domains, K°

1∞T Outside temperature for the left side (upstream), K°

2∞T Outside temperature for the right side (downstream), K°

inletu Inlet fluid velocity in X-direction, sm


66

U Velocity in X-direction, sm

*U Dimensionless velocity in X-direction, inlet* uUU =

V Velocity in Y-direction, sm

*V Dimensionless velocity in Y-direction, inlet* uVV =

X Horizontal coordinate, m

*X Dimensionless horizontal coordinate, HXX =*

Y Vertical coordinate, m

*Y Dimensionless vertical coordinate, HYY =*

Greek Symbols

θ Dimensionless temperature, ( ) ( )11 ∞∞ −−=θ TTTT i

ρ Fluid density, 3mkg

µ Dynamic viscosity, Pa.s

ν Kinematic viscosity, ρµ=ν

ω Single scattering Albedo, ( )σ+

σ

K

ξ Identified parameter, ( )

( )ω−τε−

ε1

22 o

β Mean extinction coefficient, σ+pK

oτ Optically thickness, H∗β

σ Scattering coefficient for fluid

σ Stefan-Boltzmann constant

∈ Solid domain emissivity

∆ Difference

Subscripts

i inlet

r ratio


67

rad radiation

w wall

f fluid

1∞ Outside left fluid

2∞ Outside right fluid

1. Introduction

The interest in predicting of heat transfer characteristics in laminar forced flow

in pipes and channel flows is steadily under consideration particularly when the

thermal problem is out of ordinary form. For example, in some heat transfer devices

such a fired-heater exchanger and shell-and-tube heat exchanger, the surface wall is

exposed to significant step change in the bounding fluid temperature at the exchanger

baffles/partition. This significant change in the temperature intensifies the axial wall

conduction to participate in the heat transfer modes and changes the characteristics of

the convective heat transfer problem. In particular, when this channel has a

participating medium in radiation exchange, the heat transfer characteristics are

considerably changed. This study addresses this problem and highlights the

importance of the axial wall conduction’s influence on the heat transfer parameters

such as local Nusselt number, fluid temperature distribution, and solid temperature

distribution during the presence of radiative heat exchange. In addition, according to

authors’ knowledge and the open literature survey, no specific work has been

dedicated to address this problem including the radiation effect in the gas and this is

considered the main contribution in this research. The topic of combined conduction

and radiation heat transfer in the channel/tube having participating medium has been

studied extensively in the cited literature survey. A comprehensive review of

combined conduction-radiation heat transfer investigations has been given by Siegel

[1]. Also, numerous investigations of the cited review covered a wide range of

operating conditions for the combined convective and radiative heat transfer

problems either laminar or turbulent flow, developing or developed flow, gray or

nongray participating medium, most of those investigations are listed in Modest [2]

and Viskanta [3]. Some of those researches focus on the combined radiation and

convection inside channel such as in [4-14]. The foregoing review of the available

and recent literature review reveals that the effect of axial wall conduction in the

combined radiative and convective heat transfer problem has not been extensively

investigated yet. Therefore, this work is devoted to investigate this problem and this


68

could be considered the main contribution for this study. Nevertheless, Yang et al.

[15] pioneered a research to study the effect of axial fluid conduction on laminar

forced conduction of a participating gray, absorbing and emitting medium with a

fully developed velocity entering an isothermal semi-infinite long pipe. The objective

of their study is to investigate the effect of Peclet number, conduction-radiation

parameter, and optical thickness on heat transfer behavior in the thermal entrance

region of a circular pipe.

2. Mathematical Formulation

The geometrical configuration of the investigated problem is schematically

shown in Figure 1. A fluid with an average temperature inT and velocity inu is

flowing into parallel-plate channel with height 2H with a wall thickness of t.

The assumptions which have been adopted in this model are as follows:

[1] The fluid is Newtonian and incompressible with constant physical properties

in steady-state heat transfer problem.

[2] Negligible viscous heat dissipation and fluid axial conduction effects.

[3] The flow is laminar.

[4] Hydrodynamic and thermal developing flow.

[5] The partition is adiabatic so no heat flow across it from the coolant fluids.

[6] The participating gas is a gray, emitting, isotropically scattering, and

absorbing medium.

[7] The radiation model is a P-1 approximation with a one-dimensional radiation

equation.

Figure 1. Geometrical configuration of the computational domain.


69

The conjugate heat transfer problem consisting of the flowing fluid and the tube

wall is governed by the respective energy equations in 2D form. Due to the axial

symmetry of the problem, it is sufficient to solve the lower half of the model. The

outer surfaces of the channel are exposed to coolant fluids with different

temperatures of 1∞T and 2∞T “step change boundary conditions”. The coolant fluid

in the left side is called an upstream fluid while the right side is a downstream fluid.

It is assumed that both fluids have the same convective heat transfer coefficient

which is kept uniform in the numerical model at .∞h Doing so, the mathematical

formulation of the governing equations in its derivative and non-dimenstional form is

given below.

2.1. Governing equations for fluid domain

2.1.1. Continuity equation

.0=∂

∂+

∂

∂

Y

V

X

U (1)

2.1.2. Momentum equations

,1

2

2

2

2

∂

∂+

∂

∂ν+

∂

∂

ρ−=

∂

∂+

∂

∂

Y

U

X

U

X

P

Y

UV

X

UU (2)

.1

2

2

2

2

∂

∂+

∂

∂ν+

∂

∂

ρ−=

∂

∂+

∂

∂

Y

V

X

V

Y

P

Y

VV

X

VU (3)

2.1.3. Energy equation coupled with the radiation term

.rad2

2

Y

q

Y

T

*c

k

Y

TV

X

TU

p ∂

∂−

∂

∂

ρ=

∂

∂+

∂

∂ (4)

2.1.4. P-1 radiation model

( ) .*163 3rad2

rad2

Y

TTKqKK

Y

qPRP ∂

∂=σ+−

∂

∂ (5)

By introducing the following dimensionless quantities,

,,,,, *

1in

1in

*in

*HYYHtt

TT

TTuVVuUU r ==

−

−=θ==

∞

∞

( ),

4,

4,,

31

3rad*

2in

*

ii i

f

ii

rf

wr

T

KN

TTT

QQ

K

KK

u

PP

σ

β=

−σ==

ρ=

∞


70

( )( ),1*

22,,, ω−τ

ε−

ε=ξβ=τ

σ+

σ=ωσ+=β oo

pp *H

KK

.Nuand,,Pr,Rek

Hh

K

thB

k

cUH i

wi

p==

µ=

µ

ρ= ∞

Equations (1) to (5) can be renormalized as follows:

,0*

*

*

*=

∂

∂+

∂

∂

Y

V

X

U (6)

,Re1

*

*

2*

*2

2*

*2

*

**

*

**

X

P

Y

U

X

U

Y

UV

X

UU

∂

∂−

∂

∂+

∂

∂=

∂

∂+

∂

∂ (7)

,Re1

2*

*2

2*

*2

*

**

*

**

∂

∂+

∂

∂=

∂

∂+

∂

∂

Y

V

X

V

Y

VV

X

VU (8)

,1

PrRe1

*

*

2*

2

**

**

Y

Q

NpeYY

VX

U ro

∂

∂τ−

∂

θ∂=

∂

θ∂+

∂

θ∂ (9)

( ) ( ) .1413 22

*2

YY

Qoo

r

∂

θ∂τω−=τω−−

∂

∂ (10)

2.2. Governing equation for the solid domain

Assuming steady-state conditions and constant properties of the wall, the energy

equation for 1D is adopted (neglecting the temperature gradient in y-direction) and it

is written in dimensionless form as follows:

.0*

2=

∂

θ∂

X (11)

2.3. Boundary conditions

2.3.1. At the inlet section

At the inlet section for the fluid domain, the velocity and temperature are

considered as uniform while for the solid domain the velocity is zero and the wall is

adiabatic.

.0and0at*,,,0, 4radinlet HYXTQTTPPVuU iii ≤≤=σ===== (12)


71

.0andat0 LXHYVU ≤≤=== (13)

In dimensionless form as:

.10and0at0,,1 ***2inlet

*** ≤≤==ρ

===θ= YXVu

PPQU i

r (14)

.01at0 **** HLXYVU ≤≤=== (15)

2.3.2. At the center line

At the center line the boundary condition is the symmetrical axis

.0,0at0rad LXYY

Q

Y

T

Y

V

Y

U≤≤==

∂

∂=

∂

∂=

∂

∂=

∂

∂ (16)


.0,0at0 ***

*

**

*

*

*HLXY

Y

Q

YY

V

Y

U r ≤≤==∂

∂=

∂

θ∂=

∂

∂=

∂

∂ (17)

2.3.3. At the wall

( ) .20,at1rad2

2LXHYTThQ

Y

TK

X

TtK fw ≤≤=−=+

∂

∂−

∂

∂∞∞ (18)

( ) .2,at2rad2

2LXLHYTThQ

Y

TK

X

TtK fw ≤≤=−=+

∂

∂−

∂

∂∞∞ (19)


.20,1at** ****2*

2HLXYtBKQ

NYX

tK rirro

rr ≤≤=θ=τ

+∂

θ∂−

∂

θ∂ (20)

( ) .20,1at** **2

**2*

2HLXYtBKQ

NYX

tK rirro

rr ≤≤=θ−θ=τ

+∂

θ∂−

∂

θ∂∞ (21)

And Marshak’s boundary condition [16] for a cold surface, in a simplified manner is

expressed as:

( )

.0,at0*22 rad

rad LXHYQKY

Qp ≤≤==

ε−

ε+

∂

∂ (22)

And it could be written in dimensionless form as:


72

.0,1at0* ****

*

HLXYQY

Qr

r ≤≤==ξ+∂

∂ (23)

3. Numerical Procedure and Model Validation

The system set of the governing equations and boundary conditions are

discretized numerically using a finite-difference method. Pressure-correction scheme

has been adopted to couple the pressure and velocity in the continuity and momentum

equations and Gauss-Seidel technique has been implemented as an iterative method

to solve the system of finite-difference equations. Numerous grid sizes have been

examined to achieve the mesh-independent solutions as displayed in Figure 2. It can

be seen from this Figure that after 26000 grid number there is no significant change

in the numerical results, this number has been obtained through 221 cells in the flow

direction and 118 cells in the normal direction to the flow. A computer code has been

developed to solve the governing equations simultaneously. In order to guarantee the

reliability of our self-coded program it should be assessed first before its

implementation in the parametric study. The assessment will be accomplished

through comparison with a well-known and robust program code such as ANSYS

FLUENT 12. The self-coded program is run at predefined conditions ( ,250Re =

6.0,1.0,5.0,766,01.0,2.0,5.0,767.0Pr =τ==ω===−=θ∆= ∞ orir NkBt

and )15.0=ζ and the equivalent of those conditions physically have been given to

the ANSYS FLUENT 12 program for the sake of the comparison. As it is noticed

from Figure 3, most of the predicted results of the self-coded program are mainly

coincident with those of the commercial code with a deviation of 6%.

Figure 2. Mesh-independent history.


73

Figure 3. Comparison between the self-coded program and ANSYS FLUENT 12

program.

4. Results and Discussion

A case study has been selected to highlight the effect of the axial wall

conduction on the conjugate heat transfer problem taking into account the radiation in

the participating medium. The self-coded program has been catered by the following

selected parameter ,5.0,01.0,2.0,5.0,7.0Pr,250Re =ω==−=θ∆== ∞ ir Bt

0.1,05.0,1800 =τ== or Nk and .25.0=ζ

The output results are represented in Figures 4 to 8. Figure 4 is prepared to show

the effect of the axial wall conduction and the radiation heat flux on the local Nu

along the channel. As it can be obviously seen from this Figure, that both cases “with

and without radiation effects” have a turnover point in the Nu trend at the partition

position. The enhancement in the local Nu could reach to 50% in the case without

radiation while it could reach only 8% in the second case.

The main explanation of this turnover point has been addressed in details in

Hassab et al. [17]. However, in this paper, it focuses on the effect of the radiative

medium on the conjugate heat transfer problem. The slight enhancement in the Local

Nu could be explained as follows; the high temperature difference between the two

coolant fluids "upstream and downstream fluid" at the partition plate intensifies the

magnitude of the axial wall conduction through the channel wall. However, the

radiative heat flux warms up the wall surface and attempts to decrease the gradient of

the wall surface temperature as shown in Figure 6. This warming makes the energy to

be dissipated to the outer fluid more than non-participating medium case.


74

As expected, the heat transferred from the inner fluid is clearly increased during

presence of the radiation effect as shown in Figure 7. This increase is approximately

400% compared with that in the case of non-participating medium. This is an

indicator for the importance of the radiation calculation in practice such as in boilers

and fired water heaters.

This small temperature difference between the two coolant fluids around the

partition stimulates some amount of heat to be extracted through the axial conduction

in the channel wall. This amount makes small cooling for the wall surface. This is

translated into an increase in the heat extracted from the inner fluid. Hence, the ratio

between the radiation heat flux and the total heat extracted from inner hot fluid

decreases around the partition as shown in Figure 8.

It may be interesting to observe from Figure 6 that at a location in the front

region (from-0.14 to 0.0) the wall temperature is colder than the coolant fluid itself.

In case of participating medium in the channel, the release of thermal stresses takes

place with the outer convective heat flux. So, at fouling cases and relatively low Bi

number cases, the outer convective heat flux will be limited and may cause a failure

of the tube walls.

Figure 4. Local Nu for domain of interest for 0.1,05.0,5.0 =τ==ω oN and

.0.1=ζ


75

Figure 5. The ratio between _conv_in,and_conv_out QQ ratio between delta

inconvandaxial _Q_Q_ for 0.1,05.0,5.0 =τ==ω oN and .0.1=ζ

Figure 6. Temperature surface and its corresponding outside temperature for

0.1,05.0,5.0 =τ==ω oN and .0.1=ζ

Figure 7. _conv_inQ from the inner fluid for 0.1,05.0,5.0 =τ==ω oN and

.0.1=ζ


76

Figure 8. The ratio between _rad_inQ and the total heat in from inner fluid for

0.1,05.0,5.0 =τ==ω oN and .0.1=ζ

4.1. Parametric study

The relevant parameters to the radiative heat transfer and their impacts are

discussed in this section. Those parameters are optical thickness ,oτ conduction- to-

radiation N, wall emissivity ζ and Albedo scattering factor .ω The effect of those

parameters is represented by the variation in the local Nu number; surface

temperature distribution, and heat flux. It should be noted the other parameters are

kept constant in the parametric study. Those parameters are ,250Re =

1800,01.0,2.0,7.0Pr ==== rir kBt and .5.0−=θ∆ ∞

4.4.1. The effect of the optical thickness ( )oτ

The output results are displayed from Figure 9 to Figure 14. Figure 9 is

presented to show the effect of axial conduction and radiation heat flux on the local

Nu number at different optical thickness. It has been shown that the increase in the

optical thickness is followed by an increase in the local Nu number.

This increase starts in order of unity to order of hundreds, according to the range

of the optical thickness ( )oτ as shown in Figure 9, at prescribed ( ),oτ the local Nu

remains constant with the positive direction of the flow until reaches the partition.

Around the partition, it makes turnover point as a result of axial wall conduction

impact.

This turnover has a major effect at higher optical thickness values as shown in


77

Figure 9. It is interesting to mention that the magnitude of the axial conduction

through the walls gets improved at higher values of optical thickness as shown in

Figure 10. As the optical thickness increases the heat extracted from the inner hot

fluid increases as shown in Figure 12 until the optical thickness reaches unity. After

this value, the heat extracted from the hot fluid decreases as long as the optical

thickness is getting higher. This finding is emphasized in Figure 11. The increase in

the optical thickness is followed by an increase in wall surface temperature up to

,0.1=τo then the increase in the optical thickness is accompanied by the decrease in

the wall surface temperature as shown in Figure 11. At lower optical thicknesses, the

increase in the optical thickness intensifies the magnitude of the emissive power of

the medium. Hence, the combined radiative and convective heat transfer is getting

higher at this condition and accordingly, this causes a warming in the wall surface

temperature as illustrated in Figure 11. Alternatively, at higher optical thickness

,0.1>τo the increase in the optical thickness causes a significant attenuation in the

emissive power (see Figure 13), therefore, the total heat transfer is getting smaller at

this condition, making a decrease in the wall surface temperature. Figure 14 is

presented to underline this result by showing the effect of the optical thickness

variation on the convective outer heat transfer. At the beginning, the increase in the

optical thickness leads to an augmentation in the convective heat transfer as a result

of an increase in the difference between the outer coolant fluid and wall surface

temperature. Then, this difference is getting smaller and consequently the convective

outer heat transfer decreases as shown in Figure 14.

Figure 9. Local Nu for domain of interest for 01.0,5.0,5.0 ==ω−=θ∆ ∞ N and

oτ=ζ ,1 is variable.


78

Figure 10. The ratio between delta _conv_inand_axial QQ for ,5.0=ω ,01.0=N

0.1=τo and oτ=ζ ,1 is variable.


,5.0=ω 0.1,01.0 =τ= oN and oτ=ζ ,1 is variable.

Figure 12. _conv_inQ from the inner fluid for 01.0,5.0 ==ω N and oτ=ζ ,1 is

variable.


79

Figure 13. _in_radiationQ from inner fluid for 01.0,5.0 ==ω N and oτ=ζ ,1 is

variable.

Figure 14. _conv_outQ to the outer fluid for 01.0,5.0 ==ω N and oτ=ζ ,1 is

variable.

4.4.2. The effect of conduction to radiation parameter (N)

The local Nu number trend is almost flat in the front and behind regions at lower

values of N parameter as shown in Figure 15. Additionally, the axial conduction has

an insignificant effect interrupted by the lower value of _conv_in_axial QQ at

lower value of N parameter as shown in Figure 17. As the N parameter is small the

radiation heat flux increases and warming the wall surface subsequently the gradient

of the wall temperature can be neglected as shown in Figures 16 and 18.


80

The reverse is true for higher values of the N parameter the axial heat flux

generated and transfer from the front region to the backward region making a

valuable turnover in local Nu around the partition. Figures 19 and 20 illustrate this

effect through the variation in the heat extracted from the hot fluid and the convective

outer heat transfer with respect to the variation in N parameter. As shown in those

Figures, the increase in N parameter results in a decrease in the heat extracted and

also in the convective outer heat transfer and vice versa.

Figure 15. Local Nu for domain of interest for 1.0,5.0 =τ=ω o and 0.1=ζ and

N is variable.

Figure 16. Ratio between _conv_inand_in_radiation QQ for 1.0,5.0 =τ=ω o

and 0.1=ζ and N is variable.


81

Figure 17. The ratio between delta _conv_inand_axial QQ for 1.0,5.0 =τ=ω o

and 0.1=ζ and N is variable.


1.0,5.0 =τ=ω o and 0.1=ζ and N is variable.

Figure 19. _conv_inQ from the inner fluid dimensionless for 1.0,5.0 =τ=ω o and

0.1=ζ and N is variable.


82

Figure 20. _conv_outQ to the outer fluid dimensionless for 1.0,5.0 =τ=ω o and

0.1=ζ and N is variable.

4.4.3. The effect of emissivity of the wall parameter ( )ζ

Figure 21 is presented to show the effect of axial wall conduction and the

radiation heat flux on the local Nu number at different wall emissivity ( ).ζ It has

been shown that the increase in wall emissivity, ε results in an increase in the local

Nu number as shown in Figure 21. As expected, the increase in the wall emissivity

according to Kirchoff’s law for gray opaque body is coupled by an increase in the

wall absorptivity property. In turn, the radiative heat flux is increased as the wall

emissivity increases and enhances the local Nu number as shown in Figure 21. In

addition, the increase in the wall emissivity/absorptivity wanes the effect of the axial

wall conduction, as seen in Figure 22, as a result of warming the wall surface up at

expense of an increase in the radiative heat flux towards the wall surface. This can be

evidenced by Figure 23; the hotter surface temperature is associated with the higher

wall emissivity.

Figure 21. Local Nu for domain of interest for 1.0,05.0,5.0 =τ==ω oN and ζ

is variable.


83

Figure 22. The ratio between delta _conv_inand_axial QQ for ,05.0,5.0 ==ω N

1.0=τo and ζ is variable.


1.0,05.0,5.0 =τ==ω oN and is variable.

4.4.4. The effect of Albedo parameter ( )ω

Figure 24 is introduced to show the effect of axial conduction and the radiation

heat flux on the local Nu at different Albedo scattering factors. As the Albedo

parameter ( )ω increases the local Nu number decreases. This can be explained by

that an increase in scattering coefficient relative to the absorbing coefficient leads to

vanish/disperse the radiative intensity and therefore, the radiation heat flux decreases

too. In this case the axial wall conduction reinforces his situation with decay in the

effect of the radiation as shown in Figure 25. Furthermore, as previously explained,


84

this makes the wall surface to be colder than usual in presence of weakness of the

radiation heat transfer as shown in Figure 26.

Figure 24. Local Nu for domain of interest for 0.1,05.0 =τ= oN and ζω, are

variables.

Figure 25. The ratio between delta _conv_inand_axial QQ for 0.1,05.0 =τ= oN

and ζω, are variables.


0.1,05.0 =τ= oN and ω and ζ are variables.


85

5. Conclusions

The laminar forced convection inside channels with two parallel-plates has been

studied taking into account the axial wall conduction and radiative heat transfer in

participating medium. The combined convective and radiative conjugate heat transfer

problem has been solved numerically by a self-made computer code coupling the

solid and fluid domains by reasonable boundary conditions and assumptions. The

effect of some radiation parameters on the axial wall conduction and accordingly the

heat transfer characteristics (Nu number, wall mean temperature, and heat flux

distribution) has been investigated. Those parameters are optical thickness ,oτ

conduction to radiation N, wall emissivity ζ and Albedo scattering factor .ω The

following major conclusions can be drawn:

[1] Generally, an increase in the radiation heat flux results in a decrease in the

impact of the axial conduction as the radiation warms up the solid domain.

[2] The increase in the optical thickness produces attenuation to the impact of the

axial wall conduction particularly when the optical thickness is thinner “less

than unity”. However, at optical thickness larger than unity the radiative heat

transfer decreases allowing the axial wall conduction to be significant and take

a role to enhance the Local Nu number. At 10=τo the increase in local Nu

around the partition is 14 %.

[3] As it is expected the increase in the radiative heat transfer causes an increase

in the local Nu as a result to the increase in the total heat flux received by the

wall. This is clarified by the effect of conduction to radiation parameter, N; the

increase in this parameter has two competing adverse effects; an increase in

axial wall conduction and a decrease in average Nu number. For example, the

increase in N parameter from 0.01 to 1.0 makes the Nu to drop from 100 to 3.

It was noticed that the local Nu around the partition is increased responding to

the enhancement in the axial wall conduction at higher N parameter values.

[4] In presence of the participating medium in the channel, the wall emissivity and

scattering coefficient have adverse effects on the heat transfer parameters and

axial wall conduction. The channel wall which has higher emissivity

“unpolished/oxidized material” is characterized by high efficient thermal

performance but it is much exposed to the thermal stress around the partition

as consequence of weakness of the axial wall conduction contribution to cool


86

down the wall surface. On the other hand, the increase in the scattering

coefficient results in a decrease in the thermal effectiveness but it gives an

opportunity to the axial wall conduction to protect the channel wall from the

severe thermal stress.

The computer code has been assessed via the comparison with the commercial

software “ANSYS FLUENT 12” with a good agreement between the results obtained

from the self-coded program and commercial one.

Acknowledgements

The authors would like to acknowledge Arab Academy University for their

support by licensed ANSYS FLUENT 12 software to be used to carry out the

numerical calculation for comparison.

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