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Int. J. Exergy, Vol. 6, No. 2, 2009 249

Copyright 2009 Inderscience Enterprises Ltd.

Entropy analysis of laminar-forced convectionin a pipe with wall roughness

A. Alper Ozalp

Department of Mechanical Engineering,

University of Uludag, 16059 Gorukle, Bursa, Turkey

E-mail: [email protected]

Abstract: Momentum and heat transfer rates, as well as entropy generationhave been numerically investigated for fully developed, forced convection,laminar flow in a micro-pipe. Compressible and variable fluid property

continuity, Navier-Stokes and energy equations are solved for variousReynolds number, constant heat flux and surface roughness cases; entropygeneration is discussed in conjunction with the velocity and temperatureprofiles, boundary layer parameters and heat transfer-frictional characteristicsof the pipe flow. Simulations concentrated on the impact of wall roughnessbased viscous dissipation on the heat transfer behaviour and so occurringheating/cooling activity and the resulting overall and radial entropy generation.

Keywords: laminar; surface roughness; micropipe; entropy generation; Be;Bejan number.

Reference to this paper should be made as follows: Ozalp, A.A. (2009)Entropy analysis of laminar-forced convection in a pipe with wall roughness,Int. J. Exergy, Vol. 6, No. 2, pp.249275.

Biographical notes: A. Alper Ozalp is an Associate Professor in theThermodynamics Division of the Mechanical Engineering Department ofUludag University in Turkey. His research interests include energy-exergyanalysis of thermo-fluid systems, experimental and computational fluidmechanics and heat transfer applications, compressible nozzle flows,hydrodynamic lubrication of slider bearings and engineering softwaredevelopment. Besides being an active reviewer of several SCI journals, he isthe IAESTE Delegate of Uludag University and he is also the AssociateProfessor Representative in the Engineering and Architecture FacultyCommittee.

1 Introduction

Energy and exergy analysis has recently been the topic of great interest in various

fields such as geothermal heating systems (Ozgener et al., 2006), wind energy (Sahin

et al., 2006) and fuel cells (Granovskii et al., 2006). Regardless of the type of application,

efficiency of fluid movement is evidently of high importance for the overall system

performance. Due to the widespread industrial need of fluid transportation in circular

ducts, it has been one of the fundamental research areas in engineering. Internal flow

applications are mainly based on fluid flow and heat transfer investigations; however,

issues including velocity and temperature gradients and viscous effects are also the main


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250 A.A. Ozalp

concerns due to their direct contribution in energy losses that induces entropy generation

in a thermodynamic system.Experimental and numerical investigations of heat transfer and fluid movement in

laminar viscous flows with surface roughness has attracted considerable attention over

the past few years. Wu and Cheng (2003) reported that the laminar Nusselt number (Nu)

and apparent friction coefficient increase with the increase of surface roughness,

moreover the increase rates were determined to become more obvious at larger Re.

Engin et al. (2004) determined significant departures from the conventional laminar flow

theory in microtube flows due to the wall roughness effects. Kandlikar et al. (2003)

studied the effects of surface roughness on pressure drop and heat transfer in circular

tubes and indicated a transition to turbulent flows at Re values much below 2300 during

single-phase flow in channels with small hydraulic diameters. Kohl et al. (2005)

conducted experiments to investigate discrepancies in previously published data for the

pressure drop in circular channels. Obot (2002) prepared a literature review on frictionand heat/mass transfer in micro-channels. According to the available literature on

micro-channels, the arbitrary definition of micro-channels is given with a hydraulic

diameter ofDh1000 m (1 mm). The main records are:

onset of transition to turbulent flow in smooth micro-channels does not occurif the Re is less than 1000

Nu varies as the square root of the Re in laminar flow.

Pressure drop for liquid flow through short microtubes were experimentally considered

by Phares et al. (2005). Wen et al. (2003) experimentally investigated the characteristics

of the augmentation of heat transfer and pressure drop by different strip-type inserts in

small tubes; they examined the effects of the imposed wall heat flux, mass flux and strip

inserts on the measured augmentative heat transfer. Vicente et al. (2002) presented theexperimental results carried out in dimpled tubes for laminar and transition flows; where

roughness determined to accelerate transition to critical Re down to 1400 and the

roughness-induced friction factors were 1030% higher than the smooth tube ones.

Guo and Li (2003) reported that surface friction, provoked by surface roughness, makes

the fluid Velocity Profiles (VP) flatter, leads to higher friction factors and Nu and is

responsible for the early transition from laminar to turbulent flow. Numerical and

experimental investigations of Koo and Kleinstreuer (2004) pointed out the significance

of viscous dissipation on the temperature field and on the friction factor. Celata et al.

(2006a, 2006b) indicated the role of surface roughness on viscous dissipation and the

resulting earlier transitional activity, increased friction factor values and head

loss data in temperature measurements. Morini (2005) worked on the role of the

cross-sectional geometry on viscous dissipation and the minimum Re for which viscous

dissipation effects cannot be neglected.

Since surface roughness constituted a major part in heat transfer and fluid flow

studies, some particular work concentrated on the roughness definition and roles of

roughness on the flow and heat transfer performances of various applications; such as

Cao et al.s (2006) non-equilibrium molecular dynamics simulation to investigate the

effect of the surface roughness on slip flow of gaseous argon, Pretot et al.s (2000)

natural convection investigation above a horizontal plate with various roughness

amplitudes and periods, Sheikh et al.s (2001) model to eliminate the discrepancy in the

fouling measurements by characterising the fouling as a correlated random process,


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Entropy analysis of laminar-forced convection 251

Wang et al.s (2005) regular perturbation method to investigate the influence of

two-dimensional roughness on laminar flow in micro-channels between two parallelplates, Sahin et al.s (2000) study on entropy generation due to fouling as compared to

that for clean surface tubes and Ozalps (2006a, 2006b, 2006c) numerical approach for

compressible flow in constant area ducts and converging nozzles with various surface

roughness conditions.

There have been a number of studies on the entropy generation in internal fluid

flow-related problems. Demirel and Kahraman (2000) showed the volumetric entropy

generation maps indicating the regions with excessive entropy generation due to

operating conditions or design parameters. Ratts and Raut (2004) employed the entropy

generation minimisation method to optimise a single-phase, convective, fully developed

internal flow with uniform and constant heat flux and obtained optimal Re for laminar

and turbulent flows. Sahin (1998) analytically investigated entropy generation for a fully

developed laminar viscous flow in a duct subjected to constant wall temperature anddetermined that for low heat transfer conditions the entropy generation due to viscous

friction becomes dominant and the dependence of viscosity on temperature becomes

essentially important to be considered in order to determine the entropy generation

accurately. Lin and Lee (1998) performed second-law analysis on wavy plate

fin-and-tube heat exchangers to define the effects of the fin tube spacing along

spanwise direction on the second-law performance. Zimparov (2002) reviewed the

passive heat transfer augmentation techniques to conserve the useful part of energy

(exergy) of single-phase flows.

Although numerous work exist on heat transfer and fluid flow in circular ducts,

combined effects of heat flux and surface roughness on the energy and exergy

mechanism of laminar flow were not considered. The purpose of the present study is to

perform energy and exergy analysis of laminar-forced convection in a micro-pipe with

wall roughness. Heat transfer and fluid flow in roughness-induced viscous laminar

regime are discussed through friction coefficient, mass flow rate, discharge coefficient

and Nu. To construct a base for second-law investigations, velocity and Temperature

Profiles (TPs) are presented for various heat flux, surface roughness and Re cases. Effects

of viscous dissipation are identified through energy loss and mean-temperature variations

in the flow direction. Entropy generation is explained by crosscorrelations with mass

flow rate, temperature rise, Re, friction force and Nu. Variations of frictional and thermal

entropy with respect to each other are compared for a set of surface roughness and Re

scenarios. Additionally, radial distributions of frictional and thermal entropy generations

are given in terms of Be.

2 Physical model, governing equations and computational method

2.1 Physical model

Figure 1 shows the micro-pipe analysed in the present paper. The diameter and length of

the geometry are D and L, respectively; the Re and Nu for the current problem are

defined as follows:

Re o o oT T

U D U D

= = (1a)


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252 A.A. Ozalp

/

Nur R

T

s of

T r DhD

T T

=

= = (1b)

where oand Uoare the average density and velocity of the flow at any cross-section ofthe duct; and stand for the kinematic and dynamic viscosity values with theconversion formula of = /. As the surface and mean flow temperatures are denotedby Ts and To; thermal conductivity and convective heat transfer coefficient are

characterised by f and h. Air has been selected as the working fluid in the present study,and the compressible character is handled by the ideal gas formula of equation 2(a).

It is well known that air properties, like specific heat at constant pressure (Cp), kinematic

viscosity () and thermal conductivity (f), are substantially dependent on temperature(Incropera and DeWitt, 2001); to sensitively implement the property () variations withtemperature into the calculations, the necessary air data of Incropera and De Witt (2001)

are fitted into sixth-order polynomials, which can be presented in closed form by

equation 2(b). The uncertainty of the fitted air data is less than 0.02%, the temperature

dependency is indicated by the superscript T throughout the formulation, where the

curve fitting constants of equation 2(b) are given in Table 1 for the temperature range of

1003000 K.

RT

P= (2a)

6

0

T j

jja T

== (2b)

4( ) 1 .

2.31

if z z

=

(2c)

Figure 1 (a) Schematic view of micro-pipe and (b) triangular surface roughness distribution

Table 1 Curve fitting constants of equation 2(b) for the temperature range of 1003000 K

Cp (J/kgK) v (m2/s) kf(W/mK)

a0 1.07072E+03 1.317131E07 3.98988E03

a1 5.68284E01 3.68255E09 1.35324E04

a2 1.55698E03 2.41330E10 1.28935E07

a3 1.48088E06 2.15571E13 7.74887E11

a4 7.31876E10 1.35515E16 4.72883E15

a5 1.93047E13 4.09980E20 1.09770E17

a6 2.32650E17 4.78222E24 3.02173E21


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Among the several definitions for surface roughness available in the literature,

the present model is based on the triangular structure of Cao et al. (2006) (Figure 1(b)),where the roughness amplitude and period are characterised by and , respectively.To investigate the role of surface roughness on the heat and energy transfer performance

of laminar flow, amplitude values are varied for = 150 m; however, roughnessperiodicity parameter ( = /) is kept fixed to = 2.31, which corresponds toequilateral triangle structure (Cao et al., 2006), in all computations. With the

implementation of the amplitude and period, the model of Cao et al. (2006) is

numerically characterised by equation 2(c). The Kronecker unit tensor (i) attains thevalue of i= +1 for 0 z(2.31/2)and i= 1 for (2.31/2)z2.31, and the modelfunction (f(z)) is repeated in the streamwise direction throughout the pipe length.

2.2 Governing equations

The problem considered here is steady (/t) = 0, fully developed and the flow direction

is coaxial with pipe centreline ( 0),rU U= =

thus the velocity vector simplifies to

( , )zV U r z =

denoting Uz/= 0. These justifications are common in several recent

numerical studies, on roughness-induced flow and heat transfer investigations, like those

of Engin et al. (2004), Koo and Kleinstreuer (2005) and Cao et al. (2006). With these

problem definitions and above implementations, for a laminar compressible flow with

variable fluid properties, continuity, momentum and energy equations are given as

( ) 0zUz

=

(3)

1( )z zzz rz

U PU rz z r r z

+ = +

(4)

1( )

1( )

z zz r zz

z zzrz z rz

Q UPe k U rQ

z r r z z

UU r

r r r z

+ + + + =

+ + +

(5)

where internal and kinetic energy terms are defined as e= CvT and2 / 2zk U= ,

respectively. Components of the viscous stress tensor () and heat flux terms (Q) can be

written as

Tzrz

U

r

= (6a)

4

3

T z

zz

U

z

=

(6b)

T

r f

TQ

r

=

(7a)

.Tz fT

Qz

=

(7b)


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254 A.A. Ozalp

The flow boundary conditions are based on the facts that, on the pipe wall no-slip

condition and constant heat flux exist and flow and thermal values are maximum at thecentreline. At the pipe inlet, pressure and temperature values are known and the exit

pressure is atmospheric. Denoting Uz= U(r,z) and T= T(r,z), the boundary conditions

can be summarised as follows:

in in

( ) 0 and 0 0

( ) and 0 0

0 , and 0 (Manometric).

r

T

f

Ur R f z U r

r

QT Ur R f z r

r r

z P P T T z L P

= + = = =

= + = = =

= = = = = (8)

The average fluid velocity and temperature and mass flow rate are obtained from

0

2

2 ( ) ( ) dr Rr

o

o

r U r r r U

R

=

== (9a)

0

2

2 ( ) ( ) ( ) ( ) d

( )

r R

pr

o

o o p o

r U r C r T r r r T

U C R

=

==

(9b)

02 ( ) ( ) d

r R

o or

m U A r U r r r =

== = (9c)

and the shear stress and pipe friction force are defined as

21 d

2 d

T

f o o

r R

uC U

r

=

= = (10a)

0d .

z L

fz

F D z =

== (10b)

The non-equilibrium phenomenon of exchange of energy and momentum within the

fluid and at the solid boundaries result in entropy generation, which is directly

proportional to the lost available work; this concurrence is known as the Gouy-Stodola

theorem. For a two-dimensional (r, z) compressible Newtonian fluid flow in cylindrical

coordinates, the local rate of entropy generation per unit volume (S ) is symbolised by

equation 11(a). As given in equation 11(b), the entropy generation due to finite

temperature differences ( )TS in axialzand in radial rdirections is defined by the first

term on the right side, the second term stands for the frictional entropy generation ( ).PS

Computation of the temperature and the velocity fields through equations (3)(5) on theproblem domain, will produce the input data for equation (11).

2 2 2 2

22

T Tf T T U U

Sr z T r z T

= + + +

(11a)

.T PS S S = + (11b)

Due to the existence of the velocity and temperature gradients in the flow volume,

the volumetric entropy generation rate is positive and finite. As the total and volumetric


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average entropy generation rates can be obtained by equation 12(a) and (b), respectively,

the average volumetric entropy generation, in non-dimensional form, is evaluated byequation 12(c).

0 02 d d

z L r R

z rS S r r z

= =

= == (12a)

2

SS

R L= (12b)

2

2

T

fTJ S

Q

= (12c)

Be .TS

S

=

(12d)

Petrescu (1994) and Bejan (1996) defined the dimensionless parameter of Be. The Be

(equation 12(d)) compares the magnitude of entropy generation due to heat transfer to the

magnitude of the total entropy generation. While Be > 0.5 the entropy generation due to

heat transfer dominates, for Be < 0.5 the entropy generation due to friction leads. When

the thermal and frictional generation rates are comparable Be becomes 0.5.

2.3 Computational method

Forward difference discretisation is applied in the axial and radial directions, for the

two-dimensional marching procedure. The flow domain is divided into m axial and

nradial cells (mn), where the fineness of the computational grids is examined to ensurethat the obtained solutions are independent of the grid employed. Initial runs indicated an

optimum axial cell number of m= 500, having an equal width of z, whereas, the radialdirection is divided into n= 100 cells. Since the velocity and temperature gradients are

significant on the pipe walls, the 20% of the radial region, neighbouring solid wall,

is employed an adaptive meshing with radial-mesh width aspect ratio of 1.1. The laminar

micro-pipe flow with surface roughness and heat flux governs the complete equation set

described above; however, for simultaneous handling they need to be assembled into the

three-dimensional Transfer Matrix, consisting of the converted explicit forms of the

principle equations. The sufficiently complex structure involves the highly dependent

non-linear formulations, where the solution scheme is most likely to face with

convergence problems and encounter singularities. Wu and Tseng (2001) applied Direct

Simulation Monte Carlo (DSMC) method to a micro-scale gas dynamics problem.

Since the influences of surface roughness and surface heat flux conditions are coupled

over the meshing intervals of the flow domain, DSMC implementation is an utilised

technique especially for gas flow applications with instabilities and for internal flow

problems. The benefit of DSMC becomes apparent when either the initial guesses on inlet

pressure and inlet velocity do not result in convergence within the implemented mesh or

if the converged solution does not point out the desired Re in the pipe. There exists two

types of convergence problems (singularities) such that:


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256 A.A. Ozalp

Mach number exceeds 1 inside the pipe

the exit Mach number is lower than or equal to 1 but the exit pressure deviates fromthe related boundary condition more than 0.01 Pa. (equation (8)).

Additionally, the Transfer Matrix scheme and to the DSMC algorithm are supported

by cell-by-cell transport tracing technique, which in return requires much less

specific change in programming and enables the application of different types of

boundary conditions. In order to perform accurate simulation for inlet/exit

pressure boundaries and to sensitively evaluate the balance of heat swept from the

micro-pipe walls and the energy transferred in the flow direction, the concept of triple

transport conservation is incorporated into the DSMC, where the resulting non-linear

system of equations is solved by using the Newton-Raphson method. The streamwise

variations of the three primary flow parameters (U,P, T) are investigated. In the case of a

convergence problem, U, P and T are investigated up to the singularity point,and then the local velocity is compared with that of the inlet value together

with the location of the singularity point wrt to the inlet and exit planes. DSMC method

considers the velocity variation and the corresponding pipe length, and modifies

the inlet velocity by considering the type of singularity. But if the Re does not

fit the required value then both the inlet pressure and velocity are modified in

order to either increase or decrease the Re, evaluated at the former iteration step.

Thermal equilibrium at each pressure boundary within the mesh is satisfied by

simultaneously conserving mass flux and boundary pressure matching. The convergence

criteria for the mass flow rate throughout the flow volume are in the order of 0.01% and

the deviation of exit pressure from the related boundary condition (equation (8)) is less

than 0.01 Pa.

3 Discussion

To comprehensively investigate the influences of surface roughness and heat

flux conditions on the second law of thermodynamics and momentum and energy

transport mechanisms of the fully developed laminar flow, computations are carried

out in a micro-pipe having a diameter of D= 1 mm, which is consistent with the

micro-channel definition of Obot (2002), where the pipe length is selected as L= 0.5 m.

As the inlet temperature is fixed to Tin= 278 K, computations are performed for

Re = 12500 and Q= 5100 W/m2. Surface roughness amplitude is varied within

the range of = 150 m (* = /D= 0.0010.05), where the * values are similar tothose of Wang et al. (2005) (* = 0.0050.05), Engin et al. (2004) (* 0.08) and

Sahin et al. (2000) (* 0.25). Since velocity and TPs are known to have considerablerole on entropy generation, numerical discussions are based on profile

structures, momentum and heat transfer characteristics. Detailed discussions are

presented on the radial and overall entropy generation mechanism together with

crosscomparisons among momentum and heat transfer values with those of the

second law.


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3.1 Velocity and Temperature Profiles (TPs)

Being the main designator of momentum transfer in the flow direction, VP are

presented for three Re and non-dimensional surface roughness cases in Figure 2(a).

Computations put forward that the applied heat flux on the lateral solid walls

of the microduct had no influence on the VP; thus Figure 2(a) displays the combined

impact of Re and * on the VP development in comparison with the laminar profile(equation 13(a)) and the turbulent logarithm law that is modified for roughness as

given by equation 13(b) (White, 1988), where U* is the friction velocity defined by

equation 13(c). The numerical results pointed a maximum Mach number ofM= 0.103 for

the upper Re limit of Re = 2500, where the highest density variation in the flow

direction is 3.1%. Thus the VP, given in Figure 2(a), represent the flow characteristics

for the complete pipe length. Moreover to describe the flow regime, shape factors

(H) are evaluated by equation 14(a) and compared with the characteristic laminar(Hlam= 3.36) and turbulent (Hturb= 1.70) data, resulting in the intermittency ()values (equation 14(b)). Since friction has direct contribution on entropy generation,

friction coefficient (Cf) is evaluated by equation 15(a), compared with the

traditional laminar formula (equation 15(b)) and presented in normalised form as Cf*

(equation 15(d)).

2( )

2 1o

U r r

U R

=

(13a)

( )2.44ln 8.5

*

U r R r

U

= +

(13b)

* wU

= (13c)

0

0

( )1 d

( ) ( )1 d

r R

rm

r R

rm m

U rr

UH

U r U r r

U U

=

=

=

=

=

(14a)

lam

lam turb

H H

H H

=

(14b)

2

d2

d

T

r R

f

o o

U

rCU

== (15a)

lam

16( )

RefC = (15b)

0.25

turb( ) 0.079RefC = (15c)

lam

* .( )

f

f

f

CC

C= (15d)


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258 A.A. Ozalp

Figure 2 (a) Velocity Profiles (VPs) for various Re and * cases and (b) Temperature Profiles

(TPs) for various Re, * and Qcases


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It can be seen from Figure 2(a) that at Re = 500, the lowest surface roughness of

* = 0.001 had no particular influence to distinguish the profile from that of the laminar;indeed the shape and intermittency factors attained the values of 3.29 and 0.041 implying

apparently the laminar character. However, the corresponding values for Re = 1000 and

Re = 1500 for * = 0.001 are evaluated as H= 3.22 and 3.16 and = 0.084 and 0.121showing the increasing influence of even an inconsiderable surface roughness value at

43% and 65% of the critical Re of 2300. Additionally, the *fC values showed also

augmentation with Re for the same * of 0.001 such that *f

C is evaluated as 1.028, 1.061

and 1.088 for Re = 500, 1000 and 1500, respectively. On the other hand, the highest

surface roughness in the computations (* = 0.05) caused significant and clear shifts inthe VP from that of the laminar with lowerHand higher and *

fC values even at the low

Re of 500; whereas, the impact of * on the flow pattern becomes impressive at thehigher Re of 1500 withH= 2.76, = 0.362 and * 1.35.fC = These values indicate that a

relatively high * of 0.05 can turn the flow into almost 36% transitional at the 65% of thecritical value of Re = 2300 where the corresponding friction coefficient exceeds the

laminar approach by 35%. The VPs of Figure 2(a), especially the near-wall regions

(r/R 0.75) and the centreline (r/R= 0) put forward the growing influence of surfaceroughness on the flow pattern with Re. Since friction factor is a consequence of the

velocity gradient at the wall ((dU/dr)|r=R), the growth of Cfand *fC values is an outcome

of the fuller VP with higher * and Re. Figure 2(a) additionally implies the role ofsurface roughness in laminar flow by putting forward the VP transformation from laminar

to the initial stages of transitional character even at Re = 1500 with higher Cfand andlower H and Uc/Uo values. Furthermore, the gap between the Uc/Uo ratios and the

traditional data of Uc/Uo= 2.0 (equation 13(a)) increases both with higher * and Re,which also strengths the determinations on transition.

TP development is based not only on the surface heat transfer rates but also on the

amount of energy loss (loss) due to viscous dissipation on the solid walls. Figure 2(b)displays the radial variations of temperature values, in non-dimensional form wrt to the

centreline value (Tc), for various heat flux, surface roughness and Re. The characteristics

are compared with the laminar constant heat flux formula of equation 16(a), where isthe thermal diffusivity (equation 16(b)).

4 222 d 3 1 1( )

d 16 16 4

o o

s T

U R T r rT r T

z R R

= +

(16a)

.

T

fT

T

pC

= (16b)

It can be seen from the figure that roughness implementation has no effect on the TP forthe low Re of Re = 500. It can additionally be seen that as the surface temperature (Ts)

values of the cases for Q= 50 and 100 W/m2are above the Tc, the contrary is true for

Q= 5 W/m2. The condition of Ts> Tc is expected for heat addition process where the

opposite evaluation is an outcome due to the frictional energy losses. Although heat is

added through the lateral walls of the duct with a flux of 5 W/m2, this supply is not

sufficient enough to meet the amount of energy loss due to friction. Table 2 shows the

applied flux and the corresponding total surface heat transfer values (Qs), together with

the amount of viscous energy loss (loss) data for three Re and also for the surface


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260 A.A. Ozalp

roughness range of 1 6*** 0.001( ) 0.05( ). = The total heat added through the duct

walls (Qs = 0.007854 W) is lower than the energy loss values for Re = 500 for thecomplete * range (loss= 0.0111 0.0121 W), where the gap between the Qsand lossvalues rise in the cases with higher Re. The tabulated data are projected to the plots with

significantly deviated curves from the CHF profile of equation 16(a) for Re = 1000 and

1500 at Q= 5 W/m2. The particular case of Re = 500 also shows that, although Ts> Tc

condition is valid for Q= 50 and 100 W/m2, the profiles as well imply divergence from

equation 16(a) showing significantly lower Tsvalues when compared with the analytical,

which is also a consequence of the frictional behaviour. Table 2 also indicates that, lossvalues rise with heat flux: as the velocity gradient is known not to vary with flux

(Figure 2(a)), the augmentation of shear rates, at the identical Re, can be attributed to the

augmented viscosity values at higher temperature values, which is an outcome of the

elevated heat flux values. Energy loss and temperature decrease due to friction in laminar

flow was also reported in recent work by Koo and Kleinstreuer (2004), Morini (2005)and Celata et al. (2006a, 2006b). Their common findings are wall heating due to viscous

dissipation; the dissipated energy resulted in loss of flow temperature even the surface

roughness effects were disregarded. Additionally they also indicated that, viscous

dissipation is directly related with Re where they experimentally and numerically

recorded exponential augmentations in energy loss at high Re. These results show

harmony with the present evaluations on heating/cooling TPs of Figure 2(b) and the

energy loss data of Table 2. Moreover, Ozalp (2006b, 2006c) also determined power loss

in compressible high speed converging nozzle flows due to the frictional activity. With

the increase of Re (Re = 1000 and 1500) the impact of * on TP becomes moreapparent, which can be explained by the expanded and augmented loss ranges ofTable 2. Figure 2(b) puts forward that the moderate flux of Q= 50 W/m

2is not sufficient

enough to suppress the lossvalues at Re = 1500, which resulted in the cooling contourTP. Figure 2(b) and Table 2 mutually propose that heat flux addition must be considered

by also inspecting the amount of viscous dissipation in the flow domain.

Table 2 lossvalues for various Re, * and Qcases

Q (W/m2) 5 50 100

Qs(W) 0.007854 0.07854 0.15708

loss(W) 1 6* *( )

Re = 500 0.0111 0.0121 0.0117 0.0127 0.0124 0.0134

Re = 1000 0.0468 0.0549 0.0480 0.0563 0.0494 0.0578

Re = 1500 0.1109 0.1406 0.1128 0.1429 0.1150 0.1454

3.2 Momentum and heat transfer

3.2.1 Friction coefficient (Cf)

Friction coefficient is not only an indicator parameter for frictional losses but also

an outcome of the velocity gradient at the solid wall that were discussed in

the previous section. Since the viscous dissipation plays a significant role on the entropy

generation, Cf gains higher importance especially due to its direct contribution

on the non-dimensional parameter of Be (equation 12(d)). Computational Cf values,


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for various * cases, are presented in Figure 3(a), in conjunction with the traditional

laminar theory (equation 15(b)), turbulent smooth pipe approximation (equation 15(c))and also with the available literature-based data. It can be seen from the figure that,

for Re < 500 the friction coefficient values, for the complete set of * cases (0.0010.05),are in conjunction with the laminar theory, which indicates that surface roughness

does not create a variation in the frictional characteristics of laminar flow. Indeed

the experimental data of Kohl et al. (2005), Choi et al. (1991) and Wu and Little (1983)

as well show harmony with not only the current Cf determinations but also with

equation 15(b) for Re < 500. This coinciding appearance does also verify the highly

similar VP of Figure 2(a) (Re = 500), for various * cases, with that of thelaminar. Current numerical investigations put forward shifts in Cfdata from the laminar

curve for Re > 500 indicating that the influence of surface roughness becomes

apparent with higher Re, which was also recorded by Guo and Li (2003) for microscale

flow. As the friction factor values for * = 0.001 and * = 0.05 are above the laminartheory by 6.1% and 23.1% at Re = 1000, these ratios rise to 8.8% and 34.5% at

Re = 1500 designating apparently the onset of transition. These evaluations are

similar to Vicente et al.s (2002) reports on roughness-induced friction factor

augmentation and earlier transition. Wang et al. (2005) and Engin et al. (2004)

as well reported that the role of surface roughness on friction coefficient expands

with Re. The augmentation in Cf values, together with higher flow velocity, is a

serious indicator of pressure losses and so occurring entropy rise within the flow

volume. As can be seen from Figure 3(a) that most of the available experimental

studies on laminar flows, available in the literature, reported higher friction

coefficients than those of equation 15(b) for Re > 500 (Kohl et al., 2005; Choi et al.,

1991; Wu and Little, 1983). However, the experimental friction factors of Yu et al.

(1995) were even below the laminar theory for Re < 2000. On the other hand, the

experimental records of Jiang et al. (1995), in the microchannel heat exchanger, were

considerably shifted from the laminar data and they fitted their experimental

data resulting in the analytical correlations of equation 17(a) and (b) for Re < 600

and Re > 600, respectively.

1.48410Ref

C = (17a)

0.551.36Re .f

C = (17b)

They reported that the onset of transition was around Re 600 forDh= 0.3 mm and forthe * in the range of 0.020.12. They explained this significantly early occurrence withtwo reasons:

due to microchannel behaviour

because of the sufficiently high surface roughness data.

Actually the present findings show transitional behaviour at Re > 1500 for a larger

diameter of 1 mm and lower * range, which as a result show harmony with the findingsof Jiang et al. (1995) on transition mechanism with low Re.


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262 A.A. Ozalp

Figure 3 (a) Cfvs. Re; (b) m vs.Pinand (c) Nu vs. Re for various * cases

(a)

(b)


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Figure 3 (a) Cfvs. Re; (b) m vs.Pinand (c) Nu vs. Re for various * cases (continued)

(c)

3.2.2 Mass Flow Rate ( )m

Amount of mass flow rate ( )m is directly related with the flow velocity (equation 9(c)),

thus Re and the main influencing parameter, for a fixed pipe cross-section and inlettemperature, is the inlet pressure (Pin). Figure 3(b) displays the m variation withPinfor

various * cases. The augmented Cfvalues of Figure 3(a) due to higher surface roughness

values resulted in increased friction forces counteracting the flow direction. The resistive

force resulted in lower mass flow rates at the same inlet pressures, where the difference

becomes more considerable in cases with higher m and Re, which is completely in

harmony with Cf vs. Re variations of Figure 3(a). Figure 3(b) additionally puts

forward that the computed m data are considerably lower than the isentropic operation

values due to the mentioned frictional behaviour. Defining discharge coefficient (Cd) with

equation 18(a) results in the Cdvalues of Table 3 for various Re, surface heat flux and

roughness scenarios.

real

isend

m

C m=

(18a)

0.5

0.51 Re8

d

DC

L

=

(18b)

0.5

.2

d

mC

A P

=

(18c)


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264 A.A. Ozalp

Table 3 Cdvalues for various Re, * and Qcases

Cd

Computational 1 6**( )

Q(W/m2) 5 50 100 Eq. 18(b) Eq. 18(c)

Re = 500 0.1237 0.1190 0.1216 0.1171 0.1195 0.1152 0.1250 0.1223

Re = 1000 0.1735 0.1618 0.1722 0.1607 0.1707 0.1594 0.1768 0.1696

Re = 1500 0.2110 0.1919 0.2100 0.1911 0.2089 0.1902 0.2165 0.2037

Re = 2000 0.2423 0.2160 0.2414 0.2154 0.2405 0.2147 0.2500 0.2307

Re = 2500 0.2696 0.2369 0.2689 0.2364 0.2681 0.2359 0.2795 0.2529

The tabulated data indicate not only that the Cdvalues increase with Re but also the Cdranges expand in higher Re cases revealing the enhanced role of * on the flow ratevalues. Ozalp (2006b, 2006c) reported similar findings on the Cd- relation in

compressible converging nozzle flows. On the other hand, the present numerical

values are also compared with the analytical solutions of Phares et al. (2005)

(equation 18(b)) and Sahin and Ceyhan (1996) (equation 18(c)). It can be seen that,

although quite close, the proposed equation of Phares et al. (2005) deviate from that of

Sahin and Ceyhan (1996) for Re > 1250 and Pin> 7.5 kPa. However, either of the

methods show significant similarities with the present evaluations for Re < 1000;

besides at higher Re especially the solution of equation 18(c) deviate from the

computational values. The proposed equation of Phares et al. (2005) appears to be

consistent with the numerical results for the highest roughness scenario of * = 0.05

for the complete Re range. The inconsistency of equations 18(b)(c) with the low* cases can be explained by the facts that: neither of the analytical solutions involvesurface roughness as an effective parameter on the momentum rates; moreover, in the

present work the flow turns out to be of transitional type for Re > 1500. Additionally,

Table 3 interprets the slight decrease of Cd data with heat flux values, which is a

consequence of higher viscosity, frictional forces and power loss values as discussed

through Table 2.

3.2.3 Heat Transfer (Nu)

Surface heat transfer values, in terms of Nu, are displayed in Figure 3(c) for various

Re and * cases. It can be seen that up to Re 100 neither flow velocity nor surfaceroughness produced a sensible effect on heat transfer rates and Nu attained the value of

~4.04 being between the characteristics markers of Nu = 4.36 (constant heat flux) andNu = 3.66 (constant surface temperature). Whereas for Re = 100, computations indicated

a Nu of 4.08, which is observed to augment with Re. Likewise, the heat transfer

measurements of Wen et al. (2003), for a D= 2 mm pipe, and Vicente et al. (2002), for a

D= 16 mm pipe, pointed out constant Nu of ~4 for Re < 300 (Wen et al., 2003) and Nu

of ~4.36 for Re < 700 (Vicente et al., 2002), after that an increase of Nu was recorded in

both of the studies. Since enhancement of heat transfer rates is an indicator for

transitional regime, the numerical and the experimental data of Wen et al. (2003) and


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Vicente et al. (2002) put forward that in small scale pipes transition onset is reversely

proportional to pipe diameter and can be reached at comparably lower Re than thetraditional value of Re = 2300. Figure 3(c) additionally reveals that the impact of surface

roughness on heat transfer mechanism becomes apparent for Re > 100; computations put

forward that elevates Nu, where this influence is determined to grow with Re; such thatthe Nu* = 0.05/Nu* = 0.001ratio attains the values of 1.086, 1.168 and 1.259 for Re = 500,

1000 and 1500, respectively. Kandlikar et al. (2003) also experimentally recorded

augmentations in Nu with surface roughness for the * range of 0.00180.0028, and Wuand Cheng (2003) experimentally reported the increased role of on Nu at higher Re.The experimental data of Kandlikar et al. (2003) (for Re 500), Obot (2002)

(for Re 1000) and Wu and Little (1983) (for Re 1000) are reasonably in harmony

with the current numerical outputs; however, Wu and Little (1983) reported lower heat

transfer rates than those of both the present work and other considered studies for

Re < 1000.

3.3 Entropy generation

Entropy generation is the main concern of second law of thermodynamics and the main

parameters, from the point of both momentum and heat transfer rates and also the

frictional behaviour, are the mass flow rate, flow temperature rise, Nu and friction force.

This section covers discussions of the entropy generation behaviour of the considered

laminar flow by cross-correlating the variation with other system parameters. Similar to

Demirel and Kahraman (2000) amount of entropy generation is given in non-dimensional

form (equation 12(c)) in Figure 4 and also per unit volume basis (W/m3K) in Table 4 for

various *, Qand Re cases.

Table 4 S values for various Re, * and Qcases

3(W/m K)S

Q(W/m2) 5 50 100

Re 30 2500 30 2500 30 2500

* = 0.001 0.37 2776.98 0.93 2789.68 1.79 2805.15

* = 0.01 0.37 2886.09 0.93 2898.94 1.79 2914.62

* = 0.02 0.37 2954.81 0.93 2967.16 1.79 2982.23

* = 0.03 0.37 3047.03 0.93 3058.95 1.79 3073.93

* = 0.04 0.37 3153.29 0.93 3165.33 1.79 3179.94

* = 0.05 0.37 3269.66 0.93 3281.78 1.79 3296.46

Figure 4(a) displays the variation of non-dimensional entropy generation with mass flowrate. It can be seen from the figure that if the influence of m on Jis cross-correlated,

except for the high Re of Re > 2300, effect of surface roughness on the entropy values is

ignorable, being independent of the level of heat flux. This fact can also be seen in

Table 4, where S data are identical for the * range of 0.0010.05 at the lower limit


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266 A.A. Ozalp

(Re = 30) of the tabulated Re range. However, at the upper limit (Re = 2500) the ratio of

* 0.05 * 0.001/S S = = attains the similar value of 1.18 for the complete imposed heat fluxes,

which points out an increase of up to 18% in the high Re cases. Indeed, the

roughness-induced augmentation ofJis also exposed in Figure 4(a) through the zoomed

plots for Re = 2500. Figure 4(a) additionally implies that there exist two Re intervals

regarding the role of mass flow rate on entropy generation. Up to a exergo-critical Re((Recr)ex),Jdecreases with m where these critical limits are found to increase with heat

flux such that computations revealed these boundaries as (Recr)ex6, 20 and 40 for Qof

5, 50 and 100 W/m2, respectively. Above these limits there exists direct proportion

amongJand m , which puts forward that entropy generation grows with mass flow rate.

Ratts and Raut (2004) also determined minimum entropy generation for certain Re both

in laminar and turbulent flows. Additionally, the growth of entropy with mass flow rate

was also reported by Lin and Lee (1998) for heat exchangers.

Relation of entropy generation with temperature rise (T) is plotted in Figure 4(b).For all heat flux applications, J values evaluated at higher T cases (low Re) are

significantly lower than those of the low Tflows (high Re). This outcome is completely

in harmony with the discussions on J m relation of Figure 4(a). Additionally, the

exergo-critical Re are also indicated in Figure 4(b), where J shows different variation

characters with Tas well. The roughness effect on Jis inspected by the zoomed plots

for Re = 2500, indicating emerging role of * on J at high Re. But, the zoomed plots

further put forward that the cooling rates, due to the frictional losses are higher at the low

Qof 5 W/m2with the T range records of 8.8oC (* = 0.001) to 12.6oC (* = 0.05).

For the high Qof 100 W/m2, although still cooling exits, these Tlimits rise to 5.2oC

and 9.2oC. Besides theJ T interrelation, with the aid of the zoomed plots, Figure 4(b)

interprets the augmented role of the frictional behaviour on the cooling activity,where these records show harmony with the discussions through Figure 2(b). Since

the generated entropy is composed of two terms, :TS the thermal part and :PS

the frictional part, a deeper survey by decomposing the overall generation rate

into sub-sections will be helpful in understanding the entropy mechanism. Figure 5

displays the variation of the thermal and frictional parts of entropy generation on a

logarithmic plot. The effects of * come into sight towards the end of heating period, thus

as augmentations are observed in ,PS comparably minor reductions are determined in

TS data. Although the onset of the defined region varies, the mentioned character is

valid for all heat flux applications. PS values are computed to increase continuously

with Re; since PS is directly associated with velocity, mass flow rate and Re,

augmentations become more recognisable in higher Re. As * creates a decreasing

impact on TS around the thermally criticalRe ((Recr)th400, 1300 and 1700 for Q= 5,

50 and 100 W/m2, respectively), above (Recr)thsurface roughness augments the TS data.

Figure 5 additionally displays that below (Recr)ex, TS values are comparably higher than

.PS The reason can be explained with the significantly high Tand heat transfer-based

entropy generation ( )TS values; and also because of the low frictional forces and

frictional activity-based entropy generation ( )PS due to the slow flow velocities, where

these can also be seen in Figure 4(b). Ratts and Raut (2004) also indicated augmented

thermal entropy generation for Re < (Recr)exand elevated frictional entropy generation for


19/27


Re > (Recr)ex. Supporting these determinations, Table 4 interprets the significant increase

rates in the S values at Re = 30 with higher heat flux applications, that is mainly based

on the rise of the .TS As the ratio of 50 5/Q QS S= = is 2.51 for Re = 30, it decreases to

~1.0045 for Re = 2500. The minor ratio at Re = 2500 is due to the slightly higher

temperature values (~2C Figure 4(b)) at Q= 50 W/m2, that causes minor grows in flow

viscosity, frictional activity and .PS Figure 5 shows as well that, besides the

augmentations in either of the TS and PS around Re 2500, PS becomes dominant to

TS by occupying the major portion in the overall entropy generation ( ).S Figure 5

covers essential information not only for the clarification of the (Recr)ex occurrence in

Figures 4(a) and (b), but also for the explanation of the fact that the major fraction of the

total entropy generation is due to the frictional behaviour at the high m and Re and low

Tscenarios.

3.4 Bejan number profiles

Be, as defined by equation 12(d), is the ratio of the thermal-based entropy generation rate

( )TS to the total generation ( ).S As can also be seen in Figure 5, since * plays an

augmenting role on TS in cooling cases, in Figure 6 * results in increased Be for the

complete set of cooling scenarios (Q= 5 W/m2 and Re > 400, Q= 50 W/m

2 and

Re > 1300, Q= 100 W/m2 and Re > 1700) in the entire radial domain (0 < r/R< 1).

But, for heating the impact of * on Be is opposite for the centreline (r/R= 0) and

solid wall (r/R= 1) neighbourhoods. The reason can be clarified with the high

wall temperature values in heating conditions, which cause the flow viscosity values

to rise locally. As a consequence, the frictional activity in the regions close to

the pipe walls grows, causing the PS portion to increase in the total generation.

This structure becomes identifiable especially in the cases of Re = 5001000 and

Q= 50 W/m2 and Re = 10001500 and Q= 100 W/m

2. Besides, Figure 6 points

out that, the near-wall decreasing role of * on Be expands for fixed heat flux values

with Re; additionally, the radial region of the heating-based decreasing role, also

enlarges. Such that for Q= 50 W/m2, as the defined region is 0.77 r/R1 for Re = 500,

the corresponding one is 0.54 r/R 1 for Re = 1000; whereas, at the heat flux of

Q= 100 W/m2 for Re = 500, 1000 and 1500 these radial zones become 0.79 r/R1,

0.72 r/R1 and 0.41 r/R1, respectively. On the other hand, being independent of

Re, and Q, in the entire flow scenarios Be attains very close values to 1 at the

centreline (r/R= 0) of the pipe. This not only validates that the frictional activity ( )PS

in the centreline region is negligible, but also puts forward that the major portion of thetotal entropy generation is thermal based ( ).TS However, the opposite outcome is in

effect for the wall neighbourhood, where PS dominates the entropy generation

regardless of .PS


20/27

268 A.A. Ozalp

Figure 4 (a)Jvs. m and (b)Jvs. Tfor various * and Qcases


21/27


Figure 5 TS vs. PS for various * and Qcases


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270 A.A. Ozalp

Figure 6 Be profiles for various Re, * and Qcases


23/27


4 Conclusion

Laminar, compressible, temperature-dependent property continuity, momentum and

energy equations are solved for a forced convection micro-pipe flow with wall roughness

conditions. Computations were performed to obtain the contribution of surface heat flux

and wall roughness on the energy and exergy characteristics of laminar pipe flow.

The following conclusions are attained:

a relatively high * of 0.05 is determined to turn the flow into almost 36%transitional at the 65% of the critical value of Re = 2300, where the corresponding

friction coefficient exceeds the laminar approach by 35%

the impact of * on TPs become more apparent at high Re,which is an outcome of the augmented viscous dissipation values

the shifts in Cfand Nu data, from those of the traditional laminar values,for Re > 500 and Re > 100, respectively, indicate that the influence of surface

roughness on the frictional and heat transfer behaviours become noticeable

at higher Re

there exist exergo-critical Re, below which entropy generationis determined to decrease with mass flow rate; moreover, these critical limits are

found to increase with heat flux

the influence of both the heat transfer and the frictional activity on entropygeneration is evaluated to expand with mass flow rate and Re

the effect of surface roughness on entropy generation is ignorable at low Re,

however, at the upper Re limit of 2500 the ratio of * 0.05 * 0.001/S S = = attains 1.18,pointing out an increase of up to 18% due to in high Re cases

the effects of on TS and PS come into sight towards the end of heating period,

thus as augmentations are observed in ,PS comparably minor reductions are

recorded in TS data

as Re is increased to Re 2500, PS becomes dominant to TS

by occupying the major portion in the overall entropy generation

* causes augmentations in Be for the complete set of cooling scenarios in the entireradial domain, but for heating, the impact of * on Be is opposite for the centrelineand solid wall neighbourhoods.

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274 A.A. Ozalp

Nomenclature

a Curve fit constants

A Cross sectional area (m2)

Be Bejan number

Cd Discharge coefficient

Cf Friction coefficient

*fC

Normalised friction coefficient

Cv Constant volume specific heat (J/kgK)

Cp Constant pressure specific heat (J/kgK)

D Diameter (m)

e Internal energy per unit mass (J/kg)

f(z) Surface roughness model function

Ff Friction force (N)

h Convective heat transfer coefficient (W/m2K)

H Shape factor

J Non-dimensional entropy generation

k Kinetic energy per unit mass (J/kg)

L Pipe length (m)

m Mass flow rate (kg/s)

M Mach number

Nu Nusselt number

P Static pressure (Pa)

Q Surface heat flux (W/m2)

R Radius (m), gas constant (J/kgK)

Re Reynolds number

S Total entropy generation (W/K)

S Total volumetric entropy generation (W/m3K)

PS Frictional volumetric entropy generation (W/m3K)

TS Thermal volumetric entropy generation (W/m3K)

T Temperature (K)

U Axial velocity (m/s)

V

Velocity vector

Greek letters

Thermal diffusivity (m2/s)

Roughness amplitude (mm)

* Non-dimensional surface roughness (=/D)

f Thermal conductivity of fluid (W/mK)


27/27


Greek letters

Intermittency

Dynamic viscosity (Pa.s)

Density (kg/m3)

Shear stress (Pa)

Air properties

loss Energy loss (W)

Roughness period (mm)

Subscripts

c,s Center, surface

i Dimension

in Inlet

lam Laminar

m, o Maximum, mean

r, ,z Radial, peripheral, axial

turb Turbulent

Superscripts

T Temperature dependency

_ Volumetric average

entropy conv 15

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