Convective heat transfer and pressure drop in v corrugated

ORIGINAL

Convective heat transfer and pressure drop in V-corrugatedchannel with different phase shifts

Mohamed Sakr

Received: 12 September 2013 / Accepted: 24 June 2014

� Springer-Verlag Berlin Heidelberg 2014

Abstract New energy system development and energy

conservation require high performance heat exchanger, so

the researchers are seeking to find new methods to enhance

heat transfer mechanism in heat exchangers. The objectives

of this study are investigating heat transfer performance

and flow development in V-corrugated channels, numerical

simulations were carried out for uniform wall heat flux

equal 290 W/m2 using air as a working fluid, Reynolds

number varies from 500 to 2,000, phase shifts,

0� \ Ø \ 180�, and channel heights (S = 12.5, 15.0, 17.5

and 20 mm). Governing equations of flow and energy were

solved numerically by using finite volume method. The

numerical results indicated that, wavy (V-corrugated)

channels have a significant impact on heat transfer

enhancement with increase in pressure drop though chan-

nel due to breaking and destabilizing in the thermal

boundary layer are occurred as fluid flowing through the

corrugated surfaces and the effect of corrugated phase shift

on the heat transfer and fluid flow is more significant in

narrow channel, the goodness factor (j/f) was increased

with increasing channel phase shift, the best performance

was noticed on phase shift, Ø = 180� and channel height,

S = 12.5 mm.

Abbreviations

PRESTO Pressure staggering option

RNG Renormalized group

SIMPLE Semi-implicit method for pressure-linked

equations

UDF User defined function

List of symbols

A Height of the wavy (mm)

Cp Specific heat (J/kg K)

Ce Turbulent model constant

Cl Turbulent model constant

Dh Hydraulic diameter (mm)

f Friction factor

h Heat transfer coefficient (W/m2 K)

I Turbulent intensity

j Colburn factor [Nu/(Re Pr^(1/3))]

k Thermal conductivity of the fluid (W/m K)

L Length of the domain (mm)

_m Mass flow rate (kg/s)

P Wavy pitch (mm)

p Pressure (Pa)

Q Heat transfer rate (kW)

q Heat flux (W/m2)

Pr Prandtl number (Pr = Cpl/k)

Qin Heat input rate (W)

Re Reynolds number (Re = quavDh/l)

S Channel spacing (mm)

T Temperature

u Velocity component at x-direction (m/s)

v Velocity component at y-direction (m/s)

w Width (mm)

y? Dimensionless distance from the cell center to the

nearest wall

Greek symbols

d Height of the base (mm)

e Dissipation kinetic energy (m2/s3)

M. Sakr (&)

Civil Engineering, Architecture and Building (CAB),

Faculty of Engineering and Computing, Coventry University,

Coventry CV1 5FB, UK

e-mail: [email protected]; [email protected]

123

Heat Mass Transfer

DOI 10.1007/s00231-014-1390-5

h Wavy angle

l Dynamic viscosity of the fluid (Pa s)

q Air density (kg/m3)

r Length of corrugated (mm)

rk Diffusion Prandtl number for k

ss Wall shear stress

m Kinematic viscosity of the fluid (m2/s)

Ø Phase shift

Subscripts

a Air

av Average

i Inlet

m Mean value

o Outlet

t Total

w Wall

1 Introduction

Heat exchangers are commonly used in industrial and

engineering applications such as heaters, oil cooling,

evaporator and condenser for air conditioning, petro-

chemical industry, power plants and chemical process-

ing. A lot of efforts have been done to manufacture

more efficient and compact heat exchangers by using

varies techniques of heat transfer enhancement to pro-

duce heat exchanges equipment that are less expensive.

Accurate analysis of heat transfer rate efficiency and

pressure drop should be considered when design heat

exchanger, take in account, long life performance and

economic aspect.

By enhancing surface heat transfer coefficient (a) the

volume of heat exchanger decreases and become more

compact, in addition the capital cost decrease, (b) the

pumping power decreased and (c) the overall heat transfer

coefficient of the heat exchanger (UA) increased. Higher

surface heat transfer coefficient can be achieved by

boundary layer modification or surface magnification [1]

and studying the thermo-physical parameters which have

effect on the flow behaviour such as size of the recircula-

tion, velocity and temperature.

One of the most efficient ways to improve and enhance

the performance of the heat exchanger is using a corrugated

surface heat exchanger, because channel waviness always

change the local flow directions, which lead to interrupt the

thermal boundary layer on the wall and increase flow

separation, recirculation, reattachment and thereby increase

of surface heat transfer coefficient.

Characteristics of heat transfer and pressure drop in

corrugated channel are studied by different researchers

such as Mohammed et al. [2], who numerically studied the

effects of tilt angles and height of corrugated channels on

heat enhancement rate, they recommended a wavy angle

of 60� and wavy height of 2.5 mm with channel height

17.5 mm for best heat transfer enhancement. In the same

time, Elshafei et al. [3, 4] experimentally studied the

effect of channel phase shift of V-corrugated channels on

both heat transfer and pressure drop characteristics of

flow. The results showed significant enhancement on the

average heat transfer coefficient and with plenty of pres-

sure drop depending upon the spacing and phase shift. Ali

and Ramadhyani [5] experimentally studied the heat

transfer in the entrance region for corrugation angle of 20�and laminar flow. It was found that; Nusselt number in

corrugated channel exceeded those in the parallel-plate

channel by 140 and 240 % and friction factor increased by

130 and 280 %.

Naphthalene sublimation technique used Gradeck et al.

[6] to determine the local and average transfer character-

istics for flow in a wavy surface channel. The visualization

explained the existence of a variety of complex transfer

processes and related fluid flow phenomena. These inclu-

ded secondary flows and associated spanwise mass transfer

variations; the suppression of the secondary flow by

counteracting centrifugal forces, and the destruction of the

secondary flow by the onset of turbulence.

Leonardo and Sparrow [7], Zhang and Chen [8], con-

cluded that heat transfer is always higher than those in

rectangular channel due to the mixing induced by recir-

culation in the wake of the corrugations.

Yin et al. [9] indicated that effect of phase shift on the

flow and heat transfer is more pronounced in higher Re

region than in lower Re region. Kanaris et al. [10] used

two-equation turbulence model to simulate low Reynolds

number flow (400–1,400) in corrugated channel calculated,

the results indicated that the mean heat transfer coefficient

sand friction factors are found to be in reasonable agree-

ment with the limited published experimental data. Pehli-

van [11], experimentally studied effect of sharp and

rounded corrugation peak on heat transfer and pressure

drop in corrugated channel, the obtained results indicated

that, the increase of corrugated angle and channel height

tends to increase heat transfer.

It is obvious from the literature review above that, the

enhancement heat transfer using corrugated passages is one

of the interesting methods to improve the performance of

the heat exchanger. Most of the previous research focused

on V-corrugated channels whose all configuration peaks lie

in line-phase arrangement. In the present paper, the main

objectives are studying the effect of channel heights on

convective heat transfer characteristics and pressure drop

in the channel with V-corrugated upper and lower plates

with different phase shifts 0�, 90� and 180� under constant

wall heat flux.

Heat Mass Transfer

123

2 Mathematical formulations

2.1 Channel geometry

The schematic diagram of the present problem under

investigation is shown in Fig. 1. It consists of two opposite

corrugated plates and the details of the corrugated plate are

listed in Table 1.

Air is selected as a working fluid, while the geometry is

assumed as two-dimensional because the width of the

channel is very large compared with the channel height. A

set of different phase lag or shifts (Ø) between the upper

and lower wavy plates were considered; for phase shift

Ø = 0�, the crest of the lower wall corresponds to the crest

of upper plated, Ø = 90� one of the walls has a phase–

advanced/lag of half pitch; for Ø = 180�, the crest of the

lower corrugated wall corresponds to trough of the upper

corrugated wall of the channel as shown in Fig. 2. All

corrugated channels have the same corrugation angle, h of

20�; each corrugated plate has eleven crests separated by a

distance/pitch, P = 27.27 mm.

2.2 Governing equations

Flow governing equations (velocity and momentum),

energy equation for turbulent flow and heat transfer were

considered in this simulation. The following assumptions

were employed, two dimensional flow, steady state, tur-

bulent and no slip at the wall.

Based on the above assumptions, the governing equa-

tions can be written as follows [12, 13].

2.2.1 Mass conservation

oðquiÞoðXiÞ

¼ 0 ð1Þ

where ui is flow axial velocity.

2.2.2 Momentum conservation

oðquiujÞoðXjÞ

¼ � oP

oðXiÞþ o

oðXjÞl

oui

oXj

þ ouj

oXi

� 2

3dij

oui

oXj

� ��

þ o

o Xj

� � �q0uiu0j

� ð2Þ

The symbols l, u0 and uj are the fluid viscosity, fluctu-

ated velocity and the velocity in y-direction respectively,

the term u0i u0j is the turbulent shear stress.

2.2.3 Energy equation

o

oðXjÞuiðqE þ PÞ½ � ¼ o

oðXjÞk þ Cplt

Prt

� �oT

oXj

þ uiðsijÞeff

� �

ð3Þ

Y

X

Uniform wall heat Flux, q''

Pθ

S

A

δ

Lt

Fig. 1 Schematic diagram of

the corrugated channel

Flow direction

o o=0 =90 =180o

Fig. 2 Typical configuration of

the corrugated channel

representation

Table 1 Dimensions of the corrugated channel

Height of

the wavy

(mm)

Height of

the base

(mm)

Mean

spacing

(mm)

Length of the

domain (mm)

Number

of the

wavy

5 2.5 12.5 300 11

5 2.5 15.0 300 11

5 2.5 17.5 300 11

5 2.5 20.0 300 11

Heat Mass Transfer

123

where E is the total energy, E = CpT ? P/q ? u2/2, Prt is

the turbulent Prandtl number, and (sij) is the deviatoric

stress tensor.

sij

� �eff¼ leff

ouj

oXi

þ oui

oXj

� �� 2

3leff

oui

oXj

dij ð4Þ

The most of widely two-equations turbulence model k–ewas used here, which proposed by Launder and Spalding

[14], which has ability to predict the secondary flow

motion, it consists two equation for turbulent kinetic

energy k and the dissipation rate of the turbulent kinetic

energy e. The equations for turbulent kinetic energy (k) and

rate of dissipation (e) are given by:

2.2.4 Turbulent kinetic energy (k) equation

o

oðXiÞqkui½ � ¼ o

oðXjÞlþ lt

rk

� �ok

oXj

� �þ Gk � qe ð5Þ

2.2.5 Turbulent kinetic energy dissipation (e) equation

o

oðXiÞqeui½ � ¼ o

oðXjÞlþ lt

re

� �oeoXj

� �þ C1e

ek

� Gk

þ C2eqe2

k

� �ð6Þ

In the above equations, Gk represents the generation of

turbulent kinetic energy due to mean velocity gradient, rk

and re are effective Prandtl number for turbulent kinetic

energy and rate of dissipation, respectively; C1e and C2e are

constants and lt is the eddy viscosity and is modelled as:

lt ¼qClk2

e

!ð7Þ

The empirical constants for the turbulent model are

arriving by comprehensive data fitting for a wide range of

turbulent flow [13, 14].

Cl ¼ 0:09; Ce1 ¼ 1:47; Ce2 ¼ 1:92;rk ¼ 1:0 and re ¼ 1:3

ð8Þ

3 Numerical procedure

The two-dimensional continuity, momentum and energy

equations were solved numerically. The renormalized

group (RNG) k–e turbulent was selected. The upwind and

central difference methods are used for convections and

diffusions, respectively. The FLUENT (6.3) [15] code was

used to obtain velocity and temperature distribution inside

the channel. The solution of the problem was obtained by

employing implicit method in segregated solver. The dis-

cretization is achieved by using the PRESTO method for

pressure and the second order upwind method for

momentum, energy, turbulent kinetic energy and turbulent

dissipation rate equation. The SIMPLE algorithm was used

for flow computations.

The under re-relaxation factor was selected to be 0.3 for

pressure, 0.4 for momentum, 0.8 for turbulent kinetic

energy (k), turbulent dissipation rate (e) and 1.0 for energy

equations, turbulent viscosity (lt), body force and density.

The solution convergence is met when the difference

between the residual of the algebraic equation and the

prescribed values is less than 10-6 for all variables.

3.1 Boundary and initial conditions

The solution domain was considered 2D, enclosed by inlet,

outlet and wall boundaries. On walls no-slip condition were

assumed for momentum equations. The inlet velocity val-

ues have been derived from given Reynolds numbers. To

save the computational time; the user defined function

(UDF) code was used to define hydrodynamics fully

developed flow at inlet, the outlet boundary condition is

called ‘‘pressure outlet’’, which implies a static (gauge)

pressure at the outlet boundary, which means the pressure

will be extrapolated from the flow in the interior.

The boundary conditions for a steady and two dimen-

sional flows are as follows.

3.1.1 Initial boundary conditions

At the wall:

q00 ¼ qwall ¼ 290W

m2ð9Þ

At the inlet:

The uniform profiles for all properties are as follows:

Fully developed steady flow condition,

u ¼ u yð Þ ¼ 3

2� uav 1� y2

h2

� �ð10Þ

v ¼ 0; T ¼ Tin ¼ 300:3 K; k ¼ kin; e ¼ ein ð11Þ

At the outlet:

The flow assumed as fully developed which implies

negligible stream wise gradient of all variables.

o;on¼ 0; ; ¼ u; v; k and e ð12Þ

The turbulent kinetic energy, kin and the turbulent dis-

sipation ein, at the inlet section are approximated from the

turbulent intensity I as follows [13].

kin ¼3

2ðuinIÞ2; ein ¼ C

34l

k32

Lð13Þ

In the present study, the turbulence characteristic length,

L is set to be 0.07 Dh and the turbulent intensity level I, is

Heat Mass Transfer

123

defined as the ratio of the root-mean-square of velocity

fluctuation, u0, to the mean flow velocity, u, as follows:

I ¼ u0

u� 100 % ð14Þ

3.2 Heat transfer coefficient and Nusselt number

The current numerical simulation is aimed to study the

influence of channel phase shift on heat transfer charac-

teristics and pressure drop for a fully developed flow in

narrow V-corrugated channels.

Various parameters were studied such as, geometries of

the channel including its spacing, the Reynolds number

and channel phase shift. The average reference velocity in

the channel, uav was calculated by the Reynolds number

based on the channel hydraulic diameter which is defined

as:

uav ¼Re tDh

ð15Þ

where Dh is expressed as [16]

Dh ¼4SW

2 S + Wð Þ ¼ 2S ð16Þ

Because the channel width is too large compared to

channel height W � S.

The rate of heat input Qin to channel wall is supposed to

be totally dissipated to the air flowing through the channel

passage, rising it temperature from inlet temperature Ta,i to

exit average air temperature Ta,o

Qin ¼ _mCp Ta;o � Ta;i

� �ð17Þ

and

Ta ¼R Ac

0uTadAcrR Ac

0udAcr

ð18Þ

where _m, is air mass flow rate kg/s, and Cp, specific heat of

the air with the heat input, the average heat transfer coef-

ficient along the corrugated channel hc, was calculated by

hc ¼Qin

AwðDTmÞð19Þ

where Aw, is the surface area of the corrugated plate and

the log-mean temperature difference was calculated as.

DTm ¼Tw � Ta;i

� �� Tw � Ta;o

� �ln Tw � Ta;i

� �� Tw � Ta;o

� � � ð20Þ

where Tw is average wall temperature.

Tw ¼1

r

Zr

0

Tw;xdx ð21Þ

The average heat transfer coefficient is calculated of

average Nusselt Number [2, 14] as follows:

Nu ¼ hcHrkLt

ð22Þ

where H is the half distance of the channel height, k is the

thermal conductivity of air and r is the distance from the

leading edge of the corrugated plate along the corrugated

surface.

r ¼ Lt

cosðhÞ ð23Þ

Dp

L¼

pav;o � pav;i

� �Lt

ð24Þ

where pav,o and pav,i are the surface average pressure on the

inlet and outlet of the channel. For the sake of performance

assessment of such a narrow channels, regarding their heat

transfer characteristics and pressure drop, the ratio of the

Colburn factor to friction factor, j/f is almost considered,

expressed as [4]:

j

f¼

Nu

RePr13

fð25Þ

where f is friction factors calculated by:

f ¼Dp Dh

Lt

� 12qu2

av

ð26Þ

3.3 Grid generation

Computational mesh was created using Gambit, different

structured and unstructured with several cell size were

tested by synthetically by considering simulation time

needed for specific number of iteration and the accuracy of

the solution. Because the local Reynolds number near any

wall becomes very small and owing to viscous influence, to

overcome this issue, a special a near wall modelling

approach applied to possess the accuracy of the standard

two layer approach for fine near-wall meshes, so the first

cell is placed in the laminar boundary layer, though the

laminar sub layer is valid up to y? \ 5, the nearest nodes

from the wall are located within the interval y? = 1.5–4,

where y? is the dimensional distance from the wall. The

grid independence is carried out in the analysis by adopting

different grid distribution of 3 9 104, 4 9 104, 6 9 104

and 8 9 104; the grid independence test indicated that the

grid systems of 6 9 104 ensure a satisfactory solution, this

is verified by the fact that the difference of the computed

results of an average Nusselt number with grid finer the

6 9 104 within 1 %.

Heat Mass Transfer

123

4 Result and discussion

The heat transfer and pressure drop characteristics of tur-

bulent flow in corrugated channels are discussed in the

present study, considering the effect of different channel

heights (S = 12.5, 15.0, 17.5 and 20.0 mm) and variable

phase shifts, Ø = 0�, 90� and 180� using air as working

fluid. These were carried out to study these effects on the

temperature and airflow structure. The side walls of sim-

ulated channels were uniform heat flux exposed to air flow

over a range of 500 \ Re \ 2,000. The optimal perfor-

mance of a V-shaped PHE was obtained using the surface

performance Colburn and Fanning friction factor (j/f).

To validate the numerical results of the current study,

the predicted results for average wall Nusselt number and

average surface channel temperature were compared with

the previous experimental work of Naphon [17] as shown

in Figs. 3 and 4; for channel height S = 17.5 mm, phase

shift 180� and constant wall heat flux 290 W/m2, Fig. 3

shows the comparison between the predicted average sur-

face channel temperature and measured ones. In addition,

Fig. 4 shows the comparison between average air Nusselt

number in corrugated channel with the experimental work

done by Naphon [17]. It can be clearly seen from these

figures that the values obtained from the model are con-

sistent with the experimental data and lie within ±8 %

error.

4.1 Heat transfer results

Normally, corrugated channels are used to improve and

enhance heat transfer in channels; the effect of flow pas-

sage shape and spacing are effective parameters that have

important influence on heat transfer enhancement, and

these effects on the average Nusselt number will be dis-

cussed in the following subsections.

The calculated average surface Nusselt number versus

Reynolds number for both corrugated and straight channels

were plotted in Fig. 5; this figure shows that the Nusselt

number increases with the increase the Reynolds number.

Also it is noticed that the Nusselt number for corrugated

channel is higher than for the straight channel for any value

of phase shift. The percentage of heat transfer enhancement

was increased with the increase in phase shift angle due to:

the higher fluid re-circulation swirl flow intensity in the

corrugated channel larger surface area and the transverse

vortices of the bulk flow field in the wavy wall trough. Also

the turbulent intensity is enhanced when compared to the

conventional channel. This behavior results in the higher

local wall temperature gradients [17, 18].

4.1.1 Effect of channel height

The average Nusselt number as a function of the Reynolds

number for different channel heights is presented in

Figs. 6, 7 and 8. The predicted results for different channel

height, S = 12.5, 15.0, 17.5 and 20.0 mm, while keeping

400 800 1200 1600 2000

Reynolds Number

0

20

40

60

80

100

Ave

rage

Pla

te T

empe

ratu

re(°

C)

Wavy angle=20o

Inlet air temperature=27.15 ocHeat Flux=290 W/m2

Naphon

Numerical Simulation

Fig. 3 Validation of numerical results of average corrugated surface

temperature with experimental values from Naphon [17] for different

Reynolds number and wavy angle 20�

600 800 1000 1200 1400 1600 1800

Reynolds Number

0

10

20

30

40

Ave

rage

Nus

selt

Num

ber

Wavy angle=20o

heat flux =290 W/m2

Naphon

Numerical simulation

Fig. 4 Validation of numerical results of average Nusselt number

with experimental values from Naphon [17] for different Reynolds

number and wavy angle 20�

Heat Mass Transfer

123

phase shift fixed, Ø = 0�, 90� and 180� will be discussed in

the next paragraphs.

For flow in the corrugated channel of Ø = 0� and

Ø = 90�; it is clear that the average Nusselt increases with

the decrease of the channel height and the increase of the

Reynolds number; for channel height S = 12.5 mm, the

average Nusselt number increases by about 30 % belongs

to the channel height S = 20 mm, also the results for the

channel heights S = 15.0 and 17.5 mm are close to each

other as shown in Figs. 6 and 7.

For phase shift Ø = 180� or the convergent divergent

passage, the heat transfer enhancement value is higher than

the corrugated channel with phase shift Ø = 0� and

Ø = 90� by value (25–50 %), because the V-shape waves

400 800 1200 1600 2000

Reynolds Number

0

10

20

30

40

Ave

rage

Nus

selt

Num

ber

Ø=90o

S=12.5 mmS=15 mmS=17.5 mmS=20 mm

Fig. 7 Average Nusselt number versus Reynolds number in corru-

gated channel of different height, Ø = 90�

400 800 1200 1600 2000

Reynolds Number

0

10

20

30

40

Ave

rage

Nus

selt

Num

ber

Ø=180o

S=12.5 mmS=15 mmS=17.5 mmS=20 mm



400 800 1200 1600 2000

Reynolds Number

0

4

8

12

16

20A

vera

ge N

usse

lt N

umbe

r

S=17.5 mmØ=0o

Ø=90o

Ø=180o

Parallel plate channel

Fig. 5 Averaged Nusselt Number versus Reynolds number in

corrugated of different channel phase shift, S = 17.5 mm

400 800 1200 1600 2000

Reynolds Number

0

10

20

30

40

Ave

rage

Nus

selt

Num

ber

Ø=0o

S=12.5 mmS=15 mmD=17.5 mmD=20 mmS=17.5 mmS=20 mm



Heat Mass Transfer

123

are generating swirl flow in the wavy wall trough and the

V-shaped wave has a sharp edge which has a significant

effect on the main flow distribution in the channel as shown

in Fig. 8.

4.1.2 Effect of phase shift

The effect of the different corrugated phase shifts on both

flow structure and the average Nusselt number in corru-

gated channels with different phase shifts on the average

Nusselt number for different channel heights, S = 12.5,

15.0 and 20.0 mm, are illustrated in Figs. 9, 10 and 11. It is

seen that, the average Nusselt Number has been increased

significantly with increasing the Reynolds number and the

phase shift; this is attributed to changing the phase shift of

corrugated plates changing the geometry configuration and

has significant effect on the change flow structure and the

presence of recirculation cells accompanied with corru-

gated channel. The strength of such promoted cells depends

on the Reynolds number and the channel spacing as dis-

cussed by Zhang et al. [19] and Yuan et al. [20].

Over the tested range of the Reynolds number, the

average Nusselt number generally gets higher values with

Ø = (90� and 180�) than those of phase shift Ø = 0�. The

effect of the changing phase shift for this narrow passage

on the average Nusselt number seems to be higher than

wider channels.

From the previous results, it could be concluded that the

effect of the phase shift on heat transfer in corrugated

channels is variable and depends on the value of the

channel height. Good enhancement in heat transfer was

achieved with the channel of Ø = (90� and 180�) for rel-

atively narrow channel (S = 12.5). Based on the results

reported by Zhang et al. [19], who concluded that the lat-

eral vortices and recirculation cells were generated by

wavy wall structure with convergent–divergent passage

Ø = 180�. These effects are more pronounced in narrow

400 800 1200 1600 2000

Reynolds Number

0

10

20

30

Ave

rgag

e N

usse

lt N

umbe

r

S=12.5mmØ=0o

Ø=90o

Ø=180o


gated channel of different phase shift and S = 12.5 mm

400 800 1200 1600 2000

Reynolds Number

0

4

8

12

16

20

Ave

rage

Nus

selt

Num

ber

S=15 mm

Ø=0o

Ø=90o

Ø=180o



400 800 1200 1600 2000

Reynolds Number

0

2

4

6

8

10

Ave

rage

Nus

selt

Num

ber

S=20 mm

Ø=0o

Ø=90o

Ø=180o



Heat Mass Transfer

123

channel. The other remarkable conclusion that by changing

corrugated surface phase shift, the recirculation zone

behind the V-rib moved and flow attachment on the base

wall and the recirculation eddies are formed above these

flow region.

4.2 Pressure drop results

The pressure drop is another important parameter to be

considered in designing the heat exchanger. The variation

of the pressure drop per unit length with the Reynolds

number among the tested corrugated channels at wavy

angle 20� and channel height 12.5 mm is shown in

Fig. 12.

It can be seen that the pressure drop gradually increases

with the increase of the Reynolds number also with

changing the corrugated the phase shift of the corrugated

channel, because the corrugated channel surface produced

drag forces exerted on the flow field and also produced

turbulence augmentation and rotational flow [20].

The pressure drop a function of the Reynolds number in

narrow corrugated channel (S = 12.5 mm) and in wide

channel (S = 20 mm) at different phase shift, Ø = 0�,

Ø = 90� and Ø = 180� are plotted in Figs. 13, 14 and 15

respectively. These figures illustrate that for any value of

the phase shift, the pressure drop gradually increases with

the increasing of the Reynolds number. Over the tested

range of the Reynolds number, the pressure drop increases

with the decrease of the channel heights and increase of the

phase shift. In the meantime, as the channel becomes

wider, the pressure drop gets lower values. Elshafei et al.

[3] referred that to the feature of rising vortices and eddies

in the heat of the flow in the core of the corrugated channel

leading to a significant increase of pressure drop relative to

those related with parallel flow channels.

400 800 1200 1600 2000

Reynolds Number

0

10

20

30

Pre

ssur

e dr

op p

er u

nit l

engt

h(P

a/m

)

S=12.5 mm

Ø= 0o

Ø= 90o

Ø=180o

Fig. 12 Pressure versus Reynolds number for flow in corrugated

channel of different phase shift, S = 12.5 mm

400 800 1200 1600 2000

Reynolds Number

0

4

8

12

16

20

Pre

ssur

e dr

op p

er u

nit l

engt

h(P

a/m

)

Ø=0o

S=12.5 mmS=15.0 mmS=17.5 mmS=20.0 mm


channel of different channel height, Ø = 0�

400 800 1200 1600 2000

Reynolds Number

0

4

8

12

16

20

Pre

ssur

e dr

op p

er u

nit l

engt

h(P

a/m

)

Ø=90o

S=12.5 mmS=15.0 mmS=17.5 mmS=20.0 mm



Heat Mass Transfer

123

Based on the previous simulation results and their

discussion, it appears that the variation of corrugated

channel heights and the phase shift of the corrugated

plate surface have a significant impact on the increased

both of heat transfer and pressure drop in corrugated

channels.

4.3 Performance comparison

In order to compare the thermal–hydraulic performance of

the corrugated, the performance of these corrugated chan-

nels is graphed in Figs. 16, 17 and 18 as j/f versus Rey-

nolds number for different channel height and phase shift.

400 800 1200 1600 2000

Reynolds Number

0

4

8

12

16

20

Pre

ssur

e dr

op p

er u

nit l

engt

h(P

a/m

)

Ø=180o

S=12.5 mmS=15.0 mmS=17.5 mmS=20.0 mm



400 800 1200 1600 2000

Reynolds Number

0

0.01

0.02

0.03

J/f

Ø=0o

S=12.5 mmS=15.0mmS=17.5 mmS=20.0 mm

Fig. 16 Heat transfer enhancement criterion various Reynolds num-

ber for different channel height and, phase shift Ø = 0�

400 800 1200 1600 2000

Reynolds Number

0

0.01

0.02

0.03

J/f

Ø=90o

S=12.5 mmS=15.0 mmS=17.5 mmS=20.0 mm



400 800 1200 1600 2000

Reynolds Number

0

0.01

0.02

0.03

J/f

Ø=180o

S=12.5 mms=15.0mmS=17.5 mmS=20.0 mm



Heat Mass Transfer

123

A higher good factor means that more heat transfer is

obtained with the same pumping power and heat transfer

area. These two parameters were used to detect the area of

the goodness factor for flow in such channels.

It is clear from these figures that (j/f) increases with the

decrease of the channel heights also the performance ratio

is almost constant with the increase of the Reynolds

number for all channels heights. With the increase of the

corrugated channel phase shift, the values of (j/f) are

increase for the same channel height; this means phase shift

Ø = 180� offered higher (j/f).

It is found that the maximum values of performance

were obtained by using the channel height S = 12.5 mm

and the channel phase shift Ø = 180�. This indicated that,

the channel height S = 12.5 mm is the most attractive

height from the viewpoint of energy saving compared with

other heights.

The previous study indicated that for fully developed

laminar flow of a specified fluid; (j/f) is constant for a given

surface, regardless of Reynolds number because it reveals

the influence of the cross-sectional shape regardless of the

scale of the geometry [21].

4.4 Flow visualization

Figures 19 and 20 show different variables distribution

including the velocity vector, the temperature couture and

the velocity streamlines for different corrugated channel

heights, S = 12.5 and 20.0 mm, and different phase shift,

Ø = 0�, 90� and 180� for the same Re = 1,100.

As shown in Fig. 19b, c, for S = 12.5 mm, it is clear that

the effect of the phase shift on the temperature and the flow

development in the corrugated channel, that there are smaller

recirculation region and more separation bubble regions

formed in the adjacent inlet/outlet and it increases with

increase of the phase shift, which means that the influence of

the wall on the main stream becomes greater; thus generated

more swirl flow in the wavy wall due to transfer vortices of

the bulk flow field in the wavy wall trough. These results

induced higher temperature gradient near the wavy wall.

Therefore, the net heat transfer rate from the wavy wall to the

fluid increased as shown in Fig. 19a.

Figure 20a–c show the temperature couture, velocity

vectors and velocity streamlines for different channel phase

shift and S = 20.0 mm. It can be seen that the wavy wall

has less significant effect on the temperature distribution

and on the effect on the flow structure in the core. How-

ever, we can be concluded from these figures that, as fur-

ther decreasing in the channel height, the onset and the

growth of the swirl flows encompass much of the flow

field, especially in the narrow passage. In addition, the

turbulent intensity also increases due to the sharp edge of

phase shift Ø = 180� corrugated channel, these results

were reported by Yin et al. [9] and Naphon [22].

Fig. 19 Variation of a velocity vectors, b temperature contours and c streamlines for corrugated channel height S = 12.5 mm and Re = 1,100

for different phase shifts

Heat Mass Transfer

123

5 Conclusions

Corrugated channels are one of the popular techniques that

are extensively used in compacted heat exchanger manu-

facturing. Two dimensional numerical predictions were

carried out to study the effect of the wave plate shift angle,

channel heights and Reynolds number on the thermal

performance of fluid flow and heat transfer in a corrugated

channel for low Reynolds number flow. The governing

equations were solved using finite volume methods with

certain assumptions to provide a clear environment for the

case study target. The results of the average Nusselt

number, pressure drop in different phase shifts (Ø) and

channel heights (S) were presented; also, the flow charac-

teristics have been visualized using streamlines. The fol-

lowing conclusions may be drawn:

1. Corrugated channel is a good alternative for high heat

flux applications or for more efficient heat exchange

devices used in a wide variety of engineering appli-

cations like heating and air conditioning units.

2. The average Nusselt number increased by factor 2.5 up

to 3.2 relative to that parallel plate channel depending

upon the phase shift and the spacing of the corrugated

channel due to fluid flowing through the corrugated

surface, fluid re-circulation or/and swirl flows are

generated in the corrugation troughs.

3. Thermal performance of the narrow channel is rela-

tively good in comparison with the wider channel.

4. When the phase shifts (Ø) increased, the pressure drop

in the corrugated channels increased and decreased

with the increase in channel heights(s) due to drag

forces exerted on the flow field and also producing

turbulence augmentation and rotational flow.

5. The thermal–hydraulic performance is closely related

to the distribution of velocity streamline sand breaking

and destabilizing in the thermal boundary layer which

are occurred as fluid flowing through the corrugated

surfaces.

6. The channels with phase shift Ø = 180�, the most

attractive parameters from the viewpoint of energy

saving compared to others compared with the phase

shift, Ø = 0� and 90�.

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Heat Mass Transfer

123

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Convective heat transfer and pressure drop in v corrugated

Engineering

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