1/3, the maximum rate of performance evaluation criterion is obtained and by increasing the slip velocity coefficient, the amount of PEC becomes significant.

(1)

Investigation into the effects of slip boundary condition on nanofluid flow in a double-layer microchannel

Abedin Arabpour¹^•Arash Karimipour²^•Davood Toghraie¹ ^•Omid Ali Akbari³

Received: 23 August 2017 / Accepted: 29 October 2017 / Published online: 7 November 2017 Akade´miai Kiado´, Budapest, Hungary 2017

Abstract

In this research, the laminar flow and heat transfer of kerosene/MWCNT nanofluid in a novel design of a double-layer microchannel under the influence of oscillating heat flux and slip boundary condition have been studied. This research has been investigated in the dimensionless lengths of (k

₁

) 1/3, 2/3 and 3/3 and dimen- sionless slip velocity coefficients ranging from 0.001 to 0.1. The suspension of nanoparticles in kerosene as the base fluid has been studied in Reynolds numbers of 1–100 and volume fractions of 0–8%. The results indicate that, by using novel design of double-layer microchannel in

k₁=

1/3, the maximum rate of performance evaluation criterion is obtained and by increasing the slip velocity coefficient, the amount of PEC becomes significant.

Among the studied cases, in all Reynolds numbers and volume fractions, the dimensionless length of 1/3 has the maximum amount of friction coefficient. Also, by enhancing the volume fraction of nanoparticles, slip velocity coefficient, Reynolds number and significant reduction in thermal resistance of solid wall, Nusselt number enhances.

Keywords

Double-layer microchannel Slip velocity coefficient Nusselt number Friction coefficient Thermal resistance Pressure drop

List of symbols

A

Area (m

²

)

B=b/H

Dimensionless slip velocity

C_f

Skin friction factor

C_p

Heat capacity (J kg

^-1

K

^-1

)

H

Microchannel height,

lm

K

Thermal conductivity coefficient (Wm

^-1

K

^-1

)

L

Down-layer microchannel length (lm)

L₁

Top-layer microchannel length (lm)

Nu

Nusselt number

P

Fluid pressure (Pa)

Pe=

(u

_sd_s

/a

f

) Peclet number

Pr=#_f

/a

_f

Prandtl number

q⁰⁰

(X) Oscillating heat flux (Wm

^-2

)

q⁰⁰₀

Constant heat flux (Wm

^-2

)

R

Thermal resistance (KW

^-1

)

Re=qfu_cd/lf

Reynolds number

T

Temperature (K)

(U,

V)=

(u/U

₀

,

v/U₀

)

Dimensionless velocity components in

x, y

directions

(X,

Y)=

(x/d,

y/d)

Cartesian dimensionless coordinates

u,v

Velocity components in

x,y

directions (ms

^-1

)

u_c

Inlet velocity in

x

directions (ms

^-1

)

u_s

Brownian motion velocity (ms

^-1

)

W

The width of microchannels (lm)

& Davood Toghraie

[email protected]

1 Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr 84175-119, Iran

2 Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

3 Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran

(2)

Greek symbols

b

Slip velocity coefficient (m)

u

Nanoparticles volume fraction

k=L₁

/L Dimensionless length ration

l

Dynamic viscosity (Pa s)

h=

(T–T

_C

)/DT Dimensionless temperature

q

Density (kg m

^-3

)

s

Shear stress (Nm

^-2

)

#

Kinematics viscosity (m

²

s

^-1

)

Super- and subscripts

Ave Average

c Cold

Eff Effective

f Base fluid (pure water)

H Hot

In Inlet Max Maximum Min Minimum nf Nanofluid Out Outlet

S Solid nanoparticles

Introduction

Using nanotechnology in the hydrodynamics and heat transfer sciences has created challenging progresses among the researchers interested in these topics. The transferring and rheological properties of custom fluids used in heat transfer of industrial systems can be improved by dis- tributing the oxide and non-oxide nanoparticles in the custom fluids (called nanofluid). The experimental results indicate that adding nanoparticles to the base fluid impressively enhances the heat transfer coefficient of nanofluid [1–5]. Investigating and estimating nanofluid behaviors in novel heat transfer surfaces as new experimental phases have been widely processed by the researchers [6–11]. Among the novel heat transfer methods, removing high heat fluxes by microchannel is very common in the miniature industries. Microchannel is a small and efficient heat exchanger in which by combining some properties, such as high aspect coefficient, high heat transfer coefficient, less volume of demanded fluid and small dimensions, it has become effective equipment in heat transfer fields [12–18]. The microchannel heat sink has been initially introduced by Tukemen and Pease [19].

Line et al. [20] numerically and parametrically investigated the fluid flow and heat transfer in double-layer, three-layer and four-layer microchannels made of silicon with the pure water as the cooling fluid. Their results showed that, in the multi-walled microchannel, by increasing the number of layers, the bottom wall temperature becomes uniform and

thermal resistance decreases. Leng et al. [21] numerically investigated a novel design of microchannel heat sink.

They figured out that, at the bottom wall of the truncated double-layer microchannel, the temperature distribution is more uniform and thermal resistance is lower. Also, they revealed that, in this design, the improvement in heat transfer performance is based on the weak heating effects of top layers. Bello-Ochende et al. [22] simulated a numerical optimization of a three-dimensional water fluid flow and heat transfer in a rectangular microchannel made of silicon. In this study, the geometrical optimization has been performed for determining the optimized ratio of channel dimensions in order to reduce the total maximum temperature and enhance the total thermal conductivity.

Heris et al. [23] studied the forced convective heat transfer of laminar flow of water/Al

₂

O

₃

nanofluid inside a circular tube with constant wall temperature. Their results revealed that, when the nanofluid is used as the cooling fluid, the performance of silicon microchannel heat sink improves significantly. Chen and Ding [24] studied the heat transfer performance of microchannel with water/Al

₂

O

₃

nanofluid with different volume fractions. Their results showed that the amount of thermal resistance and temperature distri- bution under the influence of inertia energy changes sig- nificantly. Hung and Yan [25], by using nanofluid and variable geometrical parameters, investigated the improvement in thermal performance of double-layer microchannel. Their results indicated that, by using water/

Al

₂

O

₃

nanofluid, the amount of heat transfer improves and

by using nanoparticles, the thermal resistance reduces. The

three-dimensional laminar flow has been numerically

studied by Xie et al. [26] using the computational fluids

dynamics. Their numerical results showed that, comparing

to the smooth microchannel, using multistage bifurcations

at the microchannel outlet causes the enhancement of the

thermal performance. In order to investigate the flow

behavior in micron dimensions, some studies have been

carried out by researchers considering the slip boundary

conditions [27–29]. Raisi et al. [30] numerically studied the

laminar flow and heat transfer in a rectangular

microchannel and figured out that, in low Reynolds num-

bers, the existence of slip on the solid walls does not have

any significant effect on heat transfer enhancement; how-

ever, in high Reynolds numbers, it causes impressive

enhancement of heat transfer. By investigating the resear-

ches about heat transfer and flow hydrodynamics in the

microchannels, it can be said that, by changing the geo-

metrics of flow and boundary conditions and increasing the

thermal conductivity of fluid, the convective heat transfer

enhances [31]. In this research, the laminar flow and heat

transfer of nanofluid in the microchannel heat sink have

been numerically studied. The double-layer microchannel

heat sink has been studied in the two-dimensional

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Cartesian coordinate and has been considered truncated.

(Due to the placing of the top wall on the bottom wall of channel, a part of the top wall has been eliminated.) Ker- osene/MWCNTs nanofluid has been used as the working fluid in the double-layer microchannel heat sink. Some investigated the issues distinguishing this study from other researches in this field are: (1) using novel design of double-layer microchannel; (2) using kerosene/MWCNTs nanofluid; (3) using slip boundary condition; (4) applying sinusoid heat flux; (5) simultaneous presentation and comparison of heat transfer and hydrodynamics results.

Problem statement

In the present study, the laminar flow and heat transfer of kerosene/MWCNTs nanofluid in the rectangular double- layer microchannel heat sink have been numerically studied. The main purpose of this study is investigating the laminar flow and heat transfer of nanofluid in a new microchannel design. By using finite volume method, this research has been numerically simulated in a two-dimensional space. In this research, the fluid and heat transfer parameters have been investigated in Reynolds numbers of 1, 10 and 100 and volume fractions of 0, 4 and 8% of nanoparticles. Figure

1

demonstrates the studied geomet- rics of this paper. According to Fig.

1, the microchannel

length is

L=

3 mm, the microchannel height is

H=

50

lm, the microchannel thickness is t=

12.5

lm

and the length of interface area to the bottom surface of channel is

L2

and variable. Considering the definition of

dimensionless length as the ratio of the top layer length to the bottom layer length (k

₁=L₁

/L), this study has been conducted in the dimensionless lengths (k

1=L1

/L) of 1/3, 2/3 and 3/3. Also, by considering the definition of dimen- sionless slip velocity coefficient as the ratio of slip velocity coefficient to the microchannel height (B

=b/H), the

numerical simulation of the present study has been performed in the dimensionless slip velocity coefficients (B

=b/H) of 0.001, 0.01 and 0.1.

In all the studied cases of this research, at the top and bottom layers, the inlet fluid enters as the contour flow with the temperature of 301 K and after heat transferring with the hot wall, fluid exits the channel. The studied microchannel has been made up of silicon and the bottom wall of microchannel with the length of L is under the influence of oscillating heat flux. All the external top and lateral walls of microchannel have been considered insu- lated. The governing boundary conditions on the problem- solving are the oscillating heat flux applied to the bottom wall of microchannel and the slip boundary condition applied to the internal walls of channel. In this numerical research, the effects of nanofluid density, the changes of dimensionless length (k

1=L₁

/L) and the dimensionless slip velocity coefficient (B

=b/H) at the range of different

Reynolds numbers have been investigated. The nanofluid properties of this simulation and the materials of microchannel wall are, respectively, indicated in Tables

1

and

2.

In this simulation, the laminar fluid flow and heat transfer are completely developed. The nanofluid proper- ties are considered constant and independent from

Insoulation surface

Solid L₁

L₂ D

U_in, T_in

A B

H C t U_in, T_in

y X

MWCNT//Kerosene Nanofluid

L

q″(X) = 2q₀″ + q₀ ″sin(

π4X) Fig. 1 Schematic of studied

geometrics in this study

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temperature. The solid–liquid suspension with low-volume fractions is modeled as single phase.

Governing equations

The governing equations on two-dimensional simulation domain are Navier–Stokes equations in Cartesian coordi- nate, which are defined as the dimensionless equations [33,

34]

Continuity equation:

oU oXþoV

oY ¼

0

ð1Þ

Momentum equation:

UoU oXþVoU

oY ¼ oP oXþ l_nf

q_nfmf

1 Re

o²U oX²þo²U

oY²

ð2Þ

UoV oXþVoV

oY ¼ oP oYþ l_nf

q_nfmf

1 Re

o²V oX²þo²V

oY²

ð3Þ

Energy equation:

Uoh oXþVoh

oY ¼anf

af

1 Re Pr

o²h oX²þo²h

oY²

ð4Þ

For non-dimensioning Eqs. (1)–(4), the following parameters are used [35]:

X¼ x

H; Y¼ y

H; V¼ v uc

; U¼ u

uc

; B¼b

H;

P¼ P

q_nfu²_c;

Pr

¼tf

af

; h¼TTc

DT ; DT¼q⁰⁰₀H kf

ð5Þ

According to Eq. (6), the bottom wall of microchannel with the length of

L

is under the influence of oscillating heat flux and the amount of

q⁰⁰₀

can be calculated by the parameters of Eq. (5),

q⁰⁰ðXÞ ¼

2q

⁰⁰₀q⁰⁰₀þq⁰⁰₀

sin

pX

4

ð6Þ

One of the parameters for investigating microchannel performance is the friction coefficient, which is calculated by the following equation [36]:

Table 1 Thermophysical properties of base fluid and nanoparticles of MWCNTs [32]

u/% q/kg m^-3 C_p/J kg^-1K^-1 k/W mK^-1 l/Pa.s

0 783 2090 0.145 0.001457

4 815 1989 0.265 0.001613

8 845 1895 0.390 0.001795

Table 2 Thermophysical properties of microchannel body [21]

k/W mK^-1 C_p/J kg^-1K^- q/kg m^-3 Pr Material

124 702 2329 – Silicon

Table 3 Study of results independency from the grid number (x9y) Parameters along the line

AB

(300990) (4009100) (6009140)

Nuave 3.13 3.29 3.37

Error 7.12% 2.373% Base mesh

Cfave 0.852 0.831 0.8036

Error 6% 3.4% Base mesh

T_max(K) 314.3 311.02 310.507

Error 1.22% 0.16% Base mesh

X

0 5 10 15 20 25 30

Nux

0 2 4 6 8

Our study Nikkhah et al. [35]

Re = 100

ϕ = 0.12

(a)

X

0 2 4 6 8 10

Us

0.3 0.4 0.5 0.6 0.7 0.8 0.9

Our study Nikkhah et al. [35]

Re = 100

ϕ = 0.12

(b)

Fig. 2 Validation with the numerical study of Nikkhah et al. [35]

(5)

Cf ¼

2

sw

qu²_in ð7Þ

The average Nusselt number is a dimensionless number and in heat transfer indicates the convective heat transfer to the conductive heat transfer and can be obtained as [37]:

Nux¼hH

k_f ! Nuave¼

1

L

Z L 0

NuxðXÞ

dX

ð8Þ

The pressure drop in the inlet and outlet sections can be obtained as:

DP¼P_inP_out ð9Þ

The performance evaluation criterion is defined as [38]:

PEC

¼

Nu_ave Nuave;u¼0

C_f C_f;u¼0

ð1=3Þ ð10Þ

The amount of thermal resistance of bottom wall of microchannel is calculated by the following equation [21]:

R¼TmaxTmin

q⁰⁰₀ A ¼TmaxTin

q⁰⁰₀ A ! A¼WR!R W

¼T_maxT_in

q⁰⁰₀ L ð11Þ

In the above equation,

T_max

,

T_min

,

A

and

q⁰⁰₀

are, respectively, the maximum temperature of bottom wall, the minimum temperature (the inlet temperature of fluid), the bottom section of microchannel with the length of (L) and the heat flux applied to the bottom wall (AB).

Governing boundary conditions on problem- solving

The thermal and hydrodynamical boundary conditions used in this problem are as follows:

0.275091 0.825272 1.37545 1.92563 2.47582 3.026 3.57618 4.12636

0 0.001 0.002 0.003

X (m)

0 0.001 0.002 0.003

X (m)

0 0.001 0.002 0.003

X (m) = 0.0

ϕ

= 0.04 ϕ

= 0.08 ϕ

Fig. 3 Changes of dimensionless temperature in Reynolds number of 1 and dimensionless slip coefficient of 0.1 in the dimensionless length of 1/3

0 0.001 0.002 0.003

X (m)

0 0.001 0.002 0.003

X (m)

0 0.001 0.002 0.003

X (m)

0.275091 0.825272 1.37545 1.92563 2.47582 3.026 3.57618 4.12636

= 0.0 ϕ

= 0.04 ϕ

= 0.08 ϕ

0 0.001 0.002 0.003

X (m)

0 0.001 0.002 0.003

X (m)

0 0.001 0.002 0.003

X (m)

0.275091 0.825272 1.37545 1.92563 2.47582 3.026 3.57618 4.12636

= 0.0 ϕ

= 0.04 ϕ

= 0.08 ϕ

(6)

U¼

1

; V ¼

0 and

h¼

0

for X¼

0 and 0:25

Y

1:25 and

X¼

60; 1:75

Y

2:75

ð12Þ

V¼

0 and

oh oX¼oU

oX

for

X¼

60 and 0:25

Y

1:25 and

X¼L₂

1:75

Y

2:75

ð13Þ V¼

0

; U¼

0 and

oh

oY¼

2q

⁰⁰₀þq⁰⁰₀

sin

pX

4

for

Y ¼

0 and 0

X

60

ð14Þ

V¼

0

; Us¼B oU

oY

and

knf

oh oY ¼ks

oh oY

for

Y¼

0:25 and 0

X

60

ð15Þ V¼

0

; U¼

0 and

oh

oY ¼

0 for

Y¼

1:5 and 0

XL₂

and

Y ¼

3 and

L₂X

60

ð16Þ V¼

0

; U_s¼ BoU

oY

and

k_nfoh

oY ¼ k_soh oY

for

Y¼

1:25 and 0

X

60

ð17Þ

0.00 0.04ϕ 0.08

Cf

0.01 0.1 1 10 100

Down Layer

0.00 0.04ϕ 0.08

Cf

0.01 0.1 1 10 100

Re = 10

Re = 100

λl= 1/3 Down Layer Re = 1

0.00 0.04ϕ 0.08

Cf

0.01 0.1 1 10 100

Down Layer

Re = 1

Re = 10

Re = 100

Re = 100 Re = 10

λl= 2/3

λl= 3/3 B = 0.001

B = 0.01 B = 0.1

B = 0.001 B = 0.01 B = 0.1

Fig. 6 Average friction factor in Reynolds number, volume fraction, slip coefficient and different dimensionless lengths at the bottom microchannel layer in the wall length of (AB)

(7)

V¼

0

; Us¼BoU

oY

and

knf

oh oY¼ ks

oh oY

for

Y¼

1:75 and

L2X

60

ð18Þ

V¼

0

; Us¼ BoU

oY

and

knf

oh oY ¼ ks

oh oY

for

Y¼

2:75 and

L2X

60

ð19Þ

Mesh study and numerical procedure

Today, due to the great cost of experimental platforms, the computation fluid dynamics (CFD) simulation and numerical method [39] have become an effective tool to predict the results in industrial and laboratory scales. In this

numerical research, in order to simulate the solving domain, by using finite volume method [40–46], the number of organized grid number has been changed from 27,000 to 84,000. For investigating the results independency from the chosen grid number, the average Nusselt number, friction coefficient and maximum wall temperature along (AB) have been compared with the results obtained from grid number of 84,000. The mentioned parameters have been investigated in the case of

k₁=

3/3 and Reynolds number of 10 in volume fraction of 4% and

B=

0.1. Comparing to the base mesh state, the changes of Nusselt number, average friction coefficient and maximum wall temperature in grid numbers of 27,000 and 40,000 are, respectively, 7.12%, 2/37, 6, 3.4, 1.22 and 0.16%. Each of the above parameters with fallacy is indicated in Table

3.

ϕ

0.00 0.04 0.08

Cf

0.01 0.1 1 10 100

Up Layer

ϕ

0.00 0.04 0.08

Cf

0.01 0.1 1 10

100 B = 0.001

B = 0.01 B = 0.1

Re = 100 Re = 10

Re = 1 Re = 1

Re = 1

λ_l= 1/3 Up Layer

ϕ

0.00 0.04 0.08

Cf

0.01 0.1 1 10 100

Up Layer B = 0.001 B = 0.01 B = 0.1

B = 0.001 B = 0.01 B = 0.1

Re = 100 Re = 10

λ_l= 2/3

λ_l= 3/3

Fig. 7 Average friction factor in Reynolds number, volume fraction, slip coefficient and different dimensionless lengths at the top layer of microchannel along the wall (CD)

(8)

According to the changes of mentioned parameters, the grid number of 84,000 has been chosen for this simulation.

In this investigation, in order to increase the accuracy of solving, SIMPLEC algorithm [47–52] has been used for discretizing the governing equations using second-order upwind method [53,

54]. The maximum residual of present

simulation is 10

^-6

[55,

56].

Results and discussion

Validation

Figure

2

indicates the validation of present research with the numerical study of Nikkhah et al. [35]. The investigation has been carried out for calculating (a) the local

Nusselt number and (b) the local slip velocity on the microchannel wall. Nikkhah et al. [35] numerically inves- tigated the laminar flow of water/MWCNTs in a rectan- gular microchannel by applying oscillating heat flux and slip velocity boundary condition on the walls. The obtained results have acceptable coincidence and negligible error.

Dimensionless temperature contours

Figures

3–5

show the dimensionless temperature contours in Reynolds number of 1 and dimensionless slip coefficient of 0.1 in the dimensionless lengths of 1/3, 2/3 and 3/3. The reduction in dimensionless temperature in different areas of microchannel indicates the dominant effect of heat transfer enhancement. By comparing the dimensionless temperature at the top and bottom layers of microchannel, it can be

X

0 10 20 30 40 50 60

Nuave

0 5 10 15 20

Re = 1

Re = 10

Re = 100

λl= 2/3

Down Layer

X

0 10 20 30 40 50 60

Nuave

0 5 10 15 20

ϕ = 0.00 ϕ = 0.04 ϕ = 0.08

Re = 1

Re = 10

Re = 100

λl= 1/3

Down Layer

X

0 10 20 30 40 50 60

Nuave

0 5 10 15 20

Re = 1

Re = 10

Re = 100

λl= 3/3

Down Layer

= 0.00

= 0.04

= 0.08

= 0.00

= 0.04

= 0.08

ϕ ϕ ϕ

Fig. 8 Local Nusselt number in Reynolds number, volume fraction and different dimensionless lengths and slip coefficient of 0.1 at the microchannel bottom layer along the wall (AB)

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said that, at the bottom layer, due to the direct contact of fluid with the hot wall, different areas of microchannel have significant temperature changes and dimensionless temperature enhancement. After entering the fluid to the microchannel and crossing the developed length and the augmentation of fluid temperature, due to the contact with wall and reduction in fluid temperature and growing the hot thermal boundary layer, the amount of heat transfer of cold fluid with hot surface decreases gradually. At the bottom layer, in addition to the mentioned factors, the main reason of dimensionless temperature changes and its maximum rate is due to the direct contact of fluid with the hot surface and heat concentration in this layer. In the top layer, by decreasing the dimensionless length due to the enhance- ment of developed length and according to the contours of top layer, the temperature changes of flow from the inlet to the outlet sections of microchannel reduce to the minimum amount, and comparing to the top layer, more uniform thermal profile has been accomplished.

Evaluating the average friction coefficient in the top and bottom layers of microchannel

Figure

6

indicates the average friction coefficient at the bottom layer of microchannel along (AB) in Reynolds numbers of 1, 10 and 100, volume fractions, slip coefficient and different dimensionless lengths. By increasing Rey- nolds number in all the studied volume fractions, the average friction coefficient reduces. By increasing Rey- nolds number, the thickness of hydrodynamical boundary layer and the changes of velocity parameter in areas in

which the fluid is in contact with wall are reduced, and consequently, the friction coefficient decreases.

In all the investigated cases, due to the existence of similar geometrical and hydrodynamical conditions of bottom layer, the friction coefficient has a uniform amount.

In Fig.

6, by increasing the dimensionless slip velocity

coefficient, on the solid wall, the velocity parameter is improved and velocity gradients are decreased. Also, in higher slip velocity coefficients, the effect of solid wall on the reduction in velocity and the effects of friction coefficient on the wall are less. Therefore, the minimum amount of average friction coefficient is obtained in the highest slip velocity coefficient. With the existence of solid nanoparticles in the cooling fluid, the effect of viscosity on the solid wall is improved, and according to Fig.

6, the max-

imum amount of friction coefficient is accomplished in the highest volume fraction of nanoparticles.

Figure

7

indicates the average friction coefficient in Rey- nolds numbers, volume fractions, slip coefficients and different dimensionless lengths at the top layer of microchannel along (CD). According to the figure, it can be concluded that, at the top layer, due to the reduction in microchannel length and enhancement of developed length, the flow at the outlet section of microchannel becomes undeveloped. This behavior causes the creation of velocity gradients in the areas near the wall and the enhancement of shear stress along (CD). Therefore, in all the studied cases, by decreasing the length of microchannel top layer and dimensionless slip velocity, the amount of friction increases insignificantly.

Among the studied cases, in all Reynolds numbers and volume fractions, the dimensionless length of 1/3 has the

X

20 30 40 50 60

Nuave

0 5 10 15 20

= 0.00

= 0.04

= 0.08

Re = 10

Re = 1 l= 2/3

Up Layer

X

40 45 50 55 60

Nuave

0 5 10 15 20

ϕ = 0.00 ϕ = 0.04 ϕ = 0.08

Re = 10

Re = 1 l= 1/3

Up Layer

ϕ ϕ ϕ λ λ

Fig. 9 Local Nusselt number in the top layer of microchannel in slip coefficient of 0.1 along the wall of (CD)

(10)

maximum amount of friction coefficient, and in the top layer with the length of 3/3, due to the development of flow on the direction of fluid motion and reduction in velocity profile changes from the developed length to lateral, the average friction coefficient has the minimum amount.

Evaluating Nusselt number behavior in the top and bottom layers of microchannel

In Fig.

8, the local Nusselt number at the bottom layer of

microchannel along (AB) in Reynolds numbers of 1, 10 and 100 with volume fractions of 0–8% of nanoparticles and dimensionless length of 2/3, 3/3 and 1/3 for slip coefficient of 0.1 has been indicated. The applying of oscillating heat flux to the solid wall influences the profile of local Nusselt number figure all over the channel. The changes of Nusselt number figure profile are obvious in

Reynolds numbers of 10 and 100 because of more heat transfer and thermal removal in this range of Reynolds number. In Reynolds number of 1, due to the low fluid velocity, the removal of heat flux from the microchannel walls influences all fluid areas and the difference of surface temperature and fluid temperature has the minimum amount; therefore, the profile of local Nusselt number figure is not under the influence of applied surface heat flux.

Figure

9

demonstrates the local Nusselt number along (CD) on the top layer of microchannel in Reynolds numbers of 1 and 10 and slip velocity coefficient of 0.1. This study has been conducted for the top layer dimensionless length of 1/3 and 2/3. By observing the behaviors of Nusselt number figures, it can be said that the increase in volume fraction causes the enhancement of Nusselt number, and the change of Nusselt number in the top layer of

ϕ

0.00 0.04 0.08

ΔP/pa

λl= 2/3 Up Layer

ϕ

0.00 0.04 0.08

ΔP/pa

1e + 2 1e + 3 1e + 4 1e + 5

Re = 100

Re = 10

Re = 1

λl= 1/3

Up Layer

ϕ

0.00 0.04 0.08

ΔP/pa

1e + 2 1e + 3 1e + 4 1e + 5 1e + 6

B = 0.001 B = 0.01 B = 0.1

λl= 3/3 Up Layer

B = 0.001 B = 0.01 B = 0.1

Re = 1

Re = 1 Re = 10

Re = 100 1e + 2 1e + 3 1e + 4 1e + 5

Fig. 10 Pressure drop of flow in the top layer of microchannel

(11)

microchannel in Reynolds number of 10, comparing to Reynolds number of 1, is more significant. Among the two studied cases, it can be concluded that, at the top layer of microchannel with the dimensionless length of 1/3, the changes of local Nusselt number are more tangible. The reason is due to the undeveloped flow in low dimensionless length.

Evaluating the average pressure drop in the top and bottom layers and the average Nusselt number at the bottom layer of microchannel

Figures

10

and

11, respectively, indicate the pressure drop

of flow at the top and bottom layers of microchannel in different cases of dimensionless length, volume fraction, Reynolds number and slip velocity coefficient. It can be

understood from the figures that the enhancement of vol- ume fraction, Reynolds number and the top layer length entail the enhancement of average pressure drop. Also, the reduction in slip velocity coefficient causes the enhance- ment of average pressure drop. The reason of this behavior is because of the reduction and depreciation of velocity parameter on solid wall surface, due to the no-slip boundary condition or low dimensionless slip velocity coefficient. In the dimensionless slip velocity coefficient of 0.1, due to the enhancement of velocity vector length on the solid wall surface and the existence of solid surface, less depreciation is created in the momentum of fluid crossing the surface. In all volume fractions and Reynolds numbers of this case, comparing to the similar cases, the amount of pressure drop has the minimum amount.

ϕ

0.00 0.04 0.08

ΔP/pa

λl= 2/3

Down Layer

ϕ

0.00 0.04 0.08

ΔP/pa

1e + 2 1e + 3 1e + 4 1e + 5 1e + 6

B = 0.001 B = 0.01 B = 0.1 Re = 100

Re = 10

Re = 1

λl= 1/3

Down Layer

0.00 0.04ϕ 0.08

ΔP/pa

λl= 3/3

Down Layer

1e + 2 1e + 3 1e + 4 1e + 5 1e + 6

B = 0.001 B = 0.01 B = 0.1

Re = 100

Re = 10

Re = 1

Re = 100

Re = 10

Re = 1

Fig. 11 Pressure drop of flow in the bottom layer of microchannel

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Figure

12

investigates the average Nusselt number at the bottom layer of microchannel according to different dimensionless lengths of the top layer, volume fraction, dimensionless slip velocity coefficient and Reynolds number. In all cases, by increasing volume fraction of nanoparticles, Reynolds number and the dimensionless slip velocity coefficient, the heat transfer mechanisms are improved and the heat transfer coefficient and Nusselt number are increased. Among the studied Reynolds numbers, Reynolds numbers of 100 and 10 have the maximum and minimum values of average Nusselt number, respectively. By increasing the amount of dimensionless slip velocity coefficient, heat transfer increases significantly and the average Nusselt number figures have higher levels.

Thermal resistance

Figure

13

describes the thermal resistance of wall at the bottom layer of microchannel along (AB) in different volume fractions, Reynolds numbers, slip coefficient and dimensionless lengths of the top layer. This factor is defined as an efficient parameter for evaluating the amount of thermal resistance in each state of double-layer microchannel. The reduction in thermal resistance of surface is as the enhancement of heat transfer from the hot wall. The increase in volume fraction of nanoparticles and Reynolds number and dimensionless slip velocity coefficient cause the enhancement of heat transfer and reduction in thermal resistance of surface. In general, the amount of thermal resistance in Reynolds number of 100, comparing to Reynolds number of 10, due to the enhancement of fluid

ϕ

0.00 0.04 0.08

Nuave

0 2 4 6 8 10 12 14

Re = 1 Re = 10

Re = 100

λ_l= 2/3

Down Layer

ϕ

0.00 0.04 0.08

Nuave

0 2 4 6 8 10 12 14

B = 0.001 B = 0.01 B = 0.1

Re = 1 Re = 10 Re = 100

λ_l= 1/3

Down Layer

0.00 0.04ϕ 0.08

Nuave

0 2 4 6 8 10 12 14

Re = 1 Re = 10

Re = 100

λ_l= 3/3

Down Layer

B = 0.001 B = 0.01 B = 0.1

Fig. 12 Average Nusselt number at the bottom layer of microchannel

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velocity and convective heat transfer coefficient, decreases significantly.

Performance evaluation criterion (PEC)

Figure

14

depicts the performance evaluation criterion (PEC) at the bottom layer of microchannel in different dimensionless slip coefficients and dimensionless lengths.

PEC factor is investigated for evaluating the heat transfer enhancement and friction coefficient caused by adding solid nanoparticles to the cooling fluid. In performance evaluation criterion figures, due to the significant heat transfer enhancement, comparing to the friction coefficient, the augmentation of volume fraction of nanoparticles and Reynolds number has an increasing progress. Among the

studied dimensionless lengths, the dimensionless length of 3/3 has the minimum amount of efficiency. Also, the dimensionless length of 1/3 has the maximum amount of performance evaluation criterion.

The main reason of this behavior is due to the optimized proportion of friction coefficient, comparing to the effect of heat transfer enhancement. Therefore, using truncated microchannel in dimensionless lengths of 1/3 and 2/3 and higher Reynolds numbers has more efficiency than the dimensionless length of 3/3. According to the behavior of performance evaluation criterion figure, it can be under- stand that the effect of heat transfer enhancement is more dominant than the friction coefficient enhancement. By increasing the performance evaluation criterion, this issue

ϕ

0.00 0.04 0.08

R*w/K/m W

0.03 0.06 0.09 0.12 0.15

Re = 10

Re = 100 λl= 2/3

Down Layer

ϕ

0.00 0.04 0.08

R*w/K/m W

0.03 0.06 0.09 0.12 0.15

Re = 10

Re = 100 λl = 1/3

Down Layer

0.00 0.04ϕ 0.08

R*w/K/m W

0.04 0.06 0.08 0.10 0.12 0.14

Re = 10

Re = 100 λl= 3/3

Down Layer B = 0.001 B = 0.01 B = 0.1

B = 0.001 B = 0.01 B = 0.1

Fig. 13 Thermal resistance of walls at the bottom layer of microchannel along the wall (AB)

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is proportional to the augmentation of dimensionless slip velocity coefficient.

Conclusions

In this study, the laminar flow and heat transfer in the double-layer microchannel with truncated top layer have been investigated. The main purpose of this study is cooling the microchannel heat sink by using kerosene/

MWCNTs. The important results of this study indicate that among the studied Reynolds numbers, Reynolds numbers of 100 and 1 have the maximum and minimum amounts of average Nusselt number, respectively. The enhancement of volume fraction of nanoparticles, Reynolds number and dimensionless slip velocity coefficient cause considerable augmentation of heat transfer and reduction in thermal resistance of surface. By increasing the Reynolds number, fluid velocity enhances and this factor prevents heat transferring from the hot surface to the central core of flow.

The dimensionless length of 1/3 has the maximum amount of performance evaluation criterion. Therefore, the truncated microchannel design with higher Reynolds numbers and dimensionless lengths of 1/3 and 2/3, comparing to the dimensionless length of 3/3, has more efficiency. The extension of this paper in nanofluid studies, according to our previous works [57–108], affords a good option for engineers to investigate the micro- and nano-simulations.

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