Heat Transfer Enhancement in Ferrofluids Flow in

Journal of Thermal Science and Engineering Applications. Received August 09, 2016; Accepted manuscript posted November 16, 2017. doi:10.1115/1.4038483 Copyright (c) 2017 by ASME

Aditi Sengupta Churchill College University of Cambridge Cambridge CB3 0DS, United Kingdom

ed ite d

HEAT TRANSFER ENHANCEMENT IN FERROFLUIDS FLOW IN MICRO AND MACRO PARALLEL PLATE CHANNELS: A COMPARATIVE NUMERICAL STUDY *

P.S. Ghoshdastidar Department of Mechanical Engineering Indian Institute of Technology Kanpur Kanpur, U.P. 208016, India

py

*Corresponding Author. Member, ASME. Membership No. 1932755. E-Mail: [email protected]

Co

ABSTRACT

Ma nu

sc rip

tN

ot

This paper presents a comparative numerical study of heat transfer enhancement in steady, laminar, hydrodynamically fully developed flow of water-based ferrofluids under no magnetic field in micro and macro parallel plate channels subjected to constant equal heat fluxes on its top and bottom, considering Brownian diffusion and thermophoresis of ferroparticles in the base fluid. While the microchannel results match very well with the experimental data for water in an equivalent microtube [18], the numerically predicted enhancement factor in ferrofluids is much below that for the same microtube. A detailed parametric study points to possible inaccuracies in the experimental results of Kurtoglu et al. [18] for ferrofluids. The nanoparticle concentration profiles in the microchannel flow reveal that (a) the nanoparticle concentration at the wall increases with axial distance; (b) the wall nanoparticle concentration decreases with increasing heat flux; and (c) the concentration profile of nanoparticles is parabolic at the exit. A comparison of thermally developing flow in microchannel and macrochannel of the same length (0.025 m) indicates that the enhancement factor at the microchannel exit is 1.089 which is only marginally higher than that at the macrochannel exit in the heat flux range of 20 - 80 kW/m2. On the other hand, for the thermally fully developed flow in both microchannel and macrochannel of the same length (0.54 m) the maximum enhancement factor for the macrochannel is 1.7, as compared to 1.1 for the microchannel, in the heat flux range of 1 - 4 kW/m2.

INTRODUCTION

ed

Ferrofluids are engineered colloids of magnetite (Fe3O4) nanoparticles in base fluid (water or oil). They are

pt

a special class of nanofluids. The main difference between a usual nanofluid and ferrofluid is that in the latter

ce

the flow and temperature fields can be altered by application of an external magnetic field [1]. Significant enhancement in thermal conductivity in magnetite nanofluids has been observed under the influence of magnetic

Ac

field [2]. While Goharkhah and Ashjaee [1] reported heat transfer enhancement under the action of alternating non-uniform magnetic field in laminar parallel plate flow subjected to constant equal heat fluxes at the top and bottom surfaces, Azizian et al. [2] showed experimentally along with supporting simulations that large enhancement in local convective heat transfer coefficient occurs by increasing magnetic field strength and gradient for laminar tube flow subjected to uniform heat flux. Ferrofluids can be used for cooling microdevices as well as in medical applications. TSEA-16-1220 Ghoshdastidar

Page 1

Downloaded From: http://thermalscienceapplication.asmedigitalcollection.asme.org/ on 11/16/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use


Choi [3] introduced to the world the concept of nanofluids which have larger thermal conductivity and single-phase heat transfer coefficient as compared to those of base fluids. The volume fraction of nanoparticles usually does not exceed 5%. Many experimental studies reporting measurement of thermal conductivities of nanofluids are found in the literature [4-9]. Several researchers attempted to explain the abnormal enhancement of thermal conductivity in terms of Brownian motion of nanoparticles and nanoconvection arising out of it [10-

ed ite d

12]. Three numerical approaches are found in the literature to model nanofluids flow and heat transfer. They are: (i) Homogeneous Flow model proposed by Choi [3] which assumes that the governing equations of mass, momentum and energy are directly applicable to nanofluids. Only the properties are modified by the bulk

py

volume fractions of nanoparticles. (ii) Dispersion Model proposed by Xuan and Roetzel [13] which is built on

Co

the assumption that augmentation of convective heat transfer originates from (a) higher thermal conductivity, and (b) the dispersion of nanoparticles. (iii) Non-homogeneous or Heterogeneous Flow Model of Buongiorno [14]

ot

which says that nanoparticles can move homogeneously with the base fluid but they also possess a slip velocity

tN

with respect to the base fluid, which is due to Brownian motion and thermophoresis. He proposed the 2component (nanoparticles + base fluid) - 4-equation (2 mass + 1 momentum + 1 energy) model. This model has

sc rip

been extensively used by many researchers for nanofluids flow and heat transfer in various geometries with promising results. Recently, Rossi diSchio et al. [15] used the Buongiorno model to simulate steady, laminar heat transfer in parallel plate flow of nanofluids with linearly varying as well as longitudinally periodic wall

Ma nu

temperature by the finite element method. It is reported that while the first case points towards weak dependence of the temperature distribution on nanoparticle concentration, the second case gives rise to nonhomogeneities in the nanoparticle concentration, and hence use of homogeneous model would be faulty.

ed

The above literature review reveals that so far no work reported calculation of enhancement of heat transfer

pt

coefficient in parallel plate flow of ferrofluids under no magnetic field. The present work is an attempt to fill

ce

this gap in the existing literature.

Ac

The objectives of the present study are: (i) to carry out a finite-difference based numerical simulation of steady, laminar, hydrodynamically fully developed flow of water-based ferrofluids under no magnetic field in micro and macro parallel plate channels subjected to constant equal heat fluxes at its top and bottom; and (ii) to calculate the enhancement in heat transfer coefficient for different volume fractions of nanoparticles. The nonhomogeneous flow model of Buongiorno [14] has been used. Three cases have been considered, namely, (a) thermally developing flow in a microchannel of 257 micron height and 0.025 m length; (b) thermally developing flow in a macrochannel of 0.003 m height and 0.025 m [TSEA-16-1220 Ghoshdastidar]

Page 2



length; and (c) thermally fully developed flow at the exit of a macrochannel of 0.003 m height and 0.54 m length.

Brownian diffusion coefficient (m2/s)

DT

Thermophoretic diffusion coefficient (m2/s)

d

Diameter (m)

h

Heat Transfer Coefficient (W/m2 K)

H

Channel Height (m)

j

Mass Flux (kg/m2 s)

k

Thermal conductivity (W/m K)

kB

Boltzmann constant (J/K)

L

Channel Length (m)

Ma nu

Le

Lewis Number

Brownian to Thermophoretic diffusivity ratio

Nu

Local Nusselt Number

ed

NBT

Sc T

ce

Re

Ac

q//

Unit normal vector

pt

n Pr

py

DB

Co

Specific heat (J/kg K)

ot

c

tN

Half-height of the channel (=H/2)

sc rip

b

ed ite d

NOMENCLATURE

Prandtl Number

Heat flux (W/m2) Reynolds Number Schmidt Number Temperature (K)

Ti

Inlet Temperature (K)

u

Velocity in the x-direction (m/s)

[TSEA-16-1220 Ghoshdastidar]

Page 3



x

Coordinate in the horizontal direction (m)

y

Coordinate in the vertical direction (m)



Thermal diffusivity (m2/s)



Non-dimensional temperature, also eq. (13)

µ

Dynamic viscosity (N s/m2)



ed ite d

Greek Letters

Kinematic viscosity (m2/s)



py

Density (kg/m3)

ρ

tN

ot

Co

Volume fraction of nanoparticles

sc rip

Subscripts avg

Average

Brownian diffusion

bf

Base Fluid

Ma nu

B

f

Fluid

i

Inlet

Nanofluid

ed

nf m

pt

p

Mean, bulk Nanoparticle

Quantities based on bulk volume fraction

T

Thermophoresis

Ac

ce

o

Superscripts *

Non-dimensional

TSEA-16-1220 Ghoshdastidar

4



PROBLEM FORMULATION Figure 1 shows the physical problem and the computational domain. The flow is hydrodynamically fully developed at the inlet of the parallel plate channel whose width is much larger than the length and the gap

ed ite d

between the plates. The temperature and nanoparticle concentration are uniform at the entrance. The top and bottom plates are subjected to constant equal heat fluxes. The assumptions in the present analysis are listed below.

py

Assumptions

Co

(i) Dilute mixture of nanoparticles in base fluid (less than or equal to 5%) (ii) Nanoparticles are in local thermal equilibrium with the base fluid

ot

(iii) No chemical reaction

(v) Laminar flow ( Re H

tN

(iv) Incompressible fluid

 1400)

sc rip

(vi) Steady 2D flow, heat and mass transfer (vii) Constant thermophysical properties

Ma nu

(viii) Negligible axial conduction and diffusion (ix) Negligible viscous dissipation Justification of Assumptions

 is less

ed

Assumption (i) is valid for most studies on ferrofluids published in open literature, in which

than 5%. Assumption (ii) is not obvious and needs an elaborate explanation in terms of time constants

pt

for heat conduction within the ferroparticles, and within the base fluid in the proximity of the Buongiorno [14] estimated these time constants as

ce

ferroparticles.

d p2 /  p ,

and

d p2 /  bf ,

Ac

respectively. For ferroparticles of 25 nm in diameter (as in the present study) in water, the heat transfer time constants are about 0.3 ns and 4.3 ns, for the ferroparticles and water around them, respectively, which are much smaller than the time constants for Brownian diffusion (~10-3 s) and thermophoresis (~10-2 s) as reported by Buongiorno [18]. Therefore, as the ferroparticles swim in the base fluid, they achieve thermal equilibrium with it very fast, which establishes assumption (ii). Since ferroparticles are chemically inert with respect to water the assumption (iii) is correct. The fluid is incompressible


5



because the Mach number involved in the study is less than 0.3. Hence, the assumption (iv) is justified. Assumption (v) and (vi) are consistent with the experimental work with which the present results are compared with.

Although heat flux on the microchannel is high (~105 W/m2) the length of the

microchannel is low (0.025 m). Hence, the wall temperature rise from the inlet to the outlet is not so high as to consider temperature-dependent thermal conductivity and viscosity in the problem

ed ite d

formulation. In the case of macrochannel (Case B) the heat flux is of the order of 10 4 W/m2 and the length is 0.025 m, and for Case C the heat flux is of the order of 10 3 W/m2 and the length is 0.54 m. Both the cases result in moderate rise in the wall temperature. The variation of properties, however,

py

with the local volume fraction of ferroparticles, has not been considered in this study for the sake of

Co

simplicity. Instead, an average value has been taken. This is the logic behind assumption (vii). The axial conduction and diffusion have been neglected (assumption (viii)) since RePr and ReSc are greater

ot

than 100. Hence, axial thermal/concentration convection dominates over axial conduction/diffusion.

Governing Equations

sc rip

flow speed is low (assumption (ix)).

tN

The viscous dissipation is neglected since the base fluid is water which has a moderate viscosity and the

The Buongiorno model [14] which emphasizes the role of Brownian diffusion and thermophoresis as slip mechanisms producing relative velocity between the nanoparticles and the base fluid has been used to obtain

Ma nu

the governing equations which are given below. Nanoparticle Continuity Equation:

 * 1  2 * 1  2 u   x* Re Sc y*2 Re ScN BT y*2

pt

Energy Equation:

ce

  *      *    1  2 1     +     Re Pr y *2 Le Re Pr  x*  x*   y *  y *  x*

Ac

u*

(1)

ed

*

1 Le Re Pr N BT

   2    2   *    *    x   y  

(2)

Non-dimensional Parameters:

x* 

 c uavg H T  Ti x * y *  u Pr  nf nf   // y  Re  *  u  knf H H  m uavg q H k nf TSEA-16-1220 Ghoshdastidar

6



k BTi 3bf d p

  0.26

N BT 

kbf 2kbf  k p

DT 

bf m  nf knf Le  Sc   p c p DBm  nf DB bf

DBTi  nf knf

ed ite d

DB 

nf Hq //

k B = 1.380650 x 10-23 J/K.

py

Note that T is in kelvin.

Co

Inlet and Boundary Conditions At the inlet (x=0 or x*= 0):

(3)

tN

ot

u *  6 y * (1  y* )

Note that since the flow is fully developed at the inlet the velocity distribution is known from the

sc rip

analytical solution [20].

T  Ti or   0

Ma nu

  m or  *  1

(4) (5)

At the walls (y = 0 and H):

No mass flux of nanoparticles, i.e., j.n  0

ed

or j p  j p , B  j p ,T    p DB    p DT

T 0 T

(6)

pt

In the dimensionless form,

ce

 1   * y N BT y *

Ac

(7)

Also,

At y = 0 (or y* =0),

kf

At y = H (or y* = 1),

kf

T   q // or *  1 y y

(8)

T   q // or *  1 y y

(9)


7



Heat Transfer Coefficient (h) At the lower wall:

 T    knf   y  y 0 h Tw  Tm

ed ite d

(10)

At the upper wall:

py

 T   knf    y  y  H h Tw  Tm

Co

Where H

ot

 u( y)T ( y)dy 0

uav H

(12)

tN

Tm 

(11)

sc rip

In the present case, h at both upper and lower walls is the same since the problem is axi-symmetric from the point of view of flow, temperature and concentration. However, the present formulation can also be used even if the top and bottom plates are subjected to unequal constant heat fluxes as the coordinate system

Ma nu

has been placed on the bottom plate (see Fig.1). Local Nusselt number (Nu)

hH knf

(13)

ed

Nu 

havg H k nf

(14)

ce

Nuavg 

pt

Average Nusselt Number

1 L hdx L 0

Ac

where havg 

(15)

Enhancement Factor

Enhancement Factor =

hnf

(16)

hbf

The subscript, bf, indicates base fluid which is water in the present study.


8



METHOD OF SOLUTION Equations (1) and (2) being of parabolic nature are solved by the pure implicit finite difference scheme. A uniform grid has been used. It may be noted that here x-coordinate behaves like time coordinate in heat conduction equation. Hence, a marching type solution procedure is applied. The image-point technique has been used to handle the thermal and concentration boundary conditions at the walls. Since eqs. (1) and (2)

guessed values of

ed ite d

are mutually coupled they are solved iteratively. At the station, i + 1, the energy equation is solved using

 . The new  values are then substituted into the nanoparticle continuity equation. The

process goes back and forth till there is no change in



and

 values at the station, i + 1. Following this

py

procedure the solution is obtained at the stations, i+2, i +3, and so on till the exit of the channel is reached.

Co

Gauss-Seidel iteration is used to solve the set of linear simultaneous equations (having tri-diagonal

tN

ot

coefficient matrix) at each x-station.

sc rip

RESULTS AND DISCUSSION Case A: Microchannel (Thermally Developing Flow)

The input data used are provided in Table 1. The dimensions of the microchannel are chosen so that it has the

Ma nu

same hydraulic diameter, length and Reynolds number as those of the microtube used in Kurtoglu et al. [18] in their experiment on heat transfer in ferrofluid under no magnetic field. The objective is to compare the present microchannel results with the microtube ones reported in Kurtoglu at al. [18] since no experimental data for

ed

microchannel flow of ferrofluids in a heated microchannel under no magnetic field are available in literature. The correlations for thermophysical properties of the ferrofluid are listed in Appendix A.

pt

Grid Independence Test

ce

Grid independence tests are performed for five different grid spacings (from very coarse to very fine) in the y-

Nu vs. x are compared with

Ac

direction keeping the number of grid points in x-direction constant. The plots of

each other for different grid sizes for this purpose. A sample grid independence test is shown in Fig.2 for

q //  7x105 W/m2 and   0.05 . It is observed that the maximum change in the Nu vs. x plots is less than 1% as the number of grid points is increased from 151x101 to 151x121. Therefore, in save on CPU time the optimum number of grid points is taken as 151 x 101, the first number indicating the number of grid points in the


9



x -direction and the second in the y -direction. The results of the grid independence test are given in Table 2 for all the parameters used in this study. Table 3 shows the non-dimensional parameters for the microchannel. Validation with the Experimental Results of Kurtoglu et al. [18] for Pure Fluid in Microtube with that in an Equivalent Microchannel Figure 3 shows a comparison of the graphs of Tw - Tm at the channel exit vs. Wall Heat Flux predicted by the

ed ite d

present numerical simulation for an equivalent microchannel with that of the experimental results of Kurtoglu et al. [18] for a microtube when the fluid is pure water (   0 ). The plots depict an excellent tally of the present numerical results with that of experimental results for microtube. Hence, it can be concluded that for the same

py

hydraulic diameter, length and Reynolds number the thermal performance of a microchannel is identical with

Co

that of a microtube under laminar condition for pure fluid.

The equivalence of the present microchannel with the microtube of Kurtoglu et al. [18] is established by

ot

forcing the hydraulic diameter and cross-sectional area of the microchannel to be equal to those of the

sc rip

The hydraulic diameter ( Dh ) of the microchannel is:

tN

microtube. The height and width of the microchannel are calculated as follows.

Dh  4 (Cross-sectional area)/(Wetted Perimeter) = 4 (W x H)/(2W) = 2H

(17)

where W is the width of the channel, considered to be infinitely wide.

m .

Ma nu

Note that the diameter of the microtube [18] is 514

Since the hydraulic diameter of the microchannel is equal to the diameter of the microtube, from eq.(8) the

514  257 m 2

pt

H=

ed

following can be written.

m .

ce

Thus, the height of the present microchannel is 257

(18)

Temperature and Concentration Profiles

Ac

Figure 4 depicts the temperature profile (T vs. y) as a function of axial distance (x) in the microchannel for the ferrofluid having a nanoparticle volume fraction of 5% at

q // = 5 x 105 W/m2. The temperature in the channel

increases along the length of the channel and the profile is symmetric as expected. The classical meniscus type profile is obtained at the tube exit.


10


Journal of Thermal Science and Engineering Applications. Received August 09, 2016; Accepted manuscript posted November 16, 2017. doi:10.1115/1.4038483 Copyright (c) 2017 by ASME Figure 5 shows the nanoparticle concentration profile (  vs. y) as a function of axial distance (x). The profiles indicate that the concentration of nanoparticles at the wall is slightly higher than that at the core implying migration of nanoparticles from the core to the wall. However, the variation of concentration with the distance normal to the wall is little indicating the flow is nearly homogeneous. The concentration at the wall also

temperature profile.

x. The concentration distribution has a parabolic shape at the exit similar to the On increasing the wall heat flux, the wall concentration decreases because of

ed ite d

slightly increases with

thermophoresis effect (the graph not shown). Temperature Difference at the Exit vs. Applied Heat Flux

py

Figure 6 reveals the wall-fluid temperature difference at the outlet increases with applied heat flux for

 = 5%

Co

for both numerical simulation and experiment. However, the comparison with the microtube [18] indicates a large quantitative deviation from the present microchannel result. The maximum temperature difference based

ot

on the numerical model is 62 K whereas the same for the experiment on microtube [18] is only 40 K.

tN

Exit Heat transfer Coefficient vs. Applied Heat Flux

with applied heat flux, for ferrofluid with

sc rip

The variation of heat transfer coefficient at the channel exit

nanoparticle volume fraction of 5% is shown in Fig.7. The computed results show that exit heat transfer coefficient is independent of the value of the wall heat flux. On the contrary, the experimental data for

Ma nu

microtube reveal an appreciable growth in heat transfer coefficient with heat flux. The maximum heat transfer coefficient in the experiment is around 19000 W/m2 K whereas the numerically predicted heat transfer coefficient is nearly 13800 W/m2 K.

ed

Enhancement Factor vs. x

Figure 8 shows the variation of enhancement factor with the axial distance in the microchannel.

The

pt

enhancement in heat is low (1.06) very near the inlet and increases to a value of 1.0875 at the exit of the

ce

microchannel. It may be noted that the flow is not thermally fully developed at the channel exit. The trend of

Ac

the graph can be explained as follows. Since

h~

kf

t

, where k f is the thermal conductivity of the fluid and

is the thermal boundary layer thickness, Enhancement Factor

~

knf  t bf

kbf  t nf

t

. From Table 3, corresponding to

q //  7x105 W/m2 and   0, Re = 998.06439 and Pr = 6.10554 and hence, RePr = 6093.7221. For

  0.05 ,

RePr = 6797.5387 for the same wall heat flux. Clearly, thermal entry length for the ferrofluid is TSEA-16-1220 Ghoshdastidar

11



higher than that of the base fluid. Hence,

Since

k nf kbf

 t bf  t nf

is greater than 1 throughout the length of the microchannel.

 1 always, the enhancement factor exceeds 1. However, very near the inlet

 t bf  t nf

is low as

ed ite d

compared to that away from the inlet and, hence the enhancement factor is low near the entrance and then it gradually increases till the exit of the channel preceded by a steep rise. Enhancement Factor vs. Applied Heat Flux

The variation of enhancement factor at the channel exit with applied heat flux, for ferrofluid with nanoparticle

py

concentration of 5% is shown in Fig.9. The computed results show an enhancement in heat transfer coefficient of

the microtube. For

  0.0165

Co

about 9% at the channel exit whereas the experimental results [18] show an enhancement of 60% at the exit of , in the same range of wall heat flux the numerically predicted enhancement is

  0.0335

, in the same range of wall heat

tN

ot

3% as compared to 20% in the experiment (figure not shown). For

flux the enhancement at the channel outlet is slightly greater than 5% as compared to 32.5% at the outlet of the

sc rip

microtube (figure not shown). All in all, for both the cases the enhancement factor increases with the volume fraction of ferroparticles, which is as expected since the conductivity of ferrofluid increases with the concentration of ferroparticles. A major point of difference is that in the numerical simulation no effect of

Ma nu

applied heat flux on heat transfer coefficient is predicted.

On the contrary, the experimental results for

microtube show an increase of heat transfer coefficient with applied heat flux for both pure and ferrofluids. It may be noted that Srivastava [21] conducted a numerical study on microtube flow of water subjected to constant

ed

heat flux and came to the conclusion that in the range of diameter of 50

m the continuum assumption is not valid and

pt

is invariant with the wall heat flux. Below the diameter of 50

m to 150 m heat transfer coefficient

m the heat transfer coefficient does not depend on wall heat flux. The numerical

ce

above the diameter of 150

Ac

results for the present microchannel which has a hydraulic diameter of 257 m are in line with the aforementioned finding. Furthermore, the high heat transfer coefficient and enhancement factor in ferrofluids flow in a microtube under no magnetic field as reported by Kurtoglu et al. [18] are in sharp contrast with Goharkhah and Ashjaee [1], and Azizian et al. [2] who clearly opined that unless a magnetic field was applied to ferrofluids a large enhancement in heat transfer coefficient was not possible.


12



Hence, the above comparison clearly points to possible inaccuracies in the experimental results of Kurtoglu et al. [18] for ferrofluids. Case B: Macrochannel (Thermally Developing Flow) The dimensions of the macrochannel are given in Table 4. The length of the channel is same as that of the microchannel. The width of the channel is much larger than the height. However, Re, Sc, Pr, and Le are same as

ed ite d

for the microchannel flow (Table 3). The heat flux range considered is 20000 - 80000 W/m2 in order to limit the fluid temperature to a value below 100oC, boiling point of water at a pressure of 1 bar. The flow is thermally developing.

py

Exit Heat Transfer Coefficient vs. Applied Heat Flux

Co

The heat transfer coefficient at the channel exit does not change with applied heat flux, for the nanoparticle volume fraction of 0.05 as observed in Fig.10 for both micro and macrochannels of the same length as expected.

ot

However, microchannel gives rise to much higher heat transfer coefficient. Since h is inversely proportional to

tN

Tw  Tm , for the same Re and Pr the microchannel flow subjected to constant and equal wall heat fluxes leads to larger heat transfer coefficient as the difference between the wall and bulk fluid temperature is smaller.

sc rip

Enhancement Factor vs. Applied Heat Flux

The variation enhancement factor at the exit of the microchannel and macrochannel with applied heat flux, for

Ma nu

ferrofluid with nanoparticle volume fraction of 5 % is shown in Fig.11. The macrochannel results show an enhancement of 7.85% in heat transfer coefficient as opposed to 8.85% in the microchannel. In both cases the channel outlet enhancement factor remains invariant with wall heat flux. Case C: Macrochannel (Thermally Fully Developed Flow at the Channel Exit))

ed

The dimensions of the macrochannel are indicated in Table 5. The length of the channel is 0.54 m which is an

pt

order of magnitude larger than that of the earlier two channels. The non-dimensional parameters remain same as

ce

before. The range of heat flux in the present case is limited to 1000 - 4000 W/m2 which is much lower than in

Ac

Case A and Case B in order to limit the fluid temperature to a value below 100 oC. The flow is thermally fully developed at the channel exit. Validation with the Analytical Solution [20] for Pure Fluid in the Macrochannel The analytical solution of the fully developed temperature profile for the heat transfer in parallel plate flow subjected to constant heat equal heat fluxes is given by the following expression [20].


13



2 4 Tw  T 35  5 3  y  1  y   θ=          Tw  Tm 17  8 4  b  8  b  

where

(19)

Tw is the upper wall temperature, Tm is the mean fluid temperature, b is the half-height of the channel,

y  0 is the centreline of the channel, and y  b is at the upper wall of the channel. The temperature

ed ite d

distribution is in the upper half of the channel. Figure 12 shows a graphical comparison of the present numerical solution with the analytical solution (eq. (19)) for the pure fluid (   0) flow between parallel plates subjected to

q //  1000 W/m2 on both sides. The plots clearly reveal an excellent match of the numerically

py

predicted temperature profile with the exact one. Furthermore, the numerically predicted Nusselt number at the

Co

channel exit has an excellent agreement with the exact solution (Nu = 4.1176). Enhancement Factor vs. Applied Heat Flux

ot

The variation of enhancement factor at the channel exit of the microchannel and macrochannel of the same

tN

length with applied heat flux, for ferrofluid with nanoparticle volume fraction of 5% is shown in Fig.13. The microchannel results show an enhancement of 11.25% in heat transfer coefficient as compared to the maximum

sc rip

value of 68% in the case macrochannel. Stronger thermophoresis effect in the case of macrochannel due to greater heating because of larger surface area may be attributed to the significant enhancement in heat transfer

Ma nu

coefficient in the case of macrochannel. It is also noted that enhancement factor slightly increases with heat flux in the case of macrochannel whereas no such thing is visible in the case of microchannel. CONCLUSIONS

The work reports a computational study of heat transfer in laminar, hydrodynamic fully developed flow of

ed

ferrofluid in a parallel plate channel subjected to constant equal heat fluxes on its upper and lower plates. Three

 m height and

pt

cases have been considered, namely (i) thermally developing flow in a microchannel of 257

ce

0.025 m length; (ii) thermally developing flow in a macrochannel of 0.003 m height and 0.025 m length; and (iii)

Ac

thermally fully developed flow at the channel exit in a macrochannel of 0.003 m height and 0.54 m length. The non-homogeneous flow model of Buongiorno [14] which takes into account Brownian diffusion and thermophoresis of nanoparticles in the base fluid is used to simulate heat and mass transfer in the nanofluid. Finite-difference method has been applied to discretize mutually coupled governing equations of continuity of nanoparticles and energy equation of nanofluid. The concentration profile reveals that the flow is nearly homogeneous in the microchannel. For all three cases, heat transfer coefficient increases with volume fraction


14



of nanoparticles (Fe3O4) and enhancement factor increases from the inlet to the outlet. The microchannel gives rise to much larger heat transfer coefficient than the macrochannels. The numerically predicted heat transfer coefficient at the channel exit in Case (i) is much lower than that reported in the experiment for ferrofluid flow in an equivalent microtube conducted by Kurtoglu et al. [18] in the heat flux range of 0.18 - 0.7 MW/m2. However, the present numerical results are in conformity with earlier works [1, 2] which clearly indicate

ed ite d

impossibility of obtaining very high heat transfer coefficients without application of magnetic field on ferrofluids. This and the comparison of the plots of enhancement factor vs. applied heat flux with those of microtube also points to possible errors in the experimental results of Kurtoglu et al. [18] for ferrofluids. The

py

results for Case (ii) reveal that the enhancement factor at the exit of the microchannel flow is only marginally

Co

higher than that for the macrochannel flow, the maximum value being about 1.089, the heat flux range being 20 – 80 kW/m2. On the other hand, for thermally fully developed flow in both microchannel and macrochannel of the

ot

same length (0.54 m) there is a considerable increase in the enhancement factor at the exit of the macrochannel,

tN

the maximum being nearly 1.7, whereas the for the microchannel it is around 1.1 for the heat flux range of 1 – 4

sc rip

kW/m2.

REFERENCES

1. Goharkhah, Mohammad, and Ashjaee, Mehdi, 2014, "Effect of an Alternating Nonuniform Magnetic

Ma nu

Field on Ferrofluid Flow and Heat Transfer in a Channel", Journal of Magnetism and Magnetic Materials, Vol. 362, pp. 80-89.

2. Azizian, R., Doroodchi, E., McKrell, T., Buongiorno, J., Hu, L. W., and Moghtaderi, B., 2014, "Effect of

ed

Magnetic Field on Laminar Convective Heat Transfer of Magnetite Nanofluids", International Journal of Heat and Mass Transfer, Vol. 68, pp. 94-109.

pt

3. Choi, S., 1995, "Enhancing Thermal Conductivity of Fluids with Nanoparticles", Developments and

ce

Applications of Non-Newtonian Flows, D.A. Siginer, and H.P. Wang, eds., ASME, New York, FED-

Ac

Vol.231/MD-Vol.66, pp. 99-105. 4. Eastman, J.A., Choi, S.U.S., Li, S., Thompson, L.J., and Lee, S., 1996, "Enhancing Thermal Conductivity Through the Development of Nanofluids", Proc. Mat. Res. Soc., Symp., Vol. 457, pp. 3-11. 5. Xuan, Y., and Li, Q., 2000, "Heat Transfer Enhancement of Nanofluids", International Journal of Heat and Fluid Flow, Vol. 21, No. 1, pp. 58-64.


15



6. Eastman, J.A., Choi, S.U.S., Li, S., Yu, W., and Thompson, L.J., 2001, "Anomalously Increased Effective Thermal Conductivities of Ethylene Glycol-based Nanofluids Containing Copper Nanoparticles", Applied Physics Letters, Vol. 78, No.6, pp. 718-720. 7.

Wen, D., and Ding, Y., 2004, "Effect on Heat Transfer of Particle Migration in Suspensions of

Nanoparticles Flowing Through Minichannels", ASME 2nd International Conference on Microchannels and

ed ite d

Minichannels, Rochester, New York, USA, June 17-19, Paper No. ICMM2004-2434, pp. 939-946.

8. Nan, C.W., Birringer, R., Clarke, D.R., and Gleiter, H., 1997, "Effective Thermal Conductivity of Particulate Composites with Interfacial Thermal Resistance", Journal of Applied Physics, Vol. 81, No.10, pp.

py

6692-6699.

Co

9. Keblinski, P., Phillpot, S.R., Choi, S.U.S., and Eastman, J.A., 2002, "Mechanisms of Heat Flow in Suspensions of Nano-Sized Particles (Nanofluids)", International Journal of Heat and Mass Transfer, Vol. 45,

ot

No. 4, pp. 855-863.

tN

10. Jang, S.P., and Choi, S.U.S., 2004, "Role of Brownian Motion in the Enhanced Thermal Conductivity of Nanofluids", Applied Physics Letters, Vol. 84, No. 21, pp. 4316-4318.

sc rip

11. Koo, J., and Kleinstreuer, C., 2004, "A New Thermal Conductivity Model for Nanofluids", Journal of Nanoparticle Research, Vol. 6, No. 6, pp. 577-588.

12. Prasher, R., Bhattacharya, P., and Phelan, P.E., 2005, "Brownian Motion-based Convective-Conductive

Ma nu

Model for the Effective Thermal Conductivity of Nanofluids", ASME Journal of Heat Transfer, Vol. 128, No. 6, pp. 588-595.

13. Xuan, Y., and Roetzel, W., 2000, "Conceptions of Heat Transfer Correlation of Nanofluids",

ed

International Journal of Heat and Mass Transfer, Vol. 43, No. 19, pp. 3701-3707.

Rossi di Schio, E., Celli, M., and Barletta, A., 2014, "Effects of Brownian Diffusion and

Ac

15.

ce

No. 3, pp. 240-250.

pt

14. Buongiorno, J., 2006, "Convective Transport in Nanofluids", ASME Journal of Heat Transfer, Vol. 128,

Thermophoresis on the Laminar Forced Convection of a Nanofluid in a Channel", ASME Journal of Heat Transfer, Vol. 136, No. 2, p. 022401. 16. Brinkman, H.C., 1952, "The Viscosity of Concentrated Suspensions and Solutions", J. Chem. Phys., Vol. 20, pp. 571-581. 17. Maxwell, J.C., A Treatise on Electricity and Magnetism, 1881, 2nd Edition, Clarendon, Oxford.


16



18. Kurtoglu, Evrim, Kaya, Alihan., Gozuacik, Devrim, Funda Yagci Acar, Havva, and Kosar, Ali, 2014, "Experimental Study on Convective Heat Transfer Performance of Iron Oxide Based Ferrofluids in MicroTubes", ASME Journal of Thermal Science and Engineering Applications, Vol. 6, p.034501. 19. Ghoshdastidar, P.S., 2012, Heat Transfer, 2nd Edition, Oxford University Press, New Delhi. 20. Burmeister, L.C., 1993, Convective Heat Transfer, 2nd Edition, John Wiley & Sons, New York.

Ac

ce

pt

ed

Ma nu

sc rip

tN

ot

Co

py

Microtube Flow, M.Tech Thesis, Indian Institute of Technology Kanpur, India.

ed ite d

21. Srivastava, Nipun, 2011, A Numerical Investigation of Convective Heat Transfer in Macro and


17



APPENDIX A

CORRELATIONS FOR THERMOPHYSICAL PROPERTIES OF FERROFLUID

Specific Heat [14]

 p c p  (1   ) bf cbf  nf

ed ite d

cnf 

Density [14]

Viscosity [16]

Co

 nf  bf (1  2.5 )

(A.3)

(A.4)

Ac

ce

pt

ed

Ma nu

sc rip

[2kbf  k p   (k p  kbf )]

tN

[2kbf  k p  2 (k p  kbf )]

ot

Thermal Conductivity [17]

k nf  kbf

(A.2)

py

 nf   p  (1   ) bf

(A.1)


18



List of Tables Table 1 Input Data for the Microchannel Table 2 Optimum Number of Grid Points based on Grid Independence Tests Table 3 Non-dimensional Parameters for the Microchannel

ed ite d

Table 4 Dimensions of the Macrochannel (Thermally Developing Flow)

Table 5 Dimensions of the Macrochannel (Thermally Fully Developed Flow at the Channel

Ac

ce

pt

ed

Ma nu

sc rip

tN

ot

Co

py

Exit)


19



List of Figures Fig. 1 Physical problem and the computational domain Fig.2 A sample grid independence for the microchannel for q //  7x10 5 W/m2 and   0.05 Fig.3 Comparison of the difference of the wall and bulk fluid temperature at the exit vs. wall heat flux predicted by the present numerical simulation with the experimental data [18] for heat

ed ite d

transfer in pure fluid (water) flow in a microtube

Fig. 4 Temperature profiles in the microchannel for q // = 5x105 W/m2 and   0.05 at three different axial locations

Co

py

Fig. 5 Concentration profiles in the microchannel for q // = 5x105 W/m2 and   0.05 at three different axial locations Fig. 6 Comparison of computed exit wall-fluid temperature difference in the microchannel

ot

with the experimental results for microtube [18] for   5%

Fig.7 Comparison of the numerically predicted channel exit heat transfer coefficient with that

tN

for microtube [18] for   5%

sc rip

Fig.8 Axial variation of enhancement factor for q // = 7x105 W/m2 and  = 0.05 in the microchannel

microtube [18] for   5%

Ma nu

Fig. 9 Comparison of enhancement factor at the exit of the microchannel with that of

Fig. 10 Comparison of microchannel and macrochannel exit heat transfer coefficients in the thermally developing flow for   5%

Fig. 11 Comparison of enhancement factor at the exit of the microchannel and macrochannel

ed

in the thermally developing flow for   5%

pt

Fig.12 Comparison of the numerically predicted fully developed temperature profile (θ vs. y)

Ac

W/m2

ce

with the analytical solution for the macrochannel flow of pure fluid (water) at q// = 1000

Fig. 13 Comparison of enhancement factor at the exit of the microchannel and macrochannel for   5% when the flow is thermally developed at the exit


20



Table 1 Input Data for the Microchannel Value

Length of channel

0.025 m

Height of channel

257x10-6 m

Inlet Temperature

298 K

Volumetric Flow Rate

0.36x 10-6 m3/s

Density of water at 298 K

997.1 kg/m3

Dynamic Viscosity of water at 298 K

0.000891 kg/m s

Specific Heat of water at 298 K

4179 J/kg K

Thermal Conductivity of water at 298 K

0.6 W/m K

Diameter of nanoparticle

25 nm

Density of nanoparticle at 298 K

5170 kg/m3

Specific heat of nanoparticle at 298 K

649.625 J/kg K

ot

Co

py

ed ite d

Input Parameter

7 W/m K

Ac

ce

pt

ed

Ma nu

sc rip

tN

Thermal conductivity of nanoparticle at 298 K


21



Table 2 Optimum Number of Grid Points based on Grid Independence Tests

No. of grid No. of grid Average points in x points in y Nusselt direction direction Number (Nuavg)

1.8x105 1.8x105 1.8x105 1.8x105

0.00 0.0165 0.0335 0.05

151 151 151 151

81 71 61 51

2.8x105 2.8x105 2.8x105 2.8x105

0.00 0.0165 0.0335 0.05

151 151 151 151

91 81 71 61

3.5x105 3.5x105 3.5x105 3.5x105

0.00 0.0165 0.0335 0.05

151 151 151 151

4.2x105 4.2x105 4.2x105 4.2x105

0.00 0.0165 0.0335 0.05

151 151 151 151

5.8x105 5.8x105 5.8x105 5.8x105

0.00 0.0165 0.0335 0.05

7.0x105 7.0x105 7.0x105 7.0x105

0.00 0.0165 0.0335 0.05

Co

25.2720 25.3673 25.4691 25.5728

111 101 91 81

25.2788 25.3731 25.4745 25.5778

151 151 151 151

121 111 101 91

25.2833 25.3774 25.4786 25.5828

151 151 151 151

131 121 111 101

25.2874 25.3811 25.4830 26.5838

sc rip

tN

ot

101 91 81 71

Ma nu

ed

25.2626 25.3605 25.4629 25.5670

Ac

ce

pt

25.2482 25.3507 25.4555 25.5604

ed ite d

Flux Nanoparticle Volume Fraction

py

Heat (W/m2)


22



Table 3 Non-dimensional Parameters for the Microchannel

Nanoparticle Re Volume Fraction

Sc

Pr

Le

NBT

1.8x105 1.8x105 1.8x105 1.8x105

0.00 0.0165 0.0335 0.05

998.06439 1024.71448 1050.05005 1072.81006

46992.23828 45770.10156 44665.75781 43718.16016

6.10554 6.16180 6.24070 6.33620

NA* 556875.62500 263404.78125 169585.14063

0.00420 0.00415 0.00409 0.00401

2.8x105 2.8x105 2.8x105 2.8x105

0.00 0.0165 0.0335 0.05

998.06439 1024.71448 1050.05005 1072.81006

46992.23828 45770.10156 44665.75781 43718.16016

6.10554 6.16180 6.24070 6.33620

NA* 556875.62500 263404.78125 169585.14063

0.00270 0.00267 0.00263 0.00258

3.5x105 3.5x105 3.5x105 3.5x105

0.00 0.0165 0.0335 0.05

998.06439 1024.71448 1050.05005 1072.81006

46992.23828 45770.10156 44665.75781 43718.16016

6.10554 6.16180 6.24070 6.33620

4.2x105 4.2x105 4.2x105 4.2x105

0.00 0.0165 0.0335 0.05

998.06439 1024.71448 1050.05005 1072.81006

46992.23828 45770.10156 44665.75781 43718.16016

5.8x105 5.8x105 5.8x105 5.8x105

0.00 0.0165 0.0335 0.05

998.06439 1024.71448 1050.05005 1072.81006

7.0x105 7.0x105 7.0x105 7.0x105

0.00 0.0165 0.0335 0.05

998.06439 1024.71448 1050.05005 1072.81006

6.10554 6.16180 6.24070 6.33620

NA* 556875.62500 263404.78125 169585.14063

0.00180 0.00177 0.00175 0.00171

46992.23828 45770.10156 44665.75781 43718.16016

6.10554 6.16180 6.24070 6.33620

NA* 556875.62500 263404.78125 169585.14063

0.00130 0.00129 0.00126 0.00124

46992.23828 45770.10156 44665.75781 43718.16016

6.10554 6.16180 6.24070 6.33620

NA* 556875.62500 263404.78125 169585.14063

0.00108 0.00106 0.00105 0.00103

ot

tN

sc rip

Ma nu

pt

ed

py

0.00216 0.00213 0.00210 0.00206

Co

NA* 556875.62500 263404.78125 169585.14063

Ac

ce

* Not Applicable

ed ite d

Heat Flux (W/m2)


23



Table 4 Dimensions of the Macrochannel (Thermally Developing Flow) Value

Length of the channel

0.025 m

Height of the channel

3x10-3 m

Ac

ce

pt

ed

Ma nu

sc rip

tN

ot

Co

py

ed ite d

Parameter


24



Table 5 Dimensions of the Macrochannel (Thermally Fully Developed Flow at the Channel Exit) Value

Length of the channel

0.54 m

Height of the channel

3x10-3 m

Ac

ce

pt

ed

Ma nu

sc rip

tN

ot

Co

py

ed ite d

Parameter


25



j.n = 0

q// T = Ti Φ = Φm

U(y)

H

ed ite d

y

x

q//

py

j.n = 0

ot

Co

L

Ac

ce

pt

ed

Ma nu

sc rip

tN

Fig. 1 Physical problem and the computational domain


26


sc rip

tN

ot

Co

py

ed ite d


Ac

ce

pt

ed

Ma nu

Fig. 2 A sample grid independence for the microchannel for q //  7x10 5 W/m2 and   0.05


27


tN

ot

Co

py

ed ite d


Fig. 3 Comparison of the difference of the wall and bulk fluid temperature at the exit vs.

sc rip

wall heat flux predicted by the present numerical simulation with the experimental data

Ac

ce

pt

ed

Ma nu

[18] for heat transfer in pure fluid (water) flow in a microtube


28


sc rip

tN

ot

Co

py

ed ite d


Ac

ce

pt

ed

Ma nu

Fig. 4 Temperature profiles in the microchannel for q // = 5x105 W/m2 and   0.05 at three different axial locations


29


tN

ot

Co

py

ed ite d


Ac

ce

pt

ed

Ma nu

sc rip

Fig. 5 Concentration profiles in the microchannel for q // = 5x105 W/m2 and   0.05 at three different axial locations


30


tN

ot

Co

py

ed ite d


Fig. 6 Comparison of computed exit wall-fluid temperature difference in the

Ac

ce

pt

ed

Ma nu

sc rip

microchannel with the experimental results for microtube [18] for   5%


31


tN

ot

Co

py

ed ite d


Fig. 7 Comparison of the numerically predicted channel exit heat transfer coefficient

Ac

ce

pt

ed

Ma nu

sc rip

with that for microtube [18] for   5%


32


tN

ot

Co

py

ed ite d


Ac

ce

pt

ed

Ma nu

sc rip

Fig. 8 Axial variation of enhancement factor for q // = 7x105 W/m2 and  = 0.05 in the microchannel


33


tN

ot

Co

py

ed ite d


Fig. 9 Comparison of enhancement factor at the exit of the microchannel with that of

Ac

ce

pt

ed

Ma nu

sc rip

microtube [18] for   5%


34


Co

py

ed ite d


tN

Ac

ce

pt

ed

Ma nu

sc rip

the thermally developing flow for   5%

ot

Fig. 10 Comparison of microchannel and macrochannel exit heat transfer coefficients in


35


Co

py

ed ite d


Fig. 11 Comparison of enhancement factor at the exit of the microchannel and

Ac

ce

pt

ed

Ma nu

sc rip

tN

ot

macrochannel in the thermally developing flow for   5%


36


ot

Co

py

ed ite d


tN

Fig. 12 Comparison of numerically predicted fully developed temperature profile (  vs.

sc rip

y ) with the analytical solution [20] for the macrochannel flow of pure fluid (water) at

Ac

ce

pt

ed

Ma nu

q // = 1000 W/m2


37


Co

py

ed ite d


ot

Fig. 13 Comparison of enhancement factor at the exit of the microchannel and

Ac

ce

pt

ed

Ma nu

sc rip

tN

macrochannel for   5% when the flow is thermally developed at the exit


38
