Thermal instability in a nanofluid saturated horizontal porous layer subjected to g-gitter

International Journal of Heat and Mass Transfer 110 (2017) 63–67

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Thermal instability in a nanofluid saturated horizontal porous layer subjected to g-gitter S. Govender ⇑ Corporate Specialist Gas Turbines and Thermo-flow Systems, Technology Engineering, ESKOM Holdings Ltd, Maxwell Drive, Sunninghill, Johannesburg 2000, South Africa School of Mechanical Engineering, University of KwaZulu Natal Durban, King George V Avenue, Durban 4001, South Africa

a r t i c l e

i n f o

Article history: Received 13 February 2017 Received in revised form 6 March 2017 Accepted 6 March 2017

Keywords: Nanofluid Porous media Rayleigh number Thermophoresis Brownian motion Heat transfer Mathieu equation g-gitter Vibration

a b s t r a c t An analytical investigation of the onset of convection in a horizontal porous saturated by a nanofluid, and subjected to g-gitter (or vertical vibration) is presented. The Darcy model is used for the porous layer and a linear stability analysis is used to determine the convection threshold in terms of the key parameters for the nanofluid in a homogenous porous medium. The results are presented for the special case when the porosity to heat capacity ratio is unity. The critical Rayleigh number and wavenumber is presented in terms of the nanofluid parameter. Ó 2017 Published by Elsevier Ltd.

1. Introduction Choi [1] defined a revolutionary fluid which consists of a base fluid and nanoparticles in suspension, known as a nanofluid, has been used to demonstrate a means of improving heat transfer in practical applications such as medicine or engineering. The experimental results by Choi [1] thus far shows substantially improved effective thermal conductivity but seem not to be independently confirmed by any other scientists working in the field. In order to validate the current model presented and previous models in open literature, it seems prudent that the key step is to commission an independent confirmation of the currently available experimental results. The present paper focuses on applications in engineering and possibility in power generation applications involving heating or cooling enhancement. As an example heat transfer enhancement in nuclear reactor applications using nanofluids has been discussed by Buongiorno [2]. This work involved flow and heat transfer in porous media and included comparative tests between quenching ⇑ Address: Corporate Specialist Gas Turbines and Thermo-flow Systems, Technology Engineering, ESKOM Holdings Ltd, Maxwell Drive, Sunninghill, Johannesburg 2000, South Africa. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.03.018 0017-9310/Ó 2017 Published by Elsevier Ltd.

metallic spheres in water and a nanofluid. The quenching process was greatly accelerated when the spheres were quenched in a nanofluid. Essentially this work concluded that nanoparticles enhances the critical heat flux limit and accelerates quenching heat transfer. In relation to water cooled nuclear reactor technologies it is proposed that sizeable power increase in the core are possible with rewards being economic gains and improved safety margins. Pioneering work on thermal instability in porous media containing nanofluids has been developed and analysed analytically by Nield and Kuznetsov [3,4] for a horizontal porous layer subjected to gravity. Later the paper Nield and Kuznetsov [3] was revised to include the zero flux boundary conditions for the nanofluid which then removed the possibility of the oscillatory mode of convection, Nield and Kuznetsov [5]. The reader is also referred to other works by Kuznetsov and Nield [5–7] for further reading. The mentioned papers presents a comprehensive formulation of the governing equations and provides numerical value ranges for the nanofluid parameters. Numerous interesting features of nanofluids has been explained and a full analytical solution is presented. The reader is referred to Tzou et al. [8] and Vadasz [9] for a review of additional applications of nanofluids. An additional application involving rotation, such as cooling of electronic equipment found in rotating radars or cooling in high speed generators was undertaken by Govender [10,11].

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S. Govender / International Journal of Heat and Mass Transfer 110 (2017) 63–67

Nomenclature Latin symbols A term in Mathieu equation Da Darcy number, equals k0
=L
DB Brownian diffusion coefficient DT thermophoretic diffusion coefficient ^ ex unit vector in the x-direction ^ ey unit vector in the y-direction ^ unit vector in the z-direction ez gravitational acceleration g
H
height of porous layer H the front aspect ratio of the porous layer, equals H
=L

k0 characteristic permeability L
the length of the porous layer. N rank for linear algebraic system of equations NA modified diffusivity ratio, NA ¼ DT ðT 1  T 0 Þ=DB ðu1  u0 Þ NB the modified particle-density increment, NB ¼ eðqcÞP u0 =ðqcÞf p dimensionless reduced pressure Pr Prandtl number, equals m
=k
Q term in Mathieu equation V dimensionless filtration velocity vector, equals ey þ w^ez u^ ex þ v ^ Va Vadasz number, equals ePr=Da Ra centrifugal based Rayleigh number, equals
Ra ¼ ð1  u
Þq
0 b
DTg
H
0 k0 =ðk
l
Þ nanoparticle based Rayleigh number, equals Ranp
Ranp ¼ ðq
P  q
0 ÞDuH
g
k0 =ðk
l
Þ R scaled gravity based Rayleigh number, equals Ra=p2 Rnp scaled rotation based Rayleigh number, equals Ranp =p2 s convection wavenumber t
dimensional time T
dimensional temperature T dimensionless temperature, equals ðT
 T 0 Þ=ðT 1  T 0 Þ T0 coldest wall temperature The author is not aware of any current studies on vibration effects in nanofluids. Gershuni et al. [12] and Gresho and Sani [13] originally used mechanical vibration in mathematical modelling aimed at increasing the stability threshold in pure fluids. Bardan and Mojtabi [14] then used a time average technique to research a confined cavity subjected to vertical vibration in porous media. Govender [15] then used the direct technique using Mathieu charts to consider stability in a horizontal porous layer subjected to vertical vibration. Later Govender [16] then used the direct technique to recover the transition from synchronous to subharmonic modes. Govender [17] then presented a study on the stability of convection in a cylinder subjected to vibration. Pedramrazi et al. [18] discusses the validity of the time averaged formulation in the Horton-Rogers-Lapwood problem using the time average and direct methods. In this work, we consider the effects of g-gitter (or vertical vibration) on the stability of convection in a nanofluid saturated horizontal porous layer. Readers are also referred to a fairly recent survey of literature on convection in porous media saturated by nanofluids presented by Nield and Kuznetsov [19]. In the current study we will use the linear stability analysis to derive the convection threshold in terms of the critical Rayleigh and wavenumbers in terms of the nanofluid parameters.

T1 u v w x X y z

hottest wall temperature horizontal x-component of the filtration velocity horizontal y-component of the filtration velocity vertical z-component of the filtration velocity horizontal length coordinate vibration amplitude in scaled Mathieu equation horizontal width coordinate vertical coordinate

Greek symbols a a parameter related to the wave number, equals s2 =p2 b
thermal expansion coefficient
d related to the vibration amplitude, d ¼ b =H
DT characteristic temperature difference e porosity c
fluid thermal conductivity g scaled Vadasz number, g ¼ Va=p2 u volume fraction of nanoparticles k
effective thermal diffusivity l
fluid dynamic viscosity p
permeability of the porous matrix q
fluid density r exponent for growth/decay of convection s scaled time t ¼ 2p þ 2s x
vibration frequency v
fluid kinematic viscosity n ratio of heat capacities, n ¼ ðqcÞm =ðqcÞf Subscripts 0 related to cold wall 1 related to hot wall ⁄ dimensional values c characteristic P related to nanoparticles

between two rigid vertical plates, spaced a distance H
apart. At z
¼ 0: T
¼ T
0 ; u ¼ u0 , and at z
¼ 1: T
¼ T
1 ; u ¼ u1 , where T
1 > T
0 , and the reference temperature is taken to be T
0 . In addition, the Boussinesq approximation extended to include the volume fraction of the nanoparticles is applied to account for the effects of the density variations. The following system of dimensional equations for continuity, momentum and energy is proposed, similar to Kuznetsov and Nield [4]:

z* g*

(

b* sin ω *t *

T=TC

z* = H* z* = 0

2. Problem formulation

x* = 0 A horizontal nanofluid saturated porous subjected to vertical vibration is presented in Fig. 1. The porous layer is constrained

)

T=TH

x*

x* = L *

Fig. 1. Nanofluid saturated porous layer subjected to vibration.

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S. Govender / International Journal of Heat and Mass Transfer 110 (2017) 63–67

r
 V
¼ 0;

ð1Þ

q
0 p
@V
p

þ V
¼
r
p
þ ðuq
P þ ð1  uÞðq
0 ð1  b
ðT
 T
0 ÞÞÞÞg
el
@t
l
þ b sinðx
t
Þ^ez ;
@u V DT þ  r
u ¼ DB r
2 u þ
r
2 T
; @t
e T0

ð2Þ

ðqcÞ
m

operator (r) twice on Eq. (6), consider the solenoidal velocity field, Eq. (5), and consider only the vertical z-component of the result yields,



 1 @ þ 1 r2 w  Rað1 þ d sin xtÞr2H T Va @t

ð3Þ

  @T
DT þ ðqcÞ
f V
 r
T
¼ c
r
2 T
þ eðqcÞ
P DB r
u  r
T
þ
r
T
 r
T
: @t
T0

þ Ranp ð1 þ d sin xtÞr2H u ¼ 0:

ð4Þ





In Eqs. (1)–(4), V is the velocity of the fluid, p is the pressure,

q
0 is the density of the nanofluid at x
¼ 0, whilst q
P is the density of the suspended nanoparticle. In addition in Eqs. (1)–(4), u is the

nanoparticle volume fraction, T
is the temperature, b
is the thermal expansion coefficient, p
is the permeability of the porous matrix, e is the porosity of the porous matrix, l
is the dynamic viscosity, ðqcÞf is the fluid heat capacity, ðqcÞP is the heat capacity of the nanoparticles and c
is the fluid thermal conductivity. In Eqs. (3) and (4), DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient of the nanoparticles, and further assume that the temperature and volumetric fraction of the nanoparticles is constant. To non-dimensionalise the governing
equations, we employ the entities k
=L
, l
k
=k0 and DT ¼ T 1  T 0 to scale the filtration velocity components ðu
; v
; w
Þ, reduced pressure, ðp
Þ, and the temperature variations T
 T 0 , where k
is
the thermal diffusivity, l
is the dynamic viscosity and k0 is a characteristic permeability associated with the porous matrix. The height of the porous layer H
is used to scale the spatial directions and offset distance as follows, x ¼ x
=H
, y ¼ y
=H
and z ¼ z
=H
. Subjecting Eqs. (1)–(4) to the dimensional analysis the following system of dimensionless steady state equations result for a homogenous porous medium, r  V ¼ 0;



 1 @ þ 1 V ¼ rpr  Rað1 þ d sin xtÞT ^ez þ Ranp ð1 þ d sin xtÞu^ez ; Va @t 1 @T V 1 NA 2 þ  ru ¼ r2 u þ r T; n @t e Le Le @T NA N N þ V  rT ¼ r2 T þ ru  rT þ A B rT  rT: @t Le Le

ð5Þ ð6Þ ð7Þ ð8Þ

The key non-dimensional parameters that emanate from the rescaling of the Eqs. (1)–(4) are the Lewis number, Le ¼ k
=DB ,


the centrifugal Rayleigh number Ra ¼ ð1  u
0 Þq
0 b
DTH
2 g
k0 =

uH
2 ðl k Þ, the nanoparticles Rayleigh number, Ranp ¼ ðq  q



g k0 =ðl k Þ, the modified diffusivity ratio N A ¼ DT ðT 1  T 0 Þ= DB ðu1  u0 Þ, the modified particle-density increment N B ¼ eðqcÞP u0 =ðqcÞf , the heat capacity ratio n ¼ ðqcÞ
m =ðqcÞ
f , and the Vadasz




P


0 ÞD




number Va ¼ ePr=Da. In Eq. (6), and the parameter d ¼ b =H
represents the dimensionless amplitude. The symbols V, T and ðpr Þ represent the dimensionless filtration velocity vector, temperature and reduced pressure, respectively, and ^ ez , is the unit vector in the z- direction. In The temperature and volume fraction boundary conditions are: at z ¼ 0: T ¼ 0; @ u=@z þ N A z ¼ 0 and at z ¼ 1: en ¼ 0 on all other walls T ¼ 1; @ u=@z þ N A @T=@z ¼ 0, and rT  ^ representing the insulation condition on these walls. In the presented boundary conditions we have followed Nield and Kuznetsov [4] and have taken the nanoparticle flux to be zero at the boundaries in addition to taking the temperature to be constant. The partial differential Eqs. (5)–(8) forms a non-linear coupled system which together with the corresponding boundary conditions accepts a basic motionless solution with a parabolic pressure distribution. The solutions for the basic temperature and flow field is given as, T B ¼ 1  z, V B ¼ 0 and uB ¼ u0 þ N A z. To provide a non-trivial solution to the system it is convenient to apply the curl

ð9Þ

The vertical Laplacian operator in Eq. (9) is defined as

r2H ¼ ð@ 2 =@x2 þ @ 2 =@y2 Þ. 3. Linear stability analysis Assuming small perturbations around the basic solution in the form V ¼ V B þ V 0 and T ¼ T B þ T 0 , u ¼ uB þ u0 and linearising Eqs. (7)–(9) yields the following linear system of equations, 

 1 @ þ 1 r2 w0  Rað1 þ d sin xtÞr2H T 0 þ Ranp ð1 þ d sin xtÞr2H u0 ¼ 0; Va @t e @T 0 eNA 2 0 e 2 0 r T þ r u;  w0 ¼ n @t Le Le   0 @T N B @T 0 @ u0 NA N B @T 0 2 0 0 w ¼r T þ  þ : @t Le @x @x Le @x

ð10Þ ð11Þ ð12Þ

In Eqs. (10)–(12), w0 ; T 0 ; u0 are the perturbations to the vertical component of the filtration velocity, the temperature and the nanoparticle volume fraction. We assume an expansion into normal modes in the x- and y-directions of the form, N X ðT 0 ; u0 ; u0 Þ ¼ hðtÞ exp½iðsx x þ sy yÞ ðAk ; Bk ; C k Þ sinðkpzÞ;

ð13Þ

k¼1

which satisfies the boundary conditions u0 ¼ 0, T 0 ¼ 0, and u0 ¼ 0 at z ¼ 0 and z ¼ 1. In Eq. (13) s2 ¼ s2x þ s2y , and k = 1, 2, 3, . . ., N. Substituting Eq. (13) in Eqs. (10)–(12) to set up residuals on the left hand sides thereby obtaining a system of 3N linear algebraic equations with 3N unknowns Ak ; Bk ; C k , k = 1, 2, 3, . . ., N. Then after allowing the determinant to vanish, one may obtain an expression for the eigenvalue Ra from the following system of algebraic equations to Rank N = 1. d  0 þ ðs2 þ p2 Þ 1 dt hB1   e ðs2 þ p2 Þ e d þ ðs2 þ p2 Þ e hA1 ¼ 0 1 Le n dt Le   Ra s2 ð1 þ dsinðxtÞÞ Ras2 ð1 þ dsinðxtÞÞ  1 d þ 1 ðs2 þ p2 Þ hC 1 np Va dt ð14Þ

Decoupling Eq. (14) and solving yields the following ordinary differential equation for the amplitude h,

 

 s2 Le e dh 2 2 ð1 þ dsin þ ðs þ p þ VaÞ  Ra 1  x tÞ 2 dt s2 þ p2 e n dt " #    2 Vas2 ðs2 þ p2 Þ Le þ 2 ð1 þ dsin xtÞ h ¼ 0;  Ra þ Ranp þ NA ðs þ p2 Þ s2 e 2

d h

ð15Þ Rescaling the above equation and considering the special case when e=n ¼ 1, yields the following form of Eq. (15), 2

dh gp4 a þ p2 ða þ 1 þ cÞ  dt ða þ 1Þ dt

    Le  ðR  R0 Þ þ R þ Rnp d sin xt h ¼ 0; þ NA

d h 2

e

ð16Þ

to Rank N = 1, where a ¼ s2 =p2 , R ¼ Ra=p2 , Rnp ¼ Ranp =p2 , g ¼ Va=p2 and R0 ¼ ða þ 1Þ2 =a  Rnp ðLe=e þ NA Þ. Eq. (16) may be cast into the

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S. Govender / International Journal of Heat and Mass Transfer 110 (2017) 63–67

canonical form of the Mathieu equation, by setting first setting t ¼ ðp=2 þ 2sÞ=x and applying the expression h ¼ X expðrsÞ to yield,

X Prime þ ðA  2Q cosð2sÞÞX ¼ 0;

ð17Þ

where, A ¼ r2 

4cFðaÞ

ðR  R Þ;

ð18Þ

0 x2      gFðaÞd Le Le R þ Rnp þ NA þ NA Q ¼ 2 ¼ 2gFðaÞjFr R þ Rnp ; ð19Þ 2 x e e p4 a ; ð20Þ FðaÞ ¼ aþ1 p 2 ða þ 1 þ cÞ r¼ : ð21Þ x

One may observe from Eq. (19) that a term of the form d=x2 (amplitude to vibration frequency ratio) appears in the definition for Q. Based on a case of high frequency and low amplitude vibration it may be appreciated that the numerical values for Q will be very small. Similarly A also assumes very small values for high vibration frequencies. A regular perturbation may be used in which Q is considered as a small parameter may be used,

X ¼ X 0 þ QX 1 þ Q 2 X 2 þ . . . ;

ð22Þ

A ¼ A0 þ QA1 þ Q 2 A2 þ . . .

ð23Þ

Fig. 2. Critical Rayleigh number versus vibration frequency.

Substituting Eqs. (22), (23) in Eq. (17) and applying a solvability condition to the results at marginal stability, r ¼ 0 yields,

A¼

Q2 ; 2

X ¼ b0

 Q 1  cos 2s ; 2

ð24Þ ð25Þ Fig. 3. Critical wavenumber versus vibration frequency.

where b0 is an integration constant. Substituting the definitions for A and Q in Eqs. (18), (19) in Eq. (24) yields,



 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R¼  F0  1  4j0 ðR0 þ F 0 Þ; 2j0 2j0

ð26Þ

where j0 ¼ 2gFðaÞx2 j2 Fr2 and F 0 ¼ Rnp ðLe=e þ NA Þ. Eq. (26) may be minimized with respect to the wavenumber a to yield the critical Rayleigh number and wavenumber respectively as a function of the key parameters. The resulting equation is to the fourth order in the wavenumber and is solved numerically. Fig. 2 shows the critical Rayleigh number as a function of the vibration frequency for various values of the nanoparticle parameter F 0 . In Fig. 2 it is also shown that for F 0 ¼ 5 and F 0 ¼ 10 the critical Rayleigh number assumes negative values. For these regions represent zones of destabilizing concentration effect that require a large temperature difference for neutral stability to be possible. However the results in Fig. 2 indicates that the vibration serves to stabilize the convection without the application of a large temperature gradient. Also indicated is the case when F 0 < 0 which presents a stabilizing concentration effect that increases the convection threshold. In Fig. 2 no real solutions exist beyond the ‘‘neck” that falls in the range x  3400—3500. Fig. 3 shows the critical wave number as a function of the vibration frequency for various values of the nanoparticle parameter F 0 . If one were to relate these findings back to Buongiorno [2], it may be appreciated that the effect of vibration is to mitigate the effect of using the nanofluid in the first place. In other words in a system where a nanofluid is used vibration should be avoided as it negates the effect of the properties of the nanofluid. Therefore when nanofluids are used in nuclear reactors as coolants, instrumentation should be in place to monitor the presence of vibration and it’s magnitude to ensure that heat transfer is as per design with no possibility of overheating and fission pro-

duct release. In principle as the frequency is increased, the wavenumber approaches zero, i.e. a ! 0. From Eq. (26), we may determine the maximum frequency for achieving absolute stabilization for high frequency and small amplitude vibration when a ! 0,

1

 3582: xmax ¼ pffiffiffiffiffiffi 2 2 2cp jFr

ð27Þ

The result in Eq. (27) corroborates the graphical results in Fig. 2. 4. Conclusion In the current study we investigate convection in a nanofluid saturated porous layer subjected to vibration. The study reveals that increasing the nanoparticle parameter F 0 serves to destabilize the convection. However vibration has a stabilizing effect on a selected value of F 0 . Typically when the critical Rayleigh number is less than zero, a large temperature difference is required to make neutral stability possible. In this work it is discovered that the effect of vibration may be used to stabilize the convection when using a nanofluid in a porous medium. References [1] S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: D.A. Springer, H.P. Wang (Eds.), Developments and Applications of non-Newtonian Flows, ASME, FED, 231/MD, vol. 66, 1995, pp. 99–105. [2] J. Buongiorno, L.W. Hu, Nanofluid heat transfer enhancement for nuclear reactor applications, in: Proceedings of ASME 2009 2nd Micro/Nanoscale Heat and Mass Transfer, December 18–21, Paper no. MNHMT2009-18062, Shanghai. [3] D.A. Nield, A.V. Kuznetsov, Thermal instability in a porous medium saturated by a nanofluid, Intl. J. Heat Mass Transf. 52 (2009) 5796–5801.

S. Govender / International Journal of Heat and Mass Transfer 110 (2017) 63–67 [4] D.A. Nield, A.V. Kuznetsov, Thermal instability in a porous medium layer saturated by a nanofluid: a revised model, Int. J. Heat Mass Transf. 68 (2014) 211–214. [5] D.A. Nield, A.V. Kuznetsov, The effect of vertical through flow on thermal instability in a porous medium layer saturated by a nanofluid, Transp. Porous Media 87 (2011) 765–775. [6] A.V. Kuznetsov, D.A. Nield, Effect of local thermal nonequilibrium on the onset of convection in a porous medium layer saturated by a nanofluid, Transp. Porous Media 83 (2010) 425–436. [7] A.V. Kuznetsov, D.A. Nield, The onset of double-diffusive nanofluid convection in a layer of a saturated porous medium, Transp. Porous Media 85 (3) (2010) 941–951. [8] D.Y. Tzou, Instability of nanofluids in natural convection, ASME J. Heat Transf. 130 (2008) 072401. [9] P. Vadasz, Heat conduction in nanofluid suspensions, ASME J. Heat Transf. 128 (2006) 465–477. [10] S. Govender, Thermal instability in a rotating vertical porous layer saturated by a nanofluid, ASME J. Heat Transf. 138 (2016) 052601. [11] S. Govender, Thermal instability of convection in a rotating nanofluid saturated porous layer placed at a finite distance from the axis of rotation, ASME J. Heat Transf. 130 (2016) 102402.

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