Nfl

Aa

b

a

ARRA

KSNNUFR

1

ctitmUahpbai

pfldtimiob

0h

J. of Supercritical Fluids 74 (2013) 115– 127

Contents lists available at SciVerse ScienceDirect

The Journal of Supercritical Fluids

jou rn al h om epage: www.elsev ier .com/ locate /supf lu

atural convection on a vertical plate with variable heat flux in supercriticaluids

li Reza Teymourtasha, Danyal Rezaei Khonakdara,∗, Mohammad Reza Raveshib

Faculty of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, IranFaculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran

r t i c l e i n f o

rticle history:eceived 13 October 2012eceived in revised form 1 December 2012ccepted 2 December 2012

a b s t r a c t

The present article investigates laminar free convection with uniform or non-uniform prescribed surfaceheat flux over a vertical flat plate in supercritical fluid, numerically. A new equation for thermal expansioncoefficient in a supercritical fluid is derived based on Redlich–Kwong equation of state as a function ofpressure, temperature and the compressibility factor. Calculated values of thermal expansion coefficienthave been compared with the experimental results which show better accuracy in comparison with van

eywords:upercritical fluidsatural convectionusselt numberniform or non-uniform heat flux

der Waals ones. The governing systems of partial differential equations were solved numerically usingthe finite difference method. The local Nusselt number was calculated and plotted as a function of thelocal Rayleigh number. Using supercritical fluids in constant heat flux free convection, decrease walltemperature in comparison with under critical fluids. It was observed that positive and negative slopes

butio

inite difference methodedlich–Kwong equation of state

of surface heat flux distri

. Introduction

In recent decades, research into enhancement of natural or freeonvection heat transfer has been carried out enormously, becausehis kind of heat transfer happened in different aspects of engineer-ng such as heat exchangers and refrigeration industries, coolingechniques for electronic components, geothermal systems, ther-

al insulation for buildings, solar collectors and space heating.sing ribs and fins to increase the surface area, ultrasonic vibrationnd applying electrical field are the most common methods, whichave been used to improve the natural convection heat transferhenomena [1–6]. Nevertheless, low heat transfer performance ofase fluids such as water, carbon dioxide, ethylene glycol, refriger-nt, and engine oil obstructs the natural convection heat transferncrement. This problem can be solved by using supercritical fluids.

A fluid heated to above the critical temperature and com-ressed to above the critical pressure is known as a supercriticaluid. Supercritical fluids can be applied in many fields such asrug delivery, chromatography, synthesis, purification and extrac-ion. Supercritical pressurized water is used as working fluidsn supercritical pressure boilers for thermal power stations. The

echanisms, which have been used to produce supercritical flu-

ds, are sometimes more expensive than the under critical fluidnes, because they involve high pressure equipment except for car-on dioxide. Between different convectional fluids, carbon dioxide

∗ Corresponding author. Tel.: +98 9113531929.E-mail address: Danyal.Rezaeikhonakdar@yahoo.com (D. Rezaei Khonakdar).

896-8446/$ – see front matter. Published by Elsevier B.V.ttp://dx.doi.org/10.1016/j.supflu.2012.12.010

n increases and decreases the heat transfer coefficient, respectively.Published by Elsevier B.V.

is the most widely used as a supercritical fluid owing to its spe-cial characteristics such as readily available, low cost solvent usedfor producing, low critical temperature, and being nontoxic andnon-flammable gas.

The phenomenon and behavior of supercritical fluids has beenthe subject of research right from 1800s. By recent developmentin industrial power system, the necessity for more accurate under-standing of the heat transfer characteristics of supercritical fluidshas increased; meanwhile a lot of investigations, both theoreticaland experimental, have been reported on heat transfer characteris-tics of these fluids. Since the physical properties of these fluids arestrongly and singularly dependent on temperature and pressure,the mechanism of heat transfer should become naturally compli-cated [7,8]. Properties, applications, advantages and also incrementof heat convection characteristics of supercritical fluids have beenintroduced by McHugh and Krukonis [9].

Many investigations have been carried out on the effectivenessof supercritical fluids in forced or mixed (forced and free) heattransfer. Forced convection of supercritical water has been exten-sively studied both experimentally and numerically [10,11]. Severalreviews of supercritical fluids forced convection have been doneby other researchers [12–14]. Kakaral and Thomas studied mixedconvection heat transfer in vertical tube of supercritical fluids. Theyevaluated the effect of flow direction at rate of heat transfer in tube[15]. The mixed convective heat transfer of carbon dioxide at super-

critical state was investigated in mini tubes [16,17], experimentally.These studies also concerned the effects of fluid’s temperatureand flow rate on the convective heat transfer in porous and minitubes.

116 A.R. Teymourtash et al. / J. of Supercri

Nomenclature

a,b EOS constants that corrects for intermolecularattractive forces (N m4/mol2)

A,B dimensionless EOS parametersCp heat capacity of the fluid (J/kg K)C0p heat capacity at the low-pressure limit (J/kg K)g local acceleration of gravity (m/s2)Gr grashof numberh heat transfer coefficient (W/m2 K)k thermal conductivity of the fluid (W/m K)k0 thermal conductivity at the low-pressure limit

(W/m K)L height of the vertical plate (m)M molecular weight of the fluid (kg/mol)Nu Nusselt numberP pressure (Pa)Pr Prandtl numberQ heat flux (W/m2)R gas constant (J/mol K)Ra Rayleigh numberT temperature (K)� density of the fluid (kg/m3)ω acentric factoru,v velocity components in the x,y direction, respec-

tively (m/s)V′ molar volume (m3/mol)u′w EOS parameterU,V dimensionless velocity components in the x,y direc-

tion, respectivelyx,y coordinates parallel and normal to the vertical plate,

respectively (m)X,Y dimensionless coordinate parallel and normal to the

vertical plate respectivelyZ compressibility factor

Greek symbols˛ thermal diffusivity

thermal expansion coefficient (1/K)v kinematic viscosity (m2/s)� inverse thermal conductivity (m K/W)� dimensionless temperature� inverse viscosity (m2/N s)

Subscriptsc critical conditionf fluidi node index in the x directionj node index in the y directionL height of the vertical plate (m)P constant pressurer reduced characteristicw wallx local distance

tpwsred

∞ outside the boundary layer

Free convection in supercritical fluids has received less attentionhan forced or mixed ones, maybe because there have been fewerressing industrial applications. At first this kind of heat transferas reviewed by Hall [13], but little further data have been reported

ince that time. On the other hand, the problem of free convectionesulting from the flow over a heated vertical plate is of consid-rable theoretical and practical interest in under critical fluid. Aetailed review of the subject can be found in the books by Bejan

tical Fluids 74 (2013) 115– 127

[18]. Continuity, momentum and energy equations for natural con-vection over vertical flat plate were solved numerically by Ostrach[19] for the first time. Muller and Estevez [20] studied mass trans-missions that are caused by natural heat convection in supercriticalfluids. They understood that heat and mass transmission are similarin most instances.

Numerical modeling for natural convection of supercritical fluidover vertical plate with variable physical properties was done forthe first time by Nishikawa and Ito [21]. However, no effect of thetemperature on the thermal expansion coefficient was accountedfor. The effect of temperature on thermal expansion coefficient wasfirst considered by Rolando [22]. In his research a new equation forthermal expansion coefficient for a supercritical fluid over isother-mal vertical plate in natural convection has been derived based onvan der Waals equation of state as a function of pressure, tempera-ture and the compressibility factor. In a large number of industrialapplications, the surface heating conditions are non-uniform. Tey-mourtash and Ebrahimi Warkiani [23] have investigated the effectof linearly variation of wall temperature on the heat transfer bynatural convection along a vertical plate in supercritical fluids.They also showed that positive and negative slope of temperaturedistribution have increased and decreased the heat transfer rate,respectively.

Many investigations have been carried out on the cases wherethe wall temperature is prescribed and the surface heat flux is tobe found. However, in many industrial applications it is much morenecessary to determine the wall temperature by knowing its sur-face heat flux. Sparrow [24] formulated the boundary layer freeconvection problem along a vertical flat plate with power law sur-face heat flux distributions by the Karman–Pohlhausen method andobtained solutions by a series expansion technique. Laminar freeconvection along vertical flat plate with power-law variation ofthe surface heat flux has been analyzed by Chen et al. [25]. Theyhave shown that an increase in slope of surface heat flux distribu-tion from negative to positive ones enhance the heat transfer rate.Lee and Yovanovich [26] developed a simple model to predict thenatural convection heat transfer characteristics of a vertical platewith arbitrarily prescribed surface heat flux variations. The analy-sis is based on the linearized approximations to the boundary layerform of the conservation equations. Surface heat flux variationsof discontinuous as well as continuous types are examined, andthe results are compared with air data obtained by using varioussolution methods including numerical simulations.

Laminar free convection with uniform or non-uniform pre-scribed surface heat flux has not received not only considerableattention in conventional fluids but also any attention in the super-critical fluids. So, the main objective of the present study is tocalculate the effect of the non-uniformity of wall heat flux on heattransfer by laminar free convection over a vertical flat plate witha linearly varying heat flux distribution into a supercritical fluid.For the first time in this study, it was proved that the Boussi-nesq approximation is applicable over a wide range of temperatureand pressure variation for supercritical fluid. Then a new equa-tion for thermal expansion coefficient for a supercritical fluid isderived based on Redlich–Kwong equation of state as a functionof pressure, temperature and the compressibility factor. Finally, anumerical model based on a finite-difference formulation has beenused to solve boundary-layer equations and a complete parametricstudy has been carried out in laminar regime for positive, zero andnegative slopes of the wall heat flux profile.

2. Boussinesq approximation and governing equations

For fluid flow under specific conditions, Boussinesq proposedthat it is possible to ignore the fluid density changes except in

A.R. Teymourtash et al. / J. of Supercri

Fd

gss

2

Be

∇Ttgc

(

(

Wsv

gic

Ipvid

P

ig. 1. The diagram of thermal expansion coefficient (ˇ) and its slope for carbonioxide at P = 10 MPa from experimental data [28].

ravitational term (�g) in 1903 [27]. The accuracy of this con-ideration and the results of them in governing equations in aupercritical fluid are as follows.

.1. Continuity equations

The general forms of equations are as follows:

1�

D�

Dt+ ∇ · U = 0 (1)

y using Boussinesq approximation, the incompressible continuityquation can be rewritten as follow:

· U = 0 (2)

his does not mean a constant value of density and neglecting theerm �−1(D�/Dt) is just due to its small scale compared to velocityradients in term � ·U. In that reference, it is cited that in someases, it is not possible to use a constant value of density:

1) When Mach number is greater than 0.3 and the flow is consid-ered to be compressible.

2) When the flow is non steady-state (transient).

hen the elevation changes of flow is as large that it is not pos-ible to neglect hydrostatic pressure change that causes densityariation.

Now consider a situation that compressibility effects are negli-ible and density variation is just due to temperature changes. Thiss the governing case in convective heat transfer problems. In thisase, the variation of density among temperature is as follows:

ı�

�= −ˇıT (3)

n these equations, is thermal expansion coefficient in constantressure and varies proportional to (1/T) for ideal gas. In Fig. 1,ariations of versus temperature are shown for supercritical flu-

ds like carbon dioxide [28]. This figure is plotted at P = 10 MPa forifferent Tr, in which Pr and Tr are:

r = P

Pc(4)

tical Fluids 74 (2013) 115– 127 117

Tr = T

Tc(5)

Pc and Tc are critical pressure and temperature, respectively. Theamounts of Pc and Tc for carbon dioxide are 7.38 MPa and 304.1 K,respectively. For water, these parameters have the amount of22.09 MPa and 647.3 K, respectively [7].

To show the reason of neglecting the term �−1(D�/Dt) in conti-nuity equations compared to velocity gradient terms, it is necessaryto use scale analysis. Using L for length, U for velocity and ıT fortemperature changes results in:

(1/�)(D�/Dt)∇ · u

≈ (1/�)u(∂�/∂x)∂u/∂x

≈ (U/�)(ı�/L)U/L

= ı�

�= −ˇıT � 1

(6)

As it is shown, if there is a small temperature difference, the firstterm can be omitted in comparison with second one in that equa-tion due to small thermal expansion coefficient. On the other hand,as shown in Fig. 1, temperature differences less than 10 K results inthe maximum value of 0.1 for ˇıT which cannot satisfy the Boussi-nesq approximation for fluids near the critical point completely.By increasing in the amount of Pr and Tr which leads to recedingfrom the critical point, the amount of thermal expansion coeffi-cient descended. For instance, the amount of for supercritical fluidat Tr = 1.2 is about 5 × 10−3 K−1. So, for the supercritical conditionfar from the critical point with more temperature differences theBoussinesq approximation is completely valid. Finally, Eq. (1) canbe reduced to Eq. (2).

2.2. Momentum equation

The general form of momentum equation is:

�DU

Dt= −∇P + �g + �∇2U (7)

2.3. Energy equation

The general form of energy equation is as follows:

�De

Dt= −∇ · q + P(∇ · U) + �ϕ (8)

For most of heat convection problems, the energy equation sim-plifies to:

DT

Dt= −˛∇2T (9)

If the fluid density (�) be assumed as a function of temperature andpressure, it is possible to expand �(P,T) and formulate it as environ-ment density �(P∞,T∞) in a second-order Taylor series expansion:

� = �∞ +(

∂�

∂T

)P

(T − T∞) + 12

(∂2�

∂T2

)P

(T − T∞)2 + · · ·

+(

∂�

∂P

)T

(P − P∞) + 12

(∂2�

∂P2

)P

(P − P∞)2 + · · ·

+(

∂2�

∂T∂P

)(P − P∞)(T − T∞) + · · · (10)

Thermal expansion coefficient equation has been expressed as

follows:

= − 1�

(∂�

∂T

)P

(11)

118 A.R. Teymourtash et al. / J. of Supercritical Fluids 74 (2013) 115– 127

FT

(((

i

(

�

ˇb

tfi

a

�

3e

caiwt

ig. 2. The magnitude of different terms of Eq. (16) for carbon dioxide at Pr = 1.02,r = 1.05 from experimental data of [28].

(∂2�∂T2

)can be achieved as:

∂�

∂T

)P

= −ˇ� (12)

∂2�

∂T2

)P

= −(

∂ˇ

∂T

)P

� − ˇ

(∂�

∂T

)P

(13)

∂2�

∂T2

)P

= −(

∂ˇ

∂T

)P

� + �ˇ2 (14)

The vertical length is small and the corresponding pressure losss small too, so:

P − P∞) ≈ 0 (15)

Replacing Eq. (15) in Eq. (10) results in:

− �∞ = �ˇ(T − T∞) + �ˇ2

2(T − T∞)2 − 1

2

(∂ˇ

∂T

)P

�(T − T∞)2 + · · ·(16)

It can be seen that if temperature differences is small, the term(T − T∞) is less than 1 and just the first term in right hand side wille remained.

Figs. 2 and 3 show the range of temperature gradient in whichhe second and third terms of Eq. (16) can be neglected. Thesegures are plotted at Pr = 1.02 and Pr = 1.05.

Finally, Eq. (16) has been reduced to Eq. (17), and the Boussinesqpproximations reliable in supercritical region.

− �∞ = �ˇ(T − T∞) (17)

. Evaluating a new equation for calculating the thermalxpansion coefficient (ˇ) in supercritical region

In most free convection problems, the thermal expansion coeffi-ient is assumed to be constant and its variation versus temperature

nd pressure has been neglected according to Boussinesq approx-mation. In supercritical fluids, this consideration will redound to

rong results. So, it is evident that determination the value of ˇhrough an appropriate equation of state is necessary. In this paper,

Fig. 3. The magnitude of different terms of Eq. (16) for carbon dioxide at Pr = 1.05,Tr = 1.05 from experimental data of [28].

by using Redlich–Kwong equation of state, a suitable equation wasderived to determine as a function of temperature, pressure andcompressibility factor (Z).

3.1. General equation for thermal expansion coefficient

Thermal expansion coefficient at constant pressure (or abbrevi-ated form of “Thermal expansion coefficient”) defines as:

ˇ = − 1�

(∂�

∂T

)P

(18)

On the other hand, from the definition of compressibility factor(Z):

� = �∗

Z(19)

In which, (�) is fluid density in pressure P and temperature T and�* is its ideal gas density in (T,P). The derivation of Eq. (19) withrespect to temperature in constant pressure is:(

∂�

∂T

)P

=(

∂(

�∗Z−1)

∂T

)p

= 1Z

(∂�∗

∂T

)P

− �∗

Z2

(∂Z

∂T

)P

(20)

By replacing Eq. (20) in Eq. (18), Eq. (21) is derived as follows:

ˇ = − 1�

(∂�

∂T

)P

= − Z

�∗

[1Z

(∂�∗

∂T

)P

− �∗

Z2

(∂Z

∂T

)P

]= (21)

− 1�∗

(∂�∗

∂T

)P

+ 1Z

(∂Z

∂T

)P

= ˇi + 1Z

(∂Z

∂T

)p

= 1T

+ 1Z

(∂Z

∂T

)p

where, ˇi is ideal gas thermal expansion coefficient. Eq. (21) is ingeneral form and is usable when the equation of state is as follow:

Z = Z(P, T) (22)

Redlich–Kwong equation of state for gases in various tempera-

tures and pressures could be stated as:

P = RT

V − b− a√

T(V(V + b))(23)

A.R. Teymourtash et al. / J. of Supercritical Fluids 74 (2013) 115– 127 119

ing he

P

wcc

Z

A

B

Z

t

A

B

Iccw

[

((

(

f

ˇ

Fig. 4. Geometry and coordinate of the system (a) linearly increas

or:

V(V2 − b2)√

T − RTV(V + b)√

T + a(V − b) = 0 (24)

here a and b are constants of equations and can be achieved fromritical properties generally. Then Eq. (24) can be rewritten by usingompressibility factor (Z) as follows:

= PV

RT(25)

= aP√T(RT)2

(26)

= bP

RT(27)

Replacing above parameters in Eq. (24) leads to:

3 − Z2 + (A − B2 − B) Z − AB = 0 (28)

The constants A and B can be obtained by Redlich–Kwong equa-ion of state:

= 0.42748(

P

Pc

) (Tc

T

)2.5(29)

= 0.08662(

P

Pc

) (Tc

T

)(30)

n Eq. (28), which is used for determination of thermal expansionoefficient as a function of temperature and pressure, the degree ofompressibility factor (Z) is three. The partial derivation of Eq. (28)ith respect to temperature in constant pressure is:

3Z2 − 2Z − B − B2 + A](∂Z

∂T

)P

− (Z + 2BZ + A)

(∂B

∂T

)P

+ (Z − B)

(∂A

∂T

)P

= 0 (31)

Considering, Eqs. (29) and (30) leads to:

∂A

∂T

)P

= −2.5A

T(32)

∂B

∂T

)P

= − B

T(33)

Replacing, Eqs. (32) and (33) in Eq. (31) results in:

∂Z

∂T

)P

= 1T

[(2.5AZ − 2ZB2 − BZ − 3.5AB

3Z2 − 2Z − B − B2 + A

)](34)

At last, by replacing in Eq. (21), the conclusion would be as

ollows:

= 1T

[1 −(

3.5AB + 2ZB2 + BZ − 2.5AZ

3Z3 − 2Z2 + Z(A − B) − B2Z

)](35)

at flux, (b) constant heat flux and (c) linearly decreasing heat flux.

The last equation states as a function of pressure, temperatureand compressibility factor, derived from Redlich–Kwong equationof state. Evaluation of has been performed in two stages. First, byfinding an appropriate root for Eq. (28), the compressibility factorhas been determined. Next by putting the determined value of Z inEq. (35), has been calculated.

4. Problem statement

As shown in Fig. 4, the steady, laminar, incompressible,two-dimensional, natural convection boundary-layer flow over avertical flat plate has been considered in this study. The coordi-nate system is chosen such that x and y measure the distance alongand normal to the plate, respectively. Far away from the plate, thetemperature of the uniform main stream is T∞. The entire surfaceof the plate is maintained at a constant and linear increasing anddecreasing heat fluxes. Auxiliary Eq. (36) is used to showing linearlyvariable heat flux on vertical flat plate surface.

Q = Q2 − Q1

Lx + Q1 (36)

where Q, Q1, Q2 and L are local heat flux, heat flux at the bottomand top of the plate and total length of the plate, respectively. Itcan be concluded that variable heat flux slopes depend upon thedifferences between top and bottom heat flux of plate. This equa-tion shows surface heat fluxes at different height of the plate. Infact, constant surface heat flux boundary condition occurs whenheat flux of top and bottom of plate are the same.

Assuming that the Boussinesq approximation is valid, theboundary-layer form of the governing equations which are basedon the balance laws of mass, momentum and energy can be writtenas:

∂u

∂x+ ∂v

∂y= 0 (37)

u∂u

∂x+ v

∂u

∂y=

∂2u

∂y2+ ˇg(T − T∞) (38)

u∂T

∂x+ v

∂T

∂y= ˛

∂2T

∂y2(39)

In which, u and v are the velocity components parallel and perpen-dicular to the plate, T is the temperature, ˇ is the thermal expansioncoefficient, v is the kinematic viscosity, � is the fluid density, g isthe acceleration due to gravity and is the thermal diffusivity.

The appropriate boundary conditions for the velocity and tem-perature of this problem are as follows:

x = 0, y > 0; u = 0, T = T∞,( )

x > 0, y = 0; u = 0, v = 0, −K

∂T

∂yy=0

= Q,

y → ∞; u → 0, T → T∞.

(40)

1 percri

f

U

U

Itr

teefic(

(

c

V

tasolllftrtE

h

20 A.R. Teymourtash et al. / J. of Su

With definition of the following dimensionless variables:

GrL = Grx=L = gˇL4Q

k2, Pr =

˛

X = x

L, Y = y

GrL−0.25L

U = uL

GrL0.5

, V = vL

GrL0.25

, � = (T − T∞)kGrL0.25

QL

(41)

Eqs. (36)–(38) are converted to the following dimensionlessorms:

∂U

∂X+ ∂V

∂Y= 0 (42)

∂U

∂X+ V

∂U

∂Y= ∂2U

∂Y2+ �GrL

−0.25 (43)

∂�

∂X+ V

∂�

∂Y= 1

Pr

∂2�

∂Y2(44)

n which, Pr and Gr are the Prandtl and Grashof number, respec-ively. In the dimensionless form, the boundary conditions can beewritten as follows:

X = 0, Y > 0; U = V = � = 0,

X > 0, Y = 0; U = V = 0,∂�

∂Y= −1,

y → ∞; U → 0, � → 0.

(45)

Eqs. (42)–(44) are coupled and non-linear differential equationso be solved under the boundary conditions given in Eq. (46). How-ver, the exact analytical solutions are not possible for this set ofquations. Hence, these equations have been solved by using thenite-difference method. The equivalent finite difference schemesorresponding to Eqs. (42)–(44) are given by:

−Vi−1,j

2Y− 1

Y2

)Ui,j−1 +

(Ui−1,j

X+ 2

Y2

)Ui,j

+(

Vi−1,j

2Y− 1

Y2

)Ui,j+1 =

U2i−1,j

X+ Gr−0.25�i−1,j, (46)

−Vi−1,j

2Y− 1

PrY2

)�i,j−1 +

(Ui−1,j

X+ 2

PrY2

)�i,j

+(

Vi−1,j

2Y− 1

PrY2

)�i,j+1 = Ui−1,j�i−1,j

X, (47)

After some derivations V values can be determined from theontinuity equation as:

i.j = Vi,j−1 −(

Y

2X

)(Ui,j − Ui−1,j + Ui,j−1 − Ui−1,j−1

). (48)

In which, the indexes i and j refer to x and y directions, respec-ively. The above equations are explicit in x-direction, while theyre implicit in y-direction. After specifying the conditions alongome initial i = 1, U values and, in the following, � values can bebtained on the i = 2 line. Then, V values are obtained on the i = 2ine. Having determined the values of all the variables on the i = 2ine, the same procedure can be used to find the values on the i = 3ine and so on. More details on the numerical procedures can beound in the textbook by Oosthuizen and Naylor [29]. A mesh sys-em with 150 × 150 nodes is proven to provide mesh-independentesults. Local heat transfer coefficient and Nusselt number, oncehe temperature field is obtained as described above, are based on

qs. (49) and (50), respectively.

x = − k

(Tw − T∞)

(∂T

∂y

)y→0

, hL = hx=L (49)

tical Fluids 74 (2013) 115– 127

NuL = Nux=L = hL × L

k(50)

5. Estimation of fluid properties

5.1. Viscosity

The procedure used to estimate the viscosity of the fluids atsupercritical conditions was given in Poling et al. [30]. This methodis recommended for dense fluids at super or near critical conditions.In this reference the product �� is presented as a function of Tr,where � is the dynamic viscosity in (Pa s) and � is defined as:

� = 0.176

(Tc

M3Pc4

)1/6

(51)

where, Tc, Pc and M are in K, bar and (g/mol), respectively. Note that� is a unique property of each fluid and does not depend on T or P.

5.2. Thermal conductivity

The method used in this work is the one proposed by Stiel andThodos [31]. Dimensional analysis has been used to obtain a corre-lation between k k0, Zc, � , and �. Finally, the following approximateanalytical expressions have been established.

k =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

k0 + 1.22 × 10−2 [exp (0.535�r ) − 1]

� Zc5

�r < 0.5

k0 + 1.14 × 10−2 [exp (0.67�r ) − 1.069]

� Zc5

0.5 < �r < 2.0

k0 + 2.6 × 10−3 [exp (1.155�r ) + 2.016]

� Zc5

2.0 < �r < 2.8

(52)

where k0 is the ideal gas or low-pressure limit of the thermal con-ductivity and �r is the reduced density, calculated in this workas:

�r = 38

PTc

ZTf Pc(53)

and � is a parameter defined by:

� = 210

(TcM3

Pc4

)1/6

(54)

where, k0, T, Pc and M are in (W/m K), K, bar, and (g/mol), respec-tively.

5.3. Heat capacity

Heat capacities of supercritical gases are related to correspond-ing values in the ideal gas or low-pressure state (at the sametemperature and composition) by the following definition:

Cp = Cp0 + Cp (55)

where, Cp is so-called the residual heat capacity. (This defini-tion can be applied to a pure gas or to a gas mixture at constantcomposition.) The residual heat capacity can be estimated from theLee–Kesler [32] correlation as a function of Tr and Pr:

Cp = Cp − Cp0 =(

Cp

)0 + ω(

Cp

)1(56)

6. Results and discussion

At first, the results of thermal expansion coefficient (ˇ) are cal-culated by the thermodynamic model based on the Redlich–Kwongequation of state. Previous work has been done by Rolando [22]

A.R. Teymourtash et al. / J. of Supercritical Fluids 74 (2013) 115– 127 121

F n ideas

wpduFa(dta

Fs

ig. 5. Comparison between thermal expansion coefficient of carbon dioxide based otudy] with reference values [28] at P = 7 MPa (b) close up.

ho used van der Waals equation of state at constant surface tem-erature. Figs. 5–7 represent three curves of versus T for carbonioxide at three different pressures for ideal-gas, reference val-es [28], van der Waals [22] and Redlich–Kwong equation of state.or carbon dioxide the thermal expansion coefficient of ideal gasssumptions was calculated as the inverse absolute temperature

= 1/Tf). It can be seen that the Redlich–Kwong equation of state,espite its simplicity, does provide an excellent representation ofhe experimental data. Also interesting is the fact that divergest the critical point.

ig. 6. Comparison between thermal expansion coefficient of carbon dioxide based on ideatudy] with reference values [28] at P = 10 MPa (b) close up.

l gas assumption, van der Waals [22] and Redlich–Kwong equation of state [present

It is well known that for liquids at low pressure increaseswith temperature, while the opposite is true for low-pressuregases. Higher-pressure isobars show a liquid-like behavior at lowertemperatures, a gas-like behavior at higher temperatures and amaximum at an intermediate temperature, close but not necessar-ily equal to Tc. At even higher temperatures, all isobars approach

the ideal-gas behavior.

As depicted, when working fluid is near the critical point, theassumption of constant thermal expansion coefficient (ˇ) is notvalid and diverges at the critical point (Tr = 1). It is important to

l gas assumption, van der Waals [22] and Redlich–Kwong equation of state [present

122 A.R. Teymourtash et al. / J. of Supercritical Fluids 74 (2013) 115– 127

Fig. 7. Comparison between thermal expansion coefficient of carbon dioxide based on ideal gas assumption, van der Waals [22] and Redlich–Kwong equation of state [presentstudy] with reference values [28] at P = 20 MPa (b) close up.

rent p

sth

eWpaFsc

oTB

Fig. 8. Thermal expansion coefficient at diffe

ay that the value of obtained from the proposed equation inhis paper has more precise values than van der Waals equation atigher pressure and temperatures.

Fig. 8(a) and (b) shows the three isobar curves of thermalxpansion coefficient for carbon dioxide and water, respectively.ith considering these figures, it can be observed that at higher

ressures for both fluids, curves will be smoother and its vari-tions’ decrease versus temperatures. By comparison betweenig. 8(a) and (b), it was observed that at the definite reduced pres-ure (Pr = 1), thermal expansion coefficient has greater value forarbon dioxide.

The main objective in this study is to determine the influencef the non-uniform heat flux distribution on supercritical fluids.his paper focused on the case of a linearly surface heat flux.oth negative and positive slope were investigated. The lack

ressures for (a) carbon dioxide and (b) water.

of experimental results in such configuration is led to comparenumerical results to experimental results that were obtained inthe case of constant surface heat flux in supercritical fluids. Thesymmetry of both configurations allows this comparison.

Nusselt number values for vertical flat plate with different con-stant heat flux are plotted as a function of the Rayleigh number forsupercritical carbon dioxide and water in Fig. 9(a) and (b), respec-tively. The results of these simulations are plotted in logarithmicscale so that the curves approach straight lines. The curve corre-sponding to experimental correlation is obtained from a relationthat was proposed by Churchill and Chu [33] and is applicable over

a wide range of Rayleigh numbers and near critical region. This is:

NuL = 0.68 + 0.67RaL1/4

[1 + (0.492/Pr)9/16]4/9

(57)

A.R. Teymourtash et al. / J. of Supercritical Fluids 74 (2013) 115– 127 123

F –Kwoe er.

I

R

nrFa

pet

ig. 9. Local Nusselt as a function of local Rayleigh at Pr = 1.05 based on Redlichxperimental correlation [33] at Qw = 5000 W/m2 for (a) carbon dioxide and (b) wat

n which, RaL is the Rayleigh number with the following definition:

aL = Rax=L = GrL · Pr = gˇQL4

˛k(58)

As it was mentioned by Incropera and Dewitt [34], the Rayleighumber must have a maximum value in the order of 109 foremaining in the laminar region to satisfy this study conditions.or this reason, height of the plate (L) and heat flux (Q) should have

limitation to prevent the turbulent occurrence.

Figs. 5–7 show that in supercritical condition (Tr, Pr 1), pro-

osed model in this study which is based on Redlich–Kwongquation of state, gives more accurate value than ideal gas assump-ion (constant thermal expansion coefficient) and van der Waals

Fig. 10. Local Nusselt as a function of local Rayleigh for Qw = 2000 W/m2 at (a) const

ng [present study], van der Waals, constant thermal expansion coefficient and

equation that was used be Rolando [22]. As shown in Fig. 9(a),this accuracy in thermal expansion coefficient estimation is ledto more precise calculation of local Nusselt number. So it can beconcluded that Redlich–Kwong equation of state has an excel-lent accuracy to predict thermal expansion coefficient and naturalconvection characteristics in supercritical region as it was usedin this paper. By comparison between Figs. 9(a) and (b), it wasobserved that at the definite Rayleigh number, Nusselt numberhas greater value for carbon dioxide. Figs. 8 and 9 show that

general behavior of water and carbon dioxide is close to eachother in supercritical condition. So for avoiding from repetitiononly carbon dioxide curves are illustrated in next parts of thispaper.

ant Tr and different values of Pr and (b) constant Pr and different values of Tr .

124 A.R. Teymourtash et al. / J. of Supercritical Fluids 74 (2013) 115– 127

Fig. 11. Effect of heat flux’s slope variation on local Nusselt number in (a) linearly increasing surface heat flux and (b) linearly decreasing surface heat flux at Pr = 1.05 andTr = 1.05.

e profi

cvscnaItn

lscc

Fig. 12. Effect of heat flux’s slope variation on (a) dimensionless temperatur

Fig. 10 shows Nux as a function of Rax for carbon dioxide at (a)onstant Tr and different values of Pr (b) constant Pr and differentalues of Tr. As it is shown, by changing supercritical fluids pres-ure (Fig. 10(a)) different Nusselt lines are closer to each other thanhanging supercritical fluids temperature (Fig. 10(b)). This phe-omenon expresses that thermal properties in supercritical fluidsre more relevant to temperature changing than pressure ones.t can be observed that supercritical fluids are more sensible toemperature changing and small variations in temperature has aoticeable effects on the local Nusselt number lines.

Fig. 11 shows the effect of surface heat flux slope’s variation on

ocal Nusselt number. Fig. 11(a) and (b) shows that by decreasingurface heat flux’s slope, local Nusselt number lines approach toonstant surface heat flux. The results of constant surface heat fluxondition have been validated before. So decreasing surface heat

les and (b) local Nusselt number for carbon dioxide at Pr = 1.05 and Tr = 1.05.

flux’s slope and approaching to constant surface heat flux condition,confirms the accuracy of presented model to estimate heat transfercharacteristics of linearly variable surface heat flux.

Fig. 12 shows the effect of changing surface heat flux slopeon dimensionless temperature profiles and local Nusselt number.Fig. 12(a) indicates that the boundary layer for linearly increasingsurface heat flux is thinner than the others, so that heat trans-fer coefficient and local Nusselt number from vertical flat platewith linearly increasing surface heat flux must be greater than con-stant and linearly decreasing surface heat flux conditions. Fig. 12(b)confirms previous deduction. It can be observed that curve cor-

respond to linearly increasing surface heat flux is above constantand linearly decreasing one. With these considerations it can beconcluded that for having greater heat transfer coefficient, linearlyincreasing surface heat flux is more useful than other kinds of

A.R. Teymourtash et al. / J. of Supercri

Ffl

hiio

wi(hiioa

f

decreases gradually and finally stop in outer region of thermal

F

ig. 13. (a) Comparison of Surface temperature distribution between supercriticaluid (Pr > 1, Tr > 1) and under critical fluid (Pr = 0.2 and Tr = 1.05).

eat fluxes. So in some special industrial process in which hav-ng a great heat transfer coefficient and local Nusselt number ismportant, linearly increasing surface heat flux condition is perfectption.

Fig. 13(a) shows surface temperatures distribution of the plateith constant heat flux which has been immersed in carbon diox-

de. Surface temperature has been calculated for under criticalPr = 0.2, Tr = 1.05) and supercritical (Pr > 1, Tr > 1) conditions. As itas been shown, at specific surface heat flux, supercritical condition

s led to intensive decrease in temperature of the plate in compar-son with under critical one. So, this figure justifies the applicationf supercritical fluids for a plate with metallurgical surface temper-

ture restrictions.

Fig. 14 shows the effect of heat flux slope variation on sur-ace temperature of the plate. As it can be observed in Fig. 14(a)

ig. 14. Surface temperature distribution of carbon dioxide for (a) linearly increasing sur

tical Fluids 74 (2013) 115– 127 125

enhancement in slope of linearly increasing heat flux has been ledto linear distribution of surface temperature. So linearly increasingwall temperature boundary condition can be achieved by specificslope of heat flux.

Two valuable results can be obtained with consideration ofFig. 14(b). These results are:

(1) For heat flux slopes steeper than specific ones, wall surface tem-perature has a maximum value at specific height of the plate.This phenomenon shows that supercritical fluids with linearlydecreasing heat flux can be used for a plate with metallurgicalsurface temperature restrictions.

For specific slope of heat flux, temperature variations along thewall decrease. So constant temperature boundary condition justabove the bottom to top of the plate can be obtained by choosingappropriate slope for linearly decreasing surface heat flux.

Fig. 15 shows surface temperature variations for constantand variable heat fluxes. It can be observed that at the bottomparts of the plate, surface temperature distribution curves for lin-early decreasing heat flux is closer to constant one. By movingupward the plate, surface temperature differences between lin-early decreasing heat flux and constant one will be had great values.Now with this consideration it may be concluded that reverse sit-uation occurs for linearly increasing surface heat flux. However, asit is shown in Fig. 15, temperature differences are enhancing frombottom of the plate to middle of it. After that, this difference isdecreasing with moving to top of the plate.

Fig. 16 shows velocity profile versus vertical distance fromplate for carbon dioxide at height of x = 0.01 m at Pr = 1.05 andTr = 1.05 with different surface heat fluxes. It can be observed thatheat flux enhances fluid temperature in thermal boundary layer.This enhancement, decrease fluids density so the ambient fluidwhich has lower temperature and higher density, pushes the insideboundary layer fluid upward. Far from the plate, fluid temperatureand density reach to ambient fluid conditions .So, fluid velocity

boundary layer. Fluid velocity can be assumed zero at the vicinityof the plate, with consideration non-sliding principle. So the fluidvelocity increasing initially to reach the maximum value, after that,

face heat flux and (b) linearly decreasing surface heat flux at Pr = 1.05 and Tr = 1.05.

126 A.R. Teymourtash et al. / J. of Supercritical Fluids 74 (2013) 115– 127

Fig. 15. Comparison of surface temperature distribution among variable increasingand decreasing heat fluxes with constant one for carbon dioxide at Pr = 1.05 andTr = 1.05.

Fs

dfc

ivatv

anatsh

Fig. 17. Contours of (a) temperature and (b) velocity of carbon dioxide at Pr = 1.05and Tr = 1.05 for Qw = 5000 W/m2.

Fig. 18. Velocity profiles of carbon dioxide at Pr = 1.05 and Tr = 1.05 for linearlyincreasing and decreasing and constant surface heat fluxes measured in height ofx = 0.01 m.

ig. 16. Velocity profiles of carbon dioxide at Pr = 1.05 and Tr = 1.05 for differenturface heat flux measured in height of x = 0.01 m.

ecreases gently. Finally, outside of thermal boundary layer and farrom the plate it reaches to zero again. Velocity curves in Fig. 16onfirm above results obviously.

As it has been shown, increasing surface heat flux, led to max-mum fluid velocity enhancing and closing to plate surface. Forarious heat fluxes one special point near the plate exists in whichll velocity curves overlap to each other. So, one special point nearhe plate has a constant velocity regardless of surface heat fluxalues.

Two contours that illustrated temperature and velocity of fluidlong the plate for constant heat flux are plotted in Fig. 17. In defi-ite vertical distance in thermal boundary layer, moving upwardlong the plate leads to an increment in the amounts of fluids

emperature as shown in Fig. 17(a) which causes decrement in den-ity of the fluid. So the ambient fluid with lower temperature andigher density, pushes the inside boundary layer fluid upward more

easily. This phenomenon results in increasing the fluid velocityalong the plate.

Fig. 18 shows velocity profile versus vertical distance from platefor constant and variable heat fluxes which has been measured inx = 0.01 m of plate. As it can be observed, velocity curves correspondto constant heat flux boundary condition is notably above variableheat fluxes. This situation is reasonable and predictable becausein constant surface heat flux boundary condition, more heat fluxesentered to the fluid in comparison with variable ones. It has beenshown that the velocity profile for linearly decreasing heat flux hashigher values than increasing one, despite of its lower heat fluxes,at the top part of the plate. This phenomenon is due to more heatflux entering to the fluid at the bottom section of the plate in lin-early decreasing heat flux, in comparison with linearly increasingone. So the fluid’s upward velocity increases and this enhancementin initial velocity at the bottom part of the plate, which leads to

velocity increment at the top part (x = 0.01 m) of the plate.

percri

7

tRcaiAuwNmttcthwsaniteo

R

[

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A.R. Teymourtash et al. / J. of Su

. Conclusion

A thermodynamic model has been developed to representhe thermal expansion coefficient of supercritical fluid based onedlich–Kwong equation of state. When developing a model ororrelation for free convention, it is normally assumed that is

constant. In view of the results presented here, when work-ng at supercritical condition this assumption is no longer valid.

numerical model was developed for the analysis of the nat-ral convection over a vertical flat plate for supercritical fluidsith constant and variable heat fluxes. The obtained values ofusselt number based on Redlich–Kwong equation have closeragnitude to experimental correlation than van der Waals equa-

ion and ideal gas assumption. According to the nature of problem,he velocity and temperature profiles make sense and a logi-al trend is observed in obtained graphs. It has been shownhat thickness of the boundary layer in linearly increasing walleat flux is much lower than constant heat flux one. The resultsere expressed in terms of dimensionless numbers as local Nus-

elt and Raleigh numbers. It was found that linearly increasingnd decreasing heat flux will increase and decrease local Nusseltumber, respectively, in comparison with the constant ones. So

n some special industrial process in which having a great heatransfer coefficient and local Nusselt number is important, lin-arly increasing surface heat flux condition is the most usefulnes.

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