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Mixed convection on a vertical plate in supercritical fluids by selecting the best equation of state
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Mixed convection on a vertical plate in supercritical fluids by selecting the best equation of state Danyal Rezaei Khonakdar a,∗ , Mohammad Reza Raveshi b a b
Faculty of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 29 May 2015 Received in revised form 12 July 2015 Accepted 13 July 2015 Available online xxx Keywords: Supercritical fluids Mixed convection Nusselt number Upward and downward flow Richardson number
a b s t r a c t The present article investigates laminar mixed convection with upward and downward flow over a vertical flat plate in supercritical fluid, numerically. At first four different equations for thermal expansion coefficient estimation in supercritical fluids are derived as a function of pressure, temperature and the compressibility factor. Calculated values of thermal expansion coefficient and Nusselt number have been compared with the experimental results and show better accuracy of Redlich–Kwong EOS in comparison with others. After that governing equations for mixed convection are solved numerically by using finite difference method. A parametric study is performed to illustrate the interactive influence of the governing parameters, specifically the Richardson number and the effect of its variation on the rate of heat transfer and Nusselt number in supercritical condition. Finally, heat transfer curves as a function of Richardson number are plotted and observed that general trend of supercritical fluids curves are similar for water and carbon dioxide. The achieved results can also be validated with shown velocity and temperature counters and vectors. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Convection along a heated vertical flat plate is one of the most common heat transfer phenomena in heat and mass transfer studies, due to its occurrence in a wide variety of heat transfer industrial equipment. If the natural convection existing in this case is accompanied by a forced flow, the combined mode of free and forced convection will exit, which is usually called as mixed convection. Depending on the forced flow direction, the buoyancy forces may aid or oppose the forced flow, causing an increase or decrease in heat transfer rates [1]. So, all mechanisms that enhance natural or forced convection can enhance mixed convection and vice versa. Using fins and ribs to increase the surface area, ultrasonic vibrating of working fluid, applying electrical field, adding surfactants or nanoparticles to the base fluid and finally, using base fluid in supercritical conditions are the most common methods, which have been used to improve one of the free and forced convection or both of them and finally enhance mixed heat transfer mechanism [2–7]. The theory of mixed convection shows that the ratio of the Grashof number to the square of Reynolds number which is known
∗ Corresponding author. Tel.: +989113531929. E-mail address: [email protected] (D. Rezaei Khonakdar).
as the non-dimensional Richardson number, has a great effect on the flow with the constant plate’s temperature boundary condition. The forced convection is dominating in small values of the Richardson number because of large values of Reynolds number; on the other hand, free convection overcomes in large values of the Richardson number because of large values of Grashof number. Reviewing the literatures shows that the mixed convection theory from a flat plate for under critical condition is well established and has been investigated by various researchers. Sparrow et al. [8] were the first to study the effect of the buoyancy forces on the forced convection flow over a vertical flat plate. The results of their study have shown that the flow can be classified by using the Richardson number. A combination of numerical integration and series expansion has been used by Merkin [9] and Wilks [10] to solve the mixed convection problem for both aiding and opposing flows in a Prandtl number of unity along a vertical plate with constant wall temperature and constant heat flux boundary condition, respectively. Lioyd and Sparrow [11] used the local similarity method for different values of the Prandtl number to discuss the mixed convection under small effect of Richardson number. Oosthuizen and Hart [12] solved the constant wall temperature and the heat flux problem over a vertical flat plate, numerically. First experimental results of the mixed convection along a vertical plate for air have been presented by Gryzagoridis [13] in a wide range of Richardson number. The comprehensive review concerning mixed convective
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Nomenclature a,b A,B Cp Cp0 g Gr h k k0 L M Nu P Pr R Ra Ri Re T ω u,v V’ u’,w U,V x,y X,Y Z
EOS constants that corrects for intermolecular attractive forces (N m4 /mol2 ) dimensionless EOS parameters heat capacity of the fluid (J/kg K) heat capacity at the low-pressure limit (J/kg K) local acceleration of gravity (m/s2 ) Grashof number heat transfer coefficient (W/m2 K) thermal conductivity of the fluid (W/m K) thermal conductivity at the low-pressure limit (W/m K) height of the vertical plate (m) molecular weight of the fluid (kg/mol) Nusselt number pressure (Pa) Prandtl number gas constant (J/mol K) Rayleigh number for the whole plate Richardson number Reynolds number temperature (K) density of the fluid (kg/m3 ) acentric factor velocity components in the x,y direction, respectively (m/s) molar volume (m3 /mol) EOS parameter dimensionless velocity components in the x and y directions respectively coordinates parallel and normal to the vertical plate, respectively (m) dimensionless coordinate parallel and normal to the vertical plate, respectively compressibility factor
Greek symbols thermal diffusivity (m2 /s) ˛ ˇ thermal expansion coefficient (1/K) kinematic viscosity (m2 /s) inverse thermal conductivity (m K/W) dimensionless temperature inverse viscosity (m2 /N s) Subscripts c critical condition fluid f i node index in the x direction j node index in the y direction P constant pressure reduced characteristic r w wall x local distance outside the boundary layer ∞
flow for under critical fluids are available in the books by Pop and Ingham [14]. Jena and Mathur [15] solved laminar mixed convection problem of a micropolar fluid from an isothermal vertical flat plate by using a finite-difference technique. They found that the velocity increased and the temperature decreased with the process of fluid’s heating. Many investigations have been done on the effectiveness of supercritical fluids in mixed convection heat transfer; however,
there are no reports about the mixed convection of supercritical fluid over a vertical flat plate. Kakaral and Thomas studied mixed convection heat transfer in a vertical tube of supercritical fluids. They evaluated the effect of flow direction at rate of heat transfer in tube [16]. Pioro and Duffey’s work [17] is considered as a very useful review literature of supercritical heat transfer phenomena until now. The mixed convective heat transfer of carbon dioxide at supercritical state in a circular vertical and horizontal tube was investigated experimentally and numerically, respectively [18,19]. These studies concerned the effects of the Richardson number on the convective heat transfer in tubes. They both used this fact that the thermophysical properties of supercritical fluids are highly sensitive to temperature. To study the mixed convection in under critical fluids with temperature-dependent properties over a vertical flat plate; at first, Kafoussias et al. [20] used a modified numerical solution to study the problem with temperature-dependent viscosity. Pantokratoras [21] assumed the temperature-dependent density to study the laminar mixed convection problem of water along a vertical isothermal plate. Mixed convection laminar boundary layer flow of water-based nanofluids along the vertical plate with temperaturedependent heat source has been investigated numerically by Rana et al. [22]. As mentioned before, in supercritical condition the properties of the fluid specially the thermal expansion coefficient are highly related to temperature and pressure. The thermal expansion coefficient (ˇ) has been assumed to be constant and its variation versus temperature and pressure has been neglected in most of previous works. In supercritical fluids, this consideration will lead to wrong results. So, it is evident that the determination of the value of ˇ through an appropriate equation of state is necessary. The effect of temperature, pressure and the compressibility factor (Z) on thermal expansion coefficient was first considered by Rolando in the problem of free convection over an isothermal vertical plate [23]. In his research a new equation for the thermal expansion coefficient has been derived based on the Van der Waals equation of state. Teymourtash and Ebrahimi Warkiani [24] have investigated the effect of linear variation of wall temperature on the heat transfer by natural convection along a vertical plate in supercritical fluids. Teymourtash et al. [25] have carried out the same problem with uniform and non-uniform surface heat flux. For the first time they proved that the Boussinesq approximation is applicable over a wide range of temperature and pressure variation for supercritical fluid. They showed that the Redlich–Kwong equation of state can estimate the supercritical fluid properties much better than Van der Waals or real gas ones. They also found that linearly increasing and decreasing heat flux will increase and decrease the local Nusselt number, respectively, in comparison with the constant ones. Among all equations of state (EOS) which used to estimate the fluid’s properties only real gas, Van der Waals and Redlich–Kwong ones have been used in previous researches for supercritical fluids [23–25]; however, there are some other useful equation of states that can be used in heat transfer estimation at supercritical condition. So, first exclusive objective of this literature is to find the best equation of state among Virial (1901), Beattie–Bridgman (1928), Redlich–Kwong (1949), Soave (1972) and Peng–Robinson (1976) [26] equations for predicting the supercritical fluid’s thermal expansion coefficient. After that, the problem of mixed convection along a vertical flat plate is solved for the first time by using a numerical model based on a finite-difference formulation and the most appropriate equation of state. Finally, numerical results are presented and effects of Richardson number on heat transfer characteristics in aiding and opposing mixed flow are investigated.
Please cite this article in press as: D. Rezaei Khonakdar, M.R. Raveshi, Mixed convection on a vertical plate in supercritical fluids by selecting the best equation of state, J. Supercrit. Fluids (2015), http://dx.doi.org/10.1016/j.supflu.2015.07.013
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2. Evaluation different equations for calculating the thermal expansion coefficient (ˇ) in supercritical region Thermal expansion coefficient at constant pressure is defined as 1 ˇ=−
∂ ∂T
(1)
Eq. (10) is the basic equation for calculation of thermal expansion coefficient as a function of pressure and temperature. Partial derivation of this equation with respect to temperature at constant pressure is as follows:
2
ˇ=
1 1 + T Z
2
3Z + (2B − 2) Z + A − 3B − B
∂Z ∂T
P
With consideration of the compressibility factor (Z) defining and referring to our previous work [25] more complete form of Eq. (1) is derived as follows:
3
∂Z ∂T
= ˇi + P
1 Z
∂Z ∂T
2
− Z + A − 2 (Z + 3B) − 2B + 3B
(2)
∂A ∂T
− (Z + B)
P
where ˇi is ideal gas thermal expansion coefficient. Eq. (2) is in general form and it can be useful when the equation of state is as follows: Z = Z(P, T )
P
∂B ∂T
P
=0
(14)
P
in which:
∂A ∂T
=− P
(3)
As mentioned before, Redlich-Kwong and Van der Waals equations of state were used in the previous works [23–25] to estimate the behavior of supercritical fluids and it was proved in the later literature [25] that Redlich–Kwong equation has a closer magnitude to experimental correlation than Van der Waals equation and ideal gas assumption. So, in this paper other common equations of state have been used for modeling thermal expansion coefficient and their results are compared with the experimental correlation and Redlich–Kwong ones. After this investigation the most appropriate equation of state to predict the thermal expansion coefficient for studying the supercritical fluid’s behavior in heat transfer phenomena will be announced.
2
2A aP ∂˛ + 2 2 T R T ∂T
2A aP =− + 2 2 T R T
∂B ∂T
=− P
2 − 2 Tr−0.5 − Tr−0.5 Tc
(15)
B T
(16)
Replacing Eqs. (15) and (16) in Eq. (14) results in
∂Z ∂T
=
P
Z2
+ A − 2 (Z + 3B) − 2B + 3B2
3Z 2 + (2B − 2) Z
∂B/∂T
+ (Z + B) ∂A/∂T
P + A − 3B2 − B
In 1976, Peng and Robinson stated general equation of state by inserting acentric factor (ω) in their equation to estimate more accurate values for gas and fluid characteristics. This equation has valuable accuracy for fluid characteristic estimation near critical region. The general form of this equation is as follows: RT a˛ − 2 V −b V + 2bV − b2
(4)
0.457235R2 Tc2
˛ = 1 + 1 − Tr0.5
(6)
2
∂Z ∂T
= ˇi + P
1 Z
∂Z ∂T
P
Z + A − 2 (Z + 3B) − 2B + 3B2 2
−B/T
P
+ (Z + B) ∂A/∂T
3Z 2 + (2B − 2) Z + A − 3B2 − B
P
(18)
Previous procedure can be done for other relevant equations of state. So, in order to summarize the discussion, other equations of state have been introduced only with their constants and final form of thermal expansion coefficient relation. 2.2. Soave modification equation
(8)
T Tc
(9)
By choosing appropriate constant it can be converted to polynomial form.
(7)
= 0.37464 + 1.54226ω − 0.26992ω2 Tr =
1 1 + T Z
= ˇi −
(5)
pc
0.077796RTc b= pc
By inserting Eq. (17) in Eq. (2) following relation for calculating thermal expansion coefficient based on Peng–Robinson equation of state can be achieved as follows: ˇ=
The constants of this equation of state are as follows: a=
P
(17)
2.1. Peng–Robinson equation of state
P=
Z 3 + (B − 1) Z 2 + A − 3B2 − 2B Z + B2 + B3 − AB = 0
(10)
in which, A, B and Z definitions are as follows:
√ In 1972, Soave replaced the 1/ T term of the Redlich–Kwong equation with a function ˛ (T,ω) involving the temperature and the acentric factor (The resulting equation is also known as the Soave–Redlich–Kwong equation). The ˛ function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials. The general form of this equation and its relevant constants are as follows: P=
RT a˛ − V −b V (V + b)
(19)
(12)
a=
0.4275R2 Tc2 pc
(20)
(13)
b=
0.08664RTc pc
(21)
A=
a˛p R2 T 2
(11)
B=
bp RT
Z=
PV RT
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˛ = 1 + 1 − Tr0.5
2
= 0.48508 + 1.5517ω − 0.1763ω2
(22)
The constant of Eq. (32) (B1 (Tr )) which has been mentioned in Ref. [26], is as follows:
(23)
B1 (Tr ) = B(0) (Tr ) + ωB(1) (Tr )
By choosing appropriate constant it can be converted to polynomial form:
Z 3 + Z 2 + B2 + B − A Z − AB = 0
(24)
in which, A, B and Z definitions are as their definitions in Eqs. (11)–(13). Finally, by inserting in Eq. (2) following relation for calculating thermal expansion coefficient based on soave modification has been achieved. 1 1 ˇ= + T Z = ˇi +
∂Z ∂T
P
1 = ˇi + Z
∂Z ∂T
B(1) (Tr ) = 0.139 − 0.172Tr−5.2
(35)
Finally, by doing the same procedure and inserting in Eq. (2), following relation for calculating thermal expansion coefficient based on the Virial equation of state has been achieved.
1 1 + T Z
P
= ˇi
(25)
3Z 2 + 2Z + B2 + B − A
(34)
ˇ= P
B(0) (Tr ) = 0.083Tr−1 − 0.422Tr−2.6
(Z + 2BZ − A) B/T + (Z + B) ∂A/∂T
in which:
(33)
1+
∂Z ∂T
P
1/Pr
0.8944ωTr−7.2 − 1.0972Tr−4.6 − 0.083Tr−3 − 0.083Tr−1 − 0.422Tr−2.6 + 0.139ω − 0.172ωTr−5.2
(36)
3. Problem statement 2.3. Beattie–Bridgman equation of state
(28)
Consider the steady, laminar, two-dimensional, mixed convection boundary-layer flow over a vertical flat plate which is shown in Fig. 1. The coordinate system is chosen such that x and y measure the distance along and normal to the plate, respectively. Far away from the plate, the velocity and the temperature of the uniform stream are U∞ and T∞ , respectively. The entire surface of the plate is maintained at a uniform temperature of Tw . As it has been mentioned in the later literature [25], the Boussinesq approximation is valid for supercritical fluids; so, the boundary-layer form of the governing equations based on the balance laws of mass, momentum and energy can be written as follows:
(29)
∂u ∂v + =0 ∂x ∂y
In 1928, this equation was proposed by the Beattie–Bridgman. The compressible form of this equation and its relevant constants are as follows:
Z =1+
B C + − 4 Tr Tr2 Tr A
A = 1284.9 B = 1.678
(26)
Pc
(27)
(RTc )2
P c RTc
C = 61.56 × 10−6
Pc
RTc4
2
Finally, with same procedure and implementing in Eq. (2), following relation for calculating thermal expansion coefficient based on Beattie–Bridgman equation of state has been achieved. 1 1 ˇ= + T Z
∂Z ∂T
P
1 = ˇi + Z
−2ATc Tr3
−BTc 4CTc + + Tr Tr5
(30)
2.4. Virial equation of state Although the Virial equation of state is not the most convenient equation of state, it is very important because it can be derived directly from statistical mechanics. This equation is also called the Kamerlingh Onnes equation. If appropriate assumptions are made about the mathematical form of intermolecular forces, theoretical expressions can be developed for each of the coefficients in this equation of state. In this case, B corresponds to interactions between pairs of molecules, C to triplets, and so on. Accuracy can be increased indefinitely by considering higher order terms. The coefficients B, C, D, etc. are functions of temperature only. Z=
(39)
PV B C D =1+ + 2 + 4 + ··· RT Vr Vr Vr
u
∂u ∂u ∂ u +v ± ˇg (T − T∞ ) = ∂x ∂y ∂ y2
u
∂T ∂T ∂ T +v =˛ ∂x ∂y ∂ y2
(40)
2
(41)
where u and v are the velocity components parallel and perpendicular to the plate, respectively. Also, T is the temperature, ˇ is the thermal expansion coefficient, v is the kinematic viscosity, is the fluid density, g is the acceleration due to gravity and ˛ is the thermal diffusivity.
(31)
For achieving the purpose of this research, the approximation form of Virial equation has been used [26]. This form of the equation is as bellows: Z=
PV = 1 + B1 (Tr ) Pr RT
(32)
Fig. 1. Geometry and coordinates of the system for (a) aiding, (b) opposing mixed flow.
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The appropriate boundary conditions for the velocity and temperature of this problem are as follows [23]: x = 0, x > 0,
y > 0;
u = U∞ ,
y = 0;
u = 0,
y → ∞;
T = T∞
v = 0,
u → U∞ ,
T = TW ,
(42)
T → T∞ .
By definition of the following dimensionless variables: u , U∞
U=
vReL1/2
V=
U∞
,
X=
1/2
x , L
Y=
yReL
,
L
=
T − T∞ . TW − T∞ (43)
Eqs. (39)–(41) are converted to the following dimensionless forms:
∂U ∂V + = 0, ∂X ∂Y
(44) 2
U
∂U ∂U ∂ U ± Ri, +V = ∂X ∂Y ∂Y 2
U
∂ ∂ 1 ∂ +V = Pr ∂Y 2 ∂X ∂Y
(45)
2
(46)
where Pr =
, ˛
gˇ (T − T∞ ) L3 , 2
Gr =
Re =
U∞ L ,
Ri =
Gr Re2
(47)
Pr is the Prandtl number; Gr is the Grashof number, Re is the Reynolds number and Ri is the Richardson number. In the dimensionless form, the boundary conditions can be written as follows: X = 0,
Y > 0;
U = 1,
X > 0,
Y = 0;
U = V = 0,
y → ∞;
U → 1,
= 0, = 1,
(48)
→ 0.
−
Vi−1,j 2 Y
+
−
Vi−1,j 2 Y
Vi−1,j 2 Y
+
−
−
Vi−1,j 2 Y
1
Y 2 −
1
Y 2
1 Pr Y 2 −
Ui,j−1 +
Ui−1,j
X
Ui,j+1 =
1 Pr Y 2
+
2 Ui−1,j
i,j−1 +
numerical procedures can be found in the textbook by Oosthuizen and Naylor [27]. A mesh system with 150 × 150 nodes is proven to provide mesh-independent results. Heat-transfer coefficient, once the temperature field is obtained as described above, is based on the following equation: Nux 1/2
Eqs. (44)–(46) are coupled nonlinear differential equations which must be solved under the boundary conditions given in Eq. (48). These equations are solved by using the finite-difference method, because the exact analytical solutions are not possible for this set of equations. The equivalent finite difference schemes corresponding Eqs. (44)–(46) are given by
Fig. 2. Comparison between thermal expansion coefficient of carbon dioxide which has been calculated from different EOS, Redlich–Kwong [25] and reference values [28] at P = 7 MPa.
X Ui−1,j
i,j+1 =
X
2
Y 2
−X W
∂ ∂Y
(52) Y =0
Supercritical fluid properties which are used in the numerical procedure can be achieved as it was mentioned in the authors’ previous work [25].
Ui,j
± Rii−1,j ,
+
ReL
=
2 Pr Y 2
(49)
i,j
Ui−1,j i−1,j
(50)
X
After some derivations V values can be determined from the continuity equation as Vi.j = Vi,j−1 −
Y 2 X
Ui,j − Ui−1,j + Ui,j−1 − Ui−1,j−1 .
(51)
Here, the index i refers to x and j to y. The above equations are explicit in x-direction, while they are implicit in y-direction. After specifying the conditions along some initial i = 1, U values and, in the following, values can be obtained in the i = 2 line. Then, V values are obtained in the i = 2 line. Having determined the values of all the variables on the i = 2 line, the same procedure can then be used to find the values in the i = 3 line and so on. More details on the
Fig. 3. Comparison between thermal expansion coefficient of carbon dioxide which has been calculated from different EOS, Redlich–Kwong [25] and reference values [28] at P = 10 MPa.
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Fig. 4. Comparison between thermal expansion coefficient of carbon dioxide which has been calculated from different EOS, Redlich–Kwong [25] and reference values [28] at P = 15 MPa.
4. Results and discussion The results of thermal expansion coefficient (ˇ) which are calculated by proposed model based on different equations of state have been illustrated here. Figs. 2–7 represent six curves of ˇ versus T for carbon dioxide and water at three different pressures for reference magnitude obtained from experimental data for CO2 [28] and H2 O [29], Redlich–Kwong [25], Peng–Robinson, Soave modification, Beattie–Bridgman and Virial equations of state. Despite the fact that the Redlich–Kwong equation is simple and it has been introduced earlier than some of other equations of state; extracted model of it has an acceptable accuracy near critical region and works more precise in comparison with others. In addition, these figures show that when the temperature increases, all curves
Fig. 5. Comparison between thermal expansion coefficient of water which has been calculated from different EOS, Redlich–Kwong [25] and reference values [29] at P = 20 MPa.
Fig. 6. Comparison between thermal expansion coefficient of water which has been calculated from different EOS, Redlich–Kwong [25] and reference values [29] at P = 30 MPa.
approach to the reference value; so, all equations of state can be used in very high temperature. It is clear that new equations of state such as Peng–Robinson and Soave, estimate thermodynamic properties more precise than Redlich–Kwong, but as it can obviously be seen in Figs. 2–7 extracted models of these equations of state cannot work properly in thermal expansion coefficient estimation. So, when consideration of all fluid’s properties with thermal expansion coefficient is needed, Redlich–Kwong equation of state is the best choice. For better comparing the accuracy of these proposed equations and their effects on heat transfer phenomenon a FORTRAN code for predicting heat transfer near critical region has been developed by using these models. Figs. 8 and 9 show the Nusselt number versus Rayleigh number for carbon dioxide and water, respectively.
Fig. 7. Comparison between thermal expansion coefficient of water which has been calculated from different EOS, Redlich–Kwong [25] and reference values [29] at P = 50 MPa.
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Following relation which was proposed by Chirchill and Chu [30], has been used to obtain the curve corresponding to experimental correlation. This curve is valid near critical region and over a wide range of Rayleigh numbers: 1/4
¯ L = 0.68 + Nu
0.67RaL
1 + (0.492/Pr)
9/16
4/9
(53)
where, RaL is the Rayleigh number with the following definition: RaL = Rax=L = GrL .Pr =
Fig. 8. Local Nusselt as a function of local Rayleigh at Pr = 1.05 based on different equation of state and experimental correlation [30] at Tw = 330 K for carbon dioxide.
Fig. 9. Local Nusselt as a function of local Rayleigh at Pr = 1.05 based on different equation of state and experimental correlation [30] at Tw = 700 K for water.
gˇQL4 ˛k
(54)
This work studies laminar regions; so, the Rayleigh number cannot exceed 109 as it was mentioned by Incropera and Dewitt [31]. One of the most useful results which can be obtained from these figures is finding the relation between thermal expansion coefficient and Nusselt number. As it can be obviously seen from Figs. 2–7, all thermal expansion coefficients, which have been obtained from different models, reported different values. So, it seems that this variation in thermal expansion coefficient magnitude may have a sensible effect on the heat transfer characteristic and Nusselt number. Figs. 8 and 9 confirm this discussion. As it can be seen in these figures thermal expansion coefficient magnitude has a direct effect on the rate of heat transfer and Nusselt number. In fact, each equations of state which reports higher values for thermal expansion coefficient leads to higher estimation of values in Nusselt number and vice versa. In addition, all thermal expansion curves converge to each other when moving upward adjacent to the plate surface. This phenomenon is due to decrease at difference between derived thermal expansions coefficient from different models. So, when moving upward, fluid temperature enhances and all thermal expansion curves converge to each other. As it can be obviously seen in Figs. 8 and 9, Nusselt curves which have been calculated by using Redlich–Kwong equation, is close to experimental curves. However, other equations of state cannot work properly in thermal expansion confident estimation in supercritical condition and lead to major errors in calculation the rate of heat transfer. So, all of the equations of state except Redlich–Kwong do not have acceptable accuracy and cannot give trustable estimations of heat transfer characteristics and Nusselt number in supercritical condition.
Fig. 10. (a) Dimensionless temperature profile, (b) Nusselt curves, of CO2 for aiding mixed flow at Pr = 1.05, Tr = 1.05.
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Fig. 11. Dimensionless velocity profile of CO2 for aiding mixed flow at Pr = 1.05 & Tr = 1.05.
Here, two different types of mixed convection are studied. The first one is aiding flow and it happens when the fluid movement direction in and out of boundary layer are the same. The other one is opposing mixed flow in which streams’ direction are opposite to each other. Due to approximate similar behavior of carbon dioxide and water in supercritical condition and prevention of figures repetition only carbon dioxide curves are illustrated in following parts of this paper. Fig. 10 shows dimensionless temperature profile and Nusselt curves for carbon dioxide at different Richardson numbers for aiding flow. As it can be obviously seen from Fig. 10(a), if the Richardson number decreases due to increase in outside velocity, the thickness of thermal boundary layer decreases and it is expected that the heat transfer coefficient and Nusselt number increases consequently. Fig. 10(b) approved the previous discussion. The curve corresponding to Ri = 10−4 has higher values than the others and this sensible distance between this curve and Ri = 10−2 shows the effect and power of forced convection in comparison with natural one. So, in mixed convection with the aiding flow for achieving greater heat transfer coefficient, increasing the outer boundary layer velocity (u∞ ) is very effective.
The vicinity of Nusselt curves in higher Richardson number shows that the portion of forced convection in total rate of heat transfer is negligible in comparison with natural one. This phenomenon is reasonable because in higher Richardson number u∞ is very small and the power of natural convection dominates the forced one. Fig. 11 shows the ratio of fluid velocity to outer boundary layer uniform velocity (u∞ ) in carbon dioxide at aiding mixed flow. Fluid velocity (u) in mixed convection depends on magnitude and direction of two different parts. First one is natural convection velocity which has been made by buoyancy effect of heated fluid near the wall in thermal boundary layer. Another one is outer boundary layer velocity (u∞ ) which is uniform and constant. When Ri number increases the ratio of fluid velocity to uniform outer boundary layer velocity decreases and vice versa. In addition, when Ri decreases (velocity increases) the maximum magnitude of velocity ratio curves decreases and it goes toward the wall. Then in Ri about 10−4 the velocity profile does not have any maximum point due to the viscous effect in the boundary layer. In fact, fluid in boundary layer decreases the outer velocity and tries to stop it. All velocity ratio curves approach to the unity with going away from the wall surface because in outside of the thermal boundary layer natural convection and viscous effect does not exist. Fig. 12 shows the dimensionless temperature profile and Nusselt curves for carbon dioxide in different Richardson numbers for opposing mixed flow. In this case, buoyancy forces have retarding effects on the flow’s outer boundary layer velocity (u∞ ); so, it is called as opposing mixed flow. As it can be obviously seen an increment in Ri results in decrement in outer velocity due to the opposing effect of the upward buoyancy-induced flow on the downward external forced flow. So, it can be predicted that the rate of heat transfer and Nusselt number should be decreased and the thickness of the thermal boundary layer should be increased by increasing the outer boundary layer uniform velocity (u∞ ). Fig. 12(b) approves the expected result. As it can be seen the curve corresponding to Ri = 10−4 has a lower Nusselt number than the others. So it can be concluded that, in opposing mixed flow at lower Richardson number (higher velocity), total rate of heat transfer increases with increasing the Richardson number. This phenomenon indicates a significant impact of forced convection on buoyancy forces.
Fig. 12. (a) Dimensionless temperature profile, (b) Nusselt curves, of CO2 for opposing mixed flow at Pr = 1.05, Tr = 1.05.
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Fig. 13. Dimensionless velocity profile of CO2 in opposing mixed flow at Pr = 1.05 & Tr = 1.05.
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Fig. 13 shows the ratio of fluid velocity to the outer boundary layer velocity (u∞ ) for carbon dioxide at opposing mixed flow. As it can be expected when Ri decreases, fluid velocity in boundary layer decreases due to retarding effect of forced flow on natural convection’s induced upward flow. In opposing mixed flow when moving perpendicularly against the wall, natural convection velocity increases and then decreases very adjacent to the wall. So, the fluid velocity decreases because of the retarding effect of natural convection velocity near to the plate surface. After that, the natural convection velocity decreases and fluid velocity decreases too. Fig. 13 confirms the above explanation. As it can be obviously seen, the ratio of fluid velocity to the uniform outer boundary layer one increases when moving vertical to the plate surface. Fig. 14(a) and (b) shows aiding and opposing flow in supercritical condition at different Ri number for carbon dioxide. As it can be observed in Fig. 14(a) and (b) at higher Ri number, Nusselt curves are closer to each other in comparison with others. In fact, increment of outer boundary layer velocity (u∞ ) leads to increment in Nusselt magnitude differences for aiding and opposing mixed flow.
Fig. 14. (a) Comparison of Nusselt number for aiding and opposing flow of CO2 in mixed convection, (b) close up view, at Pr = 1.05, Tr = 1.05.
Fig. 15. The variation of Nu/Re0.5 for CO2 and H2 O at mixed convection (a) aiding, (b) opposing flow, at Pr = 1.05, Tr = 1.05.
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Fig. 16. (a) Temperature and u Velocity contours for (a) aiding, (b) opposing flow of CO2 in mixed convection, at Pr = 1.05, Tr = 1.05 and Ri = 1.
Fig. 17. (a) v Velocity vectors for aiding flow and (b) velocity contour for opposing flow of CO2 in mixed convection, at Pr = 1.05, Tr = 1.05 and Ri = 1.
Another curve which has been illustrated in earlier references for mixed convection is Nu/Re0.5 variation when Ri increases. Here, these curves have been plotted for the first time for mixed convection in supercritical fluid conditions. Fig. 15(a) and (b) shows Nu/Re0.5 as a function of Ri number for aiding and opposing flow, respectively. As described before in aiding mixed condition for lower Ri number magnitude, increasing in Ri leads to sensible variation on rate of heat transfer and decreases Nusselt number. This phenomenon can obviously be seen in Fig. 10(b). In addition, decreasing the velocity leads to decrease in Re number, so the magnitude of Nu/Re0.5 should be approximately constant. On the other hand, in higher Ri number the Nusselt number variation is very small when Ri increasing and sometimes this variation is negligible. So, Nu/Re0.5 magnitude will be increases due to decreasing the Re number. In fact, decrement in Re number is bigger than Nusselt number when Ri increases. Fig. 15(a) confirmed above discussion and shows general behavior of carbon dioxide at Pr = 1.05 and Tr = 1.05 in the vast range of fluid’s velocity and Ri number. However, in opposing mixed flow which can be observed in Fig. 15(b) the trend of Nu/Re0.5 curves are different from aiding one and variation of Nusselt and Reynolds magnitude with Ri, leads to decrement in Nu/Re0.5 . Four different contours that illustrate temperature and velocity of fluid along the plate for aiding and opposing mixed flow are plotted in Fig. 16(a) and (b), respectively. In definite vertical distance in thermal boundary layer, moving upward along the plate
leads to an increment in the amount of fluids temperature because of decrement in density. This Phenomenon has been intensified in aiding mixed flow in comparison with opposing one. Fig. 17(a) and (b) shows v velocity vectors for aiding mixed flow and velocity contours for opposing flow of CO2 , respectively. 5. Conclusion Four different thermodynamic models have been studied and derived from Peng–Robinson, Soave modification, Beattie–Bridgman and Virial equation of state, for calculating the thermal expansion coefficient of supercritical fluid. Then, these equations have been applied to predict the rate of heat transfer from vertical flat plate. It was observed that Redlich–Kwong equation in spite of its simplicity has acceptable accuracy near critical point and reports more accurate values for thermal expansion coefficient and Nusselt number estimation rather than the others. Hence, by using extracted model from Redlich–Kwong a numerical model was developed for the analysis of the mixed convection over a vertical flat plate for supercritical fluid with constant surface temperature. According to the nature of problem, logical trend is observed at velocity and temperature profiles. In fact, obtained values of Nusselt number for aiding and opposing mixed flows show that the thickness of thermal boundary layer in the aiding mixed flow is lower than the opposing one. The same direction of induced buoyancy stream and uniform outer velocity intensifies the rate of heat transfer and Nusselt number in comparison with opposing one.
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In the next step, the effect of Ri number variations as the most famous dimensionless number in mixed convection has been studied. It was seen that for aiding mixed convections when Richardson number decreases due to an increase in uniform outer velocity, the thickness of thermal boundary layer decreases and the heat transfer coefficient and Nusselt number increases. Finally, the effect of Ri number variations on Nu/Re0.5 at mixed convection has been studied. It is concluded that for aiding mixed flow at lower Richardson number by increasing the Richardson the magnitude of Nu/Re0.5 is approximately constant. When Richardson number increases to the higher values, Nu/Re0.5 magnitude enhances. On the other hand, in opposing mixed flow Nu/Re0.5 curves are different from aiding one and the variation of Nusselt and Reynolds magnitude with Richardson number leads to decrement in Nu/Re0.5 . So, in some special industrial process in which the great heat transfer coefficient and local Nusselt number is in our major requirements, aiding mixed convection in supercritical fluids is recommended. However, the flow direction and remaining at supercritical condition should be checked; because little deviations from mentioned conditions have considerable effects on the rate of heat transfer. References [1] O. Aydın, Aiding and opposing mechanisms of mixed convection in a shear and buoyancy driven cavity, Int. Commun. Heat Mass Transfer 26 (7) (1999) 1019–1028. [2] S.U. Onbasioglu, H. Onbastoglu, On enhancement of heat transfer with ribs, Appl. Therm. Eng. 24 (2004) 43–57. [3] S. Sadri, M.R. Raveshi, S. Amiri, Efficiency analysis of straight fin with variable heat transfer coefficient and thermal conductivity, J. Mech. Sci. Technol. 26 (4) (2012) 1283–1290. [4] S.W. Wong, W.Y. Chon, Effects of ultrasonic vibrations on heat transfer to liquids by natural convection and by boiling, Am. Inst. Chem. Eng. J. 15 (2) (1969) 281–288. [5] T.B. Jones, Electrohyrodynamically enhanced heat transfer in liquids—a review, Advanced in heat transfer, in: T.F. Irvine, J.P. Hartnett (Eds.), Advances in Heat Transfer, Academic Press, New York, NY, 1978, pp. 107–148. [6] B.R. Lazarenko, F.P. Grosu, M.K. Bologa, Convective heat transfer enhancement by electric fields, Int. J. Heat Mass Transfer 18 (1975) 1433–1441. [7] K. Sadik, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (13–14) (2009) 3187–3196. [8] E.M. Sparrow, R. Eichhorn, J.L. Gregg, Combined forced and free convection in boundary layer flows, Phys. Fluids 2 (1959) 319–328. [9] J.H. Merkin, The effect of buoyancy forces on the boundary layer over a semi-infinite vertical flat plate in a uniform free stream, J. Fluid Mech. 35 (1969) 439–450.
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[10] G. Wilks, The flow of a uniform stream over a semi-infinite vertical flat plate with uniform surface heat flux, Int. J. Heat Mass Transfer 17 (1974) 743–753. [11] J.R. Lioyd, E.M. Sparrow, Combined forced and free convection flow on vertical surfaces, Int. J. Heat Mass Transfer 13 (1970) 434–438. [12] P.H. Oosthuizen, R. Hart, A numerical study of laminar combined convective flow over flat plates, J. Heat Transfer 95 (1973) 60–63. [13] J. Gryzagoridis, Combined free and forced convection from an isothermal vertical plate, Int. J. Heat Mass Transfer 18 (1975) 911–916. [14] I. Pop, D.B. Ingham, Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media, Pergaman Press, UK, 2001. [15] S.K. Jena, M.N. Mathur, Mixed convection flow of a micropolar fluid from an isothermal vertical plate, Comput. Math. Appl. 10 (3) (1984) 291–304. [16] C.R. Kakarala, L.C. Thomas, Turbulent combined force and free convection heat transfer in vertical tube flow of supercritical fluid, Int. J. Heat Fluid Flow 2 (3) (1980) 115–120. [17] I. Pioro, R. Duffey, Heat Transfer and Hydraulic Resistance at Supercritical Pressures in Power-Engineering Applications, ASME Press, New York, 2007. [18] Y. Bae, H. Kim, D. Kang, Forced and mixed convection heat transfer to supercritical CO2 vertically flowing in a uniformly-heated circular tube, Exp. Therm. Fluid Sci. 34 (2010) 1295–1308. [19] X. Lei, H. Li, S. Yu, T. Chen, Numerical investigation on the mixed convection and heat transfer of supercritical water in horizontal tubes in the large specific heat region, Comput. Fluids 64 (2012) 127–140. [20] N.G. Kafoussias, D.A.S. Rees, J.E. Daskalakis, Numerical study of the combined free-forced convective laminar boundary layer flow past a vertical isothermal flat plate with temperature-dependent viscosity, Acta Mech. 127 (1–4) (1998) 39–50. [21] A. Pantokratoras, Laminar assisting and mixed convection heat transfer from a vertical isothermal plate to water with variable physical properties, Int. J. Heat Mass Transfer 40 (2004) 581–585. [22] P. Rana, R. Bhargava, Numerical study of heat transfer enhancement in mixed convection flow along a vertical plate with heat source/sink utilizing nanofluids, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 4313–4334. [23] A. Rolando, Natural Convection Heat Transfer in Supercritical Fluid, Mechanical Engineering, University of Puerto Rico, (M. Sc. Thesis), San Juan, Puerto Rico, 2004. [24] A.R. Teymourtash, M. Ebrahimi Warkiani, Natural convection over a non-isothermal vertical flat plate in supercritical fluids, Sci. Iranica 16 (6) (2009) 470–478. [25] A.R. Teymourtash, D. Rezaei Khonakdar, M.R. Raveshi, Natural convection on a vertical plate with variable heat flux in supercritical fluids, J. Supercrit. Fluids 74 (2013) 115–127. [26] K. Annamalai, I.K. Puri, Advanced Thermodynamics Engineering, CRC Press, Boca Raton, Florida, 2002. [27] P.H. Oosthuizen, D. Naylor, Introduction to Convective Heat Transfer Analysis, McGraw-Hill, New York, NY, 1999. [28] S.B. Angus, B. Armstrong, K. Reuck, International Thermodynamic Table of The Fluid State-Carbon Dioxide, first ed., Pergamon Press, Headington Hill Hall, Oxford, England, 1976. [29] J.H. Keenan, F.G. Keyes, P.G. Hill, J.G. Moore, Steam Tables, Wiley, New York, MY, 1969. [30] S.W. Churchill, H.H.S. Chu, Correlation equations for laminar and turbulent free convection from a vertical plate, Int. J. Heat Mass Transfer 18 (11) (1975) 1323–1329. [31] F.P. Incropera, D.P. Dewitt, Fundamentals of Heat and Mass Transfer, third ed., Wiley, New York, NY, 1990, pp. 539–540.
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Contents lists available at ScienceDirect
The Journal of Supercritical Fluids journal homepage: www.elsevier.com/locate/supflu
Mixed convection on a vertical plate in supercritical fluids by selecting the best equation of state Danyal Rezaei Khonakdar a,∗ , Mohammad Reza Raveshi b a b
Faculty of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 29 May 2015 Received in revised form 12 July 2015 Accepted 13 July 2015 Available online xxx Keywords: Supercritical fluids Mixed convection Nusselt number Upward and downward flow Richardson number
a b s t r a c t The present article investigates laminar mixed convection with upward and downward flow over a vertical flat plate in supercritical fluid, numerically. At first four different equations for thermal expansion coefficient estimation in supercritical fluids are derived as a function of pressure, temperature and the compressibility factor. Calculated values of thermal expansion coefficient and Nusselt number have been compared with the experimental results and show better accuracy of Redlich–Kwong EOS in comparison with others. After that governing equations for mixed convection are solved numerically by using finite difference method. A parametric study is performed to illustrate the interactive influence of the governing parameters, specifically the Richardson number and the effect of its variation on the rate of heat transfer and Nusselt number in supercritical condition. Finally, heat transfer curves as a function of Richardson number are plotted and observed that general trend of supercritical fluids curves are similar for water and carbon dioxide. The achieved results can also be validated with shown velocity and temperature counters and vectors. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Convection along a heated vertical flat plate is one of the most common heat transfer phenomena in heat and mass transfer studies, due to its occurrence in a wide variety of heat transfer industrial equipment. If the natural convection existing in this case is accompanied by a forced flow, the combined mode of free and forced convection will exit, which is usually called as mixed convection. Depending on the forced flow direction, the buoyancy forces may aid or oppose the forced flow, causing an increase or decrease in heat transfer rates [1]. So, all mechanisms that enhance natural or forced convection can enhance mixed convection and vice versa. Using fins and ribs to increase the surface area, ultrasonic vibrating of working fluid, applying electrical field, adding surfactants or nanoparticles to the base fluid and finally, using base fluid in supercritical conditions are the most common methods, which have been used to improve one of the free and forced convection or both of them and finally enhance mixed heat transfer mechanism [2–7]. The theory of mixed convection shows that the ratio of the Grashof number to the square of Reynolds number which is known
∗ Corresponding author. Tel.: +989113531929. E-mail address: [email protected] (D. Rezaei Khonakdar).
as the non-dimensional Richardson number, has a great effect on the flow with the constant plate’s temperature boundary condition. The forced convection is dominating in small values of the Richardson number because of large values of Reynolds number; on the other hand, free convection overcomes in large values of the Richardson number because of large values of Grashof number. Reviewing the literatures shows that the mixed convection theory from a flat plate for under critical condition is well established and has been investigated by various researchers. Sparrow et al. [8] were the first to study the effect of the buoyancy forces on the forced convection flow over a vertical flat plate. The results of their study have shown that the flow can be classified by using the Richardson number. A combination of numerical integration and series expansion has been used by Merkin [9] and Wilks [10] to solve the mixed convection problem for both aiding and opposing flows in a Prandtl number of unity along a vertical plate with constant wall temperature and constant heat flux boundary condition, respectively. Lioyd and Sparrow [11] used the local similarity method for different values of the Prandtl number to discuss the mixed convection under small effect of Richardson number. Oosthuizen and Hart [12] solved the constant wall temperature and the heat flux problem over a vertical flat plate, numerically. First experimental results of the mixed convection along a vertical plate for air have been presented by Gryzagoridis [13] in a wide range of Richardson number. The comprehensive review concerning mixed convective
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Nomenclature a,b A,B Cp Cp0 g Gr h k k0 L M Nu P Pr R Ra Ri Re T ω u,v V’ u’,w U,V x,y X,Y Z
EOS constants that corrects for intermolecular attractive forces (N m4 /mol2 ) dimensionless EOS parameters heat capacity of the fluid (J/kg K) heat capacity at the low-pressure limit (J/kg K) local acceleration of gravity (m/s2 ) Grashof number heat transfer coefficient (W/m2 K) thermal conductivity of the fluid (W/m K) thermal conductivity at the low-pressure limit (W/m K) height of the vertical plate (m) molecular weight of the fluid (kg/mol) Nusselt number pressure (Pa) Prandtl number gas constant (J/mol K) Rayleigh number for the whole plate Richardson number Reynolds number temperature (K) density of the fluid (kg/m3 ) acentric factor velocity components in the x,y direction, respectively (m/s) molar volume (m3 /mol) EOS parameter dimensionless velocity components in the x and y directions respectively coordinates parallel and normal to the vertical plate, respectively (m) dimensionless coordinate parallel and normal to the vertical plate, respectively compressibility factor
Greek symbols thermal diffusivity (m2 /s) ˛ ˇ thermal expansion coefficient (1/K) kinematic viscosity (m2 /s) inverse thermal conductivity (m K/W) dimensionless temperature inverse viscosity (m2 /N s) Subscripts c critical condition fluid f i node index in the x direction j node index in the y direction P constant pressure reduced characteristic r w wall x local distance outside the boundary layer ∞
flow for under critical fluids are available in the books by Pop and Ingham [14]. Jena and Mathur [15] solved laminar mixed convection problem of a micropolar fluid from an isothermal vertical flat plate by using a finite-difference technique. They found that the velocity increased and the temperature decreased with the process of fluid’s heating. Many investigations have been done on the effectiveness of supercritical fluids in mixed convection heat transfer; however,
there are no reports about the mixed convection of supercritical fluid over a vertical flat plate. Kakaral and Thomas studied mixed convection heat transfer in a vertical tube of supercritical fluids. They evaluated the effect of flow direction at rate of heat transfer in tube [16]. Pioro and Duffey’s work [17] is considered as a very useful review literature of supercritical heat transfer phenomena until now. The mixed convective heat transfer of carbon dioxide at supercritical state in a circular vertical and horizontal tube was investigated experimentally and numerically, respectively [18,19]. These studies concerned the effects of the Richardson number on the convective heat transfer in tubes. They both used this fact that the thermophysical properties of supercritical fluids are highly sensitive to temperature. To study the mixed convection in under critical fluids with temperature-dependent properties over a vertical flat plate; at first, Kafoussias et al. [20] used a modified numerical solution to study the problem with temperature-dependent viscosity. Pantokratoras [21] assumed the temperature-dependent density to study the laminar mixed convection problem of water along a vertical isothermal plate. Mixed convection laminar boundary layer flow of water-based nanofluids along the vertical plate with temperaturedependent heat source has been investigated numerically by Rana et al. [22]. As mentioned before, in supercritical condition the properties of the fluid specially the thermal expansion coefficient are highly related to temperature and pressure. The thermal expansion coefficient (ˇ) has been assumed to be constant and its variation versus temperature and pressure has been neglected in most of previous works. In supercritical fluids, this consideration will lead to wrong results. So, it is evident that the determination of the value of ˇ through an appropriate equation of state is necessary. The effect of temperature, pressure and the compressibility factor (Z) on thermal expansion coefficient was first considered by Rolando in the problem of free convection over an isothermal vertical plate [23]. In his research a new equation for the thermal expansion coefficient has been derived based on the Van der Waals equation of state. Teymourtash and Ebrahimi Warkiani [24] have investigated the effect of linear variation of wall temperature on the heat transfer by natural convection along a vertical plate in supercritical fluids. Teymourtash et al. [25] have carried out the same problem with uniform and non-uniform surface heat flux. For the first time they proved that the Boussinesq approximation is applicable over a wide range of temperature and pressure variation for supercritical fluid. They showed that the Redlich–Kwong equation of state can estimate the supercritical fluid properties much better than Van der Waals or real gas ones. They also found that linearly increasing and decreasing heat flux will increase and decrease the local Nusselt number, respectively, in comparison with the constant ones. Among all equations of state (EOS) which used to estimate the fluid’s properties only real gas, Van der Waals and Redlich–Kwong ones have been used in previous researches for supercritical fluids [23–25]; however, there are some other useful equation of states that can be used in heat transfer estimation at supercritical condition. So, first exclusive objective of this literature is to find the best equation of state among Virial (1901), Beattie–Bridgman (1928), Redlich–Kwong (1949), Soave (1972) and Peng–Robinson (1976) [26] equations for predicting the supercritical fluid’s thermal expansion coefficient. After that, the problem of mixed convection along a vertical flat plate is solved for the first time by using a numerical model based on a finite-difference formulation and the most appropriate equation of state. Finally, numerical results are presented and effects of Richardson number on heat transfer characteristics in aiding and opposing mixed flow are investigated.
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2. Evaluation different equations for calculating the thermal expansion coefficient (ˇ) in supercritical region Thermal expansion coefficient at constant pressure is defined as 1 ˇ=−
∂ ∂T
(1)
Eq. (10) is the basic equation for calculation of thermal expansion coefficient as a function of pressure and temperature. Partial derivation of this equation with respect to temperature at constant pressure is as follows:
2
ˇ=
1 1 + T Z
2
3Z + (2B − 2) Z + A − 3B − B
∂Z ∂T
P
With consideration of the compressibility factor (Z) defining and referring to our previous work [25] more complete form of Eq. (1) is derived as follows:
3
∂Z ∂T
= ˇi + P
1 Z
∂Z ∂T
2
− Z + A − 2 (Z + 3B) − 2B + 3B
(2)
∂A ∂T
− (Z + B)
P
where ˇi is ideal gas thermal expansion coefficient. Eq. (2) is in general form and it can be useful when the equation of state is as follows: Z = Z(P, T )
P
∂B ∂T
P
=0
(14)
P
in which:
∂A ∂T
=− P
(3)
As mentioned before, Redlich-Kwong and Van der Waals equations of state were used in the previous works [23–25] to estimate the behavior of supercritical fluids and it was proved in the later literature [25] that Redlich–Kwong equation has a closer magnitude to experimental correlation than Van der Waals equation and ideal gas assumption. So, in this paper other common equations of state have been used for modeling thermal expansion coefficient and their results are compared with the experimental correlation and Redlich–Kwong ones. After this investigation the most appropriate equation of state to predict the thermal expansion coefficient for studying the supercritical fluid’s behavior in heat transfer phenomena will be announced.
2
2A aP ∂˛ + 2 2 T R T ∂T
2A aP =− + 2 2 T R T
∂B ∂T
=− P
2 − 2 Tr−0.5 − Tr−0.5 Tc
(15)
B T
(16)
Replacing Eqs. (15) and (16) in Eq. (14) results in
∂Z ∂T
=
P
Z2
+ A − 2 (Z + 3B) − 2B + 3B2
3Z 2 + (2B − 2) Z
∂B/∂T
+ (Z + B) ∂A/∂T
P + A − 3B2 − B
In 1976, Peng and Robinson stated general equation of state by inserting acentric factor (ω) in their equation to estimate more accurate values for gas and fluid characteristics. This equation has valuable accuracy for fluid characteristic estimation near critical region. The general form of this equation is as follows: RT a˛ − 2 V −b V + 2bV − b2
(4)
0.457235R2 Tc2
˛ = 1 + 1 − Tr0.5
(6)
2
∂Z ∂T
= ˇi + P
1 Z
∂Z ∂T
P
Z + A − 2 (Z + 3B) − 2B + 3B2 2
−B/T
P
+ (Z + B) ∂A/∂T
3Z 2 + (2B − 2) Z + A − 3B2 − B
P
(18)
Previous procedure can be done for other relevant equations of state. So, in order to summarize the discussion, other equations of state have been introduced only with their constants and final form of thermal expansion coefficient relation. 2.2. Soave modification equation
(8)
T Tc
(9)
By choosing appropriate constant it can be converted to polynomial form.
(7)
= 0.37464 + 1.54226ω − 0.26992ω2 Tr =
1 1 + T Z
= ˇi −
(5)
pc
0.077796RTc b= pc
By inserting Eq. (17) in Eq. (2) following relation for calculating thermal expansion coefficient based on Peng–Robinson equation of state can be achieved as follows: ˇ=
The constants of this equation of state are as follows: a=
P
(17)
2.1. Peng–Robinson equation of state
P=
Z 3 + (B − 1) Z 2 + A − 3B2 − 2B Z + B2 + B3 − AB = 0
(10)
in which, A, B and Z definitions are as follows:
√ In 1972, Soave replaced the 1/ T term of the Redlich–Kwong equation with a function ˛ (T,ω) involving the temperature and the acentric factor (The resulting equation is also known as the Soave–Redlich–Kwong equation). The ˛ function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials. The general form of this equation and its relevant constants are as follows: P=
RT a˛ − V −b V (V + b)
(19)
(12)
a=
0.4275R2 Tc2 pc
(20)
(13)
b=
0.08664RTc pc
(21)
A=
a˛p R2 T 2
(11)
B=
bp RT
Z=
PV RT
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˛ = 1 + 1 − Tr0.5
2
= 0.48508 + 1.5517ω − 0.1763ω2
(22)
The constant of Eq. (32) (B1 (Tr )) which has been mentioned in Ref. [26], is as follows:
(23)
B1 (Tr ) = B(0) (Tr ) + ωB(1) (Tr )
By choosing appropriate constant it can be converted to polynomial form:
Z 3 + Z 2 + B2 + B − A Z − AB = 0
(24)
in which, A, B and Z definitions are as their definitions in Eqs. (11)–(13). Finally, by inserting in Eq. (2) following relation for calculating thermal expansion coefficient based on soave modification has been achieved. 1 1 ˇ= + T Z = ˇi +
∂Z ∂T
P
1 = ˇi + Z
∂Z ∂T
B(1) (Tr ) = 0.139 − 0.172Tr−5.2
(35)
Finally, by doing the same procedure and inserting in Eq. (2), following relation for calculating thermal expansion coefficient based on the Virial equation of state has been achieved.
1 1 + T Z
P
= ˇi
(25)
3Z 2 + 2Z + B2 + B − A
(34)
ˇ= P
B(0) (Tr ) = 0.083Tr−1 − 0.422Tr−2.6
(Z + 2BZ − A) B/T + (Z + B) ∂A/∂T
in which:
(33)
1+
∂Z ∂T
P
1/Pr
0.8944ωTr−7.2 − 1.0972Tr−4.6 − 0.083Tr−3 − 0.083Tr−1 − 0.422Tr−2.6 + 0.139ω − 0.172ωTr−5.2
(36)
3. Problem statement 2.3. Beattie–Bridgman equation of state
(28)
Consider the steady, laminar, two-dimensional, mixed convection boundary-layer flow over a vertical flat plate which is shown in Fig. 1. The coordinate system is chosen such that x and y measure the distance along and normal to the plate, respectively. Far away from the plate, the velocity and the temperature of the uniform stream are U∞ and T∞ , respectively. The entire surface of the plate is maintained at a uniform temperature of Tw . As it has been mentioned in the later literature [25], the Boussinesq approximation is valid for supercritical fluids; so, the boundary-layer form of the governing equations based on the balance laws of mass, momentum and energy can be written as follows:
(29)
∂u ∂v + =0 ∂x ∂y
In 1928, this equation was proposed by the Beattie–Bridgman. The compressible form of this equation and its relevant constants are as follows:
Z =1+
B C + − 4 Tr Tr2 Tr A
A = 1284.9 B = 1.678
(26)
Pc
(27)
(RTc )2
P c RTc
C = 61.56 × 10−6
Pc
RTc4
2
Finally, with same procedure and implementing in Eq. (2), following relation for calculating thermal expansion coefficient based on Beattie–Bridgman equation of state has been achieved. 1 1 ˇ= + T Z
∂Z ∂T
P
1 = ˇi + Z
−2ATc Tr3
−BTc 4CTc + + Tr Tr5
(30)
2.4. Virial equation of state Although the Virial equation of state is not the most convenient equation of state, it is very important because it can be derived directly from statistical mechanics. This equation is also called the Kamerlingh Onnes equation. If appropriate assumptions are made about the mathematical form of intermolecular forces, theoretical expressions can be developed for each of the coefficients in this equation of state. In this case, B corresponds to interactions between pairs of molecules, C to triplets, and so on. Accuracy can be increased indefinitely by considering higher order terms. The coefficients B, C, D, etc. are functions of temperature only. Z=
(39)
PV B C D =1+ + 2 + 4 + ··· RT Vr Vr Vr
u
∂u ∂u ∂ u +v ± ˇg (T − T∞ ) = ∂x ∂y ∂ y2
u
∂T ∂T ∂ T +v =˛ ∂x ∂y ∂ y2
(40)
2
(41)
where u and v are the velocity components parallel and perpendicular to the plate, respectively. Also, T is the temperature, ˇ is the thermal expansion coefficient, v is the kinematic viscosity, is the fluid density, g is the acceleration due to gravity and ˛ is the thermal diffusivity.
(31)
For achieving the purpose of this research, the approximation form of Virial equation has been used [26]. This form of the equation is as bellows: Z=
PV = 1 + B1 (Tr ) Pr RT
(32)
Fig. 1. Geometry and coordinates of the system for (a) aiding, (b) opposing mixed flow.
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The appropriate boundary conditions for the velocity and temperature of this problem are as follows [23]: x = 0, x > 0,
y > 0;
u = U∞ ,
y = 0;
u = 0,
y → ∞;
T = T∞
v = 0,
u → U∞ ,
T = TW ,
(42)
T → T∞ .
By definition of the following dimensionless variables: u , U∞
U=
vReL1/2
V=
U∞
,
X=
1/2
x , L
Y=
yReL
,
L
=
T − T∞ . TW − T∞ (43)
Eqs. (39)–(41) are converted to the following dimensionless forms:
∂U ∂V + = 0, ∂X ∂Y
(44) 2
U
∂U ∂U ∂ U ± Ri, +V = ∂X ∂Y ∂Y 2
U
∂ ∂ 1 ∂ +V = Pr ∂Y 2 ∂X ∂Y
(45)
2
(46)
where Pr =
, ˛
gˇ (T − T∞ ) L3 , 2
Gr =
Re =
U∞ L ,
Ri =
Gr Re2
(47)
Pr is the Prandtl number; Gr is the Grashof number, Re is the Reynolds number and Ri is the Richardson number. In the dimensionless form, the boundary conditions can be written as follows: X = 0,
Y > 0;
U = 1,
X > 0,
Y = 0;
U = V = 0,
y → ∞;
U → 1,
= 0, = 1,
(48)
→ 0.
−
Vi−1,j 2 Y
+
−
Vi−1,j 2 Y
Vi−1,j 2 Y
+
−
−
Vi−1,j 2 Y
1
Y 2 −
1
Y 2
1 Pr Y 2 −
Ui,j−1 +
Ui−1,j
X
Ui,j+1 =
1 Pr Y 2
+
2 Ui−1,j
i,j−1 +
numerical procedures can be found in the textbook by Oosthuizen and Naylor [27]. A mesh system with 150 × 150 nodes is proven to provide mesh-independent results. Heat-transfer coefficient, once the temperature field is obtained as described above, is based on the following equation: Nux 1/2
Eqs. (44)–(46) are coupled nonlinear differential equations which must be solved under the boundary conditions given in Eq. (48). These equations are solved by using the finite-difference method, because the exact analytical solutions are not possible for this set of equations. The equivalent finite difference schemes corresponding Eqs. (44)–(46) are given by
Fig. 2. Comparison between thermal expansion coefficient of carbon dioxide which has been calculated from different EOS, Redlich–Kwong [25] and reference values [28] at P = 7 MPa.
X Ui−1,j
i,j+1 =
X
2
Y 2
−X W
∂ ∂Y
(52) Y =0
Supercritical fluid properties which are used in the numerical procedure can be achieved as it was mentioned in the authors’ previous work [25].
Ui,j
± Rii−1,j ,
+
ReL
=
2 Pr Y 2
(49)
i,j
Ui−1,j i−1,j
(50)
X
After some derivations V values can be determined from the continuity equation as Vi.j = Vi,j−1 −
Y 2 X
Ui,j − Ui−1,j + Ui,j−1 − Ui−1,j−1 .
(51)
Here, the index i refers to x and j to y. The above equations are explicit in x-direction, while they are implicit in y-direction. After specifying the conditions along some initial i = 1, U values and, in the following, values can be obtained in the i = 2 line. Then, V values are obtained in the i = 2 line. Having determined the values of all the variables on the i = 2 line, the same procedure can then be used to find the values in the i = 3 line and so on. More details on the
Fig. 3. Comparison between thermal expansion coefficient of carbon dioxide which has been calculated from different EOS, Redlich–Kwong [25] and reference values [28] at P = 10 MPa.
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Fig. 4. Comparison between thermal expansion coefficient of carbon dioxide which has been calculated from different EOS, Redlich–Kwong [25] and reference values [28] at P = 15 MPa.
4. Results and discussion The results of thermal expansion coefficient (ˇ) which are calculated by proposed model based on different equations of state have been illustrated here. Figs. 2–7 represent six curves of ˇ versus T for carbon dioxide and water at three different pressures for reference magnitude obtained from experimental data for CO2 [28] and H2 O [29], Redlich–Kwong [25], Peng–Robinson, Soave modification, Beattie–Bridgman and Virial equations of state. Despite the fact that the Redlich–Kwong equation is simple and it has been introduced earlier than some of other equations of state; extracted model of it has an acceptable accuracy near critical region and works more precise in comparison with others. In addition, these figures show that when the temperature increases, all curves
Fig. 5. Comparison between thermal expansion coefficient of water which has been calculated from different EOS, Redlich–Kwong [25] and reference values [29] at P = 20 MPa.
Fig. 6. Comparison between thermal expansion coefficient of water which has been calculated from different EOS, Redlich–Kwong [25] and reference values [29] at P = 30 MPa.
approach to the reference value; so, all equations of state can be used in very high temperature. It is clear that new equations of state such as Peng–Robinson and Soave, estimate thermodynamic properties more precise than Redlich–Kwong, but as it can obviously be seen in Figs. 2–7 extracted models of these equations of state cannot work properly in thermal expansion coefficient estimation. So, when consideration of all fluid’s properties with thermal expansion coefficient is needed, Redlich–Kwong equation of state is the best choice. For better comparing the accuracy of these proposed equations and their effects on heat transfer phenomenon a FORTRAN code for predicting heat transfer near critical region has been developed by using these models. Figs. 8 and 9 show the Nusselt number versus Rayleigh number for carbon dioxide and water, respectively.
Fig. 7. Comparison between thermal expansion coefficient of water which has been calculated from different EOS, Redlich–Kwong [25] and reference values [29] at P = 50 MPa.
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Following relation which was proposed by Chirchill and Chu [30], has been used to obtain the curve corresponding to experimental correlation. This curve is valid near critical region and over a wide range of Rayleigh numbers: 1/4
¯ L = 0.68 + Nu
0.67RaL
1 + (0.492/Pr)
9/16
4/9
(53)
where, RaL is the Rayleigh number with the following definition: RaL = Rax=L = GrL .Pr =
Fig. 8. Local Nusselt as a function of local Rayleigh at Pr = 1.05 based on different equation of state and experimental correlation [30] at Tw = 330 K for carbon dioxide.
Fig. 9. Local Nusselt as a function of local Rayleigh at Pr = 1.05 based on different equation of state and experimental correlation [30] at Tw = 700 K for water.
gˇQL4 ˛k
(54)
This work studies laminar regions; so, the Rayleigh number cannot exceed 109 as it was mentioned by Incropera and Dewitt [31]. One of the most useful results which can be obtained from these figures is finding the relation between thermal expansion coefficient and Nusselt number. As it can be obviously seen from Figs. 2–7, all thermal expansion coefficients, which have been obtained from different models, reported different values. So, it seems that this variation in thermal expansion coefficient magnitude may have a sensible effect on the heat transfer characteristic and Nusselt number. Figs. 8 and 9 confirm this discussion. As it can be seen in these figures thermal expansion coefficient magnitude has a direct effect on the rate of heat transfer and Nusselt number. In fact, each equations of state which reports higher values for thermal expansion coefficient leads to higher estimation of values in Nusselt number and vice versa. In addition, all thermal expansion curves converge to each other when moving upward adjacent to the plate surface. This phenomenon is due to decrease at difference between derived thermal expansions coefficient from different models. So, when moving upward, fluid temperature enhances and all thermal expansion curves converge to each other. As it can be obviously seen in Figs. 8 and 9, Nusselt curves which have been calculated by using Redlich–Kwong equation, is close to experimental curves. However, other equations of state cannot work properly in thermal expansion confident estimation in supercritical condition and lead to major errors in calculation the rate of heat transfer. So, all of the equations of state except Redlich–Kwong do not have acceptable accuracy and cannot give trustable estimations of heat transfer characteristics and Nusselt number in supercritical condition.
Fig. 10. (a) Dimensionless temperature profile, (b) Nusselt curves, of CO2 for aiding mixed flow at Pr = 1.05, Tr = 1.05.
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Fig. 11. Dimensionless velocity profile of CO2 for aiding mixed flow at Pr = 1.05 & Tr = 1.05.
Here, two different types of mixed convection are studied. The first one is aiding flow and it happens when the fluid movement direction in and out of boundary layer are the same. The other one is opposing mixed flow in which streams’ direction are opposite to each other. Due to approximate similar behavior of carbon dioxide and water in supercritical condition and prevention of figures repetition only carbon dioxide curves are illustrated in following parts of this paper. Fig. 10 shows dimensionless temperature profile and Nusselt curves for carbon dioxide at different Richardson numbers for aiding flow. As it can be obviously seen from Fig. 10(a), if the Richardson number decreases due to increase in outside velocity, the thickness of thermal boundary layer decreases and it is expected that the heat transfer coefficient and Nusselt number increases consequently. Fig. 10(b) approved the previous discussion. The curve corresponding to Ri = 10−4 has higher values than the others and this sensible distance between this curve and Ri = 10−2 shows the effect and power of forced convection in comparison with natural one. So, in mixed convection with the aiding flow for achieving greater heat transfer coefficient, increasing the outer boundary layer velocity (u∞ ) is very effective.
The vicinity of Nusselt curves in higher Richardson number shows that the portion of forced convection in total rate of heat transfer is negligible in comparison with natural one. This phenomenon is reasonable because in higher Richardson number u∞ is very small and the power of natural convection dominates the forced one. Fig. 11 shows the ratio of fluid velocity to outer boundary layer uniform velocity (u∞ ) in carbon dioxide at aiding mixed flow. Fluid velocity (u) in mixed convection depends on magnitude and direction of two different parts. First one is natural convection velocity which has been made by buoyancy effect of heated fluid near the wall in thermal boundary layer. Another one is outer boundary layer velocity (u∞ ) which is uniform and constant. When Ri number increases the ratio of fluid velocity to uniform outer boundary layer velocity decreases and vice versa. In addition, when Ri decreases (velocity increases) the maximum magnitude of velocity ratio curves decreases and it goes toward the wall. Then in Ri about 10−4 the velocity profile does not have any maximum point due to the viscous effect in the boundary layer. In fact, fluid in boundary layer decreases the outer velocity and tries to stop it. All velocity ratio curves approach to the unity with going away from the wall surface because in outside of the thermal boundary layer natural convection and viscous effect does not exist. Fig. 12 shows the dimensionless temperature profile and Nusselt curves for carbon dioxide in different Richardson numbers for opposing mixed flow. In this case, buoyancy forces have retarding effects on the flow’s outer boundary layer velocity (u∞ ); so, it is called as opposing mixed flow. As it can be obviously seen an increment in Ri results in decrement in outer velocity due to the opposing effect of the upward buoyancy-induced flow on the downward external forced flow. So, it can be predicted that the rate of heat transfer and Nusselt number should be decreased and the thickness of the thermal boundary layer should be increased by increasing the outer boundary layer uniform velocity (u∞ ). Fig. 12(b) approves the expected result. As it can be seen the curve corresponding to Ri = 10−4 has a lower Nusselt number than the others. So it can be concluded that, in opposing mixed flow at lower Richardson number (higher velocity), total rate of heat transfer increases with increasing the Richardson number. This phenomenon indicates a significant impact of forced convection on buoyancy forces.
Fig. 12. (a) Dimensionless temperature profile, (b) Nusselt curves, of CO2 for opposing mixed flow at Pr = 1.05, Tr = 1.05.
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Fig. 13. Dimensionless velocity profile of CO2 in opposing mixed flow at Pr = 1.05 & Tr = 1.05.
9
Fig. 13 shows the ratio of fluid velocity to the outer boundary layer velocity (u∞ ) for carbon dioxide at opposing mixed flow. As it can be expected when Ri decreases, fluid velocity in boundary layer decreases due to retarding effect of forced flow on natural convection’s induced upward flow. In opposing mixed flow when moving perpendicularly against the wall, natural convection velocity increases and then decreases very adjacent to the wall. So, the fluid velocity decreases because of the retarding effect of natural convection velocity near to the plate surface. After that, the natural convection velocity decreases and fluid velocity decreases too. Fig. 13 confirms the above explanation. As it can be obviously seen, the ratio of fluid velocity to the uniform outer boundary layer one increases when moving vertical to the plate surface. Fig. 14(a) and (b) shows aiding and opposing flow in supercritical condition at different Ri number for carbon dioxide. As it can be observed in Fig. 14(a) and (b) at higher Ri number, Nusselt curves are closer to each other in comparison with others. In fact, increment of outer boundary layer velocity (u∞ ) leads to increment in Nusselt magnitude differences for aiding and opposing mixed flow.
Fig. 14. (a) Comparison of Nusselt number for aiding and opposing flow of CO2 in mixed convection, (b) close up view, at Pr = 1.05, Tr = 1.05.
Fig. 15. The variation of Nu/Re0.5 for CO2 and H2 O at mixed convection (a) aiding, (b) opposing flow, at Pr = 1.05, Tr = 1.05.
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Fig. 16. (a) Temperature and u Velocity contours for (a) aiding, (b) opposing flow of CO2 in mixed convection, at Pr = 1.05, Tr = 1.05 and Ri = 1.
Fig. 17. (a) v Velocity vectors for aiding flow and (b) velocity contour for opposing flow of CO2 in mixed convection, at Pr = 1.05, Tr = 1.05 and Ri = 1.
Another curve which has been illustrated in earlier references for mixed convection is Nu/Re0.5 variation when Ri increases. Here, these curves have been plotted for the first time for mixed convection in supercritical fluid conditions. Fig. 15(a) and (b) shows Nu/Re0.5 as a function of Ri number for aiding and opposing flow, respectively. As described before in aiding mixed condition for lower Ri number magnitude, increasing in Ri leads to sensible variation on rate of heat transfer and decreases Nusselt number. This phenomenon can obviously be seen in Fig. 10(b). In addition, decreasing the velocity leads to decrease in Re number, so the magnitude of Nu/Re0.5 should be approximately constant. On the other hand, in higher Ri number the Nusselt number variation is very small when Ri increasing and sometimes this variation is negligible. So, Nu/Re0.5 magnitude will be increases due to decreasing the Re number. In fact, decrement in Re number is bigger than Nusselt number when Ri increases. Fig. 15(a) confirmed above discussion and shows general behavior of carbon dioxide at Pr = 1.05 and Tr = 1.05 in the vast range of fluid’s velocity and Ri number. However, in opposing mixed flow which can be observed in Fig. 15(b) the trend of Nu/Re0.5 curves are different from aiding one and variation of Nusselt and Reynolds magnitude with Ri, leads to decrement in Nu/Re0.5 . Four different contours that illustrate temperature and velocity of fluid along the plate for aiding and opposing mixed flow are plotted in Fig. 16(a) and (b), respectively. In definite vertical distance in thermal boundary layer, moving upward along the plate
leads to an increment in the amount of fluids temperature because of decrement in density. This Phenomenon has been intensified in aiding mixed flow in comparison with opposing one. Fig. 17(a) and (b) shows v velocity vectors for aiding mixed flow and velocity contours for opposing flow of CO2 , respectively. 5. Conclusion Four different thermodynamic models have been studied and derived from Peng–Robinson, Soave modification, Beattie–Bridgman and Virial equation of state, for calculating the thermal expansion coefficient of supercritical fluid. Then, these equations have been applied to predict the rate of heat transfer from vertical flat plate. It was observed that Redlich–Kwong equation in spite of its simplicity has acceptable accuracy near critical point and reports more accurate values for thermal expansion coefficient and Nusselt number estimation rather than the others. Hence, by using extracted model from Redlich–Kwong a numerical model was developed for the analysis of the mixed convection over a vertical flat plate for supercritical fluid with constant surface temperature. According to the nature of problem, logical trend is observed at velocity and temperature profiles. In fact, obtained values of Nusselt number for aiding and opposing mixed flows show that the thickness of thermal boundary layer in the aiding mixed flow is lower than the opposing one. The same direction of induced buoyancy stream and uniform outer velocity intensifies the rate of heat transfer and Nusselt number in comparison with opposing one.
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In the next step, the effect of Ri number variations as the most famous dimensionless number in mixed convection has been studied. It was seen that for aiding mixed convections when Richardson number decreases due to an increase in uniform outer velocity, the thickness of thermal boundary layer decreases and the heat transfer coefficient and Nusselt number increases. Finally, the effect of Ri number variations on Nu/Re0.5 at mixed convection has been studied. It is concluded that for aiding mixed flow at lower Richardson number by increasing the Richardson the magnitude of Nu/Re0.5 is approximately constant. When Richardson number increases to the higher values, Nu/Re0.5 magnitude enhances. On the other hand, in opposing mixed flow Nu/Re0.5 curves are different from aiding one and the variation of Nusselt and Reynolds magnitude with Richardson number leads to decrement in Nu/Re0.5 . So, in some special industrial process in which the great heat transfer coefficient and local Nusselt number is in our major requirements, aiding mixed convection in supercritical fluids is recommended. However, the flow direction and remaining at supercritical condition should be checked; because little deviations from mentioned conditions have considerable effects on the rate of heat transfer. References [1] O. Aydın, Aiding and opposing mechanisms of mixed convection in a shear and buoyancy driven cavity, Int. Commun. Heat Mass Transfer 26 (7) (1999) 1019–1028. [2] S.U. Onbasioglu, H. Onbastoglu, On enhancement of heat transfer with ribs, Appl. Therm. Eng. 24 (2004) 43–57. [3] S. Sadri, M.R. Raveshi, S. Amiri, Efficiency analysis of straight fin with variable heat transfer coefficient and thermal conductivity, J. Mech. Sci. Technol. 26 (4) (2012) 1283–1290. [4] S.W. Wong, W.Y. Chon, Effects of ultrasonic vibrations on heat transfer to liquids by natural convection and by boiling, Am. Inst. Chem. Eng. J. 15 (2) (1969) 281–288. [5] T.B. Jones, Electrohyrodynamically enhanced heat transfer in liquids—a review, Advanced in heat transfer, in: T.F. Irvine, J.P. Hartnett (Eds.), Advances in Heat Transfer, Academic Press, New York, NY, 1978, pp. 107–148. [6] B.R. Lazarenko, F.P. Grosu, M.K. Bologa, Convective heat transfer enhancement by electric fields, Int. J. Heat Mass Transfer 18 (1975) 1433–1441. [7] K. Sadik, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (13–14) (2009) 3187–3196. [8] E.M. Sparrow, R. Eichhorn, J.L. Gregg, Combined forced and free convection in boundary layer flows, Phys. Fluids 2 (1959) 319–328. [9] J.H. Merkin, The effect of buoyancy forces on the boundary layer over a semi-infinite vertical flat plate in a uniform free stream, J. Fluid Mech. 35 (1969) 439–450.
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Please cite this article in press as: D. Rezaei Khonakdar, M.R. Raveshi, Mixed convection on a vertical plate in supercritical fluids by selecting the best equation of state, J. Supercrit. Fluids (2015), http://dx.doi.org/10.1016/j.supflu.2015.07.013