Heat Transfer Characteristics of Nitrogen in Supercritical Region Using Redlich-Kwong Equation of State

Hussain Basha; G. Janardhana Reddy; N. S. Venkata Narayanan

doi:10.1515/ijcre-2018-0279

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Heat Transfer Characteristics of Nitrogen in Supercritical Region Using Redlich-Kwong Equation of State

Hussain Basha , G. Janardhana Reddy und N. S. Venkata Narayanan

Veröffentlicht/Copyright: 18. Juli 2019

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Aus der Zeitschrift International Journal of Chemical Reactor Engineering Band 17 Heft 10

Abstract

The present paper studies through numerical methods, the thermodynamic heat transfer characteristics of free convection flow of supercritical nitrogen over a vertical cylinder. In the present analysis, the values of volumetric thermal expansion coefficient (β) are evaluated based on Redlich-Kwong equation of state (RK-EOS) and Van der Waals equation of state (VW-EOS). The calculated analytical thermal expansion coefficient values using RK-EOS are very close to NIST data values in comparison with VW-EOS. A set of coupled nonlinear partial differential equations (PDEs) governing the supercritical fluid (SCF) flow are derived, converted into non-dimensional form with the help of suitable dimensionless quantities and solved using Crank-Nicolson implicit finite difference method. The simulations are carried out for nitrogen in the supercritical region. The obtained numerical data is expressed in terms of graphs and tables for various values of physical parameters. The increasing value of reduced temperature decreases the average Nusselt number and skin-friction coefficient, whereas amplifying value of reduced pressure enhance the heat transfer rate and wall shear stress in the SCF region. Present results are compared with the previous results and the two are found to be in good agreement, i. e. the numerically generated results found to be in agreement with existing results.

Keywords: supercritical nitrogen; RK-EOS; VW-EOS; compressibility factor; reduced temperature; Crank-Nicolson

Acknowledgements

The first author Hussain Basha wishes to thank Maulana Azad National Fellowship program, University Grants Commission, Government of India, Ministry of Minority Affairs, MANF (F1-17.1/2017-18/MANF-2017-18-KAR-81943) for the grant of research fellowship and to Central University of Karnataka for providing the research facilities. Further, the authors are very much thankful to all four reviewers for their valuable suggestions and comments to improve the quality of the manuscript.

A Appendix

Table 4:

Thermal expansion coefficient (β) values based on RK-EOS (Soave 1993), VW-EOS (Blundell and Blundell 2006), NIST database (NIST (National Institute of Standards and Technology)) ideal gas assumption (Poling, Prausnitz, and O’Connell 2001) for different values of reduced pressure (Pr∗) and reduced temperature (Tr∗) in SCF region.

P	5	6	7	8	9
Pr∗	1.47	1.76	2.06	2.35	2.65
T	200	230	260	290	340
Tr∗	1.58	1.82	2.06	2.29	2.69

A RK-EOS	0.199044	0.168416	0.144616	0.125790	0.095082
B	0.080472	0.083971	0.086662	0.088797	0.085206
Z	0.894724	0.933311	0.961082	0.981940	1.005412
β	0.00735783	0.00577216	0.00476230	0.00406062	0.00325637

A VW-EOS	0.247295	0.224388	0.204859	0.188191	0.154024
B	0.116128	0.121177	0.125061	0.128142	0.122959
Z	0.869767	0.906792	0.935321	0.957972	0.986259
β	0.00756008	0.00599307	0.00496363	0.00423580	0.00338444

β NIST Data	0.00697980	0.00556051	0.00462725	0.00396466	0.00319153

β Ideal Gas	0.005000000	0.00434782	0.00384615	0.003448270	0.00294117

Table 5:

Time to reach temporal maxima of flow variables, steady-state and maximum velocity at X=1.0 with various values of Pr∗ and Tr∗ for supercritical nitrogen.

		Temporal maximum (t) of		Steady-state	Maximum
Pr∗	Tr∗	U	θ	time (t)	velocity at X = 1
1.47	1.90	1.75	1.66	8.21	1.3009
1.76		1.49	1.42	7.58	1.5650
2.06		1.28	1.21	7.08	1.8219
2.35		1.16	1.08	6.67	2.0668
2.65		1.06	1.01	6.34	2.2984
1.76	1.58	0.92	0.88	5.94	2.6385
	1.82	1.33	1.24	7.19	1.7572
	2.06	1.79	1.72	8.32	1.2628
	2.29	2.34	2.23	9.33	0.9535
	2.69	3.46	3.26	10.87	0.6437

Table 6:

Average skin-friction coefficient (C¯f) and Nusselt number (Nu¯) for different values of Pr∗ and Tr∗ for nitrogen in supercritical region.

Pr∗	Tr∗	C¯f	Nu¯
1.47	1.90	4.1612	1.5603
1.76		5.4426	1.6821
2.06		6.7898	1.7926
2.35		8.1716	1.8933
2.65		9.5484	1.9845
1.76	1.58	11.8493	2.1497
	1.82	6.4584	1.7720
	2.06	3.9820	1.5393
	2.29	2.6568	1.3785
	2.69	1.5172	1.1921

B Assumption of incompressible fluid and the applicability of the Boussinesq’s approximation to supercritical fluids

For the flows satisfying certain conditions, Boussinesq’s in 1903 suggested that density changes in the fluid can be neglected except where ρ is multiplied by g. A formal justification, and the conditions under which the Boussinesq’s approximation holds for supercritical fluids, is given here (Kundu, Cohen, and Dowling 2012; Teymourtash, Khonakdar, and Raveshi 2013). The more general form of the continuity equation is as follows:

(13)1ρDρDt′+∇.U=0

With the help of Boussinesq’s approximation, the incompressible continuity equation can be rewritten in the following form:

(14)∇.U=0

This does not mean that a constant value of density. Here neglecting the term ρ−1(DρDt′) is just due to its small scale compared to velocity gradients in the term ∇.U. Based on the scaling analysis it is found that ρ−1(DρDt′) is small when compared to the velocity gradient terms in ∇.U.

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

(15)δρρ=−βδT′ where β=−1ρ(δρδT′)

In this equation, β is thermal expansion coefficient at constant pressure and varies proportional to (1T′) for ideal gas. To show the reason of neglecting the term ρ−1(DρDt′) in continuity equation compared to velocity gradient terms, it is necessary to use scale analysis. To see this, we assume that the flow field is characterized by a length scale L, a velocity scale U, and a temperature scale δT′. By this we mean that the velocity varies by U and the temperature varies by δT′ between locations separated by a distance of order L. The ratio of the magnitudes of the two terms in the continuity equation is

(16)1ρ(DρDt′)∇.U≈(1ρ)u(∂ρ∂t′)∂u∂x≈(Uρ)(δρL)UL=δρρ=−βδT′≪1

The above eq. (16) is valid since, for smaller temperature difference, the thermal expansion coefficient is smaller. Therefore the first term can be omitted in comparison with second term in the eq. (13). On the other hand, Figure 17(a) and (b) describe that, for the temperature difference of 3.78 K near the critical point gives the maximum value of 0.003 for βδT′ which satisfies the eq. (16). Also, throughout the graph far away from the critical point the above condition is valid for different temperature differences δT′. As a result, the Boussinesq’s approximation is completely valid in supercritical region. Finally, eq. (13) can be reduced to eq. (14).

$Figure 17: (a) Thermal expansion coefficient $\left( \beta \right)$(β) and its slope for nitrogen at $P_r^*\,=\,1.77$Pr∗=1.77 using RK-EOS, (b) enlarged view.$

Figure 17:

(a) Thermal expansion coefficient (β) and its slope for nitrogen at Pr∗=1.77 using RK-EOS, (b) enlarged view.

Now let’s consider the case, if the fluid density (ρ) be assumed as a function of pressure and temperature, then it is possible to expand ρ(P,T′) and formulate it as environment density ρ(P∞,T′∞) in a second-order Taylor series expansion:

(17)ρ=ρ∞+(T′−T′∞) (∂ρ∂T′)P+12(T′−T′∞)2(∂2ρ∂T′2)P+………………….+(P−P∞) (∂ρ∂P)T′+12(P−P∞)2(∂2ρ∂P2)T′+……………………+(T′−T′∞)(P−P∞)(∂2ρ∂T′∂P)+…………………

Thermal expansion coefficient (β) equation has been expressed as follows:

(18)β=−1ρ(∂ρ∂T′)P

Here (∂2ρ∂T′2)P can be obtained as:

Since from the eq. (18) we have

(19)(∂ρ∂T′)P=−βρ

Differentiating eq. (19) w.r.t T′, we get

(20)(∂2ρ∂T′2)P=−ρ(∂β∂T′)P−β(∂ρ∂T′)P

Using eq. (19) in eq. (20), we get

(21)(∂2ρ∂T′2)P=−ρ(∂β∂T′)P+ρβ2

Since it is assumed that, the vertical length is small and the corresponding pressure loss is small too, so:

(22)(P−P∞)≈0

Replacing eq. (22) in eq. (17) results in:

(23)ρ−ρ∞=ρβ(T′−T′∞)+12 ρβ2(T′−T′∞)2− 12 ρ(T′−T′∞)2(∂β∂T′)P+…….

It can be seen that if temperature differences are small, the term β(T′−T′∞) is less than 1 and just the first term in right hand side will remain in the equation. Figure 18(a) and (b), show the range of temperature gradient in which the second and third terms of eq. (23) can be neglected. These figures are plotted at Pr∗=2.09 and Pr∗=2.19. Finally, eq. (23) has been reduced to eq. (24), and hence Boussinesq’s approximation is reliable in supercritical region.

(24)ρ−ρ∞=−ρβ(T′−T′∞)

$Figure 18: The magnitude of different terms of eq. (23) for supercritical nitrogen at (a) $P_r^*\,=\,2.09$Pr∗=2.09, $T_r^*\,=\,1.40$Tr∗=1.40, (b) $P_r^*\,=\,2.19$Pr∗=2.19, $T_r^*\,=\,1.40$Tr∗=1.40 using RK-EOS.$

Figure 18:

The magnitude of different terms of eq. (23) for supercritical nitrogen at (a) Pr∗=2.09, Tr∗=1.40, (b) Pr∗=2.19, Tr∗=1.40 using RK-EOS.

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Received: 2018-10-20

Revised: 2019-03-12

Accepted: 2019-03-26

Published Online: 2019-07-18

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Schlagwörter für diesen Artikel

supercritical nitrogen; RK-EOS; VW-EOS; compressibility factor; reduced temperature; Crank-Nicolson