Document Type : Full Lenght Research Article
Author
Department of Mechanical Engineering, Qom University of Technology, Qom, Iran
Abstract
Keywords
Main Subjects
Semnan University 
Journal of Heat and Mass Transfer Research Journal homepage: https://jhmtr.semnan.ac.ir 

Thermal Behavior of Laminar Flow of Supercritical CO_{2} in
a Long Vertical MiniPipe under Constant and
Stepped Wall Heat Flux
Mahdi Mohseni [*]
Department of Mechanical Engineering, Qom University of Technology, Qom 37181 46645, Iran
Paper INFO 

ABSTRACT 
Paper history: Received: 20220731 Revised: 20230414 Accepted: 20230421 
In this study, the convective heat transfer of supercritical carbon dioxide in a long vertical minipipe has been investigated numerically. The numerical solution has been performed with the finite volume method and by developing a CFD code. The pipe has a length of 5.5 m and a diameter of 1 mm which is exposed to a constant heat flux at the wall with values of 300, 400, 500, and 600 W/m^{2} or step changes. In addition to the wall heat flux, the effects of gravity and flow direction have also been examined. Furthermore, some differences between the results of laminar and turbulent flows have been addressed. The results show that in the laminar flow, unlike the turbulent flow in the improvement regime of heat transfer, the system's thermal performance increases with increasing the wall heat flux, while in the deterioration mode, the two have similar behavior. Moreover, in part of the downward flow, reverse flow occurs, and its length can be understood by using the negative amount of wall shear stress. Furthermore, the thermal efficiency of the supercritical carbon dioxide is better at the upward flow and near the critical point than the constant property flow. In addition, from the applied stepped wall heat flux, it is concluded that the deterioration can be partially controlled or reduced by correctly determining the location of the step or any wall heat flux variations. 



Keywords: Supercritical fluid; Long mini pipe; Flow direction; Buoyancy effect; Wall temperature; Stepped wall heat flux. 


© 2022 Published by Semnan University Press. All rights reserved. 
Increasing thermal efficiency and changing some other properties of supercritical fluids, such as solubility, have led to using these fluids in many engineering systems in recent decades [1,2]. For example, the use of supercritical fluid in heat exchangers can be mentioned [3,4]. These heat exchangers are used in various industries, including extraction units or air conditioning systems of automobiles or power plants [5,6]. The design and performance of such heat exchangers have been the topic of ongoing research for many years [7,8].
Due to the complexity of supercritical fluid flow behavior and its increasing use, many studies have been conducted to investigate the thermal behavior and flow structure in this fluid. However, most of the research is related to turbulent flow, and the laminar flow has received less attention [9,10,11]. This may be because most industrial flows are turbulent, especially for normal or large pipe diameters. However, when the pipe diameter is too small, e.g., mini channels and microchannels [12,13,14], or the flow is in an annulus with a very small clearance, such as bearing [15], or the pipe contains porous material [16], the Reynolds number will naturally be small, and the flow will be in the laminar regime. Previous studies on laminar flow have investigated the effect of one or more important factors on supercritical fluid [17,18]. These factors mainly include wall heat flux, mass flux or Reynolds number, pipe diameter, and fluid type [18,19]. In some cases, the effect of channel crosssections has been examined [20]. It should be noted that almost all research on laminar flow has been performed numerically. This may be because testing with supercritical conditions in laminar flows is impossible due to pressure and temperature fluctuations. Some of the research has been reviewed in more detail in the following.
In 2010, convection heat transfer of water at supercritical pressure in a narrow annulus was numerically investigated by Hassan Zaim and Gandjalikhan Nassab [21]. Their study showed that the effect of strong buoyancy force increases velocity near the wall. As a result, the convective heat transfer enhances. In 2011, Cao et al. [20] numerically investigated the laminar convection heat transfer of supercritical carbon dioxide in horizontal circular and triangular tubes with hydraulic diameters less than 1.0 mm. It was shown that due to the strong buoyancy force, secondary flows were created in the pipe crosssections, which led to the distortion of the velocity and temperature profiles. Based on the analytical and numerical study of Peeters and T'Joen [22] in 2013, the thermally developing length in the upward flow of supercritical CO_{2} and supercritical H2O can be written as a function of a nondimensional ratio of wall heat flux and Peclet number for the specific conditions that were used. Gao et al. [23] numerically investigated the effect of flow reversal at the entry length of a vertical annulus in laminar mixed convection flow of supercritical carbon dioxide in 2018. According to their results, the dominant effect of free convection is essential for flow reversal. Viswanathan and Krishnamoorthy [18], in 2021, conducted a series of numerical tests using AnsysFluent software to investigate the effect of heat flux, tube diameter, and inlet Reynolds number in a small diameter horizontal tube. According to their results, depending on the heat flux's magnitude and the fluid temperature's proximity to the critical point, the Nusselt number finds a value lower or higher than 4.36, corresponding to flows of constant properties. Finally, they introduced a relation for the Nusselt number.
Considering the widespread use of mini and microchannels as well as porous media in recent years, the laminar flow of supercritical fluid can be the subject of further research. Therefore, in this study, the effect of flow direction, gravity, and wall heat flux on wall temperature, heat transfer, and flow structure in the flow of supercritical carbon dioxide has been investigated by developing an axisymmetric CFD code. The studied geometry is a vertical minitube exposed to constant and stepped heat fluxes in the wall. To create more realistic conditions and cover the entire region near the pseudocritical point, the length of the pipe is considered long. In addition, some differences between the laminar and turbulent flow of supercritical fluids have been noted.
In supercritical fluid flow, the changes in the fluid's thermophysical properties are very intense near its critical points. Such changes in the fluid properties significantly affect flow structure and heat transfer [24,25]. As a sample, the changes in the properties of supercritical carbon dioxide at a pressure of 8 MPa are shown in Fig. 1. As the critical point approaches, the change in fluid properties becomes more severe.
Figure 1. Variations of thermophysical Properties of supercritical CO_{2} at 8 MPa [26]
The problem under study is the flow of supercritical CO_{2} inside a long vertical minipipe with a length of 5.5 m and a diameter of 1 mm. The flow can enter from the top or bottom of the pipe and exit from the other side. The schematic of the pipe is shown in Fig. 2. The first 20 cm and the last 30 cm of the pipe are insulated, and the rest of the pipe length is exposed to a constant heat flux of 300, 400, 500, and 600 W/m^{2} or a step changes. Supercritical carbon dioxide enters the pipe at a pressure of 8 MPa, a mass flux of 40 kg/s.m^{2}, and a temperature of 27 ^{o}C (300 K). According to the mentioned conditions, the fluid flow is in the laminar regime.
Figure 2. Schematic of the geometry under consideration
According to the studied flow described in the previous section, the governing equations include the mass conservation equation, the momentum equation, and the energy equation which for the variable property flow of supercritical fluid in the steady state are as follows [27]. It should be noted that due to axisymmetric flow, the equations are written in a twodimensional cylindrical coordinate system.
The mass conservation equation

(1) 
The rmomentum equation

(2) 
The zmomentum equation

(3) 
The energy equation

(4) 
where the dissipation function is defined as

(5) 
The definitions of the variables can be found in the nomenclature. To calculate the convection heat transfer coefficient, the following Newton cooling law is used.

(6) 
Also, the local Nusselt number is defined as follows.

(7) 
The bulk enthalpy in each pipe crosssection is calculated from Eq. 8.

(8) 
The property database can also be used to obtain the bulk temperature corresponding to the bulk enthalpy.
In order to discretize the governing equations, the finite volume method is used. The SIMPLE algorithm is applied along with the staggered grid to solve the velocity and pressure coupling equations. The hybrid scheme is also used to approximate the convective terms in the equations. This scheme is based on the intermittent use of central and upwind schemes and has high stability while physically producing realistic solutions [28]. The stability of the hybrid scheme becomes more apparent when, like the present study, the aim is to model the supercritical fluid flow. This is because, to model the highvariable property flow of supercritical fluids, the numerical solution tends to diverge. The SIMPLE algorithm and the hybrid scheme applied in the present CFD code are described extensively by Versteeg and Malalasekera [28].
To reach grid independency for the results, the solutions were obtained for various numbers of nodes in the radial and axial directions. Finally, 50 and 2200 grids were used in the radial and axial directions, respectively, so increasing the number of grids does not improve the results any further.
The significant property variations of supercritical fluids cause the governing equations to become highly coupled. In these conditions, the proper choice of underrelaxation factors (URFs) is essential to the convergence of the solution. Generally, the smaller values of URFs are needed compared to the constant property flows. The suitable URFs obtain by trial and error. The following convergence criterion for termination of calculations is used to ensure that the final answers are independent of the URFs.

(9) 
where

(10) 
Due to the lack of experimental data for laminar flow, this section uses analytical solutions for the flow with constant properties to validate the numerical simulation. For this purpose, the flow of water at a pressure of 1 atm in a pipe with 1.2 m length and 1 mm diameter is considered to validate the numerical results. Other flow conditions are listed in Table 1.
Table 1. Flow conditions were used for the validation
of the CFD code.
Property 
Value 
Unit 
Inlet velocity 
0.05 
m/s 
Inlet temperature 
20 
^{o}C 
Wall heat flux 
1000 
W 
Density 
1000 
Kg/m^{3} 
Viscosity 
0.001 
Kg/m.s 
Specific heat capacity 
4182 
J/kg.^{o}C 
Thermal conductivity 
0.6 
W/m.^{o}C 
Figure 3 shows the value of the Nusselt number. As can be seen in the developed flow section, the value of the Nu is exactly 4.36, which entirely agrees with the analytical solution [29]. It should be noted that similar to the flow conditions for supercritical fluid in the next section, the beginning of the pipe is insulated, and heat transfer starts from a distance of 20 cm.
Figure 3. Variations of Nusselt number along the pipe
for water at a pressure of 1 atm
Numerical results for bulk temperature are also compared with the analytical values in Fig. 4. The fluid's analytical bulk temperature in each pipe crosssection can be obtained from the energy balance along the pipe. Due to the imposition of constant heat flux in the pipe wall, it can be written:

(11) 
Figure 4. Variations of bulk temperature along the pipe
for water at a pressure of 1 atm
It is observed from Fig. 4 that the two temperatures agree with each other. In addition, validating the present code for the turbulent flow of supercritical fluid has already been performed [30, 31].
This section investigates the effects of various factors affecting the system's thermal performance, which is explained in section 2. The results are divided into three parts as follows.
In supercritical conditions, drastic changes in fluid density produce strong buoyancy force as well as flow acceleration near the wall [32]. This leads to wall temperature and velocity profile changes compared to normal flows. Figures 5 to 7 show the velocity profiles at different pipe sections for the three situations studied in this study, i.e., upward, downward, and nogravity flows for the wall heat flux of 300 W/m^{2}.
Figure 5. Velocity profiles for three crosssections of the pipe in upward flow.
Figure 6. Velocity profiles for three crosssections of the pipe in the downward flow.
Figure 7. Velocity profiles for three crosssections of the pipe in the nogravity flow.
As can be seen from these figures, in the upward flow, the velocity profile deviates from the parabolic state in parts of the pipe and becomes like the letter M.
In other words, the maximum fluid velocity does not occur in the center of the pipe, but its location will be near the pipe wall. In the downward flow, the buoyancy force also causes the flow to return near the wall, forming a swirling flow in this region. None of these effects exist in the nogravity flow without buoyancy force.
In addition, Fig. 7 shows that the velocity continuously increases as it moves downstream of the pipe, similar to the flow in a nozzle. This is due to the decrease in the density of the supercritical fluid.
For this reason, in supercritical fluid flows, unlike normal flows, the Reynolds number is not constant and increases significantly along the length of the pipe. The flow may become turbulent if the pipe length is long enough or the inlet flow velocity is high enough.
Fig. 8 shows the changes in average velocity and Reynolds number for the conditions of the present problem.
Figure 8. Variations of average velocity and Reynolds number along the pipe for q_{w}=300 W/m^{2}.
The shear stress direction is reversed in the region where the flow returns near the wall, and its value becomes negative. Fig. 9 shows the changes in the wall shear stress for the three investigated cases. As can be seen, only in the downward flow in a part of the pipe the amount of shear stress is negative, which indicates the return of the flow in this area. Depending on the applied heat flux and the inlet flow conditions, this region shifts or disappears altogether.
Figure 9. Variations of wall shear stress along the pipe for upward, downward and nogravity flows for q_{w}=300 W/m^{2}.
When the buoyancy force is strong (for example, when the applied wall heat flux is large compared to the mass flux), one of the phenomena that may occur in supercritical fluid flow is heat transfer deterioration [24,25,32]. Because of the deterioration, the wall temperature increases suddenly, leading to an unsafe system in some conditions. To see this issue, the longitudinal changes of the wall temperature for three cases of this study are shown in Fig. 10. It should be noted that the deterioration occurs when the fluid temperature near the wall reaches the pseudocritical value at high heat fluxes [29]. This phenomenon is also the reason for the sudden decrease in wall shear stress at z=1.2 m in Fig. 9 for downward flow.
It can be seen from Fig. 10 that very little deterioration occurred, only for the downward flow. On the other hand, increasing the wall temperature means reducing the heat transfer coefficient.
For the three cases examined in this study, the changes in the heat transfer coefficients are shown in Fig. 11. It is observed that when the wall temperature is suddenly increased, the heat transfer coefficient decreases drastically. The highest heat transfer coefficient is related to the upward flow. It should be noted that the reason for the zero heat transfer coefficient at the beginning and end of the pipe is the zero heat flux in these parts, that is, the insulation of the pipe in these sections. It is interesting to note that in the turbulent flow of supercritical fluid, this phenomenon occurs in the upward flow [31,33]. This is because the codirection of flow and buoyancy force usually leads to a decrease in the flow turbulence and, thus, a decrease in the heat transfer coefficient. When turbulent flow is downward, the interaction of the buoyancy force and the flow direction increases the heat transfer.
Figure 10. Effect of flow direction and gravity on wall temperature variations for q_{w}=300 W/m^{2}.
Figure 11. Effect of flow direction and gravity on variations of heat transfer coefficients (HTC) for q_{w}=300 W/m^{2}.
One parameter that shows a heating system's performance is the Nusselt number. For this reason, Fig. 12 shows the changes in this parameter for the three studied cases. The analytical value of 4.36, corresponding to constant property flows, is also plotted for comparison. As can be seen, in the upward flow throughout the pipe and the nogravity flow, only in parts of the pipe, the value of the Nusselt number is greater than 4.36. In other words, Fig. 12 shows that the system's thermal performance is unsuitable in the downstream flow.
In supercritical conditions similar to subcritical conditions, the buoyancy effect is usually expressed in terms of the dimensionless Grashof and Reynolds numbers as follows [34,35].

(12) 
Figure 12. Effect of flow direction on Nu number for q_{w}= 300 W/m^{2} and comparison with constant property
flow conditions (Nu=4.36)
Where both b in both Grb and Reb is subscript are defined based on bulk properties. The value of n is equal to 2 and 2.7 according to the results of Shiralkar and Griffith [34] and Johnson and Hall [35], respectively. Based on the criterion of Shiralkar and Griffith [34], if B>0.01, the effect of buoyancy is important and should be considered in the calculations, while according to the criterion of Johnson and Hall [35], B should be greater than 10^{5} to consider buoyancy effect. In Fig. 13, the changes of parameter B are plotted for both upstream and downstream flows. It is seen that for both directions and based on both criteria, the effect of buoyancy is important throughout the pipe length and cannot be ignored. Moreover, the value of B is much higher for downward flow than upward flow.
Based on previous results for turbulent flow in the literature, increasing the wall heat flux in both upward and downward flows has a negative effect on the magnitude of the Nusselt number or heat transfer coefficient [31,33]. In this section, this issue is investigated for laminar flow. To do this, several simulations are performed for different wall heat fluxes from 300 to 600 W/m^{2}. The obtained results for the Nusselt number in the upward flow, the wall temperature, and the wall shear stress in the downward flow are plotted in Figures 14 to 16, respectively.
Figure 13. Variations of buoyancy parameters for upward and downward flows based on two criteria of Shiralkar [34] and Jackson [35] for q_{w}= 300 W/m^{2}.
Figure 14. Effect of wall heat flux on axial variations
of Nu number in upward flow.
Figure 15. Effect of wall heat flux on axial variations of
wall temperature in the downward flow.
Figure 16. Effect of wall heat flux on axial variations of wall shear stress in the downward flow
It can be found from Fig. 14 that in laminar flow, as opposed to turbulent flow with increasing the wall heat flux in the improvement regime, the Nusselt number increases. However, similar to turbulent flow, as shown in Fig. 15 when deterioration occurs, increasing heat flux leads to a further increase in wall temperature, which means a further reduction of Nusselt number in this area. It should be noted that, as mentioned earlier, regimes of improvement and deterioration of heat transfer in a vertical tube in the laminar flow are opposite to the turbulent flow. It can also be seen from Fig. 15 that with increasing the wall heat flux, the sudden increase in wall temperature is approaching the pipe inlet and in heat fluxes of 500 and 600 W/m^{2} due to the proximity of the inlet temperature to the pseudocritical temperature, this phenomenon appears as soon as the heat flux is applied. Similar behavior observes in Fig. 16 for the wall shear stress when the wall heat flux increases. In other words, at higher heat fluxes, the return flow begins at the inlet of the pipe at the same location where the heat flux is applied.
Because the heat flux of the wall may not be the same in all parts of the pipe where the heat transfer takes place, in this section, the effect of a step change of the wall flux on the thermal behavior of the system is investigated. The results for Nusselt number are shown in Fig. 17. It is seen that when the heat flux changes, the Nusselt number also jumps abruptly, resulting from the temperature difference at these points. Like the previous results, the Nusselt number for the upward flow near the critical point has higher values than the constant property flow. Under these conditions, the downward flow deteriorated. In fact, in the location where the second step of the wall heat flux is applied, i.e., at z=1.2 m, the wall temperature passes from the pseudocritical temperature, which in turn causes deterioration to occur at this point. This phenomenon is shown in Fig. 18. It may be concluded that the deterioration can be partially controlled or reduced by correctly determining the location of the step or any variation related to the wall heat flux. Deterioration, on the other hand, causes a sudden decrease in wall shear stress in downward flow, as shown in Fig. 19. In other words, the reverse flow starts in this region.
Figure 17. Effect of stepped wall heat flux on longitudinal variations of Nu number for both upward
and downward flows.
Figure 18. Effect of stepped wall heat flux on longitudinal variations of wall and bulk temperatures for
both upward and downward flows.
Figure 19. Effect of stepped wall heat flux on longitudinal variations of wall shear stress for both upward
and downward flows.
In this study, the laminar flow of supercritical CO_{2} in a long vertical minipipe has been numerically investigated by developing an axisymmetric CFD code. The effect of some important parameters, including flow direction, constant and stepped wall heat flux, and gravity, are examined. Severe changes in fluid properties and gravity force produce strong buoyancy force, significantly affecting the flow's structure and thermal behavior. The following are the most important results of this study.
Nomenclature
Cp 
Specific heat capacity [J/kg.K] 

d 
Pipe diameter [m] 

g 
Gravitational acceleration [m/s^{2}] 

G 
Mass flux [kg/s.m^{2}] 

h 
Heat transfer coefficient [W/m^{2}.K] 

H 
Enthalpy [J/kg] 

k_{f} 
Thermal conductivity [W/m.K] 

Nu 
Nusselt number 

p 
Pressure [N/m^{2}] 

q 
Heat flux [W/m^{2}] 


Internal Heat Source [W/m^{3}] 

r 
Radial direction of flow [m] 

Re 
Reynolds number [Re=ρVD/µ] 

T 
Temperature [^{o}C] 

u 
Velocity component in z direction [m/s] 

v 
Velocity component in r direction [m/s] 

z 
Axial direction of flow [m] 

Greek Symbols 

ϕ 
Dissipation function in energy equation kg/m.s^{3} 

µ 
Molecular viscosity [kg/m.s] 

ρ 
Density [kg/m^{3}] 

τ 
Shear stress [N/m^{2}] 

Subscript 

b 
Bulk 

i 
Inlet 

w 
Wall 

Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this manuscript. In addition, the authors have entirely observed the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication and/or submission, and redundancy.
References
[*]Corresponding Author: Mahdi Mohseni
Email: m.mohseni@qut.ac.ir