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Boundary effects on the drag coefficient and average Nusselt number of a sphere in SCW: A comparative study
Engineering Analysis with Boundary Elements 102 (2019) 1–10
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Boundary effects on the drag coefficient and average Nusselt number of a sphere in SCW: A comparative study Bo Xiong, Hao Zhang∗, Xizhong An Key Laboratory for Ecological Metallurgy of Multimetallic Mineral of Ministry of Education, School of Metallurgy, Northeastern University, Shenyang, 110819, PR China
a r t i c l e
i n f o
Keywords: Drag coefficient Average nusselt number Sphere Supercritical water Wall effects
a b s t r a c t A comparative study on cold supercritical water (SCW) passing over a hot sphere is carried out using the particleresolved direct numerical simulation (PR-DNS) to examine the wall effects. Different working conditions are considered to obtain the drag coefficient (Cd) and average Nusselt number (Nu) of hot spheres in SCW whose physical properties significantly change with temperature and pressure. The effects of wall factors on Cd and Nu under two different boundary conditions, namely a stationary one for the fluidized sphere problem and a moving one for the settling sphere problem, are compared and discussed. Numerical results show that Cd and Nu of a sphere in SCW cannot fit the previous correlations for conventional fluids well. Therefore, these correlations are improved for SCW, which are good tools to understand the basic knowledge of SCW.
1. Introduction The traditional way to burn fossil fuels for the power generation has caused seriously devastating influences on the ecological environment [1]. In order to solve this problem, a potential alternative is using the coal to produce hydrogen which has many advantages like high energy density, high cleanness and high utilization rate [2–4]. Hydrogen production based on supercritical water (SCW) has proven to be a promising technology which is drawing more and more attentions [5]. This technique uses the special thermochemical property of SCW to react with coal to generate hydrogen and the harmful component is discharged as a form of sediment rather than particle-laden gases. In the past 30 years, various SCW-based reactors were manufactured and both experimental and numerical researches were carried out [6–9]. However, the available knowledge for the design and scale-up of these devices is still limited [8], such as the determination of the drag coefficient (Cd) and Nusselt number (Nu) [10] of solid particles in SCW. Especially, these two parameters should be also able to account for the effect of the surrounding walls on the solid particles irrespective of using a tubular [5] or fluidized [11] reactor with SCW. It has been demonstrated that the wall factor (𝜆), defined as the ratio between the diameter of the container and the polar diameter of the particle, plays a great role on Cd and Nu in the conventional fluid (CF here standing for Newtonian fluids at moderate pressure and temperature) [10,12]. Therefore, there is a need to investigate Cd and Nu of the solid particles in SCW in a wide range of 𝜆 which is important but not focused before. This study aims to provide some important information to bridge this knowledge gap based on
∗
the particle-resolved direct numerical simulation (PR-DNS) [13] which has proven to be a critical tool for the study of engineering problems [14–19]. Wham et al. [20] conducted FEM simulations to study the wall effects on the drag of solid spheres in CF and both the fluidized and settling sphere problems were considered. Numerical results revealed that Re for the onset of wake formation increases whereas the length of the wake decreases with the increase of 𝜆. Song and Gupta [21] studied the wall effects on spheres in static power law fluids in cylindrical tubes and found that wall effects were less severe in power law fluids than in Newtonian fluids. Kishore et al. conducted PR-DNS to discuss the wall effects on flow and drag phenomena of spheroid particles in CF [10], heat transfer phenomena of spheroid particles in CF [12], flow and drag phenomena of spheres in shear-thickening fluids [22] and mixed convection phenomenon of a square cylinder in non-Newtonian nanofluids [23]. It was found that both Cd and Nu decreased with the increase of 𝜆. PR-DNS on SCW is also reported. The numerical results of Wei et al. claimed the critical effect of the variable thermo-physical properties of SCW in affecting the flow separation [24] and mixed [25] and free [26] convection. Wu et al. [27,28] expanded the work on a nonspherical particle in SCW and the effect of the particle shape on the drag force was examined. Cd and Nu of various spheroids in SCW were investigated by Zhang et al. [29] to discuss the effect of the spheroid aspect ratio; then they considered the effect of the incident angle of the spheroid and improved the correlations of Cd and Nu in SCW [30]. A further investigation is conducted in this study to consider the wall effects on the momentum and heat transfer phenomena of spheres in
Corresponding author. E-mail address: [email protected] (H. Zhang).
https://doi.org/10.1016/j.enganabound.2019.02.001 Received 27 December 2018; Received in revised form 29 January 2019; Accepted 14 February 2019 Available online 1 March 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.
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Engineering Analysis with Boundary Elements 102 (2019) 1–10
Nomenclature a, b, c 𝜆 Dt Cd u1 u2 Tf Ts Cdf Cds CdP Cd1 Cd2 Cp dp h Nu Nuf Nus Nu1 Nu2 Pr Re T TL TH 𝐮⃖⃗ uc Vx Vy Vz
principal semi-axes wall factor cylinder diameter, m total drag coefficien X-velocity for the fluidized sphere X-velocity for the fluidized sphere temperature for the fluidized sphere temperature for the settling sphere drag coefficient for the fluidized sphere drag coefficient for the settling sphere pressure coefficient pressure coefficient for the fluidized sphere pressure coefficient for the settling sphere specific heat capacity volume-equivalent sphere diameter, m enthalpy, J/kg average Nusselt number average Nusselt number for the fluidized sphere average Nusselt number for the settling sphere surface Nusselt number for the fluidized sphere surface Nusselt number for the fluidized sphere Prandtl number Reynolds number temperature, K low temperature in the system, inlet and initial temperature of the fluid, K high temperature in the system, constant temperature of the particle, K fluid velocity vector, m/s inlet velocity, m/s x-component of velocity, m/s y-component of velocity, m/s z-component of velocity, m/s
Fig. 1. Property of SCW in the pseudo-critical zone, P = 23 MPa.
Fig. 2. Sketch map of the calculation domain, only half of the domain is shown.
heat capacity Cp for SCW are very sensitive to the temperature and pressure near the pseudo-critical point of SCW as shown in Fig. 1. In other words, these variables will change with temperature and pressure in the numerical simulation. A sphere with the diameter dp = 0.1m is considered with constant temperature TH = 657 K and the Reynolds number is defined as Re = 𝜌 dp uc /𝜇 c where uc and 𝜇 c are inlet fluid velocity and viscosity, respectively. The calculation domain in this study is a cylinder with a length 60 dp as shown in Fig. 2. The change of the cylinder diameter (Dt) is used to describe the variation of wall effects, 𝜆 = Dt/dp . The sphere is located at (2 m, 2 m, 2 m) passed over by SCW with a temperature TL = 647 K. The key boundary conditions for the fluidized sphere problem can be summarized as follows:
Greek letters 𝜅 thermal conductivity, W/(mK) 𝜇 dynamic viscosity, Pa⋅s 𝜇c dynamic viscosity at the inlet boundary, Pa⋅s 𝜌 fluid density, kg/m3
SCW. Towards this goal, a stationary wall is set for the fluidized sphere problem while a moving one is set for the settling sphere problem with a comparative study conducted to highlight the difference. The remainder of the paper is organized as follows. Section 2 introduces the governing equations and computational issues. Section 3 verifies the model capability. Section 4 reports main numerical results with corresponding discussion. Section 5 establishes new correlations which include the wall effects on Cd and Nu of a sphere in SCW. At last, conclusions are made in Section 6.
At the inlet: Vx = uc ;Vy = 0; Vz = 0; T = TL At the surface of the particle: Vx = 0; Vy = 0; Vz = 0; T = TH Along the solid walls: Vx = 0; Vy = 0; Vz = 0; T = TL While the key boundary conditions for the settling sphere problem are set as:
2. Governing equations and computational issues
At the inlet: Vx = uc ; Vy = 0; Vz = 0; T = TL At the surface of the particle: Vx = 0; Vy = 0; Vz = 0; T = TH Along the solid walls: Vx = uc ; Vy = 0; Vz = 0; T = TL
The governing equations adopted to solve the flow field and heat transfer are: ⎧∇ ⋅ 𝜌⃖𝐮⃗ = 0 ⎪ ⎨(𝐮⃖⃗ ⋅ ∇)𝜌⃖𝐮⃗ = −∇𝑝 + ∇(𝜇∇ ⋅ 𝐮⃖⃗) ⎪(𝐮⃖⃗ ⋅ ∇)𝜌ℎ = ∇ ⋅ (𝑘∇𝑇 ) ⎩
Five Reynolds numbers (with Re changing from 10, 20, 50, 100 to 200) and five wall factors (with 𝜆 changing from 5, 6.4, 7.5, 10 to 20) are considered.
(1)
where: 𝜌-fluid density, 𝐮⃖⃗-velocity vector, p-pressure, 𝜇-dynamic viscosity, h-enthalpy, k-thermal conductivity and T-temperature. We use a commercial software package (ANSYS FLUENT15.0) to solve Eq. (1). As mentioned in previous studies [24–26,29,30], 𝜌, 𝜅, 𝜇 and the specific
3. Verification case The reliability of the current model to investigate SCW as well as the mesh system has been extensively benchmarked in our previous work 2
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figure, boundary conditions and 𝜆 have a significant effect on X-velocity. For u1 , the velocity is high at the region between the sphere and wall because stationary boudanry conditions are set on the solid in the middle. For u2 , since the wall is moving with the fluid, the fluid velocity tends to gradually decrease from the wall surface toward the particle surface. Quantitative comparison is shown in Fig. 4 which further depicts the distribution of X-velocity for a sphere in SCW at different Re and 𝜆. It is obvious to see that u1 increases from zero on the particle surface to a certain value, then decreases to zero on the surface of the solid wall. It is easy to explain this phenomenon since both the velocities at the surface of the particle and at the wall are applied with no-slip boundary condition. u2 decreasing monotonically from the surface of the wall to the surface of the particle can also be found in Fig. 4. This is because the wall is moving with the fluid. In other words, the wall has a relative velocity relative to the particle.
Table 1 Comparison of Cd between the data in reference and current results with 𝜆 = 5. Re
11.7 32.4 33
Cd Numerical [31]
Experimental [32]
Present
4.9 2.49 2.56
4.27 2.12 2.12
4.35 2.18 2.16
[29,30]. Furthermore, in this study, additional verifications are given on the wall effects on a settling sphere in CF versus previously reported data [31,32]. The working fluid is CF at 300 K and P = 0.1 MPa with 𝜆 = 5. Quantitative comparison of numerically predicted Cd at different Re are shown in Table 1. As can be seen from Table 1 that the present results are all in the acceptable region between the reference data. As Re increases from 32.4 to 33, there is no change in the experimentaly reported Cd [32]. This may be due to the low sensitivity of the experimental set to the small variation of Re. On the contrary, numerical results of Sharma and Patankar reported an increase of Cd [31] with the increase of Re from 32.4 to 33. This is not consistent with the common knowledge that Cd normally decreases with Re in CF [33,34]. Therefore, it is believed that the present results are accurate and reliable. It is safe to conduct further numerical simulations to discuss the wall effects in SCW.
4.2. Distribution of temperature Fig. 5 shows the distribution of temperature in different cases. In this subsection, Tf is defined as the temperature for the fluidized sphere problem while Ts is for the settling sphere problem. It is clearly seen that both Tf and Ts change more steeply with the increase of the distance to the center of the particle when Re is small. This is due to the fact that the thickness of the boundary layer becomes thinner as Re increases. When comparing the value of Tf − Ts , it becomes smaller when increasing Re or lifting 𝜆. This phenomenon can be explained as that the wall effect weakens as 𝜆 or Re increases, the convective heat transfer between particles and fluids is enhanced making the difference of temperature field under the two boundary conditions smaller. It is worthwhile noting that the value of Tf − Ts reaches its maximum at a certain location, which predicting that the heat transfer is similar under these two boundary conditions.
4. Results and discussions It needs to note that in the following sections, Cdf is defined as the drag coefficient for the fluidized sphere problem while Cds for the settling sphere problem. Nuf and Nus are defined in a similar way. Except for the surrounding walls, other key parameters like Re, Pr, and 𝜆, are the same in each case.
4.3. The wall effects on cd 4.1. Distribution of X-velocity Fig. 6 shows the distributions of Cdf and Cds in different cases. It is clearly seen that both Cdf and Cds increase with the decrease of the wall factor (stronger wall effects). This phenomenon is not difficult to explain since the approaching of the surrounding wall to the particle leads to the
Fig. 3 shows the distribution of X-velocity in different cases. In this subsection, u1 is defined as the X-velocity for the fluidized sphere problem while u2 is for the settling sphere problem. As can be seen from the
Fig. 3. Distribution of X-velocity of a sphere in SCW with stationary walls (top) and movings tationary walls (bottom) for different values of 𝜆 and Re. (From a to b, 𝜆 is 5,10; From 1 to 2, Re is 10,100). 3
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Fig. 4. Distribution of X-velocity of a sphere in SCW at 𝜆 = 5(a) and 𝜆 = 10(b) for different values of Re.
Fig. 5. (a) Distribution of temperature of a sphere in SCW at 𝜆 = 5 (bottom) and 𝜆 = 10 (top) for different values of Re. (b) relative deviation of Temperature.
strong retardation on the fluid field and then increases Cd. The wall effects vanish gradually when 𝜆 gets larger and further increasing 𝜆 will not create influence on Cd any longer when 𝜆 reaches one point. For a fixed value of 𝜆, both Cdf and Cds decrease with Re which is in line with the unconfined cases [33,34]. For a fixed value of 𝜆 and Re, Cdf is larger than Cds which can be attributed to the additional wall dissipation. For the settling sphere problem, there is no relative motion between the wall and fluid and thus the wall retardation effect on the fluid is smaller. It can be seen that Cds are very comparable when 𝜆=20 and 𝜆=40. However, additional dissipation takes place for the fluidized sphere problem due to the strong wall retardation effects caused by the relative motion between the fluid and wall. This fact indicates that a larger calculation domain is needed for the fluidized sphere problem than the settling one to eliminate the wall effect. Fig. 7 shows the distribution of Cdf /Cds of a sphere in SCW for different 𝜆 and Re in which combined effects of wall and Re can be observed.
Cdf /Cds is found to decrease with the increase of Re. This is because the viscosity contribution of fluid drops when lifting Re. Moreover, Cdf /Cds decreases with the increase of 𝜆. The is because the wall effect gets weaker when the wall is located further away. Therefore, the lowest value appears when Re = 200 and 𝜆 = 20. Fig. 8 shows the combined effects of 𝜆 and Re on distribution of pressure coefficient of a sphere in SCW for both fluided problem and settling problem. In this subsection and following ones, Cd1 is defined as the pressure coefficient for the fluidized sphere problem while Cd2 for the settling sphere problem. For a fixed Re and a given 𝜆, both Cd1 and Cd2 decrease from the starting point(X = −R) to the highest point of the particle surface(X = 0), and then start to rise due to the recirculating wake at the rear end of the particle surface. For a fixed 𝜆, in addition to the common decreasing trend of pressure coefficient with Re, it can be also seen that Cd1 is higher than Cd2 from the starting point(X = −R) to point C and opposite trend is found from point C to 4
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Fig. 6. (a) Distribution of Cdf and Cds of a sphere in SCW with moving walls (left half) and stationary walls (right half) for different values of 𝜆 and Re. (b) 𝜆=40 stands for the case when the wall effects vanish[26].
the ending point(X = R). At the same time, as Re increases, point C is closer to the starting point(X = −R). This is because pressure recovery improves more significantly for settling problem as Re increases. 4.4. The wall effects on Nu Fig. 9 shows the distribution of Nu of a sphere in SCW for different 𝜆 and Re. It can be seen that under both cases of the wall boundary conditions, Nu increases with the increase of Re due to the enhanced convection. In addition, more information can be found from Fig. 9. For all the Re and 𝜆, Nuf is higher than Nus . The difference between Nuf and Nus is significant when 𝜆 is small but hardly varies when 𝜆 is larger than 10. This can be due to the fact that the surrounding wall can change the fluid flow around the sphere significantly when 𝜆 is small. Considering the fact that wall effects are stronger in the fluidized sphere problem than the settling one, it can be concluded that Nu increases when the wall retardation effect is stronger. The enhancement of Nu by wall effects can also be observed from the change of Nu with 𝜆. Fig. 9a clearly shows that both Nuf and Nus decrease with the increase of 𝜆. Again, Nus with large 𝜆 is quite comparable with the unconfined particle shown in Fig. 9b. Fig. 10 describes the distributions of surface Nu of a sphere in SCW for different values of 𝜆 and Re. In this subsection and following ones,
Fig. 7. Distribution of Cdf /Cds of a sphere in SCW for different values of 𝜆 and Re.
Fig. 8. Distribution of pressure coefficient of a sphere in SCW with moving walls and stationary walls for different values of Re.(From a to c, Re is 10,50,100). 5
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Fig. 9. (a) Distribution of Nuf and Nus of a sphere in SCW for different values of 𝜆 and Re. (b) 𝜆=40 stands for the case where wall effects vanish[26].
Fig. 10. Distribution of surface Nu of a sphere in SCW with moving walls and stationary walls for different values of Re and𝜆. (From a to c, Re is 10,50,100. From 1 to 3, 𝜆 is 5,7.5,10).
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Nu1 is defined as the surface Nu for the fluidized sphere problem while Nu2 is for the settling sphere problem. For a fixed Re and a given 𝜆, both Nu1 and Nu2 fluctuate at the front slightly (from starting point (X = −R) to point a) and then decrease along the surface of the particle until the flow separation point (point b). For a fixed 𝜆, in addition to the common increasing trend of Nu1 and/or Nu2 with Re, it is worth noting that the separation point moves forward as Re increases. This can be due to the fact that the size of the recirculation vortex increases with an increase in Re [24]. Considering the fact that the separation point is closer to the starting point (X = −R) when 𝜆 is small, it can be concluded that the size of recirculation vortex becomes small with 𝜆 decreases. Regardless of Re and 𝜆, when comparing the position of the flow separation point under two boundary conditions, it is not difficult to find point b is slightly ahead for stationary wall, which implies that the recirculation vortex is larger for fluidized sphere problem under the same Re and 𝜆.
5.2. Fluidized sphere problem Via the regression analysis on the present numerical results, new coefficients in the above equation for Cdf are: {
It can be seen in Fig. 13a that these new coefficients do a good job to predict Cdf with the max and min relative deviations 11.23% and 0.33%, respectively, while the correlation coefficient is 99.82%. The good capability can be also seen from Fig. 14a. Actually, most of the relative deviations are far lower than 10%. Similar with Cdf , the regression analysis recommends below values of coefficients to use and the prediction performance for Nuf is shown in Fig. 13b. { 𝑎1 = −2.10206; 𝑎2 = −2.34165; 𝑎3 = 0.0226; 𝑎4 = 1.1067; 𝑎5 = 6.88881; 𝑎6 = 0.1746; 𝑎7 = −2.2028.
5. New correlations
Using these parameters, corresponding max and min relative deviations are 2.435% and 0.093%, respectively, while the correlation coefficient is 99.944%. The distribution of these relative deviations is shown in Fig. 14b.
5.1. Evaluation on previous correlations Due to the fact that current Cd and Nu share similar tendencies with the numerical results of Kishore and Gu [10,12] on CF though from 2D simulations, it is aimed to improve their formula for SCW. The formulas for Cd and Nu are shown as: ( ) 𝑎 𝜆 + 𝑎 2 𝑎4 𝐶𝑑 = 1 (𝑎 5 + 𝑎 6 𝑅𝑒𝑎7 + 𝑎 8 𝑅𝑒𝑎9 ) (2) 𝑅𝑒 𝜆 + 𝑎3 ( 𝑁𝑢 =
) 𝜆 + 𝑎1 (𝑎 3 𝑅𝑒𝑎4 + 𝑎 5 𝑅𝑒𝑎6 ) + 𝑎 7 𝜆 + 𝑎2
𝑎1 = 50.30808; 𝑎2 = −0.91522; 𝑎3 = −0.78761; 𝑎4 = −10.43884; 𝑎5 = −3.12144; 𝑎6 = 0.00155; 𝑎7 = 1.28438; 𝑎8 = 4.0001; 𝑎9 = 0.0629.
5.3. Settling sphere problem Similar with last subsection, determined coefficients for Cds and Nus are: { 𝑎1 = 8.01219; 𝑎2 = −1.90211; 𝑎3 = −1.83908; 𝑎4 = −7.04622; 𝑎5 = 2.26883; 𝑎6 = 0.08208; 𝑎7 = 0.92626; 𝑎8 = 0.9345; 𝑎9 = 0.33012.
(3)
and { 𝑎1 = −2.6984; 𝑎2 = −2.79934; 𝑎3 = 0.01282; 𝑎4 = 1.23509; 𝑎5 = 5.24864; 𝑎6 = 0.19642; 𝑎7 = −0.90752.
In order to test whether the correlations proposed by Kishore and Gu [10,12] are applicable to this work, we firstly compare the present numerical results with those calculated by the correlations of Kishore and Gu [10,12]. From Fig. 11a, one can find that fluctuated Cdf is observed and Nuf for CF is obviously lower than SCW as shown in Fig. 11b. Great discrepancies between the predicted results based on the correlation for CF and the present numerical results for Cds and Nus also can be seen from Fig. 12, which demonstrates the need for updating the coefficients.
It can be seen in Figs. 15 and 16 that Eqs. (2) and (3) also do well to predict Cds and Nus using these parameters. For Cds , max and min average relative deviations are 4.047% and 0.00346%, respectively, while the correlation coefficient is 99.962%. For Nus , corresponding max and min relative deviations are 2.829% and 0.1152%, respectively, while the correlation coefficient is 99.964%.
Fig. 11. Predicted (a) Cdf and (b) Nuf by the correlation [10,12] with the present numerical results for the fluidized sphere problem. 7
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Engineering Analysis with Boundary Elements 102 (2019) 1–10
Fig. 12. Predicted (a) Cds and (b) Nus by the correlation[10,12] with the present numerical results for the settling sphere problem.
Fig. 13. Predicted (a) Cdf and (b) Nuf by the proposed correlation with numerical results.
Fig. 14. Comparison between currently numerical and correlated (a) Cdf and (b) Nuf .
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Engineering Analysis with Boundary Elements 102 (2019) 1–10
Fig. 15. Predicted (a) Cds and (b) Nus by the proposed correlation with numerical results.
Fig. 16. Comparison between currently numerical and correlated (a) Cds and (b) Nus .
6. Concluding remarks
Acknowledgements
A comparative study on cold SCW passing over a hot sphere with two types of spoiler problems is carried out using the PR-DNS. The influences of different boundary conditions are highlighted and Cd and Nu at each working condition are obtained. According to the numerical results, new correlations are established. Main conclusions can be drawn as follows:
The authors sincerely acknowledge the financial supports from the National Key R & D Program of China (2016YFB0600102-4), the NSFC project (51606040) and the Jiangsu Province Science Foundation for Youths (BK20160677) on this research.
References (1) The changing trend of Cd and Nu with 𝜆 and Re is similar under the two types of spoiler problems. Discrepancies on the exact values of these two situations are due to the wall effects. (2) In order to eliminate the wall effects on Cd and Nu, a larger wall factor (𝜆) should be used for the fluidized sphere problem than the settling sphere problem. (3) New correlations of Cd and Nu for a hot sphere in SCW with two kinds of boundaries are established by considering 10 ≤ Re ≤ 200 and 5 ≤𝜆 ≤ 20 as key influencing factors with good accuracy obtained. These correlations could facilitate the phase coupling in multi-phase flow modellings of SCW. The correlations are available for the temperature between 647 K and 657 K at P = 23 MPa.
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Boundary effects on the drag coefficient and average Nusselt number of a sphere in SCW: A comparative study Bo Xiong, Hao Zhang∗, Xizhong An Key Laboratory for Ecological Metallurgy of Multimetallic Mineral of Ministry of Education, School of Metallurgy, Northeastern University, Shenyang, 110819, PR China
a r t i c l e
i n f o
Keywords: Drag coefficient Average nusselt number Sphere Supercritical water Wall effects
a b s t r a c t A comparative study on cold supercritical water (SCW) passing over a hot sphere is carried out using the particleresolved direct numerical simulation (PR-DNS) to examine the wall effects. Different working conditions are considered to obtain the drag coefficient (Cd) and average Nusselt number (Nu) of hot spheres in SCW whose physical properties significantly change with temperature and pressure. The effects of wall factors on Cd and Nu under two different boundary conditions, namely a stationary one for the fluidized sphere problem and a moving one for the settling sphere problem, are compared and discussed. Numerical results show that Cd and Nu of a sphere in SCW cannot fit the previous correlations for conventional fluids well. Therefore, these correlations are improved for SCW, which are good tools to understand the basic knowledge of SCW.
1. Introduction The traditional way to burn fossil fuels for the power generation has caused seriously devastating influences on the ecological environment [1]. In order to solve this problem, a potential alternative is using the coal to produce hydrogen which has many advantages like high energy density, high cleanness and high utilization rate [2–4]. Hydrogen production based on supercritical water (SCW) has proven to be a promising technology which is drawing more and more attentions [5]. This technique uses the special thermochemical property of SCW to react with coal to generate hydrogen and the harmful component is discharged as a form of sediment rather than particle-laden gases. In the past 30 years, various SCW-based reactors were manufactured and both experimental and numerical researches were carried out [6–9]. However, the available knowledge for the design and scale-up of these devices is still limited [8], such as the determination of the drag coefficient (Cd) and Nusselt number (Nu) [10] of solid particles in SCW. Especially, these two parameters should be also able to account for the effect of the surrounding walls on the solid particles irrespective of using a tubular [5] or fluidized [11] reactor with SCW. It has been demonstrated that the wall factor (𝜆), defined as the ratio between the diameter of the container and the polar diameter of the particle, plays a great role on Cd and Nu in the conventional fluid (CF here standing for Newtonian fluids at moderate pressure and temperature) [10,12]. Therefore, there is a need to investigate Cd and Nu of the solid particles in SCW in a wide range of 𝜆 which is important but not focused before. This study aims to provide some important information to bridge this knowledge gap based on
∗
the particle-resolved direct numerical simulation (PR-DNS) [13] which has proven to be a critical tool for the study of engineering problems [14–19]. Wham et al. [20] conducted FEM simulations to study the wall effects on the drag of solid spheres in CF and both the fluidized and settling sphere problems were considered. Numerical results revealed that Re for the onset of wake formation increases whereas the length of the wake decreases with the increase of 𝜆. Song and Gupta [21] studied the wall effects on spheres in static power law fluids in cylindrical tubes and found that wall effects were less severe in power law fluids than in Newtonian fluids. Kishore et al. conducted PR-DNS to discuss the wall effects on flow and drag phenomena of spheroid particles in CF [10], heat transfer phenomena of spheroid particles in CF [12], flow and drag phenomena of spheres in shear-thickening fluids [22] and mixed convection phenomenon of a square cylinder in non-Newtonian nanofluids [23]. It was found that both Cd and Nu decreased with the increase of 𝜆. PR-DNS on SCW is also reported. The numerical results of Wei et al. claimed the critical effect of the variable thermo-physical properties of SCW in affecting the flow separation [24] and mixed [25] and free [26] convection. Wu et al. [27,28] expanded the work on a nonspherical particle in SCW and the effect of the particle shape on the drag force was examined. Cd and Nu of various spheroids in SCW were investigated by Zhang et al. [29] to discuss the effect of the spheroid aspect ratio; then they considered the effect of the incident angle of the spheroid and improved the correlations of Cd and Nu in SCW [30]. A further investigation is conducted in this study to consider the wall effects on the momentum and heat transfer phenomena of spheres in
Corresponding author. E-mail address: [email protected] (H. Zhang).
https://doi.org/10.1016/j.enganabound.2019.02.001 Received 27 December 2018; Received in revised form 29 January 2019; Accepted 14 February 2019 Available online 1 March 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.
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Nomenclature a, b, c 𝜆 Dt Cd u1 u2 Tf Ts Cdf Cds CdP Cd1 Cd2 Cp dp h Nu Nuf Nus Nu1 Nu2 Pr Re T TL TH 𝐮⃖⃗ uc Vx Vy Vz
principal semi-axes wall factor cylinder diameter, m total drag coefficien X-velocity for the fluidized sphere X-velocity for the fluidized sphere temperature for the fluidized sphere temperature for the settling sphere drag coefficient for the fluidized sphere drag coefficient for the settling sphere pressure coefficient pressure coefficient for the fluidized sphere pressure coefficient for the settling sphere specific heat capacity volume-equivalent sphere diameter, m enthalpy, J/kg average Nusselt number average Nusselt number for the fluidized sphere average Nusselt number for the settling sphere surface Nusselt number for the fluidized sphere surface Nusselt number for the fluidized sphere Prandtl number Reynolds number temperature, K low temperature in the system, inlet and initial temperature of the fluid, K high temperature in the system, constant temperature of the particle, K fluid velocity vector, m/s inlet velocity, m/s x-component of velocity, m/s y-component of velocity, m/s z-component of velocity, m/s
Fig. 1. Property of SCW in the pseudo-critical zone, P = 23 MPa.
Fig. 2. Sketch map of the calculation domain, only half of the domain is shown.
heat capacity Cp for SCW are very sensitive to the temperature and pressure near the pseudo-critical point of SCW as shown in Fig. 1. In other words, these variables will change with temperature and pressure in the numerical simulation. A sphere with the diameter dp = 0.1m is considered with constant temperature TH = 657 K and the Reynolds number is defined as Re = 𝜌 dp uc /𝜇 c where uc and 𝜇 c are inlet fluid velocity and viscosity, respectively. The calculation domain in this study is a cylinder with a length 60 dp as shown in Fig. 2. The change of the cylinder diameter (Dt) is used to describe the variation of wall effects, 𝜆 = Dt/dp . The sphere is located at (2 m, 2 m, 2 m) passed over by SCW with a temperature TL = 647 K. The key boundary conditions for the fluidized sphere problem can be summarized as follows:
Greek letters 𝜅 thermal conductivity, W/(mK) 𝜇 dynamic viscosity, Pa⋅s 𝜇c dynamic viscosity at the inlet boundary, Pa⋅s 𝜌 fluid density, kg/m3
SCW. Towards this goal, a stationary wall is set for the fluidized sphere problem while a moving one is set for the settling sphere problem with a comparative study conducted to highlight the difference. The remainder of the paper is organized as follows. Section 2 introduces the governing equations and computational issues. Section 3 verifies the model capability. Section 4 reports main numerical results with corresponding discussion. Section 5 establishes new correlations which include the wall effects on Cd and Nu of a sphere in SCW. At last, conclusions are made in Section 6.
At the inlet: Vx = uc ;Vy = 0; Vz = 0; T = TL At the surface of the particle: Vx = 0; Vy = 0; Vz = 0; T = TH Along the solid walls: Vx = 0; Vy = 0; Vz = 0; T = TL While the key boundary conditions for the settling sphere problem are set as:
2. Governing equations and computational issues
At the inlet: Vx = uc ; Vy = 0; Vz = 0; T = TL At the surface of the particle: Vx = 0; Vy = 0; Vz = 0; T = TH Along the solid walls: Vx = uc ; Vy = 0; Vz = 0; T = TL
The governing equations adopted to solve the flow field and heat transfer are: ⎧∇ ⋅ 𝜌⃖𝐮⃗ = 0 ⎪ ⎨(𝐮⃖⃗ ⋅ ∇)𝜌⃖𝐮⃗ = −∇𝑝 + ∇(𝜇∇ ⋅ 𝐮⃖⃗) ⎪(𝐮⃖⃗ ⋅ ∇)𝜌ℎ = ∇ ⋅ (𝑘∇𝑇 ) ⎩
Five Reynolds numbers (with Re changing from 10, 20, 50, 100 to 200) and five wall factors (with 𝜆 changing from 5, 6.4, 7.5, 10 to 20) are considered.
(1)
where: 𝜌-fluid density, 𝐮⃖⃗-velocity vector, p-pressure, 𝜇-dynamic viscosity, h-enthalpy, k-thermal conductivity and T-temperature. We use a commercial software package (ANSYS FLUENT15.0) to solve Eq. (1). As mentioned in previous studies [24–26,29,30], 𝜌, 𝜅, 𝜇 and the specific
3. Verification case The reliability of the current model to investigate SCW as well as the mesh system has been extensively benchmarked in our previous work 2
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figure, boundary conditions and 𝜆 have a significant effect on X-velocity. For u1 , the velocity is high at the region between the sphere and wall because stationary boudanry conditions are set on the solid in the middle. For u2 , since the wall is moving with the fluid, the fluid velocity tends to gradually decrease from the wall surface toward the particle surface. Quantitative comparison is shown in Fig. 4 which further depicts the distribution of X-velocity for a sphere in SCW at different Re and 𝜆. It is obvious to see that u1 increases from zero on the particle surface to a certain value, then decreases to zero on the surface of the solid wall. It is easy to explain this phenomenon since both the velocities at the surface of the particle and at the wall are applied with no-slip boundary condition. u2 decreasing monotonically from the surface of the wall to the surface of the particle can also be found in Fig. 4. This is because the wall is moving with the fluid. In other words, the wall has a relative velocity relative to the particle.
Table 1 Comparison of Cd between the data in reference and current results with 𝜆 = 5. Re
11.7 32.4 33
Cd Numerical [31]
Experimental [32]
Present
4.9 2.49 2.56
4.27 2.12 2.12
4.35 2.18 2.16
[29,30]. Furthermore, in this study, additional verifications are given on the wall effects on a settling sphere in CF versus previously reported data [31,32]. The working fluid is CF at 300 K and P = 0.1 MPa with 𝜆 = 5. Quantitative comparison of numerically predicted Cd at different Re are shown in Table 1. As can be seen from Table 1 that the present results are all in the acceptable region between the reference data. As Re increases from 32.4 to 33, there is no change in the experimentaly reported Cd [32]. This may be due to the low sensitivity of the experimental set to the small variation of Re. On the contrary, numerical results of Sharma and Patankar reported an increase of Cd [31] with the increase of Re from 32.4 to 33. This is not consistent with the common knowledge that Cd normally decreases with Re in CF [33,34]. Therefore, it is believed that the present results are accurate and reliable. It is safe to conduct further numerical simulations to discuss the wall effects in SCW.
4.2. Distribution of temperature Fig. 5 shows the distribution of temperature in different cases. In this subsection, Tf is defined as the temperature for the fluidized sphere problem while Ts is for the settling sphere problem. It is clearly seen that both Tf and Ts change more steeply with the increase of the distance to the center of the particle when Re is small. This is due to the fact that the thickness of the boundary layer becomes thinner as Re increases. When comparing the value of Tf − Ts , it becomes smaller when increasing Re or lifting 𝜆. This phenomenon can be explained as that the wall effect weakens as 𝜆 or Re increases, the convective heat transfer between particles and fluids is enhanced making the difference of temperature field under the two boundary conditions smaller. It is worthwhile noting that the value of Tf − Ts reaches its maximum at a certain location, which predicting that the heat transfer is similar under these two boundary conditions.
4. Results and discussions It needs to note that in the following sections, Cdf is defined as the drag coefficient for the fluidized sphere problem while Cds for the settling sphere problem. Nuf and Nus are defined in a similar way. Except for the surrounding walls, other key parameters like Re, Pr, and 𝜆, are the same in each case.
4.3. The wall effects on cd 4.1. Distribution of X-velocity Fig. 6 shows the distributions of Cdf and Cds in different cases. It is clearly seen that both Cdf and Cds increase with the decrease of the wall factor (stronger wall effects). This phenomenon is not difficult to explain since the approaching of the surrounding wall to the particle leads to the
Fig. 3 shows the distribution of X-velocity in different cases. In this subsection, u1 is defined as the X-velocity for the fluidized sphere problem while u2 is for the settling sphere problem. As can be seen from the
Fig. 3. Distribution of X-velocity of a sphere in SCW with stationary walls (top) and movings tationary walls (bottom) for different values of 𝜆 and Re. (From a to b, 𝜆 is 5,10; From 1 to 2, Re is 10,100). 3
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Fig. 4. Distribution of X-velocity of a sphere in SCW at 𝜆 = 5(a) and 𝜆 = 10(b) for different values of Re.
Fig. 5. (a) Distribution of temperature of a sphere in SCW at 𝜆 = 5 (bottom) and 𝜆 = 10 (top) for different values of Re. (b) relative deviation of Temperature.
strong retardation on the fluid field and then increases Cd. The wall effects vanish gradually when 𝜆 gets larger and further increasing 𝜆 will not create influence on Cd any longer when 𝜆 reaches one point. For a fixed value of 𝜆, both Cdf and Cds decrease with Re which is in line with the unconfined cases [33,34]. For a fixed value of 𝜆 and Re, Cdf is larger than Cds which can be attributed to the additional wall dissipation. For the settling sphere problem, there is no relative motion between the wall and fluid and thus the wall retardation effect on the fluid is smaller. It can be seen that Cds are very comparable when 𝜆=20 and 𝜆=40. However, additional dissipation takes place for the fluidized sphere problem due to the strong wall retardation effects caused by the relative motion between the fluid and wall. This fact indicates that a larger calculation domain is needed for the fluidized sphere problem than the settling one to eliminate the wall effect. Fig. 7 shows the distribution of Cdf /Cds of a sphere in SCW for different 𝜆 and Re in which combined effects of wall and Re can be observed.
Cdf /Cds is found to decrease with the increase of Re. This is because the viscosity contribution of fluid drops when lifting Re. Moreover, Cdf /Cds decreases with the increase of 𝜆. The is because the wall effect gets weaker when the wall is located further away. Therefore, the lowest value appears when Re = 200 and 𝜆 = 20. Fig. 8 shows the combined effects of 𝜆 and Re on distribution of pressure coefficient of a sphere in SCW for both fluided problem and settling problem. In this subsection and following ones, Cd1 is defined as the pressure coefficient for the fluidized sphere problem while Cd2 for the settling sphere problem. For a fixed Re and a given 𝜆, both Cd1 and Cd2 decrease from the starting point(X = −R) to the highest point of the particle surface(X = 0), and then start to rise due to the recirculating wake at the rear end of the particle surface. For a fixed 𝜆, in addition to the common decreasing trend of pressure coefficient with Re, it can be also seen that Cd1 is higher than Cd2 from the starting point(X = −R) to point C and opposite trend is found from point C to 4
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Fig. 6. (a) Distribution of Cdf and Cds of a sphere in SCW with moving walls (left half) and stationary walls (right half) for different values of 𝜆 and Re. (b) 𝜆=40 stands for the case when the wall effects vanish[26].
the ending point(X = R). At the same time, as Re increases, point C is closer to the starting point(X = −R). This is because pressure recovery improves more significantly for settling problem as Re increases. 4.4. The wall effects on Nu Fig. 9 shows the distribution of Nu of a sphere in SCW for different 𝜆 and Re. It can be seen that under both cases of the wall boundary conditions, Nu increases with the increase of Re due to the enhanced convection. In addition, more information can be found from Fig. 9. For all the Re and 𝜆, Nuf is higher than Nus . The difference between Nuf and Nus is significant when 𝜆 is small but hardly varies when 𝜆 is larger than 10. This can be due to the fact that the surrounding wall can change the fluid flow around the sphere significantly when 𝜆 is small. Considering the fact that wall effects are stronger in the fluidized sphere problem than the settling one, it can be concluded that Nu increases when the wall retardation effect is stronger. The enhancement of Nu by wall effects can also be observed from the change of Nu with 𝜆. Fig. 9a clearly shows that both Nuf and Nus decrease with the increase of 𝜆. Again, Nus with large 𝜆 is quite comparable with the unconfined particle shown in Fig. 9b. Fig. 10 describes the distributions of surface Nu of a sphere in SCW for different values of 𝜆 and Re. In this subsection and following ones,
Fig. 7. Distribution of Cdf /Cds of a sphere in SCW for different values of 𝜆 and Re.
Fig. 8. Distribution of pressure coefficient of a sphere in SCW with moving walls and stationary walls for different values of Re.(From a to c, Re is 10,50,100). 5
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Fig. 9. (a) Distribution of Nuf and Nus of a sphere in SCW for different values of 𝜆 and Re. (b) 𝜆=40 stands for the case where wall effects vanish[26].
Fig. 10. Distribution of surface Nu of a sphere in SCW with moving walls and stationary walls for different values of Re and𝜆. (From a to c, Re is 10,50,100. From 1 to 3, 𝜆 is 5,7.5,10).
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Nu1 is defined as the surface Nu for the fluidized sphere problem while Nu2 is for the settling sphere problem. For a fixed Re and a given 𝜆, both Nu1 and Nu2 fluctuate at the front slightly (from starting point (X = −R) to point a) and then decrease along the surface of the particle until the flow separation point (point b). For a fixed 𝜆, in addition to the common increasing trend of Nu1 and/or Nu2 with Re, it is worth noting that the separation point moves forward as Re increases. This can be due to the fact that the size of the recirculation vortex increases with an increase in Re [24]. Considering the fact that the separation point is closer to the starting point (X = −R) when 𝜆 is small, it can be concluded that the size of recirculation vortex becomes small with 𝜆 decreases. Regardless of Re and 𝜆, when comparing the position of the flow separation point under two boundary conditions, it is not difficult to find point b is slightly ahead for stationary wall, which implies that the recirculation vortex is larger for fluidized sphere problem under the same Re and 𝜆.
5.2. Fluidized sphere problem Via the regression analysis on the present numerical results, new coefficients in the above equation for Cdf are: {
It can be seen in Fig. 13a that these new coefficients do a good job to predict Cdf with the max and min relative deviations 11.23% and 0.33%, respectively, while the correlation coefficient is 99.82%. The good capability can be also seen from Fig. 14a. Actually, most of the relative deviations are far lower than 10%. Similar with Cdf , the regression analysis recommends below values of coefficients to use and the prediction performance for Nuf is shown in Fig. 13b. { 𝑎1 = −2.10206; 𝑎2 = −2.34165; 𝑎3 = 0.0226; 𝑎4 = 1.1067; 𝑎5 = 6.88881; 𝑎6 = 0.1746; 𝑎7 = −2.2028.
5. New correlations
Using these parameters, corresponding max and min relative deviations are 2.435% and 0.093%, respectively, while the correlation coefficient is 99.944%. The distribution of these relative deviations is shown in Fig. 14b.
5.1. Evaluation on previous correlations Due to the fact that current Cd and Nu share similar tendencies with the numerical results of Kishore and Gu [10,12] on CF though from 2D simulations, it is aimed to improve their formula for SCW. The formulas for Cd and Nu are shown as: ( ) 𝑎 𝜆 + 𝑎 2 𝑎4 𝐶𝑑 = 1 (𝑎 5 + 𝑎 6 𝑅𝑒𝑎7 + 𝑎 8 𝑅𝑒𝑎9 ) (2) 𝑅𝑒 𝜆 + 𝑎3 ( 𝑁𝑢 =
) 𝜆 + 𝑎1 (𝑎 3 𝑅𝑒𝑎4 + 𝑎 5 𝑅𝑒𝑎6 ) + 𝑎 7 𝜆 + 𝑎2
𝑎1 = 50.30808; 𝑎2 = −0.91522; 𝑎3 = −0.78761; 𝑎4 = −10.43884; 𝑎5 = −3.12144; 𝑎6 = 0.00155; 𝑎7 = 1.28438; 𝑎8 = 4.0001; 𝑎9 = 0.0629.
5.3. Settling sphere problem Similar with last subsection, determined coefficients for Cds and Nus are: { 𝑎1 = 8.01219; 𝑎2 = −1.90211; 𝑎3 = −1.83908; 𝑎4 = −7.04622; 𝑎5 = 2.26883; 𝑎6 = 0.08208; 𝑎7 = 0.92626; 𝑎8 = 0.9345; 𝑎9 = 0.33012.
(3)
and { 𝑎1 = −2.6984; 𝑎2 = −2.79934; 𝑎3 = 0.01282; 𝑎4 = 1.23509; 𝑎5 = 5.24864; 𝑎6 = 0.19642; 𝑎7 = −0.90752.
In order to test whether the correlations proposed by Kishore and Gu [10,12] are applicable to this work, we firstly compare the present numerical results with those calculated by the correlations of Kishore and Gu [10,12]. From Fig. 11a, one can find that fluctuated Cdf is observed and Nuf for CF is obviously lower than SCW as shown in Fig. 11b. Great discrepancies between the predicted results based on the correlation for CF and the present numerical results for Cds and Nus also can be seen from Fig. 12, which demonstrates the need for updating the coefficients.
It can be seen in Figs. 15 and 16 that Eqs. (2) and (3) also do well to predict Cds and Nus using these parameters. For Cds , max and min average relative deviations are 4.047% and 0.00346%, respectively, while the correlation coefficient is 99.962%. For Nus , corresponding max and min relative deviations are 2.829% and 0.1152%, respectively, while the correlation coefficient is 99.964%.
Fig. 11. Predicted (a) Cdf and (b) Nuf by the correlation [10,12] with the present numerical results for the fluidized sphere problem. 7
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Fig. 12. Predicted (a) Cds and (b) Nus by the correlation[10,12] with the present numerical results for the settling sphere problem.
Fig. 13. Predicted (a) Cdf and (b) Nuf by the proposed correlation with numerical results.
Fig. 14. Comparison between currently numerical and correlated (a) Cdf and (b) Nuf .
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Engineering Analysis with Boundary Elements 102 (2019) 1–10
Fig. 15. Predicted (a) Cds and (b) Nus by the proposed correlation with numerical results.
Fig. 16. Comparison between currently numerical and correlated (a) Cds and (b) Nus .
6. Concluding remarks
Acknowledgements
A comparative study on cold SCW passing over a hot sphere with two types of spoiler problems is carried out using the PR-DNS. The influences of different boundary conditions are highlighted and Cd and Nu at each working condition are obtained. According to the numerical results, new correlations are established. Main conclusions can be drawn as follows:
The authors sincerely acknowledge the financial supports from the National Key R & D Program of China (2016YFB0600102-4), the NSFC project (51606040) and the Jiangsu Province Science Foundation for Youths (BK20160677) on this research.
References (1) The changing trend of Cd and Nu with 𝜆 and Re is similar under the two types of spoiler problems. Discrepancies on the exact values of these two situations are due to the wall effects. (2) In order to eliminate the wall effects on Cd and Nu, a larger wall factor (𝜆) should be used for the fluidized sphere problem than the settling sphere problem. (3) New correlations of Cd and Nu for a hot sphere in SCW with two kinds of boundaries are established by considering 10 ≤ Re ≤ 200 and 5 ≤𝜆 ≤ 20 as key influencing factors with good accuracy obtained. These correlations could facilitate the phase coupling in multi-phase flow modellings of SCW. The correlations are available for the temperature between 647 K and 657 K at P = 23 MPa.
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