Large eddy simulation of tubular reactors with spherical dimples

Chemical Engineering Journal 380 (2020) 122463

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Large eddy simulation of tubular reactors with spherical dimples Jens N. Dedeyne, David J. Van Cauwenberge, Pieter A. Reyniers, Kevin M. Van Geem , Guy B. Marin ⁎

T

Ghent University, Laboratory for Chemical Technology, Technologiepark 121, 9052 Ghent, Belgium

HIGHLIGHTS

dimples greatly enhance the convective heat transfer in tubes. • Spherical parameters have a noticeable effect on heat transfer and pressure drop. • Geometric of coke formation in steam cracking coils can be reduced. • Rate pressure drop penalty is fully compensated during the run due to slower coking. • The • Run lengths can be increased by up to 40%. ARTICLE INFO

ABSTRACT

Keywords: CFD LES OpenFOAM® Steam cracking Dimples Heat transfer Friction factor

Spherical dimples are implemented in heat exchanger channels to enhance heat transfer at a low pressure drop penalty. This combination of enhanced heat transfer and low pressure drop is also valuable for reactive applications such as steam cracking as this allows to increase product selectivity and reduce fouling. Large eddy simulations (LES) show the presence of asymmetric vortex structures inside the spherical dimples if they are implemented in a tubular reactor for steam cracking. Of all investigated geometric parameters, dimple depth and phase difference between consecutive rows is seen to have the greatest impact on the flow behavior. Spherical dimples in a tube enhance heat transfer up to 43% and increase pressure drop by only 54% relative to a bare tube. Relative to a ribbed tube, spherical dimples in tubes can reduce the pressure drop by half without impacting the heat transfer capabilities of the tube. The impact of these enhancements on product yields and run length is assessed via reactive simulations, benchmarking dimpled reactors against a non-enhanced reactor and industrially applied ribbed reactors. These simulations show the superior performance of dimpled reactors: the reduced fouling allows to increase the run length by up to 40% without a significant impact on product selectivities.

1. Introduction In recent years, interest from the chemical industry in heat transfer enhancement has increased as this allows to achieve higher efficiencies, to improve process economics and to reduce emissions, amongst others of greenhouse gasses [1]. The improvement potential for steam cracking, the dominant process for the production of light olefins such as ethene, propene and 1,3-butadiene, is significant as it is one of the most energy intensive processes in the chemical industry [2]. In this process, hydrocarbons are cracked to form light olefins, aromatic compounds and various other products at temperatures between 900 and 1200 K [2]. At the prevailing temperatures, undesired secondary reactions lead to the formation and deposition of coke on the

⁎

Corresponding author at: Technologiepark 121, B-9052 Zwijnaarde, Belgium. E-mail address: [email protected] (K.M. Van Geem).

https://doi.org/10.1016/j.cej.2019.122463

Available online 09 August 2019 1385-8947/ © 2019 Published by Elsevier B.V.

reactor wall, which decreases heat transfer and induces a higher pressure drop due to the lower cross-sectional surface area for flow. When heat transfer resistance or pressure drop becomes too large, operations are halted and coke is burned off during a decoking procedure. A decrease in the rate of coke formation can therefore increase the time between consecutive decoking operations, thereby increasing the annual production capacity of the plant. Enhanced heat transfer can significantly decrease the reactor inner wall temperature and hence the rate of coke formation, thereby improving the process economics [3]. For heat exchanger applications [4], convective heat transfer enhancement has been accomplished by implementing a variety of three dimensional (3D) elements at the wall or in the core of the flow, e.g. pin fin arrays [5], rib turbulators [6], swirl chambers [1], etcetera. Such

Chemical Engineering Journal 380 (2020) 122463

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Nomenclature

Greek symbols

Roman symbols

α α* β γ ρ τ

cp D e f I L n nspan

Heat capacity at constant pressure (J/kg/K) Tube internal diameter (m) Dimple depth (m) Fanning friction factor (–) Dimensions of a periodically repeating unit (m) Length (m) Wall normal direction (–) Number of dimples in cross section perpendicular to mean flow direction (–) Nusselt number (–) Pressure (Pa) Periodic module length (m) Prandtl number (–) Heat flux (W/m2) Tube internal radius (m) Reynolds number in cylinder with tube internal diameter as characteristic length (–) Reynolds number in square channel with channel height as characteristic length (–) Temperature (K) Time (s) Velocity (m/s) Turbulator width (m) Distance in the mean flow direction (m)

Nu p P Pr q R Re ReH T t u w z

Abbreviations CFD MERT® IHT® SFT TEF SOR

Computational Fluid Dynamics Mixing Element Radiant Tube Increased Heat Transfer Swirl Flow Tube Thermal Enhancement Factor Start-of-run

Sub- and superscripts 0 b i, j max w x τ R ~ +

elements typically increase turbulence and radial mixing of the fluid. A similar use of wall elements can be seen in the design of enhanced steam cracking coils such as mixing-element radiant tube (MERT) by Kubota [7], SCOPE by Schmidt & Clemens [8], Swirl Flow Tube (SFT) by TechnipFMC [1] or Intensified Heat Transfer Technology (IHT) by Lummus and Sinopec [9]. The implementation of these heat transfer enhancing elements induces an additional pressure drop at start-of-run (SOR) conditions, which results in a higher required compression power for heat exchangers to overcome this additional pressure drop. To weigh the benefit of the heat transfer enhancement versus the detrimental effect of the pressure drop penalty, the overall efficiency of an enhanced geometry is assessed with the Thermal Enhancement Factor (TEF), which is defined as the ratio of the Nusselt number ratio to the friction factor ratio:

TEF =

Angle between consecutive dimples (°) Normalized angle between consecutive dimples (–) Pressure gradient (Pa/m) Temperature gradient (K/m) Density (kg/m3) Stress tensor (Pa)

Bare tube value Bulk value Running indexes Maximum value Wall value Local, spatially non-averaged value Turbulent Residual Filtered/time-averaged value Non-dimensional quantity, scaled by the wall variables

exchangers [10]. Next to this, Burgess and Ligrani [11] reported that an array of these cavities typically produces multiple vortex pairs, which can improve heat transfer considerably compared to a bare tube [12]. Burgess and Ligrani [11] noticed that the dimple depth plays an important part in improving heat transfer in a rectangular channel. Heat transfer is better for deeper dimples due to two effects: first the magnitude of turbulence production and transport increases; and second, the strength and intensity of vortices and associated secondary flows ejected from the deeper dimples increase. Next to this, dimple depth has a big impact on drag, which is proportionate to the pressure drop over the dimple. Tay et al. [13] found that increasing dimple depth from 1.5% to 5% of the tube diameter can reduce the skin friction drag but that the increase in depth also results in a flow separation, increasing form drag. As both effects contribute to

( ) Nu Nu0

() f f0

1 3

(1)

In which the Nusselt number ratio is the ratio of the Nusselt number averaged over the entire length for the enhanced design to the averaged Nusselt number for the corresponding non-enhanced design. The friction factor ratio is similarly defined, with the friction factor being the Fanning friction factor, f, which can be related to the pressure drop, Δp, in a tube with radius R over a distance L by:

p=f

L ub2 R

(2)

In the case of steam cracker reactors, the pressure drop penalty also negatively affects the selectivity towards the desired products [6]. Ligrani [4] mentioned that dimpled surfaces, as depicted in Fig. 1, are notable for their low pressure drop penalty, as these elements do not protrude into the flow, which allows application of dimples in heat

Fig. 1. Cut of a dimpled tubular reactor. 2

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the total drag, the relative importance of the two effects will determine the flow behavior. Note that when the Reynolds number increases, the zone of flow separation typically decreases and, hence, will result in a drag reduction, favoring dimples in flows with high Reynolds numbers. Luo et al. [14] found that changing the dimple positions relative to pin fins in a channel can distinctly alter the flow behavior inside the channel and consequently impact the local rate of heat transfer as well as the pressure drop over the channel. These studies show that a wide range of TEFs can be reached with dimpled walls, depending on the selected values for the design parameters such as spatial configuration, depth and width of the dimples. However, the effect of the dimple dimensions or the configuration in a tubular reactor have, to the best of the authors' knowledge, not been studied before. In this work, the effect of dimple width and dimple depth, as well as the phase difference and the dimple positioning is studied with the 3D large eddy simulation (LES) framework developed by Van Cauwenberge et al. [15]. This framework is also used to study the vortex pairs reported by Burgess and Ligrani [11] and Voskoboinick [12] as these vortices and associated ejecting jets can further reduce coke deposition. Finally, the dimpled reactor is benchmarked against industrially applied ribbed reactor designs by means of reactive simulations focusing on product yields, coking rate and run length.

r=

e2 +

w2 4

2e

(3)

where e is the dimple depth and w is a measure for the dimple width. Note that w is the exact dimple width in the streamwise direction but is an approximation of the dimple width in the spanwise direction as the curvature of the cylindrical tube is not considered in this calculation. 2.2. Dimple configuration A number of additional parameters are required to uniquely define dimpled tubes with multiple spherical dimples. First, the amount of dimples in the cross section perpendicular to the mean flow direction is denoted as nspan, and will be referred to as the number of dimples in the spanwise direction. Second, the phase difference, α, is introduced to allow inline configurations shown in Fig. 1, i.e. dimples placed along lines parallel to the tube centerline, as well as staggered configurations shown in Fig. 3, i.e. dimples in row (n + 1) are placed in between two dimples in row n when looking along the centerline axis. Third, the size of a periodically repeating unit, without taking into account any phase difference, is denoted by I. For I/w = 1, the dimples are placed together as closely as possible in the streamwise direction, i.e. there is no distance in between dimples of row n and the dimples of row (n + 1). 2.3. Studied geometries

2. Geometry description

The geometrical parameters of the studied geometries are listed in Table 1. To allow more general conclusions, the parameters are scaled with respect to appropriate parameters such as the tube diameter (D), dimple width (w) and the tube circumference (πD).

2.1. Dimple dimensions A cross section of a simplified dimpled tube consisting of one single dimple is given in Fig. 2. The basis for the dimpled tube is a cylinder with radius R along a central axis perpendicular to the cross section and intersecting the cross section at the point labeled with C. The spherical dimples are formed by spheres with a radius r intersecting the cylindrical tube. The dimensions of the spherical dimple are uniquely defined by two parameters e and w as the radius r of the sphere in Fig. 2 is calculated as

3. CFD model set-up 3.1. Governing equations As in the work of Van Cauwenberge et al. [16], the flow was modeled incompressible and temperature was treated as a passive scalar, i.e. viscous dissipation and buoyancy effects were not taken into account. The filtered governing equations for conservation of mass, momentum and energy are given in Eq. (4), Eq. (5) and Eq. (6) respectively.

Fig. 2. Cross section showing a dimpled tube consisting of a cylindrical tube with radius R and one spherical dimple with radius r. Dimple width w is defined as the dimple width in the streamwise direction, due to the tube curvature an approximation of w is shown, which is valid if R ≫ r.

Fig. 3. Cut of a tubular reactor with dimples in a staggered configuration, showing geometrical parameters w, I and α with nspan = 6. 3

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Table 1 Geometry definitions for the tested cases, providing dimensionless values for the dimple depth, dimple width, the phase difference, streamwise spacing and spanwise spacing respectively. Geometry

e D

0

0

0

0

–

0

E1 E2 E3 E4

0.015 0.03 0.045 0.06

0.15 0.3 0.3 0.3

0.5 0.5 0.5 0.5

1 1 1 1

0.764 0.764 0.764 0.764

W1 W2 W3 W4 W5/A1/C1

0.03 0.03 0.03 0.03 0.03

0.1 0.16 0.2 0.24 0.3

0 0 0 0 0

1 1 1 1 1

0.127 0.204 0.255 0.306 0.382

A2 A3 A4 A5 A6 A7

0.03 0.03 0.03 0.03 0.03 0.03

0.3 0.3 0.3 0.3 0.3 0.3

0.0625 0.125 0.333 0.5 0.666 0.875

1 1 1 1 1 1

0.382 0.382 0.382 0.382 0.382 0.382

C2

0.03

0.3

0

1

0.382

w D

[ ]

. nspan

[ ]

2

[ ]

I w

[ ]

w . nspan .D

=4

[ ]

3.3. Numerical model To solve the governing equations, the open source finite volume CFD framework OpenFOAM 2.2.2 is adopted. The PISO algorithm with 2 corrector steps is used for pressure-velocity coupling. A second order backward differencing scheme is used for time integration while linear interpolation is applied to obtain the cell face values of the Gauss surface integral. Spatial discretization is done with a second order central differencing scheme. The maximum Courant number is kept below 0.8 according to the CFL condition, leading to time steps of the order of magnitude of 10−5 s. Ten flow-through times are simulated to reach fully developed turbulent flow, after which data is collected for 50 flow-through times to reach statistical steady-state. All simulations were parallelized on up to 80 cores on Intel® Xeon® E5-2670 CPUs.

(4)

• Navier-Stokes equation ui uj ui + = t x

p 1 + xi Re

2u

i x 2j

R ij

(5)

xj

• Scalar transport equation t

+

uj xj

=

3.4. Computational grid

2 qj 1 + 2 Re Pr x j xj

The commercial meshing software Pointwise V18.0 [19] is used to create the body-fitted structured grids. A butterfly topology is applied, which permits the use of high-quality hexahedral grids whilst maintaining control over the mesh spacing. This was used to ensure that all meshes allowed wall-resolved simulations, i.e. y+ between 0.6 and 0.8 for all studied cases. The computational domain corresponding to the periodic module for all geometries has a length-to-diameter ratio of 4.8. To determine the minimum grid requirements for grid-independent results, large eddy simulations are performed for the dimpled tube labeled W5/A1/C1 in Table 1, which has 8 spanwise dimples in an inline configuration for which I/w = 1; e/D = 0.03 and w/D = 0.3. Three meshes with increasing degree of refinement labeled ‘Coarse’, ‘Intermediate, ‘Fine’, were used, allowing to assess the impact of grid resolution on important parameters such as Nusselt number, friction factor and wall temperature as shown in Table 2. It is clear from these properties that the coarse mesh does not give grid independent results. The difference between the intermediate and fine mesh is less distinct. A second step, in which refinement is only applied in one direction, i.e. axial, radial or tangential, shows that additional refinement in the tangential direction further reduces the differences between the intermediate and the fine case to less than 0.2%

(6)

where ~ denotes a variable filtered with the OpenFOAM ‘simple’ LES filter, which is a simple top-hat filter for dynamic LES models. The subgrid-scale (SGS) turbulence was modelled with the dynamic kequation eddy-viscosity model [17]. 3.2. Boundary conditions As LES is computationally demanding, the computational domain is shortened by assuming fully developed flow and applying streamwise periodic boundary conditions, implying for any variable that: (7)

(x , y, z + nP , t ) = (x , y, z, t )

for every periodic module length P. For these streamwise periodic boundaries, the velocity and pressure can be written as (8)

u (x , y , z ) = u (x , y, z + nP ) p (x , y , z )

p (x , y , z + nP ) = n [p (x , y , z )

(12)

is the dimensionless temperature gradient in the streamwise direction corresponding to the imposed heat flux and β is the linear component of pressure which is adjusted to maintain a set flow rate, thus compensating for the pressure drop over the periodic module. The flow rate that is set corresponds to a Reynolds number of 11,000 for all studied geometries, i.e. the flow is sufficiently turbulent and altering the Reynolds number will have little effect on global characteristics such as Nusselt number ratio and friction factor ratio. On the fluid-solid interface, a no-slip boundary condition is imposed for the momentum equation, a zero-flux boundary condition for the mass conservation equation and an isoflux heating boundary condition for the energy conservation equation. A bulk temperature of 1000 K was imposed.

• Global continuity equation ui =0 xi

qw cp ub

p (x , y , z + P )]

(9)

However, temperature and pressure are not intrinsically periodic. To overcome this, Patankar et al. [18] proposed to separate both fields into a fluctuating periodic part and a non-periodic part with a constant gradient. This method is adopted by decomposing the pressure and temperature as

p (x , y , z ) = p ( x , y , z )

z

(10)

T (x , y , z ) = T ( x , y , z )

z

(11)

Table 2 Flow field properties for varying degrees of grid refinement.

Number of cells [–] Max y+ [–] Nu [–] f [–] Twall [K]

where 4

Coarse

Intermediate

Tangential refined

Fine

648 000 1.07 42.01 0.01235 1051.48

5 299 200 0.75 48.07 0.01434 1045.03

7 862 400 0.62 49.35 0.01485 1043.60

11 712 000 0.59 49.34 0.01483 1043.64

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for the three observed properties. The intermediate mesh with tangential refinement has approximately 35% cells less than the finer grid, resulting in faster calculation and is selected for all simulations.

4. Results and discussion 4.1. Validation against experimental data To the best of the authors’ knowledge, little experimental data is available for flow in a tube with spherical dimples. Therefore, the flow experiments and numerical results of Turnow et al. [20] in a square channel with a dimpled wall at a Reynolds number ReH = 13,042 are used as a validation case. The Laser Doppler Velocimetry (LDV) data provide information on both first order statistics and second order statistics for the three velocity components, in other words, mean and rms data is available for the axial, radial and azimuthal velocity components. Information on velocity fluctuations is important as these are crucial in the prediction of heat transfer from the wall to the center of the flow. In these experiments, the channel height is 15 mm with rows of dimples with a width of 23 mm and a depth of 6 mm, placed in a staggered arrangement. A sufficient number of dimple rows were used in the streamwise and spanwise direction to ensure fully developed flow, allowing to simulate the flow on a small domain with periodic boundary conditions in the streamwise and spanwise direction. The grid is refined similar to the refinement used by Turnow, with particular attention to the refinement at the wall to guarantee the y+ < 1 condition for the first grid point. The mean and rms profiles of the velocity from the present LES are compared in Fig. 4 to the LES and LDV profiles obtained by Turnow et al. [20]. Similar to Turnow, the simulated mean velocity magnitude near the walls is slightly higher than the experimental results. Overall, the agreement between the experimental and numerical results is satisfactory, especially in the dimpled bottom half of the channel, i.e. y/ H < 0.9. The recirculation height, i.e. the y/H-value at which the flow direction changes, is captured well. The simulated second order statistics match the experimental data well. The time-averaged flow structures inside the central dimple are visualized in Fig. 5a, showing a recirculation zone in the upstream part of the dimple and two ejecting streams inclined to the mean flow direction at an angle of approximately 45°. The notably high velocity close to the wall positively affects the heat transfer throughout the channel as it decreases the boundary layer thickness. While the time-averaged flow shows two symmetric downstream jets, a different behavior is seen for the instantaneous velocity structures depicted in Fig. 5b, where one ejecting jet is dominant over the other. It was found that the directional dominance of the vortex alternates over time.

3.5. Data reduction It was previously mentioned that the overall efficiency of an enhanced geometry can be assessed by means of the TEF, which is a combination of the Nusselt number ratio and the friction factor ratio. These ratios are calculated based on time-averaged simulation results. For all simulations, the local Nusselt number is calculated as

Nu x =

Tx n w Tw, x Tb D

(13)

To allow comparison between different designs, a spatially averaged Nusselt number is calculated as

Nu =

A

Nu x dA A

dA

(14)

The spatially averaged Nusselt number for the bare tube, Nu0, was calculated with this methodology as well rather than calculated by the use of correlations. The Fanning friction factor was calculated by the use of Eq. (2), where the pressure drop is the pressure drop over the entire simulation domain.

4.2. Frequency analysis of asymmetric vortex structures in a dimpled channel

Fig. 4. Mean (full lines) and rms (dashed lines) velocity in the averaged flow direction across the channel at ReH = 13,042. y/H = 0.0 corresponds to the reactor wall.

Voskoboinick [12] et al. reported the formation of asymmetric

Fig. 5. Time-averaged (a) and instantaneous (b) flow structures inside a dimple in a square channel. 5

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Fig. 6. Instantaneous streamwise velocity at two probe positions lying symmetrically opposite of the flow axis as a function of time (a) and corresponding spectrum for both points (b).

vortex structures inside spherical dimples in a narrow channel, a phenomenon also investigated by Isaev and Turnow [21,22]. To focus on the phenomenon, LES of flow in a narrow channel with a single dimple is performed in which the dimple width and height is fixed at 16 mm and 20 mm respectively. The channel height is fixed at 15 mm and a fully developed flow at a Reynolds number of 13,000 is imposed as inlet conditions. The instantaneous streamwise velocity is measured downstream of the dimple at two points lying symmetrically opposite of the flow axis, the result of which is shown in Fig. 6a. The measurements indeed indicate that the flow oscillates as high velocities at one probe position are generally accompanied by lower velocities at the other probe position. The frequency spectrum of these oscillations was obtained by the Fast Fourier Transform routine in MATLAB R2015a, after subtracting the time-averaged velocity from the instantaneous velocity data. The correlation of the velocity at the two probe positions is further illustrated by this spectrum as both signals show a high normalized amplitude at a frequency of around 2.4 Hz.

This corresponds to a complete cycle of around 0.4 s, which corresponds to the oscillation frequency seen in the results in the time domain. An identical procedure is applied to flow in a narrow channel with a dimple depth ranging from 6 mm to 12 mm to assess the impact of the dimple depth on this oscillatory behavior. The frequency spectra of these cases are summarized in Fig. 7, together with the spectrum of the previously discussed dimple with a depth of 20 mm. For cases with shallow dimple, no peak frequency is found, whereas with increasing dimple depth, the oscillatory behavior is more pronounced. This corresponds closely to the findings of Turnow [23] who reported that shallow dimples do not allow formation of vortices and their associated ejecting jets. Additionally, the location of the maximum in the frequency domain shifts to higher frequencies with increasing dimple depth, i.e. the deeper the dimple, the faster these oscillations occur.

Fig. 7. Spectra for flow in a narrow channel with a dimple depth of 6 mm (a), 10 mm (b), 12 mm (c) and 20 mm (d).

6

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as the ratio of the averaged Nusselt number of the enhanced tube relative to the Nusselt number of an equivalent bare tube, of 1.44 was obtained. This is an averaged value for the entire tube, as large local variations are found as shown in Fig. 9. The upstream section of the dimple is characterized by a low heat transfer coefficient. In a small region close to the upstream rim, the heat transfer rate is lower compared to that in a non-enhanced geometry. The downstream section of the dimple on the other hand has a high Nusselt number ratio as the fluid impinges on the downstream wall and is swirling at high velocities in this region. The highest heat transfer can be found at the downstream rim and downstream of the dimple as the flow is ejected out of the dimple into the bulk of the flow, efficiently transferring the heat from the wall to the bulk. This agrees with a prior study of dimpled surfaces by Rao et al. [24]. Implementing spherical dimples inevitably increases the pressure drop over the tube. This increased pressure drop is reported as an averaged Fanning friction factor (f) relative to the Fanning friction factor for a bare tube (f0) with the same diameter. For this particular case, it is found that f/f0 = 1.897.

Fig. 8. Time-averaged (a) and instantaneous (b) streamlines inside a dimple for geometry E2.

4.3. Effect of flow structures on heat transfer in dimpled tubes To illustrate the typical flow field in a dimpled tube, a single representative case is discussed. The design of this reference case, summarized as E2 in Table 1, 8 spanwise dimples with w/D of 0.3 and e/D of 0.03 were placed in a staggered configuration. A Reynolds number of 11,000 was imposed as well as a bulk temperature of 1000 K. The time-averaged local flow field depicted in Fig. 8 a shows similar vortex structures as in the narrow channel validation case. The fluid motion is symmetric in nature as flow enters the cavity at the center line of the cavity, impinges on the downstream dimple wall and breaks down into two vortices swirling in the tangential direction. The vortices are ejected in two separate jets at the downstream rim of the dimple on opposite sides of the dimple center line. The symmetry observed in the time-averaged flow field is no longer present in the instantaneous flow field, depicted in Fig. 8b. Instead, one of the vortices dominates, favoring the jet on one side of the center line over the other. This closely corresponds to the asymmetric vortex structure with a periodically changing orientation over time in a single dimple in square channels. For the considered reference case, a Nusselt number ratio, defined

4.4. Impact of geometry on dimple efficiency 4.4.1. Dimple depth The influence of dimple depth on heat transfer was studied extensively for flat dimpled plates by Burgess and Ligrani [11], showing that both heat transfer and pressure drop increase with increasing dimple depth. This was attributed to the increase in size of the vortex structures inside the dimple. To investigate the influence of this parameter on heat transfer and pressure drop in pipe flow, the dimple depth e/D is varied between 1.5% and 6.0% of the tube diameter which should contain the transition region in which the oscillatory behavior inside the dimples will start to occur. The parameters of interest, i.e. Nusselt number ratio and friction factor ratio are shown in Fig. 10, as well as the thermal enhancement factor. It can be seen that the effect of dimple height on the TEF is most distinct in the region between e/D = 0.015 and e/D = 0.03. To explain this behavior, similar to Fig. 4, the normalized timeaveraged streamwise velocity is plotted along a line between the tube axis and the dimple center in Fig. 11. It can be seen that the mean streamwise velocity inside the dimple, i.e. r/R > 1, does not become negative in the case of the shallow dimple, i.e. no vortices or recirculation zones are formed. This corresponds to the observations that shallow dimples in square channels do not have oscillating ejecting jets or swirling vortices, see Fig. 7. As these vortices and their associated ejecting jets are the major driving force for

Fig. 9. Distribution of the time-averaged Nusselt number ratio (Nu/Nu0) for the spherical dimple in a tube. White contour lines differentiate between heat transfer enhancement and heat transfer diminution relative to a bare tube, i.e. Nu/Nu0 = 1. General flow direction from bottom to top.

Fig. 10. Influence of dimple depth on Nusselt number ratio, friction factor ratio and thermal enhancement factor for Re = 11,000. Results for designs E1-E4 and 0. 7

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Fig. 13. Influence of angle between consecutive dimples on Nusselt number, Fanning friction factor and thermal enhancement factor for Re = 11,000. Results for designs A1-A7.

Fig. 11. Normalized time-averaged streamwise velocity as a function of radial coordinate for geometries E1 (blue dashed line) and E2 (yellow full line).

spanwise dimples was fixed at 4. An increase in both heat transfer and friction factor is noticed when consecutive dimples are placed closely behind each other, i.e. when the normalized angle α* is close to 0 or 1. Note that as the geometry used in these simulations consists of 4 dimples in the spanwise direction, the geometry for α* = 1 is identical to the geometry for α* = 0 and hence the same simulation results are plotted in Fig. 13 for these two configurations. For intermediate results, separate simulations were performed for configurations with α* and (1 − α*), i.e. the symmetry as found in Fig. 13 was not assumed a priori. The inline configuration leads to the highest heat transfer and pressure drop, with a remarkable decrease even for small deviations from the inline placement. A minimum, for heat transfer, pressure drop and TEF, is found for a perfectly staggered configuration similar to the configuration shown in Fig. 3.

enhanced heat transfer, low Nusselt numbers are found for this dimple depth. A minimum dimple depth, higher than 0.015 times the diameter, is required for the formation of vortices and consequently the superior heat transfer capability of dimpled reactors. 4.4.2. Dimple width To assess the influence of the dimple width, the parameter is varied between 10 and 30% of the tube diameter for a e/D of 0.03. In Fig. 12 it can be seen that an increase in dimple width is associated with an increase in both Nusselt number and pressure drop. The TEF achieves a maximum value for w/D = 0.20, after which the increase in friction factor ratio outweighs the increase in Nusselt number ratio. The effect of the dimple width is much less pronounced than the effect of the dimple depth, indicating that the vortex structures inside the dimples are mainly influenced by the dimple depth.

4.5.2. Uniformly distributed dimples compared to clustered dimple geometry The positioning of the dimples within a single row can affect performance as well. Two geometries are selected in which four dimples are placed in a single row and consecutive rows are placed in an inline configuration. The four dimples are distributed uniformly around the circumference, i.e. the angle between two dimples in a row is 90° for geometry C1, as shown in Fig. 14 (left). In geometry C2, Fig. 14 (right), the dimples are clustered at the top of the tube, leaving the bottom half without enhancements. Even though the number and dimensions of the dimples in both designs are the same, a distinct difference in performance is found. Not only is the Nusselt number ratio for geometry C2 higher than for geometry C1, i.e. 1.42 vs. 1.35, its friction factor ratio is 2% lower than for geometry C1. Overall, the TEF of geometry C1 is 1.16 while the TEF for geometry C2 is 1.23. It is clear that the relative positioning of dimples is a crucial factor to achieve an optimal performance. To better understand this remarkable difference, the time-averaged temperature difference between the wall and the bulk for geometry C2 is shown in Fig. 15, where it is normalized against the spatially and time-averaged temperature difference for a cylindrical tube under the same operating conditions. The tube wall is split into three zones. The region where dimples are placed are labeled as ‘Zone A’ in Fig. 15 and shows a similar behavior as dimpled regions in other designs, i.e. a slightly higher wall temperature in the upstream section of the dimple and a lower wall temperature throughout the rest of the domain. One could expect that the bottom half of the tube would behave similarly to a non-enhanced tube, however two distinct zones can be identified. Zone B is a transition zone, where no enhancements are located but wall temperatures remain

4.5. Impact of relative positioning on dimple efficiency 4.5.1. Phase difference between rows The asymmetric vortex structures inside dimples lead to ejecting jets which severely affect the region downstream of the spherical dimple. It is therefore of importance to investigate the influence of the position of downstream dimples as these can interact with the ejecting jets. The angle between a dimple and the downstream dimple, α, is varied between 0° and 90°, both inline configurations as the number of

Fig. 12. Influence of dimple width on Nusselt number, Fanning friction factor and thermal enhancement factor for Re = 11,000. Results for designs W1-W5 and 0. 8

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Fig. 14. Geometry with uniformly distributed dimples (geometry C1 – left) and with clustered dimples (geometry C2 – right).

below the wall temperatures for a bare tube. Zone C is characterized by wall temperatures well above the average wall temperature of a bare tube exposed to the same conditions. The existence of these zones is explained by visualizing the in-plane mean velocity for a cross section of the tube perpendicular to the axis of the tube as done in Fig. 16 (right). It is seen that non-negligible secondary flow structures develop. These flow patterns, typically referred to as secondary flows of Prandtl’s second kind [25], allow intense mixing throughout the entire cross section of the tube. The existence of these secondary flow structures are attributed to an imbalance in Reynolds stress generating vorticity. In between these two swirling zones, a zone with lower velocities is located which explains the existence of zone C in Fig. 15. This implies the lower heat transfer rate from the wall to the fluid leading to a higher wall temperature in case of an isoflux boundary condition. Close investigation of geometry C1 shows that this design also generates secondary flow structures albeit more localized due to interference with the secondary flows created by neighboring dimples. These secondary structures do not reach the center of the tube. For geometry C1, eight zones with secondary flow can be visualized, implying that each dimple generates two separate vortices, as Ligrani et al. [26] previously reported. The recirculation zones are much smaller as the swirls collide onto each other, leading to a lower heat transfer and an increased pressure drop. To fully exploit the potential of these secondary flows, these flows should be unrestricted and the lower velocity region should be kept as small as possible.

4.6. Fouling and selectivity benchmarking To assess the impact of the dimples on product yields, the 1D simulation software COILSIM1D version 3.7 was used to simulate propane cracking in an industrial Millisecond reactor. The reactor, with a length of 11 m and an internal diameter of 0.0302 m is fed 156 kg per hour of propane and 50.856 kg steam per hour at a temperature of 630 °C. The pressure at the reactor outlet was fixed at 1.7 atm, while the heat flux profile specified by Reyniers et al. [27] for the same reactor was applied and scaled to obtain a propane conversion of 80.6%. The coking rate was tuned so that the predicted run length for a bare tube corresponds to the run length predicted by Vandewalle et al. [28] who accounted for the dynamic behavior of coke growth by means of 3D CFD simulations. To assess the run length, a maximum outer wall temperature of 1100 °C and a maximum pressure drop over the coil of 2 atm were specified as constraints. The effect of coil enhancements can be considered in the 1D software by implementing correction factors for convection and friction factor, i.e. the Nu/Nu0 and f/f0 values that were previously obtained via LES. Two dimpled reactors are benchmarked against a non-enhanced tube and against two generations of ribbed tubes: a continuous helical rib (Ribbed tube) and a discontinuous helical rib (Improved ribbed tube), for which the correction factors were reported by Györffy et al. [29]. Note that these correction factors are kept constant throughout the run although these will vary due to local coking differences. For example, in ribbed tubes, it was shown by Vandewalle et al. [28] that coking mostly occurs downstream of the rib, implying a decrease in

Fig. 15. Mean temperature difference between wall and bulk normalized with corresponding temperature difference for bare tube. Mean flow direction from left to right. 9

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Fig. 16. Contours of time-averaged in-plane velocity magnitude for geometry C1 (left) and C2 (right) for Re = 11,000. White lines visualize the streamlines projected onto the cross-sectional surface.

Wall temperature profiles for all enhanced geometries are very comparable at start-of-run (SOR) conditions as they have similar heat transfer correction factors. As a consequence, the coking rate profile (Fig. 17b) is similar for the enhanced geometries. Due to the increased convection coefficient relative to the bare coil, a comparable heat flux on a comparable surface area leads to lower wall temperatures compared to a bare tube. Due to the fact that the heat flux is scaled to reach equal conversion, only small differences are found for the process gas temperature in Fig. 17d. However, the pressure drop over the reactor is quite different due to the implemented correction factors. Typically, the enhanced designs have a higher P/E ratio, i.e. the ratio of the propylene to the ethylene yield on a weight basis [30], and a measure for the cracking severity. This is because the improved heat transfer increases the selectivity towards propene, whilst the selectivity

Table 3 Correction factors for various reactor designs.

Bare tube Ribbed tube Improved ribbed tube Averaged Dimpled tube Clustered dimpled tube

Nu/Nu0

f/f0

1.00 1.40 1.40 1.42 1.42

1.00 3.00 2.10 1.74 1.54

friction factor throughout the run with a limited impact on the heat transfer capabilities of the ribs. The selected correction factors for all these designs are summarized in Table 3.

Fig. 17. External wall temperature (a), coking rate profile (b), pressure (c) and gas temperature as a function of the axial position for a Millisecond propane steam cracking reactor with different enhancements. 10

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consequence of the higher pressure drop. Due to the lower pressure drop penalty for the dimpled designs relative to other enhancements, the loss of olefin selectivity is less severe compared to the other designs. Furthermore, as the pressure drop increases slower throughout the run, the loss of selectivity relative to the bare tube decreases with increasing on stream time. Note that the 1D simulation procedure is not capable of capturing all 3D effects such as improved radial mixing, which makes process gas temperature more uniform and can improve light olefin selectivity. Another effect that is not taken into account is the effect of the oscillating jets inside dimples, these introduce high velocities and local shear rates near the wall, which can hinder deposition of coke and consequently increase the run length. The effect of higher local temperatures, e.g. in the upstream section of a dimple, and hence local differences in coking rate are not taken into account either with this simulation method. Consequently, run length estimations should be considered with care.

Table 4 Start-of-run reactor conditions, product yields and selectivities for different reactor configurations. Bare

Ribbed

Improved Ribbed

Uniform Dimples

Clustered Dimples

1176

1171

1173

1174

1175

57.5 0.60

135.8 0.62

104.0 0.61

89.4 0.61

81.8 0.61

1.54 18.13 0.40 32.79 2.99 0.95 19.69 19.41 1.01 4.11

1.48 18.49 0.32 31.99 3.05 0.94 19.77 19.41 1.08 4.56

1.50 18.30 0.34 32.28 2.99 0.97 19.76 19.41 1.06 4.45

1.51 18.22 0.36 32.41 2.98 0.97 19.76 19.45 1.04 4.35

1.52 18.27 0.37 32.56 3.00 0.94 19.71 19.37 1.04 4.25

Product selectivities [%] H2 1.91 22.49 CH4 C2H2 0.49 40.69 C2H4 C2H6 3.71 C3H4 1.18 24.43 C3H6 1,3-C4H6 1.25

1.83 22.94 0.40 39.69 3.79 1.16 24.53 1.35

1.86 22.71 0.43 40.06 3.71 1.20 24.52 1.31

1.88 22.62 0.45 40.24 3.70 1.20 24.52 1.29

1.89 22.66 0.46 40.39 3.72 1.17 24.44 1.28

Coil Outlet Temperature [K] Pressure drop [kPa] P/E ratio [wt%/wt %] Product yields [wt%] H2 CH4 C2H2 C2H4 C2H6 C3H4 C3H6 C3H8 1,3-C4H6 C4+

5. Conclusions Tubular steam cracking reactors with spherical dimples are superior to classically used bare reactors. The primary reason for enhanced heat transfer is the formation of an asymmetric vortex structure in nonshallow dimples. These vortex structures lead to the ejection of highvelocity jets of which the direction changes periodically over time. It was found that the oscillatory behavior depends on geometrical parameters, in particular dimple depth. Furthermore these geometrical parameters have a distinct impact on the overall reactor performance. A sensitivity analysis shows that for tubular reactors a minimum dimple depth is required to form vortex structures and consequently enhance heat transfer. Increasing the dimple depth further results in an unfavorable pressure drop penalty. A similar, yet less pronounced, effect was found for the dimple width. Next to the dimple dimensions, the relative positioning of the dimples has an important impact on the overall performance as an inline configuration gives better heat transfer and a higher thermal enhancement factor. It was found that a clustered circumferential distribution of the dimples can produce large secondary flow structures which further increase the performance of the dimpled reactor tubes. Simulations showed that enhanced reactors typically have a lower ethene selectivity but a higher propene selectivity at start-of-run conditions but this effect was the least pronounced for the dimpled tubes. Spherical dimples can increase the run length of an industrial steam cracking reactor by up to 40% compared to a non-enhanced reactor showing the vast potential of dimpled reactors in the field of steam cracking.

towards ethene decreases as can be seen in Table 4. Due to the decreased radial temperature gradient, the overall cracking severity is lower. Although the differences at SOR conditions are small, they become more clear throughout the run. As can be seen in Fig. 18, the initial pressure drop penalty of the dimpled designs is completely compensated after 6 to 7 days by the decreased rate of coke deposition. After this point, both TMT and pressure drop are lower for the dimpled tubes. Assuming a maximum allowable TMT of 1100 °C, Fig. 18a demonstrates that the bare tube surpasses this temperature after 10 days onstream, effectively limiting the run at 10 days. All enhanced geometries exhibit much lower TMTs at the same on-stream time and runs are not temperature limited but rather pressure limited. It can be seen in Fig. 18b that for an assumed maximum allowable pressure drop of 2 bar, end of run for the clustered dimple design is reached after approximately 14 days, leading to an increase in run length of up to 40%. The yield of valuable light olefins, defined as the sum of ethene, propene and 1,3-butadiene, throughout the run is shown in Fig. 19. A small loss of light olefin yield is predicted for the enhanced designs, as a

Fig. 18. Evolution of maximum tube metal temperature (TMT) (a) and pressure drop over the reactor (b) as a function of time on stream for several reactor enhancements. 11

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Fig. 19. Light olefin yield as a function of time on stream for several reactor enhancements.

Acknowledgments JND gratefully acknowledges financial support from the Fund for Scientific Research Flanders (FWO), in association with the Pi-Flow project and from Ghent University (Belgium) through GOA project BOF16/GOA/004. The research leading to these results has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme/ERC grant agreement n° 818607. The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation-Flanders (FWO) and the Flemish Government – department EWI. The authors would also like to acknowledge the resources provided by STEVIN Supercomputer Infrastructure at Ghent University. References [1] C.M. Schietekat, M.W.M. van Goethem, K.M. Van Geem, G.B. Marin, Swirl flow tube reactor technology: an experimental and computational fluid dynamics study, Chem. Eng. J. 238 (2014) 56–65. [2] I. Amghizar, L.A. Vandewalle, K.M. Van Geem, G.B. Marin, New trends in olefin production, Engineering 3 (2017) 171–178. [3] L.A. Vandewalle, D.J. Van Cauwenberge, J.N. Dedeyne, K.M. Van Geem, G.B. Marin, Dynamic simulation of fouling in steam cracking reactors using CFD, Chem. Eng. J. 329 (2017) 77–87. [4] P. Ligrani, Heat transfer augmentation technologies for internal cooling of turbine components of gas turbine engines, Int. J. Rotating Mach. 2013 (2013). [5] P. Promonge, Thermal augmentation in circular tube with twisted tape and wire coil turbulators, Energy Convers. Manage. 49 (2008) 2949–2955. [6] C.M. Schietekat, D.J. Van Cauwenberge, K.M. Van Geem, G.B. Marin, Computational fluid dynamics-based design of finned steam cracking reactors, AIChE J. 60 (2014) 794–808.

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