## Abstract

The number of marine scrubbers installed in industry has been on the rise over the past decade and is expected to continue in the coming years. Therefore, it is essential to ensure that the design of the scrubbers enables as an efficient operation as possible. In this study, an optimization of the exhaust cover inside an in-line scrubber was carried out. The optimization was done by combining a computational fluid dynamics model working on a simplified geometry with the method of feasible directions in order to reduce the pressure loss caused by the exhaust cover. The design is constrained in both height and width of the points making up the exhaust cover to ensure proper drainage of water and to avoid invalid designs. It was found that the optimized design reduced the pressure loss by 42% compared to the initial design. Furthermore, the scalability of the original design was investigated with the same height constraint enforced on the design variables. The result of the scalability analysis showed that the radius of the exhaust cover for the optimal designs scales linearly with the diameter of the scrubber, while the pressure loss was found to increase quadratically as the diameter of the scrubber increases.

## Introduction

With the commencement of the global maritime sulfur cap in January 2020, vessels are required to reduce their emission of sulfur, from the combustion of heavy fuel oil, through a reduction of the sulfur content in the heavy fuel oil used from 3.5% to 0.5% by mass. There are mainly two methods by which the sulfur emissions can be reduced. One is by switching to a compliant low sulfur fuel oil and the second is postcombustion cleaning of the exhaust gas by means of a scrubber [1].

Generally, wet scrubbers are used aboard vessels where either plain seawater or desalinated water with additives is used to absorb the sulfur from the exhaust gas. The primary function of a scrubber is to create a large surface area between the water and the exhaust gas whereby the sulfur can be absorbed by the water and this can be achieved either by atomizing the water, or by means of a packing material [2,3].

Two types of scrubber designs are primarily used for marine exhaust gas cleaning. One is a U-type scrubber, where one leg is the scrubber tower and the other is the exhaust gas inlet. The other is an in-line scrubber where the exhaust gas inlet is placed in the bottom of the scrubber tower. Because a marine scrubber is typically installed toward the top of the stack, it is crucial that the water does not propagate backwards in the exhaust system, as it can result in damage to the engine or other equipment. In a U-type scrubber, the exhaust gas enters the scrubber tower from the side, and water can be drained from the bottom of the scrubber tower, thereby circumventing the issue with backflow into the remaining exhaust gas system. This is however not the case for the in-line type where both the water outlet and exhaust inlet are placed at the bottom. To counteract water backflow an exhaust cover is installed above the inlet. Installing any equipment in the exhaust system is subject to stringent restrictions on the allowable pressure loss. If the resistance to flow in the exhaust system is too excessive it will limit the performance of the engine.

Earlier studies of scrubbers have generally been centered around describing global performance criteria such as sulfur removal efficiency, gas–liquid interaction, and heat transfer, which are essential parameters in the scrubber [46]. Optimization studies of scrubbers have been performed in regards to scrubbers, where especially the design of venturi scrubbers has been investigated. In these studies, the parameters have been the pressure loss, the removal efficiency, and the liquid distribution at the throat [7,8]. Only a single study has been found where the authors worked on optimizing the dimensions to comply with the tight quarters often encountered aboard marine vessels, while at the same time achieving a sulfur removal rate that ensures compliance with the global sulfur cap [9].

In this work, an optimization of the exhaust cover geometry is conducted in order to reduce the pressure loss in a packed bed in-line scrubber.

## Scrubber Design

A general overview of an in-line scrubber with its geometry and internals is shown in Fig. 1. The exhaust gas enters the scrubber at the inlet which is placed above the bottom of the scrubber to allow for the accumulation of water before it is drained. The exhaust cover is placed immediately after the inlet, and the exhaust gas is forced to make a 180 deg turn to exit under the lip of the exhaust cover. To avoid backflow of the water, the lip of the exhaust cover is placed below the inlet. When the exhaust gas has passed the exhaust cover it enters the packed bed. The packed bed is sprayed with water from the nozzles above and the water absorbs the sulfur compounds from the exhaust gas and trickles down to the bottom of the scrubber. After the exhaust gas has passed through the packed bed and past the nozzles, any droplets still entrained in the flow are removed by the demister arrangement just before it exits the scrubber at the top.

Fig. 1
Fig. 1
Close modal

### Superficial Exhaust Gas Velocity.

A driving design parameter of packed bed scrubbers is the superficial exhaust gas velocity through the packed bed. The geometry of packing and the liquid load on the packed bed will determine the maximum allowable superficial velocity. In Fig. 2 the difference in specific pressure loss for two types of metal random packing, HY-PAK-2 and IMTP-60, are shown [10]. The two types of packing have approximately the same nominal size, specific surface area, and void faction. However, as it can be seen from Fig. 2 the resultant specific pressure loss differs substantially for both dry and wet conditions.

Fig. 2
Fig. 2
Close modal

In Fig. 2 it can be seen that as more water is introduced to the packing the specific pressure loss increases. At a given liquid load and superficial velocity, the slope of the specific pressure loss curve starts to increase. This is because the packed bed is approaching the flood point, where the amount of liquid in the packing becomes large enough for the gas to start bubbling through the packed bed causing a significant increase in pressure loss [11]. The scrubber investigated in this study has a packed bed consisting of IMTP-60. From Fig. 2, it can be seen that a maximum allowable superficial velocity of 2 m/s will cover a liquid load ranging from 0 to 98 $m3m2h$ without causing flooding of the packed bed.

### Scrubber Dimensions.

When dimensioning a scrubber to a specific exhaust gas load, the diameter of the tower is the main parameter of interest, as it is directly related to the superficial gas velocity. The dimensions for the scrubber tower used in this study are given in Table 1.

Table 1

Dimensions of the scrubber casing

 rscr 1.00 m rin 0.50 m hscr 6.00 m hin 1.00 m
 rscr 1.00 m rin 0.50 m hscr 6.00 m hin 1.00 m

The points of the casing where the dimensions are used are shown in Fig. 1.

## Computational Methodology

The flow inside the scrubber is a complex two-phase flow where the exhaust gas and water interact throughout most of the scrubber.

However, as the area beneath the exhaust cover is shielded from the flow of water coming from above and it is where the majority of the interaction between the exhaust cover and exhaust gas occurs, the influence of the water can be neglected and the flow can be treated as a single phase flow. Furthermore, the shielding of the exhaust gas from the water means that the exhaust gas temperature is constant beneath the exhaust cover, thus the exhaust gas is assumed to be isothermal. As the temperature is constant and the velocity of the exhaust gas is relatively low the flow can be treated as being incompressible [12]. Considering the nature if traveling by sea, vessels often operate at constant speed for extended periods of time and this means that the exhaust gas load experienced by the scrubber is also constant for the majority of its operating hours. Therefore the flow is considered as being steady-state.

### Governing Equations.

The continuity and momentum equations for an incompressible turbulent flow are given as [12]
$∇(u)=0$
(1)
$ρ ∇(u)=−∇p+∇(μ Δ(u))+τij$
(2)

where $u$ is the velocity vector, ρ is the density, p is the pressure, μ is the dynamic viscosity and τij is the turbulent stress tensor.

The turbulence model used in this work is the $kω$-SST model which has transport equations given by the following [13]
$ρ ∇(ku)=∇((μ+σkμt)∇(k))+Pk−Dk$
(3)
$ρ ∇(ωu)=∇((μ+σωμt)∇(ω))+Pω−Dω$
(4)

where k is the turbulent kinetic energy, μt is the turbulent dynamic viscosity, ω is the specific turbulent dissipation rate, σk and $σω$ are a model constants, Pk and $Pω$ is the production of k and ω, and Dk and $Dω$ destruction of k and ω

#### Boundary Conditions.

The boundary conditions at the inlet for the turbulence model are derived based on the assumption of a fully developed pipe flow. Where k is estimated from the turbulence intensity I and the average velocity of exhaust gas at the inlet Uin, as follows [14]
$k=32(UinI)2I=0.16(ReDh)−1/8$
(5)
The turbulence intensity I, is based on the Reynolds number $ReDh$ where the characteristic length is the hydraulic diameter, Dh, of the inlet pipe. The specific turbulent dissipation rate ω is estimated from the turbulent length scale $ℓ$ and the model constant $Cμ=0.09$ and is given as follows [14]
$ω=kCμ1/4ℓℓ=0.07Dh$
(6)
The wall treatment used for the turbulence model is based on a smooth blending between the viscous sublayer and the log layer for the specific dissipation rate ω and the shear velocity $uτ$, which can be given as [15]
$ω=ω log 2+ωvis2$
(7)
$uτ=(uτ, log 4+uτ,vis4)1/4$
(8)
Finally, the velocity at the inlet has to be determined. As previously discussed the maximum superficial velocity through the scrubber is 2 m/s. With the assumption of isothermal and single-phase flow, some considerations regarding the velocity can be made. Typically the exhaust gas temperature at the scrubber inlet is approximately 250 °C. But when the gas enters the packed bed it has been cooled by the water to somewhere between 30 °C and 60 °C depending on the liquid load. Assuming that the gas is cooled to 45 °C before entering the bed, the density of the exhaust gas will be 1.1 kg/m3, whereas it will be 0.675 kg/m3 at the inlet. To account for the change in density, the volumetric flowrate of the exhaust gas through the scrubber is determined at the packed bed and then scaled according to the change in density. This can be expressed as
$Uin=AbedUbedρbedρinAin$
(9)

Where Uin and Ubed are the average velocities at the inlet and bed, Ain and Abed are the cross-sectional area at the inlet and bed and, ρin and ρbed are the density of the exhaust gas at the inlet and in the bed.

All other properties of the exhaust gas are taken at 250 °C. The remaining boundary conditions for the model, and the ones presented above, are summarized in Table 2.

Table 2

Boundary conditions for all patches and variables

BoundaryUpkω
InletEq. (9)ZGEq. (5)Eq. (6)
OutletZGFVZGZG
WallsNSZGZGEq. (7)
BoundaryUpkω
InletEq. (9)ZGEq. (5)Eq. (6)
OutletZGFVZGZG
WallsNSZGZGEq. (7)

ZG: zero gradient, FV: fixed value, and NS: no slip.

#### Numerical Setup.

The SIMPLEC algorithm is used to solve the pressure–velocity coupling, where all transport equations are Discretized using the second-order linear upwind scheme [12,16]. The convergence criteria for the model is that the initial residual falls below $10−3$ for the pressure and below $10−4$ for the remaining variables.

### Computational Mesh.

A predominantly hexagonal mesh of the full scrubber is generated using CF-mesh [17] and is shown in Fig. 3.

Fig. 3
Fig. 3
Close modal

It can be seen that the demister, nozzles, and packed bed shown in Fig. 1 are not represented in the mesh. This is to reduce the complexity of the mesh and thereby the amount of cells required to resolve these features. The simplification can be justified as these internal do not interfere with the flow characteristics near the exhaust cover. The local cell size is determined by a refinement level, which specifies the number of divisions on the edges of a cell with a global cell size. The refinement levels increase closer to the exhaust cover to ensure that the flow around the lip is properly captured. The same applies to the walls of the scrubber, where the exhaust gas is expected to flow along the wall. On all wall patches, a constant thickness boundary layer is specified to ensure that the wall functions, shown in Eqs. (7) and (8), are operating within the log-law region [15,18]. This means that y+ should be less than 500, to ensure that the wall treatment is valid [19]. In this work, the value of $y+$ is at maximum 68 on the scrubber walls and 29 on the exhaust cover while the average values are 21 and 8, respectively. Resolving the boundary layers down to the viscous sublayer to capture the flow separation that occurs after the lip of the exhaust cover and the scrubber wall where the flow adheres after the exhaust cover is beyond the scope of this work as the interest lies in the macroscopic flow structures inside the scrubber.

#### Mesh Independence.

A mesh independence test was performed using the grid convergence index (GCI) which was first introduced for structured meshes and later extended to unstructured meshes [20,21]. The GCI is based on the refinement ratio between successively finer meshes and a solution parameter which is often chosen as the main parameter in the study. The refinement ratio between two consecutive meshes ψij of size ni and nj can be expressed as [21]
$ψij=(njni)1/3 for j>i$
(10)
A global order of accuracy, P, is estimated from a solution parameter $ϕ$, which in this case is the total pressure loss over the domain. The solution parameter has to be calculated for three progressively refined meshes, to calculate the global order of accuracy which is given as
$P=1ln(ψ21)|ln(|ϕ3−ϕ2ϕ2−ϕ1|)+q(P)|q(P)=ln(ψ21P−sψ32P−s)s=sign(ϕ3−ϕ2ϕ2−ϕ1)$
(11)
Finally the solution parameter for a mesh with an infinite number of cells, $ϕext21$, is estimated using Richardson extrapolation
$ϕext21=ψ21Pϕ1−ϕ2ψ21P−1$
(12)
With the extrapolated estimate for the solution parameter the actual error $ea21$, and extrapolation error $eext21$, along with the GCI can be calculated as
$ea21=|ϕ1−ϕ2ϕ1|$
(13)
$eext21=|ϕext−ϕ1ϕext|$
(14)
$GCIfine21=1.25ea21ψ21P−1$
(15)

To perform the GCI analysis, three meshes of different sizes are required as can be seen from Eq. (11). It is suggested that the refinement ratio, ψij is greater than 1.3 between successive mesh sizes [21]. Therefore, the global mesh size of the three meshes is set such that this criteria is met. The resulting mesh size for the coarse, medium, and fine mesh are $6.6×105, 1.6×106$, and $3.9×106$, respectively. From the GCI analysis, it was found that the apparent order of accuracy P is 4.2 and the extrapolated value of the total pressure loss is 489 Pa which corresponds to $eext21=0.2%$ while the actual error $ea21=0.6%$. The pressure losses for the different mesh sizes and the result of the GCI analysis are presented in Fig. 4.

Fig. 4
Fig. 4
Close modal

It can be seen in Fig. 4 that a mesh of $3.9×106$ cells with the same refinement areas and levels is adequate to ensure a mesh-independent solution as the $GCIfine21$ is 0.19%.

## Optimization

Setting up the optimization routine requires special attention to be addressed to both the CFD simulations and the definition of design variables to ensure that the routine is robust and sufficiently fast.

### Geometry Simplification.

It was found from the GCI analysis that a mesh for a full-sized scrubber requires a mesh of $3.9×106$ cells to result in an independent solution. Evaluating a solution on a mesh with $3.9×106$ cells at each design point will require a significant amount of computational resources and time. Therefore it is investigated if a simplified domain can produce results that approximate those on the full 3D domain.

The axial symmetry of the scrubber can be used to create an axisymmetric mesh and thereby reduce the dimensions of the domain from 3D to 2D. An axial boundary condition is applied to the faces of the mesh that borders the symmetry planes. Since a new boundary condition is imposed, a comparison between the 2D approximation and full-scale solution has to be made to see if the approximation has any influence on the results.

To analyze possible differences between the full 3D model and the 2D axisymmetric model, the 3D geometry is first cut by 4 planes separated by 45 deg. Each plane is then split at the center axis to generate 8 sample planes spanning the geometry (see Fig. 5(a)).

Fig. 5
Fig. 5
Close modal

A uniformly spaced grid of sample points is placed on each sample plane such that they have the same position in each of the planes. A sketch showing a coarse version of the sample point distribution used on the sample panes can be seen in Fig. 5(b). The samples taken from the 3D model are averaged over the 8 sample planes and compared to the 2D axisymmetric solution. A contour overlay of the pressure and the velocity magnitude from the 3D averages and the 2D axisymmetric simulation are shown in Fig. 6. Here the 3D averages are shown as a filled contour while the result of the 2D axisymmetric is presented as a line contour. The contours are created with the same levels, meaning that the lines should coincide with the boundaries between the shaded levels, if the solutions were identical.

Fig. 6
Fig. 6
Close modal

It can be seen from the contour overlays that there are minor discrepancies in both the pressure and velocity magnitude. In the area under the lip of the exhaust cover, which is enlarged at the top of the contours in Fig. 6, the variation seen in the pressure and velocity magnitude is caused by a difference in the width of the predicted flow path. Here the 3D averaged solution predicts a wider path with lower velocity compared to the 2D axisymmetric solution. In the area above exhaust cover it can be seen that the difference in velocity increases as the flow approaches the outlet. This is caused by minor asymmetries in the flow in the 3D model which impacts the averaged values shown in the contours. This can also be seen in the pressure where there is a mismatch in the overlay near the vertical center of scrubber. Even with the discrepancies mentioned, the total pressure loss over the domain only varies by 0.3% between the 2D and 3D model, which is close to being insignificant. Combining this with the fact that the 2D axisymmetric model captures the main flow characteristics of the 3D model without significant differences and that the resources saved on a single evaluation are significantly reduced, as the mesh size is reduced by approximately two orders of magnitude, enables the conclusion to be drawn that the 2D axisymmetric model is suitable for optimization purposes.

### Optimization Geometry.

The geometry used for the optimization of the exhaust cover is created as a parametric model where it is generated from a series of points in a plane that are then revolved around the center axis. The resultant wedge is used for the axisymmetric model simulations. Each point $pi*$ is defined by a radius r and a height h which are measured from the origin shown in Fig. 1. The casing of the scrubber is set as fixed points that will not be modified during the optimization as these make up the walls, inlet, and outlet of the scrubber.

The points $p0*$$p2*$, shown in Fig. 7, are the points that make up the exhaust cover and it is these that will be modified during the optimization.

Fig. 7
Fig. 7
Close modal

To account for the material thickness of the exhaust cover, the points $p0*$$p2*$ are shifted accordingly. Therefore, it is only necessary to account for the points on one side of the exhaust cover as the others will follow.

#### Optimization.

The optimization method used to reduce the pressure loss caused by the exhaust cover is the method of feasible directions [22]. This method is implemented in the software package dakota which is coupled with the CFD-library OpenFOAM in order to perform the optimization [23,24]. dakota is a general toolkit containing various methods for optimization, parametrisation, and sensitivity analysis. This means that the coupling between dakota and OpenFOAM is handled through a scripted routine that dakota calls at each function evaluation.

The method of feasible directions is a gradient-based nonlinear optimization algorithm that can handle bounded and constrained optimization problems. The method of feasible directions uses the gradient of the cost function and step length determined by a one-dimensional line search to improve the guess for the design point $x$ at each iteration. If the new design point violates any of the imposed constraints or bounds the search direction is modified so that the new point is strictly feasible [25].

The gradient of the cost function is calculated from a finite difference approximation where the step size is determined by scaling each of the design variables with a factor ε. The gradient for the cost function, f, at a given point $x$ is given as
$∂f(x)∂x=f(x·(1+ε))−f(x)x·ε$
(16)

The value of ε used for the optimization in this study is $5×10−3$.

The cost function for design evaluations is created from the total pressure loss, $Δpt$, over the computational domain, which is taken as the difference between the area averaged value of the inlet and outlet. The cost function can be expressed as
$f(x)=1Ain∑k=1nface,inAkpt,k−1Aout∑k=1nface,outAkpt,k$
(17)

A convergence criterion is set, such that the optimization procedure ends when a relative change between two successive iterations is less than 0.1%.

To accelerate the evaluation of the gradient, a speculative gradient is implemented through dakota. Here the gradient of the cost function is evaluated for the current the design point concurrently to the evaluation of the design point itself.

The optimization procedure can be summarized as shown in Fig. 8.

Fig. 8
Fig. 8
Close modal

### Constraints.

During the optimization procedure it is important to ensure that an invalid geometry is not created as it can result in an optimum design which is not feasible. Therefore, constraints are imposed on the optimization problem. Many of the infeasible designs can be handled by defining the design variables on a closed interval. This is also done to account for the packed bed, which as shown in Fig. 1, is placed right above the exhaust cover. If the exhaust cover is placed too far above the inlet it will cause the height of the scrubber to increase, as the space for the packed bed has to be shifted upwards in the scrubber. This increases the amount of materials for construction and the cost of the scrubber which is undesirable. The maximum distance that the exhaust cover is allowed to be above the inlet is 0.5 m in this study. As mentioned the lip of the exhaust cover, point $p2*$, has to be below the inlet to ensure that water cannot flow back into the exhaust system. The distance at which $p2*$ must be below the inlet is set 0.05 m in this study. Another important consideration is that water must not be able to accumulate on top of the exhaust cover. This can be enforced by making each successive height $hi*$ of the points $pi*$ smaller than the one before it. The constraints used for the optimization routine are given in Eqs. (18)(22)
$hin≤h0*≤1.5$
(18)
$r0*≤r1*≤r2*$
(19)
$h0*≤h1*≤h2*$
(20)
$r1*≤r2*≤rscr$
(21)
$h2*≤hin−0.05$
(22)

The area around the inlet does however require special attention as it cannot be handled by intervals alone. Here there is a possibility that the exhaust cover could intersect the inlet if the points making up the exhaust cover are placed in the right areas and avoid these two constraints are added to the procedure. The constraints are shown in Figs. 9(a) and 9(b).

Fig. 9
Fig. 9
Close modal
The first constraint, shown in Fig. 9(a), is a distance constraint which states that the line drawn between two points in the exhaust cover must be at least a set distance d away from the edge of the inlet. The distance constraint can be formulated as
$d≥|(h2*−h1*)rin−(r2*−r1*)hin+r2*h1*−h2*r1*|(h2*−h1*)2+(r2*−r1*)2$
(23)
The second constraint, shown in Fig. 9(b), is a “safe zone” constraint that ensures that the points making up the exhaust cover are not within this safe zone. The safe zone constraint can be defined as:
$h1*≥hsz if r1*≤rsz$
(24)
$r2*≥rsz if h2*≤hsz$
(25)

The distance d in Eq. (23) used in this study is $d=0.1m$ and the safe zone in Eqs. (24) and (25) is defined as $hsz=hin+0.1m$ and $rsz=rin+0.1m$.

## Results and Discussion

The results for the optimization routine are presented in Fig. 10. It can be seen that the optimization routine converges after 21 iterations, where the initial design has a pressure loss of 488 Pa and the optimized design results in a pressure loss of 283 Pa which corresponds to a 42% reduction. Furthermore, it can be seen that the constraints for $h0*$ and $h2*$ given in Eqs. (18) and (22) are active as both have reached the upper limit of the interval on which these have been defined. Finally, the constraint for $r1*$ and $r2*$ which is defined in both Eqs. (19) and (21) can also be observed to be active several times during the optimization routine and also in the final optimized design.

Fig. 10
Fig. 10
Close modal

Comparing the initial and optimized designs, it can be seen from the velocity contours in Fig. 11 that a general reduction of the velocity magnitude is achieved. It is especially under the lip of the exhaust cover and along-side the outer wall of the scrubber where a reduction of 24% and 18%, respectively, is seen. This reduction can be attributed to the enlarged volume above the inlet which increases the width of the flow path under the exhaust cover lip from 0.11 m to 0.15 m. Furthermore, it can be observed that the recirculating zone on the outside of the exhaust cover has reduced in size in optimized design, however, there is not much difference in the flow path despite this reduction.

Fig. 11
Fig. 11
Close modal

### Scalability.

It is investigated if and how the design of the exhaust cover will change with the radius of the scrubber. If all dimensions of the scrubber casing and exhaust cover scale linearly there will not be a noticeable difference in the design. However, it is often the diameter of the scrubber that is changed if its capacity has to increase while keeping the height constant. To test if the design is applicable for larger diameter scrubbers the optimization is run for three different cases where the radius of the exhaust cover and the scrubber is scaled by 1.25, 1.5, and 1,75. The height, h, of the of the points $p0*$ to $p2*$ are limited in the same interval as before, but the radii, r, is scaled with the scrubber. All optimisations are initialized with the scaled result of the original optimization.

In Table 3 the coordinates for the optimized scaled exhaust covers are presented.

Table 3

Exhaust cover point coordinates for the initial simulation

Scale1.001.251.501.75
$h0*$1.501.501.501.50
$r1*$0.750.941.111.27
$h1*$1.501.501.501.50
$r2*$0.760.941.111.27
$h2*$0.950.950.950.95
$Δp$284303334365
Scale1.001.251.501.75
$h0*$1.501.501.501.50
$r1*$0.750.941.111.27
$h1*$1.501.501.501.50
$r2*$0.760.941.111.27
$h2*$0.950.950.950.95
$Δp$284303334365
It can seen that the constraints for $h0*, h1*$, and $h2*$ are active for all the scaled optimization runs and with the radii $r1*$ and $r2*$ being essentially equal. Generalizing this an expression for the radii and pressure loss based on the scaling factor with the radial base given in Table 1 can be determined
$r*=0.68s+0.085$
(26)
$Δp=47.8s2−22.1s+257.7$
(27)

where s is the scaling applied to the scrubber radii. These expressions are within ±1% of the scaled optimization runs.

## Conclusion

A suitable method for optimizing an exhaust cover in an in-line scrubber has been presented. By reducing the dimensionality of computational grid a sufficiently fast and accurate CFD model suitable for optimization runs was developed. The method of feasible directions was coupled with the CFD model, and constraints for the geometric parameters of the exhaust cover were imposed on the optimization routine. It was found that the optimized design of the exhaust cover resulted in a reduction of the pressure loss from 488 Pa in the initial design to 283 Pa in the optimized design. Furthermore, it was studied how the exhaust cover design varied with scrubber tower diameter while under the same height constraints as in the original optimization case. Here it was found that the radial component of the points of the exhaust cover followed a linear trend based on the scaling factor whilst the pressure loss increased quadratically.

## Funding Data

• Innovation Fund Denmark (8053-00053B; Funder ID: 10.13039/100012774).

## Nomenclature

• A =

area $(m2)$

•
• $Cμ$ =

empirical model constant

•
• Dh =

hydraulic diameter (m)

•
• Dk =

destruction of k $(kg m−1 s−3)$

•
• $Dω$ =

destruction of ω $(kg m−3 s−2)$

•
• d =

distance (m)

•
• e =

error

•
• f =

cost function

•
• GCI =

grid convergence index

•
• I =

turbulence intensity

•
• k =

turbulent kinetic energy $(m2 s−2)$

•
• $ℓ$ =

turbulent mixing length (m)

•
• n =

number of cells

•
• p =

pressure (Pa)

•
• pt =

total pressure (Pa)

•
• P =

apparent order or accuracy

•
• Pk =

production of k $(kg m−1 s−3)$

•
• $Pω$ =

production of ω $(kg m−3 s−2)$

•
• $ReDh$ =

Reynolds number

•
• r =

•
• s =

scale

•
• U =

velocity $(m s−1)$

•
• $u$ =

velocity vector $(m s−1)$

•
• $uτ$ =

shear velocity $(m2 s−1)$

•
• $x$ =

design variable vector

### Greek Symbols

Greek Symbols

• ε =

relative step size

•
• μ =

dynamic viscosity (Pa s)

•
• μt =

turbulent dynamic viscosity (Pa s)

•
• ρ =

density $(kg m−3)$

•
• τij =

turbulent stress tensor $(N m−3)$

•
• $ϕ$ =

solution parameter

•
• ψ =

mesh density ratio

•
• σ =

turbulence model constant

•
• ω =

specific turbulent dissipation rate $(s−1)$

### Subscripts

Subscripts

• a =

actual

•
• bed =

packed bed

•
• ext =

extrapolated

•
• in =

inlet

•
• log =

log-layer

•
• out =

outlet

•
• vis =

viscous-layer

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