55

ISME Journal of Thermofluids

Vol. 03, No.1 , pp. 55-67

Optimization of U-bend using Shape Function, Search Method and CFD K. Srinivasan1, V. Balamurugan1and S. Jayanti2*

1Centre for Engineering Analysis and Design, Combat Vehicles R&D Establishment Avadi, Chennai-600054, India

2 Department of Chemical Engineering, Indian Institute of Technology Madras

Chennai-600036, India.

*Corresponding author: sjayanti@iitm.ac.in

Abstract In the present work, a methodology has been presented for the automatic design of U-bend for minimum pressure loss using CFD based shape optimization. The U-tube is completely defined in two-dimensional plane if the radius of curvature and width are specified at every angle from 0o to 180o.A plot of radius and width of the U-tube is a curve in radius-width plane. This is represented using a cubic Bézier curve with two movable control points whose abscissa and ordinates are the decision variables of optimization thereby reducing the complex problem to a simple four variable optimization problem. The constrained optimization problem with minimization of pressure loss as the objective function is solved using the optimization framework by combining the flow solver and optimization solver. Numerical simulation is carried out using ANSYS FLUENT and optimization is done using Box complex method and modified Box complex method in MATLAB.The results show that a pressure drop reduction of 23% is achieved when compared to a semi-circular bend with reduced computational effort of around 30% using modified Box complex method. Keywords: Shape optimization, CFD, U-bend, Box complex method, Bézier curve 1. Introduction

Optimal shape design problems have been a topic of research for a quite long time motivated by their applications and have been investigated by several authors. Presently, optimization has become an inherent step of the design process in many of the engineering fields. There has been a tremendous interest in shape optimization in the recent years, and in particular shape optimization of curved ducts has attracted many researchers due to their ubiquitous presence in e.g. pipe fittings, air intakes, diffusers, turbine and blade passages etc [1]–[2].

Owing to the advent of affordable computers with increased speed and memory, and also with the research and code developments in computational fluid dynamics (CFD), the implementation of CFD codes in optimization is now possible. An excellent overview of CFD based optimization for a variety of applications is given in [3]. Most of the earlier applications of CFD based optimization were on aerospace and recently they are being widely used in other fields [4]–[6].

Flow patterns in curved tubes is quite different from flow in a straight tube, the former having secondary motion in the plane perpendicular to the axial direction. Flow in a bend is subjected to inertial forces and centrifugal forces due to which a complex swirling flow is developed in the bend portion. This leads to high velocity zones upstream of an inner bend portion, and low or negative velocity zones just downstream of inner bend. Significant changes in the flow near the outer bend may also occur. The flow field within the central region of the duct is disturbed considerably and a recirculating flow pattern is established with two or more counter-rotating vortices. Flow separation occurs downstream and the flow disruption caused by a bend persists for several diameters downstream. The recirculating separated flow will lead to huge pressure losses in the bend, and also velocity stratification downstream of the bend [7]–[10].

While a bend thus has deleterious effects on the flow distribution, it is an essential part of a ducting lay-out in a typical power or process plant. Experimental and numerical investigations have been carried out in the past on the curved ducts with rectangular and circular cross-sections. The subject of shape optimization is so

56

matured that an entire book is devoted for this subject [11]. Out of the many curved duct geometries studied in the past, the U-bend has caught the attention of many

researchers due to significant pressure losses. Attempts were made by varying the channel depth and width, clearance height at the tip of the turn and corner fillet radius of a U-bend with rectangular cross-section [12]. The effect of divider thickness in a 1800 bend is profound on the flow features inside and immediately after the turn [13]. Conventionally, baffles or guide vanes are used to streamline the flow. One study shows that the turning vanes have tremendous influence on pressure loss and heat transfer and could reduce the pressure drop with the same heat transfer level whereas inappropriate design of guiding vanes could even lead to increased pressure losses and reduced heat transfer [14]. But this approach may not be suitable for ducts which handle particulate matters because of problems associated with erosion of ducts and guide vanes. Therefore, it is imperative to obtain an optimal geometry without housing any internal baffles. Also, the recent papers on automated CFD based shape optimization of U-shaped ducts for reduced distortion coefficient call for variation in cross-sectional area of the duct where the duct cross-section is initially increased followed by a reduction[15]. The U-bend with varying cross-sectional area has been studied both numerically and experimentally in [16]–[17] by posing it as an optimization problem with 26 parameters. Though wide choices of geometry can be tested for optimality, the number of optimization variables is quite high and this made the CFD-based shape optimization computationally expensive and this called for the use of meta-models and accurate meta-models are to be developed with sufficient accuracy by performing large number of initial CFD simulations.

Notwithstanding the number of approaches for a systematic shape optimization of flow through curved ducts likeblack-box algorithms including a multi-objective optimizer, adjoint methods for the calculation of sensitivities and topology optimization for the optimal exploitation of available design space [18], they are not readily applicable to process or power plant ducting where the duct sizes are large and flow is turbulent. In addition to this, CFD simulations are computationally expensive due to the large number of grid points and as well as due to the iterative nature of the optimization algorithm. In CFD based shape optimization problems, the design variables of optimization are the variables that characterize the duct geometry. As the number of parameters that characterize the duct geometry increases, the number of iterations required to explore the entire parameter space for optimization increases. Thus, it is essential to represent the duct geometry with as few design variables as possible without changing the cross-sectional area of the duct and using guide vanes.

Geometry parameterization is a decisive step in conjunction with CFD based optimization. In order to keep the cost of optimization down, it is necessary to minimize the number of decision variables. But at the same time, it is necessary to have a sufficiently wide choice of shapes to enable an optimal solution to emerge. A survey of the shape parameterization techniques and shape deformations are outlined in [19]–[20].

The objective of the present work is to explore the possibility of using the shape function and the search method together with CFD to develop a robust framework for the design of U-bend with minimum pressure loss. In the present work, we report on the results obtained by using the Bézier curve [21] as the shape function and the Box method [22] as the search method.

2. Bézier curves Bézier curves are named after their inventor, Dr. Pierre Bézier, who set out in the early 1960’s to develop a

curvature formulation for use in shape design [21]. They are represented by a set of points known as “control points”. The control over the variation of the shape of the curve is carried out by moving the control points. In the present context, the control points are the design variables.

A cubic Bézier curve is shown in Fig. 1. A degree n Bézier curve has n+1 control points. A Bézier curve possesses several interesting properties out of which some are discussed here.

The curve passes through the first and the last control point and is tangent to the control polygon at those end points. Thus, with reference to Fig. 1, line segments PoP1 and P2P3 are tangents to the curve. The curve lies within the convex hull of the control points.

The equation of a Bézier curve is given by

ni

i

i

n

i tPtBtP0

1,0

(1)

iinn

i ttiin

ntB

1

!!

!

(2)

57

Fig. 1. Cubic Bézier curve with four control points

3. Proposed methodology

A schematic of the U-bend is shown in Fig. 2. In a plane, the geometry of the U-bend is completely defined if the radius of the centreline 'R' and the corresponding duct width 'W' are specified at all the angles varying from 0o to 180o. By specifying the range in which the radius of the centreline and the width of the duct vary, it is possible to generate various shapes. At θ = 0o and θ = 180o, the radius and width of the inlet and outlet ducts are known, as shown in Fig.2. If the radius of the centreline and the duct width are plotted at every angle varying from 0o to 180o, two curves will be traced. The two curves can be combined by making the angle as the parameter with abscissa as the radius of the centreline and ordinate as the width of the duct. Such a curve will be closed curve if the inlet and outlet ducts have the same width and the same centreline radius. Symmetry is neither assumed nor enforced between the left and right half of the U-bend. Typical curves are shown in Fig. 3 and the terminal values of the duct width and the centreline radius are shown in Table 1. The curve will be closed if the inlet and outlet ducts have same dimensions (Fig.3a) otherwise it will be an open curve as shown in Fig. 3b. The present method can be applied even when the inlet and the outlet ducts have different dimensions. The angles are normalized in such a way that it corresponds to the Bézier parameter 't' varying from 0 to 1. For streamlined flow, one may expect the curve to be smooth and that it can be represented by a Bézier curve of small number of degrees of freedom.

Fig. 2 Schematic of the U-bend.

R(θ)

θ = 0 θ = 180

W(θ)

RL

RR

WL WR

58

(a) (b)

Fig. 3 Schematic of the curve when the inlet and outlet ducts (a) have same dimensions (b) have different

dimensions The problem of shape optimization of U-bend is now reduced to determining the shape of the closed curve

of centreline radius vs. duct width. Obviously the curve is restricted to exist within the box defined by the range of the centreline radius and width of the ducts. Since the range of the parameters is not wide, we assume that the shape of the closed curve can be represented by a single Bézier curve defined by two control points. The abscissa and ordinate of these two control points is now the design variables of optimization. The Box complex search algorithm should attempt to determine the optimal locations of these control points that produce a U-bend geometry that has a lower reduced pressure drop compared to the base case geometry.

Table 1 Parameter Values at the Inlet and Outlet Legs of the U-bend

θ R W Bézier parameters

t = θ/180 0 Radius of the

left leg (RL) Width of the left leg (WL)

0

180 Radius of the right leg (RR)

Width of the left leg (WR)

1

Once the closed curve is constructed from its control points, the geometry can be built from the details of

the closed curve. Each value of the Bézier parameter ‘t’ uniquely corresponds to one angle and as ‘t’ varies from 0 to 1, ‘θ’ varies from 0o (inlet duct) to 180o (outlet duct). Since the values of radius of curvature and width of the duct are known at every angle, the U-bend can be unambiguously created.

4. Details of Geometry and Numerical Simulation In the present study, a base case semi-circular U-bend with centreline radius of 0.25 m and duct width of

0.05 m throughout is considered. The inlet leg and the outlet legs also have the same duct width as that of the U-bend and are separated by a distance of 10W. Flow development length 5W, where W is the width of the duct, is provided at both the inlet and the outlet. The two-dimensional numerical simulations are carried out in ANSYS FLUENT using realizable k-ε turbulence model. Uniform inlet velocity of 15 m/s is specified as the inlet boundary condition, no-slip boundary condition at the wall and the outlet boundary condition is outflow. A relative convergence residual of 10-5 is set for the continuity. The fluid medium is air under isothermal condition. The base case geometry (semicircular) with approximately 65000 cells is created and numerically simulated to determine the pressure drop across the entire duct. The pressure drop is defined as the difference between the mass-weighted-average pressure at the inlet and outlet and is estimated to be around 52 Pa.This geometry is now optimized by varying the centreline radius and the width of the U-duct for the reduced pressure drop.

(RL = RR)

(WL = WR)

Parameter of

the curve is

angle ‘θ’

Radius of the

centreline

Width of

the duct

RL

WL

Parameter of

the curve is

angle ‘θ’

Radius of the

centreline

Width of

the duct

RR

WR

59

5. Search for an Optimal Shape

a. Box Complex Method Box's complex method [22], a direct search method for solving optimization problems having inequality

constraints, begins with a number of feasible (i.e., those satisfying the constraints) points (usually twice the

number of decision variables) generated at random. In an N-dimensional space, these feasible solutions

constitute a simplex which is a figure having 2N vertices connected by straight lines and bounded by

polygonal faces. Once a set of feasible points is found, the worst point is reflected about the centroid of the

rest of the points to find a new point. Depending upon the feasibility and the function value of the new point,

the point is further modified or accepted. If the new point falls outside the variable boundaries, the point is

modified to fall on the violated boundary. If the function value of the reflected point is greater than that of the

worst of the current solutions, then the reflected point is retracted towards the already existing feasible

points. The worst point in the simplex is replaced by this new feasible point and the algorithm continues for

the next iteration. The iterative procedure of reflection and retraction is repeated until all the points approach

the same location which indicates the shrinkage of the simplex to the optimum point. An application of the

method for the optimal positioning of guide vanes in a manifold is discussed in [23]. The description of the

algorithm for the Box complex method is shown in Table 2.

Table 2 Algorithm for the Box Complex Method [24]

1. Define bounds on the decision variables x (x(L), x(U)), reflection parameter α and termination

parameter ε.

2. (a) Generate an initial set of P (usually 2N) feasible points.

(b) Evaluate f(x(p)) for p = 1, 2, 3,......,P

3. Reflection step

(a) Select xR such that f(xR) = max f(x(p)) = Fmax

(b) Calculate the centroid �̅� (of points except xR) and the new pointxm = �̅� + α(�̅� - xR)

(c) If xm is feasible and f(xm) ≥ Fmax, retract half the distance to the centroid �̅�. Continue

until

f(xm) <Fmax.

Else if xm is feasible and f(xm) <Fmax, go to step 5.

If xm is infeasible then go to step 4.

4. Check for feasibility of the solution.

If 𝑥𝑖𝑚 < 𝑥𝑖

(𝐿) set 𝑥𝑖

𝑚 = 𝑥𝑖(𝐿)

If 𝑥𝑖𝑚 > 𝑥𝑖

(𝑈) set 𝑥𝑖

𝑚 = 𝑥𝑖(𝑈)

5. Replace xR by xm. Check for termination.

(a) Calculate 𝑓̅ = 1

𝑃∑ 𝑓(𝑥(𝑝))

𝑝=𝑃𝑝=1

(b) If √∑ (𝑓(𝑥(𝑝)) − 𝑓̅)𝑝=𝑃𝑝=1

2 ≤ 𝜀 then terminate.

Else set k = k + 1 and go to step 3(a).

b. Shape Optimization using Box Complex Method The Box complex method is applied for the search of the optimal shape. The objective function is the

minimization of the pressure drop across the inlet and outlet of the U-bend. As the coordinates of the two

control points that define the closed curve are the optimization design variables, the number of optimization

variables is four. Hence the Box complex method starts with eight initial guesses as the first simplex. This

simplex is modified subsequently by the optimization algorithm till it shrinks sufficiently. At any instant of

optimization, the simplex will have eight elements that will be refined further.

The shapes explored by the optimization method before arriving at the optimal shape are shown in

Figures 4a, 4b, 4c, 4d and 4e along with their pressure drop. Figure 4a consists of eight plots of quite different

geometries that show the initial guesses i.e., geometries at simplex 1. The explored geometries at ensuing

iterations are shown in Figures 4b, 4c, 4d and 4e for simplex numbers 9, 17, 33 and 45 (simplex 45 is the final

simplex as the optimization is converged), respectively. It can be said from these plots that sufficient

searching of the shapes has been made by the optimization method before convergence.

60

Figure 5 shows the reduction in the mean value of the objective function i.e. mean pressure drop during the course of optimization. The variation of the optimization decision variables, namely the locations of the two control points defined by their 'x' and 'y' coordinates, is shown in Fig. 6 for control point 1 and in Fig. 7 for control point 2. The nomenclature, namely x-coordinate and y-coordinate, is deliberately used as these points do not have any physical significance except to define the Bézier curve which describes the centreline radius and width variation of the duct. In contrast, the curve obtained by using these control points has a physical interpretation of defining centreline radius and duct width.

As is the case with any optimization routine, the variation in the initial stages is rather chaotic but it soon settles down and the two control points exhibit the same behaviour. For the sake of convenience, the data presented in Figs. 6 and 7 are represented in Fig. 8 with iteration number as the parameter. It can be seen as the optimization routine converges, the points become clustered around the final values. The pressure drop of the optimal U-bend is around 40 Pa. There is a pressure drop reduction of 23% when compared to the baseline geometry. The optimal geometry does not show any symmetry and favours a width increase initially which then decreases to match the outlet duct width.

Fig. 4a Initial guess geometries of the U-bend with the pressure drop in Pa (Simplex 1).

Figure 4b Geometries at Simplex 9 with the pressure drop in Pa.

49.3 53.4 47.2 53.1

55.4 47.3 51. 1 54.1

86.1 53.4 47.2 53.2

68.6 68.3 63.1 54.1

61

Figure 4c Geometries at Simplex 17 with the pressure drop in Pa.

Figure 4d Geometries at Simplex 33with the pressure drop in Pa.

Figure 4e Geometries at Simplex 45 (Final simplex) with the pressure drop in Pa.

49.4 42.6 47.2 44.9

48.4 47.3 49.8 43.9

43.1 42.6 43.8 40.5

42.8 43.1 43.7 43.9

40.9 40.2 40.9 40.5

40.3 40.5 40.4 40.3

62

Figure 5 Variation of mean pressure drop in Pascal during the search process

(Base case pressure drop: 52 Pa).

Fig. 6 Evolution of x-coordinate (left) and y-coordinate of the control point 1 (right).

Fig. 7 Evolution of x-coordinate (left) and y-coordinate of the control point 2 (right).

0 5 10 15 20 25 30 35 40 4540

45

50

55

60

65

Iteration number

Mea

n p

ress

ure

dro

p (

Pa)

0 20 40 600.12

0.14

0.16

0.18

0.2

0.22

Iteration number

Contr

ol

poin

t 1 -

x-c

oord

inat

e

0 20 40 600.08

0.1

0.12

0.14

0.16

0.18

Iteration number

Contr

ol

poin

t 1 -

y-c

oord

inat

e

0 20 40 600.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

Iteration number

Contr

ol

poin

t 2 -

x-c

oord

inat

e

0 20 40 600.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Iteration number

Contr

ol

poin

t 2 -

y-c

oord

inat

e

63

Fig. 8 Coordinates of the control point 1 (left) and control point 2 (right).

Based on the optimal locations of the control points, the variation of the centreline radius and the width of

the duct of the optimal U-bend for angles varying between 0o to 180o are shown in Fig. 9. The curve is a

closed curve and if we go clockwise, we can see the reduction in the centreline radius but increase in the

width of the duct. The net effect is the increase in the cross-sectional area followed by the reduction. This type

of shape has been reported in literature for a U-bend with circular cross-section for considerable reduction in

distortion coefficient [15] for both laminar and turbulent flows.

Fig. 9 Variation of centreline radius and duct width of the optimal U-bend along the duct axis.

The velocity contours and the pressure contours for some of the non-optimal bends and as well as for the

optimal U-bend are shown in Figs.10 and 11. The pressure drop in a bend is governed by both the frictional

pressure drop and the pressure drop due to turning of the flow. These depend on the flow velocity and bend

loss coefficient, the loss coefficient depends on geometry of the pipe and as well as the bend radius. From the

pressure drop correlation for bends, it is known that, if the velocity of the fluid at the bend region is reduced,

this may result in pressure drop reduction. In a subsonic flow, the increase in the flow area i.e., expansion of

the duct is accompanied by reduction in the fluid velocity. But the expansion cannot be carried out

indefinitely as it has to be once again reduced to match with the dimensions of the outlet duct. One can see

that in the optimal solution (Figure 10), there is a small spot near the inner side of the bend entrance where

there is non-smooth variation of flow. This may be due to the fact that a cubic Bézier curve has been used to

0.1 0.15 0.2 0.250.08

0.1

0.12

0.14

0.16

0.18

Control point 1 - x-coordinate

Co

ntr

ol

po

int

1 -

y-c

oo

rdin

ate

0.2 0.25 0.3 0.350.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Control point 2 - x-coordinate

Co

ntr

ol

po

int

2 -

y-c

oo

rdin

ate

0.22 0.23 0.24 0.25 0.260.04

0.06

0.08

0.1

0.12

0.14

Centreline radius (m)

Wid

th o

f th

e duct

(m

)

64

represent the radius–width relation. Choosing a higher order Bézier curve may enable a smoother transition

at the inlet while still accounting for area change.

If the duct is over-expanded initially and/or if the contraction of the right half to match with the outlet is

not smooth, a strong recirculation zone is observed in many portions of the duct which lead to significant

pressure losses due to poor pressure recovery. Even geometries that do not have smooth variation are

eliminated by the search algorithm based on the outcome of the CFD simulation. A judicious variation of the

centreline radius and the duct of the width yields a geometry with reduced pressure drop and the same is

achieved by using a shape function, search method and CFD. The present study demonstrates that intelligence

can be built into CFD to make it a design tool which will otherwise be only an analysis tool.

m/s

Non-optimal

m/s

Non-optimal

m/s

Non-optimal

m/s

Optimal

Figure 10 Velocity contours of explored geometries.

65

Pa

Non-optimal

Pa

Non-optimal

Pa

Non-optimal

Pa

Optimal

Figure 11 Pressure contours of explored geometries.

c. Shape optimization using modified Box complex method The present authors have developed modified Box complex method with improvements made over

the original method specific to suit CFD based optimization. The primary modifications are suggested for convergence acceleration and premature breakdown of the optimization process. The details of the modified Box complex method can be found in Reference 25. The modified Box complex method is applied to the shape optimization of U-bend. The method could reach the optimal at a faster pace when compared to the original Box complex method. The minima reached by both methods are also nearly the same. A plot between the successful iteration number and the global number of iterations is shown in Figure 12 for both Box complex and modified Box complex methods. It can be concluded from the figure, that number of successful iterations and as well as number of unsuccessful iterations are considerably less in the case of modified Box complex method in comparison with the original method. Overall reduction in the computational effort is about 30% in the case of the modified method. The optimal pressure drop is nearly the same in both the methods and is around 40 Pa whereas the pressure drop of the semi-circular base case geometry is around 52 Pa.

66

Figure 12 Successful iteration number vs. global iteration number.

6. Conclusions In this paper, a methodology has been proposed for the shape optimization of U-bend for the minimization

of the pressure drop. The complexity of the problem is reduced by specifying the radius of curvature and width of the U-bend for all angles varying from 0o to 180o to define the U-bend in two-dimensions. The continuous variation of the radius of curvature and the duct width at every angle is represented by a cubic Bézier curve with four control points of which two are movable. The problem is posed as CFD-based shape optimization problem wherein the abscissa and ordinate of the movable control points are the decision variables of optimization. The optimization is carried out by using Box complex and modified Box complex methods. The proposed method has been able to evolve an optimal geometry with significant reduction in the pressure drop compared to the base case geometry which is semi-circular in shape with a reasonable amount of computational effort. The present method can be readily extended to three-dimensions for the shape optimization of U-bend with rectangular cross sections.

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20

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bal

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atio

n n

um

ber

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67

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