Multi-Objective Design Optimization of Precoolers for HypersonicAirbreathing Propulsion

Shufang Yu,∗ Trent Jones,∗ Hideaki Ogawa,† and Nitin Karwa‡

Royal Melbourne Institute of Technology, Melbourne, Victoria 3001, Australia

DOI: 10.2514/1.T4921

A precooling heat exchanger has been the key mechanism for realizing turbine-based combined-cycle engines at

high flight Mach numbers. A multi-objective design optimization coupling surrogate-assisted evolutionary

algorithms with numerical simulation has been carried out with respect to three design objectives, that is,

maximization of heat transfer effectiveness, total pressure recovery, and compactness. Physical insights into the

underlying compressible aerodynamic andaerothermal phenomena in the bare tubebankgeometryhavebeen gained

through scrutinizing the flowfields for the representative cases. With the nature of operating conditions having

relatively high inflow velocity, tube bank configurations that are potentially prone to have flow choking are removed

in the optimization process. The results from the optimization have been investigated by analyzing the selected

individuals on the Pareto-optimal front and performing sensitivity analysis with the aid of surrogate models. The

effects of uncertainties in the design parameters on the precooler performance have been examined. The

unconventional tube profiles comprising elliptic and obround sections are found to effectively reduce unfavorable

flow separation and permit smaller tube spacing ratios, yielding higher heat transfer rate per unit volume than the

conventional circular tube bank.

Nomenclature

A = area, mm2

a = elliptic major axis radius, mmb = minor axis radius, mmC = perimeter, mmCd = drag coefficientCp = specific heat capacity of air, J∕�K · kg�D = diameter, mme = ellipse aspect ratioEu = pressure loss coefficienth = heat transfer coefficient for the air,W∕�m2 · K�K = sensitivity indexL = obround flat length, mml = characteristic length, mM = Mach number_m = mass flow rate, kg∕sN = number of tubesNu = Nusselt numberP = pressure, Pa (perimeter, mm)Pr = Prandtl numberp = pitch-to-diameter ratioRe = Reynolds numberS = tube pitch, mmT = static temperature, KTt = total temperature, Kt = thickness of tube, mmU = overall heat transfer coefficientV = velocity, m/sy1 = first-layer height, my� = dimensionless wall distanceμ = dynamic viscosity, �N · s�∕m2

ρ = air density, kg∕m3

Subscripts

bank = bankd = diagonalf = skin frictioni = first-order effectin = tube bank inletL = longitudinalmax = maximumout = tube bank outlet0 = stagnationp = pressure dragQ = heat transferrow = single row of tubesT = transverset = totaltube = tubev = volumew = tube wall0 = stagnation

I. Introduction

H YPERSONIC airbreathing propulsion offers a promise offlexible and economical high-speed atmospheric transport as

well as access to space for both civilian and strategic applications[1–6]. Atmospheric air captured in the intake is compressed to beused for combustion, removing the need to carry the oxidizer and thusenabling substantial saving in weight. However, as the hypersonicairflow is decelerated by the engine intake to a subsonic velocity forsustainable combustion operation, the temperature of the airflowentering engine core can reach over 1000°C [7]. This extreme hightemperature is close to the permissible operating limit of ceramicmatrix composite materials of compressor blades [2], and moreimportantly, the thrust and specific impulse of the engine drasticallydeteriorate due to low density of hot air at the compressor inlet. Thiscritically demands the use of highly compact and effective heatexchanger technologies that can reduce the air temperature to anappropriate level near 300°C with minimum total pressure penaltybefore the flow enters the compressor stage of the core engine. Theprecooled turbojet engine being developed at the Japan AerospaceExploration Agency (JAXA), for instance, employs a bank ofcylindrical tubes as a precooler, whereas a radial configuration is usedin the Synergistic Air-Breathing Rocket Engine developed byReaction Engines Limited for the Skylon spaceplane. It has

Received 30 December 2015; revision received 24 July 2016; accepted forpublication 12 September 2016; published online 8 December 2016.Copyright © 2016 by the American Institute of Aeronautics andAstronautics,Inc. All rights reserved. All requests for copying and permission to reprintshould be submitted to CCC at www.copyright.com; employ the ISSN 0887-8722 (print) or 1533-6808 (online) to initiate your request. See also AIAARights and Permissions www.aiaa.org/randp.

*Aerospace Graduate, School of Aerospace, Mechanical and Manufac-turing Engineering (SAMME).

†Senior Lecturer, School of Aerospace, Mechanical and ManufacturingEngineering (SAMME). Member AIAA.

‡Also Senior R&D Engineer, Honeywell International, Inc., Gurgaon, India.

Article in Advance / 1

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demonstrated its capacity to cool the air entering the engine fromover1000 to −150°C [8].The design criteria for such a precooler are to achieve high heat

transfer effectiveness and high total pressure recovery withminimum total mass and volume. Because of the rather poor heattransfer characteristics of the air, a very large heat transfer area isultimately needed to achieve a high power transfer rate. Yet simplyincreasing the tube diameter, tube length, and number of tubes canconsequently increase the engine size and mass. A more compactprecooler progressively reduces its mass, dictating that a largefrontal area shall be prohibited. Nevertheless, the principalexperimental work done by Murray et al. [3] and Kays and London[9] indicates that the total pressure loss is associated with thedevelopment of compact heat exchangers. For a fixed pitch-to-diameter ratio, the smaller the tubes are, the smaller is the hydraulicdiameter of the flow passage for the external airflow. At a givenmass flow rate of air, this leads to increase in the total pressure lossthrough the tube matrix [3]. The benefits brought by a lighter andmore compact tube matrix can be outweighed by reduction inpressure available at the compressor inlet. These conflictingrelationships pose a challenge on the precooler multi-objectivedesign optimization (MDO).Considerable efforts have been made to develop understanding of

the fluid flow in crossflow tube banks due to their abundant use inengineering applications. The air speed at the front of the heatexchanger in these applications is usually limited to less than 10 m∕s[10–15]. The maximum flow velocity is reached within the heatexchanger, but the flow remains incompressible. On the other hand,the precooler for airbreathing engines would operate at much higherinlet velocity, resulting in considerably compressed flow. Previousresearch work has highlighted that the thermohydraulic performanceofmicrotubes at low air speeds is dependent on the tube and tube bankgeometry for given flow condition [14–18].Taler [12] pointed out that the recirculating flow immediately

behind the tube contributes little to the heat transfer. They observedrecirculating flow behind an oval tube with an aspect ratio of 1.86when used in an inline tube bank and air velocity of 7 m∕s. Hefound that thewake region behind the tubes contributes very little tothe heat transfer rate. Leu et al. [17] compared the performance offinned tube heat exchangers with inlet velocity of 0.5–2.6 m∕s. Theheat transfer coefficients for the oval tube are lower than those ofthe circular tube configuration, and both heat transfer and totalpressure drop for oval tube decrease with increase of axis ratios.Hasan and Sirén [13] compared the thermohydraulic performanceof a single row of oval tubes of various aspect ratios with circulartubes and air velocity of up to 17 m∕s. The oval tubes were reportedto offer much better thermohydraulic performance than circulartubes, with the performance enhanced as the tubes are streamlinedby increasing the aspect ratio. More recently, Ranut et al. [14]performed multidisciplinary optimization of the tube shape for abare tube bank at a low inlet velocity of 0.5 m∕s. They showed thatstreamlining the tube shape is necessary to maximize the heatexchanger effectiveness and minimize pressure loss. Beale andSpalding [19] performed the numerical study on pitch-to-diameterratios of 1.25, 1.5, and 2.0 for inline square, rotated square, andequilateral triangle geometries, respectively, over Reynoldsnumber between 10 and 103. The pressure loss coefficient andheat transfer factor of the airflow increasewith the reduction in tubespacing, though the trends differ with tube arrangement andReynolds number. El-Shaboury and Ormiston [16] numericallyexamined the effects of and longitudinal pitch-to-diameter ratiosindependently (1.25, 1.5, 2.0) for inline tube bank at Reynoldsnumber of 100 and 300. They reported that the length ofthe recirculation zone behind the last row tube shortens with thedecrease in transverse pitch, but their strength increases. Therecirculation zone between tubes is more susceptible to the changein longitudinal pitch. As the longitudinal pitch increases, thestrength center of the vortex in the region moves downstream.Yoo et al. [20] experimentally studied the effects of tube spacing(pitch-to-diameter ratios of 1.5, 1.75, and 2.0) for the staggeredtube bank and concluded that the local heat transfer coefficients on

each tube increase as the tube spacing decreases due to strongervortices of preceding tubes. Nevertheless, the inflow velocity of theexperiment was limited to be below 25 m∕s by the maximum airspeed in the wind tunnel.Designing a high-performance precooler crucially requires

comprehensive understanding of the underlying compressibleaerodynamics and air thermal characteristics around a bare tube bank.However, contrary to the abundant information available for tubebank flow in the incompressible regime, rather scarce literature hasprovided insight into the thermohydraulic behavior of the heatexchanger in the compressible, low-Reynolds-number regime for theclass of hypersonic airbreathing engines. Such a precooling heatexchanger optimization problem with multiple objectives wouldrepresent a formidable challenge for conventional gradient-basedoptimization approaches. Global search based on evolutionaryalgorithms is particularly suitable for the design problems that arecharacterized by nonlinear and nonsmooth design space being robustagainst the presence of local optimums [21]. The use of population-based search in aerospace design, however, would commonly entailprohibitive computational cost due to a large number of functionevaluations, e.g., computational fluid dynamics (CFD) and finiteelement analysis. Surrogate modeling can efficiently mitigate thecomputational load by replacing expensive function evaluations withapproximation from meta-analysis models [22,23]. This robustmethodology has been successfully applied to various multiple-objective optimization problems [15,24–26].In the present study, a multi-objective design optimization with

respect to three objective functions (airside heat transfer coefficient,total pressure loss, and compactness) and one constraint function formonitoring simulation convergence (flow domain mass balance) hasbeen performed to explore potential good precooler designs and toprovide more objective-oriented investigation. The surrogate-assisted evolutionary algorithms on Mathworks Matlab platformcoupled with computational fluid dynamics with ANSYS Fluent hasbeen employed. The computed Pareto-optimal front (POF) hasrevealed the correlation among the design objectives and thecharacteristics of the optimal solutions. Sensitivity analysis has beencarried out by analyzing the representative individuals on the POFand interpreting the sensitivity indices based on the predictions fromsurrogate models. The insights gained from the study are intended tobenefit the precooler designer for turbine-based combined cycleengines.

II. Numerical Modeling

A. Flow Characterization at Precooler Inlet

The freestream Mach number expected for a precooledhypersonic aircraft at 24–25 km altitude is Mach 5 [5,7]. Theambient pressure and temperature are approximately 2857 Pa and216.6 K. The total temperature of the freestream is calculated to be1300 K, which matches closely to the value reported in [4]. Theengine intake has to decelerate the incoming airflow to lowersubsonic speed (e.g., Mach 0.4–0.5) for maximum pressurerecovery and stable combustion operation in a conventional turbineengine [27]. However, Jones et al. [28] have shown that such highinlet velocities are infeasible due to the occurrence of flow choking.In the present study, the inlet Mach number M1 is taken to be 0.2.Assuming the air to be a perfect gas with a specific heat ratio of 1.4and an adiabatic compression processwith a conservative compressionefficiency of 0.8 in the intake diffuser, the static temperature andstatic pressure at precooler inlet are calculated to be 1290 K and132 kPa, respectively. The estimated inlet temperature and pressureare close to the design condition reported in [1] and scale-model testcondition in [3]. At 1290 K, the local speed of sound in air and theprecooler inlet velocity are calculated to be 720 and 144 m∕s,respectively.

B. Tube Bank Geometry

Reaction Engine Ltd is using a circular cross-section tube(Fig. 1a) of an outer diameter 0.98 mm with a wall thickness of0.04 mm for the precooler tube matrix [1]. For simplicity, a circular

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tube with diameter of 1 mm is employed to be the baseline tubeshape in the present study.With the outer diameter being larger than10 times the wall thickness, the tube can be regarded as a thin-walled cylinder according to the guideline by Den Hartog [29]. Thecross-sectional area of the tube wall of a circular cross-section tubeis approximated as

Atube � πDt (1)

whereD is the distance from the center to the tubewall midline, andt is the wall thickness.A staggered tube bank configuration is adopted by Reaction

Engine Ltd because it is more effective in heat transfer than an inlinetube bank, as shown in an earlier investigation [30]. Figure 2 showsthe tube bank configuration defined by transverse pitch ST andlongitudinal pitch SL. The tube bank arrangement is characterized bythe pitch-to-diameter ratios pL and pT . For a circular tube, it can bewritten as

pL � SLD

(2a)

pT � STD

(2b)

pL and pT in the precooler by Reaction Engines Ltd are 1.25 and2.2, respectively [3].In addition to the circular tube in a precooling system, elliptic

(Fig. 1b) and obround (Fig. 1c) cross-section tubes are considered forthe tube shape optimization. For the precooler, controlling the totalweight of the material used for manufacturing the tube bundles is ofpractical importance. For comparison of their performance, the tubewall volume is kept the same for the tubes of different shapes.

C. Computational Fluid Dynamics

A density-based solver ANSYS Fluent is employed to simulate thetwo-dimensional steady-state compressible flowfields [31]. Thedensity and dynamic viscosity of the air are calculated using the ideal

gas equation and Sutherland’s law, respectively. Thermal conductiv-

ities of the dry air over the operating temperature are calculated by

using the correlation equations provided by Kadoya et al. [32]. The

thermal conductivity was calculated, based on the static temperaturevia a power-law function, which was implemented in a user-defined

function and read in ANSYS Fluent.Because the full model of a precooler is considered axisymmetric

and the number of columns is large across the frontal area, the

modeled flow domain spans across two adjacent half-columns and

contains 10 half-tube geometries. Figure 3 shows the entire

computational domain along with boundary conditions, where thereis an auxiliary upstream zone of 10mm length and a comparable long

downstream zone. The total length of the computational domain is set

to be 50 mm. Symmetry conditions are applied in between the tubeson the top and bottom boundaries to simulate a segment of the actual

model. Thewidth of the inlet is determined by the tube shape and the

transverse pitch-to-diameter ratio of the tube bank. It should be noted

that inflow velocity and density are kept the same for all individualsimulations because the frontal area of the precooler in a full model is

unaltered with the change in tube bank geometries. Thus, it is

remarked here that the solution of the model is unaffected by the flowdomain width, ensuring a constant inflow mass flux; however, the

number of columns in the full model is higher for the more compact

configuration in the transverse direction.The Reynolds number of an incompressible flow in a tube bank is

given by

Re � ρiDVmax

μin(3)

where D is the hydraulic diameter, μi is the dynamic viscosity, and

Vmax is the maximum bulk velocity in the minimum cross section.

Vmax for an adiabatic flow through the tube bank is given as [33]

Vmax � max

�pT

pT − 1V in;

pT

pd − 1V in

�(4)

For the precooler inlet condition determined previously and the

defined baseline geometry (D � 1 mm, pL � 1.25, and pT � 2.2),the Reynolds number is estimated to be approximately 3525. Thus,the flow is in the subcritical flow regime (300 < Re < 30;000�, whichis characterized by a laminar boundary-layer detachment and a

completely turbulent wake [34]. For this flow regime, the transitionshear-stress transport four-equation model was employed as the

turbulencemodel. It has beenwidely tested to show its high fidelity in

wall shear and heat transfer predictions and demonstrated capabilityof accurately simulating the transition where the laminar boundary

layer leads to turbulent wake [31].The computational domain is represented by meshes generated

by using the ANSYS Meshing tool. Appropriate near-wall meshdensities were determined based on criterion on the dimensionless

wall distance y�, where the first-layer height (y1) is calculated

as [31]

y1 � l · y�������74

pRe

−13∕14l (5)

Row Column

Flow /2

10 rows

Fig. 2 Staggered arrangement of circular tubes.

Fig. 1 Tube prototypes considered in MDO.

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For y� � 0.8 and Reynolds number of 3500, y1 is determined tobe 3.5 μm. An inflation layer domain comprising 30 layers and agrowth rate of 1.05 is created around the tube wall. Figure 4 showsa close-up mesh for a tube bank with a 1-mm-diam cylindricalcross-section tube, pL � 1.25, and pT � 2.2. The mesh within theinflation layer has quadrilateral elements, and the outer mesh hastriangular elements. The maximum element aspect ratio hasconsistently been maintained below 3.4.To allow comparison of the performance for various tube banks,

the inlet pressure, temperature, and velocity are kept constant. Toachieve this, a pressure-inlet boundary condition with specified totalpressure and total temperature is applied at inlet of the flow domain[31]. In addition, a constant mass flux is applied at the flow domainoutlet. The solution is converged to match the prescribed mass flux atthe inlet; hence, the inlet velocity can be fixed. With these boundaryconditions, the density and velocity of airflow at the inlet remainconstant. The tube wall temperature is kept at 123.15 K (−150°C).The boundary conditions for all the simulations are summarized inTable 1.

III. Design Optimization

A. Optimization Algorithms

The multi-objective design optimization is performed in aniterative manner coupling the applications of Mathworks MATLABandANSYSWorkbench. Surrogate-assisted evolutionary algorithmsallow for optimizations that take into account multiple objectivessimultaneously and explore potential good designs in the wholesearch domain. The loop consisting of tube bank generation from thedesign variables, mesh generation (preprocessing), CFD simulation(flow/heat transfer problem solver), postprocessing (export thevalues of objective and constraint functions), and surrogate-assistedevolutionary algorithms (SAEA) is schematically described in Fig. 5.

The basic idea associated with the employed evolutionary

algorithms is analogous to Darwin’s law of natural selection.

Each configuration is considered as an individual. A population

comprising N randomly chosen individuals will evolve over

generations, where designvariables or “genes” aremanipulated using

various operators (average, crossover, mutation) to reproduce better

and better offspring (new sets of design variables). For a design

problem with conflicting objectives, the evolutionary process results

in the formation of a converged Pareto-optimal front constituted by

so-called nondominated solutions, where no further improvement

can be made for any objective function without degrading other

objective functions [15,35]. The optimization occurs over 20

generations with 96 individuals in the population pool. A simulated

crossover probability and mutation probability are 1.0 and 0.1,

respectively, to increase diversity among the population.

The optimization process is efficiently assisted by various

surrogate models that predict approximate outcomes in lieu of actual

CFD simulations to evaluate objective and constraint functions.

Multiple surrogatemodels are employed and evaluated: the quadratic

response surfacemodel [36]; artificial neural network (ANN)models

including the radial basis function network [37] and multilayer

perceptronmodel [38],which are single-layer and feed-forward types

of ANN models, respectively; and the kriging model based on

Gaussian process regression [39]. The mean-squared error (MSE) in

the actual and predicted values of the objectives and constraints is

calculated for the remaining (10%) solutions and used as themeasure

to validate the surrogate models. Prediction from the best surrogate

model with aminimum error is adopted to replaceCFD analysis, only

if the MSE is within a threshold value of 5% for all objective

functions and 20% for constraint functions and the distance to the

closest point in the archive is smaller than 5% [40]. In this study, the

minimum approximation errors for the objectives (i.e., heat transfer

coefficient, total pressure loss, and compactness) were found to be

0.588, 0.454, and 0%, yielded from the kriging model, quadratic

response surface model, and radial basis function network,

respectively, and that for the constraint function was 4.251% from

the radial basis function at the final generation (Gen � 20).Global sensitivity analysis is performed to examine how the

uncertainty in model output can be apportioned to that in the input

parameters [41]. The evaluation in the present study is made with

10,000 sample points within the design variables limit [24].

SymmetryTube Wall

Symmetry

OutletInlet

10 mm

50 mm

Flow

Fig. 3 Computational domain for flow simulation through a tube bank.

Fig. 4 Close-up of the meshing showing the inflation layer around thetube wall.

Table 1 Boundary conditions for all simulations

Location Boundary type Conditions

Cell zone conditions Fluid Density: ideal gas; viscosity: Sutherland’s law; thermal conductivity:Kadoya’s model; operating pressure: 132 kPa

Inlet Pressure inlet Total gauge pressure: 3736 Pa; total temperature: 1300 KOutlet Mass flow rate outlet Mass flux: 51.4 kg∕�m2 · s�Tube wall Wall No slip wall; fixed temperature: 123.15 KTop and bottom openings Symmetry — —

Fig. 5 Optimization loop [24].

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B. Optimization Problem

The tube shape used in the optimization is a generalized shapecomprising elliptic and obround profiles, called obelliptic tubetentatively, shown in Fig. 6. The obelliptic shape is flexible to becomea circular, elliptic, or obround tube by controlling the followingparameters: elliptic aspect ratio e and obround flat length L.The cross-sectional area of the tube wall of an obelliptic tube is

approximated as

Atube � �π b�e� 1� � 2L�t (6)

Under the constraint that the tube wall volumes of various tubeshapes are kept equal to the baseline circular tube (D � 1;t � 0.04 mm), the obround flat length can be written as a functionof the other two variables, minor radius b and elliptic aspect ratio e.The tube bank geometry to be optimized is represented by fourdesign variables: minor radius b, elliptic aspect ratio e, longitudinalpitch-to-diameter ratio pL, and transverse pitch-to-diameter ratiopT . The upper and lower bounds of the decisionvariables are shownas follows.Design variables ranges:

0.25 ≤ b ≤ 0.5

1 ≤ e ≤ 3

1.1 ≤ pL ≤ 1.75

1.6 ≤ pT ≤ 2.6

The initial population is a set of randomly chosen configurations inthe search domain defined by the limits imposed on the inputparameters; thus, the resultant obround flat length might be of anegative value, and the set of design variables is considered asgeometrically infeasible. To avoid ANSYS workbench’s stoppagedue to the geometry error, all sets of design variables are made to go

through a geometry validity check before they are written into aComma-Separated Values input file (Fig. 7). Those individuals thatfail to meet the geometry requirement are deemed as infeasible.Three characteristic parameters are chosen and used as the

objective functions to achieve the optimization goal to fulfill theprecooler design criteria. The first objective is to maximize air-sideheat transfer coefficient h because the overall thermal resistance forheat exchange between the hot airstream and the coolant is dominatedby the air-side thermal resistance. The total pressure loss betweeninflow and outflowΔP0 is able to account for losses due to form dragand viscous drag; thus, it is evaluated as the second objective tominimize. The compactness of a heat exchanger is defined by theratio between the amount of heat exchange surfaces to the totalvolume that the heat exchanger and fluids occupy. The approximateequation for calculating the circumference of the obelliptic shape,defined as

P≈π�a� b�243 �a − b�2

�a� b�2� �����������������������������������������������������

−3�a − b�2∕�a� b�2 � 4p

� 10��1

35

� 2L (7)

Also, the cross-sectional area that a single tube occupies isdefined as

Av � 2pL�L� 2a� × pT

2�2b� (8)

Because tube length perpendicular to the two-dimensional flowdomain has a unit length, the third objective to maximize theprecooler compactness is thus evaluated as

compactness � P∕Av (9)

The constraint function is to monitor if the numerical solutionreaches steady state, and it is imposed on the mass flow ratedifference between flow domain inlet and outlet Δ _m. The tradeoffhas to be made between the computation time required for eachsimulation and the accuracy of the solutions. All residuals errorchecks have lowered to 5 × 10−6, and the number of computationiterations is set to 2000. To accelerate the convergence process, theflow type in solution steering in ANSYS Fluent is chosen to besubsonic flow. The numerical solution that fails to meet the domainmass balance criterion within 2000 iterations has commonly beenassociated with a supersonic flow developed in the tube bank as aresult of the decreasing sound speed along the tube bank andflow acceleration through a convergent–divergent passage. Theinherently unsteady nature of the induced large vortex sheddingfollowing each row of tubes subsequently represented difficulty in

Fig. 6 Obelliptic tube shape used in MDO.

Fig. 7 Flowchart of SAEA-based optimization procedure.

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the convergence of the numerical simulations, which assumesteady flowfields. For such cases, the tube bank configurationhas been deemed practically infeasible. Because of the fact thatthe evolutionary algorithm was designed to be a minimizationoptimizer for all objective functions, the optimization problem canthus be stated as follows:

Minimize: �1� − h

�2�ΔP0

�3� − Compactness

Subject to: − log10�abs�Δ _m�� − 4 ≥ 0

The developed optimization procedure is schematically shownin Fig. 7.

IV. Model Validation

A. Mesh Sensitivity

A grid-independence study has been conducted to investigatethe effects of mesh refinement on the accuracy of computationalsolutions, shown in Table 2. Three mesh growth rates 1.2, 1.05,and 1.03 for the baseline model were selected. The table indicatesthat the solutions of static temperature drop and the total pressureloss between the baseline tube bank inlet and outlet areconsidered independent of the total number of elements if it isabove 66,000. The mesh growth rate of 1.05 is chosen for allsimulations to strike a balance between computation time andhigh accuracy.

B. Pressure Drop Coefficient

For the inlet Mach 0.2, the flow is locally accelerated to aboveMach 0.3, where it can no longer be assumed incompressible.Because of limited availability of literature for the pressure dropcoefficient for the compressible flow in a tube bank, the model isassessed by assuming the inlet Mach number of 0.02. Thecomputational model setup and mesh resolution described in theprevious section are used.The pressure loss coefficient data for a staggered tube bank have

been obtained by Žukauskas and Ulinskas [42]. They reported theresults in the form of the Euler number, which is given as

Eu � 2Δprow

ρ�Vmax�2(10)

whereΔprow is the pressure drop across a single row of tubes. Best-fitequations to their data in the Reynolds number range of 2 to 2 × 106

are given by Schlünder [43].For a tube bankwithD � 1 mm,pL � 1.25, andpT � 1.25with

an inlet Mach number of 0.02, the Reynolds number is about 350.The Euler number is approximately determined from the best-fitequations to be in the range of 0.7 and 0.8. The numericallydetermined Euler number for the first row is 0.74, which iswithin the range of estimated Euler number from the best-fitequations.For an inlet Mach 0.2 (Re � 3525), the Euler number from the fit

is between 0.4 and 0.5. However, the numerically determined Eulernumber is 0.66. The deviation is deemed due to the compressibilityeffects in the flow at high inlet Mach numbers.

C. Mean Heat Transfer Coefficient

The mean heat transfer coefficient was also compared with theexperimental data by Žukauskas and Ulinskas [30]. They havereported data only for a tube bank with pL � 1.25 and pT � 1.25.The correlation for the first row of staggered banks is given as

Nu � 0.8Re0.45Pr0.36f

�PrfPrw

�0.25

; 100 ≤ Re ≤ 2000 (11)

For themeanNusselt number of the inner rows of staggered banks,

Nu � 0.71Re0.45Pr0.36f

�PrfPrw

�0.25

; 40 ≤ Re ≤ 1500 (12)

where Prf is the Prandtl number in the freestream, and Prw is thePrandtl number of the boundary layer on the tube.It should be noted that the developed correlations are related

to the airflow with relatively moderate temperature variationthroughout the tube bank. The inflow static temperature in themodel is changed adaptively to 573.15 K (300°C), and it is cooledby the tubes with wall temperature kept at 273.15 K (0°C) Theoperating pressure is altered to 101,325 Pa. Also, the inflow Machnumber is reduced to 0.01, 0.02, and 0.028, such that the Reynoldsnumbers are within the valid range of the correlations. Moreover,the low-speed heat exchanger experiment by Žukauskas andUlinskas [30] was carried out in a wind tunnel, and so the inletturbulence level was kept to 1%. The temperature contour isattainable from numerical simulation, which can be used to evaluatethe heat transfer coefficient according to Eq. (15), and the thermalconductivity can be estimated from correlations provided byKadoya et al. [32]. The results are compared in Fig. 8. Thesimulation results closely match the correlations given in Eqs. (11)and (12), which were derived based on their experiment results.This validates that the numerical model is reliable for the low inletMach numbers considered in the present study. No experimentaldata are available for Mach number near 0.2, and so the modelcannot be evaluated for the conditions occurring within theprecooler.

Fig. 8 Average heat transfer for the first row of tube and inner rows.

Fig. 9 Pareto-optimal front.

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Fig. 10 Population projections on objective function planes.

Fig. 11 Box and whisker plot of optimal solutions.

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V. Analysis

The change in airflow enthalpy through a modeled tube bank

equals the heat transfer rate from the airflow to the tubes:

_mbankCp�Tin − Tout� � hAQΔT (13)

where AQ is the thermal transfer surface area in the tube bank; Tin

and Tout represent the air inlet and outlet steady-state temperature;

cp is the specific heat of air; and _mbank is the mass flow rate

calculated from the model. The log-mean temperature difference

ΔT is defined by

ΔT � �Tw − Tin� − �Tw − Tout�ln��Tw − Tin�∕�Tw − Tout��

(14)

Thus, the airside heat transfer coefficient of a tube bank can be

calculated as

h � _mCp�Tin − Tout�AQΔT

(15)

The number of transfer units (NTU) is evaluated by

NTU � UAt

_mtcp(16)

where the overall heat transfer coefficient in the numerical model is

only contributed by the external flow; thus, U � h. At and _mt are

the total thermal transfer area and total mass flow rate entering the

precooler, respectively. _mt is assumed constant regardless of

precooler configuration. Therefore, the thermal effectiveness (ε) ofthe precooler configuration can be determined using effectiveness

NTU method:

ε � 1 − exp�−NTU� (17)

Equations (18) and (19) are used to calculate the average

pressure drag coefficient and average skin friction coefficient,

where Dp is the pressure drag; Df is the skin friction; Ap is the

frontal area of a tube perpendicular to the flow direction; Af

the planform area of a tube parallel to the flow direction; and N is

the number of tubes:

Table 2 Computational results of grid-independence study

Mesh growth rateNumber ofElements

Static temperaturedrop, K

Percentagedifference, %

Total pressuredrop, Pa

Percentagedifference, %

1.2 66,514 503.56 0.234 34,447 −0.531.05 83,619 502.38 0 37,647 01.03 100,757 502.86 0.095 37,519 0.34

Table 3 Objective function values and geometry parameters of the representative tube banks

Objective functions Design parameters

Tube type h,W∕�m2 · K� ΔP0, Pa Compactness, m2∕m3 B, mm e L, mm pt pL

S1 2101 39,712 1906 0.35 1.10 0.43 1.73 1.15S2 1679 7,404 1397 0.28 2.30 0.11 2.59 1.14S3 2006 26,572 2376 0.25 1.35 0.65 1.81 1.10O1 1973 23,299 2205 0.26 1.47 0.57 1.87 1.12Baseline 2039 37,647 1215 0.50 1.00 0 2.20 1.25

Fig. 12 Flowfield plot of the representative individuals.

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Cdp � 2Dp∕NρV2

i Ap

(18)

Cdf � 2Df∕NρV2

i Af

(19)

VI. Results and Discussion

A. Pareto-Optimal Front

The optimization has been performed until converged solutionshave been achieved, and a border separating feasible solutionsfrom infeasible is illustrated. The discrete boundary is the so-called

Pareto-optimal front (POF). It is made of all nondominated solutions

that satisfy all constraints. Along the POF, decreasing thevalue of one

objective while keeping other objectives constant would move the

design point to the infeasible domain. The number of individuals that

have been evaluated by CFD is up to 1422. The whole population

pool is projected on the heat transfer coefficient, total pressure loss,

and compactness planes shown in Fig. 9. The nondominated

solutions of 96 individuals in size constitute a curved surface in the

three-dimensional space.Figures 10a and 10b show the POF projected on two-dimensional

objective planes, and the magnitude of the third objective parameter

with respect to the other two is visualized by the dots filled with

colors. Figure 10a indicates a “very strong” counteracting trend

between airside heat transfer coefficient and total pressure loss across

a precooler. Also, a precooler with comparatively higher compact-

ness tends to result in higher total pressure loss. Figure 10b

demonstrates a trend that airside heat transfer coefficient increases

gradually with the compactness.Figure 11 illustrates a box and whisker chart for output (objective

and constraint functions) and input parameters (design variables) of

the computed optimal population over 20 generations. These

columns are bounded by their respective maximum and minimum

limits. As the population evolves to the 20th generation, the optimal

solutions favor the tubes with minor radius in values between 0.25

and 0.37 mm. The median value of b is located at 0.29 mm. The

median value of the elliptic aspect ratio e for the tube shape of theoptimal individuals is at around 1.39. The relatively shorter

distance between the minimum value and median indicates that

the distribution is positively skewed: elliptic aspect ratios of half

of the optimal population are located between 1.09 and 1.39.

The transverse pitch-to-diameter ratio pT , given by the optimal

solutions, has a median value at 0.99 and spreads between 1.72 and

Fig. 13 Variation of total pressure loss with tube pitch-to-diameterratios (D � 1 mm).

Fig. 14 Comparisons of design variables and airside heat transfer coefficient.

Fig. 15 Turbulence kinetic energy contour for S1 and S3 tube banks.

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2.60. The distribution is positively skewed, and the tendency of pT

of optimal solutions is toward smaller values. As for longitudinal

pitch-to-diameter ratio pL, most values have clustered below 1.12,

and the whole distribution locates very closely to its lower bound

of 1.1.

B. Observations

1. Geometries

One of the powerful features of the Pareto-optimal front is to allow

a designer to convert the multi-objective problem into a single-objective problem by applying weighting factors on objectives. To

investigate the factors that contribute to optimal solutions, three

design points have been selected that are optimal in a single-objective

function, plus another design point that is a compromised optimal

design O1. The values of the objective functions and the geometry

parameters for the representative individuals and the baseline aredisplayed in Table 3. The individuals S1, S2, and S3 are single-

objective optimums for maximal heat transfer coefficient, minimal

total pressure loss, and maximal compactness, respectively. The

individual O1 is selected in the vicinity of the median values of all

objective functions.

2. Flowfields

Flow separation from the tube surface is indicated by therecirculation zone in the velocity streamline plot in Fig. 12. It is

observed that the optimal representatives have much smaller wake

region contrary to the baseline, the circular tube bank. The smallest

size of the recirculation zone is found to be in the tube bank S2. As thetube shape stretches along the streamwise direction and becomes less

obstructive to the flow, the flow tends to remain attached to the tube

wall over a longer distance, which significantly reduces pressure dragdue to the flow separation. In addition, it is found that, although S1has a larger tube axis ratio, its local maximum flow velocity in the

minimum cross section in the tube matrix is even higher than the

baseline tube bank. This suggests that the local flow velocity through

a tube bank is also affected by the other designvariable, the transverse

pitch-to-diameter ratio, because the longitudinal pitch-to-diameterratio is similar among the optimal individuals.

The restrictions in the tube bank induce sudden changes in flowdirection and force the flow to accelerate, whereas the statictemperature of the flow is reducing. The potential occurrence of dragdivergence Mach number (maximum local Mach number>0.65–0.75) hinders the circular tube to be compressed into asmaller volume (Fig. 13). However, the flatter tube shape such as inS1 can permit pushing the transverse pitch-to-diameter ratio to asmaller limit while maintaining the maximum local flow Machnumber below 0.685.

3. Heat Transfer Effectiveness

Provided that inflow velocity, density, and tube wall temperatureare kept constant, the cooling performance is ultimately determinedby the tube bank geometry. Heat transfer can be enhanced by havinghigher average flow velocity, larger effective heat exchange surface,and/or higher turbulence intensity developed in a tube bank [4,44].The tube bank geometry variables and the heat transfer coefficient

are normalized and compared in Fig. 14. The selected tube banks thatachieve the heat transfer coefficient from small to large areS2,O1,S3,and S1, which is in the same order that pt decreases. The narrowerflow passage can directly increase the average flow speed through thetube bank as shown in the flowfields in Fig. 12. The reduction intransverse pitch-to-diameter ratio is believed to be the main driver forthe gradual heat exchange improvement. Among the representativeindividuals, the S1 configuration attains the highest airside heattransfer coefficient. Comparatively, S1 has a minor radius of thehighest value and the smallest elliptic aspect ratio among theoptimums, which makes the tube rather closer to a standard obroundprofile with relatively low tube axis ratio. The more obroundliketubes in S1 force the flow to suddenly change its direction around thesharp corner and increase local turbulence kinetic energy from thepreceding tubes (Fig. 15). The single-objective optimumS1 improvesheat transfer coefficient by 3% compared to the baseline tube bankgeometry, which means it can use less thermal transfer surface toachieve the same cooling performance and thus reduce theprecooler mass.The heat transfer effectiveness ε and heat transfer rate per unit

precooler volume for the selected tube banks with equal number oftubes are presented in Table 4. Given that the total mass flow rate

Table 4 Comparison of heat transfer effectiveness for the selected representatives

Tube typeThermal transferarea per tube, m2

Heat transfer coefficient,kW∕�m2 · K�

Percent differencein ε, %

Heat transfer rate perunit volume, GW∕m3

Different Percentin _Q∕Volume, %

S1 0.00314 2.101 3.1 2.912 31.7S2 0.00326 1.679 −14.6 1.967 −11.0S3 0.00315 2.006 −1.3 3.081 39.3O1 0.00316 1.973 −2.7 2.925 32.3Baseline 0.00314 2.039 0.0 2.211 0.0

Fig. 16 Comparisons of design variables and total pressure loss.

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through the precoolers is constant regardless of the change in tubebank geometries, heat transfer effectiveness for the precoolers in thepresent study is only related to the airside heat transfer coefficient andthermal transfer surface area. For the given tube wall volume and aunit tube length perpendicular to the two-dimensional flow domain,the thermal transfer surface area of the unconventional tube shape isapproximated by the wall midline circumference [Eq. (7)]. Apartfrom S1, heat transfer effectiveness of the selected individuals are allless than the circular tube bank. The total pressure loss optimum S2and compactness optimum S3 have reductions in ε by 14.6 and 1.3%,respectively. This indicates that, when the precooler design isoptimized toward the other objective functions, the precooler willneed to require more tubes to provide extra thermal transfer area toachieve the heat transfer rate as the baseline model. Higher heattransfer rate per unit precooler volume demonstrates that therepresentative individuals require less volume space than the baselinemodel for a given heat transfer rate except S2. S1, which has the

improved heat transfer effectiveness and heat transfer per unitvolume, is able to considerably reduce the mass and size of theprecooler with no compromise in heat exchange power rate.

4. Total Pressure Loss

Total pressure loss through the tube bank for subsonic flow is thesum of losses caused by friction and pressure drag. Figure 16compares the total pressure loss of the selected individuals. Theconstituents of drag coefficient for the representative individuals andthe baseline model are displayed in Fig. 17. It can be seen that, for allcases, the pressure drag coefficient is responsible for over 85% of theresistance in the precooler.The total pressure loss optimal pointS2 has a relatively smallminor

radius b and the largest elliptic aspect ratio e, which makes the tubeshape rather close to the elliptic shape with relatively high tube axisratio. A more ellipticlike tube shape gives a significant reduction inthe pressure drag coefficient, leading to approximately 81.3%reduction in total drag coefficient compared to the baseline. Also, thetransverse pitch ratio of S2 is the largest among all. The consequentslower fluid motion contributes to a smaller drag force. This is alsoexemplified by the comparison of S1 and the baseline model. Theapparent increase in the total pressure loss in S1 is attributed to thedrastic reduction in transverse pitch-to-diameter ratio, although S1has a smaller tube frontal area just like other optimal individuals,which mitigates the flow blockage. Nevertheless, the S1 configurationwith a transverse pitch of 1.2 mm allows accommodating morecolumns of tubes in the transverse direction than the baseline(ST � 2.2 mm). For a given total number of tubes, the total pressureloss of the S1 configuration will still be far less than the baseline tubebank configuration with the reduced number of rows in thelongitudinal direction.

5. Compactness

Figure 18 compares the compactness of the representativegeometries, assuming that the tube length is constant. Thecompactness of S3 is improved by 108%, whereas the heat transferper unit precooler volume is improved by 39% compared to thebaseline configuration; in otherwords,S3 can significantly reduce themass and volume of the precooler structure. This is the maximumpossible improvement limit, and in reality, the geometry design has totake into account stress distribution over the tube cross section in anunconventional tube.

C. Global Sensitivity Analysis

The global sensitivity result is based on the prediction provided bythe best surrogate model with least error. The analysis employssensitivity indices bywhich the total variance in the output parameteris decomposed into a sum of the variation in input parameters [41].

Fig. 18 Compactness of precoolers comparison.

Fig. 19 Global sensitivity indices on the objective functions.

Fig. 17 Drag coefficient component and comparison.

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The first-order sensitivity index Ki represents the main effect (firstorder) of a design variable on an objective function. The totalsensitivity indexKt is the sum of all the effects of a designvariable onan objective function, including the main effect and all theinteractions (higher orders) involving the variable. The sensitivityindicesKi andKt of the objective and constraint functions are plottedin Fig. 19. The sumofKt of all input parameters that does not equal to1 indicates that the output model is not purely additive.It can be seen that the elliptic aspect ratio e, which determines the

tube adverse pressure gradient, and transverse pitch-to-diameter ratiopt exert dominant influences on the heat transfer coefficient and totalpressure loss for the external flow, whereas the minor axis radius b aswell as the transverse and longitudinal pitch-to-diameter ratios are thedeterminants of the objective function compactness, indicated bytheir first-order Ki. Moreover, the slight increase in Kt than in Ki oneach objective functionsmeans that design parameters can havemorerespective impact when interact with other variables. As for thesensitivity of the constraint function, the variation in themass balanceof the simulated flow domain is a result of the coupling effects of allinput parameters, rather than being caused by a particular parameter.It is understandable that the reason that a design solution fails to meetthe numerical convergence criterion is because the tube bankgeometry is not practically viable due to the development of transonicor even supersonic flow. The mathematical sensitivity result isconsistent with the observation that input parameters b, e, pt, and pL

are equally important factors for designing a multiple-objectiveoptimal precooler.

VII. Conclusions

In this paper, a multi-objective design optimization of a precoolingheat exchanger for hypersonic airbreathing propulsion has beenconducted with respect to three design objectives (i.e., maximizationof airside heat transfer coefficient and compactness, and minimizationof the total pressure loss for the external flow). The tube bank geometrythat is represented by the minor axis radius, elliptic aspect ratio, andtransverse and longitudinal pitch-to-diameter ratios has been optimizedbymeans of surrogate-assisted evolutionary algorithms coupled with ahigh-fidelity computational-fluid-dynamics solver for the compressibleflow, yielding a Pareto-optimal front indicative of tradeoff character-istics among the three objectives for this problem. For given constantinflow velocity and density, higher heat transfer effectiveness andtube matrix compactness are found to be associated with the highertotal pressure loss across the heat exchanger. The study with thismethodology demonstrates the potential to develop optimumconfigurations that yield greater heat transfer per unit volume andless total pressure loss, as compared to the circular tube bankconfiguration commonly adopted in industry. The tube shape as wellas tube pitch ratios are found to have significant interactive effects ondetermining the maximum flow Mach number throughout a tubebank, hence feasibility of the design.The conclusions derived from the present study must be

interpreted with care. The current approach using a two-dimensionalReynolds AveragedNavier Stokesmodel assuming steady flowfieldsin specified flow conditions at Mach 0.2 inherently imposeslimitations on the investigation of various factors that can be presentin practice including three-dimensionality, unsteady flow behavior,and real-gas effects. The influence of such factors would need to beconsidered carefully with appropriate numerical modeling orexperimental testing in the application of the insights gained from thepresent analysis to the actual design process. The influence of thevariations in theMach number and Reynolds number due to changesin the operating speed and altitude on the performance and behaviorwill also be a subject of future work.

Acknowledgments

The authors are grateful to Tapabrata Ray and Amitay Isaacs at theUniversity of New South Wales Canberra for providing the advancedmulti-objective design optimization capability developed in the group.

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