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Development of a conceptual-level thermal management system

design capability in OpenConcept

Benjamin J. Brelje∗† , John P. Jasa‡, and Joaquim R.R.A. Martins§

University of Michigan, Ann Arbor, Michigan

Justin S. Gray

NASA Glenn Research Center, Cleveland, OH

Abstract

Multiple studies have shown, somewhat unexpectedly, that thermal management constraints will be a key consideration

for hybrid- and all-electric aircraft designers. While airplane sizing and multidisciplinary analysis and design optimization

tools with support for electriﬁcation have been developed, most of these tools do not support thermal analysis and

almost all are closed-source or otherwise unavailable to the broader research community. In 2018, we introduced the

OpenConcept library, a toolkit for conceptual-level design optimization of aircraft with unconventional propulsion

built using the OpenMDAO framework. However, at that time, we had not yet implemented an eﬃcient, physics-based

thermal analysis capability or the associated numerical methods to solve the problem. In this paper, we introduce the

thermal analysis extensions to OpenConcept. We provide implementation details for an open-source, physics-based

thermal management system (TMS) analysis and design capability in OpenConcept. We develop governing equations

for component-based, air-cooled and liquid-cooled thermal management systems. We detail the implementation of

thermal mass, heat sink, heat exchanger, incompressible duct, coolant loop, and refrigeration components with analytic

derivatives in OpenMDAO. We also describe a method for computing time-dependent electrical component temperatures

throughout a mission proﬁle using OpenMDAO’s Newton solver. To illustrate thermal eﬀects, we consider a tradespace

study on a Beech King Air with a series hybrid electric propulsion system. We optimize the aircraft for minimum

fuel burn with and without thermal constraints and TMS penalties. The optimizer sizes the TMS components to keep

component temperatures within limits while minimizing the associated fuel burn penalty. We demonstrate reasonable

robustness of the thermal model across a broad range of aircraft designs and compare the optimal designs with and

without thermal constraints and TMS penalties.

I. Introduction

Electric aircraft propulsion has emerged as a widely popular topic in the aerospace research community. The initial

design studies quickly identiﬁed shortcomings in existing aircraft analysis techniques and tools. Several analysis codes

with similar levels of ﬁdelity for integrating energy used over a mission have been announced [

1

–

10

], but only one

has been open-sourced or made publicly available [

10

]. There is also signiﬁcant duplication of eﬀort in the research

community, particularly within the area of electrical system modeling and mission analysis. Despite multiple industry

and government studies demonstrating the need to include thermal constraints in analysis and optimization at the

conceptual level [

1

,

2

,

12

,

13

], no publicly-available electric propulsion mission analysis and sizing code supports

thermal analysis.

We have recently introduced OpenConcept (openconcept.readthedocs.io) a new, open source, conceptual design

and optimization toolkit for aircraft with electric propulsion [

11

]. OpenConcept consists of three parts: a library of

simple conceptual-level models of common electric propulsion components; a set of analysis routines necessary for

aircraft sizing and optimization; and several example aircraft models. All of OpenConcept’s codes compute derivatives

eﬃciently and accurately, enabling the use of OpenMDAO 2’s [

14

] Newton solver, as well as gradient-based optimization

methods.

In prior work, we performed a case study involving the electriﬁcation of existing turboprop airplanes [

11

]. We deﬁned

a series-hybrid electric propulsion architecture for the Beechcraft King Air and solved more than 750 multidisciplinary

∗PhD Candidate, Department of Aerospace Engineering

†The author is also an employee of The Boeing Company; this article is written in a personal capacity.

‡PhD Candidate, Department of Aerospace Engineering

§Professor, Department of Aerospace Engineering

1

300 400 500 600 700 800

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Fuel mileage (lb/nmi)

0.0

0.3

0.6

0.9

1.2

1.5

1.8

300 400 500 600 700 800

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Degree of hybridization (electric percent)

0

20

40

60

80

100

Fig. 1 Minimum fuel burn MDO results from our previous work [11]

design optimization (MDO) problems for diﬀerent combinations of range and speciﬁc energy (Fig. 1), demonstrating that

OpenConcept is a ﬂexible and eﬃcient way of doing trade space exploration for unconventional propulsion architectures.

In our previous version of OpenConcept, conceptual-level models of heat exchangers, heat sinks, coolant loops, heat

pumps, and associated ﬂow paths had not yet been developed, and thermal constraints were not imposed for the results

shown Fig. 1 [11].

In the broader literature, a few attempts at physics-based thermal management system (TMS) modeling of electric

aircraft have been made [

15

–

17

], but none of the codes have been publicly released or open-sourced. The primary

purpose of this paper is to describe our thermal modeling approach and the implementation of the thermal components.

We then analyze the eﬀect of thermal constraints by repeating the King Air tradespace study with TMS design variables.

II. Thermal Management System Modeling

The thermal management system of an electric (or hybrid-electric) aircraft removes waste heat from the electronic

components. Unlike conventional turbine-powered aircraft, electric aircraft have two features that signiﬁcantly increase

the magnitude of the thermal management challenge. First, while turbine engines have lower eﬃciency, they exhaust

their waste heat to the free stream and away from the aircraft. In contrast, Ohmic resistance and eddy current losses in

electrical components generate heat within the components themselves and require designers to provide a way to carry

away the heat. Second, electrical components must be kept at fairly low temperatures to operate properly, which means

their waste heat is “low-quality” and much more diﬃcult to reject from the components.

There are two general design approaches to aircraft thermal management systems: direct air cooling and liquid

cooling. The air-cooled approach uses carefully-designed heat sinks to enhance convection from each electrical

component to freestream air. The X-57 Maxwell demonstrator uses this approach [

15

,

18

]). An advantage of this

approach is system simplicity and reliability. A major disadvantage is that each electrical component requires direct

access to an air ﬂow path, increasing conﬁguration complexity and potentially increasing drag as well. The liquid-cooled

approach uses coolant loops to transfer heat from the electrical components throughout the aircraft to a heat exchanger

that can reject the heat to the air [

12

]. This approach likely reduces the number of cooling air ducts. It also provides

the option to use a refrigeration cycle or a fan to improve heat rejection at low airspeed. However, the liquid cooling

architecture is a more complex system design (with more failure modes and moving parts). Some aircraft may use a

combination of liquid cooling and direct air cooling. A notional liquid-cooled TMS architecture is illustrated in Fig. 2.

The following subsections detail the physics and numerical methods governing each component of the TMS as modeled

in OpenConcept.

2

Coolant

Reservoir

Battery

Generator

Turboshaft Motor

Heat Exchanger

Freestream

Coolant Loop

Electrical

Mechanical

Air Cycle

Machine

qout

qout

qout

Fig. 2 Example of liquid-cooled thermal management system architecture

A. Component Temperatures

All existing OpenConcept electrical components (

SimpleMotor

,

SimpleGenerator

,

SOCBattery

) have a

heat_out

output variable that computes the heat generation rate of the component at the current operating point. The

components produce heat as a fraction of operating power via an assumed eﬃciency loss, though higher-ﬁdelity heating

models could be used in the future. If the user wishes to track thermal constraints on a component, they must add an

instance of a

ThermalComponent

to the model and properly connect it to the electrical component’s heat output. The

user can either solve for quasi-steady temperatures at each analysis point, or time-accurate temperatures.

The quasi-steady formulation relies on OpenMDAO’s Newton solver to compute component temperatures that

satisfy conservation of energy. The implicit problem, implemented in ThermalComponentMassless, is:

compute Tcomp (1)

such that R(Tcomp)=qcomp −qout =0(2)

where

Tcomp

is the component temperature,

qcomp

is the heat generation rate of the electrical component, and

qout

is the

instantaneous heat rejection rate due to cooling. The heat rejection rate is computed as a function of the component

temperature (Tcomp ) and a number of other heat transfer parameters (introduced in Section II.B).

qout =q(Tcomp, . . .)(3)

The quasi-steady formulation becomes less accurate as the thermal mass increases. Even lightweight aerospace-grade

electrical components have a signiﬁcant thermal mass, and at low-speed conditions (such as the beginning of the takeoﬀ

roll), neglecting thermal mass is likely to result in unrealistic high temperatures and drive oversized TMS designs.

Therefore, we recommend using a time-accurate model, which can be expressed as

dTcomp

dt

=

qcomp −q(Tcomp, .. .)

mcompcp

(4)

Tcomp =∫tf

t0

dTcomp

dt dt (5)

where

qcomp

is the heat generated by the electrical component. Equations

(4)

and

(5)

depend on each other and hence

form an implicit cycle that can be solved in OpenMDAO using its built-in Newton solver.

3

The rate

dTcomp

dt

is computed by the

ThermalComponentWithMass

component. A numerical scheme is required to

compute the time integral in Equation

(5)

. Our

Integrator

component provides the user a choice of a fourth-order

accurate Simpson’s Rule discretization (as previously described [

11

]), or the BDF3 discretization, which sacriﬁces some

accuracy for better stability on stiﬀ systems. Both of these integration methods are solved implicitly in vectorized form

all-at-once using the Newton solver (without time marching). This means that the time integration and the implicit ODE

are solved simultaneously as one coupled nonlinear system. The user must specify an initial component temperature,

usually based on ambient conditions. Unlike the quasi-steady problem, the accuracy of the temperature proﬁle depends

on the time step chosen. A smaller time step increases the size of the OpenMDAO implicit problem that needs to be

solved and increases the computation time.

B. Component-Fluid Heat Transfer

So far, we have not addressed the question of how to compute

qout

from each component, which represents the

convective heat transfer rate from the component to a ﬂuid stream. For a liquid-cooled component, the ﬂuid stream

is a coolant like propylene glycol, whereas for an air-cooled component the ﬂuid stream comes from freestream

air. In nearly every case, designers use enhanced heat transfer surfaces, such as microchannels or ﬁnned heat sinks.

The

ConstantSurfaceTemperatureColdPlate_NTU

component implements a microchannel cold plate and is a

reasonable choice for liquid-cooled and air-cooled applications. We assume that the thermal conductivity of the

electrical component is large relative to the cooling ﬂuid resulting in a constant channel surface temperature in the

streamwise direction. We further assume that the aspect ratio of each channel is large and thus approximates the local

heat transfer properties using the theoretical result for inﬁnite parallel plates. The convective heat transfer coeﬃcient

can be computed as

h=Nu k

dh

,(6)

where

Nu

is the Nusselt number (which is set to 7.54 by default for constant temperature inﬁnite parallel plates [

19

]),

k

is the thermal conductivity of the ﬂuid, and

dh

is the hydraulic diameter of the channel. For a high aspect ratio channel,

dh=2wh

w+h(7)

where

w

is the ﬂuid channel width and

h

is the ﬂuid channel height. We neglect entrance eﬀects for this high aspect

ratio microchannel. For air cooled applications using ﬁnned heat sinks, the user may wish to modify the heat transfer

coeﬃcient to account for ﬁn eﬃciency. To compute the overall heat transfer, we ﬁrst need to compute the heat transfer

surface area as

A=2L(w+h)Nparallel,(8)

where A is the overall heat transfer surface area,

L

is the length of the microchannel in the ﬂuid ﬂow direction, and

Nparallel is the total number of individual microchannels.

Given these convective properties, we compute the actual heat transfer using the NTU-eﬀectiveness method [

19

],

which is typically used for ﬂuid-ﬂuid heat exchangers where both ﬂuids change temperature during the exchange. In this

work, we assume that the heat transfer capability of the conductive component body is “inﬁnite” for the purposes of

the NTU-

method. Therefore, the heat transfer capacity of the cold plate is governed solely by the coolant material

properties and ﬂow rate. The heat transfer capacity is computed as:

Cmin =Û

mcoolant cp,coolant,(9)

where

Û

mcoolant

is the coolant mass ﬂow rate through the entire cold plate (not just a single channel) and

cp,coolant

is the

coolant’s speciﬁc heat capacity. The number of thermal units (NTU) is computed as

NTU =Ah

Cmin

,(10)

and heat transfer eﬀectiveness is

=1−e−NTU (11)

Finally, we can compute the heat transfer as

qout =Cmin(Tcomp −Tcoolant,in),(12)

4

h

w

tf

tp

Fig. 3 Cross-sectional geometry of the oﬀset strip ﬁn heat exchanger [21]

and the coolant outlet temperature as

Tcoolant,out =Tcoolant,in +qout

Cmin

.(13)

The user is responsible for setting reasonable values for channel geometry (

L

,

w

,

h

,

Nparallel

) so that the channel

ﬂow is laminar and the inﬁnite parallel plate assumption remains reasonable, and for ensuring that the component has

suﬃcient material volume to accommodate the cooling channels. This analysis also assumes that the cooling channel

weight is accounted for in the all-up weight of the component, which may not be the case for air-cooled external heat

sinks. In practice, we have found that the thermal resistance of the component-liquid heat transfer is much smaller than

for the liquid-air main heat exchanger, and that aircraft design problems are not that sensitive to cold plate channel

design parameters. However, the detailed design of internal cooling channels in electrical components is a challenging

problem in its own right.

C. Fluid-ﬂuid Heat Transfer

After heat from electrical components is transferred into the liquid coolant loop via the cold plate, the heat must be

rejected to the atmosphere. A reasonable choice for accomplishing this is a ducted compact heat exchanger. Like the

cold plate component above, we use the NTU-eﬀectiveness method to compute the heat transfer rate,

q=U Aoverall

NTU Tin,h −Tin,c,(14)

where

U Aoverall

is the overall heat transfer coeﬃcient times the corresponding heat transfer area,

Tin,hTin,c

are the ﬂuid

inlet temperatures, the number of thermal units is computed as

NTU =U Aoverall

Cmin

,(15)

and the heat transfer eﬀectiveness is

=ΦNTU,Cmin

Cmax ,(16)

where

Cmin,Cmax

are the maximum and minimum values of the ﬂuid heat transfer capacity

Û

m cp

for the hot and cold

sides, and

Φ

is an analytical or empirical function that depends on the ﬂow arrangement of the heat exchanger (for

example, crossﬂow) [20].

For this study, we use crossﬂow plate-ﬁn heat exchangers with oﬀset strip ﬁn geometry as described by Jasa et al.

[21]

. Oﬀset strip ﬁn heat exchangers are considered “compact” heat exchangers with high heat transfer to surface area

rates [

20

]. The geometric design of a heat exchanger varies to satisfy heat transfer, pressure loss, weight, and volume

requirements. Figure 3 illustrates a cross section of oﬀset strip ﬁn channels along with a commonly-used geometric

parameterization.

We use an empirical relation from Manglik and Bergles

[22]

to compute heat transfer and pressure loss speciﬁc to the

oﬀset strip ﬁn conﬁguration. By default, OpenConcept’s

HXGroup

component uses geometric parameters representative

of a air-liquid heat exchanger, with cold-side channel width and height 1 mm, and hot-side channel width 14 mm by

1.35 mm.

5

D. Fluid Reservoir

Any liquid cooling system needs a reservoir. The thermal mass of the ﬂuid in the reservoir may signiﬁcantly aﬀect

peak temperatures. We assume perfect mixing within the reservoir (that is, ﬂuid entering the reservoir is instantaneously

mixed with the existing ﬂuid). The rate of change of temperature within the reservoir can be computed using

dTreservoir

dt

=Û

mcoolant

mcoolant

(Tin −Treservoir),(17)

where

Treservoir

is the reservoir (and reservoir outlet) temperature,

Û

mcoolant

is the coolant mass ﬂow rate,

mcoolant

is the

mass of coolant in the reservoir, and Tin is the reservoir inﬂow temperature.

The

CoolantReservoir

group combines the rate equation

(17)

with an

Integrator

to solve for reservoir

temperatures at every time point, given an initial temperature. Quasi-steady thermal analysis cannot model the eﬀect of

a ﬂuid reservoir, which is purely a thermal mass eﬀect. When

Û

m/m

becomes large due to a small coolant mass relative

to the mass-ﬂow-rate, the time constant associated with the reservoir temperature becomes small. As

m

tends to zero we

approach the quasi-steady solution. A small time constant makes the thermal ODE very stiﬀ and introduces numerical

diﬃculties in the overall time integration problem.

E. Refrigeration Cycle

A refrigeration cycle can be used to increase the temperature of “low-quality" waste heat to reject it to the atmosphere

with a smaller heat exchanger. This process works similarly to a common household refrigerator, where relatively

low-temperature waste heat is raised to a higher temperature so it can be dissipated to the ambient surroundings. For

aircraft applications, this refrigeration cycle is often an air-cycle machine (ACM), in which air is used as the working

ﬂuid.

The model used in this study was previously described by Jasa et al.

[21]

and a schematic of the work and heat ﬂow

for this simpliﬁed cycle is shown in Fig. 4. The ACM is modeled as a closed-loop Brayton cycle, where the working

ﬂuid ﬂows through a low-temperature heat exchanger and accepts input heat, shown as

Qc

. Work (

W

) is then done on

the ﬂuid in the compressor, which increases the temperature and pressure of the ﬂuid. This heated ﬂuid then ﬂows

through a high-temperature heat exchanger, where the “high-quality" waste heat is rejected, shown as

Qh

. The ﬂuid then

goes through a turbine and expands, returning to a low temperature and pressure before returning to the low-temperature

heat exchanger.

We model the ACM using a system of equations adapted from Moran et al.

[23]

to capture the relevant physics

without adding unnecessary complexity to the model. From the lifting system equations, we get the following expression

for the heat load that must be dissipated using the duct heat exchanger:

Qh=W0

1−Tc/Th

=

ηpηfW

1−Tc/Th

,(18)

where

W0

is the eﬃciency-adjusted work,

Tc

and

Th

are the temperatures of the cooling ﬂuid at the electronics and duct

heat exchangers, respectively,

W

is the work coming from the shaft,

ηp

is the shaft power transfer eﬃciency, and

ηf

is

the friction loss eﬃciency. We can then solve for the cold-side heat load, Qc, and get

Qc=Qh−2−ηfηpW.(19)

Using these equations, we can determine the amount of heat transfer on both the hot and cold sides of the lifting system

based on the work that is put in. This system is implemented in OpenConcept as the LiftingSystemComponent.

F. Coolant Duct

Ducted radiators greatly reduce cooling drag compared to ﬁnned heat sinks in the freestream [

24

,

25

]. There are

two primary mechanisms for this. First, a duct that decelerates ﬂow prior to encountering the heat exchanger element

generally undergoes a lower total pressure loss. Second, the combination of duct and heat exchanger can act as a weak

ramjet providing a further modest oﬀset to the drag of the whole arrangement. For aircraft with high-temperature

cooling loads ﬂying at relatively high speeds, a large portion of the drag can be oﬀset or potentially even produce some

positive thrust. The most famous application of this weak ramjet concept (known as the Meredith eﬀect) is the North

American P-51 Mustang’s liquid engine cooling system [24].

6

The user has two options for computing cooling drag due to ducted heat exchangers. The ﬁrst option is an

incompressible approximation. Adapting the method of Theodorsen

[25]

, we model a duct with a frontal opening,

diﬀuser, heat exchanger, and nozzle (Fig. 5). The ﬂuid density everywhere in the duct is assumed to be

ρ∞

. Let

Ahex

be

the free ﬂow passage area of the heat exchanger, and

Ae

be the exit nozzle area. Let

∆p0,hex =f(Û

m)

be the pressure loss

across the heat exchanger as a function of the duct mass ﬂow rate

Û

m

. Let

ξe

be a static pressure loss as a function of

nozzle dynamic pressure, that is

∆p0,e=ξeqe

. We assume that the nozzle expands the ﬂow back to the freestream

static pressure

p∞

, though this assumption would not hold if a variable-area exit door or cowl ﬂap were used. The total

pressure at the exit is then computed as:

p0,e=p0,∞−∆p0,hex −∆p0,e=p∞+1

2ρU2

∞−∆p0,hex −∆p0,e=pe+1

2ρU2

e(20)

Substituting ∆p0,e=ξe1

2ρU2

eand rearranging we obtain:

Ue=sU2

∞−2

ρ(pe−p∞)+∆phex

1+ξe

,(21)

By continuity:

Û

m=AeρUe.(22)

We compute net force by balancing the change in ﬂuid momentum (

Û

m∆U

) and pressure forces. To account for inlet, duct,

and nozzle losses not otherwise accounted for, we apply a factor (

Cf g =

0

.

98) to gross thrust in the drag computation

and obtain:

Fnet =Û

m(UeCf g −U∞)+AeCf g(pe−p∞).(23)

The incompressible duct computation is implemented as the

ExplicitIncompressibleDuct

component in

OpenConcept. Alternatively, the

components/ducts

package contains the

ImplicitCompressibleDuct

group, that

uses a more sophisticated 1D thermodynamic cycle modeling approach to compute drag. Isentropic relations are used to

solve for Mach numbers and ﬂow properties implicitly using OpenMDAO’s Newton solver. The compressible model

captures Mach number and heat addition eﬀects on net cooling drag. However, the additional ﬁdelity is usually not

meaningful for low-speed general aviation airplanes with moderate cooling heat loads, and the compressible relations

introduce many implicit states and some robustness issues to the overall MDO problem.

III. Case Study: Revisiting the Series Hybrid Twin

To exercise the TMS model and assess the impact of thermal constraints on the design space, we revisit our previous

MDO trade space exploration study of a series hybrid twin turboprop [

11

]. Our baseline aircraft is a Beechcraft King

Air C90GT with a drop-in replacement series-hybrid propulsion system replacing the turboprop engines.

The series-hybrid electric propulsion architecture is illustrated in Fig. 6. To enable the aircraft to continue safe

ﬂight and landing after loss of any single component on takeoﬀ, the propulsion system uses two electric motors, two

propellers, and a battery large enough to provide full takeoﬀ power in the event of engine loss. These features should

provide the same level of redundancy of the conventional twin turboprop conﬁguration. Speciﬁc power, eﬃciency, and

cost assumptions for individual powertrain components are listed in Table 1.

Table 1 Powertrain technology assumptions [11]

Component Speciﬁc Power (kW/kg) Eﬃciency PSFC (lb/hp/hr)

Battery 5.0 97% –

Motor 5.0 97% –

Generator 5.0 97% –

Turboshaft 7.15∗– 0.6

∗Not including 104 kg base weight

8

Battery

Generator

Turboshaft

Coolant

Reservoir

Motor 2

Motor 1

Heat Exchanger

Freestream

Coolant Loop

Electrical

Mechanical

Fig. 6 Systems architecture for the twin series hybrid case study.

A. Mission Analysis Methodology

To compute mission fuel burn and other performance constraint values, we perform a full mission analysis at every

MDO iteration consisting of a balanced-ﬁeld takeoﬀ (with loss of one propulsor at the

V1

speed), climb, cruise, and

descent. We use the same mission analysis methodology as our previous work [11], with the exception noted below.

OpenConcept’s balanced ﬁeld takeoﬀ length computation consists of two branched trajectories composed of ﬁve

piecewise segments:

1) Takeoﬀ roll at full power from V0to V1

2) Takeoﬀ roll at one-engine-inoperative (OEI) power from V1to VR

3) Rejected takeoﬀ with zero power and maximum braking from V1to V0

4) Transition in a steady circular arc to the OEI climb-out ﬂight path angle and speed

5) Steady climb at V2speed and OEI power until an obstacle height hois reached

We compute the balanced ﬁeld takeoﬀ by varying

V1

until the accelerate-go distance (segments 1, 2, 4, and 5) is at least

as long as the accelerate-stop distance (segments 1 and 3).

During the takeoﬀ roll (segments 1, 2, and 3), the force balance equation is:

®

dV

dt

=®

T−®

D−µ(mg−®

L).(24)

In our previous work, we had used a method that integrates segments 1, 2, and 3 with respect to velocity instead of

time [

26

]. The advantage of this method is that it exhibits good numerical stability; however, it cannot be used to

integrate general ODEs including the thermal models of Section II. During the takeoﬀ roll, the airplane is producing

maximum heat and has minimum ability to reject the heat. Therefore, instead of neglecting heating during takeoﬀ, we

changed to a time-based integration scheme capable of computing accurate time histories of all parameters. Times,

distances, and altitudes for segments 4 and 5 are computed using prescribed kinematics from Raymer

[26]

; however, we

time-integrate all the other states, including thermal loads and battery state.

The climb, cruise, and descent segments are computed using steady ﬂight equations. At each ﬂight condition, the

Newton solver sets a throttle parameter such that the following residual equation is satisﬁed:

®

Rthrust =®

T−®

D−®

mgsin(®

γ).(25)

The user speciﬁes the true airspeed and the vertical speed at each mission point, as well as one constraint per mission

segment (e.g., an altitude for top of climb or mission range for cruise). OpenMDAO then computes the segment duration

9

required to satisfy the constraints for climb, cruise, and descent using the Newton solver. OpenConcept integrates range,

altitude, fuel ﬂow, battery SOC, and all other thermal states.

Cruise drag is computed using a drag polar with constant coeﬃcients. We assumed an Oswald eﬃciency

e=

0

.

8

and matched computed range to published range for a design mission by setting

CD0=

0

.

022. Weights are computed

parametrically based on wing area, aspect ratio, MTOW, and other high-level parameters. The empty weight was

calibrated to match the King Air C90GT baseline by matching our model’s parametric operating empty weight (OEW) to

the published OEW (minus engine weight in both cases) by applying a factor of 2.0 to our model’s computed structural

weight (based on rough textbook formulas).

B. Optimization Without Thermal Constraints

We begin by re-running the series hybrid twin tradespace exploration from our previous work [

11

]. We are interested

in the optimal aircraft design at a variety of battery technology levels (quantiﬁed by the speciﬁc energy,

eb

) and design

mission ranges. Therefore, we run a grid of MDO problems formulated as follows:

minimize: fuel burn +0.01MTOW

by varying:

MTOW

Sref

dprop

Wbattery

Pmotor (rated)

Pturboshaft (rated)

Pgenerator (rated)

HE(degree of hybridization w.r.t energy)

subject to scalar constraints:

RTOW =WTO −Wfuel −Wempty −Wpayload −Wbatt ≥0

Rbatt =Ebatt,max −Ebatt,used ≥0

Rvol =Wfuel,max −Wfuel ≥0

BFL ≤4452 ft (no worse than baseline)

engine out climb gradient ≥2%

Vstall ≤81.6kts (no worse than baseline)

and vector constraints:

®

Pmotor ≤1.05Pmotor (rated)

®

Pturboshaft ≤Pturboshaft (rated)

®

Pgenerator ≤Pgenerator (rated)

®

Pbattery ≤Wbattery ·pb

The objective function was chosen in order to prioritize reducing tailpipe carbon emissions. However, certain

combinations of speciﬁc energy and range result in aircraft with zero fuel burn. Optimizing for fuel burn alone in these

cases is an ill-posed problem. Therefore, we add a small contribution of MTOW to the objective function in order

to force the optimizer to design reasonable all-electric aircraft. A potentially better objective function would be to

minimize total carbon emissions. This approach introduces location dependence into the problem, since electricity is

generated using more or less carbon-intensive methods in diﬀerent parts of the world. The vectorial constraint quantities

represent parameters tracked over time during a mission. Each entry in the vector represents an individual point in time.

Each mission segment (climb, cruise and descent) consists of 10 discrete time intervals.

We optimized one airplane at each combination of speciﬁc energy (from 250 to 800 Wh/kg) and design range (300

to 700 nautical miles). Each airplane ﬂew with the same climb, cruise, and descent speeds (both indicated airspeed

10

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Fuel mileage (lb / nmi)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

MTOW (kg)

4000

4200

4400

4600

4800

5000

5200

5400

5600

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Cruise hybridization

0.2

0.4

0.6

0.8

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Battery weight (kg)

200

400

600

800

1000

1200

1400

1600

1800

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Motor rating (kW)

340

360

380

400

420

440

460

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Engine rating (kW)

100

200

300

400

500

600

700

Fig. 7 Minimum fuel burn MDO results without thermal constraints.

and vertical speed). We used the

scipy.optimize

implementation of the SLSQP optimization algorithm to solve the

problem.

The results are similar to the previous study despite some changes to the mission analysis methods. Figure 7 exhibits

the same multimodal tradespace as we found before. At long ranges and poor

eb

, little battery is used (only enough to

provide backup power on takeoﬀ) and the airplane is essentially turboelectric. At short range and high

eb

, the mission is

ﬂown entirely on battery and no fuel is used. In between these two extremes, the optimizer prefers to use all of the

allotted maximum takeoﬀ weight until it hits the upper bound (5700 kg), above which a type rating is required in many

jurisdictions including the United States and European Union.

C. Optimization with Thermal Constraints

In this work, we modiﬁed the aircraft propulsion model to include thermal management of the motor and battery.

We added a

ThermalComponentWithMass

to the motor (lumping both motors together) and to the battery pack.

Thermal mass of both components was computed using a speciﬁc heat of 921 J/kg/K (representative of aerospace-grade

aluminum). We connected cold plates of both components in series using a liquid cooling system using a propylene

glycol and water mixture with a speciﬁc heat of 3801 J/kg/K [

27

]. The coolant loop rejects heat via a ducted heat

exchanger. We neglect the drag-oﬀsetting eﬀect of heat addition and use a

ExplicitIncompressibleDuct

to model

the air mass ﬂow and drag. We use OpenConcept’s default geometric parameters for the oﬀset strip ﬁn heat exchanger.

Finally, we include a liquid coolant reservoir upstream of the heat exchanger. We include the weight of the coolant and

heat exchanger in the empty weight of the airplane, and include the drag contribution of the duct and heat exchanger.

Figure 8 shows proﬁles of mission parameters for a single aircraft design at 250 Wh/kg and 400 nmi range. The ﬁgure

highlights the importance of time-accurate thermal analysis. During takeoﬀ and low-altitude climb, heating is at its

maximum and convective heat transfer capability is at a minimum (due to higher atmospheric temperature and lower

coolant duct mass ﬂow). A quasi-steady thermal analysis would predict very high temperatures during this part of the

mission. However, because the thermal components have considerable thermal mass, the maximum temperature is not

11

reached until the top of the climb phase. Sizing the thermal management system to a quasi-steady analysis at the most

critical condition (early in the takeoﬀ roll) would result in an oversized heat exchanger and unnecessarily high drag and

weight penalty.

We also add several design variables and constraints to the previous problem. We let the optimizer size the heat

exchanger width and area of the duct nozzle, thus allowing it to trade oﬀ weight and drag for equal heat rejection

capability. We also allow the optimizer to size the coolant reservoir. We constrain the time-accurate temperatures of the

motor and battery pack to stay within operating limits (90

°

C for the motor and 50

°

C for the battery). The full MDO

problem is as follows:

minimize: fuel burn +0.01MTOW

by varying:

MTOW

Sre f

dpr op

Wbattery

Pmotor (rated)

Pturboshaft (rated)

Pgenerator (rated)

HE(degree of hybridization w.r.t energy)

Anozzle (cooling duct outlet cross-sectional area)

nwide (number of heat exchanger cells wide)

mcoolant (coolant reservoir mass)

subject to scalar constraints:

RTOW =WTO −Wfuel −Wempty −Wpayload −Wbatt ≥0

Rbatt =Ebatt,max −Ebatt,used ≥0

Rvol =Wfuel,max −Wfuel ≥0

BFL ≤4452ft (no worse than baseline)

engine out climb gradient ≥2%

Vstall ≤81.6kt (no worse than baseline)

and vector constraints:

®

Pmotor ≤1.05Pmotor (rated)

®

Pturboshaft ≤Pturboshaft (rated)

®

Pgenerator ≤Pgenerator (rated)

®

Pbattery ≤Wbattery ·pb

®

Tmotor ≤90◦C

®

Pbattery ≤50◦C

Figure 9 shows the design variables and selected responses at the optimal points across the trade space. The motor

temperature constraint is always active at the top of climb for all the designs in the tradespace (and so is not shown in

Fig. 9). The optimizer varies the duct nozzle area (to vary cooling air mass ﬂow) and motor size (to add thermal mass)

such that the motor temperature reaches the limit at the top of the climb. The heat exchanger width converges to its

upper bound at virtually every point in the design space, while coolant mass converges to its lower bound at every point.

Figure 10 shows the diﬀerence in key variables (including fuel mileage) after accounting for thermal design and

thermal constraints. While fuel mileage worsened at every point in the design space, the impact was much larger on

certain combinations of speciﬁc energy and design range. At long range and low battery speciﬁc energy, and at short

range and high speciﬁc energy, there was little eﬀect. The long range design with low

eb

are essentially turboelectric

and beneﬁt from light weight and low battery waste heat; there is simply less overall heat to reject, thus minimizing the

12

0 50 100 150 200 250 300 350 400

range

0

5000

10000

15000

20000

25000

30000

Altitude (ft)

0 50 100 150 200 250 300 350 400

range

0

25

50

75

100

125

150

175

Indicated airspeed (knots)

0 50 100 150 200 250 300 350 400

range

500

0

500

1000

1500

Vertical speed (ft/min)

0 50 100 150 200 250 300 350 400

range

3800

3850

3900

3950

4000

4050

Weight (kg)

0 50 100 150 200 250 300 350 400

range

10

20

30

40

50

60

70

80

90

Motor temperature (C)

0 50 100 150 200 250 300 350 400

range

5

0

5

10

15

20

25

30

Battery temperature (C)

Fig. 8 Mission trajectories for a 400 nmi mission (eb=250)

13

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Fuel mileage (lb / nmi)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

MTOW (kg)

4000

4200

4400

4600

4800

5000

5200

5400

5600

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Cruise hybridization

0.2

0.4

0.6

0.8

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Battery weight (kg)

200

400

600

800

1000

1200

1400

1600

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Motor rating (kW)

350

400

450

500

550

600

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Engine rating (kW)

100

200

300

400

500

600

700

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Wing area (

m

2)

24

26

28

30

32

34

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

HX duct nozzle area (

m

2)

0.01

0.02

0.03

0.04

0.05

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Battery temp max (C)

20.0

22.5

25.0

27.5

30.0

32.5

35.0

37.5

Fig. 9 Minimum fuel burn MDO results with thermal constraints.

14

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Fuel mileage (lb / nmi)

0.2

0.1

0.0

0.1

0.2

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

MTOW (kg)

800

600

400

200

0

200

400

600

800

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Cruise hybridization

0.15

0.10

0.05

0.00

0.05

0.10

0.15

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Battery weight (kg)

400

200

0

200

400

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Motor rating (kW)

150

100

50

0

50

100

150

300 400 500 600 700

Design range (nmi)

300

400

500

600

700

800

Specific energy (Whr/kg)

Engine rating (kW)

80

60

40

20

0

20

40

60

80

Fig. 10 Diﬀerence in optimal designs due to thermal constraints (positive = thermally-constrained higher)

associated penalty. The short range designs with high

eb

use no fuel to begin with, so their fuel burn remains at zero

even as they use more energy; instead, we see the thermal management penalty as an increase in MTOW. Between

these two extreme designs, the heavy hybrid airplanes generate a large amount of waste heat and burn appreciable fuel,

making the impact of thermal constraints more signiﬁcant.

A very interesting trend emerged in the motor sizing design variable. The optimizer greatly oversized the motors in a

band in the heart of the tradespace (seen as a band of red from middle left to top right). In the rest of the tradespace, the

motor is sized by power required during climb. However, in the red band, the motor is being constrained by the thermal

problem. We suspect that this is a result of the sequencing of components in the thermal management system. We

designed the TMS architecture to provide the coldest coolant to the battery, since it has a lower operating temperature.

The consequence is that warmer coolant ﬂows into the motor. The motor inﬂow temperature varies slowly even as outside

temperature drops due to the thermal mass of the battery. The best solution available to the optimizer is to oversize the

motors to add thermal mass and avoid overheating at the critical top of climb point. Reordering the components may

result in an improvement in fuel burn in this part of the tradespace by better balancing peak temperatures between the

motor and battery.

IV. Conclusions

Thermal constraints are currently understudied compared to other disciplines in aircraft conceptual design, and

there are few publicly-available resources available for the research community to incorporate thermal constraints into

electric aircraft studies. To ﬁll this gap, we introduced thermal analysis and design capabilities within the OpenConcept

Python package. We demonstrate that thermal mass eﬀects are signiﬁcant when analyzing aircraft thermal trajectories,

particularly early in the mission when power is high and speeds and altitudes are low. Therefore, pseudo-steady thermal

models are not suﬃcient for the design of aircraft thermal management systems, because they can lead to dramatic

over-sizing. We integrated time-accurate thermal models into the mission analysis and used them to formulate constraints

15

in the aircraft design optimization problem. The time-accurate thermal analyses and derivatives were computed by the

OpenConcept package to enable eﬃcient gradient-based design optimization.

We showed that thermal constraints appreciably aﬀect the fuel burn and energy usage achievable in a series hybrid

architecture, but not uniformly throughout the tradespace. The non-uniform eﬀects make the impact of thermal

constraints on aircraft design somewhat non-intuitive and underscore the importance of including them early in the

design process. Electric aircraft architectures with a large percentage of battery power will be impacted by TMS

penalties, but because they burn little or no fuel, the penalty is seen as an MTOW and total energy increase, not a fuel

burn penalty. Conversely, turboelectric aircraft experience a modest TMS penalty due to lighter weight and lack of

battery heating. Hybrid-electric aircraft see the largest fuel burn penalty since they are heavier than turboelectric aircraft

(thus producing more motor waste heat) and use signiﬁcant quantities of batteries (producing yet more waste heat). We

also observed that the optimizer can ﬁnd creative ways to satisfy the thermal constraints (such as oversizing a motor to

add thermal mass and avoid a transient over-temperature condition).

Acknowledgements

The ﬁrst and second authors were supported by the National Science Foundation Graduate Research Fellowship

Program under Grant DGE 1256260. Any opinions, ﬁndings, and conclusions or recommendations expressed in this

material are those of the authors and do not necessarily reﬂect the views of the National Science Foundation. The

fourth author was supported by NASA ARMD’s Transformational Tools and Technologies project. This work was

also supported by the U.S. Air Force Research Laboratory (AFRL) under the Michigan-AFRL Collaborative Center in

Aerospace Vehicle Design (CCAVD), with Richard Snyder as the task Technical Monitor.

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