Development of a conceptual-level thermal management system design capability in OpenConcept

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.

Full text

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
=ΦNTU,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−ηfηpW.(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

Qc
Qh
W
Compressor Turbine
Fig. 4 The refrigeration cycle system [21]
p0,∞
Δp0,hx Δp0,e
Ahx Ae
m
.
Ue
U∞
q
Exit nozzle
Inlet Heat exchanger
Fig. 5 Ducted heat exchanger to reduce cooling drag.
7

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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