Alicia M. Zinnecker

N&R Engineering, Parma Heights, Ohio

Jeffryes W. Chapman

Vantage Partners, LLC, Brook Park, Ohio

Thomas M. Lavelle and Jonathan S. Litt

Glenn Research Center, Cleveland, Ohio

Development of a Twin-Spool Turbofan Engine 

Simulation Using the Toolbox for the Modeling and 

Analysis of Thermodynamic Systems (T-MATS)

NASA/TM—2014-218402

November 2014
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Alicia M. Zinnecker

N&R Engineering, Parma Heights, Ohio

Jeffryes W. Chapman

Vantage Partners, LLC, Brook Park, Ohio

Thomas M. Lavelle and Jonathan S. Litt

Glenn Research Center, Cleveland, Ohio

Development of a Twin-Spool Turbofan Engine 

Simulation Using the Toolbox for the Modeling and 

Analysis of Thermodynamic Systems (T-MATS)

NASA/TM—2014-218402

November 2014

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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Level of Review: This material has been technically reviewed by technical management. 
The Toolbox for the Modeling and Analysis of Thermodynamic Systems (T-MATS) is

a tool that has been developed to allow a user to build custom models of systems governed

by thermodynamic principles using a template to model each basic process. Validation of

this tool in an engine model application was performed through reconstruction of the Commercial Modular Aero-Propulsion System Simulation (C-MAPSS) (v2) using the building

blocks from the T-MATS (v1) library. In order to match the two engine models, it was necessary to address differences in several assumptions made in the two modeling approaches.

After these modifications were made, validation of the engine model continued by integrating both a steady-state and dynamic iterative solver with the engine plant and comparing

results from steady-state and transient simulation of the T-MATS and C-MAPSS models.

The results show that the T-MATS engine model was accurate within 3% of the C-MAPSS

model, with inaccuracy attributed to the increased dimension of the iterative solver solution

space required by the engine model constructed using the T-MATS library. This demonstrates that, given an understanding of the modeling assumptions made in T-MATS and a

baseline model, the T-MATS tool provides a viable option for constructing a computational

model of a twin-spool turbofan engine that may be used in simulation studies.

Nomenclature

Variables

ey

x flow error at the output of component x in model y (y = C is C-MAPSS, y = T is T-MATS)

fi

bld fractional bleed at station i  ̄fi

bld recalculated fractional bleed at station i

NR corrected shaft speed

P R pressure ratio

R-line value used to determine operating point on a compressor map

sy

x scaler for map x and model y (y = C is C-MAPSS, y = T is T-MATS)

Development of a Twin-Spool Turbofan Engine Simulation 

Using the Toolbox for the Modeling and Analysis of 

Thermodynamic Systems (T-MATS) 

Alicia M. Zinnecker 

N&R Engineering 

Parma Heights, Ohio 44130 

Jeffryes W. Chapman 

Vantage Partners, LLC 

Brook Park, Ohio 44142 

Thomas M. Lavelle and Jonathan S. Litt 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

"CTUSBDU

NASA/TM—2014-218402 1
Wf fuel flow rate (pps)

Wi flow rate at station i

Sub-/Superscripts

comp variable for compressor

cust (fixed) customer bleed

des fixed design value

map variable interpolated from component map

sc variable that has been scaled

turb variable for turbine

un variable that has been unscaled

Acronyms

BPR bypass ratio

C-MAPSS Commercial Modular Aero-Propulsion System Simulation

C-MAPSS40k Commercial Modular Aero-Propulsion System Simulation (40k)

HPC high-pressure compressor

HPT high-pressure turbine

IC initial condition

IS iterative solver (DIS = dynamic IS, SSIS = steady-state IS)

LPC low-pressure compressor

LPT low-pressure turbine

NPSS Numerical Propulsion System Simulation

pps pounds-mass per second

rpm revolutions per minute

T-MATS Toolbox for the Modeling and Analysis of Thermodynamic Systems

VBV variable bleed valve

VSV variable stator vanes

I. Introduction

There are numerous gas turbine engine simulations available, each with its own features that make it

appropriate for specific applications. Those applications for which high frequency/high fidelity information is important may require a simulation with volume dynamics, while for lower frequency applications,

simulations that only include speed and temperature dynamics, with the appropriate component performance characteristics, may suffice.1 The latter category is generally appropriate for control law design and

gas path diagnostic algorithm development.2 The Modular Aero-Propulsion System Simulation (MAPSS),3

the Commercial Modular Aero-Propulsion System Simulation (C-MAPSS),4–6 and the Commercial Modular

Aero-Propulsion System Simulation 40k (C-MAPSS40k),7, 8 fall into the latter category. Features such as

how components are modeled (lumped or stage-by-stage) or whether flows are modeled as zero- or higherdimensional also play a role in the fidelity of the simulation.9

Having a simulation readily available that captures the appropriate level of detail for a given task can

be of great benefit, but if a suitable simulation cannot be found, or if the desired model does not exist,

one may need to be created. Tools available to create customized engine models, component by component,

include the Numerical Propulsion System Simulation (NPSS),9 and the new Toolbox for the Modeling and

Analysis of Thermodynamic Systems (T-MATS).10 Although similar in modeling approach (both tools utilize

a modular, component-wise framework) and flexibility, NPSS and T-MATS offer different user experiences.

NPSS offers a platform for modeling engines through a code-based interface, while T-MATS provides a more

visual interface for model development. Additionally, T-MATS was developed as an open-source extension

for use in the MATLAB/Simulink R (The MathWorks, Inc.) environment, completely removing the need

for integration with this commonly-used simulation and modeling software, a time-consuming step of model

development with NPSS. Unlike NPSS, however, T-MATS requires the user to provide engine data (e.g.

maps and scaler) from other sources in order to run simulations. Despite the effort required to develop a new

NASA/TM—2014-218402 2
simulation, these tools offer a user the advantage of being able to define the level of fidelity in each component

(for example, developing a multistage compressor model as opposed to a lumped parameter model).

When developing a simulation using such a tool, it is important to confirm that the model is working

correctly, both to validate that the components were connected properly, and to validate the tool itself. This

paper describes the process of validating a model constructed using T-MATS that duplicates the performance

of the open-loop C-MAPSS (v2) engine model.

The engine plants used in this study are presented in Section II, with a discussion highlighting how

the T-MATS model was modified to be able to replicate the C-MAPSS engine. Section III describes the

integration of these plants with an iterative solver and validation of the steady-state and dynamic T-MATS

models against the C-MAPSS model. The process for constructing these models using T-MATS, and the

validation results, are summarized in Section IV.

II. Description of engine models

A generic twin-spool turbofan engine may be represented by the diagram in Fig. 1; this engine contains

three compressors, two turbines, a burner, and two nozzles connected through a series of ducts. The T-MATS

library contains blocks for modeling each component of an engine by providing information, such as maps

and scalers, that specifies how that component operates. In this paper, the C-MAPSS (v2) engine model,

described in Ref. 6, serves as a baseline for construction of a replica model using T-MATS. In the following

discussion of the C-MAPSS and T-MATS engine plants, the assumptions about the maps, scalers, and flow

errors in each model will be mentioned. It was important to address any differences in these assumptions

through modification of the baseline T-MATS library blocks to ensure this model adequately replicated the

C-MAPSS model. Step-by-step details of building the T-MATS engine plant model have been omitted as

they exceed the scope of the paper; they may be found in the T-MATS User’s Guide.10

A. C-MAPSS engine plant model

The C-MAPSS engine model is a computational model of a twin-spool turbofan engine that may operate

in open- or closed-loop. It is implemented in Simulink, the graphical development and simulation environment

of the widely-available MATLAB software tool, making it useful as a research platform for the controls and

health-management community.4–6 The model captures representative dynamics of a twin-spool 90, 000 lbf

thrust-class turbofan engine which, as depicted in Fig. 1, contains a fan, low-pressure compressor (LPC), and

low-pressure turbine (LPT) on one shaft, and a high-pressure compressor (HPC) and high-pressure turbine

(HPT) on the other. In addition to these five components, the engine contains a burner, two nozzles, five

ducts, and variable geometry. Locations along the engine flow path are indicated by the station numbers in

the figure; these will be used throughout the discussion of the models and simulation results.

Although it may be operated in open-loop by providing a fuel flow input profile, the C-MAPSS engine

is intended to operate in closed-loop with a controller that produces a fuel flow command to maneuver

Figure 1. Block diagram of a twin-spool turbofan engine with stations numbered as assigned in the T-MATS

engine model.

NASA/TM—2014-218402 3
the engine along a user-defined input profile without violating physical or safety limits in the engine. This

profile contains information on how the environmental conditions (altitude, Mach number, and ambient

temperature) and throttle change during a flight. In addition to fuel flow, the engine plant also requires

inputs related to the variable geometry to be provided: the position of the variable stator vanes (VSV) in

the HPC (not shown in Fig. 1) and the position of the variable bleed valve (VBV) located between the LPC

and HPC. These inputs are nominally scheduled based on the (corrected) speeds of the core shaft and fan

shaft, respectively. Off-nominal operation may be specified as part of the input profile, but is not considered

in this study for simplicity.

Operation of the C-MAPSS engine is most affected by the three compressors (fan, LPC, and HPC) and

two turbines (HPT and LPT). Each of these components is characterized by a set of maps relating (corrected)

shaft speed, (corrected) flow rate, efficiency, and pressure ratio (PR) of the component and corresponding

scalers to ‘size’ the maps based on a fixed design point. In this model, it is assumed that the (corrected) flow

rate and efficiency interpolated from the maps scale according to the relationship in Eq. (1), which holds for

both compressor and turbine maps.

(·)

un = sC

(·)(·)

map (1)

(Corrected) shaft speed, one of the variables used for interpolating the maps, scales similarly (Eq. (2)); for

the compressor map, the scaler sC

NR in Eq. (2) is replaced by sCNR/NRdes, effectively adjusting the scaler

based on a (fixed) design point, NRdes.

NRmap = sC

NRNRun (2)

The more-complex relationships in Eq. (3) are used, respectively, for unscaling (in a compressor) or scaling

(in a turbine) the PR across each component.

P Run = sC

P R(P Rmap − 1) + 1

P Rmap = sC

P R(P Run − 1) + 1 (3)

These relationships are reflected in how the component maps themselves are constructed; if a different scaling

relationship were desired, the corresponding map would have to be unscaled using the old relationship and

rescaled to the design point using the new relationship.∗ This process allows for operation of the component

to be unchanged under the new set of scaler relationships and will be utilized in construction of the T-MATS

engine plant.

The unique operating point of a compressor is defined by the (corrected) shaft speed and R-line, used

to interpolate (corrected) flow rate, efficiency, and pressure ratio from the component maps; for a turbine,

(corrected) shaft speed and PR define the operating point and are used for map interpolation. As the engine

moves through a flight profile, each compressor R-line and turbine PR must change to ensure that the flow

out of the component is the same as the flow in. An iterative solver (IS) is used by C-MAPSS to reduce these

flow errors to within a specified tolerance by adjusting the three R-lines and two PRs. The flow errors used

by the IS are calculated for the HPC, HPT, LPT, core nozzle, and the section of the flow path between the

fan and VBV. The formulation used to calculate the flow errors for the turbines indicates that the flow maps

for these components only contain information on the main flow entering the turbine (that is, at stations 40

and 48) and not any of the bleed flow injected at the inlets (stations 31 and 28). The implications of this

assumption, along with the scaling relationships for the compressor and turbine maps, will be addressed in

the discussion of matching the T-MATS plant model to this baseline model.

B. T-MATS engine plant models

The T-MATS library may be used to construct an engine plant model in a modular fashion by combining blocks representing each component of the engine. In addition to the main components (compressors,

turbines, nozzles, and burner), T-MATS contains blocks for ducts, valves, and a mechanism for ‘splitting’

the flow into a core (main) path and a bypass path (the effect of this mechanism is seen downstream of

the fan in Fig. 1). Each block can be configured for a particular application by providing the necessary

information through the block mask; this information may include performance attributes, such as maps,

scalers, pressure losses, or physical attributes such as cross-sectional area or other constants, depending on

∗The iDesign feature in T-MATS can be used to automatically generate scalers, given a design point, for a relationship

specified directly in the component code.10

NASA/TM—2014-218402 4
the component. In providing these characteristics, it is important to know the assumptions made in the

code about how the component operates using this information; this was brought to the forefront during

construction of the T-MATS engine plant model when using the maps, scalers, and other engine data from

the C-MAPSS model. Note that the model referred to as the ‘T-MATS engine plant model’ only contains

blocks for the engine components illustrated in Fig. 1; it does not contain an iterative solver.

In the T-MATS (v1) code, several assumptions are made that differ significantly from those made in the

C-MAPSS model; in particular, the scaling of the maps and the information contained in the maps were not

congruent between the models. Additionally, small differences, such as how bleed flow rates were calculated,

were enough to contribute noticeably to mismatch between the models. With a goal of obtaining an engine

plant model that was able to calculate the flow characteristics (flow rate, enthalpy, temperature, pressure,

and fuel-to-air ratio) at each station in the engine to within 1% of the C-MAPSS model, changes were needed

in the information provided for configuring each component, and to the T-MATS code itself, to account for

these assumptions.

Before considering changes at the code-level, recalculation of several maps and scalers was needed to

meet the assumed scaler relationships in T-MATS. Unlike the C-MAPSS model, where the relationship

between scaled (map) and unscaled map variables depends on the particular variable and component, TMATS assumes that each of the four map variables is scaled as in Eq. (1) (with sC

(·) replaced by sT(·))for

variables read from maps (e.g., (corrected) flow rate), and by the relationship represented in Eq. (2) (again

with sT

(·) replacing sC(·)) for variables used for interpolation (e.g., (corrected) shaft speed). While the scaler

relations for (corrected) flow rate and efficiency in the C-MAPSS model already had this form, it was

necessary to recalculate scalers for the (corrected) shaft speed in the compressor map and for the pressure

ratios in both compressor and turbine maps. The scalers for the (corrected) shaft speed were recalculated

by absorbing the design value, NRdes, into the scaler specified for the compressors in the T-MATS plant

(sT

NR = sCNR/NRdes). Modification of the scalers for PR was more involved as it required unscaling the

PR maps for the compressors and turbines and calculating new PR scalers for the unscaled maps using the

design values for each component, assuming the relationships suggested by Eqs. (1) (for the compressors)

and (2)(for the turbines).

Recalculation of the maps and scalers for the compressors and turbines addressed most of the mismatch

between the two engine plants; the small discrepancies that remained were related to the bleed flow from

the HPC to the inlet and exit of the HPT (W31 and W32, respectively, in Fig. 1). Comparison of the

code implementing the HPC in each model revealed that, while the bleed flows to the inlet and exit of the

LPT (W28 and W29, respectively) were calculated the same way in each model (W28,29 = f 28,29

bld W24, where

f 28,29

bld specifies the fraction of inflow to the HPC that bleeds to the LPT), W31 and W32 were not. In the

T-MATS HPC model, the fractional bleed rates f 31,32

bld are assumed to define the bleed flow rates to the

HPT as fractions of the inflow to the HPC, W24 (like f 28,29

bld ). In C-MAPSS, however, these bleed flow rates

are assumed to be defined as fractions of the flow in the HPC after both LPT bleeds, W28,29, and a fixed

customer bleed, Wcust, have been removed (the first line in Eq. (4)). To account for the differing definitions

of f 31,32

bld , the fractional bleed rates must be recalculated for use in the T-MATS model:

W31,32 = f 31,32

bld (W24 − W28 − W29 − Wcust)

= f 31,32

bld W24(1 − f 28

bld − f 29bld − Wcust

W24

)

=  ̄f 31,32

bld W24 − f 31,32bld Wcust

(4)

When Wcust = 0, there is a 1−2% difference in bleed flow rates W31 and W32 between the two plant models,

exceeding the stated goal of matching the models with less than 1% error in all flow characteristics at all

stations in the flow path. A closer look at the simulation results revealed that this additional error can be

attributed to the term f 31,32

bld Wcust in Eq. (4): because customer bleed is fixed, it can not be represented

as a constant fraction of W24 and incorporated into the recalculated  ̄f 31,32

bld . Analysis proceeds here under

the assumption that Wcust = 0 to avoid this discrepancy; it would be possible to modify how bleeds are

handled in the T-MATS engine plant to allow for nonzero Wcust by diverting the bleed, W31,32, from the

flow downstream of the compressor module.

The assumptions made in the models regarding the turbine maps and flow error calculations did not have

an effect on the flow characteristics in the T-MATS plant model, but are more pronounced when an iterative

solver (IS) is integrated with the plant. Although not influential in matching the T-MATS plant model to

NASA/TM—2014-218402 5
the C-MAPSS model, the modifications made to address these assumptions are included in this section.

The T-MATS code assumed the turbine maps for (corrected) flow included the main inflow (W40 or W48,

for the HPT and LPT respectively) and inlet bleed (W31 and W28, respectively), while the maps for the

C-MAPSS model captured the inflow only. This assumption is manifested in the flow error calculations made

in each model: given a flow rate interpolated from a turbine map, for a specific NRsc and P Rsc, C-MAPSS

calculates the flow error as in Eq. (5), while T-MATS uses the expression in Eq. (6).

eC

HP T,LP T = W48,50 − W40,48 − W31,28 − W32,29

= W map

40,48 − W40,48

(5)

eT

HP T,LP T = W140,48 + W32,29 − W map

40,48

= W40,48 + W31,28 − W map

40,48

(6)

The second equality in these equations holds because W48,50 = W map

40,48 + W31,28 + W32,29 in the C-MAPSS

model and W1

40,48 = W40,48 + W31,28 in the T-MATS model. The final expressions in Eqs. (5) and (6) show

that, by adjusting PR based on the flow errors, the IS drives the turbine operating condition to the case where

the (corrected) flow rate interpolated from the maps is W40,48 in the C-MAPSS model and W40,48 +W31,28 in

the T-MATS model. This difference must be accounted for at the code-level, made possible since T-MATS

has been designed to allow for low-level modifications to accommodate differing assumptions in the modeling

process.

The only assumption made about the solver architecture in the C-MAPSS engine model that was not

reproduced in the T-MATS model was related to the calculation of the flow errors in the most-upstream

region of the engine. The T-MATS engine plant requires two more solutions to be provided from the IS than

required by the C-MAPSS plant, necessitating that two additional flow errors be calculated and provided

to the IS when integrating it with the T-MATS plant. The modeling discrepancy inherent in this increased

solver dimension replaces a solver input that indicates flow is balanced for a set of steady-state conditions

(eC

fan+LP C+V BV = W20 − (W14 + W24)) with three solver inputs (eTfan, eTLP C, and eTbypass), one of which

does not indicate balanced flow for the same steady-state conditions (eT

bypass, which is between 1.26% and

3.46%). When an IS is integrated with the T-MATS plant, a steady-state condition slightly different from

the C-MAPSS model results, as will be shown in Section III.

C. Comparing engine plant models

The engine plant model constructed using the T-MATS block library was compared to the C-MAPSS

engine model in two stages to verify the modifications made in the T-MATS engine plant model. Comparison

of the two plant models was done by looking at the errors in flow characteristics (flow rate, enthalpy,

temperature, pressure, and fuel-air ratio) at each station in Fig. 1; the T-MATS plant was said to ‘match’

the C-MAPSS model when these errors were each below 1%. The validation was done by providing constant

inputs, corresponding to the steady-state flow conditions of the C-MAPSS model when simulated with the

inputs Wf = 1.67 pps, 3.33 pps, 4.17 pps, and 6.95 pps, to the T-MATS engine plant. Before validation of

the T-MATS plant as an integrated system, each engine component was validated individually at these four

steady-state conditions to check that the maps, scalers, and other engine data provided to the model were

correct. This step allowed for mismatched assumptions, such as those made on map scaling, to be identified

more readily by isolating possible sources of error to a single component. After this verification, the flow path

was established by connecting the components to form the T-MATS plant model, which closely resembled

the diagram in Fig. 1.

The T-MATS engine plant, which is not integrated with an IS, was similarly validated against the CMAPSS engine at the four steady-state flow conditions. The model inputs that would be provided by the

IS (R-lines for fan, LPC, and HPC; PRs for HPT and LPT; W20; and the calculated BPR) were specified

in the same way as fuel flow and VBV position: as constant model inputs. The flow characteristics at each

station were collected and compared to those obtained from simulation of the C-MAPSS model to verify that

the T-MATS plant was within the 1% goal. The relative error of the flow rate, temperature, and pressure

along the core flow path for Wf = 4.17 pps are shown in Fig. 2. These are representative results of the four

simulations in that all results were well within 1% of the C-MAPSS results, suggesting that the T-MATS

plant accurately replicates the C-MAPSS plant. It should be noted, however, that the flow errors calculated

for these simulations suggest that the flow is not balanced in the T-MATS plant model, even though it is

NASA/TM—2014-218402 6
20 24 30 40 45 50 90

0

0.05

T−MATS plant model: Flow characteristic errors for Wf = 4.17 pps

W error (%)

20 30 40 45 50 90

0

0.2

0.4

T error (%)

20 24 30 40 45 50 90

0

0.1

0.2

station number

P error (%)

Figure 2. Percent errors in flow rate, temperature, and pressure at stations in the core flow path, calculated

with respect to results from C-MAPSS, for simulation of the T-MATS engine plant with Wf = 4.17 pps. All

errors are less than 1%, indicating the T-MATS plant accurately replicates the C-MAPSS plant.

in the C-MAPSS model, due to the different flow error calculations. The effect of these assumptions can be

studied further by integrating an IS with the T-MATS plant and performing additional validation.

III. Evaluating the T-MATS models

The T-MATS engine plant model discussed in Section II does not include a mechanism to ensure that

mass and energy are conserved throughout a simulation, which is satisfied when the input and output flow

of each component are within a specified tolerance of each other. An IS added to the model adjusts the

operating conditions for the compressors (R-lines) and turbines (PRs) to ‘balance the flow,’ so the flow errors

meet this tolerance. The T-MATS library contains two IS’s that operate using the Newton-Raphson method:

a steady-state IS (SSIS), which minimizes the flow errors for a constant input to the engine, and a dynamic

IS (DIS), which drives the flow errors to zero at each time-step. (Mathematical detail of how these solvers

work can be found in the T-MATS User’s Guide.10) A version of the T-MATS model can be constructed

by adding one of the two solvers to the plant model discussed in Section II. The required modifications to

the structure of the Simulink model to accommodate these solvers are described in detail in the T-MATS

User’s Guide.10 Comparison of the simulation results from the T-MATS engine models with IS’s to those

from the C-MAPSS model can be used to demonstrate the utility of T-MATS in creating a Simulink-based

model given component maps, scalers, and verification data.

Because both the C-MAPSS and T-MATS models are computational models, the flow characteristics at

each station in Fig. 1 are known and can therefore be compared as part of the validation process. For this

investigation, data from simulation of each model operating at sea-level static conditions (0 ft altitude, 0

Mach, standard-day temperature) for eight fuel flow profiles were compared. Four simulations, corresponding

to the four constant fuel flow simulations listed in Table 1, were run to verify the T-MATS model when the

plant was integrated with the SSIS (the ‘steady-state T-MATS model’). The other four simulations, run with

the single-step inputs in Table 1, tested the transient performance of the plant when integrated with the DIS

(the ‘dynamic T-MATS model’). The fuel flows in Table 1 were chosen to verify the T-MATS model over a

large range of operating conditions (from 24,000 lbf to 88,000 lbf of thrust production) in both steady-state

and dynamic operation.

The IS in each model was configured as in Table 2. These settings specify limits on how much the Jacobian

is perturbed, how much the solution can change at each time-step, the maximum number of iterations that

NASA/TM—2014-218402 7
Table 1. Fuel-flow inputs provided to the C-MAPSS engine model for generation of data used to evaluate the

T-MATS engine models. Simulations included four constant fuel flows and four step-changes in fuel flow.

Simulation Initial fuel flow (pps) Final fuel flow (pps)

SS1 1.67 1.67

SS2 3.33 3.33

SS3 4.17 4.17

SS4 6.95 6.95

D12 1.67 3.33

D13 1.67 4.17

D24 3.33 6.95

D34 4.17 6.95

Table 2. Iterative solver configuration for the C-MAPSS IS and for the T-MATS SSIS. Note that C-MAPSS

uses a fixed Jacobian, while T-MATS calculates the Jacobian at the beginning of each set of iterations.

Solver setting C-MAPSS (D)IS T-MATS SSIS

Max Jacobian perturbation N/A 0.01

Max solution change per time-step (%) N/A 2

Max iterations before Jacobian recalculation N/A 100

Max iterations for while-loop 100 N/A

Termination condition (%) 2 1

can occur before the Jacobian is recalculated, and the conditions for which the solver will terminate its search

for a solution. (The termination condition for the T-MATS model is smaller than for the C-MAPSS model

because, unlike the C-MAPSS model (flow errors O(10−12)%), the solver in the T-MATS model produces

solutions with flow errors close to the termination condition.) In addition, initial conditions (IC’s) need to

be provided on a case-by-case basis due to the nonlinearity of the system. It should be noted that the IS in

the C-MAPSS model is a dynamic IS that uses a fixed Jacobian, unlike the T-MATS IS’s, which recalculate

the Jacobian if convergence is not reached after a specified number of iterations; this may contribute to some

of the mismatch observed in the simulation results presented here. In addition to presenting validation of

the T-MATS model through the simulations listed in Table 1, the sensitivity of the solver to variations in

configuration was also studied in an attempt to improve the accuracy of the T-MATS model; the results of

this investigation will be presented briefly before the transient performance of the model with the DIS is

discussed.

A. Steady-state solver results

The steady-state flow conditions resulting from simulation of the steady-state T-MATS model with the

four constant fuel flow inputs listed in Table 1 were compared to the steady-state conditions of the CMAPSS model given those same inputs. The solutions of the SSIS and the steady-state shaft speeds and

thrust production uniquely describe the operating conditions of each component in the engine plant model

and, therefore, are representative of the flow characteristics in the model. A comparison of these results

for the C-MAPSS and T-MATS models is shown in Table 3, where the error is calculated with respect to

C-MAPSS. The most accurate results (those within 1% of C-MAPSS) were the shaft speeds, turbine PRs,

and LPC R-line. Conversely, the steady-state fan R-line and BPR are the least-accurate solutions; these

solutions in particular are most influenced by the difference in solution dimension between the C-MAPSS

and T-MATS models.

The small differences between the turbine PRs and shaft speeds in the two models suggest that there

should be relatively less error in flow characteristics for the HPT and LPT (stations 40 to 50). Specifically,

the flow at these stations in the T-MATS model should be close to the flow in the C-MAPSS model since the

maps are interpolated based on (corrected) shaft speed and PR. Likewise, the large error in fan R-line, along

with the error in W20, suggests that the flow rate in the T-MATS model in upstream stations is inaccurate

NASA/TM—2014-218402 8
Table 3. Percent error in steady-state solver results, shaft speeds, and thrust produced in the T-MATS model,

calculated with respect to the C-MAPSS model. Note that W20 and BPR are calculated from simulation

results and are not part of the IS solution array in C-MAPSS.

Variable Wf = 1.67 pps Wf = 3.33 pps Wf = 4.17 pps Wf = 6.95 pps

Fan R-line 4.2402% 3.3082% 3.1824% 2.6227%

LPC R-line 0.4115% 0.8505% 0.8454% 1.9311%

HPC R-line 1.3964% 1.1804% 1.5112% 2.5628%

HPT PR 0.0382% 0.0626% 0.0592% 0.0282%

LPT PR 0.7709% 0.654% 0.5419% 0.7358%

W20 1.9178% 1.2927% 1.2474% 0.5532%

BPR 3.1275% 2.4064% 2.2044% 1.9136%

Nf (rpm) 0.7043% 0.311% 0.2625% 0.3203%

Nc (rpm) 0.2403% 0.2899% 0.3193% 0.369%

Fg,byp (lbf ) 2.6387% 1.5826% 1.3944% 0.0534%

Fg,core (lbf ) 2.3795% 1.7056% 1.4905% 0.8556%

with respect to the C-MAPSS model. The decrease in error in the LPC and HPC R-lines, combined with the

large BPR error, indicates that the error in W20 becomes concentrated in the bypass flow path downstream

of where the flow splits. Even so, the inaccuracy in W20 can be expected to affect all the stations in the

engine, and is even reflected through the error in thrust, which is calculated based on the flow into each

nozzle. These conditions are supported by the results in Fig. 3, which show the flow rate being the most

inaccurate flow characteristic (up to 2% error with respect to the C-MAPSS model) at stations 20 to 21 (the

fan), and maintaining an error around 1% at all other stations in the core flow, while the error in temperature

and pressure remained below 1% at all stations (with exception at three stations when Wf = 6.95 pps). Not

shown is the bypass flow path, which had errors in flow rates around 1.5% from the C-MAPSS results, much

larger than the error in the core flow downstream of the fan.

Because the T-MATS plant model was verified using steady-state flow conditions from the C-MAPSS

model, it was expected that integrating an IS with the T-MATS engine plant model would not cause the

steady-state results to differ significantly from the C-MAPSS steady-state results. However, from Table 3

and Fig. 3, it can be seen that this was not the case; for a given (constant) input, the steady-state solution

for the T-MATS model is close to, but not always within, 1% of the C-MAPSS solution. As discussed in

Section II, the difference in calculation (and number) of the flow errors that comprise the input to the IS

cause the T-MATS results to exceed the stated goal of 1% error. Before integration with an IS, the bypass

nozzle flow error, which in the C-MAPSS model forms a single flow error with the fan and LPC, exceeds the

error tolerance and therefore drives the IS to search the solution space for a steady-state condition where all

seven flow errors are under 1%. This is illustrated in Fig. 4, where the flow errors of the T-MATS engine

before and after the SSIS has been integrated are shown with respect to the 1% error tolerance. These

results suggest that the flow is not balanced in the T-MATS engine plant and show how the IS acts to do so

by adjusting compressor R-lines, turbine PRs, W20, and BPR to reduce the flow errors.

B. Dynamic solver results

The SSIS allows only for simulation of the system with constant inputs, as it iterates to converge the

system to a steady-state operating condition by reducing the flow errors below the specified threshold. In

order to perform simulations with inputs that change over time, the model must instead include the DIS,

which iterates at each time-step to ensure the flow remains balanced throughout the simulation. Changes

to the structure of the Simulink model necessary for integration of the DIS with the T-MATS plant model

are not detailed here, but may be found in the T-MATS User’s Guide.10 Two investigations were performed

on the dynamic T-MATS model: studying the effects of changing the solver configuration settings from

those listed in Table 2 and comparing the transient behavior of the C-MAPSS and T-MATS models using

simulation with the step inputs in Table 1.

When the DIS is configured as specified in Table 2, results from simulation of the dynamic T-MATS

model were nearly identical to those presented in Table 3 and Figs. 3 and 4, differing at most by 0.25%

NASA/TM—2014-218402 9
20 24 30 40 45 50 90

0

1

2

W error (%)

Steady−state T−MATS model: Flow characteristic errors

20 24 30 40 45 50 90

0

0.5

1

T error (%)

20 24 30 40 45 50 90

0

0.5

1

station number

P error (%)

Wf = 1.67 pps Wf = 3.33 pps Wf = 4.17 pps Wf = 6.95 pps

Figure 3. Errors in flow rate, temperature, and pressure at each station in the engine, calculated with respect

to C-MAPSS, for simulation of the steady-state T-MATS model with the constant inputs listed in Table 1.

Comparison with Fig. 2 illustrates the effect of adding an IS to the model.

0

5

Flow errors for plant and steady−state T−MATS models

flow error

(%)

0

1

2

flow error

(%)

0

1

2

flow error

(%)

plant model steady−state model

0

0.5

1

flow error

(%)

e

fan

e

LPC

e

HPC

e

HPT

e

LPT

e

core

e

bypass

Wf = 6.95 pps

Wf

 = 4.17 pps

Wf

 = 3.33 pps

Wf

 = 1.67 pps

Figure 4. Comparison of the flow errors for each engine component in the T-MATS model without (left) and

with (right) the SSIS integrated in the model. The error tolerance, here 1%, is indicated by the dashed red

line.

NASA/TM—2014-218402 10
Table 4. Iterative solver configurations studied for the dynamic T-MATS model simulation with the constantinputs listed in Table 1. The configuration chosen for simulation of the T-MATS model with step inputs is

listed in the final column.

Solver setting Variations in T-MATS DIS T-MATS DIS

Max Jacobian perturbation 0.001, 0.05 (0.0441 for SS1) 0.04

Max solution change per time-step (%) 0.5, 5, 10 5

Max iterations before Jacobian recalculation 15, 500, 1000 100

Max iterations for while-loop 100 100

Termination condition (%) 0.1, 2, 5 0.01

from the simulation results for the steady-state T-MATS model. It was of interest to study whether better

matching to the C-MAPSS model could be achieved by changing the solver configuration from these baseline

settings. A study of the eleven variations listed in Table 4 was performed, where a single change was made

from the baseline case for each simulation. The detailed implications of these choices are outside the scope

of this paper; only general trends are presented here.

The trends observed through this investigation were used to guide a choice of solver configuration that

resulted in a small improvement in how well the steady-state solution of the T-MATS model matched that

of the C-MAPSS model. Decreasing the termination condition, increasing the maximum perturbation of the

Jacobian, and increasing the maximum allowed change in the solution at each time-step generally resulted

in a steady-state condition closer to that of the C-MAPSS model. (There was little effect on the solution by

changing the maximum number of iterations between Jacobian recalculation.) Although this suggests that

the impact of changing the configuration settings of the DIS on the accuracy of the dynamic T-MATS model

was minimal, some improvement was achieved by following these trends and choosing the settings in the last

column in Table 4, which were used to run the remaining simulations to be discussed.

After verifying the steady-state behavior of the dynamic T-MATS model against the C-MAPSS model, it

was necessary to compare the transient behavior of the two models through simulation with the step inputs

listed in Table 1. Because of the time-dependence of these results, the number of stations in the engine, and

the number of flow characteristics for which data were collected, it is not practical to show all of the simulation

data here; instead, results representative of the general observations made in the model are shown, taken

from simulation of the step input from Wf = 3.33 pps to Wf = 6.95 pps. As with the steady-state T-MATS

model, looking at the IS solutions provides a way to qualitatively describe the behavior of flow characteristics

during the simulations. Figure 5 shows the comparison of these solutions for the T-MATS and C-MAPSS

models (solid and dashed lines, respectively). As observed for the steady-state T-MATS model (Table 3),

the turbine PRs and shaft speeds match well between the two models, while the compressor R-lines, inlet

flow (W20), and, especially, BPR in the T-MATS model were much less accurate. Since these solutions are

near the corresponding steady-state solution in the time ranges where Wf is constant (before the transition

at 10 seconds and from a few seconds after the transition to the end of the simulation), this mismatch may

be attributed to the differing approach to calculating flow errors. And like with the steady-state T-MATS

model, these solver results suggest that the flow rate differs more than the other flow characteristics between

the dynamic T-MATS model and the C-MAPSS model; this conclusion is supported by the results shown in

Fig. 6.

The results in Fig. 6 are similar to those in Fig. 3 in that they show pressure and temperature (generally)

remaining within 1% of the C-MAPSS results throughout the simulation, while flow rates tend to exceed

this value slightly, particularly following the transition in fuel-flow at 10 seconds. Flow characteristics at

just four stations are shown: station 23, the exit of the LPC, station 30, the exit of the HPC, station 45, the

exit of the HPT, and station 49, the exit of the LPT; many of the intermediate stations indicated in Fig. 1

are at duct exits, where the flow has only undergone a pressure reduction, so simulation data from these

additional stations do not provide additional information. One of the most obvious features in the figure is

the loss in accuracy of flow rate between stations 23 and 30 due to the presence of the VBV. This is most

clear during the 3−4 seconds following the transition in fuel flow. The likely reason for this mismatch is the

way the VBV position is specified in each model: in the C-MAPSS model, a schedule is defined that maps

the corrected core speed to the valve position, while in the T-MATS model the position is specified as an

input profile, similar to fuel flow, using the steady-state values from simulation of C-MAPSS. Consequently,

NASA/TM—2014-218402 11
0 5 10 15 20 25 30

1.4

1.6

1.8

2

Dynamic T−MATS model: Iterative solver results and shaft speeds

(step from Wf

 = 3.33 pps to Wf = 6.95 pps)

compressor

R−line

0 5 10 15 20 25 30

3.5

4

4.5

5

5.5

turbine

pressure ratio

0 5 10 15 20 25 30

8

8.5

9

bypass ratio

0 5 10 15 20 25 30

2300

2800

3300

inlet flow (pps)

0 5 10 15 20 25 30

1800

2150

2500

time (s)

fan shaft

speed (rpm)

0 5 10 15 20 25 30

8300

8700

9100

time (s)

core shaft

T−MATS

speed (rpm)

C−MAPSS

T−MATS

C−MAPSS

T−MATS

C−MAPSS

T−MATS

C−MAPSS

HPT (T−MATS)

LPT (T−MATS)

HPT (C−MAPSS)

LPT (C−MAPSS)

fan (T−MATS)

LPC (T−MATS)

HPC (T−MATS)

fan (C−MAPSS)

LPC (C−MAPSS)

HPC (C−MAPSS)

Figure 5. Comparison of DIS solutions, and shaft speed, for simulation of the dynamic T-MATS model and

the C-MAPSS model with the step input from Wf = 3.33 pps to Wf = 6.95 pps.

the valve position in the T-MATS model does not change in response to change in the corrected shaft speed;

the mismatch is most noticeable during the input transition because the prescribed VBV position changes

linearly but corrected core speed does not. Although affecting all flow characteristics downstream of stations

24 and 14, this mismatch is not enough to produce significantly different conditions in the T-MATS model

at the end of the simulation.

Additionally, it was observed that a large mismatch between the models occurred early in the simulation,

attributed to the IC’s specified for each model. The only IC’s needed by the T-MATS model are for the IS

and the integrators used for the shaft speeds, while the C-MAPSS model requires IC’s to be specified for

these and for the three inputs and six pressure and temperature sensors in the system. When possible, the

initial conditions were specified identically for the two models, but the additional initial conditions required

for the C-MAPSS model introduce the start-up transients seen in Fig. 6. After a few time-steps, the transient

dies out as flow establishes a steady-state condition prior to the fuel flow transition.

Behind the results presented in Figs. 5 and 6 is the evolution of the flow errors as the DIS balances the

flow at each time-step. From a high level, simulation of the models with a DIS progresses as follows: at the

beginning of the time-step, the IS checks whether the flow errors are within the prescribed tolerance; if not,

it iterates to find a solution that satisfies this condition before the simulation continues to the next time-step.

For the dynamic T-MATS model, which has the solver configuration specified in Table 4, this is exemplified

in Fig. 7, where the evolution of the seven flow errors is shown for the example simulation. The inset graphs

in the figure, which show the iterations for eT

HPC , eTLP T , and eTbyp at each of 10 time-steps between 10.26

and 10.395 seconds, demonstrate the operation of the DIS. The dots represent solutions of the solver at each

iteration while the solid lines illustrate how the flow error decreases on each iteration, until it is below the

0.01% tolerance (red dashed lines). The simulation then progresses to the next time-step (vertical dotted

lines), where the error initially exceeds the tolerance and the solver must iterate again. The figure shows

that, when the operating condition of the engine changes quickly (such as when provided a step change in

Wf ), several iterations are needed to balance the flow. Once the transient has died out (or Wf changes less

rapidly), the error is much closer to the tolerance at the beginning of the time-step and fewer iterations of

the solver are required.

In addition to evaluating the internal flow characteristics of the T-MATS model against the C-MAPSS

model, it is also of interest to compare the thrust production of each engine. As expected from the results in

Figs. 5 and 6, since thrust is a function of flow rate, the thrust produced by the two nozzles in the T-MATS

NASA/TM—2014-218402 12
0 5 10 15 20 25 30 −5

0

5

Dynamic T−MATS Model: Flow characteristic errors in core path

(step from Wf

 = 3.33 pps to Wf = 6.95 pps)

W error (%)

0 5 10 15 20 25 30 −5

0

5

T error (%)

0 5 10 15 20 25 30 −5

0

5

time (s)

P error (%)

station 23 station 30 station 45 station 49

Figure 6. Errors in flow rate, temperature, and pressure at stations 23, 30, 45, and 49 (the exits of the LPC,

HPC, HPT, and LPT, respectively) for the step input from Wf = 3.33 pps to Wf = 6.95 pps, calculated with

respect to the C-MAPSS simulation results. The dashed lines represent an error of 1%.

0 5 10 15 20 25 30

−0.25

0

0.25

flow error (%)

Dynamic T−MATS model: Flow error evolution through simulation

(step from Wf

 = 3.33 pps to Wf = 6.95 pps)

0 5 10 15 20 25 30

−0.25

0

0.25

flow error (%)

0 5 10 15 20 25 30 −1.5

−0.9

−0.3

0

0.3

time (s)

flow error (%)

e

core

e

byp

10.26 10.32 10.38

−0.1

0

10.26 10.32 10.38

−0.1

0

10.26 10.32 10.38

−0.2

−0.1

0

e

fan

e

LPC

e

HPC

e

HPT

e

LPT

Figure 7. Evolution of the seven flow errors in the dynamic T-MATS model for simulation with the step input

from Wf = 3.33 pps to Wf = 6.95 pps. The edge of the color region shows the flow error at the beginning of

each time step, which is reduced below the 0.01% tolerance (red dashed lines) by the DIS, as highlighted for

eT

HPC, eTLP T , and eTbyp in the inset graphs.

NASA/TM—2014-218402 13
0 10 20 30

4

5

6

7

8 x 104

Dynamic T−MATS model: Thrust production

(step from Wf

 = 3.33 pps to Wf = 6.95 pps)

time (sec)

bypass nozzle thrust (lbf

)

0 10 20 30

4000

6000

8000

10000

12000

time (sec)

core nozzle thrust (lbf

)

0 5 10 15 20 25 30 −8

−4

0

4

8

time (s)

error (%)

T−MATS

C−MAPSS

T−MATS

C−MAPSS

Fg,byp

Fg,core

Figure 8. Comparison of the calculated thrust production of the bypass and core nozzles in the dynamic

T-MATS model and the C-MAPSS model for simulation with the step input Wf = 3.33 pps to Wf = 6.95 pps.

model is within 1 − 2% of that calculated for the nozzle in the C-MAPSS model, as seen in Fig. 8.

Although the T-MATS model coupled with the IS did not meet the standard of having less than 1%

error, the results in Figs. 3 to 8 and Table 3 demonstrate that both the steady-state and dynamic T-MATS

models are still relatively accurate, within 2% of the C-MAPSS model. The main reason for this mismatch is

due to the increased size of the solution space in the T-MATS model, required by the additional information

provided by the solver to the T-MATS engine plant.

IV. Summary, conclusion, and future work

An accurate replica of the C-MAPSS twin-spool turbofan engine model, limited to open-loop operation,

has been developed using the Toolbox for the Modeling and Analysis of Thermodynamic Systems (T-MATS).

The process of constructing and validating this model emphasized the need to understand the original

assumptions during construction of the baseline model. This knowledge led to an understanding of how the

T-MATS model may need to be modified to meet a desired design approach or a specific design assumption,

as the tool was developed to be modified on an application-specific basis.10 It was not only necessary to

rescale some of the engine data provided to the T-MATS model, but also to make a code-level change to

ensure the turbine flow error calculation accurately reflected the information contained in the component

map.

Discussion of the T-MATS models focused on engine plant model and the integration of this model with

an IS. However, it was useful to begin the modeling process by validating individual component models using

the steady-state results from simulation of the baseline engine (C-MAPSS). Doing so allowed for isolation of

specific components affected by modeling assumptions that differed between the models; it was also helpful

to look at the component flow errors, both for these individual models and the plant model, in trying to

identify the effect of some of these assumptions (e.g., before modifications were made to the turbine code

in T-MATS, eT

HPT > 10% and eTLP T > 4%; these errors were reduced to < 0.5% after modifications were

made). Once these assumptions were addressed in the T-MATS plant model, validation could continue

through evaluation of this plant integrated with a steady-state or dynamic IS.

Simulation of both the steady-state and dynamic T-MATS models, with a set of constant and step fuel

flow inputs, showed that the C-MAPSS model was accurately replicated, as most of the flow characteristics

were within 1% of the C-MAPSS results for the same inputs; only the flow rate exceeded this limit, remaining

within 2% of the C-MAPSS results. This suggests that the changes made to the engine data, by rescaling some

maps and recalculating some scalers, and modification to the flow error calculation in the turbine properly

addressed the different modeling assumptions made in the two models. Validation of the dynamic T-MATS

NASA/TM—2014-218402 14
model in open-loop may be expanded to include models with nonzero customer bleed and implementation of

the VBV input thorough a scheduling scheme. Following this, the model should be validated in closed-loop

with a controller similar to that used in the C-MAPSS simulation to further demonstrate the utility of the

T-MATS tool.

References

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January 2010. 3Parker, K. I. and Guo, T.-H., “Development of a Turbofan Engine Simulation in a Graphical Simulation Environment,”

NASA/TM-2003-212543, August 2003. 4DeCastro, J. A., Litt, J. S., and Frederick, D. K., “A Modular Aero-Propulsion System Simulation of a Large Commercial

Aircraft Engine,” Proceedings of the 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, AIAA 2008-4579,

Hartford, CT, July 2008. 5Frederick, D. K., DeCastro, J. A., and Litt, J. S., User’s Guide for the Commercial Modular Aero-Propulsion System

Simulation (C-MAPSS), NASA/TM-2007-215026, October 2007. 6Liu, Y., Frederick, D. K., DeCastro, J. A., Litt, J. S., and Chan, W. W., User’s Guide for the Commercial Modular

Aero-Propulsion System Simulation (C-MAPSS), version 2, NASA/TM-2012-217432, March 2012. 7Csank, J., May, R. D., Litt, J. S., and Guo, T.-H., “Control Design for a Generic Commercial Aircraft Engine,” Proceedings

of the 46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, AIAA-2010-6629, Nashville, TN, July 2010. 8May, R. D., Csank, J., Lavelle, T. M., Litt, J. S., and Guo, T.-H., “A High-Fidelity Simulation of a Generic Commercial

Aircraft Engine and Controller,” Proceedings of the 46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, AIAA-2010-

6630, Nashville, TN, July 2010. 9Lytle, J. K., “The Numerical Propulsion System Simulation: An Overview,” NASA/TM-2000-209915, June 2000. 10Chapman, J. W., Lavelle, T. M., May, R. D., Litt, J. S., and Guo, T.-H., Toolbox for the Modeling and Analysis of

Thermodynamic Systems (T-MATS) User’s Guide, NASA/TM-2014-216638, January 2014.

NASA/TM—2014-218402 15