Rama S.R. Gorla

Cleveland State University, Cleveland, Ohio

Shantaram S. Pai and Jeffrey J. Rusick

Glenn Research Center, Cleveland, Ohio

Probabilistic Analysis of Solid Oxide Fuel Cell

Based Hybrid Gas Turbine System

NASA/TM—2003-211995

April 2003

GT–2003–38046

https://ntrs.nasa.gov/search.jsp?R=20030038813 2018-05-15T15:58:10+00:00Z
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Rama S.R. Gorla

Cleveland State University, Cleveland, Ohio

Shantaram S. Pai and Jeffrey J. Rusick

Glenn Research Center, Cleveland, Ohio

Probabilistic Analysis of Solid Oxide Fuel Cell

Based Hybrid Gas Turbine System

NASA/TM—2003-211995

April 2003

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for the

Turbo Expo 2003

cosponsored by the American Society of Mechanical Engineers

and the International Gas Turbine Institute

Atlanta, Georgia, June 16–19, 2003

GT–2003–38046
Acknowledgments

Rama S.R. Gorla is grateful to NASA Glenn Research Center for supporting this work through grant NAG3–2745.

Available from

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Available electronically at http://gltrs.grc.nasa.gov
NASA/TM—2003-211995 1 

PROBABILISTIC ANALYSIS OF SOLID OXIDE FUEL CELL BASED 

HYBRID GAS TURBINE SYSTEM 

Rama S.R. Gorla 

Cleveland State University 

Cleveland, Ohio 44115 

Shantaram S. Pai and Jeffrey J. Rusick 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

ABSTRACT 

The emergence of fuel cell systems and hybrid fuel cell 

systems requires the evolution of analysis strategies for 

evaluating thermodynamic performance. A gas turbine 

thermodynamic cycle integrated with a fuel cell was 

computationally simulated and probabilistically evaluated in 

view of the several uncertainties in the thermodynamic 

performance parameters. Cumulative distribution functions and 

sensitivity factors were computed for the overall thermal 

efficiency and net specific power output due to the uncertainties 

in the thermodynamic random variables. These results can be 

used to quickly identify the most critical design variables in 

order to optimize the design and make it cost effective. The 

analysis leads to the selection of criteria for gas turbine 

performance. 

INTRODUCTION 

The majority of global power is generated from the 

consumption of fossil fuels, which represent a finite source. 

The rapid depletion of these fossil fuel resources remains one 

of the most important problems facing future world power 

generation. This means that the way in which power is 

generated will have to change over the next century. The power 

harnessed in the fossil fuels must be used more efficiently until 

renewable sources can be developed to meet a significant 

proportion of the increasing energy requirements. The option of 

developing new technology such as fuel cells, which exceed the 

maximum efficiencies of gas turbines seems very lucrative. 

A fuel cell is a simple means of converting chemical 

energy to electricity without ignition combustion and it is 

promising to revolutionize the power generation industry. The 

chemical energy to the fuel cell is supplied on a continuous 

basis in the form of a fuel such as natural gas or synthesis gas 

while the oxidant is also supplied continuously. The fuel cell is 

not constrained by the Carnot efficiency and therefore higher 

conversion efficiencies are achievable with a fuel cell. The 

intermediate step of conversion into heat as in a heat engine is 

eliminated in a fuel cell. 

As governments around the world strive to meet their 

escalating energy demands under increasing pressure from 

environmental issues, there exists a need for clean and efficient 

energy sources. Even the most advanced gas turbine cycles 

have difficulty in reaching a thermal efficiency of 40 percent 

and have NOx emission problems due to their high operating 

temperatures. Standalone fuel cells have been manufactured 

with efficiencies of around 48 percent, producing negligible 

NOx or SOx and a reduced CO2, owing to the increased thermal 

efficiency. It has also been recognized that they could be 

symbiotically incorporated into gas turbine cycles in order to 

produce efficiencies estimated to be as high as 70 percent as 

mentioned in reference [1]. The most suitable fuel cell for this 

application is the solid oxide fuel cell (SOFC) which, on 

account of its all solid state, has the highest operating 

temperature. The fuel cell model was based on the principle of 

operation of the Westinghouse tubular SOFC design, involving 

internal reforming and preforming of methane. Its performance 

was examined under a wide range of operating conditions 

leading to exhaust temperatures in the range 370 to 935 °C. 

Fuel cell system efficiencies as high as 56 percent were 

predicted at low current densities (150 mA/cm2

), while the 

efficiency dropped to 38.7 percent at high current densities 

(600 mA/cm2

) owing to irreversibilities developed. Pressurized 

operation of the fuel cell was assessed in the range 1 to 25 atm; 

high pressure operation leads to an increased power output but 

produces a significant decrease in electrical efficiency. At full 

commercial size, the Westinghouse tubular SOFC has a 

diameter of 22 mm with an active total length of 1500 mm and 

a total active area of 834 cm2

. A 100 kW unit employs 1152 

commercial size SOFCs in its cell stack. The stack is 

maintained at a sufficiently high temperature level which leads 

to approximately 100 percent reforming of the methane to pure 

hydrogen. 

 The compressor provides the SOFC with a preheated inlet 

flow from which it produces dc power while heating to its 

operating temperature of 1000 °C. This is similar to the 

required turbine entry temperature from which the turbine can
NASA/TM—2003-211995 2 

produce ac power. The solid constructions of SOFCs mean that 

they can withstand the high operating pressures of the system. 

The complexity and performance over the past 50 years in 

gas turbine engines has increased and an array of safety nets 

was created to ensure against component failures in turbine 

engines. In order to reduce what is now considered to be 

excessive conservatism and yet maintain the same adequate 

margins of safety, there is a pressing need to explore methods 

of incorporating probabilistic design procedures into engine 

development. Probabilistic methods combine and prioritize the 

statistical distributions of each design variable, generate an 

interactive distribution and offer the designer a quantified 

relationship between robustness, endurance and performance. 

Pangalis et. al [1] developed a detailed thermodynamic 

model of a fuel cell. This model predicted the performance of a 

gas turbine and a fuel cell system integrated together in hybrid 

generation cycles. Cunnel et al [2] modeled gas turbine cycles 

integrated with a fuel cell model. Rao and Samuelsen [3] 

presented a description and application of an analysis for 

tubular SOFC based systems. Models for simulating fuel cell 

based plants were developed by Ferguson [4], Haynes [5], and 

Bessette [6]. These models were limited to systems consisting 

of ideal gases and pure steam or the models required for 

simulating many of the unit operations and processes that could 

make up a hybrid plant were not included. 

A probabilistic design system was developed by Fox [7] at 

Pratt and Whitney for the purpose of integrating deterministic 

design methods with probabilistic design techniques. Here, two 

different approaches were used for estimating uncertainty. A 

Monte Carlo approach was used on design codes that were 

judged to run relatively quickly. For more computationally 

intensive design codes, a second order response surface model 

in conjunction with Box-Behnken design experiments was used 

and then a Monte Carlo simulation was executed. Several 

researchers at NASA Glenn Research Center have applied the 

probabilistic design approaches to turbine engines and related 

systems. Chamis [8] developed a Probabilistic Structural 

Analysis Method (PSAM) using different distributions such as 

the Weibull, normal, log-normal, etc. to describe the 

uncertainties in the structural and load parameters or primitive 

variables. Nagpal, Rubinstein, and Chamis [9] presented a 

probabilistic study of turbopump blades of the Space Shuttle 

Main Engine (SSME). They found that random variations or 

uncertainties in geometry have statistically significant influence 

on the response variable and random variations in material 

properties have statistically insignificant effects. Chamis [10] 

summarized the usefulness and importance of the probabilistic 

approach, especially for turbopumps. To cost effectively 

accomplish the design task, we need to formally quantify the 

effect of uncertainties (variables) in the design. Probabilistic 

design is one effective method to formally quantify the effect of 

uncertainties. 

In the present paper, a probabilistic analysis is presented 

for the influence of a priori fixed parameter variations on the 

random variables for a gas turbine thermodynamic cycle 

integrated with a fuel cell. We focus on the integration of two 

major power generation devices, fuel cells and gas turbines. 

With the available energy resources becoming increasingly 

scarce and modern power generation as well as process plants 

becoming increasingly complex, the requirement of an efficient 

thermodynamic computation technique for design optimization 

is presently setting demanding challenges. 

PROBABILITY THEORY 

 Let X1, X2, Ö., Xn be a set of random variables defined on a 

(discrete) probability space Ω . The probability that the events 

X1= x1 X1, X2 = x2, ..., and Xn = xn happen concurrently, is 

denoted by ) 1 2 n 1 1 2 2 n n f(x , x ,....,x ) = P(X = x ,X = x ,....,X = x

for the set of desired solutions A⊆Ω. If the function 

( , ,...., ) 1 2 n f x x x is discrete, it is called the joint probability 

mass function of X1, X2, ..., Xn and has the following properties. 

0 ≤ f(x1, x2, .Ö, xn) ≤ 1 

 ∑ ∑

∈Ω

=

( , ,..., )

1 2

1 2

( , ,..., ) 1

n x x x

n f x x x (1) 

P[(X1, X2,Ö, Xn) ∈A] 

= ∑ ∑ = ⊆ Ω

∈Ω

f x x x A

n x x x

n ( , ,..., ) 1,

( , ,..., )

1 2

1 2

If f(x1, x2, .Ö, xn) is continuous it is called joint probability 

density function of X1, X2, ..., Xn and has the following 

properties: 

0 ≤ f(x1, x2, Ö, xn) 

 1 2 , , 1 2 = 1 ∫ ∫

Ω

n n ... f(x , x ... x )dx dx ...dx (2) 

P[(X1, X2,Ö,Xn) ∈A]

∫ ∫ = ⊆ Ω

A

.. f(x1 ,x2 ,..., xn )dx1dx2 ...dxn , A 

If the lower bound of A, the set of desired solutions, is equal to 

the infimum of Ω for all Xi, i.e., if A = [infi (Ω), ai], for all i = 1, 

2, Ö, n, a function F(a1, a2, Ö,an)can be defined, such that: 

F(a1, a2,Ö.,an) = [( ) ] P X1 ,X2 ,..,Xn ∈ A

∑ ∑

∈

=

A

n

n x x x

... f ( x , x ,..., x ) 1 2

( , ,..., ) 1 2

, 

A ⊆ Ω (f is discrete) (3) 
NASA/TM—2003-211995 3 

F(a1, a2, ....,an) = [( ) ] P X1 ,X2 ,..,Xn ∈ A

 = ∫ ∫ ⊆ Ω

A

.. f (x1, x2 ,..., xn )dx1dx2...dxn , A

 (f is continuous) (4) 

F is called the joint cumulative probability distribution 

function. For Ω = Rn and a continuous function f : 

F(a1, a2, Ö., an) = P (X ,X ,...,X n [ ) 1 2 ∈((ñ∞,ñ∞,Ö,ñ∞), 

 (a1, a2, Ö., an))] = ∫ ∫

− ∞ ∞ −

na a

n n f x x x dx dx dx

1

.. ( , ,..., ) ... 1 2 1 2 (5) 

The common notation 

F(a1, a2, ..., an) = P X a ,X a ,...,X a n n ( ) 1 ≤ 1 2 ≤ 2 ≤ will be 

used subsequently also. 

 The univariate probability function fXi for each criterion Xi, 

obtained from the traditional probabilistic design process, can 

also be generated with the joint probability function f. fxi is 

called marginal probability mass or density function of Xi and is 

defined by: 

1 fx = ∑ ∑

( ,..., ) ∈

2

2

... ( ,..., )

n x R x

n f x x (f is discrete) (6) 

 

1 fx = ∫ ∫ n n

R

... f (x ,..., x )dx ...dx 2 2 (f is continuous) (7) 

The joint probability function, fXY (x, y), creates the surface of a 

probability ëhumpí in the x-y-f-space, characterized by rings of 

constant probabilities. The distribution curves over the x- and 

y-axis are the aforementioned marginal probability functions 

fX(x) and fy(y), respectively. The last necessary concept to 

mention here for the development of a joint probabilistic 

formulation is the concept dependence of criteria. Two random 

variables X and Y are said to be independent, if fXY(x, y) = fX(x).

fY(y) otherwise X and Y are said to be dependent. This 

dependence is a mathematical notion and should not be 

confused with ëcasual dependenceí. For here on, mathematical 

dependence will be referred to as correlation. Correlation is 

measured by the covariance of two criteria, X and Y, defined by 

Cov(X,Y) = E[X,Y] ñ E[X]E[Y]. (8) 

It is more convenient, however, to use a covariance normalized 

by the standard deviations, σX and σY, for both criteria, called 

correlation coefficient. 

ρ = X Y

X,Y

σ σ

Cov( )

 (9) 

The correlation coefficient is defined over the interval 

[ñ1,1], indicating strongly positively correlated criteria at

values close to 1 and strongly negatively correlated criteria at 

values close to ñ1. The criteria are independent, if ρ = 0. In 

aerospace systems design ρ can be quite difficult to calculate 

by Eq. (9). It is much more effective to view the correlation 

coefficient differently for calculation purposes. Jointly 

collected data from a probabilistic or any other analysis can be 

thought of as vectors of numbers. The correlation coefficient 

measures the orthogonality, i.e., independence, of both vectors. 

ρ is simply the cosine of the angle between the two criterion 

vectors, indicating their alignment. For ρ = 1, vectors are 

parallel and point in same direction, for ρ = ñ1, vectors are 

parallel and point in opposite direction. For ρ = 0, vectors are 

orthogonal and the criteria are independent. The correlation 

coefficient plays a significant role in the formulation of joint 

probability distribution models as described in the next section. 

Probability Functions 

 Attention is now directed to the implementation of this 

probabilistic formulation in the design process. The necessary 

transition from the mathematical formulation above to a 

probabilistic model that yields the information relevant for 

multivariate decision making is described in this section. There 

are two alternatives for this task. 

Joint Probability Model 

 The first joint probability density function introduced here 

is an analytical probability model for criteria whose univariate 

distributions and their corresponding means and standard 

deviations are known. All necessary information for the model 

can be generated by the traditional probabilistic design process, 

using its output of univariate criterion distributions. A 

particular model for two criteria with normal distributions, 

represented by Eq. (10), has been introduced by Garvey and 

Tuab. Garvey further generated models for two criteria with 

combinations of normal and lognormal distributions, which are 

summarized in ref. [11]. 

2 ( )( ) ( ) ]} (10)

[( ) 2 2

1 exp{

2 1

1 ( , )

2

2

2 2

Y

Y

Y

Y

X

X

X

x

X Y

x y y

x μ f x y XY

σ

μ

σ

μ

σ

μ ρ

ρ σ π σ σ ρ

− + − − −

−

− −

=

Note that the only information needed for the Joint 

Probability Model consists of the means μX and μY, the standard 

deviations σX and σY, and the correlation coefficient ρ for the 

criteria X and Y. The model variables, x and y, are defined over 

the interval of all possible criterion values. The advantage of 

this model is the limited information needed, which makes it 

very flexible for use and application. For example, if only 

expert knowledge and no simulation/modeling is available in 

the early stages of design, educated guesses for the means, 

standard deviations, and the correlation coefficient can be used 

to execute the joint probability model. It also lends itself to use 

in combination with increasingly important fast probability 

integration (FPI) techniques. 
NASA/TM—2003-211995 4 

Implementation of Probabilistic Procedure Using FPI 

 FPI is a probabilistic analysis tool that implements a variety 

of methods for probabilistic analysis. The procedure follows the 

steps given below: 

1. Identify the independent and uncorrelated design 

variables with uncertainties. 

2. Quantify the uncertainties of these design variables 

with probability distributions based on expert opinion 

elicitation, available data or benchmark testing. 

3. It is required that there is a response function that 

defines the relationship between the response and the 

independent variables. 

4. The FPI uses the responses generated to compute the 

cumulative distribution functions (CDF)/probability 

density functions (PDF) and the corresponding 

sensitivities of the response. 

Several methods are available in the FPI to compute a probabilistic distribution. In addition to obtaining the CDF/PDF of 

the response, the FPI provides additional information regarding 

the sensitivity of the response with respect to the primitive 

variables. They provide valuable information in controlling the 

scatter of the response variable. The random primitive variable 

with the highest sensitivity factor will yield the biggest payoff 

in controlling the scatter in that particular response variable. 

Such information is very useful to the test/design engineer in 

designing or interpreting the measured data. 

DISCUSSION OF RESULTS 

This paper deals with modeling a gas turbine cycle 

integrated with a fuel cell model to produce hybrid cycle. The 

choice of hybrid configuration includes a regenerative gas 

turbine cycle with fuel cell ahead of the combustor as shown 

in Figure 1. The cycle achieved a thermal efficiency of 

64.1 percent at a pressure ratio of 14. The specific power output 

was found to be 520 W/kg s. Figures 2 and 3 illustrate the 

variation of thermal efficiency and net specific power with 

compression ratio. The combustion temperature acts to ensure 

that the turbine inlet temperature reaches a maximum of 

1300 K. The cycle is less dominated by the fuel cell, especially 

at higher compression ratios where the turbine contributes 

almost equally.

The probabilistic analysis of gas turbine field performance 

due to the uncertainties was applied to a solid oxide fuel cell 

based hybrid system. The thermodynamic random variables and 

their respective values used in this analysis are shown in 

Table 1. All the random variables were assumed to be 

independent. A scatter of ±5 percent was specified for all the 

variables. Normal distribution was assumed for all random 

variable scatters. 

The overall thermal efficiency and net specific power 

output of the hybrid gas turbine system was determined from a 

control volume analysis using the first and second laws of 

thermodynamics. The cumulative distribution functions (CDF) 

and the sensitivity factors were evaluated for the overall 

thermal efficiency response. CDF for the overall thermal 

efficiency shown in Figure 4. The sensitivity factors for the

overall thermal efficiency are plotted in Figures 5 to 7. From 

these figures, we observe that the inlet temperature of the 

compressor, cycle pressure ratio, output of the fuel cell, inlet 

temperature to the gas turbine, adiabatic efficiencies of the 

compressor and turbine and effectiveness of the regenerator 

have a lot of influence on the overall thermal efficiency. These 

thermodynamic random variables represent the most important 

indices for the gas turbine health determination. The adiabatic 

efficiencies of the compressor and turbine in the system are 

measures of irreversibilities or increase of entropy. The 

sensitivity factor for the compressor adiabatic efficiency is 

much larger than the corresponding value for the turbine. The 

sensitivity factors due to the adiabatic efficiency and inlet 

temperature of the compressor influence the most in the 

determination of the overall thermal efficiency of the system. 

Figure 8 shows the CDF for the net specific output of the 

hybrid gas turbine cycle. The sensitivity factors for the net 

specific output are plotted in Figures 9 to 11. The sensitivity 

factor due to the turbine inlet temperature influences the most 

in the evaluation of the net specific power output in the cycle. 

These results can be used to further optimize the design for 

cost effectiveness and also to assist in the health determination 

of the system. The prediction of degradation of system 

performance can be achieved from the results obtained. 

CONCLUDING REMARKS 

In this paper, a non-deterministic, non-traditional method 

has been developed to support reliability-based design. 

Probabilistic methods were applied to the thermodynamic 

analysis of a hybrid gas turbine system integrated with a fuel 

cell. The interconnection between the thermodynamic analysis 

and NESTEM codes was necessary to compute the probabilistic 

evaluation of a gas turbine field performance. Overall thermal 

efficiency and net specific power output of the gas turbine plant 

was evaluated using the thermodynamic random variables. 

Cumulative distribution functions and sensitivity factors were 

computed for the overall thermal efficiency and net specific 

power output due to the uncertainties in the thermodynamic 

random variables. Evaluating probability of risk and sensitivity 

factors will enable the identification of the most critical design 

variables in order to optimize the design, make it cost effective 

and assist in the health determination of the system. 

REFERENCES 

 [1] Pangalis, M.G., Martinez-Botas, R.F., and Brandon, N.P., 

“Integration of Solid Oxide Fuel Cells into Gas Turbine 

Power Generation Cycles, Part I: Fuel Cell 

Thermodynamic Modeling,” Proc. IME, Journal of Power 

and Energy, vol. 216, 2002, pp. 129–144. 

 [2] Cunnel, C., Pangalis, M.G., and Martinez-Botas, R.F., 

“Integration of Solid Oxide Fuel Cells into Gas Turbine 

Power Generation Cycles, Part II: Hybrid Model For 

Various Integration Schemes,” Proc. IME, Journal of 

Power and Energy, vol. 216, 2002, pp. 145–154. 

 [3] Rao, A.D., and Samuelsen, G.S., “Analysis Strategies for 

Tubular Solid Oxide Fuel Cell Based Hybrid Systems,”

ASME J. Eng. Gas Turbines Power, 124, 2002, pp. 503–

509. 
NASA/TM—2003-211995 5 

 [4] Ferguson, J.R., “Heat and Mass Transfer Aspects of 

SOFC Assemblies and Systems,” IEA Natural Gas Fueled 

Solid Oxide Fuel Cell Systems, Proc. Charmey 

Workshop, July 2–6, 1989, Swiss Federal Office of 

Energy, Berne. 

 [5] Haynes, C.L. “Simulation of Tubular Solid Oxide Fuel 

Cell Behavior for Integration into Gas Turbine Cycles,”

Ph.D. Thesis, Georgia Institute of Technology, July 1999. 

 [6] Bessette, N.F., “Modeling and Simulation for Solid Oxide 

Fuel Cell Power Systems,” Ph.D. Thesis, Georgia Institute 

of Technology, July 1994. 

 [7] Fox, E.P. 1994, “The Pratt & Whitney Probabilistic 

Design System,” AIAA–94–1442–CP. 

 [8] Chamis, C.C., 1986, “Probabilistic Structural Analysis 

Methods for Space Components,” Space Systems 

Technology Conference, San Diego, California, 

June 9–12, 1986. 

 [9] Nagpal, V.K., Rubinstein, R., and Chamis, C.C., 1987, 

“Probabilistic Structural Analysis to Quantify 

Uncertainties Associated with Turbopump Blades,”

AIAA–87–0766. 

[10] Chamis, C.C., 1986, “Probabilistic Structural Analysis 

Methods for Critical SSME Propulsion Components,”

Third Space Systems Technology Conference, AIAA, 

pp. 133–144. 

[11] Sundararajan, R, 1995, “Probabilistic Structural 

Mechanics Handbook,” Chapman and Hall, New York, 

NY. 

Table 1.Random Variables

Random Variable Mean 

Value 

Compressor inlet pressure (P1) 101.3 kPa 

Compressor inlet temperature (T1) 288 K 

Pressure ratio 14 

Turbine inlet temperature 1300 K 

Adiabatic efficiency of compressor 0.88 

Adiabatic efficiency of turbine 0.88 

Effectiveness of regenerator 0.90 

Table 2.Random Variable Labels 

Label Description 

P1 Compressor inlet pressure 

T1 Compressor inlet temperature 

PRAT Pressure ratio 

T3 Inlet temperature to the turbine 

QFC Fuel cell output 

ETAC Adiabatic efficiency of compressor 

ETAT Adiabatic efficiency of turbine 

ETREG Effectiveness of regenerator 

Inlet

Fuel cell

Exhaust Recuperator

Combustion

chamber

Single

turbine

Compressor

Figure 1.—Layout of the hybrid gas turbine system.

C/C

0

0

10

20

30

40

50

70

5 10 15 20 25 30

Figure 2.—Thermal efficiency versus compression ratio.

Compression ratio

Thermal efficiency, %

Recuperated gas turbine

FCIntegrate3

60

0

0

100

200

300

400

500

600

5 10 15 20 25 30

Figure 3.—Net specific power output versus compression

 ratio.

Compression ratio

Net specific power, kWs/kg

Recuperated gas turbine

FCIntegrate3
NASA/TM—2003-211995 6 

0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Thermal efficiency

Figure 4.—Cumulative probability of thermal efficiency. 

Cumulative probability

Figure 5.—Sensitivity factors versus random variables

 (probability = 0.001).

COMPRT

T1

T3 QFC

ETAC

ETAT

ETREG

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

Random variables

Sensitivity factors

Figure 6.—Sensitivity factors versus random variables

 (probability = 0.1).

COMPRT

T1

T3 QFC

ETAC

ETAT

ETREG

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

Random variables

Sensitivity factors

Figure 7.—Sensitivity factors versus random variables

 (probability = 0.999).

COMPRT

Random variables

T1

T3 QFC

ETAC

ETAT

ETREG

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

Sensitivity factors

0

0.2

0.4

0.6

0.8

1.0

1.2

0 50 100 150 200 250 300 350 400 450 500

Specific power output

Figure 8.—Cumulative probability of net specific power

 output. 

Cumulative probability

Figure 9.—Sensitivity factors versus random variables

 (probability = 0.001).

COMPRT

Random variables

T1

T3

QFC

ETAC

ETAT

ETREG

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

Sensitivity factors
NASA/TM—2003-211995 7 

Figure 10.—Sensitivity factors versus random variables

 (probability = 0.1).

COMPRT

Random variables

T1

T3

QFC

ETAC

ETAT

ETREG

–0.8

–0.6

–0.4

–0.2

0.2

0

0.4

0.6

Sensitivity factors

Figure 11.—Sensitivity factors versus random variables

 (probability = 0.999).

COMPRT

Random variables

T1

T3

QFC

ETAC

ETAT ETREG

–0.8

–0.6

–0.4

–0.2

0.2

0

0.4

0.6

Sensitivity factors
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13

Probabilistic Analysis of Solid Oxide Fuel Cell Based Hybrid Gas

Turbine System

Rama S.R. Gorla, Shantaram S. Pai, and Jeffrey J. Rusick

Thermodynamic analysis; Gas turbines; Probabilistic analysis; Fuel cell

Unclassified -Unlimited

Subject Category: 02 Distribution: Nonstandard

Prepared for the Turbo Expo 2003 cosponsored by the American Society of Mechanical Engineers and the International

Gas Turbine Institute, Atlanta, Georgia, June 16–19, 2003. Rama S.R. Gorla, Cleveland State University, Cleveland,

Ohio 44115; Shantaram S. Pai and Jeffrey Rusick, NASA Glenn Research Center. Responsible person, Jeffrey J. Rusick,

organization code 0510, 216–433–5375.

The emergence of fuel cell systems and hybrid fuel cell systems requires the evolution of analysis strategies for evaluating thermodynamic performance. A gas turbine thermodynamic cycle integrated with a fuel cell was computationally

simulated and probabilistically evaluated in view of the several uncertainties in the thermodynamic performance

parameters. Cumulative distribution functions and sensitivity factors were computed for the overall thermal efficiency

and net specific power output due to the uncertainties in the thermodynamic random variables. These results can be used

to quickly identify the most critical design variables in order to optimize the design and make it cost effective. The

analysis leads to the selection of criteria for gas turbine performance.