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STEADY-STATE AND TRANSIENT ANALYSIS OF A STEAM
REFORMER BASED SOLID OXIDE FUEL CELL SYSTEM

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presented by

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Sridharan Narayanan

139A

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has been accepted towards fulfillment
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Mechanical Engneering

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STEADY—STATE AND TRANSIENT ANALYSIS OF A
STEAM REFORMER BASED SOLID OXIDE FUEL CELL
SYSTEM

By

Sridharan Narayanan

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

2008

ABSTRACT

STEADY—STATE AND TRANSIENT ANALYSIS OF A
STEAM REFORMER BASED SOLID OXIDE FUEL CELL
SYSTEM

By

Sridharan Narayanan

In this thesis we perform a model-based analysis of a Solid Oxide Fuel Cell (SOFC)
system with an integrated steam reformer and with methane as fuel. The objective of
this study is to analyze the steady-state and transient characteristics of this system
for varying current demand. For the analysis, we develop a control-oriented model of
the system that captures the heat and mass transfer, chemical kinetics and electro-
chemical phenomena. We express the dynamics of the reformer and the fuel cell in
state-space form. By applying coordinate transformations to the state-space model,
we derive analytical expressions of steady-state conditions and transient behaviors of
two critical performance variables, namely utilization and steam-to-carbon balance.
Using these results, we solve a constrained fuel optimization problem in steady-state
using linear programming. Our analysis is supported by simulations. The results
presented in this thesis can be applied in predicting transient response and will be

useful in control development for SOFC systems.

To my parents

iii

ACKNOWLEDGMENTS

I would like to thank everyone who contributed towards the completion of my
thesis.

I sincerely thank my advisor Dr.l\Iukherjee for his direction, advice and support.
I cannot thank my co—advisor Dr.Tuhin Das enough for his time, guidance, patience,
insight and attention he gave me all through the process and I hereby express my
sincere gratitude for him. I really appreciate the members of my MS committee:
Dr.Ranjan Mukherjee, Dr.Tuhin Das, Dr.Andre Benard and Dr.Fang Zheng Peng for
the flexibility extended to me and for all their valuable suggestions.

I am extremely grateful to my parents whose love, blessings and encouragement
have always been a constant source of strength for me.

I would also like to thank all my fellow graduate students in the Dynamics and
Controls Laboratory for their companionship and for making the whole experience so
wonderful.

Also, my special thanks to the entire Mechanical Engineering department and

staff for making this voyage unique and memorable.

Sridharan N arayanan

TABLE OF CONTENTS

LIST OF FIGURES

1 NOMENCLATURE

2 INTRODUCTION

3 SOFC MODEL DEVELOPMENT
3.1 SOFC System Description ........................

3.2 Fundamental

Models ...........................

3.2.1 Solid Volume Model .......................

3.3 Reformer Model ..............................
3.3.1 Reformer Heat Transfer Model ..................
3.3.2 Reformer Mass Transfer and Chemical Kinetics ........
3.3.3 One Dimensional Discretized Reformer Model .........

3.4 SOFC Model

3.4.1 SOF C Heat Transfer Model ...................
3.4.2 Cell Voltage Model ........................
3.4.3 Fuel Cell Mass Transfer and Chemical Kinetics ........
3.4.4 One Dimensional Discretized Fuel cell Model ..........

3.5 Gas mixing

oooooooooooooooooooooooooooooooo

3.6 Combustor Model .............................

4 CHARACTERIZATION OF UTILIZATION
4.1 Steady-State and Transient Characteristics ...............

4.2 Simulations

5 CHARACTERIZATION OF STEAM-TO-CARBON BALANCE
5.1 Steady—State and Transient Characteristics ...............

5.2 Simulations

oooooooooooooooooooooooooooooooo

6 STEADY—STATE FUEL OPTIMIZATION
6.1 Problem Statement ............................
6.2 Optimum Fuel Operation .........................

6.3 Simulations

7 CONCLUSION

APPENDICES

vii

cocoon

10

12
12
15
16
18
19
21
22
24
27
28

32
32
34

36
36
37

39
39
39
41

43

44

Appendix 44

Formulaes to Compute Reaction Rates .................... 46
Data ....................................... 49
BIBLIOGRAPHY 52

vi

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9

4.1
5.1

6.1
6.2

LIST OF FIGURES

Schematic of the SOF C system .....................
Heat transfer into solid volume .....................
Schematic of a tubular steam reformer .................
Convective heat transfer in a reformer model ..............
Schematic of a one dimensional discretized reformer model ......
Schematic diagram of tubular SOFC unit ................
Convective and radiative heat transfers in a unit SOFC model . . . .
Schematic diagram of one dimensional discretized fuel cell model . . .
Schematic diagram of combustor .....................

Tiansient and steady-state utilization ..................
Transient. and steady-state STCR and STCB ..............

Steady-state fuel optimization ......................
Fuel optimization simulation .......................

vii

9
10
12
13
17
19
20
25
28

34

38

40
41

CHAPTER 1

NOMENCLATURE

Keo

Cross sectional area for heat transfer(1112)

Cross sectional area for heat transfer in discretized model (m2)
Specific heat(J/kg/K)

Specific heat at constant pressure (J / kg/ K)

Specific heat at constant volume (J / kg/ K)

Hydraulic diameter (m)

Catalyst ring equivalent spherical diameter (m)
Reformer bed emmisivity

Activation energy of reactions (1), (II), (III)

in Eqs.(3.17) and (3.38) (J/mol)

F araday’s constant (= 96485.34 Coulomb/mol)
Convective heat transfer coefficient(W/ m2 / K)

Molar enthalpy (J / mole)

Molar enthalpy change (J / mole)

Current draw (A)

Convective contribution to the heat transfer coeficient
in reformer(W/m2/K)

Stagnant bed conductivity (W / m/ K)

Kr

’CI, K111

K11

ICCH4a K00, KHQ
(CHQO

k

’5!

ks

L

Convective contribution to the heat transfer coeficient
in reformer(W/ m2 / K)

Equilibrium constant of reactions (1) and (III)

in Eqs.(3.17) and (3.38) (Pa2)

Equilibrium constant of reaction (II)

in Eqs.(3.17) and (3.38)

Adsorption constant for CH4, C 0, H2 (1 / Pa)
Adsorption constant for H20

Anode recirculation fraction

Fluid conductive heat transfer coefficient (W/ m / K)
Solid conductive heat transfer coefficient (W/ m / K)
Length(m)

Mass (kg)

Molecular weight (kg/mol)

Anode inlet mass flow rate (kg/ sec)

Anode exit mass flow rate (kg/sec)

Number of moles (moles)

Nusselt number

Number of cells in series

Molar flow rate of air (moles/ sec)

Molar flow rate of fuel (moles/sec)

Anode inlet flow rate (moles/ sec)

Anode exit flow rate (moles/ sec)

Number of electrons participating in electro-
chemical reaction (= 2)

Control volume pressure (N/m2)

Prandtl number

T13 7‘11, T111

0
ACT
AH

O
Ah‘f,298

KI» I£111

Partial pressure (N/rn2)

Rate of heat transfer (W)

Reynolds number along diameter

Reynold’s number along length

Universal gas constant (8.314J / mol / K)

Species rate of formation (moles/sec)

Rates of reactions (I), (II), (III)

in Eqs.(3.17) and (3.38) (mol/kg cat./sec)

Rate of electro—chemical reaction (moles / sec)
Entropy of formation at temperature T (.l / K Mol)
Steam-To—Carbon Balance (moles/ sec)
Steam-To—Carbon Ratio

Temperature (K)

Reference temperature (K)

Utilization

Volume (m3)

Velocity of fluid through a control volume (In/sex)
Net work done (W)

Species mole fraction

Reformer particle voidage

Catalyst bed void fraction

standard state Gibbs function for a unit. reaction (J / mol)
Enthalpy of reaction or adsorption (J / mol)

Enthalpy of formation at 298K and latm (J / mol)

Rate coefficient of reaction (I) and (III)

(mol PaO'5/kg cat/sec)

K11

Subscripts

a
air

asp

cat
Cb
60de
00er
des
elec

8:1}

in

177‘

red

sep

8 8

Rate coefficient of reaction (II)
( mol / kg cat / sec / Pa)
Molar flow rate (moles/sec)

Stefan-Boltzmann constant (2 5.67e-8 (l/V/mQK4)

Anode control volume

Air supply

Air supply pipe

Cathode control volume

Catalyst

Combustor control volume

Conductive heat transfer

convective heat transfer

Desired

Electrolyte

Exit condition of control volume

Gas control volume

Inlet condition of control volume

Values of 1 through 7 represent the species
CH4, C0, C02, H2, H20, N2, and 02 respectively
Preheater control volume

Reformate control volume

Radiation

Solid volume

Seperator volume

Steady—state

CHAPTER 2

INTRODUCTION

Among different fuel cell technologies, Solid Oxide Fuel Cell (SOF C) systems have
generated considerable interest. in recent years. Fuel flexibility and tolerance to impu-
rities are attractive attributes of SOF C systems. Their high operating temperatures
(800° to 1000°C) are conducive to internal reforming of fuel. The exhaust gases are
excellent means for sustaining on-board fuel reforming processes. SOFC systems are
not only tolerant to carbon monoxide but can also use it as a fuel. They also serve
as excellent combined heat and power (CHP) systems. However, the high operating
temperatures have precluded automotive applications of SOF C systems due to the
associated thermal stresses, material failure and significant start-up times.

In this thesis we perform model-based analysis of a steam reformer based SOFC
system with anode recirculation and methane as fuel. Two types of SOFCs have
typically been considered in the literature, namely the planar and the tubular config-
urations. Mathematical models of the planar version appear in [3], [5], [14] and [19],
and those of the tubular type appear in [8], [10], [12], [13], [16] and [21]. We develop
a lumped control-oriented model of a tubular SOFC system. The model serves as a
powerful simulation tool in absence of an SOFC hardware. It enables risk-free exper-

imentation and forms the basis of the analytical development presented in this thesis.

The model captures heat and mass transfer, chemical kinetics and electro—chemical
phenomena of the system. The chemical kinetics of reforming are based on experi-
mental results and observations presented in [2] and [20]. The model has similarities
with the ones presented in [10] and [16].

The transient response of a fuel cell system directly impacts its lead following
capability. Hence, characterization of the transients will be helpful in control design
for the cumulative system. One of the earlier works on transient analysis of SOFC
systems appears in [1], where the author applied dimensional analysis to characterize
voltage transients due to load changes. Transient simulations of an SOFC-Gas Tur-
bine hybrid system with anode recirculation is presented in [6]. In [17], the authors
simulate voltage response of a stand—alone SOFC plant to step changes in load and
fuel. In [15], the authors study the detrimental effects of load transients due to dif.
ferences in the response times of the SOF C, power electronics, and balance-of—plant
components. To mitigate these effects, they further investigate the effectiveness of
energy buffering devices such as battery.

In this thesis we specifically derive analytical expressions that characterize the
transient and steady-state behavior of fuel utilization (U) and steam-to—carbon bal-
ance (STCB) of an SOFC system. To the best of our knowledge, such results have not
appeared in the literature. Utilization is a critical variable in an SOFC system that in-
dicates the ratio of hydrogen consumption to the net available hydrogen in the anode.
While high utilization implies high efficiency, very high utilization leads to reduced
partial pressure of hydrogen in the fuel cell anode which can cause irreversible dam-
ages due to anode oxidation [16]. Typically, 85% is the target utilization for SOF C
systems. Steam—to—carbon ratio (STCR) is another critical variable in steam-reformer
based SOFC systems. STCR indicates the availability of steam for fuel reforming at
the inlet of the reformer. A minimum STCR, that allows stoichiometric combination

of steam and carbon, is necessary. For steam reforming of methane, a stoichiometric

mixture has an STCR value of approximately 2. A mixture lean in steam causes cata-
lyst deactivation through carbon deposition on the catalyst surfaces, [6], and therefore
must be prevented. In this thesis, instead of STCR, we analyze the transient response
of steam-to-carbon balance (STCB) due to its preferred mathematical form. This is
justified since a positive STCB automatically implies a mixture rich in steam and
hence results in a favorable value of STCR. Both U and STCR/STCB experience
dramatic transients due to step changes in load and our study focuses on predicting
these behaviors. These transients arise from the mass transfer and chemical kinetics
phenomena. Temperature variation in SOFC systems occur at a significantly slower
rate and simulations indicate that a quasi-steady thermal behavior can be assumed
with minimal loss of accuracy.

This thesis is organized as follows: In section 3, we first describe the SOF C system
under consideration. We then develop the mathematical model of the SOFC system
in three subsections. We first present the equations for fundamental gas and solid
control volumes. The following subsections elaborate on the steam reformer and
SOF C system models respectively, with emphasis on the mass transfer phenomena
and chemical kinetics. An open-loop simulation of the system model is provided next.
TYansient characterization of utilization U and STCB/STCR, due to load changes,
are carried out in sections 4 and 5 respectively, and simulation results are provided.
Based on these results, a steady-state fuel optimization problem is addressed and a
minimum fuel operating condition is derived in section 6. Concluding remarks are

provided in section 7.

CHAPTER 3

SOFC MODEL DEVELOPMENT

3.1 SOFC System Description

In this section, we describe a steam reformer based tubular SOFC system which forms
the basis of our analysis. A schematic diagram of the SOF C system is presented in

Fig.3.1. The system consists of three main components, namely,

0 Steam reformer
0 Solid oxide fuel cell

0 Combustor

The steam reformer produces a hydrogen rich gas-mixture by reforming hydrocar-
bon fuels. The solid oxide fuel cell produces electricity from electro—chemical combi-
nation of hydrogen in the anode and oxygen in the cathode. The combustor oxidizes
unused fuel to generate heat. For our system, methane is chosen as the fuel. The
hydrogen-rich gas-mixture produced by the reformer is supplied to the anode of the
fuel cell. Electro—chemical reactions occuring at the anode, due to current draw,
results in a steam-rich gas-mixture at the anode exit. A fraction A: of the anode ef-

flux is recirculated into the reformer through a mixing chamber where fuel is added.

Steam Reformer

 

 

Gas Mixer

Rccirculatcd fuel flow. kit/L,

 

 

 

 

 

 

‘

 

—>

 

‘

 

V

 

 

Fucloflow

Nf

Solid Oxide
Fuel Cell

 

 

 

 

..... 74 2 . - - - - . - <
L
Reformed
- . .{EE‘E‘EIX-‘l - fuel E
bed ' V
r ,/
- - V4 ........... l.

 

 

Exhaust

F

-- Anode nun, -

I

F
- Cathode if -

I
A'0

 

Combustor

 

 

 

 

 

{Air Supply- #7.- -

A

combustion
chambeL,»

 

 

 

 

 

Electrolyte

Prchcatcd
air

 

*—

 

 

Air flow

._I .
"cur

 

f An‘ows represent heat exchange

Figure 3.1. Schematic of the SOFC system

Steam reforming, which is an endothermic process, occurs as the fuel and steam mix-

ture flows through the reformer catalyst bed. We have considered a Nickel-Alumina

catalyst bed in our analysis. The energy required to sustain this endothermic pro—

cess is supplied from two sources, namely, the combustor exhaust gas that is passed

through the reformer, and the aforementioned recirculated flow. The recirculated

flow is routed through the reformer before being injected into the mixing chamber, as

shown in Fig.3.1. In addition to oxidizing the unused fuel, the combustor also serves

to preheat the incoming air flow. The tubular construction of individual cells causes

the preheated air to first enter the cell through the air supply tube and then reverse

its direction to enter the cathode chamber. The cathode air serves as the source of

oxygen for the fuel cell.

3.2

Fundamental Models

The thermal characteristics of the system is modeled using solid volume and gas

control volume models described below.

 

3.2.1 Solid Volume Model

The rate of change of temperature of a solid volume is dependant on the net rate
of heat transfer into the volume through the following fundamental energy balance
equation:

MSCSTS = Z Q, (3.1)

The conductive and the convective heat transfers into a solid volume is explained

through the following schematic diagram. The conductive heat transfer between

 

 

 

 

 

 

 

m—I 1n m+1 m+2

l,__-__-_-_;l,..______-,l '
i T0,m i i Tan-j-li
l | l I
I I l l
L. ________ I L ________ I

TS.m Ts,m+1

l< a

Lcond

' I Gas control volume

Solid volume
Figure 3.2. Heat transfer into solid volume

individual solid volumes is modeled using the Fourier’s law of conduction. In Fig.3.2,
the conductive heat transfer from the (m + 1)th element into the mth element is

Acond ks (Ts,m+1 " Ts,m)
Leond

 

Qcond = (3.2)

where, Acond and Lcond are the cross-sectional area and length respectively for con—
ductive heat transfer. Newton’s law of cooling is used for modeling convective heat
transfer between solid and gaseous control volumes. In F ig.3.2, the convective heat

th

transfer within the m element from the gas control volume into the solid volume

can be expressed as

Qcmw = Acoan (Tg,m _ Tag-m.) (3.3)

10

where Acmw is the area of convective heat transfer.

3.2.2 Gas Control Volume Model

The gas control volume model consists of energy and mass balance equations. Ad-
ditionally, it captures the reaction kinetics arising from reforming of fuel and elec-
trochemical reactions. The energy balance equation for a gaseous control volume
is

NgCuTg : 7li'r'tth-n. — Feathers + 2 Q9 - l”f/rttret (3'4)

The mass balance equation for individual species is constructed as follows,

(\fngg : fiierjJ-n “ Hera/jg + Rj,gs .7 = 1: 2: ‘ " ,7 (3-5)

where specific values of subscripts j, j = 1, 2, ~ - ' ,7, correspond to the species C H4,
C0, C02, H2, H20, N2, and 02 respectively, as described in the nomenclature.

From Eq.(3.5), we additionally have
7

7 7 7
Z Xjan : Z Xjrg = 1 :> Z ijq = 0 2:. 7.76;]: = 77m + Z Rj‘g (3.6)
i=1 i=1 i=1 i=1

From Eqs.(3.4) and (3.5) it is evident that, in our formulation, the states of the
gaseous control volume are T g and (VJ-,9, j = 1,2, - -- ,7. Flow is assumed to be
governed by a nominal pressure drop across each module, [16], and hence pressure is
not treated as a state variable. The gas mixture is assumed to satisfy ideal gas laws
and hence N9 in Eqs.(3.4) and (3.5) is related to Pg and T9 through the equation
N9 = Png/RuTg. In Eq.(3.4), CU, hm, and hem are related to the state variables

through the following general equations:

7
C,L,(T) = Z XijjCF) — Ru,
j=1
CpJ-(TVRU = a, + bjr + 9T? + dJ-T3 + eJ-T4, (3.7)
7 T
h. = h(T) = Z Xj( CPJ (T) dT + Ahff‘zgs)
3:1 298

11

where the coefficients aj, bj, cj, dj, ej, are given in [18]. The inlet and exite enthalpies,
hm and hem, are computed as hm = }L(T,j,,) and hem = h(Te$).In the following two

sections, details of the reformer and the fuel cell model are presented.

3.3 Reformer Model

In this section, we describe the model of a tubular steam reformer. The integrated
steam reformer provides a hydrogen-rich gas-mixture to the fuel cell anode. A
schematic diagram of the tubular steam reformer is shown in Fig.3.3. In sections
3.3.1 and 3.3.2, we present the details of a lumped steam reformer model. In section

3.3.3, we present an axially discretized version of this model.

3.3.1 Reformer Heat Transfer Model

 

 

 

 

Reformate Flow ——‘.—~—> .

......................................... I

 

 

 

 

 

 

. : Gaseous control volume a
""" ' —’I *— rrI

D Solid volume (Catalyst bed) '

 

Figure 3.3. Schematic of a tubular steam reformer

The steam reformer is modeled using three gas control volumes and one solid volume.

12

The exhaust, steam and reformate flows are modeled using gas control volumes and
the catalyst bed is modeled as a solid volume. Three convective heat transfer phenom-
ena, anvl, Qcomg, Qconv3, are captured in the model as shown in Fig.3.4. Qcoml
is the convective heat transfer between exhaust gases and the catalyst bed.Similarly,
QCOMQ is the convective heat transfer between the recirculated flow section and the
catalyst bed and (2601,03 is the convective heat transfer between the reformate and

the catalyst bed.

 

— Exhaust flow

 

——> Rcformatc flow
Catalyst
bed

 

 

 

— Recirculatcd flow

 

 

 

' Heat transfer into a gas control volume

 

 

Heat transfer into a solid volume

 

 

 

 

Figure 3.4. Convective heat transfer in a reformer model

The convective heat transfers between the annular catalyst bed and the exhaust
flow and the recirculated flow are modeled using Eq.(3.3). Here, the convection

coefficients H are calculated using

N k
712—qu

DH (3.8)

From [9] and Fig.3.4, the covective heat tranfer area Acom, and the hydraulic diamter

D H for the catalyst bed to exhaust heat transfer are derived as
Aconv : 27f7‘7‘2L’r, DH 2 2 (7‘7‘3 _ Tr2) (3'9)

13

and those for the heat transfer between the catalyst bed and the recirculated flow are
Aconv = 27r7‘1‘1Lra DH : 2""11 (3'10)

where TH and 7‘1-2 are the inner and outer diameters respectively of the catalyst bed,
rr3 is the diameter of the cylindrical exhaust can and LT is the reformer length.
The catalyst bed to reformate convective heat transfer is modeled using Eq.(3.3).
The corresponding convection coefficent is calculated using the following formula to

account for geometric contour variations in the porous catalytic bed:

2.06ReL Rel—30575 kf

6LT Pr’1/3 (3.11)

 

The Reynold’s number Re D is defined with respect to the catalyst ring equivalent

diameter (1,), and is given by

va
mm = —" (3.12)

V

Re L is defined with respect to the length of the catalyst ring Leat, and is given by

ReL = i (3.13)

 

In Eqs.(3.12) and (3.13), the empty channel velocity V is computed as

 

 

N v N L
N1-7I(TT2 _' TTI) NT
In Eqs.(3.12) and (3.13), the kinematic viscosity 1/ is computed as
7
I/ = Z Xj l/j(Tg) (3.15)
i=1

Similarly, the heat conduction coefficient kf in Eqs.(3.11) and (3.8) is computed using

7

if = 2x,- ka-(Tg) (3.16)
j=1

The prandtl number Pr is assumed to be a constant with an approximate value of

0.7.

14

3.3.2 Reformer Mass Transfer and Chemical Kinetics

In this section, we derive the mass transfer equations for the reformate control vol—
ume. The three main reactions that simultaneously occur during steam reforming of

methane are, [10], [20]:

(1)
(11)

CH4+H20 H CO+3H2
CO+H20 H COQ+H2
(III) CH4+2H20 <—+ C02+4H2

(3.17)

From Fig.3.1, the mass balance equations for CH4, C0, C02, H2 and H20 can be

written using Eq.(3.5) as follows:

Nra'qfl. = kNO/t’m —— N,.,,X1,r + 721,, + Nf

NT)?” = mfg/1’2,“ — N,,,X2,,. + R2,,

N33,, = kit/0x3,“ — [Vin/13¢ + 723,. (3-18)
Nr/I’am = kNoX/Ie — Nznxzm + 734m

Nrrf’sy = kNoX5,a — (VinXESJ + 735.7“

where N = PrVr/RuTT. Note that the reformer inlet. and exit flows shown in Fig.3.l

do not contain 02 and N2. Hence XGJ. 2 X737. = 0. From Eq.(3.17), we express R137",

 

 

 

 

 

 

j = 1, 2, - - - ,5, in terms of the reaction rates r1, 1'” and TH] as follows
- R13: _ —1 0 —1 -
Rar — T] f 1 —1 0
R/r = Gr. m = R3,, r = 7'11 G = 0 1 1 (3-19)
R4,1‘ _ T111 J 3 1 4
L ”R5,, _ —1 —1 —2 J

Since G has a rank of 2, therefore there are only two independent reaction rates

among 72”., j = 1, 2, - - - ,5. Considering the rate of formation of CH4 and CO in

the reformer to be independent variables, we can write

R3 7‘ = “721$ ‘ 722,7:
R4 r = —4R1,,— -— 732$ (3.20)
R5 r = 2R1,r + R2,r

and rewrite Eq.(3.18) as follows:

M3” = kNoX1,a — Nlej. + R1,, + Nf

Me... = wort/2,. - N...X2,.— + 722,.
NMM==rm%flAMJW—RM—Rw CH0
N34,. = max/1,. — Nam — 4R1, - 722,7.

Nr/I’ar = kNoX5,a — NinX5,r + 273m + 732;

From Eqs.(3.6) and (3.21) we deduce

7
(Vin = kNO + Nf + Z Rjfl- => Ni” = kNo + [Vf - 2R1; (3.22)
j=1

The mathematical functions representing the reaction rates r1, 7‘” and r1” are
provided in the Appendix. For the steam and exhaust control volumes which are
non-reactive, the corresponding equations can be derived using Eq.(3.5) by setting

Rm=OJ=LZ“wT

3.3.3 One Dimensional Discretized Reformer Model

The reformer model presented in sections 3.3.1 and 3.3.2 can be extended to develop
a one dimensional discretized model. Since our objective is system characterization
leading to control development, we refrain from modeling spatial dependence of the
various physical phenomena. Instead, we limit our focus to axial variations only. A
schematic diagram of a discretized reformer model is shown in Fig.3.5. The gas control

volume energy balance equations can be derived using the basic structure given in

16

 

 

|—-—— Exhaust

 

5- ——->Reformate

 

 

 

 

 

_ — - ' -— , —— —— — -

 

L- _ I Heat transfer into a gas control volume

 

 

 

L _ _l Heat transfer into a solid volume

 

Figure 3.5. Schematic of a one dimensional discretized reformer model

Eq.(3.4) and hence are not repeated here. The corresponding convective heat transfer
coefficients can be derived using Eqs.(3.8) through (3.16). The corresponding areas

of convective heat transfer are modified to Eqs.(3.9), (3.10) and (3.11).

A _ Aconv
com; ‘—

 

(3.23)

”1‘

to account for the model discretization. The mass balance equations for each gaseous
control volume can be derived from Eq.(3.5). For the reformate control volume, the
reaction rates can be modeled using the development in section 3.3.2 and Appendix A.
In the discretized model, the heat conduction in the catalytic bed between neighboring

h.

elements must be modeled. The overall heat transfer equation for the mt catalytic

17

bed element is derived from Eqs.(3.1), (3.2) and (3.3) and Fig.3.5 as follows:

'. m '
Al.s,7nCs,st,m = Z Qcorw + Z Qcond

 

 

2 Q' ()m ACOTIdkATSam-H _ TSJH) + Accmrlks(Ts,m——1 - Tam)
001w

LT /ur LT / u,»

771)

where, Z Q63”, 2 (25,70,211 1 + QUUUU 2 + QCZUU 3 and A com) is derived from Fig (3. 3)
as Acond— — 71'(r,2.2 — r7 2.1) The conductive heat transfer coefficient k3 is computed as
follows [7]:

ks = keg + kw + krad (3.24)

In Eq.(3.24), kw is the stagnant bed conductivity given by

2
k‘t k-t krrat—kf
k = k 1. 4 ——“‘— C“ — ——-—— . .
80 f 81 (kart _ kf) (log ( kf ) kcat + 0 0931 (3 25)

where km): is the catalyst bed conduction coefficient assumed to be a constant and kf

 

is the fluid conduction. coefficient computed as in Eq.(3. 16). The convection coefficient
kw is given by

km) = 0.75 kf PT RED (3.26)
where the prandtl number Pr is assumed to be a constant and kf and Re D are

computed using the formulaes in Eqs.(3.16) and (3.12). and kmd is the radiation

coefficient given by

 

1—0. 0.229 Earl T3
krad : fif— + 0‘ kt‘o km = 106 1) cut (327)

The definition of each term in Eq. (3. 27)1 IS given in the nomenclature.

3.4 SOFC Model

In sections 3.4.1 to 3.4.3, we present a lumped model of a unit tubular solid oxide

fuel cell. The fuel cell system consists of a number of such fuel cell units connected

18

electrically in series. In section 3.4.4, we present a discretized version of this model.
A schematic diagram of a unit tubular fuel cell is shown in Fig.3.6. The hydrogen-rich
reformate gas—mixture enters the anode control volume from left. It is separated from
the cathode control volume by the electrolyte which serves as a conductor of 02—
ions at temperatures above 800°C. Air is supplied to the cathode control volume via
the air feed tube (air supply pipe) as shown in Fig.3.6. The extended travel path of
air in the fuel cell is utilized to remove heat generated by the fuel cell from electro

chemical conversion of hydrogen to steam.

Anode control volume Cathode control volume

   

 

 

 

Reformate
_ _ _ _._> _ __
flow
03
1chl,o
Electrolyte Air feed [:3 Gas control volume

tube

Figure 3.6. Schematic diagram of tubular SOFC unit

3.4.1 SOFC Heat Transfer Model

A tubular unit is shown in Fig.3.6. The anode, cathode and air flows are modeled
as gas control volumes and the air feed tube and electrolyte are modeled as solid
volumes. Four convective heat transfer phenomena Qcmwl through me4 and one
radiative heat transfer de are captured in the model, as shown in F ig.3.7. The

following list gives a description of the different modes of heat transfer mentioned

19

above:

meq: Convective heat transfer between anode and electrolyte

anvg: Convective heat transfer between cathode and electrolyte

Qcm,.,,3: Convective heat transfer between cathode and air supply tube

Qc-quz Convective heat transfer between air flow and air supply tube

de: Radiative heat transfer between air supply tube and electrolyte

Anode Control

EICPU'Oll’Ie / volume

 

 

  

 

 

 

 

 

 

______ _ _ _, _ __ _ _ _ __ __ 1 —7i_
Anode ' f— | Cathode
flow — ' Q .
~ .______ ___ _C(,£(.L_-~___I flow
If fir-
.- —————— . ————————— . L.
/' growl Q d __t_ I “'3
| ra I'
c inv3 J, . 1
Cathode control '“ ’ " " — ' ‘— -------- [ _T_ {W “-0
volume ________ rm
7C; ftl i if”

 

Air control [:
volume

 

Air supply pipe

 

I Heat transfer into a gas control volume

r-‘I

 

 

 

 

. Heat transfer into a solid volume

 

Figure 3.7. Convective and radiative heat transfers in a unit SOFC model

The convective heat transfers Qconvl through me4 are modeled using Eq.(3.3). For

each case, the heat transfer coefficient ’H is assumed to be constant. The exposed

20

area for convective heat transfers are computed as follows:-

Qc011v13 Aconvl = 27f 7‘fc2,o Lfc
QC :2: A:0‘nv2 : 27f 7' :2 1L

. 0m, ( fr ,1 fc (3.28)
Qccm 113: Acoan : 27f ch1,o Lfc
Qconv/I: Aconv4 = 271' rfc1,i Lfc

The radiative heat transfer de and the corresponding exposed area for heat transfer

is computed as follows [9]:

. . 1 1 ‘— E I. . chl o 2
Q, = (m L . (T4, — Ti1 ) / + “L ’ , A. = 7r'r .
rad rad fc a p elec. EaSp Eelec Tf02,z' rad f c1,0

(3.29)

 

 

3.4.2 Cell Voltage Model

The cell voltage is modeled by subtracting the activation, ohmic, and concentration
potential from the open circuit voltage, i.e. Nernst potential. The effective cell voltage

can therefore be expressed as:

Vcell = VNernst - VAct — VConc — VOhm (3-30)
The Nernst potential is computed as

1/2
AGO R, T PH 1’0
T + u at” 2 2

 

V . = 3.31
N c rnst 11F ”F pH2O ( )
Where AC? is the Gibb’s free energy for the reaction
1
H2 + 502 —* H20 (3.32)

computed at the anode temperature Ta using Eqs.(8) and (9)given in appendix A.
Noting that p112 = PaX4,a1 p02 = PCX7,C and [21120 = 1902654,, the Nernst potential

in Eq.(3.31) can be expressed as
1/2
X4.aX7,c 1361/2

p

o,a

ln

 

A053,. + aura

VNe'rnst : TLF 71F (3'33)

21

The activation polarization loss VAct is given by [16]

7f/Accll _ 1
( 10(TC) ’ Acell ’ 27” f(:2,oLfc (3.34)

RUTC n
1117

 

 

VAct :

Where 1'0 is the exchange current density assumed to be a constant. The concentration

potential Vconc is computed using

R T ' A..
V00“, = — '5F“1n(1— —2/ ‘1“) (3.35)

7'1 im

 

where 3),,” is the fuel cell limiting current density. The ohmic voltage loss VOhm is
given by
VOhm = zfaint/40611 (336)

where Rmt is the internal resistance of each cell and is assumed to be a constant [16].
The heat generated from the electrochemical combination of hydrogen and oxygen is
expressed as the difference between the enthalpy of formation of water and the power
generated. The model assumes this heat to be generated in the electrolyte solid and

is given by the expression

Qelec 2 _ (hf(T€l€C)fi1_ _ zVcell) (3'37)

where hf(Tejec) 2 Ah? 298 + h(Telec) — h(298), presents the heat of formation of

steam at the temperature of the electrolyte solid.

3.4.3 Fuel Cell Mass Transfer and Chemical Kinetics

In this section, we derive the mass transfer equations for the anode and cathode
control volumes. For the air control volume, the corresponding equations can be
derived using Eq.(3.5) by setting Rig = 0,j = 1,2, - -- ,7.The three main reac-
tions that simultaneously occur during steam reforming of methane are, [10], [20]:

Anode control volume: The following chemical and electro-chemical reactions occur

 

22

simultaneously in the anode control volume:

(I) C71f4 i-.ff2(2 <—+ (7(2 4-13112
(II) CO+H2O H CO2+H2
(III) CH4 + 2H20 +—> C02 + 4H2

(3.33)

(IV) H2 + 02" —> H20 + 26
Steam reforming, represented by reactions I, II and III, occur in the anode due to high
temperatures and the presence of nickel catalyst. The primary electrochemical process
is steam generation from H2, described by reaction IV. Simultaneous electrochemical
conversion of C O to C02 in the anode is also possible. However, this electro—chemical
reaction is ignored since its reaction rate is much slower in presence of reactions II
and IV, as indicated in [4] and references therein. From Fig.3.1 and Eq.(3.5), the

mass balance equations for CH4, CO, C 02, H2 and H20 can be written as

Narf’m = -NoX1,a + Nina/1,1" + 7314

Nax2,a = -NoX2,a + NinX2,r + 73241

[Va/193,31 = -No?(3,a + Mat/13¢ + 733,11 (339)
[Va/1:114 = —No/I’4,a + Minx” + R4,a - ‘r‘e

Ara/Fae = —N0X5,a + (Vin/tbw + 73541 + Te

where Nu = PaVa/RuTa and re is the rate of electrochemical reaction given by

i cell
= _ 3-40
8 71F ( )

Since current 1' can be measured, the rate of electrochemical reaction re is known. As
with the reformate control volume, the anode inlet and exit flows do not contain 02
and N2. Therefore, X630 2 X7’a = O. From Eq.(3.38), we express Rim}. = 1, 2, - - - ,5,

in terms of the reaction rates TI, TI] and TH] as follows
Ra=Gr+'re[O 0 0 —1 1]T (3.41)

where Ra = [731,0 722,0 723,0, 724‘, 725347,, and G and r are given in Eq.(3.19). Since

G has a rank of 2 and re is known, therefore there are only two independent reaction

23

rates among RM, j = 1, 2, - -- ,5.

variables, we can write

Considering R141 and 7223“, to be independent

733,11 = 431,11 - 732,11.
724,“ = —4’R.1’a — R2,, — re, (3.42)
735.4. = 273111 + 722., + Te
and rewrite Eq.(3.39) as

Narf'm = -NoX1,a + Nut/1714 + 731,3

(Va/I’m = “N0X2,a + Nan/Var + 732.11

NaXBu = - 3X33; +Nmr1’3m - R1,a - 732,1. (3-43)

N444“ = -NoX4,a + (Vin/1’44 - 4R1,a - R2,a - Te

21/139,, = —N0X5,a + N,,,X5,,. + 272”, + 7224 + re

From Eqs.(3.6) and (3.43) we deduce that

7

N0 = N," + Z 72,4, :> N0 :2 Nm — 2721), (3.44)

i=1

The models of internal reforming reaction rates 7‘], 7‘1 I and TI” are shown in Ap—

pendix A.

Cathode control volume: Ionization of 02 in the cathode control volume occurs

 

through the reaction

$02 + 26 —+ 02— (3.45)

with the reaction rate as given in Eq.(3.40). Considering the mole fractions of N2 and

02 in air to be 0.79 and 0.21 respectively, the mass balance equations of the cathode

control volume can be written from Eqs.(3.40) and (3.45) as follows:

Aral-”6,0 = 0-79Nair _ (Nair 7‘ 0‘5“?) X616

N628“, = 0.21113,” — (NW — 0.5m.) 267,6. — 0.51;. (3.16)
Xjfc = 0, j=1,2,~- ,5

24

3.4.4 One Dimensional Discretized Fuel cell Model

The lumped fuel cell model presented in sections 3.4.1 through 3.4.3 can be extended
to develop a one dimensional discretized model. As in section 3.3.3, we refrain from

modeling spatial dependence of the various physical phenomena and we limit our

focus on the axial variations only.

 

 

 

 

    

 

 

 

 

 

 

Anode
flow
Electrolyte faggg/iode
4» ._
A. l ' 1 .
Pigesupp y _______ ‘37.“: f5"! _

 

Heat transfer into a gas control volume

I
L—J

I'-
I.I

 

 

L—I

 

 

I"1

Heat transfer into a solid volume

 

Figure 3.8. Schematic diagram of one dimensional discretized fuel cell model

A schematic diagram of a discretized fuel cell model is shown in F ig.3.8. The
gas control volume energy balance equations can be derived using the basic structure

given in Eq.(3.4) and hence are not repeated here. The areas of convective heat

transfer given in Eq.(3.28) are modified to

25

— A. . — A. _ A... _ A...
Acon'ul : flit: ACOII'UZ = “M1 Ac07w3 : Ma Aco7w4 : M
11ft. ufc ”f0 ’1th
(3.47)
to account for model discretization. The mass balance equations for each gaseous
control volume can be derived from Eq.(3.5). For the anode control volume, the
reaction rates can be modeled using the development in section 3.4.3 and Appendix
A. In the discretized model, the heat conduction between the neighboring elements
of the electrolyte and the air supply tube must be modeled. For each elementary unit

of the electrolyte and air supply pipe, in addition to the convective heat transfer,

the radiative and conductive components must be accounted for. The radiative heat

 

 

tiansfei between the 771‘” solid volume elements is deduced from Eq.(3.29) and Fig.3.8

as

'(771) __ / Lfc 4 1 1 — Ealp, 'rfcl,o _ 2
QMd — ‘7 171171—qu (TaSPm T641“ m)/ (Easp + E6316; 0 7702.3 , Arad _ WTfCLO

(3.48)
The overall heat transfer equation for the mth electrolyte element is desired from

Eqs.(3.1),(3.2)and (3.48) as follows:

, ' 771 7(71) (777)
A’IelecmiCelec,mTelec,m — 2 (2607311, elec+ Qrad + Z Qcond, elr c

 

 

= ZQCfov, elec + ercntdf + ' H
. Acond,eleckelec (Telec,m+l _ Telec,7n)
Lfc/ufc
Acondeleckelec (Telec,m—1 — Telec,7n.)
Lfc/ufc
where, 2 Q c on U elec =62:ko 1 + Q0321, 2, the cross-sectional area for conductive heat

transfer in the electrolyte is given by Acondfllec = 7r(r;62?0 — 76%“), and the conduc-

tive heat transfer coefficient kelec is a constant. Similarly, the over all heat transfer

26

th

equation for the m air supply pipe element is given as

' 711(711)771
AIasp.111.(yasp,-7n.711.911.71‘1 : E Qcmw (15])- 7011 +2 Qszdxmp

m 711
= E :Qwu): 1,131) _Q,(,(]) +

A(:071d,(1spkab’p (7hsp,'171.+1 _ Taspan.)

 

 

Lfc/ufc
+ Acondxzspku-S'p (TaspJn—l _ 711311.711.)
L 11/ “1 fc
'7" ° 717(771)
where ,2 (26072,. (,5,,— —Q£m'3, 3 + Q 1 (”w 4, the cross— sectional area for conductive heat

transfer in the air supply pipe is given by Acmdamfl, 2 71(7' fc1,0 — 'rfclx') and the

conductive heat transfer coefficient. kasp is a constant.

3.5 Gas mixing

The recirculated anode flow kNO and the fuel flow Nf are mixed and the mixture is
pressurized in the gas mixture, shown in Fig.(3.1). The mixing and pressurization of
the two fluid streams is achieved using an ejector or a recirculating pump [6], [11]. As
with other component models, we consider a nominal pressure differential across the
gas mixture and ignore the pressure dynamics in our model. The following energy

conservation equation is implemented to model this component:
(“111117C11,71‘1.1';1'T7711:1? : thf(Tf) + kIVOh-a(T(1) - UVf ‘l' kiNrO)h'711.i:r(YiIIi.v) (3'49)

with the enthalpy h computed as shown in Eq.(3.7) and the mole fractions of the
mixture computed as

JVf/ij + kNOXaJ

X - = . .
11111,] (Nf + kNO)

. j: 1.2.....5 (3.50)

 

27

3.6 Combustor Model

The. combustor is modeled as a lumped unit with three control volumes, namely

0 A mixing and combustion chamber where the anode and cathode exhaust gases
combine followed by combustion of unused fuel. This is modeled as a gas
control volume.

0 An air preheater control volume which models the preheating of cathode air
using the heat of combustion.

o A solid volume that acts as a separator and heat conductor between the gas

mixture in the combustion chamber and the air control volume.
A schematic diagram of the combustor is given in Fig.3.9 The reactions that occur

 

 

 

 

 

 

 

 

 

Anode exit __1'; — lV—li—xin—g—cgt bbhibustiim— _ 1
flow (l-k) No S ——:+——» Exhaust gases
Cathode exit —-:——» Qcoan I [1: Gas control volume
“W‘Nwr'O-Src) :iiiiiiiiigiiiiiiiiiié :5; soudvouune
Nair —r+ i QC011V,2 ——-> Preheated air

 

 

 

Figure 3.9. Schematic diagram of combustor

in the combustion chamber are
((1) CH4 + 202 <—+ 002 + QHQO
0)) CO + 21,02 <——> C02 (3.51)
(c) H2 + 502 H H20

We assume that each of the combustible species, namely C H4,CO and H2, have the

same selectivity for combustion. From F ig.3.9 and Eq.(3.51), we have the following

28

mass balance equations for the combustor control volume:-

Nebxlxrb = (1 — MN 0X1 (1 - llerxrbxlfib + Ric-b

Nob/Vat = (1 -k)N0X2,a - 7l(r;i:,cbX2,cb + 732w

N(.-b’l?3,cb = (1 - klN0X3 a - Tlex,c~bX3,cb + 733m

Nth/film!) = (1- klNOXda - 77(..,:,cbX4.cb + 734,4) (352)

Nab/Yaw = (1—1»)1\0X5 a — 7lex,cbX5.cb + 735w

Ncbxtixrb = (Nair - 0 5%)?(6 c -'7ie;c,cbX6,c-b + R6,Cb
(

Arr-b26161) = Nair— 0'5'r6)x7,(f — flfiil'.CbX7,(.‘b + R7,cb

Where, Ncb = Pcch-b/R-uTc-b From reactions (a), (b), (c) in Eq.(3.52), it is clear that

one molecule of C H4, CO and H2 require 2, 0.5 and 0.5 molecules of 02 for corn-
plete combustion respectively. With uniform selectivity for combustion, the oxygen
available will be proportionally distributed for each reaction. Since the flow rates of
C H4, CO, and H2 into the combustion chamber are (1 — Mil/0.13,“, (1 —— MNOXQJL
and (1 — k)N0X4,a, from Eq.(3.52) we deduce that for complete combustion, the

minimum oxygen molar flow rate is
'mf02,dcs = 2(1 _ k)1VOXl.(I. ‘l’ 0 5(1— klNoxz, a + 0 5(1—klNOX4fi (3-53)

In the combustor model, we consider two scenarios, namely, a mixture deficient in
oxygen and one with excess oxygen. For the former case, considering uniform selec-

tivity of all combustion reactions and noting that the molar flow rate of 02 at the

29

inlet of the combustion chamber is (Nair -- 0.57‘6)X7,C, we have

(1—k)N0X1.a .
—W ' (Iva/£7. — 0-576)X7,c
(l—kflVO’YQfl
REC?) : —W ' (‘Nair _ 0-5'r6)X7,c
(l—k)N0X4. . -
R4,cb = _W ' (‘Nai'r' — 0-57'6)X7,c

R)
p—d
g
I

(3.54)
733425 = —R1,cb—R2,cb
Rad) = —2R1,cb—R4,cb
Red; = 0
737.55 = '-(Naz:r-0-5‘r'e)?(7,c

with m f02,des < (Nair — 0.5're)X7,c. When the inlet combustion mixture contains
excess oxygen, i.e.

(NW — 0.5re)x7,c 2 m fozfle, (3.55)

then we have the following reaction rates

731m = —(1 - ”NO/Via
Rich Z —(1 _ klNOXZa
R3,cb = -R1,cb - Rad;
R = — l—k N X4

4,cb ( ) O ,a (3.56)
R5,Cb = —2R1,Cb _ R4,cb
7€6,cb = 0
7aid) 2 _'mf02,des 2 _l2(1 — k)NOXl,a+

o.5(1 — kwoxza + 0555/1113“)
Further more, from Eq.(3.56), we have
a n 7
7-78,. = (1 — k)NO + (NW — 0.5712) + 2 RM, (3.57)
i=1

The thermal behaviour of the combustion chamber is modeled using Eq.(3.42) as

follows:-

chbCUch : [(1 _ klNo 'l' (Iva'ir -' (151%)] hin,cb — 7'7exhex,cb + Qconvd (3-58)

30

where Q(,0.,,.,,.1 is the convective heat transfer from the separator solid to the combus-

tion chamber, expressed as

Qconvd = Acorn.’,cho-nv,l(Tsep _ Cb) (3-59)

Awm‘l, Hem-ml are the area and heat transfer coefficient respectively and T3”, is the
temperature of the separator solid. Since the air preheater is a non—reactive control
volume, there is no net change of mole fraction or molar flow rate. Therefore, the mass
transfer equations are trivial. The thermal behaviour of the air preheater volume is
given as

ArperTpT = N airhinmr — N airhegvgpr + Qcmw,2 (3'60)

Where: ANIH' = Pperr/Rqur and Qconv.2 = Arron-v2Hcowwfl(Tsc’p _ Tp7‘)- Finally) the
thermodynamics of the separator solid is modeled using the following energy balance

equation:-

A'IsepCscstep = "Qcomfl “ Qco-nv.2 (3'61)

31

CHAPTER 4

CHARACTERIZATION OF
UTILIZATION

4.1 Steady-State and Transient Characteristics

To gain understanding of the dynamics of utilization, we perform an analysis based
on the state-space models derived in previous sections. Based on the state variable
definitions in Eqs.(3.21) and (3.43), fuel utilization can be written as follows:

_ N0 (4X1’a + X20 + X41!)

U = 1 .
Nin (4X1,r + X2,7‘ + (film)

(4.1)

 

Eq.(4.l) is based on the internal reforming capability of the fuel cell anode where
a C H4 and a CO molecule can yield four and one molecules of H2 respectively, as
indicated by reaction (I), (II) and (III) in Eq.(3.38). We rewrite Eq.(4.1) with the

coordinate transformations as shown below:

' : 4X + X + X
U = 1 — N001, Cr 1,? 237‘ 4’1" (4-2)
Ni'llCT' Ca 2 4Xl,a. + X23 + X450»

 

32

Using Eqs.(4.2), (3.21) and (3.43) the following state variable descriptions for CT and

Ca can be obtained:

N' kN 4N:
' C — I « 'V . z
Z=A1Z+Bl,Z= r ,A1= 35? 1‘: ,B1= 'r (4.3)
1’ 0

Ca at? ‘13:. {7*de
It is interesting to note here that Eq.(4.3) is devoid of the reaction rates R13", 722$.
R130, and 722%,. This removes the nonlinearities associated with the reaction rates as
given in Eqs.(l) through (3). Nevertheless, Nm, N0, NT, Na are nonlinear functions

of the states, temperatures and pressures given by

5 W0: 5 M0 ,Nr=
21/13,,TAIW’, 2128me
1:

[Win , [Va = Pal/a (44)

 

 

 

 

Nin =

From Eq.(4.3), we obtain the following expression relating U, k, 2' and Nf at steady—
state

1 — k
U... = —.—— (4.5)

4nFN
__,V_.[ _. k
[.1 C

Eq.(4.5) is independent of the nonlinear variables given in Eq.(4.4). Furthermore.
since It, i and Nf are measurable and known quantities, Eq.(4.5) can be used to
exactly predict the steady—state fuel utilization for any given set of inputs.

From Eqs. (4.2) and (4.3) we note that the transient behavior of U can be predicted
from the transient response of C)», (a, No and Ni". We specifically consider the
transient response of U due to step changes in i, when k and Nf are constant. In
predicting the transient characteristics we assume that, in the process of a step change
in i, the variables Ni” and N0 can be treated as constants without significant loss of
accuracy. W'ith this assumption, Eq.(4.3) reduces to a Linear Time Invariant (LTI)

system with eigenvalues of A1 as:
Em. __
r

z o.[ (fi+.a).(

 

a2-

 

(4.6)

 

 

Since A“ E (0, 1), the eigenvalues of A1 are real and negative. The time constant for
(r, (a, and hence that for U due to a step change in i will be determined by the

maximum eigenvalue of A1. From Eq.(4.6),

N 'r 'r

+1 2 A... —+ Am... = —*"—,Q + 0.54-

1_ [Va (Na (4 7)
Art A] A _ N‘7 0 5 -
fi<1a —’ 712.a$—_7§l;1+ -Q’

4.2 Simulations

In this section we provide simulation results in support of our analysis in the previous
section. We run multiple simulations of the SOF C system with step changes in current

applied at. t = 508, as shown in F ig.4.1. The step changes in current are from 25A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

50 . . - 80
45. . . i =48 . l = 48
. ’o‘ 60?
6? 40» - '=.43 23 i = 43
o. . c
5 i E Estimated Utilization
25 O f ;— Actual Utilization
20 . . . . '; .. . ; _
4O 60 80 100 120 40 60 80 100 120

(a) (b)

time (s)

Figure 4.1. Transient and steady-state utilization

to 33, 38, 43 and 48A as shown in Fig.4.1(a). For all four simulations, the following

settings were used

NJ: 2 0.0068 moles/s, Nair = 0.0692 moles/s, k = 70%.

In F ig.4.1(b) the transient response of utilization is plotted for the four simulations.
The estimated utilization are obtained by simulating Eqs.(4.2) and (4.3) as an LTI

system with A1 evaluated at the instant t = 508. The following values were observed

34

at t = 508

N," = 0.0624 moles/s

N0 = 0.0681 moles/s A1 = —2.8347

Nr 2 0.0263 mole :> A2 = —0.1543

Na = 0.1105 mole
Both the transient response as well as the steady—state value of estimated utiliza-
tion match very closely with the non-linear model based calculation, as shown in

Fig.4.1(b). The settling time computed based on 2% error is T5. = 4 / [Agl = 25.92353

which matches well with the simulations.

35

CHAPTER 5

CHARACTERIZATION OF
STEAM-TO-CARBON BALANCE

5.1 Steady-State and Transient Characteristics

The Steam-To—Carbon-Ratio (STCR) is defined for the inlet flow of the steam re-

former and can be mathematically expressed using Fig.3.1 as

kN0X5’a
Nf + I‘u'Afo/‘l’llxz + kN0X2,a

 

STCR = (5.1)

As the name suggests and is indicated by Eq.(5.1), STCR is the ratio of the concen-
tration of steam molecules to that of carbon atoms at the inlet of the reformer. The
reactions (b) and (c) of Eq.(3.17) indicate that the stoichiometric quantity of steam
required for reforming is two moles and one mole of steam for each mole of CH4
and C 0 respectively. With this understanding, we define a new variable, namely the

Steam-To—Carbon—Balance (STCB), which is mathematically expressed as

STCB = um... — (21vf + 214N025... + max...) (5 2)
= [41% (X5. — 2X14, — x...) — 21vf
A positive value of STCB is an indication of sufficient steam at the reformer inlet

for steam reforming and hence it is an indication of a favorable STCR. \Ne rewrite

36

Eq.(5.2) with the coordinate transformations as shown below:

. . = Xv .- 2X1 — X .
STCB = kNOEa — 2Nf, 6r 0’7 ’r 2”“ (5.3)
5a = X5,a - 2Xl,a _ X2,a
Using Eqs.(5.3), (3.21) and (3.43) the state variable descriptions for {r and 5a can be

written as

N; .. m ~21)?
S=AQS+B2,S= 6’" ,A2= 1V7" Wig .132: 7’71 (5.4)
Note that, as in Eq.(4.3), the variables Er and {a in Eq.(5.4) are independent of
R1,... R2,,” Rl,aa and 7220,. N,,,, No, NT, Na are nonlinear functions of the states,
temperatures and pressures given by Eq.(4.4). From Eq.(5.4), the following steady-

state expression for STCB is obtained

 

1 ., MN.

Note, from Eqs.(4.3) and (5.4), that A2 = A1. Hence, the time constant in the tran—
sient response of STCB due to step changes in the current demand can be estimated
using the eigenvalues A13 of A1 given in Eq.(4.6). The discussion around Eqs.(4.6)

and (4.7) is also applicable for transient response of STCB.

5.2 Simulations

The simulation results provided here are continuation of those provided above for step
response of utilization in F igs.4.1(a) and (b). In Figs.5.1(a) and (b), STCR and STCB
are plotted for the four simulations described in Fig.4.1(a). In F ig.5.1(b) the transient
response of STCB are plotted. The estimated STCB are computed by considering
the system given in Eqs.(5.3) and (5.4) as an LTI system with A2 evaluated at the
instant t = 508. Both the transient response as well as the steady-state value of
estimated STCB match very closely with the non—linear model based calculation, as

shown in Fig.5.1(b).

37

STCR

 

i=48

2.5. _  
2_ » ~ i=43

//7 i=38

 

 

 

 

15»
1 ,/ .

 

4o 60 80 100 120
(a)

STCB (moles/sec)

 

r.

 

P i = 43
'3 F l: 38
' -- Estimated STCB

‘——- Actual STCB

 

i=48

 

 

 

i=33

 

 

 

60

(b)

80 100 120
time (s)

Figure 5.1. Transient and steady-state STCR and STCB

38

CHAPTER 6

STEADY-STATE FUEL
OPTIMIZATION

6. 1 Problem Statement

Using the above derived results, we address a steady—state constrained fuel optimiza-
tion problem which is stated as follows: Given that utilization and anode recircula-
tion must be constrained within ranges U33 E [Ussla U332], 0 < U831, U332 < l and

k E (k... 161,] a 0 < Ira, Ir), < 1 respectively, and given a current i

1. Determine condition(s) under which there exists a range of solutions for N f

that satisfies the constraints above and maintains STCB Z 0.

2. If a range of solutions exists, determine the minimum fuel operating conditions.

6.2 Optimum Fuel Operation

From Eq.(5.5), we note that for ensuring a steam rich inlet flow into the reformer, we

must have

 

. I'N,
STCB... 2 0 —+ Nf g 16(5n12) (6.1)

39

 

 

 

ll
iNC/(4nFU551)
lul
T ; iNC/(ZnF)
,_ iNC/(4nFU..g) 1112 . ' . :
. Z I ' :
- —————————————— : ‘i”” — ' iNC/(4nF)
1‘1 i E 5 E
lr2 : : i A
0 1* 1 i '
k ——->-
Figure 6.1. Steady-state fuel optimization
From Eq.(4.5) we have
‘ 7:JVC

USSZO—vaZk(4nF) (6.2)

and the constraints 0 < U331 S U33 S U332 < 1 are expressed as

- 4'nFU 1
Nf (W281) + (1 — U331) k g 1 (6.3)
- 4nFUL .2

Nf (ml) + (1 —- U332) k 2 1 (6.4)

Eqs.(6.1), (6.2), (6.3) and (6.4) are all linear in Nf and k and are denoted in Fig.6.l
by lrl, (7'2, [U1 and luz respectively, along with the lines k = ka and k = kb. Steady-
state constrained fuel optimization for the steam reformer based SOFC system has
thus been transformed to a problem in linear programming. From F ig.6.1 and from
Eqs.(6.1) through (6.4), we can easily deduce that a solution region exists if k), 2 k*,
where 16* is the value of k at the intersection between the lines [7'1 and [112. Hence,

from Eqs.(6.l) and (6.4) we have:

I 1
=———— k >—— 6.5
1+Uss2 —> b ( )

[3*
_ 1+ Ussg

40

From F ig.6.1 it is also evident. that if Eq.(6.5) is satisfied, then the steady-state

minimum fuel operating point is at the intersection of [U2 and k 2 kb, given by

z'NC

k. 2 (Lb, U = Uss2: Affflnin : 4'ILFU 2
ss

[1 ‘7 (1 ”' U552) kbl (6-6)

6.3 Simulations

Steady-state minimum fuel operation is demonstrated using the following simulation
results. We consider a system with 100 cells in series and i = 50A. It is desired to
attain the steady state minimum fuel operating point under the constraints, U 331 =
0.6, U332 = 0.85, ka = 0.6, kb = 0.8 and STCB... 2 0. We choose the initial operating
conditions A: = 0.65 and N f = 0.01moles/s, and set the air flow rate at Na = ION f for

the entire simulation. The simulation results are shown below in Fig.6.2. In F ig.6.2(a),

x 111-3 (a) (h)

 

 

   

 
   

Target utilization ,

 

 

 

 

 

 

 
  

   

 

 

 

 

 

 

 

N I H (“”35"
l ( / ,/V U 04 — Actual utilization I
4 1rl / /J" ' ,
l / » ' ' 0.7 .
l /_/:/n’/]r/ l l ‘ - I t% 1
0L;/ . .2 I I (‘l : . “it: _ _
0 0.2 0.4 0.6 0.8 1 0i 100 200 300 400 500
k 1 time(s)
(C)
. 6’
0.8 a;
U 4
i;

k >1 2
0.7 U , \ ; .
t5 1 51 0 ; ,12 itioxsrcs‘.
1. -;5. 1. 0.6 . i E _2 a1“ x; 1
(ii 100 300 300 400 500 0 100 200 300 400 500 0 £1100 200 300 400 f500

l . . .
time (s) time (s) time (s) 5

Figure 6.2. Fuel optimization simulation

the lines lrl, lrg, [U1 and (1&2, representing Eqs.(6.1), (6.2), (6.3) and (6.4), are plotted

and the trajectory of the Operating point in (N f vs. k) space is superimposed. The

41

initial conditon corresponds to point A in F ig.6.2(a). At this operating point, the
model is first simulated in open loop mode upto t1 = 405. Note that at A, the
conditions U331 _<_ U33 S U332 and STCB,” Z 0 are not satisfied. Specifically, at A,
U... z 0.4 and STCBSS z —0.01moles/s, as shown in Figs.6.2(b) and (0) respectively.
At It], a proportional-integral control is invoked to control utilization by varying N f.
Simultaneously, the target utilization is ramped from 0.45 at t1 to 0.85 at t3 = 2403,
as shown in Fig.6.2(b). This leads to reduction of Nf, depicted in Fig.6.2(c), from
0.01moles/s to 6.878 x 10’3moles/s. During this interval k is maintained at 0.65,
as shown in Fig.6.2(d), and the operating point in Fig.6.2(a) shifts from point A
to B. Also note from Fig.6.2(e), that STCB > 0 for t > t2. At t4 = 3008, k is
ramped from 0.65 to kb 2 0.8 at t5 = 4508. This leads to further reduction of N f to
6.705 x 10‘3moles/s, as shown in Fig.6.2(c). The corresponding shift in the operating
point from B to C is shown in Fig.6.2(a). The minimum fuel operating point as
computed using Eq.(6.6) is 6.706 x 10”3moles/s which matches very closely with that
obtained through simulation. It must be noted that the proportional-integral control
implemented here is not a proposed control strategy. It is used to automatically arrive

at the minimum fuel operating point and thus it serves to validate Eq.(6.6).

42

CHAPTER 7

CONCLUSION

In this thesis we have presented an analytical study of steady-state and transient
behaviors of an SOFC system due to changes in current demand. For the analysis
we considered a steam-reformer based tubular SOFC system with anode recirculation
and with methane as fuel. We developed a detailed control-oriented model for the
SOFC system and expressed the mass transfer and chemical kinetics phenomena of
the reformer and anode control volumes in state-space form. We derived closed-form
expressions that characterizes the steady-state and transient behaviors of utilization
(U) and steam-to—carbon balance (STCB). Our analysis was facilitated by key coor-
dinate transformations that led to elimination of non-linear reaction rate terms from
the coupled dynamic equations of the reformer and anode volumes. For predicting
the transient response we treated the molar flow rates Ni" and N0, and the mo-
lar contents NT and Na, as constants. This step was effective as it simplified the
nonlinear state-space system to a Linear-Time—Invariant form with minimal loss of
accuracy. The estimates of time constants and the steady-state values of U and STCB
matched very closely with those generated by the plant model. The results were ap-
plied to address a steady-state fuel optimization problem for the SOFC system using
the linear-programming approach. The constrained optimization problem yielded a

minimum fuel operating point. The analytical results were confirmed through simu-

43

lations.

APPENDIX A

45

APPENDIX A

Formulaes to Compute Reaction Rates

We use the following reaction rate expressions, given in [20], to model the kinetics of

steam reforming reactions in Eqs.(3.17) and (3.38). The equations below are written

for a generic gas control volume. For the SOFC system, these equations apply for

the reforming reactions in the reformate control volume as well as the anode control

 

volume.
3
Altai K1 pH2pCO __2
T1 = 2.5 pCH4pH20 — T d
pH2 I
I”. “6311 szpCOQ 2
7‘11 = 41— (PCOPH20 - 7—— 5
19112 11
4
m, = MU; p0,, ,2 _ W /62
PII; 4 H2O (C111
where
5 = 1 + ’Ccopco + 191210112 + ’CCH4PCH4 + ’CH'zopngopHZ,

pj : XjP 7 j=1,2,‘°- :5
In Eqs.(l), (2) and (3), the rate coefficients 161, it”, and K111 are given by

Eb 1 1
-——— —— , b=I,II,III,
Ru (Ty Tref,b)]

and the adsorption constants ICCO, KH2, K0114, [CH20 are given as follows

_AHq 1_ 1
Ru Tg Tref,q

where the values of Eb, Tm“), Kb’T-ref’ with b = 1,11, III, and AHq, Tref,qa IC

 

 

Kb = Harm. exp

, q = C0, H2,CH4, H20

 

 

IQ, = (C477,... f exp

 

 

(6)

(LTref’

with q = C0, H2, CH4, H20, are given in [20]. The equilibrium constants K1, K1 I

and ICU] in Eqs.(l),(2)and (3) are given by

46

 

0
IC ACT!) _
b=exp —RT , b—I,II,III (7)
u

In Eq.(7), A091 rcpresents the net Gibb’s free energy ([18]) of a reaction given by

 

pTOd TEUC
(8)
where S? is computed using the following equation:
' T
c ,T2 d .T3 e T4
298

 

47

APPENDIX B

48

APPENDIX B

Data

Gas properties:

Refer to [18], [9]

Reformer specific constants for heat transfer calculations:

Refer to [7], [16]

Parameters for computing the rates of reforming reactions:

Refer to [20]

Tubular stack parameters:

Refer to [7], [16]

Combustor parameters:

49

ch

N ”0.16
-DH.cb
Aconvl
Nu 0ng
D114)?
Aconv2
4136p
Cs

Vcb

v...

0.5 (m)

8

0.12 (m)
0.314 (m2)

8

0.079 (m)
0.314 (In?)
0.376 Kg
765(J/kg K )
5.62e-3(m3)
9.953e—3(m3)

50

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51

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lll‘lllllllllllllll