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RADIATION HEAT TRANSFER IN MULTI-TUBE,
ALKALI-METAL THERMAL-TO-ELECTRIC CONVERTER#
Jean-Michel P. Tournier and Mohamed S. El-Genk*
Institute for Space and Nuclear Power Studies
Department of Chemical and Nuclear Engineering
University of New Mexico, Albuquerque, NM 87131, USA
(505) 277 - 5442 / FAX: - 2814, E-Mail: mgenk@unm.edu
Nomenclature
Abk Radiative back face of solid element k
Afk Radiative front face of solid element k
Fkp Radiation view factor between surfaces Ak and Ap
I Total cell electrical current (A)
N Number of axial sections along BASE tubes
Nc Number of axial sections along standoff
NR Total number of radiant surface areas
NS Total number of solid elements (nodes)
Nsh Number of axial sections along shield
Nw Number of axial sections along artery
Pe Cell electrical power output (We)
q Heat flux (W / m2)
Q Heat flow (W)
RL External load resistance (Ω)
Rw Inner radius of cell (m)
T Temperature (K)
T ' Temperature correction vector (K)
T* Best estimate of temperatures (K)
Greek
β
k Flag,
β
k = 1 if element k has a radiative back face,
β
k = 0 otherwise
# paper accepted for publication as a Technical Note in the Journal of Heat Transfer (1999).
* author to whom all further correspondence should be adressed.
2
δ
kp Kronecker delta,
δ
kp = 1 if k=p,
δ
kp = 0 otherwise
ε
k Total emissivity of surface Ak
σ
Stefan-Boltzmann constant,
σ
= 5.67 x 10-8 W / m2.K4
ϕ
k Flag,
ϕ
k = 1 if element k has a radiative front face,
ϕ
k = 0 otherwise
Subscript / Superscript
air Air calorimeter
cond Condenser
hot Hot end
in Incident / input
loss Net radiant loss
out Outgoing
rods Alumina rods supporting the cell’s cold plate
Introduction
Vapor anode, multi-tube, Alkali-Metal Thermal-to-Electric Converters (AMTECs) are being
considered for a number of space missions, such as the NASA Pluto/Express (PX) and Europa
missions, scheduled for the years 2004 and 2005, respectively. These static converters can
achieve a high fraction of Carnot efficiency at relatively low operating temperatures. An
optimized cell can potentially provide a conversion efficiency between 20 and 30%, when
operated at a hot-side temperature of 1000 - 1200 K and a cold-side temperature of 550 - 650 K.
A comprehensive modeling and testing program of vapor anode, multi-tube AMTEC cells has
been underway for more than 3 years at the Air Force Research Laboratory’s Power and Thermal
Group (AFRL/VSDVP), jointly with the University of New Mexico’s Institute for Space and
Nuclear Power Studies. The objective of this program is to demonstrate the readiness of
AMTECs for flight on future U.S. Air Force space missions.
A fast, integrated AMTEC Performance and Evaluation Analysis Model (APEAM) has been
developed to support ongoing vacuum tests at AFRL and perform analyses and investigate
potential design changes to improve the PX-cell performance. This model consists of three major
components (Tournier and El-Genk 1998a and 1998b): (a) a sodium vapor pressure loss model,
which describes continuum, transition and free-molecule flow regimes in the low pressure cavity
3
of the cell; (b) an electrochemical and electrical circuit model; and (c) a radiation/conduction heat
transfer model, for calculating parasitic heat losses. This Technical Note describes the
methodology used to calculate the radiation view factors within the enclosure of the PX-cells
(Fig.1), and the numerical procedure developed in this work to determine the radiation heat
transport and temperatures within the cell cavity.
Description of PX-Series, Vapor Anode, Multi-Tube AMTEC Cell
A vapor anode, multi-tube AMTEC (Fig. 1) is a regenerative concentration cell, in which
sodium vapor expansion through the Beta”-Alumina Solid Electrolyte (BASE) is directly
converted into electricity. The BASE, whose ionic conductivity is much larger than its electronic
conductivity, separates the hot (high sodium vapor pressure) region from the colder (low sodium
vapor pressure) region. The PX cells had either 5, 6 or 7 BASE tubes each. The BASE tubes,
connected electrically in series, are brazed to a stainless steel support plate. Their low-pressure
(cathode) and high-pressure (anode) sides are covered with TiN porous electrodes and
molybdenum mesh current collectors. Sodium vapor ions diffuse through the BASE, while the
electrons circulate through the external load, then return to recombine with sodium ions at the
BASE/cathode electrode interface (Fig. 1). The neutral sodium atoms then diffuse through the
cathode electrode and traverse the vapor space to the remote condenser. Liquid sodium is then
circulated back to the cell evaporator by the capillary action of the wick structure. The cell
conical evaporator provides a larger surface area for evaporation of liquid sodium.
The heat input, Qin, is transported by conduction and radiation from the cell’s hot plate to the
BASE tubes and to the evaporator structure. The circumferential radiation shield, placed in the
cavity above the BASE tubes, reduces parasitic heat losses to the side wall. The metal stud
improves heat conduction from the hot end to the support plate, which helps achieve higher
evaporator and BASE temperatures (Fig. 1). To prevent condensation of sodium vapor and
electrical shorting of the BASE tubes in the cell, the BASE temperature is kept slightly higher (20
– 50 K) than that of the evaporator.
Modeling radiation heat transfer in the enclosure of an AMTEC cell is a challenging task
because of its complex geometry. Only a few investigators have attempted to tackle this difficult
problem. Schock and Or (1997) and Hendricks et al. (1998) have developed comprehensive
thermal models for PX-type cells, based on the SINDA Thermal Analyzer software. Shock and
4
Or (1997) used the ITAS Radiation Interchange package to calculate all radiation view factors
within the cell cavity. Hendricks et al. (1998) used the RadCAD thermal radiation analysis
software, based on Monte-Carlo ray-tracing, to evaluate all radiation view factors in the cell.
They reported their calculations of PX-type cells using 170 temperature nodes and assuming an
adiabatic cell wall. Only 12 and 5 nodes were used along the cell wall and BASE tubes,
respectively. By contrast, the evaporator standoff, BASE tubes support plate, conduction stud
and thermal heat shields were finely discretized. Results of their model compared well with
experimental data. However, calculating the radiation view factors using numerical contour
integration or Monte Carlo calculation methods is very CPU intensive. Furthermore, large
thermal analyzer codes such as SINDA/FLUINT are not particularly suitable or easily amenable to
handling the strong couplings between the many physical processes occurring in an AMTEC cell,
resulting in a relatively slow convergence.
Model Description
The present radiation/conduction thermal model was developed to calculate internal parasitic
heat losses and temperatures within the cell enclosure (Fig. 1). All surfaces that exchange radiant
energy in the interior of the cell were assumed gray and diffuse. The hemispherical total
absorptivity and emissivity of the surfaces were assumed equal, and depended only on
temperature (Siegel and Howell 1981). Even though this condition is achieved only by a limited
number of real materials, it is often the most reasonable approach, for two reasons. First, the
radiative properties are usually not known to high accuracy, especially their dependence on
wavelength and direction. Second, in many cases of practical interest, the gray-diffuse approach
is accurate enough, even in enclosures involving specular surfaces, or surfaces with both diffuse
and specular components (Schornhorst and Viskanta 1968). Two additional assumptions were
made: (a) the reflected energy is diffuse, and (b) the reflected energy is uniform over each surface
element.
The present model can easily handle 100 – 200 temperature nodes in the cell. The surface
emissivities and all thermophysical properties of structural materials and sodium are temperature
dependent. At the temperatures of interest (< 1200 K), the low pressure (<1 atm), monoatomic
sodium vapor is essentially transparent to radiation. Because of the very small density and
thermal capacity of sodium vapor, heat transport by convection is negligible compared to that due
5
to radiation and conduction.
Radiation View Factors
In this work, a radiation view factors database was developed for the various surface elements
in the complex cavity of PX-type cells (Fig. 1). This database has taken advantage of the
extensive tabulations of view factor formulas available in the literature for simple geometries
(Siegel and Howell 1981, Howell 1982). Figure 2 shows the numerical mesh used to calculate the
heat transfer in a cell without a thermal shield nor a conduction metal stud. The mesh generator
discretized the BASE tubes bundle into N sections of identical height, and the coaxial cylindrical
cavity above the BASE tubes into Nw identical axial sections. The evaporator standoff was
discretized into Nc identical axial sections, while the wick of the evaporator was discretized into
Nev = (N-Nc) axial sections of equal height. Only one axial node was used in the hot plenum wall,
since the height of the plenum was very small, typically < 2 mm.
As shown in Fig. 2, there are 2 separate enclosures for radiation heat transfer within the cell,
the high and the low vapor pressure cavities. For the purpose of calculating the radiation view
factors, the cell cavity was divided into several, radiatively coupled, elementary or sub- cavities
(Fig. 2): (a) the hot plenum baffle; (b) the evaporator standoff; (c) the inside of BASE tubes; (d)
the BASE tubes/standoff bundle; (e) the coaxial cylindrical cavity above the BASE tubes; and (f)
the annular space between the circumferential shield and the cell wall. The surfaces
β
o
, C
o
, top,
and ceil are fictitious boundaries through which radiative coupling between sub-cavities occurs
(Fig. 2). The radiation view factors for each of the sub-cavities were calculated using an
approximate analytical approach. In the artery cylindrical sub-cavity, the Nβ BASE tube tops
were approximated by an equivalent thick-circular ring of identical surface area (Fig. 1b).
Similarly, in the hot plenum sub-cavity, the Nβ BASE tube openings were approximated by an
equivalent thick-circular ring. Using these approximations, all view factors in the sub-cavities (a),
(b), (c), (e) and (f) were determined analytically using closed-form algebraic relations and
elementary flux algebra (Siegel and Howell 1981, Howell 1982).
Because of the complex geometry of the BASE tubes/standoff bundle cavity, approximate
relations of the view factors were derived, that ensured that all reciprocity )( pkpkpk FAFA = and
6
enclosure relations )1(
1
=
∑
=
R
N
kpk
F were satisfied. The reciprocity and enclosure relations were
accounted for explicitly when computing the view factors in all the cell sub-cavities. The
complete set of linear view factor relationships was solved using Gauss elimination. A complete
description of the view factor relations developed in this work cannot be included here due to
length limitation, a few examples, however, are given below.
Hottel’s crossed-string analytical method (Siegel and Howell 1981) was used, in conjunction
with the approximate cavity model of Juul (1982), to estimate the view factors between the
evaporator standoff and the BASE tubes, and between the different BASE tubes, that were
partially obstructed by the evaporator standoff or by another BASE tube (Fig. 1b). Using the
crossed-string method, the view factor between infinite, vertical cylinders 1 and 2 was obtained as
(Fig. 3a):
( ) ( ) ( ) ( )
FRRR
XRR
XX R X R
12
1 1 2 222
1
211111 1
∞ − −
= +
+
− −
−
+ − + − − −
π
sin sin , (1)
where RR
RXa
R
= =
1
2 2
and .
When the cylinders have identical radii (R=1), Eq. (1) reduces to:
F F HH H
12 21
1 2
1 1 1
∞ ∞ −
= = + − −
π
sin , (2)
where HX a
R
= =
2 2 1
, which is identical to that given by Siegel and Howell (1981).
The view factor between two infinitely-long, vertical cylinders A1 and A1’ of radii R1, partially
obstructed by another vertical cylinder A2 of radius R2, was also calculated using the crossed-
string method. For small obstruction,
β
<
ϕ
(Fig. 3b), one obtained:
( ) ( ) ( )
( )
( )
(3) . 10,
2
and , ,
2
,
,tan ,
1
sin ,tan where
,1
2
1
1
1
sin
1
11
2
11
2
1
2
2
11
22
221
1
1
<<<−+==
−
===
+
=
++
+
=
+
=
−−++−+−−+
−−+=
−−−
−
′
→
H
R
h
R
Rd
H
R
a
X
R
R
R
RH
X
RHX
R
X
RH
RRHXXRXX
X
FAA
ϕβ
π
θωβ
ωθσ
βθσ
ππ
When h=R1, then H=1,
β
=0,
γ
=
ϕ
and
σ
=
θ
. In that case, the second bracket in Eq. (3) equals
7
zero and the view factor reduces to that between two unobstructed vertical cylinders of identical
radii, Eq. (2). For large obstruction, (
β
>
ϕ
), the view factor between A1 and A1’ has the
expression (Fig. 3c):
( )( ) ( ) ( )
{
}
( ) ( )
F R X H R R X
H R
X
R
X H R
H
A A
1 1
1
21 1
10
22 2
1 1
22
→ ′
− −
= − − + + − − −
=+=−
+ +
<
πσ α
σ α
, where
and tan sin . (4)
When d = R2-R1, then H = -1, and consideration of the rectangular triangle of sides X and R−1
shows that FA A
1 1
→′ reduces to zero; the view between cylinders A1 and A1’ is completely
obstructed by cylinder A2. Although it is not directly apparent, both Eqs. (3) and (4) give
identical results when
β
=
ϕ
, that is when the cylinder A2 is in contact with the cross-strings
bd d band ′ ′ between the 2 identical cylinders (
γ
=o, and
δ
=
ϕ
, see Figs. 3b and 3c). This is
because, for these view factors derived from the crossed-string method, the length of the strings
used in the two equations is identical when
β
=
ϕ
.
Juul (1982) has developed an approximate analytical expression for the view factor between
two opposed (vertical) cylinders of finite and equal length. In the arrangement where cylinder 1 is
completely surrounded by n identical (vertical) cylinders of radius R2, and whose axes lie on a
concentric circle having a radius equal to the spacing a between cylinders 1 and 2, the view factor
between cylinders 1 and 2 can be approximated by (Juul 1982):
(
)
1012311212 21 FFFFF ×=−≅ ∞
′
∞, (5)
where F
12
∞
is the view factor between infinitely-long cylinders 1 and 2, given by Eq. (1). F10 is
the view factor between cylinder 1 and the inner surface of an opposed and concentric cylinder
having a radius Ro given by:
.sin
2
11 21
2
2
2
2
×
−−
+=
−
a
R
R
a
R
Ro
π
(6)
Juul (1982) has compared the results of this approximate method with that obtained by numerical
integration and showed Eq. (5) to be very accurate; the difference was less than 1% when R1 ≤ R2
and 3< h/R2 <10. For all cases of practical interest, the error in the view factor calculated using
Eq. (5) was less than 3%. The approximate method of Juul was extended to the case where the
8
finite cylinders 1 and 2 are separated by a height c : ,
101212 FFF ×≅ ∞ where Ao is the inner surface
of a cylinder having a radius Ro, concentric with the cylinder 1, and at the level of cylinder 2 with
a height h2.
The present model has made extensive use of reciprocity, symmetry and fractional analysis to
reduce the number of approximate factor relationships to a strict minimum and insure satisfaction
of all reciprocity and enclosure relations. For example, the view factors between the bottom ring
and BASE tube sections are simply obtained by symmetry: F F
bot top
k N k
→ →
=
+ −
β β
1. The view
factors between the first evaporator standoff section C1 and the surfaces Wk, ceil and plen, were
calculated using fractional analysis, assuming that the fractions of rays emitted by C1 and reaching
these other surfaces are unchanged, with or without the BASE tubes present (Fig. 2). These
fractions are obtained using the calculated view factors G in the annular cavity obtained when the
BASE tubes were removed. The fraction of rays leaking through the open spacings between
BASE tubes (Fig. 1) was then calculated as:
∑
−−−=
=→→→→
N
kCbotCtopCleakC k
FFFF
11111 1
β
. (7)
If we define:
G G G G
C leak C ceil C plen C W
k
N
k1 1 1 1
1
→ → → →
=
= + + ∑ , (8)
then
FG
GF
FG
GF
FG
GF k N
C ceil
C ceil
C leak
C leak
C plen
C plen
C leak
C leak
C W
C W
C leak
C leak
k
k
1
1
1
1
1
1
1
1
1
1
1
11
→
→
→
→
→
→
→
→
→
→
→
→
= ×
= ×
= × =
,
, .
, and
to
(9)
Note that the use of fractional view factors insures that the enclosure relation is satisfied for C1.
The factors FC Wk1→ can then be used to deduce all factors FC W
p k
→. By symmetry:
F F
C W C W
p k p k
→ →
=
− +
11 . (10)
9
The calculated view factors of the cell’s sub-cavities were modified to account for the
radiative coupling between them, through the fictitious boundaries
β
o
, C
o
, top, and ceil, which
were eliminated. As an illustration, let us consider the simpler case of the radiative coupling
between the hot baffle and evaporator standoff. Since the inside height of the hot baffle is much
smaller than both its diameter and the height of the first standoff axial section C
1
, all radiation
emitted by the surface
Whot
, streaming through surface C
o
, will end into surface C
1
(Fig. 2).
Therefore:
F F F
F
Whot C Whot C Whot C
Whot evap
k
→ → →
→
=
=
>
=
1 0
0
0
, for k 1, and
. (11)
FC Whot
1→ is calculated by reciprocity, and since surface C
1
sees only
Whot
and
Hot
through
C
o
, we have: F F F
C Hot C C C Whot
1 1 0 1
→ → →
=
−
. Finally, all other surfaces inside the evaporator
casing see only the hot plate through C
o
, so that:
.
and1,kfor
0
0
CevapHotevap
CCHotC
FF
FF
kk
→→
→→
=
>
=
(12)
All other factors between the hot baffle and evaporator standoff casing can be obtained by
reciprocity, or are nul.
A similar treatment is applied to the radiative coupling between the inside surfaces of the
BASE tubes and surfaces
Whot
and
Hot
of the high-pressure baffle. The only difference from
the previous treatment is that the surface areas inside a BASE tube must be multiplied by the
number of tubes in the cell, Nβ, before performing the radiative coupling treatment. In the end, all
view factors involving surfaces C
o
and
β
o
were set equal to zero.
The coupling between the tubes bundle’s sub-cavity and the artery cylindrical sub-cavity
above was the most challenging. Because of their complex geometry, in addition to the
reciprocity and enclosure relations, it was necessary to develop other approximate relations to
calculate all the radiation view factors in the cell. Again, the present model made extensive use of
reciprocity, symmetry and fractional analysis to reduce the number of unknown factors to a strict
minimum and insure satisfaction of all reciprocity and enclosure relations. However, such details
10
are beyond the scope of this Technical Note.
The heat shield (Fig. 1a) was modeled as a thin cylindrical metallic shell of radius Rsh, placed
close to the cell wall,
(
)
R
R
R
w
sh
w
−
<
<
. It was divided into Nsh identical axial sections. The
view factors for the annulus between the cell wall and the radiation shield were easily calculated
using closed-form algebraic relations. The top and bottom rings of this annular cavity were
assumed to be perfect reflectors. The top ring saw a portion of the Creare condenser surface,
which was highly reflective (
ε
~ 0.05, Tournier and El-Genk 1998a).
Numerical Solution Methodology
The energy balance and associated boundary conditions were discretized on a staggered grid
using the well-known control-volume approach proposed by Patankar (1980). The solution
obtained using this approach satisfies global conservation, even on a non-uniform grid. The multi-
tube AMTEC cell was discretized into NS ( N N N N
S w c
=
+
−
+
4 2 8 ) solid elements (Fig. 2).
There were N+Nw+1 wall sections, N+1 BASE tube elements, N evaporator tube sections, Nw+N-
Nc artery and evaporator wick sections, one each for the hot end plate, condenser, standoff
annular edge, and the evaporator surface, and 2 ring elements for the support plate.
Most solid elements k exchanged radiant energy through their front area Afk, only (fk=1 to NS).
Some elements k exchanged radiant energy in both the high-pressure, hot plenum cavity, and the
low-pressure enclosure above the BASE tubes, through both their front face, Afk, and back face,
Abk (bk= NS+1 to NR). These elements were the BASE tubes support plate nodes, bot and plen,
the BASE tubes nodes,
β
1 to
β
N+1, and the evaporator standoff nodes, C1 to CNc (Fig. 2). There
were a total of NR ( 32 ++= cSR NNN ) surface elements, which exchanged radiant energy
within the cell cavities. NR is equal to the number of solid elements, NS, minus the number of
elements that did not communicate with the vapor space, and therefore, did not exchange heat by
radiation (N-Nc evaporator wick nodes), plus the number of solid elements which also have a back
face in contact with sodium vapor (N+1 BASE tube elements, Nc evaporator standoff sections,
and 2 ring elements for the support plate). The energy balance for an element k was written as:
Λkp p
p
N
k k fk
loss
k bk
loss
T S Q Q
S
=
∑− = − −
1
ϕ β
. (13)
The terms on the left hand side of Eq. (13), arose from the well-known control-volume
11
discretization method (Patankar 1980), and include heat storage enthalpies, heat conduction
between node k and adjacent elements, convection due to liquid sodium flow in the wicks, and
source/sink terms due to condensation and evaporation of sodium at the condenser and
evaporator nodes, respectively. These terms also include the fraction of heat converted to
electrical energy in the BASE tube nodes, and heat input and losses at the outer boundaries of the
cell (hot end, side wall, and condenser).
The electric current density along the BASE TiN electrodes and the total electrical current,
calculated by the cell electrical model (Tournier and El-Genk 1998a and 1998b), were used in the
input to the present radiation/conduction model. All radiant heat exchange terms were collected
on the right hand side of Eq. (13). After the temperatures of all nodes in the cell were obtained,
Eq. (13) was used to evaluate the heat input at the hot end and the heat removed at the cold end
of the cell (condenser). In Eq. (13), the matrix coefficients, Λkp, the source terms, Sk, and the net
radiant energy loss terms, Qloss, were all functions of temperature, through the temperature-
dependent properties.
In the AMTEC cell, the mass flow rate of sodium is proportional to the cell electrical current.
However, since the heat of vaporization/condensation constitutes a large fraction of the total heat
input to the cell, it significantly affects the evaporator and condenser temperatures. Therefore, the
partial derivatives of the cell current in terms of the evaporator, BASE tubes and condenser
temperatures were obtained from the cell electrical model (Tournier and El-Genk 1998b). These
derivatives were used to linearize the dependence of the sodium flow rate on temperature in the
energy balance equations of the condenser and evaporator nodes. Results from numerical
experiments showed that this linearization was a key to obtaining an efficient, fast-converging
solution algorithm (Tournier and El-Genk 1998a).
To solve the energy balance, Eq. (13), for the solid node temperatures, the net radiant energy
loss terms, Qloss, were expressed as functions of temperatures. First, the net radiant energy losses
were expressed in terms of the rates of outgoing radiant energy (Siegel and Howell 1981):
q q F q k N
k
loss
k
out
kp p
out
p
N
R
R
= −
=
∑
1
, for = 1 to , (14)
or in matrix notation:
[
]
{
}
[
]
q q
loss out
=ℑ− F .
12
Then, the radiant energy balance for the surface Ak was expressed as (Siegel and Howell 1981):
q F q T
k
out
k kp p
out
p
N
k k
R
− − =
=
∑
( )1
1
4
ε ε σ
. (15)
Equation (15) represents an NRxNR linear system of equations, which relates the rates of outgoing
radiant energy in the cell enclosure to the surface temperatures, as:
{
}
[
]
[
]
Rq T
R F k p N
out
k k
kp kp k kp R
=
= − −
ε σ
δ ε
4
1
, where
, , = 1 to ( ) .
(16)
A solution routine was developed and tested to inverse the radiation matrix {R} using the Gauss-
Jordan elimination algorithm. Once the radiation matrix was inverted, the net radiant energy loss
terms were expressed explicitly in terms of the node temperatures, as:
[
]
{
}
[
]
{
}
{
}
[
]
[ ]
{ }
[ ]
q q T
Q T
M A R F R k p N
loss out
k k
loss
k
kp k p kp km mp
m
N
R
R
= =
= −
ℑ− ℑ− −
− −
=
∑
F F R
M
14
4
1 1
1
ε σ
σ
ε
, or
= , where
, , = 1 to .
(17)
The energy balance for the solid element k in the cell, could then be written as:
( )
Λkp p
p
N
k k fk m k bk m m
m
N
T S M M T
SR
= =
∑ ∑
− + + =
1
4
1
0
σ ϕ β
, , , k=1 to NS . (18)
The temperatures of the back faces of the solid nodes, exchanging radiant energy through
both their front and back faces, were taken equal to that of the front faces of the elements:
T T bk N N
bk fk S R
=
, = + 1 to . (19)
The matrices {
Λ
ΛΛ
Λ
} and {M}, and source term [S] in Eq. (18) were evaluated explicitly using
the best estimate of the temperatures [T*] in the cell (they depend on the values of the
temperature-dependent, thermophysical and radiative properties). One seeks a temperature
correction vector [T '], such that the vector [T ] = [T*] + [T '] satisfies the energy balance Eqs.
(18) – (19). To resolve the strong nonlinearity of radiant energy fluxes, the fourth power of the
temperature was linearized using the first two terms of the Taylor series, as:
T T T T T T T T T
m m m m m m m m m
4 4 4 3 3
4( 4= + ′≅ + ′= + ′
( ) ( ) ) ( ) [ ]
* * * * * . (20)
The linearized energy balance Eq. (18) had the final form:
13
( )
Λ
Λ
kp p
p
N
k fk m k bk m m m
m
N
k
k k kp p
p
N
k fk
loss
k bk
loss
S
T M M T T S
S S T Q Q k N
SR
S
*
,
*
,
* *
* * * * *
( ) $
$( ) ( ) , .
′+ + ′=
= − − −
= =
=
∑ ∑
∑
1
3
1
1
4
σ ϕ β
ϕ β
,
where
= 1 to
(21)
The linearized energy balance equations were solved for the temperature corrections [T '], using
Gauss elimination method with row normalization and partial pivoting. After the temperatures in
the cell cavity were corrected, an energy balance for the cell was performed again. The solution
was iterated until temperatures satisfied the convergence criterion (when the magnitude of
temperature correction vector < 0.2 K) and the cell energy balance was satisfied (within < 0.1%).
Only 2 to 4 iterations were needed to resolve the strong non-linearity of the radiant energy fluxes.
Calculated Performance of PX-5A Cell
The present radiation/conduction model was coupled to a vapor pressure loss and an AMTEC
cell electrochemical and electrical circuit models, using an efficient iterative solution procedure to
resolve the strong couplings between various physical processes in the cell (Tournier and El-Genk
1998a). The fully integrated cell model (APEAM) was used to simulate the experimental setup of
PX-series AMTEC cells at AFRL. The model results were compared with experimental
measurements of a number of PX-series cells (El-Genk and Tournier 1998). The PX-cell tests
were performed in vacuum at 5x10-6 torr (0.7 mPa). The radiation/conduction thermal model in
APEAM was modified to account for axial and radial conduction in the Min-K insulation
surrounding the cell in the vacuum tests. The test data were obtained at fixed hot and cold end
temperatures.
As an illustration, the predicted heat transfer and temperatures in the PX-5A cell, near its
peak conversion efficiency, are presented. This calculation was performed with Nc = 4, N = 10,
Nw = 12, and Nsh = 11. The resulting mesh had 79 temperature nodes and 100 surfaces
exchanging radiant energy within the cell cavity. An additional 48 temperature nodes were used
in the surrounding insulation in the test. The calculation took less than 10 s on a Pentium 200-
MHz PC; about a fourth of this time was used to calculate the view factors in the cell.
The PX-5A cell was 38 mm in diameter, and made entirely of stainless steel. It had six 40
mm-long BASE tubes, a conical evaporator, a circumferential molybdenum shield above the
14
BASE tubes, a conduction stud, 100 mm2 in cross-section area, and four 2.5 mm-thick stainless
steel rings around the evaporator standoff to enhance conduction. The PX-5A cell used a Creare
condenser (Tournier and El-Genk 1998a), whose effective emissivity was taken equal to 0.05.
Figure 4 shows the predicted heat transport and thermal energy exchange in the PX-5A cell, at
hot and cold end temperatures of 1123 K and 623 K, respectively. The conduction stud
transported 17.3 W (about 54%), and the plenum wall conducted 13.1 W (about 41%) of the heat
input to the cell, Qin = 32.2 W, to the support plate. Only 1.8 W were radiated out from the hot
end (5%). The temperature of the support plate was only 14 K lower than the cell’s hot end
temperature. About 60% of the heat conducted by the stud was consumed in the evaporation of
liquid sodium in the evaporator wick (10.3 W).
A total of 21 W = Qair + Qrods was removed at the cell condenser, which comprised: (a) the
sodium latent heat of condensation (11.1 W); (b) heat conduction up the side wall of the cell (5.4
W); (c) parasitic radiation losses to the condenser surface (1.7 W); and (d) heat conduction up
the liquid sodium return artery (2.8 W). In the low sodium pressure cavity, the hot support plate
lost 3.7 W by radiation, and the 6 BASE tubes lost 7.3 W by radiation to the cooler surfaces in
the cavity. The far end of the BASE tubes was 126 K cooler than the support plate.
The temperature of the stainless steel (SS) radiation shield varied between 863 K and 924 K,
while the cell wall temperature near the condenser was 623 K. As a result, the radiation shield
significantly reduced the parasitic heat losses through the cell wall. The PX-5A cell delivered 3.8
We at a peak conversion efficiency of 11.8%, and an average electrodes power density of 0.104
We / cm2. The predicted heat losses through the cell wall were 7.1 W, or 22% of the heat input to
the cell. As shown in Fig. 4, the predicted heat removal at the condenser end, BASE tubes cold
end temperature and cell electrical power output were in good agreement with experimental data.
Such good agreement gave confidence in the soundness of the modeling approach and the
accuracy of the radiation/conduction model.
Acknowledgments
This research was funded by the Space Vehicle Technologies Branch’s Power and Thermal
Group of the U.S. Air Force Research Laboratory, Kirtland AFB, Albuquerque, NM, under
contract F29601-96-K-0123, to the University of New Mexico’s Institute for Space and Nuclear
Power Studies (UNM-ISNPS).
15
References
El-Genk, M. S., and Tournier, J.-M., 1998, “Recent Advances in Vapor-Anode, Multi-Tube,
Alkali-Metal Thermal-to-Electric Conversion Cells for Space Power,” in: Proc. of the 5th
European Space Power Conference, SP-416, Paper No. 1046, European Space Agency.
Hendricks, T. J., Borkowski, C. A., and Huang, C., 1998, “Development and Experimental
Validation of a SINDA / FLUINT Thermal / Fluid / Electrical Model of a Multi-Tube AMTEC
Cell,” in: Proc. Space Technology and Applications Int. Forum, CONF-980103, M. S. El-Genk,
ed., American Institute of Physics, New York, NY, AIP CP No. 420, Vol. 3, pp. 1491-1501.
Howell, J. R, 1982, A Catalog of Radiation Configuration Factors, McGraw-Hill Book
Company, Inc., New York and London.
Juul, N. H., 1982, “View Factors in Radiation Between Two Parallel Oriented Cylinders of
Finite Lengths,” Journal of Heat Transfer, Vol. 104, pp. 384-388.
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Co.,
Washington, D.C..
Schock, A., and Or, C., 1997, “Coupled Thermal, Electrical, and Fluid Flow Analyses of
AMTEC Multi-tube Cell with Adiabatic Cell Wall,” in: Proc. Space Technology and Applications
International Forum, CONF-970115, M. S. El-Genk, ed., American Institute of Physics, New
York, NY, AIP CP No. 387, Vol. 3, pp. 1381-1394.
Schornhorst, J. R., and Viskanta, R., 1968, “An Experimental Examination of the Validity of
the Commonly Used Methods of Radiant Heat Transfer Analysis,” Journal of Heat Transfer, Vol.
90, No. 4, pp. 429-436.
Siegel, R., and Howell, J. R., 1981, Thermal Radiation Heat Transfer, Second Edition,
Hemisphere Publishing Corporation, New York and London, Section 9.3, pp. 283-292.
Tournier, J.-M., and El-Genk, M. S., 1998a, “AMTEC Performance and Evaluation Analysis
Model (APEAM): Comparison with Test Results of PX-4C, PX-5A, and PX-3A,” in: Proc.
Space Technology and Applications Int. Forum, CONF-980103, M. S. El-Genk, ed., American
Institute of Physics, New York, NY, AIP CP No. 420, Vol. 3, pp. 1576-1585.
Tournier, J.-M., and El-Genk, M. S., 1998b, “An Electrical Model of Vapor Anode, Multi-tube
AMTEC Cells,” in: Proc. 33rd Intersociety Energy Conversion Engineering Conference, Paper
No. 98056, American Nuclear Society, Chicago, IL.
16
Condenser
(cold end)
BASE tube
BASE tubes
support plate
Cell wall
Liquid-return
artery (wick)
Cell hot end
Low-Pressure
Sodium Vapor
Evaporator
Standoff
Hot Plenum
(High-Pressure
Sodium Vapor)
Q
in
Q
loss
Q
air
conduction
stud
thermal
shield
R
L
P
e
= R
L
I
2
I
Metal Rings
α-Alumina
Braze Section
Porous
Electrodes
(a) Elevation
Figure 1a
17
side wall
anode
current
collector
BASE tubes
tubes
connector
lead
anode lead external
load (RL)
cathode
lead
cathode
current
collector
Liquid-return
wick
(b) Plan View
Figure 1. A Schematic of Vapor Anode Multi-tube AMTEC Cell (Not to Scale).
18
Condenser
(cold end)
BASE tubes
(with Electrodes)
BASE tubes
support plate
Cell wall
Liquid-return
artery (wick)
Cell hot end
Low-Pressure
Sodium Vapor
N-N
c
cells
N
w
cells
edge
Whot Hot
plen
β
o
C
o
bot
C
1
C
2
evap
β
1
β
2
β
3
β
N
β
N+1
plen
bot
W
1
W
2
W
3
W
N
C
1
C
2
C
3
C
N
top ceil
β
1
β
2
β
3
β
N
β
N+1
ω
1
ω
2
ω
Nw
cond
a
1
a
2
a
Nw
N
c
cells
Evaporator
Standoff
Hot Plenum
(High-Pressure
Sodium Vapor)
Q
in
Q
loss
Q
air
(with N
c
= 2,
N = 5, N
w
= 4)
Figure 2. Numerical Grid Layout of Multi-tube AMTEC Cell, without Thermal Shield nor
Conduction Stud.
19
a
A1
O
ab
R1
a'
'
c
e'
d'
R2
O2
d
e
O1
α
α
α
ϕ
b'
A2
ϕ
ϕ
Figure 3a.
a
a a '
A1
'
b'
σ
γγ
e
d
e'
d'
b
f'f
g
β
ω
h
d
ϕ
ϕc
R1
A2
R2
O2
O1
R2
R1
O1
A1
'
hθ
Figure 3b.
20
a
a'
b'
'
e
d
c
g
c'
d'
R1
e'
a
α
σ
δ
δ
β
d
b
α
α
θ
O1O1
R1
R2
O2
Figure 3c.
Figure 3. Calculation of View Factor Between Infinitely-Long Parallel Cylinders Partially
Obstructed by Another Parallel Cylinder, Using Hottel’s Crossed-String Method; (a)
No Obstruction; (b) Small, Partial Obstruction ( , )
β
ϕ
≤
≤
H1 ; (c) Large, Partial
Obstruction ( , , )
β
ϕ
≥
≥
−
>
+
H X R1 1 .
21
Qin
Qloss
Qair
Qrods
Qplen Qstud
Qartery
Qwall
RL = 1.2 Ω
Pe= 3.80 We
(3.84)
I= 1.78 A
(1.79)
(623 K)
20.4 W (21.1)
0.59 W
7.1 W
(1010)
983 K
(1123 K)
13.1 W
32.2 W
17.3 W
5.4 W
668 K
775
830
865
893
921
944
973
997
1023
1061
1120 K
641
707
766
820
871
921
976
983
991
1025
1074
984
993
1007
1023
1055
863
872
887
902
915
924
2.8 W
6.8 W
7.3 W
1.9 W
1.7 W
11.1 W
0.51 W
10.3 W
1.8 W
0.4 W
2.1 W
0.1 W
1109 K
1114 K
1.1
W
3.7
W
(experimental)
conduction
radiation
phase change
temperature
node
Figure 4. Predicted Heat Transfer and Temperatures in PX-5A AMTEC Cell, at the Peak
Conversion Efficiency.
22
LIST OF FIGURES
Figure 1. A Schematic of Vapor Anode, Multi-tube AMTEC Cell (Not to Scale). (a)
Elevation; (b) Plan View.
Figure 2. Numerical Grid Layout of Multi-tube AMTEC Cell, without Thermal Shield or
Conduction Stud.
Figure 3. Calculation of View Factors Between Infinitely-Long Parallel Cylinders Partially
Obstructed by Another Parallel Cylinder, Using Hottel’s Crossed-String Method; (a)
No Obstruction; (b) Small, Partial Obstruction ( , )
β
ϕ
≤
≤
H1 ; (c) Large, Partial
Obstruction ( , , )
β
ϕ
≥
≥
−
>
+
H X R1 1 .
Figure 4. Predicted Heat Transfer and Temperatures in PX-5A AMTEC Cell, at the Peak
Conversion Efficiency.
