1. 1
LO2/LCH4 Propulsion System:
Tank Stratification Model Using MATLAB®
Alexander RivasI
Houston, TX
Abstract
Following the Vision for Space Exploration, NASA is actively researching and designing a new
generation of vehicles and systems to extend man’s presence in space. Along with the new systems, come
new choices for propellants and their tanks. The Energy Systems Division in NASA’s Johnson Space
Center has begun trade studies between Liquid Oxygen and Liquid Methane as future propellants.
Over the mission duration, the cryogenic propellant tank is subject to heat leaks which may lead to
stratification. Stratification occurs when a substance has varied temperatures in various internal locations
which may cause phase changes in the substance. Past methods to limit stratification involved installing
mixers in tank to mix the propellant and create a more homogenous body. Adding mixers or any other
device adds complexity to the system and increases the possibility of failure. Although much safer mixer
designs exists, a vivid example of this failure can be remembered on the Apollo 13 mission during which
the activation of the LO2 tank mixer ignited a damage electrical coil and caused the historic tank explosion.
In order to determine requirements for the tank design, the degree to which stratification occurs
must be evaluated. A model using MATLAB®
was created to simulate the unsteady temperature of the
propellant tank in zero-g while on a mission of 6 months. The model was then used to simulate a LO2 and
LCH4 tank at various heat leaks. This paper explains the basis for the model, follows the MATLAB®
coding logic, and presents the results for both propellants.
Introduction
Nomenclature:
r = radius, m λ = characteristic value
rmax = max radius, m n = number of characteristic values
k = thermal conductivity, W/mK θ = temp. difference (spherical Laplacian)
ρ = density, kg/m3
ψ = temp. difference (cartesian Laplacian)
cp = specific heat capacity, J/kgK ℜ = Separation of Variables eq. using radius
Q = total heat leak, W τ = Separation of Variables eq. using time
T0 = initial propellant temperature, K C = constant
Tf = final propellant temperature, K α = thermal diffusivity
I
Undergraduate Student Research Program participant, NASA Johnson Space Center, Fall 2006

2. 2
The propellant tank is modeled as a sphere with an outer radius of 1 meter, made out of an
Aluminum-Lithium alloy, typically used in aerospace applications. The tank is pressure fed with gaseous
Helium entering and pushing the liquid propellant out (see Figure 1). This tank is modeled after the tanks
used on the Lunar Excursion Module (LEM) and the tanks on future use on the Lunar Service Access
Module (LSAM) for the ascent and descent stages. The LSAM will travel with the Crew Exploration
Vehicle (CEV) to land on the lunar surface. The mission length is 6
months, during which the tank will be subjective to the conditions of
space.
This model will calculate the propellant tank temperature, as a
function of time and radius, during the mission duration of 6 months.
Some simplifying and conservative assumptions are made to model the
tank as a solid sphere experiencing a heat flux, where the ambient
temperature is differs from the exterior temperature because of the
following assumptions: 1) the conductivity of AlLi and GHe is much
larger then that of LO2 2) heat leak is spread uniformly into the propellant
because the high conductivity of AlLi spreads the heat evenly 3) no
convective motion occurs in the tank during the mission’s entirety due to
accelerations.
Model Derivation
The solid sphere of radius r having a uniform
initial temperature T0 is exposed to a temperature T∞ with a
moderate heat transfer coefficient h set to model the tank
heat leak: h=(Q/A)/(T∞-T0) (see Figure 4). The tank heat
leak includes strut heat leak and radiation, spread uniformly.
The boundary conditions in terms of θ = T-T∞ are:
Figure 1:
Propellant Tank Schematic
Figure 4:
Problem Analogue






∂
∂
∂
∂
=
∂
∂
r
r
rr
a
t
θθ 2
2
0)0,( θθ =r
0
),0(
=
∂
∂
r
tθ
),(
),(
max
max
trh
r
tr
k θ
θ
=





∂
∂
−finitet =),0(θ or
(1)
CEV
LSAM
Figure 2:
LSAM and CEV
Figure 3:
LSAM Propellant Tanks
LO2/LCH4 Tanks
Ascent Stage
Descent Stage

3. 3
By using the well-known transformation:
Eq. (1) is reduced from spherical Laplacian to cartesian Laplacian, which is expressible in terms of circular
functions. By using Eq. (2), Eq. (1) can now be expressed in terms of ψ and the condition of finite center
temperature rather than that of temperature symmetry. The result is:
Hence the problem is reduced to a problem of Cartesian geometry.
The product solution ψ (r,t) = ℜ (r)τ(t) yields
in r (radius) and t (time). The solution of Eq. (4) is
),()( rAr nnn ρ=ℜ ),sin()( rr nn λρ = (characteristic functions)
and the zeros of
)sin()1()cos()( maxmax rBirR nnn λλλ −= , (characteristic values)
where Bi = h maxr /k,
and the solution of Eq. (5) is
ra
nn
n
eCt λ
τ −
=)( .
Thus the product solution becomes
The initial value of Eq. (8) is
The coefficient an is
Finally, the unsteady temperature of the sphere is found to be
rtrtr /),(),( ψθ =
(2)
2
2
r
a
t ∂
∂
=
∂
∂ ψψ 0)0,( θψ rr =
0),0( =tψ ),()(
),(
max
max
max
tr
r
k
h
r
tr
k ψ
ψ
−=
∂
∂
−
(3)
02
2
2
=ℜ+
∂
ℜ∂
λ
r
0)0( =ℜ 0)(
1)(
max
max
max
=ℜ





−+
∂
∂ℜ
r
rk
h
r
r
02
=+ τλ
τ
a
dt
d
(4)
(5)
∑
∞
=
−
=
1
sin2
),(
n
rta
n
nn
eatr λλ
ψ (8)
∑
∞
=
=
1
0 sin
n
nn rar λθ
)cossin(
)(sin2
maxmax
maxmax0
rRr
rr
a
nnnn
nn
n
λλλλ
λλθ
−
−
=
r
r
e
rrr
rrr
TT
TtrT
n
nt
n nnn
nnn n
λ
λ
λλλ
λλλ αλ )sin(
)
)cos()sin(
)cos()sin(
(2
),( 2
1 maxmaxmax
maxmaxmax
0
−
∞
=∞
∞
∑ −
−
=
−
−
(9)
(6)
(7)

4. 4
MATLAB®
Coding
The MATLAB®
software was chosen for analysis because of its inherent nature of matrix
manipulation. The desired result was a matrix showing the Temperature values at various levels of radius
and time. Also, MATLAB®
is able to handle recursive and iterative functions, which were used to
calculate, test, and use the infinite amount of eigenvalues. MATLAB®
is also capable of working many
inputs into a function and imbedding multiple functions into a single, primary function. Lastly, 3-
dimensional and 2-dimensional plots can be created to correctly view and compare computed results.
In this tank stratification model, three MATLAB®
functions are used: Temperature.m,
estlambda.m recurnewL.mII
. The user inputs the following specifications: tank radius, propellant
conductivity, propellant density, propellant specific heat capacity, total heat leak, initial tank temperature,
ambient temperature, and number of desired eigenvalues. The Temperature.m function solves Eq. (9) by
using the eigenvalues calculated in estlambda.m. The estlambda.m function iterates to generate a list of
eigenvalues and uses the recurnewL.m function to recursively check that each newly calculated eigenvalue
is unique.
The estlambda.m function begins by solving for zeros of Eq. (7). These points are represented by
the intersection points in Figure 5. The oscillating sine and cosine functions infinitely intersect creating an
endless amount of eigenvalues. The estlambda.m function calculates the first eigenvalue and validates it
uniqueness using recurnewL.m. If the calculated eigenvalue has already been calculated, estlambda.m will
test the next whole integer. If the calculated eigenvalue is unique, that value is store into a vector created in
estlambda.m. The iteration in estlambda.m continues until the vector of eigenvalues has reached a length
‘n’ specified by the user.
When the estlambda.m and recurnewL.m functions have created the vector of eigenvalues with a
length ‘n,’ the vector is then called by the Temperature.m function. The heart of the Temperature.m
function is a double-nested loop solving Eq. (9). Because Eq. (9) is a summation involving a list of
II
See appendix for complete MATLAB®
function code.
λ1
λ2
x:[0,10] x:[0,100] x:[0,1000]
Figure 5:
Graphical Representation of Eigenvalues

5. 5
eigenvalues, radii and time levels, the double-nested loop was created to iterate through those values. The
loop begins by summating the equation using the first time level, the first radius level, and the first
eigenvalue. The inner loop then loops through the vector of eigenvalues while continuously using the first
time and radius level as inputs. Once looped through the eigenvalues, the Temperature for the first time
and radius level has been calculated. Then the function increments the radius and then loops through the
eigenvalues once again. This continues until the last radius level is inputted with the same initial time
level. After this summation, the time increments and the radius levels are again looped, which causes the
eigenvalues to also loop. This continues until the entire radius and time levels are inputted. The final result
is a matrix showing the Temperature at various times (as columns) and locations in the tank (as rows).
NOTE: r = 0 was omitted to eliminate the error of Eq. (9) in dividing by zero.
During each loop, counters are used to specify the location of each Temperature calculated to
place them in the correct radius (row) and correct time (column). The size of the matrix is specified by the
number of time and radius increments. NOTE: In order for MATLAB®
to perform vector multiplication,
the radius and time vectors must be the same length. Therefore, they must have the same number of
increments.
Results
The four tested scenarios were: 1) LO2 with Q = 4W 2) LO2 with Q = 16W 3) LCH4 with Q =
4W 4) LCH4 with Q = 16W. The specific tested properties of each propellant are shown in Table I. The
values chosen are based for cryogenic propellant storage for use on a 6-month mission in space. The
results can best be viewed and compared through various plots.
Overall, the stratification was low for all 4 tested scenarios (see Figure 6). The low stratification
yields a low possibility for the formation of slush. The highest stratification of ~25K occurred in LO2 with
Q=16, but this heat leak is purposely extremely large. Heat leaks of 4W have been already been
accomplished on previous propellant tanks. 4 watts is the nominal heat leak with a 2x factor on
performance; 16 watts is used to see the sensitivity. The stratification for LO2 was larger than LCH4,
which was also expected due to the higher conductivity and higher specific heat of LCH4 (see Table II).
LO2 LCH4
k, conductivity 0.02674 0.16 W/mK
ρ, density 1148 410 kg/m3
c specific heat capacity 1167 3500 J/kgK
Tsat, saturation temperature* 135.15 168.94 K
*at 325 psia
Table I:
Propellant Properties

6. 6
LO2 LCH4
stratification range:
Q=4 1.0 - 6.0K 0.7 - 1.0K
Q=16 6.0 - 25.0K 3.0 - 2.5K
∆T between center and rim
Q=4 5.7525K 0.8497K
Q=16 20.5339K 2.0228K
Figure 6:
Propellant Stratification
(plotted at .10 m radius increments with monthly-time increments)
Table II:
Stratification Results
LO2 with Q=4W
∆T=5.7525K
LCH4 with Q=4W
∆T=0.8497K
LO2 with Q=16W
∆T=20.5339K
LCH4 with Q=16W
∆T=2.0228K

7. 7
Another important measure is the saturation temperature, or boiling point, of the cryogenic
propellants. If the propellant temperature nears or reaches the saturation temperature, then the heated
propellant will burn off, reducing the amount available. This could seriously damper the mission. The
model shows that the temperature did not reach the stratification temperature during the 6 month period for
all tested conditions (the closest to the Tsat was LO2 with Q = 16W where the temperature reaches
125K).Figure 7 shows the propellant temperature in relation to the saturation temperature. The saturation at
a pressure of 325psia was used because this is the mean tank pressure during the mission duration.
Figure 7:
Propellant Temperature (plotted at .10 m radius increments) vs. Boiling Point
LO2 with
Tsat = 135.15 K @ 325 psia
LCH4 with
Tsat = 168.94 K @ 325 psia
LO2 with Q=16W LCH4 with Q=16W

8. 8
Conclusions
Overall, the model analysis shows that during mission duration of 6 months, propellant tanks of
LO2 and LCH4 do not experience critical temperature stratification. Therefore, a mixer may not be needed,
which will further simplify the system, reduce power consumption, and eliminate another potential source
of error. This model can be extended to model various other propellants (such as ethanol) and conditions
(such as different heat leaks). The author of this paper recommends further tests to be conducted to
continue to validate the future use of a propellant tank without a mixing system.
Acknowledgements
The author would like to thank the National Aeronautics and Space Administration and the
Johnson Space Center, the location at which the research was conducted; and the Energy Systems Division
for providing support and resources. Thank you to the Virginia Space Grant Consortium and their
Undergraduate Research Program for funding the research. Lastly, the greatest support for this model came
from Eric Hurlbert from the Energy Systems Division, JSC. He mentored the author and provided
invaluable guidance and support.
Resources
Arpaci, Vedat S. Conduction Heat Transfer. Reading, Massachusetts: Addison-Wesley Company, 1966.
287-288.
E.W. Lemmon, M.O. McLinden and D.G. Friend, "Thermophysical Properties of Fluid Systems“ in NIST
Chemistry WebBook, NIST Standard Reference Database Number 69, Eds. P.J. Linstrom and W.G.
Mallard, June 2005, National Institute of Standards and Technology, Gaithersburg MD, 20899
(http://webbook.nist.gov).

9. 9
Appendix
MATLAB®
code for Temperature.m (excluding
coding for plots):
Function [Tmatrix,deltT2]=…
Temperature(R,k,p,c,Q,To,Tf,n)
x=0:100;
a=k/(p*c);
SA=4*pi*(R^2); % m2
V=(4/3)*pi*(R^3) % m3
Mass=V*p % kg
W=Q/SA; % W/m2
h=W/(Tf-To) % W/m2*K
month=(30*24*60*60); % s
deltT=(Q*6*month)/(Mass*c) % K
Bi=h*R/k; % unitless
f = @(x)(x*R).*cos(x*R)-(1-Bi).*sin(x*R);
r=.01:1/180:1.01; % m
t=0:24*60*60:(6*30*24*60*60); % interval =
days
L=estlambda(R,Q,k,To,Tf,n);
indext=1;
indexr=1;
Tmatrix=[];
for ti=1:length(t);
for ri=1:length(r);
Tsum=0;
for i=1:n;
Tsum=Tsum+(((sin(L(i).*R)-L(i).*R.*…
cos(L(i).*R)).*exp(-a.*(L(i).^2).*t(ti))…
.*sin(L(i).*r(ri)))/(L(i).*r(ri).*(L(i)*R-sin(L(i)…
.*R)*cos(L(i).*R))));
i=i+1;
end
Tmatrix(indexr,indext)=Tf+(To-Tf)…
.*2*Tsum;
indexr=indexr+1;
ri=ri+1;
end
indexr=1;
indext=indext+1;
ti=ti+1;
end
deltT2=Tmatrix(end,end)-Tmatrix(1,end)
MATLAB®
code for estlambda.m:
function L = estlambda(R,Q,k,To,Tf,n)
x=0:100;
SA=4*pi*(R^2);
W=Q/SA;
h=W/(Tf-To);
Bi=h*R/k;
f = @(x)(x*R).*cos(x*R)-(1-Bi).*sin(x*R);
L=fzero(f,1);
for i=2:n
newL=fzero(f,ceil(L(end))+1);
if abs(newL-L(end))<0.001
newL=recurnewL(newL,L(end),R,Q,k,To,Tf);
L=[L newL];
else
L=[L newL];
end
i=i+1;
end
MATLAB®
coding for recurnewL.m:
function result = recurnewL(newL, Lend, R, Q,
k, To, Tf)
x=0:100;
SA=4*pi*(R^2);
W=Q/SA;
h=W/(Tf-To);
Bi=h*R/k;
f = @(x)(x*R).*cos(x*R)-(1-Bi).*sin(x*R);
newL2=fzero(f, ceil(Lend));
if abs(newL-newL2)<0.001
result=recurnewL(newL, ceil(Lend)+1, R, Q,
k, To, Tf);
else
result=newL2;
end