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

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

∂∂

∂∂

=∂∂

rr

rr

a

t

θθ 2

20)0,( θθ =r

0),0(=

∂∂

r

tθ),(

),(max

max trhr

trk 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

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) isra

nnneCtλτ −=)( .

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

ra

t ∂∂

=∂∂ ψψ

0)0,( θψ rr =

0),0( =tψ ),()(),(

max

max

max trr

kh

r

trk ψ

ψ−=

∂∂

−(3)

02

2

2

=ℜ+∂ℜ∂

λr

0)0( =ℜ 0)(1)(

max

max

max =ℜ

−+

∂∂ℜ

rrk

h

r

r

02 =+ τλτ

adt

d

(4)

(5)

∑∞

=

−=1

sin2

),(n

rta

nnneatr

λλψ (8)

∑∞

=

=1

0 sinn

nn rar λθ

)cossin(

)(sin2

maxmax

maxmax0

rRr

rra

nnnn

nnn λλλλ

λλθ−

−=

r

re

rrr

rrr

TT

TtrT

n

nt

n nnn

nnn n

λλ

λλλλλλ αλ )sin(

))cos()sin(

)cos()sin((2

),( 2

1 maxmaxmax

maxmaxmax

0

−∞

=∞

∞ ∑ −

−=

−

−(9)

(6)

(7)

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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

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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

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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

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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

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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).

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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