Texas Tech - Texas Space Grant Consortium

CRYOGENIC FLUID STORAGE FOR THE MISSION TO MARS

TEXAS TECH UNIVERSITY

DEPARTMENT OF MECHANICAL ENGINEERING

Kyle Chambliss

Sarah Kelly

Justin Kimble

Advisor: Dr. Darryl James

April 27, 1999

Written by Undergraduate Students Distributed ByMay 1999 Texas Space Grant ConsortiumTexas Tech University http:// www.tsgc.utexas.edu/tsgc

ABSTRACT

In the next thirty years, the National Aeronautics and Space Administration

(NASA) will conduct both manned and unmanned missions to Mars. For these missions,

cryogenic oxygen, hydrogen, and methane storage will be required. The purpose of this

project is to design tanks to store these cryogenic fluids for both the unmanned and

manned missions. The primary objectives of the tank design included minimizing heat

transfer and the weight of the tank system and finding an optimal balance between the

amount of insulation used and the capability of the cryocooler. The tank design included

inner and outer vessel design, insulation and cryocooler selection, inner and outer vessel

stiffening ring design, inner tank suspension design, external support design, and piping

and relief valve selection. The inner vessel of the tank will be constructed from Inconel,

a nickel alloy, and the outer vessel of the tank will be constructed from aluminum. A

combination of multi-layered insulation and vacuum insulation will be used to insulate

the storage vessels. Finally, spherical geometry was selected for the oxygen, hydrogen,

and methane tank designs to minimize the heat transfer and mass of the tanks.


I. INTRODUCTION

The National Aeronautics and Space Administration (NASA) is currently

coordinating several missions to Mars. Initially, the missions will be unmanned robotic

missions and will serve to send equipment to Mars that will be necessary once the

manned missions begin. The trips to Mars are scheduled to leave every two years

beginning in 2001. The manned missions are scheduled to begin in 2011 and will

continue until 2031. Three of the supplies that will be used during both the unmanned

and manned missions to Mars are liquefied hydrogen, oxygen, and methane. This project

consisted of designing storage tanks and the necessary support systems for these

cryogenic fluids. The objective of this project was to design tanks that could be used to

transfer cryogenic fluids to Mars and store them once on the surface with minimal losses

due to heat transfer.

The two reasons for designing a storage tank system for cryogenic fluids were to

have the capability to produce a combustible fuel once on the surface of Mars and to have

oxygen to use for breathing air once people inhabit the planet. The space vehicles will

most likely be powered by nuclear energy, but as an alternate fuel source for the return

trip to Earth, a combustible fuel such as methane will be available. Methane can be

produced from carbon and hydrogen. Carbon can be obtained from the atmosphere of

Mars, but hydrogen cannot. Therefore, liquid hydrogen will need to be transported from

Earth to Mars. Oxygen can also be obtained from the atmosphere of Mars. Oxygen will

be used both as the oxidizing agent for the propellant fuel for the unmanned and manned

mission return trips to Earth and for breathing air. For the unmanned missions, enough

oxygen will be produced on the surface of Mars to satisfy the oxygen requirements, and

oxygen will not have to be taken to Mars. However, for the manned missions, oxygen

will need to be transported to Mars.

The design of tanks to store cryogenic hydrogen, methane, and oxygen is crucial

for the mission. Without fuel and oxygen, the crew of the missions will not be able to

return to Earth, and the inhabitants on Mars will not be able to breathe. Also, the storage

tanks must be designed to minimize fluid loss through vaporization, minimize heat


transfer in order to minimize the weight and volume of the tanks. Loss of fluid could be

disastrous for the mission, and any unnecessary weight drastically increases the cost of

the missions.

II. OBJECTIVES

The primary objective of this project was to design a tank system that could be

used to transport cryogenic fluids to Mars and to store the fluids upon arrival on Mars

while minimizing fluid loss, heat transfer, the volume of the tanks, and the weight of the

tanks. Another objective was to determine a balance between the amount of insulation to

be used and the power requirements of the cryocooler that will further minimize the

weight of the tank system and the amount of heat transfer to the cryogenic fluids.

III. METHODOLOGY

1.0 APPROACH)

This project consisted of both research and detailed design. First, research was

conducted on the climate, atmosphere, and geology of Mars; the behavior of cryogenic

fluids; and existing technology already in use by NASA for similar projects. The project

was then divided into two major portions: the unmanned mission and the manned

mission. As part of the design, insulation for the tanks, the sizing of the tanks, support

systems for the tanks, piping and valves for the tanks, and cryocooler selection were

considered. First, materials were chosen for insulation and the tank structure. Second,

both cylindrical and spherical vessels were examined, and the tank dimensions for each

geometry were determined. Third, the support systems for the tanks and the piping and

valves for the tanks were designed and sized. Finally, the cryocooler selection and the

corresponding insulation thickness were selected. Throughout the project, heat transfer

analyses and thermodynamic analyses were conducted. A cost analysis was also

conducted to determine the most cost-effective design.


Several requirements were outlined by NASA for the design of the cryogenic

fluid storage tanks. First, the amounts of hydrogen, methane, and oxygen to be stored

were mandated for both the unmanned and the manned missions. For the unmanned

mission, 48.2 kg of hydrogen will be taken to Mars and used in the formation of methane.

Once on Mars, 108 kg of methane and 379 kg of oxygen will be produced. For the

manned mission, 4107 kg of hydrogen will be taken to Mars to be used for the production

of methane. Oxygen will also be taken to Mars to be used for breathing air during the

journey to Mars and as for backup oxygen in case the methane and oxygen production

process fails. Once on Mars, methane and more oxygen will be produced. The total

amounts of methane and oxygen required are 8360 kg of methane, 29294 kg of oxygen to

be used for propellant, and 3000 kg of oxygen to be used for breathing. The storage

tanks for both the manned and unmanned missions were designed to accommodate the

corresponding volumes for these masses of fluid. NASA also specified a design safety

factor of two to be used in all designs.

2.0 TIMELINE

This project was completed over two semesters. Initially, the plan was to

complete the design for the unmanned robotic phase of the mission during the first

semester and the design for the manned phase of the mission during the second semester.

This assumption was re-evaluated in November because it became apparent the design

for the unmanned phase would take longer than one semester and the design for the

manned mission less than a semester. The design for the manned mission took less than

one semester since the design process was established during the unmanned mission

portion of the project in the first semester.

Preliminary research of the project was conducted during September. The design

of the tanks, insulation and reliquefaction process began in October and was completed in

early February. Sarah Kelly was responsible for the insulation, support system, and

accessory design, in addition to computer simulation. Kyle Chambliss was responsible

for the actual vessel design for the oxygen, hydrogen, and methane tanks. Finally, Justin

Kimble was responsible for the design of the reliquefaction process and the arrangement

of the tanks, as well as a physical model of one of the tank systems. Heat transfer,


thermodynamic, and cost analyses were completed for both the unmanned and manned

missions. Computer simulation was begun, and a physical model of one of the tank

systems was constructed. A detailed schedule of events is shown in Appendix B.

3.0 LOCATION OF WORK

Work was performed at Texas Tech University in the library and the Mechanical

Engineering department under the advisement of Dr. Darryl James.

4.0 BACKGROUND

4.1 Mars :

Research was conducted concerning the atmosphere, geography, geology, and

climate of Mars to design a storage tank system best suited to the environment on Mars.

The atmospheric conditions on Mars are important to the design of the storage vessels in

that the vessels need to be able to withstand the weight of the fluids due to the gravity of

Mars and also the atmospheric pressure of Mars. The average surface gravity on Mars is

3.7 m/s2. Although the atmospheric pressure varies about fifteen percent throughout the

year, the average atmospheric pressure on Mars is 0.61 kPa. Additionally, the

composition of the atmosphere is important in choosing a material for the storage tanks

and also for the carbon and oxygen extraction process from the atmosphere. The

atmospheric composition is demonstrated in the following figure (Kieffer 130).

0

20

40

60

80

100

120

CO2 N2 Ar Trace

Component

Pe

rce

nt

of

Atm

os

ph

eri

c

Co

mp

os

titi

on


Figure 4.1.1 -- Atmospheric Composition of Mars

The geography of Mars also has an impact on the design of the storage vessels. The

tanks might need to be designed differently if they were to be located in a canyon as

opposed to if they were located on a plain. Also, if the plate tectonics on Mars were

active, the tank systems would need to be designed with possible earthquake activity

taken into consideration. The geography of Mars is extremely diverse. Mars contains

canyons, volcanoes, highlands, and plains. Further, Mars’ plate tectonics are not active.

Geography was assumed to not have an impact on the design of the storage tanks. The

geology of Mars should also be considered when selecting a tank design. For instance,

the material for the tanks should be compatible with the minerals on Mars. The soil of

Mars is predominantly composed of silicon dioxide (SiO2) and iron oxide (Fe2O3)

(Kieffer 31). Additionally, the climate of Mars is of extreme importance when designing

the storage tanks for the cryogenic fluids. The temperature of Mars varies from about

140K to 300K with an average temperature of about 210K. The following figure shows

the maximum and minimum temperatures for a 30 sol (Martian day) period, where Series

1 is the maximum temperature and Series 2 is the minimum temperature (Mars Pathfinder

Weather Data).

050

100150

200250300

0 5 10 15 20 25 30 35 40

Sol

Te

mp

era

ture

(K

)

Series1

Series2

Figure 4.1.2 -- Temperature Distribution for Sols 6 - 35

Mars also has strong winds. The wind speeds vary from 2 m/s to 30 m/s, which is

approximately 70 miles per hour. These winds carry a large amount of dust. The dust

particles consist of sixty percent SiO2, a mineral found in abundance in the soil. The dust


particle size ranges from 0.5 microns to 5 microns (um). On Mars, dust is moved by

suspension, saltation, and creep, where suspension occurs when fine dust particles “float”

in the air, saltation occurs when medium sized dust particles are picked up by the wind

and then dropped in a new location, and creep occurs when particles are moved when

other particles collide with them (Mutch 236). Because of the prevalence of dust in the

Martian atmosphere, physical and chemical weathering is a concern. Physical weathering

can occur by either riving, the process by which dust particles seep into abrasions in a

surface and the abrasions propagate into cracks, or by wind abrasion, which is basically

sand blasting of a surface. Chemical weathering can occur by oxidation, which occurs

when a material takes in oxygen to form oxides or higher oxidized silicates, by hydration,

which occurs when water reacts with a material to form hydroxide (OH-) or water ions,

by carbonation, which occurs when carbon dioxide reacts with a material to form

carbonates, and by solution dissolving, which occurs when a material is dissolved in

water (Kieffer 629). However, solution dissolving should not be a problem on Mars

since only trace amounts of water exist on Mars. The atmosphere, geography, geology,

and climate of Mars were considered when designing the cryogenic fluid storage tanks.

4.2 Cryogenics:

When a gas condenses at low temperatures, the gas becomes a liquid called a

cryogenic fluid. A given mass of a cryogenic fluid has a much smaller volume than the

same mass of a gas. The thermodynamic properties of a gas are needed in order to design

cryogenic tanks. Properties such as saturation temperature, saturation pressure, and

density are critical in the design of cryogenic tanks. Tables 4.2.1, 4.2.2, and 4.2.3 list the

cryogenic properties of hydrogen (H2), oxygen (O2), and methane (CH4), respectively.

The data in these tables came from Cryogenic Systems by Randall F. Barron.

Table 4.2.1 -- Thermodynamic Properties of Hydrogen

SaturationTemperature

SaturationPressure

Density

(K) (kPa) (kg/m3)14 7.88 76.8616 21.58 75.1118 48.23 73.2


20.27 101.2 70.7922 163.4 68.7224 264.6 66.0126 403.5 62.8328 587.1 58.9730 822.6 53.93

Table 4.2.2 -- Thermodynamic Properties of Oxygen


SaturationPressure

Density

(K) (kPa) (kg/m3)60 0.73 1281.770 6.22 1236.780 30.09 1190.3

90.18 101.3 1141100 254.2 1090.7110 543.2 1035.4120 1021.6 974130 1747.8 902.8140 2786.5 813.1150 4219 675.4

Table 4.2.3 -- Thermodynamic Properties of Methane


SaturationPressure

Density

(K) (kPa) (kg/m3)95 17.67 448.3100 34.5 440.4105 56.6 432.5

111.7 101.3 424.1115 132.5 419.4120 191.9 412.1125 269.3 404.7130 368 396.7135 491.3 388.4140 642.2 379.6150 1041.4 361.1160 1594 339.7170 2331 314.1180 3288.2 280


4.3 Hydrogen Tank Requirements:

Hydrogen will be needed to produce methane, CH4, on the surface of Mars. The

Martian atmosphere contains only trace amounts of hydrogen, so any hydrogen needed

must be transported from Earth. For the unmanned mission, 48.2 kg of hydrogen is

needed, and for the manned missions, 4107 kg is needed. The hydrogen tank will be full

from the beginning of the mission until the lander lands on Mars.

When the space shuttle launches, the cargo in the shuttle undergoes three G's of

force. The G forces during the landing must also be taken into consideration in the

design of the hydrogen tank. Also, since the in-flight temperature and Mars surface

temperature will be different, two heat transfer analyses were performed. The density of

cryogenic hydrogen changes by more than thirty percent from 14 to 30K. Cryogenic

fluids at lower temperatures have a greater density and a lower volume than at higher

temperatures. Because the density changes so much over a small temperature range, the

lower the temperature of the fluid, the smaller the volume of the fluid and the smaller the

size of the storage tank. The heat transfer for the cooler fluid is less because the surface

area of the tank is less, even though the temperature difference between ambient

temperature and cryogenic fluid is larger. Figure 4.3.1 is a graph of heat transfer vs.

temperature for a spherical hydrogen tank with 15.25 cm of MLI with 20 layers per inch.


Figure 4.3.1 -- Heat Transfer vs. Temperature for Spherical Hydrogen Tank

During transport, the tank experiences zero gravity. As a result, the gas and fluid will not

be separated but the cryocooler will still cool the liquid.

4.4 Methane Tank Requirements:

0.700

0.750

0.800

0.850

0.900

0.950

1.000

1.050

1.100

14 16 18 20 22 24 26 28 30

Temperature (K)

Heat

Tra

nsfe

r (W

)


The methane tank will need to store 108 kg of methane for the unmanned mission and

8366 kg for the manned mission. The methane tank will only store fluid on Mars. The

tank will be empty during the launch from Earth, and the added weight of the fluid during

the launch will not have to be taken into consideration, as in the hydrogen tank. Also,

when the tank contains fluid, the gas and liquid will be well separated due to the gravity

on Mars. The density of methane does not change much with temperature. When the

temperature and pressure of the fluid is in the higher range of cryogenic methane, the

temperature differential between ambient conditions and the fluid temperature is reduced,

and the heat transfer is minimized. Figure 4.4.1 is a graph of heat transfer vs.

temperature for a cryogenic methane tank.

Figure 4.4.1 -- Heat Transfer vs. Temperature for Spherical Methane Tank

4.5 Oxygen Tank Requirements:

0.500

0.550

0.600

0.650

0.700

0.750

0.800

0.850

0.900

80 90 100 110 120 130 140 150 160 170 180

Temperature (K)

Heat

Tra

nsfe

r (W

)


The oxygen tank design has the same requirements as the methane tank design

except for the mass of the fluid. For the unmanned mission, 379 kg of oxygen will be

needed. For the manned mission two tanks are required, one for breathing and one for

propellant. 3000 kg of breathing oxygen are required, and 29294 kg of propellant oxygen

are required.

4.6 Existing Technology:

Hydrogen is the only fluid that will be transported to Mars. The carbon dioxide

prevalent in the Martian atmosphere will be used to produce the required oxygen and

methane. The process to be employed is called the Sabatier process, named after the

Nobel prize winning chemist Paul Sabatier. The process combines carbon dioxide with

hydrogen to produce methane and water. The methane is stored, and the water is

electrolyzed to separate into hydrogen and oxygen. The hydrogen is reused in the

process and the oxygen is stored. The stored methane and oxygen are then used as a

propellant for the ascent vehicle. The chemical equation for the process is:

OHCHHCO 2422 24 +♦+ Eq. 4.6.1

Many companies produce cryocoolers, but not very many of them produce cryocoolers

for space applications. Currently, no cryocoolers are on the market that have the capacity

to cool, in a space environment, the amounts of fluids necessary for this project. One

possibility, however, is a cryocooler that has been developed at the National Institute of

Science and Technology. This cooler might be applicable, but in the event that it is not, a

cooler which meets the requirements of the tank design would need to be designed as

well.


5.0 DESIGN THEORY OF STORAGE TANKS

5.1 Insulation:

Because the hydrogen, oxygen, and methane must be kept at cryogenic

temperatures to minimize the volume and weight of their storage tanks and because the

amount of heat transfer to the cryogenic fluids should be minimized, the storage tanks

will be well insulated with a weight efficient insulation minimizing heat transfer. The

storage vessels will consist of an inner vessel containing the cryogenic fluid, a layer of

insulation, and an outer vessel.

5.1.1 Material:

The insulation for the storage tanks will consist of two parts: multi-layer

insulation and vacuum insulation. Multi-layer insulations (MLI) consist of alternating

layers of a highly reflective material, such as aluminum or mylar, and a low conductivity

spacer material, such as paper or fibrous netting. The MLI prevents radiative heat

transfer, due to the highly reflective material, and conductive heat transfer, due to the

spacer material. MLI was selected as the insulation for the storage tanks because it has

the best performance (lowest thermal conductivity) of all other insulations, including

expanded foams, gas filled powders, evacuated powders, and opacified powders. The

following table shows the thermal conductivity (k) for each of the aforementioned

insulations (Barron).

Table 5.1.1.1 -- Thermal Conductivity of Several Insulation Types

Insulation k (W/mK)

Expanded Foam 0.026

Gas Filled Powders 0.019

Evacuated Powder 0.00059

Opacified Powders 0.00033

Multilayer Insulation 0.000014


Multi-layer insulation is also low in weight but is high in cost. However, since

minimizing the weight of the storage tanks is a primary objective of this project, MLI was

chosen because of its relatively low weight in comparison to other insulations. Multi-

layer insulation having a 0.0087 mm aluminum foil layer and a 0.2 mm glass fiber paper

layer will be used to insulate the storage tanks.

Vacuum insulation will also be used. The entire region between the inner and

outer vessels of the tank will be evacuated. The MLI will not completely fill the

evacuated region; a small distance of evacuated space alone will be located between the

MLI and the outer vessel of the storage tank. This distance of evacuated space alone will

prevent conduction from the MLI to the outer shell of the vessel. A distance of 0.5 cm

will be used for each storage vessel. Vacuum insulation prevents heat transfer through

solid conduction and through gaseous convection. Further, the use of vacuum insulation

is needed to make the use of MLI effective because MLI must be evacuated to pressures

below 10mPa for MLI to be effective (Barron 396).

5.1.2 Multi-Layer Insulation:

As stated previously, multi-layer insulation prevents radiative heat transfer and

conductive heat transfer. The following equation is the Lockheed correlation for heat

loss due to radiation through MLI for any tank geometry.

q

A

SF C N T T T C T T

Nr mli c m h c r h c

s

,. . .[ * ( ) * ( )]

=− + −256 4 67 4 67ε

Eq. 5.1.2.1

where

qr,mli = radiative heat loss

A = tank surface area

SF = scale factor, accounts for non-ideal behavior

Cc = conduction coefficient

Cr = radiation coefficient

N = layers of MLI per cm

Ns = total number of layers of MLI

Th = external temperature


Tc = temperature of cryogenic fluid

Tm = mean temperature between external and cryogenic fluid temperatures

ε = layer to layer emissivity

The numbers of layers of MLI per cm (N) and the total number of layers of MLI (Ns) can

be varied to obtain the optimum amount of MLI to be used. The following figure

demonstrates the variance in heat loss as the layers of MLI per centimeter increases.

0

10

20

30

40

50

60

7.87 9.84 11.81 13.78 15.75 17.72 19.69

Layers of MLI per cm

Heat

Lo

ss

(kJ/h

r-m

^2)

Figure 5.1.2.1 : Heat Loss vs. Layers of MLI per cm

As the number of layers of MLI per cm of MLI increases, the corresponding heat loss

increases also. The following figure shows the variance of heat loss as the total number

of layers of MLI increases.

0

10

20

30

40

12.5 25 37.5 50 62.5 75

Total Layers of MLI

He

at

Lo

ss

(k

J/h

r/m

^2

)


Figure 5.1.2.2: Heat Loss vs. Total Number of Layers of MLI

As the total number of layers of MLI increases, heat loss decreases. As can be

determined from the previous two figures, a small number of layers of MLI per cm

should be used, and a large number of total number of layers of MLI, or thickness of

MLI, should be used to minimize the heat loss.

The conductive heat transfer for the insulation for a cylindrical vessel can be

determined by modeling the tank as a series of thermal resistances. The conductive heat

loss can then be given by the following equation.

LrhLk

rr

Lrh

TTq io

mlic

πππ 22

12

11

,

2

1

2

)/ln(

2

1 ++

−= Eq. 5.1.2.2

where

qc,mli = conductive heat loss in MLI

To = external temperature

Ti = cryogenic fluid temperature

h1 = internal convection coefficient

r1 = inner vessel radius

L = length of cylinder

r2 = outer vessel radius

k = thermal conductivity of MLI

h2 = external convection coefficient

The conductive heat transfer for the insulation for a spherical vessel can be

determined by modeling the tank as a series of thermal resistances. The spherical vessel

was approximated as a flat plate since the radius of curvature of the sphere was so large.

The conductive heat transfer for the spherical vessel was determined using the following

equation.


AhAk

L

Ah

TTq io

mlic

22

2

1

, 11 ++

−= Eq. 5.1.2.3

where

A = area of heat transfer

L2 = insulation thickness

k2 = thermal conductivity of insulation

The convection through the MLI is minimal and can be neglected in the analysis. The

resultant heat transfer through the MLI is the sum of the conductive and radiant heat

transfer.

5.1.3 Vacuum Insulation:

Again, as stated previously, the vacuum insulation works as a barrier against solid

conduction and gaseous convection. However, some radiant and gaseous conduction heat

transfer does occur through the insulation. The radiant heat transfer for vacuum

insulation is given in the following equation (Barron 386).

)( 44112, ioevr TTAFFq −= σ Eq. 5.1.3.1

where

qr,v = radiant heat transfer in vacuum

Fe = emissivity factor

F12 = configuration factor = 1

σ = Stefan Boltzmann constant = 5.67x10-8 W/m2K4

A1 = surface area of inner vessel

F12 equals one because the inner vessel is completely enclosed by the outer vessel. The

emissivity factor, Fe, is determined by the number of radiation shields within the vacuum.

As the number of radiation shields increase, emissivity decreases. Smaller emissivity


values result in smaller amounts of radiant heat transfer. The formulation for the

emissivity factor can be found in Appendix A.

Because an air space cannot be completely evacuated, heat can be transferred

through the residual gas in the vacuum region by gaseous conduction, or free molecular

conduction. Free molecular conduction is defined by the mean free path of the gas

molecules within the vacuum where the mean free path of a molecule is the distance that

a molecule must have in order to avoid collision with another molecule. For free

molecular conduction to occur, the mean free path of the gas molecules must be larger

than the thickness of the evacuated region. Gaseous conduction can be given by the

following equation (Barron 389).

q GpA T Tg v o i,( )= −1 Eq. 5.1.3.2

where

qg,v = gaseous conduction in vacuum

G = function of temperature, accommodation coefficient factor, gas constant,

specific heat ratio

p = pressure in evacuated region

The derivation of this expression can be found in Appendix A. The total heat transfer for

the vacuum insulation is the sum of the gaseous conduction and radiant heat transfer.

5.1.4 Insulation System:

The total amount of heat transfer occurring in the MLI and vacuum insulation

system is the sum of the heat transfer accrued in each insulation section.

q q q q qtot r mli c mli r v g v= + + +, , , , Eq. 5.1.4.1

5.2 Inner and Outer Vessel Design:


5.2.1 Materials:

The inner and outer vessels of the storage tanks will be constructed from Inconel,

a ductile nickel alloy. The piping used to drain the vessel will also constructed of

Inconel. Inconel has many properties that make it suitable for use on the Martian surface.

First, Inconel forms a thin, protective layer when subjected to oxidizing conditions, thus

making it suitable for chemical weathering conditions on Mars (Mantell). Heavily cold

worked Inconel is very corrosion resistant. Further, Inconel has a high modulus of

elasticity, improving its resistance to stress, and a low coefficient of expansion. Inconel

also has a large ultimate strength value at low temperatures. Some of the mechanical

properties of Inconel are included in the table below (Mantell).

Table 5.2.1.1 -- Properties of Inconel

Property Value unit

Density 8415 kg/m 3

Modulus of Elasticity (E) 213.7 GPaModulus of Rigidity (G) 75.85 GPa

Yield Strength (S y ) 292 MPa

Ultimate Strength (S ut) 1250 MPa

Inconel will also be used to construct the inner and outer vessel support rings and the

inner support rod brackets discussed in detail later in this report. The piping used to drain

the vessel will also constructed of Inconel.

The outer shells of the storage vessels will be constructed from Aluminum.

Although aluminum is not as strong as Inconel, aluminum weighs much less. Because

the outer shells of the storage vessels are so large and because the minimization of the

weight of the storage vessels was the primary goal for this project, aluminum was

selected for the design of the outer shells. Aluminum has a very high specific strength, or

strength to weight ratio. Aluminum also has excellent resistance to oxidation and

corrosion. When subjected to oxidation conditions, a thin aluminum oxide layer will

form on the aluminum, thus protecting it from corrosion. Additionally, aluminum can be

treated with a variety of coatings to further prevent corrosion and wear. Aluminum has a

high reflectivity, making it an ideal material for the outer shell of the storage vessels.


The high reflectivity of aluminum will help prevent radiation heat transfer to the

cryogenic fluids stored. Some of the mechanical properties of aluminum are provided in

the table below.

Table 5.2.1.2 -- Properties of Aluminum

Property Value unitDensity 2800 kg/m3

Modulus of Elasticity (E) 71.7 GPa Modulus of Rigidity (G) 26.8 GPa

Yield Strength (S y) 169 MPa

Ultimate Strength (S ut) 324 MPa

Furthermore, because two different materials will be used in the storage vessel

designs, galvanic corrosion was considered. Galvanic corrosion occurs when two metals

with vastly different oxidation potentials are in contact with one another. If two metals

with different oxidation potentials are placed in an electrolytic medium, a galvanic cell is

produced. Current is driven from one metal to the other through the electrolytic medium.

As a result, if the difference between oxidation potentials is large, corrosion will occur.

The material with the higher oxidation potential is called the anode, and this material will

be the one to corrode. The material with the lower oxidation potential is called the

cathode, and this material will not corrode. Protective coatings such as oxides or various

platings can be placed on the metals to prevent galvanic corrosion. However, aluminum

and Inconel are not significantly far apart in the Galvanic series, so galvanic corrosion

should not be a problem. A thermal expansion analysis was also performed. With a

special piping design discussed later, thermal expansion will not be a problem.

Fiberglass will be used to construct the inner support rods and the external

support system described later in this report. Fiberglass using kevlar fibers was selected

because of its low weight, high strength, and low thermal conductivity. Kevlar fiberglass

has an ultimate strength of 4.58 GPa and a thermal conductivity of 0.26 W/m2K

(Incropera).


The aluminum outer tank will be coated with an anodic coating. Anodizing the

aluminum will protect the tank from corrosion. Both the outer tank and the external

supports will be coated with a highly reflective material to further decrease radiant heat

transfer. In this design, highly polished gold film with an emissivity of 0.01 to 0.03 will

be used (Incropera 851). Weathering could damage the gold film. If this happens the

anodic coating will protect the aluminum tank from further weathering.

5.2.2 Volume:

The tank volume was determined using the density of the cryogenic fluid and the

mass of the fluid needed. The volume was calculated using Equation 5.2.2.1 where ρ is

density and m is mass.

ρm

V = Eq. 5.2.2.1

Since density changes as a function of temperature, volume is also a function of

temperature. For this reason, volume must be calculated for every temperature. A ten

percent ullage, or extra volume, factor was used in the tank designs.

5.2.3 Inner Vessel:

The inner vessel of the storage system stores the cryogenic fluid. For a spherical

vessel, the internal radius (rip) can be found using Equation 5.2.3.1.

3

4

3

πV

rip = Eq. 5.2.3.1


The inner vessel must be thick enough to withstand the pressure of the fluid within the

vessel. From the ASME Boiler Code, Section VIII, the shell thickness for a spherical

pressure vessel is given in Equation 5.2.3.2.

peS

pDt

wtup 4.0−

= Eq. 5.2.3.2

where

p = internal pressure

D = internal vessel diameter

Sut = ultimate strength of material

we = weld efficiency = 1

Once the thickness of the inner shell was determined, the mass of the shell was

calculated. The outer radius is the sum of the inner radius and the inner vessel thickness

calculated previously. The mass of the inner tank was determined using Equation 5.2.3.3.

( )33

3

4iomm rrm −= πρ Eq. 5.2.3.3

5.2.4 Outer Vessel:

The outer vessel of a cryogenic tank contains the inner vessel and the MLI

vacuum insulation. The inner radius of the outer tank, ris, was calculated using Equation

5.2.4.1, where ψ is the thickness of the MLI and ipr is the inner radius of the inner tank

and pt is the thickness of the inner pressure vessel.

ψ++= pipis trr Eq. 5.2.4.1


When designing the outer vessel of the storage tank, the external pressure is of great

importance. The external pressure could cause the vessel to collapse. The external

pressure acting on the storage tank is atmospheric pressure. From the ASME Code,

Section VIII, the thickness of the exterior spherical tank is given by Equation 5.2.4.2,

where υ is Poisson's Ratio and E is Young's Modulus.

( )[ ]

1

2

12

113

5.0−

��

�

�

��

�

�−

−=

υp

Ert iss Eq. 5.2.4.2

For a cylinder, the external shell thickness is given by Equation 5.2.4.3.

( )irE

pt 222

11

3

12

−−

��

�

�

��

�

�+√√↵

��

−= υEq. 5.2.4.3

To calculate the mass of the external tanks, equations from Section 5.2.3 were used.

Stiffening rings were added to provide additional support to the external vessel, and as a

result, the thickness of the shell was reduced. The formula for the maximum critical

pressure on the outside of the exterior tank with stiffening rings is given by Equation

5.2.4.4, where L is the distance between stiffening rings, and cp , the critical pressure, is

four times the allowable pressure.

( )

( ) ( )��

�

�

��

�

�

√√↵

��

+

−+

−

√√↵

��

+

=2

1

4

32

2

5

245.0

(21

242.2

tr

t

tr

L

tr

tE

p

ii

ic

υ

Eq. 5.2.4.4


The critical pressure must be greater than the atmospheric pressure to prevent the tanks

from collapsing. The size of the intermediate stiffening rings can be calculated from the

moment of inertia, 1I , given in Equation 5.2.4.5.

E

LDpI oc

24

3

1 = Eq. 5.2.4.5

When calculating the mass of the exterior tank and stiffening rings, the mass of the

stiffening rings needs to be added to the mass of the tank.

5.2.5 Support Systems:

The design of the storage vessels also includes various support systems. The

support systems consist of stiffening rings for the inner vessel, stiffening rings for the

outer vessel, a suspension system for the internal vessel, and an external support

mechanism. The inner vessel stiffening rings support the weight of the fluid within the

inner vessel. The stiffening rings will be formed from beams of Inconel, and the size of

the stiffening ring was determined by calculating the section modulus of the beam. The

minimum allowable section modulus was determined from the following equation

(Barron 361).

as

MZ max

min = Eq. 5.2.5.1

where

Z = section modulus

Mmax = maximum bending moment

sa = allowable stress

The maximum bending moment was derived from Roark’s elastic energy method and is

shown in Appendix A. Once the minimum allowable section modulus of the stiffening

rings was determined, the beams from which the rings will be constructed were sized to


have a greater section modulus than the minimum allowable value in order to be able to

support the maximum loading conditions.

The outer vessel stiffening rings support the weight of the inner vessel, the weight

of the insulation, and any external pressure and help maintain the shape of the outer

vessel. For short vessels, only two stiffening rings are used. For long vessels, two main

stiffening rings are used to support the weight of the vessel, and additional intermediate

rings are used to maintain the shape of the vessel. The outer vessel stiffening rings will

be formed from beams of Inconel also. The size of the stiffening rings was determined

from the area moment of inertia of the beams. The area moment of inertia used to

determine the size of the intermediate stiffening rings was determined from the following

equation (Barron 367).

Ip D L

Ec o

1

3

24= Eq. 5.2.5.2

where

I1 = area moment of inertia

pc = critical pressure (four times allowable pressure)

Do = outside diameter of outer shell

L = distance between stiffening rings

E = modulus of elasticity of ring material

The minimum allowable area moment of inertia (Imin) used to size the main support rings

is shown in the following expression (Barron 368).

as

cMII max

1min += Eq. 5.2.5.3

where

c = distance to the centroid of the tank geometry

The derivation for the calculation for the bending moment of inertia, Mmax, is shown in

Appendix A. Once the minimum allowable area moments of inertia were calculated, the

beams from which the stiffening rings will be constructed were sized to have a larger


moment of inertia than the minimum allowable value in order withstand maximum

loading equations.

The suspension system for the internal vessel suspends the internal vessel within

the outer vessel and must support the weight of the inner vessel and any loads that are

caused during transportation. The suspension system consists of support rods that

suspend the inner vessel within the outer vessel and brackets that will attach the rods to

the inner and outer vessel walls. A cross-section of a storage tank showing the

suspension system is pictured in the figure below.

Figure 5.2.5.1 -- Storage Vessel Suspension System

During transportation to Mars, the storage tank will be subjected to dynamic loads

including vertical, transverse, and longitudinal loads. The diameter and the number of

rods were selected on the basis of the maximum load in each direction, vertical,

transverse, and longitudinal. The number of rods (N) for each direction was determined

from the equation below (Barron 377).

NF

Fd

= Eq. 5.2.5.4

where

F = maximum force in each given direction (vertical, transverse, or longitudinal)

Fd = design force in each rod


The equations used to determine the dynamic loads in each direction and the design force

in each rod are shown in Appendix A. Additionally, brackets to attach the rods to the

inner and outer vessel walls were designed. The bracket is shown in the following figure.

Figure 5.2.5.2 -- Bracket Design

The brackets will be welded to the vessel walls with fillet welds. As a result, the bracket

design depended on the size of the area where the weld would be formed. The weld will

be subjected to mostly shear stresses, and accordingly, will fail in shear. The bracket was

sized using the American Welding Society expression for maximum shear. The

maximum shear (τmax) occurring in a weld is given in the following equation (Shigley

387).

bt

F

2

414.1max =τ Eq. 5.2.5.5

where

F = maximum vertical force

b = weld width

t = weld thickness

The maximum vertical force used was the maximum vertical force calculated in the

dynamic loadings used to size the support rods. The weld width and thickness are also

the width and thickness of the bracket. The length of the bracket was chosen to be


slightly larger than the diameter of the hole through which the support rod will be

positioned.

The external support serves as a base on which the tanks can be placed. The

external support will also help minimize heat transfer from the storage tank to the ground,

both in transit and on Mars. The external support will be constructed from fiberglass as

mentioned previously and will consist of two cradles on which the storage tank will rest.

A sketch of the external support is shown in the following figure.

Figure 5.2.5.3 -- External Support Structure

The external support was modeled as a simply supported beam with a distributed loading

and two end supports. The simply supported beam approximation can be seen in the

figure below.

Figure 5.2.5.4 -- Simply Supported Beam Approximation


As in a simply supported beam, the external support will deflect due to the weight of the

storage tanks. The deflection of the support structure can be determined from the

following expression (Norton 1003).

ywx Lx x L

EI=

− −( )224

2 3 3

Eq. 5.2.5.6

where

y = deflection

w = distributed load

x = any given location on beam

L = length of beam

The legs of the support structure were also designed so they would not buckle. The legs

were approximated as short Euler columns. The determining factor in the design of the

support legs was the critical pressure acting on the legs. The expression used to

determine the critical pressure (Pcr) in the legs is provided below (Norton 239).

PEI

Lcr =π 2

2 Eq. 5.2.5.7

where

L = length of leg

The critical pressure was then used to determine the allowable force (Fall) acting on each

leg, shown in the following equation.

FP

NAallcr= Eq. 5.2.5.8

where

N = safety factor

A = cross-sectional area of the leg


The dimensions of the leg were chosen so that the actual force acting in each leg was

less than the allowable force in each leg. The actual force acting in each leg is the

distributed loading, w, multiplied by the distance to the centroid of the leg. Furthermore,

the external supports must be able to support the weight of the storage tanks while being

of minimal weight themselves. For this reason, sections of material from the support will

be removed to reduce the weight of the support. When removing sections from the

support, the support must still be able to withstand the weight of the storage tank. When

a section is removed from the support structure, a stress concentration in the area of

removal results. Stress concentrations increase the stress occurring in a particular region

by a factor Kt. This relationship is shown in the following equation (Norton 231).

σ σmax = Kt nom Eq. 5.2.5.9

where

σmax = maximum stress

Kt = stress concentration factor

σnom = nominal stress (stress occurring in region without stress concentration)

The removed sections were designed so the external support would still withstand the

weight of the storage tank and so the stress concentration resulting from the removed

sections was still below the allowable stress in the support. However, since fiberglass is

a ductile material, any stress concentrations could be ignored. Stress concentrations can

be neglected in ductile materials because ductile materials yield at a point and the

material will not fail until the entire cross section of the material yields (Norton 232).

The external support system does not need to account for any wind drag effects on Mars.

Although high wind speeds occur frequently on the surface of Mars, since the atmosphere

has such a low density, the maximum wind speeds on Mars will not significantly affect

the tank structures.

5.2.6 Piping and Valves:


The design of the storage tanks for the cryogenic hydrogen, methane, and oxygen

must also include piping from which the vessels can be drained and a safety relief valve.

The length of piping inside the vessel will be as long as is reasonably possible to

minimize heat transfer. Thin-walled piping should also be used to minimize heat transfer

because it has a smaller cross-sectional area. The piping will be enclosed by a vacuum

region into the fluid in the inner vessel as shown in number 1 in Figure 5.2.6.1. The

vertical part of the piping should be as long as possible to act as a spring to keep the

stresses developed from thermal expansion from damaging the piping.

Figure 5.2.6.1 -- Various Piping Designs

The minimum wall thickness was determined from the ASA Code for Pressure Piping in

the following expression (Barron 379).

tpD

s po

a

=+2 8.

Eq. 5.2.6.1

where

t = thickness of the pipe


p = internal pressure

Do = outside diameter of pipe

sa = allowable stress of pipe material

Once the minimum pipe wall thickness was found, standard steel pipe data tables such as

the table included in Crane’s Flow of Fluids Through Valves, Fittings, and Pipe were

used to determine the pipe size and schedule.

The thickness of the vacuum jacket around the piping is found from the following

equation.

( )irE

pt 222

11

3

12

−−

��

�

�

��

�

�+√√↵

��

−= υ Eq. 5.2.6.2

where

t = thickness of the pipe

p = external pressure

ri = internal radius

The safety relief valve is a device that prevents the pressure within the storage

vessel from exceeding its design pressure. If the pressure within the vessel were to

surpass the design pressure, the safety relief valve would release the extra pressure before

the vessel was damaged. Safety relief valves are sized based on the discharge area of the

valve. The area of the discharge valve for the safety relief valve was found using the

ASME code in the following equation (Barron 383).

Am R T g M

CK pv

g u c

D

=& ( / )

max

Eq. 5.2.6.3

where

Av = discharge area of valve

mg = maximum mass flow rate through valve

Ru = universal gas constant


T = temperature of cryogenic fluid at inlet to valve

M = molecular weight of cryogenic fluid

gc = gravitational constant

KD = discharge coefficient

pmax = 1.1 x fluid pressure + atmospheric pressure

C = constant dependent on the specific heat ratio of the cryogenic fluid

5.2.7 Temperature Sensors:

In order to reduce the amount of time the cryocoolers will run, temperature

sensors will be used. The temperature sensors will be used with each individual tank to

measure the temperature of the fluid within the storage vessel. When the temperature

sensor indicates a temperature equal to the design temperature for the given cryogenic

storage vessel, the cryocooler will turn on and cool the fluid until the fluid is several

degrees below the design temperature for that particular tank. Although the loads that the

cryocoolers will have to withstand will most likely cause the cryocoolers to run

continuously, the temperature sensors will still be in place in the event the cryogenic

fluids become cool enough to enable the cryocoolers to cease running. Thus, the

temperature sensors will be utilized to help conserve power.

Temperature can be measured in a variety of ways. The temperature of a

substance can be determined by measuring the height of mercury in a capillary tube, the

electrical resistance of a platinum wire, the pressure of an ideal or near-ideal gas, the

equilibrium pressure of a gas above a boiling liquid, the difference in thermal expansion

of two metals in a composite beam, the speed of sound in a gas, and the magnetic

susceptibility of a paramagnetic material (Barron 310). When measuring temperature,

properties of the measuring instrument must be taken into consideration. The properties

of a temperature measuring device include accuracy, the deviation of the indicated

temperature from the temperature scale; sensitivity, the rate of change of the temperature

indicating property with temperature; reproducibility, the range of temperature when

several measurements were made at the same temperature; and stability, the change in the

indication of the device over a period of time.


Some commonly used temperature measuring devices employing these methods

are metallic resistance thermometers, semiconductor resistance thermometers, and

thermocouples. Metallic resistance thermometers indicate temperature by measuring the

variance of electrical resistivity of a metal with temperature. Platinum, copper, lead, and

other metals whose resistivities vary linearly with temperature are commonly used in

metallic resistance thermometers. Metallic resistance thermometers are calibrate using

( )R

RAt Bt Ct te

o

= + + + −1 1002 3 Eq. 5.2.7.1

where

Re = measured resistance

Ro = resistance at 0 °C

A, B, C = constants found by calibration at three standard temperatures

t = temperature (°C)

The sensitivity of a metallic resistance thermometer is the rate of change of resistivity

with temperature. The sensitivity, So, of a metallic resistance thermometer is

( )[ ]SdR

dTR A Bt Ct to

eo= = + + −2 4 3002 Eq. 5.2.7.2

Another commonly used type of temperature measuring device is a semiconductor

resistance thermometer. Semiconductor resistance thermometers use the same principle

as metallic resistance thermometers, except the resistivity of a semiconductor is measured

instead of the resistivity of a metal. Semiconductor resistance thermometers can be

calibrated by

( )TB R

R A R K

e

e e

=+ +

log

log log10

10

2

10

Eq. 5.2.7.3

where

T = temperature


A, B, K = calibration constants

A third type of commonly used measuring device is the thermocouple. A thermocouple

consists of a pair of dissimilar metals connected at two junctions. One junction of the

thermocouple is placed where the temperature is to be measured, and the other junction is

placed at a reference temperature location, such as an ice bath. The temperature

indication of a thermocouple is determined by measuring the electromotive force induced

in the thermocouple. Thermocouples are calibrated using

t b e b e b e b e= + + +1 22 3

44

3 Eq. 5.2.7.4

where

e = electromotive force

Metallic and semiconductor resistance thermometers are extremely effective for

measuring cryogenic temperatures while thermocouples are only moderately effective.

Many resistance thermometers appropriate for use with cryogenic temperatures are

available from many different companies.

5.2.8 Tank Connections:

A variety of connections can be used in conjunction with cryogenic tanks and

piping. Several available connectors include bayonet connections, threaded connections,

and field joint couplings. Bayonet connections are connections between two sections of

pipe that do not require welding. Bayonet connections consist of telescoping male and

female parts. The male part is inserted into the female part, and the two sections are then

clamped together. Bayonet connections have a low heat inleak and are ideal for

cryogenic applications. A bayonet connection is shown in Figure 5.2.8.1.


Figure 5.2.8.1 -- Bayonet Connection

Threaded connections are simply connections made by screwing two threaded parts

together. Threaded connections do not have as low of a heat inleak as bayonet

connections, and are not as well suited for cryogenic applications as bayonet connections.

A final type of connection is the field joint coupling. Field joint couplings are vacuum

insulated connections between two sections of pipe. However, field joint couplings

require field welding. Once the weld is made, the joint is then insulated, and the coupling

is put into place and evacuated to vacuum pressures. This type of coupling usually

requires an experienced field crew. Field joint couplings also have a low heat inleak and

are ideal for use with cryogenic fluids.

5.3 Heat Transfer Analysis:

Cryogenic fluids must be stored below the saturation temperature for the

corresponding vessel pressure. If the fluid were to reach a temperature greater than the

saturation temperature at a certain pressure, the fluid would vaporize and the tank

pressure would increase, possibly causing the tank to explode. Because of this, heat

transfer analysis is crucial in the design of cryogenic tanks. Heat transfer must be held to

a minimum. If the heat transfer through the tank were greater than the amount of energy


removed by the cryocooler, the fluid would eventually turn to a gas. Also, as heat

transfer through the tank decreases, the energy required to cool the liquid decreases.

Figure 5.3.1 is a sketch of the cryogenic tank wall.

Figure 5.3.1 -- Cryogenic Tank Wall

Figure 5.3.2 is a schematic of the heat transfer resistances of a cryogenic tank wall.

Figure 5.3.2 -- Heat Transfer Schematic of Cryogenic Tanks

The heat transfer through the MLI is given in section 5.1.2. Heat transfer through the

tank walls is also shown in section 5.1.2, as well as heat transfer through the vacuum


insulation. The heat transfer from the atmosphere to the outer surface is given in

Equation 5.3.1, where h is the convection coefficient.

( ) ( )44sss TTFTTh

A

q −+−= ××−× σ Eq. 5.3.1

5.4 Design Results:

5.4.1 Tank Designs:

The results of the calculations for the volume, inner tank, and outer tank designs

are given in table 5.4.1. The values were calculated using formulas from sections 5.2.2,

5.2.3, and 5.2.4.

Table 5.4.1 -- Design results of inner and outer tanks

O2

Unmanned

Propellant

CH3

Unmanned

H2

Unmanned

O2

Manned

Propellant

O2

Manned

Breathing

CH3

Manned

H2

Manned

Fluid Mass

(kg)

379 108 48.2 29294 3000 8366 4107

Design Temp.

(K)

90.18 111.7 14 90.18 90.18 111.7 14

Volume

(m3)

0.332 0.263 0.681 25.67 2.63 19.73 58.02

Inner Tank rI

(m)

0.90 0.94 1.10 3.66 1.71 3.35 4.67

Inner Tank t

(m)

0.0008 0.0032 0.0021 0.0007 0.0003 0.0006 0.0002

Inner Tank Mass

(kg)

9 24 22 122 12 86 41

Outer Tank rI

(m)

0.96 0.99 1.27 3.76 1.75 3.44 4.84

Outer Tank t 0.0015 0.0015 0.0020 0.0058 0.0027 0.0054 0.0075


(m)

Outer Tank Mass

(kg)

11.8 12.7 27.2 703 71 538 1500

5.4.2 Insulation:

As stated previously, for both the manned and unmanned mission cryogenic

storage vessels, both multi-layered insulation (MLI) and vacuum insulation will be used

to prevent heat transfer to the cryogenic fluid from occurring. Table 5.4.2.1 shows the

MLI design results for each of the tanks for the unmanned mission.

Table 5.4.2.1 -- Insulation Selection for the Unmanned Mission

Number of InsulationLayers Thickness (cm) Qmli (W) Qtot (W) mass (kg)

O2 12 1.5 14.9 15.1 0.71H2 60 7.6 4.4 5 5.97CH4 11 1.3 14.23 14.5 0.67

The hydrogen tank has a significantly larger number of layers of MLI and larger resulting

insulation thickness than the oxygen and methane tanks. Because the hydrogen will be

stored at a much lower temperature (14K) than the oxygen (100K) and the methane

(180K), a larger amount of insulation is needed to prevent heat transfer to the cryogenic

fluids. The heat transfer occurring due to radiation through the MLI and the total

insulation heat transfer are also shown in Table 5.4.2.1. All three storage vessels have

total heat transfers under 15 W. A cryocooler will be used with each tank to maintain the

fluid design temperature.

Similarly, the design results for each of the manned mission storage vessels are

given in Table 5.4.2.2.

Table 5.4.2.2 -- Insulation Selection for the Manned Mission


Number of InsulationLayers Thickness (cm) Qmli (W) Qtot (W) mass (kg)

O2-propellant 34 4.32 86.9 90.5 33.5O2-breathing 10 1.27 64.5 65.3 2.1H2 60 7.62 80.2 89.4 97.1CH4 28 3.56 88.4 91.4 23.1

The vessels for the manned mission have larger insulation thicknesses than do the vessels

for the manned mission. The manned mission vessels store much more cryogenic fluid

than the vessels for the manned mission. As a result, more insulation is needed to prevent

heat transfer to the cryogenic fluid. Because the weight of the vessels was a primary

concern in the design of the cryogenic storage vessels, the vessels for the manned mission

have a much greater total heat transfer due to the insulation than the tanks for the

unmanned mission. As a result, the cryocoolers will require a great deal more input

power than for the unmanned mission. If the insulation for the manned mission tanks

were selected to result in only 15 W of heat transfer, the insulation thicknesses would be

very large. As the insulation thickness increases, the size of the outer vessel also

increases. For 15 W of heat transfer due to insulation, the outer shells of the storage

tanks would be tremendously heavy. As part of the design, weight minimization was

decided to be more important than the minimization of power consumption because

power is probably more easily compensated for than is weight. If the vessels are too

heavy, the vessels may not be able to be launched into space without great difficulty or

cost. Therefore, for the manned mission, extra power will need to be produced for

cryocooler consumption in order to help minimize the weight of the storage vessels.

5.4.3 Internal Support Rings:

The internal support rings will be used to support the weight of the cryogenic

fluid within the inner vessels of the storage tanks. The dimensions for the internal

support rings for the storage vessels for the unmanned mission are shown in Table

5.4.3.1.

Table 5.4.3.1 -- Internal Support Ring Dimensions for the Unmanned Mission


b (m) h (m) l (m) Z (m3) mass (kg)O2 0.002 0.017 2.83 1.45E-07 0.81H2 0.002 0.0075 3.46 2.81E-08 0.44CH4 0.002 0.0088 2.95 3.87E-08 0.44

Since for the unmanned mission the oxygen vessel contains the greatest mass of

cryogenic fluid, the support ring for the oxygen tank has the largest cross sectional area

and largest mass. The hydrogen and methane tanks do not contain as much fluid mass as

does the oxygen tank and thus have smaller cross sectional areas than the oxygen tank.

However, the hydrogen and methane tanks are larger and as a result, have inner support

rings larger in diameter than the oxygen tank.

The dimensions for the internal support rings for the manned mission are shown

in Table 5.4.3.2.

Table 5.4.3.2 -- Internal Support Ring Dimensions for the Manned Mission

b (m) h (m) l (m) Z (m3) mass (kg)O2-propellant 0.015 0.1 11.51 3.75E-05 145.2O2-breathing 0.008 0.045 5.38 4.05E-06 16.3H2 0.008 0.058 14.68 6.73E-06 57.3CH4 0.008 0.069 10.53 9.52E-06 48.9

Because the vessels for the manned mission are much larger than the vessels for the

unmanned mission, the dimensions for the internal support rings for the storage vessels

for the manned mission are greater than those for the unmanned mission. Again, because

the propellant oxygen tank contains the largest mass of fluid, the support rings for this

tank were much larger than those for the tanks containing less fluid mass.

5.4.4 Outer Rings :

The external support rings support the weight of the inner vessel, the insulation,

and the fluid. The external support rings also help maintain the spherical shape of the

storage vessels. Two external support rings will be used for each of the tanks for both the

manned and unmanned missions. The dimensions for the external support rings for the

unmanned mission are shown in Table 5.4.4.1.


Table 5.4.4.1 -- External Support Ring Dimensions for the Unmanned Mission

N b (m) h (m) I (m4) L (m) mass (kg)O2 2 0.004 0.271 6.73E-06 2.97 30.58H2 2 0.004 0.271 6.73E-06 3.98 41CH4 2 0.004 0.271 6.73E-06 3.08 31.68

For the unmanned missions, the external support rings for the storage tanks only vary in

length. As a result, the masses are similar for all three tanks.

The dimensions for the external support rings for the manned mission are shown

in Table 5.4.4.2.

Table 5.4.4.2 -- External Support Ring Dimensions for the Manned Mission

N b (m) h (m) I (m4) L (m) mass (kg)O2-propellant 2 0.048 0.41 5.51E-04 11.85 3924.6O2-breathing 2 0.023 0.23 4.66E-05 5.52 491H2 2 0.05 0.485 9.51E-04 15.25 6224.98CH4 2 0.02 0.215 3.31E-05 10.84 784.53

Because the four tanks to be used in the manned missions will contain dissimilar masses

of fluid and are dissimilar in dimension, the dimensions and masses for the external

support rings for these tanks differ also. The propellant oxygen and hydrogen tanks have

external supports with the greatest mass and dimensions. These two tanks larger external

supports because these two tanks have the greatest total mass.

5.4.5 Internal Support Structure:

The internal support structure consists of a system of rods and brackets that

support the inner vessel within the outer vessel of each storage tank. These rods and

brackets allow the multi-layered vessels to be subjected to dynamic loadings without

damage. The support rods are connected to the inner and outer vessel walls by brackets

that are welded to the inner and outer vessel walls. The rods will fit through circular

holes in the brackets. The dimensions for the internal support rods for the unmanned

mission storage tanks are shown in Table 5.4.5.1, while the dimensions for the internal

support rod brackets for the unmanned mission are shown in Table 5.4.5.2.


Table 5.4.5.1 -- Internal Support Rod Dimensions for the Unmanned Mission

d (m) l (m) Fmax (N) Mass,each (kg) # RodsO2 0.005 0.04 4.85E+03 0.002 12H2 0.006 0.16 2.43E+03 0.008 12CH4 0.005 0.04 2.62E+03 0.002 12

Table 5.4.5.2 -- Internal Support Rod Bracket Dimensions for the Unmanned Mission

t (m) b (m) l (m) Mass,each (kg) #bracketsO2 0.002 0.01 0.02 0.004 24H2 0.002 0.01 0.02 0.004 24CH4 0.002 0.01 0.02 0.004 24

Each of the tanks for the unmanned mission will require twelve support rods. The

dimensions of the rods vary only slightly. The rods for the hydrogen tank are a little

larger in diameter than those for the oxygen and methane tanks because the hydrogen

tank has a larger mass. The length of the rods for the hydrogen tank are longer than those

for the methane and oxygen tanks since the hydrogen tank has a larger thickness of

insulation. Since each rod requires one bracket for each end where the ends are attached

to the vessel walls, 24 brackets will be needed for each tank. For the unmanned missions,

the brackets are identical in dimension.

The dimensions for the internal support rods and internal support rod brackets for

the manned missions are shown in Tables 5.4.5.3 and 5.4.5.4, respectively.

Table 5.4.5.3 -- Internal Support Rod Dimensions for the Manned Mission

d (m) l (m) Fmax (N) Mass,each (kg) # RodsO2-propellant 0.018 0.097 3.60E+05 0.244 20O2-breathing 0.006 0.035 3.69E+04 0.01 20H2 0.011 0.163 5.08E+04 0.154 12CH4 0.014 0.081 1.03E+05 0.125 12

Table 5.4.5.4 -- Internal Support Rod Bracket Dimensions for the Manned Mission


t (m) b (m) l (m) Mass,each (kg) #bracketsO2-propellant 0.009 0.07 0.04 0.17 40O2-breathing 0.003 0.022 0.02 0.017 40H2 0.0035 0.025 0.04 0.053 24CH4 0.006 0.03 0.04 0.106 24

The tanks for the manned missions have varying numbers of internal support rods and

varying dimensions for the support rods. The numbers of support rods were selected to

minimize the combined weight of the rods and brackets. The breathing oxygen tank has

the smallest internal support rods and brackets. This is because the breathing oxygen

tank has the least mass. The propellant oxygen tank has the largest internal support rods

and brackets since the propellant oxygen tank has the largest mass. Again, as stated

previously, twice as many brackets are needed as internal support rods since each rod

must have one bracket to attach each end to the internal and external vessel walls.

5.4.6 External Support Structure:

Each cryogenic storage vessel will rest on two external supports. The external

supports will be strapped to the vessels using kevlar straps to prevent the tanks from

moving off of the supports. The supports serve as a means to hold the storage tanks

stationary and also as a means to reduce heat transfer from the surface on which the tanks

rest to the tanks themselves. The dimensions for the external supports to be used for the

unmanned mission are shown in Table 5.4.6.1.

Table 5.4.6.1 -- External Support Dimensions for the Unmanned Mission

O2 H2 CH4length, L (m) 0.384 0.454 0.400thickness, t (m) 0.003 0.0032 0.003leg height, h (m) 0.036 0.042 0.039width, w (m) 0.051 0.056 0.051base , b (m) 0.044 0.048 0.044cutout radius (m) 0.012 0.015 0.013mass (kg) 0.360 0.490 0.360

The dimensions for the external supports for the unmanned mission storage vessels are

fairly similar. Again, the external supports for the hydrogen tank are larger than those for


the oxygen and methane tanks since the hydrogen tank is more massive. The supports are

all fairly lightweight, each support having a mass of less than 0.5 kg.

The dimensions for the external supports for the unmanned mission are provided

in Table 5.4.6.2.

Table 5.4.6.2 -- External Support Dimensions for the Manned Mission

O2 - Propellant O2 - Breathing H2 CH4length, L (m) 1.320 0.620 1.700 1.210thickness, t (m) 0.030 0.007 0.020 0.020leg height, h (m) 0.144 0.060 0.167 0.126width, w (m) 0.174 0.117 0.135 0.137base , b (m) 0.300 0.050 0.300 0.200cutout radius (m) 0.090 0.011 0.100 0.080mass (kg) 18.000 2.460 14.600 6.400

The dimensions for the external supports for the storage vessels for the manned mission

are much greater than the dimensions for the unmanned mission. The difference in

dimensions results because the storage tanks for the manned missions are much larger

than those for the unmanned mission. As a result, the external supports for the manned

mission are larger in mass than those for the unmanned mission. The external supports

for the propellant oxygen and hydrogen tanks are much larger and massive than those for

the breathing oxygen and methane tanks for the manned mission since the propellant

oxygen and hydrogen tanks have more mass than the breathing oxygen and methane

tanks.

5.4.7 Temperature Sensors:

Temperature sensors will be used with each of the storage vessels for both the

manned and unmanned mission to determine when the cryocoolers should run. For each


of the oxygen and methane tanks, a LakeShore PT-100 Platinum RTD (platinum

resistance thermometer) will be used. For each of the hydrogen tanks, a Scientific

Instruments Germanium Resistance Thermometer will be used. Some of the properties of

these two temperature sensors are listed in Table 5.4.7.1.

Table 5.4.7.1 -- Properties of Selected Temperature Sensors

Germanium Resistance Platinum ResistanceThermometer Thermometer

Termperature Range 1.5K - 100K 30K - 873KRepeatability +/- 0.0005 K +/- 0.01 KSensitivity, dR/dT 35000 Ohms/K nearly constantAccuracy indefinite +/- 0.35 KStability indefinite +/- 0.01 KManufacturer Scientific Instruments LakeShore

A different temperature sensor had to be selected for the hydrogen since the temperature

range of the platinum resistance thermometer does not include the design temperature for

the hydrogen. Each of the sensors has a high repeatability. The germanium resistance

thermometer repeats measurements within +/- 0.0005 K for each measurement. The

platinum resistance thermometer repeats measurements within +/- 0.01 K for each

measurement. Each of the two thermometers has a nearly constant sensitivity. A nearly

constant sensitivity means that as the electrical resistance of the material in the

thermometer changes, the temperature changes at the same rate. The platinum resistance

thermometer is accurate within +/- 0.35 K and stable within +/- 0.01 K. The germanium

resistance thermometer remains accurate and stable for an indefinitely long period of

time.


5.4.8 Piping and Valves:

The following tables provide the pipe dimensions and valve sizes for both the

unmanned and the manned missions.

Table 5.4.8 -- Pipe and Valve Sizing Results

O2

BreathingO2

PropellantH2 CH4

Nom. Pipe Size (in) 1.5 1.5 1.5 1.5Pipe Thickness (in) 0.065 0.065 0.065 0.065

Pipe OD (in) 1.9 1.9 1.9 1.9Vacuum Jacket ID (in) 2.2 2.2 2.2 2.2

Safety Valve Area (cm2) 2.2 2.2 1.7 2.5

5.4.9 Connections:

Bayonet connectors will be used for any piping connections that need to be made.

1.5” Quality Cryogenics bayonet connections will be used. All of the associated piping

for the cryogenic storage vessels and the Sabatier process will not be assembled until on

the surface of Mars. The piping will be attached to the storage tanks using these bayonet

connections. Bayonet connectors were selected because of the relative ease with which

they can be assembled. Astronauts will have to connect the piping to the tanks while

wearing cumbersome spacesuits. Bayonet connections should be fairly simple for the

astronauts to handle, even while wearing their gear.

5.4.10 Heat Transfer Analysis:

The results for the calculations of total heat transfer analysis are shown in Table

5.4.10.1. The values were calculated using equations from section 5.3.

Table 5.4.10.1 -- Heat Transfer Analysis

O2

Unmanned

CH3

Unmanned

H2

Unmanned

O2

Manned

O2

Manned

CH3

Manned

H2

Manned


Propellant Propellant Breathing

Insulation

(W)

15.0 14.0 4.5 90.5 65.3 91.4 89.4

Piping

(W)

9.1 5.7 3.3 10.3 12.5 9.6 11.5

Supports

(W)

0.6 0.3 0.4 11.0 3.4 4.3 1.3

Totals

(W)

24.6 20.0 8.2 111.8 81.2 105.3 102.3

5.4.11 Total Tank Mass:

The results for the calculations of total mass are shown in Table 5.4.11.1. The

mass values have been presented previously and are now itemized and totaled.

Table 5.4.11.1 -- Itemized Tank Masses

O2

Unmanned

Propellant

CH3

Unmanned

H2

Unmanned

O2

Manned

Propellant

O2

Manned

Breathing

CH3

Manned

H2

Manned

Inner Tank Mass

(kg)

8.67 24.07 21.66 122 12 86 41

Outer Tank Mass

(kg)

11.75 12.67 27.19 703 71 538 1500

Rods

(kg)

0.012 0.011 0.048 4.9 0.2 1.5 1.2

Brackets

(kg)

0.046 0.025 0.023 6.8 0.7 2.5 0.9


Inner Rings

(kg)

0.81 0.80 0.99 145.2 16.3 48.9 57.3

Outer Rings

(kg)

30.6 31.7 41.0 4276 739 838 6612

Insulation

(kg)

0.71 0.67 5.95 33.5 2.1 23.1 97.1

External Supports

(kg)

0.359 0.360 0.487 18.0 2.5 6.4 14.6

Piping

(kg)

2.0 2.0 2.0 2.1 2.0 2.1 2.2

Totals

(kg)

55.0 72.3 99.4 5312 846 1547 8326

5.5 Cryocooler:

5.5.1 Significance:

The cryocooler is important to the overall scope of the project in that it addresses

one of the main objectives, to determine a balance between the amount of insulation to be

added to the tanks and the cooling to be restored. Much of the other design occurring in

the project depends on the capacity of the cryocooler. The actual design of a cryocooler

is not within the scope of the project at this time. However, in the cryocooler analysis,

the cryocooler requirements for each tank design were determined.

5.5.2 Initial Questions:

This segment of the analysis in the project has been somewhat vague from the

outset. Initially, some form of condensing the vaporized fluid in the tanks was known to

be necessary. Whether design of a condensation system was required or the purchase an

existing process or mechanism accomplishing the task was not known. However,


designing a condensation system or cryocooler would extend the scope of the project too

far and, therefore, work was begun to research cryocooler technology.

Much time was spent finding what is currently available on the market in the form

of cryogenic cooling systems. During this investigation, it became apparent that there

might not be anything on the market made for cooling hydrogen, oxygen, and methane in

a space environment. However, with the advisement, the scope of the project was

changed to only include finding the input power a cryocooler would need to have to

produce a given amount of cooling.

A plot of input power as a function of cooling capacity is the tool through which

the cryocooler was evaluated. The analysis to achieve this plot was acquired from Eric

Marquardt at the National Institute of Standards and Technology in Boulder, Colorado.

5.5.3 Information from NIST:

Eric Marquardt’s name was obtained from Todd Peters at Johnson Space Center.

Marquardt had already designed a cryocooler for space applications as is shown in the

schematic in Figure 5.5.3.1.


Figure 5.5.3.1 -- Schematic of Pulse Tube Cryocooler Designed at NIST

This cryocooler is a pulse tube cryocooler in which the condensation occurs at the end of

the condenser, sometimes called a “cold finger.” Initially, this cryocooler was selected

for use in this project. However, since the scope of the project changed, the actual

selection of a cryocooler does not need to be completed. All that is necessary is a method

to evaluate how much power will have to be delivered to the cryocooler selected. The

previous schematic is a good example of what the cooler will look like when installed to

a tank, and the following equations show how to calculate the input power requirements

for a cryocooler. This cryocooler is the model upon which that method is based. That is

to say, the cryocooler evaluation was done for a cryocooler with similar efficiency.

There is not a good way to approximate the power required by the cryocooler for the

hydrogen tank because such technology has not been developed. Therefore, the

following process was conducted for the oxygen and methane tanks only.

5.5.4 Equation Derivation:

The performance of a cryocooler can be modeled as a Carnot cycle. Specifically,

the coefficient of performance of a Carnot refrigeration cycle is

Wcycle

Qc=maxβ Eq. 5.5.4.1

where Qc (Watts) is the heat transfer from the cold reservoir and Wcycle (Watts) is the net

work input to the cycle. Equation 5.5.4.1 becomes

TcTh

Tc

−=maxβ Eq. 5.5.4.2

where Tc (K) is the cold liquid temperature and Th (K) is the hot ambient temperature.

Rearranging Equations 5.5.4.1 and 5.5.4.2 give


)(max Tc

TcThQc

QcWcycle

−==β

Eq. 5.5.4.3

which can be used to calculate the input power required by the cryocooler. Qc is a

function of the amount of insulation added to the tank. Qc,max would occur in the case of

no insulation. One of the objectives of this project is to find a balance between either no

insulation or no cryocooler. These are the two extreme limits guiding the design.

The actual process for calculating Wcycle was obtained from Eric Marquardt. The

first step is to calculate what is called the Carnot power, Pc.

Tc

TcThPc

−= Eq. 5.5.4.4

The next step was to read the efficiency off of a performance curve such as the Strobridge

plot in Figure 5.5.4.2.


Figure 5.5.4.2 -- Strobridge Plot

This plot was put together from experimental data during the 1970’s. For use in

these calculations, a curve was drawn through the top data points in order to approximate

the higher efficiency of coolers in the present time. This plot is shown in Figure 5.5.4.3.


Figure 5.5.4.3. Strobridge Data for Cryocoolers Operating at 90K

The cooling capacity is a function of the amount of insulation on the tank. The efficiency

and the Carnot power are used to calculate the electrical power, Pe.

eff

PcPe = Eq. 5.5.4.5

The total power, Pt, required is calculated by dividing the electrical power by a power

conversion factor for the electronics of about 85%.

85.0

PePt = Eq. 5.5.4.6

Finally, the input power required by the cryocooler to achieve a given cooling capacity is

obtained by multiplying the total power by the amount of cooling desired.

1

10

100

2 5 15 60 100 800

Cooling Capacity (Watts)

Car

no

t E

ffic

ien

cy


5.5.5 Results:

The result of the previous equations was a curve that shows the input power

(Watts) required by a cryocooler to produce a given amount of cooling (Watts). This

relationship was investigated for four different ambient temperatures of 200, 220, 240,

and 300 K. The final product of this process is a curve which shows input power as a

function of cooling capacity. Curves of this nature are shown in Figures 5.5.5.1 and

5.5.5.2. Figure 5.5.5.1 shows the input power a cryocooler would require in order to

replace the cooling lost in the oxygen tank. Figure 5.5.5.2 shows the same relationship

for the methane tank.

Figure 5.5.5.1 -- Input Power as a Function of Cooling Capacity for the Oxygen Tank

0

500

1000

1500

2000

2500

0 50 100 150 200 250

Cooling Capacity (W)

Inp

ut

Po

wer

(W)

300K

240K

220K

200K


Figure 5.5.5.2 -- Input Power as a Function of Cooling Capacity for the Methane Tank

The requirements for the cryocoolers for both the unmanned and manned missions are

shown in Table 5.5.5.1.

Table 5.5.5.1 -- Cryocooler Requirements for Unmanned and Manned Missions.

Input Power (W)


O2 519 25CH4 130 20Total Unmanned 649

O2 (prop) 1535 111.8O2 (breath) 1350 81.2CH4 1129 105.3Total Manned 4014

0

200

400

600

800

1000

1200

1400

1600

1800

0 50 100 150 200 250


Inp

ut

Po

we

r (W

)

300K

240K

220K

200K


5.6 Cost Analysis:

The cost analysis for this project consisted of a comparison between the cost to

manufacture the design and the cost to transport the design to Mars. The purpose of the

cost analysis was to minimize the total cost for the storage vessel designs. In any

government project, reducing the cost of the project should be an issue, since the

taxpayers are basically responsible for the cost of the project. The costs for the cryogenic

storage vessel design include material costs for the vessels, the cost to construct the

vessels, and the cost to launch the vessels into space. The predominant cost for the

design is the launch cost. The cost for transporting objects into space is very high,

approximately $20,000 per kg of mass. As a result, an objective of this project was to

minimize the weight of the storage tank designs. However, at some point, minimizing

the weight of the design could have caused the cost to construct the vessels to

dramatically increase. For example, a lighter tank might cost an exorbitant amount to

construct. If the lighter tank’s construction costs cause it to cost more overall than a

heavier tank, then the launch cost benefit of the lighter tank is overruled by its high

construction costs. For this reason, the cost of the materials for the tank designs and the

cost to construct the tank designs were examined to determine whether the lightest tank

design was in fact the most cost effective design.

However, the cost analysis confirmed the original assumption that a lighter tank

would be more cost effective. Logically, the materials for the cryogenic storage vessels

cost less for smaller (lighter tanks). For smaller vessels, less material is needed, and

therefore, the materials cost less for smaller vessels. Additionally, several manufacturers

of cryogenic storage vessels, including Minnesota Valley Engineering and Cryofab,

confirmed that smaller vessels are less expensive to construct. In conclusion, the smaller

and lighter the storage vessel, the more cost effective is the design since the material,

construction, and launch costs are reduced.

The Taguchi method as described in The Engineering Design Process, by Ertas

and Jones, was also examined as a possible tool for the cost analysis. However, the

Taguchi method was not appropriate for the type of cost analysis to be completed in this

project since the Taguchi method is based on experimental data and this project produced

no experimental data.


5.7 Simulation:

The design for the cryogenic storage vessels was intended to conclude with

physical and computer simulation of the basic design created during this project. Both

physical and computer simulation of the basic cryogenic storage vessel design have been

attempted. A physical model of the storage vessel design showing how the vessel will

be constructed and how the vessel will appear internally was also attempted. Computer

simulation of the basic vessel design was begun to determine whether or not the

theoretical designs completed throughout this project could actually withstand loadings

expected to occur during the life of the storage vessels. However, due to lack of

experience with finite element analysis and simulation software and also to a shortness of

time, the computer simulations were not able to be completed. Some simulation was

completed, but the results of the simulation were inconclusive.

IV. Conclusions

At this time, the designs for the cryogenic storage vessels for both the manned

and unmanned missions have been completed. The primary objectives of this project

were to minimize the weight and volume of the storage vessels, to minimize heat transfer

to the cryogenic fluids, and to determine a balance between the amount of insulation to be

used and the power requirements of cryocooler. The objectives of this project have been

met. The designs for each of the cryogenic storage vessels minimize the weight and

volume of the tanks and minimize heat transfer. Inconel and aluminum were selected as

the materials for the inner and outer vessels of the storage vessel, respectively. Fiberglass

was selected as the material for the suspension rods and external support to help reduce

the total weight of the vessels. Multi-layered insulation and vacuum insulation were

selected to minimize heat transfer to the cryogenic fluids. A balance between the amount

of insulation and the cryocooler power was also determined and from this balance, the

insulation thickness and power requirements for the cryocooler were determined. The

only remaining work for this project is simulation and testing of the designs.


BIBLIOGRAPHY

Barron, Randall F. Cryogenic Systems. 2nd ed. New York: Oxford University Press,1985.

Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. New York: Crane Co.,1982.

Ertas, Atila and Jesse C. Jones. The Engineering Design Process. 2nd ed. New York:John Wiley & Sons, Inc., 1996.

“Exobiology Strategy Report.” World Wide Web.

Hoffman, Stephen J. and David Kaplan, eds. Human Exploration of Mars: TheReference Mission of the NASA Mars Exploration Study Team. NASA, JohnsonSpace Center, 1997.

Incropera, Frank and David P. DeWitt. Fundamentals of Heat and Mass Transfer. NewYork: John Wiley & Sons, Inc., 1996.

Kieffer, H. H., et al., eds. Mars. Tuscon: The University of Arizona Press, 1992.

Mantell, Charles L. Engineering Materials Handbook. 1st ed. New York: McGraw HillBook Company, 1958.

Moran, Michael J. and Howard N. Shapiro. Fundamentals of EngineeringThermodynamics. 3rd ed. New York: John Wiley & Sons, Inc., 1996.

Mutch, Thomas A., et al., eds. The Geology of Mars. Princeton: Princeton UniversityPress, 1976.

National Aeronautics and Space Administration. “Mars Fact Sheet.” World Wide Web.

National Aeronautics and Space Administration. Mars Pathfinder - Weather Data.World Wide Web.

National Aeronautics and Space Administration. Scientific Results of the VikingMission.

Norton, Robert L. Machine Design: An Integrated Approach. Upper Saddle River:Prentice-Hall, 1996.

Peters, Todd. Propulsion and Fluid Systems Branch, Johnson Space Center.

Shigley, Joseph Edward and Charles R. Mischke. Mechanical Engineering Design. 5th


ed. New York: McGraw-Hill, Inc., 1989.

Strobridge, T. R. "Cryogenic Refrigerators-An Updated Survey." NBS Technical Note655 (June 1974): 12 pages.

Tillman, James E. “Mars.” World Wide Web.

Weaver, David B., Michael B. Duke, and Barney B. Roberts. Mars ExplorationStrategies: A Reference Design Mission. American Institute of Aeronautics andAstronautics, Inc., 1993.

White, Frank M. Fluid Mechanics. 3rd ed. New York: McGraw-Hill, Inc., 1994.


APPENDIX A: EQUATION DERIVATION

Vacuum Insulation Radiation Analysis

The radiant heat transfer for vacuum insulation is given in the following equation.

q F F A T Tr v e o i, ( )= −12 14 4σ Eq. 1

where

qr,v = radiant heat transfer in vacuum

Fe = emissivity factor

F12 = configuration factor = 1

σ = Stefan Boltzmann constant = 5.67x10-8 W/m2K4

A1 = surface area of inner vessel

The emissivity factor was determined using the expression shown below.

1 1 11 1

21

1 11

1 2F e eN

e e ee s s s

= + −��

�↵√+ − −

��

�↵√+ + −

��

�↵√( ) Eq. 2

where

e1 = emissivity of surface 1 (inner vessel)

e2 = emissivity of surface 2 (outer vessel)

es = shield emissivity

N = number of shields

The addition of heat shields reduced the radiant heat transfer drastically. The heat shields

used in this design were the reflective layers of the MLI.

Vacuum Insulation Gaseous Conduction Analysis

Gaseous conduction also occurs in the vacuum insulation region of the tank. The gaseous

conduction is given by the following expression.

q GpA T Tg v o i,( )= −1 Eq. 3

where


qg,v = gaseous conduction in vacuum

G = function of temperature, accommodation coefficient factor, gas constant,

specific heat ratio

p = pressure in evacuated region

G is provided in the following equation.

Gk

k

g R

TFc

a=+−

��

�↵√

11 8 π

Eq. 4

where

k = specific heat ration for gas in vacuum (air)

T = temperature of air in vacuum

Fa = accommodation coefficient factor = 1

For gaseous conduction to occur, the mean free path of the gas molecules within the

vacuum must be larger than the distance between the inner and outer vessels. The mean

free path of the gas molecules (again, air) was calculated using the following equation.

λµ π

=��

�↵√

��

�↵√

p

RT

gc2Eq. 5

where

λ = mean free path

µ = gas viscosity at T

p = absolute pressure of gas in vacuum (2 mPa was used)

Inner Vessel Stiffening Rings Analysis

As discussed in the body of this report, the inner stiffening ring was sized based upon a

required section modulus which was dependent on the bending moment occurring in the

rings. The bending moment for the inner stiffening rings is shown in the following

expressions.

For 0≤φ≤θ


θφθθθπφφφπ 2sincoscossin)(sincos5.2 ++−−+=WR

MEq. 6

For θ ≤ φ ≤ π:

θφθθφφπφπ 2sincoscossin)(cos5.2 +++−−=WR

MEq. 7

where

M = bending moment

W = weight of fluid and vessel

R = radius of vessel

φ = location

θ = location angle of supports

The maximum bending moment resulting from these expressions was used to determine

the section modulus of the rings, and the section modulus was used to determine the

dimensions of the beam from which the rings would be made.

Outer Vessel Stiffening Rings Analysis

As discussed in the body of this report, the outer stiffening ring was sized based upon a

required area moment of inertia which was dependent on the bending moment occurring

in the rings. The bending moment for the inner stiffening rings is shown in the following

expressions.

For 0 ≤φ ≤θ1:

2 22

21 2 1 2 2 1 1

M

WR

πφ θ θ θ θ π θ θ π θ θ= − + − − − + −cos (sin sin ) (cos cos ) ( )sin ( )sin Eq. 8

For θ1 ≤φ ≤θ2:

2 22

21 2 1 2 2 1 1

M

WR

πφ θ θ θ θ π θ θ π φ θ θ= − + − − − + −cos (sin sin ) (cos cos ) ( )sin sin sin Eq. 9

For θ2 ≤φ ≤π:


2 22

21 2 1 2 2 1 1

M

WR

πφ θ θ θ θ θ θ θ θ= − + − + −cos (sin sin ) (cos cos ) ( sin sin ) Eq. 10

Again, the maximum bending moments were used to calculate the area moment of inertia

of the ring which was then used to size the beam from which the rings would be made.

Suspension System Support Rod Analysis

The dynamic loadings acting on the support rods are as follows:

( )[ ]

( )[ ]

( )[ ]

( )[ ]

FN W

FN W

F N W

FN W

FN W

F N W

FN W

FN W

FN W

v

g

t

g

vt g

t

g

t

g

tt g

v

g

v

g

l

g

=+

=

=

=+ −

=− −

=

=+ +

=− −

=

( )

cos

1

2

2

2

2 1 1

2

2 1 1

2

2

1 2 1

2

1 2 1

2

1

2

1

2

θ

Eq. 11 - 19

where


F Vertical Down

F Vertical Up

F Transverse I

F Transverse I

F Transverse I

F t Transverse II

F Transverse II

F Transverse II

F Longitudinal

v

t

vt

t

t

t

v

v

l

= −= −= −= −= −= −

= −= −

=

1

2

1

2

2

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