DOT/FRA/ORD-24-12 Final Report | February 2024

Modeling and Analysis of Phase Change in a DOT-113 

Tank Car Surrogate Filled with Liquid Nitrogen

i 

REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data 

sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other 

aspect of this collection of information, including suggestions for reducing the burden, to Department of Defense, Washington Headquarters Services, Directorate for Information 

Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision 

of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE

Technical Report

3. DATES COVERED (From - To)

5/11/2021 to 5/10/2022

4. TITLE AND SUBTITLE

Modeling and Analysis of Phase Change in a DOT-113 Tank Car Surrogate Filled 

with Liquid Nitrogen 

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Michael Carolan – ORCiD: 0000-0002-8758-5739

Shaun Eshraghi – ORCiD: 0000-0002-8152-0838

Aswani Krishnamurthy – ORCiD: 0000-0003-3116-7797

5d. PROJECT NUMBER

RR28A520

5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

Volpe, the National Transportation Systems Center

55 Broadway

Cambridge, MA 02142

8. PERFORMING ORGANIZATION 

REPORT NUMBER

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

U.S. Department of Transportation

Federal Railroad Administration

Office of Railroad Policy and Development

Office of Research, Development, and Technology

Washington, DC 20590

10. SPONSOR/MONITOR'S ACRONYM(S)

11. SPONSOR/MONITOR'S REPORT 

NUMBER(S)

DOT/FRA/ORD-24-12

12. DISTRIBUTION/AVAILABILITY STATEMENT

This document is available to the public through the FRA website. 

13. SUPPLEMENTARY NOTES

COR: Francisco González, III

14. ABSTRACT

As part of a series of impact tests, the Federal Railroad Administration (FRA) sponsored a research team from 

Transportation Technology Center, working with Volpe National Transportation Systems Center (Volpe), to analyze the side 

impact puncture performance of a surrogate DOT-113 tank car filled with cryogenic liquid nitrogen (LN2). This was the third 

test in a planned series of four tests on DOT-113 tank cars and surrogates. Researchers performed pre-test analyses of the 

DOT-113 surrogate filled with LN2 (Test 12), and the research team conducted the impact test on July 24, 2021. The 

surrogate tank car was impacted by a 297,200-lb ram car fitted with a 12 x 12-inch impactor traveling at 18.3 mph. The 

impact resulted in a significant amount of deformation but did not puncture the tank car. After the test, the team updated the 

pre-test finite element (FE) model to represent the measured speed of the ram car but observed some discrepancies 

between the test measurements and simulation results. The testing and modeling effort is described in detail in a separate 

FRA Technical Report; this report is focused on a phase change discrepancy noted during Test 12.

15. SUBJECT TERMS

Impact test, DOT-113 tank car, tank car performance, transportation safety, liquefied natural gas, LNG, liquid nitrogen, LN2, finite 

element analysis, FEA, hazardous materials, rolling stock, cryogenic, phase change, thermodynamics

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 

ABSTRACT

18. NUMBER 

OF 

PAGES 

142 

19a. NAME OF RESPONSIBLE PERSON

Francisco González, III a. REPORT

Unclassified

b. ABSTRACT

Unclassified

c. THIS PAGE

Unclassified 19b. TELEPHONE NUMBER (Include area code)

202-493-6076

Standard Form 298 (Rev. 8/98)

Prescribed by ANSI Std. Z39.18
ii

METRIC/ENGLISH CONVERSION FACTORS

ENGLISH TO METRIC METRIC TO ENGLISH

LENGTH (APPROXIMATE) LENGTH (APPROXIMATE) 

1 inch (in) = 2.5 centimeters (cm) 1 millimeter (mm) = 0.04 inch (in)

1 foot (ft) = 30 centimeters (cm) 1 centimeter (cm) = 0.4 inch (in)

1 yard (yd) = 0.9 meter (m) 1 meter (m) = 3.3 feet (ft)

1 mile (mi) = 1.6 kilometers (km) 1 meter (m) = 1.1 yards (yd)

1 kilometer (km) = 0.6 mile (mi)

AREA (APPROXIMATE) AREA (APPROXIMATE) 

1 square inch (sq in, in2) = 6.5 square centimeters (cm2) 1 square centimeter (cm2) = 0.16 square inch (sq in, in2) 

1 square foot (sq ft, ft2) = 0.09 square meter (m2) 1 square meter (m2) = 1.2 square yards (sq yd, yd2)

1 square yard (sq yd, yd2) = 0.8 square meter (m2) 1 square kilometer (km2) = 0.4 square mile (sq mi, mi2)

1 square mile (sq mi, mi2) = 2.6 square kilometers (km2) 10,000 square meters (m2) = 1 hectare (ha) = 2.5 acres

1 acre = 0.4 hectare (he) = 4,000 square meters (m2) 

MASS - WEIGHT (APPROXIMATE) MASS - WEIGHT (APPROXIMATE) 

1 ounce (oz) = 28 grams (gm) 1 gram (gm) = 0.036 ounce (oz)

1 pound (lb) = 0.45 kilogram (kg) 1 kilogram (kg) = 2.2 pounds (lb)

1 short ton = 2,000 pounds (lb) = 0.9 tonne (t) 1 tonne (t) =

=

1,000 kilograms (kg)

1.1 short tons

VOLUME (APPROXIMATE) VOLUME (APPROXIMATE) 

1 teaspoon (tsp) = 5 milliliters (ml) 1 milliliter (ml) = 0.03 fluid ounce (fl oz)

1 tablespoon (tbsp) = 15 milliliters (ml) 1 liter (l) = 2.1 pints (pt)

1 fluid ounce (fl oz) = 30 milliliters (ml) 1 liter (l) = 1.06 quarts (qt)

1 cup (c) = 0.24 liter (l) 1 liter (l) = 0.26 gallon (gal)

1 pint (pt) = 0.47 liter (l)

1 quart (qt) = 0.96 liter (l)

1 gallon (gal) = 3.8 liters (l)

1 cubic foot (cu ft, ft3) = 0.03 cubic meter (m3) 1 cubic meter (m3) = 36 cubic feet (cu ft, ft3) 

1 cubic yard (cu yd, yd3) = 0.76 cubic meter (m3) 1 cubic meter (m3) = 1.3 cubic yards (cu yd, yd3)

TEMPERATURE (EXACT) TEMPERATURE (EXACT) 

[(x-32)(5/9)] °F = y °C [(9/5) y + 32] °C = x °F

QUICK INCH - CENTIMETER LENGTH CONVERSION

0 1 2 3 4 5

Inches

Centimeters 0 1 2 3 4 5 6 987 1110 1312

QUICK FAHRENHEIT - CELSIUS TEMPERATURE CONVERSIO

 -40° -22° -4° 14° 32° 50° 68° 86° 104° 122° 140° 158° 176° 194° 212°

 

°F

 °C -40° -30° -20° -10° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100°

For more exact and or other conversion factors, see NIST Miscellaneous Publication 286, Units of Weights and 

Measures. Price $2.50 SD Catalog No. C13 10286 Updated 6/17/98
iii

Acknowledgements

The authors of this report gratefully acknowledge the cooperation and assistance of Hector 

Villarreal and Devangkumar Patel of Taylor-Wharton Americas for providing engineering 

drawings and design information for the DOT-113 tank car surrogate used in the modeling 

described in this report.

Additionally, inputs from Francisco González, III of the Federal Railroad Administration were 

valuable in developing the modeling described in this report and the companion testing program. 

Calculations performed by Dr. Phani Raj in Appendix E of this report were helpful as a check on 

the FE model results incorporating vapor-to-liquid phase change. The authors are grateful for Dr. 

Raj’s technical assistance during the full-scale testing and modeling of tank cars filled with 

cryogenic liquids.
iv

Contents

Executive Summary ................................................................................................................... 1

1. Introduction ............................................................................................................. 3

1.1 Background ..............................................................................................................3

1.2 Objectives................................................................................................................5

1.3 Overall Approach.....................................................................................................5

1.4 Scope .......................................................................................................................5

1.5 Organization of the Report .......................................................................................6

2. Test 12 Results and Initial Post-test Models............................................................. 7

2.1 Test 12 Results.........................................................................................................7

2.2 Initial Post-test Phase Change Modeling...................................................................8

3. Development of a Tabulated EOS for N2 Condensation .........................................11

3.1 Approach to Modeling Phase Change ..................................................................... 12

3.2 Gas Modeling in Abaqus: Ideal Gas EOS ............................................................... 15

3.3 Liquid Modeling in Abaqus: Us-Up EOS ............................................................... 17

3.4 Gas-to-Liquid Modeling in Abaqus: Tabulated EOS............................................... 18

3.5 Development of Tabulated EOS Material Models for N2........................................ 22

4. Development of a Fluid Exchange for N2 Condensation.........................................26

5. Phase Change Model Results..................................................................................28

5.1 Tabulated EOS Model Results................................................................................ 28

5.2 Fluid Exchange Model Results............................................................................... 35

5.3 Comparison of Model Results ................................................................................ 37

6. Conclusion .............................................................................................................43

7. References..............................................................................................................46

Appendix A. Description of Non-puncture FE Model................................................................48

Appendix B. Tabulated EOS Model Results..............................................................................55

Appendix C. Pneumatic Cavity with Fluid Exchange and Tabulated EOS Model Results 

Compared 105

Appendix D. Estimated Post-impact Inner Tank Volume......................................................... 115

Appendix E. Estimated Amount of Condensation in Test 12.................................................... 119

vi 

Figure 22. Comparison of Force Versus Impactor Travel Results, 92 K (left) & 96 K (right)..... 38

Figure 23. Comparison of Outage Pressure Versus Time Results, 92 K (left) & 96 K (right) ..... 39

Figure 24. Comparison of Tank Volume Versus Time Results, 92 K (left) & 96 K (right)......... 39

Figure 25. Minimum Tank Volume from Each FE Model at Various Temperatures................... 40

Figure 26. Final Tank Volume from Each FE Model at Various Temperatures.......................... 40

1 

Executive Summary

As part of a series of impact tests, the Federal Railroad Administration (FRA) sponsored a 

research team from Transportation Technology Center, working with Volpe National 

Transportation Systems Center (Volpe), to analyze the side impact puncture performance of a 

surrogate DOT-113 tank car filled with cryogenic liquid nitrogen (LN2). This was the third test 

in a planned series of four tests on DOT-113 tank cars and surrogates, as shown in Table 1. 

Table 1. Planned DOT-113 Side Impact Test Series

Test # Date Test Article Lading Report

10 Nov. 2019 Legacy DOT-113 Water [1] 

11 June 2020 Surrogate DOT-113 Water [2] 

12 July 2021 Surrogate DOT-113 LN2 [3] 

13 May 2022 New DOT-113 LN2 TBD 

Researchers performed pre-test analyses of the DOT-113 surrogate filled with LN2 (Test 12), 

and the research team conducted the impact test on July 24, 2021. The surrogate tank car was 

impacted by a 297,200-lb ram car fitted with a 12 x 12-inch impactor traveling at 18.3 mph. The 

impact resulted in a significant amount of deformation but did not puncture the tank car. After 

the test, the team updated the pre-test finite element (FE) model to represent the measured speed 

of the ram car but observed some discrepancies between the test measurements and simulation 

results. The testing and modeling effort is described in detail in a separate FRA Technical Report

[3]; this report is focused on a phase change discrepancy noted during Test 12. 

Review of the test data from Test 12 revealed that the pressure within the tank was lower after 

impact (i.e., 26 psig) than before impact (i.e., 30 psig). The volume of the tank was reduced 

through deformation during the impact, and no venting of LN2 or gaseous nitrogen (GN2) 

occurred. The team considered two physical explanations for this decrease in pressure

accompanied by a decrease in tank volume: 1) a decrease in GN2 temperature caused by mixing 

with colder LN2 during the impact, or 2) condensation (i.e., a phase change) of GN2 into LN2 

caused by pressurization as the inner tank deformed. This report documents the technical 

challenges and modeling approaches used between Tests 12 and 13 to explain the phase change 

in a tank car filled with cryogenic LN2. 

The team modified the post-test model from Test 12 to simplify several features that were not 

deemed critical to this study. This non-puncture1 FE model was updated using the initial 

conditions from Test 12. Two different approaches for accounting for GN2 to LN2 condensation 

were independently developed using techniques native to Abaqus FE software:

1. Tabulated EOS - a material model using a tabulated equation of state (EOS) featuring a 

sharp change in the pressure versus density relationship, paired with a Lagrangian mesh

1 Non-Puncture FE models do not have the capability to puncture. This is in contrast with puncture-capable FE 

models that can tear the inner and outer tanks through ductile material damage progression and element deletion. 
2 

2. Fluid Exchange - a pneumatic cavity using the ideal gas law and a prescribed pressure 

versus flow rate “vent” allowing GN2 to escape the cavity to atmosphere once the 

saturation pressure is reached

The team also developed a third approach, which did not use FE analysis, using the Test 12 data, 

a laser scan of the inner tank, and the thermodynamic properties of GN2 and LN2 to estimate the 

amount of GN2 that condensed into LN2 during the test. This third approach was treated as a 

“check” case for comparison against the amount of GN2 that was estimated to undergo phase 

change by the two approaches implemented within the FE model.
3 

1. Introduction

As part of a series of impact tests, the Federal Railroad Administration (FRA) sponsored a 

research team from Transportation Technology Center, working with Volpe National 

Transportation Systems Center (Volpe), to analyze the side impact puncture performance of a 

surrogate DOT-113 tank car filled with cryogenic liquid nitrogen (LN2). The test was performed 

in July 2021. Before performing any further testing, the research team performed a series of 

analyses to examine the modeling techniques and effects of an apparent vapor to liquid phase 

change that occurred during the impact event. This report documents the technical challenges 

and modeling approaches used in the test to explain the phase change in a tank car filled with 

cryogenic LN2.

1.1 Background

In the past decade, significant research has been conducted to analyze and improve the impact 

behavior and puncture resistance of railroad tank cars. Ultimately, the results of this research will 

be used by federal regulatory agencies (i.e., FRA and the Pipeline and Hazardous Materials 

Safety Administration (PHMSA) in the United States) to establish performance-based testing 

requirements and to develop methods to evaluate the crashworthiness and structural integrity of 

different tank car designs when subjected to a standardized shell impact scenario. A 

performance-based requirement for tank car head impact protection has already been defined 

within the current regulations [4], and an optional performance-based requirement for tank car 

shell impact resistance is applicable to DOT-117P tank cars [5]. 

To support FRA’s ongoing research program, full-scale tests are necessary to provide the 

technical information to validate modeling efforts and to inform regulatory activities. These tests 

evaluate the crashworthiness performance of tank cars used in the transportation of hazardous 

materials, including designs that comply with current regulations as well as innovative new 

designs that may improve puncture resistance. 

A DOT-113 tank car is a specialized tank car that is designed to transport cryogenic liquids2

. A 

cryogenic liquid is defined as “a refrigerated liquefied gas having a boiling point colder 

than -90 ̊C (-130 ̊F) at 101.3 kPa (14.7 psia) absolute”3

. DOT-113 tank cars are “tank-within-atank” cars, where the inner tank is in contact with the cryogenic material and resists the pressure 

exerted by the lading, while an outer tank surrounds the inner tank, with insulation between 

them, and carries the in-train forces. The article used in this test is referred to as a “surrogate” 

DOT-113 tank car and was constructed specifically for use in this shell impact test. The 

surrogate included design features representative of a specification DOT-113 tank car, including 

typical materials of construction for the inner and outer vessels, typical diameters for the inner 

and outer vessel shells, typical thicknesses for the inner and outer vessels, and typical pressure 

relief valve (PRV) arrangements. The tanks in the DOT-113 surrogate tank car were full 

diameter but were shorter in length than an actual DOT-113. The surrogate did not include 

2 Subpart F – Specification for Cryogenic Liquid Tank Car Tanks and Seamless Steel Tanks (Classes DOT-113 and 

107A) https://www.govinfo.gov/content/pkg/CFR-2018-title49-vol3/pdf/CFR-2018-title49-vol3-part179-

subpartF.pdf

3 49 CFR 173.115(g)
4 

features required of tank cars that would not influence the puncture response during a shell 

impact (e.g., couplers, trucks, brake piping, or safety appliances). The terms “surrogate” and 

“tank car surrogate” are used interchangeably throughout this report to describe the test article. 

Further details on the design of the DOT-113 surrogate are included in the full test report [3]. 

DOT-113 tank cars include several unique design features that are not found on non-pressure 

(e.g., DOT-117) or pressure (e.g., DOT-105) tank cars because of the specific properties of 

cryogenic materials. Since the inner tank of a DOT-113 tank car will be exposed to cryogenic 

temperatures, it must be constructed of either ASTM A240 Type 304 or Type 304L stainless 

steel [7], as those grades of steel maintain desirable properties at cryogenic temperatures. Since 

the inner vessel and lading must be kept at cryogenic temperatures during transit, the inner vessel 

must be surrounded by highly effective insulation. This insulation may take the form of 

expanded perlite (i.e., a granular, lightweight, natural mineral) or multiple layers of “super” 

insulating materials (i.e., multi-layer insulation, or (MLI)). Additionally, a vacuum is typically 

used in conjunction with either perlite or MLI to further reduce heat transfer into the inner 

vessel. The specification defines a maximum rate of heat transfer that is permissible through the 

insulation system, and the inner vessel and insulation must be surrounded by an external tank to 

contain the insulation, maintain the vacuum, and carry the in-train forces.

Hazardous Materials Regulations (HMR) permit the transportation of several cryogenic liquids 

via DOT-113 tank cars, including argon and ethylene. Methane, refrigerated liquid (more 

commonly referred to as LNG), was not authorized for transportation via DOT-113 tank car prior 

to 2020. PHMSA and FRA published a Notice of Proposed Rulemaking (NPRM) in October 

2019,4 permitting LNG to be transported in DOT-113 tank cars. However, after Test 12, PHMSA 

and FRA published a second NPRM in November 2021,

5

 suspending the transportation of LNG 

in DOT-113 tank cars until further research could be conducted. The Final Rule suspending 

transportation of LNG by rail was published in September 20236. 

Because the existing fleet of DOT-113 tank cars is small compared to the overall tank car fleet 

and limited accident data exist regarding the performance of these cars in derailments or 

collisions, a series of full-scale shell impact tests was conducted to provide technical information 

on the tank car’s puncture resistance. The first test in this series (Test 10) was performed in 

November 2019 [1], and the second test (Test 11) was performed in June 2020 [2]. After the 

second test, a series of finite element (FE) analyses was conducted to transition from testing and 

modeling a tank car filled with water to one filled with LN2 [6]. The third test in this series (Test 

12) was performed in July 2021 [3]. 

4 Federal Register/Vol 84, No. 206/Thursday, October 24, 2019. 

https://www.regulations.gov/document?D=PHMSA-2018-0025-0002

5 Federal Register/Vol 86, No. 213/Monday, November 8, 2021.

https://www.regulations.gov/document/PHMSA-2021-0058-0002

6 Federal Register/Vol 88, No. 169/Friday, September 1, 2023.

https://www.federalregister.gov/documents/2023/09/01/2023-18569/hazardous-materials-suspension-of-hmramendments-authorizing-transportation-of-liquefied-natural-gas
5 

1.2 Objectives

The overall objective of the modeling program was to develop FE models capable of estimating 

the impact response and puncture behavior of the outer and inner vessels of a DOT-113 tank car 

under cryogenic conditions. The modeling described in this report focused on incorporating the 

effects of gaseous nitrogen (GN2) vapor condensation on the impact response of a DOT-113 tank 

car surrogate. The team focused on improving the capability of the post-test FE model to 

accurately represent a phase change because they anticipated that a phase change would likely 

occur during the final side impact test of a modern DOT-113 tank car, which would likely affect 

the puncture resistance of the tank car.

1.3 Overall Approach

FRA has previously conducted side impact tests on DOT-105, DOT-111, DOT-112, DOT-117, 

and DOT-113 tank cars. These tests were all accompanied by companion FE analyses and 

covered a wide range of tank car designs (e.g., capacity, shell diameter, shell thickness, vintage, 

manufacturer, outage level, outage pressure, etc.) The ultimate goal of the tank car shell impact 

testing and modeling program is to understand how a particular tank car performs under a 

standardized impact scenario that is representative of typical service conditions. Typical service 

conditions for a DOT-113 tank car include carrying a cryogenic commodity within the inner 

vessel. The team used full-scale and laboratory testing with companion FE modeling to 

understand how the tank car behaves under impact conditions and the potential for performance 

improvement through design changes. Both the testing and modeling featured increasing 

complexity to ultimately represent a DOT-113 tank car under LNG service conditions subjected 

to a shell impact that punctures both inner and outer vessels. The approach included tests and 

corresponding analyses to examine the influence of different materials/thicknesses used for the 

tank shell, examine the effect(s) of modeling both the lading and the inner vessel steel using 

properties at cryogenic conditions, and ultimately modeling a DOT-113 tank car under 

“representative” conditions expected for LNG service. 

Observations, lessons learned, and data collected during the first two impact tests of a DOT-113 

tank car and tank car surrogate were used as a starting point for the technically challenging task 

of modeling a DOT-113 surrogate filled with LN2. The major differences between this third test 

and the previous two tests in this series was the use of LN2 and the temperature of the lading and 

inner tank. A further design complexity was the consideration of puncturing one or both tanks. 

Researchers collaborated with FRA and PHMSA to determine the targeted outage, pressure, and 

temperature of the DOT-113 surrogate filled with LN2. After the targeted initial conditions were 

determined, the team conducted pre-test modeling to quantify the level of uncertainty in the 

speed necessary to cause puncture of the DOT-113 surrogate. The pre-test model results were 

used to determine a targeted impact speed for each test. 

1.4 Scope

This report includes discussion of the development and execution of the FE models used in this 

test program, including developing a tabulated EOS material model for nitrogen, examining 

various mesh techniques to implement the tabulated EOS material, developing a “fluid 

exchange” model for phase change, and calibrating the volumetric flow rate for the fluid 

exchange. This report also includes a comparison between the amount of GN2 estimated to 
6 

undergo phase change from each FE implementation compared with a calculation made using 

Test 12 data.

Researchers compared several analyses with the test results to assess whether including the 

effects of GN2 condensation improve the test-model agreement compared to the pre-test FE 

models. Further details on the Test 12 procedure, results, pre-test model development, and model 

comparison are contained within the Test 12 report [3]. 

1.5 Organization of the Report

Section 2 summarizes the results of Test 12 and describes the initial post-test model used to 

investigate the likelihood of vapor to liquid condensation during the test.

Section 3 summarizes several equation of state (EOS) material models available in the FE 

modeling software. This section also describes the development of a tabulated EOS representing 

nitrogen vapor-to-liquid condensation in the post-test FE model.

Section 4 describes how the results of the Tabulated EOS model were used to calibrate a fluid 

exchange implemented within a pneumatic cavity approach to modeling the outage in Test 12.

Section 5 presents the results of FE models run using the Tabulated EOS model and the Fluid 

Exchange model and comparisons of the results obtained using these two different approaches.

Section 6 contains concluding remarks.

Appendix A describes the non-puncture FE model of the DOT-113 surrogate filled with LN2

discussed in the report. 

Appendix B contains the results of the non-puncture FE model run using the Tabulated EOS 

model at various assumed conditions.

Appendix C contains comparisons between the results of the non-puncture FE model run using 

the Tabulated EOS model and the Fluid Exchange model at various assumed conditions.

Appendix D describes the process used to estimate the post-test inner tank volume from the Test 

12 DOT-113 surrogate.

Appendix E provides a mathematical model to calculate the rise in pressure in the outage (i.e., 

vapor) volume space of a cryogenic tank car when the volume of the inner tank is compressed 

during a side impact. 
7 

2. Test 12 Results and Initial Post-test Models

This section includes a summary of the results of Test 12 supporting the hypothesis that GN2 

condensed into LN2 during the test. A complete discussion of the results of Test 12 are found in 

the full test report [3]. 

This section also includes a summary of the Test 12 post-test FE model used to investigate the 

two different implementation techniques for phase change. Several FE results using the Test 12 

FE model, but not incorporating phase change effects, are included in this section to demonstrate 

the discrepancies that led the team to incorporate phase change effects into further post-test 

models. 

2.1 Test 12 Results

Test 12 featured a 297,200-lb impactor equipped with a 12 x 12-inch impactor impacting the 

shell of the tank car at 18.3 mph. As a result of the impact, the inner and outer tanks deformed 

but neither tank punctured. No GN2 or LN2 released from the tank through any punctures, 

piping, or the PRV. Initially, the tank was filled to approximately 95 percent of its volume with 

LN2 (i.e., a 5 percent outage). As discussed in the Test 12 report [3], vapor venting prior to the 

time of the test resulted in an estimated filling level of 91 percent (i.e., a 9 percent outage) at the 

time of the test. The initial liquid and outage conditions for Test 12 are summarized in Table 2. 

Table 2. Summary of Initial Liquid and Outage Conditions in Test

Parameter Value from Test

Commodity in Tank LN2

Commodity Temperature ~ -305°F (~86 K)

Initial Tank Volume 17,900 gal

Outage Volume ~9%

Initial Outage Pressure 30 psig

The instrumentation recorded data for 30 seconds after initial contact was made by the impactor 

on the tank shell. At the end of this period, the internal pressure within the tank had settled to 

approximately 26 psig, 4 psi lower than the initial pressure. The post-test volume of the inner 

tank was estimated to be 17,300 gallons using a post-test LiDAR scan of the inner tank 

(discussed in Appendix D). A tank that did not vent any material but experienced a reduction in 

volume would be expected to have a post-test pressure higher than the initial pressure, since the 

vapor would be compressed into a smaller volume. One explanation for the measured drop in 

pressure was that some amount of GN2 had condensed into LN2 during the test, leaving the 

remaining GN2 at a lower pressure after the tank recovered its elastic energy. The phase change 

behavior during this test was highly complex. During the impact, the volume of the tank was 

decreased through deformation. The LN2 would have sloshed through the GN2, potentially 

decreasing the outage temperature at the same time deformation was increasing pressure. Both a 

temperature drop and a pressure increase on the GN2 would tend to result in condensation. After 

the impactor was brought to a stop, the tanks would recover their elastic energy and with that, 

recover some volume. An increase in tank volume would tend to reduce the internal pressure, 

potentially resulting in some LN2 vaporizing back into GN2. 
8 

2.2 Initial Post-test Phase Change Modeling

In previously conducted tank car tests using air at atmospheric or elevated pressure and ambient 

temperature, the gas in the outage could be correctly assumed to remain a gas for the duration of 

the impact event. This assumption simplified modeling the outage in those tests, as the ideal gas 

law was applicable without additional considerations. For Test 12, the gaseous N2 in the outage 

was initially at a pressure of 30 psig. At Pueblo, CO’s elevation (approximately 4,700 feet above 

sea level [8]), this corresponded to an absolute pressure of 42.3 psia [9]. At this pressure, the 

N2’s saturation temperature was approximately 87.6 K [10]. An increase in pressure also results 

in an increase in the saturation temperature (see the saturation curve in Figure 5). If the pressure 

of the gas were increased such that the temperature was now below the saturation temperature at 

the new pressure, the gas could experience a phase change by condensing into a liquid. A vaporto-liquid phase change could also occur if the pressure of the gas remained the same but the 

temperature of the gas dropped below the saturation temperature. 

A simple solution would be to keep the gas at a temperature that is much higher than the 

saturation temperature at the initial pressure of 42.3 psia. An elevated temperature of the gas 

would require a more substantial decrease in temperature or increase in pressure to cross the 

saturation curve. However, it is not necessarily possible to maintain equilibrium between GN2

and LN2 with such a large difference in temperature, as the LN2 must be kept below the 

saturation temperature at a given pressure to remain in the liquid state.

Prior to Test 12, a modeling study was conducted to examine adapting existing modeling 

techniques used for water and air-filled tank cars to GN2 and LN2 [6]. The conclusions made in 

this modeling study were then incorporated into the Test 12 pre-test models, which are 

documented in the Test 12 report [2]. The Test 12 report also documents the initial post-test 

modeling that was conducted using test conditions which led to the determination that additional 

modeling techniques for capturing the effects of GN2-LN2 condensation were needed. The posttest models from Test 12 are briefly summarized below.

The Test 12 pre- and post-test models were run using two limiting behaviors assumed for the 

GN2 in the outage. Assuming that GN2 is initially at its saturation temperature and that its 

temperature remains constant, the process of compressing the gas should be isobaric (i.e., 

occurring at a constant pressure). On the other hand, if the gas were above its saturation 

temperature it would not undergo a phase change unless the pressure increased above the 

saturation pressure at the gas’s temperature. If the GN2 remained at a constant temperature while 

being compressed, the process would be isothermal. Figure 1 contains the force-displacement 

responses from Test 12 and the post-test models that used an isothermal or isobaric assumption 

for the GN2.

These modeling efforts provided the upper and lower bound estimates of the response during the 

test. The isobaric model featured lower forces and a higher maximum impactor travel than the 

Test 12 data, while the isothermal model featured higher forces and a slightly lower maximum 

impactor travel than the Test 12 data. These outcomes implied that the actual behavior in Test 12 

was somewhere between these two extremes. 

While not being conclusive, these model results support the hypothesis that some of the GN2 in 

Test 12 condensed into LN2 during the test. If the GN2 were initially at a temperature above its 

saturation temperature at the initial pressure, then some increase in pressure would be possible as 

the impact occurred. This behavior is consistent with the isothermal response. However, the 
9 

isothermal model overpredicted the force measured during the test. If the pressure increased 

during the test but then reached the saturation pressure of the GN2, continued pressure increase 

would be prevented while the phase change occurred. In other words, the GN2 would initially 

behave more like the isothermal approximation, but once reaching the saturation pressure would 

begin to behave more like the isobaric approximation.

Figure 1. Force-displacement Responses from the Isothermal and Isobaric Post-test Models 

Compared to Test Result (Image from [2]) 

As a test of this hypothesis, a simplified Fluid Exchange model was developed and incorporated 

into the Test 12 post-test FE model. The purpose of this first investigation was simply to explore 

whether modeling this supposed isothermal/isobaric behavior could lead to closer test-model 

agreement. As the Test 12 FE model used a pneumatic cavity for the GN2, a volumetric flow rate 

versus pressure behavior was developed to simulate GN2 being “lost” due to phase change. This 

initial model was calibrated to have zero flow rate out of the cavity at pressures below 70 psia, 

and a prescribed volumetric flow rate versus pressure if pressure exceeded 70 psia. The value of 

70 psia was chosen based on review of the Test 12 pressure data and the isothermal FE model 

and cannot be considered model validation. As seen in Figure 2, the post-test model 

incorporating volumetric flow as a stand-in for phase change was in better agreement with both 

the maximum force and maximum impactor travel measured during Test 12.

Based on the improved correlation between the Test 12 measurements and the results of this first 

attempt at representing phase change within the FE model, the team undertook a further effort at 

developing a phase change material model for GN2 at a temperature above its saturation 

temperature. Since phase change occurred during Test 12, and the next planned test would use 

substantially similar conditions (e.g., LN2, outage volume, initial pressure), researchers 

reasonably concluded that phase change could also be expected to occur in Test 13. The pre-test 

models developed for Test 13 should therefore account for phase change using defensible 

modeling techniques that did not depend on using test measurements to calibrate the model 

response. 
10 

Figure 2. Force-displacement Response from Post-test Model Saturated at 70 psia (57.7 

psig) Compared to Test Result (image from [2]) 
11 

3. Development of a Tabulated EOS for N2 Condensation

The FE models developed in support of Test 12 all included a representation of the LN2 partially

filling the tank and the pressurized GN2 in the outage space. The results indicated that some 

amount of GN2 likely condensed into LN2 during the test. Since the follow-up test (Test 13) 

would also use a DOT-113 tank filled with pressurized LN2 and GN2, the team needed to 

develop a modeling technique that could predict the onset of phase change, which was expected 

to limit the pressure rise that could occur within the tank, and in turn would affect the forces 

developed between the impactor and the tank car and ultimately puncture. 

The team first examined the EOS implementations available within the Abaqus software. The 

Abaqus documentation states that EOS models “determine the pressure (positive in compression) 

as a function of the density, ρ, and the specific energy (the internal energy per unit mass), Em” 

[9]. A second approach used the results of models run using the EOS implementation to calibrate 

a volumetric leakage rate. This then could be incorporated into a pneumatic cavity representing 

the outage using a fluid exchange between the cavity and atmosphere.

The team used a non-puncture version of the Test 12 FE model (shown in Figure 3 with mesh 

and constraints visible), which is a simplified version of the model developed for Test 12 (based 

on a working draft of the CAD model from the manufacturer). The stub sills on this model were

reversed (i.e., the tapered end should be located inboard of the bolster), but this inconsistency 

was not expected to have a significant effect on the fluid response within the tank. The Test 12 

FE model was modified to remove unnecessary details, including the ability to puncture. The 

DOT-113 surrogate in Test 12 did not puncture, so a non-puncture model should be capable of 

reproducing the test results without a loss of confidence. The FE model used in the development 

of the phase change GN2 material model is described in Appendix A. 

Figure 3. Non-puncture Test 12 Model

Two different approaches to modeling the GN2 were studied. In the first approach, a GN2 

material model was developed and applied to a mesh of Lagrangian, tetrahedral elements filling 

the outage. The resulting meshes for LN2 (bottom) and GN2 (top) used in that portion of the 
12 

study are shown in Figure 4. A tied constraint connecting the bottom of the GN2 with the top of 

the LN2 is also visible in this figure.

Figure 4. Lagrangian Meshes for Lading and Outage

In a second approach to modeling GN2, the team replaced the GN2 material model and 

Lagrangian mesh with a pneumatic cavity. The cavity was assigned a prescribed leakage rate

using a fluid exchange between the cavity and atmosphere that was calibrated based on the 

saturation properties of the GN2 at the chosen temperature and the results of the corresponding 

EOS model. The team continued to model the LN2 using a Lagrangian mesh and the same 

material model used in the GN2 material model. This second approach is described in Section 4. 

3.1 Approach to Modeling Phase Change

A phase change is a complicated thermodynamic process. Implementing a numerical (e.g., FE) 

model of a phase change is an additional complication, as there are limitations to any 

commercially available software package. The goal was not to faithfully reproduce and account 

for all of the potential thermodynamic processes that governed phase change, but to develop a 

material model that accounted for the effects of the GN2 to LN2 condensation that was believed 

to occur during Test 12. Researchers employed a set of guidelines to aid in developing the 

techniques used to simulate phase change:

1) The techniques would be implemented in the Abaqus/Explicit [9] FE solver, as that was 

the solver used with the original Test 12 FE model.

2) The techniques only needed to represent condensation between vapor and liquid, as that 

was the phase change of interest during the test. Liquid to vapor evaporation or any phase 

change involving the solid state were outside the test scope.

3) The techniques should not assume that the vapor was initially saturated.

4) The techniques should be implemented using built-in features of the Abaqus FE software 

without the need to develop any user subroutines.

5) The techniques should not require complete redevelopment of the existing Test 12 FE 

models. 

6) The techniques should not result in a substantial increase in model runtime compared to 

existing Test 12 FE models.
13 

7) The techniques should be capable of representing phase change under a variety of 

unknown initial conditions, such as varied initial pressure and temperature.

8) The techniques should be adaptable to cryogenic liquids beyond just LN2. Test 12 used 

LN2 as its lading but future modeling was anticipated using other cryogenic liquids.

Figure 5 shows the saturation curve for N2 using data from the National Institute of Standards 

and Technology (NIST) [10]. The vertical axis shows the saturation pressure and the horizontal 

axis shows the saturation temperature. LN2 can only exist at pressure/temperature combinations 

on or above the saturation curve (e.g., at high pressures or low temperatures) and GN2 can only 

exist at pressure/temperature combinations on or below the saturation curve (e.g., at low 

pressures or high temperatures). Liquid and vapor can exist simultaneously for combinations of 

pressure and temperature on the saturation curve. 

Figure 5. Saturation Curve for N2 with Pressures of Interest Noted

Figure 5 also shows two horizontal dashed lines indicating two relevant pressures from Test 12.

The initial pressure of the inner tank of the tank car was consistently measured by several 

pressure transducers and mechanical gauges as 30 psig, or 42.3 psia [2]. While the exact 

temperature of the vapor in the outage was not measured during the test, knowing the initial 

pressure of 42.3 psia limits the possible temperatures at which GN2 could have existed. As 

shown on Figure 5, a pressure of 42.3 psia corresponds to a saturation temperature of 

approximately 87.6 K. For GN2 to exist at a pressure of 42.3 psia, its temperature must be no 

lower than 87.6 K. This property sets a lower limit to the temperature range of interest for any 

material model capturing GN2 phase change under Test 12 conditions.

An upper limit to the temperature range of interest can be estimated from the design of the tank 

itself. The tank used in Test 12 included a PRV with a start-to-discharge pressure (STDP) of 75 

psig, (i.e., 87.3 psia at Pueblo, CO’s elevation). If the pressure of the GN2 reached 87.3 psia 

during the test, the tank would begin to vent as designed. This PRV also effectively limits the 
14 

pressure range over which phase change is likely. Any attempt to pressurize the GN2 above the 

STDP results in activation of the PRV. N2’s saturation temperature is approximately 96.4 K at 

the STDP of the PRV used in Test 12. While GN2 can and does exist at temperatures above 96.4 

K, it was not necessary to develop a phase change EOS model for any greater temperatures 

assumed for Test 12. If the saturation temperature were above 96.4 K the PRV would activate 

and reduce pressure in the tank before the saturation pressure could be reached7

. 

Density versus pressure relationships for N2 at fixed temperatures (i.e., isotherms) between 88 K 

and 100 K are shown in Figure 6. These relationships were developed using data from the NIST 

[10] converted into the unit system used in the Test 12 FE model. The dashed vertical lines 

represent the saturation pressure at a given temperature. Densities below the dashed line

correspond to GN2 and the higher densities correspond to LN2. At 98 K and 100 K, the Test 12 

tank car would be expected to begin to vent at a lower pressure than the pressure needed to 

initiate phase change. These temperatures were included in this study to examine the sensitivity 

of the model’s response to a range of assumed initial temperatures. 

Figure 6. Isothermal Density Versus Pressure for Nitrogen (Data from NIST [10]) 

Researchers believed the sudden change in N2 density at the saturation pressure was an essential 

feature of a phase change implementation in Abaqus. For a given temperature, pressures below 

the saturation pressure would result in the gas behaving like a real gas. As the pressure increased, 

the GN2’s density would increase gradually. If the saturation pressure was reached, any further 

attempt to compress the GN2 would cause an abrupt increase in density. In a numerical 

implementation, this sudden increase in density would be achieved by a sudden reduction in 

element volume, while the mass of a Lagrangian element typically remains constant throughout 

7 In theory, a very rapid buildup of pressure that exceeds the PRV’s flow rate could occur. In this case, pressure 

within the tank could exceed the STDP of the PRV despite the PRV actively venting.











26 

4. Development of a Fluid Exchange for N2 Condensation

In pre- and post-test side impact models of the DOT-113 surrogate filled with LN2 (Test 12) [3], 

the team used a pneumatic cavity in Abaqus/Explicit to represent the pressure-volume 

relationship of the GN2 in the outage. In the pre-test models, researchers limited the pneumatic 

cavity to have either 1) no phase change and a pressure-density relationship according to the 

ideal gas relationship with a constant temperature (i.e., isothermal) or 2) only phase change and 

no pressure rise (i.e., isobaric). This approach was taken to bound the possible test outcomes 

given numerous uncertainties in the temperature and pressure of the lading before filling. The 

two bounding cases from the pre-test model succeeded in bounding the test outcome (see Figure 

1). 

After Test 12, the team added a fluid exchange to the pneumatic cavity in Abaqus to allow 

pressure rise to a certain point (i.e., the saturation pressure) and then mass from the pneumatic 

cavity was allowed to exit to atmosphere as a surrogate for a vapor-to-liquid phase change. 

While the mass from the vapor should flow into the liquid, the team determined that the mass of 

vapor condensing to liquid was insignificant compared to the total mass of liquid inside the tank. 

Based on review of the test results, researchers calibrated a fluid exchange starting at 70 psia

which gave good agreement with the test results (see Figure 2). The volumetric flow rate versus 

pressure definition used in the post-test model is shown in Figure 9. 

Figure 9. Fluid Exchange Property from Post-test 12 Model of DOT-113 Surrogate Filled 

with LN2

The team calibrated the fluid exchange definition to match the behavior observed in the test and 

demonstrate that pressure rise followed by condensation could explain the force-displacement 

and pressure-time results observed after the test. Note that this approach does not conserve mass 

in the outage as the process is non-reversible (i.e., fluid does not flow back into the cavity after 

the pressure drops). However, it was determined that this was acceptable for simulating an 

impact test as puncture of the tank car was not likely to occur after the compressed volume of the 

inner tank began to recover.

After developing a non-puncture model with a Lagrangian Tabulated EOS representation of the 

outage (Section 3), the team updated the fluid exchange definition based on the results of the 

Lagrangian Tabulated EOS model. There were several advantages to using a simple pneumatic 

cavity versus a Lagrangian Tabulated EOS model, including faster model runtime and better 
27 

model stability. To calibrate fluid exchange definitions for the pneumatic cavity from the 

Lagrangian Tabulated EOS results, the team post-processed density, volume, and pressure from 

each element in the outage of the Tabulated EOS model at 88 K, 90 K, 92 K, 94 K, 96 K, 98 K, 

and 100 K. Researchers used element density and volume to compute the total mass of GN2 

versus LN2 in the outage at each time point, as shown in Appendix B9. 

The team then computed the mass rate of change of GN2 to LN2, then converted the mass rate of 

change to volumetric flow rate for the fluid exchange definition by dividing the mass flow rate

by the saturation density of GN2 at the model temperature. Finally, the team plotted the 

volumetric flow rate against the average element pressure in the outage and fit a simple 

piecewise bi- or tri-linear curve through the results. Figure 10 shows an exemplar fit of a 

piecewise linear curve over the Tabulated EOS model result at 94 K. 

Figure 10. Fluid Exchange Piecewise Linear Fit over Tabulated EOS Results at 94 K

Figure 11 shows the resulting piecewise linear fluid exchange definitions for each temperature 

obtained using the same procedure. The team used the volumetric flow rate versus pressure 

behaviors shown in Figure 11 to simulate GN2 to LN2 condensation at various GN2 

temperatures in the FE model results reported in Section 5.2. 

Figure 11. Volumetric Flow Rate Versus Pressure for Various Temperatures, Pneumatic 

Cavity with Fluid Exchange Model







35 

5.2 Fluid Exchange Model Results

The impactor force versus impactor travel responses for the Fluid Exchange model at each 

temperature are shown in Figure 18. This figure also shows the average force-travel response 

measured in Test 12. In general, the force-travel responses are all alike. As the assumed initial 

GN2 temperature increased, the peak force estimated by the model also increased while the 

amount of indentation decreased. 

Figure 18. Impactor Force Versus Impactor Travel for Fluid Exchange Models at Various 

Temperatures

Figure 19 contains a plot of the average pressure-time history in Test 12 alongside the average 

pressure-time history from each Fluid Exchange model. The pneumatic cavity modeling 

technique results in a single average pressure over the entire volume of the cavity at each time 

increment. As the assumed initial temperature of the GN2 increased the peak pressure within the 

outage also increased. The final pressure also tended to be lower with lower initial temperatures, 

as more of the GN2 left the cavity and “condensed” into LN2 as the initial temperature was 

decreased. 

Figure 19. Pressure-time Histories for Fluid Exchange Models at Various Temperatures 
36 

The resulting tank volume versus time histories from the model run using each Fluid Exchange 

model are shown in Figure 20. The dashed line represents the nominal 17,900 gallon initial 

volume. 

Figure 20. Inner Tank Volume-time Histories for Fluid Exchange Models at Various 

Temperatures

Regardless of the assumed initial temperature of GN2 modeled, the overall shapes of the 

volume-time responses were similar. The tank volume decreased as the impact occurred. The 

volume reached a minimum, then began to increase again. The minimum tank volume 

corresponded with the approximate time the peak force was reached. As the tank recovered its 

elastic energy and the compressed GN2 expanded, its volume increased until reaching a steady 

value between the initial and the minimum value. 

Table 9 contains a summary of the amount of LN2 to be condensed by mass percent at each 

temperature. Results are reported at two different times. The first column of mass data was 

obtained at the time when the maximum pressure was reached in the cavity. The second column 

was the maximum amount of condensed LN2 that occurred during the simulation. Both sets of 

values were obtained by requesting the mass of GN2 in the pneumatic cavity (CMASS) as a 

function of time for each simulation. The mass of “condensed” LN2 was set equal to the 

difference between the initial mass of GN2 in the cavity and the mass at any subsequent point in 

time during the simulation. The percentage of LN2 is the value of this mass divided by the initial 

mass of GN2 in the cavity.

Table 12. Peak GN2 and LN2 Amounts in Outage, Fluid Exchange Models

Temperature (K) Amount of Condensed LN2 at Peak Pressure

(% of Initial GN2 Mass)

Maximum Amount of Condensed LN2

(% of Initial GN2 Mass)

88 51.2% 89.5%

90 39.3% 75.8%

92 27.7% 65.1%

94 21.3% 55.1%

96 17.1% 40.8%

98 10.0% 24.3%

100 4.7% 11.9%

38 

Figure 21 shows the force-time histories of the three FE models and Test 12 at 92 K on the left 

and 96 K on the right. At either temperature, the two models using the Tabulated EOS produced 

nearly identical results to one another. Additionally, the Fluid Exchange approach agreed with 

both Tabulated EOS results during the initial portion of the impact. After approximately 0.2 

seconds at 92 K or 0.24 seconds at 96 K, the pneumatic cavity with fluid exchange model 

exhibited a softer response than the Tabulated EOS models. At either temperature, the pneumatic 

cavity with fluid exchange approach to modeling produced a lower global peak force than the 

Tabulated EOS models.

Figure 21. Comparison of Force Versus Time Results, 92 K (left) & 96 K (right)

Figure 22 contains similar plots of impactor force versus travel at 92 K on the left and 96 K on 

the right. At either temperature, the Tabulated EOS models produced a lower maximum impactor 

travel than the pneumatic cavity with fluid exchange approach to modeling.

Figure 22. Comparison of Force Versus Impactor Travel Results, 92 K (left) & 96 K (right)

Figure 23 contains plots of average outage pressure versus time at 92 K on the left and 96 K on 

the right. The peak values of outage pressure is similar in all three models. The Fluid Exchange 

model produced a quicker drop in pressure following the peak compared to the Tabulated EOS 

models. 
39 

Figure 23. Comparison of Outage Pressure Versus Time Results, 92 K (left) & 96 K (right)

Figure 24 contains plots of tank volume versus time at 92 K on the left and 96 K on the right. 

The minimum value of outage volume is similar in all three models for a given temperature. The 

Fluid Exchange model produced a slower volume recovery compared to the Tabulated EOS 

models. 

Figure 24. Comparison of Tank Volume Versus Time Results, 92 K (left) & 96 K (right)

Figure 25 contains a plot of the minimum volume of the inner tank during each impact 

simulation from all three models run at each temperature. All the models exhibited the same 

trend, where increasing temperatures exhibited a larger minimum volume. The three modeling 

techniques also showed reasonable agreement with one another at all temperatures.

Figure 26 contains a similar plot of the final tank volume following each simulated impact. The 

final volume at a given temperature was similar between the two EOS models, but the pneumatic 

cavity with fluid exchange model exhibited a lower final volume. The difference between final 

volumes between the Tabulated EOS models and the Fluid Exchange model was generally larger 

than the difference between the minimum volumes in the different models. 
40 

Figure 25. Minimum Tank Volume from Each FE Model at Various Temperatures

Figure 26. Final Tank Volume from Each FE Model at Various Temperatures

This difference could be explained by the reversibility of each modeling technique. In the 

Tabulated EOS approach, the LN2 created from condensing GN2 could re-vaporize as the 

pressure reduced within the tank following the impact. As the LN2 re-vaporized, the tank would 

recover more of its volume. In the Fluid Exchange approach, any GN2 that underwent “phase 

change” was in fact vented from the inner tank. This material was no longer in the tank and 

could therefore not contribute to the tank’s volume recovery during impact. In this respect, the 

Tabulated EOS could be thought to represent a phase change model that was completely 

reversible, as no latent heat needs to be supplied or removed for a phase change to occur. The 

Fluid Exchange with volumetric leakage could be thought of as an irreversible phase change, 

where phase change could only happen in one direction.



44 

temperature. The Fluid Exchange approach tended to estimate a higher mass of GN2 condensing 

to LN2 at a given temperature. Both the Fluid Exchange and Tabulated EOS produced masses of 

condensed LN2 that were consistent with the results of a simplified, non-FE estimation of the 

mass of LN2 undergoing condensation. 

The range of temperatures examined using either the Tabulated EOS approach or the pneumatic 

cavity with fluid exchange approach was small, from 88 K to 100 K (-301.3 to -279.7 °F). 

However, the results of the simulations demonstrate that the GN2 temperature can have a 

significant effect on the amount of GN2 that condenses into LN2. The amount of condensing 

GN2 and the pressure at which it condenses affects the peak force reached during the simulation. 

While the models used in this study were not capable of simulating puncture of the inner or outer 

tank, the models are useful in studying the global force-displacement response of the cryogenic 

surrogate tank car. Once the models are refined, they can be used in full-scale puncture-capable 

models. 

Considering the number of inter-related variables affecting the response of the surrogate tank car

due to the complexity of LN2 and GN2 behaviors and the phase change that can occur due to 

changes in pressure and/or temperature, this effort captured the measured responses reasonably 

well. The modeling described in this report is a starting point toward future modeling of fluidstructure interaction of a cryogenic fluid undergoing a condensation phase change. As the 

technology progresses and FE software that can better represent these kinds of complex 

behaviors and interactions is developed, the level of correlation can be further improved and 

further confidence may be developed in FE models representing the behavior of the physical 

system being simulated. This is important because the same FE model can be used to analyze 

other operating conditions, different lading, different materials, and different thicknesses and 

dimensions for the tank car shells. 

At the time the phase change modeling study was performed, a future impact test (Test 13) was 

planned within FRA’s testing program. Test 13 was to feature a newly constructed DOT-113 

tank car filled with LN2 and pressurized GN2. Based on the results of this study, pre-test models 

used to plan for Test 13 incorporated GN2 to LN2 condensation. Additionally, Test 13 

instrumentation included additional sensors (e.g., thermocouples) to specifically measure the pretest LN2 and GN2 temperatures to accurately model the initial state of the fluid within the inner 

tank. 

While not examined in this study, the Tabulated EOS material model coupled with a Lagrangian 

mesh offers the opportunity to spatially vary the temperature within the GN2. While the 

Tabulated EOS materials are all assumed to be isothermal, several different material models can 

be assigned to the elements defining the GN2. For example, the bottom of the GN2 could be 

assigned a lower temperature material model than the top portion if the GN2 was thought to have 

stratified. While the temperature of each material would remain constant during the simulation 

and there would be no mixing of material, this approach would result in partial phase change 

occurring at two different pressures as each material reached its saturation pressure.

Finally, the methods described in this report are not limited to use only with LN2 and GN2. A 

similar approach could be used to develop Tabulated EOS and/or Fluid Exchange models of 

other cryogenic liquids of interest if suitable material data were available. The range of 

temperatures and pressures likely to be encountered in transportation and their relationship to the 

saturation pressures and temperatures of a different cryogenic liquid would determine whether 
45 

phase change behavior should be included in any future models that may use a cryogenic liquid 

other than LN2.
46 

7. References

[1] Trevithick, S., Carolan, M., Eshraghi, S., & Wilson, N. (2021). Side Impact Test and 

Analyses of a Legacy DOT-113 Tank Car (Report No. DOT/FRA/ORD-21/28). Federal 

Railroad Administration. 

[2] Wilson, N., Carolan, M., Trevithick, S., & Eshraghi, S. (2021). Side Impact Test and 

Analyses of a DOT-113 Surrogate Tank Car with Water (Report No. DOT/FRA/ORD21/35). Federal Railroad Administration. 

[3] Belport, S., Carolan, M. E., Trevithick, S., Eshraghi, S., & Krishnamurthy, A. (2022). Side 

Impact Test and Analyses of a DOT-113 Surrogate Tank Car with Cryogenic Lading

(Report No. DOT/FRA/ORD-22/33). Federal Railroad Administration. 

[4] Pipeline and Hazardous Materials Safety Administration (2015). Tank-head punctureresistance systems (§ 179.16). Department of Transportation. 

[5] Pipeline and Hazardous Materials Safety Administration (2015). Performance standard 

requirements (§ 179.202–12). Department of Transportation. 

[6] Eshraghi, S., Krishnamurthy, A., & Carolan, M. (2022). Finite Element Analyses of Side 

Impacts to DOT-113 Surrogate Tank Cars with Water and Liquid Nitrogen (Report No. 

DOT/FRA/ORD-22/22). Federal Railroad Administration. 

[7] ASTM International (2019). ASTM A240/A240M: Standard Specification for Chromium 

and Chromium-Nickel Stainless Steel Plate, Sheet, and Strip for Pressure Vessels and for 

General Applications. Conshocken, PA. 

[8] Colorado Department of Transportation (2022). Pueblo Memorial Airport - Colorado 

Department of Transportation. Accessed 2022 8 December.

[9] The Engineering Toolbox (2003). Air - Density and Specific Volume vs. Altitude. 

Accessed 17 March 2020.

[10] National Institute of Standards and Technology (2023). NIST Standard Reference Database 

Number 69. 

[11] Dassault Systèmes Simulia Corporation (2019). Abaqus 2019. 

[12] Yu, H., Tang, Y. H., Gordon, J., & Jeong, D. Y. (2009). Modeling the Effect of FluidStructure Interaction on the Impact Dynamics of Pressurized Tank Cars. Proceedings of 

IMECE 2009 - 2009 ASME International Mechanical Engineering Congress and 

Exposition. Lake Buena Vista, FL. 

[13] Rakoczy, P., Carolan, M., Eshraghi, S., & Gorhum, T. (2019). Side Impact Test and 

Analyses of a DOT-117 Tank Car (Report No. DOT/FRA/ORD-19/13). Federal Railroad 

Administration. 
47 

[14] The Engineering ToolBox (2004). Universal and Individual Gas Constants. Accessed 12 

March 2020.

[15] Livermore Software Technology Corporation (2018). LS-PrePost 4.6 x64. 

[16] The Engineering ToolBox (2003). Water - Density, Specific Weight and Thermal 

Expansion Coefficients. Accessed 12 March 2020.

[17] The Engineering ToolBox (2004). Water - Speed of Sound vs. Temperature. Accessed 2 

November 2021.

[18] Strobridge, T.R. (1962). The Thermodynamic Properties of Nitrogen from 64 to 300° K, 

0.1 to 200 Atmospheres. NBS Technical Note 129, Boulder Laboratories, National Bureau 

of Standards. 
48 

Appendix A. 

Description of Non-puncture FE Model

Following Test 12, a non-puncture model of the DOT-113 surrogate tank car was analyzed with 

a 9 percent N2 outage initially at 30 psig under an 18.3 mph impact condition. The non-puncture 

model is shown in Figure A1. The model contained all the features of the puncture-capable 

model used to estimate the speed to cause puncture for Test 12 except that it did not have 

puncture-capable “solid patch” regions in the outer and inner vessels. The exclusion of very 

small solid elements allowed the model to run to completion approximately eight times faster. 

This allowed for a series of models to be run examining two different approaches to simulating 

condensation of GN2 to LN2 during the impact event. 

Figure A1. DOT-113 Surrogate Non-puncture FE Model

Figure A2 shows a cut view centered on the longitudinal impact plane of the DOT-113 surrogate 

non-puncture FE model that used the pneumatic cavity GN2 modeling approach. Figure A3

shows a similar cut view of the FE model that used the Tabulated EOS GN2 modeling approach. 

Within the inner vessel, the liquid lading (grey) was represented as a Lagrangian EOS and is 

described in Appendix A.5. The gas phase was represented using either a Lagrangian mesh of 

elements with a Tabulated EOS material model (Appendix A.3) or a pneumatic cavity with a 

prescribed leakage (Appendix A.4).
49 

Figure A2. DOT-113 Surrogate Non-puncture FE Model Cut View, Pneumatic Cavity

Figure A3. DOT-113 Surrogate Non-puncture FE Model Cut View, Tabulated EOS



53 

A.4. Gaseous Nitrogen – Pneumatic Cavity with Fluid Exchange

In the second approach, a pneumatic cavity modeling technique was implemented within 

Abaqus, with a Fluid Exchange defined between the cavity and atmosphere. This Fluid 

Exchange featured a calibrated volumetric flow rate versus pressure relationship to mimic the 

effects of GN2 condensing into LN2. For pressures below a threshold value within the cavity, the 

GN2 behaved according to the ideal gas law. If the impact caused the pressure to increase to the 

threshold value, the fluid exchange allowed GN2 to exit the cavity, approximating the reduction 

in GN2 volume as it condensed into LN2. This approach does not account for the resulting 

volume increase of LN2 within the tank. This is a limitation of the pneumatic cavity modeling 

approach but was assumed by the modelers to be an inconsequential approximation. 

The pneumatic cavity modeling technique within Abaqus requires a surface to be defined that 

encloses the cavity. A reference point must also be defined within this cavity to which initial 

temperature and pressure can be assigned. 

The pneumatic cavity approach models the entire cavity with a single uniform pressure and 

uniform temperature value. The GN2 pressure within the model was allowed to change as the 

volume of the tank changed due to the impact while the temperature was held constant. The 

modeling inputs defined for the pneumatic cavity representation of the GN2 phase of the model 

are summarized in Table A2. This table includes the unit system used in the FE model.

Table A2. Properties for Nitrogen (Pneumatic Cavity)

Property Value Reference

Universal Gas Constant (R) 73.583 in-lbf/(mol-K) [13]

Molecular Weight (MW) 1.60 x 10-4 lbf⋅s2/(in⋅mol) [10]

Constant Temperature 88, 90, 92, 94, 96, 98, or 100 K

Molar Specific Heat (cp,m) 257.59 in-lbf/(mol-K) [10]

Initial Pressure 42.3 psia (30 psig)

The molar specific heat capacity at constant pressure (cp,m) for GN2 was calculated according to

Equation A1. 

Equation A1. Calculation of Molar Specific Heat

A.5. LN2

The DOT-113 surrogate in Test 12 had an initial outage volume of 9 percent and internal 

pressure of 30 psig. The liquid phase of the lading was modeled as LN2 at a temperature of 86 K. 

Within Abaqus, the Us-Up EOS model was used to describe the behavior of LN2 and applied to 

a Lagrangian mesh. The fluid was meshed with 3 inch fully integrated brick (C3D8) elements. 

The key material properties that must be input to this material model are the material’s density, 

dynamic viscosity, and bulk modulus. The bulk modulus can be calculated from the density and 

speed of sound as shown in Equation A2 The material properties were obtained from NIST
54 

Standard Reference Database Number 69 [10] at a constant pressure of 42.3 psia and temperature 

of 86 K. 

Equation A2. Calculation of Bulk Modulus

Table A3 summarizes the properties of the LN2 included in the DOT-113 surrogate FE model. 

The same material model was used in the models run using the Tabulated EOS or the pneumatic 

cavity approaches to modeling the outage. Note that the mesh size was changed between the two 

modeling approaches to maintain a similar element size between the Lagrangian meshes used for 

LN2 and GN2. These tables include the unit system used in the FE model.

Table A3. Summary of Properties for LN2

Parameter Value

Mass Density 7.16 x 10-5 lbf⋅s2/in4

Initial Pressure 42.3 psia

Initial Temperature 86 K

Sound Speed 3.00 x 104 in/s

Dynamic Viscosity 1.6987 x 10-8 psi⋅s

Bulk Modulus 6.44 x 104 psi

Mesh Implementation 3-inch Fully Integrated Brick (C3D8) Elements (Pneumatic Cavity)

5-inch C3D8 Elements (Tabulated EOS)

Distortion Control Yes, length ratio = 0.1 (default)

A.6. Membrane

An artificial surface was modeled within the inner tank to define the limits of the pneumatic 

cavity (i.e., outage). Frictionless, hard contact was assigned between the membrane and the inner 

vessel and lading. Because this surface does not correspond to any physical feature within the 

tank, modeling techniques were chosen to minimize the increase in either mass or stiffness 

introduced into the model by the membrane while not negatively impacting the FE model’s 

stability or runtime. A membrane was not included in the models using the Tabulated EOS 

approach to modeling GN2. The material properties of the membrane are summarized in Table 

A4. This table includes the U.S. customary units used in the FE model. 

Table A4. Material Properties Defined for Membrane Material

Parameter Value

Density 7.35 x 10-6 lbf-s2

/in4 

Modulus of Elasticity 1.0 x 104 psi


















































105 

Appendix C.

Pneumatic Cavity with Fluid Exchange and Tabulated EOS Model 

Results Compared

Figure C1. Comparison of Force Versus Time Results, 88 K 

Figure C2. Comparison of Force Versus Impactor Travel Results, 88 K 
106 

Figure C3. Comparison of Pressure Versus Time Results, 88 K 

Figure C4. Comparison of Tank Volume Versus Time Results, 88 K 
107 

Figure C5. Comparison of Force Versus Time Results, 90 K 

Figure C6. Comparison of Force Versus Impactor Travel Results, 90 K 
108 

Figure C7. Comparison of Pressure Versus Time Results, 90 K 

Figure C8. Comparison of Tank Volume Versus Time Results, 90 K 
109 

Figure C9. Comparison of Force Versus Time Results, 94 K 

Figure C10. Comparison of Force Versus Impactor Travel Results, 94 K 
110 

Figure C11. Comparison of Pressure Versus Time Results, 94 K 

Figure C12. Comparison of Tank Volume Versus Time Results, 94 K 
111 

Figure C13. Comparison of Force Versus Time Results, 98 K 

Figure C14. Comparison of Force Versus Impactor Travel Results, 98 K 
112 

Figure C15. Comparison of Tank Volume Versus Time Results at 98 K 

Figure C16. Comparison of Pressure Versus Time Results at 98 K 
113 

Figure C17. Comparison of Force Versus Time Results, 100 K 

Figure C18. Comparison of Force Versus Impactor Travel Results, 100 K 
114 

Figure C19. Comparison of Pressure Versus Time Results, 100 K 

Figure C20. Comparison Tank Volume Versus Time Results at 100 K 
115 

Appendix D.

Estimated Post-impact Inner Tank Volume

Following Test 12, the team contracted a third party to perform an internal and external LiDAR 

scan of the Test 12 surrogate. The resulting point cloud was produced. Using the LS-PrePost 

Software’s “Point Cloud to Mesh” feature, the portion of the point cloud corresponding to the 

inner tank’s surface was meshed using triangular elements [14]. Numerous features, such as the 

exterior tank, surrounding ground, and interior piping were removed from the point cloud to 

leave only the surface of the inner tank.

A simplified model approach was used to estimate the volume of the inner tank in its deformed 

state. The mesh was imported into the Abaqus/2019 CAE FE preprocessor [9] and assigned a 

rigid element type. The inner tank mesh had all its degrees of freedom constrained. A separate 

part was modeled as a cloud of particles surrounding the inner tank mesh, as shown in Figure 

D1. This image shows a top view of the tank and particles with a full section cut normal to the Z 

plane. These particles were assigned a material having the properties of water as summarized in

Table D1. 

Figure D1. Initial Particle Distribution, 1-inch Spacing (Section Cut)

Table D1. Properties of Water Used in Tank-filling Model

Property Value Source

Density 1.9 lbf-s2/ft/ft3 [15] 

Speed of Sound 4,814 ft/s [16] 

The model used contact between the inner tank surface and the particle cloud. Finally, a gravity 

load was applied to the model in the -X direction. Increased “G” loads were used to speed up the 

model. The water particles outside of the tank surface would fall away, but the water within the 

tank would be contained within the tank through contact. The total weight of water within the 

tank was obtained through the reaction force acting on the tank. Using the density of water 

prescribed in the model, the total volume of water in the tank could then be calculated from its 

total weight. Figure D2 shows the distribution of water within the tank after 2 seconds of settling 

under 2G.
116 

Figure D2. Final Particle Distribution, 1-ft Spacing (Section Cut)

This demonstrated that the approach was viable, but a substantial shortcoming was the obvious 

empty space toward the +X head of the tank. This “void” space is the result of the initial uniform 

grid of particles settling under gravity and conforming to the curvature of the tank. The model 

was re-run using particle spacings of 0.5, 0.25, and 0.125 ft to attempt to fill the tank more 

completely. The resulting particle distributions after standing in a 2G field are shown in Figure 

D3 through Figure D5. 

Figure D3. Final Particle Distribution, 0.5-ft Spacing (Section Cut)
117 

Figure D4. Final Particle Distribution, 0.25-ft Spacing (Section Cut)

Note that the 0.125-ft spacing model shown in Figure D5 was terminated after 1.2 seconds of 

settling time due to the model’s long runtime. 

Figure D5. Particle Distribution after 1.2 seconds, 0.125-feet Spacing (Section Cut)

Figure D6 contains a plot of the calculated volume of water within the tank versus time for each 

of the particle spacings examined. This volume was calculated by dividing the reaction force on 

the tank by the initial density of water. For each particle spacing, there is an initial overshoot that 

settles out to a stable value. This initial overshoot is likely due to dynamic forces developing 

between the water and the tank due to the sudden 2G deceleration. From this figure, it is not 

119 

Appendix E.

Estimated Amount of Condensation in Test 1213

This appendix provides a mathematical model to calculate the rise in pressure in the outage (i.e., 

vapor) volume space of a cryogenic tank car when the volume of the inner tank is compressed 

during a side impact. The cryogenic liquid is initially at a pressure above the ambient pressure 

and is assumed to be in saturated condition corresponding to its pressure. 

An equation for the mass fraction of vapor that converts into liquid is presented at the end of this 

appendix based on the pre- and post-impact (i.e., initial and final) pressure and volume with an 

assumption of saturated conditions. This equation was applied to the conditions of Test 12; it was

estimated that after the test, up to 56 percent of the mass of vapor (GN2) originally present in the 

outage space had condensed into liquid (LN2). This value may be compared with the estimated 

mass fractions of GN2 condensation from the FE models described in Section 5. 

E.1 Introduction

The physical condition of interest is shown in Figure E1 A and B. Initially, the tank car is 

partially filled with the cryogenic liquid (generally up to about 85 percent by volume) and the 

remainder of the volume is occupied by the vapor, which is assumed to be in thermodynamic 

equilibrium with the liquid (i.e., in saturated condition corresponding to the liquid 

temperature14). This condition is schematically indicated in Figure E1 A, with the initial vapor 

volume being V1.

Figure E1. A: Tank Conditions Just Prior to Impact (left) B: Reduced Vapor Volume Due 

to Deformation of Inner Tank After Impact (right)

In the scenario of interest, the double walled tank car is impacted by an impactor at some point 

along the length of the tank car and above the truck. It is normally assumed for test purposes that 

the impactor strikes at the mid-point of the tank car length and at the maximum width (i.e., 

diameter) of the cylindrical tank. First, the outer shell/tank will be deformed; depending on the 

initial kinetic energy of the impactor, this outer tank deformation will continue and impact the 

13 This appendix was developed and authored, in part, by Dr. Phani Raj of FRA’s Office of Safety.

14 This assumption differs from the assumption made in the finite element analyses, where the vapor was allowed to 

exist at a temperature above its saturation temperature.
120 

inner tank and begin to deform it. The inner tank volume will be continuously reduced if the 

impactor is still moving and the inner tank is not punctured. The inner tank may withstand the 

impact without puncturing, again depending upon the impactor’s initial energy conditions. 

Figure E1 B shows this situation schematically, without the inner tank being punctured. The final 

vapor volume, V2, is less than V1.

Since the liquid can be assumed to be an incompressible fluid, the reduction in the tank volume 

will result in the reduction of the vapor space volume only (i.e., the vapor gets compressed 

resulting in an increase of its pressure). This also results in the pressure of the liquid being 

increased by the same value. Generally, moderate compression of a large mass of liquid will not 

result in any significant (or even measurable) increase in the temperature of the liquid because of 

the large thermal capacity associated with large liquid mass. The vapor mass in a tank car is 

relatively low and the specific heat of a gas is also low compared to that of the liquid. Hence, the 

thermal inertia of the vapor in a tank car is low, resulting in the vapor attaining a higher 

temperature state (compared to liquid temperature) when compressed. The magnitude of increase 

in vapor temperature depends upon the rate of compression, the amount of compression (i.e., 

final volume/initial volume), the heat capacity ratio of the gas, and rate of cooling of the vapor 

either by the liquid mass or heat loss through the vapor wetted tank wall. In general, compression 

of a vapor mass will result in a measurable increase in its temperature, except when the rate of 

compression is very low or the vapor is in contact with a large heat sink.

Two scenarios can be postulated to determine the thermodynamic state (i.e., final pressure and 

temperature) of vapor in the outage volume of a tank car whose volume is reduced due to impact, 

assuming that the initial state of vapor is saturated at the tank’s initial pressure and 

corresponding saturation temperature. These two approaches to calculating the final 

thermodynamic state of the vapor are discussed below. 

Two scenarios are considered due to the uncertainty about the magnitude and effect of sloshing 

and agitation of liquid caused by the impact on the tank car. The sloshing of liquid may result in 

spraying small droplets of liquid into the vapor volume. 

Scenario 1 considers a situation where the rate of heating of the vapor due to compression is 

equal to the rate of cooling caused by liquid droplets mixing with the vapor. The vapor is 

assumed to remain at the initial saturation temperature of the vapor-liquid system throughout the 

impact (i.e., isothermal compression of the vapor).

Scenario 2 considers a situation where the rate of heating of the vapor due to compression is 

higher than or similar to the rate of cooling due to heat exchange with its surroundings. Scenario 

2 examines two different potential thermodynamic paths for the process of heating via 

compression, cooling due to heat exchange with the surrounding tank and/or liquid nitrogen, and 

condensation of cooled vapor into liquid.

The research team do not believe the current data from the test will be able to provide direct 

evidence for one or the other of the above scenarios. The physical phenomena that occur need to 

be surmised by comparing model results to measured extensive properties of the vapor (such as 

the pressure in the vapor space). 

E.2 Scenario 1

In Scenario 1, it is assumed that the vapor temperature remains constant (at the liquid 

temperature) due to liquid spray into the vapor space. Many small droplets in the liquid spray are 
121 

assumed to mix thoroughly with the vapor. These droplets present a significant surface area for 

heat transfer, causing part of the vapor mass to condense. In this scenario, the vapor temperature 

is always equal to the liquid temperature (which is assumed to be the initial liquid temperature). 

The liquid temperature does not change appreciably from its pre-impact value due to the large 

mass of liquid, high heat capacity, very low liquid heat of compression, and the fact that the heat 

generated due to the compression of vapor is small (due to low vapor mass) compared to the 

sensible heat of the large liquid mass. In this vapor compression scenario, the vapor temperature 

remains constant and equal to the bulk temperature of the liquid.

E.3 Scenario 2 

In Scenario 2, the phenomenon of ideal compression of vapor inside a tank car due to the entry 

of liquid mass into the vapor volume is discussed. The compression of vapor results in an 

increase in its pressure and temperature. However, the vapor is convectively and conductively

cooled by its contact with the large surface of the vapor-liquid boundary interface. The large 

mass of liquid acts as a heat sink, essentially maintained at a constant temperature. Thus, the 

vapor is simultaneously heated by compression and cooled by contact with the liquid mass. After 

compression, the vapor attains a higher pressure. As the cooling progresses, the pressure 

decreases along a constant vapor volume (i.e., constant vapor density) line until the vapor 

thermodynamic condition is the saturation condition corresponding to the then present vapor 

density. It is entirely possible that further cooling still occurs (i.e., a reduction of pressure at 

constant density). The vapor now exists in a super saturated condition. When it reaches a 

thermodynamically unstable condition, a part of the vapor condenses instantaneously, leaving the 

rest of the uncondensed vapor in a saturated condition corresponding to the new (i.e., lower) 

vapor density. The mass loss by condensation occurs entirely due to thermodynamic instability at 

a slightly later but very short time. In this scenario, this cycle is assumed to occur in infinitesimal 

steps so that at a macro scale, the thermodynamic path of the vapor is along the saturated vapor 

line with increasing pressure. 

Figure E2 shows a simplified temperature-entropy (T-S) diagram for a two-phase system. The 

figure shows the T-S15 diagram for the nitrogen system in the region around the initial state, 

using data from NIST [10]. The possible thermodynamic path of vapor during a short step of the 

compression (i.e., cooling) condensation cycle is illustrated schematically in Figure E2. The 

starting point is indicated by 1, the saturated condition of the vapor and liquid before the impact.

A situation in which the vapor is compressed due to the rapid movement of the liquid into the 

vapor space is considered. In this case, the liquid surface acts very much like the surface of a 

piston. The entire process is broken into small time steps in which the vapor is compressed 

slightly and attains a new thermodynamic condition 2, where the pressure and temperature of the 

vapor are each slightly higher than before the infinitesimal compression. The vapor is cooled by 

conduction and convection at the same time due to contact with the bulk liquid. This vapor 

cooling occurs along a constant vapor density line 2-3 (constant density because of the 

assumption of no vapor mass loss and the fact that after the infinitesimal compression the vapor 

volume remains constant over the short time step). It is entirely possible that the vapor is 

subcooled along the line 3-4 until it reaches a thermodynamic limit instability point (shown as 

15 The reference state for the entropy data is S=0 at the normal boiling point for saturated liquid nitrogen, 77.355 K 

at 1 atmosphere. 
122 

Point 4). After this, a part of the vapor instantaneously condenses, and the remaining vapor 

reaches saturated condition (i.e., pressure and temperature) corresponding to the stability limit at 

Point 4. This new condition is shown in Figure E2 as point 5. In the next small time step, the 

above compression-cooling-spontaneous condensation continues (i.e., Point 5 from the first time 

step becomes Point 1 in the second time step and Point 6 corresponds to Point 2 of the 

infinitesimal compression step). The result is that the vapor thermodynamic condition traces the 

saturated vapor line. The pressure in both the vapor and liquid will be the same at the new 

saturated vapor condition. However, the liquid will be at its original bulk temperature, which will 

be a subcooled state corresponding to its new pressure.

Figure E2. T-S Diagram for Nitrogen with the Vapor Compression Path Shown

In the above description of the vapor compression process (due to the tank car volume reduction) 

it is assumed that no liquid is sprayed into the vapor volume and therefore no condensation of 

vapor due to sudden cooling. It can be shown that the bulk liquid temperature does not change 

appreciably due to its large mass (compared to that of the vapor in the outage volume), its high 

heat capacity, its extremely low compressibility (i.e., no heat of liquid compression), and the 

very small amount of latent heat released by the spontaneous condensation (due to 

thermodynamic instability) of a very small fraction of the vapor.

This entire process may be taking place continuously as the impactor is continuing to “deform” 

the inner tank. This is because the time scales of equilibration of the vapor to the new “reduced 

vapor volume” condition will be in the order of milliseconds, whereas the movement of the 

impactor is in the order of magnitude of seconds. Therefore, one can surmise that there is a 

124 

adiabatically compressed gas rapidly loses heat to the liquid, its temperature decreasing along the 

constant vapor density line. This line intersects the vapor saturation line at Point 3. 

After the first cycle of the thermodynamic process, there are two potential scenarios that should 

be considered: 

(1) If the vapor does not condense, one can assume that the next dV compression step 

starts at Point 3 and results in reaching Point 6 isentropically and adiabatically. This cycle 

then repeats in small dV steps until the impactor comes to rest and no breach of inner 

tank has occurred. This scenario of physical and thermodynamic events could occur due 

to the assumption that the time constant for compression is of the same order of 

magnitude (i.e., seconds) as the movement of the impactor, whereas the cooling of the 

compressed gas occurs at a faster (i.e., millisecond) time frame due to the very large heat 

sink (in the form of a cold liquid) with which the vapor is in contact but not necessarily at 

temperature equilibrium. In this scenario the vapor is always assumed to have the same 

initial mass and saturated condition corresponding to the new pressure at each step and 

occupying smaller volumes as the tank deformation continues. 

(2) It is entirely possible that the vapor, in cooling from Point 2 along the constant 

density line, is subcooled below Point 3 due to paucity of nucleation sites for initiating 

condensation. This process is shown by dotted line 3-4. However, the vapor condenses at 

some supercooled condition 4, and the mass fraction of vapor that is condensed is f. In 

this case, the initial vapor loses a mass that is f multiplied by its original mass and attains 

a saturated vapor condition shown schematically in Figure E2 as Point 5. The 

compression cycle with each dV incremental reduction in volume is repeated (i.e., Point 5 

from the first increment becomes Point 1 in the second increment and Point 6 becomes 

the condition after the second infinitesimal step compression), and at every step a mass 

fraction f is removed by condensation of the vapor into a liquid. The final pressure 

reached when the impactor stops moving is less than that attained in Scenario 1.

It can be shown that if the compression is reversible (i.e., the meaning of isentropic compression) 

and that cooling along the constant density line is reversible (which is also true because it is a 

constant volume process), then irrespective of the infinitesimal steps described above, the final 

pressure in the vapor (for a specified change of the volume from its original volume) can be 

computed in a single step. This is illustrated in the mathematical analysis of the above 

thermodynamic phenomena.

E.5 Correlation for the Saturated Properties of Nitrogen

Figure E3 shows the plot of the saturated vapor pressure vs. saturated vapor temperature for 

nitrogen using data from Strobridge [18]. 







132 

Figure E6. Vapor Mass Fraction to Condense vs Initial Vapor Temperature

It should be noted that the results from Figure E6 cannot be directly compared with the results 

from the FE models presented in Section 5 because the FE models used an isothermal condition 

for the vapor and the models were not run until the inner tank was fully allowed to “springback” 

after the impact. However, it is encouraging that both the FE models and the calculations 

presented in this appendix indicated that there was a significant amount of vapor condensation 

during the side impact.
133 

Abbreviations and Acronyms

ACRONYM DEFINITION

AAR Association of American Railroads

ASTM American Society for Testing and Materials

DOF Degrees-of-Freedom

DOT Department of Transportation

EOS Equations of State

FE Finite Element

FEA Finite Element Analysis

FRA Federal Railroad Administration

kip kilopound (1000 lbf)

LN2 Liquid nitrogen

LNG Liquefied Natural Gas

MLI Multi-layer Insulation

N2 Nitrogen

NIST National Institute of Standards and Technology

PEEQ Plastic Equivalent Strain

PHMSA Pipeline and Hazardous Materials Safety Administration

PRV Pressure Relief Valve

SPH Smoothed Particle Hydrodynamic

SRB Smooth Round Bar

STDP Start to discharge pressure

T304 ASTM A240 Type 304 Stainless Steel

TC128 AAR TC128, Grade B Carbon Steel

TTC Transportation Technology Center (the site)

TTCI Transportation Technology Center, Inc. (now called MxV Rail)

Volpe Volpe National Transportation Systems Center