SECURE LIQUID NITROGEN BACKUP FOR REFRIGERATED BIOSTORAGE
By
Michael Kenworthy Balck

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
Mechanical Engineering
2010

ABSTRACT
SECURE LIQUID NITROGEN BACKUP FOR REFRIGERATED BIOSTORAGE
By
Michael Kenworthy Balck
There is need for a secure backup refrigeration system for the long-term storage of
biological material. In addition to freezer component failure natural disasters such as Tropical
Storm Allison in 2001 in Houston and Hurricane Katrina in 2005 in New Orleans have caused
power outages and flooding, leading to the loss of biological specimens when backup
refrigeration systems fail. Freezer component failure includes seal leaks, pump and compressor
breakdown, and electrical monitoring malfunction. Power outages often result from high winds
or large quantities of rain, snow, and ice that disrupt transmission lines. Floodwater, especially
from excessive amounts of rain, fills basements and ground floors where biological storage is
often located. A non-electrical backup system that can function in the presence of flooding is
needed to preserve specimens when the primary system fails.
A liquid nitrogen (LN2) refrigeration system utilizing a thermostatic expansion valve
(TEV) was designed, built, and tested for use as a backup refrigeration system. The non-electric
TEV is the key component in the system that can provide secure backup refrigeration for longterm storage of biological specimens. The passive LN2 refrigeration system does not depend on
electricity so it continues to function where common systems fail. Test results of the LN2 TEV
o

system demonstrated consistent regulation of freezer air temperature within 1 C of the set point.
Simulations show that the system’s high gain and reserve cooling capacity allows it to hold the
temperature steady even during a large increase in ambient temperature.

ACKNOWLEDGMENTS

I would like to thank my advisor, Prof. Clark Radcliffe, for his countless hours of
guidance throughout the entire process of this work. Without his help it could not have been
finished.
I am also grateful to my family, friends, and colleagues who offered constant support. I
am especially appreciative to my parents, Ralph and Nancy, who instilled in me the importance
of education and curiosity from an early age. Finally, I would like to offer a special thank you to
my wife, Margaret, whose support and encouragement awaited me every night.

iii

TABLE OF CONTENTS

LIST OF TABLES .............................................................................................................. v
LIST OF FIGURES ........................................................................................................... vi
LIST OF SYMBOLS ....................................................................................................... viii
INTRODUCTION .............................................................................................................. 1
LIQUID NITROGEN AS A REFRIGERANT ................................................................... 4
THERMOSTATIC EXPANSION VALVES ..................................................................... 8
LN2 TEV DESIGN ........................................................................................................... 11
SYSTEM ANALYSIS ...................................................................................................... 15
LINEAR FREEZER CHEST MODEL......................................................................... 18
SENSOR BULB MODEL ............................................................................................ 22
VALVE ACTUATOR MODEL ................................................................................... 23
CONTROL MODEL SIMULATION .......................................................................... 24
CONCLUSION ................................................................................................................. 31
APPENDIX A : THERMAL INSULATION CALCULATIONS.................................... 34
APPENDIX B : RESISTANCE AND HEAT CAPACITY CALCULATIONS .............. 35
APPENDIX C : SENSOR BULB HEAT TRANSFER .................................................... 40
APPENDIX D : HEAT TRANSFER RATES CALCULATION..................................... 43
REFERENCES ................................................................................................................. 46

iv

LIST OF TABLES

Table 1. Initial and final nitrogen states during the cooling process ............................................. 6
Table 2. Component properties [Incropera et al., 2007] and thermal resistance ......................... 19
Table 3. Thermal resistances........................................................................................................ 20
Table 4. Component material properties [Incropera et al., 2007] and heat capacities .................. 21
Table 5. Resistance and heat capacity summary .......................................................................... 39
Table 6. Material properties of copper at 244.5 K [Incropera et al., 2007] .................................. 41
Table 7. Thermal properties of air at 244.2 K [Incropera et al., 2007] ........................................ 41
Table 8. Thermal parameters used in the transient heat transfer rate calculation ........................ 44

v

LIST OF FIGURES

Figure 1. Nearly 50% of the he at absorbed during the cooling process occurs from
phase change. ............................................................................................................................... 5
Figure 2. Cross section of a thermostatic expansion valve, Parker model N 1/2 JW [Parker
Hannifin Corporation, 2007]........................................................................................................... 9
Figure 3. Opening and closing forces on the ball bearing at the valve seat. ................................ 10
Figure 4. Modified thermal expansion valve for use with LN 2 (LN2 TEV). For interpretation of
the references to color in this and all other figures, the reader is referred to the electronic version
of this thesis. ................................................................................................................................. 11
Figure 5. The copper seal keeps liquid nitrogen from leaking out of the valve. ......................... 12
Figure 6. The seal adapter contains a thin copper membrane. ..................................................... 13
Figure 7. The refrigeration system includes the LN2 supply tank, insulated copper tubing, LN2
TEV and the freezer chest. ............................................................................................................ 15
Figure 8. Temperature measurements show a rapid decrease and overshoot in air temperature
until steady state conditions are reached at about 3500 seconds. ................................................. 16
Figure 9. LN2 TEV refrigeration system feedback control block diagram. ................................. 17
Figure 10. LN2 TEV refrigeration system heat transfer model.................................................... 18
Figure 11. LN2 TEV non-linear cooling function with saturation. .............................................. 24
Figure 12. Simulated freezer air temperature matches the steady state experimental values and
captures the qualitative transient behavior .................................................................................... 25
Figure 13. The simulated sensor bulb temperature decreases slower than experimental
measurements but matches the steady state values. ...................................................................... 26
Figure 14. The simulation predicts the qualitative behavior of the valve as well as the correct
steady state heat transfer rate. ....................................................................................................... 26
Figure 15. Timing and overshoot is improved with alterations to the model while steady state
behavior is not significantly changed. .......................................................................................... 27

vi

Figure 16. Simulation shows that the LN2 TEV system is well suited to backup a primary
refrigeration system during the worst conditions of a power failure. ........................................... 29
Figure 17. Chest interior dimensions. .......................................................................................... 35
Figure 18. Chest exterior dimensions. ......................................................................................... 36
Figure 19. Chest insulation dimensions. ...................................................................................... 37
Figure 20. Sensor bulb geometry. ................................................................................................ 40

vii

LIST OF SYMBOLS

Area
Heat capacity
Specific heat capacity
Thermal energy
Time rate of change of thermal energy
Convection heat transfer coefficient
Thermal conductivity
Length
Mass flow rate
Heat transfer
Heat transfer rate
Specific heat transfer
Thermal resistance
Temperature
Time rate of change of temperature
Volume
Density
Time constant
COP

Coefficient of performance

LN2

Liquid nitrogen

TEV

Thermostatic expansion valve

viii

INTRODUCTION
Biostorage is growing in importance for medical and biological research. Biostorage is
the preservation of biological material in a temperature controlled environment. An example of
biostorage is the Archive for Research on Child Health (ARCH) project at the College of Human
Medicine at Michigan State University managed by the Institute for Health Care Studies (IHCS).
The purpose of the project is to connect “…the collection of biological specimens from Michigan
newborns to medical and vital records and allows for scientific investigation and research on
pregnancy and childhood health conditions” [Institute for Health Care Studies, 2010]. To do
this, the original samples must be carefully stored to protect their integrity for many decades.
Furthermore, the individual is periodically tested, with the results being recorded and added to a
file containing all previous test results. The initial samples and all test results become more
valuable as time passes because the amount of data increases.
Severe weather resulted in the loss of thousands of biological samples, costing millions of
dollars and years of research. In 2001 “…cancer, AIDS and other medical research was lost
when power to freezers failed and cell cultures were destroyed” in Baylor College of Medicine
laboratories during Tropical Storm Allison [Turner, et al., 2001]. Hurricane Katrina in 2005
affected about 300 federally funded projects at colleges and universities in New Orleans
according to a survey conducted by the National Institutes of Health [Associated Press, 2005].
Frozen blood and urine samples from thousands of donors were destroyed when electricity was
lost followed by the failure of backup generators at Tulane University [Associated Press, 2005].
Existing freezer backup systems are not reliable enough for long-term storage. They fail
in many ways including malfunction of mechanical and electrical components, power loss, and

1

flooding. Any one of these failures can lead to the destruction of biological samples. Flooding
and power loss often happen concurrently during large storms.
Complex machinery and electronics in freezers can fail over time. Seals, bearings,
electrical connections, and system batteries can all deteriorate. A common backup for this type
of failure is a second, stand alone system. This system is just as complex, expensive, and
unreliable as the system it replaces.
Existing backup systems require a second source of electrical power when a power
outage occurs. This second source can either be a gas powered generator or a battery. The
generator is used to power the mechanical refrigeration system and electrical controls.

A

generator is vulnerable to flooding and requires a large supply of fuel to power a collection of
low temperature freezers. Batteries are used to run electronic controls for the release of liquid
carbon dioxide or nitrogen. These systems have limited amounts of refrigerant on hand. Also,
batteries are not a reliable source of power. According to a freezer manufacturer’s technical
bulletin, batteries “…can develop a memory which can severely limit the reserve capacity when
needed the most” [Sanyo, 2010]. Manufacturers recommend that these batteries be replaced
approximately every three years regardless of their condition [Sanyo, 2010]

adding an

expensive, time consuming, and difficult to manage task to facility management.
Flooding is often the cause for system failures. Floodwater shorts electrical controls and
motors driving compressors and pumps that are often located close to floors due to their weight.
Water can also infiltrate generators and cause electrical failure of secondary power sources.
Non-electrical failure due to flooding includes damage to bearings and drive belts. If allowed to
contact specimens, floodwater can compromise their integrity.

2

A robust backup system would have reliable and simple components, be non-electrical,
flood resistant, and able to run for an extended period. The LN 2 TEV backup system meets all of
these criteria. The LN2 TEV has simple, low force mechanics for increased reliability. The
system is thermodynamically driven so electrical power is not needed and floodwaters do no
harm. Finally, most medical research facilities have large stores of LN 2 on site so the backup
system can be easily installed and run for an extended period.
The process of designing a reliable backup refrigeration system was motivated by the
consequences of using LN2 as the refrigerant with a TEV as the regulator. The first step was
determining the cooling ability of LN2. Next was researching how TEVs normally operate and
how LN2 temperatures affect their performance. Finally was the design and implementation of
solutions for the problems caused by low temperatures with the TEV.

3

LIQUID NITROGEN AS A REFRIGERANT
Nitrogen is a plentiful and non-toxic gas. It makes up 78% of earth’s atmosphere by
mole fraction [Moran and Shapiro, 2004]. Commercially, LN2 is a byproduct of the production
of liquid oxygen. It is a very inexpensive liquid at 20 cents per gallon in bulk because it is so
plentiful [Radcliffe, 2007]. Liquid nitrogen is stored in large tanks at a pressure of 22 psig (36.7
o

psia) and a temperature of -187 C. It can be used as a refrigerant and has the designation R-728.
A refrigerant is a substance that is used to transfer thermal energy from one space to another.
Fluids that are able to absorb and release large amounts of energy corresponding to phase
changes are often used as refrigerants.
Liquid nitrogen cools a space or object by absorbing energy from it in a complex process
involving boiling and temperature change. The three steps of cooling can be seen in (Fig. 1),
which shows the amount of energy that is absorbed by the nitrogen during the cooling process.
The process will be described by following the heat absorbed along the 36.7 psia (22 psig) line.
In the first step, liquid nitrogen warms until it reaches its boiling point. This corresponds to a
movement from point A to point B. Next the liquid nitrogen boils at constant temperature until it
is entirely gas phase, going from point B to point C. Finally the nitrogen gas warms until it
reaches an equilibrium temperature with its surroundings as there is movement from point C to
point D. Heat energy is transferred from the surroundings to the nitrogen in each step.
The cooling process involved in the LN2 TEV backup refrigeration system differs in two
ways from that shown in (Fig. 1). The LN2 is delivered in a pressure regulated tank so the initial
temperature of the LN2 is forced to be the boiling point at 36.7 psia (22 psig) because the liquid
continuously boils due to the tank’s warm environment. The temperature of the nitrogen must be
o

-187 C because pressure regulator forces the pressure to be 36.7 psia (22psig). Gauge pressure,
4

Heat Absorbed (kJ/kg)

Heat Absorbed During LN2 Cooling Process
450
400
350
300
250
E
200
150
100
50
0
A
-200

F
D

Gas Phase

14.7 psi
C

36.7 psi
Phase Change

B

Liquid Phase

-175

-150

-125
-100
-75
-50
-25
0
25
Final Nitrogen Temperature (oC)
Figure 1. Nearly 50% of the heat absorbed during the cooling process occurs
from phase change.
psig, is referenced to standard atmospheric absolute pressure, 14.7 psia. The state 36.7 psia and
o

-187 C corresponds to point B. The second difference is that there a pressure drop as the liquid
is released through the LN2 TEV orifice. As the LN2 passes through the TEV orifice, the
nitrogen's pressure changes from 36.7 psia to 14.7 psia yielding rapid boiling and phase change
from liquid to gas. The TEV causes the nitrogen to move between points B and E. The state of
the LN2 lies on the vertical portion of the dashed 14.7 psia line as the mixed phase nitrogen boils
completely. Once it is completely gas phase, the nitrogen gas warms and the state moves from
point E to point F along the 14.7 psia line.
The amount of energy absorbed per unit mass
nitrogen’s the latent heat of vaporization
capacity

, and the final temperature

from the surroundings depends on

at the boiling temperature

, its specific heat

(1).

(1)

5

Liquid nitrogen absorbs 199 (kJ/kg) to boil to a gas at 14.7 psia and constant temperature. Once
o

the nitrogen is in gas phase it continues to absorb heat at a rate of 1.048 (kJ/kg- C), its average
specific heat during the temperature rise, until an equilibrium temperature is reached. The initial
state of the LN2 in the LN2 TEV backup refrigeration system is defined by a pressure of 36.7
o

psia (22 psig) and a temperature of -187 C. The final state is atmospheric pressure, 14.7 psia,
o

o

and -29 C. The temperature rise of the nitrogen gas is 167 C yielding a heat absorption value of
175 (kJ/kg). The total heat absorption of the nitrogen is the 374 (kJ/kg) (2).
(2)
Another method of determining the energy absorbed by the nitrogen is to look at its
initial and final energy states defined by enthalpy (Table 1). The total heat absorbed by the
nitrogen is equal to the difference in the enthalpies of the initial and final states. In this case the
heat absorbed is 362 (kJ/kg), which is very close to the calculation done earlier. A value of 360
(kJ/kg) will be used for the heat absorption.
Table 1. Initial and final nitrogen states during the cooling process
[Reynolds, 1979]
o
State
Phase
Pressure (psia)
Enthalpy (kJ/kg)
Temperature ( C)
Initial
Liquid
-187
36.7
43
Final
Vapor
-27
14.7
405
The cost of cooling with LN2 can be compared to cooling with electricity in the form of
dollars per kilo-Joule of cooling [Radcliffe, 2007]. The heat absorption of LN2 in kilo-Joules per
kilogram can be converted to kilowatt-hours per gallon using the density of LN2 and unit
conversion (3).
-

6

-

-

(3)

Liquid nitrogen costs about 20 cents per gallon in bulk [Radcliffe, 2007] resulting in the cost of
cooling with LN2 to be 65.1 (cents/kW-hr) (4).
(4)
The retail cost of electricity for commercial use in Michigan is 10.11 (cents/kW-hr) [U.S. Energy
Information Administration, 2010]. Heat pumps, such as refrigerators and freezers, are known to
have a coefficient of performance (COP) of about 2.5 [Radermacher et al., 2007]. This means
that they provide 2.5 times the amount of cooling energy as they use in electrical energy.
Therefore the cost of cooling with electricity is about 4 cents per kW-hr. The cost of cooling
with LN2 is about 16 times greater than using electricity. A LN2 refrigeration system is not
intended as a primary system because of this cost. However, when an electrical power outage
occurs it is well worth the cost to use a LN2 backup refrigeration system to preserve biological
specimens.

7

THERMOSTATIC EXPANSION VALVES
Thermostatic expansion valves (TEVs) are flow controlling devices that operate
according to the temperature they sense. They are found in many commercial refrigeration and
air conditioning systems including walk-in-coolers, beverage dispensers, and ice machines
[Parker Hannifin Corporation, 2007].

A TEV is installed between the compressor and

evaporator in normal refrigeration systems. In the LN2 backup refrigeration system the TEV is
connected to the refrigerant supply line and is located in the freezer chest which acts as the
evaporator. TEVs control the cooling process and adjust the freezer temperature by regulating
the flow of refrigerant.
TEVs (Fig. 2) operate by sensing the temperature of the space it is cooling, actuating
flow regulation, and providing flow area expansion and pressure drop. When the TEV senses
that the space is too warm, refrigerant is allowed to flow to cool the space. When the TEV
senses that the space is too cold The TEV stops the flow of refrigerant, allowing the space to
warm up. The process repeats when the object again becomes too warm.
The TEV senses temperature by a thermodynamic process within the TEV’s sensor bulb.
The sensor bulb is made of copper and contains a refrigerant charge that reacts to the temperature
it is sensing. The charge boils and expands when the bulb becomes warmer than its set point. A
large pressure rise results from the phase change since limited volume is available. When the
temperature is below the set point, the charge condenses, which results in a low pressure. The
actuator reacts to this pressure signal.
The TEV’s actuator is a thin, flexible, metal diaphragm. The top side of the diaphragm is open
to the refrigerant charge in the sensor bulb. A small copper capillary tube allows the flow of the
charge between the sensor bulb and the top side of the diaphragm. The other side of the

8

Figure 2. Cross section of a thermostatic expansion valve,
Parker model N 1/2 JW [Parker Hannifin Corporation, 2007]
diaphragm is connected to a ball bearing in the valve seat through an assembly containing an
actuator pin. The diaphragm transmits a downward force on the ball bearing which acts to allow
the flow of refrigerant. The force transmitted by the diaphragm increases with rising pressure in
the sensor bulb.
A spring is used to counteract the opening force. The spring is located below the valve
seat and acts to push the ball bearing into the seat to stop the flow of refrigerant. The spring is
supported by a nut that rides inside the valve’s outlet. The nut can be turned to vary the spring’s
preload which adjusts the TEV’s temperature set point.
The TEV set point is determined by the balance of the opening and closing forces
(Fig. 3). The opening forces include the refrigerant charge pressure acting on the diaphragm area
and the refrigerant gauge pressure acting on valve orifice area. The latter force is negligible
since the pressure is relatively low and the orifice is relatively small. The closing force is the

9

Fopen=PbulbAdiaphragm

Ball Bearing
Fclose=kX
Figure 3. Opening and closing forces on the ball bearing at the valve seat.
spring force. The set point temperature is the temperature at which the force created by the bulb
pressure exactly matches the spring force in a closed position. For the Parker N 1/2 JW TEV this
o

temperature was measured to be -30 C. If the bulb temperature rises above the set point a
corresponding pressure rise occurs that overcomes the spring force and pushes the ball bearing
off the valve seat. Refrigerant is then free to flow and cool the space.

10

LN2 TEV DESIGN
Problems occur when using a conventional TEV with LN2 as the refrigerant. The very
low temperature of LN2 causes the rubber seal (Fig. 2) around the actuator pin to harden and
deform and could cause the refrigerant charge to freeze. The hardened and deformed rubber seal
allows the LN2 to leak to the underside of the diaphragm and also impedes the motion of the
actuator pin. The seal holds the actuator pin in the open position which allows constant flow of
LN2 into the freezer chest while the freezer air and valve temperatures drop to well below
o

-150 C.

The refrigerant charge on the topside of the diaphragm and in the bulb quickly
o

condenses and is believed to freeze. Its freezing temperature is believed to be close to -100 C.
This is approximately the freezing temperature of R-134a [Air Liquide, 2009], one of the
conventional refrigerants used with this TEV. TEV charges and the refrigerants used with them
have similar thermodynamic properties. A frozen charge would not allow the diaphragm to
move and would not flow between the diaphragm and sensor bulb.

Thermal Insulation
Diaphragm
Seal Adapter
Inlet
Bulb

Outlet

Figure 4. Modified thermal expansion valve for use with LN 2 (LN 2 TEV). For
interpretation of the references to color in this and all other figures, the reader is
referred to the electronic version of this thesis.

11

Thermal insulation

Extension rod

New copper seal
Seal adapter
Extended actuator pin
TEV body

Rubber seal removed

Figure 5. The copper seal keeps liquid nitrogen from leaking out of the valve.
Two modifications were made to the TEV design to allow it to be used with LN2 (Fig. 4).
The rubber seal was removed and a new seal was installed to keep the liquid nitrogen from
leaking. The actuator pin had to be lengthened to accommodate the new seal. Then thermal
insulation was inserted between the valve body and the diaphragm to keep the refrigerant charge
from freezing.
A thin copper membrane was placed between the actuator pin and the thermal insulation
extension to prevent LN2 from leaking out of the valve (Fig. 5). Copper was used for the seal
because it does not become deformed or harden like rubber does at LN2 temperature.
Corrugations were pressed into the membrane to ensure that the seal would have enough
flexibility to move without breaking. The copper seal (Fig. 6) was fabricated from 0.001” copper
shim stock and has a diameter of 0.5”. The copper membrane was soldered into a brass adapter.
The brass adapter was fabricated to easily screw into both the valve body and thermal insulation.

12

Figure 6. The seal adapter contains a thin copper membrane.
The second modification was placing thermal insulation between the diaphragm and the
valve body to keep the refrigerant charge from freezing. The insulation is a polycarbonate
cylinder that connects the diaphragm to the valve body while a polycarbonate extension rod links
the copper membrane and the diaphragm. Thermal insulation design balanced the conductive
heat transfer from the diaphragm to the valve body with the convective heat transfer from the
freezer air to the diaphragm. The goal was to make the insulation length long enough to keep the
o

diaphragm above -100 C, the suspected highest freezing temperature of the refrigerant charge.
The length was calculated to be 2.6 cm using a factor of safety of 2 [App. A]. The force needed
to buckle the extension rod was then calculated to be 410 N [App. A], much greater than the
forces involved with valve operation.
The LN2 TEV system is capable of running under flood conditions. If water
o

breaches chest seals it will freeze upon coming into contact with the -25 C interior chest walls
which helps to further seal the chest. Also the chest is pressurized to 22 psig from the LN 2

13

supply tank. For water to overcome this pressure and force itself into the bottom of the chest it
would have to be over 50 ft deep. Meanwhile, the TEV would continue to supply the chest with
LN2.

14

SYSTEM ANALYSIS
Mathematical models were developed to analyze the system, explain system behavior,
and predict system response to a power failure and rise in ambient temperature. First a linear
model for the heat transfer between the freezer chest and the surrounding environment was
developed. Then another heat transfer model was developed to describe the sensor bulb. Next a
nonlinear model was used to explain the action of the thermostatic expansion valve. Finally, a
control model was produced to predict system response.
The refrigeration system consists of the LN2 supply tank, insulated copper tubing, LN2
TEV, and the freezer chest (Fig. 7). The freezer is considered as a rectangular chest with

LN2 supply tank

Insulated copper tubing

Freezer chest
LN2 TEV

Figure 7. The refrigeration system includes the LN 2 supply tank, insulated copper
tubing, LN 2 TEV and the freezer chest.

15

insulated walls. The chest walls and bottom are made of two thin aluminum (Al) sheets with a
thick layer of urethane foam insulation in between.

The interior chest ceiling is made of

polystyrene (PS). The nitrogen is dispensed from the TEV, which is suspended in the middle of
o

the chest. Operating at room temperature (24 C), a heavily insulated, small diameter, copper
tube connects the TEV to the LN2 supply tank outside the chest.
Experiments were run to gain understanding of the system. Thermocouples were placed
in the freezer air, on the sensor bulb, interior wall, exterior wall, and in the room outside of the
chest to track their temperatures throughout the experiment (Fig. 8).The chest air and bulb
o

temperature reach steady state at about 3500 seconds and a temperature of -29 C. The interior
o

o

wall and exterior wall reach steady state temperatures of -24.5 C and 22.2 C respectively. The
o

room temperature stays near 24 C.

Temperature Measurements

Temperature, oC

30
20
10
Air

0
-10

Bulb

-20

Interior

-30

Exterior

-40

Room

0

5000

10000

15000

20000

Time, seconds
Figure 8. Temperature measurements show a rapid decrease and overshoot in air
temperature until steady state conditions are reached at about 3500 seconds.

16

The LN2 TEV refrigeration control system consists of three major components; freezer
chest, sensor bulb, and valve actuator (Fig. 9).
temperature
chest,

The heat transfer rate

and room

are inputs to the freezer chest. The result is the air temperature inside the

.

is the sensor bulb measurement of

inside the bulb that acts upon the diaphragm.

.

determines the pressure

This pressure acts against the set point

, also expressed physically as the pressure resulting from the spring’s preload

temperature,

force. The difference between

and

is the system error. The valve actuator responds

to the system error by opening and releasing LN2 or shutting off the LN2 flow. If the error is
negative the pressure in the bulb is great enough to displace the spring. This corresponds to a
measured temperature that is warmer than the set point temperature. A negative error causes the
release of LN2 and heat absorption to occur inside the chest. A positive or zero error will result
in the valve closing and no release of nitrogen. Physically this means that the closing spring
force is greater than the opening force caused by the bulb pressure.

LN2 TEV

+

(error)

∑

Valve Actuator

Freezer Chest

_

Sensor Bulb

Figure 9. LN 2 TEV refrigeration system feedback control block diagram.

17

LINEAR FREEZER CHEST MODEL
The freezer chest was separated into four thermal masses for the mathematical
model; exterior wall, foam insulation, interior wall, and freezer air (Fig. 10). Each of these
components has a thermal energy

related to a temperature

associated with the heat transfer rate

. There is a thermal resistance

between each component. There is a heat transfer rate

into the system from the room and the heat transfer rate out of system from the heat absorbed by
LN2. The model has four thermal states (
and one output (

,

,

,

), two inputs (

,

),

).

The heat transfer model for each mass is derived from an energy balance. According to
conservation of energy, each mass’s time rate of change of energy
(5)
where

are the heat transfer rates into, and out of, the associated mass. The heat transfer rate

between components of the freezer chest
(6)
is driven by the temperature difference between the components and inversely related to the

𝐿𝑁2
𝑖𝑛𝑡
𝑎𝑖𝑟

𝑤𝑖

𝑖𝑛𝑡
𝑎𝑖𝑟

𝑤𝑜

𝑖𝑛𝑡

𝑖𝑛𝑠
𝑤𝑖

𝑒𝑥𝑡
𝑒𝑥𝑡

𝑤𝑜

𝑖𝑛𝑡

𝑖𝑛𝑠

𝑟𝑜𝑜
𝑟𝑜𝑜

𝑒𝑥𝑡

Figure 10. LN 2 TEV refrigeration system heat transfer model.
Figure 10. LN2 TEV refrigeration system heat transfer model.
18

thermal resistance between the hot and cold components.
The values of the thermal resistances

and

were calculated using physical

measurements of the system components and material properties taken at steady state
temperatures. The location of the insulation temperature
wall resistance

and the exterior wall resistance

was chosen such that the interior
are equal to each other and one half of

the total resistance between the interior wall temperature
. The total resistance of the composite wall

and the exterior wall temperature
is the sum of the resistances of the two

aluminum sheets and the foam insulation. Each resistance is calculated [App. B] using (7),
where

is the thermal conductivity of the material,

is the thickness of the material, and

is

the conduction area (Table 2).
(7)
Table 2. Component properties [Incropera et al., 2007] and thermal resistance
Component

Thickness
(m)

Thermal Conductivity
(W/m-K)

Area
2
(m )

Resistance
(K/W)

-4

2.76

1.5x10

-2

2.35

1.2

Exterior Wall (Al)

176

7.5x10

Foam Insulation

0.04

7.0x10

Interior Wall (Al)

170

Interior Wall (PS)

0.13

7.5x10

-4

1.45
0.25

-6

-6

3.0x10

The total resistance of the wall
(8)
The foam insulation’s low heat transfer coefficient accounts for most of the resistance of the
wall. The value of

and

is then 0.6 (K/W).

19

The system is at steady state when temperatures cease to change. This corresponds to
each mass reaching a constant energy. If a mass is at a constant energy then the heat transfer rate
into the mass is equal to the heat transfer rate out of the mass. This implies that each heat
transfer rate is equal to one another at steady state. The value of the steady state heat transfer
rate is most accurately calculated using the calculated total wall resistance and measured steady
state temperatures of the exterior wall

and interior wall

(9).
(9)

The interior resistance
resistance

between the interior wall and freezer air, and exterior

, between the exterior wall and room air, were calculated (Table 3) using

temperature measurements, the calculated steady state heat transfer rate, and (9).
Table 3. Thermal resistances

0.12 (K/W) 0.6 (K/W) 0.6 (K/W) 0.018 (K/W)
The energy stored in a freezer component is a function of the component’s temperature
and heat capacity

. The component’s heat capacity is the product of its specific heat capacity

and mass. The time rate of change of energy stored in the component is dependent on the time
rate of change of the component’s temperature (10).
(10)
Heat capacities (Table 4) were calculated [App. B] using physical measurements for the material
volume and the material properties of density and specific heat capacity (11).
(11)

20

Table 4. Component material properties [Incropera et al., 2007] and heat capacities
Volume
Density
Specific Heat
Heat Capacity
Component
3
3
(J/kg-K)
(J/K)
(m )
(kg/m )
2
Air
0.14
1.4354
1006
2.0 x 10
-3
1.1 x 10
Interior (Al)
2770
830
3
2.7 x 10
-4
Interior (PS)
1050
1170
1.9 x 10
4
Insulation
0.16
70
1045
1.2 x 10
-3
3
Exterior
2770
871
2.1 x 10
5.1 x 10
The heat transfer model is assembled by applying (6) and (10) to (5) for each freezer
component and dividing through by the heat capacity (12a-d).
(12a)
(12b)
(12c)
(12d)
The model can then be filled in with the calculated values for the thermal resistances and heat
capacities (13a-b).
(13a)
(13b)
(13c)
(13d)
The above model can be written in matrix form,
is the input vector, and the matrices

and

𝑢, where

is the state vector, 𝑢

are given in (14) and (15) respectively.

21

(14)

(15)

The output equation (16) yields

.
(16)

SENSOR BULB MODEL
The sensor bulb is a copper cylinder that contains a refrigerant charge. Broersen [1982]
has investigated the bulb time constant

. During normal use, the bulb’s time constant can

be as short as 16.2 seconds for tightly clamped and as long as 60 seconds for loosely attached to
the refrigerant tubing of an evaporator. In the LN2 TEV application, the bulb is suspended in
open air and relies on natural convection for heat transfer causing the time constant to be much
longer.
The heat transfer dynamic model of the sensor bulb is assembled in the same manner as
the previous heat transfer models (17).
(17)
The bulb time constant

= 1100 seconds [App. C], over 18 times longer than

when attached mechanically. The sensor bulb model can be written in matrix form (18-19).
(18)

𝑦

(19)

22

VALVE ACTUATOR MODEL
The valve actuator takes the measured temperature error,

-

, and converts it to

a heat transfer rate. The system acts to minimize the temperature error by regulating the heat
transfer rate out of the freezer. Physically it does this by regulating the orifice area at the valve
seat. The area and pressure drop across the orifice are factors that determine the mass flow rate
of LN2

. The heat transfer rate out of the freezer
(20)

The valve actuator applies a cooling function to the system (Fig. 11). Saturation of

occurs

because a maximum flow area is reached. The cooling function is linear for small errors with a
o

slope of -40 (W/ C) and saturates at

o

equal to 100 (W) for errors below -2.5 C and at 0 (W)

o

o

for errors at or above 0 C. The steady state operating point occurs at an error of -1 C and 40
(W).

The saturation limit of 100 (W) was determined from experimental data and (13a)

[App. D]. The air temperature has an average slope of -0.029 (K/s) during the transient and the
average value of
value of 100 (W) for

is 12 (K). Using these values with (13a) gives a maximum
o

. The set point was determined to be -30 C by measuring the

temperature at which the valve starts to open. The steady state temperature is approximately
o

o

-29 C which gives a steady state error of -1 C and steady state heat transfer rate of 40 (W). A
o

o

linear interpolation of the points [0 C error, 0 (W)] and [-1 C error, 40 (W)] gives a slope of -40
o

(W/ C).

23

Heat Transfer Rate, W

d(QLN2)/dt As a Function of Temperature Error
120
100
80
60
40
20
0
-3

-2

-1

0

1

Temperature Error, oC
Figure 11. LN 2 TEV non-linear cooling function with saturation.
CONTROL MODEL SIMULATION
Simulation of the control model yields results close to the experimental data (Fig. 12).
The difference in the transient behavior can be attributed to the linear thermal model of the
freezer chest. The model is valid for small variations about the operating point. The transient
o

covers a temperature change of over 60 C which leads to inaccurate assumptions for material
properties and convective heat transfer coefficients. The model does predict the qualitative
behavior of the freezer that includes an overshoot of the desired operating temperature. At
steady state conditions the model matches experimental data. The difference in overshoot
o

temperature is 4.5 C, or 7%. Experimental data show the system reaching the peak overshoot
temperature at 2134 seconds. The simulation reaches the peak overshoot at 2974 seconds, 39%
o

longer. The experimental data show the air temperature rebounding to -26.4 C before finally
o

o

dropping to a steady state vale of -29 C. The simulation gives a rebound temperature of -27.0 C
o

before it reaches steady state at -29 C.

24

Temperature, oC

Experimental and Simulated Air Temperature
30
20
10
0
-10
-20
-30
-40
-50

Experiment
Simulation

0

5000

10000
15000
20000
Time, seconds
Figure 12. Simulated freezer air temperature matches the steady state
experimental values and captures the qualitative transient behavior
The initial bow of the model is attributed to a difference in predicted heat transfer rate.
The model uses an initial value of 100 (W). In reality, the system first receives a burst of warm
nitrogen gas. Then cooler nitrogen gas moves its way through the supply tube. As the tube cools
down, LN2 can begin to propagate towards the valve. This process takes a certain amount of
time which the model does not take into account.
The simulation of the bulb temperature is accurate at steady state and predicts qualitative
transient behavior well (Fig. 13). This is not surprising as the bulb model was built for steady
state conditions like the chest model. The simulation results suggest that the model predicts a
longer bulb time constant than the actual time constant.

25

Experimental and Simulated Bulb Temperature

30
Temperature, oC

20

Experiment

10

Simulation

0
-10
-20
-30
-40
0

5000

10000
Time, seconds

15000

20000

Figure 13. The simulated sensor bulb temperature decreases slower than
experimental measurements but matches the steady state values.
The simulation also predicts the heat transfer rate

(Fig. 14). The cooling function

starts out in saturation at 100 (W). Once the bulb temperature reaches the operating point
drops off dramatically as the air temperature overshoots the operating point. The steady state
value of

is 39 (W). This matches qualitative behavior of valve as well as the steady state

Heat Transfer Rate, W

Simulation of d(QLN2)/dt As a Function of Time

120
100
80
60
40
20
0
0

5000

10000
Time, seconds

15000

20000

Figure 14. The simulation predicts the qualitative behavior of the valve as well as
the correct steady state heat transfer rate.

26

heat transfer rate.
Alterations to the model can be made to produce quantitatively more accurate simulation
results during the transient. Three parameters that can be changed are

,

, and the heat

capacities of the freezer components. The transient time can be shortened by decreasing
to 360 seconds and increasing

to 180 (W). By increasing each freezer chest component’s

heat capacity by a factor of 1.75 the overshoot is controlled. While these alterations change the
transient behavior of the model, the steady state results to do not significantly change. The
alterations were made in an iterative process to better fit the experimental data.

Experimental Data and Altered Model Simulation
Simulated Air Temperature, deg C

Experimental Bulb Temperature, deg C

Simulated Bulb Temperature, deg C

Experimental Mass Flow Rate, g/min
Temperature (oC) , Mass Flow Rate (g/min)

Experimental Air Temperature, deg C

Simulated Mass Flow Rate, g/min

80
60
40
20
0
-20
-40

-60
0

2000

4000
6000
Time, seconds

8000

10000

Figure 15. Timing and overshoot is improved with alterations to the model while
steady state behavior is not significantly changed.

27

The simulation of the altered model yields results that closely match the experimental
data (Fig. 15). The simulated air temperature still decreases too quickly, however after about
160 seconds its slope closely resembles the experimental measurements. The simulated peak
overshoot is 22% more than experimental measurements, although the model predicts accurately
the time at which the peak overshoot occurs. This time matches the time at which the flow of
LN2 begins to be decrease. As the bulb temperature drops below the set point, the flow of LN 2
quickly stops. The flow of LN2 increases to its steady state value once the bulb temperature rises
to its steady state value.
The simulated mass flow rate of LN2 is calculated by dividing the simulated heat transfer
rate by the specific heat transfer of nitrogen. It is assumed that there is 50% quality flow during
the transient. It was observed that during the transient there was two-phase flow of nitrogen
through the TEV and that about half of the LN2 boils before reaching the TEV. The specific heat
transfer for 50% quality flow is 265,500 (J/kg). At steady state it was observed that only gas
phase nitrogen exits the TEV. A pure gas phase flow has a specific heat transfer of 113,200
(J/kg). The specific heat transfer for the corresponding quality of flow was used to calculate the
transient and steady state mass flow rates.

The simulated

matches very well with

averaged experimental measurements.
The model can be used to predict the system response to a power failure and a dramatic
increase in room temperature (Fig. 16). The simulation assumes excellent insulation of the LN 2
supply tank and supply line such that liquid phase nitrogen is delivered to the chest. The
o

simulation starts with the air temperature at -35 C and the conventional refrigeration system
o

running. The ambient room temperature is 23 C. A power loss occurs causing the refrigeration

28

Temperature (oC), Mass Flow Rate
(g/min)

Simulated System Response to Complete Power Loss
15
10
5
0
-5
-10
-15
-20
-25
-30
-35
-40

LN2 Mass Flow, g/min
Air, deg C
Bulb, deg C

0

5000

10000

15000

20000

25000

30000

35000

40000

Time, seconds
Figure 16. Simulation shows that the LN 2 TEV system is well suited to backup a
primary refrigeration system during the worst conditions of a power failure.
o

system to shut down. In a worst case scenario the room temperature instantly increases to 50 C
o

(122 F). After about 12 minutes the bulb temperature rises above the set point and triggers the
o

flow of LN2. At this point the air temperature has risen to -23 C. However, it is known that the
simulation overshoots by 22% more than experimental data which suggests that the air
o

temperature would rise no higher than -25 C. The air temperature would reach a steady state
o

temperature of -28.6 C with a mass flow rate of 9.1 (g/min) corresponding to a heat transfer rate
of 55 (W). The simulation shows that the LN2 TEV system quickly regulates the freezer
temperature. If the power failure lasts 7 days, the mass flow rate of 9.1 (g/min) corresponding to
30 gallons of LN2, costs $6. This an extremely low cost for the value of the biological material
being preserved.
The simulation is a worst case scenario. When power loss occurs, the room temperature
would not instantly rise to the outside temperature. Also the chest is assumed to be empty. If the
29

chest were full the freezer heat capacity would increase greatly. The higher heat capacity and the
gradual rise in room temperature would result in a very slow increase in freezer air temperature.
The increased time constant of the freezer would be many times longer than the bulb time
constant so there would be little or no overshoot of the set point temperature.

30

CONCLUSION
Experimental results of the LN2 TEV indicate successful operation. The steady state
o

o

freezer air temperature was regulated to within 1 C of the LN2 TEV set point of -30 C. Mass
flow rate measurements show proper actuation of valve motion as the flow rate dropped to a
steady value as the air temperature reached its steady state value. Increased steady state mass
flow rate and temperature overshoot in the transient were the result of a poorly insulated, small
commercial LN2 supply tank and an empty freezer chest. The tank was incapable of supplying
the TEV with liquid N2 during the low steady state flow rates because the liquid would boil upon
reaching the top of the tank. A chest full of biological material would reduce or eliminate
overshoot because of the great increase in heat capacity of the freezer. However, these issues do
not affect the functionality of the LN2 TEV.
Mathematical models were developed to describe system behavior and estimate system
parameters. Simulation of the model predicted qualitative transient behavior, as well as peak
overshoot timing and steady state temperatures and mass flow rates that matched experimental
data. Simulation of a refrigeration power loss and ambient temperature rise situation shows that
the LN2 TEV is capable of quickly regulating freezer air temperature. This is due to the LN2
TEV’s high gain and reserve cooling capacity at steady state.
Further work on the LN2 TEV should involve testing with proper insulation and tubing as
well as different TEV refrigerant charges. Vacuum insulated tubing and proper tank insulation
would better simulate a commercial application of the LN 2 TEV. Different refrigerant charges
o

would change the operating temperature. One could be used in a backup system for a -86 C
freezer, a common cryogenic appliance. Though the operating temperature set point changes

31

with bulb refrigerant charge, the valve would function in the same way as was experienced in
experiments and described by the mathematical model.
The LN2 TEV backup refrigeration system meets the need for a more reliable backup
refrigeration system for the long term storage of biological material. A common TEV was
modified to be used with LN2 by improving an important seal and insulating the valve actuator.
The TEV operates without the use of an external energy source as it converts its own
thermodynamic energy into mechanical movement.

The LN2 TEV system is capable of

operating in the presence of water with the cold and pressurized chest able to seal leaks in water
up to 50ft deep. The only external requirement for the system is a large LN 2 source which is
common in facilities that store biological material. Use of the LN2 TEV backup refrigeration
system in research and storage facilities could save thousands of biological samples, worth
millions of dollars and years of research, from future natural disasters.

32

APPENDICES

33

APPENDIX A : THERMAL INSULATION CALCULATIONS
Material properties were taken from [Incropera et al., 2007] and [Callister, 2007]. The
conduction heat transfer rate from the diaphragm to the valve body

𝐿
where

(A-1)

is the thermal conductivity of the polycarbonate extension,

area, 𝐿 is the extension length,

is the cross-sectional

is the diaphragm temperature, and

is the valve body

temperature. The convection heat transfer rate from the freezer air to the midpoint of the
extension
(A-2)
where

is the convection heat transfer coefficient,

is the extension surface area, and

is

the freezer air temperature. Setting (A-1) and (A-2)) equal to each other and solving for 𝐿

𝐿

(A-3)

Inserting property values, dimensions, and temperatures yields

𝐿

(A-4)

Carrying out the calculation and double the length for a factor of safety for 2

𝐿

𝑥

(A-5)

The extension rod of length 2.6 cm must be able to handle the forces exerted on it by the
diaphragm and spring without buckling. The maximum force that can be placed on the rod

𝐿

without buckling

𝑎

𝑥

𝑥
𝑁
34

(A-6)

APPENDIX B : RESISTANCE AND HEAT CAPACITY CALCULATIONS
Material properties were taken from [Incropera et al., 2007] and [Callister, 2007].
INTERIOR (properties evaluated at 250K)

0.65 m

0.60 m

0.21 m
m

0.25

0.42 m

0.39 m
Figure 17. Chest interior dimensions.
Area
Aluminum sides and bottom:
[(0.65 m x 0.39 m) + (0.21 m x 0.40 m)] x 2 = 0.66 m

2

2

(0.65 m x 0.42 m) x 2 = 0.54 m

2

(0.60 m x 0.42 m) x 1 = 0.25 m
Aluminum Total = 1.45 m

2

Polystyrene top:
2

(0.60 m x 0.42 m) x 1 = 0.25 m
Total Interior Area : 1.70 m

2

Resistance
-4

2

-6

Ral = 7.5x10 m/(170 W/m-K x 1.45 m ) = 3.0x10 K/W

35

-4

2

-2

Rps = 7.5x10 m/(0.13 W/m-K x 0.25 m ) = 2.3x10 K/W
-1

-1 -1

-6

Rint_wall = (Ral + Rps ) = 3.0x10 K/W
Volume
2

-4

-3

Aluminum : 1.45 m x 7.5x10 m = 1.1x10 m
2

-4

-4

3

Polystyrene : 0.25 m x 7.5x10 m = 1.9x10 m
-3

Total : 1.3x10 m

3

3

Heat Capacity
-3

3

3

-4

3

3

3

Cal = (1.1x10 m )(2770 kg/m )(830 J/kg-K) = 2.5x10 J/K
2

Cps = (1.9x10 m )(1050 kg/m )(1170 J/kg-K) = 2.3x10 J/K
3

Cint = Cal + Cps = 2.7x10 J/K
EXTERIOR (aluminum)

0.79 m

0.74 m

0.21 m
m

0.25

0.56 m

0.53 m
Figure 18 Chest exterior dimensions.
Area
2

(0.79 m x 0.53 m) x 2 = 0.84 m

2

(0.54 m x 0.21 m) x 2 = 0.22 m

36

2

(0.74 m x 0.56 m) x 2 = 0.82 m

2

(0.79 m x 0.56 m) x 2 = 0.88 m
Total Exterior Area = 2.76 m

2

Resistance
-4

2

-6

Rext_wall = 7.5x10 m/(176 W/m-K x 2.76 m ) = 1.5x10 K/W
Volume
2

-4

-3

Vext = 2.76 m x 7.5x10 m = 2.1x10 m

3

Heat Capacity
-3

3

3

3

Cext = (2.1x10 m )(2770 kg/m )(871 J/kg-K) = 5.1x10 J/K
INSULATION (urethane foam)
0.67 m

0.74 m

0.72 m

0.49 m
0.25

0.56 m

0.46 m

Figure 19. Chest insulation dimensions.
Area
2

(0.72 m x 0.46 m) x 2 = 0.66 m

2

(0.47 m x 0.21 m) x 2 = 0.198 m

37

0.21 m
m

0.65 m

0.60 m

2

(0.67 m x 0.49 m) x 2 = 0.66 m

2

(0.72 m x 0.49 m) x 2 = 0.70 m
Total Insulation Area = 2.22 m

2

Resistance
-2

2

Rins = 7.0x10 m/(0.026 W/m-K x 2.22 m ) = 1.2 K/W
Volume
Sides, top and bottom:
0.56 m x [(0.79 m x 0.74 m) – (0.65 m x 0.60 m)] = 0.11 m

3

Front and back: using result from interior area calculation
2

-2

3

0.66 m x 7x10 m = 0.05 m
Vins = 0.16 m

3

Heat Capacity
3

3

4

Cins = (0.16 m )(70 kg/m )(1045 J/kg-K) = 1.2x10 J/K
AIR
Volume
Using area result from interior front and back panel area calculation
2

3

Vair = 0.33 m x 0.42 m =0.14 m
Heat Capacity
3

3

2

Cair = (0.14 m )(1.4354 kg/m )(1006 J/kg-K) = 2.0x10 J/K

38

SUMMARY
Table 5. Resistance and heat capacity summary
Component Thermal Resistance R (K/W)
Heat Capacity C (J/K)
2
__
Air
2.0 x 10
3
Interior
0.12
2.7 x 10
4
Insulation
1.2
1.2 x 10
3
Exterior
0.018
5.1 x 10

39

APPENDIX C : SENSOR BULB HEAT TRANSFER
-2

L = 5.0 x 10 m

-2

Copper

D = 1.3 x 10 m

Bulb

Tbulb = 244.5 K

-3

t = 1.0 x 10 m
Tair = 244.2 K

Figure 20. Sensor bulb geometry.
Conservation of energy is used to model the heat transfer from the freezer air at
temperature

to the sensor bulb at temperature

energy in the bulb

(C-1). The time rate of change of

is equal to the heat transfer rate into the bulb

.
(C-1)

The change in energy stored in the bulb is equal to the product of the bulb’s heat capacitance
and the time rate of change of the bulb’s temperature

(C-2). The heat transfer rate

into the bulb is equal to the temperature difference between the air and bulb divided by the
thermal resistance

.
(C-2)

The calculation of the heat capacitance of the bulb

is begun by calculating the surface area

of the bulb (C-3). The total surface area is approximated as the surface area of a cylinder and a
sphere.
40

𝐿

𝑥

𝑥

𝑥

𝑥

(C-3)

Next the volume is calculated by multiplying the surface area by the thickness of the bulb’s wall
(C-4).

𝑡

𝑥

𝑥

𝑥

(C-4)

The mass of the bulb is the product of the volume and density (Table 6) of the copper bulb (C-5).
Table 6. Material properties of copper at 244.5 K [Incropera et al., 2007]
3
Density ρ (kg/m ) Specific Heat c (J/kg-K)
8933
369

𝑥

𝑥

(C-5)

Finally, the heat capacitance of the bulb is product of the bulb mass and the specific heat (C-6).

𝑥
The thermal resistance

between

(C-6)
and

is equal to the product of the convection heat

transfer coefficient and bulb area. The process of calculating the convection heat transfer
coefficient

𝑎 (C-7) using properties of air

starts with calculating the Raleigh number

(Table 7) [Incopera et al., 2007].
Table 7. Thermal properties of air at 244.2 K [Incropera et al., 2007]
Thermal
Density Kinematic
Thermal
Thermal
Prandtl
Expansion
ρ
Viscosity ν
Diffusivity Number
Conductivity k
Coefficient β
3
2
2
(W/m-K)
Pr
(kg/m )
(m /s)
α (m /s)
-1
(K )
1.4354

10.99x10

-6

21.8x10

-3

15.3x10

𝑎

-6

𝑣

41

0.722

-3

4.0x10

(C-7)

The Raleigh number is used in the calculation of the Nusselt number 𝑁𝑢 using the correlation
for a horizontal cylinder under natural convection (C-8) [Incropera et al., 2007].

𝑁𝑢

𝑎

𝑟

(C-8)

The convection heat transfer coefficient is directly calculated using the Nusselt number (C-9).

𝑁𝑢

(C-9)

The thermal resistance is calculated using the convection heat transfer coefficient and the surface
area of the bulb (C-10).
(C-10)
The differential equation (C-11) describing the time rate of change of the sensor bulb’s
temperature comes from rearranging (C-2).
(C-11)
The time constant of the bulb

is equal to the product of the thermal resistance and the

bulb’s heat capacity (C-12).

𝑠

42

(C-12)

APPENDIX D : HEAT TRANSFER RATES CALCULATION
Material properties were taken from [Incropera et al., 2007] and [Callister, 2007].
Steady State:
The steady state heat transfer rate into the freezer is equal to the temperature difference between
the exterior of the freezer wall and the interior of the freezer wall divided by the total thermal
resistance of the freezer wall.

𝑥

𝑥

The steady state heat transfer rate out of the freezer from the heat absorbed by the temperature
change of the nitrogen gas is equal to the product of the specific heat transfer rate of nitrogen and
the mass flow rate of nitrogen.

𝑥

𝑠

𝑥

𝑠

The two heat transfer rates are extremely close as expected for steady state conditions.

Saturation:
The linear nature of the temperature drop of the air implies a constant heat transfer rate. The
TEV is a proportional controller but includes saturation due to maximum and minimum orifice

43

areas. The minimum heat transfer rate is 0 (W) when the valve is closed. The slope of the Air
Temperature line will determine the maximum heat transfer rate. The slope is determined from
o

o

the points (1 min, 23.85 C) and (33.5 min, -33.08 C) (Table 8).

Table 8. Thermal parameters used in the transient heat tran sfer rate calculation

𝑠
2.0 x 10

2

Avg.

-2

0.12

-2.9 x 10

𝑥

44

12

REFERENCES

45

REFERENCES
Air Liquide, 2009, “Gas Encyclopaedia”, Air Liquide Group, Paris, France
http://encyclopedia.airliquide.com/Encyclopedia.asp?GasID=141
Associated Press, 2005, “Katrina floods wipe out years of research”, MSNBC, September 14,
http://www.msnbc.msn.com/id/9339191/.
Broersen, P.M.T., 1982, “Control with a thermostatic expansion valve”, International Journal of
Refrigeration, Vol. 5, No. 4, July, pp. 209-212, Butterworth and Co. (Publishers), Ltd., London,
England.
th

Callister, W.D., Jr., 2007, Materials Science and Engineering an Introduction (7 ed), pp. A5A26, John Wiley & Sons, Inc., Hoboken, NJ.
Incropera, F.P., DeWitt, D.P., Bergman, T.L., and Lavine, A.S., 2007, Introduction to Heat
th
Transfer (5 ed), pp. 840-855, John Wiley & Sons, Inc., Hoboken, NJ.
Institute for Health Care Studies, 2010, “ICHS Brochure”, College of Human Medicine,
Michigan State University, http://ihcs.msu.edu/pdf/IHCSBrochure_000.pdf.
th

Moran, M.J. and Shapiro, H.N., 2004, Fundamentals of Engineering Thermodynamics (5 ed),
pg. 592, John Wiley & Sons, Inc., Hoboken, NJ.
Parker Hannifin Corporation, 2007, “Thermostatic and Automatic Expansion Valves, Catalog
E-1”, Parker Hannifin Corporation Climate and Industrial Controls Group, Cleveland, OH,
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