Heat Transfer and Thermodynamic Modeling of a Stacked Disk Spiral Channel Counterflow Heat Exchanger for Use in Joule-Thomson Cryocoolers

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HEAT TRANSFER AND THERMODYNAMIC MODELING OF A STACKED DISK SPIRAL CHANNEL COUNTERFLOW HEAT EXCHANGER FOR USE IN JOULE-THOMSON CRYOCOOLERS

By Jonah Latham Dunham

A Thesis Submitted to the Faculty of the AEROSPACE AND MECHANICAL ENGINEERING DEPARTMENT

In Partial Fulfillment of the Requirements For the Degree of

MASTER OF SCIENCE WITH A MAJOR IN MECHANICAL ENGINEERING

In the Graduate College

University of Arizona

2005

1 STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this thesis are allowable without special permission, provided that accurate knowledge of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of this material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

Signed: ______

APPROVAL BY THESIS DIRECTOR

This thesis has been approved on the date shown below:

______

Larry D. Sobel Date Adjunct Professor in Aerospace and Mechanical Engineering

2 TABLE OF CONTENTS

ACKNOWLEDGEMENTS...... 5

DEDICATION...... 6

LIST OF FIGURES...... 7

LIST OF TABLES ...... 9

NOMENCLATURE...... 10

ABSTRACT...... 11

CHAPTER 1 ...... 12

Introduction...... 12

1.1 Background...... 12

1.1.1 Joule-Thomson Cryocoolers...... 12

1.1.2 Silicon Disk Spiral Channel Heat Exchanger...... 14

1.1.3 Joule-Thomson Expansion ...... 18

1.2 Literature Review ...... 20

1.3 Research Objectives...... 23

CHAPTER 2 ...... 25

Mathematical Formulations ...... 25

2.1 Counterflow Heat Exchanger Theory ...... 26

2.1.1 Spiral Channel Disk Heat Exchanger Theory...... 29

2.2 Joule-Thomson Expansion ...... 31

2.3 Refrigeration Capacity ...... 32

2.4 Pressure Drop ...... 33

CHAPTER 3 ...... 36

3 Numerical Formulations ...... 36

3.1 Discretization of Heat Exchanger...... 36

3.2 Linear Simultaneous Equation Solver...... 37

CHAPTER 4 ...... 40

Integration of REFPROP and Mathcad ...... 40

4.1 REFPROP...... 40

4.2 Mathcad...... 40

4.3 REFPROP and Mathcad...... 41

CHAPTER 5 ...... 43

Results...... 43

5.1 REFPROP Validation ...... 43

5.2 Numerical Study of Varying Number of Disks...... 45

5.3 Numerical Study of Varying Volumetric Flow Rate ...... 56

5.4 Discussion ...... 66

5.4.1 Varying Number of Disks ...... 66

5.4.2 Varying Volumetric Flow Rate...... 70

CHAPTER 6 ...... 73

Conclusions and Recommendations...... 73

6.1 Conclusions ...... 73

6.2 Recommendations...... 74

REFERENCE...... 76

APENDIX A...... 78

Files on Disk ...... 78

ACKNOWLEDGEMENTS

I would like to acknowledge the help of Eric W. Lemon and Chris D. Muzny, both of the

National Institute of Standards and Technology, and both of whom were instrumental in helping me understand how C++ interfaces with REFPROP (NIST Standard Database

23).

DEDICATION

I would like to dedicate this work to my family and friends for standing by me when things got tough, for never giving up on me, and for their endless love. I would like to especially dedicate this to my little buddy Chali Scooter, who got me through most of my time here at the University of Arizona. Unfortunately, he passed on before he could see me all the way to the end, but without him I never would have made it beyond my first year.

LIST OF FIGURES

Figure 1.1 Joule-Thomson cryocooler...... 13

Figure 1.2 Temperature-entropy diagram for an ideal Joule-Thomson cryocooler...... 14

Figure 1.3 Individual silicon disk spiral channel heat exchanger element...... 16

Figure 1.4 Silicon heat exchanger stack...... 17

Figure 1.5 Counterflow heat exchanger in cryostat housing ...... 17

Figure 1.6 Inversion curve for Joule-Thomson expansion ...... 19

Figure 1.7 Hampson type JT cooler ...... 21

Figure 1.8 Spiral coil heat exchanger; (a) schematic sketch of SCHE; (b) sketch of one coild...... 22

Figure 2.1 Schematic diagram of counterflow heat exchanger...... 26

Figure 2.2 Temperature profile across a counterflow heat exchanger from hot to cold fluid ...... 27

Figure 2.3 Archimedes’ spiral...... 30

Figure 3.1 Discretization process of counterflow heat exchanger...... 36

Figure 5.1 Validation of isobaric heat capacity ...... 43

Figure 5.2 Validation of dynamic viscosity...... 44

Figure 5.3 Validation of thermal conductivity...... 44

Figure 5.4 High-pressure channel Peclet number, varying number of disks...... 48

Figure 5.5 Low-pressure channel Peclet number, varying number of disks ...... 49

Figure 5.6 High-pressure channel Reynolds number, varying number of disks...... 50

Figure 5.7 Low-pressure channel Reynolds number, varying number of disks ...... 50

7 Figure 5.8 Outlet temperature of high-pressure stream, varying number of disks ...... 51

Figure 5.9 Number of transfer units, varying number of disks...... 52

Figure 5.10 Heat exchanger effectiveness, varying number of disks...... 53

Figure 5.11 Quality at state 3, varying number of disks ...... 54

Figure 5.12 Parasitic heat load, varying number of disks ...... 55

Figure 5.13 Refrigeration capacity, varying number of disks ...... 55

Figure 5.14 Pressure drop for high-pressure stream, varying number of disks...... 56

Figure 5.15 High-pressure channel Peclet number, varying standard volumetric flow rate

...... 59

Figure 5.16 Low-pressure channel Peclet number, varying standard volumetric flow rate

...... 59

Figure 5.17 High-pressure channel Reynolds number, varying standard volumetric flow rate...... 60

Figure 5.18 Low-pressure channel Reynolds number, varying standard volumetric flow rate...... 61

Figure 5.19 Outlet temperature of high-pressure stream, varying standard volume flow rate...... 62

Figure 5.20 Number of thermal units, varying standard volume flow rate ...... 62

Figure 5.21 Heat exchanger effectiveness, varying standard volume flow rate...... 63

Figure 5.22 Quality at state 3, varying standard volume flow rate...... 64

Figure 5.23 Refrigeration capacity, varying standard volume flow rate...... 65

Figure 5.24 !P/P for high-pressure stream, varying standard volume flow rate...... 66

LIST OF TABLES

Table 2.1 Property determination at each state...... 25

Table 5.1 Simulation Results for Nitrogen M = 5 SLPM, Varying Number of Disks...... 46

Table 5.2 Simulation Results for Argon M = 5 SLPM, Varying Number of Disks ...... 46

Table 5.3 Simulation Results for Krypton M = 5 SLPM, Varying Number of Disks ...... 47

Table 5.4 Simulation Results for Nitrogen, N = 6, Varying Volumetric Flow Rate ...... 57

Table 5.5 Simulation Results for Argon, N = 6, Varying Volumetric Flow Rate ...... 58

Table 5.6 Simulation Results for Krypton, N = 6, Varying Volumetric Flow Rate ...... 58

NOMENCLATURE

Latin 2 Ac cross-sectional area, m 2 AHT heat transfer surface area, m cp isobaric heat capacity, J/kg·K dh hydraulic diameter, m f friction factor 2 hht convective heat transfer coefficient, W/m ·K h enthalpy, J/kg k thermal conductivity, W/m·K L length of channel, m Lstack length of heat exchanger stack, m n twice the number of heat exchanger elements N number of disks • m mass flow rate, kg/s • M mol flow rate, mol/s M molecular weight, g/mol P pressure, psi p perimeter, m Pr Prandtl number Q standard volume flow rate, SLPM q heat transfer rate, W Re Reynolds number s entropy, J/kg·K t thickness, m U velocity, m/s x quality

Greek " heat exchanger effectiveness # angle for spiral channel, rad µ dynamic viscosity, N·s/m2 µJT Joule-Thomson coefficient, K/Pa $ kinematic viscosity, m2/s % density, kg/m3

Subscripts h high pressure stream i fluid inlet conditions l low pressure stream o fluid outlet conditions

ABSTRACT

Numerical studies of an innovative Joule-Thomson cryocooler were performed in order to characterize overall system performance. A Joule-Thomson cryocooler consists of a recuperative heat exchanger, an expansion valve, and a thermal reservoir. The technical challenge addressed in this thesis is the numerical analysis of a novel counterflow heat exchanger design recently patented by Raytheon. The heat exchanger of interest is a stack of heat exchanger disks made of silicon with spiral flow channels for the high-pressure stream. The spiral heat exchanger can be modeled as counterflow only after modifications are made to account for the spiral geometry and the asymmetry in the cooling streams. The energy balance equations associated with the heat exchanger were solved numerically in order to determine the heat exchanger outlet temperatures and pressures of the two streams. Real thermodynamic properties were used in the numerical analysis of the heat exchanger with the aid of a computer program called REFPROP that consists of a property engine and database of true thermophysical and transport properties. Numerical results were produced for the cryogens nitrogen, argon, and krypton. The analysis showed that, not surprisingly, heat exchanger effectiveness increased with both increasing number of disks and increasing flow rate, yet the analysis clearly showed a point of diminishing returns in terms of added heat transfer performance with additional heat transfer area. Overall refrigeration capacity and total flow pressure drop were both found to increase with increasing number of disks and increasing flow rate for the three fluids studied. Krypton resulted in higher refrigeration capacity and higher pressure drop as compared to nitrogen and argon at the same volumetric flow rate.

CHAPTER 1

Introduction

1.1 Background

1.1.1 Joule-Thomson Cryocoolers

The cryocooler most often used in the defense industry for missile seeker cooling systems employs cooling based on the Joule-Thomson (JT) effect, a process involving the isenthalpic expansion of high-pressure gas to create a mixed-phase bath at the desired cryogenic cooling temperature. The gas used in these systems is partially liquefied by the expansion and is then used to maintain an infrared (IR) sensing device at a fixed cryogenic temperature for the duration of the mission.

A heat exchanger is required in this device in order to reduce the temperature of compressed gas before undergoing expansion in the JT valve so that the trajectory of the gas along a line of constant enthalpy produces a 2-phase mixture of liquid and vapor. The high-pressure gas is supplied either directly from a compressor or indirectly from compressed gas stored in a pressure vessel or gas bottle. The heat exchanger and expansion valve are usually housed in an insulated infrared sensor chip Dewar package assembly. This complete system of a high-pressure gas source, counterflow heat exchanger, and isenthalpic valve is referred to as a Linde-Hampson (LH) cryocooler with

12 Joule-Thomson expansion, or more simply as a Joule-Thomson cryocooler, and is shown in Figure 1.1.

Figure 1.1 Joule-Thomson cryocooler

The fluid leaves a gas bottle at high pressure and room temperature and enters the recuperative counterflow heat exchanger. The high-pressure gas at P1 is cooled from T1 to

T2 and exits at a lower pressure P2. The gas at 2 then expands into the reservoir at a pressure P3 as a mixture of cold vapor and liquid.

The low pressure return fluid, at P4, is warmed from T4 to T5, while exchanging heat with the high pressure gas in the other channel, exiting the heat exchanger as exhaust at a lower pressure, P5. The JT effect facilitates the drop in temperature of the working fluid during the expansion of the high-pressure fluid from P2 to P3. As the reservoir is sunk to atmospheric pressure the expansion temperature T3 is the normal boiling point of that fluid. This boil-off gas then flows out to ambient through the heat exchanger, warming to nearly ambient as it simultaneously cools the high-pressure gas in the heat exchanger. A temperature-entropy diagram of an ideal JT cryocooler is shown in Figure 1.2.

Figure 1.2 Temperature-entropy diagram for an ideal Joule- Thomson cryocooler

Nitrogen, argon, and krypton are typical fluids used for this type of application, at gas bottle pressures of around 6000 psi. The initial storage pressure is required to ensure that the high-pressure cooling path in the heat exchanger is always at a pressure above the critical pressure. This guarantees that the isenthalpic expansion from state 2 to state 3 produces a 2-phase gas-liquid mixture. Maximum refrigeration capacity is obtained by having state 3 as close to saturated liquid as possible. T3 is set by the pressure maintained in the expansion reservoir, which is nearly atmospheric.

1.1.2 Silicon Disk Spiral Channel Heat Exchanger

The counterflow heat exchanger analyzed in this study has an interesting geometry. It is a heat exchanger made out of thin disks of silicon with a spiral heat transfer passage on

14 each side. Silicon was selected as the base material because of its excellent thermal properties – silicon at low temperatures has a very high thermal conductivity and thermal diffusivity – leading to high heat exchanger effectiveness. Silicon can also be easily machined with a new generation of programmable laser machining centers. A representative spiral disk is shown in Figure 1.3.

The high-pressure stream enters the heat exchanger spiral microchannel stack through a manifold at the centerline of the stack and spirals from the inner portion of the disk to the outer radius. At the maximum radius, the gas travels through a passage to the other side of the disk where it spirals inwards to the centerline again where it traverses through another passage to the next disk in series. There is a sealing disk of silicon soldered to each face of the disk with the spiral channels to seal the high-pressure gas stream. The low-pressure stream passes through the oval cutout axial vias. The use of Archimedes’ spiral geometry ensures equal spacing for the low-pressure vias.

Figure 1.3 Individual silicon disk spiral channel heat exchanger element

Each spiral disk is then stacked on top of one another to form a heat exchanger stack, creating a counterflow heat exchanger. The heat exchanger stack is shown in Figure 1.4.

A Teflon insulating disk is placed between each disk to ensure that each disk is isothermal and to minimize the parasitic heat load conducted through the silicon disks into the reservoir. The high-pressure stream spirals from the centerline to the outermost radius on one side of the disk and then from the outer radius to the inner radius on the underside of the disk and then through a hole in the Teflon insulation and ultimately to the next disk in series where the process repeats. The low-pressure stream passes through the oval cutout axial vias cooling the high-pressure gas stream in the spiral passageways.

Figure 1.4 Silicon heat exchanger stack

The heat exchanger stack is then placed in a cryostat housing that is well insulated, as shown in Figure 1.5.

Figure 1.5 Counterflow heat exchanger in cryostat housing

17 1.1.3 Joule-Thomson Expansion

For an ideal gas, the temperature change in isenthalpic expansion is zero. J.P. Joule and

William Thomson first investigated the behavior of non-ideal gases experiencing isenthalpic expansion in the early 1850’s. They expanded different gases from a high pressure initially at ambient temperature through a porous plug.

Isenthalpic constant duct area expansion results in pressure energy being dissipated in the form of heat without producing any useful work or increasing the velocity of the fluid.

Since exergy is destroyed and no work is produced, this process is thermodynamically irreversible. For a non ideal gas, this expansion results in a change in temperature across the valve.

Joule and Thomson observed that many gases experienced a decrease in temperature during this expansion process, though some gases such as hydrogen actually increased in temperature. This temperature behavior is described by the Joule-Thomson coefficient, which, according to Moran and Shapiro (2004) is defined as:

& 'T # µ JT = $ ! (1.1) % 'P "h=const

An example of an inversion curve is shown in Figure 1.6, taken from Moran and Shapiro

(2004). The green lines are isenthalps and the slope of the isenthalpic curve at any state is the JT coefficient at that state. Depending on the state, µJT may be positive, zero, or

18 negative. The inversion states are the states at which µJT is zero, with the solid black line representing the inversion curve. To the right of the inversion curve, the JT coefficient will be negative and the gas will warm on expansion. To the left of the inversion curve,

µJT will be positive, leading to a temperature drop through a JT valve. When at the inversion state, no temperature change takes place. Indeed, the inversion curve for a given substance can be thought of as a locus of states where the substance behaves as an ideal gas.

Figure 1.6 Inversion curve for Joule-Thomson expansion

The inversion temperature is typically 8-9 times larger than the substance critical temperature. For many gases, this results in an inversion temperature above ambient temperature. For other substances whose boiling point is very low, the maximum

19 inversion temperatures are less than the ambient. If these gases are expanded from high pressure and ambient temperature, they will experience an increase in temperature. It is for this reason that JT systems using these gases must be multi-staged, with the first stage used to bring the high-pressure gas below its inversion temperature at the storage pressure.

Furthermore, the more non-ideal a gas is, the more temperature drops through JT expansion and, interestingly, the more closely an isenthalpic process approximates the trajectory of an isentropic process.

1.2 Literature Review

A small number of experimental, numerical, and theoretical studies have been conducted in order to better understand the role of the counterflow heat exchanger in Joule-

Thomson systems. Little (1984), provides a thorough overview of miniature cryogenic refrigerators. He theoretically describes the considerations taken into account for both laminar and turbulent conditions inside the heat exchanger. He also discusses the effects of roughness on pressure drop through the length of each channel as compared to the classic studies of flow of fluids in smooth tubes. He describes the performance characteristics of JT systems with gases other than nitrogen or argon as well as the operation of open-cycle compared to closed-cycle refrigerators.

20 Xue, et al (2001) and Ng, et al (2002) performed both numerical and experimental studies on a Hampson-type counterflow heat exchanger. A Hampson-type counterflow heat exchanger is a widely used recuperative heat exchanger in JT systems, consisting of a tube and fin heat exchanger wound in a helix within the annular space of co-axial cylinders. This is shown in Figure 1.7, taken from Xue, et al (2001).

Figure 1.7 Hampson type JT cooler

The transient variation of temperature and pressure at the outlet of the heat exchanger were monitored experimentally for different high-pressure side inlet pressures, and heat exchanger effectiveness as a function of high-pressure side inlet pressure was determined.

The steady-state temperature and pressure profiles were determined numerically for the

JT cooler.

Chua, et al (2005) also performed numerical studies on a Hampson-type heat exchanger.

They determined the effects of load temperature, inlet pressure, and volumetric flow rate on cooling capacity. The also determined the effects of inlet pressure on coefficient of performance and effectiveness.

Bes and Roetzel (1993) and Ho, et al (1995) discuss the theory behind counterflow spiral coil heat exchangers (SCHE). A typical SCHE is shown in Figure 1.8, taken from Ho, et al (1995).

Figure 1.8 Spiral coil heat exchanger; (a) schematic sketch of SCHE; (b) sketch of one coil

22 The SCHE unit is comprised of layers of spiral coils stacked on top of one another. These spiral coils are circular-section tubes that are bent into circular spiral turns of reducing radii. A hot fluid flows radially inwards or outwards from the center core of the spiral while the cold fluid, usually liquid, flows in the coiled tubes. The cold liquid flow is countercurrent to the hot gas flow. Ho, et al compared experimental outlet temperatures of the hot and cold stream as well as heat exchanger effectiveness to theory.

1.3 Research Objectives

The types of heat exchangers described in the above section are vastly different from the spiral disk stacked heat exchanger investigated in this study, of which, no literature was found. As a result of the literature reviews, the following research objectives were established for the study:

1. Determine the heat exchanger effectiveness, expansion quality, and

refrigeration capacity as a function of number of stacked disks of a given spiral

geometry for the cryogenic fluids nitrogen, argon, and krypton.

2. Determine the pressure drop in the high-pressure channel as a function of

number of stacked disks of a given spiral geometry for the cryogenic fluids

nitrogen, argon, and krypton.

3. Determine research objectives 1 and 2 as a function of volumetric flow rate for

a fixed number of disks for each fluid of interest.

23 4. Determine if the pressure drops in the high and low-pressure channels are sufficiently low that it is safe to assume the counterflow heat exchanger operates at constant pressure.

CHAPTER 2

Mathematical Formulations

The temperature-entropy diagram illustrated in Figure 1.2 describes the thermodynamic process for the JT system studied. To address the problem of interest, each state must be determined. In order to determine a state, 2 properties must be fixed. Table 2.1 describes which states are known and which are unknown. Properties in black are known, those in red are unknown.

Table 2.1 Property determination at each state

State Property 1 Property 2

1 T1 P1

2 T2 P2

3 h3 = h2 P3

4 T4 = Tsat @ P4 = P3 P4 = P3

5 T5 P5

Table 2.1 shows that states 1 and 4 are initially defined. It also shows that if state 2 can be defined, state 3 is then automatically defined since the path from 2 to 3 is along an isenthalp. To solve the JT system, the outlet temperatures and pressures of the heat exchanger (T2, P2 and T5, P5) must be determined.

25 2.1 Counterflow Heat Exchanger Theory

The heat exchanger that cools the high-pressure gas from 1 to 2, while warming the cold exhaust gas from 4 to 5, is the component that ties this whole system together. Consider the counterflow heat exchanger illustrated in Figure 2.1. The high temperature fluid enters one end of the heat exchanger with a specified mass flow rate, inlet temperature

T1, and inlet pressure P1, and exits at lower temperature T2 and pressure P2. The low temperature boil-off fluid enters at the other end of the heat exchanger with the same mass flow rate, inlet temperature T4, and pressure P4, exiting the heat exchanger at constant pressure but at a higher temperature T5. Heat is transferred from the high temperature stream to the low temperature stream through a wall of thickness d and surface area As. The temperature profile from the hot stream to the cold stream is shown in Figure 2.2.

Figure 2.1 Schematic diagram of counterflow heat exchanger

Figure 2.2 Temperature profile across a counterflow heat exchanger from hot to cold fluid

A temperature drop !Th occurs adjacent to the wall in the hot stream, accompanied by a small temperature drop !Tw across the wall. A further temperature drop !Tl occurs adjacent to the wall in the cold stream.

If there is negligible heat transfer between the exchanger and its surroundings, as well as negligible potential and kinetic energy changes, application of the energy balance gives:

• & 'Th # dq = mh cph $ !dx (2.1) % 'x " and

• & 'Tl # ( dq = ml cpl $ !dx (2.2) % 'x "

• where q is the total heat transfer rate between the two fluids, m is the mass flow rate, T is the fluid temperature, dx is the incremental length of the heat exchanger, and cp is the

27 isobaric heat capacity of the fluid. The subscripts h and l refer to the high and low- pressure fluids. Equations 2.1 and 2.2 can be rewritten, respectively, as:

• ' (Th $ mh cph % " = hht,h ph (Tl !Th ) (2.3) & (x #

• ' (Tl $ ml cpl % " = hht,l pl (Th !Tl ) (2.4) & (x # where hht is the convective heat transfer coefficient and p is the perimeter across which heat transfer occurs. The convective heat transfer coefficient is determined from the appropriate Nusselt number, Nu:

Nu !k hht = (2.5) dh where k is the thermal conductivity of the fluid. According to Incropera and DeWitt

(1996), the Nusselt number for laminar and turbulent flow in a pipe is found be, respectively:

3.66 < Nu < 4.36 (2.6) and

4 / 5 n Nu = 0.023Re D Pr (2.7) where n = 0.4 for heating (Ts > Tm) and 0.3 for cooling (Ts < Tm), where Ts is the surface temperature of the pipe and Tm is the mean temperature of the fluid.

In heat transfer texts, it is common to define heat exchanger effectiveness as:

T ! T " = 1! 1 5 (2.8) T1 ! T4

28 where the subscripts 1, 4, and 5 refer to the state points shown in Figure 1.1. This effectiveness parameter " is a measure of how much heat was actually transferred divided by the maximum amount of heat available for transfer. According to Kays and London

(1984), the number of transfer units, NTU, for a counterflow heat exchanger is defined as:

! NTU = (2.9) 1"!

The number of thermal units is a dimensionless parameter that measures the capacity of a heat exchangers to change the temperature of the “minimum” fluid. The “minimum” fluid

• is that whose m!cp is the lower of the two fluids.

2.1.1 Spiral Channel Disk Heat Exchanger Theory

Since the stack of spiral disks functions in a manner identical to a linear path counterflow heat exchanger, equations 2.3 and 2.4 apply with some modifications to capture the spiral geometry and asymmetry in the cooling streams. For the counterflow heat exchanger described in Section 3.1, the lengths for both channels are the same. That need not necessarily be the case. In the spiral channel disk heat exchanger, the length scale for the high-pressure channel is:

Lh = 2Lspiral " N (2.10) where N is the number of disks and Lspiral is the spiral channel length per disk. The length scale !for the low-pressure return path is:

Ll = t ! N (2.11)

29 where t is the disk thickness.

The spiral geometry is that of Archimedes’ spiral, as shown in Figure 2.3.

Figure 2.3 Archimedes’ spiral

The spiral has a polar equation:

r = a" (2.12) where r is the radius of the spiral, a is an arbitrary constant, and # is the angle in radians. ! The distance between spirals is:

d = !a (2.13)

The length of a spiral of K turns is given by:

K"2# L a d (2.14) spiral = ! $ $ 0

For our spiral geometry:

5.588#10"3 a = (m) 10!

30 and

K = 5

The low pressure vias were modeled as bi-polar ellipses with major and minor axes of

610 µm and 244 µm, respectively, and center-to-center spacing of twice the minor axis.

2.2 Joule-Thomson Expansion

JT expansion is a completely irreversible, adiabatic process. The fact that the pressure change occurs too quickly for significant heat transfer to occur assures that the expansion is adiabatic and since no work is produced, the enthalpy of the expanding fluid remains constant while the entropy increases from the irreversibility of the expansion. Hence:

hi = ho (2.15) where h is enthalpy.

When this process is idealized in this manner, it is known as an isenthalpic throttling process. The temperature change in the working fluid accompanying this change in pressure is characterized by the JT coefficient, defined in equation 1.1. The slope of the isenthalpic curve at any state is the JT coefficient at that state. Depending on the state, µJT may be positive, zero, or negative.

The enthalpy at state 2 was found, and from equation 2.15, the enthalpy at state 3 was also found, and state 3 was defined. The quality at state 3 was determined by:

31 h3 ! h f x3 = (2.16) hg ! h f where h3 is the enthalpy at state 3, hf is the enthalpy at saturated liquid at P3, and hg is the enthalpy at saturated vapor at P3.

2.3 Refrigeration Capacity

Refrigeration capacity is a measure of how much cooling is available from the system. It is equal to the difference in enthalpy of the exhaust and incoming gases minus the parasitic heat load, or more formally:

• q = m(h4 ! h3 )! q parasitic (2.17) where qparasitic is the heat transferred from the top to the bottom of the heat exchanger stack and is defined as:

kTeflon Adisk !T q parasitic = (2.18) LTeflon where kTeflon is the thermal conductivity of Teflon, Adisk is the cross-sectional area of the disk, !T is the temperature difference from the top to the bottom of the heat exchanger stack, and LTeflon is the length of the stack including only the Teflon insulating disks.

"T = T1 !T4 (2.19)

" A = D 2 ! A (2.20) disk 4 disk vias and

32 LTeflon = tTeflon N (2.21)

-2 where Ddisk = 1.1176·10 m is the diameter of the silicon disk and Avias is the area of one via times the number of vias, and tTeflon is the thickness of one Teflon disk and is equal to twice the thickness of one silicon disk.

The parasitic heat load only takes into account the thermal conductivity of the Teflon because ksilicon >> kTeflon. In essence, the silicon disks can be considered to be short- circuits while the Teflon disks are very large resistors. In fact ksilicon = 148 W/m·K and kTeflon = 0.35 W/m·K, leading to an equivalent resistance of 0.35 W/m·K from:

kTeflon ! kSi kequivalent = (2.22) kTeflon + kSi

2.4 Pressure Drop

In order to accurately evaluate the thermal properties at each point along the heat exchanger, the pressure drop through the high and low-pressure channels must be determined. According to Bejan, et al, (1996), the pressure drop through a pipe is:

L !U 2 "P = f (2.23) d h 2 where f is the friction factor, L is the length of the pipe, % is the density, U is the bulk velocity of the fluid, and dh is the hydraulic diameter. The bulk velocity is found by:

• m U h = (2.24) !h Ac,h

33 and

• m U c = (2.25) !c Ac,c N via where Ac is the cross sectional area of the pipe. Ac for the low pressure via was defined above, while the high pressure cross sectional area is given by:

Ac,h = widthh ! heighth (2.26)

Hydraulic diameter is defined as:

4A d = c (2.27) h p with pl as the wetted perimeter of the low-pressure channel, and ph is the wetted perimeter of the high-pressure channel. Mass flow rate is defined as:

• • m = M ! M (2.28)

• where the mol flow rate M was determined from:

• mol M = 7.4!10"4 !Q (2.29) s where Q is standard volume flow rate in SLPM.

According to Fox and McDonald (1998), the friction factor for laminar flow in a pipe is only a function of Reynolds number and equals:

64 f = (2.30) Re

For turbulent flow in smooth pipes, Fox and McDonald (1998) suggest the Blasius correlation, valid for Re & 105:

34 0.316 f = (2.31) Re0.25 where Reynolds number is defined as:

U " d Re = h (2.32) ! where $ is the kinematic viscosity of the fluid, a strong function of temperature, which is determined by REFPROP function calls, as described in Chapter 4, at each step of the numerical integration.

CHAPTER 3

Numerical Formulations

3.1 Discretization of Heat Exchanger

In order to solve for the spiral disk heat exchanger outlet temperatures, equations 2.3 and

2.4 were discretized and solved numerically. The discretization process is shown in

Figure 3.1. The inlet temperatures Th,1 and Tl,n/2+1 are known and equal to T1 and T4 respectively. The desired results are the outlet temperatures, Th,n/2+1 and Tl,1, and will be equal to T2 and T5, respectively.

Figure 3.1 Discretization process of counterflow heat exchanger

Equations 2.3 and 2.4 are discretized as follows:

) ( mcp % & i # T T T (3.1) & # " h,i = ( l,i+1 ! h,i ) & hht,i "AHT # ' $high and

36 ! ) mcp & ' i $ T T T (3.2) ' $ # l,i = ( h,i " l,i+1 ) ' hht,i #AHT $ ( %low where AHT is the heat transfer surface area and the subscript i is a range variable from 0 to n/2-1. In this study, the azimuthal temperature variation of the low-pressure exhaust gas was not taken into account. The incremental heat transfer surface area is given by:

!AHT ,h = pHT ,h !x (3.3) and

!AHT ,l = pHT ,l !z (3.4) where !x and !z denote the incremental change in length along the high pressure and low-pressure channels, respectively, and are found from:

2L !x = h (3.5) n and

2L !z = c (3.6) n

3.2 Linear Simultaneous Equation Solver

Equations 3.1 and 3.2 create a set of linear simultaneous equations and can be written in matrix form as:

A" !T = B (3.7) where

37 &K h,0 0 0 '1 '1 '1 # $ 1 K 0 1 1 ! $ h,1 ' ' ! $ 1 1 ... '1 ! $ ! 1 1 1 K h,n / 2 1 (3.8) A = $ ' ! $ 0 0 0 0 K 1 1 1 ! $ l,0 ! $ '1 K l,2 1 1 ! $ '1 '1 0 ... 1 ! $ ! 1 1 1 0 0 K %$ ' ' ' l,n / 2'1 "! with the matrix filled in appropriately, and

& 'Th,0 # $ T ! $ ' h,1 ! $ ... ! $ ! 'Th,n / 2 1 'T = $ ( ! (3.9) $ 'T ! $ l,1 ! $ 'Tl,2 ! $ ... ! $ ! T %$ ' l,n / 2 "! and

&Tl,n / 2 'Th,0 # $T T ! $ l,n / 2 ' h,0 ! $ ... ! $ ! Tl,n / 2 'Th,0 B = $ ! (3.10) $T 'T ! $ h,0 l,n / 2 ! $Th,0 'Tl,n / 2 ! $ ... ! $ ! T T %$ h,0 ' l,n / 2 "! where Th,0 and Tl,n/2 are T1 and T4 respectively, and

" mcpi Ki = (3.11) hht,i !AHT

Equation 3.7 was solved by using Mathcad’s built in LU Decomposition solver function, outputting the !T matrix. An iteration loop compared the output to the previous iteration’s output and determined if the answer was within a specified tolerance. Within each iteration loop, the properties of the fluids at each point along the high and low pressure channels were determined using the REFPROP function calls that will be described in Chapter 4. Once the answer was within the tolerance, the temperature profiles for the high and low-pressure streams were determined and the output temperatures of the heat exchanger were recorded.

CHAPTER 4

Integration of REFPROP and Mathcad

4.1 REFPROP

REFPROP is the acronym for NIST Reference Fluid Thermodynamic and Transport

Properties, also known as NIST Standard Reference Database 23. It was developed by

Eric W. Lemon, Mark O. McLinden, and Marcia L. Huber of the Physical and Chemical

Properties Division of the National Institute of Standards and Technology (NIST).

REFPROP provides tables and plots of the thermodynamic and transport properties of industrially important fluids and their mixtures.

REFPROP is based on the most accurate pure fluid and mixture models currently available. A separate graphical user interface has been designed for the Windows operating system. A suite of FORTRAN subroutines have been developed to calculate thermodynamic and transport properties at a given (T,%,x) state. A collection of Dynamic

Link Libraries have been developed to allow linking between the FORTRAN subroutines and C++ and Visual Basic code, permitting developers to write their programs in any of the three languages mentioned above.

4.2 Mathcad

40 Mathcad was used to solve the heat exchanger model. In order for real fluid properties to be used, a way of integrating REFPROP into the Mathcad environment needed to be developed. As mentioned above, REFPROP links easily with C++ using the provided

C++ DLL. Mathcad, in turn, has capabilities that enable the user to create their own

Mathcad functions with all the functionality of Mathcad’s built in functions, such as customized error messages, interruption, and exception handling.

Mathcad user functions are created by writing a custom 32-bit Dynamic Link Library

(DLL). They must be written in a standard form that Mathcad can understand. To create a custom function, the following steps were taken:

1. Create the source code.

2. Compile the source code with a 32-bit compiler.

3. Link the object files together with the Mathcad-provided MCADUSER.LIB

library to create a DLL.

4. Place the DLL into the UserEFI subdirectory of the Mathcad install directory.

The source code was written in C++ and the 32-bit compiler used to create the REFPROP

DLL was Visual Studio .NET 2003.

4.3 REFPROP and Mathcad

Source code was written and compiled that linked REFPROP to Mathcad through a user- function DLL. Functions were written to calculate the following properties:

1. molecular weight, kg/mol

41 2. density, kg/m3

3. isobaric heat capacity, J/kg·K

4. dynamic viscosity, N·s/m2

5. kinematic viscosity, m2/s

6. thermal conductivity, W/m·K

7. Prandtl number

8. enthalpy, J/kg

9. saturation temperature, K

Most of the property calculations are performed outside of the saturation dome. Property

1 requires an input of fluid name. For properties 2 – 7, the inputs are fluid name, temperature, and pressure. For property 8, two functions were developed to accommodate the state of the fluid. One takes inputs of fluid name, temperature, and pressure for states outside the saturation dome and calculates either compressed liquid or superheated vapor enthalpy. The second requires inputs of fluid name, pressure, and phase and calculates saturated liquid or vapor enthalpy. Property 9 takes inputs of fluid name and pressure.

Thanks to Mathcad’s inherent unit conversion, the inputs of temperature and pressure can be in any units desired. Output properties, likewise, will be given in the appropriate units selected by the user.

CHAPTER 5

Results

5.1 REFPROP Validation

The Mathcad REFPROP user-defined functions were validated against thermophysical properties of matter tabulated in Incropera and DeWitt (1996). Isobaric heat capacity, dynamic viscosity, and thermal conductivity were calculated for nitrogen, oxygen, and helium as a function of temperature and compared to tabulated data. Figures 5.1, 5.2, and

5.3 show the validation results.

1.3 ) K •

g 1.2 k / J k (

y t i 1.1 c a p a c

t O2 REFPROP a 1.0 e h

O2 I&D c i r N2 REFPROP a

b 0.9

o N2 I&D s I

0.8 0 200 400 600 800 1000 1200 1400 Temperature (K)

Figure 5.1 Validation of isobaric heat capacity

43 6.0E-05

5.0E-05 ) 2 m / s • 4.0E-05 N ( y t i s o 3.0E-05 O2 REFPROP c s i v O2 I&D c i 2.0E-05 N2 REFPROP m a n N2 I&D y

D 1.0E-05 He REFPROP HE I&D 0.0E+00 0 200 400 600 800 1000 1200 1400 Temperature (K)

Figure 5.2 Validation of dynamic viscosity

0.40 ) K • m

/ 0.30 O2 REFPROP W (

O2 I&D y t i

v N2 REFPROP i t

c 0.20 N2 I&D u d

n He REFPROP o c

l He I&D a

m 0.10 r e h T

0.00 0 200 400 600 800 1000 1200 1400 Temperature (K)

Figure 5.3 Validation of thermal conductivity

The validation results show that the DLL written to interface REFPROP with Mathcad leads to correct property calculations of isobaric heat capacity, dynamic viscosity, and thermal conductivity for a variety of fluids. With this knowledge, it was with confidence that we could proceed to the next step of adding real fluid property calculation at each numerical step along the high and low-pressure channels of the heat exchanger.

5.2 Numerical Study of Varying Number of Disks

Nitrogen, argon, and krypton were selected as the cryogens with a standard volume flow rate of 5 SLPM. For each fluid, the number of silicon disks, N, was varied from 1 to 10.

For each case, the temperature profiles of the high and low pressure streams, the outlet pressures of the heat exchanger, the quality of the fluid at state 3, and the enthalpies at states 3 and 4 were calculated using Mathcad and REFPROP.

For a standard volume flow rate of 5 SLPM, the inlet temperature and pressure are, respectively, T1 = 298 K and P1 = 6000 psi. The inlet pressure P4 = 14.696 psi. The height and width of the high-pressure channel are heighth = 813 µm and widthh = 508 µm. The

-4 thickness of one disk is t = 1.27·10 m. The low-pressure inlet temperature, T4, is equal to 77.355 K for nitrogen, 87.302 K for argon, and 119.78 K for krypton. These are the normal boiling points for the respective fluid, which is defined as the saturation temperature at 1 atm. The tolerance between iterations was 1 K. The simulation results for nitrogen, argon, and krypton are presented in Tables 5.1, 5.2, and 5.3, respectively.

Table 5.1 Simulation Results for Nitrogen M = 5 SLPM, Varying Number of Disks

Temperature (K) Pressure (psi) Number of T T P P Disks 2 5 2 5 1 137.27 170.84 5999.95 14.69594 2 112.53 197.09 5999.92 14.69585 3 106.28 211.53 5999.88 14.69579 4 113.15 231.50 5999.81 14.69578 5 106.37 244.00 5999.73 14.69577 6 100.55 250.91 5999.64 14.69575 7 96.10 256.23 5999.56 14.69573 8 92.57 259.00 5999.48 14.69570 9 89.94 261.30 5999.40 14.69568 10 87.93 263.98 5999.31 14.69566

Table 5.2 Simulation Results for Argon M = 5 SLPM, Varying Number of Disks

Temperature (K) Pressure (psi) Number of T T P P Disks 2 5 2 5 1 154.79 182.59 5999.95 14.69591 2 130.67 211.52 5999.90 14.69579 3 135.26 234.06 5999.84 14.69573 4 125.06 250.64 5999.76 14.69570 5 116.39 259.40 5999.67 14.69567 6 110.02 264.92 5999.58 14.69563 7 105.28 268.09 5999.49 14.69559 8 101.60 270.45 5999.39 14.69556 9 98.60 271.93 5999.29 14.69553 10 96.64 273.65 5999.20 14.69549

46 Table 5.3 Simulation Results for Krypton M = 5 SLPM, Varying Number of Disks

Temperature (K) Pressure (psi) Number of T T P P Disks 2 5 2 5 1 206.57 207.88 5999.87 14.69585 2 164.10 236.63 5999.76 14.69563 3 149.58 248.43 5999.65 14.69545 4 143.33 255.70 5999.53 14.69529 5 139.00 260.49 5999.40 14.69514 6 136.08 263.87 5999.26 14.69503 7 129.10 266.14 5999.20 14.69473 8 133.15 269.58 5998.95 14.69482 9 130.47 259.74 5998.82 14.69425 10 129.07 260.12 5998.66 14.69430

In order to determine whether the model described in chapter 2 is valid, the Peclet number was determined. The Peclet number is defined as:

U " L convection Pe = = Re" Pr ! (5.1) # conduction

Where L is the length scale – in this case either !x or !z, depending on which stream is chosen – and ' is the thermal diffusivity defined as:

k # = (5.2) " !cp

The Peclet number is a dimensionless number relating the forced convection of a system to its heat conduction. For small Pe, conduction in the fluid stream is important. For high

Pe, convection dominates, and the temperature gradient inside the fluid can be neglected.

Peclet number decreases as the fluid flows through the high-pressure stream, while it increases through the length of the low-pressure channel. Pe is on the order of 104 for the high-pressure channel and on the order of 100 to 1000 for the low-pressure channel. This

47 is shown in Figures 5.4 and 5.5. For both channels, the Peclet number is sufficiently large that conduction between the fluid particles in each separate channel can be neglected and the model used is assumed valid.

1.00E+05 T1 = 298 K, P1 = 6000 psi nitrogen N = 4 P4 = 14.696 psi nitrogen N = 10 M = 5 SLPM argon N = 4 8.00E+04 argon N = 10

r krypton N = 4 e b krypton N = 10 m u

n 6.00E+04

t e l c e P

4.00E+04

2.00E+04 0 0.2 0.4 0.6 0.8 1

x/Lh

Figure 5.4 High-pressure channel Peclet number, varying number of disks

48 1600 nitrogen N = 4 T1 = 298 K, P1 = 6000 psi nitrogen N = 10 P4 = 14.696 psi argon N = 4 M = 5 SLPM argon N = 10 1200 krypton N = 4 r

e krypton N = 10 b m u

n 800

t e l c e P

400

0 0 0.2 0.4 0.6 0.8 1

z/Ll

Figure 5.5 Low-pressure channel Peclet number, varying number of disks

The high and low-pressure Reynolds number along the length of the channels are shown in Figures 5.6 and 5.7, respectively. The Reynolds number decreases through the length of the high-pressure channel, and was found to be both in the turbulent (Re > 2300) and laminar regimes. Low pressure Reynolds number was found to be laminar for all cases.

49 6000 T1 = 298 K, P1 = 6000 psi nitrogen N = 4 P4 = 14.696 psi nitrogen N = 10 5000 M = 5 SLPM argon N = 4

r argon N = 10 e 4000 b krypton N = 4 m u n krypton N = 10

s 3000 d l o n y e 2000 transition Re = 2300 R

1000

0 0 0.2 0.4 0.6 0.8 1

x/Lh

Figure 5.6 High-pressure channel Reynolds number, varying number of disks

2000 nitrogen N = 4 nitrogen N = 10 T1 = 298 K, P1 = 6000 psi P4 = 14.696 psi argon N = 4 M = 5 SLPM 1600 argon N = 10 krypton N = 4 r

e krypton N = 10 b

m 1200 u n

s d l o

n 800 y e R

400

0 0 0.2 0.4 0.6 0.8 1

x/Lh

Figure 5.7 Low-pressure channel Reynolds number, varying number of disks

50 ! Figure 5.8 shows the outlet temperatureT2 defined as:

! T2 T2 = (5.1) Tmin where Tmin = T4, the boil-off temperature for each fluid. Outlet temperature drops with increasing N. Numerical instabilities occur at low N for nitrogen and argon and are not related to fluid or heat transfer phenomena.

2.00 T1 = 298 K, P1 = 6000 psi P4 = 14.696 psi M = 5 SLPM )

min 1.75 T / nitrogen 2 numerical instabilities T (

argon e r

u krypton t a

r 1.50 e p m e T

t e l

t 1.25 u O

1.00 0 2 4 6 8 10 12 Number of Disks

Figure 5.8 Outlet temperature of high-pressure stream, varying number of disks

The number of transfer units is shown to increase with increasing N in Figure 5.9.

51 10 T1 = 298 K, P1 = 6000 psi P4 = 14.696 psi M = 5 SLPM 8

6 numerical instability U T N 4

nitrogen 2 argon krypton

0 0 2 4 6 8 10 12 Number of Disks

Figure 5.9 Number of transfer units, varying number of disks

Heat exchanger effectiveness increases with increased number of disks, as illustrated in

Figure 5.10. Figure 5.11 shows that the quality at state 3 decreases with increasing effectiveness, i.e. increasing number of disks. One might expect that for increasing number of disks, quality will continue to decrease, eventually equaling zero, meaning the state is 100% liquid. This may or may not be the case depending on the fluid and its melting line at 6000 psi. It the melting line is reached before entering the expansion valve, the fluid will solidify and clog up the line, ending all possibility for expansion. For nitrogen, the melting or solidification temperature at 6000 psi is 71.793 K. This is below the normal boiling point, meaning that it is possible for nitrogen to reach a quality of zero because the melting line will never be reached. Argon has a solidification temperature of

93.799 K, which is greater than its normal boiling point. This temperature, then, can conceivably be reached, meaning that there is a minimum quality at state 3. For this case,

52 the minimum enthalpy at state 3 is 0.11297. Similarly, krypton experiences a solidification temperature at 6000 psi greater than its normal boiling point. Tsolidify for krypton is 127.82 K, leading to a minimum quality at the expansion exit of 0.14285.

1.0 T1 = 298 K, P1 = 6000 psi

P4 = 14.696 psi 0.9 M = 5 SLPM

0.8 s s e n e v

i 0.7 t c

e nitrogen f f

E argon 0.6 krypton

0.5 numerical instability

0.4 0 2 4 6 8 10 12 Number of Disks

Figure 5.10 Heat exchanger effectiveness, varying number of disks

53 1.0 T1 = 298 K, P1 = 6000 psi P4 = 14.696 psi M = 5 SLPM nitrogen 0.8 argon numerical instabilities krypton 0.6 y t i l a u Q 0.4

0.2

0.0 0.4 0.5 0.6 0.7 0.8 0.9 Effectiveness

Figure 5.11 Quality at state 3, varying number of disks

Parasitic heat load decreases as N increases, as shown in Figure 5.12. The effect of number of disks on refrigeration capacity is shown in Figure 5.13. Cooling capacity increases with increasing number of disks for all fluids studied.

54 20

nitrogen

) 15 argon W (

krypton d a o l

a 10 e h

c i t i s a r

a 5 P

0 0 3 6 9 12 Number of Disks

Figure 5.12 Parasitic heat load, varying number of disks

30 T1 = 298 K, P1 = 6000 psi P4 = 14.696 psi 25 M = 5 SLPM ) W ( y t

i 20 c a p a c 15 n o i t a r e 10 g i

r nitrogen f e

R argon 5 krypton

0 0 2 4 6 8 10 12 Number of Disks

Figure 5.13 Refrigeration capacity, varying number of disks

55 Pressure drop increases for increasing number of disks, as presented in Figure 5.14. "P! is defined as:

P ! P "P# = 1 2 P1

2.5E-04 T1 = 298 K, P1 = 6000 psi P4 = 14.696 psi M = 5 SLPM 2.0E-04

1.5E-04 P / P ! 1.0E-04

nitrogen 5.0E-05 argon krypton

0.0E+00 0 2 4 6 8 10 12 Number of Disks

Figure 5.14 Pressure drop for high-pressure stream, varying number of disks

5.3 Numerical Study of Varying Volumetric Flow Rate

Nitrogen, argon, and krypton were selected as the cryogen with N = 6 geometry. For each fluid, the standard volume flow rate was varied. For each case, the temperature profiles of the high and low pressure streams, the outlet pressures of the heat exchanger, the quality

56 of the fluid at state 3, and the enthalpies at states 3 and 4 were calculated using Mathcad and REFPROP.

The inlet temperature and pressure are, respectively, T1 = 298 K and P1 = 6000 psi. The inlet pressure P4 = 14.696 psi. The height and width of the high-pressure channel are

-4 heighth = 813 µm and widthh = 508 µm. The thickness of one disk is t = 1.27·10 m. The low-pressure inlet temperature, T4, is equal to 77.355 K for nitrogen, 87.302 K for argon, and 119.78 K for krypton. The tolerance between iterations was 1 K. The simulation results for nitrogen, argon, and krypton are presented in Tables 5.4, 5.5, and 5.6, respectively. The case M = 4 for krypton is not displayed because it would not converge to the specified tolerance.

Table 5.4 Simulation Results for Nitrogen, N = 6, Varying Volumetric Flow Rate

Temperature (K) Pressure (psi)

M T2 T5 P2 P5 4 91.44 249.74 5999.74 14.69579 4.5 95.47 250.46 5999.69 14.69577 5 100.41 250.72 5999.64 14.69575 5.5 105.68 251.17 5999.60 14.69572 6 110.26 250.93 5999.56 14.69570 6.5 114.11 249.87 5999.52 14.69567 7 117.21 248.53 5999.47 14.69564

57 Table 5.5 Simulation Results for Argon, N = 6, Varying Volumetric Flow Rate

Temperature (K) Pressure (psi)

M T2 T5 P2 P5 4 98.94 260.72 5999.68 14.69570 4.5 104.10 263.09 5999.63 14.69567 5 109.58 264.45 5999.58 14.69563 5.5 115.59 265.82 5999.53 14.69560 6 120.58 266.02 5999.47 14.69557 6.5 125.78 266.42 5999.41 14.69553 7 130.38 266.41 5999.36 14.69550

Table 5.6 Simulation Results for Krypton, N = 6, Varying Volumetric Flow Rate

Temperature (K) Pressure (psi)

M T2 T5 P2 P5 4.5 136.93 266.06 5999.40 14.69510 5 135.85 263.76 5999.25 14.69504 5.5 135.56 263.20 5999.18 14.69485 6 131.62 261.08 5999.05 14.69477 6.5 130.03 259.86 5998.91 14.69469 7 129.66 259.07 5998.76 14.69461

Pe is on the order of 104 to 105 for the high-pressure channel and on the order of 100 to

1000 for the low-pressure channel. This is shown in Figures 5.15 and 5.16. For both channels, the Peclet number is sufficiently large that conduction between the fluid particles in each separate channel can be neglected and the model used is assumed valid.

58 nitrogen 4 SLPM 1.20E+05 T1 = 298 K, P1 = 6000 psi nitrogen 7 SLPM argon 4 SLPM P4 = 14.696 psi N = 6 argon 7 SLPM 9.00E+04 krypton 4.5 SLPM krypton 7 SLPM r e b m u n 6.00E+04 t e l c e P

3.00E+04

0.00E+00 0 0.2 0.4 0.6 0.8 1

x/Lh

Figure 5.15 High-pressure channel Peclet number, varying standard volumetric flow rate

2500 nitrogen 4 SLPM T1 = 298 K, P1 = 6000 psi nitrogen 7 SLPM P4 = 14.696 psi argon 4 SLPM 2000 N = 6 argon 7 SLPM krypton 4.5 SLPM r e b 1500 krypton 7 SLPM m u n t e l c 1000 e P

500

0 0 0.2 0.4 0.6 0.8 1

z/Ll

Figure 5.16 Low-pressure channel Peclet number, varying standard volumetric flow rate

59 High pressure Reynolds number along the length of the length of the channel is shown in

Figure 5.17. Reynolds number decreases along the length of the channel, starting off turbulent and exiting as laminar flow. The Reynolds number of the fluid in the low- pressure stream was found to be laminar for all cases except for krypton at 7 SLPM, as presented in Figure 5.18.

8000 nitrogen 4 SLPM T1 = 298 K, P1 = 6000 psi nitrogen 7 SLPM P = 14.696 psi 4 argon 4 SLPM N = 6 argon 7 SLPM 6000

r krypton 4.5 SLPM e

b krypton 7 SLPM m u n

s 4000 d l o n y e R 2000 transition Re = 2300

0 0 0.2 0.4 0.6 0.8 1

x/Lh

Figure 5.17 High-pressure channel Reynolds number, varying standard volumetric flow rate

60 3000 nitrogen 4 SLPM nitrogen 7 SLPM T1 = 298 K, P1 = 6000 psi argon 4 SLPM argon 7 SLPM P4 = 14.696 psi 2500 krypton 4.5 SLPM N = 6 krypton 7 SLPM r

e 2000 transition Re = 2300 b m u n

s 1500 d l o n y

e 1000 R

500

0 0 0.2 0.4 0.6 0.8 1

z/Ll

Figure 5.18 Low-pressure channel Reynolds number, varying standard volumetric flow rate

! Figure 5.19 shows that outlet temperatureT2 increases with flow rate for nitrogen and argon, while it decreases for krypton. The number of thermal units varies little for nitrogen, increases slightly for argon, and decreases with increasing flow rate for krypton, as presented in Figure 5.20.

61 1.6 T1 = 298 K, P1 = 6000 psi

P4 = 14.696 psi

) N = 6 min T / 2

T 1.4 ( e r u t

a nitrogen r e

p argon m e krypton T

1.2 t e l t u O

1.0 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Standard volume flow rate (SLPM)

Figure 5.19 Outlet temperature of high-pressure stream, varying standard volume flow rate

8 T1 = 298 K, P1 = 6000 psi

P4 = 14.696 psi nitrogen N = 6 argon krypton 6 U T N

2 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Standard volume flow rate (SLPM)

Figure 5.20 Number of thermal units, varying standard volume flow rate

62 Figure 5.21 presents the effect of number of disks on heat exchanger effectiveness while

Figure 5.22 shows that the quality at the expansion valve exit increases for nitrogen and argon while it decreases for krypton with increasing flow rate.

0.9 T = 298 K, P = 6000 psi 1 1 nitrogen P = 14.696 psi 4 argon N = 6 krypton

0.9 s s e n e v i t c e f f E 0.8

0.8 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Standard volume flow rate (SLPM)

Figure 5.21 Heat exchanger effectiveness, varying standard volume flow rate

63 0.6 T1 = 298 K, P1 = 6000 psi

P4 = 14.696 psi N = 6

0.4 y t i l a u Q

0.2

nitrogen argon krypton 0.0 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Standard volume flow rate (SLPM)

Figure 5.22 Quality at state 3, varying standard volume flow rate

Since the number of disks is constant, the parasitic heat load is constant for each fluid and equal to 4.97 W, 4.75 W, and 4.02 W for nitrogen, argon, and krypton, respectively.

Refrigeration capacity is shown to increase with increasing volumetric flow rate in Figure

5.23.

64 40 T1 = 298 K, P1 = 6000 psi

P4 = 14.696 psi

) N = 6

W 30 (

y t i

c nitrogen a

p argon a c

n krypton o i t a r e g i r f 10 e R

0 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Standard volume flow rate (SLPM)

Figure 5.23 Refrigeration capacity, varying standard volume flow rate

Pressure drop for the high-pressure fluid increases with flow rate, as shown in Figure

5.24.

65 2.5E-04 T1 = 298 K, P1 = 6000 psi

P4 = 14.696 psi 2.0E-04 N = 6 nitogen argon 1.5E-04 krypton P / P ! 1.0E-04

5.0E-05

0.0E+00 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Standard volume flow rate (SLPM)

Figure 5.24 !P/P for high-pressure stream, varying standard volume flow rate

5.4 Discussion

5.4.1 Varying Number of Disks

Peclet number is large for both the high and low-pressure streams, suggesting that the conduction within the fluid itself can be neglected. For high Peclet number, convection between the fluid and the channel wall dominates and thus the temperature gradient in the fluid is not as important.

66 Reynolds number drops through the length of the high-pressure channel at constant volumetric flow rate for all fluids studied. This is due to the nature of viscosity at high pressures. At low pressures, viscosity decreases with temperature. At high pressures, however, viscosity may increases as temperature decreases. These are known as dense gases. Thus, as heat is transferred from the high to low-pressure channel, the temperature decreases through the length of the high-pressure channel, and in turn, viscosity increases. As Reynolds number is inversely related to viscosity, Reynolds number decreases. In the low-pressure channel, temperature is increasing through the length of the channel, leading to the rise in Reynolds number through the channel.

As the number of disks increases, the percentage of flow that is turbulent decreases in the high-pressure channel. Nitrogen at 10 disks results in approximately 62% turbulent flow while N = 4 results in about 75% turbulent flow. The remainder of the flow in the channel is laminar. Argon and krypton experience a similar phenomenon. This makes sense because as the length of the channel increases, more and more heat is transferred from the high-pressure stream, and in turn, the temperature along the channel decreases even more and reaches a point of transition Reynolds number earlier in the channel.

For a larger number of disks, the outlet temperature of the high-pressure stream should approach the boil-off temperature of the evaporator. As the length of the heat exchanger

! approaches infinity, T2 should approach a value of 1. The effect on increasing number of disks for krypton is greater at lower N than for the other fluids – beyond 7 disks, T2

67 experiences little reduction in temperature. Argon and nitrogen fall off in outlet temperature more gradually and appear that for greater N, a lower T2 would occur.

As more disks are added, more heat transfer surface area is created, allowing for more heat transfer to occur. All three fluids experience a point of diminishing returns, with respect to heat exchanger effectiveness, with increasing N, though krypton reaches that point at a lower number of disks. Thus krypton requires a lower number of disks to transfer the same amount of heat to get to the 2-phase region.

The quality of the fluid exiting the expansion valve decreases for all three fluids with increased effectiveness, i.e., with increasing number of disks. As the number of disks becomes larger, the quality at state 3 becomes smaller, meaning that state 3 moves closer to the saturated liquid state at the boil-off pressure of 1 atm. There is a concern, however, that as the number of disks increases, deposition may occur, which could clog up the channel with solid nitrogen, argon, or krypton.

As N increases, the parasitic heat load decreases. This is because parasitic heat load is inversely related to the length of the heat exchanger. For a low number of disks, the overall refrigeration capacity is negative, meaning that the heat is actually transferred to the system. Not until larger number of disks does the refrigeration capacity become positive. After this point, cooling capacity increases and begins to reach an asymptotic value with a larger number of disks. The parasitic heat loss increases for larger N, contributing to the tail-off effect of cooling capacity with increased N. Krypton tails off

68 much quicker than the other fluids, suggesting that the first few disks added make the most difference in cooling capacity. Nitrogen and argon, however reach an asymptotic value slower than krypton and continue to increase in cooling capacity with increasing N.

As the number of disks increases, the pressure drop curves for nitrogen and argon become more and more linear. In fact, they appear to have a very similar slope, with argon slightly shifted in the positive !P* direction. Krypton also experiences pressure drop nearly linearly with increasing number of disks. The number of disks has a greater effect on pressure drop for krypton because it is the most viscous of the three fluids. The maximum pressure drop for krypton, at N = 10, is still on the order of one-ten thousandth of the inlet pressure of the high P stream.

Though less of the overall heat transfer for increasing number of disks is turbulent – which, of course, results in better heat transfer capabilities as compared to laminar flow – there is also an increase in heat transfer surface area. Increased surface area results in more heat being transferred from the high to low-pressure streams, and also accounts for the higher !P.

Cooling capacity reaches a point of diminishing returns for all three fluids. Beyond a certain number of disks, not much more cooling occurs. Krypton experiences little addition in refrigeration capacity beyond 5 disks, with a maximum of about 25 W.

Increasing the number of disks further only increases the pressure drop. Pressure drop increases steadily for all three fluids of interest with increased number of disks. However,

69 even for the heavier, more viscous fluid krypton, the maximum pressure drop is on the order of about one ten thousandth that of the inlet pressure. The pressure drop in both the high pressure and low-pressure channels are sufficiently low that it is safe to assume the heat exchanger operates at constant pressure. This allows for easier, faster computations when running the numerical analysis.

5.4.2 Varying Volumetric Flow Rate

Peclet number is large for both the high and low-pressure streams, suggesting that the conduction within the fluid itself can be neglected. Again, Reynolds number decreases through the length of the channel for the high-pressure stream and decreases through the channel for the low-pressure stream.

The effect of higher mass flow rate on T2 could not be shown as flow rates above 7

SLPM failed to converge even after many iterations. An interesting phenomenon is presented in this figure. Krypton decreases in temperature as flow rate increases, quite the opposite from the other two fluids. Nitrogen and argon experience no added decrease in

• T2 with increasing flow rate, suggesting that for larger flow rate, m!cp between the two fluids is large, contributing to the strange behavior.

The same can be said for heat exchanger effectiveness. Krypton decreases in effectiveness for higher mass flow rate. Nitrogen, on the other hand, experiences an

70 optimum mass flow rate of about 5.5 SLPM where effectiveness is concerned. Argon experiences a point of no added benefits in effectiveness at around 5.5 SLPM.

For nitrogen and argon, an increase in flow rate actually results in an increase in quality at the expansion exit. Krypton, on the other hand, decreases in quality at state 3 with an increase in flow rate. Nitrogen and argon are taken further from the saturated liquid at state 3 for increased flow rate. Krypton, however, becomes more and more saturated with liquid as flow rate increases. This holds to the trend of increased outlet temperature T2 and increased effectiveness with increased flow rate for nitrogen and argon and a

• decrease in both for krypton. Again, this may be due to the large difference in m!cp of the high and low-pressure streams.

All three fluids experience and increase in cooling capacity with increase flow rate.

Although the quality at state 3 increases at larger flow rates for nitrogen and argon, refrigeration capacity is also a function of flow rate as well. Flow rate dominates and makes up for the increase in quality at the expansion exit. Krypton, on the other hand, experiences an increase in cooling capacity with is nearly linear with a much higher slope than the other fluids. This is due to the fact that state 3 becomes more and more like saturated liquid while also increasing in flow rate. This results in a larger slope for the refrigeration capacity curve for krypton than for the other fluids. Increasing volumetric flow rate has a greater effect on krypton than on nitrogen and argon, though results in an increase for all three, nonetheless.

Flow rate also has a greater effect on krypton than nitrogen or argon with respect to pressure drop. Krypton is more viscous and thus, more adversely effected by an increase in flow rate. Krypton, while producing much higher refrigeration capacities than nitrogen or argon, comes with the trade-off of higher pressure drop.

The above data show that krypton is more sensitive to change in volumetric flow rate than nitrogen or argon. Krypton experiences lower high-pressure outlet temperatures, higher refrigeration capacities, and higher pressure drops with increasing flow rate.

Nitrogen and argon, however, also experience an increase in the above yet to a smaller degree. Again, the pressure drops in the low and high pressure channels are sufficiently low that one may assume the heat exchanger operates at constant pressure, speeding up the numerical analysis.

CHAPTER 6

Conclusions and Recommendations

6.1 Conclusions

1. The spiral channel stack heat exchanger can be assumed to operate at constant pressure, as the pressure drops through the heat exchanger are sufficiently low compared to the inlet pressures.

2. For all three fluids, increasing the number of disks increases overall refrigeration capacity up to a point of diminishing returns, beyond which, little addition to cooling capacity occurs with the addition of more disks.

3. Increasing volumetric flow rate increases cooling capacity for the fluids studied, though more so for krypton.

4. Krypton requires a lower number of disks to transfer the same amount of heat to get to the 2-phase region (state 3).

5. Nitrogen and argon are more sensitive to increasing number of disks, while for krypton, the first few disks added make the most difference in heat exchanger effectiveness and cooling capacity.

6. Krypton is more sensitive to changes in volumetric flow rate than the two less dense, less viscous fluids.

7. Krypton results in higher refrigeration capacity than nitrogen or argon at the same volumetric flow rate.

8. Krypton results in higher refrigeration capacity than nitrogen or argon for equal number of disks.

6.2 Recommendations

This study can be regarded as a first approximation to the analysis of a stacked disk spiral heat exchanger. For subsequent approximations, the following recommendations should be considered.

1. The optimum number of vias should be determined such that adequate heat transfer area exists while pressure drop in the low-pressure stream remains sufficiently low.

Combining equations 2.19 and 2.21, the number of vias can be found from:

• f " L " p "m Nvia = 3 (6.1) 8# " Ac !P

74 where Nvia is the number of vias, f is the friction factor, L is the length of the low pressure

• channel, p is the perimeter of one via, m is the mass flow rate, % is density, Ac is the cross sectional area of one via, and !P is the allowable pressure drop through the low pressure channel.

2. Packaging and housing issues not in the domain of this thesis must be resolved.

Sealing a stack of heat exchangers to minimize high-pressure channel leak and disk-to- disk gas flow is still a major design challenge.

REFERENCE

Bejan, A., Tsatsaronis, G., Moran, M., (1996), Thermal Design and Optimization, John

Wiley and Sons, New York, 1996

Bes, Th., and Roetzel, W., (1993), “Thermal theory of the spiral heat exchanger”

International Journal of Heat and Mass Transfer, February 1993, Vol. 36, pg 765-

773

Chua, H.T., Wang, X., Teo, H.Y., (2005), “A numerical study of the Hampson-type

miniature Joule-Thomson cryocooler” International Journal of Heat and Mass

Transfer, In Press

Fox, R.W., and McDonald, A.T, (1998), Introduction to Fluid Mechanics, Fifth Edition,

John Wiley and Sons, New York, 1996

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Incropera, F.P., and DeWitt, D.P., (1996), Introduction to Heat Transfer, Third Edition,

John Wiley and Sons, New York, 1996

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Moran, M.J., Shapiro, H.N., (2004), Fundamental of Engineering Thermodynamics, Fifth

Edition, John Wiley and Sons, New York, 2004

Ng, K.C., Xue, H., Wang, J.B, (2002), “Experimental and numerical study on a miniature

Joule-Thomson cooler for steady-state characteristics” International Journal of

Heat and Mass Transfer, January 2002, Vol. 45, pg 609-618

REFPROP, NIST Standard Reference Database 23, Version 7.0:2002

Walker, G., (1989), Miniature Refrigerators for Cryogenic Sensors and Cold Electronics,

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December 2001, Vol. 21, pg 1829-1844

APENDIX A

Files on Disk

The following pages present the files on disk for raw data in tabulated form. For (1) constant flow rate with variable number of disks, and (2) constant number of disks with variable flow rate, the following data are catalogued: fluid Cryogenic fluid

M (SLPM) Standard volume flow rate

N Number of disks x (m) Length along high-pressure channel z (m) Length along low-pressure channel

Thigh (K) Temperature along high-pressure channel

Tlow (K) Temperature along low-pressure channel

P1 (psi) Pressure at state 1

P2 (psi) Pressure at state 2

P3 (psi) Pressure at state 3

P4 (psi) Pressure at state 4

P5 (psi) Pressure at state 5 x3 Quality at expansion valve outlet

78 Files:

N2 const M.xls Nitrogen at constant M, variable N data

Ar const M.xls Argon at constant M, variable N data

Kr const M.xls Krypton at constant M, variable N data

N2 const N.xls Nitrogen at constant N, variable M data

Ar const N.xls Argon at constant N, variable M data

Kr const N.xls Krypton at constant N, variable M data