Development of a Highly Selective

Deuterium-Hydrogen Separation

Process

Daniel Mukasa

A Honors Thesis

Presented to the department

of Physics and Astronomy

of Oberlin College

Adviser: Stephen FitzGerald

April 2019

Executive Summary

For the last half century the Girdler Sulfide method has been used to produce deu-

terium gas. This process is surprisingly inefficient and yields a high cost of deuterium.

Metal organic frameworks have been presented as a possible new means of producing

deuterium gas via what is referred to as a “breakthrough experiment”. These ex-

periments will have an optimal operating temperature to maximize the effectiveness

of separation, measured by a figure of merit defined as the selectivity. This optimal

temperature is dependent on the metal organic framework used.

In this thesis we present the construction of a breakthrough experimental setup

capable of running experiments at any temperature within the range of 105 K - 200

K. This will enable experiments from which one can find the temperature which maxi-

mizes selectivity. We used primarily highly thermally conductive materials (including

copper and aluminum) in the construction of this setup, and a PID temperature con-

troller to obtain constant temperatures. A detailed analysis of PID controllers is

presented along with breakthrough experiments which exemplify the effectiveness of

this experimental setup.

2

Acknowledgements

First and foremost I would like to thank my parents for their continual support and

encouragement in my academic endeavours. Without your help I would never be here,

so many doors in life have been opened for me because of your sacrifices.

I would also like to thank Stephen FitzGerald who has been a particularly amazing

advisor over the last four years. You first exposed my to research when I was 18 and

showed you sparked my love for this career path. I’ve developed more as a critical

thinker in these last four years than I ever would have expected to prior to coming

to Oberlin. I am thankful for the intellectual development you and the entire physics

department faculty have provided me in my time here.

Finally I want to thank the FitzGerald lab, both Katie and Elizabeth, along with

every single friend and family member who have provided me an amazing support

system. Thank you all.

3

To my parents.

4

Contents

Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1 Introduction 10

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Metal Organic Frameworks (MOFs) . . . . . . . . . . . . . . . . . . . 12

1.3 Temperature Controlled Break Through Cell . . . . . . . . . . . . . . 13

2 Theory 15

2.1 PID Controller Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 PID Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Zone Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Selectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Gas Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Experimental Setup 26

3.1 Liquid Nitrogen Dipper . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Gas Loading and Absorption . . . . . . . . . . . . . . . . . . . . . . . 28

4 Methods 30

4.1 PID Tuning Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5

4.2 Breakthrough Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Results 32

5.1 PID Tuning Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Breakthrough results . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 Future Work 37

A Coding Calculations 38

A.1 Copper Specific Heat calculation . . . . . . . . . . . . . . . . . . . . . 38

A.2 Damped Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 41

B Liquid Nitrogen Dipper Schematics 43

Bibliography 46

6

List of Figures

1.1 Plot from [9] indicating the separation factor and energetic efficiency

of the Girdler sulfide method. . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Structure of Cu MOF-74 adapted from reference [18]. The red spheres

represent oxygen, blue is cobalt, black is carbon, light pink is hydro-

gen. The yellow sites are binding sites for atoms with 1 indicating the

primary open metal site, 2 the secondary site, and so on. . . . . . . . 12

2.1 A P controller with too low of a P value, resulting in steady state error

below the systems setpoint. . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 A P-I controller stabilizing onto the systems setpoint. Note, while the

setpoint is achieved, it takes some time for oscillations to dissipate. . 17

2.3 A PID controller stabilizing onto the systems setpoint. Note how the

addition of the D term allows for quickly stabilizing on the setpoint. . 18

2.4 Simulated weakly damped harmonic oscillations. Take note the simi-

larities between this system and an ideal PID temperature controller.

The code for this simulation can be found in Appendix A.2 . . . . . . 20

2.5 Simulated specific heat of copper based on Debye theory. The code

producing this plot is included in Appendix A.1 . . . . . . . . . . . . 21

2.6 A representation of a generic separation process. . . . . . . . . . . . . 22

7

3.1 Schematic of liquid nitrogen dipper: (A) liquid nitrogen container, (B)

screw legs, (C) 25Ω heater, (D) copper casing (E) aluminum sample

holder, (F) gas line, (G) Silicon diode thermometer, (H) teflon screw,

(I) height adjusting rod. . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Schematic of our gas loading setup adapted from [12] . . . . . . . . . 29

5.1 Plot showing the ultimate gain of our dipper system, Pu = 200, and

the corresponding ultimate wavelength Tu ≈ 1.3 min . . . . . . . . . 32

5.2 P controller with the optimal value P= 120 in accordance to the

Ziegler-Nichols method. Note we achieve steady state error, being an

indication that we can continue the tuning process and find an optimal

I value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.3 P-I controller with optimal values P= 120 and I= 4 which stabilizes

after approximately 960 s = 16 min . . . . . . . . . . . . . . . . . . . 34

5.4 PID controller with optimal values P= 120, I= 4 and D = 1 which

stabilizes after about 600 s = 10 min, being 6 minutes sooner than the

P-I controller in figure 5.3 . . . . . . . . . . . . . . . . . . . . . . . . 34

5.5 PID controller with optimal values from figure 5.4 across the tem-

perature range 105 K - 200 K. Note at higher temperatures we see

oscillations lasting longer as predicted . . . . . . . . . . . . . . . . . . 35

5.6 PID controller with optimal values P= 120, I= 4 and D = 1 for the

temperature 105 K - 150 K and P= 120, I= 4 and D = 8 for 160 K -

200 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.7 Breakthrough experiments run in dry ice (temperature of 187 K) and

with the dipper set to 187 K . . . . . . . . . . . . . . . . . . . . . . . 36

B.1 The Aluminum flange used to hold the dipper up with holes for the

sample holding rod, gas inputs, and a detronics 8 pin connector . . . 43

8

B.2 The front of the copper casing with the heater attached to the bottom.

The heater was epoxied into the bottom hole. . . . . . . . . . . . . . 44

B.3 The aluminum sample holding tube. 1 inch of the tube was encased in

copper while the other half in ends were exposed to allow for connec-

tions to the gas inputs . . . . . . . . . . . . . . . . . . . . . . . . . . 45

B.4 A teflon piece used as a buffer between the sample holding rod and

copper casing. This assured the sample holding rod does not drop in

temperature, and thus just an added safety feature . . . . . . . . . . 45

9

Chapter 1

Introduction

1.1 Background

Deuterium is an isotope of hydrogen, typically referred to as heavy hydrogen due

to the one added neutron in its nucleus. The isotope was originally formed in the

big bang with free neutrons and protons binding together. Most deuterium today is

bonded to oxygen, making ocean water a large natural source of the isotope in the

form of heavy water D2O. It is however very rare, having a natural abundance of

only 0.0155% [4].

Regardless of its rare nature this isotope has a number of applications. Since

it is simply a heavier version of hydrogen, researchers are exploring the prospect of

“deuterated drugs” in which every hydrogen atom is replaced with a deuterium [5].

This would yield drugs with a longer half life without increasing their toxicity [13].

Furthermore, this isotope is hoped to be used as a stable isotope tracer in biomedical

applications [17], and is used to produce clean energy in nuclear reactors [11].

Like many other isolated chemicals, deuterium is typically obtained by purifying

a more common chemical mixture. The dominant method of doing so is through the

Girdler sulfide (GS) process which produces heavy water from water [10]. This process

10

has been the dominant means of producing deuterium for the last 8 decades with

few industrially feasible alternatives. Figure 1.1 shows this method can be considered

economic, but clearly has room for improvement. Doing so could save vast amounts of

money [1] and minimize energy expenditure [8] to enable more deuterium production.

Figure 1.1: Plot from [9] indicating the separation factor and energetic efficiency ofthe Girdler sulfide method.

Any new method should have a higher separation factor and energetic efficiency

to compete with the GS process. Metal Organic Frameworks (MOFs) could be the

material that achieves this. Their highly porous structure has been widely sought after

for carbon capture applications [16], but they can also be applied to store hydrogen

and deuterium. This storage capability has led to the investigation of MOF based

separation methods, from which a selectivity as high 50 was recorded [6], making

them the ideal candidate for this challenge.

11

1.2 Metal Organic Frameworks (MOFs)

MOFs are highly porous materials, being up to 90% free volume [19], composed of

metal oxide clusters connected by organic linkers. Figure 1.2 shows this structure in

the case of copper MFU

Figure 1.2: Structure of Cu MOF-74 adapted from reference [18]. The red spheresrepresent oxygen, blue is cobalt, black is carbon, light pink is hydrogen. The yellowsites are binding sites for atoms with 1 indicating the primary open metal site, 2 thesecondary site, and so on.

This MOF has been highly characterized and is well known within isotope separation

applications due to it’s high adsorption enthalpy, which enables better separation

[14].

12

MOFs typically have open metal sites, where binding energies are the strongest, as

well as other weaker sites. The presence of these weaker binding sites will lower selec-

tivity, as described in the selectivity theory section. Which site an atom preferentially

adsorbs to is dependent on the pressure and temperature of the system. Given an

optimal operating pressure, there will exist some temperature range in which atoms

to primarily occupy the open metal site, and leave the weaker sites fairly empty.

In the case of the separation of hydrogen and deuterium one proposed method

of applying MOFs is through a breakthrough experiment. This entails flowing a

mixture of hydrogen and deuterium through a MOF at a stable temperature. Due to

deuterium having a lower zero point energy than hydrogen it would occupy binding

sites first while hydrogen flows though. One can imagine in such a situation it would

be possible to make a setup where you allow bursts of hydrogen to flow through and

retain the deuterium until a primarily deuterium mixture remains, thus achieving

isotope separation.

In such an experiment finding the aforementioned temperature range where mainly

the primary site fills is essential. It is therefore vital in the development of such a

method run breakthrough experiments at various temperatures to find the tempera-

ture range with the highest selectivity’s.

1.3 Temperature Controlled Break Through Cell

We built an experimental setup specifically for carrying out break through experi-

ments at constant temperatures within the range of 105 K and 200 K, hereby referred

to as the dipper. The goal of this device is to later be used to find optimal tempera-

tures to run breakthrough experiments at to maximize selectivity. The temperature

of this device is controlled with a PID temperature controller and tuned by following

the Ziegler-Nichols method to quickly obtain a desired temperature. A primary con-

13

cern of any such device is to assure the temperature of the thermometer is the same

as the temperature of the MOF sample. We thus constructed this setup with the goal

of maximizing its thermal conductivity.

A theoretical explanation of the PID tuning process is included along with a de-

tailed description of the construction of the dipper. Ultimately the setups capability

to bring the MOF temperature to the desired setpoint is tested by running break-

through experiments under two conditions, one in the standard dipper setup with

a setpoint of 187 K and one submerged in dry ice and acetone (with a stabilizing

temperature of 187 K).

14

Chapter 2

Theory

2.1 PID Controller Theory

A PID controller is a device commonly used to set stable temperatures in industrial

application. The controller receives a measured signal and returns a response the the

goal of that signal reaching a desired setpoint. In our application that signal is the

temperature measured by a thermometer, and the response heating power. We used

a Lakeshore 331 temperature controller for which the output signal was

u(t) = P

[e(t) + I

∫ t

0

e(τ)dτ + Dde

dt

](2.1)

[7]. Here e(t) is the error as a function of time defined as

e(t) = s− T (t). (2.2)

Where s is the setpoint and T the measured temperature. The values P, I, and

D are all proportionality constants set by the user to weigh the importance of each

contribution to the final signal.

15

The first term Pe(t), commonly referred to as the proportional gain, sends a

signal to the heater proportional to the error term. This term alone is not enough

to produce a stable temperature, as can be seen by looking at u(t) when I and D

are 0, or u(t) = Pe(t), otherwise referred to as a P controller. If for instance we

start at some stable temperature below the setpoint this term will initially heat the

sample and approach setpoint. For too large of a P value the system will over shoot

the setpoint, turn off the heater and allow the sample to cool, then rapidly heat the

sample again throwing it into an oscillatory behavior averaging below the setpoint.

For too low of a P value the heater will initially overshoot the setpoint, but later will

not be able to output enough power to sustain the setpoint.

In the case of too low of a P value the problem of too low of a P value is commonly

known as steady state error, which has been visualized in figure 2.1

Figure 2.1: A P controller with too low of a P value, resulting in steady state errorbelow the systems setpoint.

A system with steady state error can achieve its set point by including the second

term I∫ t

0e(τ)dτ integrates error with time. The addition of this term to a system

with steady state error adds an extra source of power and ideally increase the signal

16

until error goes to zero, then stabilizes by constantly outputting a signal necessary

to maintain this temperature. In practice however this is only approximately what

occurs. While the temperature does eventually stabilize to the set point with only

P and I set to the appropriate values, typically the system take time to do so. This

typically is seen as oscillations with dissipating amplitudes as visualized in figure 2.2

Figure 2.2: A P-I controller stabilizing onto the systems setpoint. Note, while thesetpoint is achieved, it takes some time for oscillations to dissipate.

This final issue is a problem in systems that aim to quickly stabilize on any given

setpoint. It is handled by the third term Ddedt

which tracks the derivative of error

with time. Such a term predicts when the system is about to overshoot the set point

during these oscillations in the PI system and cushions any sharp increase or decrease

in temperature. Such a result is visualized in figure 2.3.

2.2 PID Tuning

The main challenge in using any PID controller is finding the right parameters (P,

I, and D) to quickly reach a desired setpoint. The parameter space can be large

17

Figure 2.3: A PID controller stabilizing onto the systems setpoint. Note how theaddition of the D term allows for quickly stabilizing on the setpoint.

(with P and I ranging from 0 − 1000 and D between 0 − 200) and consequentially

overwhelming to find a start. To overcome this challenge we have implemented the

Ziegler-Nichols method [20]. This heuristic method tunes one parameter at a time,

first P then I then D, and gives rough estimates on starting points for each value.

The method typically represents their signals function as

u(t) = 0.6Pu

[e(t) +

2

Tu

∫ t

0

e(τ)d+Tu8

de(t)

dt

](2.3)

from which we can clearly see by comparison to equation 2.1 that P = 0.6Pu, I = 2Tu

and D = Tu8

. Here Pu is defined as the “ultimate gain” or the P value for which

oscillations about the setpoint begin. Similarly Tu is defined as the wavelength of

these oscillations in minutes.

This method however only gives a starting point, so the optimal parameter values

must be found through using ones understanding of PID controllers. Starting with P,

it is advised that one starts with a low P value (say 5) while I and D are both 0. The

18

previous section outlines how a low P value will yield steady state error and a large

P value will yield oscillations about the setpoint. To find the P value where these

oscillations start (being Pu) double the P value and continue to monitor temperature

with time until oscillations occur. If oscillations are too large then decrease again

until Pu is found.

After finding Pu set P = .6Pu, with I = D = 0 and assure this produces steady

state error. Now set P = 0.6Pu and I = 2Tu

. If the temperature never reaches the

setpoint I is clearly too low and must be raised by a factor of 2. If the setpoint is

hit but large oscillations occur then clearly the I term is too influential and must be

decreased by a factor of 2. Again, once a suitable I range is found change I by small

increments to find the optimal value.

Finally, to find D start with D = Tu8

and follow the same method as before. One

will know D is too low if oscillations are sustained for a long time the it takes a

comparably long amount of time to stabilize on the setpoint as if D = 0. If D is too

large however one may notice oscillations about a temperature. From here the same

method of increasing and decreasing by a factor of 2 and fine tuning until the right

value is found. While tuning it is useful to keep the following in mind.

• Oscillations about the setpoint with P and I set imply too large of a D value

• Taking a long time to stabilize on the setpoint imply too low of a D value

• Oscillations about the setpoint with only P set imply too large of an I value

• Steady state error below the setpoint with only P set imply too low of an I value

• Oscillations about the setpoint when I and D are 0 imply too large of a P value

• Steady state error below the setpoint when I and D are 0 imply too low of a P

value

19

2.3 Zone Tuning

The PID parameters found for one temperature may not be the optimal PID values

for another temperature. Thus, further zone tuning may be needed other setpoints.

We can understand why this occurs by modeling our system as a weakly damped

harmonic oscillator. A standard view of this system is a mass on a spring in a viscous

fluid. Such a system would oscillate about the springs equilibrium length with the

amplitude of these oscillations exponentially decaying with time, as simulated in figure

2.4.

Figure 2.4: Simulated weakly damped harmonic oscillations. Take note the similari-ties between this system and an ideal PID temperature controller. The code for thissimulation can be found in Appendix A.2

In our system however this equilibrium length of the spring is equivalent to the

setpoint, and the mass of the spring is equivalent to the thermal mass. Formally

thermal mass is defined as

Cth = mcv (2.4)

20

for a uniform body. It is important to note however that cv, being the specific heat,

is not constant over all temperatures. In fact it generally increases with temperature

as seen in figure 2.5.

Figure 2.5: Simulated specific heat of copper based on Debye theory. The codeproducing this plot is included in Appendix A.1

Therefore, in moving to a higher temperature we increase the specific heat, sub-

sequently increasing the thermal mass. We can imagine in reference to our spring

system this would yield a longer response time, or time of oscillations. With PID

values set for a lower temperature, longer oscillations will not be quickly handled

by the temperature controller. As a result we can expect the system to spend more

time oscillating about the setpoint as we move to higher temperatures, for which one

solution is to simply increase the D term.

2.4 Selectivity

The ideal separation process would completely extract one part of a mixture. In our

application this would be starting with a mixture of hydrogen and deuterium and

21

yielding a pure container of deuterium. This however typically does not occur as

separation processes are not perfect. One can commonly expect for some hydrogen to

still be in the final product of deuterium, and also some deuterium to be accidentally

removed from the final product. This is visualized in figure 2.6.

Figure 2.6: A representation of a generic separation process.

The goal of course is to minimize the amount of deuterium accidentally thrown

out of the and hydrogen in the product, and maximize the hydrogen thrown out and

the deuterium in the product. The more hydrogen thrown out and deuterium kept

in, the better job a separation process has done. Selectivity acts as a figure of metric

for such a process, being defined as

S =npD/npHnrD/nrH

(2.5)

where these four terms are defined as follows

npD = number of deuterium atoms in product

npH = number of hydrogen atoms in product

22

nrD = number of deuterium atoms in residue

nrH = number of hydrogen atoms in residue

Naiyuan Zhang’s honors thesis presents a more rigorous definition of this quantity

[18] however, we are only concerned with maximizing this quantity in a breakthrough

experiment. This is equivalent to maximizing the binding energy difference between

hydrogen and deuterium, as the two are related by the sum of boltzmann factors

S = Ae∆E1kBT +Be

∆E2kBT (2.6)

For a system with one binding site. Here ∆E1 and ∆E2 are the binding energy

differences of the two isotopes in the primary and secondary sites respectively, and

both A and B are pre-exponential factors with little temperature dependence relative

to the exponential term. Most hydrogen bound within copper MFU is in the ground

state [3] so ∆E is essentially the difference in ground state energies of hydrogen and

deuterium.

A common and simple model of which elucidates these isotopes ground state

energies is as a simple harmonic oscillator. Within this model it is well known the

ground state energy is given by

E0 =1

2hω (2.7)

where ω =√

km

. We can thus say quite simply

∆E1 =1

2h√k1

(1√mH

− 1√mD

)(2.8)

and

∆E2 =1

2h√k2

(1√mH

− 1√mD

)(2.9)

23

Here k1 and k2 are the spring constants for their respective sites. Since the primary

site has a greater binding energy, in the context of the simple harmonic oscillator

model, this yields a stronger restoring force and therefore a larger spring constant.

It is quite often in MOFs that secondary sites will outnumber primary sites, so if

all sites are filled in a separation process the majority of gas will be in these weakly

binding secondary sites, and thus the average binding energy difference in the MOF

will be much smaller. As a direct result selectivity of this process will be lower. If

however this process is operated at a temperature range where the gas preferentially

adsorbs to primary sites, with very little in the secondary sites, this process would in

an ideal fashion utilize the high binding energy some MOFs provide to yield a high

selectivity.

2.5 Gas Heating

One typical concern for larger breakthrough experimental is room temperature gas

heating the MOF. This becomes a concern when the rate of heat brought in by the

gas is greater than the rate of heat added by gas being adsorbed to the MOF (brought

about by the MOFs heat of adsorption). We can model the rate of heat from the gas

entering the MOF as

dQ

dt= (flow rate)cv∆T (2.10)

In our experiments we have typically used a flow rate of 0.1 sccm (standard cubic

centimeters per minute). Since we will have diatomic molecules interacting with our

MOF, we can approximate specific heat with the classical cv = 52R. Finally the

lowest temperature we ever run experiments at is 77 K, so we can define ∆Tmax =

293 K − 77 K = 216 K. Therefore the maximum power brought to our MOFs is on

the order of Pmax = 3.4× 10−4 W for an experiment lasting one minute.

24

The heat of adsorption for Cu - MFU is 32 kJmol

. In our experiments we use a mass

of 22 × 10−3 g of MFU powder. Thus under the same conditions we can calculate

the maximum amount of heat added from adsorbed hydrogen to be Padsorbed max =

1.2 × 10−2W . With Pmax being less than Padsorbed max we have concluded the heat

brought from room temperature gas will be so small that we will not cool our gas

prior to it reaching the MOF.

25

Chapter 3

Experimental Setup

3.1 Liquid Nitrogen Dipper

We have constructed a liquid nitrogen dipping apparatus specifically to run break-

through experiments at various temperatures. A schematic of this setup can be seen

in figure 3.1. It is composed of a copper casing for an aluminum sample holder, four

3-inch screws that dip into liquid nitrogen, a 25Ω heater, and a silicon diode ther-

mometer. The copper casing’s high thermal conductivity is ideal for minimizing the

possible temperature gradient in the system and assuring heat transfer to the metal

organic framework sample.

Heat is poorly transferred from one metal surface to another unless they are in

intimate contact. To achieve this, and optimize heat transfer to the sample, ther-

mal N-grease was applied to the aluminum sample holder which was clamped down

using the four screws. While aluminum has a high thermal conductivity of roughly

205 W/mK coppers is higher at 385 W/mK. Maximum heat transfer could thus be

obtained by replacing the aluminum sample holder with a copper one, however the

results shown in the experimental section indicate this would not be needed as the

MOF sample is able to quick reach the measured temperature.

26

Figure 3.1: Schematic of liquid nitrogen dipper: (A) liquid nitrogen container, (B)screw legs, (C) 25Ω heater, (D) copper casing (E) aluminum sample holder, (F) gasline, (G) Silicon diode thermometer, (H) teflon screw, (I) height adjusting rod.

The thermometer used is a DT-670C copper encased silicon diode thermometer

from Lakeshore. As previously noted, it is essential in cryogenic experimentation

to have the thermometer be the same temperature as your sample. Thus the ther-

mometer was coated with thermal N-grease and screwed onto the copper to assure

intimate contact with the dipper. Prior to use, the thermometer was calibrated using

liquid nitrogen. The thermometer is placed at the top of the copper casing purely for

geometric convenience and bolted on to assure good thermal contact.

A 25 Ω resistance heater covered with a alloy 800 sheath cartridge was placed

parallel to the aluminum sample holder to allow for heat transfer to the sample. The

heater has a maximum power output of 100 W, however the temperature controller

used has a maximum power output of only 25 W that can be used to heat the sample.

UHU Endfest Plus 300 Two Component Adhesive epoxy is used to bind the heater

to the copper casing.

27

The liquid nitrogen container is open to a room temperature environment, leading

to consistent boil off. The height adjusting lab jack is thus used to assure the screws

remain submerged in the liquid nitrogen bath. This exposure to air however plays a

significant role when heating and cooling the sample, resulting in a heating rate of 1

K every 12 minutes when the dippers screws are submerged and 1 K every 6 minutes

when out of the liquid nitrogen. A more optimal setup would instead place the sample

in a thermal isolation vacuum, but this is not necessary as a PID controller alone is

enough to sustain a stable temperature, as explained in the following section.

3.2 Gas Loading and Absorption

A Micromeritics ASAP 2020 Surface Area and Porosity Analyzer was used to load

small pressures of a hydrogen deuterium gas mixture to our MOF sample. To assure

a constant flow rate was achieved from experiment to experiment a Sierra Smart Trak

50 Mass Flow meter was used. The flow of hydrogen and deuterium was controlled by

a Swagelock needle valve which could be opened with precision on the mm scale. The

concentration of hydrogen and deuterium was measured by a triple filter quadrupole

mass spectrometer from Hiden 3F Series. Prior to running experiments the needle

valve was opened to 0.17 mm as to not overload the signal on the mass spectrometer.

A vacuum with a slight opening was placed at the end of this setup to allow for a

small flow of gas, as visualized in figure 3.2

28

Figure 3.2: Schematic of our gas loading setup adapted from [12]

29

Chapter 4

Methods

4.1 PID Tuning Procedure

The PID temperature controller was tuned following the Ziegler-Nichols method out-

lined in the theory section. This entails optimizing one parameter at a time, first P

then I then D. In finding the optimal value for each parameter the dipper was first

submerged in liquid nitrogen with the heater off until the temperature stabilized at

roughly 77 K. The height of the lab jack was then lowered until the liquid nitrogen was

roughly 1 cm from the bottom of the dipper, leaving the screws mostly submerged.

The height of the jack was never changed while collecting data, leading to liquid

nitrogen to boil off and thus decreasing cooling power. This however never became an

issue in setting temperatures as at lower temperatures the screws remained submerged

for well over 25 minutes (the length of each scan), and at higher temperatures with

faster boil off the system could still stabilize the temperature without having the

screws submerged at all.

30

4.2 Breakthrough Procedure

Two breakthrough experiments were run to verify the MOFs temperature is equal

to the measured temperature, one with the dipper set to 198 K and one with the

dipper submerged in acetone and dry ice (at 198 K). Initially the system is vacuumed

then valve 1 and the vacuum were closed off. A constant flow of a 50/50 mixture of

hydrogen and deuterium is established, at which point valve 1 and the vacuum are

opened. The mixture is directed through the MOF sample.

The time at which valve 1 is opened is assigned the value t = 0 and data is

collected until both the hydrogen and deuterium signal c level off, where c is the

time dependent concentration. The final signal is divided by c0 being the total input

concentration, such that the final signal is a 50/50 mixture of hydrogen and deuterium.

As a quantitative measure of the systems capability a selectivity is calculated from

each run and compared.

31

Chapter 5

Results

5.1 PID Tuning Results

The Ziegler-Nichols tuning method outlined in the theory section was employed to

find the optimal PID values. An ultimate gain of Pu = 200 was found with an

associated oscillation period Tu = 1.4 min. These results can be seen in figure 5.1.

Figure 5.1: Plot showing the ultimate gain of our dipper system, Pu = 200, and thecorresponding ultimate wavelength Tu ≈ 1.3 min

32

Plotting the results of the ideal P value of P= 0.6Pu = 120 we find the system

exhibiting steady state error below the setpoint as shown in figure 5.2. The optimal

I value defined using this method would be I= 21.4

= 1.3, I only has a resolution of 1

on this temperature controller so a value of I= 1 was used to start. This value was

subsequently increased to remove all steady state error, from which an optimal value

of I= 4 was found, as seen in figure 5.3

Figure 5.2: P controller with the optimal value P= 120 in accordance to the Ziegler-Nichols method. Note we achieve steady state error, being an indication that we cancontinue the tuning process and find an optimal I value

To quickly stabilize on this setpoint a starting D value of D= 1.48

= 0.175 was

found. This value was similarly increased until an optimal value of D= 1 was found.

The final results of this PID controller can be seen in figure 5.4

These values have proven to work well within the temperature range of 105 K

−140 K, but as expected outside of this range stabilization takes much longer, or is

not reached with our 25 minute experiment, as shown in figure 5.5.

In all cases the only concern is the temperature takes too long to stabilize, so D

was increased to 8, yielding faster stabilization as seen in figure 5.6

33

Figure 5.3: P-I controller with optimal values P= 120 and I= 4 which stabilizes afterapproximately 960 s = 16 min

Figure 5.4: PID controller with optimal values P= 120, I= 4 and D = 1 whichstabilizes after about 600 s = 10 min, being 6 minutes sooner than the P-I controllerin figure 5.3

34

Figure 5.5: PID controller with optimal values from figure 5.4 across the temperaturerange 105 K - 200 K. Note at higher temperatures we see oscillations lasting longeras predicted

Figure 5.6: PID controller with optimal values P= 120, I= 4 and D = 1 for thetemperature 105 K - 150 K and P= 120, I= 4 and D = 8 for 160 K - 200 K

5.2 Breakthrough results

Two breakthrough experiments were run to verify the samples temperature is equal

to that of the measure thermometer temperature, one in dry ice and a stabilized35

temperature of 187 K and another with the dipper set to 187 K. The resulting plot

for this experiment can be seen in figure 5.7.

Figure 5.7: Breakthrough experiments run in dry ice (temperature of 187 K) andwith the dipper set to 187 K

Selectivity was used as a measure of quantitatively comparing the two plots. It

is found that the dry ice experiment yields a selectivity of S = 1.14 and the dipper

yields S = 1.21, or a 6% difference between scans. This is within the variance of

typical scans we have found with this system [12], therefore we conclude the samples

temperature is accurately measured by the silicon diode thermometer.

36

Chapter 6

Future Work

We have accomplished constructing a breakthrough experimental setup capable of

running experiments at a set temperature within the range of 105 K and 200 K.

Future work will therefore be directed towards understanding how selectivity varies

with temperature for any given MOF.

One further area of investigation in this setup can be directed toward the gas

heating problem addressed in the theory section. While in theory we have shown

our pressures are so low that there is not enough thermal mass to heat the MOF

sample, experiments to verify this would be ideal. An experiment can be conducted

by drilling a thermal probe through the sample holder to make contact with the

MOF and epoxying this probe in place. Following this, one can run a break through

experiment and measure temperature as the gas adsorbs into the MOF. Other ways

of addressing this problem could also be cooling down the gas prior to it entering the

MOF, which would simply require structural adjustments to the experimental setup.

37

Appendix A

Coding Calculations

A.1 Copper Specific Heat calculation

In general specific heat is defined as

cv =1

V

dQ

dT(A.1)

For our application we will approximate this heat Q with a materials internal

thermal energy U, yielding

cv =1

V

dU

dT(A.2)

Gersten and Smith derive the internal lattice thermal energy as [2]

U(T ) =3V

2π2β4h3c3s

∫ Θd/T

0

x3

exp(x)− 1dx+ U0 (A.3)

such that the following constants are defined

β = 1/kbT

cs = speed of sound

ΘD = Debye temperature

38

U0 = zero point energy (which is independent of T)

x = βhω (unitless)

This would yield cv in the units Jm3K

, so we have introduced a conversion factor 1ρ

where ρ is the atomic mass density. Thus we have as our final answer

cv =3

2π2β4h3c3sρ

d

dT

∫ Θd/T

0

x3

exp(x)− 1dx (A.4)

#we will use the numpy to include arrays in our calculation of c_v

#Sympy is imported for the numerical evaluation of difficult

#integrals and derivatives

#Matplotlib will be used to plot the results of this calculation

import numpy as np

from sympy import *

import matplotlib.pyplot as plt

#here well define each of the constants in the first term of c_v

#(with units to their right)

k_b = 1.38065*10**(-23) #J/K (joules per kelvin)

h_bar = 1.0545718*10**(-34) #J s

c_s = 340.27 #m/s

o_D = 347 #K (copper Debye temperature)

rho = (8.96*10**(6)) #m^3/g

def c_v(T):

#The c_v(T) function will act as a mean of finding the value

#of c_v at any given temperature T

#we will declare x to be a symbol, or variable for sympy

39

x = symbols(’x’)

#same for the temperature t, which acts as a substitute term for T

t = symbols(’t’)

#we accordingly define the constants beta

# and define the outer term of c_v

beta = 1/(k_b*T)

outer_term = 3/(2*(np.pi**2)*(beta**4)*(h_bar**3)*(c_s**3))

#the integral will be defined Int

#the integral is taken with respect to x with bounds 0 and o_D

Int = Integral((x**3)/(exp(x)-1),(x,0,o_D/t))

#we then take the derivative of this integral with respect to the

#temperature t

derivative = diff(Int,t)

#now switch the substitute term with the desired temperature

second_term = float(derivative.subs(t,T))

return -outer_term*second_term/(rho*1000) #return c_v

#Define an x axis Temp_range

Temp_range = np.linspace(.0001,300, 300)

#Define an array for the y values

specific_heat_array = np.zeros(len(Temp_range))

for i in range(len(Temp_range)):

40

#loop through each element of the array to input

#c_v at every temperature in Temp_range

specific_heat_array[i] += c_v(Temp_range[i])

#plot the outcome

plt.figure(figsize =(20,10))

plt.plot(Temp_range,specific_heat_array)

plt.show()

A.2 Damped Harmonic Oscillator

Our system acts most like a weakly damped harmonic oscillator. John Taylor derives

the motion of such a system to be [15]

x(t) = Ae−βtcos(ω1t− δ) (A.5)

such that ω1 =√ω2

0 − β2 and ω0 =√

km

. Furthermore k is the spring constant,

β is the damping constant, m the mass attached to the spring, delta a phase factor,

and A the amplitude of oscillations.

#again we import numpy for numerical calculations

#and matpotlib for plotting purposes

import numpy as np

import matplotlib.pyplot as plt

#below we define the constants in this calculation

#all of which were chosen more or less arbitrarily

#to produce a wave similar to that seen in the PID

41

# tuning process

beta = 8

A = 1

B = 1

k = 1000

delta = 0

def x(t,m):

#here we define the x(t) function, for which

#mass and time are kept as variables

w_0 = np.sqrt(k/m)

w_1 = np.sqrt(w_0**2 - beta**2)

return A*np.exp(-beta*t)*np.cos(w_1*t+delta) #return x(t)

#define the x axis as time

time = np.linspace(0,.8, 1000)

#plot the results

plt.figure(figsize =(20,10))

#we plot -x(t) to resemble the PID system

plt.plot(time,-x(time,1), label = ’m = 1’)

#we also include a 0 line for refrence

plt.plot(time,np.zeros(len(time)))

plt.show()

42

Appendix B

Liquid Nitrogen Dipper Schematics

Below are the schematics for the parts used in the construction of the dipper setup

Figure B.1: The Aluminum flange used to hold the dipper up with holes for thesample holding rod, gas inputs, and a detronics 8 pin connector

43

Figure B.2: The front of the copper casing with the heater attached to the bottom.The heater was epoxied into the bottom hole.

44

Figure B.3: The aluminum sample holding tube. 1 inch of the tube was encased incopper while the other half in ends were exposed to allow for connections to the gasinputs

Figure B.4: A teflon piece used as a buffer between the sample holding rod and coppercasing. This assured the sample holding rod does not drop in temperature, and thusjust an added safety feature

45

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