research article thermal expansion measurements in fresh ... · meter along with a sample holder,...

Research ArticleThermal Expansion Measurements in Fresh andSaline Ice Using Fiber Optic Strain Gauges and MultipointTemperature Sensors Based on Bragg Gratings

Aleksey Marchenko1 Ben Lishman2 David Wrangborg1 and Torsten Thiel3

1The University Centre in Svalbard 9171 Longyearbyen Norway2Institute for Risk and Disaster Reduction University College London London WC1E6BT UK3Advanced Optics Solutions (AOS) GmbH 01139 Dresden Germany

Correspondence should be addressed to Aleksey Marchenko alekseymarchenkounisno

Received 22 November 2015 Revised 9 February 2016 Accepted 14 February 2016

Academic Editor Xingwei Wang

Copyright copy 2016 Aleksey Marchenko et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper describes the use of Fiber Bragg Grating (FBG) sensors to investigate the thermomechanical properties of saline iceFBG sensors allowed laboratory measurements of thermal expansion of ice samples with a range of different sizes and geometriesThe high sampling frequency accuracy and resolution of the FBG sensors provide good quality data across a temperature rangefrom 0∘C to minus20∘C Negative values of the effective coefficient of thermal expansion were observed in ice samples with salinities6 ppt 8 ppt and 94 ppt A model is formulated under which structural transformations in the ice caused by temperature changescan lead to brine transfer from closed pockets to permeable channels and vice versaThis model is compared to experimental dataFurther in experiments with confined floating ice heating as well as thermal expansion due to vertical migration of liquid brinecaused by under-ice water pressure was observed

1 Introduction

Saline ice is a composite material a solid ice matrix con-taining liquid and gas inclusions with a structure whichchanges under the influence of thermal andmechanical loadsSaline ice consists of pure ice grains grouped in columnsor platelets and brine pockets and channels It has beenthought that sea ice is impermeable to brine transport whenthe liquid brine content is less than 5 and is permeablewhen the brine content is greater than 5 [1] In factsea ice always contains porous channels and liquid brinecan migrate through these under the influence of pressuregradients [2] Thermal changes influence the permeability ofsea ice significantly as brine freezes into ice on cooling andice melts into water on heatingThus the final state of a salineice sample subjected to thermal changes is determined notonly by initial and final conditions but also by the historyof temporal variations and salt fluxes at the boundaries This

means that the thermomechanical properties of saline icesamples depend on their size and geometry

Fresh ice and brine both exhibit typical thermal behaviorthey expand when heated and contract when cooled Salineice however can behave atypically because of phase changesWhen brine freezes this leads to a mean density reduction(since ice is less dense than water) and so cooling can leadto expansion Similarly and by the inverse process heatingcan lead to contraction This unusual behavior of salineice was studied in laboratory experiments and describedby Pettersson [3] and Malmgren [4] Their experimentswere conducted by immersing a sea ice sample in fluidand measuring the changes in fluid volume as the sampletemperature changed Fluid volume changes were then usedto calculate thermal volume expansion coefficients Johnsonand Metzner [5] note that this procedure assumes that noadditional fluid is added to the volumeof the immersion fluidThis assumption has been found to be invalid since brine can

Hindawi Publishing CorporationJournal of SensorsVolume 2016 Article ID 5678193 13 pageshttpdxdoiorg10115520165678193

2 Journal of Sensors

leak out of the sample and be expelled into the immersionfluid

Theoretical models for the calculation of the coefficientof thermal expansion of saline ice have been developed byMalmgren [4] and Cox [6] Malmgren and Cox made oppos-ing assumptions when formulating the basic equationsMalmgren assumed that brine is never expelled from thesample and that all salts are trapped in brine pockets insidethe sample Temperature changes therefore cause internal icemelt or brine refreezing Cox contrastingly assumed that thebehavior of saline ice is analogous to a pure ice cup filledwith liquid brine both are free to expand and contract inde-pendently and hence phase changes have no effect on thethermal expansion of the ice

Johnson and Metzner [5] used a Michelson Interferom-eter to measure linear thermal expansion of cylindrical icesamples taken from a sheet of first-year congelation sea ice inHarrison Bay AlaskaTheir apparatus included the interfero-meter along with a sample holder temperature control unitand computer-controlled data acquisition unit The diameterof their samples was 38mm and the length was 7125mmfor 2 ppt ice and 6933mm for 4 ppt ice Optically flat targetmirrors were frozen to the front surface of the samplestrimmed parallel with a microtome and used to control thesample length with the interferometer They were able toachieve a displacement resolution of 3164 nm correspondingto a strain resolution of the order of 5 times 10minus6

Mean and instantaneous values of the thermal expansioncoefficient were found to be similar to the fresh ice ther-mal expansion coefficient of 5 times 10minus5∘C Their 4 ppt iceshowed hysteresis when cooled and rewarmed after the initialwarming test Similar effects were observed in the experi-ments of Butkovich [7] with fresh water ice where the coeffi-cient of linear thermal expansion decreased with succeedingruns Their conclusion was that their results agreed with theanalysis of Cox [6]

Systematic investigations of the thermomechanical be-havior of saline ice samples were performed using fiberoptic strain and temperature sensors based on Fiber BraggGratings in the cold laboratories of the University Centre inSvalbard Norway and University College London UK from2012 to 2015 [8ndash11] The FBG system used was designed byAdvance Optic Solutions GmbH (Germany) The flexibilityof the FBG system allowed experiments to be conducted withmany different sizes and geometries of ice sample and withfloating ice

The present paper describes the instrumentation andexperimental setups and summarizes experimental resultson the measurement of thermal expansion of saline ice Weformulate a new model of the thermal expansion of salineice assuming the possibility of structural changes in theice associated with gradual transformation of closed brinepockets into permeable brine channels under the influenceof temperature changes Mass changes in the ice with closedbrine pockets are estimated using the experimental dataThermal expansion of saline ice caused by the migrationof liquid brine through the ice is also discussed in thepaper

2 Instrumentation

The fiber type used for this kind of optical sensing istypically standard telecom single-mode fibers (SMF) that arewidely used in communications with an overall diameterof 025mm and a core diameter of only about 001mmA Fiber Bragg Grating is a periodical index change in therefractive index 119899 along this optical silica fiberrsquos core formedby an interference pattern of two UV laser beams that thefiber is exposed to There are various techniques to generatethe two coherent beams (Figure 1(a)) [12] The index changeresults from a certain photosensitivity of the fiberrsquos corewhich is due to the presence of some chemical dopantsinside the silicarsquos structure typically germanium oxide Theinterference pattern being generated by two coherent laserbeams consists of a periodic UV power modulation along thefiber determined by the incident angle of the two beamsThisUV pattern migrates to the fiber corersquos index change patternby the said photosensitive mechanism

Areas of higher 119899 alternate with areas of lower 119899 witha period of less than 10minus6mm while Δ119899 is relatively smallquantity (Δ119899 lt 0001 and 119899 = 145) Due to the index changeeach ldquograting linerdquo reflects a very small portion of the lightwave propagating along the fiber back to the light sourceAlthough a single reflected portion is negligible compared tothe transmitted power the effect becomes noticeable becausethe amount of ldquograting linesrdquo in a conventional FBG is about4000mm and a typical FBG with 10mm length consists of40 thousand reflections If the lightrsquos wavelength matches thecondition

120582Bragg = 2 sdot 119899eff sdot Λ (1)

where 119899eff is the fiberrsquos effective index 120582Bragg is the lightrsquoswavelength and Λ is the index changersquos period then allreflected light wave portions are propagating ldquoin-phaserdquo andinterfere constructively Depending on the actual Δ119899 thisresults in a typical narrow-band power peak in the reflec-tion spectrum at 120582Bragg and vice versa in a loss in transmis-sion (Figure 1(b)) According to (1) the reflected wavelengthchanges with modifying either 119899eff or Λ

It is easy to see that 120582Bragg will change when the fiber isstrained or compressed (Figure 1(c)) whereas the effectiverefractive index is a material property and thus 119899eff is sensi-tive to changes in temperature This sensitivity of the peakwavelength with respect to thermal and mechanical loadsallows the usage of FBG as strain and temperature sensors[12 13] This is now a common field of optical sensingSeveral features make the FBG system highly appropriate forgeophysical measurements [14 15] These features include

(i) long-term zero-point stability due to the inscribedphysical structure

(ii) transmission of the sensorrsquos signal over distances upto tens of miles

(iii) a form factor which allows FBG to be embedded intostructures

(iv) good immunity against electromagnetic radiation

Journal of Sensors 3

Laser beam

Laser beam

Cladding

Core

GratingInterference pattern

Fiber

Λ

120583m

120583mempty9

empty125

(a)

ReflectionTransmission

minus40

minus30

minus20

minus10

0

Loss

(dB)

1572 1574 1576 15781570Wavelength (nm)

(b)

Light sourceFBG

sensor

Reflected light Transmitted light

(c)

Figure 1 Interference pattern of laser beams in the fiber core during FBG generation (a) Reflected and transmitted signals after the passingof broad band light travelling along the fiber with an FBG inside (b) and a simplified illustration of the strain detection mechanism with abroad band light source containing a continuous spectrum of wavelengths (ie colours)

Figure 2 An FBG thermistor string and strain sensor

(v) ability to cascade and multiplex large numbers ofsensors in a network with a lower complexity thanwith electrical equivalents

In our experiment FBG sensors are practical for measuringthe thermal expansion of large samples because in contrastto standard dilatometers the fiber can be embedded directlyinside the ice sample An FBG thermistor string with 12 dis-tributed thermistors and strain sensor are shown in Figure 2

The neighbour thermistors extend from each other on 1 cmdistance Typical strain resolution for FBG systems is 10minus6(1 120583strain) or better and the accuracy is typically 5 sdot 10

minus6

(5 120583strain) These characteristics are comparable to Michel-son Interferometer Laser Dilatometers as used by Johnsonand Metzner [5] and discussed above

The variation (Δ120582Bragg) of the peak wavelength caused bythe extension (Δ119871119871) and the change of the temperature (Δ119879)of the sensor is described by the equation

Δ120582

120582

= GF sdot

Δ119871

119871

+ 119879119870 sdot Δ119879

(2)

where the gauge factor GF = 0719 and a linear temperaturecoefficient 119879119870 = 55 sdot 10

minus6 are the constants obtained from acalibration cycle for our FBG sensors in standard SMF fiberwithin a temperature range from minus20∘C to 0∘C

The variation of the peak wavelength Δ120582 is measuredwith a spectrometer that receives the reflected signal fromthe FBG sensor For the calculation of the strain (Δ119871119871)according to formula (2) it is necessary to measure thetemperature change (Δ119879) at the strain sensorrsquos position inorder to compensate 119879119870 The temperature measurementscan easily be performed with another FBG sensor protected


(a) (b)

Figure 3 Installation of FBG strain sensor and thermistor string in ice sample (a) Ice sample inside plastic housing (b)

from mechanical deformation or alternatively with a ther-mometer In our experiments we used FBG strain sensorswith a reference peak wavelength in the vicinity of around1534 nm and FBG temperature sensors with a reference peakwavelength in the vicinity of around 1565 nmThe strain andtemperature sensors were cascaded in one optical fiber TheFBGmeasurement systemrsquos nominal resolution and accuracyin our experiment were 008∘C and 04∘C respectively

3 Experiments

31 Thermal Expansion of Unconfined Ice Experiments wereperformed in the cold laboratory of the University Centre inSvalbard (UNIS) Ice was formed in the UNIS ice tank from amixture of sea water pumped from a nearby fjord and freshwater The salinity of the ice samples varied from 0 to 10 pptand the ice had columnar structure The ice samples had arectangular shapewith the long dimension around 20 cmandthe shorter dimensions 5ndash10 cm The optical fibers with FBGstrain sensors had an integrative working length of 20 cmthat was defined by two brass anchor bolts with nuts andwashers The optical fibers were placed into 2mm wide cutsthat were sawn in the ice samples and were fastened at theedges of the ice samples with nuts and washers (Figure 3(a))The ice blockrsquos thermal expansion or shrinkage is thereforetransferred to the optical fiber with the FBG inside (this fiberis prestrained to around 03 by adjusting the nuts accord-ingly) The FBG fiber was not frozen into place since thiswould allow localized shear around the FBG itself to distortthe measurements The FBG temperature sensor (thermistorstring with 12 thermistors) was encapsulated in a 1mm stain-less steel capillary tube and inserted into a drilled hole in theice sample so that one thermistor registered air temperaturenear the FBG strain sensor Other thermistors measuredtemperature inside the ice sample

The equipment used in our measurement setup includesa broadband SLED light source with a central wavelengthof 1550 nm and a bandwidth (FWHM) of sim90 nm (AOSGmbH Germany) a NIR spectrometer ldquoI-MON 512E-USB20 interrogation monitorrdquo with a detectable spectral widthof 1510ndash1595 nm (Ibsen Photonics SA Denmark) a PCand two FBG sensors one for strain measurement and

Computer

LEDIMON

Ice s

ampl

e

Stress sensor

Thermistor string

Figure 4 Schematic of experiment for measuring of thermalexpansion

one for temperature measurement (both from AOS GmbHGermany) The Ibsen monitor is distributed with operatingsoftware including a LabView source codeThe experimentalschematic is shown in Figure 4 All electronic devices theLED source the spectrometer and the PC were installedoutside the cold laboratory Regular optical single-modecableswere used to connect the equipment to the FBG sensorsembedded into the ice sample in the cold laboratory

The ice sample was covered by a plastic housing to avoidevaporation (Figure 3(b)) The temperature in the cold labo-ratory (25 times 25 times 18m) was changed in increments of 2∘Cfrom minus20∘C to a nominal 0∘C programmed and displayedon the laboratoryrsquos control system Temperature control isprovided by a Frigobase Carel system with temperatureprecision better than 1∘C and set point resolution of 01∘CTime intervals for the temperature increments were set to 2hours or greater Temperature and extension are measuredat 1 samples During the experiment we realized that theactual air temperature close to the ice samples was lower thanthe displayed temperature by several degrees We thereforeswitched off the cooling system to allow the room to warmresulting in maximal environmental temperature of minus3∘C

Ice salinity was measured with a Mettler Toledo SevenPro conductivity meter SG7 with resolution 001 ppt The


salinity of the ice samples was measured before and afterthe experiments In some experiments multiple ldquodummyrdquosamples are kept under the same conditions as the strain-gauged sample these samples can then be used to measurechanges in salinity during the experiments

32 Thermal Expansion of Confined Ice A series of exper-iments were undertaken in the cold rooms of UniversityCollege LondonThe FBG strain sensor was used to measurethe linear strain in ice samples of cylindrical shape undervarious conditions The ice was formed between two steelpipes an outer pipewith inside diameter of 11 cmand an innerpipe with outside diameter of 2 cm Fresh ice was made fromLondon tap water saline ice was made from the same with8 ppt NaCl addedThe samples were formed in layers so thatup to 1 cm depth of water was added and allowed to freezebefore the next layer was formed No evidence was seen ofsupercooled water or large bubbles within these layers Theexperiments are conducted in air

Ice samples were tested in three conditions uncon-strained constrained within one pipe and constrainedbetween two pipes (in the first two cases pipes were removedafter forming by briefly warming)These conditions are illus-trated in Figure 5 The pipe has length of 180mm and thesamples are milled 5mm from either end of the pipe suchthat the initial ice sample has two flat parallel ends 190mmapart On these flat ends we place an aluminium spacer ofwidth of 4mm which supports the FBG strain sensor so thatexpansion in the ice in the along-pipe (119911) direction stretchesthe sensor

Temperatures are measured using the FBG 12-thermistorstring inserted through a drilled hole in the ice (or iceand pipe) such that temperatures are recorded throughthe sample in the 119909-direction The temperature used forcalibration of the strain sensor is that closest to the sensor inthis case thermistor 7 For calculation of thermal expansionwe use the average of thermistors 1ndash5 which gives a meanvalue through the ice but eliminates the thermistors in theair which respondmuch faster to temperature changes in theroom Temperature and extension aremeasured at 1 samplesThe entire apparatus is shown in a photograph in Figure 6

33 Thermal Expansion of Floating Confined Ice Laboratoryexperiments were conducted in the UNIS cold lab ice tank toinvestigate the possibility of ice expansion under the influ-ence of under-ice water pressure without external heatingor cooling This physical mechanism is not discussed in thescientific literature It can occur when liquid brine migratesthrough the ice from a relatively warm region to a relativelycold region under the influence of a pressure gradient Lateralconfinement provides a vertically directed shear force at theice edgesThis shear force is in balance with the pressure forceapplied to the ice bottom (Figure 7(a))

Saline ice of 8 cm thickness was frozen on the surfaceof sea water of 1m depth with initial salinity 12 ppt Theice salinity was about 6 ppt The ice was not floating inhydrostatic equilibrium because it cohered to the tank wallsOverpressure in the tank was created by gradually pumpingair into a submerged car inner tube using an electrical pump

The pump was powered by a laboratory power supply withvariable voltage so that the pumping speed could be regulated(Figure 7(a)) The increase of water pressure in the tankcaused ice creep in the vertical direction However our maininterest was in measuring any extension or compression ofthe ice in the horizontal plane

This extension or compression in the horizontal planewasmeasured with one FBG fiber optic strain sensor mounted onsteel angle brackets frozen into the ice (Figures 7(b) and 8(a))Another FBG strain sensor was mounted on similar bracketsfixed to the tank wall (Figures 7(b) and 8(a)) This sensorrecorded deformation of the tank wall under ice action TheFBG thermistor string was used to measure temperature inthe air above the ice and the temperature profile in the ice Atemperature and pressure sensor SBE-39 was used to recordwater pressure below the ice during the experiment Theresolution of the pressure measurements was 4sdot10minus4 dbar Allsensors were synchronized and provided measurements withsampling interval of 1 s

4 Results of Experimental Studies

41 Thermal Expansion of Unconfined Ice Temperature vari-ations in the cold laboratory cause the ice to expand andcontract The laboratory temperature is programmed into anexternal controller and three programs are used

(i) In the first the control temperature is changed by 2∘Cevery 2-3 hours or rarely over a period of several daysThese are referred to as long-term tests

(ii) In the second the control temperature is changed by15ndash20∘C up and down in each test twice These teststake 4ndash6 hours and are referred to as short-term tests

(iii) In the third set the room temperature began atminus20∘Cand the cooling system was then switched off Thisled to an increase in temperature which was moremonotonic than the other tests (where the coolingsystem caused sinusoidal temperature fluctuationsabout a long-term trend)These tests take 8 hours andare referred to asmonotonic tests

The coefficient of volumetric thermal expansion (VCTE) of amaterial is determined by the formula

120581 =

1

120575119881

119889120575119881

119889119879

(3)

where 120575119881 is the infinitesimal volume of the material and 119889120575119881

is a change of the volume caused by the temperature changefrom 119879 to 119879+119889119879 If the sample mass 120575119898 = 120588120575119881 is conserveddefinition (3) can be formulated as

120581 = minus

1

120588

119889120588

119889119879

(4)

VCTE of fresh ice is calculated with formula (4) where the icedensity is calculated by the formula [16]

120588

119894= 9168 (1 + 153 sdot 10

minus4

119879)

minus1

(kgm3)

(5)


I

2 cm

11 cm

(a)

II

18

cm

(b)

III

(c)

Figure 5 Schematic of experimental configurations unconstrained ice sample (a) toroidal ice sample constrained by external steel pipe andfree to expand into central cavity (b) toroidal ice sample constrained by external and internal steel pipes (c)

FBG thermistor Inner Icetorus

Outerpipe

FBGstraingauge

Supportsand

spacerspipe

Figure 6 Full experiment photo schematic and exploded schematic

VCTE of fresh ice is around 153 sdot 10

minus4 Kminus1 Linear coefficientof thermal expansion of fresh ice (CTEFI) is around 51 sdot

10

minus5 Kminus1In general when saline ice includes permeable channels

filled by liquid brine the mass of a saline ice sample is not aconstant and formula (4) cannot be used for the calculationofVCTE for saline iceHerewe assume that saline ice consists

of ice with closed brine pockets (ICP) and permeable brinechannels ICP is not permeable by brine and the ICP salinityis constant Therefore VCTE for ICP is calculated accordingto formula (4) as follows

120581icp = minus

1

120588icp

119889120588icp

119889119879

(6)


LED

IMON

PC

Air balloon

ElectricalpumpFBGTS

SBE 39 Water

Ice

1

12 Pressureforce

Shearforce

(a)

FBGTS 05

mFBGI

FBGW

1m

02m

(b)

Figure 7 Scheme of the experiment on ice extension due to the water pressure increase in the ice tank (a) Locations of FBG strain sensorson the ice surface (FBGI) and on the tank wall (FBGW) (b)

(a)

Ice

Air

Water10

8

6

4

2

0

Dep

th (c

m)

minus10 minus6 minus4 minus2 0minus8

Temperature (∘C)

(b)

Figure 8 Laboratory experiment in the ice tank (a) Temperature profile in model ice measured with FBG thermistor string (b)

ICP density is calculated by the formula [17]

120588icp =

(1 minus ]) 120588119908120588

119894119878

120590icp (120588119894 minus 120588

119908) + 120588

119908119878 (1 minus 120590icp)

(7)

where 120588

119908= 999 kgm3 is the fresh water density 120590icp is the

ICP ice salinity and 119878 is the fractional salt content of the brineFor sea water brine the fractional salt content is calculatedwith the formulas

119878 = 120572119879 120572 = minus00182Kminus1 119879 isin (0

∘C minus82

∘C)

119878 = 0149 + (119879 + 82) 120572

1015840

120572

1015840

= minus001

∘Cminus1 119879 isin (minus82

∘C minus23

∘C)

(8)

The ICP salinity is equal to the mass of salt in the liquid brineincluded in closed brine pockets in a unit mass of ICP VCTEcalculated with formulas (6) and (7) coincides with VCTEintroduced by Malmgren [4] who assumed that permeablechannels are absent in sea ice

Since saline ice includes ICP and permeable channelsfilled by liquid brine the total salinity is equal to a sum

120590si = 120590icp + 120590pc where 120590pc is a mass of salt in permeablebrine channels inside a unit volume of saline ice Liquid brinelocated in the permeable channels does not influence thethermal expansion of saline ice [6] The salinity of the ICPis constant but the ICP mass is changing since some closedbrine pockets can transform into permeable channels due tothe temperature changes Thermal expansion of saline ice isdetermined by volumetric changes of the ICP which dependon both the thermal and mass changes

Let us assume that themass of an infinitesimal ICP sampleis 120575119898icp = 120588icp120575119881icp when the ICP temperature is 119879 The ratioof the ICPmasses related to different temperatures is equal to

120575119898icp

120575119898icp0=

120588icp

120588icp0

120575119881icp

120575119881icp0 (9)

where the subscript ldquo0rdquo is related to the values of the massdensity and volume at 119879 = 119879

0 Assuming ice isotropy and

small deformations we can rewrite formula (9) in the form

120576

120575119898=

120588icp

120588icp0(1 + 3120576

120575119871) minus 1 (10)


where the relative change of the ICP mass 120576120575119898

and the lineardeformation 120576

120575119871are determined by the formulas

120576

120575119898=

120575119898icp minus 120575119898icp0

120575119898icp0

120576

120575119871=

120575119871icp minus 120575119871icp0

120575119871icp0

(11)

where 120575119871icp is a linear dimension of the sample and 120575119871icp0 isits initial value

The experiments were performed with ice samples offinite sizes FBG sensors registered the temperature 119879 andlinear deformation 120576

119871of the samples depending on the time

The linear deformation of a sample is determined by theformula

120576

119871=

119871 (119905) minus 119871

0

119871

0

(12)

where 119871 is the sample length 1198710is its initial value and 119905 is the

timeThe effective coefficient of linear thermal expansion

(ECTE) of an ice sample is determined by the formula

120581si119871 =119889120576

119871

119889119905

(

119889119879

119889119905

)

minus1

(13)

where the temperature 119879(119905) is registered in a point insidethe sample Temperature gradients within the sample canmake it difficult to analyze (13)Therefore in our experimentswe changed the air temperature slowly and measured thetemperature with FBG thermistor string at several pointswithin an ice sample to control the temperature gradient overthe sample

A representative time of temperature equilibration over asample is given by the formula

119905

lowast=

120588si119888si119896si

119871

2

lowast

(14)

where 120588si = 920 kgm3 is typical value of sea ice density 119896si =2W(msdotK) is a typical value of the thermal conductivity ofsea ice 119871

lowast= 01m is representative length of the sample and

119888si is the specific heat capacity of saline ice The specific heatcapacity of fresh ice is equal to 2 kJ(kgsdotK) Formula (14) thengives a time of temperature equilibration in fresh ice samplesof 25 hours

The specific heat capacity of saline ice increases withtemperature according to the formula derived by Schwerdt-feger [17] This is because there is a greater degree of phasechange and thus a greater latent heat expenditure at highertemperatures For example the specific heat capacities ofsaline ice with salinities 4 ppt and 8 ppt reach respectively104 kJ(kgsdotK) and 187 kJ(kgsdotK) when the temperature isminus3∘C With these salinities the times of temperature equili-bration reach respectively 13 hours and 234 hours

The relative change of the ICP mass 120576

119898in a sample is

estimated with a formula similar to (10)

120576

119898=

120588icp

120588icp0(1 + 3120576

119871) minus 1 (15)

Results of the long-term tests with fresh ice samples areshown in Figure 9 The temperature change interval was 4-5 hours Figure 9(a) shows temperature variations in theice samples as measured with the FBG thermistor stringThe insert graph on Figure 9(a) shows temporal variationsinduced by the cooling cycle of the fansThe data are averagedover 15min intervals to minimize the influence of the fancycle Calculated values for ECTE for fresh ice are shown inFigure 9(b) and these are close to the CTEFI shown by thedashed line with deviation less than 10

Two mixtures of sea water and fresh water were madewith respective salinities 94 ppt and 23 ppt These twomixtures were frozen and prepared as ice samples results forthese samples are shown in Figures 10ndash12 The experimentslasted for 190 h and 415 h respectively The temperatureand strain are shown as a function of time in Figure 10Strain is initially defined as zero In Figure 10(a) the strainand temperature gradients have opposite signs between 50and 100 h In Figure 10(b) the gradients have the same signthroughout The opposite gradients of strain in Figure 10(a)show the temperature range in which saline ice exhibitsabnormal thermal expansion

The dependency of strain on temperature is shown inFigures 11(a) and 11(b) The circles indicate the beginningof the experiments Abnormal thermal expansion (the icecontracts with increasing temperature) is seen above aroundminus10∘C in Figure 11(a) (which shows the test conducted onice made with water of 94 ppt salinity) The less salinesample also shows a slightly nonlinear but still positive ECTE(Figure 11(b)) In both of the experiments hysteresis wasobserved in the strain-temperature curves at the points wherethe cycle switches from heating to cooling similar effectswere observed by Butkovich [7] and Johnson and Metzner[5] Figures 11(c) and 11(d) show the relative changes of theICPmasses 120576

119898in the ice samples calculatedwith formula (15)

One can see that the ICPmass decreases with the temperatureincrease in the sample with salinity 94 ppt The ICP massof the sample with salinity 23 ppt changes in the oppositedirection Repeating thermal loading leads to the loss of theICP mass in both cases This suggests that thermal cyclingtends to promote open channels in the ice in favour of closedbrine pockets

Dependencies 120576

119871(119879) calculated from the experimental

data were approximated by polynomial fit on different timeintervals to allow the calculation of the ECTE by formula(13) Examples of polynomial approximations are shown inFigure 12(a) and Figure 12(b) shows the ECTE versus thetemperature The black squares show the ECTE calculated byJohnson and Metzner [5] with an ice salinity of 2 ppt

ECTEs calculated with using of the data of the long-term(the temperature change interval is 2-3 hours) short-termand monotonic tests with saline ice samples are compared inFigure 13(a) In all these tests ice samples were made from amixture of sea and fresh water of 6 ppt salinity Averaging ofthe strain-temperature curves over 15min and polynomialinterpolation of these averages was used to calculate ECTEon specific time intervals Their duration was 11 hours in thelong-term test 2 hours in the short-term test and 8 hoursin the monotonic test The long-term test shows abnormal


385 390 395 400

minus1318

minus1322

minus1326

minus1330380

10 20 30 400t (h)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

T(∘

C)

(a)

CTEFI

times10minus5

minus6 minus5 minus4 minus3minus7

46

48

50

52

54

ECTE

(Kminus1)

T (∘C)

(b)

Figure 9 Temperature (a) and the coefficient of thermal expansion versus the time (b) for fresh ice sample in long-term test

T

S

50 100 1500t (h)

minus25

minus20

minus15

minus10

minus5

0

3120576 Lmiddot104

T(∘

C)

(a)

T

S

100 200 3000t (h)

minus20

minus15

minus10

minus5

05120576 Lmiddot104

T(∘

C)

(b)

Figure 10 Temperature (119879) and strain (119878) versus the time in ice samples with salinities 94 ppt (a) and 23 ppt (b) Long-term tests

thermal expansion with a negative ECTE when the icetemperature is higher than minus6∘C Short-term and monotonictests show normal thermal expansion with a positive ECTEdecreasing with the temperature increase Figure 13(b) showssimilar increase of the ICP mass of the samples with thetemperature increase in all three tests

42Thermal Expansion of Confined Ice In total 27 tests wereconducted on confined ice across the three geometries shownin Figure 5 and for both fresh and saline ice (ie 6 differentexperimental configurations) These experiments took placein the cold rooms of University College London Due tothe complications of sharing working spaces and runninglong experiments (often gt 24 h) tests were not conductedaccording to a rigorous schedule but rather on an ad hocbasis which allowed for both warming and cooling to beinvestigated in all configurations

Figure 14 shows the results of a typical experiment Themeasured strain and temperature are calculated directlyfrom the fiber Bragg wavelengths using known calibrationcoefficients and measured wavelengths for the thermistors

submersed in 0∘C iced waterThese measurements are shownin Figures 14(a) and 14(b) In the temperature plot we can seetwo minor sources of error variations in the air temperaturedue to the cold room fan cycle (duration of 5 minutes) anda sudden cooling (just after 0200) when the door to anadjoining cold room was opened Neither of these processeshas a strong effect on the observed in-ice temperatures (thelowest six lines on the top right plot) but both will have someeffect on overall results

Calculated ECTEs are shown in Figures 14(c) and 14(d)For these ice temperature and strain are averaged over 600 speriods If the temperature difference in this window is lessthan 01∘C then the measurement is ignored since slightresidual strains at constant temperatures can lead to highanomalous expansion coefficients The results shown aretherefore those for which the temperature difference acrossthe measurement window was sufficiently high These arethen shown as a function of time and temperature

Figure 15 shows results combined across all experimentsFor fresh ice (Figure 15(a)) there is a range of positive values ofECTE with a few negative outliersWe see no clear trendwith


times10minus4

minus5

minus4

minus3

minus2

minus1

0

120576 L

minus20 minus15 minus10minus25

T (∘C)

(a)

times10minus4

minus12

minus10

minus8

minus6

minus4

minus2

0

120576 L


T (∘C)

(b)

times10minus4


minus6

minus4

minus2

0

2

120576 m

T (∘C)

(c)

times10minus4

minus20

0

20

40120576 m


T (∘C)

(d)

Figure 11 Strains versus temperature in ice samples with salinities 94 ppt (a) and 23 ppt (b) and the relative changes of the ICP masses ofthe samples with salinities 94 ppt (c) and 23 ppt (d) versus temperature Long-term tests

times10minus4

94ppt 23ppt

minus6

minus4

minus2

0

120576 L

minus15 minus10minus20

T (∘C)

(a)

CTEFI

0

2

4

6times10

minus5

minus2

minus4

minus6

minus8

94ppt

23ppt

ECTE

(Kminus1)


T (∘C)

(b)

Figure 12 Polynomial approximations of the strain-temperature dependencies of ice samples (a) and ECTE versus the temperature calculatedfor the same samples (b) The curves are constructed with initial ice salinities 94 ppt and 23 ppt Long-term tests


1

2

3

times10minus5

ECTE

(Kminus1)

minus10

minus5

0

5CTEFI

minus6minus7 minus5minus9 minus4minus8

T (∘C)

(a)

1

2

3

times10minus4


0

5

10

15

20

25

30

35

120576 m

T (∘C)

(b)

Figure 13 ECTE (a) and the ICP masses (b) versus the temperature in long-term (1) short-term (2) and monotonic (3) tests with saline icesamples The ice salinity is 6 ppt

times10minus4

1 2 3 4 5 60t (h)

00

05

10

15

20

25

120576 L

(a)

minus12

minus10

minus8

minus6

minus4

1 2 3 4 5 60t (h)

T(∘

C)

(b)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)

05 10 15 20 2500t (h)

(c)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)

minus95 minus90minus105minus110minus115 minus100

T (∘C)

(d)

Figure 14 Results from a typical thermal expansion experiment Strain (a) and temperature (b) measurements are shown versus the timeCalculated ECTEs are shown as a function of time (c) and temperature (d) The results shown are for unconstrained fresh ice

temperature Average ECTEs are 469 times 10minus5 Kminus1 for uncon-strained fresh ice 269 times 10minus5 Kminus1 for ice constrained by anexternal pipe and 643 times 10minus5 Kminus1 for ice constrained by bothinternal and external pipesThe average for the unconstrainedpipe is within 10 of the literature value For saline ice with

no pipe (average value 524 times 10minus5 Kminus1) and with 1 pipe(average 506 times 10minus5 Kminus1) we see similar behavior For salineice (Figure 15(b)) with two pipes the values over the firsteight hours of the test are similar (average 542 times 10minus5 Kminus1)but then we see a clear trend towards negative ECTE with


0

5

10

15times10

minus5EC

TE (K

minus1)

minus5

minus10

minus6minus8minus10minus12minus14minus16

T (∘C)

(a)

0

5

10

times10minus5

ECTE

(Kminus1)

minus5

minus10minus18 minus8minus10minus12minus14minus16

T (∘C)

(b)

Figure 15 Experimentally measured ECTE plotted as a function of temperature for fresh ice (a) and for saline ice (b) I unconstrainedsample shown by unshaded circles II sample with one pipe shown by shaded circles and III sample with two pipes

200 400 600 800

00

02

Pressure dbar1 ice strain millistrain2 tank wall strain millistrain

1

2

t (s)

minus02

minus04

minus06

(a)

200 300 400 500 600 700 800

00

01

02

0

8

4

t (s)

minus01

minus02

minus03

Tem

pera

ture

var

iatio

ns (∘

C) Depth = 1 cm

(b)

Figure 16 Strains in ice and tank walls induced by the increase of the water pressure below the ice (a) Temporal variations of the icetemperature induced by brine migration through the ice (b)

rising temperature (above around minus10∘C) It may be that thedual pipes prevent brine drainage and the phase changesassociated with trapped brine drive the negative thermalexpansion coefficient The conditions at the surface of the icecan affect thermal expansion by constraining the ice as wellas by constraining the brine We believe that the high scatterseen in our results may be due to residual stresses whichbuild up and relax in the ice over time particularly as it sticksagainst the pipesThese residual stresses may be released laterthan they build up which would mean that the correlationbetween observed ice strain and instantaneous temperatureis less clear than if all strains are realized immediately

43 Thermal Expansion of Floating Confined Ice The effectsof under-ice pressure on a confined ice sheet are shown inFigure 16 As the pressure was gradually increased in thewater the strain in the ice also increased (Figure 16(a)) At the

same time the strain measured on the inside face of the walldecreased indicating that the wall bent outwards slightlyThetemperature at 1 cm depth and deeper increased as a resultof water migrating through the ice sheet (Figure 16(b)) Thiswater migration was also visually observed The decrease inthe surface temperature is related to the overall temperaturechanges in the cold lab This effect is also picked up by thethermistor at a depth of 1 cm after around 500 s

Figure 16(a) shows positive thermal expansion of icecaused by the vertical migration of liquid brine through theice (which has a vertical temperature gradient) Under-icepressure drives the brine upwards The brine temperature isinitially equal to the temperature of its surrounding ice sincethe brine is in local thermodynamic equilibrium with iceTherefore as brine migrates upward warmer brine from thebottom of the ice (where the ice is warmest) heats up thetop layers of the ice From Figure 16 ice deformation reaches


about 2 sdot 10

minus4 when the temperature increases by about 01∘CThis deformation is caused by thermal expansion of ice andice compression between the tank walls The conclusion hereis that if brine is forced vertically upwards through the ice thisis likely to warm the ice (since the sea water and lower brineare warm compared to the brine near the upper surface)leading to an expansion

5 Conclusions

FBG sensors are a productive tool for laboratory measure-ments of the thermomechanical properties of saline ice It waspossible to perform experiments with samples of differentsizes and geometriesThe high sampling frequency accuracyand resolution of the FBG sensors provided good quality dataacross a temperature range from 0∘C to minus20∘C The sensorsand their associated hardware and software were stable androbust The main complication in these experiments was indeveloping techniques tomount the FBG sensors onto the icesamples

In this paper we have compared values of the coeffi-cients of linear thermal expansion calculated from laboratoryexperiments (ECTE) to predictions based on a model ofice as an impermeable medium (to liquid brine) and on afresh ice model Negative values of ECTE were found whenthe ice samples had salinities of 6 ppt 8 ppt and 94 pptAbnormal thermal expansion contraction with increasingtemperature was observed at temperatures higher than minus8∘C(6 ppt and 8 ppt experiments) or minus11∘C (94 ppt experiments)Hysteresis effects similar to those described in earlier workwere observed Surface area to volume ratio and confinementof the ice appear to affect the ECTE In experiments withconfined floating ice we observed the heating and thermalexpansion of ice due to the vertical migration of liquid brinethrough the ice under the action of water pressure beneaththe ice

We formulated a new model of saline ice consisting ofthe ice with closed brine pockets (ICP) and permeable brinechannels and assumed that closed brine pockets can trans-form into permeable channels under temperature changesThis processmay influence the fraction of themass containedin closed pockets (the ICP mass) Using experimental datawe estimated changes of the ICP mass of the ice sampleswith different salinity It was discovered that the ICP massdecreases with increasing temperature in ice samples ofhigh salinity (94 ppt) and the ICP mass increases withincreasing temperature in ice sampleswith salinities 6 ppt and23 ppt Repeated or cycling temperature changes influencethe decrease in ICP mass with time in all samples

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors wish to acknowledge the support of the ResearchCouncil of Norway through SFI SAMCoT

References

[1] W F Weeks and S F Ackley ldquoThe growth structure and prop-erties of sea icerdquo in The Geophysics of Sea Ice N UntersteinerEd pp 9ndash164 Plenum Press New York NY USA 1986

[2] K M Golden H Eicken A L Heaton J Miner D J Pringleand J Zhu ldquoThermal evolution of permeability andmicrostruc-ture in sea icerdquo Geophysical Research Letters vol 34 no 16Article ID L16501 2007

[3] O Petterson ldquoOn the properties of water and icerdquo in Vega-Expeditionens Veisnskapliga Iakttagelser Nordenskiold Ed vol2 pp 247ndash323 F amp G Beijers Forlag Stockholm Sweden 1883

[4] F Malmgren ldquoOn the properties of sea icerdquo in The NorwegianNorth Polar Expedition with the ldquoMaudrdquo 1918ndash1925 vol 1 no 51927

[5] J B Johnson and R C Metzner ldquoThermal expansion coeffi-cients for sea icerdquo Journal of Glaciology vol 36 no 124 pp 343ndash349 1990

[6] G F N Cox ldquoThermal expansion of saline icerdquo Journal ofGlaciology vol 29 no 103 pp 425ndash432 1983

[7] T R Butkovich ldquoThermal expansion of icerdquo Journal of AppliedPhysics vol 30 no 3 pp 350ndash353 1959

[8] A Marchenko T Thiel and S Sukhorukov ldquoMeasurements ofthermally induced deformations in saline ice with fiber Bragggrating sensorsrdquo in Proceedings of the 21st IAHR InternationalSymposium on Ice ldquoIce Research for a Sustainable EnvironmentrdquoZ Li and P Lu Eds Dalian University of Technology PressDalian China June 2012

[9] A Marchenko D Wrangborg and T Thiel ldquoUsing distributedoptical fiber sensors based on FBGs for the measurement oftemperature fluctuations in saline ice andwater on small scalesrdquoin Proceedings of the 22nd International Conference on Port andOcean Engineering under Arctic Conditions (POAC rsquo13) p 11Espoo Finland June 2013

[10] B Lishman and A Marchenko ldquoAn investigation of relativethermal expansion and contraction of ice and steelrdquo in Proceed-ings of the 22nd IAHR International Symposium on Ice paper1173 Singapore 2014

[11] D Wrangborg and A Marchenko ldquoMeasurement of loadsexerted by sea ice on the quay at kapp Amsterdam on svalbardrdquoTech Rep POAC15-141 Trondheim Norway 2015

[12] A Othonos and K Kalli Fiber Bragg Gratings Fundamentalsand Applications in Telecommunications and Sensing ArtechHouse 1999

[13] Y J Rao ldquoIn-fibre Bragg grating sensorsrdquoMeasurement Scienceand Technology vol 8 no 4 pp 355ndash376 1997

[14] P Ferraro andG DeNatale ldquoOn the possible use of optical fiberBragg gratings as strain sensors for geodynamical monitoringrdquoOptics and Lasers in Engineering vol 37 no 2-3 pp 115ndash1302002

[15] B Mueller J Meissner and T Thiel Results of Continuous In-Situ Stress Measurement with Optical Strain Sensors Taylor ampFrancis London UK 2006

[16] E R PounderThe Physics of Ice Pergamon Press Oxford UK1965

[17] P Schwerdtfeger ldquoThe thermal properties of sea icerdquo Journal ofGlaciology vol 4 no 36 pp 789ndash807 1963

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



leak out of the sample and be expelled into the immersionfluid

Theoretical models for the calculation of the coefficientof thermal expansion of saline ice have been developed byMalmgren [4] and Cox [6] Malmgren and Cox made oppos-ing assumptions when formulating the basic equationsMalmgren assumed that brine is never expelled from thesample and that all salts are trapped in brine pockets insidethe sample Temperature changes therefore cause internal icemelt or brine refreezing Cox contrastingly assumed that thebehavior of saline ice is analogous to a pure ice cup filledwith liquid brine both are free to expand and contract inde-pendently and hence phase changes have no effect on thethermal expansion of the ice

Johnson and Metzner [5] used a Michelson Interferom-eter to measure linear thermal expansion of cylindrical icesamples taken from a sheet of first-year congelation sea ice inHarrison Bay AlaskaTheir apparatus included the interfero-meter along with a sample holder temperature control unitand computer-controlled data acquisition unit The diameterof their samples was 38mm and the length was 7125mmfor 2 ppt ice and 6933mm for 4 ppt ice Optically flat targetmirrors were frozen to the front surface of the samplestrimmed parallel with a microtome and used to control thesample length with the interferometer They were able toachieve a displacement resolution of 3164 nm correspondingto a strain resolution of the order of 5 times 10minus6

Mean and instantaneous values of the thermal expansioncoefficient were found to be similar to the fresh ice ther-mal expansion coefficient of 5 times 10minus5∘C Their 4 ppt iceshowed hysteresis when cooled and rewarmed after the initialwarming test Similar effects were observed in the experi-ments of Butkovich [7] with fresh water ice where the coeffi-cient of linear thermal expansion decreased with succeedingruns Their conclusion was that their results agreed with theanalysis of Cox [6]

Systematic investigations of the thermomechanical be-havior of saline ice samples were performed using fiberoptic strain and temperature sensors based on Fiber BraggGratings in the cold laboratories of the University Centre inSvalbard Norway and University College London UK from2012 to 2015 [8ndash11] The FBG system used was designed byAdvance Optic Solutions GmbH (Germany) The flexibilityof the FBG system allowed experiments to be conducted withmany different sizes and geometries of ice sample and withfloating ice

The present paper describes the instrumentation andexperimental setups and summarizes experimental resultson the measurement of thermal expansion of saline ice Weformulate a new model of the thermal expansion of salineice assuming the possibility of structural changes in theice associated with gradual transformation of closed brinepockets into permeable brine channels under the influenceof temperature changes Mass changes in the ice with closedbrine pockets are estimated using the experimental dataThermal expansion of saline ice caused by the migrationof liquid brine through the ice is also discussed in thepaper

2 Instrumentation

The fiber type used for this kind of optical sensing istypically standard telecom single-mode fibers (SMF) that arewidely used in communications with an overall diameterof 025mm and a core diameter of only about 001mmA Fiber Bragg Grating is a periodical index change in therefractive index 119899 along this optical silica fiberrsquos core formedby an interference pattern of two UV laser beams that thefiber is exposed to There are various techniques to generatethe two coherent beams (Figure 1(a)) [12] The index changeresults from a certain photosensitivity of the fiberrsquos corewhich is due to the presence of some chemical dopantsinside the silicarsquos structure typically germanium oxide Theinterference pattern being generated by two coherent laserbeams consists of a periodic UV power modulation along thefiber determined by the incident angle of the two beamsThisUV pattern migrates to the fiber corersquos index change patternby the said photosensitive mechanism

Areas of higher 119899 alternate with areas of lower 119899 witha period of less than 10minus6mm while Δ119899 is relatively smallquantity (Δ119899 lt 0001 and 119899 = 145) Due to the index changeeach ldquograting linerdquo reflects a very small portion of the lightwave propagating along the fiber back to the light sourceAlthough a single reflected portion is negligible compared tothe transmitted power the effect becomes noticeable becausethe amount of ldquograting linesrdquo in a conventional FBG is about4000mm and a typical FBG with 10mm length consists of40 thousand reflections If the lightrsquos wavelength matches thecondition

120582Bragg = 2 sdot 119899eff sdot Λ (1)

where 119899eff is the fiberrsquos effective index 120582Bragg is the lightrsquoswavelength and Λ is the index changersquos period then allreflected light wave portions are propagating ldquoin-phaserdquo andinterfere constructively Depending on the actual Δ119899 thisresults in a typical narrow-band power peak in the reflec-tion spectrum at 120582Bragg and vice versa in a loss in transmis-sion (Figure 1(b)) According to (1) the reflected wavelengthchanges with modifying either 119899eff or Λ

It is easy to see that 120582Bragg will change when the fiber isstrained or compressed (Figure 1(c)) whereas the effectiverefractive index is a material property and thus 119899eff is sensi-tive to changes in temperature This sensitivity of the peakwavelength with respect to thermal and mechanical loadsallows the usage of FBG as strain and temperature sensors[12 13] This is now a common field of optical sensingSeveral features make the FBG system highly appropriate forgeophysical measurements [14 15] These features include

(i) long-term zero-point stability due to the inscribedphysical structure

(ii) transmission of the sensorrsquos signal over distances upto tens of miles

(iii) a form factor which allows FBG to be embedded intostructures

(iv) good immunity against electromagnetic radiation


Laser beam

Laser beam

Cladding

Core


Fiber

Λ

120583m

120583mempty9

empty125

(a)


minus40

minus30

minus20

minus10

0

Loss

(dB)

1572 1574 1576 15781570Wavelength (nm)

(b)

Light sourceFBG

sensor


(c)






minus6



Δ120582

120582

= GF sdot

Δ119871

119871

+ 119879119870 sdot Δ119879

(2)





(a) (b)



3 Experiments



Computer

LEDIMON

Ice s

ampl

e

Stress sensor

Thermistor string




















120581 =

1

120575119881

119889120575119881

119889119879

(3)



120581 = minus

1

120588

119889120588

119889119879

(4)


120588

119894= 9168 (1 + 153 sdot 10

minus4

119879)

minus1

(kgm3)

(5)


I

2 cm

11 cm

(a)

II

18

cm

(b)

III

(c)



Outerpipe

FBGstraingauge

Supportsand

spacerspipe




10




120581icp = minus

1

120588icp

119889120588icp

119889119879

(6)


LED

IMON

PC

Air balloon

ElectricalpumpFBGTS

SBE 39 Water

Ice

1

12 Pressureforce

Shearforce

(a)

FBGTS 05

mFBGI

FBGW

1m

02m

(b)


(a)

Ice

Air

Water10

8

6

4

2

0

Dep

th (c

m)


Temperature (∘C)

(b)



120588icp =

(1 minus ]) 120588119908120588

119894119878

120590icp (120588119894 minus 120588

119908) + 120588

119908119878 (1 minus 120590icp)

(7)

where 120588




∘C minus82

∘C)

119878 = 0149 + (119879 + 82) 120572

1015840

120572

1015840

= minus001


∘C minus23

∘C)

(8)





120575119898icp

120575119898icp0=

120588icp

120588icp0

120575119881icp

120575119881icp0 (9)




120576

120575119898=

120588icp

120588icp0(1 + 3120576

120575119871) minus 1 (10)





120576

120575119898=

120575119898icp minus 120575119898icp0

120575119898icp0

120576

120575119871=

120575119871icp minus 120575119871icp0

120575119871icp0

(11)





120576

119871=

119871 (119905) minus 119871

0

119871

0

(12)




120581si119871 =119889120576

119871

119889119905

(

119889119879

119889119905

)

minus1

(13)



119905

lowast=

120588si119888si119896si

119871

2

lowast

(14)








120576

119898=

120588icp

120588icp0(1 + 3120576

119871) minus 1 (15)






Dependencies 120576





385 390 395 400

minus1318

minus1322

minus1326

minus1330380

10 20 30 400t (h)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

T(∘

C)

(a)

CTEFI

times10minus5


46

48

50

52

54

ECTE

(Kminus1)

T (∘C)

(b)


T

S

50 100 1500t (h)

minus25

minus20

minus15

minus10

minus5

0

3120576 Lmiddot104

T(∘

C)

(a)

T

S

100 200 3000t (h)

minus20

minus15

minus10

minus5

05120576 Lmiddot104

T(∘

C)

(b)









times10minus4

minus5

minus4

minus3

minus2

minus1

0

120576 L


T (∘C)

(a)

times10minus4

minus12

minus10

minus8

minus6

minus4

minus2

0

120576 L


T (∘C)

(b)

times10minus4


minus6

minus4

minus2

0

2

120576 m

T (∘C)

(c)

times10minus4

minus20

0

20

40120576 m


T (∘C)

(d)


times10minus4

94ppt 23ppt

minus6

minus4

minus2

0

120576 L


T (∘C)

(a)

CTEFI

0

2

4

6times10

minus5

minus2

minus4

minus6

minus8

94ppt

23ppt

ECTE

(Kminus1)


T (∘C)

(b)



1

2

3

times10minus5

ECTE

(Kminus1)

minus10

minus5

0

5CTEFI


T (∘C)

(a)

1

2

3

times10minus4


0

5

10

15

20

25

30

35

120576 m

T (∘C)

(b)


times10minus4

1 2 3 4 5 60t (h)

00

05

10

15

20

25

120576 L

(a)

minus12

minus10

minus8

minus6

minus4

1 2 3 4 5 60t (h)

T(∘

C)

(b)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)

05 10 15 20 2500t (h)

(c)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)


T (∘C)

(d)





0

5

10

15times10

minus5EC

TE (K

minus1)

minus5

minus10


T (∘C)

(a)

0

5

10

times10minus5

ECTE

(Kminus1)

minus5


T (∘C)

(b)


200 400 600 800

00

02


1

2

t (s)

minus02

minus04

minus06

(a)

200 300 400 500 600 700 800

00

01

02

0

8

4

t (s)

minus01

minus02

minus03

Tem

pera

ture

var

iatio

ns (∘

C) Depth = 1 cm

(b)







about 2 sdot 10


5 Conclusions






Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Laser beam

Laser beam

Cladding

Core


Fiber

Λ

120583m

120583mempty9

empty125

(a)


minus40

minus30

minus20

minus10

0

Loss

(dB)

1572 1574 1576 15781570Wavelength (nm)

(b)

Light sourceFBG

sensor


(c)






minus6



Δ120582

120582

= GF sdot

Δ119871

119871

+ 119879119870 sdot Δ119879

(2)





(a) (b)



3 Experiments



Computer

LEDIMON

Ice s

ampl

e

Stress sensor

Thermistor string




















120581 =

1

120575119881

119889120575119881

119889119879

(3)



120581 = minus

1

120588

119889120588

119889119879

(4)


120588

119894= 9168 (1 + 153 sdot 10

minus4

119879)

minus1

(kgm3)

(5)


I

2 cm

11 cm

(a)

II

18

cm

(b)

III

(c)



Outerpipe

FBGstraingauge

Supportsand

spacerspipe




10




120581icp = minus

1

120588icp

119889120588icp

119889119879

(6)


LED

IMON

PC

Air balloon

ElectricalpumpFBGTS

SBE 39 Water

Ice

1

12 Pressureforce

Shearforce

(a)

FBGTS 05

mFBGI

FBGW

1m

02m

(b)


(a)

Ice

Air

Water10

8

6

4

2

0

Dep

th (c

m)


Temperature (∘C)

(b)



120588icp =

(1 minus ]) 120588119908120588

119894119878

120590icp (120588119894 minus 120588

119908) + 120588

119908119878 (1 minus 120590icp)

(7)

where 120588




∘C minus82

∘C)

119878 = 0149 + (119879 + 82) 120572

1015840

120572

1015840

= minus001


∘C minus23

∘C)

(8)





120575119898icp

120575119898icp0=

120588icp

120588icp0

120575119881icp

120575119881icp0 (9)




120576

120575119898=

120588icp

120588icp0(1 + 3120576

120575119871) minus 1 (10)





120576

120575119898=

120575119898icp minus 120575119898icp0

120575119898icp0

120576

120575119871=

120575119871icp minus 120575119871icp0

120575119871icp0

(11)





120576

119871=

119871 (119905) minus 119871

0

119871

0

(12)




120581si119871 =119889120576

119871

119889119905

(

119889119879

119889119905

)

minus1

(13)



119905

lowast=

120588si119888si119896si

119871

2

lowast

(14)








120576

119898=

120588icp

120588icp0(1 + 3120576

119871) minus 1 (15)






Dependencies 120576





385 390 395 400

minus1318

minus1322

minus1326

minus1330380

10 20 30 400t (h)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

T(∘

C)

(a)

CTEFI

times10minus5


46

48

50

52

54

ECTE

(Kminus1)

T (∘C)

(b)


T

S

50 100 1500t (h)

minus25

minus20

minus15

minus10

minus5

0

3120576 Lmiddot104

T(∘

C)

(a)

T

S

100 200 3000t (h)

minus20

minus15

minus10

minus5

05120576 Lmiddot104

T(∘

C)

(b)









times10minus4

minus5

minus4

minus3

minus2

minus1

0

120576 L


T (∘C)

(a)

times10minus4

minus12

minus10

minus8

minus6

minus4

minus2

0

120576 L


T (∘C)

(b)

times10minus4


minus6

minus4

minus2

0

2

120576 m

T (∘C)

(c)

times10minus4

minus20

0

20

40120576 m


T (∘C)

(d)


times10minus4

94ppt 23ppt

minus6

minus4

minus2

0

120576 L


T (∘C)

(a)

CTEFI

0

2

4

6times10

minus5

minus2

minus4

minus6

minus8

94ppt

23ppt

ECTE

(Kminus1)


T (∘C)

(b)



1

2

3

times10minus5

ECTE

(Kminus1)

minus10

minus5

0

5CTEFI


T (∘C)

(a)

1

2

3

times10minus4


0

5

10

15

20

25

30

35

120576 m

T (∘C)

(b)


times10minus4

1 2 3 4 5 60t (h)

00

05

10

15

20

25

120576 L

(a)

minus12

minus10

minus8

minus6

minus4

1 2 3 4 5 60t (h)

T(∘

C)

(b)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)

05 10 15 20 2500t (h)

(c)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)


T (∘C)

(d)





0

5

10

15times10

minus5EC

TE (K

minus1)

minus5

minus10


T (∘C)

(a)

0

5

10

times10minus5

ECTE

(Kminus1)

minus5


T (∘C)

(b)


200 400 600 800

00

02


1

2

t (s)

minus02

minus04

minus06

(a)

200 300 400 500 600 700 800

00

01

02

0

8

4

t (s)

minus01

minus02

minus03

Tem

pera

ture

var

iatio

ns (∘

C) Depth = 1 cm

(b)







about 2 sdot 10


5 Conclusions






Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) (b)



3 Experiments



Computer

LEDIMON

Ice s

ampl

e

Stress sensor

Thermistor string




















120581 =

1

120575119881

119889120575119881

119889119879

(3)



120581 = minus

1

120588

119889120588

119889119879

(4)


120588

119894= 9168 (1 + 153 sdot 10

minus4

119879)

minus1

(kgm3)

(5)


I

2 cm

11 cm

(a)

II

18

cm

(b)

III

(c)



Outerpipe

FBGstraingauge

Supportsand

spacerspipe




10




120581icp = minus

1

120588icp

119889120588icp

119889119879

(6)


LED

IMON

PC

Air balloon

ElectricalpumpFBGTS

SBE 39 Water

Ice

1

12 Pressureforce

Shearforce

(a)

FBGTS 05

mFBGI

FBGW

1m

02m

(b)


(a)

Ice

Air

Water10

8

6

4

2

0

Dep

th (c

m)


Temperature (∘C)

(b)



120588icp =

(1 minus ]) 120588119908120588

119894119878

120590icp (120588119894 minus 120588

119908) + 120588

119908119878 (1 minus 120590icp)

(7)

where 120588




∘C minus82

∘C)

119878 = 0149 + (119879 + 82) 120572

1015840

120572

1015840

= minus001


∘C minus23

∘C)

(8)





120575119898icp

120575119898icp0=

120588icp

120588icp0

120575119881icp

120575119881icp0 (9)




120576

120575119898=

120588icp

120588icp0(1 + 3120576

120575119871) minus 1 (10)





120576

120575119898=

120575119898icp minus 120575119898icp0

120575119898icp0

120576

120575119871=

120575119871icp minus 120575119871icp0

120575119871icp0

(11)





120576

119871=

119871 (119905) minus 119871

0

119871

0

(12)




120581si119871 =119889120576

119871

119889119905

(

119889119879

119889119905

)

minus1

(13)



119905

lowast=

120588si119888si119896si

119871

2

lowast

(14)








120576

119898=

120588icp

120588icp0(1 + 3120576

119871) minus 1 (15)






Dependencies 120576





385 390 395 400

minus1318

minus1322

minus1326

minus1330380

10 20 30 400t (h)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

T(∘

C)

(a)

CTEFI

times10minus5


46

48

50

52

54

ECTE

(Kminus1)

T (∘C)

(b)


T

S

50 100 1500t (h)

minus25

minus20

minus15

minus10

minus5

0

3120576 Lmiddot104

T(∘

C)

(a)

T

S

100 200 3000t (h)

minus20

minus15

minus10

minus5

05120576 Lmiddot104

T(∘

C)

(b)









times10minus4

minus5

minus4

minus3

minus2

minus1

0

120576 L


T (∘C)

(a)

times10minus4

minus12

minus10

minus8

minus6

minus4

minus2

0

120576 L


T (∘C)

(b)

times10minus4


minus6

minus4

minus2

0

2

120576 m

T (∘C)

(c)

times10minus4

minus20

0

20

40120576 m


T (∘C)

(d)


times10minus4

94ppt 23ppt

minus6

minus4

minus2

0

120576 L


T (∘C)

(a)

CTEFI

0

2

4

6times10

minus5

minus2

minus4

minus6

minus8

94ppt

23ppt

ECTE

(Kminus1)


T (∘C)

(b)



1

2

3

times10minus5

ECTE

(Kminus1)

minus10

minus5

0

5CTEFI


T (∘C)

(a)

1

2

3

times10minus4


0

5

10

15

20

25

30

35

120576 m

T (∘C)

(b)


times10minus4

1 2 3 4 5 60t (h)

00

05

10

15

20

25

120576 L

(a)

minus12

minus10

minus8

minus6

minus4

1 2 3 4 5 60t (h)

T(∘

C)

(b)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)

05 10 15 20 2500t (h)

(c)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)


T (∘C)

(d)





0

5

10

15times10

minus5EC

TE (K

minus1)

minus5

minus10


T (∘C)

(a)

0

5

10

times10minus5

ECTE

(Kminus1)

minus5


T (∘C)

(b)


200 400 600 800

00

02


1

2

t (s)

minus02

minus04

minus06

(a)

200 300 400 500 600 700 800

00

01

02

0

8

4

t (s)

minus01

minus02

minus03

Tem

pera

ture

var

iatio

ns (∘

C) Depth = 1 cm

(b)







about 2 sdot 10


5 Conclusions






Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























120581 =

1

120575119881

119889120575119881

119889119879

(3)



120581 = minus

1

120588

119889120588

119889119879

(4)


120588

119894= 9168 (1 + 153 sdot 10

minus4

119879)

minus1

(kgm3)

(5)


I

2 cm

11 cm

(a)

II

18

cm

(b)

III

(c)



Outerpipe

FBGstraingauge

Supportsand

spacerspipe




10




120581icp = minus

1

120588icp

119889120588icp

119889119879

(6)


LED

IMON

PC

Air balloon

ElectricalpumpFBGTS

SBE 39 Water

Ice

1

12 Pressureforce

Shearforce

(a)

FBGTS 05

mFBGI

FBGW

1m

02m

(b)


(a)

Ice

Air

Water10

8

6

4

2

0

Dep

th (c

m)


Temperature (∘C)

(b)



120588icp =

(1 minus ]) 120588119908120588

119894119878

120590icp (120588119894 minus 120588

119908) + 120588

119908119878 (1 minus 120590icp)

(7)

where 120588




∘C minus82

∘C)

119878 = 0149 + (119879 + 82) 120572

1015840

120572

1015840

= minus001


∘C minus23

∘C)

(8)





120575119898icp

120575119898icp0=

120588icp

120588icp0

120575119881icp

120575119881icp0 (9)




120576

120575119898=

120588icp

120588icp0(1 + 3120576

120575119871) minus 1 (10)





120576

120575119898=

120575119898icp minus 120575119898icp0

120575119898icp0

120576

120575119871=

120575119871icp minus 120575119871icp0

120575119871icp0

(11)





120576

119871=

119871 (119905) minus 119871

0

119871

0

(12)




120581si119871 =119889120576

119871

119889119905

(

119889119879

119889119905

)

minus1

(13)



119905

lowast=

120588si119888si119896si

119871

2

lowast

(14)








120576

119898=

120588icp

120588icp0(1 + 3120576

119871) minus 1 (15)






Dependencies 120576





385 390 395 400

minus1318

minus1322

minus1326

minus1330380

10 20 30 400t (h)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

T(∘

C)

(a)

CTEFI

times10minus5


46

48

50

52

54

ECTE

(Kminus1)

T (∘C)

(b)


T

S

50 100 1500t (h)

minus25

minus20

minus15

minus10

minus5

0

3120576 Lmiddot104

T(∘

C)

(a)

T

S

100 200 3000t (h)

minus20

minus15

minus10

minus5

05120576 Lmiddot104

T(∘

C)

(b)









times10minus4

minus5

minus4

minus3

minus2

minus1

0

120576 L


T (∘C)

(a)

times10minus4

minus12

minus10

minus8

minus6

minus4

minus2

0

120576 L


T (∘C)

(b)

times10minus4


minus6

minus4

minus2

0

2

120576 m

T (∘C)

(c)

times10minus4

minus20

0

20

40120576 m


T (∘C)

(d)


times10minus4

94ppt 23ppt

minus6

minus4

minus2

0

120576 L


T (∘C)

(a)

CTEFI

0

2

4

6times10

minus5

minus2

minus4

minus6

minus8

94ppt

23ppt

ECTE

(Kminus1)


T (∘C)

(b)



1

2

3

times10minus5

ECTE

(Kminus1)

minus10

minus5

0

5CTEFI


T (∘C)

(a)

1

2

3

times10minus4


0

5

10

15

20

25

30

35

120576 m

T (∘C)

(b)


times10minus4

1 2 3 4 5 60t (h)

00

05

10

15

20

25

120576 L

(a)

minus12

minus10

minus8

minus6

minus4

1 2 3 4 5 60t (h)

T(∘

C)

(b)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)

05 10 15 20 2500t (h)

(c)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)


T (∘C)

(d)





0

5

10

15times10

minus5EC

TE (K

minus1)

minus5

minus10


T (∘C)

(a)

0

5

10

times10minus5

ECTE

(Kminus1)

minus5


T (∘C)

(b)


200 400 600 800

00

02


1

2

t (s)

minus02

minus04

minus06

(a)

200 300 400 500 600 700 800

00

01

02

0

8

4

t (s)

minus01

minus02

minus03

Tem

pera

ture

var

iatio

ns (∘

C) Depth = 1 cm

(b)







about 2 sdot 10


5 Conclusions






Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










I

2 cm

11 cm

(a)

II

18

cm

(b)

III

(c)



Outerpipe

FBGstraingauge

Supportsand

spacerspipe




10




120581icp = minus

1

120588icp

119889120588icp

119889119879

(6)


LED

IMON

PC

Air balloon

ElectricalpumpFBGTS

SBE 39 Water

Ice

1

12 Pressureforce

Shearforce

(a)

FBGTS 05

mFBGI

FBGW

1m

02m

(b)


(a)

Ice

Air

Water10

8

6

4

2

0

Dep

th (c

m)


Temperature (∘C)

(b)



120588icp =

(1 minus ]) 120588119908120588

119894119878

120590icp (120588119894 minus 120588

119908) + 120588

119908119878 (1 minus 120590icp)

(7)

where 120588




∘C minus82

∘C)

119878 = 0149 + (119879 + 82) 120572

1015840

120572

1015840

= minus001


∘C minus23

∘C)

(8)





120575119898icp

120575119898icp0=

120588icp

120588icp0

120575119881icp

120575119881icp0 (9)




120576

120575119898=

120588icp

120588icp0(1 + 3120576

120575119871) minus 1 (10)





120576

120575119898=

120575119898icp minus 120575119898icp0

120575119898icp0

120576

120575119871=

120575119871icp minus 120575119871icp0

120575119871icp0

(11)





120576

119871=

119871 (119905) minus 119871

0

119871

0

(12)




120581si119871 =119889120576

119871

119889119905

(

119889119879

119889119905

)

minus1

(13)



119905

lowast=

120588si119888si119896si

119871

2

lowast

(14)








120576

119898=

120588icp

120588icp0(1 + 3120576

119871) minus 1 (15)






Dependencies 120576





385 390 395 400

minus1318

minus1322

minus1326

minus1330380

10 20 30 400t (h)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

T(∘

C)

(a)

CTEFI

times10minus5


46

48

50

52

54

ECTE

(Kminus1)

T (∘C)

(b)


T

S

50 100 1500t (h)

minus25

minus20

minus15

minus10

minus5

0

3120576 Lmiddot104

T(∘

C)

(a)

T

S

100 200 3000t (h)

minus20

minus15

minus10

minus5

05120576 Lmiddot104

T(∘

C)

(b)









times10minus4

minus5

minus4

minus3

minus2

minus1

0

120576 L


T (∘C)

(a)

times10minus4

minus12

minus10

minus8

minus6

minus4

minus2

0

120576 L


T (∘C)

(b)

times10minus4


minus6

minus4

minus2

0

2

120576 m

T (∘C)

(c)

times10minus4

minus20

0

20

40120576 m


T (∘C)

(d)


times10minus4

94ppt 23ppt

minus6

minus4

minus2

0

120576 L


T (∘C)

(a)

CTEFI

0

2

4

6times10

minus5

minus2

minus4

minus6

minus8

94ppt

23ppt

ECTE

(Kminus1)


T (∘C)

(b)



1

2

3

times10minus5

ECTE

(Kminus1)

minus10

minus5

0

5CTEFI


T (∘C)

(a)

1

2

3

times10minus4


0

5

10

15

20

25

30

35

120576 m

T (∘C)

(b)


times10minus4

1 2 3 4 5 60t (h)

00

05

10

15

20

25

120576 L

(a)

minus12

minus10

minus8

minus6

minus4

1 2 3 4 5 60t (h)

T(∘

C)

(b)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)

05 10 15 20 2500t (h)

(c)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)


T (∘C)

(d)





0

5

10

15times10

minus5EC

TE (K

minus1)

minus5

minus10


T (∘C)

(a)

0

5

10

times10minus5

ECTE

(Kminus1)

minus5


T (∘C)

(b)


200 400 600 800

00

02


1

2

t (s)

minus02

minus04

minus06

(a)

200 300 400 500 600 700 800

00

01

02

0

8

4

t (s)

minus01

minus02

minus03

Tem

pera

ture

var

iatio

ns (∘

C) Depth = 1 cm

(b)







about 2 sdot 10


5 Conclusions






Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










LED

IMON

PC

Air balloon

ElectricalpumpFBGTS

SBE 39 Water

Ice

1

12 Pressureforce

Shearforce

(a)

FBGTS 05

mFBGI

FBGW

1m

02m

(b)


(a)

Ice

Air

Water10

8

6

4

2

0

Dep

th (c

m)


Temperature (∘C)

(b)



120588icp =

(1 minus ]) 120588119908120588

119894119878

120590icp (120588119894 minus 120588

119908) + 120588

119908119878 (1 minus 120590icp)

(7)

where 120588




∘C minus82

∘C)

119878 = 0149 + (119879 + 82) 120572

1015840

120572

1015840

= minus001


∘C minus23

∘C)

(8)





120575119898icp

120575119898icp0=

120588icp

120588icp0

120575119881icp

120575119881icp0 (9)




120576

120575119898=

120588icp

120588icp0(1 + 3120576

120575119871) minus 1 (10)





120576

120575119898=

120575119898icp minus 120575119898icp0

120575119898icp0

120576

120575119871=

120575119871icp minus 120575119871icp0

120575119871icp0

(11)





120576

119871=

119871 (119905) minus 119871

0

119871

0

(12)




120581si119871 =119889120576

119871

119889119905

(

119889119879

119889119905

)

minus1

(13)



119905

lowast=

120588si119888si119896si

119871

2

lowast

(14)








120576

119898=

120588icp

120588icp0(1 + 3120576

119871) minus 1 (15)






Dependencies 120576





385 390 395 400

minus1318

minus1322

minus1326

minus1330380

10 20 30 400t (h)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

T(∘

C)

(a)

CTEFI

times10minus5


46

48

50

52

54

ECTE

(Kminus1)

T (∘C)

(b)


T

S

50 100 1500t (h)

minus25

minus20

minus15

minus10

minus5

0

3120576 Lmiddot104

T(∘

C)

(a)

T

S

100 200 3000t (h)

minus20

minus15

minus10

minus5

05120576 Lmiddot104

T(∘

C)

(b)









times10minus4

minus5

minus4

minus3

minus2

minus1

0

120576 L


T (∘C)

(a)

times10minus4

minus12

minus10

minus8

minus6

minus4

minus2

0

120576 L


T (∘C)

(b)

times10minus4


minus6

minus4

minus2

0

2

120576 m

T (∘C)

(c)

times10minus4

minus20

0

20

40120576 m


T (∘C)

(d)


times10minus4

94ppt 23ppt

minus6

minus4

minus2

0

120576 L


T (∘C)

(a)

CTEFI

0

2

4

6times10

minus5

minus2

minus4

minus6

minus8

94ppt

23ppt

ECTE

(Kminus1)


T (∘C)

(b)



1

2

3

times10minus5

ECTE

(Kminus1)

minus10

minus5

0

5CTEFI


T (∘C)

(a)

1

2

3

times10minus4


0

5

10

15

20

25

30

35

120576 m

T (∘C)

(b)


times10minus4

1 2 3 4 5 60t (h)

00

05

10

15

20

25

120576 L

(a)

minus12

minus10

minus8

minus6

minus4

1 2 3 4 5 60t (h)

T(∘

C)

(b)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)

05 10 15 20 2500t (h)

(c)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)


T (∘C)

(d)





0

5

10

15times10

minus5EC

TE (K

minus1)

minus5

minus10


T (∘C)

(a)

0

5

10

times10minus5

ECTE

(Kminus1)

minus5


T (∘C)

(b)


200 400 600 800

00

02


1

2

t (s)

minus02

minus04

minus06

(a)

200 300 400 500 600 700 800

00

01

02

0

8

4

t (s)

minus01

minus02

minus03

Tem

pera

ture

var

iatio

ns (∘

C) Depth = 1 cm

(b)







about 2 sdot 10


5 Conclusions






Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













120576

120575119898=

120575119898icp minus 120575119898icp0

120575119898icp0

120576

120575119871=

120575119871icp minus 120575119871icp0

120575119871icp0

(11)





120576

119871=

119871 (119905) minus 119871

0

119871

0

(12)




120581si119871 =119889120576

119871

119889119905

(

119889119879

119889119905

)

minus1

(13)



119905

lowast=

120588si119888si119896si

119871

2

lowast

(14)








120576

119898=

120588icp

120588icp0(1 + 3120576

119871) minus 1 (15)






Dependencies 120576





385 390 395 400

minus1318

minus1322

minus1326

minus1330380

10 20 30 400t (h)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

T(∘

C)

(a)

CTEFI

times10minus5


46

48

50

52

54

ECTE

(Kminus1)

T (∘C)

(b)


T

S

50 100 1500t (h)

minus25

minus20

minus15

minus10

minus5

0

3120576 Lmiddot104

T(∘

C)

(a)

T

S

100 200 3000t (h)

minus20

minus15

minus10

minus5

05120576 Lmiddot104

T(∘

C)

(b)









times10minus4

minus5

minus4

minus3

minus2

minus1

0

120576 L


T (∘C)

(a)

times10minus4

minus12

minus10

minus8

minus6

minus4

minus2

0

120576 L


T (∘C)

(b)

times10minus4


minus6

minus4

minus2

0

2

120576 m

T (∘C)

(c)

times10minus4

minus20

0

20

40120576 m


T (∘C)

(d)


times10minus4

94ppt 23ppt

minus6

minus4

minus2

0

120576 L


T (∘C)

(a)

CTEFI

0

2

4

6times10

minus5

minus2

minus4

minus6

minus8

94ppt

23ppt

ECTE

(Kminus1)


T (∘C)

(b)



1

2

3

times10minus5

ECTE

(Kminus1)

minus10

minus5

0

5CTEFI


T (∘C)

(a)

1

2

3

times10minus4


0

5

10

15

20

25

30

35

120576 m

T (∘C)

(b)


times10minus4

1 2 3 4 5 60t (h)

00

05

10

15

20

25

120576 L

(a)

minus12

minus10

minus8

minus6

minus4

1 2 3 4 5 60t (h)

T(∘

C)

(b)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)

05 10 15 20 2500t (h)

(c)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)


T (∘C)

(d)





0

5

10

15times10

minus5EC

TE (K

minus1)

minus5

minus10


T (∘C)

(a)

0

5

10

times10minus5

ECTE

(Kminus1)

minus5


T (∘C)

(b)


200 400 600 800

00

02


1

2

t (s)

minus02

minus04

minus06

(a)

200 300 400 500 600 700 800

00

01

02

0

8

4

t (s)

minus01

minus02

minus03

Tem

pera

ture

var

iatio

ns (∘

C) Depth = 1 cm

(b)







about 2 sdot 10


5 Conclusions






Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










385 390 395 400

minus1318

minus1322

minus1326

minus1330380

10 20 30 400t (h)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

T(∘

C)

(a)

CTEFI

times10minus5


46

48

50

52

54

ECTE

(Kminus1)

T (∘C)

(b)


T

S

50 100 1500t (h)

minus25

minus20

minus15

minus10

minus5

0

3120576 Lmiddot104

T(∘

C)

(a)

T

S

100 200 3000t (h)

minus20

minus15

minus10

minus5

05120576 Lmiddot104

T(∘

C)

(b)









times10minus4

minus5

minus4

minus3

minus2

minus1

0

120576 L


T (∘C)

(a)

times10minus4

minus12

minus10

minus8

minus6

minus4

minus2

0

120576 L


T (∘C)

(b)

times10minus4


minus6

minus4

minus2

0

2

120576 m

T (∘C)

(c)

times10minus4

minus20

0

20

40120576 m


T (∘C)

(d)


times10minus4

94ppt 23ppt

minus6

minus4

minus2

0

120576 L


T (∘C)

(a)

CTEFI

0

2

4

6times10

minus5

minus2

minus4

minus6

minus8

94ppt

23ppt

ECTE

(Kminus1)


T (∘C)

(b)



1

2

3

times10minus5

ECTE

(Kminus1)

minus10

minus5

0

5CTEFI


T (∘C)

(a)

1

2

3

times10minus4


0

5

10

15

20

25

30

35

120576 m

T (∘C)

(b)


times10minus4

1 2 3 4 5 60t (h)

00

05

10

15

20

25

120576 L

(a)

minus12

minus10

minus8

minus6

minus4

1 2 3 4 5 60t (h)

T(∘

C)

(b)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)

05 10 15 20 2500t (h)

(c)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)


T (∘C)

(d)





0

5

10

15times10

minus5EC

TE (K

minus1)

minus5

minus10


T (∘C)

(a)

0

5

10

times10minus5

ECTE

(Kminus1)

minus5


T (∘C)

(b)


200 400 600 800

00

02


1

2

t (s)

minus02

minus04

minus06

(a)

200 300 400 500 600 700 800

00

01

02

0

8

4

t (s)

minus01

minus02

minus03

Tem

pera

ture

var

iatio

ns (∘

C) Depth = 1 cm

(b)







about 2 sdot 10


5 Conclusions






Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










times10minus4

minus5

minus4

minus3

minus2

minus1

0

120576 L


T (∘C)

(a)

times10minus4

minus12

minus10

minus8

minus6

minus4

minus2

0

120576 L


T (∘C)

(b)

times10minus4


minus6

minus4

minus2

0

2

120576 m

T (∘C)

(c)

times10minus4

minus20

0

20

40120576 m


T (∘C)

(d)


times10minus4

94ppt 23ppt

minus6

minus4

minus2

0

120576 L


T (∘C)

(a)

CTEFI

0

2

4

6times10

minus5

minus2

minus4

minus6

minus8

94ppt

23ppt

ECTE

(Kminus1)


T (∘C)

(b)



1

2

3

times10minus5

ECTE

(Kminus1)

minus10

minus5

0

5CTEFI


T (∘C)

(a)

1

2

3

times10minus4


0

5

10

15

20

25

30

35

120576 m

T (∘C)

(b)


times10minus4

1 2 3 4 5 60t (h)

00

05

10

15

20

25

120576 L

(a)

minus12

minus10

minus8

minus6

minus4

1 2 3 4 5 60t (h)

T(∘

C)

(b)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)

05 10 15 20 2500t (h)

(c)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)


T (∘C)

(d)





0

5

10

15times10

minus5EC

TE (K

minus1)

minus5

minus10


T (∘C)

(a)

0

5

10

times10minus5

ECTE

(Kminus1)

minus5


T (∘C)

(b)


200 400 600 800

00

02


1

2

t (s)

minus02

minus04

minus06

(a)

200 300 400 500 600 700 800

00

01

02

0

8

4

t (s)

minus01

minus02

minus03

Tem

pera

ture

var

iatio

ns (∘

C) Depth = 1 cm

(b)







about 2 sdot 10


5 Conclusions






Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

2

3

times10minus5

ECTE

(Kminus1)

minus10

minus5

0

5CTEFI


T (∘C)

(a)

1

2

3

times10minus4


0

5

10

15

20

25

30

35

120576 m

T (∘C)

(b)


times10minus4

1 2 3 4 5 60t (h)

00

05

10

15

20

25

120576 L

(a)

minus12

minus10

minus8

minus6

minus4

1 2 3 4 5 60t (h)

T(∘

C)

(b)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)

05 10 15 20 2500t (h)

(c)

times10minus5

45

50

55

60

65

70

75

80

ECTE

(Kminus1)


T (∘C)

(d)





0

5

10

15times10

minus5EC

TE (K

minus1)

minus5

minus10


T (∘C)

(a)

0

5

10

times10minus5

ECTE

(Kminus1)

minus5


T (∘C)

(b)


200 400 600 800

00

02


1

2

t (s)

minus02

minus04

minus06

(a)

200 300 400 500 600 700 800

00

01

02

0

8

4

t (s)

minus01

minus02

minus03

Tem

pera

ture

var

iatio

ns (∘

C) Depth = 1 cm

(b)







about 2 sdot 10


5 Conclusions






Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

5

10

15times10

minus5EC

TE (K

minus1)

minus5

minus10


T (∘C)

(a)

0

5

10

times10minus5

ECTE

(Kminus1)

minus5


T (∘C)

(b)


200 400 600 800

00

02


1

2

t (s)

minus02

minus04

minus06

(a)

200 300 400 500 600 700 800

00

01

02

0

8

4

t (s)

minus01

minus02

minus03

Tem

pera

ture

var

iatio

ns (∘

C) Depth = 1 cm

(b)







about 2 sdot 10


5 Conclusions






Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










about 2 sdot 10


5 Conclusions






Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article thermal expansion measurements in fresh ... · meter along with a sample holder,...

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