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Applied Thermal Engineering 73 (2014) 140e151

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

Mathematical simulation of the freezing time of water in smalldiameter pipes

Andr�e McDonald a, *, Ben Bschaden a, Erik Sullivan a, Rick Marsden b

a Department of Mechanical Engineering, 4-9 Mechanical Engineering Building, University of Alberta, Edmonton, Alberta T6G 2G8, Canadab Cenovus Energy Incorporated, 500 Centre Street SE, Calgary, Alberta T2P 0M5, Canada

h i g h l i g h t s

� Mathematical simulation of the freezing of water in circular pipes.� A one-dimensional, finite-length scale, transient heat conduction model approach was developed.� Comparison of predictions of finite-length scale model and experiments were conducted.� Model results compared well to experimental results.� Application of the model to practical scenarios was considered and explored.

a r t i c l e i n f o

Article history:Received 11 January 2014Accepted 17 July 2014Available online 25 July 2014

Keywords:Ice thicknessMoving boundary problemSeparation of variablesSolidificationWater

* Corresponding author. Tel.: þ1 780 4922675; fax:E-mail address: andre.mcdonald@ualberta.ca (A. M

http://dx.doi.org/10.1016/j.applthermaleng.2014.07.041359-4311/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

The cooling and freezing of stagnant water in cylindrical pipes were investigated under conditions ofvarying ambient air temperatures. Experiments were conducted to measure the transient temperature ofwater in a 50.8 mm nominal Schedule 80 steel pipe that was exposed to stagnant and forced air attemperatures ranging from �25 �C to �5 �C. There was a 10 vol.% air gap in the pipe to avoid excessiveincrease of fluid pressure during the freezing process. A one-dimensional transient heat conductionmathematical model was developed to estimate the freezing time of the liquid in the pipe. The modelwas based on the separation of variables method for a finite length-scale solidification problem, and wasverified with the experimental transient temperature data. It was found that the model predicted thesolidification time of the water in the pipe to within 15% of that which was measured by experiments formost of the tests that were conducted. The model was applied to predict the cooling and freezing be-haviors of water in steel and copper pipes of various inner diameters and in insulated pipes. The resultsof the model and their agreement with experimental data suggest that a separation of variables methodfor a finite length-scale heat conduction problemmay be applied to moving or free boundary problems topredict phase change phenomena.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction as dendritic ice, which forms in advance of the annular solid ice.

The phenomenon of solidification of stagnant water in cylin-drical pipes has received attention from numerous investigatorsgiven the detrimental effects of freezing such as the possibleblockage and bursting of the pipes. When pipes are exposed to low-temperature environments, the water will experience cooling,supercooling of the liquid phase below the fusion temperature,solidification at the fusion temperature, and further cooling of thesolid ice [1]. Gilpin [2] has shown that supercooled water in pipes ischaracterized by the growth of thin plate-like solid crystals known

þ1 780 4922200.cDonald).

6

The consequence of the formation of either dendritic ice or solidannular ice in the pipe is the possibility of flow blockage andbursting of the pipe due to the generation of large internal pres-sures. It has been shown that dendritic ice formation may beresponsible for early blockage of the water pipes under certainconditions [2]. Sugawara et al. [3] and Oiwake et al. [4] conductedboth numerical analyses and experimental studies to quantify thechanges in internal pipe pressure during the freezing process, whileGordon [1] presented experimental results of pressures required tocause bursting of a variety of pipes used in building applications.The numerical analyses of Sugawara et al. [3] and Oiwake et al. [4]and the experimental work of Gordon [1] included the case wherethe pressure increase in the freezing fluid depressed the fusionpoint. They admitted that the continuous freezing point depression

Nomenclature

cp specific heat capacityD diameter (m)h average heat transfer coefficient (W m�2 K�1)H heightJ0 Bessel function of order 0 of the first kindJ1 Bessel function of order 1 of the first kindk thermal conductivity (W m�1 K�1)L latent heat of fusion (J kg�1)Pr Prandtl number, Pr ¼ cpm=kr radial coordinateR radiusR00 resistanceRaDo

Rayleigh number, RaDo¼ gbðTs � ToÞD3

oPr=y2

ReDoReynolds number, ReDo

¼ rVDo=m

t time (s)T temperature (�C)U overall heat transfer coefficient (W m�2 K�1)V velocity (m s�1)Y0 Bessel function of order 0 of the second kindY1 Bessel function of order 1 of the second kind

Greek symbolsa thermal diffusivity (m2 s�1)b eigenvalue (m�1)

c ice thickness (m)d thickness (m)3 emissivityf function dependent on r, onlyl separation constant (m�1)m dynamic viscosity (kg m�1 s�1)r density (kg m�3)s StefaneBoltzmann constant, 5.67 � 10�8 W m�2 K�4

t function dependent on t, onlyJ function dependent on r and t∞ ambient< function dependent on r, only

Subscriptsf freezingi initialin interfaceinn innerinsul insulationk numberL liquidn numbero outers solidsur surfacesuper supercooled

Fig. 1. Schedule 80 steel pipe with thread-o-lets, plugs, and installed thermowell.

A. McDonald et al. / Applied Thermal Engineering 73 (2014) 140e151 141

rendered analysis of the problem difficult, forcing the use of in-cremental time steps and numerical code.

Further experimental studies on the freezing of stagnant waterin pipes have explored the effects of internal free convectivemotionduring the solidification process. Due to the decrease in the densityof water as it freezes [5], complicated flow patterns may emerge,which may have an effect on the rate of formation of dendritic andsolid annular ice [6e8]. Gilpin [8] has shown specifically that forquiescent water with supercooling less than 2 �C, free convectionproduces non-uniform ice distribution. For greater amounts ofsupercooling, relatively uniform ice distribution is formed since thevelocity of the advancing dendritic ice front exceeds that of the freeconvective motion.

The modeling of solidification phase change problems has beenthe subject of extensive research investigations [9e16]. Initialmodel studies focused on this phase change in slabs with semi-infinite extent [9] or in liquids that were initially at the fusiontemperature [9,10], with the intent of finding the temperaturedistribution in thematerial during phase change and the location ofthe moving boundary. The non-linearity of the interface energyequation made analytical solutions of the temperature distributiondifficult to obtain. This has resulted in computer-aided numericalsolutions of the problem, and, as outlined in an early review byMuehlbauer and Sunderland [11], the use of several mathematicaltechniques such as the variational technique, the heat-balance in-tegral technique, and the RiemanneMellin contour integral tech-nique. More recently, model studies have focused on thedetermination of the time required for freezing of materials. Pham[12] has extended Planck's equation [13] to include sensible heateffects during the freezing of food-based materials with simpleshapes. Other investigators such as Cleland et al. [14] and Hung [15]used numerical methods, coupled with experimental data, to pre-dict the freezing time of irregularly shaped bodies of food. Whilethese studies on the prediction of freezing time through the use ofeither simple models or numerical techniques are useful for solid

foodmaterial, they do not enable the determination of the transienttemperature distribution in the material or estimation of thetransient solidification front during freezing.

The objectives of this study were to: (1) develop an analyticalheat conduction model to estimate the freezing time of liquids inpipes; (2) conduct experiments to validate the model; and (3)extend the use of the model to cases of practical interest.

2. Experimental method

The freezing of water in cylindrical pipes was used as a basis togather data for validation of the mathematical model that wasdeveloped in this study. In order to mitigate the risk of brittlefractures, all piping material and components were rated for lowtemperature service. A 50.8 mm (2 in) Schedule 80 carbon steelpipe (ASTM A333 Grade 6) that was approximately 305 mm (12 in)long was used for all the tests and is shown in Fig. 1. Thread-o-letswere welded at each end of the pipe as well as one in the middle tocreate a closed system and to allow insertion of instrumentation fortemperature measurements. The total system volume wasapproximately 550 mL (18.6 fl oz).

A thermowell with a T-type thermocouple (PP-T-20, Omega,Montr�eal, QC, Canada) was inserted along the centerline of the

A. McDonald et al. / Applied Thermal Engineering 73 (2014) 140e151142

pipe through one of the thread-o-lets and was used to measurethe transient temperature of the water and ice during the coolingand freezing events. Thermal paste (8610 Non Silicone HeatTransfer Compound, M.G. Chemicals, Burlington, ON, Canada) wasapplied to the thermocouple junction to reduce contact resistancebetween the thermocouple and the thermowell. The thermowellhad an outer diameter of 12.7 mm (0.50 in) and extended insidethe pipe approximately 127 mm (5 in) (see Fig. 1). The other twoorifices were plugged using 19 mm (0.75 in) NPT stainless steelplugs and Teflon tape. The ambient temperature within thefreezer was measured using a T-type thermocouple. The temper-ature inside the duct was recorded using a J-type rigid probe-stylethermocouple (TJ36-ICSS-14U-6-CC, Omega, Montr�eal, QC, Can-ada). Tap water was used as the process fluid. For all the tests, a10 vol.% air gap was introduced into the pipe to avoid excessiveincrease of fluid pressure during the freezing process. The tran-sient temperature data was collected with a data acquisitionsystem (SCXI-1600, National Instruments, Austin, TX, USA) and ameasurement and automation explorer software for data pro-cessing (National Instruments, Austin, TX, USA) was used. The datasampling interval was 1 Hz (1 sample per second). Once recorded,the data was then exported into a Microsoft Excel file for furtheranalysis.

Fig. 2. a) Components and b) dimensio

All the cooling and freezing tests were conducted in an 18.2 m3

(640 ft3) cold room freezer (Foster Refrigerator USA, Kinderhook,NY, USA). Five tests in which the ambient air was held stagnant attemperatures of �5 �C, �10 �C, �15 �C, �20 �C, and �25 �C wereconducted in a closed duct to reduce the effects of circulating aircurrents external to the pipe. Several tests were also conducted tosimulate forced air cooling and freezing, over a temperature rangeof�25 to�5 �C, in increments of 5 �C. In order to control the flow ofair over the pipe, an experimental assembly was constructed of a254 mm � 305 mm (10 in � 12 in) galvanized sheet metal duct, a0.25 kW (0.33 hp) direct-drive tube-axial fan (DDA-12-10033B,Leader Fan Industries, Toronto, ON, Canada), a galvanized sheetmetal 305 mm (12 in) round to 254 mm � 305 mm (10 in � 12 in)rectangular reducer, and a 41 mm (1.625 in) slotted C-channeltubing (McMaster-Carr, Aurora, USA) (see Fig. 2a). The overall di-mensions of the assembly were 2 m � 0.66 m � 0.48 m(80 in� 25.9 in� 18.9 in) as shown in Fig. 2b. A triangular hole wascut into the duct to allow access for the test pipe. The entire as-semblywas elevated at approximately 30� to ensure that the air gapin the pipe remained at one end of the pipe. A digital manometer(475-1-FM, Dwyer Instruments, Michigan City, USA) was used tomeasure the air speed after exit from the fan andwithin the vicinityof the pipe.

ns of the forced air test apparatus.

A. McDonald et al. / Applied Thermal Engineering 73 (2014) 140e151 143

3. Mathematical models

3.1. Cooling time

Cooling of the liquid in a freezer with ambient temperature, T∞,from an initial temperature (Ti) to its melting point, Tf, will occurbefore the initiation of freezing. The time required to cool theliquid from its initial temperature to the freezing point can bedetermined by considering a one-dimensional transient heatconduction model in cylindrical co-ordinates for the cooling of thestagnant fluid in the pipe. Assuming constant properties, thegoverning equation is

1r

v

vr

�rvTvr2

�¼ 1

aL

vTvt; 0 � r � Rinn: (1)

The boundary and initial conditions are.

vTð0; tÞvr

¼ 0; (2)

�kLvTðRinn; tÞ

vr¼ U½TðRinn; tÞ � T∞�; (3)

Tðr;0Þ ¼ Ti � T∞: (4)

€Ozisik [17] has solved Eq. (1) with the boundary and initialconditions of Eqs. (2)e(4) to give

Tðr; tÞ ¼ T∞ þ 2uðTi � T∞ÞRinn

X∞n¼1

J0ðbnrÞ�b2n þ u2

�J0ðbnRinnÞ

exp��aLb

2nt�;

(5)

where u ¼ U/kL and the eigenvalues are

bnJ1ðbnRinnÞ ¼ uJ0ðbnRinnÞ: (6)

The overall heat transfer coefficient (U) during this single-phaseheat transfer problem can be estimated by utilizing correlationequations that are available from the work of others. The pipematerial, insulation, and outside heat transfer coefficient (ho)constituted the overall heat transfer coefficient at the outer surfaceof the pipe. In general

1U

¼ R00 þ 1

ho;convectionþ 1

ho;radiation: (7)

The thermal resistance of the pipe material and any insulation isgiven as

R00 ¼R

00pipeþR

00insul ¼

ln

Do;pipe

Dinn;pipe

!

2kpipeDo;pipeþ

ln

Do;insulDinn;insul

!

2kinsulDo;insul: (8)

Free convection and radiation will occur at the outer surfaceof the pipe during the stagnant air tests. The orientation of thepipes in this study will be horizontal. The free convective heattransfer coefficient for horizontal pipes is given by Churchill andChu [18] as

ho;convection¼kairDo

266640:6þ

0:387Ra1=6Doh1þ

�0:559Pr

�9=16i8=27377752

; RaDo�1012:

(9)

Eq. (9) is valid for laminar free convective flow over cylinderswithin an infinite fluid medium for uniform surface temperature oruniform heat flux. In this study, the tests were conducted in aclosed channel in which the cylinder occupied approximately 25%of the height of the channel. It is possible that free convectivemotion occurred within the gap between the channel wall and thecylinder, increasing the heat transfer coefficient beyond the valuepredicted by Eq. (9). It is also possible that with high values ofRayleigh number, the free convection motion would be turbulentwithin the gap, further increasing the heat transfer coefficient.There are no correlation equations for the heat transfer coefficientsfor this specific case; however, verification of the Rayleigh numberwould confirm the possible occurrence of free convective currents.Ostrach [19] has shown that the critical Rayleigh number that isrequired to induce free convective motion in enclosed spacesbounded by rigid planes is 1708. Turbulence will first appear at aRayleigh number that is approximately 5830. The length scale inthe Rayleigh number is the gap distance between the planes. Understagnant air conditions, the approximate radiation heat transfercoefficient is [20]

ho;radiationz3pipes

�T4sur � T4∞

�Tsur � T∞

; (10)

where the StefaneBoltzmann constant is s ¼ 5.67 � 10�8

W m�2 K�4 and T is in Kelvin.Due to the difficulties in estimating the overall heat transfer

coefficient for the stagnant air tests, the models developed in thisstudy, coupled with the experimental results, will be used to esti-mate values of the overall heat transfer coefficients. Mcquistonet al. [21] has also suggested that the determination of overall heattransfer coefficients over bare pipes exposed to free convection iscomplicated due to the lack of knowledge of the transient surfacetemperature of the pipe upon which the heat transfer coefficientswill depend. Experimental data for the overall heat transfer co-efficients for horizontal bare steel pipes that were presented byMcquiston et al. [21] will also be used in this study.

Forced convection will dominate the heat transfer at the outersurface of the pipe during the forced air tests. Perkins and Leppert[22] have shown that the average heat transfer coefficient over thepipe will be affected by flow in both the laminar boundary layerregion and in the separated region over the pipe, with significantincreases in the local heat transfer coefficient in the separated re-gion of flow. Provided that the Reynolds number is lower than thecritical value for laminar-to-turbulent transition for flow over cy-lindrical bodies (2 � 105), the correlation equation for the averageheat transfer coefficient is [22]

ho;convection ¼ kairDo

h0:31Re0:50Do

þ 0:11Re0:67Do

iPr0:40: (11)

Perkins and Leppert [22] accounted for the variation of dynamicviscosity across the boundary layer by using a ratio of the dynamicviscosity at the cylinder wall to the bulk dynamic viscosity. Theratio was assumed to be unity for this study since the variation ofdynamic viscosity of gases with temperature is small.

In this study, the cooling and freezing experiments were con-ducted in a closed rectangular duct system (see Fig. 2). It has beenshown that when the diameter of a pipe occupies a portion of thecross-sectional area of a ducted channel, blockage of the gas flowwill occur [23]. Blockage will affect the pressure and velocity dis-tributions around the cylindrical pipe, which will in turn affect thelocation of the separation point on the cylindrical pipe and theaverage heat transfer coefficient [22]. To that end, Robinson et al.[23] proposed an empirically determined blockage correction

A. McDonald et al. / Applied Thermal Engineering 73 (2014) 140e151144

factor for the heat transfer coefficient for flow over a non-isothermal cylinder in cross flow in a channel as ð1þ ffiffiffiffiffiffiffiffiffiffiffiffi

Do=Hp Þ.

Therefore, Eq. (11) becomes

ho;convection ¼ kairDo

1þ

ffiffiffiffiffiffiDo

H

r !h0:31Re0:50Do

þ 0:11Re0:67Do

iPr0:40;

(12)

whereH is the height of the channel, and it is 254mm (10 in) in thisstudy (see Fig. 2b).

The use of the blockage correction factor that was developedfrom experimental work by Robinson et al. [23] for cylinders withnon-isothermal surfaces will most likely relate closely to the truethermal condition of the pipes that were used in this present study.The properties of the air in external flow over the pipe weredetermined at the film temperature, Tfilm. The surface temperatureof the pipewas assumed to be the average of the initial temperatureand the fusion temperature of the water in the pipe.

3.2. Freezing time

A one-dimensional transient heat conduction model, in cylin-drical co-ordinates, was used to estimate the freezing time of theliquid in the pipes. Fig. 3 shows a schematic of the model used inthe analysis. The pipe was of finite length and was filled withquiescent liquid at the freezing point. In order to induce solidifi-cation, the surrounding ambient air temperature was less than thefreezing point of the liquid. It was assumed that the growth of thesolid layer was axisymmetric and that all properties were constant.

The governing equations of the temperature distribution in thesolid and liquid phases are

TLðr; tÞ ¼ Tf ; 0 � r< rinðtÞ; (13)

1r

v

vr

�rvTsvr2

�¼ 1

as

vTsvt

; rinðtÞ< r � Rinn: (14)

The boundary and initial conditions are

Tsðrin; tÞ ¼ Tf ; (15)

Fig. 3. Schematic of solidification in a circular pipe.

�ksvTsðRinn; tÞ

vr¼ U½TsðRinn; tÞ � T∞�; (16)

Tsðr;0Þ ¼ Tf ; (17)

rinð0Þ ¼ Rinn: (18)

Conservation of energy at the solideliquid interface will yieldanother boundary condition, the interface energy equation, whichestablishes the problem as a free boundary problem, with rin(t)representing the transient location of the interface. The solid-eliquid interface energy equation is

ksvTsðrin; tÞ

vr¼ rsL

drindt

: (19)

The initial condition of Eq. (17) suggests that the axisymmetricsolid forms instantaneously at t ¼ 0. This assumption serves tosimplify the mathematical development of the model. However,Gilpin [2] has shown that ice will form initially as plate-like den-dritic crystals that are distributed throughout the water and will beeventually engulfed by solid as the freezing event proceeds.

This is a finite length-scale problem in which the pipe is boun-ded at a radius of r¼ Rinn (see Fig. 3). The governing equation of thetemperature distribution in the solid phase (Eq. (14)) was solved byusing the separation of variables method [17]. Given the non-homogeneity of Eqs. (15) and (16), the solution for Ts(r,t) wasassumed to be the sum of two functions, one (J) depending on rand t for the homogeneous solution, and the other, f, on r only, forthe particular solution, so that

Tsðr; tÞ ¼ Jðr; tÞ þ fðrÞ: (20)

Eq. (20) was substituted into Eq. (14), and after separation gives

1r

v

vr

�rvJ

vr

�¼ 1

as

vJ

vt; (21)

1r

ddr

�rdfdr

�¼ 0: (22)

Substitution of Eq. (20) into the boundary and initial conditionsof Eqs. (15)e(17), followed by separation will produce homoge-neous boundary conditions forJ from Eqs. (15) and (16). Since f isa function of r, only, Eq. (17) will remain non-homogeneous aftersubstitution of Eq. (20).

The solution for J(r,t) was assumed to be the product of twofunctions, with < depending on r only, and t, on t only, so that

Jðr; tÞ ¼ <ðrÞtðtÞ: (23)

Eq. (23) was substituted into Eq. (21), and after separation

d2<k

dr2þ 1

rd<k

drþ l2k<k ¼ 0; (24)

dtkdt

þ asl2ktk ¼ 0: (25)

Integration of Eqs. (24) and (25) gives

<kðrÞ ¼ AkJ0ðlkrÞ þ BkY0ðlkrÞ; (26)

tkðtÞ ¼ Ck exp��asl

2kt�; (27)

where Ak, Bk, and Ck are integration constants.

Table 1General properties of solid ice [5].

Property Value

Melting point, Tf 0 �CThermal diffusivity, as 1.194 � 10�6 m2 s�1

Specific heat capacity, cp,s 2010 J kg�1 K�1

Thermal conductivity, ks 2.2 W m�1 K�1

Density, rs 917 kg m�3

Latent heat of fusion, L 334,000 J kg�1

A. McDonald et al. / Applied Thermal Engineering 73 (2014) 140e151 145

Application of the boundary condition at r ¼ rin (Eq. (15)) forJ(rin,t) ¼ 0 to Eq. (26) gives

Bk ¼ �AkJ0ðlkriÞY0ðlkriÞ

: (28)

The boundary condition of Eq. (16) at r ¼ Rinn for�ksðvJðRinn; tÞÞ=vr ¼ UJðRinn; tÞ is applied to Eq. (26) in order tofind an expression for the eigenvalue, lk

lk

�J1ðlkRinnÞ � J0ðlkriÞ

Y0ðlkriÞ Y1ðlkRinnÞ�

�J0ðlkRinnÞ � J0ðlkriÞ

Y0ðlkriÞ Y0ðlkRinnÞ� ¼ U

ks: (29)

One of the values of the separation constant is lk ¼ l0 ¼ 0.Substitution of lk ¼ 0 into Eqs. (24) and (25) and application of theboundary conditions of Eqs. (15) and (16) to the solution of Eq. (24)showed that<0ðrÞ ¼ 0 and J0ðr; tÞ ¼ <0ðrÞt0ðtÞ ¼ 0. Therefore

Jðr; tÞ ¼X∞k¼1

ak

�J0ðlkrÞ �

J0ðlkriÞY0ðlkriÞ

Y0ðlkrÞ�exp

��asl

2kt�; (30)

where ak ¼ AkCk.The solution of the particular problem that is governed by Eq.

(22) is

fðrÞ ¼ E ln r þ F; (31)

where E and F are constants of integration.Application of the boundary conditions at r ¼ rin (Eq. (15)) for

f(rin) ¼ Tf and r ¼ Rinn (Eq. (16)) for �ksðdfðRinnÞ=drÞ ¼ U½fðRinnÞ �T∞� to Eq. (31) gives

E ¼ UTf � T∞

U ln

�rinRinn

�� ks

Rinn

; (32)

F ¼ Tf �UTf � T∞

U ln

�rinRinn

�� ks

Rinn

ln rin: (33)

Finally, the temperature distribution in the solid phase is

Tsðr;tÞ�Tf ¼UTf �T∞

U ln

�rinRinn

�� ks

Rinn

ln�

rrin

�þX∞k¼1

ak

�J0ðlkrÞ�

J0ðlkrinÞY0ðlkrinÞ

�Y0ðlkrÞ�exp

��asl

2kt�:

(34)

Eq. (24) is a SturmeLiuoville equation and the boundary con-ditions of Eqs. (15) and (16) are homogeneous, after transformationin terms ofJ. Therefore, orthogonality can be applied to determineak [17]. The initial condition of Eq. (17) was used to produce

U�Tf � T∞

�U ln

�rinRo

�� ks

Ro

ln�

rrin

�þX∞k¼1

ak

�J0ðlkrÞ �

J0ðlkrinÞY0ðlkrinÞ

Y0ðlkrÞ�¼ 0:

(35)

It is uncertain if the ½J0ðlkrÞ � ððJ0ðlkrinÞÞ=ðY0ðlkrinÞÞÞY0ðlkrÞ�term in Eq. (35) is orthogonal. Therefore, orthogonality cannot beapplied implicitly to find ak. However, a numerical check wasconducted for lk, which showed that only one of the values of l1, l2,

l3, l4, l5, or l6 were such that the ½J0ðlkrÞ � ððJ0ðlkrinÞÞ=ðY0ðlkrinÞÞÞY0ðlkrÞ� term was very large for either k ¼ 1, 2…, or 6,compared to other values of k. It is important to note that any one ofthe values could be used for a given solution since only one value ofk will produce a relatively large value of the ½J0ðlkrÞ � ððJ0ðlkrinÞÞ=ðY0ðlkrinÞÞÞY0ðlkrÞ� term. So, one value of ak will apply. Therefore,the integral of each term in Eq. (35) over the region rin(t) < r � Rinnof the solid was determined to give

UTf � T∞

U ln

�rinRo

�� ks

Ro

ZRo

ri

ln�

rrin

�dr þ ak

ZRo

ri

�J0ðlkrÞ �

J0ðlkrinÞY0ðlkrinÞ

� Y0ðlkrÞ�drz0:

(36)

Eq. (36) will yield approximate values of ak. In addition, theexpression was solved numerically since there are no closed formanalytical solutions of the integrals of the Bessel functions of zeroorder, J0(lkr) and Y0(lkr).

Knowledge of the transient location of the solideliquid inter-face, rin(t), during the freezing event will enable estimation of theliquid freezing time within a pipe of inner radius, Rinn. Differenti-ation of Eq. (34) at r ¼ rin and substitution into the solideliquidinterface energy equation of Eq. (19) produced

ks

2664 U

�Tf �T∞

�rin

�U ln

�rinRinn

�� ks

Rinn

�þX∞k¼1

aklk

�J0ðlkrinÞY0ðlkrinÞ

Y1ðlkrinÞ� J1ðlkrinÞ�

�exp��asl

2kt�3775zrsL

drindt

:

(37)

The ice thickness as a function of time is

cðtÞ ¼ Rinn � rinðtÞ: (38)

A MATLAB (MathWorks, Inc., Natick, MA, USA) code was used tosolve the expressions for the first 6 values of lk (Eq. (29)). By dis-cretizing the spatial domain, values of ak in Eq. (36) were calculated.The values of time that corresponded to each rin value werecalculated using Eq. (37). The differential equation in Eq. (37) wasestimated by a first-order finite difference approximation. Theinfinite series summation in Eq. (37) was found to be insignificant,with a dominant first term on the order of approximately 10�70.This simplified the expression to

ksUTf � T∞

rsLrin

�U ln

�rinRinn

�� ks

Rinn

�zdrindt

: (39)

Eq. (39) gives a solution for the time corresponding to theequally-spaced discretized ice thickness, c. Awatereice systemwas

Table 2Temperature and cooling rate data during cooling of water in 50.8 mm nominaldiameter (49.3 mm inner diameter) Schedule 80 steel pipes e stagnant air tests.

T∞, �C Ti, �C Tsuper, �C Cooling rate, �C/s

�5.3 ± 1.2 21.6 �3.2 1.8 � 10�3

�9.1 ± 1.2 21.9 �2.5 3.1 � 10�3

�13.7 ± 0.9 21.0 �2.3 3.8 � 10�3

�16.1 ± 3.5 21.3 �2.2 5.1 � 10�3

�22.2 ± 0.7 21.4 �1.8 6.3 � 10�3

A. McDonald et al. / Applied Thermal Engineering 73 (2014) 140e151146

investigated in this study. General properties of solid ice that wereused in the model are shown in Table 1 [5].

4. Results and discussion

4.1. Temperature traces

Transient temperature traces were generated for the variousconditions of ambient air temperature for both the stagnant andforced air tests of cooling and freezing of water. Fig. 4 shows thetransient temperature traces when the average ambient air tem-perature was approximately �14 �C to �16 �C, where the set-pointtemperature was �15 �C. The ambient air temperature fluctuateddue to cycling of the fans in the cold room freezer. The traces show aregion of liquid cooling from approximately 21 �C, a distinct so-lidification plateau, and after complete solidification, further cool-ing of the solid ice. The temperature profile of Fig. 4a for thestagnant air test shows evidence of supercooling of the liquid withthe amount of supercooling on the order of 2.3 �C. During super-cooling, dendritic ice growth occurs. As shown in the figure,at �2.3 �C, the water temperature returns to approximately 0 �C, atwhich point, an annulus of solid ice begins to form at the inner wallsurface of the pipe.

An increase in the cooling rate (change in temperature per unittime) by exposing the liquid and pipe to forced ambient air pro-duced a transient temperature profile that was devoid of evidenceof significant supercooling (see Fig. 4b). This is expected sinceincreased cooling rates would enable rapid development of thesolid annular ice in the pipe. The solid ice would quickly engulf the

Fig. 4. Transient temperature traces of water in 50.8 mm nominal diameter Schedule80 steel pipes under a) stagnant and b) force air tests.

thin layer of dendritic ice growth, significantly reducing theamount of supercooling. Tables 2 and 3 show the average airtemperatures, the initial water temperatures, supercooled watertemperatures, and the approximate cooling rates that were ob-tained for the stagnant and forced air tests in this study. As thecooling rate increased due to a decrease in the ambient air tem-perature (see Table 2) or the use of forced air (see Table 3), theamount of supercooling decreased. The standard deviation waspresented with the average ambient temperature (T∞). The numberof data points that were used to generate this averagewas in excessof 5000.

The increase in the cooling rate of the water during the forcedair tests was due primarily to the forced convection heat transfer atthe outer surface of the pipe. The speed of the free-stream air flowwas measured to be 31 km/h (19.3 miles/h). This free-stream airvelocity was chosen since the American Society of Heating,Refrigeration, and Air-conditioning Engineers (ASHRAE) have sug-gested that outdoor wind speeds in the winter will typically beabout 24 km/h (15 miles/h) [24]. In the forced air tests of this study,the velocity around the pipes would be higher due to blockage inthe duct channel. With the blockage correction factor from Rob-inson et al. [23] that is shown in Eq. (12) and data on the outer pipediameter and duct height, the air velocity over the pipe may rangefrom 56 to 68 km/h (35e43 miles/h) as the flow moves fromlaminar to laminar-separated flow over the cylindrical pipe.

4.2. Cooling and freezing times e predicting the experimentalresults

The cooling and freezing times of liquids in pipes will depend onthe transient temperature distributions of the liquid and solidphases. The temperature distribution models of Eqs. (5) and (34)were used to estimate the cooling and freezing times, respec-tively. The overall heat transfer coefficient (U) is a major parameterthat needs to be specified in order to determine the temperaturedistributions and the cooling and freezing times. Table 4 shows thevalues of the heat transfer coefficients that were used to estimatethe cooling and freezing times. The overall heat transfer coefficientfor the stagnant air tests with free convection was determined byfitting the experimental data results for the cooling of water in anambient at �5.3 �C and freezing at �6.6 �C. It was assumed that theoverall heat transfer coefficient of 15.4 Wm�2 K�1 was constant forall the other stagnant air tests in the study. For free convection of airover a 20 mm outer diameter horizontal cylinder, €Ozisik has

Table 3Temperature and cooling rate data during cooling of water in 50.8 mm nominaldiameter (49.3 mm inner diameter) Schedule 80 steel pipes e forced air tests.

T∞, �C Ti, �C Tsuper, �C Cooling rate, �C/s

�6.4 ± 0.9 21.0 �1.7 1.4 � 10�2

�11. 6 ± 0.8 19.1 �1.3 1.7 � 10�2

�16.0 ± 0.7 22.4 0.1 2.2 � 10�2

�21.2 ± 0.6 21.2 0.1 2.0 � 10�2

�26.0 ± 0.6 19.7 0.1 1.8 � 10�2

Table 4Overall heat transfer coefficients, U.

Stagnant air Forced air

T∞, �C U, W m�2 K�1 T∞, �C U, W m�2 K�1

�6.2 15.4 �6.4 98.8�10.5 15.4 �11.5 99.8�14.9 15.4 �16.2 99.7�18.6 15.4 �21.2 99.5�23.3 15.4 �26.1 99.3

A. McDonald et al. / Applied Thermal Engineering 73 (2014) 140e151 147

suggested that the heat transfer coefficient be on the order of101 W m�2 K�1 [17]. Mcquiston et al. [21] have also shown that for50.8 mm diameter horizontal bare steel pipes, the combined effectsof free convection and radiation would produce an overall heattransfer coefficient on the order of 13 W m�2 K�1. The predictedvalue of the overall heat transfer coefficient in this study for thestagnant air falls within the order-of-magnitude that was found byother investigators [17,21].

Eqs. (7) and (12) were used to estimate the overall heat transfercoefficient for the forced air tests. Due to the forced convectivenature of the tests, the contributions of radiation and the thermalresistance of the metal pipe wall to the overall heat transfer coef-ficient was small. As expected, the overall heat transfer coefficientsin the forced air tests were nearly an order-of-magnitude greaterthan those of the stagnant air tests. Eqs. (11) and (12) will be validfor the case of laminar flow over the pipe, when the Reynoldsnumber is lower than the critical value for laminar-to-turbulenttransition for flow over cylindrical bodies (2 � 105). The Reynoldsnumber will be based on the undisturbed, free-stream air velocity.

Fig. 5. Plots of a) cooling time, b) freezing time, and c) total time versus temperature of s

In this study, and for a free-stream air velocity of 31 km/h, theReynolds number was approximately 4.0 � 104, which is less thanthe critical Reynolds number, indicating laminar flow. Small vari-ations in the Reynolds numbers occurred due to variations in thefilm temperature. In industrial practice, ASHRAE have suggestedoutdoor wind speeds of 24 km/h [24]. At this speed, the flowwouldbecome turbulent if the pipe outer diameter, including the thick-ness of any insulation, was approximately 390 mm (15.4 in) orlarger.

The cooling and freezing times, as determined from Eqs. (5) and(34), respectively are shown in Fig. 5 for the stagnant tests atvarious ambient air temperatures. In all the tests in this study, athermowell with a 12.7 mm outer diameter was inserted in thecenter of the pipe. Therefore, the cooling time was determined atthe outer surface of the thermowell, where r ¼ 18.3 mm and thefreezing time was determined as the time required for ice growthfrom the inner wall of the pipe to the outer surface of the ther-mowell. The cooling time of the water in the pipe was taken to bethe time required for the water temperature to decrease from theinitial temperature, Ti to the supercooled temperature, Tsuper. Fig. 5shows that as the ambient air temperature increases, the coolingand freezing times increase. Deviations between the experimen-tally measured cooling times and those predicted by the modelcould be as high as 25% (see Fig. 5a). This is most likely due to theassumptions that were made during estimation of the heat transfercoefficient. For free convection, coupled with radiation, as is thecase in this study, the heat transfer coefficient will depend on thesurface temperature of the pipe, which changes during the coolingprocess and will vary for different ambient air temperatures.However, it was assumed that the heat transfer coefficient was

tagnant ambient air for water in 50.8 mm nominal diameter Schedule 80 steel pipes.

A. McDonald et al. / Applied Thermal Engineering 73 (2014) 140e151148

constant during each test. Further, Fig. 4 shows that during coolingof the liquid, the ambient air temperature is not constant, butdecreased to the desired set-point temperature. The model of Eq.(5) assumed a constant ambient temperature, which will producedeviations in the model predictions and the experimental resultsfor cooling time. During freezing of the water in the pipe, it is mostlikely that the surface temperature of the pipe was close to themelting point of water (0 �C). Therefore, any variation in the heattransfer coefficient would be due primarily to changes intemperature-dependent properties. In addition, the ambient tem-perature, though it fluctuated slightly during the freezing period ofthe water, significant decreases were not observed (see Fig. 4).Fig. 5b shows that the maximum deviation between the experi-mentally determined freezing time and the model prediction is nomore than 13% for any ambient temperature that was used in thestudy. The total time required to cool and freeze the water in thepipes is shown in Fig. 5c. There is good agreement between thepredictions of the models and the experiments for the total time,with a maximum deviation of no more than 7%.

The novelty of this study is such that a separation of variablesmethod is used to provide estimates of the freezing time in amoving or free boundary problem. Other investigators havedeveloped alternate models to predict the freezing time of mate-rials such as foodstuffs. Pham [12] has modified Planck's equation[13] for predicting freezing times by incorporating the sensible heateffects into their model. To that end, Fig. 5b shows the predictionsof freezing times from the model developed by Pham [12]. Thefigure shows that the model over-predicts the freezing times byapproximately 25e45%, supporting the need for a new model torefine the predictions of freezing time.

Fig. 6. Plots of a) cooling time, b) freezing time, and c) total time versus temperature of

The models for the estimation of the cooling and freezing timeswere applied to the case where forced air was used to drive thecooling and freezing processes of the water in the pipe. Fig. 6 showsthat the cooling and freezing times are significantly reduced whenforced air cooling is used, in lieu of free convection and radiationwith stagnant air (see Fig. 5). The deviations in the liquid coolingtime between the model predictions and the experimental resultscan be as high as 50%, as shown in Fig. 6a. These deviations decreaseas the ambient air temperature decreases. The deviations are alsolower for the freezing times (see Fig. 6b), and are less than 8% whenthe total of the cooling and freezing times are considered (seeFig. 6c). The large deviations observed for the prediction of theliquid cooling times may be attributed to the non-constant ambientair temperatures at the beginning of the forced air tests.

4.3. Cooling and freezing e practical scenarios

The cooling and freezing of liquids in pipes are of particularimportance in applications that involve heating, ventilating, andair-conditioning (HVAC) or the transport of liquids outdoor in in-dustrial applications. When the fluids become quiescent in thepipes that are exposed to cold ambient air conditions, the stagnantfluids will cool and freeze more rapidly than if they were flowingthrough the pipe. For the case of bare, uninsulated horizontal pipes,Mcquiston et al. [21] have provided data on the overall heat transfercoefficients during free convection and radiation for pipes withouter diameters of 12.7 mm (0.5 in) to 127 mm (5 in), where thetemperature difference between the surrounding air and the pipesurface varies from DT¼ 28 �C (50 �F) to DT¼ 278 �C (500 �F). Fig. 7presents a graphical summary of the maximum overall heat

forced ambient air for water in 50.8 mm nominal diameter Schedule 80 steel pipes.

Fig. 8. Model prediction of the total (cooling þ freezing) time in stagnant ambient airfor water in a) bare and b) insulated pipes as functions of pipe inner diameter.

A. McDonald et al. / Applied Thermal Engineering 73 (2014) 140e151 149

transfer coefficients that were estimated from the data fromMcquiston et al. [21] for pipes filled with liquid at an initial tem-perature of 20 �C and exposed to ambient air temperaturesfrom �5 �C to �25 �C. The surface of the pipe was assumed to be20 �C to produce temperature differences that ranged fromDT¼ 25 �C to DT¼ 45 �C. Overall heat transfer coefficients obtainedat temperature differences below 28 �C were approximate since nodatawas provided byMcquiston et al. [21]. The figure shows that asthe pipe size increased and the temperature difference decreased,the overall heat transfer coefficient decreased. The values reportedin Fig. 7 represent the maximum overall heat transfer coefficientssince as the fluid cools, the surface temperature of the pipewill alsodecrease, reducing the temperature difference and the overall heattransfer coefficient.

Fig. 8 shows curves of the model prediction of the total timerequired for the cooling and freezing of water in completely filledmetal pipes of various inner diameters. The results of Fig. 8a forbare, uninsulated pipes show that for pipes with inner diametersless than 40mm, the total times required for cooling and freezing ofthe water were close when the stagnant ambient air temperatureis �15 �C or less. The curves nearly coalesce for smaller pipe innerdiameters at low stagnant ambient air temperatures. Similar to themodel results of Figs. 5a and 6a, the cooling time of the water in thepipewas taken to be the time required for thewater temperature todecrease from the initial temperature (20 �C) to the supercooledtemperature (�2 �C). The supercooled temperature of �2 �C waschosen based on the results observed in Table 2, where the averagesupercooled temperature was �2.4 �C. In practical scenarios, therewill be no thermowell inserted in the center of the pipe. Therefore,the cooling timewas determined at the pipe center and the freezingtime was determined as the time required for ice growth from theinner wall of the pipe to the center of the pipe (r ¼ 0). Given thehigh thermal conductivity of metal pipes, their resistance to heattransfer was neglected. This will enable the use of the curves ofFig. 8 with a variety of metal pipes, including copper and steelpipes. The model results presented in Fig. 8 were generated underthe assumption that the internal pressure in the pipe is relieved asthe water froze. It is possible that in practical applications, as thewater freezes, the pipe may be blocked, causing an increase in theinternal pressure in the pipe through the freezing event. The Cla-peyroneClausius equation will show that as the internal pressureon the water increases, the fusion temperature of the water willdecrease. Therefore, no distinct solidification plateau, similar tothat shown in Fig. 4, will be present. The effects of internal pres-surization will form the basis of future work beyond this study.

Fig. 7. Maximum overall heat transfer coefficients for bare horizontal metal pipes instagnant ambient air [21].

Insulation may be used to increase the total time required forthe cooling and freezing of the water in the pipe. There are a varietyof insulation materials that are available on the market for use inindustrial and HVAC applications. However, ASHRAE Standard 90.1-2013 [25] for energy efficiency in buildings specifies that for pipeswith less than 100 mm nominal diameters that contain hot waterbetween 41 �C and 60 �C or cold water between 4 �C and 16 �C, theinsulation thickness should be between 15 and 40 mm and have athermal conductivity of 0.030e0.040 W m�1 K�1. For this study, itwas assumed that a 25 mm thick insulation is used to protect thepipes thermally, and the thermal conductivity of the insulation is0.035 W m�1 K�1. Eq. (8) shows the expression to calculate theresistance due to insulation R

00insul for cylindrical pipes and tubes.

Table 5 presents typical values of the inner and outer diameters ofstandard tubing and pipe insulation based on the pipe outerdiameter for a 25 mm nominal insulation thickness [24]. The valuesobtained from Eq. (8) for the resistance due to insulation will beused in Eq. (7) to estimate the overall heat transfer coefficient. Theheat transfer coefficient due to the combination of free convectionand radiation can be obtained from Fig. 7. The total times requiredto cool and freeze water in insulated pipes are shown in Fig. 8b. Thefigure shows clearly that the use of insulation increases the totaltime required to cool and freeze the water by nearly an order ofmagnitude when compared to bare, uninsulated pipes. While thisobservation is intuitive, it has been observed experimentally byother investigators [1]. The extended times due to the use ofinsulation will reduce the cooling rates of the water in the pipes.

Table 5Typical values of the inner and outer diameters of standard tubing and pipe insu-lation (25 mm thick) [24].

Do,pipe, mm Dinn,insul, mm Do,insul, mm R00insul, m

2 K W�1

12.7 13 60 1.315.9 16 73 1.622.2 23 73 1.228.6 29 73 1.034.9 35 89 1.241.3 42 89 1.054.0 55 102 0.966.7 68 114 0.879.4 80 127 0.8114 115 168 0.9

A. McDonald et al. / Applied Thermal Engineering 73 (2014) 140e151150

Fig. 4 and the results presented in Tables 2 and 3 suggest that lowercooling rates will increase supercooling and the volume fraction ofdendritic ice that is formed. It is expected that in insulated pipes,the amount of supercooling may be larger than that observed inbare, uninsulated pipes.

In industrial practice, pipes will also be installed outdoors, andin most cases, they will be insulated. Exposure to winter winds atspeeds on the order of 24 km/h [24] or greater are possible. Fig. 9shows graphical results of the model prediction of the total cool-ing and freezing times of water in bare and insulated 50.8 mmnominal Schedule 80 steel pipes exposed to cold air with a speed of24 km/h. The outer diameter of the steel pipe is 60.3 mm, and as per

Fig. 9. Model prediction of the total cooling and freezing times of water in a) bare andb) insulated 50.8 mm nominal Schedule 80 steel pipes exposed to cold air with a speedof 24 km/h.

ASHRAE [24], for a 25 mm thick insulation, the inner and outerdiameters of the insulation are 61 mm and 114 mm, respectively.The figure shows a similar trend to that observed in Fig. 8, in whichthe insulation significantly increased the total time required forcooling and freezing of water in the pipe.

5. Conclusions

This study focused on the estimation of the cooling and freezingtime of water in cylindrical pipes that were exposed to stagnantand forced air at temperatures that ranged from�25 �C to�5 �C, inincrements of 5 �C. A model that was based on the separation ofvariables method was developed and used to estimate the freezingtime of water in both bare and insulated pipes. The model wasvalidated with experimental transient temperature data of waterduring the cooling and freezing processes in a bare 50.8 mmnominal Schedule 80 steel pipe.

The estimates of the cooling and freezing times of the water thatwere obtained from the model were in good agreement with theexperimental measurements. Larger deviations were observed inthe cooling times due to the assumption of constant ambient airtemperature in the cooling phase in the model, while in the ex-periments, the ambient temperature decreased from an initialvalue to the set-point temperature at the initial stage of that phase.A further source of discrepancy in the forced air test was likely dueto uncertainties in the estimates of the heat transfer coefficientexternal to pipe.

Future work and extensions of this study and model arepossible. In particular, the effects of internal pressurization on thefreezing time could be modeled analytically. This extension of themodel could be coupled with pressure vessel analysis to estimatethe burst pressures to predict failure in piping where freezing mayoccur. Other liquid material systems could also be explored. Wateris a unique material in that its density decreases as it experiences aphase change from liquid to solid. Materials such as oils and non-Newtonian fluids may pose other unique issues as they cool,freeze, and are pressurized in the cylindrical pipes.

Acknowledgements

The authors gratefully acknowledge the technical assistance ofA. Palomino with fabrication and assembly of the experimentalduct system. Funding for this project was provided by the NaturalSciences and Engineering Research Council of Canada and CenovusEnergy Incorporated.

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