Frazil ice evolution on open channels

LICENTIATE THESIS ^{1994:36 L}

DIVISION OF WATER RESOURCES ENGINEERING

ISSN 0280 — 8242

ISRN HLU - TH - L - - 1994/36 - L - - SE

Frazil Ice Evolution on Open Channels

LARS HAMMAR

September 1994

TEKNISKA

HÖGSKOLAN I LULEÅ

LULEÅ UNIVERSITY OF TECHNOLOGY

FRAZIL ICE EVOLUTION IN OPEN CHANNELS

Lars Hammar

Division of Water Resources Engineering

Lulea University of Technology, S-971 87 Lulea, Sweden September 1994

Preface

The paper presented in this licentiate thesis has been submitted for publication in the Journal of Hydraulic Research. It is an extension and deepening of two earlier papers.

Hammar, L. and Shen, H.T. A Mathematical Model for Frazil Ice Evolution and Transport in Channels, Proceedings, "6th Workshop on the Hydraulics of River Ice", October 23-25, 1991, Ottawa, Ontario, Canada.

Hammar, L. and Shen, H.T. Frazil Transport and Evolution in Channels, Proceedings, "7th Workshop on the Hydraulics of River Ice", September 23-25, 1993, Saskatchewan, Saskatoon, Canada.

This work has been carried out at the Division of Water Resources Engineering at Luleå University of Technology and Vattenfall Utveckling in Älvkarleby. It was funded by Vattenfall and the COLDTECH Foundation, Luleå. I wish to acknowledge and thank these organisations for their support.

I would like to thank Anders Sellgren for being my supervisor and supporting my work. I am very grateful to Professor Hung Tao Shen of Clarkson University, Potsdam, U.S.A. for his patience and cooperation in this and other related works. I am indebted to my family for their tremendous understanding and endurance during many evenings at home.

Summary

In cold climats, running waters may become supercooled. This supercooling leads to the production of frazil ice. Frazil ice is formed, mainly at the free water surface. It forms, as singular so called primary nuclei. These nuclei are then convected and mixed with the supercooled water. The crystals grow during the transport, under release of latent heat. When ice concentration is high enough, mixing will cause flocculation of singular crystals to larger particles. This flocculation is the dominant factor for growth in size. Collisions in between particles is regarded as the single most important source for new particles, secondary nucleation - multiplication.

Frazil ice is the most important form of river ice during the freeze-up period. It is the origin of all other forms of river ice. The formation of frazil ice leads to the surface ice run, which eventually may develop into an ice cover. Frazil that are entrained to the underside of an ice cover may accumulate into hanging dams, causing head loss and flooding. Frazil crystals suspended in the turbulent supercooled water can attach to the bed material and contribute to the growth of anchor ice. The attachment of frazil may cause damage on underwater cables and blocking of trash-racks for hydro-power stations and fresh- water intakes.

The aim of this project, is to develop a model for frazil ice evolution in rivers, so that a better understanding of the phenomenon can be obtained. The model will be used for studies on frazil-ice related problems.

In the model, the transport and mixing processes are simulated by a turbulence model with a K-E formulation for turbulence, utilizing PHOENICS as an equation solver. The model simulates the temperature and ice particle distribution in the water, together with the size distribution of ice particles.

The primary nuclei are provided through seeding at the free surface. The ice crystals will then grow both in size and in numbers. The growth in size is governed by the heat transfer between individual ice crystals and the ambient, super-cooled turbulent water, together with collision induced flocculation.

Flocculation of crystals is related to differences in buoyant velocities between different sized particles, as well as the turbulent shear. The particle size distribution is handled by subdividing of the size range into fractions represented by mean diameters. The change of concentration in the different size fractions is based on increases of frazil particle sizes due to thermal growth and flocculation, as well as the multiplication of frazil crystals. The multiplication of crystals is due to secondary nucleation, i.e. small ice fragments being shedded off in collisions between larger particles or flocculates. The amount of secondary nucleei produced is dependent on the colission energy involved. The distribution of the particle fractions in the water is then governed by convection, turbulent diffusion and differences in buoyant velocity. Small particles, < 100 pm, tend to be evenly distributed in the flow, while larger particles are concentrated to the upper layer.

The model is formulated for three-dimensional unsteady flows. The analysis presented is, however, limited to two-dimensional, steady flows in wide channels. Comparisions with experimental results of Carstens (1966) shows the capacity of the model to reproduce measured temperature curves. Since there is no existing method, at the present, for measuring frazil concentrations and size distributions, the model provides a useful tool for examining the phenomenon of frazil evolution, in addition to be able to predict frazil production and transport in river channels.

A sensitivity analysis of major parameters led to the following conclusions:

Collisions is a significant factor for frazil ice growth.

The existence of larger particles is the single most important factor for secondary nucleation, as the collision energy is strongly dependent on the particle sizes.

The description of the turbulence characteristics is very important, as turbulence to a large extent governs the temperature and particle distribution as well as it strongly influences the collision frequencies.

Sammanfattning

Vid kall väderlek kan snabbt strömmande vatten bli underkylt. Denna underkylning leder till produktion av kravis (frazil ice). Kravisen bildas vid vattenytan i form av enskilda sk primära kristaller. Dessa transporteras och blandas med det underkylda vattnet. Kristallerna växer, och avger latensvärme. Vid tillräckligt höga koncentrationer av ispartiklar orsakar omblandningen sammanslagningar (flocculation) av kristaller till större partiklar. Detta är den dominerande faktorn för tillväxt i storlek. Kollisioner mellan partiklar anses allmänt vara den främsta källan till nybildning av kristaller (secondary nucleation,multiplication).

Kravisbildning är i regel ursprunget till andra former av is i floder och älvar. Kravisproduktion leder till tillväxt av istäcken och att is kan aysättas under befintliga istäcken varvid sk hängdammar bildas med översvämmningsskador som följd. Kravis orsakar även tillväxt av bottenis som kan leda till skador på kablar etc. Igenfrysning av intagsgrindar till vattenkraftverk och friskvattenintag är exempel på andra problem orsakade av kravis.

Målet för detta projekt är att utveckla en modell för kravistillväxt i älvar. Syftet är att modellen skall användas för studier av olika problem relaterade till kravisproduktion.

Kravismodellen baseras på att den turbulenta strömningen i en kanal beskrivs med en K-E ansats, och nyttjande av ekvationslösaren PHOENICS. Modellen simulerar temperaturutvecklingen i vattenmassan och fördelningen av partikelkoncentrationer i flödet, samt storleken av partiklar i intervallet 4µm- 1,4mm. Förekomst av primära kristaller simuleras i modellen genom att ett partikelflöde föreskrivs i ytskiktet. Därefter tillväxer primärkristallerna genom termiskt utbyte och via kollisioner.

Sammanslagningen av kristaller relateras till skillnader i stighastighet mellan olika partikelstorlekar och till turbulenta skärspänningar. Partiklarnas olika storlekar hanteras genom en uppdelning i fraktioner representerade av en medeldiameter. Förändringen av koncentration i storleksfraktionerna baseras på termisk tillväxt från en lägre till en högre fraktion och på sammanslagningar under bevarande av befintlig massa. I samband med kollisioner bryts delar av iskristallerna loss, och bildar nya sekundära kristaller i den lägsta storleksfraktionen. Mängden sekundära kristaller som bildas vid varje individuell kollision mellan två partiklar baseras på kollisionsenergin. Fördelningen av partikelfraktionerna i vattenmassan styrs av konvektion, turbulent diffusion och skillnader i partiklarnas stighastighet. Mindre partiklar , <100 gm, tenderar att vara relativt jämnt fördelade i vattenmassan medan övriga partiklar koncentreras mot ytskiktet.

Modellen är formulerad för tredimensionella icke-stationära flöden. De fiesta applikationer är dock hitills tvådimensionella, stationära i en oändligt bred kanal. En jämförelse med Carstens (1966) laboratorieresultat visar på modellens kapacitet att reproducera uppmätt temperaturfördelning vid kravisbildning. Hitills har simulerade partikelfördelningar ej kunnat jämföras med uppmätta. Det finns idag inga tillförlitliga mätresultat eftersom det är mycket komplicerat att mäta iskoncentration och partikelfördelning.

Studierna inkluderar känslighetsanalys av centrala parametrar. Genomförda simuleringar resulterar i följande slutsatser.

Kollisioner spelar en avgörande roll för tillväxt av kravis.

Förekomsten av större partiklar är den väsentligaste faktorn för sekundär partikelbildning, eftersom kollisionsenergin stiger dramatiskt med partikelstorleken.

Beskrivningen av flödets turbulenskaralcteristik är mycket viktig, då turbulensen i stor utsträckning styr temperaturutbredningen och partikelfördelningen samt starkt påverkar kollisionsfrekvenserna.

FRAZIL EVOLUTION IN CHANNELS

Lars Hammar

Division of Water Resources Engineering

Luleå University of Technology, S-971 87 Luleå, Sweden and

Hung Tao Shen

Department of Civil and Environmental Engineering Clarkson University, Potsdam, New York, 13699-5710 U.S.A

SUMMARY

A simulation model for the formation and evolution of frazil in open channels is developed. In this model, the primary nucleation is assumed to be due to mass exchange of seeding crystals at the free surface. The model of frazil crystals growth is based on the rate of heat transfer between crystals and the ambient turbulent flow. Secondary nucleation and flocculation are simulated based on binary collisions of frazil particles.

The model is validated with existing experimental data. It is capable of simulating the variation of water temperature during the frazil formation period and the evolution of size and concentration distributions of frazil in the flow. Effects of the surface heat exchange, the rate of seeding, and the flow condition are examined.

RÉSUMÉ

Un modMe mathariatique d&rivant la formation et l'volution du frasil dans un &oulement a surface libre a 6te &veloppi. Dans ce mockle, la nucl&tion primaire est attribut& a. Mchange de masse rsultant de l'introduction de cristaux a la surface de l'eau. La simulation de la croissance des cristaux de frasil est bas& sur le taux de transfert de chaleur entre les cristaux individuels et l'&oulement turbulent. La nucl&tion secondaire et la floculation sont simul&s a partir de collisions binaires entre les particules de frasil.

Le modMe est validé a. l'aide de donn&s exp6rimentales existantes. Il est capable de simuler la variation de la temp&ature de l'eau pendant la p&iode de formation du frasil., ainsi que la croissance et la distribution de la concentration du frasil dans fecoulement. Les effets de f&hange de chaleur å la surface, le taux d'ensemencement et les conditions d'icoulement sont 6tudies.

I INTRODUCTION

Frazil ice is the most important form of river ice during the freeze-up period. It is the origin of almost all other forms of river ice. The formation of frazil ice can lead to surface ice runs, which eventually develop into the ice cover. Frazil granules that are entrained to the underside of the ice cover may accumulate into hanging dams. Frazil crystals suspended in the turbulent supercooled water can attach to the bed material and contribute to the growth of anchor ice. All of these processes can have significant impacts on physical and biological conditions of the river. Although the process of frazil evolution is not completely understood, it is generally accepted that frazil ice forms in supercooled turbulent river water. The turbulence condition is necessary to effectively maintain the heat transfer from the river over its depth to the atmosphere. Turbulence is also required for the downward mixing of frazil crystals on the water surface to prevent the formation of a static surface ice cover, which will inhibit the surface heat loss and the frazil production. In addition to turbulence and a small degree of supercooling, the presence of seed ice crystals is also required. These seed ice crystals or nuclei can come from a number of sources, including supercooled snow or ice particles, cold airborne dust falling down to the water surface, and ice particles from the border ice (Osterkemp 1978, Daly 1984). Once seeded, ice crystals will grow and multiply, in a dynamic balance between latent heat released by the growing ice crystals and heat exchanges with the atmosphere and the channel bed. The multiplication of frazil crystals is due to secondary nucleation. The secondary nucleation is caused by the shedding of small ice fragments from large frazil crystals when they collide with other particles or solid boundaries. The growth and multiplication processes generally cause the water temperature to rise to near 0° C to balance the latent heat realeased by frazil crystals.

Mathematical models have been used to simulate frazil and grease ice formation in surface waters.

Omstedt and Svensson (1984), and Omstedt (1985a,b) developed a series of numerical models for the formation of frazil ice and grease ice in the upper layers of the ocean. The vertical distributions of ice concentration and temperature were simulated. In these models, the ice crystals were considered to have a constant size, with a constant heat exchange rate between the water and ice crystals. The secondary nucleation was assumed to be directly related to the thermal growth of ice crystals. The flocculation of frazil crystals were modelled, but the effect of differential rising was neglected. Nyberg (1984) following the model of Omstedt (1985b), developed a two-dimensional model for river channels. Mercier (1984), extending the work of Daly (1984), formulated a kinetic model of frazil growth. Mercier simulated frazil formation in channels using a Monte Carlo technique. Hammar and Shen (1991) developed a two-dimensional model for frazil formation in channel by generalizing Omstedt's formulation to variable crystal sizes. Distribution of water temperature, ice concentration, and crystal size were examined. In this paper, the model of Hammar

and Shen (1991) is modified and extended to include the kinetics of secondary nucleation and flocculation. The present model simulates the variation of water temperature and the evolution of frazil size and concentration. The frazil formation is initiated by seeding small ice nuclei on the water surface. Secondary nucleation and flocculation are simulated based on the collision of frazil particles. The model is validated with existing flume data. Effects of surface heat exchange, seeding rate, rate of secondary nucleation, and the flow condition are examined.

2 MODEL FORMULATION

The present mathematical model uses either prescribed values on turbulence parameters or a standard k-E model to simulate the flow condition. A general equation solver PHOENICS for heat and mass transfer problems (Spalding 1981) is used to solve the governing flow- and transport equations. The frazil particles are assumed to be thin circular disks with a constant ratio of 1/10 between the thickness and diameter, based on the flume data obtained by Daly and Colbeck (1986).

The size distribution of frazil particles is described by logarithmically spaced size groups. Seeding crystals as well as those produced by secondary nucleation are assumed to be in the lowest size group.

2.1 Mean flow equations

The governing equations for the mean flow, frazil concentration, and water temperature are:

au, au, a p a au, at +u-i — ax. ---i;— ax,+— ax.(üt — a.)+gl

aT " aT a fit aT,

at ax,

ack ack

at

1 a,

a er, ack (--) ax, ac ax,

ac,

LiCk ""floc,k 0X3

(3)

in which p is the density of the mixture, pwl-(p,-pw)EkCk ; p, and py, are densities of ice and water, (1)

(2)

respectively; U. is the ith component of the mean velocity; P is the mean pressure,

et

kinematic eddy viscosity; g, is the ith gravity component; a, and 6, are Prandtl/Schmidt numbers;

T is water temperature; ^Ck is the volumetric concentration of frazil in the kth size fraction; cOk is the frazil buoyant velocity of kth size fraction; Sb and ^Sf are source terms due to heat loss at the channel boundaries and the latent heat released from frazil growth; ^Sck is the source term due to the thermal growth of frazil; and ^Sfl„_,k is the source/sink term due to secondary nucleation and flocculation.

2.2 Frazil evolution

The frazil ice size distribution is divided into eight logarithmically spaced size groups. The sizes of these groups range from 4 gm to 1.435 mm. Seed crystals and secondary nuclei are assumed to be in the lowest size group. In the following sections the formulation for frazil evolution including thermal growth, secondary nucleation, flocculation, and the treatment of source/sink terms will be discussed. Turbulent shear can cause flocculated particles to break up. However, since particle breakup is limited to large, weakly bounded aggregates under high shear (Mercier 1984), particle breakup will not be considered.

2.2.1 Thermal growth of frazil crystals

The rate of growth of an ice crystal depends on the rate of transfer of latent heat from the crystal to the ambient turbulent flow. The rate of heat transfer per unit ice surface area, q, can be expressed in terms of the heat transfer coefficient, h, as:

q=h(Ti -Tw)

where T, is the ice surface temperature and Tw is the ambient water temperature. The heat transfer coefficient, h, can be expressed in its dimensionless form as a Nusselt number defined by:

N.= hl

where 1 is a characteristic length of the ice crystal, and kw is the thermal conductivity of water. In this study, the face radius calculated based on the surface area of the ice crystal, As, i.e. (As/4701/2, is used as the characteristic length. Combining Eqs. 4 and 5 gives the following equation for the rate of heat transfer:

(4)

Nk

q- u

l w(Ti-Tw)

(6)

The Nusselt number depends on the flow condition and the particle size. In this study, the following formulation developed by Batchelor (1980) and Wadia (1974), as summarized by Daly (1984) is used:

Nu=(—)+0.17e2

;

i

if in*< P112 ^(7a)

and,

p N _[(J ;.)

+0.55(--L)113]

in m *

• if , 1 <m*<10

p 1/2

(7b)

in which, m4=rt1, is the ratio between the face radius of the ice crystal and the Kolmogorov length scale.

For large particles, i.e. m*>1:

0.035 P^r 1

Nu=1.1[(- 1

)+0.80a T (—) i

] ;

a 7m*4/3<1000

m*

and,

0.24 1

Nu=1.1[(- 1

)+0.80a T

a 7m *413

m*

in which, aT=-\/(21c/U) is the turbulence intensity, and U is the mean flow velocity. In the transition region, i.e. 1<m*<10, the variation of Nu is not well defined. In this study, Eq 8 is used when m4

>1. It is of interest to note that when m* increases, the Nu number decreases. Therefore, the thermal growth rate of frazil particles decreases rapidly with the increase in particle size.

2.2.2 Secondary nucleation

The model for the kinetics of secondary nucleation developed by Evans et al. (1974) and Mercier (1984) is adopted. This formulation considers that the collision breeding is the primary mechanism for secondary nucleation. The number of nuclei produced due to collisions between particles in size classes v, and vj is:

I(v.v.)- f vf+' f v-i'ZC (v. v.)dv.dv.

Vi_1/2

where I(v„vi) is the number of nuclei produced per unit time, v, and vj are volumetric sizes of the colliding particles, Z is the number of nuclei produced per unit collision energy and, CE(v„vj) is the rate of collisional energy transfer to the crystals per unit volume of fluid. The parameter Z may be considered as a material constant for the small supercooling levels in rivers, when the impurity concentration is low. The function CE(v„vj) can be expressed in terms of collision frequency and collision efficiency. The main contributors to these in open channel flows are turbulent shear and differential rising. The collision frequency function is the probability that two non-interfering particles of sizes v, and vj will collide in a unit time. For turbulent shear this is, (Mercier 1984):

13sh(vovi) =0.39 0. /2

<E>

1

1/2 1/3 1/3,3

(v. +1' _t ⁽¹⁰⁾

in which (0= mean energy dissipation rate; and 0= kinematic viscosity of water. The collision frequency function due to differential rising is (Findheisen 1939):

13 (v v

)= 0

dr ^{19 j} KV.

+1,•12

11) p w

Combining these equations, and assuming that the collisions are inelastic, yields:

(9)

1

-1⁾

Viv;

pi -1 [b(vi 1/3

+vi

( <E> )3/2E

— sh

2 vi +vi.

g

+0.00076(

Iv;

where b=0.0066/K113/4, Ku=kurtosis of the velocity derivative (Mercier 1984), and g(v1) are the number density functions. The collision efficiency functions for differential rising and turbulent shear ^Edr and ^{Esh ,} are introduced to account for particle interference effects. These functions are evaluated according to the procedure outlined by Pearson et al. (1984). The value of Ed, decreases when the difference in particle size increases.

2.2.3 Flocculation

The instantaneous expected number of collisions between all particles in the ith size class and jth size class per unit volume per unit time is:

F.= ß(vpvi)E(vi,vAit

in which, (1), and 1:1)j = the number concentration of the ith and jth-size particles. Each collision per unit volume will reduce the local number concentration of i- and j-particles by one. The colliding particles will merge into a new particle of volume v,,,„ge=v,+vi-ny 1, where nv, is the volume contributed to secondary nuclei production. The size vmerge will not correspond exactly to the prescribed size groups, but will fall between vk and vic+1 ^. In the simulation scheme, the merged particles will be distributed to the two neighbouring size groups k and k+1. Based on the volume conservation method of Lawler et al. (1980), a fraction

f

v -

Vk+l —Vk ⁽¹⁴⁾

of F„ is assigned to the kth size group, and a fraction (1-f) is assigned to the (k+l)th size class.

When the merged particle size is larger than the size of the last size class m, then the fraction (12)

f

Vm

of Fa is assigned to the mth size class.

2.2.4 Source terms

The source term, ^Sf, due to frazil growth, in Eq 2, and the corresponding source term, Sck, in Eq 3 can be determined from the heat transfer between the ice crystal and the ambient turbulent flow.

The term ^Sf can be written as:

S f=E k

q k(clik p

where, dfl, is the mean face diameter of particles in the kth size group. Correspondingly, the rate of ice production per unit volume due to the thermal growth of ice mass in the kth size group is

sk =4 ckqk(dfic

where, L is the latent heat of fusion of ice. The net increase on ice concentration of the kth size group due to thermal growth and flocculation then becomes

S

S _Ck i

`

AV_{k1 AVk}

(18)

in which Avk=vk+l-Vk. The term on the right hand side of Eq. 18 represents the increase in ^Ck, contributed by the growth in the (k-1)th size group, and the decrease in Ck due to growth in the kth size group.

2.3 Turbulence modelling

In a well-mixed flow, the depth-averaged turbulence parameters may be estimated from the observed flow data. Assuming smooth boundaries, the friction factor, f, can be determined from

(15)

(16)

(17)

-21og(Re ¹

/1/2)-0.8

in which Re=4URPI), and R=hydraulic radius. The shear velocity u*=U(f/8)112. Assuming the vertical distribution of the energy dissipation rate is

3

(20) icy h

The depth-averaged value of e(y) can be determined from Eq. 20 by assuming the thickness of the viscous sublayer equals to 191u*. The turbulent kinetic energy can be approximated by

2 v

0.3 h

(21)

and the depth-average value of k(y) determined in a similar way as that of e(y).

In an open channel flow, parameters u,k and E are depth dependent. A standard k-e model is used to simulate the variations of these turbulence parameters over the flow depth.

3 MODEL VERIFICATION

The present model is validated using experimental data of Carstens (1966). The experiments were conducted in a racetrack shape recirculating flume in a cold room. The flow, which was driven by a propeller, has a cross section of 0.2 m by 0.2 m. The tests were performed at a temperature of about -10°C in a cold room. Water temperatures were measured at a point located at 0.05 to 0.1 m below the water surface. Due to the mixing effect of the propeller, the vertical temperature gradient was found to be negligible. Because of this, the well-mixed condition will be imposed in the numerical model for calibrations with the experimental data. The turbulence parameters are calculated from the measured flow conditions.

Two cases presented by Carstens are calibrated. The first case corresponds to Case A in Carstens' figure 6. The second case corresponds to figure 7 of Carstens' paper. Table 1 summarizes flow parameters and heat loss rates of these experiments.

Case 1 0 Case 2

Table 1. Parameters of calibration simulations

Case U, m/s k, 11-12/s2 t, 1712/s3 Q, J/m3-s

1 0.5 0.00096 0.0012 1,400

2 0.33 0.00048 0.00038 600

In Table 1, the heat loss rate, Q, is estimated from the initial slope of the water temperature curve. The parameter Z and the initial seeding rate, Io, are calibrated to be 3 x 1015 nuclei/J and 104 nuclei/m3-s.

Figure 1 compares the simulated and measured water temperatures for both cases. These comparisions confirm the validity of the simulation model.

0.010

0.000

-0.010

-0.020

-0.030 -0.040 -0.050 -0.060 -0.070 -0.080 -0.090 -0.100

0 1 2 3 4 5 6 7 8 9 10

Time(min)

Fig 1. Comparision of measured and simulated water temperatures.

Figure 2 shows the evolution of particle size distribution. A comparision of these figures shows that initially the relative concentration of large particles increase continuously when the water temperature is steadily decreasing. The increase in large particles is due partly to the rapid rate of thermal growth of small particles, and partly to flocculation. As the particle concentration increases, especially for larger particles, a massive increase in secondary nuclei emanating from particle collisions occurs. This leads to the rapid increase in small particles and the recovery of water temperature. Thereafter, the concentrations of small size particles continue to increase

++++ 1 min xxxx 2 min

**** 3 min

EIDOD 4 min

0000 5 min

°"' 6 min

,..n. a 8 min

mainly due to secondary nucleation, while concentrations of large size particles continue to increase through thermal growth and flocculation of smaller particles. The water temperature remains essentially constant as a result of the balance of the heat loss through the water surface and the latent heat released from the growth of ice particles.

o ^{1 2 3 4 5} 6 7 8

Size groups (1-8)

Fig 2. Variations of concentration in different size groups, Case 1.

4 SENSITIVITY ANALYSIS

To examine the sensitivity of the process to various model parameters, additional simulations are performed for Case 1. Figure 3 shows the effects of different parameters. As expected, a larger heat loss rate leads to a higher degree of supercooling and a faster temperature recovery. A larger seeding rate increases the number of smaller particles and hence faster water temperature recovery. The dissipation rate reflects the level of turbulence intensity, which directly affects the collision frequency, as can be seen from Eq. 10. A higher dissipation rate leads to a higher rate of secondary nucleation and hence a faster rate of water temperature recovery. A larger Z value corresponds to a higher rate of collisional breeding, and a faster recovery of water temperature.

Concentration

Heat loss rate, J/rd-s 1400 700 2800

0 1 2 3 4 5 6 7 8 C)

Dissipation rate, rd/s3 0.012 0.0012

0.12

0 1 2 3 4 5 6 7 8

d)

Seeding rate, nuclei/m3-s 104 0.5x104 2x104 0.000

-0.020

-0.040 C

cr) -0.060

E

E—i" -0.080 -0.100 -0.120

0.000

-0.020

-0.040

E ^-0.060

-0.080

-0.100

-0.120

0 1 2 3 4 5 6 7 8

Time(min)

0.000

-0.020

-0.040

-0.060

-0.080

-0.100

-0.120

0.000

-0.020

-0.040

-0.060

-0.080

-0.100

-0.120

Parameter Z, nuclei/J 3x1015 3x1014 3x1016

0 1 2 3 4 5 6 7 8

Time(min) Fig 3. Effects of a) heat loss rate; b) seeding rate; c) dissipation rate;

and d) parameter Z, on water temperature.

5 SAMPLE SIMULATIONS

In order to examine variations of water temperature and frazil concentration over the flow depth, two-dimensional simulations are made. In these simulations, the flow and turbulence conditions are simulated with the k-E model. Results of two sample simulations are presented in this section. Example I corresponds to Case 1 of the well-mixed case used in the model verification.

Figur 4 shows the flow and turbulence conditions.

1 . 0 - 0.9 0.8 -

0.0 0.7 - 0.6 - 0.5 - 0.4 - 0.3 - 0.2 - 0.1 -

Example I Example II

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 u/U, k/u.2, cH/<10u.3>

Fig 4. Vertical distributions of turbulence parameters for the channel flows.

Figure 5 shows the mean water temperature and the water temperature distribution at 10 equally spaced depths between 1 cm from the water surface and 1 cm from the channel bottom. The temperature drop goes a bit lower and the maximum supercooling occurs roughly 1 min later then in the one-dimensional well-mixed case. This result is to be expected, and can be explained by the fact that the k-E model predicts the maximum levels on k and e close to the bottom of the flume, where the concentration of particles are at their minimum. The collision breeding is less than the well mixed case.

Mean water temperature 0.000

-0.020

el.,' ^-0.040 E

-0.060

-0.080 - 0.100

0 2 4 6

Time(min)

8 10

0.010 -

0.000

0.010 -

-0.020 -

-0.030

-0.040

C17

-0.050 -

E a.)

-0.060

-0.070 -

-0.080 -

-0.090 -

-0.100

1 ^{2 3 4} 5 ⁶ 7 ⁸ 9 10

Time(min)

Fig 5. Variations of water temperature with travel time and depth, Example I.

Figures 5 and 6 shows that the two-dimensional k-c simulation results in a depth-varying temperature profile with a change of roughly 0.01 °C over the depth of the flow.

-0.000 -

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0.160

0.180

0.200

0.010

Temp(°C)

Fig 6. Variation of vertical temperature profiles for different travel times, T„ along the channel, Example I.

In Fig 7 the corresponding relative vertical concentration profiles for particles in different size

3 min

•

0.010 -0.030 -0.050 -0.070 -0.090

classes are presented. The relative vertical concentration for C, is defined as

cie,,

-0.00 - C6 C3

0.02

0.04 -

i../7 0.06 - /7 ...)(1,/

(I' .:1 il 10-5

i I 7. /III 0.08

14 ^{10 6}

.= 0.10 _, ^si 10 7

e, lih!

Ce5) ^II I

›, 10 a 0.12

/ i i i "Cj 1 0 ⁹ I. I t

fs i c) 1

0.14 - •'t I! (....1, 10

I %

i M ;

I it; 10 1

C2 C7

0 1 2 3 4 5 6 7 8

Size Groups

0.18 -: / / [-Ix\

0.20

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Relative Concentration

Fig 7. Relative vertical concentration profiles at 1.1.4 min, Example I.

As can be seen in this calculation, concentration gradients exist over the depth even in shallow flumes. The notable increase in relative concentration of the small sized fractions, C1 and ^C2, close to the channel bottom in the flume can be explained by the large values of k and E close to the bottom, which favours secondary nucleation. The small sized and large sized fractions are generally at their maximum concentration close to the surface due to seeding of small sized crystals on the surface and the larger buoyancy effect on larger sized particles. The intermediate sized particles are more evenly distributed.

In Example H, a simulation is made for a uniform, wide rectangular channel with a flow depth of 5 m. The bottom roughness has an equivalent roughness height of 0.1 m. The average flow velocity was kept at 0.5 m/s together with the same values on the other parameters as in Case 1. Simulation results are presented in Figs. 8-10. As expected, the water temperature evolution is much slower and the degree of supercooling is smaller. Water temperature profiles are much

0.005

0.000

-0.005 -

-0.010 -

-0.015 -

10 20 30 40 50 60 70

Time (min)

5 10 15 20 25 30 35 40 45 50 55 60

Time (min)

Mean water temperature 0.000

-0.010 LI

-0.020 E ci..) -0.030

-0.040

-0.060 LI

-0.020

E- -0.025

-0.030 -

-0.035

-0.040

-0.045

-0.050

more pronounced. The frazil is distributed with a pronounced concentration maximum close to

the water surface for all particle sizes.

Fig 8. Variation of water temperature with travel time and depth, Example II

3.500 -

4.000 -

4.500 -

5.000 0.000 -

0.500 -

1.000 -

1.500

2.000

2.500 -

3.000

0.005 -0.005 -0.015 -0.025 -0.035 -0.045

Temp (°C)

Fig 9. Variation of vertical temperature profiles for different times, T,, along the channel, Example II.

10 0.00

0.50 1.00 1.50 2.00

3.00 3.50 4.00 4.50

C, C, C, C,

• ,"/

•

,,-

iii

• 7/Z' 10 ^-6

10 ,

//j.1/: ₁₀ ^e

.111

Ili

10 .!.i

10

0 1 2 3 4 5 6 7 8

Size Groups

5.00

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Relative concentration

Fig 10. Relative vertical concentration profiles at T,•=5 min, Example H..

6 CONCLUSIONS

In this paper, a mathematical model for formation and evolution of frazil ice in turbulent channel flows is developed. Both distributions of frazil concentration and temperature are simulated. The frazil formation is initiated by the mass exchange of seeding crystals at the water surface. The increase in the number of frazil crystals in the ice-water mixture due to secondary nucleation is modelled. The increase in particle size due to thermal growth as well as flocculation are also modelled. Secondary nucleation and flocculation are modelled by considering binary collisions between frazil particles. The model simulates the size distribution of frazil particles size in the turbulent channel flow. The size distribution is a result of crystal seeding, thermal growth and flocculation of frazil particles. The model is validated with existing laboratory data, and used to study the intricate thermal-mechanical processes involved in frazil evolution. The simulated results, which illustrate processes involved in the evolution of frazil ice in turbulent channel flows, are presented and discussed. Effects of the surface heat loss rate, the rate of seeding, the flow condition, and secondary nucleation parameters are examined.

ACKNOWLEDGEMENT

The first author would like to acknowledge the support provided by the COLDTECH Foundation, Lulea, Sweden.

REFERENCES

Ashton, G. D. (Editor), 1986. River and lake ice engineering, Water Resources Publications, Littleton, Co.

Batchelor, G. K., 1980. Mass transfer from small particles suspended in turbulent fluid. Journal of Fluid Mechanics, 98(3): 609-623.

Carstens, T., 1966. Experiments with supercooling and ice formation in flowing water. Geofysis ke Publikasjoner, 26(9): 3-18.

Daly, S. F., 1984. Frazil dynamics, CRREL Monograph 84-1, USA Cold Regions Research and Engineering Laboratory, Hanover, NH.

Daly, S. F. and S. Colbeck, 1986. Frazil ice measurements in CRREL's flume facility, Proc. JAHR Symposium on Ice, Iowa City, IA.

Evans, T.W., G. Margolis and A.F. Sarofim, 1974. Models of secondary nucleation attributable to crystal-crystalizer and crystal-crystal collisions. A.I.Ch.E. Journal, 20(5):959-966.

Findheisen, W., 1939. Zur frage der regentrop-fenbildung in reinem wasserwolken. Meteor. Z., 56:365-368.

Hammar, L. and RT. Shen, 1991. A mathematical model for frazil evolution and transport in channels. Proc. 6th Workshop on the Hydraulics of River Ice, Ottawa, 201-216

Lawler, D.F., C.R. O'Melia and J.E. Tobiason, 1980. Integral water treatment plant design: From particle size to plant performance. In: Particulates in Water, M.C. Kavanaugh and J.O.

Leckie (Editors), Advances in Chemistry Series, No. 189, American Chemical Society, Washington, D.C., 353-388.

Mercier, R.S., 1984. The reactive transport of suspended particles: Mechanics and Modeling. Ph.D.

Dissertation, Joint Program in Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA.

Nyberg, L., 1984. Ice formation in rivers. In: Numerical Simulation of Fluid Flow and Heat/Mass Transfer Processes, C.A. Brebbia and S.A. Orsag (Editors). Springer-Verlag.

Omstedt, A. and U. Svensson, 1984. Modeling supercooling and ice formation in a turbulent Ekman layer. Journal of Geophysical Research, 89(C1):735-744.

Omstedt, A., 1985a. On supercooling and ice formation in turbulent sea water. Journal of Glaciology, 31(109): 263-271.

Omstedt, A., 1985b. Modelling frazil ice and grease ice in the upper layers of the ocean. Cold Regions Science and Technology, 11:87-98

Osterkemp, T.E., 1978. Frazil ice formation: A Review. Journal of the Hydraulics Division.

ASCE.104(HY9): 1239-1255.

Pearson, H.J., I.A. Valioulis and E.J. List, 1984. Monte Carlo simulation of coagulation in discrete particle-size distributions. Part I. Brownian motion and fluid shearing. Journal of Fluid

Mechanics, 143:367-385.

Spalding, D.B., 1981. A general purpose computer program for multi-dimensional one- and two- phase flow. In: Mathematics and Computers in Simulation, XXIII, North-Holland Publishing Co., 267-276.

Wadia, P.H., 1974. Mass transfer from spheres and discs in turbulent agitated vessels, Ph.D.

Dissertation, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA.

N„

Sbf

Sck

Sflockc =

UI NOTATION

Pi,P. = Gt,c =

=

1-)t

COk

gi

Ck

k„,

1

m =

turbulence intensity

rate of dissipation of turbulent energy

kinetic energy of turbulence density of ice and water Prandtl/Schmidt numbers molecular kinematic viscosity eddy viscosity

frazil buoyant velocity gravitational constant in x,

direction

heat transfer coefficient

volumetric concentration of frazil in kth size fraction

thermal conductivity of water characteristic length of ice crystal ratio between face radius of ice crystal and Kolmogorov length scale

Nusselt number mean pressure Prandtl number

rate of heat transfer per unit area heat loss/gain source terms (boundaries and frazil latent heat) source term due to thermal growth of frazil

source/sink term due to secondary nucleation and flocculation

temperature

temperature of ice and water mean, fluctuating velocity component in x, direction

LIST OF FIGURES

Fig 1. Comparision of measured and simulated water temperatures.

Fig 2. Variations of concentration in different size groups.

Fig 3. Effects of a) heat loss rate; b) seeding rate; c) dissipation rate; and d) parameter Z on water temperature.

Fig 4. Vertical distribution of turbulence parameters for the channel flows.

Fig 5. Variations of water temperature with travel time and depth, Example I.

Fig 6. Variation of vertical temperature profiles for different travel times, Tt, along the channel, Example I.

Fig 7. Relative vertical concentration profiles at TF4 min, Example I.

Fig 8. Variations of water temperature with travel time and depth, Example II.

Fig 9. Variation of vertical temperature profiles for different travel times, Tt, along the channel, Example II.

Fig 10. Relative vertical concentration profiles at TF5 min, Example II.

/9a/-7/•"c3,G

-

- -

HÖGSKOLAN I MC

LULEÅ UNIVERSITY, SWEDEN

ISBN ISSN ISRN

Institution/Department Upplaga /Number of copies

Avdelning /Division Datum /Date

K274/c-72/2/2/./

Titel/Title

/ /cr . Æ- izaZ/h o77 .

Författare/Author(s)

///72/-72"---

Uppdragsgivare/Commissioned by Typ/Type

O Doktorsavhandling/Ph.D. thesis Licentiatuppsats/Licentiate thesis o Forskningsrapport/Resarch report O Teknisk rapport/Technical report o Examensarbete/Final project report o Övrig rapport/Other report

Språk/Language

0 Svenska/Swedish 0 Engelska/English 0 Sammanfattning, högst 150 ord/Abstract, max 150 words

A simulation model for the formation and evolution of frazil in open channels is developed. In this model, the primary nucleation is assumed to be due to mass exchange of seeding crystals at the free surface. The model of frazil crystals growth is based on the rate of heat transfer between crystals and the ambient turbulent flow. Secondary nucleation and flocculation are simulated based on binary collisions of frazil particles. The model is validated with existing experimental data. It is capable of simulating the variation of water temperature during the frazil in the flow. Effects of the surface heat exchange, the rate of seeding, and the flow condition are examined.

Nyckelord, högst 8/Keywords, max 8

frazil, ice-particles, collisions, growth, thermal exchange

Underskrift av granskare/handledare / Signature of examiner/supervisor

Wzi