C - /

FRAZIL ICE FORMATION IN TURBULENT FLOW

by Andreas Muller

Sponsored by U.S. Army Corps of Engineers Cold Regions Research and Engineering Laboratory, Hanover, N.H. Grant No. DACA89-77-0536

IIHR Report No. 214

Iowa Institute of Hydraulic Research The University of Iowa Iowa City, Iowa

TC June 1978 24 .164 no.214 1978 LIBRARY

OCT 4 1978

Bureau of Reclamation D e n '"" '■-'inrario BUREAU OF RECLAMATION DENVER LIBRARY 92072386 A 92072386

FRAZIL ICE FORMATION IN TURBULENT FLOW

/ by Andreas Muller/

Sponsored by U.S. Army Corps of Engineers Cold Regions Research and Engineering Laboratory, Hanover, N.H. Grant No. DACA89-77-0536

IIHR Report No. 214

Iowa Institute of Hydraulic Research The University of Iowa Iowa City, Iowa 4 June 1978 i ACKNOWLEDGEMENTS

This study was conducted by the author while he was on leave from the Swiss Federal Institute of Technology (ETH) Zurich at IIHR. The invitation by Dr. John F. Kennedy and a grant from the ETH which made this stay possible are gratefully acknowledged. The research project was funded by the U.S. Army Corps of Engi­ neers, Cold Region Research Engineering Laboratory, Hanover, N.H. under Grant No. DACA89-77-0536. The apparatus used in the investigation, was built and installed by shop personnel of IIHR with their admirable skill. Helpful discussions with Dr. M.A. Larson, Dr. T.E. Osterkamp, Dr. J.R. Glover, Dr. J.C. Tatin- claux and members of the staff of IIHR, CRREL and ETH are sincerely appre­ ciated. TABLE OF CONTENTS

Page

LIST OF TABLES v LIST OF FIGURES vi LIST OF SYMBOLS viii ABSTRACT xi I. INTRODUCTION 1 II. NUCLEATION OF ICE PARTICLES IN SUPERCOOLED WATER 2 A. Critical radius for particle growth 3 B. Nucléation of crystals from melts and 5 supersaturated solutions III. HEAT TRANSFER 6 A. Heat conduction 6 B. Heat convection in turbulent flow 12 IV. EXPERIMENTAL TECHNIQUES 17 A. Experimental setup 17 1. Turbulence jar 17 2. Cooling system 21 3. Temperature measurement 21 4. Velocity measurement 22 5. Schlieren system 25 Experimental methods 25 1. Temperature in the cold room 25 2. Cooling 25 3. Seeding 26 4. Typical procedure of an experiment 26 5. Estimate of the heat balance 28 6. Particle count 30 30 V. RESULTS A. Evaluation of data 30 B. Nucléation 46 1. Initial breeding 46 2. Collision breeding 51 C. Heat Transfer 51 1. Accuracy of measurement 51 2. Particle size 52 3. Turbulent heat transfer 52 D. Turbulent data 56 1. Velocity measurement 56 2. Estimate of the dissipation rate 56 3. Kolmogorov scales of turbulence 58 4. Mean Nusselt number 60 VI. SUGGESTIONS FOR FUTURE RESEARCH 61 A. Basic research problems 61 B. Test of the conclusions ofthe present Study 62 in flume experiments C. Continuation of the present study 63 1. Simulation of natural conditions 63 2. Multiplication process at low supercoolings 63 3. Heat transfer at low turbulence levels 63 4. Velocity measurement 63 VII. CONCLUSIONS 65 LIST OF REFERENCES 66 APPENDIX 1 Input data and results 69 APPENDIX 2 Estimate of the turbulence parameter 82 and the local Nusselt number in the jar APPENDIX 3 Computer program 88 LIST OF TABLES Page No.

4 Table 1 List of physical constants 32 Table 2 List of experiments 52 Table 3 Measurement of mean Nusselt number Nu 57 Table 4 Estimate of dissipation rate and heat transfer from velocity data Appendix 1: Input data and results 70 Table 1 Experiment no. E03 71 Table 2 Experiment no. E05 73 Table 3 Experiment no. E07 74 Table 4 Experiment no. E08 75 Table 5 Experiment no. E09 76 Table 6 Experiment no. Ell 77 Table 7 Experiment no. E12 78 Table 8 Experiment no. E13 79 Table 9 Experiment no. E14 80 Table 10 Experiment no. E15 81 Table 11 Experiment no. E16

Appendix 2: Estimate of turbulence parameter and local Nusselt number in the jar 83 Table 1 Dissipation rate E(z,t) 84 Table 2 Kolmogorov length scale M(z,t) 85 Table 3 Kolmogorov velocity scale u^(z,t) 86 Table 4 Kolmogorov time scale 87 Table 5 Nusselt number Nu (z,t)

v LIST OF FIGURES Page No 9 Figure 1 Transient temperature field around a sphere of temperature Tg brought into contact with a fluid of temperature TL at t = 0 (equation 3.8) Figure 2 Transient heat transfer rate q(R,t/xc) from a sphere of 10 temperature TE brought into contact with a fluid of tempera­ ture Tg at t = 0 (equation 3.11)

Figure 3 Transient heat loss Q(R,t/xc) of a sphere of temperature Tg 11 brought into contact with a fluid of temperature TL at t = 0 (equation 3.12) 15 Figure 4 Estimate of the Nusselt number Nu (tg/xc) based on exchange time te (equation 3.19). (For comparison, the transient heat transfer shown in figure 2 is depicted) 16 Figure 5 Nusselt number Nu (R,te) as a function of exchange time te and of sphere radius R 18 Figure 6 Cross sectional view of turbulence jar 19 Figure 7 Grid to generate turbulence and ice stick to seed the flow 19 Figure 8 Overall view of experimental set-up 20 Figure 9 Circulating pump and piping 23 Figure 10 Schlieren system 24 Figure 11 Light source of schlieren system 27 Figure 12 Measurement of heat gain PG (figure 12.1) and total heat gain PT (figure 12.2). (P-g is the sum of Pg and the power PG intro­ duced by the stirring motion of the grid (equation 4.5)) 29 Figure 13 Measurement of the time constant of residual cooling 33 Figure 14 Figures 14.03 to 14.16 present the experimental results. The last two digits of the figure number denote the number of the experiment

Nomenclature: TT the total increase of temperature T-g Ti the temperature increase Tg due to latent heat of fusion PD the power of density Pp due to latent heat of fusion DN the particle density DN TIME time t after start of the experiment

Figure 15 Series of photographs showing development of frazil ice. t is the time after seeding of the supercooled water. The disc crystals are best visible on figure 15.8. Figure 16 Power density Pp and particle density as functions of time 48 and initial supercooling TgQ (figure 16.1: Re = 1500, figure 16.2: Re = 2300) Figure 17 For the same initial supercooling TL0 the particle density 50 increases with increasing Reynolds-number Reg Figure 18 Probability density D of the Nu number measured in the jar 53 (figure 18.1: Experiments with "equilibrium" cooling; figure 18.2: Experiments with "non equilibrium" cooling) Figure 19 Phase mean u of the vertical velocity and rms of turbulent 54 fluctuations u' normalized by maximum velocity UQ of the grid. Curve A is measured at a point behind the edge of the grid rod, curve B behind the center of a grid mesh. (g (t/x0) are mean values used in equation 5.18) (figure 19.1: Re = 1500, figure 19.2: Re = 2300) g g

vn LIST OF SYMBOLS

A Area of the grid dA change in surface area of a crystal 3 C constant in equation = C u' /L heat capacity of water CL d diameter of grid rods D probability density D, particle density N £ frequency of the eccentric g(t/T0) normalized velocity fluctuations at z=0 G free enthalpy spatial frequency kn K number of periods for phase mean L latent heat of fusion turbulent length macroscale m number of particles (counted on photograph) NC number of particles in volume agitated by grid n e n t number of particles in jar Nu Nusselt-number Nu measured mean Nusselt-number Nusselt-number estimated from velocity data Num Ap pressure drop across grid P power due to latent heat of fusion heat gain by conduction PC power density due to latent heat of fusion PD power introduced by the grid power in equation 6.1 H P interpolated for the time of particle count PIN mean heat transfer per particle PN residual cooling of heat exchanger PR total heat gain PT Pe Peclet-number Pr Prandtl-number q(R,t) heat flux at sphere surface heat flux per unit area % Q(R,t) heat loss of a sphere water discharge Ql Qs ice discharge r radial coordinate R radius of particle critical radius of particle Rcrit. mean radius of particles radius of outer sphere R1 Re Reynolds-number based on grid-rod diameter g Re Reynolds-number based on particle diameter P s fluid shear S' entropy AS entropy change per unit mass for melting t time At time step t exchange time e T temperature temperature increase due to other source than latent heat of fusion surface temperature of a growing ice crystal temperature increase due to latent heat of fusion T interpolated for the time of particle count water temperature water temperature at t=0 temperature of ice point air temperature in cold room total increase of water temperature

u vertical flow velocity in jar u mean flow velocity u' rms of flow velocity fluctuations Kolmogorov microscale of velocity \ U(t) velocity of grid at time t amplitude of grid velocity ice volume in jar

IX mean ice volume of particles water volume in jar ice volume z(t) location of grid at time t Z amplitude of grid motion a coefficient in equation 4.4 e dissipation rate e mean dissipation rate r\ Kolmogorov microscale of length X thermal conductivity- y chemical potential Ay change in chemical potential y chemical potential of water L y chemical potential of ice s y o viscosity V kinematic viscosity density of water density of ice s a surface tension at ice-water interface standard deviation of Nu time scale of heat conduction Kolmogorov microscale of time Tk period of the excentric time constant of residual cooling radial frequency of the eccentric

x ABSTRACT

Turbulence affects frazil ice formation in three ways: - it prevents stratification and formation of surface ice - it is involved in production and transport of ice nuclei - it controls the transfer of the latent heat of fusion from the growing ice particle to the supercooled water. To study ice nucléation and heat transfer, an experiment was conceived which allowed frazil ice to be produced under controlled condi­ tions. Turbulence was generated by a moving grid in a turbulence jar, where water could be cooled below the freezing point. Frazil was observed and photographed by means of a schlieren optics. Several experiments were conducted for super-coolings between -#Q5°C and -.30°C and two velocities corresponding to Re = 1500, 2300 based on the diameter of the grid rod. The rate of ice production and the total ice mass was calculated from the in­ crease of temperature. The number of ice particles were counted on the pho­ tographs. No frazil ice, regardless of turbulence and foreign material, was observed unless the water was seeded with ice nuclei. The number of particles grew during the experiment; the growth rate increased with greater supercooling and higher velocity of the grid. This indicates a multiplica­ tion process induced by secondary nucléation. The heat transfer per particle normalized with supercooling and the size of the particles was constant in all experiments within the accuracy of measurement. From these observations, it can be concluded that the total ice production is predictable if the heat transfer per particle can be estimated from turbulence data and if the number of particles can be calculated. A nu­ cléation theory is, however, not available and is regarded as the crucial question.

xi I. INTRODUCTION

In the realm of ice engineering many of the key problems re­ main unsolved. Among these, none is more prominent than the development of reliable predictors relating the occurrence and rate of formation of frazil ice in turbulent flows to water temperature, rate of cooling and flow properties. In the absence of reliable engineering tools for making an accurate determination of the flow conditions under which frazil will form and of the amount of frazil to be expected, designers are compelled to adopt expensive measures (in many instances unnecessary) for controlling or sup­ pressing frazil in structures which must pass supercooled water: notably intakes for power plants, domestic water supplies and emergency core cooling systems of nuclear reactors. Frazil ice forms in water of a river or a lake which is cooled to below the freezing point, but which is too well mixed for a surface ice- sheet to develop. In these flows, the ice occurs initially not as a monolith, but as small discs dispersed throughout the fluid. These particles often aggregate to form a bulky mixture of liquid water and ice which can produce ice jams in river channels and block the inlets to water-diversion struc­ tures. Osterkamp (1977) gives an excellent up-to-date review of re­ search on frazil ice formation. Michel (1963) studied frazil ice formation in an outdoor flume and measured the variation of the water temperature during supercooling and ice formation. Coldroom experiments are reported by Carstens (1966), Schmidt (1975) and Hanley (1977). Carstens (1966) observed a strong influence of turbulence induced by the pump and gives a heat bal­ ance equation based on heat transfer considerations. Schmidt (1975) measured the density of the frazil particles by means of modified Laser- Doppler-Anemometer. Hanley (1977) correlated water temperature and the mass of frazil slush. The objective of the research reported here was to study the role of turbulence in the initial stage of frazil ice formation. It turns out that the problem requires an inter-disciplinary approach. Results of research in chemical engineering, ice physics and fluid mechanics can be

1 2

transferred to get a better understanding of frazil ice formation. Turbulence affects frazil ice formation in three ways. 1. Turbulent mixing inhibits the stratification of the flow and prevents the formation of a surface ice sheet. The whole bulk of water can be supercooled which is a necessary condition for frazil ice formation. This effect of turbulence is not treated in the present study. 2. Ice crystals can only grow in supercooled water if they ex­ ceed a certain critical size. By collision with the wall and other solid particles or by fluid shear, such crystals are produced in great quantities. Turbulent motion is involved in both of these nucleation mechanisms, as well as in the transport of ice nuclei. The nucleation of ice crystals is dis­ cussed in Chapter II of this report. 3. The latent heat of fusion which is released during frazil ice formation heats the supercooled water back up to 0°C degrees. Turbulent heat transfer exceeds heat conduction in a fluid at rest by an order of mag­ nitude. This topic is discussed in Chapter III. To compare the theoretical concepts with the actual process, an experiment was conceived which allowed frazil ice to be produced under controlled conditions. Turbulence was generated by a moving grid in a tur­ bulence jar where water could be cooled below the freezing point. Frazil ice was observed and photographed by means of a schlieren system.

II. NUCLEATION OF ICE PARTICLES IN SUPERCOOLED WATER

As water can be cooled below the freezing point if no ice is present, a barrier must exist which prevents the formation of ice. The state of supercooled water is therefore metastable. During the formation of a crystal, a new interface must be built up. The energy required is proportional to its area. As the gain of energy due to the phase change is proportional to the volume, the balance of these two forms of energy is negative for a very small crystal, therefore, it can­ not grow and ice formation cannot start. A crystal is only able to grow if it exceeds a certain critical radius. In this chapter the theory describing this effect is summarized and mechanisms are outlined which can nucleate 3

crystals in melts and supersaturated solutions.

A. Critical radius for particle growth When an ice nucleus is growing in supercooled water of temp­

erature T t, the change in free enthalpy dG is 2.1 dG = (y. yL) ps dVs + odA where y and yL are the chemical potentials of the solid and the liquid phase, PgdVg is the mass increase of the ice crystal, a the surface tension of the liquid-solid interface and dA is the change in surface area (see

e.g. Hobbs (1974), Chapter 7). One can get an approximation of Ay = (hg yL) by integrating the relation dAy _ dS' 2.2 dT " dn

from the ice point Tq to the supercooled temperature (see e.g. Van Hook (1961)). Here, T is the temperature and dS'/cln is the change of entropy per unit mass if the mass dn of water is frozen to the crystal. This ap-

proach leads to

L AS’dT r - (T ) (T - T ) 2.3 Ay (T ) - Ay (T ) = - / An An o L o

and 2.4 Ay (Tl) = - AS (To - Tl)

Ay (T ) is zero at the ice point Tq and AS = - AS'/An is the increase of entropy per unit mass for melting of ice. As the first term on the right of equation (2.1) is negative, it represents a gain term of free enthalpy which is proportional to dV. The second term is positive and represents the consumption of energy due to the increase dA in surface area. Only if the gain term exceeds the consumption, the system becomes unstable, i.e. dG < 0 and the particle can grow. Equilibrium exists at a critical radius R where dG = 0. If the ice nucleus is supposed to be spherical, equation crit (2.1) can be rewritten as 2.5 0 = - AS (T - T t ) p 4tt R dR + 8tt oRdR O L ^ and the critical radius becomes -8 0cm 2 o 5.9 10 2.6 R T - T. crit AS (T V ps o L Table 1: List of Physical Constants

Ice Water Density (0°C) 0.917-103 (1) 0.9999 •103 (1) kg 3 m (2) Heat Capacity 2.094 4.2177 CD k Joule kj^C Thermal Conductivity 2.2 (2) 0.565 (1) Watt m°C Index of refraction 1.309/1310 (1) 1.333 (1) X = 5893 % (2) (2) Entropy at 0°C 2.119 3.343 k Joule kg°C (2) Latent Heat of Fusion 333.6 k Joule kg (2) Change of Entropy 1.224 k Joule for Melting of Ice kg°C •3 (2) Surface Energy 33-10" Joule 2 m (1) Handbook of Chemistry and Physics (1975)

(2) Hobbs (1974) 5

for the numerical values given in table 1. Equation (2.6) is the Kelvin equation and states that Rcrit varies inversely proportional to Tq - T L - This theory is valid in an isothermal isobaric system and does not describe the microscopic effects of crystal growth.

B. Nucleation of crystals from melts and supersaturated solutions In the literature three types of nucleation are described (see e.g. Hobbs (1974), Chapter 7, Strickland-Constable (1972)). 1. Homogeneous nucleation occurs if, as a consequence of sta­ tistical fluctuations, a sufficient number of molecules find sufficient order to form a crystal. This type of nucleation is only observed in water cooled below -40°C and it can, therefore, be excluded as a source of frazil

ice. 2. Heterogeneous nucleation occurs when the ice crystal grows on a foreign particle. Most materials which act as ice nuclei are active only at temperatures several degrees below the ice point (Roberts and Hallet (1968), Thijssen (1968)). In the present study no heterogeneous nucleation of frazil ice was observed at supercoolings up to 1 C. 3. Secondary nucleation occurs if for some reason crystals of supercritical size are produced and brought in supercooled melts of super­ saturated solutions. It is the main nucleation process m industrial crys­ tallization and hence of special interest to chemical engineers. They dis­ tinguish four different mechanisms of secondary nucleation. 3.1 Initial breeding where a seed crystal is introduced in a supersaturated solution of a supercooled melt. This may be crystal dust washed off from a large crystal or microscopic crystals fallen into the melt, as Osterkamp (1974) has observed it. 3.2 Needle and polycrystalline breeding occurs at high super­ saturations where crystals grow imperfectly and produce needles or poly­ crystal conglomerates. If the solution is stirred, needles breat off and polycrystals break up to form secondary nuclei. 3.3 Fluid shear on the surface of a large crystal can produce new nuclei, as observed by Sung (1973) and Estrin (1975). The shear s used in their experiments was of the order of 3000 sec 6

3.4 Collision breeding is thought to be the dominant process of secondary nucleation. If a crystal collides with a solid, a great number 4 of nuclei is produced. Measurements of several 10 nuclei per collision are reported by Bauer (1974). The size of these nuclei is of the order of 30 ym. Also sliding of a crystal along a wall gives rise to nuclei (Gar- abedian, 1972). The process by which collision breeding acts is still under discussion. Solid parts may be broken off from the crystal, or a partially ordered boundary layer is dislodged from the crystal surface. See also: Huige (1972), Youngquist (1972), Garabedian (1974), Larson (1976). Consequences for the nucleation of frazil ice will be discussed in section VB.

III. HEAT TRANSFER

While frazil ice is formed in supercooled water, latent heat of fusion is released and raises the water temperature. Frazil ice forms at supercooling AT of less than 0.1°C and the water cannot be heated above 0°C without stopping ice production. From the heat balance »5VS L = “ T 3 a a lower limit of the ratio of the volume Vg of ice and the volume of the heated water can be estimated by asstiming the temperature of the volume VL is raised uniformly by AT. L is latent heat of fusion and C the^spe-

cific heat of the water. For a supercooling AT = 0.1°C, the ratio tt- is S equal to V b„L 's L 725 3.2 V p tc a t S L i This large ratio indicates that heat transfer must be one of the processes controlling the growth of the ice particles. In the following, an attempt is made to estimate the heat trans­

fer by conduction and convection.

A. Heat Conduction In a fluid at rest, heat transfer occurs only by heat conduction. 7

The temeprature field T(r,t) around an ice particle is a solution of the dif fusion equation 3T = JL u 2t 3,3 3t pC

and the heat flux per unit area qQ is equal to q = -XVT 3,4

If a particle is growing and heat transfer controls the growth, the surface of the particle is essentially at the equilibrium temperature Tg, the temperature at which the latent heat of fusion is released. The temp­ erature for large r reaches the temperature TL of the supercooled water. The only length scale given is a length R characterizing the size of the narticle. The corresponding time scale is p£ _ 3.5 Tc X

Obviously, eq. 3.3 applies only if R is large compared with the

scale of the Brownian motion. The stationary temperature field around a circular ice disc was studied by Fujioka (1974), who was interested up to what diameter

the disc shape was stable. Here, a spherical symmetry is assumed to get an analytical form for the stationary as well as the transient solution. These solutions are considered, to be sufficiently accurate to get estimates of heat transfer, even if the actual shape of the particle is neglected. The stationary solu­

tion is given by (x - T ) R 3.6 T(r) = ---E + Tt

where R is now the radius of the spherical particle. The corresponding

heat flux at the surface of the particle is

q(R) = _ 4itR2X f| CR) = 4ttR(Te - Tl )x 3’7

Because the ice nucleus is growing, the radius R is not a con­ stant, as assumed in the stationary solution. In order to estimate the heat loss due to transient, an extreme case can be calculated. Consider that a sphere of radius R, the surface temperature of which is kept constant at Tg, is immersed in supercooled water of temperature TL at t = 0. The spherical 8

solution for the temperature field around the sphere can be represented as a Fourier series in a domain R < r < R^, where R^ >> R (see e.g. Carslaw (1959)). T \ ( R1 fR R , - 2 Sinkn (r ' R) T(r,t) = Tl + (TE V j R -R (r " R J ‘n=l rnr r/R 3.8 exp [-»■*>' y; The solution is found by separating the space and time variables in equa­ tion 3.3 and developing the deviation of theinitial condition from the stationary solution in a series of functions sin(kn (r - R)) T (k ) = (t exp -(k„R) 3.9 n v nJ E r/R •y , _ nn n R1 - R

These functions form a complete set of orthogonal functions in the interval 2 R < r < Rl with a weighting function r . Equation 3.8 fulfills the boundary conditions

T(R,t) = Te T(Rr t) = Tl and the initial condition 3.10 IT, r = R T(r,0) = R < r ;S Rj

The first term represents the stationary solution, the second term the transient. From the temperature field, the heat flux q(R,t) and the inte­ grated heat loss Q(R,t) at the surface of the particle can be calculated and represented in dimensionless form

2 • q (R ,t) exp 4ttR (Te - Tl) • x -»„W2 7 ]| 3.11 2Q(R,t) 2 Q(R, t)

4"R(TE - V x Tc R3 p c c t e - i p )

2 f 2R i r 3.12 n=l Rx - R (knR)2 \_l Transient temperature field around a sphere of temperature Tp which is brought into Figure 1. contact with a fluid of temperature T L at t - 0 (equation 3.8)

<£> +

o

12

The first terms in equations 3.11 and 3.12 represent the constant heat flux, and the linearly increasing heat loss, respectively, predicted by the stationary solution. The contribution of the transient to the heat flux is infinite at t = 0 and damps out as t increases. The contribution of the transient to the heat loss converges toward a constant value, i.e. the energy which is needed to heat the fluid around the particle from TL to the temperature predicted by the stationary solution. Figure 1 shows the solution for T(r,t) (equation 3.8) for various values of the Fourier number t/x^. The calculations were performed for - 20R. From the heat flux q(R,t) shown in figure 2, it can be seen that the steady state value is reached between 1 and 10 time steps x . In figure 3, the integrated heat loss Q(R,t) is compared with the heat loss due to the

stationary term alone. 4ir 3 The total heat that is released by a sphere of ice is - j - R psL

which corresponds to a value of

2 Q(R,t) 483 3.13 3 4^- R3pCAT

if the supercooling AT is assumed to be 0.1 C. This heat loss is reached at a time of 110 x which is long compared with the time needed to reach the steady state heat transfer at the surface. It is, therefore, concluded that the steady state solution is a sufficient approximation for the heat con­ duction from a growing ice crystal to a fluid at rest.

B. Heat convection in turbulent flow In a turbulent flow, heat is also transported away from the ice particle by fluid motion. The heat conduction is still proportional to the local temperature gradient, but the temperature field itself is altered by mixing. The blob of warmer fluid around the particle which is described by T(r,t/x ) in figure 1, is constantly removed, leading to temperature pro­ files corresponding to much shorter times t/xFor such times, figure 2

shows a much larger heat transfer. This turbulent heat exchange between the fluid and the small ice particles depends on the ratio of the microscale of turbulence to the size 13

of the particle. If the microscale is large compared to the dimensions of the particle, the ice nuclei are convected with the fluid without sensing the turbulence. The relative motion and, therefore, the heat convection are small. As the microscale of turbulence decreases, the rate of shear and the motion of the particle relative to the fluid will increase. This way, the heated blob around the particle gets more and more eroded. The microscales are defined within the theory of turbulence, see e.g. Tennekes and Lumley (1972). It is assumed that turbulent energy is transferred from large length scales, where it is produced, to smaller and smaller scales until the gradients become so large that the turbulent energy is dissipated by viscous forces. The microscales are, therefore, determined by the viscosity v and the rate of energy dissipation e. With these two parameters, one can form a length scale n, a velocity scale u^ and a time scale x k* 1/2 1/4 = (ve) 3.14 TÌ ■ ( 4 f . Tk = These scales are referred to as the Kolmogorov microscale of length, time and velocity. It follows from the concept of the energy cascade that the energy dissipation e at small scales is equal to the energy input at large scales. This energy input is of the order of energy u'2 of the large eddies divided by the time scale L/u' where u' and L are the macroscales of velocity and

length. -z u' 3.15 e = C

C is a constant which should be of order one. The question, how convective heat and mass transfer can be de­ scribed, how the flow around particles, drops or bubbles can be modelled, is treated in numerous papers. Acrivùs (1962) calculated heat transfer based on Stokes flow, Lee (1968) used a boundary layer solution around a particle in a parallel stream. Pearson (1968) measured the rate of mass exchange around a drop in a parallel flow. Turbulent exchange models based on eddy concepts were used by Lamont (1970) and Theofanous (1976) to describe gas absorption at gas liquid interfaces which are large compared with the turbulent 14

scales. Schwartzberg (1968) introduced a slip velocity to describe the ex­ change in a stirred tank. Frankel (1968) studied heat and mass transfer of small spheres in shear flow, but he could not give an exact solution of the exchange for the limiting case of large shear. As long as a frazil ice particle is not much larger than the Kol­ mogorov length scale, it is convected with the flow and will sense the turbulent motion as a variable shear. From this concept, there follows a simple model which can give a rough estimate of the increase in heat transfer. If a spherical particle of radius R is exposed to a flow of shear s, the maximum velocity vM of the fluid at the vortex of the sphere relative to its center is v,. = R • S 3.16 M

The time it takes for the fluid to move one particle diameter is a measure for the exchange time t it takes to exchange the fluid around the particle.

If s is given by the microscales as

3.18

the exchange time becomes

3.19

and is independent of R. The exchange of the fluid around the particle during t has the effect that the transient part of the heat flux equation 3.11 becomes domi­ nant. To model this exchange based on the concept of an exchange time, one can assume that after each time step t the transient process described by equation 3.8 starts again at t = 0. The heat flux q is then approximated by

q (R,t /t ) = 3.20

Fig. 4 shows values of q(t /x ) normalized with one half the heat flux of the stationary solution corresponding to usual practice. This value cor­ responds to a Nusselt number Nu of F 2q(R,t /t c) 3.21 NU W = ---q(R,«) Figure 4. Estimate of the Nusselt number Nu (t /t ) based on an ex­ change time te (equation 3.19). (For comparison, the tran­ sient heat transfer shown in figure 2 is depicted) Figure 5. Nusselt number Nu (R^t ^) as a function of the exchange time and of the radius R of the sphere

O' 17

It follows from equations 3.5 and 3.17 that the ratio te/tc is proportional to the inverse of a Peclet number Pe

Pe = — = I -Ü4- = lr Pr • Re 3.22 pe t 2 e yQ/p 2 P for shear flow. The Reynolds number Repof the particle is

R r 2s 3.23 p v For s = 1300 sec"1 and R = 0.03 cm, values as they are observed in the experiment, Rep is equal to 65. The Prandtl number Pr

Pr = V 3.24

is constant as long as the temperature of the water is not varied. The Nusselt number Nu is, therefore, only a function of the particle Reynolds number, i.e. it depends on the size of the particle and the turbulent shear.

It converges to 2 for large values of t^/t or Re + 0- Fig- 5 shows the de­ pendence of Nu on R and x^. In the model presented here, the boundary layer around the par­ ticle is neglected and for this reason it is expected to overestimate the

heat transfer, (Frankel (1968)). In the initial stage of frazil ice formation, the particles do not interfere, so that the total heat transfer is equal to the particle heat trans­ fer multiplied by the number of particles.

IV. EXPERIMENTAL TECHNIQUES

A. Experimental Setup The experiment designed to produce frazil ice under controlled con­ ditions was installed in the cold room of the Iowa Low Temperature Flow Facili­ ty. It allowed visual observation of the ice formation by means of a schlieren system, velocity measurements by Laser-Doppler-Anemometry and control of heat

transfer processes. 1. Turbulence jar. Figure 6 shows a cross sectional view of the

jar. The inside dimensions were 17.2 x 12 cm and the depth was 20 cm. The 10 cm I------—— I

Figure 6. Cross sectional view of the turbulence jar Figure 7. Grid to generate turbulence and ice stick to seed the flow

Figure 8. Overall view of the experimental set-up 20

Figure 9. Circulating pump and piping 21

bottom and the smaller side walls were constructed of .32 cm thick stainless steel plates, and .95 cm thick glass walls formed the other side walls. Turbulence was generated by a grid of .62 cm diameter plastic rods spaced 2.54 cm apart (fig. 7). The grid was connected to an overhead eccentric with variable speed control capable of achieving 1/2 to 2 Hertz oscillation of the grid in the vessel. The eccentric and turbulence jar were mounted on a common frame. Figure 8 shows an overall view of the ap­ paratus in place. The amplitude Z of the grid motion was 7 cm, leaving a- bout 2 cm freedom at both bottom and top, when water was placed in the ves­ sel. The water depth was usually maintained between 18 and 18.5 cm. 2. Cooling System. The cooling liquid was supplied to the jar from the large cooling tank of the cold room. Temperature control of the coolant was accurate to about +_ 0.50°C. The lower parts of the stainless steel walls and the bottom of the jar were incorporated into a heat ex­ changer, where coolant could be circulated by a pump. Cooling of the lower parts of the jar produced an unstable stratification of the water below 4 C and the water was easily mixed by moving the grid at a low speed. Figure 9 shows the circulating pump and the piping. A diffuser chamber of approxi­ mately .5 liters was located down-stream of the pump and three plastic tubing lines of 16 mm I.D. connected the diffuser and the three walls, one each per wall. Each cooler wall had a single return line as well as one air vent for purging the system. Two valves on the suction side of the pump allowed either to supply coolant from the cooling tank or to recirculate the coolant

already in the circuit. 3. Temperature Measurement. Measurement of temperature of cool­ ant, water and room air was accomplished with linear type thermistors (manu­ factured by Yellow Springs Instrument, YSI #44018) with a known temperature dependence. The absolute values of the air and coolant thermistors were call brated in an ice bath to an accuracy of +_ 0.05°C, while the water thermistor was kept within +_ 0.02°C. Temperature variations in the air and in the water

could be measured to *_ 0.02, and 0.002°C, respectively. The time constant of the water thermistor was of the order of 10 sec. The coolant thermistor was mounted in the diffuser chamber, while the water thermistor was attached to the grid in the turbulence jar, the 22

sensor being nearly in line with the plastic rods. Consequently, the water temperature measurement was an integrated profile over the entire distance traversed by the grid as it moved inside the jar. The room temperature was measured near the turbulence jar, and its variation was approximately +_ 1°C for a given temperature setting, because of lag times in the chiller units at each end of the room. As previously mentioned, the coolant temperature also varied from a given setting by + 0.5°C. 4. Velocity Measurements. To measure flow velocities u(t) in the turbulence jar, a Laser-Doppler-Anemometer was used. The TSI optics (Thermo Systems, Inc.) was equipped with a double Bragg cell from OEI (Op- tische Elektronische Instrumente), to shift the frequency of the incoming beams. A TSI tracker demodulated the signal. The output signal which was proportional to the instantaneous vertical velocity, passed a low pass filter set at 200 Hz before it was digitized by the IBM 1800 computer of the IIHR. As the turbulence producing grid was driven by an eccentric, the velocities were periodic with the period tq of the eccentric. A magnetic transducer was used to give a synchronization signal when the grid reached the maximum velocity. Programs were available to calculate phase means of the mean flow velocity u and the rms of the turbulent fluctuations u'

1 K_1 u(n • At) = Y Z u(i t + n At), n = 1...96 4.1 v o K i=0 K-l ,|!/2 u' (nAt) = ||- Z^ [u (i tq + nAt) - u^(nAt) n=l...96 4.2

The time step At was chosen to be .005 x K was equal to 100 and n o n max was equal to 96. This way, a phase mean over 100 half periods was measured. The measurements were taken in the plane of maximum velocity. Two measurement points were selected, one at the edge of a grid rod, the other at the center of a grid mesh. The Re number is based on the velocity ampli- S tude UQ of the grid and the diameter d of the grid rod

d U Re 4.3 T = 0°C Lamp Condenser Camera Lens 60 cm F. L.

------1---- i *------1 76 cm 285 cm

Figure 10. Schlieren system

NJ O-i 24

Figure 11. Light source of the schlieren system 25

Two values of velocity were chosen Uq = .66 m/sec and Uq = .44 m/sec, corresponding to Re = 2300 and 1500 respectively. The corresponding fre­ quencies of the eccentric were 1.5 Hz and 1 Hz. 5. Schlieren System. A schlieren system was set up to observe frazil ice formation visually. Photographs were taken to determine the number of ice particles. The amount of light scattered by frazil ice discs depends strongly on their orientation. As the index of refraction for ice is about 2% below the index of refraction for water, the discs can reflect all the light by total reflection at low angles of incidence. If the discs are oriented perpendic­ ular to the illuminating beam, only .01% of the light is reflected from the plane surface and the discs are hardly visible. A schlieren system which is able to detect small changes in the index of refraction, is well suited to observe the frazil particles. Figure 10 shows a sketch of the set-up. A projector lamp (General Electric G-E DLR, 21.5 V, 250 Watts) is driven by a Variac to give an adjustable light output. A condenser fo­ cuses the light on a beam stop which acts as the light source to illuminate a spherical mirror with 40 cm focal length. This mirror forms the collimated beam to illuminate the turbulence jar (fig. 11). On the receiving side, a lens (diameter 15 cm, focal length 60 cm) focuses the collimated beam onto the knive edge. The image of the scattered light is found in the image plane of the turbulence jar. Photographs were taken by placing the camera without objective into the image plane, or by means of an opaque screen. The ex­ posure time was 0.001 sec with the power of the lamp reduced to 90 Watts.

B. Experimental Methods 1. Temperature in the Cold Room. During all the experiments, the room temperature was kept above 0°C to melt ice crystals blown out of the air cooling system. In this way, it was assured that the supercooled water was not seeded by such nuclei. 2. Cooling. The temperature of the coolant was set at -2.5°C. Two procedures were followed to supercool the water: a) The supply of coolant from the cooling tank was stopped when the temperature of the water was about 0.7°C above the desired supercooling 26

and the coolant remaining in the system was recirculated. Due to friction and heat loss to the water, the temperature of the coolant increased. The pump was stopped the moment the temperatures of the coolant and of the water were equal and the experiment was started. The state achieved in such a way was called "equilibrium condition". b) When the temperature of the water was approximately 0.5°C above the desired supercooling, the pump was stopped. The water temperature still decreased until there was an equilibrium between heat loss to the coolant re­ maining in the heat exchanger and the heat gain from the room. The experi­ ment was started when the minimum temperature was reached. As there was still a heat flow to the coolant, this was called the "non equilibrium condition". Neither of these methods allowed an exactly repeatable setting of the final supercooling. By trial and error, an interval of supercooling be­ tween -0.3°C and -0.07°C was covered in the series of experiments. 3. Seeding. No frazil ice was observed down to temperatures of _1°C without seeding the supercooled water with ice nuclei. The water was seeded with a cylindrical piece of ice frozen to a wooden stick (fig. 7). At the beginning of each experiment, the plastic rods of the grid were touched several times with this ice stick which was removed afterwards. Only in experiment #03, was the ice stick hit continuously against the grid. No special care was taken to keep the ice surface free of dust. 4. Typical Procedure of an Experiment. After cooling and seeding as described in the previous sections, the water was agitated by the moving grid. To reduce the relative humidity in the Cold Room, the chiller units were turned on, so that there was no condensation on the glass walls of the jar. The temperature of the water was recorded continuously and the room temperature was taken at regular intervals and linearly interpolated. Series of photographs were taken from the schlieren image during frazil ice for­ mation. The first ice particles were visible after 5 to 20 seconds and the number of particles then increased depending on supercooling and speed of the grid. In experiments with "non equilibrium" cooling, some ice was forming on the heat exchanger. The temperature of the water increased due to the latent heat of fusion and heat gain from the room. The experiment was stopped when the water temperature reached the ice point. 27

Figure 12. Measurement of the heat gain Pc (figure 12.1) and the total heat gain PT (figure 12.2). PT is the sum of Pc and the power PG introduced by the stirring motion of the grid (equation 4.5) 28

After each experiment, the water was heated up to about .5°C to make sure that all the ice was melted. 5. Estimate of the Heat Balance. An attempt was made to esti­ mate the heat balance of the supercooled water, in order to determine the total mass of ice in the jar. Apart from the latent heat of fusion, the ' water was heated by heat conduction through the walls and by viscous dissi­ pation of turbulent energy introduced by the grid. When the water was cooled by the "non equilibrium" method, there was also some residual cooling through the heat exchanger. It was assumed that the heat gain by conduction P is proportional V-l to the temperature difference T - T. of the room and water. It can be K L measured while the grid is moved at a low speed.

pc = a (tr - tl> 4-4

If the grid is moved with frequencies f = 1 Hz and 1.5 Hz, a constant power P is added. The total heat gain P„ is, therefore, b I

PT = PG (f) + a (TR - Tl) 4.5

Fig. 12 shows measured values of P_ and P in function of T - T. . Values b I K L of a and P^ (f) were calculated using least square approximation, a = 0.68 Watt/°C P„ (1 Hz) = .09 Watt + 0.21 b — Pn (1.5 Hz) = 1.38 Watt + 0.25 b — The residual cooling P in the nnon equilibrium” condition was assumed to K decrease exponentially with time

PR = PR (t = 0) • exp (-t/Tr) 4.6

The time constant was determined from the exponential rate of change of the water temperature after the pump was stopped (fig. 13). The mean value for the three runs was 240 secs. As the temperature T t reached a minimum at t = 0, the constant P (t = 0) can be assumed to be equal to the R heat gain P^ (equation 4.5) at this moment. K) KO 30

Radiation and condensation were not included in this analysis. The gain of radiant energy is of the order of .1 Watt if a temperature difference 2 of 3°C and a radiant surface of 100 cm is assumed. The effect of conden­ sation cannot be estimated, as the relative humidity was not measured. It was felt that the humidity decreased markedly when the chiller units of the room were cooling. It is possible that condensation can explain the scatter of the measurements shown in fig. 12. 6. Particle Count. On the pictures, particles were counted in a square corresponding to 14 cm 2 on the glass walls. The accuracy of this method is limited for two reasons: Despite filtering the water before cooling, there were dust particles in the water which could not be distinguished from ice particles. The depth of field of the lens image was marginal, so that small particles off the focal plane did not give a visible trace. Whenever the particle density became so high that the images superimposed, a count was no longer possible.

V. RESULTS

A. Evaluation of Data The results of 12 experiments, listed in table 2, are summarized in the tables of appendix 1 and depicted in figures 14.03 to 14.16. The tables list as input data T. the supercooling of the water L i T the room temperature R Nr the number of particles counted on the photographs as a Li function of time after starting the experiment. Interpolated values of TD are underlined. K As results, the following values were calculated: T^ the total increase of temperature T„ the temperature increase due to other sources than the latent Li heat of fusion, as approximated by equations 4.4 to 4.7

5.1 ipr * V dt Tc pLtV hL rL Io 31

Tj the temperature increase due to the latent heat of fusion

5.2 Ti ■ tt - Tc

Tj is proportional to the volume Vj of ice

PLVLCL ;Jl m 51e4 cm3 5.3 vi - Ti

P the power due to the latent heat of fusion which is cal­ culated from the rate of change of „ dTI Watt sec . dTI c; 4 p = plv lc l a r = 15700 — ^c— -jr the power density corresponding to P(= P/water volume) As output data related to the number of particles, the following values are given: X Tj interpolated for the time of the particle count

Dn the particle density N the number of particles in the volume agitated by the grid P interpolated for the time of the particle count IN PIN the power per particle P^ = -jq— 5.5 PN the mean voltane of ice particle M 5.6 V. / N , (N total number of particles) M 'I ' ”T * ^ T the mean radius of a particle 3V_m \1/3 5.7 "m 4ir Nu the mean NusseIt number 2 P, N 5.8 Nu = 4itPm Ct e - t l > x

Nu. an error of Nu is estimated on the basis of equation 5.9 which shows the functional dependence of Nu on the measured values. P/Nc 5.9 N = const u (te - V ANu The relative error is, therefore, Nu Table 2: List of Experiments

Experiment No. t l (t = 0) Re Cooling

E03 -.127 2300 non equilibrium

E05 -.137 2300 non equilibrium

E06 -.108 1500 non equilibrium

E07 -.288 1500 equilibrium

E08 -.185 1500 non equilibrium

E09 -.142 1500 non equilibrium

Ell -.168 1500 equilibrium

E12 -.092 1500 equilibrium

E13 -.148 2300 equilibrium

E14 -.067 2300 equilibrium

E15 -.105 2300 equilibrium

E16 -.099 2300 equilibrium Figure 14. Figures 14.03 to 14.16 present the experimental results. The last two digits of the figure number denote the number of the experiment

Nomenclature

TT total increase in temperature TT Tl temperature increases Tj due to latent heat of fusion

PD Power density Pq due to latent heat of fusion DN Particle density TIME time t after start of experiment co ®1 TT io- TI

r» CM

E03

Figure 14.03. Re 2300, "non equilibrium" cooling g DN CCM^-35 PD CKWATT/ÌT33 CO u>- ©n m 0.0 0.8 1.6 2.4 3.2 4.0 0.0 , 4.0 , 8.012.0 „B.B . 0.1 .0’ ¿. *0T l' *I00T0 ¿0.0 ’* 0 ‘. E05 .0'* .0'* iue 40. e 2300,equilibrium" ="noncooling Re 14.05.Figure ¿aÀ TT • a I T *0' (s.o 0.0 5.0 0. ¿ ¿08.0 ¿50.‘0 ¿00.‘0 (‘so.'o 1*00.'0

50 ‘ ¿00 5. ¿00 ^ ¿00.0 ¿50.0 ¿00.0 .‘0 IE CSEO TIME IE CSEO TIME 50 50.0 .'0 ‘tae.'e ‘tae.'e .'0 ^ 00 .'0 450.0 4 s 0.0 3 5 36

» © 1 • TT a TX

0.0 ¿0.0 1i00B.'0 |TS0.'0 200.0 250.0 ¿00.0 350.0 ^00.0 450.0 TIME C S E O

G>

E06

Figure 14.06. Re 1500, !fnon equilibrium*1 cooling g 37 to

E07

Figure 14.07. Re 1500, "equilibrium" cooling g 38

(0

r \ N

E08

Figure 14,08. Re 1500, "non equilibrium" cooling o 39

»

E09 Figure 14.09. Re = 1500, "non equilibrium" cooling 40

co

A CSI

El 1

Figure 14.11. Re 1500, "equilibrium" cooling g co

o

cs>

E12

Figure 14.12. Re = 1500, "equilibrium" cooling 42 m

E13

Figure 14.13. Re 2300, "equilibrium" cooling g 43

(0 • TT * TI

©

E14

2300, "equilibrium" cooling Figure 14.14. Re6 44

A 01

E15

Figure 14.15. Re 2300, "equilibrium" cooling g 45

«

E16

Figure 14.16. Re 2300, "equilibrium" cooling g 46

AT ANu II 2 ANC 5.10 Nu 3 N„ II

The contribution of each term is based on the following consideration. AP the largest error of the power due to ice production is given by the inaccurate estimate of the other terms in the heat balance equation. Figure 12 indicates that an error^of

1.2 Watts is adequate. This corresponds to an error in — ¿ r •5r- O ^ i & L of 7 • 6 10 x.e. an error of 0.003°C in AT^ for At sec = 40 sec.

A(Tt T 1 This error is equivalent to the absolute accuracy of T^ equal EJ to + _ .02°C. AT If the heat balance is accurate to AP = 2 Watts, the error of II the temperature increase due to ice production is proportional to the time integral of AP

1.2 Watt dt = 7•6•10 •t 5.11 4T,rII - J / 15700 Watt sec o ------t t z —

ANr The error of the particle count is assumed to be Æ 7 Li L» The figures 14.03 to 14.16 show T^, T^, P^ and as a function of time. Appendix 3 contains a listing of the program which calculates the results and gives the details of the numerical methods. Figures 15.1 to 15.10 show the sequence of photographs taken during experiment 16. The ice particles grow as disc crystals. At lower tempera­ tures, the discs became unstable (Arakawa, (1954), Sekerka, (1958)) and de­ veloped into star-like crystals.

B. Nucleation 1. Initial Breeding. No ice formation was observed down to super­ coolings of 1°C regardless of turbulence or foreign particles. This finding suggests that there must be a kind of initial breeding to produce frazil ice at supercoolings of less than .1°C. As frazil ice occurs in nature only in situations where the air temperature is well below the freezing point, there are always sources of ice available which can act as ice nuclei. Osterkamp 47

15.1: t = 5 sec

15.7: t = 150 sec 15.8: t = 180 sec 15.9: t = 240 sec

5 cm f *

Figure 15. Series of photographs showing the develop­ ment of frazil ice. t is the time after seeding of the supercooled water. The disc crystals are 15.10: t = 380 sec best visible on figure 15.8. DN CChT-33 PD CKWATT/ÌT35 8.0 0.0 1.0 2.4 3.2 <4.0 _0.0 <4.0 0.0 12.0 Figure 16. The power density PD and the particle density DN density particle the and PD density power The 16. Figure e 10, iue 62 R 20) LU 16.1: 2300) = (figure Re _ T 16.2: figure supercooling 1500, = Re initial the on depend F gr 16.1 igure &>

Figure 16.2

Figure 16. The power density PD and the particle density Dm depend on the initial supercooling Ty} (figure 16.1: Re = 1500, figure 16.2: Re = 2300) DN CChT-3? 0.8 0.0 1.6 2.4 3.2 4.0 iue 7 Frte aeiiil uecoigT the supercooling sameinitial FortheT 17.Figure particle density increasesincreasingdensitywith particle Reynolds-numberRe

5 0 51

(1974) observed airborne crystals at the water air interface. Snow and hoar­ frost can be blown into the water (Osterkamp, 1976), or floating ice can pro­ duce nuclei by collision breeding. Special care has to be taken in labora­ tory experiments where ice crystals may be blown out of the chiller units of a Cold Room and seed the flow. In all experiments, the particle density was in the range of 0.1 3 to 0.3 particle/cm after the experiments had run for 10 to 30 seconds. This indicates that the seeding with the ice stick provided about the same initial conditions. The only exception is experiment: #03, where the ice stick was hit continuously and produced more nuclei. 2. Collision Breeding. The particle density normally increased during the experiment. Therefore, there must be a multiplication process which creates new nuclei. There is no direct evidence from the experiment on the cause of nucleation. Ice particles may collide with the grid or undergo collisions among themselves. The increase of the particle density as depicted in fig. 16.1, 2 is showing a dependence on the initial supercooling for both Re. No multiplication was observed in experiments #6, 12, and 14. The multiplication also depends on the velocity of the grid, as shown in fig. 17. Preliminary experiments with a lower velocity (Re = 750) showed that no multiplication took place. It seems that the multiplication starts only after the particles have grown for some time (experiments 14-16). No information can be gained from the experiment, as to whether a multiplication process also takes place in nature. It is, therefore, essen­ tial to get more knowledge on the origin of nuclei in natural streams.

C. Heat Transfer 1. Accuracy of measurements. An interpretation of the heat trans­ fer data must take into account the limited accuracy of measurements. The increase of temperature T^ due to ice production is of the order of only several millidegrees in 30 seconds, while the accuracy of meas­ urement is at most +_ 2 millidegrees. This leads to an error of more than 100% if the ice production is low, as in experiments nos. 6, 12, and 14. More­ over, T^, i.e. the heat gain from other sources, can dominate T^ by a factor of two to four, as in experiments nos. 14 and 15. The values of T^, are only 52

estimated by equations 4.4 to 4.7 which further reduces the accuracy. 2. Particle Size. The particle size distribution could not be measured from the photographs. The only information on the particle size is a mean volume which is equal to the total ice volume divided by the total number 3 of particles in the 3720 cm of water. The values listed for the mean volume show a surprisingly low scatter and the size does not change significantly in most experiments with "equilibrium" cooling. In experiment #14, where the number of particles remains constant, there is an increase in volume by a factor of 5. It is somewhat problematic to calculate a mean radius from the mean volume (equation 5.6). The frazil ice particles are not spheres and the relations between mean volume, mean radius and particle size distribution are not linear. But it was felt to be the only way to calculate a length scale for the particles.

3. Turbulent Heat Transfer. P N is the heat which must be trans- ported from the surface of the particle to the fluid. The Nusselt number Nu is equal to P^ normalized with one half of the heat flux due to conduction and depends only on the particle Reynolds number Re (equation 3.23).

The mean particle radius Rj^ and the Kolmogorov time scale did not vary signi­ ficantly from experiment to experiment, so that Nu is expected to be approxi­ mately constant. Figure 18 shows a probability density D of all measurements of Nu separately for "non equilibrium" and "equilibrium" cooling. Numerical values are given in table 3. Nu is about a factor of two larger with "non equilibrium" cooling and shows a larger scatter. This may be the effect of the ice forma­ tion on the walls of the heat exchangers which simulates an increased heat transfer from the particle. Therefore, the "equilibrium" cooling gives more reliable results. Table 3: Measurement of the Nusselt number Nu __ "equilibrium" "non equilibrium" cooling Mean value Nu 9.5 16 Standard deviation a, 4.7 6 Nu 53

i i Nu i 1 i M g 15.1: "equilibrium" cooling i 0.3- i i r* l # i 0.2- i i i i i i i 0 .1- i i i i 1i 1 | Nu i1 I ------1------I ■ ------► 10 20 30

Figure 18. Probability density D of the Nu number measured in the jar (figure 18.1: Experiments with "equilibrium" cooling; figure 18.2: Experiments with "non equilibrium" cooling) 54

Figure 19.1 ff/U0

Figure 19. Phase mean u of the vertical velocity and rms of tur­ bulent fluctuations u' normalized with the maximum velocity U of the grid. Curve A is measured at a point behind the edge of the grid rod, curve B behind the center of a grid mesh. g(t/x0) are the mean values used in equation 5.18 (figure 19.1: Re = 1500, figure 19.2: Re = 2300) g 55

Figure 19. Phase mean u of the vertical velocity and rms of turbu­ lent fluctuations u' normalized with the maximum velocity UQ of the grid. Curve A is measured at a point behind the edge of the grid rod, curve B behind the center of a grid mesh, g (t/Tq ) are the mean values used in equation 5.18 (figure 19.1: Re = 1500, figure 19.2: Re = 2300) § o 56

The small variation of the Nusselt number in all experiments is one of the main results of the study. It shows that the total ice production is described by the product of the number of particles and the particle heat transfer. The large change in measured power densities between .2 kWatt/m2 3 in experiment #14 and 12 kWatt/m in experiment #7 is, therefore, only the consequence of the increased number of ice particles (fig. 16. 1,2).

D. Turbulence Data 1. Velocity measurements. The velocity data measured by LDA are depicted in figure 19.1, 2. Phase mean of mean velocity and rms value for one half period are normalized with the velocity amplitude U of the grid. The curve A is measured at a point behind the edge of the grid rod, B is measured at the center of a grid mesh. Both measuring points were in the plane z = 0, the plane of maximal grid velocity, t/t = 0 corresponds to the moment the grid was at z = 0. t q is the period of the eccentric. The positive sign of themean velocity corresponds to the direction of the grid motion. As the ratio of the projected area of the grid elements to the total cross sectional area was .437, there was a marked counterflow through the grid mesh to fulfil continuity. This jet-like counterflow produces the nega­ tive velocity peak shown in curve B. Curve A instead starts with a positive peak corresponding to the wake flow behind the bar. The results show a com­ plicated three-dimensional instationary structure which is a combination of a jet and wake flow. The turbulent fluctuations are somewhat more homogenous, especially for larger values of t/TQ . One problem is inherent in this type of data reduction. As u' (t/t ) is the variance of all velocity measurements taken at t/x during 100 periods, there is no possibility to distinguish fluctuations with a time scale smaller than the period and "fluctuations of the mean" with time scales larger than the period. 2. Estimate of the Dissipation Rate. The mean dissipation rate e was determined in two ways: Naudascher (1970) reports a drag coefficient of 1.08 for the same grid geometry and Re = 6600. Assuming this value is valid also for lower and non stationary velocities, the pressure loss AP Table 4: Estimate of the Dissipation Rate and the Heat Transfer from Velocity Data

Re 1500 2300 g .43 .658 m/sec Uo

.40 1.34 Watt)

e 1375 4667 cm2 ^ calculated from pressure drop sec 3)

.09 +_ .21 1.38 + .25 Watt) measured from PG — ) temperature ) increase

e/C 45.2 280 cm2 ) estimated 3 ) sec ' from velocity data

C 30 16 ) ) 20 30 ) Nu m 58

across the grid is

AP = 1.08 p/2 U (t) 5.13 where

U(t) = UQ cos wt = Z(j) C O S U)t 5.14 is the velocity and

z(t) = Z sin ut 5.15 is the location of the grid. Z is the amplitude of the eccentric. The power Pg introduced by the grid into the flow is

AP AU 5.16 where A is the grid area. The mean dissipation rate e is the time mean PG of PG divided by that mass of the water which is in the volume 2*Z’A. The assumption of a constant drag coefficient fails for Re < 100, but the error will be small, as the velocities are small. Numerical values are given in table 4. A second method to determine the dissipation rate is the measurement of the temperature increase due to the moving grid (equa­ tion 4.5, fig., 12). This measurement of PG corresponds surprisingly well with the value calculated from the drag coefficient. 3. Kolmogorov Scales of Turbulence. Both methods described in the previous section give an integrated value of the dissipation rate. In order to determine local microscales, an estimate of e as a function of time and space is necessary, e is related to the macro scales of turbulence by .3 e = C u 5.17 m so one needs an estimate of local and u'. Assuming C is a constant, it can be determined if e is calculated from equation 5.17 and compared with the e of table 4. The following, again rather crude model, is used to estimate u': In the plane z = 0, u'/U0 = g (t/xQ) is measured. To smooth the spatial 59

inhomogenity horizontally, a mean value between curve A and B is adopted (fig. 19.1,2). The values of u'/U as a function of z are assumed to be equal to

u'(z,t) = U(z) • g(t/xQ) 5-18

This assumption has to be taken with care for several reasons. The two passages of the grid are no longer at t/xQ = 0 and t/xQ = .5 but happen within one half period. The error contributed to a mean value e , however, will be small, as g(t/xQ) is nearly constant for t/xQ > .15. More serious is the fact that the diffusion of turbulent energy is neglected, so that u' will be under­ estimated for z / 0. A value of L is taken from a paper by Bearman (1973), who studied m 4 . the wake of a pair of cylinders (Re = 2.5 10 ) and measured a correlation length L = 3.7 d (d = diameter of the cylinder). For a distance of the two cylinders of 3 d, he did not find that the length scale is changed by the presence of the second cylinder. With these approximations, the mean dissipation rate divided by

C is

u |5(z,t) z_ 1 dz dt 5.19 C 2-Z-xo 3.7d Numerical values of e/C are given in table 4. A comparison with e calculated from pressure drop yields C = 16 for Re = 2300, C = 30 for Re = 1500. g g As the diffusion of turbulent energy is neglected, one assumes that the energy input by the grid is dissipated in the same layer where it was pro­ duced. This leads to values of C which are a factor, 4 and 7 respectively, larger than the value of C = 4 used by Rao and Brodkey (1972) in a mixer, e is expected to be too large around z = 0 and too low around z = + Z. On the other hand, errors in determining e have limited effect on the micro ^ scales (equations 3.14), as n is proportional to £ > and Tk t0 e 60

Keeping the drawback of the model in mind, one can calculate the local dissipation rate (equation 5.17) and the micro scales to get the order of magnitude. In tables 1 to 4 of appendix 2, values of e(t,z), n (t,z), u (t,z) and x (t,z) are listed for Re = 2300 assuming C = 16. The hori- K K g zontal line corresponds to the time (in seconds) the vertical one to the z- axis (in cm). The minimum microscale (table 2) n = 40ym is still 2 orders of mag­ nitude larger compared to the critical particle radius of .5ym (equation 2.5) for a supercooling of .1°C. The minimum time scale (table 4) x^ is small compared to x , the time scale of heat conduction, if the mean radius of approximately .3mm is used.

pC „2 i -t c i x = — R., = 1.19 sec 5.20 c x M -1 The maximal shear s is the inverse time scale, s = 1300 sec max This shear is still smaller than the value of s = 3000 sec-* used by Sung (1973) to produce ice nuclei by fluid shear. 4. Mean Nusselt Number. Based on the model discussed in section III B, a mean Nusselt number can be calculated which should correspond to the measured value. The heat transfer per particle is equal to

1 PN (z,t) = ^ Nu ( - ^ _ - ■) 4IIR(Te - t l) x 5.21

The local Nusselt number is a function of the time scale x^and the exchange time t which was assumed to be 2 x, . As the distribution of the radius R e k is not known, it will be approximated by R^^. Values for Nu can be calculated from fig. 4 which gives Nu as a function of t /x . In table 5 of appendix 6 C 2, local Nu are listed as functions of z and t. The total mean heat transfer due to ice production is then + z 5.22 pi = / / ° DNPN Cz,t:) dZ dt

is the density of the particles. In equation 5.8, the mean Nu^ is equal to Pj normalized by the number of particles N^, the temperature difference and the radius R^. As A*2Z is the number of particles, it follows 61

2xk (z,t) Nu C— ---- ) dz dt K T c

5.23

In table 3, numerical values of Ni^ are given assuming C = 16. They cor­ respond to within a factor of 2 to 3 with the values observed. The physical content of this result may be summarized as follows. As long as the level of turbulence is sufficient for the time scale xk to be short compared to the time scale xc the heat flux from the frazil ice par­ ticle to the fluid will be an order of magnitude higher than the heat con­ duction if the fluid were at rest.

VI. SUGGESTIONS FOR FUTURE RESEARCH

A. Basic Research Problems 1. The question of how turbulent transport and diffusive trans­ port superimpose, when a particle of the size of the micro scale is floating in a fluid, is quite general. Beside heat transfer, this situation is present when dissolved gas is transported to a growing bubble or dissolved salt to a

growing crystal. In Chapter III, an attempt was made to estimate the heat transfer from an ice particle to the supercooled fluid in a turbulent flow with quite crude models. The question to be answered is how such a particle senses the turbulence. If it only senses shear from the flow, it would be useful to know the flow field and the temperature field around a heated sphere m a

shear flow. Research in this direction was done on suspensions (Jeffrey (1922),

Goldsmith (1962), Cox (1968), TBzeren (1977)). As the actual forms of frazil particles are discs or needles, an attempt should be made to include also such geometries. It would be more difficult to get an insight in the flow around the particle if the size were no longer small compared to the micro scales. The limit, where Brownian motion starts to be important, should also be included in the study. 62

2. The nucleation theories are applied to isothermal and isobaric systems. When frazil ice grows in a turbulent flow, the temperature will fluctuate, because the frazil particles act as point sources of heat, and the pressure around the particle fluctuates due to vertical displacement (hydrostatic pressure) and turbulent pressure fluctuations. A theory which includes these effects would be useful. 3. The result of the present study seems to indicate that the num­ ber of nuclei present in the supercooled water is the factor with the largest variation. The question, therefore, arises, what sources of nuclei are ac­ tive in a river. Do all particles originate from outside the flow, or is there a multiplication process within the flow which produces new nuclei? Possible mechanisms are collision with the bed and collisions be­ tween particles or fluid shear. (Similar multiplication processes have been observed in the atmosphere (Koenig (1968)). The present experiment, where turbulence is produced by a grid, does not separate the two processes. An experiment could be designed where the flow is produced by a rotating cylinder. Such a flow produces wall tur­ bulence and is more similar to natural conditions. Collisions with the wall could occur only if the flow forces ice particles to collide with the wall. A schlieren system with its axis parallel to the cylinder axis can be used to observe ice particles. The size distribution of the nuclei would be of interest and can be measured by a Coulter Counter (Larson (1976)). Probably, a method must be found to prevent the nozzle of the Coulter Counter to freeze up when supercooled water is withdrawn.

B. Test of the Conclusions of the Present Study in Flume Experiments The conclusions of this study are that turbulence increases both the production of frazil nuclei as well as the heat transfer. If this is true, frazil ice production in an open channel should depend on the height above the bed in the same way as the turbulent velocity fluctuations. Near the bed, more frazil ice should be produced than in the bulk of the fluid, increased roughness should also increase frazil ice formation. It seems possible to measure these effects in a laboratory flume where frazil ice is not nucleated by the pump. In field measurements, it 63

may be more difficult to measure these effects as presence of frazil ice is due to production as well as transport. A vertical profile of frazil ice may also be dominated by buoyancy. While in a flume a formation zone may be distinguished, in a river this may be possible only in certain locations. Two methods to detect frazil ice are suggested: 1. With a schlieren system, similar to the one used in the present study, frazil ice particles can be observed. Water ice mixture can be with­ drawn from different depths of the channel. Care must be taken that the temperature of the withdrawn mixture does not change and that the suction pump does not get clogged with frazil ice. 2 . As the latent heat of fusion is large, it should be possible to measure the heat which is needed to melt the ice. Water which is with­ drawn from the flow would pass a heater to melt the ice. From the temperature difference AT before and after the heater, the energy input PH and the dis­ charge of water, the discharge of ice Qg can be calculated using the heat balance equation.

H PSQSL * Plqlcl4T

C. Continuation of the Present Study 1. Simulation of natural conditions. The time scale character­ izing the present experiments was of the order of 200 to 600 seconds, i.e. the time until the water reached the ice point. The supercooling was of the order of .1°C. In a natural stream Osterkamp (1975) measured supercooling of only several hundredths of a degree, but for several hours. A rough estimate of the turbulence intensities shows that his Kolmogorov time scale is larger than in the present experiments. It would be of interest to extend the experimental parameters in the direction of these more natural conditions, i.e. lower supercooling, lower turbulence and longer observation time. In order to make such experiments possible, the following modifi­ cations of the apparatus are suggested: To keep the low supercooling constant over a longer time, the heat gain from the room must be reduced. Therefore, the air temperature around 64

the jar and the temperature of the coolant must be stabilized. The temperature of the coolant can be controlled by a control valve in the bypass of the coolant (see figure 9). As the air temperature in the Cold Room varies + 1°C, the jar should be packed in temperature controlled boxes. If the air temp­ erature in the Cold room is kept below the freezing point, the temperature inside the box can be controlled by a heater. This box has to be built in such a way that the flow still can be seeded during the experiment and that the schlieren system is not obstructed. A special problem for the observation of the frazil ice with the schlieren optic was the presence of dust in the water. Special care should be taken to remove dust from the air and it should be possible to filter the water before the experiment. 2. Multiplication process at Low Supercoolings. The present ex­ periments indicate that new nuclei are produced in the turbulence jar at supercooling larger than .09°C and .14°C for Re = 2300 and 1500 respectively. At lower supercooling, only the initial seed crystals grow. It would be of interest to know whether at higher speeds of the grid, multiplication takes place, even at lower supercooling. To increase the turbulence, the number of grids should be increased. To allow continuous velocity measurements by LDA, the new grids should be built in two parts, so that the optical beams are not interrupted when the grid passes. The eccentric mechanism must be equipped with roller bearings to reach higher velocities. 3. Heat Transfer at Low Turbulence Levels. In the present experi­ ments, only a limited range of Re was covered, as no multiplication process was observed at lower speeds of the grid. The dependence of the Nusselt number on Re, therefore, was not recognized. If a sufficient number of seed crystals is introduced by repeated contact of the ice stick with the grid, it should be possible to measure also the heat transfer at lower Re. This type of experiment is similar to the ones proposed under C.l. 4. Velocity Measurement. The estimate of the micro scales of tur­ bulence (section V.C.3) and of the turbulent heat transfer (section V.C.4) was based on crude assumptions on the turbulent velocity u f. Careful measurements of u f as a function of space and time can replace the theoretical assumptions 65

and lead to more accurate results.

VII. CONCLUSIONS

1. No frazil ice was observed at supercoolings of the order of .1°C, unless the flow was seeded with ice crystals. This suggests that in natural situations, the supercooled water is also seeded by some external source of ice and is not nucleated within the flow. 2. The number of ice particles increased during the experiments, therefore, a multiplication process produces new nuclei. From the experi­ ment, no information on the mechanism of this process can be gained. 3. The total ice production can be described as a product of the number of particles and the heat transfer per particle. The number of par­ ticles is given by the multiplication process which-intensifies with increasing supercooling and increasing speed of the grid. The heat transfer per particle normalized with the supercooling, and the size of the particle is constant for all experiments within the errors of measurement (Nu = 10). 4. The normalized heat transfer per particle can be estimated

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YOUNGQUIST G.R. § RANDOLPH A.D. 1972 Secondary Nucleation in a Class II System: Ammonium Sulfate-Water. AIChE J. 18, 421-429 Appendix 1: Input data and results EXPERIMENT # E03 70 INPUT DATA : RE = 2300. NON EQUILIBRIUM COOLING TIME TL TR TIME NC SEC C c SEC 0. -0.127 2.1 A 20. 206 30. -0.117 2.0,2 AQ. 360 60. -0.107 1 .91 60. AAA 90. -0.061 1.87 120. -0.035 1 .8 3 150. -0.018 1.8 A 180 . -0.012 1 .85

RESULTS : TIME TT TC T I P PD SEC C C C WATT KWATT/M“3 0 . 0.000 0.000 0.000 0.0 0.00 30. 0.010 0.000 0.010 5.0 1 .33 60. 0.020 0.001 0.019 1 A. 2 3.81 90. 0.066 0.002 0.0 6 A 18.2 A.88 1 20 . 0.092 O.OOA 0.088 10.A 2.79 150. 0.109 0.005 0.10A 5.0 1 . 3A 180. 0.115 0.007 0.108 0.0 0.00

TIME Til DN NE PIN PN VM RM NU NUE SEC C C M “ - 3 WATT MWATT MM"3 MM PERCENT 20. 0.006. 1 .23 35 A3 3.3 0.93 0.07 0.26 8. 57. AO. 0.013 2.1 A 6192 8.0 1.30 0.08 0.27 12. A2. 60. 0.019 2.6 A 7636 1A .2 1.85 0.10 0.29 17. 38. EXPERIMENT # E 05 71 INPUT DATA : RE = 2300. NON EQUILIBRIUM COOLING TIME TL TR TIME NC SEC C C SEC 0 . -0.137 3.77 15. 27 30. -0.137 3.60 45. 46 60. -0.136 3.27 90. 90 90. -0.133 2.98 120. 197 120. -0.121 2.71 150. 398 150. -0.093 2.40 180. 475 180. -0.058 2.17 240. 335 210. -0.030 1 .96 240. -0.018 1.81 270. -0.012 1.69 300 . -0.007 1 .57

RESULTS : TIME TT TC T I P PD SEC C C C WATT KWATT/M“3 0 . 0.000 0.000 0.000 0.0 0.00 30. 0.000 0.000 0.000 -0.1 -0.02 60. 0.001 0.001 - 0.000 0.4 0.11 90. 0.004 0.002 0.002 3.3 0.88 120. 0.016 0.004 0.012 9.7 2.61 1 50. 0.044 0.005 0.039 15.7 4.21 180. 0.079 0.007 0.072 15.6 4.20 210. 0.107 0.008 0.099 9.5 2.56 240 . 0.119 0.010 0.109 3.7 0.99 270. 0.125 0.01 2 0.113 1 .8 0.48 300. 0.130 0.014 0.116 0.0 0.00

TIME TII D N NE PIN PN VM RM NU NUE SEC C CM~-3 WATT MWATT MM ~ 3 MM PERCENT 15. 0.000 0.1 6 464 -0.0 -0.06 0.00 45.-Q.000 0.27 791 0.2 0.23 - 0.00 ______90. 0.002 0.54 1548 3.3 2.12 0.04 0.22 21. 150. 120. 0.012 1.17 3388 9.7 2.87 0.15 0.33 20. 52. 150. 0.039 2.37 6845 15.7 2.29 0.23 0.38 18. 40. 180. 0.072 2.83 8170 15.6 1.91 0.35 D.44 21. 50. 240. 0.109 1.99 5762 3.7 0.64 0.75 0.56 18. 151. experiment # E 06 72 INPUT DATA : RE = 1500 NON EQUILIBRIUM COOLING TIME TL TR TIME NC SEC C c SEC 0. -0.108 4.57 60. 39 30. -0.108 4.25 120. 52 60. -0.107 3.80 180. 56 90. -0.105 3.46 300. 59 120. -0.102 3.14 420. 51 150. -0.099 2.85 540. 91 180. -0.097 2.60 210. -0.093 2.37 240. -0.090 2.26 270. -0.087 2.02 300 . -0.084 1.99 330 . -0.079 1 .98 360. -0.075 2.27 390. -0.068 2.79 420. -0.064 2.91 450. -0.059 3.09 480. -0.053 3.27 510. -0.045 3.52 540. -0.042 3.65 570. -0.035 3.70 600 . -0.030 3.96

RESULTS : TIME TT TC TI P PD SEC C C C WATT KWATT/M“3 0. 0.000 0.000 0.000 0.0 0.00 30. 0.001 0.000 0.000 0.1 0.02 60. 0.001 0.001 0.000 0.4 0.11 90. . 0.003 0.001 0.002 1 .0 0.27 120. 0.006 0.002 0.004 1.2 0.33 150. 0.009 0.003 0.006 0.9 0.25 180. 0.011 0.003 0.008 1.2 0.32 210. 0.015 0.004 0.011 1 .4 0.37 240. 0.018 0.005 0.013 1 .1 0.29 270. 0.021 0.006 0.015 1.1 0.28 300. 0.024 0.007 0.017 1.5 0.40 330. 0.029 0.008 0.021 1 .5 0 .42 360. 0.033 0.010 0.023 1.7 0.46 390. 0.040 0.013 0.027 1 .4 0.38 420. 0.044 0.016 0.028 0.7 0.19 450. 0.049 0.01 9 0.030 1 .1 0.28 480. 0.055 0.023 0.032 1 .6 0.44 510. 0.063 0.027 0.036 0.7 0.18 540. 0.066 0.031 0.035 0.3 0.09 570. 0.073 0.035 0.038 0.7 0.19 600. 0.078 0.040 0.038 0.0 0.00

TIME Til DN NE PIN PN VM RM NU NUE SEC C CM “-3 WATT MWATT MM“3 MM PERCENT 60. 0.000 0.23 670 0.4 0.61 0.01 0.15 11 . 324. 120. 0.004 0.31 894 1.2 1.38 0.18 0.35 11 . 145. 180. 0.008 0.33 963 1 .2 1.22 0.32 0 .42 8. 152. 300. 0.017 0.35 '1014 1.5 1.46 0.67 0.54 9. 127. 420. 0.028 0.30 877 0.7 0.81 1.29 0.68 5 . 219. 540. 0.035 0.54 1565 0.3 0.20 0.89 0.60 2. 442. EXPERIMENT ft E 07 INPUT DATA : RE = 1500. E QUILIBRIUM COOLING TIME TL TR TIME NC SEC C C SEC 23 0. -0.288 0.00 30. 60. 27 30. -0.283 -0.12 90. 454 60. -0.276 -0.14 61 . -0.276 -0.14 90. -0.233 -0.15 120. -0.101 -0.16 150. -0.038 -0.12 180. -0.021 -0.08 » o O 210. -0.015 •

RESULTS : TIME TT TC TI P PD KWATT/M“: SEC C C C WATT 0.00 0. 0.000 0.000 0.000 0.0 3.0 0.80 30 . 0.005 0.000 0.005 3.5 0 .94 60. 0.012 0.001 0.011 22.4 6.01 61 . 0.012 0.001 0.012 46.4 12.48 90. 0.055 0.001 0.054 50.9 13.70 120 . 0.187 0.001 0.186 20.9 5 .62 150. 0.250 0.001 0.249 1 .61 180. 0.267 0.001 0.266 6.0 0.29 210. 0.273 0.001 0.272 1 .1 NUE TIME Til DN NE PIN PN VM RM NU PERCENT SEC C CM “ -3 WATT MWATT MM“3 MM 0.48 15. 61 . 30. 0.005 0.14 395 3.0 7.54 0.47 0.62 12. 61 . 60. 0.011 0.16 464 3.5 7.56 0.98 18. 17. 90. 0.054 2.70 7808 46.4 5.94 0.28 0.40 EXPERIMENT # EOS INPUT DATA : RE = 1500. NON EQUILIBRIUM COOLING TIME TL TR TIME NC SEC C C SEC 0 . -0.180 3.71 20. 57 30. -0.185 3.72 60. 75 60 . -0.177 3.72 90. 210 90. -0.153 3.73 105. 260 120. -0.117 3.75 120. 367 1 50. -0.073 3.77 150. 428 180 . -0.045 3.79 180. 238 210. -0.028 3.81 240. -0.018 3.87

RESULTS : TIME TT TC TI P PD SEC C C C WATT KWATT/M“3 0 . 0.000 0.000 0.000 0.0 0.00 30. 0.000 0.000 -0.000 1 .8 0.48 60. 0.008 0.001 0.007 7.8 2.09 90. 0.032 0.003 0.029 14.8 3.99 120. 0.068 0.004 0.064 19.9 5.34 150. 0.112 0.007 0.105 17.6 4.73 180. 0.140 0.009 0.131 10.4 2.79 210 . 0.157 0.01 2 0.145 5.5 1 .47 240. 0.167 0.01 5 0.152 1.1 0.00

TIME T11 DN NE PIN PN VM RM NU NUE SEC C C M “ - 3 WATT MWATT MM“3 MM PERCENT 20.-0.000 0.34 980 1 .2 1.21 -0.01 60. 0.007 0.45 1290 7.8 6.02 0.21 0.37 26. 46. 90. 0.029 1.25 3612 14.3 4.11 0.33 0.43 18. 31 . 105. 0.046 1.55 4472 17.3 3.88 0.42 0.46 17. 31 . 120. 0.064 2.1 8 6312 19.9 3.15 0.40 0 .46 17. 31 . 150. 0.105 2.55 7361 17.6 2.39 0.57 0.52 18. 41 . 180. 0.131 1.42 4093 10.4 2.53 1.28 0.67 23. 64. 75 experiment n E09 input DATA : RE = 1500. NON EQUILIBRIUM COOLING TIME TL TR TIME NC SEC C C SEC 0 . -0.136 4.77 20. 41 36 30. -0.142 4.75 60. 107 60. -0.144 4.73 120. 136 90. -0.141 4.72 140. 185 120. -0.129 4.72 165 . 220 150. -0.112 4.72 180. 224 180. -0.081 4.73 240. 210. -0.058 4.75 240. -0.039 4.78 270 . -0.027 4.78 300. -0.017 4.78 330. -0.012 4.78

RESULTS : TIME TT TC TI P PD k w a t t /f t : SEC C C C WATT 0.00 0 . 0.000 0.000 0.000 0.0 -0.11 30. 0.000 0.000 - 0.000 -0.4 0.01 60. 0.000 0.001 -0.001 0.0 0.77 90 . 0.003 0.003 - 0.000 2.9 1 .69 120. 0.015 0.005 0.010 6.3 2.96 150. 0.032 0.008 0.024 11 .0 180. 0.063 0.011 0.052 12.4 3.32 2.43 210 . 0.086 0.01 5 0.071 9.0 240. 0.105 0.019 0.086 6.0 1 .61 0.94 270. 0.117 0.023 0.094 3.5 0.41 300. 0.127 0.027 0.100 1.5 0.42 330. 0.132 0.032 0.100 1 .5

DN NE PIN PN VM RM NU NUE TIME Til PERCENT SEC C CM~-3 WATT MWATT MM ~ 3 MM 20.-0.000 0.24 705 -0.3 -0.37 -0.01 60.-0.001 0.21 619 0.0 0.07 -0.10 57. 120. 0.010 0.64 1840 6.3 3.40 0.21 0.37 20. 0.81 2339 9.4 4.03 0.32 0.43 23. 47. 140. 0.019 44. 165. 0.038 1.10 3182 11.7 3.67 0.47 0.48 22. 22. 46. 180. 0.052 1 .31 3784 12.4 3.27 0.55 0.51 81 . 240. 0.086 1 .33 3852 6.0 1.55 0.89 0.60 19. EXPERIMENT ft E1 1 76 INPUT DATA : RE = 1500. EQUILIBRIUM COOLING TIME TL TR TIME NC SEC C C SEC 0. -0.168 2.83 20. 43 30. -0.160 2.80 60. 59 60. -0.153 2.77 110. 120 90. -0.148 2.47 135 . 170 120. -0.138 2.17 150. 270 150. -0.127 1 .87 180. 350 180. -0.112 1.58 210. -0.091 1.32 240 . -C.070 1 .09 270. -0.052 0.8 6 300. -0.040 0.63 330 . -0.030 0.47 360. -0.023 0.31 390. -0.017 0.15 420. -0.013 0.15

SULTS : TIME TT TC TI P PD SEC C C C WATT KWATT/M“ 0. 0.000 0.000 0.000 0.0 0.00 30. 0.008 0.004 0.004 1 .8 0.49 60. 0.015 0.008 0.007 1.2 0.32 90. 0.020 0.012 0.008 2.2 0.58 120. 0.030 0.01 5 0.015 4.0 1 .06 150. 0.041 0.01 7 0.024 5.5 1 .47 180. 0.056 0.020 0.036 8.3 2.23 210. 0.077 0.022 0.055 10.0 2 .70 240. 0.098 0.023 0.075 9.4 2.53 270. 0.116 0.025 0.091 7.2 1 .95 300. 0.128 0.026 0.102 5.3 1 .42 330. 0.138 0.026 0.112 4.1 1 .10 360. 0.145 0.02 7 0.118 3.2 0.85 390. 0.151 0.027 0.124 2.4 0.66 420. 0.155 0.028 0.127 0.7 0.19

TIME T U DN NE PIN PN VM RM NU NUE SEC C CM “ -3 WATT MWATT MM“3 MM PERCENT 20. 0.003 0.26 739 1.2 1.66 0.14 0.33 9. 1 20. 60. 0.0C7 0.35 1014 1 .2 1.16 0.28 0.40 5. 135. 110. 0.013 0.71 2064 3.4 1.63 0.25 0.39 8. 68. 135. 0.019 1 .01 2924 4.7 1.61 0.27 0.40 9. 57. 150. 0.024 1 .61 4 644 5.5 1.18 0.20 0.36 7. 55. 180. 0.036 2.08 6020 8.3 1.38 0.24 0.39 9. 46. 77 EXPERIMENT # E 1 2 INPUT DATA : RE = 1500. EQUILIBRIUM COOLING TIME TL TR TIME NC SEC C r SEC 0. -0.092 1 .38 30. 7 30. -0.086 1.20 90. 10 60. -0.083 1 .02 180. 18 90. -0.078 0.78 240. 17 120. -0.075 0.47 420 . 14 150. -0.071 0.74 180. -0.067 0.94 210 . -0.064 1 .12 240. -0.062 1.32 270. -0.058 1 .50 300 . -0.054 1 .75 330. -0.050 1.79 360. -0.047 2.14 390. -0.043 2.26 420 . -0.040 2.39 4 50. -0.037 2.51

RESULTS : TIME TT TC T I P PD KWATT/M'3 SEC C C C WATT 0. 0.000 0.000 O.i000 0.0 0.00 30. 0.006 0.002 O.l004 1.5 0.40 60. 0.009 0.003 0. 006 1.4 0.37 90. 0.01 4 0.005 0. 009 1 .6 0.42 120. 0.017 0.005 0. 012 1.3 0.35 150. 0.021 0.007 0. 014 1.4 0.38 180. 0.025 0.008 0. 017 1 .0 0.27 210. 0.028 0.01 0 0. 01 8 0.4 0.10 240. 0.030 0.01 2 0. 018 0.5 0.13 270. 0.034 0.014 0. 020 0.9 0.24 300. 0.038 0.01 6 0. 022 0.8 0.21 330. 0.042 0.019 0. 023 0.4 0.10 360. 0.045 0.022 0. 023 0.2 0.06 0.04 390. 0.049 0.025 0. 024 0.1 420. 0.052 0.028 0. 024 -0.2 -0.05 450 . 0.055 0.032 0. 023 1.1 1 .60 NUE TIME Til DN NE PIN PN VM RM NU PERCENT SEC C CM '-3 WATT MWATT MM ~ 3 MM 12.29 1.40 0.69 58. 13G. 30. 0, .004 0.04 120 1.5 9.03 2.19 0.81 40. 132. 90. 0..009 0.06 172 1 .6 n 3.30 2.20 0.81 17. 172. 180 . U «.017 0.11 309 1.0 300. 240. 0..01 8 0.10 292 0.5 1 .72 2.52 0.84 9. -0.83 3.96 0.98 -6. -520. 420. 0..024 0.08 240 -0.2 EXPERIMENT # E 1 3 INPUT DATA : RE = 2300. EQUILIBRIUM COOLING TIME TL T R TIME NC SEC c C SEC 0. -0 . 1 4 8 1 .31 15. 62 20. -0 . 1 4 2 1 .31 60. 242 40. - 0 . 1 3 8 1.11 75. 375 60. -0 .1 30 0.91 90. 625 80 . - 0 . 1 1 7 0.70 100. - 0 . 0 9 5 0.55 120. - 0 . 0 6 7 0 . 30 140 . -0 . 0 4 7 0.22 160. - 0 . 0 3 0 0.15 o 00 o 180. -0.021 • 200 . - 0 . 0 1 5 0.00

R E S U L T S : TIME TT TC T I P PD SEC C c C WATT KWATT/M 0. 0 . 0 0 0 0 . 0 0 0 0. 00 0 0.0 0.00 20. 0.006 0. 00 3 0.003 1.8 0 . 49 40. 0 . 0 1 0 0.005 0.005 2.8 0.74 60. 0.018 0. 00 8 0.010 6.4 1 .73 80 . 0.031 0. 01 0 0.021 12.1 3.24 100. 0. 05 3 0. 01 2 0.041 18.1 4 . 87 120. 0.081 0. 01 4 0 . 0 6 7 17.4 4 . 69 140 . 0.101 0.01 5 0.086 13.2 3 .55 160. 0. 11 8 0.01 7 0.101 8.9 2.40 180. 0 . 1 2 7 0. 01 9 0. 10 8 4.7 1 .26 200. 0. 13 3 0.02 0 0. 11 3 0.8 0.21

TIME T 1 1 DN NE PIN PN VM RM NU NUE SEC C C M “-3 WATT MW ATT MM “ 3 MM PERCENT 15. 0. 00 2 0.37 1066 1 .4 1 .29 0.09 0.28 9. 110. 60. 0 . 0 1 0 1 .44 4162 6.4 1.55 0.10 0 . 29 12. 43. 75. 0.018 2.23 6450 10.6 1 .65 0.11 0.30 13. 37. 90. 0.031 3.72 10750 15.1 1.40 0.12 0 . 30 12. 34. 79 EXPERIMENT # E14 INPUT DATA : R F = 2300. EQUILIBRIUM COOLING TIME TL TR TIME NC SEC C C SEC 0. -0.067 3.00 35. 67 20. -0.060 2.94 60. 67 40 . -0.056 2.88 90. 60 60. -0.051 2.83 120. 67 80. -0.047 2.53 150. 58 100. -0.043 2.23 210. 56 120 . -0.038 1 .93 270. 57 140. -0.036 1 .76 160. -0.030 1 .59 180. -0.028 1 .41 200. -0.025 1 .24 220 . -0.022 1 .07 240. -0.018 0.99 260. -0.016 0.72 280 . -0.013 0.55 300. -0.011 0.38

RESULTS : TIME TT TC TI P PD SEC C C C WATT KWATT/M“3 0. 0.000 0.000 0.000 0.0 0.00 20 . 0.007 0.004 0 .003 1.1 0.30 40. 0.011 0.008 0.003 0.4 0.-10 60. 0.016 0.012 0 .004 0.5 0.13 80. 0.020 0.016 0.004 0.3 0.08 100. 0.024 0.01 9 0.005 0.9 0.24 120. 0.029 0.023 0 .006 0.3 0.08 140. 0.031 0.026 0.005 0.8 0.22 160. 0.037 0.029 0.008 0.9 0.25 180. 0.039 0.031 0.008 -0.1 -0.04 200. 0.042 0.034 0 .008 0.4 0.10 220. 0.045 0.036 0 .009 0.9 0.23 240. 0.049 0.039 0 .010 0.6 0.16 0.09 260. 0.051 0.041 0. 010 0.4 280. 0.054 0.043 0 .011 0.5 0.13 300. 0.056 0.045 0.011 1.1 0.00

TIME Til DN NE PIN PN VM RM NU NUE PERCENT SEC C CM “ - 3 WATT MW ATT MM“ 3 MM 35. 0.003 0.40 11 52 0.6 0.48 0.10 0.29 8. 261 . 60. 0.004 0.40 1152 0.5 0.42 0.13 0.32 7. 306. 90. 0.004 0.36 1032 0.6 0.59 0.17 0.34 11 . 274. 120. 0.006 0.40 1152 0.3 0.25 0.22 0.38 5. 501 . 238. 150. 0.007 0.35 997 0.9 0.86 0.28 0.40 18. 210. 0.008 0.33 963 0.6 0.64 0.35 0.44 18. 318. 270. 0.011 0.34 980 0.4 0.42 0.44 0.47 17. 467. 80 E XPERIMENT ■# E1 5 INPUT DATA : RE = 2300. EQUILIBRIUM COOLING TIME TL TR TIME NC SEC C C SEC 0 . -0.105 1 .00 30. 28 20. -0.099 1 .24 60. 60 AO . -0.095 1 .35 120. 81 60 . -0.090 1 .47 150. 107 30. -0.087 1 .60 180 . 140 100. -0.083 1 .72 210. 146 120. -0.079 1 .85 240. 168 140. -0.075 1 .90 270. 265 160. -0.072 1 .98 300. 300 180. -0.067 2.10 200. -0.063 2.17 220 . -0.058 2.24 240. -0.052 2.32 260 . -0.047 2.40 280. -0.041 2.47 300. -0.036 2.53 320. -0.030 2.58 340. -0.026 2.63 360. -0.021 2.68

SULTS : TIME TT TC T I P PD r- SEC C V- C WATT KWATT/ M " 3 0 . 0.000 0.000 0.000 0.0 0.00 20. 0.006 0.003 0.003 1.8 0.48 40. 0.010 0.005 0 .005 1.3 0.36 60. 0.015 0.008 0.007 0.9 0.23 80. 0.018 0.011 0.007 0.4 0.10 100. 0.022 0.014 0 .008 0.7 0.19 120. 0.026 0.017 0 .009 0.6 0.17 140. 0.030 0.021 0.009 0.2 0.05 160. 0.033 0.024 0 .009 0.5 0.14 180. 0.038 0.027 0.011 0.9 0.23 200. 0.042 0.051 0.011 0.8 0.22 220 . 0.047 0.034 0 .013 1.5 0.42 240. 0.053 0.038 0.015 1.5 0.40 260. 0.058 0.041 0.017 1 .4 0.39 280. 0.064 0.045 0 .019 1.4 0.38 300. 0.069 0.049 0 .020 1 .4 0.37 320. 0.075 0.053 0 .022 0.9 0.25 340. 0.079 0.056 0 .023 0.5 0.14 360. Q.084 0.060 0 .024 0.0 0.00

TIME Til DN NE PIN PN V M RM NU NUE SEC C CM “-3 WATT MWATT MM “3 MM PERCENT 30. 0.004 0.17 481 1 .6 3.26 0.33 0 .43 22. 110. 60. 0.007 0.36 1032 0.9 0 .83 0.26 0.40 7. 178. 120. 0.009 0.48 1 393 0.6 0.45 0.25 0.39 4. 234. 130. 0.009 0.64 1840 0.4 0.20 0.20 0.36 2. 383. 180. 0.011 0.83 2408 0.9 0.35 0.18 0.35 4 . 195. 210. 0.012 0.87 2511 1 .2 0.47 0.19 0.36 6. 162. 240. 0.015 1 .00 2889 1.5 0.52 0.21 0.37 8. 1 44. 270. 0.018 1 .58 4558 1.4 0.31 0.16 0.33 6. 1 54. 300. 0.020 1 .79 5160 1 .4 0.27 0.16 0.33 6. 167. EXPERIMENT U E16 INPUT DATA : RE = 2300. EQUILIBRIUM COOLING TIME TL TR TIME NC SEC C C SEC 82 0. -0.099 1 .61 5. 88 20. -0.092 1 .70 10. 4 0 . -0.089 1 .79 15. 86 60. -0.086 1.88 20. 82 102 80. -0.081 1 .97 60. 110 100. -0.077 2 .04 100 . 120. -0.073 2.11 150. 148 200 140. -0.G67 2.19 180. 332 160. -0.062 2.26 240. 180. -0.055 2.33 200. -0.050 2.40 220. -0.042 2.48 240. -0.036 2.53 260. -0.029 2.58 2 80. -0.024 2.63 300 . -0.019 2 .68

RESULTS : TIME TT TC TI P PD KWATT/M“3 SEC C C C WATT 0 . 0.000 0.000 0 .000 0.0 0.00 20. 0.007 0.003 0 .004 1.5 0.40 -0.04 40. 0.010 0.006 0. 004 -0.1 60. 0.C13 0.009 0 .004 0.6 0.16 0.25 80 . 0.01 8 0.01 3 0 .005 0.9 0.13 100. 0.022 0.016 0 .006 0.5 0.33 1 20 . 0.026 0.01 9 0 .007 1.2 140. 0.032 0.023 0 .009 1.6 0.42 0.52 1 60 . 0.037 0.026 0 .011 1 .9 180. 0.044 0.030 0 .014 1 .9 0.51 0.60 200. 0.049 0.034 0 .015 2.2 0.69 220. 0.057 0.037 0 .020 2.6 240. 0.063 0.041 0 .022 2.2 0.58 0.47 260. 0.070 0.045 0 .,025 1.7 0.25 2 80. 0.075 0.049 0 .026 0.9 0.37 300. 0.080 0.052 0 .-028 1 .4

DN NE PIN PN V M RM NU NUE TIME TII PERCENT SEC C C M “ -3 WATT MWATT MM “ 3 MM 349. 5. 0.001 0.49 1 410 0.4 0.27 0.03 0.19 4 . 6. 214. 10. 0.002 0.52 1 513 0.7 0.49 0.05 0.23 1 70. 15. 0.003 0.51 1 479 1 .1 0.76 0.08 0.27 9. 20. 0.004 0.49 1410 1.5 1 .06 0.11 0.30 11 . 148. 289. 60. 0.004 0.61 1754 0.6 0.34 0.08 0.27 4. 317. 100. 0.006 0.65 1 892 0.5 0.26 0.13 0.31 3. 9. 1 36. 150. 0.C10 0.88 2545 1 .7 0.69 0.16 0.33 130. 180. 0.014 1.19 344 0 1.9 0.55 0.16 0.34 8. 1 33. 240. 0.022 1 .98 5710 2.2 0.38 0.15 0.33 9. 82

Appendix 2 Estimate of the turbulence parameter and the local Nusselt number in the jar RE = 2300. C = 16-

DISSIPATION RATE (l*T2/SEC“3)

0.00 0.00 0 . 00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.01 . 01 0.74 0.58 0.29 0.14 0.09 0.07 G .04 0.03 0.01 0

0.03 0.02 0 . 01 1 .75 1 .38 0.68 0.34 0.22 0.17 0.1 Q 0.07 0.03 0. 02 2.64 2.07 1 .03 0.51 0.33 0.26 0.15 0.11 0.05

0.06 0.04 , ,02 3.22 2.53 1 .26 0.62 0.40 0.32 0.18 0.13 0 0.07 0.04 0 .02 3.42 2.69 1 .34 0.66 0.43 0.34 0.19 0.14 0.04 0 .02 3.22 2.53 1.26 0.62 0.40 0.32 0.18 0.13 0,06 0.03 0 .02 2.64 2.07 1.03 0.51 0.33 0.26 0.15 0.11 0.05 0.03 0.02 0 .01 1 .75 1.38 0.68 0.34 0.22 0.17 0.10 0.07 0.01 0 .01 0.74 0.58 0.29 0.14 0.09 0.07 0.04 0.03 0.01 0.00 0.00 0 .00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Table 1 Dissipation rate & (z,t)

0 0 w RE 2300 C 16

KOLMOGOROV LENGTH SCALE (MICRONS)

00 co 00 00 00 00 00 00 00 oo 00

53. 56. 67. 80. ■89. 95. 109. 117. 142. 159. 182

43. 45. 54. 65. 72. 76. 87. 95. 114. 128. 147

39. 41 . 49. 58. 65. 69. 79. 85. 103. 116. 133

37. 39. 46. 55. 62. 66 . 75. 81 . 98. 110. 126

36. 38. 46. 55. 61 . 65. 74. 80. 97. 109. 124

37. 39. 46. 55. 62. 66. 75. 81 . 98. 110. 126

39. 41 . 49. 58. 65. 69. 79. 85. 103. 116. 133

43. 45. 54. 65. 72. 76. 87. 95. 114. 128. 147

53. 56. 67. 80. 89. 95. 109. 117. 142. 159. 182

00 00 00 00 00 00 00 00 00 00 oo

Table 2 Kolmogorov length scale (Z /t)

0 0 RE = 2300. c = 16.

KOLMOGOROV VELOCITY SCALE CCM/SEC)

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1 .53 1 .27 1 .13 0.99 3.40 3.20 2.68 2.25 2.02 1 .90 1 .66 1 .90 1 .58 1 .40 1 .22 4.21 3.97 3.33 2 .79 2.51 2.36 2.06 2 .11 1 .55 1 .36 4.67 4 .40 3.69 3 .09 2.78 2.61 2.28 1.74 2.21 1 .83 1 .63 1.42 4.91 4 .62 3.83 3.25 2.92 2.75 2.40 2.25 1.86 1 .66 1.45 4.98 4.69 3.94 3.30 2.96 2.79 2.43 2.21 1 .83 1.63 1.42 4.91 4.62 3.88 3.25 2.92 2.75 2.40 2.11 1 .74 1 .55 1.36 4.67 4.40 3.69 3.09 2.78 2.61 2.28 2.06 1 .90 1.58 1 .40 1.22 4.21 3.97 3.33 2.79 2.51 2.36 1 .53 1.27 1 .13 0.99 3.40 3.20 2.68 2.25 2.02 1 .90 1 .66 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Table 3 Kolmogorov velocity scale i^(z,t)

0 0 on RE 2300. C 16.

KOLMOGOROV TIME SCALE (MSEC)

00 00 00 00 00 00 00 00 00 oo 00 N* O 1 .56 1 .76 2.50 3.56 4.41 4.98 6.54 7.66 11.17 i 18.50

1.01 1 .1 A 1.62 2.31 2.87 3.23 4.25 4.98 7.25 9.14 12.02

0.83 0.93 1.32 1 .89 2.34 2.64 3.46 4 .06 5.92 7.45 9.80

0.75 0.84 1 .20 1 .71 2.11 2.38 3.13 3.67 5.35 6.74 8.87

0.73 0.82 1.16 1 .65 2.05 2.31 3.04 3.56 5.19 6.5 4 8.60

0.75 0.84 1 .20 1 .71 2.11 2.38 3.13 3.67 5.35 6.74 8.87

0.83 0.93 1.32 1 .89 2.34 2.64 3.46 4.06 5.92 7.45 9.80

1.01 1 .14 1.62 2.31 2.87 3.23 4.25 4.98 7.25 9.14 12.02

1.56 1 .76 2.50 3.56 4.41 4.98 6.54 7.66 11.17 14.07 18.50

00 oo 00 "00“ " 00 00 00 00 00 00 00

Table 4 Kolmogorov time scale

0 0 O' RE = 2300. c = 16.

NUSSELT-NUMBER

2 2 . 2 . 2 . 2. 2. 2. 2. 2. 2. 2. 23. 22. 19. 17. 14. 43. 40. 36. 32. 29. 27. 29. 27. 22. 21 . 19. 53. 51 . 42. 36. 34. 33. 32. 30. 23. 22. 20. 56. 54. 47. 38. 36. 35. 31 . 25. 23. 21 . 57. 56. 50. 41 . 37. 36. 33, 34, 32. 26. 23. 21. 53. 56. 50. 42. 37. 36. 33 31. 25. 23. 21 . 57. 56. 50. 41 . 37. 36. 30. 23. 22 . 20. 56. 54. 47. 38. 36. 35. 32 27. 22. 21 . 19. 53. 51 . 42. 36. 34. 33. 29 22 19. 17. 14. 43. 40. 36. 32. 29. 27. 23 . 2 . 2 . 2 2 . 2. 2 . 2. 2. 2. 2. 2 .

Table 5 Nusselt num ber Nu (z,t)

00 ^ 4 Appendix 3: Computer Program 89

c FRAZIL c CALCULATES FROM NUMBER OF TEMPERATURE MEASUREMENTS c N n y NUMBER OF PARTICLE-COUNT C RE REYNOLDS NUMBER OF GRID C ZOO TIME OF TEMPERATURE MEASUREMENT MEASURED WATER TEMPERATURE c TLMOO c TL (K ) CORRECTED WATER TEMPERATURE c TR (K) ROOM TEMPERATURE (DEGREE CELSIUS) c ZN (L) TIME OF PARTICLE COUNT c NC (L) NUMBER OF PARTICLES COUNTED IN 14 CM**2 c THE FOLOWING QUANTITIES TOTAL INCREASE OF TEMPERATURE c TT ((C) TEMPERATURE INCREASE DUE TO HEAT CONDUCTION c T C ( K ) TEMPERATURE INCREASE DUE TO LATENT HEAT OF FUSION c TI ((C) c TLI (L) WATER TEMPERATURE INTERPOLATED AT ZN(L) TEMPERATURE INCREASE DUE TO LATENT HEAT OF FUSION c T 11 ( L ) c INTERPOLATED AT ZN(L) c NT (L) TOTAL NUMBER OF PARTICLE IN THE JAR c DN ( L) PARTICLE DENSITY (CM**-3) c NE (L) NUMPER OF PARTICLE PARTICIPATING EXCHANGE POWER DUE TO ICE PRODUCTION AT ZOO (WATT) c P (K) OCWATT/M**3) POWER DENSITY c PD (IC) (WATT) PIN(L) POWER DUE TO ICE PRODUCTION AT ZN(L) c (MWATT) c PN (L) P(L)/NE(L) MEAN VOLUME OF PARTICLES ( M M * * 3 ) c VM (L) (MM) c RM (L) MEAN RADIUS OF PARTICLE c PNU(L) NUSSELT -NUMBER c P N U E ( L ) PERCENTERROR OF NUSSELT NUMBER c NAME NAME OF DATA-FILE c EQU E QU = Q : NON EQUILIBRIUM COOLING EGU=1: EQUILIBRIUM COOLING c c ZCON ST TIME CONSTANT OF RESIDUAL COOLING c A______ON Z(25),TLM(25),TL(25),TR(25),NC(25),TT(25),TC(25),TI(25)

¡ S i » NT(25)*N£(25),P(25),PN(25),VM(25),RM(25),PNU(25),NAME(6) DIMENSION T11 ( 1 5) ,ZN(1 5),PIN(15),DN(15),TLI(15),PNUE(15) LOG IC AL *1 A (9) DATA IPD/2H.D/,IAT/2HAT/,IPR/2H.R/,IES/2HES/ DATA IBL/2H /,IPP/2H .P/,ILT/2HLT/ ZC0NST=238. CALL DATE(A) TYPE 10,A . 1 0 FORMAT(11EXPERIME NT WITH FRAZIL-ICE ,/9Al) 1 5 TYPE 20 2D FORMAT(//'SEXPERIMENT # ') ACCEPT 30,NAME 3 0 FORMAT(6A 2) NAME(5)=IPD NAME(6) = IAT CALL CLOSE(5) 90

C C INPUT 0F DATA F ROM DISC FILE C CALL ASSIGNd , NAME,12,IER) READ(1,40)N,M,RE,EQU 40 F0RMAT(216,2E15.7) DO 50 K = 1,N 50 READ (1,60)Z(K),TLM(K),TR(K) 60 F0RMAT(3E15.7) DO 63 1=1,M 63 READ(1,65)ZN(L),NC(L> 65 FORMAT(E15.7,16) CALL CL OS E (1 ) C C CALCULATIONS OF RESULTS C T C(1)=0 P < 1) = 0 C CORRECTION OF TLM(K) IF MINIMUM IS NOT AT K=1 KMIN=1 DO 66 K=1,N 66 TL(K)=TLM(K) DO 68 K = 2,N 68 IF(TL(K) .LE.TL(K-1))KMIN=K DO 69 K=1,KWIN 69 TL(K)=TL(KMIN) C CALCULATION OF TCCK) IF (RE .EQ.1500.)G0T080 DO 70 K = 2,N 70 TC(K)=TCCK-1) + ((.439E-4*(-TL(K)+TR(K)) + .74E-4)*(Z OO - Z (K-1 ) ) ) GOTO 100 8 0 DO 90 K = 2,N 9 0 TC (K)=TC(K-1 >+((.439E-4*(-TL(K>+TR(K)>+.035E-4>*(Z(K)-Z(K-1)) ) 100 IF (EQU)103,103,108 103 TC2=TC(2) DO 105 K = 2,N 105 TC(K)=TC(K)-(ZC0NST*TC2*(1-EXP(-Z(K>/ZC0NST)))/(Z(2)-Z(1)) C 108 DO 110 K = 1,N TT(K)=(TL(K)-TL(1)) 110 TI(K)=(TT(K)-TC(K)) N1=N-1 DO 115 K=2,N1 P ( K ) = ( (TI (K + 1 ) — TI (K-1 )) / (Z(K + 1)”Z (K-1.) )) *1 5677. 115 PD(K)=P(K)*.269 91

DO 14D L = 1,M DO 12G K = 1 ,N 120 I F (Z.N (L) .GE.Z (N + 1-K) ) GOTO 130 1 30 K =N +1-K PIN(L)=P(K)+(( P ( K +1)-P(K ))/(Z(K + 1)-Z(K)))*(ZN(L)-Z(K) ) TII(L)=TI (K) + ((TI (K + D-TI ( K ) ) / (Z ( K + 1 )-Z (K ) ) ) * ( Z N ( L > ~Z (K ) ) TL1(L)=TL(K)+((TL(K+1)-TL(K))/(Z(K+1)-Z(<)))*(ZN(L)-Z(K)) NT(L)= NC(L)*22.11 NE(L)=NC(L)*17.2 DM(L)=NT(L)/3715. PN(L)=(PIN(L)/NE(L)>*1000. VM(L)=TII(L)*51400./NT (L) RM(L)=(VM(L)/4.19)**.333333 PNU(L)=2*(PN(L)/(“TL1(L)*RM(L)*.566*4.*3.14159)) 1 40 PNUE(L)=100.*(1.2/PIN(L)+0.Q2/(-TLI(L))+(2.55E-5*Z(L))/TII(L) 1+.66/SQRT(FL0AT(NC(L)))) C C OUTPUT OF DATA C NAME( 5 ) = I P R N A w E(6)=1ES CALL ASSIGNC1,NAME,12,IER) NAME(5)=IBL NAME(6)=I8L WRITE(1,145)NAME 145 FORMATC EXPERIMENT # 1,6A2/1X,'INPUT DATA :') IF (EQU)147,147,152 147 WRITE(1,150)RE FORMATI'RE •,F6.0,3X, 'NON EQUILIBRIUM COOLING'/ 1 50 NC ' / 11X,' TIME TL TR',1 OX,' TIME 21X,' SEC C C',10X,' SEC') GOTO 154 1 52 WRITE (1,153)RE FORMAT( ' RE = •,F6.0,7X,'EQUILIBRIUM COOLING'/ 1 53 NC ' / 11X,' time TL TR',1 OX,' TIME 21X,' SEC C C',10X,' SEC') 154 DO 155 K=1,M 1 55 WRITE(1,160)Z (K),TLM(K),TR(K),ZN(K),NC(K) 1 60 FCRMAT(1X,F10.0,F10.3,F10.2,10X,F10.0,I10) M1=M+1 DO 165 K =M1,N 1 65 WRITE (1,170)Z(K),TLM(K),TR(K) 1 70 FORMAT(1X,F10.0,F10.3,F10.2) WRITE (1,190) F 0 R M A T ( / ' RESULTS :'/ 190 PD ' 11X* TIME TT TC TI P 2/1X,' SEC C C 3 C WATT KWATT/M-3' ) WRITE(1,200)(Z(K),TT(K),TC(K),TI(K),P(K),PD(K),K-1,N) 2 00 F0RMAT(1X,F10.0,3F10.3,F10.1,F10.2) WRITE (1,210) RM 210 FORMAT(/1X,' TIME TII DN NE PIN PN VM 1 NU NUE') 92

WRITE(1,215) 215 FORMATdX,' SEC C CM“-3 WATT MW ATT MM “3 MM • 2,8X,'PERCENT') WRITE(1,220)(ZN(L),TII(L),DN(L),NE(L),PIN NAME(5)=IBL NAME(6)=IBL CALL SYMB0L(0.,-5.,.5,NAME,U. CALL PLOT(0.,G.,999) GOTO 15 END UNCLASSIEIED SECURITY CL ASSI PIC ATI OH OP THIS P AG E fW><«n P R » READ INSTRUCTIONS /J report documentation page BEFORE COMPLETING FORM 2. R E C IP IE N T 'S c a t a l o o h u m e e n 3 2. OOVT ACCESSION HO. T t m Report Nfry 214 | ( t y p e o> p r r e p o r t a p e r i o d ci o v e r e d 4. TITLE (And SubtUlm) (Final Report Dec . 1977 - May 191978). ^Frazil Ice Formation in Turbulent Flow S. PERFORMING ORO. REPORT NUMBER IIHR Report No. 214 FA. ¿ONTRAéf O k ORANT NUMBER!*; . AUTHOR^ DACA89-77-0536 Andreas Müller

Institute of Hydraulic Research ^ The University of’Towa^,. * 12. R EPO R T D A T E I I. controllino o p p i c e n a m e a h d a d d r e s s January 1978^

U.s. Army Cold Regions Research $ Eng. Lab. IS. NUMBER OP PAGES P 0 Box 282, Lyme Road 95 ______Hanover. New Hampshire 03755— __ IS. SECURITY CLASS. ( ° l M a MONITORING AGENCY HAME A ADORES*» *— Controlling Of/le ) UNCLASSIFIED

-IS«. OECLASSIFICA+ION>DOWNORADINO SCHEDULE

s. DISTRIBUTION STATEMENT («I A li * » « < )

Distribution Unlimited

17. D IS TR IB U TIO N s t a t e m e n t io t t t.. HapaH}

IS. HEY WORDS fConl Im h on » « « • »IB* I t n * e * « M > r • " * U m lU r Ay block mm tit)

Experiment; Frazil ice; Growth; Ice Seeding; Turbulent Flow

20. ABSTRAC T (Continu» an mrmm oldo U •mmmmr m * làmtllf h, AJoufc «»I'D Turbulent affects frazil ice formation in three ways: - it prevents stratification and formation of surface ice - it is involved in production and transport of ice nuclei - it controls the transfer of the latent heat of fusion from t e growing ice particle to the supercooled water. To study ice nucléation and heat transfer, an experiment was conceive w li­ ai lowed frazil ice to be produced under controlled conditions. Turbulence was

EDITION OP * NOV Et IS OMOLCTE yn yn ! ro"M JAN 73 g /N 0 1 0 2 *0 1 4 " 4401 I ITNfIT.ASSTFTF.n ______.LIUK1TV CLASSIFICATION OF t h i s PAGEOry»«« Put* Knfr+d)

generated by'a moving grid in a turbulence jar, where water could be cooled below the freezing point. Frazil was observed and photographed by means of a schlieren optics. Several experiments were conducted for between - 05°C and -.30°C and two velocities corresponding to Re - 1500 23UU based on ¿he diameter of the grid rod. The rate of ice productron and the total ice mass was calculated from the increase of temperature. The number of ■j pp ■na.T'ticlcs wexe counted on the ph.otogx3.plis. # , i PNo frazil ice, regardless of turbulence and foreign material, was observed! unless the water was seeded with ice nuclei. The number of particles grew during the experiment; the growth rate increased with greater supercooling a higher velocity of the grid. This indicates a multiplication process induce bygsecondary nucleation. The heat transfer per particle normalized with supe ■ cooling and the size of the particles was constant in all experiments within

the " F r r t L L ' observations, it can be concluded that the total ice produc- tion Is predictable if the heat transfer per particle can be estimated from turbulence data and if the number of particles can be calculate^ A "uclea tion theory is, however, not available and is regarded as the crucial question

SECURITY CLASSIFICATION OF THIS PAOEfPMf» Dmrn «totor*O