University of New Hampshire University of New Hampshire Scholars' Repository
Doctoral Dissertations Student Scholarship
Summer 1974 AN APPROXIMATE SOLUTION TO THE MOVING BOUNDARY PROBLEM ASSOCIATED WITH THE FREEZING AND MELTING OF LAKE ICE STEPHEN DANFORTH FOSS
Follow this and additional works at: https://scholars.unh.edu/dissertation
Recommended Citation FOSS, STEPHEN DANFORTH, "AN APPROXIMATE SOLUTION TO THE MOVING BOUNDARY PROBLEM ASSOCIATED WITH THE FREEZING AND MELTING OF LAKE ICE" (1974). Doctoral Dissertations. 1064. https://scholars.unh.edu/dissertation/1064
This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. INFORMATION TO USERS
This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted.
The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction.
1.The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity.
2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame.
3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete.
4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced.
5. PLEASE NOTE: Some pages may have indistinct print. Filmed as received.
Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 75-1655 FOSS, Stephen Danforth, 1941- AN APPROXIMATE SOLUTION TO THE MOVING BOUNDARY PROBLEM ASSOCIATED WITH THE FREEZING AND MELTING OF LAKE ICE. University of New Hampshire, Ph.D., 1974 Engineering, chemical
Xerox University Microfilms, Ann Arbor, Michigan 48106
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. AN APPROXIMATE SOLUTION TO THE MOVING BOUNDARY PROBLEM ASSOCIATED WITH THE FREEZING AND MELTING OF LAKE ICE
by
STEPHEN D. FOSS B.S., University of New Hampshire, 196-4 M.S., University of New Hampshire, 1965
A THESIS
Submitted to the University of New Hampshire In Partial Fulfillment of The Requirements for the Degree of
Doctor of Philosophy Graduate School Engineering Ph.D. Program Transport Phenomena July, 1974 This thesis has been examined and approved.
S J 'Yo j a z. Thesis direq ;or, Stephen S. T. Fan, As so. Prof. of Chemical Engineering
William Mosberg, Asso. Prof. of Mechanical En leering
£ C Shan S. Kuo, Prof. of Mathematics
Russell L. Valentine, Asso. Prof. of Mechanical Engineering
Francis Hall, Prof., Institute of Natural and Environmental Resources
Q julQu u a ^ ZJ., Date '
ii TABLE OF CONTENTS
ACKNOWLEDGEMENTS ...... V
LIST OF TABLES ...... vi
LIST OF FIGURES ...... vii
ABSTRACT...... ■___ viii
I. INTRODUCTION...... 1
II. DESCRIPTION OF THE PROBLEM...... 13
A. Solution of Moving Boundary Problem Utilizing a Quasi-Steady State Assumption...... 16
B. Comparison to Westphal’s Solution...... 18
III. ONE^-DMENSIONAL FREEZING OF ICEWITH CONSTANT HEAT FLUX IN THE WATER L A Y E R ...... 24
A. Factors Contributing to Heat Flux ...... 26
B. Fundamental Equations ...... 27
C. Solution Utilizing Quasi-Steady State Assumption...... 28
D. Specific Examples ...... 29
1. Case I Step drop in ambient air temperature ...... 30 2. Case II Sinusoidal variation in ambient air temperature ...... 30
IV. DEVELOPMENT OF THE MOVING BOUNDARYPROBLEM TO SIMULATE THE FREEZING AND MELTING CONDITIONS ON A L A K E ...... 35
A. Factors Considered in Developing the M o d e l ...... 35
B. Fundamental Equations ...... 36
C. Determination of the Characteristic Heat Flux in the Water layer...... 4l
V. LAKE ICE FORMATION EXPERIMENT...... 44
A. Great East Lake ...... 44
iii B. Description of the Experiment ...... 44
C. Supplemental D a t a ...... ,...... 54
VI. ICE DEPTH PREDICTION...... 65
A. Prediction During January Observation Period...... 65
B. Prediction of Heat Flux Frcm February Through March D a t a ...... 65
C. Heat Flux Prediction...... 67
D. Numerical Procedure ...... 72
VII. CONCLUSIONS ...... 74
VIII. NOMENCLATURE...... 78
IX. BIBLIOGRAPHY ...... 80
X. APPENDICES...... 86
A. EXACT ANALYTICAL SOLUTION OF WESTPHAL...... 87
B. SENSIBLE HEAT CONSIDERATIONS...... 90
C. EFFECTIVE THERMAL CONDUCTIVITY OF S N O W ..... 91
D. AMBIENT AIR TEMPERATURES TAKEN DURING THE FIRST OBSERVATION PERIOD OF JANUARY 7th TO JANUARY 20th, 1974 ...... 92
E. DEFINITION OF CORRELATION COEFFICIENT...... 104
F. COMPUTER PROGRAMS OF ICE MO D E L ...... 106
iv ACKNOWLEDGSVENTS
The author is in deep gratitude to his advisor, Professor
Stephen S. T. Pan, for his continued guidance and sincere interest throughout the development of this research. Professor Pan’s helpful suggestions for improving the final manuscript was greatly appreciated.
The author also wishes to thank his wife, Maureen, for her patience and understanding during the many difficult years of my PhD career. Por this I am deeply indebted.
The author wishes to thank Mrs. Brenda Cavanagh for her excellent typing of this manuscript. LIST OF TABLES
Table Page
I. Deviation of Ice Depth for 'Various Stefan Numbers ...... 23
II. Extinction Factors ...... 42
III. Ice Depth Measurements at Great East Lake Wakefield, N.H. - Winter 1971! ...... 52
IV. Snow Depth Observations ...... 53
V. Average Daily Air Temperatures ...... 55
VI. Concord Weather lata During First Observation Period 60
VII. Air Temperatures during Second Observation Period...... 6l
VIII. Concord Weather lata During Second Observation Period .... 64
vi LIST OP FIGURES
Figure Page
1. Simplified Illustration of Air-Ice-Water System ...... 14
2. Comparison of .Approximate Ice Depth Prediction versus Westphal’s Exact Solution ...... 22
3. Air-Ice-Water System with Constant Heat Flux Existing in the Water Phase ...... 25 2 4. 5 vs e Step Change Air Temperature with Constant Heat Flux in Liquid Phase ...... 31 2 5. 6 vs 0 Periodic Air Temperature with Constant Heat Flux in Liquid Phase ...... 33
6. Air-Snow-Ice-Water Problem...... 37
7. New Hampshire and Maine ...... 45
8. Great East Lake Wakefield, New Hampshire ...... 46
9. Ice Cover on Great East Lake (March 8, 1974) ...... 47
10. Ice Depth Measurement on Great East lake (March 8, 1974) ...... 47
11. Ice Depth Measurement on Great East Lake (March 8, 1974) ...... 49
12. Thermally Regulated Cabinet for Temperature Recorder 49
13. Sketch of Ice Depth Measuring Device ...... 50
14. Scatter Diagram of Lake Air Temperature and Concord Air Temperature From January 7th to January 2 0 t h ...... 56
15. Scatter Diagram of Lake Air Temperature and Portland Air Temperature Fran January 7th to January 20th ...... 57
16. Snow-Ice Cover on Great East Lake Together with Predicted Ice Depth Frcm Mathenatical Model Assuming No Heat Flux (First Observation Period) ...... 66
17. Snow-Ice Cover on Great East Lake Together with Predicted Ice Depth Frcm Mathematical Model Assuming No Heat Flux (Second Observation Period) ...... 68
18. Snow-Ice Cover on Great East Lake Together with Predicted Ice Depth Using Calculated Heat Flux (Second Observation Period) ...... 71 vii ABSTRACT
AN APPROXIMATE] SOLUTION TO THE MOVING BOUNDARY
PROBLEM ASSOCIATED WITH THE FREEZING AND MELTING OF LAKE ICE
by
STEPHEN D. FOSS
In this thesis, an approximate solution Is developed for the
moving boundary problem associated with the freezing and melting of
lake Ice. The difficulty encountered in obtaining solutions to moving
boundary problems is due to the fact that the differential equations
describing the phenomena are coupled and non-linear. Mathematicians and
engineers through the years have applied both analytical and numerical
techniques in solving moving boundary problems, however, due to the
complexity of the basic phenomenological equations, many of these solu
tions have been restricted to systems with relatively simple boundary
conditions. Boundary conditions encountered in natural phenomena involving phase changes are often rather complex.
The mathematical model developed made use of a quasi-steady
state assumption for the ice temperature profile. This assumption eliminated the need to solve the parabolic energy equation describing the heat flow within the ice layer. The validity of this assuimption was dependent on two criteria. First, the ambient air temperature must vary slowly with time. Secondly, the Stefan number (defined as the ratio of the driving force for freezing to the resistance to freezing) must be small. The solution obtained by utilizing the assumption was compared with an exact analytical solution obtained by Westphal (16)
viii with simple boundary conditions. Frcm the comparison, a Stefan number range was determined for which reasonable accuracy could be achieved using the approximate solution.
The quasi-steady state assumption was then incorporated in developing a mathematical model describing ice growth and decay on a lake. To verify the mathematical model, an experiment was conducted at
Great East Lake in Wakefield, New Hampshire. The purpose of the exper iment was twofold. First, climatological data was obtained to be used as system parameters and variables in the mathematical model. Secondly, ice depth measurements were made to compare with the model. Additional data was obtained from the U. S. weather stations at Concord, N.H. and
Portland, Maine. This data supplemented climatological data taken at the lake site. The ice depths observed during the experimental study agreed quite closely with the predictions made using the mathematical model.
Application of the quasi-steady state technique to the moving boundary problem dealing with alloy solidification or glass melting exists. However, care must be exercised to ensure that the basic criteria are met. 1
I. INTRODUCTION
This thesis considers a problem which falls into a broad
classification conmonly referred to as Stefan-like problems, moving-
boundary problems, or fhee-boundary problems. The essential feature
of these problems is the existence of a moving boundary, the location
of which is dependent on the diffusion or phase-changing process itself.
A few familiar examples are the following: growth or collapse of a
bubble in a one or two-component system; process of freeze-drying;
diffusion in a two-phase medium with a moving interface; conductive
drying; evaporation or condensation; progressive freezing or melting.
The first known solutions to problems of these types date back
to the l860's when Franz Neumann presented analytical solutions of a
semi-infinite region either all solid or liquid, subjected to a step
temperature change at the boundary. Most of these solutions are listed
in Carslaw and Jaeger (1). In 1891, Stefan (2) simplified Neumann’s
solution to the case in which the liquid is initially at the fusion temperature. Since then, numerous solutions to the problem have been presented by both mathematicians and engineers. These solutions can be classified as follows:
(1) exact analytical
(2) approximate analytical
(3) numerical
A number of papers has been published proving the existence, uniqueness and stability of a solution to the Stefan problem (3,^,5j6).
In all of these papers, although uniqueness and existence were determined, no solutions were given. Several review articles have been published 2
In the last decade concerning heat conduction with melting and freezing (7,8,9,10,11,12).
Carslaw and Jaeger (1) extended the exact solution of Neumann to include radial freezing of infinitely long cylinders and spheres.
A more recent contribution, that of Portnov (13) gives the exact solution to the freezing problem with arbitrary temperature variation of a fixed boundary. Portnov in his work used a form of Poisson integral to find the exact solution. Jackson (14) shows that the integral derived by Portnov will also provide exact solutions for other important problems involving phase change in finite slabs. Langford
(15) presented a set of new analytical solutions of the heat equation for the case in which both the temperature and heat flow rate are prescribed at a single fixed boundary. Westphal (16) utilized the method due to Portnov to solve the problem of finding the location of the progressing phase-change front under the assumption that the radiation boundary condition is imposed at the fixed surface.
Variational techniques were developed by Biot (17). The method was developed using Qnsager's theorem, which is a reciprocity law of coupled irreversible or nonequilibrium processes. Because of Biot’s extensive contributions to the field, the technique is usually referred to as Biot's variational principle.
By integrating the energy equation over the region of interest, the equation can be reduced to an ordinary differential equation in time. The resulting integral quantity was called the heat balance integral by Goodman (18). The integral is analogous to the manentum integral developed by Von Karmen and Pohlausen for boundary layer theory (19). The solutions assumed that the temperature profile is 3
expressible as a polynomial in the distance variable with undetermined coefficients which are a function of time. When the equation was substituted into the heat equation and integrated, a set of ordinary differential equations for the coefficients was produced. These coefficients were determined by applying the boundary conditions. The basic disadvantage to the method is that it yields only average values.
Snyshlyaev (20) made use of an approximate integral solution to solve the problem of heat propagation and sublimation of the outer surface of a plate of finite thickness. Rathjen and Jiji (21) derived a nonlinear singular integro-differential equation for the solid-liquid interface to solve a heat conduction problem with melting or freezing in a two- dimensional comer. Elmas (22) uses an integral technique to develop an approximate solution to the moving boundary problem associated with dip coating in fluidized beds. A note by Savino and Siegel (23) showed how Elmas’s solution can be made analytic.
A number of papers was published giving approximate analytical solutions to moving boundary problems. landau (24) eliminated the moving boundary by introducing a transformation that is now called the
Landau transformation. By use of the transformation the problem was put into a form involving only one parameter. For the case of a semi-infinite melting solid with its face heated at a constant rate, he derived a quasi-steady state solution. Kcmori and Hirai (25,
26) obtained an approximate analytical solution of the heat conduction problem with change of phase for a slab. In their paper they compared the approximate solutions of Neumann and Stefan to experimental results obtained in freezing soy bean curds. A number of approximate analytical solutions dealing with the freezing of saturated liquids 4
Inside or outside of cylinders and spheres was published (27,28,29)<
Iheofanous and Lim (30) make use of an approximate analytical solution
to solve the diffusion equation for non-planar moving boundary problems.
A number of numerical solutions has been published. Pechoc
(31) presented a generalized numerical solution of the heat equation
with moving boundary. Tao (32) presented a generalized numerical
solution of freezing a saturated liquid in cylinders and spheres.
In a later paper Tao (33) derived a generalized solution of freezing
a saturated liquid in a convex container. Teller and Churchill
(34) presented a numerical solution for the freezing of a liquid
outside a sphere. Seider and Churchill (35) studied the effect of
insulation on a one dimensional freezing front. Lazaridis (36)
presented a numerical solution of the multi-dimensional solidification
(or melting) problem.
A number of papers have been published utilizing the
perturbation technique to solve freezing and melting problems. Lock,
et. al. (37) presented a paper to study one-dimensional ice formation near the surface of a semi-infinite domain. They developed approximate
solutions by perturbation expansion. A boundary condition existing in
the problem was found to be incorrect and was corrected by Lock (38).
Pedroso and Domoto (39) obtained the exact solution by perturbation for a planar solidification of a saturated liquid with convection at the wall.
Chuang and Szekely (40,41) presented papers utilizing Green’s functions for solving melting or freezing problems. Their technique combined analytical and numerical methods. 5
In addition a number of papers have been written dealing with approximate solutions. Chou and Sunderland (42) presented an approximate solution to the phase change problem of spherical bodies. In their solution they assumed that the one-dimenslonal solution of Stefan can be applied for the spherical system. Lim and Theofanos (43) presented a pseudo-steady state solution for unsteady state moving boundary problems for non-planar geometries.
Many other papers dealing with the moving boundary problem have been written. To list only a few. Boley (44) presented a paper utilizing the "embedding technique" to obtain a starting solution for the melting and solidifying of a slab. Later Yogoda and Boley
(45) published a paper using the same technique to obtain starting solutions for melting of a slab under plane of axisyrnmetric hot spots. Lin (46) used a transformation technique to calculate the solution to a one-dimensional freezing or melting process in a body with variable cross-section. Gringerg (47) presented a paper based on the expansion of solutions into series in terms of
"instantaneous" eigen-functions of the corresponding problems.
Selmin (48) in his PhD thesis solved the moving boundary problem in finite regions by using a finite integral-transform technique.
He applied the method to various geometries and used the method to analyze the diffusion controlled penetration of a reactant into a spherical ion exchange particle. A large number of papers have been published with specific application to moving boundary problems. Krasnikov and Snimov (49) formulated and solved the b
conjugate problem of heat conduction in relation to conductive drying. Their approximate solution was analyzed and compared with experimental data. The agreement between analytical and experimental results indicated that the mechanism was properly understood.
Demchenko and Koslitina (50) solved the diffusion problem with a moving interface numerically. Their method was based on a integro- interpolation method. Dyer and Sunderland (51) presented a paper with application to the freeze drying of bodies exposed to thermal radiation boundary conditions. In their study they made use of a quasi-steady state assumption. Theofanous, et. al. (52) by using an integral method solved the problem of convective diffusion bubble dissolution. Olsen and Hilding (53) studied both analytically and experimentally the transient process of simultaneous condensing and freezing of a pure vapor upon an initially dry very cold infinite horizontal plate. A short note has been written by Kehoe (.54) describing a moving boundary problem with variable diffusivity. The method used was one of successive approximation. Kcmori and Hirai (55) discussed the application of Stefan's problem to the freezing of cylindrically shaped food. Muehlbauer, et. al. (56) using approximate solutions determined temperature distribution and rate of phase change for the transient one-dimensional solidification of a finite slab of a binary alloy.
A number of recent papers have been published dealing with the estimation of ice growth on inland waters and sea ice. The typical moving boundary problem is ccmplicated in the ice growth studies of 7
lakes and ponds. Typically, problems arise from the following sources; transient ambient air temperature variations, effect of wind velocity on heat transfer, snow cover on ice, surface melting, slush from melting snow and/or rainfall, solar radiation, heat flux existing in the water layer beneath the ice, density inversion of water at 4°C and variation in the physical properties. Because of these complications it is extremely difficult to apply exact analytic or sophisticated numerical techniques to solving the moving boundary problem. As a result, many papers have been published utilizing approximate solutions to the moving boundary problem. The approximations are based on physical intuition of the heat transfer mechanism. Lee and Simpson (57) presented an operational method of forecasting ice growth. The forecast of ice growth based on the ice potential can only be used during the period when ice thickness is increasing. Bilello (58) presented a paper dealing with the formation, growth, and decay of sea ice. In this paper Bilello reviews the methods of previous investigators. Equations relating to the accrea- tion of sea ice to meterological data are derived empirically frcm obser vations. later Bilello (59) published a paper predicting river and lake ice formation. In this paper Bilello made use of an empirical relation to determine the date of ice formation. Adams, et. al. (60) published a paper on the formation of sea ice. In this paper they utilized a quasi steady state assumption for the temperature profile in the solid phase.
No effect of existing heat flux in the water layer was presumed. They did however account for the effect of solar radiation and wind velocity on the heat transfer coefficient. Williams (61) calculated the effect of thermal radiation on the heat flux in the water phase. In this paper, he investigated possible sources of heat generation in the water layer 8
beneath the ice. He concluded by stating that during the melting
period of lake ice radiation effects were far predominant. In a
recent paper by Fertuck, et. al. (62) estimations of ice growth were
made numerically as a function of air temperature, wind speed and snow
cover. In this paper a finite difference technique was used. However,
no account of heat flux in the water phase was given.
In this thesis, a mathematical model is developed to simulate
the periodic freezing and melting of lake ice. Geometrically the
problem is considered to be a function of one space dimension and time.
Previously both analytical and numerical techniques have been developed
for solving the freezing and melting problem associated with the ice- water system. However, most of these methods have been applied to relatively simple and idealized boundary conditions. The boundary
conditions considered were either a fixed temperature maintained at the
stationary boundary or the radiative boundary condition applying at the air-ice interface. Furthermore, it was also assumed that the water temperature was at the melting point of ice, thereby eliminating heat flux considerations within the water phase. The boundary conditions ccmmonly associated with the freezing and melting of lake ice depend on many system parameters and variables that may vary as a function of time. These parameters and variables are wind velocity, ambient air temperature, solar radiation, snow cover, snow density, effective thermal conductivity of snow, rainfall, slush, white-ice formation and lake bottom characteristics. Variation in these parameters and variables with time presents far more complicated boundary conditions than the idealized boundary conditions applied to many of the previous solutions. 9
lb illustrate the complexity involved in obtaining an analytical
solution to an ice-water system involving simple boundary conditions, the
solution of Neumann (1) will be briefly discussed. The system considered
is a liquid system contained within the semi-infinite region x > 0. The
temperature of the liquid region is initially constant, T^, which is
greater than the melting point. The surface temperature at x = 0 is
subsequently maintained at zero temperature. Neumann assumed all
physical properties such as density, specific heat and thermal conductivi
ties, etc. constant.
Suppose that is the latent heat of fusion of the substance
(BTU/lb) and that Tq is its melting point, then if the surface of separ
ation between the solid and liquid phase is at b(t), one boundary
condition to be satisfied at the surface is that the temperature of the
solid phase equals the temperature of the liquid phase is equal to the
melting point of the solid
T_ = T, = T , when x = b(t) (1) S _L O
A second boundary condition concerns the absorption or liberation of
latent heat at this surface. The amount of heat liberated at the
boundary must be removed by conduction. This requires
3T, aT db k - k — §■ = P x — (2) 9x 9x dt where k is the thermal conductivity and p is the density.
For linear flow of heat, the temperatures Tg and T^ in the
solid and liquid region must satisfy
a2T 1 9T — 2^ = ------(3) ax a at s 10
9 Tx 1 3TX (4) 9x where a = k/pc .
Finally the boundary condition existing at the fixed boundary is that the solid temperature is at its melting point temperature, T .
T = T when x = 0 (5)
The solutions to Equation 3 and 4 as given in Cars law and Jaeger (1) are
T T = erf s (6) erf 3 2 (a t)1/2
T, - T x b o T, = T, ------T ~ 7 o — erfc 172 (7) b erfc [3(— ) ] 2 (a-jt) al
rx — C*- p where 3 is given by the transcendential equation, erfx = 2/v/JF de and erfc x = 1 - erfx
,2 i / o -a.^/a. 1/2 h »s (Tb - V e 172: (8) erf 3 ks a 1172 Tq erfc C3(as/a1) ] cPs T
Inserting Equations 6 and 7 into Equation 2 yields an ordinary differential equation which when integrated gives the position of the interface as a function of time. EUe to the fact that many of the system variables are time dependent, the system studied in this thesis is more 11
complicated. Furthermore, the appearance or disappearance of snow
layers, white ice formation, and heat flux existing within the water
layer greatly complicates the problem. The mathematical model
developed in this case makes use of a quasi-steady state assumption.
This assumption determines the form of the temperature profile which
simplified the solution to the moving boundary problem. The simplified
solution which was developed, is compared with an exact analytical
solution of Westphal (16). The purpose for the comparison was to
determine the validity of the quasi-steady state assumption in satisfying
the energy equation. It was noted that the two solutions agreed very
well for small Stefan numbers (c (T - T )/X„) . However, as the P O cl I Stefan number increases, the deviation between the two solutions becomes
apparent. The magnitude of the Stefan number was found to be a signifi
cant parameter in determining the validity of the quasi-steady state
assumption.
A mathematical model was then developed of the periodic freezing
and melting on a lake or pond. The model has made use of the quasi
steady state assumption. In conjunction with the development of the
model, a lake ice experiment was conducted during the winter months of
early 197^ on a lake in east central New Hampshire. The main purposes of
the experiment were the following: (1) to obtain climatological data
during the freezing and melting period to be used as parameters in the mathematical model and (2) to obtain ice depth measurements in order to
verify the validity of the model. Tb supplement the data taken at the
lake site, climatological data frcm the United States weather stations in
Concord, New Hampshire and Portland, Maine was also obtained. The 12
agreement realized between Ice depth predictions determined by applica tion of the mathematical model and the actual Ice depth observed at the lake site was very good considering the uncertainties In some of
the climatological data and physical property parameters used. II. DESCRIPTION OF THE PROBLEM
The moving boundary problem under consideration can be described
most simply by examining the following simplified statement of the
problem.
Consider initially, for t < 0, a system consisting of air and
water in thermal equilibrium. The temperature of the air is maintained
at the freezing point of water and the temperature of the water is
uniform also at its freezing point. For t > 0, the air temperature is
subjected to a temperature fluctuation that is a function of time,
TCl (t). For the freezing problem it is assumed that the air temperature is less than or equal to T , where TQ is the melting point of ice. As
supercooling effects are neglected, ice begins to form at a thickness b(t) for t > 0. Figure 1 illustrates the moving boundary air-ice-water
system under investigation. In the foregoing discussion, it is assumed
that the heat transfer process is one-dimensional. Heat transfer effects in the lateral directions have been neglected. For systems such as lakes and ponds where end effects can be neglected, this assumption is reason able.
The fundamental equations described in this system are: For one-dimensional heat conduction within a solid phase medium with no internal energy sources, the energy equation describing the heat conduction process within the ice is given by 14
boundary conditions
ks-JT °hg(T8-Ta (t)) x=0, t> 0
x=b(t), „ dT. t e O * r r r * 0 a x
P Xr^2- - k £!L-k x=b(t), ps^dt ks ax kiTx t> 0 initial conditions \ S T0 t< 0
Ta =Ta lt) t > 0
b(0)=0
Air Tfl (t) x*0
x«b(t)
*
Rgure L Simplified Illustration of Air-lce-Wbter System 15
where p - density of ice, s c = specific heat of ice, ps ’ k = thermal conductivity of ice, s Tg = ice temperature,
x = position variable,
t = time,
b(t) = ice thickness.
The boundary conditions associated with Equation 9 are the
following: It is assumed that Newtons law of cooling applies as a
convective boundary condition at the air-ice interface. This'equation
relates the rate of heat conduction in the ice to the rate of heat
convection to the air.
3T k — = h (T - T ) x = 0, 3 9x g S a t > 0 (10)
where h =? heat transfer coefficient between air g and ice
T& = ambient air tanperature.
At the ice-water interface it is assumed that the ice and water are in thermal equilibrium and at the melting point T , therefore
Ts = Tf = Tq x = b(t), t > 0. (11)
The boundary condition that describes the movement of the ice- water interface is given by
db 3T 3T, p Af — = k kn --- x = b(t), dt 3x x 3x t > 0. (12) 16
where Af = latent heat of fusion of Ice.
Ihis expression related the progress of the phase moving front to the
difference between the heat conducted out of the water and that conducted
out of the Ice.
The initial conditions to be applied to the problem are that
Initially the system consist of air and water, whereby the water temp
erature is maintained at its freezing point T .
b(o) = 0 , T = T , t = 0 (13) 1 o
Since T^CxjO) = Tq for t = 0 and T^Mt),!) = Tq for t > 0,
then T^(x,t) = Tq for x > b(t) and t > 0. Therefore, for this problem
3T — = 0 , X = b(t) , t > 0 (14) 8x
A. Solution of Movlng-Boundary Problem Utilizing a Quasi-
Steady State Assumption
The boundary conditions applied to the problem of freezing and melting on a lake or pond often complex. It is often necessary to simplify the solution to the energy equation in order to arrive at a solution to the freezing process. Furthermore, a simplified solution if proven valid may become more adaptable than the more sophisticated solution. To simplify Equation 9, the energy equation, a quasi-steady state assumption is imposed. It has been assumed that the ice temperature varies linearly with distance in the form given by,
Tg(x,t) = a^(t) + a£(t) x 0 < x ^ b(t) (15) 17
where and a2 are parameters that depend on time. The validity of
this assumption for the solution to the energy equation in the ice layer depends upon two existing satisfying criteria.
First, the ambient air temperature must vary slowly with time.
This criterion is a necessary condition that must be met in order t< satisfy quasi-steady state heat conduction regardless of the existence of a moving boundary. In stating this criterion differently, the rate controlling mechanism of heat transfer exist at the boundaries.
Secondly, it is necessary to show that the Stefan number is a meaningful parameter in establishing the validity of a linear temperature profile in the ice. Physically the Stefan number is the ratio of the driving force for freezing c„(tQ - T& ) to the resistance of freezing and 25 presents a measure for the relative speed of movement of the freezing front. For small Stefan numbers the movement of the ice is rate controlling, therefore, the validity of the assumption is most strongly dependent on the first criterion. These criteria can be stated briefly as follows:
(1) Slowly varying ambient air temperature
(2) Snail Stefan number, where Ste = c (t - T0 )/A„, c = p o a i jo specific heat of the ice, Tq = melting point temperature
of ice, T = ambient air temperature and A„ = heat of cl X fusion of ice.
To determine the validity of the quasi-steady state assumption, compari son is made with the exact analytical solution by Westphal (16). Westphal developed a series solution to a one-dimensional freezing problem for the case whenNewton’s law ofcooling holds at a fixed boundary. He uses the Method of Portnov (13) expanding theposition of thephase- change front progress in a power series in powers of \ZTT (Appendix A). 18
Applying the convective boundary condition, Equation 10, to the
quasi-steady state assumption, Equation 15 gives
9T k — s = k a?(t) = h [a, Ct) - T (t)] (16) 9x x=0 S d S 1 a
implying the melting point boundary condition Equation 11 to
Equation 15 gives
Tq = a-j^Ct) + a2(t) bCt) (17)
Substituting Equations 16 and 17 into Equation 15 yields the following
ice temperature profile relation
h [x - b(t)] [T - T (t)] Ts (x,t) = T0 + f ------h m ---a--- (18) ks 1 + — ---- k s
Applying Equation 10 to the ice-water interface, Equation 12, yields
db 9T_ h [T - T (t)]
ps Af — = ks — “ " V t o 7 ~" (19) dt 9x X=b(t) x
ks
Equation 19 describes the movement of the ice-water interface as a func
tion of time.
B. Comparison to Westphal*s Solution
Westphal illustrates his method for the case whereby the ambient air temperature is subjected to a step drop. Such would be the case 19
associated with freezing of water in a refrigerator. For this case
Tn Ct) = T , t > 0 (20) a d. where T < T . He assumes that the physical properties of the ice 3. O together with the hea.t transfer coefficient remain constant. Making use of these same conditions and integrating Equation 19 with the initial condition b(0) = 0 gives
2k 2k b (t) + — b(t) + — [T - T ]t= 0 (21) 1 . cl O hg ps Xf or
k k „ 2k i/p b(t) = - — + [(— )2 --- — (T - T )t] h i . a, U g g Ks f
Since Westphal expressed his variables in dimensionless forms, it is necessary to rewrite Equation 21 in terms of the following same dimensional groups so that a meaningful comparison can be made.
h_ b(t) 6 = - S , k s l/p 6 = 2hn.('t/Po1g S CoCOo) S Ps 311(1
Ste = 0p(T0 - Ta)/Xf
2 5 can be considered as a dimensionless ice thickness parameter, 0 a dimensionless time parameter and Ste the Stefan number. By inserting 20
these groups into Equation 21, the following expression results
02Ste 1/2 6 = -1 + [1 + ] . (22) 2
Equation 22 expresses the dimensionless depth parameter as a function of dimensionless time for a given step change Stefan number.
Hie validity of Equation 22 depends largely on the assumption of the linear depth dependence of T as a function of time. A step s change boundary condition should be viewed as a severe test, since a step change is clearly not a slowly varying temperature fluctuation.
However, since b(0) = 0 the need for a slowly varying temperature condition is virtually eliminated. Equation 22 is solved using the following physical properties of ice (Af = 144 BHJ/lb, cp = 0.5 BTU/lb/°P and Tq = 32°P). In testing the validity of the model the following four values of AT were used: -5, -10, -15 and -20°C. These temperature differences were identical to those used by Westphal in his conputation hence the results can be meaningfully compared. It is of interest to note that the temperature range illustrated in this comparison is well suited to the temperatures experienced in southern New Hampshire during the winter season. Southern New Hampshire is the location of the lake ice experiment . In corrparing the results of Equation 22 with that of
Westphal, it is necessary to make a transformation. In Westphal's derivation, p.. was used in Equation 12 rather than p . Therefore, in j. s order to make a meaningful comparison between the two results, it is necessary to modify Equation 22 by replacing 0 with 0*, which is defined as Hence we have
(6 + l)2 = 1 + j 0*2Ste (23)
2 2 when (6 + 1) versus 6* is plotted, comparison of the two results can be made (Figure 2). The two solutions agree remarkably well for small Stefan numbers and as the Stefan number increases the nonlinearity of Westphal’s solution becomes apparent, liable 1 compares the deviations produced by using the approximate solution to that of Westphal. Fran
Ihble 1 it can be noted that for = -20°C one can expect an error of less than 5 % in the ice depth predictions. Such errors are tolerable for many engineering calculations. 2.0 -
1. 8 - -
1.6 ■■ ,oO
1.4 .. oO
Figure 2. Comparison of Approximate ice Depth Prediction Versus Westphai's Exact Solution TABLE 1
Deviation of Ice Depth for Various Stefan Numbers
Ste T C°C) 0*' cl
0.0 0.0314 -5 16
0.0 0.0627 -10 16
0.033 0.0938 -15 16
0.05 0.1254 -20 16
note
- 5F o s s ^Westphal* ~ ^Westphal 24
III. ONE-DIMENSIONAL FREEZING OF ICE WITH CONSTANT
HEAT FLUX IN THE WATER LAYER
In the development of the fundamental equations in the previous section no account was made of any heat flux existing in the water layer beneath the ice. The reason for this was that the system had to be idealized so that a meaningful comparison could be made of the approximate solution to that of the exact analytical solution of
Westphal (16) in order to check the validity of the quasi-steady state assumption on the temperature profile in the solid phase. Once the validity of the quasi-steady state assumption has been established, together with the necessary constraints on the Stefan number to justify the assumption, attempts can be made to apply the model to cases involving more realistic boundary conditions.
In this section, the fundamental equations are developed by making use of the quasi-steady state assumption for the case where a constant heat flux exists in the water layer. Figure 3 is an illustra tion of the air-ice-water system under investigation. Previously investigators have assumed that the temperature of the water layer remains constant at the melting point of ice. Such a condition totally eliminates the existence of a heat flux in the water phase, thereby, simplifying the problem considerably. However, neglecting the existence of a heat flux within the water layer during the freezing or melting of lake ice could lead to considerable error. Generally lakes and ponds have associated with them temperature differentials that are a function of water depth. Due to the density inversion property of water at 4°C, an
Idealized representation of an ice-water system may be seen to consist 25
boundary conditions
ks 4 i r =hg - Ta » » x=0, t> 0 d T I •= constant x=b(t), t>0 x*b(t), V V To t> 0
* f -g jr = ks-| ~ - -constant x=b(t), t> 0 initial conditions Ta =T0 t<0 V Ta(,) t >0 b(0)=0
x= b (t) Water T, =Tj (x,t)
Figure 3. Air-Ice-Water System with Constant Heat Flux Existing in the Water Phase 26
of two distinct zones. The upper zone consists of a stable region where the density of water increases as a monotonic function with depth. In this zone conduction is the major contributing factor for heat transfer.
In the lower zone convection is the principal factor in heat transfer.
Due to the density inversion property of water at i)°C, the density of water decreases monotonically with increasing temperature above 4°C.
As a result, gravitational forces create natural convection currents in the lower zone.
The problem can be described as follows: initially for t < 0 the system consists of air and water, and the air temperature is assumed to be at T , the freezing point of water. Beginning at time t = 0 the air is subjected to a temperature fluctuation which is a prescribed func tion of time, T = T (t). With the formation of ice, the heat flux, c i cl Q^, between the ice and the water phase is assumed constant. The assump tion of constant heat flux in the water phase is justified if the water conditions beneath the ice are dynamically and thermally stable. It is still assumed that the ambient air temperature fluctuation is a slowly varying function of time so that a linear temperature dependence with ice depth can be applied. Also for the solution to the freezing problem T (t) is less than or equal to Tq . Supercooling effects will be neglected.
A. Factors Contributing to Heat Flux
During the freezing of lake ice, the water layer beneath the ice usually experiences a temperature gradient that results in a heat flux for melting which can be expressed as
s, - “l ™ 27
The main factors that contribute to the existence of the melting heat
flux are as follows:
(1 ) solar radiation penetrating the ice cover,
(2 ) residual sensible heat contained within the lake,
(3 ) springs and bottom characteristics,
(4) convection currents.
Williams (6l) considers these effects during the melting period on
White Lake, Ontario. He noted that the water temperatures during the thaw period was as high as 7-5°C. He attributed solar radiation as the major contributing factor resulting in the heat flux formation.
B. Fundamental Equations
The following appropriate boundary conditions are applied to the quasi-steady state assumption, Equation 15: Newtons law of cooling applies at the air-ice interface.
3T k — - = h [T - T (t)], x = 0, t > 0 (25) s 9x g s a
At the ice-water interface thermal equilibrium is assumed to exist between the water and the ice. Therefore, the ice and water temperatures are at the melting temperature Tq .
T = T = T , x = b(t) , t > 0 (26) S o
The velocity of the ice-water interface is governed by the difference between the heat conducted into the solid and that conducted out of the liquid. Therefore, Ihe following Initial conditions are seen to exist by the statement of the problem
b(t) = 0, T-.(0,t) = T (t) = T for t = 0 -L Q , 0 (28)
Equation 19 representing the temperature profile In the ice for the case when no heat flux exists in the water layer is still applicable for this solution. Ihe reason for this is because boundary conditions
(25) and (26) are similar to those applied previously. Therefore, applying Equation 19 to Equation 27 yields
db h [T - T (t)] _ g o a (29) dt 1 + -5
C. Solution Utilizing Quasi-Steady State Assumption
For the special case when average values can be used for the physical properties and the heat transfer coefficient, it is convenient to introduce the following dimensionless parameters and variables so as to generalize the relationships.
6 = h8 b(t> k
0 = 2 h (t/Cpkc ) )1/2 6 P s 29
Ste ' °PS (To ' V t))/Xf
S* - Q„ =ps/hg Xf
Ihe new dimensionless heat flux parameter Q* can be seen to be the
product of the Stefan number defined by Cps(Tg - Tq )/Ap and the ratio
of the heat conducted out of the liquid to the heat convected away at
the surface. By substituting these dimensionless parameters and variables
into Equation 29, the following dimensionless equation is obtained:
d6 e Ste Ce) ------Q * GO) de 2 1 + 5
Equation 30 relates the dependence of dimensionless ice depth to dimen
sionless time with the Stefan number and heat flux acting as governing parameters.
D. Specific Examples
ihe following two examples serve to illustrate the use of
Equation 30. First consider the case where the ambient air temperature is subjected to a step change. This example is well suited for studying ice growth or decay over relatively short time periods. An example of this case would be the freezing of water in a refrigerator when the water temperature is initially above the freezing point. In the second case, a system is considered where the ambient air temperature is assumed to vary sinusoidally. This example illustrates hypothetically how the seasonal air temperature may vary with time. Also imposed on the temperature profile is a daily periodic sinusoidal variation that tends to account for daily temperature fluctuations. 30
1. Case I Step drop in the ambient air temperature
For this case the Stefan number and Q* are constant, ihe
initial conditions imposed on the air temperature are
T (t) = T , t < 0 a o
Ta (t) = Ta , t > 0
where T < T a o
Integrating Equation 30 for the initial conditions 5 = 0 when 0 = 0
yields
2 Ste Ste - Q* 6 0 = 4 C— p m (— :------) --- ] (31) Q* Ste - Q* (1+6) Q*
o Figure 4 presents a plot of 6 versus 8 for various values of the
Stefan number holding Q* constant. Asymptotic values of 6 can be obtained by equating Equation 30 to zero, therefore,
Ste 6 = 1 . (3 2 ) max Q*
These values are presented in Figure 4 as upper limits to 6 for a given step change Stefan number.
2. Case II Sinusoidal variation in ambient air temperature
Case II is an attempt to simulate seasonal as well as daily air temperature fluctuations to determine the effect on ice depth formation 8 max
- 8 max 8
8max
20
Figure 4. Step Change Air Temperature with Constant Heat Flux in Liquid Phase 3 2
and decay. In order to simulate the seasonal and daily air temperature
variations, the ambient air temperature is assumed to vary sinusoidally
with time with a seasonal angular frequency of three months and a
maximum temperature perturbation of AT^ of 30°C. The air also experiences
daily fluctuations with an angular frequence a>2 of one day and a maximum
temperature perturbation of AT2 of 5°C.
Ta (t) = To ’ * * 0 A T A T T (t) -• T + --- (1 - cos oint) + --- (1 - cos u)0t) 2 2
t i O (33)
Substituting Equation 33 into Equation 30 and integrating numerically by the method of Runge Kutta gives the results shown in Figure 5.
The upper and lower bounds of the daily ambient air temperature fluctuations are also given in Figure 5. The air temperature is pres cribed to fluctuate sinusoidally, with a period of one day, between these bounds.
An expected time lag exists between the minima of the air temper ature, the input variable, and the maxima of the dimensionless ice thickness, the response variable. Arranging Equation 30 into a linear form involving 6 leads to the following expression
dS t — 5- = 6 - Q* x (34) de where 6 (1+6) T = ------Ste Q =.0127 'L=-B(l-cos(.OO27902)) -25(l-cos(25702)) -3 0
8 upper -20 bound
lower bound -10
20
2 j0
Figure 5. 8 vs 6Z Periodic Air Temperature Fluctuations with Constant Heat Flux in Liquid Phase & 34
x is coninonly referred to as the time constant and is associated with linear ordinary differential equations. Equation 34 is nonlinear; however, it does provide an estimate for the time lag within the system.
Evaluating Equation 34 when T is at its minimum (6 = 8, T = -30°C) c L cl gives x = 380. This value is in reasonable agreonent with the observed value of 300, considering the nonlinearity associated with the equation. 35
IV. DEVELOPMENT OF THE MOVING BOUNDARY
PROBLEM TO SIMULATE IHE FREEZING AND MELTING
ON A LAKE
Ihe quasi-steady state assumption has been adopted for solution
of the moving boundary problem associated with the freezing and melting of lake ice. Use of this assumption presumed the form of the ice temperature profile, thereby simplifying the problem considerably.
Boundary conditions associated with lake ice growth and decay are often complex. Making use of the quasi-steady state assumption greatly simplifies the expression relating the rate of movement of the phase front to the heat conduction process within the ice layer. Therefore, the resultant phase front expression becomes readily adaptable to the boundary conditions ccmmonly associated with freezing and melting lake ice. It was seen in comparing the approximate solution utilizing the quasi steady state assumption with that of Westphal (16) that for small
Stefan numbers the approximate solution agrees closely with Westphal1s exact solution. Weather conditions for southern New Hampshire in the vicinity of lattitude 43° and longitude 71° seldom experience temperatures below -30°F which would yield a Stefan number equal to .218. Under these conditions the maximum error introduced by using the quasi-steady state assumption would be under 1 0 % .
A. Factors Considered in Developing the Model
Ihe following factors are considered in developing the lake ice model:
(!) variation of ambient air temperature with time
(2 ) variation of the heat transfer coefficient with changes 36
in wind velocity
(3) snow cover on lake ice surface
(a) variation in snow density with time
(b) variation in effective thermal conductivity of snow
(4) heat flux existing beneath the ice
(a) solar radiation penetrating the ice cover
(b) springs and bottom characteristics
(5) precipation in the form of rainfall
The following factors are not considered in the development of the lake ice model:
First sensible heat terms within the ice have been neglected.
These terms generally appear in the same order of magnitude as the
Stefan number. Mams, et. al. (60) attribute a maximum of 7 percent error by not considering these terms. This error would result under severe conditions, temperatures below -30°F and ice depths greater than one foot (APPENDIX B).
White ice is defined as ice formed by the freezing of slush, melted snow, melted surface ice and rainfall. Due to the complexities involved during white ice formation and extreme difficulty in accounting for its effect on the total ice growth, periods of excessive white ice formation are not studied. Periods during which white ice formation can be accounted for are considered. Sublimation from the ice surface as noted by Fertuck, et. al. (62) is only about .3 inches during a winter and has been ignored. Figure 6 illustrates the air-snow-ice-water problem under investigation.
B. Fundamental Equations
Utilizing the quasi-steady state assumption for the snow and ice 37
boundary conditions
ksn “I t 0 =hg CferTa x =-S(t), d x t> 0 ^Tsn _ . ^Ts -Ken , ~ — Re * x=0, 31 ax s ax t20’
Tsn = Ts *= 0 ,
^ f d T =ksl ? ' Qtt) t£°
initial conditions b(o)= b ,, S(t)=S,
Ts =Ts (x), *^n =Tsn(x)
Air T „ (t) T X=-S(t), _r'_r Snow(xjt) — _ r _ r x=0 Ice^TsCx^)^ x=b(t)
Water (x,t)
Figure 6. Air-Snow-lce - Water Problem layer It Is assumed that both the snow and the Ice temperature vary
linearly with distance, therefore,
T = a_(t) + b,(t) x -S(t) < x < 0 sn _L -L t > 0 (35)
Ts = a2 ^ + b2 ^ x t > 0 (36)
Ihe convective boundary condition which occurs at the air-snow Interface
Is given by Newtons law of cooling
t > 0 (37)
where k = effective thermal conductivity of snow, sn h = film coefficient between air and snow, g kov, is found to be a function of snow density. Pitman (63) published O il values of effective thermal conductivity of snow at -5°C as a function of the density of snow within the the range .1 < p < .6 g/cc. Ihe data was fit to a linear equation within the range of interest in this study (.1 £ p ^ .3 g/cc) as follows: o i l
ksn = -029 + -39 Psn (APPENDIX C),
where k [=] BTU hr-^ ft-1 °P-1 and p [=] g/cc. sn Ksn 39
Mams (60) has noted that in still air, the transfer of heat
from the top surface is largely by turbulent natural convection.
Although there is significant transfer by radiation, the latter is difficult to access without precise knowledge of the effective sink temperature in the sky, T , and the total emissivity of ice, e.
1.71 e A 4 h - — g------* C V ' V ] <38) 10 (T - T ) s o where Tg and are absolute temperatures of surface ice and the sky.
Due to the difficulty in obtaining h the total heat transfer coefficient hg = hQ + h^ was determined experimentally, where h, equals the heat transfer coefficient due to convection. The heat transfer coefficient, h , was found by previous investigations to be strongly dependent on g air velocity, U. Fertuck, et. al. (62) give values of h ranging from g Q h = .29U to h = .37U. Mams, et. al. (60) suggest h = .62U' over g g o ice when U > 5 mph. They also present experimental evidence that h =2.05 BTU hr-^ ft-^ °F-^ in still air over ice at -40°F. In this g study h for snow covered lakes is assumed equal to .37U when U > 5mph g and 2.05 when U < 5 mph. For snow free lakes h is assumed equal to o o .62U' for U > 5 mph and 2.05 when U < 5 mph.
It is assumed that thermal equilibrium exists between the snow- ice interface, therefore, the boundary conditions that apply at the interface are that the snow temperature and ice temperature are equal and the energy flux across the interface is equal.
Tsn <*•*> = Ts ( t ’V x = 0, t > 0 (39) At the ice-water Interface, thermal equilibrium is also assumed to exist. Therefore, the ice and water temperatures are seen to be at the melting point of ice T .
T (x,t) = T x = b(t),t £ 0 (41) o O
The movement of the ice-water interface is related to the difference in the heat removed from the liquid and that conducted into the solid.
db 9T
PS xf TTQu ' ks oX— ■ 9w(t) t > 0 (42)
Q^Ct) refers to the heat flux existing in the water phase at x equal to b(t). This heat flux may be due to conduction, convection and radiation and may vary as a function of time.
Utilizing the two quasi-steady state equations, Equation 35 and
36 together with the appropriate boundary conditions, Equations 37,
39 s 40 and 41 and then substituting into Equation 42 yields 41
where
h = .37U when S(t) > 0, U > 5 inph S
h = .62U'^ when S(t) = 0, U > 5 mph §
h = 2.05 when U < 5 mph o ksn = .029 + .39 p sn
Af = 144 BTU/lb
P O = 57-5 lhm/ft3
k = 1.27 BTU hr"1 ft"1 °F"1 s
T = 32°P o
It remains to determine the characteristic heat flux Q^(t) in the water
phase.
C. Determination of the Characteristic Heat Flux in the Water
Layer
Williams (61) in his paper attributes thefollowing sources to
contributing toward a heat flux in the water layer below melting ice:
(1) snow melt runoff
(2) lake bottom residual heat
(3) ground-water flow through the bottom sediments
(4) lateral flow of heat under the ice from open water near
the shoreline
(5) solar radiation penetrating the ice cover
William’s (6l) study was made during the spring melt period. The lake that was studied was fairly shallow, from 3 to 9 meters. He noted that solar radiation is the major contribution factor for heat flux formation in the water layer. Williams showed that the radiation penetrating the 42
ice cover may be estimated from the following formula
Qjj. = Q0 exp (-Ax) (44)
where
Q^. is the solar radiation heat flux at depth x (ft),
BTU hr-1 ft-2
Qq is the solar radiation absorbed at the ice surface,
BTU hr"1 ft"2
A is the extinction coefficient, ft"1
Table II gives a list of extinction values for various types of ice and
snow of interest.
TABLE II
Extinction Factors
Type A (ft'1 )
clear field ice Q.305
white field ice 1.78
snow 6.10
In this model, the following semi-empirical equation is used to account for the heat flux existing in the water layer.
^(t) = AQ (t) exp (-Asn set) - Agb(t)) + Bo (45)
where AQ Ct) is the amount of absorbed radiation at the snow or icer-air interface. This factor is dependent upon the albedo of the ice surface and upon the total irradiation received at the interface. B is a o 43
constant which, accounts for all other heat flux sources. Since Equation
45 Is seml-emplrlcal the parameters Aq and Bq have to be determined from experimental data. 44
V. LAKE ICE FORMATION EXPERIMENT
A. Great East lake
A number of lakes and ponds in southeastern New Hampshire
(Figure 7) were considered before a site was selected. The criteria
used in selection were the availability of electric power, accessibility
and lack of comnercial and public activity. Great East Lake, located
40 miles north of Durham, New Hampshire, in Wakefield, New Hampshire, met the above requirements. Ihe lake is approximately 5 miles long and
2.5 miles wide (wides point) and has a maximum depth of approximately
100 feet (Figure 8). The location of the ice experiment was approxi mately 200 yards off-shore on the northwest quardrant of the lake.
Water depths at this location measured between 35 and 40 feet. The lake has no natural inlets, except for a few minor streams. Its chief source is from bottom springs of which there are many. The lake is almost completely pollution free and the water is so clear that it is considered by many to be quite drinkable. During the past 10 winters at Great East
Lake the maximum ice thickness has varied between 2 and 3 feet with an average maximum of around 30 inches.
B. Description of the Experiment
During 1973-74 the winter was relatively mild in New Hampshire producing at most a maximum of 14.5 inches of ice on Great East Lake.
Air temperatures seldom went below -20°F and then only for a short duration. Snowfall was light producing at most a 6 to 8 inch snow cover on the lake at one time. Figure 9 is a picture of the ice cover on
Great East Lake March 8, 1974. The ice was cut by means of a 7 inch ice auger. Several holes were cut in the vicinity of 45
New Hampshire Maine
Great East Lake Portland
Concord
Figure 7. New Hampshire and Maine location of experiment X
4 } 41 44 £7 «2 BO it » u 74
7 B SO rr 42 fii 2 0 H 43 M 74 23 34 30 102 4040 n 34 40 40 M 22 42 •7 60 54 22 45 54 30 82 42 22 64 n 49 36 94 43 40 40 94 40 M 30 90 20 •4 46 47 90 20 23
» 92
GREAT EAST LAKE
ACTON TWP YORK CO MAINE,
WAKEFIELD TWP, CARROLL CO, NEW HAMPSHIRE
AREA I76fl ACRES
0 t 2 3 4 FOURTHS OF MILE
o figure 8. Great East Lake Wakefield, New Hampshire 47
Figure 9. Ice Cover on Great East Lake (March 8, 1974)
Figure 10. Ice Depth Measurement on Great East Lake (March 8, 1974) 48
the experiment site and an average value was recorded as the ice depth.
A flat arm measuring device with a calibrated scale (centimeters) was used to determine the ice thickness. Figures 10 and 11 are pictures of an ice depth measurement made on March 8, 1974. Snow cover on the ice surface was measured by a standard ruler. The density of the snow was determined by melting a known volume of snow and recording the melted volume. The ambient air temperature was monitored continuously with an
Esterline Angus temperature recorder. The recorder was housed in a temperature controlled cabinet (Figure 12). A 5000 BTU heater was regulated to keep the cabinet temperature in the vicinity of 60°F„ The cabinet and equipment was stored in a boathouse mainly for security reasons. The probe of the thermocouple extended out of the boat house and was shielded from direct sunlight.
The following data was measured and recorded at Great East Lake during the winter months of January through March 1974.
(1) ice depth measurements
(2) ambient air temperature
(3) snow fall
(4) snow density
Ice depth measurements were made by cutting 7 inch holes with an ice auger and then measuring the ice depth by use of a flat arm measuring device. Figure 13 is a sketch of the ice-thickness measuring device.
Each measurement was made with the time of measurement being recorded.
Measurements were made at frequent intervals (4 to 5 days) so as to pro duce fairly continuous data.
Ttoo observation periods were conducted during the months of
January through March. The reason for the lack of continuity between 49
Figure 11. Ice Depth Measurement on Great East Lake (March 8, 1974)
Figure 12. Thermally Regulated Cabinet for Temperature Recorder 50
Support Rod Scale
ce
r
7in. lot Arm
Figure 13. Sketch of Ice Depth Measuring Device 51
periods was due to a considerable amount of rainfall during the latter
part of January which resulted in considerable slush formation. This
slush in turn produced a noticeable amount of white ice. The lack of
continuity of ice formation resulted in two distinct observation periods
with a 15 day gap between periods. The first continuous observations
were taken fran January 7th through January 20th. The second period
began on February 6th and ended March 8th. The experimental ice depth
data recorded at Great East Lake during these periods are listed in
Thble III. The air temperature was taken using a copper-constant an
thermocouple. The range of the recorder using this thermocouple was
quite adequate for the purpose of the experiment. The chart speed on
the recorder was variable and was set at 3 inches per hour, its
lowest setting. Data was recorded between January 7th and January 20th,
(first observation period). Unfortunately the temperature data obtained
by the recorder during the second observation period was found unaccept
able. The reason for this was due to slipage frcm the chart drive
sprocket over extended periods of time. Air temperature data obtained
from nearby U.S. Weather Bureau stations were used to supplement the air
temperature data during the second period. APPENDIX D lists the temper
ature data taken during the January 7th to January 20th observation
period.
Snow depth measurements were recorded during the time of ice
depth measurements. Snow depth was measured by means of a ruler and
recorded in Thble IV. The snow density was deteimined by melting a known volume of snow and recording the melted volume. TABLE I I I
Ice Depth Measurements at Great East Lake
Wakefield, N.H. - Winter 197-4
Date Ice Thickness (cm)
1/7/7^ - 3 pm 8.8
1/9/74 - 11 am 13.0
1/9/74 - 4 pm 14.0
1/13/74 - 12 am 17-3
1/20/74 - 12 am 18.1
3 days rain and slush
1/25/74 - 12 am 31.0
2/6/74 - 11 am 25.0
2/10/74 - 11 am 26.8
2/17/74 - 1 pm 33.0
2/25/74 - 2 pm 36.8
3/2/74 - 11 am 36.2
3/2/74 - 3 pm 35.6
3/8/74 - 10 am 27.0 TABLE IV
Snow Depth Observations
Date Snow Depth (inches)
1/7/74 no snow
1/9/74 - 13. am 1.75 snowing
1/9/74 - 4 pm 5.38 snowing
1/13/74 - 12 am 3.5
1/20/74 - 12 am 6.5
2/6/74 - 11 am no snow
2/10/74 - 11 am 1.5
2/17/74 no snow
2/25/74 no snow
3/2/74 no snow
3/8/74 no snow 5k
C. Supplemental Data
To supplement the Great East Lake data, weather data from the
U.S. Weather Bureau was also cited. The reason for obtaining this data
is as follows: First, the data observed at Great East did not provide
all of the essential data necessary to make predictions using the
existing model. Secondly, even though the time interval, 3 to 5 days, proved quite satisfactory for the ice depth measurements, the interval was too large for many of the system parameters such as snow depth, wind velocity, sunshine, etc. Interpolation of conditions within these
intervals would be very risky. The nearest locations were Concord, New
Hampshire, (70 miles south west) and Portland, Maine, (45 miles east).
The climatological data frcm these locations provided the following necessary information.
(1) ambient air temperature (3 hour intervals)
(2) standing snow depth
(3) snow fall per day
(4) water equivalent of snow
(5) wind velocity
(6) periods of sunshine
It is of interest to compare the ambient air temperature data of
January 7, 1 9 7 k , through January 20th at Great East Lake with that of
Portland, Maine, and Concord, New Hampshire. Table V lists the air temperature data on a daily average for the three locations. Figures 14 and 15 give the correlations of the Great East Lake data versus the
Concord and Portland data. From a correlation analysis (APPENDIX E), a correlation coefficient r = .93 was obtained between the Great East Lake air temperature data and the Portland data. The average temperature 55
TAKLiH. V
Average Daily Air Temperatures
Date Concord.... Portland.... Lake*
1/7/74 17 °F 23 21.4°F
8 9 12 14.2
9 3 5 5.1
10 15 13 11.0
11 20 18 17.0
12 21 15 12.9
13 1 3 5.4
14 4 10 7-4
15 34 33 26.3
16 23 16 12.2
17 1 0 -3.1
18 -5 -1 -2.4
19 22 23 18.5
20 7 9 6.8
Average temperature during 14 day observation period are
% lake . = 10.9°F
TnConcord = 12.3°F Tl . = 12.8°P Portland
*Daily average air temperatures are determined from hourly
readings between 1 am and 12 pm. 56
40
3 0
20
-10
-20 -10 0 20 3 0 4 0 Tc CF)
Figure 14. Scatter Diagram of Lake Air Temperature and Concord Air Temperature from January 7 to January 20 57
40
3 0
20
-10
-20 -10 20 3 0
Figure 15. Scatter Diagram of Lake Air Temperature , and Portland Air Temperature from January 7 to January 20
* 5S
during that period was TQE = 10.9°F and TpQrt = 12.8°F. The correlation
coefficient for the Concord data when compared to the Great East Lake
air temperature data was r = .90. The average temperature at Concord
during that period was = 11.7°F. The Portland data was used to
estimate the Great East Lake air temperature during the second observa
tion period. The reason for the choice was primarily because of the
closeness of the Portland weather station to Great East Lake. Secondly,
the Portland air temperature data gave a better correlation with that
of Great East Lake than that of Concord with Great East as can be noted
when comparing Figures 14 and 15.
The equation used to describe the Great East Lake air tenperature
is the following:
TGE - TPort-1-9°P
Equation 46 was obtained by first determining the mean air temperatures frcm January 7th to January 20th for both Great East Lake and Portland,
Maine. In correcting for the offset noted in Figure 15, the difference between the two means was used as a correction factor. This is essentially assuming a one to one correspondence between the Great East Lake and
Portland air temperatures with an offset of 1.9°F, Portland being at the higher temperature. The difference between the two means was used during the second observation period.
Tbtal snow depth on the surface of the ice was only measured during periods of ice observation. In order to analyze the freezing problem as accurately as possible it is necessary to supplement the snow depth data with predicted snow depth estimates. It may be noted that 59
the effective thermal conductivity of snow is an order of magnitude smaller than the thermal conductivity of ice. Therefore, knowledge of the snow conditions on the surface of the lake during the interim period between observations is most essential. These estimates were obtained frcm the Concord data. It was assumed that even though the Portland air temperature data gave a better correlation with the Great East Lake data
(probably due to its closeness) that the Concord weather station was further inland and the snow conditions would generally be more similar to that at the lake location. Therefore, standing snow depth was estimated in the interim during observations at Great East Lake with the Concord data.
Snow depth per day and water equivalent per day was used to estimate the density of the snow during a snowfall. Density of snow has been shown earlier to correlate with the effective thermal conduc tivity of snow. Again the Concord data is used for these estimates.
Wind velocity effects the rate of heat transfer on the air-ice or air-snow interface. Wind velocities can fluctuate widely during a day and as a result it was found best to use a daily average. The wind velocity used in this model was that of Concord.
Estimates on the periods of sunshine are necessary in order to account for solar radiation effects on the heat flux beneath the ice cover. Sunshine should have a high degree of correlation within the relatively short distances between locations. Again the Concord data was used.
Table VI lists the supplemental climatological data from Concord during the first observation period. Table VII lists the temperature data observed at Concord and Portland during the second observation 60
TABLE 71
Concord Weather Data During
First Observation Period
Wind Water Day Velocity Equivalent Snow Sunshine
Jan 7 4.5 mph 0 In 0 In 7-3 hrs
8 11.5 0 0 9-2
9 2.9 .22 6.4 0
10 .9 .27 3-0 0
11 2.7 • 32 2.7 0
12 11.2 0 0 8.5
13 4.3 0 0 9.3
14 5.8 0 0 8.0
15 7.1 T 3-8
16 5.5 .17 2.0 0.1
17 9.5 0 0 9.1
18 3-3 TT 6.3
19 7-3 .06 0.4 6.0
20 2.0 0 0 8.0
* T = Trace TABLE VII
Air Temperatures During Second Observation Period
^Concord ^Portland
6 18 °P 22°P
7 19 20
8 7 15
9 8 19
10 6 13
11 22 21
12 19 20
13 29 32
14 20 22
15 12 17
16 14 20
17 30 31
18 19 24
19 23 28
20 38 42
21 35 35
22 32 39
23 34 36
24 27 28
25 28 30
26 23 30
27 25 26
28 38 37 62
TABLE VII (continued)
1 41 45
2 33 35
3 32 34
4 43 47
5 49 48
6 41 46
7 47 53
8 36 36 63
period. Equation 46 is used to estimate the daily average Great East
Lake air temperatures during the second observation period. Table VIII lists the supplemental climatological data from the Concord weather bureau for the second observation period. 64
TABLE VIII
Concord Weather Data During
Second Observation Period
Wind Water lay Velocity Equivalent Snow Sunsh:
Feb. 6 10.5 nph 0 in 0 in 10.1 7 4.6 .4 4.4 0 8 0 0 0 3.2 9 4.2 0 0 9.8 10 2.5 0 0 9.2 11 4.2 .01 .3 1.5 12 5.6 0 0 10.4 13 4.3 0 0 8.1 14 10.6 0 0 9.1 15 7-5 0 0 9.8 16 4.8 0 0 7.8 17 9.2 .01 .2 6.9 18 13-5 .02 .4 10.7 19 4.2 . 6 1.7 1.3 20 12.8 .01 0 2.7 21 9.8 0 0 10.8 22 1.6 1.2 0 0 23 18.6 .07 0 7.1 24 6.5 0 0 11.0 25 5.9 0 0 5.1 26 10.5 0 0 10.4 27 4.5 0 0 11.1 28 5.6 0 0 5.7 March 1 9.4 0 0 7.4 2 4.9 0 0 2.2 3 6 . 9 0 0 0 4 2.3 0 0 1.3 5 10.1 0 0 2.5 6 7.9 0 0 9.2 7 9.1 •o 0 9-5 8 11.9 0 0 8.3 65
VI. ICE DEPTH PREDICTION
A. Prediction of Ice Depth During the January Observation
Period
Ihe snow-ice profile observed during the January observation
period is shown in Figure 16 . Also superimposed in the figure is the
predicted ice depth determined from Equation 43 using the observed
and supplemental climatological data taken during that period. Equation
43 is used assuming no heat flux present in the water phase. During
the freezing period, cold snow has a very high albedo thereby reflecting
a great deal of the incident solar radiation. Therefore, the contribu
tion of radiation to producing a melting heat flux is very msall during
the freezing period. It can be observed by comparing the predicted ice
depth versus the actual data that the reliability of the model is very
good before the onset of snow cover. Ihe physical properties of the
ice-water system are well established during this period. However, after
the onset of snow cover the variation of physical and thermal properties,
namely changes in the effective thermal conductivity of the snow, becomes rather substantial with changes in snow density. Therefore,
errors introduced due to lack of knowledge of snow conditions during
the interfals between observations reduced the accuracy of the model.
Furthermore, the effective thermal conductivity of snow is an order
of magnitude smaller than the thermal conductivity of ice. As a result, a one inch layer of snow has the insulating property of one foot of
ice. This points out the important effect of snow depth during the interval between observations.
B. Prediction of Heat Flux From February through March data 66
1.4
1.2
1.0
Snow Cover b(t) fe e t
C lea r Ice
A observed and estimated snow cover
• predicted ice depth assuming no heat flux
O observed ice depth
7 12 17 22 January 1974
Figure 16. Snow-lce Cover on Great East Lake Together with Predicted Ice Depth from Mathematical Model Assuming No Heat Flux (First Observation Period) Heat flux predictions can be made from the February and March
data. First it is instructive to determine the ice growth profile as
a function of time assuming that no melting heat flux exist. Equation
43 together with the local climatological data can be used to determine
the progress of the phase moving front as a function of time. Figure
17 represents the; snow-ice profile observed on Great East Lake during the February 6th through- March 8th freezing-melting period. Also in
Figure 17 is a plot of the predictions made by integrating Equation 43 along with the climatological data. It can be noted that during the freezing period little, if any, melting due to a melting heat flux exists. Pounder (64) explains this fact as follows: the albedo of snow varies from .6 to .9. Snow, therefore, reflects a large fraction of solar radiation incident on it. However, in the spring the intensity of radiation increases and enough is absorbed to cause the snow to melt.
The albedo of wet snow is much lower, about .45, so the absorption of radiation is further increased once melting starts. When snow on the ice cover melts, the wet ice also has an albedo of about .5. The melting period began on February 28th through March 8th. As a result, a discrepancy exists between the model and the observed ice depth.
C. Heat Flux Prediction
Since radiation is considered to be the major factor in forming an appreciable ice-water melting heat flux, it is assumed that the heat flux takes on the form given by Equation 45. Equation 45 can be put in an alternate form
•Sun(t) ( A S(t) Ap(b xwi-i:?+-ei) ^qXwhite^ Q^(t) = Aq ----- e 1 white 3 white +B^ 68
Rain producing .14 feet White ice
White Ice
b (t) fe e t Snow Clear Ice
A observed and estimated snow cover • prediction assuming no heat flux
O observed ice depth
26 February 1974 March 1974
Figure 17. Snow-lce Cover on Great East Lake Together with Predicted loe Depth from Mathematical Model Assuming No Heat Flux (Second Observation Period) 69
where
Ao , EL o = constants to be determined
Sun(t) = hours of sunshine per day
X = extinction factor for snow 1
S(t) = depth of snow cover
X = extinction factor for clear field ice 2
= extinction factor for white ice 3
= white ice thickness xwhite
During the freezing period little if any heat flux within the water
layer exist. This can be seen by comparing the results of the model with the observed data in Figure 17. Without heat flux considerations
the mathematical model explains quite adequately how the ice thickness
increases during the freezing period. It will be assumed, therefore,
that Bq is equal to zero and the only source contributing to a melting heat flux is from solar radiation. Based on the data obtained on
March 8th, Aq can be determined as follows: Inserting Equation 47 into
Equation 43 yields
Sun(t) -(x1S(t)-X2(b(t)-x1 wnite ^ 3 xwhite ^
(48)
This equation relates the rate of melting of lake ice during the melting period as a function of the climatological conditions. The right hand side of Equation 48 can be seen to consist of two terms. The first expression describes the rate of ice decay assuming no heat flux 70
existing in the water layer and the second tern representing the contri
bution due to an existing solar heat flux in the water layer.
Since only one data point existed during the melting period,
that point taken on March 8, 1974, will be used to predict A . The predicted ice depth that is used, b , is the prediction made on March Jr 8th, assuming no heat flux in the water layer. The observed ice depth on March 8th is denoted by bQ . At the onset of the melting period,
February 28, 1974, as noted in Figure 17, bQ is equal to b . Therefore integrating Equation 48 from February 28 to March 8 yields
Ao f tz (-A s(t)-J ('b -x ) b = b ° Sun(t) exp 1 J ° waLt;e o p 20 ps Xf \
- V w h l t e 5 dt (1|9)
where t^ = February 28, t2 = March 8
Therefore, Aq can be estimated from Equation 49
(bp - bo ) 2D ps xf A° ° f= 2 - ... (-XxSttJ-X ( b ^ )-X^C ) dt <50) Sun(t) exp J
By using the climatological data from February 28 through Mhrch 8, Aq —1 —2 was determined from Equation 50 to be 56.85 BTU hr ft . The integral appearing in Equation 50 was evaluated by the method of Runge Kutta.
Substituting this value of Aq together with Bq = 0 into Equation 47 yields an expression for the melting heat flux during the melting period. Equation 48 was integrated numerically by the method of Runge
Kutta using the necessary climatological data obtained during the second observation period. The results of this computation is illustrated in Figure 18. 71
1.3
1.2 Rain producing .14 feet White ice White Ice
b it) Snow Clear ice feet
A observed and estimated snow cover • prediction with calculated heat flux O observed ice depth
26 March 1974
Figure 18. Snow-lce Cover on Great East Lake Together wsth Predicted Ice Depth Using Calculated Heat Flux (Second Observation Period) 72
D. Numerical Procedure
Ihe numerical integration technique used to evaluate Equations
^3 and 48 In predicting the ice depth, as a function of time was a
fourth order Runge Kutta formula. The integration procedure is as
follows: Given an ordinary differential equation of the form
dy — = f(x,y) (51) dx where f (x,y) is a function of x and y, then the integration formula is
yx+h = + 6 + 2k2 + 2k3 + V (52) where = hf(x,y) , k, k2 = hf(x+ 2> y + 2 ")
h kp k = hf(x+ - y + — ) o 2 2
k^ = hf(x+h, y+k^)
h equals the interval of integration of the independent variable.
During the first and second observation periods, the interval of integration was one day. Tnerefore, daily average values of the following system variables were used in the model:
(1) ambient air temperature
(2) wind velocity
(3) hours of sunshine
(4) standing snow depth 73
(5) snow density
The need to refine the increment of integration to less than one day is not deemed necessary. Ihe reason being that errors introduced by utilizing climatological data from Concord and Portland to estimate data at Great East Lake were larger than the errors introduced by using an integration step of one day. The numerical computation was accomplished using an IBM 360/50 digital computer. Typical execution time was less than 6 seconds for evaluating ice depth predictions for the second observation period. Appendix F lists the computer programs used in this thesis. 74
VII. CONCLUSIONS
A simplified solution to the moving boundary problem associated
with the melting and freezing of ice based on the assumption of linear
temperature profile was presented. The mathematical model was tested
with actual ice formation measurement data and the predicted results
compared quite satisfactorily with the observed data. The system that was studied was the periodic growth and decay of lake ice during the winter months of 1973-74. The complexities inherent in this type of a
system that make pure analytic solutions impossible and many classical
approximate techniques impractical include the existence of complex and
involved boundary conditions. Simplification in the mathematical
formulation of the problem due to the quasi-steady state assumption has made it possible to handle complex boundary conditions with no computa tional difficulty. Furthermore, errors introduced by adopting the quasi
steady state assumption are often of the same order of magnitude or less than the errors introduced when using the climatological data in the model.
Errors associated with the climatological data generally arise from the following sources: (1) Supplementary data recorded at
Concord, New Hampshire and Portland, Maine, was obtained 70 miles and
45 miles respectively frcm the lake site. (2) The equation expressing the dependence of heat transfer coefficient, h , with wind velocity § neglect variations in h^ with changes in e as a function of snow or ice wetness. (3) The semi-empirical expression relating the dependence (t) with time fails to take into account the variation in the snow and ice albedo with time. What has been assumed 75
is that during the freezing period the albedo of snow or ice is equal
to 1.0 and during the melting period the albedo is incorporated into
A and is constant. (4) Errors introduced during the formation of
white ice are the most difficult to determine and account to a great
extent in the discrepancies with the observed and calculated ice
depths.
It is important to re-emphasize the necessary criteria
that must be met in order to justify the quasi-steady state
assumption. First the temperature boundary conditions inposed on
the ice layer should be slowly varying with time. This restric
tion is a necessary requirement for quasi-steady state heat
conduction regardless of the existence of a moving boundary. Ambient
air temperatures during the winter months in New Hampshire seldom
fluctuate more than two degrees Fahrenheit per hour. Such rate of
fluctuations justifies a quasi-steady state assumption on heat
conduction in a solid system such as ice. The second important criteria
imposed on this system is that the Stefan number must be relatively
small. It was noted that for Stefan numbers less than 0.2, errors of
less than 10# can be expected in the ice depth prediction. For weather
conditions experienced in this study, Stefan numbers produced satisfied
this second requirement.
Ihe possibility of generalizing the results of this work to
other moving boundary problems other than the freezing or melting of
ice exists. Problems involving heat conduction and phase change in which the two restricting criteria are satisfied may be solved in a
similar manner. Examples of possible extension of this technique would be the process of glass formation or alloy solidification. In the 76
process of glass formation, the latent heat of fusion of silicon dioxide
is large. As a result the Stefan number associated with the melting or
solidification of glass may be relatively small. However, glass has. the
complication in the fact that it does not have a distinct melting temp
erature and the present analysis must be modified to account for this property. In the solidification of many metal alloys the Stefan number associated with the process is relatively small making the quasi-steady
state approach applicable.
In problems other than heat conduction, such as cases involving mass as well as heat transfer, application of a quasi-steady state approach has to be attempted with caution. The reason is due to the need of satisfying another criterion with respect to the conservation of mass or continuity equation. An example of a moving boundary problem involving heat and mass transfer would be the conductive drying process associated with paper or fibrous material. A simplified model assumes that the system is composed of two regions. One region being the dry region where the moisture is in vapor form. In the other region, the moisture is in a mixed (vapor and liquid) form. Thus, the problem leads to simultaineous solution of pure diffusion problem in one region and the solution of an unsteady state coupled problem of heat and mass transfer with moving boundary in the other region.
There are a number of possible direct applications for the existing model. A direct application of the existing model could be in the prediction of the thickness of ice layer on a lake or pond leading to a determination on safety for human portage. Such predictions would serve as a tool for state and local safety corrmissions in being able to forecast ice safety hazards. Another direct application will be predicting the growth and decay of sea ice and would therefore be a useful means for 77 accessing and navigability of Arctic or similar waters.
Ihe mathematical model as developed has its highest reliability during the freezing period. However, during the melting period two additional complexities arise. First, surface melting occurs. Because of this phenomenon, a water layer develops on top of the ice. Ihe problem is therefore complicated by the additional moving boundary at the ice surface. Secondly and possibly of considerable importance is the existence of an internal heat source within the ice due to the absorption of radiant energy by ice during the melting period. 78
V I I I . NOMENCLATURE
a,b system parameters in quasi-steady state assumption
A , B = heat flux parameters o’ o b = ice depth3 ft
c = specific heat, BTU lbm""^°F P f = function
h = heat transfer coefficient, BTU hr_^ft-^°F_'1' 6 k = thermal conductivity % = heat flux, BTU hr ^ft ^ V = correlation coefficient
S = snow depth, ft
Sun = hours of sunshine per day
Ste = Stefan number
t = time, hr
T = temperature, °P
U = wind velocity, mph
X = distance variable, ft
= thickness of white ice, ft xwhite otters
a = thermal diffusivity, ft'Vhr
3 = constant
6 = dimensionless ice thickness
A = difference
e = emissivity
X = extinction coefficient, ft-1
= latent heat of fusion of ice, BTU/lb *f p = density, lb /ft^
t = time constant 2 0 = dimensionless time parameter
go = angular time frequency
Subscripts
a = ambient
C = Concord, New Hampshire
GE = Great East Lake
1 = liquid
max = maximum
o = melting
Port = Portland, Maine
s = solid (ice)
sn = snow
Superscript
# revised parameter 80
IX. BIBLIOGRAPHY
1. H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids (2nd ed.: Oxford Clarendon Press, 1959), PP. 282-296.
2. J. Stefan, "Uber die Theorie der Eisbildung, Inbesondere Uber die Eisbildung im Polarmeere," Annalen der Physik und Chemji 42: 269-286, 1891.
3. J.R. Cannon and C.D. Hill, "Existence, Uniqueness, Stability and Monotone Dependence in a Stefan Problem for the Heat Equation," Journal of Mathematics and Mechanics, 17.' 1-20, 1967.
4. J.R. Cannon, J. Douglas and C.D. Hill, "A Multi-Boundary Stefan Problem and the Disappearances of Phases," Journal of Mathe matics and Mechanics, 17:: 21-33, 1967.
5- B.A. Boley, "Uniqueness in a Melting Slab with Space- and Time- Dependent Heating," Quarterly of Applied Mathematics, 70: 48l- 487, 1970.
6. B. Sherman, "A General One-Phase Stefan Problem," Quarterly of Applied Mathematics, 70: 377-382, 1970.
7. S.G. Bankoff, "Heat Conduction or Diffusion with Change of Phase," Advances in Chemical Engineering, vol. 5 (New York: Academic Press, 1964), pp. 75-150.
8. R.W. Ruddle, Ihe Solidification of Castings, (London: Institute of Metals, 1957).
9. B. Chalmers, Principles of Solidification, (New York: John Wiley & Sons, 1964).
10. B.A. Boley, "The Analysis of Problems of Heat Conduction and Melting," High Temperature Structures and Materials: Proceedings of Third Symposium on Naval Structural Mechanics (New York: Columbia University Press, 1963), pp. 292-296.
11. J.C. Muehlbauer and J.E. Sunderland, "Heat Conduction with Freezing or Melting," Applied Mechanics Reviews, 18: 951-959, 1965.
12. J.A. Bruno, Mathematical Study of Seme Solid-Liquid Melting and Freezing Problems with Moving Interfaces, PhD thesis, University of Pittsburg, 1970, 282 pages.
13. I.G. Portnov, "Exact Solution of Freezing Problem with Arbitrary Temperature Variation of Fixed Boundary," Soviet- Physics-Doklady, 7(3): 186-189, 1962. 81
14. F. Jackson, "The Solution of Problems Involving the Melting and Freezing of Finite Slabs by a Method due to Portnov," Proceeding Edinburgh Mathematical Society, 14(2): 190-128, 1964/65.
15. D. Langford, "New Analytical Solutions of the One-Dimensional Heat Equation for Temperature and Heat Flow Rate Both Prescribed at the Same Fixed Boundary (with Application to the Phase Change Problem)," Quarterly of Applied Mathematics, 24: 315- 322, 1966.
16. K.O. Westphal, "Series Solution of Freezing Problem with the Fixed Surface Radiating into a Medium of Arbitrary Varying Temperature," International Journal of Heat Mass Transfer, 10: 195-205, 1967.
17. M.A. Biot, "Variational Principle in Irreversible Thermodynamics with Application to Viscoelasticity," Physical Review, 97: 1463-1479, 1955. 18. T.R. Goodman, "The Heat-Balance Integral and Its Application to Problems Involving a Change of Phase," Transactions of the ASME. 80: 335-342, 1958.
19. H. Schlichting, Boundary Layer Theory, (New York: McGraw Hill Book Company, 1955), chapter 12.
20. P.P. Smyshlyaev, "Approximate Solutions of the Heat Conduction Equation for a Two-Layer Plate with Sublimation at the Outer Surface," Journal of Engineering Physics, 12(4): 451-459, 1967.
21. K.A. Rathjen and I.M. Jiji, "Heat Conduction with Melting or Freezing in a Comer," Journal of Heat Transfer, Transaction of ASME, 101-109, 1971.
22. M. Elmas, "On Heat Transfer with Moving Boundary," International Journal of Heat Mass Transfer, 13: 1625-1627, 1970.
23. J.M. Savino and R. Siegel, "A Discussion of the Paper on the Solidification of a Warm Liquid Flowing over a Cold Wall," International Journal of Heat Mass Transfer, 14: 1225-1226, 1971.
24. H.G. Landau, "Heat Conduction in a Melting Solid," Quarterly of Applied Mathematics, 8: 81-94, 1950.
25. E. Hirai and T. Kamori, "A Proposal to Solve Neumann’s Problem Approximately, Taking Account of Bulk Dilatation by Phase Change - A Semi-Infinite Solid," Journal of Chemical Engineering Japan, 4(1): 96-99, 1971* 82
26. T. Karaori and E. Hirai, "Solutions of Heat Conduction Problem with Change of Phase - a Slab," Journal of Chemical Engineering Japan, 5C3): 242-248, 1972.
27. Y. Shih and S. Y. Tsay, "Analytical Solutions for Freezing a Saturated Liquid Inside or Outside cylinders," Chemical Engineering Science, 26: 809-816, 1971.
28. Y. Shih and T.C. Chou, "Analytical Solutions for Freezing a Saturated Liquid Inside or Outside Cylinders," Chemical Engineering Science, 26: 1787-1793, 1971.
29. J.P. Gupta, "An Approximate Method for Calculating the Freezing Outside Spheres and Cylinders," Chemical Engineering Science, 28: 1629-1633, 1973-
30. T.G. Theofanous and H. C. Lim, "An Approximate Analytical Solution for Non-Planar Moving Boundary Problems," Chemical Engineering Science, 26: 1297-1300, 1971.
31. V. Pechoc, "Generalized Numerical Solution of Heat Equation with Moving Boundary Condition," Collection Czech. Chem. Camrnum., 36: 3222-3235, 1971.
32. L.C. Tao, "Generalized Numerical Solutions of Freezing a Saturated Liquids in Cylinders and Spheres," AICHE Journal, 13: 165-169, 1967.
33. L.C. Tho, "Generalized Solution of Freezing a Saturated Liquid in a Convex Container," AICHE Journal, 14: 720-721, 1968.
34. A.S. Teller and S.W. Churchill, "Freezing Outside a Sphere," Chemical Engineering Progress Symposium, 61(59): I85-I89, 1965.
35. W.D. Seider and S.W. Churchill, "The Effect of Insulation on Freezing Front Motion," Chemical Engineering Progress Symposium, 61(59): 179-184, 1965.
36. A. Lazaridis, "A Numerical Solution of the Multidimensional Solidification (or Melting) Problem," International Journal of Heat Mass Transfer, 13: 1459-1477, 1970.
37- G.S. Lock, J.R. Gunderson, D. Quon and J. K. Donnelly, "A Study of One-Dimensional Ice Formation with Particular Reference to Periodic Growth and Decay," International Journal of Heat Mass Transfer, 12: 1343-1352, 1969.
38. G.S. Lock, "On the Perturbation Solution of the Ice-Water Layer Problem," International Journal of Heat Mass Transfer, 14: 642-644, 1971. 83
39. R.I. Pedrosco and G.A. Dcmoto, ''Exact Solution by Perturbation Method for Plannar Solidification of a Saturated Liquid with Convection at the Wall," International Journal of Heat Mass Transfer, 16: 1816-1819, 1973.
40. Y.K. Chuang and J. Szekely, "Use of Green's Function for Solving Melting or Solidification Problems," International Journal of Heat Mass Transfer, 14: 1285-1294, 1971-
41. Y.K. Chuang and J. Szekely, "Use of Green's Function for Solving Melting or Solidification Problems in the Cylindrical Coordinate System," International Journal of Heat Mass Transfer, 15: 1171-1174, 1972.
42. S.H. Cho and J.E. Sunderland, "Phase Change of Spherical Bodies," International Journal of Heat Mass Transfer, 13: 1231-1233, 1970.
43. H.C. Lim and T.G. Theofanous, "On Unsteady State Moving Boundary Problems for Non-Plane Geometries," Journal of Chemical Engineering Japan, 4: 100-101, 1971.
44. B. Boley, "A General Starting Solution for Melting and Solidifying Slabs," International Journal Engineering Science, 6: 89-111, 1968.
45. H.P. Yagoda and B.A. Boley, "Starting Solutions for Melting of a Slab Under Plane or Axisymmetric Hot Spots," Quarterly Journal Mechanics of Applied Mathematics, 23: 225-246, 1970.
46. S. Lin, "Che-Dimensional Freezing or Melting Process in a Body with Variable Cross-Sectional Area," International Journal of Heat Mass Transfer, 14: 153-156, 1971.
47. G.A. Grinberg, "A Method of Approach to Problems of the Theory of Heat Conduction, Diffusion and the Wave Theory and Other Similar Problems in Presence of Moving Boundaries and its Application to Other Problems," Journal of Applied Mathematics and Mechanics, 31: 215-224, 1967.
48. M.S.M. Selmin, Heat and Mass Transfer with a Moving Boundary PhD Thesis, Iowa State University, 1970, 213 pages.
49. V.V. Krasnikov and M.S. Smirnov, "Application of Conjugate Problems of Heat Conduction with a Moving Boundary to the Analysis of Drying Processes," Journal of Engineering Physics, 14(5): 407-410, 1968.
50. V.F. Demchenko and S.S. Kozlitina, "Numerical Solution of the Diffusion Problem in a Two-Phase Medium with a Moving Inter face," Journal of Engineering Physics, 15: 985-989, 1968. «4
51. D.F. Dyer and J,E. Sunderland, "Freeze^Drying of Bodies Subject to Radiation Boundary Conditions,'-' Journal of Heat Transfer, 93: 427-431, 1971.
52. T.G. Theofanous and H.C. Llm, "Approximate Analytical Solution for Non-Planar Moving Boundary Problems," Chanical Engineering Science, 26C8 ): 1297-1300, 1971.
53- W.A. Olsen and W.E. Hllalng, "Three Phase Heat Transfer: Transient Condensing and Freezing from a Pure Vapor on to a Cold Horizontal Plate - Analysis and Experiments," International Journal of Heat Mass Transfer, 15CIO) 1887-1903, 1972.
54. P.Kehoe, "A Moving Boundary Prlblem with Variable Diffusivity," Chemical Engineering Science, 27: 1184-1185, 1972.
55* T. Komori and E. Hlrai, "An Application of Stefan's Problem to the Freezing of Cylindrical Foodstuff," Journal Chanical Engineering Japan, 3: 39-44, 1970.
56. J.C. Muehlbauer, J.D. Hatcher, D.W. Lyons and J.E. Sunderland, "Transient Heat Transfer Analysis of Alloy Solidification," Transaction ASME, Journal of Heat Transfer, 95: 324-331, 1973.
57. O.S. Lee and L.S. Simpson, "A Practical Method of Predicting Sea Ice Formation and Growth," Technical Report, U.S. Navy Hydrographic Office, 1954, 27 pages.
58. M.A. Bilello, "Formation, Growth and Decay of Sea Ice in the Canadian Arctic Archipelago," Journal of the Arctic Institute of North America, 14(1): 1-24, 1961.
59. M.A. Bilello, "Method for Predicting River and Lake Ice Formation," Journal of Applied Meterology, 3: 38-44, 1964.
60. C.M. Adams, D.N. French and W.D. Kingery, "Solidification of Sea Ice," Journal of Glaciology, 3: 745-760, i960.
61. G.P. Williams, "Water Temperature During the Melting of Lake Ice," Water Resources Research, 5(5): 1134-1138, 1969.
62. L.J. Fertuck, J.W. Spyker and W.H.W. Husband, "Numerical Estimation of Ice Growth as a Function of Air Temperature, Wind Speed and Snow Cover,',' Transactions Canadian Society for Mechanical Engineers, Engineering Journal, 54: 1-6, 1971.
63. D. Pitman and B. Zuckerman, "Effective Thermal Conductivity of Snow at -88°, -27°, -5°C," Journal of Applied Physics, 38: 2698-2699, 1967. 85
64. E.A. Pounder, The Physics, of ice, (Oxford; Pergamon Press, 19651.
65. C.M. Mams, Jr., '"Thermal Considerations in Freezing,’’ (In liquid Metals and Solidification. Cleveland, Ohio, American Society for Metals, 1957).
66. W. Yolk, Applied Statistics for Engineers, (New York: McGraw- Hill, 1 9 5 $ ) • 86
APPENDICES 87
APPENDIX A
EXACT ANALYTICAL SOLUTION OF WESTPHAL
Westphal, in his solution, utilized the method due to Portnov
(13) to determine the position of the phase-change front as a function of time. Essentially Portnov’s technique in application to the ice-water system involves solving the following simultaneous equations
3 ^ d 2 T ± (1=1, 0 < x < b) (1-A) = “i 2 ~ at ax (i = 2, b < x < °°) with supplementary conditions
3T k-, — = h (Tx - T (t)) ax
T2 (x ,0) = g(x) > 0 , b(0) = 0 (2-A)
and the conditions at the free boundary
T^b.t) = T2(b,t) = 0
db a ^ pX^, — k, dt x 3x x = bCt) (3-A)
Portnov obtains a formal solution to equation (1-A) without satisfying any boundary conditions and is given by OB
b2(y-n)2 T(y,t) { exp [- ] } (L (bn) — — ■ dn \fn ■ 4t 1 2 \/t
b2(y-n)2 { exp [- ] } (L(bn) — p d n (4-A) V/”tt 4t 2
where y = x/b(t). The functions (f)^, (J)2 and bCt) are unknown and have
to be determined using the boundary condition (2-A) and (3-A). To
determine these three unknown functions ^(w), (j>2(w) and b(t), one
assumes that these functions together with the ambient air temperature
T (t) can be expanded in power series £1
0
(5-A)
b(2 \jt) = E bn (2\/t)n
Ta (2 ][t) = E Tan (2 /t)n
where the expansion of b(2 t) already satisfied the initial condition.
By substituting the formal solution. Equation 4-A, into the boundary condition, Equations 2-A, 3-A, and utilizing the power series expansions,
Equation 5-A, the coefficients, (J)^, (J)^, bn and T&n can be determined with some difficulty. Westphal upon obtaining the coefficients bn 89
was then able to determine the location of the phase front, b, as a
function of time. 90
APPENDIX B
SENSIBLE HEAT CONSIDERATIONS
Adams (65) considers the effect of sensible heat content of
the ice on the freezing process. He derives an expression to account
for the additional heat that must be removed to lower the temperature
of the solid ice. This increase in heat flow is given by the following
expression
, ~ 1/2 1 + [ 1 + ^ Ste (1 - ZJ)] (1-B)
k where Z = --- — h + k g s
Consider, for example, the severe condition when the ice thickness, b, is greater than one foot and the Stefan number is equal to .218, produced when the ambient air temperature is -30°F. The error introduced by not considering the sensible heat effect (from equation
1-B) is 6 .3%. 91
APPENDIX C
EFFECTIVE THERMAL CONDUCTIVITY OF SNOW
The effective thermal conductivity of snow was determined frcm
data taken from Pitman (63) which related the dependence of effective
thermal conductivity of snow to snow density.
It is assumed that a linear equation is suitable for inter
polation purposes between the snow density range of 0.1 g/cc to 0.3
g/cc. Frcm Pitman’s data the effective thermal conductivity of snow
varies from O.O63 to 0.146 BTU hr ^ ft ^ °F within this range. The
data of this experiment was recorded at -5°C. Therefore, the calculated
dependence of the effective thermal conductivity of snow and snow
density is given by the following linear relation
ksn = 0 . 2 9 + .39 psn > .1 < Psn < .3 g/cc (1-C)
For settled snow the effective thermal conductivity of snow is taken as k = 0.123 BTU hr-1 ft-1 °F_1. sn 92
APPENDIX D
AMBIENT AIR TEMPERATURES TAKEN DURING THE FIRST
OBSERVATION PERIOD OF JANUARY 7 TO JANUARY 20, 1974
rate Time Temp. (°F) Observed Data
1/7/74 5 pm 22.0 time 2:35 to 3:00 pm (8.3, 8.7, 9.1, 8.9, 8.7)cm water temp. 33, 34, 34°F
6 pm 22.0
7 pm 22.0
8 pm 20.5
9 pm 22.0
10 pm 22.0
11 pm 21.0
12 pm 20.0
1/8/74 1 am 20.0
2 am 19-7
3 am 17.0
4 am 16.5
5 am 14.5
6 am 12.5
7 am 9-5
8 am 9.0
9 am 16.0
10 am 22.0
11 am 31.0
12 am 32.0 93
1/8/74 1 pm 23.Q
2 pm 21.0
3 pm 18.5
4 pm 14.9
.5 pm 12.5
6 pm 9.0
7 pm 6.0
8 pm 4.0
9 . pm 4.0
10 pm 4.0
11 pm 3.5
12 pm 0.0
1/9/74 1 am 3.0
2 am 1.0
3 am 3.0
4 am 4.0
5 am 6.0
6 am 6.0 Ice snow time depth Com) depth (In) 7 am 5.0 10:50 am 13.0 1.75 12:25 am 3.0 8 am 4.0 13.5 1:50 pm 13.7 4.0 2:00 pm 13-8 9 am 3-5 3:14 pm 14.0 5.0 4:10 pm 14.0 5.38 10 am 3-0
11 am 4.0 psn 0.0833 g/cc
12 am 6.0
1 pm 5-0
2 pm 5.0
3 pm 6.0 94
1/9/74 4 pm 7.5
5 pm 6.0
6 pin 6.0
7 pm 6.0
8 pm 6,0
9 pm 6.0
10 pn 7.0
11 pm 7.0
12 pm 7.0
1/10/74 1 am 7.0
2 am 7.0
3 am 7.0
4 am 7.0
5 am 8.0
6 am 8.0
7 am 7.5
8 am 8.0
9 am 7-5
10 am 8.5
11 am 10.0
12 am 13-0
1 pm 13-5
2 pm 14.0
3 pm 14.0
4 pm 14.0
5 pm 14.0
6 pm 14.0 95
1/10/74 7 pm 13.0
8 pn 13.0
9 pm 14.0
10 pm 14.0
11 pm 14.0
12 pm 14.5
1/11/74 1 am 14.5
2 am 15.5
3 am 16.0
4 am 16.0
5 am 16.0
6 am 16.0
7 am 16.0
8 am 16.0
9 am 16.0
10 am 17.0
11 am 17.5
12 am 17.5
1 pm 18.5
2 pm 18.0
3 pm 19.0
4 pm 18.0
5 pm 18.0
6 pm 18.0
7 pm 17.0
8 pm 17-5
9 pm 18.0 96
1/11/74 10 pm 17.5
11 pm 17-5
12 pm 17-5
1/12/74 1 am 16.0
2 am 15-0
3 am 10.5
4 am 8.0
5 am 7.0
6 am 6.0
7 am 5.5
8 am 4.0
9 am 6.0 10 am 16.0
11 am 32.0
12 am 28.0
1 pm 18.0
2 pm 26.0
3 pm 23-0
4 pm 21.0
5 pm 16.5
6 pm 12.0
7 pm 9-0
8 pm 8.0
9 pm 9.0
10 pm 6.0
11 pm 4.0
12 pm 3.0 97
1/13/74 1 anj 2,0
2 am 0.5
3 am -1.0
4 am -7-0
5 am -7.0
6 am -2.0
7 am -11.0
8 am -11.0
9 am +1.0
10 am 20. Q
11 am 25.0 ice snow 12 am 23.0 time depth, (cm) depth, Cin)
1 pm 16.0 12:00 am 17-3 3-5
2 pm 17.0
3 pm 17.0
4 pm 16.0
5 pm 12.0
6 pm 7.0
7 pm 5.0
8 pm 6.0
9 pm 5.0
10 pm 3.5
11 pm -0.5
12 pm 0.0
1/14/74 1 am -2.0
2 am -6.0
3 am -4.0 98
i/wm 4 915 -4,0
5 am -10.0
6 am -10.0
7 am -10.0
8 am -14.0
9. am -3.0
10 am +7.0
11 am 4.0
12 am 18.0
1 pm 19..5
2 pm 19.0
3 pn 20.0
4 pm 19.5
5 pm 18.5
6 pm 17.0
7 pm 17.0
8 pm 19.0
9 pm 20.0
10 pm 20.0
11 pm 21.0
12 pm 21.0
1/15/74 1 am 22.0
2 am 22.5
3 am 24.0
4 am 26.0
5 am 28.0
6 am 28.5 99
1/15/74 7 am 28.5
8 am 29.5
9. am 30.5
ao am 32.5
11 am 33.0
12 am 33.0
1 pm 32.0
2 pm 31.0
3 pm 32.0
4 pm 30.0
5 pm 28.5
6 pm 28.5
7 pm 28.0
8 pm 28.0
9 pm 28.0
•10 pm 28.0
11 pm 28.0
12 pm 28.0
1/16/74 1 am 23-5
2 am 22.5
3 am 22.5
4 am 20.5
5 am 20.0
6 am 20.0
7 am 19.0
8 am 17.0
a am 15.0
10 am 13.0 100
1/16/74 11 am 13.0
11 am 11.0
12 am 10.0
1 pm 8.0
2 pm 8.0
3 pm 8.0
4 pm 8.0
5 pm 8.0
6 pm 7.0
7 pm 6.5
8 pm 6.0
9 pm 5.5
10 pm 5-5
11 pm 5.0
12 pm 3*5
1/17/74 1 am 3.0
2 am 2.0
3 am 0.5
4 am -1.5
5 am —6.0
6 am —9.0
7 am -12. Q
8 am -14.0
a am -9-0
10 am —6.0
11 am +16.0
12 am 13-0 101
1/17/74 1 pm 16. Q
2 p 12.0
3 pm 10.0
4 pm 7-0
5 pm 3.0
6 pm -3,0
7 pm -9.0
8 pm -15.0
9_ pm -16 .0
10 pm -17.0
11 pm -19.0
12 pm —20.0* * lower limit; of recorder
1/18/74 1 am -20.0*
2 am -20.0*
3 am -20.0*
4 am -20.0*
5 am -20.0*
6 am -20.0*
7 am -20.0*
8 am -20.0*
9 am -9.0
10 am -1.0
11 am +2.0
12 am 9.0
1 pm 9.0
2 pm 13.0
3 pm 10.0 102
1/18/74 4 pm 1Q.0
5 pm 8.0
6 pm 8.0
7 pm 7-5
8 pm 7-5
9 pm 7.5 10 pm 7-0
11 pm 7.0
12 pm 7.5
1/19/74 1 am 8.0
2 am 8.0
3 am 9.0
4 am 1Q.0
5 am 11.0
6 am 11.0
7 am 12.0
8 am 13.0
9 am 17.0
10 am 35.0
11 am 41.0
12 am 47-0
1 pm 36.0
2 pm 33.0
3 pm 32.0
4 pm 28.0
5 pm 24.5
6 pm 19.5 103
1/19/74 7 pm 17-0
8 pm 14.0
9 pm 11.0
10 pm 8.0
11 pm 6.0
12 pm 4.0
1/20/74 1 am 2.0
2 am 0.0
3 am -3.0
4 am . -4.0
5 am -6.0
6 am -4.0
7 am -10.0
8 am -11.5
9 am -4.0 Ice snow 10 am +3.5 time depth(cm) depth(in)
11 am 11.0 11:30 am 18.1 6.5
12 am 22.0
1 pm 20.0
2 pm 19.3
3 pm 14.0
4 pm 10.5
5 pm 12.5
6 pm 13.0
7 pm 12.5
8 pm 13.0
9 pm 13.5 10 pm 13.0 11 pm 13 • 0 12 pm 13-0 104
APPENDIX E
DEFINITION OF CORRELATION COEFFICIENT
It was assumed from Figures 14 and 1 5 that a linear correlation
exists between the Concord and Great East L a k e air temperature data
and also between the Portland and Great East Lake air temperature data.
If a linear correlation exists, the correlation coefficient defined
by Volk (66) as
n | (x - x) (y - y)
~ n p n p -I /o (1“E [£ (x - x)2 e (y - y) ] i i
_ n _ n where x = E x,/n , y = E y./n , n = n u m b e r of data points • jL • 1 1 1 r is a meaningful parameter to determine the correlation between variable x and variable y. When a perfect positive correlation exists between x and y, there is no residual deviation of y frcm y (y=a+bx)
and r equals 1.0. When there is no correlation between x and y, r
equals zero. 105
C PROGRAM TO DETERMINE CORRELATION BETWEEN GE , PORTLAND AND COMCL C DATA DIMENSION Y(20),X(20,2),XBAR(2),R(2) READ(5,2)N R E A D ! 5,1) (Y(I ) ,I = 1,N) DO 10 J=1,2 10 R EA D( 5 , 1 ) (X(I ,J) ,1 = 1,N ) X B AR ( 1 ) =0 XBAR ( 2 )•= 0 YBAR=0 DO 15 I=1,N XBAR(1)=XRAR(1)+X(I,1)/N XBAR{2)=XBAR(2)+X(I,2)/N 15 Y B A R= YB AR +Y (I )/N SUMTOP =0 SUMB1=0 S U M B 2 = 0 DO 20 J = l,2 DO 20 I=1,N SUMTOP = SUMTOP+(X(I,J )-XBAR(J ) )*(Y(I)-YBAR) SUMBl = SUMRl+( X ( I , J )-XBAR (J )•)•**2 SUMB2 = SU M R 2 + (Y( I )-YBAR)**2 20 R ( J ) = SUM TOP /SORT ( SUM B1 *SUMB2 ) W R I T E (6,3) W R I T E ( 6 , 4 )(R(J),J=1»2) W R I T F (6,5 ) WRI TF( 6,6)YBAR,XBAR( 1),XBAR!2) 1 FORMAT!16F5.0) 2 F O R M A T (15) 3 FORMAT!1H1,1 CORRELATION COEFFICIENTS',/,' R1 AND R2',1X) 4 FORMAT!2E14.7) 5 FORMAT!' YBAR, X1 BAR, X2BAR',1X) 6 FORMAT!3F14.7) CALL EXIT END 106
APPENDIX P
COMPUTER PROGRAMS
C PROGRAM TO PREDICT LAKE ICE GROWTH AND DECAY COMMON TA!50),WVEL(50),TSNUW(50),SNOWF(50),WE(50),SUNSHN!50) X L 1 = • 0 1 * 2 . 5 4* 12 . XL2=.07*2.54*12. BQBS = .88 5 I M = 23 FUNC=0 XLAMDA=142. R H 0 S = 5 7 .5 R EAD(5,1)NDAYS,BO,BR,IR WRITE!6,4) READ!5,2)(TA(I),I=1,NDAYS) READ!5,2)(WVEL(I),1=1,NDAYS) READ(5,2)(TSNOW(I),1=1,NDAYS) READ!5,2)(SNOWF!I),I=1,NDAYS) R EA D (5,2) (WE (I ),1 = 1,NDAYS) READ!5,2)(SUNSHN(I),1=1,NDAYS) DEPTH = B0/(2.54*12. ) DO 10 1=1,NDAYS T A {I )=T A(I )-1.9 SDEPTH = TS N O W (I )/12.‘ „ - • DAY = I IF(I.EO.IR) DEPTH=DEPTH+BR WRITF(6,3)DAY,DEPTH,SDEPTH CALL RIJNGEtDAY,DEPTH,ANS, I ) IF!I.GE.IM)FUNC = FUNC+EXP(-XL1*!ANS-BR)-XL2*BR)*SUNSHN! I)/24. 10 DEPTH=ANS A0=(ANS-BOBS)/FUNC/24.*XLAMDA*RHOS WRITE(6,6)A0 6 FORMAT(1 A 0=',E 14.7,1BTU/HR/FT**21,1X ) 1 FORMAT!15,2F5.0,15) 2 F ORMAT( 16F 5.0) 3 F0RMAT(F5.0,F10.5,5X,F5.1) 4 FORMA T(1H1,1 DAY',2X, ' DEPTH(FT) ',2X,•SNOW!FT) •,IX) CALL EXIT END SUBROUTINE RUNGE(DAY,DEPTH,AMS,I) COMMON TA!50 ) ,WVEL(50),TSNOW(50),SNOWF(50),WE(50) ,SUNSHN(50) COMMON /DATA/ XKSNOW,HG,0,K IF(WVEL(I).LE.5) GO TO 5 IF(TSNOW(I).GT.O.) GO TO 10 HG=.62*WVEL(I)**.8 GO TO 15 5 H G = 2 .05 GO TO 15 10 HG = •37*WVE L(I) 15 IF!SNOWF(I).EO.O. ) GO TO 11 XKSNOW=.029+.39*(WE(I)/SNOWF(I)) GO TO 12 11 X KSNOW=.12 3 107
12 I F ( I . G F . 2 3 ) (J = 56 . H 5* S UN SH N ( I ) /2 4. * EX P ( -. 0 1*2 . 54* 1 2 . * ( DFPT H~. 14) 1 .07*2.54*12.*.14) I F ( I . L T . 2 3 ) (0 = 0 K = I H=1 H2=.5 Y=DEPTH X = D A Y T 1 = H* FI )N ( X,Y ) T2=H* FU N (X+H2 t Y + T1/2. ) T3=H*FUN(X+H2»Y+T2/2. ) T4 = H* FI IN ( X +H »Y+ T3 ) A N S = Y+ ( T 1 + 2 . * ( T2 + T3)+T4)/6. RETURN END FUNCTI UN FUN(X,Y) CfiMNUN T A { 50) , WVFI. ( 5U ) , TSNUN (50) , SNOWF ( 50 ) T WE ( 50 ) , SUNSHN { 50 ) COMMON /O A T A / X KSiMt IW, HG , U , I X L AMDA = 142. RFIf )S = 57 .5 X K I C F = 1 . 2 7 FUN = HG / ( RHOS*X L AND A ) 4 24 . * ( 32 . -T A ( I ) ) / ( 1. + FIG* ( Y / XK I CE + T S iMUW ( I ) /> lOW/12. ) )-0*24./(RHUS-XLAMDA) RFTURM END - • 108
C PROGRAM TO PREDICT ICE DEPTH FORMATION ASSUMING NO HEAT FLUX IN COMMON TA(30,24)>SWIND(30,8),HSN0W(30),WEQUIV(30) COMMON /DATA/TAIR, XKSNOW, HG , S DEPTH , C) DIMENS ION TOTSNO(30) ,SUNSHN!30) C INITIALIZATION TI M E = 0 XKSMOW=.17 2 C DATA INPUT READ(5»1)NDAYS 1 F O R M A T (15) DO 10 1=1,NDAYS 10 R E A D (5,2)(T A (I ,J )» J= 1» 24) 2 FORMAT(16 F 5 . 1) DO 20 1=1,NDAYS 2 0 READ(5,3)(SWIMD(I,J),J=l,8) 3 FORMAT(8F3.0) READ(3,4)(HSNOW(I),I=1,NDAYS) 4 F 0 R M A T (16F5.0) READ(5,5) (WEOUIV(I ) ,I=1,N D A Y S ) 5 FORMAT(16F5.2) READ(5,4)(TOTSNO(I),I=1,NDAYS) READt 5,4) (S UN SH N (I ),1 = 1,NDAYS) READ(5,12)DEPTH,JO 12 FORMAT(F5.0,I5) DEPTH = DEP T H / (2.54*12. ) W R I T E (6,7) 7 FORMAT( 1H1, 1 DAY',2X,1 ICE DEPTH',2X,'SNOW DEPTH',IX,' ALL IN FE 1', IX) C INTEGRATION BY RUNGE KUTTA DAY = 0 SDEPTH=TOTSNO( 1 )/12. WRITE(6,6)DAY,DEPTH,SDEPTH DO 30 1=1,NDAYS SDEPTH = TOTSN O( I )/12 . DO 40 J =J O ,24 CALL RIJNGE( 1. , TI ME , D EP TH , ANS , I , J, SUNSHN ( I ) ) TIME=TIME+1 40 DEPTH=AMS J0=1 DAY=TIME/24. 30 WRITE(6,6)DAY,DEPTH,SDEPTH 6 F O R M A T (3 F10.5) CALL EXIT END SUBROUTINE RUNGE(H ,X,Y,ANS,I,J,S) COMMON TA(30,24)»SWIND(30»8),HSN0W(30),WEQUIV(30) COMMON /DATA/TAIR,XKSNOW,HG,SDEPTH,Q C TAIR=DEG F TAIR=TA(I,J) C CALCULATION TO DETERMINE HEAT TRANSFER COEFFICIENT 109
k=( j- n / 3+1 IF(K.EQ.O.OR.K.EO.8)GO TO 10 C VELOCITY OF WIND = MPH VEL = ( ( SW I MD ( I ,K+ 1)- S W I N D ( I * K ) ) /3 .* ( J -.1 -3*( K- 1) )+SWIND( I , K ) ) * 1. 1 5 GO TO 15 10 IF(K.FQ.O)VEL = SWIND( 1,1) #1.15 IF(K.E0.8)VEL = SW I ND (I ,8)*1 . 15 15 IF(VFL.LE.5.5) GO TO 5 C HG = BTIJ/HR/SOFT/DEG F HG=.37*VEL GO TO 20 5 HG = 2.05 20 CONTINUE I F(HSN0W (I).E0.0.) GO TO 11 C THERMAL CONDUCTIVITY = BTU/HR/FT/DEG F XKSNOW=.0 29+.39*(WEQUIV(I)/HSNOW(I)) GO TO 12 11 XK S NOW = .123 12 CONTINUE DEPTH=Y 0=S/24.*39.8*EXP(-6.1*SDEPTH-.305*DEPTH) H2=H/2 T1=H* FUN (X »Y) T2=H*FUN(X+H2,Y+Tl/2.) T3 = H*FUN ( X+ H 2 , Y + T2/2. ) * * TA = H* F UN (X + H,Y + T 3) ANS = Y + ( Tl+2.*( T2+T3 ) + TA-)-/6. RETURN END FUNCTION FUN(X »Y ) COMMON /DATA/TAIR,XKSNOW»HG,SDEPTH,Q XLAMDA=142. RHOS=57.5 XKIC E= 1.27 FUN=HG/(RHOS#XLAMDA)*(32.-TAIR)/(l.+HG*(Y/XKICE+SDEPTH/XKSNOW)) RETURN END