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ABSTRACT

TRANSPORT PHENOMENA IN POROUS MEDIA
WITH EMPHASIS ON WATER MOVEMENT IN SOILS

BY

Lawrence Thomas Novak

Unrestricted dumping of domestic and industrial
waste water into the waterways will be a thing of the past
in the United States. Stringent federal standards will
require that waste water be treated before being returned
to the waterways. One method of treating this waste water
is by spray irrigation of the soil mantle.

In order to successfully design and manage a
spray irrigation facility, mathematical models are required
which will predict the movement of water and chemical
compounds in the soil. This work addresses itself to
modeling the water movement in soils.

One objective of this work is to develOp a general
model for water movement in soils. This model is then
compared to less general models which have previously
been proposed to describe water movement in soils. Another
objective is to test the models by comparing the simulation

results with the experimental results.

-Lawrence Thomas Novak

The general model was derived by making material
balances on water in the liquid and vapor form, and by
making an energy balance on the soil. This resulted in a
set of transport equations which describe the movement
of water and energy in the soils. The novel feature of
this model is that the condition of interphase vapor-liquid
equilibrium is not assumed. This equilibrium assumption
was the basis of an earlier model for water movement in
soils which was developed by Philip and DeVries.

The unsteady state numerical solution was develoPed
for the general model, Philip and DeVries model, and the
isothermal equation. An explicit finite difference
technique was used in the numerical solutions. The
unsteady state solution to the Philip and DeVries model
has not been obtained prior to this work. With the
numerical solution to the above models, soil drying
simulations were run and the results were compared to
experimental results.

It was found that even with dry soils the inter-
phase vapor-liquid assumption is valid. Furthermore,
the isothermal equation and Philip and DeVries model
give essentially the same drying saturation profiles
when identical irreducible saturations are used in the
models. Good agreement was obtained between calculated

and experimental drying saturation profiles.

Lawrence Thomas Novak

The solution to the Philip and DeVries model developed
in this work should be useful in problems involving drying
of soils or porous media where a predictive capability is
needed. The general model will also be useful in these
types of problems to evaluate the interphase vapor-liquid
equilibrium assumption upon which the Philip and DeVries
model is based.

For soils which are above the irreducible saturation,
the isothermal equation for water movement would be the
most expedient model to use if soil temperature informa-
tion was not desired. In particular, the isothermal
equation would be a good base for structuring models to
describe the movement of chemical compounds in soils

subjected to spray irrigation with liquid wastes.

TRANSPORT PHENOMENA IN POROUS MEDIA

WITH EMPHASIS ON WATER MOVEMENT IN SOILS

BY

Lawrence Thomas Novak

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemical Engineering

1972

\Jf‘x ,

DEDICATION

In memory of my father, John A. Novak

ii

ACKNOWLEDGMENTS

The author would like to express his appreciation
to his major professor, Dr. George Coulman, for his
interest and guidance in this work. Appreciation is
also extended to Drs. Myron Chetrick, Carl CooPer, and
William COOper for their participation in this work.

The author is greatly indebted to the Diamond
Shamrock Foundation and the National Science Foundation
(Grant GI-20) for financial support.

Special thanks are due to my wife Donna for her
moral support throughout this work and for her help in

preparation of this thesis.

iii

TABLE OF CONTENTS

DEDICATION '. . . . . . . . . . . . .
ACKNOWLEDGMENTS. . . . . . . . . . . .
LIST OF TABLES . . . . . . . . . . . .
LIST OF FIGURES. . . . . . . . . . . .
NOMENCLATURE. . . . . . . . . . . . .
Chapter

I INTRODUCTION . . . . . . . . . .

II THEORY . . . . . . . . . . . .

III

IV

Derivation of the Macrosc0pic Equation of
Continuity for Liquid Phase Water in the
SOil. O O O I O I O I O O O O

Derivation of the Macrosc0pic Equation of
Continuity for Water Vapor in the Soil .

Derivation of the Macrosc0pic Equation of
Energy for the Soil. . . . . . . .

Model Parameters. . . . . . . . .
General Model for Water Movement in Soils
under the Condition of Vapor-Liquid

Equilibrium . . . . . . . . . .
NUMERICAL SIMULATION . . . . . . .
RESULTS AND DISCUSSION. . . . . . .

CONCLUSIONS 0 I O O O O O O O 0

iv

Page
ii
iii
vi
vii

ix

13

15

l6

I9

38
41
51

108

REFERENCES I I I I I I I I I I I I
APPENDIX
A CAPILLARY POTENTIAL AS A FUNCTION OF

SATURATION, TEMPERATURE, HISTORY, AND
COMPOSITION. . . . . . . . .

ANALOGY BETWEEN THE ISOTHERMAL EQUATION

OF

GARDNER AND THE UNSTEADY STATE DIFFUSION

EQUAT ION I I I I I I I I I I

THREE DIMENSIONAL EQUATIONS OF CHANGE FOR

POROUS MEDIA . . . . . . . .

LIQUID WATER VAPOR PRESSURE AS A FUNCTION

OF TEMPERATURE, SATURATION, AND
COMPOSITIONI I I I I I I I I

INTERFACIAL AREA MODEL (SIMPLE CUBIC PACKING

OF RODS) AND COMPUTER PROGRAM. . .

COMPUTER PROGRAM SUBROUTINES CONTAINING
PHYSICAL PROPERTY DATA . . . . .

COMPUTER PROGRAM FOR NUMERICAL SOLUTION
THE GENERAL MODEL. . . . . . .

COMPUTER PROGRAM FOR NUMERICAL SOLUTION
THE PHILIP AND DE VRIES MODEL. . .

COMPUTER PROGRAM FOR NUMERICAL SOLUTION
THE ISOTHERMAL EQUATION. . . . .

OVER ALL WALL HEAT TRANSFER COEFFICIENT
CALCULATION. . . . . . . . .

SURFACE FILM COEFFICIENTS. . . . .
CALCULATION OF PART OF THE CAPILLARY

POTENTIAL-SATURATION FUNCTION FOR
VALENTINE SAND. . . . . . . .

COMPARISON OF CONDUCTIVE ENERGY FLUX AND

CONVECTIVE ENERGY FLUXES . . . .

COMPUTER PROGRAM SUBROUTINES CONTAINING

NATURAL WEATHER TIME VARYING BOUNDARY
CONDITIONS I I I I I I I I I

V

T0

T0

TO

Page

112

116

119

121

123

127

142

154

165

176

184

188

191

193

196

10.
11.

12.

13.

14.

LIST OF TABLES

Page
Legend for figure 12 . . . . . . . . 44
Simulation problems. . . . . . . . . 52
Simulation parameters for simulation number
1 I I I I I I I I I I I I I I 56
Simulation parameters for simulation number
2 I I I I I I I I I I I I I I 57
Simulation parameters for simulation numbers
3'4, and 6 I I I I I I I I I I I 63
Simulation parameters for simulation number
5 I I I I I I I I I I I I I I 7i
Material and energy fluxes during drying
(simulation number 5) . . . . . . . 78
Simulation parameters for simulation number
7 I I I I I I I I I I I I I I 81
Simulation parameters for simulation number
8 I I I I I I I I I I I I I I 85
Simulation number 8 results . . . . . . 86
Simulation parameters for simulation number
9 I I I I I I I I I I I I I I 87
Simulation parameters for simulation number
lOI I I I I I I I I I I I I I 89
Simulation parameters for simulation number
llI I I I I I I I I I I I I I 94
Maximum daily soil temperature changes over
an annual cycle at Argonne, Ill.. . . . 107

vi

Figure

12.

13.

14.

15.

16

17.

18.

LIST OF FIGURES

Differential element in a soil column . . . .
Interfacial area model . . . . . . . . .

Interfacial area per volume versus soil satura-
tion I I I I I I I I I I I I I I

Hydraulic conductivity versus soil saturation .
Hydraulic conductivity versus soil saturation .
Hydraulic conductivity versus sOil saturation .
Capillary potential versus soil saturation . .
Capillary potential versus soil saturation . .
Capillary potential versus soil saturation . .
Capillary potential versus soil saturation . .

Effective thermal conductivity and volumetric
heat capacity versus soil saturation. . . .

Diagram of sand column for drying problems . .

Soil saturation versus depth of soil (simulation
numbers 1 and 2) . . . . . . . . .

Soil temperature versus depth of soil (simula-
tion number 2) . . . . . . . . . . .

Equilibrium mole fraction of soil water vapor
versus depth of soil (simulation number 2). .

Soil saturation versus depth of soil (simulation
nulrlber 3) I I I I I I I I I I I I I

Soil saturation versus depth of soil (simulation
number 6). . . . . . . . . . . . .

Cumulative evaporation versus time (simulation
numbers 3,4, and 6) . . . . . . . . .

vii

Page

25

27
29
3O

31

34
35

36

39

58

59

6O

64

66

67

Figure.

19I

2()I

21.

22.

23.

24.

25.

26.

27I

28.

29.

30,31

32,33

34,35

36.

37.

Soil saturation versus depth of soil (simula—
tion numbers 4 and 5). . . . . . . . .

Soil temperature versus depth of soil (simula-
tion number 5) . . . . . . . . . . .

Equilibrium mole fraction of soil water versus
depth of soil (simulation number 5) . . . .

Surface vapor flux versus time (simulation
number 5) I I I I I I I I I I I I I

Surface mole fraction of soil water vapor versus
time (simulation number 7) . . . . . . .

Surface mole fraction of soil water vapor versus
time (simulation number 9) . . . . . . .

Surface mole fraction of soil water vapor versus
time (simulation number 10). . . . . . .

Boundary conditions for simulation number ll
(solar radiation flux versus time) . . . .

Boundary conditions for simulation number 11
(ambient temperature versus time). . . . .

Boundary conditions for simulation number 11
(ambient relative humidity versus time). . .

Boundary conditions for simulation number 11
(wind speed versus time). . . . . . . .

Soil temperature versus time, z=0 (simulation
number 11) . . . . . . . . . . . .

Soil temperature versus time, z=9 cm. (simula-
tion nurrlber 11) I I I I I I I I I I I

Soil temperature versus time, 2:18 cm. (simula—
tion number 11). . . . . . . . . . .

Soil saturation versus depth of soil (simula-
tion number 11) I I I I I I I I I I I

Interfacial area model (simple cubic packing of
rods) . . . . . . . . . . .

viii

Page

68

73

74

77

83

92

93

95

96

97

98

100-101

102-103

104-105

106

129

02 C191 3’

O!
£=*£

NOMENCLATURE

Cross sectional area of soil column (cmz)
Interfacial area per volume (cm-l)
Molar concentration of gas phase (gm.mole/cm3)

Molar concentration of water vapor (gm.mole/cm3)

Equilibrium molar concentration of water vapor
(gm.mole/cm3)

Volumetric heat capacity of soil (cal./cm3-OK)
Heat capacity (cal./gm.—OK)

Diffusion coefficient for water vapor in soil
(cmZ/day)

Ordinary diffusion coefficient for water vapor
in air (cmZ/day)

Inside column diameter (cm)
Outside column diameter (cm)
Particle or droplet diameter (cm)

Outside column wall diameter (cm)

Parameters in the equations of Philip and De Vries
Interphase evaporation rate (gm./cm3-day)
Enthalpy of vaporization for water (cal./gm.)

Film coefficient for heat transfer (cal/cmz—day—OK)

ix

WWW

m,s

N 7’. W

28
2 £3 1:"

28

w,c

Surface film coefficient for heat transfer
(cal./cm2-day-0K)

Thermal conductivity (cal./cm-day9K)

Effective thermal conductivity (cal./cm—day-°K)
Film coefficient for mass transfer (cm/day)

Surface film coefficient for mass transfer (cm/day)
Hydraulic conductivity (cmz/day)

Over all interphase mass transfer coefficient
(cm/day)

Column length (cm)
Molecular weight of water (gm./gm.mole)_

Molar flux of water vapor with respect to fixed
coordinates (gm.mole/cmZ—day)

Convective molar flux of water vapor at the
soil-atmosphere interface (gm.mole/cmZ-day)

Partial pressure of water in the gas phase (atm.)
Total pressure (atm.)

Total gas phase pressure (atm.)
Capillary pressure (cm) 1

Vapor pressure of liquid water (atm.)
Total energy flux (cal./cm2-day)
Convective energy flux (cal./cm2-day)
Radiative energy flux (cal./cm2-day)
Volumetric flow rate (cm3/day)

Radius of capillary (cm)

Gas law constant (atm.cm3/gm.mole—°K)

Saturation = volume of liquid water per volume
of gas and liquid in soil (dimensionless)

X

(0|

0

<: <2
2"“;sz 2 <: <: <6

NC"

A

Nu=

Nu =

Pr=

Saturation = volume of water per volume of gas
and liquid in soil (dim.)

Time variable (day)
Temperature (OK)

Ambient temperature (OK)
Internal energy (ca1./gm.)

Over all column wall heat transfer coefficient
(cal./cm2-day-0K)

Molar volume (cm3/gm.mole)

Volumetric flux of liquid water (cm/day)
Volumetric flux of water (cm/day)

Molar average velocity (cm/day)

Velocity of component i with respect to fixed
coordinates (cm/day)

Width of soil column (cm)
Mole fraction of water in the gas phase (dim.)

Equilibrium mole fraction of water in the gas
phase (dim.)

Ambient mole fraction of water in the gas phase
(dim.)

Spatial variable (cm)

Dimensionles Number

 

Particle Nusselt number

Nusselt number

Prandtl number

——EE—- Particle Reynolds number

xi

Re

SC

VL

aw

= Reynolds number

= Schmidt number

 

-—-11J3 Particle Sherwood number

Sherwood number

Greek Symbols

 

Porosity or void fraction (dim.)
Total potential (cm)

Capillary potential (cm)

Water content
soil (dim.)

liquid water volume per volume of

Water content = water volume per volume of soil (dim.)
Concentration of component i (gm. mole/cm3)

Viscosity (gm./cm—day)

Density (gm./cm3)

Effective conductivity and part of the distillation
effect--see the equations of Philip and De Vries

Surface tension (or free energy) (dyne/cm)
Shape factor (dim.)
Channeling factor (dim.)

Conversion factor (1.01324 x 106 dyne/cmZ—atm.)

Subscript

 

Vapor-liquid
liquid
air-water

gas

xii

CHAPTER I

INTRODUCTION

There is a growing desire today to achieve a
cleaner aquatic environment by cleaning up domestic and
industrial waste water. One method of treating waste
water is by spray irrigation of the soil mantle. This
method is gaining popularity in areas where inexpensive
land is available.

The soil mantle can act as a living filter. As
waste water trickles down through the soil, some chemical
compounds are adsorbed onto soil particles. Others are
absorbed by the roots of surface vegatation. Also,
microbial life in the soil feed on suspended solids and
soluble chemical compounds in the water and convert them
into microbial biomass and other chemical compounds which
are excreted. Chemical compounds which are not accumulated
into the biomass of surface vegetation or soil organisms,
will accumulate in the soil. As this accumulation continues,
the soil becomes saturated and these chemical compounds
eventually wash downward into the water table. A manage?
ment goal of a spray irrigation facility would be to
minimize the amount of chemical compounds entering the

water table.

It can be seen from the above discussion, that
the water acts as a carrier of chemical compounds in the
soil. In order to successfully design and manage a soil
mantle Spray irrigation facility, one would like to be
able to predict the outcome of various spraying policies.
To do this a model is required which could accurately
predict water movement in soils over a range of operating
conditions. The water model would be the foundation upon
which other models could be built. For example, a model
to describe phosphorous movement in soils would recognize
that soluble phosphorous compounds move with the soil water
and are adsorbed or desorbed from the soil particles and
also taken up by the roots of terrestrial vegetation.

The objectives of this work are to develOp a general
model for water movement in soils and to compare this model
to less general models which have been prOposed to descibe
water movement in soils. Another objective is to test the
models by comparing the simulation results with experimental
results.

The first model to describe water movement in soils
was prOposed by Darcy in 1856. Darcy studied the flow of
water through the sand filters of Dijon, France and con-
cluded that the volumetric flow rate of water could be

predicted by the following empirical equation [1].

(1.1) Q = VOA = - KA(A—IP:—)

Equation (1.1) is called Darcy's law and states
that the volumetric flow rate of water is prOportional to
the pressure drOp (AP) across a sand column of length L.
The prOportionality constant is the product of the hydraulic
conductivity (K) and the column cross sectional area (A).
Hydraulic conductivity is a function of the water viscosity,
the particle sizes, and the particle size distribution of
the porous media. Darcy's law has been found valid for

low flow rates in porous media.

DQD

= ___B
(1.2) Rep Au <1

This is because at low flow rates, the frictional forces
outweigh the inertial forces [2].

The type of flow problem studied by Darcy is
described by the terms saturated, isothermal, and steady
state flow. A saturated porous media or soil consists of
two phases: liquid and solid. When the volumetric flux
(V0) of water through a constant temperature soil does not
change with time, the flow is said to be steady state and

isothermal.

Water flow in soils also occurs when a gas phase is
present in the soil. This type of flow is called unsaturated.
Collis-George experimentally verified that Darcy's law
holds for unsaturated flow if the driving force is changed

from a pressure gradient to a potential gradient [3].

O

(1.3) V = -KV®
where,
(1.4) Q = W - z

The potential (¢) includes a capillary potential
(9) and a gravitational potential (-z). The capillary
potential is equivalent to the other literature terms
soil potential, suction, and tension. The potential (a)
has been shown to be the free energy per weight of water
of the soil water at a given depth 2 [4]. Appendix A
illustrates the dependence of capillary potential on
saturation, temperature, history, and composition.

A practical problem of interest is the distribution
of water in the soil during and following irrigation [5,6].
The water flow in this type of problem is called unsteady
state. That is, the volumetric flux of water (V0)
changes with time. To handle this type of problem, Gardner

pr0posed the following model[7].

(1.5)

where,

(1.6)

(1.7)

in soils is a material balance
Equation (1.5) states that the

water (volume) per bulk volume

0)

Q)
r1.

0

CD1

| A

“3172
82

Se

E i 1.0

Gardner's equation for

isothermal water movement
on water in the soil.
rate of accumulation of

of soil is equal to the

rate of gain of water (volume) per bulk volume of soil

due to convective flow.

The amount of water per bulk

volume of soil is called the water content (5) and is

defined in Equation (1.6) as the product of saturation

(S) and porosity (e).

the drying of a soil

flow in soils during

A phenomenon

and unsteady state.

phase by water vapor

capillary movement.

were by engineers.

which would predict the drying rates of porous media.

which is more difficult to model is
or other porous media. The water
drying is unsaturated, non-isothermal,

Water movement can occur in the gas

diffusion and in the liquid phase by

The first attempts to handle the drying phenomenon

Their main concern was to develOp models

They

did not concern themselves with models based on the detailed

i

mechanisms of water movement in porous media during drying.
As a result, their models were a blend of theory and
empiricism. The engineering drying literature has been
reviewed in references 8 and 9. To describe the saturation
profiles in‘a drying porous media, a diffusion equation was
used with a constant diffusivity [10]. This approach did
not fit the experimental data. In Appendix B, it is shown
that Gardner's equation and Darcy's law can be combined to
yield a diffusion equation with a variable diffusivity.

A new model to describe the drying phenomenon was
pr0posed by Philip and De Vries [11]. They assumed that
the water vapor was in equilibrium with soil water. In
this situation, the soil air is saturated with water vapor.
The main point is that for the first time a model was
proposed which mathematically described water movement
occuring in two phases: gas and liquid.

The model of Philip and De Vries is given below.

A reader interested in a detailed derivation is referred

to Reference 11.

as _ . . - 3K

(1.8) 5-1:- - V (DTVT) + V (Dave) + —3Z

(1 9) c LT = V°(>\VT) - H mm v6)
' t vap 8

Equations (1.8) and (1.9) neglect the effect of soil water
composition on the capillary potential and hence no
composition variable appears in these equations. The
energy balance (1.9) contains a conductivity (1) which
includes the thermal distillation effect [11].

At a later date, Taylor and Cary used the thermo-
dynamics of irreversible processes to develOp a model
for unsaturated, nonisothermal, and unsteady state flow [12].
Their model also assumes vapor-liquid equilibrium and the
equations are of the same form as the equations of Philip
and De Vries.

To solve practical problems using models for water
movement in soils, the model equations must be solved.
For unsteady state problems, the equations are nonlinear
partial differential equations.

Gardner's equation has been solved numerically for
infiltration, redistribution, and some drying problems.
A literature survey on numerical solutions to these unsteady
state problems has been given in Reference 13.

The drying of soils like many porous media
consists of a constant drying rate and falling drying rate
period. Covey studied the constant drying rate period
using Gardner's equation. The model parameters were varied
and saturation profiles were calculated [14]. Later,
Klute gt_al. studied the falling rate period using Gardner's
equation [15]. Neither of these studies compared the

numerical results with experimental results.

A study by Fritton et al. used Gardner's equation
to simulate the drying of soil and compared computed results
with experimental results [16]. Gardner's equation fits
the cumulative evaporation versus time data but did not
fit the saturation profiles. The experimental saturation
profiles had an inflection point whereas the numerical
results did not.

The equations of Philip and De Vries have not
previously been used to simulate unsteady state water
movement in soils or other porous media. Their usage has
been in calculating the water vapor, liquid, and energy
fluxes from experimental saturation and temperature profiles

at some instant of time.

CHAPTER II

THEORY

This chapter will be concerned with the development
of a general model for water movement in soils. Although
develOped specifically for soils, it is general enough to
apply to other porous media such as crystalline catalysts,
wood, fibers, and paper. Before developing this theory,
some important concepts and terminology will be discussed.

The general model for water movement in soils will
be a set of equations which describe the transport of
fluids and energy through the soil. One fluid will be
the liquid phase and the other fluid will be the gas phase.
Both phases are fluid mixtures because the fluids contain
more than one component.

A theory of fluids is conveniently divided into
equilibrium and nonequilibrium properties of fluids. The
equilibrium properties are the thermal equation of state
(P = P(T,Vl,...,Vn)] and caloric equation of state
(U = U(T,Vl,...,Vn)). These equations relate the local
pressure and internal energy of the fluid to the local
temperature and volume occupied by the components of the

fluid. The equations of state along with the equations

10

of change and apprOpriate boundary conditions describe

the transport pehnomena of the fluid. Transport phenomena
is concerned with the fluxes of mass, momentum, and energy
which occur in fluids not at equilibrium.

The equations of change are the equation of continuity,
theequation of motion, and the equation of energy. In their
most general form, they are written in terms of fluxes of
mass, momentum, and energy. These equations have been
derived for transport phenomena in fluids by Bird et_al. [17].

The equations of continuity and energy are mathe-
matical statements of the conservation laws of science
applied to a fluid: conservation of mass and conservation
of energy. An early statement of the conservation of mass
principle was given by the Roman poet Lucretius (ca. 96-55 BC),
contemporary of Julius Caesar: "Things cannot be born from
nothing, cannot when begotten be brought back to nothing" [18].
The conservation of mass principle was modified much later
by Albert Einstein in his theory of relativity. For systems
in which the velocity of mass approaches the speed of
light, the conservation of mass principle is obeyed if
relativistic mass is used. The principle of conservation
of energy says that energy may be transformed from one
kind to another, but it cannot be destroyed. This principle
and the principle of conservation of mass are generaliza-
tions of man's experience, so far not contradicted by

observation of nature.

11

The equation of motion is a statement of Newton's
second law of motiOn applied to a local fluid element.
Newton's second law says the unbalanced force acting on
a body is equal to the time rate of change of motion (or
momentum) of the body.

There are three levels of analysis which can be
applied to develop the theory of transport phenomena in
soils: molecular, microscopic, and macroscopic. The
molecular level of analysis develops theories of transport
based on the movement of molecules [19]. The movement of
molecules is caused by intermolecular forces. On this
level of analysis even a fluid is discontinuous as there
are voids between molecules. This difficulty is overcome
by using statistical distribution functions, and the
equations of change derived by molecular considerations
are partial differential equations with continuous
functions [19].

The microscopic level views a fluid as a continuum
and is not concerned with the individual movement of
molecules. It is concerned with the microscopic prOperties
of fluids such as local temperature, pressure, and velocity.
Transport in soils is then described by applying the
microsc0pic equations of change to the regions of the soil
where fluids are present such as the macrOpores and micro-
pores. The difficulty of this approach is the complexity

of the boundary cdhditions between the phases present.

12

This difficulty is circumvented in the macroscopic
analysis as the complex microscopic boundary conditions are
incorporated into the macrosc0pic equations of change.
Furthermore, the macroscopic approach is concerned with
macroscopic properties of soils which are directly measure-
able such as temperature and saturation.

The general model for water movement in soils will
be the macrosc0pic equations of change. The macrosc0pic
equations of change for soils can be derived by two methods.
The first method involves volume averaging the microsc0pic
equations of change over a small macrosc0pic volume of soil.
Bird et al. have used this method to develop the macrosc0pic
equations of change for a general nondistributive flow
system [20]. The second method involves definition of
macroscopic quantities and derivation of the macroscopic
equations of change in terms of these quantities.

The following derivation of the macroscopic equations
of change will utilize the second method since it is more
simple and straightforward. The nomenclature is defined
on pages ix-xii and Appendix C contains the three dimensional

form of the macroscopic equations of change.

13

Derivation of the Macroscopic Equation of
Continuity for Liquid Phase Water in the Soil

 

A mass balance on liquid phase water in a differ-
ential element of soil (see Figure 1) yields the macro-
sc0pic equation of continuity for liquid water. Liquid
water enters the differential element at position 2 with
a volumetric flux V0 and leaves the element at position
2 + dz with a volumetric flux V0. Liquid water may evaporate
or water vapor may condense in the element also. The mass
rate of water evaporation per bulk volume of soil will be
represented by the letter E.

The rate of accumulation of liquid water mass in
the differential element is equal to the rate of gain of
liquid water mass by convective flow plus the rate of
gain of liquid water mass by condensation of water vapor.

That is,

So 8Adz

o o _ w
(2.1) pwV A12 - pwV Alz+dz - EAdz — ——-——3t .

where, ow mass density of liquid phase water

6 = liquid water content
S = saturation (liquid water volume per volume of
voids)
and, 8 = 86

Since the cross sectional area is a constant, we can divide
by Adz and then take the limit as dz approaches zero.

Equation (2.1) becomes

14

 

 

/

 

 

 

 

z ‘ z+dz

volume of soil particles in the differential clement

 

W“---

 

 

 

Vanna(fiflflnddIHWQrinAUmrdflflcnumidladammm

volume of soil air in.thc differential element

A = soil column cross sectional
area.

6 i soil porosity (or void
fraction)

S '1nmunufion

' (1-5)“!
' SeAdz

- (1-S)elds

Figure l.--Differential element in a soil column.

15

 

 

(2.2) -

Macroscopic Equation of
Continuity for Liquid Phase Water

 

 

The rate of evaporation (E) will be defined mathe-

matically later in this chapter.

Derivation of the Macroscopic Equation
of—Continuity for Water Vapor in the Soil

 

 

A molar balance on water vapor in a differential
element of soil (Figure 1) results in the macroscopic
equation of continuity for water vapor. Water vapor
enters the differential element at z with a molar flux
NW and leaves the element at z + dz with a molar flux NW.
The flux NW is with respect to a fixed coordinate system
and contains convective and diffusional flow contributions.
Water vapor may also enter or leave the vapor state by
liquid evaporation or vapor condensation.

The molar rate of accumulation of water vapor in
the differential element is equal to the molar rate of
gain of water vapor by convective flow and diffusion plus

the molar rate of gain of water vapor due to liquid water

evaporation. That is,

~ ~

3(C e(l-S)Adz)
(2.3) NwA] - N A EAdz "‘ w

z w lz+dz M _' at
w

 

 

Where, Mw = molecular weight of water

16
and

(2.4) C = —— = ———

Dividing by Adz and taking the limit as dz approaches zero,
Equation (2.3) becomes

3(Cw€(l-S))
3t

8N
w E _
W

82

 

MacroscoPic Equation of
Continuity for Water Vapor

 

 

In Equation (2.4), the molar concentration of water
vapor is given by the ideal gas law. The partial pressure
of water, the total gas pressure, and the mole fraction of
water vapor present in the gas phase has been represented

by the symbols, pw, p, and Xw respectively.

Derivation of the Macroscopic
Equation of Energy for the Soil

 

 

By making an energy balance on the differential
element of soil (see Figure 1) we obtain the macroscopic
equation of energy for the soil column. Energy enters
the differential element at z with a conductive energy
flux q and leaves the element at z + dz with a conductive
energy flux q. Sensible heat transferred by the fluxes
of liquid water and water vapor also contribute to an energy
flux. This flux has been assumed negligible compared to

the conductive energy flux and will be justified by the

t

l7

simulation calculations later in this work. A cooling or
heating effect is also present in the element due to
evaporation or condensation of water.

The rate of accumulation of enthalpy in the differ—
ential element is equal to the rate of gain of conductive
energy plus the rate of gain of enthalpy due to condensation

of water vapor.

That is,
_ CBT Adz
(2.6) quz quz+dz - AHvapEAdz — (_§E)
where, C = volumetric heat capacity of the soil
AHvap = enthalpy of vaporization of water

Dividing by Adz and taking the limit as dz approaches zero,

Equation (2.6) becomes

.99. = E.
(2.7) 32 AHvapE C at

Macrosc0pic Equation of Energy
for the Soil

 

 

Thus far we have derived the macrosc0pic equations
of change in one dimensional form. An implicit assumption
in these derivations has been that fluxes, concentrations,
and temperature are constant over the cross sectional column

area (A).

18

The macroscopic equation of motion has been derived
by Raats for a soil. He has shown that if inertia is
negligible, the equation of motion yields Darcy's law [21].
Darcy's law can be used to describe the volumetric flux
of liquid water in the soil (see Equations (1.3) and (1.4)).
Darcy's law is written in terms of a potential driving force
(¢). Appendix A indicates the dependence of capillary
potential (W) on saturation, temperature, history, and
composition.

The expression for the flux of water vapor in soils

is dependent on the number of components diffusing and

the convective flux of the gas phase. In general,
“' ~ de ~ 2...
(2.8) NW = “ CDaW€(l-S)-E_ + XW 1 Ni
where,
(2.9) cv* = Z c.v. = Z N.
l i 1 l 1
~ — 2 ~ — P
(2.10) C - i Ci — if
and,
V* = molar average velocity with respect to fixed
coordinates
Vi = average velocity of molecules i with respect
to fixed coordinates
D = diffusion coefficient of water in soil air

aw

19

When water is the only component moving and it is moving

by diffusion, Equation (2.8) reduces to

~ ~ €(l-S)de
(2.11) Nw = - CDaw (l-Ew)dz

 

The energy flux is written in terms of an effective

conductivity.
(2.12) q = - k ——

Model Parameters

Rate of Evaporation (E):
Interphase Mass Transfer

Evaporation and condensation of water in soils
involves the transfer of water molecules across the gas-
liquid interface present in the soil. This phenomenon
is referred to as interphase mass transfer. The theory of
interphase mass transfer is developed in a textbook by
Treybal [22]. In summary, the rate of water mass transfer
per unit area of gas-liquid interface is equal to the
product of an over—all mass transfer coefficient (K0) and

~

* ~
a concentration driving force (Cw - Cw).
The evaporation rate (E) can now be written as

~

”-1:
(2.13) E = Koa(Cw — cw)Mw

Where ,

20

~*
2 4 ~* Pvap wa
and a = gas-liquid interfacial area per bulk volume
of soil (Adz).
Pvap = water vapor pressure

It should be noted that the water vapor pressure
(Pvap) is dependent on saturation, temperature, and water
phase composition. This mathematical dependence is indicated
in Appendix D.

When the liquid phase contains only trace amounts
of components other than water, the main resistance to
water mass transfer is in the gas phase. An exception to
this statement would be if a trace component is a surface

active agent which creates a small film of resistance at

the interface. When the gas phase is the main resistance,

(2.15) KO ; km

where,
km = gas phase film coefficient for mass transfer
%— = gas phase resistance to mass transfer
m

For concentrated water solutions, the liquid phase resistance
may become significant and must be included in calculating'
the over all mass transfer coefficient (see Treybal Reference

22).

21

To allow mass transfer calculations using Equation
(2.13), the mass transfer coefficient and interfacial area
need to be quantified. The gas-liquid operations such as
humidification, absorbtion, and stripping also require
knowledge of the volumetric mass transfer coefficient
(Koa). This coefficient is determined experimentally. Since
no one has previously considered a general water movement
in soils model of the form presented here, the volumetric
mass transfer coefficient has not been measured.

The model presented here could be used to determine
the volumetric mass transfer coefficient from a drying
experiment. In addition to the model physical parameters,
the data required are the saturation (8), temperature (T),
and the gas phase water mole fraction (2w). No one has
measured all of these quantities together. As a result,
this work will theoretically estimate the volumetric mass
transfer coefficient.

Soils (porous media) are packed beds of small
particles, so earlier work on heat and mass transfer in
packed beds will be useful for determining the film mass
transfer coefficient (km). Since packed beds are composed
of particles, some early investigations of mass transfer
from single particles are pertinent. Fréssling studied
evaporation from liquid drops and proposed the following

dimensionless correlation [23].

\
)

22

 

(2.16) Sh = 2.0 + 0.55 Reg/ZScl/3
where,
k D
(2.17) Sh = EE—E-
aWi
D V p
(2.18) Re = —E—E—
P u
' _ u
(2.19) Sc — DD .
aWi
and, Sh = Sherwood number
Rep = Particle Reynolds number
So = Scmidt number
km = gas phase film coefficient for mass transfer
Dp = liquid droplet diameter
Dawi = ordinary diffusion coefficient of component
wi in air
u = air viscosity
p = air density
Vg = air velocity

Later, Ranz and Marshall also studied evaporation

from liquid drops and proposed the following correlation [24].

1/2 1/3

(2.20) Sh = 2.0 + 0.60 Rep Sc

23

It can be seen that Equations (2.16) and (2.20)
predict Sherwood numbers of 2.0 when the air is stagnant.
This number is a theoretical value for a sphere in an
infinite stagnant medium. However, a recent mass transfer
study using the collapsing bubble method verified the
theoretical value of the Sherwood number (Sh = 2.0) for
an oxygen bubble in stagnant water (Rep = 0.0) [25].

At low Reynolds numbers (Rep<10), most of the
experimental data in packed beds are from heat transfer
studies. Kunii and Suzuki predict values of Sherwood
numbers much less than 2.0 at low Reynolds numbers.

Their experimental data substantiates their model [26].

(J

_ s
(2.21) Nu -- m (RepPr)

0

S

(2.22) Sh = m (RepSc)

where,
hD
(2.23) Nu =TE
C u
(2.24) Pr = {—
(2.25) 05 = 1.0 (shape factor)
5 = 0.4 (porosity or void fraction)
C = 10.0 (ratio of average channeling length

to particle diameter)

>24

and, Nu = Nusselt number

Pr = Prandtl number

h = gas phase film coefficient for heat transfer
k = gas phase thermal conductivity
Cp = gas phase heat capacity

Later, Littman gt_al. used frequency response
techniques to study heat transfer from packed beds of
small particles (porous media). They have claimed Kunii's
method for calculating the film heat transfer coefficient
from experimental data is incorrect. They have presented
their own data and data of others at low Reynolds numbers
to show that the Nusselt number asymptotes to a value of
two [27]. The conclusions of Littman gt_al, will be used
to calculate the film mass transfer coefficient (km).

In order to determine the gas-liquid interfacial
area per volume of soil (a), the model illustrated in
Figure 2 was used. The soil is viewed as a packed bed of
cylindrical rods arranged in simple cubic packing. This
model predicts a porosity of 0.22 and a drainage saturation
of 0.37.

To calculate interfacial area per volume of soil (a)
from the model, something must be said about the manner
in which the water distributes itself at a given saturation.
In this model, the water interface is assumed to have a

constant radius of curvature. Also, for saturations up to

25

 

simple cubic packing of rods

front View

porosity

drainage saturation

 

 

 

 

Figure 2.--Interfacial area model.

26

the drainage point the interface is tangent to the rod at
the point of intersection. When surface tension measure—
ments are made with capillary tubes, it is also common to
assume that the gas-liquid interface is tangent to the tube
surface at the point of contact.

The drainage point is reached during wetting when
adjacent interfaces first touch one another. This occurs
where the square diagonals intersect the rods (see front
view, Figure 2). The saturation at the drainage point is
calculated to be 0.37.

Beyond the drainage point, it is assumed that
drainage is slow enough so that the water will distribute
itself as shown in the front View of Figure 2. The radius
of curvature beyond drainage is taken to be the radius of
curvature of a capillary tube which has the same capillary
potential as experimental capillary potential data on a
soil.

The mathematical development and computer program
for this model are given in Appendix E. Figure 3 contains
the calculated relationship between interfacial area and
saturation. At the irreducible saturation of the soil,
the interfacial area must be zero. In Figure 3 the

irreducible saturation is zero.

i 27

 

1000. V
Data from simulation of simple
cubic rod packing model: Program
Area 3, Run No. 2/1/72.
10. 03. 03.
I
500. ‘_

Interfacial Area Per Volume (cm

 

 

 

0.5 1.0

Soil saturation (dimensionless)

Figure 3.--Interfacial area per volume versus soil saturation.

28

Other models for interfacial area were considered.
A model consisting of a simple cubic packing of spheres
arrangement resulted in a porosity of 0.48 and a drainage
saturation of 0.18. With a body centered packing of spheres
arrangement, the porosity is 0.32. At a saturation of
0.12, the body centered packing of spheres was near drainage.
Both of these models predict drainage saturations much
lower than actually occur for sandy soils. For example,
a sandy soil on the south campus swine waste disposal site
had a drainage saturation of 0.35. For this reason and
the computational difficulties involved with the sphere
models, the simple cubic packing of rods model was chosen
for this work.

The over all mass transfer coefficient and inter-
facial area are two physical parameters which have just
been discussed. Other physical parameters in the general
soil water model are hydraulic conductivity, capillary
potential, water vapor diffusion coefficient, liquid water
vapor pressure, liquid water enthalpy of vaporization,
effective thermal conductivity, and volumetric heat capacity.

Hydraulic Conductivity»

 

Experimental data on hydraulic conductivity on
Uplands sand is given in Figures 4-6. For high saturations,
hydraulic conductivity is determined by steady flow experi-
ments and Darcy's law. For lower saturations, hydraulic

conductivity is determined from unsteady state flow

29

Data reference: W.J. Staple, Soil Sci. Proc.,
33 (840) 1969.
Uplands sand (78% 30-100 mesh)

0 Experimental data points

. Least squares data points

Hydraulic Conductivity (cm./day)

 

0.2 0.4 0.6 0.8 1.0

Soil saturation (dim.)

Figure 4.--Hydraulic conductivity versus soil saturation.

 

30

Data reference: W.J. Staple, Soil Sci. Proc.,
33 (840) 1969.

Uplands sand (78% 30-100 mesh)

0 Experimental data points

. Least squares data points

Hydraulic Conductivity (cm./day)

 

0.2 0.4 0.6 0.8 1.0
Soil saturation (dimensionless)

Figure 5.--Hydrau1ic conductivity versus soil saturation.

31

Data reference: W.J. Staple, Soil Sci. Proc.,
33 (840) 1969.

Uplands sand (78% 30-100 mesh)
0 Experimental data points
. Least squares data points

Hydraulic Conductivity (cm./day)

 

0.2 0.4 0.6 0.8 1.
Soil saturation (dim.)

Figure 6.--Hydraulic conductivity versus soil saturation.

 

32

experiments and the equations of Gardner and Darcy (i.e.,
Equations (1.5) and (1.3)). The data contained in

Figures 4-6 was curve fitted using a least squares technique
so that the data could be used in computer calculations.
Appendix F contains the documented subroutine for this

data.

Capillary Potential

 

Figures 7-8 contain the capillary potential data
on Uplands sand. At high saturations the capillary potential
is determined by making measurements with a manometer.
For lower saturations vapor pressure measurements are
used to determine capillary potential (see Appendix A) [2] .
Appendix F contains the documented subroutine which is a
capillary potential curve fit modified to correct for
temperature. Comparison of literature data on hydraulic
conductivity and capillary potential demonstrates that
these parameters are strongly dependent on particle size
and particle size distribution. Figures 9-10 contain the
capillary potential data for the Valentine sand used by
Hanks et;al,[30]. The calculation of some of the data
points is explained in Appendix L. Appendix F contains
the documented subroutine which is the Valentine sand
capillary potential curve fit modified to correct for

temperature.

33

Data reference: W.J. Staple, Soil Sci. Proc.,
33 (840) 1969.

o Drying curve for Uplands sand
(78% 30-100 mesh)
x Least Square fit of the experimental points (0 ).

-103

-102

Capillary Potential

+w

-101

 

-10
0.2 0.4 0.6 0.8 1.0
Soil saturation (dim.)

FiIJLIre 7.—-Capillary potential versus soil saturation.

34

Data reference: W.J. Staple, Soil Sci. Proc.,
33 (840) 1969.

(D Drying curve for uplands sand
(78% 30—100 mesh) -- Experimental

X Least squares fit of the experimental
points ( G) ).

-107

-106

Capillary Potential (CM.)

[1):

-10

 

-10 _
0.2 0.4 . . 0.6 0.8 1.0
Soil saturation (dim.)

Figure 8.--Capillary potential versus soil saturation.

35

Data reference: R.J. Hanks, et.a1., Soil Sci. Proc.,
31 (594) 1967

C) Drying curve for Valentine sand
(experimental values)

£3 Drying curve for Valentine sand
(calculated from saturation profile at the start
of Hanks drying experiment-—see appendix L)

——— Least squares computer curve fit

 

-104
-103
2'
.9
H
.3
u -102
C
(D
u
0
0.:
>~
H
(U
H
H
'6.‘
l
{3 -10
ll
9.
+
-100

0.2 0.4 V 0.6 0.8 1.0

Soil saturation (dimensionless)

Figure 9.--Capillary potential versus soil saturation.

36

Data reference: R.J. Hanks, et.a1., Soil Sci. Proc.,
31 (594) 1967

C) Drying curve for Valentine sand
(experimental values)

 

Least squares computer curve fit

+ w = Capillary Potential (cm.)

 

0.2 0.4 0.6 0.8 1.0

Soil saturation (dimensionless)

Figure 10.--Capillary potential versus soil saturation.

37

Water Vapor Diffusion Coefficient

 

In this study the water vapor diffusion coefficient
in soil has been assumed equal to the ordinary diffusion
coefficient of water vapor in air. Philip and De Vries
have suggested a value of two thirds of the ordinary
diffusion coefficient [11]. Clearly the ratio of water
vapor diffusion coefficient in soil to the ordinary diffusion
coefficient will be a function of saturation and not
constant. It has been shown that the assumption just made
is an upper bound on the water vapor diffusion coefficient
(Daw) in soil [28]. The ordinary diffusion coefficient
for water vapor in air has been fitted as a function of
temperature using an empirical equation from the International

Critical Tables.

T(°K) 1.75

(2.26) .5 (27 .15)

aw IT(°K) = ”flaw '0 (0C)

Liquid Water Vapor Pressure
and Enthalpy of Varpofiiation

 

 

Data on liquid water vapor pressure and enthalpy
of vaporization was obtained from the Handbook of Chemistry
and Physics. Vapor pressure and enthalpy of vaporization
were curve fit as functions of temperature and vapor
pressure was corrected for the effect of saturation described
in Appendix D. The documented subroutines for these

parameters are contained in Appendix F.

38

Thermal Parameters: Effective
ThermaILConductivity and
Vqumetric Heat Capacity

 

 

 

The thermal parameters of effective thermal
conductivity and heat capacity were obtained on Ottawa
sand from Moench [29]. This data is presented in Figure 11,
and Appendix F contains the documented subroutines which
are curve fits of this data. The method of thermal con-
ductivity measurement and volumetric heat capacity calcula-
tion is contained in Moench's thesis [29]. In summary,
the thermal probe method was used to measure the apparent
thermal conductivity. The theory of Philip and De Vries
was used to estimate the energy flux due to the distillation
effect. The effective thermal conductivity is then obtained
by subtracting the energy flux due to the distillation effect
from the apparent thermal conductivity.

General Model for Water Movement

in Soils Under the Condition of
Vapor-Liquid Equilibrium

 

 

Thus far the general model for water movement in
soils has been presented along with the physical parameters
contained in the model. If vapor-liquid equilibrium exists,
the general model reduces to the equations of Philip and
De Vries. The equations of continuity for the liquid phase
water (2.2) and water vapor (2.5) are added to give

3p 8

w _ _ §_ 0 ~
(2'27) '_§E—'- 32 (pwV + Nwa)

39

Effective Bulk Thermal Conductivity X 104 (cal./cm. sec. deg. C.)-ke

 

 

 

Data reference: A.F. Moench, Ph.D. thesis,
69-12533, pp. 74—80, Univ. Microfilms, '
Inc., Ann Arbor, Mich.
70,-~ Thermal conductivity measurement ‘”
by the cylindrical thermal
probe method.
600 -_ .0-
50. *- +
40. “ 3‘
I
30. a" t
20. .1— “(-
10. 3: “
1 l 1 : p
0.2 0.4 . 0.6 0.8 1.0

Figure ll.-—Effective thermal conductivity and volumetric

Soil saturation (dimensionles

heat capacity versus soil saturation.

S)

Volumetric Heat Capacity (cal./cm.3 deg. C.)

40

The energy balance (2.7) is modified to include the dis-

tillation effect.

(2.28) c—=-——-(—-"1AH M

In the next chapter, the general model for water
movement in soils will be tested. A drying problem will
be described and the numerical solution of the general
model and earlier models will be developed. Using the
numerical methods, some drying problems will be run using
the new general model developed in this work, the theory
of Philip and De Vries, and the isothermal equation of

Gardner.

CHAPTER III

NUMERICAL SIMULATION

In this chapter, a general drying problem will
be stated mathematically followed by the development of
the numerical methods for simulation of the drying
problem. The theory develOped in Chapter II, the theory
of Philip and De Vries, and the isothermal water movement
equation will be used to simulate some drying problems.
In Chapter IV, a number of drying problems will be defined
and the simulation results will be presented and discussed.

Hanks eg_al. have published experimental data
obtained from drying of Valentine sand by radiation [30].
The saturation and temperature profiles published are I
averages of two replicate eXperiments. Attempts by this
author to obtain their original data were unsuccessful
because they no longer had this data.

The complete set of boundary conditions for their
drying experiments on Valentine sand were not published.
In their experiments, ambient temperature was controlled
at 2522 0C. but there was no control on humidity. Radiant
energy was supplied by heat lamps and they attempted to

minimize surface turbulence.
/

41

42

The following general drying problem is for a sand
column of the same type and dimensions as the columns used
by Hanks gt_al. Figure 12 is a diagram of the sand column
with boundary conditions illustrated. Table 1 contains
the legend for Figure 12.

In order to simulate a drying problem, the initial
conditions and boundary conditions for the problem must be
stated. The surface boundary conditions say that water can
only be transported across the top surface of the column
in the vapor state. Furthermore, the water flux is given
by the product of a surface film mass transfer coefficient
(km,s) and a concentration driving force. The driving
force is the difference in water concentration between the
atmosphere and soil surface. The film coefficient is a
function of wind velocity and is calculated in Appendix K-
The energy boundary condition contains a radiative flux
(qr) and a convective flux (qc). The convective energy
flux is given by a surface film heat transfer coefficient
(hs) and a temperature driving force. The temperature
driving force is the temperature difference between the

atmosphere and soil surface. Again, the film coefficient is

a function of wind velocity and is calculated in Appendix K.

43

Anuhuw,mxmdthnm

 

 

 

 

 

 

 

 

 

 

 

 

 

_ Q1 1"” 2"" T“
2 2T I .
1 §
* \
R
x qwm- UO(TA-T)
L ‘ “—91 k
.\ D k
(I .5: ll
-1

See Table l for the legend to Figure 12.

Figure 12.—-Diagram of sand column for drying problems.

44

TABLE l.-—Legend for Figure 12.

 

Materials

L\\\\‘

UIUJIH
[:1
E

Column Dimensions

Styrofoam insulation
Plexiglass column wall

Valentine sand

 

Column bottom; composed of a layer of plexiglass and wood

top column surface

= convective flux of energy at the column walls (radial direction)

tOp column surface

LI = 45 cm. (Column length)

Di = 10 cm. (Column inside diameter)

Dw = 11 cm. (Outside diameter of column wall)

D0 = 27 cm. (Outside diameter of column)
Definitions of Variables

qr = radiant flux of energy from heat lamps

qC = convective flux of energy at the

qwall

Nw’c = molar flux of water vapor at the

TA = ambient temperature

~w,A = ambient mole fraction of water vapor

Initial Conditions

 

 

(3.1) S(z,t) = 8(2)
2 e[O,L]
(3.2) T(z,t) = T(z) '
~ ~ t = 0
(3.3) Xw(z,t)= Xw(Z)
Boundary7Conditions
At the tOp column surface,
(3.4) v°(o,t) =
(3.5) Nw(O,t) = w,c = km,sP (Xw,A_Xw(O’t)) t Z 0
RT
(3.6) q(O,t) = qr + qC = qr + hS(TA-T(O,t))

At the bottom column surface,

(3.7) v°(L,t) = 0

~ t 3 o
(3.8) Nw(L,t) = o
(3.9) q(Lot) = 0

 

45

The bottom boundary conditions say the column bottom
is impervious to water flow and is a perfect insulator.

Due to a lack of published information by Hanks gt_al., a
column bottom heat transfer coefficient cannot be estimated.
The validity of the perfect insulator assumption will have
to be demonstrated by computer simulation.

The general model for water movement in soils
(Equations (2.2), (2.5), (2.7)) is the set of equations of
change written in terms of fluxes of mass and energy. These
fluxes of mass and energy can be written in terms of
macrosc0pic prOperties of the soil (Equations (1.3), (2.11),
(2.12), (2.13)). The equations of change will now be
written in terms of the macrosc0pic properties of the soil.

When the soil porosity and liquid water density
are constant for the drying problems considered, Equations

(2.2), (1.3, (1.4), (2.13) can be comined to give

~* ~
0 Koa PMw (Xw Xw)

(3'10) 3t — e 32 p eRT
W
0 — _ fl _

When the gas phase pressure changes with time and
position are negligible in comparison with other terms in
the equation of continuity for the water vapor, Equations

(2.5), (2.11), (2.13) combine to give

46

 

— ~* ~
(3 12) 29 = _ BEE.+ 52:,(XW-xw)
' 3t 32 eT

_ Xw(l-S)
(3.13) C = -—T__—_'

_. D (l-S) 3x
(3.14) N = — aw W

w 32

T(l-Xw)

The energy balance equation with a term to
include radial losses through the soil column becomes

(see Appendix J for calculation of U0)

K a 4Do
3_T__9_<1_ _9_ .._9_
(3.15) C at — az AHvap RT PMw(Xw- Xw) D 2 (T- -TA )UO
1
__ AT.
(2.12) q — ke 32

It can be seen that the three coupled partial
differential equations are highly nonlinear. This fact
precludes the use of successive linearization and implicit
numerical techniques. Based on these considerations and
the nature of the boundary conditions, it was decided to
use the following explicit technique.

A first order forward difference formula on time

was used.

Si.j+1' 1.1

(3.15) at

 

47

 

 

c. . - . . -

1.3+1 1.3 z 39
(3'17) 0t ‘ atl .
1,]

T. . -T. .

(3 18) lIJ+l 1!] 2'. El
' 6t 3t .
1,]

On the spatial derivatives of the fluxes, a first order
forward difference formula was used. A first order back-
ward difference formula was then used on the spatial partial

derivatives in the flux terms.

 

 

 

 

 

O 0
v 0 I-V I I O
_ 1+1 J 1!) 2 _ 3V
(3.19) ( dz ) - FE—l.
ll
W1 j‘Wi-igj o
_ I I _ g
(3.20) Ki’j( 52 1) v Ii,j
fi -fi -
w. . w. . 8N
(3.21) —( 1+1'3 l'3) ” - 53!).
OZ 1!]
x -x
_ Daw‘l'si.1( ‘ W1,j wi-1.j’ ~ -
(3.22) ( ~ dz ) Nwl
Tl .(1-x ) l.j
.3 W- -
1:3
q. .-q. .
_ 1+l,j _1 j 3 _ fig
(3.23) ( 52 ' ) — azl

 

48

 

T. .-T._1 .
(3.24) ‘ke ( 1’3 1 '3) E ql- .
1,3' 62 1'3
If Equation (3.20) were substituted into Equation (3.19)
and the hydraulic conductivity were constant, the following

second order difference formula would result.

2
Q)
*6

W. .-2W. .+W. .
(3.25) K( 1+111. 2111 l'lIJ) = K
52 33 ilj

The first order difference formulas on space were so
chosen to obtain this type of a result. Equation (3.25)
can be recognized as the second order difference formula
for the second partial derivative of W with respect to
z.

The numerical approximation to Equations (2.12),
(3.lO)-(3.15) is obtained by combining these equations
with Equations (3.16)-(3.24). The resultant set of
algebraic equations can then be solved on the digital
computer. Appendix G contains the documented computer
program which uses the above numerical techniques to
solve the general model equations.

If vapor-liquid equilibrium exists, the general
water model reduces to the equations of Philip and De Vries
((2.27), (2.28)). These equations subject to the constraints
of constant porosity, liquid water density, and gas phase '

pressure reduce to
l

49

~

 

- N M
as _ _ l 3 o w w
”-2“ a)? ' E I)?” + pw’
' N~ 4 D
8T _ _ Bq _ 3 w _ o _
(3°27) C "E - 82 (_32)AHvapr 7 Uo(T TA)
i

~

where, gas phase pressure (P) appearing in Nw is treated
as a constant.
The initial conditions for the Philip and De Vries

equations are:

(3.28) §(z,t) §(z)

(3.29) T(z,t) T(z).

The boundary conditions are the same as those given by
Equations (3.4)-(3.9). The explicit numerical techniques
used for the general problem (Equations (3.16), (3.17)-(3.24))
were used to transform the equations of Philip and De Vries
into a set of algebraic equations which could be solved
on the digital computer. The computer program which
solves the equations of Philip and De Vries is listed in
Appendix H.

The simplest water model for unsteady state flow
in soils is the isothermal equation of Gardner. For a
constant porosity Gardner's equation and Darcy's law

combine to give

50

as 1 W0
(3.30) "E- " E- 32

-0 _ _ 3! -
(3.31) V - K(82 1)

These equations cannot predict the constant drying rate
during the constant rate period, and so to run a simulation
experimental data must be available to determine the
constant rate. If the boundary conditions at the soil
surface were known, the constant drying rate could be cal-
culated from the controlling transfer process: that is
heat transfer or mass transfer.

The initial conditions for the drying problem using

the isothermal equation are

(3.28) §(z,t) = §(z) t = o
The boundary conditions are
(3.32) VG = constant, during the constant rate drying period.

When the surface saturation reaches the irreducible satur—
ation, the drying rate begins to decrease and the boundary

condition becomes

(3.33) S(O't) = SIrreducible

Appendix I contains a listing of the documented computer
program which solves the isothermal equation for a drying

problem.

 

CHAPTER IV

RESULTS AND DISCUSSION

Using the numerical method outlined in Chapter III, [

a number of drying problems were simulated. Table 2 con-

_ 1 ‘5' ”4'“;-
1 (

tains a listing of the simulated drying problems. It

can be seen from Table 2 that a number of simulations were
run for each of the models: (1) isothermal Equation (3.30),
(2) Philip and De Vries model (3.26, 3.27), and (3) general
model (3.10, 3.12, 3.15). For simplicity, in the future

the above models will be denoted as: (1) IE, (2) P&DVM,

and (3) GM respectively. All of the simulations are for
the soil column described in Figure 12 and Table l.

A A numerical solution to any of these models brings
up questions of stability and convergence. A numerical
solution is stable if the difference between the real
solution and numerical solution does not have large
oscillations or grow large with time. A convergent
solution is one where the difference between the real solution
and numerical solution is very small.

In this work the experimental drying data of
Hanks gt_§1. gives an indication about the general shape

of the solution. ‘This information is useful for a stability

51

 

52

case

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Aumcpumov HmfiumnuomH

mampoz wumum mpmwumca

 

 

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53

I o.m Noo.o o.m Noo.o Army umnEsz poo3umzm
um; I I I I msoHUHpcoo HmHuHcH
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x x x x Axm>ozv
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mampoz mumum xpmmumca

 

Ha 0H m m h Hmnfidz coflumHDEHm

.UmSCfluCOUII.N mqmdh

54

criterion. If a solution deviates largely from the general
solution shape it can be judged unstable.

The real solution will be the solution converged to
by adjustment of the finite difference sizes used in the
numerical solutions. In this work, the procedure used to
arrive at stable, convergent solutions was the following.

A space increment of 9 cm. was chosen and 5 space increments
were used for the 45 cm. column. The time increment was
then changed until a stable solution was obtained. Using
the stable time increment as a base value, the time incre-

ment was decreased until the solutions converged. By

changing the time increments while holding the space increment

fixed the ratio of time increment to space increment was
changed. It was found that time increments differing by a
couple orders of magnitude gave solutions which agreed out
to 7-10 significant figures. For the long simulation runs,
the larger time increments which gave convergent solutions
were used to minimize computation cost. It was felt that
7-10 significant figure agreement was sufficient.

The Michigan State University CDC-6500 computer was
used to simulate the drying problems. A number of compiler
options were available and the FTN compiler was chosen
since it is guaranteed against system errors by CDC and has
about 13 significant figures in single precision. In
contrast the RUNT compiler which generates a poorer code
matched only 9 significant figures of some sample FTN

calculations and overlooked a fortran program syntax error.

55

A drying eXperiment was chosen to be an "acid test"
for the different models. This is because water movement
would occur in two phases and there would be temperature
gradients to affect the water movement. The porous media
chosen was a sand.

A literature search did not turn up any references
which contained the saturation and temperature profiles for
a drying experiment and a complete set of physical prOperties
data. However, the drying data on Valentine sand by
Hanks et_§1. [30] did contain the saturation and temperature
profiles and some capillary potential data. Since the data
needed on Valentine sand could not be found in the litera-
ture, data on other sands was found to supplement the
information provided by Hanks gt_al. Hydraulic conductivity
and capillary potential for Uplands sand was published by
Staple [6]. Effective thermal conductivity and volumetric
heat capacity was published by Moench for Ottawa sand [29].
With this data simulation numbers 1 and 2 were run. Table 2,

3, and 4 and Figures l3-15 contain the simulation parameters

and results [33].

The unsteady state solution of the P&DVM has not
previously been obtained. However, the constant and falling
.rate drying periods have been studied numerically for the
IE [14,15]. In the P&DVM it was found that the material
lmalance on water determined the maximum allowable time

increment. During simulation numbers 1 and 2 it was found

56

 

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57

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58

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“.50. H30. 60 spawn

59

 

 

 

 

 

 

 

 

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61

that the time increment could be increased with time.
For example the time increment used between 0 and 1 day
was 1 sec. After 1 day a 30 sec. time increment was
used.

Simulation numbers 1 and 2 used the hydraulic
conductivity and capillary potential (drying curve) for
Uplands sand. From Figure 13 it is obvious that the
simulation results do not fit the experimental drying data
(Figure 16) on valentine sand. This gives an idea of how
sensitive the hydraulic parameters are to the specific
soil or porous media type. Uplands sand is a soil in which
liquid water has a high capillary conductivity. This type
of soil has a very slight S-shape saturation profile during
drying because the liquid water movement is so rapid.

The IE gives only the saturation profiles whereas
the P&DVM gives the saturation, temperature, and equilibrium
mole fraction of water vapor profiles. The saturation
profiles calculated by both models were found to be the same
and these results are contained in Figure 13. The equilibrium
mole fraction of water vapor profile contains a peak
(Figure 15). This is because the temperature is monotonically
decreasing with depth and the saturation is monotonically
increasing with depth. The result is that although the
surface temperature is the largest and hence would yield
the largest free water vapor pressure, the surface saturation

is low enough to depress the vapor pressure far enough

7.

62

below the free water vapor pressure to give the profile

peak (see Appendix D).
Since the Uplands sand data did not fit the
experimental drying data, the capillary potential function

for Valentine sand was obtained. The capillary potential

was measured by Hanks at low saturations and as discussed
in Appendix L, the capillary potential could be calculated

at the higher saturations from data published by Hanks et al.

[30].
Simulation number 3 was run with the IE to illustrate
the effect of the capillary potential function on the drying

:saturation profiles. Tables 2 and 5 show that simulation

rnxmber 3 contains slightly different initial conditions

Also, the Valentine sand capillary

than simulation 1 .
Figure 16

potential function is used in simulation number 3.
demonstrates that the Valentine capillary potential function

results in a less water conductive soil and a more pronounced

S-shape saturation profile. It also takes less real time

in ssiJnulation number 3 to reach the irreducible saturation.
Ever: vvith the Valentine sand capillary potential function,

the simulation did not fit the experimental drying data

on Valentine sand.
Since the hydraulic conductivity of Uplands sand

aPPEared to be too large, simulation numbers 4 and 6 were run

 

 

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64

 

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65

to investigate decreasing the hydraulic conductivity by

0.1 and 0.01 respectively. Tables 2 and 5 contain the
simulation parameters and Figures 17-19 contain the
simulation results compared with the experimental data.
Simulation number 4 fits the experimental drying saturation
profiles the best and also fits the cumulative evaporation
data the best. By decreasing the hydraulic conductivity

of Staple, the over all drying rate is decreased and the
S-shape of the saturation profile becomes more pronounced
and skewed. The S-shape saturation profiles obtained
during the drying simulations are very similar in shape

to the drying saturation profiles for sands and clays that
are presented by Larian [8]. The data presented by Larian
is experimental data taken by engineers concerned with the
drying phenomenon back in the 1930's. Unfortunately no
capillary potential or hydraulic conductivity data was taken
on these porous media.

The Blake-Kozeny equation for saturated flow in
porous media at low particle Reynolds numbers indicates
that the saturated hydraulic conductivity is proportional
to the effective particle diameter squared. This implies
that by cutting the hydraulic conductivity by 0.1, the
effective particle diameter of the porous media is cut
by 1//I—1 Since sands range in particle sizes from 2.0 mm.
through 0.10 mm. in diameter and Staple's saturated

hydraulic conductivity of 188 cm/day corresponds to an

66

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67

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68

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69

 

effective particle diameter of .62 mm. we have the following:
By cutting Staple's hydraulic conductivity by 0.1, the
effective particle diameter turns out to be 0.20 mm. which

is still in the sand classification. Then, the hydraulic
conductivity in simulation number 4 does correspond in
magnitude to a sand.

The data reported by Hanks et_al. on drying of
Valentine sand [30] are an "average" of two replicate drying
runs. The saturation measurements were by y-ray transmission.
The work of Fritton et_al, indicates that the accuracy of
the saturation profiles measured by this technique is not
too good. There are differences between gravimetric and
y-ray measurements which are probably due to slight inhomo-
geneities in the soil column compared to the soil used to
standardize the y-ray measurements [16]. Considering the
scatter in the saturation profiles of Fritton et_al.,

Hanks §E_al. have probably published an eyeball curve fit
of the actual data. Attempts by this author to obtain more
experimental data on Valentine sand were unsuccessful
[because the original data had been lost. With the accuracy
of the y-ray measurements, simulation 4 is considered to be
a good fit.

1 As in simulation numbers 1 and 2, it was found in‘
simulation numbers 3-6 that the time increment could be

increased as the simulation proceeded. In simulation numbers

70

3, 4, and 6, a time increment of 6 secs. was used to 1 day
and after 1 day a 60 sec. time increment was used.

Using the capillary potential and hydraulic con-
ductivity data which gave a good fit of the experimental
data in simulation number 4, simulation number 5 was run
to compare the IE and P&DVM. In addition to the hydraulic
parameters required for the IE, the P&DVM requires the
effective thermal conductivity and volumetric heat capacity
data. The data on Ottawa sand by Moench [29] was used.

It was assumed that this thermal data would not vary much
from one sand type to another. Tables 2 and 6 contain

the simulation parameters for simulation number 5. The
size increment between 0 and 1 day was 6 sec. and the time
increment after 1 day was 240 sec.

The saturation, temperature, and equilibrium mole

fraction of water vapor profiles from simulation number 5

can be found in Figures 19, 20, and 21 respectfully. After '

5 days of drying there is good agreement between the IE
and P&DVM. However, after 40 days of drying, there are

some differences between the IE (No. 4) and P&DVM (No. 5)
(refer to Figure 19). The irreducible saturation in simula-
tion number 4 was set at S=0.015 which is the value reported
by Hanks gt_al. for the drying of Valentine sand. The P&DVM

calculates an irreducible saturation of S=0.00977 at 40 days

of drying time. If the P&DVM is restrained to S=0.015, the

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71

 

 

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75

saturation profiles obtained are in good agreement with those
obtained in simulation number 4 using the IE. And so the
differences at 40 days in Figure 19 between the IE and

P&DVM are due mainly to the differences in the irreducible
saturation used in the IE and calculated in the P&DVM.

Since the irreducible saturation appears in the
surface evaporation rate Equation (3.5), there is the
following explanation for why the P&DVM results in a lower
irreducible saturation and greater overall drying rate.
First, the ambient humidity used by Hanks et_al. is not
known. If a higher humidity was used in simulation number
5, the higher irreducible saturation would have been
calculated in the P&DVM. Second, if the capillary potential
function for Valentine sand is on the low side at S=0.015, the
irreducible saturation calculated would turn out to be
lower than S=0.015.

The temperature profiles calculated in simulation
number 5 have shapes similar to the experimental temperature
profiles. However, the calculated temperatures are larger
than the experimental temperatures. For example, at a
drying time of 40 days the surface temperature calculated
is 5 deg.C. larger than the eXperimental value. The cal-
culated values are larger because the zero heat flux boundary
condition at the column bottom apparently was not met during
the drying experiment. As mentioned earlier, the temperature
profiles have very little influence on the calculated satura—

tion profiles. This was demonstrated by comparison of the

76

calculated saturation profiles using the IE and P&DVM
with identical irreducible saturations.

Since the calculated temperature profile was similar
in shape to the experimental temperature profile, the
equilibrium mole fraction of water vapor profile in Figure
21 is expected to be correct in shape. As a result the
water vapor flux by diffusion as calculated in the P&DVM
should be correct.

Figure 22 contains the surface evaporation rate as
a function of time. It is interesting to note that the
P&DVM predicts the sartup until the "constant rate" period
is reached and then it predicts the falling rate period.

Table 7 contains a tabulation of the material and
energy fluxes during drying (simulation number 5). This
data illustrates what is happening during the drying of
porous media. During the first hour of drying, the evapora-
tion-rate increases to the so called constant rate. During
the constant rate period the liquid and vapor fluxes in the
tOp 18 cm. of the column increase. The energy flux at the
surface decreases due to convective losses to a lower
temperature environment. The energy fluxes into the greater
depths begin to increase.

Between the third and fourth day, the falling rate
period begins and the evaporation rate drOps rapidly during

the next day, as shown in Figure 22. It can be seen from

77

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(Kep-gmo/‘m6)

78

 

 

 

TABLE 7.--Material and energy fluxes during drying (simulation number 5).
Tine days) Liquid Flux Profile Vapor Flux Profile Energy Flux Profile
(gm/cmZ-day) (gm/cmZ-day) (cal/cmZ-day)
1/24 = 1 hour 0 -.l4129E+01 .85801E+03 2:0 cm.
-.46369E+00 .63111E-04 .23667E+02 z=9 cm.
-.25982E+00 .15354E-04 .101883+02 z=18cm.
—.l39lSE+OO .18587E-05 .31995E+Ol 2=27cm.
-.65285E-02 .93506E-07 .75314E+00 z=3ocm.
l -.l4388E+Ol .85335E+03 2:0 cm
—.88040E+OO .72146E-04 .118858+02 z=9 cm.
-.3OOZlE+OO .18900E-04 .85283E+01 2=chm.
—.l9946E+00 .66912E-05 .55225E+Ol 2:27cm.
—.11651E+00 .58193E—06 .27264E+01 z=36cm.
2 -.l4409E+Ol .85289E+03
— 10207E+Ol .ll7OOE-O3 .lOl4OE+02 etc.
—.42355£+00 .22522E—04 .70477E+Ol
—.l9684E+00 .795253-05 .44487E+01
-.99388E-01 .18328E-05 .21548E+01
3 —.12793E+01 .82407E+O3
—.11182E+Ol -.l3005E-02 .45391E+02
-.70782E+00 .l779OE-03 .24730E+02
-.l8273E+00 .30663E—O4 .12806E+02
-.79696E-Ol .71885E-05 .53402E+Ol
4 0 -.4l64lE+00 .56079E+03
—.38723E+00 -.l6920E-Ol .23402E+03
-.3lll4E+00 .237lOE-02 .lS406E+O3
-.99323E-01 .386768-03 .99201E+02
-.348l9E-Ol .lOZlOE-O3 .47797E+02
3 0 - -.21450E+00 .46602E+O3
-.18575E+00 ‘.25756E-01 .25299E+03
-.l7023E+00 .34746E-02 .16180E+03
-.67437E-Ol .554688—03 .1039lE+03
-.18150E-Ol .l4136E—03 .50424E+02
4O -.36044E-01 .37104E+03
-.lO89ZE-02 -.34923E-Ol .24895E+03
-.35984E-Ol .49758E-02 .15033E+03
—.23704E-01 .l7778E-02 .93742E+02
~.62395E-02 .22893E-03 .45435E+02

 

See Tables 2 and 6 for simulation conditions.

79

Table 7 that during this period the liquid flux in the tOp
18 cm. drOps rapidly. This is accompanied by a sharp
decrease in the surface vapor flux. However, the vapor
fluxes at the z=9 and 18 cm. levels begin to increase
rapidly. The energy fluxes at depths of z=9 cm. or more
begin to increase at a rate faster than before the start
of the falling rate period.

It should be noted that the difference in magnitude
between the liquid and vapor fluxes is usually an order of
magnitude or more throughout the simulation (Table 7). This
coupled with the fact that the effect of temperatures on
capillary potential is small explains why good agreement is
obtained between the IE and P&DVM. The IE is a one phase
model and the P&DVM demonstrates that it is the liquid flux
which is controlling the drying process in simulation number
5.

In Chapter II it was assumed that the convective
flow energy fluxes were negligible in comparison to the
conductive energy flux. The data in Table 7 was used in
calculations contained in Appendix M to justify this
assumption. The assumption was found valid.

Simulation numbers 7-10 contain the results of
simulations using the general model (GM) developed in
Chapter II. These simulations were run to investigate the
interphase vapor-liquid equilibrium assumption which is

the basis for the P&DVM. As mentioned in Chapter II, there

80

is some disagreement in the literature about the value of
the particle Sherwood number at very low particle Reynolds
numbers. Since the interphase mass transfer coefficient is
contained in the dimensionless particle Sherwood number and
will have an affect on the calculated value of the gas phase
mole fraction of water vapor, it was decided to investigate
how sensitive the gas phase humidity (2w/2;) was to the
particle Sherwood number.

Simulations 7-10 used initial conditions for a
dry soil. The drying simulation results from the P&DVM
supplied the saturation and temperature profile initial
conditions for the general model (GM). The GM initial
condition for the gas phase mole fraction of water vapor
was taken to be the equilibrium mole fraction of water vapor
calculated by the P&DVM. Simulations 7-10 then show the
unsteady state response of the gas phase mole fraction of
water vapor from the initial conditions. A larger deviation
from the interphase vapor-liquid equilibrium assumption
would be found in a dry soil as opposed to a wet soil.

Tables 2 and 8 contain the simulation parameters
for simulation number 7. A particle Sherwood number of
0.002 was used as opposed to the generally accepted value
of 2.0. A lower value of the particle Sherwood number
would result in a lower gas phase humidity. Figure 23
shows the simulation number 7 results. It can be seen

that the surface mole fraction of water vapor in the gas

 

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83

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84

 

phase quickly approaches a pseudo steady state value close
to 100% humidity. At lower column depths, the gas phase
humidity is even closer to 100%. This means that even
under dry soil conditions, the interphase vapor-liquid
equilibrium assumption is valid. ‘Also, the dynamics of
the gas phase mole fraction of water vapor are very fast.

Simulation number 8 was run with the generally
accepted value for the particle Sherwood number of 2.0.
Tables 2 and 9 contain the simulation parameters and Table
10 contains the results. These results show that at all
.column depths the gas phase humidity is very close to 100%.
Also the gas phase equilibrium mole fraction dynamics are
much faster than in simulation number 7. Evaporation is
taking place near the surface and condensation is taking
place near the column bottom. This is to be expected from
the profiles in Figures 15 and 21. These figures indicate
that water vapor diffuses away from the peak, toward the
surface, and toward the column bottom. This type of result
has not previously been obtained either by eXperiment or
theory.

Simulation numbers 9 and 10 are analagous to numbers
7 and 8 and were run to investigate what affect the soil
hydraulic properties might have on the interphase vapor-.
liquid equilibrium assumption. Tables 2, 11, and 12 contain

the simulation parameters for simulation number 9 and 10.

85

 

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86

TABLE lO.-—Simulation number 8 results.

 

Mole Fraction of Soil Water Vapor vs. Time
Particle Sherwood Number = 2.0

Time = 0.0 Secs.

 

Mole Fraction of Equilibrium Mole Fraction

 

 

POSltlon Soil Water Vapor of Soil Water Vapor
0. .llOS4655E-Ol .llOS4655E-Ol
.900E+Ol .65715938E-Ol .65715938E-Ol
.180E+02 .60620075E-01 .60620075E-Ol
.270E+02 .54257406E-Ol .54257406E-01
.360E+02 .51391514E-01 .51391514E-01

Time = 0.002

Secs.

0. .11054643E-Ol .11054655E-01
.9OOE+01 .65715919E-01 .65715938E-01
.180E+02 .60620075E-01 .60620075E-01
.270E+02 .54257407E-Ol .54257406E-01
.360E+02 .51391515E-01 .51391514E-01

Time = 0.196

Secs.

0. .11054642E-01 .llOS4655E-Ol
.900E+01 .65715916E-01 .65715935E-01
.180E+02 .60620075E-01 .60620075E-Ol
.270E+02 .54257407E-01 .54257405E-Ol
.360E+02 .51391515E-01 .51391514E-01

 

See Tables 2

and 9 for simulation conditions.

 

 

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88

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90

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91

The results of these simulations are contained in Fig-
ures 24 and 25. Again, as in simulation numbers 7 and 8,
the gas phase humidity is close to 100% and the interphase
vapor-liquid assumtion is valid.

The factor controlling the allowable time incre-
ment in simulation numbers 7-10 is the mass transfer
coefficient. For lower particle Sherwood numbers the
interphase mass transfer coefficient is smaller, and a
larger time increment can be used. Although it is not
economical to run the GM for long periods of real time,
it can be run for a few seconds to check out the inter-
phase vapor-liquid equilibrium assumption. If the
assumption is valid, then the P&DVM is valid and can be
run for long periods of real time. If the assumption is
not valid, the P&DVM is only an approximation and the GM
shOuld be run. It may be possible to develop a numerical
procedure for the GM which would be more economical than

the existing numerical procedure.

With the interphase equilibrium assumption validated,

it was decided to use the P&DVM to simulate the response
of the soil column to time varying boundary conditions.
This type of simulation also has not been run previous to
this work.

The simulation parameters are listed in Table 13
and the time varying boundary conditions are given in

Figures 26-29. Natural weather conditions existing in

l.j.4-

 

 

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94

TABLE 13.--Simulation parameters for simulation number 11.

 

Numerical Information

 

Size of Time Increment

Column Length

Size of Space Increment

Number of Space Increments

.69444444E-04 for 0-1 day real time
.27777777E-02 for greater than 1 day
.45000000E+02
.90000000E+Ol
.50000000E+Ol

 

The Following is a List of Physical Parameters for the System

 

Column OD (Cm.)
Column ID (Cm.)

Porosity (dim.)

.26000000E+02
.10000000E+02
.30500000E+00

Ordinary Diffusion Coef. Air—Water (cmZ/day) .19ooooooz+os

Over All Mass Transfer Coef.

Pressure (Atm.)
Molecular Weight
Gas Constant (CC.

(l/day) .38000000E+06
.10000000E+01

of Water .18000000E+02
—Atm/Gm.Mole-Deg.K.) .82057000E+02

 

The Following is

a Listing of the Initial Conditions

 

 

Increment Position Saturation Temperature Mole Fraction of Water Vapor
(Cm.) (dim.) (Deg.C.) (dim.)
O .90E+01 .5715511E+00 .2500000E+02 .3117211E-01
2 .18E+02 .8424127E+00 .2500000E+02 .3ll7231E-Ol
3 .27E+02 .9057420E+00 .2500000E+02 .3117271E-Ol
4 .36E+02 .9916412E+00 .2500000E+02 .3117300E—Ol

 

95

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96

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99

Lansing, Michigan on June 11, 1971 were used to construct
a daily weather cycle which was repeated day after day
for the duration of the simulation.

The temperatures at the soil surface, 9 cm., and
18 cm. soil depths were plotted as functions of time in
Figures 30-35. The saturation profiles are given in
Figure 36. From Figures 30-36 it can be seen that between
the second and third day the falling rate period begins
and the surface saturation is near the irreducible satura-
tion.

The amplitude of the temperature oscillations
increases with time at all depths. For example, before the
falling rate period, the magnitude of the surface varia-
tion is about 15 deg. C., whereas during the falling rate
period the magnitude increases to about 30 deg.C. At
lower depths the magnitude of the temperature variation
decreases. Also, there is a lag between temperature peaks
in comparison to the surface temperature peak. The
temperature peak at the surface occurs at about 1300 hrs.,
whereas the temperature peak for z=9 cm. occurs at about
1330 hrs. At a depth of z=18 cm., the temperature peak
occurs at about 1500 hrs.

Simulation number 11 illustrates the moderating
affect that soil water has on the soil temperature variations

and magnitude of the soil temperature. During the nighttime

100

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('mrp) uorqeanes IIOS

107

hours the soil temperature dropped below the ambient
temperature when the soil surface saturation was high.
This was because evaporation was taking place at night
with the subsequent cooling of the surface.

Although the bottom column boundary condition and
radial column energy loss do not represent exactly the
boundary conditions in an open field of soil, simulation
number 11 does give a reasonable estimate of the tempera-
ture variations. Cary has reported data on maximum daily
variations in soil temperatures at various depths for
Argonne, Illinois [32]. Table 14 shows these variations.
If the soil is wet, these simulation results are close

to those reported for Argonne daily maximums.

TABLE l4.-—Maximum daily soil temperature changes over an
annual cycle at Argonne, Ill.

 

Maximum daily soil temperature

 

Depth (cm.) variation (deg.C.)
l 12
10 9
20 3
50 0.5

 

If the soil is very dry the simulation results are off by a

factor of two on the large side.

CHAPTER V

CONCLUSIONS

The objectives of this work, outlined in Chapter I,
have been met. A general model to describe water movement
in soils has been deve10ped. This new model and earlier
models for water movement in soils have been used in
digital computer simulations. The different models have
been compared with one another and with experimental drying
data.

Using the general model developed in Chapter II, it
was shown that even under dry soil conditions, the inter-
phase vapor-liquid equilibrium assumption was valid. This
assumption is the foundation for the Philip and De Vries
model and has not previously been justified by soil air
humidity measurements, or by theoretical analysis. The
general model can indicate the validity of the assumption
for a given situation defined by the following information:
(1) saturation profile, (2) temperature profile, and (3)
boundary conditions.

The unsteady state solution of the Philip and

De Vries equations has been obtained for both constant and

108

109

time varying boundary conditions. Previous to this work,
the Philip and De Vries equations have not been solved
for unsteady state problems. Simulations on identical
drying problems using the Philip and De Vries model and
the isothermal equation has shown the following. When
the irreducible saturations (during the drying falling
rate period) used in both models are the same, there is
good agreement between the saturation profiles calculated
by each model.

An advantage of the Philip and De Vries model is
that it can be used to calculate saturation, temperature,
and equilibrium mole fraction of water vapor profiles
during the drying of soil or porous media. With the
hydraulic and thermal prOperties of the porous media, the
irreducible saturation can be calculated for a given set
of boundary conditions. The isothermal equation cannot
predict the irreducible saturation. This is because the
surface temperature is not known since this model does not
predict temperature.

Current use of the isothermal equation for drying
has mainly been in fitting experimental cumulative evapora-
tion data. Experimental information on the values of
irreducible saturation is used in the simulations. Now for
the first time the Philip and De Vries model has been solved

in this work to yield a predictive model.

110

A time varying boundary condition problem was run
with the Philip and De Vries model. Natural weather
conditions were used as the surface boundary conditions.

The temperature variations in the soil which were calculated
were reasonably close to the experimental data. The
simulations also showed that soil water acts to minimize

the maximum soil temperature reached in a daily cycle.

Also a dry soil has a daily temperature variation almost
double that of a wet soil under the weather conditions studied.

Literature physical property data was used with the
isothermal equation and the Philip and De Vries model in an
attempt to fit experimental drying data. The saturation
profiles calculated fitted the experimental saturation.
profiles. Although the calculated temperature profiles were
similar in shape to the experimental temperature profiles,
the calculated temperatures were larger than the experimental
temperatures. This is because the column bottom boundary
condition on energy flux was not a close enough approximation
to the drying experiment boundary condition.

The solution to the Philip and De Vries model developed
in this work should be useful in problems involving drying
of soils or porous media where a predictive capability is
needed. The general model will also be useful in these.
types of problems to evaluate the interphase vapor-liquid
equilibrium assumption upon which the Philip and De Vries

model is based. The following is a brief listing of some

111

problems where the Philip and De Vries model and the
general model would be useful.

(1) Drying of a porous media where a certain

temperature or dryness must not be exceeded
in the porous media.
(2) Management strategies for irrigation practices.
(3) Management strategies for insect pest control
(example, cereal leaf beetle) where the insect
hatching is strongly dependent on soil moisture
and temperature.

(4) Meteorological studies.

For soils which are above the irreducible saturation,
the isothermal equation for water movement would be the
most expedient model to use if soil temperature information
was not desired. In particular, the isothermal equation
would be a good base for structuring models to describe
the movement of chemical compounds in soils subjected to

spray irrigation with liquid wastes.

REFERENCES

112

10.

11.

REFERENCES

Darcy, H. "Les Fontainer Publiques de la ville de Dijon,"
Dalmont, Paris, (1856).

Collins, R.E. Flow of Fluids Through Porous Materials.
New York: Reinhold Publishing Corp., 1961, 47.

 

Collis-George, N., Childs. E.C. "The Permeability of
Porous Materials." Proc. Royal Soc. (London),
Vol. 210A (1950), 392.

 

Irmay, S. "A Model of Flow of Liquid-Gas Mixtures in
Porous Media and Hysteresis of Capillary Potential."
Paper 23c, 68th National Meeting of the AIChE,
Houston, Texas, 1971.

Philip, J.R. "The Theory of Infiltration: l. The
Infiltration equation and Its Solution." Soil Sci.,
83:345(1957). '

Staple, W.J. "Comparison of Computed and Measured
Moisture Redistribution Following Infiltration."
Soil Sci. Proc., 33:84o(l969).

 

Gardner, W., Widtsoe, J.A. "The Movement of Soil
Moisture." Soil Sci., 11:215(l921).

 

Larian, M.G. FundamentalsJof Chemical Engineering
Operations. EngleWood Cliffs, N.J.: Prentice-
Hall, Inc., 1958, Chapter 8.

 

Treybal, R.E. Mass—Transfer Operations, 2nd edition.
New York: McGraw-Hill Book Co., 1968, Chapter 12.

 

Ceaglske, N.H., Hougen, O.A. "Drying Granular Solids."
Ind. & Eng. Chem., 29:805(1937).

 

Philip, J.R., De Vries, D.A. "Moisture Movement in
Porous Materials under Temperature Gradients."
Trans. Amer. Geophysical Union, 38:222(l957).

113

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.
23.

24.

114

Taylor, S.A., Cary, J.W. "Thermally Driven Liquid
and Vapor Phase Transfer of Water and Energy in
Soil." Soil Sci. Proc., 26:413(1962).

 

Irrigation of Agricultural Lands, ed., Hagan, R. M.,
Haise, H.R., Edminster, T.W. Series of Agronomy
No. 11. MadiSon, Wisconsin: Amer. Soc. of Agronomy,
1967, 200.

 

Covey, W. "Mathematical Study of the First Stage of
Drying of Moist Soil." Soil Sci. Proc., 27:130(1963).

 

Klute, A., Whisler, F.D., Scott, E.J. "Numerical
Solution to the Nonlinear Diffusion Equation for
Water Flow in a Horizontal Soil Column of Finite
Length." Soil Sci. Proc., 29:353(l965).

 

Fritton, D.D., Kirkham, D., Shaw, R.H. "Soil Water
Evaporation, Isothermal Diffusion, and Heat and
Water Transfer." Soil Sci. Proc., 34:183(1970).

 

Bird, R.B., Stewart, W.E., Lightfoot, E.N. Transport
Phenomena. New York: John Wiley and Sons, Inc.,
1960, Chapters 3, 10, 18.

 

Lucretius. "De Rerum Natura," Classics of Western
Thoughts, ed. Nulle, S.H., Vol. 1. New York:
Harcourt, Brace and World, Inc., 1964.

 

Hirschfelder, J.O., Curtiss, C.F., Bird, R.B.
Molecular Theory of Gases and Liquids. New York:
John Wiley and Sons, Inc., 1964.

 

Bird, R.B. "The Equations of Change and the Macrosc0pic
Mass, Momentum, and Energy Balances." Chem. Eng.
Sci., 6:123(1957).

Raats, P.A.C., Klute, A. "Transport in Soils: The
Balance of Momentum." Soil Sci. Proc., 32:452(1968).

Treybal. Mass-Transfer Operations, Chapter 5.

 

Frossling, N. Ger. Beitr. Geophys., 52:170(l938).

 

Ranz, W.E., Marshall, W.R. "Evaporation from DrOps:
Part 1." Chem. Eng. Progr., 48(3):l4l(1952).

 

25.

26.

27.

28.

29.

30.

31.

32.

33.

115

Wise, D.L., Wang, D.I.C., Matelles, R.I. "Increased
02 Mass Transfer Rates from Single Bubbles at Low
Reynolds Numbers." Biotech. & Bioeng., 11:647(l969).

 

Kunii, D., Suzuki, M. "Particle to Fluid Heat and
Mass Transfer in Packed Beds of Fine Particles."
Int. J. Heat & Mass Transfer, 10:845(l967).

 

Littman, H., Barile, R. G., Pulsifier, A.H. "Gas—
Particle Heat Transfer Coefficients in Packed Beds
at Low Reynolds Numbers." Ind. & Eng. Chem. Fund.,
7:554(1968).

 

Schechter, R.S. The Variational Method in Engineering.
New York: McGraw-Hill Book Co., 1967, 213.

Moench, A.F. "An Evaluation of Heat Transfer Coefficients
in Moist Porous Media." Ph.D. Dissertation.
University Microfilms Order No. 69-12533, (1969).

Hanks, R.J., Gardner, H.R., Fairbourn, M.L. "Evaporation
of Water from Soils as Influenced by Drying with
Wind or Radiation." Soil Sci. Proc., 31:593(l967).

Weber, H.C., Meissner, H.P. Thermodynamics fgr Chemical
Engineers. New York: John Wiley & Sons, Inc., 1959.

 

Cary, J.W. "Soil Moisture Transport Due to Thermal
Gradients: Practical Aspects." Soil Sci. Proc.,
30:428(l966).

Novak, L. T., Coulman, G. A. "A Simulation of Two
Phase Water Movement in Soil." Proc. 1972 Summer
Simulation Conf., San Diego, California, p. 940.

 

APPENDIX A
CAPILLARY POTENTIAL AS A FUNCTION OF
SATURATION, TEMPERATURE, HISTORY,

AND COMPOSITION

116

117

 

=0

 

 

= +w (called
capillary
potential)

I
l

porous plate

 

 

 

 

 

The free energy of water per weight of water in soil
is a function of saturation, temperature, and composition.

It can be calculated as follows:

(A.l) d? = —)dS + (3T)dT

(3
as T.¢i s ¢i

()dcbi
ig—d’ISJ

where, W = free energy per weight of water = dyne-cm/dyne
Y°= standard state (S = 0, T,¢l,¢2,...,¢n) = 0
An analogy has been made between the soil drawing

water as shown above, and a capillary tube drawing water.

 

9L.

 

 

118

The capillary tube analogy appears to be the source for
calling W the capillary potential.

When the capillary potential (V) is due to surface
phenomenon such as curved gas-liquid interfaces in soil
capillaries, the effect of temperature can be calculated

from the surface tension (or free energy) as follows:

(A.3)

Integration of Equation (A.3) gives capillary potential

as a function of temperature. This integration is contained
in the physical property subroutine for capillary potential
(see Appendix F).

It was mentioned that capillary potential is also a
function of history. This is because there is a hysteresis
in the capillary potential—-saturation function which is
dependent on the drying--wetting history of the soil. From
the discussion in Appendix E on the interfacial area model,
it is evident that the way in which the water distributes
itself beyond the drainage point will affect the interfacial
area--saturation function. The water distribution phenomenon
likewise affects the capillary potential-~saturation function,
and hence the hysteresis. For a drying process, there is
no hysteresis as long as the soil saturation is monotonically
decreasing. In this case a unique curve exists for the

capillary potential--saturation function.

 

APPENDIX B
ANALOGY BETWEEN THE ISOTHERMAL EQUATION
OF GARDNER AND THE UNSTEADY STATE

DIFFUSION EQUATION

119

The isothermal equation of Gardner and Darcy's
law can be combined to give an equation of a form similar
to the unsteady state diffusion equation. Assuming
capillary potential (V) is only a function of saturation,

Darcy's law can be written

Ull

0 av a§ a

(13.1) v =-K(§§-§E—-1)=—D +K.

O)
N

Substituting Equation (D.l) into Gardner's equation for

constant porosity (e),

3?. __
(B.2) € 'a-E' - '3‘;(D 52‘) 5';

In the absence of gravity Equation (B.2) is analogous in
form to the unsteady state diffusion equation written with

a variable diffusivity.

120

APPENDIX C
THREE DIMENSIONAL EQUATIONS OF CHANGE

FOR POROUS MEDIA

121

The

three dimensional equations of change can be

obtained by making material and energy balances which

include the

directions.

possibility for variation in three spatial

The equations of continuity (Equations (2.2),

 

(2.5), (2.7) in three dimensional form are:
3pw6 0
(Col) T = -V°pwV - E
(C.2) at w M
w

(C.3) Cél=-V°q-VH E

at vap
where,

.. a a 9..
(C.4) V — (5;, 5?: 32)

In the above

are vectors.

equations the flux quantities (V0, NW, q)

122

APPENDIX D
LIQUID WATER VAPOR PRESSURE AS A FUNCTION
OF TEMPERATURE, SATURATION,

AND COMPOSITION

123

The vapor pressure of water in soil is dependent
on temperature, saturation, and water phase composition.
Based on free energy considerations, it can be shown that
the vapor pressure--temperature relationship should obey

the Clapeyron equation [31].

d In Pva AHva
(13.1) B = " T—E

(1%)

This equation can be intergrated to give

 

(D 2) ln/PvaP(T2) = .. £52134; _ .1__)
° (P (T 7 R T T '
vap 2 l
where, AHvap = constant over [T1,T2]

This equation is not accurate over a wide range of
temperatures because the enthalpy of vaporization is a
function of temperature. For purposes of this study a

quadratic relationship between ln PV and (%) was used

3P
to obtain a good fit of vapor pressure over a range of
Saturation is another variable which affects the

vapor pressure of water in soils. If a wet soil is dried

under isothermal conditions, the vapor pressure is reduced

124

125

as the soil dries out. This is due to capillary and
adsorption forces. The adsorption forces predominate in
very dry soil. The phenomenon can be quantified based on

free energy considerations.
Consider the following states for soil water.

State 1 - pure water in soil at 25°C., saturation :8 =1.0

State 2 — pure water in soil at 25°C., saturation =S

The change of free energy of soil water between states one

and two is

P (82)

_ _ va
(D.3) AG — RT ln F_—BT—_T"

vap S1

The free energy change can also be written as
(D.4) AG = VAP = V é’W)

where, AP = -Y

The combination of Equations (D.3) and (D.4) give the
relationship between capillary potential (and saturation)
and water vapor pressure. This equation is used to
determine capillary potential under low moisture conditions

in the soil by making water vapor pressure measurements.

126

Vpgw

(0.5) P (82) = P RTY

vap ap(sl) EXP(

V ), wio

When the water in soil is not pure, the soluble
compounds affect the water vapor pressure. Methods for
handling this situation will not be covered here as they

are covered in standard thermodynamics textbooks [31].

APPENDIX E
INTERFACIAL AREA MODEL (SIMPLE CUBIC

PACKING OF RODS) AND COMPUTER PROGRAM

127

Figure 37 on the next page defines the terms in
the following equations which are not defined in the

nomenclature.

The solid and void fractions are calculated as follows for

the simple cubic packing of rods.

2
(E01) 1-8 =1TDL=1
4D§L 4

(B.2) e = 1 - % = 0.21460

Before drainage is reached:

 

 

x* x*
(E.3) Se = §%_ f (y*-§) dx = §§ f (y*-§) dx
D L o D o
x* * x* _
(3.4) a = 3%E-+ §%— f \/:+ (§§.)2 dx - §%—J 1 + (gfioz dx
D L D L o D L 0. _

The equation of a rod cross section:

(3.5) (x-R)2 + y2 = R2

The equation of the gas-liquid interface:

2 2

(E.6) x + (y-k)2 = r

128

129

 

simple cubic packing of rods y axis

front vie

 

 

 

 

 

 

Rod diameter = D = 2R

Rod length = L
r = radius of curvature of liquid-gas interface

*
y = y distance from x axis to the gas-liquid interface

§ = y distance from x axis to the rod surface

Figure 37.--Interfacial area model (simple cubic packing
of rods).

so,
(3.7) §
(3.8) Y*

+R

130

2 _ (x-R)2

- rZ-x2+k

Before drainage the liquid interface is tangent to the

solid interface at the gas—liquid-solid contact point

(x=x*).

so,

(E.9) x*
(E.10) r
(E.ll) k

R(1-cose)

 

2 1

R(-l + V1 + tan 6 ) 03034

R tane

Combining Equations (E.3)-(E.8) and integrating we get

(3.12) s

(E.l3) a

*
g . [kx* _ 15[x*\/r2-X*2 + r2 sin ICES—)3

D e

*
—k [(x*-R)\/R2-(x*-R)2 + 32 sin—1(x ”R)J

 

R
+5 32 sin-l(-l) ]

*

 

% + ;% [r[sinfl(¥—)-sin-l(0)]
-R[sin—l(x;-R)-sin_l(-l)J]

131

After drainage the following water distribution is assumed:

Drainage occurs at 6 =45° or xl=x2=§::
2
_w=P_.2_O_

r corresponds to
exptl. capillary
potential

 

 

The equations for saturation and interfacial area after

drainage are given below.

x
8L 1*- 2 *_
(3.14) S = —§— f (y -y) dx + (y*-y) dx
DL 0 I1

 

_8L .1 d*2
(E.15) a — —§— 7°‘dh + (a§—) dx
D L
where,

(3.16) (y*-y) = k - \Irz-x2 - \[RZ-(x-R)2

(3.17) (fit-5;) = -(x-R) - fiZ-(x-R)2

 

 

(3.18) y = -(x-R)

132

Substitution of Equations (E.l6)-(E.18) into Equations
(E.l4) and (E.lS) and integration give the following

equations which apply after drainage.

 

x
(E.l9) S_= 2:3[kxl-%(Xfl/r2—xi + rzsin—l(fi))-%(x§—x§—2(x2-xl)R)
x -R
-%[(x2—Rh/2x2R-x§ + R2(sin_l( 2R )-sin_l(-l))]]
x
(3.20) a = E; [sin'1(—-l—)-sin‘l(0)]
D r

To find x1, consider the intersection of the line (y=-(x-R))

with the circle (x2+(y-k)2=r2). Then,

 

(E.21) x1 = -%(k-R)iv;r2-k(k-R)2 , where r = 35(k—R)2

The location of the radius of the radius of curvature
(r) on the y axis will be at y = k. The following conditions
must be satisfied by this location.

1. The saturation calculated for a given k must be
equal to the experimental saturation associated
with the experimental capillary potential
represented by the radius of curvature (r).

2. A solution must exist for a given k. That is
the radius of curvature must intersect the rod

or the line y = -(x-R).

133

The existence of a solution is also dependent on
the rod radius. The following is the documented
computer program which calculates the interfacial area
versus saturation curve over the entire saturation range.
Beyond drainage, logic statements are used to insure a
solution exists and a Newton-Raphson interation is used

to locate the radius of curvature at y = k.

134

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APPENDIX G

COMPUTER PROGRAM FOR NUMERICAL SOLUTION

TO THE GENERAL MODEL

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TO THE PHILIP AND DE VRIES MODEL

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TO THE ISOTHERMAL EQUATION

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APPENDIX J

OVER ALL WALL HEAT TRANSFER

COEFFICIENT CALCULATION

184

 

Radial energy losses from the soil column (see
Figure 10) involves heat conduction through the soil, the
column wall, and the insulation. An energy balance on

the column wall or insulation gives the following equation.

3T __ 3
(J.l) pCp '—t-- 32(k

3T 1 8

82 r 3r

(rk 3;)

when Equation (J.l) can be written as Equation (J.2),

3 3T _
(J.2) fika 5; - 0

the over all heat transfer coefficient can be calculated

in the following manner. The radial heat flows are:

 

 

 

 

 

 

 

 

 

 

 

 

(J.3) qfiDOdz = UonDOdz(TA-T) = hnDodz(TA-T3)
/ I
I-I
. I :/
Insulation‘—' . : ///
Wall — T
, _ _ / A
8011 i /I_/L / H
In”
lé—Dw‘“)
o ’l
TTT T
12 3

185

 

186

 

 

 

 

 

 

 

( 4) Do (T3‘T2) (Tz’Tl)
J. q2n—— dz = -k.2wdz = -k 2ndz
2 l ln(Do/Dw w ln(Dw/Di)
(T -‘I‘)
_ l _ _ fQE
_ keZNdzln(Di7D*7 - kiZNdz IQE
r
Equation (J.3) and (J.4) can be combined to give
(J 5) U = 1
° 0 D 1n(D./D*) ln(D /D.7 ln(D /D )
_g E 1 + w i + o w + 2
2 k k k. hD
e w i o

The diameters and materials of construction for the soil
column in Figure 12 have been given in Table 1. The thermal

conductivities for these materials are:

ke = 8.5 x 10 4 cal/cm-sec-OC(for saturation = 0, data of
Moench)

4 cal/cm-sec-°C(for p1exig1ass--acrylic resin,
data from Polymer Handbook,
eds., J. Brandrup, E. Immergut,
Interscience, N.Y., 1966).

k = 5 x 10'
W

k. = 8.26 x 10'5

l cal/cm-sec-OC(for styrofoam, data from

Principles of Polymer Systems,
F. Rodriquez, McGraw Hill,
NoYo’ 1970).

2 4

for 9* = 9.2 cm and h = 1 btu/hr-ft -OF (1.355 x 10’

cal/cmzvsec-OC), the over all heat transfer coefficient is

6

U0 = 6.35 x 10- cal/cm-sec-OC = 0.548 cal/cmZ-day-OC

187

The value of the film coefficient (h) is reasonable for

low wind velocities and the over all coefficient is not

 

that sensitive to the film coefficient. The resistance

controlling the over all coefficient is the insulation as

is shown below.

 

(J.5*) U0 = 4 41 4 4 = 6.35 x 10'6 1
.13x10 + .257x10 + 14.62x10 + .74x10 i.

soil wall insulation film coeff.

 

 

 

APPENDIX K

SURFACE FILM COEFFICIENTS

188

The wind velocity at the soil surface has an effect
on the surface film mass transfer and heat transfer
coefficients. To estimate these coefficients for the
soil column in Figure 12, some heat and mass transfer

correlations for laminar flow past a flat plate will be used. *

 

 

i;
(K.1) Nu = 0.664 ReLJ’EPrl/3 Reference: Bennett, C.O. Myers, E
J.E.,Momentum, Heat, and Mass -~
8 1/3 Transfer, McGraw—Hill, N.Y.,
(K.2) Sh = 0.664 ReL Sc 1962, p. 302, 476.
where,
km SL
(K.3) Sh = ’
.an
hsL
(K.4) Nu = —k—-
V Lp
(K.5) ReL = —3——
0

Using the properties of air at room temperature and a

wind velocity of four miles per hour and a dimension of

L = 27 cm, the surface film heat transfer coefficient

is calculated to be hS = 1.8 btu/hr-ftZ-OF = 21.1 cal/cmz-
day-0C. By combining Equations (K.1) and (K.2) the surface

film mass transfer coefficient can be calculated to be

189

190

(K.6) k

Nu S 1/3
m,s Pr

4100' hS (cm/day).

86510 cm/day

APPENDIX L
CALCULATION OF PART OF THE CAPILLARY
POTENTIAL--SATURATION FUNCTION

FOR VALENTINE SAND

191

It can be seen from Equation (1.5) that at steady
state the volumetric flux of water is constant. Before the
actual drying of valentine sand, Hanks, et al. wet the sand
covered it, and allowed it to redistribute for a week.

The resultant distribution was a saturation profile
corresponding to the initial conditions for simulation
numbers 3-6 (see Figures 16,17). Assuming that during the
week an equilibrium saturation distribution was reached,
the capillary potential-—saturation function can be cal-
culated over the range of saturations in the initial

condition saturation profiles (Figures 16,17).

(L.l) v° = —K(§§ - 1) = o, w=w<s>
then,
(L.2) ‘y(5)|z -‘¥(S)Iz == (z-zo)

O

The data points in Figure 9 identified as being calculated
were calculated from Equation (L.2) and the initial condition
data of Hanks et al. The capillary potential at saturation

was taken to be

(L.3) 4(5 = 1)]zO= 45 = —1 cm.
It is not uncommon for air bubbles to cause capillary potentials

of this magnitude when the saturation is essentially one.
192

APPENDIX M

COMPARISON OF CONDUCTIVE ENERGY

FLUX AND CONVECTIVE ENERGY FLUXES

193

From data obtained from simulation number 5 and
contained in Table 7, the conductive energy flux can be
compared with the convective energy fluxes which represent
the energy transfer by the movement of liquid and vapor

water.

When time = 40 days, we have the following:

at z=0
(M.1) q = +37l.04 cal/cmz-day
_ _ 0: _ _ 2 _
(M.2) qL — OLCpL(T Tref)v 1.0(48.5 25.0)0 — 0.0 cal/cm day
_ ~ _ —_ _ —-_ 2—
(M.3) qV — Nwapr(T Tref)‘ .0360(48.5 25.)— .845 cal/cm day

where, q,qL, and qV are the conductive, liquid convective,

and vapor convective energy fluxes respectively.

at z=9

(M.4) q = +248.95

(M.5) qL = 1.0(39.8—25.)(-.10892E-02)=-.01615
(M.6) = -.34923E-01(39.8—25.)=-.516

qv
From looking at the fluxes in Table 7 at 40 days it can be

seen that the energy fluxes due to convective flow at
lower depths are also negligible compared to the conductive

energy fluxs.

194

195

At t = 3 days, the convective flow energy terms in
comparison to the conductive energy term seems to be about

the largest. This is the start of the falling rate period.

at z=0

(M.7) q = 824.07

(M.8) qL = 1.0(26.8-25.)(0) = 0.0

(M.9) qv = -.14409E+01(26.8—25.) = —2.45
at z=9

(M.lO) q = 45.391
(M.ll) qL = l.0(26.0+25.)(-.11182E+Ol)=—l.1182

(M.lZ) = -.13005E—02(26.0-25.)=-.013005

qv
comparison between q and qL and qv:

qL+qV

(M.13) % = x 100 = -113.1/45.391 = 2.49

 

At other depths for t=3 days, the ratio of convective flow
to conductive energy flux will be smaller than 2.49%.

At the beginning of the falling rate period the
convective energy fluxes can be as high as 2%% of the
conductive flux at z=9 cm. At other depths this per cent
falls to about %%. Also after the falling rate period is
underway the maximum % convective to conductive flux occurs
at z=9 cm. and the % is about %. As an example see

Equations (M.l-M.6).

APPENDIX N

COMPUTER PROGRAM SUBROUTINES CONTAINING

NATURAL WEATHER TIME VARYING

BOUNDARY CONDITIONS

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