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SOIL TEMPERATURE REGIME AND. ITS
INTERACTION. WITH SUBMERGED TILE

Thesis for the Degree of M. S.
L" MICHIGAN STATE UNIVERSITY
.j SIROUS HAJI-DIAFARI
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ABSTRACT

SOIL TEMPERATURE REGIME AND ITS INTERACTION
WITH SUBMERGED TILE

By

Sirous Haji—Djafari

The purpose of this study was to investigate the
effect of thermal environment on submerged drain tile and
the conditions causing the cracking of clay tile due to
the freezing of part of water within the tile. The loca-
tion of interest for the investigation was near Saginaw,
Michigan. A new technique was used to obtain soil tem-
peratures from air temperatures.

In order to ascertain the behavior of‘a crack
resulting from freezing the water within a drain tile, an
experiment was conducted in the laboratory. The result
of this experiment led to the determination of realistic
conditions for solving the mathematical model of the
freezing. In order to investigate the effect of sub-
merged drain tile on the Ebrmal environment, a computer
model was developed and utilized to study four, six and
eight inch submerged drain tile burried at 36, 42 and 48

inches depth.

Sirous Haji-Djafari

The investigation led to following results: the
average soil temperature varies as a function of depth but
the average soil temperature is always greater than the
average air temperature. When the soil temperature sur-
rounding a submerged tile is reduced to less than 32°F for
a period of time, the water inside the tile will freeze.
Cracking will occur when about 1/3 of the water has frozen

and the location will be at the interface of the ice water

Approved Aégéfiifl,fl/ é? ;EZ%Z;%”“\

Ma or Professor

Approved 45 A W

Head of Department

phase and the tile wall.

SOIL TEMPERATURE REGIME AND ITS INTERACTION
WITH SUBMERGED TILE

By

Sirous Haji-Djafari

A THESIS

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

MASTER OF SCIENCE
Department of Agricultural Engineering

1972

ACKNOWLEDGMENTS

The author wishes to express his sincere apprecia-
tion to Dr. George E. Merva, who served as his Major
Professor for the entire graduate program. His experience
and inspiration were a constant source of assistance during
the study. It has been a great privilege to work with him.

The author is indebted to Professor E. H. Kidder
of the Agricultural Engineering Department, Michigan State
University, for his unfailing guidance.

Sincere acknowledgment is due to Professor
J. E. Adney, Mathematics Department, Michigan State
University, for his cooperation and assistance.

Appreciation is also extended to Associate Profes-
sor J. V. Beck, Mechanical Engineering Department, Michi-
gan State University, for his helpful suggestions.

The author expresses his thanks to the Hancock
Brick and Tile Company, Finley, Ohio, for furnishing the
required funds to complete this project.

The author is also thankful to his wife, Azizeh,
for all the efforts put forth by her during the period of

this study.

ii

TABLE OF CONTENTS»

ACKNOWLEDGMENTS . . . . . . . . . . . . .
LIST OF TABLES
LIST OF FIGURES . . . . .

INTRODUCTION

Chapter
I. REVIEW OF BASIC POINTS . . . . . . . . . . . .

1.1 Introduction . . .

1.2 Source and Amount of Heat .
1. 2.1 Latitude and Slope of Land .
1.2.2 Vegetative Cover . . . . . .

1.3 Thermd.Pr0perties of Soils
1.3.1 Heat Capacity of Soils
1.3.2 Thermfl.Conductivity and

Soil Factors Affecting It
1. 4 General Differential Equation
for the Temperature Field .‘. .
1.5 Daily and Seasonal Changes in Soil
Temperature . . . . . . . . . . .
1. 5.1 Damping Depth . . . . . . . . .

II. DEFINITION OF THE PROBLEM

2.1 Background. . . .
2.1.1 Tile for Over- size Trenches
2.1.2 Clay Tile and Frost Action
2.2 EXperimental Set-up . . .
2.2.1 Procedure
2.2.2 Observations .
2.3 Mathematical Model . .
2.3.1 General Assumptions
2.3.2 Specific Assumptions
2.3.3 Determination of Initial
and Boundary Conditions . . . . . .
2.4 Solution of the Model . .
2.4.1 Formula and Method of Solution.
2.4.2 Numerical Solution .
2.4. 3 Applying the Boundanrand Initial
Conditions in Solution of Related
Formulas

iii

Page

ii

vii

I-IH
oooouxza 4:-

19

22
25

28

28
28

31
31
32
35
35
37

41
57
59
59

62

Chapter

III.
3.1 Results . . . . . . . .
3.1.1 Results of Experiments
3.1.2 Results of Soil Temperature
Study . . . . . . . . . . . . . .
3.1.3 Results of Computer Modeling
3.2 Discussion of Results . . . . .
IV. CONCLUSIONS . . . . . . . . .

V. RECOMMENDATIONS FOR FURTHER STUDIES

LIST OF REFERENCES

Appendices
I. FOURIER SERIES MODELING OF PERIODIC
PHENOMENA . . . . . . . .

II. COMPUTER MODELING FOR STUDY OF SOIL
TEMPERATURE WITH SUBMERGED DRAIN TILE

III. CONVERSION FACTORS

iv

Page

66
66
66

67

70
77

79
81
83

85

92
108

Table

1-3

1-4

1-5

1-6

2-1

2-3

2-4

2-5

LIST OF TABLES

Specific heatvalues<3 densities p, and
specific heat per unit volume C of several
soil minerals and rock materials

Specific heat values c, densities p, and
specific heat per unit volume C of several
organic soils and soil materials . . . .

Specific heat values c, densities p, and
specific heat per unit volume C of several
soils after Kersten (1949) . . . . . .

ThermalprOperties of some typical soils and
of the common soil constituents (at 10°C)

Comparison of thermalproperties of frozen
and unfrozen soil

Damping depths oftemperature fluctuation
having various periods for several values
of thermal diffusivity

Recorded daily soil temperature (°FJ at
different depths in East Lansing Horti-
culture Farm in 1963 . . . . . . . .

Annual average of soil temperature (°F.)
for East Lansing Horticulture Farm
Station .

Annual average soil temperature (°F.) at
the University of Minnesota, St. Paul
Station. Soil is under sod cover

Effect of soil coverage on yearly average
soil temperature (°F.) at different
depths O O O O o O O O 0

Difference between air temperature and soil
temperature at different depths in °F.

Effect of soil cover on. soil temperature
at different depths in the cold season

Page

12

13

14

20

21

27

42

43

43

46

48

69

Table Page
3-2 Effect of soil cover on soil temperature at
different depths in the warm season . . . . 69
3-3 Time and required duration of submergance
for cracking to take place in clay drain

tile . . . . . . . . . . . . . . . . . . . 76

I-l The calculated fourier coefficients for 8
inches depths of soil . . . . . . . . . . . 88

I-2 The calculated fourier coefficients for 20
inches depths of soil . . . . . . . . . . . 89

I-3 The calculated fourier coefficients for 40
inches depths of soil . . . . . . . . . . . 90

I-4 The calculated fourier coefficients for 80
inches depths of soil . . . . . . . . . . . 91

vi

Figure

1-1

2-2

2-3

2-8

LIST OF FIGURES

Extrinsic factors which influence frost
action .

Intrinsic factors which influence frost
action

Frost penetration in bare and grass-covered
soils.

Heat conductivity and diffusivity in coarse
' quartz sand

Linear heat conduction .

Location of thermocouples in the experi-
mental setup . . . . . .

A picture of the cracked tile showing the
general appearance. Note the thermo-
couple wires entering from the right

Temperature variation for each thermo-
couple . . . . .

The shape of ice causing cracking of a
drain tile. The location of the crack
is shown in the picture at the lower
right of the tile

Grid system

Effect of soil coverage on soil temperature.

Variation of monthly average air and soil
temperature by depth

The average monthly air temperature for
East Lansing, Saginaw and Evart in 1968.

Air temperature of East Lansing and
Saginaw for the year 1963

vii

Page

18
19

33

33

34

36
39
45

47

50

51

Figure

2-9

2-10

2-11

2-12

3-1

3—4

3-6

Calculated soil temperature for Saginaw,
1963 .

Recorded soil temperature for East Lansing,
Horticulture Farm Station, 1963

Recorded soil temperature for East Lansing,
Horticulture Farm Station, 1968

Approximation of drain tiles in grid system
for cumputer model

Frost penetration in bare and sod surface
soil

Temperature history in the soil around a
four inch submerged drain tile burried
at 36, 42, and 48 inch depths

Temperature history in the soil around 6
inch drain tile burried at 36, 42, and
48 inch depths .

Temperature history in the soil around
eight inch submerged drain tile burried
at 36, 42, and 48 inch depths

The effect of water within the four-inch
drain tile on soil temperature. (Iso-
thermal lines are shown in the figure)

A photograph of an 8 inch clay tile removed

from a field located near Saginaw,
Michigan..

viii

Page

54

55

56

58

68

71

72

73

74

78

INTRODUCTION

Drainage plays an important role in agricultural
production, both in arid regions as well as in regions where
the primary role of drainage is in the removal of excess
ground water from the plant root zone.

A well-designed drainage system requires high
quality material which, for a plastic conduit system is
synonymous with a material capable of supporting the over-
burden load. For a rigid conduit system a crack-free tile
is required which is capable of supporting the loading
which may occur due to trench and/or over-burden, since a
cracked tile, if disturbed, may collapse, thus causing
blockage of the tile line and failure of the System.

Although progress has been made toward providing
safe design criteria, a source of damage to buried tile
lines remains uninvestigated, i.e., the cracking of tile
due to water freezing within the conduit. Such a condi-
tion may occur in areas where good tile outlets are not
available and pump drainage is utilized. Under these con-
ditions, tile lines are often installed with minimum cover
and submerged outlets are used. A large portion of the
drainage system may be submerged during the cold season,

and penetration of severe cold weather can cause freezing

of a portion of the water within the lines and subsequent
cracking of the buried clay conduit. A knowledge of the
condition conductive to freezing damage would lead to the
establishment of design criteria which would ultimately
increase the reliability of the drainage installed under
these conditions.

In this thesis, a study has been made to investi-
gate the effect of thermal environment on submerged drain
tile and the conditions causing cracking of clay tile due
to the freezing of part of the water within the tile. In
this study the location of interest is near Saginaw, Michi-
gan, where extensive areas of soil had been laid bare
during the cold months. These areas are in large part
pump drained and the drainage system pumps may not be
Operative during the winter season.

For this study soil temperatures were calculated
from air temperatures based on a new technique which has
been presented in the body of this thesis. The data re-
quired to support the hypothesis leading to the technique
was obtained from United States Weather Bureau records
from stations in Michigan, Minnesota, and Wisconsin.

It is felt that if it is demonstrated experimen-
tally that cracking does occur under certain temperature
conditions, and if a similar set of conditions can be
predicted for a field location through application of the

selective modeling technique, then one could expect that

tile drainage due to cracking can be anticipated for the
location of interest. An experiment was conducted in the
laboratory. The result of this experiment led to a deter-
mination of realistic conditions for solving the mathematical
model of tile freezing. In order to investigate the effect
of submerged drain tile on thermal environment, a computer

model was developed and utilized in the theoretical analysis.

CHAPTER I
REVIEW OF BASIC POINTS
1.1 Introduction

In studying soil temperature and frost action around
drain tile, many influencing factors must be considered.

The factors which influence the soil temperature are divided
into two groups: extrinsic and intrinsic factors
(Kavianpour 1971).

Extrinsic factors are those which determine the
ambient conditions. The extrinsic factors are summarized
in the block diagram of Figure 1—1.

Intrinsic factors are those inherent to the soil
material and include the thermal properties of soil, such
as; the thermal conductivity, k, the specific heat, c, and
the thermal diffusivity, a. The intrinsic factors are
given in Figure 1-2.

In order to develOp the basic concepts relating to
thermal properties of soil, it is necessary to review some
of the work which has been carried out with respect to the

major factors and their influence on the overall thermal

diffusivity of the soil. This is the objective of Chapter I.

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1.2 Source and Amount of Heat

According to Baver, 1965, the temperature of the
soil is primarily dependent upon the amount of radiant
energy that is received from the sun. .The quantities of
heat reaching the earth's surface by conduction from within
the earth, or resulting from chemical and biological proces-
ses, are so small that such sources have a negligible
effect upon soil temperature.

The amount of radiation reaching the earth from the
sun depends upon the amount emitted by the sun and the
absorption of the radiation by the atmosphere.

In addition to the atmospheric conditions, there
are several characteristics of the earth's surface that
greatly affect the amount of radiation that is retained.

(Baver 1965). These may be grouped as follows:

1.2.1 Latitude and slope of land

 

The angle at which the sun's rays meet the earth
greatly influences the amount of radiation received per
unit area. Radiation reaching the earth at an angle is
scattered over a wider area than the same radiation
striking the earth's surface perpendicularly. Conse-
quently, in the former case the amount of heat received
per unit area is decreased in proportion to the increase

in area covered. The amount of radiation reaching the

earth per unit area is proportional to the cosine of the
angle made between the perpendicular to the surface and the
direction of the incoming radiation. Therefore, the radia-
tion received per unit area decreases with an increase in

this angle.

1.2.2 Vegetative cover

 

The major effect of vegetation of soil temperature
is the insulating quality of plant cover on temperature
fluctuations. Bare soil is unprotected from the direct
rays of the sun and becomes very warm during the hottest
part of the day. When cold seasons arrive, unprotected
soil rapidly loses its heat to the atmosphere. In winter
the vegetation acts as an insulating blanket to reduce the
rate of heat loss from the soil. Consequently, a pro-
tected soil is cooler in summer and warmer in winter than
one that is bare.

The investigation of Petit (see Baver, 1965,
pp. 365-366) has shown that frost penetrates deeper and
disappears slower under bare conditions than under grass
or surface mulches. This is illustrated in Figure 1-3.

It may be seen that the sod cover decreased the rate and
depth of penetration of frost as compared with bare soil.
When thawing occurred, frost disappeared from the pro-

tected soils sooner than from the bare soil owing to the

fact that the latter soil was frozen to a greater depth.

Temperature - °C
I
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Figure

Grass Cover

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1—3. Frost penetration in bare and grass-covered

soils [Petit (1893)]. Redrawn from Baver
1965 p. 366.

10

1.3 Thermal Properties of Soils

The temperature of a soil depends not only on the
quantity of energy transported to it, but also on its
ability to absorb this energy (Geiger 1965). The basic
factors which control soil temperature are the thermal
prOperties of the soil. Other factors such as intensity
of absorption of energy (related to soil color) also
affect this ability.

Two independent thermal prOperties enter into a
quantitative description of the heat transfer by conduc-
tion, viz., the thermal conductivity, k, and the heat
capacity per unit volume, C = pc. In many equations the
ratio of these two quantities appears. The ratio is called

the thermal diffusivity and is denoted by the symbol

77 pc'

DeVries (van Wijk, et a1. 1963), Geiger (1965), and
Baver (1965) present a detailed discussion relating to

thermal pr0perties of soils.

1.3.1 Heat capacity of soils

 

The specific heat of any substance is defined as the
number of calories of heat required to raise one gram of the
substance one degree on the centigrade scale (Baver 1965).
The heat capacity per unit volume of a given material C as

defined above is equal to its specific heat, c, times its

11

mass density, p. The specific heat of water is 1.00 calorie
per gram per °C at 16°C (one BTU/lbm per °F at 60°F). All
other constituents of soils have much lower specific heats.

It may be seen from the data in Tables 1-1 and 1-2
that quartz has the lowest specific heat of the major soil
constituents and humus the highest, excepting water. The
aluminosilicate kaolin has a slightly higher specific heat
than quartz. Since the major constituents in most soils
are quartz, aluminosilicates, water ahd humus, it is evi-
dent that humus and water will affect the heat capacity
considerably.

The heat capacity per unit volume of soil can be
found by adding the heat capacities of the different soil
constituents in one cm3 (DeVries, see van Wijk, et al.,
1963). Thus, if x5, xw, and x3 denote the volume fractions
of solid material, water (or ice) and air, respectively,
one has

C = x C + x C + x C
s s w w a a

The third term in the right-hand side can usually

be neglected. The value of CW = 1.00 cal cm-3 °C'1 in the
3 -1

case of water, 0.45 cal cm- °C in case of ice at 0°C and

.43 cal cm.3 °C-1

for ice at -20°C. The specific heats of
twelve different mineral soils and material were measured
by Kersten (van Wijk, et al., 1963) (see Table 1-3). The

found that the specific heat of most soil minerals varied

12

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14

 

 

 

Table 1-3. Specific heat values c, densities p, and
specific heat per unit volume C of several
soils after Kersten (1949).*
c1 . p C
(cal g 1 $23: 1 (cal cm-S
Soil °c 1) (°C) (g cm 3) °c 1)
Northway fine sand 0.197 61.0 2.76 .543
Northway sand 0.171 - 6.7 2.74 .468
0.185 18.8 --
0.191 60.2 --
Northway silt loam 0.168 -10.4 2.70 .453
0.176 20.4 --
0.193 60.4 --
Chena river gravel 0.194 61.0 2.70 .523
0.196 60.4 --
Fairbanks silt loam 0.164 - 8.4 2.70 .442
0.183 18.8 --
0.194 61.3 --
Graded Ottawa sand 0.157 - 9.5 2.65 .416
0.164 18.5 --
0.176 37.7 --
0.189 60.3 --
20-30 Ottawa sand 0.183 37.8 2.65 .48
0.189 59.9 --
Lowell sand 0.159 - 9.5 2.67 .420
0.188 19.7 --
0.188 60.9 ~-
Crushed quartz 0.190 60.9 2.65 .503
Crushed trap rock 0.193 59.9 2.97 .573
Crushed feldspar 0.190 59.3 2.56 .486
Crushed granite 0.161 -13.3 2.67 .429
0.174 19.4 --
0.189 60.9 --

 

(Adopted from van Wijk, et al., 1963 page 212)

*See Appendix III for conversion factors from metric

to British units.

15

-1

linearly from .16 i .01 cal cm.3 °C at -l8°C to .19 i

.01 cal cm.3 °C-1 at 60°C. Since the specific mass of

these minerals is about 2.7 g cm.3 an average value of
GS of about .46 holds for a mineral soil at 10°C.
The specific heat of soil organic matter was

determined by several authors (see van Wijk, et al., 1963),

the most probable value being .46 cal g'1°C-1 (extremes

1

are .42 and .48 cal g'1 °C- ). An average value of spe-

cific mass of the organic materials in soil is 1.3 g cm.3
and therefore, GS = .60 is a good average value in the
case of organic soils.

DeVries concludes that if the volume fractions of

soil minerals and of organic matter are denoted by xm and

x0 respectively, the heat capacity per unit volume.equals

3 1

C = .46 x + .60 x0 + xw cal cm C .

1.3.2 Thermal conductivity and
soil factors affecting it

 

Geiger (1965) defines the thermal conductivity k
as the amount of heat in calories that will flow through
a one cm. cube of substance in one second, when the tem-
perature difference between opposite faces is one degree
centigrade, and there are no variations in temperature.

In natural soils, k varies not only from place to
place, but also in one place as the water content of the

soil changes (van Wijk 1965).

16

1.3.2.1 Soil composition
and porosity

Von Schwarz and others (Baver 1965) have shown that
the heat conductivity of different soils follows the order:
sand > loam > clay > peat.

The degree of packing and porosity of the soil
seems to be the major factor determining the thermal trans-
fer. Smith and Byers (1938) arrived at the following
approximate expression for the thermal conductivity of a

dry soil,
k = kzp + k1 (l-p)

where k2 and k1 are the conductivities of dry air and of

the soil material respectively, and p is porosity. This
expression shows that the conductivity of the soil decreases
as the porosity increases. Skaggs and Smith (1968) con-
clude that the thermal conductivity of different soils can

be calculated in terms of soil porosity.

1.3.2.2 The influence of soil
moisture in thermal conductivity

Earlier investigations have shown that the heat
conductivity of soils and soil materials increases with
the moisture content. Patten (see Baver, 1963, pp. 376-
379) studied the conductivity of various soils at differ—
ent moisture contents. He found that the conductivity of

dry quartz particles, as well as of soil, was only about

l7

one-half to one-third that of water (.005 calorie per sq.
cm. per second per degree change in temperature gradient)
and about one-fifteenth to one-twentieth that of a solid
quartz block. Thus, the conductivity of quartz, for ex-
ample, is greatly decreased when it is divided into parti-
cles. The reduction in conductivity is due to the small
amount of surface contact between particles through which
heat will readily flow. The presence of water between the
particles increases the conductivity to values higher than
that of pure water. The presence of a water film at the
points of contact of the particles replaces the air which
is a poor heat conductor and thus improves the thermal
contact between the particles.

A typical example of Patten's curves is given in
Figure 1-4. It is seen that heat conductivity increases
with moisture content. It is also to be noted that the
greatest rate of increase in conductivity takes place at
the lower moisture contents. On the other hand, with fine
quartz powder and fine-textured soils, Patten's original
curves indicate that the greatest rate of change of heat
conductivity apparently occurred at the higher moisture
contents.

Also, it can be seen from Figure 1-4 that the
diffusivity a = EE increases rapidly at first with in-
creasing moisture to a maximum, and then decreases. This

is due primarily to a greater rise in conductivity at the

60 ~

55 P

 

 

 

 

18

 

10
5 I-
0 I 1 l I
10 20 30 40 50
Moisture Percentage
l = Heat conductivity, K, c.g.s. x 104 plotted against
H20 in per cent by wet weight
2 = Same as l but with H20 expressed in per cent by volume
3 = Diffusivity, K/C x 102, plotted against H20 in per
cent by wet weight
4 Same as 3 but with H20 expressed in per cent by volume
5 = Total porosity in per cent by volume
Figure 1-4. Heat conductivity and diffusivity in Coarse

quartz sand . (Redrawn from Bevar, 1965, p. 377.)

19

lower moisture contents, as compared with the increase in
heat capacity. As the moisture content becomes larger,
however, the value of c becomes greater and diffusivity
decreases. Thermal properties of some typical soils and
the common soil constituents are given in Tables 1-4 and

1-5.

1.4 General Differential Equation
for the Temperature Field

The basic law which quantitatively defines heat
conduction is generally attributed to the French mathema-
tician Jean Fourier (1768-1830). With reference to

Figure 1-5

 

Figure 1-5. Linear heat conduction.

the one dimensional form of the Fourier law states that the
quantity of heat dQ conducted in the x-direction of a homo-

genous solid in time dt is a product of the conducting

20

.mNHa: gmNpHNm

ou oHNuoE Eonm mNouomm :OHmNo>coo Now HHH wanomm< meme

HmomH .Hth am> Eoumv

 

 

 

o.o NN.m om.H Na. N.H N. N.
H.o Nm.m mm.H Nm. N. N. N.

N.m No.4 NH.H NH. NH. N. o. Beam
N.NH HN.HN NN.m N. N.N o. N.
N.NH N.NN o.m m. N.N o. N.

e.N N N m. o. c. o. NmHu
0.4H om.om N.N N. N.m o. N.
N.mH o.mm N.N m. N.N o. N.

o.N mm.m NN.N m. N. o. o. eemm

NN.©N om.HH me. N.m Hooov ouN

cow NON mooo. co. NH<

Ne.m NN.H oo.H NN.H page:

e H o. c. Nopume Uchmuo

NN.Nm Nm.NH Ne. N mHeNosz NmHu

o.mNH mN.NN NN.OH HN Nppmso

mmewmw m-OH x_lmw m-OH x wmm HoovmeU AvoVomm so mmwwme Noumz oocmpmHSm
So N N Hmo HmoHHHHE mo mGOHuomNm oasHo>
a a a U 2 mx 3x

 

 

«.HUOOH New muemspHpmaou
HHOm coEEoo may mo paw mHHOm Hmonxw osom mo mothomoum HmENoae

.q-H oHHMH

21

.Oo>HNoO ohm whoauo map mmhmuosmawm somoco prcowmomowcH ohm pomzoO pm: One
.ucoucoo ousumHoE .pmo: onHoomm oHNuoESHo> .>HH>Huo:O:oo Hweuone "muoposmnmm««

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.HO .m .ch HmHsouHo >0>Nsm HmonoHooO Echmv

 

v.NO
O.H

NHOO.
OO.H
N.H

OOH

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Nu

N.Om OOH

mo
mo
m.mm
v.Om
NHO. mOO.
me. mm.
v.m m.N

OOH
O.H

OOO.
mm.
O.v

NHH NHH
w.H O.H

O.HN
OOO. NHO.
um. Ne.
v.m O.m

How so

Hem nHO zuHmcoO p02

Hmso hon EwO kuHmcoO pm:
HoEDHo> kn uqoo

-NomO unopcoo ounpmHoz
HucmHoz HMO “coo

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uanoz we: pcoo

-Nomv pcognoo ousumHoz
Hmso you HmuO ossHo>

.ans you new; unopmH
Hoom you NEUO

NpHsHmsmmNe HNEN0;N
five mEo Hog Hmov

“mo: onHoomm oHNNoesHo>
HOo oom Eu Non HmoEO

kuH>Huoswaoo Heapoge

 

U03N£H

:oNohm Oozmgh

:oNonm

 

Oaxaca coNon

 

 

ooH\Noumz

Hucoonmm moO HmHu

Hpcmuemm NNH-NHO NHHm

««NouoEmNmm

 

 

¢.Hflom GONOHMHS fifim GONOHM

mo moHuNomonm HwENoHp mo :omHNmmEou .m-H oHan

22

area A normal to the flow path x, the temperature gradient
%; along the path, the thermal conductivity k.

Expressed analytically:

d _ dT ’

35%- - kAa-i- [1.1]

The Equation [1.1] defines the transient heat flow
in a linear conductor, and from this is derived the general
partial differential equation satisfied by the transient
temperature field in a three-dimensional volume.

Schneider (1955) derives the equation for three-

dimensional and obtains:

finer-511%)+g§CkT§§3+g§CkT§§3+é=cp§§ [1.21
where O = rate of generation or utilization of energy per
unit volume.

The most general partial-differential equation for
heat conduction is the same form, but with kT, c, p, and O
replaced by k(x,y,z,T), c(x,y,z,T), p(x,y,z,T) and
O(x,y,z,T,t) to include spatial and temporal as well as

temperature dependence.

1.5 Daily and Seasonal Changes
in Soil Temperature

The movement of heat into or out of the soil depends
upon the difference in temperature between the surface of

the soil and lower layer of the soil, i.e., the soil

23

temperature gradient. Much information regarding the tem-
perature regime of the soil can be obtained by solving the
differential equation of heat flow for the soil.

For a homogenous and isotropic 5011, assuming no
significant generation or utilization of heat occurs in the

soil, equation [1.2] can be written (Merva 1970)

32T 32T 32 _

+ + -
311' 357’ 327

[1.3]

9 he
wlo:
H-I-I

where:
T = temperature
x,y,z = Cartesian coordinates, 2 increasing downward
q = thermal diffusivity = k/pc
t = time
With the further assumption that no gradient of tem-
perature exists in the x,y directions, equation [1.3]

reduces to

2

8 1 8T
——7 = —'—- p [1.4]
82 a at

The solution of equation [1.4] will describe the
behavior of the temperature in the soil mass.

Merva (1970) solves this equation with the following
initial and boundary conditions:

T(z,t) = some quasi-steady state condition (a)

T(o,t) = a known periodic function of time (b)

T (w,t) = a finite value (c)

24

and obtains:

: : /nn 2

- at

= Znnt _ nn
T(z,t) A0 + n=le [En COS( E?~z)

 

T

~ 2nnt nn
+ bn 51n C-FF_" /E?'z)] [1.5]

where:

r is the maximum period of time for which the solution is
valid. For a diurnal temperature variation T is 24 hr. or
86,400 sec. while for the annual temperature variation, 1 =
365 days.

2 = depth of 5011.

A0, an, and bn are Fourier series coefficients that can be

obtained from

 

 

T
_ 1
A0 - 1F] f(t)dt . [1.63]
o
T 2
_ 2 nwt.
an - ?’/ f(t) cos 't dt [1.6b]
o
T
_ 2 a Znnt
bn _ TEff“) s1n 1: d1; [1.6c]
o

where f(t) is the surface temperature. In many cases the

surface temperature data is available from ESSA, Weather

Bureau.

25

For any soil A0 is the average temperature of the
soil surface for a period of time equal to one complete
cycle of the longest periodic component of the Fourier
Series solution, and in practice AO would be determined by
averaging temperatures over a length of time equal to the
length of record used to obtain for the Fourier series
representation. The interpretation of Equation [1.5] as

Merva states is:

The exponential multiplier therefore tells us to

what depth a fluctuation with a frequenc of 2nn/r will
enetrate. It is immediately apparent that the lowest
requency components of the Fourier Series expansion
influence temperature at greater depths than do high
frequency components. Thus, one would expect the

yearly temperature fluctuation to be detectable at

far greater depths than the daily fluctuations.

Also from the Equation [1.5] we note that the
argument of the sinusoidal fluctuation contains an
angular frequency which is a function of time, and a
phase shift which is a function of depth. The result
of the phase shift is that the time of occurrence of
She maximum or minimum temperature is shifted with

ept .

1.5.1 Damping depth

The depth to which fluctuation will penetrate to
soil mass before being reduced by a factor of e'1 is called
the damping depth and can be calculated (van Wijk et al.,
1963) from:

D = -—- [1.6]

26

The physical explanation of the damping and retarda-
tion of the temperature variation with depth lies in the
fact that a certain amount of heat is stored or released in
a layer when the temperature in that layer increases or
decreases respectively.

The damping depth depends on the period of tempera-
ture variation. It is /365 z 19 times larger for the annual
variation than for the diurnal variation in a given soil.
Table 1-6 presents damping depths for several values of

thermal diffusivities.

27

Table 1-6. Damping depths of temperature fluctuation
having various periods for several values
of thermal diffusivity.*

 

 

Damping Depth D (cm)

 

Dif2::?31ty Period of Fluctuation
in cmZ/sec hourly daily yearly
.001 1.07 5.24 100.19
.002 1.51 7.42 141.69
.003 1.85 9.08 173.54
.004 2.14 10.49 200.38
.005 2.39 11.73 224.03
.006 2.62 12.85 245.42
.007 2.83 13.87 265.08
.008 3.03 14.83 283.38
.009 3.21 15.73 300.57
.010 3.39 16.58 316.83
.011 3.55 17.39 0 332.30
.012 3.71 18.17 347.07
.013 3.86 18.91 361.24
.014 4.01 19.62 374.88
.015 4.15 20.31 388.04
.016 4.28 20.98 400.76
.017 4.41 21.62 413.10
.018 4.54 22.25 425.07
.019 4.67 22.86 436.72
.020 4.79 23.45 448.07

 

*See Appendix III for conversion factors from
metric to British units.

CHAPTER II

DEFINITION OF THE PROBLEM

2.1 Background

Drain tiles in common use are clay, concrete or
plastic. A well-designed drainage system requires high-
quality material as well as proper installation to insure
a long-lasting drain for the more severe exposure condi-
tions that are likely to be encountered in ordinary farm

drainage.

2.1.1 Tile for over-size trenches

Many tests have been made (Manson, in Luthin, 1957)
to determine experimentally and theoretically the loads to
which drain tile are subjected in service. These tests
look to design of tile systems which will avoid cracking
of the tile from overloading after installation. The effect
of numerous factors have been studied, such as soil type,
depth of cover, width of trench, and bedding conditions of
the tile as laid. Based on these studies, tables (for
example see Luthin, p. 316 or Standards for Drainage, p. 34)
have been prepared to show safe allowable depths of trench
for drain tile of different diameters.

28

29

2.1.2 clay tile and frost action

The frost resistance of clay products is largely
dependent on the quality and handling of the raw clay or
shale previous to, and during burning.

In order to obtain factual information regarding
the durability of clay tile under actual service condi-
tions in a cold region, Miller and Manson (Luthin, 1957,
pp. 321-322) dug up some tile which had been installed
for an average of 33 years and concluded that:

1. The shale tile examined were all in first-class
condition even where the depth of cover averaged
but 1.7 feet.

2. Many of the surface clay tile were in poor condi-
tion where the depth of cover was 2.00 feet and
less.

Frost penetration is an important factor in tile
line freezing. There are a number of factors other than
depth of penetration that influence the effeCts of frost
action on drain tile. Manson summarizes them as:

1. Quantity and source of water carried by the tile

during cold weather.

Vegetative cover.

Physical condition of the soil cover--fa11 plowed,
etc.

Depth and duration of snow cover.
Frequency and duration of winter temperatures.

U14:- CNN

Considerable attention should be paid to factor
one, especially in areas where good tile outlets are not
available and pump drainage is utilized, because tile lines
are often installed with minimum cover and submerged outlets

are used. Thus a large portion of the drainage system may

30

be submerged and close enough to the surface that severe
cold weather can cause freezing of the water within the
lines, with subsequent cracking of the buried conduct if
the pump is not operated during the winter period.

In this thesis, a study has been made to investi-
gate the temperature history around submerged drain tile
to determine the reasonable depth at which cold weather
will not result in freezing of the water inside the drain
tile. It is noted that the presence of water within the
tile is expected to inhibit the cooling action and thus the
freezing of the water. To determine if freezing will occur,
therefore, it is necessary to ascertain the penetration and
duration of the 32° front in soil at the desired location.
It is felt that if it can be demonstrated experimentally
that cracking does occur under certain temperature condi-
tions, and, if a similar set of conditions can be predicted
for a field location through selective modeling techniques,
then one could expect that tile damage due to cracking can
be anticipated for the selective location. In the present
study the location of interest is near Saginaw, Michigan,
where extensive areas of soil lay bare during the cold
months. These areas are in large part pump drained and the
drainage system pumps may not be operative during the winter

season.

31
2.2 Experimental Set-up

In order to ascertain the behavior of a crack result-
ing from freezing of the water inside a drain tile, an
experiment was conducted in the Physical Properties Labora-
tory at Michigan State University. The result of this
experiment led to a determination of realistic conditions

for solving the mathematical model of tile freezing.

2.2.1 Procedure

Two four-inch clay drain tiles were selected for
the experimental phase of the study. The ends were blocked
with wood sealed to the tile with silicon rubber sealer.
Four thermocouples located midway between the ends of the
tile were placed one inch apart on a diameter inside one
of the tiles, while a second experiment was performed
using only three thermocouples. When the silicon rubber
seal was dry, the tiles were soaked in water for 24 hours
and were then filled with 40°F. water.

A No. 40 Constantan wire was wrapped around the
second tile and the ends of wire were fixed with a nail
to the wood used to block the tile ends. The wire was
attached to a strain gauge amplifier indicator in order
to determine the exact time of cracking so that when the
tile cracked, the wire broke and the meter on the amplifier

tvent to full scale.

32

Both tiles were embedded in a box of dry soil and
additional thermocouples were located on each of four sides
of the tiles as shown in Figure 2-1. The whole system was
placed in a freezer with the average ambient temperature
maintained at 0°F. The thermocouples were connected to a
self-balancing potentiometer recorder. In addition the
temperatures of the thermocouples were checked with a
Millivolt potentiometer.

The instruments used were:

1. Strain gage amplifier from Daytronic, model 300 D,
with type 90 strain gage input model.'

2. Self-balancing potentiometer from Texas Instrument,
point Multiriter.

3. Millivalt potentiometer from Leeds 8 Northrup Company

Catalog No: 8686.

2.2.2 Observations

 

The temperature readings of thermocouples corre-
sponding to the points 1,2,3,4 are given in Figure 2-2.
The other experiment showed a similar behavior.

Since the tOp surface of tile was exposed to
ambient air with average temperature zero degrees F.,
points 1,2,3 cooled in this order respectively. Point 4
cooled faster than other points because the thermal dif-

fusivity of the soil was greater than that of water.

33

Thermocouples

 

4 inchVClay Drain Tile

    

 

 

 

Figure 2-1a. Location of thermocouples in the
experimental setup.

 

 

Figure 2-1b. A picture of the cracked tile showing
the general appearance. Note the thermo-
couple wires entering from the r1ght.

34

 

 

 

 

 

 

 

4:00PM-
Crack occurred at.
3:35 PM
-3:00PM-
~---- Point 1
— —.— Point 2
2:00PMF- Point 3
,1 ,/ _. .. Point 4
i’.’ I
3 :
o .
E 1:00PM - /
2% I
:1 E
I...
12:00PMr
11:00AM~
10:00MW-
9:00AM ‘

30 31 32 33 34 35 36 37 38 39 40 41 42
Temperature in °F

Figure 2-2. Temperature variation for each thermocouple.

35

When the temperature of point one reached 32°F. it
remained constant at this value. The temperature of points
2,3 decreased respectively until they reached 32°F. and
then they also remained constant. The temperature of
point 4 dropped below 32°F. When the temperature of all
of the water inside the drain tile became 32°F., the forma-
tion of ice started. Ice was formed in the upper portion
of the tile. The formation was crest-shaped and occupied
about one-third of the volume inside the tile at the time
of cracking. The shape of the ice and location of the
crack are shown in Figure 2-3. Thus, it has been demon-
strated that cracking of a clay tile due to freezing of
water within the tile is possible provided the tile system
remains at 32°F. for a sufficiently long duration of time.
It is probable that the condition indicated will occur if
the temperatures surrounding the tile drop to less than

32°F. on all sides.

2.3 Mathematical Model

2.3.1 General assumptions

 

Heat is transported in the soil mainly by conduc-
tion, i.e., the transfer of thermal energy on a molecular
scale. To facilitate the mathematical treatment the fol-

lowing general assumptions are made:

36

 

 

 

Figure 2-3. The shape of ice causing cracking of a
drain tile. The location of the crack is
shown in the picture at the lower right of
the tile.

1. The soil is almost saturated in the winter,
therefore, the thermal conductivity is uniform throughout
the soil mass. This assumption has a deficit in that the
thermal properties of frozen soil are different from those
of unfrozen soil. Consequently, when the soil starts
freezing there will be a moving boundary within the soil
mass across which the thermal properties differ with time
and temperature. Since we are interested only in a critical
temperature around a drain tile for simplification we assume

that the soil is homogenous and isotropic and therefore the

37

thermal properties are independent of space and time coor-
dinates. The thermal diffusivity of frozen soil is used in
the calculations since frozen soil has the higher value.
The resulting temperatures will thus be conservative.

2. The latent heat of fusion of water can be
ignored in the model during the phase change.

3. The medium is at rest, i.e., that no motion of
matter on a macroscopic scale occurs.

4. The heat flow is in the vertical and horizontal
directions only.

5. No heat is generated or utilized within the

soil medium.

2.3.2 Specific assumptions

The surface of the soil is subjected to a seasonal
variation in temperature. The temperature fluctuation is
sinusoidal and its daily and yearly fluctuation decreases
with depth. At the damping depth the temperature is almost
constant for a period of one cycle (see Sec. 1.5). The
damping depth is a function of the thermal diffusivity and
differs for different soils. The damping depth for daily
temperature fluctuations for most soils normally is less
than 12 inches (see Table 1-6).

The upper coordinate boundary for the present solu-
tion of the mathematical model is located at 13 inches

below the soil surface so that daily temperature

38

fluctuations can be ignored (see Figure 2-4). For a soil
having a different thermal diffusivity, this depth should
be calculated and the upper coordinate boundary chosen
accordingly.

It is assumed for the purposes of the model that
T(x,z,t) represents the temperature of the soil at depth
2 and distance x from the center of the tile at time t.
In the solution, for the upper boundary it is assumed that
one inch above the first x coordinate line the temperature
is constant at some value G. The assumption is necessary
to facilitate solving of the tridiagonal matrix associated
with the solution. In the computer solution it is possible
to change the value of G at every time increment, thus, a
slow temperature variation of the upper boundary is possible.

Since the soil is a semi-infinite body, it is
assumed that at infinite depths soil temperature remains
constant, i.e., T(x,m,t) = constant for each period of time
where m represents the maximum grid point, which in the
present problem is 68 inches. Since the soil temperature
at the yearly damping depth usually is nearly constant, it
is desirable that the value of m be equal or greater than
the yearly damping depth.

It is assumed that at the right hand side of the
soil system there exists a boundary beyond which the tem-
perature of the water in the tile will not effect the soil

temperature. This point is taken 68 inches from the center

39

 

 

8011 Surface‘*

j—t

 

 

    

.------....C..........f.'.d............'............

0 n o

. u

. u

o o

. a

. n

. o

. c

. o

. u

. o

. o

. u

. .

c n

. o

. o

. o

. c

1 . .

+ . -

OJ C H

X H .

........... —. .‘.

.J . o
x IIIIIIIIII JO‘IO

. .
II ....... ‘.~..

1 . .

. . .

. .

OJ C .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

O .

,
0 1 1
. +
.1 .1
Z Z
.1
Z

ll. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

m,m

 

Grid system.

Figure 2-4.

40

of the drain tile. Mathematically, the assumption can be
stated as T(m,z,t) = T(z,t), a known temperature at every
depth and time.

Due to symmetry, the vertical line passing through
the center of the tile is an insulated boundary, i.e.,
aT/ax = 0 for T(0,z,t).

Finally it was assumed that the tile line was sub-
merged by 32°F. water, and that the temperature in the tile
remains constant at 32°F. until a point one inch below the
tile line reaches 32°F. (see Sec. 2.2). Formation of ice
will begin from the coldest point and will extend downward
in the tile as was demonstrated by experiment. Because of
the complexity of solution for a two phase, moving boundary,
ice-water problem, the entire system is considered as a
single phase system. It is assumed the cracking will occur
when the temperature one inch below the bottOm of the tile
reaches 32°F. The assumption is justifiable since experi-
mental evidence shows only about one-third of the water
must turn to ice to initiate cracking of the tile. It is
thought that the time required to attain 32°F. at one inch
below the bottom of the tile is more than adequate to allow

for conversion of one-third of the water to ice.

41

2.3.3' Determination of initial
anddboundarygconditions

 

 

2.3.3.1 Determination of
soil temperature

The soil temperature readings to a depth of 36
inches in Miami fine sandy loam soil under fescue grass at
the East Lansing, Horticulture Farm were obtained from the
U.S. Weather Bureau Climatological records for 1963, the
coldest year of record at this station for the past 15
years.

The temperature of the soil for eleven days (from
February 21 until March 4, 1963) was chosen (Table 2-1)
and by interpolation the temperature for every inch of
soil depth was estimated and used as the daily temperature
data.

Because the annual average of soil temperature and
soil cover and soil characteristics differs from location
to location, the temperature reading of the East Lansing
Station can not be assumed to be representative of the
soil temperature for other locations. Since the recorded
temperature for an arbitrary location is not available,
the soil temperatures were estimated using a Fourier series
technique as follows.

The annual average soil temperature A0 is not equal
throughout the soil profile, it varies with depth. Tables

2-2 and 2-3 give annual average values of soil temperatures

.zmegon .mumm HmonoHouweHHO .swohzm Hogumoz .m.: Scum :oxmu ohm mama

.O ogon mo oonwom .mmmhw odomom.HoO:= EmoH szmm oaHm HEmHz

 

42

 

:oHpmHommNuxm
NN NN . mm mm NN NN NN NN NN NN NN NN menuaH NN
NN NN mm mm mm mm NN HN NN HN HN NN NoeucH NN
NN HN HN HN HN HN NN NN NN NN mm mm mogueH 4N
HN NN NN NN NN NN NN NN HN NN NN NN NoguaH NH
NN NN NN NN NN NN NN NN NN NN NN NN mogueH N
NN NN NN NN NN NN NN NN NN NN HN NN mogueH N
NN NN N H- NH N N- N NH N N H- .nHe .NENN
NN NN NN NN NN NH NH NN NN 4N NH HH .xee HHN
NNN NNN NNN HNN NNNN NNNN NNNN NNNN HNNN NNNN NNNN HNNN gamma

mxmm

 

 

.mcmH :H sham ohouH50HpNo: wchcwH pmmm
:H maumow uqonommHO Hm H.moO ouzpmyomEou HHON HHHHO wouhooom .H-N oHan

43

Table 2-2. Annual average of soil temperature (°F.) for
East Lansing Horticulture Farm Station.

 

 

 

Depth 1967 1968 1969 1970
Air 46.7 47.9 47.4 48.1
2 in. 53.3 53.2 53.2 52.4
4 in. 51.8 51.5 51.3 51.0
8 in. 50.1 49.6 49.9 50.1
20 in. 49.3 49.1 49.5 49.5
40 in. 49.3 49.2 49.4 49.5
80 in. 49.1 49.0 49.4 49.1

 

Data from U.S. Weather Bureau, Climatological Data
for Michigan.

Table 2-3. Annual average soil temperature (°F.) at
University of Minnesota, St. Paul Station.
Soil is under sod cover.

 

 

 

Depth 1963 1964 1965
Average air temp. 45.9 46.9 43.6
1 cm. 51.2 51.7 47.9

g S cm. 50.6 51.0- 47.2
10 cm. 49.8 50.2 46.4
20 cm. 48.2 48.7 45.4
40 cm. 47.9 48.0 45.0
80 cm. 47.2 48.2 45.1
120 cm. 47.5 48.2 45.6
160 cm. 47.7 -- 46.4
320 cm. 47.3 -- 46.9
480 cm.‘ 47.5 -- 47.1
640 cm. 48.0 -- 47.9
800 cm. 48.7 —- 47.7
960 cm. 48.0 48.2 47.8
1120 cm. 47.8 48.1 47.7
cm. 48.0 47.9 47.8

1280

 

for Minnesota.

Data from U.S. Weather Bureau, Climatological Data

44

for the East Lansing, Horticulture Farm Station from 1967-
1970 and annual average soil temperatures for different
depths and related annual air temperatures for the University
of Minnesota, St. Paul Station, respectively. The recorded
soil temperatures reveal that the soil surface has the
highest yearly average temperature.

Annual average soil temperature rapidly decreases
with increasing depth to a certain depth. Beyond this
depth gradually A0 increases as shown in Table 2-3. At
any depth the yearly average soil temperature is greater
than annual average air temperature for a given location.

Soil temperature is also highly affected by surface
cover. AC has a greater value for bare soil than for grass
or sod-covered soil as is shown in Figure 2-5 and Table 2-4.

As discussed above, since each depth of soil has a
different annual average temperature, the annual air tem-
peratures can not be used in a Fourier series to obtain the
boundary conditions to be substituted for the soil surface
temperatures to obtain a solution using Equation [1.5].
Indeed, if one analyzes data for the soil temperature, the
air temperature and the temperature one inch above the
ground, the results show that the annual average temperature
at one inch above the ground is greater than the annual air

temperature. This fact is illustrated in Figure 2-6.

H.coHpmum Hsmm .um .mwomoncHz mo kuHmHo>HGD .mumw HmonoHoumEHHu .HmomHO :mopsm
Hoaumoz .m.: Eoum :onp «puma .ohnumhomfiop HHom co ommHo>oo HHom mo uoommm .m-N oNSMHm

.ooO .>oz .uoo .mom .w:< HHSO mash. km: .HQ< .Hmz .nom .qu

T l 1 I d d- 4 I I I I J o

 

A

OH

     

, eunumpomEoH

LON
ommho>< Hmsqc<

 

1
O
L0

45

 

do exniexedmal

mangmuomsoh
HH< ommpo><

 

ON
mmmho Nova:

 

 

HHom ohmm News:

 

Table 2-4. Effect of soil coverage on yearly average soil

temperature (°F.) at different depths.

 

 

under bare

under soybean

 

Depth under sod soil covered soil

Air 45.9°F.

1 centimeter 51.2 55.8 53.1

5 centimeters 50.6 55.8 51.6
10 centimeters 49.8 53.0 50.8

20 centimeters 48.2 49.9 48.9
40 centimeters 47.9 48.1 47.2
80 centimeters 47.5 47.5 47.1

 

Data from U.S. Weather Bureau, 1963. Climatological
Data, University of Minnesota, St. Paul Station, Minnesota.

In order to calculate the soil temperature at dif-
ferent depths, a Fourier series was fit to the soil tempera-
tures at each depth as recorded at the East Lansing
Horticulture Farm. Such a technique gives a function which
describes the temperature fluctuation at each depth in the
soil for the above mentioned station. In order to obtain
the soil temperatures for another location in Michigan it
was postulated that the same general nature of temperature
fluctuation would be present; however, the mean soil tem-
peratures A0 for each depth would be different depending
upon whether the corresponding average annual air tempera-
tures were greater or less than the average annual air

temperature at the East Lansing location. Thus, the

47

.Hm< .Nmz .nom .cmO

.numow >9 mop: «HomEou HHom One NHN ommpo>m zchcoE mo :oHumHHm>

mo

OH

NOOH
.ooO .>oz

.uoo .mom .m:< >H3O .csw Hm: .HQ< .Hmz .nom .cmO

 

a

 

q

.\

t \‘0/

d

d d 1‘ J

‘-

ohspmhomEms

ohzumhomEoH
ommho>< Hmsc<

HH<-

d

 

4

 

 

 

o>onm

O:90Nm
.:H H

.N-N 6NsNHN

ON

om

OQ

om

O0

O5

OO

 

do einieiedmal

48

correct value of Ao must be determined. At the present time,
calculating A0 is difficult, because soil temperature is

recorded in only a few locations in every state, and the

relation between soil temperature and-air temperature has
not been determined. Therefore, to get an estimate of
annual average soil temperature at different depths, the

following procedure was applied.

The difference between the annual average air and
soil temperatures at selected depths was calculated from
Tables 2-2 and 2-3 and the results are given in Table 2-5
for the Minnesota and Michigan data.

Table 2-5. Difference between air temperature and soil

temperature at different depths in °F.

 

 

University of Minnesota East Lansing

 

 

St. Paul Horticulture Farm
Depth 1963 1964 1965 average 1968 1969 1970 average

cm. 6.3 5.8 4.3 5.5 -- —- -- --

cm. 4.7 3.6 4.1 5.3 5.8 4.3 5.1

10 cm. 3.9 2.8 3.3 3.6 3.8 2.9 3.4

20 cm. 2.3 1.8 2.0 ' 1.9 2.5 2.0 2.1

40 cm. 2.0 1.4 1.5 -- -- -- --

50 cm. - -- -- -- 1.2 2.1 1.4 1.6

80 cm. 1.3 1.3 1.5 1.4 -- -- -- --

100 cm. - -- -- -- 1.3 2.0 1.4 1.6
120 cm. 1.6 1.3 2.0 1.6 -- -- -- --
200 cm. - -- -- -- 1.8 2.0 1.0 1.6

 

49

As can be seen from Table 2-5, the annual average
soil temperature at 8 inches is about two degrees F. higher
than air temperature for any given year, and between 20-80
inches depth the difference reduces to 1.5 degrees F. It
is worthwhile to note that this difference depends on soil
coverage as well as kind of soil, and soil moisture content.
For example, for bare soil the variation is greater than
sod-covered soil (Table 2-4).

Based upon the above reasoning it is felt that
adding two F. degrees to the annual average air tempera-
ture, will with good accuracy yield an estimate of the
annual average soil temperature at 8 inches for a soil that
has a sod cover. Similarly, adding 1.51h degrees to the
annual average air temperature will give an estimate of the
average annual soil temperature between 20-80 inches depth.

The temperature fluctuations for every soil depth
are dependent upon weather conditions, surface coverage,
type of soil and soil moisture content. It is assumed in
the present study that if the daily air temperature varia-
tion of a region has a similar pattern in comparison with
another region, then the soil temperature fluctuation for
the two regions with the same kind of soil will also have
a similar shape.

The daily air temperature variations for East
Lansing, Saginaw and Evart (Central Lower Michigan) for

1968 are given in Figure 2-7. In Figure 2-8 a similar

H.2meHOHz .mumO HNOHmoHoumEHHo swohsm Hosumoz .m.: EONH coxmu apnea .OOOH

 

 

 

 

 

 

 

:H ppm>mm One .Bmchmm .qumcmH pmmm pom oysumhomsop NHN HHaucoE ommpo>m one .N-N oHSNHm
Hm HOMO: ON >9m39nom Hm zhmscmh
om mN ON mH OH m om mN ON mH OH . m Om mN ON mH OH
- q q - H d u I- H. H- - I-I. . T I H H OHI
O
upm>m N
n. zmchmm \ OH
5 l
9
.w
ONG
I
9
ON m.
mchcmH umwm N m
NHL.

O
U)

 

51

 

 

 

 

.mOOH Ham; map How smaHmmm Ocm mchcmH ummm mo oNSHNNoQEou NH< .O-N oNSNHm
Huge: L- thspnom NAHNHENO
m mN ON mH OH -fi mN ON mH OH m mN ON mH OH
d d 1 J a a W d a a “I u d-I I— JI CHI.
> . N-
/ 1 O
\\ 1 m
I I- - , m1
ZNCHmmm. \// \\ , . OHw
\
\\> \N 7 x x, . NH m
//< x .. ON m
7., \/ . N m
\ I .I.\ 40” .10
N I \ .d
NN /\ mchcmH ummm . mm
N -
N O...
N -me

 

 

O
U)

52

graph is given for the air temperature of East Lansing and
Saginaw for 1963. It can be seen from Figures 2—7 and 2-8
that air temperature of Saginaw has almost the same shape
as for East Lansing, except for a few days in February 1963
(from 14-25), which were much cooler.

Accordingly, the reasoning discussed above was
applied to calculate the soil temperature for Saginaw. To
obtain the temperature fluctuations, the Fourier coeffi-
cients were calculated for each depth at which the tempera-
ture was recorded and available. Since the recorded daily
soil temperature in East Lansing is not complete for 1963,
the recorded soil temperature at depths 8, 20, 40 and 80
inches of 1968 were used to calculate the Fourier coeffi-
cients. For accuracy the first 73 coefficients for 8, 20,
40 and 80 inches were calculated and the results are given
in Appendix I.

In the present study the temperature distribution
for Saginaw for 1963 at 8, 20, 40 and 80 inches depths were
estimated, based on average values of soil temperature as
calculated above, and soil temperature fluctuations which
are characteristic of Central Lower Michigan as determined
from the East Lansing data.

The annual average air temperature at Saginaw in
1963 was 45.5 degrees F. The annual average of soil tem-

perature at 8 inches was taken to be equal to 47.5 degrees F.

53

and at 20, 40 and 80 inches depths the calculated value of
AC was found to be 47.0°F.

The calculated Fourier coefficients for 8, 20, 40
and 80 inches and the assumed annual average value of soil
temperature estimated as above were used in the Fourier
series model to estimate the daily temperatures for the
above depths. The calculated temperatures for Saginaw are
given in Figure 2-9 and the recorded temperatures for East
Lansing in 1963 and 1968 at different depths are given in
Figures 2-10 and 2-11.

By interpolation, the temperature for every inch
of soil depth was estimated and used as the daily tempera-
ture data where such information was required as initial

condition to the computer solution.

2.3.3.2 Thermal diffusivity

Since the temperature readings in East Lansing are
obtained in fine sandy loam, a thermal diffusivity was
chosen equal to .006 cmz/sec and with interpolation a =
.009 cmz/sec is used for frozen soil (see Table 1-5). In
calculating the soil temperature for Saginaw, a = .009

2 . . .
cm /sec was used assum1ng a frozen s1lt loam $011.

.mOOH .chmem Now ohsumhomEou HHom OopmHsonu

ON

 

cope:

ON mH

mN ON

L H . .

 

 

Huangnom -

mH

Hm
Ahmscmh

ON

 

Om mN

b h

mH

’

 

OHM

.N-N opsNHN

OH

 

54

Agnew

‘I ‘

NNNNN .cH N

 

Npgoe .:H on

ON
.mN
.ON
-NN
.ON
-ON
.Om
.Hm
er
.mm
-vm
.mm
.om
.hm
-wm

 

-Om

go ainiexedmel

55

.coHpmum Ehmm oHSHHsoHuHo:

Hm

.mOOH
.mchamH ummm Now oNSHNNomEop HHom Oowhouom .OH-N onstm

Hm
thshnom - . Humzcmh

 

 

 

- seem:

 

NN NN NN NH NH

‘Em

 

ON
0w

«m

N ON mH OH m mm mN ON mH OH

 

 

ICIIIII

H ‘-

 

 

 

: L
fi' m
m M

“19.1

 

meow .:H «N -4 KN :E:» -I- < .0m 6

 

meow .cH cm L ..... H r. mm

A I
O 03
Q' N)

M
q.

 

.OOOH

Hm

m ON

ON
4

.coHpmpO ENmm ONDHHsquHo: .OchsmH pmmm Now oNSHNNomEou HHom OmONoooO

Ouemz

OH

I. Q. NHOSHnom

mH

xhwsch
mN

OH

.HH-N whszm

 

-I7./

N . ...-....u. N../L

Qmmfi 4.3.” ON III-III.-.-

momw .c

 

H

cede-0.0.0.. \n- I

OO

N) >.

....»<
N.....

ON mH

I‘II‘I I

.CH ON /

.,%/

-.N.../I |.N.:.

 

L

ON
ON
ON
NN
ON
ON

 

-Om

Nm
mm
«m
mm

om
mm
Om
Om
ON
HO
NO
ON

 

 

v

elnieieuma;

do

57

2.3.3.3 Depth and diameter
of drain tile

The most often used drain tile in drainage systems
are four, six, and eight inch tile. 'Depths vary depending
upon the type of soil and drainage conditions. The most-
recommended depth for drainage of Michigan soils is from
36 to 48 inches (Standards for Drainage of Michigan Soils,
1963). In the computer solution the tile boundaries are
delineated by grid points as shown in Figure 2-12. Solu-
tions obtained by applying the model using the boundary
and initial conditions determined above were examined for
four, six and eight inch tile for depth of 36 in., 42 in.,

and 48 in.

2.4 Solution of the Model

Based on the concepts advanced in preceeding
sections, the temperature distribution around a tile and
the cracking of a submerged tile due to freezing of the
water inside the drain can be studied using the following
approach.

The cross section of soil which is our problem
Space is covered with a grid as shown in Figure 2-4. The
left boundary of this grid passes through the center of the
tile. Since the solution is symmetrical about a vertical
line passing through the center of the tile, only the right

half of the solution need to be computed. The grid lines

58

 

 

 

 

 

 

 

 

1 3 4

1 r I J———’
31- I: 2. 33
32r- 33 ’
33- i 34-
34- 35'

V
35» 36 '
36- 37 ‘
37f "’( Four inch tile (48 in.
Six inch tile

(48 in. depth) depth)

 

I

29
30—
31w

32-

T

1“33

34

T

35

36

 

 

37w

 

Eight inch tile (48 in. depth)

Approximation of drain tiles in grid
system for computer model.

59

are spaced one inch apart and cover a square of 68-inch
width and depth. Grid points corresponding to the tile
boundary vary depending upon the diameter and depth of the

tile.

2.4.1 Formula and
method of solution

The general heat conduction equation is (see section

1.4):

a ’ 8T 3 8T 3 3T _ 8T
'a‘i(kT ‘53?) + fickr '5?) + Vickr "57) - DC "55 [2'1]

With the assumptions stated above, the equation can be

written:
32T+32T=1_B_T_ [2 2]
8x Oz a at
where:
T = Temperature

z,x = Cartesian coordinates, z increasing downward,
and x increasing horizontally from the center
of the tile to the right

Thermal diffusivity of the soil

Q
II

time

H
II

2.4.2 Numerical solution

The analytical solution of Equation [2.2] is com-

plicated. A numerical approximation to the solution of

60

Equation [2.2] may be obtained by the step-wise solution of
an associated difference equation. Three approaches are
possible. They are:

a. The explicit difference method, which yields
equatiOns that are simple to solve, but require an uneco-
nomically large number of time steps of limited size.

b. The implicit difference method, which yields
equations that do not possess a singularity to limit the
time step, but which require at each time step the solu-
tion by iteration of large sets of simultaneous equations.

c. The alternating-direction implicit method
(Cranahan, 1969) has been presented by Peaceman and
Rachford (1955), and Douglas (1955) and is used in this
paper.

The alternating-direction implicit procedure re-
quires the line-by-line solution of small sets of
simultaneous equations that can be solved by a direct,
non-iterative method.

Essentially, the principle is to employ two dif-
ference equations which are used in turn over successive
time-steps, each of duration At/Z. The first equation is .
implicit only in the x-direction and the second is implicit
only in the y-direction. Let T(i,j,n) represent T(z,x,t)
where Z = iAz and X = ij (Figure 2-4). Thus if Ti’j is

an intermediate value at the end of the first time-step

(see Appendix II for detail), we have:

61

*
-T.

+21 1* *
1-l,j (T + 3 Ti,‘ ’ T

J i+l.j

followed by:

-T

1 a t
3

i'lej J i+10j

where:

a = Thermal diffusivity;

At Time increment;

Ax

1
i,j-1,n+l * ZCX * 1) Ti,j,n+l ‘

_ 1
' Ti,j-1,n + 2(7 ‘ 1)
Ti.j.n * Ti.j+1.n [2'3]
Ti,j+1,n+l =
[2.4]
[2.5]

Distance-increment in the horizontal direction;

A2 = Depth increment in the vertical direction.

2(% + 1)

Let b

and f = 2c% - 1)

[2.6a]

[2.6b]

Equations [2.3] and [2.4] can then be written in the form:

t

at k
-T + bT. . - T

+ T

i-1.j 1.: 1+1.j = Ti.j-1.n

+ fT. .
1:3,“

i.j+l.n [2.7]

62

and

'Ti,j-l,n+l + bTi,j,n+l ‘ Ti,j+1,n+1 =

* f* 3‘:

Ti-1.j * Ti,j * Ti+l,j- [2.8]

In general the equation [2.7] is solved for the
*
intermediate values T, which are then used in Equation

[2.8], thus leading to the solution T. at the end of

19j9n+1
the whole time interval At. A tridiagonal coefficient

matrix is used in the solution of Equations.[2.7] and

[2.8] (Carnahan, et al., 1969, pp. 441-446).

2.4.3 Applying the boundary
and initial conditions in
solution of related fOrmulas

Based on the preceeding sections, the behavior of
soil surrounding a submerged tile will be predicted by

solving Equations-[2.7] and [2.8]:

9:

* 1:
'Ti-1.j * bTi.j ‘vTi+1.j = Ti.j-1.n * fTi.j.n

* Ti,j+l,n [2.7]

63

-T + bT - T

i,j-1,n+l i,j,n+1 i,j+1,n+1 =

*

* *
Ti-l,j * fTi,j * Ti+1,j N [2.8]

t
where T and T refer to temperature at the beginning and end
of half time step At/2; and with the following boundary and

initial conditions:

Time Temperature Condition

at t=0 and

at multiples T = known throughout the

of 24 hours soil mass
T = known 2 = -1 0 < x i m

at t > 0 T = known z = m 0 < x < m
T = constant x = m 0 > z 2 m
3T - 0 1 h 'd - 0
5;-- a ong t e 51 e x-0 s z < m

The temperature of water in the tile will be held constant
at 32°F., for t > 0, until a grid point one inch below the

tile line becomes 32°F.

64

Equation [2.7] was applied to each point i = 1,2,
---m-l in the j column and the following tridiagonal system

for the j column was obtained:

 

 

 

 

 

1 i" b'I‘ i“ d
= - + . - =
1 0.3 1.1 2.1 l
2 31": b'I‘ i" d
= - . + — . =
1 1.1 2.1 3.1 2
""""""""""""""""" [2.7a]
- T + bT T - d I
1 - 1 - 1'19j lej i+193 - 1
* b * - d
1 " "“2 ' Tm-3,j m-2,3 Tm-—1,3 ‘ m-z
* bT - d I
1 - m-l - Tm-2,j m-l,j m-l )
with
1‘ W
d0 = 2T0,1 + me0 + G for 1 = 0
1>for j = 0
d1 = 2Ti,l + fTi,0 for 1 - l,2,---m-l“
d0 - T0,j-1 + fT0,j + T0,j+l + G for 1 = 0
d1 = Ti,j-l * fTi,j + Ti,j+1 for 1 = 1,2,3---m-1 [2.7b]
for j = 1,2,---m-l
d0 = 2T0,m-1 + fTO, + G for i = 0
for j = m
d1 = 2Tim-l + fT1,m for i = l,2,---m-l

 

65

Comparing the coefficients of Equation [2.7a] with
coefficients of the tridiagonal matrix (see Carnahan, et al.,
1969, pp. 441-446) it can be seen that a = -1, b = b, c = -1
except for the first equation where a = 0, and for the last
equation where c = 0.

This procedure is valid except for points that have
a constant temperature (for example water in the tile). At
such locations the tridiagonal matrix must be modified.

This has been done through consultation with Dr. J. V. Beck,
Associate Professor of Mechanical Engineering, Michigan State
University, as follows.

When one of the unknowns in the tridiagonal system
becomes known and constant, the value of the b coefficient
is given a magnitude of unity and the other coefficients
are taken equal to zero in the equation. The known value
was substituted instead of the value d (value of the right
hand side of Equation [2.7a].

The other coefficients in the column of the tri-
diagonal matrix related to this.point will be zero. For
detail see sample computer program in Appendix II.

The procedure is repeated for successive columns
j = 0,1,---,n-l until all the Ti’j are found at the end of
the first half time-step. The temperatures at the end of
the second half time step are found similarly, by applying
EquatiOn [2.8] with related boundary and initial conditions,

to each point in a row (j = 0,1,---,n-1), for successive

rows (1 = 0,1,---,n-l).

CHAPTER III

3.1 Results

The results of this research have
been outlined in three headings:

3.1,1 Results of Experiments

An experiment was conducted in order to deter-

mine the realistic condition for solving the math-

ematical model of freezing water inside a drain

tile. The results of this experiment can be sum-

marized as follows:

1.

In the process of cooling the water inside the
drain tile there was a gradient between the
water and the ambient temperature and it was
greatest between the water and the upper sur-
face of the tile.

Water started to freeze when the temperature
throughout the water inside the tile reached
32° F., and the temperature surrounding the
tile was less than 32° F.

The conversion of only a portion of the water

to ice caused the cracking of the tile.

66

4.

67

The location of the crack was at the water—ice
interface on the interior tile surface in the

lower third of the tile (see Figure 2-3).

3.1.2 Results of Soil Temperature Study

1.

The annual average soil temperature is not
equal throughout the soil profile; it varies
with depth.

Annual average soil temperature rapidly de-
creases with increasing depth to a certain
depth. Beyond this depth gradually its value
decreases. At any depth the yearly average
soil temperature is greater than annual aver-
age air temperature for a given location.

The highest annual average temperature has
been observed at one inch above the ground of
given location.

Annual average temperature of bare soil at
certain depth is greater than sod-covered

soil (see Figure 3-1) and the frost penetra—
tion is deeper in bare 5011 than sod covered
soil.

In the winter the bare soil is much colder than
sod or grass covered soil. In the summer the
bare soil is warmer than soil under sod. This

fact is illustrated in Tables 3-1, 3—2.

68

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69

Table 3-1. Effect of soil cover on soil temperature
at different depths in the cold season.

 

 

 

 

 

January February
Ave e 'r T .
rag al emp 6.2°F 14.8°
Average Soil under under under under under under
tem at sod bare soybean sod bare soybean
p. soil soil

1 centimeter 24.5 9.9 18.2 22.0 19.7 19.8
5 centimeters 24.9 11.2 18.7 21.9 20.5 19.4
10 " 25.7 12.0 19.4 21.9 19.7 19.5
20 " 26.8 12.2 20.6 24.1 18.0 19.5
40 " 30.4 14.5 23.9 27.5 17.3 20.4
80 " 34.2 22.5 30.7 32.8 20.1 24.4

 

Source: Data are taken from Climatological Data,
University of Minnesota, St. Paul, 1963.

Table 3-2. Effect of soil cover on soil temperature at
different depths in the warm season.

 

 

 

 

 

June July
Average air Temp.

70.4°F 73.5
Average Soil under under under under under under

temp at sod bare soybean sod bare so bean
° soil soil Y

1 centimeter 73.5 89. 89.1 80.6 94.6 84.8
5 centimeters 72.5 87.1 82.7 79.3 93.1 79.9
10 " 70.5 81.5 79.4 76.8 87.8 78.6
20 " 66.5 74.0 73.3 72.6 82.0 75.8
40 ” 62.3 68.5 66.2 68.2 76.9 71.8
80 ” 57.5 61.8 59.8 63.6 71.5 67.0

 

Source: Data are taken from Climatological Data,
University of Minnesota, St. Paul, 1963.

70

3,1,; Results of Computer Modeling

In order to investigate the effect of thermal en-
vironment on submerged drain tile 3 computer model was
developed. The calculated soil temperature for Saginaw,
Michigan for 1963 was used in the model, and the boundary
and initial conditions were based on the experimental re-
sults.

The results of the computer program modeling soil
temperature around four, six and eight inch submerged
drain tile buried at 36, 42 and 48 inch depths are given
in Figures 3-2, 3-3, and 3-4.

Figure 3-2 represents the temperature history of
the soil around four inch drain tile. It can be observed
that on February 9, the soil temperature at the 31 inches
depth was 32° F. and at 37 inches depth the temperature
was 32.4° F. On this day for the purposes of the model,
water was assumed to have entered the drain tile at a
constant temperature equal to 32° F. The effect of water
within the four-inch drain tile on soil temperature has
been shown in Figure 3-5. The model results indicate that
the presence of water inside the drain tile inhibited rapid
cooling of the soil around the tile, and 7 days were re-
quired before soil one inch below the drain tile reached 32°
F. or less. On this day the soil one foot from the drain
tile at the 37 inch depth had a temperature equal to 31.54° F.

Since the ambient soil temperature was less than 32° F. it is

71

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32.
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44

Figure 3-5.

2 3 4 5 6 7 8 9 10 11 12 13 14
Distance from Center of Tile in Inches

The effect of water within the four-inch drain
tile on soil temperature. (Isothermal lines are
shown in the figure.)

75

felt, in accordance with the experimental evidence, that
the water inside the drain tile would begin to freeze. At
a point one inch below the four inch drain tile buried at
42 inch and 48 inch depths the temperature reached 32° F.
or less on February 21 and on February 25, respectively.

One inch above the six inch drain tile the tempera-
ture was 32° F. on February 10, while one inch below the
tile the temperature was 32.22 degrees F. and on February
19 (after 9 days) one inch below the six inch tile the
temperature attained a value of 32° F. or less. On Feb-
ruary 22, and March 1, the temperature one inch below the
six inch tile buried at 42 and 48 inch depths reached 32° F.
or less, respectively.

Similar observations were noted for the eight inch
drain tile buried at 36, 42, and 48 inch depths except more
time was required to cool the eight inch tile. February 20
and February 23 and March 3 were the dates at which the temv
perature one inch below the drain tile became 32° F. or less
respectively. Other results are summarized in Table 313 and
can be stated as follows:

1. The tile which had been buried deeper, froze

later in the season than shallow-buried tile.

2. Tile which had less diameter froze faster than

big diameter tile.

76”

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77

3.2 Discussion of Results

The results of this research give a useful tool
for determining the behavior of water inside submerged
drain tile as it may relate to the cracking of clay tile.
This research investigates the variation of soil temper-
ature and its relation with air temperature as a means of
determining the soil temperature at every depth. As in-

dicated previously, the results are conservative, because:

1. The recorded soil temperatures for Saginaw
were not available, therefore the soil temper-
ature for Saginaw for 1963, the coldest year
on record, had to be calculated based on assump-
tions relating soil temperature and air temper-
ature, and the general nature of soil tempera-
ture fluctuations.

2. The latent heat of fusion of water and varia-
tion of thermal diffusivity of frozen and un-
frozen soil and also ice and water were not
taken into account due to the complexity of

the required model.

However, it is believed the results of the pres-
ent study are logical and in agreement with field observa-
tions. Figure 3-6 is a photograph of an 8 inch clay tile
removed from a field located near Saginaw. The tile had

been buried at 48 inches. Note that the position of the

78

cracks are in agreement with the general location of cracks
noted in the experimental investigation. Also, the fact

that the model suggests that freezing can occur in such

tile buried at 48 inches gives support to the hypothesis

that cracking of this tile resulted from freezing of water
since our model indicates that for 1963 at Saginaw, Michigan,
especially for bare soil, the soil temperature up to 48 inch
depth had reached 32° F. or less, and cracking due to freez~

ing could result.

 

Figure 3-6. A photograph of an 8 inch clay tile removed
from a field located near Saginaw, Michigan.

Compare the location of cracks with that
shown in Figure 2-3.

CHAPTER IV

CONCLUSIONS

The following conclusions are based on the inves—

tigation conducted on soil temperature and effect of ther-

mal environment on submerged drainage conduits.

l.

The annual average soil temperature is not
equal throughout the soil profile; it varies
with depth. Annual average soil temperature
is always greater than annual average air
temperature.

When the temperature of soil surrounding a
submerged drain tile is reduced to less than
32° F. for some period of time, the water in-
side the drain tile will freeze and as a result
of the phenomena when about 1/3 of the water
has been changed to ice the clay drain tile
will crack. The primary crack will occur at
the water-ice interface on the interior tile
surface, which usually is located in the lower

half of the tile.

79

80

The tile which has been buried deeper will
freeze later in the season than shallow—

buried tile.

Tile of less diameter will freeze faster than
big diameter tile.

The time which is required to freeze the water
inside the drain tile in order to crack the

clay tile depends upon the temperature gradient
between water and ambient. When the temperature
gradient increases the water will freeze faster

than with a low temperature gradient.

CHAPTER V

RECOMMENDATIONS FOR FURTHER STUDIES

1. This study is the first step in evaluating
the effect of environment temperature and frost penetra—
tion on submerged drain tile. Further research is needed
in order to utilize the result of this study in developing
a complete model for predicting soil temperature around
drain tile.

2. Soil temperature reading should be extended
for more locations in every state under different soil
cover, such as bare soil, grass covered soil, and sod
covered soil. 4

3. The process of data recording should be accom-
panied by the measurement of the solar radiation, which is
a governing factor in increasing the soil surface temper-
ature. Solar radiation data are required in studying the
relationship between surface and air temperature.

4. The soil temperature measurement should be ac-
companied by the measurement of the moisture content of
soil. Variation of soil moisture is required in the esti-
mation of thermal properties of soil as well as the study

of the migration of moisture in the soil.

81

82

5. Investigation should be continued to complete
the mathematical model in order to determine the soil tem-
perature at every depth with respect to soil surface tem-
perature or air temperature. This study should include
effect of the latent heat of fusion, and the variation of
thermal diffusivity in moving media (i.e., frozen and un-
frozen soil as well as water and ice) in determining the
soil temperature, and water inside the drain tile in differ-

ent seasons .

LIST OF REFERENCES

LIST OF REFERENCES

Barakat, H. Z. and J. A. Clark, 1966. On the solution of
the diffusion equation by numerical methods.

Journal of Heat Transfer, Transactions of ASME,
Series C. 88:421-427.

Baver, L. D., 1965. Soil physics. 3rd ed. John Wiley
and Sons, Inc., New York. 362-384. ,

Carnahan, B., H. A. Luther and J. O. Wilkes, 1969. Applied
numerical methods. John Wiley and Sons, Inc.,
New York. 429-761.

1963. Climatological data of Michigan. U.S. Department of
Commerce. Monthly Summaries.

1967-1970. Climatological data of Michigan. U.S. Depart-
ment of Commerce. Annual Summaries.

1963. Climatological data of Minnesota. U.S. Department
of Commerce. Monthly Summaries.

1963-1965. Climatological data of Minnesota. U.S. Departs
ment of Commerce. Annual Summaries.

1962-1963. Climatological data of Wisconsin. U.S. Depart-
ment of Commerce. Annual Summaries.

Douglas, J. Jr. 1955. On the numerical integration of
2 2
3 U + 3 u = 33 by implicit methods. J. Soc. Indust.
3X2 3y 3t
App. Math. 3:42-65.

 

Geiger, R. 1965. The climate near the ground. Harvard
University Press, Cambridge. 26-33.

Kavianpour, A. 1971. Analytical estimation of thermal
properties and variation of temperature in asphaltic
pavements. Ph.D. Thesis, Michigan State University,
East Lansing. 6-9.

83

84

Lachenbruch, A. H. 1970. Some estimates of the thermal
effects of a heated pipe line in permafrost.
Geological Survey Circular. 6-32.

Luthin, J. N. 1957. Drainage of agricultural lands.
American Society of Agronomy. Madison. 315-339.

Merva, G. E. 1970. Physical principles of the plant en-
vironment. Unpublished class notes. Agricultural
Engineering Dept., Michigan State University, East
Lansing. 95-121 and Appendix I.

Peaceman, D. W. and J. H. H. Rachfor. 1955. The numerical
solution of parabolic and ellyptic differential
equations. J. Soc. Indust. App. Math. 3:28-41.

1963. Recommended standards for drainage of Michigan soils.
Bulletin of Agricultural Experiment Station and
Cooperative Extension Service.

Schneider, P. J. 1955. Conduction heat transfer. Addison-
Wesley Publishing Company, Inc., Cambridge. 1-5.

Skaggs, R. W. and E. M. Smith. 1968. Apparent thermal con-

ductivity of soil as related to soil porosity.
Transactions of the ASAE. 11(4):504-507.

Smith, W. O. and H. G. Byers. 1938. The thermal conduc-
tivity of dry soils for certain of the great soil
groups. Soil Sci. Soc. Am. Prod. 3:13-19.

van Wijk, W. R., et a1. 1963. Physics of plant environ-
ment. North-holand Publishing Company. Amsterdam.
102-132 and 210-213.

van Wijk, W. R. 1965. Soil microclimate, its creation,
observation and modification. Meterological Mono-
graphs. 6(28):59-73.

APPENDICES

APPENDIX I

FOURIER SERIES MODELING

OF PERIODIC PHENOMENA

APPENDIX I

FOURIER SERIES MODELING

OF PERIODIC PHENOMENA

In engineering analysis and design it is often
necessary to work with environmental parameters which
change with time such as the temperature of the earth's
surface or the relative humidity within a plant canopy.

The parameters may vary continuously and the function

which describes the process, i.e., the mathematical model,
may not be known. As a first step in the description of
the process one has instead a sequence of data obtained
over some interval of time, for example, the series of
average daily temperatures for each day of the year. In
many cases, much information could be obtained about a
process from a continuous mathematical model. (Merva 1970).

For most environmental parameters (temperature,
relative humidity, partial pressure of C02’ etc.) directly
related to the plant environment, there exists a consider-
able amount of periodicity. The presence of periodicity
suggests that some type of periodic function could be con-
structed to fit the observed data. Such a function would
be useful in that it would serve to model the behavior of
these parameters. Fourier Series will furnish such a func-

tion.

85

86f

A Fourier series approximation of this function

can be written:

k

anCoswnt + Zn=1

f(t) = A0 + 2:: bnsinwnt (I-l)

l
where f(t) = T = temperature at some point of interest;

A an and bn are all constant scaler coefficients.

0’
It is necessary to determine the coefficients A0,
an and bn and the arguments wnt up to a value of k such
that the function is sufficiently accurate. To determine
the arguments wnt it is necessary only that as t varies
from zero to t where c is the interval of time over which
the model is desired,L the,argument must vary from zero to
some integer multiple of Zn. The necessary conditions are
fulfilled if (on = 3&1 .
Equation (I-l) for a function being modeled over

the interval t can be written:

f(t) = A0 + 2k 1 (ancos 3%1 t + bn sin Q31 t) (1-2)
n:
AO , an and bn can be obtained from
1 .
A0 = «E 7‘; f(t) dt (1-3)
t 2 H
an = g [0 f(t) Cos 1% t dt (1-4)
1: . 2 11
bn = t3- IO f(t) sm Ti} t dt (1-5)

The function f(t) can now be approximated to the
desired accuracy by performing the integrations indicated

in equations (1—3), (I-4), and (I-S). In most cases only

87

A0, a1, a2 and b1 and b2 must be found to obtain a reason-
able approximation to the function f(t). Determining more
coefficients will usually give a better representation of
f(t) although not necessarily a better model for the fluc-
tuations of the parameter being investigated. The accur-
acy will depend on what one uses for f(t) in the integra-
tion. -

The first 73 Fourier coefficients for 8, 20, 40, 80
inches depth of soil have been computed by equations I-3,
I-4, and I-5 and the results are given in Tables (I-l),

(I-Z), (1-3), and (1-4). (See section 2.3.3.1).

88

TABLE I-l

THE CALCULATED fOURlER COEFFICIENTS FOR 8 INCHES DEPTHS OF SOIL

N0 A COEFF. B COEFF. NO A COEFF. B COEFF.
1 “180552 “110084 38 '0041 0004
2 .451 1.179 39 .074 .296
3 1.018 .759 40 .134 .086
4 0163 10335 41 -0000 -0091
5 “.510 .278 42 .190 “.130
6 “0239 -0932 43 -0132 0101
7 .169 .224 44 .069 .240
8 “.767 .464 45 .257 “.420
9 0379 -0507 46 '0151 -0046

10 -9248 .420 47 '0479 0408

11 10626 “0266 48 0568 .016

12 “.972 0299 49 0147 -0169

13 '0316 “0113 50 0061 -0001

14 0519 0375 S1 -0148 “0104

15 “.098 “.418 52 .027 .022

16 ’0024 -0460 53 ’0017 -0094

17 -0397 0275 54 -0116 -0159

18 .013 “.004 55 “.128 .135

19 .261 .546 56 .201 .031

20 .357 “.517 57 “.121 .085

21 -0417 0020 58 0104 0009

22 -0345 . -0011 59 -0005 "0081

23 0097 -0008 60 -0046 .052

24 '0042 0234 61 0019 0017

25 .454 -0122 62 0312 -0105

26 “.282 “.222 63 “.149 “.094

27 0096 0090 64 -0240 -0054

28 .199 .285 65 .060 .182

29 “.357 “.361 66 .154 “.033

30 0180 0002 67 -0108 0075

31 -0185 0003 68 0080 -00Q8

32 0115 0098 69 -0037 .005

33 0048 “0109 70 0149 ‘0187

34 “.068 .065 71 “.142 .037

35 -0201 “0172 72 .-0248 -0217

36 “.108 .151 73 .039 .172

37 -0115 0119 '

89

TAEUEI-Q

THE CALCULATED FOURIER COEFFICIENTS FOR 20 INCHES DEPTHS OF SOIL

NO A COEFF. B COEFF. NO A COEFF. B COEFF.
1 -150583 -110904 38 ‘ 0007 0038
2 .293 1.043 39 .017 .262
3 0560 0884 40 -0054 0115
4 “.212 .856 41 “.001 “.096
5 “.614 “.103 42 .179 “.022
6 0089 “0704 43 -0015 -0033
7 0181 0475 44 0042 0167
8 -0683 0299 4S 0408 -0203
9 0387 -0257 46 ”0010 -0054

10 “.255 .252 47 “.305 “.014

11 1.106 .230 48 .268 .171

12 -0737 0030 49 0015 -0103

13 “.076 “.187 50 .041 .070

14 0389 0264 51 -0021 “0100

15 0047 -0273 52 0089 0122

16 0081 -0249 53 -0052 -0060

17 -0375 0104 54 0019 -0209

18 .037 “.035 55 “.003 .007

19 “.017 .458 56 .037 .059

20 0320 -0218 57 -0099 -0040

21 -0265 -0066 58 0028 '0036

22 -0036 -0143 59 .075 -0047

23 0041 0088 60 -0074 -0124

24 “.281 .184 61 “.037 .039

25 .290 .136 62 .068 .153

26 -0076 -0189 63 -0014 -0028

27 0084 0106 64 0045 '0054

28 -0080 0203 65 0031 0002

29 “0149 -0232 66 0052 0018

30 0164 0120 67 -0048 0045

31 “.074 .065 68 .037 .093

32 -0064 0192 69 ’0044 -0030

33 .051 .054 70 .114 “.038

34 0001 -0070 71 -0068 0046

35 “0043 -0175 72 0034 ‘0103

36 “.055 .123 73 ' .090 .011

37 -0092 0027

90

TABLE I-3

THE CALCULATED.FOURIER COEFFICIENTS FOR 40 INCHES DEPTHS 0F SOIL

NO A COEFF. B COEFF. NO A COEFF. B COEFF.
1 ”110044 “120678 38' -0004 -0017
2 .399 .750 39 .020 .094
3 .289 .763 40 “.104 “.006
4 -0330 0736 41 0098 -0059
5 “.421 “.174 42 “.004 .035
6 0343 -0452 43 -0063 ”0161
7 .077 .295 44 .034 .109
8 “.520 “.037 45 .158 “.044
9 0282 -0053 46 0001 0036

10 -0228 “0060 47 0027 -0147

11 .548 .416 48 “.059 .047

12 -0295 ”0232 49 0015 -0022

13 ' -0052 -0093 50 -0045 0051

14 .030 .301 51 .085 “.070

15 -0028 0006 52 -0005 -0001

16 .082 .026 53 .060 .001

17 -0180 ‘0082 54 0018 -0026

18 .067 “.115 55 .005 .032

19 ’0108 0029 56 1"’001'34 0020

20 0122 0108 S7 '0062 -0039

21 -0072 ‘0031 58 0025 -0013

22 0023 -0066 59 -0001 -0009

23 .019 .180 60 “.017 “.111

24 -0143 -0077 61 0036 -0043

25 -0014 0083 62 -0063 0027

26 0110 -0125 63 “0045 -0063

27 -0034 -0022 64 -0036 0073

28 “.097 “.081 65 “.040 .025

29 0114 0027 66 -0068 -0029

30 .010 .148 67 “.023 .008

31 “.092 .017 68 .010 “.017

32 -0045 0141 69 0004 0031

33 -0029 0017 70 0032 0024

34 0010 -0047 71 -0043 -0010

3S 0062 -0086 72 , 0033 0046

36 -0003 -0053 73 “0044 -0063

37 .023 “.025

I11 11.....-

 

91

TABLE I-h

THE CALCULATED,FOUR1ER COEFFICIENTS FOR 80 INCHES DEPTHS OF SOIL

N0 A COEFF. 8 COEFF. N0 A COEFF. B COEFF.
1 ”70894 ”120433 38' 0066 0010
2 0017 0747 39 0020 ”0036
3 ”0186 0615 40 ”0047 0000
4 ”0424 0398 41 0048 ”0004
S ”0426 ”0269 42 ”0045 0058
6 .323 “.221 43 “.030 “.025
7 ”0026 0206 44 ”0044 0066
8 ”0310 ”0014 45 ”0025 0072
9 .221 .040 46 .003 .070

10 ”0068 ”0042 47 0067 ”0039

11 0215 0306 48 ”0037 0053

12 ”0076 ”0208 49 0054 0013

13 ”0041 ”0166 50 ”0027 0027

14 “.122 .119 51 .016 “.002

15 .061 .067 52 .041 .023

16 ”0036 0102 53 0075 0020

17 “.045 ”0063 S4 ”0069 ”0067

18 0048 0086 55 0037 ”0084

19 ”0058 ”0017 S6 ”0070 ”0043

20 0081 0073 S7 ”0023 ”0018

21 ”0118 ”0026 S8 0088 ”0005

22 0015 0015 59 0042 ”0058

23 “.063 .194 60 .062 “.054

24 ”0021 ”0125 61 0029 ”0008

25 ”0004 0045 62 ”0007 ”0129

26 0127 0044 63 ”0017 ”0061

27 ”0043 ”0004 64 0015 0052

28 “.097 “.086 65 “.004 .060

29 .012 .095 66 .103 .006

30 0045 0010 67 0000 ”0027

31 0024 ”0021 68 0061 ”0058

32 0034 ”0047 69 ”0063 ”0033

33 ”0012 0007 70 ”0029 ”0030

34 ”0071 ”0100 71 ”0039 0020

35 ”0055 0012 72 ”0018 0010

36 0050 ”0071 73 ”0047 ”0037

37

”0017 ”0064

APPENDIX II

COMPUTER MODELING FOR STUDY OF
SOIL TEMPERATURE WITH

SUBMERGED DRAIN TILE

APPENDIX II

COMPUTER MODELING FOR STUDY OF
SOIL TEMPERATURE WITH

SUBMERGED DRAIN TILE

II.l Differential Equation

Unsteady heat conduction equation in the soil is

given by
aZT + BZT = 1 TL (11.1)
8x2 322 a at

Let T(i,j,n) represent T(z,x,t), where Z = iAz and x = ij
(Figure 2-4). The finite difference approximate derivative

of partial differential equation by the Taylor's expansion

 

 

 

is given
T. . -T. .

31 = 1,Jip+1 1.1 + 0(At) (II.Z.a)
3t At

2 - . . .
3_; = Ti.j'1;n 2T13J9n+T143+19n + O[(AX)2] (II.2.b)
3x (Ax)2

2 . . -2T. . + . .

§_l = 1‘1’J:n 1,1,n» 1*1zJ:n + O[(Az)2] (II.Z.c)
822 (A2)2

Ifihere O ( ) represents the discretization error.

For convenience let define

 

 

6: Tiaj,n = i0j'1LEEZTiaj;E+Ti:j+l:n (II-3-a)
(13:02
and
2 T. . -2T. . +T. .
62 Ti,j,n = 1-1iJLn 111)“ 1+1,J,n (11.3.b)
(A2)2

92

93

Essentially, the principle of the implicit alter«
nating-direction method (Cranahan, 1969) has been used in
this paper to employ two difference equations which are
used in turn over successive time-steps, each of duration
At/Z. The first equation is implicit only in the x-direc-
tion and the second is implicit only in the y-direction.
Thus, if Ti j is an intermediate value at the end of the

9

first time-step, the Equation (II-l) can be written

 

 

i. . - T- - 2 T. . Z T. . (11 4 a)
1’] 1,3,1}. = 6X1’J 62 1,3,n ° °
At/Z
followed by:
T i
i’j’n+1 iij = 2 i. . + 2 T. .
At/Z 5x 1:3 52 1,J,n+1 (II.4.b)

Written in full and rearranged, with x = z for simplicity,

the equations (2-3) and (2-4) can be obtained.

II.2 Expansion of the Formula and
Applying Related Boundary

and Initial Conditions

The Equations (2-7) and (2-8) with the related
boundary and initial conditions are used in computer model-
ing. The expansion of the Equation (2-7) is given in sec-
tion (1.2.4.3). The expansion of the Equation (2-8) in the

tridiagonal system for the 1 rows is as following.

94

 

j = l "Ti,0 + bTi,1 ‘ Ti,2 = d1
J = 3 —T 1 + bTi 2 - T1 3 = dz
__ ......... (II.5.a)
J = 3 “T1 J-l + bTi J - T1 3+1 = dj
j = m-Z 'Ti,m-3 + bTi,m-2 - Ti,m-1 = dm_2
j = m-l “Ti,m-1 + bTi,m‘1 = dm'l
with
* * .
d0 = T1,j + fT(o,J) + G
k 1: 1':
dj - Ti-1’j + fTi,j + Ti+1,j (II.5.b)
d T fT * *
_ = '_ ° + - -
m 1 1 1:3 1,3 + Ti+1,j + Ti,m

The coefficients of the tridiagonal matrix (see

Cranahan, et al., pp. 441-446) are:

a1 = O and ak = -l k 2,3,--e,m

bk = b k

1’32-"9m

cm = O and ck = -1 k = l,---,m-l (II.6)

2 (% + 1) (See equation 246a).

where b

95

When one or two or more points attain a constant
temperature the tridiagonal matrix should be modified.
This condition happens when the temperature inside the
drain tile remains constant for a period of time.

For simplicity let us assume that only one point,
for example point (9.5), has a known constant temperature,
say Tc'

In tridiagonal system for j = 5 column the equa«

tions are

. * * * ’

1 = 8 -T7,5 + bT8,5 ‘ T 9,5 =d8 (II.7.a)
* * * ’

i = 9 ‘T8 5 + ng 5 ' T10 5 =d9 (II.7.b)

. * * * ’

1 = 10 -T9,5 + leO,5 F T11,5 =d10 (II.7.C)

1:
Since the T9 5 is constant, the condition of (II-6) no longer
’

can be used in determining the coefficients of Equations

(II-7).
Equation (II,7.a) can be written
a * _ d’ *
‘T7,5 + bT8,5 ‘ 8 + T9,5 d8
hence c8 = 0; d8 = Tc+d8 = Tc + (T8,4+fT8,5+T8,6)

I

. . . * * _ * _
81m11arly 1n Equation (11.743 leO,S - Tll,5 - d10 + T9,5-d10

hence alo = 0; d8 = Tc+di = TC+(T10,4+fT10,5+T10,6)

and

The

96

finally the Equation (II-7.b) can be modified

equations of tridiagonal system for i = 9 row are

= 4 -T9,3 + bT9,4 — T9,5 = d4 (II.8.a)
= s -T9,4 + bT9,5 - 19,6 = d5 (11.8.b)
= 6 ‘T9,5 * bT9,6 ' T9,7 ’ d7 (II.8.c)

A similar procedure should apply to Equation (II-8).

2-11

For two or more constant temperatures (see Figure

) more equations are involved (see computer program).

11.3 List of Principal Variables

ProgramSymbol Definition

 

 

A,B,C,D coefficient vectors of tridiagonal matrix

DTAU

time step At

DX space increment, Ax (one inch)

F f = 2(1/A-1)

I,J raw and column subscripts, i,j

ICOUNT counter on the number of time—steps

IFREQ number of time-steps between successive
printings of temperatures

G constant temperature at one inch above x

coordinate

M

RATIO

TAU
TMAX

TPRIME

TRIDAG

TSTAR

ZERO

97

number of space increment (68)

A=a At/(Ax)2

matrix of temperatures T at each grid
point

Time, t.

Maximum time for successive runs (24 hr)
Vector for temporary storage of tempera«
tures computed by TRIDG. These values
T' are then placed in the appropriate
row of T or column T

Subroutine for solving a tridiagonal
system of simultaneous equations (see
Carnahan, g£_al., 1969)

Matrix of temperatures T at the end of
the first half timeastep

Constant temperature (32° F.)

Because of FORTRAN limitations all subscripts are

advanced by one, when they appear in the program; e.g.,

T T
m

1,0

m became T(2,l) and T(M + l, M + 1), respectively.

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APPENDIX III

CONVERSION FACTORS

APPENDIX III

CONVERSION FACTORS

Length:
12 in. = 1 ft = 30.48 cm
Mass:
1 kg = 1000 g = 2.205 1bIn
454 g = 1 lbm
l lbm = .45359 kg
Energy:
1 Btu = 252 cal
1 cal = 3.9657 x 10"3 Btu

1 hp—hr = 2545 Btu
Specific heat:
1 cal/g°C = .999346 Btu/1b°F
Thermal conductivity:
1 cal/sec-cm—°C = 242 Btu/hr—ft-°F = 5806 Btu/day-ft-°F
Thermal diffusivity:
1 cmZ/sec = 3.875 ftz/hr
Volumetric heat capacity:

1 cal/cm3-°C = 62.48 Btu/ft3—°F

108

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h‘ --. 0" ' ‘~— 4""- ’-04 o'.

HICHIGQN STATE UNIV.

LIBRQRIE

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