MODELING HEAT GENERATION DUE TO SOIL AMENDMENT
DURING SOIL SOLARIZATION
A Thesis
Presented to the faculty of the Department of Mechanical Engineering
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
in
Mechanical Engineering
by
Duff Ralston Harrold
SPRING
2013
© 2013
Duff Ralston Harrold
ALL RIGHTS RESERVED
ii
MODELING HEAT GENERATION DUE TO SOIL AMENDMENT
DURING SOIL SOLARIZATION
A Thesis
by
Duff Ralston Harrold
Approved by:
__________________________________, Committee Chair
Timothy Marbach
__________________________________, Second Reader
Dongmei Zhou
____________________________
Date
iii
Student: Duff Ralston Harrold
I certify that this student has met the requirements for format contained in the University format
manual, and that this thesis is suitable for shelving in the Library and credit is to be awarded for
the thesis.
__________________________, Graduate Coordinator ___________________
Akihiko Kumagai
Date
Department of Mechanical Engineering
iv
Abstract
of
MODELING HEAT GENERATION DUE TO SOIL AMENDMENT
DURING SOIL SOLARIZATION
by
Duff Ralston Harrold
Soil solarization is a non-chemical treatment of agricultural soil for the control of soil-borne pathogens and
pests. To be most effective, the solarization process is undertaken when ambient temperatures are highest,
often causing growers to lose the most productive time of year. Pre-solarization amendment of soil with
organic matter has been shown to raise peak soil temperatures and possibly shorten the time required for
effective soil treatment. Reliable predictive tools are necessary to characterize the solarization process and
to minimize the opportunity cost incurred by farmers due to growing season abbreviation but current
models do not accurately predict temperatures for soils with internal heat generation due the microbial
breakdown of the soil amendment. To address the need for a more robust model, a first-order source term
was developed in the course of this thesis to model the internal heat source during amended soil
solarization. This source term was then incorporated into an existing “soil only” model and validated
against data collected from amended soil field trials conducted at the Kearney Agricultural Center. The
expanded model outperformed both the existing stable-soil model and a constant source term model,
predicting daily peak temperatures to within 0.1˚C during the critical first week of solarization.
_______________________, Committee Chair
Timothy Marbach
_______________________
Date
v
ACKNOWLEDGEMENTS
There is a long list of people without whom this thesis would not have been completed:

Professor Jean VanderGheynst, whose support, ideas, suggestions, and seemingly
bottomless-pit of patience made the following chapters possible;

My parents, who always encouraged me to continue without ever asking, “so how
much longer are you going to be in school?” – thanks in particular to my mom who
never dissuaded me from daydreaming…and to my dad for making this particular
daydream even possible;

My Sacramento State Advisor Timothy Marbach, and ME department chair Sue
Holl who saved me from Civil Engineering (oh my!) and have always had my
educational/professional best interest at heart;

Professor Dongmei Zhou for enthusiastically trudging through this thesis, giving
thoughtful comments and even gently pointing out my bad grammar;

Jeffery Meyer, Joel Switzer, and Javier Gonzales-Rocha, who were my faithful coconspirators in scholarly crime and who provided many memorable moments of
inspiration and motivation even when I “just don’t want to learn anymore”;

Blanca Ruelas for showing me the true meaning of perseverance; and,

Professor Irwin Segel, my first true scientific mentor and who will always be my
educational hero – his particular genius in taking ordinary lecture points and turning
them into “aha!” moments would be commemorated in countless folksongs if I
knew the first thing about writing folksongs.
vi
TABLE OF CONTENTS
Page
List of Tables………………………………………………………………………………………………. viii
List of figures ................................................................................................................................................. ix
Nomenclature .................................................................................................................................................. x
Chapter
1
2
3
4
INTRODUCTION AND RESEARCH OBJECTIVES ........................................................................... 1
1.1
Background......................................................................................................................................1
1.2
Amendment enhanced solarization ..................................................................................................5
1.3
Research Objectives ........................................................................................................................6
MODEL DEVELOPMENT ..................................................................................................................... 8
2.1
Introduction .....................................................................................................................................8
2.2
Previous modeling work ..................................................................................................................8
2.3
Using the Marshall stable soil model ............................................................................................. 11
2.4
The amended soil model ................................................................................................................ 21
MODEL VALIDATION ....................................................................................................................... 26
3.1
Introduction ................................................................................................................................... 26
3.2
Materials and Methods .................................................................................................................. 27
3.3
Results ........................................................................................................................................... 34
3.4
Discussion...................................................................................................................................... 42
CONCLUSIONS ................................................................................................................................... 45
Appendix A.1 MATLAB Code for general soil model ............................................................................... 48
Appendix A.2 MATLAB pdepe function and Initial Conditions ................................................................ 57
Appendix A.3 Unmodified Marshall MATLAB code ................................................................................. 58
References ..................................................................................................................................................... 71
vii
LIST OF TABLES
Tables
Page
2.1: Governing equations for amended soil temperature model. ............................................................... 15
2.2: Summary of results for case 0 (Stable soil; source term, q = 0)........................................................ 18
2.3: Summary of results for case 1 (Amended soil; source term, q = 0) .................................................. 19
2.4: First-order source term parameters and variables ............................................................................... 25
3.1: Amended soil temperature profile simulation cases .......................................................................... 26
3.2: Site dependent model parameter values. ............................................................................................ 30
2.3: Summary of results for case 1 (source term, q = 0) .......................................................................... 35
3.3: Summary of results for case 2 (1˚ source term with Q’ CO2 = 300 J/g-CO2) ........................................ 38
3.4: Summary of results for case 3 (Constant source term, q = 10 W/m3) ............................................. 40
3.5: Mineralization Parameters .................................................................................................................. 41
3.6: Summary of Results for days 2-8 ....................................................................................................... 43
3.7: Summary of Results for entire solarization period ............................................................................. 43
viii
LIST OF FIGURES
Figures
Page
2.1: Energy balance for tarp and stable soil surface (adapted from [22]). .................................. 10
2.2: Case 0 predicted vs. measured temp. for soil-only at 12.7 cm depth................................... 18
2.3: Case 1 predicted vs. measured temp. for amended soil at 12.7 cm depth ............................ 19
2.4: Energy Balance for tarp and amended soil surface .............................................................. 21
2.3: Kearney field trial predicted and measured temperatures for amended plots ...................... 35
3.1: Optimization curve for the apparent CO2-linked heat yield ................................................ 37
3.2: Case 2 amended soil temperature profile (Predicted vs. Measured) .................................... 38
3.3: RMSE for solarization days 2-8 vs. constant source term, q .............................................. 39
3.4: Case 3 amended soil temperature profile (Predicted vs. Measured) .................................... 40
ix
NOMENCLATURE
Symbol
Description
Units
𝐴1
First order pre-exponential
days −1
𝐵
Constant B = ρdry Q CO2 k1 C0
W/m3
𝐶
Concentration of mineralizable carbon
g CO2-C/g-dry soil
𝐶0
Initial mineralizable carbon content
g CO2-C/g-dry soil
𝐶𝑟
Mineralizable carbon remaining in the soil at any time
g CO2-C/g-dry soil
𝐶𝑠
Soil volumetric heat capacity
J/m3 ˚C
𝐶𝑡
Tarp volumetric heat capacity
J/m3 ˚C
𝐶𝑈𝐸
Critical Use Exemption
𝐷
Constant D = ρdry k1 C0
𝑑𝑡
Tarp thickness
𝐸𝑎
Activation energy
𝑓𝑐,max
g − CO2 /m3 s
m
J/mol
Maximum fraction of tarp area with condensation
𝑘
Soil thermal conductivity
W /m˚C
𝑘1
First order rate constant
days −1
𝐾𝐴𝐶
Kearney Agricultural Center
𝑀𝐴𝐸
Mean Absolute Error
𝑀𝐵𝐸
Mean Bias Error
𝑚𝐶𝑂2
Molecular weight of carbon dioxide
𝑀𝑒𝐵𝑟
Methyl Bromide
g/mol
x
𝑚𝑂2
Molecular weight of oxygen
𝑂𝑀
Organic Matter
𝑞̇
g/mol
Volumetric heat generation rate
W/m3
Latent heat transfer due to evaporation
W/m2
𝑞𝑐,𝑠−𝑡
convection heat transfer from soil to tarp
W/m2
𝑞𝑐,𝑡−𝑎
convection heat transfer from tarp to ambient air
W m-2
𝑄𝐶𝑂2
𝐶𝑂2 -linked heat yield
J/g-CO2
′
𝑄𝐶𝑂
2
apparent 𝐶𝑂2 -linked heat yield
J/g-CO2
𝑞𝑔𝑒𝑛
rate of heat generation per unit mass of dry soil
W/g-dry soil
𝑞𝐻2 𝑂
Latent heat transfer due to drop-wise condensation
W/m2
𝑄𝑚
Total heat generation from mineralization
J/cm3
𝑄𝑂2
𝑂2 -linked heat yield
J/g-O2
𝑞𝑟,𝑠−𝑠𝑘𝑦
radiation heat transfer from soil to sky
W/m2
𝑞𝑟,𝑠−𝑡
radiation heat transfer from soil to tarp
W/m2
𝑞𝑟,𝑡−𝑠𝑘𝑦
radiation heat transfer from tarp to sky
W/m2
𝑞𝑠,𝑠𝑏
beam solar radiation absorbed by soil
W/m2
𝑞𝑠,𝑠𝑑
diffuse solar radiation absorbed by soil
W/m2
𝑞𝑠,𝑡𝑏
beam solar radiation absorbed by tarp
W/m2
𝑞𝑠,𝑡𝑑
diffuse solar radiation absorbed by tarp
W/m2
Simulated total heat generation
J/cm3
𝑞𝑐,𝑒
𝑄𝑠
xi
𝑅
Universal gas constant
J/mol ∙ ˚C
𝜌𝑑𝑟𝑦
Dry soil bulk density
g/cm3
𝜌𝑤𝑒𝑡
Wet soil bulk density
g/cm3
𝑅𝑀𝑆𝐸
𝑆𝑂𝑀
Root mean squared error
Soil organic matter
𝑡
time
𝑇
temperature
˚C
𝑇𝑐
weather station soil temperature
˚C
𝑇𝑚
Measured temperature
˚C
Maximum daily temperature
˚C
𝑇𝑠
Simulated temperature
˚C
𝑇𝑡
Tarp temperature
˚C
𝑈
Utilization fraction of mineralizable carbon
𝑤
Percent water content (dry basis)
𝑇𝑚𝑎𝑥
s / min / days
%
xii
1
1
INTRODUCTION AND RESEARCH OBJECTIVES
1.1
1.1.1
Background
Agricultural pests and pathogens
Among the challenges of agriculture, particularly large-scale commercial
monoculture, are the problems posed by plant disease, weeds and pests. Soil-borne
pathogens, nematodes, and invasive weeds can lead to significant decreases in crop
production. Historically, there have been two basic soil “disinfestation” approaches: the
application of chemical fumigants (from 1869) and heating by steam (from 1893) [1].
1.1.2
Chemical solutions
Methyl bromide (MeBr) is the most widely used chemical fumigant for control of
soil borne pests and is believed to be the largest anthropogenic source of atmospheric
MeBr [2]. It grew in popularity during most of the 20th century and continues to be used
today; however, it is also a potent agent of ozone depletion [3, 4].
This catalytic
destruction of stratospheric ozone has led an international agreement for the restriction of
production and use of MeBr [2]. Under the Montreal Protocol, MeBr use in the United
States was banned in 2005 and developing countries (e.g., Mexico) will ban its use by
2015; however, a critical use exemption (CUE) allows for the continued use if there are
no technically and economically feasible alternatives or substitutes available to the user
[5]. Chemical alternatives to MeBr, such as chloropicrin, have been developed but bring
with them their own environmental and health concerns [6].
2
1.1.3
Non-chemical alternatives
The use of steam to achieve this thermal deactivation was first demonstrated in
Germany in 1888 and then soon after employed on a commercial scale in the United
States [7]. It was noted that low-temperature steam treatment (60 – 70˚C) had distinct
advantages over the traditional high-temperature (100˚C) treatments. The biological
vacuum left by the latter led to high susceptibility of rapid pathogen re-infestation where
the lower temperature treatments left significant levels of pathogen antagonistic
thermophilic saprophytes. Also, soil exposed to high temperatures exhibited phytotoxic
levels of Manganese not present in untreated or low-temperature treated soils [7, 8].
1.1.4
The solarization alternative
Solarization, a hydrothermal process for disinfestation of soil [9], is a relatively
recent alternative to chemical fumigation. There have been a number of early attempts to
harness solar energy for pest control.
Hagan [10] used cellophane mulching (the
covering of the soil surface with organic or inorganic materials) during the growing
season to control nematodes. Grooshevoy [11] describes the use of solar radiation on
tobacco seed-bed soil in cold frames to kill chlamydospores of Thielaviopsis basicola.
However, solarization in its present form, as a pre-planting soil treatment utilizing clear
plastic mulch for controlling soil-borne pathogens, was first described in 1976 in Israel
[12] and has since been studied in over 60 countries throughout the world [13] including
the United States, Northern Africa, Spain, Italy, Japan, and many countries in the Middle
East.
3
1.1.4.1 Solarization advantages
Many of the advantages of using soil solarization in place of chemical
fumigants are immediately apparent. The elimination of health and safety issues as
well as the associated logistical complications (e.g., required public buffer zones for
application chemical application, air quality regulations, etc.) [14]. Solarization
poses no such public health threats and could even be used within urban areas.
Another clear advantage of solarization is the reduction of dependence on ozone
depleting chemicals such as methyl bromide. Implementation of this passive solar
heating system may also reduce material cost of soil treatment to the grower as well
as reducing the reliance of agriculture on fossil fuels.
1.1.4.2 Solarization disadvantages and limitations
The principle disadvantage of solarization is the requirement of growers to
solarize during the hottest part of the year.
For locations in the northern
hemisphere this would typically be in July/August and is prime growing season for
many commercial crops. This may cause an interruption of the growing season and
a resulting opportunity cost to the farmer; however, multiple season effects of a
single solarization treatment could help mitigate this challenge [15]. Control of soilborne pathogens for three successive years after solarization treatment has been
reported for a variety of crops [16, 17] possibly due to shifts microbial populations
and induced soil suppressiveness (i.e., resistance to pathogen growth) [13].
4
Geography presents another challenge to the widespread use of solarization.
Since pathogen lethality is highly temperature dependent, the most effective solar
treatments would be expected in latitudes with relatively high ambient
temperatures and favorable angles of incident solar radiation. Direct thermal killing
via solarization can only happen in certain regions at certain times of year; however,
it was realized early on that the rate of pathogen control was frequently higher than
could be explained by the level of lethality due to high temperature. These effects
have been reported in deep soil layers (where temperatures are low) as well as in
climatically marginal regions. The above-mentioned effects that allowed for
multiple-season benefits may also play a part in these sub-lethal effects of
solarization though the exact mechanisms are unclear [13]. Additionally, in regions
where lethal temperatures are not achievable, the combined use of solarization and
fumigants at very low doses have shown great promise[18] .
5
1.2
Amendment enhanced solarization
In order to address these limitations, efforts have been made to shorten the
required duration, and expand the geographic feasibility of treatment by increasing
the temperatures achieved during solarization. One such strategy includes the
addition of organic matter amendment to the soil prior to solarization.
This
increase in mineralizable carbon content results in a sharp increase in thermophilic
microbial growth and respiration accompanied by the generation of heat.
1.2.1
Advantages of amended solarization
It has been shown that relatively small increases in temperature can have a
disproportionate effect on the time necessary for inactivating pathogens [16] and
weed propagules [19].
Gamliel and Stapleton [20] reported an increase in
temperature (2-3˚C) in soils amended with chicken compost versus non-amended
soils during solarization, as well as increased crop yield, improved control of rootknot nematodes, and increased soil suppressiveness for pathogenic fungi and
bacteria.
Some of these benefits were attributed directly to the temperature
increase though complimentary effects of increased levels of beneficial thermophilic
microbes and release of volatile organic compounds (VOCs) were considered. The
increased temperature due to exothermic microbial degradation of organic matter
may also allow for treatment at deeper levels than afforded by solarization alone.
6
1.2.2
Potential challenges of amended solarization
Amendment of soil with organic matter can significantly increase solarization
soil temperatures, however during the stabilization process soils can become
phytotoxic (i.e., poisonous to plants) due to the evolution of the same (VOCs) that
may contribute to the enhanced effectiveness of amended solarization.
The
solarization treatment must be long enough to allow sufficient stabilization of the
soil and adequate dissipation of the VOCs prior to the planting of crops. However,
Simmons et al. [21] showed that by the end of 22 days of solarization, the remaining
biological activity did not produce sufficient levels of phytotoxic compounds to
significantly decrease seedling germination and growth compared to plants grown
in soil alone.
1.3
Research Objectives
Accurate modeling of the soil solarization process will be important in bringing this
technology to its full potential. With mounting environmental concern and pressure
on growers to transition to more sustainable agricultural practices, the need for
reliable models also increases.
These models would allow growers to predict
outcomes, plan for them, and therefore more easily adopt the new practice by
minimizing the impact of transition. The Marshall model developed for stable soil
[22] is one such tool but it does not take into account heat generated from the
breakdown of organic matter that may be incorporated into the soil prior to
solarization.
In order to accurately predict amended soil temperatures, the
development of a more general solarization model is needed, ideally accounting for
7
the quantity and type of amendment used. Therefore, the goals of this thesis
research are to:

Demonstrate the need for a soil amendment source term by using the
existing model to predict amended soil temperatures and comparing to field
trial data (chapter 2);

Develop a model for a soil amendment source term to predict the heat
generated during solarization (chapter 2);

Incorporate the source term into the solarization model resulting in a general
soil solarization model (chapter 2);

Validate the resulting general solarization model against field trial data
(Chapter 3)
8
2
2.1
MODEL DEVELOPMENT
Introduction
The modeling of heat transfer in soil has long been an active field of research
[23-25]. Important soil processes such as evaporation, water penetration, microbial
growth, pest emergence, plant growth, nutrient mineralization, anti-nutrient
formation and permafrost freezing/thawing are all functions of soil temperature.
With the growing emphasis on high efficiency and sustainable agricultural practices,
tools to predict soil dynamics have become even more important. Solarization is
one such area that would benefit greatly from continued model development.
2.2
Previous modeling work
Early soil modeling efforts focused solely on developing models for the
prediction of evaporative water loss in bare and cultivated soils [26, 27]. Though
these models took into account heat and mass transfer, the outputs were purely
surface water loss. The interest in determining soil temperature profiles grew after
several field studies [12, 28, 29] showed that the use of plastic mulch raised soil
temperatures, increased vegetable yields, decreased weed growth, and could be
used to control pathogens. The first temperature model applied to solarization, as
we know it today [30] used a 1-D soil conduction model for both covered and
uncovered plots and showed the strong combined effect of soil moisture and plastic
mulching. Chung and Horton [31] developed a 2-D coupled soil heat and water
transfer model for bare soil with partial surface mulch.
They found that for
intermittent ground covering of widths less than one meter, the tarp edge effects
9
were significant and a 2-D model accounting for the horizontal energy transfer was
necessary.
Marshall [22] developed a 1-D model using wide mulch (> 1.5 m) and
treating the soil as a semi-infinite solid, thus neglecting edge effects. This stable soil
model accounts for conduction in the soil; convection both above and below the
tarp; direct and diffuse solar radiation downward; radiation from the soil upwards;
and latent heat transfer below the tarp in the form of evaporation and condensation
(fig. 2.1). The model uses as inputs, soil properties, field moisture content, weather
data, and tarp properties and outputs soil temperature as a function of depth and
time.
Marshall was able to reliably predict temperature profiles at two California
locations (UC Davis and Kearney Agricultural Center, Parlier, CA) with distinct soil
types. For both locations, the root mean squared error (RMSE) comparing predicted
vs. measured temperatures was less than or equal to 1.25 ± 0.13˚C when modeled
without latent heat effects [22]. Surprisingly, it was reported that the model with
evaporation/condensation at the soil-tarp interface did not predict solarization soil
temperatures as well as that without latent heat transfer terms (i.e., q c,e and q H2 O ).
10
Figure 2.1: Energy balance for tarp and stable soil surface (adapted from [22]).
Refer to table 2.1 for variable definitions
11
2.3
Using the Marshall stable soil model
For the purposes of developing an amended soil solarization model, Simmons
et al. [21] conducted field trials during the summer of 2011 in Parlier, CA at the
Kearney Agricultural Center (KAC).
These trials compared the solarization
temperature profiles of non-amended, stable soil (SS), and soil amended with
undigested organic matter (AS). AS plots showed a substantial increased
temperature profile for all time points relative to the SS plots. This was attributed
to the heat generation associated with the decomposition of amended organic
matter. The effect was particularly notable within the first week of solarization.
Since the Marshall model was well-validated using data from KAC, it seemed to be
an appropriate starting point for the development of a more general model that
would include a soil heat generation source term.
To this end, the following preliminary simulations were performed:
1. Test case (Case 0): Validation of the Marshall model against SS field
trial data;
2. Base case (Case 1): Comparison of the stable soil model to AS field
trial data.
The expectations were that case 0 would perform with results similar to those
reported in Marshall (2012), and case 1 would show markedly poorer correlation
with measured data. These two cases together frame the performance goals of this
research: the formulation of a source term that would allow accurate prediction of
both AS and SS temperature profiles.
12
2.3.1
Setting up the model
1-D, non-steady conduction was assumed to be the sole mode of heat transfer
in the soil with generation. It is presented as equation 2.1 in table 2.1. The inputs
W
J
required were soil conductivity, k (m ˚C), volumetric heat capacity, CS (m3 ˚C), and a
W
source term, q̇ (m3 ), which was set to zero in the stable soil model.
Heat capacity was calculated using the soil component volumetric fractions
(i.e., sand, silt, clay, organic matter, and water) and their respective heat capacities
[22]. Since the heat capacity of air was negligible relative to that of the other soil
components, its contribution to soil heat capacity was not considered. Thermal
conductivity was calculated as described in Marshall [22].
Cs and k were considered to be temperature independent at depths greater
than 0.5 cm.
In the original Marshall model (figure 2.1), both values were
recalculated in the upper most soil layer as the moisture level changed due to
evaporation (moisture levels in the lower layers are considered to be constant). If
latent heat effects were neglected, as reflected in table 2.1 above and figure 2.4
below, C𝑠 and k were considered to be constant throughout. Heat generation is
discussed in section 2.4.
Boundary Conditions
The lower boundary (eq. 2.3) was defined as the depth at which soil
temperature remains unaffected by surface heat flux (i.e., no temperature gradient
exists). At the upper boundary (eq. 2.4), between the soil surface and the tarp,
seven heat transfer modes were identified; the sum of which, were equal to the soil
13
conduction. The tarp was assumed to accumulate no energy (eq. 2.5). These
conditions are summarized in table 2.1.
Convection
One component of the heat transfer between the tarp and ambient air was
convection (𝑞𝑐,𝑡−𝑎 ), which can be forced convection due to forced bulk flow of air
(i.e., wind) and/or natural convection due to buoyant forces (i.e., heated air rising
from tarp surface). Forced convection was modeled as fully turbulent flow across a
flat plate and natural convection used correlations for the upper surface of a heated
horizontal plate. The equations describing convection heat transfer are as described
in table 2.4 of Marshall [22] and were not modified for the current study.
Convection between the tarp and soil surface comprises sensible and latent
heat transfer. Sensible heat transfer (𝑞𝑐,𝑠−𝑡 ) was modeled as natural convection in a
horizontal rectangular cavity with bottom heating and the air gap between soil and
tarp was assumed to be saturated. Mass transfer through the tarp (water loss) was
assumed to be negligible. Equations describing the sensible heat transfer are as
described in table 2.5 of Marshall [22] and were not modified for the current study.
Marshall found that the model performed better without considering latent heat
transfer effects (i.e., evaporation (𝑞𝑐,𝑒 ), and condensation (𝑞𝐻2 𝑂 )) so for the current
study latent heat transfer was neglected. This was achieved in the simulation by
setting fc,max = 0 in table 3.2 presented below in chapter 3.
14
Radiation
Radiation heat transfer comprised long-wave radiation (λ ≥ 3 μm) from the
soil, tarp and sky (𝑞𝑟 ), and short-wave radiation (λ < 3 μm) from the sun (𝑞𝑠 ).
Objects near ambient temperature emit long-wave radiation and the equations
modeling this are as presented in table 2.7 of Marshall [22]. Solar radiation is
composed of beam radiation, which has not been scattered by the atmosphere and
diffuse, or scattered, radiation. Incident angles for beam radiation were determined
as described in table 2.8 of Marshall [22]. Available CIMIS weather data [32]
typically provide total solar radiation; however, for this model the separate
components were needed and resolved as summarized in table 2.9 of Marshall[22].
All radiation modeling was unchanged for the present study.
15
Table 2.1: Governing equations for amended soil temperature model.
Eq. #
Function
2.1
1-D heat
conduction
2.2
Initial
conditions
2.3
Boundary
condition
at x = 5 m
2.4
Boundary
condition
at x = 0 m
2.5
Energy
balance
on tarp
Equation
𝑘
𝜕 2 𝑇𝑠
𝜕𝑇𝑠
+
𝑞̇
=
𝐶
𝑠
𝜕𝑥 2
𝜕𝑡
𝑇𝑠 [(0 𝑚 ≤ 𝑥 ≤ 0.1 𝑚), 0 𝑠] = 𝑇𝑐 (0.15 𝑚, 0 𝑠) + 5˚𝐶
𝑇𝑠 [(0.1 𝑚 ≤ 𝑥 ≤ 0.14 𝑚), 0 𝑠] = 𝑇𝑚 (0.127 𝑚, 0 𝑠)
𝑇𝑠 [(0.127 𝑚 ≤ 𝑥 ≤ 0.2 𝑚), 0 𝑠] = 𝑇𝑐 (0.15 𝑚, 0 𝑠)
𝑇𝑠 [(𝑥 > 0.2 𝑚), 0 𝑠] = 𝑇̅𝑐 (0.15 𝑚)
−𝑘
−𝑘
𝐶𝑡 𝑑𝑡
𝜕𝑇𝑠 (5, 𝑡)
=0
𝜕𝑥
𝜕𝑇𝑠 (0,𝑡)
𝜕𝑥
= 𝑞𝑠,𝑠𝑏 + 𝑞𝑠,𝑠𝑑 − 𝑞𝑐,𝑠−𝑡 − 𝑞𝑟,𝑠−𝑡 − 𝑞𝑟,𝑠−𝑠𝑘𝑦
𝑑𝑇𝑡
= 𝑞𝑠,𝑡𝑏 + 𝑞𝑠,𝑡𝑑 + 𝑞𝑐,𝑠−𝑡 − 𝑞𝑐,𝑡−𝑎 + 𝑞𝑟,𝑠−𝑡 − 𝑞𝑟,𝑡−𝑠𝑘𝑦 = 0
𝜕𝑡
Cs, soil volumetric heat capacity (J m-3 ˚C-1);
Ct, tarp volumetric heat capacity (J m-3 ˚C -1);
dt, tarp thickness (m);
k, soil thermal conductivity (W m-1 ˚C -1);
𝐪̇ , volumetric heat generation rate (W m-3);
qc,s-t, convection heat transfer from soil to tarp (W m-2);
qc,t-a, convection heat transfer from tarp to ambient air (W m-2);
qr,s-sky, radiation heat transfer from soil to sky (W m-2);
qr,s-t, radiation heat transfer from soil to tarp (W m-2);
qr,t-sky, radiation heat transfer from tarp to sky (W m-2);
qs,sb, beam solar radiation absorbed by soil (W m-2);
qs,sd, diffuse solar radiation absorbed by soil (W m-2);
qs,tb, beam solar radiation absorbed by tarp (W m-2);
qs,td, diffuse solar radiation absorbed by tarp (W m-2);
Tc, weather station soil temperature (˚C);
Tm, measured soil temperature (˚C);
Ts, predicted soil temperature (˚C);
Tt, tarp temperature (˚C); x, soil depth (m).
t, time (s);
16
Table 2.1 summarizes the stable soil model as developed by Marshall [22]
with the following modifications:
1. The governing equation (2.1) reflects the addition of the source term, q̇ ,
which will be set to zero for the both the test and base cases;
2. Initial conditions specifications were altered to reflect the Simmons [21] field
trial temperature sensor configuration.
Marshall used two temperature
sensors for each test plot (@ 5 cm and 15 cm) where Simmons used a single
thermistor at 5 inches of depth (~12.7 cm). Both trials incorporated the
Kearney Agricultural Center (KAC) weather station sensor temperature data
collected for 15 cm of depth. The initial surface condition (x = 0 cm) was set
to 5˚C above the Parlier weather station soil temperature sensor (x = 15 cm)
as a reasonable estimate.
3. The boundary conditions and energy balance equation were simplified by
eliminating the latent heat effects (i.e. convective mass transfer due to
evaporation, q c,e, and condensed water dripping from the tarp surface back
to the soil q H2 O ) (ref. figure 2.1)
17
2.3.2
Testing the stable-soil model: preliminary simulations
Field data [21] comprise temperature measurements for soil-only and
amended microcosms recorded at 12.7 cm depth and at 10-minute intervals for 22
days. The average standard deviation for all measurements at a given time point
was approximately 0.57˚C. The predicted vs. measured temperature profiles were
analyzed but only days 2-8 were used for graphical comparison throughout this
study since: a) it is more visually informative than the full 22-day plot; and b) the
first week is considered to be the most critical, since peak temperatures for
amended soil would be expected within this interval. The point-by-point differences
in temperature were then used to calculate the errors in the predicted values for
both the soil only (table 2.2) and the amended soil cases (table 2.3). Maximum daily
temperature differences for days 2-8 were also compared for each case.
Case 0: Validation of the Marshall model against field trial data for stable soil
The stable soil model was used to predict soil-only plot temperatures and
visually showed good agreement (fig. 2.2) with measured temperature data. A
summary of results for the entire solarization period is given in table 2.2 below.
Case 1(base case): Application of the stable soil model to field trial amended soil
The stable-soil model was then used to predict the temperatures of soil
amended with 10% added organic matter (compost and wheat bran) and, as
expected, the model showed significantly poorer visual correlation to measured
values (figure 2.3) than for that of the test case. This effect is presumably due to
18
microbial degradation of organic matter and associated generation of heat. Table
2.3 summarizes the results for the base case.
Predicted
Measured
44
soil only temperature (oC)
42
40
38
36
34
32
30
28
1
2
3
4
5
6
7
time (d)
Figure 2.2: Case 0 predicted vs. measured temp. for soil-only at 12.7 cm depth
(note: The first day of solarization is designated as “day 0”)
Table 2.2: Summary of results for case 0 (Stable soil; source term, 𝑞̇ = 0)
Days 2-8
Days 9-22
Days 2-22
1.8
1.6
1.3
RMSE (˚C)
1.3
1.3
1.2
MAE (˚C)
1.1
0.8
0.1
MBE (˚C)
1993
3000
1007
Number of points, N
43.4
Avg. Max Tm (˚C)
43.5
Avg. Max Ts (˚C)
+0.1
Avg. ΔTmax (˚C)
RMSE = root mean squared error, MBE = mean bias error, MAE = mean absolute error
8
19
Predicted
Measured
44
amended soil temperature (oC)
42
40
38
36
34
32
30
28
1
2
3
4
5
6
7
8
time (d)
Figure 2.3: Case 1 predicted vs. measured temp. for amended soil at 12.7 cm depth
Table 2.3: Summary of results for case 1 (Amended soil; source term, 𝑞̇ = 0)
Days 2-8
Days 9-22
Days 2-22
1.5
1.4
1.4
RMSE (˚C)
1.2
1.2
1.2
MAE (˚C)
1.1
0.4
-1.0
MBE (˚C)
1993
3000
1007
Number of points, N
44.1
Avg. Max Tm (˚C)
43.1
Avg. Max Ts (˚C)
-1.0
Avg. ΔTmax (˚C)
RMSE = root mean squared error, MBE = mean bias error, MAE = mean absolute error
20
2.3.3
Discussion
The root mean squared errors (RMSE), the mean absolute errors (MAE), and
the mean bias errors (MBE) fell within one standard deviation when comparing the
two cases, with the exception of the MBE for days 2-8. The soil only microcosms
showed a very slight average over-prediction (MBE = + 0.1˚C), while the amended
microcosms (MBE = -1.0) showed a nearly 1˚C under-prediction for the initial 7-day
period. All absolute values were within 2˚C implying that the model performed
relatively well on average for both cases.
Since one of the critical factors in pathogen and weed seed inactivation is the
maximum temperature reached, a comparison of measured and predicted maximum
daily temperature may provide a more informative measure of model’s ability to
predict solarization effectiveness. The model predicted very well the soil-only
microcosm maximum temperatures for days 2-8, differing only by 0.1˚C on average.
The amended plots, on the other hand, showed an average under-prediction of
approximately 1˚C (43.1˚C vs. 44.1˚C). This difference was due to the heat added by
microbial action on amended soil, which is not accounted for in the stable soil
model; a properly modeled source term is necessary to accurately predict, not only
temperature profiles on average, but daily maximum temperatures as well.
21
2.4
The amended soil model
To date, there has been no published modeling work that includes the effects
of the addition of organic matter amendment on soil solarization. These effects are:

Altered soil heat capacity and conductivity due to changes in basic soil
properties (i.e., soil color, bulk density, porosity, water-holding
capacity, relative content of silt, sand, and clay);

Heat generation within soil boundaries associated with the microbial
decomposition of the organic soil amendment.
Figure 2.4 reflects the modifications made to the Marshall model (fig. 2.1)
Figure 2.4: Energy Balance for tarp and amended soil surface
22
2.4.1
Possible forms of source term
Heat generation from the degradation of soil organic matter (SOM) is the
result of microbial action and is mechanistically linked to the rate at which unstable
carbon-based compounds (i.e., cellulose, hemi-cellulose, lignin, etc.) are converted to
stable carbon molecules such as CO2. Therefore, mathematical modeling of energy
generation associated with microbial activity can be done indirectly by modeling
this carbon mineralization rate, specifically the rate at which CO2 is evolved from the
soil. Mineralization kinetics for sewage systems with variable microbial population
density have been successfully described using variety of models including first
order, and Monod kinetic models [33]. For the case of carbon mineralization in
compost-amended soil, a first-order model showed as good or better fit to validation
data than other models including second order and Monod kinetic models [34].
High-solid aerobic microbial degradation processes, such as compost stabilization,
have been estimated to produce 9500 kJ per kg of O2 consumed during
decomposition [35]. Since for every mole O2 consumed one mole of CO2 is evolved,
the estimated heat yield becomes 6909 kJ/kg CO2 evolved
23
2.4.2
Formulation of first order source term
The rate equation for mineralizable carbon in compost can be modeled as a
first order degradation:
𝑑𝐶
= −𝑘1 𝐶
𝑑𝑡
(2.6)
where C is the concentration of mineralizable carbon in grams CO2-C/gram dry soil
and k1 (s −1 ) is the first order mineralization rate constant. Assuming k1 is constant
over time, integration gives an expression for Cr (t), the mineralizable carbon
remaining in the soil at any time:
Cr (t) = C0 e−k1 t
(2.7)
where 𝐶0 is initial mineralizable carbon content (g CO2-C/g dw). The first order
rate constant, 𝑘 (s −1 ) is defined by equation 2.8:
𝑘1 = 𝐴1 𝑒𝑥𝑝 [−
𝐸𝑎
]
𝑅𝑇
(2.8)
where A1 is an experimentally determined pre-exponential (time−1), Ea is the
activation energy ( J ⋅ mole−1), R is the universal gas constant ( J ⋅ mole−1 ∙ K −1 ) and
T is the temperature (K) [36]. The parameter k1 is clearly temperature dependent
and so, in fact, would not be constant over time, especially near the surface where
the temperature fluctuations are the largest.
Equation 2.7 only holds if k1 is
constant with respect to time and therefore an estimated average temperature is
used in equation 2.8.
24
Differentiating equation 2.8 yields the degradation rate in terms of initial
concentration:
𝑑
𝐶 (𝑡) = −𝑘1 𝐶0 𝑒 −𝑘1 𝑡
𝑑𝑡 𝑟
(2.9)
Multiplying this result by the heat yield, QCO2 (J/g-CO2 evolved), gives the rate of
W
heat generation per unit mass of dry soil, q gen (
gdw
):
𝑞𝑔𝑒𝑛 = 𝑄𝐶𝑂2 𝑘1 𝐶0 𝑒 −𝑘1 𝑡
(2.10)
W
Multiplying by the dry-weight bulk density and letting B = ρdw QCO2 k1 C0 (m3 ) gives
W
the volumetric heat rate, q̇ (m3 ), represented in equation 2.11:
𝑞̇ = 𝜌𝑏 𝑞𝑔𝑒𝑛 = 𝑩𝒆−𝒌𝟏 𝒕
(2.11)
The variables required for the calculation of B and k1 were either obtained
directly from a reference or calculated based on a referenced value as summarized
in table 2.4. The amount of heat generated in an aerobic degradation process is
coupled to the oxygen uptake, and VanderGheynst [35] estimated this value to be
Q O2 = 9500 J/g of O2 consumed. Simmons [21], on the other hand, measured
mineralizable carbon content based on CO2 released from amended soil reactors.
Aerobic degradation of organic matter releases approximately one CO2 molecule for
every O2 consumed, so the carbon dioxide-linked heat yield was calculated as
follows:
𝑄𝐶𝑂2 = 𝑄𝑂2 (
𝑚𝑂2
𝐽 32 𝑔/𝑚𝑜𝑙
) = 𝟔𝟗𝟎𝟗 𝑱/𝒈 𝑪𝑶𝟐 𝒓𝒆𝒍𝒆𝒂𝒔𝒆𝒅 (2.12)
) = (9500 ) (
𝑚𝐶𝑂2
𝑔 44 𝑔/𝑚𝑜𝑙
25
Table 2.4: First-order source term parameters and variables
Symbol
Parameter
Value
Units
Source
𝑨𝟏
First orde preexponential
𝑙𝑛(A1 ) = 15.5
𝑑𝑎𝑦𝑠 −1
[34]
𝑬𝒂
Activation
energy
45400
𝐽/𝑚𝑜𝑙
[34]
𝑹
Gas Constant
8.314
𝐽/𝑚𝑜𝑙 ∙ 𝐾
[36]
16.9
𝑚𝑔
𝑔𝑑𝑤
[21]
9500
𝐽
𝑔 𝑂2 𝑐𝑜𝑛𝑠𝑢𝑚𝑒𝑑
𝑪𝟎
𝑸𝑶𝟐
Total
Mineralizable
Carbon
Heat Generation
(O2 − linked)
[35]
𝑸𝑪𝑶𝟐
Heat Generation
(CO2 − linked)
6909
𝐽
𝑔 𝐶𝑂2 𝑒𝑣𝑜𝑙𝑣𝑒𝑑
calculation
𝝆𝒅𝒓𝒚
Soil Bulk Density
1.7 ∗ 106
𝑔/𝑚3
[21]
26
3
MODEL VALIDATION
3.1
Introduction
The 1-D amended-soil heat transfer model developed in chapter 2 is
described by equation 2.1:
𝜕 2 𝑇𝑠
𝜕𝑇𝑠
𝑘
+
𝑞̇
=
𝐶
𝑠
𝜕𝑥 2
𝜕𝑡
(2.1)
W
Note that if the source term, q̇ (m3 ), is set to zero, 2.1 becomes equivalent to that
described by Marshall [22] for stable soils. Validation of the amended soil model
consisted of comparison with two independent field data sets:
1. actual measured field temperature profiles
2. measurements of organic matter mineralization potential
3.1.1
Temperature profiles:
The simulated field temperature profiles with the first-order source term
(𝑞̇ = 𝐵𝑒 −𝑘1 𝑡 ) were validated against field trial data [21]. These results were also
compared to zero source term (𝑞̇ = 0) and constant source term (𝑞̇ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
simulation results. The validation cases are summarized below (table 3.1).
Table 3.1: Amended soil temperature profile simulation cases
Case
Source term
Description
Case 1 (base case)
𝒒̇ = 𝟎
Amended soil profile using the stable-soil model
Case 2
𝒒̇ = 𝑩𝒆−𝒌𝟏𝒕
Amended soil profile using first order source term
Case 3
𝒒̇ = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
Amended soil profile using constant source term
27
3.1.2
Heat Generation/Mineralization Potential:
As a secondary validation method, the simulated heat generation over the
solarization period (i.e., Qs = ∫ q̇ dt) was compared to the experimentally
determined mineralization potential from amended soils collected from field site
[21]. During the soil stabilization process via microbial respiration, shown below,
soil microbes convert mineralizable organic matter to carbon dioxide and heat, Qm ,
is released in the process:
𝑚𝑖𝑐𝑟𝑜𝑏𝑖𝑎𝑙 𝑟𝑒𝑠𝑝𝑖𝑟𝑎𝑡𝑖𝑜𝑛
𝐴𝑚𝑒𝑛𝑑𝑒𝑑 𝑠𝑜𝑖𝑙 + 𝑂2 ⇒
𝑆𝑡𝑎𝑏𝑙𝑒 𝑠𝑜𝑖𝑙 + 𝐶𝑂2 + 𝑄𝑚
Using CO2 emission data, Qm can be calculated to validate the modeled heat
generation.
3.2
3.2.1
Materials and Methods
Data Collection
The VanderGheynst lab conducted all data collection used for validation
purposes in this study. Field methods and laboratory methods are described in
detail by Simmons [21]and Reddy, Jenkins et al. [37].
Temperature profile data: Temperature data were collected at the Kearney
Agricultural Center (KAC) in Parlier, CA.
Soil microcosms were constructed
consisting of either 100% KAC soil or KAC soil amended with 10% organic matter; a
temperature recording thermistor was placed at 12.7 centimeters from the surface
of each microcosm.
These microcosms were then placed in the ground and
solarized for 22 days during mid-summer [21]. Results for each plot type were
averaged and these values were used for validation (average standard deviation for
28
all temperature measurements collected during the solarization period
was
approximately 0.57˚C.)
Mineralization potential: Samples of KAC field trial soil were analyzed both before
and after solarization to determine mineralizable carbon content.
Respiration
measurements [37] performed on amended soil mixtures prior to and following
field solarization revealed that 85% of potential respiration was exhausted during
the 22-day treatment (i.e., utilization, U = 0.85) [21]. From these data, generated
heat was calculated using the method described below (sec. 3.2.2.4).
3.2.2
Model Set-up and Data Analysis
The 1-D soil heat transfer model (eq. 2.1) requires three inputs: thermal
conductivity, 𝒌; heat capacity, 𝑪𝒔 ; and the source term, 𝒒̇ . Both non-source term
constants 𝑘 and 𝐶𝑠 are calculated as previously discussed (table 2.3) and require
site-dependent user inputs (table 3.2) [22].
3.2.2.1 Model inputs: Non-source Terms
Site-dependent user inputs fall into three categories: experimental, soil, and
tarp parameters (table 3.2). Between stable and amended soils, the only differences
are found in the soil parameters since location, time and tarp qualities were
identical.
29
Experimental Variables: The maximum time in minutes was defined based on the
length of the field trials used for model validation. The location KAC was identified
in terms of latitude and longitude as well as in reference to the nearest standard
meridian for solar time adjustment.
Soil parameters: The introduction of organic matter to the soil affected bulk density,
porosity, water-holding capacity, degree of saturation, and relative content of sand
silt and clay.
These variations in turn affected the thermal conductivity and
volumetric heat capacity. Mass fractions of water and organic matter (OM), as well
as soil bulk density are on a total soil dry weight basis. Mass fractions of sand, silt
and clay were based on dry weight of pure mineral soil.
Plastic parameters: The thickness and dimensions of the plastic for the Simmons
[21] field trials varied from those of the Marshall field trials but all other parameters
remained consistent. As discussed above, latent heat effects were determined to be
negligible, and this was reflected by the max. fraction of tarp area with condensation
being set to zero.
30
Table 3.2: Site dependent model parameter values.
Input Variables:
experimental variables
maximum time (min)
latitude of field site (˚)
longitude of field site (˚W)
longitude, local standard meridian (˚W)
soil parameters
mass fraction, water
mass fraction, om
mass fraction, sand
mass fraction, silt
mass fraction, clay
bulk density (dry g/cm 3)
soil type
soil color value, moist
soil color value, dry
mass fraction, water, field capacity
mass fraction, water, air dry
plastic parameters
plastic thickness (µm)
air gap height (cm)
plastic dimension, wind, L (m)
plastic dimension, W (m)
condensation thickness (mm)
max. fraction of tarp area with
condensation
emittance, long-wave
transmittance, long-wave
Soil Only
Amended
Source
31440
36.6
119.5
120
31440
36.6
119.5
120
0.1248
0.0049
0.71
0.23
0.06
1.60
2
4
6
0.15
0.006
0.22
0.1034
0.71
0.23
0.06
1.39
2
4
6
0.22
0.006
[21]
17.8
0.45
8.5344
1.8288
0.5
17.8
0.45
8.5344
1.8288
0.5
[21]
0
0.15
0.75
0
0.15
0.75
[31]
[21]
CIMIS
CIMIS
CIMIS
[21]
[31]
[31]
[31]
[21]
[31]
[31]
[31]
[31]
[31]
[31]
[21]
[21]
[31]
[31]
[31]
31
3.2.2.2 Inputs to Model: Source Term
In order to evaluate the performance of the first order amended soil model, three
distinct source term simulation cases were examined:
Case 1: Zero source term  𝒒̇ = 𝟎
The model was run assuming that there was no heat generation. The only
differences between case 1 and the stable soil test case 0 (Ch. 2.3) are in the
user defined soil parameters (table 3.2). This is the base case described in
section 2.3.
Case 2: First-order source term (eq. 2.28)  𝒒̇ 𝒈𝒆𝒏 = 𝑩𝒆−𝒌𝟏 𝒕
The values for B and k1 were determined as described in chapter 2.4.2 and
summarized in table 2.4. The calculated value of QCO2 = 6909 J/g (CO2 ) was
used for the initial simulations and further simulations were conducted to
find a value of QCO2 that optimized the temperature profile prediction
results.
Case 3: Constant source term  𝒒̇ = 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭
Various constant values were used for the heat generation term to serve as a
comparison for the first-order source model. The attempt was made to
optimize model performance for the first, most crucial week of solarization.
32
3.2.2.3 Model Outputs
Temperature Profiles: The expanded soil heat transfer model presented in chapter 2
was solved numerically using MATLAB 7.14.0 (The MathWorks, Inc., Natick, MA).
The complete code is attached in appendix A.1. Standard MATLAB functions used
to obtain a solution include:

fzero – Used to solve the energy balance on the tarp (table 2.1; eq. 2.5).
This function finds the zero of a continuous, single-variable function.

pdepe – Used to solve the parabolic partial differential equation (eq. 2.1)
with associated initial conditions (eq. 2.2) and boundary conditions (eqs.
2.3 and 2.4).
Discretization as prescribed in Marshall [22] was used
throughout.

cumtrapz – Numerical integration function used to determine total heat
generation over solarization period (eq. 3.4)
33
3.2.2.4 Simulation Data Analysis
Temperature profile simulation
Standard Error Metrics: To assess overall model accuracy, predicted temperatures
generated by model simulation (Ts) were compared to measured field temperatures
(Tm) by calculating the following error values (eq. 3.1-3.3):

root mean squared (RMSE) – descriptor of average model performance;

mean bias error (MBE) – indicator of long term over or under-prediction;

mean absolute error (MAE) – additional descriptor of average model
performance based on potential RMSE ambiguities [38].
𝑛
2
1
2
1
𝑅𝑀𝑆𝐸 = ( ∑(𝑇𝑠,𝑖 − 𝑇𝑚,𝑖 ) )
𝑛
(3.1)
𝑖=1
𝑛
1
𝑀𝐵𝐸 = ∑(𝑇𝑠,𝑖 − 𝑇𝑚,𝑖 )
𝑛
(3.2)
𝑖=1
𝑛
1
𝑀𝐴𝐸 = ∑|𝑇𝑠,𝑖 − 𝑇𝑚,𝑖 |
𝑛
(3.3)
𝑖=1
Maximum Daily Temperature Error: Since the maximum temperature reached is a
key factor in pathogen and weed seed inactivation, and since the above functions
take into account all temperature disparity along the entire profile, it is useful to
compare day by day maximum temperatures regardless of the exact time of
occurrence. Although equations 3.1-3.3 give insight into how well the model fits the
system overall, comparing the daily maximum temperatures reached may provide a
34
better measure of the model’s ability to predict the effectiveness of solarization
treatment.
Total Heat Generation
Simulated total heat generation, 𝑄𝑠𝑖𝑚 ( 𝐽/𝑐𝑚3 ), is obtained by numerically
integrating the source term over the period of solarization (eq. 3.4):
𝑡
(3.4)
𝑄sim = ∫ 𝑞̇ 𝑑𝑡
0
Mineralization potential
The volumetric heat generated, 𝑄𝑔𝑒𝑛 ( 𝐽/𝑐𝑚3 ), from mineralization of organic
soil amendment was calculated from respiration data using equation 3.5,
′
𝑄𝑔𝑒𝑛 = 𝑈 ∙ 𝐶0 ∙ 𝜌𝑑𝑟𝑦 ∙ 𝑄𝐶𝑂
2
(3.5)
where 𝑈 is utilization (ref: section 3.2.1), 𝐶0 is the theoretical maximum amount of
′
𝐶𝑂2that can be evolved (𝑔 𝐶𝑂2 /𝑔 𝑑𝑟𝑦 𝑠𝑜𝑖𝑙), 𝜌𝑑𝑟𝑦 is the density of dry soil, and 𝑄𝐶𝑂
is
2
the apparent 𝐶𝑂2 -linked heat yield (eq. 2.12).
The dry basis density of soil was
calculated according to equation 3.6,
𝜌𝑑𝑟𝑦 = 𝜌𝑤𝑒𝑡 (1 − 𝑤)
3.3
3.3.1
(3.6)
Results
Predicted vs. Measured Temperature Profile
Case 1: Zero source term  𝒒̇ = 𝟎
As shown in section 2.3.2, when used to predict the temperature profile of
amended soil, the stable soil model showed decreased correlation with field data
than when modeling stable soil. Figure 2.3 and table 2.3 are reproduced below.
35
Predicted
Measured
44
amended soil temperature (oC)
42
40
38
36
34
32
30
28
1
2
3
4
5
6
7
8
time (d)
Figure 2.3: Kearney field trial predicted and measured temperatures for amended plots
Table 2.3: Summary of results for case 1 (source term, 𝑞̇ = 0)
Days 2-8
Days 9-22
Days 2-22
RMSE (˚C)
1.4
1.5
1.4
MAE (˚C)
1.2
1.2
1.2
MBE (˚C)
-1.0
1.1
0.4
Number of points (N)
1007
1993
3000
Avg. Max Tm (˚C)
44.1
Avg. Max Ts (˚C)
43.1
Avg. ΔTmax (˚C)
-1.0
RMSE = root mean squared error, MBE = mean bias error, MAE = mean absolute error
36
Though the RMSE, MAE, and MBE show small errors in on aggregate and
suggest a slight over-prediction trend, the average maximum temperature
difference during the first week of solarization reflects an approximately 2.4%
under-prediction. One of the principal aims of the current research is to minimize
the error in maximum temperature prediction.
Case 2: First order source term  𝒒̇ = 𝑩𝒆−𝒌𝟏 𝒕
When the model was run using parameter values as described in table 2.4,
the model did not converge on a solution temperature profile. Since the model did
converge for the zero source term (i.e., when B = 0), the first order equation was
rewritten as,
′
𝑞̇ = 𝐵𝑒1−𝑘𝑡 = 𝑄𝐶𝑂
𝐷𝑒 −𝑘1 𝑡
2
(3.7)
′
where 𝑄𝐶𝑂
is the apparent 𝐶𝑂2-linked heat yield (J/g-CO2 ) and 𝐷 comprises the
2
remaining pre-exponential constants from equation 2.11 (i.e., 𝐷 = ρdw k1 C0 ) with
𝑔−𝐶𝑂2
units of (
𝑚3 𝑠
). An optimal value between 0 and 6909 J/g-CO2 was sought. It
seemed reasonable to vary the value of 𝑄𝐶𝑂2 since this value was determined for
purely aerobic systems with forced aeration [35] where the field soil system
described by our model is likely not purely aerobic. The model was run using values
of heat yield values, at regular intervals up to 6909 J/g-CO2 , and plotted against each
corresponding RMSE value as a measure of model performance over days 2-8. An
optimal value of approximately 300 J/g-CO2 emerged (figure 3.1).
37
1.8
1.7
1.6
1.5
RMSE
1.4
1.3
1.2
1.1
1
0.9
0.8
0
50
100
150
200
250
300
350
400
450
500
550
600
Apparent Heat Yield, Q'(J/g-CO2)
Figure 3.1: Optimization curve for the apparent 𝐶𝑂2-linked heat yield
′
The resulting temperature profile for this optimal 𝑄𝐶𝑂
(figure 3.2) shows a
2
very tight correlation for the first week (RMSE = 0.97 ˚C), a slightly decreased
correlation for the final 2 weeks (RMSE = 1.9 ˚C), and an overall RMSE of 1.65 ˚C.
There was a very slight general over-prediction for the first week (MBE = 0.02)
however inspection of the high points of figure 3.2 shows a slight under-prediction
by the model. The average measured maximum daily temperature, 𝑇𝑚 , was 44.13 ˚C
where the corresponding average predicted maximum, 𝑇𝑠 , was 44.12 ˚C.
This
indicates that during the critical first week of solarization, the predicted daily
maximum temperatures given by the model with a first order source term showed
38
near perfect correlation to measured daily maximum temperatures. Table 3.4
provides a summary of these results.
Predicted
Measured
44
amended soil temperature (oC)
42
40
38
36
34
32
30
28
1
2
3
4
5
6
7
8
time (d)
Figure 3.2: Case 2 amended soil temperature profile (Predicted vs. Measured)
Table 3.3: Summary of results for case 2 (1˚ source term with Q’CO2 = 300 J/g-CO2)
Days 2-8
Days 9-22
Days 2-22
RMSE (˚C)
1.0
1.9
1.7
MAE (˚C)
0.9
1.7
1.4
MBE (˚C)
> 0.1
1.7
1.1
Number of points (N)
1007
1993
3000
Avg. Max Tm (˚C)
44.1
Avg. Max Ts (˚C)
44.1
Avg. ΔTmax (˚C)
> 0.1
RMSE = root mean squared error, MBE = mean bias error, MAE = mean absolute error
39
Case 3): Constant source term  𝒒̇ = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
The objective of this case study was to determine if a constant heat
generation model could be optimized to outperform the first-order model over days
2-8 of the solarization period. As with case 2, an optimal value was sought by
implementing several constant source terms into the model and plotting the
resulting RMSE values (figure 3.3); these data suggest an optimal value of 𝑞̇ =
10 𝑊/𝑚3.
2
1.9
1.8
1.7
RMSE
1.6
1.5
1.4
1.3
1.2
1.1
1
0
5
10
15
20
25
30
Constant Source Term (W/m3)
Figure 3.3: RMSE for solarization days 2-8 vs. constant source term, 𝑞̇
The resulting temperature profile for this optimal constant source term
presented below (figure 3.4) shows a slightly weaker correlation for the first week
(RMSE = 1.11 ˚C), with a marked increase in error for the final 2 weeks (RMSE = 2.96
40
˚C), and an overall RMSE of 2.50 ˚C (table 3.5). The relative error in predicted
maximum daily temperatures is within 1%.
Predicted
Measured
44
amended soil temperature (oC)
42
40
38
36
34
32
30
28
1
2
3
4
5
6
7
8
time (d)
Figure 3.4: Case 3 amended soil temperature profile (Predicted vs. Measured)
Table 3.4: Summary of results for case 3 (Constant source term, 𝑞̇ = 10 𝑊/𝑚3)
Days 2-8
Days 9-22
Days 2-22
RMSE (˚C)
1.1
3.0
2.5
MAE (˚C)
0.9
2.8
2.2
MBE (˚C)
-0.2
2.8
1.8
Number of points (N)
1007
1993
3000
Avg. Max Tm (˚C)
44.1
Avg. Max Ts (˚C)
43.7
Avg. ΔTmax (˚C)
-0.4
RMSE = root mean squared error, MBE = mean bias error, MAE = mean absolute error
41
3.3.2
Validation of total heat generated
′
′
The model results for case 2 (i.e., for 𝑞̇ = 𝑄𝐶𝑂
𝐷𝑒 −𝑘1 𝑡 where 𝑄𝐶𝑂
= 300 𝐽/𝑔2
2
𝐶𝑂2) were analyzed to determine the total volumetric heat generated. Equation 3.4
was integrated numerically using the trapezoid method and resulted in a net heat
generation of 𝑸𝒔𝒊𝒎 = 𝟔. 𝟎 𝑱/𝒄𝒎𝟑.
Equation 3.5 was then evaluated using the
parameter values shown below (table 3.5) with a resulting heat generation, 𝑸𝒈𝒆𝒏 =
𝟔. 𝟓 𝑱/𝒄𝒎𝟑 .
Table 3.5: Mineralization Parameters
Symbol
U
ρwet
w
ρdry
𝐶0
𝑄𝐶𝑂2
Parameter
Value
Utilization
0.85
Wet basis density
1.7
Water content
18.2%
Dry basis density
1.39
Max. mineralization 16.9 × 10−3
𝐶𝑂2 -linked heat yield
300
Units
-g/cm3
g/g
g/cm3
g-CO2 /g dw
J/g-CO2
Source
[21]
[21]
[21]
Calculated from ρwet
[21]
Optimized via simulation
42
3.4
3.4.1
Discussion
Model Performance
These results (as summarized in table 3.6) confirm that a first order source
term can be used to model soil heating associated with organic matter stabilization
via microbial decomposition of organic amendment during the solarization process.
Based on RMSE (and a point by point comparison), case 2 appears to give the best
overall fit for the initial solarization period (days 2-8) with RMSE = 1.0; however, all
other RMSE values are within one standard deviation. More indicative of the first
order model performance over the initial period is the prediction of the daily high
temperatures. The Marshall model (stable-soil) [22] performed well for soil-only
(ΔTmax = 0.1˚C) but not as well for amended soil (ΔTmax = 1.1˚C). Implementation of
the first order source term resulted in tight correlation of daily maximum
temperatures (ΔTmax > 0.1˚C), as did a constant source term (ΔTmax = 0.4˚C).
The difference between the first order and constant source terms can be
clearly seen in the overall performance as presented in table 3.7. Case 3, though
comparable to case 2 over the short term, clearly shows poor average performance
during the final 2 weeks (RMSE = 2.5).
Moreover, the constant source over-
estimates the total source heat generated relative the calculated field value (Qsim =
19 J/cm3 vs. Qgen = 6.5 J/cm3). The first order model showed only a slight overall
under-estimate of generated heat (Qsim = 6.0 J/cm3).
43
Table 3.6: Summary of Results for days 2-8
Case 0
Case 1
Case 2
Case 3
RMSE
(˚C)
Avg. ΔTmax
(˚C)
%Error
ΔTmax
1.3
1.4
1.0
1.1
0.1
1.1
< 0.1
0.4
0.33%
2.4%
< 0.2%
1.0%
Table 3.7: Summary of Results for entire solarization period
RMSE
(˚C)
Case 0
Case 1
Case 2
Case 3
3.4.2
1.6
1.4
1.7
2.5
Qgen
Qsim
(J/cm3) (J/cm3)
--6.5
6.5
--6.0
19
% Error
Qsim
---7.7%
190%
Error Values
It should be noted that, in all cases, there is a slight phase shift of the
simulated temperature profile relative to the measured profile. This means that the
maximum and minimum temperatures of the overlaid data sets do not correspond
to the same time measurement. This shift has a slight effect on the values of RMSE,
MAE, and MBE. Furthermore these phase-shifts are not uniform from case to case
so, without a phase correction, small differences in these values should be taken
with a grain of salt. The daily maximum temperature comparison takes into account
this phase shift and may provide a more informative measure of model’s ability to
predict solarization effectiveness.
44
3.4.3
Heat Yield Reduction
The initial estimate for the heat yield (i.e., the expected heat released due to
aerobic digestion of mineralizable biomass) of 6909 J/g-CO2 released had to be
reduced in order for the model to converge on a solution. The optimized value of
300 J/g-CO2 (i.e. a 23-fold reduction in expected heat yield) indicates that the
decomposition is not completely aerobic and is likely due in part to the much less
exothermic anaerobic digestion process.
3.4.4
Amendment discontinuity
Another possible contributor to such a large reduction in effective heat yield
may lie in vertical discontinuity of amendment level. The model assumes that the
soil amendment is continuous and consistent between the boundaries (i.e., 0 < 𝑥 ≤
2 𝑚) whereas the actual amended depths in these field trials did not exceed 17.4 cm
(i.e., the depth of the soil microcosm).
The effect, if any, of the horizontal
discontinuity of soil amendment (due to the use of microcosms as opposed to evenly
amended plots) is also unknown.
45
4
CONCLUSIONS
Solarization as a means of disinfestation of agricultural soil from pathogens
and pests holds promise as an alternative to fumigation with methyl bromide. The
health and environmental hazards associated with the use of methyl bromide could
be avoided or, at the vary least, mitigated by employing solarization technology.
The principal challenge to the feasibility of widespread implementation of
solarization is that it needs to be carried out before planting and during the hottest
time of the year thereby requiring an interruption of the most productive part of the
growing season for many crops.
One strategy for decreasing the duration of the solarization treatment is to
amend the soil with organic matter to provide, via microbial action and concomitant
heat generation, an internal heat boost and elevation of soil temperatures. At
already elevated temperatures, the effects of solarization are non-linear, thus an
additional few degrees of temperature can give disproportionate results on the
deactivation of pathogens and weed seeds. In order to optimize the solarization
process it is important to be able to accurately model the soil temperature profiles,
allowing farmers to determine the minimum duration of solar treatment for the
required disinfestation and reducing the economic impact of growing season
interruption.
Previously, a model was developed for stable soils (i.e., soils with low levels
of organic matter) that provided accurate temperature profiles for a given soil type and
moisture content, at any geographic location using available local climatic data. The
46
current research has expanded this predictive capability by modeling the heat generated
within soil that has been amended with mineralizable organic matter (i.e., unstable soils).
The energy, released from microbial breakdown of organic matter in the soil was
modeled as a first-order degradation (eq. 2.11). The solarization model that included
first-order decomposition showed strong correlation with available field
temperature data for amended soil, out-performing the original stable soil (i.e.,
source term = 0) model as well as an alternative constant source term model.
Reduction of the expected heat yield was required to make the model converge on a
solution suggesting that the metabolic breakdown of the soil amendment was not a fully
aerobic process. Optimal fit with field data was obtained using a heat yield value of 300
J/g-CO2 released contrasting with 6909 J/g-CO2 for completely aerobic heat yield in highsolids degradation.
The first order source term was able to predict maximum daily temperatures to
within 0.1˚ C for the first 8 days with an RMSE for all points of approximately 1 ˚C. The
22-day performance of the model, though showing a slightly weaker correlation to field
values than the initial solarization period, was still good giving an RMSE of
approximately 1.7˚C.
The total source heat generated over the 22-day treatment, as predicted by
the first-order model was then validated against amended soil microbe respiration
data and the associated heat generation. The model showed better than 92%
𝑘𝐽
𝑘𝐽
agreement with measured data (𝑄𝑔𝑒𝑛 = 6.5 𝑐𝑚3 𝑣𝑠. 𝑄𝑠𝑖𝑚 = 6.0 𝑐𝑚3 ).
47
These results, taken together, showed that that first order model was the
most accurate and robust of the models studied here.
Refining the Model:
The notable decrease in model performance beyond the critical first week of
solarization may be due to the fact that the model does not account for fact that the
soil was not uniformly amended throughout the modeled volume (i.e., only the top
17.4 cm of the soil system was amended). This discontinuity in soil amendment
with accompanying changes in heat capacity, conductivity, porosity, water capacity,
etc. may affect the resulting temperature profiles. This provides an opportunity for
model improvement in the future.
Another aspect of the model that could be improved is in the treatment of the
first-order rate constant 𝑘1 . The current model assumes a constant 𝑘1 at all time
points and depths by using an approximated average temperature.
Since the
temperature is a function of time and depth and 𝑘1 is strongly dependent on
temperature, it follows that 𝑘1 is, in reality, a function of time and depth, as well.
This is the first time a source term has been included in a model to predict
temperature during solarization of unstable soil. Despite the simplifications made
to the model source term, the simulated temperature compared quite well to field
temperature measurements.
The results support future efforts to develop
combined soil amendment, stabilization and solarization processes and models to
assist farmers in managing these processes. Irwin Segel.
48
APPENDIX A.1: MATLAB Code for general soil model
function soilT_pdepe6_revK
% program to predict soil temperatures during solarization using CIMIS
% climate data and file of input variables - soil conduction equation
% solved using pdepe solver, tarp energy balance solved using fzero
% revK has soil type choice (soil or amend), revised output with error
% values (RMSE, MAE, MBE) and source term equation, and prints MATLAB
%temp s
% profile plots onto sheet 1 of the output excel folder.
% "7day" shows the 7-day profile on sheet one instead of full
solarization
% period
clear all
global i param ki Ci Vwat Vwai sigma R MW Ti Tsave v Tair ...
truns TskyK Tt_fluxes Ts_fluxes cum_mev cum_mdrip dc fc fc_max ...
field Tsoili Tamendi Tcimisi type T0 B b Q TF rho_b
tic; % Start timer
type = input('enter microcosm type (soil or amended): ','s');
source = input('Enter source term? ','s');
% assign columns depending on type of microcosm
col_soil = 2;
col_amended = 3;
% set constants
sigma = 5.6697e-8; % Stefan-Boltzmann constant (W/m2-K4)
R = 8.315; % universal gas constant (J/mol-K)
MW = 18.015; % molecular weight of water (g/mol)
% initialize counters
i = 1; % set counter for printing out Tt and heat fluxes
cum_mev = 0; % cumulative flux of evaporated water (g/m2)
cum_mdrip = 0; % cumulative flux of dripping water (g/m2)
fc = 0; % fraction of tarp area with condensation
% read in user-defined input variables and field temperature data
param = xlsread('input variables_revG.xlsx', type);
field = xlsread('field temps_revF.xlsx');
fieldtime = field(:,1); % field time (d)
Tsoil = field(:,col_soil); % field temperatures, soil only (oC)
Tamend = field(:,col_amended); % field temperatures, amended soil (oC)
Tsoili = Tsoil(1); % initial temp for soil only
Tamendi = Tamend(1); % initial temp for amended soil
% set initial soil temp "T0" depending on microcosm of interest
% to be used in function "u0" (line 212)
TF = strcmp(type,'soil'); % This is a string comparison test ==> if
type = soil it returns '1'
if TF == 1
T0 = Tsoili;
else
T0 = Tamendi;
end
49
% read in condensation parameters
fc_max = param(24,:) % maximum fraction of tarp area with condensation
dc = param (23,:)/1e3; % condensation thickness (m)
% calculate initial volumetric moisture content and soil heat transfer
% properties
Mwa = param(7,:); % moisture content (dry mass basis)
rho_b = param(12,:); % bulk density (g/cm3)
rho_wa = 1; % water density (g/cm3)
Vwai = Mwa*rho_b/rho_wa; % initial volumetric moisture content
[ki,Ci] = k_C_Vwa(Vwai); % ki, thermal conductivity (W/m-K)
% Ci, volumetric heat capacity (J/m3-oC)
Vwat = Vwai; % volumetric moisture content of upper soil
% layer at time t
% calculate soil radiation properties, based on moisture content
soil_rad(Vwai);
% set plastic radiation properties
tarp_rad;
% read in environmental data from CIMIS
cimis = xlsread('CIMIS_revC.xlsx');
jday = cimis(:,6); % julian date
t24h = cimis (:,5)/100; % time of day (h)
t0 = t24h(1); % initial time point
trunh = 24*(jday-jday(1))+t24h-t0; % accumulated time (h, = 0 at t0)
truns = trunh*3600; % accumulated time (s)
G = cimis(:,12); % solar radiation (W/m2)
Tair = cimis(:,16); % air temperature (C)
rh = cimis (:,18)/100; % relative humidity, fraction
v = cimis(:,22); % wind speed (m/s)
soil_T = cimis(:, 26); % soil temperature (oC)
Tcimisi = soil_T(1); % Initial condition from CIMIS @ 15 cm
valid = find(isfinite(soil_T) == 1);
Tsave = mean(soil_T(valid)); % average soil temperature (oC)
Ti = soil_T(1); % first CIMIS soil temperature (oC)
% calculate dew point temperature (oC) using Tair and rh CIMIS data
Tdew = dewpoint(Tair, rh);
% calculate sky temperature (K) according to Duffie and Beckman as a
% function of Tair (oC), Tdew (oC), and time from midnight (h)
TskyK = (Tair+273.15).*(0.711 + 0.0056*Tdew + 0.000073*Tdew.^2 + ...
0.013*cosd(15*t24h)).^0.25;
% calculate incidence angle, diffuse radiation, and beam radiation
inc_Gd_Gb(jday, t24h, G, rh);
% define time points at which solution is requested
% tmax = param(2,:);
tmax = param(2,:);
tm = 0:10:tmax; % look at temperature at these times (min)
ts = tm*60; % look at temperature at these times (s)
% define delta x that pdepe uses (delta x = 0.5 cm from 0-50 cm, 1 cm
from
% 51 cm = 2 m, and 2 cm from 2.02-5 m)
xmax = 5;
delx1 = .5/100; delx2 = 1/100; delx3 = 2/100;
xm1 = 0:delx1:50/100; xm2 = 51/100:delx2:2; xm3 = 2.02:delx3:xmax;
xm = [xm1 xm2 xm3];
50
% use pdepe to solve PDE system; set m = 0
% for slab geometry; set max time step = 4 min (Davis), 6 min (Kearney)
m = 0;
timestep = 6;
options = odeset('MaxStep', timestep*60);
status = 'running model'
% Solve
sol = pdepe(m, @pde_soil, @pde_soil_IC, @pde_soil_BC, xm, ts, options);
% put solutions for soil temperature into a 2D array called Ts
Ts = sol(:,:,1);
% xm in m, to graph in cm, *100
% ts in s, to graph in d, /86400
% Display post calculation input values (param vector values)
dgap = param(20,:)
% Plot results
% calculate error terms (rmse = root mean square error, mbe = mean bias
% error, mae = mean absolute error) - only include points from 1-number
of days specified at the start
% and eliminate blank field temperature measurements
% make new matrix to store solution in Excel (t in d, x in cm, T in oC)
time = (ts/86400)';
Ts_results = [time Ts(:,1:26)];
diff_results = [time Tsoil Ts(:,26) (Ts(:,26)-Tsoil) ...
Tamend Ts(:,26) (Ts(:,26)-Tamend)];
Ts_plot = Ts(:,26);
if TF == 1
figure(1)
plot(time, Ts_plot, 'b--', fieldtime, Tsoil, 'k-')
axis([1 8 28 45])
xlabel('time (d)')
ylabel('soil only temperature (oC)')
lg1_1 = ['Predicted'];
lg1_2 = ['Measured'];
legend(lg1_1, lg1_2)
figure(2)
plot(time, Ts_plot, 'b--', fieldtime, Tsoil, 'k-')
axis([1 22 28 48])
xlabel('time (d)')
ylabel('soil only temperature (oC)')
lg1_1 = ['Predicted'];
lg1_2 = ['Measured'];
legend(lg1_1, lg1_2)
51
figure(3)
plot(time, Ts(:,26)-Tsoil, 'mo-')
xlabel('time (d)')
ylabel('predicted-measured soil T (oC)')
lg4a = ['Soil Only'];
legend(lg4a)
% calculate error for days 2-8
index5soil_7 = find((isfinite(Tsoil) == 1) & (fieldtime >= 1) &
(fieldtime < 8));
rmse5soil_7 = ((sum((Ts(index5soil_7,26)Tsoil(index5soil_7)).^2))/length(index5soil_7))^0.5;
mbe5soil_7 = (sum(Ts(index5soil_7,26) Tsoil(index5soil_7)))/length(index5soil_7);
mae5soil_7 = (sum(abs(Ts(index5soil_7,26) Tsoil(index5soil_7))))/length(index5soil_7);
points5soil_7 = length(index5soil_7);
% calculate error for days 2-22
index5soil_21 = find((isfinite(Tsoil) == 1) & (fieldtime >= 1) &
(fieldtime < 22));
rmse5soil_21 = ((sum((Ts(index5soil_21,26)Tsoil(index5soil_21)).^2))/length(index5soil_21))^0.5;
mbe5soil_21 = (sum(Ts(index5soil_21,26) Tsoil(index5soil_21)))/length(index5soil_21);
mae5soil_21 = (sum(abs(Ts(index5soil_21,26) Tsoil(index5soil_21))))/length(index5soil_21);
points5soil_21 = length(index5soil_21);
% calculate error for days 9-22
index5soil_final = find((isfinite(Tsoil) == 1) & (fieldtime >= 8) &
(fieldtime < 22));
rmse5soil_final = ((sum((Ts(index5soil_final,26)Tsoil(index5soil_final)).^2))/length(index5soil_final))^0.5;
mbe5soil_final = (sum(Ts(index5soil_final,26) Tsoil(index5soil_final)))/length(index5soil_final);
mae5soil_final = (sum(abs(Ts(index5soil_final,26) Tsoil(index5soil_final))))/length(index5soil_final);
points5soil_final = length(index5soil_final);
else %(type == 'amended') In this case the strcmp test (line 126) would
have returned a '0' for false
figure(1)
plot(ts/86400, Ts(:,26), 'k-', fieldtime, Tamend, 'b--')
axis([1 8 28 45])
xlabel('time (d)')
ylabel('amended soil temperature (oC)')
lg2_1 = ['Predicted'];
lg2_2 = ['Measured'];
legend(lg2_1, lg2_2)
52
figure(2)
plot(ts/86400, Ts(:,26), 'k-', fieldtime, Tamend, 'b--')
xlabel('time (d)')
axis([1 22 28 48])
ylabel('amended soil temperature (oC)')
lg2_1 = ['Predicted'];
lg2_2 = ['Measured'];
legend(lg2_1, lg2_2)
figure(3)
plot(ts/86400, Ts(:,26)-Tamend, 'gx-')
xlabel('time (d)')
ylabel('predicted-measured soil T (oC)')
lg4b = ['Amended'];
legend(lg4b)
% calculate error for first days 2-8
index5amend_7 = find((isfinite(Tamend) == 1) & (fieldtime >= 1) &
(fieldtime < 8));
rmse5amend_7 = ((sum((Ts(index5amend_7,26)Tamend(index5amend_7)).^2))/length(index5amend_7))^0.5;
mbe5amend_7 = (sum(Ts(index5amend_7,26) Tamend(index5amend_7)))/length(index5amend_7);
mae5amend_7 = (sum(abs(Ts(index5amend_7,26) Tamend(index5amend_7))))/length(index5amend_7);
points5amend_7 = length(index5amend_7);
% calculate error for days 2-22
index5amend_21 = find((isfinite(Tamend) == 1) & (fieldtime >= 1) &
(fieldtime < 22));
rmse5amend_21 = ((sum((Ts(index5amend_21,26)Tamend(index5amend_21)).^2))/length(index5amend_21))^0.5;
mbe5amend_21 = (sum(Ts(index5amend_21,26) Tamend(index5amend_21)))/length(index5amend_21);
mae5amend_21 = (sum(abs(Ts(index5amend_21,26) Tamend(index5amend_21))))/length(index5amend_21);
points5amend_21 = length(index5amend_21);
% calculate error for days 9-22
index5amend_final = find((isfinite(Tamend) == 1) & (fieldtime >= 8) &
(fieldtime < 22));
rmse5amend_final = ((sum((Ts(index5amend_final,26)Tamend(index5amend_final)).^2))/length(index5amend_final))^0.5;
mbe5amend_final = (sum(Ts(index5amend_final,26) Tamend(index5amend_final)))/length(index5amend_final);
mae5amend_final = (sum(abs(Ts(index5amend_final,26) Tamend(index5amend_final))))/length(index5amend_final);
points5amend_final = length(index5amend_final);
end
53
figure(4)
plot(ts/86400, Ts(:,1), 'mo-', ts/86400, Ts(:,end), 'gx-',
Tt_fluxes(:,2),...
Tt_fluxes(:,3), 'bd-')
xlabel('time (d)')
ylabel('temperature (oC)')
lg3_1 = ['x = ',num2str(xm(1)*100),' cm'];
lg3_2 = ['x = ', num2str(xm(end)), ' m'];
lg3_3 = ['tarp'];
legend(lg3_1, lg3_2, lg3_3)
if TF == 1
xlswrite('tempdata_soil.xlsx', Ts_results, 'Ts_results', 'B2')
xlswrite('tempdata_soil.xlsx', Ts_fluxes, 'Ts_fluxes', 'B2')
xlswrite('tempdata_soil.xlsx', Tt_fluxes, 'Tt_fluxes', 'B2')
xlswrite('tempdata_soil.xlsx', diff_results, 'diff_results', 'B2')
else
xlswrite('tempdata_amend.xlsx', Ts_results, 'Ts_results', 'B2')
xlswrite('tempdata_amend.xlsx', Ts_fluxes, 'Ts_fluxes', 'B2')
xlswrite('tempdata_amend.xlsx', Tt_fluxes, 'Tt_fluxes', 'B2')
xlswrite('tempdata_amend.xlsx', diff_results, 'diff_results', 'B2')
end
% calculate cumulative predicted soil heat flux, MJ m-2, between days
2-8
index_soil_7 = find((Ts_fluxes(:,2) >= 1) & (Ts_fluxes(:,2) < 8));
cum_soil_7 = (trapz(Ts_fluxes(index_soil_7,1),
Ts_fluxes(index_soil_7,3)))/1e6
% calculate cumulative predicted soil heat flux, MJ m-2, between days
2-22
index_soil_21 = find((Ts_fluxes(:,2) >= 1) & (Ts_fluxes(:,2) < 22));
cum_soil_21 = (trapz(Ts_fluxes(index_soil_21,1),
Ts_fluxes(index_soil_21,3)))/1e6
% calculate cumulative predicted soil heat flux, MJ m-2, between days
9-22
index_soil_final = find((Ts_fluxes(:,2) >= 8) & (Ts_fluxes(:,2) < 22));
cum_soil_final = (trapz(Ts_fluxes(index_soil_final,1),
Ts_fluxes(index_soil_final,3)))/1e6
54
% Set up Excel sheet entries
% Set up input values for writing to excel
input_headings(1:26,1)={
'experimental variables'
'maximum time (min)'
'latitude of field site (o)'
'longitude of field site (oW)'
'longitude, local standard meridian (oW)'
'soil parameters'
'mass fraction, water'
'mass fraction, om'
'mass fraction, sand'
'mass fraction, silt'
'mass fraction, clay'
'bulk density (dry g/cm3)'
'soil type'
'soil color value, moist'
'soil color value, dry'
'mass fraction, water, field capacity'
'mass fraction, water, air dry'
'plastic parameters'
'plastic thickness (um)'
'air gap height (cm)'
'plastic dimension, wind, L (m)'
'plastic dimension, W (m)'
'condensation thickness (mm)'
'max. fraction of tarp area with condensation'
'emittance, long-wave'
'transmittance, long-wave'};
col_header={'days 2-8' 'days 9-22' 'days 2-22'};
row_header(1:5,1)={'RMSE', 'MAE', 'MBE','Flux','N'}; % Column cell
array (for row labels)
run_heading = {type};
source_heading = {source};
% Input_Output page for soil case
if TF == 1
xlswrite('tempdata_soil.xlsx',input_headings,'Input_Output','A1'); %
Write input headings to excel
xlswrite('tempdata_soil.xlsx',param,'Input_Output','B1'); % Write input
values to excel
xlswrite('tempdata_soil.xlsx',run_heading,'Input_Output','E1');
xlswrite('tempdata_soil.xlsx',source_heading,'Input_Output','E2');
xlswrite('tempdata_soil.xlsx',col_header,'Input_Output','F3'); % Write
column header
xlswrite('tempdata_soil.xlsx',row_header,'Input_Output','E4'); % Write
row header
%Output of calculated values
% Enter target values into vertical arrays for output table on
values1(1:5,1)={rmse5soil_7,mae5soil_7,mbe5soil_7,cum_soil_7,points5soi
l_7};
values2(1:5,1)={rmse5soil_final,mae5soil_final,mbe5soil_final,cum_soil_
final,points5soil_final};
55
values3(1:5,1)={rmse5soil_21,mae5soil_21,mbe5soil_21,cum_soil_21,points
5soil_21};
% Write to Excel file
xlswrite('tempdata_soil.xlsx',values1,'Input_Output','F4'); % Write
data
xlswrite('tempdata_soil.xlsx',values2,'Input_Output','G4'); % Write
data
xlswrite('tempdata_soil.xlsx',values3,'Input_Output','H4'); % Write
data
% Use Excel Active X to format Excel sheet
% Use active x to paste plots to excel sheet and format
Excel = actxserver('Excel.Application');
Workbooks = Excel.Workbooks;
Workbook = invoke(Workbooks,
'Open','C:\Users\Duff\Dropbox\Solarization\MATLAB\RevK\tempdata_soil.xl
sx');
Sheets = Excel.ActiveWorkBook.Sheets;
%Paste Fig.1 @sheet1/A2
sheetHndl = get(Sheets, 'Item', 1);
print(figure(1),'-dmeta');
sheetHndl.Range('A1').PasteSpecial;
%Past Fig.2 on same sheet @G10
print(figure(2),'-dmeta');
sheetHndl.Range('J1').PasteSpecial;
print(figure(3),'-dmeta');
sheetHndl.Range('A22').PasteSpecial;
print(figure(4),'-dmeta');
sheetHndl.Range('J22').PasteSpecial;
%Format Input_Output\column 1
sheetHndl = get(Sheets,'Item',8);
sheetHndl.Columns.Item(1).columnWidth=42;
set(Excel,'Visible',1); %If I want to actually open the sheet
Workbook.Save
% Input_Output page for amended case
elseif TF == 0
preexp(1,1:2)={'B = ' num2str(B)};
power(1,1:2)={'b = ' num2str(b)};
ratio (1,1:2)={'Q = ' num2str(Q)};
xlswrite('tempdata_amend.xlsx',input_headings,'Input_Output','A1'); %
Write input headings to excel
xlswrite('tempdata_amend.xlsx',param,'Input_Output','B1'); % Write
input values to excel
xlswrite('tempdata_amend.xlsx',run_heading,'Input_Output','E1');
xlswrite('tempdata_amend.xlsx',source_heading,'Input_Output','E2');
xlswrite('tempdata_amend.xlsx',preexp,'Input_Output','E10');
xlswrite('tempdata_amend.xlsx',power,'Input_Output','E11');
xlswrite('tempdata_amend.xlsx',ratio,'Input_Output','E13');
xlswrite('tempdata_amend.xlsx',col_header,'Input_Output','F3'); % Write
column header
xlswrite('tempdata_amend.xlsx',row_header,'Input_Output','E4'); % Write
row header
56
% Output of error and flux values
% Enter target values into vertical arrays
values1(1:5,1)={rmse5amend_7,mae5amend_7,mbe5amend_7,cum_soil_7,points5
amend_7};
values2(1:5,1)={rmse5amend_final,mae5amend_final,mbe5amend_final,cum_so
il_final,points5amend_final};
values3(1:5,1)={rmse5amend_21,mae5amend_21,mbe5amend_21,cum_soil_21,poi
nts5amend_21};
% Write to Excel file
xlswrite('tempdata_amend.xlsx',values1,'Input_Output','F4'); % Write
data
xlswrite('tempdata_amend.xlsx',values2,'Input_Output','G4'); % Write
data
xlswrite('tempdata_amend.xlsx',values3,'Input_Output','H4'); % Write
data
% Use Excel Active X to format Excel sheet
% Use active x to paste plots to excel sheet and format
Excel = actxserver('Excel.Application');
Workbooks = Excel.Workbooks;
Workbook = invoke(Workbooks,
'Open','C:\Users\Duff\Dropbox\Solarization\MATLAB\RevK\tempdata_amend.x
lsx');
Sheets = Excel.ActiveWorkBook.Sheets;
%Paste Fig.1 @sheet1/A2
sheetHndl = get(Sheets, 'Item', 1);
print(figure(1),'-dmeta');
sheetHndl.Range('A1').PasteSpecial;
%Past Fig.2 on same sheet @G10
print(figure(2),'-dmeta');
sheetHndl.Range('J1').PasteSpecial;
print(figure(3),'-dmeta');
sheetHndl.Range('A22').PasteSpecial;
print(figure(4),'-dmeta');
sheetHndl.Range('J22').PasteSpecial;
%Format Input_Output\column 1
sheetHndl = get(Sheets,'Item',8);
sheetHndl.Columns.Item(1).columnWidth=42;
set(Excel,'Visible',1); %If I want to actually open the sheet
Workbook.Save
end
if TF == 0
Heat_term = Q
Preexponential = B
Rate_constant = b
Critical_ratio = B/b
end
total_cpu_minutes = toc/60
% compare final and initial temperatures at xmax
Ts(end, end)
Tsave
disp('Program concluded')
57
APPENDIX A.2: MATLAB pdepe function and Initial Conditions
%---------------------------------------------------------------------function [c,f,s] = pde_soil(x, t, u, DuDx)
global ki Ci Vwat b B S Q TF rho_b
if TF == 1
S = 0;
elseif TF == 0
Q = 300; % (6909 J/g-CO2 evolved) calculated from (9500 J/g-O2
evolved) VanderGheynst 1997. Model does not converge...need to downtweak this number  optimized at 300 J/g-CO2
C0 = .0169; % (gCO2-C/gdw) Simmons 2012
rho = rho_b*10^6; % Convert bulk dry density to per m^3 basis from per
cm^3
Ea = 45400; % (J/mol)from Aslam 2008
Tbar = 37+273; % (Kelvin) estimate of avg. temp
A1=exp(15.5); % (day^-1)from Aslam 2008
R=8.314;
% (J/mol*K)Universal gas constant in J/mol-K
b=A1*exp(-Ea/(R*Tbar)); % 1st order rate contant k from Aslam 2008
B=Q*C0*rho*b;
S = B*exp(-b*t); % For first order source term runs
%S = 10; % For constant source term runs
end
%if (x <= .5/100)
%[k,C] = k_C_Vwa(Vwat);
%else
k = ki;
C = Ci;
%end
c = C; % C, volumetric heat capapcity (J/m3-oC)
f = k*DuDx; % k, thermal conductivity (W/m-oC)
%s = 0; % No source term
% s = 10^8*exp(-10^0*t); % source term
s = S; % source term
%---------------------------------------------------------------------function u0 = pde_soil_IC(x)
global Tsave Tcimisi T0
% set initial soil temperatures using field measurements
if (x>=0) && (x<=10/100)
u0 = T0+5;
elseif (x>10/100) && (x<=14/100)
u0 = T0;
elseif (x>14/100) && (x <=16/100)
u0 = (Tcimisi+T0)/2;
elseif (x>16/100) && (x<=20/100)
u0 = Tcimisi;
end
58
APPENDIX A.3: Unmodified Marshall [22] MATLAB code
%---------------------------------------------------------------------function [pl, ql, pr, qr] = pde_soil_BC(xl, ul, xr, ur, t)
global i param Vwat Vwai sigma R MW eSl pSl aSs_d pSs_d eTl tTl ...
pTl aTs_d tTs_d pTs_d aCs_d tCs_d pCs_d inc Gd Gb v Tair ...
truns TskyK Tt_fluxes Ts_fluxes cum_mev cum_mdrip dc fc fc_max
% function to define boundary conditions
% BC1 - energy balance on mulched soil surface
% BC2 - heat flux (at x = xmax) = 0
% calculate current moisture content on mass basis, to write to Excel
rho_b = param(12,:); % bulk density (g/cm3)
rho_wa = 1; % water density (g/cm3)
Mwat = Vwat*rho_wa/rho_b; % moisture content (mass basis) at time t
% interpolate CIMIS data so that it matches up with all the times
% at which pde is being solved
iTair = interp1(truns, Tair, t, 'linear');
iGb = interp1(truns, Gb, t, 'linear');
iGd = interp1(truns, Gd, t, 'linear');
iv = interp1(truns, v, t, 'linear');
iTskyK = interp1(truns, TskyK, t, 'linear');
iinc = interp1(truns, inc, t, 'linear');
ulK = ul + 273.15; % soil temperature, x = 0 (K)
[tTs, pTs, aTs] = tarp_rad_inc(iinc);
[tCs, pCs, aCs] = tarp_rad_cond_inc(iinc);
% calculate effective tarp radiation properties based on fraction of
%tarp covered with condensation
aTCs = fc*aCs + (1-fc)*aTs;
pTCs = fc*pCs + (1-fc)*pTs;
tTCs = fc*tCs + (1-fc)*tTs;
aTCs_d = fc*aCs_d + (1-fc)*aTs_d;
pTCs_d = fc*pCs_d + (1-fc)*pTs_d;
tTCs_d = fc*tCs_d + (1-fc)*tTs_d;
% solar radiation absorbed by tarp (W/m2)
q_solar_tb = iGb*(aTCs+(pSs_d*tTCs*aTCs_d)/(1-pSs_d*pTCs_d));
q_solar_td = aTCs_d*iGd*(1+(pSs_d*tTCs_d)/(1-pSs_d*pTCs_d));
% solar radiation absorbed by soil (W/m2)
q_solar_sb = aSs_d*tTCs*iGb/(1-pTCs_d*pSs_d);
q_solar_sd = aSs_d*tTCs_d*iGd/(1-pTCs_d*pSs_d);
% assume there is no energy accumulation in plastic, so solve algebraic
% equation for Tt using fzero - use range of 2 values for TtK_guess,
%one value must give positive result, and the other a negative result
% try 275 as lower bound for kac (lower ambient air temperatures at end
%of experiment) - 280, 395 for davis
TtK_guess = [275 395];
TtK = fzero(@tarp_temp, TtK_guess, [], iTair, iTskyK, ulK, iv,...
q_solar_tb, q_solar_td);
59
% calculate convection coefficients and latent heat of vaporization
% for first iteration, set Tt = Tair
if (i == 1)
[hci, hmci, hfg] = inside_conv(ulK, iTair + 273.15);
hco = outside_conv(iTair+273.15, iTair+273.15, iv);
else
[hci, hmci, hfg] = inside_conv(ulK, TtK);
hco = outside_conv(TtK, iTair+273.15, iv);
end
% calculate rate of evaporation and dripping - if fc_max is set to 0 by
% user, latent heat transfer is eliminated from model - also check that
no
% moisture drips off tarp (m_drip > 0 or m_ev < 0) when tarp is dry (fc
=
% 0)
if (fc_max == 0)
m_ev = 0;
m_drip = 0;
elseif ((fc_max > 0) && (fc < fc_max) && (fc > 0))
m_ev = (hmci*MW/R)*(psat (ulK)/ulK-psat (TtK)/TtK);
m_drip = 0;
elseif ((fc_max > 0) && (fc == fc_max))
m_ev = (hmci*MW/R)*(psat (ulK)/ulK-psat (TtK)/TtK);
if (m_ev < 0)
m_drip = 0;
else
m_drip = m_ev;
end
elseif ((fc_max > 0) && (fc == 0))
m_ev = (hmci*MW/R)*(psat (ulK)/ulK-psat (TtK)/TtK);
m_drip = 0;
if (m_ev < 0)
m_ev = 0;
end
end
% calculate remaining terms in soil surface energy balance
q_ce = hfg*m_ev;
q_ci = hci*(ulK - TtK);
q_r_st = (1-fc)*sigma*(eTl*eSl/(1-pTl*pSl))*(ulK^4 - TtK^4)...
+fc*sigma*(ulK^4-TtK^4)/(1/eSl+1/0.96-1);
q_h2o = m_drip*cpf(TtK)*(TtK-273.15);
q_r_ssky = (1-fc)*sigma*eSl*tTl*(ulK^4-iTskyK^4)/(1-pSl*pTl);
% soil surface energy balance boundary condition
pl = q_solar_sb + q_solar_sd - q_ci - q_ce - q_r_st - q_r_ssky + q_h2o;
ql = 1;
% set lower boundary condition to flux = 0
pr = 0;
qr = 1;
60
% calculate tarp terms and prepare matrix to be written to Excel
q_co = hco*(TtK-(iTair+273.15));
q_r_tsky = sigma*eTl*(TtK^4 - iTskyK^4);
Tt_fluxes(i,:) = [t t/86400 TtK-273.15 q_solar_tb q_solar_td q_ci q_ce
...
-q_h2o -q_co q_r_st -q_r_tsky m_ev m_drip fc];
Ts_fluxes(i,:) = [t t/86400 pl q_solar_sb q_solar_sd -q_ci -q_ce q_h2o
...
-q_r_st -q_r_ssky Mwat];
% calculate cumulative mass of water evaporated and mass of water
dripping
if (t > 0)
cum_mev = trapz(Tt_fluxes(:,1), Tt_fluxes(:,12));
cum_mdrip = trapz(Tt_fluxes(:,1), Tt_fluxes(:,13));
end
% calculate fraction of tarp area with condensation, constrain fraction
to
% be less than maximum
fc = (cum_mev-cum_mdrip)/(1e6*dc);
if (fc > fc_max)
fc = fc_max;
elseif (fc < 0)
fc = 0;
end
% re-calculate moisture content of uppermost soil layer, update soil
% radiation parameters
Vwat = Vwai - ((cum_mev-cum_mdrip)/(1e6*0.5/100));
soil_rad(Vwat);
i = i + 1;
%---------------------------------------------------------------------function TtK_fun = tarp_temp(TtK, iTair, iTskyK, ulK, iv,...
q_solar_tb, q_solar_td)
global i sigma R MW eSl pSl eTl pTl fc fc_max
% function that defines tarp energy balance
% calculate convection coefficients and latent heat of vaporization
% for first iteration, set Tt = Tair
if (i
[hci,
hco =
else
[hci,
hco =
end
== 1)
hmci, hfg] = inside_conv(ulK, iTair + 273.15);
outside_conv(iTair+273.15, iTair+273.15, iv);
hmci, hfg] = inside_conv(ulK, TtK);
outside_conv(TtK, iTair+273.15, iv);
61
% calculate rate of evaporation and dripping - if fc_max is set to 0 by
% user, latent heat transfer is eliminated from model - also check that
% no moisture drips off tarp (m_drip > 0 or m_ev < 0) when tarp is dry
%(fc = 0)
if (fc_max == 0)
m_ev = 0;
m_drip = 0;
elseif ((fc_max > 0) && (fc < fc_max) && (fc > 0))
m_ev = (hmci*MW/R)*(psat (ulK)/ulK-psat (TtK)/TtK);
m_drip = 0;
elseif ((fc_max > 0) && (fc == fc_max))
m_ev = (hmci*MW/R)*(psat (ulK)/ulK-psat (TtK)/TtK);
if (m_ev < 0)
m_drip = 0;
else
m_drip = m_ev;
end
elseif ((fc_max > 0) && (fc == 0))
m_ev = (hmci*MW/R)*(psat (ulK)/ulK-psat (TtK)/TtK);
m_drip = 0;
if (m_ev < 0)
m_ev = 0;
end
end
% calculate terms in tarp energy balance
q_ce = hfg*m_ev;
q_ci = hci*(ulK - TtK);
q_r_st = (1-fc)*sigma*(eTl*eSl/(1-pTl*pSl))*(ulK^4 - TtK^4)...
+fc*sigma*(ulK^4-TtK^4)/(1/eSl+1/0.96-1);
q_h2o = m_drip*cpf(TtK)*(TtK-273.15);
q_co = hco*(TtK-(iTair+273.15));
q_r_tsky = sigma*eTl*(TtK^4 - iTskyK^4);
TtK_fun = q_solar_tb + q_solar_td + q_ci + q_ce - q_co - q_r_tsky...
+ q_r_st - q_h2o;
%--------------------------------------------------------------------function [k,C] = k_C_Vwa(Vwa)
global param
% function to calculate thermal conductivity and volumetric heat
capacity
Mom = param(8,:); % organic matter content (g/dry g)
sand = param(9,:); % sand fraction, particle size analysis
silt = param(10,:); % silt fraction, particle size analysis
clay = param(11,:);% clay fraction, particle size analysis
rho_b = param(12,:);
class = param(13,:); % soil type, for empirical parameters
% particle size analysis sums 3 mineral fraction to 1 - need to adjust
to
% account for organic matter in solid fraction
Msa = (1-Mom)*sand; % sand content (g/dry g)
62
Msi = (1-Mom)*silt; % silt content (g/dry g)
Mcl = (1-Mom)*clay; % clay content (g/dry g)
rho_si = 2.75; % silt density (g/cm3)
rho_cl = 2.75; % clay density (g/cm3)
rho_sa = 2.65; % sand (quartz) density (g/cm3)
rho_om = 1.5; % organic matter (peat) density (g/cm3)
rho_wa = 1; % water density (g/cm3)
% convert mass fractions to volume fractions using bulk density and
% component density
Vom = Mom*rho_b/rho_om;
Vsa = Msa*rho_b/rho_sa;
Vsi = Msi*rho_b/rho_si;
Vcl = Mcl*rho_b/rho_cl;
V = 1 - Vsa - Vsi - Vcl - Vom; % soil porosity
% volumetric heat capacity (J/m3-C)
C = 1.92e6*(Vsa+Vsi+Vcl)+2.51e6*Vom+4.18e6*Vwa;
% thermal conductivities of soil components (W/m-oC)
k_om = 0.25; % organic matter (peat)
k_si = 2.90; % silt
k_cl = 2.90; % clay
k_sa = 7.69; % sand (quartz)
k_w = 0.6; % water
% saturated soil thermal conductivity from Cote and Konrad
Sr = Vwa/V; % equivalent to Nw/n*(rho_b/rho_wa)
ksat = (k_om^(Vom))*(k_si^(Vsi))*(k_sa^(Vsa))*(k_cl^(Vcl))*(k_w^(V));
% empirical constants for various soil classifications
X = [1.70 0.75 0.75 0.30];
N = [1.80 1.20 1.20 0.87];
K = [4.60 3.55 1.90 0.60];
% dry soil and normalized thermal conductivities
kdry = X(class)*(10^(-V*N(class)));
kr = K(class)*Sr/(1+(K(class)-1)*Sr);
k = (ksat-kdry)*kr + kdry;
%---------------------------------------------------------------------function soil_rad(Vw)
global param eSl pSl aSs_d pSs_d
eSl = 0.90 + 0.18*Vw; % long-wave emittance, soil
pSl = 1-eSl; % long-wave reflectance, soil
% calculate short-wave soil reflectance as function of soil color and
% moisture content
colorm = param(14,:);
colord = param(15,:);
Mwa_fc = param(16,:);
Mwa_ad = param(17,:);
rho_b = param(12,:);
rho_wa = 1;
Vwa_fc = Mwa_fc*rho_b/rho_wa;
Vwa_ad = Mwa_ad*rho_b/rho_wa;
% this relationship was developed at incidence angles near 60o,
provides
63
% soil reflectance for diffuse radiation
pm = 0.069*colorm-0.114;
pd = 0.069*colord-0.114;
if (Vw < Vwa_ad)
pSs_d = pd;
elseif (Vw > Vwa_fc)
pSs_d = pm;
else
pSs_d = pd + (pm-pd)*(Vw-Vwa_ad)/(Vwa_fc-Vwa_ad);
end
aSs_d = 1 - pSs_d;
%---------------------------------------------------------------------function tarp_rad
global param eTl tTl ...
pTl aTs_d tTs_d pTs_d aCs_d tCs_d pCs_d
% read in long-wave radiation properties from input file
eTl = param(25,:);
tTl = param(26,:);
pTl = 1-eTl-tTl;
% calculate short-wave properties for diffuse radiation
[tTs_d, pTs_d, aTs_d] = tarp_rad_inc(60);
[tCs_d, pCs_d, aCs_d] = tarp_rad_cond_inc(60);
%---------------------------------------------------------------------function [t, p, a] = tarp_rad_inc(in)
global param
% function to calculate short-wave optical properties of dry tarp
na = 1; % refractive index, air
nt = 1.51; % refractive index, tarp
ref = asind((na/nt)*sind(in)); % refraction angle (deg.)
% calculate reflection loss coefficient
if (ref == 0)
r = (na-nt)^2/(na+nt)^2;
else
r = 0.5*((sind(ref-in))^2/(sind(ref+in))^2 +...
(tand(ref-in))^2/(tand(ref+in))^2);
end
dt = param (19,:)/1e6; % tarp thickness (m)
Kt = 165; % LDPE extinction coefficient (m-1)
tt = exp(-Kt*dt/cosd(ref)); % transmission coefficient
t = (1-r)^2*tt/(1-(r*tt)^2);
p = r +((r*tt^2*(1-r)^2)/(1-(r*tt)^2));
a = 1 - t - p;
64
%---------------------------------------------------------------------function [tC, pC, aC] = tarp_rad_cond_inc(in)
global param dc
% function to calculate short-wave tarp optical properties (full
coverage
% of condensation)
na = 1; % refractive index, air
nt = 1.51; % refractive index, tarp
nw = 1.33; % refractive index, water
% calculate refraction angles (deg.)
reft = asind((na/nt)*sind(in));
reftd = asind((na/nt)*sind(60));
refwd = asind((nt/nw)*sind(60));
refad = asind((nw/na)*sind(40.5));
% calculate reflection loss coefficients
if (reft == 0)
rt = (na-nt)^2/(na+nt)^2;
else
rt = 0.5*((sind(reft-in))^2/(sind(reft+in))^2 +...
(tand(reft-in))^2/(tand(reft+in))^2);
end
rtd = 0.5*((sind(reftd-60))^2/(sind(reftd+60))^2 +...
(tand(reftd-60))^2/(tand(reftd+60))^2);
rwd = 0.5*((sind(refwd-60))^2/(sind(refwd+60))^2 +...
(tand(refwd-60))^2/(tand(refwd+60))^2);
rad = 0.5*((sind(refad-40.5))^2/(sind(refad+40.5))^2 +...
(tand(refad-40.5))^2/(tand(refad+40.5))^2);
dt = param (19,:)/1e6; % tarp thickness (m)
Kt = 165; % LDPE extinction coefficient (m-1)
tt = exp(-Kt*dt/cosd(reft)); % transmission coefficients, tarp
ttd = exp(-Kt*dt/cosd(reftd));
% values in 19-term transmission function by Tsilingiris (1988)
% Kw (1/m) = extinction coefficient for wavelength band
% lam = amplitude for wavelength band
Kw = [0.058 0.039 0.025 0.018 0.026 0.038 0.055 0.081 0.137 0.205
0.255...
0.324 0.425 1.33 2.2 2.9 5.17 42.5 1800];
lam = [0.0466 0.029 0.0345 0.0408 0.0413 0.04 0.039 0.0375 0.0375
0.0367...
0.036 0.035 0.0327 0.0629 0.0548 0.0476 0.0263 0.153 0.1676];
twd = 0;
for j = 1:length(Kw)
twd = twd + lam(j)*exp(-Kw(j)*dc/cosd(refwd));
end
c1 = 1 - rwd*rad*twd^2;
c2 = c1*(1-rtd*ttd*rwd*tt)-(1-rwd)^2*ttd*tt*twd^2*rad*rtd;
tC = (1-rad)*(1-rwd)*(1-rt)*twd*tt/c2;
pC = rt + ((1-rtd)*(1-rt)*ttd*tt/c2)*(rwd*c1+(1-rwd)^2*twd^2*rad);
aC = 1 - tC - pC;
65
%---------------------------------------------------------------------function inc_Gd_Gb(julian, timeh_24, Gtotal, humid)
global param inc Gd Gb Tair
% function calculates incidence angle and splits total solar radiation
% into beam and diffuse components
lat = param(3,:); % site latitude (deg.)
long = param(4,:); % site longitude (deg.)
longref = param(5,:); % standard meridian (deg.)
% equation of time (min)
B = (julian-1)*360/365;
EoTmin = 229.2*(7.5e-5+1.868e-3*cosd(B)-3.2077e-2*sind(B) ...
-1.4615e-2*cosd(2*B)-4.089e-2*sind(2*B));
tsolar = timeh_24 + (4*(longref-long) + EoTmin)/60; % solar time
omega = 15*(tsolar-12); % hour angle (deg.)
decl = 23.45*sind(360*(284+julian)/365); % declination angle (deg.)
% incidence angle (deg.)
inc = acosd((cosd (lat).*cosd (decl).*cosd(omega)+sind
(lat).*sind(decl)));
% extraterrestrial radiation (W/m2) and clearness index
Go = 1367*(1+0.033*cosd(360*julian/365)).*cosd(inc);
kt = Gtotal./Go;
% calculate fraction of total radiation which is diffuse based on
% clearness index value
GdG = zeros(size(kt));
kt0 = find(kt > 1);
kt(kt0) = 1;
kt1 = find((kt >=0) & (kt <= 0.3));
kt2 = find((kt > 0.3) & (kt < 0.78));
kt3 = find(kt >= 0.78);
GdG(kt1) = 1.000-0.232*kt(kt1)+0.0239*sind(90-inc(kt1))...
-0.000682*Tair(kt1)+0.0195*humid(kt1);
if (GdG(kt1) > 1)
GdG(kt1) = 1;
end
GdG(kt2) = 1.329-1.716*kt(kt2)+0.267*sind(90-inc(kt2))...
-0.00357*Tair(kt2)+0.106*humid(kt2);
if (GdG(kt2) < 0.1)
GdG(kt2) = 0.1;
elseif (GdG(kt2) > 0.97)
GdG(kt2) = 0.97;
end
GdG(kt3) = 0.426*kt(kt3)-0.256*sind(90-inc(kt3))+0.00349*Tair(kt3)...
+0.0734*humid(kt3);
if (GdG(kt3) <= 0.1)
GdG(kt3) = 0.1;
end
% cleaning up
nosun = find(inc > 90);
GdG(nosun) = 0;
inc(nosun) = 90;
Go(nosun) = 0;
kt(nosun) = 0;
Gd = GdG.*Gtotal; % diffuse radiation (W/m2)
Gb = Gtotal - Gd; % beam radiation (W/m2)
66
%---------------------------------------------------------------------function Tdp = dewpoint(Ta, RH)
% from ASAE standard D271 .2 APR1979 (R2005) - 'Psychrometric Data'
% calculate dewpoint temperature (oC) as function of air temperature
(oC)
% and relative humidity fraction
for m=1:length(Ta)
pvs(m)=psat(Ta(m)+273.15);
end
p = pvs'.*RH;
if ((p < 620.52) | (p > 4688396))
fprintf('out of range - dewpoint calculation')
end
a(1) = 19.5322;
a(2) = 13.6626;
a(3) = 1.17678;
a(4) = -0.189693;
a(5) = 0.087453;
a(6) = -0.0174053;
a(7) = 0.00214768;
a(8) = -0.138343e-3;
a(9) = 0.38e-5;
TdpK = 255.28;
for n = 1:9
TdpK = TdpK+a(n)*(log(0.00145*p)).^(n-1);
end
Tdp = TdpK - 273.15;
%---------------------------------------------------------------------function ps = psat(TK)
% from ASAE standard D271 .2 APR1979 (R2005) - 'Psychrometric Data'
% calculate saturated vapor pressure (Pa) as a function of temperature
(K)
r = 22105649.25;
a = -27405.526;
b = 97.5413;
c = -0.146244;
d = 0.12558e-3;
e = -0.48502e-7;
f = 4.34903;
g = 0.39381e-2;
if ((TK >= 273.16) && (TK <= 533.16))
lnps_r = (a+b*TK+c*TK^2+d*TK^3+e*TK^4)/(f*TK-g*TK^2);
ps = r*exp(lnps_r);
elseif ((TK >= 255.38) && (TK < 273.16))
lnps = 31.9602-(6270.3605/TK)-0.46057*log(TK);
ps = exp(lnps);
else
fprintf('out of range - saturated vapor pressure calculation')
end
67
%---------------------------------------------------------------------function [h_ci, hm_ci, hfg_Jg] = inside_conv(T1, T2)
global param
% calculate heat and mass transfer coefficients between soil and tarp
Tave = (T1+T2)/2; % average temperature (K)
beta = 1/Tave; % expansion coefficient (1/K)
g = 9.807; % gravitational constant (m/s2)
dgap = param (20,:)/100; % height of air gap (m)
% calculate properties of moist air at Tave
[k_m, p_m, cp_m mu_m] = satair_props(Tave);
% calculate Rayleigh number
Ra = g*beta*p_m^2*cp_m*(T1-T2)*(dgap^3)/(k_m*mu_m);
% calculate Nusselt number
if (Ra == 0)
Nu = 0;
elseif (T1 < T2)
Nu = 1;
else
A = 1-1708/Ra;
B = ((Ra/5830)^(1/3))-1;
if A < 0
A = 0;
end
if B < 0
B = 0;
end
Nu = 1 + 1.44*A + B;
end
h_ci = Nu*k_m/dgap; % convection heat transfer coefficient (W/m2-oC)
D = 21.7e-6*((Tave/273.15)^1.88); % diffusivity, water vapor in air
(m2/s)
Le = k_m/(p_m*cp_m*D); % Lewis number
hm_ci = (h_ci*D*Le^(1/3))/k_m; % convection mass transfer coefficient
(m/s)
% interpolate to determine heat of vaporization (J/g) as a function of
% temperature (from Incropera and Dewitt)
TK_table2 = [273.15 275 280 290 300 310 320 330 340 350 360 370 380 390
400];
hfg_table = [2502 2497 2485 2461 2438 2414 2390 2366 2342 2317 2291
2265 ...
2239 2212 2183];
hfg_Jg = interp1(TK_table2, hfg_table, Tave, 'linear');
68
%---------------------------------------------------------------------function h_co = outside_conv(T1, T2, vel)
global param
% calculate heat transfer coefficient between tarp and ambient air (T1,
T2
% in K, vel in m/s)
Tave = (T1+T2)/2; % average temperature (K)
beta = 1/Tave; % expansion coefficient (1/K)
g = 9.807; % gravitational constant (m/s2)
L = param(21,:); % tarp dimension, in predominant wind direction (m)
A = L*param(22,:); % tarp surface area (m2)
P = 2*L + 2*param(22,:); % tarp perimeter (m)
% calculate properties of dry air at Tave
[k_d, p_d, cp_d mu_d] = dryair_props(Tave);
% calculate Reynolds and Rayleigh numbers
Re = vel*L*p_d/mu_d;
Ra = g*beta*p_d^2*cp_d*(T1-T2)*((A/P)^3)/(mu_d*k_d);
% calculate natural convection coefficient, hn (W/m2-oC)
if (Ra < 1e4)
Nun = 0;
elseif ((Ra > 1e4) && (Ra < 1e7))
Nun = 0.54*Ra^0.25;
elseif ((Ra >= 1e7) && (Ra < 1e11))
Nun = 0.15*Ra^(1/3);
else fprintf('out of range - Nusselt number, natural convection, tarpair');
end
hn = Nun*k_d/(A/P);
% calculate forced convection coefficient, hf (W/m2-oC)
Nuf = 0.037*(Re^(4/5))*((cp_d*mu_d/k_d)^(1/3));
hf = Nuf*k_d/L;
% select larger coefficient
if (hf >= hn)
h_co = hf;
else h_co = hn;
end
69
%---------------------------------------------------------------------function cp = cpf(TK)
% determine specific heat of water (J/g-oC) at temperature TK (K) by
% interpolating from table
if ((TK < 275) || (TK > 400))
fprintf('out of range - specific heat of water')
end
TK_table = [275 280 285 290 295 300 305 310 315 320 325 330 335 340 345
...
350 355 360 365 370 375 380 385 390 400];
cpf_table = [4.211 4.198 4.189 4.184 4.181 4.179 4.178 4.178 4.179 ...
4.180 4.182 4.184 4.186 4.188 4.191 4.195 4.199 4.203 4.209 4.214 ...
4.220 4.226 4.232 4.239 4.256];
cp = interp1(TK_table, cpf_table, TK, 'linear');
%---------------------------------------------------------------------function [k_da, p_da, cp_da, mu_da] = dryair_props(TK)
% interpolate properties of dry air from tables at temperature (K)
if ((TK < 250) || (TK > 400))
fprintf('out of range - dry air properties')
end
% from Cengel heat transfer text
% p (kg/m3), cp (J/kg-oC), k (W/m-oC), mu (kg/m-s or Ns/m2)
TK_table = [250 280 290 298 300 310 320 330 340 350 400];
k_table = 1e-2*[2.23 2.46 2.53 2.59 2.61 2.68 2.75 2.83 2.90 2.97
3.31];
p_table = [1.413 1.271 1.224 1.186 1.177 1.143 1.110 1.076 1.043 1.009
0.883];
cp_table = [1003 1004 1005 1005 1005 1006 1006 1007 1007 1008 1013];
mu_table = 1e-5*[1.61 1.75 1.80 1.84 1.85 1.90 1.94 1.99 2.03 2.08
2.29];
k_da = interp1(TK_table, k_table, TK, 'linear');
p_da = interp1(TK_table, p_table, TK, 'linear');
cp_da = interp1(TK_table, cp_table, TK, 'linear');
mu_da = interp1(TK_table, mu_table, TK, 'linear');
70
%---------------------------------------------------------------------function [k_ma, p_ma, cp_ma, mu_ma] = satair_props(TK)
% calculate/interpolate properties of moist air (rh=1) at temperature
(K)
if ((TK < 275) || (TK > 400))
fprintf('out of range - moist air properties')
end
TC = TK - 273.15; % temperature (oC)
patm = 101325; % atmospheric pressure (Pa)
pv = psat(TK); % saturated vapor pressure (Pa)
[k_da, p_da, cp_da, mu_da] = dryair_props(TK);
% from Incropera and Dewitt heat transfer text
% properties for saturated water vapor, cp (J/kg-oC)
TK_table = [275 280 285 290 295 300 305 310 315 320 325 330 335 340 345
...
350 355 360 365 370 375 380 385 390 400];
cpg_table = 1e3*[1.855 1.858 1.861 1.864 1.868 1.872 1.877 1.882 1.888
1.895...
1.903 1.911 1.920 1.930 1.941 1.954 1.968 1.983 1.999 2.017 2.036 ...
2.057 2.080 2.104 2.158];
cpg = interp1(TK_table, cpg_table, TK, 'linear');
H = 0.6219*pv/(patm-pv); % humidity ratio (kg H2O/kg dry air)
xw = pv/patm; % mole fraction (mol water vapor/mol air)
p_ma = (patm-pv)/(287*TK); % air density (kg dry air/m3)
cp_ma = cp_da + H*cpg; % specific heat, moist air (J/kg dry air-oC)
% interpolate viscosity of moist air based on mole fraction and
temperature
% (oC) - from Table 9 in Mason and Monchick (1965), kg/m-s
xw_table = 0:0.1:1;
TC_table = 0:20:140;
mu_table = 1e-8*[1717 1701 1666 1614 1546 1464 1369 1263 1146 1020 885
1815 1796 1759 1705 1635 1551 1453 1344 1225 1095 957
1908 1888 1849 1793 1722 1635 1536 1425 1303 1170 1029
2000 1977 1937 1879 1806 1718 1617 1504 1380 1245 1101
2089 2064 2022 1963 1889 1800 1698 1583 1457 1320 1174
2174 2148 2104 2044 1969 1879 1776 1660 1533 1394 1246
2258 2230 2185 2124 2048 1957 1853 1736 1608 1468 1318
2340 2310 2264 2203 2126 2034 1929 1812 1682 1541 1390];
mu_ma = interp2(xw_table, TC_table, mu_table, xw, TC, 'linear');
%interpolate thermal cond. of moist air based on mole fraction and
% temperature (oC) - from Table 10 in Mason and Monchick (1965), W/m-oC
k_table = (1e-5/0.2390)*[573 577 573 563 547 526 501 474 444 410 374
610 614 611 601 585 564 539 512 481 447 410
648 651 648 638 622 601 576 549 518 483 446
684 688 684 675 659 638 613 586 555 520 482
720 724 720 711 696 675 651 624 592 557 519
755 759 756 747 732 712 688 661 630 595 557
789 793 790 782 768 748 724 698 667 632 594
822 827 825 817 803 784 761 735 705 670 633];
k_ma = interp2(xw_table, TC_table, k_table, xw, TC, 'linear');
71
References:
[1] DeVay, J. E., 1991, "Use of Soil Solarization for Control of Fungal and Bacterial
Plant Pathogens Including Biocontrol," FAO Plant Production and Protection Paper,
[2] Warwick, N., Pyle, J. and Shallcross, D., 2006, "Global Modelling of the
Atmospheric Methyl Bromide Budget," Journal of atmospheric chemistry, 54 (2) pp.133159
[3] Ristaino, J. B. and Thomas, W., 1997, "Agriculture, Methyl Bromide, and the Ozone
Hole: Can We Fill the Gaps?," Plant Disease, 81 (9) pp.964-977
[4] Serca, D., Guenther, A., Klinger, L., Helmig, D., Hereid, D. and Zimmerman, P.,
1998, "Methyl Bromide Deposition to Soils," Atmospheric Environment, 32 (9) pp.15811586
[5] Goodhue, R. E., Fennimore, S. A. and Ajwa, H. A., 2005, "The Economic
Importance of Methyl Bromide: Does the California Strawberry Industry Qualify for a
Critical Use Exemption from the Methyl Bromide Ban?," Applied Economic Perspectives
and Policy, 27 (2) pp.198-211
[6] Gan, J., Yates, S., Ernst, F. and Jury, W., 2000, "Degradation and Volatilization of
the Fumigant Chloropicrin after Soil Treatment," Journal of environmental quality, 29 (5)
pp.1391-1397
[7] Baker, K., 1962, "Principles of Heat Treatment of Soil and Planting Material,"
Journal of the Australian Institute of Agricultural Science, 28 (2) pp.118-126
[8] Olsen, C. and Baker, K., 1968, "Selective Heat Treatment of Soil, and Its Effect on
the Inhibition of Rhizoctonia Solani by Bacillus Subtilis," Phytopathology, 58 pp.79-87
72
[9] Stapleton, J., 1991, "Thermal Inactivation of Crop Pests and Pathogens and Other
Soil Changes Caused by Solarization," FAO Plant production and protection paper,
[10] Hagan, H. R., 1933, "Hawaiian Pineapple Field Soil Temperatures in Relation to the
Nematode Heterodera Radicicola (Greef) Muller," Soil Science, 36 (2) pp.83-95
[11] Grooshevoy, S., Levykh, P. and MALBIEVA, M. E. I., 1940, "Methods of
Disinfecting Seed-Bed Soil by Natural Sources of Heat," The AI Mikoyan pan-Soviet sci.
Res. Inst. Tob. and Indian Tob. Ind.(VITIM)], Rostoff-on-Don, (Publ. 141) pp.49-61.
[12] Katan, J., Greenberger, A., Alon, H. and Grinstein, A., 1976, "Solar Heating by
Polyethylene Mulching for the Control of Diseases Caused by Soil-Borne Pathogens
[Verticillium Dahliae, Fusarium Oxysporum, Tomatoes, Eggplants, Fungus Diseases],"
Phytopathology, 66
[13] Katan, J. and Gamliel, A., 2009, "Soil Solarization – 30 Years On: What Lessons
Have Been Learned? ," Recent Developments in Management of Plant Diseases, Plant
Pathology in the 21st Century, Springer, Netherlands, pp.265-283, Chap.19
[14] Carpenter, J., Lynch, L. and Trout, T., 2001, "Township Limits on 1, 3-D Will
Impact Adjustment to Methyl Bromide Phase-Out," California Agriculture, 55 (3) pp.1218
[15] Davis, J., 1991, "Soil Solarization: Yield and Quality Benefits for Potato in a
Temperate Climate, Short-and Long-Term Effects and Integrated Control," FAO Plant
Production and Protection Paper,
73
[16] Pullman, G., DeVay, J. and Garber, R., 1981, "Soil Solarization and Thermal Death:
A Logarithmic Relationship between Time and Temperature for Four Soilborne Plant
Pathogens," Phytopathology, 71 (9) pp.959-964
[17] Tjamos, E. and Paplomatas, E., 1988, "Long‐ Term Effect of Soil Solarization in
Controlling Verticillium Wilt of Globe Artichokes in Greece," Plant pathology, 37 (4)
pp.507-515
[19] Egley, G. H., 1990, "High-Temperature Effects on Germination and Survival of
Weed Seeds in Soil," Weed Science, pp.429-435
[20] Gamliel, A. and Stapleton, J., 1993, "Effect of Chicken Compost or Ammonium
Phosphate and Solarization on Pathogen Control, Rhizosphere Microorganisms, and
Lettuce Growth," Plant Disease, 77 (9) pp.886-891
[21] Simmons, C. W., Guo, H., Claypool, J., Marshall, M. N., Perano, K. M., Stapleton,
J. J. and VanderGheynst, J. S., 2012, "Managing Compost Stability and Amendment to
Soil to Enhance Soil Heating During Solarization," University
[22] Marshall, M. N., 2012, "Development and Validation of Predictive Models of Soil
Heating and Pathogen Inactivation During Soil Solarization," Ph.D Dissertation
University of California, Davis
[23] Duan, X. and Naterer, G., 2009, "Heat Conduction with Seasonal Freezing and
Thawing in an Active Layer near a Tower Foundation," International Journal of Heat and
Mass Transfer, 52 (7-8) pp.2068-2078
[24] FitzGerald, D., 1886, "Evaporation," Transactions of the American Society of Civil
Engineers, 15 (1) pp.581-646
74
[25] Horton, R., Chung, S., Hanks, J. and Ritchie, J., 1991, "Soil Heat Flow," Modeling
plant and soil systems., pp.397-438
[26] Penman, H. L., 1948, "Natural Evaporation from Open Water, Bare Soil and Grass,"
Proceedings of the Royal Society of London. Series A. Mathematical and Physical
Sciences, 193 (1032) pp.120-145
[27] Van Bavel, C. and Hillel, D., 1976, "Calculating Potential and Actual Evaporation
from a Bare Soil Surface by Simulation of Concurrent Flow of Water and Heat,"
Agricultural Meteorology, 17 (6) pp.453-476
[28] Waggoner, P. E., Miller, P. M. and DE Roo, H., 1960, "C. Plastic Mulching:
Principles and Benefits," Bull. Conn. Agric. Exp. Stn., 634
[29] INADA, K., 1973, "Photo-Selective Plastic Film for Mulch," Jpn. Agric. Res. Q, 7
pp.253-256
[30] Mahrer, Y., 1979, "Prediction of Soil Temperatures of a Soil Mulched with
Transparent Polyethylene," Journal of applied meteorology, 18 (10) pp.1263-1267
[31] Chung, S. O. and Horton, R., 1987, "Soil Heat and Water Flow with a Partial
Surface Mulch," Water Resources Research, 23 (12) pp.2175-2186
32] DWR, 2006, "California Irrication Management Information System (Cimis),"
California Department of Water Resources, http://www.cimis.water.ca.gov
[33] Simkins, S. and Alexander, M., 1984, "Models for Mineralization Kinetics with the
Variables of Substrate Concentration and Population Density," Applied and
Environmental Microbiology, 47 (6) pp.1299-1306
75
[34] Aslam, D. N., VanderGheynst, J. S. and Rumsey, T. R., 2008, "Development of
Models for Predicting Carbon Mineralization and Associated Phytotoxicity in CompostAmended Soil," Bioresource technology, 99 (18) pp.8735-8741
[35] VanderGheynst, J. S., Walker, L. P. and Parlange, J. Y., 1997, "Energy Transport in
a High‐ Solids Aerobic Degradation Process: Mathematical Modeling and Analysis,"
Biotechnology progress, 13 (3) pp.238-248
[36] McQuarrie, D. A. and Simon, J. D., 1997, Physical Chemistry: A Molecular
Approach, Univ Science Books.
[37] Reddy, A., Jenkins, B. and VanderGheynst, J., 2009, "The Critical Moisture Range
for Rapid Microbial Decomposition of Rice Straw During Storage," Transactions of the
ASAE, 52 (2) pp.673-677
[38] Willmott, C. J. and Matsuura, K., 2005, "Advantages of the Mean Absolute Error
(MAE) over the Root Mean Square Error (RMSE) in Assessing Average Model
Performance," Climate Research, 30 (1) pp.79