This is commonly known as the Fourier number. Its magnitude is a measure of
the degree that heating or cooling effects have penetrated through the solid
(or soil). If a/22 is small, a large t value is required before significant
temperature changes occur through the solid, if a/2? is large, the reverse is

true.
D. GENERALIZATION OF EXPERIMENTAL DATA

Note that the role played by the dimensionless numbers in the preceding
example is to group all rectangles (Example 1) and all systems of heat con-
ductions (Example 2). The grouping is accomplished by the dimensionless
numbers Ky and Fo respectively. Within each of the two groups of phenomena
the individual members differ by only a scale factor; that is, all processes

within each group are "similar".

For the case of Example 2, one can state the problem of similarity ina
converse manner; namely, if the temperature distribution in two geometrically
similar bodies is the same, then Fo for these processes must also be equal,
regardless of the individual values a, t and 2. The dimensionless combination
is a generalized variable or criterion of similarity. One can regard the
Fourier number as the generalized time, transforming one process into a simi-

lar process.

Instead of computing the values of all like points of two similar sys-
tems, it is sufficient to compare dimensionless numbers that contain the mean
values of the quantities in the range under consideration. This is funda-
mental in considering experimental data. In an experiment related to Example
2, it is necessary to measure all the values which are parts of the dimension-
less criterion Fo. However, the results of the experiment must be presented
not in terms of a relationship between separate values of a, t, 2, t and x,
but in terms of temperature as a function of the Fourier number and a coor-

dinate system x j.e.,

t= fG , Fo) (8)

12


These processes illustrate an imbalance in both energy and mass exchange. For
example, from 10 to 20 kg of carbon contained in 30-80 kg of COg may be lost
by a hectare of soil. An additional 100-200 g of nitrogen may be either in-
corporated or lost.

The presence of plants will alter the picture further. Green plants
affect the energy and mass transfer processes in a complex manner: carbon from
atmospheric COj and water and mineral nutrients from the soil are rapidly
incorporated into the photosynthetic organs. These elements, and also hydro-
gen from photochemically decomposed water, combine to form organic molecules
under the action of sunlight; simultaneously oxygen is liberated into the
atmosphere.

Formally, this process is partially "reversible": about 25-30% of the
organic substances produced during photosynthesis are oxidized as a result of
respiration, a suitable amount of oxygen is absorbed, C02 is evolved, and a
corresponding amount of energy stored during photosynthesis is released as
heat. However, photosynthesis and respiration are spacially and temporally
separated from each other. Thus, photosynthesis occurs only in the presence
of light and in special photosynthesizing organs, whereas respiration proceeds
jin all living cells, and is not reversed photosynthesis.

This account is given to show that "plant-cover soil system" photosyn-
thesis proceeds with an active balance. Its total mass increases, as does the
amount of bound energy. The amount of COg absorbed per hectare of plants
during a day of active photosynthesis may reach 800 or even 1000-1200 kg. The
amount of oxygen evolved is 640-960 kg, the amount of newly formed plant mass
reaches 400, perhaps 500-600 kg per day and the amount of energy stored by the
system is 1.6 to 2.4 kcal.

The situation is further complicated as grazing insects and animals are
added to the system. In a microcosm environment introduction of toxicants
adds yet a new dimension. We will now be dealing with behavioral problems as
well as animal life activities such as "tunneling" and constant "working" of


radiation, the depth for heat transfer averages from 50 to 100 cm, (Kreith and
Sellers, 1975). For the moisture, it extends to the impervious layer of the
soil where groundwater flows horizontally.

The net radiation, which is the difference between the incoming and
outgoing streams of solar and longwave sky or terrestrial radiation, must be
balanced by (1) the sensable heat flux between the surface and air, (2) the
evaporative heat flux between the surface and the air, and (3) the total heat
flux between the surface and the soil column itself.

The net radiation is a function of incoming solar radiation, surface
reflectivity, surface emissivity, surface temperature, air temperature and

vapor pressure.

The sensable heat flux is directly proportional to the temperature
difference between the surface and air. The important quantity entering the
proportionality is the coefficient of surface heat transfer a, It varies
strongly with turbulence of the ambient air. The evaporative flux likewise
introduces into the problem a mass transfer coefficient denoted by On It
relates the flux with vapor pressure of the soil and air.

The preceding discussion is a very brief account of heat and mass trans-
fer processes that take place in abiotic systems of soil environment. These
are uncoupled from biotic processes only in the sense that major impacts of
Plant and animal activities must be combined and included in an appropriate
way into one or more of the parameters outlined above. Boundary conditions
have been taken at the air soil surface and no complications caused by "edge
effects" etc., that are common in microcosms are considered. The fgl lowing
sections include a more specific account of the rules and criteria that scale
these processes from one system into another.

B. PROPOSED CRITERIA FOR SCALING MICROCOSM PROCESSES

The complete boundary conditions suitable for a microcosm environment
have not been analyzed as a part of the processes of heat and mass transfer.

16


FOREWORD

Effective regulatory and enforcement actions by the Environmental Protection
Agency would be virtually impossible without sound scientific data on pollu-
tants and their impact on environmental stability and human health. Respon-
sibility for building this data base has been assigned to EPA's Office of
Research and Development and its 15 major field installations, one of which
is the Corvallis Environmental Research Laboratory (CERL).

The primary mission of the Corvallis Laboratory is research on the effects
of environmental pollutants on terrestrial, freshwater, and marine ecosystems;
the behavior, effects and control of pollutants in lake systems; and the de-

velopment of predictive models on the movement of pollutants in the biosphere.

This report presents theoretical criteria for scaling of microcosms used
for screening toxic substances.

James C. McCarty
Acting Director, CERL

tii


The functional form of equation (8) can be determined from experimental data.
Once this is done, the group representation will be narrowed down to a single
or some specified experiments.

This example demonstrates the power of Theory of Similarity and dimen-
sional reasoning. In more complex systems with tens of variables the number
of dimensionless criteria increase. Treating experimental data with dimen-
sionless criteria permits a rational design of experiments and a condensed
presentation of results.

13


two systems. An appropriate form of Lu may also be used to account for fil-
tration, i.e., tu, = ay/a, where a is filtration diffusivity. Lu increases
with moisture content of a system. Thus, when the spread of mass transfer

potential is greater than temperature, Lu>]. For this reason, the filtration

number Lu, could be 10 to 100 times greater than Lu.
Criterion Bi

This is commonly known as Biot Number. It relates the heat and mass
transfer between the soil and air. Its numerical value is aR/a, for Big and
a R/ay, for Bi, As we know from the earlier discussion 4 and a, are the
convective transfer coefficients for heat and for vapor, respectively. Again,
the connection between time coordinate of the system and transfer of heat
(mass) from and to the soil is established by Criterion Bi. In a like manner
Bi, can be defined for radiative transport which includes surface reflectivity

and emissivity.
Criterion ¢€

We have discussed this dimensionless number, which is purely a function
of the soil composition, and it can be interpreted as the resistance (1/e) to
moisture or heat transports through the soil matrix.
Criterion Ko

This is the ratio of heat (rAu) expended in evaporation of incremental
mass Au of liquid water in heating the wet soil to an equivalent amount (c,At).

This is one way of bringing the specific heat of the soil fq and the heat of
evaporation r into account for the transformation.

Criterion Pn

As explained earlier, mass transfer in either vapor or liquid phase can
take place within the soil as a direct consequence of temperature gradient.

19


liquid potential. The coefficient that links the liquid potential with the
soil property « and mass flow is the diffusivity coefficient ay. This is a
function of moisture content because the cross sectional area available for
the flow of the liquid changes with e. One also must expect a rapid change in
the moisture diffusity as the liquid changes from a discrete to continuous

phase. The moisure diffusity an depends on soil types, temperature, and pres-
sure.

(c) Both vapor and mass transport may take place under isothermal condi-
tions primarily by diffusion. The vapor mass transport under temperature
gradient may contain convection, because the air in the soil may no longer be
at rest. The coefficient of vapor diffusion ay relates the non-convective
vapor mass transport in the soil with the soil properties and the gradient of
capillary vapor potential. The moisture diffusivity an, of vapor and liquid
can be used to account for both phases when they occur simultaneously.

(d) Energy transport in the soil will occur by heat conduction in the
solid and liquid (heat conduction in the air can be neglected) and by trans-
port of the heat of evaporation r. When the air in the soil is moving,
energy also will be transported by convection. The coefficient of thermal
diffusivty aq connects the heat flux to the soil properties and the gradient
of temperature in the soil. Among the soil properties of importance are soil
heat capacity fq and bulk density.

For temperatures as they generally occur (in the shade), the convection
transport within the body of the soil contributes to the total heat flux ina
minor way. However, the contribution increases rapidly with increasing temp-
erature, so that around 60°C (when the soil is saturated) the convection
contribution could equal the contribution from conduction (Eckert and Pfender
1978).

(e) Energy and mass transfers must be considered for the soil column

extending from the top of the vegetation to the soil depth where vertical
transfer of energy and mass is negligible. For the daily cycle of input solar

15


By attaching different values to the coefficient a whole series of fig-
ures is obtained (Fig. 1-b). These figures are similar among themselves
because their sides are proportional, i.e.

BL TOE Fre = Ke )

Therefore, on multiplying the sides of the basic figure by some quantity

Ky (which can be given any arbitrary value, but the same for both sides), a

group of similar figures is obtained. The values Ky are called the transfor-
mation multipliers or dimensionless numbers.

By so constructing a group of figures, every one of them differs from the
others within a given group only by its scale; distortions of geometric form
do not occur. In addition, every point on one figure corresponds to a similar
point on the other figures. In this type of geometry the transformations of
the figures are called similar. This term is used also in the Theory of
Similarity.

Sometimes it is difficult to maintain exact geometric similarity. For
example, if the depth of the soil is transformed by a factor Ky, all linear
dimensions such as soil grain size must also be similarly transformed other-
wise the transformation is distorted. The formal Theory of Similarity does
not deal with the problem of distortion, even though distorted models are
widely used (Bogardi 1974). In these cases it is endeavored to ensure the
similarity of the dominating processes and phenomena.

Example 2
Consider the partial differential equation:
azt _ at
aoe = bt (2)

which is generally referred to as the one-dimensional diffusion equation.
Depending on the physical significance attached to the variables, equation (2)
is representative of a wide class of physical phenomena. For example, if we
define t as temperature, a as thermal diffusivity, t as time and choose x as a

10


SECTION 2
A STATEMENT OF THE PROBLEM

When microcosms are used as tools to screen toxicants under a variety of
circumstances and conditions, they will produce consistent results only when
they are applied with uniform criteria. Criteria must be established not only
for the construction and composition of microcosm system components, but also
for the level of accuracy in conducting measurements and interpreting final
results. When these precautions are observed then toxicant screenings will
reduce the risk of either approving some potentially hazardous materials or
needlessly preventing the use of some valuable ones.

Criteria must be based on our scientific understanding of physical and
biological processes taking place in a microcosm. Criteria should be used to
attain consistency and uniformity of test results under generally similar test
conditions. But, criteria for establishing similarity of test conditions
should not be too stringent lest the cost of duplicating specific procedures
and exactly similar systems become prohibitive. Also, in practice it might be
necessary to deviate slightly from certain prescribed requirements in order to
more closely match a given facility, operation or hypothesis (Gillett and
Witt 1977).

In establishing the protocols or criteria, the question should not be how
closely these have been followed. Obviously, general rules of similarity must
be adhered to, but primarily we must be prepared to interpret and compare
results of different microcosm systems when several components and procedures
have deviated somewhat from an established baseline. In other words, the
criteria of uniformity of results must be established not on the basis of an
absolutely identical system, but on the basis of general rules for similarity
of components and conditions, leaving some flexibility in selection of compo-
nent sizes and properties as well as in environmental conditions of the
laboratory.


Luikov, A. V. 1975. "Systems of Differential Equations of Heat and Mass
Transfer in Capillary-Porous Bodies (Review). International J. of Heat
and Mass Transfer. Vol. 18:1-14 Pergammon Press, NY.

Nichiporovich, A. A. 1975. “Energy and Mass Transfer in Plant Communities"
Heat and Mass Transfer in the Biosphere, D. A. deVries and N. H. Afgan,
editors. Script Book Co. Washington, DC.

25


dt*/dF,

0.8 p

EXAMPLE

* Lu do*/dF dt*/dF,
0.6 5 32 -26
0.6 15 £13 .03

—

x] °
UID

1.0

Connection Between Heat-Mass Transfer Potentials and Their Rates of Change With Lu.


1)
2)
3)

4)

5)
6)
7)

CONTENTS

OBJECTIVES. © 6 we ee ee 1
A STATEMENT OF THE PROBLEM. - ~~ - ee eee ee eee ee eet 2
GENERAL APPROACH AND EXAMPLES - - - 2 - + ee ee ee ee ee ees 5
A) UNCOUPLED ABIOTIC PROCESSES. - - +--+ ee eee ee eee 5
B) FUNDAMENTAL THEORY OF SIMILARITY - - +--+ e+e ee eee eee 7
C) DIMENSIONLESS NUMBERS AS SCALING CRITERIA. -- +--+ ++ ee = 9

EXAMPLE 1. 6 ee ee ee ee 9

EXAMPLE 2 2 6 6 ee ee ee ee 10
D) GENERALIZATION OF EXPERIMENTAL DATA- - ~~ +--+ +++ ee eee 12
SPECIFIC APPROACH . 2. 1 ee ee ee ee ee 14
A) MAJOR PROCESSES OF HEAT AND MASS TRANSFER IN TERRESTRIAL SYSTEMS 14
B) PROPOSED CRITERIA FOR SCALING MICROCOSM PROCESSES. - +--+ - + 16
DISCUSSION. © 6 we 21
CONCLUSION AND RECOMMENDATION . - - - eee ee tee ee ee es 23
REFERENCES . 1 1 wwe ee ee 24


ABSTRACT

Theoretical developments based on heat and moisture transfer in soil lead
to dimensionless numbers that describe important processes taking place in
porous media. It is proposed that these numbers can be used as preliminary
scientific criteria for scaling the results from microcosms both as a

means of comparing two generally similar but non-identical systems as well as
for better understanding the real world.


coordinate directed away from an infinite plane, equation (2) describes the
propagation of heat in an infinite solid, or soil when the soil is suddenly
heated to a temperature exceeding that of the surroundings. Again, we might
define t as a velocity of a flow parallel to a plate, choose a as kinematic
viscosity, let x be the normal distance from the plate and t be time. Equa-
tion (2) then describes the velocity variation of the flow if the wall is
suddenly set into motion. Diffusion of vorticity in a fluid, slowing down of
neutrons in matter, etc., are also represented by equation 2. Each of these
is a class of processes.

Now, assume we have two systems describing heat transfer into soil. One
is defined by the value of its parameters t', t', x' and a' and the other by
tisk, t! ; wSK t! ; xtak x! > and al'=K,a'. The second system is derived from the
first one by corresponding transformation multipliers. To establish simi-
larity for these two systems equation 2 must remain valid on transformation
from the first to the second system. Hence:

2+! 1
e2t" ot"
"5ymz = “By @

By expressing the prime quantities in terms of double prime quantities, we

obtain: .
KK a OK '
a @tto=¢ & (5)
x T

By comparing equation (5) with (3), it is seen that they can be simultaneously
true only if:

KK Ke KK =] 6
[rn cn (6)
x Tt x

If we insert the physical values for the multipliers in equation (6), we

obtain the dimensionless number:

Fo = = 7)

11


SECTION 1
OBJECTIVES

Microcosms serve at least two useful purposes: (1) to screen toxicants
in a safe and economical way, and (2) to enable simulation and the study of
ecological processes under controlled environments. The objectives of this
proposal are to address both items, namely, (a) to establish scientific cri-
teria for interpreting and comparing results from generally similar but not
absolutely identical microcosms, and (b) to establish theoretical and empir-
ical scaling transformations which link certain important processes occurring
in the microcosm with those in the real world.

Valid, uniform criteria are necessary to reduce the risk of approving
potentially harmful substances and to needlessly prevent the use of some valu-
able ones. Unfortunately, the opportunities for committing errors of this
type are not imagined, but very real. While it is readily admitted that
microcosms are used to “compress'' the time scale of the actual process, it is
not known how this impacts the results. It is the objective of this proposal
to show that definitive, although preliminary progress can be made to provide
some of these answers. However, to do this we must work with an area of
research where frontiers of knowledge in many disciplines must be integrated
to achieve a significant contribution. The following discussion presents a
basic road map to get us there.


Notwithstanding, we propose to begin with an analysis that otherwise most
closely approximates microcosms and use this scaling criterion as a prelim-
inary step. Appropriate modifications will be introduced in the course of the
study as opportunities for further progress become evident.

Luikov (1966) in his celebrated book, and more recently in a review paper
(1975) derived the systems of differential equations of heat and mass transfer
in capillary porous media. He then presented a comprehensive solution for an
infinite plate with impermeable bottom layer and an exposed top. We shall
make extensive use of his analysis without recounting the mathematics, which
is somewhat detailed and involved. For a review of a less complex treatment,
but also less general, the paper by Eckert and Pfender (1978) should be con-
sulted.

The criteria for scaling heat and mass transfer are dimensionless numbers
that transform space and time distribution of temperature t* and moisture 0*
from one system into another. Both t* and 6* are nondimensionalized with
respect to the initial states of temperature and moisture differences, respec-
tively, in the soil and the air. These functions, expressed implicitly in
terms of scaling criteria are:

t*

f (f Fo, Lu, Big, Bij, &s Ko, Pn)

OF = 9 (Ra Fos Lu, Bigs Bigs es Koy Pn)

We shall give the physical meaning of these criteria and rely on the
reader's intuition by recalling the previous two sections on similarity anal-
ysis to establish general acceptability of the following statements. It will
suffice to note from example 2 given earlier that t* can be written as a
function of zt and Fo, where x is the distance in the soil, R total depth of
the soil and Fo the Fourier number. We recall that the simplicity of this
result stems from the fact that the soil was considered completely dry. homo-
genous, and there was no mass transfer whatsoever. With respect to the pro-
posed, more complex system, we will, of course, use additional criteria. It

W7


United States Environmental Research EPA-600 3-79-017
Environmental Protection Laboratory Februar y 1979
Agency Corvallis OR 97330

Research and Development

SEPA Development of
Scaling Criteria for
Terrestrial
Microcosms

> P >


Hence, as a second objective of this proposal we seek to establish theoretical
transformations linking certain important processes in microcosms to the real
world.


If we accept this premise as reasonable, then how do we interpret results
of tests from one microcosm in terms of another? Even if we should insist
upon adherence to stringent requirements, then how closely should we demand
that these requirements be followed? What would be the impact of minor de-
viations on the results?

To summarize, if we anticipate successful uses of microcosms for toxicity
screening purposes at the outset, we should establish scientific criteria for
interpreting and comparing results obtained from two generally similar but
non-identical microcosm systems. The development of such criteria is the
statement of the first objective of this proposal.

When microcosms are used to simulate real processes taking place in the
natural ecosystem with the objective of better understanding them, we are no
longer comparing two generally similar systems as in objective one. In all
probability we are dealing with one real system which we do not understand
completely and one artificial system which is designed to mimic the real
world, even when we do not understand it well.

Haefner and Gillett (1976) proposed mathematical modeling of the micro-
cosm system as a step toward better understanding of the real world. The
feasibility of this strategy depends entirely upon our ability to relate some
of the important processes from the microcosm to the real world. Mathematical
models cannot perform magic. If it were possible to use mathematics to faith-
fully duplicate important processes in a microcosm, then our understanding
would be advanced enough to by-pass the microcosm altogether. It is true,
however, that a closely controlled process such as a microcosm is easier to

manipulate, observe, and understand.

It is also assumed that by going to a microcosm, an otherwise extremely
long-term dose-response process can be "compressed" in time and conducted
within weeks instead of years. The assumption is that time scales are not the
same. It is precisely for these reasons that one must establish some rules

that scale down the real process to a microcosm process, or vice versa.


REFERENCES

Bogardi, J. 1974. Sediment Transport in Alluvial Streams. Akadémiai Kiadd

Budapest.

Eckert, E. R. G. and Pfender. 1978. "Heat and Mass Transfer in Porous Media
with Phase Change. Proceedings of the 6th International Heat Transfer

Conference, Toronto, Canada.

Gillett, J. W. and Witt, J. M. 1978. "Chemical Evaluation: Projected Appli-
cation of Microcosm Technology". Presented at the Symposium on "Microcosms
in Ecological Research" Augusta, GA. Nov 8-10, 1978 for the Savannah
River Ecology Laboratory.

1977. "The Proceedings of the Workshop on Terrestrial Microcosms,
Symposium on Terrestrial Microcosms and Environmental Chemistry" held at
Oregon State University, 1977.

Haefner, J. W. and Gillett, J. W. 1976. "Aspects of Mathematical Models and
Microcosm Research. Proceedings of Environmental Protection Agency
Conference, Environmental Modeling and Simulation, Wayne Ott, editor.
EPA-600/9-76-016. pp. 624-28.

Hansen, A. G. 1964. Similarity Analysis of Boundary Value Problems in Engin-

eering. Prentice-Hall, NY.

Kreith, F. and Sellers. W. D. 1975. "General Principles of Natural Evapora-
tion" Heat and Mass Transfer in the Biosphere, D. A. deVries and N. oH.

Afgan, editors. p. 207. Script Book Co. Washington, DC.

Luikov, A. V. 1966. Heat and Mass Transfer in Capillary Porous Bodies.
Translated by P.W.B. Harrison, W. M. Pur, editor, Pergammon Press, NY.

24


SECTION 6
CONCLUSIONS AND RECOMMENDATIONS

It was demonstrated in this proposal first by discussion of fundamentals
of Theory of Similarity, second by physical reasoning, third by theoretical
analysis of specific problems and finally by a specific typical example, that
two apparently similar microcosm systems could produce totally divergent
results, for example with respect to mass and heat transfer.

These differences were calculated for the abiotic processes. We do not
know the extent or exact nature of all impacts of these differences on biotic
processes. Undoubtedly, the impact of these differences on the chemodynamics
of toxic substances in the abiotic portion would be to change chemical concen-
tration of the toxicant with respect to space and time. To the extent that
biological response is controlled by concentration of toxicants, the system
will differ more or less markedly in response to exposure to toxicants. The
accuracy, sensitivity and even practicability of measurements (and consequent
extrapolation to the real world) of toxic substance impact therefore depend on
sound and uniform similarity criteria applied to individual test systems.

23


ACKNOWLEDGMENT
The author is indebted to Jim Gillett of the Corvallis Environmental Research

Laboratory for his persistence in seeking help from a “systems modeler".
Without his stimulus my ideas on the subject would have remained dormant.

vi


SECTION 4
SPECIFIC APPROACH
A. MAJOR PROCESSES OF HEAT AND MASS TRANSFER IN TERRESTRIAL SYSTEMS

To introduce the criteria for scaling the processes of heat and mass
transfer in a real soil environment, Jet us explore the nature of these pro-
cesses and identify some key physical parameters of (a) the soil, (b) liquid
water transport, (c) vapor transport, (d) heat transport, and (e) boundary
conditions.

(a) Soil may be classified as clay, silt, fine and coarse sand, or gravel
according to average grain size d. The values of d distinguishing the four
soil types are .002, 0.02, 0.2 and 2 mm. Each soil type has a distribution of
particle sizes around the mean. Detailed knowledge of this distribution leads
to pore size distribution in the soil, which one can estimate by €, a dimen-
sionless parameter. The quantity 1/e represents resistance to diffusion of
vapor inside the soil. It shows how many times the coefficient of vapor
diffusion in air, ay. is greater than the coefficient of vapor diffusion
within the body of the soil. In a similar manner, the resistance of heat
diffusion in the solid parts of the soil must be obtained. These coefficients
change dramatically with the moisture content. Therefore, for each soil type,
they must be determined experimentally for several moisture contents.

The soil is saturated when water completely fills the space between the
particles. When unsaturated, parts of the space contain air and water vapor.
Under all conditions except at very high moisture content the air is in a con-
tinuum phase. When the continuity is interrupted, air exists in the form of
bubbles distributed throughout the continuous liquid phase.

(b) To consider liquid flow within the soil, one must be satisfied with a

continuous approach which disregards the particulate structure. The liquid
mass transport is thus readily linked with the gradient of the capillary

14


Pn links this interaction. As mass is transferred from one point to another
under its own gradient, its potential drops. Pn expresses this drop in terms
of incremental mass transfer Au and temperature drop At. Thus, pn = gar,
where 6 is a constant of thermal gradient coefficient. Just as in the case of

Ko, Pn is used to bring another thermodynamic property, 6, into account.

Since ec, Cp> r and 6 represent mechanical and thermodynamic properties, a
generalized criterion containing all these properties is often encountered in

analysis. It is Fe = or also obtained from the product of the last 3 cri-

teria, i.e., Fe = cKoPn. This number is independent of the heat and mass
transfer potentials.

20


DISCLAIMER

This report has been reviewed by the Corvallis Environmental Research
Laboratory, U.S. Environmental Protection Agency, and approved for
publication. Mention of trade names or commercial products does not
constitute endorsement or recommendation for use.

Vi


the soil. Outside the soil environment plant life interacts with wind motion
damping its turbulence, depleting its CO,, etc.

From this discussion it is obvious that development of complete scaling
rules for a microcosm containing all the physical, chemical and behavioral
factors is an impossible task. It may not even be required! We must follow
the suggestion made by Haefner and Gillett (1976) that the biotic and the
abiotic system are more or less "uncoupled", at least in the Ist order of
approximation. For example, the energy imbalance of photosynthesis is about
2-5% of incident radiation. This amount can either be ignored or included as
a “sink" without disrupting the major processes of heat transfer. Also, the
net exchange of C02, 02, No, etc. with air can be approximately accounted for
if the total mass of air is not appreciably changed in the process. The
impact of active root systems, tunneling, etc. can be included as a part of
the physical character of the soil, affecting heat and mass transfer.

In other words, the rules for uncoupled systems can be developed in a
formal theoretical manner but the utility is confirmed or denied by the
evidence of carefully conducted experiments, either existing or planned
(Gillett and Witt 1978).

Again, as a general guideline, an attempt will be made to account for
those biotic processes that tend to significantly alter mass, momentum and
energy transfer in the microcosm. In this manner, even though we would not
fully understand the completely coupled system, we would increase the pro-
bability of successfully simulating a system, uncoupled, but reasonably
realistic.

B. FUNDAMENTALS OF THE THEORY OF SIMILARITY

The concept and the formal statement of the general Theory of Similarity
is somewhat abstract, but it has a far-reaching practical application in the
conduct of scientific experiments and treatment of data. One statement of

this theory ascribed to Kirpichev-Gukhman is "Two phenomena are similar if


The Theory of Similarity combines the advantages of a purely theoretical
approach with those of a primarily empirical approach into a single powerful
analytical tool.

C. DIMENSIONLESS NUMBERS AS SCALING CRITERIA

To demonstrate the essential role of the Theory of Similarity, we provide
two examples each using a multiplier that transforms results from a class of

Processes or phenomena to a group of processes or phenomena.

Example 1
Examine the process of constructing a group of geometric figures as in

Figure 1. We can assume that the rectangles belong to a whole class of plane
shapes with the common property of all angles being right angles. To dis-
tinguish one single rectangle from the class, the numerical values. of the
sides 2, and 2. must be given: In this case 2, and 2g are the conditions of
single-valuedness. But a distinct group of figures will be obtained if we
assume the ratio of sides are a constant equal to the coefficient Kp.

fo
by i (a)
23 ag?
Qt £5°7
| el] (»)
1

Figure 1. Class and group of plane rectangular figures: a) class of figures
(sides have arbitrary length and b) group of similar figures (the ratio of
sides is a constant).


EPA-600/3-79-017
February 1979

DEVELOPMENT OF SCALING CRITERIA
FOR TERRESTRIAL MICROCOSMS

by

Mostafa A. Shirazi
Freshwater Systems Division
Corvallis Environmental Research Laboratory
Corvallis, Oregon 97330

Corvallis Environmental Research Laboratory
Office of Research.and Development
U.S. Environmental Protection Agency
Corvallis, Oregon 97330


will help if the reader establishes a mental picture of relations between the
criteria, the processes they each represent, and the processes they each
interact with.

Criterion x

This is the dimensionless space coordinate. It is normalized with re-
spect to the original depth of the soil R. It does not matter what the abso-
jute value of x might be, nor in what units it is presented, it is only neces~
sary to know the relative depth expressed in terms of a fraction of the total
depth in the two systems being compared. In this problem ; is the only
coordinate system. It means that ©* and t* are invariant with other dimen-
sions. It means there is no edge effect or major non-homogeneity in the soil,
in either the horizontal or the lateral directions. With respect to a con-
fined system such as a microcosm, it means that attention must be focused to
those central points of the microcosm that are immune to wall effects. Other-
wise, the problem must be reworked completely to comply with the principle of

single-valuedness discussed earlier.
Criterion Fo

This is the familiar Fourier number. This parameter translates the time
scale from one system to another. Numerically, it is equal to agt/R?, that
is, a measure of thermal diffusivity, aq with time t as heat moves across a
unit cross section of the soil.

Criterion Lu

This is simply the ratio of mass diffusivity for moisture to thermal dif-
fusivity for heat, i.e., an/ Ag: In the transformation from one system to
another, if we choose to associate thermal diffusivity with the time coordi-
nate as we have done by choosing the Fourier number, the connection of mass
transfer with the time coordinate will be established by knowing Lu for the

18


RESEARCH REPORTING SERIES

Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields
The nine series are

Environmental Health Effects Research

Environmental Protection Technology

Ecological Research

Environmental Monitoring

Socioeconomic Environmental Studies

Scientific and Technical Assessment Reports (STAR)
Interagency Energy-Environment Research and Development
“Special” Reports

9 Miscellaneous Reports

This report has been assigned to the ECOLOGICAL RESEARCH series This series
describes research on the effects of pollution on humans, plant and animal spe-
cies, and materials. Problems are assessed for their long- and short-term influ-
ences. Investigations include formation, transport, and pathway studies to deter-
mine the fate of pollutants and their effects. This work provides the technical basis
for setting standards to minimize undesirable changes in living organisms in the
aquatic, terrestrial, and atmospheric environments.

@PnNanFoONn=

This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161


TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)

7. REPORT NO. 2.

EPA-600/3-79-017

3. RECIPIENT'S ACCESSION NO.

4. TITLE AND SUBTITLE

Microcosms

IS. REPORT DATE

February 1979

Development of Scaling Criteria for Terrestrial

6, PERFORMING ORGANIZATION CODE

7. AUTHOR(S)
Mostafa A. Shirazi

Ig. PERFORMING ORGANIZATION REPORT NO.

Ig. PERFORMING ORGANIZATION NAME AND ADDRESS:

Ecosystems Modeling & Analysis Branch
Corvallis Environmental Research Laboratory
U.S. Environmental Protection Agency
200 S.W. 35th Street -- Corvallis, OR 97330

70. PROGRAM ELEMENT NO.

TAA602

71, CONTRACT/GRANT NO.

12, SPONSORING AGENCY NAME AND ADDRESS
Corvallis Environmental Research Laboratory
Office of Research & Development
U.S. Environmental Protection Agency
200 S.W. 35th Street -- Corvallis, OR 97330

13. TYPE OF REPORT AND PERIOD COVERED
inhouse

14, SPONSORING AGENCY CODE

EPA/600/02

15. SUPPLEMENTARY NOTES

16. ABSTRACT

for better understanding the real world.

Theoretical developments based on heat and moisture transfer in soil lead

to dimensionless numbers that describe important processes taking place in
porous media. It is proposed that these numbers can be used as preliminary
scientific criteria for scaling the results from microcosms both as a

means of comparing two generally similar but non-identical systems as well as

p7. KEY WORDS AND DOCUMENT ANALYSIS

fa. DESCRIPTORS >. OENTIFIERS/OPEN ENDED TERMS |c. COSATI Field/Group

Microcosm
similitude
soil moisture

toxicity screening

8. DISTRIBUTION STATEMENT 19. SECURITY GLASS (This Report) 21, NO. OF PAGES
Unclassified
release unlimited 20. SECURITY CLASS (This page) 22. PRICE
Unclassified

EPA Form 2220~1 (Rev. 4-77)

26

US. GPO: 1979—-698-230/136



SECTION 5
DISCUSSION

The foregoing presentation demonstrates the concept of transferability
from one system to another by using analytically-derived transformations based
on similarity criteria. According to these criteria, two systems are similar
only if the corresponding criteria governing those processes are numerically
equal.

An example will demonstrate this conclusion. Consider Figure 2 repro-
duced from Luikov (1966 - p 299). In this figure, Luikov plots the rates of
change of t* and 0* with respect to Fourier number for several Lu numbers.
Since, according to the previous discussion, the Fourier number is essentially
the time coordinate for the process, these figures actually depict the time
rate of change of moisture potential and temperature at a given point in the
system.

Suppose now that we have two systems, I and II, for which Fo; = Forr but
Luy = 0.5 and Luyy
is to choose the soil properties (say, by compaction, etc.) in system I, so
that its diffusivity a, differs by a factor of 0.5/0.15 = 3.3 from system II.

Now the Fourier numbers in the two systems can be made equal since the diffu-

= 0.15. This is entirely possible, since all we need to do

sivity for heat is not changed.

With these inputs we use Figure 2 to calculate the appropriate rates in
the two systems at a point where 0* = 0.6. The latter choice is entirely
arbitrary. We find from these theoretical calculations that the rate of
temperature change differs by a factor of 8.6 and moisture by a factor of 2.5,
respectively, between systems I and II.

This is a dramatic change in time rate of important abiotic processes caused
by an apparently innocent variation of one soil property. The analysis can be
used to evaluate the relative importance of other system properties, since
they are not all equally important.

21


SECTION 3
GENERAL APPROACH AND EXAMPLES
A. UNCOUPLED ABIOTIC PROCESSES

Plant-cover together with the soi] medium comprise a living system in
which energy and mass transfer differ radically from those of dead bodies
(Nichiporovich, 1975).

Imagine a soil column in a lysimeter. The column is isolated from the
surrounding soi] medium and the soil is sterile; there are no microorganisms
in the soil and its top layer is devoid of plant life. Assume also that the
soil is dried to a state of equilibrium with the water potential of the air
and is protected from rainfall. Exchange of energy in this case must account
for the fact that soil moisture varies reversibly with variation of water
conditions of the air medium. Moreover, with variation of temperature a
certain amount of gas will physically be evolved or absorbed in a reversible
manner. Transfer of energy and mass will be balanced during a 24 hour day and
the mean amount of heat and mass of the soil will remain constant.

The situation changes when liquid water is added to this soil. Now,
transfer of water in the body of the soil and evaporation from its exposed
surface take place. The heat balance will be altered. In the absence of
filtration to the subsoil and removal of matter, the material balance of the

soil will remain constant, neither its dry nor total weight will change.

If the soil is inhabited by microorganisms and it contains a sufficient
amount of organic substances, heat will evolve from the microorganisms during
their life activity and the decomposition of organic compounds. However, this
contribution to the soil heat balance is generally negligible. Many chemical
transformations of substances in the soil may occur and mass exchange with the
environment should now include evolution of C02 and absorption of oxygen. In
some cases uptake of nitrogen and release of other gases may be involved.


they are described by one and the same system of differential equations and
have similar conditions of single-valuedness" (Luikov 1966). Because of
direct relevance of this theory to the proposed research, we shall interpret
this statement in some detail with the help of several examples.

In relation to the problems of microcosms, the system of differential
equations referred to in the theorem is derived from the application of
conservation of mass, momentum and energy. Since in their general form, the
equations explain a whole class of phenomena, the integration of such systems
yields innumerable different solutions. The specific conditions which iden-
tify an individual problem from among a whole class of problems are known as
the conditions of unambiguity or single-valuedness. These conditions include
(a) geometrical properties of the system, (b) all physical constants of the
bodies which are essential for the phenomena under consideration, (c) initial
conditions describing the state of the system at the initial instant, and (d)
conditions of interaction of the system with the surrounding medium, ij.e.,
boundary conditions.

Even though the Theory of Similarity has been used to obtain solutions
for certain specific physical problems (Hansen 1964) with a minimum of re-
course to experimentation, it does not appear to be of direct help in con-
sidering our microcosm problem. Nevertheless, once the system of equations
and conditions of single-valuedness are written for the problem at hand, the
theory can be applied to translate from a whole class of problems to a smaller
sub-class or group of problems. Then with experimentation guided by the
theory, solution to an individual problem can be readily obtained. This
occurs without actually solving the full system of equations.

The advantages of direct experimentation are obvious. They relate
directly to the problem at hand and their accuracies can be established to
meet the requirements of the problem. The drawbacks are also obvious. Re-
sults from single experiments cannot be extended to other problems without
some theoretical foundation.