United States
Environmental Protection
Agency
«>EPA Research and
Development
SIMPLIFIED MODELING OF
AIR FLOW DYNAMICS IN
SSD RADON MITIGATION SYSTEMS
FOR RESIDENCES WITH GRAVEL BEDS
EPA-600/R-92-090
May 1992
Prepared for
Office of Radiation Programs
Prepared by
Air and Energy Engineering Research
Laboratory
Research Triangle Park NC 27711
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before compter'
1. REPORT NO, 2.
EPA-600/R-92-090
3 PB9 2-195635
4, TITLE AND SUBTITLE
Simplified Modeling of Air Flow Dynamics in SSD
Radon Mitigation Systems for Residences with
Gravel Beds
5. REPORT DATE
May 1992
6. PERFORMING ORGANIZATION CODE
F.PA/ORD
7. AUTHOR(S>
T.A.Reddy, K.J. Gadsby, H. E, Black, III,
D. T. Harrje, and R. G. Sextro
8. PERFORMING ORGANIZATION REPORT NO.
PU/CEES Report No. 246
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Princeton University
Center for Energy and Environmental Studies
Princeton, New Jersey 08544
10. PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO.
CR814673
12. SPONSORING AGENCY NAME AND ADDRESS
EPA, Office of Research and Development
Air and Energy Engineering Research Laboratory
Research Triangle Park, North Carolina 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final; 8/89 - 2/91
14. SPONSORING AGENCY CODE
EPA/600/13
15. supplementary notes AEERL project officer is Ronald B. Mosley, Mail Drop 54, 919/
541-7865.
16. abstract jn an a)_t,crript to better understand the dynamics of subslab air flow, the
report suggests that subslab air flow induced by a central suction point be treated as
radial air flow through a porous bed contained between two impermeable disks.
(NOTE: Many subslab depressurization systems, those now considered most effec-
tive for mitigating residences for radon, do not perform entirely satisfactorily,
even when designed and installed by professionals.) The report shows that subslab
air flow is most likely to be turbulent under actual field situations in houses with
subslab gravel beds, but remains laminar when soil is present under the slab. The
physical significance of a model is discussed, and simplified closed-form equations
are derived to predict pressure and flows at various distances from a single central
depressurization point. A laboratory apparatus was built to verify the model and
experimentally determine the model coefficients of the pressure drop vs. flow for
commonly encountered subslab gravel materials. These pressure drop coefficients
can be used in connection with the simplified model as a rational way to assess sub-
slab communication in houses. Preliminary field verification results in a house with
gravel under the basement slab are presented and discussed.
17. KEY WORDS AND DOCUMENT ANALYSIS
a. descriptors
b.identifiers/open ended terms
c. coSATt Field/Group
Pollution Mathematical Models
Radon Air Flow
Slabs Dynamics
Pressurizing
Residential Buildings
Gravel
Pollution Control
Stationary Sources
Subslab Depressuriza-
tion
13 B 12 A
07B 20D
13 M, 13 C 2 OK
14 G
08G
18. DISTRIBUTION STATEMENT
Release to Public
19. SECURITY CLASS (This Report}
Unclassified
21. NO. OF PAGES
80
20. SECURITY CLASS (This page)
Unclassified
22. PRICE
EPA Form 2220-1 (9-73)
i
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EPA-600/R-92-090
May 1992
SIMPLIFIED MODELING OF AIR FLOW DYNAMICS IN SSD RADON MITIGATION
SYSTEMS FOR RESIDENCES WITH GRAVEL BEDS
by
T A K T fladeW FT F TTT
X xvcUtiy j Xv.iJ i VjcACZfcL»Jp y JL JL«X_j• OiltAvi^) JL.LJLy
D.T. Harrje, and R.G. Sextro
Center for Energy and Environmental Studies
Princeton University
Princeton, NJ 08544
EPA Cooperative Agreement CR-814673
EPA Project Officer
Ronald B. Mosley
Radon Mitigation Branch
Air and Energy Engineering Research Laboratory
Research Triangle Park, NC 27711
Prepared for
U. S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RESEARCH AND DEVELOPMENT
WASHINGTON, DC 20460
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EPA REVIEW NOTICE
This report has been reviewed by the U.S. Environmental Protection Agency, and
approved for publication. Approval does not signify that the contents necessarily
reflect the views and policy of the Agency, nor does mention of trade names or
commercial products constitute endorsement or recommendation for use.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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ABSTRACT
The technique presently considered most effective for mitigating residences for radon
is the subslab depressurization technique. Given that a large number of such mitigation
systems designed and installed by the professional community do not perform entirely to
satisfaction, there is a need to better understand dynamics of subslab air flow. In this
report, it is suggested that subslab air flow induced by a central suction point be treated
as radial air flow through a porous bed contained between two impermeable disks. Next,
we show that subslab air flow is most likely to be turbulent under actual field situations
in houses with subslab gravel beds, while remaining laminar when soil is present under
the slab. The physical significance of this model is discussed and simplified closed-form
equations are derived to predict pressure and flows at various distances from a single
central depressurization point. A laboratory apparatus was built in order to verify our
model and experimentally determine the model coefficients of the pressure drop versus flow
for commonly encountered subslab gravel materials. These pressure drop coefficients can
be used in conjunction with our simplified model as a rational means of assessing subslab
communication in actual houses. Preliminary field verification results in a house with
gravel under the basement slab are presented and discussed.
ii
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TABLE OF CONTENTS
Page
Abstract ii
List of Figures iv
List of Tables vi
Glossary vii
Metric Equivalents ix
Nomenclature x
Acknowledgments xii
1. Introduction 1
2. Conclusions 4
3. Specification of the problem....... 5
4. Mathematical model for radial flow 12
5. Laboratory apparatus 20
6. Experimental results and analysis of radial flow 23
7. Field verification 32
8. Graphical representation 39
9. Pressure drop considerations 41
10. Future work 46
References............. 47
Appendix A. Brief review of scientific theory relating to flow through
porous media... 49
Appendix B. Experiments to determine porosity and equivalent diameters
of gravel.... 58
Appendix C. Quality Assurance and Quality Control (QA/QC) Statement 63
iii
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Figures
Fig, no. Page
1 Schematic of subslab air flows in a house with a gravel bed when the
house is being mitigated following the subslab depressurization
technique. (Note that part of the air flowing through the gravel
bed originates from the basement and the rest from the ambient
air.) 6
2 Schematic of subslab air flow in a house without a gravel bed when
the house is being mitigated following the subslab depressurization
technique 6
3 Expected Reynolds number for air flows through subslab gravel beds
of houses being mitigated for radon. A radial cylindrical disk flow
with impermeable boundaries is assumed with disk spacing = 0.1 m
(4 in.), diameter of gravel = 0.012 m (1/2 in.) and porosity of bed
= 0.4. (Reynolds numbers above 10 indicate turbulent flow while
those below 1 correspond to laminar flow.)... 9
4 Expected Reynolds number for air flows through subslab soil beds of
houses being mitigated for radon. A radial cylindrical disk flow
with impermeable boundaries is assumed with disk spacing = 0.1 m
(4 in.), flow rate = 2.36 1/s (5 cfm) and porosity of bed = 0.4.
(Reynolds numbers below 1 indicate laminar flow while those between
1 and 10 correspond to transition range.) 9
5 Variation in Reynolds number of air flow through a porous media
for different superficial air velocities. The porous media is
assumed to have a porosity of 0.4, and the three different values
of particle diameter relate to gravel beds 11
6 Variation in Reynolds number of air flow through a porous media
for different superficial air velocities. The porous media is assumed
to have porosity of 0.4, and the three different values of particle
diameter relate to sand beds...... 11
7 Schematic of a model to duplicate flow conditions occurring beneath
the concrete slab of a residence when induced by a single suction point.
The air flow is assumed to be radial flow through a homogenous
porous bed of circular boundary 16
8 Cross-section of the experimental laboratory apparatus 21
9 Layout of the test holes to measure static pressures in the porous bed...... 21
10 Data from Exps. A1 and A2 versus model with exponent b = 2
[eq. (11)] 26
11 Data from Exp. A3 versus model with exponent b = 2 [eq. (11)] 26
iv
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Figures (continued)
Fig, no. Page
12 Data from Exps. A1 and A2 versus model with exponent b = 1.6
[eq. (9)] 28
13 Data from Exp, A3 versus model with exponent b = 1.4 [eq. (9)] 28
14 Log-log plot of the observed pressure drop values of Exps. 1 and 2,
in meters of water, versus those of the second term on the left hand
side of eq. (15). This figure serves to illustrate the fact that the
intercept, which corresponds to the permeability of the porous bed,
cannot be estimated very accurately from regression of the data
points at hand 31
15 Plan of the basement of House 21 showing the relative positions of
the various subslab penetrations. The suction hole of the mitigation
system is marked as a + 33
16 Comparison of observed and computed pressure drops for different
subslab penetrations using coefficients of 0.012 m gravel (b = 1.6
and k = 9.4x109 m2). Data of hole 12 not included 35
17 Same as Fig. 16 but with distance from suction hole 36
18 Same as Fig. 16 but using coefficients of 0.019 m gravel (b = 1.4
and k = 3.4xl0'9 m2) 38
19 Pressure drop in a sand bed with radial airflow between two
impermeable disks 40
20 Pressure drop in a gravel bed with radial airflow between two
impermeable disks. The correction factor F can be determined from
Fig. 21 40
21 Correction factor F for gravel beds to be used in Fig. 20.... 42
v
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Tables
Table No. Page
1 Summary of the different experiments performed with the
laboratory apparatus using river-run gravel ........... 24
2 Results of regression using a quadratic model for pressure drop
[eq.(H)] 24
3 Results of regression using an exponential model for pressure drop
[eq.(9)]. The underlined values of b correspond to those yielding
the highest R2.... 27
4 Results of regressing experimental data from House 21
using eq. (10) 27
5 Determination of the pressure loss coefficient at the throat of
the mitigation suction pipe in House 21 44
6 Relative pressure drops in the mitigation system of House 21... 44
B1 Experimental observations in order to determine porosity of the
river-run gravel of 3/4 in. (0.019 m) nominal diameter. Numbers
indicate volume of water in cubic centimeters poured in (1) and
drained out (2) from a total volume of 1 liter. 59
B2 Experimental observations in order to determine porosity of the
river-run gravel of 1/2 in. (0.012 m) nominal diameter. Numbers
indicate volume of water in cubic centimeters poured in (1) and
drained out (2) from a total volume of 1 liter 59
B3 Results of the experiment to determine the mean equivalent
particle diameter of the river-run gravel. Total volume of the
sample was 1 liter 61
CI Monitoring instruments 65
vi
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GLOSSARY
Aggregate - crushed stone, stone, or other inert material or combinations thereof having
hard, strong, durable pieces.
Air permeability (sub-slab) - a measure of the ease with which air can flow through a
porous medium. High permeability facilitates air movement, and hence generally
facilitates the implementation of soil-depressurization.
Basement - A Type of house construction where the bottom livable level has a slab (or
earthen floor) which averages 3 ft or more below grade level on one or more sides of the
house and is sufficiently high to stand in.
Coefficient of Determination (R2) - This fractional measure represents the proportion of the
total variability of the response variable that is explained by the relationship between the
response variable and the exogenous variables.
Contractor - a building trades professional licensed by the state.
Crawlspace - an area beneath the living space in some houses, where the floor of the
lowest living area is elevated above grade level. This space (which generally provides only
enough head room for a person to crawl in), is not living space, but often contains utilities.
Depressurization - In houses, a condition that exists when the air pressure inside the
house or in the soil is slightly lower than the air pressure outside. The lower levels of
houses are almost always depressurized during cold weather, due to the buoyant force on
the warm indoor air (creating the natural thermal stack effect). Houses can also be
depressurized by winds and by appliances which exhaust indoor air.
Detached houses - Single family dwellings as opposed to apartments, duplexes, townhouses,
or condominiums. Those dwellings which are typically occupied by one family unit and
which do not share foundations and/or walls with other family dwellings.
Entry routes - Pathways by which soil gas can flow into a house. Openings through the
flooring and walls where the house contacts the soil.
HAC system - A heating and air conditioning system. Typically residential because there
is no intentional ventilation connected to the distribution system.
House air - Synonymous with indoor air. The air that occupies the space within the
interior of a house.
Indoor air - That air that occupies the space within the interior of a house or other
building.
Livable space - Any enclosed space that residents now use or could reasonably adapt for
use as living space.
Mean (arithmetic) - The mean of a set of quantitative data is equal to the sum of the
measurements divided by the number of measurements.
vii
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Mitigation - The act of making less severe, reducing or relieving. For the purposes of this
standard, a building shall not be considered as mitigated until it has been demonstrated
to comply with applicable limits of indoor radon concentration.
Outside air - air taken from the outdoors and, therefore, not previously circulated through
the system.
Permeability - (see air permeability).
Picocurie (pCi) - A unit of measurement of radioactivity. A curie is the amount of any
radionuclide that undergoes exactly 3.7 x 1010 radioactive disintegrations per second. A
picocurie is one trillionth (10 ") of a curie, or 0.037 disintegrations per second.
Picocurie per liter (pCi/L) - A common unit of measurement of the concentration of
radioactivity in a fluid, A picocurie per liter corresponds to 0.037 radioactive
disintegrations per second in every liter of air.
Radon - The only naturally occurring radioactive element which is a gas. Technically, the
term "radon" can refer to any of a number of radioactive isotopes having atomic number
86. In this document, the term is used to refer specifically to the isotope radon-222, the
primary isotope present inside houses. Radon-222 is directly created by the decay of
radium-226, and has a half-life of 3.82 days. Chemical, symbol Rn-222.
Soil depressurization system - a system designed to withdraw air below the slab through
means of a vent pipe and fan arrangement (active) or a system designed to lower sub-
slab air pressure by use of a vent pipe to the outside but relying solely on convective air
flow of upward air in the vent (passive).
Soil gas - Gas which is always present underground, in the small spaces between particles
of the soil or in crevices in rock. The major constituent of soil gas is air with some
components from the soil (such as radon) added.
Stack effect - The upward movement of house air when the weather is cold, caused by the
buoyant force on the warm house air. House air leaks out at the upper levels of the
house, so that outdoor air (and soil gas) must leak in at the lower levels to compensate.
The continuous exfiltration upstairs and infiltration downstairs maintain the stack effect
air movement, so named because it is similar to hot combustion gases rising up a fireplace
or furnace flue stack.
Standard deviation - This is statistical measure of dispersion, defined as the positive
square root of the mean of squares of deviations from the mean.
Ventilation - the process of supplying or removing air, by natural or mechanical means,
to or from any space. Such air may or may ot have been conditioned.
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METRIC EQUIVALENTS
Metric
Multiply bv
Yields nomnetric
centimeter (em)
0.39
inch (in)
centimeter (cm)
0.033
foot (ft)
meter (m)
3.28
foot (ft)
square meter (m2)
10.76
square foot (ft2)
liter (L)
0.35
cubic ft (ft3)
cubic meter (m3)
35.31
cubic ft (ft3)
liter per second (L/sec)
2.11
cubic foot per minute (cfm)
Pascal (Pa)
0.004
inch of water column (in WC)
Becquerel per cubic meter
0.027
picocurie per liter (pCi/L)
(Bq/m3)
degree Centigrade (°C)
(9/5°C)+32
degree Fahrenheit (°F)
ix
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Nomenclature
A
cross-sectional area of flow
a
parameter representive of the resistivity to flow of the porous bed
b
exponent appearing in eq.(3b)
c
dimensionless constant appearing in eq.(3a)
d
diameter
dv
equivalent diameter of pebbles
F
correction factor given by equation (1Tb)
f
friction factor
g
acceleration due to gravity
h
thickness of the porous bed
K
parameter representative of the conductivity to flow of the porous bed
Kp
pressure loss coefficient at entry to suction pipe
k
permeability of porous bed
L
length of pipe
n
exponent appearing in eq,(3a)
Ap
pressure drop
P
pressure
P.
atmospheric pressure
q
total volume flow rate
R2
coefficient of determination of regression
Re
Reynolds number
r
radial distance from center of the suction hole
r0
outer radius of the laboratory apparatus
SEM
standard error of the mean of the regression estimate
V
air velocity
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x distance along flow
a opening angle of porous bed through which radial flow occurs
p density
v kinematic viscosity
H dynamic viscisity
$ porosity of porous bed
Suffix
a air, ambient
b porous bed
ent entrance
f fluid
p pipe
w water
xi
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ACKNOWLEDGMENTS
The assistance of R. Gafgen and R. de Silva during the experimental phase of this
study is acknowledged as also is that of D. Hull for his insight into certain statistical
aspects. Fruitful discussions and insights into fundamental hydrological concepts by Prof.
G.F. Finder and J, Guarnaccia were invaluable during the initial stages of this study.
Critical comments, encouragement, and indepth reviews of the manuscript by R. Mosley,
D. Sanchez, and L. Sparks of U.S. EPA, AEERL, are acknowledged.
xii
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1. Introduction
The presence of radon 222, a colorless, odorless gas naturally present in trace
concentrations in soil gas and underground water has been found to be a serious health
hazard in many American houses. Generally, radon-contaminated soil gas enters a home
through cracks and openings in the slab being driven by pressure differences between the
subsoil and the basement and walls, or between the subsoil and the living area in case of
a slab-on-grade house construction. These pressure differences occur naturally, either
because of stack (i.e. temperature difference) or wind effects, or due to zonal depressuri-
zation due to the effect of the heating and air-conditioning system (HAC), The
Environmental Protection Agency (EPA) and other health organizations have recognized
the health risk of elevated indoor radon concentrations and have set a guideline of 148
Bq/m3 (4 pCi/L) as a radon level beyond which mitigation is recommended. It is estimated
that up to 10 million US homes have elevated indoor radon concentrations. Thus there
is a need to ensure the effectiveness and proper design of mitigation systems.
An EPA sponsored workshop was held at Princeton University in order to summarize
available knowledge on various radon diagnostic techniques [1]. Four phases of radon
diagnostics were defined:
(a) Radon problem assessment diagnostics involving radon source strength, location,
house characteristics, and house occupancy characteristics.
(b) Pre-mitigation diagnostics entailing the selection of the best mitigation system for the
particular building taking into account radon source strength and location,
particularly at the substructure.
(c) Mitigation installation diagnostics used during installation of mitigation systems in
order to assure proper operation.
(d) Post-mitigation diagnostics to assure that the radon guidelines have been met and
1
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that the mitigation system is adjusted properly.
The emphasis of the workshop was on diagnostics since each home, housing division,
and region has different radon characteristics which require that special attention be paid
to system design in order to maximize mitigation performance and minimize cost. This
issue was of particular importance since it was found that a large number of houses which
had been mitigated still had elevated radon levels. In fact, a recent study [2] found that
64% of the homes in New Jersey where post-mitigation radon measurements have been
made, remained above 148 Bq/m3 (4 pCi/L). Diagnostics are therefore crucial for providing
information relevant and necessary to the successful design and implementation of a radon
mitigation system.
Many participants of the workshop felt that radon mitigation via subslab
depressurization (SSD) was the best approach for houses with a gravel bed. Surveys
indicate that systems based on this technique account for more than 50% of all installed
systems [3], (Another promising technique involves subslab pressurization. Since the two
techniques are similar in terms of subslab dynamics of induced air flow, they can both be
treated in the same scientific framework.) In the pre-mitigation diagnostic phase, the
degree of communication under the slab as well as the permeability characteristics of the
subslab medium must be determined before appropriate depressurization conditions can
be determined. Proper attention to these aspects will ensure that reasonable flows, and
hence the desired degree of depressurization, will prevail at all points under the slab.
Lowering the pressures at all points of the subslab to values below those of the
basement/crawl-space/living area will subsequently reduce radon-rich soil-gas from moving
or seeping into the building by convection.
Parallel with the above aspect is the concern that presently mitigators tend to over-
design SSD systems in order to err on the safe side. In so doing, more radon from the
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soil is removed and vented to the ambient air than would have occured naturally. There
is thus a prudent need to downsize current overly robust SSD mitigation systems and
decrease emission exhaust quantities of radon while simultaneously ensuring that indoor
radon levels do not exceed the desired concentrations.
The Center for Energy and Environmental Studies of Princeton University is
currently involved in the formulation and verification of a rapid diagnostic protocol for
subslab and wall depressurization systems designed to control indoor radon levels [4]. It
is hoped that the protocol would lead not only to the ability to distinguish between homes
that are hard or easy to mitigate, but also to the articulation of a more rational and
scientific approach which would be especially useful to the ever-increasing body of
professional radon mitigators. Other researchers have or are also addressing this issue
of optimal SSD design [5 - 7].
Our approach to the formulation of the diagnostics protocol consists of:
(1) a practical component, in that specific guidelines would be provided so that the
effectiveness of the engineering design of the radon mitigation system would be
enhanced, and
(2) scientific studies at a more fundamental level which would validate and provide
quantitative guidelines more rationally.
The scientific component would involve improved understanding of those factors
governing the pressure-induced air flow beneath the concrete slab in order to predict and
then optimize the pressure field extension patterns induced by single and multiple suction
points. The present study specifically addresses the former aspect, while the latter would
be dealt with in a subsequent study.
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2. Conclusions
The important features of this study are as follows:
(a) We have outlined the general problem of radon mitigation system design and
discussed the scope and limitations of prior studies in both this aspect and also at
a more fundamental aerodynamical level. The first problem should be to determine
the nature of air flow below the concrete slab and how this is likely to affect the
pressure drop versus flow correlation for given subslab conditions.
(b) Next we give arguments to support the suggestion of a prior study (Ref, [8]) that flow
under the slab of a house during mitigation using the subslab depressurization (and
the pressurization) technique be likened to radial flow between two impermeable
disks.
(c) We point out that subslab air flow under actual operation of mitigation systems is
likely to be turbulent if a gravel bed is present and laminar in the presence of soil.
(d) We then present a mathematical treatment to analytically predict the pressure field
in homogeneous circular porous beds when subjected to a single central suction hole.
(e) A laboratory apparatus constructed so that it can specifically duplicate conditions
which occur in practice under slabs of real houses being mitigated for radon using
the depressurization (or the pressurization) technique) is then described. The
experimental procedure followed in order to measure the pressure field of turbulent
air flow is outlined from which the regression coefficients of the pressure drop versus
flow correlation can be determined.
(f) Preliminary field verification results of our modeling approach in a house with gravel
under the basement slab are presented and discussed. A striking conclusion of our
study is that even a visual inspection of the porous material under the slab may be
an indicator good enough for a sound engineering design, if used in parallel with our
4
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modeling approach and given a table containing the aerodynamic pressure drop
coefficients of commonly found subslab material.
(g) How closed form equations can be used to generate design figures useful to the
practical mitigator has also been illustrated,
(h) Practical notions regarding proper piping design of the mitigation system in view of
pressure drop considerations have finally been addressed.
3. Specification of the Problem
In terms of modeling the induced subslab pressure fields, one could conceptually
divide the present housing stock construction into broadly three groups: (i) those with a
gravel bed under the concrete slab, (ii) those without, in which case soil is the medium
under the slab, and (iii) those houses which have both. In the case of (ii), the subslab
permeabilities are much lower than for case (i) requiring more careful design of the
mitigation system. In New Jersey, houses less than about 30 years old typically have
gravel beds of about 0.05-0.1 m (2 in.-4 in.) under the slab. However other states seem
to have very different construction practices: for example, houses in Florida and New
Mexico are built directly on compacted fine-grained soil which offers high resistance to air
flow [5-7],
Fig. 1 schematically depicts the type of construction and the expected air flow paths
one would typically expect in a house with a gravel bed when a single suction point
through the slab is used to induce a pressure field. In case of a radon mitigation system
using subslab pressurization, one could, to a good approximation, simply assume similar
aerodynamic effects with the direction of air flows reversed. Since the permeability of
the gravel bed is usually very much higher than that of the soil below, one could assume,
except for the irregular pattern around the footing which would occur over a relatively
small length, that the subslab air flow is akin to radial flow between two impermeable
5
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Fig. 1 Schematic of subslab air flows in a house with a
gravel bed when the house is being mitigated following
the subslab depressurization technique. Note that
part of the air flowing through the gravel bed
originates from the basement and the rest from the
ambient air.
Fig. 2 Schematic of subslab air flow in a house without a
gravel bed when the house is being mitigated following
the subslab depressurization technique.
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circular disks with a spacing equal to the thickness of the gravel bed. Note that this
model equally accounts for the leakage of air from the basement which essentially occurs
from the perimeter cracks.
In case of a house without a gravel bed, suction applied at a simple penetration
through the slab (as in Fig. 1) is no longer practical in low permeability soils since the
area of depressurization is small. In order to enhance mitigation effectiveness, the
contemporary thinking is to increase this area of depressurization by either digging a
gravel pit below the concrete slab as shown in Fig. 2, or more simply, by hollowing out
a hemisphere of about 0.3-0.45 m (12 in.-18 in.) radius underneath the suction hole. Even
under such conditions, and provided the soil underneath is free of major obstructions like
concrete footings, duct work, piping, and large rocks, one could view air flow as occurring
between two impermeable circular disks with a spacing equal to either the depth of the
gravel pit or the radius of the hollow hemisphere.
The above discussion was intended primarily to suggest that flow underneath the slab
be visualized as occurring in radial streamlines terminating at the central suction point.
Note that such a representation would perhaps be too simplistic or even incorrect for a
house with a partial-basement (case (iii) above). In the present study, we shall limit
ourselves to understanding the flow and pressure drop characteristics through a
homogeneous bed (of either gravel or sand) with uniform boundary conditions, the obvious
case to start with being a circular configuration.
The first question to be addressed relates to the nature of the flow, i.e., whether the
flow is laminar or turbulent, and whether there is a transition from one regime to another.
As outlined in Appendix A, where a brief overview of the basic scientific theory of
flow through porous beds is given, the Reynolds number gives an indication of the flow
7
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regime. Though there is an inherent ambiguity in the definition of the quantity
characterizing the length dimension, we shall adhere to the following definition:
q 1 d,
Re = . . (1)
A va 4
where q - total volume flow rate,
A - cross-sectional area of flow (in the case of radial flow
through a circular bed of radius r and thickness h, the
area = 2jtrh),
v„ - kinematic viscosity of air,
d, - equivalent diameter of pebbles, and
$ - void fraction or porosity of the gravel bed.
Let us first look at flow through a gravel bed. Some typical values of the above
parameters could be assumed:
h = 0.1 m (4 in.), d, = 0.0125 m (1/2 in.), va (at 15°C) = 14.6x10* m/s, and $ = 0.4.
The values of q encountered in practice range from 9.4 x 10"3 to 47.2 x 10"3 m3/s
(20-100 cfm). Under these conditions, the resulting Reynolds number for radial flow at
different radii can be determined from eq. (1) or Fig.3. From Appendix A, we note that
a safe lower limit for turbulent flow is when Re >10, and a safe upper limit for laminar
flow is when Re < l.1 Since basements do not generally exceed 6 m (20 feet) in radius,
we note from Fig.3 that subslab flow would tend to be largely turbulent when a gravel bed
is present. This by itself is an important finding since explicit recognition does not seem
'The common held conception that turbulent flow occurs at Reynolds numbers of
several thousands is valid only for flow inside tubes and ducts and not for flow in packed
beds.
8
-------
Fig. 3 Expected Reynolds number for air flows through subslab
gravel beds of houses being mitigated for radon. h
radial cylindrical disk flew with impermeable
boundaries is assumed with disk spacing = 0.1 m (4
in.), diameter of gravel = 0.012 m (1/2 in.) and
porosity of bed = 0.4. (Reynolds numbers above 10
indicate turbulent flow while those below 1 correspond
to laminar flow.)
Fig. 4 Expected Reynolds number for air flows through subslab
soil beds of houses being mitigated for radon. A
radial cylindrical disk flow with impermeable
boundaries is assumed with disk spacing = 0.1 m (4
in.), flow rate =2.36 1/s (5 cfm) and porosity of bed
- 0.4. ^ (Reynolds numbers below 1 indicate laminar
flow while those between 1 and 10 correspond to
transition range.)
-------
to have been made of this aspect in earlier studies.
The usefulness of Fig.3 can be extended to cover other types of circular
configurations. Thus if one would like to estimate Re numbers for radial flow through a
slice with impermeable sides and with an opening angle a instead of an entire
pie-configuration, one needs simply to use the following correction:
Re a = Re^o • ( 360/a) (2 a)
where Re^ is read from Fig.3.
Also if the disk spacing is not 0.1 m (4 in.) but say h', the Reynolds number can be
obtained from Fig.3 corresponding to an effective radius r in meters given by:
r' = r • ( 0.1 / h' ) (2b)
Similar corrections can be made to other parameters as well. Thus, in conclusion,
we should expect turbulent flow conditions to prevail through subslab gravel beds during
normal operation of mitigation systems using the subslab depressurization (or
pressurization) technique. On the other hand, in a house with soil as the subslab medium,
this observation is no longer valid. Grain diameters of sand range from 0.06 to 2 mm
(0.0024-0.08 in.) [9J while volume flow rates in corresponding mitigation systems are
typically lower, about 0.83 x 10s-6.2 x 103 m3/s (2-15 cfm). Assuming some typical values
of h = 0.1 m (4 in.),
-------
Reproduced from
best available copy.
tr
ui
E
D
2
in
9
c
10 13 20 25 30 35 -W 45 53
VELOCITY (m/s)
Fig. 5 Variation in Reynolds number of air flow through a porous media for different
superficial air velocities. The porous media is assumed to have a porosity of
0.4, and the three different values of particle diameter relate to grave! beds.
it-
u
ID
a
D
in
a
10 15 20 25
VELOCITY (isn/s)
Fig. 6 Variation in Reynolds number of air flow through a porous media for different
superficial air velocities. The porous media is assumed to have porosity of 0.4,
and the three different values of particle diameter relate to sand beds.
11
-------
and 4. These have been computed from eq.(l) for different values of particle diameters
corresponding to both gravel and sand beds. We note that even with very small velocities
of the order of a few millimeters per second, flow is likely to be turbulent.
4. Mathematical Model for Radial Flow
At the onset, let us mention that there are essentially two different problems involved
with modeling the inflow of radon-enriched air into a residence. One problem is associated
with flow through openings in the slab resulting from small pressure differences impressed
across the concrete slab due to environmental driving forces (stack effect, wind and HAC
operation). The other problem involves modeling the air flow under the concrete slab when
relatively large pressure differentials are applied at one, or several, perforations through
the slab (conditions that arise when houses are being mitigated). Though both these
problems essentially involve modeling the air flow through the ground under the slab, the
difference is that the dynamics and entry paths are entirely different: flow paths and flow
regimes (both in soil and into the basement) and external factors causing flow will be
widely different in both. The pressure differences in the former (i.e., flow occurring
naturally in the absence of a suction pressure) are so small that the assumption of a
laminar flow is usually valid, and effects like diffusion through soil and via small capillary
cracks into the basement have to be considered [15]. In the case of the latter, the flow
regime in gravel beds will most probably be turbulent and the flow paths of radon
enriched air will be predominantly towards the suction hole.
As a result of these differences, studies or models pertaining to one problem cannot
be applied as such to the other. There are a number of studies which have addressed the
first problem (e.g., Refs. [10-14]) while studies relating to the latter are very few. We
could only find two relevant studies; one numerical study using a finite difference
computer model [7], and the other an empirically based study [8]. The principal
-------
drawbacks of Refs. [7,14] are that laminar flow is assumed and the computer code may
be difficult to use and to transfer to other research groups. Moreover, a certain amount
of effort is required in order that the results of such codes be useful to practitioners.
The core of any model is the formulation of the structure of a correlation between
pressure drop and Reynolds number. There is an abundance of literature relating to flow
through porous media as evidenced by the large number of books and monographs on this
topic (e.g., [16-21]). A major portion of this literature relates to laminar flow where
Darcy's Law holds (see Appendix A for explanation). This fact is not too surprising since
flow characteristics in petroleum and gas fields, or water flow in subsoils, or
decontamination of subsoil acquifers, were some of the problems which historically led to
the scientific treatment of flow through porous media. Subsequently, this was extended
to various other problems such as flow through fluidized and packed beds, both in chemical
kinetics and in areas involving drying of cereal grains and sensible heat storage. We have
made a preliminary search through such literature and find that one cannot directly adopt
a particular correlation as such to the present problem though insight into the type of
needed model structure can be gained.
Consequently, (this is further discussed in Appendix A), we have adopted in the
present study the following simple model structure for the onset of turbulent flow.
Re • f = c (3a)
where f is the friction factor of the porous bed and c and n are empirical
dimensionless coefficients.
In Appendix A (see eqs.(A16 and A17)), we show that this model structure is identical
to the following model:
13
-------
1 dp
a
(3b)
Pfg dx
where the left-hand side is the pressure drop in head of the flowing fluid per unit
bed length. The parameter a can be loosely interpreted as the resistivity of the porous bed
to the flow of the particular fluid.
Theory predicts a value of unity for the exponent b when the flow is laminar (we
then have the Darey Law), and a value nearer to 2 when the flow is turbulent. Given the
irregularities both in shape and size prevalent in gravel beds below the concrete slab, the
coefficients should be determined from regression to actual data obtained by experiments
on the specific bed material. The real objective of this study is thus to gauge the accuracy
of such a model structure when applied to house-like conditions. Note also that the
effective permeability2 k of a porous bed [see eqs.(A9-A12) of Appendix A] can be easily
deduced from the coefficient a of eq.(3b), since
where va is the kinematic viscosity of air.
Ref. [8] also uses a model like that of eq.(3) with, however, the parameters
transposed, i.e. (q/A) expressed as a function of the pressure drop gradient. It is trivial
to go from one form to another but we find it more convenient to work with an equation
like eq.(3b).
"We use the term 'effective permeability' instead of permeability, only because of the
uncertainty involved in the extrapolation from the present experimental data (done under
turbulent flow) to Darcy flow (i.e., laminar flow conditions) under which permeability is
conventionally defined. This aspect is further discussed in Section 6.
v,
1
k
(4)
g
a
14
-------
We shall now seek to derive a mathematical expression for the pressure field when
air flows radially through a circular homogeneous gravel bed when suction is applied at
the center of the circle. The laws of conservation of matter and of energy must be
satisfied in any hydrodynamic system. By setting viscosity terms to zero in the
Navier-Stokes equations we get the Euler equation of motion. For the suction pressures
encountered in this particular problem, air can be assumed to be an incompressible fluid
and we have Bernouilli's equation, which in differential form is:
p(r) \
where p, is the density of air, V the superficial velocity and p(r) the pressure of
air at a radial distance r from the center. Strictly, the distance should be taken from
the outer edge of the suction pipe (r' in Fig. 7). The diameter of the suction pipe is
typically so small that one could neglect this difference without any error in the
subsequent analysis.
Assuming that energy or pressure drop (in units of head of air) lost as a result of
viscous drag by the gravel bed can be simply treated as an additive term, we have
Total pressure = Pressure drop due + Pressure drop due
drop to changing cross-section to viscous drag
Assuming a simple model such as eq.(3b) yields
W) + = 0
S"Pa j
(5)
dr ^2g
(6)
Integrating eq.(6), we have
constant
(7)
15
-------
q
F^• 7 3chema.tc of 3 model to duplicate flew condit^ons
occurring beneath the concrete slab of a residence
when induced by a single suction point. The air flow
is assumed to be radial flow through a homogenous
porous bed of circular boundary.
16
-------
The constant of integration can be found from the boundary conditions.
At the outer fringe of the cylindrical disk:
r = r0 , p = pa (the atmospheric pressure).
Introducing this into eq.(7), the following expression for the pressure drop is obtained
p(r)-pa
P„.g
f q Y 1 1 "\
. (r^-r^) - „
(1-b) 2g
v2jlh/
v2jth/ v
r2 rJ
(8)
Note that the expression [p(r) - pj is a negative quantity which represents the
suction pressure, i.e., the pressure below the ambient pressure.
Since the pressure drop is often measured in units of head of water, it is more
convenient to modify the above equations into:
p(r)-p. Pa
= a . _
Pwg P,
(q 1 1 1 p. / q V fl
. (r,-b-r0'-b) - _
d-b) 2g pw
^2jlhy
v2lth/
1
7 'T-
(9)
We note that the second term on the R.H.S. of eq.(9) relates to the pressure drop
arising simply as a result of decreasing cross-sectional area in the direction of flow (which
is radially inward) while the first term accounts for the viscous drag due to the gravel bed.
In case (and this will normally be so, as we shall see in the next section) the viscous drag
term is very much larger than the former effect, the above equation simplifies into:
p(r)-pa
Pw-g
k g Pw
( q >
2jih
(1-b)
. (r -r„ )
(10)
Note that k is an effective permeability defined in eq.(4).
17
-------
The above equation predicts the pressure field for a pre-speeified total air flow rate
q. In ease one wishes to predict the resulting flow rate for an imposed suction pressure,
the above equation can still be used by simply rearranging the appropriate terms.
It is clear that the derivation is easily modified in case one wishes to either assume
another model structure for the pressure drop correlation [i.e., eq.(3)], or even when the
flow is through a homogeneous circular bed with impermeable sides and segmented into
an angle a as against 360°. The expression (q/2rth) simply needs to be modified
appropriately.
For the special case of b=2, eq.(9) transforms into:
p(r)-pa
-
f i i >
1
f1 11
-
Pa
r q t
= -
2. .
-
+ _ .
-
,
Pwg
_
r r
\l '¦"J
2g
r2 r02
_
p"w
^2ihy
(11)
On the other hand, during laminar flow, Dairy's Law holds and the exponent b=l.
Moreover during such cases, the pressure drop due to changing cross-section is essentially
negligible as compared to the pressure drop due to viscous drag offered by the particles
of the porous bed (see for example, [16]). Under these circumstances, the expression for
the radial pressure drop in a circular porous bed is given by
p(r)
a
dr
pR.g 2r.h _
which, on integration and on introducing the boundary conditions yields
p(r)-pa
(12)
Pw-g
P. q fr \
= a . . . In
pw 27th
v rv
(13a)
18
-------
1 p„ q fr \
. _ . . In
k pw 27th
Vr° J
(13b)
It is easy to modify these equations to apply to outward radial flow as one would
encounter in houses where the subslab pressurization technique is used. The boundary
conditions are still the same but now the pressure at the throat of the suction pipe is
higher than ambient pressure and the quantity [p(r) - pj is positive and represents the
pressure above the ambient pressure. The final expression analogous to eq.(9) is:
P(r)-p.
Pw-g
27th
1 1 pa
. (r^-r.14) - _ . _
(1-b) 2g pw
fq >
2
( 1
2tth y
pr
(14)
Another instance where our approach is directly applicable is when the porous bed
consists of two or more types of porous material. For example, one could come across
a house construction where the subslab gravel bed consists of two horizontally distinct
layers of gravel of different sizes. The above equations can be easily modified to apply
to such cases as well.
The practical implications of the parameters k and b are that if they are really
constant for a given bed material and can be determined by actual experiments in the
field, they will serve as indices by which a mitigator will be able to assess how much of
the area from the suction hole he can hope to access for a given suction pressure.
The irregular boundary conditions that arise in practice are however not easily
tractable with a simple expression such as eq.(9), and resorting to a numerical computer
code may be the only proper way of proceeding in order to predict resulting pressure fields.
Another problem in applying an approach such as the above to practical situations may
be the drastic departure from homogeneity in certain subslab gravel beds (as also in the
19
-------
case of subslab soil beds). How influential these problems are in real situations is
addressed to some extent in section 7.
5. Laboratory Apparatus
One needs to evaluate the soundness of the mathematical derivation presented above
and also to determine the numerical values of the empirical coefficients of eq.(3b), To this
end, a laboratory model consisting of a 2.4 m (8 ft) diameter circular section that is 0.15
m (6 in.) deep was constructed as is schematically shown in Pig. 7. The top and bottom
impermeable disks were made from 0.02 m (3/4 in.) thick plywood, and a wire mesh at the
outer periphery of the disks was used to contain the gravel between the two disks (Pig.
8), The apparatus allowed experiments to be conducted with a maximum disk spacing (or
depth of gravel bed) of 0.095 m (3.75 in.). An open-cell foam sheet 0.025 m (1 in.) thick
was glued to the underside of the top plywood disk. It was hoped that during experiments
heavy weights placed on top of the plywood disk would effectively eliminate gaps that may
exist between the disk and the gravel top that could cause short-circuiting of the air flow.
In so doing, we hoped to guarantee that air flow occurs through the bed and not over it.
A 0.038 m (1.5 in.) diameter hole was drilled at the center of the top disk to serve
as the suction hole. Nine holes, whose layout is shown in Fig. 9, were drilled on three
separate rays of the top disk and a PVC pipe of 0.012 m (1/2 in.) inner diameter with
chamfered ends was tightly squeezed into these holes. Pressure measurements at these
nine holes would then yield an accurate picture of the pressure field over the entire bed.
The total volume of the packed bed is approximately 0.43 m3 (16 ft3) which, for
river-run gravel, translates into a net weight of about 700 kg (1530 lb).
Equipment needed for the experiments included:
20
-------
Attachenent to measure
flow and pres sure
Suction tube
Top plywood disk
.Foam cover
Gravel bed
Bottom plywood disk
Floor
Fig, 8 Cross-section of the experimental laboratory
apparatus.
Fig. 9 Layout of the test holes to measure static pressures
in the porous bed.
21
-------
(i) an industrial vacuum cleaner capable of sucking 45 x 103 ms/s (95 cfm) of air through
a 0.05 m (2 in.) diameter orifice under 1.9 m (75 in.) of water static vacuum pressure;
(ii) a speed control and an air by-pass adapter (which is simply a perforated length of
plastic pipe). Both these are needed in order to vary the air flow rate through the
porous bed;
(iii) a 3 mm stainless steel pitot-tube (Dwyer No. 166-6) to measure velocities from 0,05
to 15 m/s (10 to 3000 ft/min). Tables for different pipe diameters (as described in
Ref. [4]) enabled the corresponding volume flow rate to be deduced;
(iv) an electric digital micromanometer (EDM) (Neotronics Model EDM-1) which can
measure pressures with a resolution of 0.025 x 10 3m (10 3 in.) of water or 0.25 Pa,
and having a maximum range of up to 0.5 m (20 in.) of water. This is also
described in Ref [4J.
Other apparatus included two mounting devices: (i) a 0.038 m (1.5 in.) outer diameter
brass pipe to connect the suction hole to the vacuum hose with arrangements to attach the
pitot tube [called the Flow Pressure Tube (FPT.ij, and also the EDM, and, (ii) a 0.019 m
(0.75 in.) stainless steel pipe to mount the EDM in order to measure the pressure at each
of the nine different taps. These devices have already been described in detail in a
previous report!4]. All measurements performed were in accordance with the approved QA
project plan [22], Certain details are given in Appendix C.
The experimental procedure for the apparatus filled with a certain type of porous
medium entailed fixing the 0.038 m (1.5 in.) FPT device (inner diameter of 0.035 m. or
1.36 in.) above the central suction hole of the top plywood disk and connecting it to the
suction hose of the vacuum cleaner. A pressure tap at this pipe (see Fig. 7) placed 5
diameters above the suction hole permitted the static vacuum pressure to be measured as
well, which could then be used to get an estimate of pressure losses due to changing
direction and cross-section, and also due to turbulence at entry into the pipe.
-------
An experimentation run consisted first of selecting a certain total air flow rate and
gradually changing the speed of the vacuum cleaner in order to achieve this flow rate.
The pressure measurements (representative of the corresponding static pressure inside
the porous bed) at each of the nine taps were then taken in turn with all other taps
closed. This completed a series of readings pertaining to one run. In subsequent runs,
the total air flow rate was set to another predetermined value and the readings were
repeated.
6. Experimental Results and Analysis of Radial Flow
Table 1 summarizes the different experiments performed using the laboratory
apparatus. For example, Experiment A involved river-run gravel of nominal diameters of
0.012 and 0.019 m (1/2 in. and 3/4 in.) which we shall refer to as small and large gravel,
respectively. Experiments A1 and A2 differ only in the spacing between the plywood
disks, i.e., the thickness of the gravel bed was altered. Experiment A1 involved three
separate runs each with a different total volume flow rate, the specific values of which
are also given in Table 1. The flow regime (based on the corresponding Reynolds number)
was found to be turbulent throughout the radial disk.
The specific values of the mean gravel diameter and the porosity of the bed are
required for computing the Reynolds number [given by eq.(l)]. We have performed porosity
measurements and also computed the mean equivalent diameter of the various porous bed
materials chosen in the present study; these are desciibed in detail in Appendix B.
Also, in all the analyses involving regressions which are discussed below, the values
of the static pressure at hole 10 were not included since it is very likely that the
turbulence due to entrance effects would still be present that close to the outer periphery
of the disk.
23
-------
Table 1. Summary of the different experiments
performed with the laboratory apparatus using
river-run gravel
Experiment Gravel Disk No. of Total
size spacing runs flow
(nominal rate
diameter) (cfm) 1/s
Al
1/2 in.
3 in.
3
43.5
20.5
(0. 012 m)
(0.075 m)
63. 8
30.1
79.0
37.3
A2
1/2 in.
3 ,75in.
2
46.8
22,1
(0.012 m)
(0.10 m)
66.5
31.4
A3
3/4 in.
3.7 5in.
4
23.7
11.2
(0.019 m)
(0.10 m)
32.2
15.2
37.4
17.6
44.0
20.8
Table 2. Results of regression using a quadratic
model for pressure drop [eq.(11)]
Experimental No. of Model Model Parameter 'a'
run data without with Mean SEM
points intercept intercept
Al 24 0.964
A2 16 0.975
A1+A2 40 0.968
A3 28 0.967
344.7 13.82
356.2 14.64
0.953 350.1 10.13
0.914 303.4 10.72
24
-------
From Table 1, we note that the flow is likely to be turbulent. Hence we start by
regressing the observed data following eq.(ll). Table 2 assembles the subsequent results.
Since caution is recommended when the soundness of a regression model is to be judged
according to its coefficient of determination (i.e. the R2 value) [23], we have also run the
model with an intercept and found no appreciable discord between the two sets of R2
values (see Table 2). Consequently, in our present study, we can assume the R2 value
to be indicative of the goodness-of-fit of the no-intercept model. We also note from Table
2 that the estimated values of the constant £a' differ only by about 5%. This is how one
would expect it to be since permeability of the bed (inversely related to the coefficient 'a'
as given by eq.(4)) should not alter with change in the thickness of the porous bed. This
suggests that the foam attached to the top plywood disk of our laboratory apparatus does
seem to do a satisfactory job of eliminating short-circuiting of air flow. In order to get
average estimates, we have treated the observations of Exps.Al and A2 together and
estimate the 'mean' parameters. The corresponding value is also shown in Table 2. In
general, the Standard Error of the Mean (SEM) values are 5% or less of the mean value,
a gratifying observation.
We note that R2 values are very high despite which, as depicted by Figs. 10 and 11,
the fits could be improved. Consequently, we have rerun the regression using eq.(9)
wherein both the constant 'a' and the exponent 'b' are to be determined by least square
errors. Instead of simply determining the optimal value of the exponent b we have
performed several runs the results of which are summarized in Table 3. Such an approach
yields insight into the sensitivity of the parameters a and b on the regression. We note
that Rz values have improved significantly and one cannot realistically expect better fits
(given the measurement errors in our readings, we may in fact be overfitting in the sense
that we are trying to assign physical meaning to random errors), as compared to assuming
a quadratic exponent (see Table 2). This is also seen in Figs. 12 and 13 where we note
25
-------
RADIUS(m)
Fig. 10 Data from Exps. A1 and A2 versus model with exponent
b = 2 (eg. (11)) .
RADIUS(m)
Fig. 11 Data from Exp. A3 versus model with exponent b * 2
(eq. (11)),
26
-------
Table 3. Results of regression using an exponential
model for pressure drop [eq.(9)]. The underlined
values of b correspond to those yielding the highest R2.
Experimental No. of b R2 Parameter 'a' Effective
run data Mean SEM permeability
points of porous bed
(m8)
A1+A2
40
1.3
0.984
86.
71
1.78
1.4
0.991
106
.8
1.59
1.5
0.996
130
.6
1.30
1.6
0.998
158
.5
1.15
1.7
0.997
191
.2
1.75
A3
28
1.2
0.994
23,
60
0.34
1.3
0.996
32.
35
0.39
1.4
0, 997
44.
22
0.50
1.5
0.996
60.
26
0 .74
1.6
0. 994
81.
90
1.22
9.4 x 10"
34 x 10"
Table 4. Results of regressing experimental data from House 21
using eq.(10)
Trial
run
k
(m2)
SEM R
(%>
Remarks
1.6
1.7
9.3 x 10"'
7.5 x 10"®
6-7
0.80
0.00
With all data points
1.6
1.7
7.1 x 10~s
5.8 x 10"'
0.96
0.97
With data of holes 11
and 12 removed
1.6
1.7
10.0 x 10"
7.3 x 10"9
0.88
0.87
With data of holes 9,
10, 11, and 12 removed
27
-------
RADIUS (m)
Fxg. a2 Dats from Exps. A1 and A2 versus model with exponent-
b - 1.6
-------
that the fit between model predictions and observed pressure drop has improved.
The value of the exponent b that yields the highest R2 value is underlined in
Table 3, Note also that the SEM values for the estimate of 'a' are relatively small.
Other aspects need specific mention as well:
(a) We find that for the type of river run gravel used and for the range of flow conditions
investigated, the effect of pressure drop in the flow arising as a result of changing
cross-sectioned area [i.e. the second term on the RHS of eq.(9)l corresponds to less
than a percent of the total pressure drop. Thus eq.(10) is the appropriate one to use
in order to predict pressure fields in porous gravel beds under flow conditions akin
to those encountered during operation of practical radon mitigation systems.
(b) We note that the optimal value of b is not too well determined, since R2 values only
change in the third decimal point when b is varied in steps of 0.10 (Table 3). It is
highly unlikely that the experimental accuracy of our readings can lead us to place
this much faith in the exact or best value of b identified by regression. Consequently,
one should rather think in terms of a certain range for the b values and not try to
attach undue importance to physical interpretation of the exact value of the exponent
b.
(c) The study referred to earlier [8] found values of the exponent b to be 1.56 for the
cylindrical disk model. This is generally bourne out in the present study where we
find b=1.6 for the small river-run gravel and b=1.4 for the large gravel.
(d) There is however a serious drawback in our ability to accurately determine the values
of the bed permeabilities from the present experiments. This is because the present
data were collected during turbulent flow regimes, and though the estimate of
parameters a and b identified by regression are perfectly satisfactory for predicting
pressure drops at flow regimes in the range within which the present experiments
29
-------
were performed, these estimates may yield misleading and erroneous predictions when
used outside this range of flow conditions. In other words, parameter sets identified
by regression are not generally accurate for extrapolation purposes.
In order to make this point clearer, let us rewrite eq.(10) as follows:
In
p(r)-p.N
( 1 va >
" P„
'q 1
= In
+ In
- .
Pwg ;
Ik I ,
. Pw
2ih,
. (ri b-r0, b)
(1-b)
(15)
If we were to plot the term on the left-hand side on the y-axis and the second term
on the right-hand side of the above equation on the x-axis, the intercept of such a line
would give us an estimate of the permeability. Figure 14 shows such a representation for
the experimental data using small gravel (Experiments A1 and A2) and b=1.6. It is now
clear that since the data points essentially lie in a region far away from the x-axis» the
intercept term is bound to be ill-defined from a subsequent statistical regression. The
values of effective permeability of the porous bed calculated following eq.(4) are included
in Table 3 and show a three fold difference between small and large gravel sizes. The
numerical values do seem to correspond to those cited in the radon literature [9, 24]. If
more accurate values of permeability are to be determined, the experimental design of our
apparatus needs to be suitably modified.
There is another purely statistical limitation in identifying parameters from a least
square regression such as the present, where a mathematical model without an intercept
term is fitted to quantities which vary drastically like the pressure drop quantities do as
one moves radially outward (Pigs. 10-13 indicate an order of magnitude variation). The
regression would favor the larger values of pressure drop since it tries to capture as much
of the variation in observed values of pressure drop in terms of absolute variation from
zero. Consequently the model parameters estimated will not be very sound because the
30
-------
10
p-
o
M
•c
(1)
n
3
CO
«
-------
regression is unduly influenced by a relatively small number of observations. One
possibility is not to evaluate goodness-of-fit between model and observed data based on the
R2 statistic but rather on the Chi-square statistic [23], Though this test would overcome
the above mentioned limitation, other problems (beyond the scope of this report) are likely
to arise. Another possibility could involve performing more measurements at higher
pressure drops in order to avoid such uncertainty in the estimated parameters. These
issues will have to be addressed in subsequent studies.
7. Field Verification
The irregular boundary conditions and the non-homogeneity in subslab beds that arise
in practice are however not easily tractable with a simple expression such as eqs. (10) and
(13), and resorting to a numerical computer code may be the only rigorous way to proceed
in order to predict pressure fields under actual situations [7,14], We shall show in this
section that our simplified approach nevertheless has practical relevance in that it could
be used as a means of determining which areas under the slab have poorer connectivity
as compared to the rest.
The house under investigation (H21) has a partial basement with a gravel bed under
the basement slab. As shown in Fig. 15, the basement (though rectangular) is close to
being square (6.45m x 7.60m). It has two sides exposed to the ambient air above grade,
while the other two sides are adjacent to slab-on-grade construction. The 0.1m suction
hole is situated roughly at the center of the basement slab. Though 19 holes were drilled
across the slab, (Fig. 15), two of them (holes 11 and 12) were found to be blocked beneath
the slab. Consequently, pressure data measured by the EDM from only 17 holes have been
used in this study. This blockage was later found to be due to the presence of an oversized
footer for a support column.
32
-------
Fig. 15 Plan of the basement of House 21 showing the relative
positions of the various subslab penetrations. The
suction hole of the mitigation system is marked as a
+.
33
-------
All measurements performed in this house were in accordance with the approved
QA/PP project plan [22]. Certain details are given in Appendix C, Three sets of runs were
carried out which we have termed as follows depending on the air flow rate through the
FPT device described earlier:
(i) 28 l/s - High flow,
(ii) 23.4 L/s - Medium flow
(iii) 18.1 L/s - Low flow
Note that our analytical expression for the pressure field under turbulent flow given
by eq. (10) is strictly valid for a circular disk with the boundary conditions being that at
r = t0, p = pa. We approximate the rectangular basement by a circle of 3.5m mean radius.
We need to also include the extra path length of ambient air flowing down the outer
basement wall, going under the footer, and then flowing through the subslab gravel into
the suction hole. We estimate this to be about 2m. Consequently, we find r0 = 5.5m. The
thickness of the subslab gravel bed h has been found to be about 5cm.
The gravel under the slab, though river-run, were found to be highly heterogeneous
in size and shape. In general, their average size was slightly less than 0.012m, However,
we decided to use the properties of the 0.012m gravel determined experimentally in the
laboratory (See Table 1).
Figure 16 shows the observed and calculated pressure drops for the high and low flow
rates. Readings from holes 13 and 14 are lower and we suspect poorer connectivity to
these holes, i.e., some sort of blockage in this general area. We note that the agreement
between model and observation is indeed striking given the simplification in our model and
also the various assumptions outlined above.
The previous figures indicate which areas under the slab are non-uniform. A better
way of illustrating how well the model fares against actual observations is shown in Fig.
17. The solid line represents the model predictions while observations are shown by
-------
HOLE NUMBER
HOLE NUMBER
16 Comparison of observed and computed pressure drops
for different subslab penetrations using coefficients
of 0.012 m gravel (b = 1.6 and k = 9.4xlCTs m2) . Data
of hole 12 not included.
35
-------
DISTANCE (m)
DISTANCE (m)
Fig. 17 Same as Fig, 16 but with distance from suction hole.
36
-------
discrete points. We note again the satisfactory predictive ability of this modeling approach
and also the fact that certain holes have pressure drop values higher than those predicted
by the model.
In order to illustrate the fact that our approach is sensitive to the selection of type
of gravel bed, Fig. 18 presents the experimental observations plotted against model
predictions with gravel bed coefficients taken to be those that correspond to the 0.019m
gravel. We note the very large differences between model prediction and observed pressures
over the entire basement, thereby suggesting that our approach has enough sensitivity to
be of practical relevance.
An alternate approach to the one adopted here and described above, would be not to
assume specific gravel bed coefficients but to determine these from regression. This entails
using eq.(10) along with the data set of actual observations and determining the
parameters k and b by regression. Since b is not a parameter that varies greatly [8], we
have chosen two different values of b (1.6 and 1.7) to see what difference this leads to in
terms of the coefficient of determination (R2) and in the values of k.
Regression results are summarized in Table 4. We have performed three different
trial runs. The first uses all data points. In trial 2, pressure drop observations from hole
12 (hole that is blocked) have been removed. We note that the R2 improves dramatically,
from 0.80 to 0.96. For trial run 3, holes 9, 10, and 12 have been removed in order not to
bias the regression since these holes have high pressure drop values. We note that the R2
of trial run 3 is 0.88, an improvement over that of trial 1.
Other than the very high R2 values found, the most striking feature is that regression
yields a value of k which is practically identical to that of the 0.012m gravel determined
experimentally in our laboratory apparatus. This suggests that even a visual inspection
37
-------
o
ti
X
CL
O
cc
a
Ui
sz
13
rs
rs
W
K
CL
O
w
X
8L
O
IE
a
W
ec
D
«
ti
iii
k
o.
0,5
-D- OBSERVED
-x~CALCU_ATED
TOTAL FLOW
•28 (l/s)
HOLE HUMBER
e-OBSERVED
«-CALCULATED
TOTAL FLOW
18 (l/s)
Hon: NUMBER
Fig. 18 Same as Fig. 16 but using coefficients of 0.019 m
gravel (b=1.4 and k=3. 4x10"® m*) .
38
-------
of the porous material under the slab can be an indicator good enough for a mitigator to
select a standard bed material and using the physical properties of the material get a
sound estimate of what the suction pressure ought to be in order to generate a certain
pressure field under the slab. The need to categorize commonly found subslab material,
deduce their aerodynamic pressure drop coefficients in laboratory experiments, and thence
tabulate these in handbooks seems to be worth investigating.
8. Graphical Representation
The approach developed here will serve to illustrate how closed form solutions for the
pressure drop in porous beds can be represented in graphical form suitable for professional
radon mitigators. Let us illustrate our approach using the simplest case of a circular
porous bed with radial inflow between two impermeable disks. From the discussion in the
above section, it would seem that we could equally apply our model to square basements
and also to houses with a partial basement.
Equation (13) is valid for laminar flow which would prevail where soil is the subslab
material. It can be written as:
Aph(r) • k = (v./g) • (p,/pj • (l/27t) • (q/h) • ln(r/r„) (16)
where Aph is the pressure drop in head of water and is equal to lp(r)-p»]/(pw-g).
Four curves have been plotted in Fig. 19 for four different values of (q/h);
1.0 • 103 m2/s, 5 * 103 m2/s, 1 ~ 102 m2/s, 2 • 102 m2/s. Thus, if the values of (r/r„), (q/h),
and k are known, Aph(r) can be obtained from this graph.
In the case of gravel under the slab, the flows will probably be turbulent and the
pressure drop is given by eq. (10) which can be rewritten as:
39
-------
Fig. 19 pressure drop in a sand bed with radial airflow between
two impermeable disks.
FRACTIONAL RADIAL DISTANCE,r/rQ
Fig. 20 Pressure drop in a gravel bed with radial airflow between
two impermeable disks. The correction factor F can be
determined from Fig. 21.
40
-------
Aph(r) • F = (1/k) • (va/g) • (pK/pJ ¦ [(l/2jt)h)] • [l/(l-b)l - I(r/rJlb-l] (17a)
where F = [(r«-b)) - (q/hTf (17b)
Figure 20 shows plots for (Apk(r)-F) vs. (r/rj. Two curves have been plotted for the
two different values of b and k. Figures 21a, 21b, and 21c show plots for the correction
factor F for different values of r0 and (q/h) values of 0.05 m2/s, 0,5 m2/s, and 7.5 m2/s. Each
graph has two curves, each representing a different value of b (1.4 or 1.6). It is easily seen
that these graphs can be utilized to obtain values for Aph(r) for given values of r, r0, and
q/h.
Figures 19, 20, and 21 have been presented here in order to illustrate how, from a
closed-form equation, figures can be generated which would be useful to practitioners. The
figures are not meant to cover the entire gamut of conditions which may arise in actual
practice.
9. Pressure Drop Considerations
There are basically three different sources of pressure drops in the mitigation system.
APwtai = Apbed + Apent + Appipe (18)
where Apb6d = pressure drop in porous bed,
Ap„,t = pressure drop due to change of direction and that associated with
entrance effects into the mitigation pipe,
App:pe = pressure drop in the mitigation pipe.
The pressure drop in the subslab bed is given by equations akin to eqs. (10) or (13).
The pressure drop at the entrance to the suction pipe involves accounting for the following
effects; (i) change in flow direction, (ii) change in cross-sectional area, (iii) entrance effects
41
-------
» [„]
Ml [in]
m m
Fig. 21 Correction factor F for gravel beds to be used in
Fig. 20.
42
-------
at the throat of the suction pipe. From an engineering viewpoint, it is more convenient to
treat these together. In accordance with actual practice [25], we propose the following
simplified empirical equation for the head loss:
(Ap,n/pw-g) = Kp • 11-(A/A,)]2 • (Vp72g) (19)
where Kp is the dimensionless pressure loss coefficient which should not depend on
the velocity or the bed thickness, and is a constant for a specific type of bed material,
Ap is the cross-sectional area of the suction pipe,
Ab is the surface area of a cylinder of diameter equal to that of the suction pipe and
height equal to the thickness of the porous bed, and
Vp is the velocity of air in the suction pipe.
If dp is the diameter of the suction pipe, then:
Ay/Ah = (jtdpz/4) • (l/jidph) = (dp/4h) (20a)
and
Vp = q/Ap = 4q/7idp2 (20b)
Table 5 assembles the results of determining the entry pressure loss coefficient for
three different flow rates. We note that Kp values are exactly the same, a gratifying result.
This enables us to place a certain amount of confidence in our model for the entrance
losses.
The pressure drop in the piping includes losses due to elbows, fittings, as well as
straight pipe. Following Ref. [25], losses in the straight pipe is given by:
ApPiPS = f • (L/dp) • (q/A)72g (21)
43
-------
Table 5. Determination of the pressure loss coefficient at the
throat of the mitigation suction pipe in House 21
Suction
Suction
Total
pressure
pressure
Air flow
before
after
Run
{L/s)
entry
entry
Kp
(cit. water)
(err. water)
1
18,3
1.300
1.664
0. 053
2
23.4
1.938
2.540
0. 053
3
28.4
2.700
3.589
0.053
Table 6. Relative pressure drops in the
mitigation system
of House 21
Total
Air flow
^Pbad ^Pent
APpy*
Hydrodynamic
Run
(L/s)
{cm water) {cm water)
(cm Water) effectiveness
(%)
1
18.3
1.30 0.363
8,0 x 10"3
77.8
2
23.4
1.94 0.602
13.0 x 10~3
75.9
3
26 .4
2.70 0.669
17.7 x 10~3
74 . 9
44
-------
Pressure losses in bends and fittings are normally expressed in terms of an
equivalent pipe diameter. For example, a 90° elbow has the same pressure drop as a
straight pipe of length equal to about 25 times the pipe diameter [25],
Since the primary objective of the mitigation system is to create a suction pressure
under the slab only, we can define a hydrodynamic effectiveness of the mitigation system
based on these three pressure drops.
Hydrodynamic effectiveness = Ap^ / Aptola) (22)
We have computed these various pressure drop values for the case of H21 in order
to get an idea of the relative magnitude of these pressure drops. The mitigation system
(with one suction hole only) in H21 has about 7m of straight pipe of 0.1m diameter and
three 90° elbows. This translates into a total length of {7+3-25-0.1) = 14.5 m.
Table 6 assembles the various pressure drops in the three elements of the mitigation
system. While Apbed and Ap,nthave been measured, Appjpe has been calculated from eq. (21).
The hydrodynamic effectiveness defined by eq. (22) is also given.
We note from Table 6 that Applpe is negligible compared to Ap^, while Apenl is about
30% of Ap^. The hydrodynamic effectiveness is close to being independent of the flow rate
and is about 75%. Thus 75% of the energy used up by the suction fan goes directly into
creating the subslab depressurization while the rest can be regarded as being redundant
expenditure of energy. Though present knowledge does not permit us to suggest a
particular value for the optimal hydrodynamic effectiveness, we suggest that future
engineering guidelines dealing with mitigation system design specify a working range for
this index.
45
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10. Future Work
Logical extensions of this study would involve applications of this methodology to
houses with (i) homogeneous beds but with irregular boundaries, and (ii) non-homogeneous
porous beds. One approach is to develop a computer program using numerical methods
(either finite element or finite difference could be used) to solve the basic set of
hydrodynamic and mass conservation equations [7,14]. Pressure fields under the slab for
practically any configuration could be thereby predicted. An optimization algorithm could
then be attached to the above program in order to obtain the optimal layout and the
number of mitigation suction points for the particular subslab conditions such that certain
well-defined and physically relevant constraints are satisfied.
Our present line of thinking is that though the above approach offers great flexibility,
it is not easy to use by non-experts. Developing engineering guidelines for practitioners
based on such a code demands a certain amount of effort and practical acumen. It would
be wiser to define a few "standard" cases of basement shape, subslab conditions and
mitigation pipe locations; try to develop simplified closed-form solutions of these cases; and
then see how well these solutions fare with respect to actual measurements taken in the
field. If such an approach does give satisfactory engineering accuracy, its subsequent
use to mitigation system design would be relatively simple and straightforward.
46
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References
1. D.T. Harrie and L.M. Hubbard, Proceedings of the Radon Diagnostics Workshop,
April 13-14, 1987, EPA-600/9-89-057 (NTIS PB89-207898), June 1989.
2. J. Wang and M. Cahill, Radon reduction efforts in New Jersey, presented at the
Annual Meeting of the National Health Physics Society, Boston, MA, July 4-8, 1988.
3. D.C. Sanchez, Technical issues related to emission releases from subslab radon
mitigation systems, presented at ASCE National Conference on Environmental
Engineering, Austin, TX, July 9-12, 1989.
4. K.J. Gadsby, L.M. Hubbard, D.T. Harije, and D.C. Sanchez, Rapid Diagnostics:
Subslab and Wall Depressurization Systems for Control of Indoor Radon; in
Proceedings: the 1988 Symposium on Radon and Radon Reduction Technology, Volume
2, EPA-600/9-89-006b (NTIS PB89-167498), March 1989.
5_ B. H. Turk, J. Harrison, R. J. Prill, and R, G. Sextro, Soil gas and radon
entry potentials for substructure surfaces, in Proceedings! the 1990
International Symposium on Radon and Radon Reduction Technology, Volume 2,
EPA-60Q/9-9l-026b (PB91-234450), July 1991,
6. C.S. Fowler, A.D. Williamson, B.E. Pyle, F.E. Belzer, and R.N. Coker, Engineering
Design Criteria for Sub-Slab Depressurization Systems in Low-Permeability Soils,
EPA-600/8-90-063 (NTIS PB90-257767), August 1990.
7. J.M. Barbar and D.E. Hintenlang, Computer Modeling of Subslab Ventilation Systems
in Florida, 34th Annual Meeting of Health Physics, Abstract No. TAM-E8,
Albuquerque, NM, 1989.
8. T.G. Matthews, D.L. Wilson, P.K. TerKonda, R.J. Saultz, G. Goolsby, S.E. Burns,and
J.W. Haas, Radon diagnostics: subslab communication and permeability
measurements, in Proceedings: the 1988 Symposium on Radon and Radon Reduction
Technology, Volume 1, EPA-600/9-89-006a (NTIS PB89-167480), March 1989.
9. W.W. Nazaroff, B.A. Moed, and R.G. Sextro, 'Soil as a source of indoor radon
generation, mitigation and entry,' Chap.2, Radon and Its Decay Products in Indoor
Air, W.W. Nazaroff and A.V. Nero (Eds.), John Wiley and Sons, NY 1988.
10. A.G. Scott, 'Preventing radon entry,' Chap.10, Radon and Its Decay Products in Indoor
Air, W.W. Nazaroff and A.V. Nero (Eds.), John Wiley and Sons, NY 1988.
11 R.C. Bruno, Sources of indoor radon in houses, J. Air Pollution Control Association,
Vol.33, pp.105-109, 1983.
12, W.W. Nazroff, B.A. Moed, R.G. Sextro, K.L. Revson , and A.V. Nero, ^Factors
Influencing Soil as a Source of Indoor Radon: a Framework for Geographically
Assessing Radon Source Potentials,' report LBL-20645, Lawrence Berkeley Laboratory,
Berkeley, CA 1985.
13. R.J. Mowris, 'Analytical and Numerical Models for Estimating the Effect of Exhaust
Ventilation on Radon Entry in Houses with Basements or Crawl Spaces,' M.S. Thesis,
Lawrence Berkeley Laboratory, Berkeley, CA 1986.
47
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14. C. de Oliveira Loureiro, 'Simulation of the Steady-state Transport of Radon from Soil
into Houses with Basement under Constant Negative Pressure/ LBL-24378, Lawrence
Berkeley Laboratory, Berkeley, CA 1987.
15. v.C. Rogers and K.K. Nielson, Benchmark and application of the RAETRAD
¦odel, in Proceeding!: the 1990 International Symposium on Radon and
Radon Reduction Technology, Volume 2, EPA-6QQ/9-91-026b, July 1991.
16. M. Muskat, The Flow of Homogeneous Fluids through Porous Media, McGraw-Hill,
1937.
17. R.E. Collins, Flow of Fluids Through Porous Materials. Reinhold Publishing Co., 1961.
18. A.E. Scheidegger, The Physics of Flow Through Porous Media, Univ. of Toronto
Press, 3rd Edition, 1974.
19. P.S. Huyakorn and G.F. Pinder, Computational Methods in Subsurface Flow,
Academic Press, 1983.
20. J. Bear, Dynamics of Fluids in Porous Media. Dover Publications, New York 1988.
21. J.D. Gabor and J.S.M. Bote rill, 'Heat Transfer in Fluidized and Packed Beds,' Chap.6,
Handbook of Heat Transfer Applications, W.M. Eoshenow, J.P, Hartnett, and E.N.
Game (Eds.), 2nd Edition, McGraw-Hill,1973.
22. C.S. Dudney, et al., Investigation of Radon Entry and Effectiveness of Mitigation
Measures in Seven Houses in New Jersey, EPA-600/7-90-016 (NTIS DE89016676),
August 1990.
23. G.E.P. Box, W.G. Hunter, and J.S. Hunter, Statistics for Experimenters: An
Introduction to Design, Data Analysis and Model Building. John Wiley & Sons, New
York 1978.
24. P.K. Hopke (Ed.), Radon and Its Decay Products, American Chemical Society, 1987.
25. ASHRAE, Handbook of Fundamentals, American Society of Heating, Refrigeration
and Air-Conditioning Engineers, Atlanta, 1985.
48
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Appendix A: Brief Review of Scientific Theory
Relating to Flow through Porous Media [16-211
A porous medium is defined as a solid containing holes or voids, either connected or
unconnected, dispersed within it in either a regular or random manner such that holes
occur relatively frequently within the solid [17]. In this study, we are specifically
interested in unconsolidated isotropic beds such as sand or gravel and our discussion will
be limited to such material. Since the structure of such beds is so irregular and random
that it can be described only in statistical terms, the prevalent approach is to treat such
beds on a macroscopic basis, analogous to the approach followed in the kinetic theory of
gases. Thus, on a macroscopic scale the system can be defined in terms of a few
determinable quantities from which phenomena like fluid flow or heat transfer can be
accurately predicted for engineering purposes.
Al. Definitions of geometrical quantities
All the following properties are bulk properties in that they pertain to a unit total
volume of the bed. Note that as such they have significance only for samples of porous
beds containing a relatively large number of pores.
(a) Porosity ()
The porosity or void fraction of a material is the fraction of the bulk volume of the
total material occupied by voids. Thus
Volume of pores or void volume Vp
$ = = _ (Al)
Total or bulk volume Vx
The void fraction for randomly packed beds of uniformly sized spheres in containers
49
-------
of diameters about 50 times the particle diameter is in the range of 0.36-0.43 [21J.
(b) Equivalent diameter (dv)
Porous unconsolidated material such as gravel beds are made up of pebbles with
varying sizes and of irregular shape. The equivalent diameter is defined in terms of a
mean spherical particle having the same volume. Thus
d-
¦v
1/3
(A2a)
where V, is the volume of n particles selected randomly.
Alternatively, since V, is not an easily measurable quantity, we can use the following
expression to estimate dv:
( 6
_ • (14) • vT
nit
(A2b)
Note that dv is the mean diameter. For a more accurate treatment, the distribution
of the particle diameters have to be determined experimentally for which purpose sieving
is done using different sizes of screens.
(c) Particle shape factor (s)
The shape factor is important as it affects the surface area per unit volume and is
usually defined in terms of a spherical particle which has the minimum surface area per
unit volume. Thus
surface area of a sphere per unit volume
(A3)
surface area of the particle
50
-------
(d) Effective diameter {d8)
For purposes of friction drop or heat transfer calculations, it is the surface area (A,)
of the particles per unit volume of material which is the influencing parameter. Thus
nndv2 Tcdv3n 6 6
As = = . = (l-4») • _ (A4)
VT 6 VT.ds ds
Following the hydraulic radii theory, the effective diameter can be computed as:
4 (void volume) 2
ds = = _ . . dv (A5)
wetted area 3 (1-0)
A2, Reynolds number
The Reynolds number, which is the ratio of inertia force to viscous force, gives an
indication of the type of flow: whether laminar or turbulent. This is of crucial importance
since pressure drop as well as heat transfer characteristics of the porous bed are greatly
influenced by the regime under which flow occurs.
The Reynolds number is defined as
qp V.l
Re = .1 = (A6)
Ap. v
where V is the apparent or surface velocity, q the volumetric flow rate, p the fluid
density, p. the dynamic viscosity, v the kinematic viscosity, A the cross-sectional area of
the porous bed and 1 a characteristic length dimension.
Note that V is not the actual velocity in the pores but is a velocity obtained by
measuring the discharge q through an area A in the absence of the porous bed. It is often
51
-------
referred to as the superficial velocity.
The question that arises is what to choose as the length dimension of the bed which
should determine the nature of the flow. Different researchers seem to have chosen this
dimension differently leading to a certain inconsistency in the corresponding Reynolds
numbers thereby computed [16-18], The most common definitions of the characteristic
length for porous beds are;
(i) 1 = d,
(ii) 1 = dv
(iii) 1= (dy4>)
It must be pointed out that usually in flow through pipes, a dimension of the flow
channel is chosen for 1. However for porous beds it is difficult to measure pore or void
diameter and consequently the particle or pebble diameter is preferred. However, this by
itself does not explain all the characteristics of an actual porous bed with heterogeneous
and irregular particles. Consequently in the present study we have opted to follow
definition (iii). By including the porosity as well, the definition of the Reynolds number
will actually correspond to the fluid velocity in the pores since this is given by (V/).
For the purposes of this study, we shall assume the following safe limits for the flow:
laminar for Re < 1 and turbulent for Re> 10. Since the particles are irregular and of
different sizes, the transition from laminar to turbulent flow is not abrupt at a single
critical Reynolds number as is the flow through pipes. Instead the transition is rather
gradual and therefore there exist studies which report laminar flows at Re values close to
5 and above, while others report the onset of turbulence at Re numbers close to 5. Thus
the critical values stated above should be viewed as inciicators for establishing the regime
in an approximate manner rather than strict numerical cut-off values.
52
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A3. Flow dynamics
The flow of fluids through porous media is complicated by the fact that the flow is
highly irregular and tortuous. Despite this, the analogy to flow through pipes is used to
study both laminar and turbulent regimes by starting explicitly with a correlation between
Reynolds number and the friction factor. Consequently for porous media, the friction factor
f can be defined as [17J:
dv • g •
r*0AN
v i y
dp
dx
(A7)
where dp is the pressure drop differential (Pa in SI units) across a porous bed length
differential dx.
During the laminar regime, i.e. Re 1., the product (Re * f) is assumed to be
essentially equal to a constant c. Thus, from eq.(A7) and the definition of Re, we have
q <(>.dv2 dp
_ = g . • _ (A8)
A c.v dx
where v is the kinematic viscosity of the fluid flowing through the porous bed.
This equation is referred to as Darcv's law following the experimental
scientist who originally proposed it. We can recast eq.(A8) as
q dp
_ = K . (A9)
A dx
where K = g . ($.dv3) / (c.v) (AlO)
53
-------
and is a coefficient representative of the conductivity of the porous bed to the
particular fluid.
The parameter K depends on characteristic parameters of the porous bed as also
(provided of course that the flow remains laminar) on those of the fluid (because of the
inclusion of v). In order to separate these, we define the permeability (sometimes also
referred to as the 'intrinsic permeability') of the porous bed as
4. dv2
k = (All)
c
The conductivity K and the permeability are correlated as follows:
K • c
k = (A12)
g
Thus the permeability can be defined as the volume of a fluid per unit viscosity
passing through a unit cross section of the medium in unit time under the action of a unit
pressure gradient [16]. It has units of area and is determined only by the structure of the
porous bed and is entirely independent of the nature of the fluid. It is thus a constant
for a bed made up of a specific porous material.
Finally, the following aspects need to be spelled out explicitly:
(i) The above derivation is intended more as a heuristic guide to understanding flow
behavior rather than a formal proof of Darcy's law which most text books derive from
the classical hydrodynamical equations of Navier-Stokes [20].
(ii) Equation (All) is actually an operational definition of k. This is because the
heterogeneity and irregularities of commonly encountered porous beds do not permit
k to be accurately computed from basic properties of the bed which have to be
54
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deduced experimentally. Thus the form of the equation suggests that a statistical
averaging as against strict accounting for variation in flow in the individual pores has
been adopted. Consequently, only on a macroscopic sense is the velocity of a fluid
flowing through a porous medium directly proportional to the pressure gradiant acting
on the fluid.
Darcy's law is no longer valid when the flow becomes partially or completely
turbulent. Under such conditions, the literature contains several empirical models
proposed by different researchers to treat flow in porous media. These are addressed
briefly below,
(a) Linear dependence of (Re • f) on Re [16]:
This approach starts with the assumption that
Re • f = c, + c2 • Re (A13)
where Cj and c2 are dimensionless constants which depend on the properties of both
fluid and porous media.
From the above, we obtain the relation
dp q fq \2
= Cj _ + c2
dx A
vA /
(A 14)
where c', and c./ are given by
c,.v c2
and c2' = (A15)
g.dv24 g.dv.
Such a treatment originally proposed by Forchheimer, has a certain amount of appeal
55
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since it can simultaneously account for different types of flow while yielding the relative
contributions of each on the total pressure drop. With c'2=0, we get back Darey's law while
with c'j =0, we obtain the quadratic exponent found for turbulent flow in pipes according
to Fanning's equation. Thus we can view this approach as treating actual pressure drop
as consisting of a pressure drop resulting from laminar flow added to a pressure drop
occurring from turbulent flow.
(b) White and later Missbach [18] have suggested the following model;
Re • f! = c
(A16)
which is analogous to the following
dp fq \
_ = a . _
dx [A ,
(A17)
where
a =
( c.v
Vdv"*'-.0&1 y
and b =
2n-l
11
(A18)
It is clearly seen that for laminar flow b would be equal to 1 while for turbulent flow
it would be close to 2. For mixed flow, the exponent would be between 1 and 2, the exact
value being dependent on the circumstances specific to the particular case. Unlike the
Darcy equation (eq.(A9)) where the interpretation of the constant k is unambiguous, it is
difficult to assign a rigorous interpretation to the coefficients of eqs.(A13 & A16). However
loosely speaking the coefficient a of eq.CAl?) can be considered to represent the resistivity
to flow offered by the porous media to the particular fluid.
Several other studies have proposed either variants of the above two approaches or
56
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more complex empirical correlations, either between the Reynolds number and the friction
factor, or directly for the pressure drop in the porous bed against parameters describing
both material and flow conditions. We shall not discuss these here given the more
advanced nature of these models and the inappropriateness of resorting to such models in
the framework of the type of practical application we have in mind.
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Appendix B: Experiments to Determine Porosity
and Equivalent Diameters of Gravel
The porosity of a gravel bed ($) and the equivalent diameter (dv) of porous materials
have been defined in Appendix A, Though these parameters do not explicitly figure in the
pressure drop versus flow models outlined in the main portion of this report as also in
Appendix A, they do implicitly influence such models through the permeability term of the
porous bed. Thus a knowledge of these parameters would indeed be useful. Consequently
we have undertaken an experimental determination of u and dv, the results of which are
presented and discussed below.
Bl. Determination of bed porosity (0)
Experimental techniques to determine porosity of porous beds are well known (see
for example, Refs. [16,17]. Perhaps the simplest technique is to choose a certain volume
of the bed material and then measure the volume of the voids by measuring the volume
of a liquid (for example, water) needed to completely saturate the porous bed. Either the
volume of liquid poured in or drained out (or both) could then be used to directly estimate
the porosity.
Tables Bl and B2 present the experimental observations relating to the volume of
water both poured in and then drained out from a total volume of the porous material of
1 liter. Note that the observations of the first run have to be discarded due to errors
arising as a result of initial wetting of the gravel. Repetitions both in number of samples
and runs for each sample assure the determination of a sounder and more representative
value of <(s.
We note that within-sample variance is smaller than that across samples for larger
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Table B1. Experimental observations in order to determine
porosity of the river-run gravel of 3/4" <0.019 m) nominal
diameter. Numbers indicate volume of water in cubic
centimeters poured in (1) and drained out (2) from a total
volume of 1 liter.
Run
Sample A
Sample
B
Sample C
Sample
D
1
2
1
2
1
2
1
2
1*
439
414
440
420
454
435
462
440
2
424
425
415
415
447
447
427
426
3
417
420
410
413
437
440
420
425
4
412
415
414
416
434
439
422
425
5
409
410
416
418
438
436
422
424
Mean
416
. 5
414 .
6
439
.7
423.
9
423.7
SEM
2.
16
CO
* .3
4
1.
71
0.8
3
5.72
The values of thxs run are discarded due to inxtxal wettxricf
of the gravel
Table B2. Experimental observations in order to determine
porosity of the river-run gravel of 1/2" (0.012 m) nominal
diameter. Numbers indicate volume of water in cubic
centimeters poured in (1) and drained out (2) from
a total volume of 1 liter.
Run Sample A Sample B Sample C Sample D Sample E
1 21 21 21 21 2
1*
442
465
433
376
435
389
416
365
419
365
2
376
384
376
381
394
392
375
374
374
375
3
370
374
374
376
385
385
37 0
373
362
363
4
368
370
372
375
376
377
371
374
366
368
5
370
366
374
375
380
380
365
373
365
368
Mean 372.5 375.4 383.6 371.9 367.6 374.2
SEM 2.10 0.92 2.35 1.14 1.14 2.66
{*) The values of this run are discarded due to initial wetting
of the gravel.
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gravel which suggests that the errors of our experimental procedure are lower than the
variations associated with taking different samples. Though this is not so for the smaller
gravel, the magnitude of these values are small and can be confidently overlooked for our
purpose.
We summarize the values of the porosity estimated for river run gravel:
0.012 m (1/2 in.) nominal diameter: = 0.374
0.019 m (3/4 in.) nominal diameter: = 0.424.
B2. Determination of equivalent diameter (dv)
The most convenient way of deducing dv is from eq.{A2b). Since we already have an
estimate of the porosity, all that remains is to determine the number of gravel stones in
a given volume.
Again we chose a total volume of 1 liter and counted the number of stones, the
results of which are shown in Table B3. It would have been better to get an estimate of
the particle size distribution as well, but this could not be done due to lack of appropriate
screening sieves. Thus only an estimate of the mean diameter has been obtained in this
study.
The mean number of pebbles in both sizes of gravel are given in Table B3 along with
the SEM values. We note that for large gravel the SEM is almost 5% while for smaller
gravel it is less: something which is to be expected given that experimental errors are
larger for larger gravel.
The mean values of dv computed from the experimental readings of Table B3 are:
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Table B3. Results of the experiment to determine the mean
equivalent particle diameter of the river-run gravel.
Total volume of the sample was 1 liter.
Large pebbles Small pebbles
Nominal diameter Nominal diameter
= 3/4" CO.019 m) =1/2" {0.012m)
Sample No. No. of pebbles No. of pebbles
1 209 1458
2 179 1688
3 174 1529
4 181
Mean 185.75 1558.3
SEM 7.89 68.0
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River-run gravel Nominal diameter
Estimated equivalent diameter
Large 0.019 m (3/4 in.) 0.018 m (0.71 in.)
Small 0.012 m (1/2 in.) 0.009 m (0.36 in.)
We note that deviation between nominal and equivalent diameters is small for the
large river-run gravel. As for the smaller sized gravel, the important difference is likely
to be as a result of the fact that this gravel type contained a relatively large number of
very small pebbles thereby decreasing the estimated effective mean value. Thus we
attribute this departure from the nominal diameter to have arisen as a result of improper
or non-rigorous sieving separation process adopted by the supplier rather than as a result
of the pebbles being systematically smaller. A fact to be retained is that in practical
situations, the large variation in the distribution of particle diameters even when a specific
nominal diameter is specified would lead to a loss of scientific predictability or
reproducibility in the pressure drop versus flow relationships when these are estimated
from actual experimental measurements.
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Appendix C: Quality Assurance and Quality Control (QA/QC) Statement
Data, in general, have been collected in accordance with the data quality goals set
forth in the QAPP section of Ref. 22, We shall, however, address in this appendix the
measurements specifically relevant in the framework of this particular study.
Two types of experiments were performed: one in the laboratory setting (described
in Section 5) and another in an actual research house (described in Section 7). Both these
types of experiments essentially involved the use of the same types of instruments and of
data gathering techniques. Consequently, the discussion that follows applies to both these
experimental settings.
Equipment used are described below:
(i) an industrial vacuum cleaner capable of sucking 45 x 10 3 m3/s (95 cfm) of air through
a 0.05 m (2 in.) diameter orifice under 1.9 m (75 in.) of water static vacuum pressure;
(ii) a speed control and an air by-pass adapter (which is simply a perforated length of
plastic pipe). Both these are needed in order to vary the air flow rate through the
porous bed;
(iii) a 3 mm stainless steel pitot-tube (Dwyer No. 166-6) to measure velocities from 0.05
to 15 m/'s (10 to 3000 ft/min). Tables for different pipe diameters (as described in
Ref. [4]) enabled the corresponding volume flow rate to be deduced;
(iv) an electric digital micromanometer (EDM) (Neotronics Model EDM-1) which can
measure pressures with a resolution of 0.025 x 10 3m (103 in.) of water or 0.25 Pa,
and having a maximum range of up to 0.5 m (20 in.) of water. This is also described
in Ref. [4].
(v) two mounting devices: (a) a 0.038 m (1.5 in.) outer diameter brass pipe to connect the
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suction hole to the vacuum hose with arrangements to attach the pitot tube [called
the Flow Pressure Tube (FPT)3, and also the EDM, and (b) a 0.019 m (0.75 in.)
stainless steel pipe to mount the EDM in order to measure the pressure at each of
the nine different taps. These devices have already been described in detail in a
previous report [4].
Details of the accuracy, precision and completeness as well as the calibration details
are given in Table Cl. The data gathered was discrete in nature (as against data
electronically measured and stored continuously in a data logger). For the laboratory
experiments, three sets of experiments were conducted (see Table 1). In all, 9 discrete
runs of data gathering were involved, with each run entailing one air flow rate
measurement and 9 pressure difference measurements. During field experiments in the
research house, only 3 discrete runs were performed, each run requiring the measurement
of one air flow rate and 19 pressure difference measurements. Care was taken in both
experimental settings that all data gathered were in accordance with the particular QAPP
and met those requirements. How the data collected was analyzed is described in Sections
5 and 7 of the main portion of this study.
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Table CI. Monitoring Instruments
Frequency
Instrument Range Accuracy Precision Completeness Calibration of Calibration
Standard Zero Span
1. Pitot-tube 0.05-15 m/s 5% 5% 95% Factory Before each 1/6 months
Dwyer No. 166-6 measurement
2. Electric Digital 0-5000 Pa 0.25 Pa 0.25 Pa 95% Factory Before each 1/6 months
Micromanometer, measurement
Neotronics Model EDM-1
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