Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
SOLVING TURBULENT FLOW DYNAMICS OF
COMPLEX, MULTIPLE BRANCH RADON MITIGATION
SYSTEMS
L. Moorman, Ph.D.
Radon Home Measurement and Mitigation, Inc.
Fort Collins, CO
ABSTRACT
Active radon mitigation systems with a large flexibility in their complexity are simulated
via computer calculations assuming incompressible gas flow dynamics. When designing
more complex, multiple branch systems, insight into the turbulent flow dynamics inside
the systems vent pipes may be of advantage in design and optimization of pipe sizing and
radon ventilator characteristics. I will show how their effects can be simulated and
assessed during the installation of real systems.
For an N-branch, active, non-looped, radon mitigation system I will show how to
construct the complete set of non-linear equations with its independent variables and
demonstrate how these can be uniquely solved and applied to optimize a system under
design. A number of real systems were used to compare with calculations.
I will show a number of applications using this program:
1) The effects of sub-slab material and its resistance on the system.
2) The effects of turbulent flow through vent pipes of various diameters.
3) The effects of varying ventilator characteristics.
4) Calculations of multiple pipe branches in real time.
INTRODUCTION
The equation that governs the pressure distribution for the air in a single branch radon
system in its simplest form consists of three components, the pressure loss through the
soil and the radon extraction cavity, the pressure loss due to the resistance of the air with
the pipe system and the pressure boost by the action of an operating ventilator somewhere
in the pipe system. I will first discuss each of these components separately and indicate
in each case which theoretical functional behavior approximates the description of the
various pressure differences best. After that I will put together the components and
describe how the equations for complete radon mitigation systems can be derived for
multiple branches. Numerical evaluation of the solutions will result in gaining
quantitative insight into the use of the equivalency of pipe sizes in complex radon
systems.
SUB SLAB MATERIAL AND CAVITY RESISTANCE PARAMETERIZATION.
A dimensionless parameter that characterizes the flow for a gas through an opening or
capillary is the Knudsen number. This is defined as the ratio of the mean free path of
particles in the gas divided by a characteristic dimension such as the diameter of the pipe:
!
Kn =
(1)
d
In the movement of atoms or molecules of gas through small capillaries of the sub-slab
medium, such as soil, rock or concrete, the flow may be more determined by the
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
collisions with the walls of the pores in the porous material than by collisions among the
gas particles. In that case, the viscosity of the gas (which is a parameter for the friction
inside the gas) does not play a large roll in the dynamics of the system. Therefore, on the
one hand, very slow air movement inside a medium when no active soil depressurization
exists, direct diffusion of radon gas through small capillaries may be the largest mode of
transportation for radon. On the other hand, when enough small cracks and openings
exist, or if the pore size in the material is large enough, convective movement of air
which drags radon with it through these openings is more likely to dominate the mode of
transportation for radon.
The sub-slab material plus extraction cavity resistance (friction) for air movement when
an active soil depressurization system operates can be measured in situ by placing a fan
with relevant power and flow characteristics directly over the final excavated extraction
cavity and measuring the pressure across the fan while running the fan at various speeds.
When varying pressure differences, ΔPf, and measuring corresponding flow rate s, F, the
curve that is generated appears in practice sufficiently linear in the relevant flow range
for active mitigation systems. Fig. 1 shows an example of these parameters that were
measured in existing homes during mitigations.
Fig 1: Flow measured as a function of pressure difference across ventilator in the configuration described
in text for various residences and different branches in one residence.
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
Flows were measured by timing the capture of air from the end of the fan into a well
determined size bag of known volume. Accuracy was verified by employing the same
method to measure the fan curves as advertised by manufacturers for various fans. Flow
measurements were found to be somewhat lower than the manufacturer’s specification,
but sufficiently accurate to be able to use this method for this purpose.
The resulting slope in Fig 1 is independent of which ventilator is used on a radon
extraction cavity for the measurement. The observed linearity allows us to define a (local)
linear friction Rg for the combined soil with extraction cavity through the following
relationship:
!Pf = PFR g
(2)
This equation can also be understood by interpreting the right hand side as the combined
soil pressure drop inside the cavity (ΔPs=PFRg) which must counteract the pressure boost
(ΔPf) by the fan in this simple model. The experimental linearity may be related to the
fact that air velocities further away from the extraction cavity, representing the majority
of the sub-slab material, are small. Deviations from linearity are expected at higher flow
rates.
By dividing both sides of the Eq. (2) by pressure and flow rate I obtain an equation for
the friction:
!Pf
Rg =
(3)
PF
The unit of resistance is independent of the units used for pressure. If flow is expressed
in cubic feet per minute (cfm) this resistance expresses how many minutes it takes for
one cubic foot air to move through the extraction cavity (min/cf) [in SI units the flow
would be expressed in m3/s and the resistance would be in s/m3].
In Fig. 2 a graph is shown of a number of measured soil resistances graphed against the
relative fan depressurization, which is the measured depressurization across the fan
compared to the maximum fan depressurization at zero air flow.
What can be seen in this “landscape of resistances” is a correlation of low soil resistance
with low relative fan depressurization, as in large gravel, and a correlation of high soil
resistance with high relative fan depressurization, as is the case in very tight clay. Other
sub-slab material types found in between are designated as pea gravel, sand, looser clay,
hard sand and clay. It must be emphasized here that the measurement of the resistance is
unique to the specific residence and includes not only the sub-slab material type
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
Fig 2: Resistances measure of excavated radon cavities in existing homes following method of Fig 1.
but also any inhomogeneities in the sub-slab material and any leaks due to openings in
the slab. The measured resistance defined here is a good quantification of the effective
resistance that a radon mitigation system will be subject to via this radon extraction
cavity. In this sense, I have found a quantification for the resistance of the extraction
cavity with sub-slab material which is also valid when dealing with a multi-branch radon
mitigation system. Thus this information can be used in simulations that include the
complete system with all branches as will be shown later.
VENT PIPE MODELING
Laminar, viscous flow is indicated by values of the Knudsen parameter, defined in Eq.
(1), smaller than 0.01 indicating that collisions between the particles of the gas are much
more abundant and are determining the character of the flow rather than particle-to-wall
collisions. This is the situation in air moving through a pipe at small speed and flow rates.
At small flow rates, for viscous, laminar flow through a cylindrical tube the velocity of a
fluid with viscosity η can be shown to be a parabolic (quadratic) function of the distance
from the axis with the largest velocity on the central axis of the tube, and a zero velocity
near the wall. The volume rate of fluid crossing any section of the tube with radius r for
this flow is given by:
#r 4
F=
!P
(4)
8"L
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
This is known as Poiseuille’s formula and is important in medicine since it gives a
qualitative understanding of blood flow through our arteries and veins. In addition, by
reading Eq. (4) from right to left it can be interpreted that under steady state flow
conditions the pressure loss along any section of a cylindrical pipe of length L is a linear
function of the flow rate through a cross section of the pipe:
8"L
#P = 4 F
(5)
!r
However, when the flow velocity of a fluid becomes sufficiently large, laminar flow
breaks down and turbulence sets in.
The critical point for the onset of turbulence in a fluid can be characterized by a
dimensionless parameter referred to as the Reynolds’ number, which is the ratio of the
shear stress in the fluid due to turbulence and the shear stress due to viscosity given by:
2r"v
NR =
(6)
!
In this equation ρ is the density of the fluid and v the velocity. Experiments have shown
that the flow will be laminar when the Reynolds’ number is below 2000 and the flow will
be turbulent for values above 3000. In between these values the flow can be either
laminar or turbulent and can go back and forth.
When the air speed increases through a pipe, the same model applies and when the
Reynolds’ number increases beyond its critical value the air flow becomes turbulent and
the resistance increases. For such situations the linear relationship between pressure loss
along the pipe and flow rate is no longer valid and the new relationship is a nonlinear
function of flow rate that must go through zero. Thus I can extend the pressure loss
behavior to a more general form with two parameters which may be different for each
diameter of the pipe:
!P( F ) = cF a
(7)
The variables for the pressure drop along 100 ft of the pipe for different diameters are
known from the heating and ventilation industry and values for 2”, 3”, 4” and 6”round
tubes are known.
The derivative of this pressure loss to flow can be written as:
"#P
( F ) = caF a !1
"F
The length must be scaled appropriately to the total equivalent length of the pipe where
90 degree elbows and 45 degrees elbows have a certain equivalent length in air resistance
to straight section of pipe of the same diameter. This equivalent length approximation is
used in the heating and ventilation industry for ducts and assumes that the nonlinearity
(power of F) of the components is the same for connections as for straight pipe. If this is
not the case a different equivalent length would have to be determined for each
component at each air flow rate. In the following I will use the assumption that the
equivalent length calculation can be made.
VENTILATOR CURVE MODELING
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
The modeling of the ventilator curve (fan curve) can be done sufficiently accurate for our
goal by three parameters. I assign a new function with three parameters to each ventilator
and have verified it can be accurately described for each fan (f) by:
(8)
g ( f , F ) = q1 + q 2 F b2
This means the derivative to the flow rate of the fan curve is:
"g
( f , F ) = q 2 b2 F b2 !1
"F
In our model of multiple branch radon mitigation systems I will allow for the possibility
that any pipe section between nodes (pipe TEE’s) can have a fan inserted.
SINGLE BRANCH RADON MITIGATION SYSTEM
For the calculation of a single Branch radon mitigation system, I can now add the
individual components keeping in mind that all pressure losses due to sub-slab material
and cavity resistance and resistance in pipe system together are counteracted by the
pressure boost from the ventilator, resulting in the nonlinear one-dimensional equation:
a
PF1 R g + c1 F1 ! g ( f , F1 ) = 0
(9)
Once the resistance for the sub-slab material (Rg) is measured, it can be shown that a
unique solution exists for each fan and pipe diameter.
By dividing the equation by pressure and flow I find:
a !1
F1
g ( f , F1 )
(10)
!
=0
P
PF1
This can be employed in a graphic solution even during the mitigation as is shown in the
fan resistance response curve in Fig. 3.
R g + c1
For any given ventilator, relevant calculations can be made that will give us insight in the
flow dynamics of the system by numerically evaluating the solution to Eq. (9) as a
function of equivalent length for various cavity resistances. The resulting curves for one
commonly used fan are shown in Fig 4. It shows that the relative change of volume rate
through the system is very different depending on the resistance. For low cavity
resistance the flow rate depends strongly on the equivalent length for short equivalent
lengths only whereas for high resistances the flow rate is nearly independent of the
equivalent length.
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
Fig 3: Fan Resistance Response curve: Method of graphically solving Eq. (10). For 4” diameter pipe
system and 175 ft equivalent length the red squares indicate the resistance for all flow rates. To solve the
equation find the crossing with the fan curve (green and blue squares) indicated by a red circle and shift
the circle up by the extraction cavity friction (heavy red solid line) along the fan curve to find the value
inside the second circle. The arrow points at a flow rate of approximately 80 cfm which is the steady state
flow that this ventilator in this mitigation system will support. For a 3” diameter pipe it is indicated at the
second arrow that the flow will go down to approximately 50 cfm., thus loosing 38% of the flow rate
through this system. The dashed horizontal line indicates the resistance of a different extraction cavity that
was measured with a higher resistance which leads to a flow rate solution even with a 4” pipe of
approximately 50 cfm.
Similarly I have evaluated this information given in figure 4 for different diameter pipes,
which is most relevant for 3” and 2” pipes. I can than evaluate the ratio for flow rate of
the 3” pipe with 4” pipe. This ratio is given in Fig. 5 as a function of equivalent length for
each cavity resistance. It can be seen that suppression effect of flow for smaller pipe
sizes are strongest for low cavity resistances (crawlspace, gravel) and weakest for large
cavity resistances (hard sand, or clay). This can be qualitatively reformulated by the
statement that high flow rates are easier choked by a smaller diameter pipe than low flow
rates, and that when the flow rate is small the effect of pipe diameter is not very
important.
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
Fig4: For a FR150 fan with 4-inch pipe for a single branch system the volume rate is numerically
calculated for several extraction cavity resistances as function of equivalent length.
The information obtained from these calculations and for similar calculations for 2”
diameter vent pipes can be summarized by defining a criterion for flow rate loss. I
choose to extract from Fig. 5 a critical equivalent length that corresponds to a flow rate
loss of 20% for each resistance. These critical equivalent lengths can be graphed against
the sub-slab material and cavity resistance as has been done in figure 6 which in turn
allows for the following interpretation. The regions divided by the solid and dashed lines
are the regions in which there is a certain equivalence of system. It can be seen that in
some cases less than 20% loss is expected by switching to a smaller pipe diameter. This
is indicated by the notation: 3” or 4” inside the graph. Similarly, there is a region at large
cavity resistances and shorter length systems where the flow dynamics for even 2” pipe
would be similar to 3” and 4” pipe diameters. On the other hand it is clear that at small
cavity resistances only for very short systems, up to 10 ft equivalent length, the 3” and 4”
systems are similar and at 25 ft equivalent lengths one can expect already as much as
30% flow loss when switching to a 3” vent pipe. Nevertheless, the correct interpretation
of this graph is that it shows equivalence of systems but does not show whether either of
the equivalent systems will work to reduce radon levels.
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
Fig. 5: For a FR 150 fan for a single branch system the ratio between the volume rate employing 3 inch
and 4 inch piping in the system is shown.
Fig 6: Equivalent Dynamics Graph (Fan specific, here for FR150). Flow reduction data points for 20%
flow loss are connected with straight lines in the graph. The areas in between the lines are marked for
which pipe diameters are equivalent within the chosen criterion of the flow loss. The open symbols with
dashed lines are for 30% loss.
TWO BRANCH RADON MITIGATION SYSTEM
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
In deriving the equations for a 2-Branch system I can develop a set of three equations that
are similar to the Kirchhoff equations in electronic circuits with the one important
difference that the equations here are nonlinear in the flow and that it is not immediately
clear that only a single flow solution exists.
i)
Fan equation: !P2 + !P1 = !Pf
ii)
iii)
Kirchhoff Rule through the two branches: !P2 = !P3
Flow conservation: F1 = F2 + F3
(11)
By substituting iii) in i) the set of equations can be written as two nonlinear equations in
two independent variables (F2 and F3 ) only:
PF2 R g 2 + c 2 F2a 2 + c1 ( F2 + F3 ) a1 ! g ( f , F2 + F3 ) = 0
i’)
(12)
ii’)
PF2 R g 2 + c 2 F2a 2 ! PF3 R g 3 ! c3 F3a3 = 0
The equations can be written in a vector formulation as
r r
r
(13)
A( F ) = 0
and solved numerically by finding the root of the two dimensional equation.
The solution of Eq. (13) can be found numerically by use of an assumed reasonable
r
starting value of the flow vector, F0 , and, by iteration, finding the small root deviation
vector from the previous approximate solution. For iteration number n I find:
"1
(
%
r
r
. /Ai +
&
# A ( F ) = " L"1 A ( F )
))
d n = "! ,,
(14)
!
q
n
pq q
n
&- /Fk * r #
q
q
&'
Ft #
$ pq
In which I defined the Jacobian matrix:
rr & 'A #
(15)
L = $$ i !!
% 'Fk "
and the inverse of this matrix is:
1 & L22 ' L12 #
$
!
(16)
L'1 =
det L $% ' L21 L11 !"
I than find the next iteration of the solution vector in the two dimensional “flow space”
from the previous approximate solution by
r addition:
r
r
(17)
Fn +1 = Fn + d n
r
I then choose to continue the iteration until a certain stopping criterion d n " ! , for a
sufficiently small value of ε, is reached, at which time the closest approximation to the
r
solution is Fn for final value of t reached. The method described here to solve Eq. (13) is
mathematically known as the Newton–Raphson method of root finding for a nonlinear
system of functions with multiple independent variables.
N-BRANCH RADON MITIGATION SYSTEM
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
More general solutions have been derived for 3- and 4-Branch radon mitigation systems
with a possibility of up to 7 ventilators in the various branches of the mitigation system.
From this, it is possible to formulate an N-Branch mitigation system with (2N-1)
ventilator insertion points. In a non-looped system as is shown schematically in Fig. 7
this would be one ventilator for each of the N branches, one ventilator in the common
upper stack of the system (Branch 1) and (N-2) ventilators for the interconnecting pieces
between the nodes of the N pipe branches.
As an example, when deriving the equation for 4 branches I write down the fan equation,
three Kirchhoff equations and the continuity equation. This leads to four non-linear
equations with four independent unknown variables. The four independent variables are
the flow values in each of the four branches that connect to the ground:
& F2 #
$ !
r $ F3 !
(18)
F =$ !
F4
$ !
$F !
% 5"
Fig. 7: Schematics of a 4-branch radon mitigation system with the parameter definitions used in the
derivations.
and the 4-dimensional mitigation system equation thus is:
r r
r
(19)
A( F ) = 0
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
Similar to Eq. (15) I can derive the 4-dimensional Jacobian matrix:
In this:
&/ + .
rr & *A # $ .
$
L = $$ i !! = $
% *Fk " $ 0
$ 0
%
/
/
/ #
!
'- ' )
')
' )!
', ' ( ' (!
!
0
,
' + !"
(20)
( = a ! f1
' = b ! f2
& = c ! f3
% = d ! f4
$ = e ! f5
# = A ! FA
" = B ! FB
in which the flow resistance components of the upper stack are::
a = PF1(2345) = c1 a1 ( F2 + F3 + F4 + F5 ) a1 !1
A = PA(3B) = c A a A ( F3 + F4 + F5 ) a A !1
B = PB(45) = c B a B ( F4 + F5 ) aB !1
and in which the flow resistance components of the branches with sub-slab material
connections are:
b = RP 2 = PRg 2 + c2 a2 F2a2 !1
c = RP3 = PRg 3 + c3 a3 F3a3 !1
d = RP 4 = PRg 4 + c4 a4 F4a4 !1
e = RP5 = PRg 5 + c5 a 5 F5a5 !1
In addition the active ventilator boost derivatives in the common upper part of the vent
stack is given by:
f1 = f1 (2345) = q1b1 ( F2 + F3 + F4 + F5 ) b1 !1
and for the individual branches i = 2, 3, 4, or 5 these ventilator boost derivatives are given
by:
f i = f i (i ) = qi bi Fi bi !1
For the interconnecting sections A and B between the various node points the ventilator
boost derivatives are given by:
f A = q A b A ( F3 + F4 + F5 ) bA !1
f B = q B bB ( F4 + F5 ) bB !1
Solving the set of equations defined in Eq. (19) is similar to the approach of solving Eq.
(13), as was done in Eq. (17) with use of Eq. (14), following the Newton-Raphson
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
method for a set of non-linear equations with multiple independent variables. In this case
the 4x4 Jacobian matrix In Eq. (20) must be inverted. Although there is an analytical
expression for this inversion, the numerical evaluation of this is cumbersome. Instead, a
numerical evaluation of the inverted 4x4-matrices was applied directly via the Quattro
Pro matrix inversion function and solutions of the set of equations were calculated with
an iterative approach by using three embedded macros. Verification of the validity of the
inversion process was done by multiplication with the original matrix and by testing the
convergence of the solution. With fairly generic starting values of the flows, the four
dimensional calculations converged most of the time in less than 8 steps to a result of the
final flow that is close to the accuracy of the computer.
The equations presented contain all of the physics that was discussed before. Additional
physical effects that can be included are gravitational effects which can be responsible for
a fan to act differently when moving air horizontally over 25 ft compared to moving air
over a vertical lift of 25 ft, as well as the lift due to temperature differences of air inside
the pipe compared to outside air (the stack effect for a passive system). Also certain
aspects of the Bernoulli effect can be included in these equations, and have not yet been
included in the above formulation. Finally I must mention that these equations are for
incompressible flow, which in its defense is a sufficiently accurate approximation for the
air parameter regime in which radon systems generally operate.
In Fig 8 I show an example of a calculation for a house where we installed a three branch
system with the HP220 ventilator in the attic of the garage 2 ft under the discharge point.
Slab areas were 600, 1050, and 100 sf to be serviced by the three branches 2, 3 and 4,
respectively.
The lower half of this figure shows the vacuum (negative) pressures that the computer
program calculated to solve Eq. (19) for the cavity parameters measured and equivalent
lengths of pipes calculated. The horizontal axis gives relevant observation points along
the system with the Sky-limit pressure on the right and the Sub-slab limit pressure on the
left, which will always be the same and at zero pressure.
At the mark “P-hole” the pressures in each of the three extraction cavities is given and
“P1-below fan” is the pressure in the top stack of the system immediately below the
ventilator in the attic of the garage. The program allows for placing the system at further
distances away from the discharge point for instance when installing an outside system..
At the marker P_45B the pressure is given at the node where branches 4 and 5 come
together and the air flows into pipe B. The same is the case for the other node points.
The top half of the figure shows the flow rates indicated on the right vertical axis of the
numerical solution that the computer calculated for each marker along the horizontal axis.
In this case an HP220 fan was simulated, the top stack was 45 equivalent feet long and
branch 2, 3, and 4 each 65 equivalent feet long with 2, 3, and 4 inch diameter pipe,
respectively and branch 5 was 40 equivalent feet long with 3 inch diameter pipe. A and
B were 10 and 20 equivalent feet of 4 inch diameter pipe.
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
Cavity resistances were for cavity 2: 2×10-5 min/cf, for cavity 3: 8×10-5 min/cf and for
cavity 4: 1×10-3 min/cf and cavity 5 was essentially deactivated in this calculation by
giving it a virtual resistance of 100 min/cf.
It is interesting to see that the flows indicated at P_hole through branch 2, even though
the vacuum pressure is small, is almost as large as through branch 3, despite the fact that
branch 2 only uses 2 inch piping and branch 3 has 3 inch piping. This is due to the fact
that the cavity resistance of branch 2 was a factor four lower than the cavity resistance for
branch 3.
Fig. 8: The numerical program shows the resolution in a graph that can be conveniently read of. The
vacuum pressures in the three radon extraction cavities under the slab are given at P_hole, as are the three
airflows in the individual branches.
When calculating Reynolds numbers the program showed that branch 5 and B were
operating in the laminar regime, rather than the turbulent regime as is the case for the
other branches as shown in Fig. 9.
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Proceedings of the American Association of Radon Scientists and Technologists 2008 International
Symposium Las Vegas NV, September 14-17, 2008. AARST © 2008
Fig. 9: Reynolds numbers in the various pipe branches indicating the flow dynamics applicable.
CONCLUSION
A theory for single-branch radon mitigation systems was developed in which
experimental data were used to support the theory that the sub-slab material and
extraction cavity resistance can be sufficiently well described by a single resistance
parameter. The differences between laminar and turbulent flow through round pipes
were discussed and the roll of the Reynolds’ number as a criterion for the onset of
turbulence in the air flow. A method utilizing the Fan Resistance Response graph was
developed to aid in graphically solving the 1-branch mitigation system equation. By
numerically solving the single branch radon mitigation equation for many different
situations I have derived criteria for the equivalence of different size piping under various
circumstances, and developed a quantitative insight in those circumstances when pipe
sizes cannot be considered equivalent. These insights were graphically summarized in
the Equivalent Dynamics Graph which can be helpful during installation of radon
mitigation systems possibly leading to more appropriate pipe size choices in the future.
As an example of an N-Branch system, for a four-branch, active, non-looped, radon
mitigation system I derived the complete set of non-linear equations with its independent
variables and demonstrated how these can be solved numerically and applied to optimize
a system under design. A three-branch example was discussed in detail showing that this
system has simultaneously turbulent and laminar flow in different branches.
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