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OCR for page 183
APPENDIX C
DETERMINATION OF MANNING'S N
A parameter that describes the hydraulic resistance of the pavement surface must be
known in order to predict the water film thickness that occurs on a pavement surface during
sheet flow. Manmng's n is commonly used for this purpose and was used for this ouroose in
ache equations that were selected and developed during this study for use in the PAVDRN
model. Values for M~nning's n must be determined experimentally by measuring water flow
depths under either natural or artificial rainfall. During the course of this study, the hydraulic
resistance of three different types of pavement surface were determined:
Portland cement concrete
Porous asphalt
Densegraded asphalt concrete
A great deal of experimental data that could be used to determine Manning's n values
was available in the literature, and these data were used during this project to verify values of
Mann~ng's n established by previous researchers (2931, 36, 399. Additional data were also
obtained as part of this project for porous pavements and for long Dow paths on Portland
cement concrete. Both sets of data were used in the analyses to give the values of MaTniing's n
used in PAVDRN. These analyses are described in this appendix.
C1
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GENERAL APPROACH
Mamiing's n was determined by first expressing the flow depth, y, as a function of
independent variables such as the pavement slope (S), mean texture depth (MID), flow path
length (L), and rainfall rate (i):
y = f(S, MID, L, i)
(C1)
Data from the literature, as well as data obtained during this study, were used with regression
analyses to determine functional forms and coefficients for equation Cal. Variables that were
not statistically significant were eliminated during the regression analyses. Separate regression
equations for y were established for each of the pavement surfaces. Once the relationships for
y were determined, the equations for y were substituted into the kinematic wave equation,
equation C2 (also see equation 9), thereby eliminating the flow depth as a variable in the final
expression for M~ng's n. The kinematic wave equation is expressed as (35~:
~ 36 . ~ S D 5
where
0.6
y  Hydraulic How depth (mm)
n  M~nmng's hydraulic resistance variable
Lo  Drainage path length (m)
C2
(C2)
OCR for page 183
.
1
Rainfall intensity (mm/h)
S = Drainage surface slope (minim)
Since MaTm~g's n is surface specific, the regression analyses must be considered
separately for each of the surfaces of interest, in this case Portland cement concrete, dense
graded asphalt concrete, and porous asphalt. The analyses follow for each of these surfaces.
PORTLAND CEMENT CONCRETE SURFACES
In a previous study, Reed and Stong (35) performed experiments using the artificial
rainfall simulator at Penn State to develop an expression for Manning's n for Portland cement
concrete surfaces. The experiments were performed on three brushed PCC surfaces with MID
values of 0.25, 0.70, and I. 12 mm and rainfall rates of approximately 25, 50, and 75 mm/in
using the same artificial rainfall facility as used in this project. The PCC surfaces were 0.30 m
wide by 7.3 m long. Testing was conducted with slopes of 0.5, I.5, and 2.5 percent. The
facility and the test protocols are described in chapter 3 of this report. Reed and Stong (35)
collected 2,367 data points, which represented 789 observations; each observation represented
the average of three data points. Dunng this study, 1,656 data points representing 552
observations were collected for the Reynold's number ranging from approximately 200 to
A,.
C3
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VALII)ATION OF DATA
The kinematic wave approx~nation is valid only when certain criteria are met
· ~
requiring that (30):
The flow must be fully rough,
The slope of the energy gradient (Sf) must be approximately equal to the surface
slope (S) of the pavement, and
Certain requirements must be met with respect to the Froude number.
In order to satisfy the requirement of full roughness, the data must satisfy the following
relationship:
where
n ,/i~f 2 1.05x103
R _
Si _
Hydraulic radius (mm)
Slope of the energy gradient (m/m)
~ this case the hydraulic radius is equal to the flow depth, y (30). A total of 141
observations were eliminated from the data set based on this criterion. All 141 observations
that were el minated were for either high rainfall intensities, long drainage lengths, large flow
C4
(C3)
OCR for page 183
rates, shallow pavement slopes, or combinations thereof. Under these conditions, pelting rain
could add turbulence and cause a fully rough flow that is not ensured by equation C3.
Elimination of these 141 data points ensured fully rough conditions and agreement with the
Thematic wave assumption.
The second criterion that must be satisfied for Me kinematic wave equation to be valid
is that the energy gradeline slope, Sf be approximately equal to the surface slope, S. of the
pavement. This criterion is satisfied if NF2K is greater than five (309:
NF K = S
y
where
NF
K =
S _
Froude number
Kinematic wave number
Hydraulic flow depth (m)
Runoff surface slope (m/m)
Drainage path length (m)
The Froude number, NF, is defined as:
NF =
v
CS
(C~)
(CS)
OCR for page 183
where
v = Flow velocity (rn/s)
g =
Gravitational constant, 9.81 m/s2
One additional criterion must be met for subcritical flow when the Froude number is
less than 0.5 (50,511:
NF K = S ~ > 5
y
where the variables are as previously described. A total of 76 of 1,341 observations were
eliminated because they did not meet the criterion of equation C6.
Manning's n for 500 < NR < 1,000
Before an equation for Mann~ng's n could be established, it was necessary to establish
(C6)
the relationship between the flow depth and the experimental variables. The flow depth for the
1, 124 valid data points was regressed versus Me flow rate, q, Me mean texture depth, MID,
and the surface slope, S. with the following results (Et2
I=
0.926):
122.43 q0308 MTDO.0316
S 0.286
C6
(C7)
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Mean texture depth, runoff surface slope, and flow rate were the only statistically significant
parameters. Removing MTD from equation C7 resulted In equation C8 with an R2 of 0.92.
126~66 qO3~2
y=,
S 0.285
(C8)
Similar results were observed by Reed and Stong (351. On the basis of these results, it appears
that MTD has little effect on flow depth. However, hydraulic flow depth, y, is the sine of the
water film thickness and mean texture depth. Therefore, the MTD parameter was retained In
the relationship and equation C8 was used In the work that follows.
The next step was to combine equation C8 with equation C2. In equation C2, S is
raised to a very small power and, according to equation C2, appears to not be very important
In determining the Manning's n value over a range of Wpical drainage slope values.
Therefore, the smallest and largest slopes from the data set, 0.005 and 0.025 respectively,
were used to calculate an average value of S00Q5, 0.90. The maximum error introduced by
replacing S002s in this manner was 2. ~ percent. The kinematic viscosity, v, was also replaced
with a constant value. Since the data from Reed and Stong were measured at a temperature of
approximately 18 °C (v= 1.057 x 10~ m2/s) and the data taken from this study were measured
at 9 °C (v = 1.35 x 10~ m2/s), a weighted average was used for the kinematic viscosity. In
other words, 714 of I, 124 total data points were from the study by Reed and Stong (35), and
fiche kinematic viscosity used In equation C8 was weighted more toward a ~ of 1.057 x 10~
OCR for page 183
m2/s. Upon replacing S and v in equation C8 with the appropriate constants, equation C9
was obtained:
n=
0.319
N 0.480
R
(C9)
Figure C] is a logIog plot of equation C9 and the calculated M~ng's n values from
I, 124 experimental observations, and there is a good degree of scatter around equation C9.
On the other hand, the figure shows that as the Reynold's number increases, the experiments
data appear to be underestimated, especially as the Reynold's number approaches 1,000.
Therefore, because of the apparent dependency of Manning's n on the Reynold's number,
equation C9 was reevaluated by considering Mann~ng's n independently within several
different ranges in the Reynold's number.
Mann~ng's n for Reynold's Numbers Less Than 500
Equation C9 underestimates Manning's n as the Reynold's number approaches 1,000,
as shown in figure Cal. This was a major reason for the experimentation that was done on
Portland cement concrete surfaces dunug this study. The authors also reviewed the transition
region for open channel flow, which is commonly accepted to occur within the range of
Reynold's numbers of 500 to 1,000. It is Important to note the commonly assumed transition
region is for flow without pelting rainfall. Two regions were considered, NR < 500 and
500 < NR < 1,000. The 500 value was am important dividing point because below 500 flow is
C8
OCR for page 183
0.1  .
In
~ 
oo1
 ~ ~
i ~ 1
l l
~
ll

a
rTr
1 n=~.413/Nr^0.535 ~  
_
_
[
_ .
. J lo
1 C 1
I n ~

~ _
'1
Cal _ O 5
~CL°,
~ ]L 1
O l ~
l _ 11
to ~
. ~1
O j
~1
i 3
bloc 1
my ~
{:e~
~0
~D D
~ . ~ ~
01
10 100 1000
Reynold's Number, NR
Figure C1. Mann~ng's n on PCC surfaces for Reynold's numbers less than 1,000 (Eq.C9).
C9
OCR for page 183
traditionally assumed to be lam~nar. However, in the case of shadow flow with pelting rain, as
is the case on a pavement surface, the flow is disturbed, possibly creating turbulent flow at
Reynold's numbers less than 240.
The first step was to reduce the data set so that it contained only Reynold's numbers
less than 500. A total of 1,070 of I, 124 observations had Reynold's numbers less than 500
and were used In the subsequent analysis. The procedure applied was identical to the one
implemented for Reynold's numbers less than 1,000. A regression equation was performed
and MTD, S. and v were removed as before. The result was the following, with R2 _ 0.916:
0.345
NR
(CIO)
Figure C2 is a logIog plot of equation CIO and the calculated Manmng's n values
from 1,070 observations. Once again, there is degree of scatter around equation C10 In
Figure C2. On the over hand, equation C10 still appeared to overestimate M~ming's n as
the Reynold's number approaches 500. Also of concern was the manner in which the data
points appear to level off prior to a Reynold's n~nber of 500. Figure C2 shows that the
M~ning's n values appear to begin to level off at approximately a Reynold's number of 240.
C10
OCR for page 183
1
 ~
I o.1
ce
0.01
_ ~
_
1 1 1 ! !1
! 1
. ~ i 1
^1
of ill l
to /
1 i l
1 w _
At
. . 111
o
1~11
'I ~
[I
.
1 ~ ~ 1
.
1 1 ~ 1111 .
' ~1 1 ~ 1 1 _ .
n = 0.3451 NR ^0.502
0
of ~
. ~ .
100
Reynold's Number, NR
1 ! ~ 1 1~
1 1 1 1 1_1
rim 1 1
LO
. i
L1
.. .

1000
Figure C2. Manning's n on PCC surfaces for Reynold's numbers less than 1,000 (Eq. C10).
C11
OCR for page 183
Manning's n for NR<240
The previous procedures were applied to a data set for Reynold's numbers less than
240. A maximum Reynold's number of 240 was chosen because it appears as though
Mann~ng's n attains a constant value when NR is less Shari 240 (see figure C21. The entire
data set was first reduced to a data set of 940 observations with Reynold's numbers less than
240. A regression equation for hydraulic flow depth was completed and, with systematic
substitution and elimination of variables, produced
n=
0.388
N0.s3s
win an R2 of 0.887. Figure C3 is a logIog plot of equation C1 ~ and the calculated
(Cafe)
Manning's n values from 940 observations. The points in figure C3 appear to be consistently
scattered around equation C] I . Also, equation C1 I, shown in figure C3, does not
overestimate Manmng's n as the Reynold's number approaches 240. It is important to
remember that equation C1 ~ was developed under certain parameter restrictions. Equation C
~ ~ is for rainfall runoff on PCC surfaces with MTDs between 0.25 mm and ~ . 12 Ann, runoff
surface slopes between 0.005 and 0.025 m/m, rainfall intensities between 25 and 75 mm/in, and
drainage path lengths up to 40 m.
C12
OCR for page 183
i ~ i 1 1 1 ; ! ~ ~ , i i ! ; I _
~1 ~ ~1 ~ l 1 1 1 · 1 1 1 ~1
i ' ~ I I I I I I I ~
1 1 1 1 1 1 ~ 11 1 ~ 1 1 ! 1
~ ~ 1 ~ 1 1 1 1
_ _
~211 1 1 ~ 1 111111 1 1 1 1
l 11111 1 11111
1~R ~ , , ,i 1:
:_1 1 1 1 1 1 1 1 1 ~ 1 1~ D:V
I ~ ~ ~ ~ ~ ~ ~,_~
~ jl~'ll I
i~ 1
_

i ' _: · .
Tl I ~ ~
lit ~o'D~°~Il a' I I I I ;
11111 1~7~1 1 1 1
_ _ _' o ~ , / _ _ _

~ S,~UIUUB~

o
do
Figure C3. Manmng's n on PCC surfaces for Reynold's numbers less than 1,000 (Eq. Cll).
OCR for page 183
Summary of Mar~n~ng's n on Portland Cement Concrete
The results of the previous analyses produced two observations. These observations
maybe explained as follows.
1. As the upperlimit Reynold's number was reduced from 1,000 to 500 to 240, so
were the R2 values for subsequent regressions. Since data with higher Reynold's
numbers were eliminated, an increased emphasis on lower Reynold's numbers was
Inevitable. Data with lower Reynold's numbers or hydraulic flow depths were prone to
have more error due to the precision of the point gauge. The increased scatter of data
for smaller Reynold's numbers can be seen In figure C~.
2. As the upperlimit Reynold's number was reduced from 1,000 to 500 to 240,
the exponent of S systematically decreased from equation CIO to equation
Cat.
Constant Mann~ng's n for Reynold's Numbers Greater Than 500
The experimental Mann~ng's n values appear to level off and approach a constant value
beginning at a Reynold's number of approximately 500. For Reynold's numbers greater than
500, a constant value can be assumed for Mann~ng's n, as evidenced in figure C4 with a value
of 0.017, based on one surface with a MTD of 0.91 Inn.
C~4
OCR for page 183
o
o
 So
~ me
o
l
1 ~I 1
° 1 1 1
~ 1
1
o
 I
· l
so
o
l
o 
o
K/
/ 1
7
o
o
l
o
8
0
0
o so
ooo
O ~
~ Z
,_
O ~
v,
~ Z

O
lo:
lo
1^
o
o o
. ~
~ 0
o
~ 0
0
c
0

o
A
lo
lo
U S,8UlUUB
V'
g
lo
Figure Cat. Manr ing's n on PCC surfaces for Reynold's numbers greater than 500.
C15
o
u~
o
o
OCR for page 183
Equations Used In PAVDRN
The equations that were chosen for predicting Mann~ng's n on PCC surfaces are:
NR (50O ~ NR < 10003 (C12)
NO.502 (240 ~ NR < 500) (C13)
n 0.38,5 (NR < 240) (C14)
For the Reynold's numbers equal to or greater than 1,000, hydraulic flow resistance or
Manmng's n on PCC surfaces is a constant of 0.012, the value of n from equation C12 for a
Reynold's number of 1,000 and a traditional value of n for concretelined channels.
MANNING'S N FOR POROUS PAVEMENTS
First, the criterion for fully rough flow was tested as described earlier, and of the 1,495
data points, only two sets did not satisfy this criterion. It was observed that the fully rough
conditions did not prevail at high depth values with a tow rainfall intensity. This enhances the
C16
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theory that the impact of the pelting rain causes internal turbulence in the flow and adds to the
fully rough condition.
The second criterion requires that the friction slope be equal to the surface slope. This
is satisfied if the kinematic wave number is greater than 20. All of the data sets satisfied this
criterion. Additionally, for Froude numbers less than 0.5, the criterion as described by
equations C4 through C6 must be satisfied. Analysis of the data revealed that 47 data sets
did not satisfy these criteria. These data sets included low depths on shallow slopes with a
high rainfall intensity, possibly reflecting the difficulty In measuring the smaller depths. Upon
examination of the data, a total of 49 data sets were eliminated from Me total of i,493 data sets
leaving a total of 1,4421 data sets remaining for further analysis.
A regression of the data sets was performed to give:
Y = 0.1590 MTD0~50 Hi ~0~5 S°al~ %AV0.072
where
y  Depth of flow (mmy
MTD
S 
%AV
Mean texture depth (mm)
Rainfall intensity (mm/h)
Drainage path length (m)
Channel slope (m/m)
Percent air void content of the mixture
C17
(C15)
;
OCR for page 183
with an R2 = 0.799. The values of percent air void content were utilized due to the fact that
the permeability measurements of the mixtures were not available. With the coefficient of the
air void content appearing to render the parameter negligible, a subsequent regression was
performed and showed:
y = 0.~995 MTDO~29 ~ ~0345 SO~1
with an R2 _ 0.797
A subsequent regression of the data showed:
y = 0.2080 (I L)0.346 s0.~6
with an R2 0.790, a negligible decrease in the correlation.
(C16)
(C17)
Equation C15 was then equated to the kinematic wave equation, as was done for the
PCC surfaces previously and solved for Mar~ng's n to produce:
n=
N 0.424
C18
(C18)
OCR for page 183
An attempt was made to furler simplify the relationship; however, the slope term
appears to be a significant parameter and was retained in Me relationship. Figure C5
illustrates the observed data points and the relationship shown In equation C18. The three
curves displayed are equation C5 solved for each of the three slopes used in the
experimentation, namely 0.005, 0.015, and 0.025 mim. It is evident from the figure that the
slope term plays a significant role in the relationship. The applicability of equation C18 is
limited to porous surfaces with the following criteria: mean texture depth between 1.25 and
2. 13 mm, Reynold's numbers less than 550, slopes less than 0.025 m/m, and void contents
between 20 and 33 percent.
M=NING'S N FOR DENSEGRADED ASPHALT CONCRETE
Recently, Reed, Warner, and Huebuer (52) developed a similar expression for dense
graded asphaltic concrete surfaces. The data were obtained from four DGAC surfaces of the
Gallaway et al. (3) study win MTD values of 0.23, 0.48, O.S1, and 0.99 mm. The slopes
ranged from 0.5 percent to 4 percent, with rain rates applied up to 150 mm/in and drainage
lengths up to 7.3 m. A regression analysis of the data was performed and combined with the
kinematic wave approximation, as described earlier for the PCC and porous pavement
surfaces. This resulted with:
0.0823
n=
N0174
R
This relationship yielded an R2 of 0.88 and is bounded by a maximum NR Of 230 and the
experimental ranges, as Indicated previously.
C19
(C19)
OCR for page 183
1.00
In
.' 0.10
ce
a
s=~.025
s = 0.0~5
o s = 0.005
s=0.025
~ :
\
_ s=~s:
of. I"
n=1.490S°~°6
N 0.424
R2 = (~_79
I ~ 3~^
m
3
~1 0 1 00 ~ 000
Reynold's Number, NR
Figure CS. Manning's n for porous asphalt surfaces.
C20