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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 Dense-graded 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 (29-31, 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. C-1
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) (C-1) 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 C-2 (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) C-2 (C-2)
. 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,. C-3
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 1.05x10-3 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 (C-3)
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 C-3. 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 C-S (C~) (C-S)
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 C-6. Manning's n for 500 < NR < 1,000 Before an equation for Mann~ng's n could be established, it was necessary to establish (C-6) 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 q0-308 MTDO.0316 S 0.286 C-6 (C-7)
Mean texture depth, runoff surface slope, and flow rate were the only statistically significant parameters. Removing MTD from equation C-7 resulted In equation C-8 with an R2 of 0.92. 126~66 qO3~2 y=, S 0.285 (C-8) 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 C-8 was used In the work that follows. The next step was to combine equation C-8 with equation C-2. In equation C-2, S is raised to a very small power and, according to equation C-2, 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 S0-02s 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 C-8 was weighted more toward a ~ of 1.057 x 10~
m2/s. Upon replacing S and v in equation C-8 with the appropriate constants, equation C-9 was obtained: n= 0.319 N 0.480 R (C-9) Figure C-] is a log-Iog plot of equation C-9 and the calculated M~ng's n values from I, 124 experimental observations, and there is a good degree of scatter around equation C-9. 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 C-9 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 C-9 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 C-8
0.1 - . In ~ - o-o1 - ~ ~ 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 C-1. Mann~ng's n on PCC surfaces for Reynold's numbers less than 1,000 (Eq.C-9). C-9
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 (C-IO) Figure C-2 is a log-Iog plot of equation C-IO and the calculated Manmng's n values from 1,070 observations. Once again, there is degree of scatter around equation C-10 In Figure C-2. On the over hand, equation C-10 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 C-2 shows that the M~ning's n values appear to begin to level off at approximately a Reynold's number of 240. C-10
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 C-2. Manning's n on PCC surfaces for Reynold's numbers less than 1,000 (Eq. C-10). C-11
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 C-21. 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 C-3 is a log-Iog plot of equation C-1 ~ and the calculated (Cafe) Manning's n values from 940 observations. The points in figure C-3 appear to be consistently scattered around equation C-] I . Also, equation C-1 I, shown in figure C-3, does not overestimate Manmng's n as the Reynold's number approaches 240. It is important to remember that equation C-1 ~ 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. C-12
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 C-3. Manmng's n on PCC surfaces for Reynold's numbers less than 1,000 (Eq. C-ll).
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 upper-limit 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 upper-limit Reynold's number was reduced from 1,000 to 500 to 240, the exponent of S systematically decreased from equation C-IO 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
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. C-15 o u~ o o
Equations Used In PAVDRN The equations that were chosen for predicting Mann~ng's n on PCC surfaces are: NR (50O ~ NR < 10003 (C-12) NO.502 (240 ~ NR < 500) (C-13) n 0.38,5 (NR < 240) (C-14) 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 C-12 for a Reynold's number of 1,000 and a traditional value of n for concrete-lined 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 C-16
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 C-4 through C-6 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 C-17 (C-15) ;
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 S-O-~1 with an R2 _ 0.797 A subsequent regression of the data showed: y = 0.2080 (I L)0.346 s-0.~6 with an R2 0.790, a negligible decrease in the correlation. (C-16) (C-17) Equation C-15 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 C-18 (C-18)
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 C-5 illustrates the observed data points and the relationship shown In equation C-18. The three curves displayed are equation C-5 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 C-18 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 DENSE-GRADED 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. C-19 (C-19)
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 C-S. Manning's n for porous asphalt surfaces. C-20
