Not for Sale

• #### Appendix D Model Evaluation 203-214

The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001

Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 45
CHAPTER 3 SELECTION AND DEVELOPMENT OF MODELS FOR PAVDRN In order to develop guidelines for Improving the drainage of water from pavement surfaces, it was necessary to select a mode! for predicting the depth of sheet flow on pavement surfaces. Since the maximum allowable water fun thickness is the min~murn flow depth at which hydropl~nmg is Initiated, it was also necessary to select a model that can be used to predict the onset of hydroplaning as a function of water fun thickness. A literature search was conducted to identify and evaluate water film thickness and hydroplaning models ~ the existing literature. From the literature search, a one-~mensional, steady-state form of the kinematic wave equation was selected for the prediction of the depth of sheet flow. Additional development of the mode! was also undertaken as cart of this protect. , . ~ The models arid predictive equations that are contained ~ PAVDRN are described In this section. A more comprehensive discussion of surface flow models and the development of values for Manmng's n can be found In Appendices B and C and the references cited therein. WATER lILM THICKNESS MODEL Before discussing the water film thickness model, the terms water fihn thickness (WFT), mean texture depth (MTD), and water depth, y, must be clearly be defined (see figure I). Water depth is defined as the total thickness of the water film on the surface of the 45

OCR for page 45
pavement. It is the sum of the MTD and Me WFT. The water below the MTD is trapped in the macrotexture and is considered immobile and does not contribute to the flow depth, y. The WFT is the thickness of Me water fihn above the tops of the asperities on the pavement surface. The total depth of flow, y, is the thickness of the WFT plus the MTD. There are two general types of models that can be used to detenn~ne water film thickness: (~) an analytical mode} and (2) an empirical model. Bow types have been developed and used for this purpose. Empirical Mode! A one~mensional, steady-state model was developed by Gallaway f2) as presented in equations 3 and 4: WFT = 0.003726 L0519 i0.s62 MTD0.l2s s0-364 and where - MTD y = WET + MTD (3) (4) WFT = Water film thickness (in) (1 In 25.4mm) L Plane length of Towpath (ft) (1 It = 305 mm) 46

OCR for page 45
i = Rainfall intensity (in/h) MID = Mean texture depth (in) Pavement slope (mlm) Depth of water contributing to flow (in) (1 inlh = 25.4 mmJh) (1 in = 25.4 mm) (1 in = 25.4 mm) (1 in = 25.4 mm) In equations 3 and 4, the flow path, L, is presumed to be over a simple, planar surface. Gallaway's equation is based on an extensive set of water depth data for a variety of pavement types. The equation, however, does not contain a variable, such as M~nning's n, to describe the hydraulic resistance of the pavement surface. The equation was developed by combining data from pavements with different types of surfaces in a single regression analysis. Since Gallaway's equation does not include a term for the hydraulic resistance of the pavement surface, does not differentiate between different surfaces, and is empirical, the equation was not considered for use ~ this project. One-Dimensional Analytical Models Two analytical models were considered during the project. The first is a fully dynamic model based on the principles of conservation of mass and momentum. The model has been used by many investigators to represent the equations of state for shallow wave motion, equations 5 and 6. Equations 5 and 6 represent spatially varied, unsteady flow in one · ~ dlmenslon. 47

OCR for page 45
dh flu Sh + h- + u = ~ - f = ~5' at fix fix where Depth of flow u - Spatially averaged velocities (x - directions . 1 I Rainfall rate over the domain Infiltration rate Rainfall rate adjusted for infiltration em + u ~ + g ~ = g(Sox Sfx) h h (6, where u, h, i, and f are the same as described In equation 5 and g - Acceleration due to gravity (32.2 ft/s 2 or 9.~1 m/s 2) Sax Sex vr Slope of the flow path In the x-direction Slope of the energy grade Ime In the Erection Terminal rainfall velocity Ox - Angle of rainfall input with respect to the x-axis The last term on the right-hand side of equation 6 represents the momentum due to the angle of incidence of rainfall velocity In the Indirection. The term By is often taken to be 48

OCR for page 45
negligible; Pus costly can be assumed equal to zero, and the right-most tenn is dropped from the equation. Under steady-state conditions where the friction slope, Sf,, , is equal to the slope of the flow plane, SOx, equations 5 and 6 simplify to a form known as the kinematic wave equations as presented in equations 7 through 9: hen = ~ teq teq = where (7) ( I) ( ~ I) u = ahm~1 (8) (9) he = Equilibrium water flow depth (ft)(1 It = 305 man) i = Excess rainfall rate (ft3/s/ft2)(1 ft3/s/ft2 = 305 mm3/s/mTn2) to = Time to equilibrium (s) L = Plane length (ft)(1 It = 305 mm) a = Friction loss coefficient m = Friction loss exponent u = Velocity of flow (ft/s)(1 ft/s 305 mm/s) Equations 7 through 9 represent the kinematic wave solution for steady-state, one~'mensional flow. 49

OCR for page 45
Two-Dimensional Analytical Models An analytical two-~mensional flow mode! written by Zhang and Cundy (33) was evaluated as part of this study. Their mode] is representative of two-~mensional models and illustrates the difficulties that are typical of two-dimensional models. The primary limitation of their mode} is the lack of stability and convergence In the solutions. Stability and convergence are relatively easy to obtain with sets of linear equations, but difficult to acquire win ache non linear sets of equations implicit in the Zhang and Cundy model. The computational stability and convergence for nonlinear sets of equations are usually obtained by assuming linearity and using the linear solution as a conservative first approximation of the nonlinear case. As a consequence, extremely small tune steps are used that result In dramatically Increasing execution times as ache geometric complexity of the flow surfaces augments. For a multilane pavement, this may result in hundreds of thousands of iterations to reach equilibrium conditions. Therefore, no further use was made of two~mensional models, which are discussed in Appendix B. Mode} of Choice The one~imensional kinematic model, represented by equations 7 through 9, was chosen as the preferred model for predicting water film thickness. This mode} is based on theories and includes a variable, M~nmng's n, that accounts for the hydraulic effect of surface roughness on water depths. The one~mensional, steady-state form of me model was used 50

OCR for page 45
developing the surface drainage guidelines and the PAVDRN program. The selection of the one~mensional flow equation was determined by its computational stability and efficiency of solution. The flow domain, i.e. flow paths and channels, for a one-~unensional model must be defined by the user or must be established by an analysis of the topography of the surface prior to applying equations that determine flow, depth, and/or velocity. For example, PAVDRN analyzes the topography of a section and determines the longest flow path length before applying the kinematic wave equation to determine water film thickness at points along the flow path. The longest flow path is determined from geometric conditions as the path from the point where Me water falls on the pavement to the pout where it exits the surface of the pavement. where n L 1 WFr = The following equation is used In PAVDRN to calculate the water fiDn thickness: - MID 36.1 Sash Mann~ng's roughness coefficient Drainage path length (in) Rainfall rate (in/h) 51 (10) (1 in = 25.4 mm) (1 in/in = 25.4 mm/in)

OCR for page 45
S = Slope of drainage path (mm/mm) MID = Mean texture depth (in) (1 ~ = 25.4 mm) Values of water film thickness calculated according to equation 10 are presented In figure 8. Subsurface Flow Mode} For porous asphalt surfaces, flow within the porous asphalt layer parallel to the pavement surface must be considered. Two options were explored for the subsurface flow model. The first was based upon ache consideration of three-dunensional, fully saturated flow, represented by equation ~ ~ . The second, a one~unensional model, is discussed In the following. where _ (Kxx She fix fix h _ Kss = _ (Kyy ah) + ~ (Kzz ah) W-S ah (11) BY Liz Liz s St Piezometric head or potential Hydraulic conductivity of the porous material in the direction of the . · . prmc~pa1 axes W = Sources arid sinks of water Ss = Storage coefficient or specific storage of the porous material 52

OCR for page 45
- Portland cement concrete, MTD = 0.91 mm, Manning's n = 0.031, - ~ Dense graded asphalt concrete, MTD = 0.91 mm, Manning's n - 0.0327, Open graded asphalt concrete, MTD = 1.5 mm, Manning~s n = 0.0355 100 90 80 E - ~n to t' 50 - e~ ~ 40 a) - ~ 30 70 60 20 10 O- i '.~k MY , ad/ it,' a. ~ Slope of flow plane: 2%, Rainfall intensity: 40 mm/in 1 - 0 5 10 15 20 25 Drainage path length, m Figure 8. Water film thickness versus distance along flow path for several pavement surfaces as calculated using PAVDRN. 53

OCR for page 45
The solution of equation ~ ~ for the quantity of flow can Include the full dynamic nature of the effects of hydroplaning In the porous asphalt drainage layer. However, the computational effort required to arrive at a solution for either a two- or three-~mensional mode! is unjustified when the desired solution is the surface water film thickness. In this case a one-~vmensional mode] is sufficient. In some cases, the multidimensional models do not converge and require operator intervention In selecting different boundary conditions or, in the case of transient analysis, varying time steps and computational grids. The other option for flow through porous asphalt, a steady-state, one~mensional model, has several advantages. Like the one-~mensional surface How model, it is unconditionally stable. As a consequence, little or no operator intervention is necessary to arrive at a solution. Appropriate material properties can be quantified with a reasonable level of effort. Therefore, the one-~nnensional surface flow model, which was modified as described in the following, was chosen to compute water film thicknesses on porous pavements. For dete~n~ng water film thickness on porous pavement sections it is necessary to modify equation 10 to account for the infiltration rate. The addition of a term to account for infiltration rate results In equation 12. This form of the equation was used In PAVDRN to estimate the water film thickness for porous surfaces: DIVEST . n L T L 36.1 3.6 - MTD 54 (12)

OCR for page 45
where and I . 1 f f = MANNING'S N n = Mann~ng's roughness coefficient (i -f) -Excess rainfall rate (in/h) Rainfall rate fiItracion rate or permeability of pavement (iffy) (1 in/in = 25.4 mm/in) (1 inA1 = 25.4 mm/in) (1 inlh = 25.4 mm/in) In order to predict the water fihn thickness that occurs on a pavement surface during sheet flow, as presented In equations 10 and 12, Maurung's n must be known. The hydraulic roughness of a surface, Mann~g's n, can be expressed In terms of an e-value as used above and defined by Manning (34J. ManT~ng's e-value is surface-specific, requiring different expressions for different surfaces. During the course of this project, the hydraulic resistance of three different types of pavement surfaces was determined: . Portland cement concrete · Porous asphalt . Dense-graded asphalt concrete 55

OCR for page 45
experunental data and the regression of the data with various forms of equations 13 through 15, were developed and used In PAVDRN to determine Manning's n: I. Portland cement concrete surfaces: n 0 319 (NR ~ 1000) (16) NOSO2 (NR ~ 50O) (17) 2. Dense-graded asphalt concrete : n = 0.0823 N~0~74 (~) 3. Porous asphalt concrete: 1.490 S 0 306 N 0.424 (19) where NR = q 58 (20)

OCR for page 45
and NR - Reynold's number q = Quantity of flow per unit width (m31slm) v = Kinematic viscosity of water Equations 16 through 19 are presented graphically In ISgures 9 through 12. HYDROPLANING SPEED MODEL The hydroplaning model selected for the study is based upon the work of Gallaway and his colleagues (4) and as further developed by others (37, 38). On the basis of the work reported by these authors, for water hum thicknesses less than 2.4 mm (0.095 ill), the hydroplaning speed is determined by: HPS = 26.04 WFI-0259 where (21) HPS = Hydroplaning speed (mi/h) WFr - Water film thickness (in) (1 mi/h = 1.61 1an/h) (1 in 25.4 mm) For water film thicknesses greater than or equal to 2.4 mm (0.095 in), the hydroplaning speed IS: 59

OCR for page 45
. 0.12 0.10 0.08 to ·= 0.06 0.04 0.02 0.00 500 ~ NR ~ 1 000, Water temperature = ~ SAC. Ago. lb - ~ ., - _, ~ ., R ain fall intensity = ~ mcn/h R ainfall intensity = ~ O mm/in - R ainfall intensity = 20 mcn/h \ \ - b R ainfall intesity = 40 mm/in R ainfall intensity = 60 mm/in - 10 30 50 70 90 110 Length of flo w path, m Figure 9. Manning's n versus length of flow path for various rainfall rates, Portland cement concrete, 500 < NR < 19000. 60

OCR for page 45
0.12 0.10 0.08 0.06 w 0.04 0.02 0.00 10 l NR ~ 50O, Water temperature = ~ SAC. \ -\ ~ \ \ ~ - · - - Rainfall intensity = 5 mm~h Rainfall intensity = 10 mm~h - - - - - - Rainfall intensity = 20 mm~h Rainfall intensity = 40 mm~h - Rainfall intensity = 60 mm~h - 30 50 70 Length of Towpath, m 90 110 Figure 10. Mannir g's n versus length of flow path for various rainfall rates, Portland cement concrete, NR < 500 61

OCR for page 45
0.06 0.05 0.04 a) ·' 0.03 ce 0.02 0.01 0.00 - . - - Water temperature = 1 0°C. - - - Rainfall intensity = ~ mm~h Rainfall intensity = JO mm/in Rainfall intensity = 20 mm/in Rainfall intensity = 40 mm/in Rainfall intensity = 60 mm/in ~ ~ ~ ~ ~a c I ~ _ _ __ ~ ~ 1 0 30 50 70 90 1 1 0 Length of Towpath, m Figure ~ ~ . Mar mr~g's n versus length of flow path for various rainfall rates, dense-graded asphalt concrete. 62

OCR for page 45
0.25 0.20 0.15 v, - ._ ~5 0.10 0.05 0.00 Water temperature = 1 0°C | Rainfall intensity= ~ mm/in | \ --- Rainfall intensity = 10 mm/in \ ~ Rainfall intensity = 20 mmfh \ ---- Rainfall intensity = 40 mm/in \ \ \ ~ Rainfall inte laity = 60 mmlh .-, ____~~ , ~ . , ,,,, _ _ __ _ _, ~ - . _, - - - ~ · - - - - - _ · - -._. _, - - _ . _ _._ _._ _._ . 1 0 30 50 70 90 1 1 0 Length of flow path, m Figure 12. Manning's n versus length of flow path for various rainfall rates, porous asphalt concrete. 63

OCR for page 45
HPS = 3.09 A where A is the greater of the values calculated using equations 23 and 24: 10.409 ~ 3 507 FIT 0 06 or (22) 28.952 7 817 WET 0.06 - (23) MID 0.14 (24) The model predicts the onset of hydroplaning on the basis of the water film thickness where water hen thickness is the thickness of the wafer film above the mean tenure depth, as presented In figure 1. The results of equations 21 through 24 are shown graphically ~ figure 13. The mean texture depth can be determined from sand patch or micro profile measurements. Suggested levels of macrotexture are presented In the guidelines for design situations where acme measurements are not available. Although the hydroplaning prediction models (equations 2 through 24) are empirical, they represent the state of the art: The development of rational equations was considered too ambitious an undertaking given the complexity of We problem, the resources required, and the extent of the data that would be needed for a rational model. 64

OCR for page 45
CD ~ O · . . to to 11 11 11 ~ Q Q C~e ~ C - e C~e t I I I \ee ( - e c~et \e C~] (~ (-e ~ ~ ~ cr cr t ~e lLi l]J E ~ ~ ~ ~ · . . · C ~C~e C\e N V ~ ~ ~ ~ r~ e I 1 1 e ~ I I : / 1 ~ e 1 1 e e 1: 1 a e e e 1: 1 ~ e e e ~ : 1 4' · e de. ~ 1:1 e e lli! ee ~ ~ 11: 1 lee ~ .1. e 11: 1 e. .1e . ICI t. ~ . Ir1 te e. . 1~1 ~e ~ Itl 1e · ~ ~ lll t ~ . ~ e lll ~ e e er . 1:11 C~te ~n a ~ Y C) · - - · - ~ - e_ C~ _ C~ O O O ~( ~C~e O O O O O O ~ 0 ~~ ~ w) 4/~ Spasds Ou!ueldo!ep/H Figure 13. Hydroplaning speed versus water film thickness. 65

OCR for page 45
Unfortunately, experimental test data needed to verify the prediction of water depths on the surface of porous asphalt pavement was not obtained during this study and such data are not available In the literature. In order to obtain reliable experimental data, very high rainfall rates are needed and the experimental section must be well isolated so that the entire flow is contained within the boundaries of the test sections. In simple terms, very high, uniform rainfall rates are needed on a "leak-proof" test section. These conditions could not be satisfied with the available equipment, neither In the laboratory nor In the Users. Capturing a natural event was considered but the idea was discarded because of the coordination effort and costs associated with capturing such an event. Therefore, it was not possible to validate equations 21 Trough 24 for porous asphalt surfaces. Rainfall Intensity The AASHTO highway design guides Include an equation for relating rainfall intensity, vehicle speed, and maximum allowable sight distance as follows: i - [80,000/(SV-Vi)] 147 where i -Rainfall intensity (in/h) sv Sight distance (ft) V; Vehicle velocity (mi/h) 66 (25) (1 in/in-25.4 mm/in) (1 ft _ 0.305 mm) (1 m~/h 1.61 l~n/h)

OCR for page 45
This relationship is depicted In figure 14. Although it appears In equation 25 that intensity is a function of sight distance, sight distance is a function of Intensity. As Intensity increases, sight distance decreases. Likewise, vehicle velocity decreases with increased intensity and decreasing sight distance. 67

OCR for page 45
- -- - V ehicIe speed = 60 knn/h - --V chicle speed = 80 krr/h V chicle speed = 100 k m/h Vehicle speed = 120 krn/h ----Vehicle speed =160 k m/h 80 s ~ 60 - ~n 0 - - ce ._ ce 40 20 100 200 300 Sight distance Sv, m 400 500 Figure 14. Rainfall ~ntemit~r versus sight distance for various vehicle speeds. 68