**Suggested Citation:**"Appendix D Model Evaluation." Transportation Research Board. 1998.

*Improved Surface Drainage of Pavements: Final Report*. Washington, DC: The National Academies Press. doi: 10.17226/6357.

**Suggested Citation:**"Appendix D Model Evaluation." Transportation Research Board. 1998.

*Improved Surface Drainage of Pavements: Final Report*. Washington, DC: The National Academies Press. doi: 10.17226/6357.

**Suggested Citation:**"Appendix D Model Evaluation." Transportation Research Board. 1998.

*Improved Surface Drainage of Pavements: Final Report*. Washington, DC: The National Academies Press. doi: 10.17226/6357.

**Suggested Citation:**"Appendix D Model Evaluation." Transportation Research Board. 1998.

*Improved Surface Drainage of Pavements: Final Report*. Washington, DC: The National Academies Press. doi: 10.17226/6357.

**Suggested Citation:**"Appendix D Model Evaluation." Transportation Research Board. 1998.

*Improved Surface Drainage of Pavements: Final Report*. Washington, DC: The National Academies Press. doi: 10.17226/6357.

**Suggested Citation:**"Appendix D Model Evaluation." Transportation Research Board. 1998.

*Improved Surface Drainage of Pavements: Final Report*. Washington, DC: The National Academies Press. doi: 10.17226/6357.

**Suggested Citation:**"Appendix D Model Evaluation." Transportation Research Board. 1998.

*Improved Surface Drainage of Pavements: Final Report*. Washington, DC: The National Academies Press. doi: 10.17226/6357.

**Suggested Citation:**"Appendix D Model Evaluation." Transportation Research Board. 1998.

*Improved Surface Drainage of Pavements: Final Report*. Washington, DC: The National Academies Press. doi: 10.17226/6357.

**Suggested Citation:**"Appendix D Model Evaluation." Transportation Research Board. 1998.

*Improved Surface Drainage of Pavements: Final Report*. Washington, DC: The National Academies Press. doi: 10.17226/6357.

**Suggested Citation:**"Appendix D Model Evaluation." Transportation Research Board. 1998.

*Improved Surface Drainage of Pavements: Final Report*. Washington, DC: The National Academies Press. doi: 10.17226/6357.

**Suggested Citation:**"Appendix D Model Evaluation." Transportation Research Board. 1998.

*Improved Surface Drainage of Pavements: Final Report*. Washington, DC: The National Academies Press. doi: 10.17226/6357.

**Suggested Citation:**"Appendix D Model Evaluation." Transportation Research Board. 1998.

*Improved Surface Drainage of Pavements: Final Report*. Washington, DC: The National Academies Press. doi: 10.17226/6357.

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APPENDIX D MODEL EVALUATION Sensitivity analyses were performed for PAVDRN and for the water film thickness and hydroplaning speed models. The purpose of the sensitivity analysis was to determine ache sensitivity of PAVDRN and the models to the various input parameters, to make certain that the models behaved in a reasonable manner, and to establish the sensitivity of the response variables (predicted values) to the input parameters (design parameters). The results of the sensitivity analyses are given In this appendix. ANALYSIS OF THE PAVDRN MODEL The sensitivity analyses for the PAVDRN program were conducted by varying three variables: cavemen" slope, S; rainfall intensity, i; and mean texture depth, MID. Each of these input variables was assigned three values-a low, an intermediate, and a high value as given in table D-~. The analyses were conducted by fixing two of the input variables (S and i, S and MID, or i and MTD) at the intermediate value and allowing ache other two variables (MID, i, and S. respectively) to assume their low, intermediate, and high values. The calculations were conducted for the four pavement types: dense-graded asphalt concrete, plain Portland cement concrete, grooved Portland cement concrete, and open-graded porous asphalt concrete. The values for the rainfall intensity, i, pavement slope, S. and mean texture depth, MID, used In these calculations are given in table D-~. Note that the values for rainfall D-1

intensity and pavement slope are the same for each of Me pavement types, but ache MID is different for the open-graded asphalt concrete. Table Dot. Values used In PAVDRN sensitivity analysis. Variable Minimum Intermediate Maximum Rainfall' intensity' i 1.00 (25.4) 3.0 (76) 6.0 (152) in/in (mm/h) Slope' S 0.0050 (0.0050) 0.015 (0.015) 0.030 (0.030) ft/ft (m/m) Mean texture depth' MTD 0.01 (.25) 0.02 (0.51) 0.05 (1.14) in (mm)3 DGAC and PCC(1) 1 `"DGAC =den se-graced asphalt concrete, PCC = plain or grooved Portland cement concrete, and OGAC open-graded asphalt concrete. Values of water film thickness versus flow paw length for the four pavement types are shown In figures D-1 through Dot. The figures show how the water film thickness varies for each of the variables when ache others are held at their extreme values. D-2

- rnPnC=~nr=H=H ~cr~h:~t ~nn~r=~` 5 4 ~ 2 1 it, ~ O 0 5 a- MTD = 0.254mm ~ MTD = 0.50Smm - · - MTD = 1.143mm 1 flow path distance, m - ~ 3 - 2 1 O 0 5 10 15 20 Flow path distance, m · Cross-siope = O.58/0 Cross-siope = 1.50/0 Cross-siooe = 3.0°/0 25 30 - ~ 3 - 2 o ~_ 0 5 · RF] = 25.4 much .- RF! = 76.2 mmlh *-- RF] = 152.4 mm~h 10 15 20 Flow oath distance. m 25 .. 30 ~ ~- . _·' ~_1 4~ _ =, a- ~ __ ~ 1~__= ~it_ _ ~u 1 concrete pavement. D-3

E 3 ~ -- (Portland cement concrete) -I MTD = 0~254mm MID = O.508mm MTD - 1 ~ 143mm Flow path distance' m 25 30 4 3 - 2 o ~ : : 10 15 20 25 Flow path distance, m Cross-slope = 0.5°/0 Cross-siope = 1.58/0 Cross-siope = 3.0% 0 5 30 - ~ 3 - 2 o 10 15 20 Flow path distances m RFt = 25.4 mmfh RFt = 76.2 mmlh · RFi = 1 52 4 mmfh 25 . 30 Figure ~2. Values for water film thlc~ess versus flow path length for Portland cement Dot

4 3 E 2 - ~ 1 - . O o Resee ~ concrete) _ ~MTD=1.0mm MTD = 2.5mm ~ MTD = 4.0mm 5 1015 Flow path distance, m 3 ~ Cross-siope = 0.58XO .- Cross-siope = 1.S8/o - ~ 2 - ~ 1 O 0 5 10 · Cross-slope = 3.0% ____ it= - . , . _ 15 20 25 30 Flow path distance, m ~ RF! = 25.4 mm/h 1 10 15 20 Flow path distance, m . . Figure D-3. Values for water film thiclmess versus flow path length for open-graded asphalt concrete pavement (no internal flow). D-5

4 3 - ~ 2 - ~ 1 o _ _ _ (Open~raded aspha = 1 ~ 1 L _~ MTD=1.0mm, Permeability = O.75mmIs · MTD = Grimm, Per7neability = 2.1 mmJs --- MTD = 4.0mm, Pe~eabili~ = 3.5mmIs 0 5 10 15 20 25 30 Flow path distance, m ~ 2 - 1 -it Cross-sIope = O.5°/O Cross-siope - 1.5/ Cross-sIope = 3.0% it____ ~ __________________ , _~ , _ 0 5 10 15 Flow path distance, m ! I 20 25 30 3 -~RFI= 25.4 much ~RFI= 76.2 mm~h ~I ~ RFI = 152 4 mmlh 0 5 10 15 20 Flow path distance, m 25 30 Figure >4. Values for water film thickness versus flow path length for ppen-g~aded asphalt concrete pavement (internal rioter). D-6

SENSITIVITY ANALYSIS OF THE WATER FILM THICKNESS MODEL The following equation is used in PAVDRN to calculate the water film thickness: y ~ 42.32 n q\06 ~ s0.5 where Y - n q - S - Flow depth (in) Manning's roughness coefficient Flow (ft3/s/ft) Slope of the drainage path (ft/ft) with Me values and coefficients In English units. The water film thickness is predicted in PAVDRN using the following relationship: (D-~) WFT = [ n L i s ]06 _ MID (D-2) where n = Manning's roughness coefficient ~= Drainage path length (in) i = Ra~nfaB rate (in/h) D-7

S = Slope of drainage path (mm/mm) MTD = heart texture depth (in) and the values and coefficients are In English units. The sensitivity analyses were conducted using procedures outlined earlier. Low, intermediate, and high values for n, q, and S are given In table D-2, and the analyses were conducted for several flow path lengths, as given In the table. Table D-2. Values used In kinematic wave equation sensitivity analysis. Variable Minunum Intermediate Maximum Manning's n 0.01 0.025 o.o5 Rainfall, intensity, i, 1.00 (25.4) 3.0 (76) 6.0 (152) in/in (mm/h) Slope, S. 0.0050 (0.0050) 0.015 (0.015) 0.030 (0.030) ft/ft (m/m) Drainage path length, L, 3.0 (0.92) 24 (7.3) 48 (14.6) ft (m, Flow, q, 6.94 x 10~5 3.37 x 10~3 6.67 x 10~3 ft3/s/ft~m3/h/myt~, (6.45 x10-6) (3.13x104) (6.20x104) The values of the flow rate, q, are the result of multiplying the drainage path length, L, by the rainfall ~tensi~9 i, and convening Me units as appropriate. D-8

SENSITIVITY OF THE: MODEL TO VARIATIONS IN N The partial derivative of the kinematic wave equation witch respect to n is: By 5.677 qO 6 = (D-3) On n04 S03 Using n 0.01, q 0.00000667 ft2/s, (0.00223m2/hr) aIld S 0.005 ft/ft (0.005 mm/mm) to obtain the highest By/On : By 5.677 (0.006667~°6 = ~ figs in <220.6 mm) (D-4) fin (O 011° 4 (O 005~° 3 Multiplying this value by the highest n (= 0.05) yields the highest change In y _ 0.434 In (~.0 my. Using n 0.05, q 0.0000694 fF/s, (0.0232 in2/hr) and S - 0.03 ft/ft (0.03 mm/mm) to obtain the lowest value of By/8n : By 5.677 (0.00006944)°6 0 ~723 in (4 3~3 error) (D-5) an to 05~°4 LO 03~°3 Multiplying this value by the lowest n ~ 0.01) yields the lowest change in y-0.001723 in (0.0438 loamy. D-9

SENSITIVITY OF THE MODEL TO VARIATIONS IN Q The partial derivative of the kinematic wave equation with respect to q is: BY = en 5.677 n06 q04 S03 Using n-0.05, q-0.0000694 ft2/s, (0.0232 huh) and S = 0.005 ft/ft (0.005 mm/mm) to obtain the highest Gy/6q : BY = ~q 5.677 (o.oS)06 (0.00006944)° 4 (O.Oo5)o 3 = 212ft2/s/in Multiplying this value by the highest value of q, 0.00666 ft2/s, (0.00223 iIl2/h) yields the highest change in y per UIlit Bow. (D-6) (D-~ Using n = 0.01, q-0.00000666 ft2/s, and S = 0.03 ft/ft to obtain the lowest Oy/6n BY = Be 5.677 (0.01)°6 _ - (0.006667)o 4 (0 o3)o.3 7.61ft2/s/in . (D-8) Multiplying this value by the lowest q, 0.0000694 ft2/s, (0.0232 huh) yields the lowest change in y per unit flow . - D-10

SENSITIVITY OF THE: MODEL TO VARIATIONS IN S The partial derivative of the ~nematic wave equation with respect to S is: ~Y _ ~ _ ~S -2~838 (n q)0 6 sl.3 (D-9) Us~ng n 0.05, q 0~00667 ft2/s~ and S-0~005 ft/ft (0.005 m=/r~rn) to obtain the highest By/6se ~y ~ = ~S . 2e 838 [(0 05) (Oe0066697)] (o.oo5)1 3 22 e 808 (D-10) Multiplying this value by the highest Se' OeO3 ftlft, yields the highest change in IYI = 0e684 1n (17e4 mm) e Us~ng n -OeOl ~ q - OeO000694 ft2/s~ (OeO232 ~n2/h) and S = 0eO3 ft/fl (OeO3 rnmirurn) to obtain the lowest dy/6n : ~y ~S = 2 e 838 [(O e O 1) (0 ~ 00006944)] I (o.03)l 3 = ~ 0~0546 (D-11) Multiplying this value by the lowest S = 0.005 ft/ft (0.005 rnm/rr~m) yields the lowest change in 1 Y 1 o e 000273 it1 (O e 00694 mm) ~ D-11

ANALYSIS OF RESULTS For the values of the highest change In the flow depth, y (see table D-3), the order of sensitivity is: q > S > n Table D-3. Variables affecting changes In flow depth. Changes in y, in (mm) Variable High Low M~nning's n 0.434 (~.0) 0.00172 (0.0438) Flow, q, (ft2/s) (cm3/s) 1.416 (36.0) 0.000528 (0.0134) Slope, S (ft/ft) (mm/mm) 0.684 (17.4) 0.000273 (0.00694) This shows that for high flows, the flow is the major factor that determines the flow depth. At high rainfall rates and tong drainage path lengths, the quantity of the flow has the largest effect on the depth. For the lowest values of the change in y, the order is: n > q > S This shows that at low flows, the resistance of the pavement, which is characterized by the Manning roughness coefficient, is the most dominant factor affecting the depth of flow D-12 . -