APPENDIX A:  TABLES


TABLE A-1   PROPERTIES OF WATER IN SI UNITS
Temperature
(°C)
Specific
Gravity
Density
(g/cm3)
Heat of
Vaporization
(cal/g)
Viscosity Vapor Pressure
Absolute
(cp)
Kinematic
(cs)
(mm Hg) (mb) (g/cm2)
0 0.99987 0.99984 597.3 1.790 1.790 4.58 6.11 6.23
5 0.99999 0.99996 594.5 1.520 1.520 6.54 8.72 8.89
10 0.99973 0.99970 591.7 1.310 1.310 9.20 12.27 12.51
15 0.99913 0.99910 588.9 1.140 1.140 12.78 17.04 17.38
20 0.99824 0.998211 586.0 1.000 1.000 17.53 23.37 23.83
25 0.99708 0.99705 583.2 0.890 0.893 23.76 31.67 32.20
30 0.99568 0.99565 580.4 0.798 0.801 31.83 42.43 43.27
35 0.99407 0.99404 577.6 0.719 0.723 42.18 56.24 57.34
40 0.99225 0.99222 574.7 0.653 0.658 55.34 73.78 75.23
50 0.98807 0.98804 569.0 0.547 0.554 92.56 123.40 125.83
60 0.98323 0.98320 563.2 0.466 0.474 149.46 199.26 203.19
70 0.97780 0.97777 557.4 0.404 0.413 233.79 311.69 317.84
80 0.97182 0.97179 551.4 0.355 0.365 355.28 473.67 483.01
90 0.96534 0.96531 545.3 0.315 0.326 525.89 701.13 714.95
100 0.95839 0.95836 539.1 0.282 0.294 760.00 1013.25 1033.23
Source:  Linsley, R. K. et al. (1982). Hydrology for Engineers. 3d. ed., New York: McGraw-Hill.


TABLE A-2   PROPERTIES OF WATER IN U.S. CUSTOMARY UNITS
Temperature
(°F)
Specific
Gravity
Density
(lb/ft3)
Heat of
Vaporization
(Btu/lb)
Viscosity 1 Vapor Pressure
Absolute
(lbs/ft2)
Kinematic
(ft2/s)
(in Hg) (mb) (lb/in2)
32 0.99986 62.418 1075.5 3.746 1.931 0.180 6.11 0.089
40 0.99998 62.426 1071.0 3.229 1.664 0.248 8.39 0.122
50 0.99971 62.409 1065.3 2.735 1.410 0.362 12.27 0.178
60 0.99902 62.366 1059.7 2.359 1.217 0.522 17.66 0.256
70 0.99798 62.301 1054.0 2.050 1.058 0.739 25.03 0.363
80 0.99662 62.216 1048.4 1.799 0.930 1.032 34.96 0.507
90 0.99497 62.113 1042.7 1.595 0.826 1.422 48.15 0.698
100 0.99306 61.994 1037.1 1.424 0.739 1.933 65.47 0.950
120 0..98856 61.713 1025.6 1.168 0.609 3.448 116.75 1.693
140 0.98321 61.379 1014.0 0.981 0.514 5.884 199.26 2.890
160 0.97714 61.000 1002.2 0.838 0.442 9.656 326.98 4.742
180 0.97041 60.580 990.2 0.726 0.386 15.295 517.95 7.512
200 0.96306 60.121 977.9 0.637 0.341 23.468 794.72 11.526
212 0.95837 59.828 970.3 0.593 0.319 29.921 1013.25 14.696
1 To obtain values of viscosity, multiply values shown in Table by 10-5.
Source:  Linsley, R. K. et al. (1982). Hydrology for Engineers. 3d. ed., New York: McGraw-Hill.



APPENDIX B:  DERIVATION OF THE NUMERICAL DIFFUSION COEFFICIENT
OF THE MUSKINGUM-CUNGE METHOD


Space-time discretization of kinematic wave equation

Figure B-1  Space-time discretization of kinematic wave equation.

Expanding the grid function Q( jΔx,nΔt ) (Fig. B-1) in Taylor series about point ( jΔx,nΔt ) leads to:

                                  ∂Q                   1      ∂2Q
Q j n+1  =  Q j n  +  [ _____ ] j  Δt  +  ___ [ ______ ] j  Δt 2  +  ot 3)
                                  ∂t                     2       ∂t 2
(B.1)

                                       ∂Q                       1      ∂2Q
Q j+1n+1  =  Q j+1 n  +  [ _____ ] j+1  Δt  +  ___ [ ______ ] j+1  Δt 2  +  ot 3)
                                        ∂t                        2       ∂t 2
(B.2)

                                 ∂Q                     1      ∂2Q
Q j+1n  =  Q j n  +  [ _____ ] n  Δx  +  ___ [ ______ ] n  Δx 2  +  ox 3)
                                 ∂x                      2      ∂x 2
(B.3)

                                       ∂Q                         1      ∂2Q
Q j+1n+1  =  Q j n+1  +  [ _____ ] n+1  Δx  +  ___ [ ______ ] n+1  Δx 2  +  ox 3)
                                        ∂x                         2       ∂x 2
(B.4)

Substituting Eqs. B.1 to B.4 into Eq. 9-61 and neglecting third-order terms yields:

           ∂Q                    1         ∂2Q
X  { [ ____ ] j  Δt  +   ____   [ ______ ] j  Δt 2 }
            ∂t                     2          ∂t 2
 

                         ∂Q                        1         ∂2Q
+  (1 - X )   { [ _____ ] j+1  Δt  +   ____   [ _____ ] j+1  Δt 2 }
                          ∂t                         2          ∂t 2
 

       C          ∂Q                       1          ∂2Q
+  ____  { [ _____ ] n   Δx  +   ____   [ ______ ] n  Δx 2 }
       2           ∂x                        2          ∂x 2
 

       C          ∂Q                           1          ∂2Q
+  ____  { [ _____ ] n+1   Δx  +   ____   [ ______ ] n+1  Δx 2 }  = 0
       2           ∂x                            2          ∂x 2
(B.5)

in which C = c (Δtx) is the Courant number.

Expressing the derivatives at grid point [( j + 1)Δx, (n + 1)Δt ] in terms of the derivatives at grid point ( jΔx, nΔt ) by means of Taylor series:

    ∂Q                     ∂Q                  ∂2Q
[ _____ ] j+1  =   [ _____ ] j    +  [ ______ ] j,n   Δx   +  ox 2)
     ∂t                       ∂t                  ∂xt
(B.6)

    ∂Q                      ∂Q                     ∂2Q
[ _____ ] n+1  =    [ _____ ] n    +  [ _______ ] j,n   Δt   +  ot 2)
     ∂x                       ∂x                    ∂xt
(B.7)

    ∂2Q                      ∂2Q                   ∂3Q
[ ______ ] j+1  =    [ ______ ] j    +  [ _______ ] j   Δx   +  ox 2)
     ∂t 2                      ∂t 2                  ∂t 2x
(B.8)

    ∂2Q                       ∂2Q                    ∂3Q
[ ______ ] n+1  =    [ ______ ] n    +  [ _______ ] n   Δt   +  ot 2)
     ∂x 2                      ∂x 2                   ∂x 2t
(B.9)

Substituting Eqs. B.6 to B.9 into B.5 and neglecting third-order terms:

            ∂Q                      1         ∂2Q
X  { [ _____ ] j   Δt  +   ____   [ ______ ] j  Δt 2 }
             ∂t                       2          ∂t 2
 

                         ∂Q                        ∂2Q                            1         ∂2Q
+  (1 - X )   { [ _____ ] j   Δt  +   [ ______ ] j,n  Δx Δt  +  ____   [ ______ ] j  Δt 2 }
                          ∂t                        ∂xt                            2          ∂t 2
 

       C          ∂Q                       1          ∂2Q
+  ____  { [ _____ ] n   Δx  +   ____   [ ______ ] n  Δx 2 }
       2           ∂x                        2          ∂x 2
 

       C          ∂Q                         ∂2Q                            1         ∂2Q
+  ____  { [ _____ ] n   Δx  +  [ ______ ] j,n  Δx Δt  +  ____   [ ______ ] n  Δx 2 }  = 0
       2           ∂x                         ∂xt                           2          ∂x 2
(B.10)

In Eq. B.10, dividing by Δt and simplifying:

    ∂Q                     ∂Q                Δt       ∂2Q             c Δx      ∂2Q
[ _____ ] j  +   c [ _____ ] n    +  ____ [ ______ ] j  +  ______ [ ______ ] n
     ∂t                      ∂x                  2        ∂t 2                2         ∂x 2
 

                               C           ∂2Q
+ Δx  { ( 1 - X ) +  ____ }  [ ______ ] j,n  = 0
                               2           ∂xt
(B.11)

The first two terms of Eq. B.11 constitute the kinematic wave equation, Eq. 9-18. The remaining terms are the error R of the first-order-accurate numerical scheme:

         Δt       ∂2Q             c Δx      ∂2Q                                           C            ∂2Q
R =  ____ [ ______ ] j  +  ______ [ ______ ] n  + Δx  { ( 1 - X )  +  _____ }  [ ______ ] j,n  = 0
          2        ∂t 2                2         ∂x 2                                           2           ∂xt
(B.12)

From Eq. 9-18:

 ∂Q                 ∂Q
____    =  - c   ____
 ∂t                   ∂x
(B.13)

Therefore:

  ∂2Q                   ∂2Q
______    =  - c   ______
 ∂x ∂t                   ∂x 2
(B.14)

 ∂2Q                   ∂2Q
______    = c 2   ______
  ∂t 2                    ∂x 2
(B.15)

Substituting Eqs. B.14 and B.15 into B.12 and simplifying:

                           1        ∂2Q
R = c Δx ( X  -  ___ )  _____
                           2         ∂x 2
(B.16)

Comparing Eq. B.16 with the right-hand side of the diffusion wave equation, repeated here:

  ∂Q             ∂Q              ∂2Q
 ____  + c   _____  = νh  _______
   ∂t              ∂x                ∂x 2
(B.17)

it follows that the numerical diffusion coefficient of the Muskingum-Cunge method is:

                    1
νh = c Δx ( ___  -  X )
                    2
(B.18)


140815

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