CE 498/698 and ERS 685 (Spring 2004) Lecture 111 Lecture 11: Control-Volume Approach (steady-state) CE 498/698 and ERS 685 Principles of Water Quality Modeling

CE 498/698 and ERS 685 (Spring 2004) Lecture 112 Things are changing… with respect to –time –space Partial differential equations CONTROL VOLUME APPROACH Steady-state: changing only with space (this lecture)

CE 498/698 and ERS 685 (Spring 2004) Lecture 113 Completely Mixed Lake Model For volume i: 012  i-1ii+1  n-1n 0

CE 498/698 and ERS 685 (Spring 2004) Lecture 114 For volume i (centered difference): Notes: Mixing length for E is avg between adj cells k is temperature dependent U can change 012  i-1ii+1  n-1n

CE 498/698 and ERS 685 (Spring 2004) Lecture 115 For volume i (centered-difference): i-1ii  i-1ii+1  n-1n

CE 498/698 and ERS 685 (Spring 2004) Lecture 116 For volume i (backward difference): i-1ii  i-1ii+1  n-1n

CE 498/698 and ERS 685 (Spring 2004) Lecture 117 For volume i (centered-difference): i-1ii  i-1ii+1  n-1n if Q is constant

CE 498/698 and ERS 685 (Spring 2004) Lecture 118 For volume i (centered-difference): 012  i-1ii+1  n-1n At steady-state: where

CE 498/698 and ERS 685 (Spring 2004) Lecture 119 For volume i (centered-difference): 012  i-1ii+1  n-1n At steady-state:n equations n+2 unknowns c i-1 c i+1 Boundary conditions Dirichlet boundary conditions Neumann boundary conditions

CE 498/698 and ERS 685 (Spring 2004) Lecture 1110 For loading at volume 1: 01  nn+1 open boundaries

CE 498/698 and ERS 685 (Spring 2004) Lecture 1111 For loading at volume 1: 01  nn+1 For loading at volume n: n equations n unknowns solve for c’s open boundaries

CE 498/698 and ERS 685 (Spring 2004) Lecture 1112 For loading at volume 1: For loading at volume n: n equations n unknowns solve for c’s pipe boundaries n  1 Q 0,1 c 0 QncnQncn

CE 498/698 and ERS 685 (Spring 2004) Lecture 1113 Numerical dispersion Example 11.1: Backward differences Example 11.3: Centered differences Figure E Figure E11.3

CE 498/698 and ERS 685 (Spring 2004) Lecture 1114 Numerical dispersion Taylor-series expansion Backward difference

CE 498/698 and ERS 685 (Spring 2004) Lecture 1115 Numerical dispersion Taylor-series expansion Backward difference numerical dispersion: does not occur w/ centered difference

CE 498/698 and ERS 685 (Spring 2004) Lecture 1116 Positivity See Example 11.5 Diagonal Super- diagonal Subdiagonal positive solution subdiagdiagsuperdiag

CE 498/698 and ERS 685 (Spring 2004) Lecture 1117 Positivity must be negative to have positive solution substitute: and for centered differences: segment length limit

CE 498/698 and ERS 685 (Spring 2004) Lecture 1118 Positivity for backward differences: always negative!

CE 498/698 and ERS 685 (Spring 2004) Lecture 1119 Summary of constraints Centered difference Backward difference Positivity Numerical dispersion Method

CE 498/698 and ERS 685 (Spring 2004) Lecture 1120 Physical dispersion what you measure what you put in your model numerical dispersion due to numerical model

CE 498/698 and ERS 685 (Spring 2004) Lecture 1121 Physical dispersion Critical segment size For backward differences, leads to, so or