Page 114 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Unit Process Principles 69

TABLE CD4.2
(a) Comparison between Finite Difference and Mathematical Solutions for Continuous
Input of Salt Starting at t ¼ 0
Finite Difference Solution
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Dt (s) t (s) V (m ) Q (m =s) u ¼ V=Q (s) t=u C in,t (kg=m ) C t (kg=m ) C tþDt (kg=m )
0.01 0.00 1000 1000 1 0.00 1000 100.000000 109.000000
0.01 0.01 1000 109.000000 117.910000
0.02 0.02 1000 117.910000 126.730900
Mathematical Solution
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e( t=u) (C in C t¼0 ) (kg=m ) (C in C) (kg=m ) C (kg=m ) Diff. (kg=m )
1.00000 900 900.000000 100.000000 0.000000
0.99005 900 891.044850 108.955150 0.044850
0.98020 900 882.178806 117.821194 0.088806
Notes: (1) Dt is by trial and error.
(2) t ¼ t þ Dt
(3) V is a design input.
(4) Q is the flow through the reactor.
(5) q is the detention time.
(6) C in is the specified salt concentration flowing into reactor.
(7) C t is the reactor concentration at time, t Dt.
(8) C tþDt is the reactor concentration at time, t þ Dt, calculated by finite difference equation.
(9) Mathematical solution—dimensionless.
(10) The quantity, (C in C t ) is calculated as e t=u (C in C t¼0 ).
(11) C t ¼ C in,t (C in,t C t ).
(12) Difference ¼ C t (numerical) C t (mathematical).

TABLE CD4.2
(b) Solution Finite Difference Mass Balance Equation—Continuous Input
(0.1 t=u 0.5) of Salt a
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Dt (s) t (s) V (m ) Q (m =s) u ¼ V=Q (s) t=u C in,t (kg=m ) C t (kg=m ) C tþDt (kg=m )
0.01 0.00 1000 1000 1 0.00 100 100.000 100.000
0.01 0.01 100 100.000 100.000
0.02 0.02 100 100.000 100.000
Notes: (1) Dt is from ‘‘calibration’’ with mathematical solution t þ Dt.
(2) t ¼ t þ Dt.
(3) V is a design input.
(4) Q is the flow that must be processed.
(5) u is the detention time.
(6) C in is the specified salt concentration flowing into reactor.
(7) C t is the reactor concentration at time, t Dt.
(8) C tþDt is the reactor concentration at time, t þ Dt, calculated by finite difference equation.
a
This table is a copy of Table CD4.2 (a) with changes to reflect the pulse loading.