Lab manual - Civil & Environmental Engineering

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53Response curves for theadvective dispersion equation areshown in Figures 5-5 and 5-6. Theexit age curve is highly skewed forlow Pe because tracer that is in thereactor longer has more time fordispersion. As the Peclet numberincreases the dispersion decreasesand the response becomes closer toplug flow.Flow through porous media(such as groundwater through soil)is a type of flow with dispersion.The above equations can be appliedby recognizing that the relevantwater velocity is the pore watervelocity. The pore water velocity isQU = where A is the crossAεsectional area of the porous mediaand ε (volume of voids/totalvolume) is the porosity of theporous media.The form of equation 5.21 isexactly like the normal distributioncurve:3.012.50.82.0E0.6E 1.5FF1.0t* at F=0.1 0.40.50.840.20.000.0 1.0 2.0 3.0t*Figure 5-5 Exit age (E) and cumulativeexit age (F) curves for the advectivedispersion equation with Pe of 100.0.710.60.80.5E0.4F0.6EF0.3t* at F=0.10.40.20.10.530.20.000.0 1.0 2.0 3.0t*Figure 5-6 Exit age (E) and cumulativeexit age (F) curves for the advectivedispersion equation with Pe of 4.where2CA 1 ⎡ −x⎤= exp ⎢ 2 ⎥M σ 2π⎣4σx ⎦x5.21Or in dimensionless form:σ = 2Dt5.222xdσ 2x = *2tPe5.232L2The variance in concentration over space ( σx) is the variance in concentrations takenfrom many different positions in the reactor at some single moment in time, t. Thevariance in x ( σ ) has dimensions of length squared.2xClosed Boundary Conditions: Completely Mixed Flow Reactor in SeriesCompletely mixed flow reactors are sometimes connected in series to create a reactorsystem with flow characteristics in between CMFR and PFR. CMFRs in series increaseReactor Characteristics
54overall process efficiency because the reactants are at higher concentrations in the firstreactors than they would be in a single large CMFR. Thus the total reactor volume can besmaller with a series of CMFRs. Similarly the reaction process can be made moreefficient given the same reactor volume it if is divided into series CMFRs.Non ideal reactors with mixing levels between that of CMFRs and PFRs and reactorscontaining baffles can also be modeled as a series of CMFRs. In this case the mixinglevel for the system is represented by the number of CMFRs that are used to represent thenon-ideal reactor. The equation describing the concentration of a conservative tracer inthe N th reactor is given in equation 5.24.()⎛ ⎞N⎜CNt N tM ( N 1! )e θ ⎟∀ ⎝ ⎠⎛ ⎞= ⎜ ⎟− ⎝θ⎠The non-dimensional form of equation 5.24 is given in equation 5.25N −1− NtN( ) ( ) ( ) *( NtN1 )*N −*t e −=EN tN −1!It can be shown that equations 5.18 and 5.25 converge for large N when5.245.25Pe = 2N5.26Equation 5.24 is limited to integral values of N. The model could be extended toinclude non integral values ofE0.80.70.60.50.40.30.20.10.00.2600.0 1.0 2.0 3.0t*EFt* at F=0.10.6F0.4Figure 5-7 Exit age (E) and cumulative exit age(F) curves for series CMFR with N of 2.N if the factorial functionwere replaced with theΓ function. Note that the Γfunction is not defined as anextension of the factorialfunction in Excel. Instead usethefunctionsEXP(GAMMALN(N)) tocalculate the Γ function.Response curves for theadvective dispersion equationare shown in Figure 5-7.Notice that the effluentconcentration begins to risemuch more quickly in seriesCMFRs than in the advectivedispersion model. This distinction between the two models lessens as the number ofreactors increases and the Pe increases.Reactor StudiesMeasuring Reactor CharacteristicsOne of the easiest methods to determine the mixing (dispersion) characteristics of areactor is to add a spike input of a conservative material and then monitor the10.80.2CEE 453: Laboratory Research in Environmental Engineering Fall 2006
Page 2 and 3: 49defined as a section of along rea
Page 4 and 5: 51∂C∂C=−U∂t∂xThe velocity
Page 8 and 9: 55concentration of the material in
Page 10: 5722θPe = 5.35σ2tMass Conservatio
Page 13 and 14: 60The flow characteristics of a baf
Page 15 and 16: 62into place (so the putty on the o
Page 17: 64Table 5-2.Equipment listDescripti

Lab manual - Civil & Environmental Engineering

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