Unity Mach number axial dispersion model for heat exchanger design

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4.3 Evaluation procedureThe inlet and outlet signals can only be measured in the time domain. For the proposed method theyhave to be Laplace transformed into the frequency domain according toT∞∫z=0( s) T ( z) exp( − sz) dz= . (21)Figure 2. Example of inlet and outlet signals of an adiabatictransient experiments. Numerical integration according toequation (22) with denoted discrete measuring points.The experimental data will consist of a number of n equidistant discrete measuring points (figure2). Therefore, the integration according to equation (21) becomes a summation and the ratio betweenoutlet and inlet signal of equation (20) is approximatelyT ′′T ′( s)( s)≈1T′′1exp( −sz1)+21T′1exp( −sz1)+2n−1∑i=2n−1∑i=21T′′iexp( −szi)+ T′′nexp( −sz21T′iexp( −szi)+ T′nexp( −sz2nn). (22))Now equation (20) can be evaluated for different Laplace variables s. Since in general thedispersion model will not exactly fit, the dispersive Peclet number will not be the same for differentLaplace variables s, i.e.Pe = Pe(s) ≠ const. (23)For practical applications a mean dispersive Peclet number Pe m is proposed. Equation (24) requiresthe evaluation of equation (20) for four different values of s, table 2:
1Pem=23⎡ 1 1 ⎤ 1 ⎡ 1 1⎢ + ⎥ − ⎢ +⎢⎣Pe1 2Pe−1 2 ⎥⎦6 ⎣Pe+1Pe+ −1⎤⎥⎦(24)Table 2. Evaluation points for chosen value of s 1PesPe +1 s 1Pe +1/2 s 1 /2Pe -1/2 -s 1 /2Pe -1 -s 14.4 ExampleInstead of an adiabatic thermal experiment Balzereit [14] performed tracer experiments which areanalogous to adiabatic thermal experiments. The inlet signal is a Dirac pulse of concentration whichhas been realized with great experimental effort. The outlet signal for the shell side of a shell and tubeheat exchanger with 11 flow baffles is shown in figure 3.Figure 3. Experimental (•) and calculated (⎯) system responses to a Dirac pulse for the shell side of ashell and tube heat exchanger with 11 flow baffles; dimensionless outlet concentration as function ofdimensionless time: (a) unity Mach number dispersion model, (b) cascade model and (c) parabolicdispersion model.The evaluation procedure after section 4.3 yields a dispersive Peclet number Pe m = 38.6. For theparabolic dispersion model (M = 0) as in the original work of Balzereit [14] the dispersive Pecletnumber is 37.6, confirming the following relationship which can be derived from a comparison of bothmodels:(M=0) 2(M=1) [Pe ]Pe = (M=0)(M=0)Pe −1+exp[ −Pe](25)Finally, the heat exchanger can also be modeled as a cascade of stirred tanks in series [4]. Thetransfer function relating the outlet signal to the inlet signal in the frequency domain isT ′′T ′( s)1=( s) ( 1+s N ) N. (26)
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Unity Mach number axial dispersion model for heat exchanger design

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