Concentration wave velocity

Table VII shows that, for the methanation reactor model, the dynamic response of the gas temperatures and CO and C02 concentrations should be much faster (by two orders of magnitude) than the response of the catalyst and thermal well temperatures. This prediction is verified in the dynamic responses shown in Figs. 18 and 19 and the previous analysis of the thermal and concentration wave velocities.

Obviously, the velocity with which the concentration C . moves down the column (the concentration wave velocity) is less than that of the interstitial liquid. If one could ride with the wave having a concentration C, one will only witness around oneself therefore, for any wave having a fixed value of specific concentration Cg, dCg = 0. Now, Cg is a function of the independent variables z and t therefore, for any small change in z and t, the change in Cg is given by... 

For any particular concentration Cg, the concentration wave velocity must satisfy dCa = 0. Compare a reformulated (7.1.10) under such a condition. 

Ca) is fixed. The velocity is variously called the concentration wave velocity of species i, the migration rate of species i or the concentration front propagation velocity of species i (Sherwood et al., 1975). A more formal method of arriving at expression (7.1.12a) is given below. 

The analysis carried out earlier to arrive at the concentration wave velocity expression (7.1.12a) for i/J.,- is also valid here. Therefore, for equilibrium nondispersive operation of an ion exchange colunm, we can write, for ionic species A,... 

Here we define the system in terms of the binary mbcture of counterions A and B and ignore the presence of any co-ion Y on other ions. Therefore, the concentration wave velocity i/J-a of ionic species A through the column is given hy (for a given concentration, say Caw or mole fraction Xa ,)... 

Wankat (1990, pp. 465-470) for removal of K from a solution using an ion exchange bed in the form of Na. Langmuir isotherm describes this ion exchange with Kab = 1-54, where A = IC " and B = Na. The mole fraction of potassium in the 0.2 N feed solution Xkm, = 0.7. Determine the concentration wave velocity of and determine when this shock wave exits the column of length 30 cm and diameter 2 cm if the volumetric flow rate of the solution is 20 cm /tnin. 

Following the column mass balance based development of (7.1.15c), one can determine the concentration wave velocity of the shockwave at time t when the exit concentration jumps from C 2 = 0 to C, the incoming concentration ... 

Equation (7.1.73) provides the velocity of propagation of any y,. through the bed and represents the characteristics. During the half-cycle when the bed is hot (T = Thot), the concentration wave velocity is given by... 

During the half-cycle when the bed is colder (T = rcoid). the concentration wave velocity is given by... 

If the interstitial velocity of the liquid in the packed bed is 53 cm/min, calculate the migration velocity (concentration wave velocity) of species 1 whose equiUbrium relation between tbe soUd-pbase concentration of solute Cii and the mobile/liquid-phase concentration C12 is given by Cn = 4Ci2- Compare it with the simation where Ep = 0. (Ans. 7.11 cm/min 7.57 cm/min.)... 

Let us reflect on the net directional movement of species 1 and species 2 in this countercurrent column. It is clear that the lighter species 1, H2, moves up the column and the heavier species 2, N2, moves down the counter-current column. One could argue that, in effect, the concentration wave velocity, i j, of species 1 is positive vis-a-vis the z-coordinate in Figure 8.1.1, whereas that for species 2, N2, is negative. We illustrate the quantitative criterion for this type of species movement in a counter-current system of two immiscible phases in Section 8.1.1.3, where we illustrate also that only two species may be separated in such a countercurrent column having steady state flow. 

This equation has an exactly similar form to equation (7.1.9) (the de Vault equation). Therefore, following a procedure similar to that used to obtain equations (7.1.12a) and (7.1.12e), we can obtain tbe following expression for tbe concentration wave velocity, i , of species i (Fisb et al., 1989) ... 

If we recall the development of equation (8.1.35), we obtain the following expression for the concentration wave velocity "a for species i ... 

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