Mass fluxes

Mow consider a second experiment, in which substances 1 and 2 are interconverted by chemical reaction in a Wicke-Hugo cell of the type shown In Figure 10.2. Then the net mass flux must vanish, since mass is conserved in the chemical reaction, so... 

The rate of mass transfer (qv) depends on the interfacial contact area and on the rate of mass transfer per unit interfacial area, ie, the mass flux. The mass flux very close to the Hquid—Hquid interface is determined by molecular diffusion in accordance with Pick s first law ... 

Interfacial Contact Area and Approach to Equilibrium. Experimental extraction cells such as the original Lewis stirred cell (52) are often operated with a flat Hquid—Hquid interface the area of which can easily be measured. In the single-drop apparatus, a regular sequence of drops of known diameter is released through the continuous phase (42). These units are useful for the direct calculation of the mass flux N and hence the mass-transfer coefficient for a given system. 

Fanning friction factor Darcy friction factor mass flux... 

Gm Molar gas-phase mass flux kmol/(s-m ) (lbmol)/(h-fF)... 

Mass flux of A by diffusion with respect to the mean mass velocity kmoP(m -s) or moP(cm -s) lbmol/(fF-h)... 

The first equahty (on the left-hand side) corresponds to the molar flux with respect to the volume average velocity while the equahty in the center represents the molar flux with respect to the molar average velocity and the one on the right is the mass flux with respect to the mass average velocity These must be used with consistent flux expressions for fixed coordinates and for Nc components, such as ... 

Tbe mass-transfer coefficients k c and /cf by definition are equal to tbe ratios of tbe molal mass flux Na to tbe concentration driving forces p — Pi) and (Ci — c) respectively. An alternative expression for tbe rate of transfer in dilute systems is given by... 

T] Low M.T. rates. Low mass-flux, constant property systems. Ns, % L local k. Use with arithmetic difference in concentration. Coefficient 0.323 is Blasius approximate solution. 

S] Low mass-flux with constant property system. Use with arithmetic concentration difference. 

E. Laminar, fully developed parabolic velocity profile, constant mass flux at wall... 

Solution of Eq. (6-114) for G and differentiation with respect to p reveals a maximum mass flux = P2VMJ RT) and a corresponding exit velocity and exit Mach number Mo = L/. This... 

Example 7 Flow through Frictionless Nozzle Air at po and temperature To = 293 K discharges through a frictionless nozzle to atmospheric pressure. Compute the discharge mass flux G, the pressure, temperature, Mach mimher, and velocity at the exit. Consider two cases (1) po = 7 X 10 Pa absolute, and (2) po = 1.5 x 10 Pa absolute. 

The exit Mach number Mo may not exceed unity Mo = 1 corresponds to choked flow sonic conditions may exist only at the pipe exit. The mass velocity G in the charts is the choked mass flux for an isentropic nozzle given by Eq. (6-118). For a pipe of finite length. 

Once the mass flux G has been determined. Fig. 6-21r or Q>-2 h can be used to determine the pressure at any point along the pipe, simply by reducing 4fL/D and computing p From the Figures, given G, instead of the reverse. Charts for calculation between two points in a pipe with known flow and known pressure at either upstream or downstream locations have been presented by Loeb (Chem. Eng., 76[5], 179-184 [1969]) and for known downstream conditions By Powley (Can. J. Chem. Eng., 36, 241-245 [1958]). 

Equation (6-128) does not require fric tionless (isentropic) flow. The sonic mass flux through the throat is given by Eq. (6-122). With A set equal to the nozzle exit area, the exit Mach number, pressure, and temperature may be calculated. Only if the exit pressure equals the ambient discharge pressure is the ultimate expansion velocity reached in the nozzle. Expansion will be incomplete if the exit pressure exceeds the ambient discharge pressure shocks will occur outside the nozzle. If the calculated exit pressure is less than the ambient discharge pressure, the nozzle is overexpanded and compression shocks within the expanding portion will result. 

Choked and unchoked flow situations arise in pipes and nozzles in the same fashion for homogeneous equihbrium flashing flow as for gas flow. For nozzle flow from stagnation pressure po to exit pressure pi, the mass flux is given by... 

Pipe Flow For steady-state flow through a constant diameter duct, the mass flux G is constant and the governing steady-state momentum balance is ... 

G = mass flux V = specific volume of stream P = stream pressure f= friction factor D = duct diameter L = flow length g = acceleration due to gravity b = angle of inclination from vertical... 

Superficial vapor mass flux, Ib/s fr k Capacity coefficient... 

FIG. 26-63 Discharge mass flux for highly suhcooled water (20 C) from orifice and4-mm pipe of various lengths. (Data of Uchida and Narai, 1966 reproduced hy peimission of ASME.)... 

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