Ind. Eng. Chem. Process Des. Dev. 1985, 24, 961-966
HK = heavy key component i = on the ith tray LK = light key component o = operating line
961
King, C. J. "Separation Processes"; McGraw-Hill: New York, 1960; pp 798ff. Nishida, N.; Stephanopolous, G.; Westerberg, A. W. AIChE J . 1981, 2 7 , 321. Smoker, E. H. Trans. AIChE 1938, 3 4 , 165. Treybal, R. E. "Mass Transfer Operations"; McGraw-Hill: New York, 1966.
Literature Cited Fauth, G. F.; Shinskey, F. G. Chem. Eng. Prog. 1975, 7 1 , 49. Guthrie, K. M. Chem. Eng. 1969, 76, 114. Hengstebeck, R. J. Chem. Eng. 1969, 76, 115.
Received f o r review January 23, 1984 Revised manuscript received November 1, 1984 Accepted November 23, 1984
Entropy Generation and Work due to the Jetting Air Stream in Air Mixers Tetsuo Aklyama" and Isao Tada Department of Chemical Engineering, Shizuoka University, Hamamatsu, 432, Japan
The flow system of an isentropic nozzle followed by adiabatic pipe flow prior to an air mixer was considered, and the entropy generation and work due to the jetting air stream were calculated. The mixing efficiency of air mixers was then evaluated on the assumption that the smaller the entropy generation, and the larger the work, the higher the mixing efficiency would be. The calculations indicated that the negative-pressure air mixer (NPAM) can be more advantageous than the positive-pressure air mixer (PPAM) in the sense that air is being used more effectively for mixing in the former than in the latter. This confirmed the results of the previous work (Akiyama and Tada, 1984) that was done without the adiabatic pipe and had the work alone as a criterion for comparison. The effect of scale factor and the number of air injections were examined both theoretically and experimentally, which suggested that a lesser number of air injections with a larger column (or air tank) would be preferable from a practical viewpoint. In theoretical calculations of the flow through the nozzle and pipe, a quasi-steady-state assumption was made for mechanical energy and mass balance equations. This assumption was proven valid, through experiments, in claiming the theoretical conclusions reported herewith.
Numerous studies have been reported concerning solid mixing in fluidized and spouted beds (Davidson and Harrison, 1971; Mathur and Epstein, 1974; Davidson and Keairns, 1978; Cheremisinoff and Cheremisinoff, 1984),but studies of the transient-type air mixer are scarce. The negative-pressure air mixer (NPAM) was introduced by Akiyama et al. (1982) to overcome the drawback (necessity of installing filter bags) of the conventional type of air mixer, or the positive-pressure air mixer (PPAM). The NPAM, as well as some PPAM, mixes solid particles with repetitive air injections to secure the desired degree of mixing. Quantitative comparisons regarding the mixing efficiency between the NPAM and PPAM have been made both experimentally and theoretically (Akiyama et al., 1983; Akiyama and Tada, 1984). It was shown experimentally that the NPAM can become more advantageous than the PPAM in the sense that air is being used more effectively in the NPAM than in the PPAM. To explain this experimental finding, the work due to the jetting air stream through an isentropic nozzle was computed. A quasi-steady-state assumption was made in the computation. It was found that a unit mass of air performs larger work in the NPAM than in the PPAM in the range of practical use. This was interpreted to give a theoretical ground for supporting the experimental finding. The objectives of the present investigation are 4-fold. The first objective is to extend the theoretical analysis to a more realistic flow system than the previous one (an adiabatic pipe was considered after an isentropic nozzle). The second objective is to calculate the entropy generation due to the jetting air stream as another criterion to evaluate the potential use of the air stream for mixing. The 0196-4305/85/1124-0961$01.50/0
entropy generation results as a direct consequence of energy dissipation with a flow system. Under the assumption of isentropic flow in the nozzle region, it is mainly the flow through the pipe that contributes to the entropy generation, whereas the mixing occurs in the column (after air flows through the pipe). Hence, low entropy generation means high mass jet velocity into the column, which ought to improve mixing. Thus, the entropy generation can be considered as a more objective criterion than the work in evaluating the effective use of air flow. The third objective is to examine, both theoretically and experimentally, the effect of scale factor as well as the number of air injections on solids mixing. The fourth objective is to examine the validity of a quasi-steady-state assumption for the mechanical energy and mass balance equations by carrying out experiments of the jetting air stream and comparing experimental and theoretical values of entropy generation and work.
Theory Governing Equations and Computational Scheme. We consider the flow system of an isentropic nozzle followed by adiabatic pipe flow prior to an air mixer as shown in Figure 1. The corresponding schematics of the experimental apparatus for the NPAM and PPAM are shown in Figures 2 and 3, respectively. The numbers in these figures match the subscript numbers in Figure 1. The mass velocity through a well-rounded nozzle alone, if we assume ideal isentropic flow, is (Shapiro, 1953)
0 1985 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
962
Atmowhere
rs" I
-
u--u Solenoid osdlograph
Amp1if ier
Pressure sensor
4 Valve
Air
Compressor
t
Ressure regulator P S Air filter
(@
Cone 2
valve Sdcnoid valve
Gasinlet Valve
u
Air tank (Reservoir)
4
Vacuum pump
%
I
Sdmoid owibqraph
Figure 3. Schematic of the experimental apparatus of the PPAM.
TraD
Figure 2. Schematic of the experimental apparatus of the NPAM.
where m refers to molecular weight and y = 1.4 for air. Through the pipe alone, adiabatic flow is represented by
__ Levenspiel's doto
I
I
02
0.L
0.6
0.8
10
12
G/Gu 1-1
Figure 4. Comparison of flow rates between transient and steadyflow systems.
where the Mach number U
Ma= -
c
C=yRT/m
(3)
the pipe resistance factor
B = 4fL/d
(4)
and 7-1 2
Y j = 1 + -Mat
j = 1,2
(5)
The present study used the following formulas for the friction factor f = 16/Re for Re I2100 f = 0.0791/R,0.25 for Re > 2100
Other pertinent relationships are isentropic relationship Pip;? = pp-Y = constant where subscript i refers to the initial state, and
where A, = 2.04 X 10'' m2 is the cross-sectional area of the nozzle. V, stands for the volume of the air mixer column (we call it column hereafter) minus that of particles and V , the volume of the air tank. The smaller the subscript number, the upper the position of air flow. To find the flow rate from the reservoir through a pipe into the column, a trial-and-error scheme is needed. Lapple (1943) prepared a chart, treating B as a parameter for the quick solution of these equations (excluding eq 11 and 12). An error in Lapple's work was pointed out by Levenspiel (1977),who recalculated and redrew the Lapple charts. The present system differs from that of Lapple and Levenspiel because the mass balance equations are required to account for the transient state. The flow rate in the PPAM was compared with Levenspiel's data in Figure 4. The difference between the two works becomes apparent in the region outside of the choking flow (P3
