EVALUATION OF MIXING PERFORMANCE IN A MAGNETIC MIXER BASED ON INFORMATION ENTROPY

Transcription

1 14 th European Conference on Mixing Warszawa, 1-13 September 212 EVALUATION OF MIXING PERFORMANCE IN A MAGNETIC MIXER BASED ON INFORMATION ENTROPY Rafał Rakoczy, Stanisław Masiuk, Marian Kordas a West Pomeranian University of Technology, Institute of Chemical Engineering and Environmental Protection Process, al. Piastów 42, Szczecin, Poland rrakoczy@zut.edu.pl Abstract. The mixing plays a fundamental role in domains as fluid dynamics and chemical engineering. Several types of reactors or mixers are used in chemical engineering industries. Static or alternating magnetic field might be applied to augment the process intensity instead of mechanical mixing. The novel approach to a mixing process is based on the application of a transverse rotating magnetic field. In the following discussion, a new method based on information entropy to evaluate the mixing equipment (magnetic mixer) on the basis of the transient response method is presented. Keywords: magnetic mixer; transverse rotating magnetic field (TRMF); residence time distribution (RTD); Informational mixing capacity 1. INTRODUCTION The simulation of chemical engineering processes is connected with the knowledge about hydrodynamics, transfer processes and chemical reactions. The residence time distribution (RTD) is one of the most informative characteristics to obtain hydrodynamic information. This systemic approach is allowed to establish the flow patter in the various types of mixers or chemical reactors. The RTD technique may be treated as a quantitative method to derive simple hydrodynamic models in the case of complex engineering systems. For design and scale-up purposes, it is essential to have information on the RTD. Many experimental investigations and theoretical considerations are reported on the RTD of impinging chemical engineering devices [1-6]. Several types of reactors or mixers are used in chemical engineering industries [7,8]. Static or alternating might be applied to augment the process intensity instead of mechanical mixing. The novel approach to a mixing process is based on the application of a transverse rotating magnetic field [9]. It is obvious that this kind of magnetic field may be induced a complex flow. There is, therefore, a need to investigate the influence of a transverse rotating magnetic field (TRMF) on the hydrodynamic conditions. From the practical point of view, the evaluation of mixing performance of engineering system may be done by using the stimulus-response technique. This method is based both on the analysis of the response curve (the response is the recording tracer leaving the vessel) and physical information about the process itself. In the following discussion, a new method based on information entropy to evaluate the mixing equipment (magnetic mixer) on the basis of the transient response method is presented. 395

2 2. THEORETICAL BACKGROUND The foundation of information theory is a quantification of information. In information theory, the Shannon entropy, H(X), is a measure of the uncertainty associated with a random variable. The informational entropy of a discrete random variable, X, is defined as follows [1] H ( X ) = p ( xi) log 2 p ( xi) dx [ bit] (1) where p(x i ) is the probability mass function of X. When then injection of the tracer is expressed in the form of a delta function, the concentration change in the tracer with time at the outlet becomes RTD. In this case, the mixing performance of the equipment may be evaluated from the point of view of informational entropy (see Eq.(1)) based on the uncertainty regarding the amount of time taken by the tracer element to reach the outlet. It is convenient to treat the residence time as discontinuous time. From this, the informational entropy for RTD may be defined in the following form [11] where E ( θi ) θ interval ( θ, θ θ) i m i 2 i (2) i= 1 ( ) ( ) log ( ) H θ = E θ Δθ E θ Δθ Δ is the probability that the observed element reaches the outlet in the time i +Δ ; Δ θ - dimensionless residence time interval. The minimum and maximum values of the informational entropy (see Eq.(2)) are presented in Table 1. Table 1. The minimum and maximum values of H ( θ ) Value E ( θ ) H ( θ ) minimum E ( θi ) = θi a E ( θ ) = ( 1 Δθ ) H ( θ ) min log i θ i = a Description = Δθ perfect mixing flow maximum i E ( θi ) = e θ H( θ ) max = log e log Δθ perfect mixing flow Remarks: a denotes some specific dimensionless residence time H ( θ ) max when the concentration of tracer in the equipment is perfectly homogenous (the concentration of tracer at the outlet decreases exponentially) H ( θ ) min when the tracer does not disperse in the equipment completely (the concentration of tracer at the outlet shows the form of the delta function) Taking into consideration the above definition of informational entropy (see Eq.(2)), the mixing capacity may be defined as follows { } { ( θ) ( θ) } ( θ) ( θ) M = H H H H (3) where M 1. In the case of these considerations, the mixing capacity M may be expressed as min 396 max

3 m M = E Δ E Δ e i= 1 ( θi) θ log2 ( θi) θ { log } (4) because the time interval, Δ θ, is very small value and may be omitted. In the context of mixing process, a M of zero would imply complete piston flow while non-zero M implies the mixing flow (M=1 for the perfect mixing flow). 3. EXPERIMENTAL DETAILS 3.1 Experimental apparatus The schematic diagram of experimental set-up is graphically presented in Fig.1. The experimental apparatus is consisted of a cylindrical plexiglas column (1) (inner diameter.1 m; outer diameter.11 m; length 1 m); inlet (2) and outlet (3) spouts; a spout for introduction of trace into the fluid entering the vessel (4); inlet (5a) and outlet (5b) conductivity electrodes; temperature sensors (6a and 6b); generator of a TRMF (7); a.c. transistorized inverter (8); multifunction computer meter CyberScan PCD 65 Eutech (9) and personal computer (1). Fig.1. The sketch of experimental set-up Investigations were realised with the temperature of the working liquid (tap water) variation between 2 and 25 o C. The volumetric flow rate of liquid was varied from.84 to.167 dm 3 s -1. The working volume (volume of TRMF generator) was equal to m 3. Mixing was provided by means of the modified three-phase stator of a induction squirrel cage electric motor, which parameters are compatible with Polish Standard PN-72/E-6. The magnetic induction of TRMF was controlled by mans of the alternating current frequency equal to the frequency of the TRMF. In these measurements, this stator was supplied with 5 Hz three-phase alternating current. The a.c. transistorized inverter was used to change the frequency of the TRMF. In this experimental procedure, this frequency was varied in the 397

4 range of f = 1-2 Hz. The more detailed description of this kind of magnetic field measurements can be found in the relevant literature [12]. Moreover, the obtained values in this article suggest that the averaged values of magnetic induction may be analytically described by the following relation: Bavg.5 ftrmf = e (5) where f TRMF is frequency of transverse rotating magnetic field (s -1 ). 3.2 Experimental procedure The mixing process under the action of a TRMF was evaluated from residence time distribution measurement using the stimulus-response technique. This technique is the simplest and most direct way of finding the residence time distribution (RTD) curves. In the case of this experimental work, the RTD is obtained by injecting a tracer instantaneously (a pulse input) at the inlet of flow system, and then measuring the tracer concentration, c(t), at the outlet as a function of time. The stimulus-response technique demands injecting the tracer within a very short time compared to the flow residence-time scale. This impulse can be = t τ described by the Dirac delta function (δ-dirac function), that is δ ( t τ ) such t = τ that δ( t τ ) = dt 1. RTD experiments were realized with a saturated solution of brine (25wt% NaCl). In order to describe the dynamic behavior of the magnetic mixer were performed for different sections of the installation. The continuous measurements of conductivity were collected for the liquid phase by introduction of the tracer solution under the effect of TRMF (conductivity electrode 5b) and without the effect of this kind of magnetic field (conductivity electrode 5a). Once the operating conditions are stable (or considered so by constant value of B avg and V ), the tracer is introduced into the inlet of the plexiglas column (spout for introduction of trace 4). Samples are then collected every 1 s until the tracer disappearance in the tested magnetic mixer. The mean time of duration, t d., is equal about 3 s. The typical examples of conductivity measurements under the effect of TRMF and without the effect of TRMF for t,1 are presented in Fig.2. d ( ) a) b) Fig.2. Typical examples of experimental results: a) B avg =.69 mt (f TRMF =1Hz); b) B avg = 8.89 mt (f TRMF =2Hz); 398

5 4. RESULTS AND DISCUSSION The RTD technique is served as a reasonable indicator of the type and extent of mixing. Response curves from the RTD experiment may be treated as the quantitative description of the mixing system. For the tracer input method the RTD (or exit age-distribution) function, E(t), may be related to the outlet tracer concentration, c(t), by the following function of time: () () () ( ) ( ) ( ) E t = c t c t dt or E t = c t c t Δt (6) i i i i= Taking into account the RTD function (Eq.(6)), the statistical parameter such as the mean residence time, t m, may be obtained by using the following relationships: t = t E t dt E t dt t = t E t dt or t = t E t Δt () () () ( ) (7) m m m i i i The E ( θ ) curve may be given as follows [13] E ( ) t E( t) θ = (8) Basing on the above definitions (see Eqs.(6-8)), the informational entropy (Eq.(2)) and the informational mixing capacity (see Eq.(4)) may be calculated. It should be noticed that the mixing capacity M may be described by means of the relative simple relationship as follows m * * M = f ( B, V ) M = f B avg B avg, V V max max (9) where B avg 3-1 = 8.89mT and V.167 dm s max = max The obtained values of the informational mixing capacity M are given in Fig.3. Fig.3. The influence of the volumetric flow rate and the magnetic field intensity on the informational mixing capacity Fig.3 demonstrates that the informational mixing capacity increases with the increase of the magnetic field intensity. It is found that as the volumetric flow rate increase, the parameter 399

6 M under the action of TRMF increases. Moreover, the calculated points presented in Fig.3 suggest the following expression 5. CONCLUSION * { (( ) ( )) (( ) ( )) } 2 * 2 M =.75exp.5 B V In the present report, the influence of the TRMF on the RTD was studied systematically. RTD of magnetic mixer was determined experimentally by impulse method. Based on these investigations, the informational mixing capacity was developed in order to describe the mixing process under the action of the TRMF. Form tests it was found that this parameter above evaluates the performance of mixing on the basis of not only the range of distribution but also the characteristic of the tailing parts of RTD. ACKNOWLEDGEMENT This work was supported by the Polish Ministry of Science and Higher Education from sources for science in the years under Inventus Plus project 6. REFERENCES [1] Adeosun J., Lawal A., 29. Numerical and experimental studies of mixing characteristics in a T-junction microchannel using residence time distribution, Chemical Engineering Science, 64, [2] Christensen D., Nijenhuis J., van Ommen J., Coppens M.-O., 28. Residence times in fluidized beds with secondary gas injection, Powder Technology, 18, [3] Gao Y., Vanarase A., Muzzio F., Ierapetritou M., 211. Characterizing continuous powder mixing using residence time distribution, Chemical Engineering Science, 66, [4] Guo Q., Liang Q., Ni J., Xu S., Yu G., Yu Z., 28. Markov chain model of residence time distribution in a new type entrained-flow gasifier, Chemical Engineering and Processing, 47, [5] Hornung Ch., Mackley M., 29. The measurements and characterisation of residence time distribution for laminar liquid flow in plastic microcapillary arrays, Chemical Engineering Science, 64, [6] Madhurabthakam Ch., Pan Q., Rempel,G., 29. Residence time distribution and liquid holdup in kenics KMX static mixer with hydrogenated nitrile butadiene rubber solution and hydrogen gas system, Chemical Engineering Science, 64, [7] Masiuk S., Rakoczy R., 27. Power consumption, mixing time, heat and mass transfer measurements for liquid vessel that are mixed using reciprocating multiplates agitator, Chemical Engineering and Processing, 46, [8] Rakoczy R., Masiuk S., Kordas M., Grądzik P., 211. The effects of power characteristics on the heat transfer process in various types of motionless mixing devices, Chemical Engineering and Processing. Process Intensification, 5, [9] Rakoczy R., Masiuk S., 211. Studies of a mixing process induced by a transverse rotating magnetic field, Chemical Engineering Science, 66, [1] Shannon C., A mathematical theory of communication, Bell System Technical Journal, 27, [11] Ogawa K., 27. Chemical Engineering. A new perspective, Elsevier, Amsterdam. [12] Rakoczy R., Masiuk S., 29. Experimental study of bubble size distribution in a liquid column exposed to a rotating magnetic field, Chemical Engineering and Processing. Process Intensification, 48, [13] Levenspiel O., Chemical Reactor Engineering, Wiley, New York. 4 (1)