ARTICLE pubs.acs.org/IECR
Numerical Assessment of Hydrodynamic Cavitation Reactors Using Organic Solvents Peeush Kumar and V. S. Moholkar* Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati, 781 039 Assam, India
bS Supporting Information ABSTRACT: This paper attempts to give a physical insight into the behavior of cavitation bubbles in a hydrodynamic cavitation reactor, with organic liquids employed as the bulk medium. The diffusion limited model of cavitation bubble dynamics has been coupled to hydrodynamics of flow through an orifice. Two kinds of bubbles, namely, air and argon, and two organic liquids, namely, n-hexane and toluene, have been chosen as the model system. The effect of four parameters on the dynamics of cavitation bubbles, namely, cavitation number, orifice to pipe diameter ratio, downstream recovery pressure, and initial bubble radius, has been investigated. Simulations show interesting results that dynamics of cavitation bubbles in organic media is relatively insensitive to operating parameters. These results have been attributed to physical properties of the organic media. Relatively, the intensity of the collapse of bubbles in toluene is higher than in hexane; however, the temperature peaks attained at the transient collapse of bubbles in both liquids are too low to allow formation of radical species in appreciable quantities. Hydrodynamic cavitation reactors with organic media are unsuitable for sonochemical reactions, although some physical processes or chemical reactions with external addition of source of radicals/ions may well be carried out.
1. INTRODUCTION Several new technologies have emerged in the past two decades that offer safer, easier, and more efficient means of energy introduction in the system. One among these is the hydrodynamic cavitation technology in which the cavitation phenomenon, that is, radial motion of a tiny gas/vapor bubble (comprising nucleation, growth, and transient collapse), is used for inducing the physical/chemical transformation. Hydrodynamic cavitation has been extensively used for intensification of various physical and chemical processes such as degradation of dyes,1-3 hydrolysis of oils,4,5 degradation of recalcitrant pollutants and wastewater treatment,6-8 enhancement of anaerobic digestion,9 nanosynthesis,10-12 potable water disinfection,13,14 and microbial cell disruption.15,16 The literature on application of hydrodynamic cavitation for process intensification is quite vast, and references cited above are only a few representative studies. For greater details on literature on hydrodynamic cavitation, we refer to the reader to state-of-the-art reviews.17-19 Several authors have also addressed issues of modeling and optimization of hydrodynamic cavitation.20-24 The liquid medium employed in the physical/chemical processes with hydrodynamic cavitation is mostly water. Maximum cavitation intensity is achieved in water due to its low vapor pressure, high surface tension, low viscosity, and high density. However, in many physical processes such as extraction of oil from seeds or chemical processes such as nanosynthesis with hydrophobic precursors, water may not be the suitable medium. In such cases, organic solvents such as hexane, toluene, decane, dodecane, and so forth have been used. In order to assess the suitability of these solvents toward given physical/chemical process, one needs to assess the behavior of cavitation bubble in these liquids. In this paper, we have addressed this issue with numerical simulations. The sonochemical effect essentially r 2011 American Chemical Society
involves generation of high reactive radical species through dissociation of the vapor molecules entrapped in a bubble at the moment of transient collapse. For organic solvents, formation of radical species such as •CH3 and •H (for bubbles of inert or monatomic gases) in addition to OH, O, and HO2 (for oxygen or air bubbles) is expected. Using the diffusion limited model for the cavitation bubble dynamics, we have tried to assess the behavior of two kinds of bubbles, namely, air and argon in two representative organic solvents, namely, hexane and toluene.
2. MATHEMATICAL MODEL The most general treatment of the problem of radical generation through cavitation bubbles was given by Storey and Szeri,25 who showed that the vapor transport in the bubble was a diffusion limited process. During the final moments of bubble collapse, the radial motion of the bubble becomes extremely fast. Not all of the solvent vapor that has evaporated into the bubble can escape during collapse by condensation at the gas-liquid interface, as the time scale of oscillation of bubble becomes smaller than the time scale of diffusion of vapor and time scale of condensation at gas-liquid interface. The entrapped vapor in the bubble is subjected to extremes of temperature and pressure generated in the bubble during transient collapse, at which the vapor molecules undergo thermal dissociation. In view of the results of Storey and Szeri,25 Toegel et al.26 developed a diffusion limited model based on boundary layer approximation, which has been used in this paper. This model has been described extensively in our earlier paper.22 Hence, only the key features of the Received: December 15, 2010 Accepted: February 17, 2011 Revised: February 12, 2011 Published: March 10, 2011 4769
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Table 1. Summarization of the Diffusion Limited Bubble Dynamics Formulationa variable
equation
initial values
Equation 1b 1a. radius of the bubble (R)
dR=dt d2 R 3 dR=dt dR 2 1R 2 þ 1c dt 2 3c dt 1 dR=dt R dPi dR=dt 2σ - 4ν ¼ 1þ ðPi - Pt Þ þ FL c FL c dt R FL R
1b. Bubble wall velocity (dR/dt)
at t = 0, R = Ro and dR/dt = 0
other parameters internal pressure in the bubble: Pi ¼
Ntot ðtÞkT ½4πðR3 ðtÞ - h3 Þ=3
pressure in bulk liquid medium: 1 1 Pt ¼ FL Uo 2 þ Pvc - FL ðUtnew Þ2 2 2 new where Ut ¼ Ut þ u0 sinð2πfT tÞ Equation 2 2. number of solvent (hexane or toluene) molecules in the bubble (NS)
dNS DCS ¼ 4πR 2 DS dt Dr
4πR 2 DS r¼R
CSR - CS ldiff
at t = 0, NS = 0
other parameters instantaneous diffusive penetration depth: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ldiff ¼ minð RDS =jdR=dtj, R=πÞ Equation 3 3. heat transfer through bubble (Q)
at t = 0, Q = 0
dQ DT ¼ 4πR 2 λ dt Dr
r¼R
T0 - T 4πR 2 λ lth
other parameters thermal diffusion length: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lth ¼ minð Rk=jdR=dtj, R=πÞ Equation 4 4. temperature of the bubble (T)
CV , mix dT=dt ¼ dQ =dt - Pi dV =dt þ ðhS - US Þ dNS =dt
at t = 0, T = To
other parameters mixture heat capacity: CV , mix ¼ CV , i Ni
∑
molecular properties of solvent:
hS ¼ ð1 þ fi =2ÞkTo
enthalpy
3
US ¼ NS kT 3 þ internal energy
heat capacity (Ar)
!
∑ θi =T i ¼ 1 expðθ =TÞ - 1 i
CV ¼ 3kNAr =2
heat capacity of other species ði ¼ O2 =N2 =C6 H12 =C6 H5 - CH3 Þ : CV , i ¼ Ni kðfi =2 þ
∑ððθi =TÞ2 expðθi =TÞ=ðexpðθi =TÞ - 1Þ2 ÞÞ
Notation: R radius of the bubble; dR/dt, bubble wall velocity; c,velocity of sound in bulk liquid medium; FL, density of the liquid; ν, kinematic viscosity of liquid; σ, surface tension of liquid; λ, thermal conductivity of bubble contents; κ, thermal diffusivity of bubble contents; θ, characteristic vibrational temperature(s) of the species; NS, number of solvent molecules in the bubble; t, time, DS, diffusion coefficient of solvent vapor; CS, concentration of solvent molecules in the bubble; CwR, concentration of solvent molecules at the bubble wall or gas-liquid interface; Q, heat conducted across bubble wall; T, temperature of the bubble contents; To, ambient (or bulk liquid medium) temperature; k, Boltzmann constant; NAr, number of Ar molecules in the bubble; Ni, number of molecules of any other species (oxygen/nitrogen/n-hexane/toluene) in the bubble; fi, translational and rotational degrees of freedom; CV,i, heat capacity at constant volume; Ntot, total number of molecules (gas þ vapor) in the bubble; h, van der Waal’s hard core radius; Po, ambient (bulk) pressure in liquid; Uo, mean velocity of the flow at orifice; Pvc, mean pressure in flow at vena contracta; Ut, mean velocity in the flow at any location for time t > 0; u0 , mean turbulent fluctuation velocity in the cavitating flow. b Equation 1 can be split into two simultaneous equations by substituting dR/dt = s. The instantaneous penetration depths for mass and thermal diffusion have been determined through dimensional analysis. For greater details refer to ref 22. a
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Table 2. Thermodynamic Properties of Gas and Solvent Speciesa Lennard-Jones force constants degrees of freedom (translational þ rotational) (fi)
σ (10-10 m)
N2
5
3.68
92
O2
6
2.65
380
Ar
3
3.42
124
C6H12
6
5.75
386
1036, 1932, 1987, 2104, 4193
C6H5-CH3
6
5.82
451
1004, 1056, 2112, 2153, 2326, 4224, 4375
species
a
ε/k (K)
characteristic vibrational temperatures θ (K) 3350 2295, 5255, 5400
Sources: refs 28-30, 35, and 36.
model have been described here. This model comprises of a set of four ODEs, which are summarized in Tables 1 and 2 along with thermodynamic data. The physical properties of the two solvents are given in the Supporting Information. The main components of the diffusion limited model are (1) Keller-Miksis equation for the radial motion of the bubble;27 (2) equation for the diffusive flux of water vapor; (3) equation for heat conduction through the bubble wall; and (4) overall energy balance treating the cavitation bubble as an open system. The transport parameters for the heat and mass transfer (thermal conductivity and diffusion coefficient) are determined using Chapman-Enskog theory using Lennard-Jones 12-6 potential at the bulk temperature of the liquid medium.28-30 The vapor pressures of the two solvents have been determined using Antoine type of correlations. The model ignores diffusion of gases across the bubble wall during radial motion as the time scale for gas diffusion is much higher than that for the radial motion of bubble. 2.1. Determination of the Pressure Recovery Profile in Orifice Flow. Hydrodynamic cavitation can be generated simply by throttling discharge of a pump through a constriction such as an orifice. At the downstream of the orifice, the velocity of the flow increases at the expense of reduction in pressure. At this moment, the dissolved gas in the medium can get released, generating the cavitation bubbles. These bubbles move with the flow. As the pressure in the flow recovers, these bubbles undergo volume oscillations giving rise to high temperature and pressures inside the bubble. For the solution of the bubble dynamics model, we need an expression for the instantaneous pressure (Pt) in the bulk medium. For this, we have used a 4-step algorithm based on Kolmogoroff’s hypothesis that the rate at which large eddies supply energy to the smaller eddies is proportional to reciprocal of the time scale of the larger eddies, described as follows: (1) In view of the experimental results of Yan et al.,31 we use a simple expression to calculate the mean pressure in the flow at any time t: Pt ¼ Pvc þ
ðP2 - Pvc Þ t τ
ð5Þ
(2) Applying the Bernoulli equation between vena contracta and at any point downstream of the orifice, the instantaneous mean velocity (Ut) corresponding to Pt is Ut ¼ ½ðPori þ FL Uo 2 =2 - Pt Þ=ðFL =2Þ1=2
ð6Þ
(3) Approximating the turbulent velocity fluctuations as a sinusoidal function superimposed over mean velocity, we
Table 3. Simulation Parametersa parameters set number P2 (atm) Ro (μm)
β
Ci
τ (s)
u0 (m/s) fT (kHz)
Toluene set 1
1
200
0.6 1
0.0396
2.081
0.64
set 2 set 3
1 1
100 200
0.6 1 0.5 1
0.0396 0.0431
2.081 2.148
0.64 0.705
set 4
1
200
0.5 1.1 0.0452
2.114
0.694
set 5
1.2
200
0.5 1
0.0393
2.355
0.772
set 1
1
200
0.6 1
0.0371
2.332
0.717
set 2
1
100
0.6 1
0.0371
2.332
0.717
set 3
1
200
0.5 1
0.0404
2.406
0.79
set 4 set 5
1 1.2
200 200
0.5 1.1 0.0424 0.5 1 0.0363
2.369 2.65
0.78 0.869
Hexane
a
A pipe size of 2 in. is taken for all simulations. Liquid temperature is assumed to be 20 °C.
as get new value of mean velocity Unew t Utnew ¼ Ut þ u0 sinð2πfT tÞ
ð7Þ
Calculation of turbulent fluctuation velocity, u0 , and its steady state frequency, fT, is explained in greater detail in the Supporting Information provided with this paper. is (4) The bulk pressure in the flow corresponding to Unew t recalculated as 1 1 Pt ¼ FL Uo 2 þ Pvc - FL ðUtnew Þ2 2 2
ð8Þ
2.2. Numerical Solution. The set of ODEs given in Table 1 can be solved using Runge-Kutta fourth order-fifth order with adaptive step size control.32 As the cavitation bubble is in a highly turbulent flow, the instabilities are maximum and a bubble is likely to fragment or “collapse” at the first compression during radial motion. Therefore, we have considered the first compression of the bubble as a condition for bubble collapse.25 Various parameters required for the simulation of the radial motion of the cavitation bubble are (1) recovery pressure downstream from the orifice (P2), (2) orifice to pipe diameter ratio (β), (3) cavitation number (Ci), and (4) initial (or equilibrium) bubble radius (Ro). We have devised five sets of parameters for simulations, described in Table 3by permutation-combination of various values of P2, β, Ci, and Ro. 4771
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Figure 1. Simulation results for the air bubble in hexane for parameter set 3 (P2 = 1 atm, Ro = 200 μm, β = 0.5, Ci = 1). Time variation of (A) normalized bubble radius (R/Ro); (B) number of hexane molecules in the bubble; (C) temperature in the bubble; and (D) pressure in the bubble.
The composition of the bubble contents at transient collapse has been calculated using software FACTSAGE that uses the SOLGASMIX algorithm for minimization of Gibbs free energy proposed by Eriksson.33 In this approach, we assume that the bubble contents, that is, the species generated from thermal dissociation of solvent vapor, are in thermodynamic equilibrium.22,34
3. RESULTS AND DISCUSSION Representative simulation results have been depicted in Figures 1 and 2. A representative summary of entire simulations for hexane as the liquid medium is given in Table 4. The entire set of results from the simulations have been provided in the Supporting Information. Prior to discussion of the results of the simulations, we would like to point out some peculiar features of the bulk pressure profile in the flow, which essentially drives the radial motion of
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Figure 2. Simulation results for the argon bubble in toluene for parameter set 5 (P2 = 1.2 atm, Ro = 200 μm, β = 0.5, Ci = 1). Time variation of (A) normalized bubble radius (R/Ro); (B) number of toluene molecules in the bubble; (C) temperature in the bubble; and (D) pressure in the bubble.
the cavitation bubble. The bulk pressure in the flow has two components: (1) mean pressure gradient, approximated as ≈[1/(2L)]FL(1 - β2)Uo2, and (2) turbulent pressure gradient, approximated as ≈[1/(2L)]Fu0 2. The mean pressure gradient has a linearly increasing trend, while the turbulent components are oscillatory. The mean pressure gradient would always tend to compress a cavitation bubble due to its continuous rising profile. The turbulent pressure gradient, however, tends to expand the bubble during its negative oscillation. The expansion of the bubble is thus proportional to the intensity of the turbulent pressure gradient, which in turn varies inversely with the orifice to pipe diameter ratio. The greater 0 the permanent pressure head loss, the higher the magnitude of uh. The mean pressure gradient depends on two factors, namely, (1) cavitation number (Ci), which governs the minimum pressure reached at vena contracta, and (2) final recovery pressure (P2) that depends on the upstream pressure Pups and the orifice to pipe diameter ratio β. 4772
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Table 4. Simulation Results for Hexane as the Liquid Medium (A) Results for Air Bubble parameter set for simulation species
set 1
set 2
set 3
set 4
set 5
T = 630 K
T = 493 K
T = 839 K
T = 528 K
T = 972 K
P = 7.6 atm NS = 5.27 1014
P = 2.2 atm NS = 1.91 1014
P = 36.7 atm NS = 1.01 1015
P = 3.2 atm NS = 3.92 1014
P = 93.3 atm NS =1.18 1015
C2H6 C2H4
1.30 10-5
2.54 10-6
5.85 10-5
3.58 10-6
9.48 10-5 2.84 10-6
CH4
8.22 10-1
9.40 10-1
7.78 10-1
8.06 10-1
7.13 10-1
3.23 10
-4
1.44 10
-6
1.03 10-3
1.16 10
-4
4.48 10
-4
1.06 10-4
1.68 10
-1
1.63 10
-2
2.39 10-1
1.82 10
-2
6.09 10
-2
1.49 10-2
3.44 10
-2
1.16 10
-1
2.97 10-2
9.72 10
-4
3.83 10
-4
1.35 10-3
Conditions at First Compression of the Bubble
CO CO2 H2 H2O N2 NH3
2.06 10
-5
3.64 10
-4
4.92 10
-2
4.39 10
-2
8.38 10
-2
6.15 10
-4
-5
2.39 10
-2
1.10 10
-2
1.69 10
-2
3.22 10
-4
1.74 10
(B) Results for Argon Bubble parameter set for simulation species
set 1
set 2
set 3
set 4
set 5
T = 712 K P = 8.0 atm
T = 576 K P = 4.8 atm
T = 847 K P = 23.4 atm
T = 664 K P = 5.3 atm
T = 971 K P = 50.5 atm
NS = 4.25 1014
NS = 1.84 1014
NS = 6.71 1014
NS = 3.20 1014
NS = 7.38 1014
1.81 10-5
8.46 10-6
4.24 10-5
1.16 10-5
5.87 10-5
-1
9.27 10
-1
-1
-1
6.36 10-1
3.10 10
-2
-2
3.06 10-1
Conditions at First Compression of the Bubble
C2H6 CH4 H2
7.59 10
-1
1.17 10
7.23 10
-1
2.14 10
8.45 10
2.51 10-6
C2H4
The radial motion of a cavitation bubble will be a function of the resultant of the two components of the bulk pressure in the flow. 3.1. Recovery Pressure (P2). Comparison of results of sets 3 and 5 gives us an idea of the effect of P2 on bubble dynamics for hexane or toluene, respectively. With rising P2, bubble collapse becomes more intense, with higher temperature and pressure peaks attained. Moreover, we find that for hexane the temperature peaks reached for both air and argon bubbles are practically the same for sets 3 and 5. This is a surprising result as one would expect higher temperature for argon as it is a monatomic gas. This anomaly is attributed to large evaporation of hexane into the bubble, as a result of which the bubble gets supersaturated with hexane during collapse. The population of hexane molecules in the bubble is higher than in the gas molecules. In this situation, the peak temperature reached in the bubble is governed by heat capacity of vapor molecules and not gas molecules. Thus, both air and argon bubbles give same peak temperature. For toluene as well we see a similar result, and explanation for this can be given along similar lines as for hexane. Moreover, we find that temperature and pressure peaks for any set (3 or 5) are higher for toluene than hexane. We attribute this result to higher vapor pressure and higher vapor entrapment of hexane in a cavitation bubble that lowers the temperature peak.
7.47 10
3.2. Orifice to Pipe Diameter Ratio (β). Comparison of the results of sets 1 and 3 reveals the effect of β on bubble dynamics. For the air bubble in hexane, the temperature peak at collapse shows ∼33% rise with β reducing from 0.6 to 0.5, while for an argon bubble this rise is marginal (∼15%). For toluene, radial dynamics of both air and the argon bubble shows almost insensitivity toward β. We attribute these results to rather low surface tension and low density of toluene and hexane. As β reduces, the permanent pressure head loss in the flow increases, with concurrent increase in turbulent fluctuating velocity. However, as the density of two liquids is small, the magnitude of the turbulent fluctuations is small, and hence, the overall pressure profile does not change significantly so as to affect the dynamics of the cavitation bubble. 3.3. Initial Bubble Radius (Ro). Comparative evaluation of sets 1 and 2 gives the effect of Ro. The smaller bubble (100 μm) in set 1 is found to undergo less intense collapse in hexane. This result is consistent for air and argon bubbles. On the other hand, for toluene, both air and argon bubbles reach same condition of temperature and pressure at collapse, irrespective of Ro. We attribute this result to low surface tension of the two liquids. Initial bubble size essentially manifests its effect over radial motion of cavitation bubble through Laplace pressure (2σ/Ro). 4773
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Industrial & Engineering Chemistry Research The initial pressure inside the bubble at inception is Pνc þ (2σ/ Ro), where Pvc is pressure at vena contracta. Therefore, smaller bubbles have higher Laplace pressure, which results in greater expansion and more intense collapse the cavitation bubble. However, if the surface tension is small, this effect is not pronounced, and expansion of the bubble is almost similar for 100 and 200 μm initial size. Due to higher inertia, compression of the larger bubble (and hence the temperature peak attained) in the collapse phase is higher. In the case of toluene, the intensity of turbulent pressure fluctuations is relatively higher and it dominates the bubble motion. As this factor is the same for both sets 1 and 2, the dynamics of both 100 and 200 μm bubbles are similar, and hence, the overall cavitation effect is insensitive to Ro. 3.4. Cavitation Number (Ci). The higher the Ci, the higher the pressure at vena contracta. The bubbles generated in the flow are in mechanical equilibrium with the flow and pressure inside them is Pνc þ (2σ/Ro). The pressure inside the bubble varies directly with Ci. Higher pressure inside the bubble cushions the collapse of the bubble. Secondly, rise in the mean pressure in the flow in the immediate vicinity of the vena contracta reduces the effect of negative oscillation of the turbulent pressure gradient, causing lesser expansion of the bubble and reduction in the intensity of collapse. Reduction in the cavitation effect with increasing Ci is more pronounced for hexane. As seen from comparison of the results of sets 3 and 4, the peak temperature at bubble collapse was reduced by one-third for both air and argon bubbles with Ci changing from 1 to 1.1. Quite interestingly, cavitation intensity in toluene is relatively insensitive to Ci, as the collapse temperatures reduce only marginally (
