Blocking filtration model of microfiltration membrane separation for non-ewtonian fluid solution Son-il Ri a*, Zhen-jia Zhang a, Li-na Chi a, Wei-li Zhou a a School of Environmental Science and Engineering,Shanghai Jiaotong University,Shanghai, 4, China ABSTRACT Microfiltration (MF) mechanism for non-ewtonian fluid solution is one of the key problems to be resolved in membrane filtration process. To depict the mathematic model of MF rate variation for non-ewtonian fluid solution, Darcy equation was extended to non-ewtonian fluid solution and the basic membrane filtration equation was established. MF rate variation for non-ewtonian fluid based on Darcy equation was derived by introducing the blocking filtration constant. Under the constant pressure, three kinds of MF flat membranes (sulfonated polyethersulfone MF, SPES,.45μm; cellulose acetate MF, CA,.μm and polyamid MF,PA,.μm) were used for the filtration test and the on-line diagnoses of the time dependences of MF rate and the filtrate mass have been carried out to characterize the above model. Experimental results show that the measurement data fit well with the established mathematic model. The MF rate variation rule suggested in this paper could be used for the study on fine filtration including MF together with the conventional fine filtration theory for non-ewtonian fluid based on Poiseuilles equation. Keywords: on-ewtonian fluid, microfiltration membrane, mechanism model, differential equation 1. ITRODUCTIO
Filtration is required for many non-ewtonian slurry fluids, e.g. macromolecular solution, synthetic fiber solution, paper pulp solution, dope, crude oil, grease, suspensions including some solid particles, emulsions, milk and so on in the organic macromolecular and the food industries. The main objects of the study on filtration of the non-ewtonian fluids are considered to be organic macromolecular solutions, which are the non-ewtonian fluids [1-13]. With the development of industry and the demands of both environment protection and energy, the filtration characteristics for non-ewtonian fluid have become more and more interested at present. Grasping the micro-filtration mechanism for non-ewtonian fluid is one of key problems to be resolved for membrane filtration process. The filtration characteristics for non-ewtonian fluids are different from those for ewtonian fluids, thus non-ewtonian characteristics of the filtration fluid must be taken into account. The flowing courses of non-ewtonian fluids are accompanied with many special phenomena different from the case of ewtonian fluids. The previous report showed the flowing characteristic of polymer solution in multi-porous medium and modified Poiseuille equation to capillary flowing equation for non-newtonian fluid [14]. 8( + 3) 1/ Δ p = 4 K T ( ) (1) + 3 π Where Δp is the pressure difference for the filtration (Pa);K T is the consistency index (Pa s - ); is the power-law exponent. While non-newtonian fluid went through globose particles, the relations between the traversing rate and pressure as followed [14]: Q = πr (3 + 1)( 3+ (1/ ) p 1/ K T ) Δp ( ) L 1/ () Where Q is flux (m 3 /s); is the number of capillaries; r p is the radius of a capillary (m); L is length of a capillary (m). Shirato et al. put forward the mechanism judging equation for cake filtration of non-newtonian fluid [15]: 1 1 K q ( ) ( ) c = (3) J J
Where K c is the constant of cake filtration (s /m 1+ ); q is filtration quality for unit filtration area (m); J is filtration rate (m/s); J is initial filtration rate (m/s). The macromolecular polymer solution which belongs to non-ewtonian fluid obeying a power law was adopted in this study in order to establish the Darcy equation for non-ewtonian fluid and to derive the corresponding rule for MF rate variation.. THEORY.1 Extended Darcy equation At present, microfiltration theory are thought to come from Poiseuille equation and Darcy law [16,17]. In the capillary model based on Poiseuille equation, a great deal of capillaries perpendicular to surfaces are considered to exist in deposited substances or filtrated media, and the filtration rate of suspensions can be expressed using Poiseuille equation: dv dt n rp Δp = (4) 8μL π 4 dv Where is the filtration rate (m 3 /s); V is the filtrated volume ( m 3 ); t is filtration time (s); μ dt is the viscosity of the filtration fluid (Pa s). In the terrene mechanics, the pervasion rate of a liquid penetrating through the sand layer can be calculated by using the experiential Darcy law. dv Δp = K (5) dt L Where K is a coefficient; L is the thickness of the sand layer (m). In the pervasion model based on Darcy law, the structure of the filtrating cake or the filtrated media is considered to have the structure of the couch of multi-porous particles similarly with the sand layer, but the only difference between them is that the filtrating cakes are much thinner and the particles are further finer. Referred to Poiseuille equation, the velocity of a liquid flowing through a capillary is inversely proportional to its viscosity, and modifying Darcy law, the equation for the filtration rate can be rewritten as: dv Fdt K Δp = p (6) μl Where F is the filtration area (m ); K p is the permeate coefficient (m ). 3
The pervasion coefficient K p is dependent on the dimension and shape of a solid particle and the structure of the filtrating cake, it being experimentally measured. Equation (4) and Equation(6) show that the filtration rate and pressure intensity are likely to be dependent on the viscosity of the filtrated liquid and the characteristics of the filtrating cake. Although the physical parameters in these equations are not the same, all of them can be found to imply the fundamental idea that the filtration rate is equal to the ratio of the impellent force to the resistance. As the membrane technique such as the membrane filtration, the bio-reaction of the membrane further proceeds, the research on the filtration theory of the micro-porous membrane based on Darcy law is sure to be of great significance. Using Darcy law to derive the filtration process theory further approximating to real filtration processes than introducing capillary law, which may be attributed to the fact that two factors, that is, the inner diameter and the length of a capillary presented in the capillary filtration model cannot be considered to be surely constant and they are usually different from position to position even for the same filtration medium while the hydraulic radius of the whole filtration area shown in Darcy model can be expressed as the pore rate and the specific surface area and the pore rate can more precisely depict the space structure of membrane filtration medium than capillary factor. Darcy law has been originated from the water permeate equation obtained when tap water going through the sand layer, thus it can be applied for the deduction of flow characteristics of low viscosity liquid, while flow equation of non-ewtonian fluid can not be directly applied to Darcy equation. In order for Darcy equation to be widely used in the industrial fluids, it must be extended to the more general type representing the flow characteristics of non-ewtonian fluid as well. Establishing the force equilibrium equation for the flowing liquor in a round type pipe, the relationship between force equilibrium and pressure is expressed as follows: πrlτ = πr Δp (7) w Where R is the radius of a pipe (m); L is the thickness of the filtration medium; τ w is the mean cut stress on the wall surface located at the radius R in a pipe (/m ). 4
ΔpR τ w = (8) L In the case of flowing in a multi-porous medium layer, replacing the hydraulic radius R H for a multi-pore medium layer with the hydraulic radius R in Equation (8), the relationship between the cut force and the pressure for a multi-porous medium layer can be written as follows: ΔpR τ H w = (9) L Equation (9) denotes the relationship between the flow pressure difference and mean cut stress, and the relationship between the power-law fluid flow and the ewtonian fluid one is as follows: Where γ& w 3 + 1 & γ w = ( ) & γ w, (1) 4 is the cut rate at the wall surface located at the radius R in a pipe (m -1 ) for the ewtonian fluid; γ& w, is the cut rate at the wall surface located at the radius R in a pipe for the non-ewtonian liquid. The relationship between the cut rate of power-law fluid flow in a pipe with an arbitrary figure of cross section and the flow rate of a fluid J can be expressed as follows [18]: J & γ w, = ξ (11) R H Where ξ is the constant relating to the configuration of the cross section of a pipe line, which is considered to be ξ= for a round pipe,ξ=3 for a quadrangle pipe and ξ=1 for a multi-porous medium. The hydraulic radius R H of a multi-pore medium layer can be expressed by using the permeate coefficient K p [18]: R H = K p (1) Incorporating Equation (9) and (1) into the fluid equation of a power- law fluid du dr τ W = K & Tγ W = KT ( ), and combining Equation(11), then the following equation can be written: 5
1 4 Δp 1+ J ( ) K p 3 + 1 KTL = (13) Equation (13) can be considered as Darcy equation extended to the power- law fluid in a multi-pore medium layer. Equation (13) not only contains the characteristic constants, K p and L of a membrane filtration medium, but also the flow constants of the fluid characteristics, K T and, and therefore, it can be applied to the power- law fluid in the multi-pore medium. If incorporating K T =μ and =1 into Equation (13), then it can be changed into Darcy equation for the ewtonian fluid. In the following section, it will be introduced into the deduction the filtration theory of micro-filtration membrane for the power- law fluid.. Part blocking filtration The blocking model for the incomplete membrane pores has been suggested from three mechanisms for the filtration of the micro-porous membrane based on the theory of Hermans-Bredee blocking filtration (Equation (14)) [13,16,17], and the model for the macroscopic change of the microfiltration volume in the deposited layer through which the particles of micro-sizes pass has been considered. However, some unknown courses of the membrane filtration have been still found to exist during the practical filtration by the micro-porous membrane, which do not belong to the three mechanisms for the membrane filtration mentioned above. d t dv dt n = Kn ( ) (14) dv Where K n is a constant. Because there are three mechanisms of the blocking during the practical filtration, the values of n are not the same. Generally, the early stage of the filtration is related mainly to the mechanical blocking, and the late stage to the filtration of the filtrating cake. In our study, hydraulic radius R H and initial porosity ε of filtration medium are regarded as the key factor deciding filtration mechanism in order to establish a more practical model according to MF process. Part blocking filtration means particles in feed are gradually rejected and adhered inside pore of porous medium, and thus regarded as the gradual decreased process for hydraulic radius R H and initial porosity in porous medium. 6
In case where every channels of membrane are connected, namely network is formed, particulates are rejected inside membrane due to physical and adsorption function. As shown in Figure 1, particulates in feed adhered on internal surface of membrane medium with increasing filtration time, so the decreasing of membrane porosity results in the decrease of hydraulic radius of fluid. During practical membrane filtration process, the particulates of feed are smaller than membrane s pore or during initial filtration, filtration carry out on part blocking mechanism. Fig. 1. Inner membrane rejection for partly blocking filtration.3 Deduction of part blocking filtration constant During the filtration process, the decrease with filtration time of porosity and hydraulic radius of filtration medium due to particulates adhered or rejected by porous medium result in the increase of the filtration resistance. Therefore, filtration velocities gradually decrease during constant pressure filtration, whilst pressure increase during constant flux process. When filtration mass q is obtained, the variation in membrane porosity is as followed: V xq x x ε f = = ε q = ε(1 - q) (15) V V ε V amely, εf ( q)= ε(1 Aq) (16) x x Where A = =, x is the rejected volume for unit permeate volume (m 3 /m 3 );V is the ε V V membrane volume for unit filtration area(m 3 /m ); V is the pores volume of unit filtration area(m 3 /m ); ε is the initial porosity and ε f is the changing porosity of membrane during filtration process. V Hydraulic radius of porous medium is defined as R H, =, where S S is the surface area of multi-pore medium for unit filtration area (m /m ). Point contacting is supposed among 7
particulates, namely the surface loss is neglected, then the variation of hydraulic radius responding to permeate volume q is as followed: V xq x x x RH, f = = RH, q = RH (1 q) = RH (1 q),, (17) S S R S V amely RH, f q) = RH, (1 Aq) (18) ( H, Where, RH, f is the varying hydraulic radius (m) of membrane along with filtration process. Because the permeate constant of filtration medium continuously vary during filtration procedure, a blocking filtration constant K, namely the constant that varies continuously during f filtration, is introduced in order to distinguish the initial permeate constant K p and the permeate constant of the filtration medium which remains variable. By combining Kozeny-Carman equation with Darcy equation, membrane permeate factor equations based on the hydraulic radius and pore ratio factors can be obtained: K p = R H, ε k (19) Where, k " is Kozeny constant Permeate factor K f varying with filtration can be obtained from Equation (16), (18) and (19): K R ( 1 Aq) ε( 1 Aq) RHεf H, H, 3 f = = = ( 1 Aq) () k K k 3 f ( q) K( p 1 Aq) R ε k = (1) Equation (1) means that with the feed permeating through the membrane, the filtration 3 volume increases, while the permeation coefficient decreases proportionally with( 1 - Aq )..4 Differential equation of partial blocking filtration of power-law fluid and flux variation equation Substituting the membrane permeation factor K f (Equation (1) ) depending on the filtration process for the permeation factor K p in Darcy Equation extended to the power-law fluid (Equation (13)), then the differential equation for the MF partial blocking filtration of the power- law fluid can be written as follows: 8
1 1+ dq 4 Δp 3 J = ( ) ( Kp(1 Aq) ) dt 3 + 1 KTL = () Where, t is the filtration time (s). Substituting K T =μ and =1 into Equation (), Eq () is changed into the differential equation for the partial blocking filtration of the non-ewtonian fluid: J = dq dt KpΔp = ( ) (1 Aq) μl 3 (3) Separating and coordinating the variables in the differential equation Equation (), then the dependence of the permeation quality on the filtration time can be given by: q 3+ 3 4 Δp 1+ t ( 1 Aq) dq = ( ) K p + dt (4) 3 1 K L + q T 1 3+ (1 (1 ) 3 = + AJ t ) (5) A 1 4 Δp 1+ J K ( ) p 3 + 1 KTL = (6) From Equation () and (5), the relationship between the filtration rate and time can be written as follows: 3+ 3 3+ 3+ J = J 1+ AJt) 1 ( (7) Differentiating the differential Equation () by the filtration quality q, the rule of the total dr resistance variation = f (R) is given by: dq d t 3+ 5 3 + 3 dt 3 3 AJ 3+ 3 + ( ) dq = (8) dq As can be seen in Equation (8), the total resistance variation of the filtration quality is 3 + 5 directly proportional to power of the total resistance in the course of the filtration by the 3+ 3 MF membrane partial blocking of the power-law fluid. Substituting K T =μ and =1 into Equation (5), Equation (5) is changed into the change rule of the total resistance for the partial blocking filtration of the ewtonian fluid: 9
dr dq 1 3 3AJ 4 3 d t dt = = ( ) (9) dq dq Comparing the general equation for the partial blocking filtration of the ewtonian fluid Equation (9) with the general equation for the Hermans-Bredee blocking d t filtration dv dt n = Kn ( ), it can be seen that the behaviors of the resistance change are quite dv similar with each other but the orders of the equation are not the same, that is, the mechanism index for the partial blocking filtration is equal to 4/3. Based on Equation (), the equation distinguishing the MF membrane partial blocking mechanisms of the power-law fluid under the constant pressure of the MF membrane is given by: J 3(1+ ) 3(1+ ) 3(1+ ) = J J Aq (3) The relationship of J 3(1+ ) versus q in Equation (3) has the linear characteristic. If the plot of the MF rate of the power-law fluid versus the permeation quality measured under the condition of the constant pressure shows the linear behavior, then this kind of filtration belongs to the partial blocking micro-filtration, and as mentioned above, the equations containing the several operational factors can be employed. 3. EXPERIMETAL 3.1 Apparatus and materials The experimental apparatus for testing the constant-pressure filtration of the MF equation of the power- law fluid is shown in Figure. A pilot-scale cross-flow filtration unit equipped with a flat suction type membrane module with the effective diameter of 5cm and the active area of 19.6 cm was used in our experiment. Three kinds of MF flat membranes used were followed: sulfonated polyethersulfone MF (SPES,.45μm), cellulose acetate MF (CA,.μm) and polyamid MF (PA,.μm).Carboxymethylcellulose (CMC, China medicine chemical reagent limited corporation) solutions with different concentrations were employed in this experiment and the rheological constants of the CMC solutions were determined using DJ-1 revolving viscosity meter. 1
Fig.. Constant TMP filtration system 1-Feed tank; -Pump; 3-Membrane module; 4-Agitator ;5-Pressure gauge ; 6-Magnetic stirring mill ; 7-Valve; 8-Vaccum gauge; 9-Filter tank; 1-Buffer slot; 11-Vaccumpump; 1-Electric balance;13-pc. 3. Results and discussion Darcy equation extended to the power-law fluid was proved through MF membrane permeate experiment under the constant pressure. First, the permeation rate of CMC solution into a membrane operated for a certain time Δt, dq=δv/f was measured under a defined pressure, and then the filtration rate, J=dq/dt was calculated. In Figure.3 the values of the permeation rate determined through our experiments are compared with those calculated according to the mathematical model Equation (13). Darcy law for the ewtonian fluid (Equation(6)) shows that the velocity of the fluid flowing through the filtrated medium is directly proportional to its pressure intensity, while the penetrating rate of the power-law fluid derived in this paper is proportional in the power of 1/ to its pressure. Generally, the flowing constant of the power-law fluid is ranged from to 1, so the index 1/ in the Darcy equation expanded to the power-law fluid is greater than 1, that is, the plot of the penetration rate versus pressure intensity has the nonlinear characteristic. It can be clearly seen from Figure 3 that the permeation rate is dependent on the pressure, and that the permeation rate is directly proportional to n power (n>) of the transmembrane pressure. 11
Fig. 3. Membrane permeate flux vs. pressure on the MF of.5% CMC solution using the CA membrane (the membrane thickness is.µm) The data determined under the different experiment conditions are consistent with the mathematical model and the results of the error analysis are shown as Table 1. Table 1. Error analysis for the filtration rate Membrane o. J Exp (L m - h -1 ) J Cal (L m - h -1 ) Average relative error ( %) CA1 5.99 57.34 8. CA 49.93 45.68 8.5 PA1.38.56 1.7 PA 61.15 58.14 4.9 SPES1 91.7 89.55.4 SPES 87.88 83.7 5.5 It can be seen from Table 1 that the maximum discrepancy between the values of the membrane rate obtained from the experiment and the model is smaller than 1.7% when the membrane materials are SPES, CA and PA, respectively. The MF rate model Equation(7) was proved through MF membrane filtration test under the constant transmembrane pressure. The comparison of the curve of the flow rate variation measured on line with that calculated using the mathematical model Equation (7) is presented in Figure 4. 1
Fig. 4. Membrane permeate flux vs. time on the MF of.5% CMC solution using the SPES membrane (the membrane thickness is.45µm) As shown in Figure 4, the filtration flow rate gradually decreases as the filtration proceeds forward, which may be interpreted by the fact that the longer the filtration time, the more the residual microscale particles in the internal pores of the membrane, resulting in the increase of the total resistance, which may cause the decrease of the filtration flow rate. The calculated values on the basis of the mathematical model tend to be fitted with the experiment values. The judging curve shown as Figure 4 can be obtained using the equation (3) and the data from Figure 5. Fig. 5. The judging curve of filtration model on the MF of.5% CMC solution using the SPES membrane (the membrane thickness is.45µm) 13
According to Figure 5, the judging curve of the partial blocking filtration shows the linear property, which indicates that the MF process accords with the partial blocking filtration mechanism. Figure 6 presents the judging curves for both the partial blocking filtration resulting from the treatment of the measuring data for the microfiltration of the tap water and the conventional cake filtration. Fig. 6. The judging curve of filtration model on the MF of tap water using the PA membrane (the membrane thickness is.µm) As can be seen in Figure 6, the judging curve for the cake filtration has the nonlinear characteristic, whereas that for the partial blocking filtration shows the linear property, which illustrates that this microfiltration satisfies the filtration mechanism of the partial blocking filtration. Substituting K T =μ and =1 into Equation (3),then the partial blocking MF judging equation for newtonian fluid as followed : 1 1 1 3 3 3 J = J J Aq (31) The cake filtration juding coordinate is given as followed [16,17]: dt/dv V This suggests that the membrane filtration has another blocking filtration mechanism except the three kinds of Hermans-Bredee filtration mechanism (the mechanism index n =, 3/, 1), that is, another mechanism to the power of n=4/3. 14
4. COCLUSIO Equation (13) could be used for the study on the MF of the power-law fluid because Darcy equation extended to the non-ewtonian fluid contains not only the membrane filtration medium constants K P and L but also the flowing constants of the flowing characteristics K T and. The MF rate of the macromolecular solution is comparatively low, which may be attributed to the fact that it has high values of both surface tension and viscosity; the network structure is easy to be formed in it; the pore size of the MF membrane much smaller than that of the normal filtration material. Under the same transmembrane pressure, the filtration rate of non-ewtonian fluid is much lower than that of ewtonian fluid with low viscosity, and even the dead-end filtration can not be performed. Membrane filtration can be realized only by using the cross-flow filtration method where the cut force is relatively stronger. The membrane filtration has another blocking filtration mechanism except the three kinds of Hermans-Bredee filtration mechanism (the mechanism index n =,3/,1), that is, another mechanism to the power of n=4/3. List of symbols A constant(m -1 ) F filtration area (m ) J J k filtration rate(m/s) initial filtration flow rate(m/s) constant K c constant of cake filtration (s /m 1+ ) K p permeate coefficient(m ) K k " K f K n constant Kozeny constant blocking filtration constant, namely membrane permeation factor with filtration going on(m ) constant K T consistency index (Pa s - ) L length of a capillary (m) or thickness of filtration medium(m)or 15
thickness of the sand layer (m) n Q q the power-law exponent. the number of capillaries number of capillaries or exponent denoting the blocking mechanism flux (m 3 /s) filtration quality for unit filtration area(m) Δp pressure difference (Pa ) R H R H, R H,.f r p hydraulic radius of mul-pore medium layer(m) hydraulic radius of membrane before filtration Hydraulic radius of membrane varying with filtration procedure(m) radius of a capillary (m) S surface area of mulpore medium for unit filtration area(m /m ). t filtration time (s ) V V filtrated volume ( m 3 ) the pores volume for unit filtration area x rejected volume for unit permeate volume (m 3 /m ) ε ε ε f τ w γ& w γ& w, ξ μ pore ratio of membrane the initial pore ratio of membrane the varying pore ratio with filtration procedure the mean cut stress on the wall surface located at the radius R in a pipe (/m ). cut rate at wall surface mean cut stress on the surface located at the radius R in a pipe for ewtonian fluid (m -1 ) cut rate on wall surface located at the radius R in a pipe for non-ewtonian fluid(m -1 ) constant relating to the configuration of pipe cross section permeate viscosity(pa s) REFERECES [1] S. Agashichev, Modelling non-ewtonian behaviour of gel layers at membrane surfaces 16
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