STUDY THE AXAIL DISPERSION MODEL IN ION EXCHANGE COLUMN Al-Khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal, Vol. 5, No. 4, PP 1-12 (2009) Study the Axail Dispersion Model in Ion Exchange Column Iman Hahider Mahdi Department of Chemical Engineering/ Collage of Engineering/ Al- Nahreen University (Recived 31 January 2008; accepted 3 March 2009) Abstract A mathematical model is developed which predicates the performance of cylindrical ion exchange bed involving comparing of axial dispersion model for cation exchange column with different assumption, this model permits the performance to predicate the residence time within the bed with the variance, axial dispersion and Pecklet No. to indicated deviation from plug flow model. Two type of systems are chosen for positive ions first with divalent ions (Ca +2 ) to exchange with resin of Na +1 form used as application in water softener units and second with monovalent ions (Na +1 ) to exchange with resin of H +1 form used as application in deionize water units with two parameters effected in the operation of the two systems different flow rate and different feed concentration by their effect on axial dispersion in each one where the system are compare from the performance which describe in term of breakthrough time other parameters (the time equivalent to usable capacity "tu" , time equivalent to equilibrium capacity "tt" and length of unused bed "HUNB") are calculated in the operation condition that with axial dispersion are very important in predicated the best condition to operation the cation exchange column as a softener and deionize units. Keywords: Ion exchange, axil dispersion, Ca +2 ion, Na +1 ion, Pe No., breakthrough time, height of unused bed (HUNB), residence time, variance 1. Introduction Over the past decades the ion exchange processes has found increased applications in a range of diverse fields. Most of these applications use cylindrical column ion exchange beds in which the electrolyte solution to be treated is injected in one end and the effluent is collected at the other such as water softner when treated a solution of divalent ion (Ca +2 ,Mg +2 ) with using cation exchange column with Na-form to exchange divalent , other application is the dionized water with using resin in H-form to treated all the divalent and monovalent positive ion where most of these application are conventional one- dimensional. Linear ion exchange column has been well investigated, and many models have been developed to predict ion exchange performance in vertical flow, however most models assume no axial dispersion model. Important to investigated a model that account axial dispersion since the axial dispersion coefficient lumped together all mechanisms leading to axial mixing in packed bed that its account not only for molecular diffusion and conversion mixing but also for non uniformities in the fluid velocity across the packed bed . Several workers studied dispersion with porous media, the first who deal with axial dispersion O. Levenspiel 1958 when he study the commercial application of step experiment and he design chart to ease the calculation of axial dispersion, H. Dunck 1966 study he effect of axial dispersion on the elution band of nonlinear equilibrium chromatography and the effect change in axial dispersion coffin with elution velocity , O .Levenspiel 1972 found the c-curve in closed vassal for various extend of back mixing as predicated the dispersion model , C.Y. Wen, T. Tfan l.t. 1975 point that the axial dispersion provides satisfactory description only of mixing model that not deviated significantly from plug flow, R. Shinnar 1987emphasized that the application of axial dispersion model for Pe No. value for less than 10 is not advised, Iman Hahider Mahdi Al-Khwarizmi Engineering Journal, Vol. 5, No. 4, PP 1-12 (2009) 2 C. T. Tan1987 the value of dispersion coefficient are either assumed or calculated from molecular diffusivity and radial pore velocity prior to incorporation into the model the radial dispersion R. Y. A. Ofoli, 1991 reported that axial dispersion model is suitable for analysis of reactor up to 30% conversion , Y. Tsaur and D. Shallcross 1997 developed a model to simulate ion exchange performances with fixed annular bed which take into account the radial dispersion in effect in both solution and exchange phases, however the model assumes instantaneous equilibrium between ions in the solution and exchange phases existed throughout the exchanger, M. A. Barros and Aszpla 2002 analyze the Cr +3 ion to exchange in NaA zeolite in presence of K + ,Cu +3 ion in binary and ternary cation exchange with evaluate the dimensionless variance of Breakthrough curve which useful to estimate the axial dispersion of the bed, M. Doung and D. Shallcross 2002 developed theoretical model of fixed bed ion exchange and obtained the model for pure dispersion without ion exchange performance, I. M. Abu-Reesh and B.l F. Abu- Sharkh 2003 make a comparison of axial dispersion and tank-in-series models for simulating the performance reactors they found that tank-in-series models presents computational advantages compared to dispersion model , Doung M. and Shallcross D.2005 made a series of experiment in wedge-shape cell ion exchange for both Na-and H- form bed with different flow rate lead to indicated dispersion coefficient function of pore velocity and radial distant of the wall. 2. Dispersion and Axial Dispersion Model Dispersion is a general term which is used to describe the various types of self-induced mixing processes occurs during the flow of a fluid through a pipe or vessel. Dispersion arise from the effects of molecular diffusion or as a result of the flow pattern existing within the fluid, the important consequence of dispersion is that the flow in a packed bed reactor, the truth that dispersion reflects the back mixing which accurse with flow is a trendy for both axial and radial dispersion since the bed diameter is normally greater than the particle diameter so its comments to have effects of radial dispersion but prevalence of plug and axially dispersed that it can reduce the efficiency of separation. Axial dispersion is caused by the twin effects of molecular diffusion and turbulent mixing which a rises from splitting and recombination of flow around the particle therefore it is important to describe the behavior across axial dispersion, the model characterizes mass transfer in axial direction in term of an effective or apparent longitudinal diffusivity DL is superimposed on the plug flow velocity. In order to predicate axial dispersion many assumption are suggest the model assumes that the fluid velocity and reactant concentration are constant across the tube diameter. The magnitude of the dispersion is assumed to be independent of position within the vessel, so there will be no stagnant regions and no by passing or short-circuiting of fluid in the model column also eliminating the assumption of instantaneous equilibrium through the bed By changing the magnitude of the dispersion parameter, one may vary the performance of the reactor from that of plug flow ( uL LD = 0) to that of a single continuous stirred tank reactor ( uL LD =∞). Since these two types of phenomena are characterized by a single parameter, and since we force the model to fit the form of Fick's law of diffusion, DL should be regarded as an effective dispersion coefficient having the units of an ordinary molecular diffusivity (length 2 /time). However, it is significantly greater in magnitude because of turbulence effects. Consider the binary system involving exchange between cations A and B. We assume initially that all the exchangeable sites of the resin bed are in the B-form and the solution phase within the pores of the resin bed is free of cations A. Initially, i.e., at t = 0, a solution containing cations A is injected into the top of the resin bed at a volumetric rate, FL, which is constant with time and uniform across the top of the resin bed. The feed solution concentration is constant for t >0 with time. A solution is distributed at a rate uniform across the resin bed. For some time after injection begins, A cations do not appear in the effluent. As the resin gradually converts to the A form, the concentration of A cations in the effluent increases from zero to the feed concentration. Plotting the concentration of A cations in the effluent against the produced effluent volume gives the breakthrough curve. The ion exchange process involving A and B cations may be presented by: aR − B b+ + bA a+  bR- A a+ +a B b+ …(1) Where, R is the solid cation exchange resin, a is the valance of A and b is the valance of B and the process is considered reversible. Iman Hahider Mahdi Al-Khwarizmi Engineering Journal, Vol. 5, No. 4, PP 1-12 (2009) 3 A thin but finite slice of the bed having the thickness ∆Z at position Z is considered now. For t >0, CA and CRA will be volumetric concentration of cations A in the solution phase and in the solid resin phase respectively. JA is the dispersion flux of the A cations. a material balance for A cations over the section may be written as  dt dC ZA dt dC ZA RA cb A cb )1(  ZZAZALZZAZAcb CCFJJA   ()( …(2) In this equation the terms on the left hand side arise from the accumulation of A cations in the solution and resin phases. The first term on the right hand side arises from the dispersion terms and the second results from the convection terms then taking limits as ∆z. →0 Z C A F Z J t C t C A cb LAAR b bA             1 …(3) Fick’s Law can be used to relate the dispersion flux to the concentration gradient: Z C DJ A LA    …(4) where, the axial dispersion coefficient, DLis assumed to be independent of CA and constant with respect to Z The substitution of the derivative of the axial dispersion flux into equation (3) yields Z C A F Z C D C t C A cb LA L RA b bA              2 2 1 …(5) Equation (5) is the basic of the theoretical model for ion exchange performance in a fixed bed. Several assumptions are implicit in equation (5):  The resin bed is homogeneous with uniform porosity  The bulk phase is plug flow with axial dispersion and mass transfer between phases is accounted for by film diffusion  Fluid flows at a rate constant with time, under isotherm condition and is uniformly distributed  The axial dispersion is independent of chemical concentration  The bed is chemically inert and there is no mass transfer in the solid film that CA= CpA r =Rp that mean t C RA   = 0 …(6) Z C A F Z C D t C A cb LA L A          2 2 …(7) by expressing in dimensionless form equation (7) May be more easily solved u = c L A F , Pe= bL D uL  …(8) z = (ut + x)/L , θ = t t = tu/ …(9) then equation (7)become z C z C Pe C         2 2 1  …(10) that axial dispersion model is characterized by the Pecklet number (Pe No.) which can be regarded as the ratio PeNo.= rate of transport by convection /rate of transport by diffusion or dispersion. axial dispersion model z C z C uL DC Lb         2 2  …(11) where the dimensionless group ( uL D L ) is call vessel dispersion number which is measures the extent of axial dispersion where  uL D L 0 negligible dispersion (plug flow)  uL D L ∞ large dispersion (mixed flow) This model represents quite satisfactorily flow that deviates not too greatly from plug flow. Equation (11) may be more easily solved using numerical techniques with initial and boundary Iman Hahider Mahdi Al-Khwarizmi Engineering Journal, Vol. 5, No. 4, PP 1-12 (2009) 4 condition, since the resin is initially in the B-form and the solution within the pores of the bed contains B cations only, the initial condition for the partial differential equation (11) may be written C (0,z) = 0 …(12) At t = 0, there is a step change in the composition of the solution injected into the bed, the boundary conditions that apply to this equation depend on characteristics of the system at the tracer injection and monitoring stations according to these points the bed is assumed as "open," depending on plug flow into the test section that open boundary is one at which the same dispersion parameter characterizes the flow conditions within and adjacent to the test section. There are then three different possible sets of boundary conditions depending on whether a completely open vessel or a closed-open vessel or an open-closed vessel is assumed different solutions will be obtained for different boundary conditions. if we now consider concentration in the feed to an "open" tube that can be regarded as extending to infinity in both directions from the injection point, that the boundary condition are c(θ,0)=1, 2 2 z c   (θ,1) = 0 …(13) The basic method of estimating DL relating the measured mean residence times ( t ) variances Pecklet number or dispersion parameter for the test section where ( t ) calculated by the equation (14) ) )( ( 0 dt dt tdF tt    …(14) By using F =C/C0 Then determined the variance of the response to a pulse input which is measures the spread of the distribution about the mean defined as       0 0 2 2 )( )( )( dt dt tdF dt dt tdF tt t  …(15) Where σ 2 t is the variance of the response to a pulse input, in order determination of the reactor dispersion parameter        222 )(8 2 )( uL D uL D t LL t  …(16) 3. Experimental Work A series of experiment were conduct to determined the axial dispersion by examine the breakthrough curve for ion exchange bed using a strong acid resin first with Amberlite IR-120- styrene-DVB gel type first with Na +1 ionic form of a total capacity of 1.9-2.15 meq/ml to treated Ca +2 ion in CaCl2 solution secondly with same resin type but with H +1 ionic form of 1.8 eq/l total capacity to treated ion in NaCl solution. The both kind of resin is packed in a glass column of inside diameter (2.45cm) and (1m)long, the column is packed with 44.5 cm resin bed depth which supported inside the column by a glass disk welded to the wall of the column on the bottom section. Fig.1. Schemed Diagram for the System of Experimental Work The column is corporate with glass valve of shut of type located to coordinated the different concentration (500,450,350,250 ppm) first with a solution of CaCl2 then with a solution NaCl of which represent the feed to be treated, a dosing pump with a rotameter is used to control the different flow injected (4, 2, 1.25, 0.9 l/hr) which is connected to the column as down flow operation condition. First with a solution of CaCl2 Iman Hahider Mahdi Al-Khwarizmi Engineering Journal, Vol. 5, No. 4, PP 1-12 (2009) 5 then with a solution NaCl six experiment in each feed type were cared, the effluent was collected then analysis using Ethylen Diamine Tetracetic Acide (EDTA) titration method to determined the out put concentration for Ca +2 ion and PH value method for determined the out put concentration for Na +1 in both type the experiments is continuous until (Cout = Co) figure (1) show a schemed diagram for the system of experimental work. 4. Result 4.1 The Breakthrough time The breakthrough time of Ca +2 -Na +1 exhaustion and Na +1 -H +1 runs effected by the CaCl2, NaCl solution concentration and the flow rate in these curves three zone can be distinguished the first zone is almost saturated where the exist concentration is practically zero then the exchange take places that concentration is increases until the concentration limited value or where it exhausted where the zone is call mass transfer zone which its width measuring the efficiency of ion exchange column, the third zone is almost free of ionic spaces. The result in tables (1) and (2) shows that at high CaCl2, NaCl solution concentration feed input reducing the breakthrough time and lead the effluent concentration equal to initial concentration in shorter time at the same flow rate and same bed depth because the concentration of ions in the solution increases which raises the mass transfer rate through the film until it exceed the diffusion rate through the resin bed then the diffusion become controlling kinetics which is slower the process , figures (2) and (3) shows the breakthrough curve for different CaCl2, NaCl solution (500,350,250ppm) at the same flow rate (1.25 l/hr)and same bed depth (44.5cm). Table 1, Experiments Runs for CaCl2 Salt at Cation Exchange with Na- Form Exp no. Salt Con. (ppm) Flow rate l/hr Break-through time(min) tt tu HUNB 1 450 4 582 453.75 397.42 12.415 2 450 2 1470 1366.5 1289.74 5.7 3 450 0.9 2310 2183.02 2070.7 5.145 4 500 1.25 540 419.7 361 13.8 5 350 1.25 885 744.74 629.718 15.45 6 250 1.25 972 892.8 779.054 12.7 Table 2, Experiments Runs for NaCl salt at Cation Exchange with H-Form Exp no. Salt Con. (ppm) Flow rate l/hr Break-through time(min) tt tu HUNB 1 450 4 400 232.95 125.988 45.92 2 450 2 1080 774.115 364.4 52.925 3 450 0.9 1761 1421.29 1162 18.243 4 500 1.25 300 182.4 122.36 32.92 5 350 1.25 700 504.39 404.65 19.80 6 250 1.25 740 612.4 322.66 47.312 The result in tables (1) and (2) shows that high CaCl2, NaCl solution flow rate input reducing the breakthrough time and lead the effluent concentration equal to initial concentration also in shorter time at the same concentration and same bed depth that lower flow rate there will be sufficient contact time ions exchanges that the equilibrium is established soon, as the flow rate increases equilibrium is no longer reached and the mass transfer through the film rises until the diffusion through the resin becomes the controlling factor and the system is said to exhibit particle controlled kinetics, figures (4) and (5) shows the breakthrough curve for different CaCl2 ,NaCl flow rate(4, 2, 1.25, 0.9 l/hr) at the same concentration (450 ppm)and same bed depth (44.5 cm). Iman Hahider Mahdi Al-Khwarizmi Engineering Journal, Vol. 5, No. 4, PP 1-12 (2009) 6 Fig.2. Breakthrough Time for CaCl2 Solution at Constant (1.25 l/hr) Flow Rate Fig.3. Breakthrough Time for NaCl Solution at Constant (1.25 l/hr) Flow Rate Fig.4. Breakthrough Time for CaCl2 Solution at Constant Concentration (450ppm) Flow Rate at constant CaCl2 flow rate (1.25 l/hr) 0 0.2 0.4 0.6 0.8 1 1.2 0 500 1000 1500Time (min) c /c o Ca (500 ppm) Ca (350 ppm) Ca (250 ppm) constant NaCl flow rate (1.25 l/hr) 0 0.2 0.4 0.6 0.8 1 1.2 0 200 400 600 800 1000 Time (min) c /c o Na (500 ppm) Na (350 ppm) Na (250 ppm) cnstant CaCl2 Con.(450ppm) 0 0.2 0.4 0.6 0.8 1 1.2 0 500 1000 1500 2000 2500 3000 Time (min) c /c o Ca(4l/hr) Ca(2l/hr) Ca(0.9l/hr) Iman Hahider Mahdi Al-Khwarizmi Engineering Journal, Vol. 5, No. 4, PP 1-12 (2009) 7 Fig. 5. Breakthrough Time for NaCl Solution at Constant Concentration (450ppm) It important to notice that breakthrough time of Ca +2 -Na +1 exhaustion for CaCl2 solution is greater than Na +1 -H +1 exhaustion for NaCl solution in the same condition of salt concentration and flow rate that the exchange of divalent with monovalent can go much completely because the concentration of the monovalent ions is squared in Ca +2 -Na +1 exchange that effected on the selectivity of the resin with taking in a count the decreasing of size of hydrated ions from Na +1 to Ca +2 which make the exchange capacity for divalent ions is grater than for monovalet ion . The time equivalent to usable capacity (tu) and the time equivalent to total stoichiometric capacity of packed bed (tt) if the entire bed attains to equilibrium are easily to determined by eq. (17) where tb is the time related to C = 0.05Co …(17) The length of unused bed is determined by equation (18) that (tu/tt) is the fraction of the total bed capacity or length utilized to the breakpoint. L t t H t u UNB        1 …(18) Where L is the total bed length The HUNB values is listed in tables (1)and(2) for CaCl2 and NaCl ion exchange operation solution at different feed concentration and different flow rate that the small value means that the breakthrough curve is close to an ideal step with negligible mass transfer resistance. It can be notice that HUNB depends on concentration of the ion to be exchange, ions types, the temperature and fluid velocity while it independent of column length which allows the design and scale-up from small scale laboratory to be determined that the full-scale column obtained by simply adding the HUNB to the length of bed needed. It's important to taken into account the axial dispersion because differences in the extent of axial dispersion in small-scale laboratory and full- size column may caused discrepancies. 4.2 Axial Dispersion Residual time axial dispersion for Ca +2 -Na +1 and Na +1 -H +1 ions exchange is calculated and listed in tables (3) and (4), the values of dispersion number for Ca +2 -Na +1 and Na +1 -H +1 exhaustion is increasing with increasing flow rate at the same salt concentration that at height flow rate the bed is consist of a series of mixing cell the dimension of each of which is of the same as the size of the bed particles forming while at low flow rate the effect of molecular diffusion predominate and the cell mixing contributes relatively little to dispersion. For Ca +2 -Na +1 figure(6) show small extend of dispersion ( uL D L <0.01) which mean small deviation from plug flow that the spreading tracer curve does not significantly change in shape as it passes( during the time it is measuring it)which gives symmetrical curves.    0 )1( dt C C t o t  bt o u dt C C t 0 )1( constant NaCl Con.(450) 0 0.2 0.4 0.6 0.8 1 1.2 0 500 1000 1500 2000 Time (min) c /c o Na(4l/hr) Na(2l/hr) Na(0.9l/hr) Iman Hahider Mahdi Al-Khwarizmi Engineering Journal, Vol. 5, No. 4, PP 1-12 (2009) 8 Table 3, Experiments Runs for CaCl2 Solution at Cation Exchange with Na- Form Exp no. Salt Con. (ppm) Flow rate l/hr Residence time (min) αө 2 uL DL DL Pe NO. 1 450 4 449.4 0.145 0.0588 77.36 45.98 2 450 2 1453.19 0.0306 0.0145 9.538 229.885 3 450 0.9 2291 0.00596 0.00294 0.87 1133.78 4 500 1.25 441 0.0878 0.00432 1.806 771.6 5 350 1.25 756.05 0.00726 0.00358 1.47 915.36 6 250 1.25 899.18 0.00294 0.00145 0.596 2298.85 Table 4, Experiments Runs for NaCl Solution at Cation Exchange with H-Form Exp no. Salt Con. (ppm) Flow rate l/hr Residence time (min) αө 2 uL DL DL Pe NO. 1 450 4 273.959 0.296 0.1044 137.359 31.928 2 450 2 841.03 0.1514 0.061 40.125 54.64 3 450 0.9 1684.92 0.0066 0.0295 8.79 112.765 4 500 1.25 195.966 0.0239 0.01145 4.707 291.12 5 350 1.25 516.316 0.017 0.00824 3.387 404.53 6 250 1.25 659.88 0.035 0.01648 6.7757 74.57 Fig.6. Axial Dispersion for CaCl2 Solution at Constant Flow Rate (1.25l/hr) But with Na +1 -H +1 ion exchange figure (7) show small extend of dispersion ( uL D L >0.01) which mean small deviation from plug flow that resulting S-shape response curves is not symmetrical that mean the flow is undistributed as it passing the entrance and exit boundaries .  uL DL 0 uL DL Iman Hahider Mahdi Al-Khwarizmi Engineering Journal, Vol. 5, No. 4, PP 1-12 (2009) 9 Fig.7. Axial Dispersion for NaCl Solution at Constant Flow Rate (1.25l/hr) The values of vessel dispersion number for Ca +2 -Na +1 and Na +1 -H +1 is increasing as the CaCl2 , NaCl salt concentration increasing for the same flow rate as shown in tables (3) and (4)that when concentration differences exist the rate of transfer(exchange) is proportional to the product of the molecular diffusivity and the concentration gradient where the fluid at centre moves with twice the mean velocity this can rise the dispersion since elements of fluid will take different times to traverse the length of the column. Figures (8) and (9) show small extend of dispersion ( uL D L <0.01) which mean small deviation from plug flow for Ca +2 -Na +1 and Na +1 - H +1 ion exchange. The dimensionless variances of breakthrough curve is calculated and listed in table (3) and(4) for the both case of ion exchange(Ca +2 ,Na +1 ) that it value of σ 2 ≈ 0 means the packed bed close to an ideal flow reactor with negligible axial dispersion Fig.8. Axial Dispersion for CaCl2 Solution at Constant Concentration (450ppm)  uL DL 0 uL DL  uL DL 0 uL DL Iman Hahider Mahdi Al-Khwarizmi Engineering Journal, Vol. 5, No. 4, PP 1-12 (2009) 10 Fig.9. Axial Dispersion for NaCl Solution at Constant Concentration (450ppm) Pe No. represent all of the effects that cause deviations from ideal plug flow behavior such as nonuiform velocity profiles or eddies listed in table (3) and(4) for the both type ion exchange (Ca +2 ,Na +1 ) its mean increasing as the flow rate increasing and decreasing with increasing (Nacl,CaCl2) salt concentration that as Pe No. increasing from 0 to ∞ the flow pattern in the column changes from complete mixing (CSTR) to no mixing, with low Pe No. high conversion with low residence time accurse because molecular diffusion strongly effects the rate of dispersion in laminar flow means at low flow rate it promotes dispersion but at higher flow rate it has opposite effect . 5. Conclusion A mathematical model has been developed to simulate ion exchange with fixed bed; the axial dispersion model is applied with experimental observation for both Ca +2 and Na +1 ion in cation exchange column. The breakthrough time is decreasing with increasing flow rate and feed concentration and its greater for divalent ions from monovalent ions. The axial dispersion model show small extend from plug flow for Ca +2 ions exchange both exchange in both increasing flow rate and increasing salt concentration While its show large extend from plug flow for Na +1 ions with increasing flow rate and salt concentration. Nomenclature a valence of cation A Ac circular cross-sectional area b valence of cation B CSTR continuous stirrer tank reactor C A solution phase concentration CRA resin phase concentration C0 total normality in the solution CsA surface phase concentration in particle CpA pore phase concentration in particle DL solution phase dispersion coefficient d column inner diameter εb bed porosity F C/C0 FL flow rate HUNB length of unused bed JA dispersion flux of A cations in s o solution L bed length Pe Peclet number θ dimensionless time R solid cation exchange resin Rp resin radius t time t mean residence times tb time related toC = 0.05Co tt time equivalent to total stoichiometric  uL DL 0 uL DL Iman Hahider Mahdi Al-Khwarizmi Engineering Journal, Vol. 5, No. 4, PP 1-12 (2009) 11 capacity tu time equivalent to usable capacity u velocity for linear flow z dimensionless column distance σ 2 t variance of the response to a pulse input 6. References [1] Bros A. S. D. and Zola A. 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Res. 1997, 36, 2359-2367 12-1، صفحة 4 ، العذد 5 مجلة الخوارزمي الهىذسية المجلذ ايمان حيذر مهذي (200 9 ) 12 دراسة التشتث المحوري لعمود الطبقة الثابته مه المبادالت االيووية ايمان حيذر مهذي جايؼة انُٓسيٍ / كهية انُٓدسة/ قسى انُٓدسة انكيًيأية الخالصة جى جطٕيس يٕديم زياضي نٕصف اداء ػًٕد اسطٕاَي َٕع انطثقة انثاتحّ يٍ انًثادالت االيَٕية يحضًٍ يقازَة يٕديم انحشحث انًحٕزي تؼدد يٍ . انفسضيات حيث اٌ ْرا انًٕديم يسًح تححيد االداء نهؼًٕد ٔ حساب انٕقث انًقيى نححديد االَحساف ػٍ ػًٕد انجسياٌ انًضغٕط نيحثادل يغ زشٌ تصيغة ايٌٕ (ايٌٕ انكانسيٕو انًٕجة ) جى اخحياز ًَٕذجيٍ يٍ اَظًة انحشغيم ناليَٕات انًٕجثة جى انؼًم أال يغ ايَٕات ثُائية انحكافؤ نيحثادل يغ زشٌ تصيغة ايٌٕ انٓيدزٔجيٍ ٔانري (ايٌٕ انصٕديٕو انًٕجة)انصٕديٕو ٔانري يطثق تٕحدات انًاء انسائق ٔانثاَي يغ ايَٕات احادية انحكافؤ . يطثق في ٔحدات انًاء اناليَٕي جى اخحياز ػايهيٍ نكم ٔاحد يٍ انُظاييٍ ًْا اخحالف يؼدل جسياٌ انًغري ٔاخحالف جسكيص انًهح ػٍ طسيق جاثيسْى في انحشحث انؼًٕدي نكم حانة انصيٍ انًكافئ نهسؼة انًسحؼًهة ٔانصيٍ انًكافئ نسؼة االجصاٌ )تٕاسطة يقازَة االداء ٔانري يؼثس ػُّ تصيٍ االسحُصاف حيث جى حساب يحددات اخسٖ حسثث خالل ظسٔف انؼًم الٌ ْرِ انًحددات يغ انحشحث انًحٕزي يًٓة جدا نححديد افضم ظسٔف نحشغيم ػًٕ د انطثقة (ٔطٕل انؼًٕدٖ غيس انًسحؼًم . انثاتحّ نهًثادل االيَٕي انًٕجة كٕحدات انًاء انسائق ٔٔحدات انًاء اناليَٕي