1 Introduction The clamour for energy-saving techniques in almost all branches of industry has acted as a spur in the development of thermal separation equipment The design and process engineering improvements that have ensued entail that feedstocks are subjected to less severe treatment and can thus be optimally exploited They also entail production under ecologically favourable conditions (cf Fig 1.1) A typical example is provided by low-pressure-drop packing in the vacuum rectification of mixtures that are unstable to heat and that necessitate a large number of theoretical stages for their thermal separation The attendant decrease in the total pressure drop and operation under vacuum ensure that the temperature at the bottom of the column is comparatively low Hence, decomposition products that are detrimental to the environment can be largely avoided, i.e atmospheric pollution is reduced and less residues have to be disposed of Another advantage is that the reduction in the average column pressure brought about by vacuum operation increases the average relative volatility of the components in the mixture and thus reduces energy consumption Low-pressure-drop, high-performance packing is an essential requirement in the economic design of an integrated separation plant, because it permits heat pumps to be installed and a number of columns to be linked together Fig 1.1 Relationships established by separation techniques between energy consumption, processing and environmental protection Packed Towers in Processing and Environmental Technology Reinhard Billet Copyright © 1995 VCH Verlagsgesellschaft mbH, Weinheim ISBN: 3-527-28616-0 Introduction From this point of view, it is not surprising that packing designed to conserve energy has been the subject of many new developments on the part of equipment manufacturers However, before any one particular type of packing can be selected for a given separation task, adequate knowledge must be available on the performance characteristics, e g the separation efficiency, the pressure drop, the liquid holdup, the capacity, and the costs A physical model is required to describe the hydraulics and the mass transfer efficiency in a given separation column and thus to allow the main dimensions to be calculated and the process engineering performance to be predicted The parameters that affect the design, the capacity, and the specific properties of the product and inlet streams must be known in order to devise the model, and the only means of acquiring these data is by experiment Consequently, the aim of this book is to supplement the theoretical considerations by the results of relevant studies that were performed in the author's laboratories and pilot plants and were scaled up to meet practical requirements It is known that packed columns, whether random or stacked, allow lower pressure drops per theoretical stage than plate columns and are thus better suited to meet demands on optimum energy consumption in thermal separation plants The main applications for packed columns are the separation of vapour-liquid or gas-liquid systems, e.g in rectification, absorption, and desorption In many cases, they have also proved to be superior to conventional plate columns for liquid-liquid extraction processes Greatest significance from the aspect of saving energy is attached to their use in rectification, a subject to which particular attention has been devoted in the course of this book In future, these separation processes will not be confined to petroleum refineries and the chemical and allied industries They will also be adopted on a wider scale in ecological engineering for purifying off-gas streams and for water treatment, and the demand for the necessary equipment, including packing, will grow accordingly Rectification, absorption, desorption, and liquid-liquid extraction processes consist essentially of passing two countercurrent phases through a packed column (cf Fig 1.2) In rectification, the vapour produced in the distillation section of the column flows countercurrent to the liquid formed in a condenser Contact between the two phases is thus established, with the result that the low-boiling component flows upwards and the high boiler accumulates at the bottom of the column Physical absorption processes consist of mass transfer from the gas into the liquid phase, i e into a solvent that selectively absorbs the desired component from the gas stream In desorption - often referred to as stripping - mass transfer proceeds in the opposite direction, i.e from the liquid into the gas phase In the extraction of a component from a mixture of liquids by means of a selective solvent, mass transfer takes place between two liquid phases It is taken for granted here that the reader is already acquainted with these separation processes and their thermodynamic fundamentals together with the literature on the subject and the standard terminology The scope of this book has therefore been restricted to a comprehensive treatment of packed-bed technology and its application to separation processes / Fundamental operating characteristics of packed columns for gas-liquid systems Packed towers for continuous Rectification Absorption Solvent I A (absorbent)J { b a s Feed Liquid = A Loaded feed : | gas Extraction Feed j f Extract * Bottom product Loaded solvent - Feed Fig 1.2 Desorption y DesorbedTTcarrier liquid t t / ™ -gas — Auxiliary product |f Raff mate ^ I Solvent End product Applications for packed columns in thermal separation processes 1.1 Fundamental operating characteristics of packed columns for gas-liquid systems The optimum choice of a packed column for a given separation task would be impossible without a sound knowledge of the characteristic parameters and physical quantities that permit process engineering evaluation and comparison The gas or vapour load in a packed column is usually expressed by the capacity factor, which is given by Fy = Uy (1-1) where uv is the superficial velocity, o v is the density of the gas or vapour, and QL is the density of the liquid At low or moderate pressures, QV is small compared to QL and can thus be neglected Hence, the vapour capacity factor, which is often resorted to in practice as a measure of the dynamic load on the column, is the product of the superficial velocity uv and the square root of the vapour density, i.e Fy = Uy \J~Qy~ (1-2) Introduction The maximum permissible value for the capacity factor also depends on a dimensionless flow parameter that allows for the ratio of the liquid to the vapour flow rate LIV and is given by (1-3) In practice, the capacity of packed columns is frequently described by the following function, which has to be determined by experiment: Fy (1-4) yfoL A crucial factor in evaluating packed columns, especially those intended for vacuum rectification, is the resistance offered by the bed of packing to the flow of vapour A major criterion in selecting packing for absorption processes in which gas is forced through the column by a compressor is the pressure drop Ap in the bed, because it largely governs the compressor rating The pressure drop per unit height of bed Ap/H depends on the gas uv and liquid uL loads (cf Fig 1.3) It is usually expressed as a function of the capacity factor F v for a given liquid load uL or a given liquid-gas ratio, i.e (1-5) Another factor of great importance in evaluating packed columns is the pressure drop per unit of separation efficiency, which is defined by the number of theoretical stages nt in rectification or by the number of transfer units NTU in absorption and desorption A general term, i.e the number of separation units N required for a given process, may be taken to cover both cases Thus, Gas or vapour Liquid PT Ap Gas or vapour I H=N T ? Liquid Fig 1.3 Flowchart for a packed column 1.1 Fundamental operating characteristics of packed columns for gas-liquid systems N = nt or N = NTU (1-6) The total pressure drop in the gas or vapour is given by &P=PB-PT where (1-7) pB is the pressure at the lower column inlet and pT is the pressure at the upper column outlet In analogy to Eqn (1-5), the total pressure drop per separation unit can be expressed as a function of the capacity factor for a given ratio LIV of the liquid to the vapour flow rate, i.e ~ - = HFy) (1-8) The following relationship thus applies: ^ (1-9) If the vapour pressure at the top of the column is kept constant during rectification, that at the bottom of the column - and thus the boiling point - become less as the pressure drop per unit efficiency for the specific bed of packing decreases Low boiling points are an absolute necessity in the vacuum rectification of thermally instable mixtures They also entail larger differences between the temperature of the heating medium and that of the mixture, i.e more effective heat transfer Moreover, if a large number of theoretical stages is required in a given vacuum rectification, a low pressure drop allows entire installations to be designed from the aspect of optimum energy consumption The number of theoretical stages nt required to separate a binary feed with a molar flow rate F can be determined diagrammatically by the McCabe-Thiele method, which is based on the concept of an equilibrium stage, i.e a stage in which the ascending vapour is in phase equilibrium with the descending liquid An example of the corresponding diagram is shown in Fig 1.4, in which x¥ is the mole fraction in the feed of the more volatile component that has to be concentrated to a mole fraction xD in the overhead product (distillate) with a molar flow rate D; and xB is the mole fraction of this volatile fraction that remains in the bottom product (molar flow rate B) The equilibrium curve is the locus of the points x, y The two operating lines derived from material balance equations - BI for the stripping zone and DI for the enrichment zone - intersect at the point / on the g-line, which is the locus of all points of intersection of the two operating lines for any given feed stream The number of theoretical stages nt is represented by the steps that link the equilibrium curve with the two operating lines between the points x B and xD It is the sum of the number of stages in the enrichment zone ntez and the number in the stripping zone ntsz, i e nt = nt,ez + nt,sz (1-10) Introduction Reflux condenser^ Number of theoretical stages n t Equilibrium curve Operating *i V xB xF xD Liquid mole fraction x Fig 1.4 Determination of the the number of theoretical stages in a continuous rectification column, taking a binary mixture as an example If xF, xD and xB are given, nt depends on the phase equilibrium in the mixture, i e on the shape of the equilibrium curve for the binary system, and on the position of the two operating lines BI and DI The position of the intersect / can be obtained from the material balance at the inlet cross-section of the column, i.e The factor / in this equation is a measure for the thermal state of the feed and is given by /=!-" h'F-hF Ahv (1-12) where hF is the molar enthalpy of the feed at the inlet temperature, h'F is the molar enthalpy at the operating temperature in the inlet cross-section of the column, and Ahv is the molar condensation enthalpy of the vapour stream in the inlet cross-section If the individual components of the mixture have roughly the same molar evaporation enthalpy, the molar flow rates of both the vapour and liquid will remain practically constant along the respective flow paths both above and below the inlet, i.e in both the enrichment and stripping zones In this case, the relationship between the vapour y and the liquid x mole fractions of the more volatile component in the enrichment zone is given by the following linear equation: y = r + 1• x + r+ (1-13) 1.1 Fundamental operating characteristics of packed columns for gas-liquid systems where r is the ratio of the reflux flow rate L to the overhead product flow rate D, i.e r=|- d-14) The corresponding equation for the stripping zone is where b is the ratio of the flow rate of liquid L in the stripping zone to that of the bottom product B, i e b = -j (1-16) If the values for xF and / are given and those for xD and xB are specified, the operating lines for the enrichment and stripping zones in Fig 1.4 can be easily plotted Inserting x = in Eqn (1-13) yields the value for y0 at the intercept with the axis of ordinates, i.e d-17) where xD is the mole fraction of the more volatile component in the overhead product The reflux ratio r at the head of the column is a factor that greatly affects the economics of rectification If the molar evaporation enthalpies for the low-boiling A/z, and the high-boiling Ahh components differ significantly, a linear relationship no longer exists between the mole fractions v and x of the more volatile component in the vapour and in the liquid respectively In this case, allowance for the difference between Aht and Ahh is made by the expression K- —^— Ahh - Ahi (1-18) The equation for the enrichment zone is thus no longer linear, i.e r+1 y= r+1 ^ r+1 # The intercept >;0 with the axis of ordinates in this case is given by — (1-20) Introduction Whether its position is higher or lower than the corresponding intercept formed by the linear relationship {cf Eqn (1-17)} depends on which of the two components - the highboiler or the low-boiler - has the greater molar evaporation enthalpy Likewise, the relationship for the stripping zone in the column is also nonlinear if the molar evaporation enthalpies of the components differ In this case, the following equation applies: / yA *—r (1 21) " where the parameter mI is given by xF-xB m, = ^ r,+f ^ (1-22) the reflux ratio r7 at the inlet cross-section, by = and yt and xt are the coordinates of the intersect / (cf Fig 1.4) An alternative concept to the number of theoretical stages nt for the evaluation of separation efficiency is the number of transfer units NTU If the molar flow rates for the liquid L and the vapour V are kept constant and there is no mixing in the axial direction, the NTU for steady-state operation can be expressed as follows in the terms of the concentration difference y* - y in the vapour (cf Fig 1.5): —£- (1.24) where y* is the phase equilibrium concentration of the more volatile component in the vapour in contact with the liquid of concentration x* at the phase boundary in any given horizontal cross-section of the column; and x and y are the fractions that correspond to the concentrations of the more volatile component in the bulk of the vapour and in the liquid respectively It may often be assumed that the resistance to mass transfer in rectification is predominantly in the vapour phase, i e x —» x* In this case, the surface mass transfer coefficient in the liquid phase is |3L —» °o and that in the vapour phase fV is identical to the overall mass transfer coefficient on the vapour side kov Accordingly, NTUV = NTUOV The height of a transfer unit is then defined by 1.1 Fundamental operating characteristics of packed columns for gas-liquid systems HTUOV = H NTUn H (1-25) dy ye-y If it is assumed that kov is constant over the height of the column in mass transfer controlled by the vapour phase, a mass balance yields HTUOV = (1-26) ds kov aph where aph is the phase contact area per unit column volume, ds is the column diameter, V is the molar flow rate of the vapour, and HTUOV is usually determined by experiment Hence the following relationship exists between the number of theoretical stages per unit height nt/H and the height of a transfer unit HTUOV: V H HTUn (1-27) \n[myxy- (p v ) x * = x =k o v "it y*dy x+dx 0&Z7/Z2 y Liquid mole fraction x ^o x0 J a=- Fig 1.5 Determination of the number of transfer units by the two-film theory 10 Introduction In other words, the number of transfer units in systems with a low relative volatility is given by (NTUov)a = smal, = n, (1-28) Likewise, the height of a transfer unit can be equated to the height equivalent of a theoretical stage, i.e {HTUOy) a , small = HETS = — (1-29) The stripping factor X for a given section of the column is defined as the product of the mean slope myx of the equilibrium curve and the molar vapour/liquid ratio VIL, i e \ = myxj- (1-30) Once the height of a transfer unit HTUOV and the number of transfer units NTUOV are known, the height of the column required for the relevant separation process can be obtained from their product Hence, cy H = HTUQV ' NTUOV = HTUOV ^— (1-31) The figure thus derived is identical to that obtained from the number of theoretical stages nt and the separation efficiency ntIH for the packing concerned {cf Eqn (1-27)}, i e H = An analogous analysis applies for mass transfer in the liquid phase This procedure for the determination of column height is referred to in the literature as the HTU-NTU concept It is applied in Chapters 3, and 14 The efficiency, expressed as the number of separation units per unit height NIH, can be assessed graphically from its relationship to the column load or capacity factor, which can be written as NIH = i{Fv) (1-33) A knowledge of this function is essential in determining the load relationships for the pressure drop per unit height AplH {Eqn (1-5)} and the pressure drop per separation unit Ap/TV {Eqn (1-8)} These relationships allow the column volume vv per unit of vapour flow rate and unit of separation efficiency to be determined in terms of the capacity factor /v{Eqn (1-34)} or the pressure drop per unit of separation efficiency ApIN {Eqn (1-35)}, i e 1.2 Theoretical column efficiency 11 vv=i(Fv) (1-34) vv=f(Ap/N) (1-35) The specific column volume vv is an important factor in determining the capital investment costs and is defined by where H is the effective height of the column, N is the number of theoretical separation units, and uv is the superficial vapour velocity 1.2 Theoretical column efficiency The relative volatility a, a term that was introduced in Eqns (1-28) and (1-29), is a measure for the ease with which a mixture can be separated It expresses the relationship between the molar fraction y of the more volatile component in the vapour and that of the liquid x with which it is in phase equilibrium; and it is defined by Its magnitude governs the shape of the equilibrium curve (cf Fig 1.4), which is defined by the following equation: y =f w = i a + ( /i)x ^-38) If a is constant, the curve assumes the form of an equilateral hyperbola The greater the value of a, the greater the value of y for a given value of x As a rule, the relative volatility a of the two components of a mixture, and thus the ease with which they can be separated, decreases with a rise in pressure Thus, if the value of a for a given mixture is comparatively small, the number of theoretical stages nt required for separation increases with the specific pressure drop Ap/nt within the packing, particularly during vacuum operation If the pressure at the head of the column pT is kept constant, the pressure drop Ap governs the rise in pressure up to the value at the bottom of the column pB, i.e [^y (1-39) It follows that the reflux ratio required for separation in the enrichment zone of a rectification column rises In other words, the energy consumption must increase In view of the following relationship, the liquid-vapour ratio also increases: 12 Introduction L_ V (1-40) r+ Hence, the number of theoretical stages required for a given separation task, as defined by Eqn (1-41), is greater in beds of packing with high values of Ap/n, than in those with low values (cf Fig 1.6): Ap Ap V\ n, (1-41) If this aspect is taken into consideration in the planning stages of a separation plant, it will be seen that the maximum feasible load entails not only the smallest column diameter but also the largest number of theoretical stages, and thus the greatest height Hence, if separation is under vacuum with a large number of theoretical stages, the optimum load will be less than the maximum to an extent that depends on the pressure drop characteristic for the packing selected Accordingly, the design load in separation tasks of this nature depends on the optimum pressure drop per separation stage and is associated with the minimum total costs Particular attention must therefore be devoted to the pressure drop per theoretical stage Ap/nt in selecting packing for rectification processes involving a large number of theoretical stages As Ap/nt increases, the relative volatility a of the mixture decreases in the direction from the top to the bottom of the column Consequently, the theoretical number of separation Liquid/Vapour ratio L/V Reflux ratio r Fig 1.6 Qualitative effect of the pressure drop per theoretical stage on the relationship between the number of theoretical stages and the reflux ratio 1.2 13 Theoretical column efficiency stages nt ought to be higher than that in an isobaric column (ntiso), i.e a column in which there is theoretically no pressure drop, and the ratio ntisolnt {Eqn (1-42)} can be regarded as the theoretical column efficiency r\c {Eqn (1-43)}: (nt) Ap/nt = (nt) Ap/m > = nttiso (1-42) n t f [pT, a r Ap/nt> nt(L/V, (1-43) In other words, the theoretical column efficiency depends on the operating pressure pT and relative volatility aT at the top of the column, the pressure drop per separation stage Ap/nt, and the number of theoretical stages nt The value of r\c thus varies from the one separation task to another It is comparatively difficult to calculate x\c accurately from phase equilibrium relationships, and a graphicalmathematical method based on the qualitative method shown in Fig 1.7 would therefore be more expedient for the evaluation of Eqn (1-41) This plot is analogous to that proposed by Gilliland for isobaric (theoretically Ap/nt = 0) columns The ordinates in the diagram are referred to as stage-number parameters and are defined by Eqn (1-44) for isobaric and by Eqn (1-45) for nonisobaric columns, i e _ ft t, iso n ^t,min,iso , 1L t,iso ~ ll - (1-44) ll t S _ - t,min,iso „ Fig 1.7 Determination of the theoretical column efficiency - S Wnt (1-45) > o(v-D v rrmm,iso • e=- v g rmin,iso+ Energy parameter The abscissae are referred to as energy parameters and are defined by Eqn (1-46) for isobaric and Eqn (1-47) for nonisobaric columns, i e 14 Introduction riso-rmin.iso (v-l)rminjso = e ^ = o ( j ^+1 46) vg rw/n^/50 + According to Eqn (1-46), the reflux ratio riso in the isobaric column exceeds the minimum rminJso by a factor v, i e Similarly, according to Eqn (1-47), the reflux ratio r in the real column exceeds the minimum in the isobaric column rminiso by a factor vg, i.e f = V g rmin,iso (!- 49 ) If the factor v by which the actual reflux ratio exceeds the minimum rmin in the real column is the same as that for the isobaric column, as determined by Eqn (1-47), the following equation would apply: r = v rmin (1-50) In this case, the factor vg for the real column is given by Eqn (1-51), which is obtained by combining Eqns (1-49) and (1-50); and the reflux ratio r for the real column, by Eqn (1-52), which is obtained by combining Eqns (1-48) and (1-50), i e (1-51) r =ris0^shL- (1-52) 'min, iso This method allows the number of theoretical stages nt and the reflux ratio r for a real column to be determined from the corresponding values ntminiso and rminiso for an isobaric column The theoretical column efficiency, as defined by Eqn (1-42), can then be obtained quite easily in the light of the qualitative diagram in Fig 1.7 The following relationships apply and can be evaluated by alternative means: t, iso iso s-i co\ r] c = — —^—J v\c = nt,iso (1-53) - n t,min,iso = d-54) ' ^ _Lz±_ numinAso + siso *- ^iso n t,min,iso ' 1.2 15 Theoretical column efficiency An example of a diagram that can thus be obtained is shown in Fig 1.8 It allows the actual values for the reflux ratio and the number of theoretical stages to be obtained in both isobaric and real columns, i.e for both Ap/nt = and Aplnt > 0, from the corresponding minimum values for an isobaric column, i.e a column with no pressure drop In this case, the theoretical column efficiency r\c will also be known It is evident from Fig 1.9 that a low theoretical column efficiency, corresponding to a large number of theoretical stages, can be expected if the energy requirements are high in rectification columns with a moderate to comparatively high pressure drop per theoretical stage Ap/nt This applies in this case particularly to columns that are operated with a lower reflux ratio, as defined by (1-56) Hence, an urgent requirement in separation processes of this nature is packing with a very small pressure drop per theoretical stage, e.g < Ap/nt < mbar The advantages of packing with small values of Ap/nt are demonstrated in Fig 1.10 They apply particularly to operation under vacuum with a large number of theoretical stages nt 0.7 0.6 0.5 V\ \ A V V\ \ \A \ \ \s s \ 0.4 N \ n s t-ntmin.iso r s s 0.3 X 0.2 nnin.iso 'Vg ~ ' ) Vg ^m i n i s o "*"' 0.1 0.1 0.2 0.3 0.4 \ ^\ \ 0.5 0.6 0.7 0.8 0.9 1.1 Energy parameter e Fig 1.8 Diagram for the determination of the number of theoretical stages in nonisobaric vacuum rectification Plotted for vacuum systems with relative volatilities between 1.1 and 3.6 and overhead pressures between 20 and 250 mbar 16 Introductior eon column effi cien 1.0 Ap/n t = 6.5 fnbar \ f 0.8 ^ —' l , "C — ^-— 0.6 T" r~~ k-7 i—• "120 f)pm = 53 ] = 0.947 f =1 Ap/nt = 13mbar I 0.2 • • PT =15C mb(Jl / 0.4 x B = 0 ppm »» — • 90 100 110 120 Reflux ratio r 130 140 150 Fig 1.9 Theoretical column efficiency as a function of the reflux ratio in the rectification of a mixture of thionyl chloride and ethylene chloride too = , = = :?- • / 0.96 \ 0.92 0.88 0.84 / / 0.80 g - 0.76 / / / 0.72 0.68 10 / A I / / \ V j / / / * / /I J A 'J7 / / 20 1/ / 40 60 x D =0.9995, a T = \M ^tmin.Jso 100 = ^L 200 400 600 1000 Pressure at head of column pT [mbar] Fig 1.10 Theoretical column efficiency as a function of the overhead pressure 1.3 Energy consumption and number of theoretical stages 17 1.3 Energy consumption and number of theoretical stages One of the most important tasks in planning rectification plants is the exact determination of the relationship expressed by Eqn (1-41), i e between the number of theoretical stages nt and the reflux ratio r that corresponds to a constant pressure pT at the top of the column and that is required to separate a mixture with low-boiling mole fractions of xF, xD and xB in the feed, distillate and bottoms This relationship forms the basis for all optimization calculations by fixing the mutual dependence of capital expenditure and the costs of utilities This is demonstrated qualitatively in Fig 1.11 The number of theoretical stages under otherwise identical conditions is theoretically higher in the separation of systems with a low relative volatility a than of systems with a high one This relationship can be expressed in thermodynamic terms and, as has been demonstrated in Section 1.2, is of particular importance in planning rectification processes with a large number of theoretical stages designed to separate mixtures whose relative volatilities are comparatively small and, furthermore, become even less at higher pressures The economics in cases of this nature dictate the use of packing with a low pressure drop per theoretical stage Ap/nt This requirement is illustrated qualitatively in Fig 1.6, which shows the number of theoretical stages nt as a function of the reflux ratio r for columns with low and high values of Ap/nt A diagram of this nature is characteristic for the vacuum rectification of a mixture whose relative volatility a is comparatively small at a column overhead pressure pT and becomes progressively smaller as a result of the rise in pressure that occurs as the mixture descends through the column Economic limits are imposed on the pressure drop per theoretical stage Ap/nt in separation tasks of this nature, because the minimum total costs CT>min, i.e the sum of the capital investment C7 and the operating costs Co, shifts towards high reflux ratios r and towards large numbers of theoretical stages The relationship is shown qualitatively in Fig 1.12 The consequence of the shift is a pronounced rise in the minimum total costs CT>min, as is evident from the qualitative diagram presented in Fig 1.13 The only economic solution for vacuum rectification in this case is to install low-pressure-drop, highly efficient packing, because it allows considerable savings in costs \ CO CD CO x F , x D , xB = const ) • CO 1D3I \ \ man! o I ZD Fig 1.11 Qualitative relationship between the number of theoretical stages and the reflux ratio with the relative volatility as parameter rmini ^ ^ " ^ ^ Reflux ratio r 18 Introduction Theoretically, the curve that joins the minima M7, M2, in the set of total costs curves presented in Fig 1.13 passes through a minimum itself This case would arise in vacuum rectification with a large number of theoretical stages when the progressive increase in the pressure drop per theoretical stage Ap/nt is initially associated with a drop in capital investment costs; and subsequently, with a rise The drop could be caused by the reduction in column diameter; and the rise, by the cumulative effect of the growing number of theoretical stages Hence packing of this nature has an optimum value for the pressure drop per theoretical stage (Ap/nt)opt, at which the total costs CT pass through a minimum The parameters that are given by the definition of the task are the molar flow rate F of the mixture fed with a molar fraction of xF of low-boiling components and the specified purities of the distillate and bottoms, which flow at rates of D and B and with molar fractions of xD and xB, respectively, of the low-boiling component The rate of heat flow Q in a rectification plant operated along the lines of the flow chart shown in Fig 1.4 is governed by the reflux ratio r, the characteristic molar liquid enthalpies hD' and hB of the distillate and bottoms at their respective boiling points, the molar liquid enthalpies hF and hF of the feed at its boiling point and its inlet temperature, respectively, and the molar evaporation enthalpy Ahvof the overhead product vapour, i.e _ xD - xB h The parameter that decisively affects the rate of heat consumption is thus the reflux ratio r It must always be higher than the minimum value rmin that is associated with an infinitely large number of theoretical stages and would therefore raise the costs to an infinite extent (cf Fig 1.11) In binary systems the minimum reflux ratio rmin is comparatively easy to determine (cf Fig 1.14) Thus, if flow is equimolar, ~ x ° ymin 1- x ° ~yi yl-x] n ^sn (1 58) " If the molar vaporization enthalpy A/i, of the low-boiling component differs from that of the high boiler Ahh, r r """ = K yi ~ K-xD XD X ~ ' - yi-x, (1-59) (159) where K is the factor given by Eqn (1-18) in Section 1.1 In Fig 1.14, ymin is the point where the enrichment line, which passes through the common intersect / of the g-line and the equilibrium curve, intersects the axis of ordinates The boundary case, in which hardly any overhead product is withdrawn arises if the reflux ratio r is infinitely large The curves for the rectification and stripping zones then coincide with the diagonals in the ylx diagram This case corresponds to a minimum number of theoretical stages ntmin, which can be calculated from the values of xD and xB for the lowboiling fractions in the overhead product and the bottoms by means of the Fenske equation, if it is assumed that the relative volatility is constant, i.e 1.3 Energy consumption and number of theoretical stages 00