Reboiler Separation Efficiencies for Binary Systems - Industrial

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Reboiler Separation Efficiencies for Binary Systems Branislav M. Jaćimović, Srbislav B. Genić, and Nikola B. Jaćimović* Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11000 Belgrade, Serbia ABSTRACT: Reboilers are essential for the proper work of distillation and stripping columns. Hereby the intensity of axial mixing along the reboiler (Peclet number) is used for the reboiler separation efficiency modeling and estimation for binary systems. Two fully worked examples show that the separation (Murphree) efficiency varies significantly depending on the adopted reboiler flow model. Murphree efficiency ranges between 1.40 and 0.663 in example 1, and between 1.962 and 1 in example 2. The general conclusion is that the reboiler performance must be considered carefully in the case of design of the new plants,as well as in the case of “brown field” projects.

1. INTRODUCTION In the engineering practice both reboilers and partial condensers are usually modeled as equilibrium or theoretical stages (TS), which is admissible for preliminary design purposes. However, for the plant final design it is necessary to have a more accurate reboiler mass transfer capabilities prediction. This can also be applied in the case of “brown field” projects or in other cases of revamp, when the existing equipment performances are to be simulated for newly requested operating conditions. Although the aforementioned approximation can sometimes be used, in most cases reboilers are not equivalent to an equilibrium stage. In other words, some reboiler types, such as kettle type reboilers, have separation effects which correspond to more than one theoretical stage. Some other types, such as recirculating reboilers, usually present less than one equilibrium stage (partial mass transfer stage, PMTS). This nonideal behavior of the reboilers strongly affects the performance of the connected distillation/stripping column, especially in cases when there are only a few separation stages in the column. Obviously, such nonideal behavior, as well as the reboiler construction details, is especially significant for the thermal design of the reboiler. It is clear from all the above that an accurate calculation method for prediction of the reboiler separation efficiency is needed for the reboiler design when the reboiler and the distillation/stripping column are in the final stages of the design. In the open literature, an excellent overview of all the reboiler types with all application possibilities can be found.1,6 However, the problem of the reboiler separation efficiency is rarely discussed. Process parameters that define the reboiler are separation efficiency (mass transfer capability) and reboiler vapor production, and hereby the binary systems are investigated. Mass transfer capability is usually expressed as stage efficiency (Murphree efficiency). In the case that vapor and liquid leaving the reboiler are in equilibrium, it is considered that the reboiler efficiency is equal to the efficiency of the TS. As there is no vapor entry into the reboiler, it is not possible to define Murphree tray efficiency for the gas phase, so the reboiler separation efficiency is expressed using the Murphree efficiency for the liquid phase x in − xout xN − x W EML = = x in − x*(yout ) xN − x*(yN + 1) (1) © 2012 American Chemical Society

where the usual nomenclature is used for tray numbering from top to bottom (for column with N trays top tray is i = 1 and bottom tray is i = N). In eq 1, x, kmol/kmol, represents the mole fraction of the more volatile component in the liquid, while the subscripts in, out, N and W represent the reboiler inlet, reboiler outlet, Nth tray, and bottom product, respectively. In the same equation y, kmol/kmol, represents the mole fraction of the more volatile component in the vapor, and a superscripted asterisk (∗) indicates the state of equilibrium. Vapor production of the reboiler is usually defined as vapor− liquid ratio using one of the three basic terms which can be found in the references1−6 (notation for all three definitions is the same as in Figures 1 and 2): (1) Boilup ratio (BR) BR =

s ĠN + 1 Ġ 1 = = ̇ ̇ ̇ ̇ W Lout (L /G)ss − 1

(2)

(2) Circulation rate (CR) s ̇ s = Ġ /L̇ in CR = ĠN + 1/LN

(3)

(3) Outlet fraction of vapor (EF) s EF = ĠN + 1/L̇ rec

(4)

s In eqs 2−4, the following nomenclature was used: (i) Ġ N+1 , kmol/h, mole flow rate of the vapor that leaves the reboiler; (ii) Ẇ , kmol/h, mole flow of the bottom product; (iii) L̇ Ns , kmol/h, mass flow rate of the liquid that enters the reboiler; (iv) L̇ rec, kmol/h, mass flow of the recycle stream; (v) (L̇ /Ġ )ss is the slope of the operating line of stripping section. The subscript ss and superscript s denote the stripping section of the column. The boilup ratio, the circulation rate, and the outlet vapor fraction are usually expressed using the corresponding mass flow rates (all in kg/h): ∼s ∼s ̇ CR mas = ĠN + 1/LN (2a)

Received: Revised: Accepted: Published: 5793

September 26, 2011 March 10, 2012 March 23, 2012 March 23, 2012 dx.doi.org/10.1021/ie202193m | Ind. Eng. Chem. Res. 2012, 51, 5793−5804

Industrial & Engineering Chemistry Research

Article

Table 1. Process parameters for various types of reboilers

Figure 1. Schematic representation of the connection between the column and the reboiler: the case when the reboiler has separation effects which are equal to the efficiency of one theoretical stage (1a) and the case when the reboiler has no separation effects (1b).

∼s BR mas = ĠN + 1/W͠ ̇

(3a)

∼s ∼ EFmas = ĠN + 1/L̇ rec

(4a)

For the specific type of the reboiler the vapor−liquid ratio is usually defined by using two different values, which are mutually dependent. For the oncethrough reboilers, there is the following relationship between CR and BR: CR =

The literature data considering reboiler separation efficiency and vapor production are presented in Table 1. 5794

s s ĠN + 1 ĠN + 1 BR = = s ̇s ̇ ̇ LN BR +1 GN + 1 + W

(5)

dx.doi.org/10.1021/ie202193m | Ind. Eng. Chem. Res. 2012, 51, 5793−5804


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Figure 1b does not show the flow of the recycle stream (L̇ rec), which exists in reality. The real flow schematic is shown in Figure 2 (for recirculating and forced circulation reboilers), and is described in later text using eqs 8−18.

2. SEPARATION EFFICIENCY OF RECIRCULATING AND FORCED CIRCULATION REBOILERS The liquid that leaves the Nth tray (L̇ Ns ;xN) is mixed with the recycle stream (L̇ rec;xrec) which leaves the reboiler. This results in a mixture at the bottom of the column which has the composition xW. This liquid at the bottom of the column is divided into two flows: the bottom product (Ẇ ) and the liquid that enters the s reboiler (L̇ rec+Ġ N+1 ). After passing through the reboiler this liquid partially vaporizes, so that at the reboiler outlet there is s both vapor (Ġ N+1 ;yN+1) and liquid (L̇ rec;xrec) which are in equilibrium. For the recirculating and forced circulation reboilers, separation effects are limited because on the one hand Figure 2. Representation of the basic fluid flows in recirculating and forced circulation reboilers.

For the recirculating and forced circulation reboilers there is the following relationship between the CR and EF: CR =

s ĠN + 1

s ĠN + 1 + L̇ rec

=

EF EF + 1

(8)

s ̇ s xN = (L̇ rec + ĠN ̇ W L̇ recx rec + LN + 1)x W + Wx

(9)

which yields ̇ s xN = L̇ recx W + LN ̇s x W L̇ recx rec + LN

(6)

(10)

and on the other hand s s (L̇ rec + ĠN + 1)x W = L̇ recx rec + ĠN + 1yN + 1

In this case, CR does not depend on the slope of the operating line of the stripping section (L̇ /Ġ )ss, but solely on the available reboiler energy, that is, of the capacity of the installed pump. For the kettle and internal reboilers there is a relationship between CR and BR:

(11)

If the equilibrium line can be approximated with a straight line with slope m y* = mx + n

s ĠN + 1

Ġ BR CR = = s = ̇ ̇ ̇ L in BR + 1 GN + 1 + W

s ̇ s = (L̇ rec + ĠN L̇ rec + LN + 1) + Ẇ

(12)

s and if the vapor (Ġ N+1 ) and liquid (L̇ rec) leaving the reboiler are in equilibrium (tie line t2=const in Figure 3), then considering that the mixing of the phases which flow through the pipes is intensive yields yN + 1 = mx rec + n (13)

(7)

Unlike the recirculating and forced circulation reboilers, CR of the kettle and internal reboilers depends on the slope of the operating line of the stripping section (L̇ /Ġ )ss. In literature, reboilers are schematically presented (in respect to separation efficiency) as shown in Figure 1, in which Q̇ reb, W, represents the heat duty of the reboiler. Figure 1a shows a reboiler which is considered to have the efficiency which is equal to the efficiency of one theoretical stage (oncethrough, kettle and internal reboiler).1,9 Using the example of the kettletype reboilers, Treybal9 states: “The kettle reboiler, with the heating medium inside the tubes, provides a vapor to the tower essentially in equilibrium with the residue product and then behaves like a theoretical stage”. Figure 1b schematically presents a reboiler which is considered to have no separation effects. This means that the mass transfer in recirculating and forced circulation reboiler can be disregarded and Treybal9 explains it as follows: “The vertical thermosyphon reboiler with the heating medium outside the tubes, can be operated as to vaporize all the liquid entering it to produce vapor of the same compositions as the residue product, in which case no enrichment is provided.” By all means this approach is conservative as the real fraction of the more volatile component in the bottom product is certainly less than the estimated value (the estimated separation efficiencies are lower than the real values), which will be shown in further text.

In Figure 3 p (Pa) is the system pressure, t (°C) is the temperature, h (kJ/kmol) represents the specific enthalpy, and the subscripts Lin, Lout, and Gout, denote the liquid at the reboiler inlet, liquid at the reboiler outlet, and vapor at the reboiler outlet, respectively. Considering that the ratio between the flow rates of the liquid leaving the Nth tray and the recycle stream is s ̇s ̇ s ĠN LN LN +1 = (L̇ /Ġ)ss EF = s ̇ ̇ ̇ L rec GN + 1 L rec

(14)

by solving the system of eqs 10, 11, and 13 the unknown compositions of bottom product, vapor, and recycle stream are obtained xW =

(L̇ /Ġ)ss xN (1 + EFm) − n (L̇ /Ġ)ss (1 + EFm) + m − 1

yN + 1 = x rec = 5795

(L̇ /Ġ )ss [(mxN + n) + EFmxN ] − n (L̇ /Ġ )ss (1 + EFm) + m − 1

(L̇ /Ġ)ss [EF(xN − n) + xN ] − n (L̇ /Ġ )ss (1 + EFm) + m − 1

(15)

(16)

(17)



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required to have separation in the reboiler which is equivalent to one theoretical stage.11 These high ratios are common with forced circulation reboilers. This means that the forced circulation reboilers can have separation effects which are nearing to a theoretical stage. Figure 3 shows the recirculating and forced circulation reboiler process in equilibrium and enthalpy-concentration diagrams. In the case when EML < 1, the line 3−2 in the phase diagram has greater slope than the isotherm of the heterogeneous region which passes through point 4 (t4=const).9 Note: in the enthalpy−concentration diagram there are four streams: stream number 1 represents boiling (saturated) liquid which leaves the Nth tray; stream number 2 is saturated vapor which leaves the reboiler (which comes to the Nth tray); stream number 3 is column bottom product; stream number 4 is the mixture which would be formed by streams 1 and 2 by using the lever rule. The tie line t2=const, which determines the compositions of liquid and vapor which leave the reboiler, passes through the point which is in the heterogeneous region and has the S coordinates (xW; hW + Q̇ reb/(L̇ rec + Ġ N+1 )).

3. SEPARATION EFFICIENCY OF KETTLE REBOILERS If the axial mixing of liquid is present in the reboiler, the mathematical model is based on setting the balance equations for an elementary surface (dS) L̇ = dĠ + (L̇ + dL̇)

as shown in Figure 4 where the following notation was used: DAM, (m2/s) coefficient of axial mixing; ρL, (kg/m3) liquid density; ML, (kg/kmol) liquid molar mass; AL, (m2) crosssectional area of the liquid flow path; z, (m) length coordinate; ZL, (m) length of the liquid flow path. The mass balance equation for the more volatile component (according to Figure 4) is obtained by using the presumption that in the observed cross section there is the ideal mixing of liquid which means that the vapor which is extracted locally is in equilibrium with the liquid (y = y*(x)).

Figure 3. The recirculating and forced circulation reboiler process shown in equilibrium and enthalpy composition diagrams.

It has to be noted that eq 15 can be used for the inverse task: determination of xN when xW is known. Reboiler efficiency can be defined as the Murphree efficiency for the liquid phase (mxN + n) − xN (L̇ /Ġ)ss EF[(mxN + n) − xN ] + (mxN + n) − xN 1 = 1 + (L̇ /Ġ)ss EF

(19)

EML =

⎛ ρ ∂x ⎞ Lẋ + ⎜ −DAM L ⎟AL = dGẏ * + (L̇ + dL̇)(x + dx) ML ∂z ⎠ ⎝ ⎡ ρ ∂⎛ ∂x ⎟⎞⎤ ⎜x + dz ⎥AL +⎢ −DAM L ML ∂z ⎝ ∂z ⎠⎦ ⎣ (20)

(18)

In this case the reboiler efficiency is EML < 1, as given in ref 1 and explained, but not clearly quantified in ref 10. It is obvious that the efficiency will be EML → 1 for EF → 0, that is, when L̇ rec → ∞. Very high recycle to bottom flow rate ratios are

Figure 4. Schematic representation of continuous evaporation with axial mixing along liquid flow. 5796



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the steam condensation temperature (thot ≈ tcond). Hence, it can be concluded that the aforementioned assumption q̇(z)=const is proven in engineering practice. It must be noted that there are rare cases when this assumption is not valid. When heat transfer flux is constant then the liquid evaporation is uniform, and it can be shown that z ̇ ̇ ) = L̇ in − ς(L̇ in − Lout ̇ ) L̇ = L̇ in − (L in − Lout ZL

Equation 19 yields dĠ = −dL̇

(21)

On the basis of eqs 20 and 21 the following is obtained: DAM ρ L d2x dx 1 dL̇ + AL − (y* − x) = 0 2 ̇ L ML dz dz L̇ dz

(22)

By using the transformation z = ZL ς

(30)

(23)

which can be prearranged into

the following is obtained:

⎛ BR ⎞ ⎟ L̇ = L̇ in⎜1 − ς ⎝ 1 + BR ⎠

DAM ρ L d2x dx 1 dL̇ * + AL − (y − x ) = 0 2 2 ̇ L ML ZL dς ZL dς L̇ ZL dς

and

(24)

dL̇ BR = −L̇ in dς 1 + BR

where ς is the dimensionless length. Now the Peclet number, which defines the axial mixing intensity along the reboiler, can be introduced ̇ LML LZ Pe = DAM AL ρ L

1 d2x dx 1 − + BR Pe dς2 dς L̇ in 1 − ς

(25)

(

1 d2x dx − − A(y* − x) = 0 2 Pe dς dς

(27)

(33)

(34)

where Pe =

(28)

̇ LML LZ DAM AL ρ L

= L̇ in

To obtain an integral solution to the hereby discussed problem, the constant heat flux assumption along the reboiler (q̇(z)=const) needs to be introduced. In the engineering practice, this assumption is practically always valid, but it has to be elaborated in more detail. Local heat flux equals q ̇ = k(t hot − tcold)

)

And finally, the following linear nonhomogenous second-order differential equation with variable coefficients is obtained

(26)

that is

̇ Lout 1 = ̇ L in 1 + BR

1 + BR

⎛ BR ⎞ ⎜ − L̇ ⎟ ⎝ in 1 + BR ⎠

× (y* − x) = 0

Considering that for the kettle type reboilers the boilup ratio is ̇ L̇ − Lout L̇ Ġ BR = = in = in − 1 ̇ ̇ ̇ Lout Lout Lout

(32)

Replacing eqs 31 and 32 into eq 26 derives

and thus eq 24 becomes 1 d2x dx 1 dL̇ * − + (y − x ) = 0 Pe dς2 dς L̇ dς

(31)

ZLML ⎛ BR ⎞ ⎜1 − ς ⎟ DAM AL ρ L ⎝ 1 + BR ⎠

⎛ BR ⎞ ⎟ = KPe⎜1 − ς ⎝ 1 + BR ⎠

ZLML DAM AL ρ L

(36)

BR 1 + BR − ςBR

(37)

KPe = L̇ in

(29)

where k, W/(m2·K), represents the overall heat transfer coefficient, and thot and tcold, both in °C, represent hot and cold fluid temperatures, respectively. Because of the change of liquid composition and flow rate along the heat exchanger, all of its significant properties change (i.e., bubble and dew point, density, heat capacity, viscosity, etc). This means that the overall heat transfer coefficient changes along the reboiler, but in a relatively small range (i.e., k ≈ constant). In most cases the less volatile component (component B) prevails in the cold fluid and the fraction of the more volatile component changes over a very small range. Therefore, the cold fluid temperature is practically constant (tcold = tBP ≈ tB,BP) along the heat transfer surface. Hot fluid is usually saturated steam, which means that its temperature changes only due to the pressure drop. Since the pressure drop has little or no influence on the change of tcond, the hot fluid temperature can be regarded constant and equal to

A=

(35)

By considering that the axial liquid mixing coefficient (DAM) is constant along the reboiler and that it is obtained for the averaged liquid flow, it can be shown that the Peclet number is linearly dependent on the dimensionless coordinate ς. Equation 34 can be solved with the following boundary conditions: • at the point where liquid enters the reboiler L inx in +

DAM AL ρ L ⎛ dx ⎞ ⎜ ⎟ = L inx(z = 0) ⎝ dz ⎠ z = 0 ML

(38)

or in the dimensionless form ⎛ dx ⎞ 1 + x in = x(ς = 0) ⎜ ⎟ (Pe)ς= 0 ⎝ dς ⎠ς= 0 5797

(39)



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• at the point where liquid leaves the reboiler ⎛ dx ⎞ ⎜ ⎟ =0 ⎝ dz ⎠ z = z

case, the discriminate of the characteristic equation is always greater than 0 and is 2 D = Peav + 4Aav (m − 1)Peav

(40)

L

The solution of this differential equation is

or in the dimensionless form

⎛ dx ⎞ =0 ⎜ ⎟ ⎝ dς ⎠ς= 1

x(ς) = C1exp(r1ς) + C2exp(r2 ς) + C3

dx(ς) = C1r1 exp(r1ς) + C2r2 exp(r2 ς) dx

If the equilibrium line is approximated with a straight line (y* = mx + n), a second-order linear ordinary differential equation is obtained

d2x(ς) dx 2

(42)

⎛ dx ⎞ =0 ⎜ ⎟ ⎝ dς ⎠ς= 1

(43)

1

∫0

1 (C1r1 + C2r2) + x in = C1 + C2 + C3 Peav

(54)

C1r1 exp(r1) + C2r2 exp(r2) = 0

(55)

x (m − 1) + n C1 = in m−1

(44)

×

x (m − 1) + n C2 = − in m−1

⎛ BR ⎞ 2 + BR ⎟d ς = K KPe⎜1 − ς Pe ⎝ 1 + BR ⎠ 2(1 + BR)

r1 exp(r1 − r2) r2 × r r 1 − 1 exp(r1 − r2) − 1 [1 − exp(r1 − r2)] r2 Pe

1

BR dς 1 + BR − ςBR 0 d(1 + BR − BRς) = 1 + BR − ςBR 1

∫0

C3 = −

∫

(47)

B=

and its characteristic equation is = (48)

From here, it is obvious that the roots of the characteristic equation are Pe ⎡ r1/2 = av ⎢1 ± 2 ⎢⎣

A (m − 1) ⎤ ⎥ 1 + 4 av Peav ⎦⎥

(59)

In this case, Murphree efficiency for the liquid phase is calculated according to eq 1. By replacing

In this case, the differential eq 42 becomes

1 2 r − r − Aav (m − 1) = 0 Peav

(58)

xout = x(ς = 1) = C1 exp(r1) + C2 exp(r2) + C3

(46)

1 d2x dx − − Aav (m − 1)x = Aav n dς Peav dς2

n m−1

(57)

The fraction of more volatile component in the liquid at the outlet is

= ln(1 + BR − ςBR)|10 = ln(1 + BR)

1 r r 1 − 1 exp(r1 − r2) − 1 [1 − exp(r1 − r2)] r2 Peav (56)

(45)

Aav =

(53)

Now, the constants of integration can be obtained

Equation 42 can be solved numerically, after which the reboiler efficiency EML can be determined based on known values for xin and xout = x(ς = 1). This problem can be simplified and it can come down to solving a linear second-order differential equation with constant coefficients if the variables Pe and A can be averaged (subscript av) along the reboiler. These averaged values would be Peav =

= C1r12 exp(r1ς) + C2r2 2 exp(r2 ς)

(52)

Boundary conditions are

with the following boundary conditions: ⎛ dx ⎞ 1 + x in = x(ς = 0) ⎜ ⎟ (Pe)ς= 0 ⎝ dς ⎠ς= 0

(51)

And the first and second derivatives of the general solution are

(41)

1 d2x dx − − A(m − 1)x = An dς Pe dς2

(50)

y*(xout) − xout y*(x in) − x in r exp(r1) − 1 exp(r1) r 2

r r 1 − 1 exp(r1 − r2) − 1 [1 − exp(r1 − r2)] Pe r2

(60)

and bearing in mind that x (BR + 1) − xout yout = in BR

(49)

(61)

reboiler efficiency becomes

The case which will be regarded in this paper is the case when m > 1, which is always fulfilled in the case of distillation when the reboiler is located underneath the stripping column. In this

EML = λ reb 5798

1−B λ reb − (1 + BR) + B

(62)



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where the reboiler stripping factor equals λ reb = mBR = m

Ġ ̇ Lout

(63)

Reboiler efficiency can be determined in two ways: (1) by solving the ordinary differential equation with variable coefficients, eq 42, which yields an exact solution and (2) by solving the ordinary differential equation with averaged (constant) coefficients, eq 47, which yields an approximate solution. Reboiler efficiency depends on three parameters: slope of the operating line, m; Peclet number, Peav; boilup ratio, BR. In general, smaller deviations of the approximate solution in regards to the accurate solution are gained with greater values of the slope coefficients of the equilibrium line m, greater values of the Peclet numbers Peav, and smaller values of the boilup ratio BR. It has to be noticed that in the case of plug flow when Peav → ∞, the ordinary differential equation with averaged coefficients yields the correct solution for reboiler efficiency for every m and every BR. In general, for slope coefficients m < 5 it is recommended to use the ordinary differential equation with variable coefficients, regardless of Peav and BR. In other cases it is possible to use the ordinary differential equation with constant coefficients, but it is necessary to consider the following facts: (1) for Peav < 1 and m = 5 the calculation using the ordinary differential equation with the averaged (constant) coefficients yields the reboiler efficiency deviations of up to 15% in regards to the exact solution (the solution obtained using the ordinary differential equation with variable coefficients); (2) for Peav ≥ 1 and m = 5 the deviations can be up to 10%; (3) for Peav ≥ 1 and m > 5 the deviations can be up to 5%, and therefore in this case it is recommended to use the ordinary differential equation with averaged coefficients as it is a much simpler method. Figure 5 shows the kettle reboiler process in equilibrium and enthalpy−concentration diagrams. In the case when EML > 1 the line 2−3 in the phase diagram has a smaller slope than the tie line t4=const of the heterogeneous region, which passes through point 4. This case has not been presented and discussed in the available literature. Although it is not explicitly noted in Figures 5 and 6, the line 1−4 in these figures represents the ratio Q̇ reb/L̇ in, same as in Figure 3. In this case for every Pe > 0 Murphree stage efficiency is EML > 1. In addition, the efficiency rises as the axial mixing intensity declines, that is, as the Peclet number rises (more favorable separation effects are achieved with greater Peclet numbers). It is considered that for this type of reboilers EML = 1,1,9 while it is shown in this paper that the separation effects for this type of reboilers are in every case greater than the separation effects of one theoretical stage. In cases when Pe → ∞, the fluid flow in the reboiler acts as a plug flow model. In case of the plug flow model, the greatest separation effects are achieved. For shell-and-tube heat exchangers the high values of Peclet numbers (Pe > 40) have nearly the same effect as an infinite value of Pe, and the plug flow model can be applied.8 In case of the kettle reboilers for BR ≥ 1 and Pe ≈ 30 and above the flow has the separation effects which are consistent with the plug flow model. For BR < 1 this occurs for Pe ≈ 40 and above. In case when Pe → 0, the flow starts to act according to the ideal mixing model, when the Murphree stage efficiency

Figure 5. The kettle reboiler process shown in equilibrium and enthalpy−concentration diagrams.

is EML = 1. For shell-and-tube heat exchangers the ideal mixing model can be used for Pe ≤ 0.1. For shell-and-tube heat exchangers it is shown that there is no significant difference between configurations with 11, 17, and 25 baffles for the Peclet number at equal fluid flow rates. Reducing the number of baffles to 7, however, leads to a decrease of the Peclet number for the whole range of the fluid flow rates.7 The conclusion is that it is necessary to provide such a geometrical configuration of the heat exchanger which will prevent the intensive axial mixing of the liquid, which will provide more favorable separation effects. It is possible to achieve this by introducing a greater number of baffles along the pipe bundle. Further experiments will be needed to determine exactly the dependence of the Peclet number on the geometry. The Peclet number depends only on the geometry and not on the Reynolds number. However, this must be verified by further experiments.8 3.1. Plug Flow Model. The boundary separation effects are achieved in the case of the plug flow of liquid. In this case, the reboiler has the greatest separation efficiency, which will be shown in the further text. It should also be stressed that the separation effects in this case are the boundary separation effects regardless of the type of the reboiler. 5799



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Integration of eq 65 yields x(ς) =

⎞m − 1 y*(x in) − x in ⎛ BR n ⎜1 − ς⎟ − ⎝ ⎠ m−1 1 + BR m−1 (67)

Integration of eq 66 along the reboiler yields x in

1 dx BR = dς = ln(1 + BR) 0 1 + BR − ςBR out y* − x

∫x

∫

(68)

By combining eqs 66 and 68 it is clear that ln

y*(x in) − x in * y (xout) − xout

m−1

= ln(1 + BR)

(69)

The following coefficient can be introduced BPF =

y*(xout) − xout 1 = y*(x in) − x in (1 + BR)m − 1

(70)

and finally, the reboiler efficiency for the plug flow of liquid is EML,PF = λ reb

1 − BPF λ reb − (1 + BR) + BPF

(71)

Ideal Solution. For the ideal solutions the phase equilibrium is defined by using the relative volatility α αx y* = 1 + (α − 1)x (72) Considering eqs 68 and 72 ⎡⎛ 1/ α− 1 ⎛ 1 − xout ⎞α / α− 1⎤⎥ x in ⎞ dx ⎢ = ln ⎜ ⎟ ⎜ ⎢⎝ x ⎟⎠ ⎥ xout y* − x ⎝ 1 − x in ⎠ ⎣ out ⎦

∫

Figure 6. The continuous evaporation process when the liquid is ideally mixed shown in equilibrium and enthalpy−concentration diagrams.

(73)

In the case of the plug flow of liquid (Pe → ∞) a linear nonhomogenous first-order differential equation is obtained dx BR − = (y* − x) dς 1 + BR − ςBR

x in

it is obvious that ⎛ x in ⎞⎛ 1 − xout ⎞α ⎜ ⎟⎜ ⎟ = (1 + BR)α− 1 ⎝ xout ⎠⎝ 1 − x in ⎠

(64)

(74)

which yields

or dx BR dς − = * 1 + BR − ςBR y −x

(1 − xout)α (1 − x in)α = (1 + BR)α− 1 xout x in

(65)

By using the method of trial and error, on the basis of eq 75 the outlet fraction of more volatile component in the liquid (xout) can be determined. Experimentally Defined Equilibrium Line. In general, if the solution is not ideal and if the equilibrium line cannot be approximated by a straight line, then xout can be determined based on the experimental data by using eq 68. Finally, on the basis of the known liquid compositions at inlet and outlet (xin and xout) it is possible to determine the composition of vapor:

Three different ways of defining the system equilibrium will be considered in this paper: equilibrium line is straight; ideal solutions; equilibrium line determined experimentally. Straight Equilibrium Line. If the equilibrium line is straight, y*(x) = mx + n, then

∫x

x in

x in dx dx = (mx + n) − x x y* − x

∫

ln =

=

y*(x in) − x in y*(x) − x

x (BR + 1) − xout yout = in BR

m−1 y*(x in) − x in ln (m − 1)x + n

m−1

(75)

(76)

as well as the reboiler efficiency by using eq 1. 3.2. Ideal Mixing Model. If there is the ideal mixing of the liquid in the reboiler, then the separation effects are equal to

(66) 5800



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Figure 7. Schematic representation of oncethrough reboiler (a) and internal reboiler (b).

Example 1. A mixture of methanol (more volatile component A) and water (component B) which contains xin = 0.02 kmol A/kmol (A+B) of water at the pressure of p = 1.0 bar enters a reboiler. Mole flow rate of liquid at the reboiler inlet is L̇ in = 260 kmol/h. The slope of the operating line of the stripping section of the column is (L̇ /Ġ )ss = 2.0376. In the region in which the liquid changes its composition the equilibrium line can be approximated with a straight line y* = 6.7x (m = 6.7). Determine the following: (a) composition and flow rate of vapor at the outlet as well as the reboiler efficiency for the plug flow of liquid, as a best separation effect; (b) same as previous, but for the ideal mixing of liquid; (c) same as previous, but for the axial mixing of liquid and Peav = 5, (d) show the change of the composition of liquid for all three cases along the reboiler; (e) for these working conditions show the influence of the intensity of the axial mixing on the Murphree efficiency; (f) determine the efficiency of the recirculating reboiler, compositions, and flow rates of the characteristic phases, if the outlet fraction of vapor is EF = 0.25. Solution. a. Plug Flow of Liquid. Boilup ratio is directly dependent on the slope of the operating line of the stripping column

the ones of the theoretical stage. In this case, as shown in Figure 6, the line that connects points 2 and 3 is a tie line, and the outlet point of the process “Out” is located on the equilibrium line. This process is shown in Figure 6. In case when EML = 1, the line 2−3 coincides with the tie line of the heterogeneous region which passes through point 4, because there is the ideal mixing of the liquid inside the reboiler. The situation shown in Figure 6 is consistent with the separation effects which are equal to those of a theoretical stage and which are valid for the oncethrough (Figure 7a) and internal reboiler (Figure 7b). L̇ y*(xout) − x in Ẇ 1 − out = − s =− = ̇ ̇ G BR xout − x in GN + 1

(77)

which yields x (BR + 1) − n BR x (BR + 1) − n BR xout = in = in λ reb + 1 BRm + 1 (78)

In general, the slope of the straight line that connects the inlet (In) and outlet (Out) points (inlet and outlet streams) is y − x in L̇ Ẇ 1 − out = − s =− = out Ġ BR xout − x in ĠN + 1 1 = ⎛ xin dx ⎞ ⎟ 1 − exp⎜∫ ⎝ xout y * −x ⎠

BR =

in the case of the plug flow of liquid, the composition of the liquid at the reboiler outlet is determined using eq 67 (79)

xout = x(ς = 1)

For an ideal solution with constant relative volatility, eq 79 becomes y − x in L̇ Ẇ − out = − s = out Ġ xout − x in ĠN + 1 1 = 1/ α− 1 ⎛x ⎞ ⎛ 1 − xout ⎞α / α− 1 ⎜ ⎟ 1 − ⎜ in ⎟ ⎝ 1 − xin ⎠ ⎝ xout ⎠

1 = 0.9637 (L̇ /Ġ )ss − 1

=

⎞m − 1 y*(x in) − x in ⎛ BR n ⎜1 − ς⎟ − ⎝ m−1 1 + BR ⎠ m−1

= 0.000427 kmol A/kmol (A+B)

Vapor composition at the reboiler outlet can be determined using eq 76 (80)


4. EXAMPLES The following examples illustrate the proposed models.

= 0.040309 kmol A/kmol (A+B) 5801



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at the reboiler inlet ς = 0 and thus the mole fraction of the component A at the inlet is

Mole flow rates of the liquid and vapor that leave the reboiler are, respectively, ̇ = Lout

L̇ in = 132.4 kmol/h 1 + BR

x(ς = 0) = 0.013249 kmol A/kmol (A + B)

and according to eq 59, the mole fraction of the component A in the liquid at the outlet is

̇ = 127.6 kmol/h Ġ = L̇ in − Lout

xout = x(ς = 1)

Finally, according to eqs 63, 70, and 71, the exchanger efficiency in case of the plug flow of liquid equals

= C1 exp(r1) + C2 exp(r2) + C3

1 − BPF EML,PF = λ reb = 1.40 λ reb − (1 + BR) + BPF

= 0.001386 kmol A/kmol (A+B)

and according to eq 62, the mole fraction of the component A in the vapor at the outlet is

where

λ reb = mBR = 6.4571 BPF =

1 (1 + BR)m − 1


= 0.021351

= 0.039314 kmol A/kmol (A+B)

b. Ideal Mixing of Liquid. In this case, the mole fraction of the more volatile component (A) at the outlet streams can be determined using eq 80 and the fact that the vapor is in equilibrium with the liquid, so xout = 0.005267 kmol A/kmol (A+B) and yout = 0.035287 kmol (A+B). Reboiler efficiency in case of the ideal mixing of liquid is EML,IM = 1. c. Partial Mixing of Liquid (Axial Mixing). Since in this case Peav = 5 (greater than 1), and m = 6.7 (greater than 5), as mentioned beforehand, the error that is made by using the averaged values in regards to the more complex solving of the ordinary differential equation with the variable coefficients is less than 5%. Therefore, the ordinary differential equation with the constant (averaged) coefficients will be used. According to eq 46, in this case the averaged dimensionless coefficient A is

Coefficient B can be determined using eq 60 and equals B=

r exp(r1) − 1 exp(r1) r 2

r r 1 − 1 exp(r1 − r2) − 1 [1 − exp(r1 − r2)] r2 Pe

= 0.069315

and finally, according to eq 63, the reboiler efficiency equals EML = λ reb

1−B = 1.317 λ reb − (1 + BR) + B

d. Composition Change. Figure 8 shows the change of the mole fraction of the more volatile component in the liquid along the reboiler.

Aav = ln(1 + BR) = 0.6749

According to eq 49, the roots of the characteristic equation will be r1/2 =

Peav ⎡ ⎢1 ± 2 ⎢⎣

A (m − 1) ⎤ ⎧ 7.5481 ⎥=⎨ 1 + 4 av ⎥⎦ ⎩−2.5481 Peav

According to eqs 56, 57, and 58, the constants of integration C1, C2, and C3 are, respectively, x (m − 1) + n C1 = in m−1 ×

1 r r 1 − 1 exp(r1 − r2) − 1 [1 − exp(r1 − r2)] r2 Peav −7

Figure 8. Change of the composition of liquid along the reboiler in case of axial mixing using the ordinary differential equation with the constant coefficients.

= 1.84424 × 10

x (m − 1) + n C2 = − in m−1 ×

r 1− 1 r2

C3 = −

e. Influence of the Intensity of the Axial Mixing on the Murphree Efficiency. The influence of the axial mixing on the reboiler efficiency is shown in Figure 9. It is obvious that the reboiler efficiency approaches the efficiency for the plug flow of liquid, as mentioned beforehand. f. Efficiency of the Recirculating Reboiler. Mole flow rate of the vapor that leaves the reboiler is

r1 exp(r1 − r2) r2 = 0.013248 r exp(r1 − r2) − 1 [1 − exp(r1 − r2)] Pe

n =0 m−1

Now, the change of the mole fraction of the more volatile component in the liquid along the reboiler can be expressed using eq 51

s ĠN + 1 =

x(ς) = C1 exp(r1ς) + C2 exp(r2 ς) + C3 5802

̇s LN = 127.6 kmol/h (L̇ /Ġ)ss dx.doi.org/10.1021/ie202193m | Ind. Eng. Chem. Res. 2012, 51, 5793−5804


Article

Example 2. A mixture of benzene (more volatile component A) and toluene (component B) is treated in a distillation column at the atmospheric pressure (p = 1.0 bar). The liquid flow rate at the reboiler inlet is L̇ in = 212.5 kmol/h, and the equation of the operating line of the stripping section of the column is y = 1.415x − 0.012. The mixture in question can be regarded as an ideal mixture with the relative volatility α = 2.42. Determine the (a) composition and flow rate of liquid at the outlet as well as the flow rate of the vapor at the reboiler outlet; (b) the composition of the liquid at the reboiler inlet, composition of the vapor at the reboiler outlet, and reboiler efficiency for the plug flow of liquid; and (c) same as pervious, but for the ideal mixing of liquid. Solution. a. Composition and Flow Rate. Composition of liquid at the reboiler outlet (bottom product) can be determined on the basis of the equation of the operating line:

Figure 9. Influence of the intensity of the axial mixing on the Murphree efficiency.

xout = 1.415xout − 0.012 → xout = 0.029682 kmol A/kmol (A+B)

The slope of the operating line of the stripping column is (L̇ / Ġ )ss = 1.415, and so according to eq 2 the boilup ratio equals BR = 2.409639. Now the liquid and vapor flow rates at the reboiler outlet can be easily determined L̇ out = 62.3kmol/h and Ġ = 150.2 kmol/h. b. Plug Flow. On the basis of eq 75, which is valid for the plug flow of liquid and ideal solutions, the following is obtained:

Considering eq 4, the mole flow rate of the recycle stream is L̇ rec =

s ĠN + 1 = 510.4 kmol/h EF

The mole fraction of component A in the bottom product can be determined by using eq 15 (L̇ /Ġ)ss xN (1 + EFm) − n (L̇ /Ġ)ss (1 + EFm) + m − 1

xW =

(1 − xout)α (1 − x in)α = (1 + BR)α− 1 xout x in

= 9.776 × 10−3 kmol A/kmol (A+B)

and using the trial and error method xin = 0.130066 kmolA/ kmol(A+B). According to eq 77 the composition of vapor at the outlet is yout = 0.171725 kmolA/kmol(A+B) and finally, using eq 1, the reboiler efficiency is EML = 1.962. The composition of liquid which is in equilibrium with the outlet vapor can be determined by using eq 72, as the equilibrium for the ideal mixtures is defined by using the relative volatility:

and according to eq 16 the mole fraction of component A in the vapor that leaves the reboiler is yN + 1 =

(L̇ /Ġ )ss [(mxN + n) + EFmxN ] − n (L̇ /Ġ)ss (1 + EFm) + m − 1

= 0.031 kmol A/kmol (A+B)

According to eq 17 the mole flow rate of the more volatile component in the recycle stream is x rec =

x*(yout ) =

= 0.078912 kmol A/kmol (A+B)

c. Ideal Mixing. In the case of ideal mixing, the composition of liquid which enters the reboiler can be determined using eq 79:

(L̇ /Ġ )ss [EF(xN − n) + xN ] − n (L̇ /Ġ)ss (1 + EFm) + m − 1

= 4.568 × 10−3 kmol A/kmol (A+B)

−

Finally, the reboiler efficiency can be calculated using eq 18 EML =

yout α − (α − 1)yout

y*(xout) − x in 1 = BR xout − x in

where

1 = 0.663 ̇ 1 + (L /Ġ )ss EF

y*(xout) =

Summary. The composition of the bottom product in each of the cases is as follows: for the plug flow of liquid, xW = xout = 0.427 × 10−3 kmolA/kmol(A+B); EML,PF = 1.40; for the ideal mixing, xW = xout = 5.267 × 10−3 kmolA/kmol(A+B); EML,IM = 1; for axial mixing, xW = xout = 1.386 × 10−3 kmolA/kmol(A+B); EML = 1.317; for the recirculating reboiler, xW = 9.776 × 10−3 kmolA/kmol(A+B); EML = 0.663. It is obvious that the worst separation effects are achieved in the case of the ideal mixing of liquid, while the best separation effects are achieved in case of the plug flow of liquid. The most veritable method for prediction of the reboiler efficiency is the axial mixing model, which is between the two boundary models ideal mixing and plug flow. In any case, the recirculating reboiler has much worse separation effects than kettle type reboiler.

αxout = 0.068925 kmol A/kmol (A+B) 1 + (α − 1)xout

Again, by using the method of trial and error, the composition of liquid at the inlet is obtained: x in = 0.057416 kmol A/kmol (A+B)

Composition of vapor at the reboiler outlet is determined as yout = 0.068925 kmolA/kmol(A+B) by using eq 76 and the composition of liquid which is in equilibrium with the outlet vapor is x*(yout) = 0.029682 kmolA/kmol (A+B). Finally, the reboiler efficiency is EML = 1.

5. CONCLUSION Murphree stage efficiency, despite all of its shortcomings, is commonly used for expressing the rate of mass transfer in stage 5803



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ς = dimensionless coordinate

unit operations. It has been shown in this paper that the common belief (which is particularly true for kettle type reboilers) that the reboiler equals to one theoretical stage is not always valid. It has been shown, both theoretically and numerically in two examples that the reboiler efficiencies can vary. The boundary effects are achieved for two models: plug flow (the best separation effects) and ideal mixing (the worst separation effect). However, the truth is somewhere in between, and is up to date best described with the model of axial mixing. It has also been shown on an example that as the axial mixing increases (Peclet number decreases), the separation effects inside the reboiler approach the ideal mixing model, and vice versa as the axial mixing decreases (Peclet number increases), the separation effects approach the plug flow model. Although this paper has studied the binary mixtures, the same principle can easily be adopted for any multicomponent system.

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Subscripts

av = averaged BP = boiling point cold = cold fluid G = gas (vapor) hot = hot fluid IM = ideal mixing in = reboiler inlet L = liquid N = number of trays out = reboiler outlet PF = plug flow reb = reboiler rec = recycle stream ss = stripping section of the column W = bottom product Superscripts

AUTHOR INFORMATION

■

Corresponding Author

*Fax: +381 11 3370364. Tel.: +381 11 3302360. E-mail: nikola. [email protected].

* = state of equilibrium S = stripping section of the column

REFERENCES

(1) Design Practices Committee Fractionation Research Inc. Reboiler Circuits for Trayed Columns. Chem. Eng. 2011, 2011 26−35; www. che.com. (2) Buckley, P. S.; Luyben, W. L.; Shunta, J. P. Design of Distillation Column Control Systems; Instrument Society of America: Huntington Beach, CA, 1985. (3) Couper, J. R. Chemical Process Equipment: Selection and Design; Gulf Professional Publishing: Oxford, U.K., 2005. (4) Aleksandrov, I. A. Rektifikacionie i Absorbcionie Apparati; Himia: Moscow, 1971. (5) Cao, E. Heat Transfer in Process Engineering, McGraw Hill Professional: New York, 2009. (6) Tammami, B. How to Select the Best Reboiler for Your Processing Operation. Hydrocarbon Process. 2008, 87 (3), 91−94. (7) Roetzel, W.; Balzereit, F. Axial Dispersion in Shell-and-Tube Heat Exchangers. Int. J. Therm. Sci. 2000, 39, 1028−1038. (8) Roetzel, W.; Lee, D. Experimental Investigation of Leakage in Shell-and-Tube Heat Exchangers with Segmental Baffles. Int. J. Heat Mass Transfer 1993, 36, 3765−3771. (9) Treybal, R. E. Mass Transfer Operations, 3rd ed; McGraw-Hill Inc.: Singapore, 1981. (10) Furzer, I. Vertical Thermosyphon Reboilers. Maximum Heat Flux and Separation Efficiency. Ind. Eng. Chem. Res. 1990, 29, 1396− 1404. (11) Keskinen, K. I.; Nyman, T.; Björk, J.; Aittamaa, J. Considering The Non-Ideality of Reboilers in the Calculation and Design of Distillation Columns. AIChE Annual Meeting, Indianapolis, November 3−8, 2002.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the Ministry of Science and Technological Development of Serbia for a partial support to this study through the Project of Energy Efficiency.

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NOTATION A = dimensionless coefficient AL = cross-sectional area of liquid flow path, m2 B = dimensionless coefficient BR = boilup ratio CR = circulation rate C = constant of integration D = coefficient of axial mixing, m2/s D = discriminant of equation EML = Murphree stage efficiency for liquid phase EF = outlet fraction of vapor Ġ = vapor mole flow rate, kmol/h ∼ Ġ = vapor mass flow rate, kg/h h = enthalpy, J/kmol k = overall heat transfer coefficient, W/(m2·K) KPe = constant of the Peclet number m = slope of the equilibrium line M = molar mass, kg/kmol p = pressure, bar Pe = peclet number Q̇ = heat duty, W q̇ = heat transfer flux, W/m2 t = temperature, °C Ẇ = mole flow rate of bottom product, kmol/h x = mole fraction in liquid, kmol/kmol y = mole fraction in vapor, kmol/kmol z = length coordinate, m ZL = length of liquid flow path in the reboiler, m α = relative volatility λ = stripping factor ρ = fluid density, kg/m3 5804


Reboiler Separation Efficiencies for Binary Systems - Industrial

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