Mathematical Modeling of the Sorption of Volatile Components in

24 Mathematical Modeling of the Sorption of Volatile Components in Newtonian, HighViscous Liquids with the Aid of Bubbling

Downloaded by RUTGERS UNIV on May 18, 2017 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1974-0133.ch024

JOHANNES P. ROOS Corporate Research Department, Akzo Research Laboratories, Arnhem, The Netherlands

With the aid of capillaries gas is bubbled through a viscous liquid contained in a vessel to add or extract some volatile components to or from the viscous liquid. A general mathematical model is presented for this sorption. Theoretical and experimental litera ture data on bubble formation, velocity of rise, and mass transfer, especially with regard to viscous liquids are discussed. The results of lab-scale experiments for bubble behavior in polyisobutene (viscosity: 15-50 kgrams/msec) agree well with theoreti cal predictions. The mathematical model was used to calculate the desorption rates of water and lactam from a nylon 6 melt, and these results agree well with data from pilot-plant measure ments.

T

his paper deals with a general steady-state sorption process. A diagram of a continuously operating, gas-liquid contacting reactor is shown in Figure 1. Liquid is fed at a volumetric flow rate V; gas is injected through η capillaries at a total volumetric gas flow rate G; and the temperature is constant through out the reactor. The total amount of mass absorbed into or desorbed from the liquid is small compared wih the liquid mass. Thus, the liquid density and the liquid flow rate can be assumed to be constant. Consider a number of I components indicated as i = 1, 2, . . . , I. The concentrations in the liquid are c . The liquid is described as an ideal mixer because in the lab-scale experiments (discussed below) liquid circulations induced by the rising bubbles were visible. The circulation time (about 1 min) is shorter than the liquid residence time (about 1 hr). If necessary, other types of flow patterns can easily be incorporated into the model. Let c ^ be the concentration of component i in the liquid feed, and let Q (0) and Ç * ( H ) be the molar flow rates of component i in the bubble phase entering and leaving the liquid, respectively. The overall mass balances can be written as: l

1

l

c» V +

(H) = c-

m

V + Q (0) {

303

Hulburt; Chemical Reaction Engineering—II Advances in Chemistry; American Chemical Society: Washington, DC, 1974.

(1)

CHEMICAL REACTION ENGINEERING

Π

Gas outlet ι

>

Liquid feed

>

il 0


0 0

\ ο

0 0

W7 Gas feed

Liquid outlet Figure 1.

Continuous gas-liquid reactor

-e-e-e-Bubble growth Φ

1

m ι

ι

φ

.

1

e

^-e-^G-^-e/\ ι /\ ι \ ι \ ι \ ι \

J

ι

2

ι

/

ι

\

\

9 \

I

\

/

I I I

Bubble growth

\

\ \ \

• h - O - G - Q — O - Q - O —I t(?Mxi) Figure 2.

x - coalescence

xi

Rising bubbles with growth and horizontal coalescence


24.

ROOS

Sorption of Volatile Components

305

Mass Balances in the Bubble Phase At the orifices N bubbles are formed per unit time. As a bubble rises, its volume changes because of hydrostatic pressure, mass exchange with the liquid, and coalescence with a neighboring bubble. Because of coalescence the num ber of bubbles N(x) passing through a horizontal plane per unit time varies with the height χ (see Figure 2); N(0) = N . Since coalescence between two bubbles is fast, we assume that it occurs instantaneously—i.e., at particular heights. Thus, both N(x) and the molar mass M of component i in a bubble are discontinuous functions of x. However, the molar flow rate Q' of com ponent i in the bubble phase, which satisfies 0

0

1


1

= N(x) M*

(2)

is continuous. Let ν be the velocity of rise of a bubble; the number of bubbles per unit length then equals N(x)/v(x). Further, let Φ (χ) be the molar flow rate of component i passing from the liquid into one bubble having its center at height x. The amount of component i transferred per unit time from the liquid into the bubbles between the heights x and x is given by ι

1

2

./"* w*« φί

The mass balance for component i in the bubble phase now reads: Q (*«) - Q (χι) [

{

χ

f*

2

Φ (*)

dx

1

or, as a differential equation:

ψ-Φ'ϊ

(3)

dx ν We define the coalescence function f (x) : c

/.(*)-^)

(4)

No

This function is discussed in detail under Coalescence. It can be assumed that the physical mass transfer between liquid and bubbles is determined by its resistance in the liquid phase ( I ) . Let c be the concentration of component i in the liquid at the bubble interface, d be bubble diameter, and k be the coefficient of mass transfer; then l

int

l

Φ* =fc*χ d (c 2

(5)

chm)

1

It is assumed that at the interface between bubble and liquid the concentration in the gas phase ( c ^ ) is proportional to that in the liquid phase (c ) : l

int

i c'int

i =

m

1

i c'gas

i =

6

M

i

m

1

/ A \ (6)

This equation defines the distribution coefficient ra*. Its relationship to the saturated vapor pressure P* of the pure component i can easily be derived from the ideal gas law and Henry's law:


306


pi = yi Pi £i

Π

(7)

where p is the partial vapor pressure, γ* is the activity coefficient of component i in the liquid, and x is the mole fraction of component i in the liquid. Thus, 1

l

where p is the liquid density, M is the molecular weight of the liquid, R is the gas constant, and Γ is the absolute temperature. Let Q be the total molar flow rate in the bubble phase—i.e., Q(x) = Σζ) (χ) (sum over all gases and vapors) ; the following equation relates bubble diameter d and total gas flow rate Q : Downloaded by RUTGERS UNIV on May 18, 2017 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1974-0133.ch024

ι

Q - Ν

I

P b

(9)

where p is the molar gas density. Using Equations 2, 4, 5, 6, and 9, Equation 3 can be written as: b

2 -/·*·* *?( - " »Ι?) ί

,

β,

,

,

(10)

The molar gas density p depends on position x; from the ideal gas law and the hydrostatic pressure (see Figure 1) it follows that b

Β + Pb

=

ç>g(H-x) RT

where Β is the pressure above the liquid, ρ is the liquid density, g is the accel eration from gravity, and H is the distance between the orifices and the liquid level. The amount of component i fed through the orifices per unit time is known. The initial conditions for the system of differential equations (see Equation 10) therefore read: x « 0

Qi = Qi (0) = known

(12a)

Comparing Equations 9 and 12a, we note that

0(0) = EQi (0) = G = N l do 9b

3

o9h

(12b)

where d is diameter of the bubble formed at an orifice. Since we wish to describe sorption especially in high-viscous liquids, our discussion on bubble behavior is directed to viscous liquids. 0

Bubble Formation Since narrow capillaries are used, the gas flow rate in a capillary does not vary with time because of the large pressure drop in the capillary [constant-flow conditions according to Davidson and Schiiler (2)]. There are three main physical models that describe bubble formation at an orifice. They have been developed by Van Krevelen and Hoftyzer (3), Davidson and Schiiler (2), and Kumar, Kuloor, and co-workers (4, 5,6). Kumar and Kuloor's model gives a good description of bubble formation under widely varying conditions (4, 5, 6). It leads to the following equation


24.

ROOS

307

Sorption of Vohtile Components

for the diameter d of a bubble formed at an orifice in a liquid with a viscosity higher than 0.5 kg/msec: 0

d


0

= 2.36

f-V

4

(13)

where G / n is the volumetric gas flow rate per orifice and ν the kinematic liquid viscosity. The coefficient 2.36 in Equation 13 has the value of 2.42 according to Van Krevelen and Hoftyzer (3) and 2.31 according to Davidson and Schiiler (2). For high-viscous liquids no literature data are available. Table I gives some experimental data on bubble behavior in molten polyisobutene. Some experimental details are given in the Appendix. The fifth column of Table I gives a value of 2.3 for the coefficient in Equation 13, which is in good agree ment with the theoretical one. The diameter of a bubble formed at an orifice in a high-viscous liquid such as polyisobutene is two to four times larger than that of a bubble formed in a low-viscous liquid at the same gas flow rate. In other words, the production at a sufficiently high rate of small bubbles in highviscous liquids, which is desirable from the point of view of mass transfer, is not possible with the use of a simple capillary. Table I.

Experimental Data on Bubble Behavior in Polyisobutene (see Appendix)

Expertment

(kg/ msec)

G/n (cm / sec)

do (cm)

Con stant"

b (cm/ sec)

Ratio

fc

2B 3 4 5 6 7 8 9 10 11 12 13 14 15

40.0 47.5 24.0 25.3 31.0 30.5 15.5 16.5 31.2 31.1 31.0 31.0 28.0 45.0

2.18 2.32 2.37 2.37 2.35 4.58 2.40 1.12 2.35 8.01 11.30 13.75 0.92 0.94

2.26 2.37 1.99 2.05 2.15 2.54 1.85 1.65 2.16 2.91 3.19 3.21 1.73 2.05

2.27 2.24 2.22 2.26 2.26 2.26 2.30 2.44 2.26 2.25 2.26 2.16 2.35 2.47

1.61 1.43 2.30 2.30 2.01 2.72 3.12 1.87 2.06 3.91 4.97 4.77 1.35 0.73

1.07 1.04 1.19 1.19 1.15 1.10 1.21 0.97 1.18 1.23 1.29 1.23 1.08 0.67

0.39 0.36 0.46 0.46 0.44 0.36 0.50 0.50 0.46 0.35 0.39 0.32 0.49 0.48

M

3

V

c

° Constant in Equation 13, calculated. Velocity of rise after one coalescence. Ratio between the measured velocity of rise and the velocity calculated with the aid of Hadamard's Equation 18b. b

c

Bubble Behavior during Rising Since we are dealing with high-viscous liquids, we restrict ourselves to laminar flow. This means that the Reynolds number (Re) defined by Re = ^

(14)

V

is less than about 2. In low-viscous liquids bubbles behave like solid spheres: velocity of rise and mass transfer are described by the corresponding equations derived for solid spheres (see Equation 18a and 19a); these bubbles are called rigid interface bubbles. The same holds for sufficiently small bubbles in a high-


308


II

viscous liquid (7, 8, 9). Larger bubbles in a high-viscous liquid, however, are internally mobile. They are still spherical, but the gas inside them circulates (9); they are called free interface bubbles. Bond and Newton (7) have established a critical diameter d*: d* -

2jL y

(15)


99

where ρ is the liquid density and σ is the surface tension; they suggested that bubbles with a larger diameter (d > d*) have a free interface, and those with a smaller diameter have a rigid interface. Note that d* is about 0.1-0.5 cm for any liquid. Experimental data of Garner and Hammerton (9), however, show that Equation 15 can only be used as a first approximation. The Reynolds number corresponding to the critical diameter d* must be less than 2 so that the transition from rigid to free interface can take place under laminar flow conditions. This means that the fluidity number M satisfies: M « ^ > 10" (16) ça where μ is the dynamic viscosity; this corresponds to a viscosity of 0.1 kg/msec or higher. The interface mobility of bubbles in laminar flow can thus be characterized as: 2

6

M < 10~ : 2

rigid interface

M > 10" , d < d*: rigid interface 2

(17)

M > 10~, d > d*: free interface 2

In the application discussed below the bubbles have a free interface, M being about 10 and the bubble diameter exceeding 1 cm. 8

Velocity of Rise The velocity of rise ν of a rigid interface bubble satisfies the Stokes* equation: (18a) The gas density has been neglected because it is small compared with the liquid density. Details on the derivation of Equation 18a are given by Schlichting (10). For a free interface bubble (Hadamard-Rybczynski) the velocity of rise satisfies (11): (18b) There is enough experimental evidence to show that Equation 18a holds for low-viscous liquids. For liquids with viscosities of 0.1 kg/msec or higher, however, Equation 18b applies as shown in Table II. Table I gives some experimental data for molten polyisobutene. Column seven shows the ratio between the measured velocities and those calculated by Equation 18b. The measured velocity is too high, perhaps because of liquid circulations induced by the rising bubbles. This effect of liquid circulations on the velocity of rise of bubbles (12, 13) is not taken into consideration.


24.

ROOS

Sorption of Volatile Components

309

Table II. Experimental Data on the Relation between Velocity of Rise and Diameter for Rubbles Rising in a Liquid with a Viscosity of 0.1 kg/msec or higher Liquid Sugar syrup


Oil (unspecified)

V

cm /sec) 2

122 122 8.81 9.95 8.87

do

(cm)

ν (cm/sec)

0.4 1.2

0.11 0.97

1 1

0.059 0.088 0.119

0.027 0.048 0.102

0.83 0.75 0.79 0.77 1.00 0.94 0.99 1 1 1 1 1 1 1 1 1

Glycerol

10.0 12.1 12.1 12.1

0.42 0.36 0.48 0.24

1.1 0.87 1.45 0.37

Dextrose/water

10.5 21.6 21.6 23.0 23.0

0.64 0.37 1.04 0.54 1.09

3.25 1.41 4.13 1.05 4.22

Glycerol/water

1.46 1.46 6.17 6.17

Ratio"

0.21 0.38 0.61 0.87

2.5 8 5 10

Reference (7) (U) (9)

(15)

(1β)

Ratio between the measured velocity of rise and the velocity calculated with the aid of Hadamards Equation 18b. a

Mass Transfer Levich (II) has derived the following relations for mass transfer between bubbles and liquids in laminar flow. For rigid interface bubbles (Stokes' regime) : Sh = 0.65 P e ' 1

3

(19a)

For free interface bubbles (Hadamard's regime): Sh = 1.01 Pei/

2

(19b)

where Sh is the Sherwood number (kd/D), Pe is the Peclet number (vd/D), and D is the diffusivity. Other workers have derived similar equations with slightly different values for the coefficients (17, 18). The mobility of the bubble interface is characterized by Equation 17. Generally, Pe is large ( 1000 or higher), so that mass transfer at a free interface is five times higher than that at a rigid interface. Recent experiments (15, 16) show that mass transfer between bubbles and liquids with viscosities of 0.1 kg/msec or higher does satisfy Hadamard's relation (Equation 19b). Coalescence Some studies of coalescence in liquids with viscosities up to 0.1 kgram/ msec have been reported (19, 20). As far as we know, however, they have not yet resulted in a description of coalescence which can be extrapolated to a viscous melt. Although it has been assumed that at any height the bubbles have the same diameter and velocity of rise, these quantities vary in practice. One


310


Π


Liquid streamlines

Figure 3.

Initiation of vertical coa lescence

cause of coalescence is that the rise of a bubble is accelerated by the wake of a neighboring bubble( see Figure 3). The rise of bubble 2, which is attracted by the wake of bubble 1, is accelerated, and after some time the two bubbles coalesce. Lab-scale experiments with polyisobutene have shown that the first coalescence occurs at about 5-10 cm from the orifices, the second coalescence at about 70-80 cm from the orifices, followed by a third, a fourth and some times a fifth coalescence. Figure 4 shows the relation between residence time and height of rise. The dots, where the velocity shows a jump, represent coalescence. These experiments show that coalescence is influenced by the gas flow rate and the viscosity. Since coalescence is complex and since the total rate of sorption is impor tant, it seems reasonable to take coalescence into account by using an average value of the function f (x), defined in Equation 4. Let x , x , . · . , x be the heights χ at which the 1st, 2nd, . . . , c coalescence occurs, and let x = 0, x = H, then c

x

2

c

th

0

c+1

/

c

(x) = 1

XQ

« i

=

Mathematical Modeling of the Sorption of Volatile Components in

Recommend Documents