Bird R.B., Stewart W.E., Lightfoot E.N. Transport Phenomena

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s22.9 Matrix Approximations for Multicomponent Mass Transport

717

variations are not too large. Multicomponent analyses of this sort have been presented

by many investigators, for quiescent media2 and for forced-convection systems."'

We begin with the species continuity equations as given in Eq. 19.1-15, and apply

them to an N-component gas system with N

1 independent mole fractions

and an

equal number of independent diffusion fluxes

J:.

Let

[XI

and

[J"]

denote, respectively,

the column arrays of independent mole fractions x,,

. .

x,-,

and independent diffusion

fluxes

J:,

JG-,;

then approximating the molar density

in Eq. 19.1-15 by a reference

value

c,,,

gives the linearized equation system

for laminar or turbulent flows free of homogeneous chemical reactions.

For multicomponent ordinary diffusion, the flux expression may be written either as

a matrix generalization2f4 of Fick's first law (Eq.

of Table 17.8-2),

or as a matrix statement3t5 of the Maxwell-Stefan equation (Eq. 17.9-1):

The matrices

[Dl

and

[A]

must be (N

1) and nonsingular to give the stated

number of independent fluxes (in Eq. 22.9-2), and of independent mole fractions (in

Eq.

22.9-3). Consistency of these two equations then requires that

[Dl

[Alp'

at any given

state.

In the moderate-density gas region, the elements of the matrix

[A]

are predictable

accurately from Eq. 17.9-1, giving

in which the divisors

BOP

are the

binary

diffusivities of the corresponding pairs of

species. In the first approximation of the Chapman-Enskog kinetic theory of gases, the

coefficient for a given pair a,

depends only on

and

as in Eq. 17.3-11. These simple

expressions lead us to prefer Eq. 22.9-3 over Eq. 22.9-2, unless measurements of

[Dl

are

available at the desired conditions. Formally similar equations may be written in mass-

or volume-based compositions and fluxes, after appropriate transformation of the coeffi-

cient matrix

[A]

[Dl.

Mass units are preferred if the equation of motion is included in

the problem formulation, since the mass average velocity is then essential as indicated in

919.2.

Onsager,

Ann.

N.Y.

Acad. Sci.,

46,241-265 (1948);

Dunlop and

J. Gosting,

Phys.

Chem.,

63,86-93 (1959);

Kirkaldy,

Can.

Phys.,

37,30-34 (1959);

de Groot and

Mazur,

Non-Equilibrium Thermodynamics,

North-Holland, Amsterdam

(1961);

Kirkaldy, D. Weichert, and

Zia-U1-Haq,

Can.

Phys.,

41,2166-2173 (1963);

Cussler, Jr., and

Lightfoot,

AKhE Journal,

10,

702-703,783-785 (1963);

H. T. Cullinan,

Ind. Eng. Chem. Fund.,

4,133-139 (1965).

Prober, PhD thesis, Univ. of Wisconsin

(1961).

Toor,

AKhE

Journal,

10,460465 (1964).

Stewart and

Prober,

Ind. Eng. Chem. Fund.,

3,224-235 (1964).

Tambour and

Gal-Or,

Physics of Fluids,

19,219-225 (1976).

718

Chapter

Interphase Transport in Nonisothermal Mixtures

For multicomponent systems (N

31, each of these flux expressions normally has a

nondiagonal coefficient matrix, giving a coupled system of diffusion equations. Equation

22.9-3 can be decoupled by use of the transformation

[PI -'[A] [PI

]

(22.9-5)

AN-

" "

in which [PI is the matrix of column eigenvectors of [A], and A,,

. .

A,-,

are the corre-

sponding eigenvalues. These eigenvalues, the roots of the equation det[A

A11

are

positive at any locally stable state of the mixture; they are also invariant to similarity

transformations of [A] to other composition units. Here

is the unit matrix of order

The matrix [Dl, when used, is reducible in like manner with the same matrix [PI, and its

eigenvalues Dl,

. .

DN-, are the reciprocals of A,,

AN-,. For economy of effort, [A]

(or

[Dl)

and the arrays derived therefrom will always be evaluated at reference property

values, so will not need the subscript

,,*;

however, a subscript

will be added in [A],

[Dl,

[PI, and [PI-' when these arrays are based on quantities in mass units.

Equation 22.9-5 suggests that the following transformed compositions and trans-

formed diffusion fluxes should be useful:

Hereafter, an accent

(')

will

be placed on such transformed yariabks and on the corre-

sponding diagonal matrix elements, including the eigenvalues

and D,. Premultiplication

Eq.

22.9-3 by [PI-' and use of Eqs. 22.9-5 through 9 then gives uncoupled flux equations

formally equivalent to Fick's first law for

binary systems. The multicomponent

continuity equation 22.9-1 correspondingly transforms to

Thus, the transformed compositions

and fluxes

for each

satisfy the continuity and

flux equations of a binary problem with the same

function (laminar or turbulent) as the

multicomponent system, and with a diffusivity

91AB

equal to the eigenvalue

/A,.

The initial and boundary conditions on [?I and [j'] are obtained from those on

1x1

and

[

J"1 by application of Eqs. 22.9-6 and

The resulting quasi-binary problems may

then be solved, using theory or correlations of experiments, and the results combined5

via Eqs. 22.9-7 and 9 to get the multicomponent solution in terms of [XI and

[

J"].

Local mass transfer rates in binary systems are expressible in the form

as indicated in Eq. 22.1-7 and s22.8. The notation

. .

after

9,,

stands for any additional

variables (such as

of 922.8) on which the binary mass transfer coefficient

may de-

pend. The corresponding set of equations in the notation of Eqs. 22.9-10 and

522.9 Matrix Approximations for Multicomponent Mass Transport

719

or in matrix form,

Transformation of this result into the original variables gives the interfacial diffusion

fluxes

J:,,

J&,,,

into the gas phase as

or the composition differences for given fluxes Jao as

Here

[kl

is the diagonal matrix shown in Eq. 22.9-14, and

[h-'

is formed from the reci-

procals of the same diagonal elements.

As for binary systems, further information is needed to calculate the species fluxes

N,, relative to the interface, which give the local transfer rates.

flux ratio r

NAo/NBo

was specified in Eq.

21.1-9

to solve for N,,; analogous specifications are required for

multicomponent systems. The calculation of the fluxes N,, from diffusion fluxes

and

relative transfer rates is called the "bootstrap problem,"',7 and is treated well in Ref. 1.

This problem becomes simpler if Eq. 22.9-14 is rewritten as follows, using the array [No]

of interfacial molar fluxes N,,,,

. .

NN-l,O relative to the interface,

to allow direct insertion of relations among the species transfer rates. The corresponding

result for the array [no] of interfacial mass fluxes n,,,,

. . .

n,-,,, relative to the interface

is:

Several special forms of these results will now be given.

For systems with no net molar interfacial flux, the N-term summation in Eq. 22.9-17

vanishes, and this equation takes the convenient form

in which the diagonal array [k] needs no net-flux correction. This result can be extended

to moderate net molar interfacial flux by approximating each transfer coefficient kx(Da,

$,,I

in Eq. 22.9-14 as a linear function of the net molar interfacial flux, using the tangent line

of the 8-curve in Fig. 22.8-2 for the chosen mass transfer model. This gives the

linear equation system8

for the stagnant-film model given in 322.8. In the same manner, one obtains

Krishna and

Standart,

Chem.

Eng.

Commun.,

3,201-275

(1979).

Stewart,

AlChE Journal,

19,398400 (1973); Erratum, 25,208 (1979).

720

Chapter 22 Interphase Transport in Nonisothermal Mixtures

for the penetration model given in g20.4 and 522.8, and

for the limit

in laminar boundary layers, shown in Figs. 22.8-5,6 and valid for non-

separated boundary layers in three-dimensional steady flows.9

In systems with

net mass interfacial pux, as

steady-state solid-catalyzed reac-

tions, Eq. 22.9-18 reduces to

The elements of the matrix

can be predicted from expressions for the binary Sherwood

number or

factor as defined for mass-based units in Table 22.2-1, with eigenvalues

fim

inserted in place of binary diffusivities

a,,.

For a given flow field, the product [~][k,l[~l~' in Eqs. 22.9-19 through 22 is a func-

tion of the matrix [A]. This matrix triple product, here called [k,], is non-diagonal for

whereas [kl is diagonal as noted above. A simple, efficient method for approxi-

mating such functions has been developed by Alopaeus and Nord6n.lo Let f be a scalar

real-valued function defined on the eigenvalues of a matrix [A], in which the diagonal el-

ements are dominant as in Eq. 22.9-4. The proposed approximations to the elements of

the matrix [B]

[A] are then as follows:

for diagonal elements,

Bii

f(Aii)

(22.9-24)

df(Aii)

d~,

A~~

for off-diagonal elements, Bij

f(AJ

f(Aji)

(22.9-25)

Aij

A-A,.

otherwise.

Alopaeus and Nordenlo tested these approximations to mass-transfer coefficient matrices

[k,] of the form b[DI1-P or the form b[AlP-', and to the corresponding fluxes N,,, in sys-

tems of 3 to 25 gaseous species. Exponents

from 0.25 to 0.66 were used; values from 0 to

0.5 appear in the mass transfer expressions of this chapter. Comparisons were made

against exact calculations of elements

kxap

and

N,,

via Eq. 22.9-19, and against a film model

given by Krishna and Standart" in which each element

kXop

is calculated independently

with the corresponding binary diffusivity

gap.

The calculations from Eqs. 22.9-24 and 25

were

times quicker than those with Eq. 22.9-19 and proved quite accurate (relative

errors typically less than 1% and seldom as large as

lo%),

especially when done directly

from the diagonally dominant Stefan-Maxwell matrix [A] rather than from its inverse,

[Dl.

Calculations with the Krishna-Standart film model were slower than those with Eqs. 22.9-

24 and 25, and the typical errors were several times as large. Therefore, Eqs. 22.9-24 and 25

are recommended as practical approximations to the elements of the product matrix

[B]

[~l[k,][P]-' in Eqs. 22.9-19 through 22 whenever

Eq.

22.9-4 is used. This approxima-

tion may be used in Eq. 22.9-23 also, with [Bl transformed at the end into mass-based

units; however, Eq. 22.9-20 or 22 will be more convenient and comparably accurate at the

moderate net molar fluxes normally encountered in heterogeneous catalysis.

The accuracy of the linearized solutions depends on the choice of the reference prop-

erty values, especially when the property variations are large. In the following discus-

sion all properties are evaluated at a common reference state, with composition given

a mole fraction

[xref1

a,[xb1

a,)[xol

(22.9-26)

Stewart,

AlChE

Jouvnal,

9,528-535 (1963).

Alopaeus and

Norden,

Computers

Chemical Engineering,

23,1177-1182 (1999).

Krishna and

Standart,

MChE

Journal,

22,383-389 (1976).

Questions for Discussion

721

or a mass fraction

[wref1

ao[mb1

a,)[w,I

(22.9-27)

Note that

[x,,,]

remains open to choice even for Eq. 22.9-20, 21, or 22, since the average

compositions shown there provide net-flux corrections and not physical property values.

Equations 22.9-17, 18 and several other approximations for multicomponent mass

transfer have been tested" against detailed variable-property integrations for isothermal

systems. The conclusions from this study were as follows:

For twenty problems of unsteady-state gaseous diffusion, covering a wide range

of net mass transfer rates, linearization in molar units approximated the exact so-

lutions best. Rates of isobutane evaporation and condensation, for the system

C4H,,-N2-H,

in the geometry of Example 20.1-1, were approximated with a

standard deviation of 1.6% by Eq. 22.9-17, using reference mole fractions calcu-

lated from Eq. 22.9-26 with

0.5. Linearization in mass-based units, via Eq.

22.9-18, proved inferior because of the large variations in

and

[A,].

This

method, with its preferred

value of 0.8, gave a standard deviation of 3.8% for

the interfacial fluxes

N,,

of the single transferable species (isobutane). Quasi-

steady-state film approximations proved less accurate; use of correction factors

Ox,

$,,/(exp$,,

(as given by Stewart and Prober5 for the film model of

522.8)

gave

standard deviation of 7.88% with

optimized to 1.0. The film

model of Krishna and Standart,'' which does not use linearization, gave a stan-

dard deviation of 14.3% independent of

and

a,.

These results favor the use of

Eq. 22.9-17 (or, for moderate transfer rates, Eq. 22.9-21) with

0.5 for the gas

phase in transfer operations described by a penetration model.

For twenty problems of momentum and mass transfer in laminar gaseous bound-

ary layers of Hz, Nz and

CO,

on a porous flat plate, solved accurately by Prober;

linearization in mass-based units approximated the exact solutions best. The de-

tailed variable-property solutions for

n,,

were approximated12 for all three

species with a standard deviation of 0.55% by Eq. 22.9-18, using mass transfer co-

efficients

predicted via

Eq.

20.2-47 and 22.9-27 with

optimized to

0.4.

The

film models of Stewart and Prober5 and of Standart and ~rishna'~ gave standard

deviations of 4.78% (with

1.0)

and 8.25%, respectively, for the species trans-

fer rates.

The methods presented here are coming into widespread use in the engineering of

multicomponent separation processes. Advances in computing technology have facili-

tated the use of these methods and stimulated investigations toward better ones, to deal

with nonlinear phenomena including complex chemical reactions.

QUESTIONS FOR DISCUSSION

Under what conditions can the analogies in Table 22.2-1 be applied? Can they be applied in

systems with chemical reaction?

Why is the heat transfer coefficient in

Eq.

22.1-6

defined differently from that in

Eq.

14.1-1-or

is it?

Some of the mass transfer coefficients in this chapter have a superscript

and others

have

superscript

Explain carefully what these superscripts denote.

What conclusions can you draw from the analytical calculations

mass transfer coefficients

§22.2?

Young

and

Stewart,

Ind.

Eng.

Chenz. Res.,

25,476-482

(1986).

722

Chapter 22 Interphase Transport in Nonisothermal Mixtures

PROBLEMS

22~.1.

What is the significance of the 2 in Eqs. 22.3-20 and 21?

What is the meaning of the subscripts 0,

and

in §22.4?

What is meant by the term "model insensitive"?

In what way does surface tension have an influence on interphase mass transfer? How is sur-

face tension defined? How does surface tension depend on temperature?

Discuss the physical basis for the film model, the penetration model, and the boundary layer

model for heat and mass transfer.

How are the heat and mass transfer coefficients affected by high mass-transfer rates across

the interface?

Prediction of mass transfer coefficients in closed channels. Estimate the gas-phase mass

transfer coefficients for water vapor evaporating into air at 2 atm and

25"C,

and a mass flow

rate of 1570 IbJhr, in the systems that follow. Take

a,,

0.130 cm2/s.

(a) A 6-in. i.d. vertical pipe with a falling film of water on the wall. Use the following correla-

tion' for gases in a wetted-wall column:

Sh,,,

0.023 Re0.83~~0.44 (Re

2000) (22A.1-1)

(b) a 6-in.-diameter packed bed of water-saturated spheres, with

100

Calculation of gas composition from psychrometric data.

stream of moist air has a wet-

bulb temperature of 80°F and a dry-bulb temperature of 130°F, measured at 800 mm Hg total

pressure and high air velocity. Compute the mole fraction of water vapor in the air stream.

For simplicity, consider water as a trace component in estimating the film properties.

Answer

x,,

0.0158 (using n

0.44 in

Eq.

22.3-38)

Calculating the inlet air temperature for drying in a fixed bed. A shallow bed of water-satu-

rated granular solids is to be dried by blowing dry air through it at 1.1 atm pressure and a su-

perficial velocity of 15 ft/s. What air temperature is required initially to keep the solids at a

surface temperature of 60"F? Neglect radiation. See 514.5 for forced-convection heat transfer

coefficients in fixed beds.

Rate of drying of granular solids in a fixed bed. Calculate the initial rate of water removal in

the drying operation described in Problem 22A.3, if the solids are cylinders with

180 ft-l.

Evaporation of a freely falling drop. A drop of water, 1.00 mm in diameter, is falling freely

through dry, still air at pressure of

atm and a temperature of 100°F with no internal circula-

tion. Assume quasi-steady-state behavior and a small mass-transfer rate to compute (a) the

velocity of the falling drop, (b) the surface temperature of the drop, and (c) the rate of change

of the drop diameter in cm/s. Assume that the film properties are those of dry air at 80°F.

Answers: (a) 390 cm/s; (b) 54°F; (c) 5.6

cm/s

Effect of radiation on psychrometric measurements. Suppose that a wet-bulb and dry-bulb

thermometer are installed in a long duct with constant inside surface temperature

and that

the gas velocity is small. Then the dry-bulb temperature Tdb and the wet-bulb temperature TWb

should be corrected for radiation effects. We assume, as in Example 22.3-2, that the ther-

mometers are so installed that the heat conduction along the glass stems can be neglected.

(a) Make an energy balance on a unit area of the dry bulb to obtain an equation for the gas

temperature T, in terms of T,,,

T,,

h,,,

e,,, and adb (these last two are the emissivity and absorp-

tivity of the dry bulb).

Gilliland and

Sherwood,

Ind.

Eng.

Chem.,

26,516-523 (1934).

Problems

723

(b) Make an energy balance on a unit area of the wet bulb and obtain an expression for the

evaporation rate.

(c)

Compute

x,,

for the pressure and thermometer readings of Example 22.3-2, with the ad-

ditional information that

15 ft/s,

130°F,

edb

adb

eWb

aWb

0.93, dry-bulb diame-

ter

0.1 in., and wet-bulb diameter

0.15 in. including the wick.

Answer:

(c)

xAm

0.0021

22B.3.

Film theory with variable transport properties.

(a) Show that for systems in which the transport properties are functions of

Eqs. 19.4-12

and 13 may be integrated to give for

respectively,

(b) Make the corresponding changes in Eqs., 19.4-16 and

as well as in Eqs. 22.8-5 and

Then verify that Eqs. 22.8-7 and

remain valid. Thus it is not necessary to work with the inte-

grals in calculating transfer rates if

hloc

and

kX,,,,

can be predicted.

(c)

Show that

h,,,

and

kx,loc

have to be evaluated

terms of the physical properties and flow

regime (laminar or turbulent) that prevail at the conditions for which

hi,,

and

k&,

are desired.

22B.4.

An evaporative ice maker. Consider a circular shallow dish of water 0.5 m in diameter and

filled to the brim, resting on an insulating layer, such as loose straw, and in a windless area.

At what air temperature can the water be cooled to freezing if the relative humidity of the air

is 30%? Make the following assumptions: (i) neglect radiation, (ii) consider radiation to a

night sky of effective temperature 150K, and (iii) assume that the dish has a lip around the

edge 2 mm high.

228.5.

Oxygen stripping. Calculate the rate at which oxygen transfers from quiescent oxygen-

saturated water at 20°C to a bubble of pure nitrogen

mm in diameter, if the bubble acts as

a rigid sphere. Note that it will first be necessary to determine the bubble velocity of rise

through the water.

22B.6.

Controlling diffusional resistance. Water drops 2 mm in diameter are being oxygenated by

falling freely through pure oxygen at 20°C and a pressure of

atm. Do you need to know the

gas-phase diffusivity to calculate the rate of oxygen transport? Why? The solubility of oxy-

gen under these conditions is 1.39 mmols/liter, and its diffusivity in the liquid phase is

about 2.1

cm2/s.

Curve

fit:

log

~H~O

0.6715

0.030

0.0000798

Fig.

22B.7.

Water vapor pres-

sure under its own vapor data

from

Lange's Handbook of

Chemistry

Dean, ed.),

15th edition, McGraw-Hill,

Temperature,

New York (1999).

724

Chapter 22 Interphase Transport in Nonisothermal Mixtures

22B.7. Determination of diffusivity (Fig. 22B.7). The diffusivity of water vapor in nitrogen is to be

determined at a pressure of 1 atm over the temperature range from 0°C to 100°C by means of

the "Arnold experiment" of Example 20.1-1. It will, therefore, be necessary to use the correc-

tion factor

OA,

to the penetration model. Calculate this factor as a function of temperature. The

vapor pressure of water in this range may be obtained from Fig. 22B.7 or calculated from

where

p~~~

is the vapor pressure in mm Hg, and

is the temperature in degrees centigrade.

22B.8. Marangoni effects in condensation of vapors. In many situations the heat transfer coeffi-

cient for condensing vapors is given as

k/S,

where

is the thermal conductivity of the

condensate film, and 6 is the film thickness. Correlations available in the literature are nor-

mally based on the assumption of zero shear stress at the free surface of the film, but if the

surface temperature decreases downward, there will be a shear stress

duldz, where u is

the surface tension, and

is measured downward, that is, in the direction of flow. How much

will this effect change a heat transfer coefficient of 5000 kcal/hr

C for a water film? The

kinematic viscosity of water may be assumed to be 0.0029 cm2/s, the density is 0.96 g/cm3,

the thermal conductivity 0.713 kcal/hr. m

C, and

du/dT

-0.2 dynes/cm

for the pur-

poses of this problem.

(p:i2)(

)

Partial Answer:

p(v,)

I+--

The term in

.r,

represents the effect of surface tension gradients, and when this term is

small, its denominator will be near the value for no gradient. For the conditions of this prob-

lem, pg6

14.3 dyn/cm2. Surface tension effects will thus be small for systems such as the

one under consideration, where the surface tension increases downward. In the opposite

case, however, even small gradients can cause hydrodynamic instabilities and thus can

have major effects.

22B.9. Film model for spheres. Derive the results that correspond to Eqs. 22.8-3,4 for simultaneous

heat and mass transfer in a system with spherical symmetry. That is, assume a spherical mass

transfer surface and assume that

and

depend only on the radial coordinate

Show that

Eqs. 22.8-7 and 8 do not need to be changed. What difficulties would be encountered if one

tried to use the film theory to calculate the drag on a sphere?

22B.10. Film model for cylinders. Derive the results that correspond to Eqs. 22.8-3,

for a system

with cylindrical symmetry. That is, assume a cylindrical mass transfer surface and assume

that

and

depend only on

Verify that Eqs. 22.8-7,8 do not need to be changed.

22C.1. Calculation of ultrafiltration rates. Check the accuracy of the predictions shown in Fig. 22.8-9

for the following data and physical properties:

Physical system:

Rotation rate of disk filter

273 rpm

Bovine serum albumin at p,

2.2 g/100 ml

Diffusivity in phosphate buffer (at pH 6.7)

7.1

cm2/s

Kinematic viscosity of buffer

0.01 cm2/s

Partial specific volumes of protein and buffer are 0.75 and 1.00 ml/g, respectively

Hydraulic permeability,

0.0098 cm/min psi

Effect of protein concentration:

Solution density p

0.997

0.224~~ in g/ml

Protein-buffer diffusivity ratio '9ps(0)/9ps(pp)

21.34,/tanh(21.34$, where

w~~~/(w~+~

uses)

is the volume fraction of protein, with

and

being the par-

tial specific volumes of protein and solvent

Protein-buffer viscosity ratio ~(O)/p(p,)

1.11

0.054p,

0.00067p$, with pp in g/100 ml

Osmotic pressure

.rr

0.013& in psi (100 rnl/g)'

Problems

725

The operating data are as follows:

Transmembrane

pressure difference,

(PO

psh

psi

4.0

Percolation velocity

vfir

cmlmin

0.032

0.049

0.061

0.066

0.074

0.078

0.079

0.081

0.082

Chapter

Macroscopic Balances

for

Multicomponent

Systems

523.1 The macroscopic mass balances

523.2' The macroscopic momentum and angular momentum balances

523.3 The macroscopic energy balance

523.4 The macroscopic mechanical energy balance

523.5

Use of the macroscopic balances to solve steady-state problems

523.6O Use of the macroscopic balances to solve unsteady-state problems

Applications of the laws of the conservation of mass, momentum, and energy to engi-

neering flow systems have been discussed in Chapter

(isothermal systems) and Chap-

ter

(nonisothermal systems). In this chapter we continue the discussion by

introducing three additional factors not encountered in the earlier chapters: (a) the fluid

in the system is composed of more than one chemical species;

(b)

chemical reactions may

be occurring, along with changes of composition and production or consumption of

heat; and (c) mass may be entering the system through the bounding surfaces (that is,

across surfaces other than planes

and

2).

Various mechanisms by which mass may

enter or leave through the bounding surfaces of the system are shown in Fig.

23.0-1.

Fig. 23.0-1. Ways in which mass may enter or leave

Water

Heated

CH4,

and

through boundary surfaces:

(a)

benzoic acid enters

Air

NH,

H,O

combustion products

system

dissolution of the wall;

(b)

water vapor en-

Surface

ters the system, defined as the gas phase, by evapo-

ration, and ammonia vapor leaves by absorption;

(c)

oxygen enters the system by transpiration

through a porous wall.

Surface

Aqueous

Water benzoic

Surface

acid

Cold

CH4

NH~

Aqueous

ammonia

out

(a)

(b)

Surface