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On gas absorption into a turbulent liquid
Chemical Engineering Science, 1967, Vol. 22, pp. 1163-l 176. Pergamon Press Ltd., Oxford.
Printed in Great Britain.
On gas absorption into a turbulent liquid G. E. FORTESCUE~ and J. R. A. PEARSON Department
of Chemical Engineering, Pembroke Street, Cambridge (Received 27 January 1967)
Abstract-Mass transfer across the surface of a turbulent flowing liquid is here assumed to be determined by relatively large scale eddies. Their length scale is taken to be the integral scale of the turbulence in the bulk of the fluid, while their intensity is also taken to correspond to that of the bulk turbulent flow field. Calculations have been carried out for a simple cellular model and have been compared with experimental results for absorption of CO2 into water in the flow behind a grid placed across the entrance to an open channel. Agreement has been remarkably good. It is emphasized that there are no free parameters in the model. The original objective of the work was to provide a model for the process of re-aeration of polluted rivers and streams.
1. INTRODUCTION GAS ABSORPTION by turbulent liquids is a process that remains difficult to predict numerically, whilst not being easy to measure except as a gross overall Mass transfer into the fluid is phenomenon. clearly a combination of diffusion and convection, but the effects are coupled and unsteady. In principle, a complete knowledge of the kinematics of any flow field should allow the mass transport to be deducible from it, using standard mass conservation equations and boundary conditions. Unfortunately, the kinematics of turbulent flow fields is usually imperfectly known, and then only in terms of certain statistically relevant mean quantities. Models for predicting gas absorption have therefore to replace the exact physical situation by a much simpler one. The value of such models depends on how generally they can be applied, how accurate they are, and the extent to which they involve arbitrary parameters that vary from system-to-system : if the parameters correspond to reasonably well defined physical quantities, and can be deduced or measured a priori, then the models tend to be plausible; if the models are accurate and can be applied widely, then they are also useful. The classical model is the film model of WHITMAN [I]. This supposes that resistance to mass transfer
t Present address: Colonial Sugar Refining Corporation,
occurs across a thin layer of stagnant fluid, of thickness h,, situated at the surface, and that the bulk fluid below is well mixed. Calculation of the absorption rate is very simple, and is given by k,=Dlh,,
(1)
where k, is the mass transfer coefficient and D is the diffusion coefficient, provided h, is known. Either h, can be obtained experimentally or attempts can be made to deduce it from physical reasoning. Using the first method, there are doubts whether results will apply over a wide range of values of D. Using the second, it is difficult to envisage the existence of a true layer unless some surface-active or insoluble material actually overlies the surface!. One is therefore forced to treat the layer model as a sophisticated simplification, in which h, may be a function of all the variables in the problem. The next model developed was the penetration model of HIGBIE [2] who based his approach on a surface-renewal argument. The role of the turbulence was to bring up elements of bulk fluid to the $ It is perhaps unfortunate that the most commonly used fluid, water, is particularly prone to form rigid surface layers, and so give an obvious visual appearance of surface immobility; hence the prevalence of the film approach. Sydney, N.S.W., Australia.
1163
G.
E. FORTESCUE and J. R. A. &ARSON
surface, where unsteady absorption, as into an infinite stagnant liquid, takes place for a prescribed time TH, after which the element returns to the bulk and is replaced by another. Again calculation of mass transfer is simple, giving
DAVIES[16]) retain one or more arbitrary constants, the only way in which comparison with experimental results can serve to distinguish between them is by holding everything constant except D. Whitman predicts ka D, HigbieDanckwerts ka D* and Kishinevskii ka D” where 0 < n <+. Unfortunately it is difficult to vary D over k,=2JDlnT,. (2) a wide range, whilst it is easy to elaborate models sufficiently to match available experimental DANCKWERT~ [3] modified this model to allow for a information. random distribution of surface ages, consequent on We shall present here a new model that attempts a fractional rate of replacement of surface elements, to retain the most obvious physical characteristics s,. This yields the relation of a deforming (plane) surface such as is observed at the interface between a turbulent liquid and a k,=JG. (3) gas and uses only such parameters as can be readily correlated with measurable kinematic quantities However, once again the predictions involve characteristic of the liquid flow field. All theories arbitrary parameters, TH or sn, that must either be agree that processes close to the surface determine determined experimentally or deduced from physi- rates of gas absorption and that mixing and thus cal arguments. Although a turbulent flow field can mass transfer in the bulk of the fluid is rapid by yield characteristic times, there have been few comparison. A valid model must therefore pay particular attention to the flow very near the attempts to relate these directly to TH or s,. surface. It is in this regard that earlier models can Both the Whitman and Higbie-Danckwerts models have been elaborated (HANRAT~Y [4]; be faulted. The details of our quasi-steady largeeddy model are given in the next section, together PERLMUTTER[5]) or combined and elaborated (TOORand MAKCHELLO [6], DOBBINS[7], HARRIOT~ with the relevant transport equations. Section 3 describes experimental apparatus that [8], MARCHELLOand TCX~R[9], RUCKENSTEIN [lo], DOBBINS[ 1l]), but progressively improved agree- was used to test the model, and discusses difficulties ment with observation has only been achieved at that were encountered in making measurements. Section 4 compares, briefly, experimental and theothe expense of increasing complexity. retical results that were obtained for absorption An alternative steady-state phenomenological approach has been advanced by I&G~INEVSKII [12, into a laminar flow, which could be regarded as a 131 in which the turbulent convective effects give simple large eddy. This preliminary work was undertaken to ensure that later results would be rise to an eddy diffusivity, D,, which is obtained from experiment. In highly agitated flows he meaningful. The main results on absorption into a argues that D,>> D and so the mass transfer co- turbulent flow are given in Section 5. Further details can be found in FORTESXJE[17]. efficient kK obeys the relation Application of the results to rates of re-aeration for polluted streams and rivers is given briefly in k,aJD+D,+i& (4) the concluding section. The turbulent diffusivity concept can be refined, so that D, is taken to be a function of position, usually increasing with distance from the surface, being zero at the surface; in this case no surface renewal need be postulated. Because most of these models (for exceptions see O’CONNOR and DOBBMS [ 141, LEVICH [15],
2. THE LARGE-EDDYMODEL The convective effect of turbulence on the flow field near the absorbing surface is here represented by eddies sweeping fresh liquid to the vicinity of the surface and then removing solute-enriched liquid to be dissipated in the bulk of the absorbing
1164
On gas
absorption into a turbulent liquid
medium. The surface acts as a constraint on possible motions, in that no normal velocity is allowed at the surface; but tangential surface velocities are permitted?. The velocity structure will obey the equation of mass conservation but not of momentum conservation, since we are not concerned with dynamic aspects of the flow. Although turbulent flows contain a spectrum of eddies of constantly varying nature, we shall suppose that their mean mass-transfer properties can be modelled by means of a regular sequence of steady square roll cells touching the surface, moving as a whole with the local mean surface velocity. These are shown in Fig. I. Direction
Thus we shall take the length scale, A, of our eddies to be the integral length scale of the underlying turbulent fiowfield. [See Eq. (7) below]. For convenience the eddies are taken to be square, and we shall without further justification impose the following arbitrary velocity pattern within them: u = A sin(nx/A)cos(ay/A)
,
u = -A cos(nx/A)sin(ny/A)
,
(5)
w=o. where A is a so far unspecified constant, and the co-ordinate system is defined by Fig. 2. The alternating nature of the vorticity within the eddies is shown on Fig. 1. It is worth observing that these simulate a general pattern of upwelling along the planes x=2NA, where N is an integer, with down currents along the planes x =(2N+ l)A. By making A a function of the third co-ordinate, z, a slightly
of mean flow
Bulk of fluid FIG. 1. Distribution of roll cells near the surface as proposed in the large-eddy model.
We have now to prescribe the size of the cells, and the motions taking place within them. Unfortunately there is not a great deal of direct experimental information relating to the velocity structure of turbulent flows near a free surface. ELLISON [18] has pointed out that the liquid surface in open channel flow appears to be disturbed by large, low frequency eddies, as anyone sitting at a river bank in time of spate can see for himself. The nearly fixed surface position that is imposed by gravity might, on general grounds, be expected to damp out small rather than large eddies, while the lack of any significant additional dissipating or turbulence-generating mechanism at the surface suggests that the general eddy pattern will not be substantially different from that relevant in the bulk of the field of motion. Furthermore, all available evidence (see for example HINZE [19] or TOWNSEND [20]) seems to suggest that it is the large eddies that are dominant in transfer processes. 7 The film model allows no fluid velocity whatever in the neighbourhood of the surface, while the surface renewal models appear to allow discontinuous normal velocities of the actual surface, both physically unrealistic.
surface
A
A
FIO.2. A large eddy showing co-ordinate system and length scale.
more plausible model is obtained, but since this does not materially affect later calculations, we shall not introduce this complication here. We might have added a velocity component, w, in the z-direction, but we have no evidence that the ensuing complications would lead in the end to any substantial difference in our results, whilst there is evidence (TOWNSEND [20]) that two-dimensional roll eddies do in fact predominate in shear flows downstream of grids and cylinders.
1165
G. E. FORTESCUE and J. R. A. PEARSON
The mean kinetic energy associated cell [Eq. (5)] is given by
with the roll
;(a2 +u2+w2)&dy
= $A2.
(6)
It is generally true that the main energy-containing eddies have length scales comparable with the integral length-scale of the turbulence given by A=
?&(x)u,(x s
u;(x) )
+ r)dr
(7)
i-
0
where u, is the velocity as a first approximation
in the r-direction, we put
and so
‘4=2(@+@+33)f,
(8)
i.e. twice the root-mean-square turbulent intensity. This we expect to be an upper bound. The mean mass transfer coefficient, l&c, given by
I;,,+
A -ac
s(> 0
dy y=o
Finally we must choose a suitable condition y=A. Because we have already implied mixing in the bulk is rapid, we shall take c=
)
aty=A.
(12c)
Clearly the sudden discontinuity in &/c?y at y=A, which is implied by (12c), cannot be thought of as having any physical counterpart. A more plausible picture is obtained by considering only the upper half (y c $12) of the roll cell to be physically relevant, as shown in Fig. 3, with smaller scale eddies increasing in importance for y > +A, thus achieving a rapid rise in mass transport within the bulk. The bottom of the cell is thus to be interpreted as a virtualsink for solute, with the strictly anti-symmetric structure about y=+A only introduced for computational convenience. This interpretation can be justified a posteriori by reference to concentration profiles. Rising
dx
Cbulk
for that
1 jet
Falling
’
jet
(9) ,\jl
where c is the point solute concentration, is then obtained from a solution of the steady masstransfer equation (v.V)c = DV’c ,
I t-
A
-I
I
FIG. 3. Physical interpretation of large-eddy model. Only the top half of eddy is involved in physical transfer process, which is dominated by jets of rising and falling liquid.
(10)
which in our roll-cell case becomes A sin (7rx/A) cos (ny/A) &/ax -A
cos (7cx/A) sin (ny/A)
&/8y
= D(d2c/8x2 + d2c/dy2).
(11)
Our remaining task is to prescribe boundary conditions on c. If we assume no gas-phase resistance to mass transfer, then we can assume saturation at the surface, i.e. c=c*aty=O. Because of the symmetry require that
of the cell structure,
&/ax =0 at x = NA, Nan integer.
(12a) we
(12b)
We note that Eq. (11) and boundary conditions (12) can be made dimensionless using (c* - cb”rk) as a characteristic concentration, A a length and A a velocity. Only one dimensionless parameter remains, the P&let number P= AA/D.
(13)
If P is large, then molecular diffusion will only be significant in a region close to the surface (compared with A), and our model will be dominated by “turbulent convection”; under these circumstances we might expect it to give good agreement with experiment. However, if P is small, then mass transfer will be as in the Whitman model, i.e. by molecular diffusion across a film of thickness A; we do not expect the model to be valid in this range.
1166
On gas absorption
into a turbulent liquid
3. EXPERIMENTAL ARRANGEMENTS
I.21
Having decided on a model for turbulent absorption, we wished to make measurements in a situation where the statistical properties of the turbulent flow field were well known. This ruled out the traditional stirred cell; indeed only one possibility suggested itself: the flow downstream of a square-mesh grid. Detailed investigations in air by BATCHELORand TOWNSEND[21] and in water by GIBSON and SCHWARZ [22] provided sufficient information for our purposes. Figure 4 shows the schematic layout of the apparatus used. A continuous recirculating water flow passed through a rotameter R, a distributor D, consisting of a packing of JJ in. ceramic balls followed by a 64 mesh wire screen, a calming section C, consisting of a closed duct 3 ft long, 12 in. wide and 13 in. deep; it then flowed through one of three square grids G made of cylindrical brass rods placed at the downstream end of the calming section, into the test section T consisting of an open rectangular channel 1 ft wide and 2 ft 3 in. long. The water level in the test section was maintained at between 14 and 2 in.; an even flow could only be achieved by flowing over a weir W (shown inset). The inlet and outlet arrangements both introduced undesirable end-effects on the flow profiles near the surface; however, direct observation with talc particles showed that these were confined to less than 10 per cent of the total surface area of absorption, and allowance could be made for them in predicting absorption rates. A
FIG. 4. Schematic diagram of absorption
apparatus.
0
20
IO
(a) laminar flow,
f, min Re = 1500.
I
b f,
;o min
(b) turbulent flow, Re=4500; grid No. 1, x0=18 cm. FIG. 5. Typical forms of experimental observations for absorption rate R as a function of time r;
much more serious source of error was introduced by the high rates of absorption taking place in the weir: these could neither be neglected nor estimated, and so were separately measured by covering up most of the absorbing layer with a raft. The space above the test section was filled with CO, and the rate of absorption was measured with a soap-film meter. Figure 5 shows the differing results obtained for laminar flows (a) and turbulent flows (b). In the laminar case, absorption occurs in a series of steps, with cburk changing discontinuously, whereas in the turbulent case, cbu& appears to decrease continuously_i. Assuming that t The straight line obtained on a semi-logarithmic plot in the turbulent case was in accordance with the concept of a fully mixed fluid medium. The period of the discontinuities in the laminar case appeared to correlate with the time required to recirculate the total water volume.
1167
G.
E. FORTESCUE and J. R.
cbuk was initially zero, R, corresponding to &ik=O was obtained by extrapolation to zero time. The water used was tap water de-aerated by spraying into a partial vacuum. Experiments were carried out at atmospheric pressure. Before an experiment the space above the test section was purged with CO,. The amount of gas absorbed during this time was minimized by having a stagnant water surface, and was calculated to lead to less than 3 per cent of saturation. No steps were taken during a run to control temperature, which was however measured and never rose by more than 2°C. Absorption rates were always corrected to a standard temperature of 20°C by the relation R020’=ROT”(~*D~)20”/(~*Df)P.
A. PEARSON
found to agree in the laminar case, then it could justifiably be assumed that the experimental methods and observations were reliable. Turning then to the turbulent case, comparison of experimental observations and predictions based on the large-eddy model of Section 2 could then be regarded as a true test of the model. The first requirement for our calculations was a knowledge of the velocity field. The only relevant region is that close to the surface, because absorbed CO2 only “penetrates” a short distance into the water. The surface velocity field was measured photographically using talc particles as tracers. It was found that for our purposes the surface velocity vector could be written (15)
(14)
The assumption of a Df dependence war suggested by KILNER'S[23] results and justified (I posteriori.
4. RESULTSFOR LAMINAR FLOW Experiments were first carried out under conditions of laminar flow largely in order to decide whether the apparatus could be used to give predictable results at all. In particular, it was moet important to discover whether the large absorption in the weir downcomer could be satisfactorily measured. This was studied by measuring absorption rates with and without a raft placed on the surface; the effect of the raft was to reduce the length of the test section leading to the downcomer. Because the amount of absorption at the flat free surface was expected to be much less in laminar than in turbulent flow, experiments undertaken with and without the raft would need to be much more sensitive in the former than the latter case, to achieve comparable proportional accuracy in the difference of the two observations. Furthermore, calculations of gas absorption in the laminar case could be undertaken without postulating a model; the total actual steady velocity field could be measured and so used directly in the massIf therefore, experimental transfer equation. observations and numerical predictions were
Here x is a co-ordinate pointing downstream along the surface, z cross-stream along the surface with origin along the centre-line of the test section, and y vertically downwards from the surface. d is the half-width of the channel. Figure 6(a) shows a representative plot of f(z/d) for laminar flow experiments, while Fig. 6(b) is a typical plot of position (x) as a function of time (t) for a tracer particle. Care had to be taken to obtain U,, and hence dU,,/dx from such plots, because numerical differentiation is a notoriously unreliable process. Details of how smooth functions were obtained are given in FORTESCUE [17]. To obtain u for use in Eq. (lo), it was assumed that v=(U,,-ydu,/ax,o) (16)
It was also assumed that #c/$~~~~~c/~x~, a2c/az2. These are both consistent with the notion of a very thin layer near the surface in which c is substantially different from cb”rk. Hence the masstransfer equation becomes
1168
au ac Da2c 2-C "ZY ax ay ay2'
Uac
(17)
On gas absorption into a turbulent liquid
with boundary conditions c=c*at y=O,
(18)
c4Oas y+co, and initial condition c=O
0.6. 0 0 0
\
0.4
ac -=-
ay 00
0.2 -
Ap 0.2
0.4
0.6
0.6
(19)
These correspond to fresh water entering the test section at x=0 and a saturated surface. We note that if a solution of (17) is obtained for US= USO, the solution for all other USis obtained by a suitable scaling of y; in consequence
0
0
al
I.0
r/d
Fro. 6. (a) Surface velocity profile as a function of cross-channel position for laminar flow: -, mean curve: 0, from observations.
ac ay
z~otfw~l*.
(20)
If therefore we carry out all our absorption calculation in terms of a plane flow model l_JJo,the true absorption can be obtained by multiplication by a factor 1 [f(z/d)]+dz
$ I
-0-84,
(21)
d
as was in fact done. Equation (17) was solved by standard finitedifference approximation methods. Being a parabolic problem no unusual difficulties were encountered; near z=O, U,,(x) was taken to be a smooth monotonic positive function, thus avoiding any computational difficulties?. Total absorption was calculated in terms of a mean absorption coefficient.
40
20 SCC
(b) Typical position gether nomials
plot of (0) as with (-).
surface tracer particle a function of time, toapproximating polyRe=3400.
a,
where L is the effective length of the absorbing section. This method was employed to calculate li, both without and with the raft in position. L could not be zero in the latter case because otherwise the flow in the downcomer would not have been closely similar in the two cases. t Little error was introduced in this way, because. the “attached eddy” at the entrance point was small. 1169
G. E. FORTEKXIE and J. R. A. For purposes of comparison solutions obtained for the “simple penetration equation
ac a% ‘Jsoz = Da7
PEARSON
were also theory”
(23)
3
(23)
with boundary conditions (18) and initial conditions (19) as before. Dimensionless flow rates will be given in terms of the Reynolds number
i&l
3X0
2500
EC0
Re
Re=4Q/vw,
FIG. 8. Laminar flow experiments at various Re. 0, Absorption rate without raft, observed; 0, absorption rate with raft, observed; A, absorption in weir calculated on the basis of Eq. (17); x , absorption in weir calculated on the basis of Eq. (23); Broken and full curves denote “best fits”, from which absorption rates in test section could be calculated.
(24)
where Q is the total flow rate, v the kinematic viscosity and w the wetted perimeter of the duct. Values of D were taken from CULLEN [24], and c* from SEIDELL [25]. Figure 7 shows results obtained at Re =5000, with a raft in four separate positions, together with a correction calculated according to Eq. (17), boundary conditions (18) and initial conditions (19). It seems reasonable to assume that the broken line refers fairly accurately to absorption in the downcomer alone, in that it lies closely parallel to the axis and that the absorption is independent of conditions upstream of the weir provided the raft is not too close to it. The Reynolds number chosen was at the upper end of the the arguments held a fortiori “laminar” range; for lower values.
5
00
I
0
7’
i’- _a__,,‘bx 4
3
xx
x
0
0 0
xx x
xx A’
x x
.I’ c’ __dH
2-
__-__--
3.0 -
________-----
FIG. 9. Mass-transfer <
2.5/
2.o.l 0
/
/
coefficient as a function of Reynolds number. 0, observed results corrected according to Eq. (17) (See Fig. 8); X, observed results corrected according to Eq. (23) (See Fig. 8); --, calculated according to (17) ; - - -, calculated according to (23).
/
4
2 Clearance,
6
cm
FIG. 7. Effect of raft clearance (i.e. distance between raft and weir) on absorption rate, R. 0, experimental values; x, experimental values less calculated correction; - - - -, estimated absorption in weir; -, estimated absorption in weir plus calculated correction. Re=5000.
Figure 8 shows total absorption rates with and without a raft in position, as a function of Reynolds number. The full lines are merely smooth “experimental” curves. The broken lines are curves for absorption in the weir calculated according to Eq. (17), as in Fig. 7, and also according to simple 1170
On gas absorption
into a turbulent liquid
penetration theory, Eq. (23). The estimated absorption in the test section is obtained by difference. It is at once apparent that the results of simple penetration theory and of our more elaborate approach are significantly different. Figure 9 shows as points the results obtained by difference from Fig. 8, replotted as a mass-transfer coefficient vs. Reynolds number. The upper points are those obtained using Eq. (17) for the end correction while the lower follow from Eq. (23). The upper full line represents the mean masstransfer coefficient calculated for a free surface up to the weir, with no raft, using Eq. (17). The lower full line represents a similar calculation based on Eq. (23). As might be expected, the predictions of the former are vastly better than those of the latter though they overestimate by some 10 per cent. In a qualitative sense, it can be seen that a positive value of dU,/dx involves a volume flow of water towards the free surface, which steepens the concentration gradient and hence increases the absorption. We may expect this simple effect to be a significant cause of increased absorption rates in other flows.
TABLE1 Rod diameter d (in.)
Mesh spacing M (in.)
Mid
1
+
Q
5.0
2
6
i?5
4.1
3
*
*
5.0
Grid No.
Equation (11) was solved numerically for various values of P. Over the range that was relevant, the mean mass-transfer coefficient, defined in Eq. (9), was closely approximated by
liRC = 1*46(0,/2/A)+.
Figure 10 shows typical concentration profiles. The values of turbulent intensity chosen are at the low end of the range of interest; even so, the Peclet number P is of the order of 100-1000, and so as expected the vertical concentration gradient is very sharp over most of the surface of the eddy.
5. RFSULTSFOR TURBULENT FLOW
4
A ‘known’ turbulent velocity field was generated by inserting square grids made from circular cylindrical rods at right angles to the mean flow direction near the end of the calming section. The details of the meshes are given in Table 1. All had values of M/d close to that used by BATCHELOR and TOWNSEND [21]. We have therefore adopted their results for U2 and A, which were
0.2
x/n 0.6
I,
c $
_
t
JS_ u
0.086 (x/M-10)”
0.6 -
(25)
and $=(O.l+=+.
(27)
(26)
The Reynolds numbers used for the main flow ranged from 4000 to 10,000, i.e. rather larger than those used in the laminar flow experiments, but small enough for the turbulence described by (25) and (26) to dominate the mass-transfer process. 1171
FIG. 10. (a) Calculated concentration contour for large-eddy model with JiS=O.O3 cm/ se-c, h=0.5 cm; numbers within the cell refer to values of (C-a&/(C* Cbdk). Arrows indicate general direction of eddy flow.
I.0
G. E. FORTESCUEand
FIG. 10. (b) Calculated concentration profiles for various values of x/A near the top, middle and bottom of the cell. -, Jiiz=O*lO cm/set; ---, ,/iiz=O*O3 cm/set; h=0*5 cm.
The absorbed gas is carried into the bulk of the fluid (see Fig. 3) largely by the layers of “surface” fluid that plunge downwards near the edge of the eddy. Details of the numerical methods used are to be found in FORTESCUE[17+except for artificially small values of P, solution was not carried out simultaneously over the entire square cell, but rather in the manner adopted for the laminar flow case described above; the elliptic problem was replaced by an approximately-valid parabolic one, and so a step-by-step solution from x =0 was employed with suitably chosen boundary conditions. It is worth comparing the result (27) with that predicted by (2), the crude Higbie model, when T, is taken to be A/,fS2, which gives kH = 142(4/&Q3,
J.R.A.PEARSON In interpreting experimental observations, an end correction had to be obtained to eliminate the absorption in the weir and downcomer. This was done by using a raft as before; a laminar flow calculation for the short distance between the end of the raft and the weir was carried out as in the previous section. Although at the high Reynolds numbers involved, some “natural” turbulence could be expected, experiments carried out without a grid in position suggested that the error involved in using a laminar flow approximation would not be unacceptably large-nevertheless an error is involved. The adjustable variables in the experiments were Reynolds number (Re), mesh size (M) and distance of the grid from the entry to the test section (x0). Variation of predicted E, with M proved to be small. Thus for x0 = 18 cm, and at all Re, E, (grid 2) = 144 IE, (grid 3) =0*95 I;, (grid 1). Figure 11 shows a plot of all experimental values expressed as (exp li,/predE,) against Re. It is seen that they lie about l&15 per cent below the values predicted by (27), but are greater than the values predicted by (28). There is no systematic dependence on mesh size. Agreement with the large eddy model could be improved by supposing that only 70 per cent of the total turbulent intensity ‘resided’ in the large model eddies; as was suggested earlier, our calculations were expected to provide an upper bound for absorption. The
(28)
a value roughly 20 per cent lower. Relation (27) is, of course, a function of x. Using (25) and (26), and integrating over the test length L gives a mean value
E, =;
s
LER&x)dx 0
(29)
that can be used for comparison with experimental results. We note that in this turbulent case, the absorption due to any variation in the mean surface velocity has been neglected by comparison with that due to the eddies, which are themselves taken to be those characteristic of the turbulence observed behind square grids in closed channel flows. 1172
--_-------_
=
10 ___~________
_____-_________ \
8. eco0
Penetmlion
theory
Boo0 Re
RG. 11. Comparison between observed and prex0=18 cm. dicted mass-transfer coefficients. Theoretical kT, Eq. (27) Grid no. Symbol 21 0.85 0.81 Ret x 10-4 3 0.77 Ret x 10-4 -, expected upper bound using (27); - - -, predicted by penetration theory using (28). 2 0
m,cQJ
On gas absorption
into a turbulent liquid
0.2 t
loo0
am0
6ocQ
Re
FIG. 14. Complete range of mass-transfer coefficients as observed and calculated as functions of Reynolds number. corrected experimental values for laminar 0, flow; corrected experimental values for grid A, No. 3, x0=18 cm; x, corrected experimental values for grid No. 2, x0=10 cm; , predicted values from large-eddy model; ---, predicted values based on Eq. (17).
FIG. 12. Variation of I& with grid configuration: 0, grid No. 3 and x0=18 cm; x,gridNo.2andxo=lOcm.
effect of varying x,, is shown on Fig. 12; different grids were chosen, so as to maximize the different absorption rates. x0 could not be varied over a wider range because it either led to too low values of turbulent intensity in the test section, or involved a range of values of x in the test section for which Eqs. (25) and (26) are not valid. Figure 13 shows a logarithmic plot of some experimental values of li, against Re. Best fitting straight lines through the somewhat scattered observations have slopes lying between 0.50 and 0.53, entirely in agreement with the model. Finally, Fig. 14 shows both laminar and turbulent measurements of IE plotted against Re together with predicted values. The agreement looks most convincing when displayed in this manner, i.e. logarithamically (!). I
, 3000
7ow
5ocQ
OXQ
6. DISCUSSION
Re
F’IG. 13. Mass-transfer coefficient number.
vs. Reynolds
0, grid No. 3 and x0=18 cm; x,gridNo.2andxo=10cm; -, best fits to corrected experimental values.
The interpretation proposed above for turbulent absorption has been tested under the most favourable circumstances. Nevertheless, the success of a simple calculation scheme under both laminar and turbulent flow conditions suggests that the method
1173
G. E. FORTESCUEand J. R. A. PEAR%IN
may be more generally applied. In either case, all that is really required is information about the surface velocity distribution, whether it be steady or only statistically defined. For all circumstances where the relevant Peclet number is large, it can be assumed that the component of the velocity vector parallel to the surface is a function of surface position only, and so the component normal to the surface can be obtained from the equation of continuity, as in Eq. (16). Just as the mean masstransfer coefficient given by Eq. (19) was obtained by integration over a known spatial distribution of eddies, so the absorption for a known statistical distribution of different eddy sizes could be obtained, for example, by integration over the size spectrum, using the correlation function as weighting function. An advantage of this method is that, with suitable refinement, it could be used in circumstances where both random turbulent eddies and steady secondary flows were significant in promoting mass transfer across a free surface. This situation often arises when vigorous stirring occurs well beneath the surface. It is however worth noting that the method has one limitation. It requires the free surface to remain essentially flat. Cases where large surface waves arise, possibly with some breaking, often lead to rapid increases in the mass-transfer coefficient. This point is discussed in more detail in FORTFSCUE[17], where reasons are given for supposing that these difficulties were absent in our experiments. This is a non-trivial matter because, in practical circumstances where very large absorption rates are required, the establishment of very large interfacial areas is of prime importance. The large-eddy approach could also be extended to cover the case of absorption with reaction, by addition of a reaction term in our basic Eq. (10). If the reaction rate were a function only of c, then a numerical solution could usually be obtained. However, if the reaction term involved the concentration of one or more other reactants, which were significantly depleted, then coupled sets of equations would be required and a more elaborate scheme would be needed. In these circumstances, the difference between the large-eddy model and
the Higbie model might well be greater than occurs in this case of physical absorption. The original impetus for this work came from a desire (PEARSONand PEARSON[26]) to predict reaeration rates for polluted streams and rivers, as in the case of DOBBINS[ll]. By using the same criteria as O’CONNORand DOBBINS[14], i.e. ,/ti” equal to 10 per cent of the mean velocity and A to be 10 per cent of the stream depth, our predictions turn out to be rather better than those given by penetration theory. Their results are replotted in Fig. 15. Clearly the scatter precludes the drawing
Reported
k,
ft/doy
FIG. 15. Comparison of reported and estimated reareation rates. Estimates based on Eq. (27). 0 x V n 0 @ +
Ohio River Elk River Clarion River Illinois River Tennessee River Brandywine River San Diego Bay
of firm conclusions, though it is perhaps worth observing that information on the surface velocity distribution of flowing streams should be much easier to obtain than on the entire turbulent structure. It is to be hoped that such information may be made available.
On gas absorption
Acknowledgment-One of us (G.E.F.) wishes to acknowledge the financial support provided by The Colonial Sugar Refining Co. Ltd., while the work reported here was being carried
out.
into a turbulent
liquid
separation vector rate of surface renewal ; surface renewal time US surface velocity vector surface centre-line velocity ur velocity in r-direction u, 0, w velocities in x-, y-, z- direction field velocity vector V n co-ordinate vector x, Y, z Cartesian co-ordinates A integral length scale V kinematic viscosity. r
v,o
NOTATION velocity scale field concentration saturation concentration d width of channel D diffusion coefficient E mean kinetic energy of roll cell velocity distribution function ; film thickness k mass-transfer coefficient k mean k L length of test section A4 mesh size P Peclet number R rate of absorption Re Reynolds number A
cz
_ _ Subscripts D Danckwerts H Higbie K
Kishinevskii in test section roll cell turbulent Whitman 0 initial
L RC T W
REFERENCES LEWIS W. K. and WHITMAN W. G., Ind. Engng Chem. 1924 16 1215. HIGBIE R., Trans. Am. Inst. Chem. Engrs 1935 35 365. DANCKWERTS P. V., Ind. Engng Chem; 195143 1460. HANRATTY T. J.. A.Z.Ch.E. JZ 1956 2 359. PERLMUTTERD. ‘D., Chem. Engng Sci. 1961 16 287. TOOR H. L. and MARCHELLOJ. M., A.Z.Ch.E. JI 1958 4 97. DOBBINS W. E., Water PolI. Res. Inst. Conf. 1962 2 70. HARRIOTT P., Chem. Engng Sci. 1962 17 149. MARCHELLOJ. M. and Toot H. L., Ind. Engng Chem. Fundls 1963 2 8. RUCKENSTE~NE.. Chem. Engne Sci. 1963 18 233. DOBBINS W. E., j. San. En&g Div., Proc. A.S.C.E. 1964 53. KISHINEVSKII M. K., J. Appl. Chem. U.S.S.R. 1955 28 881. KISHINEVSKII M. K. and SEREBRYANSKIIV. T., J. Appl. Chem. U.S.S.R. 1956 29 29. O’CONNOR D. J. and DOBBINS W. E., Trans. A.S.C.E. 1958 123 641. LEVICH V. G., Physic0 Chemical Hydrodynamics. Prentice Hall 1962. DAVIES J. T., Advances in Chemical Engineering, Vol. 4, Academic Press 1963. FOR~ESCUE G. E., Ph.D. Thesis, University of Cambridge 1965. ELLI~~N T. H., J. Fhdd Mech. 1960 S 33. H~NZE J. 0.. Turbulence. McGraw-Hill 1959. TOWNSENDA. A., The Structure of Turbulent Shear Flows, Cambridge University Press 1956. BATCHELORG. K. and TOWNSENDA. A., Proc. R. Sot. 1948 193A 539. GIBBON C. H. and SCHWARZ W. H., J. Fluid Mech. 1963 16 357,365. KILNER A. A., Ph.D. Thesis, University of Cambridge 1963. CULLEN E. J., Ph.D. Thesis, University of Cambridge 1956. SEIDELL A., Solubilities of Inorganic and Metallic Organic Components, Van Nostrand 1940. PEARSONC. R. and PEARSONJ. R. A., A.Z.Ch.E.-I.Chem. E. Symp. Ser. 1965 9 50.
R&me-On suppose que le transfert de masse a travers la surface d’un liquide a flot turbulent est determine par des remous a relativement grande kchelle. Leur kchelle de longueur est consider&e etre I’tkhelle integrale de la turbulence dans l’ensemble du fluide, tandis que leur intensite est prise aussi pour correspondre a eelle du champ de l’ensemble du courant turbulent. Les calculs ont ete faits pour un simple modele cellulaire et compares aux resultats experimentaux pour une absorption de CO2 dans l’eau dans la partie du courant sit&e derriere une grille p1ac.k en travers de I’entrke d’un canal ouvert. L’accord de ces calculs et& remarquablement satisfaisant. 11faut souligner qu’il n’y a pas de parametres dans le modele. A l’origine le but du travail dtait de foumir un modele pour le processus d’abation des rivieres et des tours d’eau pollues.
1175
G. E. FORTESCUE and J. R. A. PEARSON 2usanun~Von der Stoffiibertragung quer zur OberNche einer turbulent stromenden Fliissigkeit wird (bier) angenommen, dam sie durch verhilltnism&sig grosse Strudel bestimmt wird. Deren Liingenmassstab wird als Integrahnassstab der Turbulenz im Hauptvolumen der Fltissigkeit angesehen, wahrend ihre Intensitit ausserdem als der des Wirbelstriimungsfeldes der Hauptmassen enstprechend gedacht ist. Ftir em einfaches Zellenmodell wurden Rerechnungen durchgefiihrt und mit den experimentellen Ergebnissen fti COz-Absorption in Wasser innerhalb der Striimung-hinter einem quer zum Emgang eines offenen Leitkanals angebrachten Gitters-verglichen. Die *reinstimmung war bemerkenswert. Es wird ausdrtlcklich darauf hingewiesen, dass es im Model1 keine freien Parameter gibt. Das ursprtlngliche Ziel der Arbeiten war die Erstellung eines Modells fur die Rehiftung von verunreinigten Fltissen und Stromen.
1176
Printed in Great Britain.
On gas absorption into a turbulent liquid G. E. FORTESCUE~ and J. R. A. PEARSON Department
of Chemical Engineering, Pembroke Street, Cambridge (Received 27 January 1967)
Abstract-Mass transfer across the surface of a turbulent flowing liquid is here assumed to be determined by relatively large scale eddies. Their length scale is taken to be the integral scale of the turbulence in the bulk of the fluid, while their intensity is also taken to correspond to that of the bulk turbulent flow field. Calculations have been carried out for a simple cellular model and have been compared with experimental results for absorption of CO2 into water in the flow behind a grid placed across the entrance to an open channel. Agreement has been remarkably good. It is emphasized that there are no free parameters in the model. The original objective of the work was to provide a model for the process of re-aeration of polluted rivers and streams.
1. INTRODUCTION GAS ABSORPTION by turbulent liquids is a process that remains difficult to predict numerically, whilst not being easy to measure except as a gross overall Mass transfer into the fluid is phenomenon. clearly a combination of diffusion and convection, but the effects are coupled and unsteady. In principle, a complete knowledge of the kinematics of any flow field should allow the mass transport to be deducible from it, using standard mass conservation equations and boundary conditions. Unfortunately, the kinematics of turbulent flow fields is usually imperfectly known, and then only in terms of certain statistically relevant mean quantities. Models for predicting gas absorption have therefore to replace the exact physical situation by a much simpler one. The value of such models depends on how generally they can be applied, how accurate they are, and the extent to which they involve arbitrary parameters that vary from system-to-system : if the parameters correspond to reasonably well defined physical quantities, and can be deduced or measured a priori, then the models tend to be plausible; if the models are accurate and can be applied widely, then they are also useful. The classical model is the film model of WHITMAN [I]. This supposes that resistance to mass transfer
t Present address: Colonial Sugar Refining Corporation,
occurs across a thin layer of stagnant fluid, of thickness h,, situated at the surface, and that the bulk fluid below is well mixed. Calculation of the absorption rate is very simple, and is given by k,=Dlh,,
(1)
where k, is the mass transfer coefficient and D is the diffusion coefficient, provided h, is known. Either h, can be obtained experimentally or attempts can be made to deduce it from physical reasoning. Using the first method, there are doubts whether results will apply over a wide range of values of D. Using the second, it is difficult to envisage the existence of a true layer unless some surface-active or insoluble material actually overlies the surface!. One is therefore forced to treat the layer model as a sophisticated simplification, in which h, may be a function of all the variables in the problem. The next model developed was the penetration model of HIGBIE [2] who based his approach on a surface-renewal argument. The role of the turbulence was to bring up elements of bulk fluid to the $ It is perhaps unfortunate that the most commonly used fluid, water, is particularly prone to form rigid surface layers, and so give an obvious visual appearance of surface immobility; hence the prevalence of the film approach. Sydney, N.S.W., Australia.
1163
G.
E. FORTESCUE and J. R. A. &ARSON
surface, where unsteady absorption, as into an infinite stagnant liquid, takes place for a prescribed time TH, after which the element returns to the bulk and is replaced by another. Again calculation of mass transfer is simple, giving
DAVIES[16]) retain one or more arbitrary constants, the only way in which comparison with experimental results can serve to distinguish between them is by holding everything constant except D. Whitman predicts ka D, HigbieDanckwerts ka D* and Kishinevskii ka D” where 0 < n <+. Unfortunately it is difficult to vary D over k,=2JDlnT,. (2) a wide range, whilst it is easy to elaborate models sufficiently to match available experimental DANCKWERT~ [3] modified this model to allow for a information. random distribution of surface ages, consequent on We shall present here a new model that attempts a fractional rate of replacement of surface elements, to retain the most obvious physical characteristics s,. This yields the relation of a deforming (plane) surface such as is observed at the interface between a turbulent liquid and a k,=JG. (3) gas and uses only such parameters as can be readily correlated with measurable kinematic quantities However, once again the predictions involve characteristic of the liquid flow field. All theories arbitrary parameters, TH or sn, that must either be agree that processes close to the surface determine determined experimentally or deduced from physi- rates of gas absorption and that mixing and thus cal arguments. Although a turbulent flow field can mass transfer in the bulk of the fluid is rapid by yield characteristic times, there have been few comparison. A valid model must therefore pay particular attention to the flow very near the attempts to relate these directly to TH or s,. surface. It is in this regard that earlier models can Both the Whitman and Higbie-Danckwerts models have been elaborated (HANRAT~Y [4]; be faulted. The details of our quasi-steady largeeddy model are given in the next section, together PERLMUTTER[5]) or combined and elaborated (TOORand MAKCHELLO [6], DOBBINS[7], HARRIOT~ with the relevant transport equations. Section 3 describes experimental apparatus that [8], MARCHELLOand TCX~R[9], RUCKENSTEIN [lo], DOBBINS[ 1l]), but progressively improved agree- was used to test the model, and discusses difficulties ment with observation has only been achieved at that were encountered in making measurements. Section 4 compares, briefly, experimental and theothe expense of increasing complexity. retical results that were obtained for absorption An alternative steady-state phenomenological approach has been advanced by I&G~INEVSKII [12, into a laminar flow, which could be regarded as a 131 in which the turbulent convective effects give simple large eddy. This preliminary work was undertaken to ensure that later results would be rise to an eddy diffusivity, D,, which is obtained from experiment. In highly agitated flows he meaningful. The main results on absorption into a argues that D,>> D and so the mass transfer co- turbulent flow are given in Section 5. Further details can be found in FORTESXJE[17]. efficient kK obeys the relation Application of the results to rates of re-aeration for polluted streams and rivers is given briefly in k,aJD+D,+i& (4) the concluding section. The turbulent diffusivity concept can be refined, so that D, is taken to be a function of position, usually increasing with distance from the surface, being zero at the surface; in this case no surface renewal need be postulated. Because most of these models (for exceptions see O’CONNOR and DOBBMS [ 141, LEVICH [15],
2. THE LARGE-EDDYMODEL The convective effect of turbulence on the flow field near the absorbing surface is here represented by eddies sweeping fresh liquid to the vicinity of the surface and then removing solute-enriched liquid to be dissipated in the bulk of the absorbing
1164
On gas
absorption into a turbulent liquid
medium. The surface acts as a constraint on possible motions, in that no normal velocity is allowed at the surface; but tangential surface velocities are permitted?. The velocity structure will obey the equation of mass conservation but not of momentum conservation, since we are not concerned with dynamic aspects of the flow. Although turbulent flows contain a spectrum of eddies of constantly varying nature, we shall suppose that their mean mass-transfer properties can be modelled by means of a regular sequence of steady square roll cells touching the surface, moving as a whole with the local mean surface velocity. These are shown in Fig. I. Direction
Thus we shall take the length scale, A, of our eddies to be the integral length scale of the underlying turbulent fiowfield. [See Eq. (7) below]. For convenience the eddies are taken to be square, and we shall without further justification impose the following arbitrary velocity pattern within them: u = A sin(nx/A)cos(ay/A)
,
u = -A cos(nx/A)sin(ny/A)
,
(5)
w=o. where A is a so far unspecified constant, and the co-ordinate system is defined by Fig. 2. The alternating nature of the vorticity within the eddies is shown on Fig. 1. It is worth observing that these simulate a general pattern of upwelling along the planes x=2NA, where N is an integer, with down currents along the planes x =(2N+ l)A. By making A a function of the third co-ordinate, z, a slightly
of mean flow
Bulk of fluid FIG. 1. Distribution of roll cells near the surface as proposed in the large-eddy model.
We have now to prescribe the size of the cells, and the motions taking place within them. Unfortunately there is not a great deal of direct experimental information relating to the velocity structure of turbulent flows near a free surface. ELLISON [18] has pointed out that the liquid surface in open channel flow appears to be disturbed by large, low frequency eddies, as anyone sitting at a river bank in time of spate can see for himself. The nearly fixed surface position that is imposed by gravity might, on general grounds, be expected to damp out small rather than large eddies, while the lack of any significant additional dissipating or turbulence-generating mechanism at the surface suggests that the general eddy pattern will not be substantially different from that relevant in the bulk of the field of motion. Furthermore, all available evidence (see for example HINZE [19] or TOWNSEND [20]) seems to suggest that it is the large eddies that are dominant in transfer processes. 7 The film model allows no fluid velocity whatever in the neighbourhood of the surface, while the surface renewal models appear to allow discontinuous normal velocities of the actual surface, both physically unrealistic.
surface
A
A
FIO.2. A large eddy showing co-ordinate system and length scale.
more plausible model is obtained, but since this does not materially affect later calculations, we shall not introduce this complication here. We might have added a velocity component, w, in the z-direction, but we have no evidence that the ensuing complications would lead in the end to any substantial difference in our results, whilst there is evidence (TOWNSEND [20]) that two-dimensional roll eddies do in fact predominate in shear flows downstream of grids and cylinders.
1165
G. E. FORTESCUE and J. R. A. PEARSON
The mean kinetic energy associated cell [Eq. (5)] is given by
with the roll
;(a2 +u2+w2)&dy
= $A2.
(6)
It is generally true that the main energy-containing eddies have length scales comparable with the integral length-scale of the turbulence given by A=
?&(x)u,(x s
u;(x) )
+ r)dr
(7)
i-
0
where u, is the velocity as a first approximation
in the r-direction, we put
and so
‘4=2(@+@+33)f,
(8)
i.e. twice the root-mean-square turbulent intensity. This we expect to be an upper bound. The mean mass transfer coefficient, l&c, given by
I;,,+
A -ac
s(> 0
dy y=o
Finally we must choose a suitable condition y=A. Because we have already implied mixing in the bulk is rapid, we shall take c=
)
aty=A.
(12c)
Clearly the sudden discontinuity in &/c?y at y=A, which is implied by (12c), cannot be thought of as having any physical counterpart. A more plausible picture is obtained by considering only the upper half (y c $12) of the roll cell to be physically relevant, as shown in Fig. 3, with smaller scale eddies increasing in importance for y > +A, thus achieving a rapid rise in mass transport within the bulk. The bottom of the cell is thus to be interpreted as a virtualsink for solute, with the strictly anti-symmetric structure about y=+A only introduced for computational convenience. This interpretation can be justified a posteriori by reference to concentration profiles. Rising
dx
Cbulk
for that
1 jet
Falling
’
jet
(9) ,\jl
where c is the point solute concentration, is then obtained from a solution of the steady masstransfer equation (v.V)c = DV’c ,
I t-
A
-I
I
FIG. 3. Physical interpretation of large-eddy model. Only the top half of eddy is involved in physical transfer process, which is dominated by jets of rising and falling liquid.
(10)
which in our roll-cell case becomes A sin (7rx/A) cos (ny/A) &/ax -A
cos (7cx/A) sin (ny/A)
&/8y
= D(d2c/8x2 + d2c/dy2).
(11)
Our remaining task is to prescribe boundary conditions on c. If we assume no gas-phase resistance to mass transfer, then we can assume saturation at the surface, i.e. c=c*aty=O. Because of the symmetry require that
of the cell structure,
&/ax =0 at x = NA, Nan integer.
(12a) we
(12b)
We note that Eq. (11) and boundary conditions (12) can be made dimensionless using (c* - cb”rk) as a characteristic concentration, A a length and A a velocity. Only one dimensionless parameter remains, the P&let number P= AA/D.
(13)
If P is large, then molecular diffusion will only be significant in a region close to the surface (compared with A), and our model will be dominated by “turbulent convection”; under these circumstances we might expect it to give good agreement with experiment. However, if P is small, then mass transfer will be as in the Whitman model, i.e. by molecular diffusion across a film of thickness A; we do not expect the model to be valid in this range.
1166
On gas absorption
into a turbulent liquid
3. EXPERIMENTAL ARRANGEMENTS
I.21
Having decided on a model for turbulent absorption, we wished to make measurements in a situation where the statistical properties of the turbulent flow field were well known. This ruled out the traditional stirred cell; indeed only one possibility suggested itself: the flow downstream of a square-mesh grid. Detailed investigations in air by BATCHELORand TOWNSEND[21] and in water by GIBSON and SCHWARZ [22] provided sufficient information for our purposes. Figure 4 shows the schematic layout of the apparatus used. A continuous recirculating water flow passed through a rotameter R, a distributor D, consisting of a packing of JJ in. ceramic balls followed by a 64 mesh wire screen, a calming section C, consisting of a closed duct 3 ft long, 12 in. wide and 13 in. deep; it then flowed through one of three square grids G made of cylindrical brass rods placed at the downstream end of the calming section, into the test section T consisting of an open rectangular channel 1 ft wide and 2 ft 3 in. long. The water level in the test section was maintained at between 14 and 2 in.; an even flow could only be achieved by flowing over a weir W (shown inset). The inlet and outlet arrangements both introduced undesirable end-effects on the flow profiles near the surface; however, direct observation with talc particles showed that these were confined to less than 10 per cent of the total surface area of absorption, and allowance could be made for them in predicting absorption rates. A
FIG. 4. Schematic diagram of absorption
apparatus.
0
20
IO
(a) laminar flow,
f, min Re = 1500.
I
b f,
;o min
(b) turbulent flow, Re=4500; grid No. 1, x0=18 cm. FIG. 5. Typical forms of experimental observations for absorption rate R as a function of time r;
much more serious source of error was introduced by the high rates of absorption taking place in the weir: these could neither be neglected nor estimated, and so were separately measured by covering up most of the absorbing layer with a raft. The space above the test section was filled with CO, and the rate of absorption was measured with a soap-film meter. Figure 5 shows the differing results obtained for laminar flows (a) and turbulent flows (b). In the laminar case, absorption occurs in a series of steps, with cburk changing discontinuously, whereas in the turbulent case, cbu& appears to decrease continuously_i. Assuming that t The straight line obtained on a semi-logarithmic plot in the turbulent case was in accordance with the concept of a fully mixed fluid medium. The period of the discontinuities in the laminar case appeared to correlate with the time required to recirculate the total water volume.
1167
G.
E. FORTESCUE and J. R.
cbuk was initially zero, R, corresponding to &ik=O was obtained by extrapolation to zero time. The water used was tap water de-aerated by spraying into a partial vacuum. Experiments were carried out at atmospheric pressure. Before an experiment the space above the test section was purged with CO,. The amount of gas absorbed during this time was minimized by having a stagnant water surface, and was calculated to lead to less than 3 per cent of saturation. No steps were taken during a run to control temperature, which was however measured and never rose by more than 2°C. Absorption rates were always corrected to a standard temperature of 20°C by the relation R020’=ROT”(~*D~)20”/(~*Df)P.
A. PEARSON
found to agree in the laminar case, then it could justifiably be assumed that the experimental methods and observations were reliable. Turning then to the turbulent case, comparison of experimental observations and predictions based on the large-eddy model of Section 2 could then be regarded as a true test of the model. The first requirement for our calculations was a knowledge of the velocity field. The only relevant region is that close to the surface, because absorbed CO2 only “penetrates” a short distance into the water. The surface velocity field was measured photographically using talc particles as tracers. It was found that for our purposes the surface velocity vector could be written (15)
(14)
The assumption of a Df dependence war suggested by KILNER'S[23] results and justified (I posteriori.
4. RESULTSFOR LAMINAR FLOW Experiments were first carried out under conditions of laminar flow largely in order to decide whether the apparatus could be used to give predictable results at all. In particular, it was moet important to discover whether the large absorption in the weir downcomer could be satisfactorily measured. This was studied by measuring absorption rates with and without a raft placed on the surface; the effect of the raft was to reduce the length of the test section leading to the downcomer. Because the amount of absorption at the flat free surface was expected to be much less in laminar than in turbulent flow, experiments undertaken with and without the raft would need to be much more sensitive in the former than the latter case, to achieve comparable proportional accuracy in the difference of the two observations. Furthermore, calculations of gas absorption in the laminar case could be undertaken without postulating a model; the total actual steady velocity field could be measured and so used directly in the massIf therefore, experimental transfer equation. observations and numerical predictions were
Here x is a co-ordinate pointing downstream along the surface, z cross-stream along the surface with origin along the centre-line of the test section, and y vertically downwards from the surface. d is the half-width of the channel. Figure 6(a) shows a representative plot of f(z/d) for laminar flow experiments, while Fig. 6(b) is a typical plot of position (x) as a function of time (t) for a tracer particle. Care had to be taken to obtain U,, and hence dU,,/dx from such plots, because numerical differentiation is a notoriously unreliable process. Details of how smooth functions were obtained are given in FORTESCUE [17]. To obtain u for use in Eq. (lo), it was assumed that v=(U,,-ydu,/ax,o) (16)
It was also assumed that #c/$~~~~~c/~x~, a2c/az2. These are both consistent with the notion of a very thin layer near the surface in which c is substantially different from cb”rk. Hence the masstransfer equation becomes
1168
au ac Da2c 2-C "ZY ax ay ay2'
Uac
(17)
On gas absorption into a turbulent liquid
with boundary conditions c=c*at y=O,
(18)
c4Oas y+co, and initial condition c=O
0.6. 0 0 0
\
0.4
ac -=-
ay 00
0.2 -
Ap 0.2
0.4
0.6
0.6
(19)
These correspond to fresh water entering the test section at x=0 and a saturated surface. We note that if a solution of (17) is obtained for US= USO, the solution for all other USis obtained by a suitable scaling of y; in consequence
0
0
al
I.0
r/d
Fro. 6. (a) Surface velocity profile as a function of cross-channel position for laminar flow: -, mean curve: 0, from observations.
ac ay
z~otfw~l*.
(20)
If therefore we carry out all our absorption calculation in terms of a plane flow model l_JJo,the true absorption can be obtained by multiplication by a factor 1 [f(z/d)]+dz
$ I
-0-84,
(21)
d
as was in fact done. Equation (17) was solved by standard finitedifference approximation methods. Being a parabolic problem no unusual difficulties were encountered; near z=O, U,,(x) was taken to be a smooth monotonic positive function, thus avoiding any computational difficulties?. Total absorption was calculated in terms of a mean absorption coefficient.
40
20 SCC
(b) Typical position gether nomials
plot of (0) as with (-).
surface tracer particle a function of time, toapproximating polyRe=3400.
a,
where L is the effective length of the absorbing section. This method was employed to calculate li, both without and with the raft in position. L could not be zero in the latter case because otherwise the flow in the downcomer would not have been closely similar in the two cases. t Little error was introduced in this way, because. the “attached eddy” at the entrance point was small. 1169
G. E. FORTEKXIE and J. R. A. For purposes of comparison solutions obtained for the “simple penetration equation
ac a% ‘Jsoz = Da7
PEARSON
were also theory”
(23)
3
(23)
with boundary conditions (18) and initial conditions (19) as before. Dimensionless flow rates will be given in terms of the Reynolds number
i&l
3X0
2500
EC0
Re
Re=4Q/vw,
FIG. 8. Laminar flow experiments at various Re. 0, Absorption rate without raft, observed; 0, absorption rate with raft, observed; A, absorption in weir calculated on the basis of Eq. (17); x , absorption in weir calculated on the basis of Eq. (23); Broken and full curves denote “best fits”, from which absorption rates in test section could be calculated.
(24)
where Q is the total flow rate, v the kinematic viscosity and w the wetted perimeter of the duct. Values of D were taken from CULLEN [24], and c* from SEIDELL [25]. Figure 7 shows results obtained at Re =5000, with a raft in four separate positions, together with a correction calculated according to Eq. (17), boundary conditions (18) and initial conditions (19). It seems reasonable to assume that the broken line refers fairly accurately to absorption in the downcomer alone, in that it lies closely parallel to the axis and that the absorption is independent of conditions upstream of the weir provided the raft is not too close to it. The Reynolds number chosen was at the upper end of the the arguments held a fortiori “laminar” range; for lower values.
5
00
I
0
7’
i’- _a__,,‘bx 4
3
xx
x
0
0 0
xx x
xx A’
x x
.I’ c’ __dH
2-
__-__--
3.0 -
________-----
FIG. 9. Mass-transfer <
2.5/
2.o.l 0
/
/
coefficient as a function of Reynolds number. 0, observed results corrected according to Eq. (17) (See Fig. 8); X, observed results corrected according to Eq. (23) (See Fig. 8); --, calculated according to (17) ; - - -, calculated according to (23).
/
4
2 Clearance,
6
cm
FIG. 7. Effect of raft clearance (i.e. distance between raft and weir) on absorption rate, R. 0, experimental values; x, experimental values less calculated correction; - - - -, estimated absorption in weir; -, estimated absorption in weir plus calculated correction. Re=5000.
Figure 8 shows total absorption rates with and without a raft in position, as a function of Reynolds number. The full lines are merely smooth “experimental” curves. The broken lines are curves for absorption in the weir calculated according to Eq. (17), as in Fig. 7, and also according to simple 1170
On gas absorption
into a turbulent liquid
penetration theory, Eq. (23). The estimated absorption in the test section is obtained by difference. It is at once apparent that the results of simple penetration theory and of our more elaborate approach are significantly different. Figure 9 shows as points the results obtained by difference from Fig. 8, replotted as a mass-transfer coefficient vs. Reynolds number. The upper points are those obtained using Eq. (17) for the end correction while the lower follow from Eq. (23). The upper full line represents the mean masstransfer coefficient calculated for a free surface up to the weir, with no raft, using Eq. (17). The lower full line represents a similar calculation based on Eq. (23). As might be expected, the predictions of the former are vastly better than those of the latter though they overestimate by some 10 per cent. In a qualitative sense, it can be seen that a positive value of dU,/dx involves a volume flow of water towards the free surface, which steepens the concentration gradient and hence increases the absorption. We may expect this simple effect to be a significant cause of increased absorption rates in other flows.
TABLE1 Rod diameter d (in.)
Mesh spacing M (in.)
Mid
1
+
Q
5.0
2
6
i?5
4.1
3
*
*
5.0
Grid No.
Equation (11) was solved numerically for various values of P. Over the range that was relevant, the mean mass-transfer coefficient, defined in Eq. (9), was closely approximated by
liRC = 1*46(0,/2/A)+.
Figure 10 shows typical concentration profiles. The values of turbulent intensity chosen are at the low end of the range of interest; even so, the Peclet number P is of the order of 100-1000, and so as expected the vertical concentration gradient is very sharp over most of the surface of the eddy.
5. RFSULTSFOR TURBULENT FLOW
4
A ‘known’ turbulent velocity field was generated by inserting square grids made from circular cylindrical rods at right angles to the mean flow direction near the end of the calming section. The details of the meshes are given in Table 1. All had values of M/d close to that used by BATCHELOR and TOWNSEND [21]. We have therefore adopted their results for U2 and A, which were
0.2
x/n 0.6
I,
c $
_
t
JS_ u
0.086 (x/M-10)”
0.6 -
(25)
and $=(O.l+=+.
(27)
(26)
The Reynolds numbers used for the main flow ranged from 4000 to 10,000, i.e. rather larger than those used in the laminar flow experiments, but small enough for the turbulence described by (25) and (26) to dominate the mass-transfer process. 1171
FIG. 10. (a) Calculated concentration contour for large-eddy model with JiS=O.O3 cm/ se-c, h=0.5 cm; numbers within the cell refer to values of (C-a&/(C* Cbdk). Arrows indicate general direction of eddy flow.
I.0
G. E. FORTESCUEand
FIG. 10. (b) Calculated concentration profiles for various values of x/A near the top, middle and bottom of the cell. -, Jiiz=O*lO cm/set; ---, ,/iiz=O*O3 cm/set; h=0*5 cm.
The absorbed gas is carried into the bulk of the fluid (see Fig. 3) largely by the layers of “surface” fluid that plunge downwards near the edge of the eddy. Details of the numerical methods used are to be found in FORTESCUE[17+except for artificially small values of P, solution was not carried out simultaneously over the entire square cell, but rather in the manner adopted for the laminar flow case described above; the elliptic problem was replaced by an approximately-valid parabolic one, and so a step-by-step solution from x =0 was employed with suitably chosen boundary conditions. It is worth comparing the result (27) with that predicted by (2), the crude Higbie model, when T, is taken to be A/,fS2, which gives kH = 142(4/&Q3,
J.R.A.PEARSON In interpreting experimental observations, an end correction had to be obtained to eliminate the absorption in the weir and downcomer. This was done by using a raft as before; a laminar flow calculation for the short distance between the end of the raft and the weir was carried out as in the previous section. Although at the high Reynolds numbers involved, some “natural” turbulence could be expected, experiments carried out without a grid in position suggested that the error involved in using a laminar flow approximation would not be unacceptably large-nevertheless an error is involved. The adjustable variables in the experiments were Reynolds number (Re), mesh size (M) and distance of the grid from the entry to the test section (x0). Variation of predicted E, with M proved to be small. Thus for x0 = 18 cm, and at all Re, E, (grid 2) = 144 IE, (grid 3) =0*95 I;, (grid 1). Figure 11 shows a plot of all experimental values expressed as (exp li,/predE,) against Re. It is seen that they lie about l&15 per cent below the values predicted by (27), but are greater than the values predicted by (28). There is no systematic dependence on mesh size. Agreement with the large eddy model could be improved by supposing that only 70 per cent of the total turbulent intensity ‘resided’ in the large model eddies; as was suggested earlier, our calculations were expected to provide an upper bound for absorption. The
(28)
a value roughly 20 per cent lower. Relation (27) is, of course, a function of x. Using (25) and (26), and integrating over the test length L gives a mean value
E, =;
s
LER&x)dx 0
(29)
that can be used for comparison with experimental results. We note that in this turbulent case, the absorption due to any variation in the mean surface velocity has been neglected by comparison with that due to the eddies, which are themselves taken to be those characteristic of the turbulence observed behind square grids in closed channel flows. 1172
--_-------_
=
10 ___~________
_____-_________ \
8. eco0
Penetmlion
theory
Boo0 Re
RG. 11. Comparison between observed and prex0=18 cm. dicted mass-transfer coefficients. Theoretical kT, Eq. (27) Grid no. Symbol 21 0.85 0.81 Ret x 10-4 3 0.77 Ret x 10-4 -, expected upper bound using (27); - - -, predicted by penetration theory using (28). 2 0
m,cQJ
On gas absorption
into a turbulent liquid
0.2 t
loo0
am0
6ocQ
Re
FIG. 14. Complete range of mass-transfer coefficients as observed and calculated as functions of Reynolds number. corrected experimental values for laminar 0, flow; corrected experimental values for grid A, No. 3, x0=18 cm; x, corrected experimental values for grid No. 2, x0=10 cm; , predicted values from large-eddy model; ---, predicted values based on Eq. (17).
FIG. 12. Variation of I& with grid configuration: 0, grid No. 3 and x0=18 cm; x,gridNo.2andxo=lOcm.
effect of varying x,, is shown on Fig. 12; different grids were chosen, so as to maximize the different absorption rates. x0 could not be varied over a wider range because it either led to too low values of turbulent intensity in the test section, or involved a range of values of x in the test section for which Eqs. (25) and (26) are not valid. Figure 13 shows a logarithmic plot of some experimental values of li, against Re. Best fitting straight lines through the somewhat scattered observations have slopes lying between 0.50 and 0.53, entirely in agreement with the model. Finally, Fig. 14 shows both laminar and turbulent measurements of IE plotted against Re together with predicted values. The agreement looks most convincing when displayed in this manner, i.e. logarithamically (!). I
, 3000
7ow
5ocQ
OXQ
6. DISCUSSION
Re
F’IG. 13. Mass-transfer coefficient number.
vs. Reynolds
0, grid No. 3 and x0=18 cm; x,gridNo.2andxo=10cm; -, best fits to corrected experimental values.
The interpretation proposed above for turbulent absorption has been tested under the most favourable circumstances. Nevertheless, the success of a simple calculation scheme under both laminar and turbulent flow conditions suggests that the method
1173
G. E. FORTESCUEand J. R. A. PEAR%IN
may be more generally applied. In either case, all that is really required is information about the surface velocity distribution, whether it be steady or only statistically defined. For all circumstances where the relevant Peclet number is large, it can be assumed that the component of the velocity vector parallel to the surface is a function of surface position only, and so the component normal to the surface can be obtained from the equation of continuity, as in Eq. (16). Just as the mean masstransfer coefficient given by Eq. (19) was obtained by integration over a known spatial distribution of eddies, so the absorption for a known statistical distribution of different eddy sizes could be obtained, for example, by integration over the size spectrum, using the correlation function as weighting function. An advantage of this method is that, with suitable refinement, it could be used in circumstances where both random turbulent eddies and steady secondary flows were significant in promoting mass transfer across a free surface. This situation often arises when vigorous stirring occurs well beneath the surface. It is however worth noting that the method has one limitation. It requires the free surface to remain essentially flat. Cases where large surface waves arise, possibly with some breaking, often lead to rapid increases in the mass-transfer coefficient. This point is discussed in more detail in FORTFSCUE[17], where reasons are given for supposing that these difficulties were absent in our experiments. This is a non-trivial matter because, in practical circumstances where very large absorption rates are required, the establishment of very large interfacial areas is of prime importance. The large-eddy approach could also be extended to cover the case of absorption with reaction, by addition of a reaction term in our basic Eq. (10). If the reaction rate were a function only of c, then a numerical solution could usually be obtained. However, if the reaction term involved the concentration of one or more other reactants, which were significantly depleted, then coupled sets of equations would be required and a more elaborate scheme would be needed. In these circumstances, the difference between the large-eddy model and
the Higbie model might well be greater than occurs in this case of physical absorption. The original impetus for this work came from a desire (PEARSONand PEARSON[26]) to predict reaeration rates for polluted streams and rivers, as in the case of DOBBINS[ll]. By using the same criteria as O’CONNORand DOBBINS[14], i.e. ,/ti” equal to 10 per cent of the mean velocity and A to be 10 per cent of the stream depth, our predictions turn out to be rather better than those given by penetration theory. Their results are replotted in Fig. 15. Clearly the scatter precludes the drawing
Reported
k,
ft/doy
FIG. 15. Comparison of reported and estimated reareation rates. Estimates based on Eq. (27). 0 x V n 0 @ +
Ohio River Elk River Clarion River Illinois River Tennessee River Brandywine River San Diego Bay
of firm conclusions, though it is perhaps worth observing that information on the surface velocity distribution of flowing streams should be much easier to obtain than on the entire turbulent structure. It is to be hoped that such information may be made available.
On gas absorption
Acknowledgment-One of us (G.E.F.) wishes to acknowledge the financial support provided by The Colonial Sugar Refining Co. Ltd., while the work reported here was being carried
out.
into a turbulent
liquid
separation vector rate of surface renewal ; surface renewal time US surface velocity vector surface centre-line velocity ur velocity in r-direction u, 0, w velocities in x-, y-, z- direction field velocity vector V n co-ordinate vector x, Y, z Cartesian co-ordinates A integral length scale V kinematic viscosity. r
v,o
NOTATION velocity scale field concentration saturation concentration d width of channel D diffusion coefficient E mean kinetic energy of roll cell velocity distribution function ; film thickness k mass-transfer coefficient k mean k L length of test section A4 mesh size P Peclet number R rate of absorption Re Reynolds number A
cz
_ _ Subscripts D Danckwerts H Higbie K
Kishinevskii in test section roll cell turbulent Whitman 0 initial
L RC T W
REFERENCES LEWIS W. K. and WHITMAN W. G., Ind. Engng Chem. 1924 16 1215. HIGBIE R., Trans. Am. Inst. Chem. Engrs 1935 35 365. DANCKWERTS P. V., Ind. Engng Chem; 195143 1460. HANRATTY T. J.. A.Z.Ch.E. JZ 1956 2 359. PERLMUTTERD. ‘D., Chem. Engng Sci. 1961 16 287. TOOR H. L. and MARCHELLOJ. M., A.Z.Ch.E. JI 1958 4 97. DOBBINS W. E., Water PolI. Res. Inst. Conf. 1962 2 70. HARRIOTT P., Chem. Engng Sci. 1962 17 149. MARCHELLOJ. M. and Toot H. L., Ind. Engng Chem. Fundls 1963 2 8. RUCKENSTE~NE.. Chem. Engne Sci. 1963 18 233. DOBBINS W. E., j. San. En&g Div., Proc. A.S.C.E. 1964 53. KISHINEVSKII M. K., J. Appl. Chem. U.S.S.R. 1955 28 881. KISHINEVSKII M. K. and SEREBRYANSKIIV. T., J. Appl. Chem. U.S.S.R. 1956 29 29. O’CONNOR D. J. and DOBBINS W. E., Trans. A.S.C.E. 1958 123 641. LEVICH V. G., Physic0 Chemical Hydrodynamics. Prentice Hall 1962. DAVIES J. T., Advances in Chemical Engineering, Vol. 4, Academic Press 1963. FOR~ESCUE G. E., Ph.D. Thesis, University of Cambridge 1965. ELLI~~N T. H., J. Fhdd Mech. 1960 S 33. H~NZE J. 0.. Turbulence. McGraw-Hill 1959. TOWNSENDA. A., The Structure of Turbulent Shear Flows, Cambridge University Press 1956. BATCHELORG. K. and TOWNSENDA. A., Proc. R. Sot. 1948 193A 539. GIBBON C. H. and SCHWARZ W. H., J. Fluid Mech. 1963 16 357,365. KILNER A. A., Ph.D. Thesis, University of Cambridge 1963. CULLEN E. J., Ph.D. Thesis, University of Cambridge 1956. SEIDELL A., Solubilities of Inorganic and Metallic Organic Components, Van Nostrand 1940. PEARSONC. R. and PEARSONJ. R. A., A.Z.Ch.E.-I.Chem. E. Symp. Ser. 1965 9 50.
R&me-On suppose que le transfert de masse a travers la surface d’un liquide a flot turbulent est determine par des remous a relativement grande kchelle. Leur kchelle de longueur est consider&e etre I’tkhelle integrale de la turbulence dans l’ensemble du fluide, tandis que leur intensite est prise aussi pour correspondre a eelle du champ de l’ensemble du courant turbulent. Les calculs ont ete faits pour un simple modele cellulaire et compares aux resultats experimentaux pour une absorption de CO2 dans l’eau dans la partie du courant sit&e derriere une grille p1ac.k en travers de I’entrke d’un canal ouvert. L’accord de ces calculs et& remarquablement satisfaisant. 11faut souligner qu’il n’y a pas de parametres dans le modele. A l’origine le but du travail dtait de foumir un modele pour le processus d’abation des rivieres et des tours d’eau pollues.
1175
G. E. FORTESCUE and J. R. A. PEARSON 2usanun~Von der Stoffiibertragung quer zur OberNche einer turbulent stromenden Fliissigkeit wird (bier) angenommen, dam sie durch verhilltnism&sig grosse Strudel bestimmt wird. Deren Liingenmassstab wird als Integrahnassstab der Turbulenz im Hauptvolumen der Fltissigkeit angesehen, wahrend ihre Intensitit ausserdem als der des Wirbelstriimungsfeldes der Hauptmassen enstprechend gedacht ist. Ftir em einfaches Zellenmodell wurden Rerechnungen durchgefiihrt und mit den experimentellen Ergebnissen fti COz-Absorption in Wasser innerhalb der Striimung-hinter einem quer zum Emgang eines offenen Leitkanals angebrachten Gitters-verglichen. Die *reinstimmung war bemerkenswert. Es wird ausdrtlcklich darauf hingewiesen, dass es im Model1 keine freien Parameter gibt. Das ursprtlngliche Ziel der Arbeiten war die Erstellung eines Modells fur die Rehiftung von verunreinigten Fltissen und Stromen.
1176