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The flow of thin films of a viscous liquid down nearly horizontal surfaces
Chemical Engineering Science, 1966, Vol. 21, pp. 715-717. Pergamon Press Ltd., Oxford.
Printed in Great Britain.
The flow of thin films of a viscous liquid down nearly horizontal surfaces (Received 15 March 1966) IN A RECENT paper G~~DRIDGE and GARTSIDE[l] comment that in view of the boundary conditions involved, the Nusselt equation [2] relating flow rate to film thickness in a viscous fluid flowing down a plane inclined at c( to the horizontal m,s =
Other forces acting on the control volume parallel to the plane are the weight of the fluid pgmsx sin a and the shear force on the plane due to the viscous drag
au
0
708x where 70 = p -
ay y=~.
_3eY gsm a
(0
In order to evaluate 70 we must assume a velocity profile in the liquid. If the Reynolds number is low there is considerable evidence both in the paper by G~~DRIDGE and GARTSDE [l] and elsewhere 131 that the velocity profile is
cannot be expected to hold in the limit as er --f 0. This assertion which is backed up with experimental evidence deserves to be considered in more detail. Let us consider a film of a viscous fluid flowing down an inclined plane of slope a. If the film has not reached a steady state the film thickness m will vary with distance along the plane, x, measured positive in the direction of flow. We will dm . confine our attention to circumstances where Z 1s small but
parabolic
and hence u = g
(2my - y2). In which case
70=$. . The sum of these forces will cause the fluid to accelerate and hence by a momentum balance we can say
not necessarily small compared with a.
6M=pgmsina&---1;--
3pQSx
pgm cos a8m -
where M is the momentum flux, i.e. M= ‘1
Hence by substitution term in (Twe obtain dm
s
6Q2P ompu2 dy = 5m.
of M into Eq. (2) and neglecting the 3Qv-amasina
;i;;=&j2
-
T-gm3cosa
For any given system and flow rate there are clearly two values of m of particular significance, that given by 3Qv m3 = T, i.e. the Nusselt film thickness mo, and that g sin a
FIG. 1 The pressure approximation, the atmospheric pressure on the
at the point (x, y) see Fig. 1, is to a first given by (m - y) pg cos a + PO, where PO is pressure. Hence the force exerted by the surface AA’ of the control volume denoted
by the chain-dotted
line in Fig. 1 is p&U’)
Similarly the force on BB’ is p&U’)
+ pg
additional
term equal to am8 g (
>
6Q2
+ pg $ cos cc. Cm + 6rnj2 2 cosa
and hence the net force on the film due to pressure is pgmsm cos a. In this analysis the effect of surface tension has been neglected. If the curvature of the surface is not negligible an should be included.
(3)
denoted by ml. given by m3 = -hereafter 5g cos a It will of course be realised that for values of m near to dm ml, Z becomes large, a condition which is prohibited by our assumptions, particularly the omission of the surface tension term. Hence Eq. (3) is not valid in the vicinity of m = ml, but this in no way invalidates the use of ml as a correlating parameter. Equation‘(3) can be used to investigate the approach of the film to its equilibrium thickness. The behaviour under these circumstances depends crucially on whether mo or ml is the larger.
715
Shorter Communications
dm dx +vefor
It will be seen that rno > ml if 3Qv 6Q2 --;->7,i.e.if~tana<1. 5g sm a gsm a where Re = Q/V Let us consider first the case where Re large. In this case 2
is -I-ve if m <
m
>
mo
5-veformo>m>ml dx ml
>
i.e. a and/or
mo,
dm
;r;+veforml>m
ma
dmis-veifmO
dm -is dx
fve
dm -+Oasm+mo. ZF
if ml < m
dm
andx+Oasm-+mO. That is to say if a film is produced at a thickness less than the thickness changes as the fluid moves downstream and tends asymptotically to mo, the Nusselt film thickness. For the case of a = 90”, i.e. a vertical surface, ml = co and Eq. (3) reduces to that found by WILKESand NEDDERMAN[3]. If m > mr Eq. (3) predicts that m will increase without ml
limit.
dm
This is obvious for the case of m large where ‘;i;;
becomes tan a, that is to say, the liquid surface becomes horizontal and the velocity tends to zero. The case of ml < mo, i.e. a and Re small, is more relevant to the subject matter of this paper. Inspection of Eq. (3) gives
The principal point of interest here is that whilst m = mo is an equilibrium film thickness, it represents an unstable equilibrium, since any change in the tihn thickness is amplified. Films of thickness less than mo tend to ml rather than the value predicted by the Nusselt equation. The approach to ml gets progressively more rapid and ml is reached after a finite length of travel. This implies that such flows can only be produced on a finite plane. This behaviour is entirely analogous to the well known phenomenon of choking of turbulent flows in open channels and the conditions m > ml and m < ml correspond to the tranquil and rapid flow regimes respectively. It will be noted that the critical condi5 cos a tions occur when 2 = * This is a criterion of the 6 gm3 form of a critical value of a Froude Number. Again for m > mo the film becomes thicker with distance downstream and in the limit becomes horizontal as m tends to cc.
+
030t 025-
x
I
005 -
0
IO
eb
Results
of
ond
Gartside
30
40
Distance
Q-0437
Goodridge
I
/
50 from
cm
2/set
a=O”
end
60 of plote,
FIG. 2 716
I
I
70
I30 cm
I
90
100
I
110
)
Shorter Communications Since for the case of mo > ml the rate of departure from the Nusselt film thickness is controlled, the film will have the Nusselt film thickness at some point only if the film thickness downstream of that point is held at some suitable value by a weir or similar device at the end of the plane. In the case of ml > ma the rate of approach to the Nusselt film thickness is controlled and hence this will be obtained at some point provided the film at the distributor is of suitable thickness. Thus we see that for high Reynolds numbers or high angles of inclination the film will tend to the Nusselt film thickness irrespective of the conditions downstream but for low Reynolds numbers and low angles of inclination the Nusselt film thickness only occurs if the exit device controls the film thickness at the downstream end at a suitable value. The conditions in this case are independent (within limits) of the upstream conditions. In GARTSIDB'S [4] results the film thickness at the end of the plane seems to be controlled by surface tension effects as the liquid falls round the sharp end. As the liquid approaches the end of the plane the thickness departs from the Nusselt value and tends to this thickness. The rate of departure from the Nusselt film thickness is compared with the predictions of Eq. (3) integrated graphically for the case of Q = O-892 cma/sec, a = 20.8 min and integrated algebraically to 6Q2m gm4 ---zz 3Qvx for Q = 0.437 cma/sec, a = 0. 4 5
I) x
0 20
Results
of
and
NOTATION
‘Thelreticol
0=0,892 a=20,8
curve
cm %ec min
t
I
IO
acceleration due to gravity film thickness Nusselt film thickness critical film thickness atmospheric pressure velocity along the plane distance along the plane distance perpendicular to the plane momentum flux d Q flow rate per unit width of plane Re Reynolds Number = Q/V a angle of inclination of the plane to horizontal p absolute viscosity kinematic viscosity p” density D surface tension 9
x
I O-05 L
A theory has been presented to give the variation of film thickness with distance as a viscous liquid flows along a nearly horizontal surface. It is shown that in this case, in contrast with the case of flow down steep surfaces, the film thickness is controlled primarily by the conditions at the downstream end of the plate. The predictions are compared with the measurements of GOODRIDGEand GART~IDBand satisfactory agreement is found. R. M. NBDDBRMAN Dept. of Chemical Engineering Pembroke Street Cambridge
Goodridge
Flow
0
CONCLUSION
Gartside
x
-
The results are plotted in Fig. 2 and 3 and it can be seen that the agreement is satisfactory in view of the difficulty involved in measuring the small depth and angles. An additional error is also involved in the case of Q = 0.437 cma/sec, a = 0 as these results were obtained in an apparatus where the edge effects were appreciable and hence the value of Q is somewhat in doubt. The agreement in Fig. 3 suggests that the change in film thickness is due to the acceleration effects described in this paper and not due to a hydraulic jump as suggested by GART~IDE[4]. For a hydraulic jump to occur the upstream depth must be less than ml. In this case ml = 0.098 cm compared with the least depth measured of O-156 cm.
1
I
I
I
I
20
30
40
50
60
Distance
from
FIG. 3
end of plate,
cm
m m0 ml PO u X
&TBRBNCBS
111 GOODRIDGEF. and GARTSIDEG., Trans. Znstn. Chem. Engrs., 1965 43 T62. NU~~ELTW., Z. Per. Dt. Zng., 1916 60 569. WILKESJ. 0. and NBDDERMAN R. M., Chem. Engng Sci., 1962 17 177. 141 GARTSIDEG., Ph.D., Thesis, Durham University, 1962.
:;
717
Printed in Great Britain.
The flow of thin films of a viscous liquid down nearly horizontal surfaces (Received 15 March 1966) IN A RECENT paper G~~DRIDGE and GARTSIDE[l] comment that in view of the boundary conditions involved, the Nusselt equation [2] relating flow rate to film thickness in a viscous fluid flowing down a plane inclined at c( to the horizontal m,s =
Other forces acting on the control volume parallel to the plane are the weight of the fluid pgmsx sin a and the shear force on the plane due to the viscous drag
au
0
708x where 70 = p -
ay y=~.
_3eY gsm a
(0
In order to evaluate 70 we must assume a velocity profile in the liquid. If the Reynolds number is low there is considerable evidence both in the paper by G~~DRIDGE and GARTSDE [l] and elsewhere 131 that the velocity profile is
cannot be expected to hold in the limit as er --f 0. This assertion which is backed up with experimental evidence deserves to be considered in more detail. Let us consider a film of a viscous fluid flowing down an inclined plane of slope a. If the film has not reached a steady state the film thickness m will vary with distance along the plane, x, measured positive in the direction of flow. We will dm . confine our attention to circumstances where Z 1s small but
parabolic
and hence u = g
(2my - y2). In which case
70=$. . The sum of these forces will cause the fluid to accelerate and hence by a momentum balance we can say
not necessarily small compared with a.
6M=pgmsina&---1;--
3pQSx
pgm cos a8m -
where M is the momentum flux, i.e. M= ‘1
Hence by substitution term in (Twe obtain dm
s
6Q2P ompu2 dy = 5m.
of M into Eq. (2) and neglecting the 3Qv-amasina
;i;;=&j2
-
T-gm3cosa
For any given system and flow rate there are clearly two values of m of particular significance, that given by 3Qv m3 = T, i.e. the Nusselt film thickness mo, and that g sin a
FIG. 1 The pressure approximation, the atmospheric pressure on the
at the point (x, y) see Fig. 1, is to a first given by (m - y) pg cos a + PO, where PO is pressure. Hence the force exerted by the surface AA’ of the control volume denoted
by the chain-dotted
line in Fig. 1 is p&U’)
Similarly the force on BB’ is p&U’)
+ pg
additional
term equal to am8 g (
>
6Q2
+ pg $ cos cc. Cm + 6rnj2 2 cosa
and hence the net force on the film due to pressure is pgmsm cos a. In this analysis the effect of surface tension has been neglected. If the curvature of the surface is not negligible an should be included.
(3)
denoted by ml. given by m3 = -hereafter 5g cos a It will of course be realised that for values of m near to dm ml, Z becomes large, a condition which is prohibited by our assumptions, particularly the omission of the surface tension term. Hence Eq. (3) is not valid in the vicinity of m = ml, but this in no way invalidates the use of ml as a correlating parameter. Equation‘(3) can be used to investigate the approach of the film to its equilibrium thickness. The behaviour under these circumstances depends crucially on whether mo or ml is the larger.
715
Shorter Communications
dm dx +vefor
It will be seen that rno > ml if 3Qv 6Q2 --;->7,i.e.if~tana<1. 5g sm a gsm a where Re = Q/V Let us consider first the case where Re large. In this case 2
is -I-ve if m <
m
>
mo
5-veformo>m>ml dx ml
>
i.e. a and/or
mo,
dm
;r;+veforml>m
ma
dmis-veifmO
dm -is dx
fve
dm -+Oasm+mo. ZF
if ml < m
dm
andx+Oasm-+mO. That is to say if a film is produced at a thickness less than the thickness changes as the fluid moves downstream and tends asymptotically to mo, the Nusselt film thickness. For the case of a = 90”, i.e. a vertical surface, ml = co and Eq. (3) reduces to that found by WILKESand NEDDERMAN[3]. If m > mr Eq. (3) predicts that m will increase without ml
limit.
dm
This is obvious for the case of m large where ‘;i;;
becomes tan a, that is to say, the liquid surface becomes horizontal and the velocity tends to zero. The case of ml < mo, i.e. a and Re small, is more relevant to the subject matter of this paper. Inspection of Eq. (3) gives
The principal point of interest here is that whilst m = mo is an equilibrium film thickness, it represents an unstable equilibrium, since any change in the tihn thickness is amplified. Films of thickness less than mo tend to ml rather than the value predicted by the Nusselt equation. The approach to ml gets progressively more rapid and ml is reached after a finite length of travel. This implies that such flows can only be produced on a finite plane. This behaviour is entirely analogous to the well known phenomenon of choking of turbulent flows in open channels and the conditions m > ml and m < ml correspond to the tranquil and rapid flow regimes respectively. It will be noted that the critical condi5 cos a tions occur when 2 = * This is a criterion of the 6 gm3 form of a critical value of a Froude Number. Again for m > mo the film becomes thicker with distance downstream and in the limit becomes horizontal as m tends to cc.
+
030t 025-
x
I
005 -
0
IO
eb
Results
of
ond
Gartside
30
40
Distance
Q-0437
Goodridge
I
/
50 from
cm
2/set
a=O”
end
60 of plote,
FIG. 2 716
I
I
70
I30 cm
I
90
100
I
110
)
Shorter Communications Since for the case of mo > ml the rate of departure from the Nusselt film thickness is controlled, the film will have the Nusselt film thickness at some point only if the film thickness downstream of that point is held at some suitable value by a weir or similar device at the end of the plane. In the case of ml > ma the rate of approach to the Nusselt film thickness is controlled and hence this will be obtained at some point provided the film at the distributor is of suitable thickness. Thus we see that for high Reynolds numbers or high angles of inclination the film will tend to the Nusselt film thickness irrespective of the conditions downstream but for low Reynolds numbers and low angles of inclination the Nusselt film thickness only occurs if the exit device controls the film thickness at the downstream end at a suitable value. The conditions in this case are independent (within limits) of the upstream conditions. In GARTSIDB'S [4] results the film thickness at the end of the plane seems to be controlled by surface tension effects as the liquid falls round the sharp end. As the liquid approaches the end of the plane the thickness departs from the Nusselt value and tends to this thickness. The rate of departure from the Nusselt film thickness is compared with the predictions of Eq. (3) integrated graphically for the case of Q = O-892 cma/sec, a = 20.8 min and integrated algebraically to 6Q2m gm4 ---zz 3Qvx for Q = 0.437 cma/sec, a = 0. 4 5
I) x
0 20
Results
of
and
NOTATION
‘Thelreticol
0=0,892 a=20,8
curve
cm %ec min
t
I
IO
acceleration due to gravity film thickness Nusselt film thickness critical film thickness atmospheric pressure velocity along the plane distance along the plane distance perpendicular to the plane momentum flux d Q flow rate per unit width of plane Re Reynolds Number = Q/V a angle of inclination of the plane to horizontal p absolute viscosity kinematic viscosity p” density D surface tension 9
x
I O-05 L
A theory has been presented to give the variation of film thickness with distance as a viscous liquid flows along a nearly horizontal surface. It is shown that in this case, in contrast with the case of flow down steep surfaces, the film thickness is controlled primarily by the conditions at the downstream end of the plate. The predictions are compared with the measurements of GOODRIDGEand GART~IDBand satisfactory agreement is found. R. M. NBDDBRMAN Dept. of Chemical Engineering Pembroke Street Cambridge
Goodridge
Flow
0
CONCLUSION
Gartside
x
-
The results are plotted in Fig. 2 and 3 and it can be seen that the agreement is satisfactory in view of the difficulty involved in measuring the small depth and angles. An additional error is also involved in the case of Q = 0.437 cma/sec, a = 0 as these results were obtained in an apparatus where the edge effects were appreciable and hence the value of Q is somewhat in doubt. The agreement in Fig. 3 suggests that the change in film thickness is due to the acceleration effects described in this paper and not due to a hydraulic jump as suggested by GART~IDE[4]. For a hydraulic jump to occur the upstream depth must be less than ml. In this case ml = 0.098 cm compared with the least depth measured of O-156 cm.
1
I
I
I
I
20
30
40
50
60
Distance
from
FIG. 3
end of plate,
cm
m m0 ml PO u X
&TBRBNCBS
111 GOODRIDGEF. and GARTSIDEG., Trans. Znstn. Chem. Engrs., 1965 43 T62. NU~~ELTW., Z. Per. Dt. Zng., 1916 60 569. WILKESJ. 0. and NBDDERMAN R. M., Chem. Engng Sci., 1962 17 177. 141 GARTSIDEG., Ph.D., Thesis, Durham University, 1962.
:;
717