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Velocity distribution in a liquid film flowing over a rotating conical surface
Chemical Engineering Science, 1969, Vol. 24, pp. 1647-1654.
Pergamon Press.
Printed in Great Britain.
Velocity distributions in a liquid film flowing over a rotating conical surface S. BRUINt Food Technology Department, Agriculturaí University, Wageningen, The Netherlands (Fint received4 Februaty 1969; in revisedform 14 July 1969) Abstract- An analysis is given of radiai, tangential and meridional velocity profiles in a liquid film flowing over a rotating conical disc. In the Navier-Stokes equations, the pressure gradient in radial direction is neglected. The velocity profiles were obtained using a “complex function method”. The equations ensuing from this approach are considerably simpler than the more exact equations obtained independently by Nikolaev [SI, and yet of the same accuracy. Numerical results for velocity distributions are given.
2. DERIVATION OF THE VELOCITY PROFILES
1. INTRODUCTION
VELOCITY profiles in liquid films flowing over rotating conical surfaces are of considerable interest in industry. Efficiency of equipment like centrifugal disc atomisers and centrifugal film evaporators [l, 21 is greatly influenced by the nature of the velocity distributions. In the past few years a number of studies was devoted to this subject[3,6]. The first analysis, valid for cones rotating at very high angular velocities, was due to Hinze and Milborn[3] who derived equations for the radial velocity profile and thickness of the liquid film in flow over a rotating conical surface. In the present study also meridional and tangential velocity profiles as wel1 as the pressure distribution in the film are calculated. The second type of velocity profiles, derived in this study, is suitable for high as wel1 as intermediate angular velocities of the tone. Solutions for radial, tangential and meridional velocity profiles in such flow situations were derived independently by Nikolaev [5]. However the equations derived in the present study have a more simple form without substantial loss of accuracy. To distinguish which of the two types of velocity profiles wil1 be representative in a specific situation, a dimensionless criterion (CL)is introduced.
2.1 Description of the model In Fig. 1 the Aow situation in the tone of, e.g. a centrifugal film evaporator is sketched. The tone rotates at angular velocity of w radians per second. The liquid is brought up to the tone surface in a horizontal plane at a radius R,,. Spherical coordinates are the best suited to the model. However some simplifications can be made in the coordinate system due to the fact that the film wil1 be very thin. Moreover the flow wil1 show axial symmetry with respect to the coordinate 4. It is assumed that the following condition holds: 60/Ro 6 1. (1) This condition says that the film thickness Cao)is much smaller than the radius RW The condition implies that the velocity in the 8-direction is much smaller than both the velocity in radial direction and the velocity in the +direction. Moreover it is assumed that the flow within the layer is completely viscous. Introducing a coordinate system which rotates at an angular velocity W, the velocity in tangential direction with respect to the new coordinate system becomes:
tPresent address: Laboratory for Physical Technology, TechnicaJ University, Eindhoven, The Netherlands.
1647
cESVd24No.
Il-A
S. BRUIN
ui=-v@+orsinO.
(2)
tot 8
-
(4)” --w2rsin8cos8+20vá,cos8 r
With these assumptions the Navier-Stokes equations and the continuity equation can be simplified to:
=gsinO-~$+~~+2~] $+2q++$=f).
v 8% ; v9avr (4)” - o2 sin2 8 + 20~; sin 8 + ar ra0 r = -g
cos cd$+;$
-~,?$-~~-Wi+2wv,sin~ r
(3)
=__2
Y
r2
(5)
a2d ae2 (4)
(6)
Further simplifications can be made in this system of equations by recognising that the pressure term in (3) and the viscous term in (3) are one order smaller in magnitude than the corresponding terms. A coordinate s instead of 8 is introduced, related by: - ds = rd0.
(7)
The trigonometrie functions of 8 appearing in (3-5) are replaced by the same function of the angle p (the tone has an angle 2B). Introducing the following dimensionless parameters: 02r sin /3
Fr=
g
(8)
’
Qo+=
Qo
(9)
25~sin~pr~(01,)~‘~’
Fig. l(a). (y+
=
0 1’2&
(11)
;
uc’/’
wr sinp’ v= ())
0
cT=s
-
u
P=
_(J&
w=
&&,
(12)
112
(13)
Pl=f, 0
P-Po
(14)
p(wv)+orsin@
the equations of motion and the continuity equation become: u2+r)uu,-Figs. 1 (a, b). Sketch of rotating conical disc.
1648
V
sin /3
u,-IP-1+2w =-- totFr p + UIT,,
(15)
Velocity distributions in a liquid film flowing over a rotating conical smface
V W,+2U= sin /3
2uw+77uw,--
W,,,
-WZ--1+2W=~+tanPP~, qu,+3u--
(16)
(17)
1 v,=o. sin p
If the velocities are made dimensionless the average radial velocity ( v,],.,~J :
using
fore the second derivatives arises only through the definition of the u-coordinate in Eq. ( 13). These Eqs. (15-17) or (15a-17a) are thought to be sufficiently accurate to describe the flow in the tone. Equations (15-18) are not the same as Nikolaev[5] gives, because here the pressure term is dropped from ( 16) as a consequente of the assumption (1). Boundary conditions are: (a) U=V=W=Ol=s7j~=0 (b) U,= W,=O ICr)
1~77
(19) .
0
(12a)
Eqs. ( 15- 18) become: uo2+rjwo
(3 1,-&j r)
+2Rr)Wo=-(n2772-
- wo2-f12q2+2LR~wo
(Uo),- wo2-n2q2 tot B Fr +fi~(U~)~~
(15a)
tan j3
= c12?-j2F
+ R2q2 tan /3 . P,
( 17a)
(Vo), = 0; (~/o)“~/(R~sinB).
(18a)
Equations (15a-17a) illustrate the use of the criterion 0 in simplifications of the equations. When R > 1 the terms with f12 and fi tend to dominate, which means that the terms non linear in the velocity can be neglected. When R %=1 (so that f12 9 R, R 3 20 say) the terms at the L.H.S. of the equations linear in0 can be neglected in comparison with the terms quadratic inR. At the R.H.S. al1 terms must be retained, the R be-
1
The first boundary condition expresses the familiar no-slip condition, the second implies that no momentum flux takes place across the free surface of the liquid film. The third condition expresses that the pressure in the liquid at its free surface equals the pressure in the gas phase. The last expression assets constancy of the volumetric overall liquid flow rate through the film. 2.2 Solution of the momentum balance equations (i) The case R 3 1. If the criterion Q is so large that its square is much higher than itself (a2 % Cn),fi 3 20 say, we can drop at the L.H.S. of (15a) al1 terms except the one with f12. In ( 16a) at the L.H.S. only the last term is retained and in Eq. (17a) only the second term at the L.H.S. remains. This set is readily solved, the solution, (2Oa), was already given by Hinze and Milborn [3]. In our notation U, V, Wand P are given by: (a) U= (l-~)~+o-~02) (b) V=
(
3sinp-2%
(c)
w=-~(l-y)(8++~,4+2(8+)3~)
(d)
P = (cotP++-)(G+-a).
1649
>(
~S+a2-~03
1
(20)
S. BRUIN
From (20) one can derive a relation for the film thickness, using Eq. (19d): /
in.+
\1/3
(21)
From this relation one can conclude that 6 Qo1j3, 6 - u+lj3 and 6 - r-2’3 when the gravity forces can be neglected. This wil1 be allowed when the angular velocity is high, making the Froude number also large, and the numerator in (2 1) being very nearly unity. Returning to the condition 0 s 1, one can conclude that the Eqs. (20a-d) can be expected to be accurate enough when Q0 is small, R. is large and the tone has a very high angular velocity. This is expressed by the following dimensionless group (fI 3 20): sin4/3 In practica1 cases, where R. and the angle at the vertex are fixed by the construction of the conical surface, dynamically similar flows are obtained by leaving ov/Q,* unaltered. A large angle at the vertex wil1 be very favourable for satisfaction of condition (22). From (20d) one sees that the pressure distribution is linear over the film. In the next section it wil1 be shown that, if the term containing W is retained, this is no longer the case. In Fig. 2 velocity distributions are sketched. (ii) The case CR> 1. In this case the term linear in fi must be retained in Eqs. (15a-17a). The following set of equations remains:
1 -2u= -1+2w=-
wmU tan /3 Fr +tanP&.
eous solution is possible as wil1 be shown. Once the solutions for U and W are obtained, the pressure distribution follows from (24) and the 13component of the velocity can be calculated by use of the continuity equation. Defining: F(7)) = 1-9; Eqs. (23a) and (23b) become:
Ca)u, -2w=-F(r)), (b) w,,+
*=
(23)
The continuity equation remains unchanged. Now Eqs. (23a) and (23b) are coupled, simultan-
U-iW.
of the
(26)
(25a) and (25b) are now combined to *c,-2i*=-F(q).
1
(24)
(25)
2~ = 0.
This set can be solved by introduction complex function T:
Equations give:
(a) -1+2W=-z+U_ (b)
Fig. 2. Radial, tangential and meridional velocities from Eq. (20).
(27)
The solution of (27) which satisfies the boundary conditions is: q = -#S(q)
1650
[tanh (S+d(2i)) + l-cash
sinh (ad(2i)) (ad(2i))].
(28)
Velocity distributions in a liquid fìlm flowiug over a rotating couical surface
The velocities U and V now follow from (28) by taking its real and imaginary part respectively: U = .%e{W},
W = -.Fm{*}.
(2%
Performing the separation on (28) gives: U=bF(v)
[-sinhosino+&],
W=-+‘(n)
[l+&-coshocosc];
(30) (31)
where:
5r=
sin 28+ sinh v cos
(304 ---+s+=sC~,
and: ti
e
Fig. 3. The integral 1,(6+); experimental points of Nikolaev[S].
sinh 2W sinh u cos o - sin 2S+ cash u sin u cosh26++cosh26+
Ola) The velocity in the Mirection can be calculated from the continuity equation. Because of the differentiation to the q-coordinate however, first the film thickness as a function of radius must be determined. Again Eq. ( 19d) can be used for this purpose. Evaluation of the integral gives: 4Qo+ = F(r1M,(~+),
(32)
where: Zo
=
(32a) is a support to our view that these velocity profiles are of sufficient accuracy. In the analysis of Nikolaev, complex equations for the velocity distribution were derived, also yielding (32a) as a close approximation. Notwithstanding their complexity, apparently no substantial additional accuracy is obtained with those equations. Equating the L.H.S. of (32) and the first term of (32a), gives a formula which reduces to the film thickness predicted by (21) case (i) in the limit 6+ + 0.
sinh 26+ - sin 26+ + cash 6+ sin 6+ - sinh 6+ cos 6+ cash 26+ + cos 26+ + sinh S+ sin 6+( sin 26+ + sinh 26+) - cash 6+ cos S+ (sinh 26+ - sin 26+) cash 26++cos 26+ sinh 26+ - sin 26+ (32a)
= cash 26+ + cos 2S+’
Relation (32) provides, albeit implicite, a unique relation between the film thickness and the radius. In Fig. 3 the function Z,,(6+) is sketched, from this graph the film thickness 6+ can be determined as a function of Q0+ by equating R.H.S. and L.H.S. of (32) graphically. The curve is confirmed by experimental data as Nikolaev reported. The fact that the velocity distribution formulae (30) and (3 1) give the result
Now that the relation between radius and film thickness has been established, a relation for the velocity in the 8-direction wil1 be derived. Integration of the continuity equation gives: 6 V=
1651
(33)
S. BRUIN
The partial differential of (tT) with respect to ôf follows straightforward
from Eq. (3Oa):
2 (sin2S + sinh u cos W+ sinh 2S+ cash u cos u) (sin 2S+ - sinh 2S+) &?r (cash 2S++cos 2S+)2 HaS+ o,q = +
cos 2S+ sinh u cos C+ cash 2S+ cash v sin u cash 2S+ + cos 2S+
The derivative of the film thickness with respect to the radius follows from (32). Implicite differentiation gives for the derivative: z
(35)
= 4-#J/(d9.
The derivative of Z,,(S+) follows again straightforward from (32a): dZ,(S+) -= dS+
cash 2S+- cos 2S+ cash 2S+ + cos 2S+ -
1.
sinh 2S+ - sin 2S+ 2 (35a) cash 2S++ cos 2W
In Figs. 4 and 5 the radial and tangential velocities U and W are given for four different values of S+. The profiles in radial direction show a maximum when the film is not too thin, which dis-
Fig. 4(b). Figs. 4(a, b). Radial velocity profile predicted by Eq. (30) for some values of dimensionless film thickness S+.
05
Fig. 5. Tangential velocity profile predicted by Eq. (31) for some values of dimensionless film thickness 6+.
Fig. 4(a).
1652
Velocity distributions in a liquid film flowing over a rotating conical smface
appears when the film becomes thinner (Fig. 4). This maximum is also found in the radial velocity profiles near a disk rotating in an infìnite domain of liquid Schlichting[7, p. 831, and for condensation on a rotating disk Sparrow and Gregg [8].
r Ra s
3. CONCLUSIONS
Summarising the results, one can say that: (i) The assumption concerning the pressure term in the radial component of the NavierStokes equation is proved to be sound; (ii) An alternative solution for the velocity profiles is obtained by introduction of a complex velocity. These velocity profiles are given by more simple formulae than Nikolaev gives, while the loss of accuracy is negligible. Acknowledgments-
Many stimulating discussions with Mr. Van Gent and Mr. Verhagen (both from the Netherlands Ship Model Basin, Wageningen, The Netherlands) are gratefully acknowledged. NOTATION
F(q)
function defined in (25) Froude number gravitational acceleration, Lt+ d-1 ZO@+; film thickness function defined in (32a) P pressure, ML-V2 Po pressure at free surface of liquid film, ML-‘t-2 P dimensionless pressure Fr g
U, U, z), ve u9 V, V, W, Wo
volumetric flow We L3t-’ dimensionless volumetric flow rate radial coordinate, L distance from apex at which conequid enters tone, L coordinate perpendicular to tone surface, L dimen sionle s s radial velocitie s radial velocity, Lt-’ meridional velocity, Lt-’ tangential velocity, Lt-’ dimen sionle s s meridional velocitie s dimensionale s s tangential velocitie s
Greek symbols half the angle at apex of tone, radians film thickness (perpendicular to conical surface) dimensionless film thickness dimensionles s radial coordinate tone angle in spherical coordinate system kinematic viscosity, L2t-’ functions defined in (30a) and (3 la) liquid density, ML-3 dimensionle s s s-coordinate polar angle in spherical coordinates, rad. complex velocity function angular velocity of tone, te1 dimensionles s angular velocity
REFERENCES [l] MAUTNER M., Int. Fed. Fruitjuice Prod., Rep. Sc. Techn. Commission, Wädenswill, Switzerland 1961. [2] BROMLEY L. A., Ind. Engng Chem. 1958 50233. [3] HINZE J. 0. and MILBORN H.,J. uppl. Mech. 1950 145. [4] ZINNATULIN N. G., Rep. S.M. Kirou Chemico-Technological Insr. Kazan, No. XXXZI, Mech. Ser. 1961. 1.51NIKOLAEV V. S., Int. chem. Engng 1967 7 595. 161 VACHAGIN K. D., Inl. chem. Engng 1966 6 228. 171 SCHLICHTING H., Grenzschichtfheorie. Braun 1958. [8] SPARROW E. M. and GREGG J. L., J. Heat Transfer 1959 113.
Résumé - On donne une analyse des profils de vélocité radiale, tangentielles
et médiane d’une pellicule liquide s’écoulant sur un disque R rotation conique. Dans les équations de Navier-Stokes, le gradient de pression dans le sens radial est négligé. Les courbes de vélocité sont obtenues à partir d’une “méthode de fonction complexe”. Les équations qui s’en suivent sont considérablement plus simples que les équations les plus exactes obtenues indépendamment par Nikolaev C.S.[5], et cependant de la même précision. Les résultats numériques des distributions de vélocité sont donnés.
1653
S. BRUIN Znsammenfassung-Es wird eine Analyse von radialen, tangentiallen und meridionalen Geschwindigkeitsprofilen in einem Fliissigfilm, der iiber eine rotierende, konische Scheibe fliesst, beschrieben. In den Navier-Stokes Gleichungen wird der Druckgradient in der radialen Richtung vernachIassigt. Die Geschwindigkeitsprofìle sind unter Venvendung einer “Komplexfunktionsmethode” erhalten worden. Die auf diesem Wege erhaltenen Gleichungen sind wesentlich einfacher als die unabhängig davon durch Nikolaev C.S.[5] erhaltenen exakteren Gleichungen, sind jedoch von gleicher Genauigkeit. Es werden numerische Resultate fur die Geschwindigkeitsverteilungen angegeben.
1654
Pergamon Press.
Printed in Great Britain.
Velocity distributions in a liquid film flowing over a rotating conical surface S. BRUINt Food Technology Department, Agriculturaí University, Wageningen, The Netherlands (Fint received4 Februaty 1969; in revisedform 14 July 1969) Abstract- An analysis is given of radiai, tangential and meridional velocity profiles in a liquid film flowing over a rotating conical disc. In the Navier-Stokes equations, the pressure gradient in radial direction is neglected. The velocity profiles were obtained using a “complex function method”. The equations ensuing from this approach are considerably simpler than the more exact equations obtained independently by Nikolaev [SI, and yet of the same accuracy. Numerical results for velocity distributions are given.
2. DERIVATION OF THE VELOCITY PROFILES
1. INTRODUCTION
VELOCITY profiles in liquid films flowing over rotating conical surfaces are of considerable interest in industry. Efficiency of equipment like centrifugal disc atomisers and centrifugal film evaporators [l, 21 is greatly influenced by the nature of the velocity distributions. In the past few years a number of studies was devoted to this subject[3,6]. The first analysis, valid for cones rotating at very high angular velocities, was due to Hinze and Milborn[3] who derived equations for the radial velocity profile and thickness of the liquid film in flow over a rotating conical surface. In the present study also meridional and tangential velocity profiles as wel1 as the pressure distribution in the film are calculated. The second type of velocity profiles, derived in this study, is suitable for high as wel1 as intermediate angular velocities of the tone. Solutions for radial, tangential and meridional velocity profiles in such flow situations were derived independently by Nikolaev [5]. However the equations derived in the present study have a more simple form without substantial loss of accuracy. To distinguish which of the two types of velocity profiles wil1 be representative in a specific situation, a dimensionless criterion (CL)is introduced.
2.1 Description of the model In Fig. 1 the Aow situation in the tone of, e.g. a centrifugal film evaporator is sketched. The tone rotates at angular velocity of w radians per second. The liquid is brought up to the tone surface in a horizontal plane at a radius R,,. Spherical coordinates are the best suited to the model. However some simplifications can be made in the coordinate system due to the fact that the film wil1 be very thin. Moreover the flow wil1 show axial symmetry with respect to the coordinate 4. It is assumed that the following condition holds: 60/Ro 6 1. (1) This condition says that the film thickness Cao)is much smaller than the radius RW The condition implies that the velocity in the 8-direction is much smaller than both the velocity in radial direction and the velocity in the +direction. Moreover it is assumed that the flow within the layer is completely viscous. Introducing a coordinate system which rotates at an angular velocity W, the velocity in tangential direction with respect to the new coordinate system becomes:
tPresent address: Laboratory for Physical Technology, TechnicaJ University, Eindhoven, The Netherlands.
1647
cESVd24No.
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S. BRUIN
ui=-v@+orsinO.
(2)
tot 8
-
(4)” --w2rsin8cos8+20vá,cos8 r
With these assumptions the Navier-Stokes equations and the continuity equation can be simplified to:
=gsinO-~$+~~+2~] $+2q++$=f).
v 8% ; v9avr (4)” - o2 sin2 8 + 20~; sin 8 + ar ra0 r = -g
cos cd$+;$
-~,?$-~~-Wi+2wv,sin~ r
(3)
=__2
Y
r2
(5)
a2d ae2 (4)
(6)
Further simplifications can be made in this system of equations by recognising that the pressure term in (3) and the viscous term in (3) are one order smaller in magnitude than the corresponding terms. A coordinate s instead of 8 is introduced, related by: - ds = rd0.
(7)
The trigonometrie functions of 8 appearing in (3-5) are replaced by the same function of the angle p (the tone has an angle 2B). Introducing the following dimensionless parameters: 02r sin /3
Fr=
g
(8)
’
Qo+=
Qo
(9)
25~sin~pr~(01,)~‘~’
Fig. l(a). (y+
=
0 1’2&
(11)
;
uc’/’
wr sinp’ v= ())
0
cT=s
-
u
P=
_(J&
w=
&&,
(12)
112
(13)
Pl=f, 0
P-Po
(14)
p(wv)+orsin@
the equations of motion and the continuity equation become: u2+r)uu,-Figs. 1 (a, b). Sketch of rotating conical disc.
1648
V
sin /3
u,-IP-1+2w =-- totFr p + UIT,,
(15)
Velocity distributions in a liquid film flowing over a rotating conical smface
V W,+2U= sin /3
2uw+77uw,--
W,,,
-WZ--1+2W=~+tanPP~, qu,+3u--
(16)
(17)
1 v,=o. sin p
If the velocities are made dimensionless the average radial velocity ( v,],.,~J :
using
fore the second derivatives arises only through the definition of the u-coordinate in Eq. ( 13). These Eqs. (15-17) or (15a-17a) are thought to be sufficiently accurate to describe the flow in the tone. Equations (15-18) are not the same as Nikolaev[5] gives, because here the pressure term is dropped from ( 16) as a consequente of the assumption (1). Boundary conditions are: (a) U=V=W=Ol=s7j~=0 (b) U,= W,=O ICr)
1~77
(19) .
0
(12a)
Eqs. ( 15- 18) become: uo2+rjwo
(3 1,-&j r)
+2Rr)Wo=-(n2772-
- wo2-f12q2+2LR~wo
(Uo),- wo2-n2q2 tot B Fr +fi~(U~)~~
(15a)
tan j3
= c12?-j2F
+ R2q2 tan /3 . P,
( 17a)
(Vo), = 0; (~/o)“~/(R~sinB).
(18a)
Equations (15a-17a) illustrate the use of the criterion 0 in simplifications of the equations. When R > 1 the terms with f12 and fi tend to dominate, which means that the terms non linear in the velocity can be neglected. When R %=1 (so that f12 9 R, R 3 20 say) the terms at the L.H.S. of the equations linear in0 can be neglected in comparison with the terms quadratic inR. At the R.H.S. al1 terms must be retained, the R be-
1
The first boundary condition expresses the familiar no-slip condition, the second implies that no momentum flux takes place across the free surface of the liquid film. The third condition expresses that the pressure in the liquid at its free surface equals the pressure in the gas phase. The last expression assets constancy of the volumetric overall liquid flow rate through the film. 2.2 Solution of the momentum balance equations (i) The case R 3 1. If the criterion Q is so large that its square is much higher than itself (a2 % Cn),fi 3 20 say, we can drop at the L.H.S. of (15a) al1 terms except the one with f12. In ( 16a) at the L.H.S. only the last term is retained and in Eq. (17a) only the second term at the L.H.S. remains. This set is readily solved, the solution, (2Oa), was already given by Hinze and Milborn [3]. In our notation U, V, Wand P are given by: (a) U= (l-~)~+o-~02) (b) V=
(
3sinp-2%
(c)
w=-~(l-y)(8++~,4+2(8+)3~)
(d)
P = (cotP++-)(G+-a).
1649
>(
~S+a2-~03
1
(20)
S. BRUIN
From (20) one can derive a relation for the film thickness, using Eq. (19d): /
in.+
\1/3
(21)
From this relation one can conclude that 6 Qo1j3, 6 - u+lj3 and 6 - r-2’3 when the gravity forces can be neglected. This wil1 be allowed when the angular velocity is high, making the Froude number also large, and the numerator in (2 1) being very nearly unity. Returning to the condition 0 s 1, one can conclude that the Eqs. (20a-d) can be expected to be accurate enough when Q0 is small, R. is large and the tone has a very high angular velocity. This is expressed by the following dimensionless group (fI 3 20): sin4/3 In practica1 cases, where R. and the angle at the vertex are fixed by the construction of the conical surface, dynamically similar flows are obtained by leaving ov/Q,* unaltered. A large angle at the vertex wil1 be very favourable for satisfaction of condition (22). From (20d) one sees that the pressure distribution is linear over the film. In the next section it wil1 be shown that, if the term containing W is retained, this is no longer the case. In Fig. 2 velocity distributions are sketched. (ii) The case CR> 1. In this case the term linear in fi must be retained in Eqs. (15a-17a). The following set of equations remains:
1 -2u= -1+2w=-
wmU tan /3 Fr +tanP&.
eous solution is possible as wil1 be shown. Once the solutions for U and W are obtained, the pressure distribution follows from (24) and the 13component of the velocity can be calculated by use of the continuity equation. Defining: F(7)) = 1-9; Eqs. (23a) and (23b) become:
Ca)u, -2w=-F(r)), (b) w,,+
*=
(23)
The continuity equation remains unchanged. Now Eqs. (23a) and (23b) are coupled, simultan-
U-iW.
of the
(26)
(25a) and (25b) are now combined to *c,-2i*=-F(q).
1
(24)
(25)
2~ = 0.
This set can be solved by introduction complex function T:
Equations give:
(a) -1+2W=-z+U_ (b)
Fig. 2. Radial, tangential and meridional velocities from Eq. (20).
(27)
The solution of (27) which satisfies the boundary conditions is: q = -#S(q)
1650
[tanh (S+d(2i)) + l-cash
sinh (ad(2i)) (ad(2i))].
(28)
Velocity distributions in a liquid fìlm flowiug over a rotating couical surface
The velocities U and V now follow from (28) by taking its real and imaginary part respectively: U = .%e{W},
W = -.Fm{*}.
(2%
Performing the separation on (28) gives: U=bF(v)
[-sinhosino+&],
W=-+‘(n)
[l+&-coshocosc];
(30) (31)
where:
5r=
sin 28+ sinh v cos
(304 ---+s+=sC~,
and: ti
e
Fig. 3. The integral 1,(6+); experimental points of Nikolaev[S].
sinh 2W sinh u cos o - sin 2S+ cash u sin u cosh26++cosh26+
Ola) The velocity in the Mirection can be calculated from the continuity equation. Because of the differentiation to the q-coordinate however, first the film thickness as a function of radius must be determined. Again Eq. ( 19d) can be used for this purpose. Evaluation of the integral gives: 4Qo+ = F(r1M,(~+),
(32)
where: Zo
=
(32a) is a support to our view that these velocity profiles are of sufficient accuracy. In the analysis of Nikolaev, complex equations for the velocity distribution were derived, also yielding (32a) as a close approximation. Notwithstanding their complexity, apparently no substantial additional accuracy is obtained with those equations. Equating the L.H.S. of (32) and the first term of (32a), gives a formula which reduces to the film thickness predicted by (21) case (i) in the limit 6+ + 0.
sinh 26+ - sin 26+ + cash 6+ sin 6+ - sinh 6+ cos 6+ cash 26+ + cos 26+ + sinh S+ sin 6+( sin 26+ + sinh 26+) - cash 6+ cos S+ (sinh 26+ - sin 26+) cash 26++cos 26+ sinh 26+ - sin 26+ (32a)
= cash 26+ + cos 2S+’
Relation (32) provides, albeit implicite, a unique relation between the film thickness and the radius. In Fig. 3 the function Z,,(6+) is sketched, from this graph the film thickness 6+ can be determined as a function of Q0+ by equating R.H.S. and L.H.S. of (32) graphically. The curve is confirmed by experimental data as Nikolaev reported. The fact that the velocity distribution formulae (30) and (3 1) give the result
Now that the relation between radius and film thickness has been established, a relation for the velocity in the 8-direction wil1 be derived. Integration of the continuity equation gives: 6 V=
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(33)
S. BRUIN
The partial differential of (tT) with respect to ôf follows straightforward
from Eq. (3Oa):
2 (sin2S + sinh u cos W+ sinh 2S+ cash u cos u) (sin 2S+ - sinh 2S+) &?r (cash 2S++cos 2S+)2 HaS+ o,q = +
cos 2S+ sinh u cos C+ cash 2S+ cash v sin u cash 2S+ + cos 2S+
The derivative of the film thickness with respect to the radius follows from (32). Implicite differentiation gives for the derivative: z
(35)
= 4-#J/(d9.
The derivative of Z,,(S+) follows again straightforward from (32a): dZ,(S+) -= dS+
cash 2S+- cos 2S+ cash 2S+ + cos 2S+ -
1.
sinh 2S+ - sin 2S+ 2 (35a) cash 2S++ cos 2W
In Figs. 4 and 5 the radial and tangential velocities U and W are given for four different values of S+. The profiles in radial direction show a maximum when the film is not too thin, which dis-
Fig. 4(b). Figs. 4(a, b). Radial velocity profile predicted by Eq. (30) for some values of dimensionless film thickness S+.
05
Fig. 5. Tangential velocity profile predicted by Eq. (31) for some values of dimensionless film thickness 6+.
Fig. 4(a).
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Velocity distributions in a liquid film flowing over a rotating conical smface
appears when the film becomes thinner (Fig. 4). This maximum is also found in the radial velocity profiles near a disk rotating in an infìnite domain of liquid Schlichting[7, p. 831, and for condensation on a rotating disk Sparrow and Gregg [8].
r Ra s
3. CONCLUSIONS
Summarising the results, one can say that: (i) The assumption concerning the pressure term in the radial component of the NavierStokes equation is proved to be sound; (ii) An alternative solution for the velocity profiles is obtained by introduction of a complex velocity. These velocity profiles are given by more simple formulae than Nikolaev gives, while the loss of accuracy is negligible. Acknowledgments-
Many stimulating discussions with Mr. Van Gent and Mr. Verhagen (both from the Netherlands Ship Model Basin, Wageningen, The Netherlands) are gratefully acknowledged. NOTATION
F(q)
function defined in (25) Froude number gravitational acceleration, Lt+ d-1 ZO@+; film thickness function defined in (32a) P pressure, ML-V2 Po pressure at free surface of liquid film, ML-‘t-2 P dimensionless pressure Fr g
U, U, z), ve u9 V, V, W, Wo
volumetric flow We L3t-’ dimensionless volumetric flow rate radial coordinate, L distance from apex at which conequid enters tone, L coordinate perpendicular to tone surface, L dimen sionle s s radial velocitie s radial velocity, Lt-’ meridional velocity, Lt-’ tangential velocity, Lt-’ dimen sionle s s meridional velocitie s dimensionale s s tangential velocitie s
Greek symbols half the angle at apex of tone, radians film thickness (perpendicular to conical surface) dimensionless film thickness dimensionles s radial coordinate tone angle in spherical coordinate system kinematic viscosity, L2t-’ functions defined in (30a) and (3 la) liquid density, ML-3 dimensionle s s s-coordinate polar angle in spherical coordinates, rad. complex velocity function angular velocity of tone, te1 dimensionles s angular velocity
REFERENCES [l] MAUTNER M., Int. Fed. Fruitjuice Prod., Rep. Sc. Techn. Commission, Wädenswill, Switzerland 1961. [2] BROMLEY L. A., Ind. Engng Chem. 1958 50233. [3] HINZE J. 0. and MILBORN H.,J. uppl. Mech. 1950 145. [4] ZINNATULIN N. G., Rep. S.M. Kirou Chemico-Technological Insr. Kazan, No. XXXZI, Mech. Ser. 1961. 1.51NIKOLAEV V. S., Int. chem. Engng 1967 7 595. 161 VACHAGIN K. D., Inl. chem. Engng 1966 6 228. 171 SCHLICHTING H., Grenzschichtfheorie. Braun 1958. [8] SPARROW E. M. and GREGG J. L., J. Heat Transfer 1959 113.
Résumé - On donne une analyse des profils de vélocité radiale, tangentielles
et médiane d’une pellicule liquide s’écoulant sur un disque R rotation conique. Dans les équations de Navier-Stokes, le gradient de pression dans le sens radial est négligé. Les courbes de vélocité sont obtenues à partir d’une “méthode de fonction complexe”. Les équations qui s’en suivent sont considérablement plus simples que les équations les plus exactes obtenues indépendamment par Nikolaev C.S.[5], et cependant de la même précision. Les résultats numériques des distributions de vélocité sont donnés.
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S. BRUIN Znsammenfassung-Es wird eine Analyse von radialen, tangentiallen und meridionalen Geschwindigkeitsprofilen in einem Fliissigfilm, der iiber eine rotierende, konische Scheibe fliesst, beschrieben. In den Navier-Stokes Gleichungen wird der Druckgradient in der radialen Richtung vernachIassigt. Die Geschwindigkeitsprofìle sind unter Venvendung einer “Komplexfunktionsmethode” erhalten worden. Die auf diesem Wege erhaltenen Gleichungen sind wesentlich einfacher als die unabhängig davon durch Nikolaev C.S.[5] erhaltenen exakteren Gleichungen, sind jedoch von gleicher Genauigkeit. Es werden numerische Resultate fur die Geschwindigkeitsverteilungen angegeben.
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