STABILITY OF FLOW IN PARALLEL CHANNELS-EFFECTS OF BUOYANCY
R. Žitný
Czech Technical University in Prague, Department of Process Engineering, Technická 4,
166 07 Prague 6 , CZECH REPUBLIC
Abstract. The effect of flow asymmetry was observed experimentally in lateral parallel
channels of continuous direct ohmic heater. While the flow is uniform at isothermal
conditions, flowrate in parallel channels differs in case of heating. The effect can be explained
by buoyancy and theoretical analysis predicts existence of two solutions, symmetric and
asymmetric distribution of flowrates, which can be stable within a certain range of
temperatures and flowrates. Experimental verification was based upon a) flow visualisation
(injection of a coloured tracer and monitoring the tracer by Canon MV-100 camera), b)
measurement of temperature profiles (11 thermometers Pt100), and on c) stimulus response
experiments using KCl as a tracer for conductivity methods (2 Pt conductivity probes) and
Tc99 as a radioisotope tracer (collimated scintillation detectors). Results confirm predicted
influence of operational parameters and geometry (diameter of channels) on the stability of
flow.
INTRODUCTION
Parallel flows are typical for many important apparatuses of process industries, e.g.
flows in shell&tube or plate heat exchangers, heaters, reactors. Sometimes flow irregularities,
instabilities or just only non-uniform distribution of flow in parallel channels occur. These
undesirable phenomena can be caused by natural convection if the apparatus operates at nonisothermal conditions, which is typical for heat exchangers or heaters.
Coloured
tracer
KMnO4
3
1
2
1
Figure 1. Continuous ohmic heater (scheme and photograph showing asymmetry of flows). 1lateral channels;2-central channel,3-electrodes.
The effect of flow asymmetry was observed experimentally in lateral parallel channels
of continuous direct ohmic heater, with two planar electrodes (electrical current flows directly
through the heated liquid), see Fig.1. Liquid enters the top of heater and flows downwards
through two rectangular channels where liquid is preheated only by warm electrodes. At the
bottom of heater the two parallel streams join and liquid flows upward in a nearly uniform
electrical field between electrodes (distance 0.036 m, voltage 220 V, 50 Hz). In order to
suppress the electrode fouling, a perforation of electrodes was suggested, assuming that the
d:\doc\!apl\ohmic\nancy01\nannew.doc
1
cold cross-flow could displace overheated substance moving slowly along the electrode
surface.
While the flow in parallel channels is uniform at isothermal conditions, a nonuniform
distribution of flowrate and different temperature profiles are developed in parallel channels
at heating and also the cross flow is changed. These phenomena can be explained by the effect
of buoyancy and can be detected by measuring of RTD.
STABILITY OF PARALLEL FLOWS-CONSTANT WALL TEMPERATURE
Parallel laminar flows in lateral channels of continuous ohmic heater lose symmetry in
case of heating: one stream is delayed and even stopped or reversed if the temperature
increase is too high. A similar situation occurs in a simpler and probably more frequent case
when two vertical parallel streams are separated by wall having a constant temperature Te, see
Fig.2. It is assumed that
 the liquid having temperature T0 enters two identical rectangular channels (cross section H
x B, length L) where is heated from the wall; it does not matter whether all four or just
only one side of channel are hold at temperature Te – the only difference is heat transfer
surface. Heat transfer coefficient  [W.m-2.K-1] is constant.
 Flow in parallel channels is laminar and internal recirculation due to nonuniform
transversal temperature profile is negligible.
 Heat exchanger is perfectly insulated.
pl(0) pr(0)
T0 Te T0
x
L
Ql
H
Insulation
Q-Ql
H
pl(L)= pr(L)
Tl(L)  Tr(L)
Figure 2. Parallel flows heated by wall at constant temperature Te.
Distribution of flow-rate will be expressed in terms of relative flowrate in the left
channel 
Ql  Q ,
Qr  (1   )Q .
(1)
The value =0.5 corresponds to the symmetric distribution, while =0, =1 corresponds to
the completely stopped flow in the left or in the right channel.
The following analysis is based upon the fact that the pressure difference p(L)-p(0)
must be the same in the left and in the right channel at a steady state. We shall consider only
two contributions to the pressure difference: the first represents viscous forces (written for
flowrate Ql =Q [m3.s-1] in the left channel)
d:\doc\!apl\ohmic\nancy01\nannew.doc
2
p f (0)  p f ( L) 
12 Lf
Q
BH 3 l
(2)
where B, H [m] are dimensions of rectangular cross-section (depth and width of channel
respectively), and the coefficient f equals 1 for fully developed laminar flow between infinite
plates or
f 
1
192 H
1
nB
1 5

5 tgh
 B n1,3,5,... n
2H

HQl
24 BL
(3)
for laminar flow in a rectangular channel B x H. The first term in Eq.(3) represents correction
to the finite ratio of B/H (rectangular channel) and the second term is a correction for dynamic
pressure (p  0.5 u2).
While the component of pressure pf(x) accordant with friction forces decreases in the
direction of flow, the hydrostatic pressure pb(x) increases depending upon the axial
temperature profile. For a constant value of the heat transfer coefficient  [W.m-2.K-1] and
constant heat capacity cp [J.kg-1.K-1] the axial temperature profiles are given by
x
x

Tl  Te
 e Wl L ,
T0  Te

Tr  Te
 e Wr L ,
T0  Te
Wr  (1   )W ,
where Wl  W ,
(4)
and
W
Q0cp
BL
.
(5)
Assuming linear temperature dependence of density (coefficient proportionality  [K-1]) and
the exponential temperature profiles Eq.(4), the contribution of hydrostatic pressure can be
expressed as
L
pb (0)  pb ( L)   0 g  [1   (Tl  T0 )]dx   0 gL{1   (T0  Te )[1  Wl (1  e

1
Wl
)]} (6)
0
Summing pressure differences corresponding to the friction forces (2) and buoayant
forces (6) we can express equilibrium of forces in the left (lower index l) and right (index r)
channel by equation
12 Lf
(Ql  Qr )  0 gL (Te  T0 )[Wl (1  e 1/Wl )  Wr (1  e 1/Wr )] .
BH 3
(7)
Eq.(7) can be rearranged to a dimensionless form by introducing Grashoff and Reynolds
numbers
Z  (Te  T0 )
gH 3 B0
Gr

12fQ
96 Re
Gr 
02 g (Te  T0 )(2 H ) 3
Q0
,Re 
2

B
(8)
giving
2  1  ZW [ (1  e

1
W
)  (1   )(1  e
d:\doc\!apl\ohmic\nancy01\nannew.doc

1
( 1  ) W
)]
(9)
3
The balance of forces Eq.(9) is satisfied by the symmetric solution =0.5 for any combination
of parameters Z, and W, however an asymmetric solution can also exist for sufficiently high
values of Z, see Fig.3
1.5
Z
0
0.001
0.005
0.01
Region of only
symmetric flow
1.0
0
10
W
20
30
Figure 3. Z(W) for asymmetric distributions of flow-rate =0,0.001,0.005,0.01, Eq.(9) .
The curve corresponding to =1 or 0 (all liquid flows in the one channel only) is described by
equation following immediately from Eq.(9)
1  ZW (1  e 1/W ) .
(10)
For the values of Z bellow this curve (for example for any Z<1) the symmetric solution must
be stable because it the only solution satisfying the balance Eq.(7). Nevertheless, the
symmetric solution could be possibly stable even for higher value of Z. To analyse this, let us
assume a small disturbance of flowrate Ql+Q, Qr-Q, i.e. slightly increased flowrate in the
left channel and properly decreased flowrate in the right channel. Then the pressure difference
in the left channel will be changed by increment (see Eq.(7)),
02 c p g
W  1 1/Wl
12 Lf
pl  pl (0)  Q[
(1  l
e
)]
3  ( Te  T0 )
BH
B
Wl
and similarly in the right channel
02 c p g
W  1 1/Wr
12 Lf
pr  pr (0)  Q[
(1  r
e
)] .
3  ( Te  T0 )
B
Wr
BH
(11a)
(11b)
In the case that pl>pr the pressure at the inlet to the left channel will be slightly higher than
the pressure in the right channel and this difference induces transversal flow towards the right
channel. This redistribution of flow acts against the disturbance Q, which means that the
flowrates Ql,Qr will be stable. The stability condition pl-pr>0 for Q>0 can be rewritten
using Eqs. (11a) and (11b) into dimensionless form
d:\doc\!apl\ohmic\nancy01\nannew.doc
4
1  ZW (1 
Wr  1 1/Wr Wl  1 1/Wl
e

e
)
2Wr
2Wl
(12)
This general inequality (12) can be applied to the symmetric solution when Wr=Wl=W/2,
giving:
1  ZW (1 
W  2  2 /W
e
).
W
(13)
This inequality, together with Eq.(10) is presented in Fig.4
5
Eq.(13)
Z
4
3
regionasymmetric
of conditional
stability of symmetric
distribution of flow-rates
stable symmetric
2
Eq.(10)
1
0
W
5
10
Figure 4. Regions of unconditionally and conditionally stable symmetric solution
This Figure demonstrates that there exists a rather wide range of Z where a symmetric flow
distribution could, but need not exist
1
1
.
1/W  Z 
W (1  e )
W  (W  2)e 2/W
(14)
For W>5 this region can be characterised by simple inequality
4
2Z
gH 3 B 2 L

 (Te  T0 )
 1.
2W  1 W
6fc p Q 2
(15)
A similar stability analysis can be performed also for the case of asymmetric solution,
e.g. for the case of very small <<0.1. In this case the Eq.(9), balance of forces, reduces to
1
1  ZW[1 
1    (1 )W
e
]
1  2
(16)
and the inequality (12) to
1
W (1   )  1  (1 )W
1  ZW[1 
e
].
2W (1   )
(17)
Combining (16) and (17)
d:\doc\!apl\ohmic\nancy01\nannew.doc
5
(1   ) 2  (1   ) 2 

2
(18)
we arrive to the conclusion that the asymmetric solution cannot be stable for any positive .
This conclusion casts a new light to the physical meaning of the asymmetric solution: It
represents a magnitude of the flow-rate disturbance which is necessary to make the symmetric
solution unstable in the range of Z given by inequalities (14).
STABILITY OF PARALLEL FLOWS-VOLUMETRIC HEAT SOURCE IN CENTRAL
CHANNEL
A similar procedure can be applied for the case of ohmic heater shown in Fig.1, not
considering cross flow through perforated wall. The only principal difference is in
temperature profiles in parallel channels, because the wall temperature Te is no longer a
constant. The axial temperature profiles follow from the following assumptions:
 Temperature depends only on axial coordinate x (or dimensionless coordinate =x/L).
 Heat transfer coefficient  [W.m2.K-1] comprises thermal resistances of liquid layers in
the lateral and central channels and also thermal resistance of wall (of electrode separating
lateral and central channel). This coefficient is the same in the both lateral channels.
 The heater is perfectly insulated.
 Two streams flowing out of the lateral channels are ideally mixed at the bottom of heater
and flow upwards between electrodes, see Fig.5. There is a uniform volumetric source of
heat in the central channel characterised by intensity of heat generation G [W/m3].
T0
pl(0)
x
T0
pr(0)
Ql
Q
Qr
Tl
T
Tr
Insulation
L
H
H
pl(L)= pr(L)
Tl(L)  Tr(L)
Fig. 5 Parallel flows in a heater with volumetric heat source
Temperature profiles in lateral channels (cross-section H x B) are described by
equations
dTl
 B (T  Tl )
dx
dT
Qr c p r  B (T  Tr )
dx
Ql c p
(19)
(20)
while the temperature in the central channel (cross-section Hc x B) is governed by equation
d:\doc\!apl\ohmic\nancy01\nannew.doc
6
 (Ql  Qr ) c p
dT
 B (Tl  Tr  2T )  GHc B
dx
(21)
This system of equations can be solved analytically giving temperature profiles in lateral
channels in the form
TG
2
[(Wr  M (1  Wr )  M 2 )(1  e  / M )  (1  M  Wr )  ]
F
2
T
2
Tr  T0  G [(Wl  M (1  Wl )  M 2 )(1  e  / M )  (1  M  Wl )  ]
F
2
Tl  T0 
where
TG 
GHc

,
M
F  Wl 2  Wr2 ,
WW
l r (Wl  Wr )
,
Wl 2  Wr2

(22)
(23)
x
,
L
(24)
and the meaning of Wl, Wr is the same as previously, see Eq.(5).
These temperature profiles enable to express pressure differences corresponding to
buoyancy (using similar procedure and assumptions as in Eq.(6))
pb (0)  pb ( L)   0 gL{1  
1

TG
1  M  Wr 1
[(Wr  M (1  M  Wr ))(1  M (1  e M )) 
 ]} (25)
F
2
6
Friction forces are the same as previously so that we can immediately write balance of forces
in the left and right lateral channel (we need not take into account the central channel)
TG
12 Lf
1
(Wl  Wr )[(1  M )(1  M (1  e 1/ M ))  ] .
3 ( Ql  Qr )  0 gL
BH
F
2
(26)
This equation can be rewritten into dimensionless form
1  ZG
(1  M )[1  M (1  e 1/ M )] 
 2  (1   ) 2
where M is a function of W and 
1
2,
(27)
M
WW
 (1   )
l r (Wl  Wr )
W 2
,
2
2
Wl  Wr
  (1   ) 2
(28)
ZG 
gTG B 2 H 3 L
12 fc p Q 2
(29)
and the dimensionless number ZG
reminds Z/W in Eq.(15), the only difference is that the temperature TG substitutes the
temperature difference Te-T0. For practical calculations it is possible to use specific power G
[W.m-3], total power P [W] or corresponding adiabatic temperature increase Tmax-T0 in the
definition of ZG
d:\doc\!apl\ohmic\nancy01\nannew.doc
7
gTG B 2 H 3 L gGH c B 2 H 3 L
g0 (Tmax  T0 ) BH 3
gPBH 3
ZG 



.
12 fQ
12 fc p Q 2
12 fc p Q 2
12 fc p Q 2
(30)
The Eq.(27) and the region of W, TG where the asymmetric solution can exist is
represented in graphical form in Fig.6
0
0.02
0.05
0.1
0.2
0.49
ZG
3
2
Region of stable
symmetric flow
1
0
5
10
W
Figure 6. ZG(W) for asymmetric distributions of flow-rate =0,0.02,0.05,0.1,0.2,0.49 Eq.(27)
The solutions of Eq.(27) for =0, and =0.5 give limits of conditional stability of symmetric
solution
1
W
2  ZG 
 .
(31)
W
2
(2  W )[1  (1  e  2 /W )]  1
2
This statement, inequality (31), can be proved rigorously repeating the stability
analysis applied for derivation Eq.(12). Thus we arrive to the general stability constraint
2
1

ZG
M
{
2
(
1

m
)[
1

M
(
1

e
)]  1 
 2  (1   ) 2

1
1

(1  2 ) 2 (1  2 M  2 M 2 )e M  2 M 2
M
[

2
(
1

M
)[
1

M
(
1

e
)]  1]}
 2  (1   ) 2
 (1   )
(32)
which reduces to (31) for =0.5, i.e. for the case of stability of the symmetric flow.
STABILITY OF PARALLEL FLOWS - OTHER CONFIGURATIONS OF FLOW
There are several other possibilities of the parallel flows arrangement – volumetric
heating in parallel channels (with and without heat transfer between parallel streams), co-
d:\doc\!apl\ohmic\nancy01\nannew.doc
8
current flow in the central channel and so on. We confine oneself to the summary of the two
previous cases and the case of volumetric heating in parallel channels in the following table
Case
Constant
temperature
of wall Te
Criterion
Stability limits of symmetric flow
1
Z
W  (W  2)e  2 /W
gH B0
12fQ
3
Z  (Te  T0 )
Te
L
H
Volumetric
heat source
in central
channel
ZG 
G
gGH c B 2 H 3 L
ZG 
12 fc p Q 2
L
1
W
(2  W )[1  (1  e  2 /W )]  1
2
H
Volumetric
heat source
in lateral
L
channels
(insulated wall)
G
gGB 2 H 4 L
ZH 
12 fc p Q 2
G
ZH 
(33)
1
2
(34)
H
NATURAL CONVECTION IN LATERAL CHANNELS
The analysis of parallel flows was based on assumption, that the effect of internal
recirculation in lateral channels can be neglected, however, this restriction is to be quantified.
The following estimates are rather speculative and crude, because they are based upon
assumption of only one dimensional velocity profile, characterising internal recirculation in a
slim vertical channel with one side held at a constant temperature Te and other sides insulated,
see Fig.7:
H
y
0.03
Te
0.02
0.01
T0
0
-0.01 0
0.002
0.004
0.006
-0.02

0.008
-0.03
Fig.7 Transversal temperature and velocity profiles
Assuming linear temperature profile near the heat transfer surface we can solve the Navier
Stokes equation
d:\doc\!apl\ohmic\nancy01\nannew.doc
9
p
 2u
0     2  0 g[1   (T  T0 )]
x
y
(35)
giving the cubic velocity profile, shown in Fig.7. Pressure gradient dp/dx in Eq.(35) is
adjusted so that the net flow-rate is zero, because the velocity profile should characterise only
internal recirculation flow in the lateral channel. Now we want to compare this velocity
profile with the velocity profile corresponding to forced axial flow Q/2. As a measure of
comparison the velocity gradients at wall can be used. For the gradient of recirculating flow
follows from the cubic velocity profile
u
 2 (4 H  9 )
2LB
|y  H   0 g (Te  T0 )
  g (Te  T0 )
2
y
24 H 
3c p Q
(36)
assuming sufficiently thin thermal boundary layer
2 
4LBH
 H 2 .
0c p Q
(37)
The gradient (36) can be related to the velocity gradient of fully developed axial flow
(parabolic velocity profile in laminar flow between infinite plates)
2LB
u
|circulation g (Te  T0 )
3c p Q
2LB 2 H 2
y

 g (Te  T0 )
 R  1
u
3Q
9 c p Q 2
|
y axial
BH 2
(38)
and this is the criterion (R<<1) ensuring validity of simplified analysis of parallel flows. Let
us assume that the stability criterion (15) predicts the upper bound of stability of parallel
flows for the case with constant wall temperature,
(Te  T0 )
gH 3 B 2 L
 1.
6fc p Q 2
(39)
Substituting Eq.(39) to the inequality (38) follows
4
4

 1 ,
3H 3 Nu
Nu 
H
,

(40)
and this inequality can be satisfied only for thermally developing flows when Nu is
sufficiently large.
RTD EXPERIMENTS AND RESULTS
Experiments were performed with different thickness of lateral channels (Hl=Hr= 18,
11 and 7.8 mm). Temperature at lateral channels was recorded by two Pt100 probes and
d:\doc\!apl\ohmic\nancy01\nannew.doc
10
optical fibre probes Nortech TP-21-M02 which prove to be a sensitive indicator of flow
instabilities, however results have not completely evaluated yet. Visualisation using KMnO4
as a colour tracer indicates that the asymmetric flow could exist within a certain range of
operational parameters, see the following table:
Texper.
[K]
20
Te-T0
Eq.(11)
58
65
39
83
22
31
14
42
15.3
14
18
72
17
18
68
18
H
[mm]
8
Q
[ml/s]
76
8
8
18
Flow pattern
Evaluation
agree
10.3
symmetric,
stable
asymmetric
unstable
symmetric,
stable
asymmetric ?
9.2
unstable
acceptable
agree
acceptable
Eq.(11)
unstable
agree
Experiments were carried out with water, with well known thermophysical properties. Based
upon data Kast (1974) we have evaluated temperature dependence of 
(41)
  ( 2.5  0.9T  0.004T 2 ) 105
and temperature dependence of Gr/Re as

 (0.04T 2  2T  8)  106
c p
(42)
see Figs.8,9
Water temperature expansion
0.00100
0.000100
0.00010
 [1/K]
3
-1
-1
/(.c) [s .kg .m ]
0.001000
0.00001
0.000010
1
10
T [C]
100
0
10
20
30
40
50
T [C]
60
Fig.8 Coefficient of volumetric expansion
Fig.9 Temperature dependence Gr
Residence time distribution was identified by using KCl and Tc-99 as a tracer [4]. Results
obtained with these tracers are very similar, however the radioisotope is a better tracer at
heating because solution of Tc99 has no effect upon direct ohmic heating (this is not true for
solution of KCl which increases power and temperature when passing between electrodes).
Selected results of RTD measured in the system with narrow and wide lateral channels
with and without heating and simulation by presented model for different width of
perforation h are shown in Figs. 4a, b, c and d.
d:\doc\!apl\ohmic\nancy01\nannew.doc
11
umin/umax
1
0.5
0
-25
-15
-5
5 ZG-ZGcrit
1
V/Vtheor
0.9
0.8
0.7
-25
-15
-5
5
ZG-ZGcrit
15
CONCLUSION
Buoyancy has undesirable effects in heaters with downwards oriented parallel flows (nonuniform distribution and instability of flow), which can be suppressed by
 Decreasing width of channels (this is the most efficient way)
 Increasing viscosity
 Cross-flow (perforation of walls) has also positive effect improving stability.
This research has been subsidized by the Research Project of Ministry of Education of the
Czech Republic J04/98:21220008
REFERENCES
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[1] Žitný,R.,Šesták, J., Dostál M., Zajíček M., Continuous direct ohmic heating of liquids,
International Conference CHISA’98 , Prague, 23-28 August (1998)
[2] Žitný,R., Thýn, J., Verification of CFD predictions by Tracer Experiments,
International Conference CHISA’2000, Prague, 28 - 31 August (2000)
[3] Thýn, J., Žitný, R., Klusoň, J., Čechák,T.,Analysis and Diagnostics of Industrial
Processes by Radiotracers and Radioisotope Sealed Sources, Ed. ČVUT, Prague (1999)
[4] Žitný,R., Thýn, J.,Parallel flow asymetries in continuous heaters, CVUT Workshop
2001, Praha,
[5] Joye D.D., Wojnovich M.J.: Aiding and opposing mixed convection heat transfer in a
vertical tube: loss of boundary condition at different Grashof numbers, Int.J.Heat and
Fluid Flow, Vol.17, No.5, 1996, 468-473
[6] Kast,W., Krischer O., Reinicke H., Wintermantel K.: Konvektive Wärme und
Stoffűbertragung, Springer Berlin 1974.
14-16 February (2001)
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