Journal of Materials Processing Technology 120 (2002) 215±225
Power law ¯uids and Bingham plastics ¯ow models
for ceramic tape casting
Sunil C. Joshi*, Y.C. Lam, F.Y.C. Boey, A.I.Y. Tok
School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore
Received 1 June 2001
Abstract
A generalized pressure ¯ow is used as a basis for developing a ¯ow model for ceramic tape casting with different types of ¯uids such as
Newtonian, power law and Bingham plastics. The slurry ¯ow is modeled as part of a pressure ¯ow through an imaginary parallel channel.
Analytical equations for the ¯ow ®eld are presented. Equations for obtaining velocity pro®les and ¯ow rates are included. These can be used
to estimate the thickness of the ceramic tape to be cast. The formulations were validated by means of published data, the results of which
are included in the paper. Finally, the effect of various process parameters on the size of the imaginary ¯ow channel is studied.
# 2002 Elsevier Science B.V. All rights reserved.
Keywords: Ceramic tape casting; Imaginary ¯ow channel; Perovskites; BaTiO3; Bingham plastics
1. Introduction
With increasing usage of multi-layer packages and
capacitors, the ceramic tape-casting process has gained
signi®cance over the conventional ceramic processing techniques such as dry pressing and slip casting, for forming
thin, ¯at sheets of ceramics. With this process, ¯at ®lms of
thickness ranging from a few microns to a few millimeters
can be laid precisely. These ¯at packages are then bound and
sintered together in the required layered form.
A schematic diagram of the tape-casting process is shown
in Fig. 1. In this process, a container with a rectangular
opening is ®rst ®lled with an especially formed ceramic
slurry. Once the casting head, on which the container is
mounted, is in motion, slurry ¯ows through the opening and
a thin tape is formed on the ¯at carrier provided underneath.
Uniformity in tape thickness is maintained by maintaining
constant the viscosity of the slurry, the hydrostatic pressure
in the slurry container, the geometry of the rectangular
opening and the speed of the casting head, throughout the
process.
Researchers have modeled successfully the ¯ow behavior
of ceramic slurry during tape casting as a one-dimensional
problem of ¯uid ¯ow in a parallel channel. The ¯ow was
*
Corresponding author. Tel.: ‡65-790-6948; fax: ‡65-791-1859.
E-mail address: mscjoshi@ntu.edu.sg (S.C. Joshi).
assumed as fully developed at the exit of the channel under
the combined effect of hydrostatic pressure in the slurry
chamber and the drag due to the relative velocity between
the casting head and the ¯at platform. Chou et al. [1]
considered the ¯ow of a Newtonian slurry as a combination
of pressure and drag (Couette) ¯ow between two parallel
plates. They further extended their ¯ow calculations to
estimate the thickness of a dried ceramic tape. The analytical
results were in good agreement with the actual measurements. Pitchumani and Karbhari [2] pointed out that ceramic
slurries exhibit non-Newtonian behavior with higher solid
contents. They presented a ¯ow model for Oswald±de Waele
power law ¯uids. They used generalized planer Couette ¯ow
as a basis for developing the model. Several parameters were
derived as a function of Reynolds number and Froude
number. Finally, equations for estimating the dry tape
thickness were derived. They studied the in¯uence of the
physical parameters of a BaTiO3 slurry and the geometrical
dimensions of the casting head on tape thickness. Ring [3]
attempted the modeling of Bingham plastics ¯ow using the
shear rate as a yield criterion to separate the plug ¯ow zone
from the rest of the ¯ow ®eld. Later, Huang et al. [4] showed
that Ring's model was inaccurate, stating that the shear
stress instead of the shear rate should form the yield criterion
for modeling Bingham plastics ¯ow. In their work, they
assumed the slurry ¯ow as a generalized Couette ¯ow. The
fact that the velocity gradient changes, from negative to zero,
and to positive, as the hydrostatic pressure in the slurry
0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 4 - 0 1 3 6 ( 0 1 ) 0 1 0 6 5 - 2
216
S.C. Joshi et al. / Journal of Materials Processing Technology 120 (2002) 215±225
Nomenclature
C
g
h
H
K
L
n
P
Q
t
u
x, y, z
constant of integration
gravitational acceleration
height or depth
height of slurry column in the slurry chamber
apparent viscosity
length of the slurry channel or width of the
Doctor's blade
power law exponent
pressure
slurry flow rate
time
velocity
orthogonal axes system and associated variables
Greek letters
a
correction factor for tape width accounting for
side flow
b
correction factor for weight loss during aging
of the tape
d
tape thickness
Z
dynamic viscosity
l
constant indicating the relative size of the
imaginary flow channel
r
density
t
yield stress
Operators
@=@y
partial differentiation with respect to y
Subscripts
c
casting head
l
lower half of the flow channel
p
plug flow region
s
slurry
tp
dry tape
u
upper half of the flow channel
x, y
component along the respective axis in
Cartesian system
0
actual tape-casting flow channel
1, 2
numeric variable descriptors
Superscripts
1, 2
numeric variable descriptors
chamber increases, was used as a derivation criterion. After
grouping the process parameters into several dimensionless
numbers, two critical pressure gradients, at which the sign
of the velocity gradient changed, were identi®ed. These
gradients were utilized subsequently for obtaining the ®nal
velocity pro®les in a casting channel. Loest et al. [5] carried
out numerical simulation of Bingham plastics ¯ow using
®nite element methods. Their work was focussed mainly on
the formation of vortices in the slurry reservoir and the
design of a vortices-free chamber.
Fig. 1. Schematic diagram of the ceramic tape-casting process.
In this paper, unlike that of previous research, the generalized pressure ¯ow is used as a basis for developing a ¯ow
model for the ceramic tape-casting process. The slurry ¯ow
is represented as a part of the pressure ¯ow in a parallel
channel of imaginary height or depth. Analytical formulations for determining the size of the imaginary channel for
¯ows of Newtonian, power law ¯uids, and Bingham plastics
under prescribed tape-casting conditions are presented.
Mathematical expressions for obtaining velocity pro®les
and ¯ow rates are included, which are then used to estimate
the thickness of a dried ceramic tape. The developed ¯ow
models are validated using published experimental data and
analytical models. The effect of various process parameters
on the size of the imaginary ¯ow channel is also studied.
2. Governing and constitutive equations
The generalized Navier±Stokes equation of ¯uid in
motion in the x-direction may be written as [6]
2
@u
@u
@P
@ u @2u @2u
r
‡u
‡ rgx ‡ Z
‡
‡
ˆ
@t
@x
@x
@x2 @y2 @z2
(1)
If the fluid flow is steady …@u=@t ˆ 0†, fully developed
@u=@x ˆ 0 and @ 2 u=@x2 ˆ 0† and gravity and flow in the
z-direction are negligible (i.e. rgx 0 and @ 2 u=@z2 0),
the above equation may be written for non-Newtonian fluids
as
@P @
@u
‡
Z
0ˆ
(2)
@x @y
@y
Eq. (2) can be solved with the appropriate boundary
conditions to define the flow behavior of different types
of viscous formulations in tape-casting processes. The viscous formulations can be classified into the following four
categories based on the constitutive relationship between
shear stress and shear rate [7]:
Newtonian fluids :
tˆZ
@u
@y
(3a)
S.C. Joshi et al. / Journal of Materials Processing Technology 120 (2002) 215±225
n
@u
Pseudo-plastics :
tˆK
; n<1
@y
n
@u
Dilatants :
tˆK
; n>1
@y
Bingham plastics :
t ˆ ty ‡ K
@u
@y
217
(3b)
(3c)
(3d)
3. Development of flow models
3.1. Newtonian fluids
The velocity pro®le of a Newtonian ¯ow may be obtained
by integrating Eq. (2), twice, with respect to y, such that
1 @P y2
uˆ
‡ C1 y ‡ C2
(4)
Z @x 2
When fluid flows between two parallel plates, as shown in
Fig. 2, with one of them stationary and the other moving, the
constants of integration in Eq. (4) may be determined from
the boundary conditions: u ˆ 0 at y ˆ 0, and u ˆ uc at
y ˆ h0 . Here, uc represents the relative velocity of the
casting head, and h0 refers to the depth of the actual flow
aperture.
The resulting equation for the velocity is [6]
1 @P
uc y
uˆ
(5)
y2 yh0 † ‡
2Z @x
h0
and for the flow rate is
Z h0
h30 @P
uc h0
u dy ˆ
‡
Qˆ
12Z @x
2
0
(6)
In the above equations @P=@x ˆ P=L, where P represents
the hydrostatic pressure in the slurry chamber (i.e.
P ˆ rs gH) and L refers to the length of the Doctor's blade
in the tape-casting unit. The negative sign indicates that the
pressure drop is in the flow direction.
It may be noted from Eq. (6) that the ®rst term, on the lefthand side of the equation, represents the contribution of the
pressure ¯ow, and the second term accounts for the drag
effect. Thus, the pressure and drag effects are additive in
Newtonian ¯ow. Therefore, Chou et al. [1] could simulate
the tape-casting process by linearly superimposing these two
effects, without having to solve for the analytical solution
directly.
Fig. 2. Developed flow between parallel plates.
Fig. 3. Pressure flow in an imaginary channel for slurry flow in the tapecasting process.
As shown in Fig. 3, the actual velocity distribution
observed in a tape-casting process may be assumed as a
part of the velocity pro®le of the pressure ¯ow in an
imaginary channel of a similar geometry. The main advantage of this concept is that, unlike the actual tape-casting
process (shown in Fig. 2), both boundaries of the imaginary
¯ow channel are stationary (shown in Fig. 3). This results in
a symmetrical velocity pro®le about the centerline of the
channel depth and gives an additional boundary condition
based on @u=@y to solve for velocity in a simpli®ed and
straightforward manner.
Once the velocity pro®le of an imaginary ¯ow channel is
known, the velocity pro®le for a tape-casting process can be
obtained by mapping the real aperture onto the imaginary
opening, as shown in Fig. 3. The stationary boundaries of the
actual and imaginary apertures are taken as the ®rst mapping
boundaries. The casting head velocity is taken as the other
mapping condition. The magnitude of the casting velocity
matches with the two velocity vectors, one from the lower
and the other from the upper half of the imaginary velocity
pro®le, due to symmetry. However, the true mapping location of the casting head onto the ¯ow ®eld of the imaginary
channel can be determined easily by relating the size of the
imaginary ¯ow channel with the size of the real aperture and
the actual casting head velocity.
In the following derivations, it is assumed that the ratio
between the cross-sectional areas of the imaginary ¯ow
channel and the actual ¯ow aperture is equal to 2l, where
l is a constant to be determined mathematically from the
tape-casting process parameters. Since the actual and the
imaginary ¯ow channels are of the same width, 2l also
represents the ratio of the depth of the channels. Thus, the
depth of an imaginary ¯ow channel is always equal to 2lh0.
It may be noted that when l ˆ 0:5, the sizes of the
imaginary and real ¯ow channels are equal. This is possible
only when uc ˆ 0 and both the velocity pro®le are exactly
the same. At l ˆ 1, the depth of the imaginary ¯ow channel
is twice (2h0) that of the depth of the actual aperture and the
218
S.C. Joshi et al. / Journal of Materials Processing Technology 120 (2002) 215±225
maximum velocity vector for the imaginary ¯ow ®eld (at the
imaginary plane of symmetry) is equal to uc. Any further
increase in the magnitude of uc leads to an increase in the
magnitude of l. Thus, depending upon the conditions of tape
casting, l may vary between 0.5 and 1.
By applying the above concept of the imaginary channel
and the boundary conditions: u ˆ 0, at y ˆ 0 and at
y ˆ 2lh0 , to Eq. (4), we get
1 @P y2
lh0 y
(7)
uˆ
Z @x
2
However, it can be seen from Fig. 3 that u ˆ uc at
y ˆ 2lh0 h0 ˆ h0 …2l 1†. Substituting this condition in
Eq. (7), the casting head velocity is given by
h20 @P
uc ˆ
1 2l†
(8)
2Z @x
For any given tape-casting process conditions, the parameters in Eq. (8), except for l, are known a priori. These can
be used to determine l. Once l is known, the velocity profile
may be obtained using Eq. (7). The flow rate Q may be
calculated as
Z 2lh0
h3 @P
Qˆ
u dy ˆ 0
1 3l†
(9)
6Z @x
h0 …2l 1†
Alternatively, Eq. (9) can also be derived by substituting
Eq. (8) into Eq. (6), indicating that the flow rate obtained
using the present concept is exactly the same as is calculated
using the analytical solution.
3.2. Power law fluids
For analytical purposes, pseudo-plastics (Eq. (3b)) and
dilatants (Eq. (3c)) can be grouped as power law ¯uids. By
comparing these equations with the de®nition of ¯uid viscosity (i.e. t ˆ Z…@u=@y†), the viscosity of a power law ¯uid
can be written as
n 1
@u
ZˆK
(10)
@y
Substituting Eq. (10) into Eq. (2), we get
@P
@ @u n
ˆK
@x
@y @y
(11)
Integrating the equation twice with respect to y, we get
1=n
@P
@u
C1
y
ˆ K 1=n
(12)
@x
@y
and
y…@P=@x† C1 †1=n‡1
ˆ K 1=n u ‡ C2
@P=@x†…1=n ‡ 1†
(13)
Similar to Newtonian fluids, the concept of imaginary flow
channel can be applied to power law fluids (see Fig. 3). This
leads to three different boundary conditions, viz. at y ˆ
lh0 ; @u=@y ˆ 0 and u ˆ 0 at y ˆ 0 and y ˆ 2lh0 . These
boundary conditions are used to solve Eqs. (12) and (13) to
obtain the final relationship for u as
uˆ
lh0 …@P=@x††1=n‡1 … lh0 …@P=@x††1=n‡1
K 1=n …@P=@x†…1=n ‡ 1†
(14)
y…@P=@x†
Since u ˆ uc at y ˆ h0 …2l
uc ˆ
1†, Eq. (14) reduces to
1=n‡1
‰h0 …@P=@x†…l 1†Š
lh0 …@P=@x††1=n‡1
K 1=n …@P=@x†…1=n ‡ 1†
(15)
Eq. (15) can be used to determine l. Once l is known, the
rate of flow through the actual aperture can be obtained as
Qˆ
1
K 1=n …@P=@x†…1=n
(
‡ 1†
lh0 …@P=@x††1=n‡2 ‰h0 …l 1†…@P=@x†Š1=n‡2
@P=@x†…1=n ‡ 2†
)
@P 1=n‡1
h0
lh0
(16)
@x
However, it may be noted that Eq. (15) is rendered indeterminate if l > 1:0 and the index 1/n is not a whole number.
Therefore, Eqs. (15) and (16) cannot be applied to the whole
variety of process conditions for power law fluids.
To avoid the above dif®culties, a new set of equations can
be derived by shifting the origin of the coordinate system to
the middle of the imaginary channel, as shown in Fig. 4. This
results in separate velocity and ¯ow rate formulations for the
upper and the lower halves of the axes system, leading to an
increased number of equations to describe the ¯ow ®eld for
the tape-casting process.
While deriving the new set of equations, two out of the
three boundary conditions (i.e. at y ˆ 0; @u=@y ˆ 0, and at
y ˆ lh0 ; u ˆ 0) should be selected appropriately for the
respective halves of the ¯ow channel to obtain the constants
of integration for Eqs. (12) and (13).
Since @P=@x is negative, choosing a boundary condition
with a negative value of y results in a set of equations which
are determinate under all possible process conditions. These
equations provide the ¯ow ®eld only for the lower half of the
imaginary channel, but the ¯ow ®eld for the upper half is
exactly the same (mirror image) due to symmetry. The ¯ow
®eld for the entire imaginary channel can be obtained by
summing the ¯ow ®elds of the lower half and the upper half
(mirror image of the lower half).
Therefore, Eqs. (12) and (13) are solved using the negative y value for the derivation and by applying the boundary
conditions: @u=@y ˆ 0 at y ˆ 0, and u ˆ 0 at y ˆ lh0 . The
resulting equation for velocity may be written as
uˆ
y…@P=@x††1=n‡1 … lh0 …@P=@x††1=n‡1
K 1=n …@P=@x†…1=n ‡ 1†
(17)
S.C. Joshi et al. / Journal of Materials Processing Technology 120 (2002) 215±225
219
Fig. 4. Newtonian fluid flow in an imaginary channel for three limiting cases with the coordinate system shifted to the center of the channel:
(a) l ˆ lu > 1:0; (b) l ˆ lp ˆ 1:0; (c) 0:5 < l ˆ ll < 1:0.
Since the above equation is valid only within the range
lh0 < y < 0, it is necessary to study the following three
cases to ascertain the location of the casting head velocity
vector and to determine the corresponding value of l.
Case 1. The actual aperture is smaller than half of the
imaginary aperture (Fig. 4(a)).
This implies that l ˆ lu > 1:0 with u ˆ uc at y ˆ
lu h0 ‡ h0 ˆ h0 …lu 1†. Substituting this condition into
Eq. (17), we get
uc ˆ
‰ h0 …lu
1†…@P=@x†Š1=n‡1 … lu h0 …@P=@x††1=n‡1
K 1=n …@P=@x†…1=n ‡ 1†
(18)
The corresponding flow rate is
Z h0 …lu 1†
u dy
Qu ˆ
l h
( u 0
)
‰ h0 …lu 1†…@P=@x†Š1=n‡2 … lu h0 …@P=@x††1=n‡2
ˆ
K 1=n …@P=@x†2 …1=n ‡ 1†…1=n ‡ 2†
(
)
h0 … lu h0 …@P=@x††1=n‡1
(19)
K 1=n …@P=@x†…1=n ‡ 1†
Case 2. The real aperture is exactly half of the imaginary
aperture (Fig. 4(b)).
This case is when l ˆ lp ˆ 1:0, and u ˆ uc at y ˆ 0. For
this condition, the resulting expression for uc may be derived
from Eq. (17) as
uc ˆ
‰ lp h0 …@P=@x†Š1=n‡1
1=n
K …@P=@x†…1=n ‡ 1†
(20)
The corresponding flow rate is
Z 0
1
Qp ˆ
u dy ˆ 1=n
K …@P=@x†…1=n ‡ 1†
lp h0
(
1=n‡1 )
‰ lp h0 …@P=@x†Š1=n‡2
@P
‡ lp h0
lp h0
@x
@P=@x†…1=n ‡ 2†
(21)
Case 3. The real aperture is larger than half of the imaginary
aperture (Fig. 4(c)).
This is possible only when 0:5 < l ˆ ll < 1:0. In this
case, u ˆ uc at y ˆ h0 …1 ll †. With this condition,
Eq. (17) reduces to
uc ˆ
‰ h0 …1
ll †…@P=@x†Š1=n‡1 … ll h0 …@P=@x††1=n‡1
K 1=n …@P=@x†…1=n ‡ 1†
(22)
The corresponding flow rate can be obtained by adding the
flow fields for the upper half …Q1l † and a part of the lower half
Q2l † of the imaginary channel as
Z 0
Z 0
u dy ‡
u dy
(23)
Ql ˆ Q1l ‡ Q2l ˆ
ll h0
h0 …1 ll †
220
S.C. Joshi et al. / Journal of Materials Processing Technology 120 (2002) 215±225
where
Q1l ˆ Qp
the same as Eq: …21†; but with lp ˆ ll †
(23a)
and
Q2l ˆ
1
K 1=n …@P=@x†…1=n ‡ 1†
(
‰ h0 …1 ll †…@P=@x†Š1=n‡2
@P=@x†…1=n ‡ 2†
1=n‡1 )
@P
‡ h0 …1 ll †
ll h0
@x
(23b)
Thus, for a given set of process parameters, such as the type
of slurry, the casting head velocity, the size of the casting
aperture and the pressure drop, Cases 1±3 will result in three
different values of l: lu, lp and ll, derived using Eqs. (18),
(20) and (22), respectively. As stated earlier, lu should
always be greater than 1, lp ˆ 1 and 0:5 < ll < 1:0. For
any given process conditions, only one among the three
values lu, lp and ll will fall within its prescribed limits and
that will be the correct value of l. After l is determined, the
appropriate equation can be selected from Eqs. (19), (21)
and (23) to obtain the corresponding flow rate.
3.3. Bingham plastics
When a Bingham plastic ¯ows between two stationary
parallel plates, the relationship between the shear stress and
the pressure gradient can be written from force equilibrium
shown in Fig. 5 as
@P
tˆy
(24)
@x
As per the definition of Bingham plastics, @u=@y would be
non-zero if t ty , otherwise the fluid will form a plug at the
center. Therefore, as a general case, the velocity profile
shown in Fig. 6(a) may be expected with this type of fluid.
The height of the plug, hp, can be estimated from Eq. (24)
as
hp ˆ
2ty
@P=@x
(25)
Fig. 6. Pressure flow in a channel with parallel boundaries for Bingham
plastics.
Comparing Eqs. (3d) and (24), @u=@y is given by
@u 1 @P
ˆ
y ty
@y K @x
(26)
When one of the stationary plates starts to move, the
resulting flow represents the tape-casting process. Similar
to power law fluids, the flow of Bingham plastics in the tapecasting process is assumed as a part of the pressure flow
through two stationary parallel plates separated by a distance
of 2lh0 (see Fig. 6(b)).
It may be seen from Fig. 6(a) that the ¯ow ®eld for
Bingham plastics is symmetrical about the mid-depth of the
channel. Therefore, a coordinate system with its x-axis
coinciding with the axis of symmetry was used for
developing the ¯ow model. Unlike power law ¯uids, the
constitutive equation for Bingham plastics includes no
exponent and the solution with either positive or negative
y-coordinates will never be indeterminate. The present
model is derived using the positive (upper) half of the velocity pro®le for the imaginary channel. The complete ¯ow
®eld is then obtained by adding the solution of the positive
half and its mirror image for the negative half as shown in
Fig. 6(b). However, since @P=@x is negative, the negative
sign convection is followed for the corresponding ty so that
hp, determined using Eq. (25), is positive.
The resulting equation for the velocity within a non-plug
zone is obtained by integrating Eq. (26) as
Z 0
Z lh0 1
@P
du ˆ
y ty dy
(27)
K
@x
u
y
Upon simplification, the above equation can be written as
uˆ
Fig. 5. Force equilibrium for a developed pressure flow between parallel
plates.
@P=@x 2
‰y
2K
lh0 †2 Š
ty
‰y
K
lh0 Š
(28)
Similar to the flow model for power law fluids, three limiting
cases are studied for determining the correct value of l. Each
of the cases requires a separate set of equations, which are
derived as follows.
S.C. Joshi et al. / Journal of Materials Processing Technology 120 (2002) 215±225
Fig. 7. Flow in an imaginary channel for three limiting cases for Bingham plastics: (a) lu > 1:0 and lu h0
and lp h0 ‡ 12 hp h0 ; (c) 0:5 < ll < 1:0 and ll h0 ‡ 12 hp < h0 .
Case 4. The real aperture is smaller than half of the
imaginary aperture and is contained within the non-plug
zone (Fig. 7(a)).
This means l ˆ lu > 1:0 such that lu h0 12 hp > h0 . The
value of uc can be calculated by substituting y ˆ lu h0
h0 ˆ h0 …lu 1† in Eq. (28). Thus
@P=@x 2
‰h0 …1
uc ˆ
2K
t y h0
2lu †Š ‡
K
The corresponding flow rate is
Z lh0
@P=@x 3
Qu ˆ
‰h0 …1
u dy ˆ
6K
h0 …lu 1†
(29)
ty h20
3lu †Š ‡
2K
(30)
‰ty
lp h0 …@P=@x†Š2
2K…@P=@x†
Qp ˆ Q1p ‡ Q2p ˆ
(31)
The flow rate can be calculated by summing the flow from
non-plug zone …Q1p † and a part of the flow from the plug
Z
lp h0
hp =2
u dy ‡ up
In simplified form
"
@P=@x 3
1
lp h0 †2 hp
Qp ˆ
6K
2
hp 2
ty
lp h0
‡
2K
2
Q2p ˆ
Case 5. The real aperture lies within the plug zone, but may
or may not extend beyond half of the imaginary aperture
(Fig. 7(b)).
This case is when l ˆ lp such that lp h0 12 hp h0 and
lp h0 ‡ 12 hp h0 . In such a situation, u ˆ uc ˆ up at y ˆ
1
2 hp ˆ ty =…@P=@x†. Thus
uc ˆ u p ˆ
zone …Q2p † as
‰ty
221
1
2 hp
hp
2
h3p
8
lp h0 …@P=@x†Š2
‰2h0 …lp
4K…@P=@x†
1
2 hp
> h0 ; (b) lp h0
h0
h0 …lp
1†
(32)
#
2…lp h0 †
3
(32a)
1†
hp Š
(32b)
Case 6. The real aperture is larger than half of the imaginary
aperture and extends beyond the plug zone (Fig. 7(c)).
It this case 0:5 < l ˆ ll < 1:0 with ll h0 ‡ 12 hp < h0 .
Then, at y ˆ h0 …1 ll †; u ˆ uc . From Eq. (28), we get
@P=@x†h20 ty h0
(33)
uc ˆ
1 2ll †
2K
K
The corresponding flow rate is a sum of the flow rates from
three regions: the non-plug zone from the upper half …Q1l †,
the plug zone …Q2l † and a part of the lower non-plug zone
222
S.C. Joshi et al. / Journal of Materials Processing Technology 120 (2002) 215±225
Q3l † may be written as
Z
1
2
3
Ql ˆ Ql ‡ Ql ‡ Ql ˆ
ll h0
hp =2
Z
u dy ‡ up hp ‡
h0 …1 ll †
hp =2
Table 1
Tape-casting process parameters for perovskite ceramic slurry [1]
Parameter
u dy
(34)
Upon simplification, we obtain
Q1l ˆ Q1p
the same as Eq: …32a†; but with lp ˆ ll †
(34a)
hp ‰ty ll h0 …@P=@x†Š2
(34b)
2K…@P=@x†
"
#
3
hp
@P=@x 3 hp
3
2
3
h0
Ql ˆ
‡ 2…ll h0 †
3…ll h0 † h0
6K
8
2
"
#
ty h2p
ll h0 …hp † ‡ 4ll h20 3…ll h0 †2 h20 (34c)
‡
2K 4
Q2l ˆ
For a given set of process parameters, solution to Eqs. (29),
(31) and (33) will result in three different values of l (i.e. lu,
lp and ll). Among these, only one value will satisfy the
prescribed boundary conditions, which are
for lu :
lu > 1:0 and
for lp :
lp h0
for ll :
0:5 < ll < 1:0
1
2 hp
lu h0
1
2 hp
> h0
and lp h0 ‡ 12 hp h0
h0
Value
2
and ll h0 ‡ 12 hp < h0
Once the correct size of the imaginary channel is known, the
velocity profile can be found using Eq. (28) and the flow rate
can be calculated using the appropriate equation from
Eqs. (30), (32) and (34).
4. Validation of the models
4.1. Newtonian fluids
Chou et al. [1] veri®ed their ¯uid ¯ow model by conducting experimental studies on the casting of perovskite
slurry at different speeds. The process parameters that they
used are listed in Table 1. By applying the principle of mass
conservation to the amount of slurry ¯owing out of the
chamber and the ®nal geometry of the casted tape, they
deduced a formulation to estimate the thickness of the aged
Z (N s/m )
rs (kg/m3)
rtp (kg/m3)
a
h0 (m)
DP (Pa)
L (m)
b
1.5
2030
3440
0.89
0.40210 3
188
1.5910 2
0.6
tape. Based on the same principle, the tape thickness can be
calculated using the present model as
dtp ˆ
abrs Q
rtp uc
(35)
For the data presented in Table 1, the results of the above
equation were compared with the model of Chou et al. [1]
and their experimental measurements in Table 2. The present
model agrees exactly with the model of Chou et al. The
experimental and analytical results are in close agreement.
The value of parameters l and Q determined using the
present model (Eqs. (8) and (9)) are tabulated in Table 2. The
large values of l indicate that the actual ¯ow aperture was
much smaller than half of the imaginary ¯ow channels.
4.2. Power law fluids
The present model was validated against Pitchumani and
Karbhari's model [2] for a BaTiO3 slurry and related process
parameters. The casting head geometry used was: h0 ˆ
300 mm …1 mm ˆ 10 6 m†, L ˆ 0:01 m and H ˆ 0:05 m. P
was taken as 981 Pa with rs ˆ 2000 kg=m3 and rs =rtp ˆ
0:58. a and b were the same as given in Table 1. The casting
speed was varied from 0.01 to 0.1 m/s. Two viscosity
models, one with n ˆ 0:59 and another with n ˆ 1:09, were
studied with K ˆ 2:7 N s=m2 . After ®nding l and the corresponding Q from Eqs. (18)±(23), Eq. (35) was used to
estimate the tape thickness. The results and their comparison
with Pitchumani and Karbhari's model are presented in
Fig. 8. As seen in this ®gure, the same values of the tape
thickness were obtained using both the models.
Table 2
Comparison of predicted and measured tape thickness for Newtonian (perovskite) ceramic slurry
uc (10
2
m/s)
Present model
Q (10
0.440
1.277
1.621
2.059
2.988
4.396
0.927
2.609
3.301
4.181
6.049
8.879
6
3
m /s)
dtp (10
6
m)
l
Present model
Chou's model [1]
Experimental data [1]
3.95
10.52
13.23
16.66
23.96
35.01
66.4
64.4
64.2
64.0
63.8
63.6
66.4
64.4
64.2
64.0
63.8
63.6
71.1
66.0
63.5
63.5
63.5
62.2
S.C. Joshi et al. / Journal of Materials Processing Technology 120 (2002) 215±225
223
Fig. 8. Effect of casting speed on tape thickness for power law fluids.
Fig. 9. Velocity profiles for Bingham plastic ceramic formulation under different pressure conditions.
4.3. Bingham plastics
5. Effect of various process parameters on k
Three cases were selected from the work of Huang et al.
[4] to test the formulations presented in Eqs. (24)±(34). In
these case studies, P was varied from 1 to 6 units with
ty ˆ 0:5 and uc ˆ Z ˆ L ˆ h0 ˆ 1. The comparison between the various velocity pro®les, obtained using the
present model and those presented in Ref. [4], is shown
in Fig. 9. The results from both models are in excellent
agreement.
It may be noted that at P ˆ 1; l ˆ 2:0. This indicates that
the real aperture was smaller than half of the imaginary ¯ow
channel. For the remaining two cases, l < 1, and the depth
of the imaginary channel was less than double the depth of
the actual ¯ow opening.
As seen in the previous section, the relative size of the
imaginary ¯ow aperture varies with the casting head geometry and the slurry properties. In order to study these
effects, various plots of l were obtained as a function of
the different process parameters listed in Table 3. The plots
for Newtonian, power law ¯uids and Bingham plastics are
presented in Figs. 10, 11 and 12, respectively.
Fig. 10(a) shows the effect of h0 on the values of l. As
the actual ¯ow opening becomes deeper, the value of l
starts to drop nonlinearly, and eventually it falls asymptotically towards a minimum value of 0.5. This happens fairly
quickly at smaller openings when the casting velocity is
smaller. This indicates that drag ¯ow is dominant at smaller
Table 3
Various casting head parameters and slurry properties used for studying variations in l (Figs. 10±12)
6
Figure
h0 (10
10a
10b
10c
11
12a
12b
50±550
300
300
300
300
300
m)
P (kPa)
Z or K (N s/m2)
n
ty (Pa)
uc (cm/s)
Model used
1.0
0.5±1.5
1.0
1.0
1.0
0.5±1.0
2.7
2.7
2.2±3.2
2.7
2.7
2.7
±
±
±
0.25±1.25
±
±
±
±
±
±
0±10
5
0.5±2.5
0.5±2.5
0.5±2.5
0.5±2.5
0.5±2.5
0.0±10.0
Newtonian
Newtonian
Newtonian
Power law fluid
Bingham plastic
Bingham plastic
224
S.C. Joshi et al. / Journal of Materials Processing Technology 120 (2002) 215±225
Fig. 10. Effect of: (a) the depth of the casting channel; (b) the hydrostatic pressure; (c) the viscosity on l at different casting velocity, for a Newtonian fluid.
Fig. 11. Effect of the power law exponent on l at different casting velocity for a power law fluid.
S.C. Joshi et al. / Journal of Materials Processing Technology 120 (2002) 215±225
225
Fig. 12. Effect of: (a) the yield stress on l at different casting velocity; (b) the casting velocity on l at different hydrostatic pressures, for Bingham plastics.
openings. Similarly, a decrease in l, but more gradual, is
observed in Fig. 10(b), as the hydrostatic pressure in slurry
chamber is reduced. The relationship between P and l
became increasingly nonlinear as the velocity of the casting
head is increased. In contrast, as shown in Fig. 10(c), l is a
weak function of Z, and the relationship is linear at different
casting velocity.
In the case of power law ¯uids, as seen from Fig. 11, l
exhibits a nonlinear behavior when the power law exponent
changes. The nonlinearity became pronounced with an
increase in casting velocity.
ty in the Bingham plastics model has very little effect on
the relative size of the imaginary ¯ow channel (see
Fig. 12(a)). As seen from Fig. 12(b), l is directly proportional to the casting velocity, its value increasing with the
velocity.
It may be noted that the power law model can be used for
Newtonian ¯uids with n ˆ 1 and K ˆ Z. Similarly, the
results of the Bingham plastics and the Newtonian ¯ow
models are the same at ty ˆ 0.
It may be seen from Fig. 12(a) that l ˆ 0:5 at uc ˆ 0. At
this condition, the sizes of the real and imaginary apertures
are the same and the slurry ¯ows under hydrostatic pressure
only. This limiting case cannot be analyzed using the models
of other researchers [2,4], the reason being that they used the
generalized Couette ¯ow as the basis of their formulations,
which become indeterminate when the casting head velocity
uc ˆ 0. In a similar way, the present models become indeterminate at P ˆ 0. However, P ˆ 0 physically represents a
situation when the casting chamber is empty, and as a result,
the ¯ow of slurry is no longer a physical reality.
6. Conclusions
The ¯ow of different types of slurry formulations was
modeled successfully as a generalized pressure ¯ow
between parallel plates. The developed models include only
one unknown geometric parameter (l). The procedure for
estimating the parameter is the same for different ¯uids such
as Newtonian, power law and Bingham plastics and can be
implemented easily. In addition to its use in ¯ow rate
calculations, l can be used as a guide to check whether
the pressure or the drag effects are dominant. The developed
models also provide a solution to a situation when ceramic
slurry is allowed to ¯ow under hydrostatic pressure only.
References
[1] Y.T. Chou, Y.T. Ko, M.F. Yan, Fluid flow model for ceramic tape
casting, J. Am. Ceram. Soc. 70 (10) (1987) C280±C282.
[2] R. Pitchumani, V.M. Karbhari, Generalized fluid flow model for
ceramic tape casting, J. Am. Ceram. Soc. 78 (9) (1995) 2497±2503.
[3] T.A. Ring, A model of tape-casting Bingham plastics and Newtonian
fluids, in: M.F. Yan, et al. (Eds.), Advances in Ceramics, Vol. 26, 1989,
pp. 569±576.
[4] X.Y. Huang, C.Y. Liu, H.Q. Gong, A viscoelastic flow modeling
of ceramic tape casting, Mater. Manuf. Process. 12 (5) (1997) 935±
943.
[5] H. Loest, R. Lipp, E. Mitsoulis, Numerical flow simulation of
viscoplastic slurries and design criteria for tape-casting unit, J. Am.
Ceram. Soc. 77 (1) (1994) 254±262.
[6] M.C. Potter, D.C. Wiggert, Mechanics of Fluids, Prentice-Hall,
Englewood Cliffs, NJ, 1991, pp. 254±258.
[7] G.J. Sharpe, Non-Newtonian fluids: two phase flow, Solving Problems
in Fluid Dynamics, Wiley, New York, 1994, pp. 191±199 (Chapter 7).