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Prediction of temperature profiles in twin screw extruders
Journal of Food Engineering 12 (1990) 145-164
Prediction of Temperature Profiles in Twin Screw Extruders Ibrahim 0. Mohamed Agricultural
Engineering
Department,
University of Gezira, PO Box 20. Wad Medani, Sudan
Robert Y. Ofoli” 205 A. W. Farrall Hall. Michigan State University. East Lansing, Michigan 48824- 1323. USA (Received 11 April 1989; accepted 14 March 1990)
ABSTRACT A model incorporating viscous dissipation effects and a heat transfer coej@ient based on the Brinkman and Graetz numbers is presented for predicting the temperature proflIes of non-Newtonian food doughs in a twin-screw extruder, assuming uniform product temperatures in the direction normal to the screw shafts. Experimental measurements were obtained to evaluate the model for three screw configurations: 30” forwarding paddles (3OF), feed (two-start or double-flighted) screws and single-start (single-flighted or single-lead) screws. Model predictions were well within engineering accuracy for 30F paddles and feed screws under all experimental conditions. Predictions for single-start screws were inaccurate, with deviations of up to 50%. In general, results indicate that the one-dimensional energy equation is suficient for heat transfer analysis of extruder sections configured with mixing paddles and feed screws, particularly at high flow rates, high RPM, or combinations of the two variables. However, the level of mixing provided by single-start screws over the RPM and flow rates used in this study does not justify the assumption of uniform temperatures in the transverse direction. For these screws, at least a two-dimensional formulation of the energy equation must be used. *To whom correspondence
should be addressed. 145
Journal of Food Engineering 0260-8774/90/$03..50 Publishers Ltd, England. Printed in Great Britain
- 0
1990
Elsevier
Science
146
I. 0. Mohamed, R. Y. Ofoli
NOTATION
Ai
Ax Br CP D E” E,* Gz
h k 4,
L hi MC r1 *c z 9 Qh R T Ti r,, Tw
W X
Z
Area available for flow (m’) Cross-sectional area of screw channel ( m2) Brinkman number (dimensionless) Specific heat (kJ kg- ’ K- ’ ) Barrel diameter (m) Rate of viscous dissipation of mechanical energy (kJ h- ’ ) Rate of viscous dissipation of mechanical energy per unit volume (kJ h-’ m-j) Graetz number (dimensionless) Average heat transfer coefficient (W m-Z “C - ’ ) Thermal conductivity (W m- ’ K- ’ ) Consistency coefficient at reference temperature (Pa s”) Length of the filled zone (m) Throughput (kg h- ’ ) Moisture content (decimal) Flow behavior index (dimensionless) Number of parallel channels Number of screw tips Screw rotational speed (RPM) Heat flux at the barrel (W m-‘) Heat transfer through the boundary (W ) Gas constant (cal [g-mole K]-I) Product temperature (“C) Inlet product temperature (“C) Reference temperature (“C (40°C used in this study)) Barrel temperature (“C) Width of screw channel (m) Dimension in the axial direction (m) Direction along screw helix (m)
Average shear rate (s - ’ ) Activation energy (cal g-mol _ ’ ) Non-Newtonian average apparent viscosity (Pa s) Dimensionless temperature Shear stress (Pa) Screw helix angle (degrees) Axial coordinate (dimensionless) Angle shown in Fig. 1 (degrees)
Prediction of temperature profiles in twin screw extruders
147
INTRODUCTION Heat transfer in twin screw extruders is of great industrial importance. An understanding of its nature and mechanisms during extrusion cooking is a requirement for proper control and optimization of the cooking process. The predominant heat sources in an extruder are heat transfer into the barrel, and viscous energy dissipation within the barrel. The latter constitutes a major portion of the overall heat energy required for cooking, depending on extruder design, operating conditions and moisture content (Rossen & Miller, 1973). It should, therefore, be an important component of heat transfer analyses in both single- and twinscrew extruders. Adequate accounting of viscous dissipation in twinscrew extruders is, however, very difficult because of the complexities associated with assessing the shear rate inside the extruder. There is also a general lack of data on the heat transfer coefficient. When these complications are added to the already complex flow dynamics inside twin-screw extruders, analysis becomes difficult. The problem can be reduced, somewhat, by adopting a one-dimensional strategy for both heat transfer and flow dynamics in the extruder. Yacu (1985), van Zuilichem et al. (1985), Bouvier et al. (1987), and Davis (1988) are among recent authors who have used one-dimensional formulations for heat transfer analysis in single- and twin-screw extruders. These studies have demonstrated, to some extent, that reasonable accuracy can be obtained by the one-dimensional approach. No studies were found in the technical literature, however, which provide a complete analysis of heat transfer during twin-screw extrusion of foods and which incorporate the effect of viscous dissipation under varying shear rate conditions. In addition, no literature sources have provided anything beyond an assumed constant heat transfer coefficient. The objectives of the present study were to: (1) Develop an a priori heat transfer model for the axial temperature profile in twin-screw extruders, incorporating the effects of viscous dissipation. (2) Conduct experiments to evaluate the model’s accuracy in predicting the temperature of the extrudate at the end of three screw configurations: kneading discs staggered at 30 degree forwarding (30F), single-start (single-flighted or single-lead) screws, and feed (two-start or double-flighted) screws. (3) Assess the suitability of using the one-dimensional heat transfer equation for analysis in twin-screw extruders.
I. 0. Mohamed, R. Y. Ofoli
148
THEORETICAL
CONSIDERATIONS
For single-start and feed screws, the energy equation was developed to account for temperature variation along the screw helix, and then transformed to describe the variation in the axial direction. For kneading discs, a macroscopic energy balance was performed to derive a differential equation to describe the temperature variation in the axial direction. Assumptions
The following assumptions were made governing differential equations: (1)
(2)
(3) (4) (5)
(6) (7)
in the development
of the
The process is at steady state. Thermal properties (specific heat and thermal conductivity) are constant. While it is recognized that these are functions of temperature, over the moderate ranges of temperature encountered in this exercise, these may be assumed relatively constant. The fluid is incompressible and homogeneous. Inertial and gravitational forces are negligible. Fluid viscosity is independent of strain history and time-temperature history. This assumption is made for mathematical simplicity in analysis. Those who must account for the two variables are referred to Morgan et al. ( 1989) or Mackey et al. (1989), who have provided comprehensive models for protein doughs and starch doughs, respectively, accounting for the two variables in addition to temperature, moisture content and shear rate. Temperature gradients at right angles to the screw shafts are negligible. Heat losses via screw shafts are negligible.
The energy equation for feed and single-start screws
If the entire extruder is considered as a system, then the energy balance can be written in terms of the change in the thermal energy of the extruded material, heat input into (output out of) the extruder, and the energy expended inside the extruder as viscous dissipation. Mathematically, &,A
T= Qh + E,
Prediction of temperature profiles in twin screw extruders
149
In a viscous system, the rate of viscous dissipation is the product of the non-Newtonian viscosity and the square of the velocity gradient. If the velocity gradient is represented by the average shear rate, then the viscous dissipation in the differential volume A,Az of a twin-screw extruder becomes E, = A, q,, f;Az
(2)
The heat transfer from a control volume bounded by z and (z +Az) can be written as Q,,=qWAz
(3)
Substituting eqns (2) and (3) into eqn ( 1) yields IziC,,A T= q WAz +A,r],j;Az Dividing the above approaches zero,
equation
by AZ, and
(4) taking
the limit as AZ
(5)
hjC d7‘=qW+A.,rj,l;; ’ dz
The effect of temperature can be incorporated into the viscosity model by an Arrhenius expression. For power law fluids, the expression is
The heat flux at the wall can be expressed in terms of the average heat transfer coefficient by q = I;(
T- T,,.)
(7)
If the shear rate in the extruder is expressed in terms of an average value, then eqns (6) and (7) can be substituted into eqn (5) to yield tic;, F=
Wh( T- T,,)+A,,K,,j;;+‘)
(8)
From eqn (8), the energy equation in the axial direction becomes
MC,
sin Q,
z=
Wh( T- Tw)+ A ,-K~~~~+”exp[y
(3.11
(9)
I. 0. Mohamed, R. Y. O$oS
150
To make eqn (9) dimensionless,
define
and x=x
(lob)
L
Incorporating
the dimensionless
nic
P
terms into eqn (9),
(L - T,,)sin@ z= L
Wh( T- T,)
AE
+A,&3p+‘Jexp
R[O( T, - T,,)+ T,,]I
(11)
Equation ( 11) can be divided by k( T, - T,,) to give MC --9sin
?=
Wh(T-L)
dx
k(Tw - To) +A.,K,,j’;+”
AE
k( T,, - T,,) exp
W(
Xv - T,,)+ &I 1
(12,) The last term of eqn ( 12) may be multiplied and divided by L* to yield dO
W&(0- 1)
Gzsinaz=k
-Br$exp
AE i NW L - 6,) + T,,]I
(13)
where Gz=~
(14)
and Br=
KoL2j’;+‘1 k( T,, - T,) exp
(15)
151
Prediction of temperature profiles in twinscrew extruders
Equation (13) is a first order nonlinear differential equation which can be solved numerically, subject to the boundary condition @=‘-
T. - T,,
atX=O
(16)
Tw- To where T, is the temperature
at the inlet of the filled zone.
The energy equation for kneading discs The development of this equation also begins with eqn (1). For kneading discs, the heat transfer term in eqn ( 1) may be calculated from the dimensions of the cross-section of the extruder barrel (Fig. 1) and a differential axial length bounded by x and x + Ax:
$4 Ax
Q,,= d2W-
(17)
The equivalent of eqn (2) for kneading discs is E, = A,q,j;Ax
(18)
Substituting eqns ( 17) and ( 18) into eqn ( 1) gives &K’,AT=q[2D(n-
11,)]Ax +A;q,f;Ax
(19)
Dividing eqn (19) by Ax, taking the limit as Ax approaches using the definition of the derivative,
zero, and
Equation (6) may be substituted into eqn (20) to give ~~~~=q[2U(n-yl)]+A,K,~y:““)exp Incorporating MC
P
R 7-r iAE i’ ‘rjl
(21)
eqns ( 1 Oa) and ( 1Ob) into eqn (2 1) yields
(T”-T,))dO-2qD(x_~)+A.~ L
dx
j(n+l)exp 1 0 a
AE
I
R[O( T,,,- To)+ 7-J
-AE (22)
I. 0. Mohamed, R. Y. Ofoli
152
which may be divided by k( T, - T,) to give A&r dO kL
2qD(n-
q)
dX - k( Tw- T,,) + A;& p’f + ’ ’
1
AE
k( TM- T,,) exp
R[O( T, - T,,)+ T,,]
(23)
By multiplying and dividing the last term by L’, eqn (23) becomes
GLd@_%Ww4
A,
I
1
AE
k( T,>,- T,,) - Br L’ exp R[O( T,. - T,,) + T,,]
dX
(24)
where the Graetz and Brinkman numbers are as defined in eqns (14) and ( 15). respectively.
Incorporating eqns (7) and ( I Oa) into the first term on the right hand side of eqn (24). one obtains
I
AE
Gz~=21)‘~-W’~(~_l)-Br~exp
W(
Equation (25) can be solved numerically, condition in eqn ( 16).
MATERIALS
T,,.- T,) + T,,lI
subject
(25)
to the boundary
AND METHODS
A Baker Perkins (MPF-SOD) co-rotating twin-screw extruder (APV Baker, Grand Rapids, Michigan, USA) was used for this work. Three
Fig. 1.
Cross-section
of twin-screw extruder barrel.
Prediction of temperature profiles in twin screw extruders
153
screw configurations were used: feed (twin-flighted) screw, single-start, and kneading discs staggered at 30” forwarding (30F). Each of the screw configurations covered a barrel length-to-diameter (L/D) ratio of 15. Two three-hole dies of length 258 cm and diameter O-3175 cm each were mounted on a twin-hole die head, and used for all runs. The three screw types used are shown in Fig. 2, along with their screw dimensions. Other details are given in Table 1. Soy polysaccharide (SPS) donated by the Ralston Purina Company (St. Louis, Missouri, USA) was used as test material. The material was extruded at a moisture content of 70% (wet basis) at three flow rates: 33, 46, and 60 kg h- ‘. For each throughput, the extruder was run at screw i.5
2.6
mm
ml
(a)
9.5
mm
(b)
r
I 26.6
-
50.2
mm
4
26.0
Fig. 2.
=
Dimensions of Baker-Perkins screw elements used in this study. (a) Feed (double-fitted) screws; (b) single-start screws; (c) paddles.
I. 0. Mohamed, R. Y. ofoli
154
Dimensions
of Extruder
TABLE 1 Barrel and Screws used in this Study”
Barrel dimensions:
Barrel diameter Distance between screw shafts Angle shown in Fig. 1
50.8 mm 40.0 mm 38-O”
Single-start screw dimensions:
Channel depth Length of tip along helix Length of root along helix Tip area Root area Flange area Volume Barrel volume Barrel surface area Wetted volume Wetted area
10.8 mm 633.4 mm 342.4 mm 16.7 cm? 18.0 cm’ 140.6 cm? 5 1.5 cm3 193.2 cm3 127.8 cm’ 00.2 cm3 478.6 cm2
Feed screw dimetlsiorls:
Channel depth Length of tip along helix Root and flange area Screw tip area Volume Screw root area Wetted area Wetted volume
IO.4 mm 165.8 mm 105.6 cmi 4.9 cm? 52.4 cm-’ 18.0 cm’ 348.9 cm’ 88.4 cm3
3OFpaddle dimensions:
14.0 cm1 81.2 cm3 1.3 cm: 1.9 cm? 14.5 cm? 265.3 cm’
Disc volume Wetted volume Disc tip area Disc flange area Disc side area Wetted area “All measurements of 5.08 cm.
and calculations
based on an axial length
speeds of 200, 300 and 400 RPM. The flour was fed dry into the extruder with a K-Tron feeder, and immediately mixed with water fed through an injection port to form a dough. The barrel temperature, T,, was set at 10°C to cool the product. During a selected number of extrusion runs, the extruder was ‘dead-stopped’, the barrel dismantled quickly, and the length of the filled zone measured.
Prediction of temperature profiles in twin screw extruders
An important part of this work was the evaluation the model in predicting extrudate temperatures at the configuration (prior to the die). This temperature inserting a thermocouple with a long start into the before it touched the tip of the rotating screw.
15.5
of the accuracy of end of each screw was measured by extruder until just
RESULTS AND DISCUSSION Solution of the differential equations A numerical procedure employing a fourth order Runge-Kutta method was used to solve the nonlinear first order differential equations. A computer program was written in Fortran to perform all necessary calculations. The thermal conductivity was calculated by (Andersen, 1950): k=k,MC+(l
-MC)
k,
(26)
The thermal conductivity of water, k,, was evaluated at 20°C (@597 W/(M”K)); the thermal conductivity of the solid, k,, was estimated as 0.259 W/(M”K). A specific heat of 2.05 kJ kg-’ K- * was used. At moderate to high shear rates, the power law model below (Howkins, 1987) may be used to characterize the temperaturedependent viscosity of SPS at 70% moisture: rl = 6.7 x
1()3j-0.75
exp
1
(27:1
The activation energy in the above equation is 4520 cal (g-mole)-’ (Howkins, 1987). The geometric data for the feed screws, single-start screws, and kneading discs used in the solution of eqns ( 13) and (25) are: (a) feed screws: W= 0.025 m A, = 0.0003 m’
I. 0. Mohamed, R. Y. ofoli
156
(c) 30F paddles: Ai=0*001596
m*
The number of parallel channels formed by intermeshing n, small tips is (Booy, 1980):
screws with
It, = 2n, - 1 The number of parallel channels for a pair of two-start feed screws is, therefore, three. The boundary condition of eqn ( 16) requires that the inlet temperature to the filled zone is known. Since there are no reliable models available to predict the temperature at the end of the partially filled zone (which, in this study, corresponds to the inlet of the filled zone), the inlet temperatures were obtained by experimental measurements, using thermocouples with tips protruding into the food material (Fig. 3). The temperature registered by the first thermocouple in the filled zone was taken as the inlet temperature. The location of this thermocouple was determined by measuring the filled length after ‘dead-stopping’ the extruder. The filled length was found to be 23.5 cm for kneading discs, and 10.1 cm for feed and singlestart screws. To estimate the average heat transfer coefficient, the length of the longest filled zone (23.5 cm) was used as the characteristic dimension to calculate the Graetz and Brinkman numbers for all screw configurations. The choice of this length was made to standardize the analysis for all screw configurations, and for convenience. Alternatively, the true filled length of each configuration could have been used. The computational scheme below was used to integrate eqns ( 13) and (25) over the dimensionless length x = O-0 to x = 1.0: Input the mass flow rate (kf ), the RPM (N), the barrel temperature (T,.), and the temperature at the inlet of the filled zone (T;). (2) Calculate the average shear rate from the following expressions (Mohamed et al., 1990): ( 1)
Single-start screws: 3 = 1 9*4N0’4’3 p = 28.0N0.539 Feed screws: 30F paddles:
p=
42.8NO.448
(29) (30) (31)
(3) Calculate the Graetz and Brinkman numbers, using eqns (14) and ( 15), respectively. (4) Estimate the average heat transfer coefficient from (Mohamed & Ofoli, 1989)
Prediction of temperature profiles in twin screw extruders
Fig. 3.
Location of thermocouples
157
in extruder barrel.
6 = 0.04 GZl~4Rr().85
(32)
(5) Calculate the temperature at the current axial location. (6) Increase x by Ax, where Ax is an appropriate incremental length. (Since the viscosity depends on temperature, it is obvious that greater accuracy will be obtained with as small an incremental value as possible. This must be weighed against the availability and cost of computing time.) (7) Repeat steps (5) and (6) until the entire profile has been obtained. For each screw configuration, temperature profiles were calculated for all throughputs (33, 46, and 60 kg hh ‘) and at all three screw speeds (200,300, and 400 RPM).
Predicted versus measured temperature profiles Comparison of simulated and experimental results for the three screw configurations are given in Table 2. Each of the values in this table represents the temperature measured at the end of the respective configuration. In general, there is close agreement between the predicted and experimental values for 30” forwarding paddles. The largest deviation from experimental data for this configuration is less than 8%.
I. 0. Mohamed, R. Y. OfoIi
1.58
Comparison
TABLE 2 of Predicted and Measured Tip Temperatures
Flow rate (kg h - ‘)
RPM
Tip temperature
Difference (“C)
Measured
Predicted
200 300 400
60.5 70.0 77.8
57.5 64.5 75.4
3.0 5.5 2.4
46
200 300 400
60.0 70.5 79.4
57.8 67.5 78.0
2.2 3.0 1.4
60
200 300 400
57.2 68.0 73.3
57.2 67.4 74.0
0.0 0.6 1.3
200 300 400
52.2 62.2 71.1
47.5 53.5 70.0
4.7 8.7
46
200 300 400
48.9 59.4 66.1
45.7 56.8 65.7
3.2 2.6 0.4
60
200 300 400
394 44.4 55.5
37.9 46.8 54.2
1.5 2.4 1.3
200 300 400
62.8 70.5 75.5
36.0 36.5 38.0
20.1 34.0 37.5
(a) 30” Forwarding paddles 33
(b) Feed (double-jlighted) screws 33
(c) Single-start screws 46
1.1
Since increasing the shear rate is generally considered as conducive to improved mixing, greater accuracy in the prediction of the temperature at the end of the screw section would be expected with increasing flow rates and/or screw rotational speeds. This is generally the case, with greater accuracy in prediction with increased rotational speed (Table 2). In addition, at a given RPM, predictions became more accurate with increasing flow rate. Since the model is based on a well-mixed stream (negligible temperature gradients in the transverse direction), both of these tendencies are to be expected.
Prediction of temperature proji/es in twin screw extruders 80.0-
o
_ .
q
-_
_ A 0
6D.D-
...-
400 400 300 300 200 200
RPM RPM RPM RPM RPM RPM
(meosured) (predicted) (measured) (predicted) (measured) (predicted)
[ 0.2
.
159
0
@ a e EL E F 40.0-
20.01.. 0.0
t
I
I,,
I 0.4
Dimensionless
Fig. 4.
Typical
I,
I.. 0.6
axial
, 0.8
.
a., 1 .o
length
predicted versus measured temperature profiles for 30F paddles. (These profiles were obtained at a flow rate of 46 kg h- I.)
A typical temperature profile for staggered 30” forwarding kneading discs is shown in Fig. 4. The simulation results for feed screws show tendencies similar to those observed for kneading paddles. At the lowest flow rate (33 kg h- ‘), the prediction of the temperature at the end of the screw configuration is relatively inaccurate at 200 and 300 RPM, even though within generally accepted engineering accuracy. At the same flow rate, however, there is close agreement between the experimental and the predicted temperature at 400 RPM f l-5% error). Predictions were much more accurate at all rotational speeds when the flow rate was increased to 46 kg h-l. At this throughput, the prediction at 400 RPM remained accurate, and the simulation at the lower rotational speeds unproved greatly, with the maximum difference between simulated and experimental temperatures being 3*2”C. When the flow rate was increased to 60 kg h- I, close agreement with the experimental data was obtained at all rotational speeds. As was the case for 30F paddles, the increase in mixing implied by higher RPM and greater throughputs significantly improved the predictive capability of the model. A typical temperature profile for feed screws is shown in Fig. 5. All model predictions for single-start screws were extremely inaccurate at all flow rates and at all rotational speeds. Rotational speeds appeared to make little difference to the accuracy of prediction. For
160
I. 0. Mohamed, R. Y. ofoli
o a
20.0
0.0
400 RPM (measured) 400 RPM (predicted) 300 RPM (measured)
1 I I , I 1 0.2
, . * . , . I . , I . . ,
Dimensionless
Fig. 5.
Typical
0.8
0.6
0.4
axial
1 .o
length
predicted versus measured temperature profiles for feed screws. (These Profiles were obtained at a flow rate of 46 kg h- I.)
80.0 o -
v
i
60.0
a - A ---.
400 400 300 300 200 200
RPM RPM RPM RPM RPM RPM
(measured) (predicted) (measured) (predicted) (measured) (predicted)
0 cl
A
:0
0 A
20.01
0.0
I I I , 1 I . , . 1 . , I , . , , . I ( 0.2
0.4 Dimensionless
Fig. 6.
0.8
0.6 axial
1.0
length
Typical predicted versus measured temperature profiles for single-start screws. (These profiles were obtained at a flow rate of 46 kg h- ’ .)
example, at the medium flow rate (46 kg h-l), model predictions differed by 375”C, 34.OT, and 26WC at 400 RPM, 300 RPM and 200 RPM, respectively, becoming greater with increasing roJationa1 speeds (Fig. 6). Similarly large differences between simulated and experimental values
Prediction
of temperature profiles in twin screw extncders
161
were observed at the low and high flow rates. Of the three configurations studied, single-start screws appeared to provide the least degree of mixing, as determined qualitatively by examining the uniformity of extrudates. Given incomplete mixing, it would be expected that significant temperature gradients could exist in both the radial and tangential directions, in contrast to the assumptions on which the theoretical development was based. The one-dimensional model is, therefore, inadequate for single-start screws. It is also possible that, at the relatively low shear rates characteristic of single-start screws (Mohamed et al., 1990), SPS at 70% moisture content may not behave like a power law fluid, and may have a yield stress. For doughs which have a yield stress, a plug flow region would exist at the root of the screw in the absence of mixing, causing material in the region closer to the barrel to be subjected to a greater shear rate. The net result is a greater viscous energy dissipation, which would produce a higher temperature in the region closer to the barrel surface. Secondly, the relatively wide screw tip of single-start screws generate more viscous dissipation at the contact surface between the barrel and the screw tip. In the absence of significant levels of mixing, this energy raises the temperature of the relatively small amount of material in the region of the barrel, resulting in higher temperatures at the end of the screw assembly.
70.0_ -
_ ---
33 kg hr-t 46 kg hr-1 60 kg hw
60.0-
30.0
0.0
.
.
.
, 0.2
I
I
7
, 0.4
Dimensionless
Fig. 7.
Simulated temperature
.
.
.
, 0.6
axial
I
.
I
, 0.8
8
.
.
, 1 .o
length
profiles for feed screws at 400 RPM and various mass flow rates.
162
I. 0. Mohamed, R. Y. Ofoli
In summary, at a given throughput, increasing the screw speed increases the shear rate (hence, also the Brinkman number) and results in greater viscous dissipation. This results in higher product temperatures. lead to larger Graetz At a given screw speed, higher throughputs numbers, which increases the average heat transfer coefficient. These effects combine to produce a smaller temperature difference between the product and the barrel, as can be seen from the slopes of the temperature profiles in Fig. 7. While the trend in Fig. 7 is partially the result of residence time, it is also an indication of greater convective cooling at increased throughputs. At high screw speeds and low throughputs, the extruder’s capacity for convective cooling may not be enough to overcome the rate of viscous energy generation in situations where it is necessary to keep the final product temperature within a specific range.
CONCLUSIONS A model incorporating viscous dissipation effects and a heat transfer coefficient based on the Brinkman and Graetz numbers has been developed for predicting the temperature profiles of non-Newtonian food doughs in a twin-screw extruder, assuming uniform product temperatures in the direction normal to the screw shafts. The model was based on a one-dimensional treatment of the energy equation. Experiments were conducted to evaluate the accuracy of the model in predicting the temperature at the end of three screw configurations: 30” forwarding paddles (30F), feed (double-flighted) screws, and single-start screws. Close agreement was obtained with experimental data for 30F paddles under all experimental conditions except at 300 RPM at a flow rate of 33 kg h-l, where the predicted and experimental results differed by 55°C. Model predictions at all other conditions were within 3.O”C of the experimental values. For feed (double-flighted) screws, predicted values for 33 kg h-l at 200 and 300 RPM and for 46 kg h- ’ at 200 RPM differed from experimental values by 4*7”C, 8.7”C and 3*2”C, respectively; the largest difference for all other conditions using feed screws was 2-6°C. Predictions for single-start screws were inaccurate, with differences of up to 37.5”C between the simulated and experimental values. The most plausible reason for this lack of accuracy is that the degree of mixing provided by single-start screws over the RPM and flow rates used in this study does not justify the assumption of negligible transverse temperature gradients.
Prediction of temperature profiles in twinscrew extruders
163
The one-dimensional energy equation appears to be adequate for the analysis of heat transfer in the mixing paddle and feed screw regions of extruders, particularly at larger flow rates, high RPM, or combinations of the two variables which ensure thorough mixing. For single-start screws, however, it is recommended that at least a two-dimensional form of the energy equation be used. The results of this study confirm what is generally known about extruders: at high screw speeds and low throughputs, the extruder’s capacity for convective cooling may not be sufficient to overcome the rate of viscous energy generation in situations where it is necessary to keep the final product temperature within a specific range. The authors recognize that the moisture content used in this study is impractical for industrial applications. This moisture content was selected, however, to ensure that the test material was a power law fluid over the shear rate range of this study. Due to the hygroscopic nature 01 SPS, moisture contents below 70% result in a material which deviates significantly from power law behavior. In spite of this, it is hoped that the type of information provided by studies such as this would be useful in the design, control and optimization of the extrusion cooking process.
REFERENCES Andersen, S. A. (1950). Automatic Refrigeration. Maclaren and Son Ltd, Nordborg, Denmark. Booy, M. L. ( 1980). Isothermal flow of viscous liquid in co-rotating twin screw devices. Polymer Engineering and Science, 20 ( 18), 1220-g. Bouvier, J. M., Fayard, G. & Clayton, J. T. (1987). Flow rate and heat transfer modeling in extrusion cooking of soy protein. Journal of Food Engineering, 6, 123-41.
Davis, W. M. (1988). Heat-transfer in extruder reactors. Chemical Engineering Progress, November 1988,35-42. Howl&s, M. D. ( 1987). A predictive model for pressure drop in food extruder dies. MS Thesis, Michigan State University, East Lansing, Michigan. Mackey, K. L., Ofoli, R. Y., Morgan, R. G. & Steffe, J. F. (1989). Rheological modeling of potato flour during extrusion cooking. Journal of Food Process Engineering, 12 ( 1), 1 - 11.
Mohamed, I. 0. & Ofoli, R. Y. ( 1989). Average heat transfer coefficients in twin screw extruders. Biotechnology Progress, 5 (4), 158-63. Mohamed, I. O., Ofoli, R. Y. & Morgan, R. G. (1990). Modeling the average shear rate in a co-rotating twin screw extruder. Journal of Food Process Engineering,
12 (3), 227-46.
Morgan, R. G., Steffe, J. F. & Ofoli, R. Y. ( 1989). A generalized viscosity model for extrusion of protein doughs. Journal of Food Process Engineering, 11, 55-78.
164
I. 0.
Mohamed,R. Y. ofoli
Rossen, J. L. & Miller, R. C. (1973). Food extrusion. Food Technology, 27, 46-53. Yacu, W. ( 1985). Modelling a twin screw co-rotating extruder. Journal of Food Process Engineering, &l-2 1. van Zuilichem, D. J., Tempel, W. J., Stoip, W. & van? Riet, K. ( 1985). Production of high-boiled sugar confectionery by extrusion-cooking of sucrose: liquid glucose mixtures. Journal of Food Engineering, 4,3 7-5 1.
Prediction of Temperature Profiles in Twin Screw Extruders Ibrahim 0. Mohamed Agricultural
Engineering
Department,
University of Gezira, PO Box 20. Wad Medani, Sudan
Robert Y. Ofoli” 205 A. W. Farrall Hall. Michigan State University. East Lansing, Michigan 48824- 1323. USA (Received 11 April 1989; accepted 14 March 1990)
ABSTRACT A model incorporating viscous dissipation effects and a heat transfer coej@ient based on the Brinkman and Graetz numbers is presented for predicting the temperature proflIes of non-Newtonian food doughs in a twin-screw extruder, assuming uniform product temperatures in the direction normal to the screw shafts. Experimental measurements were obtained to evaluate the model for three screw configurations: 30” forwarding paddles (3OF), feed (two-start or double-flighted) screws and single-start (single-flighted or single-lead) screws. Model predictions were well within engineering accuracy for 30F paddles and feed screws under all experimental conditions. Predictions for single-start screws were inaccurate, with deviations of up to 50%. In general, results indicate that the one-dimensional energy equation is suficient for heat transfer analysis of extruder sections configured with mixing paddles and feed screws, particularly at high flow rates, high RPM, or combinations of the two variables. However, the level of mixing provided by single-start screws over the RPM and flow rates used in this study does not justify the assumption of uniform temperatures in the transverse direction. For these screws, at least a two-dimensional formulation of the energy equation must be used. *To whom correspondence
should be addressed. 145
Journal of Food Engineering 0260-8774/90/$03..50 Publishers Ltd, England. Printed in Great Britain
- 0
1990
Elsevier
Science
146
I. 0. Mohamed, R. Y. Ofoli
NOTATION
Ai
Ax Br CP D E” E,* Gz
h k 4,
L hi MC r1 *c z 9 Qh R T Ti r,, Tw
W X
Z
Area available for flow (m’) Cross-sectional area of screw channel ( m2) Brinkman number (dimensionless) Specific heat (kJ kg- ’ K- ’ ) Barrel diameter (m) Rate of viscous dissipation of mechanical energy (kJ h- ’ ) Rate of viscous dissipation of mechanical energy per unit volume (kJ h-’ m-j) Graetz number (dimensionless) Average heat transfer coefficient (W m-Z “C - ’ ) Thermal conductivity (W m- ’ K- ’ ) Consistency coefficient at reference temperature (Pa s”) Length of the filled zone (m) Throughput (kg h- ’ ) Moisture content (decimal) Flow behavior index (dimensionless) Number of parallel channels Number of screw tips Screw rotational speed (RPM) Heat flux at the barrel (W m-‘) Heat transfer through the boundary (W ) Gas constant (cal [g-mole K]-I) Product temperature (“C) Inlet product temperature (“C) Reference temperature (“C (40°C used in this study)) Barrel temperature (“C) Width of screw channel (m) Dimension in the axial direction (m) Direction along screw helix (m)
Average shear rate (s - ’ ) Activation energy (cal g-mol _ ’ ) Non-Newtonian average apparent viscosity (Pa s) Dimensionless temperature Shear stress (Pa) Screw helix angle (degrees) Axial coordinate (dimensionless) Angle shown in Fig. 1 (degrees)
Prediction of temperature profiles in twin screw extruders
147
INTRODUCTION Heat transfer in twin screw extruders is of great industrial importance. An understanding of its nature and mechanisms during extrusion cooking is a requirement for proper control and optimization of the cooking process. The predominant heat sources in an extruder are heat transfer into the barrel, and viscous energy dissipation within the barrel. The latter constitutes a major portion of the overall heat energy required for cooking, depending on extruder design, operating conditions and moisture content (Rossen & Miller, 1973). It should, therefore, be an important component of heat transfer analyses in both single- and twinscrew extruders. Adequate accounting of viscous dissipation in twinscrew extruders is, however, very difficult because of the complexities associated with assessing the shear rate inside the extruder. There is also a general lack of data on the heat transfer coefficient. When these complications are added to the already complex flow dynamics inside twin-screw extruders, analysis becomes difficult. The problem can be reduced, somewhat, by adopting a one-dimensional strategy for both heat transfer and flow dynamics in the extruder. Yacu (1985), van Zuilichem et al. (1985), Bouvier et al. (1987), and Davis (1988) are among recent authors who have used one-dimensional formulations for heat transfer analysis in single- and twin-screw extruders. These studies have demonstrated, to some extent, that reasonable accuracy can be obtained by the one-dimensional approach. No studies were found in the technical literature, however, which provide a complete analysis of heat transfer during twin-screw extrusion of foods and which incorporate the effect of viscous dissipation under varying shear rate conditions. In addition, no literature sources have provided anything beyond an assumed constant heat transfer coefficient. The objectives of the present study were to: (1) Develop an a priori heat transfer model for the axial temperature profile in twin-screw extruders, incorporating the effects of viscous dissipation. (2) Conduct experiments to evaluate the model’s accuracy in predicting the temperature of the extrudate at the end of three screw configurations: kneading discs staggered at 30 degree forwarding (30F), single-start (single-flighted or single-lead) screws, and feed (two-start or double-flighted) screws. (3) Assess the suitability of using the one-dimensional heat transfer equation for analysis in twin-screw extruders.
I. 0. Mohamed, R. Y. Ofoli
148
THEORETICAL
CONSIDERATIONS
For single-start and feed screws, the energy equation was developed to account for temperature variation along the screw helix, and then transformed to describe the variation in the axial direction. For kneading discs, a macroscopic energy balance was performed to derive a differential equation to describe the temperature variation in the axial direction. Assumptions
The following assumptions were made governing differential equations: (1)
(2)
(3) (4) (5)
(6) (7)
in the development
of the
The process is at steady state. Thermal properties (specific heat and thermal conductivity) are constant. While it is recognized that these are functions of temperature, over the moderate ranges of temperature encountered in this exercise, these may be assumed relatively constant. The fluid is incompressible and homogeneous. Inertial and gravitational forces are negligible. Fluid viscosity is independent of strain history and time-temperature history. This assumption is made for mathematical simplicity in analysis. Those who must account for the two variables are referred to Morgan et al. ( 1989) or Mackey et al. (1989), who have provided comprehensive models for protein doughs and starch doughs, respectively, accounting for the two variables in addition to temperature, moisture content and shear rate. Temperature gradients at right angles to the screw shafts are negligible. Heat losses via screw shafts are negligible.
The energy equation for feed and single-start screws
If the entire extruder is considered as a system, then the energy balance can be written in terms of the change in the thermal energy of the extruded material, heat input into (output out of) the extruder, and the energy expended inside the extruder as viscous dissipation. Mathematically, &,A
T= Qh + E,
Prediction of temperature profiles in twin screw extruders
149
In a viscous system, the rate of viscous dissipation is the product of the non-Newtonian viscosity and the square of the velocity gradient. If the velocity gradient is represented by the average shear rate, then the viscous dissipation in the differential volume A,Az of a twin-screw extruder becomes E, = A, q,, f;Az
(2)
The heat transfer from a control volume bounded by z and (z +Az) can be written as Q,,=qWAz
(3)
Substituting eqns (2) and (3) into eqn ( 1) yields IziC,,A T= q WAz +A,r],j;Az Dividing the above approaches zero,
equation
by AZ, and
(4) taking
the limit as AZ
(5)
hjC d7‘=qW+A.,rj,l;; ’ dz
The effect of temperature can be incorporated into the viscosity model by an Arrhenius expression. For power law fluids, the expression is
The heat flux at the wall can be expressed in terms of the average heat transfer coefficient by q = I;(
T- T,,.)
(7)
If the shear rate in the extruder is expressed in terms of an average value, then eqns (6) and (7) can be substituted into eqn (5) to yield tic;, F=
Wh( T- T,,)+A,,K,,j;;+‘)
(8)
From eqn (8), the energy equation in the axial direction becomes
MC,
sin Q,
z=
Wh( T- Tw)+ A ,-K~~~~+”exp[y
(3.11
(9)
I. 0. Mohamed, R. Y. O$oS
150
To make eqn (9) dimensionless,
define
and x=x
(lob)
L
Incorporating
the dimensionless
nic
P
terms into eqn (9),
(L - T,,)sin@ z= L
Wh( T- T,)
AE
+A,&3p+‘Jexp
R[O( T, - T,,)+ T,,]I
(11)
Equation ( 11) can be divided by k( T, - T,,) to give MC --9sin
?=
Wh(T-L)
dx
k(Tw - To) +A.,K,,j’;+”
AE
k( T,, - T,,) exp
W(
Xv - T,,)+ &I 1
(12,) The last term of eqn ( 12) may be multiplied and divided by L* to yield dO
W&(0- 1)
Gzsinaz=k
-Br$exp
AE i NW L - 6,) + T,,]I
(13)
where Gz=~
(14)
and Br=
KoL2j’;+‘1 k( T,, - T,) exp
(15)
151
Prediction of temperature profiles in twinscrew extruders
Equation (13) is a first order nonlinear differential equation which can be solved numerically, subject to the boundary condition @=‘-
T. - T,,
atX=O
(16)
Tw- To where T, is the temperature
at the inlet of the filled zone.
The energy equation for kneading discs The development of this equation also begins with eqn (1). For kneading discs, the heat transfer term in eqn ( 1) may be calculated from the dimensions of the cross-section of the extruder barrel (Fig. 1) and a differential axial length bounded by x and x + Ax:
$4 Ax
Q,,= d2W-
(17)
The equivalent of eqn (2) for kneading discs is E, = A,q,j;Ax
(18)
Substituting eqns ( 17) and ( 18) into eqn ( 1) gives &K’,AT=q[2D(n-
11,)]Ax +A;q,f;Ax
(19)
Dividing eqn (19) by Ax, taking the limit as Ax approaches using the definition of the derivative,
zero, and
Equation (6) may be substituted into eqn (20) to give ~~~~=q[2U(n-yl)]+A,K,~y:““)exp Incorporating MC
P
R 7-r iAE i’ ‘rjl
(21)
eqns ( 1 Oa) and ( 1Ob) into eqn (2 1) yields
(T”-T,))dO-2qD(x_~)+A.~ L
dx
j(n+l)exp 1 0 a
AE
I
R[O( T,,,- To)+ 7-J
-AE (22)
I. 0. Mohamed, R. Y. Ofoli
152
which may be divided by k( T, - T,) to give A&r dO kL
2qD(n-
q)
dX - k( Tw- T,,) + A;& p’f + ’ ’
1
AE
k( TM- T,,) exp
R[O( T, - T,,)+ T,,]
(23)
By multiplying and dividing the last term by L’, eqn (23) becomes
GLd@_%Ww4
A,
I
1
AE
k( T,>,- T,,) - Br L’ exp R[O( T,. - T,,) + T,,]
dX
(24)
where the Graetz and Brinkman numbers are as defined in eqns (14) and ( 15). respectively.
Incorporating eqns (7) and ( I Oa) into the first term on the right hand side of eqn (24). one obtains
I
AE
Gz~=21)‘~-W’~(~_l)-Br~exp
W(
Equation (25) can be solved numerically, condition in eqn ( 16).
MATERIALS
T,,.- T,) + T,,lI
subject
(25)
to the boundary
AND METHODS
A Baker Perkins (MPF-SOD) co-rotating twin-screw extruder (APV Baker, Grand Rapids, Michigan, USA) was used for this work. Three
Fig. 1.
Cross-section
of twin-screw extruder barrel.
Prediction of temperature profiles in twin screw extruders
153
screw configurations were used: feed (twin-flighted) screw, single-start, and kneading discs staggered at 30” forwarding (30F). Each of the screw configurations covered a barrel length-to-diameter (L/D) ratio of 15. Two three-hole dies of length 258 cm and diameter O-3175 cm each were mounted on a twin-hole die head, and used for all runs. The three screw types used are shown in Fig. 2, along with their screw dimensions. Other details are given in Table 1. Soy polysaccharide (SPS) donated by the Ralston Purina Company (St. Louis, Missouri, USA) was used as test material. The material was extruded at a moisture content of 70% (wet basis) at three flow rates: 33, 46, and 60 kg h- ‘. For each throughput, the extruder was run at screw i.5
2.6
mm
ml
(a)
9.5
mm
(b)
r
I 26.6
-
50.2
mm
4
26.0
Fig. 2.
=
Dimensions of Baker-Perkins screw elements used in this study. (a) Feed (double-fitted) screws; (b) single-start screws; (c) paddles.
I. 0. Mohamed, R. Y. ofoli
154
Dimensions
of Extruder
TABLE 1 Barrel and Screws used in this Study”
Barrel dimensions:
Barrel diameter Distance between screw shafts Angle shown in Fig. 1
50.8 mm 40.0 mm 38-O”
Single-start screw dimensions:
Channel depth Length of tip along helix Length of root along helix Tip area Root area Flange area Volume Barrel volume Barrel surface area Wetted volume Wetted area
10.8 mm 633.4 mm 342.4 mm 16.7 cm? 18.0 cm’ 140.6 cm? 5 1.5 cm3 193.2 cm3 127.8 cm’ 00.2 cm3 478.6 cm2
Feed screw dimetlsiorls:
Channel depth Length of tip along helix Root and flange area Screw tip area Volume Screw root area Wetted area Wetted volume
IO.4 mm 165.8 mm 105.6 cmi 4.9 cm? 52.4 cm-’ 18.0 cm’ 348.9 cm’ 88.4 cm3
3OFpaddle dimensions:
14.0 cm1 81.2 cm3 1.3 cm: 1.9 cm? 14.5 cm? 265.3 cm’
Disc volume Wetted volume Disc tip area Disc flange area Disc side area Wetted area “All measurements of 5.08 cm.
and calculations
based on an axial length
speeds of 200, 300 and 400 RPM. The flour was fed dry into the extruder with a K-Tron feeder, and immediately mixed with water fed through an injection port to form a dough. The barrel temperature, T,, was set at 10°C to cool the product. During a selected number of extrusion runs, the extruder was ‘dead-stopped’, the barrel dismantled quickly, and the length of the filled zone measured.
Prediction of temperature profiles in twin screw extruders
An important part of this work was the evaluation the model in predicting extrudate temperatures at the configuration (prior to the die). This temperature inserting a thermocouple with a long start into the before it touched the tip of the rotating screw.
15.5
of the accuracy of end of each screw was measured by extruder until just
RESULTS AND DISCUSSION Solution of the differential equations A numerical procedure employing a fourth order Runge-Kutta method was used to solve the nonlinear first order differential equations. A computer program was written in Fortran to perform all necessary calculations. The thermal conductivity was calculated by (Andersen, 1950): k=k,MC+(l
-MC)
k,
(26)
The thermal conductivity of water, k,, was evaluated at 20°C (@597 W/(M”K)); the thermal conductivity of the solid, k,, was estimated as 0.259 W/(M”K). A specific heat of 2.05 kJ kg-’ K- * was used. At moderate to high shear rates, the power law model below (Howkins, 1987) may be used to characterize the temperaturedependent viscosity of SPS at 70% moisture: rl = 6.7 x
1()3j-0.75
exp
1
(27:1
The activation energy in the above equation is 4520 cal (g-mole)-’ (Howkins, 1987). The geometric data for the feed screws, single-start screws, and kneading discs used in the solution of eqns ( 13) and (25) are: (a) feed screws: W= 0.025 m A, = 0.0003 m’
I. 0. Mohamed, R. Y. ofoli
156
(c) 30F paddles: Ai=0*001596
m*
The number of parallel channels formed by intermeshing n, small tips is (Booy, 1980):
screws with
It, = 2n, - 1 The number of parallel channels for a pair of two-start feed screws is, therefore, three. The boundary condition of eqn ( 16) requires that the inlet temperature to the filled zone is known. Since there are no reliable models available to predict the temperature at the end of the partially filled zone (which, in this study, corresponds to the inlet of the filled zone), the inlet temperatures were obtained by experimental measurements, using thermocouples with tips protruding into the food material (Fig. 3). The temperature registered by the first thermocouple in the filled zone was taken as the inlet temperature. The location of this thermocouple was determined by measuring the filled length after ‘dead-stopping’ the extruder. The filled length was found to be 23.5 cm for kneading discs, and 10.1 cm for feed and singlestart screws. To estimate the average heat transfer coefficient, the length of the longest filled zone (23.5 cm) was used as the characteristic dimension to calculate the Graetz and Brinkman numbers for all screw configurations. The choice of this length was made to standardize the analysis for all screw configurations, and for convenience. Alternatively, the true filled length of each configuration could have been used. The computational scheme below was used to integrate eqns ( 13) and (25) over the dimensionless length x = O-0 to x = 1.0: Input the mass flow rate (kf ), the RPM (N), the barrel temperature (T,.), and the temperature at the inlet of the filled zone (T;). (2) Calculate the average shear rate from the following expressions (Mohamed et al., 1990): ( 1)
Single-start screws: 3 = 1 9*4N0’4’3 p = 28.0N0.539 Feed screws: 30F paddles:
p=
42.8NO.448
(29) (30) (31)
(3) Calculate the Graetz and Brinkman numbers, using eqns (14) and ( 15), respectively. (4) Estimate the average heat transfer coefficient from (Mohamed & Ofoli, 1989)
Prediction of temperature profiles in twin screw extruders
Fig. 3.
Location of thermocouples
157
in extruder barrel.
6 = 0.04 GZl~4Rr().85
(32)
(5) Calculate the temperature at the current axial location. (6) Increase x by Ax, where Ax is an appropriate incremental length. (Since the viscosity depends on temperature, it is obvious that greater accuracy will be obtained with as small an incremental value as possible. This must be weighed against the availability and cost of computing time.) (7) Repeat steps (5) and (6) until the entire profile has been obtained. For each screw configuration, temperature profiles were calculated for all throughputs (33, 46, and 60 kg hh ‘) and at all three screw speeds (200,300, and 400 RPM).
Predicted versus measured temperature profiles Comparison of simulated and experimental results for the three screw configurations are given in Table 2. Each of the values in this table represents the temperature measured at the end of the respective configuration. In general, there is close agreement between the predicted and experimental values for 30” forwarding paddles. The largest deviation from experimental data for this configuration is less than 8%.
I. 0. Mohamed, R. Y. OfoIi
1.58
Comparison
TABLE 2 of Predicted and Measured Tip Temperatures
Flow rate (kg h - ‘)
RPM
Tip temperature
Difference (“C)
Measured
Predicted
200 300 400
60.5 70.0 77.8
57.5 64.5 75.4
3.0 5.5 2.4
46
200 300 400
60.0 70.5 79.4
57.8 67.5 78.0
2.2 3.0 1.4
60
200 300 400
57.2 68.0 73.3
57.2 67.4 74.0
0.0 0.6 1.3
200 300 400
52.2 62.2 71.1
47.5 53.5 70.0
4.7 8.7
46
200 300 400
48.9 59.4 66.1
45.7 56.8 65.7
3.2 2.6 0.4
60
200 300 400
394 44.4 55.5
37.9 46.8 54.2
1.5 2.4 1.3
200 300 400
62.8 70.5 75.5
36.0 36.5 38.0
20.1 34.0 37.5
(a) 30” Forwarding paddles 33
(b) Feed (double-jlighted) screws 33
(c) Single-start screws 46
1.1
Since increasing the shear rate is generally considered as conducive to improved mixing, greater accuracy in the prediction of the temperature at the end of the screw section would be expected with increasing flow rates and/or screw rotational speeds. This is generally the case, with greater accuracy in prediction with increased rotational speed (Table 2). In addition, at a given RPM, predictions became more accurate with increasing flow rate. Since the model is based on a well-mixed stream (negligible temperature gradients in the transverse direction), both of these tendencies are to be expected.
Prediction of temperature proji/es in twin screw extruders 80.0-
o
_ .
q
-_
_ A 0
6D.D-
...-
400 400 300 300 200 200
RPM RPM RPM RPM RPM RPM
(meosured) (predicted) (measured) (predicted) (measured) (predicted)
[ 0.2
.
159
0
@ a e EL E F 40.0-
20.01.. 0.0
t
I
I,,
I 0.4
Dimensionless
Fig. 4.
Typical
I,
I.. 0.6
axial
, 0.8
.
a., 1 .o
length
predicted versus measured temperature profiles for 30F paddles. (These profiles were obtained at a flow rate of 46 kg h- I.)
A typical temperature profile for staggered 30” forwarding kneading discs is shown in Fig. 4. The simulation results for feed screws show tendencies similar to those observed for kneading paddles. At the lowest flow rate (33 kg h- ‘), the prediction of the temperature at the end of the screw configuration is relatively inaccurate at 200 and 300 RPM, even though within generally accepted engineering accuracy. At the same flow rate, however, there is close agreement between the experimental and the predicted temperature at 400 RPM f l-5% error). Predictions were much more accurate at all rotational speeds when the flow rate was increased to 46 kg h-l. At this throughput, the prediction at 400 RPM remained accurate, and the simulation at the lower rotational speeds unproved greatly, with the maximum difference between simulated and experimental temperatures being 3*2”C. When the flow rate was increased to 60 kg h- I, close agreement with the experimental data was obtained at all rotational speeds. As was the case for 30F paddles, the increase in mixing implied by higher RPM and greater throughputs significantly improved the predictive capability of the model. A typical temperature profile for feed screws is shown in Fig. 5. All model predictions for single-start screws were extremely inaccurate at all flow rates and at all rotational speeds. Rotational speeds appeared to make little difference to the accuracy of prediction. For
160
I. 0. Mohamed, R. Y. ofoli
o a
20.0
0.0
400 RPM (measured) 400 RPM (predicted) 300 RPM (measured)
1 I I , I 1 0.2
, . * . , . I . , I . . ,
Dimensionless
Fig. 5.
Typical
0.8
0.6
0.4
axial
1 .o
length
predicted versus measured temperature profiles for feed screws. (These Profiles were obtained at a flow rate of 46 kg h- I.)
80.0 o -
v
i
60.0
a - A ---.
400 400 300 300 200 200
RPM RPM RPM RPM RPM RPM
(measured) (predicted) (measured) (predicted) (measured) (predicted)
0 cl
A
:0
0 A
20.01
0.0
I I I , 1 I . , . 1 . , I , . , , . I ( 0.2
0.4 Dimensionless
Fig. 6.
0.8
0.6 axial
1.0
length
Typical predicted versus measured temperature profiles for single-start screws. (These profiles were obtained at a flow rate of 46 kg h- ’ .)
example, at the medium flow rate (46 kg h-l), model predictions differed by 375”C, 34.OT, and 26WC at 400 RPM, 300 RPM and 200 RPM, respectively, becoming greater with increasing roJationa1 speeds (Fig. 6). Similarly large differences between simulated and experimental values
Prediction
of temperature profiles in twin screw extncders
161
were observed at the low and high flow rates. Of the three configurations studied, single-start screws appeared to provide the least degree of mixing, as determined qualitatively by examining the uniformity of extrudates. Given incomplete mixing, it would be expected that significant temperature gradients could exist in both the radial and tangential directions, in contrast to the assumptions on which the theoretical development was based. The one-dimensional model is, therefore, inadequate for single-start screws. It is also possible that, at the relatively low shear rates characteristic of single-start screws (Mohamed et al., 1990), SPS at 70% moisture content may not behave like a power law fluid, and may have a yield stress. For doughs which have a yield stress, a plug flow region would exist at the root of the screw in the absence of mixing, causing material in the region closer to the barrel to be subjected to a greater shear rate. The net result is a greater viscous energy dissipation, which would produce a higher temperature in the region closer to the barrel surface. Secondly, the relatively wide screw tip of single-start screws generate more viscous dissipation at the contact surface between the barrel and the screw tip. In the absence of significant levels of mixing, this energy raises the temperature of the relatively small amount of material in the region of the barrel, resulting in higher temperatures at the end of the screw assembly.
70.0_ -
_ ---
33 kg hr-t 46 kg hr-1 60 kg hw
60.0-
30.0
0.0
.
.
.
, 0.2
I
I
7
, 0.4
Dimensionless
Fig. 7.
Simulated temperature
.
.
.
, 0.6
axial
I
.
I
, 0.8
8
.
.
, 1 .o
length
profiles for feed screws at 400 RPM and various mass flow rates.
162
I. 0. Mohamed, R. Y. Ofoli
In summary, at a given throughput, increasing the screw speed increases the shear rate (hence, also the Brinkman number) and results in greater viscous dissipation. This results in higher product temperatures. lead to larger Graetz At a given screw speed, higher throughputs numbers, which increases the average heat transfer coefficient. These effects combine to produce a smaller temperature difference between the product and the barrel, as can be seen from the slopes of the temperature profiles in Fig. 7. While the trend in Fig. 7 is partially the result of residence time, it is also an indication of greater convective cooling at increased throughputs. At high screw speeds and low throughputs, the extruder’s capacity for convective cooling may not be enough to overcome the rate of viscous energy generation in situations where it is necessary to keep the final product temperature within a specific range.
CONCLUSIONS A model incorporating viscous dissipation effects and a heat transfer coefficient based on the Brinkman and Graetz numbers has been developed for predicting the temperature profiles of non-Newtonian food doughs in a twin-screw extruder, assuming uniform product temperatures in the direction normal to the screw shafts. The model was based on a one-dimensional treatment of the energy equation. Experiments were conducted to evaluate the accuracy of the model in predicting the temperature at the end of three screw configurations: 30” forwarding paddles (30F), feed (double-flighted) screws, and single-start screws. Close agreement was obtained with experimental data for 30F paddles under all experimental conditions except at 300 RPM at a flow rate of 33 kg h-l, where the predicted and experimental results differed by 55°C. Model predictions at all other conditions were within 3.O”C of the experimental values. For feed (double-flighted) screws, predicted values for 33 kg h-l at 200 and 300 RPM and for 46 kg h- ’ at 200 RPM differed from experimental values by 4*7”C, 8.7”C and 3*2”C, respectively; the largest difference for all other conditions using feed screws was 2-6°C. Predictions for single-start screws were inaccurate, with differences of up to 37.5”C between the simulated and experimental values. The most plausible reason for this lack of accuracy is that the degree of mixing provided by single-start screws over the RPM and flow rates used in this study does not justify the assumption of negligible transverse temperature gradients.
Prediction of temperature profiles in twinscrew extruders
163
The one-dimensional energy equation appears to be adequate for the analysis of heat transfer in the mixing paddle and feed screw regions of extruders, particularly at larger flow rates, high RPM, or combinations of the two variables which ensure thorough mixing. For single-start screws, however, it is recommended that at least a two-dimensional form of the energy equation be used. The results of this study confirm what is generally known about extruders: at high screw speeds and low throughputs, the extruder’s capacity for convective cooling may not be sufficient to overcome the rate of viscous energy generation in situations where it is necessary to keep the final product temperature within a specific range. The authors recognize that the moisture content used in this study is impractical for industrial applications. This moisture content was selected, however, to ensure that the test material was a power law fluid over the shear rate range of this study. Due to the hygroscopic nature 01 SPS, moisture contents below 70% result in a material which deviates significantly from power law behavior. In spite of this, it is hoped that the type of information provided by studies such as this would be useful in the design, control and optimization of the extrusion cooking process.
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