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A new model framework is proposed to describe the
expansion of extrudates with extruder operating conditions
based on dimensional analysis principle. The Buckingham pi
dimensional analysis method is applied to form the basic
structure of the model from extrusion process operational
parameters. Using the Central Composite Design (CCD)
method, whole wheat flour was processed in a twin-screw
extruder with 16 trials. The proposed model can well
correlate the expansion of the 16 trials using 3 regression
parameters. The average deviation of the correlation is 5.9%.

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Hongyuan Cheng and Alan Friis

Food Production Engineering, National Food Institute

Technical University of Denmark

Søltofts Plads, Building 227, DK-2800, Lyngby, Denmark

E-mail: hyc@bio.dtu.dk

which affects the extent of biopolymer modification and

Expansion, Extrusion, whole wheat flour, modelling.

finally the expansion and texture of extruded products.

Although starch is known to be the key biopolymer in

extrusion cooking, other ingredients of cereal-based food

systems such as proteins, fat or fiber also influence the

A new model framework is proposed to describe the

system and product characteristics, e.g. by competing with

expansion of extrudates with extruder operating conditions

starch for water (Moraru and Kokini, 2003). Due to these

based on dimensional analysis principle. The Buckingham pi

compositional and structural complexities of the raw material

dimensional analysis method is applied to form the basic

as well as the large number of operational parameters

structure of the model from extrusion process operational

involved in the extrusion process, obtaining the desired

parameters. Using the Central Composite Design (CCD)

extrudate properties, e.g. optimal expansion and texture, is a

method, whole wheat flour was processed in a twin-screw

challenging task and often depends on trial and error

extruder with 16 trials. The proposed model can well

experience. The empirical experience is often valid only for

correlate the expansion of the 16 trials using 3 regression

the specific extrusion equipment that has been used to

parameters. The average deviation of the correlation is 5.9%.

generate the good operation conditions.

In practical food industrial applications, one often

needs to run some trials in a pilot extrusion process in a new

food product development stage. Through the trials, a set of

Diets with high amounts of whole grains may help

specific operating parameters can be established for the new

achieve significant weight loss, and also reduce the risk of

recipe material extrusion. The procedure sometimes is out of

chronic diseases such as diabetes and cardiovascular disease.

control in the time or raw materials cost frame. In this work,

Epidemiological studies have shown that increased intakes of

we present an engineering procedure to find out the suitable

whole grain products are associated with reduced risks of

process parameters to reach correct expansion for whole

diabetes mellitus, hypertension, and cardiovascular disease

wheat flour extrusion in a pilot plant, which includes

(Jones, 2000, Slavin, 2004, Katcher et al. 2008). However,

experimental design and process parameter correlation.

most cereal products available in Europe and the United

States are produced from highly refined grains, which lead to

the loss of many potentially beneficial micronutrients,

antioxidants,

minerals,

phytochemicals,

and

fiber.

Subsequently, the consumption of whole grain is far less than

The extrusion experiments design was setup by a

the three servings on a daily basis as suggested by the food

traditional way. With considering the extruder limitation, a

pyramid of the United States Department of Agriculture

Central Composite Design (CCD) (Esbensen, 2000) was used

(Slavin, 2004). Whole grain breakfast cereals might

in the study, which was based on five levels of three

contribute to satisfy the recommended daily intake, since

variables (Table 1). The independent extrusion variables

they fit to the increasingly fast-paced nature of consumer

considered were barrel temperature in different zones, feed

lifestyles. Breakfast cereals are convenient to prepare and

water content and screw speed. All other parameters were

easily consumed. In addition, they appeal to consumers of all

kept constant. Operating ranges and five standardized levels

income levels (Jones, 2000)

were established by preliminary study of each variable.

According to the CCD, the experimental plan comprised 15

Extrusion cooking has become a well-established

trials (8 factorial points, 6 axial points and 1 central point).

industrial technology by offering continuous and flexible

processes which allow producing breakfast cereals with

Table 1 Coded levels for the central composite design

diverse textures and shapes and ultimately reducing the costs

Variables

Levels

of the final products. The extrusion cooking process can be

analyzed in terms of operational parameters, system and

101

111

125

139

149

product characteristics. By changing the operational

9.2

10.5

12.5

14.5

15.8

parameters it is possible to influence the time-temperature-

208

245

300

355

393

shear history of the grain flour in the extruder. System

parameters such as specific mechanical energy (

generally used to describe the time-temperature-shear history,

In Table 1,

solid theoretic background and is the direction to develop a

die), °C,

model to describe the extrusion process operation. However,

w is the water content, weight percent, %,

screw speed, rpm, ? equals to 1.682.

the mechanism-based model often needs accurate food fluids

physical property correlations to support its prediction and

In the investigation, the extrudate expansion was set

estimation for the extrusion process behaviors. Because the

as the objective of the extrusion process operation. Through

food fluids belong to non-Newtonian fluid and have very

the experiments, it was expected to develop a quantitative

complicated behaviors, the development of a precise physical

correlation for extrudate expansion and extrusion operating

property calculation model for such food fluid is very

conditions. With the help of the correlation model, the

difficult.

number of runs could be reduced for a similar recipe food

extrusion using the whole wheat flour.

In this work, a dimensional analysis based model is

proposed to correlate the extrudate expansion and extruder

operating conditions. Dimensional analysis method is a

classical way in industrial applications to setup a model. In

the applications of fluid mechanics and fluid heat transfer,

Whole wheat grain was milled to obtain the whole wheat

the dimensional analysis method has obtained tremendous

flour. The whole wheat flour was processed in a Werner &

successful achievements. However, this method is seldom

Pfleiderer Continua 37 co-rotating twin-screw extruder. The

used to setup the correlation between extrudate expansion

CCD table, i.e. Table 1, was used to set up 15 experimental

and extrusion operating parameters.

runs. First, 15 trials were carried out to search the optimal

extrusion conditions for maximum expansion. After the 15

Extrudate expansion is a reflection of extrusion

trials, one more run was carried out to obtain the maximum

equipment design and process operation conditions. The

expansion. The last trial conditions were estimated from the

equipment design may include different pre-mixing methods,

response surface methodology. The trial capacity was in 22-

various screw structures, die design, etc. Because the process

27kg/hr levels. In the experimental work, the wheat flour,

operation conditions are the adjustable process control

water and additives are directly fed into extruder without pre-

parameters for an existing production line, we will only study

mixing. The extrudates were dried at 110°C for about 10

the correlation between process operation conditions and

minutes in a continuous processing oven. The extrudate bulk

extrudate expansion in this work. In an extrusion process,

density was measured during the extrusion operations using

many process parameters have influence on the extrudate

weight method for 1 liter extrudates.

expansion, e.g. different zone temperatures, die temperature

and pressure, process capacity, screw speed, torque, water

content, fluid viscosity, specific mechanical energy, etc.

Many researchers have used different methods to correlate

the process parameters with extrudate expansion (bulk

In decades, many investigations have been carried

density) and achieved their successes. In this work, the

out for the relationship between extrudate expansion and

dimensional analysis method will be applied to analysis and

operating conditions (Alvarez-Martinez, et al. 1988, Cai and

correlate these parameters.

Diosady, 1993, Moraru and Kokini, 2003). As the food

market is very volatile, food producers have to change their

Historically, the dimensional analysis methods

recipe all the time. Sometimes the maximum expansion is the

include the Rayleigh method and the Buckingham pi method

target. In other cases, mild extrusion conditions are expected

(Buckingham, 1914, Rayleigh, 1915, Perry and Green,

in order to improve the nutritional quality of products (Singh

1999). In this work, the Buckingham pi method is applied to

et al. 2007). A quantitative correlation for the extrudate

construct a model. For an extrusion process, we can find out

expansion and operating condition will significant benefit the

a set of key variables and their dimensions in the engineering

food extrusion applications.

system as below:

In these correlations and models, some are based on

empirical regression from operating conditions (Ding, et al.

2006). Others use the models from a sort of theories or

?

mechanisms (Fan and Mitchell, 1994). In the empirical

regressions, a model is often constructed with linear and

? =density, mass/volume,

quadratic terms according the statistical significant for a

Here, the units

specific extrusion process. This kind of model construction

temperature, mass, time, force and length. Among the

normally results in many regression coefficients to be

variables, ? is the bulk density of extrudates, g/liter,

B

w is

determined from experimental data. No doubt, the empirical

the water flowrate added into the extruder, kg/hr,

equation has played an important role in extrusion product

T is the

total flowrate of all feed materials (wheat flour, water and

development. However, sometimes the empirical models

additives), kg/hr,

contain many operating parameters and product properties.

d is the die temperature, °C,

temperature (25°C), which is also the raw material initial

The uncertainty of the measurement of the product properties

could be very diverse. Thus, the uncertainty can transfer into

temperature,

the empirical correlation model. The mechanism model has a

From the key extrusion process parameters, a

in Figure 2. As shown in Figure 2, the bulk density

variable and units matrix is formed as shown in Table 2.

estimation errors are evenly distributed.

Table 2 Selected variables and units matrix

Table 3 Coefficients of equation (1)

Unit

?

Coefficient

?

?

0

?

1

1

0

0

0

0

0

0

Value

13508

0.7774

-0.2882

0

0

1

1

0

0

0

1

0

0

-1

-1

-1

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

-2

1

-3

In the formation of Table 2, we assume that the fluid density

just before flash-out from die linearly proportions to the bulk

density.

Through calculation, it can be found out that the

rank of the matrix shown in Table 2 is 5. The physical

variables in Table 2 are 8. From the Buckingham pi theorem,

three dimensionless groups can be established from the

Figure 1: Correlation results of bulk density at different runs

matrix, which are given as follows:

?

,

? ?? ?

0

In the studied extrusions process, the term

the water content of processing materials. The term

the temperature changes of processing materials from its

initial condition to the vicinity of flash-out from extruder.

The term

represents the energy added into the

? ?? ?

processing materials. In fact, the term ??

widely used specific mechanical energy (SME). From the

three dimensionless groups, different model expressions can

be constructed. In this work, a model is obtained as

?

?

?

? =

Figure 2: Relative deviations of equation (1) in correlation

B

(

1)

?

?

?

? ??

?

?

?

? ?

with bulk density and extusion operating conditions

0

?

where ? is the bulk density of extrudates, g/liter,

B

w is the

In Figure 2, the relative deviation (

water content of feed material and equals to

exp

cal

?

? ?

fraction.

%

(3)

exp

be determined from experimental data.

?

From the equation (1), it can be seen that the

equation lacks of physical properties of food fluids. Thus it is

only suitable to the specific whole wheat flour extrusion.

However, the equation is simple and contains only

In engineering applications, a simple and reliable

correlation coefficients. The simple format model meets the

model often can help engineers to reach an optimal solution

engineering applications.

through observing the interactions of different operational

parameters. In this work, the model construction is based on

Using the data from the 16 runs for whole wheat

the engineering principle. A dimensional analysis method is

flour extrusion, the model coefficients are determined as

used to build a model with the key process parameters. From

shown in Table 3. The average absolute deviation (AAD) of

the model one can quantitatively estimate the extrudate bulk

the model correlation with experimental bulk density data is

density changes with different control parameters and the

5.9%, where AAD is calculated as.

interactions of these parameters.

The model estimation results show that the proposed

1

exp

cal

?

?

?

? ?

AAD =

??

?%

(2)

can successfully represent the extrudate bulk density in

exp

?

?

different extruder operating conditions. The average absolute

?

?

In equation (2),

deviation in the estimation of extrudate bulk density is 5.9%.

runs. The model correlation results for the extrudate bulk

density are shown in Figure 1. The estimation error

distribution of the model for extrudate bulk density is shown

We acknowledge Hanne T. Pedersen, Jørgen Busk

from Danish Technological Institute and Danish

Technological Institute for providing its pilot plant, raw

materials and technical support in this research work.

Alvarez-Martinez, L.; K. P. Kndury and J. M. Harper 1988. “A

general model for expansion of extruded products.”

of Food Science

Cai, W. and L. L. Diosady. 1993. “Modeling of expansion and

water solubility index of wheat starch during extrusion

cooking”

Buckingham, E. 1914. “On physically similar systems; illustrations

of the use of dimensional equations".

Ding, Q.-B.; P. Ainsworth; A. Plunkett; G. Tucker and H. Marson.

2006. “The effect of extrusion conditions on the functional

and physical properties of wheat-based expanded snacks”.

Esbensen, K.H. 2000, “Multivariate Data Analysis-in Practice” 4th

Ed. CAMO ASA, Oslo, Norway

Fan, J.; J.R. Mitchell; J. M. V. Blanshard. 1994. “A computer

simulation of the dynamics of bubble growth and shrinkage

during extrudate expansion”.

23, 337-356.

Jones, J. M., 2000. “Cereal nutrition”. In: Fast, R. B.; Caldwell, E.

F.: Breakfast Cereals and how they are made. St. Paul:

American Association of Cereal Chemists. p.411-442.

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cardiovascular disease risk factors in men and women with

metabolic syndrome”.

Nutrition

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during extrusion and microwave heating of cereal foods”.

120-138.

Perry, R.H and D. W. Green. 1999.

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Singh, S.; S. Gamlath and L. Wakeling. 2007. “Nutritional Aspects

of Food Extrusion: a Review”

Science and Technology

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