- Email: [email protected]
Cutting force of fibrous materials
Journal of Food Engineering 66 (2005) 57–61 www.elsevier.com/locate/jfoodeng
Cutting force of fibrous materials Andrzej Dowgiallo Sea Fisheries Institute, 81-332 Gdynia, Kollataja 1, Poland Received 15 July 2003; accepted 23 February 2004
Abstract A model describing the relationship between cutting force and cutting speed during cutting operation of fibrous materials was developed. The model was based on analysis of their rheological model. The experimental data, obtained during cutting different fibrous materials, were used to estimate the parameters of the model using non-linear parameter estimation. According to the developed model the unit power of the cutting operation of a given fibrous material with a knife of defined geometric parameters is a constant value. 2004 Elsevier Ltd. All rights reserved. Keywords: Cutting; Force; Model; Fibrous material
1. Introduction The known theories of size reduction (Rittinger’s, Bond’s and Kick’s laws) apply to materials with either a friable or crystalline structure which require reduction to granular or powdered forms. In general, these materials are disintegrated by crushing using compressive (e.g., crushing rolls) or impact (e.g., hammer mills) forces. They cannot, however, be applied to foodstuffs with a fibrous structure like fresh vegetables, fruits or meat, which are usually cut in order to obtain a product of a desired shape and size (slicing, dicing, shredding). In general, the authors of numerous books on food processing limit their discussion of this cutting operation to a review of the machines used to do it (for example, Earle, 1985; Fellows, 1996; Singh, 1986; Singh & Heldman, 1993; Valentas, Levine, & Peter Clark, 1991). Only Loncin and Merson (1979) report that ‘‘. . . it is not possible, at present, to describe a food material completely or to furnish the parameters needed to design a size-reduction operation. These parameters must be determined experimentally’’. Brennan, Butters, Cowell, and Lilley (1990) mention that ‘‘. . . little work has been carried out on the energy requirement for a cutting operation of fibrous materials’’.
E-mail address: [email protected] (A. Dowgiallo). 0260-8774/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2004.02.034
Thus, there is no cutting theory for fibrous materials nor is there a mathematical formula, which would allow for the calculation of cutting resistance. The goal of the current work was to use theoretical analysis and the experiments performed to develop such a theory.
2. Analysis of the cutting operation of fibrous materials A force P acts during knife cutting operations, submerging through a material at a constant speed, further referred to as the cutting speed. This force is comprised of the following: Pe is expended on elastic and plastic deformation; Pf is the force, which overcomes friction; Pd is the disintegration force expended by the cutting edge on the structure of the material. The value of force P , which is P ¼ 0 at the moment the edge comes into contact with the material, increases to a value of Pmax ¼ Pd þ Pe þ Pf , which refers to stabilized cutting conditions, and is further referred to as the cutting force Pc . Fig. 1 presents the changes in force P which were recorded by an Instron testing machine during a cutting operation. Fibrous materials are not less stiff than friable or crystalline materials. This is why a considerable part of the cutting force can be used to deform the material. When there is plastic deformation, the material can either be torn or bruised especially when blunt cutting
58
A. Dowgiallo / Journal of Food Engineering 66 (2005) 57–61
where S, the surface of the material deformation zone, c1 , the coefficient. The size of surface S decreases as resistance R of the material to deformation increases: c2 ð2Þ S¼ ; R where R, the resistance of material to deformation, c2 , the coefficient. Inserting (2) to (1) results in: c3 ð3Þ Pc ¼ ; R
Fig. 1. Changes in the value of force P acting on the knife during cutting as recorded by the Instron testing machine: zone I––knife submerges into the material; zone II––cut stabilization; zone III––knife emerges from the material.
edges are used. In order to avoid this, cutting should be conducted under conditions which ensure that the deformation of the material is minimal. Creating such conditions is dictated by the property of the material which ‘‘. . . can vary with the rate with which the stress is applied; some materials are plastic and ductile if the stress is applied slowly but can be elastic or brittle if the stress is applied as by impact’’ (Loncin & Merson, 1979). Practice has shown that this property also applies to fibrous materials. Such conditions can be described based on the analysis of a rheological model, for example the Burgers model, which is regarded as producing a good description of the behavior of organic materials (Steffe, 1992). The dashpots, which are the component of the model that imitate the behavior of intracellular fluids, describe the resistance strength dependent on the relative speed of the dashpot elements. Thus, the higher the speed versus the applied load, the greater the resistance to deformation is; the modeled material stiffens and deformation decreases. As the cutting speed is increased, the zone in which the pressure of the knife causes deformation is reduced. In effect, in order to elicit the force to destroy the material, and thus to cut it, a decreasing amount of force is necessary. In border cases, the stress can be concentrated along the theoretical line of the blade and the decreasing power of the cut decreases as the stiffness of the material increases to achieve the minimal value. This phenomenon is also supported by the slow speed of the approaching stress wave in organic materials. Based on the preceding, it can be assumed that the more the material is deformed during cutting, the greater is the force necessary to cut it. This dependency can be described by the following general function: Pc ¼ c1 S;
ð1Þ
where c3 ¼ c1 c2 , the coefficient.Keeping in mind that the relationship between the resistance to deformation and the speed of deformation is generally R ¼ c4 vn , Eq. (3) can be transformed as follows: Pc ¼
C ; vn
ð4Þ
where parameter C ¼ c3 =c4 is the constant of a given material and the knife of known geometrical parameters––thickness and blade angle. Morrow and Mohsenin (1966) stated that in agricultural products with high water content the cells behave in a nearly elastic manner and the cellular fluids represent the viscous element (dashpot) in their rheological models. Since the resistance against deformation during impact stressing depends on the first relative power speed elements of the dashpot, it can be presumed that the value of n in Eq. (4) will be n ¼ 1. Tests were conducted to verify Eq. (4) and to determine the values of coefficients C and n.
3. Materials and methods The experiment measured the cutting force on fibrous vegetable and fish materials (potatoes, beets, carrots and trout). The cuboidal samples, on a stand equipped to an Instron 1122 testing machine, were cut with a flat knife with a thickness of b ¼ 0:7 mm which was sharpened on one side (b ¼ 22). The length of the cutting line was a ¼ 20 mm. Fig. 2 presents the scheme of the experiment during which the range of speeds applied was as follows: • • • •
v 2 ½0:175; 3:75 [m s1 ]––for beets, v 2 ½0:175; 4:04 [m s1 ]––for carrots, v 2 ½0:175; 4:04 [m s1 ]––for potatoes, v 2 ½0:35; 2:5 [m s1 ]––for trout.
The slowest cutting speed produced the smoothest cut in the trout samples without any trace of tearing. The upper speed limit was that at which the decreases in the cutting force value was negligible. The lower speed limit for cutting the vegetables was established arbitrarily and the upper limit in the same way as for the trout samples.
A. Dowgiallo / Journal of Food Engineering 66 (2005) 57–61
Measurements of the cutting force were repeated from 4 to 7 times for the various speed values.
59
linear regression) in order to obtain the parameters for Eq. (4), which was transformed into: k ; ð5Þ vn where, according to Eq. (1), parameter k ¼ C=a is the constant for a given material and the knife of known geometrical parameters––thickness and blade angle. The following values were obtained from the operation:
pj ¼ 4. Results and discussion After the cutting force was converted into unit cutting force, which corresponded to the length of the cutting line ðpj ¼ Pc =aÞ, they were analysed statistically (non-
for beets : pj ¼ 0:034v0:97 ; R2 ¼ 98:96%; standard error of n ¼ 0:0230; for carrots : pj ¼ 0:039v1:00 ;
R2 ¼ 99:65%;
standard error of n ¼ 0:0154;
ð7Þ
for potatoes : pj ¼ 0:016v0:92 ; standard error of n ¼ 0:0160; for trout : pj ¼ 0:032v
0:92
;
ð6Þ
R2 ¼ 96:84%; ð8Þ 2
R ¼ 97:48%;
standard error of n ¼ 0:0323:
ð9Þ
Independent of the type of material cut, the values of parameter n are close to n ¼ 1, as anticipated. This is why Eq. (5) was modified to: pj ¼ kv1 :
The results were again analysed statistically and the following were obtained:
Fig. 2. Scheme of cutting samples.
0.1
0.01
0.001
Carrots 0.1
0.01
0.001 0.1
0.1
1 v [m.s-1]
0.1
0.1
Potatoes
1 v [m.s-1]
Trout
pj [N .mm-1]
p [N .mm-1] j
1
Beets
pj [N.mm-1]
p [N.mm-1] j
1
ð10Þ
0.01
0.001
0.01 0.1
1 v [m.s-1]
1 v [m.s-1]
Fig. 3. Plot and measurement points of the dependence pj ¼ f ðvÞ for beets, carrots, potatoes and trout.
60
A. Dowgiallo / Journal of Food Engineering 66 (2005) 57–61 Beets
Carrots 12 10
12
frequency
frequency
15
9 6 3
8 6 4 2 0 0.03
0 26
29
32
35 k
38
41 44 (x 0,001)
0.034 0.038 0.042 0.046 k Trout
12
12
10
10 frequency
frequency
Potatoes
0.05
8 6 4 2
8 6 4 2
0
0 12
14
16
18
k
20
22
28
30
(x 0,001)
32
34
k
36
38
(x 0,001)
Fig. 4. Distribution of k for beets, carrots, potatoes and trout.
Table 1 Average values of the k coefficient and statistical tests Raw material
Average value
Standard deviation
Confidence interval k for a ¼ 0:05
Beet Carrot Potato Trout
0.0331 0.0391 0.0165 0.0321
0.0023 0.0039 0.0020 0.0018
0.0331 ± 0.0045 0.0391 ± 0.0076 0.0165 ± 0.0039 0.0321 ± 0.0035
for beets : pj ¼ 0:033v1 ;
R2 ¼ 98:90%;
for carrots : pj ¼ 0:039v1 ;
R2 ¼ 99:65%;
1
2
for potatoes : pj ¼ 0:014v ; 1
for trout : pj ¼ 0:031v ;
R ¼ 95:16%; 2
R ¼ 97:27%:
ð11Þ ð12Þ ð13Þ ð14Þ
Dependencies (11)–(14) and the measurement points are presented in Fig. 3. Based on the results of these experiments, it can be assumed that, of the cases presented, formula (10), which describes the dependence between the unit cutting force and cutting speed, is true. This means that the product pj v for a given material, sized N m=mm s, and therefore W=mm, is a constant value. It describes the power necessary to cut, in a way as presented in Fig. 2, a unit length of a given material with a knife of defined geometric parameters. It is also a material indicator of k, which describes the resistance of the material to cutting. Obviously, the empirical values of the k coefficient are scattered. However, according to the results of the statistical tests (frequency histogram for k is presented in Fig. 4), we cannot reject the idea that k for beets, carrots, potatoes and trout come from a normal distribu-
tion with 90% or higher confidence. Their average standard deviations and the interval ranges, which cover 95% of the expected values, are presented in Table 1. Considering that the mechanical properties of raw materials of organic origin vary significantly, it can be stated that the k coefficient obtained during the studies quite precisely describes the unit power necessary to cut a unit length with a blade of predetermined parameters. This coefficient is also an indicator, which characterizes the raw material’s resistance to cutting. It would be prudent to emphasize again that the minimum speed has to be such that the cut layers will accelerate to the point at which their inertia and resistance to crushing or bending will be less than that to being cut. 5. Conclusions The following hypotheses can be formulated based on the results of the experiments performed: The unit power of the cutting operation of a studied fibrous material with a knife of defined geometric parameters is a constant value.
A. Dowgiallo / Journal of Food Engineering 66 (2005) 57–61
During the cutting operation it is essential that the knife exerts enough acceleration on the layers of the material so that their inertia and resistance to crushing or bending will be less than that to being cut. These hypotheses require verification on a much wider experimental scale which takes into consideration a greater number of materials and friction during cutting, the impact of the knife’s geometric parameters, the angle of the blade to the cut, etc. References Brennan, J. G., Butters, J. R., Cowell, N. D., & Lilley, A. E. V. (1990). Food engineering operations (third ed.). London: Elsevier Applied Science.
61
Earle, R. L. (1985). Unit operations in food processing (second ed.). New York: Pergamon Press. Fellows, P. J. (1996). Food processing technology. Principles and practice (second ed.). Chichester, West Sussex: Ellis Horwood. Loncin, M., & Merson, R. L. (1979). Food engineering. Principles and selected applications. New York: Academic Press. Morrow, C. T., & Mohsenin, N. N. (1966). Consideration of selected agricultural products as visco-elastic materials. Journal of Food Science, 31. Singh, R. P. (1986). Energy in food processing. New York: Elsevier. Singh, R. P., & Heldman, D. R. (1993). Introduction to food engineering. San Diego: Academic Press. Steffe, J. F. (1992). Rheological methods in food process engineering. East Lansing: Freeman Press. Valentas, Kenneth J., Levine, L., & Peter Clark, J. (1991). Food processing operations and scale-up. New York: Marcel Dekker.
Cutting force of fibrous materials Andrzej Dowgiallo Sea Fisheries Institute, 81-332 Gdynia, Kollataja 1, Poland Received 15 July 2003; accepted 23 February 2004
Abstract A model describing the relationship between cutting force and cutting speed during cutting operation of fibrous materials was developed. The model was based on analysis of their rheological model. The experimental data, obtained during cutting different fibrous materials, were used to estimate the parameters of the model using non-linear parameter estimation. According to the developed model the unit power of the cutting operation of a given fibrous material with a knife of defined geometric parameters is a constant value. 2004 Elsevier Ltd. All rights reserved. Keywords: Cutting; Force; Model; Fibrous material
1. Introduction The known theories of size reduction (Rittinger’s, Bond’s and Kick’s laws) apply to materials with either a friable or crystalline structure which require reduction to granular or powdered forms. In general, these materials are disintegrated by crushing using compressive (e.g., crushing rolls) or impact (e.g., hammer mills) forces. They cannot, however, be applied to foodstuffs with a fibrous structure like fresh vegetables, fruits or meat, which are usually cut in order to obtain a product of a desired shape and size (slicing, dicing, shredding). In general, the authors of numerous books on food processing limit their discussion of this cutting operation to a review of the machines used to do it (for example, Earle, 1985; Fellows, 1996; Singh, 1986; Singh & Heldman, 1993; Valentas, Levine, & Peter Clark, 1991). Only Loncin and Merson (1979) report that ‘‘. . . it is not possible, at present, to describe a food material completely or to furnish the parameters needed to design a size-reduction operation. These parameters must be determined experimentally’’. Brennan, Butters, Cowell, and Lilley (1990) mention that ‘‘. . . little work has been carried out on the energy requirement for a cutting operation of fibrous materials’’.
E-mail address: [email protected] (A. Dowgiallo). 0260-8774/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2004.02.034
Thus, there is no cutting theory for fibrous materials nor is there a mathematical formula, which would allow for the calculation of cutting resistance. The goal of the current work was to use theoretical analysis and the experiments performed to develop such a theory.
2. Analysis of the cutting operation of fibrous materials A force P acts during knife cutting operations, submerging through a material at a constant speed, further referred to as the cutting speed. This force is comprised of the following: Pe is expended on elastic and plastic deformation; Pf is the force, which overcomes friction; Pd is the disintegration force expended by the cutting edge on the structure of the material. The value of force P , which is P ¼ 0 at the moment the edge comes into contact with the material, increases to a value of Pmax ¼ Pd þ Pe þ Pf , which refers to stabilized cutting conditions, and is further referred to as the cutting force Pc . Fig. 1 presents the changes in force P which were recorded by an Instron testing machine during a cutting operation. Fibrous materials are not less stiff than friable or crystalline materials. This is why a considerable part of the cutting force can be used to deform the material. When there is plastic deformation, the material can either be torn or bruised especially when blunt cutting
58
A. Dowgiallo / Journal of Food Engineering 66 (2005) 57–61
where S, the surface of the material deformation zone, c1 , the coefficient. The size of surface S decreases as resistance R of the material to deformation increases: c2 ð2Þ S¼ ; R where R, the resistance of material to deformation, c2 , the coefficient. Inserting (2) to (1) results in: c3 ð3Þ Pc ¼ ; R
Fig. 1. Changes in the value of force P acting on the knife during cutting as recorded by the Instron testing machine: zone I––knife submerges into the material; zone II––cut stabilization; zone III––knife emerges from the material.
edges are used. In order to avoid this, cutting should be conducted under conditions which ensure that the deformation of the material is minimal. Creating such conditions is dictated by the property of the material which ‘‘. . . can vary with the rate with which the stress is applied; some materials are plastic and ductile if the stress is applied slowly but can be elastic or brittle if the stress is applied as by impact’’ (Loncin & Merson, 1979). Practice has shown that this property also applies to fibrous materials. Such conditions can be described based on the analysis of a rheological model, for example the Burgers model, which is regarded as producing a good description of the behavior of organic materials (Steffe, 1992). The dashpots, which are the component of the model that imitate the behavior of intracellular fluids, describe the resistance strength dependent on the relative speed of the dashpot elements. Thus, the higher the speed versus the applied load, the greater the resistance to deformation is; the modeled material stiffens and deformation decreases. As the cutting speed is increased, the zone in which the pressure of the knife causes deformation is reduced. In effect, in order to elicit the force to destroy the material, and thus to cut it, a decreasing amount of force is necessary. In border cases, the stress can be concentrated along the theoretical line of the blade and the decreasing power of the cut decreases as the stiffness of the material increases to achieve the minimal value. This phenomenon is also supported by the slow speed of the approaching stress wave in organic materials. Based on the preceding, it can be assumed that the more the material is deformed during cutting, the greater is the force necessary to cut it. This dependency can be described by the following general function: Pc ¼ c1 S;
ð1Þ
where c3 ¼ c1 c2 , the coefficient.Keeping in mind that the relationship between the resistance to deformation and the speed of deformation is generally R ¼ c4 vn , Eq. (3) can be transformed as follows: Pc ¼
C ; vn
ð4Þ
where parameter C ¼ c3 =c4 is the constant of a given material and the knife of known geometrical parameters––thickness and blade angle. Morrow and Mohsenin (1966) stated that in agricultural products with high water content the cells behave in a nearly elastic manner and the cellular fluids represent the viscous element (dashpot) in their rheological models. Since the resistance against deformation during impact stressing depends on the first relative power speed elements of the dashpot, it can be presumed that the value of n in Eq. (4) will be n ¼ 1. Tests were conducted to verify Eq. (4) and to determine the values of coefficients C and n.
3. Materials and methods The experiment measured the cutting force on fibrous vegetable and fish materials (potatoes, beets, carrots and trout). The cuboidal samples, on a stand equipped to an Instron 1122 testing machine, were cut with a flat knife with a thickness of b ¼ 0:7 mm which was sharpened on one side (b ¼ 22). The length of the cutting line was a ¼ 20 mm. Fig. 2 presents the scheme of the experiment during which the range of speeds applied was as follows: • • • •
v 2 ½0:175; 3:75 [m s1 ]––for beets, v 2 ½0:175; 4:04 [m s1 ]––for carrots, v 2 ½0:175; 4:04 [m s1 ]––for potatoes, v 2 ½0:35; 2:5 [m s1 ]––for trout.
The slowest cutting speed produced the smoothest cut in the trout samples without any trace of tearing. The upper speed limit was that at which the decreases in the cutting force value was negligible. The lower speed limit for cutting the vegetables was established arbitrarily and the upper limit in the same way as for the trout samples.
A. Dowgiallo / Journal of Food Engineering 66 (2005) 57–61
Measurements of the cutting force were repeated from 4 to 7 times for the various speed values.
59
linear regression) in order to obtain the parameters for Eq. (4), which was transformed into: k ; ð5Þ vn where, according to Eq. (1), parameter k ¼ C=a is the constant for a given material and the knife of known geometrical parameters––thickness and blade angle. The following values were obtained from the operation:
pj ¼ 4. Results and discussion After the cutting force was converted into unit cutting force, which corresponded to the length of the cutting line ðpj ¼ Pc =aÞ, they were analysed statistically (non-
for beets : pj ¼ 0:034v0:97 ; R2 ¼ 98:96%; standard error of n ¼ 0:0230; for carrots : pj ¼ 0:039v1:00 ;
R2 ¼ 99:65%;
standard error of n ¼ 0:0154;
ð7Þ
for potatoes : pj ¼ 0:016v0:92 ; standard error of n ¼ 0:0160; for trout : pj ¼ 0:032v
0:92
;
ð6Þ
R2 ¼ 96:84%; ð8Þ 2
R ¼ 97:48%;
standard error of n ¼ 0:0323:
ð9Þ
Independent of the type of material cut, the values of parameter n are close to n ¼ 1, as anticipated. This is why Eq. (5) was modified to: pj ¼ kv1 :
The results were again analysed statistically and the following were obtained:
Fig. 2. Scheme of cutting samples.
0.1
0.01
0.001
Carrots 0.1
0.01
0.001 0.1
0.1
1 v [m.s-1]
0.1
0.1
Potatoes
1 v [m.s-1]
Trout
pj [N .mm-1]
p [N .mm-1] j
1
Beets
pj [N.mm-1]
p [N.mm-1] j
1
ð10Þ
0.01
0.001
0.01 0.1
1 v [m.s-1]
1 v [m.s-1]
Fig. 3. Plot and measurement points of the dependence pj ¼ f ðvÞ for beets, carrots, potatoes and trout.
60
A. Dowgiallo / Journal of Food Engineering 66 (2005) 57–61 Beets
Carrots 12 10
12
frequency
frequency
15
9 6 3
8 6 4 2 0 0.03
0 26
29
32
35 k
38
41 44 (x 0,001)
0.034 0.038 0.042 0.046 k Trout
12
12
10
10 frequency
frequency
Potatoes
0.05
8 6 4 2
8 6 4 2
0
0 12
14
16
18
k
20
22
28
30
(x 0,001)
32
34
k
36
38
(x 0,001)
Fig. 4. Distribution of k for beets, carrots, potatoes and trout.
Table 1 Average values of the k coefficient and statistical tests Raw material
Average value
Standard deviation
Confidence interval k for a ¼ 0:05
Beet Carrot Potato Trout
0.0331 0.0391 0.0165 0.0321
0.0023 0.0039 0.0020 0.0018
0.0331 ± 0.0045 0.0391 ± 0.0076 0.0165 ± 0.0039 0.0321 ± 0.0035
for beets : pj ¼ 0:033v1 ;
R2 ¼ 98:90%;
for carrots : pj ¼ 0:039v1 ;
R2 ¼ 99:65%;
1
2
for potatoes : pj ¼ 0:014v ; 1
for trout : pj ¼ 0:031v ;
R ¼ 95:16%; 2
R ¼ 97:27%:
ð11Þ ð12Þ ð13Þ ð14Þ
Dependencies (11)–(14) and the measurement points are presented in Fig. 3. Based on the results of these experiments, it can be assumed that, of the cases presented, formula (10), which describes the dependence between the unit cutting force and cutting speed, is true. This means that the product pj v for a given material, sized N m=mm s, and therefore W=mm, is a constant value. It describes the power necessary to cut, in a way as presented in Fig. 2, a unit length of a given material with a knife of defined geometric parameters. It is also a material indicator of k, which describes the resistance of the material to cutting. Obviously, the empirical values of the k coefficient are scattered. However, according to the results of the statistical tests (frequency histogram for k is presented in Fig. 4), we cannot reject the idea that k for beets, carrots, potatoes and trout come from a normal distribu-
tion with 90% or higher confidence. Their average standard deviations and the interval ranges, which cover 95% of the expected values, are presented in Table 1. Considering that the mechanical properties of raw materials of organic origin vary significantly, it can be stated that the k coefficient obtained during the studies quite precisely describes the unit power necessary to cut a unit length with a blade of predetermined parameters. This coefficient is also an indicator, which characterizes the raw material’s resistance to cutting. It would be prudent to emphasize again that the minimum speed has to be such that the cut layers will accelerate to the point at which their inertia and resistance to crushing or bending will be less than that to being cut. 5. Conclusions The following hypotheses can be formulated based on the results of the experiments performed: The unit power of the cutting operation of a studied fibrous material with a knife of defined geometric parameters is a constant value.
A. Dowgiallo / Journal of Food Engineering 66 (2005) 57–61
During the cutting operation it is essential that the knife exerts enough acceleration on the layers of the material so that their inertia and resistance to crushing or bending will be less than that to being cut. These hypotheses require verification on a much wider experimental scale which takes into consideration a greater number of materials and friction during cutting, the impact of the knife’s geometric parameters, the angle of the blade to the cut, etc. References Brennan, J. G., Butters, J. R., Cowell, N. D., & Lilley, A. E. V. (1990). Food engineering operations (third ed.). London: Elsevier Applied Science.
61
Earle, R. L. (1985). Unit operations in food processing (second ed.). New York: Pergamon Press. Fellows, P. J. (1996). Food processing technology. Principles and practice (second ed.). Chichester, West Sussex: Ellis Horwood. Loncin, M., & Merson, R. L. (1979). Food engineering. Principles and selected applications. New York: Academic Press. Morrow, C. T., & Mohsenin, N. N. (1966). Consideration of selected agricultural products as visco-elastic materials. Journal of Food Science, 31. Singh, R. P. (1986). Energy in food processing. New York: Elsevier. Singh, R. P., & Heldman, D. R. (1993). Introduction to food engineering. San Diego: Academic Press. Steffe, J. F. (1992). Rheological methods in food process engineering. East Lansing: Freeman Press. Valentas, Kenneth J., Levine, L., & Peter Clark, J. (1991). Food processing operations and scale-up. New York: Marcel Dekker.