Mathematical modeling of whey foam mat drying

This study aimed to fit mathematical models to whey foam drying processes in different air temperature conditions, to determine the effective diffusion coefficient and to obtain the activation energy. For foam formation, 5.0% Emustab^® was added to the serum in mass and subjected to stirring in a domestic shaker for 15 min. After, it was spread onto trays to form a thin layer of about 1 cm in which the drying conditions were: 40, 45, 50, 55 and 60 °C, 5.6 m s^-1 and 60%. The Logarithm and Midilli models describe the kinetics of serum foam drying; the effective diffusion coefficient increased with the elevation of drying temperature and activation energy of 50.84 kJ mol^-1.

Keywords:
activation energy; critical moisture content; drying rate; foam

Authorship

Fernanda Machado Baptestini ^**Corresponding author: fernanda.baptestini@ufes.br

Universidade Federal do Espírito Santo, Departamento de Engenharia Rural, Alegre, Espírito Santo, Brazil. fernanda.baptestini@ufes.brUniversidade Federal do Espírito SantoBrazilAlegre, Espírito Santo, Brazil Universidade Federal do Espírito Santo, Departamento de Engenharia Rural, Alegre, Espírito Santo, Brazil. fernanda.baptestini@ufes.br

http://orcid.org/0000-0002-9505-6220

Paulo Cesar Corrêa

Universidade Federal de Viçosa, Departamento de Engenharia Agrícola e Ambiental, Viçosa, Minas Gerais, Brazil. copace@ufv.br; juliana.zeymer@ufv.br; jdbustosv@misena.coUniversidade Federal de ViçosaBrazilViçosa, Minas Gerais, Brazil Universidade Federal de Viçosa, Departamento de Engenharia Agrícola e Ambiental, Viçosa, Minas Gerais, Brazil. copace@ufv.br; juliana.zeymer@ufv.br; jdbustosv@misena.co

Juliana Soares Zeymer

Rildo Araújo Leite

Instituto Federal do Norte de Minas Gerais, Arinos, Minas Gerais, Brazil. rildo.leite@ifnmg.edu.brInstituto Federal do Norte de Minas GeraisBrazilArinos, Minas Gerais, Brazil Instituto Federal do Norte de Minas Gerais, Arinos, Minas Gerais, Brazil. rildo.leite@ifnmg.edu.br

Jaime Daniel Bustos-Vanegas

Gheila Corrêa Ferres Baptestini

Instituto Federal de Espírito Santo, Santa Maria de Jetibá, Espírito Santo, Brazil. gheilacf@yahoo.com.brInstituto Federal de Espírito SantoBrazilSanta Maria de Jetibá, Espírito Santo, Brazil Instituto Federal de Espírito Santo, Santa Maria de Jetibá, Espírito Santo, Brazil. gheilacf@yahoo.com.br

*Corresponding author: fernanda.baptestini@ufes.br

This work is part of the first author Post-Doctoral report.

The authors declare that there is no conflict of interest in the conduction and publication of the work

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (4)
Tables (5)
Formulas (8)

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Figure 1:
Whey foam drying curves at 40, 45, 50, 55 and 60 °C.

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Figure 2:
Drying rate curves of whey foam at 40, 45, 50, 55 and 60 °C.

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Figure 3:
Moisture ratio curves of whey foam drying at 40, 45, 50, 55 and 60 °C

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Figure 4:
Representation of Ahrrenius.

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Table 1:
Mathematical models used to model the whey foam drying

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Table 2:
Relative and estimated mean error, coefficient of determination and residual distribution of model adequacy given by Equation 1 for 40, 45, 50, 55 and 60 °C

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Table 3:
Critical moisture content and critical time

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Table 4:
Relative and estimated mean error, coefficients of determination, residue distribution and parameter estimation of whey foam drying curves at 40, 45, 50, 55 and 60 °C

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Table 5:
Effective diffusion coefficient (m² s^-1) of whey foam dried at 40, 45, 50, 55 and 60 °C, regression equation as a function of the temperature (T) and its coefficient of determination

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Figure 1: Whey foam drying curves at 40, 45, 50, 55 and 60 °C.

Figure 2: Drying rate curves of whey foam at 40, 45, 50, 55 and 60 °C.

Figure 3: Moisture ratio curves of whey foam drying at 40, 45, 50, 55 and 60 °C

Figure 4: Representation of Ahrrenius.

Table 1: Mathematical models used to model the whey foam drying

Model	Model designation
$M R = a \exp (- k t) + (1 - a) \exp (- k a t)$	Exponential of two terms (*Sharaf-Eldee et al., 1980*Sharaf-Eldeen YI, Blaisdell JL, Hamdy MY (1980) A model for ear corn drying. Transactions of the American Society of Agricultural Engineers, 23:1261-1265.)	(7)
$M R = a \exp (- k t)$	Henderson and Pabis (Henderson & Pabis, 1961Henderson SM & Pabis S (1961) Grain drying theory: temperature effect on drying coefficient. Journal of Agricultural Engineering Research, 6:169-174.)	(8)
$M R = a \exp (- k t) + b$	Logarithmic (*Yagcioglu et al., 1999*Yagcioglu A, Degirmencioglu A & Cagatay F (1999) Drying characteristics of laurel leaves under different conditions. In: 7th International Congress on Agricultural Mechanization and Energy, Adana. Proceedings, Cukurova University. p. 565-569)	(9)
$M R = a \exp (- k t) + b t$	Midilli (Midilli, 2002Midilli A, Kucuk H & Yapar Z (2002) A New model for single layer drying. Drying Technology, 20:1503-1513.)	(10)
$M R = \exp (- k t)$	Newton (Lewis, 1921Lewis WK (1921) The drying of solid materials. Journal of Industrial & Engineering Chemistry, 13:427-433.)	(11)
$M R = \exp (- k t^{n})$	Page (Page, 1949Page GE (1949) Factors influencing the maximum rates of air drying shelled corn in thin layers. West Lafayette, Purdue University. 44p.)	(12)

Where: t - drying time, min; k - drying rate, min^-1; a, b, n - model parameters, dimensionless

Table 2: Relative and estimated mean error, coefficient of determination and residual distribution of model adequacy given by Equation 1 for 40, 45, 50, 55 and 60 °C

Temperature (°C)	MRE (%)	EME (decimal d.b.)	R² (%)	RD
40	7.992	0.547	98.60	B
45	5.185	0.251	99.61	B
50	7.289	0.151	99.83	B
55	7.286	0.241	99.63	B
60	4.905	0.076	99.97	B

Where, R - random; B - biased.

Table 3: Critical moisture content and critical time

Temperature (°C)	U_c (kg_w kg_db ^-1)	t_c (min)
40	2.123	1050
45	2.152	750
50	2.051	480
55	2.139	375
60	2.244	300

Table 4: Relative and estimated mean error, coefficients of determination, residue distribution and parameter estimation of whey foam drying curves at 40, 45, 50, 55 and 60 °C

40 °C
Model	MRE (%)	EME (decimal d.b.)	R² (%)	RD	Estimation of parameters
7	14.626	0.027	99.39	B	a = 1.829; k = 0.010
8	27.940	0.049	99.75	B	a = 1.049; k = 0.007
9	5.309	0.013	99.76	R	a = 1.412; k = 0.004; b = -0.410
10	3.863	0.013	99.89	R	a = 0.992; k = 0.003; n = 1.126; b = -0.0004
11	30.987	0.051	99.67	B	k = 0.975
12	11.284	0.024	99.50	B	k = 0.001; n = 1.312
45 °C
Model	MRE (%)	EME (decimal d.b.)	R² (%)	RD	Estimation of parameters
7	10.130	0.017	99.80	B	a = 1.768; k = 0.016
8	21.166	0.039	98.92	B	a = 1.043; k = 0.012
9	3.632	0.008	99.96	R	a = 1.261; k = 0.008; b = -0.251
10	1.534	0.004	99.99	R	a = 0.999; k = 0.006; n = 1.112; b = -0.0004
11	24.311	0.042	98.57	B	k = 0.011
12	8.994	0.016	99.82	B	k = 0.004; n = 1.232
50 °C
Model	MRE (%)	EME (decimal d.b.)	R² (%)	RD	Estimation of parameters
7	11.190	0.023	99.55	R	a = 1.819; k = 0.019
8	23.875	0.044	98.25	B	a = 1.052; k = 0.014
9	6.186	0.017	99.77	R	a = 1.304; k = 0.008; b = -0.294
10	4.620	0.016	99.83	R	a = 0.994; k = 0.006; n = 1.159; b = -0.001
11	27.111	0.047	97.83	B	k = 0.013
12	8.917	0.021	99.62	R	k = 0.004; n = 1.287
55 °C
Model	MRE (%)	EME (decimal d.b.)	R² (%)	RD	Estimation of parameters
7	13.287	0.024	99.54	B	a = 1.833; k = 0.024
8	28.480	0.048	98.13	B	a = 1.048; k = 0.017
9	8.370	0.018	99.77	R	a = 1.302; k = 0.011; b = -0.293
10	5.068	0.016	99.85	R	a = 0.993; k = 0.007; n = 1.185; b = -0.001
11	31.770	0.050	97.76	B	k = 0.017
12	9.334	0.021	99.64	B	k = 0.005; n = 1.309
60 °C
Model	MRE (%)	EME (decimal d.b.)	R² (%)	RD	Estimation of parameters
7	7.586	0.019	99.77	R	a = 1.865; k = 0.019
8	20.372	0.052	98.26	B	a = 1.036; k = 0.013
9	4.654	0.015	99.89	R	a = 1.405; k = 0.007; b = -0.398
10	1.487	0.009	99.85	R	a = 0.999; k = 0.005; n=1.196 b = -0.00005
11	21.845	0.050	98.00	B	k = 0.013
12	5.116	0.015	99.86	R	k = 0.003; n = 1.329

Where, R: random; B: biased.

Table 5: Effective diffusion coefficient (m² s^-1) of whey foam dried at 40, 45, 50, 55 and 60 °C, regression equation as a function of the temperature (T) and its coefficient of determination

Temperature (°C)
40	45	50	55	60
2.751x10^-10	3.394x10^-10	5.277x10^-10	6.616x10^-10	8.525x10^-10
$D_{e f} = - 9 \times 1 0^{- 10} + 3 \times 1 0^{- 11} \times T$			R² = 97.99%

M R = a \exp (- k t) + (1 - a) \exp (- k a t)

M R = a \exp (- k t)

M R = a \exp (- k t) + b

M R = a \exp (- k t) + b t

M R = \exp (- k t)

M R = \exp (- k t^{n})

D_{e f} = - 9 \times 1 0^{- 10} + 3 \times 1 0^{- 11} \times T

D_{e f} = 8.15 \times 1 0^{- 2} \exp (\frac{50840.11}{R T_{a}})

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