Modeling the Equilibrium Moisture Sorption Data for Some Nigerian Foods

Abstract

The water sorption isotherms of thirty different Nigerian food products at various temperatures were obtained from literature. The collected data were divided into three groups namely cassava-based products, grains, and others. The data were used to test the applicability of nine sorption models using three statistical parameters as the criteria. The GAB model fits most of the data well when either a mean percentage deviation or standard error estimate was used as the criteria. With the coefficient of determination as the criterion, the Exponential, Bradley, and Smith models were the best for cassava-based products, grains, and others respectively.

Keywords:

• Water sorption
• Sorption data
• Sorption models
• Nigerian foods

Introduction

Nigeria is a nation with many agricultural products. Such products include rice, sorghum, wheat, plantains, yams, okra, and cassava. Most of these products are available on a seasonal basis and normally require a short time to mature. They are, therefore, abundant during harvest with the supply tapering off as the next harvest approaches. Consequently, a good percentage of these farm products spoil during the abundant period. It should be noted that cassava alone provides the major source of dietary calories; it is used as a cheap source of carbohydrate food for man and livestock.[1] For human consumption, it can either be cooked or eaten raw as a vegetable or it can be grated and roasted into flour and eaten as gari. A fundamental property of biological material, which influences dehydration, shelf life prediction, and storage stability, is the water sorption characteristics.[2]

Many authors have presented the sorption isotherms of foods that can be found in Nigeria. While some of these authors have modeled their data, others have not. The authors have also used different criteria to assess the suitability of the model to fit their sorption data. The main objectives of this work are to use a set of nine models to predict the sorption isotherm for some Nigerian foods and to use three different criteria to assess their suitability. The sorption data have been obtained from literature.

Data Sources and Analysis

Water Sorption Data

A total of ninety-one sorption isotherm data sets for different Nigerian foods have been obtained from literature. The sources are summarized in Table 1. The collected data are divided into three groups: cassava based, grains, and others. With the exception of Onayemi and Oluwamukomi[3] who presented the sorption data in tabular form, the data are in graphical form. The actual values used for this work are extracted from these graphs. A minimum of six pairs of equilibrium moisture content (m) and water activity (a _w ) were extracted from each graph. The ranges for m and a _w for each product are also included in Table 1; the actual values are shown in Table 2.

Table 1 Sources and range of the various sorption data*

S/N	Food products	Temperature (°C)	a w range	m range	Source
Cassava based
1	Cassava chips	25, 45, 60	0.16–0.88	5.9–19.7	Salgado Cerventes et al.[4]
2	Fufu	25, 32, 45	0.11–0.96	2.8–14.4	Sanni et al.[2]
3	Fufu (cabinet dried)	25, 32, 45	0.11–0.96	3.6–13.8	Sanni et al.[5]
4	Fufu (rotary dried)	25, 32, 45	0.11–0.96	3.8–14.6	Sanni et al.[5]
5	Fufu (sun dried)	25, 32, 45	0.11–0.96	3.6–14	Sanni et al.[5]
6	Gari	25, 32	0.13–0.73	4.5–11.2	Onayemi and Oluwamukomi[3]
7	Lafun	25, 32, 45	0.11–0.96	2.9–14.2	Kuye and Sanni[6]
8	Soy fufu	25, 32, 45	0.11–0.96	3.3–14.8	Sanni and Kuye[7]
9	Soy lafu	25, 32, 45	0.11–0.96	3.8–15.1	Sanni and Kuye[7]
10	Tapioca	25, 32, 45	0.11–0.96	1.7–14.6	Sanni et al.[2]
Grains
11	Maize adsorption	20, 25, 40	0.1–0.98	4–24.5	Sopade and Ajisegiri[8]
12	Maize desorption	20, 25, 40	0.1–0.98	5.6–24.4	Sopade and Ajisegiri[8]
13	Millet dough A	25, 35	0.1–0.84	2–34.17	Jideani et al.[9]
14	Millet dough B	25, 35	0.13–0.85	5.5–33.75	Jideani et al.[9]
15	Millet dough C	25, 35	0.13–0.85	1.4–17.5	Jideani et al.[9]
16	Rice	40, 50, 60, 70	0.04–0.81	2.5–17.1	Aguerre et al.[10]
17	Sorghum adsorption	20, 25, 40	0.1–0.98	4–24	Sopade and Ajisegiri[8]
18	Sorghum desorption	20, 25, 40	0.1–0.98	5.6–24.5	Sopade and Ajisegiri[8]
19	Wheat	45, 60	0.1–0.86	2.2–23.3
20	Wheat paste	40, 50, 60, 70, 80, 90	0.1–0.93	0.6–26.3	Andrieu et al.[11]
Others
21	Kuka adsorption	34, 37, 45	0.1–0.98	2.31–47	Ajisegiri et al.[12]
22	Kuka desorption	34, 37, 45	0.1–0.98	3.28–47	Ajisegiri et al.[12]
23	Okra	20, 28, 35, 50	0.11-0.95	6.4–83.3	Ihekoronye and Ngoddy[13]
24	Plantain (ripe)	40, 50, 60, 70	0.1–0.7	5–35	Ajibola[14]
25	Plantain (unripe)	40, 50, 60, 70	0.1–0.89	1.2–29.8	Ajibola[14]
26	Red pepper	15, 25, 40	0.11–0.95	6–36	Kim et al.[15]
27	Soyabean flour	25, 32, 45	0.11–0.96	4.7–14.98	Kuye and Sanni[6]
28	Yam flour	25, 32	0.125–0.73	4.6–12.5	Onayemi and Oluwamukomi[3]
29	Yam flour (elubo)	25, 32	0.125–0.73	4–11.1	Onayemi and Oluwamukomi[3]
30	Yogurt	20, 35, 50	0.112–0.85	1–20.43	Kim and Bhowmik[16]

*Except where otherwise stated, the data are for desorption isotherms.

Table 2 Raw sorption data for the different products

a w	m	a w	m	a w	m	a w	m	a w	m	a w	m
Cassava chips						Fufu
25°C		45°C		60°C		25°C		32°C		45°C
0.16	7	0.16	6.5	0.16	5.9	0.11	4.9	0.11	3.7	0.11	2.8
0.26	8.75	0.26	8.25	0.27	7.5	0.23	5.7	0.23	4.9	0.23	3.8
0.35	9.25	0.35	9.25	0.37	8.5	0.35	6.4	0.35	5.6	0.35	4.6
0.46	10.71	0.46	10.36	0.47	9.75	0.46	7	0.46	6.3	0.46	5.4
0.57	13	0.57	11.78	0.57	11.07	0.55	7.8	0.55	7	0.55	6.2
0.65	14.5	0.65	12.5	0.67	11.78	0.64	8.7	0.64	7.8	0.64	6.9
0.75	16.56	0.75	14.05	0.75	13.25	0.75	10	0.75	9	0.75	8
0.79	17.81	0.79	15.55	0.82	14.5	0.86	12.5	0.86	12	0.86	9.6
0.85	19.68	0.86	16.81	0.88	16.11	0.96	14.4	0.96	13.6	0.96	11.2
Fufu (cabinet dried) at 25°C						Fufu (rotary dried) at 25°C
25°C		32°C		45°C		25°C		32°C		45°C
0.11	5	0.11	4.4	0.11	3.6	0.11	5	0.11	4.6	0.11	3.8
0.23	5.8	0.23	5.2	0.23	4	0.23	6	0.23	5.6	0.23	4.6
0.35	6.2	0.35	5.4	0.35	5	0.35	6.4	0.35	6	0.35	5.4
0.46	7.2	0.46	6.6	0.46	6.2	0.46	7.4	0.46	7	0.46	6.6
0.55	8.2	0.55	7.4	0.55	6.8	0.55	8.6	0.55	8	0.55	7
0.64	9	0.64	8	0.64	7.6	0.64	9.6	0.64	8.6	0.64	7.8
0.75	9.8	0.75	9.2	0.75	8	0.75	10.2	0.75	9.6	0.75	8.8
0.86	12.8	0.86	12	0.86	11.2	0.86	13	0.86	12.2	0.86	11.6
0.96	13.8	0.96	13.6	0.96	13.4	0.96	14.6	0.96	14.2	0.96	13.8
Fufu (sun dried)						Gari
25°C		32°C		45°C		25°C		32°C
0.11	4.6	0.11	3.6	0.11	3.6	0.13	5	0.13	4.5
0.23	5.4	0.23	4.8	0.23	4.2	0.22	5.8	0.22	5.1
0.35	6	0.35	5.8	0.35	4.8	0.36	6.5	0.36	6.2
0.46	7.2	0.46	6.8	0.46	6	0.44	7.5	0.44	6.2
0.55	8	0.55	7.2	0.55	6.6	0.5	7.9	0.5	7.7
0.64	8.6	0.64	7.8	0.64	7.2	0.63	9.5	0.63	9.1
0.75	9.4	0.75	9	0.75	8	0.73	11.22	0.73	10
0.86	12.4	0.86	11.8	0.86	11
0.96	14	0.96	13.6	0.96	13.2
Lafun						Soy fufu
25°C		32°C		45°C		25°C		32°C		45°C
0.11	4.82	0.11	3.66	0.11	2.91	0.11	5.66	0.11	4.83	0.11	3.25
0.23	5.64	0.23	4.88	0.23	3.81	0.23	6.58	0.23	5.41	0.23	4.73
0.35	6.4	0.35	5.72	0.35	4.6	0.35	7.31	0.35	6.51	0.35	5.38
0.46	7.01	0.46	6.3	0.46	5.4	0.46	8.18	0.46	7.15	0.46	6.23
0.55	7.85	0.55	7.13	0.55	6.24	0.55	8.94	0.55	8.23	0.55	7.45
0.64	8.89	0.64	7.83	0.64	7	0.64	9.64	0.64	9.38	0.64	8.18
0.75	10.01	0.75	9	0.75	8	0.75	11.22	0.75	10.14	0.75	9
0.86	12.5	0.86	12.1	0.86	9.6	0.86	13.54	0.86	11.31	0.86	10.11
0.96	14.23	0.96	13.6	0.96	11.2	0.96	14.81	0.96	12.67	0.96	11
Soy lafu						Tapioca
25°C		32°C		45°C		25°C		32°C		45°C
0.11	5.91	0.11	5.1	0.11	3.8	0.11	3.6	0.11	2.3	0.11	1.7
0.23	6.83	0.23	5.91	0.23	4.68	0.23	6	0.23	4.2	0.23	3.1
0.35	7.34	0.35	6.75	0.35	5.66	0.35	7	0.35	5.8	0.35	4.2
0.46	8.3	0.46	7.41	0.46	6.6	0.46	7.7	0.46	6.7	0.46	5.1
0.55	9.09	0.55	8.62	0.55	7.85	0.55	8.2	0.55	7	0.55	5.4
0.64	9.83	0.64	9.7	0.64	8.61	0.64	8.8	0.64	7.7	0.64	5.8
0.75	11.41	0.75	10.53	0.75	9.84	0.75	9.8	0.75	8.6	0.75	6.8
0.86	13.78	0.86	11.8	0.86	10.71	0.86	12	0.86	10.3	0.86	8.3
0.96	15.05	0.96	13.24	0.96	11.83	0.96	14.6	0.96	12.4	0.96	9.8
Maize adsorption						Maize desorption
20°C		25°C		40°C		20°C		25°C		40°C
0.1	5.56	0.1	5.2	0.1	4	0.1	6.7	0.1	6.3	0.1	5.55
0.2	7.5	0.2	6.8	0.2	5.63	0.2	8.71	0.2	8.4	0.2	7.75
0.3	8.5	0.3	8	0.3	7	0.3	10.1	0.3	9.75	0.3	9.17
0.4	9.65	0.4	9.5	0.4	8.4	0.4	11.38	0.4	11.38	0.4	10.75
0.5	11	0.5	10.7	0.5	9.63	0.5	12.93	0.5	12.6	0.5	12.2
0.6	12.5	0.6	11.38	0.6	11	0.6	15.68	0.6	14.45	0.6	13.85
0.7	14	0.7	13.39	0.7	13	0.7	16.48	0.7	16.48	0.7	15
0.8	15.75	0.8	14.4	0.8	14.9	0.8	17.8	0.8	17.8	0.8	17.45
0.9	19	0.9	18.5	0.9	17.8	0.9	18.75	0.9	18.75	0.9	18.4
0.98	24.5	0.98	24.5	0.98	23.7	0.98	24.35	0.98	24.35	0.98	23.5
Mllet dough A				Mllet dough B				Mllet dough C
25°C		35°C		25°C		35°C		25°C		35°C
0.1	5	0.1	2	0.13	7.5	0.13	5.5	0.13	2.73	0.13	1.4
0.22	7.5	0.22	4	0.22	9.28	0.22	7.2	0.22	4.3	0.22	2
0.31	8.33	0.31	5.03	0.33	10.7	0.33	8.57	0.33	5.4	0.33	2.8
0.42	10	0.42	7.5	0.43	13.5	0.43	10	0.43	6.4	0.43	4
0.52	10	0.52	8.33	0.57	17.5	0.57	14.25	0.57	8.6	0.57	4.5
0.75	16.11	0.75	15	0.75	26	0.75	22.5	0.75	12.5	0.75	12.5
0.84	23	0.84	34.17	0.85	33.75	0.85	26	0.85	17.5	0.85	9.5
Rice								Wheat
40°C		50°C		60°C		70°C		45°C		60°C
0.19	6.25	0.11	3.33	0.04	1.25	0.17	2.5	0.1	7.1	0.06	2.2
0.22	6.88	0.31	6.88	0.11	2.86	0.2	3.21	0.26	9.8	0.11	3
0.31	7.7	0.51	9.6	0.16	3.54	0.23	3.21	0.44	12	0.19	4
0.34	8.2	0.64	11.42	0.2	4.16	0.29	3.32	0.56	13.8	0.28	5.42
0.52	11.87	0.74	13.75	0.25	4.64	0.49	6.13	0.7	16	0.38	6.67
0.66	13.75	0.82	14.64	0.3	5	0.63	7.5	0.83	19	0.48	8.07
0.73	15	0.49	7.5	0.74	9.64	0.56	10
0.81	17.08	0.63	9	0.8	10.25	0.66	11.5
0.74	10.83	0.75	14
0.81	12.77	0.79	17.5
0.82	19.4
0.86	23.3
Sorgum adsorption						Sorgum desorption
20°C		25°C		40°C		20°C		25°C		40°C
0.1	6	0.1	5.65	0.1	4	0.1	6.8	0.1	6.73	0.1	5.6
0.2	7.5	0.2	7	0.2	6.1	0.2	8.62	0.2	8.4	0.2	7.7
0.3	8.9	0.3	8.5	0.3	7	0.3	10	0.3	9.92	0.3	9.2
0.4	10	0.4	9.88	0.4	8.5	0.4	11.67	0.4	11.67	0.4	10.9
0.5	11.5	0.5	11	0.5	10	0.5	13.01	0.5	12.86	0.5	12.4
0.6	12.75	0.6	12.3	0.6	11.5	0.6	14.88	0.6	14.47	0.6	14
0.7	14	0.7	13.8	0.7	13	0.7	17.25	0.7	17.25	0.7	15.7
0.8	14.9	0.8	15	0.8	14.6	0.8	17.85	0.8	17.36	0.8	17.2
0.9	18.3	0.9	18.3	0.9	17.45	0.9	18.9	0.9	18.9	0.9	18.45
0.98	24	0.98	24	0.98	23.4	0.98	24.5	0.98	24.5	0.98	23.8
Wheat paste
40°C		50°C		60°C		70°C		80°C		90°C
0.04	4.44	0.06	3.5	0.06	2.2	0.06	1.5	0.06	1	0.04	0.63
0.1	6	0.1	4.44	0.11	3	0.13	2.2	0.13	1.5	0.16	1.5
0.17	6.88	0.17	5.42	0.19	4	0.22	3.5	0.22	2.2	0.23	2.1
0.28	7.5	0.28	6.67	0.28	5.42	0.3	4.44	0.31	3.4	0.32	3
0.38	9.5	0.38	7.92	0.38	6.67	0.4	5.8	0.41	5	0.43	4.42
0.47	10.5	0.47	9.58	0.48	8.07	0.51	8.13	0.51	7.14	0.52	6.65
0.57	12.14	0.56	11.11	0.56	10	0.6	9.58	0.61	9.5	0.6	8.13
0.67	13.75	0.66	13	0.66	11.5	0.68	11.1	0.69	10.5	0.71	10.5
0.74	16	0.74	15	0.75	14	0.77	13.75	0.77	13	0.77	12.14
0.88	26.25	0.79	18.75	0.79	17.5	0.82	17.5	0.82	16.1	0.82	15
0.86	23.75	0.82	19.4	0.93	19.4	0.93	22.5	0.93	21.7
0.86	23.3
Kuka adsorption						Kuka desorption
34°C		37°C		45°C		34°C		37°C		45°C
0.1	4.5	0.1	2.5	0.1	2.31	0.1	4.5	0.1	3.3	0.1	3.28
0.22	5	0.22	4	0.22	2.31	0.22	8	0.22	7.5	0.22	5
0.32	6.83	0.32	5	0.32	4	0.32	10	0.32	9.97	0.32	6.25
0.42	8.16	0.42	5.8	0.42	5	0.42	11.8	0.42	11	0.42	7.5
0.52	9.16	0.52	7.5	0.52	6.25	0.52	14.54	0.52	12.6	0.52	10
0.63	10.83	0.63	11	0.63	7.6	0.63	17.27	0.63	16.33	0.63	12.5
0.75	15	0.75	13	0.75	11.25	0.75	20	0.75	20	0.75	16.25
0.8	18.9	0.8	18.57	0.8	15	0.8	25	0.8	23.32	0.8	20.71
0.92	28	0.92	25.63	0.92	21.43	0.92	28	0.92	28.75	0.92	23.57
0.98	47	0.97	47	0.97	46.5	0.98	47	0.97	47	0.97	46.5
Plantain (ripe)								Yam flour (elubo)
40°C		50°C		60°C		70°C		25°C		32°C
0.12	10	0.11	7.5	0.1	6	0.1	5	0.13	4.5	0.13	4
0.22	11	0.21	9	0.21	7.8	0.2	7	0.22	5.2	0.22	5
0.32	12.55	0.31	11	0.31	9	0.3	9.5	0.36	6.5	0.36	6.3
0.48	14.37	0.47	12.55	0.5	11	0.5	9.7	0.44	8.5	0.44	8.1
0.61	26.43	0.6	20	0.6	21	0.6	15.23	0.57	9	0.57	8.3
0.7	35	0.7	27.86	0.7	22.86	0.69	16.01	0.63	9.5	0.63	9
0.73	11.1	0.73	10.3
Plantain (unripe)								Yam flour
40°C		50°C		60°C		70°C		25°C		32°C
0.12	4.96	0.07	1.25	0.12	4.95	0.07	1.18	0.13	4.8	0.13	4.6
0.22	8	0.22	7	0.22	4.95	0.2	4.95	0.22	5.8	0.22	5
0.32	10	0.32	9	0.31	7	0.31	6	0.36	6.2	0.36	6
0.52	10.25	0.47	10	0.5	9	0.5	7	0.44	7.6	0.44	7
0.6	11.88	0.6	11	0.6	10	0.6	9	0.57	8.6	0.57	8.2
0.71	17.8	0.74	15.08	0.69	13	0.69	11	0.63	8.9	0.63	8.5
0.89	29.8	0.89	26.8	0.88	22.95	0.77	13.5	0.73	12.5	0.73	11.6
Red pepper						Soybean flour
15°C		25°C		40°C		25°C		32°C		45°C
0.11	7.5	0.11	7	0.11	6	0.11	6.85	0.11	5.81	0.11	4.74
0.23	9	0.23	8	0.23	7	0.23	7.19	0.23	6.11	0.23	5.68
0.32	10.1	0.32	9	0.32	8	0.35	8.31	0.35	7.25	0.35	6.71
0.44	12	0.44	12	0.44	9	0.46	9	0.46	8.15	0.46	7.85
0.58	16	0.58	15	0.58	11.5	0.55	10.58	0.55	9.11	0.55	8.93
0.68	18	0.68	17	0.68	14	0.64	11.42	0.64	10	0.64	9.95
0.77	21	0.77	19.5	0.77	16	0.75	12.34	0.75	11.42	0.75	11
0.95	36	0.95	33	0.95	27	0.86	13.9	0.86	12.83	0.86	12.1
0.96	14.98	0.96	13.9	0.96	13.23
Okra
20°C		28°C		35°C		50°C
0.1	10.76	0.1	9.09	0.1	8.18	0.1	6.36
0.23	17.69	0.23	15.38	0.23	13.08	0.23	10.77
0.33	20.66	0.33	18.46	0.33	16.15	0.33	13.08
0.48	26	0.48	22.67	0.48	20	0.48	15.38
0.58	32	0.58	28.67	0.58	24.67	0.58	16.15
0.76	50	0.76	45.71	0.76	39.33	0.76	27.33
0.95	83.3	0.95	80	0.95	74	0.95	60
Yogurt
20°C		35°C		50°C
0.11	2.5	0.11	2	0.11	1
0.24	3.13	0.22	2.4	0.18	2
0.33	3.75	0.32	2.5	0.3	2.4
0.45	5	0.44	3.75	0.42	2.5
0.58	7.5	0.57	5	0.57	3.78
0.68	10	0.63	6.25	0.75	8.75
0.75	14.3	0.74	10.7	0.81	14.3
0.85	20.43	0.83	18	0.83	18

Sorption Models

Available models are either dependent or independent of temperature. Those that are independent of temperature, in general, give a better fit for the sorption data at any given temperature. They are either linear or can easily be linearized. For this work, we have used nine such models to fit the extracted sorption data. The linear forms of these models are shown in Table 3 where m _o is the monolayer moisture content; c and k are related to the monolayer and multilayer properties respectively; A and B are empirical constants; m _o , c, and k must be positive to have any physical meaning.[4] Also, k and c are both temperature dependent and are related to the interaction energy between water and food and the interaction energy between the multiple layers of water, respectively.[15]

Table 3 Linear forms of selected sorption models

Model	Linear form
BET[17]
Bradley[18]	ln(−ln a w ) = ln(−A) + m ln B
Caurie[19]	ln m = A − 100/(B a w )
Exponential[20]	ln m = ln A + B a w
GAB21–23
Iglesias and Chirife[24]	ln (m + (m ^2 + m 0.5)^0.5) = A + B a w
Kuhn[25]	m = A/ln a w  + B
Oswin[26]	ln m = ln A + B ln[a w /(1 − a w )]
Smith[27]	m = A − B ln(1 − a w )

As can be seen from Table 3, the GAB and Iglesias-Chirife models are three-parameter equations while the others are two-parameter equations. The GAB and BET models have theoretical bases[2] while the others are empirical. When the parameter k is 1, the GAB equation reduces to the BET equation. For any particular product, the numerical values of m _o and c obtained using the BET and GAB models are usually different. It has also been observed that most models can only predict the sorption for a particular range of activities. For example, the BET model is known to give a good fit for a _w up to 0.43.[15] However, our use of the nine models is to ascertain whether any of them would predict the sorption data for the whole range of interest. It should be noted that apart from Sanni et al.[2] who used five of the models presented, other authors have used at most three such models.

Assessing Goodness of Fit

Different statistical parameters have been used in the literature to assess the suitability of the models for predicting the sorption data. Some of these parameters are[6] [28] [29]: coefficient of determination (r ²), standard error of the estimate for water activity (SE), mean relative percentage deviation (P %), Root mean square deviation (B), χ ²—test, and plots of residuals.

From Table 3, it is apparent that the linear forms of the models are transformed sorption data variables. Hence, a value of r ² close to 1 does not necessarily imply that the model fits the actual data very well. From a practical point of view, P permits a direct visualization of the fitting ability of a given model. The residual plot is useful for detecting inconsistencies between experimental and calculated values. A random distribution of the residuals around zero indicates that the model correctly represents the particular set of data. For comparison of the various models listed in Table 3, we have used only r ², SE, and P.

Results and Discussions

For each product and at a given temperature, the model that gave the best fit using r ², SE, and P as criteria are shown in Table 4. It can be seen that, except for few cases, the three criteria do not give the same best fit model. Using P as the criterion, it can be seen that the GAB model fits about 60% of the data used for this study while the Caurie model was not suitable for any of the data studied. A closer look at Table 4 will also reveal that the GAB model fits 76% of the cassava products, 58% of the grain data, and 48% of the others. When SE is used as the criteria, a similar trend as that for P is obtained but with lower percentages. With r ² as the criteria, the Exponential model fits 38% of the cassava-based data; the Bradley model fits 35% of the grains data, and Smith model fits 35% of the others data. The GAB, BET, and Caurie models gave poor fits if r ² is used as the fitness criteria. The r ² for the GAB model varies from 0.9291 to 0.9971 for cassava-based products, 0.4821 to 0.9998 for grains, and 0.018 to 0.9931 for others. This shows that although the GAB model was not the best for cassava-based products, it was also not the worst when r ² is used as the criterion. The BET model is known to be a good fit for a _w up to 0.43[15]; this explains why it was a poor fit for our data since the a _w s are higher than 0.50 (see Table 1). It would appear that the Caurie model is not suitable for the data sets used for this work.

Table 4 Best model using the different criteria and the GAB parameters

Criteria			GAB parameters
Sl. No.	Food products	P	r ^2	SE	m o	C	k
1	Cassava chips at 25°C	GAB	Oswin	GAB	7.772	30.7	0.7277
2	Cassava chips at 45°C	GAB	Bradley	Bradley	7.995	28.3	0.6241
3	Cassava chips at 60°C	GAB	Bradley	Bradley	7.607	22.6	0.6105
4	Fufu (cabinent dried) at 25°C	GAB	Exponential	GAB	5.161	85.7	0.6674
5	Fufu (cabinent dried) at 32°C	GAB	Exponential	GAB	4.443	103.7	0.7121
6	Fufu (cabinent dried) at 45°C	Exponential	Exponential	GAB	4.147	34.2	0.7215
7	Fufu (rotary dried) at 25°C	Exponential	Exponential	GAB	5.437	61.8	0.6644
8	Fufu (rotary dried) at 32°C	GAB	Exponential	GAB	4.900	84.2	0.6864
9	Fufu (rotary dried) at 45°C	GAB	Exponential	GAB	4.387	44.5	0.7139
10	Fufu (sun dried) at 25°C	GAB	Exponential	GAB	4.979	65.1	0.6781
11	Fufu (sun dried) at 32°C	GAB	Igle-Chiri	GAB	4.711	33.6	0.6863
12	Fufu (sun dried) at 45°C	GAB	Bradley	GAB	3.907	53.4	0.7361
13	Fufu at 25°C	GAB	GAB	GAB	4.857	205.1	0.6971
14	Fufu at 32°C	GAB	Exponential	GAB	4.389	49.4	0.7160
15	Fufu at 45°C	GAB	Igle-Chiria	GAB	4.237	18.6	0.6639
16	Gari at 25°C	GAB	Smith	GAB	4.969	101.2	0.7658
17	Gari at 32°C	Igle-Chiri	Exponential	Exponential	4.668	55.3	0.7553
18	Lafun at 25°C	GAB	GAB	GAB	4.988	99.5	0.6857
19	Lafun at 32°C	GAB	Exponential	GAB	4.474	43.6	0.7099
20	Lafun at 45°C	Igle-Chiri	Igle-Chiri	GAB	4.227	19.8	0.6648
21	Soy fufu at 25°C	GAB	Exponential	GAB	5.883	107.7	0.6405
22	Soy fufu at 32°C	Igle-Chiri	Igle-Chiri	Igle-Chiri	5.963	32.7	0.5711
23	Soy fufu at 45°C	GAB	Igle-Chiri	GAB	5.908	16.2	0.5251
24	Soy lafu at 25°C	GAB	GAB	GAB	5.920	159.7	0.6447
25	Soy lafu at 32°C	Igle-Chiri	Igle-Chiri	Igle-Chiri	6.092	39.6	0.5790
26	Soy lafu at 45°C	Igle-Chiri	Igle-Chiri	GAB	5.940	17.8	0.5569
27	Tapioca at 25°C	GAB	Bradley	GAB	5.845	26.7	0.6160
28	Tapioca at 32°C	GAB	Bradley	GAB	6.354	9.4	0.5320
29	Tapioca at 45°C	GAB	Bradley	GAB	4.713	8.4	0.5701
30	Maize adsorption at 20°C	GAB	Bradley	Bradley	7.394	36.5	0.6993
31	Maize adsorption at 25°C	Bradley	Bradley	Bradley	6.869	37.6	0.7138
32	Maize adsorption at 40°C	GAB	Bradley	Bradley	6.777	15.6	0.7207
33	Maize desorption at 20°C	GAB	Igle-Chiri	GAB	10.258	24.1	0.5754
34	Maize desorption at 25°C	GAB	Igle-Chiri	GAB	9.814	23.1	0.5964
35	Maize desorption at 40°C	GAB	Igle-Chiri	GAB	9.411	19.4	0.6038
36	Millet dough A at 25°C	Oswin	Kuhn	Kuhn	6.183	52.3	0.8516
37	Millet dough A at 35°C	BET	Oswin	Kuhn	4.944	5.6	0.9846
38	Millet dough B at 25°C	Smith	Smith	Smith	9.124	18.1	0.8715
39	Millet dough B at 35°C	Exponential	Exponential	Exponential	7.448	13.3	0.8713
40	Millet dough C at 25°C	GAB	Smith	Oswin	5.106	7.8	0.8410
41	Millet dough C at 35°C	Igle-Chiri	Exponential	Igle-Chiri	3.215	4.4	0.8761
42	Sorgum adsorption at 20°C	Bradley	Bradley	Bradley	7.675	42.7	0.6700
43	Sorgum adsorption at 25°C	Bradley	Bradley	Bradley	7.441	34.7	0.6841
44	Sorgum adsorption at 40°C	GAB	Bradley	Bradley	7.087	15.7	0.6985
45	Sorgum desorption at 20°C	GAB	Igle-Chiri	GAB	10.198	24.3	0.5821
46	Sorgum desorption at 25°C	GAB	Igle-Chiri	GAB	9.952	24.9	0.5904
47	Sorgum desorption at 40°C	GAB	Igle-Chiri	GAB	9.719	18.1	0.5939
48	Sorgum desorption at 45°C	GAB	Igle-Chiri	GAB	9.719	18.1	0.5939
49	Wheat at 45°C	GAB	Bradley	GAB	9.353	36.9	0.6255
50	Wheat at 60°C	Oswin	Oswin	GAB	5.084	10.6	0.9035
51	Wheat paste at 40°C	GAB	Smith	GAB	6.284	69.6	0.8471
52	Wheat paste at 50°C	GAB	Oswin	GAB	5.809	23.2	0.8702
53	Wheat paste at 60°C	Oswin	Oswin	GAB	5.084	10.6	0.9035
54	Wheat paste at 70°C	GAB	Igle-Chiri	Igle-Chiri	5.787	5.0	0.8103
55	Wheat paste at 80°C	Igle-Chiri	Igle-Chiri	Smith	4.897	3.2	0.8816
56	Wheat paste at 90°C	Igle-Chiri	Smith	Smith	3.922	3.4	0.9257
57	Kuka adsorption at 34°C	GAB	Smith	Oswin	4.980	34.7	0.9059
58	Kuka adsorption at 37°C	GAB	Oswin	Oswin	4.356	10.0	0.9318
59	Kuka adsorption at 45°C	Oswin	Oswin	Kuhn	3.338	9.2	0.9517
60	Kuka desorption at 34°C	GAB	Bradley	Bradley	9.608	9.9	0.7782
61	Kuka desorption at 37°C	GAB	Smith	Smith	9.507	6.8	0.7866
62	Kuka desorption at 45°C	Oswin	Oswin	Oswin	5.832	9.9	0.8799
63	Okra at 20°C	GAB	Smith	GAB	17.965	13.0	0.8373
64	Okra at 28°C	GAB	Smith	Smith	15.786	11.8	0.8558
65	Okra at 35°C	GAB	Smith	Smith	13.294	13.1	0.8709
66	Okra at 50°C	Oswin	Oswin	GAB	9.065	26.6	0.8874
67	Pepper at 15°C	Smith	Smith	Smith	8.196	41.1	0.8135
68	Pepper at 25°C	Smith	Smith	Smith	7.950	28.1	0.8004
69	Pepper at 40°C	Smith	Smith	GAB	6.082	73.7	0.8175
70	Plantain (ripe) at 40°C	GAB	Kuhn	GAB	−6.937	0.96	29.5985
71	Plantain (ripe) at 50°C	GAB	Kuhn	GAB	−6.642	0.99	147.49
72	Plantain (ripe) at 60°C	Kuhn	Exponential	Kuhn	5.838	204.3	1.0908
73	Plantain (ripe) at 70°C	Oswin	BET	GAB	6.776	27.8	0.8624
74	Plantain (unripe) at 40°C	GAB	Smith	GAB	6.202	56.8	0.8856
75	Plantain (unripe) at 50°C	GAB	Smith	Smith	21.369	2.4	0.4440
76	Plantain (unripe) at 60°C	GAB	Smith	GAB	5.017	38.1	0.8892
77	Plantain (unripe) at 70°C	Bradley	Bradley	Bradley	10.670	4.3	0.4881
78	Rice at 40°C	Igle-Chiri	Bradley	GAB	8.672	11.5	0.6531
79	Rice at 50°C	GAB	Bradley	GAB	8.329	8.5	0.6124
80	Rice at 60°C	GAB	Bradley	GAB	5.399	10.6	0.7380
81	Rice at 70°C	GAB	Bradley	Bradley	4.809	6.0	0.7424
82	Soyabean flour at 25°C	Exponential	Exponential	Igle-Chiri	7.504	52.5	0.5405
83	Soyabean flour at 32°C	Exponential	Exponential	Igle-Chiri	6.444	44.7	0.5814
84	Soyabean flour at 45°C	Igle-Chiri	Igle-Chiri	GAB	6.862	22.6	0.5349
85	Yam flour (elubo) at 25°C	Igle-Chiri	Bradley	Bradley	6.369	18.4	0.6266
86	Yam flour (elubo) at 32°C	Bradley	Oswin	GAB	6.730	15.6	0.5443
87	Yam flour at 25°C	Kuhn	BET	Kuhn	−4.422	0.99	308.83
88	Yam flour at 32°C	GAB	Kuhn	Kuhn	4.213	270.8	0.8518
89	Yogurt at 20°C	BET	Kuhn	BET	2.942	15.8	1.0355
90	Yogurt at 35°C	GAB	Kuhn	Kuhn	1.895	47.6	1.0950
91	Yogurt at 50°C	GAB	Kuhn	GAB	1.564	17.3	1.0953

^aIgle-Chiri: Iglesias-Chirife (see [24]).

The GAB parameters are shown in Table 4. For any given product, each of the parameters varies with temperature. This confirms the earlier results[2] [16] which showed that the influence of temperature on GAB parameters can be calculated with an Arrhenius form of equation. Table 4 also shows that m _o is negative for ripe plantain at 40 and 50°C and for yam flour at 25°C despite their respective P and SE values being the lowest. Since a negative m _o has no physical meaning, it follows that the GAB model cannot be used for these data.

The 95% confidence interval for the average values of m_o, c, and k (ignoring the negative values) is computed for each type of food products, shown in Table 5. The average values for k are less than 1 and are approximately constant (two decimal place accuracy) for each of the food product types. Our data, therefore, validate the fact that, theoretically, values of k should be less than 1. The confidence intervals for m _o and c are much larger than that for k. It would, therefore, appear that, within the 95% confidence limit, only m _o and c vary with temperature.

Table 5 95% confidence intervals for the GAB parameters

m o	c	k
Cassava based	5.34 ± 0.07	57 ± 3	0.659 ± 0.004
Grain	7.21 ± 0.15	22 ± 1	0.755 ± 0.009
Others	7.61 ± 0.26	36 ± 3	0.795 ± 0.011
All	6.74 ± 0.07	39 ± 1	0.738 ± 0.003

A two-way analysis of variance was also used to compare the values of P, r ², and SE obtained for the nine models when applied to the various sorption data. The results are summarized in Table 6. Note that F(P), F(r ²), and F(SE) are the calculated F values for P, r ², and SE data respectively while F _crit represents the critical F-Statistic values at the specified probabilities (p = 0.05 and p = 0.01). Table 6 shows that the computed F values are significantly greater than the critical F values at the 0.05 or 0.01 levels. We can, therefore, conclude that there is significant difference among the nine models considered, irrespective of the criteria used. The variability is highest when r ² is used as the criterion, followed by P while SE is the lowest.

Table 6 Summary of the analysis of variance for the P, r ² and SE data

F(P)	F(R ^2)	F(SE)	F crit( p  = 0.05)	F cri( p  = 0.01)
All	16.7475	101.7127	8.6648	1.9512	2.5360
Cassava based	18.0949	110.4289	4.9541	1.9913	2.6137
Grains	6.8232	37.4152	4.5060	1.9831	2.5977
Others	5.3865	28.2626	3.0365	1.9674	2.5671

The ability of one or more criteria to predict the same best model is shown in Table 7. This table shows that none of the three criteria gave the same best model for 7% of the data sets used while the three criteria gave the same best model for 15% of the data sets. It would also appear that using either P or SE would give the same best model for a majority of the data sets used. This is probably because the linear form is a transformation of m and/or a _w . Thus, the r ² is an assessment of how linear the transformed variable is rather than the actual variable. P and SE are dependent on the actual variables, and these probably explain why these two criteria gave similar results for most of the foods. The results for the worst model (not shown here) also follow similar trend.

Table 7 Ability of the different criteria to predict the same best model

Average deviation	Correlation coefficient	Standard error	No of data sets
≤	≤	≤	14
≤	≤	NA	13
≤	NA	≤	39
NA	≤	≤	19
NA	NA	NA	6

Conclusion

The sorption isotherm data for thirty Nigerian foods at various temperatures have been modeled by using nine different equations. The goodness of fit for each model was assessed by using three statistical parameters. It was found that one parameter is not good enough to assess the predictive ability of the models. Using either the mean percentage deviation or standard error of the estimate as the criterion as the parameter, the GAB model fits roughly 60% of the data. For cassava-based products, the GAB model gives the best fit for 75% of the data sets. With the coefficient of determination as the criterion, the GAB model is not as good as other models such as the Exponential, Bradley, or Smith models. Except for very few cases, the three criteria did not produce the same best model for any particular data set.