VERC - Some Simple Methods for the Estimation of Surface Area and Volume of Thompson Seedless Grapes

- Research Notes -

Some Simple Methods for the Estimation of Surface Area and Volume of Thompson Seedless Grapes
by
Ming Cheung and Matthew Yen

CATI Publication #960902
Š copyright September 1996, all rights reserved

I. Introduction

The application of microwave heating in a vacuum environment is a promising dehydration process for fruits and other foods. It offers quality final products by preserving their shape, color, flavor and nutritional value. As in the other dehydration processes, researchers are trying to establish functional relationships between final product attributes such as physical properties and moisture content, and the controlling variables, such as power, pressure, time, mass, etc. The objective of this research project, undertaken at the Dried Foods Technology Laboratory at California State University, Fresno, was to examine the relationship of surface area and volume to dehydration characteristics of grapes in the microwave vacuum drying system (MIVAC).

One phase of this project was to estimate the dehydration time of a specific product for a set of given conditions. Operators of the MIVAC unit can estimate the dehydration time of certain food products based on their years of experience. However, in the attempt to establish a meaningful prediction of dehydration time, there is a need to determine the surface area and volume of the product being processed. This paper presents some simple methods for estimating surface area and volume of Thompson Seedless grapes which are used for the production of Grape Puffs^TM.

II. SIGNIFICANCE OF SURFACE AREA AND VOLUME IN DEHYDRATION PROCESS

In general, food dehydration involves the removal of moisture from the food by applying heat. Moisture is constantly removed from the product surface by evaporation. Moisture concentration at the surface of the product is always at a low point relative to the moisture concentration inside the product. Thus, there exists a moisture concentration gradient throughout the product as long as there is moisture residual inside it.

The rate at which evaporation occurs at the product surface can be mathematically written as:
     dw/dt =  k_mA (H_w - H_a) 	  
where w denotes the moisture concentration, k_m is the mass-transfer coefficient describing moisture transfer to the surrounding air, A is the product surface area, and H_w and H_a are absolute humidity at wet-bulb and air-bulb conditions. As it is shown in this expression, the dehydration rate is proportional to the product surface area. It must be measured or estimated for any meaningful analytical works.

Product volume is directly related to the product moisture content and the dehydration time. Although the product volume can be simply obtained by dividing the product mass with its density, volume calculations with other methods can be a good means to validate the measurements and the surface area calculations.

III. METHODS

Thompson Seedless grapes were selected for this study. Accurate surface area measurement presents a challenging task. It can be done by peeling the skin off the berry and measuring the skin area on a grid paper. Or, it can be measured by the method of wax dipping [Kamffer, 1989]. Both methods are not only tedious but also time consuming. Geometrically speaking, the general form of the Thompson Seedless berry can be classified as in between a ellipsoid and a cylinder. In order to decide which geometrical shape is most suitable, we will evaluate the berry's surface area and volume both (i) as an ellipsoid, and (ii) as a cylinder. Ten berries were randomly chosen for such calculations. Surface area calculations were further cross evaluated by comparing with direct measurements and volume calculations. Each method is described as follows:

(i) Grape berry is approximated as an ellipsoid. Figure 1 shows an ellipse placed in a x-y coordinate. a and b are the half-lengths of the major axis and minor axis of the ellipse, respectively. Applying the principles of calculus, the surface area can be obtained by revolving the ellipse around the x-axis into a ellipsoid (see Figure 2). The differential element can be written as:
    dA = 2 pi y dx  
Integrating the differential element from -a to a, one obtains the expression of surface area:

The volume of a berry as an ellipsoid can be obtained by integrating the differential volume form -a to a:

(ii) Grape berry is approximated as a cylinder. Figure 3 shows a cylinder with a height of 2a and a radius of b. the area of this cylinder is simply:
   A	= 2 pi b x 2a + 2 pi b²   
		= 4 pi ab + 2 pi b²       (3)
The volume of this cylinder is:
   V	= pi b² x 2a  
		= 2 pi ab²                (4) 
(iii) Direct measurements. Skin was peeled off from each berry and placed on a sheet of grid paper. Surface areas were measured by counting the number of grids that were covered by berry skin. Volume of a berry can also be measured. Place a cup of water on a balance and record its weight. Then each berry is gently placed into the water by suspending it with a thin wire. The amount of weight increased is equivalent to the buoyant force. Since buoyant force is simply the volume of the berry multiplied by the density of water, therefore,
    Volume of the berry = increased weight/density of water.  
IV. RESULTS AND DISCUSSIONS

A. Surface calculations with both methods and measurements are given in Table 1. The average surface areas are 11.53 cm² by grid paper counting method; 12.24 cm² by estimating as an ellipsoid; and 20.91 cm² by estimating as a cylinder. Assuming the grid paper measurement is the actual surface area, the ellipsoid method would, on an average, overestimate it by 6.3% and the cylinder method would overestimate it by 82%. It can be seen that the ellipsoid method is accurate to within 15% of the measured values, while the cylinder method has percentage errors between 55% to 93%.

It should also be noted that the standard deviations for grid paper measurements, ellipsoid calculations and cylinder calculations are 1.25, 1.20, and 1.98, respectively. This reflects the fact that there exist inevitable measurement errors with the grid paper counting method. This can be attributed to (i) surface area at the stem of each berry was omitted; and (ii) grape skin is curved when torn off by hand. So this method is also an approximated estimation of the berry surface area.

The last column is the ratio of the berry weight to its surface area, F. This ratio has a value of around 0.50 g/cm². This value is surprisingly consistent in spite of the inherent inaccuracy of the surface area measurement. This provides a useful short-cut to estimate the berry surface area once the weight of a berry is known, i.e.,
    berry surface area = berry weight/F  
or simply,
    berry surface area (cm²) = 2.0 (cm² /g) x berry weight (g).
The reciprocal of F multiplied by the berry density is simply the ratio of surface area to volume and has some physical significance. It is a good measure of the ability of moisture uptaking as well as the ease of dehydration for a particular berry. The larger the ratio, the easier the food to be dehydrated, and vice versa.

B. Table 2 shows the volume data based on three methods and their errors. It is interesting to note that the ellipsoid method would, on an average, underestimate the volume at 11.2%; while the cylinder method would overestimate it by 33.1%. The ellipsoid data are accurate between 3.75% to 24.36% of the measured values, and the cylinder data has a range of percentage error between 13.54% to 44.46%.

As a point of reference, the density of a Thompson Seedless grape is also computed with measured data by using the formula:
     density of a berry = weight of a berry/volume of a berry.   
The results are consistently yielding a value of 1.06 g/cm³. This consistency is also warranted by relatively small standard deviations of the measured volumes and measured weights, i.e., 0.599 and 0.637, respectively. In other words, weight measurements and volume measurements are much more accurate than surface area measurements.

C. A regression analysis was also performed to correlate the measured surface areas and volumes with the lengths of the major axis and minor axis. The resulting regression relationships are given in Table 3 and Table 4. They are summarized as follows:
     A = 1.1 + 8.414 ab  
and  V = 1.74 + 3.194 ab² 
Comparing these with the ellipsoid equations (1) A = 9.86 ab and (2) V= 4.187 ab², as well as with the cylinder equations (3) A = 12.56 ab + 6.28b² and (4) V = 6.28 ab², one can readily see that the regression analysis is more consistent with the ellipsoid estimations.

V. CONCLUSIONS

From the above results, one may conclude that in the absence of the actual measurements of surface area, the ellipsoid method can be a fairly satisfactory estimator. While the appearance of the grape berry is more close to a cylinder, the actual calculations have demonstrated that the cylinder method would lead to greater errors on estimating the berry's surface area and volume.

For practical applications, we have found that the surface area can be readily estimated by multiplying the berry weight with a factor of 2.0 (cm² /g). By doing so, we can estimate the surface area at an average accuracy of 0.4%. This is a surprisingly consistent finding! Besides, it is the product of the reciprocal of F value and the density, e.g., 2.0 (cm² /g) x 1.06 (g/cm³) gives a value of 2.12 (cm² /cm³). This is a good measure of the ease of dehydration. For example, assume volume = 1 cm³, the S/V ratio for sphere is 4.14, cube is 6, and cylinder is 4.55, etc. It is recommended that this value be compared with those of other products and used as an index for dehydration processes.

The methods investigated and the experimental data gathered in this study can be used in the study of the MIVAC dehydration process, and more specifically, in the analysis of the dehydration rate and dehydration time. Similar methods can be applied to estimate the surface areas or volumes of other fruits or vegetables with soft skin and relative small sizes. These methods can also be extended for the analytical work other the MIVAC or dehydration process, such as evaluation of the berry maturity or effectiveness of vineyard growing practices, etc.

AUTHOR'S NOTE

This publication contains preliminary results and has not undergone peer review.

REFERENCES

F.H. Kamffer, S. DE Meillon, H.A. Van De Venter and H.L. Gaigher. 1989. A simple method for the determination of the surface area of seeds. Plant Varieties and Seeds. Vol.2, 105-108.

Edward J. Cogan, Robert Z. Norman. 1963. Calculus, Difference and Differential Equations. Second Edition, Prentice-Hall, Inc.

{ page top }
{ CATI , VERC , VERC - Research Publication }

Copyright Š 1996. All rights reserved.
CALIFORNIA AGRICULTURAL TECHNOLOGY INSTITUTE - CATI
College of Agricultural Sciences and Technology
California State University, Fresno