Theoretical Modeling of Fruit Settling Depth in Water

Abstract

Settling depth is one of the hydrodynamic properties of fruits and vegetables that is important in hydraulic handling. In this research, settling depth of an irregular-shaped fruit with a density lower than density of water was theoretically modeled. Based on the model, the effective parameters on settling depth were dropping height and some fruit characteristics, such as density, volume, and shape factor.

Keywords:

• Settling depth
• Theoretical modeling
• Fruits and vegetables
• Sorting
• Hydrodynamic properties
• Physical characteristics

INTRODUCTION

Hydrodynamic properties are very important characters in hydraulic transport and handling as well as hydraulic sorting of agricultural products. Kheiralipour et al.[¹] determined and compared some hydrodynamic properties of two apple varieties: Redspar and Delbarstival. The properties were terminal velocity, coming up time, bounce, and drag forces related to the varieties in water. Jordan and Clark[²] theoretically modeled going-down terminal velocity and experimentally modeled dropping time of kiwi fruit (with the density higher than water density) in water. Kheiralipour et al.[^3,4] theoretically and experimentally modeled upward terminal velocity of apple (with the density lower than the density of water) and that of kiwi fruit, downward, with the density higher than the density of water.

Settling depth of fruits is one of the hydrodynamic properties important to determine the depth of water required to provide a cushion bed for the falling fruits. The cushion is needed to reduce or eliminate the impact forces exerting due to contact with the container or the channel bottom.[⁵] Settling depth of a fruit or vegetable in water, with the density lower than the density of water, is a depth the target will settle in water due to its initial velocity. After reaching the depth, the fruit returns to the water surface.

One of the effective parameters on settling depth of fruits is the dropped height. For example, the settling depth of a fruit was 7 in. when dropped from a height of 2.5 in., whereas the fruit was settled to a depth of 18 in. after dropping from a height of 36 in.[⁵] No detailed studies concerning settling depth modeling of fruits have yet been performed. So, the purpose of the present research was to model the settling depth of irregular-shaped fruit or vegetables in water.

MATERIALS AND METHODS

The target is a fruit falling in air that will come in contact with a water surface after dropping a height as d₀. After that, the fruit will descend in the water to a depth of d (Fig. 1). Suppose that fruit mass is m, fruit volume is V, fruit density is (= m/V), fruit diameter is D, and the largest cross-sectional area of the fruit is A_p.

Figure 1 Travelling a fruit in air and water.

Four forces will act on the fruit falling down in air: gravitational force (downward), buoyant, drag,[⁶] and frictional forces (upward). The fruit can reach an acceleration as follows:

(1)

where F_w is gravitational force, F_d is drag force, F_b is buoyant force, F_f is frictional force, and is fruit acceleration in air.

Frictional force has a low value and can be ignored. Gravitational, buoyant, and drag forces are expressed by Eqs. (2),[⁷] (3),[⁸] and (4),[⁹] respectively:

(2)

(3)

(4)

where g is gravitational acceleration, C_D is drag coefficient, v is fruit velocity and ρ_a is density of air.

By substituting Eqs. (2)–(4) in Eq. (1):

(5)

where is fruit acceleration in air.

Equation (5) can be changed as dividing by :

(6)

Drag coefficient is a function of Reynolds number (N_R). For N_R > 1:[¹⁰]

(7)

and

Then,

(9)

where D is the fruit diameter, μ_a is the static viscosity of the air, and k₁ is a constant factor. By replacing Eq. (9) in Eq. (6):

(10)

where k₂ is a constant factor.

For real spherical objects, A/V can be computed directly as a function of the diameter. But, most fruits have non-spherical shapes, so according to Jordan and Clark,[²] A/V can be separated into two parts: a dimensionless shape factor (S_f), and a pure size (S):

(11)

We can assume the following equation:

(12)

where k₃ and k₄ are constant factors. So, Eq. (10) can be:

(13)

where k₅ is a constant factor. By neglecting for a small height, fruit acceleration can be:

(14)

As acceleration in Eq. (14) is constant, the final velocity () of the fruit after dropping from a height as d₀, can be calculated using the following equation:[¹]

(15)

The above equation for the mentioned fruit, with , , and , will be:

(16)

By replacing acceleration from Eq. (16) in Eq. (13):

(17)

A large buoyant force will exert on the fruit when contacting (with velocity of ) with the water surface. Eq. (13) for fruit dropping in water will be:

(18)

where is the fruit acceleration in water, is water density, and is static viscosity of the water. In this equation, fruit acceleration in water () is dependent on fruit velocity () with power of (2 - n). The solution of Eq. (18) to obtain fruit velocity and then displacement is complicated. For this purpose, we calculate the mean of fruit acceleration in water and then use Eq. (15). If density of the fruit is less than density of water (), it descends to a depth of d (settling depth) and its velocity reaches to zero (). Then, the fruit will come up to the water surface. Thus, initial acceleration of the fruit () is:

(19)

Thus, its final acceleration can be obtained by replacing = 0:

(20)

Mean acceleration of the fruit dropping in water is:

(21)

where d can be derived from Eq. (15):

(22)

By replacing from Eq. (22) in Eq. (17):

(23)

So, d will be:

(24)

RESULTS AND DISCUSSION

Settling depth of a fruit with an irregular shape and with density less than density of water was theoretically modeled (Eq. 24). In this model, settling depth of fruit depends on some fruit characteristics, such as density, volume (or mass), and shape factor, and some water characteristics, such as density and static viscosity, and dropping height of the fruit. In Eq. (24), it absolutely can be seen that settling depth increases by increasing of the fruit density, because the numerator of the equation increases whereas the denominator decreases due to decreasing of . Also, the settling depth increases by increasing of the fruit volume. But it is clear that the effect of fruit volume (or fruit mass) on settling depth is less than the effect of fruit density.

The settling depth will increase by increasing of the falling height, because the power of the falling depth (d₀) in the numerator of the model is higher than that in the denominator. It can be told that the effect of the falling height is higher than that of the fruit density, because both the falling depth value and its power are higher than corresponding values for fruit.

CONCLUSIONS

In this research, settling depth of a fruit with density less than density of water was theoretically modeled. Based on the model, the fruit settling depth depends on dropping height, density volume, and shape factor of fruit. The author suggests experimental attempts to determine fruits’ settling depth with density less than density of water and investigate the ability of settling depth in fruit sorting.