Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer

*Le Anh Duc and Keum Dong Hyuk*

#### **Abstract**

Mathematical modeling for rapeseed drying on concurrent-flow dryer was built based on energy and mass transfer balances. The fourth-order Runge–Kutta method was used for solving four ordinary differential equations. A computer simulation program for circulating concurrent-flow rapeseed dryer was developed using these models. A pilot-scale concurrent-flow dryer was used to verify the fitness of simulation program. Two drying experiments were conducted. The output parameters of the simulation program were compared and analyzed with experiment data. The RMSE of simulated moisture contents ranged from 0.334 to 0.506%w.b. with the coefficient of determinations ranged from 0.994 to 0.997. The RMSE of simulated rapeseed temperatures during drying process ranged from 1.15 to 1.77°C with the R2 ranging from 0.904 to 0.925. The experimental drying rates were 2.38 and 2.80% w. b./h. In comparison with simulated values, the difference between simulated value and measured value of drying rate were 5.04 and 5.08%; drying time were 7.14 and 0.47%; and germination ratio were 1.87 and 0.47%. The simulated fuel energy consumption for drying were 4.62 and 8.57% lower than the experimental values. The analytic results showed that the simulation results have good fitness with experimental data.

**Keywords:** concurrent-flow dryer, mathematical modeling, simulation, drying rate, grain temperature, moisture content

#### **1. Introduction**

There are many different grain dryer designs on the market. Basically, the classical configurations of the moving bed dryers fall into four categories according to the relative directions of seed and air flows: the mixed flow dryer, the cross flow dryer, the concurrent-flow dryer, and the countercurrent flow dryer. The configuration with parallel concurrent-flow displays some advantages, such as obtaining of more homogeneous products as well as a better energy usage.

Concurrent-flow drying is a relatively new grain drying technology. In a concurrent-flow dryer, both the grain and drying air are moving in the same direction. This type of dryer has the advantage of using very high drying air temperatures without affecting grain quality and does not suffer the variation in grain moisture contents. Besides, energy efficiency of this type dryer is high.

The concurrent-flow drying principle was introduced the first time in 1955 by Öholm. Thompson et al. [1] developed simulation models of concurrent-flow dryer for corn drying. Now most of the drying simulation model was implemented based on the model presented by Thompson et al. that was developed for simulating high temperature corn drying. Felipe and Barrozo [2] studied the simultaneous heat and mass transfer between air and soybean seeds in a concurrent moving bed dryer, based on the application of a two-phase model to the drying process. Keum et al. [3] studied on circulating concurrent flow for rice drying with the drying temperature from 98 to 126°C, and air flow rate from 28.5 to 57.1 cmm/m2 . The study results showed that drying rate ranged from 1.09 to 2.2% d.b./h, and energy consumption ranged from 6224 to 6992 kJ/kg-water.

From these advantages, the use of concurrent-flow drying principle for drying of rapeseed has been recommended owing to:

Mathematical modeling and computer simulation can be used to predict the moisture and temperature of the products during the drying process and the energy consumption and drying capacity of the different drying system. Many different models have been proposed to describe the drying process in the basic types of convective grain dryer. However, no previous research was found in literature that study on dynamic simulation of concurrent flow for rapeseed.

content of rapeseed and the temperature and relative humidity of drying air were

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer*

to the next thin layer. As the drying air absorbs moisture, its temperature is decreased, and its ability to pick up more moisture (drying potential) decreases. A deep bed consisting of a number of thin layers is, therefore, simulated by calculating the air and moisture changes as the drying air passes from one thin layer of grain to the next layer. Each layer dries to equilibrium conditions for short time intervals, and the exhaust air from one layer is used as the input drying air to the next. The drying procedure continues for a number of drying time intervals until the desired

final moisture content of the material is achieved.

The thin layer drying models is used in simulation of deep-bed dryer, in which the average changes in moisture content and temperature on a thin layer of grain are calculated over a discrete time interval Δt. In the simulation, to solve mathematical models of deep-bed drying process, the depth bed was divided into nth thin layers with a thickness of Δx each, and dynamic heat and mass balances were set up in each section, and then the model consisted in a set of partial differential equations (**Figure 2**). Drying is achieved by continuously passing hot air through the static grain bed in one direction, from the top to the bottom section. From the first thin layer at the top section, the air evaporates moisture from the grain and carries it

predicted.

**65**

*Concurrent-flow drying model.*

**Figure 2.**

**Figure 1.**

*Block diagram of rapeseed concurrent-flow dryer.*

*DOI: http://dx.doi.org/10.5772/intechopen.91036*

#### **2. Concurrent-flow drying model**

#### **2.1 Selection of drying principle**

The drying process of rapeseed in concurrent-flow dryer was described in **Figure 1**.

The hot air and rapeseed are moving the same direction in the drying chamber. At the end of cycle drying, hot air is exhausted to ambience by suction fan; rapeseed is continued to be circulated on the drying chamber for the next cycle drying until it reaches the desired final moisture content.

#### **2.2 Mathematical model**

The mathematical model used in the study consists of a set of four partial differential equations in four independent variables: air humidity, air temperature, grain temperature, and grain moisture content.

Based on the theory of energy and mass transfer, a concurrent-flow rapeseed drying model was developed. By using this mathematical models, performance of concurrent-flow rapeseed dryer can be predicted; the temperature and moisture

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer DOI: http://dx.doi.org/10.5772/intechopen.91036*

**Figure 1.** *Block diagram of rapeseed concurrent-flow dryer.*

The concurrent-flow drying principle was introduced the first time in 1955 by Öholm. Thompson et al. [1] developed simulation models of concurrent-flow dryer for corn drying. Now most of the drying simulation model was implemented based on the model presented by Thompson et al. that was developed for simulating high temperature corn drying. Felipe and Barrozo [2] studied the simultaneous heat and mass transfer between air and soybean seeds in a concurrent moving bed dryer, based on the application of a two-phase model to the drying process. Keum et al. [3] studied on circulating concurrent flow for rice drying with the drying temperature

showed that drying rate ranged from 1.09 to 2.2% d.b./h, and energy consumption

From these advantages, the use of concurrent-flow drying principle for drying

• Energy saving: energy efficiency of this type dryer is about 30–40% better than

• It is possible to use a high drying temperature (up to 130°C) without increasing the grain temperature excessively because grain are exposed to drying air in a

• The drying air flow is parallel to the grain flow leads to more homogeneous moisture content and temperature distributions because all grains are exposed to the same temperatures, therefore guaranteeing the quality of the dried grains.

Mathematical modeling and computer simulation can be used to predict the moisture and temperature of the products during the drying process and the energy consumption and drying capacity of the different drying system. Many different models have been proposed to describe the drying process in the basic types of convective grain dryer. However, no previous research was found in literature that

The drying process of rapeseed in concurrent-flow dryer was described in

The mathematical model used in the study consists of a set of four partial differential equations in four independent variables: air humidity, air temperature,

Based on the theory of energy and mass transfer, a concurrent-flow rapeseed drying model was developed. By using this mathematical models, performance of concurrent-flow rapeseed dryer can be predicted; the temperature and moisture

The hot air and rapeseed are moving the same direction in the drying chamber. At the end of cycle drying, hot air is exhausted to ambience by suction fan; rapeseed is continued to be circulated on the drying chamber for the next cycle drying until it

. The study results

from 98 to 126°C, and air flow rate from 28.5 to 57.1 cmm/m2

ranged from 6224 to 6992 kJ/kg-water.

*Current Drying Processes*

**2. Concurrent-flow drying model**

reaches the desired final moisture content.

grain temperature, and grain moisture content.

**2.1 Selection of drying principle**

**2.2 Mathematical model**

**Figure 1**.

**64**

of rapeseed has been recommended owing to:

a cross flow type dryer without heat recovery.

short time, leads to high drying rate (1.5–2% w.b./h).

study on dynamic simulation of concurrent flow for rapeseed.

**Figure 2.** *Concurrent-flow drying model.*

content of rapeseed and the temperature and relative humidity of drying air were predicted.

The thin layer drying models is used in simulation of deep-bed dryer, in which the average changes in moisture content and temperature on a thin layer of grain are calculated over a discrete time interval Δt. In the simulation, to solve mathematical models of deep-bed drying process, the depth bed was divided into nth thin layers with a thickness of Δx each, and dynamic heat and mass balances were set up in each section, and then the model consisted in a set of partial differential equations (**Figure 2**). Drying is achieved by continuously passing hot air through the static grain bed in one direction, from the top to the bottom section. From the first thin layer at the top section, the air evaporates moisture from the grain and carries it to the next thin layer. As the drying air absorbs moisture, its temperature is decreased, and its ability to pick up more moisture (drying potential) decreases. A deep bed consisting of a number of thin layers is, therefore, simulated by calculating the air and moisture changes as the drying air passes from one thin layer of grain to the next layer. Each layer dries to equilibrium conditions for short time intervals, and the exhaust air from one layer is used as the input drying air to the next. The drying procedure continues for a number of drying time intervals until the desired final moisture content of the material is achieved.

Some of these assumptions are made to simplify the mathematical model. So, for developing these mathematical models, the following assumptions were made:

3.For the humidity of the air (humidity balance):

change of moisture content in the grains.

*DOI: http://dx.doi.org/10.5772/intechopen.91036*

*dM*

where:

dryer.

**67**

using these models.

data [4]. In this category fall:

*dM*

a: specific surface area of grain (m2

ca: specific heat of dry air (kJ/kg�K). cp: specific heat of dry grain (kJ/kg�K). cv: specific heat of water vapor (kJ/kg�K). cw: specific heat of water in grain (kJ/kg�K).

Ga: air flow rate (kg/h�m2

vp: grain velocity (m/h).

Gp: grain flow rate (kg/h�m2

H: enthalpy of dry air (kJ/kg).

*dx* ¼ *Mo* � *Me vp*

).

hc: convection heat transfer coefficient (kJ/h�m2

). hfg: vaporization latent heat of water within grain (kJ/kg).

The amount of water in and out the differential volume is equal to the rate of

moisture transferred ¼ moisture in–moisture out

*dx* ¼ an appropriate thin layer drying equation

�*PtQ*�1

Eqs. (1) and (2) represent the respective energy balances. Eqs. (3) and (4) result from mass balances applied for both fluid and solid phases of a concurrent-flow

The equation for the moisture content is obtained from the empirical thin layer equation for rapeseed. The four differential equations from (1)–(4) constitute the concurrent-flow drying model. A computer simulation program was developed

To solve these differential equations, the initial and boundary condition of grain and the drying air must be known and furnished to the simulation program as input

The initial conditions of the air humidity, grain moisture, and both air and grains temperatures were assumed constant at the dryer inlet, resulting in the following

> Tx¼0 ¼ Tin (5) θx¼0 ¼ θo (6) Hx¼0 ¼ Hin (7)

• The initial or inlet temperature and moisture content of grain

• The initial or inlet temperature and absolute humidity of the drying air

model boundary conditions (x = 0) for concurrent-flow dryer which are:

/m3 ). *dM*

*dx* (3)

*Q* exp �*PtQ* (4)

� o C).

*dH dx* ¼ � *Gp Ga*

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer*

4.For the moisture content of the grain (thin layer drying equation):

Energy balances and mass balances are written on a differential volume (S�Δx) located at an arbitrary location in the grain bed. There are four unknowns in this problem:

T(x,t): the air temperature.

θ(x,t): the grain temperature.

H(x,t): the humidity ratio.

M(x,t): the grain moisture content.

Therefore, four equations for material and energy balances must be made in order to calculate of T, θ, H, and M. Resulting from the balances are four Eq. (4):

1.For the enthalpy of the air (air enthalpy balance):

The change in sensible heat of air that results due to heat transfer by convection in time. The air enthalpy balance over the differential volume:

energy out ¼ energy in – energy transferred by convection

$$\frac{dT}{d\mathbf{x}} = \frac{-h\_c a}{G\_a c\_a + G\_a c\_v H} (T - \theta) \tag{1}$$

2.For the enthalpy of the grain (grain enthalpy balance):

The enthalpy from the air to the grains due to convection heat transfer over the control volume is equal to the required heat to evaporate water inside the grains and to heat water vapor extracted from the grains and the rate of accumulated heat inside the grains. The grain enthalpy balance over the differential volume:

energy transferred ¼ change in internal grain energy – energy for evaporation

$$\frac{d\theta}{d\mathbf{x}} = \frac{h\_c a}{\mathcal{G}\_p \left(c\_p + \mathcal{M}c\_w\right)} (T - \theta) - \frac{h\_{\rm f\tilde{g}} + c\_v (T - \theta)}{\mathcal{G}\_p \left(c\_p + \mathcal{M}c\_w\right)} \mathcal{G}\_a \frac{dH}{d\mathbf{x}} \tag{2}$$

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer DOI: http://dx.doi.org/10.5772/intechopen.91036*

3.For the humidity of the air (humidity balance):

The amount of water in and out the differential volume is equal to the rate of change of moisture content in the grains.

moisture transferred ¼ moisture in–moisture out

$$\frac{dH}{dx} = -\frac{G\_p}{G\_d} \frac{dM}{dx} \tag{3}$$

4.For the moisture content of the grain (thin layer drying equation):

$$\frac{dM}{dx} = \text{an appropriate thin layer drying equation}$$

$$\frac{dM}{dx} = \frac{M\_o - M\_\varepsilon}{v\_p} \left(-Pt^{Q-1}Q\right) \exp\left(-Pt^Q\right) \tag{4}$$

where:

Some of these assumptions are made to simplify the mathematical model. So, for

• The heat capacities of moist air and of grain are constant during the short time

Energy balances and mass balances are written on a differential volume (S�Δx) located at an arbitrary location in the grain bed. There are four unknowns in this

Therefore, four equations for material and energy balances must be made in order to calculate of T, θ, H, and M. Resulting from the balances are four Eq. (4):

The change in sensible heat of air that results due to heat transfer by convection

*Gaca* þ *GacvH* ð Þ *T* � *θ* (1)

*hfg* þ *cv*ð Þ *T* � *θ Gp cp* þ *Mcw Ga* *dH*

*dx* (2)

energy out ¼ energy in – energy transferred by convection

The enthalpy from the air to the grains due to convection heat transfer over the control volume is equal to the required heat to evaporate water inside the grains and to heat water vapor extracted from the grains and the rate of accumulated heat inside the grains. The grain enthalpy balance over the differential volume:

energy transferred ¼ change in internal grain energy – energy for evaporation

developing these mathematical models, the following assumptions were made:

• Grain shrinkage is negligible during the drying process.

• Particle-to-particle conduction is negligible.

• Initial moisture content of grain is uniform.

• Airflow and grain flow are plug-type and constant.

• No temperature gradients exist within each grain particle.

• The dryer walls are adiabatic and heat losses are negligible.

• Operation is in the steady state.

*Current Drying Processes*

• The solids flow rate is uniform.

T(x,t): the air temperature. θ(x,t): the grain temperature. H(x,t): the humidity ratio.

*dθ*

**66**

*dx* ¼ *hca*

*Gp cp* þ *Mcw*

ð Þ� *T* � *θ*

M(x,t): the grain moisture content.

1.For the enthalpy of the air (air enthalpy balance):

in time. The air enthalpy balance over the differential volume:

*dT*

2.For the enthalpy of the grain (grain enthalpy balance):

*dx* ¼ �*hca*

periods.

problem:

a: specific surface area of grain (m2 /m3 ). ca: specific heat of dry air (kJ/kg�K). cp: specific heat of dry grain (kJ/kg�K). cv: specific heat of water vapor (kJ/kg�K). cw: specific heat of water in grain (kJ/kg�K). Ga: air flow rate (kg/h�m2 ). Gp: grain flow rate (kg/h�m2 ). hfg: vaporization latent heat of water within grain (kJ/kg). H: enthalpy of dry air (kJ/kg). hc: convection heat transfer coefficient (kJ/h�m2 � o C). vp: grain velocity (m/h).

Eqs. (1) and (2) represent the respective energy balances. Eqs. (3) and (4) result from mass balances applied for both fluid and solid phases of a concurrent-flow dryer.

The equation for the moisture content is obtained from the empirical thin layer equation for rapeseed. The four differential equations from (1)–(4) constitute the concurrent-flow drying model. A computer simulation program was developed using these models.

To solve these differential equations, the initial and boundary condition of grain and the drying air must be known and furnished to the simulation program as input data [4]. In this category fall:

The initial conditions of the air humidity, grain moisture, and both air and grains temperatures were assumed constant at the dryer inlet, resulting in the following model boundary conditions (x = 0) for concurrent-flow dryer which are:

$$\mathbf{T\_{x=0}} = \mathbf{T\_{in}} \tag{5}$$

$$
\theta\_{\mathbf{x}=\mathbf{0}} = \theta\_{\mathbf{o}} \tag{6}
$$

$$\mathbf{H}\_{\mathbf{x}=\mathbf{0}} = \mathbf{H}\_{\text{in}} \tag{7}$$

$$\mathbf{M}\_{\mathbf{x}=\mathbf{0}} = \mathbf{M}\_{\mathbf{o}} \tag{8}$$

ln *Pv* ¼ *hfg hfgo*

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer*

each level of moisture content as the slope of the straight line obtained

Using MATLAB simulation program, the coefficients were found.

hc: convection heat transfer coefficient of grain bed (kJ/h�m2

*hfg hfgo*

The ratio of heat of vaporization of water in grain to the heat of vaporization of saturated water on a logarithmic scale gives the ratio of latent heat *hfg=hfgo* at

This result is fairly similar to the result of Cenkowski et al. [9]: A = 0.5; B = 14.5.

*μa*

μa ¼ 0*:*06175 þ 0*:*000165∙T (15)

*hc* ¼ 0*:*277 � *Ga* � *d* � *Ga*

Equilibrium moisture content of rapeseed was determined using Modified Hal-

Thin layer drying equation of rapeseed was determined using model of Page:

MR ¼ exp *:* �*P* � *t*

P = 0.02246 + 3.2428∙RH + 0.0006308∙T2 – 2.01481∙RH2 – 0.06077∙T∙RH.

Eqs. (1)–(4) and the boundary conditions Eqs. (5)–(8) were solved using the

In numerical analysis, the Runge–Kutta methods are an important family of iterative methods for the approximation of solutions of ordinary differential equations (ODEs). These techniques were developed around 1900 by the German mathematicians Runge C. and Kutta M.W. One member of the family of

*M* ¼ ½ � exp ð Þ �4*:*9758 � 0*:*0132 � *T* 1*=*1*:*8755ð Þ � ln *RH* �1*=*1*:*8755 (16)

where C is a constant of integration.

*DOI: http://dx.doi.org/10.5772/intechopen.91036*

Convection heat transfer coefficient [5]:

d: geometric mean diameter of rapeseed (m).

where the drying constants were determined:

Q = 0.60932 – 1.72018∙RH2 + 0.02529∙T∙RH.

μ a: dynamic viscosity of air (kg/h�m).

T: drying temperature (K). Equilibrium moisture content [8]:

Thin layer drying [10]:

**2.4 Numerical solution**

Runge–Kutta methods.

**69**

from Eq. (12):

A = 0.3734. B = 14.2442.

where:

sey equation:

ln *Ps* þ *C* (12)

(14)

*Q* (17)

�K).

¼ 1 þ *A* � exp ð Þ �*B* � *M* (13)

where: Tin: inlet air temperature (°C). θo: initial grain temperature (°C). Hin: inlet humidity ratio of drying air (kg/kg). Mo: initial moisture content of grain (dec., d.b.)

#### **2.3 Related equations**

The related equations used for simulation such as specific surface area, latent heat, convection heat transfer coefficient, equilibrium moisture content, and thin layer drying equation of rapeseed were taken from specific studies.

#### *2.3.1 Specific surface area of rapeseed*

Specific surface area of rapeseed was determined in Eq. (9) [5]:

$$a = \frac{\mathsf{G}(\mathsf{1} - \varepsilon)}{d} \tag{9}$$

where:

ε: void fraction was calculated based on bulk density and true density of rapeseed, ε = 0.389.

d: diameter of rapeseed was determined in our previous study, d = 2.21 � 10�3 m [6]. Then, the specific surface area of rapeseed is a = 1659 m2 /m3 .

#### *2.3.2 Latent heat*

Gallaher [7] established the following equation Eq. (10) to determine the dependence of the latent heat of vaporization of water from the product on its moisture content:

$$\mathbf{h\_{fg}} = \mathbf{h\_{fgo}} \bullet [\mathbf{1} + \mathbf{A} \bullet \exp \cdot (\mathbf{-B} \bullet \mathbf{M})] \tag{10}$$

hfg: latent heat of vaporization of water in grain (kJ/kg-water). hfgo: latent heat of vaporization of free water (kJ/kg-water).

$$\mathbf{h\_{fgo}} = (2502.2 \text{--} 2.386 \text{\textdegree T})$$

T: drying air temperature (°C).

M: rapeseed moisture content (decimal, d.b.)

A, B: coefficients.

The coefficients A and B were determined based on equilibrium moisture content of rapeseed (Modified Halsey equation) [8].

The values of the relative humidity can be replaced by equilibrium relative humidity (ERH) obtained from the equilibrium moisture content versus ERH relationships described in above section. Then:

$$\mathbf{P\_v = P\_s} \mathbf{\bullet} \mathbf{ERH} \tag{11}$$

The relationship between vapor pressure and latent heats of two substances at the same temperature is as follows [5]:

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer DOI: http://dx.doi.org/10.5772/intechopen.91036*

$$\ln P\_v = \frac{h\_{\rm f\bar{g}}}{h\_{\rm f\bar{g}o}} \ln P\_s + C \tag{12}$$

where C is a constant of integration.

The ratio of heat of vaporization of water in grain to the heat of vaporization of saturated water on a logarithmic scale gives the ratio of latent heat *hfg=hfgo* at each level of moisture content as the slope of the straight line obtained from Eq. (12):

$$\frac{h\_{\rm fg}}{h\_{\rm fg}} = \mathbf{1} + A \cdot \exp\left(-B \cdot \mathbf{M}\right) \tag{13}$$

Using MATLAB simulation program, the coefficients were found. A = 0.3734.

B = 14.2442.

Mx¼0 ¼ Mo (8)

*d* (9)

/m3 .

hfg ¼ hfgo∙½ � 1 þ A∙ exp *:*ð Þ –B∙M (10)

Pv ¼ Ps∙ERH (11)

where:

*Current Drying Processes*

where:

seed, ε = 0.389.

*2.3.2 Latent heat*

moisture content:

T: drying air temperature (°C).

A, B: coefficients.

**68**

M: rapeseed moisture content (decimal, d.b.)

tent of rapeseed (Modified Halsey equation) [8].

tionships described in above section. Then:

the same temperature is as follows [5]:

**2.3 Related equations**

Tin: inlet air temperature (°C). θo: initial grain temperature (°C).

*2.3.1 Specific surface area of rapeseed*

Hin: inlet humidity ratio of drying air (kg/kg). Mo: initial moisture content of grain (dec., d.b.)

The related equations used for simulation such as specific surface area, latent heat, convection heat transfer coefficient, equilibrium moisture content, and thin

*a* ¼ 6 1ð Þ � *ε*

ε: void fraction was calculated based on bulk density and true density of rape-

d: diameter of rapeseed was determined in our previous study, d = 2.21 � 10�3 m

Gallaher [7] established the following equation Eq. (10) to determine the dependence of the latent heat of vaporization of water from the product on its

hfgo ¼ ð Þ 2502*:*2–2*:*386∙T

The coefficients A and B were determined based on equilibrium moisture con-

The relationship between vapor pressure and latent heats of two substances at

The values of the relative humidity can be replaced by equilibrium relative humidity (ERH) obtained from the equilibrium moisture content versus ERH rela-

hfg: latent heat of vaporization of water in grain (kJ/kg-water). hfgo: latent heat of vaporization of free water (kJ/kg-water).

layer drying equation of rapeseed were taken from specific studies.

Specific surface area of rapeseed was determined in Eq. (9) [5]:

[6]. Then, the specific surface area of rapeseed is a = 1659 m2

This result is fairly similar to the result of Cenkowski et al. [9]: A = 0.5; B = 14.5. Convection heat transfer coefficient [5]:

$$h\_c = 0.277 \cdot G\_a \cdot \left(\frac{d \cdot G\_a}{\mu\_a}\right) \tag{14}$$

where:

hc: convection heat transfer coefficient of grain bed (kJ/h�m2 �K). d: geometric mean diameter of rapeseed (m). μ a: dynamic viscosity of air (kg/h�m).

$$
\mu\_{\text{a}} = 0.06175 + 0.000165 \text{\AA} \tag{15}
$$

T: drying temperature (K).

Equilibrium moisture content [8]:

Equilibrium moisture content of rapeseed was determined using Modified Halsey equation:

$$M = \left[ \exp \left( -4.9758 - 0.0132 \cdot T \right) \right]^{1/1.8755} \left( -\ln RH \right)^{-1/1.8755} \tag{16}$$

Thin layer drying [10]:

Thin layer drying equation of rapeseed was determined using model of Page:

$$\mathbf{MR} = \exp\left(-P \cdot t^Q\right) \tag{17}$$

where the drying constants were determined:

P = 0.02246 + 3.2428∙RH + 0.0006308∙T2 – 2.01481∙RH2 – 0.06077∙T∙RH. Q = 0.60932 – 1.72018∙RH2 + 0.02529∙T∙RH.

#### **2.4 Numerical solution**

Eqs. (1)–(4) and the boundary conditions Eqs. (5)–(8) were solved using the Runge–Kutta methods.

In numerical analysis, the Runge–Kutta methods are an important family of iterative methods for the approximation of solutions of ordinary differential equations (ODEs). These techniques were developed around 1900 by the German mathematicians Runge C. and Kutta M.W. One member of the family of

Runge–Kutta methods that is so commonly used is the fourth-order Runge–Kutta method or also called as RK4, meaning that the error per step is on the order of h5 , while the total accumulated error has order h4 [11].

Let an initial value problem be specified as follows:

$$\mathbf{y}' = f(\mathbf{y}, \mathbf{t}), \mathbf{y}(\mathbf{t}\_0) = \mathbf{y}\_0 \tag{18}$$

**3. Simulation program**

*DOI: http://dx.doi.org/10.5772/intechopen.91036*

The numerical solution for four ordinary differential equations was obtained by

Concurrent-flow Drying Simulation, version LAD), was built for drying simulation. The concurrent-flow dryer simulation model was programmed with the

The program of concurrent model terminates in one of two ways:

• When the moisture content within the dryer reaches a specified level

However, condensation or absorption is not simulated in the concurrent model since it does not occur in a properly designed dryer. Equations used by more than one model (e.g., psychrometric equations) are programmed as separate subroutines of function subprogram. A computer simulation program was built using these models. This program was used in predicting the performance and temperature

> Drying time (h) Number of pass

Final moisture content (% w.b.) Drying rate (% w.b./h) Water removal rate (kg/m2

)

Total energy consumption (kJ/kg-water)

Fan static pressure (Pa) Fan power (kW/m2

Fan energy (kJ/kg-water) Fuel energy (kJ/kg-water) )

using MATLAB code programs based on fourth-order Runge–Kutta methods. MATLAB is an interactive program and technical computing environment with numeric computation and data visualization. It provides integrated numerical analysis, matrix computation, signal processing, and graphics in an easy-to-use environment where problems and solutions are easily expressed without complicated programming. MATLAB-based software, entitled RCDSim-LAD (Rapeseed

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer*

**3.1 Main program**

sequence:

• Input data

• Initialize arrays

• Evaluate constants

• Output when appropriate

profile within the grain bed.

• Initial grain temperature (°C)

• Drying air flow rate (cmm/m2

• Grain flow velocity (m/h) Drying and ambient air condition: • Drying air temperature (°C) • Ambient air temperature (°C) • Ambient air relative humidity (dec.)

• Initial grain moisture content (dec, w.b.)

• Desired final moisture content (dec,w.b.)

*Input and output data in the simulation program.*

Initial grain condition:

Dryer specification: • Capacity (kg)

**Table 1.**

**71**

• Solve four ordinary differential equations

• When condensation or absorption is detected

**Input data Output data**

)

where *f*(t,y) is a function of y and t and the second equation is an initial condition.

In order to calculate yn+1 with a known value of yn, integrate Eq. (18) in the interval tn ≥ t ≥ tn+1 to yield.

$$\boldsymbol{\gamma}\_{n+1} = \boldsymbol{\gamma}\_n + \int\_{t\_n}^{t\_{n+1}} \mathbf{f}\left(\mathbf{y}, \mathbf{t}\right) d\mathbf{t} \tag{19}$$

The RK4 method is derived by applying a numerical integration method to the right side of Eq. (19). Then, the general form of the RK4 method for this problem is given by the following equations:

$$y\_{n+1} = y\_n + \frac{1}{6}(k\_1 + 2k\_2 + 2k\_3 + k\_4) \tag{20}$$

$$\mathbf{t\_{n+1}} = \mathbf{t\_n} + \mathbf{h} \tag{21}$$

for n = 0, 1, 2, 3, …

$$k\_1 = hf(\mathcal{y}\_n, t\_n) \tag{22}$$

$$\hbar k\_2 = hf\left(\mathcal{Y}\_n + \frac{k\_1}{2}, t\_n + \frac{h}{2}\right) \tag{23}$$

$$k\_3 = hf\left(\mathcal{Y}\_n + \frac{k\_2}{2}, t\_n + \frac{h}{2}\right) \tag{24}$$

$$k\_4 = h\_{\cdot} f\left(\mathbf{y}\_n + k\_3, \mathbf{t}\_n + h\right) \tag{25}$$

h: size of the interval

Thus, the next value (yn+1) is determined by the present value (yn). The slope is a weighted average of slopes:

k1 is the slope at the beginning of the interval.

k2 is the slope at the midpoint of the interval, using slope k1 to determine the value of y at the point (tn + h/2) using Euler's method.

k3 is again the slope at the midpoint, but now using the slope k2 to determine the y-value.

k4 is the slope at the end of the interval, with its y-value determined using k3.

In averaging the four slopes, greater weight is given to the slopes at the midpoint:

$$slope = \frac{1}{6}(k\_1 + 2k\_2 + 2k\_3 + k\_4) \tag{26}$$

To solve mathematical models of drying process, the depth bed model was divided into 10 thin layers, and the dynamic heat and mass balances were set up in each section and calculated over a discrete time interval Δt = 0.01 h.

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer DOI: http://dx.doi.org/10.5772/intechopen.91036*

#### **3. Simulation program**

#### **3.1 Main program**

Runge–Kutta methods that is so commonly used is the fourth-order Runge–Kutta method or also called as RK4, meaning that the error per step is on the order of h5

� �,y tð Þ¼ 0 y0 (18)

6 ð Þ *k*1 þ 2*k*2 þ 2*k*3 þ *k*4 (20) tnþ1 ¼ tn þ h (21)

*k*4 ¼ *h:f yn* þ *k*3, *tn* þ *h* � � (25)

6 ð Þ *k*1 þ 2*k*2 þ 2*k*3 þ *k*4 (26)

� � (22)

(23)

(24)

� �dt (19)

y' ¼ *f* y, t

*yn*þ1 ¼ *yn* þ

1

*k*2 ¼ *h:f yn* þ

*k*3 ¼ *h:f yn* þ

*yn*þ1 ¼ *yn* þ

where *f*(t,y) is a function of y and t and the second equation is an initial

In order to calculate yn+1 with a known value of yn, integrate Eq. (18) in the

ð*tn*þ1

f y, t

*tn*

The RK4 method is derived by applying a numerical integration method to the right side of Eq. (19). Then, the general form of the RK4 method for this problem is

*k*1 ¼ *h:f yn*, *tn*

*k*1 2 , *tn* þ *h* 2

*k*2 2 , *tn* þ *h* 2

Thus, the next value (yn+1) is determined by the present value (yn). The slope is

k2 is the slope at the midpoint of the interval, using slope k1 to determine the

k3 is again the slope at the midpoint, but now using the slope k2 to determine

k4 is the slope at the end of the interval, with its y-value determined using k3. In averaging the four slopes, greater weight is given to the slopes at the

To solve mathematical models of drying process, the depth bed model was divided into 10 thin layers, and the dynamic heat and mass balances were set up in

� �

� �

while the total accumulated error has order h4 [11]. Let an initial value problem be specified as follows:

condition.

*Current Drying Processes*

interval tn ≥ t ≥ tn+1 to yield.

given by the following equations:

for n = 0, 1, 2, 3, …

h: size of the interval

a weighted average of slopes:

the y-value.

midpoint:

**70**

k1 is the slope at the beginning of the interval.

value of y at the point (tn + h/2) using Euler's method.

*slope* ¼ 1

each section and calculated over a discrete time interval Δt = 0.01 h.

,

The numerical solution for four ordinary differential equations was obtained by using MATLAB code programs based on fourth-order Runge–Kutta methods. MATLAB is an interactive program and technical computing environment with numeric computation and data visualization. It provides integrated numerical analysis, matrix computation, signal processing, and graphics in an easy-to-use environment where problems and solutions are easily expressed without complicated programming. MATLAB-based software, entitled RCDSim-LAD (Rapeseed Concurrent-flow Drying Simulation, version LAD), was built for drying simulation.

The concurrent-flow dryer simulation model was programmed with the sequence:

The program of concurrent model terminates in one of two ways:

However, condensation or absorption is not simulated in the concurrent model since it does not occur in a properly designed dryer. Equations used by more than one model (e.g., psychrometric equations) are programmed as separate subroutines of function subprogram. A computer simulation program was built using these models. This program was used in predicting the performance and temperature profile within the grain bed.

#### **Table 1.**

*Input and output data in the simulation program.*

**Table 1** listed input data and output data which was simulated by a simulation program. The flow chart of the simulation program is shown in **Figure 3**. For the convenience, the interface graphical user interface (GUI) was built in **Figure 4**.

#### **3.2 Energy consumption**

*3.2.1 Fuel energy consumption*

$$EnFuel = \frac{G\_d \cdot (c\_d + c\_v \cdot h) \cdot (T - T\_{amb}) \cdot t}{0.85 \cdot \left(M\_o - M\_f\right) \cdot \rho\_g \cdot \varkappa} \tag{27}$$

where:

EnFuel: fuel energy consumption (kJ/kg-water). Ga: air flow rate (kg/h�m2 ). ca: specific heat of dry air (kJ/kg�K). cv: specific heat of water vapor (kJ/kg�K). h: absolute humidity (kg-water/kg-dry air). T: drying air temperature (°C). Tamb: ambient air temperature (°C). t: drying time (h). Mo: initial moisture content (decimal, d.b.) Mf: final moisture content (decimal, d.b.) ρg: dry grain bulk density (kg/m3 ). x: grain layer thickness (m).

*3.2.2 Fan power*

$$PFan = \frac{\Delta P \cdot \text{g}\_a \cdot A}{60 \cdot e\_f \cdot 1000} \tag{28}$$

where: PFan: fan power (kW). ΔP: pressure drop (Pa). A: cross-section area of grain bed (m2 ). ga: air flow rate (m3 /min�m2 ). ef: efficiency of the fan and motor (usually a value of about 0.5 is used).

*3.2.3 Fan energy consumption*

$$EnFan = \frac{PFan \cdot t \cdot 3600}{\left(M\_o - M\_f\right) \cdot \rho\_g \cdot \varkappa} \tag{29}$$

where:

EnFan: fan energy consumption (kJ/kg-water).

*3.2.4 Total energy consumption*

$$\text{EnTot} = \text{EnFuel} + \text{EnFan} \tag{30}$$

**Figure 3.**

**73**

*Flow chart of the simulation program.*

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer*

*DOI: http://dx.doi.org/10.5772/intechopen.91036*

where:

EnTot: total energy consumption (kJ/kg-water).

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer DOI: http://dx.doi.org/10.5772/intechopen.91036*

**Figure 3.** *Flow chart of the simulation program.*

**Table 1** listed input data and output data which was simulated by a simulation program. The flow chart of the simulation program is shown in **Figure 3**. For the convenience, the interface graphical user interface (GUI) was built in **Figure 4**.

> *EnFuel* ¼ *Ga* � ð Þ� *ca* þ *cv* � *h* ð Þ� *T* � *Tamb t* 0*:*85 � *Mo* � *Mf*

> > ).

*PFan* ¼ Δ*P* � *ga* � *A*

).

ef: efficiency of the fan and motor (usually a value of about 0.5 is used).

*EnFan* ¼ *PFan* � *t* � 3600 *Mo* � *Mf* � *ρg* � *x*

60 � *ef* � 1000 (28)

EnTot ¼ EnFuel þ EnFan (30)

EnFuel: fuel energy consumption (kJ/kg-water).

).

� *ρg* � *x*

(27)

(29)

**3.2 Energy consumption**

*Current Drying Processes*

where:

*3.2.1 Fuel energy consumption*

Ga: air flow rate (kg/h�m2

t: drying time (h).

*3.2.2 Fan power*

where:

where:

where:

**72**

ca: specific heat of dry air (kJ/kg�K). cv: specific heat of water vapor (kJ/kg�K). h: absolute humidity (kg-water/kg-dry air).

Mo: initial moisture content (decimal, d.b.) Mf: final moisture content (decimal, d.b.)

T: drying air temperature (°C). Tamb: ambient air temperature (°C).

ρg: dry grain bulk density (kg/m3

x: grain layer thickness (m).

PFan: fan power (kW). ΔP: pressure drop (Pa).

ga: air flow rate (m3

*3.2.3 Fan energy consumption*

*3.2.4 Total energy consumption*

A: cross-section area of grain bed (m2

/min�m2

EnFan: fan energy consumption (kJ/kg-water).

EnTot: total energy consumption (kJ/kg-water).

).

#### **Figure 4.**

*Interface of the simulation program RCDSim-LAD.*

#### **3.3 Psychrometric properties**

#### *3.3.1 Saturated vapor pressure*

$$P\_s = r \frac{a + b \cdot T\_k + c \cdot T\_k^2 + d \cdot T\_k^3 + e \cdot T\_k^4}{f \cdot T\_k - g \cdot T\_k^2} \tag{31}$$

Using MATLAB 7.3.0., Simulated results were exported to Excel (**Figure 5**) by clicking the button "Simulate and export data to Excel" on the interface of the

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer*

*DOI: http://dx.doi.org/10.5772/intechopen.91036*

In excel interface, besides the output data in the simulation program were listed in **Table 1**, the drying air temperature, rapeseed temperature, and moisture content of rapeseed in the drying process versus drying time are also displayed and automatically represented by graphs, in which the rapeseed temperature at the top layer, middle layer, bottom layer, and the maximum rapeseed temperature in the

In order to verify a fitness of simulation program, a pilot-scale concurrent-flow rapeseed dryer with capacity of 200 kg/batch was designed and manufactured. The dimension of the dryer is shown in **Figure 6**, and the structural principle of the dryer is shown in **Figure 7**. The pilot-scale rapeseed dryer includes a drying tower with grain inlet section, plenum section, and drying section; burner; drying fan; variable speed discharge augers; and bucket elevator for circulating

The hot air is supplied by a kerosene jet burner; after going through the mixed chamber, the drying air will enter the dryer through plenum section. Rapeseed and drying air are moving the same direction until drying the air out by force of suction

simulation program (**Figure 4**).

*Simulated data with excel interface.*

**Figure 5.**

**4. Model validation**

the grains.

**75**

**4.1 Pilot-scale concurrent-flow dryer**

drying chamber versus drying time are also displayed.

where:

Ps: saturated vapor pressure (Pa).  RH: relative humidity (dec). Patm: atmospheric pressure, Patm = 101,325 (Pa).

#### *3.3.3 Specific volume*

$$ws = 287 \cdot T\_k \cdot \frac{0.6219 + h}{0.6219 \cdot P\_{atm}} \tag{33}$$

vs.: specific volume (m3 /kg).

#### *Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer DOI: http://dx.doi.org/10.5772/intechopen.91036*

**Figure 5.** *Simulated data with excel interface.*

**3.3 Psychrometric properties**

*Current Drying Processes*

*Interface of the simulation program RCDSim-LAD.*

*3.3.1 Saturated vapor pressure*

d = 0.12558 � 10�3

*3.3.2 Absolute humidity*

*3.3.3 Specific volume*

**74**

where:

**Figure 4.**

*Ps* ¼ *r*

h: absolute humidity (kg-water/kg-dry air).

Patm: atmospheric pressure, Patm = 101,325 (Pa).

/kg).

RH: relative humidity (dec).

vs.: specific volume (m3

Ps: saturated vapor pressure (Pa).

(32)

(33)

Using MATLAB 7.3.0., Simulated results were exported to Excel (**Figure 5**) by clicking the button "Simulate and export data to Excel" on the interface of the simulation program (**Figure 4**).

In excel interface, besides the output data in the simulation program were listed in **Table 1**, the drying air temperature, rapeseed temperature, and moisture content of rapeseed in the drying process versus drying time are also displayed and automatically represented by graphs, in which the rapeseed temperature at the top layer, middle layer, bottom layer, and the maximum rapeseed temperature in the drying chamber versus drying time are also displayed.

#### **4. Model validation**

#### **4.1 Pilot-scale concurrent-flow dryer**

In order to verify a fitness of simulation program, a pilot-scale concurrent-flow rapeseed dryer with capacity of 200 kg/batch was designed and manufactured. The dimension of the dryer is shown in **Figure 6**, and the structural principle of the dryer is shown in **Figure 7**. The pilot-scale rapeseed dryer includes a drying tower with grain inlet section, plenum section, and drying section; burner; drying fan; variable speed discharge augers; and bucket elevator for circulating the grains.

The hot air is supplied by a kerosene jet burner; after going through the mixed chamber, the drying air will enter the dryer through plenum section. Rapeseed and drying air are moving the same direction until drying the air out by force of suction

**Figure 6.** *Technical drawing of pilot-scale concurrent-flow dryer.*

centrifugal fan through five exhaust air ducts. Rapeseed flow rate is controlled by two variable speed discharge augers. Rapeseed is out the dryer by discharge augers. Then, the grains are circulated by bucket elevator from the top of the dryer and flow down the vertical drying chamber.

The dimensions of the pilot-scale concurrent-flow dryer are shown in detail in **Figure 6**.

#### **4.2 Experimental design**

During the experiment the drying air temperature and grain temperature are continuously measured by temperature sensors (Thermocouple T-type, Omega, USA). The total of 16 temperature sensors was arranged at necessary positions inside and outside the dryer (**Figure 7**). Data from sensors were transferred to data logger system. Two computers were used to record the temperature data from the data logger (Datascan 7327, UK).

In the plenum section, drying air temperature input was measured by three sensors No. 1–3. In exhaust air ducts, two sensors at upper ducts (No. 10–11) and two sensors at lower ducts (No. 12–13) were arranged to measure exhaust air temperature. In the drying chamber, six sensors (No. 4–9) were arranged at six positions in the cross-section of the drying chamber to measure the temperature of grains. In two discharge augers, two sensors (No. 14–15) were arranged above the two discharge augers to measure discharge rapeseed temperature (rapeseed temperature after drying). To measure ambient air temperature, one temperature sensor (No. 16) was arranged outside the dryer (**Table 2**).

Two hygrometers (MTH4100, Sanyo, UK, 10 – 99%, 2.5%) were installed to record the ambient air relative humidity (No. 17) and exhaust air relative humidity

*Design layout for the temperature and relative humidity sensor locations in the pilot-scale concurrent-flow*

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer*

*DOI: http://dx.doi.org/10.5772/intechopen.91036*

**Temperature measurement Sensors** Drying air temperature Inlet 3 No. 1–3

Rapeseed temperature Drying chamber 6 No. 4–9

Ambient air temperature 1 No. 16

Ambient air relative humidity 1 No. 17 Exhaust air relative humidity 1 No. 18

*Temperature sensors distribution and hygrometer used in the experiment.*

Outlet 4 No. 10–13

Discharge 2 No. 14–15

(No. 18) (**Table 2**).

**Table 2.**

**77**

**Relative humidity measurement**

**Figure 7.**

*dryer.*

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer DOI: http://dx.doi.org/10.5772/intechopen.91036*

**Figure 7.**

centrifugal fan through five exhaust air ducts. Rapeseed flow rate is controlled by two variable speed discharge augers. Rapeseed is out the dryer by discharge augers. Then, the grains are circulated by bucket elevator from the top of the dryer and

The dimensions of the pilot-scale concurrent-flow dryer are shown in detail

During the experiment the drying air temperature and grain temperature are continuously measured by temperature sensors (Thermocouple T-type, Omega, USA). The total of 16 temperature sensors was arranged at necessary positions inside and outside the dryer (**Figure 7**). Data from sensors were transferred to data logger system. Two computers were used to record the temperature data from the

In the plenum section, drying air temperature input was measured by three sensors No. 1–3. In exhaust air ducts, two sensors at upper ducts (No. 10–11) and two sensors at lower ducts (No. 12–13) were arranged to measure exhaust air temperature. In the drying chamber, six sensors (No. 4–9) were arranged at six positions in the cross-section of the drying chamber to measure the temperature of grains. In two discharge augers, two sensors (No. 14–15) were arranged above the two discharge augers to measure discharge rapeseed temperature (rapeseed temperature after drying). To measure ambient air temperature, one temperature sen-

flow down the vertical drying chamber.

*Technical drawing of pilot-scale concurrent-flow dryer.*

in **Figure 6**.

**76**

**Figure 6.**

*Current Drying Processes*

**4.2 Experimental design**

data logger (Datascan 7327, UK).

sor (No. 16) was arranged outside the dryer (**Table 2**).

*Design layout for the temperature and relative humidity sensor locations in the pilot-scale concurrent-flow dryer.*

#### **Table 2.**

*Temperature sensors distribution and hygrometer used in the experiment.*

Two hygrometers (MTH4100, Sanyo, UK, 10 – 99%, 2.5%) were installed to record the ambient air relative humidity (No. 17) and exhaust air relative humidity (No. 18) (**Table 2**).

In both experiments, grain flow velocity was set up at 5 m/h; this value is equivalent to the mass of circulated rapeseed which is 1000 kg/h. The grain flow velocity is controlled by two discharge rollers. The rotation of discharge rollers was 3.5 rpm. The rotation of discharge rollers was controlled by an inverter (S500, Mitsubishi, Japan).

After the experiment was completed, dried rapeseed samples were sealed in double-layer polythene bags for 24 h to reach ambient conditions [14]. The samples were then tested for germination. The germination tests were conducted according

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer*

**Figure 8.**

**Figure 9.**

**Figure 10.**

**79**

*Average variation of drying air temperature during drying process.*

*DOI: http://dx.doi.org/10.5772/intechopen.91036*

*Rapeseed temperature during drying process at discharge augers.*

*Temperature of exhaust drying air in test 1 and test 2.*

A centrifugal suction fan 1 HP with air flow rate 30 cmm/m2 [12] was used for sucking the exhaust drying air. An anemometer (Velocicalc-Plus, TSI, USA) was used to measure the drying air velocity.

The jet burner using kerosene (OL-3, Daewon, Korea) was used for heating drying air. The burner can be raised to the temperature of drying air up to 140°C. A temperature sensor (PT-100 Ω) was installed at the influx duct to control the burner, and a temperature control equipment (HSD-V2, Hansung, Korea) was used. An electric balance (A-200, Cass, Korea, accuracy 0.01 kg) was used to weigh the mass of Kerosene loss by drying process.

The dimension of drying chamber (height length width) is 0.5 m 0.7 m 0.5 m. Height of tempering section is 0.5 m. In both experiments, rapeseed samples using are Spring rapeseed, variety Sunmang F1-hybrid, were harvested in June in Jeonnam-do, Yeonggwang-gun. The samples of 200 kg were cleaned and stored in a refrigerator at a temperature of 4°C [13]. The initial moisture content of samples in Test 1 is 23.0% and in Test 2 is 23.2% (**Table 3**).

#### **4.3 Experimental results**

In both experiments, there are a difference in drying air temperature in the plenum chamber. The drying air temperature is highest at the position in front of the plenum, and lowest at back of plenum. In Test 1, the average temperature of drying air in the plenum section is 96.9, 84.2, and 83.9°C at the front, middle, and back of the plenum, respectively. In Test 2, the average temperature of drying air in plenum section is 128.1, 111.3, and 106.1°C at front, middle, and back of plenum, respectively (**Table 4**). The average temperature of drying air in the plenum chamber during drying process is 89.4 and 116.8°C for Test 1 and Test 2, respectively (**Figure 8**).

The temperature of rapeseed during drying at discharge augers (**Figure 9**) and the temperature of air at exhaust ducts (**Figure 10**) for both Test 1 and Test 2 are fairly uniform. Detailed drying conditions in Test 1 and Test 2 shown in **Table 5** and results of rapeseed drying in a pilot-scale dryer were summarized in **Table 6**.

#### **Table 3.**

*Initial rapeseed conditions.*

#### **Table 4.**

*Drying air temperature in the plenum chamber.*

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer DOI: http://dx.doi.org/10.5772/intechopen.91036*

After the experiment was completed, dried rapeseed samples were sealed in double-layer polythene bags for 24 h to reach ambient conditions [14]. The samples were then tested for germination. The germination tests were conducted according

**Figure 8.** *Average variation of drying air temperature during drying process.*

**Figure 9.** *Rapeseed temperature during drying process at discharge augers.*

**Figure 10.** *Temperature of exhaust drying air in test 1 and test 2.*

In both experiments, grain flow velocity was set up at 5 m/h; this value is equivalent to the mass of circulated rapeseed which is 1000 kg/h. The grain flow velocity is controlled by two discharge rollers. The rotation of discharge rollers was 3.5 rpm. The rotation of discharge rollers was controlled by an inverter (S500,

A centrifugal suction fan 1 HP with air flow rate 30 cmm/m2 [12] was used for sucking the exhaust drying air. An anemometer (Velocicalc-Plus, TSI, USA) was

The jet burner using kerosene (OL-3, Daewon, Korea) was used for heating drying air. The burner can be raised to the temperature of drying air up to 140°C. A temperature sensor (PT-100 Ω) was installed at the influx duct to control the burner, and a temperature control equipment (HSD-V2, Hansung, Korea) was used. An electric balance (A-200, Cass, Korea, accuracy 0.01 kg) was used to weigh the

The dimension of drying chamber (height length width) is 0.5 m 0.7 m

In both experiments, there are a difference in drying air temperature in the plenum chamber. The drying air temperature is highest at the position in front of the plenum, and lowest at back of plenum. In Test 1, the average temperature of drying air in the plenum section is 96.9, 84.2, and 83.9°C at the front, middle, and back of the plenum, respectively. In Test 2, the average temperature of drying air in plenum section is 128.1, 111.3, and 106.1°C at front, middle, and back of plenum, respectively (**Table 4**). The average temperature of drying air in the plenum chamber during drying process is 89.4 and 116.8°C for Test 1 and Test 2, respec-

The temperature of rapeseed during drying at discharge augers (**Figure 9**) and the temperature of air at exhaust ducts (**Figure 10**) for both Test 1 and Test 2 are fairly uniform. Detailed drying conditions in Test 1 and Test 2 shown in **Table 5** and results of rapeseed drying in a pilot-scale dryer were summarized in **Table 6**.

**Rapeseed Test 1 Test 2**

Test 1 96.9 84.2 83.9 Test 2 128.1 111.3 106.1

**Test no. Drying air temperature (°C)**

200 23.0 22.8

**Front Middle Back**

200 23.2 24.7

 0.5 m. Height of tempering section is 0.5 m. In both experiments, rapeseed samples using are Spring rapeseed, variety Sunmang F1-hybrid, were harvested in June in Jeonnam-do, Yeonggwang-gun. The samples of 200 kg were cleaned and stored in a refrigerator at a temperature of 4°C [13]. The initial moisture content

of samples in Test 1 is 23.0% and in Test 2 is 23.2% (**Table 3**).

Mitsubishi, Japan).

*Current Drying Processes*

used to measure the drying air velocity.

mass of Kerosene loss by drying process.

**4.3 Experimental results**

tively (**Figure 8**).

Initial weight (kg)

*Initial rapeseed conditions.*

**Table 3.**

**Table 4.**

**78**

Initial moisture content (% w.b.) Initial grain temperature (°C)

*Drying air temperature in the plenum chamber.*

to protocols described for the standard germination test [15]. The average germination percentages of Test 1 and Test 2 are 94.7 and 84.5%, respectively [16]. simulated moisture content during drying process for Test 1 is shown in **Figure 11**

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer*

The analytical results showed the good fitness between simulated moisture content and measured moisture content for both Test 1 and Test 2. This result showed the good agreement of the simulation program for predicting the moisture content

The simulated temperature of rapeseed during drying process has a good correlative with the experimental data. The R2 of rapeseed temperature are 0.904 and 0.925 in Test 1 and Test 2, respectively. The RMSE of rapeseed temperature are 1.15 and 1.77°C in Test 1 and Test 2, respectively. The comparison of the measured and simulated temperature of rapeseed for Test 1 is shown in **Figure 13** and for Test 2 shown in **Figure 14**. The analytical results showed that simulated values have a very

The discharge rapeseed temperature of the simulation program tends to be higher than the measured values during drying process. However, the average differences between measured and simulated values are small. It showed a good fitness between the values of the model and the values of the experiments.

rapeseed after drying were investigated. The comparison of the measured and

simulated results of both Test 1 and Test 2 was listed in **Table 7**.

The drying time, drying rate, fuel energy consumption, and germination ratio of

and for Test 2 shown in **Figure 12**.

*DOI: http://dx.doi.org/10.5772/intechopen.91036*

of rapeseed in concurrent-flow dryer.

**Figure 12.**

**Figure 13.**

**81**

*Measured and simulated moisture content in test 2.*

*Measured and simulated temperature of rapeseed in test 1.*

good fitness to measured values by experiment.

#### **4.4 Model validation**

The simulation program was validated by comparison results of numerical model with experimental data of pilot-scale concurrent-flow dryer as described. The input data of simulation program were entered in accordance with the data of the actual experiment, such as initial rapeseed conditions, dryer specification, and drying air and ambient air conditions. The fitness of simulated results with measured results was evaluated based on the coefficient of determination (R2 ) and the root mean square error (RMSE).

The R2 of moisture content versus drying time in Test 1 and Test 2 were 0.994 and 0.997, respectively. The RMSE of moisture content in Test 1 and Test 2 were 0.334 and 0.506%w.b., respectively. The comparison of the measured and

#### **Table 5.**

*Drying conditions for drying tests.*

#### **Table 6.**

*The results of rapeseed drying in pilot-scale dryer.*

**Figure 11.** *Measured and simulated moisture content in test 1.*

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer DOI: http://dx.doi.org/10.5772/intechopen.91036*

simulated moisture content during drying process for Test 1 is shown in **Figure 11** and for Test 2 shown in **Figure 12**.

The analytical results showed the good fitness between simulated moisture content and measured moisture content for both Test 1 and Test 2. This result showed the good agreement of the simulation program for predicting the moisture content of rapeseed in concurrent-flow dryer.

The simulated temperature of rapeseed during drying process has a good correlative with the experimental data. The R2 of rapeseed temperature are 0.904 and 0.925 in Test 1 and Test 2, respectively. The RMSE of rapeseed temperature are 1.15 and 1.77°C in Test 1 and Test 2, respectively. The comparison of the measured and simulated temperature of rapeseed for Test 1 is shown in **Figure 13** and for Test 2 shown in **Figure 14**. The analytical results showed that simulated values have a very good fitness to measured values by experiment.

The discharge rapeseed temperature of the simulation program tends to be higher than the measured values during drying process. However, the average differences between measured and simulated values are small. It showed a good fitness between the values of the model and the values of the experiments.

The drying time, drying rate, fuel energy consumption, and germination ratio of rapeseed after drying were investigated. The comparison of the measured and simulated results of both Test 1 and Test 2 was listed in **Table 7**.

**Figure 12.** *Measured and simulated moisture content in test 2.*

**Figure 13.** *Measured and simulated temperature of rapeseed in test 1.*

to protocols described for the standard germination test [15]. The average

**4.4 Model validation**

*Current Drying Processes*

**Test No.**

**Table 5.**

**Table 6.**

**Figure 11.**

**80**

root mean square error (RMSE).

**Average drying air temperature (°C)**

*Drying conditions for drying tests.*

Drying time (h) Drying rate (% w.b./h) Fuel consumption (kJ/kg-water) Initial germination rate (%)

Initial moisture content (% w.b.) Final moisture content (% w.b.)

*The results of rapeseed drying in pilot-scale dryer.*

*Measured and simulated moisture content in test 1.*

germination percentages of Test 1 and Test 2 are 94.7 and 84.5%, respectively [16].

The simulation program was validated by comparison results of numerical model with experimental data of pilot-scale concurrent-flow dryer as described. The input data of simulation program were entered in accordance with the data of the actual experiment, such as initial rapeseed conditions, dryer specification, and drying air and ambient air conditions. The fitness of simulated results with measured results was evaluated based on the coefficient of determination (R2

The R2 of moisture content versus drying time in Test 1 and Test 2 were 0.994 and 0.997, respectively. The RMSE of moisture content in Test 1 and Test 2 were 0.334 and 0.506%w.b., respectively. The comparison of the measured and

Test 1 89.4 30 25.4 (24.2–26.4) 71.6 (67.1–75.3) Test 2 116.8 25 28.6 (26.7–31.4) 63.4 (60.8–68.0)

**Experimental results Test 1 Test 2**

**Ambient air temperature (°C) Ave. (min–max)**

> 23.0 13.8 3.90 2.38 4915 94.7

**Air flow rate (cmm/m2 )**

) and the

23.2 11.4 4.22 2.80 4831 84.5

**Ambient relative humidity (%) Ave. (min– max)**

of a concurrent-flow dryer. Energy balances and mass balances are written on a differential volume located at an arbitrary location in the grain bed. The mathematical model consists of a set of four partial differential equations in four independent variables including air humidity, air temperature, grain temperature, and

*Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer*

The simulation program can predict the drying time, drying rate, drying air humidity and temperature, grain temperature and moisture content during drying process, water removal rate, drying fan parameters, germination ratio, fuel energy,

with a capacity of 200 kg/batch was designed, manufactured, and tested. Two drying experiments were conducted. The output parameters of the simulation pro-

gram were compared and analyzed with experiment data.

drying were 4.62 and 8.57% lower than the experimental values.

rapeseed drying in circulating concurrent-flow dryer.

\* and Keum Dong Hyuk2

1 Nong Lam University, Ho Chi Minh City, Vietnam

2 SungKyunKwan University, Suwon, South Korea

provided the original work is properly cited.

\*Address all correspondence to<EMAIL_ADDRESS>
© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

A computer simulation program for circulating concurrent-flow rapeseed dryer was developed using these models along with a detailed description of the program.

To evaluate a fitness of simulation program, a pilot-scale concurrent-flow dryer

The RMSE of simulated moisture contents ranged from 0.334 to 0.506%w.b. with the coefficient of determinations ranging from 0.994 to 0.997. The RMSE of simulated rapeseed temperatures ranged from 1.15 to 1.77°C with the coefficient of determinations ranging from 0.904 to 0.925. The experimental drying rates were 2.38 and 2.80% w.b./h. The difference between simulated value and measured value of drying rate were 5.04 and 5.08%; drying time were 7.14 and 0.47%; and germination ratio were 1.87 and 0.47%. The simulated fuel energy consumption for

The analytic results showed that the simulation results have good fitness with experimental data. So, the mathematical modeling and the simulation program were proved their reliability and were shown to be a convenient tool for simulation of

grain moisture content.

**Author details**

Le Anh Duc1

**83**

and total energy consumption.

*DOI: http://dx.doi.org/10.5772/intechopen.91036*

**Figure 14.**

*Measured and simulated temperature of rapeseed in test 2.*

#### **Table 7.**

*Measured and simulated results of test 1 and test 2.*

The difference between measured value and simulated value of final moisture content were 2.2 and 5.17%; drying rate were 5.04 and 5.08%; drying time were 7.14 and 0.47%; and germination ratio were 1.87 and 0.47%. The simulated values of fuel energy consumption for drying were 4.62 and 8.57% lower than the measured values for Test 1 and Test 2, respectively. The differences are derived from the sequential changes in drying air temperature, ambient temperature, and drying air humidity of the simulation model and experiment conditions. In general, there is a good fitness between measured values by experiment and simulated values from the simulation program for all of the output parameters such as final moisture content, drying time, drying rate, fuel energy, and germination ratio at the different experiment conditions.

#### **5. Conclusions**

Mathematical modeling for rapeseed drying on concurrent-flow dryer was built based on energy and mass transfer balances applied for both fluid and solid phases

#### *Mathematical Modeling and Simulation of Rapeseed Drying on Concurrent-Flow Dryer DOI: http://dx.doi.org/10.5772/intechopen.91036*

of a concurrent-flow dryer. Energy balances and mass balances are written on a differential volume located at an arbitrary location in the grain bed. The mathematical model consists of a set of four partial differential equations in four independent variables including air humidity, air temperature, grain temperature, and grain moisture content.

A computer simulation program for circulating concurrent-flow rapeseed dryer was developed using these models along with a detailed description of the program. The simulation program can predict the drying time, drying rate, drying air humidity and temperature, grain temperature and moisture content during drying process, water removal rate, drying fan parameters, germination ratio, fuel energy, and total energy consumption.

To evaluate a fitness of simulation program, a pilot-scale concurrent-flow dryer with a capacity of 200 kg/batch was designed, manufactured, and tested. Two drying experiments were conducted. The output parameters of the simulation program were compared and analyzed with experiment data.

The RMSE of simulated moisture contents ranged from 0.334 to 0.506%w.b. with the coefficient of determinations ranging from 0.994 to 0.997. The RMSE of simulated rapeseed temperatures ranged from 1.15 to 1.77°C with the coefficient of determinations ranging from 0.904 to 0.925. The experimental drying rates were 2.38 and 2.80% w.b./h. The difference between simulated value and measured value of drying rate were 5.04 and 5.08%; drying time were 7.14 and 0.47%; and germination ratio were 1.87 and 0.47%. The simulated fuel energy consumption for drying were 4.62 and 8.57% lower than the experimental values.

The analytic results showed that the simulation results have good fitness with experimental data. So, the mathematical modeling and the simulation program were proved their reliability and were shown to be a convenient tool for simulation of rapeseed drying in circulating concurrent-flow dryer.

#### **Author details**

The difference between measured value and simulated value of final moisture content were 2.2 and 5.17%; drying rate were 5.04 and 5.08%; drying time were 7.14 and 0.47%; and germination ratio were 1.87 and 0.47%. The simulated values of fuel energy consumption for drying were 4.62 and 8.57% lower than the measured values for Test 1 and Test 2, respectively. The differences are derived from the sequential changes in drying air temperature, ambient temperature, and drying air humidity of the simulation model and experiment conditions. In general, there is a good fitness between measured values by experiment and simulated values from the simulation program for all of the output parameters such as final moisture content, drying time, drying rate, fuel energy, and germination ratio at the different

**Test 1 Measured Simulated Difference (%)** Final moisture content (% w.b.) 13.8 13.5 2.2 Drying rate (% w.b./h) 2.38 2.26 5.04 Fuel energy (kJ/kg-water) 4915 5153 4.62 Drying time (h) 3.9 4.2 7.14 Germination ratio (%) 94.7 96.5 1.87 **Test 2 Measured Simulated Difference (%)** Final moisture content (% w.b.) 11.4 10.8 5.17 Drying rate (% w.b./h) 2.80 2.95 5.08 Fuel energy (kJ/kg-water) 4831 4417 8.57 Drying time (h) 4.22 4.2 0.47 Germination ratio (%) 84.5 92.5 8.64

Mathematical modeling for rapeseed drying on concurrent-flow dryer was built based on energy and mass transfer balances applied for both fluid and solid phases

experiment conditions.

**5. Conclusions**

**82**

**Figure 14.**

*Current Drying Processes*

**Table 7.**

*Measured and simulated temperature of rapeseed in test 2.*

*Measured and simulated results of test 1 and test 2.*

Le Anh Duc1 \* and Keum Dong Hyuk2

\*Address all correspondence to<EMAIL_ADDRESS>
© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Thompson TL, Peart RM, Foster GH. Mathematical simulation of corn drying—a new model. Transactions of ASAE. 1968;**24**(3):582-586

[2] Felipe CAS, Barrozo MAS. Drying of soybean seeds in a concurrent moving bed: Heat and mass transfer and quality analysis. Drying Technology. 2003; **21**(3):439-456

[3] Keum DH, Han JG, Kang SR, Kim OW, Kim H, Han JW, et al. Development of rice circulating concurrent-flow dryer (I) performance teset of pilot scale dryer. Journal of Biosystems Engineering. 2005;**10**(2):97-106

[4] Han JW, Keum DH, Kim W, Duc LA, Cho SH, Kim H. Circulating concurrentflow drying simulation of rapeseed. Journal of Biosystems Engineering. 2010;**35**(6):401-407

[5] Keum DH. Simulation of Agricultural Products and Foods Process Engineering. South Korea: SungKyunkwan University Publisher; 2005

[6] Duc LA, Han JW, Hong SJ, Choi HS, Kim YH, Keum DH. Physical properties of rapeseed (I). Journal of Biosystems Engineering. 2008;**33**(2):101-105

[7] Gallaher GL. A method of determining the latent heat of agricultural crops. Agricultural Engineering. 1951;**32**(1):34-38

[8] Duc LA, Hyuk KD. Equilibrium moisture content isotherm characteristics of rapeseed. Asia Pacific Journal of Sustainable Agriculture, Food and Energy. 2016;**4**(1):10-14

[9] Cenkowski S, Muir WE, Jayas DS. Simulation of canola and barley drying in deep bed. Journal of Food Process Engineering. 1989;**12**:171-190

[10] Duc LA, Han JW, Keum DH. Thin layer drying characteristics of rapeseed (*Brassica napus* L.). Journal of Stored Products Research. 2011;**47**(1):32-38

**Chapter 5**

**Abstract**

**1. Introduction**

**1.1 Preface**

bed dryer.

**85**

Determination on Fluidization

*Bui Trung Thanh and Le Anh Duc*

ity types in designing a continuous fluidized bed dryer.

**Keywords:** refined salt, solid particles, aerodynamic, minimum fluidization velocity, homogeneous fluidization, bed fraction, fluidized bed dryer

The phenomenon in which solid particles float in a gas stream and have a liquidlike property is called a fluidized bed. This phenomenon of fluidization in gas or liquid flow was discovered by Fritz Winkler in the 1920s [1]. This one was investigated by Lewis et al. and had been raised to fluidized theory [2]. The first commercial fluidized bed dryer was installed in USA in 1948 [3]. The fluidized bed technology is used for drying of bulk materials, includes examples such as: vibrating fluidized bed dryer, normal fluidized bed dryer without vibrating device and pulsed fluidized

Mathematical modeling and computer simulation of grain drying are now widely used and become an important tool for designing new dryers, for analyzing existing drying systems and for identifying drying conditions [4]. Identifying the drying conditions is necessary to establish the optimal protocol for ensuring seed quality [5]. To solve the simulation models, equations concerning aerodynamic properties such as the gas stream velocity and particle velocity are the most important components. The aerodynamic properties are affected by shape, density and size of particles [6].

Velocity Types of the Continuous

Refined Salt Fluidized Bed Drying

After the centrifugation stage, refined salt particles have rather high moisture content; therefore, the moist salt particles in contact with each other will stick together in a short time. In particular, the moist salt particles will stick together faster and tighter and form a larger unit when they are exposed to drying hot air. For this reason, the refined salt was dried by rotary drum dryers with vibrating balls distributed along the drum or a vibrating fluidized bed dryers. These drying methods make poor product sensory quality, low product recovery efficiency, while also lead to an increase of heat and electricity energy consumption. In order to increase the efficiency of refined salt drying technology by conventional continuous fluidized bed dryers, the chapter focuses on the study of aerodynamic properties of refined salt grains in the continuous fluidized particle layer. The content of the chapter presents theoretical and empirical methods to determine fluidization veloc-

[11] Fausett LV. Applied Numerical Analysis Using Matlab. Upper Saddle River, New Jersey: Prentice Hall Inc.; 1999. p. 07458

[12] Hong SJ, Duc LA, Han JW, Kim H, Kim YH, Keum DH. Physical properties of rapeseed (II). Journal of Biosystems Engineering. 2008;**33**(3):173-178

[13] Cassells JA, Caddick LP, Green JR, Reuss R. Isotherms for Australian canola varieties. In: Proceedings of the Australian Postharvest Technical Conference. 2003. pp. 59-63

[14] ANSI/ASAE S448.1. Thin-layer drying of agricultural crops. In: ASAE Standards 51st Edition. 2004. pp. 598-600

[15] Association of Official Seed Analysis. Rules for testing seeds. Journal of Seed Technology. 1993;**16**:1-113

[16] Duc LA, Han JW. The effects of drying conditions on the germination properties of rapeseed. Journal of Biosystems Engineering. 2009;**34**(1): 30-36

#### **Chapter 5**

**References**

*Current Drying Processes*

**21**(3):439-456

2005;**10**(2):97-106

2010;**35**(6):401-407

2005

**84**

Products and Foods Process Engineering. South Korea:

[7] Gallaher GL. A method of determining the latent heat of agricultural crops. Agricultural Engineering. 1951;**32**(1):34-38

moisture content isotherm

and Energy. 2016;**4**(1):10-14

[8] Duc LA, Hyuk KD. Equilibrium

characteristics of rapeseed. Asia Pacific Journal of Sustainable Agriculture, Food

[9] Cenkowski S, Muir WE, Jayas DS. Simulation of canola and barley drying in deep bed. Journal of Food Process Engineering. 1989;**12**:171-190

[1] Thompson TL, Peart RM, Foster GH.

[10] Duc LA, Han JW, Keum DH. Thin layer drying characteristics of rapeseed (*Brassica napus* L.). Journal of Stored Products Research. 2011;**47**(1):32-38

[11] Fausett LV. Applied Numerical Analysis Using Matlab. Upper Saddle River, New Jersey: Prentice Hall Inc.;

[12] Hong SJ, Duc LA, Han JW, Kim H, Kim YH, Keum DH. Physical properties of rapeseed (II). Journal of Biosystems Engineering. 2008;**33**(3):173-178

[13] Cassells JA, Caddick LP, Green JR, Reuss R. Isotherms for Australian canola

varieties. In: Proceedings of the Australian Postharvest Technical Conference. 2003. pp. 59-63

[14] ANSI/ASAE S448.1. Thin-layer drying of agricultural crops. In: ASAE

Standards 51st Edition. 2004.

[15] Association of Official Seed

Analysis. Rules for testing seeds. Journal of Seed Technology. 1993;**16**:1-113

[16] Duc LA, Han JW. The effects of drying conditions on the germination properties of rapeseed. Journal of Biosystems Engineering. 2009;**34**(1):

pp. 598-600

30-36

1999. p. 07458

[2] Felipe CAS, Barrozo MAS. Drying of soybean seeds in a concurrent moving bed: Heat and mass transfer and quality analysis. Drying Technology. 2003;

Mathematical simulation of corn drying—a new model. Transactions of

[3] Keum DH, Han JG, Kang SR, Kim OW, Kim H, Han JW, et al. Development of rice circulating concurrent-flow dryer (I) -

performance teset of pilot scale dryer. Journal of Biosystems Engineering.

[4] Han JW, Keum DH, Kim W, Duc LA, Cho SH, Kim H. Circulating concurrentflow drying simulation of rapeseed. Journal of Biosystems Engineering.

[5] Keum DH. Simulation of Agricultural

SungKyunkwan University Publisher;

[6] Duc LA, Han JW, Hong SJ, Choi HS, Kim YH, Keum DH. Physical properties of rapeseed (I). Journal of Biosystems Engineering. 2008;**33**(2):101-105

ASAE. 1968;**24**(3):582-586


**5. Therapeutic strategy and decision-making algorithm**

As with all other meningiomas, the decision-making process for SOM must be tailored to each patient. Mass effect of the tumor, age, general condition, comorbidities, symptomatology, its impact on daily life, and the patient's wishes must be taken into account. In cases with absent or mild symptoms without mass effect on imaging, simple clinical and radiological monitoring can be chosen initially, with patient follow-up on a regular basis (every 3–6 months). In contrast, the presence of optic neuropathy, severe neurological symptoms, significant proptosis, or serious mass effect warrants surgical operation. Although a subject of debate, optimal

#### **Figure 3.**

*Decision-making algorithm of first-line treatment for spheno-orbitary meningiomas, in accordance with the 2016 EANO guidelines [16]. The choice of radiation treatment is mainly based on the tumor volume, stereotactic radiosurgery being preferred for smaller tumors and radiotherapy being preferred for larger tumors.*

#### *Spheno-Orbital Meningiomas DOI: http://dx.doi.org/10.5772/intechopen.101983*

surgical resection remains the current reference treatment for SOM, in accordance with the general EANO (European Association of Neuro-Oncology) guidelines for the management of meningiomas published in 2016 [16]. If the patient is in a fragile state of health or categorically refuses the operation, radiation treatment may be offered as an alternative. The choice of technique is then mainly based on the tumor volume, stereotactic radiosurgery being preferred for smaller tumors and radiotherapy being preferred for larger tumors. To summarize this reasoning, we propose a simple algorithm highlighting the main points to be taken into account during decision-making in cases of SOM (**Figure 3**).


**5. Marginal response in bovines**

The weight gain in growing bovines in pasture in the dry season is curvilinear as a function of supplement supply, based on corn and soybean meal, in which the supplement conversion (kg of supplement/kg of accretion in weight gain) becomes worse with increase in the supplementation (Lana et al., 2005; Keane et al., 2006; Lana, 2007b) (Figure 6).

The milk production by supplemented cows in pasture or in feedlot is also curvilinear as a function of increase in the concentrate supply, based on corn and soybean meal (Figure 7A), in which the marginal increase in milk production per kg of concentrate decreases with increase in the amount of concentrate (Bargo et al., 2003; Pimentel et al., 2006a; Sairanen et al., 2006; Lana et al., 2007a,b), as shown in Figure 7B, and in some studies the milk response to concentrate was satisfactory only up to 2-4 kg of concentrate/animal/day (Fulkerson et al., 2006).

Rationality in the Use of Non Renewable Natural Resources in Agriculture 277

0.1 2.1 2.0 2.9

0.2 0.9 1.3 3.0

0.6 2.0 2.9

2.3 3.3 4.0

1.3 1 Considering US\$0.966/kg of N, US\$1.208/kg of P2O5 and US\$0.36/kg of bean, it is necessary 2.68 and 3.36 kg of bean to pay 1 kg of N or P2O5. Efficiency worse than 2.68 or 3.36:1 for N or P2O5 is not economically desirable. These calculations can be used to choose the level of fertilization. 2 Level of

Table 4. Efficiency of use of fertilizers (kg of bean/kg of fertilizer) calculated with base in the

Fig. 6. Body weight gain (BWG) of growing bovines in pasture during the dry season, as a function of daily intake of supplement with 24% CP, in which the values among parenthesis represent the differential in kilograms of supplement given daily divided by the differential

in weight gain, in relation to the previous treatment (Lana, 2005; Lana et al., 2005)

Efficiency of use of fertilizers (kg of grains/kg of fertilizer) 1 50 2 100 150 200 250 300

> 0.0 0.4 0.4 0.6

> 0.0 0.2 0.3 0.7

> 0.1 0.4 0.6

> 1.1 1.2 2.1

0.0 0.3 0.2 0.4

0.0 0.1 0.2 0.5

0.1 0.2 0.4

0.8 0.9 1.7

0.0 0.8 1.6 2.4 3.2 Supplement (kg/animal/day)

0.0 0.2 0.2 0.3

0.0 0.1 0.1 0.3

0.0 0.2 0.3

0.6 0.6

0.0 0.8 0.8 1.1

0.1 0.4 0.5 1.3

0.2 0.7 1.1

1.5 1.9 2.9

> 0.0 0.2 0.4 0.6 0.8 1.0

ADG (kg/animal/day) .

Fertilizer (kg/ha)

N

N

P2O5

P2O5

equations of Table 3

0.0 0.2 0.4 0.6 0.8 1.0

(1.5:1)

ADG (kg/animal/day) .

Second factor

N (kg/ha) 0 30 60

N (kg/ha) 0 50 120

1.1 10.9 11.5 14.2

> 0.6 2.8 4.5 7.9

5.4 12.2 15.3

> 4.0 6.9 6.0

fertilizer (kg/ha) - N in the first two cases or P2O5 in the last two cases.

(40:1) (7.0:1)

0.0 0.8 1.6 2.4 3.2 Supplement (kg/animal/day)

The curvilinear response can also be verified with specific nutrients, such as the observed positive curvilinear response in milk production and negative curvilinear response in the efficiency of use of nitrogen by increasing the dietary crude protein content from 11 to 19% in cows with mean production of 38 kg of milk/day (Baik et al., 2006). In the third experiment of Figure 7, in addition to decreasing response in milk production, there was decreasing response in body weight variation with increase in the concentrate level (0.20, 0.12, and 0.095 kg of body weight gain per additional kilogram of concentrate intake; Teixeira et al., 2006).

1 Kg of fertilizer/ha; 2 Ton of grain/ha; 3 1 = Bolsanello et al. (1975) and Oliveira et al. (1982), p.155; 2 = Malavolta (1989), p.273.

Table 3. Changes in constants of linear regression of reciprocal of bean production (x1,000 kg/ha) as a function of reciprocal of amount of fertilizer (kg/ha/year), by a second factor

According to the Biotechnology and Biological Sciences Research Council (1998), formerly known as AFRC (Agricultural and Food Research Council), all currently feed systems calculate the dietary requirements of energy and protein to meet the animals needs for maintenance and production. However, in practice, the situation is different, because there is no need for the farmer to meet the cow's nutritional requirements if it is against the economical interest. So, it is evident that studies in animal response to increasing levels of concentrate or specific nutrients are needed, as suggested by Lana (2003; p.87).

Although the animal's responses to nutrients are curvilinear, the daily weight gains estimated by the level 1 of NRC (1996) of beef cattle are linear as a function of intakes of metabolizable energy and protein (Figure 8A). In the same way, the milk production estimated by the model CNCPS 5.0 as a function of intakes of metabolizable energy and protein, and model NRC (2001) of dairy cattle as a function of intakes of net energy for lactation and metabolizable protein, were linear by using increasing levels of concentrate

The curvilinear response can also be verified with specific nutrients, such as the observed positive curvilinear response in milk production and negative curvilinear response in the efficiency of use of nitrogen by increasing the dietary crude protein content from 11 to 19% in cows with mean production of 38 kg of milk/day (Baik et al., 2006). In the third experiment of Figure 7, in addition to decreasing response in milk production, there was decreasing response in body weight variation with increase in the concentrate level (0.20, 0.12, and 0.095 kg of body weight gain per additional kilogram of concentrate intake;

Coefficient

2.794 8.516 6.718 8.205

183.36 46.235 15.623 28.056

6.7411 8.8241 10.186

180.38 90.586 132.74

(b) r2 ks

0.82 1.00 0.99 1.00

0.23 0.78 0.99 0.98

0.98 1.00 1.00

0.95 1.00 1.00

5 11 15

136 87 199

1 kmax 2 Sorce of

0.5 1.3 1.4 1.6

0.1 0.4 0.7 1.0

0.7 1.2 1.5

0.8 1.0 1.5 data 3

1

2

1

2

(a)

2.044 0.782 0.710 0.630

10.764 2.5229 1.3364 0.9539

1.3812 0.8181 0.6842

1.3257 1.0402 0.6684

concentrate or specific nutrients are needed, as suggested by Lana (2003; p.87).

1 Kg of fertilizer/ha; 2 Ton of grain/ha; 3 1 = Bolsanello et al. (1975) and Oliveira et al. (1982), p.155; 2 =

Although the animal's responses to nutrients are curvilinear, the daily weight gains estimated by the level 1 of NRC (1996) of beef cattle are linear as a function of intakes of metabolizable energy and protein (Figure 8A). In the same way, the milk production estimated by the model CNCPS 5.0 as a function of intakes of metabolizable energy and protein, and model NRC (2001) of dairy cattle as a function of intakes of net energy for lactation and metabolizable protein, were linear by using increasing levels of concentrate

Table 3. Changes in constants of linear regression of reciprocal of bean production (x1,000 kg/ha) as a function of reciprocal of amount of fertilizer (kg/ha/year), by a second factor According to the Biotechnology and Biological Sciences Research Council (1998), formerly known as AFRC (Agricultural and Food Research Council), all currently feed systems calculate the dietary requirements of energy and protein to meet the animals needs for maintenance and production. However, in practice, the situation is different, because there is no need for the farmer to meet the cow's nutritional requirements if it is against the economical interest. So, it is evident that studies in animal response to increasing levels of

Teixeira et al., 2006).

(kg/ha/year) Second factor Intercept

N (kg/ha) 0 30 60

N (kg/ha) 0 50 120

Fertilizer

N

N

P2O5

P2O5

Malavolta (1989), p.273.

1.3 1 Considering US\$0.966/kg of N, US\$1.208/kg of P2O5 and US\$0.36/kg of bean, it is necessary 2.68 and 3.36 kg of bean to pay 1 kg of N or P2O5. Efficiency worse than 2.68 or 3.36:1 for N or P2O5 is not economically desirable. These calculations can be used to choose the level of fertilization. 2 Level of fertilizer (kg/ha) - N in the first two cases or P2O5 in the last two cases.

Table 4. Efficiency of use of fertilizers (kg of bean/kg of fertilizer) calculated with base in the equations of Table 3

Fig. 6. Body weight gain (BWG) of growing bovines in pasture during the dry season, as a function of daily intake of supplement with 24% CP, in which the values among parenthesis represent the differential in kilograms of supplement given daily divided by the differential in weight gain, in relation to the previous treatment (Lana, 2005; Lana et al., 2005)

Rationality in the Use of Non Renewable Natural Resources in Agriculture 279

Up to 150 6 117 37 56 14 151-300 12 238 70 73 28 301-600 14 451 106 94 43 601-1200 14 821 111 159 66 1201-2400 3 1667 316<PHONE_NUMBER>-4800 1 4000 300 1800 300

Up to 150 0.38 3.17 8.33 2.08 0.25 151-300 0.40 3.40 8.57 3.27 0.38 301-600 0.41 4.27 10.53 4.77 0.45 601-1200 0.59 7.41 12.51 5.15 0.41 1201-2400 0.42 5.27 12.66 2.63 0.21 2401-4800 1.00 13.33 13.33 2.22 0.17 Table 5. Number of producers, daily mean milk production by producer, area for the herd, number of animals and milking cows in the herd, and productivity indexes per area and per

The milk production in farmer level (first case) ranged from 60 to 4000 kg/producer/day. The increase in milk production was highly correlated with the number of milking cows (r = 0.94), followed by moderate correlation with the size of pasture (r = 0.67) and, surprisingly, the productivity per cow and per unit of area did not correlate with the milk production per producer (r = 0.11 and 0.06, respectively; Table 6). In the nation level (second case), the result repeated, in which there was high correlation of milk production/state/year with the total of milking cows in relation to productivity of milk/km2/year and milk/cow/year (r = 0.95,

Parameter Correlation (r) Parameter Correlation (r)

herd 0.94 Milk (kg/ha) 0.13

Total milking cows 0.93 Milk (kg/cow/day) 0.11 Area for the herd (ha) 0.67 Milking cows/ha 0.06

Table 6. Linear correlation of daily milk production by producer with: total of animals in the herd, total of milking cows, area for the herd, farmer size and some productivity indexes (daily milk production per hectare and per cow, milking cows per hectare and daily milk

Therefore, the milk production by Brazilian farmers is more dependent on the farmer size and pasture extension, than on productivity indexes, and similar effect occur with crop production (Lana, 2009). Agricultural showed the same results verified with dairy cattle, in

Total area in the farm 0.20 Milk (kg/total of

Area for the herd (ha)

Milk (Kg/milking cow/day)

Herd (number of animals)

Milk (Kg/total herd/day)

animals/day) -0.11

Milking cows (number)

Milking cows/total herd

Production per producer (Kg of milk/day)

Milk (Kg/ha/day)

Production (Kg of milk/ producer/day)

Kg of milk/ producer/day

Number of producers

Milking cows/ha

cow, as a function of production levels

0.55 and 0.51, respectively; Table 7).

production per total animals in the herd)

Total animals in the

(Figure 8B), as suggested by Lana (2005; p.290-291) and Lana (2007b; p.39 a 43). Therefore, in order to these systems be compatible with the tropical conditions, in which it is more evident the curvilinear responses to nutrients, it is necessary modifications in future versions, by adopting models of saturation kinetics.

Fig. 7. Production of milk (A) and efficiency of use of concentrate (B) as a function of intake of increasing level of concentrate in three experiments (Pimentel et al., 2006b, 2006c; Teixeira et al., 2006)

#### **6. Production versus productivity**

Two studies were conducted to evaluate the factors that affect milk production in Brazil, on farmer level or by state of federation (Guimarães et al., 2008; Lana et al., 2009). In the first case, data were collected from fifty producers that sell milk for a dairy plant in the south region of Rio de Janeiro state, including data of daily milk production by producer, with the respective data of production per cow and per hectare, farmer size and size area designated to the herd, total of milking cows and herd size, and breed (Table 5). In the second case, data were collected from EMBRAPA and IBGE in the years of 2004-2006, in which the emphasis was in milk production per state instead of production per producer.

Fig. 8. Mean daily weight gain of steers in pastures, observed and estimated by level 1 of NRC (1996) as a function of intake of metabolizable energy and protein in the supplement (A); and observed milk production (mean of data of Figure 7A) and estimated by CNCPS 5.0 and NRC (2001) as a function of intakes of metabolizable energy or net energy of lactation, respectively, and metabolizable protein (B)

(Figure 8B), as suggested by Lana (2005; p.290-291) and Lana (2007b; p.39 a 43). Therefore, in order to these systems be compatible with the tropical conditions, in which it is more evident the curvilinear responses to nutrients, it is necessary modifications in future

Efficiency, kg of milk/ .

Fig. 7. Production of milk (A) and efficiency of use of concentrate (B) as a function of intake of increasing level of concentrate in three experiments (Pimentel et al., 2006b, 2006c; Teixeira

Two studies were conducted to evaluate the factors that affect milk production in Brazil, on farmer level or by state of federation (Guimarães et al., 2008; Lana et al., 2009). In the first case, data were collected from fifty producers that sell milk for a dairy plant in the south region of Rio de Janeiro state, including data of daily milk production by producer, with the respective data of production per cow and per hectare, farmer size and size area designated to the herd, total of milking cows and herd size, and breed (Table 5). In the second case, data were collected from EMBRAPA and IBGE in the years of 2004-2006, in which the emphasis

A

Fig. 8. Mean daily weight gain of steers in pastures, observed and estimated by level 1 of NRC (1996) as a function of intake of metabolizable energy and protein in the supplement (A); and observed milk production (mean of data of Figure 7A) and estimated by CNCPS 5.0 and NRC (2001) as a function of intakes of metabolizable energy or net energy of lactation,

Milk (kg/cow/day) .

0123456 Concentrate (kg/cow/day)

Obs milk Est milk

B

kg of concentrate

0 0.2 0.4 0.6 0.8 1 1.2

> 01234567 Concentrate, kg/animal/day

Exp1 Exp2 Exp3 B

Exp1 Exp2 Exp3 A

versions, by adopting models of saturation kinetics.

01234567 Concentrate, kg/animal/day

was in milk production per state instead of production per producer.

Observed ME allowed MP allowed

0 0.8 1.6 2.4 3.2 Supplement, kg/animal/day

**6. Production versus productivity**

0 0.3 0.6 0.9 1.2 1.5

respectively, and metabolizable protein (B)

ADG, kg/animal/day .

Milk, kg/animal/day .

et al., 2006)

Table 5. Number of producers, daily mean milk production by producer, area for the herd, number of animals and milking cows in the herd, and productivity indexes per area and per cow, as a function of production levels

The milk production in farmer level (first case) ranged from 60 to 4000 kg/producer/day. The increase in milk production was highly correlated with the number of milking cows (r = 0.94), followed by moderate correlation with the size of pasture (r = 0.67) and, surprisingly, the productivity per cow and per unit of area did not correlate with the milk production per producer (r = 0.11 and 0.06, respectively; Table 6). In the nation level (second case), the result repeated, in which there was high correlation of milk production/state/year with the total of milking cows in relation to productivity of milk/km2/year and milk/cow/year (r = 0.95, 0.55 and 0.51, respectively; Table 7).

Table 6. Linear correlation of daily milk production by producer with: total of animals in the herd, total of milking cows, area for the herd, farmer size and some productivity indexes (daily milk production per hectare and per cow, milking cows per hectare and daily milk production per total animals in the herd)

Therefore, the milk production by Brazilian farmers is more dependent on the farmer size and pasture extension, than on productivity indexes, and similar effect occur with crop production (Lana, 2009). Agricultural showed the same results verified with dairy cattle, in

Rationality in the Use of Non Renewable Natural Resources in Agriculture 281

The agriculture progress is based in improvements of animals and plants productivity per unit of area, which is only applicable when land is the limiting factor, but other factors are

Models of saturation kinetics are important tools to improve the efficiency and decrease costs of utilization of non renewable natural resources in agriculture, allowing the conservation of these resources for the future generations, and decreasing the negative

Fig. 9. Effect of cultivated area and productivity on production of some main cultures (coffee, sugarcane, corn and bean), in municipalities of Zona da Mata and Central of Minas

Http://scotaaron.com/resources2.html

inc., ISBN 0-8247-8994-6, New York, NY, United States

Aaron, S. (2005). *Some Statistics on Limited Natural Resources*, 31.07.2006, available from

Angus, J.F. (1995). Modeling N Fertilization Requirements for Crops and Pasture, In:

*Nitrogen Fertilization in the Environment*, P.E. Bacon, (Ed.), 109-127, Marcel Dekker

emerging as limiting, such as water, fertilizer and petrol.

**7. Conclusions**

impacts in the environment.

Gerais state, Brazil.

**8. References**

which cultivated area generally presents more than 90% correlation with crop production (Table 8 and Figure 9). Then, the concepts about agricultural production need to be revised, facing the actual problems related with the inadequate use and depletion of the non renewable natural resources, and environmental pollution.

Table 7. Linear correlation (r) of annual milk production by Brazilian states with total of milked cows, liters of milk/km2/year, liters of milk/cow/year and surface of the state (in km2).

Source: Lana & Guimarães (2010); n = number of municipalities; 1a = first harvest; 1b = mean of second and third harvest; 2 = racemes instead of ton (1,000 kg); 3 = number (x1000) of oranges instead of ton (1,000 kg); 4a = rice planted in wet land; 4b = rice planted in dry land, without or with irrigation; 5a,b,c = harvest 1, 2 and 3, respectively; Source of data: www.cidadesnet.com.br (year of 2003).

Table 8. Linear correlation (r) among some variables related to agricultural production, in municipalities of Zona da Mata and Central of Minas Gerais state, Brazil.