4.3 R-fields

The heat transfer phenomena in the thermal processes are modelled by the R elements and according to the causality attributed we determine the variable effort (e) and flux (f) by using the following relation:

$$f = \varphi\_{\mathbb{R}}^{-1}(\mathfrak{e})\tag{15}$$

Note: the thermal resistance R is equal to the inverse of the heat transfer coefficient h (R = 1/h).

Convective heat transfer phenomena between hot air and agricultural product are modelled by Rc1, where the convective heat flux is expressed by:

$$
\dot{Q}\_{c1} = \frac{1}{R\_{c1}} \left( T\_{ha} - T\_{pr} \right) A\_{pr} = h\_{c1} \left( T\_{ha} - T\_{pr} \right) A\_{pr} \tag{16}
$$

Apr and hc<sup>1</sup> are respectively the area of product and the convective heat transfer coefficient.

$$h\_{c1} = h\_c = Nu \frac{\lambda\_a}{D} \tag{17}$$

Nu is the Nusselt number determined by the Reynolds number (Re), which provides information on the flow regime.

λ<sup>a</sup> and D are respectively the thermal conductivity of the air and the characteristic diameter of the layer of the product.

The Reynolds number is given by this relation:

$$\text{Re} = \frac{U\_a \, l}{v} \tag{18}$$

The airflow will certainly be turbulent in the dryer, to calculate the number of Nusselt we use the following correlation [21]:

$$\text{Nu} = 0, \text{O23.} \,\text{Re}^{0.8}. \text{Pr}^{0.4}, \text{Pr} = \text{0.7} \,(\text{Prandtl number}). \tag{19}$$

Rc2 models the convection heat transfer phenomena between the agricultural product and the moist air in the chamber, the convective heat flow is given by:

$$
\dot{Q}\_{c2} = \frac{1}{R\_{c2}} (T\_{pr} - T\_{ma}) A\_{pr} = h\_{c2} (T\_{pr} - T\_{ma}) A\_{pr} \tag{20}
$$

hc<sup>2</sup> ¼ hc is the convective heat-transfer coefficient.

The phenomena of heat transfer by convection between the humid air and the internal wall of the chamber are modelled by the element Rc3 in which the convective heat flux is expressed by:

$$
\dot{Q}\_{c3} = \frac{1}{R\_{c3}} (T\_{ma} - T\_{wa}) A\_{wa} = h\_{c3} (T\_{ma} - T\_{wa}) A\_{wa} \tag{21}
$$

where Awa is the area of the wall and hc<sup>3</sup> ¼ hc is the convective heat transfer coefficient between the moist air and the inner wall, taking as Nusselt number [22]:

$$Nu = 0,036. \,\text{Re}^{4/5}. \text{Pr}^{1/3} \tag{22}$$

Rcd models the conduction heat transfer phenomena the chamber walls and the external environment through the insulation, the conduction heat flow is given by:

$$
\dot{Q}\_{cd} = \frac{1}{R\_{cd}} (T\_{uu} - T\_{ma}) A\_{uu} = h\_{cd} (T\_{uu} - T\_{ma}) A\_{uu} \tag{23}
$$

hcd is the conductive heat-transfer coefficient across the insulation and estimated by:

$$h\_{cd} = \frac{\lambda\_i}{d\_i} \tag{24}$$

λ<sup>i</sup> is the thermal conductivity of the insulation and di is the average mean thickness of the insulation.

Revap models the evaporation heat transfer phenomena from the agricultural product and the moist air in the drying chamber, the evaporation heat flow is given by:

$$\dot{Q}\_{enap} = \frac{1}{R\_{enap}} \left( T\_{pr} - T\_{ma} \right) A\_{pr} = h\_{enap} \left( T\_{pr} - T\_{ma} \right) A\_{pr} \tag{25}$$

$$h\_{evap} = 0.016 h\_c \frac{\left[P(T\_{pr}) - \chi\_{ma} P(T\_{ma})\right]}{\left(T\_{pr} - T\_{ma}\right)}\tag{26}$$

hevap is the evaporative heat transfer coefficient [23] and γ is the relative decimal humidity and P(T) the saturated vapour pressure given by Jain and Tiwari [22]:

$$P(T) = \exp\left[25.317 - \frac{5144}{T + 273.15}\right] \tag{27}$$

Revac models the phenomenon of discharge of humid air to the outside through the chimney for a natural type of flow [24], the corresponding heat flow is given by:

$$
\dot{Q}\_{enac} = c\_d A\_e \sqrt{2g\Delta H} \Delta P \tag{28}
$$

ΔP and ΔH are the difference in partial pressure and the difference in pressure head (m), respectively.

$$
\Delta P = \left[ P(T\_{ma}) - \chi\_{am} P(T\_{am}) \right] \tag{29}
$$

$$
\Delta H = \frac{\Delta P}{\rho\_d \mathbf{g}} \tag{30}
$$

#### 4.4 (0.1)-junctions

Using the mathematical properties for the junctions (0.1):

( P i ei ¼ 0) for 1-junctions.

( P i fi ¼ 0) for 0-junctions.

The energy flow balances equations are:

• energy balance equation of the product

$$
\dot{Q}\_{pr} = \dot{Q}\_{c1} - \dot{Q}\_{c2} - \dot{Q}\_{enap} \tag{31}
$$

• energy balance equation of the moist air in the drying chamber

$$
\dot{Q}\_{\text{mat}} = \dot{Q}\_{c2} + \dot{Q}\_{enap} - \dot{Q}\_{c3} - \dot{Q}\_{enac} \tag{32}
$$

Dynamic Modelling by Bond Graph Approach of Convective Drying Phenomena DOI: http://dx.doi.org/10.5772/intechopen.91276

• energy balance equation of the wall of the drying chamber

$$
\dot{Q}\_{\text{out}} = \dot{Q}\_{c3} - \dot{Q}\_{cd} \tag{33}
$$

The above equations determined by bond graph elements can be used to develop the detailed equations for the energy flow balance:

• energy balance equation of the product

$$C\_{pr}\frac{dT\_{pr}}{dt} = h\_{c1}(T\_{ha} - T\_{pr})A\_{pr} - \left(h\_{c2} + h\_{evap}\right)(T\_{pr} - T\_{ma})A\_{pr} \tag{34}$$

• energy balance equation of the moist air in the drying chamber

$$\begin{split} \mathbf{C}\_{ma} \frac{dT\_{ma}}{dt} &= \left( h\_{c2} + h\_{enap} \right) \left( T\_{pr} - T\_{ma} \right) A\_{pr} \\ &- h\_{c3} (T\_{ma} - T\_{uu}) A\_{uu} - c\_d A\_c \sqrt{2g\Delta H} \Delta P \end{split} \tag{35}$$

• energy balance equation of the wall of the drying chamber

$$\mathbf{C}\_{wa}\frac{dT\_{wa}}{dt} = h\_{c3}(T\_{ma} - T\_{wa})A\_{uu} - h\_d(T\_{uu} - T\_{am})A\_{uu} \tag{36}$$
