Values of parameters used in numerical simulation.
Drying operations play an important role in food industries. They are often the last operation of the process of manufacturing a product, with a strong influence on the final quality. The processes are numerous and depend on the type and amount of product to be dried and water to be evaporated, the desired final quality, or the desired functionality for the dried product. In this chapter, we present a modeling study of heat transfer during drying a moist agricultural product placed in a hot air flow in a tunnel dryer with partial solar heating. The bond graph approach has been used for system modeling, and it is an object-oriented graphical approach based on an energetic description between subsystems. Some drying tests have been carried out on tomatoes and the experimental results are compared with the theoretical results for the validation of the developed model.
- tunnel dryer
- bond graph modelling
- thermal transfer
- moisture content
Dehydration and drying involves the partial or complete elimination of the water contained in the food. Due to low water activity (
The shape or physical state of the product to be dried can help the industrialist to choose the right way or technique to practice drying. This requires a high heat input to cause the water to be easily removed from the wet product.
The drying device is a partial solar heating tunnel dryer ( Figures 1 and 2 ); it uses solar energy as a source of renewable energy that represents an available market, furthermore to be practically free. Heat and mass transfers were investigated with the development of a mathematical model of the thin layer solar drying of food products, in a convective tunnel dryer.
Multiple numerical and experimental studies on convective drying were realized [4, 5]. On the one hand, they focus on understanding the phenomena that govern the internal migration of moisture, heat and mass transfer at the level of the air-product contact [6, 7]. On the other hand, they studied the physical analysis of the dryers and the optimization of their behaviour [8, 9, 10].
We recall that the drying is a complex phenomenon, and it includes many other phenomena that emerge from fluid mechanics, thermodynamics and heat and mass transfer [6, 11]. These phenomena are in turn playing a leading role in the drying process. The modelling tool used in this work is the bond graph approach . It can model multi-disciplinary systems  whose their comportment is nonlinear. This is explained by the diversity of physical phenomena (thermal, mechanical, electrical, thermodynamic, chemical, etc.). The bond graph approach is a modelling tool that provides both the behaviour and the necessary analysis of models. It is a causal and modular energy modelling approach allowing the generation of ordinary differential equations. The bond graph approach can be used as a solution for the design and understanding of physical phenomena in complex industrial systems.
In process engineering, the use of the pseudo-bond graph approach is often widespread for reasons that have been widely justified [13, 14, 15], but for this approach, the product of effort and flow has not the dimension of power.
The first part of this work focuses on the development of a graphical model and the deduction of mathematical equations that describes the phenomena of heat and mass transfer for convective drying process. In the second part, we show an analysis of the results with the influence of different aerothermal parameters.
2. Drying kinetics
The theory of drying is described by Lewis , especially during thin film drying.
The most important parameter used in process drying is the drying rate equation :
3. Experimental device
The device studied is an indirect convective solar dryer composed of a tunnel and four solar collectors. The tunnel is of dimensions: 4 m × 2 m × 1.8 m, consisting of a drying chamber and an auxiliary heating system. The width of the drying chamber is 0.9 m with a chimney on the roof used for the evacuation of humid air. The chamber is made of galvanized sheet metal, the side walls of which are insulated with a layer of polyurethane 0.15 m thick ( Figures 1 and 2 ).
The drying chamber contains perforated iron trays. The solar module is formed by four air collectors placed with an inclination between 36.7° and 45° for optimal operation. The ambient air was sucked in by the fans and warmed up then mixed by the solar air forming the drying air. The latter comes into contact with dry agricultural products placed in the trays ( Figures 1 and 2 ).
The auxiliary heating system consists of a gas burner and an exchanger that transfers the heat produced by the burner into the incoming air. The tunnel dryer has two fans in the centre that blow hot air into the drying chamber.
The agricultural products placed on a tray are brought into contact with a flow of hot air; the moist air accumulates in the drying chamber and then escapes through the chimney due to the difference in vapour pressure between the room and the exterior for a type of non-forced flow.
For the study of this drying process, we take into consideration the following hypotheses:
Dry air is often a mixture of solar air and auxiliary air for a hybrid drying operation ( Figure 3 ).
The contact surfaces between air and product remain constant during drying.
The agricultural product to be dried is estimated as a thin layer of water.
The phenomenon of heat transfer by conduction between the grains of products is neglected.
The phenomena of condensation and radiation transfer are neglected inside and outside the drying chamber.
The products to be dried are characterized by their surface temperature.
The phenomena that will be studied:
Convective heat transfer and evaporation between air and product.
Heat transfer by convection between the air and the inner walls of the chamber.
Heat exchange by conduction between the internal air and the external environment.
Evacuation of humid air to the outside through the chimney.
4. Modelling technique
The dynamic behaviour of the thermal processes is generally described by the nonlinear differential equations. Their formulation and resolution by the classic numerical methods are limited . These equations are really associated with the physical phenomena such as storage and energy dissipation. The bond graph approach allows by their graphical description to show these energy exchanges in the system.
The bond graph modelling approach is a unified and causal approach applied to all types of dynamical systems; it allows the modellers to obtain the mathematical model in the form of a state equation easier than the classical modelling methods. Moreover, to provide information on the structural properties of the studied system.
The first modelling step is to divide the global system into subsystems that exchange power with each other; this is the word-pseudo bond graph ( Figure 4 ). The effort and flow variables are marked at the input and output of each subsystem. Depending on the physical phenomena that occur during drying and using the properties of the bond graph approach, the words are replaced by their corresponding elements, which lead to the complete models shown in Figure 5 .
The pseudo-bond graph model of the studied tunnel dryer was developed based on the equations derived from the energy balances of the system.
The variables of effort and flux are respectively temperature (
4.1 Effort sources
The two sources of effort used in this pseudo-bond graph model are
The energy storage phenomena are modelled by the
On the surface of the product modelled by the element
is the thermal heat flow accumulated on the surface of the product and is thermal capacity of products.
with being the mass and being the specific heat of the product.
is the thermal heat flow accumulated in the chamber and is the thermal capacity of the moist air.
, and are respectively the density of the moist air, the volume of the chamber and the specific heat of the moist air.
The accumulation of energy on the inner wall of the chamber is modelled by the
and are the thermal heat flow and the thermal capacity of the inner wall.
, and are respectively the density, the volume and the specific heat of inner walls.
The heat transfer phenomena in the thermal processes are modelled by the
Note: the thermal resistance R is equal to the inverse of the heat transfer coefficient
Convective heat transfer phenomena between hot air and agricultural product are modelled by
and are respectively the area of product and the convective heat transfer coefficient.
The Reynolds number is given by this relation:
The airflow will certainly be turbulent in the dryer, to calculate the number of Nusselt we use the following correlation :
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
where is the area of the wall and is the convective heat transfer coefficient between the moist air and the inner wall, taking as Nusselt number :
is the conductive heat-transfer coefficient across the insulation and estimated by:
is the thermal conductivity of the insulation and is the average mean thickness of the insulation.
are the difference in partial pressure and the difference in pressure head (
Using the mathematical properties for the junctions (0.1):
The energy flow balances equations are:
energy balance equation of the product
energy balance equation of the moist air in the drying chamber
energy balance equation of the wall of the drying chamber
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
energy balance equation of the moist air in the drying chamber
energy balance equation of the wall of the drying chamber
5. Results and discussion
For the numerical evaluation of the thermal performance of the model developed for the tunnel dryer, the calculations were performed using the system parameters ( Table 1 ). The [20-sim] software has been used for all simulations and is dedicated to bond graph simulation. Its use is quite simple and straightforward.
The measurements made were recorded in different operating variables following a series of experiments carried out in order to determine the performance of the studied tunnel dryer. This section presents the experimental results for several tomato drying operations and also a comparison with the theoretical results. The interpretation of the results depends on the influence of the aerothermal parameters (velocity and temperature of drying air).
5.1 Influence of the hot air temperature
In a first step, we consider that the speed of drying air is constant and we only vary its temperature.
Figures 6 and 7 show the evolution of the product temperature and the temperature of the humid air, they reach after a certain time the temperature of the drying air. With the same conditions, we also represent the evolution of the moisture content of the product ( Figure 8 ). Increasing the drying air temperature from 55 to 75°C is accompanied by a reduction in drying time. This is due to a potential increase to water evaporation.
Experimentally, for a temperature of 75°C, an air velocity of 2 m/s is sufficient 125 minutes to dry tomatoes. A decrease in temperature of 10° results in an increase in drying time of 150 minutes to reach this content. Still with a lower temperature of 55°C, the drying time increases up to 180 minutes.
5.2 Influence of hot air velocity
For a constant drying air temperature 55°C and an increase in air velocity beyond 2 m/s does not show a good influence on the variation of the product temperatures and the humid air temperature in terms of reducing the drying time, this is clear in Figures 9 and 10 .
The evolution of the moisture content is shown in Figure 11 we see that the influence of the drying air velocity is less important than the drying air temperature because an increase in the air velocity has brought a small decrease in drying time [27, 28].
The predicted values of the different variables are in good agreement with the experimental values. The quality of the fit was determined using the Root Mean Square Errors (RMSE).
We present in Table 2 below the values of the mean squared error.
||16.1 kg kg−1 (
|Variables||Influence of drying air temperature (Ua = 2 m/s)||Influence of drying air velocity (Tac = 55°C)|
|Tac = 55°C||Tac = 65°C||Tac = 75°C||Ua = 1 m/s||Ua = 2 m/s||Ua = 3 m/s|
|Internal air temperature||0.8034||0.9499||0.8665||0.8589||0.8034||0.9063|
In this chapter, the bond-graph approach has been used for modelling a drying system with partially solar heating. This method provides reliable estimates of temperature distributions in the product and the moist air, also the moisture distributions in the product.
The geometry of the dryer, the physical properties of building materials, agricultural product and air are taken into account.
The influence of two aerothermal factors was studied to evaluate the performance of the dryer. The developed model can be adapted to other wet agricultural products as well as to other drying processes.
The challenge for the engineering designer is now to define optimal dryers, which provide a product of constant good quality. For this, the derived model of the tunnel dryer described by equations (1), (34), (35) and (36) will be used subsequently for the control of heat and mass transfer in drying process, which is important to enhance product quality such as colour and flavour.
This study was supported by the Research Program of Tunisian Ministry of High Education and Scientific Research.
surface area (
thermal capacity (
specific heat (
characteristic diameter of the layer of the product (
mean thickness of the insulation (
gravitational acceleration (
heat-transfer coefficient (
drying constant (
characteristic length (
saturated water vapour pressure (
heat flow plate (
air absolute temperature (K)
product moisture content (
drying rate (
kinematic viscosity of air (
thermal conductivity (
decimal relative humidity
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