Page 170 - Biosystems Engineering

Models for Heat Transfer in Heated Substrates 149

Therefore, a model that can be applied under conditions of artificial
substrate heating is needed to improve environmental conditions for
crop development. There are four ways in which such a model can be
distinguished from models subject to natural conditions:
First, the geometry of the system to be modeled is more complex
because of the heat supplied by the heating system and its distribu-
tion within the substrate. Second, thermal properties of heated sub-
strates show greater variability than the thermal properties of soils
under environmental conditions, which depend on other physical
properties such as texture, degree of compaction, or water content
(de Vries 1963; Ochsner et al. 2001). Factors that affect thermal
properties can depend on two variables: space and time. In addition,
studies by other authors (Campbell et al. 1994; Tarnawski et al. 2000)
have reported a variation in apparent thermal conductivity depend-
ing on temperature. Considering the thermal gradients within heated
substrates, the variation of apparent thermal conductivity cannot be
disregarded in thermal analysis.
Third, the temporal variation of temperatures requires thermal
analysis to be performed in a transient state, which allows modeling
for long periods, which are determined by the crop growth cycle. It
also allows modeling at short-time intervals, which can be applied as
a base for control (Challa and van Straten 1993). Finally, the models
should be accessible to potential users, mainly, greenhouse crop pro-
ducers, technicians, and suppliers of heating installations.
A number of models that can be applied to temperature predic-
tion in heated substrates have been developed. Alvarez et al. (1996)
analyzed the thermal behavior of greenhouses with an underground
heat source at a fixed depth, based on the modeling of the different
energy processes involved. They developed a one-dimensional ana-
lytical model by using a Green’s function solution in the Laplace
domain and the fast Fourier transform (FFT) to solve the model.
Model parameters were optimized by using a SIMPLEX algorithm
during a 3-day simulation that showed the ability of the method to
estimate greenhouse soil temperature. The method provided a rea-
sonable description of heat flux in greenhouse substrates heated by
warm-water pipes.
In this model, the soil boundary conditions were soil surface tem-
perature and heating pipe temperature. Surface temperature was esti-
mated from outdoor temperature data, net solar radiation on the green-
house, and wind speed outside the greenhouse. The soil was considered
as a continuous, uniform medium with constant properties at various
depths. The depth of the heating system was designed as a variable
that could be modified. An initial adjustment was performed in order
to obtain the experimental parameters required for good performance
of the model. As a result, temperature data were obtained at two soil
depths (10 cm and 20 cm) at hourly or lower frequencies.