Page 174 - Biosystems Engineering

Models for Heat Transfer in Heated Substrates 153

of a minimum number of boundary conditions: initial distribution of
temperatures in the system formed by the substrate, insulator, and
greenhouse air temperature, substrate surface temperature, and heating-
cable temperature.
Under natural conditions, the model showed good agreement
with experimental data, regardless of the values and methods used to
determine the thermal properties of the substrate, and of the intro-
duction of substrate properties according to depth (compaction,
moisture, and presence of a differentiated surface layer). However,
when an artificial heat source was present, simulation accuracy
depended on several factors. After having tested different methods to
estimate the thermal properties of the substrate, the authors found
that the method by Campbell et al. (1994) generated the best response
and that considering the variation of substrate properties at depth
affected the accuracy of simulation results.
The final model was validated by using an experimental test con-
sisting of nine geometric configurations for substrate electric cable
heating using a sand substrate (Fernandez et al. 2005b). Temperature
was measured at nine locations and moisture content was measured
at three locations inside the substrate. The experimental validation
was conducted after having simulated each geometric configuration
during 3 days, under various environmental and operational condi-
tions. Predictions for the root zone (0 to 150 mm depth) were gener-
ally acceptable with slight overestimations. The mean value of the
RMSE at the vertical of the heating cable was 0.30°C, whereas its
maximum value did not exceed 0.61°C. For locations between cables,
prediction was less exact, and RMSE values ranged from 0.12 to
1.00°C, with a mean value of 0.57°C. Overestimations in the root zone
occurred mainly when room temperatures were high. Drying of the
surface layer could account for such overestimations. Drying would
produce a sharp decrease in the effective thermal conductivity of the
upper layer of the substrate, which was not considered in the model.
The coefficient of determination reached values close to 1. Errors
were acceptable, with maximum values observed in zones different
from the root development zone, which was less relevant for the
study. Therefore, the model properly described the thermal state of
heated substrates and provided a useful tool for designing and mon-
itoring this type of substrate-heating system.
By combining dimensional analysis with the FEM-based physical
model described above, Fernandez and Rodriguez (2006) expanded
the range of application of the dimensioning expressions used for
electric cable–heating systems and achieved cable spacings of 350 mm
for a 225-mm depth. Their research was structured in two parts,
according to the mathematical model used: (1) application of a FEM-
based physical model to simulate different configurations of the heat-
ing system and (2) application of dimensional analysis to the results of
the model in order to obtain the relevant dimensioning expressions.