ABSTRACT
The present thesis in order to obtain the doctoral degree is outlined in frame
of the thermal mathematical models research lines within the Interdisciplinary
Modelling Group Intertech [8], [9]. In particular, this thesis presents some mathematical
models for the heat transmission in industrial flat grinding [10], [11],
using deeply various special functions from Physics-Mathematics, such as Hypergeometric
functions, Hermite functions or Bessel functions [12].
The thesis begins with a chapter devoted to the derivation of the heat equation,
starting from first principles. The presence of a convective term will be
shown to be equivalent to the application of a coordinate transformation [13].
This heat equation will be used to model heat transfer in flat grinding. Next we
will study the Green function for the heat equation in the absence of convection,
both in the 1-dimensional and in the 3-dimensional case. From these results, the
following chapter presents the Jaeger model of flat grinding [14], [15]. This latter
model uses a moving, infinite linear strip sliding on the workpiece surface.
The properties of Bessel functions will turn out to be crucial in order to derive
a compact analytic expression for the temperature field within the workpiece
[16], [17]. Next we turn our attention to the SV (Samara-Valencia) model of flat
grinding [18], [19]. In the SV model one solves the heat equation in the presence
of a convective term, with variable initial and boundary conditions. This flexibility
in the initial and boundary conditions renders the SV model more general
than the Jaeger model, thus allowing for the inclusion of lubricating fluid and
also for intermittent grinding wheels. Moreover, they also allow one to obtain
the temperature field in the transient regime, and not just in the stationary
state. Having done this, the Jaeger and the SV models are compared in the
stationary state, in the case of continuous dry grinding. The solution to the SV
model will be given by the sum of two terms, denoted T (0) and T (1). In the
first place, we will prove analytically that T (0) equals the Jaeger solution for
an infinite workpiece. Particularizing this equality to the workpiece surface will
allow us to obtain the value of an improper integral that has not appeared tabulated
anywhere yet [20]. In the second place, we will obtain an expression for
T (1) and prove that the surface values of T (0) and T (1) are equal. This will lead
to an expression for the Dirac delta function that has also not been published
anywhere yet [21]. Since the solutions to the Jaeger and the SV models are each
unique [22], this will yield the value of one more improper integral that also does
not appear in the standard tables [20]. Following this, a quick method will be
presented to compute the maximal surface temperature, with numerical results.
Finally, the temperature field for a titanium VT20 alloy workpiece [23], [24] will
be computed numerically, both following the Jaeger model and the SV model.
This will provide a numerical check of the equivalence between the two models,
previously proved analytically. The MATLAB programmes used for numerical
work will be presented in the appendix. In the final chapter, the SV model will
be used to determine the transient temperature field in the case of dry grinding,
and also in the simplest case of wet grinding. This latter case assumes a constant
heat transfer coefficient between the workpiece surface and the lubricating fluid.
From the solutions for the temperature field we will conclude that wet grinding
merely adds a correction term to dry grinding, the correction being proportional
to the heat transmission coefficient. Next we will develop a method to obtain
the maximal temperature in the case of wet grinding, as well as a criterion to
determine the relaxation time (both in wet and in dry grinding). Graphs will
be presented that show the time evolution of the temperature field in all cases
studied. The corresponding MATLAB programmes will also be given in the
appendix.