SAMPE Journal - July/August 2022 - 48

FEATURE / HIGH-RATE AIRCRAFT MANUFACTURING
a two- or three- dimensional heat transfer finite
element approach would provide a more precise
and detailed
thermal profile
in the long-run.
The governing equation (Equation 1) for onedimensional
heat transfer problem can be stated as
(1)
where a is thermal diffusivity as a function
of material characteristics such as thermal
conductivity, density and specific heat; T is the
temperature through the thickness direction y; V is
the roller head speed, and t and x and are the time
and location on the tape surface, respectively. The
boundary conditions on the problem include the
heat flux on the tape top surface, and an insulation
boundary on the far side of the substrate.
The substrate tape layers built up on the tool
can be considered sufficiently large compared
to the thin thermoplastic incoming tape that is
placed as a new substrate layer. Therefore, the
substrate layers can be idealized as a semi-infinite
solid for seeking a solution to the thermal model of
the heated substrate tape. The predominant mode
of heat transfer into the tape is by conduction12
.
Therefore, a closed form solution can be readily
obtained for the temperature on the tape surface
as well as the temperature through the thickness of
the substrate y (Equation 2), for any time duration
of laser exposure14
at the given heat flux on the tape
surface.
Where T is the temperature and ax and ay
(2)
where Ti is the initial temperature of the material;
T is the final temperature at depth y and time t; k
and a are the thermal conductivity and diffusivity
of the material; ierfc is the complementary error
function; and the laser heat flux qo
the laser power and absorptance of the material.
The heat flux on the tape surface at the roller nip
point can be calculated12
for a known incident
angle of the laser beam on the substrate. A onedimensional
transient solution is sought for the
thermal response of the tape in the vicinity of
roller nip point. The heat penetration depth in
this semi-infinite substrate model of undefined
depth is seen as a function of the square root of the
thermal diffusivity and elapsed time. This means
that the thermal penetration depth depends on the
time of exposure of the material beyond its initial
temperature. For a certain fraction of exposure time
there exists a critical depth at which the substrate
will not feel the effect of laser heat moving at the
speed of the roller head.
(4)
and
Figure 6. Representation of the finite difference modeling approach.
are the thermal diffusivity coefficients in x and y,
chosen based on the heat capacity and directiondependent
thermal conductivity coefficients as
well as the grid spacing in x and y.
In this method, a 2D grid is created for each
'' is a function of
thermal domain T, Figure 6. The size of the grid
elements and the timestep are determined to
maintain stability for the Euler method according
to the criteria for each dimension and each region:
However, a sharp increase in tape surface
temperature can be expected with a high intensity
laser. As the tape section approaches the roller
nip point, a section would no longer be under the
laser spot, and the absence of heat flux results in
immediate cooling of the tape. The temperature
sustained in the tape after a fraction of exposure
time just before undergoing consolidation at the
nip point is critical to the formation of crystallinity
and bonding of the incoming tape. Although, more
detailed consideration of thermal contact of the
tape with the roller and substrate at the nip point
are possible, the cooling response of a tape is
readily obtained from a simplified solution14
heat transfer differential equation for conduction.
of the
Finite Difference Thermal Model
A thermal model was developed using a twodimensional,
explicit finite difference method,
and Euler method for solving the time-dependent
heat equation over a two-dimensional grid. This is
perhaps the simplest numerical method for solving
the 2D time dependent heat equation, chosen
for its straightforward implementation (enabling
easy modification and control over boundary
conditions) and its well-defined stability criteria.
The time-dependent heat equation (Equation 3)
can be modeled as:
(3)
48 | SAMPE JOURNAL | JULY AUGUST 2022
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