A factory would like to produce plain carbon steel strips with pieces of
polyethylene plastic film bonded on them. The bonding operation will use a
laser that is already available to provide a constant heat flux of q′′0= 85, 000 W/m2for a specified period of time, ∆ton, across the top surface of the thin
adhesive-backed film, to be affixed to the metal strip as shown in the sketch.
The metal strip has a thickness D = 1.25 mm, length L = 600 mm, and width
W = 600 mm. The plastic film is perfectly centered on the metal strip and has
thickness d = 0.1 mm, length l = 44 mm, and width w = 500 mm. The heat
flux is applied over the strip’s complete width of 600 mm. The strip is initially
at the ambient temperature of 25◦C and located on a conveyor belt made of an
insulating open mesh material so that the upper and lower surfaces of the strip(including the plastic film) are exposed to air blowing as shown along the length of the plate at 10 m/s.
In order for the film to be satisfactorily bonded it must be cured above 90◦C for 10 s and the plastic film will degrade if a temperature of 200◦C is exceeded. Determine the minimum period of time ∆ton necessary for proper curing and thus optimize productivity of the metal strips, since each strip will have to remain stationary under the laser during the bonding operation.
All modes of heat transfer must be considered, and any assumption must
be justified. If a computer program is necessary, the accuracy of the program
as well as the results need to checked. For example, it may be possible to
check the program by comparing numerical results using different resolutions
to show grid convergence, and against analytical results, obtained for some
limiting situations (e.g. steady state), to show correctness of the program. Your
report will be graded on the basis of the physical understanding exhibited, the execution of the project, and the clarity of the writing and reasoning presented.
To determine the minimum period of time required for proper curing of the plastic film on the carbon steel strip, we need to consider heat transfer mechanisms and calculate the temperature profile of the strip.
First, let's identify the different modes of heat transfer that are present in this scenario:
1. Conduction: Heat transfer through the solid carbon steel strip via conduction.
2. Convection: Heat transfer from the surface of the strip to the surrounding air via forced convection due to the air blowing along the length of the plate.
3. Radiation: Heat transfer through radiation from the surface of the strip to the surroundings.
To solve this problem, we can use the following steps:
Step 1: Calculate the heat flux absorbed by the plastic film
The heat flux provided by the laser, q′′0, is 85,000 W/m². Since the heat flux is applied over the entire width of the strip, we can calculate the heat absorbed by the plastic film using the following equation:
Q = q′′0 * w * l
Where:
Q is the heat absorbed by the plastic film,
q′′0 is the heat flux,
w is the width of the plastic film, and
l is the length of the plastic film.
Step 2: Calculate the temperature rise of the plastic film
Assuming no heat loss to the surroundings during the heating process, we can use the specific heat capacity of the plastic film to calculate the temperature rise, ∆T, using the following equation:
Q = mc∆T
Where:
m is the mass of the plastic film (density * volume),
c is the specific heat capacity of the plastic film, and
∆T is the temperature rise.
Step 3: Calculate the heat transfer from the plastic film to the carbon steel strip
Assuming the heat transfer from the plastic film to the carbon steel strip occurs only through conduction and neglecting any heat loss from the strip's bottom surface, we can calculate the heat transfer using Fourier's law of heat conduction:
q = -k * (dT/dx)
Where:
q is the heat transfer rate from the plastic film to the strip,
k is the thermal conductivity of the steel, and
(dT/dx) is the temperature gradient across the film thickness.
Step 4: Calculate the temperature distribution across the carbon steel strip
Using the heat transfer rate determined in Step 3, we can solve the heat conduction equation considering the strip's geometry and boundary conditions, such as the strip's initial temperature and the convective heat transfer from the strip's surface. This can be solved either analytically or numerically using a finite element method.
Step 5: Determine the minimum time required for proper curing
The minimum time required for proper curing is the time needed for the highest temperature on the plastic film to exceed 90°C and remain below 200°C for at least 10 seconds. This can be determined from the temperature distribution calculated in Step 4.
By following these steps, it is possible to determine the minimum period of time necessary for proper curing of the plastic film on the carbon steel strip.
It is important to note that the above steps provide a general approach to solving the problem. However, the specific calculations and assumptions may vary depending on the actual variables provided in the problem. Additionally, an accurate numerical or analytical solution would require more detailed information and the use of appropriate software or techniques.