University of Notre Dame AME 30334 Design Project
University of notre dameHeat Transfer
Design Project
Joshua Szczudlak
5/4/2012
Scientists study the world as it is; engineers create the world that has never been.
-Theodore Von KarmanTable of Contents
1 Problem Statement 2
2 Discussion 2
2.1 Assumptions 2
3 Analysis 3
4 Results and Conclusions 6
5 References 7
List of Tables
Table 1. Given Parameters 3
Table 2. Material Properties 3
List of Figures
Figure 1. Geometric representation of the problem 2
Figure 2. COMSOL model of the metal strip and plastic film 4
Figure 3. Graphical representation of the creation of the boundary layer 5
Figure 4. Maximum and minimum temperature history of the film 6
1 Problem Statement
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 for a specified period of time across the top surface of the thin adhesive-backed film to affix it to the metal strip. 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. The problem is to determine the minimum period of time 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.
2 Discussion
For such as sensitive of a manufacturing process as this is, accuracy becomes very important. It is useful to consider as many modes of heat transfer as possible. Therefore, the following modes of heat transfer will be considered: (1) Radiation from all surfaces (2) Convective cooling of the top and bottom surface of the strip (3) Conduction from the film to the strip and (4) Radiative heating of the film by the laser.
2.1 Assumptions
We will make a few initial assumptions; additional assumptions will be made as the discussion of the problem develops. The initial assumptions are: (1) Constant properties, (2) The only heat source is the heat of the laser and that is totally absorbed not reflected. A representation of the system as given in the problem statement is show in Figure 1.
Figure 1. Geometric representation of the problem
Table 1 shows the parameters given in the problem statement.
Table 1. Given Parameters
Parameter / ValueStrip Thickness / D=1.25 mm
Strip Width / W=600 mm
Strip Length / L=600 mm
Film Thickness / d=0.1 mm
Film Width / w=500 mm
Film Length / l=44 mm
Ambient Temperature / T∞=25℃
Free-stream Velocity / u∞=10 m/s
Constant Heat Flux / qo''=85,000 W/m2
Minimum Cure Temperature / Tmin=90℃
Maximum Cure Temperature / Tmax=200℃
In order to run an accurate model of the problem additional parameters needed to be supplied to COMSOL. Table 2 gives a list of these parameters and their assumed values. For a list of references used in obtaining these values see the References section below.
Table 2. Material Properties
Property / Assumed ValueAir Prandtl Number / Prair=0.713
Emissivity of Plastic / εplastic=0.91
Emissivity of Steel* / εsteel=0.70
Conductivity of Plastic / kplastic=0.45 W/(m∙K)
Conductivity of Steel* / ksteel=43 W/(m∙K)
Conductivity of Air / kair=0.0257 W/(m∙K)
Density of Air / ρair=1.205 kg/m3
Kinematic Viscosity of Air / υair=15.11 x 10-6 m2/ K
*Type 310 Rolled Steel
3 Analysis
The majority of the analysis was done numerically using the finite element program COMSOL Multiphysics with accompanying analytical solutions to support the findings. The base model used was Heat Transfer in Solids. This model included all of the aforementioned modes of heat transfer such as radiation from all surfaces, convective cooling of all exposed surfaces, conduction of the film to the strip, and the radiative heating of the film by the laser. Figure 1 shows the COMSOL model.
Figure 2. COMSOL model of the metal strip and plastic film
One thing that COMSOL does not handle well is turbulence. If the flow over the strip crosses into the turbulent regime the values predicted by the model may be off. Therefore it is a beneficial exercise to compute analytically the boundary layer characteristics of the problem. The first step in any boundary layer calculation is to compute the relevant Reynolds numbers. The local Reynolds number can be computed using the equation,
Rex=u∞xυ (1)
where u∞ is the free stream velocity, υ is the dynamic viscosity, and x is the position along the metal plate. The Reynolds numbers of interest are the total Reynolds number over the whole plate and the Reynolds number at the leading edge of the plastic film because they will give us insight into the boundary layer. The Reynolds number at the leading edge of the film is ReLE = 1.84 x 105 and the Reynolds number over the length of the plate is ReL = 3.97 x 105. In both cases the flow is lower than the assumed critical Reynolds number of 5 x 105 and is therefore laminar.
The next step in the analysis is to determine if the film is thick enough to trip the laminar boundary layer to turbulent. This is found by first determining the boundary layer thickness at the leading edge of the film with the equation,
δ=5xRex (2)
where δ is the boundary layer thickness. The boundary layer thickness at the leading edge of the film is 3.24 mm. This means that the film thickness is only 3.1 % of boundary layer thickness and is therefore negligible and will not trip the boundary layer.
Figure 3. Graphical representation of the creation of the boundary layer
For the remainder of the analytic support, two additional assumptions need to be made: (3) The convection across the top and bottom surfaces is uniform, and (4) The mass and thermal resistance of the film are negligible, and (5) The temperature of the plate is can be considered constant at all positions, which shall be proved later.
We use this knowledge of the boundary layer to estimated a value for the convection heat transfer coefficient, h, using the equation for the average Nusselt number under laminar flow.
Nux=hxk=0.664Rex1/2Pr1/3 / (3)Using Equation 3 the average Nusselt number over the whole plate was 373.76. This gave an approximate convective heat transfer coefficient of 16.01 W/(m2∙K).
Bi=hLck / (4)where Lc is the characteristic length of the body. By neglecting the mass and thermal resistance of the film we can assume the entire system can be modeled by the metal strip. The Biot number of the metal strip is then Bi=2.33 x 10-4 which allows us to make a lumped capacitance assumption. It is for this reason that we were able to make assumption (5) that the temperature of the plate was constant.
These assumptions allow us to estimate the increase in temperature per unit time of the film/strip system. Using an energy balance it can be shown that,
qtot=qlaser-qconv-qrad-qcond / (5)where qconv=h∆TSAstrip, qrad=εσAT4, and qlaser=85,000 W/m2∙KSAfilm. Also, because of the assumption of lumped capacitance qcond=0. Additionally,
qtotΔt=ρcpVΔT / (6)Using Equation 5 and Equation 6 as well as given parameters and assumed values a temperature gradient can be found. This temperature gradient is approximately 12℃/s.
4 Results and Conclusions
The COMSOL model was the major tool used to determine the amount of time laser time required. Figure 3 shows the volume maximum and volume minimum temperature history of the film.
(a) ∆Ton=8 s / (b) ∆Ton=9 sFigure 4. Maximum and minimum temperature history of the film
The laser time was chosen through an iterative process. The first ∆Ton chosen was 10 s which stayed over the 90℃ limit for approximately 25 s. The next choice was to decrease the ∆Ton to 8 s. Data from this iteration is plotted in Figure 3 (a). Although maximum temperature in the film volume is well below the upper cure limit, the minimum temperature in the volume does not stay above the lower limit for 10 s. The obvious next step was to step up the laser time to 9 s. This gave the desired results. The temperature of the film stayed above the 90℃ limit for just over 10s. However for monetary reasons it would be beneficial to decrease this laser time as much as possible. By iterating between the two cases presented in Figure 3 the approximate minimum laser temperature, ∆Ton, is 8.75 s. This allows for a cure time of over 10 s but leaves enough extra time in the curing process to account for any minor impurities in the film.
The COMSOL model can be verified against the approximate analytical temperature gradient by estimating the temperature gradient in the 8 s laser model presented in Figure 3 (a). The temperature gradient of the volume minimum was approximately 8.33℃/s and the temperature gradient of the volume maximum was approximately 16.1℃/s. This means that the approximate average temperature gradient is 12.2℃/s, which corresponds quite nicely with the analytical model.
5 References
[1] http://www.engineeringtoolbox.com/emissivity-coefficients-d_447.html
[2] http://www.omega.com/temperature/z/pdf/z088-089.pdf
[3] http://www.engineeringtoolbox.com/thermal-conductivity-d_429.html
[4] http://www.nd.edu/~paolucci/AME30334/Design_Project/NP.pdf
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