UNIVERSITY OF NOTRE DAME
Heat 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 Karman
University of Notre Dame
AME 30334 Design Project
Table 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
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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.
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Table 1. Given Parameters
Parameter
Strip Thickness
Strip Width
Strip Length
Film Thickness
Film Width
Film Length
Ambient Temperature
Free-stream Velocity
Constant Heat Flux
Minimum Cure Temperature
Maximum Cure Temperature
Value
𝐷 = 1.25 mm
𝑊 = 600 mm
𝐿 = 600 mm
𝑑 = 0.1 mm
𝑤 = 500 mm
𝑙 = 44 mm
𝑇∞ = 25℃
𝑢∞ = 10 m/s
𝑞𝑜′′ = 85,000 W/m2
𝑇𝑚𝑖𝑛 = 90℃
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
Air Prandtl Number
Emissivity of Plastic
Emissivity of Steel*
Conductivity of Plastic
Conductivity of Steel*
Conductivity of Air
Density of Air
Kinematic Viscosity of Air
*Type 310 Rolled Steel
Assumed Value
𝑃𝑟𝑎𝑖𝑟 = 0.713
𝜀𝑝𝑙𝑎𝑠𝑡𝑖𝑐 = 0.91
𝜀𝑠𝑡𝑒𝑒𝑙 = 0.70
𝑘𝑝𝑙𝑎𝑠𝑡𝑖𝑐 = 0.45 W/(m ∙ K)
𝑘𝑠𝑡𝑒𝑒𝑙 = 43 W/(m ∙ K)
𝑘𝑎𝑖𝑟 = 0.0257 W/(m ∙ K)
𝜌𝑎𝑖𝑟 = 1.205 kg/m3
𝜐𝑎𝑖𝑟 = 15.11 𝑥 10−6 m2 / K
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.
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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∞ 𝑥
𝜐
(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=
5𝑥
√𝑅𝑒𝑥
(2)
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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 =
ℎ𝑥
𝑘
1/2
= 0.664𝑅𝑒𝑥 𝑃𝑟
1/3
(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 =
ℎ𝐿𝑐
𝑘
(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)
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where 𝑞𝑐𝑜𝑛𝑣 = ℎ∆𝑇(𝑆𝐴)𝑠𝑡𝑟𝑖𝑝, 𝑞𝑟𝑎𝑑 = 𝜀𝜎𝐴𝑇 4 , and 𝑞𝑙𝑎𝑠𝑒𝑟 = 85,000 W/(m2 ∙ K)(SA)film. Also,
because of the assumption of lumped capacitance 𝑞𝑐𝑜𝑛𝑑 = 0. Additionally,
(6)
𝑞𝑡𝑜𝑡 Δ𝑡 = 𝜌𝑐𝑝 𝑉Δ𝑇
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
200
200
180
180
160
160
140
140
Temperature [oC]
Temperature [oC]
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.
120
100
80
100
80
60
60
Minimum Data
Maximum Data
Minimum Cure Temperature
Maximum Cure Temperature
40
20
0
120
20
0
0
2
4
6
8
10
time [s]
12
(a) ∆𝑇𝑜𝑛 = 8 s
14
16
18
Minimum Data
Maximum Data
Minimum Cure Temperature
Maximum Cure Temperature
40
20
0
2
4
6
8
10
time [s]
12
14
16
18
(b) ∆𝑇𝑜𝑛 = 9 s
Figure 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
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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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