Saying "FEA", or Finite Element
Analysis, brings a glazed look to most eyes, but call it heat
transfer analysis and thermographers sit up and take note.
This paper will present FEA for heat transfer analysis applications
of interest to thermographers. Examples will be presented
that show how FEA can be applied to make thermographers more
efficient, more effective, and more informed of the implications
of their results. Other applications will be outlined so that
thermographers will be aware of when to consider having FEA
done for their projects.
What is FEA?
FEA, or Finite Element Analysis,
is a method for performing calculations of complex systems.
It derives its name from the multiple, small elements used
to break up the problem for analysis. It can be used where
explicit solutions are not possible. It is most often used
for mechanical system calculations, such as stress and strain,
but is equally applicable to heat transfer, as well as other
types of phenomena that can be described by the appropriate
types of equations. These can include low and high frequency
electromagnetic systems, dynamic response of objects or assemblies
to vibration and shock, fatigue, and others. One branch of
FEA, which is sometimes addressed by other calculation methods,
is Computational Fluid Dynamics, or CFD. CFD is also of interest
in heat transfer calculations where fluids are involved. CFD
has not been used in the examples discussed here, although
at least one of them would have benefited from it. CFD is
often run in combination with heat transfer calculations,
to give what is called a conjugate analysis.
While it can be set up and done
by hand, FEA is usually performed by computer. There are many
commercial codes for FEA available. Some are highly specialized,
while others are more general. The examples presented here
were done with general purpose programs that have the capability
of doing stress/strain, heat transfer, dynamics, non-linear
materials, and other types of calculations. Included in the
heat transfer capability are conduction, convection, and radiation
calculations. There is also a module that can be used for
CFD.
Why do heat transfer analysis?
The processes and objects that
are looked at by infrared thermographers are undergoing heat
transfer and heat related phenomena. If they were not, there
would be no reason for thermographers to look at them. By
performing an analysis of their heat transfer behavior a better
understanding of their characteristics is obtained. This improved
understanding includes explanation of the significance of
measured temperatures, determination of temperatures that
are not available for measurement, determination of the time
characteristics of a thermal event, and the prediction of
the response to changes in the thermal process. Examples of
these will be presented.
HowFEA
and heat transfer analysis affect thermography
By understanding the characteristics
of the objects and thermal processes that they examine, thermographers
can get better results, faster results, more reliable results,
better understanding of their results, and, in some cases,
results which would not be otherwise available. Thermographers
can save time, money, and effort by using FEA at the right
time and place. This makes them more effective, more productive,
more efficient, more profitable, and more useful to their
customers. FEA should be applied when the information it supplies
will help the thermographer or their customers. The three
examples in this paper point to some of these applications.
Finite Element Analysis as a Tool for Thermography,
Thermosense XXI, 19991 presents two other cases:
estimation of the surface temperature above a buried, hot
pipe as a guide to the ability to find it using IR, and suggestive
of the ability to find a leak in it using IR; and estimation
of the surface temperature response on a sample with an internal
flaw in response to a square wave thermal pulse. The thermographer's
world includes steady state processes, such as hot electrical
components or buried pipes, as well as transient or time dependent
processes, such as solar heating and NDT using active IR.
All of these can be modeled and examined with FEA. The results
of the FEA examination can be used to judge when to work,
to judge whether to work, to judge how to work, and to understand
the results. They will tell the thermographer about things
that cannot be seen.
What is required to perform
FEA
The most basic requirement to
accomplish FEA easily is a good FEA program. Many FEA programs
include a physical modeling capability, or a CAD program.
Some of them can import CAD information from other software
packages. The FEA programs used here were Cosmos/M and Cosmos/Works.
They were used in conjunction with SolidWorks, a 3D modeling
program.
In addition, information about
the system being simulated is needed. This includes the physical
properties of the materials and information about the boundary
conditions of the problem. The properties required are thermal
conductivity for both steady state and transient problems,
and heat capacity and density for transient problems. Boundary
conditions specify how the objects in the simulation interact
with their environment, for example, by convection to air,
by receiving solar heating, or by radiating to a night sky.
Almost anyone can learn to use
an FEA program, but not everyone should. In order to benefit
from the FEA answers it is useful to have some assurance that
they are correct. This will stem, in part, from an understanding
of heat transfer phenomena, material properties, and system
behavior on the part of the person performing the FEA. It
is also a good idea to be able to afford the FEA system and
generate enough use out of it for it to remain a familiar
tool and one with good payback. A consulting and services
firm or a large corporation are more logical places for the
FEA analysis to reside than a strictly IR services firm. The
IR firm can more profitably use FEA as provided from outside.
Examples
Both steady state and transient
examples will be presented. In some cases, they have been
simplified. In general, they have been prepared for presentation
here, rather than as part of a specific project. The properties
and boundary conditions have therefore been approximated where
convenient.
Roof Moisture Survey
IR is used for qualitative surveys
of roof moisture. ASTM standard C1153-902 describes
desirable procedures and conditions for carrying out such
surveys. One of the recommendations that it makes is that,
in the event of overcast conditions, the roof survey should
not be done unless the building is heated to 18° F above
the outside temperature. FEA simulations were done to examine
the impact of this requirement. The cases examined were:
Wet and dry roof comparison for clear
conditions, unheated buildings,
Wet and dry roof comparison for cloudy
conditions, unheated buildings, and
Wet and dry roof comparison for cloudy
conditions, heated buildings.
Conditions were taken as winter in
NY. Solar flux data 3 and assumed air and sky temperatures
are shown in the following figures. The air temperature was
applied to convection from the outside surface of the roof
for all cases and to the inside surface for unheated cases.
A constant 65° F inside air temperature was used for the
heated case.
Solar Loading on Horizontal
Surface, 40N Lat., mid January
Assumed Air Temperature,
40N Lat, mid-Jan
Sky Temperature, 40N Lat.,
mid-July
The roof characteristics were based on two
inches (2") of expanded polystyrene molded bead insulation
board. The contribution of the membrane, roof deck, and air
spaces between the roof deck and the insulation were all ignored
as being minor. The wet section of roof was treated as polystyrene
plus 10% by weight of water. The properties of the wet section
were then approximated as being equal to the properties of
the polystyrene plus 0.10 times the property of water. This
gave, for example, a dry density of 1.0 lbm/ ft 3 and
a wet density of 1.0 + 0.10* 62.4 = 7.24 lbm/ ft3
. The density is the most strongly affected property, because
it has the greatest spread between the two materials. The
others were treated similarly.
The response of the roof under the three
conditions, clear sky, cloudy sky, and cloudy sky, but heated
building, is shown in the following three figures. Note that
the first 1-2 hours of the simulation should not be used,
since that is the period during which the simulation is building
its "starting" conditions. The swings shown in the
temperatures during that time period are not real, merely
a numerical artifact.
Under clear sky conditions, the thermal contrast
between the dry and wet areas of the roof is more than sufficient
to be seen using a standard IR camera. At mid-day, the dry
areas are 2-3° F warmer than the wet areas. At night,
the usual time that roof moisture surveys are conducted, from
about 4PM to about 6AM the wet areas are 1-2° F warmer
than the dry areas. The cloudy sky response is quite different.
It still shows the surface of the roof getting warm during
the day, only less so. It exhibits only a small thermal contrast
between the dry and wet areas. At night the difference is
less than 0.5° F. In the daytime, the maximum difference,
occurring in the late morning, is less than 1.0° F. This
would make detection of the wet areas much more difficult
under these conditions. With the same cloudy environmental
conditions, but with a heated building, the thermal contrast
improves. At night, the contrast generally exceeds 1.5°
F. The daytime contrast is not very good, with a maximum of
about 0.5° F. It is interesting to note that the contrast
in the cloudy, heated building case is always negative. That
is, the wet areas are always warmer than the dry areas. The
wet areas are losing more heat to the outside due to their
greater surface temperature. In other words, this demonstrates
that wet insulation does not work as well as dry.
The shape of the roof temperature response may not be exactly
what the intuitive observer would expect. Why does it show
the shoulders near sunrise and sunset? This can be explained
by plotting the response together with the forcing functions,
i. e., the time curves for flux and temperature, and by doing
some brief calculations of the relative importance of the
different factors. The following figure shows the clear sky
case with the input data.
Comparison of Driving
Forces and Roof Response, 40N Lat., mid-Jan., Clear Sky
The events that occur around sunrise are
changes in sky temperature, air temperature, and solar flux.
Their timing is tabulated below.
Calculations of the heat flux
to the roof (where positive values are roof gains) for each
of the types of heat transfer: convective, radiative, and
direct solar flux, were made and are tabulated below.
At 6 hours
or 6AM, the sky temperature starts to rise and the roof responds.
At 7, the sky temperature levels off and the air temperature
starts to rise. The roof responds more strongly to the leveling
of the sky temperature, but rises slowly due to the air temperature's
increasing. At 7.5, the solar flux begins, reaching 14 BTU/
h ft 2 °F by 8, and the roof temperature again
begins to climb strongly in response. Note that the roof temperature
soon exceeds the air temperature and the direction of convective
heat transfer reverses, so that the roof starts heating the
air.
One other item of interest is available from
the FEA results with little additional effort. The behavior
of the underside and center of the roof as it cycles through
the day can also be examined. Note that the center of the
roof would not be available to thermographic inspection, so
this is a purely calculated result that might be of use to
the thermographer, the owner, or the roofing
materials salesman or other thermographic customer. These
are presented for the unheated building, clear sky case in
the following figure.
Response through the Roof,
40N Lat., mid-Jan, clear sky
Extending the FEA Use for Roof Moisture
An item that
thermographers should consider is the potential for additional
value to be gained by applying FEA to roof moisture surveys.
For example, if a well documented survey were conducted in
which the environmental conditions were measured, the roof's
actual temperatures were measured (rather than only imaged),
and a limited number of moisture samples were taken, it should
be possible to calibrate the FEA model and then, based on
observed temperatures of the roof surface, actually report
moisture content!
CMU wall
A common application for IR is to examine
CMU walls for the presence or absence of pilasters. IR can
detect the pilasters because they respond differently than
the rest of the wall to the varying heat loads imposed on
the wall. The heat loads that are of interest are: the solar
load on the (outside) of the wall; the convection from each
face of the wall; the radiative interchange between the wall
and its surroundings, especially the sky and, in particular,
the night sky; and the heat load represented
by any building heat that may be in place. The building heat
is combined in the convective boundary condition on the inside
of the wall. As with the roof moisture survey, essentially
all of these loads vary over time, both on a daily basis and
on a seasonal basis. The solar loading is very different in
December than it is in July. Solar loading is also affected
by location (latitude) and sky conditions, and also by the
orientation of the wall. It can also be affected by the surroundings
of the wall - trees, buildings, etc.
Diurnal temperature
variations are also different, both seasonally and, to a lesser
extent, day to day. What this means is that the wall's observable
thermal behavior varies in response to a great many inputs.
It is sometimes enough to know where the wall is and its orientation
in order to know the best time of day to get a good image
and good thermal contrast between the pilasters and empty
cells, especially if a wall like it, near it, and at the same
time of year as it, has been examined before. If not, the
thermographer may find himself, or herself, showing up at
9 PM to get the same great imaging that had been obtained
before and being two hours late or five hours early.
FEA can provide an
estimate of the wall's behavior before going into the field.
By using FEA ahead of time, the thermographer can arrive at
the right time, get the best images, and leave in less time
than might have been spent just waiting. They can certainly
do it in less time than if they got there two hours late and
had to wait eight to ten hours for good conditions to recur,
or had to leave and come back. FEA is economical, and makes
the thermographer more efficient, more effective, more productive,
and more profitable.
The CMU example is based on a wall with alternate
cells filled with grout and a single thickness of block. Properties
for the grout and block were approximated from literature
values. Conditions were chosen to approximate central NC in
mid-July in order to compare the FEA results with values published
by Gregory Stockton and Lee Allen in Using Infrared Thermography
to Determine the Presence & Correct Placement of Grouted
Cells in Single-Width Concrete Masonry Unit (CMU) Walls
at Thermosense XXI, 1999 4 . The exterior of the
wall was treated as facing south.
Since the wall is essentially infinite in
the vertical direction, that is, the heat transfer is in and
out of the wall and to the left and right in the plane of
the wall, but does not occur significantly in the vertical
direction, only a small slice of a single block needs to be
modeled. The next figure shows the block as modeled. The slice
is one inch (1") thick in the vertical direction. The
thermographer can only see the long faces of the block that
are 1" thick, either the exterior or the interior. As
shown, the exterior of the wall is the lower left face, so
that the filled cell is on the thermographer's left when viewing
the exterior wall and on the thermographer's right when viewing
the interior wall. (The model was initially set up to work
on the exterior face, but the results did not show a good
match to the reported data. In conversations with Greg Stockton,
he pointed out that the measurements had been made from the
interior.)
The boundary conditions (BCs)
for the model are:
Convection with the outside air from the
exterior face to the outdoor air temperature,
Convection with the interior air from
the interior face to the interior air temperature,
Solar flux on the exterior face,
Radiation between the exterior face and
the sky at the sky temperature, and
Radiation between the interior face and
the interior of the building, taken as equal to interior
air temperature.
To use the BCs described above requires
information about the environmental conditions: outside air
temperature, inside air temperature, solar flux, sky temperature,
and sky condition, all as a function of time of day. It was
necessary to make assumptions about most of these. The solar
flux has been based on literature values 5 for
32° North latitude, mid-July, for a southern elevation.
Time plots of these input conditions are shown below.
Adjusted ASHRAE Solar
Heat Gain Factor Data, 32N Lat., mid-July, South Elevation
Air Temperatures, assumed
Clear
Sky Temperature, assumed
The data extracted from
the plot in Stockton & Allen is shown below. Its lack
of smoothness is due to the difficulty in reading precise
values from the graph.
Stockton's Data
The results for the FEA simulation are shown
below for the interior wall and again with the values extracted
from Stockton & Allen for comparison. There is fairly
good agreement between the simulation and the field data.
The chief differences
between the simulation and the data are:
There is an offset in time between the
occurrences of the peaks in the data vs. the simulation,
and
The simulation shows a smaller range of
thermal contrast (filled-empty) than the data.
The offset is easily explained, since
several factors contribute to it. The two primary factors
are the orientation of the wall and the difference between
solar time, as used in the FEA, and clock time, as reported
by Stockton & Allen. If the actual wall faced the SW rather
than due S, it would exhibit later peaks. In addition, the
time used here is solar time, as per ASHRAE. The data from
the reference paper was corrected from daylight savings time
back to standard time, but no correction was applied for longitude,
which would affect the actual local solar time relative to
the clock time. In addition, the time curves assumed for the
environmental characteristics, especially temperatures will
affect the timing of the calculated wall response. Obtaining
actual values for the information, rather than using assumptions,
would address these issues. Even with the assumptions and
known problems, the FEA results are quite good and definitely
valuable. They show when the separation between the filled
and empty cell temperatures is at a maximum and would be imaged
most easily and reliably. They could be readily extended to
the other walls of the building to allow the thermographer
to plan a "field campaign" while still in the office.
The plan would address which walls to image at what times,
and in what sequence.
The following two figures present the results
for the behavior of the exterior wall in combination with
the results already seen for the interior. The exterior exhibits:
Much wider temperature swings,
Much higher temperatures,
Much higher thermal contrast between the
filled and empty cells, and
Different timing for the temperature
peaks.
This makes sense. The exterior
is directly exposed to the primary loading, the solar flux.
It receives this loading as soon as it occurs, whereas the
interior wall experiences a lag due to the heat transfer through
the wall. By performing this type of analysis for each wall
in a structure, the best program for examining the entire
structure could be determined before going into the field.
CMU Response, 32N
Lat., mid-July, interior and exterior walls
CMU Response, 32N Lat.,
mid-July, contrasts
The FEA program can present its results
graphically in a form that strongly resembles an IR thermograph.
Images of this type are presented below in four-hour intervals
starting at 24 hours, or midnight of the solar day Ð second
cycle, and extending through the second day to 48 hours. Two
types of thermal images are presented. The first is an isometric
view of the concrete block, including its internals, which
the thermographer would not be able to see. The second is
the view of the one inch slice of the interior wall that was
modeled and which corresponds to the view that the thermographer
would have. Temperature scales have been adjusted for each
image.
Time
Isometric Image Thermal Step
= 96
24 hours, midnight at start of
second cycle
Time
Isometric Image Thermal Step
= 112
28 hours, 4AM
Considerable contrast between the filled and empty
cell locations on both exterior and interior surfaces.
The filled is hotter in both cases.
Time
Isometric Image Thermal Step
= 128
32 hours 8 AM
Time
Isometric Image Thermal Step
= 144
36 hours, noon
Almost no contrast on the interior surface. Considerable
contrast on the exterior, with the empty cell much hotter.
Time
Isometric Image Thermal Step
= 160
40 hours, 4 PM
High contrast on the exterior, with the empty cell
much hotter than the filled. Moderate contrast on the
interior, with the empty cell cooler than the filled.
Time
Isometric Image Thermal Step
= 176
44 hours, 8 PM
Time
Isometric Image Thermal Step
= 192
48 hours, midnight
Time
Interior Wall - Thermographer's
View Images
Reference image from the interior, showing the cell
locations, empty on the left, filled on the right
Time
Thermal Step = 98
24 hours, midnight at start of second cycly
Time
Thermal Step = 112
28 hours, 4 AM
Time
Thermal Step = 128
32 hours, 8 AM
Time
Thermal Step = 144
36 hours, noon
Time
Thermal Step = 160
40 hours, 4 PM
Time
Thermal Step = 176
44 hours, 8 PM
Time
Thermal Step = 192
48 hours, midnight
Internal Temperature of An Electrical
Component
This is an example of calculating temperatures
at surfaces or points that cannot be observed directly, but
which are of interest because they represent the true conditions
of the object. It is carried out as a steady state analysis,
because it is assumed that there is no (short term) time component
involved. The case presented is a sealed electrical component
that shows mild heating on the surface when examined thermographically.
The thermographer cannot directly measure the contact's temperature
in the component because it is sealed. The owner would much
prefer making a shut down or keep running decision based on
the contact's temperature as opposed to the case temperature.
The thermographer would like to oblige, but the camera will
not help. FEA will.
For the example, a very simplified component
was created, as shown below.
Fully Assembled Electrical
Component, as would be seen in place by a thermographer
Fully Assembled Electrical
Component, as seen in place
Interior of the Electrical
Component
For the sake of expediency,
some simplifications were made in preparing the FEA model.
While they will certainly affect the values of temperatures
obtained, they do not affect the principles demonstrated,
i. e., that FEA will allow a calculation of temperatures unavailable
for measurement. The simplifications made include:
Radiation was omitted.
Air circulation in the case cavity was
omitted, i. e., a CFD module was not included in the modeling.
The lack of air circulation in the case tends
to under-represent the flow of heat from the interior components
to the inside of the case wall. This error was partially offset
by increasing the thermal conductivity of the air.
The thermal image of the component, as seen
by the thermographer, resembles the FEA generated images below.
Full Contactor-case 2-5W :: Thermal
Time Step: 1
Units : Fahrenheit
Exterior of Component,
as would be seen by thermographer, with a wide T range.
Full Contactor-case 2-5W :: Thermal
Time Step: 1
Units : Fahrenheit
Exterior of Component,
as would be seen by thermographer, with a narrow T range.
The ambient temperature used in this example
is 70° F. The component shows a 15 to 25° F rise over
ambient. Some localized heating is seen on the side of the
case at the location of the contacts. Were air movement in
the interior included, this hot spot would be less distinct
and the rest of the case would be slightly warmer. The temperature
rise of the exposed surfaces, when combined with areas of
the surfaces, the ambient temperature, and typical heat transfer
coefficients, provide an estimate of 5 watts of heat dissipation
from the component. By careful measurement of these values,
the thermographer would be able to provide this heat loss
estimate to the customer based strictly on thermographic results.
By then applying FEA, and by assigning the heat generation
to the contact face between the two contact blocks shown as
shades of blue in the interior figure, the temperature of
the contacts can be calculated. This was done with the results
shown below.
Full Contactor-case 2-5W :: Thermal
Time Step: 1
Units : Fahrenheit
Cut Section of
Electrical Component with 5W heat generation at contact
faces. Includes internal air.
Full Contactor-case 2-5W :: Thermal
Time Step: 1
Units : Fahrenheit
Cut Section of
Electrical Component with 5W heat generation at contact
faces.Internal air not shown.
The contacts are at 170° F, or 100°
F above ambient. This is significantly higher than the exterior
temperature rise of 15-25° F. The value obtained is sensitive
to materials of construction of the component, as well as
the configuration of the component. For a real case in the
field to be accurately calculated, the materials and the configuration,
which is likely to be considerably more complicated, would
be needed and would be reflected in the model. What is important
in this example is the fact that using FEA allows the calculation
of the interior, non-measurable, temperatures of the component.
The fact that this particular configuration shows an incremental
rise of 75° F over the exterior case temperature is almost
incidental.
The same approach can, and should, be applied
to other objects. These could be electrical components, mechanical
components, process equipment, or structures. The application
of FEA in this manner will make thermographers more effective.
It will increase the level, type, and quality of information
that they can provide.
Summary and Recommendations
The application of FEA for heat transfer
calculations in both steady state and transient situations
has been demonstrated. Used this way, FEA will provide information
of value to the thermographer and the thermographer's customer.
The information can include estimates of temperatures that
are not accessible to measurement, guidance as to the significance
of measurements, and descriptions of temporal behavior of
systems of interest. Practicing thermographers should be aware
of these capabilities so that they can take advantage of them
to improve their business, their profitability, the scope
of jobs that they can perform, the type of customer that they
can satisfy, and the time they spend providing results. FEA
of this type
can be outsourced from various consulting organizations.
References
1 Kleinfeld, Jack M., Finite Element
Analysis as a Tool for Thermography , Thermosense XXI, SPIE,
Bellingham WA, 1999
2 ASTM C 1153-90 Standard Practice
for the Location of Wet Insulation in Roofing Systems Using
Infrared Imaging, ASTM, Philadelphia PA, 1990
4 Stockton, Gregory R. and Lee
R. Allen, Using Infrared Thermography to Determine the Presence
& Correct Placement of Grouted Cells in Single-Width Concrete
Masonry Unit (CMU) Walls, Thermosense XXI, SPIE, Bellingham
WA, 1999
