ASNT NDT Handbook Volume 3 Infrared and Thermal Testing

view as required for maintenance Emissivity changes can cause severe
inspections. Single chip arrays of errors in radiometric detection methods,
128 X 128, 256 X 256, 512 X 512, unless something is done to keep
640 X 512 , 800 X 500, 1200 X 1000 and emissivity constant. This has been
larger elements apiece are for full field successfully accomplished primarily by
imaging systems. The most common array coating the test surface with materials
detectors are made of platinum silicide or that possess uniform, high emissivity
indium antimonide; mercury cadmium values (typically c = 0. 7 to 0.9). Uniform
telluride, common for single element emissivity is necessary for temperature
detectors, has also been introduced for measurement accuracy while high
multiple arrays. emissivity is desirable to provide a larger
radiant intensity (per the
Other Radiometers. A great variety of Stefan-Boltzmann lmv). It was once
radiometer designs and readouts exist. thought that coatings needed to be dark
One radiometer detects a thermographic in color, such as lampblack suspended in
image by means of the a polymeric binder. However, transparent
evaporation/condensation nature of a polymeric coatings also have been
thin oil film on a membrane at the focal successful. 1\•fetal surfaces especially must
plane. Some scanning radiometers be coated and nonmetal surfaces to a
produce C~scan and multiple A~scan lesser degree; however, high emissivity
recordings of thermographic information coating will help. After testing, the
rather than the optical image. This type of coatings are usually removed from the
readout gives an accurate value for material surface.
radiation intensity at each scan point.
High speed nne scanners with a scope The emissivity problem can also be
readout (instead of the slower mechanical reduced by special design of the
recorder) display the temperature profile radiometer system. One quite ingenious
of a "line" across the surface. method uses two radiometers, where the
infrared signal from one radiometer is
The infrared microscope scans an delayed, a constant signal added to it and
extremely small spot (only about 25 pm the sum divided by the other radiometer's

fl x 10-3 in.] in diameter) over the TABLE 1. Total radiation emisslvities at all wavelengths.

surface of very small materials. The Material Emissivity Range
resulting image is magnified and (0 to 1.0)
displayed on some form of cathode ray
tube, such as a television tube. Infrared Metallic
nondestructive testing microscopy is
suited for intricate surfaces in specialized Highly polished aluminum, silver, gold, brass, tin 0.002 to 0.04
applications such as the inspection of
integrated circuits. Nonmkroscopic Polished brass, copper, steel, nickel, 0.03 to 0.08
systems are also common. chromium, platinum, clean mercury

Emissivity Variables Dull, smooth, dean aluminum and alloys, copper, 0.08 to 0.20
brass, nickel, stainless steel, iron, lead, zinc
Infrared thermography is most successful
for surfaces with high emissivity. High Rough ground or smooth machined castings; 0.15 to 0.25
emissivity provides several important
effects. First, as seen in the steel mill products, sprayed metal, molten metal
Stefan-Boltzmann law, surfaces with a
high emissivity e emit a higher intensity Sooth, slightly oxidized aluminum, copper, brass, 0.20 to 0.40
radiation at a given temperature, thereby lead, zinc
providing a larger signal for the infrared
detector. Second, high emissivity surfaces Bright aluminum, gilt, or bronze paints 0.30 to 0.55
are, by definition, poor reflectors. Low
emissivity surfaces tend to reflect Heavily oxidized and rough iron, steel, copper, 0.60 to 0.85
radiation from other sources. The infrared aluminum
detector thus senses energy unrelated to
the temperature of the object being Nonmetallic
examined. This contributes to the noise of
the test and reduces the sensitivity to White or light colored paint, plaster, brick, tile, 0.80 to 0.95
details of interest. Third, high emissivity porcelain, plastics, asbestos
surfaces also absorb more radiant energy.
Radiant sources can therefore be effective Snow at 263 K (-1 0 "'Cor +14 °F) 0.85
in inducing a thermal gradient in the test
sample, a condition required for many Red, brown, green, buff and other colors of paint, 0.85 to 0.95
infrared thermal tests. Table 1 provides a tile, inks, clays or stone; glass and translucent
list of typical surface emissivities for a plastics; glass fiber composites; ice crystals; oil;
variety of materials in a temperature range varnish
of about 273 to 313 K (0 to 40 oc; 32 to
104 °P).2·' White bond paper, sand, wood (planed oak) 0.90

Carbon black, asphalt, carbon fiber composites, 0.90 to 0.97
matte black paints, tar

Concrete 0.92

Brick (common red) 0.93

Water 0.96

Human skin 0.98

36 Infrared and Thermal Testing

output. Sensing delay time between the Melting point coatings melt at some
two radiometers is accomplished by specific temperature. Anomalies are
scanning one ahead of the other. The usually associated with a temperature
result is an emissivity independent increase, so the materials melt first over
readout (even on very rough surfaces) for anomalies. Melting point compounds also
regions of constant temperature. Another arc comparatively insensitive and require
method uses a single radiometer that relatively high surface temperatures.
samples the reflected and emitted infrared
radiation. Although considerably more Liquid aystals are cholesteric liquids
simple, this design is suited only for very whose optical properties cause them to
smooth surfaces. reflect vivid spectral colors for
temperature changes. Their adjustable
Thermosensitive Indicators response is quite sensitive and can be
made to change from red to blue over a
As mentioned above, the other basic temperature gradient as small as 1 K
method for observing thermograms
involves the application of (1 oc = 1.8 °F). An additional feature is
thermosensitive materials directly on the
surface. This approach conducts the that their responses can be adjusted to
thermal pattern on the test surface into occur at temperatures only slightly al10ve
the thennosensitive material, usually a ambient. Tests on honeycombs and
thin coating sprayed, painted or held laminated structures \\'ith liquid crystals
against the surface. A variety of coatings have been quite successful. Figure 5 shows
have been developed, including the invisible thermogram of irregularities
permanent color change coatings, in a light gaged honeycomb structure.
phosphor coatings, melting point coatings Notice that the thermal gradient can be
and liquid crystals. intensified by contact heating one surface
while cooling the near surface ·with cool
Perrnanent color change coatings, such as air.
creosote, react chemically and change
color permanently over temperature Both passive and active infrared
gradients; these materials are nondestructive testing find a variety of
comparatively insensitive and normally applications throughout diverse
require that the surface be heated weH industries. The development of
above the ambient temperature. satisfactory (sensitive) thermograms
depends on several important factors. It
Pllvspllor coatings are materials whose must be remembered that, because of
fluorescent radiation intensity, under transient heat flow, a thermogram of a
ultraviolet light, is a function of given area changes with respect to
temperature. observation time (heating time). \.Yhen
heat is applied to an inspection surface, as
FIGURE 5. liquid crystal testing of light gaged honeycomb shown in the common setup in Fig. 6, a
structure. thermogram will develop that is a
function of the material, the nature of the
Cool air discontinuity, the observation time and
the heat intensity. A complicated
relationship exists among these variables.
However, a practical summary follows.

Green FIGURE 6. Common radiometer setup for
active testing.

Radiometer

Infrared lamp Infrared lamp

Aluminum honeycomb with Black coating
0.3 mrn (0.013 in.) thkk skin Test material
Thermogram

Fundamentals of Infrared and Thermal Testing 37

I.Transient temperature differences over given depth, infrared testing is less
discontinuities in materials having sensitive to smaller discontinuities
higher thermal diffusivities (metals) than to larger ones.
are shorter m duration and lower in S.Discontinuities parallel to the test
magnitude than they are in materials surface are usually easier to detect
having lower thermal diffusivities than ones perpendicular to the
(nonmetals). surface.

2.The magnitude of the surface Emissivity
temperature difference over a
discontinuity decreases with the depth Emissivity is a variable defined as a ratio
of the discontinuity. Near surface of the total enefb'Y radiated by a given
anomalies are much easier to detect surface at a given temperature to the total
than deep ones. energy radiated by a blackbody at the
same temperature. A blackbody is a
3.The observation time required for a hypothetical radiation source that yields
temperature difference to reach its the maximum radiation energy
maximum value over a discontinuity theoretically possible at a given
increases as the thermal diffusivity temperature. Also a blackbody will absorb
decreases, as the depth of the all incident radiation falling upon it.
discontinuity increases and the size Blackbodies have an emissivity of 1.0 and
increases. At a given depth, small aH real materials have emissivities
discontinuities reach their peak between 0 and 1.0.
intensities on the thermogram before
larger discontinuities reach their peak Figure 7 represents the
intensities. Stefan-Boltzmann law for blackbodies at
various intensities. Note that the
4.The duration of the transient wavelength envelope shifts toward the
temperature difference over a visible range for increasing blackbody
discontinuity decreases as the temperature, according to \.Yien's
discontinuity size decreases and as the displacement la'~N. As shown by \.Yien's
discontinuity depth increases. At a equation (Eq. 1), the wavelength of
maximum intensity is computed simply
fiGURE 7. Stefan~Boltzmann radiation law by dividing 2897 by the temperature of
for blackbodies (total energy E= e7", the surface:

maximum" 3000· r-1). b

T

where b is the ·wien displacement constant
(2897 pm·K-1), Tis temperature (kelvin)
and },1nax is maximum wavelength
(micrometer).

The effect of emissivity on the
radiation curve is given in Fig. 8. As

fiGURE 8. Emissivity effect on radiation from surface of
emissivity c: with hypothetical intensity.

,.-, c = 1 (blackbody)

I\ £ = 0.9 (graybody)
I\ '\
I\
I '''

I r varies

(not

graybody)

10 100 Wavelength (relative unit)

Wavelength j.Jm

legend
1. 300 K ;;+27 °C "'-t80 °F

2. 195 K;; -78 oc;; -108 °F
3. 126 K = -147 oc = -233 Of

38 Infrared and Thermal Testing

shown, it acts somewhat like a filter or In the most general case, incident
valve. Graybody materials have emissivity infrared waves are reflected and
values that are less than that of a transmitted in addition to being absorbed.
blackbody at all temperatures and Because the whole must equal the sum of
wavelengths. Some materials, called its parts, the fractions of absorbed,
spectral radiators, have a spectral reflected and transmitted energy must
emissivity that varies in a characteristic equal the total incident energy:
fashion over the range of emitted
wavelengths. (3) a + r + 1

Emissivity is a surface phenomenon where a= absorptivity coefficient;
depending on the surface condition and r = reflectivity coefficient; and
composition. Smooth materials have t = transn1issivity coefficient.
lower emissivities than rough materials.
Freshly polished metals have lower If the transmissivity is low enough to
ernissivities than oxidized or corroded be neglected (as is usually the case with
metal surfaces. Nonmetals usually have infrared testing), then a = 1 - r and
higher emissivities than metals. r = 1 - a. Another important relationship,
Lampblack or certain metallic powders known as Kirchoff's law, states that the
yield very high emissivities. Special ratio of radiation intensities for two
blackbody cavities for calibrating surfaces iS equal to the ratio of their
radiation equipment produce the highest absorptivities. This means that
absorptivity equals emissivity or
emissivity available, within a fraction of a emissivity equals 1 minus reflectivity.
Several deductions from the preceding
percent of that for a blackbody. radiation laws are useful: (1) efficient
Because the object of infrared testing is emitters arc efficient absorbersi
(2) inefficient absorbers are inefficient
to measure surface temperature changes, emitters; (3) efficient reflectors are
emissivity can be an uncontrolled inefficient emitters; and (4) inefficient
variable. Variations of emissivity across reflectors arc efficient absorbers.
the surface of a material can cause false
indications. \Vhen the emissivity
decreases in a localized region, the
radiation intensity decreases, falsely
indicating a localized reduction in
temperature and vice versa. Also, surfaces
with low emissivity values, such as
polished n1etals1 are more difficult to test
than high emissivity surfaces.

Because all materials are continuously
radiating infrared energy, it seems they
might eventually cool down to absolute
zero. This would happen if materials did
not pick up energy from other sources by
means of radiation, conduction or
convection. All materials are continuously
and simultaneously radiating and
absorbing infrared energy. \Nhen a
material is wanner than its surroundings,
its radiation emission will exceed its rate
of absorption1 causing the temperature of
the material to drop. The opposite is true
for material cooler than its surroundings.
\-\7hen the material and its surroundings
reach the same temperature, the thermal
equilibrium point, each body is emitting
radiation at the same rate that it absorbs
it (without a net change in energy).

The basic relationship describing the
above radiation interchange is defined by
Prevost's l<lW of exchanges. For the case of
one body surrounded by the interior walls
of another body, Prevost's law yields:

where£ is emissivity (of materials 1
and 2); K11 is Boltzmann's constant;
Tis absolute temperature (K); and H' is
net gain or loss of radiation intensity.

Fundamentals of Infrared and Thermal Testing 39

1.

PART 2. General Approaches and Techniques of
Infrared and Thermal Testing6

Thermography is one of several Emissivity is a unitless surface property
techniques used to see the umee11. ?,8 As the that describes the ability to emit energy. It
name implies, it uses the distribution is a unitless quantity and on a scale from
(~graph)') of surface temperatures (thermo-) 0 to 1, where E = 1 for a blackbody.
to assess the structure or behavior of what Generally, emissivHy E depends on
is under the surface. Traditionally the wavelength 'A, temperature T, viewing
term thermography has denoted a contact angle 8 and surface conditions such as
technique to record a distribution of roughness, oxide layers and physical and
surface temperatures whereas infrared chemical contamination. Objects whose
thermography is a contactless technique emissivity is independent of the
with distinct advantages. In the 1970s, wavelength are called gray bodies while
the term thermography came to usually colored bodies refer to full dependence
mean noncontact, infrared thermography. emissivity.
Contact thermograph}' can be deployed
with liquid crystal paints applied on the The fundamental equation of infrared
surface of interest and monitored with <l thermography relates the irradiance Ncam
conventional video camera: cholesterol (that is, spectral radiant po·wer incident
esters, under temperature effect change on a surface per unit area) received by the
orientation and reflect colored light from camera to the radiance emitted from the
red to violet when illuminated with white surface under consideration Nsur at a given
light. Deployment with an array of temperature T, neglecting the atmosphere
thermocouples is also possible but not contribution as in most nondestructive
practical \\'hen numerous grouped testing applications:
readings are needed. In nondestructive
testing, thermography is generally (6) Ncam = ENsur + {I - c) N{'Jl\'
deployed contactless.
with Nenv being the radiance emitted by
Physical Basis the surrounding environment considered
as a blackbody. If emissivity of the surface
Planck's law describes the distribution of is high, Eq. 6 reduces to Ncam = Nmr and
the spectral radiance L'} b the rate at which knowledge of the calibration curve of the
energy is emitted by a blackbody bat a camera linking environmental radiance
given temperature T, per unit surface, per Nsur to temperature T permits retrieval of
unit of solid angle and as function of the the surface temperature, assuming there is
wavelength k no radiometric distortion. Various
techniques can help to solve low or
where h is Planck's constant (6.626076 x uneven emissivity problem: (1) covering
the inspected surface with a high
w-34 }s), cis the speed of light emissivity paint, (2) a reflecting cavity's
artificially increasing the emissivity
(-3 x 10s m·s-1) and KB is Boltzmann's through multiple reflections,
constant (1.381 x J0-23 U·K-')). A (3) performing simultaneous observation
blackbody is also an instrument that, as a of the surface in different spectral bands
perfect absorber, totally absorbs energy (as in two color pyrometry), (4) relying on
coming from any direction and from any thermal imprint transfer on a high
wavelength while, as a perfect radiator, it emissivity material on which the
follows Kirchkoff's laws and reemits this observation is performed and (S) taking
energy until the thermodynamic into account in Eqs. 4 and 6
equilibrium is reached with the measurements of local emissivity values
surrounding environment. For normal and radiance from the surrounding
bodies, Eq. 4 becomes Eq. 5 with environment.
correction factor~ (emissivity):

40 Infrared and Thermal Testing

Instrumentation detectors in which photon
interactions either change
Infrared thermography uses two different
kinds of infrared cameras: scanning conductivity (photoconductive) or
radiometers and focal plane arrays generate voltage. Detectors that
available either in one or two dimensions generate voltage are called plwtuelectric
(one-dimensional is useful when or plwtovoltaic.
inspecting moving objects, the second 2. In pltotoemissive photonic detectors,
dimension then provided by the
displacement). Infrared images are called the signal observed is constituted by
lhennogmms. measurement of an electron current i).

Scanning radiometers are generally Because no heating is needed as for
equipped with an internal temperature
reference seen by the detectors during thermal detectors, response time is short.
scanning so that the output signal is
directly calibrated in temperature. Because of their solid state structure, these
However, the electromechanical scanning
by the detectors through prisms or detectors are compact, reliable, robust and
mirrors limits the frame rate and
sometimes corrupts the output signal. In much employed. Common materials in
focal plane arrays or starring arrays,
because the infrared image of the scene is photoelectric devices (photodiodes and
directly imaged on the detector matrix,
no scanning is needed. All pixels are phototransistors) are silicon, indium
acquired simultaneously and high speed
frmne rate is achieved but temperature arsenide, indium antimony and cadmium
calibration is done externally- for
instance, by pointing the focal plane array mercury telluride. Quantum detectors
camera to a blackbody of known
temperature. require cooling to reduce the noise to an

'f\\'o families of infrared detectors exist. acceptable level. The most common
In thermal detectors the incoming
radiation heats the surface and this approach is liquefied gas stored in a
heating affects a property of the heated
material that then translates into vacuum vessel called a dewar. For
variations of the signal output. In
bolometers, the electrical conductivity example, liquid nitrogen provides cooling
changes. Thermal detectors do not require
cooling and their response is independent at a temperature of 77 K (-196 "C ~
of the wavelength so that an interference
filter is added in the optic window to -321 oF). Other cooling approaches are
limit spectral sensitivity. Because detector
temperature changes are required, joule-thompson gas expansion, stirHng
response is slow. Recently,
micromachining technology has made cycle engines or thermoelectric elements
microbolometer arrays available in focal
plane arrays and especially aimed at based on the peltier effect.
qualitative applications. Thermopiles and
thermocouples generate a voltage Because the atmosphere lacks perfectly
difference through thermoelectric
flat spectral transmission properties, the
thompson effect. In pyroelectric detectors
electric charges are generated by incident selection of the operating wavelength
radiation absorption (heating),
pyroelectric elements can be made as band is conditioned by the application
point or image detectors. Pyroelectric
tubes are similar to standard vidicon and detector type. Among the important
television cameras except for the face
plate and pyroelectric target material. ~riteria are operating distance,

In photonic detectors, the signal is mdoor/outdoor operation, temperature
obtained by measuring directly the
excitation generated by the incident and emissivity of bodies of interest.
photons. Heating of the sensitive surface
is unnecessary. Photonic detectors are of Following Planck's Jaw (Eq. 4), high
two types: (1) quantum (photoconductive
and photoelectrk/photovoJtaic) and temperature bodies emit more in the short
(2) plwtoemissh'e.
wavelengths, thus long •wavelengths ·will
I. Quantum detectors are solid state
be of more interest to observe near room

temperature objects. These are also

preferred for outdoor operation where

signals are less affected by solar radiation.

At operating distances restricted to a few

meters (about ten feet) in absence of fog

or water droplets, atmospheric absorption

has little effect. lvfost common bands in

infrared thermography are 3 to 5 pm

(short waves) and 8 to 12 pm (long waves)

because these match the <Itmospheric

transmission bands.

Another important point to consider is

the detectivity D (or normalized

detectivity D*) of the d°Cete=c~to3r21usoeFd/ for
instance a 77 K (~196 cooled

indium antimony detector operating in

the 3 to S pm range has a detectivity

seven times higher than does a 77 K

(~ 196 °C = -321 oF) cooled cadmium

mercury tellmium detector operating in

the 8 to 12 pm range. Detailed studies

have concluded that for temperatures

from 263 to 403 K (-10 "C to +130 "C;

14 to 266 oF), measurements can be done

·without much difference in both the 3 to

5 pm band and the 8 to 12 pm band, ·with

however a slight preference for 3 to 5 pm

devices for which the errors in

Fundamentals of Infrared and Thermal Testing 41

temperature measurements are generally speed sewing of seat cushions and airbags
smaller. For particular applications (such in the automobile industry. Modeling
as mi1itary), bispectral cameras operating makes it possible to optimize sewing
simultaneously in both bands are operations through needle redesign and
deployed to characterize target thermal needle cooling- with significant
signatures more accurately. economic and quality benefits because of
the million of products sewn daily.

Passive Thermography Active Thermographic
Techniques
The first Jaw of thermodynamics expresses
the principle of energy conservation and Pulsed Thermography
states that an important quantity of heat
is released by any process consuming Pulsed thermography is one of the most
energy because of the law of entropy common thermal stimulation techniques
(Eq. 4). Temperature is thus an essential in infrared thermography. One reason for
parameter to measure in order to assess this test's popularity is its quickness. It
proper operation (Fig. 9). relies on a short thermal stimulation pulse
lasting from a few ms for high
In passive thermography, abnormal conductivity materials such as metals to a
temperature profiles indicate a potential few seconds for low conductivity
problem and key words are the specimens (such as plastics and graphite
temperature difference with respect to the epoxy laminates). Brief heating at a few
surrounding, often referred to as the degrees above initial component
delta T (t.'J) or the hot spot. A t.T of temperature prevents damage to the
I to 2 K (I to 2 oc ;c 2 to 4 oF) is generally component.
found suspicious while a 4 K (4 °C =: 7 ol:)
value is a strong evidence of abnormal Basically, pulsed thermography consists
behavior. Generally, passive thermography of briefly heating the specimen and then
is rather qualitative because the goal is recording the temperature decay curve
simply to pinpoint anomalies. However, (Fig. 9). Qualitatively, the phenomenon is
some investigations provide quantitative as follows. The temperature of the
measurements if thermal modeling is material changes rapidly after the initial
available so that measured surface thermal pulse because the thermal front
temperature (isotherms) can be related to propagates, by diffusion, under the
specific behaviors or subsurface surface and also because of radiation and
discontinuities. convection losses. The presence of a
discontinuity modifies the diffusion wte
Dedicated modeling helps process
control researchers, for instance, to
understand needle heating during high

FIGURE 9. Schematic setup for infrared thermographic nondestructive evaluation. Thermal stimulation is only
needed in active procedures. Drawing shows reflective scheme.

Subsurface structures of interest

Thermal stimulation control ~· ~ . Specimen
in active procedures
euc '~
O::::LJControl
.•u,
>igo•l> IPo»iv•""'hoiq"'
0
.----~>-1 lockin technique 6..2j 1 - r - - - - - - - /
~
~·· ~
~ .•
e
.... •~· 3e
Digital image recording Infrared camera ~ ~E

Computer (visualization, ~
processing) .... .·.•5.,
•~

lamp or other thermal
stimulation devices such
as hot air jets in active

procedures

42 Infrared and Thermal Testing

so that, when the surface temperature is monitored during the application of a
step heating pulse (Fig. 9). Step he~ling
observed, discontinuities appear as areas thermography finds many apphc<ttw_ns
of different temperatures with respect to 1 such as for coating thickness evaluatiOn

surrounding sound area, once the thermal (including multilayered coatings),
front has reached them. In a first
approximation, observation time t is inspection of coating-.to-substrate b?nd or
function of the square of the depth z and evaluation of composite structures. !I
the Joss of contrast C is proportional to
the cube of the depth: Lockin Thermography

(7) Thermal waves were already investigated
by Fourier and Angstrom in the
(8) c 1 nineteenth century. Lockin thermography
is based on such waves generated inside
z' the inspected specimen and detected
remotely.IO \'\lave generation, for instance,
where a is the thermal diffusivity of the is performed by periodically depositing
material. heat on the specimen surface (for
example, through sine modulated lamp
These relations indicate two limitations heating) while the resulting osciHating
of infrared thermography: observable temperature field in the stationary regime
discontinuities are generally shallow and is remotely recorded through its thermal
contrasts are generally weak. An empirical infrared emission (Fig. 9).
rule of thumb says that tile radius of the
smallest detectable discontinuity should be at Lockin thermography is also called
least as large~ and preferabl)' two or more photothermic radiometry'. The word lockin
times larger, than its depth under the surface. refers to the necessity to monitor the
This rule is valid for homogeneous exact time dependence of the heating,
isotropic material. In case of anisotropy, it modulated between the output signal and
is more constrained. the reference input signal. This is done
with a lockin amplifier in a point-by-point
Various configurations are possible: laser heating or by computer in full field
point, line or surface inspection. Pul~ed deployment so that both phase and
heating is achieved by laser beam, h1gh magnitude images become available.
power photographic flashes, lamps with Phase images are related to the
mechanical shutter and hot air jets. In propagation time and, because they are
some instances, a cool pulse is preferred, relatively insensitive to local optical
for instance if the temperature of the part surface features (such as nonuniform
to inspect is already higher than ambient heating), they are interesting for
temperature due the manufacturing nondestructive testing. Modulation
process. In that case, a cold thermal images are related to the thermal
source such as a line of air jets is used diffusivity and found some uses for the
while such a cold thermal source docs not characterization of electromagnetic fields
induce spurious thermal reflections into in antennas. 11
the infrared camera as in the case of a hot
thermal source (Eq. 6). Recording of four raw thermograms 51
to 54 located equidistantly on the sine
Obsen'ation is possible either in modulation cycle yields to the phase
reflection mode, ·where the thermal source $(ro)and magnitude A(m) images:
and detector located on same side of the
inspected part (Fig. 9) or in transmission (9) $(w) ~ s, - s,
mode, where the source and detector are
on opposite sides of the part. The atan · .
reflection approach is best suited to detect 54 - 52
discontinuities close to the heated surface
while the transmission approach permits The depth range of magnitude image is
detection of discontinuities close to the
rear surface. The transmission approach is roughly given by thermal diffusion
not always possible- sometimes the rear r;2-k
surface is not accessible. Also, in length !t:
transmission mode, the discontinuity (11) l'
depth can not be estimated because the
travel distance is the same regardless of \ OlpCp
discontinuity depth.

Step Heating Thermography with thermal conductivity k, density p,

The step heating technique is sometimes specific heat CP and modulation .
referred to as long pulse tlzennogmplly or frequency (1}, In the case of phase unages,
time resolved infrared radiumel1y. Here the
increase of surface temperature is the depth range is about twice larger.

Equation 11 indicates that higher
modulation frequencies restrict the
analysis in a near surface region.

Fundamentals of Infrared and Thermal Testing 43

Loss Angle Lockin Thermography. Other within the field of view, the temporal
stimulations are possible. For instance, an
ultrasonic transducer (shaker) can be decay f(x) is extracted from the
attached to the specimen, or the specimen thermogram sequence, where x is the
can be partly immersed in an ultrasonic index ill the thermogram sequence. Next
bath. In these cases, the high frequency from ((x) the discrete Fourier transform
ultrasonic signal (typically about 40 kHz)
is modulated with a low frequency signal. F(w) is computed, where {I) is the
The lmv frequency modulation creates a
thermal wave of desired wavelength as in frequency variable. Finally, from the real
photothermal lockin thermography while R(w) and imaginary /(co) components of
the high frequency acts as a carrier F(w), the amplitude A(w) and phase ¢(ro)
delivering heating energy inside the are computed:
specimen.12 This technique is referred to
as the loss ansle lockin thermography and (12) A(ro)
has been applied with success for
detection of corrosion, vertical cracks and (13) ¢(w) a t a n - 1-Rl((ww-))
delaminations.
In pulsed phase thermography as in
Pulsed Phase Thermography. An ideal lockin thermography, it is possible to
Dirac pulse has a flat frequency spectrum, explore the various frequencies. However
thus a thermal pulse in pulsed in pulsed phase thermography the
thermography launches under the analysis is performed in the transient
specimen surface a mix of frequencies mode while in lockin thermography, the
that can be unscrambled by performing signal is recorded in the stationary mode.
the Fourier transform of the temperature
decay on a pixel-by-pixel basis. This Vibrothermography
enables computation of phase images as
in lockin thermography. Such a Vibrothemography is an active infrared
processing technique that combines thermographic technique where, under
somehow advantages of both pulsed the effect of mechanical vibrations (20 to
thermography and lockin thermography SO Hz) induced externally to the structure,
is called pulsed phase thermograplly. 13 because of direct conversion from
mechanical to thermal energy, heat is
The process is as follows. After pulsed released by friction precisely at
heating the specimen, the temperature discontinuities such as cracks and
decay is recorded and for each pixel (i,j)

\ •'-.u-.--·

TABlE 2. Applications of infrared thermographic techniques.

Technique Process Control Discontinuity Detection Material Characterization
Passive
Carton sealing line inspection, automobile brake Walls, moisture evaluation, roofs, Glaze thickness on ceramics,
thermography system efficiency, heat dissipation of electronic assemblies crush tests investigation
modules, recycling process identification,
lockin printed circuit boards, glass industry (bottles, liquid level in tanks
thermography bulbs), welding process, metal (steel) casting
(active) Crack identification, disbanding, Adhesion strength, anisotropic
Bearings, fan and compressors, pipelines, steam impact damage in carbon fiber material characterization,
traps, refractory lining, rotating kilns, turbine reinforced plastics coating thickness in ceramics,
blades, electric installations, gas leaks moisture evaluation
Metal corrosion, crack detection,
Aircraft structural component inspection, loose disbanding, impact damages in Depth profile of thermal
bolts detection carbon fiber reinforced plastics, conductivity or diffusivity
turbine blades, subsurface
Plastic pipe inspection defect characterization (depth, Thermophysical properties
size, properties) in composites, (diffusivity etc.), underalloyed
Radar absorbing structure investigation wood, metal, plastics and overalloyed phases in
coatings on steel, moisture,
Pulsed Aircraft structural component inspection, solder Defects in adhesive and spot anisotropic material
thermography quality of electronic components, spot welding welded lap joint characterization
(active) inspection
Coating wear, fatigue test, closed Thermal conductivity
Water entrapment in buildings and fresco crack detection measurement in c.:1rbon fiber
delamination reinforced plastics

Step heating Degradation of erasable programmable read only Coating thickness mea~urement
thermography memory chips
(active) Variations in viscoelasticity and
Paper structure (cockling) emissivity

Vlbrothermography Failure analysis
(active)

44 Infrared and Thermal Testing

FIGURE 10. Application of active pulsed thermography to detect subsurfac:;:e discontinuities in

carbon fiber reinforced plastic specimens. Thermogram sequence reveals impact damage on
bottom right corner. Progression of the thermal contrast is clearly seen after initial thermal
pulse (flashes are fired at t = 0 s). Maximum thermal contrast image clearly shows extent of
delaminated area.

t=O.s I"" 0.98 s t"" 1.25 $ t= 1.51 .s

f=l.78s I= 2.06 s t=2.32s t;; 2.87 s

1=3.42s t=4.5ls Maximum contrast at
1""' 1.44 s(see Fig. 11)

delaminations. Discontinuities are excited (focal plane arrays with more than
at specific mechanical resonances: local 512 x 512 pixels), greater pixel
subplates formed from delaminations response uniformity (> 99.5 percent)
presence resonate independently of the and faster acquisition, no longer tied
rest of the structure at particular to video standards of 25 or 30Hz.
frequencies. 14 Consequently, by changing Some infrared cameras, for example,
(increasing or decreasing) the mechanical have a 30 kHz frame rate.
excitation frequency, local thermal 2.Computer hardware and sofhvare can
gradients might appear or disappear. be improved to process the infrared
signaL For example, wavelet transform
Common Applications and processing could be used with both
Limitations time and frequency capability, and
thermograms could be evaluated
Common applications of infrared automatically.
thermographic techniques are listed in 3.lnfrared machine vision applications
Table 2. Their limitations and capabilities ofinfrared thermography include
are listed in Table 3. Figures 10 and 11 integrated processing and, for active
show results from active pulsed procedures, particular heating

thermography on a carbon fiber schemes. ror example, lateral heating

reinforced plastic specimen: an impact may be applied to detect cracks on
damage specimen is detected. Dedicated concrete structures, and performance
processing such as thermal contrast from uncooled microbolometer based
computing permits extraction of infrared cameras may be improved for
quantitative information about the continuously monitoring various
discontinuity. industrial processes.

Future developments can be
anticipated in three directions.

!.Hardware can be improved to acquire
the infrared signaL For example,
infrared cameras with low noise
(< 20 mK), higher spatial resolution

Fundamentals of Infrared and Thermal Testing 45

TABLE 3. Advantages and limitations of infrared thermographic techniques.

Technique Advantages Disadvantages

All thermographic Fast, surface inspection Variable emissivity
techniques
Ease of deployment Cooling tosses (convection/radiation causing perturbing contrasts)
Deployment on one side only
Safety (no harmful radiations) Absorption of infrared signals by the atmosphere (especially for distances
Ease of numerical thermal modeling greater than a few meters [about 10ft])

Ease of interpretations of thermograms Difficulty to get uniform heating (active procedures)
Great versatility of applications (see
Transitory nature of thermal contrasts requiring fast recording infrared
Table 2) cameras
Sometimes unique tool (corrosion around
Need of straight viewing corridor between infrared camera and target
rivets) (although it could be folded through first surface mirrors)

limited contrasts and limited signal to noise ratio causing false alarms-
measurement of a few degrees above background at around 300 K
(27 'C ~ -80 'F)

Observable defects generally shallow

Passive No interaction with specimen Works only if thermal contrasts naturally present
thermography No physical contact

lockin No physical contact Require modulated thermal perturbation
thermography large inspected surface- several m2 Require observation for at least one modulation cycle (longer observation
(active)
(30 ft2) simultaneously with respect to pulsed thermography)
Pulsed Thickness of inspected layer under the surface
thermography Phase and modulation images available related to the modulation frequency (unknown defect depth might require
(active)
Modulated ultrasonic heating (for some multiple experimentations at different frequencies)
applications, might require physical
contact or bath immersion) Requires apparatus to induce the pulsed thermal perturbation
Computation of thermal contrasts require a priori knowledge of defect free
No physical contact
zone in field of view
Quick (pulsed thermal stimulation: cooling Inspection surface limited (~0.25 m2 maximum).
or heating)

Phase and modulation images available
with frequency processing (as in pulsed
phase thermography)

Step heating No physical contact Require apparatus to induce the thermal perturbation
thermography Reveal dose cracks Risk of overheating the specimen
(active)
Difficulty to generate mechanical loading
Vibrothermography Thermal patterns appear only at specific frequencies
(active) Physical contact to induce thermal stimulation

FIGURE 11. Logarithmic progression of thermal contrast over
white line passing through center of delaminated area of
Fig. 10. Contrast value over discontinuity free areas is 1.

1.5 Time (s)

2c 1.4

~

·~ 1.3
g 1.2

,e;; 1.1

~c

u0

0.9
150

Pixel position
(increment)

46 Infrared and Thermal Testing

PART 3. Calibration for Infrared
Thermography1s

Correction of Vignetting transmit oblique rays and a part of the
Effect light cone may be cut off, causing an
amplitude reduction at the edges of the
Before quantitative analysis of image. In Fig. 12, the central bright area
thermograms, it is necessary to convert corresponds to the portion of sensor fully
the raw image sequence into temperature reached by radiation and the dark area is
images. This involves generally one or two caused by a loss of radiation caused by the
steps such as image restoration (for limited lens aperture (the three dots in
example, correction for vignetting effect Fig. 12 correspond to locations of three
and/or noise suppression) and conversion reference points in Fig. 14). Vignetting is
of raw pixel values to temperature also more severe if expansion rings restrict
following a calibration procedure. In the the field of view because of the limited
case of focal plane arrays, the image effective aperture obtained in this case.17
restoration is generally limited to the
vignetting effect (if present) because the As predicted by the theory,ls
noise level is low. Here, uniform heating experiments carried out with the focal
and emissivity are assumed. plane array camera shmved that this effect
depends both on pixel location and
An example of vignetting is shown on temperature difference between the target
Fig. 12. This phenomenon is explained as and the ambient. Figure 13 clearly
follows. 16 If a cone of rays from a point in illustrates the vignetting effect on three
the object space, limited by the thermograms recorded at uniform
diaphragm of the lens, is formed and temperature, respectively 282, 295 and
intercepted with the image plane
perpendicular to the lens axis, the 323 K (9' 22 and so °C; 48, 72 and 122 °F)
intercept is a circle if the object lies on the for af band c. Belmv ambient temperature
optical axis and more generally an ellipse
if the object is laterally displaced. (295 K [22 oc = 72 °F]), vignetting has an
!vforeover, for many lenses, the front and
rear apertures are too small to fully opposite behavior than above ambient
temperature: the curvature direction
FIGURE 12. Vignetting effect is visible in image of uniform changes. At ambient temperature,
vignetting is not visible (Fig. 13b). In
temperature target (323 K[50 oc = 122 °F]) acquired with Fig. 13, the left drawing in each pair is an
uncorrected plot whereas the right
focal plane array camera. Three squares correspond to drawing is obtained after correction for
reference points (see Figs. 13 and 14). vignetting.

In Fig. 14, the difference between the
signal at the central reference point
(corresponding to the center of the
brightness area in Fig. 12) and three
points placed at different distances from
it, is shmvn (starting at the center of the

FIGURE 13. Side view w·lre frame representations of raw and
corrected images showing vignetting effect: (a) 282 K

(9 oc =48 of), (b) 295 K (22 oc = 72 of), (c) 323 K

(50 oc = 122 oF). Left drawings are before correction for
vignetting; right drawings, after correction.

(a) (b) (c)

Fundamentals of Infrared and Thermal Testing 47

image, the three dots shown in Fig. 12 optical elements.20 Because in infrared
correspond to the*, +, o plots of Fig. 14).
This difference is expressed by: thermography (depends on the
temperature, a software approach is
(14) ti;,;(G},y) = Cil,r ~ G/;
necessary. The idea is to create an
where d;,;(G},.) is the difference in gray (M x N x 2) matrix file (where M x N is the
level at reference G}.r; G/; is the gray level
of the pixel i,j at the temperature t; and image format) containing the coefficients
G},,.is the gray level of the reference at
temperatme t. a;,j and b;,; for every location (i,j) in the

In the case of a sequence of image.
thermograms taken at different By combining Eqs. 14 and 15, solving
temperatures, this difference is a linear
function of the temperature in gray level for Gf,jrr and adding the effect of
which may then be expressed as
temperature derived with GJ~~~v.o, the
(15) d;,;(c.!._,.) = a;.; c.!,, + b;,; correction formula then becomes:

where a;,; and bi,j are constant coefficients ,,, ,,,,.o(16)
for every location (i,j) in the image. , (crdG.c.~m = .Ten\'~ (1'ref ) + bij
b; '1· ~a;,·
The ambient temperature for this test '
corresponds to gray level 985. For such a 1'1 I - a·· l' '
level the correction is negligible and does
not depend on the position in the image where GJ~~· is the gray level at the
(see Fig. 14, around gray level 985, where reference point location corresponding to
the three curves get close together). the actual ambient temperature (at the
time the correction is computed) and
In the visible spectrum, a possible GJ~~·.o is the temperature (in gray level) of
hardware solution to this vignetting the room when the correction matrix was
problem is to add an additional lens in created. As an illustration of the
front of the objective. 19 If the original effectiveness of this procedure, it was
optics of the camera introduces a applied to the left plots of Fig. 13, and the
plots on the right of Fig. 13 are obtained
r-Jdistortion {then using a lens with a after application of Eq. 15. The
improvement obtained for Fig. 13b (the
distortion function corrects tht' ambient temperature case) is attributed to
response of the global system. Of course the noise filtering effect of the fitting
such an approach would attenuate the process with Eq. 16.
signal. This approach is possible if the
Noise Evaluation and
function f does not depend on the Temperature Calibration

features of the scene, such as temperature The technique proposed by Haddon21 and
because of emission originating from
Lee22 and reported in Maldague2:l was

used to chawcterize the noise content

present in infrared images. Two images, A

and B, of the same scene were recorded

and a third image C was obtained by

fiGURE 14. Linear behavior of signal loss due to vignetting subtraction of A and R. Because the scene
for three reference points of Fig. 12.
in A and ll is the same, image C

represents the noise introduced by the

camera. The standard deviation is

120 computed with the formula:

"'- 100 1[ · · - · · · · - - -
2}:(i,i) - Cav]'
~ (17) oc ~ ·. I

c ZMN

~ 80
a_ 60

70 where C;;Jv is the average of C,
M =maximum row, N =maximum
~ 40 column, i = (0, 1, ..., M-1), and j = (0, 1,

~ ..., N-1).
~
l listograms of image C at about 296 K
>, 20
(2:l oc ~ 73 °F) for the three different
~
s"' 0 infrared cameras 'i\'ere computed (see
Table 4 for figures about the three
gw -20 cameras). Newer advanced technology
cameras {focal plane array) exhibit a noise
-40 1000 1500 2000 2500 3000 content restricted in contrast to the frame
500 grabbing system (whose technolot.'Y dates
back to the 1980s). To compute the
Gray levels at reference position

48 Infrared and Thermal Testing

standard deviation in degret's celsius, the For the other cameras, calibration
calibration function for each camera is functions were obtained directly from the
also needed (see Eq. 18 below). manufacturers' data sheets.
Concerning the focal plane array camera,
experiments have been carried out to It is important to point out that such a
estimate this function and to verify its calibration procedure is only valid for a
stability in time; measurements performed specific experimental setup. If the
through one month showed variations experimental conditions change, it is
Jess than one percent. necessary to repeat the process. For
instance, such changes may involve a
The next step consists of comierting change of the objective or a change of the
gray level values into temperature. operating range of the camera (see the
Figure 1S shows calibration curves for case in Fig. 15 for the stirling cooled
three different infrared cameras. For camera (camera 3) at 273 to 523 K (0 to
instance, in the case of the focal plane 250 "C; 32 to 480 "F) and 123 to 353 K
array, the following relationship is (-150 to +80 "C; -240 to +175 "F). As
obtained where sis the gray level value mentioned before, focal plane arrays do
(linear best fit with a third order not have internal temperature references;
polynomial function): it can then be of interest to include a
blackbody in the field of vie\\' to
(18) 1tc) ~ - 13.4 + o.o5 g recalibrate the focal plane array as the
experiment progresses.
- 1.6 X to-S g2
Table 4 indicates for the three tested
+ 2.2 X 1()-9 g3 cameras, the standard deviation (at
ambient temperature) according to Eq. 17
TABlE 4. Comparison of infrared cameras. in both gray levels crG and temperatures aT
(with images converted to temperature
Standard Deviations Noise using a calibration function such as the
Equivalent one of Eq. 18 for the focal plane array).
Gray Material
The reference to camera 2 with a digital
Infrared Camera Level crT at T~mb Loss board in Table 4 means that the digital
(percent) signal is obtained directly from the
oc K or <>c "F 5 to 6 camera electronic unit through a
Camera 1, focal plane array dedicated board (single analog-to-digital
1.5 0.04 0.072 conversion inside the electronic unit)
whereas the acquisition through the
Camera 2 with frame grabber 4.3" 0.17h 0.306b 18 to 19 frame grabber for the same camera is done
Camera 2 with digital board 2.03 O.Q8b 0.144b 9 to 10 from the analog video output of the
electronic unit by a frame grabber_ Thus
Camera 3, stirling coo!ingc 1.3 0.13 0.234 14tol5 this makes reference to a triple acquisition
that degrades the signal as indicated
a. Thermal range= 10. (analog to digital to analog to digital). See
b. Assuming a linear function temperature (gray level value). elsewhere24 for additional details on this
digital board.
c. Range 273 to 523 K (O to 250 ~c = 32 to 482 ~f).

FIGURE 15. Comparison among temperature calibration TABLE 5. Noise standard deviation for focal plane array.
functions for three tested infrared cameras. For focal plane
array camera, experimental points are plotted too. Temperature Standard Deviation
O"c O"T (K or 0 C) O"T (<>F)
433 (160) (3201 ~----

413 (140) (2841 (K) ("C) ("F)

[L 393 (120) [248} 282.4 9.2 48.6 1.47 0.054 0.097
284.8 11.6 52.9 1.43 0.050 0.090
G'-..' 373 (100) (2121 290.8 17.6 63.7 1.49 O.D45 0.081
296.2 23.0 73.4 1.50 0.039 0.070
"-' 353 (80) (1761 300.8 27.6 81.7 1.48 0.033 0.059
304.2 31.0 87.8 1.48 0.029 0.052
(60) (1401 307.3 34.1 93.4 1.54 0.027 0.048
309.5 36.4 97.5 1.62 0.025 0.045
313 (40) (1041 _ J_ __,__ _L _ __t__ _j 311.3 38.2 100.7 1.62 0.023 0.041
293 (20) (681 1000 1500 2000 2500 312.9 39.8 103.6 1.61 0.022 0.039
273 (0) (321 3000 314.7 41.5 106.7 1.61 0.020 0.036
315.9 42.8 109.0 1.60 0.019 0.034
253 (-10) HI 317.4 44.2 111.6 1.60 0.019 0.034
318.4 45.3 113.5 1.63 0.019 0.034
233 (-20) (-40j t___ 319.4 46.2 115.2 1.64 0.019 0.034
323.7 50.5 122.9 1.69 0.021 O.D38
soo

Gray levels (arbitrary units)

legend

- - - - "'900 rilnge (0.250}
---a- =focal plane array camera
------ =900range(-150.80)

Fundamentals of Infrared and Thermal Testing 49

In column 2, the values of standard
deviation in gray level for the focal plane
array camera and stirling cooled camera
are very close, \\'hereas the corresponding
values in column 3 exhibit a threefold
difference. This difference is explained by
the different temperature responses from
the cameras which have calibration curves
·with very different slopes, as seen in
Fig. IS.

VVith respect to noise, the focal plane
array camera was further studied and
standard deviation was computed, in gray
levels G and celsius degrees T, in a range
of temperature from 282.4 to 323.7 K (9.2
to 50.5 oc [48.5 to 123 oF]) (Table 5).

This discussion illustrates issues
addressed in calibration of equipment for
infrared thermographic signal acquisition.

SO Infrared and Thermal Testing

References

1. Section 6, "Thermal and Infrared 11. Balageas, D. and P. Levesque. uEMII{:

Nondestructive Testing." A Photothermal Tool for
Nondestructive Testing Handbook,
second edition: Vol. 9, Special Electromagnetic Phenomena
Nondestructive Testing Methods. Characterization.'1 International
Columbus, OH: American Society for journal of Thermal Sciences- Revue
Nondestructive Testing (1995). GCm?rale de Thermique. Vol. 371
No. 317. New York, NY: Elsevier
2. Wolfe, W.L. and G.J. Zissis. 1"1le Science (September 1998): p 725-739.
lnfi·ared Handbook. \'\7ashington, DC:
Office of Naval Research, Department 12. Rantala1 ]., D. VVu and G. Busse.
"Amplitude Modulated Lockin
of the Navy (1985).
3. Cohen,]. Elements of Thermography for Vibrothermography for NDE of
Polymers and Composites." Research
Nondestructh-e Testing. NRS Technical in Nondestructit'e Evaluation. VoL 7,
Note 1177. \'Vashington, DC: National No. 4. Columbus, OH: American
Society for Nondestructive Testing
Bureau of Standards (1983). (1996): p 215-228.
4. ASTM E 1316, "Standard Terminology
13. lvialdague, X. and S. Marinetti. upuJse
for Nondestructive Examinations:
Section], Infrared Examination." Phase Infrared Thermography."
Annual Book ofASTM Standards: journal of Applied Physics. Vol. 79,
Vol. 03.03. \Vest Conshohocken, PA: No. 5. College Park1 MD: American
Institute of Physics (1996):
American Society for Testing and p 2694-2698.
Materials.
14. Tenek1 L.H. and E.G. Henneke. 11Flaw
5. Bolz, R.E. and G.L. Tuve, eds. Dynamics and Vibro-Thermographic
Hand/Jook of Tables for Applied Thermoelastic NDE of Advanced
E11gineering Science. Cleveland, OH: Composite Materials.11 Tlzermosense
CRC Press (1973): p 211. Xlll. Proceedings SPIE Vol. 1467.

6. Maldague, X. "Thermographic Bellingham, VVA: International
Techniques for NDT.'' Encyclopedia of Society for Optical Engineering
Materials Science a11d Teclmolog}'. (1991): p 252-263.
Oxford, United Kingdom: Elsevier
Science Limited (to be published). 15. Marinetti, S., X. :Maid ague and

7. Maldague, X.P.V. Nondestructive M. Prystay. "Calibration Procedure for
Evaluation ofMaterials by Infrared 1:ocal Plane Array Cameras and Noise
Thermography. London, United
Kingdom: Springer-Verlag (1993). Equivalent Material Loss for
Quantitative Thermographic NDT."
8. Maldague, X.P.V., ed.lnfrared Materials Evaluation. Vol. 55, No. 3.
Metllodologr and Teclmolog)'. Columbus, OH: American Society for
langhorne, PA: Gordon and Breach Nondestructive Testing (March 1997):
(1994): p 525. p 407-412.
16. Ballard, D.H. and C.M. Brown.
9. Osiander, R., ].W.M Spicer and Computer l'ision. Upper Saddle River,
].C. Murphy. 11Analysis Method for NJ: Prentice Hall (1982).

Full-Field Time Resolved Infrared 17. Maldague, X., ).-C. Krapez and
Radiometry." Thermoset1se XVIII. SPIE P. Cielo. "Temperature Recovery and

Vol. 2766. Bellingham, WA: Contrast Computations in NDE
International Society for Optical
Thermographic Imaging Systems,"
Engineering (1996): p 218-227. journal ofNondestructive EwJluation.
Vol. 10, No. !,january 1991, p 19-30.
10. Busse, G. "Nondestructive Evaluation 18. Kingslake, R. Applied Optics and
of Polymer Materials." Nondestructive Optical Engineering. Vol. 2. New York,
'!!'sting and Evaluatiou International. NY: Academic Press (1965):
Vol. 27, No. 5. Oxford, United p 212-213.
19. Cicio, P. Optical Techniques fOr
Kingdom: Elsevier Science Limited Industrial lmpection. San Diego, CA:
(1994): p 253-262. Academic Press (1988).

Fundamentals of Infrared and Thermal Testing 51

20. Hamrelius, T "Accurate Temperature
Measurement in Thermography: An
Overvie'w of Relevant Features,
Parameters and Definitions."
Tllerrnosense XIII. Proceedings
SPIE Vol. 1467. Bellingham, WA:
International Society for Optical
Engineering (1991): p 448-457.

21. Haddon, j.F. "Generalised Threshold
Selection for Edge Detection.'1 Pattem
Recognition. Vol. 21, No. 3. Oxford,
United Kingdom: Pergamon Press for
Pattern Recognition Society (1988):
p 195-203.

22. Lee, D.)., 1~F. Krile and S. Mitra.
"Digital Registration Technique for
Sequential Fundus Images."
Applications o(Digital Image
Processing X. SPIE Vol. 829.
Bellingham/ V\7A: International
Society for Optical Engineering
(1987): p 293-300.

23. Maldague, X. Non-Destmctive
Evaluation o( Materials by Infrared
Thermography. London, United
Kingdom: Springer Verlag (1993).

24. Jalbert, L. and X. Maldague. "Design
of an Integrated Conversion/Control
System for Infrared Image and Signal
Acquisition." Vision Interface 1995
{Quebec, Canada]. Toronto, Canada:
Canadian Image Processing and
Pattern Recognition Society (1995):
p 148-152.

52 Infrared and Thermal Testing

CHAPTER

Heat Transfer

Vladimir P. Vavilov, Tomsk Polytechnic University,
Tomsk, Russia (Parts 1, 2 and 3)
Roberto Li Voti, University of Rome, Rome, Italy
(Part 4)
Mario Bertolotti, University of Rome, Rome, Italy
(Part 4)
Douglas D. Burleigh, San Diego, California (Parts 1, 2
and 3)
Grigore L. Liakhou, Technical University of Moldavia,
Kishinau, Moldavia (Part 4)
Stefano Paoloni, University of Rome, Rome, Italy
(Part 4)
Concita Sibilia, University of Rome, Rome, Italy (Part 4)

PART 1. Fundamentals of Heat Transfer

Heat Transfer Mechanisms where 7~ is surface temperature, T1 is fluid
temperature and hr is convection heat
Heat transfer occurs in a medium or transfer coefficient (\·V·nr2·K-').
between bodies in three different ·ways:
conduction, com•ecliuu and radiation The maximum radiation flux emitted
(Fig. 1). Conduction is the propagation of by a blackbody is given hy the
heat energy whenever a temperature Stefan-Boltzmann law:
difference exists between two solid bodies
in contact or among parts of a body. where a is the Stefan-Boltzmann constant
Convection involves the mass movement (5.67 x JQ---8 \'V·m-2 ·K-4 ). Graybodies are
of gas or liquid molecules over large characterized ·with an emissh'it)' E that
distances. Two solid bodies will exchange varies from zero to unity and determines
energy by convection if they are in the energy emitted by radiation:
contact with a fluid. Radiation is a process
of heat transfer and is characteristic of all FIGURE 1. Heat transfer mechanisms:
matter at temperatures higher than (a) conduction; (b) convection; (c) radiation.
absolute zero. Radiated energy may be (a)
transported over large distances through
gases or a vacuum with no conduction or Heat
convection medium.
(b)
Heat energy is transferred from an
object with a higher temperature to an Fluid
object with a lower temperature. fn a
general case, the temperature distribution
in a body depends on three coordinates
X,)~Z and on time t. The temperature
distribution is expressed T(X,)~z,t). Heat
transfer occurs bet\veen bodies until
thermodynamic equilibrium is reached.
Temperature differences in a medium or
in a body tend to decrease with time
because of heat transfer.

The analysis of three-dimensional heat
transfer problems is extremely difficult. It
is common to simplify the mathematics
by considering a less complicated
two-dimensional or one-dimensional case.

In a one-dimensional case, the
conductive heat transfer 0cd (\·V·m-2) is
given by the Fourier lav·::

(1)

where k is a material property called ~(c)
tllrrmal conductivity (\·V·m-1·K-1). Tlms, in a
t. t. t r,
plate of thickness L with the stationary
r, c=J
surface temperatures T1 and T2 where '12 >
legend
T1, the rate of heat transfer is equal to
T = temperature
(2) Q,d Tz-1j
R r, =fluid temperature

where thermal resistance R "" L·k 1 T, = surface temperature
(m2·W-I.!().

Convection heat transfer is described
with Newton's law of cooling:

54 Infrared and Thermal Testing

Two bodies 1 and 2 emit radiant Heat Transfer in Gas Filled
energy independently. The resulting heat Voids
flux between them is of the form:
~vfany typical voids in materials, especially
where G is a dimensionless geometric voids in composite laminates, can be
factor dependent on the shape and approximated as thin, gas filled gaps. The
orientation of two bodies {for two large conductive heat transfer that occurs in
size parallel planes G = 1). Over relatively such voids because of a differential
large distances, radiation energy may be temperature T 1-T2 on opposite surfaces of
partially absorbed in the gaseous medium the void is given by Eq. 2. Convection
bet\veen two bodies. In thermal may be neglected if the product of grashof
nondestructive testing, the distances are number Grand prandtl number Pr fits the
small and radiation energy losses are also following condition:
small.
(10) Gr· Pr < 1000
In the case of a single surface, heat
exchange between a warm body '/~ and a The grashof number has the same role
cold ambient Tamb occurs by combination in free convection that the reynolds
of convection and radiation: number has in forced convection. The
grashof number indicates the ratio of the
(7) Q QC\.+Qrd buoyancy force relative to the viscous
force acting on the fluid. The prandtl
hcv (Ts ~ Tamb) number relates the relative thicknesses of
the hydrodynamic and thermal boundary
+ crG(£1~4 ~ £·Ta~th) layers. It is a ratio of the (molecular)
momentum diffusivity to the thermal
In thermal nondestructive testing, a diffusivity.
temperature difference (T5 - Tamh) is
typically small and emissivities are Verification of Eq. 10 for temperature

rtypically high (< ~ 1): differences T1 ~ T2 < 100 oc yields the

(8) ll,d (T, -J;,nb) 4aG ( T, +Jemb £ maximum detectable void thickness
typical in materials evaluation:
X (J~-1~mb)
(11) d < 6 mm (0.25 in.)

In thin voids of large lateral size, the
radiation heat flux is approximated:

\h/ith this assumption, Eq. 7 is identical TABLE 1. Recommended values of the
to Eq. 3 with 11 = ll,v + llrd combining both
convection and radiation. Typically, the surface heat exchange coefficient (E = 0.9;
value of 11 is not known with high
accuracy because it depends on the body's T,mb = 293 K (20 "C = 70 "F); h,v =
shape and orientation, composition and
roughness, as well as on the temperature 1.7 x (T5 - Tamb) 113; hro is determined
difference T5 - Tamb that varies with time. by Eq. 8. Values for hare in W·m-2.K-1.
The recommended values of II arc
presented in Table 1, where E = 0.9; Tamb = Ts- Ta hiT h,d IJ

293 K (20 oc = 68 °F); ll,v is determined by K or oc ('F)

(9) hcv = 1.7~1~-Tamb 1 (1.8) 1.7 5.2 6.9
5.3 8.2
and 11rct is determined by Eq. 8. 5 (8) 2.9 5.4 9.1
Nonadiabalic heat transfer involves all 5.7 10.3
10 (18) 3.7 6.0 11.3
three mechanisms above. In active 6.3 12.1
thermal nondestructive testing, an 20 (36) 4.6 6.6 12.9
external heat source is used to provide a 6.9 B.6
heat flux to a test body. If an external 30 (54) 5.3 7.3 14.3
heat flux is much more powerful than the 7.6 14.9
heat flux between a body and the 40 (72) 5.8 8.0 15.6
environment, the case is referred to as 8.4 16.3
adiabatic. Only conduction must then be 50 (90) 6.3
considered. Additionally, the
corresponding analytical adiabatic 60 (108) 6.7
solutions have a simpler form.
70 (126) 7.0

80 (144) 7.3

90 (162) 7.6

100 (180) 7.9

Heat Transfer 55

where Tis defined by Eq. 13: The iriternal heating component H' in
Eqs. 15 and 16 is usually not needed for
(13) T ~ the analysis of tlwrmal nondestructive
testing, so it is assumed that w = 0.
Hence, the ratio between a conductive
and a radiation component of heat Internal heat generation is not a part of
transfer in a thin gas filled void is as the fourier equation:

follows: I aT
u. at
(14) Q,d - k
Q,d - 4dcrT3 A steady state heat flow with no heat
generation is described by the laplace
for air (f5il7leodcv~oid13s 5(k°F~);0q.0_3d.qW,ct·-n1r>1·1K7-1f; oTr ~ equation:
330 K
(18) a"T + ~0
d < 0.5 mm [0.02 in.J). The consideration ax2

above implies that, in typical material

voids, heat transfer occurs mostly by Steady state techniques are not used for
thermal nondestructive testing, because
conduction; the contribution from the temperature disturbances that exist in
a solid around interilal discontinuities
radiation is negligible. This would be diminish because of the diffusion of heat
through the surrounding material. As a
predicted merely by the fact that the J'o...T is result, no thermal signature is generated
on the surface that indicates the presence
low and the ambient temperature is low. of a subsurface discontinuity.

Differential Equation of For the transient process of thermal
Heat Conduction nondestructive testing, the most
appropriate equation is the fourier
The theoretical simulation of thermal equation (Eq. 17), which is typically
nondestructive testing scenarios requires
the prediction of three·dimensional analyzed.
temperature distributions in a solid body.
This is achieved by analyzing the dynamic Material Thermal
heat conduction governed by the Properties
following differential equation:
The basic material thermal properties are
(15) cPaaTr thermal conductivity k, specific heat C
and density p (see Table 2). 1•2 Thermal
+ (a)z (k7. a()Tz ) + w(x,y,z) conductivity is a material heat transport
characteristic responsible for the
where w{X,}~Z) is rate ofenergy generation attenuation of heat flux~ especially in
per unit volume in the medium (W·m-3), the steady state case. Thermal
Cis specific heal U·kg-1·K-1) and pis density conductivity values in 1~1ble 2 depend on
(kg·m-3), Equation 15 is a general the processing parameters of the
manufacturer and are given for reference
expression of the conservation of energy only.

for the medium in which heat is The nvo dynamic characteristics are
thermal diffusivity a and thermal
generated and propagated \Vhile efhisivity e:
experiencing diffusion and absorption.
(19) e ~ ~kCp
Equation 15 accounts for the
anisotropy in the thermal properties of See Table 2. Thermal diffusivity is a
measure of the rate at which heat is
materials such as composites and is diffusing through a material. Generally,
expressed with three different thermal materials that have high thermal
conductivity k also have high thermal
conductivities k.vkpkr The coordinates diffusivity and respond more quickly to
changes IY.T in their thermal environment
correspond to the three principal than do materials with low thermal
directions. In the case of an isotropic conductivity.

material, in ·which kx = ky = k1, the Thermal effusivity e is a measure of the
expression \'•till be a simpler form of ability of a material to increase its
Eq. 15. Using the fact that"~ k(Cp)· 1 is temperature as a response to a given
energy (heat) input. It is often referred to
thermal dif(ush'il)' (m2·s··l), Eq. 15 becomes: as thermal inertia and is used in calculating

I aT a'T a'T
ax'- + Oyz
(16)
a ilt

+ a'T + -w
az2
k

56 Infrared and Thermal Testing

a thermal mismatch factor r that Thermal diffusion length J.l is related to

characterizes a thermal contact between the heat stimulation frequency f (Hz).

two bodies, 1 and 2: This parameter is used in the theory of
thermal waves and expresses the fact that
(20) r ~ el - ez lower frequency thermal waves penetrate
deeper into a material than high
e1 + e2 frequency thermal waves. Conversely a
thermal wave of a fixed frequency will
r == 0 means no thermal mismatch, that propagate deeper into a material of higher
diffusivity than into a material of lower
is, the interface of two materials is not diffusivity. In thermal testing, thermal
diffusion length serves as an estimate of
detected on the surface; r == 1 specifies the the depth at which a buried discontinuity
could be detected in a one-sided test by
case when the second material is a perfect applying heat stimulation of different
frequencies.
conductor; conversely r = -1 is realized
One more thermal parameter is thermal
when the second material is a perfect resistance R:

insulator. (22) R ~ L
Another dynamic thermal parameter is k

thermal di{fi1sion lmgth ~~ (meter):

(21) ~

- ...,,-

TABlE 2. Thermal properties of common materials in order of Increasing thermal conductivity. 1•2 Values
depend on manufacturer's processing parameters and are given for illustrative purposes only.
Conversions: 1 kg ~ 2.2 lb; 1 m ~ 39.4 in.; 1 mm ~ 0.039 in.; 1 K ~ 1 "C ~ 1.8 "F.

Density Thermal Dlffusivity a Effusivity e
p (kg·m-3)
Material Specific Heat Conductivity (1o-<' m'·s-1) [W·>I(s)·m-2·K-1]
C (J·kg-1·K-1) k (W·m-1·K-1)

Air (thin gaps) 1.2 1005 0.070 58.0 9.19
Rubber (soft) 1100 2010 0.130 0.0588 536
Po!yisoprene 913 1905 0.134 0.0770 483
Polyvinyl chloride 0.140 0.11 422
Polyaramld (1.) 1330 1047 0.142 0.102 445
Polyaramid (II) 1330 1047 1.69 0.121 485
Pine (II) 2512 0.174 0.126 490
Epoxy resin 550 1700 0.200 0.09 667
Glass fiber plastic (1.) 1300 1200 0.30 0.13 832
Glass fiber plastic (I!) 1900 1200 0.38 0.17 922
Plaster 1900 1005 0.233 0.211 507
Water 1100 4193 0.586 0.140 1570
Graphite epoxy (.1) 1000 1200 0.64 0.52 888
Graphite epoxy (II) 1600 1200 1.28 1.04 1260
Zirconia 1600 0.65 0.219 1390
Brick (red) 5100 582 0.755 0.505 1060
Glass 1700 879 0.879 0.430 1340
Concrete 2442 837 1.51 0.752 1740
Nickel superalloy 2400 837 9.5 2.60 5890
Steel AISI 316 8300 440 13.4 3.47 7190
Silicon nitride 8240 468 16.0 9.65 5150
Titanium 2400 691 21.9 9.32 7170
Zirconium 4500 522 22.7 6440
Aluminum oxide 6570 278 46.0 12.4 11800
Bronze 3970 765 52.0 15.2 13900
Steel AISI 1010 8800 420 63.9 14.1 14700
Graphite 7830 434 116 18.8 13400
Silicon 2300 670 148 75.3 15700
Tungsten 2330 712 174 89.2 21100
Aluminum 2024-T6 alloy 19300 132 177 68.3 20700
Gold 2770 875 313 73.0 28600
Copper 19450 134 365 120.0 36500
Silicon carbide 9000 406 490 100.0 32300
Diamond 3160 675 660 230.0 34100
3516 502 374.0

Heat Transfer 57

The heat flux Q through a wall of
thickness L is related to the overall
temperature difference across the wall ['}.T:
(23) Q ~ /',T

R
Equation 23 is analogous to Ohm's law
in electric circuit theory. An analogy to
electricity is often used to solve complex
problems in heat conduction. In thermal
nondestructive testing, a discontinuity is a
thermal resistance site across which
temperature experiences a significant
change. This is the same as an electrical
resistance causing a drop in voltage.
Similarly, a multilayer wall behaves as N
thermal resistances in steady state in a
series:

N

"L..1k2·

ioo} /

58 Infrared and Thermal Testing

PART 2. Heat Conduction in Sound Solids

Basic Models and on a surface, where z:::::: 0, this is
simplified to:
In the classical heat conduction theory, a
solid is modeled with one of four simple (26) T(z ~ O,t)
geometric bodies: a semiinfinite body, a
plate, a cylinder or a sphere (l'ig. 2). where H' is the absorbed heat energy
A heat pulse or function can be simulated
with (1) an imtantaneous (Dirac) pulse, O·m-2), z is the depth into the material
(2) a square pulse, (3) a st<1' function or (4)
a periodic (harmonic) function (Fig. 3). and Eq. 27 expresses the fourier
Taking into account that surface heat number Fo:

*transfer could be either adiabatic (h ::: 0) (27) Fo,,

or nonadiabatic (11 0), this yields 32 (4 x Square pulse heating of a material as a

4 x 2) possible combinations for the function of depth z is described by:

thermal testing of solids. Those most (28) T(z,t) = ~'[2k'r~,
useful for thermal nondestructive testing
are discussed below. Details of other
combinations are available elsewhere.3

Semiinfinite Body x exp(-1- )
(Thermally Thick 4Foz
Specimen)
- 2 Foz -Fozh
The solutions of the fourier equation for n
the adiabatic heating of a thermally thick
specimen are as follows. Instantaneous 1
pulse heating of a material is described as x exp( 4Fo,- Fo,, )
a function of depth z:
- Ertc----b-
z,rFoz

exp(--(25) T(z,t) ~ Wa 1 1 + Erfc 1 ]
kz )n·FOz4Foz )
2~Fo,,

FIGURE 2. Solid models: (a) semiinfinite body; (b) plate; FIGURE 3. Types of heating: (a) instantaneous pulse;
(c) disk; (d) sphere. (b) square pulse; (c) step function; (d) periodic function.

(a) (c) (a) (c)

[~_]

(b) (d)

( )(b) (d)

Heat Transfer 59

On a surface where l = 0, this is (36) T(z,t) g [Erf2c-1-~
simplified:
lr
(29) T{z ~ O,t) ~ 2;L (ft - h )
e,l n - exp(Biz+Bi~Fo2 )

Here Q is the absorbed heat power xEric(2ft·o, + m,~)]
(W-rn-2) and
On the surface, where z = 0, Eq. 36
(30) Fo,
becomes:

Step function heating of a material is (37) T(z ~ O,t) ~ [1 - exp(H2at)
described as a function of depth z:

(31) T(z,t) Qz [2 ~.I:_F:IoT , X Erfc(HFt)]

k

The biot criterion Biz is defined by Eq. 38:

(38) Bi, lrz
k

H expresses the ratio of heat transfer
coefficient h to thermal conductivity k:

and on a surface, where z = 0, this is (39) H ~ 1_1_
simplified to:
k
/I(32)
T(z ~ O,t) ~ 2Q The contribution of surface heat
e ~n transfer to a temperature signal could be
evaluated by the ratio of transient
Harmonic heating (the alternating nonadiabatic ('/~1ad) and adiabatic (Tad)
solution component only) of a material is temperatures on the surface of a
described as· a function of depth z if z > 0. semiinfinite body given by Eqs. 32 and 37
for step function heating:
(33) T(z,t) ~ _zeQ,J_;
-[;. l-exp(x2)Erfcx
x ex+~) 2X

x ex+(wt-;- %)] where

(41) X ~ H .,fai

where e is thermal effusivity. On a surface The ratio is shown in Fig. 4 as a
where z = 0, this is simplified to: function of x. The divergence between the
two solutions is less than one percent if
Q H\l(at) < 0.011. As an example, in a
zej;(34) T(z~O,t) ~
graphite epoxy composite, the analysis
expHwt-%)]
time in laboratory conditions must be
within about. 1 s to neglect surface heat

exchange with a one percent loss of
accmacy.

The variable ro is angular modulation Plate
frequency (snr1) and Q is defined by:
The solution of the fourier equation for
(35) Q ~ ~n [1 + cos(cot)j the adiabatic heating of a solid plate by
an instantaneous pulse as a function of
where ~n is maximum absorbed heat depth z is:
power (\'V·m-2).

The solution of the fourier equation for
IWtWdiabatic step function heating is given
by the following expression:

60 Infrared and Thermal Testing

(42) T(z,t) lVa + [lVa (44) )nro

Lk tk

Lx 2 cos(nnZ) t,[1 2x + exp(-n2 n2 ro)]

11=1 where the subscripts pi and sib denote a
plate and a semiinfinite body respectively.
x exp(-n2 .n2 ·Fo)] The latter function is shown in Fig. S to
clearly illustrate the fact that the
and on a front (heated) surface, where divergence between the temperatures of a
z = 0, this becomes: plate and a semiinfinile body made of the
same material diminishes for thicker
(43) lVa + [ -W-o: plates and at earlier times. To ensure that
Lk Lk such divergence is less than one percent,
the fourier number must be less than
x 2I 0.19. As an example, for a graphite epoxy
composite, the time of analysis must be
11"'1 less than 1.5 s for a 2 mm (0.08 in.) thick
plate and less than 9 s for a S mm
x exp(-n2 .rt2 ·Fo)J (0.2 in.) thick plate.

Here Lis the plate thickness and Z = z.L-1
is the dimensionless in~depth coordinate.

Plate versus Semiinfinite FIGURE 5. Comparison of adiabatic solutions for plate and
Body Solutions semiinfinite body with instantaneous pulse heating.

The solutions of the fourier equation for a 1.25 7
semiinfinite body are simpler than those -
for a plate. Jt is obvious that the 1.2
divergence between these solutions I
increases in time, with more heat '·,_:" 1.15
penetrating into the material. It is I
convenient to show the corresponding >--"
condition as a function of time and 1.1 1/
thermal properties in the case of
instantaneous pulse heating. Combining 1.05
Eqs. 26 and 43 produces:

0.2 0.4 0.6 0.8 1.0

FIGURE 4. Comparison of nonadiabatic and adiabatic Fo
solutions for semiinfinite body (step function heating).
legend
~ Fo-= fourier number (Ht ·L-1), where Lis material thickness (rnm), tis time
(second), and u is thermal diffusivity (ml·s--1)
Tpl = temperature of plate
T,,b"' temperature of semiinfinite body

0.98 ~

',u_, 0.96 f'-.._

~ -.......

he ~

0.94 ~

0.92

0.02 0.04 0.06 0.08 0.1

legend

h =heat transfer coefficient (W·m-l.K-1)
k = thermal conductivity (W·m-l.K-1)

r.d = adiabatic temperature

Tmd = transient nonadiabatic temperature
a = thermal diffusivity (m2·s-1)
t = time (second)

Heat Tran~fer 61

PART 3. Heat Conduction in Solids with Buried
Discontinuities

Coating on Substrate (48) T"t• r.- [I 2~Qc ~ + :f(-r)"

Heat transfer in solids with buried 11=1
discontinuities is not easily understood.
Some useful information may be obtained l FJ]x(exp-t-•
by modeling the in-depth propagation of
pulsed thermal energy as the 1
superposition of thermal waves. Then, by
analogy 1.vHh optical waves, refraction and ~-n1t-• Erfc'/l
reflection phenomena may be introduced
at the boundary between two materials. Here t* = n2·L2·ac-1 is the thermal
The simplest solutions arise for a coating
on a substrate material, with the latter transit time for the nth reflection of a
being represented by a semiinfinite body.4
thermal wave, the subscript c specifies a
For instantaneous pulse heating:
coating and a thermal mismatch factor r
(45) T F(t) ~ ~W
is determined for the interface between a
en/nt coating and a substrate.

+ [__!:I'_ The thermal mismatch factor r is a
e,.{rrt
good indicator of the thermal contact
x 2IJ-r)"
between two materials. \'\1hen r = 0, this
Jlo=]
indicates no thermal mismatch between
x exp(<)]
two identical materials. Cases \Vhere r > 0
For square pulse heating:
correspond to a more effusive substrate;
(46) T'(t) ~
values of r < 0 mean that a substrate is of

lower effusivity. The graphical
presentation of these cases is shown in

Fig. 6 for step function heating. During
the initial period, the whole structure

behaves as a semiinfmite body and the
corresponding function is a straight line
in a graph ofT versus ~t. During this time
the interface between a coating and a

substrate will not be detected. During

thermal transit time, the surface
temperature of the coating starts to

+ 2jt ,L}r)" Ft FIGURE 6. Influence of substrate thermal properties on
behavior of surface temperature (step function heating of
11=1 thermally thick sample).

2Jl-0, Insulating coating -------:;.•••.....
... .... ... "...'...Thermally thick specimen
where: X I,(-r)" r(t - th )] •••' ~'

(47) F(t) 11=1 • .,.. """ ..,. - -

t:)Exp(_ •••• *"" '

p- (:t;-rrt-* Erfc ~t* Conducting

t \1 t coating

For step function heating: Thermal transit tinlf•

Square root time (\1s)

62 Infrared and Thermal Testing

deviate from the temperature that would This restriction limits the choice of a
appear in the absence of a coating. The temporal step, once a spatial ·step is
sign of the difference between a real and fixed. In two-dimensional and
expected temperature depends on thermal three-dimensional models, this may lead
mismatch factor magnitude. to a very long computation time. The
other type of calculation scheme is called
Numerical Modeling implicit and makes it possible to overcome
Principles this difficulty.'

Analytical solutions are of little help Transition Criteria for
when simulating practical test situations One-Dimensional to
where a material is anisotropic and, more Three-Dimensional Models
importantly, where discontinuities have
complex shapes. The discontinuities that may be
analytically simulated must be very large
With growing computer power, laterallyi otherwise lateral heat diffusion
numerical techniques are becoming will reduce the temperature signature that
increasingly common in the analysis of would reveal a discontinuity and reduce
three-dimensional discontinuities. As the length of time during which the
required by the geometry of a test part, discontinuity may be observed. Lateral
the heat diffusion equation may be heat diffusion is affected by the type of
written either in cartesian coordinates as the material, the discontinuity depth and
in Eqs. 15 to 18 or in cylindrical or time. The ratio rd·f-1 between
spherical coordinates. 'r\rith a transient discontinuity radius and depth appears in
numerical technique, the second partial a rule of tlzumb to define whether a
derivative with respect to coordinates and discontinuity may be regarded as
the first partial derivative with respect to one-dimensional (infinite) or
time are replaced with their finite three-dimensional (finite). Approximate
difference substitutes: rd·I-1 values are given in Table 4 and Fig. 7
for two different materials and two
a2 r 1i+l + 1f-t - 2T;
(49) ax2
(Lix)z

(50) ar rt+1 - rP FIGURE 7. lateral discontinuity size:
at ' ' (a) one-dimensional cartesian model;
M (b) two-dimensional cylindrical model.

where subscript i denotes spatial mesh (a)
points (the x dimension), superscript p
specifies time mesh points (current time),
t1X is spatial step and At is time step.

Equations 49 and 50 may be
substituted into the one-dimensional

presentation of Eq. 51:

Discontinuity

This simple numerical scheme allmvs the (b) It
calculation of the temperature in the
mesh point T1at a time t + At if all ~ !I I
temperatures at a time tare known. An
extension of Eqs. 50 and 51 to two and @--:----!o-,,
three dimensions can easily be made.s I~

Equation 51 specifies an explicit legend
calculation scheme that for mathematical
reasons requires stability: d = discontinuity thickness
L = sample thickness (meter}
(L1x)2 I = discontinuity depth (meter)
Q = heat transfer or heat flux (W·m-2)
(52) $ 2 rd = radius of discontinuity
aM

Heat Transfer 63

different heating times. It is evident that discontinuity to.Tm and the time of its
one-dimensional discontinuities in metals optimum observation tm(to.Tm) and (2) the
must be laterally larger than similar maximum rwming contrast
discontinuities in materials whose thermal
conductivity is lower. Also, lateral heat (53) c~~n =
diffusion is less at earlier times ~ it
increases with time. and the time of its optimum observation
tm(C,Rm). The maximum temperature
Moreover, such a criterion can be signal could be expressed in the form of
generalized in the case of anisotropic the normalized contrast:
materials with the weighting ratio Y(krkr1)
where k1 and kr are thermal conductivities where Tmax is the maximum surface
in the I and r directions, respectively. The temperature that occurs above a
criterion ratio thus becomes discontinuity free area at the end of
(rd·I-1H(kl'k,-1). Note that such extension heating. Both running and normalized
is also valid for isotropic materials such as contrasts are dimensionless and
steel because in such a case k1 ;= kr. independent of heating power because
both aT and Tare linearly proportional to
The material properties in Tables 3 Q. (Note that, in transient thermal
and 4 were obtained for an air filled nondestructive testing, Tis not the
discontinuity (void) in a material where absolute temperature but is the
discontinuity depth I= 0.1 L and where differential temperature, the temperature
discontinuity thickness d = 0.1 L. Lis wall in excess of the ambient or initial
thickness. temperature.)

Basic Features of Transient Discontinuity Detection Criteria
Thermal Nondestructive
Testing Generally, a temperature signal can be
detected if one of its characteristics, such
Informative Parameters as amplitude, exceeds the noise level.
Often this is expressed with the simple
Two pairs of iu{ormative parameters are of formula:
practical interest in transient thermal
nondestructive testing: (1) the maximum (55) s > 1
temperawre signal above or under a
An expression of this form can be
TABLE 3. Material properties for air filled written for any chosen informative
discontinuity. parameter or combination of them. In
thermal nondestructive testing,
Conductivity K Dlffusivlty a discontinuities are typically detected by
evaluating temperature signals f1T(t) that
Material (W·m-'·K-1) (m'·s-') occur over discontinuities. Then Eq. 55
can be written as follows:
Steel 32.0 7.3 X lQ--6
0.64 5.2 X 10-7 (56) > 1
Graphite epoxy 0.07 5.8x10-5
Air defect

TABLE 4. Approximate values of lateral where <Jnd is the standard deviation of the
defect size corresponding to transition temperature in an area without anomalies
from two-dimensional cylindrical model and K is the coefficient that determines
to one-dimensional cartesian model in the reliability of decision making
material where discontinuity depth (typically 1 ~ K ~ 3).
I= 0.1 L and where discontinuity
thickness d = 0.1 L. Assume that there are both additive
and multiplicative noise:
Material Heating Critical rd·l-1Values
Thickness L The magnitude of additive noise <Jadd does
Time Graphite not depend on a measured parameter
mm (in.) (s) Epoxy Steel such as sample temperature. In thermal
nondestructive testing, this type of noise
. . v··~· --~-.-~, -----.,-·~-~----· -~~--'- -~>·---~~~ ~-~--- is typically determined by the infrared

1.0 (0.04) 0.01 23 10
1.0 (0.04) 100.00 30 10
5.0 (0.20) 23 10
5.0 (0.20) 0.01 32 10
100.00

64 Infrared and Thermal Testing

system temperature resolution !lTres and Time Evolution of Temperature

some reflective noise O'ref(t) (in the ideal Signals

case cradd = !lTres)· Multiplicative noise is The variation of temperature versus time
proportional to a measured signal. For in transient thermal nondestructive
testing is of particular interest because the
example1 variations .6e in emissivity analysis of the temporal peculiarities of
influence useful temperature signals so the signal is important in reducing noise
that output infrared signals =otleT(t), where and in characterizing subsurface
discontinuities.
T(t) is the excess specimen temperature
(here it is assumed T(t) is not dependent Figure 8 is composed of six plots that
show the temperature evolution on the
on local absorptivity). This means that the front and rear surfaces of a 5 mm (0.2 in.)
magnitude of the multiplicative noise can thick graphite epoxy specimen heated
with a 0.01 s heat pulse. The specimen
be approximated as <imult = CnmnT(t)l contains an air filled circular
where C11tun is the noise running contrast discontinuity that has a radius of 5 mm
independent of time. Then, the simple (0.2 in.) and a thickness of 0.1 mm
model of the signal-to-noise ratio (SNR) in (0.004 in.), at a depth of 0.25 mm
(0.01 in.), except for Figs. Be and Bf. The
transient thermal nondestructive testing thermal properties of the graphite epoxy
are treated as isotropic. Tables 3 and 4
can be presented: show discontinuity parameters.

(58) SNR tlT(t) I~igures 8a and 8b show temperature
versus time for the front and rear surfaces,
['""()j2 both over a discontinuity and over sound
material (without a discontinuity). W'ith
K ~1LlTn2; + cr2nf(t) + C., T t short heating pulses, a noticeable
difference between Tnd and Td appears in
Equation 58 shows that the the cooling stage. For longer pulses or
discontinuities nearer the surface, the
signal-to-noise ratio varies in time/ difference may. also appear during the
reaching a maximum value at a particular heat pulse.
time that is the best experitnental
observation time: tm(s ~ maximum). The In Figs. 8c and 8d, the temperatures
maximum signal-to-noise ratio that can AT= Tn- Tnct for the front and rear
surfaces are shown in comparison with
be achieved in a thermal nondestructive the running temperature contrasts:

test is: (60) c'""

(59) SNRmax (Below this point in the text the
superscript nm will be omitted).
It is controlled by the temperature
These informative parameters reveal
resolution of the infrared thermographic the maximum temperatures for both
sample surfaces but they occur at different
equipment used. In modern infrared observation times lm. On the front surface
(Fig. Sc), after heating, the maximum
thermog r0a.0p0h1icoicm. aWgehresr1e.6tThreesrmcaanl be as of temperature signal ATm occurs earlier than
small as noise the maximum contrast Cm. Conversely, if
both maximums occur during a heat
the environment is higher than 0.1 K pulse, Cm surpasses 11Tm· Typically1 the
maximum signal-to-noise ratio occurs at
(0.1 oc = 0.18 °F), temperature resolution the time tm(Cm) that may be regarded as
the optimum obseroation time in a
is not a limiting factor. In fact1 reflective one-sided test. On the rear surface, the
additive noise can be very high, maximum contrast appears very early
when the temperature signal is rather
partiCularly if temperature is being small. Therefore, in a two-sided test
(Fig. 8d), the optimum observation time
measured during heating. Even after occurs when the temperature signal
exceeds the temperature resolution of a
heating, the residual heat energy can recording device and meets the conditions
of Eq. 55. Notice that, because of the
cause significant indications of reflective accepted convention for .6T, rear surface
temperature signals are negative.
nature. However, an organized test can
From here on the analysis of
often reduce the ratio of the discontinuities will primarily be

multicomponent noise to surface noise

caused by variations in optical properties

such as endssivity and absorptivity.

Unpainted objects max have quite a high

level of noise with Cr~~~tse reaching 20 to

100 percent for corroded and greasy

metals.6 Nonmetallic materials are

characterized with noise of about
C1~1~se""' 4 to 6 percent. The best way to
reduce noise (down to one to two percent)

is generally to apply a flat black high

emissivity coating to the surface. The

coating is commonly applied by using

black spray paint or other coatings. Water

washable paint is available and normal

aerosol spray paint may be removed using

a solvent such as acetone.

Heat Transfer 65

FIGURE 8. Temperature evolutions on front and rear surfaces of 5 mm (0.20 in.) thick graphite epoxy

specimen: (a) front surface excess temperature versus time; (b) rear surface excesS temperature versus time;
(c) front surface temperature and running contrast cruo versus time; (d) rear surface temperature and running

contrast cruo versus time; (e) front surface temperature for three discontinuity depths, versus time; (f) rear

surface running contrast for three discontinuity depths, versus time. Parameters: discontinuity thickness d =
0.1 mm (0.004 in.); discontinuity depth I~ 0.25 mm (0.01 in.); heat transfer Q = 100 kW.m-2; discontinuity
radius rd = 5 mm (0.2 in.); heat pulse duration th = 0.01 s.

(a) (d)

9" 2.S 0 \• /'[run
c15
\•. 9" -0.05 \I I
,_~E.
2.0 \ 15 -0.10 v ······..........................
,_'
1.S c,._~E. -0.15 /\
-0.20 J !J.Tx 100
\ \\D··..
'h -0.25
1.0
-0.30
0.5
............. 0 5 10 1$

Timet (s)

0 2 3
0
Timet (s)
(e)

0.8

(b)

0.16 /. ~D 0.6

9" 0.12 /" . ... ~ 0.4

15 / 10 15 0.2

c 0.08 5 0
,._~E. 0

',_ 0.04

0
0

Time t (s) 2 3
Time t (s)

(c) (f)

1.2 0

!\

i :,
0.8

0.6 ,~·· .....~ -0.004

~ ·········........ ~ ~

0.4 -0.008 5 10 15
Timet (s)
0.2 -0.012
0
0
0 2 3
Time l (s)

legend

cwn "' running contrast
D "' having measured discontinuity
J = defect depth (meter)

ND = having no measured discontinuity
T ~ temperature

Tamb = ambient temperature

66 Infrared and Thermal Testing

FIGURE 9. Temperature evolutions on front surface of 5 mm (0.20 in.) graphite epoxy
specimen: (a) front surface temperature in sound area versus time (80 cycles); (b) front

surface temperature in both sound and discontinuity areas versus time (three cycles); (c) front
surface temperature signal and contrast versus time (three cycles). Parameters: air filled
discontinuity thickness d ~ 0.1 mm (0.004 in.); stimulation frequency f ~ 0.2 Hz, close to
optimal; air filled discontinuity at depth I~ 1.0 mm (0.04 in.); heat transfer Q ~ 10 kWm-2;

discontinuity radius rd = 5 mm (0.2 in.); modulated heating.

(a)

200 ;~

9 ~----
0 Sound area
200 300 400
~
f-,0

<1(,

~<;.; 100
/-~6ge
~u /

1 /;":J"o

~

~

0.
E0
~ 100

Timet (s)

(b) , - - - - - Q"' (Oml2) [1 ~cos (2nfi)J

9 //"'·.. Phase lag

0 30 /\
~
f-,C
20
<1u2
~t\
c~
.~c"-

u§

~u
;":J"o
~

~ 10

0.

E

~

0 ,• 15
5 10

Timet (s)

(c)

,

3.0 ' ' ' ........ ' ' '

1.5

0 10 15
5
Heat Transfer 67
Timet (s)

Legend
cnm "' running contrast
Td = temperature in area with measured discontinuity
T00 =temperature in area without measured discontinuity
tJ.T =temperature change

performed with cylindrical geometry to (Fig. 9b). On the surface of a semiinfinite
permit reasonable accuracy in simulating body this phase lag is -45 degrees. The
real discontinuities while keeping the phase difference between sound and
analysis relatively simple. discontinuity areas enhances
discontinuity detectability because a
Figure Be shows the front surface higher signal·to·noise ratio appears in the
temperature difference versus phase domain. Because a phase lag can be
expressed in terms of time, the thermal
discontinuity depth, and Fig. Bf shows the wave approach uses the advantages of
rear surface running contrast versus phase analysis in the same manner that
discontinuity depth. Figures 8e and 8f the pulsed approach takes advantage of
show the temperature evolutions when time domain treatment.
the discontinuity is at three different
depths, 0.25, 0.5 and 1.0 mm (0.01, 0.02 The concept of an absolute
and 0.04 in.). Because of the diffusive temperature signal and dimensionless
nature of heat conduction, the front contrast is also valid for thermal wave
surface temperature versus time behavior techniques. Examples of time evolution of
is very sensitive to the depth of 11T and Care shown in Fig. 9c. Both
discontinuities. Deeper discontinuities informative parameters experience
produce lower amplitude temperatures at oscillations. The constant component of
later times (Fig. Be). This difference may the temperature signal increases in time,
be explained by using thermal wave at least within the first wave periods.
theory, in which the harmonic Meanwhile the temperature contrast
components of a heat pulse propagates in reaches a maximum at about 5 s after
depth depending on wave frequency turning on a heat source that produces a
while also experiencing phenomena of
interference, reflection and attenuation. FIGURE 10. Optimal detection parameters in 5 mm (0.2 in.)
Rear surface signals do not vary as much
with discontinuity depth (Fig. 8f), for a thick graphite epoxy specimen versus discontinuity depth:
heat pulse must pass through the whole (a) front surface; (b) rear surface. Parameters: discontinuity
specimen. Moreover, the highest thickness d = 0.1 mm (0.004 in.); heat transfer Q = 106
temperature signals on the rear surface W·m-2; discontinuity radius rd = 5 mm (0.2 in.); heat pulse
occur for discontinuities located in the
middle of the specimen (see the curve for duration th = 0.01 s.
I= 2.5 mm (0.1 in.) in Fig. 8f).
(a) 20
When using a thermal wave technique,
a heating function is often described: 3.0

......

%"(61) Q = ~ 2.4 / 16

[1 - cos(2nft)j "c' ~ \ lm... ..··/ 12 3

~u 1.8 ..!'
.Co 1.2
0.6 ~
Figure 9 is composed of three a~ 0f\8Tm\u"0 ....······
additional plots that show the ..···· 8E
temperature versus time for the same c ........····.······.x:.._..·1..-..-··-···· F
material and discontinuity type, with the ~._!'
discontinuity at a depth of 1.0 mm 4
(0.04 ln.). The surface temperature in both E"<l
materials with and without ~ 0
discontinuities reflect both the sample
thickness and the material. Typically, a 0 2 34 5
thermal wave technique known as a
classical photothermal technique is applied (b) 0.16 Discontinuity depth I (min) 8~
to rather thin materials requiring about 5
to 10 cycles to reach a steady state ~ ..· .......l.m...... .... ·Ep
condition where amplitude and phase
characteristics of surface thermal waves "o"' 0.12 2 34 7 c
are further analyzed. Recently, this Discontinuity depth I (min)
technique has been realized as a lockit1 .Co 0
thermograpllic technique based on ·~
recording an alternating thermal wave u"
component over large areas. In thermal i:~
nondestructive testing of rather thick ~0 ~~
materials, such as a 5 mm (0.20 in.) 6
graphite epoxy specimen, even the 80 l~~"' 0.08 .g.s
cycles shown in Fig. 9a are not enough to
get rid of influence of the constant E<l 0.04 E
component. ~ ~
5
The most remarkable feature of a 0 E
surface thermal wave is its phase Jag 0 K
relative to a periodic heating function 0
4

5

legend

tm = optimum observation time (second)
llTm = maximum temperature signal above (a) or under

(b) discontinuity

68 Infrared and Thermal Testing

thermal wave having a frequency of power is Q = 106 W·m-2 and corresponds
0.2 Hz. This frequency is close to optimal to a specimen excess temperature of
when detecting an air filled discontinuity -110 K (-110 °C = -200 ol') at the end of
in carbon fiber reinforced plastic at the heating. As expected1 the optimum
depth of 1 mm (0.04 in.). observation time increases with
discontinuity depth reaching tm ~ 10 sat
Discontinuity Depth I ~ 3 mm (0.125 in.).

An illustration of how discontinuity The corresponding graphs for the rear
depth affects optimal detection surface are symmetric in regard to a
parameters is shown in Fig. 10 for a 5 mm discontinuity depth of I= 2.5 mm
(0.2 in.) graphite epoxy specimen. On the (0.10 in.) (Fig. lOb). Therefore, a
front surface1 the temperature signal two-sided test is optimal for detecting
decays rapidly for deeper discontinuities discontinuities in the middle of a
(Fig. lOa). If the temperature resolution of specimen.
infrared equipment is assumed to be 0.1 K
(0.1 oc = 0.2 °F) the depth limit for Discontinuity Thickness
detected discontinuities will be about
3 mm (0.125 in.). Note that the absorbed The thicker a discontinuity is, the higher
the relative temperature above it will be,
FIGURE 11. Optimal detection parameters in 5 mm (0.2 in.) and the later the optimum observation
thick graphite epoxy specimen versus discontinuity thickness: time will be. This statement is Hlustrated
(a) front surface; (b) rear surface. Parameters: discontinuity in Hg. 111 where the data are p'resented
depth I= 2.5 mm (0.1 0 in.); heat transfer Q = 1O' W-m-2; for an air filled delamination located in a
5 mm (0.2 in.) thick graphite epoxy
discontinuity radius rd = 5 mm (0.2 in.); heat pulse duration specimen at a depth of 2.5 mm (0.10 in.).
In this case, the temperature profiles are
th = 0.01 s. similar for both the front and rear
surfaces.
0.6 .:>-- 11
10 For discontinuity characterization in
0.5 perturbing temperature distributions1 the
..... .....l.m.......... discontinuity thickness is less important
0.4 .k" ~.............. ... than the discontinuity thermal resistance:
0.3 93
....··"/ (62) R,
/0.2 8 .f
~ Discontinuity lateral Size and
I0.1 E Configuration
0
7F The diffusive nature of heat conduction
0 0.2 0.4 0.6 makes detection parameters rather
6 sensitive to variations in discontinuity
size. Both discontinuity thickness and
s lateral dimensions influence the
magnitude of the disturbance in heat flow
0.8 caused by the presence of a discontinuity.
As discontinuity thickness increases/ the
Discontinuity thickness d (mm) surface temperature signals increase and
appear at later times (l'ig. 11). Lateral
(b) discontinuity dimensions are often
confronted with discontinuity depth. It is
P' 0.36 ·y -------I-- 1m 9.S 3 intuitively obvious that a sufficiently large
............. discontinuity could make heat flow
0 0.30 ....... 9.0 .f one-dimensional. This is illustrated in
Fig. 12 where variations in tlTm and tm are
~ '/_...········· ··········· ..E~, shown in as a function of rd·f-1. It may be
observed that, for graphite epoxy1 lateral
...!' 0.24 8.S c heat diffusion stops if 2rd·l-1 > 5. For the
more isotropic aluminum/ this condition
<1 ·g0 becomes Zrd·/-1 > 10. The accepted rule of
thumb states that, in a one-sided thermal
~ 8.0 nondestructive test1 the lateral
;-··0c> 0.18 discontinuity size should be twice as large
~ as the discontinuity depth (2r,ri-I > 2).
.~c .lS Fig. 12b demonstrates the validity of this
u 0.12 rule of thumb. Because time parameter tm
~ l 7.S 0 is stable compared to variations in lateral

iel 0.06 E
a~.
E0 7.0 ·Eg"_
,". 0 0.2 0.4 0.6 0.8
6.S 0

Discontinuity depth I (min)

legend

tm "' optimum observation time (second)
ATm =maximum temperature signal above or under discontinuity

Heat Transfer 69

discontinuity size, domain treatment is are illustrated by Fig. 13, which shmvs
preferable in characterizing discontinuity thermal images taken at three different
depth.
times. The three discontinuities simulate a
Another aspect of lateral heat diffusion 50 percent reduction in thickness caused
is how temperature distributions are
influenced by the planar configuration of by corrosion in a 2 mm (0.08 in.) thick
discontinuities and how neighboring
discontinuities interact. These phenomena steel specimen and each has the same area
of 25 mm2 (0.06 in. 2) (Fig. 13a). The most
,--.,-:,
accurate representation of discontinuity
FIGURE 12. Optimal detection parameters versus lateral shape and the best resolution of the
discontinuity size on front surface, where separation of neighboring discontinuities
llTm(rd'l-1)/llTm(rd'l-1 ==):(a) schematic; (b) maximum
temperature signal versus rd.J-1; (c) optimum observation occur in the thermal images taken at
time versus rd.J-1. earlier times (Fig. 13b) because of weak

(a) lateral heat diffusion. At the optimum
observation time (Fig. 13c), some
temperature signals reach maximum

values but the effect of heat diffusion
becomes noticeable. At this time, a five

fold change in discontinuity size causes
about a five fold change in fiT1111 even if
the discontinuity area is constant

(compare the signals for discontinuities 1
and 3). Increased heat dissipation at later

times will significantly distort the thermal
signatures of the discontinuities
(Fig. 13d).

td==O.l mm

(b) 0.75 Graphite epoxy FIGURE 13. Influence of discontinuity configuration on
0.50 surface temperature distribution. Detection of 50 percent
~ corrosion in 2 mm (0.08 in.) steel specimen when Q = 1Q6
0.25 W-m-2 and t11 = 0.01 s: (a) discontinuity location where
II discontinuity 1 measures 1 x 25 mm2 (= 25 mm2 =
0 0.04 in. 2), discontinuity 2 measures 2.5 x 10 mm2
I. 0 5 10 1S (= 25 mm2 = 0.04 in.2) and discontinuity 3 measures
5 x 5 mm2 (= 25 mm 2 = 0.04 in.2); (b) temperature
;"!' distribution at 0.1 s; (c) 0.5 s; (d) 1 s.
(a) (c)
,;:\
2
~ 3

;"!' D

<l (b) (d)

20

Ratio of discontinuity size to distance, 2rd·f-1

(c)

0.75 f/!5t,mln"m

· Graphite epoxy
0.50

0.25

0 ' 20
0 s 10 1S

Ratio of discontinuity size to distance, 2rd·J-1

legend

d == discontinuity thickness
I = discontinuity depth (meter)
th = heat pulse duration (second)

70 Infrared and Thermal Testing

Heating Process same brief contrast rise occurs at the end
of any heating pulse. Notice that, at later
Each discontinuity type may require the times, both the front surface temperature
optimization of a heating process that will signal and the contrast may become
best reveal that type of discontinuity. The negative because of faster cooling of the
signal-to-noise ratio can serve as a material above a discontinuity (Fig. 14).
practical means for comparing the
effectiveness of different types of heating. To summarize, three points should be
This usually leads to maximizing the kept in mind.
temperature contrast C. Because the
contrast value is independent of Q, the 1. Heat pulse duration should be short
absorbed heat energy must meet the enough to ensure an adequate
requirements of Eq. 55: temperature contrast C above a
discontinuity, to allow detection.
(63) ti.T(t) ~ si\ T,"
2. Total heat pulse energy must be high
The heat must not damage the specimen, enough to create a detectable
or the test would not be nondestructive. temperature signal!'!T.
The maximum front surface temperature
Tmax occurring at the end of the heating 3. Total heat pulse energy should not
process must be less than the destruction damage the surface.
threshold temperature Tdestr·
The comparison between pulsed
Some aspects of the heating technique heating and thermal wave testing is
are illustrated with Fig. 14 for a one-sided presented in Table 5 for a one-sided
pulsed heating test. The highest contrast thermal test of an air filled discontinuity
is achieved by a short duration heating located I mm (0.04 in.) deep in aS mm
pulse (I" ~ 0.01 s). Following a longer (0.2 in.) thick graphite epoxy specimen. A
heating pulse T11 = 25 s, the maximum flash test results in higher contrast. In
contrast appears at about 7 s. Stopping practice, hmvever, other factors may be
the heating pulse at 7 s causes a small rise important to the optimization of a test
in the contrast whereas the absolute procedure, such as hardware limitations
sample temperature starts to drop. This and interference between a heater and
infrared camera.
FIGURE 14. Influence of heat pulse duration on time
evolution of running contrast crun in graphite epoxy Type of Material
specimen. Parameters: discontinuity thickness d = 0.1 mm
(0.004 in.); sample thickness L ~ 5 mm (0.2 in.); Theoretically, subsurface discontinuities
discontinuity depth I~ 1.0 mm (0.04 in.); discontinuity are detected best in materials that create
radius rd ~ 5 mm (0.2 in.). maximum temperature contrasts. It is
clear that, for materials of low isotropy,
heat does not penetrate deeply into the
structure and lateral diffusion is minimal.
On the other hand, for materials of high
isotropy, heat will penetrate more deeply
into the material. Also, lateral heat
diffusion becomes significant and may

th"' 0.01 s

1'. TABLE 5. Front surface maximum

0.20 temperature contrasts and their optimum

I\ observation conditions in inspection of air

l \c I \ filled discontinuity in graphite epoxy,
I \ :'•,
u2 where L ~ 5 mm (0.2 in.), I~ 1 mm
t; I \
I (0.04 in.), d ~ 0.1 mm (0.004 in.) and
~ 0.15 \:
rd = 5 1momc (0.2 in.). Conversion:
~c 1K ~ 1.8 OF.
K.._I '.4u0 I \i th = 7 s ~
l"
I
~
~ Optimum
I Observation
~ Time tm(Cm)
a~. 0.10
(s)
--......E

ij~
p ;I \ ...•, \\ Heating Temperature
Time Contrast Cm
\ '\
(K ~ oq

0.05 ' ' •,·.. '•, Pulsed Heating 0.230 2.6
I th := 0.01 s 0.185 8.5
I ' " .... '•'•, ···..... 0.127 7.0
th""7s
I; .... th""25s 0.181 9.7
0 Thermal Wave Heat 0.208 5.3
t::o:O.l Hz 0.167 6.2
0 10 20 t"" 0.2 Hz
t"" 0.5 Hz
Time l (s)

Heat Transfer 71

FIGURE 1S. Dependence of optimal detection parameters on type of material: (a) schematic;
(b) maximum temperature contrast versus thermal conductivity; (c) optimum observation

time versus thermal diffusivity. Parameters: heat transfer Q = 1os W·m-2; heat pulse duration

th = 0.01 s. 1 mm = 0.04 in.

(a)

(b)

2

~. /

\,) v

gt: / ~-- --A---· --.
v --/
c 100 300
~ -·,..o-~ ~-
8
1-o--
~
0.5 5 10 50
il

~

~

0.

E

~

E

~

:·E•;;;

0
0.1

Thermal conductivity kin W·m-1·K-1

(c)
10

3 I
I
.1'
I
..~E,
Q
.cg \

~5 \',

1l ~ ~:':a.. ' ' ' ' ' !:- -------- --..

0

E

~

·Eg_

0

0

10 100 1000

Thermal diffusivity o: (1 Q-7 m2.s-1)

legend

d = discontinuity thickness
I = discontinuity depth

rd = radius of discontinuity

ll. = rubber
0 = graphite epoxy
0 =concrete
.6. = titanium

• =steel
• =aluminum
-=l=lmm
- - = 1=2.5 mm

72 Infrared and Thermal Testing

cause a loss of resolution in the image of thermal signature fade quickly. As a result,
the discontinuity. In practice, a thermal if testing thin aluminum is necessary, a
nondestructive test procedure should be snapshot thermographic camera with
optimized by evaluating the high frame rate must be used to monitor
signal-to-noise ratio. Therefore, in some the high amplitude but brief signals.
materials, such as metals, high Furthermore, unpainted aluminum is
temperature contrasts may be hidden by
high noise. Also, the best observation characterized by high reflectance and low
time should be within the performance emissivity; this results in a thermal image
capability of the infrared equipment used.
Thus, rear surface corrosion in thin with considerable noise. Therefore, to
aluminum panels produces significant obtain valid results when testing
temperature signals but the high thermal unpainted aluminum, the radiation
diffusivity of aluminum makes the properties of the aluminum must be
enhanced and made uniform by applying
a high emissivity coating, such as paint.

The influence of the type of material 0 '-! . -·· ... ,. ·;·'.;·· - \·;
on optimal detection parameters is
illustrated in Fig. 15. It is seen that the TABLE 7. Thermal testing parameters in the detection of
greatest thermal contrasts appear in the
materials having higher thermal material loss In steel. Steel properties: a= 7.3 x 10--6
conductivity (Fig. 15b) although the m2.s-1; k::: 32 W·m-l·K·l. Conversions: 1 m = 39.4 in.;
maximum contrast occurs at relatively
short times (Fig. lSc). This tendency is 1 mm = 0.039 ln.; 1 K = 1 oc = 1.8 °F.
independent of discontinuity depth
although for deeper discontinuities lateral Maximum Optimum
heat diffusion is more significant in
materials of higher thermal conductivity. Specimen Material Defect Running Observation
Thickness L loss Radius fd Contrast c:r,un Time lm·Cf:t"
Heating Power and Surface Heat (K = oq
Flux (mm) (percent) (mm) (s)

The surface temperature signal above a Short pulse heating: th = 0.01 s; Tmax o. 8.4 K for Q = 106 W·m-2;
subsurface discontinuity is Jinearly maximum specimen temperature at end of flash heating is
proportional to the absorbed thermal weakly dependent on specimen thickness
energy, whereas the temperature contrast
is independent of it. Typically, more 1 90 5 8.79 0.11
powerful heating provides a better image 10 8.89 0.15
of discontinuities because of a higher 20 8.89 0.15
signal·to.noise ratio SNR = 11T·11Tres 0.99 0.14
determined by the noise level of an 50 5 1.00 0.19
infrared detector. However, the 10 1.00 0.19
signal·to·noise ratio that appears because 20 0.11 0.16
of variations in surface optical properties 0.11 0.22
is a not straightforward function of 10 5 0.11 0.23
heating power. It may increase because of 10
strong reflections or fluctuate because of 20
some nonlinear phenomena. In general, it
is recommended that the heating should 5 90 5 4.56 0.61
be sufficiently powerful to create a 4.78 1.23
maximum 1\T signal but not sufficient to 10 8.51 2.05
destroy the surface of the material. 0.58 1.06
20 0.89 1.57
The magnitude of the surface heat flux 0.99 2.51
impinging on the front and rear surface is 50 5 0.055 1.49
described by using coefficients hF and hR. 0.097 1.97
Temperature contrasts remain at the same 10 0.11 2.96
level within a wide range of 11 for thin
materials that have a high or medium 20
thermal conductivity and that meet the
condition: 10 5

(64) Bi ~ IlL < 0.1 10
k
20
For a one.sided test of thick materials
with low thermal conductivity, increasing 10 90 5 2.24 0.86
the fronl surface heat flux slightly 4.68 2.42
decreases temperature contrasts. In a 10 7.52 4.73
two·sided test, rear surface contrast is 0.24 3.14
practically independent of h. 20 0.58 4.26
0.89 6.25
Examples of Optimal 50 5 0.017 4.70
Detection Parameters 0.053 5.51
10 0.093 7.82
As discussed above, one of the main
purposes of theoretical analysis is to 20
optimize a thermal nondestructive testing
process and to increase the probability of 10 5
discontinuity detection. During heating,
any discontinuity located at a particular 10
depth in a material of specified thermal
properties will produce a maximum value 20
of a chosen thermal parameter at a
long pulse heating: th = 1 s

10 90 5 1.47 5.22
4.12 5.97
10 7.21 7.88
0.19 6.61
20 0.53 7.60
0.86 9.38
50 5 0.014 7.88
0.049 9.06
10 0.090 10.90

20

10 5

10

20

Thermal wave heating at quasi optimum frequency f (Hz) for
particular material loss

10 90 (I= 4) 5 8.60 0.26
10 8.78 0.26
20 8.78 0.26
0.19 7.34
50(1=0.14)5 0.50 7.59
10 0.72 14.69
20 0.013 10.43
0.043 10.93
10(1=0.1) 5 0.073 11.26
10
20

74 Infrared and Thermal Testing

particular time. Three examples of TABLE 8. Thermal testing parameters in detection of
thermal nondestructive testing procedures
are described in Tables 6 to 8. Many I material loss in aluminum. Aluminum properties:
previously discussed thermal testing a= 8.6 X lo~s m2·s-1; k = 210 w.m-1·K-1• Conversions:
scenarios are illustrated by data in these
tables. 1m~ 39.4 in.; 1 mm ~ 0.039 in.; 1 K ~ 1 •c ~ 1.8 "F.

In a graphite epoxy composite, the Maximum Optimum
lateral dimensions of discontinuities do
not affect ~Tand C values, at least in thin Specimen Material Defect Running Observation
Thickness L loss Radius rd
samples (Table 6). The best observation Contrast c~m Time tm(q~n)
time is primarily a function of (mm) (percent) (mm)
discontinuity depth whereas the (K = 0 C) (s)
temperature contrast depends on both
discontinuity depth and thickness. Short pulse heating: th = 0.01 s. .8.Tmal("' 5.57 K for Q = 106 W·m-1

In the evaluation of back surface 1 90 5 8.27 0.02
material loss caused by corrosion in steel
or aluminum (Tables 7 and 8), neither the 10 8.88 0.03
discontinuity depth nor discontinuity
thickness provide accurate physical 20 8.91 0.04
representations of the material loss.
Hmvever, the basic test features remain 50 5 0.91 0.02
the same: discontinuity lateral dimensions
have the greatest influence on 10 0.99 0.04
temperature contrasts but not on
observation times. Also, the data in 20 1.00 0.05

Table 7 illustrate the fact that each of the 10 5 0.10 0.03
three common thermal wave techniques
(pulsed, step function and harmonic 10 0.11 0.04
heating) provides similar detection
parameters that reflect the underlying 20 0.11 0.06
basic physical principles.
Short pulse heating: th = 0.01 s. 8.Tmax"" 4.S4 K for Q = 106 W·m-2
Note that the data in Tables 7 and 8
2 90 5 7.48 0.03
(short pulse heating) agree well with the 8.78 0.05
one-dimensional predictions on the 10 8.91 0.07
relationship between material loss and 0.83 0.04
dimensionless contrast:7 20 0.98 0.06
1.00 0.09
50 5 0.088 0.04
0.108 0.07
10 0.110 0.10

20

10 5

10

20

t!.L c
(65) L 1+ c

(66) c M
L

1-M
L

It is important to note that Eqs. 65 and
66 are independent of material thermal
properties. For the three levels of material
loss (90 percent, SO percent and 10

percent) presented in Tables 7 and 8, the
corresponding running contrasts are
predicted to be 9, 1, and 0.11 respectively.
These values are close to those obtained
numerically.

Heat Transfer 75

PART 4. Heat Diffusion in Periodic Regime

Thermal Waves where the analogy between Eq. 71 and
the helmholtz wave equation has been
The term thermal 1vave commonly refers to
the typical ·wavelike temperature field put into evidence by introducing the
induced by a harmonic heating process. wave number~= -l(jw-D-1). Consequently,
These waves were used by Lord Kelvin and
A.]. Angstrom to investigate the thermal the well knmvn theory of wave physics
diffusivity of bodies. Only since the early
1980s, have the waves been called may be applied to the harmonic thermal
thermal, arousing debatc8·11 in the field.
scientific community.
As an example it is useful to study the
The fundamental point to be Green function, the solution of Eq. 71 for
understood is how the heat conduction
could show wave behavior. Heat the unitary harmonic point source in the
conduction is well known in the classical origin 0:1•3;12
case of a diffusive process, as shown by
the fourier equation: (72) G2(r,m) exp[-~r]

4nkr

exp[-{~]

4nkr

where k is the thermal conductivity of the (73) G2 (r,t) )jRe[G2 (r,w) exp(iwt

medium, D is the thermal diffusivity of !...)exp[-{] cos(wt-
the medium and w -is the heat supplied
per unit time per unit volume. The green 4nkr f
function solution of Eq. 67 shows clearly
where r is the distance from the origin,
the diffusive behavior for a single heating
pulse at the origin 0 at time t = 0 where and £ is the thermal diffusion length:
pc is the heat capacity per unit volume

and t "?. 0:3,12

1 ~ ~(74) { = 2

(68) G1(x,)',z,t)

A different situation is found in the On first inspection, it may be noticed

case of harmonic heating. If the quantity that the temperature field G2 in Eqs. 72
w is harmonic in time, with the period and 73 behaves as a spherical wave of
2n·or1, the temperature field Tis forced to
wavelength 2Jtt and for this reason has
be harmonic with the same periodicity. By been called thermal wave. The main
characteristic of such a wave is that it
introducing into Eq. 67 the complex
decays strongly, moving away from the
quantities Wand T as found in Eqs. 69 heating source (see the exponential term
in Eqs. 72 and 73). In practice the thermal
and 70, Eq. 71 provides the fourier wave vanishes within some wavelengths,
equation in the harmonic regime: 1,3,12 so scientists disagree on the term wave.
The damping is driven by the wave
w number ~ = (l+j)·>-1 . It is a complex
k quantity with the same real and

imaginary part. As a consequence the
thermal diffusion length I plays a double

role: from one side it is proportional to
the thermal wavelength 2rrf; from the

other side it represents the extinction
length at which the exponential term in
Eqs. 72 and 73 is reduced to e-1 of its

initial value.
Once introduced, the spherical thermal

wave G2 can be used to describe any

76 Infrared and Thermal Testing

harmonic field T that may result from an the z direction, like a forward damped
appropriate superposition of spherical wave. Note that, at anx time, the
thermal waves emitted by the heat source. temperature oscillates ,\,ithin the two
Analogously the plane thermal wave can be exponential envelopes ±exp {-z·£-1), that
limit the thermal wave range. The
introduced when the heat source is difference between spherical and plane
thermal waves is only geometric and does
planar. By harmonically heating the not affect the general properties just
medium along the plane x,y the shown; in particular any field T may
always be also decomposed in terms of
temperature T may be represented by a plane thermal waves propagating in various
plane thermal wave propagating in the z directions.
direction: 1•3•12
Thermal Wave Reflection and
(75) f(z,m) Aexp(-Pi') Refraction
Aexp(-~·z)
Besides the thermal wave generation and
Aexp[-(l-+fi-)z] propagation in a homogeneous medium,
other basic phenomena should be
(76) T(z,t) Re [f(z,m) exp(imt)j observed when at least two media are
involved: the reflection and the
A exr(-7)cos(rot-f) re(raction.1· 13•14 These phenomena are
discussed below for plane thermal waves
where A is the amplitude at z == 0 and tis in general.
the wave vector in the direction of
propagation. The plane thermal wave in When a plane thermal wave
approaches the interface (the plane z = 0)
Eqs. 75 and 76 is plotted in Fig. 16 as a between two media, it is partially reflected
function of the distance z, for different back and refracted beyond. By imposing
the continuity at the interface of both the
values of the time: in particular the curves
field f and the vertical heat flux
a,b,c,d correspond to rot= 0, rr./3, rr./2, rr.. As
the time goes on, the maximum [-kaf.(az)-1] yields Eq. 77:
temperature rise is reduced but moves in
(77)

FIGURE 16. In plane thermal wave, temperature rise is plotted where z = 0 and where the fields in the
versus distance z (expressed in units t of thermal diffusion first medium (1\) and in the second
medium (f2) have been decomposed in
length) at different times: curves a, b, c, d respectively terms of plane thermal waves:

correspond to rot= 0, n/3, rr./2, n. Two envelopes represent f,(x,z) = A e-P,[>in(o,)x + m>(O,)zj

maximum and minimum values of temperature rise. (78) + rAe~~{sln(o;)x-cos(o;)z]

A Positive envelope +e-=J f 2(x,z) tA e-P,[>in(o,)x+ws(o,JzJ

Ec" Negative envelope ~e--f where A, rA and tA are respectively the
amplitudes of the incident, reflected and
~ 2 3
refracted waves and 81 , 8{ and 82 are
·."5' AJ2 Distance z from the source respectively the angles between the
directions of the waves and the normal to
i (unit l "' thermal diffusion length) the interface. By combining Eq. 77 with

.rJ 0 Eq. 78 the Snell relationships for the
angles coefficients are obtained:
i'!

"'~ -A/2

Q.

E
~

-A
0

Legend e,

A "'amplitude k-(79)
a=:curvetot=O { r1::- . (ee;) = sin(e 2 )
b =curve tot= n/3 sm 1 =
c =curve tot= n/2 vD, vD2
d =curve (J)t""' rr.
e =root of natural logarithms

Heat Transfer 77

and the relationships for the reflection r Thermal Wave Mirror
and the transmission t coefficients are
obtained: If the effusivity mismatch between the
media is very high, a relevant reflection
(80) r ~ e1cos(e!) - e2 cos(Bz) phenomenon takes place (see Fig. 17),
e1cos(e,) + e2cos(Bz) whether in phase for very small e2-e1- 1 or
in opposition to phase for very large
(81) t r+1 e2·ec1•15 In both cases the second medium
behaves as a thermal wave mirror (r ~ ±1);
where e ~ k·V(IF1) is the thermal but the transmission (always t = r + 1) in
one case tends to 2, in the other tends to
effusivity. Note that in the case of D2 > D 1 0. The interface between a generic gas and
Snell's law is valid o :S arcsin a generic solid provides an example.
nly until e1 Because any solid is always at least 100
(v[D 1·D2- 1]) (see Eq. 79). For larger values times more effusive than any gas, the
effusivity mismatch is extremely high and
of 81 the thermal waves in the second the solid consequently behaves as a
thermal wave mirror for the gas and vice
medium become heterogeneous; the versa for reciprocity but with the
follmving differences.
planes at constant phase differ by the
1. If the incident wave is coming from
planes at constant amplitude.t4 Although the gas, the effusivity ratio is
e2·ec1 >100, the reflection r ~ -1, and
such waves have been not yet observed, the transmission t ~ 0. In synthesis a
destructive interference takes place
the theory predicts for them an between incident and reflected waves
in the gas. At the interface the
anomalous attenuation larger than the temperature rise is kept to zero.

one described in Eqs. 75 and 76, as will be 2. If the incident ·wave is coming from
the solid, then the effusivity ratio
discussed later. Equations 80 and 81 give e2 -e1- 1 < O.OL the reflection r ~ 1 and

the coefficients rand t and consequently the transmission t----) 2. In synthesis a

the efficiency of the heat transfer through constructive interference between
incident and reflected waves takes
the two media. Generally the reflection r place in the solid. At the interface
there is the n1aximum temperature
and transrniss·!On t depend mainly on the rise.
just to conclude this theoretical
effusivity mismatch. However, they discussion, it is of practical interest to
quantify the appropriate depth for a
depend also on the diffusivity mismatch. thermal wave mirror, which is the lower
limit for the thickness of a material and
In fact the term cos(82) in Eqs. 80 and 81 beyond which the reflection properties are
is linked to diffusivity by Eq. 79; one lost. An advanced study on layered
structures has recently shown how the
exception is for normal incidence reflection properties of a slab change with
its thickness. If R= is the reflection for an
(81 ~ 82 ~ 0) where rand t depend only on
effusivity. In Fig. 17 both coefficients are R.·.'·"infinite thick slab, the effective reflection

plotted versus the effusivity ratio e2·ec1, R for a finite slab differs by
just for normal incidence.

FIGURE 17. Thermal reflection and transmission coefficients
between medium 1 and medium 2 are plotted versus
thermal effusivity ratio e2·ec1 for normal incidence.

2

"cU 0 0.1 (82) R

~ where d is the thickness of the slab. Note
once again the perfect analogy between
c Eq. 82 and the equations for the
0 Fabry-Perot interferometer in wave
-~ physics. In thermal radiation, thermal
10 100 wave interference occurs in the slab,
~ -I which may either enhance (constructive
interference) or inhibit (destructive
0.01 interference) the effective reflection R
\\'ith respect to JL. To study the efficiency
legend of such a thermaJ·wave mirror, R·R,;;1 is

e = thermal effusivity
r = reflection coefficient
t = transmission coefficient

78 Infrared and Thermal Testing

plotted in Fig. 18 as a function of the distance (L = 1 mm [0.04 in.]) until it
normalized thickness defined as
rf.1· 1 -IRJ·(I-R~)-1, for different values of approaches the aluminum foil, which acts
IR~I = 0.2, 0.5, 0.9, 0.99 and 0.999. Note as a thermal wave mirrbr. The foil forms
that all curves merge together in two an oblique angle 8 with the x axes. As a
opposite limits. consequence the reflected wave forms an

1. For a high normalized thickness the angle 28 with the incident one, and the
slab is thermally thick and behaves as temperature rise in air where the waves
an infinite medium, so R ~ R'"' (mirror are superimposed is calculated:l4
regime).
(83) f,1,(x,z) A exp(-p,1,z)
2. For a low normalized thickness the
slab is thermally too thin, the incident + RA exp{~air
thermal wave is transmitted beyond
the slab, without a relevant reflection x [cos(2e)z-sin(ze)x]}
R ~ 0 (transparent window).
where A is the amplitude of the incident
Note also that the transition from \\'ave in the origin 0, and R is given by
transparent to mirror regime occurs when Eqs. 80 and 82. The detection of the
the normalized thickness is practically thermal field in Eq. 83 may be achieved
unitary, which for a high effusivity by using the nzirage technique.Is-2z A
mismatch (IRJ ~ I) corresponds to an probe beam is sent along the )'axis, in air,
extremely small thickness d ~ 1·(1 - R~).
For this reason, even extremely small close to the origin 0. The thermal
discontinuities, cracks and delaminations
behave as thermal wave mirrors and may gradients encountered along the path
be easily detected by thermal wave produce the beam deflection measured by
interferometry. 16 , l 7
a remote position sensor. In particular the
Evidence of Thermal Wave beam orientation is chosen to detect the
Reflection
FIGURE 19. Scheme of thermal wave cavity for detection of
A simple experiment to prove the reflected thermal wave.
reflection of thermal waves is shown in
Fig. 19. A plane thermal ·wave is generated
in air by heating periodically a thin
absorbing layer with a wide pumped laser
beam (S = 4 mm [0.16 in.]). The plane
wave propagates in air along z for a short

,_.,f;;:-,~·-;_j'-'i-:<,._ ;·~':'?-'• ;-,·.;·.~ -~;.·•::~c·,·

FIGURE 18. Normalized thermal reflection IR·R.-11 of slab is

plotted versus normalized thickness for different values of
interface reflection coefficient R. (0.2, 0.5, 0.9, 0.99, 0.999).
Normalized thickness is defined as J.[-1.1R~I·(l-R.2)-1 where I

is discontinuity thickness and l is thermal diffusion length of
slab.

1.1

"·"' 1.0 0.999 Incident wave
Mirror regime
~ 0.9
~
~ 0.8
Transparent fr
"·"a. 0.7 window Pump beam
5
E
~
0.6 Absorbing layer
c
·~ 0.5
"" 0.4

~

.," 0.3

.~ 0.2
E 0.1
z0 0 legend

O.ol 0.1 10 100 L "" cavity length
0 =mirror center
IR..INormalized thickness = !__. 5 = beam width
x,y,z =axes
{ 1- R.! e =- angle of incidence

Heat Transfer 79

components of the deflection angle along where the tilt angle 9 between the mirror
the x and z directions:l4,15 and the absorbing layer has been set at
9 = 11 degrees. Both components ibx and
(84) <l>x J_!__<liz_ aT,,, dy iPz have been measured and their ratio .Q
has been compared ·with Eq. 89. The
11 dT ilx results have been repeated by changing
the modulation frequency {(for example,
)' fair)· In Fig. 20 the amplitude ratio IQ1 is
plotted as a function of if; the symbols
euax-111 dn L dTair represent the experimental data and the
dT full Jines are the theoretical values by
Eq. 89 for the tilt angles e reported on the
(85) <1>, f_!_<liz_ ilT,;, dy right scale (0, 5, 10, 11 and 15 degrees).
Note that, although there is some noise,
II dT ilz all experimental points lie near the Hne at
9 = 11 degrees at all frequencies. This
)' agreement demonstrates the following
points.
;1; ddnf L -d-T-a11;;-,
1. The reflection phenomenon takes
eff place as described by Eq. 83.

where 11 is the refractive index of air, 2. The aluminum foil acts as a thermal
dll·(dT)-1 is the optothermal coefficient of wave mirror for a wide spectrum lair of
air and Lerr is the effective length useful the thermal wavelength.
for the beam deflection. Leer depends on
the lateral dimensions S of the incident Evidence of Thermal Wave
thermal wave. By combining Eqs. 83 to Refraction
and 85 Eqs. 86 and 87 are obtained:
Besides the proof of the thermal wave
(86) <i>x c{Rsin(ze) reflection, an experiment may be set up to
prove the refraction of the thermal waves
X ,.,,!cm(2o),- >in(zo)xJ} as shown in Fig. 21. The pumped laser
beam propagates along y, is modulated in
_ c[Rsin(ze)] time and is focused onto a solid sample
(medium 1) by means of a cylindrical
(87) <1>, c[e-~•'•'-Rcos(ze) lens. The dimensions of the ellipsoidal
X e~••lw>(20)Hin(2o)xJ] pumped beam spot in the plane x,z (see

_ c[1- Rcos(ze)J

where e is the root of natural logarithms FIGURE 20. Amplitude ratio .Q between deflection
components along x and z is plotted versus modulation
and C is a constant defined as: frequency square root: open squares (D) represent
experimental data; full lines are calculated by Eq. 89 for
CI(dll)(88)~ L,rrP, 1, A different values of tilt angle e.
-- -
II dT Air thermal diffusion length, mm (1 Q-3 in.)

Note that the simplifications in the 2.0 1.5 1.0 0.8 0.6 0.5 0.4
exponential terms in Eqs. 86 and 87 are (80)(60)(40) (32) (24) (20) (16)
allowed if the probe beam travels close to
the origin 0, at a distance shorter 0.3 ..15
than lair- In this case the two components 0 •~
are proportional to each other and their .,.0 0.2 "~'
ratio Q depends only on the angle 8 and .:.le~ .r!rr>crJ • rJ•rJrlrJ •• "'• 11
on the reflection coefficient of the mirror ~ 10 ~
R as follows: • • ••
~
0,
'0 •c
<~>x cv~
(89) Q ~ Rsin(ze) il s5 ~
1 - R cos(ze) 'a_
<1>, :v2
«E 0.1

-sin(ze) ~ -tg(e) 0 ,, ,, 0
0
1 + cos(ze)

Jn Eq. 89 it is assumed that R"' -1 because Frequency square root .Y(Hz)
the aluminum foil acts as a thermal wave
mirror for any e (see Eqs. 80, 81 and 82). legend
An experiment has been performed in the 0 =experimental datum
thermal wave cavity described in Jlig. 19,
- =calcula_tion with Eq. 89 for degree of incidence on scale to right

80 Infrared and Thermal Testing

Fig. 19) are adjusted to provide a line '!fwave 2 forms an angle (82 - 81) with the
heating source along x. The generated
z axis (see Fig. 21 and Eq. 90). The
thermal wave propagates along z inside 1
detection of the thermal field in Eq. 90 is
the solid until it reaches the interface accomplished by the mirage technique, as
with the second medium (air), which is at
an oblique angle to the line source (the in the previous experiment. The probe
normal to the interface 11 forms the angle beam is placed in the second medium
81 with the z axis). The theory predicts
that the incident wave is reflected back near the interface, where the heat flux is
and refracted in the secorid medium greater. The two components of the
according to Eqs. 78 and 79. Equation 90 deflection angle along x and z are given
is written for the thermal field refracted in by combining Eqs. 84, 85 and 90:14
the second medium: t4
(91) <i>_, c(tsin{Bz-91}
(90) f 2(x,z) tA cxp(-Pzi')
x cxp{-P2 [sin(B,-B1)x
lA cxP{-P2 [sin (B2 -B1)x
)z]})+ cos(e2-B1
Jzl}+ cos(e,-e1
(92) ci\, c(tcos{B2 -e1}
where A is the amplitude of the incident
x exp{-P2 [sin(B2 -B1)x
J:wave in the origin 0, t is the transmission
)z]})+ cos(B2-B1
coefficient given in Eq. 81, is the
Note that in this case the two
position \Vith respect to 0, p2 is the wave
components are proportional to each
vector that points the direction of the other everywhere, regardless of the values
refracted therm_!!l wave, 82 is the refraction of x and z, and their ratio depends only
angle between ~2 and the normaln to the on the incident and refracted angles

interface. In Eq. 90 the quantity 72 is

expressed as a function of reference axes x
and z. Note that in such a reference
system the direction of the refracted

FIGURE 21. Schematic setup for detection of refracted thermal The analogy between Eqs. 89 and Eq. 93
wave.
suggests that the amplitude ratio Q is
Medium 2
useful not only to detect the thermal
Refracted wave wave but also to reveal its direction. The

~Rotation physical reason is in the basic principle of
the mirage technique: the deflection angle
.J stage
is not related directly to the temperature
Incident wave
field t· but rather to its gradient Vf, which
Pump beam spot
5 points the direction of the heat flux.
Therefore If the direction of a thermal
Medium 1
wave is of interest, the knowledge of the
legend
n = incidence direction normal to interface scalar function T could be useless, even if
S "' beam width
it is known for '!_ny plane, whereas the
x,y,z =axes
j3 = wave vector vectorial field VT immediately gives such
l:l1 = angle of incidence information, point by point. Therefore
01 = angle of refraction
the mirage represents the most appropriate
technique for this purpose. In particular

the ratio between the components filx and
ibz gives, as previously shown, the local

information on the angle between ~ and

the z axis. Coming back to the differences

between Eqs. 89 and 93, an important
point is the sign of Q. In the reflection

experiment it is always negative whereas
in the refraction experiment it depends

on the sine of €12 and 81, which is fixed by
Snell's law (see Eq. 79). As a result if

D2 > 1J1, then 82 > q 1 and consequently
Q > 0; whereas if D2 < D1 then Q < 0. The
sine of Q helps to discriminate one case

from the other. Knowledge of Q permits
calculation of the refraction angle 82 by
solving Eq. 93 as follows:

Heat Transfer 81

The results obtained with the setup phosphorus) more diffusive than air. Note
that in the first case the linear behavior is
shown in Fig. 21 provide an example. An broken around 91 "" 30 degrees, which
corresponds to the limit angle:
argon laser illuminates a thin solid sample
(medium 1) not far from one edge. The (95) elim = arcsin~ Dt
Dz
pumped beam is modulated in time and is
Beyond the limit angle Eqs. 79 and 94
focused by a cylindrical lens only in the z become useless and consequently the

direction and so has a line source 6 mm experimental points measured for e1 > 30
(0.25 in.) wide along x and a few
degrees, for the stainless steel sample only,
micrometers (-1.4 x to-4 in.) wide along are meaningless. By using the least
squares technique to calculate the slopes
z. A helium neon laser probe coHinear (see full lines in Fig. 23) the diffusivity
with the pump argon laser is placed in air ratios are obtained: D111r·Dair-1 = 2.2 ± 0.1
and DFeNI'Dair-1 = 0.25 ± 0.02, where FeNi
(medium 2) close to the sample edge. A refers to the stainless steel and InP refers
rotation stage allows the movement of the to the indium phosphorus material. These
diffusivity ratios lead to the values D 111r =
sample in the x,z plane to change the 0.44 cm2 ·s-1 and DFeNi = 0.05 cmZ.s-1,
when it is assumed that Dalr = 0.2 cm2·s-l,
orientation of the edge. The pump and in perfect agreement with the values
the probe are fixed. As a consequence the given in the literature.23 Has a ne\\'
methodology been introduced here to
incidence angle el is changed by the determine the thermal diffusivity of
materials? The question is reasonable but
rotation ~tage. 1!_1. Fig. 22 the amplitudes difficult to answer. It is useful to
of both <Px and cfl2 are shown versus 91 for remember three points.
a semiconductor wafer of indium
phosphorus. 1. This technique based on Snell's Jaw
makes it possible to determine the
From these experimental data the thermal diffusivity of one medium if
that of the other one is well known.
ratio Q is calculated and eventua1ly, by
using Eq. 94, the refraction angle e2 is 2. This technique works whether the
calculated. Finally plotting the quantity heat source is in the first or second
sin (82) versus sin (81) provides the most medium. Therefore it is suitable even
appealing proof of Snell's law; in fact a
straight line is expected, with the slope

equal to Y(D2·D1-') (see Eq. 79). In Fig. 23
this procedure is reported for two

materials: a stainless steel (36 percent
nickel) less diffusive than air and a

semiconductive material (indium

FIGURE 22. Amplitude of deflection signal (~V) is plotted FIGURE 23. Sin 92 is plotted versus sin e1 for two different

versus incidence angle (degree). Sample is indium materials: nickel iron alloy sample (D) and indium
phosphorus wafer. Pump power is 800 mW; modulation phosphorus semiconductor wafer (+). Full lines are derived
by least squares method.
frequency I= 16 Hz. Plotted squares (0) are data measured
for component along x; plotted plus signs(+) are data for Incidence angle (degrees)
component along z.
DD
90 ,-~~~~~~~~~~~~~~-.
0.5
80

70 •• •
60 +++ +

50

40

30

D

20

10 ~·~·~·~·~·-·-·_·_·_i·_·____~----~----~----~ 0
0
0 0.2 0.4 0.6
0 10 20 30 40 50 60 Incidence angle (sin 91) 0.8

Incidence angle (degree) legend
e 1 =angle of incidence
legend e2 "' refracted angle

+ "' datum in z direction 0 =datum from staintes5 steel (36 percent nickel) sample
0 =datum in x direction + =datum from indium phosphorus semiconduclor

82 Infrared and Thermal Testing

for measuring the diffusivity of FIGURES 24. Numerical simulations for amplitude of thermal
nonabsorbing materials.
3. Preliminary results indicate that this field. Contour plots of amplitude of 'fare calculated as
technique guarantees the same
accuracy as the other well known function of coordinates p, Swhen stainless steel sample with
techniques. 24 ·27 thermal diffusivity D1 = 5 mm2·s~1 is heated by source at

Heterogeneous Thermal Wave oblique incidence 81: (a) homogeneous thermal waves at
p <p4<f241 t8,,1 20 degrees; (b) heterogeneous thermal waves
Snell's law for thermal waves establishes at = 70 degrees; (c) heterogeneous thermal
that 1-(YD-1) has the same role as the =
refractive index has for electromagnetic e,
waves. This analogy suggests the simple
question of whether the same analogy waves at p > 6(2, 81 = 70 degrees, amplitude increasing
pertains to total reflection for thermal with height 1;.
waves. The answer is in the refracted
thermal field when the incident angle is (a}
9t > 8nm·· The refracted thermal wave is
still planar but no longer
homogeneous: 13•14

where B is a constant, p is the variable (b) \ Heating tine
parallel to the interface and I; is the
variable vertical to the interface. In Eq. 96 Heating line

two different solutions are included (c)
depending on the sign ± in the
-2f1
exponential term. Both of them satisfy legend
the wave equation Eq. 77 but the
r2 "' air thermal diffusion length (mm)
smoeluantiionnglfe2ssmianytsheeemspatcoebSe>ph0y, sbieccaalluyse it p =variable parallel to interface
is amplified in the direction C- ~=variable vertical to interface

To clarify this point it is helpful to

study the amplitude of the field T2 for a

simple system made of air and a low

diffusivity material (stainless steel with
36 percent nickel). The air is in the half

space for S> 0 whereas the stainless steel
is for C< 0. The heating source is an

oblique line inside the material; the line

begins from the origin 0 and forms an

angle e, with I;= 0.

The amplitude of the field T(p,l;),
calculated by a numerical simulation, is

reported in the contour plots of Fig. 24 for
two different values of the angle 81 chosen
to induce homogeneous (Fig. 24a) or
heterogeneous (Figs. 24b and 24c)

refracted thermal waves. In particular
in Fig. 24a the incidence angle is

8t = 20 degrees< ellm = 30 degrees.
Note that plane thermal waves depart

from the heating line inside the material,
as it is pointed out by the arrows. The
wave propagating toward the air-to-steel

interface is partially reflected back and
also refracted in air. Consequently a

thermal interference occurs just below the
surface, where the incident and reflected
wave are superimposed, as is revealed by

the strong distortion of the wave front. In
air the refracted plane thermal wave

Heat Transfer 83

changes direction according to Snell's law
(82 = 43°), as pointed out by the arrows in
Fig. 24a. Of course, because of the finite
dimensions of the heating source, the
refracted field is too distant to be a planar
wave close to the origin 0.

These boundary effects vanish ·within a
thermal diffusimllength {'2,28 In other
words the refracted wave becomes plane
at a suitable distance p > 12• A different
case is reported in fi.gs. 24b and 24c
where 81 = 70 degrees > Sum = 30 degrees.
As the incident wave approaches the
air-to-steel interface it is reflected back
giving rise to the usual interference
phenomenon. In air the thermal field is
now too far to be a plane wave (see
Fig. 24b). However it is still possible to
recognize a restricted region, close to the
interface, lNhere the amplitude tends to
maintain a plane wave front (see the
arrows in Fig. 24c)..: In such a region the
un~table solution T2 + takes place instead
ofT2 -. As a consequence, the amplitude
increases greatly with the height l;. The
physical reason is that in this zone the
main heat flux comes from the higher air
layers rather than from the inside material
(see the arrows ill Fig. 24c). Unfortunately
this heterogeneous wave could be
observed far from the origin (p > 6£2)
where the wave is too weak to he
detected.

84 Infrared and Thermal Testing

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Heat Transfer 85