Sensing Thermal Underground Transients of Buried Wired Structures

8/12/2019 Sensing Thermal Underground Transients of Buried Wired Structures

1/14

Journal of Earth Science and Engineering 3 (2013) 730-743

Sensing Thermal Underground Transients of Buried

Wired Structures

Ira Kohlberg1, William Szymczak

2and

Camille DAnnunzio

3

1. Kohlberg Associates, Inc, 11308 South Shore Road Reston, VA 20190, USA

2. Naval Research Laboratory, Physical Acoustics Branch, 4555 Overlook Ave SW, Washington DC 20375, USA

3.Northrop Grumman ES, Automated Sensor Exploitation Technology Center, 1550 W. Nursery Road, Linthicum MD 21090, USA*

Received: October 15, 2013 /Accepted: October 30, 2013 / Published: November 25, 2013

Abstract: Mathematical investigations of the dynamic response of buried systems to thermal and/or electromagnetic stimulation

continues to be of great importance. The size of such systems can range from the microelectronic scale to large underground

structures. Stimulation can occur from unwanted electromagnetic signals entering the buried system, and for assessing the operating

state of a buried system that is not usually physically accessible. In both cases detecting damage or status can be accomplished by

examining the time dependence of the resultant surface temperature. This study shows how to determine surface temperature for a

hypothetical thermal-plus-systems using a combination of Fourier-space and Laplace-time transform techniques. The hypothetical

model can be generalized from scaling the relevant relationships.

Key words: Electromagnetic heating, buried wire, surface temperature, Laplace transform.

1. Statement of ProblemThe goal of this analysis is to examine the

conditions where a buried wire is heated enough by

the electric field so as to be detected at the surface. In

the absence of all other heating effects detecting the

wire would only depend upon ambient conditions and

the sensitivity of the instruments. Unfortunately,

detection may be hindered to some extent by direct

heating of the ground from the electric field. The

primary concern of this paper is direct heating and an

estimate of the temperature rise as a function of

ground parameters and the electric field is sought.Fig. 1 is a sketch of the problem to be solved

analytically. The model consists of a bare (no insulation)

copper wire of length,L, radius a, located a distance d

below the earths surface. The wire is parallel to the

Corresponding author: Ira Kohlberg, president, Ph.D.,

research fields: electromagnetic theory, information theory,communications theory and fluid dynamics. E-mail:

[email protected].* this work on this paper was done independently and has no

association with NGES.

earths surface which is they-zplane located at dx .

The wire is assumed to be straight and points in the

positive z direction. An electromagnetic plane wavestrikes the earths surface at oblique incidence and

propagates into the ground. Both a, and dare much less

than the wavelength air of the incident wave in free

space and the wavelength in the ground ground , which

is less than air . The wires lengthL will also affect the

interaction but in this exploratory analysis the length is

assumed to be infinite. A real system may be comprised

of numerous wires that under circumstances may

interact with each other. However, the basic techniques

for solving the dynamic behavior system will involve

the fundamental concepts this study.

In summary, an expression for the surface

temperature is derived as a function of time and

distance from the buried wire, temporal behavior of

the incident electric field, and other relevant physical

constants. Using Fourier-space and Laplace-time

transform techniques the surface temperature is

found to be a sum of two terms: explicit ground heating

DAVID PUBLISHING

D


2/14

Sensing Thermal Underground Transients of Buried Wired Structures 731

Fig. 1 Temperature sensing of RF heating of buried wire.

produced by the sources ground-penetrating electric

field and thermal generation in the wire and its

transport to the earths surface. A closed-form

expression in terms of the Laplace transform variable

is derived for the surface temperature. The time

dependent surface temperature is computed from the

inverse Laplace transform using asymptotic

approximations.

2. Summary of Approach

An analytic description of the

physical-mathematical model for solving the problem

is rendered in Section 3. In this section an overview of

the physical ideas and mathematical approach is

provided. For all modes of polarization: TE

(Transverse Electric), TM (Transverse Magnetic),

TEM (Transverse Electromagnetic), circular and

elliptical and angles of incidence, electromagnetic

waves are propagated into the ground in a completely

predictable way [1, 2]. This electromagnetic energy

gets coupled into the wire, heating it up. Again, thecoupling theory for this is well known. The

temperature of the wire is raised above its

surroundingsi.e., the ground, and conversely the

wire cooled by heat transfer to the ground. Eventually

this temperature effect is sensed at the earths surface

above the wire. Fig. 2 is a sketch of how

electromagnetic energy heats the wire.

A critical feature of Fig. 2 is the shaded region; this

is the skin depth. Even though the radius of the wire is

actually quite small, the skin depth is very thin. If)(tI is the total current flowing in thez-direction, athe

radius of the wire, the skin depth )( a , and the conductivity, the resistance per unit length is

)2/1(' aR eff which is clearly much greaterthan the dc resistance per unit length

)/1('2aR dc . The Joule heating per unit length

of wire is )(')(2 tIRt effW . Because the incident

energy source is time dependent so will be the

temperature of the wire and the temperature at the

earths surface above the wire. This elevated ground

temperature will exist over a modest surface area

(measured by the lateral dimension, y) in the

neighborhood above the wire.

Fig. 3 is a sketch of how thermal energy travels

from the wire to the surface. As the wire gets heated

the thermal energy initially enters the ground by

thermal conduction, physical contact between wires

surface and ground. This conduction process is

described by diffusion [3] as is the transport processfrom the wire to the surface. The flow of heat is

symmetric about the vertical axis. It is assumed that

the heat transfer to free space from the ground is

negligible so that the plane dx is an insulatingboundary. Any thermal energy reaching this boundary

is reflected.

Even though diffusion is not, strictly speaking, like

wave propagation, directionality can still be attributed


3/14

Sensing Thermal Underground Transients of Buried Wired Structures732

Fig. 2 Electromagnetic heating of the buried wire.

Fig. 3 Heat traveling from wire to surface.

to the flow of heat and in some ways treated as

wave-like. This is shown by the arrows in Fig. 3. At

the beginning of the heating process the wire is

unaware of the reflecting boundary. Thermal energy

has not had enough time (in a diffusion sense) to

travel from the center of the wire to the boundary, get

reflected, and influence future heat transfer between

the wire and ground. This being the case, the

interaction between the wire and ground proceeds as if

in an infinite medium, and is independent of angle

around the wire.

Consider heat traveling along ray A which strikes

the surface at normal incidence. There will be a

meaningful reflection at dx , but that reflection

wont reach the wire until the roundtrip diffusion time:

GD DdT /2 , GD is the ground diffusion constant.

For the conditions considered DT can be in the

seconds to tens of second range. By extending this

line of reasoning the reflection from ray B will arrive

back at the wire later than from A and be even weaker,

C will arrive back at the wire later than from B and be

even weaker, etc.. By extending this semi-quantitative

argument it can be seen that 50% of the heat from the

wire (rays D, E, F) can never return and the rest have

diffuse weak reflection from the assumed insulating

boundary and have extremely long transport delays.

Based on these arguments it is assumed that the

interchange between the wire and ground is modeled


4/14


as if it occurs in an infinite medium (this is easily

modified if necessary).

Since the radius of the wire is significantly much

smaller than the other dimensions the heat source into

the ground derived from the wire-ground interaction,

)(t , is modeled as a two dimensional spatial delta

function, )()()( yxt . The last element that

needs to be addressed is the direct heating of the

ground from the electric field. This is given by the

expression: )(''

2

1)( 2

0

0 tEtG

.

When all the pieces are put together the resulting

equation becomes

)()()(),(2

2

2

2

yxttyy

T

x

TK

t

TC G

GGG

GG

The solution of this equation gives the surface

temperature as a function of time and space.

Quantitative details are rendered in the next section.

3. Theoretical Model and Solution

3.1 Wire Heating

Heating the wire occurs in the skin depth. Althoughthe current is confined to the skin depth the heat itself

is rapidly transferred over the entire cross section by

diffusion. The diffusion time is

WD DaT /2 (1)

Using 05.0a cm and a wire diffusion constantof 3.1WD cm

2/s gives

34 109.1)3.1/()1025( DT s (2)

based on the following data for copper:

4

103.1/

WWW CKD m2

/s thermal diffusivity401WK watts/m-deg K thermal conductivity

61045.3 WC Joules/m3-deg K specific heat

per unit volume

The heat transport equation is

)(1

22

2

tQr

T

rr

TK

t

TC W

WWW

WW

(3)

2

)()(

a

ttQ WW

w/m

3

)()( 2 tIRt effW w/m

which is transformed into

)(1

22

2

tr

T

rr

TD

t

TW

WWW

W

(4)

W

W

W

WW

Ca

t

C

tQt

2

)()()(

Eq. (4) is solved in cylindrical coordinates using

Bessel functions. This is facilitated greatly by

introducing the function

)(),(),( tTtrTtr WSW (5)

where, ),()( tarTtT WWS is the surface

temperature of wire located at ar . It is not known

ahead of time, determined as part of solution. The

important feature of Eq. (4) is that it sets up the

boundary condition

0)(),(),( tTtaTta WSW (6)

Eq. (6) enables us to efficiently use the orthogonal

properties of the Bessel functions. Substituting Eq. (5)

into Eq. (4) gives

)(1

22

2

trrr

Dt

WW

(7)

dtdTa

Ct

adtdTtt WS

W

WWSWW

22

)(1)()(

Writing

n

n

nn rJtAtr

1

0 )()(),( (8)

n

n

nnW rJtBt

1

0 )()()( (9)

it follows that

)()()(

2)(

1

tptqJq

tB WnWnn

n

(10)

)(

2

1 nnn

qJqp (11)

where, na qn is the nth

root of the Bessel function.

Inserting Eqs. (8)-(11) into Eq. (7) gives

)(tpAdt

dAWnnn

n (12)


5/14


D

nnwnwn

T

q

a

qDvD

2

2

22 (13)

If 0)0( tAn , then

t

wnnn dtptA

0

)())(exp()(

t

wnn dtp

0

)()exp( (14)

Because n is so large

)()( t

ptA w

n

nn

(15)

Taking the Laplace transform, denoted by ~ and

Laplace transform variable denoted by s gives the

following:

)(~~~

spAAs Wnnnn (16)

)(~

)(~~

sp

ss

pA W

n

nW

n

nn

(17)

Consider calculating the heat transfer to the ground.

It is given by the expression

ar

W

W r

trTaKt

),()2()(

ar

Wr

traK

),()2( (18)

ar

n

n

nn

r

rvJtA

r

tr

0

0 )()(),(

nqx

n

n

nnx

xJvtA

0

0 )()( (19)

)()(),( 10

n

n

nnn qJvtA

r

tr

(20)

nn qav (21)

)()(

10 xJx

xJ

(22)

)()()2()( 10

n

n

n

nnW qJvtAaKt

)()()2( 10

n

n

n

nnW qJqtAK

(23)

)()(~

)2()(~

1

0

n

n

n

nnW qJvsAaKs

)()(~

)2( 10

n

n

n

nnW qJqsAK

(24)

)()(~

)2()(~

1

0

n

n

n

nnW qJvsAaKs

)()(~

2 10

n

n

n

n

n

nWW qJq

s

psK

(25)

When

n

n

n

n p

s

p

(26)

it follows that

)()(~

2)(~

1

0

n

n

n

n

n

nWW qJq

psKs

(27)

qnpnJ1(qn )

n

qn 2a2CWJ1(qn )

qnJ1(qn )KWqn

2

2a2CW

KW

qn

2 (28)

n

n n

WWq

saCs0

2

2 1)(~

4)(~

(29)

Using the result [4]

n

nnq02 4

11 (30)

gives

WSWW TsCass2

)(~

)(~

(31)

3.2 Electric Field Heating of Ground

It is assumed that the wavelength and attenuation

length of the radiation is much larger than depth of

penetration. Ground heating is then time dependent

but not space dependent in the domain of interest. The

heat generation for a harmonic waveform at radian

frequency and with time dependent waveform

)(tE is


6/14


)(''

2

1)( 2

00 tEtG

w/m

3 (32)

When the electric field is constant over time,

0)( EtE , so that

s

EsG

2

0

0

0

''

2

1)(

~

If G is the specific heat per cubic meter the

temperature rise is

t

GE dttE

CtT

0

2

0

0 ')'(''

2

1)(

(33)

s

sE

CsT

GE

)(~

''21)(~

2

0

0

(34)

The < > indicates that its the Laplace transform

of the square of the electric field which is calculated.

If2

0

2 )( EtE it follows that

2

20

0

0 ''

2

1)(

~

s

E

CsT

GE

4. Computation of Surface Temperature

The time and space equation for the ground surface

temperature requires solution of time-space diffusion

in ground including ground heating. The equation is

1

2

2

2

2

y

T

x

T

t

T

D

GGG

G

)()()(),(11

yxKtKty GGG (35)

Taking the spatial Fourier transform in the

y-direction of Eq. (35) gives

dyyjtyxTtxT GG )exp(),,(),,( (36)

dyjtxTtyxT GG )exp(),,(2

1),,( (37)

1 22

2

Tx

T

t

T

D G

GG

G

)()(),( 11 xKtKt GGG (38)

The laplace transform

~~

~2

2

2

TxTT

Ds GGG

G

)()(~

),(~

11 xKsKs GGG (39)

is re-expressed using the form

G

GG

sC

ssxFsxT

),(~

),,(~

),,(~

G

G

sC

ssxF

)()(~

2),,(

~ (40)

In deriving Eq. (40) it was assumed that ground

heating was spatially uniform.

The diffusion equation for ),,(~

sxF is solved

using the insulating boundary condition between the

earth and air.

)()(~~

~~

12

2

2

xKsFx

FF

D

sG

G

(41)

The insulating boundary condition is

0

~

dxx

F (42)

When dx0 , (41) becomes

(43)

)exp(),()exp(),(~

xsBxsAF (44)

2s /DG (45)

The insulating boundary condition (42) and (44) yield

)2exp(),(),( dsBsA (46)

When 0x (41) may be expressed as

(47)

)exp(),(~

xsCF (48)


7/14


Matching at 0x gives

),(),(),( sBsAsC (49)

Integrating over the delta function yields

(50)

GK

s

x

F

x

F )(~~~

0

00

(51)

GK

ssCsBsA

)(~

),(),(),(0

(52)

GK

ssBsAsBsA

)(~

),(),(),(),(0

(53)

GKssB

2)(

~

),( (54)

)2exp(2

)(~

),( dK

ssA

G

(55)

)2exp(12

)(~

),( dK

ssC

G

(56)

The total temperature is

(57)

The surface ground temperature is

G

GG

sC

ssdFsdT

)()(~

2),,(

~),,(

~ (58)

)exp(),(),,(~

dsAsdF

)exp(),( dsB (59)

)exp()2exp(),(),,(~

ddsBsdF

)exp(),( dsB (60)

)exp()(

~

)exp(),(2),,(~

dK

sdsBsdF

G

(61)

(62)

G

WSWWG d

K

sTsCassdxT )exp(

)(~

)(~

),,(~ 2

G

G

sC

s )()(2 (63)

The next step is to convert Eq. (63) back to y

space using the equation

dyjsdTsydT GG )exp(),,(~

2

1),,(

~(64)

GDs /2 (65)

G

G

G

G

Ds

dDs

K

ssydT

/

)/exp(

)(~

),,(~

02

2

G

G

sC

sdy

)(~)cos( (66)

Using the general formula [4]

)()cos()exp( 22

0

022

22

aKdxax

x

x (67)

gives

G

WSWWG K

K

sTsCassydT

)(~

)(~

),,(~

0

2

G

GG

sC

sdyDs

)(~

)/(22 (68)

)(0K is Bessel function of the second kind andy

is the lateral distance from center of wire ( 0y is

right over center line of wire).

Examination of Eq. (68) shows that in order to

complete the analysis it is necessary to determine the

dynamic temperature at the wires surface, )(sTWS ,

and then take the inverse Laplace transform.

Determining )(sTWS is in itself a problem within a

problem. As discussed in Section 2, a determination of

exactly how the heat transfer between the wire and

ground occurs is needed. Considering the model of

Section 3.1, the equations that couple )(sTWS to the

ground are examined.

This is a two-region diffusion problem. The inner

region is the wire which extends in the range


8/14


ar0 and the outer region (ground) is the range,ra . The upper limit of infinity is chosen

because for the geometry and time scale of this

problem theres not much chance for thermal energy

returning to the wire. The boundary conditions

between the wire and ground are: continuity of

temperature, )(sTW , and heat flux,

)()(~

)(~ 2

sTsCass WSWW at ar .The equations in the ground are

)(1 2

2

2

tr

T

rr

TK

t

TC G

GGG

GG

(69)

GGGG

GG

Ksr

T

rr

T

TD

s

/)(

~~

1~

~ 2

2

2

(70)

)(''

2

1)( 2

00 tEtG

(71)

To satisfy the boundary conditions the Bessel

function of the second kind, 0K , is considered,

which solves the equation

00

2

20

2 10 K

D

s

r

K

rr

K

G

(72)

The solution of Eq. (70) is then

GGGG sCrDsKsAsrT /~

)/()(),(~

0 (73)

where, )(s is a constant to be determined from the

boundary conditions. Applying continuity at ar

gives

WSGGGG TsCaDsAKarT~

/~

)/(),(~

0 (74)

WSEG TTaDsAK

~~)/(

0 (75)

GGE sCT /~~ (76)

The energy balance equation is

ar

GGWSWW

r

TaKsTsCass

~

2)(~

)(~

)(~ 2

ar

GG

r

rDsKAaK

/(2

0 (77)

Using the relationship

)/()/(

10

aDsKD

s

r

rDsKG

Gar

G

(78)

gives

)(~

)(~

)(~ 2 sTsCass WSWW

)/()(2 1 aDsKsAD

saK G

G

G (79)

Eqs. (75) and (79) provide solutions for the two

unknowns: )(sA and )(~

sTWS . From Eq. (74) it

follows that

)/(

)(~

)(~

)(0 aDsK

sTsTsA

GEWS

(80)

Inserting the foregoing expression in Eq. (79) and

working through the algebra gives

sCaP

TPssT

W

EWWS 2

~)(

~

)(~

(81)

G

GD

saKP 2 (82)

)/(

)/()/(

0

1

aDsK

aDsKaDs

G

GG (83)

sTsCass WSWW2 )(

~)(

~)(

~

sCaP

TPssCas

W

EWWW 2

2

~)(

~

)(~

(84)

Inserting Eqs. (81)-(84) into Eq. (68) give the

complete expression for the surface temperature in

Laplace transform space.

)(~

)(~

)(),,(~

2

K

sTsCasssydT

G

EWWG

)(~

)/( 220 sTdyDsK EG (85)

sCaKKDsaK

KKDsaKs

WGG

GG

210

10

)/(/2

)/(/2)(

(86)


9/14


5. Results and Sample Calculations

As shown in Eq. (85) there are two contributions to

the surface temperature: the contribution from the

wire (the first term on the right hand side) and thecontribution from ground heating (the second term on

the right hand side). The second term is

straightforward and given by

s

sE

CsT

GE

)(~

''

2

1)(

~2

0

0

(87)

If 202 )( EtE

2

20

0

0 ''

2

1)(

~

s

E

CsT

GE

(88)

On the other hand, for a ramp pulse where

max

2max

2 )(t

tEtE (89)

then

3max

2max

0

0 ''

2

1)(

~

st

E

CsT

GE

(90)

For completeness permittivity information is

provided in Table 1.

It is interesting to examine and interpret the

components of the first term in Eq. (85). Consider the

expression: )(~

)(~ 2 sTsCas EW .

This term refers to the heating of the wire. Where

)(~

sW came from is known, but where did the

expression involving the temperature due to ground

heating come from? This term arises from the fact that

the surface of the wire is in contact with the ground

and ground heating controls in part the ability to

transfer heat to the ground. Realistically, in order to

detect the heated wire with high probability to have

the condition )(~

)(~ 2 sTsCas EWW is

required. Assuming a constant current:2

0

2 )( ItI gives

Table 1 Some values for /

0

and

/ 0.

/ 0 / 0

Sandy Sandy

Dry 2.55 Dry 0.02

Wet 20.00 Wet 2.60Loamy Loamy

Dry 2.44 Dry -----

Wet 20.00 Wet 2.4

Clay Clay

Dry 2.27 Dry 0.03

Wet 11.30 Wet 2.83

s

IRs

eff

W

2

0)(

~ (91)

The term, )(s , is the most complex component. It

pertains to the interaction between the wire and

ground and involves Eqs. (69) to (86). From (86)

)/(/2

)/(/2)(

210

10

WGG

GG

sCaKKDsaK

KKDsaKs

)(2

)(1

1

0

12

CCG

CW

sTKsTK

sTsKCa

(92)

where, the hybrid diffusion time scale naturally

occurs. The hybrid characterization is useful because

it befits the interaction involving a feature of the wire,

namely a and the ground diffusion constant.

GC

D

aT

2

(93)

Recalling that GGG CKD / yields the

simplified form

)(2

)(1

1)(

0

1

CG

CCW

sTKC

sTKsTCs

(94)

It is interesting to note from Table 2 (compiled

from previous sections) that the ratio of specific heats

is about 2 and hence )(s is approximated by


10/14


Table 2 Summary of Thermal Physical Constants.

401WK w/m-deg K Thermal conductivity

61045.3 WC Joules/m3-deg K specific heat per

unit volume

4103.1/ WWW CKD m2/s thermal diffusivity

GK Thermal conductivity of ground w/m-deg K

Typical values 86.15.1 GK ; representative value

7.1GK

GC Specific heat per unit volume of ground Joules/m3-deg K

Typical values610)45.24.1( GC ; Representative

value6102GC

61085.0 G

GG

C

KD m2/s

)(

)(1

1)(

0

1

C

CC

sTK

sTKsTs

(95)

What is the time behavior that corresponds to the

inverse Laplace transform of )(s ? That function is

)(

)(1

1)()(

0

1

11

C

CC

sTK

sTKsTLsLtx (96)

The inverse transform of Eq. (96) does not appear

to be available from existing sources. The principal

difficulty is the term: )(/)( 01 CC sTKsTK .

Numerical techniques are required to compute the

inverse transform for all times. This term is examined

numerically in Appendix A.

For early times, characterized by very large s,

asymptotic expansions for )(1 CsTK

and

)(0 C

sTK can be used. The general result for

complex argument is Ref. [5]:

.....

8

11)exp(

2

)(0

zz

zzK (97)

.....

8

31)exp(

2

)(1

zz

zzK (98)

.....

2

11

)(

)(

0

1

zzK

zK (99)

For the exploratory model it follows that

1)(

)(

0

1 zK

zK (100)

Therefore

sT

LTsT

LsLtx

C

CC1

11

1

1)()(

111(101)

2/1)/()/exp(

111)( CC

CC

TtErfcTt

TtT

tx

(102)

2/12/1 )/(1)/( CC TtErfTtErfc (103)

0

12

0

2

)12(!

)1(

2)exp(

2

n

nnx

nn

xdxErf (104)

Using Eq. (105) in the limit as 0t yields

CCC tTTtTtx

11

11)(

(105)

When2/1)/( CTt becomes very large it follows [4]

CCC Ttt

TTtErfc /exp

1)/(

2/1

2/1

(106)

0

11

11)(

2/1

t

T

TtTtx C

CC

(107)

The term, )/(22

0 dyDsK G , is a basic

structure of diffusion theory and relates how heat

travels and spreads in the vertical and horizontal

directions. Its inverse Laplace transform is

)4/exp(2

1)()( 2/12/10

1 tbt

sbKLtf (108)

GD

dyb

22 (109)


11/14


An example of a time dependent solution by

starting from Eq. (85) is given by dropping the terms

involving ground heating and using a constant current.

In this case

)(

)(~

)(),,(~

0 sbKK

sssydT

G

WG

)(1

1

10

2

0sbK

sK

IR

sT G

eff

C (110)

Using the convolution theorem and Eqs. (102) and

(108) the time behavior of the surface temperature can

be determined.

'

00

20

)()'('

),,(tt

G

eff

G dftxdtK

IRtydT (111)

For illustrative reasons onlyif it is assumed that the

wire/ground interaction could be neglected (this

assumes that somehow the heat energy pumped into

the wire goes directly into the groundthe theoretical

best case) then

11

1

CsT (112)

t

G

eff

G dttbtK

IRtydT

0

20

')'4/exp('2

1

),,( (113)

)(2

1)exp(

'

1

2

1')'4/exp(

'2

11

0

Eduuu

dttbt

t

(114)

tD

dy

G4

22 (115)

duuu

E )exp('

1)(1 Exponential Integral [5]

)(2

),,( 1

20

EK

IRtydT

G

eff

G (116)

6. Conclusions

In this study a closed-form expression in Laplace

transform space was derived for the surface

temperature of buried wire systems ranging from

microelectronic size to large underground being

heated from an unwanted external electromagnetic

source. It was determined that buried heated wire

systems can produce detectible thermal signatures that

can provide information of system performance and

status. An expression was provided for the surface

temperature as a function of time and distance from

the buried wire, temporal behavior of the incident

electric field, and other relevant physical constants.

Using Fourier-space and Laplace-time transform

techniques the surface temperature was found to be a

sum of two terms: explicit ground heating produced

by the sources ground-penetrating electric field andthermal generation in the wire and its transport to the

earths surface. A closed-form expression in terms of

the Laplace transform variable was derived for the

surface temperature. The time dependent surface

temperature was computed from the inverse Laplace

transform using asymptotic approximations.

References

[1] J. R. Wait, Excitation of currents on a buried insulatedcable, J. Appl. Phys. 49 (2) (1978) 876-880.

[2] D. Poljak, F. Rachidi, K. Drissi, K. Kerroum, S.V.Tkachenko, S. Sesnic, Generalized form of telegraphers

equations for the electromagnetic field coupling to buried

wires of finite length, IEEE. Trans. Electromagn. Compat

51 (2) (2009) 331-337.

[3] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids,Oxford University Press, 2nd ed., Oxford University,

1986.

[4] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Seriesand Products, Academic Press, Inc. New York, 1980.

[5] M. Abramowitz, I. Stegun, Handbook of MathematicalFunctions, U.S. Department of Commerce NBS 53, 1964.

[6] K.J. Hollenbeck, INVLAP.M: A matlab function fornumerical inversion of Laplace transforms by the de

Hoog algorithm, 1998, [on line],

http://www.isva.dtu.dk/staff/karl/invlap.htm.

[7] F.R. De Hoog, J.H. Knight, A.N. Stokes, An improvedmethod for numerical inversion of Laplace

transforms, S.I.A.M. J. Sci. and Stat. Comput. 3 (1982)

357-366.


12/14

Journal of Earth Science and Engineering 3 (2013) 730-743

Appendix ANumerical Evaluation of x t

Beginning with (94) and using the transformation w sTC and GW

C

C

C 2 yields

)(

)(1

1),(

0

1

wK

wKwC

CwFs

X (A1)

where

C is a constant used here as a parameter

K0 and K1 are the modified Bessel functions (of the second kind).

Consider the the inverse transform

CC TxTCwFLCf ,,1 (A2)

where

C

T

tf

Ttx

CC

,1

. (A3)

The inverse transform function f ,C is approximated using [6] which is an implementation of the algorithm [7] based onaccelerating the convergence of Fourier series approximations. Figure 4 displays the graph of f ,C for values of C rangingfrom 0.1 to 8.

Fig. 4 Plots of f ,C as a function of for different values of C.


13/14


A log-log plot of the graph shown in Fig. 5 reveals the asymptotic behavior of f as a function of for different values of C.

In particular it can be seen that f ,C 1/ 2 for sufficiently small and f ,C 4 /3 for large values of .

Fig. 5 Log plots of f ,C as a function of for different values of C.

It is also interesting to note the dependence on the parameter Cof the approximate transition times from one form of asymptotic

behavior to the other. Fig. 6 shows the details of the log plots of f ,C for small values of clearly showing that thepersistence of the rate

1/ 2 depends on the parameter C. A closer examination of the values shown in Fig. 6 reveals

CCf

1

),( for3

10

(A4)

Fig. 6 Early time log plots of f ,C as a function of for different values of C.


14/14


For large values an examination of Fig. 5 suggests f ,C C4 / 3. Combining these asymptotic behaviors, a non linearleast squared fit was performed using four parameters for the function

CpCpC

pCg

p4

3/42

1

3

,

(A5)

minimized using the relative norm

M

i i

ii

f

gf

ME

1

22 1

(A6)

where the values fi f j,Ck are those evaluated for 104 j 102 and 101 Ck101 as shown in Figs 4-6.The parameters obtained using this minimization yielded p1 0.5624, p2 3.6904, p3 0.7446, and p4 1.3747

resulting in the relative error E 0.0558. A comparison of f ,C to the fit g,C as functions of Cfor 10n with4 n 2 is displayed in Fig. 7, clearly demonstrating the accuracy of the fit.

Fig. 7 Log plots of Cf , and fit values Cg , as a function of Cfor different values of .

Consider the comparison between the function fit (A5) and the estimate given in (105). Using the function fit (A5) together with

the transformation (A3) gives

CCpCCCC TCtpTtCpTtCT

pC

T

tf

Ttx

///,

1

43/4

2

1

3

. (A7)

Setting 1C , (A7) yields CtT

ptx 1 for sufficiently small t, in agreement with (105) if ...564189.0

11 p

The least squares fit derived above for p1 is within 0.32% of this asymptotic value.

Sensing Thermal Underground Transients of Buried Wired Structures

Documents

Transcript of Sensing Thermal Underground Transients of Buried Wired Structures