Time-domain reflectometry method and its application for measuring moisture content in porous materials: A review

Measurement 42 (2009) 329–336

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Review

Time-domain reflectometry method and its application for measuring moisture content in porous materials: A review Robert Cˇerny´ * Department of Materials Engineering and Chemistry, Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 166 29 Prague 6, Czech Republic

a r t i c l e

i n f o

Article history: Received 12 March 2008 Received in revised form 22 August 2008 Accepted 29 August 2008 Available online 6 September 2008

Keywords: Time-domain reflectometry Moisture content Porous media

a b s t r a c t The time-domain reflectometry (TDR) method is based on the analysis of dielectrics’ behavior in time-varying electric field which can be utilized in the determination of permittivity and electrical conductivity of a wide class of materials. Measurement of moisture content in porous materials belongs to its most frequent applications; after several decades of use TDR method became one of the most recognized techniques in this field. In this review, a short historical overview of the TDR method applications is introduced, the function of a typical TDR device is described and the main TDR data interpretation techniques are surveyed first. Then, the methods for determination of moisture content in porous media from the measured relative permittivity are analyzed and prospective ways of their application are identified. Finally, some recommendations for further developments in TDR measurement of moisture content in porous materials are formulated. Ó 2008 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3.

4.

5.

Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TDR devices – fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TDR data interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Basic theoretical principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Travel time analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Frequency-domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Individual probe calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of moisture content in porous materials using the measured relative permittivity . . . . . . . . . . . . . . . . . . . . . . . 4.1. Empirical conversion functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Dielectric mixing models – homogenization techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Historical overview The application of time-domain reflectometry (TDR) in engineering and natural sciences has a relatively long * Tel.: +420 2 2435 4429; fax: +420 2 2435 4446. E-mail address: [email protected] 0263-2241/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2008.08.011

329 330 331 331 332 332 333 333 333 334 335 335 335

history. Already in 1930s it became a recognized technique in cable testing. The physical principal was quite simple. It is well known from the electromagnetic wave theory that if any change of impedance appears along the transmission line of the waves there is a partial or total reflection on it. In a cable such a reflection means a fault. If the time between launching the waves into the cable and detection

330

´ / Measurement 42 (2009) 329–336 R. Cˇerny

of reflections is measured the spatial location of the fault can be determined using the known propagation velocity of the waves. The original version of the method was suitable for conducting materials mostly where the electrical properties were usually well known. Later, a straightforward modification of the method appeared, using quite opposite way of data evaluation. The principle of this modification consisted in using the information on the spatial location of the reflection point. If this location was known, the propagation velocity of the waves in the material could be calculated which could then be utilized for determination of the electrical properties. Fellner-Feldeg [1] was probably the first who used the TDR methodology for this purpose, analyzing the dielectric properties of alcohols held in coaxial cylinders. The fact that liquids were the first materials used in measuring dielectric properties by TDR was not a coincidence. They were homogeneous and could achieve a very good contact with the measuring cell. Therefore, the electromagnetic field within the cell was not disturbed by secondary interfacial reflections and the main reflections on the head and the end of the probe were relatively easy to distinguish. So, in the first applications the main users of the TDR measuring technology were researchers in the field of chemistry and physics of liquids. In 1980s the investigation of the electrical properties of liquids still remained the most frequent topic among the TDR-based studies (see e.g. [2–5]) but the application range of the method was spread to several other fields of research. A fast development of the TDR technology was initiated in soil science where the method found an increasing use in soil moisture measurement (see e.g. [6–8]). TDR also became a recognized technique in the determination of electrical properties of liquid crystals (e.g. [9–11]). First applications of TDR measuring technology appeared in the field of biomaterials [12]. Since 1990s TDR already could be considered a wellestablished technique that was widely used in tens of scientific laboratories. Its application for measurement of the electrical properties of liquids and loose or soft solid materials was still dominant (see e.g. [13–20]). However, first experimental setups suitable for compact solid materials began to appear. In 1996 the method was first used for wood [21], in 1997 for Portland cement concrete [22]. During the last couple of years the number of ISI references of TDR measurements was stabilized at approximately 120–130 per year and its increase was not as dramatic as during 1990s. Besides the well-established measurements on traditionally analyzed materials mentioned before, some additional applications appeared. TDR was used for the dielectric characterization of coals [23], compact building materials [24–27], compact rock materials [28,29], foods and agricultural materials [30,31] or human skin [32]. Following the traditional cable testing procedures, some advanced applications of TDR also appeared in extensometry [33] and in crack propagation monitoring [34]. Improvements in theoretical concepts aimed at the interpretation of TDR-provided data and development of advanced methods for complex TDR applications were an-

other topics pursued by many investigators. An efficient numerical tool for electric line simulation in complex configurations was proposed in [35]. A novel inverse transmission-line method was studied in [36]. Advanced models of electromagnetic wave propagation along transmission lines were formulated and solved in [37,38]. Low-frequency dielectric measurements using TDR were reported in [39]. Combined approach based on time-domain reflectometry and frequency-domain analysis was proposed in [40]. A method for determination of water content profiles using TDR reflection data was presented in [41]. Simultaneous measurement of dielectric properties and levels of liquids was analyzed in [42–44]. The current state of the TDR measuring technology can be characterized as very advanced but its development is still far from completed. The experimental setups used in various scientific laboratories are being continuously improved and refined to achieve higher accuracy and to get more information from the measured data. Also, the investigation of new materials leads to further adjustments of the technology. It can be anticipated that the development of the method will continue rapidly in the near future and its application will be spread to other fields of research and other materials. 2. TDR devices – fundamentals A device based on the TDR principle (see e.g. [1], in more details [45]) launches electromagnetic waves and then measures the amplitudes of the reflected waves together with the time intervals between launching the waves and detecting the reflections. The fundamental element in any TDR equipment used for the determination of moisture content in porous materials is a metallic cable tester. This usually consists of four main components: a step-pulse generator, a coaxial cable, a sampler and an oscilloscope. The step-pulse generator produces the electromagnetic waves. The electric part of the electromagnetic waves consists of sine waves covering a large frequency range, but which frequencies the step-pulse generator produces is not arbitrary. If a sine wave is superimposed on harmonic sine waves, where the highest frequency tends towards infinity, the result will be a perfect periodic square wave. This is what happens in the step-pulse generator. The periodic square wave is commonly called ‘‘voltage step”. In the cable tester Tektronix 1502B, which is often used in soil science (see e.g. [45]), the voltage steps are produced by superimposing a sine wave with a ground frequency of 16.6 kHz on harmonic sine waves up to 1.75 GHz. One voltage step is transmitted over a period of 10 ls, and then there is a pause in the transmission lasting 50 ls. This pause ensures that standing waves die out before new waves are launched from the cable tester. The rise time of the voltage steps, which depends on the highest frequency of the sine waves (for an infinite frequency it would be equal to zero), is for this particular case about 200 ps. The coaxial cable connects the step-pulse generator and the sampler. The shield of the coaxial cable is connected to earth and its electric potential is 0 V. The electromagnetic

´ / Measurement 42 (2009) 329–336 R. Cˇerny

waves produced by the step-pulse generator are launched into the conductor in the coaxial cable with a voltage drop of several tenths of a volt (for Tektronix 1502B it is 0.225 V) between the conductor and the shield. The TDR probe itself is conductively connected to the coaxial cable in such a way that the cable is open ended and the probe forms this open end. In principle, the coaxial cable and the probe differ only in the type of dielectric. While the cable has usually polyethylene as a dielectric, the dielectric of the probe is the measured porous material. The sampler detects the electromagnetic waves launched by the step-pulse generator and transmitted by the coaxial cable-TDR probe system. It generally consists of two main components, a high precision timing device and a high precision voltmeter. When the electromagnetic waves launched by the generator are detected by the sampler, the sampler starts to measure the voltage between the shield and the conductor at a certain time interval. The set of data obtained consists of voltage as a function of time. The oscilloscope displays the simultaneous measurements of time and voltage obtained by the sampler on a liquid crystal display, or the data in a digital form can be directly sent to a PC and displayed there. This generates a curve called the trace. The evaluation of data obtained by a cable tester is based on the following basic principles. Any change of impedance in the cable–probe system causes a partial or total reflection of the waves. Therefore, one reflection will be on the cable/probe interface, where the dielectric is suddenly changed, and therefore the impedance must also be changed, while the second reflection is on the open end of the probe, where the impedance tends towards infinity and the wave is reflected in phase. The reflected waves are superimposed on the waves transmitted from the metallic cable tester. The voltmeter in the sampler detects a change in the voltage between the conductor and the shield, and the timing device in the sampler registers the time interval between the start of the transmission of the waves and the detection of the reflection. Reflected waves can be either in phase with the incoming waves, which happens in the case when the electromagnetic waves meet an increase in impedance, or in counter phase, when a decrease of impedance is met. Therefore, in the case of a cable–probe system, the reflection on the cable/probe interface can cause either a decrease in the amplitude (if the impedance of the probe is smaller than the impedance of the cable, which is most often the case) or its increase in the opposite case (reflection on the open end always results in an amplitude increase). Thus, three characteristic times can be recognized on the trace. The first is the time when the front of the initial voltage step (always positive), induced by the step-pulse generator, reaches the sampler. The second one is the time when the voltage step induced by the reflection on the cable/probe interface (mostly negative) reached the sampler. The third characteristic time is the time when the voltage step induced by the reflection on the open end, i.e. on the end of the probe (always positive) reached the sampler. The length of the probe is known. Thus the velocity of the electromagnetic waves in the

331

probe can be easily calculated. Alternatively, the entire waveform can be analyzed by means of the dielectric spectroscopy methods. 3. TDR data interpretation There are two basic methods of the trace analysis. The first one is based just on the identification of the times of reflections at the head and the end of the probe. In this way, the apparent permittivity of the probe (not dependent on frequency) can be determined. This method is usually called travel time analysis or time-domain analysis. The other one assumes that the model of dielectric dispersion and relaxation in the probe is known and tries to identify the unknown model parameters using the Fourier transform of the measured waveforms. This method is mostly called frequency-domain analysis or dielectric spectroscopy analysis. Both methods have their advantages and disadvantages, and a choice between them depends on the actual aims of the experimental work being done. 3.1. Basic theoretical principles From the physical point of view, propagation of the TDR signal in the probe can be considered as propagation of electromagnetic waves in an absorbing medium. The simplest solution of Maxwell equations for the electric field vector E is that of a plane, time harmonic wave, which for a wave propagating in the z-direction can be expressed (see e.g. [46,47]) as follows 

E ¼ E0  eiðxtk

zÞ

ð1Þ

; 

where x is the angular frequency, k is the complex wave number, 

k ¼

x c

¼x

pffiffiffiffiffiffiffiffi ffi le ;

ð2Þ

c is the complex velocity of the wave propagation in the given medium, l is the magnetic permeability, e the complex permittivity. Assuming a non-ferromagnetic medium, i.e. l  l0, l0 is the magnetic permeability of vacuum, we can write

1 1 c0 c ¼ pffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffi ;  

l0 e

l0 e0 er

ð3Þ

er

where e r is the complex relative permittivity, e0 is the permittivity of vacuum, c0 is the wave propagation velocity in vacuum. Combining (2) and (3) we obtain 

k ¼

x pffiffiffiffiffi  er ;

and using the notation k = k0 + ik00 ,

x pffiffiffiffiffiffi er1 x pffiffiffiffiffiffi 00 er2 ; k ¼ 0

k ¼

ð4Þ

c0

c0

c0

where

er ¼ e0r þ ie00r then ð5Þ ð6Þ

´ / Measurement 42 (2009) 329–336 R. Cˇerny

332

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 ðe0r Þ2 þ ðe00r Þ2 þ e0r 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 ¼ ðe0r Þ2 þ ðe00r Þ2  e0r : 2

er1 ¼

ð7Þ

er2

ð8Þ

E ¼ E0  e

e 

iðxtk0 zÞ

ix tcz

¼ AðzÞ  e



f

¼ E0  e

zcx

0

pffiffiffiffiffi

er2



ix tcz

e

0

pffiffiffiffiffi

er1



ð9Þ

;

where A(z) is the amplitude of the wave,

AðzÞ ¼ E0  e

zcx

0

pffiffiffiffiffi

er2

ð10Þ

;

and cf is an equivalent to the phase velocity of a wave propagating in a non-absorbing medium,

c0 cf ¼ pffiffiffiffiffiffi :

ð11Þ

er1

If an electromagnetic impulse is propagating through an absorbing medium of a known thickness Dz, we can measure basically two quantities. The first is the travel time Dt of the impulse, i.e. the time necessary to pass the distance Dz, and the second is its attenuation a after passing Dz,

a¼

AðDzÞ : Að0Þ

Using (10) and (11) we obtain

ln a ¼ 

x pffiffiffiffiffiffi er2  Dz

ð12Þ

c0

2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   c 0 Dt ðe0r Þ2 þ ðe00r Þ2 þ e0r ¼ Dz 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 c0  ln a ; ðe0r Þ2 þ ðe00r Þ2  e0r ¼ 2 x Dz 1 2

which leads to the following relations for

e ¼

0

l is the length of the probe (the signal has to pass it two times before being detected by the sampler) and Dt is the time between the detection of reflection on the cable/ probe interface (probe head) and the reflection on the end of the probe. This treatment involves two very important simplifications that put relatively strict limits to its application. First, the signal attenuation in the probe is supposed to be negligible. Second, the frequency dependence of the permittivity is not taken into account. Therefore, the travel time analysis does not result in the determination of dielectric permittivity as it is defined exactly in the electromagnetic wave theory but in a kind of apparent physical quantity that is usually called ‘‘apparent permittivity”, er,a. So, the basic relation in the travel time analysis of TDR waveforms reads

er;a ¼

 2 c 0 Dt : 2l

ð21Þ

As it follows from the above written, the travel time analysis is suitable for materials with a low attenuation (i.e. low dielectric relaxation and low electric conductivity) and low frequency dependence of the permittivity. On the other hand, the analysis of the measured trace is very simple which is its main advantage.

ð13Þ

Substituting (7) and (8) into (12) and (13) we finally arrive to a system of two algebraic equations for the unknown real and imaginary parts of the complex relative permittivity, e0r , e00r in the form

 c 2

ð20Þ

3.3. Frequency-domain analysis

Dz c0 ¼ pffiffiffiffiffiffi : Dt er1

0 r

The common travel time analysis of a TDR signal (see e.g. [6]) is based on the application of Eq. (18) where

Dz ¼ 2l;

Substituting (5)–(8) into (1) we arrive at k00 z

3.2. Travel time analysis

" 2

 2 # ln a

 ðDtÞ  Dz x  c 2 Dt e00r ¼ 2 0   ln a: Dz x

ð14Þ ð15Þ

e0r , e00r ð16Þ ð17Þ

For very small attenuation of the electromagnetic wave in the material, i.e. ln a ? 0, the relations 16,17 can be simplified into the form

 2 c 0 Dt Dz c0 pffiffiffiffi0 ¼ 2  er  ln a: x Dz

e0r ¼

ð18Þ

e00r

ð19Þ

The frequency-domain analysis of TDR waveforms is based on a comparison between the incident signal, V+(t), and the signal reflected by the sample, V(t) (see e.g. [23]). Any discontinuity in the coaxial line gives rise to a reflection that can be characterized by the complex reflection coefficient,

CðxÞ ¼

ZðxÞ  Z 0 ; ZðxÞ þ Z 0

ð22Þ

where Z(x) is the line impedance and Z0 the characteristic line impedance in vacuum. C(x) can be determined experimentally, departing from the measurements of V+(t) and V(t). Using Fourier transform, V(x) and V+(x) can be found, and their quotient gives the complex reflection coefficient,

CðxÞ ¼

V  ðxÞ : V þ ðxÞ

ð23Þ

Taking into account (22) together with the relationship between the line impedance in vacuum and the charged line impedance,

Z0 ZðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; er ðxÞ we arrive at

ð24Þ

´ / Measurement 42 (2009) 329–336 R. Cˇerny

er ðxÞ ¼

 2 1  CðxÞ : 1 þ CðxÞ

ð25Þ

The most common model of the dielectric relaxation and dispersion is the Debye model, given by

er ðxÞ ¼ e0r ðxÞ  je00r ðxÞ ¼ e1 þ

es  e1 ; 1 þ jx s 0

ð26Þ

where e0r ðxÞ is the dielectric dispersion, e00r ðxÞ the dielectric absorption and s0 the relaxation time. A comparison of relations (25) and (26) makes it possible to identify both e0r ðxÞ and e00r ðxÞ, so also two of the model parameters es, e1 and s0. 3.4. Individual probe calibration

333

4. Determination of moisture content in porous materials using the measured relative permittivity There are three basic approaches to the determination of moisture content from measured relative permittivity. The first possibility is utilization of empirical conversion functions generalized for a certain class of materials. The second is application of dielectric mixing models, which assumes knowledge of the relative permittivities of the material matrix, water, air and other parameters, that cannot be measured directly but have to be determined by empirical calibration of the model. The third method for evaluation of moisture content from measured relative permittivity consists in empirical calibration for the particular material using a reference method, such as the gravimetric method. This method is the most reliable until now but the most time consuming one.

For high-accuracy measurements, the TDR probes require individual calibration. The main reason is the fact that in Eq. (21) the physical length of the probe l is not identical with the probe electrical length (sometimes called as the characteristic probe length) lp which depends on a particular experimental setup. Consequently, the real travel time in the sensor Dtm defined as the time interval between reflection points identifying the beginning and the end of the sensor is not identical with the travel time reading obtained in the experiment. So, in a real experiment, instead of Eq. (21) we have

Empirical conversion functions are very popular particularly in soil moisture measurements. For many years, the third-order polynomial relation by Topp et al. [6] was considered the best solution for the determination of moisture content of soils using TDR measurements. This relation, which can be expressed as

pffiffiffiffiffiffiffi

h ¼ 5:3  102 þ 2:92  102  eeff  5:5  104

er;a ¼

c Dt m ; 2lp

ð27Þ

where the travel time in the sensor itself, Dtm, can be expressed as

Dt m ¼ t probe  tref ;

ð28Þ

tref is the reference time (sometimes called as the travel time of the signal in the sensor head), and tprobe is the particular travel time reading on the trace. The two unknown parameters of every probe, tref and lp, can be determined using two materials with known relative permittivity. Usual choices are either water and air (see e.g. [48]) or water and benzene (see e.g. [49]). In any case, the permittivities of both materials should differ as much as possible to achieve highest possible accuracy of the parameter determination. For instance, performing consecutive experiments with water and benzene on the same TDR probe we obtain

pffiffiffiffiffiffi

c ðtw  t ref Þ 2lp pffiffiffiffiffi c eb ¼ ðtb  tref Þ 2lp

ew ¼

ð29Þ ð30Þ

where ew is the relative permittivity of water, eb the relative permittivity of benzene, tw the travel time reading from calibration in water, tb the travel time reading from calibration in benzene. Using Eqs. (29) and (30), the reference time tref and the characteristic probe length lp can be determined as follows:

pffiffiffiffiffiffi pffiffiffiffiffi ew  tb  eb  tw pffiffiffiffiffiffi pffiffiffiffiffi ew  eb c tw  tb lp ¼  pffiffiffiffiffiffi pffiffiffiffiffi : 2 ew  eb

t ref ¼

ð31Þ ð32Þ

4.1. Empirical conversion functions

 e2eff þ 4:3  106  e3eff ;

ð33Þ

where eeff is the effective relative permittivity and h the moisture content in the porous body [m3/m3], had for the originally studied materials standard error of estimate 0.0468 and was proposed for materials having the bulk density close to 1500 kg/m3. Some other formulas were proposed in the subsequent years but none of them achieved the frequency of application of Eq. (33). Presently, the normalized conversion function proposed by Malicki et al. [50],

pffiffiffiffiffiffiffi h¼

eeff  0:819  0:168  q  0:159  q2 7:17 þ 1:18  q

ð34Þ

which takes into account the changes in the bulk density of dry material q, and had for the originally studied materials standard error of estimate 0.0269, is considered by many researchers active in soil science as universal for different types of materials. The utilization of empirical conversion functions is very easy and straightforward, thus their high frequency of application and many citations in the scientific literature. However, their major shortage is a limited possibility of extrapolation outside the moisture range of the original set of experiments. Even simple mathematical analysis of Eqs. (33) and (34) shows immediately that for eeff ? 1 we have h < 0. Furthermore, in Eq. (34) for lower q values one can get eeff < 1 if h ? 0. So, both these relations (as well as many other relations of this type, particularly those developed for organic soils) should be used with care. Possible extension to different types of materials is a second major problem with empirical conversion functions. Their limited applicability for building materials

´ / Measurement 42 (2009) 329–336 R. Cˇerny

334

was demonstrated in several cases, for instance in [51] for cellular concrete where the Topp et al. [6] formula failed completely and the Malicki et al. [50] relation failed surprisingly in the range of higher moisture. Generally, it can be stated that empirical conversion functions used in current research for TDR data conversion are anything but universal. These are always limited to specific groups of materials. 4.2. Dielectric mixing models – homogenization techniques In terms of homogenization, a porous material can be considered as a mixture of three phases, namely the solid, liquid and gaseous phases. The solid phase is formed by the materials of the solid matrix. The liquid phase is represented by water and gaseous phase by air. In the case of dry material, only the solid and gaseous phases are considered. The volumetric fraction of air in porous body is given by the measured total open porosity. In case of penetration of water, a part of the porous space is filled by water. For the evaluation of relative permittivity of the whole material (i.e. the effective relative permittivity), the permittivities of the particular constituents forming the porous body have to be known. The effective relative permittivity of a multi-phase composite cannot exceed the bounds given by the relative permittivities and volumetric fractions of its constituents. The upper bound is reached in a system consisting of plane-parallel layers disposed along the electric field vector. The lower bound is reached in a similar system but with the layers perpendicular to the electric field. These bounds are usually called Wiener’s bounds, according to the Wiener’s original work [52] and can be expressed by the relations

eeff ¼ f1

1

f2 f3 e1 þ e2 þ e3

eeff ¼ f1 e1 þ f2 e2 þ f3 e3 ;

ð35Þ ð36Þ

where (35) represents the lower limit and (36) the upper limit of effective relative permittivity (fj is the volumetric fraction of the particular phase, ej its relative permittivity). The mixing of phases resulting in effective relative permittivity functions falling between the Wiener’s bounds can be done using many different techniques. We will give couple of characteristic examples of mixing formulas in what follows which were successfully applied by various scientists for dielectric mixing in the past. Only self-consistent formulas will be accounted for which allow to model the material behavior in sufficiently wide moisture range. The Lichtenecker’s equation [53]

ekeff ¼ f1 ek1 þ f2 ek2 þ f3 ek3

ð37Þ

is a straightforward generalization of Wiener’s formulas. The parameter k in Eq. (37) varies within the [1, 1] range. Thus, the extreme values of k correspond to the Wiener’s boundary values. The parameter k may be considered as describing a transition from the anisotropy at k = 1.0 to another anisotropy at k = 1.0. Another mixing treatment was introduced by Rayleigh [54] and a little bit later, with a somewhat different theo-

retical justification, by Maxwell Garnett [55]. It consists in perception of a continuous phase 1 (in the particular case of a wet porous medium it is the solid matrix) containing randomly distributed spherical scattering particles of discontinuous phases 2 and 3 (in the above mentioned case it is air and water, respectively). The formula by Rayleigh, which can be considered as a simple generalization of the classic Lorentz treatment of the electric field in a cavity, can be expressed (in a simple extension from the original 2 to 3 phases) as



3 eeff  1 X ej  1 f ¼ eeff þ 2 j¼1 j ej þ 2

 ð38Þ

The formula by Maxwell Garnett (extended to the threephase system again) can be written as





3 eeff  e1 X ej  e1 ¼ f : eeff þ 2e1 j¼2 j ej þ 2e1

ð39Þ

The derivation of Maxwell Garnett’s formula (39) is based on the assumption that the basic electric field of the composite is that of the solid matrix. Bruggeman [56] made a further step towards generalization of this treatment and assumed that the basic electric field is the electric field of the mixture. The resulting formula reads





3 X eeff  e1 ej  e1 : ¼ f eeff þ 2eeff j¼2 j ej þ 2eeff

ð40Þ

Later, a variety of mixing formulas appeared which reflected the various shapes and topologies of liquid and gaseous phase inclusions within the porous medium. In one of the most popular models of this type Polder and van Santen [57] extended the Bruggeman formula to elliptical inclusions and formulated its three useful simplifications (given in somewhat different algebraic form). The first of them, the original one, is valid for spherical inclusions, the second for needle-shape inclusions and the third for their disc shape. The resulting mixing formulas can be written as

eeff ¼ e1 þ

3 X

fj ðej  e1 Þ 

3eeff 2eeff þ ej

ð41Þ

fj ðej  e1 Þ 

5eeff þ ej 3eeff þ 3ej

ð42Þ

fj ðej  e1 Þ 

2ej þ eeff : 3ej

ð43Þ

j¼2

eeff ¼ e1 þ

3 X j¼2

eeff ¼ e1 þ

3 X j¼2

The large difference between the relative permittivity of free and bound water in porous medium led in the applications of the above dielectric mixing formulas to extensions of the three-phase models to four phases. Dobson et al. [58] extended the Lichtenecker’s power-law formula [53] and arrived at the relation

h¼

eaeff  hbw ðeabw  eafw Þ  ð1  wÞeas  weaa ; eafw  eaa

ð44Þ

where hbw is the amount of water bonded on pore walls [m3/m3], ebw the relative permittivity of bound water (3.1), efw the relative permittivity of free water (79 at

´ / Measurement 42 (2009) 329–336 R. Cˇerny

20 °C), ea the relative permittivity of air, w the total open porosity, and a is an empirical parameter. De Loor [59] used the Polder–van Santen model for disc inclusions [57] and formulated its four-phase extension in the form

h¼

3ðes  eeff Þ þ 2hbw ðebw  efw Þ þ 2wðea  es Þ   eeff eefws  eeas þ 2ðea  efw Þ     eeff hbw eefws  eebws  eeff w eeas  1   : þ eeff ees  eeas þ 2ðea  efw Þ

ð45Þ

fw

Dielectric mixing models were tested in many practical applications and their perspectives for further use seem to be better than those of the empirical conversion functions. Even three-phase models based on Eqs. (44) and (45) were found to give reasonable approximations of empirical data for soils as it was presented for instance in an extensive comparison of various formulas in [60]. Four-phase models (44) and (45) were successfully tested for several building materials, for instance in [51] their application for cellular concrete led to very prospective results. 5. Concluding remarks The time-domain reflectometry method can be considered as a very prospective technique for measuring moisture content in porous media. In contrast to most other methods commonly used for that purpose, it does not require calibration for every material, in general. The TDR probe calibration can be done in advance all once for every single probe. Another advantage of the TDR method is that it is well applicable for the materials with higher salt content, where an application of methods such as the resistance method or the capacitance method is impaired by a significant loss of accuracy. The TDR method has a high potential not only in laboratory measurements. Its field version makes possible an easy determination of moisture content in situ. Contrary to the common gravimetric method, TDR enables continuous long-term non-destructive monitoring of moisture content in soils, rocks and building materials. This makes it useful for instance for monitoring water condensation processes in building envelopes (particularly during wintertime) so that it can serve as a good assessment method for design solutions made by civil engineers. However, it should be noted that in the current state of research common applications of TDR technique for monitoring moisture content in all porous materials regardless of their type are not possible yet. The method requires further investigations particularly in the data acquisition process. General formulas for determination of moisture content from measured relative permittivity are not yet available. For instance, formulas commonly used in soil moisture monitoring are mostly not suitable for building materials. The presence of a significant amount of bound water in many building materials seems to be the critical factor in this respect. Building materials – contrary to most soils – often contain a considerable amount of hygroscopic moisture and its inclusion into the relations for calculation

335

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