Decontamination of Radionuclides from Concrete by Microwave ...

Decontamination of Radionuclides from Concrete by Microwave Heating. I: Theory Zdeneˇk P. Bazˇant, F.ASCE,1 and Goangseup Zi2 Abstract: The paper analyzes a proposed scheme of decontamination of radionuclides from concrete structures, in which rapid microwave heating is used to spall off a thin contaminated surface layer. The analysis is split in two parts: 共1兲 the hygrothermal part of the problem, which consists in calculating the evolution of the temperature and pore pressure fields, and 共2兲 the fracturing part, which consists in predicting the stresses, deformations and fracturing. The former is assumed to be independent of the latter, but the latter is coupled to the former. The heat and moisture transfer governing the temperature and pore pressure fields induced by the decontamination process is analyzed using an improved form of Bazˇant and Thonguthai’s model for heat and moisture transfer in concrete at high temperatures. The rate of the distributed source of heat due to the interaction of microwaves with the water contained in concrete is calculated on the basis of the standing wave normally incident to the concrete wall. Since the microwave time period is much shorter than the time a heating front takes to propagate over the length of microwave, and since concrete is heterogeneous, the ohmic power dissipation rate is averaged over both the time period and the wavelength. The reinforcing bars parallel to the surface are treated as a smeared steel layer. The recently developed microplane model M4 serves as the constitutive model for nonlinear deformation and distributed fracturing of concrete. Application of the present model in numerical computations is relegated to a companion paper which follows. DOI: 10.1061/共ASCE兲0733-9399共2003兲129:7共777兲 CE Database subject headings: Concrete; Microwaves; Contamination; Contaminants; Heating; Diffusion; Thermal stresses; Pore pressure.

Introduction Concrete is ubiquitous in nuclear facilities. As a consequence of their longtime operation, various radionuclides, such as strontium, cesium, cobalt, uranium, etc. 共Spalding 2000兲, have gradually diffused from the environment into a surface layer of concrete. Although the radionuclide concentrations are very small, the exposure to radiation over many years could be hazardous to human health. Typically, the contaminated layer is only 1–10 mm thick 关Fig. 1共a兲兴 共White et al. 1995兲, and so a demolition of the whole structures is unnecessary. Nevertheless, to guarantee a safe longtime work environment, the contaminated layer needs to be removed and properly disposed of as nuclear waste. Possible decontamination techniques include removal of the contaminated layer by hammer and chisel, by high-pressure water jet, and by various thermal treatments. This paper deals with the last, which can be of two types: 共1兲 heat conduction from a heated surface, and 共2兲 heating generated by microwaves throughout the volume of concrete. The former type has been studied for a long time with regard to fire resistance of buildings, and even more 1 McCormick School Professor and W. P. Murphy Professor of Civil Engineering and Materials Science, Northwestern Univ., 2145 Sheridan Rd., Evanston, IL 60208. E-mail: [email protected] 2 Research Associate, Dept. of Civil Engineering, Northwestern Univ., 2145 Sheridan Rd., Evanston, IL 60208. E-mail: [email protected] Note. Associate Editor: Franz-Josef Ulm. Discussion open until December 1, 2003. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on March 12, 2002; approved on October 29, 2002. This paper is part of the Journal of Engineering Mechanics, Vol. 129, No. 7, July 1, 2003. ©ASCE, ISSN 0733-9399/2003/7-777–784/$18.00.

deeply with regard to the effects of hypothetical nuclear reactor accidents 共e.g., Bazˇant and Thonguthai 1978, 1979; Ahmed and Hurst 1997; Gawin et al. 1999兲. As a result, the material characteristics needed to calculate the temperatures and pore pressures due to heat conduction from a heated surface are known relatively well 共Harmathy 1970; Harmathy and Allen 1973; Bazˇant and Kaplan 1996; Neville 1997; Voda´k et al. 1997兲. The recent studies of the decontamination process have emphasized microwave heating, which allows a much faster removal of the contaminated layer 共within only about 10 s; White et al. 1995兲. The driving force of the spalling is the microwave heat source which is distributed through the volume of concrete and is generated by microwaves emitted from a powerful applicator 关Fig. 1共a兲兴. Microwave heating has already been applied in various civil engineering problems. For instance, 共1兲 microwaves have been extensively used for nondestructive evaluation of materials. 共2兲 Microwave ovens 共instead of the traditional ovens兲 have been used for effective drying of porous geomaterials 共Wei et al. 1985兲. 共3兲 Microwave heating has been used to accelerate the curing process of concrete at early ages 共Watson 1968a; Moukwa et al. 1991兲. 共4兲 Microwaves of low frequency and low power density were shown capable of heating properly insulated concrete specimens to an almost uniform temperature 共Hertz 1981, 1983兲, which made it possible to measure the effect of temperature on compression strength in the absence of temperature gradients. 共5兲 Microwaves of high frequency, which are suitable for the present decontamination process, have been shown capable of generating a localized field of high stress that can serve as a demolition tool 共Watson 1968b; Wace et al. 1989; White et al. 1995兲. Some valuable investigations of the microwave decontamination of concrete have already been undertaken. Li et al. 共1993兲 JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003 / 777

plication. The constitutive, fracture, thermal, and diffusion models of concrete applied here are of course known. Nevertheless, since these models exist in different variants, and since some minor modifications were made in them in this project, they are briefly described in the Appendices of parts I and II of this study.

Heat Generation by Transverse Electromagnetic Waves The microwaves represent electromagnetic waves of frequency 300 MHz–30 GHz. The energy carried by electromagnetic waves through surface S is ⫺

冕

S

P•dS⫽⫺

冕

S

EÃH•dS⫽

⳵ ⳵t

冕

V

共 w e ⫹w m 兲 dV⫹

冕

V

p ␴ dV. (1)

Fig. 1. 共a兲 Sketch of microwave power decontamination system; 共b兲 transmission 共t兲 and reflection 共r兲 of transmission electron microscopy wave at interfaces of different media; 共c兲 typical layout of concrete wall and wave reflection by reinforcing bars and aggregates; and 共d兲 partition of concrete body into reinforced segments and unreinforced segments

analyzed one-dimensional temperature profiles using a linear heat transfer model. Lagos et al. 共1995兲 extended the heat transfer model to two dimensions and calculated the heat generation rate based on the standing wave normally incident to a homogeneous concrete wall. They smeared the reinforcing bars into an infinitely thin layer whose reflection factor was determined according to the area–ratio of the bars. They assumed the dielectric properties of concrete to be constant over the thickness of concrete during the decontamination process. However, they could not study the development of pore pressures because they did not model the moisture transfer coupled to the heat transfer. They assumed the surface layer to spall off when the compressive stress in the direction parallel to surface under a perfect restraint in that direction exhausts the compressive strength of the concrete. They did not take into account the deformation of the body surrounding the heated zone. The practical objective of this study, which was summarized at a recent conference 共Bazˇant and Zi 2001兲, is twofold: 共1兲 to present a model-based mathematical analysis of microwave heating and spalling of concrete; and 共2兲 to apply it to the decontamination process that takes into account not only the thermal deformation and surface layer restraint but also the moisture transfer, pore pressures, and overall deformation of the structure. The theory will be explained in this paper, while the companion paper that follows 共Zi and Bazˇant 2003兲 will present the numerical ap778 / JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003

共e.g., Cheng 1983兲 where P⫽Poynting vector characterizing the power density of the electromagnetic wave, E⫽electric field strength vector; H⫽magnetic field strength vector; dS⫽ndS, where dS⫽surface element and n⫽its unit normal; V⫽volume of body; ⑀⫽⑀ ⬘ ⫺i⑀ ⬙ ⫽complex dielectric permittivity; ␮⫽␮ ⬘ ⫺i␮ ⬙ ⫽complex magnetic permittivity; ␴⫽␻⑀ ⬙ ⫽dielectric conductivity; ␻⫽2␲ f ⫽angular velocity; f ⫽frequency, w e ⫽␧E 2 /2 ⫽electric energy density; w m ⫽␮H 2 /2⫽magnetic energy density; p ␴ ⫽␴E 2 ⫽ohmic power dissipation; t⫽time; and V⫽volume. Because the heat generation rate is a function of the electric field strength, one needs to solve the electric field strength vector E to obtain the heat source. On exit from the microwave applicator, the waves are guided and simple. But farther away the electromagnetic field can become complicated 关Fig. 1共a兲兴. An accurate solution would have to be obtained numerically from the Maxwell equations, which is not a simple affair. For our purpose, however, an approximate solution can be obtained by using the solution of a standing electromagnetic wave, particularly the solution of a transverse electromagnetic wave normally incident to a half space of a dielectric material, the concrete. The heat source calculated in this manner needs of course some further adjustment to obtain the proper power distribution 共Thue´ry 1992兲.

Electric Strength of Standing Electromagnetic Waves Let us now review the solution of the transverse electromagnetic waves, which form a standing wave pattern. The propagation of electromagnetic waves is governed by the Maxwell equations in which the electric field strength and magnetic field strength are coupled. Because the concentration of dielectric sources due to ferromagnetic materials in concrete is usually negligible 共Li et al. 1993兲, the electromagnetic wave generation inside concrete may be neglected, which means that the Maxwell equations become decoupled 共von Hippel 1954兲; “ 2 E⫽⑀␮

⳵ 2E ⳵t 2

(2a)

“ 2 H⫽⑀␮

⳵ 2H ⳵t 2

(2b)

and

The transverse electromagnetic waves may be considered to be parallel, and their incidence to the concrete surface to be normal.

Therefore, aside from time t, the dielectric field depends only on coordinate x normal to the surface 关Fig. 1共b兲兴. Eqs. 共2a兲 and 共2b兲 are simply solved as E⫽E0 e i␻t⫺␥x

(3a)

H⫽H0 e i␻t⫺␥x

(3b)

and

where ␥⫽complex propagation factor. At the interface of two different media, the electromagnetic wave is partially reflected and partially transmitted as a refracted wave. The magnitude of refracted wave is given by Fresnel’s equation, and the reflection and refraction angles by Snell’s law 共e.g., von Hippel 1954兲. If the wave is incident to the interface of two dielectric media in normal direction, the wave forms a standing wave pattern. Since only a normal incidence is considered here, the wave can be calculated simply from the continuity condition at each interface 共e.g., Cheng 1983; Wait 1985兲. Consider the concrete wall, sketched in Fig. 1共b兲, to be subdivided into parallel strips normal to the surface which either contain steel reinforcement or not 关Fig. 1共d兲兴. Medium 0 represents air and media k⫽1,2, ¯M the layers of concrete with different water contents 关Fig. 1共b兲兴. Medium M ⫹1 represents air in the case of unreinforced strips, or steel in the case of reinforced strips in Fig. 1共d兲. By analogy with the transmission line theory 共which is used to calculate the voltage and current in an electric circuit兲, the wave solutions in medium k are obtained as follows 共Wait 1985兲:

to advance through one wavelength of the electromagnetic wave, it is meaningful to average the heat generation rate over period T; I 共ave h 兲⫽

H 共 k 兲 ⫽ 共 C k /␩ k 兲共 e

⫺R k e

兲

for 0⬍x⬍l

(4)

where C k ⫽transmission factor; R k ⫽reflection factor; and ␩ k ⫽intrinsic impedance of medium k. Here ␩ k ⫽␥ k /i␻⑀ k ⫽ 冑␮ k /⑀ k . The transmission factor of air is given as the initial electric strength E 0 . The reflection factor between air and the first concrete layer is denoted as R 0 R 0 ⫽ 共 Z 1 ⫺␩ 0 兲 / 共 Z 1 ⫹␩ 0 兲

(5)

where Z 1 ⫽transmission impedance of medium 1. The impedance of medium k is obtained from the transmission line theory as Z k ⫽␩ k 共 Z k⫹1 ⫹␩ k tanh ␥ k l k 兲 / 共 ␩ k ⫹Z k⫹1 tanh ␥ k l k 兲

(6)

Z M ⫹1 ⫽␩ k 共 ␩ M ⫹1 ⫹␩ M tanh ␥ M l M 兲 / 共 ␩ M ⫹␩ M ⫹1 tanh ␥ M l M 兲 (7) where l k ⫽x k ⫺x k⫺1 ⫽width of each layer. For unreinforced strips, ␩ M ⫹1 ⫽␩ 0 . For reinforced strips, ␩ M ⫹1 ⫽0, which implies an almost 100% reflection by reinforcing bars. From the condition of continuity at each interface, R k and C k are obtained as follows: R k ⫽e ⫺2␥ k x k⫺1 关 E k⫺1 共 x k⫺1 兲 ⫺H k⫺1 共 x k⫺1 兲 ␩ k 兴 / 关 E k⫺1 共 x k⫺1 兲 ⫹H k⫺1 共 x k⫺1 兲 ␩ k 兴 C k ⫽ 关 E k⫺1 共 x k⫺1 兲兴 / 关 e ⫺␥ k x k⫺1 ⫹R k e ␥ k x k⫺1 兴

T⫽2␲/␻

0

␴ 关 Re共 E 兲兴 2 dt

(8) (9)

As we have seen, the solution of the electric field strength E k of medium k 关Fig. 1共b兲 关Fig. 1共b兲, Eq. 共4兲兴 is a complex function. The real part of E k is taken as the actual solution. The rate of volumetric heat generation by the transverse electromagnetic waves can be obtained from Eq. 共1兲. Because the wave period T ⫽1/f is far shorter than the time that the thermal heat front takes

(10)

where C⫽C ⬘ ⫹iC ⬙ ⫽transmission factor; R⫽R ⬘ ⫹iR ⬙ ⫽reflection factor; ␤⫽phase factor; and ␣⫽attenuation factor 共1/␣ represents the depth through which the field strength decays to 1/e⫽0.368 of its original value兲; here ␥⫽ 冑i␮␻(␴⫹i⑀␻) ⫽␣⫹i␤⫽complex propagation factor. A typical layout of a reinforced concrete wall is depicted in Fig. 1共c兲. The wave is reflected and scattered by steel reinforcing bars as well as the aggregates. Although the arrangement of the bars is three dimensional, they may be approximately treated one dimensionally and thus their location will be characterized just by the depth d below the surface 共Fig. 1兲. Therefore, due to the heterogeneity of concrete wall, we may take the spatial average of Eq. 共11兲 over the wave number 2␲/␤. As a result, the averaged heat generation rate is obtained as I 共 h 兲 ⫽ 21 ␴ 储 C 储 2 共 e ⫺2␣x ⫹ 储 R 储 2 e 2␣x 兲

(12)

Note that, when the second term, which represents power reflection by rebars, is neglected, Eq. 共13兲 becomes the well-known Lambert’s law: I Lambert⫽I 0 e ⫺2␣x

and ␥kx

冕

1 ⫽ ␴ 储 C 储 2 兵 e ⫺2␣x ⫹ 储 R 储 2 e 2␣x ⫹ 关 R ⬘ cos 2␤x⫺R ⬙ sin 2␤x 兴 其 2 (11)

E 共 x 兲 ⫽C k 共 e ⫺␥ k x ⫹R k e ␥ k x 兲 ⫺␥ k x

1 T

(13)

where I 0 ⫽heat generation rate at the surface. The advantage of Lambert’s law, which is widely used in low-temperature food engineering 共Metaxas and Meredith 1983; Taoukis et al. 1987; Meredith 1998兲, is that it is simple and easy to understand.

Effect of Reinforcing Bars on Microwave Penetration As an approximation, the reinforced concrete may be considered for our purposes as a parallel combination of unreinforced and reinforced strips in the direction normal to surface, labeled by subscripts U and R; Fig. 1共d兲. The overall average volumetric heat generation in the reinforced concrete wall can be obtained as the average of the heat generations in two adjacent strips I 共 h 兲⫽

再

共 1⫺p 兲 I U ⫹pI R

for x⬍d

共 1⫺p 兲 I U

for x⭓d

(14)

where p⫽area fraction of the steel reinforcing bars on a plane parallel to concrete surface; and I U , I R ⫽rate of heat generation in unreinforced and reinforced strips normal to surface, respectively. Both I U and I R are obtained from Eq. 共13兲. Note that, because of different boundary conditions, the electric strength of E U is different from E R ; it is assumed that E U is transmitted into air at the opposite surface of the wall, and E R is perfectly reflected at the location of the steel bars d. For the sake of simplicity, the foregoing analysis neglects calculation of the three-dimensional diffraction and scattering of the electromagnetic waves due to steel bars. The diffraction and scattering are surely much less significant than the wave reflection, because of the steep power decay in the concrete cover 共see Part II兲. JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003 / 779

ditions. The strain vectors on the microplanes are assumed to be the projections of the continuum strain tensor, and the stress tensor is related to the microplane stress vectors through a variational principle. For a detailed description of the microplane model and the history of development, with a literature review, see Bazˇant et al. 共2000b兲, and for various related aspects also Brocca and Bazˇant 共2000兲.

Thermal Degradation of Concrete

Fig. 2. Relative dielectric properties of concrete depending on water content 共Hasted and Shah 1964兲

Dielectric Properties of Concrete The properties of dielectric materials generally depend on moisture contents and temperature changes 共Metaxas and Meredith 1983兲. The relative dielectric permittivity ␬ ⬘ ⫽␧ ⬘ /␧ 0 and the relative dielectric loss ␬ ⬙ ⫽␧ ⬙ /␧ 0 of concrete quadratically increase as a function of volume fraction of water v w 共Hasted and Shah 1964; Shah et al. 1965兲 (␧ 0 ⫽dielectric permittivity of air⫽8.86 ⫻10⫺12 F/m). The dielectric constants ␬ ⬘ and ␬ ⬙ change significantly when v w exceeds about 20%. When v w is less than 20%, the changes can be approximated by linear functions 共Fig. 2兲. The concretes that need to be decontaminated are usually old concretes, in which the volume fraction of water is around 7%. Because the water content is a function of pore pressure and temperature, as described by the constitutive laws 关Eq. 共17兲兴, the dielectric properties of concrete depend not only on the water content but also, indirectly, on temperature. Regarding a direct effect of temperature on the dielectric properties, no information is available and probably this effect is negligible.

Application of Microplane Constitutive Model M4 To determine whether a given microwave source will achieve spalling and predict the depth of spalling, a good constitutive model relating the stress and strain in concrete is needed. Li et al. 共1993兲 considered a restrained one-dimensional elastic bar and estimated its stress simply as ␴⫽⫺E␣⌬T. This estimate, however, ignores the nonlinearity of deformation on approach to spalling and the confining effect of the body surrounding a hot spot heated by microwaves. A three-dimensional constitutive model needs to be used. For this purpose, version M4 of the microplane model 共briefly described in Appendix III of Part II兲 has been adopted 共Bazˇant et al. 2000b; Caner and Bazˇant 2000兲. The microplane model is a powerful explicit model that yields the best data fit over broad range of nonlinear triaxial behavior, softening damage and tensile cracking of concrete 共Bazˇant et al. 2000b; Caner and Bazˇant 2000兲. Fracture propagation can be handled with model M4 most easily in the sense of the crack band model. The microplane model differs from the classical tensorial models based on plasticity by the fact that the constitutive law is expressed in terms of vectors rather than tensors. The vectors are the stress and strain vectors on planes of all possible orientations within the material. Thanks to all possible microplane orientations, the model automatically satisfies tensorial invariance con780 / JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003

Besides the triaxial and strain softening behavior, concrete also undergoes thermal degradation and transient creep. Both the compressive strength f c and the Young’s modulus E degrade as the temperature increases. The degradation is typically determined from tests of the residual mechanical properties of concrete exposed for some time 关typically 12 h, 共Felicetti and Gambarova 1998兲兴 to various controlled temperature histories. The surface temperature, as calculated in Part II of this study 共Zi and Bazˇant 2003兲, reaches less than 400°C during the 10 s of heating envisaged for decontamination. If this temperature were sustained for many hours, the compressive strength of concrete would degrade to about 85% of the original value 共Bazˇant and Chern 1987兲. But what if this temperature lasts for only 10 s? The degradation is caused by dehydration of the calcium silicate hydrates in cement paste. Since the chemical reactions of dehydration at high temperature cannot happen instantly and probably take much longer than the desired 10 s duration of the decontamination process, the degradation will be neglected. The temperature increase near 400°C will intensify creep as well as the apparent effect of strain rate on the elastic modulus of concrete. This could be taken into account by introducing on the microplanes a creep law based on the Maxwell or Kelvin chain 共Zi and Bazˇant 2001兲. According to finite element simulations, the maximum strain rate in the heated concrete prior to spalling, which occurs near the surface, is about ␧⫽1⫻10⫺4 s⫺1 . Compared to the typical loading rate in quasi-static laboratory tests, which is about 5⫻10⫺6 s⫺1 , the rate effect would increase the apparent Young’s modulus to approximately 120% 共Bazˇant et al. 2000a兲 of the normal value, but only near the place of maximum strain rate. Elsewhere the rate effect will be significantly less. The rate effect will be neglected in the present finite element computations. It might seem that the rate effect might get offset somewhat by the effect of thermal degradation. However, thermal degradation is a far slower process 共Ulm et al. 1999兲, which is surely negligible within the duration of 10 s.

Hygrothermal Strain The thermal expansion coefficient ␣ T of cement mortar changes significantly with temperature, which is caused by moisture effects. But for concrete, the change of ␣ T with temperature is much less pronounced. This is explained by the restraining effect of aggregates in concrete, which are usually very stable chemically 共Harmathy 1970; Bazˇant and Kaplan 1996; Neville 1997兲. Therefore the thermal expansion coefficient ␣ T will simply be taken as constant (⬇␣ T ⫽10.0⫻10⫺6 K⫺1 ) in the temperature range of the decontamination process. Thus the strain rate ␧˙ T due to thermal dilatation is expressed as ␧˙ T ⫽␣ T T˙

Regarding shrinkage, two kinds must be distinguished: 共1兲 the average shrinkage of the cross section of a concrete member, which is not a constitutive property but a property of the whole cross section, with an inevitably complex mathematical description 共Hansen and Almudaiheem 1987; ACI 1994; Bazˇant and Baweja 1995; Bazˇant and Baweja 2000兲; and 共2兲 the shrinkage at a point of the continuum approximating concrete, which is a constitutive property. Unfortunately, the latter is next to impossible to measure directly 共direct measurements have been made only on cement paste shells 0.75 mm in thickness, which is the maximum thickness needed to ensure that the humidity profile across the wall would remain almost uniform during a programmed linear decrease of environmental relative humidity at the rate 3%/h; Bazˇant and Najjar 1972兲. Therefore the constitutive shrinkage of the material had to be inferred indirectly—by fitting the finite element solutions of test specimens to the measured deformations and adjusting the shrinkage model until a good fit is obtained 共Bazˇant and Chern 1987; Bazˇant and Xi 1994; Bazˇant et al. 1997兲. The conclusion from these studies is very simple: ␧˙ h ⫽␬ s h˙ where ␬ s ⫽shrinkage coefficient 共taken as ␬⫽0.5⫻10⫺3 , according to model B3; Bazˇant and Baweja 2000兲.

Conclusions 1.

2.

3.

4.

The paper presents a mathematical formulation for analyzing a proposed technique of decontamination of concrete walls from radionuclides residing in a thin surface layer, which is to be spalled off by rapid microwave heating. The formulation consists of 共1兲 a model for heat generation in the bulk of concrete by microwave power dissipation; 共2兲 a model for heat and moisture transfer with buildup of pore pressure; 共3兲 a constitutive model for nonlinear triaxial behavior and fracturing of concrete; and 共4兲 numerical solution. The heat and moisture transfer is based on the model of Bazˇant and Thonguthai 共1978兲, which is improved by introducing a more realistic rapid increase of the magnitude of the moisture permeability upon exceeding 100°C. An ohmic heat source term representing the rate of heat generation by microwaves is included in the formulation. A simple analytical expression for the heat generation rate is developed. The heat generation rate caused by normally incident transmission electron microscopy waves is averaged over both the frequency and the wavelength. The microwave power reflection by steel reinforcing bars is taken into account in the resulting formula. The special case for no reinforcement agrees with the Lambert’s law used in food engineering. The heat generation rates are determined separately for two kinds of strips normal to wall surface: one representing an unreinforced concrete and the other a concrete containing reinforcement bars at which the microwave is 100% reflected. The recently developed version M4 of microplane model for concrete is adopted for the analysis of stresses and fracturing. Shrinkage and swelling due to changes in water content are taken into account, but creep is neglected because the duration of the spalling process is very short 共only about 10 s兲.

Appendix: Heat and Moisture Transfer in Concrete Governing Equations The mass conservation equation and heat conservation equation are ⳵w ⫹“"J⫽I 共 w 兲 ⳵t

(15a)

⳵ 共 ␳CT 兲 ⫹“"q⫽I 共 h 兲 ⳵t

(15b)

and

Here “⫽gradient operator; T⫽temperature; ␳⫽mass density of concrete; C⫽specific heat of concrete; w⫽specific water content; C w ⫽specific heat of water; J⫽water flux vector; q⫽conductive heat flux vector; I (h) ⫽distributed source of heat; and I (w) ⫽distributed source of water, due to release of chemically bound water 共Bazˇant and Thonguthai 1978; Bazˇant and Kaplan 1996兲; subscripts 共w兲 and 共h兲 are labels for water and heat. The heat capacity of oven dried concrete can be used for C since the latent heat due to heating induced chemical decomposition of hydrates in cement paste is relatively small, and negligible for concrete 共Harmathy and Allen 1973兲, due to its large volume fraction of aggregates. Although the apparent heat capacity of concrete depends on the water content, the mass fraction of water is usually so small 共less than 6% of total mass of concrete, even at saturation兲 that this dependence may be neglected, for simplicity 共Bazˇant and Thonguthai 1978兲. The water flux J and the heat flux q may be expressed in terms of the gradient of water content w and the gradient of temperature T, respectively a “P g

(16a)

q⫽⫺k“T

(16b)

J⫽⫺ and

where a⫽permeability; g⫽gravity acceleration; P⫽pore pressure; and k⫽heat conductivity. Heat is also transferred by the movement of water inside of concrete, which is described by the convection term qcv⫽C w TJ. In concrete, however, this term is negligible, because the diffusivity to pore water is about 3 orders of magnitude smaller than the heat diffusivity.

Equation of State of Pore Water Except for temperatures above the critical point of water 共374.15°C兲, one must distinguish the vapor from the liquid water in the pores of concrete. These two phases of water can be assumed to be locally always in thermodynamic equilibrium. Bazˇant and Thonguthai’s 共1978, 1979兲 model based on this hypothesis was shown to give acceptable match of the test data. Because of the complexity of the pore system, and especially the role of water adsorbed in the nanopores of hydrated cement paste, the formulation of the sorption isotherms of concrete, i.e., the curves of specific water content w versus relative humidity of water vapor in the pores, h⫽ P/ P s (T) 关where P s (T)⫽saturation pore pressure at temperature T兴, must be semiempirical. The isotherms are described as

冉 冊

w w1 ⫽ h c c

1/m 共 T 兲

for h⭐0.96

(17a)

JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003 / 781

Fig. 3. Change of permeability by change of humidity and temperature

and w⫽ 共 1⫹3␧ V 兲 n/␯ for h⭓1.04

(17b)

where T⫽temperature in °C; T 0 ⫽25°C; c⫽mass of anhydrous cement per unit volume of concrete; w 1 ⫽saturation water content at reference temperature T 0 共⫽25°C兲; m(T)⫽1.04⫺ 关 T ⬘ /(22.34 ⫹T ⬘ ) 兴 ; T ⬘ ⫽(T⫹10) 2 /(T 0 ⫹10) 2 ; d␧ V ⫽d␴ V /(3K)⫹␣ T dT; ␴ V ⫽n P; ␧ V ⫽linear volumetric strain; K⫽bulk modulus of concrete; n⫽porosity accessible to water; ␯⫽␯(T, P)⫽specific volume of water; and ␣ T ⫽coefficient of linear thermal dilatation of concrete. The porosity 共pore volume accessible to water兲, which was estimated from considerations of weight loss, is expressed as n⫽ 关 n 0 ⫹␳ ⫺1 0 w d (T) 兴 ␺(h) for h⭓1.04, where n 0 ⫽reference porosity at 25°C; w d (T)⫽weight loss 关obtained from thermogravimetic measurements 共Harmathy and Allen 1973兲兴; ␳ 0 ⫽specific weight of water; and ␺(h)⫽1⫹0.12(h⫺1.04).

Permeability and Conductivity The permeability of concrete is a complex property. Because the capillaries in good quality concretes are not continuous, water molecules must pass through the nanopores in the hardened cement paste. Because the width of such pores 共from 0.5 nm up兲 is much smaller than the mean free path of vapor molecules 共about 80 nm at 25°C兲, water molecules cannot pass through the nanopores in a vapor state but must become adsorbed on the pore walls and migrate along adsorption layers. So the nanopores control the permeability, which explains the extremely low values of the permeability of concrete at normal temperatures. But this is not the case at high temperatures. When the temperature is increased above 100°C, the permeability jumps sharply up 共Bazˇant and Thonguthai 1978兲. This can be explained by heat-induced changes in the structure of the smallest pores, particularly elimination of the narrowest necks, of nanometer dimensions, on the passages through cement paste. Based on data fitting, the permeability was inferred to jump up about 200⫻, but the present optimum fits of test data shown later indicates that the permeability jumps up, around 100°C, only about 6.5⫻ 共Fig. 3兲. The initial trend of function f 3 (T), which represents the permeability increase upon exceeding 100°C, is the same as proposed in Bazˇant and Thonguthai’s 共1978兲 work, but the magnitude of the jump needs to be scaled down, as represented by the following function: a⫽a 0 f 1 共 h 兲 f 2 共 T 兲 for T⭐100°C

(18a)

and 782 / JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003

Fig. 4. Fits of 共a兲 Bazˇant and Thonguthai’s experiments 共1978兲; 共b兲 England and Ross’ experiments 共1970兲; and 共c兲 by model proposed; symbols represent experimental data and solid lines represent fitting

a⫽a 0 f 2 共 100兲 f 3 共 T 兲 for T⬎100

(18b)

where a 0 ⫽reference permeability at 25°C. Function f 1 (h) characterizes the permeability corresponding to moisture transfer along the adsorbed water layers. The temperature dependence of permeability below 100°C is given by an Arrhenius-type equation f 2 (h); f 1 共 h 兲 ⫽␣⫹

1⫺␣ 1⫺h 1⫹ 1⫺h c

冉 冊

4

, for h⭐1; f 1 共 h 兲 ⫽1, for h⭓1 (19)

冋 冉 冊册

f 2 共 T 兲 ⫽exp

Q

1

R ¯T 0

⫺

1

(20)

¯T

where ␣⫽1/关 1⫹0.253(100⫺min(T,100°C)) 兴 ; h c ⫽0.75, ¯T ⫽absolute temperature; Q⫽activation energy for water migration; and R⫽gas constant. Based on data fitting, the value Q/R ⫽2, 700 K was recommended 共Bazˇant and Najjar 1972兲. Function f 3 (T), which describes the abrupt increase of permeability near 100°C, is revised as f 3 共 T 兲 ⫽5.5

再

冎

2 ⫺1 ⫹1 1⫹exp关 ⫺0.455共 T⫺100兲兴

(21)

which is found by fitting the data used in Bazˇant and Thonguthai 共1978兲, England and Ross 共1970兲, and Zhukov and Schenchenko 共1974兲 共Fig. 4兲. The thermal conductivity of cement, too, depends on the changes of temperature and moisture content significantly. However, the thermal conductivity k is for concrete is much less sensitive to the changes of temperature and moisture than it is for the

hardenened cement paste and concrete. The reason for this difference is that the mineral aggregates, which represent most of the volume of concrete and are usually chemically stable materials 共Harmathy 1970兲, conduct heat no less than the cement paste but do not transfer moisture significantly. In general, the thermal conductivity depends on the volume fraction of aggregate and its type 共Bazˇant and Kaplan 1996; Neville 1997兲.

for brick is assumed for concrete. In the absence of vacuum, both the surface heat transfer, Eq. 共22兲, and the radiation, Eq. 共23兲, take place simultaneously. They may be conveniently characterized as

Distributed Sources of Water and Heat

Why Not Liquid – Gas Transport Model?

When concrete is heated, the chemically bound water becomes free and gets released into the pores. This is reflected in the source term I (w) of the mass conservation condition in Eq. 共15兲. The amount of dehydrated water is obtained experimentally, by 105 weight loss measurements; w d (T)⫽w 105 h f d (T), where w h ⫽the hydrate water content at 105°C. The values of f d (T) are interpolated using the experimental data by Harmathy and Allen 共1973兲. At temperatures below 100°C, the phenomenon must be reversed because of further hydration of cement. The increase of the hydrate water content w h below 100°C may be described as w h (t e )⬇0.21c 关 t e /(␶ e ⫹t e ) 兴 1/3, where t e ⫽equivalent hydration period and ␶ e ⫽23 days 共see Bazˇant and Kaplan 1996兲. Then the distributed water source is I (w) ⫽w˙ d ⫺w˙ h . The distributed heat source term in Eq. 共15b兲 is absent when concrete is heated by conduction from the surface. But when concrete is heated by microwaves, the heat source, Eqs. 共13兲 and 共14兲, generated by ohmic heat dissipation within the concrete volume, is significant.

One might wonder why the recent model of Mainguy et al. 共2001兲, or some other multiphase transport model, has not been adopted. In that model, all the mobile water is assumed to consist of liquid 共capillary兲 water and water vapor contained in the gas 共air兲, the pressure gradient of which is regarded as one driving force of transport. From this starting hypothesis, it is deduced that the water transport is controlled by the flow of liquid 共capillary兲 water and involves evaporation and condensation at gas-liquid interfaces. The starting hypothesis, however, ignores the fact that the capillaries in normal hardened cement paste are not continuous 共unless the water-cement ratio were abnormally high兲. While diffusing, the water molecules must pass through nanopores in calcium silicate hydrates only about 1–3 nm wide, which cannot contain liquid water and thus cannot allow its passage. Moreover, since the mean free path of water molecules in a vapor phase is about 80 nm, there is no chance of vapor molecule passage through the tortuous nanopores 共they would bounce of pore walls far more often than of each other, and have about equal chance to be reflected forward or backward兲. Therefore, water molecules can move through such passages only along adsorption layers which fill these pores. Such water transport is controlled not by viscosity of liquid water, but by the lingering times of the adsorbed water molecules on the surface of calcium silicate hydrates 共Bazˇant 1972, 1975兲. For these reasons, the theory of Mainguy et al. 共2001兲 and numerous other multiphase transport theories are not applicable 共the separation of transport of air in Mainguy et al.’s model might be relevant, although this has not been worked out兲.

Boundary Conditions Heat and mass are transferred at the surface to the surrounding environment. Physically accurate modeling of the environment near the surface, which would call for nonlinear hydrodynamics, is not necessary. For heat transfer, it suffices to use Newton’s law of cooling 共e.g., Chapman 1987兲 and a similar law for moisture transfer. Thus the boundary conditions simply are n"J⫽B w 共 P S ⫺ P am兲 ,

(22a)

n"q⫽B T 共 T S ⫺T am兲

(22b)

and

(24)

where B eq⫽equivalent heat transfer coefficient⫽B T ⫹␥␴(T 2S ⫹T 2am)(T S ⫹T am).

References

where B w ⫽moisture transfer coefficient and B T ⫽heat transfer coefficient; n⫽unit outward normal of the boundary surface; P am⫽ambient partial pressure of water vapor; T am⫽ambient temperature; P S ⫽partial vapor pressure at the surface 共i.e., in the capillary pores adjacent to the surface兲; and T S ⫽surface temperature. A perfectly sealed 共or insulated兲 surface is a limiting case for B w ⫽0 共or B T ⫽0), and perfect moisture transmission 共or heat transmission兲 is a limiting case for B w →⬁ 共or B T →⬁). In the case of free convection of air near the surface, B T is in the range of 5–25 J/m2 s°C 共e.g., Chapman 1987兲 and B w ⬇⬁ 共Bazˇant and Thonguthai 1978兲. The heat radiation from the surface, which is the only mechanism of heat loss in a vacuum, is described by Stefan’s radiation law 4 n"qr ⫽␥␴ 共 T 4S ⫺T am 兲

n"q⫽B eq共 T S ⫺T am兲

(23)

where qr ⫽radiation heat flux; ␴⫽Stefan coefficient⫽5.67 ⫻10⫺8 J/m2 s K4 ; and ␥⫽heat emissivity, which varies in the range of 0–1. For a perfectly black surface, ␥⫽1, while for brick, ␥⫽0.9 共Jones 2000兲; because of lack of data, and also because the precise value is not very important, the same emissivity ␥ as

Ahmed, G. N., and Hurst, J. P. 共1997兲. ‘‘Coupled heat and mass transport phenomena in siliceous aggregate concrete slabs subjected to fire.’’ Fire Mater., 21, 161–168. American Concrete Institute. 共ACI兲. 共1994兲. ‘‘Prediction of creep, shrinkage, and temperature effects in concrete structures.’’ ACI manual of concrete pratice Part 1: Materials and general properties of concrete, ACI 209R-92, Detroit. Bazˇant, Z. P. 共1972兲. ‘‘Thermodynamics of interacting continuua with surfaces and creep analysis of concrete structures.’’ Nucl. Eng. Des., 20, 477–505. Bazˇant, Z. P. 共1975兲. ‘‘Theory of creep and shrinkage in concrete structures: A precis of recent developments.’’ Mechanics today, S. NematNasser, ed., Vol. 2, Pergamon, New York, 1–93. Bazˇant, Z. P., and Baweja, S. 共1995兲. ‘‘Justification and refinement of Model B3 for concrete creep and shrinkage. 1. Statistics and sensitivity.’’ Materials and structures, Vol. 28, RILEM, Paris, 415– 430. Bazˇant, Z. P., and Baweja, S. 共2000兲. ‘‘Creep and shrinkage prediction model for analysis and design of concrete structures: Model B3— short form.’’ Adam Neville Symposium: Creep and shrinkage— structural design effects, ACI SP-194, A. Al-Manaseer, ed., American Concrete Institute, Farmington Hills, Mich., 85–100. Bazˇant, Z. P., Caner, F. C., Adley, M. D., and Akers, S. A. 共2000a兲. ‘‘Fracture rate effect and creep in microplane model for dynamics.’’ J. Eng. Mech., 126共9兲, 962–970. JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003 / 783

Bazˇant, Z. P., Caner, F. C., Carol, I., Adley, M. D., and Akers, S. A. 共2000b兲. ‘‘Microplane model M4 for concrete I: Formulation with work-conjugate deviatoric stress.’’ J. Eng. Mech., 126共9兲, 944 –953. Bazˇant, Z. P., and Chern, J. 共1987兲. ‘‘Stress-induced thermal and shrinkage strains in concrete.’’ J. Eng. Mech., 113共10兲, 1493–1511. Bazˇant, Z. P., Hauggaard, A. B., Baweja, S., and Ulm, F. 共1997兲. ‘‘Microprestress-solidification theory for concrete creep. I: Aging and drying effects.’’ J. Eng. Mech., 123共11兲, 1188 –1194. Bazˇant, Z. P., and Kaplan, M. F. 共1996兲. Concrete at high temperatures, Longman, London. Bazˇant, Z. P., and Najjar, L. J. 共1972兲. ‘‘Nonlinear water diffusion in nonsaturated concrete.’’ Materials and Structures, Vol. 5, RILEM, Paris, 3–20. Bazˇant, Z. P., and Thonguthai, W. 共1978兲. ‘‘Pore pressure and drying of concrete at high temperature.’’ J. Eng. Mech. Div., Am. Soc. Civ. Eng., 104共5兲, 1059–1079. Bazˇant, Z. P., and Thonguthai, W. 共1979兲. ‘‘Pore pressure in heated concrete walls: Theoretical prediction.’’ Mag. Concrete. Res., 31共107兲, 67–76. Bazˇant, Z. P., and Xi, Y. 共1994兲. ‘‘Drying creep of concrete: constitutive model and new experiments separating its mechanisms.’’ Mater. Struct., 27, 3–14. Bazˇant, Z. P., and Zi, G. 共2001兲. ‘‘Spatial and temporal scaling of concrete response to extreme environments.’’ Proc., 3rd Int. Conf., Concrete Under Severe Conditions, N. Banthia, K. Sakai, and O. E. Gjørv, eds., Univ. of British Columbia, Vancouver, BC, 3–10. Brocca, M., and Bazˇant, Z. P. 共2000兲. ‘‘Microplane model and metal plasticity.’’ Appl. Mech. Rev., 53共10兲, 265–281. Caner, F. C., and Bazˇant, Z. P. 共2000兲. ‘‘Microplane model M4 for concrete II: Algorithm and calibration.’’ J. Eng. Mech., 126共9兲, 954 –961. Chapman, A. J. 共1987兲. Fundamentals of heat transfer, Macmillan, New York. Cheng, D. K. 共1983兲. Field and wave electromagnetics, Addison–Wesley, London. England, G. L., and Ross, A. D. 共1970兲. ‘‘Shrinkage, moisture and pore pressure in heated concrete.’’ Proc., American Concrete Institute Int. Seminar on Concrete for Nuclear Reactors, West Berlin, Germany, Special Publication No. 34, 883–907. Felicetti, R., and Gambarova, G. 共1998兲. ‘‘Effects of high temperature on the residual compressive strength of high siliceous concretes.’’ ACI Mater. J., 95共4兲, 395– 406. Gawin, D., Majorana, C. E., and Schrefler, B. A. 共1999兲. ‘‘Numerical analysis of hygrothermal behaviour and damage of concrete at high temperature.’’ Mech. Cohesive-Frict. Mater., 4, 37–74. Hansen, W., and Almudaiheem, J. A. 共1987兲. ‘‘Ultimate drying shrinkage of concrete-Influence of major parameters.’’ ACI Mater. J., 84共3兲, 217–223. Harmathy, T. Z. 共1970兲. ‘‘Thermal properties of concrete at elevated temperature.’’ J. Mater., 5共1兲, 47–75. Harmathy, T. Z., and Allen, L. W. 共1973兲. ‘‘Thermal properties of selected masonry unit concrete.’’ ACI J., 70共15兲, 132–142. Hasted, J. B., and Shah, M. A. 共1964兲. ‘‘Microwave absorption by water in building materials.’’ Br. J. Appl. Phys., 15, 825– 836. Hertz, K. 共1981兲. ‘‘Microwave heating for fire material testing of concrete—A theoretical study,’’ Institute of Building Design Rep. No. 144, Technical Univ. of Denmark, Lyngby, Denmark. Hertz, K. 共1983兲. ‘‘Microwave heating for fire material testing of concrete—an experimental study.’’ Institute of Building Design, Rep. No. 164, Technical Univ. of Denmark, Denmark. Jones, H. R. N. 共2000兲. Radiation heat transfer, Oxford Univeristy Press, Oxford. Lagos, L. E., Li, W., and Ebadian, M. A. 共1995兲. ‘‘Heat transfer within a concrete slab with a finite microwave heating source.’’ Int. J. Heat Mass Transf., 38共5兲, 887– 897.

784 / JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003

Li, W., Ebadian, M. A., White, T. L., and Grubb, R. G. 共1993兲. ‘‘Heat transfer within a concrete slab applying the microwave decontamination process.’’ J. Heat Transfer, 115, 42–50. Mainguy, M., Coussy, O., and Baroghel-Bouny, V. 共2001兲. ‘‘Role of air pressure in drying of weakly permeable materials.’’ J. Eng. Mech., 127共6兲, 582–592. Meredith, R. J. 共1998兲. Engineer’s handbook of industrial microwave heating, The Institution of Electrical Engineers, London. Metaxas, R. C., and Meredith, R. J. 共1983兲. Industrial microwave heating, IEEE Power Engineering Series 4, Peter Peregrinus Ltd., Exeter, England. Moukwa, M., Brodwin, M., Christo, S., Chang, J., and Shah, S. P. 共1991兲. ‘‘The influence of the hydration process upon microwave properties of cements.’’ Cem. Concr. Res., 21, 863– 872. Neville, A. M. 共1997兲. Properties of concrete, 4th Ed, Wiley, New York. Shah, M. A., Hasted, J. B., and Moore, L. 共1965兲. ‘‘Microwave absorption by water in building materials: Aerated concrete.’’ Br. J. Appl. Phys., 16, 1747–1754. Spalding, B. 共2000兲. ‘‘Volatility and extractability of Strontium-85, Cesium-134, Cobalt-57, and Uranium after heating hardened portland cement paste.’’ Environ. Sci. Technol., 34, 5051–5058. Taoukis, P., Davis, E. A., Davis, H. T., Gordon, J., and Talmon, Y. 共1987兲. ‘‘Mathematical modeling of microwave thawing by the modified isotherm migration method.’’ J. Food. Sci., 52共2兲, 455– 463. Thue´ry, J. 共1992兲. Microwaves: industrial, scientific, and medical applications, Artech House, Boston, 104. Ulm, F.-J., Coussy, O., and Bazˇant, Z. P. 共1999兲. ‘‘The ‘‘chunnel’’ fire I: chemoplastic softening in rapidly heated concrete.’’ J. Eng. Mech., 125共3兲, 272–282. Voda´k, F., Cˇerny´, R., Drchalova´, J., Hosˇkova´, Sˇ., Kapicˇkova´, O., Michalko, O., Semera´k, P., and Toman, J. 共1997兲. ‘‘Thermophysical properties of concrete for nuclear-safety related structures.’’ Cem. Concr. Res., 27共3兲, 415– 426. von Hippel, A. R. 共1954兲. Dielectric materials and applications, MIT Press, Cambridge, Mass. Wace, P. F., Harker, A. H., and Hills, D. L. 共1989兲. ‘‘Removal of concrete layers from biological shield by microwaves.’’ Rep. No. EUR 12185, Nuclear Science and Technology, Commission of the European Communities, Brussels, Belgium. Wait, J. R. 共1985兲. Electromagnetic wave theory, Harper & Row, New York. Watson, A. 共1968a兲. ‘‘Curing of concrete.’’ Microwave power engineering, E. C. Okress, ed., Vol. 2, Academic, New York. Watson, A. 共1968b兲. ‘‘Breaking of concrete.’’ Microwave power engineering, E. C. Okress, ed., Vol. 2, Acdemic, New York. Wei, C. K., Davis, H. T., Davis, E. A., and Gordon, J. 共1985兲. ‘‘Heat and mass transfer in water-laden sandstone: microwave heating.’’ AIChE J., 31共5兲, 842– 848. White, T. L., Foster, D., Jr., Wilson, C. T., and Schaich, C. R. 共1995兲. ‘‘Phase II microwave concrete decontamination results.’’ ORNL Rep. No. DE-AC05-84OR21400, Oak Ridge National Laboratory, Oak Ridge, Tenn. Zhukov, V. V., and Schenchenko, V. I. 共1974兲. ‘‘Investigation of causes of possible spalling and failure of heat-resistant concretes at drying, first heating and cooling.’’ Zharostoikie betony (Heat-resistant concretes), K. D. Nekrasov, ed., Stroizdat, Moscow, 32– 45. Zi, G., and Bazˇant, Z. P. 共2001兲. ‘‘Continuous relaxation spectrum for concrete creep and its incorporation into microplane model M4.’’ J. Eng. Mech., 128共12兲, 1331–1336. Zi, G., and Bazˇant, Z. P. 共2003兲. ‘‘Decontamination of radionuclides from concrete by microwave heating. II: Computations.’’ J. Eng. Mech., 129共7兲, 785–792.

Decontamination of Radionuclides from Concrete by Microwave Heating. II: Computations Goangseup Zi1 and Zdeneˇk P. Bazˇant, F.ASCE2 Abstract: Based on a mathematical model developed in the preceding Part I of this study, a numerical analysis of the process of decontamination of radionuclides from concrete by microwave heating is conducted. The aim is to determine the required microwave power and predict whether and when the contaminated surface layer of concrete spalls off. As customary, the finite element method is used for the stress and fracture analysis. However, as a departure from previous studies, the finite volume method is adopted to treat the heat and moisture transfer, in order to prevent spurious numerical oscillations that plagued the finite element response at moving sharp interface between the saturated and unsaturated concrete, and to deal accurately with the jumps in permeability and in sorption isotherm slope across the interface. The effects of wall thickness, reinforcing bars, microwave frequency, and power are studied numerically. As a byproduct of this analysis, the mechanism of spalling of rapidly heated concrete is clarified. DOI: 10.1061/共ASCE兲0733-9399共2003兲129:7共785兲 CE Database subject headings: Computation; Concrete; Microwaves; Contamination; Contaminants; Heating; Diffusion; Thermal stresses; Pore pressure.

Introduction Based on the model developed in the preceding Part I of this study 共Bazˇant and Zi 2003兲, we will now conduct numerical analysis of the evolution of the fields of temperature, pore pressure, and stress during rapid microwave heating and analyze the implications for the process of radionuclide decontamination of concrete. For stress and deformation analysis, we will employ the finite element method 关Fig. 1共b兲兴. On the other hand, for the heat and moisture transfer, we will adopt the finite volume method 关Fig. 1共a兲兴, which will be a departure from the previous practice in nuclear reactor safety research and fire research 共Bazˇant and Thonguthai 1978, 1979; Bazˇant and Kaplan 1996; Ahmed and Hurst 1997兲. Thanks to enforcing exactly the local mass and heat conservation even though the water and heat fluxes are only approximate, this method 共Patankar 1980兲 can suppress the spurious numerical oscillations that have previously been experienced with finite elements in rapid heating problems characterized by a sharp moving interface between saturated and unsaturated regions, associated with jumps in permeability and sorption isotherm slope across the interface 共Bazˇant and Thonguthai 1978, 1979; Celia et al. 1990兲. For discretization in time, we will employ Picard’s method, in which the differential equation coefficients are kept constant during each iteration process, in order to cope with the 1 Research Associate, Dept. of Civil Engineering, Northwestern Univ., 2145 Sheridan Rd., Evanston, IL 60208. E-mail: [email protected] 2 McCormick School Professor and W. P. Murphy Professor of Civil Engineering and Materials Science, Northwestern Univ., 2145 Sheridan Rd., Evanston, IL 60208. E-mail: [email protected] Note. Associate Editor: Franz-Josef Ulm. Discussion open until December 1, 2003. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on March 12, 2002; approved on October 29, 2002. This paper is part of the Journal of Engineering Mechanics, Vol. 129, No. 7, July 1, 2003. ©ASCE, ISSN 0733-9399/2003/7-785–792/$18.00.

severe nonlinearity of the model. The analysis by finite volume method will be simplified as axisymmetric, which is doubtless an adequate approximation. The material subroutine for the nonlinear triaxial stress–strain relation with strain-softening will be based on microplane model M4 recently developed at Northwestern University. The strains will be assumed to be small. To avoid using too small time steps because of numerical stability problems, we will use an implicit rather then explicit finite element formulation. On the other hand, since the calculation of the tangential stiffness matrix of the microplane model M4 is quite computationally demanding, the initial stiffness matrix will be used in an iterative solution of the system of nonlinear equilibrium equations.

Effects of Power Pattern The power of the electromagnetic field produced by a point source placed in a free space 共having no dielectric loss兲 decays in space in proportion to r ⫺2 , where r⫽distance between the source and a point where the power is measured. This decay is reflected in the Poynting vector 共e.g., Cheng 1983兲. The power decay in space is affected by the composition of the medium, the type of antenna and other factors. The effects of various power patterns must be solved from the Maxwell equations 关see Eqs. 共2a兲 and 共2b兲 in Part I兴. Both the applicator 共antenna兲 共Fig. 1 in Part I兲 and concrete need, in general, to be included in the analysis of electromagnetic wave propagation. The power may be approximately considered to flow within a cone 共called the solid angle兲 of a certain constant slope k 共Kraus 1988兲. The effective power flux reduction due to the spreading of the cross section of this cone with distance s from the applicator may be approximated by the ratio f 共 x 兲 ⫽y 20 / 共 y 0 ⫹kx 兲 2

(1)

where y 0 ⫽equivalent radius of the applicator at x⫽0. Taking y 0 ⫽6 cm and k⬇0.5, one finds the power reduction due to cross JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003 / 785

Fig. 1. 共a兲 Finite volume mesh for analysis of heat and moisture transfer and 共b兲 finite element mesh for analysis of mechanical deformation in which the size of one element of heated zone is set to 5 mm

section spreading to be about f (x)⫽90% at x⫽2 cm. But at that distance the heat generation rate for f ⫽18.0 GHz is reduced to almost zero 共Fig. 2兲. Therefore, the spreading of the effective cross section of the power flux can be neglected in practice.

Applicator Efficiency A more important aspect of the decontamination process is the efficiency of the applicator. Obviously, the maximum efficiency is obtained for surface reflection coefficient R 0 ⫽0 关Eq. 共5兲 in Part I兴, which means that 100% of the power input gets transmitted from the applicator into the concrete. In that case, the intrinsic impedance of the applicator, ␩ a , is equal to that at concrete surface, ␩ 1 . Introducing an empirical efficiency parameter ␾ 苸 关 0,1兴 , one may write ␩ a ⫽␾␩ 1 ⫹ 共 1⫺␾ 兲 ␩ 0

Fig. 3. Effect of reinforcing bars in 10 cm wall where 19% of reinforcing bars are located in 2.5 cm from heating surface: 共a兲 without reinforcing bars 共circles兲 and 共b兲 with reinforcing bars 共solid dots兲

three times higher than that without any optimization 关i.e., ␩ a ⫽␩ 0 ; Fig. 2共a兲兴. It thus becomes clear that the efficiency ␾ of the applicator is a very important parameter, overwhelming others. Since the efficiency has not been measured experimentally, ␾ ⫽50% is assumed for the studies that follow.

(2)

where ␩ 0 ⫽intrinsic impedance of air. Here it is assumed that the intrinsic impedance of the applicator is in the range of 关 ␩ 0 ,␩ 1 兴 . The heat generation rates with the maximum efficiency 共i.e., ␩ a ⫽␩ 1 ) are plotted in Fig. 2共b兲 for different frequencies. For frequency f ⫽18.0 GHz, the surface heat generation rate is almost

Validity of Spatial Averaging of Heat Generation The simple heat generation in Eq. 共12兲 in Part I is obtained by means of the temporal and spatial averaging of the ohmic heat dissipation of Eq. 共1兲 in Part I. The heat generation by temporal and spatial averaging 关Eq. 共12兲 in Part I兴 is compared to the heat generation by temporal averaging 关Eq. 共11兲 in Part I兴 in Fig. 2共a兲 for two frequencies: f ⫽2.45 and 10.6 GHz. The former is a smooth exponential function while the latter oscillates along the radiation direction. However, the difference is negligible for f ⫽2.45 and visually indistinguishable for f ⫽10.6 GHz. This shows that the averaging over both space and time is justified for the practical frequency range.

Effect of Reinforcing Bars on Pore Pressures and Temperatures

Fig. 2. Examples of volumetric heat generations calculated by Eq. 共12兲 in Part I; 共a兲 with zero applicator efficiency, in which heat generations for f ⫽2.45,10.6 GHz are compared to heat generations averaged only in time, Eq. 共11兲 in Part I; 共b兲 with maximum efficiency; here, reinforcing bars are located 2.5 cm under surface to which transmission electron microscopy waves are incident 786 / JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003

As mentioned in Part I 共Bazˇant and Zi 2003兲, the power carried by microwaves gets reflected from conductive materials, such as the steel reinforcing bars. To investigate this effect, the temperature and pore pressure profiles after 10 s of heating that are obtained in the absence of steel bars 关Fig. 3共a兲兴 are compared to the profiles with reinforcing bars 关Fig. 3共b兲兴 for three different microwave frequencies, f ⫽2.45, 10.6, and 18.0 GHz, after 10 s of heating. The area fraction of the steel bars 共in a projection on the

Fig. 5. Contour plot of development of pore pressure 共left; MPa兲 and temperature 共right; °C兲 after 10 heating with f ⫽18.0 GHz and P 0 ⫽1.1 MW/m2

Fig. 4. Effect of different frequencies on pore pressure and temperature profiles; solid dots represent the case with reinforcing bars and circles represent the case without reinforcing bars

The effect of pore pressure is in fact even weaker since the foregoing estimate is the maximum possible pore pressure if the additional pore space created by the formation of microcracks has been neglected. Taking it into account, an even smaller tensile volumetric stress in an unrestrained element of concrete would be indicated. So, although the effect of pore pressure is not completely negligible, it cannot be the main cause of spalling. Fig. 5 shows the contour plot of pore pressure and temperature that develops after 10 s of heating at frequency f ⫽18.0 GHz and power input P 0 ⫽1.1 MW/m2 . As we can see, the heated zone is localized very near the heated surface.

Stress Fields and Triggering of Spalling surface of the wall兲 is considered to be about 19%. The location of the bars is marked by the dashed lines in the figure. The center of the reinforcing bars in concrete structures is located typically at 2.5– 4 cm below the surface. At that depth, the electromagnetic power is almost exhausted by the power dissipation 共Fig. 2兲. Therefore, it appears that the existence of steel bars is not important for the decontamination process. However, note that this argument is not true in general. It holds true only for the high-power decontamination process and typical reinforced concrete structures. If much lower frequencies were used or if a conductive material, the steel, were located closer to the concrete surface, this effect could get important.

Effect of Microwave Frequencies on Pore Pressures and Temperatures The effects of different frequencies, f ⫽2.45, 10.6, and 18.0 GHz, are plotted in Fig. 4. The same initial power density, P 0 ⫽1.1 MW/m2 , is considered for every case. As the frequency increases, the location of the peaks of pore pressure and temperature shift toward the heated surface. The reason is that the energy dissipation rate is higher at a higher frequency. The maximum pore pressure for f ⫽18.0 GHz after 10 seconds of heating is P max,10⫽2.0 MPa at 7.5 mm below the surface. If this pore pressure acted on an unrestrained element of concrete, it would produce in concrete the tensile volumetric 共hydrostatic兲 stress ␴ V ⬇0.1⫻2.0⫽0.2 MPa, where the value 0.1 is adopted for the typical porosity of concrete. Compared to the tensile strength of ordinary concrete, f t⬘ ⬇4 MPa, this value of tensile volumetric stress is only about 5% of f t⬘ .

Because the distributed cracking described by the microplane model represents strain softening, the stress and deformation analysis must employ some localization limiter 共Bazˇant 2002兲, which could have the form of either the crack band model or some of the nonlocal models 共Bazˇant and Ozˇbolt 1990, 1992; Ozˇbolt and Bazˇant 1996; Jira´sek and Bazˇant 2002兲. The former has been adopted for the present purpose. Thus the finite element size l c is fixed as a material property. The proper value of l c depends on the postpeak softening slope of the uniaxial tensile response given by the constitutive model. The constitutive model gives a good match of material test data when the finite element in the test specimens size is about 5 mm. The same finite element size l c is, therefore, adopted here, although this is less than the typical maximum aggregate size in concrete. The thickness of the layer that spalls off within about 10 seconds as a result of microwave heating is also about 5 mm, as revealed by experimental trials of microwave induced spalling 共White et al. 1995兲. Unfortunately, the data from these trials cannot be used to verify the present theoretical predictions because the material properties were not documented 共not even the compressive strength f c⬘ of the concrete in the wall subjected to microwave heating was measured兲. Only the average depth of the removed surface layer was recorded 共White et al. 1995兲. Therefore, the typical properties of concrete used in the nuclear facilities had to be assumed for the present numerical simulations. Fig. 6 shows the available experimental data 共Jansen and Shah 1997兲 and their numerical fitting using the present theoretical model. Fig. 7 depicts the constant-value contour plots of the computed strain field after 10 s of microwave heating; it shows the mechanical strain, i.e., the total 共elastic plus viscoelastic兲 strain JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003 / 787

Fig. 6. Uniaxial compression data of 10.17 cm⫻20.03 cm cylinder tested by Jansen and Shah 共1997兲 共circles兲 and its optimal fitting 共solid line兲 by finite element simulation in which axisymmetric quadrilateral elements of size 5 mm by 5 mm are used; E⫽48.5 GPa, k 1 ⫽0.000125, k 2 ⫽160.0, k 3 ⫽6.4, and k 4 ⫽150.0

minus the hygrothermal strain 共strain produced by changes of temperature and water content兲. It is found that the maximum principal mechanical strain in the surface layer exceeds 0.005 in tension and the strain state is essentially biaxial 共Fig. 8兲. This strain value is much higher than the typical strain at peak in uniaxial tension 共about 0.0002兲. It follows that the concrete must undergo postpeak softening and suffer disintegration by cracking. The compressive stress induced by the temperature increase is resisted not only by radial and circumferential biaxial compression generated by the resistance of the cold concrete mass surrounding the heated zone, but also by tensile stress in the circumferential direction of the axisymmetric mesh caused by radial expansion 共a situation similar to that analyzed by Ulm et al., 1999兲.

Is Spalling Triggered by Pore Pressure or Compressive Thermal Stress? The question of the main cause of spalling of the surface layer of a rapidly heated concrete wall has been the subject of many debates. One school of thought, initiated by Harmathy 共1965, 1970兲, Harmathy and Allen 共1973兲, Li et al. 共1993兲, and Lagos et al. 共1995兲, holds that the pore water cannot escape fast enough 共a phenomenon called ‘‘moisture clog’’兲 and thus develops high vapor pressures which cannot be resisted by the tensile strength of

Fig. 8. 共a兲 Distribution of radial stress with respect to height and 共b兲 evolution of radial stress at center of heated zone by time

concrete. Another school of thought 共Bazˇant 1997兲 is that the thermal expansion of the saturated heated zone, resisted by the cold concrete mass that surrounds the heating zone, leads in the surface layer to very high compressive stresses parallel to surface which either crush concrete, or cause the compressed surface layer to buckle, or both. The relative significance of these two mechanisms must of course depend on the type of problem, and can be different for microwave heating in the bulk of concrete and for conductive heating by fire. In the present problem, the highest pore pressure calculated has the value 2.0 MPa, which causes in concrete a hydrostatic tension of about 0.2 MPa. This value is not enough to initiate spalling concrete. Besides, as soon as cracks start to form, the volume available to pore water rapidly increases 共by orders of magnitude兲, which must cause a rapid drop of pore pressure before the cracks can become large and open widely 共Bazˇant 1997兲. So it appears that the pore pressure development cannot be the main cause of spalling, although it is not a negligible factor in the triggering of spalling. The main cause must be the compressive stresses along radial lines emanating from the center of the heated zone. These stresses, engendered by the resistance of cold concrete to the thermal expansion of the heated zone, reach values as high as about 50 MPa 关Fig. 8共b兲兴, according to the present analysis. This is certainly enough to cause compressive crushing as well as buckling of the compressed layer.

Application of Finite Volume Method to Heat and Water Transfer in Concrete Since the finite volume method 共Eymard et al. 1998, 2000兲 has not been used for the coupled heat and water transfer problems of concrete, its application will now be described. In this method, the domain is divided into discrete control volumes 共Fig. 9兲. The interfaces 共or boundaries兲 of a control volume are placed midway between adjacent representative points 共which is generally accomplished by Voronoi tessellation, although that approach is not needed for the regular node arrangement used here兲. The discretization equations are derived by integrating Eqs. 共19兲 and 共20兲 of Part I over the control volume shown in Fig. 9共a兲, and over the time interval from t to t⫹⌬t Fig. 7. Contour plot of mechanical strain ␧ 1 after 10 s heating, in which deformation is exaggerated by 100 788 / JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003

⫺w i0 兲 ⫹c x 共 J e ⫺J w 兲 i⫹1,m⫹1 ⫹c y 共 J n ⫺J s 兲 i⫹1,m⫹1 共 w i⫹1,m⫹1 0 ⫽I 共i⫹1,m⫹1 w兲

(3)

Fig. 9. 共a兲 Two dimensional discretization of finite volume where W,E,S, N⫽labels for West, East, South, and North; shaded rectangle represents control volume and 共b兲 difference in pressure 共or temperature兲 profiles implied by finite element and finite volume methods

⫺T i0 兲兴 ⫹c x 共 q e ⫺q w 兲 i⫹1,m⫹1 关共 ␳C 兲 0 共 T i⫹1,m⫹1 0 ⫹c y 共 q n ⫺q s 兲 i⫹1,m⫹1 ⫽I 共i⫹1,m⫹1 h兲

(4)

where i⫽label for time step (i⫽1,2, . . . ); m⫽label for iteration number in an implicit scheme (m⫽1,2, . . . ); 0⫽label for current control volume 关the center point in Fig. 9共a兲兴; s,e,n, w⫽labels for south, east, north, and west interfaces, respectively, of a square control volume in Fig. 9; c x ⫽⌬t/⌬x and c y ⫽⌬t/⌬y for a twodimensional plane flow problem; c y ⫽⌬t/y 0 ⌬x if y is the radial direction in the axisymmetric flow problem, where y 0 is the radial distance of point 0 from the radial center; and ⌬x, ⌬y⫽sizes of the current control volume 共Fig. 9兲. Due to severe nonlinearity of the problem, modified Picard’s iteration 共Celia et al. 1990兲, in which the differential coefficients of the fluxes are taken as constants during each iteration, is adopted to solve the partial differential equations. To calculate the fluxes at the control–volume interfaces, linear distributions of the state variables between the representative points are assumed. This simplifies the calculation of the flux at an interface. For example, J w ⫽⫺

a w P 0⫺ P W g x 0 ⫺x W

(5)

where J w ⫽water flux through the interface W 共west兲; a w ⫽interface permeability at interface W; P 0 , P W ⫽pore pressures at points 0 and W 共west兲; and x 0 , x W ⫽distances of the points 0 and W from the center 关Fig. 9共a兲兴. Note that, in the finite volume method, the flux J w at the west interface of current control volume 0 关Fig. 9共b兲兴 is exactly equal to the flux J e at the east interface of the adjacent control volume lying to the west, even though the flux values are only approximate. Therefore, the condition of local mass balance 共as well the condition of local heat balance兲 is satisfied in the finite volume method exactly. This is a well-known advantage of the finite volume method, important for avoiding spurious oscillation in highly nonlinear problems with high local gradients and sharp fronts. The advantage of the finite volume method for the analysis of moisture transfer in concrete was recognized already by Eymard et al. 共1998兲 and was explored by Mainguy and Coussy 共2000兲 in the problem of calcium leaching from concrete, although in the absence of heat transfer. Mainguy et al. 共2001a兲 showed an effective application of the finite volume method to drying of porous materials.

Fig. 10. Explanation why small error in water content caused by lack of exact mass balance can cause enormous error in pore pressure

In the finite element approach, by contrast, the local mass balance cannot be satisfied exactly. For example, if standard finite elements with linear shape functions are used 关Fig. 9共b兲兴, then the flux at the west boundary of the current control volume 0 is generally different from the flux at the east boundary of the adjacent control volume lying to the west 关note the different slopes adjoining the interface in Fig. 9共b兲兴. Nevertheless, for a special case of mass lumping 共achieving pressure interpolation imitating the finite volume method兲, in low-order finite elements and for implicit time integration applied to coupled diffusion–dissolution problems, the finite element method gives the same results as the finite volume method and provides an accurate oscillation-free determination of sharp fronts 共Mainguy et al. 2001b兲. The reason why exact local mass balance is needed to avoid spurious oscillation of pressure and concentration 共water content兲 is explained by Fig. 10, showing a sharp change of slope of the sorption isotherm of relative pore pressure h 共humidity兲 versus water concentration w 共specific pore water content兲. At the transition from unsaturated to saturated state (h⫽1), a very small error ⌬w in water concentration, which is insignificant for the pressures in the unsaturated states, is seen to cause an enormous change of pressure in the saturated state 共because the slope of the pressure concentration isotherm drops above h⫽1 by orders of magnitude兲. This means that if the control volume with outflux ⫹ J⫺ w is unsaturated and the adjacent control volume with influx J w ⫺ is saturated, even a very small error in J w would cause a very large pressure change in the saturated control volume. The heat generation I (h) depends on the water contents of all the control volumes along the wave propagation path. The water contents, in turn, depend on the pore pressures and temperatures produced by the microwave heating. To avoid the complication that stems from this mutual coupling, the heat generation I (h) is in each time step calculated explicitly, based on the water contents of the control volumes in the last previous time step. The error caused by this one-step delay in the calculation of heat generation is very small if a small time step is used. After taking the first-order Taylor expansion of the terms superscripted by m⫹1 and collecting the terms that contain the variations ␦ P and ␦T from one iteration to the next 共in the same time step兲, one obtains the following system of algebraic linear equations: M M M M M KM P ␦ P 0 ⫹K P ␦ P W ⫹K P ␦ P S ⫹K P ␦ P E ⫹K P ␦ P N ⫹K T ␦T 0 0

W

S

E

N

⫹K TM ␦T W ⫹K TM ␦T S ⫹K TM ␦T E ⫹K TM ␦T N ⫽R M W

S

E

N

0

(6)

JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003 / 789

K HP ␦ P 0 ⫹K HP ␦ P W ⫹K HP ␦ P S ⫹K HP ␦ P E ⫹K HP ␦ P N ⫹K TH ␦T 0 0

W

S

E

N

0

⫹K TH ␦T W ⫹K TH ␦T S ⫹K TH ␦T E ⫹K TH ␦T N ⫽R H W

S

E

N

共18兲, too. For example, when the surface emissivity is infinite, the interface permeability of the west boundary surface is a⫽a 0 .

(7)

Conclusions

where i⫹1,m /␦x w 兲 ⫹c y 共 a i⫹1,m /␦x s 兲 ⫹c x 共 a i⫹1,m /␦x e 兲 KM P ⫽c x 共 a w s e

1.

0

⫹c y 共 a i⫹1,m /␦x n 兲 n

(8)

i⫹1,m /␦x w 兲 , KM P W ⫽⫺c x 共 a w i⫹1,m KM /␦x s 兲 , P ⫽⫺c y 共 a s S

(9)

2.

i⫹1,m /␦x e 兲 , KM P E ⫽⫺c x 共 a e i⫹1,m /␦x n 兲 KM P ⫽⫺c y 共 a n N

(10)

K TM ⫽ 共 ⳵w/⳵T 兲 i⫹1,m ⫺ 共 ⳵I 共 w 兲 /⳵T 兲 i⫹1,m ,

4.

0

K TM ⫽K TM ⫽K TM ⫽K TM ⫽0 W S E N

3.

(11)

⫺w i0 兲 ⫹c x 共 J e ⫺J w 兲 i⫹1,m R M ⫽⫺ 关共 w i⫹1,m 0 ⫹c y 共 J n ⫺J s 兲 i⫹1,m ⫺I 共i⫹1,m w兲 兴

(12)

K HP ⫽K HP ⫽K HP ⫽K HP ⫽K HP ⫽0

(13)

K TH ⫽␳C

(14)

0

W

S

E

N

0

K TH ⫽⫺c x 共 k wi⫹1,m /␦x w 兲 ,

6.

W

/␦x s 兲 , K TH ⫽⫺c y 共 k i⫹1,m s S

5.

(15)

K TH ⫽⫺c x 共 k i⫹1,m /␦x e 兲 , e E

/␦x n 兲 K TH ⫽⫺c y 共 k i⫹1,m n N

(16)

R H ⫽⫺ 关 ␳CT i⫹1,m ⫹c x 共 q e ⫺q w 兲 i⫹1,m ⫹c y 共 q n ⫺q s 兲 i⫹1,m ⫺I 共 h 兲 兴 (17) Here ␦x, ␦y⫽distances between the representative points of adjacent control volumes 关Fig. 9共b兲兴; M ⫽label for mass; H⫽label for heat; R⫽residual which is to be reduced to almost zero by the iteration; S,E,N, W⫽labels for control volumes adjacent to the current control volume 0 in the direction of south, east, north, and west, respectively, in Fig. 9共a兲; a⫽interface permeability; and k ⫽interface heat conductivity. The differential coefficients of the fluxes need to be multiplied by the radial distance of the corresponding interfaces if an axisymmetric problem is considered. The interface permeability is easily computed from the mass conservation condition. For example, if a steady state flow is considered, the mass flux on the interface w measured with respect to west control volume must be equal to the mass flux measured with respect to the current control volume 0 关see Fig. 9共a兲兴. Therefore, the equivalent permeability of the interface is a w⫽

a Wa 0 f a W ⫹ 共 1⫺ f 兲 a 0

7.

The computations confirm that the power and efficiency of the microwave applicator is a key factor for the proposed decontamination process. For the maximum power efficiency considered, the heat generation per unit volume of the wall is almost three times greater than it is for zero efficiency. Therefore the efficiency should be accurately measured. In view of the high microwave frequencies considered, averaging of the heating rate over the spatial wavelength and the time period causes no appreciable error. The calculations confirm that a 5 mm thick surface layer of typical concrete can be spalled off within 10 s of microwave heating of frequency 18.0 GHz and power 1.1 MW/m2. Calculations show that the thickness of the concrete wall has a negligible effect on the evolution of pore pressure and temperature. The reason is the short heating duration which is of the order of 10 s only. For long heating durations, differences would of course be obtained, due to microwave reflection and heat loss at the opposite surface of wall. The electromagnetic power carried by the microwaves is almost exhausted when the waves reach the location of the reinforcing bars in typical concrete structures. Therefore, the same decontamination process can be used for both unreinforced and reinforced concrete walls. The pore water pressure caused by heating is not negligible but is not a major factor. The main cause of spalling is high compressive stress parallel to surface along the radial lines emanating from the heated zone and high tensile stress along the circumferential lines, both produced by thermal expansion of the heated zone confined by cold concrete. The present computational experience confirms that the finite volume method is preferable to the finite element method for simultaneous heat and mass transfer in concrete heated to high temperature. Adoption of this method helps to eliminate spurious oscillations of response caused by propagation of a sharp interface with order-of-magnitude jumps in pore pressure and permeability 共the interface separates a saturated zone from a nonsaturated zone, or a cold zone of small permeability from a hot zone of a high permeability兲.

Acknowledgments Grateful appreciation is due to the Department of Energy for supporting both parts of this study under Grant No. DE-FG0798ER45736 to Northwestern University. Thanks are due to Dr. Brian Spalding of the Oak Ridge National Laboratory for valuable consultations and constructive monitoring of the progress of this research.

Appendix I: Input Data for Analysis (18)

where f ⫽constant representing the location of interface; a W ⫽representative permeability of west control volume; and a 0 ⫽representative permeability of current control volume. The interface heat conductivity is calculated by Eq. 共18兲 similarly. One can calculate the interface permeability at boundaries from Eq. 790 / JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003

The input data for the parametric studies of microwave heating in Figs. 3, 4, 5, and 11 are as follows: age of concrete⫽20 years; initial relative humidity in the pores h 0 ⫽0.6; T 0 ⫽25.0°C; saturation water content⫽135 kg/m3 ; apparent mass density of concrete⫽2288 kg/m3 ; heat capacity of concrete⫽1000 J/kg °K; reference permeability⫽40.5⫻10⫺12 m/s; heat conductivity

Appendix III: Review of Microplane Constitutive Model for Concrete

Fig. 11. Effect of wall thickness on pore pressure and temperature profiles in which f ⫽10.6 GHz is used; 共top兲 rapid heating with high power 共1.1 MW/m2兲; 共bottom兲 slow heating with low power 共0.01 MW/m2兲. Solid dots represent profiles across 30 cm wall and circles represent profiles across 10 cm wall.

⫽3.67 J/m2 s °C; amount of unhydrous cement⫽227 kg/m3 ; mass transfer coefficient of water flux⫽⬁; heat transfer coefficient of heat flux⫽10.0 J/m2 s °C; relative dielectric permittivity ⑀ ⬘ /⑀ 0 of concrete⫽5.0; surface emissivity⫽0.9; the range of frequencies applied⫽2.45– 18.0 GHz; the initial power density ⫽1.1 MW/m2 for the rapid heating and 0.011 MW/m2 for the slow heating; the area fraction of reinforcing bars in a plane parallel to surface⫽19%; distance of the centroid of reinforcing bars from concrete surface⫽at least 2.5 cm. The input data for the mechanical deformation analysis are: E⫽48.5 GPa; k 1 ⫽0.000125, k 2 ⫽160, k 3 ⫽6.4, and k 4 ⫽150.0 for the microplane model; the thermal expansion coefficient ⫽␣ T ⫽12.0⫻10⫺6 .

Appendix II: Effect of Wall Thickness on Pore Pressures and Temperatures Fig. 11共a兲 shows the pore pressure and temperature profiles for two different wall thicknesses, 10 and 30 cm, after 5, 10, and 15 s of heating with the power density P 0 ⫽1.1 MW/m2 , which is quite high and equals the power intensity used by White et al. 共1995兲. For the short durations 共about 10 s兲, contemplated for the decontamination process, the profiles obtained for two different wall thicknesses are almost identical 共White et al. 1995兲 共and doubtless also about the same as in a half space兲. If the wall were heated for a relatively long time with a low power density, appreciable differences in the temperature profiles would be observed because of the power reflected from the wall surface opposite to the heated surface; see Fig. 11共b兲; P 0 ⫽0.01 MW/m2 . A wall 10 cm thick would be heated a little faster than a wall 30 cm thick. But the differences are still very small.

For readers’s convenience, model M4, already mentioned and referenced to in Part I will now be summarized 共Bazˇant et al. 2000; Caner and Bazˇant 2000兲. The strain vectors on the plane of any orientation within the material, called the microplane, are assumed to be the projections of the continuum 共macroscopic兲 strain tensor ␧ i j 共where the subscripts, i⫽1,2,3 refer to Cartesian coordinates x i ). This is called the kinematic constraint. Thus the component of the strain vector ␧ nj on any microplane is ␧ nj ⫽␧ jk n k 共Bazˇant and Prat 1988兲 where n i are the direction cosines of the normal to the microplane. The normal strain vector is ␧ N i ⫽n i n j n k ␧ jk , and its magnitude is ␧ N ⫽n j ␧ nj ⫽n j n k ␧ jk ⫽N i j ␧ i j , where N i j ⫽n i n j 共the repeated Latin lowercase subscripts indicate summation over 1, 2, 3兲. The magnitude of the strain vector on the microplane is 储 ␧ nj 储 ⫽ 冑␧ nj ␧ nj . The shear strain components in two orthogonal 共suitably chosen兲 directions m i and l i tangential to the microplane 共normal to n i ) are ␧ M ⫽m i (␧ i j n j ) and ␧ L ⫽l i (␧ i j n j ). Because of the symmetry of ␧ i j , ␧ M ⫽M i j ␧ i j , ␧ L ⫽L i j ␧ i j , where M i j ⫽(m i n j ⫹m j n i )/2 and L i j ⫽(l i n j ⫹l j n i )/2 共Bazˇant and Prat 1988兲. Since the kinematic constraint relates the strains on the microplanes to the macroscopic strain tensor, the static equivalence can be enforced only approximately. This is done by means of the virtual work theorem which is written for the surface of a unit hemisphere 共Bazˇant 1984兲 2␲ ␴ ␦␧ ⫽ 3 ij ij ⫽

冕 冕

⍀

⍀

共 ␴ N ␦␧ N ⫹␴ L ␦␧ L ⫹␴ M ␦␧ M 兲 d⍀

(19)

共 ␴ N N i j ⫹␴ L L i j ⫹␴ M M i j 兲 ␦␧ i j d⍀

(20)

The normal stress and normal strain are split into their volumetric and deviatoric parts, and if Eq. 共20兲 is written separately for the volumetric and deviatoric components, one has ␴ i j ⫽␴ V ␦ i j ⫹␴ D ij

␴D i j⫽

3 2␲

冕冋 冉 ⍀

␴D Ni j⫺

冊

(21)

册

␦ij ⫹␴ L L i j ⫹␴ M M i j d⍀ 3

(22)

The elastic increments of the stresses in each microplane over the time step 共or load step兲 are written as ⌬␴ V ⫽E V ⌬␧ V , ⌬␴ D ⫽E D ⌬␧ D , and ⌬␴ T ⫽E T ⌬␧ T , where E V ⫽E/(1⫺2␯); E D ⫽5E/(2⫹3␩)(1⫹␯), and E T ⫽␩E D . Here ␩ is a parameter that can be chosen; the best choice is ␩⫽1 共Carol et al. 1991; Bazˇant et al. 1996; Carol and Bazˇant 1997兲. The volumetric–deviatoric split makes it possible to reproduce the full range of Poisson ratio ⫺1⭐␯⭐0.5 in elastic analysis. The term ⫺␦ i j /3 in Eq. 共22兲 ensures that ␴ Dkk ⫽0 even when 兰 ⍀ ␴ D d⍀⫽0. The integration is conducted numerically according to an optimal Gaussian quadrature integration formula for a spherical surface, characterized by discrete directions ␮ and associated N weights w ␮ ; ␴ i j ⫽(3/2␲)s i j ⬇6 兺 ␮⫽1 w ␮ s (␮) where s i j ij , ⫽ 兰 ⍀ (␴ N N i j ⫹␴ L L i j ⫹␴ M M i j )␦␧ i j d⍀ and N⫽the number of the microplanes. The inelastic behavior with fracturing damage, modeled as strain softening, is characterized on the microplanes in terms of strain-dependent yield limits for the components ␴ V , ␴ D , ␴ N , ␴ L and ␴ M , called the stress–strain boundaries. These boundaries can approximately reflect various physical mechanisms such as frictional slip in a certain direction, progressive growth of microJOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003 / 791

cracks of a certain orientation, axial splitting, lateral spreading under compression, etc. The constitutive model is explicit. In each time step, and at each integration point of each finite element, the newly calculated strain tensor is supplied as the input to the microplane constitutive subroutine. From this, the subroutine first calculates the strain components on all the discrete microplanes 共whose number, from experience, must be at least 21, for acceptable accuracy兲. From those components, the stresses are calculated first elastically, and if any stress component exceeds the value on the stress–strain boundary for the given strain, the stress value is dropped onto the boundary at constant strain. From the stresses on the microplane, the continuum stress tensor is calculated as the output, which is then used by the finite element program to calculate the nodal forces.

References Ahmed, G. N., and Hurst, J. P. 共1997兲. ‘‘Coupled heat and mass transport phenomena in siliceous aggregate concrete slabs subjected to fire.’’ Fire Mater., 21, 161–168. Bazˇant, Z. P. 共1984兲. ‘‘Microplane model for strain controlled inelastic behavior.’’ Mechanics of engineering materials, C. S. Desai and R. H. Gallagher, eds., Chap. 3, Wiley, London, 45–59. Bazˇant, Z. P. 共1997兲. ‘‘Analysis of pore pressure, thermal stress and fracture in rapidly heated concrete.’’ Proc., Int. Workshop on Fire Performance of High-Strength Concrete, NIST Special Publication 919, L. T. Phan et al., eds., National Institute of Standards and Technology, Gaithersburg, Md., 155–164. Bazˇant, Z. P. 共2002兲. Scaling of structural strength, Hermes-Penton, London. Bazˇant, Z. P., Caner, F. C., Carol, I., Adley, M. D., and Akers, S. A. 共2000兲. ‘‘Microplane model M4 for concrete I: Formulation with work-conjugate deviatoric stress.’’ J. Eng. Mech., 126共9兲, 944 –953. Bazˇant, Z. P., and Kaplan, M. F. 共1996兲. Concrete at high temperatures, Longman, London. Bazˇant, Z. P., and Ozˇbolt, J. 共1990兲. ‘‘Nonlocal microplane model for fracture, damage and size effect in structures.’’ J. Eng. Mech., 116共11兲, 2484 –2504. Bazˇant, Z. P., and Ozˇbolt, J. 共1992兲. ‘‘Compression failure of quasi-brittle material: Nonlocal microplane model.’’ J. Eng. Mech., 118共3兲, 540– 556. Bazˇant, Z. P., and Prat, P. C. 共1988兲. ‘‘Microplane model for brittle plastic material: I. Theory.’’ J. Eng. Mech., 114共10兲, 1672–1688. Bazˇant, Z. P., and Thonguthai, W. 共1978兲. ‘‘Pore pressure and drying of concrete at high temperature.’’ J. Eng. Mech. Div., Am. Soc. Civ. Eng., 104共5兲, 1059–1079. Bazˇant, Z. P., and Thonguthai, W. 共1979兲. ‘‘Pore pressure in heated concrete walls: Theoretical prediction.’’ Mag. Concrete Res., 31共107兲, 67–76. Bazˇant, Z. P., Xiang, Y., and Prat, P. C. 共1996兲. ‘‘Microplane model for concrete. I. Stress–strain boundaries and finite strain.’’ J. Eng. Mech., 122共3兲, 245–254; 123共3兲, E411–E411. Bazˇant, Z. P., and Zi, G. 共2003兲. ‘‘Decontamination of radionuclides from concrete by microwave heating. I: Theory.’’ J. Eng. Mech., 129共7兲, 777–784.

792 / JOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2003

Caner, F. C., and Bazˇant, Z. P. 共2000兲. ‘‘Microplane model M4 for concrete II: Algorithm and calibration.’’ J. Eng. Mech., 126共9兲, 954 –961. Carol, I., and Bazˇant, Z. P. 共1997兲. ‘‘Damage and plasticity in microplane theory.’’ Int. J. Solids Struct., 34共29兲, 3807–3835. Carol, I., Bazˇant, Z. P., and Prat, P. C. 共1991兲. ‘‘Geometric damage tensor based on microplane model.’’ J. Eng. Mech., 117共10兲, 2429–2448. Celia, M. A., Bouloutas, E. T., and Zarba, R. L. 共1990兲. ‘‘A general mass-conservative numerical solution for the unsaturated flow equation.’’ Water Resour. Res., 26共7兲, 1483–1496. Cheng, D. K. 共1983兲. Field and wave electromagnetics, Addison–Wesley, London. Eymard, R., Galloue¨t, T., and Herbin, R. 共2000兲. ‘‘The fine volume methods.’’ Handbook of numerical analysis, P. G. Ciarbet and J. L. Lions, eds., North Holland, Amsterdam. Eymard, R., Galloue¨t, T., Hilhorst, D., and Slimane, Y. N. 共1998兲. ‘‘Finite volumes and nonlinear diffusion equations.’’ RAIRO-Mathematical Modelling and Numerical Analysis—Mode´lisation Mathe´matique et Analyse Nume´rique, 32共6兲, 747–761. Harmathy, T. Z. 共1965兲. ‘‘Effect of moisture on the fire endurance of building materials.’’ Moisture in Materials in Relation to Fire Tests. ASTM Special Technical Publication No. 385, American Society of Testing Materials, Philadelphia, 74 –95. Harmathy, T. Z. 共1970兲. ‘‘Thermal properties of concrete at elevated temperatures.’’ J. Mater., 5共1兲, 47–74. Harmathy, T. Z., and Allen, L. W. 共1973兲. ‘‘Thermal properties of selected masonry unit concrete.’’ ACI J., 70共2兲, 132–142. Jansen, D. C., and Shah, S. P. 共1997兲. ‘‘Effect of length on compressive strain softening of concrete.’’ J. Eng. Mech., 123共1兲, 25–35. Jira´sek, M., and Bazˇant, Z. P. 共2002兲. Inelastic analysis of structures, Vol. 735, Wiley, London. Kraus, J. D. 共1988兲. Antennas, MacGraw–Hill, New York. Lagos, L. E., Li, W., and Ebadian, M. A. 共1995兲. ‘‘Heat transfer within a concrete slab with a finite microwave heating source.’’ Int. J. Heat Mass Transf., 38共5兲, 887– 897. Li, W., Ebadian, M. A., White, T. L., and Grubb, R. G. 共1993兲. ‘‘Heat transfer within a concrete slab applying the microwave decontamination process.’’ J. Heat Transf., 115, 42–50. Mainguy, M., and Coussy, O. 共2000兲. ‘‘Propagation fronts during calcium leaching and chloride penetration.’’ J. Eng. Mech., 126共3兲, 250–257. Mainguy, M., Coussy, O., and Baroghel-Bouny, V. 共2001a兲. ‘‘Role of air pressure in drying of weakly permeable materials.’’ J. Eng. Mech., 127共6兲, 582–592. Mainguy, M., Ulm, F.-J., and Heukamp, F. H. 共2001b兲. ‘‘Similarity properties of demineralization and degration of cracked porous materials.’’ Int. J. Solids Struct., 38, 7079–7100. Ozˇbolt, J., and Bazˇant, Z. P. 共1996兲. ‘‘Numerical smeared fracture analysis: Nonlocal microcrack interaction approach.’’ Int. J. Numer. Methods Eng., 39, 635– 661. Patankar, S. V. 共1980兲. Numerical heat transfer and fluid mechanics, McGraw–Hill, New York. Ulm, F.-J., Coussy, O., and Bazˇant, Z. P. 共1999兲. ‘‘The ‘‘Chunnel’’ fire. I: Chemoplastic softening in rapidly heated concrete.’’ J. Eng. Mech., 125共3兲, 272–282. White, T. L., Foster, D., Jr., Wilson, C. T., and Schaich, C. R. 共1995兲. ‘‘Phase II microwave concrete decontamination results,’’ ORNL Rep. No. DE-AC05-84OR21400, Oak Ridge National Laboratory, Oak Ridge, Tenn.