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7.3. Experimental section 99[ ( ) ( )]2 θ 0.5 − θ 2.094S 2 (θ) ≈ exp − √π . (7.6)1 − θ 0.670In case the beam consists of rock, which has a much higher modulus than gel, the compressibilityeffects have to be taken into account. The parameter A is then replaced by[129]( ) ( 1 − 2νp1 − K ) 2p3 K SA =1 − K pK S+ (1 − ρ)(KpK L− K pK S), (7.7)in which K p is the bulk modulus of the drained network, K S is the bulk modulus of thesolid constituting the porous network, K L is the bulk modulus of the pore fluid, and ρ isthe volume fraction of the solid content. The hydrodynamic relaxation time τ R is relatedto the permeability D and the viscosity of the pore fluid η L by [129]{ 2(1 + νp )τ R =3K p+ 1 − ρ + 1 [ρ − 4(1 + ν p)−K L K S 3( 1 − 2νp3) ( )]} ( )Kp ηL a 2. (7.8)K S DThe bulk modulus K L is known for most common liquids, and the bulk modulus K p equalsE p /3(1 − 2ν p ). The approximation K p /K S ≈ ρ 2 can be used to estimate K S [129].Many materials, including aged gels that still contain a certain amount of water,show viscoelastic (VE) behavior. When the characteristic time of viscoelastic relaxationis at least an order of magnitude longer than the characteristic time of hydrodynamicrelaxation, the total load relaxation W (t)/W (0) is adequately represented by the productof the hydrodynamic relaxation function R(t), as described above, and an uniaxial stressrelaxation function, Ψ V E : [133]W (t)W (0) = R(t)Ψ V E(t). (7.9)The function Ψ V E is essentially an empirical function and can be inferred from experimentaldata. However, for many systems the VE relaxation can be adequately describedby a stretched exponential:[ ( ) α ] tΨ V E = exp − , (7.10)τ V Ewhere τ V E is the VE relaxation time and α is a parameter (0 ≤ α ≤ 1).7.3 Experimental section7.3.1 MaterialsThe Bentheim sandstone was obtained from a Bad-Bentheim quarry in Germany. Rectangularplates were sawed from a single block. The length and the width of the plateswere 160 ± 5 mm and 25.2 ± 0.4 mm, respectively. The thickness of the plates varied
100 7. Permeability reduction in porous materials by in situ formed silica gelbetween 8.05 and 10.10 mm. For each plate the thickness and width were checked atseveral positions on the plate surface. The precision of the thickness and width was about0.05 mm. The sawed plates were dried at 60 ◦ C for five days.Bentheim is a quasi-homogeneous sandstone without distinct bedding planes or otherform of anisotropy. Mercury intrusion porosimetry (MIP) on a same batch of Bentheimsandstone indicated that the sandstone has a narrow pore size distribution with an averagepore size of 33 ± 10 µm and a total porosity of about 0.22. The permeability of therock with respect to water, which was measured on several cylindrical cores using acustom-made core injection set-up, was found to be 1.6 ± 0.3 µm 2 . Bentheim sandstoneis quartz-rich with little clay content. To test the presence of clays, small rock slabswere soaked in water and their length was monitored in a dynamic mechanical analyzer(DMA, type Perkin Elmer DMA 7e). In addition, the rock was analyzed in a scanningelectron microscope (SEM, type FEI XL30 FEG-SEM) equipped for energy-dispersiveX-ray spectroscopy (EDX, type PGT-IMIX PTS EDX). A fractured surface was coatedwith gold. The coating is needed to prevent the accumulation of static electric fields atthe sample during imaging due to the electron irradiation. Another piece of rock wasground and analyzed using X-ray diffraction (XRD) in a Rigaku Miniflex apparatus.TMOS was obtained from Acros Organics (99% pure). Deionized water was usedfor the hydrolysis reaction. The homogenization chemical used was methanol (FisherScientific, 99.9% pure). Ethanol (Pharmco, dehydrated, 200 Proof) was used to exchangewith the mother liquid of the gels after curing. For the beam bending measurementson the sandstone samples the pore liquid was exchanged with glycerol (Acros Organics,99+% pure).Buffer solutions of the water were made by dissolving the following chemicals. Theacid solution (pH = 3.1) contained 8.47 g of citric acid, 3.22 g of NaOH and 2.18 g ofHCl per liter. The alkaline solution (pH = 9.25) contained 0.1 M of ammonium-hydroxideand 5.12 g per liter of ammonium-chloride (about 0.1 Molal). The latter was diluted withdeionized water (with a ratio of 1:9) to yield a second alkaline solution with a pH of 8.8.7.3.2 Preparation of gels and rock samplesThe sol-gel solutions were prepared by first mixing TMOS with methanol in various ratiosto form homogeneous mixtures. The water (acid or alkaline) was added gradually to eachmixture. The water/TMOS molar ratio was always equal to 4:1. The corresponding totalvolume fraction of TMOS in solution varied between 0.35 and 0.55. The solutions werehomogenized by stirring during one minute. The solutions were cast in cylindrical Teflonmolds with an inner diameter of 11.5 ± 0.1 mm and a length of about 13 cm. Afterseveral days of curing at room temperature the gel rods were removed from the moldsand immersed in ethanol with an excess volume of about 10 to 1 for at least one day,after which the gels were put into a second bath of fresh ethanol for several days. Thisprocedure was applied to maximize the exchange of the mother liquid in the gels withethanol, so that the viscosity of the pore liquid is known, and in order to minimize furtheraging of the gels and minimize the degree of viscoelasticity. Scherer showed that whenthe pore liquid is exchanged, using various alcohols, the measurement of the permeability
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TWO-PHASE REACTIVE TRANSPORTOF AN O
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Dit proefschrift is goedgekeurd doo
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4. The effect of pH on mass transfe
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Samenvatting . . . . . . . . . . .
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2 1. Introductionproducerbarrierlow
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4 1. IntroductionTwo inter-European
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Part ITwo-phase bulk systems7
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10 2. Coupled mass transfer and sol
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32 3. Interfacial effects during re
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0+,44 4. The effect of pH on mass t
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46 4. The effect of pH on mass tran
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Page 151 and 152: -142 Appendix A, - C D *, - D
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150 Appendix CwhereΩ A (θ) = −2
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152 Appendix C
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154 Appendix D
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156 Appendix Eand equal to zero in
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158 Appendix EAs the saturations, d
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160 Bibliography[12] B. Vinot, R. S
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162 Bibliography[40] M. D. Mantle a
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164 Bibliography[70] F. J. M. van d
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166 Bibliography[97] K. Yoshida, A.
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168 Bibliography[127] G. W. Scherer
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170 Bibliography
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172 Summaryand gelation rates. The
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174 Samenvattingwaterfase. De T 1 v
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176 List of publications
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178 Dankwoordimmer opgewekte stemmi
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Prior to the speech, Harald Cramér
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