Theoretical Estimation of Diffusion Coefficients for Binary Gas Mixtures

Transcription

1 52 Appendix A Theoretical Estimation of Diffusion Coefficients for Binary Gas Mixtures The diffusion coefficient D 2 for the isothermal diffusion of species through constantpressure binary mixture of species and 2 is defined by the relation J D 2 c, (A.) where J is the flux of species and c is the concentration of the diffusing species. Mutual-diffusion, defined by the coefficient D 2, can be viewed as diffusion of species at infinite dilution through species 2, or equivalently, diffusion of species 2 at infinite dilution through species 2. Self-diffusion, defined by the coefficient D, is the diffusion of a substance through itself. There are different theoretical models for computing the mutual (self) diffusion coefficient of gases. For non-polar molecules, Lennard-Jones potentials provide a basis for computing diffusion coefficients of binary gas mixtures [3]. The mutual diffusion coefficient, in units of cm 2 /s is defined as D T 3/2 M + M 2 M M 2 f D pσ2 2 Ω, (A.2) D where T is temperature of the gas in units of Kelvin; M and M 2 are molecular weights of species and 2 ; p is the total pressure of the binary mixture in units of atmospheres; f D is the second-order correction, usually between. and.3; σ 2 is the Lennard-Jones force constant for the gas mixture, defined by σ 2 /2 (σ + σ 2 ); Ω D is the collision integral

2 53 defined by Ω D.636 (T ) exp ( T ) exp ( T ) exp (3.894 T ), (A.3) where T T/ɛ 2, is the Boltzman gas constant, ɛ 2 (ɛ ɛ 2 ) /2 and ɛ 2 ɛ ɛ 2. Values of σ (2),Ω D and ɛ (2) are tabulated for most naturally occurring gases [3]. The self-diffusion coefficient of a gas can be obtained from Eq. A.2, by observing that for a one-gas system: M M 2 M, ɛ ɛ 2 and σ σ 2.Thus, f D 2 D.858 T 3/2 M pσ 2 Ω. (A.4) D It is useful to define observable diffusion, D obs, which is diffusion that one observes in an experiment. Observable diffusion os species in the binary mixture of species and species 2 is D obs, p / (p + p 2 ) D (p atm)/ (p + p 2 ) + p 2 / (p + p 2 ) D 2 (p atm)/ (p + p 2 ) p D (p atm) + p 2 D 2 (p atm) D (p p ) + D 2 (p p 2 ). (A.5) Equation A.5 has a simple physical explanation when applied to gases. The observable diffusion rate of gas in the mixture of gases and 2 is equal to the diffusion rate of one atom of gas through the rest of atoms of gas, plus the diffusion rate of one atom of gas through the atoms of gas 2. Equation A.5 enables the estimation of the diffusion coefficient for the binary mixture of 29 Xe-nitrogen and 3 He-nitrogen.

3 54 A.. Observable Diffusion Constant for a Mixture of Xe-29 and Nitrogen The relevant parameters [3] are: σ Xe 4.47 ɛ Xe / 23. M Xe 3.4 σ N ɛ N2 / 7.4 M N2 28 At T (33 ± ) K and p (p Xe + p N2 )atm, σ Xe N σ Xe Xe 4.47 ɛ Xe N T ɛ Xe N Ω D.83 ɛ Xe Xe 23 T ɛ Xe Xe.333 Ω D (A.6) The above parameter values yield D Xe N (p Xe + p N2 ) m 2 /s (A.7) D Xe Xe (p Xe + p N2 ) m 2 /s. (A.8) The observable diffusion rate for a mixture of 29 Xe and Nitrogen gas is therefore D obs p Xe m 2 /s + p N m 2 /s. (A.9) The cell used in Xenon experiments had the following pressures: p Xe (.48 ±.) atm and p N2 (.4±.) atm. The theoretical estimation of the observable diffusion constant is thus D obs (.8 ±.8) 5 m 2 /s.

4 55 A..2 Observable Diffusion Constant for a Mixture of He-3 and Nitrogen The relevant parameters [3] are: σ He 2.55 ɛ He /.22 M He 4 σ N ɛ N2 / 7.4 M N2 28 At T (38 ± ) K and p (p He + p N2 )atm, σ He N σ He He 2.55 ɛ He N T ɛ He N2.587 Ω D.726 ɛ He He.22 T ɛ He He Ω D.623. (A.) The above parameter values yield D He N (p He + p N2 ) m 2 /s (A.) D He He (p He + p N2 ) m 2 /s. (A.2) The observable diffusion rate for a mixture of 3 He and Nitrogen gas is therefore D obs p He m 2 /s + p N m 2 /s. (A.3) The cell used in Helium experiments had the following pressures: p He (.75 ±.) atm and p N2 (.±.) atm. The theoretical estimation of the observable diffusion constant is thus D obs (.77 ±.2) 4 m 2 /s.

5 56 Appendix B Supplement on Fourier Transforms The Fourier Transform of e 2π x, where 2π /T 2,isgivenby: [ F e 2π x ] + e 2π x e 2πix dx e 2πix e 2π x dx + e 2πix e 2π x dx [cos (2πx) i sin (2πx)] e 2π x dx [cos (2πx) i sin (2πx)] e 2π x dx Let u x so that du dx, then: [ F e 2π x ] + 2 π 2 + 2, [cos (2πu)+i sin (2πu)] e 2π u du [cos (2πu) i sin (2πu)] e 2π u du cos (2πu)e 2π u du which is a Lorentzian function, with: FWHM 2 /πt 2.

6 57 Appendix C Imaging Parameters The following are the descriptions of some of the most common parameters in MR imaging:. Bandwidth (BW): Anti-aliasing filter bandwidth of the receiver. 2. Sampling Period ( t): Sampling period of the A/D converters. 3. Acquisition Time or Readout Interval (T AcqT ime ): Time interval during which the signal is acquired. 4. Field-of-View (FOV x, FOV y ): Image size along the x and y-coordinates. 5. Matrix Size (N x N y ): Number of pixels along the readout and phase-encode directions. 6. Spatial Resolution ( x, y): Resolution in image space. 7. Raw Data Resolution ( x, y ): Resolution in -space. 8. Readout Amplitude (G x ): Amplitude of the readout gradient. 9. Maximum Amplitude in Y-Gradient (G max y ): Maximum amplitude of y-gradient used in imaging.. Incremental Amplitude in Y-Gradient ( G y ): Incremental amplitude of y- gradient used in imaging.. Phase Encode Interval (t Gy ): Time interval during which the phase encode gradient is applied.

7 58 Below, is a set of formulas which define and connect these parameters: t /BW (C.) T AcqT ime t N x (C.2) G max y G y N y (C.3) FOV x / x (C.4) FOV y / y (C.5) x FOV x /N x (C.6) y FOV y /N y (C.7) x γ 2π G x t (C.8) y γ 2π G y t Gy (C.9)