6. Transport phenomena. Basel, 2008

Transcription

1 6. Transport phenomena Basel, 2008

2 6. Transport phenomena 1. Introduction 2. Phenomenological equations for transport properties 3. Transport properties of a perfect gas 4. Diffusion in a fluid 5. Measurement of the diffusion coefficients, viscosity and thermal conductivity References: Supplementary material: German version for this chapter (Prof. Huber lecture from 2007). P. Atkins, J. de Paula, Atkins Physical Chemistry, Oxford Univ. Press, Oxford, 8th ed., 2006, Chapter 21. I. Tinoco, K. Sauer, J.C. Wang, J.D. Puglisi Physical Chemistry, Principles and applications in biological sciences, Prentice- Hall, New Jersey, 4th ed., 2002, Chapter 6.

3 6.1 Introduction Transport properties of a substance: the ability of the substance to transfer matter, energy or some other properties, from one place to another. Diffusion: migration of matter down a concentration gradient. Thermal conduction: migration of energy down a temperature gradient Electric conduction: migration of electric charges along an electric potential gradient Viscosity: migration of linear momentum down to a velocity gradient. Transport properties are expressed by phenomenological equations (empiric equations obtained from a summary of experiments and observations)

4 6.2 Phenomenological equations Flux, J represents the quantity of a property passing through a given area in a given interval of time/(area and duration of time). Types of flux: - Matter flux: if the matter is flowing through the area in the interval of time > number of molecules/(m 2 s). - Energy flux: if energy is flowing through the area in the interval of time > J/(m 2 s). Experimental observations on transport properties show that the flux of a property is proportional to the first derivative of a related property (gradient of the property).

5 6.2.1 Fick first law of diffusion Fick first law of diffusion: the flux of matter which diffuse parallel with the direction Oz in a container, is proportional to the concentration gradient. J ( matter) dν dz (6.1) N - number density of molecules z distance where there is a flux of matter [J] = nr. /m 2 s - If the concentration is changing steeply with position > diffusion process is fast. - If the concentration is uniform > flux of matter is 0 > no diffusion.

6 Fick first law of diffusion J > 0 J < 0 flux towards +Oz direction flux towards -Oz direction Fick first law of diffusion for the molecules of a perfect gas: J ( matter) = D dn dz (6.2) D diffusion coefficient; [D] = m 2 /s

7 6.2.2 Thermal conductivity The rate of thermal conduction (flux of energy associated with thermal motion) is proportional to the temperature gradient: J ( energy) dt dz (6.3) [J] = J/(m 2 s) where J = Joule! Similar to the diffusion relation, the flux of energy due to the thermal motion is : J ( energy) = k dt dz (6.4) k = coefficient of thermal conductivity [k] = J/(K m s) where J = Joule!

8 6.2.3 Viscosity Hypothesis: A Newtonian fluid is formed by a series of layers moving past one another, in a tube/container. - the layer next to the wall is stationary - the velocity of the succesive layers depends on the distance from the wall. Molecules move between layers and bring a x-component of linear momentum they have in their original layer (initial layer) to the layer in which they move (final layer). The final layer is accelerated or retarded, dependng on the linear momentul of the molecule.

9 Viscosity Viscosity: net retarding effect of molecules to different layers. It depends on the transfer of x-component of linear momentum (flux in the Oz direction) into the layer of interest. J ( x comp momentum) dv dz x (6.5) J Similar to the diffusion relation, the flux of energy due to the thermal motion is : (6.6) η = coefficient of viscosity (or viscosity) ( x comp momentum) dv = η dz x [η ] = kg/(m s) or Poise (P) 1P = 10-1 kg/m s

10 Thermal conductivity and viscosity

11 6.3 Transport properties of a perfect gas Model: simple kinetic theory of gases Diffusion coefficient, D: (6.7) 1 D = λc c 3 λ- mean free path of particles - mean speed of the particles in a gas λ decreases as the pressure in the gas is increasing > the diffusion coefficient is decreasing. Gas molecules diffuse more slowly than the pressure is high. c increases with the temperature > diffusion coefficient is increasing with the temperature. Molecules in a hot sample diffuse more quickly than in the case of a cool sample. D is greater for small molecules, than for large molecules (because the mean speed of the particles is inverse proportional to the mass of the particles).

12 6.3.1 Thermal conductivity perfect gas Coefficient of thermal conductivity, k : k 1 = λ cc 3 v, m [ A] (6.8) λ- mean free path of particles c - mean speed of the particles in a gas [A] - molar concentration of the gas molecules C v,m λ decreases as the pressure in the gas is increasing molar heat capacity at V constant. λ decreases as the molar concentration of the gas is increasing k f(pressure) k is higher for gases with a high heat capacity because a gradient of temperature corresponds to a higher variation of energy.

13 6.3.2 Viscosity perfect gas Viscosity, η: 1 η = Mλc [ A] 3 (6.9) λ- mean free path of molecules c - mean speed of molecules in a gas [A] - molar concentration of the gas molecules M molar mass of molecules c increases when the temperature increases (T 1/2 ) increasing as temperature increases (for gases). λ decreases as the pressure in the gas is increasing [A] increases as the molar concentration of the gas is increasing viscosity is η f(pressure)

14 6.3.3 Transport properties for gases c (m s -1 ) k ηc v k

15 Transport properties for gases Units m 2 s -1 J K -1 m -1 s -1 kg m -1 s -1

16 Diffusion coefficients for various media Substance and diffusion medium H 2 in H 2 Xe in Xe H 2 in air I 2 in air NaCl in H 2 O Na + NaCl in H 2 O Cl - ethanol in H 2 O Ag in Cu (6.55 mol%) Cu in Zn (75 mol%) P=1bar