# Analytical Methods for Problems of Molecular Transport

Binary Gas Mixtures.

- Analytical methods for problems of molecular transport.
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Discussion of the Slip Coefficient Results. Appendix 1. Bracket Integrals for the Planar Geometry Appendix 2.

## Analytical Methods for Problems of Molecular Transport

Bracket Integrals for Curvilinear Geometries Approximate Expressions for the Special Functions. Appendix 3. Bracket Integrals for Polynomial Expansion Method Calculation of the Bracket Integrals of the First Kind. Appendix 4. The Variational Principle for Planar Problems Some Definitions and Properties for Integral Operators. The Variational Principle. Appendix 5. Some Definite Integrals Some Frequently Encountered Integrals. Some Integrals Encountered in Boundary Problems. Table page Transformation of the various derivatives to new variables.

Functions for calculating the second-order transport coefficients. Values of the slip-flow coefficient. The slip-flow coefficient. The velocity defect at the wall. The dimensionless velocity defect. The mean velocity. The thermal-creep coefficient.

Numerical values of the velocity defect. Velocity defect data obtained during the solution of Problem 9. The isothermal-slip coefficient for different values of the tangential momentum. Comparison of experimental and theoretical values of drag on a sphere. The reduced thermal force. The reduced thermal force on NaCl aerosol particles. Expressions for some bracket integrals. Values of the reduced heat flux ratio. Analytical and numerical values of reduced torque on a rotating sphere.

Values of the transport coefficients and of the slip coefficients for a selection of binary gas mixtures obtained using the rigid-sphere potential model and the first- and second-order Chapman-Enskog approximations. Values of the transport coefficients and of the slip coefficients for a selection of binary gas mixtures obtained using the Lennard-Jones potential model and the first- and second-order Chapman-Enskog approximations.

Relevant parameters for two of the most commonly used intermolecular potential models; the rigid-sphere model and the Lennard-Jones potential model. The tangential momentum accommodation coefficients in a binary gas mixture chosen for calculation of the slip coefficients and slip factors. A comparison of diffusion-slip coefficient values. Values of the J-integrals for use in evaluating bracket integrals containing two.

Values of the I-integrals for use in evaluating bracket integrals for planar problems. Numerical values of the spherical special functions of the first- and second-kind. Numerical values of the cylindrical special functions of the first- and second-kind. Numerical values of the integrals of the cylindrical special functions of the first-.

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Values of the bracket integrals of the first-kind for the polynomial expansion. Values of the commonly encountered i-integrals. Values of some reduced omega-integrals. A Mathematica program to compute the reduced omega-integrals for selected.

Figure page Passage of molecules across an arbitrary surface element. Components of the pressure tensor. Geometry of a generic encounter. Parameters of a generic encounter.

Similarly, dialysis is the transport of any other molecule through a semipermeable membrane due to its concentration difference. Both osmosis and dialysis are used by the kidneys to cleanse the blood. Osmosis can create a substantial pressure. Consider what happens if osmosis continues for some time, as illustrated in [link].

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Water moves by osmosis from the left into the region on the right, where it is less concentrated, causing the solution on the right to rise. This movement will continue until the pressure created by the extra height of fluid on the right is large enough to stop further osmosis. This pressure is called a back pressure. The back pressure that stops osmosis is also called the relative osmotic pressure if neither solution is pure water, and it is called the osmotic pressure if one solution is pure water. Osmotic pressure can be large, depending on the size of the concentration difference.

For example, if pure water and sea water are separated by a semipermeable membrane that passes no salt, osmotic pressure will be This value means that water will diffuse through the membrane until the salt water surface rises m above the pure-water surface! One example of pressure created by osmosis is turgor in plants many wilt when too dry.

Turgor describes the condition of a plant in which the fluid in a cell exerts a pressure against the cell wall. This pressure gives the plant support. Dialysis can similarly cause substantial pressures.

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Reverse osmosis and reverse dialysis also called filtration are processes that occur when back pressure is sufficient to reverse the normal direction of substances through membranes. Back pressure can be created naturally as on the right side of [link]. A piston can also create this pressure. Reverse osmosis can be used to desalinate water by simply forcing it through a membrane that will not pass salt. Similarly, reverse dialysis can be used to filter out any substance that a given membrane will not pass.

One further example of the movement of substances through membranes deserves mention. We sometimes find that substances pass in the direction opposite to what we expect. Cypress tree roots, for example, extract pure water from salt water, although osmosis would move it in the opposite direction. This is not reverse osmosis, because there is no back pressure to cause it. What is happening is called active transport , a process in which a living membrane expends energy to move substances across it. Many living membranes move water and other substances by active transport.

The kidneys, for example, not only use osmosis and dialysis—they also employ significant active transport to move substances into and out of blood. The study of active transport carries us into the realms of microbiology, biophysics, and biochemistry and it is a fascinating application of the laws of nature to living structures.