Kedem- Katchalsky equations

Nonequilibrium thermodynamics provides a second approach to combined convection and diffusion problems. The Kedem-Katchalsky equations, originally developed to describe combined convection and diffusion in membranes, form the basis of this approach [6,7] ... 

Kedem-Katchalsky equations Phenomenological equations for combined convection and diffusion, derived from nonequilibrium thermodynamics. See Eqs. (19) and (20). 

Manufacture of bacterial cellulose with desired shape Development of Kedem-Katchalsky equations of the transmembrane transport for binary nonhomogeneous non-electrolyte solutions Honeycomb patterned bacterial cellulose... 

The osmotic contribution to the drug transport is described by the so-called Kedem-Katchalsky equations (based on nonequilibrium thermodynamics) [50,51]. A simplified version is... 

Based on Donnan s [18, 19] and Onsager s [41, 42] fundamental works, the theories for Donnan dialysis systems were developed [20-26, 32-36]. The BAHLM system could be considered as two DD systems, operating in consecutive order, continuously in one module (see Fig. 6.2) the first is composed of feed/LM and the second is composed of LM/strip compartments, separated by ion-exchange membranes. Therefore, the Kedem-Katchalsky equations [43, 44] can be applied to our case ... 

L. Axel, Flow limits of Kedem-Katchalsky equations for fluid... 

The processes whereby water passes back and forth across the capillary wall are called filtration and absorption. The flow of water depends upon the relative magnitude of hydraulic and osmotic pressures across the capillary wall and is described quantitatively by the Kedem Katchalsky equations (the particular form of the equations applied to capillary water transport is referred to as Starling s Law). Recently, the physical mechanism of Starling s Law has been re-assessed [54]. Overall, in the steady state there is an approximate balance between hydraulic and osmotic pressures which leads to a small net flow of water. Generally, more fluid is filtered than is reabsorbed the overflow is carried back to the vascular system by the lymphatic circulation. The lymphatic network is composed of a large number of small vessels, the terminal branches of which are closed. Flap valves (similar to those in veins) ensure unidirectional flow of lymph back to the central circulation. The smallest (terminal) vessels are very permeable, even to proteins that occasionally leak from systemic capillaries. Lymph flow is determined by interstitial fluid pressure and the lymphatic pump (oneway flap valves and skeletal muscle contraction). Control of interstitial fluid protein concentration is one of the most important functions of the lymphatic system. If more net fluid is filtered than can be removed by the lymphatics, the volume of interstitial fluid increases. 

Same as a peripheral microvessel, the wall of the BBB can be viewed as a membrane. The membrane transport properties are often described by Kedem-Katchalsky equations derived from the theory of irreversible thermodynamics [47] ... 

Note that this equation is identical to the Kedem-Katchalsky model and does not imply a linear concentration gradient as it is frequently reported. It may be expressed as well as [7,8]... 

There are a number of other models of transport of solvent and solute through a reverse osmosis membrane the Kedem-Katchalsky model, the Spiegler-Kedem model, the frictional model, the finely porous model, the preferential sorption-capUlary flow model, etc. Most of these models have heen reviewed and compared in great detail hy Soltanieh and GiU (1981). We will restrict ourselves in this hook to the solution-diffusion and solution-diffusion-imperfection flux expressions for a number of reasons. First, the form of the solution-diffusion equation is most commonly used and is also functionally equivalent to the preferential sorption-capiUary flow model. Secondly, the solution-diffusion-imperfection model is functionally representative of a number of more exact three-transport-coefficient models, even though the transport coefficients in this model are concentration-dependent... 

Note Here Cj, tt and P correspond to infinitely thin solutions in equilibrium with the local section of the membrane therefore C, is the molar concentration of solute i in a solution of osmotic pressure w.) There are three parameters here Qjj (the intrinsic hydraulic permeability), P (the local solute permeahility coefficient) and <7,- (the local solute reflection coefficient). When these two equations are integrated across a membrane of thickness assuming Qsi, P and <7 to be essentially constant across the membrane thickness, one obtains, for the whole membrane, two equations for the Spiegler-Kedem model (based on the Kedem-Katchalsky model) ... 

The transport of both solute and solvent can be described by an alternative approach that is based on the laws of irreversible thermodynamics. The fundamental concepts and equations for biological systems were described by Kedem and Katchalsky [6] and those for artificial membranes by Ginsburg and Katchal-sky [7], In this approach the transport process is defined in terms of three phenomenological coefficients, namely, the filtration coefficient LP, the reflection coefficient o, and the solute permeability coefficient to. 

We may also use g to calculate an equivalent pore radius, r, according to equation (9) derived by Kedem and Katchalsky (15) ... 

In addition to the approach using phenomenological equations for modelling ion transport in soils, the theory of irreversible thermodynamics may be adapted to soils [26], as for the case of ion-exchange membranes. Spiegler [251 and Kedem and Katchalsky [27,28] are the prime examples of this approach to transport models. The detailed review by Verbrugge and Pintauro contains a number of other references to mathematical approaches for modelling the fundamental electrokinetic phenomena. 

The procedure proposed by Patlak et al. in order to recover the intrinsic non linearity of transport across thick membranes seems a very useful one, because it recognizes the limits of the integral practical equations derived by Kedem and Katchalsky and utilizes these equations in a local form, a situation where the limits pointed out by several Authors vanish. As demonstrated by Axel, such a treatment leads to an integral equation for solute flow coincident with the one derived by hydrodynamic methods. 

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