PDF Publication Title:
Text from PDF Page: 008
2.4. Boundary conditions In this part we will give the boundary conditions for the feed channel (including the unsupported elec- trolyte zones) shown in Fig.2(a), which will be combined with the depth-averaged equations to complete the model. At the inlet, we assume Note that since the inlet zone is not charged, zkcin = 0 should be satisfied. At the outlet, we assume in in ∂ψv u = u , ck = ck , ∂x = 0. (32) k v ∂ck ∂ψv p =pa, ∂x =0, ∂x =0, (33) where pa is the atmospheric pressure. On all the non-penetrating walls shown in Fig.2(a) (solid black lines or blocks, e.g., the splitter), apply u·n=0, Jk ·n=0. (34) where n is the normal vector of the walls pointing to the outside of the channel. On these walls the viscous boundary layer thickness should be comparable to the channel depth, which is much smaller than the x and y dimensions, so the non-slip boundary condition is not applied here. Finally, we need to give the upper and lower boundary conditions of the charged channel. When only two species exist (one anion, one cation), the previous work [13, 20] just assumes Jy,− = 0 (ideal CEMs) and ψv = ±V/2 at the anode and cathode side membranes. However, this leads to almost perfect ion removal, which cannot be used to predict the magnitude of outlet concentration. Furthermore, these two boundary conditions are not enough for a multi-component system. A possible solution is to assign artificial transference numbers to all the ions, but it turns out this could be too rigid of a boundary condition since transference numbers can depend on electric current. In this paper, we choose to directly simulate the transport in the membranes and electrode channels, so we will give boundary conditions in the electrode channel instead of on the upper and lower boundaries of the charged channel. We will use the depth-averaged equations in the limit of thick EDLs for transport in the membranes. However, since the membrane is porous, we need to modify the Dk to Deff. In this paper, we simply assume Deff/Dk = 0.1 for any k. For the kk electrode channels, which are not charged, we use the homogenized equations and assume plug flow with u = uE. At the inlet, we assume ∂ψv = 0, and ck = cA for the anode and ck = cC for the cathode stream. ∂x k k At the outlet, we assume ∂ck = 0 and ∂ψv = 0. On the non-penetrating walls, also assume Eq.(34). On ∂x ∂x 2.5. Non-dimensionalization In this part, we will non-dimensionalize the PB equation (Eq.(15)), the depth-averaged equations (Eq.(23a)- (23c), (22)-(29)) and the boundary conditions. The scales we choose to non-dimensionalize variables are ˆ summarized in Table 1. We will use the hat accent (f) to represent the dimensionless averaged quantities, ̃ and use the tilde accent (f) to represent all the other dimensionless quantities. Except for the accents, the dimensionless equations and boundary conditions look the same as the dimensional form. Also note that by definition, the integrals αEO, αDO, βP , βEO, βDO, δk themselves are dimensionless. Though we have the electrode surface, assume u · n = 0, Jk · n = 0 for k = 1,2,··· ,Ns, and ψv = ±V/2. Here we assume fast reaction on electrodes so the reaction rate is limited by ionic flux of water ions. Note that V may differ from the true voltage between electrodes due to surface overpotentials. kkkk used h and z to calculate the integrals, their dimensions cancels each other and the two variables can be substituted by 1 and z ̃ = z/h respectively to get the fully dimensionless form. To compare with experiments, we also scale the total current by the advection of positive charges z+c+FuinHiohio, where Hio and hio are respectively the width and depth of the inlet and outlet for the feed. 8PDF Image | Theory of shock electrodialysis
PDF Search Title:
Theory of shock electrodialysisOriginal File Name Searched:
2012-05431.pdfDIY PDF Search: Google It | Yahoo | Bing
NFT (Non Fungible Token): Buy our tech, design, development or system NFT and become part of our tech NFT network... More Info
IT XR Project Redstone NFT Available for Sale: NFT for high tech turbine design with one part 3D printed counter-rotating energy turbine. Be part of the future with this NFT. Can be bought and sold but only one design NFT exists. Royalties go to the developer (Infinity) to keep enhancing design and applications... More Info
Infinity Turbine IT XR Project Redstone Design: NFT for sale... NFT for high tech turbine design with one part 3D printed counter-rotating energy turbine. Includes all rights to this turbine design, including license for Fluid Handling Block I and II for the turbine assembly and housing. The NFT includes the blueprints (cad/cam), revenue streams, and all future development of the IT XR Project Redstone... More Info
Infinity Turbine ROT Radial Outflow Turbine 24 Design and Worldwide Rights: NFT for sale... NFT for the ROT 24 energy turbine. Be part of the future with this NFT. This design can be bought and sold but only one design NFT exists. You may manufacture the unit, or get the revenues from its sale from Infinity Turbine. Royalties go to the developer (Infinity) to keep enhancing design and applications... More Info
Infinity Supercritical CO2 10 Liter Extractor Design and Worldwide Rights: The Infinity Supercritical 10L CO2 extractor is for botanical oil extraction, which is rich in terpenes and can produce shelf ready full spectrum oil. With over 5 years of development, this industry leader mature extractor machine has been sold since 2015 and is part of many profitable businesses. The process can also be used for electrowinning, e-waste recycling, and lithium battery recycling, gold mining electronic wastes, precious metals. CO2 can also be used in a reverse fuel cell with nafion to make a gas-to-liquids fuel, such as methanol, ethanol and butanol or ethylene. Supercritical CO2 has also been used for treating nafion to make it more effective catalyst. This NFT is for the purchase of worldwide rights which includes the design. More Info
NFT (Non Fungible Token): Buy our tech, design, development or system NFT and become part of our tech NFT network... More Info
Infinity Turbine Products: Special for this month, any plans are $10,000 for complete Cad/Cam blueprints. License is for one build. Try before you buy a production license. May pay by Bitcoin or other Crypto. Products Page... More Info
CONTACT TEL: 608-238-6001 Email: greg@infinityturbine.com (Standard Web Page)