Chapter 7. Principles of Unsteady - State and Convective Mass Transfer

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1 Suppleental Material for Tranport Proce and Separation Proce Principle hapter 7 Principle of Unteady - State and onvective Ma Tranfer Thi chapter cover different ituation where a tranfer i taking place, when the condition are changing with tie and where there i a fluid trea that contribute to thee a tranfer procee. The following exaple illutrate convective and tranient a tranfer in fuel cell yte Diffuion and heical Reaction in the node haber of a Direct Methanol Fuel ell Diffuion of O and O through tagnant itrogen in a Solid Oxide Fuel ell Daniel López Gaxiola 1 Student View Jaon M. Keith

2 Principle of Unteady State and onvective Ma Tranfer Exaple 7.5- Diffuion and heical Reaction in the node haber of a Direct Methanol Fuel ell n aqueou 40 ole % ethanol olution i entering the anode of a direct ethanol fuel cell. The fuel i diffuing through the ga diffuion layer (GDL) with a thickne of c [1]. The diffuion coefficient of the fuel in the GDL i etiated by García et al. [1] to be The governing equation for ethanol in the GDL i given by: 9 d dz = 0 with the following boundary condition: t z = 0 : = b t z = δ : = k 1 The rate contant for the cheical reaction occurring at the catalyt layer located at z = δ i 6 k = Deterine the olar fraction of ethanol at z = δ and teady tate, if the initial 1 ol concentration b of ethanol i 500. Strategy The olar fraction at the catalyt layer can be obtained by olving the governing differential equation. Solution We can tart by olving the given differential equation given in the proble tateent, a hown in the following tep: d dz = 0 = c 1 (1) d = c dz 1 = Daniel López Gaxiola Student View Jaon M. Keith

3 Suppleental Material for Tranport Proce and Separation Proce Principle 1. García,.L., Sethuraan, V.., Weidner, J.W., White, R.E., Dougal, R., Journal of Fuel ell Science and Technology, 1, 4 48 (004). pplying the firt boundary condition at z = 0, we have: b = c = Subtituting c into Equation yield: = () t z = δ: = () The equation for ethanol flux through the GDL i decribed by Fick Law, given by: d D dz = (4) t z = δ, the olar flux of ethanol i equal to the reaction rate. Therefore: z =δ = (5) Since thi proce i at teady tate, we can equal Equation 4 and 5 to get: d D = (6) dz Fro Equation 1, we have that c 1 =. Subtituting thi into Equation 6 give: D = (7) We can ubtitute Equation into Equation 7 to olve for c 1, a hown in the following tep: 1 1 ( ) Dc = k Dc = k c δ + k b 1 ( ) c = c 1 = (8) Daniel López Gaxiola Student View Jaon M. Keith

4 Principle of Unteady State and onvective Ma Tranfer ow we can ubtitute thi equation for c 1 into Equation to yield: = + (9) b Equation 9 can be evaluated at the boundary condition for z = δ to obtain the olar fraction at thi point. Hence, = Reducing thi equation and writing in ter of the olar fraction of ethanol, we have: = ( D + ) (10) where i the overall concentration of the fuel entering the fuel cell. The overall concentration can be obtained by dividing the feed concentration of ethanol b by the feed olar fraction of 0.4. Subtituting nueric value into Equation 10 give: ol H OH ol H OH = ol H OH ( ) ol + x ol c = 4 + ( ) x = Daniel López Gaxiola 4 Student View Jaon M. Keith

5 Suppleental Material for Tranport Proce and Separation Proce Principle Exaple 7.5-4: Diffuion of O and O through tagnant in a Solid Oxide Fuel ell olid - oxide fuel cell operating at a teperature of 9.15 K and a preure of 1.9 at, i producing O fro an electrocheical reaction of O with oxygen fro air. The partial preure of each ga at the ga diffuion layer the bipolar plate channel, located 0.79 away, are given in the following table. Label Ga Partial Preure (at) at Ga Diffuion Layer Partial Preure (at) at ipolar Plate O O Deterine the olar flux of O and O in non diffuing. The diffuion coefficient are given below: D = D = D = Where: = O, = O and =. Thee diffuion coefficient were etiated uing Fuller et al. ethod decribed in Section 6.E of Geankopli. The following equation decribe ulticoponent diffuion for two coponent diffuing in tagnant []: P P + = ln D D RT ( z z1 ) P 1. Geankopli,.J., Ma Tranport Phenoena, Holt, Rinehart and Winton Inc., ew York, 197. Daniel López Gaxiola 5 Student View Jaon M. Keith

6 Principle of Unteady State and onvective Ma Tranfer D D + + D D P P + P DP D D D D + = ln RT( z z1 ) D D + + D D P P 1 + P D D D D In thi proble, the flux will be aued poitive fro the bipolar plate to the ga diffuion layer. Strategy We can deterine the fluxe and by iultaneouly olving the equation given in the proble tateent. Solution Firt, we can ubtitute the preure given and the diffuivity coefficient, a well a the operating condition of the fuel cell. Thu, ln L at ( 9.15K )( ) ol K 1.9 at at + = 5 5 at We can olve thi equation for by following the next tep: 1.9 at at = ln at 1.1 at ( 9.15K )( ) ol K ol = 0.61 (1) So far we have obtained one of the iultaneou equation for the diffuion proce occurring in thi fuel cell. However, in order to olve thi proble, we need a econd equation, obtained fro the equation for +, a hown below: Daniel López Gaxiola 6 Student View Jaon M. Keith

7 Suppleental Material for Tranport Proce and Separation Proce Principle ( 1.9at) + = at 4 ( 9.15K)( ) ol K ( at) ( 0.01 at) + ( 1.9 at) ln ( 0.1 at ) ( at) + ( 1.9 at) In thi equation, it can be een that oe ter have the factor expreion can be iplified a follow: in coon. Thu, thi + ( )( 0.47 at ) ( at ) ( )( 1.9 at) ol + + = ln + ( )( at) ( 0.47 at ) + ( )( 1.9 at) We can further iplify thi equation by reoving the olar flow of oxygen in the denoinator a hown in the following tep: ( ) ( ) ol ln = () ow we can ubtitute Equation 1 into Equation to get: ( + ) + ( ) ( ) ( ) ol ( ) = ln Siplifying iilar ter and oving all ter to the left ide, we have: ln = 0 + Daniel López Gaxiola 7 Student View Jaon M. Keith

8 Principle of Unteady State and onvective Ma Tranfer Thi equation can be olved by trial and error or uing coputer oftware to obtain the olar flow rate of carbon dioxide to be: ol = We can enter thi value into Equation 1 to deterine the flux of oxygen a hown below: ol ol = ol = The following figure illutrate the diffuion proce occurring in the fuel cell cathode. = 0 Ga Diffuion Layer ipolar Plate Daniel López Gaxiola 8 Student View Jaon M. Keith