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1 Chapte.4 iffusion with Chemical eaction Example fluiize coal eacto opeates at 45 K an atm. The pocess will be limite by the iffusion of oxygen countecuent to the cabon ioxie, CO, fome at the paticle suface. ssume that the coal is pue soli cabon with a ensity of 80 g/m 3 an that the paticle is spheical with an initial iamete of m. i (% O an 79% N ) exists seveal iametes away fom the sphee. The iffusivity of oxygen in the gas mixtue at 45 K is m /s. If a quasi-steay state pocess is assume, calculate the time necessay to euce the iamete of the cabon paticle to m. (ef. Funamentals of Momentum, Heat, an Mass Tansfe by Welty, Wics, an Wilson, 4 th Eition, 00, pg. 496.) Solution The eaction at the cabon suface is C(s) + O (g) CO (g) We have iffusion of oxygen () towa the suface an iffusion of cabon ioxie (B) away fom the suface. The mola flux of oxygen is given by N, c mix y + y (N, + N B, ) In this equation, is the aial istance fom the cente of the cabon paticle. Since N, N B,, we have N, c mix y The system is not at steay state, the mola flux is not inepenent of since the aea of mass tansfe 4π is not a constant. Using quasi steay state assumption, the mass (mole) tansfe ate, 4π N,, is assume to be inepenent of at any instant of time. W 4π N, 4π c mix y constant y, y,inf -3

2 t the suface of the coal paticle, the eaction ate is much faste than the iffusion ate to the suface so that the oxygen concentation can be consiee to be zeo: y, 0. Sepaating the vaiables an integating gives y, 4π c mix y 0 W W 4π c mix y, > W 4πc mix y, Since one mole of cabon will isappea fo each mole of oxygen consume at the suface W C W 4πc mix y, Maing a cabon balance gives ρc 4 3 π M C t 3 ρc M C 4π t 4πc mix y, Sepaating the vaiables an integating fom t 0 to t gives t t 0 ρc M c C y mix, f i ρc t M c C y mix, ( i f ) The total gas concentation can be obtaine fom the ieal gas law c P T (0.0806)(45) mol/m3 Note: m 3 atm/mol o K The time necessay to euce the iamete of the cabon paticle fom m to m is then ( ) ( ) 5 5 (80) t 4 ()(0.006)(.3 0 )(0.) 0.9 s -4

3 Example Pulveize coal pellets, which may be appoximate as cabon sphees of aius mm, ae bune in a pue oxygen atmosphee at 450 K an atm. Oxygen is tansfee to the paticle suface by iffusion, whee it is consume in the eaction C(s) + O (g) CO (g). The eaction ate is fist oe an of the fom ɺ " C O whee 0. m/s. This is the eaction ate pe unit suface aea of the cabon pellets. Neglecting change in, etemine the steay-state O mola consumption ate in mol/s. t 450 K, the binay iffusion coefficient fo O an CO is m /s. (ef. Funamentals of Heat Tansfe by Incopea an ewitt.) Solution We have iffusion of oxygen () towa the suface an iffusion of cabon ioxie (B) away fom the suface. The mola flux of oxygen is given by N, c y + y (N, + N B, ) In this equation, is the aial istance fom the cente of the cabon paticle. Since N, N B,, we have N, c y The system is not at steay state, the mola flux is not inepenent of since the aea of mass tansfe 4π is not a constant. Using quasi steay state assumption, the mass (mole) tansfe ate, 4π N,, is assume to be inepenent of at any instant of time. W 4π N, 4π c y constant y, y,inf The oxygen concentation at the suface of the coal paticle, y,, will be etemine fom the eaction at the suface. The mole faction of oxygen at a location fa fom the pellet is. Sepaating the vaiables an integating gives y, 4π c y y, W W 4π c (y, y, ) > W 4πc ( y, ) -5

4 The mole of oxygen aive at the cabon suface is equal to the mole of oxygen consume by the chemical eaction W 4π ɺ " 4π C O 4π c y, 4πc ( y, ) 4π c y, ( y, ) y, > y, + " y, The total gas concentation can be obtaine fom the ieal gas law. (Note: m 3 atm/mol K) c P T (0.0806)(450) mol/m3 The steay-state O mola consumption ate is W 4πc ( y, ) 4π( )( )( 0.63)(0-3 ) W mol/s Example biofilm consists of living cells immobilize in a gelatinous matix. toxic oganic solute (species ) iffuses into the biofilm an is egae to hamless poucts by the cells within the biofilm. We want to teat 0. m 3 pe hou of wastewate containing 0. mole/m 3 of the toxic substance phenol using a system consisting of biofilms on otating is as shown below. Waste wate fee steam biofilm C 0 C (z) Inet soli suface Well-mixe contacto C 0 Teate waste wate biofilm z0 etemine the equie suface aea of the biofilm with mm thicness to euce the phenol concentation in the outlet steam to 0.0 mole/m 3. The ate of isappeaance of phenol (species ) within the biofilm is escibe by the following equation -6

5 c whee 0.09 s - The iffusivity of phenol in the biofilm at the pocess tempeatue of 5 o C is m /s. Phenol is equally soluble in both wate an the biofilm. (ef. Funamentals of Momentum, Heat, an Mass Tansfe by Welty, Wics, an Wilson, 4 th Eition, 00, pg. 496.) Solution The ate of phenol pocesse by the biofilms, W, is etemine fom the mateial balance on the pocess unit W 0. m 3 /h(0. 0.0) mol/m mol/h W is then equal to the ate of phenol iffuse into the biofilms an can be calculate fom c W S N,z S z 0 In this equation, S is the equie suface aea of the biofilm an N,z is the mola flux of phenol at the suface of the biofilm. The mola flux of (phenol) is given by N,z c y + y (N,z + N B,z ) Since the biofilm is stagnant (o noniffusing), N B,z 0. Solving fo N,z give N,z ( y ) c y The mole faction of phenol in the biofilm, y, is much less than one so that c can be consiee to be constant. Theefoe N,z c y c Biofilm Soli suface N,z Maing a mole balance aoun the contol volume S z gives z -7

6 S N,z z S N,z z+ z + S z 0 iviing the equation by S z an letting z 0 yiels N, z Substituting N,z c (E-) c into equation (E-) we obtain c c > c c (E-) The solution to the homogeneous equation (E-) has two foms ) c C exp ) c B sinh z z + C exp + B cosh z z The fist exponential fom () is moe convenient if the omain of z is infinite: 0 z while the secon fom using hypebolic functions () is moe convenient if the omain of z is finite: 0 z δ. The constants of integation C, C, B, an B ae to be etemine fom the two bounay conitions. We use the hypebolic functions as the solution to Eq. (E-). c B sinh z + B cosh z (E-3) t z 0, c c s c 0 B t z δ, Theefoe c 0 B cosh δ + B sinh δ sinh B B cosh δ δ sinh c 0 cosh δ δ Equation (E-3) becomes -8

7 sinh c c 0 cosh δ δ sinh z + c 0 cosh z cosh δcosh z sinh δsinh z c c 0 cosh δ Using the ientity cosh( B) cosh()cosh(b) sinh()sinh(b) we have c c 0 δ cosh δ cosh ( z) c z 0 c 0 sinh ( δ z) cosh δ z 0 c 0 tanh δ The mola flux of phenol at the biofilm suface is given by N,z c c0 δ z 0 tanh δ δ The imensionless paamete δ Fo this poblem we have epesents the atio of eaction ate to iffusion ate m s 0 m 0 s δ 9.49 This value inicates that the ate of eaction is vey api elative to the ate of iffusion. The flux of phenol into the biofilm is then -9

8 N,z 0 (0.0)( 0 ) 0.00 (9.49) tanh(9.49) mol/(m s) The equie suface aea of the biofilm is finally S W N, z (3.9 0 )(3600) 57.0 m Example Consie a spheical oganism of aius within which espiation occus at a unifom volumetic ate of C. That is, oxygen (species ) consumption is govene by a fist-oe, homogeneous chemical eaction. (a) If a mola concentation of C () C,0 is maintaine at the suface of the oganism, obtain an expession fo the aial istibution of oxygen, C (), within the oganism. (b) Obtain an expession fo the ate of oxygen consumption within the oganism. (c) Consie an oganism of aius 0.0 mm an a iffusion coefficient fo oxygen tansfe of 0-8 m /s. If C, mol/m 3 an 0 s -, what is the mola concentation of O at the cente of the oganism? What is the ate of oxygen consumption by the oganism? Solution (a) If a mola concentation of C () C,0 is maintaine at the suface of the oganism, obtain an expession fo the aial istibution of oxygen, C (), within the oganism. + Figue E- Illustation of a spheical shell 4π The one-imensional mola flux of is given by the equation " N C (E-) pplying a mole balance on the spheical shell shown in Figue E- yiels fo steay state 4π N " 4π " N + 4π

9 iviing the equation by the contol volume (4π ) an taing the limit as 0, we obtain ( " N ) + 0 (E-) Fo a fist oe eaction, C an substituting the mola flux fom equation (E-) into the above equation, we have C C 0 C C 0 (E-3) In this equation, an ae constants inepenent of. We want to tansfom this equation into the fom y α y 0 (E-4) Let α, we can tansfom equation (4.6-3) into the fom of equation (E-4) by the following algebaic manipulations C α C 0 C C + α C 0 C C + α C 0 Since becomes ( C ) C C + C + C C +, the above equation ( C ) α C 0 Let y C, the equation has the same fom as equation (E-4) with the solution y B sinh(α) + B cosh(α) o C B sinh(α) + B cosh(α), whee α -3

10 The two constants of integation B an B can be obtaine fom the bounay conitions t 0, C finite o C 0 t, C C 0 (a nown value) pplying the bounay at 0 yiels 0 B pplying the bounay at yiels C 0 C B sinh(α) B sinh( α) Theefoe the concentation pofile fo species within the oganism is C C 0 sinh( α) sinh( α) (E-5) α t the cente of the oganism, the concentation is given by C ( 0) C 0 sinh( α) (b) Obtain an expession fo the ate of oxygen consumption within the oganism. ate of oxygen consumption within the oganism. 4π ( C ) The oxygen concentation within the oganism is given by equation (E-5) C C 0 sinh( α) sinh( α) (E-5) C C sinh( α) 0 α sinh( α ) + cosh( α) C C sinh( α) 0 α cosh( α) sinh( α) C C 0 [ ] ( α ) coth( α) ) -3

11 / Let φ α Thiele moulus fo a fist oe eaction. Ignoing the minus sign, the ate of oxygen consumption within the oganism is then ate of oxygen consumption 4π C 0 (φ cothφ - ) ate of oxygen consumption 4π C 0 (φ cothφ - ) (c) Consie an oganism of aius 0.0 mm an a iffusion coefficient fo oxygen tansfe of 0-8 m /s. If C, mol/m 3 an 0 s -, what is the mola concentation of O at the cente of the oganism? What is the ate of oxygen consumption by the oganism? α t the cente of the oganism, the concentation is given by C ( 0) C 0 sinh( α) α / / m α / 4 ( ) / 4.47 α C ( 0) C 0 sinh( α) sinh(4.47) mol/m 3 ate of oxygen consumption 4π C 0 (φ cothφ - ) ate 4π(0-4 )(0-8 )( ) [4.47 coth(4.47) - ] mol/s The following Matlab pogam plots the concentation of oxygen within the oganism as a function of position. % Example.4-4 % alfa4.47e4; % m e-4; % m alfa4.47; C05e-5; % mol/m3 o(:50)/50; *o; -33

12 CC0*sinh(alfa*)./(o*sinh(alfa)); plot(o,c) gi on xlabel('/');ylabel('c_(mol/m^3)') Figue E.4-4 Oxygen concentation pofile in a spheical oganism. We now consie the iffusion of species into a spheical catalyst paticle whee homogeneous fist oe chemical eaction occus. The concentation pofile fo species within the spheical catalyst paticle is then C C sinh( α) sinh( α) (.4-) In this equation C is the concentation of species at the suface of the catalyst paticle an α is efine by the expession α, whee is the fist oe ate constant an is the iffusivity of in the paticle. t the cente of the bea, the concentation is given by α C ( 0) C sinh( α) (.4-) -34

GRAVITATION. Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., New Delhi -18 PG 1

GRAVITATION. Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., New Delhi -18 PG 1 Einstein Classes, Unit No. 0, 0, Vahman Ring Roa Plaza, Vikas Pui Extn., New Delhi -8 Ph. : 96905, 857, E-mail einsteinclasses00@gmail.com, PG GRAVITATION Einstein Classes, Unit No. 0, 0, Vahman Ring Roa