Mass Transfer Between a Sphere and an Unbounded Fluid

Transcription

1 Mass Tansfe Between a Sphee and an Unbounded Fluid R. Shanka Subamanian Depatment of Chemical and Biomolecula Engineeing Clakson Univesity When a single-component liquid dop evapoates into ai, o when a solid, modeled as a singlecomponent sphee, dissolves in a liquid o sublimes into a gas, we can constuct a simple model of the diffusive tanspot that occus between the object and the suounding fluid. The model can help us calculate the ate of mass tansfe, and eventually the ate of change of the adius of the sphee with time. a ssumptions. The sphee contains a pue component ; theefoe, we need to conside the mass tanspot pocess only in the suounding fluid.. The fluid is unbounded in extent and quiescent. It contains only the diffusing species and a non-tansfeing species B. 3. The motion aising fom diffusion can be neglected. This equies that eithe the mixtue in the fluid be dilute in species, consisting pimaily of the non-tansfeing species B, o that the ate of mass tanspot be small. 4. The poblem is spheically symmetic. This means that in a spheical pola coodinate system (, θφ, ) thee ae no gadients in the pola angula coodinate θ, o in the azimuthal angula coodinate φ. 5. fte an initial tansient, steady state is assumed to pevail. This implies that the change in size of the sphee due to mass tansfe occus on a time scale that is vey lage compaed with the time scale fo the diffusion pocess fo a given adius of the sphee to each steady state. 6. Thee ae no chemical eactions.

2 Because thee ae no gadients in the θ and φ diections, and thee is no time-dependence, the flux N depends only on. t steady state the ate at which species entes the spheical shell shown at location must equal the ate at which species leaves the shell at +. a Using the standad symbol fo the mola flux of at these two locations, we can wite the steady state consevation of mass statement as π π + + = 4 N 4 N Dividing though by the facto 4π, and eaanging yields ( ) N ( ) N + + = Taking the limit as leads to the odinay diffeential equation d N d = We can integate this immediately to obtain N C = C, which can be ecast as N = whee C is an abitay constant of integation. Now, we poceed to use Fick s law. dx N = x N + NB cdb d

3 The fist tem in the ight side coesponds to convective tanspot, which can be neglected in this poblem because of assumptions and 3. Thus, we obtain the following fist ode odinay diffeential equation fo the mole faction of species in the fluid. dx C = d c D B Integation of this equation is staightfowad, and leads to the following solution. C x = + C cd B Thee ae two abitay constants that need to be evaluated. Theefoe, we must wite two bounday conditions. t the suface of the sphee, we can assume equilibium to pevail between the two phases. Fo example, if species is evapoating into a gas, the patial pessue of species in the gas phase at the inteface can be assumed to be equal to its equilibium vapo pessue at the pevailing tempeatue. If the gas mixtue is assumed ideal, then the mole faction of species in the gas phase at the inteface is the atio between this equilibium vapo pessue of and the pevailing total pessue in the gas phase. In non-ideal cases, a coesponding esult can be used to obtain the equilibium mole faction of species in the gas phase at the inteface. Likewise, fo a solid dissolving in a liquid, o subliming into a gas, the equilibium mole faction of species in the fluid at the inteface can be obtained. = x a x Fa fom the sphee, we can assume the composition to appoach that in the fluid in the absence of the sphee. Thus, x ( ) = pplication of these two bounday conditions pemits us to evaluate the constants C and C as C = cd ax C = B Substituting these esults in the solution leads to the following esult fo the adial distibution of the mole faction of species in the fluid. x a = x The flux of species is given by 3

4 = N cd ax B so that the mola ate of mass tansfe at the suface of the sphee can be witten as W a N a c D a x D ac = 4π = 4π B = 4π B whee we have used the fact that the poduct cx = c, the mola concentation of in the fluid at the inteface. ssuming that the mola ate of tanspot is elatively small, we can use a mass balance on the sphee to deduce the ate of change of its size with time. Let the molecula weight of be M, and the density of the sphee be. Then, we can wite d 4 da dt 3 π = π = dt π 3 a 4 a 4 DB M ac which leads to a diffeential equation fo the time-dependence of the adius of the sphee. da a dt = DB M c If the adius at time zeo is a, then the solution can be witten as D B M c a t = a t The Quasi-Steady State ssumption Note that we assumed steady state to pevail in the diffusion poblem, which, stictly speaking, equies the size of the sphee to emain unchanged. s stated in assumption 5, this only equies that the time scale ove which the sphee changes appeciably in size be lage compaed with the time scale ove which the diffusion pocess aound a sphee of constant size eaches steady state. Then, the ate of mass tansfe fom the sphee to the fluid can actually be used to calculate the time evolution of the size of the sphee. This type of assumption is called a quasi-steady state assumption. We can make a judgment about whethe it is a good assumption in a given situation by compaing these two time scales. The time needed fo the diffusion pocess aound a sphee of adius a to each steady state is appoximately of the same ode of magnitude as a / D B. By estimating the time it takes fo the sphee to completely dissolve in the fluid, we can get an idea about the time scale fo the size to change appeciably. Fom the equation fo the adiustime histoy of the sphee, this time scale is found to be of the ode of magnitude of a, whee we have discaded the facto, because this is only an ode of magnitude D M c B 4

5 estimate. Using these esults, the following estimate can be obtained fo the atio of the two time scales. Time fo diffusion pocess to attain steady state a / D M c = = Time fo sphee to change in size appeciably a / D M c B ( B ) Theefoe, in this poblem, assumption 5 would be valid when the dimensionless goup M c. ( ) / The quasi-steady state assumption is invoked commonly in poblems whee thee ae two vey diffeent time scales involved. Fo example, we can use the quasi-steady assumption in the poblem of calculating the ate of change of the height of liquid in a lage stoage tank though a small pipe at the bottom. To calculate the velocity of flow out of the pipe and theefoe the volumetic flow ate, we can assume the level of the fluid in the tank to emain constant. fte obtaining such a volumetic flow ate fom a steady-state model, it can be used in an unsteady mass balance on the contents of the tank to calculate the ate of change of height of the liquid in the tank. 5