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Drying of spherical granule

  1. Nov 26, 2014 #1

    Maylis

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    1. The problem statement, all variables and given/known data
    Fig. below shows the cross-section of a porous spherical granule of radius a. The pores are initially saturated with water. The granule dries in air at pressure P and temperature T. The drying rate is controlled by diffusion of water vapor through the dry region B; the shrinking core A has the original moisture content ##\rho## kmol per unit granule volume.

    Using the equation within the region B, ##N_{1}c_{2} - N_{2}c_{1} = -DC \frac {dc_{1}}{dr}## show that the drying
    time is
    $$ \frac {\frac {\rho a^{2} R T}{6DP}}{ ln \Big(\frac {P}{P-P^{*}} \Big)} $$
    where p* is the vapor pressure.

    For a granule 10 mm in diameter, the drying time at 25 °C is 20 hours, the initial
    water content being 50 mg; P = 1 bar, p* = 0.032 bar. Estimate D.
    2. Relevant equations


    3. The attempt at a solution
    I am having a lot of problems solving these mass transfer problems, I am lacking intuition for the problem

    So I start with the equation
    $$N_{1}c_{2} - N_{2}c_{1} = -DC \frac {dc_{1}}{dr}$$
    I am thinking 1 refers to water and 2 refers to air. I think the flux of air into the water, ##N_{2}##, is negligible since air is not very soluble in water, thus I am saying that term is zero, leaving me with
    $$N_{1}c_{2} = -DC \frac {dc_{1}}{dr} $$
    substituting ##c_{2} = C - c_{1}##
    $$N_{1}(C - c_{1}) = -DC \frac {dc_{1}}{dr} $$
    rearranging and integrating,
    $$ \int_{\rho}^{0} \frac {dc_{1}}{C-c_{1}} = - \frac {N_{1}}{DC} \int_0^a dr $$
    $$ln \Big(\frac {C}{C - \rho} \Big) = - \frac {N_{1}}{DC}a$$

    From here I don't know where I will get those pressure terms
     

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    Last edited: Nov 26, 2014
  2. jcsd
  3. Nov 26, 2014 #2
    You're good up to here. Nice job and nice reasoning.

    But, N1 is a function of r in your problem (so the integrations are incorrect). This is very similar to the problem you were doing the other day for evaporation of a drop. The above equation applies in the region between r = x and r = a. You need to take into account the spherical geometry, just like the problem the other day. 4πr2N1 is constant in this region. You need to express the C and c in terms of the total pressure P and the partial pressure of water vapor (which is equal to the equilibrium vapor pressure at r = x), respectively. You need to solve for 4πr2N1, which is the rate at which water is leaving the inner core.

    Chet
     
  4. Nov 27, 2014 #3

    Maylis

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    Gold Member

    Thanks, solved it now
     
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