Handbook of Solvents - George Wypych - ChemTech - Ventech!

374 Semyon Levitsky, Zinoviy Shulman
where:
2 −1 2
2 −1
( 6/ π) ( 6π) ( 1 1)
, ε α[ 1 ( R) ] , α ( 0 R)
/ ( 1 R ) [7.2.54]
h = Ja = Le Di + M Di = K + f k K = k −k −k
Ja Jacob number, Ja = cfΔTf(εl) -1
ΔTf superheat of the solution with respect to the interface, ΔTf =Tf0 -TfR Le Lewis number, af /D0 Kα mass fraction of the evaporated liquid 46
Here M1 follows certain cumbersome equation, 52 including f(k). The approximation
Ja>>1 corresponds to the case of a thin thermal boundary layer around the growing bubble.
Since, for polymeric solutions Le >> 1, the condition of small thickness of the diffusion
boundary layer is satisfied in this situation as well.
We start the analysis of the solution [7.2.54] from the approximationf=0that corresponds
to D ≈ D0 = const. Then from [7.2.54] it follows:
( )
−1
K Le c l Δ T
[7.2.55]
α =
f
Because of the diffusion resistance, the solvent concentration at the interface is less
then in the bulk, k R k 0 -k R. It permits to assume in [7.2.56] k R ≈k 0 and, hence, to
find easily the vapor temperature.
In the diffusion-equilibrium approximation (i.e. Le → 0) ΔT =ΔT*. When the diffusion
resistance increases, the actual superheat ΔT lowers and, according to [7.2.56], at
Le →∞ ΔT→0. However, in the latter case the assumptions made while deriving [7.2.56],
are no longer valid. Indeed, the Ja number, connected with the superheat of the solution with
respect to the interface, is related to the Ja 0 value, corresponding to the bulk superheat, by
Ja=Ja 0(ΔT/ΔT*). Since the ratio ΔT/ΔT* varies in the range (0, 1), then, at small diffusion coefficients,
it may be that Ja > 1. In this case, the asymptotic solution of
the problem takes the form 46 h = Ja, and, for thin diffusion boundary layer, it can be received
instead of [7.2.54]:
−1
2 ( 6 ) ( 1 )
h = Ja = / π Le Di + M
[7.2.57]
1
Finally, at Di