Molecular transport term

The two terms on the right-hand side of this expression appear in closed form. However, the molecular transport term vV2 (Ut) is of order Re 1, and thus will be negligible at high Reynolds numbers. 

The molecular transport term vV2(m m ) is closed, but negligible (order ReL 1) in high-Reynolds-number turbulent flows. The production term... 

The first term on the right-hand side of this expression is the molecular transport term that scales as Sc Re 1. Thus, at high Reynolds numbers,26 it can be neglected. The two new unclosed terms in (3.88) are the scalar flux (u.ja), and the mean chemical source term (Sa(chemical reacting flows, the modeling of (Sa(0)) is of greatest concern, and we discuss this aspect in detail in Chapter 5. 

Thus, the closure problem reduces to finding an appropriate expression for the scalar flux (Ujtp). In high-Reynolds-number turbulent flows, the molecular transport term is again negligible. Thus, the scalar-flux term is responsible for the rapid mixing observed in turbulent flows. 

The triple-correlation term (u xix/)") and the molecular-transport term T -, defined by... 

The molecular-transport term rV2( 2) will be negligible at high Reynolds number. The scalar-variance-production term V4, is defined by... 

The estimate of (U X > will also contain statistical error. However, at high Reynolds numbers, the molecular transport term in (6.178) will be small, and thus noise in this term is less problematic. 

The Reynolds nenlogy assumes that E = D aed neglects the molecular transport terms that is, it assumes thel the transport resistances reside wholly in the turbulent region. The result of combining Eqs. (2.4-25a) aed (2.4-25b) is... 

Fig. 19.1. Temperature dependencies of the primary nucleation rate (I) (A) and the linear crystal growth rate (G) (Q) for poly(ethylene succinate) (PEISU) [14] with a molecular weight (M) of 8,770. The solid and broken lines are results from the best fitting procedure for G based on Eq. (19.2) and for I based on Exj. (19.11) by the Arrhenius and the WLF expressions of the molecular transport term, respectively...

Fig. 19.3. Temperature dependence of common logarithm of crystal growth rate from the melt for a variety of crystalline materials Glycreine [1], Li20-2Si02 [9, PBO-2B2O3 [12] and isotactic poly(styrene) [19] (i-PS). Solid lines are best fitting by Arrhenius expression of the molecular transport term...

The ratio of Tg and T°j is well known to be 2/3 for the most of polymers [44 6]. That is, To comes near to (l/2)T°j. This result is much used for the best fitting procedure for crystal growth data. Anyhow, the activation energy for molecular transport term can be expressed by either or Arrhenius... 

AY [)mr were measured by a Rheovibron (Toyo Seiki) at a heating rate of 2°C/min and a frequency of lOH over the temperature range from 150 C to 200° C. These phenomenological results suggest that the activation energy for the molecular transport term in polymer crystallization is associated with molecular diffusion in the super-cooled melt above T. This indicates that the activation process in molecular transport in polymer crystalUzation could be similar to that in the confined molecular motion in the rubbery state above T... 

Fig. 19.6. Natural logarithm of G vs. crystallization temperature (Tc) (.4) and super-cooling (T /TziT) dependence of natural logarithm of G plus the molecular transport term of ziE/RT (Q) for PE [53]. Broken lines indicate morphological transition from truncated lozenge to leticular crystal...

Fig. 19.12. Temperature dependence of nucleation rate for PESU with M = 9,150 [59]. Open circle is the nominal nucleation rate reduced by the initial view area and open triangle is the real nucleation rate reduced by the real effective area during crystallization. Solid line is the best fitting by Arrhenius expression of the molecular transport term, respectively...

K, is generally expressed as Kj= n<7e<7 /4H, . An application to nucleation rate based on Eq. (19.11) causes that a molecular transport term is... 

The pre-exponential factor of Gq in Eq. (19.2) can be expressed as a function of molecular weight as given by The molecular transport term... 

Gmax is presented by only one rate at a given molecular weight and can be formulated by equating to zero the derivative of Eq. (19.2), either Arrhenius or WLF expression of molecular transport term [75]. The crystal growth rate (G) can be formulated as a function of the maximum crystal growth rate and the reduced super-cooling (Z) based on Eq. (19.2) with the Arrhenius expressions in the molecular transport term, as follows ... 

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