Frozen layer

Calculate the gap-wise velocity distribution. The thiekness of frozen layer is determined by the effective zero velocity region (Fig. 3.2). This calculation requires an accmate prediction of viscosity ranging from the processing temperature down to the temperature of the material at the mold wall. 

Use the concept of a no-flow temperature which indicates the temperature at which the stagnation of flow occurs (see, for example, Janeschitz-Kriegl 1979 Kennedy 1995). Although most reported studies simply assume a constant value of the no-flow temperature, the no-flow temperature is related to processing. This is especially true for semicrystalline materials. It would be more appropriate to use a no-flow temperature depending on crystallization. 

Use the increase of viscosity relative to the viscosity at the initial melt temperature as a no-flow condition. Janeschitz-Kriegl (1977) defines an increase of viscosity by a factor of 2.7 as a criterion of solidification. 

Both methods 1 and 3 involve using of constitutive model for viscosity prediction. Care should be taken to ensure the realistic behavior of the viscosity model chosen. Kennedy (1995) has shown that, while some existing models can deal with fluid behavior above the melting point, the extrapolation to low temperatures does not predict the sharp increase in viscosity for the semicrystaUine polymer. To produce a sudden rise in viscosity, we may need to incorporate the crystallization kinetics in rheology. This will be discussed in more detail in the next chapter. 

---

The next two figures, Figs. E13.2g and E13.2h, show the thickness of the fraction of frozen layer, and regions of trapped air. Both are important for designing a good mold. 

Polymer orientation varies through the thickness of the injection-molded part owing to the fountain flow of the melt in the mold cavity. The flow at the center of the cross-section is deformed through extension and the highly stretched flow front rolls up to the cold mold surface, where orientation is frozen in a thin surface layer. The rest of the melt required to fill the cavity flows under this stationary frozen layer in more or less a plug fashion, with minimum orientation. Surface orientation in an injection-molded part can be significantly different from that in the core of the part. 

We often cut a watermelon in half and pul it into the freezer to cool it quickly. But usually we forget to check on it and end up having a watermelon with a frozen layer on the top. To avoid this potential problem a person wants to set the timer such that it will go off when the temperature of the exposed surface, of the watermelon drops to 3 C. 

Figure 9 (a) Schematic representation of fountain flow, showing the velocity and shear rate profiles and the deformation of a cubic element of melt as it approaches the flow front, (b) Model for growth of the frozen layer in a mould cavity... 

Sublimation always starts at an open surface and then moves inwards into the sample. After some of the ice is sublimated, the sample exhibits two distinct regions, namely the dry layer (from which ice crystals have sublimated) and the frozen layer (where ice crystals are still present). These two regions meet at the so-called ice interface , sublimation interface , freeze-drying interface or, simply, interface . 

This paper presents the application of a model based predictive control strategy for the primary stage of the freeze drying process, which has not been tackled until now. A model predictive control framework is provided to minimize the sublimation time. The problem is directly addressed for the non linear distributed parameters system that describes the dynamic of the process. The mathematical model takes in account the main phenomena, including the heat and mass transfer in both the dried and frozen layers, and the moving sublimation front. The obtained results show the efficiency of the control software developed (MPC CB) under Matlab. The MPC( CB based on a modified levenberg-marquardt algorithm allows to control a continuous process in the open or closed loop and to find the optimal constrained control. 

To carry out the objective of this paper, we consider a one-dimensional freeze drying model based on the work of Liapis and Sadikoglu [8]. During the primary drying, the vial contains two regions a dry layer, in which the majority of water was sublimated and a frozen layer. These two areas are separated by a moving interface called the sublimation front. In this work, it is assumed that [5] ... 

Where Ti is the dried layer temperature, T2 is the frozen layer temperature, is the mass transfer flux of the water vapour, Cj is the bound water and H is the sublimation interface. The different parameters of the model are presented in [12], In this work, we use a simplified equation to describe the dynamic of the mass flux based on the diffusion equations of Evans. The equation is given by the following expression ... 

As a result of the temperature difference between the two fluids heat will flow from the liquid subject to freezing, across the metal wall and into the coolant. The rate of heat removal will of course be dependent upon the resistance to heat flow provided by the fluids themselves, the metal wall, the solid frozen layer, and any fouling resistance on the coolant side. 

For steady state conditions (i.e. constant frozen layer thicknesses) it may be assumed that the temeprature of the liquid/frozen layer interface is the freezing temperature 2 - and the only heat removal is the heat transferred across the resistances from the "hot" liquid to the coolant. Referring to Fig. 9.3 the wall temperatures of the hot and cold fluids are T and 7 respectively. Under these conditions of steady state the heat flux. 

Rearranging gives the value of the equilibrium frozen layer thickness... 

Equation 9.9 indicates the effect of the variables on the thickness of the solid layer and confirms some intuitive observations. Increasing the liquid and coolant temperatures T and Z f e heat transfer coefficient will also reduce the frozen layer thickness. Opposite changes in the variables will have the converse effect. 

When no frozen layer exists x, = O so that equation 9.9 reduces to... 

It is also possible to define an dimensionless frozen layer thickness as... 

---