Melting conduction without melt removal

As pointed out in the previous section, melting can often be modeled in terms of simple geometries. Here we analyze the transient conduction problem in a semi-infinite solid. We compare the solutions of this problem, assuming first (a) constant thermophysical properties, then (b) variable thermophysical properties and finally, and (c) a phase transition with constant thermophysical properties in each phase. These solutions, though useful by themselves, also help demonstrate the profound effect of the material properties on the mathematical complexities of the solution. 

The equation of thermal energy (Eq. 2.9-16) for transient conduction in solids without internal heat sources reduces to 

If the thermal conductivity k and the product pCp are temperature independent, Eq. 5.3-1 reduces for homogeneous and isotropic solids to a linear partial differential equation, greatly simplifying the mathematics of solving the class of heat transfer problems it describes.1 

Example 5.2 Semi-infinite Solid with Constant Thermophysical Properties and a Step Change in Surface Temperature Exact Solution The semi-infinite solid in Fig. E5.2 is initially at constant temperature Tq. At time t — 0 the surface temperature is raised to T. This is a one-dimensional transient heat-conduction problem. The governing parabolic differential equation 

This transformation follows from general similarity solution methods, and is a similarity transformation. The term similar implies that profiles of the variable T — T. , t) (at different coordinates x) differ only by a scale factor. The profiles can be reduced to the same curve by changing the scale along the axis of ordinates. Problems that lack a characteristic length are generally amenable to this solution method. 

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We shall see later that this combination of the key variables is also characteristic of conduction heating with phase transfer. The heat flux is infinite at t = 0, but quickly drops with the inverse of t1/2. Thus after 10 s it is only 30% of the flux at 1 s, and after 60 s, it is only 13% of the heat flux at 1 s The obvious conclusion is that conduction melting without melt removal becomes inefficient for anything but short times. 

The preceding examples discuss the heat-conduction problem without melt removal in a semi-infinite solid, using different assumptions in each case regarding the thermophysical properties of the solid. These solutions form useful approximations to problems encountered in everyday engineering practice. A vast collection of analytical solutions on such problems can be found in classic texts on heat transfer in solids (10,11). Table 5.1 lists a few well-known and commonly applied solutions, and Figs. 5.5-5.8 graphically illustrate some of these and other solutions. 

The thermal conductivity of metals is much greater than that of all polyurethanes. The impact of this is that in the case of metals the heat is readily removed by the coolant into the stock and machine. The lower thermal conductivity of the polyurethane means that the heat remains near the surface of the part. Without proper care, the part can melt quite easily. Most polyurethanes start softening at about 135°C and melt to a gummy state by 180°C. 

It is clearly essential to separate the formed product from the mandrel without damage to the product and, if possible, to the mandrel since it can then be re-used. This is generally accomplished by using a cathode which is covered by a natural or chemically induced thick oxide layer. Suitable materials include titanium, chromium and steel. For some products, their shape predetermines that a permanent mandrel cannot be used. Non-permanent mandrels have to be constructed of a material which can be removed from the inside of the product and several techniques have been used. The non-permanent mandrel may be made from a low-melting-point metal (e.g. Zn, A1 or their alloys), a metal which may be removed by chemically etching or a non-metallic material (e.g. perspex, PVC or epoxy resins) soluble in organic solvents and plated by electrodeless deposition with a layer of silver or copper to make it conducting. 

The concept of low-molecular-weight imide prepolymers can be viewed as an alternative route to enhanced processability. The development of such systems has been conducted on the basis of three fundamental requirements. First, the prepolymers should be of low molecular weight, allowing for the possibility of a low melting point and low viscosity. Second, imide groups should be present in the prepolymer so as to remove the particularly troublesome polyamic acid to imide conversion process mentioned previously. Third, the prepolymers should have reactive terminal groups capable of reaction by an addition mechanism so as to convert the molten prepolymer to a cross-linked polymer without the harmful evolution of volatiles. 

Figure 7.94 shows the temperature profiles at three different screw speeds with and without conduction the heat flux is -10,000 W/m. At a screw speed of 10 rev/sec (600 rpm), the temperature drops about 2°C after a distance of 10 D. At a screw speed of 3.34 rev/sec (200 rpm), the melt temperature drops about 6.5°C at 0.83 rev/sec (50 rpm) the melt temperature drops about 27°C. Clearly, at low screw speeds, the melt temperature can be affected significantly by conduction through the barrel. The reason for the large effect of barrel cooling at low screw speed is because the residence time of the polymer increases with reducing screw speed. As a result, more time is available to remove heat from the polymer melt at low screw speed. 

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