Flow analysis network

As referenced in Fig. 8.52, at any given point during the mold filling process there are five different types of nodes. These are, 

The Hele-Shaw model was used to describe the flow in the FAN formulation. For a Newtonian case, the flow is described by 

The continuity equation is satisfied by performing a mass (volume) balance around each control volume, i. Using the notation found in Fig. 8.53 the mass balance is written as 

If we perform a mass balance on all the control volumes in the system, we can write 

Note that all other entries in martrix [a] are zero, ain = 0. This matrix is symmetric and banded. The size of the band, if the cells are properly numbered, is very small compared to the size of a problem. We will discuss matrix storage, manipulation and solution in more detail in Chapter 9 of this book. Once the matrix system has been assembled, we can store and re-use it every time step. Every time step we apply the boundary conditions by setting all the pressures of the empty and partially filled nodes to zero. If the pressure on node i 

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Despite the widespread use of this machine for compounding an extensive range of polymer-based formulations, only very limited analytical work has been reported on its operational performance. In one report, a modified flow analysis network method of simulation was used to describe flow of a Newtonian... 

Z. Tadmor, E. Broyer, and C. Gutfinger. Flow analysis network (fan) - a method for solving flow problems in polymer processing. Poly. Eng. Sci., 14(9) 660, 1974. 

Based on the flow analysis network, Wang etal., [18] and Osswald [11] developed the finite element/control volume appproach (FEM-CVA) for injection and compression molding, respectively. Similar to FAN, FEM-CVA assigns a fill factor to every nodal point or nodal control volume. The nodal control volumes are constructed by connecting element centroids to element midsides, as shown in Fig. 9.28. 

The boundary conditions are defined in the same way as with the flow analysis network. The nodes whose control volumes are empty or partially filled are assigned a zero pressure, and the gate nodes are either assigned an injection pressure or an injection volume flow rate. Just as is the case with flow analysis network, a mass balance about each nodal control volume will lead to a linear set of algebraic equations, identical to the set finite element formulation of Poisson s or Laplace s equation. The mass balance (volume balance for incompressible fluids) is given by... 

C. Gutfinger, E. Broyer, and Z. Tadmor, Analysis of a Cross Head Film Blowing Die with the Flow Analysis Network (FAN) Method, Polym. Eng. Sci., 15, 385-386 (1975). 

Example 15.3 The Flow Analysis Network Method Clearly Eq. E15.2-22 is identical to Eq. E15.2-21. This is the basis for the flow analysis network (FAN) method developed by Tadmor et al. (30) to solve two-dimensional steady or quasi-steady state flow problems in injection molds and extrusion dies. In two-dimensional flows the pressure distribution is obtained by dividing the flow region into an equal-sized mesh of square elements... 

B. David, T. Sapir, A. Nir, and Z. Tadmor, Twin Rotor Mixers and Extruders, The Extended Cartesian Flow Analysis Network Method, Int. Polym. Process., 5, 155 (1990). 

The numerical procedure we are using is the FAN (Flow Analysis Network) technique devised by Tadmor, Broyer and Gutfinger [18] and applied by them to the analysis of injection molding in Cartesian coordinate. This represents a simplified form of the present problem where both surfaces are stationary and the intersurface distance H is largely constant. 

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