The parallel plate geometry

The large gap sizes available can be used to overcome the limitations encormtered using the cone-and-plate geometry, such as its sensitivity to 

Form factors for the parallel plate geometry, in terms of the apparent or Newtonian shear stress and the shear rate at r = are given below  

---

The samples were cured with 0.2, 0.4, 0.8 and 1.6 wt.% dicu-myl peroxide. In this way, we obtained twelve different networks with great variations in relaxation Intensities. Dynamic mechanical measurements were performed In torsion in the linear region (deformations smaller than 5 %) with a mechanical spectrometer, using the parallel-plate geometry. The frequency ranged from 0.01 to 15 Hz and the temperature was usually between 300 and 435 K. 

Such a generalization must reproduce, for example, the known results for the Casimir effect in the case of the parallel plates geometry at finite temperatures. Since energy is an additive quantity, we expect to have L- and T-dependent contributions plus a mixed (LT-dependent) contribution representing the interference of the two effects. In the next Section, we will show that the proper extension of expressions (16) and (21), for this case, is... 

To determine the potential at the midplane, the Poisson-Boltzmann equation must be solved for the parallel plate geometry. For a z—z electrolyte, the Poisson-Boltzmann equation is given by [18]... 

The linear viscoelastic properties of all samples were characterized by dynamic shear measurements in the parallel-plate geometry. Experimental details have been previously published [9]. Using time-temperature equivalence, master curves for the storage and loss moduli were obtained. Fig. 1 shows the master curves at 140°C for the relaxation spectra and Table 3 gives the values of zero-shear viscosities, steady-state compliances and weight-average relaxation times at the same temperature. 

The parallel disk geometry, also called the parallel plate geometry, consists of two disks with radius ro separated by the gap h (Figure 3-8). Assuming steady, laminar, and isothermal flow, the expression for shear rate is ... 

For file periodic case, we just change the definition of file dimensionless time variable source term, as follows for the parallel-plates geometry ... 

However, the use of the parallel plate geometry is not recommended for viscosity measurements, because the shear strain rate variation along the gap between the plates is larger than that experienced in concentric cylinder systems.However, there might be advantages when using the geometry in oscillatory studies. ... 

These different geometries present advantages and disadvantages. For example, the cone-and-plate arrangement offers a constant shear rate across the sample y, whereas in the parallel plate geometry, though it is easy to set and vary the gap, the shear rate is zero at the center and a maximum at the rim. 

Fluid inertia effects have been found to be very small for the cone-and-plate geometries typically supplied with these instruments. While inertial corrections are foimd to be imimportant for the parallel plate geometries, for shearing gaps of the order of 2 mm or less (except possibly for very thin fluids), they must be taken into accoimt in the concentric cylinder geometry (especially for high-density, mobile fluids). Evaluation methods are available for p, in the case of cylindrical and plane Couette flow, taking into account fluid inertia [Aschoff and Schummer, 1993]. 

Numerous variants of this technique have been created to adapt it for more practical applications. The most popular of these is the parallel-plate apparatus. Here the melt is placed between two parallel disks, 1-2 mm apart. The shear rate varies from the center to the circumference of the disk and corrections need to be performed [1]. The parallel-plate geometry is less sensitive to errors in gap and is also more able to handle filled materials. In contrast, cone-and-plate geometries are not useful in cases where the filler dimension is of the same order of magnitude as the gap. Parallel plates are also recommended in situations where rheology is measured as a function of temperature, where tool thermal expansion would otherwise affect the accuracy of the measurement. 

The parallel plate geometry [1,3] offers uniform current density and potential distribution. The incorporation of this electrode geometry into a plate and frame cell body, particularly in a modular filter-press format, provides a versatile workhorse for many electrochemical reactors. Many developments start with a small, single cell before being scaled-up by increasing the electrode area and then by designing a multiple cell stack in a filter press stmcture. Parallel plate cells have many advantages ... 

In Table 4 the value assumed by Fmax. d. and/Rcfd are quoted for this geometry as a function of the aspect ratio It is evident that, the circular geometry can be considered as the limit case for = 0 and the parallel plate geometry for = 1. 

In this contribution, we focus on results obtained using the parallel plate geometry shown in Figure 6 in which the structure probed is that which lies in the plane defined by the flow vector and the vorticity vector. 

The parallel-plate geometry is a popular and convenient choice of reactor for a diverse range of processes. Reasons for this include ... 

At low and medium frequencies (/ < IGHz), dielectric spectra are commonly taken in the parallel-plate geometry (cf Fig. 6) that consists of two flat elecfrodes of the area A that sandwich a sheet or film of the material under investigation with thickness d. From the complex capacitance, the dielectric permittivity spectrum can then be derived by... 

Hence, the velocity profile inside the parallel plate geometry is... 

In this section we discuss two examples of mixing in simple geometries parallel plates and concentric cylinders. The parallel plate geometry is used as the basis of the analysis of the single-screw extruder and concentric cylinders are used as part of the design of rotational dies. The flow kinematics are calculated and mixing in terms of striation thickness reduction is evaluated. 

Solution. Figure 6.11 shows the parallel plate geometry. Initially, the particles of the minor component can be simulated by idealized geometries such as cubes, spheres, and rectangular parallelepipeds (with the long axis along the z or... 

The effect of the flow geometry and flow type (Couette and Poiseuille) is shown in Figure 6.16. Note that the first trace of a tagged material does not come out of Poiseuille flow in parallel plate geometry earlier than 67% of the mean residence time. Finally, Poiseuille flow of Newtonian fluids in a circular pipe (CPPF) exhibits the same F function as Couette flow of Newtonian fluids in the parallel plate geometry (PCF see Problem 6A.13). 

---