The Dynamics of a Roll Press Nip

The problem presented concerned the dynamics of a roll press nip, a crucial component in a paper-making machine. Modern commercial paper-making machines are huge items of equipment. They may be as long as a football field and cost many millions of dollars each. Integrity of the process is extremely important; the paper in such a machine travels so fast (up to 20 km/sec) that a break is viewed as a major calamity and may take many man-hours and dollars to recover from. The size and speed of the machines means that it is not easy to make measurements as the paper passes through. The hostility of the environment therefore dictates that a thorough theoretical understanding of the important parts of the process is crucial if the processes involved are to be optimized. In this study we do not seek to answer anyone specific question, but rather wish to propose a general framework for modeling the flow and deformation under a roll press nip. Because of the difficulty of making measurements in the nip region and the need to closely control the process, the distributions of pressure, velocity and felt porosity within the nip have traditionally been subjects of great debate. Previous treatments have included lubrication theory models and "Bernoulli" based models. Although some progress may be made using thin layer theory, we shall show the required modeling does not take the form of standard lubrication theory. As far as models based on Bernoulli's equation are concerned, we simply note that the discussion below shows that the drag force exerted by the felt on the liquid is a key physical component of the flow process. Clearly, a full three-phase flow treatment is required. In this study we will thus address the following questions: (i) Is it possible to propose a general theoretical treatment of the roll press nip? (ii) What determines the physics of the water movement within the paper and felt in the roll press nip and how is this connected to the details of the air movement and the deformation of the felt? (iii) When a general model has been proposed, is it possible exploit the geometry within the nip to generate some simple exact solutions? (iv) What are the key non-dimensional parameters in the problem and how large are they likely to be for realistic paper-making machines? A further matter of interest concerns the influence that the size, shape and separation of the rollers have on the whole process. We approach the modeling from a rather general point of view, beginning by including all effects that might be important and then making clearly defined assumptions to simplify the equations. In this way it is possible to make changes to the model if circumstances change

Oblique shock reflection from an axis of symmetry: shock dynamics and relation to the Guderley singularity

Oblique shock reflection from an axis of symmetry is studied using Whitham's theory of geometrical shock dynamics, and the results are compared with previous numerical simulations of the phenomenon by Hornung (2000). The shock shapes (for strong and weak shocks), and the location of the shock-shock (for strong shocks), are in good agreement with the numerical results, though the detail of the shock reflection structure is, of course, not resolved by shock dynamics. A guess at a mathematical form of the shock shape based on an analogy with the Guderley singularity in cylindrical shock implosion, in the form of a generalized hyperbola, fits the shock shape very well. The smooth variation of the exponent in this equation with initial shock angle from the Guderley value at zero to 0.5 at 90° supports the analogy. Finally, steady-flow shock reflection from a symmetry axis is related to the self-similar flow

Numerical shock propagation using geometrical shock dynamics

A simple numerical scheme for the calculation of the motion of shock waves in gases based on Whitham's theory of geometrical shock dynamics is developed. This scheme is used to study the propagation of shock waves along walls and in channels and the self-focusing of initially curved shockfronts. The numerical results are compared with exact and numerical solutions of the geometrical-shock-dynamics equations and with recent experimental investigations

Parallel Computation of Three-Dimensional Flows using Overlapping Grids with Adaptive Mesh Refinement

This paper describes an approach for the numerical solution of time-dependent partial differential equations in complex three-dimensional domains. The domains are represented by overlapping structured grids, and block-structured adaptive mesh refinement (AMR) is employed to locally increase the grid resolution. In addition, the numerical method is implemented on parallel distributed-memory computers using a domain-decomposition approach. The implementation is flexible so that each base grid within the overlapping grid structure and its associated refinement grids can be independently partitioned over a chosen set of processors. A modified bin-packing algorithm is used to specify the partition for each grid so that the computational work is evenly distributed amongst the processors. All components of the AMR algorithm such as error estimation, regridding, and interpolation are performed in parallel. The parallel time-stepping algorithm is illustrated for initial-boundary-value problems involving a linear advection-diffusion equation and the (nonlinear) reactive Euler equations. Numerical results are presented for both equations to demonstrate the accuracy and correctness of the parallel approach. Exact solutions of the advection-diffusion equation are constructed, and these are used to check the corresponding numerical solutions for a variety of tests involving different overlapping grids, different numbers of refinement levels and refinement ratios, and different numbers of processors. The problem of planar shock diffraction by a sphere is considered as an illustration of the numerical approach for the Euler equations, and a problem involving the initiation of a detonation from a hot spot in a T-shaped pipe is considered to demonstrate the numerical approach for the reactive case. For both problems, the solutions are shown to be well resolved on the finest grid. The parallel performance of the approach is examined in detail for the shock diffraction problem

Homogenization of the Equations Governing the Flow Between a Slider and a Rough Spinning Disk

We have analyzed the behavior of the flow between a slider bearing and a hard-drive magnetic disk under two types of surface roughness. For both cases the length scale of the roughness along the surface is small as compared to the scale of the slider, so that a homogenization of the governing equations was performed. For the case of longitudinal roughness, we derived a one-dimensional lubrication-type equation for the leading behavior of the pressure in the direction parallel to the velocity of the disk. The coefficients of the equation are determined by solving linear elliptic equations on a domain bounded by the gap height in the vertical direction and the period of the roughness in the span-wise direction. For the case of transverse roughness the unsteady lubrication equations were reduced, following a multiple scale homogenization analysis, to a steady equation for the leading behavior of the pressure in the gap. The reduced equation involves certain averages of the gap height, but retains the same form of the usual steady, compressible lubrication equations. Numerical calculations were performed for both cases, and the solution for the case of transverse roughness was shown be in excellent agreement with a corresponding numerical calculation of the original unsteady equations