Force balance calculations

The yield stress may then be calculated on the basis of a simple force-balance calculation from the following equation ... 

Many of the uncertainties associated with the above tests arise from the model used to calculate the stress or strain transferred between the matrix and fibre. (Laser Raman or Fluorescence Spectroscopy) LRS is a technique for determining directly the strain in the fibre in analogy to an embedded strain gauge. The strain is probed by recording a Raman (or fluorescence) spectrum of the reinforcing fibre at increments of applied strain. The shift in the frequency at which a stress-sensitive peak occurs can be used to estimate the strain profile in fhe reinforcing fibre. The rate of strain development in the fibre can be used to assess the quality of the interface. Quantification still requires an appropriate stress-transfer model although a simple force-balance calculation can be used to estimate the interfacial shear stress. 

By applpng a force balance, calculate the incipient fluidizing velocity for a system with particles of particle density 5000 kg/m and mean volume diameter 100 xm and a fluid of density 1.2kg/m and viscosity 1.8x10 Pas. Assume that the voidage at incipient fluidization is 0.5. 

Equations 3 to 7 indicate the method by which terminal velocity may be calculated. Erom a hydrodynamic force balance that considers gravity, buoyancy, and drag, but neglects interparticle forces, the single particle terminal velocity is... 

For the gravity discharge case, the height of the fluid at maximum vacuum, which is the point at which air would begin to backflow into the tank, is determined by Eq. (26-54). Equation (26-55) calculates the corresponding vacuum in the tank s headspace at this hquid height. Since the drain nozzle is open to the atmosphere, this solution is a static force balance that is satisfied when the sum of the internal pressure and the remaining fluid head is equal to the atmospheric pressure. 

An advantage of this approach to model large-scale fluidized bed reactors is that the behavior of bubbles in fluidized beds can be readily incorporated in the force balance of the bubbles. In this respect, one can think of the rise velocity, and the tendency of rising bubbles to be drawn towards the center of the bed, from the mutual interaction of bubbles and from wall effects (Kobayashi et al., 2000). In Fig. 34, two preliminary calculations are shown for an industrial-scale gas-phase polymerization reactor, using the discrete bubble model. The geometry of the fluidized bed was 1.0 x 3.0 x 1.0 m (w x h x d). The emulsion phase has a density of 400kg/m3, and the apparent viscosity was set to 1.0 Pa s. The density of the bubble phase was 25 g/m3. The bubbles were injected via 49 nozzles positioned equally distributed in a square in the middle of the column. 

The Eulerian gas velocity field required in both the mass balance and the above transport equation for nh is found by an approximate method first, the complete field of liquid velocities obtained with FLUENT is adapted downward because the power draw is smaller under gassed conditions next, in a very simple way of one-way coupling, the bubble velocity calculated from the above force balance is just added to this adapted liquid velocity field. This procedure makes a momentum balance for the bubble phase redundant this saves a lot of computational effort. 

Substitution of Equation (3.62) into Equation (3.60) gives the relative zero shear viscosity. When the shear rate makes a significant contribution to the interparticle interactions, the mean minimum separation can be estimated from balancing the radial hydrodynamic force, Fhr, with the electrostatic repulsive force, Fe. The maximum radial forces occur along the principle axes of shear, i.e. at an orientation of the line joining the particle centres to the streamlines of 6 = 45°. This is the orientation shown in Figure 3.19. The hydrodynamic force is calculated from the Stokes drag, 6nr 0au, where u is the particle velocity, which is simply... 

Knowing the force balance bubble volume Vfb and the other system charctaeristics, the final bubble volume VF can be directly calculated from Eq. (61). 

Campbell et al. and Yamamuro et al. assume a form for (or F ) and then determine the solids conveying angle (p. Hyun and Spalding calculated q> using a force balance and two torque balances. Campbell and Dontula and Yamamuro et al. did not use a torque balance in their analysis. The following is a list of assumptions from all the models discussed. Additional assumptions will be added as needed. Assumptions that do not have keys were used by all authors ... 

In the second approach shown in Fig 3.12(b), a force is applied continuously using a Vickers microhardness indenter to compress the fiber into the specimen surface (Marshall, 1984). For ceramic matrix composites where the bonding at the interface is typically mechanical in nature, the interface shear stress, Tf, against the constant frictional sliding is calculated based on simple force balance (Marshall, 1984) ... 

The above process description allows t to be calculated, if the properties of the ice bed and brine are known. A force balance for steady-state operation yields the following relations for any inlet section geometry ... 

The calculation of velocity along the axis has been carried out by solving a force balance for a spherical particle arising along the spout ... 

With Eq. (6.4), the radial distance of the position where the maximum radial velocity of gas flow appears, rmax, can be calculated to be 1.974 times the diameter of the accelerating tube, R.dC. Assume the particles are carried out of the impingement zone at r = 1.974R.dC with an initial radial velocity of zero. From the force balance, the movement equations of sphere particles after leaving the impingement zone can be obtained as... 

Clarke and co-workers developed a model to calculate the thickness of the amorphous film observed in polycrystalline ceramics.37,38 The model is based on a force balance between an attractive van der Waals dispersion force that acts across the grain boundaries, any capillary forces present, and repulsive disjoining forces (such as steric forces and electrical double-layer forces) in the amorphous film.37,38 The repulsive steric force is based on the... 

At any radial position r, l/d(r) can be calculated from l/g(r) and e(r) by using Model LR through a trial /(A, r), which may be approximated by polynomials or other adequate functions with the parameter vector A, to be optimized with respect to Nst = min. Now, the problem becomes how to find K(r), that is, how to determine A, in order to satisfy energy minimization, force balance, continuity and the given boundary conditions. Therefore, Model OR is simplified for the PFC regime as the K-Radial (KR) model ... 

A simple use of mass-balance considerations that shows this net transformation of chlorinated ethenes is the observed accumulation of chloride in contaminant plumes. For example, in a study of a large plume of TCE-contaminated groundwater at Dover Air Force Base, Delaware, Witt et al. (2002) showed that the concentration of chloride increased as TCE concentrations decreased, suggesting the net transformation of the chlorine in TCE to chloride. This chloride tracer of chlorinated ethene biodegradation is of use for mass-balance calculations to demonstrate biodegradation processes in the held. These held observations formed the basis of experimental studies under controlled laboratory conditions that documented the many and varied microbial processes that transform and destroy chlorinated ethenes in groundwater systems. 

Figure 28-9 When two atoms i and j are separated by infinite distance, there are no interactions between them. As two nonbonded atoms approach one another, two forces have to be considered. Attractive dispersion forces (London forces) result from the interaction of instantaneous dipoles on each atom i and j. As the nonbonded atoms continue to approach one another, a repulsive interaction overwhelms the attractive interaction. and the energy curve rises sharply. The two nonbonded atoms can reach an equilibrium position where repulsive and attractive forces balance. Different mathematical relationships have been used in force field calculations to reproduce the nonbonded steric interactions.

---