Maximum separation force

However, in steady elongational flow, the maximum separating force in the connector is obtained when the dumbbell is aligned in the direction of flow and, again, for the case of two beads in contact is given by -... 

If the mixing device generates a simple shear flow, as shown in Fig. 3.23, the maximum separation forces that act on the particles as they travel on their streamline occur when they are oriented in a 45° position as they continuously rotate during flow. However, if the flow field generated by the mixing device is a pure elongational flow, such as shown in Fig. 3.24, the particles will always be oriented at 0° the position of maximum force. 

Eq. 106 describes the force as the separation between the surfaces is decreased. Let us now assume that, after reaching some minimum separation distance do, a force is applied to separate the surfaces. In this case, the force will differ slightly from that given in Eq. 106, as the asperities are extended above their original height up to a maximum extension 5s, at which point the asperity will separate from the substrate. In order to calculate the separation force Ps, one must change the lower limit of integration in Eq. 106 to... 

Example 2.7 A nylon ring with a nominal inside diameter of 30 mm, an outer diameter of SO mm and a width of S mm is to be made an interference fit on a metal shaft of 30 mm diameter as shown in Fig. 2.17. The design condition is that the initial separation force is to be 1 kN. Calculate (a) the interference on radius needed between the ring and the shaft and (b) the temperature to which the nylon must be heated to facilitate easy assembly. What will be the maximum stress in the nylon when it is in position on the shaft The coefficient of friction between nylon and steel is 0.2S. The short-term modulus of the nylon is 1 GN/m, its Poisson s ratio is 0.4 and its coefficient of thermal expansion is 100 X 10- °C- . 

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... 

Of course, when multiple pairs of electrons participate in double or triple covalent bonds, those electrons stay within the same bonding axis. Lone pairs repel other lone pairs more strongly than they repel bonding pairs, and the weakest repulsion is between two pairs of bonding electrons. Two lone pairs separate themselves as fcir apart as they can go, on exact opposite sides of an atom if possible. Electrons involved in bonds also separate themselves as far apart as they can go but with less force than two lone pairs. In general, all electron pairs try to maintain the maximum mutual separation. But when an atom is bonded to many other atoms, the ideal of maximum separation isn t always possible because bulky groups... 

The number of forces separating the particles is smaller. Repulsive forces act between particles with the same electrostatic charge. The mixing of fiuids leads to the development of shear forces, which try to separate the particles. The maximum hydrodynamic force acting on spheres in a uniform shear field can be expressed as [22,24] ... 

The bond between the two atoms of a diatomic molecule is characterised by a force constant of lOOON/m. This bond is responsible for a vibrational absorption at 2000 cm Accepting that the energy of radiation is transformed into vibrational energy, estimate a value for the length of the bond at the maximum separation of the two atoms. 

If the reactor flow is dominated by forced convection and free of laminar eddies, the same expression (equation 25) can be used to estimate the maximum separation between the mixing point and the growth interface. For a given mass flow, the Peclet number, Pe, is independent of pressure, whereas the linear velocity increases with decreasing pressure therefore, the maximum allowable length increases with the square of the decreasing pressure, and operation at reduced reactor pressures is advantageous when sharp interfaces are desired. 

Calendering problem with a Newtonian viscosity polymer. A calender system with R = 10 cm, w = 100 cm, h0 = 0.1 mm operates at a speed of U =40 cm/s and produces a sheet thickeness In = 0.0218 cm. The viscosity of the material is given as 1000 Pa-s. Estimate the maximum pressure developed in the material, the power required to operate the system, the roll separating force and the adiabatic temperature rise within the material. 

Figure 6.22 depicts schematically the flow configuration. Two identical rolls of radii R rotate in opposite directions with frequency of rotation N. The minimum gap between the rolls is 2H0. We assume that the polymer is uniformly distributed laterally over the roll width W. At a certain axial (upstream) location x = X2 (X2 < 0), the rolls come into contact with the polymeric melt, and start biting onto it. At a certain axial (downstream) location x A), the polymeric melt detaches itself from one of the rolls. Pressure, which is assumed to be atmospheric at X2, rises with x and reaches a maximum upstream of the minimum gap location (recall the foregoing discussion on the pressure profile between non-parallel plates), then drops back to atmospheric pressure at X. The pressure thus generated between the rolls creates significant separating forces on the rolls. The location of points A i and X2 depends on roll radius, gap clearance, and the total volume of polymer on the rolls in roll mills or the volumetric flow rate in calenders. 

Calendering of Polymers The Newtonian Haskell Model A 0.2-m-diameter, 1-m-wide, equal-sized-roll calender operates at a speed of 50 cm/s. At a gap separation of 0.02 cm, it produces a 0.022-cm-thick film. Assuming a Newtonian viscosity of 104 poise, calculate in the last nip (a) the maximum pressure (b) the separating force and (c) estimate the mean temperature rise. 

Separating Force between Rolls in an Experimental Calender A cellulose acetate-based polymeric compound is calendered on a laboratory inverted, L-shaped calender with 16-in-wide rolls of 8 in diameter. The minimum gap between the rolls is 15 mil. The sheet width is 15 in. Calculate the separation force and the maximum pressure between a pair of rolls as a function of exiting film thickness, assuming that film thickness equals the gap separation at the point of detachment. Both rolls turn at 10 rpm. The polymer at the calendered temperature of 90°C follows a Power Law model with m = 3 x 106 dyne.s"/cm2 and n = 0.5. [Data based partly on J. S. Chong, Calendering Thermoplastic Materials, J. Appl. Polym. Sci., 12, 191-212 (1968).]... 

The relaxation force is zero when the centers of charge of the ion and its cloud coincide, and it is nonzero when they are separated. So let it be assumed in this approximate treatment that the relaxation force is proportional to d, i.e., proportional to the distance through which the ion has moved from the original center of charge of the cloud. On this basis, the relaxation force will be given by the maximum total force of the atmosphere on the central ion, i.e., z multiplied by the... 

DLVO theory predicts one maximum separating two minima for the disjoining pressure-distance isotherm (30), which suggests the possible existence of two types of films 1) Common thick film achieved by the balance of electrostatic and dispersion forces and 2) Newton film (thin film) with approximately a bilayer structure. Increasing the electrolyte concentration depresses the electrostatic repulsive forces and causes transition from the thick to the thin film. The electrostatic effect will of course vary with difference in the structure of the double layer. 

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