Slip equation

Mixture conservation of mass equation Mixture conservation of momentum equation Mixture conservation of energy equation Slip equation (concerning the difference in velocity)... 

In addition, Turner and Trimble defined a slip equation of state combination as the specification of mass flux, momentum flux, energy density, and energy flux as single-valued functions of the geometric parameters (area, equivalent diameter, roughness, etc.) at any z location, and of mass flux, pressure, and enthalpy,... 

An example of a slip equation of state combination is for thermodynamic equilibrium with a slip ratio of... 

A finite correlation may replace the slip equations (6-8). With assumptions (1) and (2) and use of Eqs. (6-8)--(6-13), instead of two Eqs. (6-5), the pressure term is present only in Eq. (6-13), which may be solved separately. With assumptions (1) and (3),the phase energy equation (6-9) becomes equivalent to the phase mass conservation equation (6-3), thus reducing the order of the set. 

Equation (3.11) gives the first-order approximation to the temperature jump if it is assumed that the temperature gradient at the wall is the same as that at y = k. To obtain the higher-order approximation, the same approaeh is applied as that whieh was used to obtain the seeond-order velocity slip equation [2]. This results in (with 0 = T/Treference) ... 

The factor fj was measured experimentally (Carman, 1956) to fall between 0.8 and 1. Taking the average value of 0.9, the ratio of the Knudsen flux to the slip flux is about 1.4, meaning that the Knudsen flux under the molecular flow regime is higher than the flux predicted by the extra slip on the viscous flow. The reason for this is because the slip equation is an extension of the viscous flow, that is the fluid is still in its bulk state to induce viscosity. Therefore, it does not predict correctly the observed flux when Kn l, that is when the true molecular flow is dominant. 

As can be seen from the above, there is currently no clear consensus as to how liquids may slip against lyophobic solids. All of the proposed models are however essentially based on the same fluid-like principle - of enhanced mobility or reduced viscosity of an interfacial layer, be it a monolayer, a thicker layer of liquid or a gas layer. Probably because of this, they all predict a broadly similar form of slip equation, with a constant slip length. This implies that in all cases some degree of... 

For interactions between two quadmpolar molecules which have and 0g of the same sign, at a fixed separation r, the angular factor in equation (Al.5.131 leads to a planar, T-shaped stmcture, 0 = 0, 0g = nil, (ji = 0, being preferred. This geometry is often seen for nearly spherical quadmpolar molecules. There are other planar (ij) = 0) configurations with 0 = jr/2-6g that are also attractive. A planar, slipped parallel stmcture,... 

Clearly then, the continuum approach as outlined above is faulty. Furthermore, since our erroneous result depends only on the non-slip boundary condition for V together with the Navier-Stokes equation (4.3), one or... 

Equations (3.59) and (3.60) are recast in terms of their components and solved together. After algebraic manipulations and making use of relations (3.61) slip-wall velocity components are found as... 

G is a multiplier which is zero at locations where slip condition does not apply and is a sufficiently large number at the nodes where slip may occur. It is important to note that, when the shear stress at a wall exceeds the threshold of slip and the fluid slides over the solid surface, this may reduce the shearing to below the critical value resulting in a renewed stick. Therefore imposition of wall slip introduces a form of non-linearity into the flow model which should be handled via an iterative loop. The slip coefficient (i.e. /I in the Navier s slip condition given as Equation (3.59) is defined as... 

After the imposition of no-slip wall boundary conditions the last term in Equation (5.64) vanishes. Therefore... 

This problem requires use of the microscopic balance equations because the velocity is to he determined as a function of position. The boundary conditions for this flow result from the no-slip condition. AU three velocity components must he zero at the plate surfaces, y = H/2 and y = —H/2. 

For gas flow through porous media with small pore diameters, the slip flow and molecular flow equations previously given (see the Vacuum Flow subsec tion) may be applied when the pore is of the same or smaller order as the mean free path, as described by Monet and Vermeulen (Chem. E/ig. Pi og., 55, Symp. Sei , 25 [1959]). 

Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the sohd surface is called the boundary layer The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundaiy layer thickness is conventionally t en to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

During start-up or at high slips, the value of will be too high compared to / i and equation (1.3) will modify to... 

During a run, if the supply voltage to a motor terminal drops to 85% of its rated value, then the full load torque of the motor will decrease to 72.25%. Since the load and its torque requirement will remain the same, the motor will star to drop speed until the torque available on its speed-torque curve has a value as high as 100/0.7225 or 138.4% of T to sustain this situation. The motor will now operate at a higher slip, increasing the rotor slip losses also in the same proportion. See equation (1.9) and Figure 1.7. 

The higher the full load slip, the higher will be the rotor losses and rotor heat. This is clear from the circle diagram and also from equation (1.9). An attempt to limit the start-up current by increasing the slip and the rotor resistance in a squirrel cage motor may thus jeopardize the motor s performance. The selection of starling current and rotor resistance is thus a compromise to achieve optimum performance. 

From equation (1.9), slip loss = S P. If the full-load slip is S and the speed varied to slip, Sj the additional slip loss due to the increased slip... 

To achieve a better torque, the slip-ring rotors arc normally wound in star, in which case the rotor current is v3 time more than in delta for the same output. Also since the torque is proportional to the rotor current equation (1.1), the torque developed will be greater in this case. 

The inverter may be a current source inverter, rather than a voltage source inverter (.Section 6.9.4) since it will be the rotor current that is required to be vtiried (equation (1.7)) to control the speed of a wound rotor motor, and this can be independently varied through the control of the rotor current. The speed and torque of the motor can be smoothly and steplessly controlled by this method, without any power loss. Figures 6.47 and 6.48 illustrate a typical slip recovery system and its control scheme, respectively. 

If the field excitation is also lost, the generator will run as an induction motor again driving the primer mover as above. As an induction motor, it will now operate at less than the synchronous speed and cause slip frequency current and slip losses in the rotor circuit, which may overheat the rotor and damage it, see also Section. 1.3 and equation (1.9). A reverse power relay under such a condition will disconnect the generator from the mains and protect the machine. 

Grady and Asay [49] estimate the actual local heating that may occur in shocked 6061-T6 Al. In the work of Hayes and Grady [50], slip planes are assumed to be separated by the characteristic distance d. Plastic deformation in the shock front is assumed to dissipate heat (per unit area) at a constant rate S.QdJt, where AQ is the dissipative component of internal energy change and is the shock risetime. The local slip-band temperature behind the shock front, 7), is obtained as a solution to the heat conduction equation with y as the thermal diffusivity... 

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