Uniform pressure assumption

When the inlet assumptions state that water is entering the GDL at its interface with the catalyst layer, further clarification must be made between what has been called the uniform flux assumption and the uniform pressure assumption. The imiform flux assumption includes an individual source of liquid water for every inlet throat along the GDL/catalyst layer interface, while the uniform pressure assumption includes only a single source of liquid water that is connected to each inlet throat along the GDL/catalyst layer interface. Pltysically, the uniform pressure assumption assumes that there is a water cluster outside the GDL with negligible hydraulic resistance from one side to the other. Due to the microstracture of the catalyst layer, this scenario would approximate reality only if a pocket of liquid water could form between the catalyst layer and the GDL. Conversely, the uniform flux assumption assumes no hydraulic connectivity outside of the GDL between inlet locations. A compromise between these two assumptions was made by Hinebaugh and Bazylak in a 2D stmctured pore network model of GDL invasion, where the first row of pores and throats within the GDL is initialized as fully saturated. Similar to the uniform flux assumption, a liquid water source was... 

Constant pressure means constant f ext) because we define our constraints with respect to the surroundings, where we control the process. In the last step, we assume mechanical equilibrium between the system and the surroundings. This is an excellent assumption for slow processes with systems with movable boundaries, for which we will generally take f ext — P- It is not a good assumption for processes in which there is a sudden change of constraints, such as an explosion or the removal of a stop which secures a piston. In such cases, the system may not even have a uniform pressure. 

Although Eq. (91) was found to be in approximate agreement with the experimental data on which it was based, it does not encompass the observation of previous workers that Tfm first increases before it starts to decrease with increasing particle size. This limitation of Eq. (91) has been attributed by Lefroy and Davidson in the case of very fine particles to breakdown of the assumption of uniform pressure across a horizontal section of the annulus, which is essential to the formulation of Eq. (90). 

The microscopic contour of a meniscus or a drop is a matter that presents some mathematical problems even with the simplifying assumption of a uniform, rigid solid. Since bulk liquid is present, the system must be in equilibrium with the local vapor pressure so that an equilibrium adsorbed film must also be present. The likely picture for the case of a nonwetting drop on a flat surface is... 

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

A useful simphfication of the total energy equation applies to a particular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state conditions, and mass flow at a rate m through a single planar entrance and a single planar exit (Fig. 6-4), to whi(m the velocity vectors are perpendicular. As with Eq. (6-11), it is assumed that the stress vector tu is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + pgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. 

We will cover a simple drying model to examine the radiation drier of coated paper. We assume there are no major temperature or humidity variations in the direction of the paper web thickness, and that temperature T and humidity u are constant in the direction of thickness. This assumption requires that the capillary action be ignored, and the pressure gradient of water is zero on the assumption hu/dx = dT/dx = 0. How is it possible that the humidity distribution remains uniform ... 

If the range of the channel height is limited to be above 10 pm, then the no-slip boundary condition can be adopted. Furthermore, with the assumptions of uniform inlet velocity, pressure, density, and specified pressure Pout at the outlet, the boundary conditions can be expressed as follows ... 

The present model takes into account how capillary, friction and gravity forces affect the flow development. The parameters which influence the flow mechanism are evaluated. In the frame of the quasi-one-dimensional model the theoretical description of the phenomena is based on the assumption of uniform parameter distribution over the cross-section of the liquid and vapor flows. With this approximation, the mass, thermal and momentum equations for the average parameters are used. These equations allow one to determine the velocity, pressure and temperature distributions along the capillary axis, the shape of the interface surface for various geometrical and regime parameters, as well as the influence of physical properties of the liquid and vapor, micro-channel size, initial temperature of the cooling liquid, wall heat flux and gravity on the flow and heat transfer characteristics. 

Equation 8.19 is based on the following assumptions the approach velocity v is uniform and parallel, the streamlines are horizontal above the weir, there is atmospheric pressure under the nappe, and the flow is frictionless. 

The vessel is subjected to an internal pressure from the compressed air, which we shall designate as p. The internal pressure is uniformly distributed over the internal surfaces of the vessel, giving rise to both circumferential stress, also known as hoop stress, and longitudinal stress, (see Figure 8.7). We will examine each of these stresses independently before we begin the material selection process. In our development, we will make the following assumptions ... 

The design of this distributor is as follows. The approach is based on turbulent flow into the distributor, and thus for Ren >2100 (based on the distributor diameter and the liquid velocity at the inlet of the distributor). Furthermore, the diameters of the distributor openings as well as the distance between them are considered to be uniform throughout its length. Under these assumptions, the pressure drop across the distributor is (Perry and Green, 1999)... 

In summary, the first-derivative conditions (5.10) imply uniform values of the derivative (intensive) properties of S throughout the system. In this way, the system-wide uniformity of temperature, pressure, and other intensive properties is obtained from the Gibbs criterion of equilibrium as a deduction, not an assumption. 

Equations (10.23) and (10.24) hold for the /3-phase as well and could be inserted into Eqn. (10.22). The additivity of pt with respect to the elastic and electric potential is based on 1) the assumption of linear elastic theory (which is an approximation) and 2) the low energy density of the electric field (resulting from the low value of the absolute permittivity e0 = 8.8x10 12 C/Vm). In equilibrium, V/i, = 0 and A V, = df-pf = 0. Therefore, in an ionic system with uniform hydrostatic pressure, the explicit equilibrium condition reads Aa/fi=A)... 

Assuming that the second process is rapid, we obtain the following standard picture of adsorption on a uniform surface the equilibrium concentration q, which depends on the pressure of the gas, is determined by the Langmuir isotherm. The only difference from the standard picture is that the statistical sum for all states of the adsorbed molecule in a potential hole must be replaced by a combination of two statistical sums for all states of the adsorbed molecule and for all possible states of the surface element. This, of course, has no effect on the form of the Langmuir equation. Under very simple assumptions the kinetics of establishment of equilibrium will also not differ from those on a uniform surface. Thus, the initial velocity is proportional to the pressure and approaches equilibrium exponentially. 

The second assumption concerns the pressure and concentration gradients in the membrane. The solution-diffusion model assumes that when pressure is applied across a dense membrane, the pressure throughout the membrane is constant at the highest value. This assumes, in effect, that solution-diffusion membranes transmit pressure in the same way as liquids. Consequently, the solution-diffusion model assumes that the pressure within a membrane is uniform and that the chemical potential gradient across the membrane is expressed only as a concentration gradient [5,10]. The consequences of these two assumptions are illustrated in Figure 2.5, which shows pressure-driven permeation of a one-component solution through a membrane by the solution-diffusion mechanism. 

Consider a bubble rising in a fluidized bed. It is assumed that the bubble is solids-free, is spherical, and has a constant internal pressure. Moreover, the emulsion phase is assumed to be a pseudocontinuum, incompressible, and inviscid single fluid with an apparent density of pp(l — amf) + pamf. It should be noted that the assumption of incompressibility of the mixture is not strictly valid as voidage in the vicinity of the bubble is higher than that in the emulsion phase [Jackson, 1963 Yates et al., 1994]. With these assumptions, the velocity and pressure distributions of the fluid in a uniform potential flow field around a bubble, as portrayed by Fig. 9.10, can be given as [Davidson and Harrison, 1963]... 

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