Boundary conditions insulator

Studies of double carrier injection and transport in insulators and semiconductors (the so called bipolar current problem) date all the way back to the 1950s. A solution that relates to the operation of OLEDs was provided recently by Scott et al. [142], who extended the work of Parmenter and Ruppel [143] to include Lange-vin recombination. In order to obtain an analytic solution, diffusion was ignored and the electron and hole mobilities were taken to be electric field-independent. The current-voltage relation was derived and expressed in terms of two independent boundary conditions, the relative electron contributions to the current at the anode, jJfVj, and at the cathode, JKplJ. 

Reactor wall thermal boundary conditions can have a strong effect on the gas flow and thus the deposition. Here, for example, we indicate how cooling the reactor walls can enhance deposition uniformity. We consider the results of three simulations comparing the effects of two different wall boundary conditions. Figure 4 shows how the ratio of the computed susceptor heat flux to the onedimensional heat flux varies with the disk radius for the different conditions (the Nusselt number Nu is a dimensionless surface heat flux). In two cases the reactor walls are held at 300 K (0 = 0), and in one case the walls are insulated ( 0/ r —... 

Equation (20) indicates that the total heat flow (Q) is constant over each concentric sphere within the insulating layer. Application of Eq. (20) to the problem described in Figure 5 gives Boundary conditions ... 

The theory of thermally thin ignition is straightforward and can apply to (a) a material of thickness d insulated on one side or (b) a material of thickness 2d heated symmetrically. The boundary conditions are given as... 

For a fully developed fire, conduction commonly overshadows convection and radiation therefore, a limiting approximation is that h hk, which implies Tw T. This result applies to structural and boundary elements that are insulated, or even to concrete structural elements. This boundary condition is conservative in that it gives the maximum possible compartment temperature. 

A unit temperature difference is imposed to the top wall of Fig. 34 and the conductance is computed. The boundary conditions are T= 1 at the upper side, T= 0 at the bottom side and insulation for all the other sides. The gas conductivity is considered through the gas phase heat balance equation so is not taken into account here. Taking into account the gas conductivity to compute the transverse effective conductivity for the DPF structure is not correct. 

The boundary condition (4.4.52c) states that the normal component of the cationic flux at the membrane vanishes at the insulating portion of the membrane surface and is equal to a given constant i throughout the conducting site. The boundary condition (4.4.52d) asserts the vanishing of the normal component of the anionic flux at the membrane, corresponding to the ideal permselectivity of the latter. Finally, the boundary conditions (4.4.52e,f) state that the surface r == 0, 0 < x < 6 is that of symmetry. We observe that c(r,x), [Pg.150]

Several approaches to solving this expression for various boundary conditions have been reported [25,26]. The solutions are qualitatively similar to the results at a hemisphere at very short times (i.e., when (Dt),y4 rD), the Cottrell equation is followed, but at long times the current becomes steady-state. Simple analytical expressions analogous to the Cottrell equation for macroplanar electrodes or Equation 12.9 for spherical electrodes do not exist for disk electrodes. For the particular case of a disk electrode inlaid in an infinitely large, coplanar insulator, the chronoamperometric limiting current has been found to follow [27] ... 

As described in the introduction, submicrometer disk electrodes are extremely useful to probe local chemical events at the surface of a variety of substrates. However, when an electrode is placed close to a surface, the diffusion layer may extend from the microelectrode to the surface. Under these conditions, the equations developed for semi-infinite linear diffusion are no longer appropriate because the boundary conditions are no longer correct [97]. If the substrate is an insulator, the measured current will be lower than under conditions of semi-infinite linear diffusion, because the microelectrode and substrate both block free diffusion to the electrode. This phenomena is referred to as shielding. On the other hand, if the substrate is a conductor, the current will be enhanced if the couple examined is chemically stable. For example, a species that is reduced at the microelectrode can be oxidized at the conductor and then return to the microelectrode, a process referred to as feedback. This will occur even if the conductor is not electrically connected to a potentiostat, because the potential of the conductor will be the same as that of the solution. Both shielding and feedback are sensitive to the diameter of the insulating material surrounding the microelectrode surface, because this will affect the size and shape of the diffusion layer. When these concepts are taken into account, the use of scanning electrochemical microscopy can provide quantitative results. For example, with the use of a 30-nm conical electrode, diffusion coefficients have been measured inside a polymer film that is itself only 200 nm thick [98]. 

On the surface of the tubes, furnace, etc., the boundary conditions require that the flow of any substance across a material surface be equal to zero so that the normal surface of the component of the corresponding vector of the flow is zero. These conditions are identical for all the substances. The boundary conditions for temperature will be the same as the boundary conditions for a,..., h, if we do not remove heat from the plume, i.e., if only thermally insulated, not thermally radiating surfaces are introduced or if all walls with temperature T0 are located where the gas temperature is equal to T0. 

Effect of Thermal Boundary Conditions. When the side walls are cooled instead of being insulated, there is no critical Rat number, and any transverse temperature gradient will lead to a buoyancy-driven secondary flow. Compared with the previous example (Figure 8b), the rolls are reversed and now rotate outward. These examples demonstrate the strong influence of the thermal boundary conditions on CVD reactor flows. 

The boundary conditions for the momentum balance were ux = uy = 0 on the upper and lower plates, p = 105,000 Pa at the left wall (x = 0) and p = 0 at the right wall (x = 0.015m). The thermal boundary conditions were T = 200°C on the upper and lower plates, and insulated boundary conditions (dT/dx=0) on both... 

Here, E n = 0 on Sp (Neumann type boundary condition), where n is the unit outward normal from the pore region, and T> is compact. E can be interpreted as the microscopic electric field induced in the pore space when a unit macroscopic field e is applied, assuming insulating solid phase and uniform conductivity in the pore fluid. Its pore volume average is directly related to the tortuosity ax ... 

Problem definition requires specification of the initial state of the system and boundary conditions, which are mathematical constraints describing the physical situation at the boundaries. These may be thermal energy, momentum, or other types of restrictions at the geometric boundaries. The system is determined when one boundary condition is known for each first partial derivative, two boundary conditions for each second partial derivative, and so on. In a plate heated from ambient temperature to 1200°F, the temperature distribution in the plate is determined by the heat equation 8T/dt = a V2T. The initial condition is T = 60°F at / = 0, all over the plate. The boundary conditions indicate how heat is applied to the plate at the various edges y = 0, 0[Pg.86]

Laplace s equation governs the potential distribution [Eq. (21)]. Since overpotential is ignored, the potential immediately adjacent to the electrodes is constant. At insulated surfaces the normal potential gradient must be zero. These two requirements dictate the boundary conditions for the differential equation. 

Prior to the tests, all the samples were dried in a vacuum oven at 80°C for at least 72 h to minimize the moisture effect and then transferred to a desiccator. Measurements were carried out on a cone calorimeter provided by the Dark Star Research Ltd., United Kingdom. To minimize the conduction heat losses to insulation and to provide well-defined boundary conditions for numerical analysis of these tests, a sample holder was constructed as reported in [14] with four layers (each layer is 3 mm thick) of Cotronic ceramic paper at the back of the sample and four layers at the sides. A schematic view of the sample holder is shown in Figure 19.12. Three external heat fluxes (40, 50, and 60kW/m2) were used with duplicated tests at each heat flux. 

The necessity to solve Laplace s equation requires formulating all boundary conditions, and at this point the cell geometry becomes important. Generally, there are two types of boundary conditions that come into play. Any electrically insulating cell wall is mathematically described by zero-flux or von Neumann boundary conditions ... 

Consider the insulation around a cylindrical system, as shown in Fig. 3.8. The boundary conditions are the first kind at r = rj and the third kind at r = rc. 

The boundary condition that iJ/ must satisfy at an interface can vary between the conductive and insulating cases mentioned earlier. A conductive surface infers a constant surface potential leading to a Dirichet boundary condition,... 

Another special area is the insulating plane outside the disk, defined by Z = 0, R > 1. Here, the boundary condition is usually given as in the set (12.18), zero gradient with respect to Z. This is expressed as a four-point first derivative, as... 

This leaves the boundary conditions. The equation for the insulating plane is given in (12.44), producing four matrix entries. The remaining points are now those on the disk surface itself, and the points outside the diffusion space. On the disk surface, for the Cottrell-like simulation, we have zero concentrations, and at the outer points all concentrations are unity. These produce single entries in the matrix. 

To explain the behavior of the current flowing through the sample, the functional dependence ja(x) must be introduced as a boundary condition into the model describing injection currents in insulators. 

Until now, a recombination velocity of thermalized as well as hot carriers has been directly introduced in various models as a boundary condition independent of the carrier position in an insulator. This corresponds to the assumption jr(x) = const, and from Eq. (196), js =j0aiexp(—ax x), for the exponential character of the primary injection current jp = jo exp( —of x), follows. The source carriers are being thermalized with a probability v (per unit time) and rate eu N(x) equal to the carrier injection rate,... 

Some surfaces are commonly insulated in practice in order to minimize heat loss (or heal gain) through them. Insulation reduces heat transfer but does not loialf eliipinate it unless its thickness is infinity. However, heat transfer through a properly insulated surface can be taken to be zero since adequate insulation reduces heat transfer through a surface to negligible levels. Therefore, a well-insulated surface can be modeled as a surface with a specified heat flux of zero, llien the boundary condition on a perfectly insulated surface (at X - 0, for example) can be expressed as (Fig. 2-30)... 

FIGURE 2-30 A plane wall with insulation and specified temperature boundary conditions. 

Steam flows through a pipe shown in Fig. 2-35 at an average temperature of = 200 C. The inner and outer radii of the pipe are ri = 8 cm and = 8,5 cm, respectively, and the outer surface of the pipe is heavily insulated. If I the convection heat transfer coefficient on the inner surface of the pipe is /) = 65 W/m " K, express the boundary conditions on the inner and outer sur- ] faces of the pipe during transient periods. 

SOLUTION The flow of steam through an insulated pipe is considered. The boundary conditions on the inner and outer surfaces of the pipe are to be obtained. 

C How is the boundary condition on an insulated surface expressed mathematically ... 

Z-Xl Water flows through a pipe at an average temperature of VO C. The inner and outer radii of the pipe are rf = 6 cm and rj = 6.5 cm. respectively. The outer surface of the pipe is wrapped with a thin electric heater that consumes 300 W pet m length of the pipe. The exposed surface of the heater is heavily insulated so that the entire heal generated in the heater is transferred to the pipe. Heat is transferred from the inner surface of the pipe to the water by convection with a heat transfer coefficient of h = 85 W/m K. Assuming constant thermal conductivity and one-dimensionat heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of the heal conduction in the pipe during steady operation. Do not solve. 

---