Order-of-magnitude analysis

The engineer is offered a large variety of flow-modeling methods, whose complexity ranges from simple order-of-magnitude analysis to direct numerical simulation. Up to now, the methods of choice have ordinarily been experimental and semi-theoretical, such as cold flow simulations and tracer studies. 

Based on the assumptions and the order of magnitude analysis described in Section IV.A.2, and following the similar procedure of solution, the averaged vapor-flow velocity is given by... 

Order-of-magnitude analysis, 11 145 Order-of-magnitude estimates, 9 529 Order parameters, of liquid crystalline materials, 15 82-85 Ordinary differential equation (ODE), 25 281... 

Such relationship can be obtained using the approaches of rigorous modeling, order-of-magnitude analysis, or black box analysis as suggested by Wibowo and Ng [5], Examples of how SA relates to OV are given in Table 12. Note that due to the complex phenomena involved in these unit operations, shortcut empirical models that have some physical basis are often the most practical to describe the relationship. Unfortunately, such models are rarely available, making it difficult to quantify the relationship. For this reason, this part of the procedure is not emphasized in this article. 

To test the validity of the extended Pitzer equation, correlations of vapor-liquid equilibrium data were carried out for three systems. Since the extended Pitzer equation reduces to the Pitzer equation for aqueous strong electrolyte systems, and is consistent with the Setschenow equation for molecular non-electrolytes in aqueous electrolyte systems, the main interest here is aqueous systems with weak electrolytes or partially dissociated electrolytes. The three systems considered are the hydrochloric acid aqueous solution at 298.15°K and concentrations up to 18 molal the NH3-CO2 aqueous solution at 293.15°K and the K2CO3-CO2 aqueous solution of the Hot Carbonate Process. In each case, the chemical equilibrium between all species has been taken into account directly as liquid phase constraints. Significant parameters in the model for each system were identified by a preliminary order of magnitude analysis and adjusted in the vapor-liquid equilibrium data correlation. Detailed discusions and values of physical constants, such as Henry s constants and chemical equilibrium constants, are given in Chen et al. (11). 

From an order-of-magnitude analysis, when the mean-free path of a molecule is less than 0.01 times the pore radius, bulk diffusion dominates, and when it is greater than 10 times the pore radius, Knudsen diffusion dominates. This means that Knudsen diffusion is significant when the pore radius is less than about 0.5 fim. For reference, a typical carbon gas-diffusion layer has pores between 0.5 and 20 /rm22 229 in radius, and a microporous layer contains pores between 0.05 and 2 Thus, while Knudsen... 

In the above expression, the first term represents the accumulation and convective transport of enthalpy, where is the heat capacity of phase k. The second term is energy due to reversible work. For condensed phases this term is negligible, and an order-of-magnitude analysis for ideal gases with the expected pressure drop in a fuel cell demonstrates that this term is negligible compared to the others therefore, it is ignored in all of the models. 

We will start with equation (4.23) and perform an order of magnitude analysis to reduce the terms to those that are important to this situation. First, the transfer is from the flowing water into the sediments, so the appropriate width of the flow would be the depth of the flow, h. We will orient the z-direction so that it is vertical upward, and then the characteristic length scale would also be h. 

The flux is in the z-direction, so the gradients in the z-direction are much greater than those in the x- and y-directions. The order-of-magnitude analysis of the second derivatives typically follows those of the first derivatives, so the first two terms in brackets are much smaller than the third term. 

You will be measuring velocity profiles in a fully developed open-channel flow (no change with longitudinal distance). To get an idea of which parameters you need to measure accurately, you need to perform an order-of-magnitude analysis on equation (4.26). 

An order of magnitude analysis will tell us that d jdr d" jdx. In addition, by definition, dCjdr = 0. Now, if we put everything that is known on the left-hand side of equation (6.27), the result will be... 

The natural convection velocity is given by Eq. (68b). The order-of-magnitude analysis provides the following algebraic equation ... 

The conclusion of this study is that semiquantitative (order of magnitude) analysis is possible for broad spectrum analysis of resin lots cleaned by a consistent method and blanked with Milli-Q water. The concentrations of the resin impurities will vary as a function of the following ... 

For the purpose of an order-of-magnitude analysis to compare the relative magnitudes of the two terms, consider a one-dimensional flow in which the velocity in given as u. In this case, rearranging Eq. 3.69, we seek situations for which... 

The second is to express the original equations in terms of dimensionless variables. The order of magnitude analysis results in a scaled conduction given by,... 

We start the derivation with an order of magnitude analysis of the continuity equation... 

Next, an order of magnitude analysis is performed to simplify the momentum balance. This is illustrated using the x-component of the equation of motion in terms of stress... 

Adiabatic Compression Heating Melting of polymers by adiabatic compression has been shown to be feasible for processes such as injection molding (2). Discuss this method, in principle, in terms of an order-of-magnitude analysis of the terms of the thermal energy balance for an amorphous (PS) and a semicrystalline polymer (LDPE). Use the data in Appendix A. 

It is assumed that the radius of the pipe is relatively small compared to the values of z being considered. In this case, the same type of order of magnitude analysis as used in deriving the boundary layer equations indicates that v u and that... 

To show how one might proceed to analyze a new problem to obtain an important functional relationship from the differential equations, consider the problem of determining the hydrodynamic-boundary-layer thickness for flow over a flat plate. This problem was solved in Chap. 5, but we now wish to make an order-of-magnitude analysis of the differential equations to obtain the functional form of the solution. The momentum equation... 

It will be noted that the viscous-dissipation term is omitted from the energy equation for the present. In accordance with the order-of-magnitude analysis of Sec. 6-1, ... 

The assumption (d) imposed in deriving the model appears to be valid for both types of combustors since the feed rate of coal is relatively small under normal operating conditions. The order of magnitude analysis shows that the convective uc, is... 

The order-of-magnitude analysis of the magnetophoretic mobility of weakly paramagnetic particles, 10 /Ltm in diameter, acted on by the body forces available for the magnetic separation, returns the following... 

The energy conservation equation is intimately linked to momentum conservation equations via the fourth and fifth terms. For most reacting systems, the contribution of energy released or absorbed by chemical reactions usually dominates the other terms originating from pressure and viscous effects. For highly viscous flows with low heats of reaction, it may be important to consider the viscous heating terms. An order of magnitude analysis is often used to examine the relative importance of different terms. 

With the above assumptions, the total activation energy calculated from equation (4) should be about 19 Kcal/jmole. The discrepancy can be attributed to inaccuracies in the estimation of the heat of reaction (1) and (2). This order of magnitude analysis indicates that the model can roughly account for the third order dependence on hydrogen pressure and first order dependence on benzene pressure. 

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