Sensitivity analysis, steady-state

A one-at-a-time sensitivity analysis was performed to measure the sensitivity of steady-state current production, y, and the fraction of current due to diffusion-based EET, d, to the model input parameters. The sensitivity was evaluated by calculating the resulting average elasticity given a 25% change in each parameter. Elasticity, <5 (unitless), also called relative sensitivity, measures the proportional effect of a change... 

Once the model was complete, it was adjusted to a steady state condition and tested using historic carbon isotope data from the atmosphere, oceans and polar ice. Several important parameters were calculated and chosen at this stage. Sensitivity analysis indicated that results dispersal of the missing carbon - were significantly influenced by the size of the vegetation carbon pool, its assimilation rate, the concentration of preindustrial atmospheric carbon used, and the CO2 fertilization factor. The model was also sensitive to several factors related to fluxes between ocean reservoirs. 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

We have studied the steady-state kinetics and selectivity of this reaction on clean, well-characterized sinxle-crystal surfaces of silver by usinx a special apparatus which allows rapid ( 20 s) transfer between a hixh-pressure catalytic microreactor and an ultra-hixh vacuum surface analysis (AES, XPS, LEED, TDS) chamber. The results of some of our recent studies of this reaction will be reviewed. These sinxle-crystal studies have provided considerable new insixht into the reaction pathway throuxh molecularly adsorbed O2 and C2H4, the structural sensitivity of real silver catalysts, and the role of chlorine adatoms in pro-motinx catalyst selectivity via an ensemble effect. 

The last issue that remains to be addressed is whether the MBL results are sensitive to the characteristic diffusion distance L one assumes to fix the outer boundary of the domain of analysis. In the calculations so far, we took the size L of the MBL domain to be equal to the size h - a of the uncracked ligament in the pipeline. To investigate the effect of the size L on the steady state concentration profiles, in particular within the fracture process zone, we performed additional transient hydrogen transport calculations using the MBL approach with L = 8(/i — a) = 60.96 mm under the same stress intensity factor Kf =34.12 MPa /m and normalized T-stress T /steady state distributions of the NILS concentration ahead of the crack tip are plotted in Fig. 8 for the two boundary conditions, i.e. / = 0 and C, =0 on the outer boundary. The concentration profiles for the zero flux boundary condition are identical for both domain sizes. For the zero concentration boundary condition CL = 0 on the outer boundary, although the concentration profiles for the two domain sizes L = h - a and L = 8(/i - a) differ substantially away from the crack tip. they are very close in the region near the crack tip, and notably their maxima differ by less than... 

Calculate and plot the time-dependent species profiles for an initial mixture of 50% H2 and 50% Cl2 reacting at a constant temperature and pressure of 800K and 1 atm, respectively. Consider a reaction time of 200ms. Perform a sensitivity analysis and plot the sensitivity coefficients of the HC1 concentration with respect to each of the rate constants. Rank-order the importance of each reaction on the HC1 concentration. Is the H atom concentration in steady-state ... 

In order to understand the fundamental concept of the cause of temperature sensitivity, in the analysis described in this section it is assumed that the combustion wave is homogeneous and that it consists of steady-state, one-dimensionally successive reaction zones. The gas-phase reaction occurs with a one-step temperature rise from the burning surface temperature to the maximum flame temperature. 

Equation (48) e ees with experimental results in some circumstances. This does not mean the mechanism is necessarily correct. Other mechanisms may be compatible with the experimental data and this mechanism may not be compatible with experiment if the physical conditions (temperature and pressure etc.) are changed. Edelson and Allara [15] discuss this point with reference to the application of the steady state approximation to propane pyrolysis. It must be remembered that a laboratory study is often confined to a narrow range of conditions, whereas an industrial reactor often has to accommodate large changes in concentrations, temperature and pressure. Thus, a successful kinetic model must allow for these conditions even if the chemistry it portrays is not strictly correct. One major problem with any kinetic model, whatever its degree of reality, is the evaluation of the rate cofficients (or model parameters). This requires careful numerical analysis of experimental data it is particularly important to identify those parameters to which the model predictions are most sensitive. 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

Existence and uniqueness of the particular solution of (5.1) for an initial value y° can be shown under very mild assumptions. For example, it is sufficient to assume that the function f is differentiable and its derivatives are bounded. Except for a few simple equations, however, the general solution cannot be obtained by analytical methods and we must seek numerical alternatives. Starting with the known point (tD,y°), all numerical methods generate a sequence (tj y1), (t2,y2),. .., (t. y1), approximating the points of the particular solution through (tQ,y°). The choice of the method is large and we shall be content to outline a few popular types. One of them will deal with stiff differential equations that are very difficult to solve by classical methods. Related topics we discuss are sensitivity analysis and quasi steady state approximation. 

An analysis of chemical reactor stability and sensitivity-XIV The effect of the steady state... 

It is clear from Si-29 studies that Si-29 is a useful tool for measuring the steady-state oligomerizations but not for studying the actual oligomerization process in real time. In general, Si-29 sensitivities are not useful for analysis of solutions below 0.5 wt % or for measuring reactive half-lives for weight percents less than 90%. 

We have chosen the steady state with Yfa = 0.872 and FCD = 1.0 giving a dense phase reactor temperature of Yrd = 1.5627 (Figure 7.14(b) and (c)) and a dense-phase gasoline yield of x-id = 0.387 (Figure 7.14(a)). This is the steady state around which we will concentrate most of our dynamic analysis for both the open-loop and closed-loop control system. We first discuss the effect of numerical sensitivity on the results. Then we address the problem of stabilizing the middle (desirable, but unstable) steady state using a switching policy, as well as a simple proportional feedback control. 

Firstly, the concentrations of HMs in effluent were determined. At least three samples of 40 mL were acidified with 10 mL of concentrated HNO3 to decompose MOs. These samples were diluted to the sensitivity range of the atomic absorption spectrophotometer (AAS) (Model Philips 9200X-AAS) (diluted 5 times for Cu2+ analysis, and 25 times for Zn2+ analysis) and then analyzed. The average of these measurements yielded the total HM concentration in the effluent, / j (mg/L), for that specific experimental run. To convert (mg metal/L) to (mg metal/kg sludge), the measured metal concentration (as, mg metal/L) at each HRT was divided by the steady state MLSS concentration (as, kg TSS/L). 

There are also two factors that have already been noted in the numerical analysis of the kinetic model of CO oxidation (1) fluctuations in the surface composition of the gas phase and temperature can lead to the fact that the "actual multiplicity of steady states will degenerate into an unique steady state with high parametric sensitivity [170] and (2) due to the limitations on the observation time (which in real experiments always exists) we can observe a "false hysteresis in the case when the steady state is unique. Apparently, "false hysteresis will take place in the region in which the relaxation processes are slow. 

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