Reaction In Continuous Stirred Tank Reactor Research Articles

Rate expression model from thermodynamics application and optimal kinetic parameters determination for urea synthesis and production process

Urea, an essential organic fertilizer, enhances soil fertility by providing 0.466 nitrogen for maximum crop yield. In this study, urea is synthesized from NH3 and CO2 in an equilibrium reaction process adhering to Le Chatelier's principle, maintained under process conditions: flow rate of 63.5 kg/s, temperature of 184 °C, and pressure of 160 kg/cm2. A new rate expression model, formulated in terms of extent of reaction and mole fraction, was developed based on mass action relations and thermodynamic models. Two industrial reactors were considered: a plug flow reactor (PFR) at Notore and a continuous stirred tank reactor (CSTR) at Indorama plants. Transient reactor models, based on material and energy balance conservation principles, were numerically resolved using MATLAB version 2020 with specified input conditions. A non-linear regression statistical optimization model was employed to refine kinetic parameter values, ensuring optimal and high-quality urea yield. Model validations were conducted using literature data, revealing higher urea yields of 0.726 and 0.7032 for the CSTR and PFR, respectively. Deviations (0.134, 0.10 to 1.135 and 0.635, 0.326 to 0.850) and root mean square errors (RMSE) (0.043, 0.033 to 0.193 and 0.137, 0.087 to 0.162) were observed when validated against plant and literature values for the CSTR and PFR respectively. The refined kinetic parameters (activation energies, Arrhenius constants, and rate constants) exhibited negligible deviations (0.0004–0.0466 and 0.0004 to 0.0491) and RMSE (0.0228, 0.0055, and 0.0256 and 0.0241, 0.0096, and 0.0269) when validated against plant data, significantly enhancing urea yield in CSTR and PFR reactors respectively.

A Mechanistic Model on Catalyst Deactivation by Coke Formation in a CSTR Reactor

A mechanistic model on catalyst deactivation by coke formation in a continuous stirred tank reactor (CSTR) has been developed in the paper. Catalyst deactivation by coke formation was treated as a surface reaction. Four reaction mechanisms representing coke formation through different routes were proposed. The evolved system of ordinary differential equations (ODEs) was solved numerically using MATLAB. This approach was validated by applying it to the skeletal isomerization of 1-pentene over ferrierite. Simulation results were compared qualitatively to those obtained from the literature. Simulation results indicated that coke formation is an extremely rapid process with fast formation of coke components on the strongest acid sites leading to final coke. The coke deposition is slower at higher residence times resulting in more stable product formation and weaker deactivation. The results obtained from this work revealed that the developed model is indeed able to successfully demonstrate the most essential features of catalyst deactivation by coke formation and are in agreement with the findings in the literature. Future work is aimed to extend the study to different reactors such as a plug flow reactor, in addition to analysis of the reaction system’s sensitivity to variables such as temperature and pressure.

Exploration of sulfate reducing sludge granulation with real domestic sulfate-laden wastewater

Extending sulfate reducing sludge granulation to real sulfate-laden wastewater was trialed in this study. Two types of reactors, namely sulfate reducing upflow sludge blanket (SRUSB) reactor and continuous stirred tank reactor (CSTR), were used for granulation. Sulfate-reducing granules were observed after 5 months' operation of stepwise hydraulic retention time (HRT) shortening, recirculation cycle adjustment, and air scouring in the SRUSB reactor. Comparatively, granular sludge was achieved in the CSTR after 6 months of cultivation with an average size of 220 μm. With the process of granulation, the HRTs of the two types of reactors were reduced to 2.2 and 2.8 h, respectively for the SRUSB and CSTR reactors. The abundance of sulfate-reducing bacteria (SRB) related genera reached 29.3 and 45% in the SRUSB and CSTR reactors, respectively. Incomplete organic oxidizing SRB were dominant in the CSTR and SRUSB is predominated by complete organic oxidizing SRB, which facilitated a higher organic loading rate. Considering the operating merits, the CSTR is preferable as a pretreatment unit (e.g., for acidification), while the SRUSB reactor can be used for maximum organics removal or sulfide production. Due to the high suspended solids in the influent, full granulation was not achieved by using real sulfate-laden wastewater, and fine particles accumulation in the reactor was also a concern particularly for the long-term operation.

Linear Model Predictive Control for a Coupled CSTR and Axial Dispersion Tubular Reactor with Recycle

This manuscript addresses a novel output model predictive controller design for a representative model of continuous stirred-tank reactor (CSTR) and axial dispersion reactor with recycle. The underlying model takes the form of ODE-PDE in series and it is operated at an unstable point. The model predictive controller (MPC) design is explored to achieve optimal closed-loop system stabilization and to account for naturally present input and state constraints. The discrete representation of the system is obtained by application of the structure properties (stability, controllability and observability) preserving Cayley-Tustin discretization to the coupled system. The design of a discrete Luenberger observer is also considered to accomplish the output feedback MPC realization. Finally, the simulations demonstrate the performance of the controller, indicating proper stabilization and constraints satisfaction in the closed loop.

Adsorption of Metalworking Fluids in CSTR Reactor by Modified Sugarcane Bagasse with Aluminium Sulphate as Adsorbent

Metalworking fluids (MWFs) are widely used in the metal forming industrials. It was used for reducing of the friction and cooling of the mechanical processes. In this research, the modified sugarcane bagasse (MSB) with aluminium sulphate (Al2(SO4)3) was used as an adsorbent for removal of the metalworking fluid in Continuous Stirred Tank Reactor (CSTR). The point of zero charge of sugarcane bagasse and modified sugarcane bagasse were pH 6.1 and 3.7, respectively. The effect of initial concentration of metalworking fluid was studied. When used the reactor 3 L and flow rate at 0.055 L/min, the % removals of MWF at initial concentration 6,480-41,513 g/m3 were 98.0-56.7%, respectively. The significant uptake of metalworking fluid was demonstrated by FT-IR spectroscopy.

Accelerated two-stage bioprocess for hydrogen and methane production from palm oil mill effluent using continuous stirred tank reactor and microbial electrolysis cell

This paper investigates the production of hydrogen (H2) and methane (CH4) from palm oil mill effluent (POME) using an integrated approach of thermophilic continuous stirred tank reactor (CSTR) and mesophilic microbial electrolysis cell (MECs). CSTR reactor was operated at pH 5.5, 80 rpm, 2 days HRT, 60 g COD L-1 d-1 organic loading rate (OLR) and 55 °C temperature with the given hydrogen yield of 205 ml H2 gCOD-1 along with acetic, butyric, propionic, and lactic acid as by-products. Continuous, single-chambered MECs fed with dark fermentation effluents were operated at an applied voltage of 0.5 V at 37 °C to obtain methane yield and production rate (MPR) of 290 ml CH4 gCOD-1 and 2700 ml CH4 L-1 at 8 days of hydraulic retention times (HRT). The overall process led to total energy recovery of 92.72% with 91% COD removal efficiency. Microbial community analysis reveals Thermoanerobacterium sp dominated in CSTR whereas exoelectrogens of Methanobacterium formicicum and Methanobacterium beijingense were found to be the chief dominant microbial species on anodic electrode of MECs.

The effectiveness of anaerobic digestion process by thermal pre-treatment on food waste as a substrate

Thermal pre-treatment is necessary for the feedstock of Anaerobic Digester (AD) to improve the degradation process. This study aims to investigate the effect of thermal pre-treatment on Food Waste (FW) as a substrate through Biochemical Methane Potential (BMP) test followed by pilot scale using a continuous stirred tank reactor (CSTR) digester. TS, VS, COD, SCOD, and solubilization samples were measured during on CSTR reactor. FW was pretreated by heating at 70 °C, 90 °C, and 120 °C for 45 minutes. Results showed that thermal pre-treatment at 70 °C had the most effective digestion process with TS, VS, COD reduction of 8.26%; 2.54%; and 14% and increase SCOD and solubilization of 10.2% and 24.7%, respectively compared to thermal pre-treatment at 90 °C and 120 °C (p<0.05). Therefore, thermal pre-treatment at 70 °C was chosen for AD pilot scale in this study. Meanwhile, pilot scale experiment showed the results that thermal pre-treatment at 70 °C was obtained significant differences of 31% TS; 38% VS; 42% COD; 23% SCOD; and 30% solubilization compare with without thermal pre-treatment (p<0.05). Pre-treatment with thermal on FW as a substrate may help in increasing the degradation process at the hydrolysis phase in order to increase the effectiveness of the process in AD reactor.

Carbon and Nitrogen Removal Characteristics of ABR Decarbonization-CANON Coupling Process

If municipal wastewater can be treated by the completely autotrophic nitrogen removal over nitrite (CANON) process, it will greatly reduce the energy consumption of municipal wastewater treatment. The CANON reactor with a fiber carrier was started up by seeding nitrosation sludge and anaerobic ammonia oxidation (ANAMMOX) sludge in the continuously stirred tank reactor (CSTR). An ABR decarbonization system was added to the front of the CANON system to build the ABR decarbonization-CANON coupling process to examine carbon and nitrogen removal characteristics of the whole system. The high throughput sequencing technology of MiSeq was also employed to analyze the structure of the microbial community before and after the reactivation. The results showed that mixing nitrosation sludge and ANAMMOX sludge in the CSTR reactor under controlled parameters (DO of 0.5-2 mg·L-1; HRT for 6 h; pH of 8) allowed the CANON system to successfully start within 55 d, with a TN removal rate of 81%-87% and ammonia nitrogen load of 0.195 kg·(m3·d)-1. The effluent COD concentration of the ABR decarbonizing system did not adversely affect the subsequent CANON system, and the TN removal rate of the ABR decarbonization-CANON process was between 74% and 87%. Additionally, the average concentration of COD in the effluent was 40 mg·L-1. At the same time, the Proteobacteria gate significantly improved after the CANON system began, and the proportion of Sphingobacteria decreased to 6.8%. Nitrifying bacteria and anaerobic ammonia oxidizing bacteria in the CANON system continuously eliminated the inferior bacterial groups to become the dominant group in the reactor. The integrated ABR decarbonization-CANON process had a positive effect on the denitrification and decarbonization of urban sewage.

Enriched ammonia-tolerant methanogenic cultures as bioaugmentation inocula in continuous biomethanation processes

Many ammonia-rich biomass sources, such as manures and protein-rich substrates, are potential inhibitors of the anaerobic digestion (AD) process. It was previously demonstrated that bioaugmentation of Methanoculleus bourgensis MS2T in an ammonia inhibited process in a continuous stirred tank reactor (CSTR), resulted in up to 90% recovery of the methane production compared to the uninhibited production. However, cultivation of pure strains has practical difficulties due to the need of special growth media and sterile conditions. In contrast, acclimatized enriched cultures have minor sterility requirements. In the current study, an enriched ammonia-tolerant methanogenic culture was bioaugmented in a CSTR reactor operating under ammonia-induced, inhibited-steady-state. The results demonstrated that bioaugmentation, completely counteracted the ammonia toxicity effect. This indicates that a commercial application of bioaugmentation could improve up to 36% the methane production, the greenhouse gas reduction efficiency and the gross revenue of ammonia inhibited full scale biogas reactors. 16S rRNA gene sequencing showed that bioaugmentation changed the microbial composition of the reactors resulting in higher bacterial and lower archaeal community diversity. The bioaugmented reactor showed a fourfold increase of the abundance of the bioaugmented methanogens compared to the control reactor. This indicates that ammonia-tolerant methanogens established well in the ammonia-inhibited reactor and dominated over the domestic methanogenic population. Finally, this study showed that the enriched culture alleviated ammonia toxicity 25% more efficiently than the previously used pure culture.

Full Scale Process Design for Energy Recovery from Swine Manure

The aim of this study is to investigate biogas reactor designs to recover energy from swine manure and utilize that energy for a small scale pig farm in Norway. Continuous stirred tank reactor (CSTR) and upflow anaerobic sludge blanket (UASB) reactor were the design alternatives investigated. Simulations were based on the anaerobic digestion model no.1(ADM1) implemented in Aquasim software.&#x0D; The model was calibrated based on a series of laboratory batch reactors. The batch reactor with the highest biogas yield was first simulated to obtain the organic matter concentration in the feed manure. The resulting calibrated model was used to simulate CSTR and UASB reactors for a manure feed flow rate of 2m3/day. Different CSTR volumes in the range of 10- 60 m3 and UASB volumes in the range of 2-20 m3 were assessed using simulations. At low reactor volumes overloaded conditions were observed. Maximum energy production of 128 kWh/day at a biogas production rate of ~20 m3/day (68-71 % methane content in the biogas) was simulated for the reactor volumes for CSTR and UASB of 30m3 and 2.5m3, respectively. An efficient biogas production (e.g. to cover farm energy needs) can be obtained from a UASB reactor that can be constructed and installed at a reasonable cost.

An investigation of two‐stage thermophilic and mesophilic fermentation process for the production of hydrogen and methane from palm oil mill effluent

The two‐stage process of combined hydrogen and methane production with the recirculation of digestate sludge using palm oil mill effluent as substrate was investigated. During the first stage, an up‐flow anaerobic sludge blanket reactor (UASB) was operated at thermophilic conditions [hydraulic retention time (HRT)−2 days, organic loading rate −75 gCOD L−1d−1] for hydrogen production. The operation performance at short HRT and low pH was useful to separate the acidogenesis from methanogenesis. The effluents from UASB reactor containing mainly acetate and butyrate were fed into a continuous stirred tank reactor (CSTR) for methane production under the mesophilic temperature at two different HRTs (5 and 3 days). Both UASB and CSTR reactors were operated for 120 days continuously and a stable production of the hydrogen and methane was obtained simultaneously. The total hydrogen and methane yields obtained were 215 L H2 kgCOD−1 and 320 L CH4 kgCOD−1, respectively with the total COD removal efficiency of 94% in the two stage process. © 2017 American Institute of Chemical Engineers Environ Prog, 36: 895–902, 2017

Direct integration of CSTR-UASB reactors for two-stage hydrogen and methane production from sugarcane syrup

This study attempted to directly integrate a continuous stirred tank reactor (CSTR) with an up-flow anaerobic sludge blanket (UASB) reactor for two-stage hydrogen and methane production. CSTR was used to produce hydrogen from a 25 g-COD/L sugarcane syrup. The hydrogenogenic effluent from CSTR was directly fed to UASB for methane production. The working volumes of CSTR and UASB reactors were 1 and 24 L, respectively. Optimization of hydraulic retention times (HRT) for two-stage hydrogen and methane production after an integration were examined. A maximum hydrogen production rate of 17.5 L/L.d and methane production rate of 2.25 L/L.d were achieved at optimal HRTs of 3 h in the CSTR and 3 d in the UASB with a total energy production rate of 270 kJ/L.d. The two-stage reactors performed well in producing hydrogen and methane over 200 days with a total COD removal of 97.5%. The natural microflora in sugarcane syrup was greatly affected by HRT but HRT did not affect the archaea community. The volatile fatty acids to alkalinity ratio in the UASB reactor was below the critical value of 0.4 at every HRT, indicating stability of a long term methane production process with direct feeding of non-pretreated hydrogenogenic effluent.

Process enhancement of hydrogen and methane production from palm oil mill effluent using two-stage thermophilic and mesophilic fermentation

The present study investigates the technical possibilities of hydrogen and methane production from palm oil mill effluent (POME). The production was carried out in two stage (thermophilic and mesophilic) continuous phase with recirculation of the digestate sludge. The reactors used for the present study, up-flow anaerobic sludge blanket reactor (UASB) and continuous stirred tank reactor (CSTR) were operated under thermophilic and mesophilic conditions, respectively. The UASB reactor was operated at 2 days hydraulic retention time (HRT) and 75 kgCOD m3 d−1 organic loading rate (OLR) for hydrogen production. The effluents from UASB reactor containing mainly with acetate and butyrate were directly fed into CSTR for methane production and 5 days HRT was maintained. Both UASB and CSTR reactors were operated for 120 days continuously, and a stable production of the hydrogen and methane was obtained in the separate reactors. The maximum hydrogen and methane production rate achieved was 1.92 L H2 L−d−1 and 3.2 L CH4 L−1 d−1, respectively. The cumulative hydrogen and methane yields were 215 L H2/kgCOD−1 and 320 L CH4/kgCOD−1, respectively with the total COD removal efficiency of 94%. Thermoanaerobacterium species was dominant in hydrogen reactor, while methane reactor was dominated with Methanobrevibacter sp.

Anaerobic Co-digestion of Agricultural Byproducts with Manure for Enhanced Biogas Production

Biogas is extensively promoted as a promising renewable energy. Therefore, the search of appropriate co-substrates has come into focus. In this study, we examined the potential of using agricultural byproducts as alternative co-substrates for increased biogas production. The biochemical methane potential (BMP) of six agricultural organic byproducts were tested. Consecutively, the byproduct with the highest BMP was used as a co-digestion substrate with manure, in a continuous stirred tank reactor (CSTR). Meadow grass had the highest BMP value [388 ± 30 NmL of CH4 g–1 of volatile solids (VS)] among all mono-substrates tested. On the basis of BMP, the substrates ranked as follows: meadow grass > spring barley, winter wheat, winter barley, ryegrass > rapeseed > manure. Co-digestion of manure with byproducts resulted in only an additive and not synergistic methane production. Continuous co-digestion of 34 g L–1 raw meadow grass with manure increased the methane production rate of the CSTR reactor by 114% compa...

Continuous production of β-cyclodextrin by cyclodextrin glycosyltransferase immobilized in mixed gel beads: Comparative study in continuous stirred tank reactor and packed bed reactor

Cyclodextrin glycosyltransferase (CGTase) from Bacillus sp. C26 was highly stable when it was immobilized in mixed alginate–gelatin gel beads. Enzymatic synthesis of β-cyclodextrin (β-CD) from starch by immobilized CGTase was performed in a continuous stirred-tank reactor (CSTR) and a packed-bed reactor (PBR). A comparative study on reaction kinetics found that the apparent maximum production rate (Vmax,app) for reaction in CSTR (0.5742gL−1h−1) was higher than that in PBR (0.33gL−1h−1) likely due to the higher mass transfer rate in CSTR. The lower apparent Michaelis–Menten constant (Km,app) in PBR (31.88gL−1) compared to that in CSTR (58.3gL−1) indicate that the enzyme in PBR required lower amount of substrate to become half-saturated. The appropriate dilution rate for operation in both reactors was the same at 0.75h−1. To effectively produce β-CD, an integrated CSTR-PBR was developed. Through this integrated system, the final β-CD concentration and yield were improved from 6.10gL−1 and 15.3% in a single CSTR up to 10.6gL−1 and 26.5%, respectively. Moreover, the immobilized CGTase had good operational stability for 96h of continuous use. This study has shown that the integrated CSTR-PBR was effective for continuous production of β-CD from starch and may greatly contribute to developing industrialized production of β-CD.

Bioaugmentation as a solution to increase methane production from an ammonia-rich substrate.

Ammonia-rich substrates inhibit the anaerobic digestion (AD) process and constitute the main reason for low energy recovery in full-scale reactors. It is estimated that many full-scale AD reactors are operating in ammonia induced "inhibited steady-state" with significant losses of the potential biogas production yield. To date there are not any reliable methods to alleviate the ammonia toxicity effect or to efficiently digest ammonia-rich waste. In the current study, bioaugmentation as a possible method to alleviate ammonia toxicity effect in a mesophilic continuously stirred-tank reactor (CSTR) operating under "inhibited steady state" was tested. A fast growing hydrogenotrophic methanogen (i.e., Methanoculleus bourgensis MS2(T)) was bioaugmented in the CSTR reactor at high ammonia levels (5 g NH4(+)-N L(-1)). A second CSTR reactor was used as control with no bioaugmentation. The results derived from this study clearly demonstrated a 31.3% increase in methane production yield in the CSTR reactor, at steady-state, after bioaugmentation. Additionally, high-throughput 16S rRNA gene sequencing analysis showed a 5-fold increase in relative abundance of Methanoculleus spp. after bioaugmentation. On the contrary to all methods used today to alleviate ammonia toxicity effect, the tested bioaugmentation process performed without interrupting the continuous operation of the reactor and without replacing the ammonia-rich feedstock.

Experimental and modeling of a non-isothermal CSTR to find out parameter regions and conditions causing input multiplicity for acid catalyzed hydrolysis of acetic anhydride

The continuous stirred tank reactor (CSTR) presents challenging operational problems due to its complex open-loop non-linear behavior such as input and output multiplicities, ignition/extinction, parametric sensitivity, non-linear oscillations and perhaps even chaos. Many researchers have studied the non-linear dynamic behavior of CSTR for several reaction schemes from a theoretical stand point. The present experimental studies are carried out to identify the existence of input multiplicity in a non-isothermal CSTR. The sulfuric acid catalyzed hydrolysis of acetic anhydride reaction system has been chosen and the regions of existence of input multiplicity are theoretically identified. The reactor showed the same steady state temperature for two input flow rates of reaction mixtures into the CSTR reactor and thus confirming the existence of input multiplicity for the sulfuric acid catalyzed hydrolysis of acetic anhydride reaction system. It was observed that the time taken by the reactor was too long at a low feed flow rate of reaction mixture than at a higher feed flow rate reaction mixture. The model simulations of steady state temperatures were in close agreement with experimental data for the above feed flow rates of reaction mixture conditions.

Comparison of ASBR and CSTR reactor for hydrogen production from palm oil mill effluent under thermophilic condition

Hydrogen production from palm oil mill effluent (POME) by Thermoanaerobacterium thermosaccharolyticum PSU-2 was investigated both in batch and continuous reactors using anaerobic sequencing batch reactor (ASBR) and continuous stirred tank reactor (CSTR). The hydrogen production determined from batch experiment of POME at an inoculum size of 0%, 10%, 20% and 30% (v/v) was 161, 201, 246 and 296 mL H2/g-COD with COD removal efficiency of 21%, 23%, 23% and 23%, respectively. Continuous hydrogen production was start-up with 30% (v/v) inoculum in both ASBR and CSTR reactors and more than 30% COD removal could be obtained at HRT of 4 days, corresponding to OLR of 11.3 g COD/ L·day. Similar hydrogen production rates of 2.05 and2.16 LH2/L. day were obtained from ASBR and CSTR, respectively. COD removal efficiency of ASBR was 37.7%, while it was 44.8% for CSTR. However, ASBR was stable in term of alkalinity, while the CSTR was stable in term of hydrogen production, soluble metabolites concentration and alkalinity. Therefore, the CSTR was found to be more stable in hydrogen production than ASBR under the same OLR.

Continuous Biohydrogen Production with CSTR Reactor under High Organic Loading Rate Condition

A continuous stirred-tank reactor (CSTR) was used to produce biohydrogen gas from organic wastewater. The hydrogen producing reactor was operated under high organic loading rate of 21 kgCOD/m3·d, and molasses wastewater was used as substrate. Hydrogen production rate, pH value, sugar utilizing rate and fermentative products in effluent were investigated in continuous fermentation. When Organic Loading Rate was controlled at 21 kgCOD/m3·d, the average concentrations of acetic acid, ethanol, propionic acid, butyric acid and valeric acid in liquid fermentative products were 833, 748, 482, 484 and 256mg/L respectively. There is not any fermentation product playing dominant role absolutely in hydrogen production fermentation. The pH value in effluent was about 4.7~4.9, the average utilizing rate of sugar reached 92.1%, most of the sugar in molasses wastewater was utilized. The biogas production rate in hydrogen producing fermentation was from 21.2 to 27.1L/d, and the average biogas production rate was about 25.1L/d. The hydrogen content was about 37%.

OBSERVER-BASED INPUT-OUTPUT LINEARIZATION CONTROL OF A MULTIVARIABLE CONTINUOUS CHEMICAL REACTOR

The goal of this paper is to develop a nonlinear observer- based control strategy for a multi-variables continuous stirred tank reac- tor (CSTR). A new robust nonlinear observer is constructed to estimate the whole process state variables. The observer is coupled with a non- linear controller, designed based on the input-output linearization for controlling the concentration and reactor temperature. The closed loop system is shown to be globally asymptotically stable based on Lyapunov arguments. Finally, computer simulations are developed for showing the performance of the proposed controller. Continuous stirred tank reactors (CSTR) are ones of the most important plants in chemical industry. In practice, however, the control of these reac- tors poses a number of challenging problems. These problems arise from the presence of severe nonlinearities of these chemical reactors, as well as for mon- itoring and control applications, only a few measurements are available, either because the measuring devices do not exist or are too expensive. Therefore, we can deduce that the main difficulties arising in the control of CSTR's reactors arrive from two main sources: the process complexity and the difficulty to have reliable measurements of state variables ((1), (16)). In recent years, a various design methods of nonlinear control strategies have been proposed. Most of these based on differential geometric concepts ((15)-(18), (23)). This method allows a certain class of systems to be linearized using state feedback and co- ordinate transformations ((12), (14)). Extensions of the method like adaptive linearization ((21), (25)), robust linearization (24), and asymptotically exact linearization (13) account for small model-plant-mismatch.