Steady states of a biotechnological process for producing lactic acid at a given substrate concentration in the inlet stream

Relationships are derived that form the domain in which the multiplicity of steady states exists for the continuous process of lactic acid production in a fermenter with stirring. Algorithms are described for estimating the technological characteristics of the process under the conditions of multiplicity. Methods and results of calculations are presented for different variants of specifying the concentration of a substrate in the inlet stream, and examples of numerical calculations are given.     

同时发生初次和二次成核且为聚结控制的沉淀过程多定态及稳定性分析

The possibility of multiplicity in an isothermal continuous mixed suspension-mixed product removal crystallizer is explored using the bifurcation theory.A process involving agglomeration controlled precipitation is considered in which secondary nucleation occurs simultaneously with primary nucleation.The determinant equations for the existence of multiple steady states are developed and the multiplicity boundaries dependent on the physical and kinetic properties and operational parameters of the process are obtained by resolving these determinant equations.The number of steady states in the precipitator for various multiplicity regions is determined and the linear stability of these steady states is analyzed by using the Routh criterion.     

SUDDEN DESTABILIZATION OF CONTROLLED CHEMICAL PROCESSES

Input multiplicity occurs when more than one set of inputs can produce the same set of outputs. Input multiple steady states are divided into compatible steady states having process gains of similar sign, and opposed steady stales with process gains of opposite sign. For a system controlled with reset action, only the compatible steady states satisfy the necessary condition for stability. Any disturbance which drives the controlled system from the designed steady state to a less stable or unstable compatible steady state can cause sudden destabilization of the process. Several examples are given of the possible types of behavior resulting from this phenomenon.

Multiple steady states also occur for systems with proportional controllers. For single-input-single-output systems with continuous process characteristics, whether or not reset action is used, two steady states positioned next to each other cannot both be stable under closed-loop control. However, under proportional control, opposed steady states for which 1 + K_cK_p is positive can be stable.     

MSMPR连续结晶器的多定态

对混合悬浮混合排料(MSMPR)连续结晶器的定态方程组进行了剖析,应用非线性方程组的延拓求解法,求得了定态方程组的分岔图,即MSMPR连续结晶器状态参数与操作参数的依赖关系.利用分岔理论对MSMPR连续结晶过程的定态数目及相应的参数范围进行了预测,导出了连续结晶过程存在多定态或唯一定态的判据。该判据仅与过程的动力学参数和操作参数有关,而与过程的状态参数无关。最后应用Routh-Hurwitz准则,确定了连续结晶过程各定态的稳定性。     

Multiplicity of steady states in FCC units: effect of operating conditions

The FCC dynamics has been studied to determine the effect of operating conditions on multiplicity of steady states. The proposed model is capable to analyze several dynamic features to characterize the FCC process. Three combustion modes are analyzed: Partial combustion, total combustion and transition mode. At least three steady states have been determined in both partial and complete combustion modes. Furthermore, it is established in this paper that operating conditions are incapable to change the number of steady states. It is believed that a similar analysis should be applied to identify the effect on the multiplicity of physical parameters such as oxidation rates.     

MULTIPLICITY AND STABILITY ANALYSIS OF AGGLOMERATION CONTROLLED PRECIPITATION

The possibility of multiplicity in continuous isothermal MSMPR precipitators has been explored for agglomeration controlled conditions and general criteria, independent of nucleation kinetics, are developed for stability and multiplicity of the steady states. For the Volmer model of primary nucleation and the magma dependent power law model of secondary nucleation, parameter regions are determined in which multiple steady states exist, and their linear stability is analyzed. The analysis holds in general for all types of agglomeration kernel. For the Volmer nucleation kinetics three steady states exist in the region of multiplicity with the “middle” one always being unstable. The analysis for magma dependent power law model showed multiplicity regions having as many as four steady states, the number depending on magma and kinetic order. Unlike the case of molecular growth control, limit cycle behaviour is not possible, and the approach to the steady state is always asymptotic     

MULTIPLICITY AND STABILITY ANALYSIS OF AGGLOMERATION CONTROLLED PRECIPITATION

The possibility of multiplicity in continuous isothermal MSMPR precipitators has been explored for agglomeration controlled conditions and general criteria, independent of nucleation kinetics, are developed for stability and multiplicity of the steady states. For the Volmer model of primary nucleation and the magma dependent power law model of secondary nucleation, parameter regions are determined in which multiple steady states exist, and their linear stability is analyzed. The analysis holds in general for all types of agglomeration kernel. For the Volmer nucleation kinetics three steady states exist in the region of multiplicity with the “middle” one always being unstable. The analysis for magma dependent power law model showed multiplicity regions having as many as four steady states, the number depending on magma and kinetic order. Unlike the case of molecular growth control, limit cycle behaviour is not possible, and the approach to the steady state is always asymptotic     

Multiple steady states for isothermal catalytic gas—solid reactions with a positive reaction order

The multiplicity of steady states of solid catalysed gas reactions induced by multicomponent diffusion is described, and a necessary condition for existence is given. Two examples of such multiple steady states are discussed and their stability analyzed.     

Steady-state multiplicity in a solid oxide fuel cell: Practical considerations

Steady-state multiplicity in a solid oxide fuel cell (SOFC) in three modes of operation, constant ohmic external load, potentiostatic and galvanostatic, is studied using a detailed first-principles lumped model. The SOFC model is derived by accounting for heat and mass transfer as well as electrochemical processes taking place inside the fuel cell. Conditions under which the fuel cell exhibits steady state multiplicity are determined. The effects of operating conditions such as convection heat transfer coefficient and inlet fuel and air temperatures and velocities on the steady state multiplicity regions are studied. Depending on the operating conditions, the cell exhibits one or three steady states. For example, it has three steady states: (a) at low external load resistance values in constant ohmic external load operation and (b) at low cell voltage in potentiostatic operation.     

Analysis of multiple steady states of complex biochemical reactions

A general theory for the identification of criteria within which an enzymic reaction can exhibit multiplicity (i.e. multiple steady states) is reviewed. Application of the theory to four types of kinetic models encountered in biochemical reactions is illustrated and figures are presented to delineate the region of multiplicity. The stability of the steady states is analysed for small perturbation about the steady state.     

Modelling of chemical reactors-XXXII: Nonisothermal Nonadiabatic Tubular Reactor. One Dimensional Model-detailed Analysis

A detailed analysis of the region of multiple steady states in nonisothermal-nonadiabatic systems is performed. It is shown that in a certain region of Damköhler numbers five steady states are possible. These steady states are referred to as the kinetic, diffusion and intermediate regimes. Numerical procedures for calculation of regions of multiplicity are described. Numerical analysis of the problem indicates that with higher value of the cooling parameter β multiplicity is suppressed. For higher values of the Peclet number multiple steady states disappear. The analysis has shown that in nonadiabatic systems Frank-Kameneckij approximation of the temperature dependence of the reaction rate must be used with some care.     

等温CMSMPR结晶器定态及瞬态特性分析(Ⅱ)分岔行为及瞬态特性

应用非线性方程组的延拓解法 ,对同时发生初级成核和二次成核的等温连续混合悬浮混合排料(CMSMPR)结晶器的定态方程组进行了求解 ,考察了结晶过程的定态在不同参数区域之间的分岔行为 ,确定了各参数区域内结晶过程的定态数目 ,并应用Routh -Hurwitz准则对各定态的稳定性进行了分析 .采用四阶Runge -Kutta法求解CMSMPR结晶器的动态方程组 ,分析了在多定态区域以及持续振荡区域中结晶过程的瞬态特性     

The regions of multiple stable steady states of a biological reactor with wall growth,utilising inhibitory substrates

The stirred tank biological reactor with wall growth and an inhibitory substrate is analysed. The necessary and sufficient conditions for the existence and stability of a multiplicity of steady states are given. Diagrams are presented showing the effect of wall growth, reactor feed concentration and a kinetic inhibition parameter on conversion and on the existence of multiple steady states.     

Thermal regimes of exothermic processes in continuous stirred tank reactors

Thermal regimes of a CSTR are analysed from the standpoint of the thermal explosion theory. Critical phenomena and time characteristics of the process are considered. It is shown that for appropriately chosen dimensionless parameters the critical conditions of the thermal explosion and induction period in a CSTR slightly differ from those for non-flow (batch) systems. These corrections may be performed by using an additional parameter which is characteristic for flow systems. In the region of multiplicity of steady states, the critical condition of thermal explosion is characterized by a jump of self-heating, while in the non-flow systems such phenomenon occurs only for a zero order reaction. For a unique steady state the transition through the critical condition is continuous.     

Modeling the continuous biotechnological process of lactic acid production

The results of the mathematical modeling of a continuous biotechnological process for producing lactic acid are considered. The model is based on the Luedeking-Piret kinetic equation, which includes the concepts of the maximum specific rates of biomass growth, substrate consumption, and product formation. The possibility of the existence of optimum conditions for productivity with respect to the target product and the possibility of the existence of a multiplicity of steady states that ensure the same value of productivity are shown. An algorithm is described for calculating the characteristics of the process for different concentrations of a substrate in the inlet stream and different dilution rates.     

Modeling of the Nitration of Amyl in a Continuous Stirred Tank Reactor and a Tube Reactor

Parametric analysis of mathematical models of amyl nitration in a continuous stirred tank reactor and a tube reactor was erformed. Curves of steady states versus control parameters, multiplicity and neutrality curves of the steady states, and parametric and phase portraits of the system were plotted. The critical values of the input concentration and the heat-transfer coefficient leading to disturbance of the steady-state nitration of amyl were obtained for the tube reactor.     

Steady-state multiplicity analysis of two-stage-riser catalytic pyrolysis processes

New two-stage-riser fluidized catalytic pyrolysis (TSRFCP) for maximizing propylene yield technology is considered as an efficient route to moderate the propylene demand/supply gap and to lower the propylene price. The possibility of existence of complex nonlinear behavior associated with the TSRFCP process puts limitations on the supervision of this system. Based on the developed and validated model for the TSRFCP process, this paper focuses on the elucidation of multiple steady states and relevant (in) stability characteristic over a wide range of operating condition. First, graphic analysis of heat generation/removal curves demonstrates that the TSRFCP process has at least one steady state and a maximum of three output steady states under the considered operating conditions and uncertainties such as cooling water flow rates and Conradson carbon residue. Then, operating maps revealing topologies between important input and output variables can disclose detailed nonlinear behavior (input/output multiplicity). Moreover, depending on the choice of the input variable and the relevant operating/design condition, input multiplicity may exist. In short, these results can guide the succeeding control structure selection for realistic TSRFCP processes.     

Bifurcation and its implications for a novel autothermal circulating fluidized bed membrane reformer for the efficient pure hydrogen production

The present investigation is a prelude to the experimental exploration of Static Bifurcation Behavior (SBB) in a novel Autothermal Circulating Fluidized Bed Membrane Reformer (ACFBMR) for pure hydrogen production by steam reforming of heavy hydrocarbons. Important impacts of a wide range of design and operating parameters on the reformer performance are explored with two reformer configurations. One is with the catalyst regeneration before the gas-solid separation and another is with the catalyst regeneration after the gas-solid separation. For both configurations there are three steady states (multiplicity of the steady states, static bifurcation behavior). The system behavior in the bifurcation region is quite complex and defies the simple logic of non-autothermal processes. For the first configuration, on the branch of upper temperature steady state the carbon formation and deposition on the nickel catalyst is highest, while the net hydrogen yield is highest on the branch of lower temperature steady state. For the second configuration, the conversion of heptane is always 100%. In the multiplicity region, the order of net hydrogen yield from high to low is the middle, upper and lower temperature steady states, while the order of reformer exit carbon flow rate from high to low is the lower, middle and upper temperature steady states. Efficient production of pure hydrogen for fuel cells requires fundamental and practical understanding of their bifurcation behaviors.     

Multiplicity of steady states in an isothermal batch fluidized bed reactor

An investigation of uniqueness and multiplicity of the steady states in an isothermal fluidized bed reactor and continuous stirred tank reactor is presented. Material balance equations for the two phase model of Davidson and Harrison and the modified Partridge and Rowe model, are transformed into a single equation which is of the same form as the material balance equation for an isothermal continuous stirred tank reactor. Exact uniqueness and multiplicity criteria are derived, and numerical examples illustrating the influence of some parameters on the steady state multiplicity are presented.