What Is Energy Homeostasis?

Multiple steady state phenomena in the rectification process.

For any chemical process, it can be regarded as a white box or black box or gray box model (depending on how well we master the process laws). Parameters related to process feed logistics, equipment specifications, size parameters, and process operating parameters can be used as process input variables; parameters such as flow, temperature, and composition of the process product logistics that we care about can be used as process output variables (Output variables); and in the process, we usually do not care about the amount, such as the flow of the intermediate logistics, composition, and temperature and composition distribution in the equipment can be used as internal state variables of the process (internal state variables). The state variables of chemical processes with multiple steady states can also be divided into the above three types. There are three different types of multi-stable states, which are output multiplicity, input multiplicity, and internal state multiplicity. Gani and Jørgensen [28] gave definitions of these three different types of multiple steady states.

(1) Output multiplicity: On the premise of meeting the requirements of the degree of freedom of the process, for a given set of input variables, there are multiple solutions for the specified output variable. For example, in the normal distillation process, we specify the feed conditions (flow, temperature, pressure, composition), the number of trays or packing height, the type of reboiler and condenser, the feed position, and the profile of the pressure distribution of the entire column. , The output of the tower top and the return flow, but the product composition is different.

(2) Input multiplicity: On the premise of meeting the requirements of the degree of freedom of the process, when specifying the value of a specified set of output variables, there are multiple solutions for a set of input variables we specify. Typical examples are during reactive distillation, when we specify the feed conditions, the number of trays or packing height, the type of reboiler and condenser, the position of the feed, the profile of the pressure distribution of the whole column, the return flow, and the overhead column. When the purity of the kettle product, the thermal load of the reboiler has two or more solutions.

(3) Multiplicity of internal states: On the premise that the degree of freedom requirements of the process are met, when a set of specified input and output variables are specified, there are multiple sets of solutions for describing the state variables inside the process. For example, in the rectification process, when we formulate our specified feed conditions, number of trays or packing height, reboiler and condenser type, feed location, pressure profile of the whole column, top recovery and At the time of return flow, the product composition is also the same at this time, but the temperature and gas-liquid phase composition distribution of the whole column are different. ^[1]

There are multiple steady-state phenomena in many physical and chemical processes, such as chemical reaction processes and mass transfer unit operation processes (rectification, extraction, absorption, crystallization, membrane separation, etc.). Multiple steady-state phenomena can exist on a single process device, or in a set of devices, or even in the entire chemical process system. It is precisely the existence of multiple steady-state phenomena that has led researchers to explore their causes.

(1) The root cause of multiple steady state phenomena

For a chemical process, we usually use a system of equations, a mathematical model of the process, to describe the characteristics of the process. Model equations include linear algebraic equations, nonlinear algebraic equations, and differential equations. Therefore the process model can be described by the system of equations f (x) = 0 composed of these equations. If all the equations describing the process are linear equations, when the number of unknown quantities and the number of independent equations are the same, and there is only a unique solution, there is no possibility of multiple steady-state phenomena. However, due to its own complexity, many actual physical and chemical processes have some nonlinear algebraic equations and differential equations in the system of equations used to describe the process. When the number of unknown quantities and the number of independent equations are the same, there may be multiple solutions and multiple Steady state phenomenon. Of course, since the state variables of a process generally have their practical significance, they must be in the feasible region . When the solution is outside the feasible region, the solution has no practical physical meaning. When there are two or more solutions in the feasible region , the process has multiple steady-state phenomena. Therefore, the root cause of multiple steady-state phenomena is the non-linearity of the process.

(2) Specific causes of multiple steady-state phenomena during rectification

Many researchers have analyzed the specific chemical process and explored the causes of its multiple phenomena, and the results show that the specific reasons for the multiple steady-state phenomena caused by different processes are different. As far as the rectification process is concerned, it is found that there are multiple steady-state phenomena in two-component and multi-component ordinary distillation, azeotropic distillation, and reactive distillation. In addition, for thermally coupled distillation columns and internally connected distillation columns There are also multiple steady-state phenomena in the system.

The specific causes of multiple steady-state phenomena in the distillation process are mainly the following:

(A) Non-linear conversion between mass flow or volume flow and molar flow

Jacobsen investigated the two-component ordinary distillation, and found that under the condition of constant molar flow, when the flow is expressed by mass flow or volume flow (not molar flow), there are multiple steady-state phenomena.

(B) Interaction between molar flow and composition in the tower

Jacobsen found that when the energy balance equation was introduced into the model equation, there were multiple steady-state phenomena in the process, although the return flow and the rising steam flow in the tower kettle were expressed in molar flow.

(C) Thermodynamic behavior of the system

Bekiaris investigated the multiple steady-state phenomena of systems with different thermodynamic behaviors (including VLE and VLLE) during homogeneous azeotropic distillation and heterogeneous azeotropic distillation from the perspective of residual curves and distillation curves. According to the phase diagram of the system, / analysis is used to give the necessary and sufficient conditions for the existence of this kind of multiple steady-state phenomena; cold

The influence of condenser on multiple steady-state phenomena was explored. Güttinger and Esbjergde also used / analysis to reveal multiple steady-state phenomena in homogeneous azeotropic distillation and heterogeneous azeotropic distillation sequences, respectively.

(D) Nonlinearity of reaction kinetics

Due to the nonlinear relationship between reaction rate and temperature and composition during reactive distillation, multiple steady-state phenomena may occur during reactive distillation.

(E) Process nonlinearity caused by the connection of rectification column and other unit equipment

Chavez and Lin found multiple steady-state phenomena in the internally connected rectification column system, and this multiple steady-state did not exist in a single distillation column. Recently, Kano and Wang found that there are multiple steady-state phenomena in thermally coupled distillation column systems.

It is worth mentioning here that / analysis is a very effective tool for analyzing multiple steady-state phenomena caused by the thermodynamic behavior of the system. The so-called / analysis is to analyze the performance of the system when it is assumed that the number of trays in the distillation column is infinite or the height of the packing is infinitely high, and the distillation column is in full reflux operation. Although the number of plates or packing height is limited under practical operating conditions, and the distillation column is also operated at an economically feasible reflux ratio, Bekiaris pointed out that if there are multiple steady-state solutions under the limit condition of / , Under actual conditions, as long as the number of trays is sufficient or the packing height is high enough, the reflux ratio is large enough, there will still be multiple steady-state phenomena. The number of trays or packing height is reduced, the reflux ratio is reduced, and the probability of multiple steady-state phenomena is reduced. Therefore, when we find that there are multiple steady-state phenomena under the limit condition of / , then there may be multiple steady-state solutions under actual operating conditions. ^[1]

As stated by Doherty and Perkins, although the rectification process model has serious nonlinearities, the literature on rectification holds that the rectification model has only one singular point of global asymptotic stability. For a long time, it has been generally believed that there is only one steady state in the rectification process. Rosenbrock also proved that for an ideal two-component system, only a unique solution can exist under the constant molar flow assumption.

The earliest discovery of multiple steady-state phenomena in the rectification process is due to the well-known Soviet scholar Petlyuk. Petlyuk found that multiple steady-state phenomena may exist during the distillation of ternary non-ideal systems based on the constant molar flow assumption. He also proposed / analysis, which was adopted by researchers such as Bekiaris. Shewchuk simulated the distillation process of the ethanol-water-benzene system and found that different initial values were assigned to the process, and the ethanol dehydration column converged to obtain two different solutions. Magnussen also simulated the distillation process of the ethanol-water-benzene system and found three steady-state solutions. The liquid-liquid phase separator was not considered in the simulation calculation. Instead, a stream was introduced from the top of the tower. The results showed that the entire liquid phase composition corresponding to all the solutions was located in the homogeneous phase zone, and there was no phase separation. This result has caused many researchers to pay attention to the multiple steady-state phenomenon during azeotropic distillation. Different researchers have different results for the same system because the assumptions used in the model are different.

For the ordinary distillation process of the two-component ideal system, Jacobsen found multiple steady-state phenomena in the following two different situations: Under the assumption of constant molar flow, the nonlinear relationship between different input variables may cause multiple steady-state phenomena , Such as mass flow or volume flow and molar flow, non-linear relationship between heat load and molar flow; When the energy balance equation is introduced into the model, the constant molar flow assumption is abandoned, even if the flow is expressed in molar flow, due to the molar flow The non-linear interaction between the composition and composition may also cause multiple steady-state phenomena. Jacobsen found that when the model uses different types of independent variables, multiple steady-state phenomena may not occur. When the energy balance equation is introduced into the model, five steady-state solutions are found when the reflux mass flow rate Lw and the column rising steam volume V are taken as independent variables; and when the overhead distillate mass flow rate Dw and the tower rising steam No multiple solutions were found for the quantity V as an independent variable. The author also makes a detailed analysis of the stability of the solution. Jacobsen studied the effect of multiple steady states on process operation and control for the same system. For manual operation, the unstable operating point cannot be reached due to the multiplicity and instability of the solution; and this unstable operating point can be reached by single-point control of composition or temperature.

Bekiaris used the / analysis tool to investigate the multiple steady-state phenomena of the ternary system homogeneous azeotropic distillation process in detail. Under the limit of infinite number of trays in the distillation column or infinitely high packing height, and under the condition of full reflux operation, we can plot the output of overhead distillation as the bifurcation parameter according to the phase diagram of the system Corresponding bifurcation diagram. Through rigorous theoretical analysis, sufficient and necessary conditions for the emergence of this type of multiple steady state are proposed. In addition, the author confirmed through simulation that the multiplicity obtained under the limit operating conditions is also useful for practical operation of the distillation column, that is, if there are multiple steady-state solutions under the limit condition of / , This multiple steady state phenomenon may occur in a practical distillation column. Based on the results of this research, Bekiaris also explored the multiple steady-state phenomena in the ternary system of heterogeneous azeotropic distillation. The condenser examined the multiplicity of process solutions. P

Subsequently, some researchers have further explored the multiple steady-state phenomenon of azeotropic distillation based on theoretical analysis and simulation.

It is worth noting that the multiple steady state phenomena mentioned above regarding the rectification process are theoretically analyzed and simulated based on certain assumptions. As for whether the multiple steady-state phenomenon actually exists in the rectification process, it has not been experimentally verified. Some researchers have carried out relevant experiments for specific systems. Kienle's methanol-n-propanol binary system for the first time confirmed the first type of multiple steady-state phenomena reported by Jacobsen: if the reflux ratio and the tower heat load were used as independent input variables, when the reflux volume flow was the same, the Three different reflux ratios were obtained. Koggersb's experiments on the methanol-isopropanol binary system confirmed once again that multiple steady-state phenomena do indeed exist.

Güttinger tested the ternary homogeneous system of methanol-methyl butyrate-toluene on a rectification column used for industrial pilot tests. The results showed that when the feed flow, composition and operating conditions of the column were the same, two A stable steady-state solution; and the results agree well with the / analysis and simulation results. Wang [69] found two different profiles of temperature distribution in the homogeneous azeotropic distillation process of the ternary system of isopropyl alcohol-cyclohexane-water. For the ethanol-water-cyclohexane system, Müller70 verified for the first time experimentally the existence of multiple steady-state phenomena in heterogeneous azeotropic distillation. The experimental results agree well with the / analysis and simulation results. These experiments

The results verify the multiple steady-state phenomena caused by the VLE and VLLE systems reported by Bekiaris, Güttinger, and Esbjergde. With the enthusiasm for reactive distillation research in the 1980s, some researchers also discovered multiple steady-state phenomena in reactive distillation. As far as we know, Pisarenko first reported multiple steady-state phenomena in reactive distillation. His theoretical analysis of a reactive distillation column with a single product stream showed that, for the same feed and operating conditions, three different steady-state solutions were obtained. Nijhuis and Jacobs reported multiple steady-state phenomena during MTBE reactive distillation and gave corresponding explanations. Their research results have attracted the interest of many researchers, and a large number of papers on the multiple steady-state phenomena of reactive distillation have emerged in the following ten years. During this period, due to advances in computer technology, some researchers' research on multiple steady-state phenomena began to gradually shift from traditional steady-state simulation to dynamic control. ^[1]