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Section “Modeling of physical, mechanical and thermal processes in machines and devices”
MATHEMATICAL MODEL OF CONDENSER-EVAPORATOR OF AN AIR SEPARATION INSTALLATION
V. V. Chernenko, D. V. Chernenko
Siberian State Aerospace University named after Academician M. F. Reshetnev
Russian Federation, 660037, Krasnoyarsk, ave. them. gas. "Krasnoyarsk worker", 31
Email: [email protected]
A mathematical model of the condenser-evaporator of cryogenic air separation plants, based on the joint solution of the equations of hydrodynamics and heat transfer for tubular devices, is considered.
Key words: condenser-evaporator, mathematical model, design, optimization.
MATHEMATICAL MODEL OF AIR SEPARATION PLANT EVAPORATOR-CONDENSER
V. V. Chernenko, D. V. Chernenko
Reshetnev Siberian State Aerospace University 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation E-mail: [email protected]
The mathematical model of evaporator-condenser of cryogenic air separation plants, based on the simultaneous solution of hydrodynamics and heat exchange equations for the tubular devices.
Keywords: evaporator-condenser, mathematical model, design, optimization.
Condenser-evaporators in air separation units (ASU) are used to condense nitrogen due to the boiling of oxygen, i.e. They are heat exchangers with a change in the state of aggregation of both media involved in the heat exchange process.
The efficiency of the condenser-evaporator largely determines the efficiency of the entire installation. For example, an increase in the temperature difference between heat-exchanging media by 1 °K leads to an increase in energy consumption for air compression to 5% of the total energy costs. On the other hand, a decrease in temperature pressure below the limit value leads to the need for a significant increase in the heat transfer surface. Taking into account the high energy consumption and metal consumption of ASU devices, the need to optimize each of their elements, including the condenser-evaporator, becomes obvious.
The most appropriate method for studying and optimizing such large and expensive objects is mathematical modeling, since it allows you to objectively consider and compare many different options and select the most acceptable one, as well as limit the scale of the physical experiment by checking the adequacy of the model and determining the numerical values of the coefficients that cannot be obtained analytically way.
ASU condenser-evaporators operate in natural circulation mode; accordingly, they have a complex relationship between the thermal and hydraulic characteristics of the evaporation process. Heat transfer from the boiling liquid is determined by the circulation rate, which, in turn, can be found from hydraulic calculations with known values of heat flows and geometric dimensions of the heat exchange surface, which are the target function of the optimization problem. In addition, the boiling process occurs simultaneously with the condensation process, which imposes restrictions on the ratio of heat flows and temperature pressures of both processes. Thus, the model should be built on the basis of a system of equations that describe the circulation of boiling liquid and heat transfer processes on both sides of the heat transfer surface.
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The presented model, the diagram of which is shown in Fig. 1 includes the most typical cases encountered in the design and operation of evaporator condensers. The calculation method is based on the principle of successive approximations.
The following input factors are used: the value of the total thermal load; pressure on the boiling side; pressure on the condensing side; concentration of evaporating vapors in terms of O2; condensate concentration by N2; height, outer and inner diameters of pipes.
The block of pre-selected parameters includes determination of the boiling and condensation temperatures of working media, taking into account impurities, as well as a preliminary assessment of the values of the available temperature pressure and the average specific heat flux on the active surface of the heating section from the side of the boiling liquid, necessary to start the hydraulic calculation.
The purpose of the hydraulic calculation is to determine the circulation rate, the length of the economizer zone, pressures and temperatures in characteristic sections of the channel. For the calculation, a traditional circuit diagram with natural fluid circulation is used (Fig. 2).
1 Input factors /
Pre-selection of parameters
Hydraulic calculation
Thermal calculation
Heat transfer during condensation
Recoil temperature when boiling
Convergence of calculation results and selected - _ values
Output parameters
Rice. 1. Design diagram of the ASU condenser-evaporator model
Rice. 2. Hydraulic model of the condenser-evaporator ASU: I - pipe length; 1op - length of the lower part; /ek - length of the economizer part; 4ip - length of the boiling part; 1р - working length; ω0 - circulation speed
The task of thermal calculation is to clarify the value of the heat flux density in the active section of the pipe based on the results of hydraulic calculation, as well as to clarify the available temperature pressure taking into account hydrostatic and concentration temperature depression. The condensation calculation module uses a heat transfer model for the condensation of a single-component vapor on a vertical wall with a laminar flow of the condensate film. The boiling calculation module is based on a model of heat transfer to a two-phase flow in a pipe.
Section “Modeling of physical-mechanical and thermal processes in machines and devices”
Hydraulic and thermal calculations are repeated in the same sequence if the preliminary and calculated values of the heat flux density differ by more than 5%. The calculation accuracy, as a rule, turns out to be sufficient after the second approximation.
The output parameters are the heat exchange surface area, the diameter of the central circulation pipe, the number and layout of pipes in the tube sheet and the diameter of the apparatus casing.
1. Narinsky G. B. Liquid-vapor equilibrium in oxygen-argon, argon-nitrogen and oxygen-argon-nitrogen systems // Proceedings of VNIIKIMASH. 1967. Vol. eleven ; 1969. Vol. 13.
2. Grigoriev V. A., Krokhin Yu. I. Heat and mass transfer devices of cryogenic technology: textbook. manual for universities. M.: Energoizdat, 1982.
3. Air separation using deep cooling method. 2nd ed. T. 1 / ed. V. I. Epifanova and L. S. Axelrod. M.: Mechanical Engineering, 1973.
© Chernenko V.V., Chernenko D.V., 2016
| The most important: An electrical capacitor can store and release electrical energy. At the same time, current flows through it and the voltage changes. The voltage across the capacitor is proportional to the current that passed through it over a certain period of time and the duration of this period. An ideal capacitor produces no thermal energy. If an alternating voltage is applied to a capacitor, an electric current will arise in the circuit. The strength of this current is proportional to the frequency of the voltage and the capacitance of the capacitor. To estimate the current at a given voltage, the concept of capacitor reactance is introduced. The variety of types and types of capacitors allows you to choose the right one. |
A capacitor is an electronic device designed to accumulate and subsequently release an electrical charge. The performance of a capacitor is directly related to time. Without considering the change in charge over time, it is impossible to describe the operation of a capacitor.
Unfortunately, errors are periodically found in articles; they are corrected, articles are supplemented, developed, and new ones are prepared.
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When studying the dynamics of turbine control, the change in pressure pg in the condenser is usually not taken into account, assuming lg = kp £1pl = 0. However, in a number of cases the validity of this assumption is not obvious. Thus, during emergency control of heating turbines, opening the rotary diaphragm can quickly increase the steam flow through the LPC. But at low flow rates of circulating water, characteristic of conditions of high thermal loads of the turbine, the condensation of this additional steam can proceed slowly, which will lead to an increase in pressure in the condenser and a decrease in power gain. A model that does not take into account the processes in the capacitor will give an overestimated efficiency of the noted method of increasing injectivity compared to the actual one. The need to take into account processes in the condenser also arises when using a condenser or its special compartment as the first stage of heating network water in heating turbines, as well as when regulating heating turbines operating at high thermal loads using the method of sliding back pressure in the condenser and in a number of other cases.
The condenser is a surface-type heat exchanger, and the above principles of mathematical modeling of surface heaters are fully applicable to it. Just as for them, for a capacitor one should write down the equations of the water path either assuming the parameters are distributed [equations (2.27) - (2.33)], or approximately taking into account the distribution of parameters by dividing the path into a number of sections with lumped parameters [equations (2.34) - ( 2.37)]. These equations must be supplemented with equations (2.38)–(2.40) for heat accumulation in the metal and equations for the vapor space. When modeling the latter, one should take into account the presence in the steam space, along with steam, of a certain amount of air due to its influx through leaks in the vacuum part of the turbine unit. The fact that the air does not condense determines the dependence of the pressure change processes in the condenser on its concentration. The latter is determined both by the amount of inflow and by the operation of the ejectors, pumping air out of the condenser along with part of the steam. Therefore, the mathematical model of the vapor space should, in essence, be a model of the “condenser vapor space - ejectors” system.
Resistor
The mathematical model of the resistor (Fig. 2.1) is described by Ohm’s law:
U R =IR, or I=gU R, where g=1/R.
In the first case, the voltage drop U R across the resistor is specified, and the desired value is the current I through the resistor. In the second case, the current I is specified through a resistor, and the desired value is U R across the resistor.
nominal resistance value R N;
resistance tolerance R;
temperature coefficient TCR.
Tolerance R is the limit of resistance deviations from the nominal value that arise during the manufacturing process of resistors:
in this case, the resistances of resistors during their production can take on the following values:
If the resistance value R is less than the nominal R H , then the relative deviation R/ R H 0, otherwise R/ R H 0.
Usually the tolerance R is specified as a percentage.
Temperature coefficient TKR sets the resistance value for the current temperature T:
where T N – nominal temperature value taken equal to 27 0 C.
Thus, TKR is equal to the relative deviation of the resistance from the nominal value when the temperature changes by 1 0 C. Sometimes TKR is set in propromil (ppm) :
TKR ppm = TKR 10 6 .
Capacitor
The mathematical model of the capacitor (Fig. 2.2) is written as:
or
In the first case, the given value is the voltage drop U C (t) across the capacitor, and the desired value is the current through the capacitor I(). In the second case, the given value is the current through the capacitor I(t), and the desired value is the voltage drop U C (t).
Parameters of the mathematical model:
nominal value of capacitance CH;
capacity tolerance С;
temperature coefficient TKC.
The concept of tolerance and temperature coefficient was given when describing the resistor model.
Inductor
The inductor (Fig. 2.3) is described by two mathematical models:
or
The parameters of the mathematical model are L H , L , TKL, the contents of which are similar to those considered for the resistor and capacitor.
Real models of a resistor, capacitor, and inductance are more complex than those discussed here.
Thus, models of even the simplest components can be quite complex if a high degree of adequacy of the parameters of a physical object and its mathematical model is required.
Double winding transformer
The transformer (Fig. 2.4) can be represented as the following mathematical model:
where L 1, L 2 are the inductances of the windings,
M 12 – mutual inductance.
The model parameters are the values of L 1, L 2 and the coupling coefficient
The value of K SV ranges from zero to one. The value of K SV = 1 indicates the presence of a rigid connection between the windings, which is typical for matching and power transformers and for output transformers of amplifiers. K value NE<1 говорит о наличии в трансформаторе индуктивности рассеяния, что приводит к уменьшению коэффициента передачи на высоких частотах. Такие трансформаторы используются в резонансных контурах фильтров.
Sometimes the following parameters are specified:
In addition to the listed parameters, you need to indicate the method of switching on the windings - consonant or counter.
The industrial production of elemental sulfur by the Claus method is based on the partial oxidation of hydrogen sulfide in the original acid gas with atmospheric oxygen and sulfur dioxide.
As is known, the composition of acid gas, in addition to H 2 S, usually includes: CO 2, H 2 O and hydrocarbons. This causes side chemical reactions to occur that reduce the yield of sulfur.
The amount of each component from this set of impurities influences the choice of one or another modification of the Claus process.
In our case, the original acid gas consists of approx. 95%Vol. H2S; 3.5% vol. H2O; up to 2% vol. hydrocarbons.
In world practice, acid gases of this composition are processed into sulfur according to the most rational “direct Claus process”.
In the thermal stage of the process, reactions of partial oxidation of hydrogen sulfide both into sulfur and sulfur dioxide occur. And also the interaction reactions of the components present in the system, for example:
2H 2 S + O 2 = S 2 + 2H 2 O + 37550 kcal/kmol H 2 S
2H 2 S + 3O 2 = 2SO 2 + 2H 2 O + 125000 kcal/kmol H 2 S
2H 2 S + SO 2 = 3S + 2H 2 O
H 2 S + CO 2 = COS + H 2 O - 6020 kcal/kmol COS
CH 4 + 2O 2 = CO 2 + 2H 2 O + 192000 kcal/kmol CH 4
When leaving the thermal stage in the gas, in addition to the target product - elemental sulfur - there are also other components present: H 2 S, CO 2, COS, CS 2, CO 2, H 2 O, CO, H 2 and N 2.
The degree of conversion (conversion) of the initial hydrogen sulfide into sulfur in the thermal stage of the process can reach a value of about 70%.
Ensuring a total conversion of more than 70% for the installation is achieved by sequentially connecting several catalytic stages to the thermal system. In the latter, operating conditions for the process are maintained in which all sulfur-containing components of the process gas enter into chemical reactions with the release of sulfur, for example:
2H 2 S + SO 2 = 3/N S N + 2H 2 O + Q 1,
2COS + SO 2 = 3/N S N + 2CO 2 + Q 2, where N=2-8
In addition to the described Claus chemical transformations, processes of sulfur condensation and the capture of fog- and droplet-like liquid sulfur occur.
Condensation occurs in devices specially designed for this purpose - condenser-generators when the gas is cooled below the dew point of sulfur vapor.
Condensation is preceded by the association reaction of sulfur polymers into the S8 form.
8/N S N -> S 8 + Q 3
S 8 (gas) -> S 8 (liquid) + 22860 kcal/kmol
the droplet collection process occurs in the outlet chambers of the condensers, which are equipped with mesh bumpers. On these bumpers, sulfur mist and droplets coagulate, which are then removed from the gas flow under the influence of gravitational and inertial forces; in addition, a special apparatus, a sulfur trap, installed after the last-stage condenser-generator serves the same purpose.
Calculation of basic technological devices.
The mathematical model is characterized by the following main parameters:
a) name of the object: sulfur production plant, including a thermal reactor, a catalytic reactor, a sulfur condenser, a furnace heater, and a mixer.
b) method of modeling an object: mathematical modeling of individual devices and the entire installation. Calculation of equations of phase and chemical equilibrium, material and heat balances of devices. Connection of devices into technological schemes and calculation of their material and heat balances.
c) name of the parameter: 1. Component composition, 2. Temperature, 3. Pressure, 4. Enthalpy of flows of the technological scheme of installations for the production of elemental sulfur.
d) estimation of object parameters: relative error between calculated and experimental data<= 5%.
Summary: the developed model allows
1. Calculate technological schemes of various modifications (any number of catalytic stages, “1/3 -2/3”, etc.),
2. Solve inverse problems of mathematical modeling, including ensuring the desired characteristics of flows (ratio H 2 S+COS/SO 2 = 2, temperatures at any point in the process flow diagram), etc.
Calculation of the installation apparatus is carried out using a package of application programs compiled according to mathematical models based on the principles of chemical thermodynamics. The composition of mathematical models is determined by the devices included in the technological scheme of the sulfur production plant, the main ones of which are the following:
Reactor-generator;
Catalytic converter;
Process gas heater;
Mixer;
Energy-technological equipment (sulfur capacitors);
The basis of the mathematical software is made up of models of these devices. In mathematical software, the computational methods of Newton, Wolf, Wegstein, and “secants” are widely used, which implement iterative calculations of material and heat balances of individual devices and the technological scheme as a whole.
Currently, the operation of application programs for calculating sulfur production plants is carried out under the control of the problem-oriented Comfort language, using a bank of physical and chemical properties of substances.
Mathematical models of basic devices.
The developed models of apparatus for sulfur production plants are based on the principles of thermodynamics. Equilibrium constants of physicochemical processes are calculated through reduced Gibbs potentials using data contained in standard thermodynamic tables.
Technological schemes of sulfur production plants are complex chemical-technological systems consisting of a set of devices interconnected by technological flows and operating as a single whole, in which the processes of H 2 S oxidation, sulfur condensation, etc. take place. Each device corresponds to one or several software modules built on a block principle. Each block is described by a system of equations reflecting the relationship between the physicochemical and thermodynamic parameters of processes, flow rates, compositions, temperatures and enthalpies of input and output flows.
For example, the technological diagram of a three-stage sulfur production plant can be represented as follows:
P I - I-th flow of the technological scheme,
And J is the J-th block (apparatus) of the technological scheme.
To simulate technological schemes of sulfur production plants, a unified structure of flows connecting blocks (devices) has been introduced, which includes:
Component composition of the first stream [mol/hour]
Temperature [deg.C]
Pressure [atm]
Enthalpy [J/hour]
For each apparatus of the technological scheme, the above flow parameters are determined.
Below is a description of the calculation of the circuit in the Comfort system:
Reactor-generator furnace model (REAC)
The mathematical model describes the oxidation process of acidic, hydrogen sulfide-containing gas in a thermal reactor and in furnace heaters. The model is built by considering the chemical, phase and thermal equilibrium of the outgoing flows and the overall temperature. These parameters are found from solving a system of nonlinear equations of material and heat balances, chemical and phase equilibrium. The equilibrium constants included in the balance equations are found through changes in the Gibbs energy in the reactions of substance formation.
The calculation results are: component composition, pressure (specified), temperature, enthalpy and output flow rate.
Catalytic converter model (REAST).
To describe the processes occurring in the catalytic converter, the same mathematical model was adopted as for describing furnaces operating on acid gas.
Model of capacitor-generator (economizer) (CONDS).
The mathematical model is based on determining the equilibrium pressure of sulfur vapor at a given temperature in the apparatus. The parameters of the outgoing stream are determined from the condition of thermodynamic equilibrium of the reactions of sulfur transition from one modification to another.
The condenser model includes equations of material and thermal balance and equations of phase equilibrium of sulfur vapor in the apparatus.
The system of equations for the mathematical model of a capacitor has the following form.
The equilibrium of sulfur vapor content is determined from the equilibrium condition:
YI=PI(T)/P at T< T т.р.
(I+1)/2 (I-1)/2 YI=KI*YI*P at T>T t.r.
where T t.r. - sulfur dew point temperature. The content of UI inerts is determined by the balances:
The amount of sulfur at the input and output is interconnected by balances:
V SUM(I+1) XI=W SUM(I+1) YI +S,
where S is the amount of condensed sulfur.
The total gas flow rate at the outlet is determined from the condition
SUM UI + SUM YI=1
Mixer model (MIXER).
The model is intended to determine the component-wise flow rates of a flow obtained as a result of mixing several flows. The component composition of the outlet flow is determined from the material balance equation:
XI - XI" - XI"" - XI""" =0 , where
XI - consumption of the I-th component in the output stream,
XI"-XI""" - expenses of the I-th component in the input flows.
The temperature of the outlet flow is determined by the “secant” method from the condition of maintaining thermal balance:
H(T)-H1(T)-H 2 (T)-H3(T)=0, where
H(T) - enthalpy of the output flow
H1(T) -H3(T) - enthalpies of input flows.
Model for calculating real (non-equilibrium) parameters (OTTER).
The mathematical model is based on a comparison of experimental data and calculated values of compositions and other parameters of installations to determine the degree of deviation of real indicators from thermodynamic equilibrium ones.
The calculation consists of solving a system of algebraic equations. The result of the calculation is the new (nonequilibrium) composition, temperature and enthalpy of the flow.
Below are the results of the circuit calculation
