Course code: KKEK 3152 Modeling of Chemical Processes Equations are required to design equipments
Top left: Distillation Column Top Right: Steam Boiler Bottom Left: Fluidized Bed Reactor Bottom Right: Natural Gas Compressor Modeling of Chemical Processes
CHAPTER ONE Introduction and modeling principles 1- Definition of Model 2- Application of Process Models 3- Mathematical Models and Terms Definition 4- A Systematic Approach for Developing Dynamic Models 5- Classifying Mathematical Models 6- Types of Mathematical Models 7- Process Modeling Fundamentals Introduction and modeling principles
1- Definition of model: A model of a system is: a representation of the essential aspects of the system in a suitable (mathematical) form that can be experimentally verified in order to clarify questions about the system 2- Application of Process Models: Applications of models in engineering can be found in: Research and Development. This type of model is used for the interpretation of knowledge or measurements. An example is the description of chemical reaction kinetics from a laboratory set-up. Process Design. These types of model are frequently used to design and build (pilot) plants and evaluate safety issues and economical aspects. Planning and Scheduling. These models are often simple static linear models in which the required plant capacity, product type and quality are the independent model variables. Process Optimization. These models are primarily static physical models although for smaller process plants they could also be dynamic models. Prediction and Control. Application of models for prediction is useful when it is difficult to measure certain product qualities, such as the properties of polymers, for example the average molecular weight. Process models are also used in process control applications, especially since the development of model-based predictive control. These models are usually empirical models, they can not be too complex due to the online application of the models. 3- Mathematical models and terms definition
A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. The variables represent some properties of the system, for example, measured system outputs often in the form of signals, timing data, .etc. One or more assumptions are imposed on the model. These assumptions limit the functionality of the model. They are put to simplify the formulation and/or solution of the model. State the modeling objectives and the end use of the model. They determine the required levels of model detail and model accuracy. 3-2 Terms Definition State Variables Input variables Parameters
A state variable is a variable that arises naturally in the accumulation term of dynamic material or energy balance.A state variable is measurable ( at least conceptually) quantity that indicate the stateof a system. For example, temperature is the common state variable that arises from a dynamic energy balance. Concentration is a state variable that arises when dynamic component balances are written. Input variables An input variable is a variable that normally must be specified before a problem can be solved or a process can operated. Input variables typically include: Flow rates of streams Compositions or temperatures of streams entering a process. Input variables are often manipulated (by process controllers) in order to achieve desired performance. Parameters A parameter is typically a physical or chemical property value that must be specified or know to mathematically solve a problem.Examples include density, viscosity, thermal conductivity, heat transfer coefficient, and mass-transfer coefficient. 4- A Systematic Approach for Developing Dynamic Models
Draw a schematic diagram of the process and label all process variables. List all of the assumptions that are involved in developing the model. The model should be no more complicated than necessary to meet the modeling objectives. Determine whether spatial variations of process variables are important. If so, a partial differential equation model will be required. Write appropriate conservation equations (mass, component, energy, and so forth). (continued) Introduce equilibrium relations and other algebraic equations (from thermodynamics, transport phenomena, chemical kinetics, equipment geometry, etc.). Perform a degrees of freedom analysis to ensure that the model equations can be solved. Simplify the model. It is often possible to arrange the equations so that the dependent variables (outputs) appear on the left side and the independent variables (inputs) appear on the right side. This model form is convenient for computer simulation and subsequent analysis. Classify inputs as disturbance variables or as manipulated variables (for process control). 5- Classifying mathematical models
5-1 Linear vs. nonlinear 5-2 Deterministic vs. probabilistic (stochastic) 5-3 Static vs. dynamic 5-4 Lumped parameters vs. distributed parameters 5-1 Linear versus nonlinear models
Mathematical models are usually composed by variables, which are abstractions of quantities of interest in the described systems, and operators that act on these variables, which can be algebraic operators, functions, differential operators, etc. If all the operators in a mathematical model present linearity, the resulting mathematical model is defined as linear. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model. Linear ordinary differential equations (ODE)
continue Linear ordinary differential equations (ODE) Nonlinear ODE 5-2 Deterministic versus probabilistic (stochastic)
A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables. Therefore, deterministic models perform the same way for a given set of initial conditions. Conversely, in a stochastic model, randomness is present, and variable states are not described by unique values, but rather by probability distributions. Familiar examples of processes modeled as stochastic time series include: stock market exchange rate fluctuations, signals such as speech, audio and video, random movement such as Brownian motion or random walks. 5-3 Static versus dynamic models
A static (steady-state) model does not account for the element of time, while a dynamic model does. Static model: Static models are usually used for determining the final state of the system. Steady state: No further changes in all variables No dependency in time: No transient behavior Can be obtained by setting the time derivative term zero Continue Dynamic model Describes time behavior of a process due to changes in input, parameters, initial condition, etc. Described by a set of differential equations (DE), - ordinary (ODE), partial (PDE) Mostly used in safety, process control and real time simulation 5-4 Lumped vs. distributed parameters models
If the model is homogeneous (consistent state throughout the entire system) the parameters are lumped. If the model is heterogeneous (varying state within the system), then the parameters are distributed. Distributed parameters are typically represented with partial differential equations. When the spatial effects are of less importance or do not vary considerably, a lumped parameters model is used. On the other hand a distributed parameter model will be used to account for these variations Case A. Continuous Stirred-Tank Reactor
If the tank is well-mixed, the concentrations and density of the tank contents are uniform throughout. This means that the outlet stream properties are identical with the tank properties, in this case concentration CA and density . The balance region can therefore be taken around the whole tankas in fig below. The total mass in the system is given by the product of the volume of the tank contents V (m3) multiplied by the density (kg/m3), thus V (kg). The mass of any component A in the tank is given in terms of actual mass or number of moles by the product of volume V times the concentration of A, CA (kg of A/m3 or kmol of A/m3), thus giving V CA in kg or kmol. Case B. Tubular Reactor In the case of tubular reactors, the concentrations of the products and reactants will vary continuously along the length of the reactor, even when the reactor is operating at steady state. This type of behavior can be approximated by choosing the incremental volume of the balance regions sufficiently small so that the concentration of any component within the region can be assumed approximately uniform. The basic concepts of the above lumped parameter and distributed parameter
systems are shown in Fig. below. 6- Types of Mathametical Models
Theoretical models (based on physicochemical law) Advantage provide physical insight into process behavior applicable over wide ranges of conditions Disadvantage expensive and time consuming to develop complex processes typically include some model parameters which are not readily available, such as reaction rate coefficients, physical properties, or heat transfer coefficients. Empirical models (obtained by fitting experimental data) Easer to develop than theoretical models but they have a serious disadvantage which is typically do not extrapolate well, i.e., should be used with caution for operating conditions that were not included in the experimental data used to fit the model. Semi-empirical models (combined approach) can be extrapolated over a wide range of operating conditions than empiricalmodels. require less development effort than theoretical models. Therefore semi-empirical models are widely used in industry. 7- Process Modeling Fundamentals
7-1 System States. 7-2 Mass Relationship for Liquid and Gas 7-3 Energy Relationship 7-4 Composition Relationship 7-1 System States Conservation Laws
To describe a process system we need a set of variables that characterize the system and a set of relationships that describe how these variables interact and change with time. The variables that characterize a state, such as concentration, temperature and flow rate, are called state variables. They can be derived from the conservation balances for mass, component, energy and momentum. Open system mass balance
Component balance Energy balance : result of the first law of thermodynamics. Momentum balance : result of general caseof Newtons second law 7-2 Mass Relationship for Liquid and Gas
7-2-1 Mass balance This equation relates the rate of change in mass m to the difference between inlet mass flow (Fm,in) and outlet mass flow (Fm,out): For N inlet flows and M outlet flows: When volumetric flow is used instead ofmass flow: If the density i and volumetric flow Fv,i are measured variables The density in Eqn. above is defined as the mass per unit volume at a certain pressure, temperature and composition Accumulation term The rate of change in the mass of a system can be described by: Liquid Accumulation If there is only one inlet flow and one outlet flow, the accumulation in a liquid vessel can be written as: If the temperature and pressure effects can be neglected If the inlet and outlet density are the same, the equation becomes: Gas Accumulation For an ideal gas volume it holds that: in which: n number of moles, M molecular weight (kg/mole), V volume (m3), P absolute pressure (N/m2), R gas constant (N.m/mole.K), T absolute temperature (K) 7-2-2 Properties of Liquid and Gas Mass Transfer
Characterization of mass transport Liquids and gases (generally fluids) are not capable of passing on static pressures. If a fluid is subject to shear stress as a result of flow, the shear stress will lead to a continuous deformation. For gases and so called Newtonian fluids at constant pressure and temperature, the viscosity is independent of the shear stress: in which: shear stress, N/m2 dynamic viscosity, kg/m.s dv/dy velocity gradient, s1 v flow velocity, m/s The flow pattern inside a body or along a body with diameter d (for example a tube) depends on the flow velocity v and can be characterized by the Reynolds number: density, kg/m3 d characteristic flow dimension, m Resistance to flow Figure below shows a pipe with a flow restriction in the form of an orifice. The flow through the orifice is turbulent. The velocity will increase from point A to point B. Ares is the area of the opening of the restriction, and Ccor is a correction factor between 0 and 1, depending on the type of opening. As can be seen from above, the flow (F) can be determined by measuring the pressure drop and taking the square root. 7-3 Energy Relationship (Energy Balance)
Energy y Balance:Analogous to the mass balance with N inlet and M outlet mass flows, the energy balance for a system can be described as: E is the total energy, which is equal to the sum of internalenergy U, kinetic energy KE and potential energy PE. E is the total energy per unit mass The terms on the right-hand side of the energy balance refer to entering convective energy flows, the leaving convective energy flows, the net heat flux Q that enters the system and the net amount of work W that acts upon the system with: in which: WS applied mechanical work, J/s and WE expansion energy, J/s. If the pressure is constant, we may write for the expansion energy: Temperature dependency
In most thermal applications, the energy balance can be further simplified: KE 0 because the flow velocities are often small, the contribution of the kinetic energy can be ignored. PE 0 because differences in height are often small, the contribution of the potential energy can be ignored. d(P/)/dt 0 For many liquids, because pressure differences are often small The result of these simplifications is an enthalpy balance. This balance does not account for mechanical changes but is valid for most thermal systems: Temperature dependency The specific enthalpy H i of a substance i depends on the temperature T with the specific heat capacity cP: The absolute specific enthalpy at a certain temperature is related to a reference temperature Tref according: However, not the absolute enthalpy, but only the contribution of the enthalpy flux is of interest in the energy balance.: Phase dependency Example
If a liquid mass flow Fm of a component i with a constant specific heat cP,iis heated up from a initial Tinit to an operating temperature T, then the enthalpy flux can be written as: Phase dependency If in the temperature trajectory a transfer of phase is included, for instance, from liquid to vapor at boiling temperature Tbp with a heat of vaporization Hi,vap , then the absolute specific enthalpy becomes: Only the contribution of the enthalpy change is of interest in the energy balance. If a liquid mass flow Fm of a component i with a constant specific heat cP,i is heated up from Tinit to the boiling point Tbp and evaporated, then the enthalpy flux can be written as: 7-3 Energy Relationship (Thermal Transfer Properties)
Convective heat transfer At an interface between gas and liquid or solid or between liquid and solid, convective heat transfer can take place when those media have a temperature difference. It can be in the form of free convection, such as in the case of a central heating radiator, or it can be forced convection, for example, an air flow from a blower. The left situation will occur at gas-liquid or liquid-liquid interfaces. Both media show a gradient. When in a gas or liquid temperature differences exist, natural convection flow will raise and eliminate these differences. The right situation will occur at the interface between gas or liquid and a conductive solid. The heat flow Q convection per unit area A at a temperature difference T on the boundary layer can be given by: The thermal resistance for heat transfer is 1/. Thermal conduction Conduction takes place within stagnant gas or liquid layers (layers, which are sufficiently thin that no convection as a result of temperature gradients can occur), solids or on the boundary between solids. The model describes the heat transport Q in terms of the thermal conductivity and the temperature gradient dT/dx Thermal radiation Heat transport can also take place from one object to another object through radiation. The wavelength of electromagnetic heat radiation is in the infrared range. An object radiates energy proportional to the fourth power of its absolute temperature: in which A is the surface area of the object and the Stefan-Boltzmann constant (56.7 109 W/m2K4). 7-4 Composition Relationship
7-4-1 Component Balance The component balance of component k for a considered process system or phase can be described as a concentration balance with concentration Ck [mole.m3] and volumetric flows Fv: where i = 1, N is the number of inlet flows and j = 1, M the number of outlet flows. As a partial mass balance with mass fraction xk [kg.kg1] and mass flows Fm this equation can be written as: 7-4-2 Component Equilibria
Liquidliquid or liquidgas equilibria The stationary distribution of a component j between two phases can be described by a distribution coefficient Kj which is a function of the temperature. xj is the molar or weight fraction in one of the phases and yj the fraction in the other. The line which describes the relationship between y and x is called the equilibrium curve. This relationship is often linear over a certain range. Vaporliquid equilibria at boiling point The Antoine equation gives the vapor pressureof the pure component j. A, B and C are constants. For a binary mixture, it is sometimes permissible to assume constant relative volatility. When the pressure P and the liquid compositions x1 and x2 or vapor compositions y1 and y2 are known, the relative volatility is defined as: 7-4-3 Component Transfer Properties
Interface component transfer When components are transferred across a liquid-liquid or a liquid-gas interface A, in each phase, owing to the resistance to component transfer, a concentration gradient will occur. In the model that describes this interfacial transfer, it is assumed, as shown in the figure, that the concentration gradient restricts itself to a boundary layer. The flow J phase of component j per unit area A at a composition difference on the boundary layer can be given by: in which xj and yj are the bulk concentrations of phase-1 and phase-2, xj interface and yj interface are the concentrations at the interface. Component diffusion Diffusion takes place within stagnant gas or liquid layers (layers in which no convection occurs). The component flow per unit area as a result of conduction is determined by the Ficks law: The model describes the component transport J in terms of the diffusivity D and the composition gradient dCj /dx of component j. If no steady state is established, no constant concentration gradient exists. Then the change of the concentration with time has to be considered as represented by the equation: Reaction kinetics Chemical reactions produce or consume components. The transformation is expressed in moles, because of the stoichiometric relationship between produced and consumed components. The reaction rate rj for component j is defined as the moles of component j produced or consumed per unit of time and per volume. This reaction rate is positive when a component is produced and negative when it is consumed. When component i is consumed or produced k times faster than component j, then The reaction rate depends on the temperature and component concentrations, according to: The order of the reaction equals the sum of the coefficients (, ,). For a first-order reaction the reaction rate equation becomes: Example of a component system: A CSTR
In a stirred reactor as shown in Figure, component A is converted to component B according the stochiometric relationship: The reaction is exothermal and the components are dissolved. The following assumptions are made: the reactor is well mixed and can be considered as a lumped system in the operating range the density and the heat capacities do not vary as a function of temperature and composition Mass balance The general equation for the mass balance is Since the reactor has one entering and one effluent flow, the balance can be simplified to: and since the density is constant (1) Component balances The general equation for the component balance of component k is Since for the reaction it holds that: the component equation for component A can be simplified to: Homework Substitution of the mass balance derived previously Equ (1):
gives after elimination of equal terms: A similar component equation can be derived for component B: Homework Energy balance A simple approach is to consider the specific enthalpy H not as a function of the temperature and composition H(T,C)but only as a function of temperature H(T). Then the reaction heat has to be accounted for as a separate term. For the reactor with one entering flow and one effluent flow, the basic Eqn. can be written as: which can be modified to: (2) Rewriting of the left-hand side term gives: Replacing of dV/dt (according to equ. 1) and remember that dH =cp dT Therefore dH(T)/dt is equal to (3) substitution of equ 3 in equ 2 results in:
Elimination of equal terms gives: The terms cP Fv,in (Tin T ) and rAB V HAB are contributions of enthalpy changes. The first indicate the heating-up of the feed and the second the reaction heat by the conversion, and for the cooling term (Qcool Example of hierarchy of the model
As mentioned previously, the real purposes of the modeling effort, the scope and depth of these decisions will determine the complexity of the mathematical description of a process. To further demonstrate the concept of model hierarchy and its importance in analysis, let us consider a problem of heat removal from a bath of hot solvent by immersing steel rods into the bath and allowing the heat to dissipate from the hot solvent bath through the rod and thence to the atmosphere . For this elementary problem, it is wise to start with the simplest model first to get some feel about the system response. Level 1 In this level, let us assume that:
(a) The rod temperature is uniform, that is, from the bath to the atmosphere. (b) Ignore heat transfer at the two flat ends of the rod. (c) Overall heat transfer coefficients are known and constant. (d) No solvent evaporates from the solvent air interface. The many assumptions listed above are necessary to simplify the analysis (i.e., to make the model tractable). Let T0 and T1 be the atmosphere and solvent temperatures, respectively. The steady-state heat balance (i.e., no accumulation of heat by the rod) shows a balance between heat collected in the bath and that dissipated by the upper part of the rod to atmosphere where T is the temperature of the rod, and L1 and L2 are lengths of rod exposed to solvent and to atmosphere, respectively. Obviously, the volume elements are finite (not differential), being composed of the volume above the liquid of length L2 and the volume below of length L1. Solving for Tfrom Eq. above yields where Equation above gives us a very quick estimate of the rod temperature and how it varies with exposure length. Level 2 Let us relax part of the assumption (a) of the first model by assuming only that the rod temperament below the solvent liquid surface is uniform at a value T1. This is a reasonable proposition, since the liquid has a much higher thermal conductivity than air. The remaining three assumptions of the level 1 model are retained. Next, choose an upward pointing coordinate x with the origin at the solvent-air surface. The figure shows the coordinate system and the elementary control volume. Applying a heat balance around a thin shell segment with thicknessx gives where the first and second terms represent heat conducted into and out of the element and the last term represents heat loss to atmosphere. We have decided, by writing this, that temperature gradients are likely to exist in the part of the rod exposed to air, but are unlikely to exist in the submerged part. Dividing previousEq.by R2 x and taking the limit as x > 0 yields : Substitution of the heat flux along the axis is related to the temperature according to Fourier's law of heat conduction yields: Equation above is a second order ordinary differential equation, and to solve this, two conditions must be imposed. One condition was stipulated earlier: The second condition (heat flux) can also be specified at x = 0 or at the other end of the rod, i.e., x = L2. Level 3 In this level of modeling, we relax the assumption (a) of the first level by allowing for temperature gradients in the rod for segments above and below the solvent-air interface. Let the temperature below the solvent-air interface be T1 and that above the interface be T11. Carrying out the one-dimensional heat balances for the two segments of the rod, we obtain Level 4 Let us investigate the fourth level of model where we include radial heat conduction. This is important if the rod diameter is large relative to length. Setting up the annular shell shown in Figure and carrying a heat balance in the radial and axial directions that leads to following equation: Here we have assumed that the conductivity of the steel rod is isotropic and constant, that is, the thermal conductivity k is uniform in both x and r directions, and does not change with temperature.