Mudrock composition

Mudrocks, by definition, consist of at least fifty percent mud-sized particles. Specifically, mud is composed of silt-sized particles that are between 1/16 – 1/256 of a millimeter in diameter, and clay-sized particles which are less than 1/256 millimeter.

Mudrocks contain mostly clay minerals and quartz ($S_{i}O_{2}$)and feldspars (a group of minerals including aluminum silicates of soda (sodium oxide), potassium (potassium oxide), or lime (calcium oxide)). They can also contain the following particles at less than 63 micrometres: calcite ($CaCO_{3}$), dolomite ($CaMgCO_{3}$), siderite ($FeCO_{3}$), pyrite ($FeS_{2}$), marcasite ($FeS_{2}$), heavy minerals, and even organic carbon.

The term "mudrock" allows for further subdivisions of siltstone, claystone, mudstone, and shale. For example, a siltstone would be made of more than 50-percent grains that equate to 1/16 - 1/256 of a millimeter. "Shale" denotes fissility, which implies an ability to part easily or break parallel to stratification. Siltstone, mudstone, and claystone implies lithified, or hardened, detritus without fissility.

Type	Min grain	Max grain
Claystone	0 µm	4 µm
Mudstone	0 µm	64 µm
Shale	0 µm	64 µm
Siltstone	4 µm	64 µm
Slate	na	na

Turbulence Models

Turbulence:
Due to the high degree of mixing, turbulent flow exists. The heat and mass transfer rates are high and can not be explained by molecular transport only. It was seen as a random process in attempt to be specified in statistical parameters. Yet, it is not completely random.

Turbulence is normally expressed in terms of velocity. So there are extras terms in N-S equations, called Reynolds stress, described in terms of turbulence viscosity. This eddy viscosity is a strong function of the distance from the wall, where the eddy diminishes.

In turbulent flow, the kinetic energy imparted to the flow is dissipated into smaller eddies, and finally all the energy dissipated as thermal energy. Two sets of equations in CFD simulation are added to address the conservation of turbulence kinetic energy, $k$, and the turbulent energy dissipation rate, $\varepsilon$.

Prandtl Mixing Length:
When two eddies come closer, they are mixed. The mean distance between eddies is called the "mixing length", which analogous to the mean-free path in gases. The mixing length is a function of distance to the wall.

Model and Simulation: Laws of Conservation

Equation of Continuity:
For 2D incompressible fluid flow from mass conservation,

\[ \frac{\partial v_{x}}{\partial x}+\frac{\partial v_{y}}{\partial y}=0
\]

Thus,

\[ v_{y}=\int_{0}^{y}\frac{\partial v_{x}}{\partial x}dy
\]

Cauchy-Riemann Equation:
Define velocity by new variables, $Psi$, stream function and $phi$, velocity potential,

\[ v_{x}=-\frac{\partial\Psi}{\partial y}
\]

\[ v_{y}=\frac{\partial\Psi}{\partial x}
\]

\[ v_{x}=-\frac{\partial\phi}{\partial x}
\]

\[ v_{y}=-\frac{\partial\phi}{\partial y}
\]


An analytical function $w(z)$ of a complex variable, z, may be chosen such that

\[ w(z)=\phi(z)+i\Psi(z)
\]

satisfying above equations. Velocities can be obtained as the real and imaginary parts of the equation,

\[ \frac{dw}{dz}=-v_{x}+iv_{y}
\]

Equations of continuity, stream function and velocity function in different coordinates are shown below.

Navier-Stokes Equations:

Momentum transport:

\[ \rho\left(\frac{\partial v_{x}}{\partial t}+v_{x}\frac{\partial v_{x}}{\partial x}+v_{y}\frac{\partial v_{x}}{\partial y}+v_{z}\frac{\partial v_{x}}{\partial z}\right)=-\frac{\partial p}{\partial x}+\mu\left[\frac{\partial^{2}v_{x}}{\partial x^{2}}+\frac{\partial^{2}v_{x}}{\partial y^{2}}+\frac{\partial^{2}v_{x}}{\partial z^{2}}\right]+\rho g_{x}
\]

\[ \rho\left(\frac{\partial v_{y}}{\partial t}+v_{x}\frac{\partial v_{y}}{\partial x}+v_{y}\frac{\partial v_{y}}{\partial y}+v_{z}\frac{\partial v_{y}}{\partial z}\right)=-\frac{\partial p}{\partial y}+\mu\left[\frac{\partial^{2}v_{y}}{\partial x^{2}}+\frac{\partial^{2}v_{y}}{\partial y^{2}}+\frac{\partial^{2}v_{y}}{\partial z^{2}}\right]+\rho g_{y}
\]

\[ \rho\left(\frac{\partial v_{z}}{\partial t}+v_{x}\frac{\partial v_{z}}{\partial x}+v_{y}\frac{\partial v_{z}}{\partial y}+v_{z}\frac{\partial v_{z}}{\partial z}\right)=-\frac{\partial p}{\partial z}+\mu\left[\frac{\partial^{2}v_{z}}{\partial x^{2}}+\frac{\partial^{2}v_{z}}{\partial y^{2}}+\frac{\partial^{2}v_{z}}{\partial z^{2}}\right]+\rho g_{z}
\]

Chemical species in incompressible fluids:

\[ \rho\left(\frac{\partial C_{A}}{\partial t}+v_{x}\frac{\partial C_{A}}{\partial x}+v_{y}\frac{\partial C_{A}}{\partial y}+v_{z}\frac{\partial C_{A}}{\partial z}\right)=D_{AB}\left[\frac{\partial^{2}C_{A}}{\partial x^{2}}+\frac{\partial^{2}C_{A}}{\partial y^{2}}+\frac{\partial^{2}C_{A}}{\partial z^{2}}\right]-r_{A}

\]

Heat transfer:

\[ \rho\left(\frac{\partial T}{\partial t}+v_{x}\frac{\partial T}{\partial x}+v_{y}\frac{\partial T}{\partial y}+v_{z}\frac{\partial T}{\partial z}\right)=k\left[\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial z^{2}}\right]-\triangle H_{rxn}
\]

1D momentum balance for an unsteady state model gives,

\[ \rho\left(\frac{\partial v_{x}}{\partial t}+v_{x}\frac{\partial v_{x}}{\partial x}\right)=-\frac{\partial p}{\partial x}+\mu\left[\frac{\partial^{2}v_{x}}{\partial x^{2}}+\frac{\partial^{2}v_{x}}{\partial y^{2}}\right]+\rho g_{x}
\]

Using the incompressible fluid solution of equation of continuity,

\[ \rho\frac{\partial v_{x}}{\partial t}=-\frac{\partial p}{\partial x}+\mu\frac{\partial^{2}v_{x}}{\partial y^{2}}+\rho g_{x}

\]

The solution of 1D model equations are given below.

Boundary Conditions:
Differential equations of second order in spatial coordinates and first order in time require two boundary conditions and one initial condition to describe the system.

BC of the first kind:
The temperature or concentration of chemical species are specified at boundaries. This is also known as Dirichlet condition.

BC of the second kind:
The heat transfer or mass transfer flux is constant, also known as Neumann condition.

\[ q=-k_{f}\frac{\partial T}{\partial x}\mid_{x=0}
\]


BC of the third kind:
The flux is not constant, but the transfer coefficient is constant. Usually described in terms of heat or mass transfer coefficient.

\[ -k_{f}\frac{\partial T}{\partial x}\mid_{x=0}=h(T_{s}-T_{\infty})
\]

Model and Simulation: Physical Laws

Equation of State:
Ideal gas: $pV = RT$
Real gas: Peng-Robinson EOS
\[ p=\frac{RT}{(V-b)}-\frac{\alpha a}{V^{2}+2Vb-b^{2}}
\]

$ a = \frac{0.457235\,R^2\,T_c^2}{p_c} $
$ b = \frac{0.077796\,R\,T_c}{p_c} $
$ \alpha = \left(1 + \kappa \left(1-T_r^{\,0.5}\right)\right)^2 $
$ \kappa = 0.37464 + 1.54226\,\omega - 0.26992\,\omega^2 $
$ T_r = \frac{T}{T_c} $

In terms of compressibility, $Z=PV/(RT)$,  and in polynomial form:

$ A = \frac{a\alpha p}{ R^2\,T^2} $
$ B = \frac{b p}{RT} $
$ Z^3 - (1-B)\ Z^2 + (A-2B-3B^2)\ Z -(AB-B^2-B^3) = 0 $

where $\omega$ is the acentric factor of the species, R is the universal gas constant. Possible roots, 1 three real roots (largest, gas volume; smallest liquid volume); 2 one real (true volume) two imaginary.

The fugacity of pure component is given by,

\[ \ln\phi=\ln\frac{f}{p}=Z-1-\ln(Z-B)-\frac{A}{2\sqrt{2}B}\ln\left(\frac{Z+2.414B}{Z-0.414B}\right)

\]

Mixing rules:
Parameters of PR equation for the mixture of N simple fluids where molar percent of the i-th component is zi are defined by mixing rules

\[ a=\sum_{i}\sum_{j}z_{i}z_{j}a_{ij}
\]

\[ b=\sum_{i}z_{i}b_{i}
\]

where, $a_{ij}=(1-\delta_{ij})\sqrt{a_{i}a_{j}}$

Here aij is expressed through empirically determined binary interaction coefficient δij characterizing the binary formed by component i and component j.

Some δij are tabulated (e.g. table 4.2 in Pedersen and Christensen(2006)). For those one which are not tabulated formula of Chueh and Prausnitz (1967) could be used:

\[ 1-\delta_{ij}=\sqrt{\frac{2V_{c,i}^{1/6}V_{c,j}^{1/6}}{V_{c,i}^{1/3}V_{c,j}^{1/3}}}
\]


Formula for fugacity coefficient of the mixture, calculated via PR equation and standard mixing rules, has expression (see Peng and Robinson (1976) or equation 4.65 in Pedersen and Christensen (2006) for more detail)

\[ \ln\phi_{i}=\ln\frac{f_{i}}{z_{i}p}=\frac{b_{i}}{b}(Z-1)-\ln(Z-B)-\frac{A}{2\sqrt{2}B}\left(\frac{2}{a}\sum_{j}^{N}z_{j}a_{ij}-\frac{b_{i}}{b}\right)\ln\left(\frac{Z+2.414B}{Z-0.414B}\right)
\]

The enthalpy departure of the a fluid is given by,

\[ H-H^{*}=RT(Z-1)+\frac{T\frac{da}{dT}-a}{2\sqrt{2}b}\ln\left(\frac{Z+2.414B}{Z-0.414B}\right)
\]


Henry's Law:
The mole fraction or partial pressure of ith component, $p_i$ is proportional to the mole fraction of ith component in the liquid phase, $x_i$,

\[ p_{i}=H_{i}x_{i}
\]

For ideal gas,

\[ p_{i}=y_{i}P
\]


Thus, a linear equilibrium relationship is obtained,

\[ \frac{y_{i}}{x_{i}}=\frac{H_{i}}{P}
\]

For pure gas adsorption at pressure P, the number of moles adsorbed on the surface, n, is given by Henry's law for adsorption,

\[ n=kP
\]


$H_i$ is Henry constant; $P$ is total pressure; $y_i$ is mole fraction of ith component in vapor phase.

Newton's Law of Viscosity:
The shear stress is proportional to shear rate,

\[ \tau_{yx}=-\mu\frac{dv_{x}}{dy}

\]

All the gases and liquids and solutions of low-molecular-weight compounds exhibit Newtonian behavior. Non-Newtonian behaviors are shown below,



Some dimensionless numbers, such as Prandtl and Schmidt numbers can be obtained for empirical correlations for heat and mass transfer.

Fourier's Law:
Heat transfer rate of conduction due to temperature difference is written as,

\[ \dot{q}=kA\frac{(T_{2}-T_{1})}{\triangle z}=-kA\frac{dT}{dz}

\]

The differentiation form may be used if heat transfer area is a variable,

\[ \frac{d}{dz}\left(A\frac{dT}{dz}\right)=0

\]

Convective heat transfer at low temperature by Newton's law of cooling,

\[ \dot{q} = hA(T_2 - T_1)
\]

$k$ is the thermal conductivity; and $h$ is the heat transfer coefficient.

Fick's First Law:
For steady-state diffusion without convection, Fick's first law applies. The molar flux is described as,

\[ J_{A}=-CD_{AB}\nabla x_{A}\underrightarrow{C\,is\,constant}-D_{AB}\nabla C_{A}\underrightarrow{1-D}-D_{AB}A\left(\frac{dC_{A}}{dz}\right)

\]


where, $D_{AB}$ is the diffusion coefficient of A in B; $J_A$ is the molar flux of A; $C_A$ is the molar concentration of A; $x_A$ is the mole fraction of A.

Or the Fick's law gives the diffusive flux relative to the average velocity of fluid mixture. In absence of convection, the diffusive flux and the total flux are the same. For diffusion relative to fixed/stationary coordinates, the molar flux becomes,

\[ N_{A}=J_{A}+vC_{A}=-C_{A}D_{AB}\nabla x_{A}+v^* C_{A}
\]

where, average velocity, $v^*$ is $x_A v_A + x_B*v_B$.

\[ N = N_A + N_B = Cv^* = C_A v_A + C_B v_B
\]

Since the mass transfer results in some amount of convection (diffusional flux), the first law is valid only for dilute solutions or in cases of diffusion in solids. In concentrated solutions, the diffusion coefficient depends on the concentration.

Fick's Second Law:
It describes how diffusion causes the concentration to change with time.

\[ \frac{\partial C}{\partial t}=D_{AB}\frac{\partial C^{2}}{\partial z^{2}}

\]

Film Model:
The assumptions for film theory:

1. The entire variation of concentration or temperature takes place within a thin fluid film adjacent to the interface. While, in the bulk fluid, the concentration or temperature is almost uniform.
2. Within the film, the mass or heat transfer takes place by steady-state 1-D diffusion or conduction, respectively.

The mass transfer flux for species A is written in terms of diffusion coefficient, concentrations at the two faces of the fictitious film and film thickness,

\[ N_{A}=D_{AB}\frac{\left(C_{A2}-C_{A1}\right)}{\delta_{m}}

\]

Note that $C_{A2}$ is same as the concentration in bulk flow.

But in terms of mass transfer rate, $k$, which is measurable by experiment,

\[ N_{A}=k(C_{A2}-C_{A1})

\]

Differently, the mass transfer coefficient depends on the hydrodynamics, i.e. the fluid flow field in the bulk fluid.

Thus, the film thickness can be obtained,

\[ \delta_{m}=\frac{D_{AB}}{k}

\]

Two-Film Theory:
For two-phase flow, the mass transfer across the interface can be simplified:

1. The resistance to mass transfer lies in two different fictitious films adjacent to the interface on both of its sides.
2. The concentration of the chemical species in both phase at the interface is in equilibrium (insignificant mass transfer across the interface).

The total mass transfer coefficients $K_g$ and $K_l$ can be expressed in term of the individual mass transfer coefficient (when Henry's law applies):

\[ \frac{1}{K_{g}}=\frac{1}{k_{g}}+\frac{H}{k_{l}}
\]

\[ \frac{1}{K_{l}}=\frac{1}{k_{l}}+\frac{1}{Hk_{g}}

\]

If the mass transfer in the liquid phase is controlling, then kg>>kl and hence $K_l=k_l$. Vice versa.

Similarly, this also applies to heat transfer, where constant temperatures are canceled.

\[ \frac{1}{U}=\frac{1}{h_{1}}+\frac{1}{h_{2}}
\]


Arrhenius' Law:
The chemical reaction rate can be the product of two terms, one depends on concentration (associated with pressure terms) and one depends on temperature only.

\[ -r_{A}=f\left(C_{A},C_{B},...\right)k^{'}(T)
\]

The temperature term can be written as follows, Arrhenius' Law,
\[ k^{'}(T)=k_{0}^{'}e^{-E/RT}
\]

$k^{'}$ is the kinetic rate expression; $k_0^{'}$ is the constant, E is the activation energy of the reaction.

Adsorption Isotherms:
The equilibrium relationship relating $q$ (the mount adsorbed on the solid) and $C_A$ (the concentration of the adsorbate in the fluid)  is known as Isotherm. Sometimes, isotherms are given in terms of $K = K_a / K_d$ (rates of adsorption and desorption) and $\theta = q / q_0$ ($q_0$ is the adsorption capacity of adsorbent). Commonly used adsorption isotherms are,

• Langmuir (1918), no interaction between the adsorbed molecules

\[ p=\frac{1}{K}\left(\frac{\theta}{1-\theta}\right)
\]

• Brunauer-Emmett-Teller (1938), $p_{r}=p/p_{s}$, $q_m$ = constant

\[ \frac{q}{q_{m}}=\frac{K_{B}p_{r}}{(1-p_{r})(1-p_{r}+K_{B}p_{r})}
\]

• Freundlich (1926) Empirical

\[ q=k_{F}C^{1/n_{F}}

\]

• Raddke and Prausnitz (1972) combines Freundlich isotherms and Henry's law

\[ q=\frac{1}{\left(\frac{1}{K_{Hp}}+\frac{1}{k_{F}p^{1/n_{F}}}\right)}

\]

References:
Verma, Ashok Kumar. 2014. Process Modelling and Simulation in Chemical, Biochemical and Environmental Engineering. CRC Press.

http://en.wikipedia.org/wiki/Equation_of_state

http://kshmakov.org/fluid/note/3/

Model and Simulation: Simplifications in Model

Model Simplification:

Continuum: Laws of conservation hold in continuous phase. For dispersed bubble or solid particles in liquid, if the main mechanism is away from the interface of the dispersed and continuous phases, it may be convenient to consider the entire dispersion to be a continuous phase, e.g. study on the pipe wall and pressure loss. If the main mechanism takes place at the gas-liquid interface, continuum is meaningless, e.g. mass transfer between the bubble and liquid.

Combining simple and rigorous models: Two ways to model dispersed bubble in liquid, 1. build rigorous cell model for the dispersed bubble and surrounding liquid, and combine them in a simple manner. 2. Apply simple thin film model at the surface of the bubble and combine with the mass transfer and bubble-size distribution in a rigorous manner.

Uniform Probability Distribution: Useful in averaging the properties assuming the individual entity follow uniform distribution law, e.g. in adsorption, the probability of occupying the empty space on the surface is the same.

Parallel Mechanisms: The mechanism of  heat transfer is conduction, convection and radiation due to molecular motion, bulk flow and infrared radiation. The parallel mechanism allows us to treat different mechanisms separately, but under the assumption that there is no interaction between different mechanisms.

Analogy to Electrical Circuits: Many transport mechanisms are analog to the flow of current in electrical circuits.
Current flow: Ohm's Law;
Heat transfer: Fourier's Law;
Mass transfer: Fick's Law.
(Heat transfer due to radiation is described by Stephan-Boltzman's Law, i.e., the rate of heat transfer is proportional to the difference of the fourth power of the temperature.)

Film Model and Boundary Layer Approximation: The film model uses the assumption that in the small region adjacent to the wall, the variation is much larger than that at the center. Thus the concept of thin film at the wall is developed and accounts for all the variations. The central cylinder is considered with no variation.
But the boundary layer model allows the layer thickness to vary with time and distance. It is more likely an approximation to the reality, the experimental results.

Order of magnitude Approximation: If the ratio of the contribution from one mechanism and that from others is high enough, then the other mechanisms can be neglected to reduce the complexity.

Quasi-Steady State: The assumption for unsteady-state problem, that during a small time interval, the process is at steady-state is known as "quasi-steady state".

Finite and infinite dimensions: Many mathematical equations approach certain value when one or some of its parameters goes to infinity or zero, which is not possible in physical experiments. Thus, it can be assumed that when the parameter value is large or small enough, it is close to infinity or zero.

References:
Verma, Ashok Kumar. 2014. Process Modelling and Simulation in Chemical, Biochemical and Environmental Engineering. CRC Press.

Model and Simulation: Equations

Equations describe the interrelations between various parameters.

Algebraic Equation:

• Linear equation, solved by matrix inversion
• Non-linear equation, solved by linearisation or non-linear optimization

A non-linear equation $ F(\vec{x})=\vec{0}$ can be written by

\[ F(\vec{x})=\vec{R}
\]

where, $\vec{x}$, $\vec{0}$ and $\vec{R}$ are vectors.

R are the residuals and the objective function becomes $|\vec{R}|$, which is to be minimized.

By linearisation, the set of non-linear equations can be converted to,

\[ F_{i}(\vec{x})=-\sum_{j}J_{ij}(\vec{x})\triangle x_{j}
\]

where, $J_{ij}(\vec{x})=\frac{\partial F_{i}(\vec{x})}{\partial x}$.

An initial guess is required to start this iterative solving process. The Jacobian matrix is estimated using the values of x from the previous iteration until the right-hand side becomes less than the tolerance.

Various optimization methods can be used to search the optima within a given range. The problems can be

1. The obtained optima is local optima or there are multiple optima -- Genetic Algorithm, Simulated Annealing
2. The solution won't converge or stick at particle values -- try different methods or change the initial guess

Again, the choice of initial guess is very important.

Differential Equations:
If only one type of derivative is involve, the equation is called ODE, ordinary differential equation, as opposed to PDE, partial differential equation.

A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions which are given on a hypersurface in the domain. A Cauchy problem can be an initial value problem (given initial value, how the system evolves with time) or a boundary value problem (specifying the boundary condition instead), but it can be none of them. Cauchy boundary conditions require both the function value and normal derivative on the boundary of the domain (a Dirichlet and a Neumann boundary condition) to ensure a unique solution.

Sometimes, PDEs are converted into simultaneous ODEs or algebraic equations (a large system of coupled ODEs with derivatives with respect to time). When the phenomena vary with time, the spatial variables are discretised and time is considered a continuous variable. Such as parabolic and hyperbolic equations in unsteady-state models. This method is termed a "method-of-line".

In steady state models, spatial discretisation results in algebraic equations. Such as elliptic equations.


Differential-Algebraic Equations:

DAEs, these types of mixed equations contain unsteady-state material and energy balance equations, equilibrium relationships, consistency equations etc.


Other Equation Type Classification:

Homogeneous Function:
If x is the solution of the function, so is $\lambda x$.
\[ f(\lambda x)=\lambda^{n}f(x)
\]


Homogeneous linear differential equations:
 If  $\phi(x)$  is a solution, so is  $c \phi(x)$. In another word, each term in a linear differential equation must contain y or any derivative of y, such as $ \sin(x) \frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + y = 0 $.

Second order differential equations:

\[ Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0
\]

If $B^2 - AC > 0$, hyperbolic partial differential equation;
If $B^2 - AC = 0$, parabolic PDE;
If $B^2 - AC < 0$, elliptic PDE.

The most famous wave equation is a hyperbolic PDE. In one spatial dimension, this is $u_{tt} - u_{xx} = 0$.

Source of Equations:
The source of the equations depends on the field of application and the knowledge of the modeler.

Empirical Equations:
These can be obtained by fitting the experimental data in terms of important parameters with somem simple functions. The relation can explicit or implicit (friction factor by McCable et al. 1993).

Based on theoretical concepts:
Such as the calculation/estimation of thermodynamic properties.

Consistency Equations:
Based on the fact that the fractional composition of each component should add up to 1, in terms of mole and mass.

Law of conservation:
Such as equation of continuity and the momentum-balance equation with considerations of energy/mass generation or depletion in chemical reaction, viscous dissipation, electrical or electrochemical effects on the flow and other phenomena. The generalized equation of change is known as the Navier-Stokes equation.

Integration/Averaging:
Flow rate is obtained by integrating the velocity over the cross-sectional area;
Average velocity is then obtained by division of the total area.
Also, number-averaged particle diameter, porosity...

Population Balance:
Discrete event model considers the birth and death of population (breakage, agglomeration or coalescence).

References:
Verma, Ashok Kumar. 2014. Process Modelling and Simulation in Chemical, Biochemical and Environmental Engineering. CRC Press.

Models and Simulation - Types of Models

Types of Models:
Three major model types are deterministic model, discrete-event model and stochastic model.

• Deterministic Models: No random variable;
• Lumped Parameter Models: ODE, all spatial distributions are lumped into one variable;
• Distributed Parameter Models: PDE, consisting of spatial derivatives;
• Steady-State/Static Models: No temporal derivative;
• Unsteady-State/Dynamic Models: With temporal derivatives;
• Stochastic Models: Random variables involved (probability distribution function/pseudo-random numbers)
• Population Balanced Models: In multiphase system, discrete/dispersed phases are present as particles, droplets or bubbles. Similar to the "birth and death" of individuals, the particles can agglomerate or break and the bubbles and droplets can break and coalesce. Thus, the number of identities of discrete phases, called population, and the size of the individuals change and form a particle/droplet/bubble-size distribution.
• Agent-Based Models: Each part of the model , called an agent, is capable of making decisions with strong human behavior involved;
• Discrete-Event Models: The states, even time, are considered discrete and NON-continuous;
• Artificial Neural Network-based Models: A black-box approach to correlate various parameters and then predict outcomes from information available. No exact relation between the inputs and outputs is almost possible to establish. No contribution to the understanding of the phenomena.
• Fuzzy Models: based on fuzzy mathematics or fuzzy logic involving fuzzy variables. Fuzzy, not well quantified, such as small, large, very large.

-- The general-purpose language/software can handle various types of models/problems, but require coding of the model equations using the syntax and semantics of the software. While, more dedicated software generates an appropriate set of models of equations for the users to use, but the scope is limited.

-- Stochastic Models:

The random numbers are characterized by their moment. The first moment is known as "average" or "expectation". A model may consider only average properties, thus avoiding the complexity of the process due to the randomness of the properties. Many models consider random behavior of the parameters, although the outputs of the model are average properties. These models also are probabilistic in nature. Various moments of the random variables may be determined analytically (Coulomb's Law?) or by Monte Carlo simulation. The random variables follow a probability distribution function. The choice of the distribution is determined from the assumption involved.

References:

Verma, Ashok Kumar. 2014. Process Modelling and Simulation in Chemical, Biochemical and Environmental Engineering. CRC Press.