How is This Change in Body Temperature Described by the Continuity Equation Applied to Heat

Continuity Equation

27th European Symposium on Computer Aided Process Engineering

Qi Zhang , ... Ignacio E. Grossmann , in Computer Aided Chemical Engineering, 2017

3.4 Continuity Constraints

Continuity equations are required at the boundaries of each season in order to maintain mass balance and feasible transitions:

(3a) $y_{imh,0}=y_{imh,\left|{\overline{T}}_h\right|}\forall i,m\in M_i,h$

(3b) $z_{im{m}^{\prime }ht}=z_{im{m}^{\prime }h,t+\left|{\overline{T}}_h\right|}\forall i,\left(m,m^{\prime }\right)\in T{R}_i,h,-{\theta }_i^{\text{max}}+1\le t\le -1$

(3c) $y_{imh,\left|{\overline{T}}_h\right|}=y_{im,h+1,0}\forall i,m\in M_i,h\in H\backslash \left\{\left|H\right|\right\}$

(3d) $z_{im{m}^{\prime }h,t+\left|{\overline{T}}_h\right|}=z_{im{m}^{\prime },h+1,t}\forall i,\left(m,m^{\prime }\right)\in T{R}_i,h\in H\backslash \left\{\left|H\right|\right\},-{\theta }_i^{\text{max}}+1\le t\le -1$

(3e) ${\overline{Q}}_{jh}=Q_{jh,\left|{\overline{T}}_h\right|}-Q_{jh,0}\forall j,h$

(3f) $Q_{jh,0}+n_h{\overline{Q}}_{jh}=Q_{j,h+1,0}\forall j,h\in H\backslash \left\{\left|H\right|\right\}$

(3g) $Q_{j,\left|H\right|,0}+n_{\left|H\right|}{\overline{Q}}_{j,\left|H\right|}\ge Q_{j,1,0}\forall j.$

The cyclic schedules are enforced by applying Eqs. (3a)–(3b) while Eqs. (3c)–(3d) match the state in which the system is at the end of one season to the beginning of the next. Also, by adding xEqs. (3e)–(3g), we allow excess inventory $\left({\overline{Q}}_{jh}\right)$ accumulated over the course of a season to be carried over to the next.

Finally, the objective is to minimize the total annual cost, which consists of capital and operating expenses, for which piecewise-linear approximations have been incorporated.

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Proceedings of the 8th International Conference on Foundations of Computer-Aided Process Design

Dung A. Pham , ... Sun-Keun Lee , in Computer Aided Chemical Engineering, 2014

2.1 Continuity equation

The continuity equation is expressed as follows:

(1) $\frac{\partial \rho }{\partial t}=-\nabla \cdot \left(\rho \overrightarrow{\mu }\right)$

where ρ is the density (kg/m³), and $\overrightarrow{u}$ is the velocity vector. The continuity equation means the overall mass balance. The Hamiltonian operator (∇) is a spatial derivative vector. The independent variables of the continuity equation are t, x, y, and z. The first term of Eq. (1) is the accumulation term of the total mass within a controlled volume. The second term denotes the convection term of the total mass. If a fluid is incompressible like a liquid or gas under a mild temperature and pressure (normally an absorber with structured-packing bed is operated at about 25–100 °C, and the pressure drop per a pack unit is about 24–36Pa. The Mach number of the flow is therefore below 0.3(Durst, 2008)), the density may be constant with time and space. At this case, Eq. (1) is simplified into

(2) $\nabla \cdot \overrightarrow{u}=0$

In the porous medium with the porosity ε, the incompressible gas-liquid system is expressed as:

(3) $\begin{array}{c}\epsilon \frac{\partial (1-{\alpha }_L){\rho }_G}{\partial t}=-\epsilon \nabla \cdot ((1-{\alpha }_L){\rho }_G{\overrightarrow{\mu }}_G)+\epsilon r_{GL}\\ \epsilon \frac{\partial {\alpha }_L{\rho }_L}{\partial t}=-\epsilon \nabla \cdot ({\alpha }_L{\rho }_L{\overrightarrow{\mu }}_L)-\epsilon r_{GL}\end{array}$

where α_L is the liquid hold-up which changes with time and space, and r_GL (kg/m³/h) is the total mass transfer rate per unit volume from gas to liquid.

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The wave equation and solutions

Leo Beranek , Tim Mellow , in Acoustics (Second Edition), 2019

2.2.3 The continuity equation

The continuity equation is a mathematical expression stating that the total mass of gas in a deformable "box" must remain constant. Because of this law of conservation of mass, we are able to write a unique relation between the time rate of change of the incremental velocities at the surfaces of the box.

Figure 2.2. Change in volume of the box with change in position. From (a) and (b) it is seen that the incremental change in volume of the box is τ  =   (∂ξ _x /∂x) Δx Δy Δz.

One-dimensional derivation	Three-dimensional derivation
Refer to Fig. 2.2. If the mass of gas within the box remains constant, the change in volume τ depends only on the difference of displacement of the air particles on the opposite sides of the box. Another way of saying this is that, unless the air particles adjacent to any given side of the box move at the same velocity as the box itself, some will cross into or out of the box and the mass inside will change.	If the mass of gas within the box remains constant, the change in incremental volume τ depends only on the divergence of the vector displacement. Another way of saying this is that, unless the air particles adjacent to any given side of the box move at the same velocity as the side of the box itself, some will cross into or out of the box and the mass inside will change; so
In a given interval of time the air particles on the left-hand side of the box will have been displaced ξ x . In this same time, the air particles on the right-hand side will have been displaced
${\xi }_x+\frac{\partial {\xi }_x}{\partial x}\Delta x\text{.}$
The difference of the two quantities above multiplied by the area ΔyΔz gives the increment in volume τ
(2.11a) $\tau =\frac{\partial {\xi }_x}{\partial x}\Delta x\Delta y\Delta z$ or	(2.11b) $\tau =V_0\text{div}\xi =V_0\nabla ·\xi$
(2.12) $\tau =V_{\text{0}}\frac{\partial {\xi }_x}{\partial x}\text{.}$
Differentiating with respect to time yields,	Differentiating with respect to time yields,
(2.13a) $\frac{\partial \tau }{\partial t}=V_0\frac{\partial u}{\partial x},$ where u is the instantaneous particle velocity.	(2.13b) $\frac{\partial \tau }{\partial t}=V_0\nabla ·q,$ where q is the instantaneous particle velocity.

Example 2.1. In the steady state, that is,

$\partial u/\partial t=j\omega \tilde{u}=\sqrt{2}{u}_{rms},$

determine mathematically how the sound pressure in a plane progressive sound wave (one-dimensional case) could be determined from measurement of the particle velocity alone.

Solution. From Eq. (2.4a) we find in the steady state that

$-\frac{\partial p_{rms}}{\partial x}=j\omega {\rho }_0{u}_{rms}\text{.}$

Written in differential form,

$-\Delta p_{rms}=j\omega {\rho }_0{u}_{rms}\Delta x\text{.}$

If the particle velocity is 1   cm/s, ω is 1000   rad/s, and Δx is 0.5   cm, then

$\begin{array}{l}\Delta p_{rms}=-j0.005\times 1000\times 1.18\times 0.01\hfill \\ =-j0.059\text{Pa}\text{.}\hfill \end{array}$

We shall have an opportunity in Chapter 5 of this text to see a practical application of these equations to the measurement of particle velocity by a velocity microphone.

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Fully Coupled Solver for Incompressible Navier-Stokes Equations using a Domain Decomposition Method

Jerome Breil , ... Tadayasu Takahashi , in Parallel Computational Fluid Dynamics 2002, 2003

2.3 Pressure Equation

Instead of the continuity equation we use an equation for pressure by modifying the well-known "Poisson equation for pressure" obtained from (1);

(9) $\mathrm{\Delta }\mathit{p}+\mathit{\rho }\left(\mathbf{\nabla }\cdot \mathit{C}\left[\mathbf{u}\right]-\mathbf{f}\right)=0\text{.}$

The equation (9) can be balanced by multiplying by μ and adding the divergent equation

(10) $-\mathit{\nu }\mathbf{\Delta }\mathit{p}+\frac{\mathrm{\gamma }}{\mathit{\rho }}\mathbf{\nabla }\cdot \mathbf{u}=\mathit{\nu \rho }\mathbf{\nabla }\cdot \left(\mathit{C}\left[\mathbf{u}\right]-\mathbf{f}\right)\text{,}$

where γ is a parameter. This form of pressure equation has been proposed in [2]. The formulation with equation for pressure is equivalent to the original system only if the continuity equation is satisfied also on the boundary, namely the following additional boundary condition has to be used

(11) $\mathbf{\nabla }\cdot {\left.\mathbf{u}\right|}_{\partial \mathit{\Omega }}=0\text{.}$

Now the problem is equivalent to the original equations.

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Special Volume: Computational Methods for the Atmosphere and the Oceans

Bennert Machenhauer , ... Peter Hjort Lauritzen , in Handbook of Numerical Analysis, 2009

3.2.1 Explicit HIRLAM-DCISL

The explicit continuity equation for moist air is solved for each model layer as described in Section 3.1.3. (see Eq. (3.21)). Hybrid trajectories determine the irregular upstream departure area δ _k Aⁿ , and an "upstream integration" determines the horizontal mean of Δ_k $\overline{p}$ ⁿover the departure area δ _k Aⁿ (3.29). Here Δ_k $\overline{p}$ ⁿis defined as

(3.62) ${\Delta }_k{\overline{p}}^n={\overline{p}}_{k+1/2}^n-{\overline{p}}_{k-1/2}^n.$

The departure cells are the same for all tracers, including water vapor, and Lagrange interpolations between the hybrid trajectory departure points determine the departure points for temperature Tand the velocity components uand v. In HIRLAM-DCISL, two alternative upstream integration methods are available, the method of Nair and Machenhauer [2002] and that of Nair, Scroggs and Semazzi [2002]. The mean top pressures of the arrival cells ${\overline{\widehat{p}}}_{k-1/2}^{n+1}$ are determined hydrostatically from Eq. (3.22), i.e., from the Lagrangian pressure thicknesses δ_k ${\overline{\widehat{p}}}^{n+1}$ in Eq. (3.33). Together with Eq. (3.62), these values determine a mean value of the vertical pressure velocity ω = dp/dtalong the trajectory (Eq. (3.34)). This ω is consistent with the hydrostatic assumption and the horizontal flow, contrary to the inconsistent vertical velocities, based on partly Eulerian solutions to the continuity equation, which are applied in traditional semi-Lagrangian models such as HIRLAM. ω is used in the thermodynamic equation (Eq. (3.2)) in the energy conversion term αω/c_p= R_dT_v/c_pω/p, which is approximated with

(3.63) $\Delta t{\left[{\left(\frac{R_d{T}_{\upsilon }}{c_p}\frac{\omega }{p}\right)}_k\right]}^{n+1}=\frac{R_d}{c_p}{\left[T_{\upsilon }^n+{\tilde{T}}_{\upsilon }^{n+1}\right]}_k\left[\frac{{\overline{\widehat{p}}}_k^{n+1}-{\overline{\left({\overline{p}}_k^n\right)}}_{*}^{\delta }}{{\overline{\widehat{p}}}_k^{n+1}+{\overline{\left({\overline{p}}_k^n\right)}}_{*}^{\delta }}\right].$

The hydrostatic mean surface pressure (Eq. (3.23)) is the weight of all NLEV model layers above the surface:

(3.64) ${\overline{p}}_s^{n+1}=\sum _{l=1}^{\text{N}\text{L}\text{E}\text{V}}{\delta }_k{\overline{\widehat{p}}}^{n+1},$

determining the top pressure of Eulerian cells (Eq. (3.24))

(3.65) ${\overline{p}}_{k-1/2}^{n+1}=A_{k-1/2}+B_{k-1/2}{\overline{p}}_s^{n+1}.$

The explicit continuity equations for passive tracers (Eq. (3.40)) and water vapor (Eq. (3.43)) are

(3.66) ${\left({\overline{\stackrel{⌢}{q}}}_i^{\delta }\right)}_k^{n+1}{\delta }_k{\overline{\widehat{p}}}^{n+1}\Delta A={\overline{{\left({\overline{\stackrel{⌢}{q}}}_i^{\Delta }\right)}_k^n{\Delta }_k{\overline{p}}^n}}^{\delta }\delta A_k^n$

and

(3.67) $\begin{array}{lll}{\left({\overline{\stackrel{⌢}{q}}}_{\upsilon }^{\delta }\right)}_k^{n+1}{\delta }_k{\overline{\widehat{p}}}^{n+1}\Delta A& =& {\overline{{\left({\overline{\stackrel{⌢}{q}}}_{\upsilon }^{\Delta }\right)}_k^n{\Delta }_k{\overline{p}}^n}}^{\delta }\delta A_k^n\\ & +& \Delta t{\overline{{\left({\overline{\stackrel{⌢}{P}}}_{q_{\upsilon }}^{\Delta }+{\overline{\stackrel{⌢}{K}}}_{q_{\upsilon }}^{\Delta }\right)}_k^{n+1/2}{\delta }_k{\overline{p}}^{n+1/2}}}^{{\delta }_{n+1/2}}\delta A_k^{n+1/2},\end{array}$

respectively, determine updated specific concentrations, ${\left({\overline{\stackrel{⌢}{q}}}_i^{\delta }\right)}_k^{n+1}$ and ${\left({\overline{\stackrel{⌢}{q}}}_{\upsilon }^{\delta }\right)}_k^{n+1}$ , in Lagrangian arrival cells (δV= δ $\stackrel{\wedge }{p}$ ΔV) from ${\left({\overline{\stackrel{⌢}{q}}}_i^{\Delta }\right)}_k^n$ and ${\left({\overline{\stackrel{⌢}{q}}}_{\upsilon }^{\Delta }\right)}_k^n$ plus Eq. (3.62). Finally, the updated specific concentrations, ${\left({\overline{\stackrel{⌢}{q}}}_i^{\Delta }\right)}_k^{n+1}$ and ${\left({\overline{\stackrel{⌢}{q}}}_i^{\delta }\right)}_k^{n+1}$ , in the Eulerian cells (ΔV= Δp ΔA) are determined from ${\left({\overline{\stackrel{⌢}{p}}}_i^{\delta }\right)}_k^{n+1}$ and ${\left({\overline{\stackrel{⌢}{p}}}_{\upsilon }^{\delta }\right)}_k^{n+1}$ by 1D vertical remappings.

The discretized explicit momentum and thermodynamic equations are straightforward grid-point semi-Lagrangian and finite difference approximations to Eqs. (3.1) and (3.2), respectively (see KÄllén [1996] and Undén et al. [2002]), except that in the thermodynamic equation the consistent energy conversion term is approximated consistently with (3.63). Regarding the addition of the physics in Eq. (3.67): since DMI-HIRLAM adds the physics at the arrival level (no averaging along the trajectory), that procedure was also adopted in HIRLAM-DCISL. Of course, it should ideally be done as indicated in Eq. (3.67).

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Governing Equations for CFD: Fundamentals

Jiyuan Tu , ... Chaoqun Liu , in Computational Fluid Dynamics (Third Edition), 2018

3.2.2 Physical Interpretation

Let us examine the physical meaning of the continuity equation of Eq. ( 3.12) as applied to an infinitesimal small control volume for the two-dimensional case of the fluid flow between two parallel plates to illustrate the fundamental physical principle. Two situations are considered.

Consider the first situation, if ∂u/∂x  >   0, then the velocity at the surface at x  +   Δx is greater than the velocity at the surface x, that is u(x  +   Δx)   > u(x). Since more fluid is physically leaving the control volume than entering along the x direction, there should be more fluid entering than leaving along the y direction. Here, ∂v/∂y  <   0 and the velocity at the surface y  +   Δy are less than the velocity at the surface y, that is v(y  +   Δy)   < v(y).

Alternatively, for the second situation, if ∂u/∂x  <   0, then the velocity at the surface at x  +   Δx is less than the velocity at the surface x, that is u(x  +   Δx)   < u(x). Since more fluid is physically entering the control volume than leaving along the x direction, there should be more fluid leaving than entering along the y direction. Here, ∂v/∂y  >   0, and the velocity at the surface y  +   Δy is greater than the velocity at the surface y, that is v(y  +   Δy)   > v(y).

Both situations satisfy the continuity equation: ∂u/∂x  +   ∂v/∂y  =   0 (mass conservation).

Example 3.1

Consider a laminar boundary layer that can be approximated as having a velocity profile u(x)   = U _∞ y/δ where δ  = cx ^1/2, c is a constant, U _∞ is the free-stream velocity, and δ is the boundary layer thickness. With reference to the two-dimensional fluid flow over a flat plate as shown in Fig. 3.1.1 below, determine the velocity v (vertical component) inside the boundary layer.

Fig. 3.1.1. Two-dimensional flow over a flat plate.

Solution: As the boundary layer grows downstream, the horizontal velocity u is gradually slowed down due to viscous effect and the no-slip condition at the surface of the flat plate. In order to satisfy the continuity equation, the vertical velocity v should be positive and acting to remove the fluid away from the boundary layer.

We begin the analysis by substituting the velocity profile u(x) into Eq. (3.12), yielding

$\frac{\partial }{\partial x}\left(\frac{U_{\infty }y}{c{x}^{1/2}}\right)+\frac{\partial v}{\partial y}=0\Rightarrow -\frac{U_{\infty }y}{2c{x}^{3/2}}=-\frac{\partial v}{\partial y}$

Integrating the vertical velocity v with respect to y, the equation becomes

$v=\frac{U_{\infty }{y}^2}{4c{x}^{3/2}}=\left(\frac{U_{\infty }y}{c{x}^{1/2}}\right)\left(\frac{y}{4x}\right)=\frac{uy}{4x}$

Discussion: This physically means that the velocity ratio v/u  = y/4x increases away from the surface at a fixed x location, that is it decreases further downstream at a fixed y location. At the edge of the boundary layer, y  = δ  = cx ^1/2, the velocity ratio v/u equals to c/4x ^1/2. If the constant c is assumed unity, the boundary layer thickness δ and the velocity ratio v/u as a function of the horizontal distance x from the leading edge of the flat plate can be described, and they are illustrated in Figs 3.1.2 and 3.1.3. The latter figure further illustrates the decrease of the velocity ratio v/u further downstream of the fluid flow over the flat plate. At some downstream distance x, the change in the horizontal u velocity is appreciably small. Here, ∂u/∂x  →   0 that also leads to ∂v/∂y  →   0 hence satisfying the continuity equation (3.12).

Fig. 3.1.2. Boundary layer thickness δ as a function of the horizontal distance x.

Fig. 3.1.3. Velocity ratio v/u as a function of the horizontal distance x.

Example 3.2

Consider the CFD case in Chapter 2 for the steady two-dimensional incompressible, laminar flow between two stationary parallel plates with the following dimensions: height H  =   0.1   m and length L  =   0.5   m. Using CFD, plot the velocity vector along the channel length. Discuss the physical meaning of the continuity equation by plotting the velocity components u and v close to the bottom wall surface along the channel length with the working fluid taken as air and a uniformly distributed velocity profile of 0.01   m/s applied at the channel inlet (u_in ).

Solution: The problem is described as follows:

Fig. 3.2.1. Two-dimensional laminar flow between two stationary parallel plates.

From the CFD simulation, the flow field along the channel length is illustrated in Fig. 3.2.2. It is observed that the flow gradually changes from a uniform profile at the inlet surface to a parabolic profile as it travels downstream along the channel.

Fig. 3.2.2. Velocity profile distribution along the channel length.

The resultant velocity profiles of u and v close to the bottom wall surface along the channel length are given in Figs 3.2.3 and 3.2.4.

Fig. 3.2.3. Horizontal velocity u profile along the channel length.

Fig. 3.2.4. Vertical velocity v profile along the channel length.

Discussion: It is observed that within the hydrodynamic entrance region, that is x  <   3H, the horizontal velocity u decreases along the channel length. This means that ∂u/∂x  <   0 close to the wall surface. The airflow is slowed along the x direction due to the no-slip boundary condition imposed near the wall as it flows over the wall surface. More fluid is therefore physically entering than leaving the flow domain along the x direction. At the same time, there should be more fluid leaving than entering along the y direction; the vertical velocity v increases, which implies that ∂v/∂y  >   0, in order to conserve the mass. On the other hand, when the flow is in the fully developed region, that is x  ≥   3H, it is observed that the horizontal velocity u does not appreciably change along the x direction, that is ∂u/∂x  =   0. In accordance with the continuity equation, ∂v/∂y must also be zero within this flow region. This is clearly reflected by the constant vertical velocity v as shown in Fig. 3.2.3 in the fully developed region. The physical meanings of the hydrodynamic entrance region and fully developed region are further discussed in Example 3.5.

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Some Advanced Topics in CFD

Jiyuan Tu , ... Chaoqun Liu , in Computational Fluid Dynamics (Third Edition), 2018

Review Questions

9.1

Simplify the general continuity equation below to a steady incompressible flow equation:

$\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho u\right)}{\partial x}+\frac{\partial \left(\rho v\right)}{\partial y}+\frac{\partial \left(\rho w\right)}{\partial z}=0$

9.2

How is the pressure term used to satisfy the continuity equation in the marker-and-cell (MAC) method?

9.3

Discuss briefly the idea behind the fractional-step procedure.

9.4

What types of applications and situations involve compressible flows?

9.5

What difficulties arise from modelling a transient supersonic flow around an aerofoil?

9.6

What is the biggest difficulty that has to be overcome with compressible flows?

9.7

What techniques can be used to minimize oscillations that occur in compressible flow due to discontinuities at the shock front?

9.8

How does a higher-order scheme such as a fifth-order scheme deal with discontinuities?

9.9

Under what circumstances would adaptive meshing be used? What would happen if a fixed mesh is employed instead?

9.10

Discuss briefly the concept behind the r-refinement grid adaptive technique.

9.11

What kinds of applications commonly use moving grids?

9.12

Explain how a moving grid can be applied to simulate a screw supercharger shown below. What parts would need to remain stationary and what parts would be allowed to move?

9.13

What is the main advantage in the numerical solution of using multigrid methods in terms of the handling of high- and low-frequency errors?

9.14

Explain what domain decomposition and load balancing are in parallel computing.

9.15

What is the immersed boundary method, and how is this different from using a boundary-fitted grid?

9.16

What is a direct numerical simulation (DNS)? How does it differ from a Reynolds-averaged Navier–Stokes (RANS) approach in terms of its handling of turbulence?

9.17

What are the Kolmogorov microscales? How do these scales impact on the mesh design?

9.18

Why can't DNS be used to solve high Reynolds number flows at the moment?

9.19

What is the main concept behind LES in turbulent modelling?

9.20

What are the subgrid-scale (SGS) models in LES? How are they used to define small scales of turbulence?

9.21

In the RANS–LES coupling approach, in which region of the mesh would you apply the RANS model and which region would you apply the LES model?

9.22

Why would you use a RANS–LES coupling approach to model high Reynolds number turbulent flows?

9.23

What is the difference in one-way coupling and two-way coupling in multiphase flows?

9.24

Would you use an Eulerian–Eulerian or Eulerian–Lagrangian for a multiphase flow that had a high mass loading for the secondary phase (i.e. not the continuous phase)?

9.25

Combustion is a complex phenomenon to model. What types of considerations must be made in modelling combustion?

9.26

Explain fluid–structure interaction (FSI) modelling. In what applications can this be used?

9.27

What is the key requirement (and the most complex) that enables the interaction between the fluid and structure in FSI?

9.28

What advanced techniques would be required to simulate airflow through the respiratory system into the lungs? What about pulsating blood flow through veins and arteries?

9.29

Briefly discuss the concept of the lattice Boltzmann method.

9.30

Briefly discuss the concept of the Monte Carlo method.

9.31

Briefly discuss the concept of the particle method.

9.32

Briefly discuss the concept of the discrete element method.

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Polymers

Denis Constales , ... Guy B. Marin , in Advanced Data Analysis & Modelling in Chemical Engineering, 2017

10.3 Macroscale Modeling Techniques

In general, the continuity equations of the reaction components depend on the selected reactor configuration. Industrial-scale polymerizations are commonly carried out in batch, tubular, or continuous stirred-tank reactors. Usually, relatively simple reactor models are used and thus ideal flow patterns in the reactor (eg, without backmixing) and isothermal temperature profiles are assumed. Deviations from the ideal flow pattern and from spatial isothermicity can lead, however, to significantly different local kinetics resulting in different local apparent rates and thus in different reactor averaged apparent rates as compared to those predicted assuming an ideal flow pattern and isothermicity.

The simplest way to model nonideal flow patterns and temperature profiles is to divide the reactor into a discrete number of compartments and include exchange/recycle streams between the compartments (Topalos et al., 1996; Zhang and Ray, 1997). For each compartment, a kinetic scheme is applied and the conversion profile and the polymer properties are obtained from averaging the properties of each compartment with respect to their size.

An important first "compartment model" is based on the division of each compartment into a discrete number of perfectly mixed segments. For example, in Fig. 10.13, for each compartment of an industrial-scale reactor for a polymerization process still in nondispersed medium, two small perfectly mixed segments and one large perfectly mixed segment are considered. The small segments reflect imperfect mixing (eg, of the initiator) and temperature gradients (eg, hot spots) at the inlet, whereas the large segment reflects the bulk zone of the compartment with an internal recycle to both small segments.

Fig. 10.13. Example of a compartment model with three perfectly mixed segments (two small and one large) to account for macromixing; V_i,k , volume of segment k in compartment i; q _Vi,k , exit volumetric flow rate for segment k in compartment i; q _V0, volumetric feed flow rate; f_l , recycle ratio (l  =   1,2); subscript V in the flow rates is not shown in order not to overload the figure.

Denoting the volume of the kth segment as V_i,k , q _Vi,k as its exit volumetric flow rate, R _Mi,k as the corresponding net monomer production rate, [M] _i,k as the monomer concentration, f_l as the lth recycle rate from the third segment, and q _V0 as the volumetric feed flow rate with monomer concentration [M]₀, the following continuity equations can be written down for the monomer in each segment k of a compartment i:

(10.67) $V_{i,1}\frac{d{\left[\mathrm{M}\right]}_{i,1}}{dt}={\left[\mathrm{M}\right]}_0{q}_{V0}+{\left[\mathrm{M}\right]}_{i-1,3}{q}_{Vi-1,3}+f_1{\left[\mathrm{M}\right]}_{i,3}{q}_{Vi,3}+R_{\mathrm{M}i,1}{V}_{i,1}-{\left[\mathrm{M}\right]}_{i,1}{q}_{Vi,1}$

(10.68) $V_{i,2}\frac{d{\left[\mathrm{M}\right]}_{i,2}}{dt}={\left[\mathrm{M}\right]}_{i,1}{q}_{Vi,1}+f_2{\left[\mathrm{M}\right]}_{i,3}{q}_{Vi,3}+R_{\mathrm{M}i,2}{V}_{i,2}-{\left[\mathrm{M}\right]}_{i,2}{q}_{Vi,2}$

(10.69) $V_{i,3}\frac{d{\left[\mathrm{M}\right]}_{i,3}}{dt}={\left[\mathrm{M}\right]}_{i,2}{q}_{Vi,2}+R_{\mathrm{M}i,3}{V}_{i,3}-f_1{\left[\mathrm{M}\right]}_{i,3}{q_V}_{i,3}-f_2{\left[\mathrm{M}\right]}_{i,3}{q}_{Vi,3}$

in which it is assumed that the volumetric flow rates are balanced:

(10.70) $q_{Vi,1}=q_{V0}+q_{Vi-1,3}+f_1{q}_{Vi,3}$

(10.71) $q_{Vi,2}=q_{Vi,1}+f_2{q}_{Vi,3}$

(10.72) $\left(1+f_1+f_2\right)q_{Vi,3}=q_{Vi,2}$

Analogously, overall sth order moment equations can be written down. For example, assuming that no reaction occurs in the feed, the total concentration of dead polymer in the first segment of compartment i follows from

(10.73) $V_{i,1}\frac{d{\mu }_{0i,1}}{dt}={\mu }_{0i-1,3}{q}_{Vi-1,3}+f_1{q}_{Vi,3}{\mu }_{0i,3}+R_{{\mu }_{0i,1}}{V}_{i,1}-{\mu }_{0i,1}{q}_{Vi,1}$

in which μ _0i,1 and $R_{{\mu }_{0i,1}}$ are the zeroth-order moment for the dead polymer molecules and the corresponding net production rate in segment 1 of compartment i. The net production rates can be calculated with the method of moments, as explained previously, assuming a given reaction scheme per segment. In addition, a temperature variation can be accounted for. Taking again the first segment, from an energy balance it follows that

(10.74) $\rho c_{\mathit{p}}\frac{d{T}_{i,1}}{dt}{V}_{i,1}=Q_0+Q_{i-1,3}+f_1{Q}_{i,3}+Q_{\mathrm{r}i,1}-Q_{i,1}$

in which T _i,1 is the temperature of the first segment of compartment i, ρ, and c _p are the corresponding density and specific heat capacity at constant pressure, and the subscript "r" is used to distinguish the net heat production by reaction from the flow contribution terms. The reaction term is given by

(10.75) $Q{\prime }_{i,1,\mathrm{r}}=\left(-{\mathrm{\Delta }}_{\mathrm{r}}H\right)k_{\mathrm{p}}{\left[M\right]}_{i,1}{\lambda }_{0,i,1}{V}_{i,1}$

in which Δ_r H is the propagation reaction enthalpy and λ _0i,1 is the zeroth-order moment of the living polymer molecules. Note that it is assumed that the generated heat is only due to propagation. The energy flow terms are proportional to the temperature of the flow while taking into account the contributions of the monomer and the polymer to obtain the correct density and specific heat capacity.

A second important compartment model comprises compartments consisting of a perfectly mixed segment and a plug-flow segment, which in turn can be represented by a series of perfectly mixed segments (see Fig. 10.14). This compartment model is selected in case segregation is important. Equations analogous to those for the compartment model of Fig. 10.13 can be derived.

Fig. 10.14. Example of a compartment model with one perfectly mixed segment and one plug-flow segment to account for macromixing; the plug-flow segment can be represented by a series of perfectly mixed segments (here three); V_i,k , volume of segment k in compartment i; q _Vi,k , exit volumetric flow rate for segment k in compartment i; q _V0, volumetric feed flow rate; f _1,k , recycle ratio; in the figure subscript V has been omitted.

Instead of using compartment models, the flow pattern in the reactor also can be calculated via computational fluid dynamics (CFD). However, when using CFD, relatively small reaction networks are often used to reduce the computational cost. An exception is gas-phase polymerization, such as the production of low-density polyethylene). For more details on the application of CFD calculations for polymerization processes, the reader is referred to Asua and De La Cal (1991), Fox (1996), Kolhapure and Fox (1999), and Pope (2000).

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21st European Symposium on Computer Aided Process Engineering

Mehdi Berreni , Meihong Wang , in Computer Aided Chemical Engineering, 2011

2 Mathematical modeling and model validation

The mathematical model consists of a continuity equation for each chemical component, equations for energy balance, heat transfer and pressure drop. Coke buildup on the internal tube wall was also included. The 1D peudo-dynamic model is steady-state regarding mass and energy balances, but dynamic regarding coke build-up. In order to validate the model, data (i.e. frequency factor, activation energy, reactor design and operating parameters) from Sundaram and Froment (1979) is used. More details regarding modelling and model validation were described in Berreni and Wang (2011).

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Introductory Article: Electromagnetism

N.M.J. Woodhouse , in Encyclopedia of Mathematical Physics, 2006

Conservation of Charge

To see the connection between the continuity equation and charge conservation, let us look at the total charge within a fixed V bounded by a surface S. If charge is conserved, then any increase or decrease in a short period of time must be exactly balanced by an inflow or outflow of charge across S.

Consider a small element d S of S with outward unit normal and consider all the particles that have a particular charge e and a particular velocity v at time t. Suppose that there are σ of these per unit volume (σ is a function of position). Those that cross the surface element between t and t + δt are those that at time t lie in the region of volume

$|\mathit{v}\cdot \mathit{n}\text{d}S\delta t|$

shown in Figure 1 . They contribute eσ v · d S δt to the outflow of charge through the surface element. But the value of J at the surface element is the sum of eσ v over all possible values of v and e. By summing over v , e, and the elements of the surface, therefore, and by passing to the limit of a continuous distribution, the total rate of outflow is

${\int }_S\mathit{J}\cdot \text{d}\mathit{S}$

Figure 1. The outflow through a surface element.

Charge conservation implies that the rate of outflow should be equal to the rate of decrease in the total charge within V. That is,

[31] $\frac{\text{d}}{\text{d}t}{\int }_V\rho \text{d}V+{\int }_S\mathit{J}\cdot \text{d}\mathit{S}=0$

By differentiating the first term under the integral sign and by applying the divergence theorem to the second integral,

[32] ${\int }_V\left(\frac{\partial \rho }{\partial t}+\text{div}\mathit{J}\right)\text{d}V=0$

If this is to hold for any choice of V, then ρ and J must satisfy the continuity equation. Conversely, the continuity equation implies charge conservation.

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