Drift Term in the Continuity Equation

Averaged Continuity Equation

Then the depth-averaged continuity equation under the assumption of shallow-water flow takes the following form(4.13)∂η∂t+∂qx∂x+∂qz∂z=0 Alternatively, the continuity equation can be written in terms of the total depth, h, and the depth-averaged velocities, u¯ and w¯, as follows(4.14)∂h∂t+∂∂x(u¯h)+∂∂z(w¯h)=0 where the depth-averaged horizontal velocities are defined as follows(4.15)u¯=qxhw¯=qzh

From: Free-Surface Flow: , 2019

27th European Symposium on Computer Aided Process Engineering

Rodrigo G.D. Teixeira , Argimiro R. Secchi , in Computer Aided Chemical Engineering, 2017

2 Mathematical Modelling

Developing averaged continuity, momentum and energy equations individually for each phase results in the so called Two-Fluid (Separated) Model (TFM). Common simplifying assumptions, such as neglecting viscous and Reynolds stresses and covariance terms, lead to the following field equations for phase k (Morales-Ruiz et al., 2012):

(1) $\frac{\partial \left[{\alpha }_k{\rho }_k\right]}{\partial t}+\frac{\partial \left[{\alpha }_k{\rho }_k{\nu }_k\right]}{\partial x}={\Gamma }_k$

(2) $\begin{array}{l}\frac{\partial \left[{\alpha }_k{\rho }_k{v}_k\right]}{\partial t}+\frac{\partial \left[{\alpha }_k{\rho }_k{v}_k^2\right]}{\partial x}\hfill \\ =-{\alpha }_k\frac{\partial P}{\partial x}-{\alpha }_k{\rho }_{k}g\mathrm{s}\mathrm{e}\mathrm{n}\left(\theta \right)-F_{kw}+{\Gamma }_k{v}_{ki}+F_{ki}+F_{kMV}\hfill \end{array}$

(3) $\frac{\partial \left[{\alpha }_k{\rho }_k{h}_k\right]}{\partial t}+\frac{\partial \left[{\alpha }_k{\rho }_k{\nu }_k{h}_k\right]}{\partial x}={\alpha }_k\frac{\partial P}{\partial t}+Q_{kw}+{\Gamma }_k{h}_{ki}+Q_{ki}$

These six equations of change must be solved for pressure P, one of the void fractions α_k , phasic velocity v_k and enthalpy h_k , which require previous knowledge of the dependence of the latter, and also of densities p_k , upon pressure and temperature T.

Alternatively, both equations in each of these three sets can be added together to yield three new field equations governing the flow of the two-phase mixture as a whole:

(4) $\frac{\partial {\rho }_m}{\partial t}+\frac{\partial \left[{\rho }_m{\nu }_m\right]}{\partial x}=0$

(5) $\frac{\partial \left[{\rho }_m{v}_m\right]}{\partial t}+\frac{\partial \left[{\rho }_m{v}_m^2\right]}{\partial x}=-\frac{\partial P}{\partial x}-{\rho }_{m}g\mathrm{s}\mathrm{e}\mathrm{n}\left(\theta \right)-F_{mw}-\frac{\partial }{\partial x}\left[\frac{{\alpha }_v{\rho }_v{\rho }_l}{{\alpha }_l{\rho }_m}{\nu }_{\nu }^{df{t}^2}\right]$

(6) $\begin{array}{l}\frac{\partial \left[{\rho }_m{h}_m\right]}{\partial t}+\frac{\partial \left[{\rho }_m{\nu }_m{h}_m\right]}{\partial x}=\frac{\partial P}{\partial t}+Q_{mw}-\frac{\partial }{\partial x}\left[\frac{{\alpha }_{\nu }{\rho }_{\nu }{\rho }_l}{{\rho }_m}\left(h_v-h_l\right)v_v^{dft}\right]\hfill \\ +\left[v_m+\frac{{\alpha }_{\nu }\left({\rho }_l-{\rho }_v\right)}{{\rho }_m}{v}_v^{dft}\right]\frac{\partial P}{\partial x}\hfill \end{array}$

The Drift-Flux Model (DFM), which is a Mixture Model, is based on the solution of Eqs. (4), (5) and (6) for P, v_m and h_m , complemented by vapour-phase continuity Eq. (1) for α_v and also by a kinematic constitutive equation for the mean drift velocity v_v ^dft , which accounts for the relative motion between the liquid and vapour phases (Ishii and Hibiki, 2011). The latter was calculated in this study by means of the correlations developed by Bhagwat and Ghajar (2014).

Even though most recent two-phase flow simulation studies are based on either the Two-Fluid or the Drift-Flux Model, the recommendation of simpler pressure-gradient steady-state equations based on the assumption of homogeneous (equal phase velocities) flow is seen to persist in other references. Setting v_v ^dft   =   0 in Eq. (5) in steady-state indeed yields Eq. (7), which is entirely analogous to the well-known macroscopic mechanical energy balance for single-phase flows (Bird et al., 2002):

(7) $\frac{dP}{dx}=-F_{mw}-{\rho }_{m}g\mathrm{s}\mathrm{e}\mathrm{n}\left(\theta \right)-{\rho }_m{\nu }_m\frac{d{\nu }_m}{dx}$

The empirical correlations of Beggs and Brill (1973) were used in this study to predict the friction, elevation and acceleration contributions on the right-hand side of Eq. (7).

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GENERALIZED GOVERNING EQUATIONS FOR MULTIPHASE SYSTEMS: AVERAGING FORMULATIONS

Amir Faghri , Yuwen Zhang , in Transport Phenomena in Multiphase Systems, 2006

Continuity Equation

The area-averaged continuity equation is

(4.148) $\frac{\partial 〈\rho 〉}{\partial t}+\frac{\partial }{\partial z}(〈\rho 〉〈\omega 〉)=0$

where 〈ρ〉 and 〈w〉 are, respectively, the area-averaged density and axial velocities of the multiphase mixture. The product of the deviations of density and velocity is neglected. For a mixture of liquid and vapor, it can be expressed as

(4.149) $〈\rho 〉=\alpha {〈{\rho }_v〉}^v+(1-\alpha ){〈{\rho }_{\ell }〉}^{\ell }$

where α is the vapor or holdup void fraction, α = ɛ _v , which represents the time-averaged volumetric fraction of vapor in a two-phase mixture.

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Multiphase stirred reactors

Chao Yang , Zai-Sha Mao , in Numerical Simulation of Multiphase Reactors with Continuous Liquid Phase, 2014

3.2.1 Governing equations

Based on the Eulerian–Eulerian two-fluid model, the Reynolds-averaged continuity and momentum equations for phase k are written as

(3.1) $\frac{\partial ({\rho }_k{\alpha }_k)}{\partial t}+\frac{\partial ({\rho }_k{\alpha }_k{u}_{kj}+{\rho }_k\overline{{{\alpha }^{\prime }}_k{u^{\prime }}_{kj}})}{\partial x_j}=0$

(3.2) $\begin{array}{l}\frac{\partial ({\rho }_k{\alpha }_k{u}_{ki})}{\partial t}+\frac{\partial ({\rho }_k{\alpha }_k{u}_{ki}{u}_{kj})}{\partial x_j}=-{\alpha }_k\frac{\partial P}{\partial x_i}+\frac{\partial ({\alpha }_k\overline{{\tau }_{kij}})}{\partial x_j}+F_{ki}+{\rho }_k{\alpha }_k{g}_i-\\ {\rho }_k\frac{\partial }{\partial x_j}({\alpha }_k\overline{{u^{\prime }}_{kj}{u^{\prime }}_{ki}}+u_{ki}\overline{{{\alpha }^{\prime }}_k{u^{\prime }}_{kj}}+u_{kj}\overline{{{\alpha }^{\prime }}_k{u^{\prime }}_{ki}}+\overline{{{\alpha }^{\prime }}_k{u^{\prime }}_{kj}{u^{\prime }}_{ki}})\end{array}$

(3.3) $\sum {\alpha }_k=1.0$

where α _k is the phase volume fraction.

The correlation term of phase holdup and velocity fluctuations $\overline{{{\alpha }^{\prime }}_k{u^{\prime }}_{ki}}$ in both continuity and momentum equations represents the transport of both mass and momentum by dispersion. Since the influence of the dispersed phase on turbulence structure is not well understood, a simple gradient assumption can be adopted to model $\overline{{{\alpha }^{\prime }}_k{u^{\prime }}_{ki}}$ , which is given by

(3.4) $\overline{{u^{\prime }}_{ki}{{\alpha }^{\prime }}_k}=-\frac{{\nu }_{k\text{,t}}}{{\sigma }_{\text{t}}}\frac{\partial {\alpha }_k}{\partial x_i}$

where σ _t is the turbulent Schmidt number for the phase dispersion. The value of this number depends on the size of the dispersed phase and the scale of turbulence. It was found that the simulation results were sensitive to σ _t in solid–liquid flow simulation and a value between 1.0 and 2.0 was suggested (Shan et al., 2008). In gas–liquid systems, the value of 1.0 was recommended (Ranade and Van den Akker, 1994), but Wang and Mao (2002) suggested a value of 1.6 was suitable.

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Deforming Grid Methods

Nikolaos D. Katopodes , in Free-Surface Flow, 2019

10.6.6.2 Non-Hydrostatic Pressure

The predictor velocity field, $U^{⁎}$ and $w^{⁎}$ , satisfies the depth averaged continuity equation, but not local continuity. To achieve local continuity, the non-hydrostatic pressure field must be computed. This is accomplished by correcting the horizontal and vertical velocity fields with a non-hydrostatic pressure correction, $p_d^c$ , as follows

(10.224) $U_{j,k}^{n+1}=U_{j,k}^{⁎}-\mathrm{\Delta }t(\frac{{p_d^c}_{G_{2j},k}-{p_d^c}_{G_{2j+1},k}}{D_j})$

and

(10.225) $w_{i,k}^{n+1}=w_{i,k}^{⁎}-2\mathrm{\Delta }t(\frac{{p_d^c}_{i,k}-{p_d^c}_{i,k-1}}{\mathrm{\Delta }{z}_{i,k}+\mathrm{\Delta }{z}_{i,k-1}})$

The non-hydrostatic pressure field is then updated, as follows

(10.226) $q_{i,k}^{n+1/2}=q_{i,k}^{n-1/2}+{p_d^c}_{i,k}$

The pressure correction field needs to be computed implicitly. Integrating the continuity equation at time step $n+1$ , over a cell with $N_s$ sides, we obtain

(10.227) $A_i(w_{i,k+1}^{n+1}-w_{i,k}^{n+1})+\sum _{m=1}^{N_s}{U}_{m,k}^{n+1}\mathrm{\Delta }{z}_{m,k}^{uw}{N}_{m}d{f}_m=0$

Next, substituting the pressure corrector terms of Eqs. (10.224) and (10.225) for the velocities at the new time step into Eq. (10.227) we obtain

(10.228) $\begin{array}{cc}\hfill & \frac{1}{\mathrm{\Delta }t}[A_i(w_{i,k+1}^{⁎}-w_{i,k}^{⁎})+\sum _{m=1}^{N_s}{U}_{m,k}^{⁎}\mathrm{\Delta }{z}_{m,k}^{uw}{N}_{m}d{f}_m]\hfill \\ \hfill & =\sum _{m=1}^{N_s}(\frac{{p_d^c}_{N{e}_m,k}-{p_d^c}_{i,k}}{D_m})\mathrm{\Delta }{z}_{m,k}^{uw}d{f}_m+\frac{2A_i}{\mathrm{\Delta }{z}_{i,k}+\mathrm{\Delta }{z}_{i,k-1}}q{c}_{i,k-1}\hfill \\ \hfill & -2A_i(\frac{1}{\mathrm{\Delta }{z}_{i,k}+\mathrm{\Delta }{z}_{i,k-1}}+\frac{1}{\mathrm{\Delta }{z}_{i,k}+\mathrm{\Delta }{z}_{i,k+1}})q{c}_{i,k}+\frac{2A_i}{\mathrm{\Delta }{z}_{i,k}+\mathrm{\Delta }{z}_{i,k+1}}q{c}_{i,k+1}\hfill \end{array}$

Finally, solving Eq. (10.228), the pressure corrector field is obtained, and the horizontal velocity field at time $n+1$ can be computed from Eq. (10.224). The vertical velocity field can be computed starting at the lower boundary with $w_{i,1}^{n+1}=0$ , and using the discretized continuity equation, (10.227), to compute the velocity field with

(10.229) $w_{i,k+1}^{n+1}=w_{i,k}^{n+1}-\frac{1}{A_i}\sum _{m=1}^{N_s}{U}_{m,k}^{n+1}\mathrm{\Delta }{z}_{m,k}^{uw}{N}_{m}d{f}_m$

The non-hydrostatic pressure field can then be updated from Eq. (10.226).

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Decay of Round Turbulent Jets with Swirl

D. Ewing , in Engineering Turbulence Modelling and Experiments 4, 1999

2.1 Momentum Equations

Using the standard thin-shear layer assumptions, it is straightforward to show that the averaged continuity and momentum equations in an incompressible, high-Reynolds-number, turbulent, round jet flow with swirl can be reduced to

(1) $\frac{\partial U}{\partial x}+\frac{1}{r}\frac{\partial rV}{\partial r}=0$

(2) $U\frac{\partial U}{\partial x}+V\frac{\partial U}{\partial r}=-\frac{1}{\rho }\frac{\partial P}{\partial x}-\frac{1}{r}\frac{\partial r\overline{uv}}{\partial r}\text{,}$

(3) $-\frac{W^2}{r}=-\frac{1}{\rho }\frac{\partial P}{\partial r}+\frac{\overline{w^2}}{r}-\frac{1}{r}\frac{\partial r\overline{v^2}}{\partial r}\text{,}$

and

(4) $U\frac{\partial W}{\partial x}+\frac{v}{r}\frac{\partial rW}{\partial r}=-\frac{1}{r^2}\frac{\partial r^2\overline{vw}}{\partial r}\text{,}$

Where (U, V, W) and (u, v, w) are the mean and fluctuating velocity components in the axial, x, radial, r, and azimuthal, θ, directions respectively (v. figure 1 ). It is assumed here that the flow is statistically homogeneous in the azimuthal direction.

Figure 1. Geometry of the jet

These equations can be integrated over a plane downstream of the jet outlet to yield integral momentum equations given by

(5) $\frac{d}{dx}\left(2\pi {\displaystyle \underset{o}{\overset{\infty }{\int }}\left[U^2-\frac{W^2}{2}+\overline{u^2}-\frac{\overline{v^2}+\overline{w^2}}{2}\right]rdr}\right)=0$

and

(6) $\frac{d}{dx}\left(2\pi {\displaystyle \underset{o}{\overset{\infty }{\int }}\left[UW+\overline{uw}\right]}{r}^{2}dr\right)=0\text{.}$

It follows that the axial and angular momentum fluxes are conserved as the jet evolves downstream; i.e.,

(7) $2\pi {\displaystyle \underset{o}{\overset{\infty }{\int }}\left[U^2-\frac{W^2}{2}+\overline{u^2}-\frac{\overline{v^2}+\overline{w^2}}{2}\right]rdr=M_o}$

and

(8) $2\pi {\displaystyle \underset{o}{\overset{\infty }{\int }}\left[UW+\overline{uw}\right]r^{2}dr=G_{\mathrm{\theta }}\text{.}}$

The strength of the swirling jet is characterized by the swirl number given by

(9) $S=\frac{G_{\mathrm{\theta }}}{M_{o}R}\text{,}$

where R is the radius of the jet outlet.

Chervinski and Chiger[5] argued that the contribution of turbulent stresses in the momentum equations are negligible and hypothesized the velocity profiles in the jet have self-similar solutions of the form

(10) $U\left(x,r\right)=U_s\left(x\right)f\left(\eta \right)$

Where η=r/δ(x). They found the reduced equations had self-similar solutions where the scales for the mean velocities are given by U _s ∝(x−x _o )⁻¹ W _s ∝(x−x _o )⁻² to first order. These self-similar solutions are at most asymptotic solutions for the flows generated from finite sources. It is evident, however, that W ²/U ²∼(x−x _o )⁻² to first order while measurements indicate that $\overline{u^2}/U^2,\overline{v^2}/U^2\text{,}$ and $\overline{w^2}/U^2$ are constant as x→∞ [4,7,8]. Thus, the solutions deduced from the reduced equations are not consistent with the solutions for the full equations because the neglected Reynolds stresses make a lower order contribution to the axial momentum equation than the retained term, W² /2, as x→∞.

It is straightforward to show, however, that the full equations do not have internally consistent self-similar solutions indicating some of the terms in the equation must be higher order if self-similar solutions exist. It is assumed here that the swirl term, W² /r, in the radial momentum equation is negligible. It follows the radial momentum equation can be integrated and substituted into the axial momentum equations yielding the same first-order axial momentum equations as in the non-swirling jet[9]; i.e.,

(12) $U\frac{\partial U}{\partial x}+v\frac{\partial U}{\partial r}=-\frac{1}{r}\frac{\partial r\overline{uv}}{\partial r}$

and

(13) ${\displaystyle \underset{o}{\overset{\infty }{\int }}\left[U^2+\overline{u^2}-\frac{\overline{v^2}+\overline{w^2}}{2}\right]rdr=M_o\text{.}}$

It is generally recognized that this assumption is only valid in weakly swirling flows but, as it will be shown later, these equations are the first order equations in the far field of flows with both small and large initial swirl number. The turbulent stresses are retained in equation13 because Hussein et al.[9] showed these terms were necessary to accurately evaluate the momentum flux in a non-swirling jet.

It is hypothesized that the averaged momentum equations have self-similar solutions of the form given in table 1. Here, η=r/δ(x) is the similarity variable and δ(x) is the characteristic length scale in the flow. Following George[10], the x-dependent scales for the Reynolds stresses are not chosen as U _S ²(x) a priori but are instead determined from the constraints in the equations. Substituting these solutions into the momentum equations yields

Table 1. Similarity solutions and constraints for moments in momentum equations.

Moment	Solution	Constraint	Moment	Solution	Constraint
U(x, r)	U S (x)f(η)	U S ∝M o 1/2 δ −1	W(x,r)	W S (x)f θ (η)	$W_S\propto U_S\frac{G_{\theta }}{\delta M_o}$
$\overline{uv}\left(x,r\right)$	R s (x)g(η)	$R_s\propto U_s{W}_S\frac{d\delta }{dx}$	$\overline{vw}\left(x,r\right)$	R vw (x)g vw (η)	$R_{vw}\propto U_S^2\frac{d\delta }{dx}$
$\overline{u^2}\left(x,r\right)$	K u (x)k u (η)	K u ∝U S 2	$\overline{uw}\left(x,r\right)$	R uw (x)g uw (η)	R uw ∝U S W S
$\overline{v^2}\left(x,r\right)$	K v (x)k v (η)	K v ∝U S 2
$\overline{w^2}\left(x,r\right)$	K w (x)k w (η)	K w ∝U S 2

(14) $\left[U_S\frac{d{U}_S}{dx}\right]f^2-\left\{\left[U_S\frac{d{U}_S}{dx}\right]+2\left[\frac{U_S^2}{\delta }\frac{d\delta }{dx}\right]\right\}\frac{1}{\eta }\frac{df}{d\eta }{\displaystyle \underset{o}{\overset{\eta }{\int }}\tilde{\eta }f\left(\tilde{\eta }d\tilde{\eta }=-\left[\frac{R_S}{\delta }\right]\frac{1}{\eta }\frac{d\eta g}{d\eta }\text{,}\right.}$

(15) $\left\{\left[U_S\frac{d{W}_S}{dx}\right]+\left[\frac{U_S{W}_S}{\delta }\frac{d\delta }{dx}\right]\right\}f{f}_{\theta }-\left\{2\left[\frac{U_S{W}_S}{\delta }\frac{d\delta }{dx}\right]+\left[U_S\frac{d{W}_s}{dx}\right]\right\}\frac{1}{{\eta }^2}\frac{d\eta f_{\theta }}{d\eta }{\displaystyle \underset{o}{\overset{\eta }{\int }}\tilde{\eta }}fd\tilde{\eta }=-\left[\frac{R_{vw}}{\delta }\right]\frac{1}{{\eta }^2}\frac{{\eta }^2{g}_{vw}}{d\eta }\text{,}$

(16) $\left[U_S^2{\delta }^2\right]{\displaystyle \underset{o}{\overset{\infty }{\int }}{f}^2\tilde{\eta }d\tilde{\eta }+\left[K_u{\delta }^2\right]{\displaystyle \underset{o}{\overset{\infty }{\int }}{k}_u\tilde{\eta }d\tilde{\eta }}-\frac{1}{2}\left\{\left[K_v{\delta }^2\right]{\displaystyle \underset{o}{\overset{\infty }{\int }}{k}_v\tilde{\eta }d\tilde{\eta }}+\left[K_w{\delta }^2\right]{\displaystyle \underset{o}{\overset{\infty }{\int }}\tilde{\eta }{k}_{w}d\tilde{\eta }}\right\}=M_o\text{,}}$

and

(17) $\left[U_s{W}_s{\delta }^3\right]{\displaystyle \underset{o}{\overset{\infty }{\int }}{\tilde{\eta }}^2}f{f}_{\theta }d\tilde{\eta }+\left[R_{uw}{\delta }^3\right]{\displaystyle \underset{o}{\overset{\infty }{\int }}{\tilde{\eta }}^2{g}_{uw}d\tilde{\eta }=G_{\theta }\text{.}}$

These self-similar solutions are consistent with the momentum equations if the x-dependent portion of the terms (contained in square brackets) in each equation are proportional. These terms are proportional when the scales for the solutions satisfy the constraints given in table 1.

The results show that there are two different velocity scales in this flow that are not proportional, in agreement with the analysis of Chervinski and Chiger[5]. One scale describes the decay of the axial velocity component while the second describes the decay of the tangential velocity component. One consequence is that the Reynolds stresses are not all proportional to U _s ² or proportional to each other as commonly assumed a priori in analyzing many free-shear flows. In particular, the Reynolds stresses in the angular momentum equation, such as $\overline{vw}\text{,}$ are not proportional to the Reynolds stresses in the axial momentum equation, such as $\overline{uv}\text{.}$

It is important to note that the jet's growth rate has not yet been uniquely determined from the analysis of the momentum equations. The growth rate could be determined from measurements and input into the analysis as it was in the previous analyses [5,6]. However, analyses of other shear flows have shown that the growth rate of these flows could be uniquely determined by extending the analyses to the transport equations for the turbulent Reynolds stresses[10]. This approach also provides confirmation that the results deduced from the momentum equations are consistent with the transport equations for the Reynolds stresses.

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Design tools related to engineering

Hans F. Burcharth , ... Alberto Lamberti , in Environmental Design Guidelines for Low Crested Coastal Structures, 2007

13.4.4.4. LIMCIR (Universitat Politècnica de Catalunya)

The LIMCIR code is an advanced Q-3D circulation model, developed at the Universitat Politècnica de Catalunya (Cáceres, 2004), solving the depth and time averaged continuity and momentum equations while recovering a depth averaged undertow. The resulting partial differential equations are solved with a staggered grid and an Alternating Direction Implicit method that allows, at the end of each iteration, to obtain a centered scheme in space and time. The closure sub models are based on state of the art formulations.

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Bed shear stresses are obtained according to Madsen (1994) in the presence of waves.

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Roller model is based on Dally and Brown (1995).

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Eddy viscosity is evaluated based on Nielsen (1985) formulation to consider the bottom turbulence and Osiecki and Dally (1996) to consider the roller turbulence. It can also employ the Smagorinsky model.

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Wave induced mass flux can be obtained from De Vriend and Stive (1987) or Fredsoe and Deigaard (1992).

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Wind stress is considered using the Yelland and Taylor (1996) formulation.

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The overtopping term can be obtained following Owen (1980), Hedge and Reis (1998), Van der Meer and Janssen (1995), or Allsop et al. (1995) considering sloping or vertical structures.

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ENTROPY GENERATION IN POROUS MEDIA

A.C. BAYTAŞ , A.F. BAYTAŞ , in Transport Phenomena in Porous Media III, 2005

8.3 GOVERNING EQUATIONS

In principle, the equations in thermal science describe the various transport phenomena and may be written at the microscopic level. The description and solution of a transport problem at the microscopic level is not practical and sometimes also impossible. A different level of description is needed, namely the macroscopic level, at which measurable, continuous and differentiable quantities may be determined and boundary value problems can be stated and solved. The representative elementary volume, or REV, is a conceptual space unit. When the measuring volume is at least of REV, measurable characteristics of the porous medium become continuum quantities. Volume averaging is a method that makes the measurable quantities continuum properties based on the REV concept. The continuum or macroscopic, governing equations are derived based on the microscopic governing equations. The porosity is the most important property of a porous medium and it affects most of the physical properties of the medium. Measurement of porosity is made by using several techniques, such as imbibition, mercury injection and gas injection methods give an effective porosity value. The true porosity can be measured by using direct, optical and gamma-ray attenuation methods, see Kaviany (1995). The local porosity can be measured directly along a porous column in the gamma-ray attenuation method, see Baytas and Akbal (2002) and Ishakoglu and Baytas (2002). Using the definitions of the porosity and the volume average of a quantity and the local (microscopic) quantity of fluid (see Liu and Masliyah, 1999), the volume-averaged velocity vector, v, and the intrinsic phase-averaged velocity vector, v_e , are related by the expression v = ɛv_e which is also known as the Dupuit–Forchheimer relationship, see Ingham (2004). Volume averaging of the governing equations can be performed by averaging over the REV, term by term.

8.3.1 Continuity equation

The continuity of mass equation can be written as follows, see Bejan (1995):

(8.1) $\epsilon \frac{\partial \rho }{\partial t}+\nabla \cdot (\rho v)=0,$

where ρ is the density of the fluid and ɛ is the porosity. One can observe that the volume averaged continuity equation is very similar to the continuity equation for a clear fluid.

8.3.2 Momentum balance equation

The volume averaged Navier–Stokes equations can be written as follows:

(8.2) $\rho \left[\frac{1}{\epsilon }\frac{\partial v}{\partial t}+\frac{1}{{\epsilon }^2}(v\cdot \nabla )v\right]=-\nabla P+{\mu }_e{\nabla }^{2}v-\frac{\mu }{K}v-c\rho \left|v\right|v+\rho g,$

where P is pressure, K is permeability of porous medium, $c=c_e/\sqrt{K}$ and c_e is the form-drag constant, g is gravity, t is time, μ is the viscosity of fluid, and μ _e is the effective fluid viscosity and it depends on the geometry of the permeable medium. The permeability, K, is the measure of the flow conductance of the porous medium and it is defined by the Darcy law and can be written by using porosity as follows, see Ergun (1952):

The third and fourth terms on the right-hand side of equation (8.2) are the viscous drag and form drag, respectively. Equation (8.2) is the volume averaged Navier–Stokes equations for homogenous fluid flow in an isotropic porous media and it is known as the Hazen–Dupuit–Darcy model, see Lage (1998).

For steady Newtonian fluid flow in a porous medium of constant porosity, equation (8.2) can be reduced to the famous Darcy equation by neglecting the inertial and viscous forces and the form drag terms, as follows:

(8.4) $v=-\frac{K}{\mu }(\nabla P+\rho g),$

where the terms v and ∇P are the Darcy velocity and the pressure gradient vectors, respectively. For anisotropic porous medium, the permeability, K, is a general second-order tensor.

Henry Darcy (1856) introduced a one-dimensional empirical model, a modern refinement (popularized by Muskat (1937)) of which is equation (8.4), for single phase Newtonian fluid flow in porous media based on the unidirectional water permeation in a fountain. When the flow is weak, or at low discharge fluid rates, the pressure drop is linearly related to the flow discharge rate. The Darcy model ignores the boundary effects on the flow and this assumption is not valid when the boundaries of the porous medium are taken into account. To overcome this problem, then the Brinkman model is usually employed, see Ingham and Pop (1998, 2002) and Nield and Bejan (1999).

In the presence of a magnetic field, a body force term has to be added to equation (8.2) as follows:

(8.5) $\rho \left[\frac{1}{\epsilon }\frac{\partial v}{\partial t}+\frac{1}{{\epsilon }^2}(v\cdot \nabla )v\right]=-\nabla P+{\mu }_e{\nabla }^{2}v-\frac{\mu }{K}v-c\rho \left|v\right|v+\rho g+\frac{\gamma }{\epsilon }(v\times B_0)\times B_0,$

where γ is the electrical conductivity of the fluid and B ₀ is the constant externally imposed magnetic field. For more details, see, for example, Nield (1999), Tasnim et al. (2002) and Ingham (2004).

8.3.3 Energy equation

The energy equation for an homogeneous porous medium can be derived by using the first law of thermodynamics and for local thermal equilibrium and isotropic porous media it is given as follows:

(8.6) ${(\rho c_p)}_f\left(\sigma \frac{\partial T}{\partial t}+v\cdot \nabla T\right)=k{\nabla }^{2}T+q^{‴}+\frac{\mu }{K}{v}^2,$

where q″′ is the volumetric heat source strength and the last term on the right-hand side of equation (8.6) is the viscous dissipation effect appropriate for the Darcy model. For more details, see Bejan (1995), Kaviany (1995), Hossain and Pop (1997), Nield (2000, 2002) and Ingham (2004), and for the thermal non-equilibrium model Baytas (2003). Also, in equation (8.6), k is the thermal conductivity of the porous medium and it is a combination of the conductivities of the two constituents, namely

(8.7) $k=\epsilon k_f+(1-\epsilon )k_s,$

where the subscripts f and s represent the fluid and solid phases, respectively, σ is the capacity ratio of two constituents and it is given by the following:

(8.8) $\sigma =\frac{\epsilon {(\rho C_p)}_f+(1-\epsilon ){(\rho C_p)}_s}{{(\rho C_p)}_f}.$

If the porous media is subjected to a hydromagnetic effect, then a new energy production term due to the magnetic effect has to be added to the equation (8.6), namely we obtain the following:

(8.9) ${(\rho c_p)}_f\left(\sigma \frac{\partial T}{\partial t}+v\cdot \nabla T\right)=k{\nabla }^{2}T+q^{‴}+\frac{\mu }{K}{v}^2+\frac{\gamma }{\epsilon }{B}_0^{2}v.$

8.3.4 Entropy generation

Non-equilibrium conditions, due to the exchange of energy and momentum within the fluid and at the solid boundaries, cause a continuous entropy production in the flow field. This entropy generation is due to the irreversible nature of heat transfer and viscosity effects within the fluid and at the solid boundaries. The property entropy is a measure of the molecular disorder or randomness of a system, and the second law of thermodynamics states that entropy can be created but it can never be destroyed. Based on the increase of entropy principle for any system, the local entropy generation rate per unit volume can be estimated writing the second law of thermodynamics in a differential form as an open system as follows, see Arpaci and Larsen (1984):

(8.10) $S_{\text{gen}}^{‴}=\frac{k}{T_0^2}{(\nabla T)}^2+\frac{\mu }{T_0}\Phi +\frac{q^{‴}}{T_0},$

where T ₀ is the absolute boundary temperature and Φ is the viscous dissipation. The first term on the right-hand side of equation (8.10) represents the entropy generation due to the heat transfer irreversibility, while the second term is the entropy generation associated with fluid friction, and the last term is the entropy generation due to volumetric heat generation.

The second law of thermodynamics can be applied to the homogeneous porous medium to yield the volumetric entropy generation rate, ${\dot{S}}_{\text{gen}}^{‴}$ , as follows:

(8.11) ${\dot{S}}_{\text{gen}}^{‴}=\frac{k}{T_0^2}{(\nabla T)}^2+\frac{\mu }{K{T}_0}{v}^2+\frac{\mu }{T_0}\Phi +\frac{\gamma B_0^2}{\epsilon T_0}v,$

where the second term on the right-hand side of equation (8.11) represents the viscous dissipation (irreversibility) term for porous media and it is important for the Darcy flow model, and the third term on the right side of equation (8.11) is the extra viscous dissipation term for the non-Darcy flow model and the last term on the right-hand side of the equation represents the entropy generation due to hydromagnetic effects. On the Brinkman model, the extra viscous dissipation as used by some authors, Φ, can be written in a two-dimensional Cartesian coordinate system given as follows:

(8.12) $\Phi =2\left[{\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial v}{\partial y}\right)}^2\right]+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2,$

where u and v are the fluid velocity components in x- and y-directions, respectively.

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Virtual erosion prediction, design optimization and combustion system integration of high pressure fuel injector systems

M. Gouda , ... V. Srinivasan , in Fuel Systems for IC Engines, 2012

2.2 Numerical Framework of Applied CFD Method

AVL CFD methodology considers each fluid as a continuum media and applies the conservation laws. An ensemble averaging is affected to remove the microscopic interfaces, resulting in macroscopic conservation equations which are analogous to their single-phase counterparts but differ in that a new variable called volume fraction, and new terms referred to as the interfacial exchange terms, Ishii [2], and Lahey and Drew [3]. The averaged continuity and momentum equations implemented follow from the theoretical work of Drew and Passman [4].

One of the key elements of cavitation modelling is vaporization rate, or the interfacial mass exchange. In the current implantation the Rayleigh equation for the dynamics of single-bubble growth had been coded. In general bubbles undergo extensive break-up and coalescence processes; the bubble sizes are not uniform. Therefore, the ability to model the poly-dispersion of bubbles becomes important. This is especially true for the simulation of erosion damage due to bubble collapses. Wang et al. [5] applied the moment method leading to a set of modelled transport equations for bubble number density and interfacial area. The source terms associated with break-up and coalescence of bubbles are obtained from the work of Ishii's group (Ishii et al. [6], Sun et al. [7]), which divide individual mechanism according to two size groups. To capture the turbulence in the system a recently developed k-ζ-f turbulence model (Basara [8], Hanjalic et al. [9]) which is particularly suited for predicting near wall turbulence is extended for multiphase implantation.

Figure 1 is a demonstration of the applied numerical method described above. The graph shows a comparison between numerical and experimental results on capturing cavitation phenomena on a physical and a virtual test rig.

Figure 1. Lab measurement results (top left) vs. CFD results (bottom left) on prediction of cavitation in the test geometry. Simulation/measurement comparison on pressure drop over the system as function of operated flow rate (right). [12]

The erosion model follows the work of Berchiche et al. [10] and Franc & Riondet [11]. The driving-force for the erosion mechanism is the bubble collapse on the material surface. If the collapses occur repeatedly at the same location, the material yields after a certain time and the erosion process is started. After removal of the hardened layer the erosion process accelerates until a steady material removal rate is reached. The applied model estimates the so-called mean depth of penetration rate (MDPR) under steady material removal conditions. Equation (1) shows the formulation of the erosion rate estimation. The difficulty in modeling erosive effects lies in combining the fluid dynamics with material related aspects. Estimation of flow aggressiveness is critical. Only when the bubble-collapse is aggressive enough the erosion damage will be observed. [1]

(1) $\mathit{MDPR}=NS\text{Δ}\mathit{L}=NSL\left[{\left(\frac{\epsilon \text{'}}{{\epsilon }_U}\right)}^{1/\theta }-1\right]$

Incubation time, Ti, is given in equation (2).

(2) $T_i=\frac{1}{NS}{\left[\frac{{\sigma }_U-{\sigma }_Y}{\sigma -{\sigma }_Y}\right]}^{\frac{1+\theta }{\mathit{n\theta }}}\frac{\left(1+n\right)\left(1+\theta +\mathit{n\theta }\right){\sigma }_Y+\left(1+\theta \right)\left({\sigma }_U-{\sigma }_Y\right)}{\left(1+n\right)\left(1+\theta +\mathit{n\theta }\right){\sigma }_Y+\left(1+\theta \right)\left(\sigma -{\sigma }_{Y)}\right)}$

In equations (1) and (2) N and S represent the impacts per unit area and the size of the impact loads respectively. The rest of the variables are material related parameters and coefficients.

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White in Time Scalar Advection Model as a Tool for Solving Joint Composition PDF Equations: Derivation and Application

V. Sabel'nikov , O. Soulard , in Engineering Turbulence Modelling and Experiments 6, 2005

RANS/EMC solver

In the expression of the PDF equation (4), it has been supposed that the mean density 〈ρ〉, the Favre averaged velocity $\tilde{\mathit{U}}$ the turbulent diffusion coefficient Γ _T and the turbulent mixing frequency 〈ω_c 〉 were known. A RANS solver can be used to compute these quantities. Namely, the Favre averaged continuity and momentum equations are solved and a standard k – ϵ model is used to compute the turbulent stresses, with k the turbulent kinetic energy and ϵ the turbulent dissipation.

(22) $\begin{array}{lll}\mathit{Continuity}\hfill & :\hfill & \frac{\partial 〈\mathit{\rho }〉}{\partial \mathit{t}}+\frac{\partial }{\partial {\mathit{x}}_{\mathit{j}}}\left(〈\mathit{\rho }〉{\tilde{\mathit{U}}}_{\mathit{j}}\right)=0\hfill \\ \mathit{Momentum}\hfill & :\hfill & \frac{\partial 〈\mathit{\rho }〉{\tilde{\mathit{U}}}_{\mathit{i}}}{\partial \mathit{t}}+\frac{\partial }{\partial {\mathit{x}}_{\mathit{j}}}\left(〈\mathit{\rho }〉{\tilde{\mathit{U}}}_{\mathit{j}}{\tilde{\mathit{U}}}_{\mathit{i}}\right)=-\frac{\partial 〈\mathit{\rho }〉}{\partial {\mathit{x}}_{\mathit{j}}}+\frac{\partial {\mathit{\sigma }}_{\mathit{ij}}}{\partial {\mathit{x}}_{\mathit{j}}}\hfill \\ \mathit{Turbulent}\mathit{kinetic}\mathit{energy}\hfill & :\hfill & \frac{\partial 〈\mathit{\rho }〉\mathit{k}}{\partial \mathit{t}}+\frac{\partial }{\partial {\mathit{x}}_{\mathit{j}}}\left(〈\mathit{\rho }〉{\tilde{\mathit{U}}}_{\mathit{j}}\mathit{k}\right)=\frac{\partial }{\partial {\mathit{x}}_{\mathit{j}}}\left(〈\mathit{\rho }〉\frac{{\mathit{\nu }}_{\mathit{t}}}{\mathit{P}{\mathit{r}}_{\mathit{k}}}\frac{\partial \mathit{k}}{\partial {\mathit{x}}_{\mathit{j}}}\right)+{\mathit{P}}_{\mathit{k}}-{\mathit{d}}_{\mathit{k}}\hfill \\ \mathit{Turbulent}\mathit{dissipation}\hfill & :\hfill & \frac{\partial 〈\mathit{\rho }〉\mathrm{ϵ}}{\partial \mathit{t}}+\frac{\partial }{\partial {\mathit{x}}_{\mathit{j}}}\left(〈\mathit{\rho }〉{\tilde{\mathit{U}}}_{\mathit{j}}\mathrm{ϵ}\right)=\frac{\partial }{\partial {\mathit{x}}_{\mathit{j}}}\left(〈\mathit{\rho }〉\frac{{\mathit{\nu }}_{\mathit{t}}}{\mathit{P}{\mathit{r}}_{\mathrm{ϵ}}}\frac{\partial \mathrm{ϵ}}{\partial {\mathit{x}}_{\mathit{j}}}\right)+{\mathit{P}}_{\mathrm{ϵ}}-{\mathit{d}}_{\mathrm{ϵ}}\hfill \end{array}$

σ_ij models the turbulent stresses with an eddy viscosity hypothesis:

(23) ${\mathit{\sigma }}_{\mathit{ij}}=-\frac{2}{3}〈\mathit{\rho }〉\mathit{k}{\mathit{\delta }}_{\mathit{ij}}+〈\mathit{\rho }〉{\mathit{\nu }}_{\mathit{t}}\left(\frac{\partial {\tilde{\mathit{U}}}_{\mathit{i}}}{\partial {\mathit{x}}_{\mathit{j}}}+\frac{\partial {\tilde{\mathit{U}}}_{\mathit{j}}}{\partial {\mathit{x}}_{\mathit{i}}}-\frac{2}{3}\frac{\partial {\tilde{\mathit{U}}}_{\mathit{k}}}{\partial {\mathit{x}}_{\mathit{k}}}{\mathit{\delta }}_{\mathit{ij}}\right)$

P_k and d_k (respectively P_ϵ and d_ϵ ) are the production and dissipation terms of the turbulent kinetic energy (resp. dissipation). Standard expressions are used for these terms, as found for instance in Pope (2000). The eddy viscosity is given by ${\mathit{\nu }}_{\mathit{t}}={\mathit{C}}_{\mathit{\mu }}\frac{{\mathit{k}}^2}{\mathrm{ϵ}}$ . Standard values are chosen for the k – ϵ model constants, as given in Pope (2000).

The statistics of species mass fractions Y_k and total enthalpy h_t are computed with an EMC solver using SPDEs derived with the procedure detailed in section :

(24) $\begin{array}{l}\begin{array}{lll}\mathit{Mass}\mathit{fraction}\hfill & :\hfill & \frac{\partial {\mathit{Y}}_{\mathit{k}}}{\partial \mathit{t}}\mathit{dt}+\left({\tilde{\mathit{U}}}_{\mathit{j}}-\frac{1}{2}\frac{\partial {\mathrm{\Gamma }}_{\mathit{T}}}{\partial {\mathit{x}}_{\mathit{j}}}-\frac{1}{〈\mathit{\rho }〉}\frac{\partial 〈\mathit{\rho }〉}{\partial {\mathit{x}}_{\mathit{j}}}{\mathrm{\Gamma }}_{\mathit{T}}\right)\frac{\partial {\mathit{Y}}_{\mathit{k}}}{\partial {\mathit{x}}_{\mathit{j}}}\mathit{dt}+\sqrt{2{\mathrm{\Gamma }}_{\mathit{T}}}\frac{\partial {\mathit{Y}}_{\mathit{k}}}{\partial {\mathit{x}}_{\mathit{j}}}\circ \mathit{d}{\mathit{W}}_{\mathit{j}}\left(\mathit{t}\right)\hfill \\ \hfill & \hfill & =-{\mathit{\omega }}_{\mathit{k}}\left({\mathit{Y}}_{\mathit{k}}-{\tilde{\mathit{Y}}}_{\mathit{k}}\right)\mathit{dt}+\mathit{S}(\mathit{Y},{\mathit{h}}_{\mathit{t}})\mathit{dt}\hfill \end{array}\\ \begin{array}{lll}\mathit{Total}\mathit{enthalpy}\hfill & :\hfill & \frac{\partial {\mathit{h}}_{\mathit{t}}}{\partial \mathit{t}}\mathit{dt}+\left({\tilde{\mathit{U}}}_{\mathit{j}}\frac{1}{2}\frac{\partial {\mathrm{\Gamma }}_{\mathit{T}}}{\partial {\mathit{x}}_{\mathit{j}}}-\frac{1}{〈\mathit{\rho }〉}\frac{\partial 〈\mathit{\rho }〉}{\partial {\mathit{x}}_{\mathit{j}}}{\mathrm{\Gamma }}_{\mathit{T}}\right)\frac{\partial {\mathit{h}}_{\mathit{t}}}{\partial {\mathit{x}}_{\mathit{j}}}\mathit{dt}+\sqrt{2{\mathrm{\Gamma }}_{\mathit{T}}}\frac{\partial {\mathit{h}}_{\mathit{t}}}{\partial {\mathit{x}}_{\mathit{j}}}\circ \mathit{d}{\mathit{W}}_{\mathit{j}}\left(\mathit{t}\right)\hfill \\ \hfill & \hfill & =-{\mathit{\omega }}_{\mathit{h}}\left({\mathit{h}}_{\mathit{t}}-{\tilde{\mathit{h}}}_{\mathit{t}}\right)\mathit{dt}\hfill \end{array}\end{array}$

In the enthalpy equation, a unity Lewis number assumption has been made and acoustic interactions, viscous dissipation, and body forces were neglected under a low Mach number assumption. In particular, the material derivative of pressure has been neglected. In these equations, the turbulent diffusivity is defined by Γ _T   = ν_t/Sc_t , where Sc_t is a turbulent schmidt number supposed to be of order unity. The mixing frequencies are defined by ${\mathit{\omega }}_{\mathit{k}}={\mathit{\omega }}_{\mathit{h}}={\mathit{\omega }}_{\mathit{c}}={\mathit{C}}_{\mathit{\varphi }}\frac{\mathit{\epsilon }}{\mathit{k}}$ where C_ϕ is a constant supposed to be equal to 0.7. The Favre averaged values of the species mass fraction and total enthalpy are computed by: ${\tilde{\mathit{Y}}}_{\mathit{k}}=\frac{1}{\mathit{N}}\mathrm{\Sigma }{\mathit{Y}}_{\mathit{k}}$ and ${\tilde{\mathit{h}}}_{\mathit{t}}=\frac{1}{\mathit{N}}\mathrm{\Sigma }{\mathit{h}}_{\mathit{t}}$ , where the sums are taken over N realisations of the stochastic fields. (Favre - and not Reynolds - averages are obtained from these sums, because the stochastic fields directly represent the Favre PDF.)

Information is transmitted from the RANS solver to the EMC solver via 〈ρ〉, $\tilde{\mathit{U}}$ , Γ _T and ω_c . The influence of the EMC solver on the RANS solver is achieved through the mean pressure: the Favre averaged temperature is computed from ${\tilde{\mathit{Y}}}_{\mathit{k}}$ , ${\tilde{\mathit{h}}}_{\mathit{t}}$ and $\tilde{\mathit{U}}$ . Then, the mean equation of state is used to compute the mean pressure $〈\mathit{P}〉=〈\mathit{\rho }〉\mathit{r}\tilde{\mathit{T}}$ .

From a numerical point of view, the RANS equations(22) are solved with an explicit Osher procedure and a second order Essentially Non-Oscillatory interpolation. The stochastic equations(24) are solved as described in the previous section.

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Medical Biotechnology and Healthcare

S. Zhou , ... Z. Cui , in Comprehensive Biotechnology (Second Edition), 2011

5.24.3 Cell Proliferation/Death

Chondrocytes are the architects of cartilage. Their survival and proliferation are of key importance to cartilage tissue engineering. Therefore, cell growth in engineered constructs has been actively modeled. In the early study of cartilage tissue engineering conducted by Freed et al., chondrocytes were seeded in polyglycolic acid scaffolds of different thickness and cultured for up to 30 days. Total cell number was measured by DNA assay at different time points. Cell density as a function of culture time was fitted into polynomial equations. This could be regarded as a very simple model for cell growth. The relationships of growth rate versus scaffold thickness, growth rate versus culture time, and growth rate versus cell density were investigated. Cells in thinner scaffolds were found to proliferate more rapidly [3].

Galban and Locke developed a series of biphasic models for oxygen diffusion and cell proliferation in heterogeneous polymer matrix. The first one used a moving boundary approach. This model dealt with two cases. One of them assumed that a layer of cells would form at the outer wall of the constructs and grow toward the center (this may be far from reality, considering the seeding method commonly used and the porous structure of the scaffold). The other case assumed that cells would form thin bands in the direction vertical to the surface of the scaffold. These bands would then grow toward each other until they merge together. Parameters such as diffusivity and growth rate were adjusted to fit Freed's experimental data (as mentioned above) into the model. The second model used a volume-averaging method. The average volume consisted of two phases: the cell colony and the void phase, which included polymer matrix and nutrient fluid. Effective diffusivity and reaction rate were derived as a function of the cell volume fraction, in order to enable a single-averaged continuity equation for the nutrient. Again, results were compared to Freed's experimental data. Both models considered glucose as the only rate-limiting substrate for cell proliferation. However, gradients of glucose were not presented; spatial distribution of cell density was not given, either. The third model was an improvement to the second one. The same volume-averaging method was used. However, the spatial cell variation was presented in this study. Cell population and metabolic product were regarded as inhibiting factors to cell proliferation [4].

Kino-Oka et al. used an automaton approach to model chondrocyte growth in atelocollagen gel. The 3D scaffold was divided into tiny cubes, each of which could accommodate just one cell. When cell division happened, the daughter cell could move to any of the 26 neighboring cubes, provided it was empty. Oxygen was regarded as the rate-limiting factor. The relationship between division rate and oxygen concentration adopted in this study was very similar to the M–M equation used for oxygen consumption rate; that is, when oxygen dropped, division rate decreased in a nonlinear manner. Two initial cell densities (2 and 20 million/ml) were assumed in the model and the result was compared with experimental data. Cell proliferation was more active with lower seeding density. However, after 588   h of culture, the cell growth curve reached a plateau for both cases and the saturated cell density was about 30 million/ml. Steep oxygen gradient was expected across the scaffold. Cell mass was much denser near the surface due to sufficient nutrient supply in this region [7].

Lewis et al. modeled the heterogeneous proliferation of chondrocytes in the same engineered constructs (cylindrical polyethylene glycol terephthalate (PEGT)/polybutylene terephthalate (PBT) scaffolds) as reported previously by Malda et al. [10, 11]. Cell proliferation rate was assumed to be a function of only oxygen. However, unlike the M–M-type relationship between proliferation and oxygen used by Kino-Oka, as mentioned above, a linear relationship was assumed in this study. Another assumption made was that oxygen consumption was entirely due to proliferation. Obviously, this is not true, because even unproliferating chondrocytes in native cartilage consume oxygen. The results were compared to the cell distribution data obtained by Malda et al. [11]. The microelectrode used by Malda, as aforementioned, was employed to measure oxygen distribution. The side and bottom of the construct were blocked so that oxygen diffused only one-dimensionally. Overall, this model matched experimental data well, except for a few discrepancies. First, it predicted lower cell density in the outer periphery on day 3 of culture. This may be explained by the uneven initial cell seeding. Another discrepancy was that on day 14, the oxygen gradient in the deep zone of the construct predicted by the model was steeper than what was measured. The authors argued that this might be caused by the decrease in oxygen consumption rate in the center of the construct [9].

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