Abstract
Our interest now is to study chemical reactions in the presence of fluid motion so that fluid mixing effects can be incorporated in the determination of the product composition. In the study of transport phenomena in moving fluids, the fundamental laws of motion (conservation of mass and Newton’s second law) and energy (first law of thermodynamics) are applied to an elemental fluid known as the control volume (CV).
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- 1.
The N–S equations have been derived in many textbooks. Interested reader may see Ref. [30], for example.
- 2.
Turbulent stresses arise out of Reynolds averaging of instantaneous forms of NS Equations 6.1 and 6.2. The instantaneous velocities and pressure are given the symbols \(\hat{u}\) and \(\hat{p}\). The same will apply to any other scalar, such as h or \(\omega \). Thus, any instantaneous quantity \(\hat{\Phi }\), say, is written as \(\hat{\Phi } = \Phi + \Phi ^{'}\) where \(\Phi ^{'}\) represents fluctuation (positive or negative) over time-mean value \(\Phi \). Time-averaging thus implies
$$ \overline{\hat{\Phi }} = \frac{1}{T \rightarrow \infty }\,\int _{0}^{T}\,\hat{\Phi }\, dt = \Phi + \frac{1}{T \rightarrow \infty }\,\int _{0}^{T}\,\Phi ^{'}\, dt = \Phi \,\,.$$Thus, \(\overline{\Phi ^{'}} = 0\). One important consequence of this definition is that the time-averaged product of two instantaneous quantities are given by
$$\overline{\hat{\Phi _{1}}\,\hat{\Phi _{2}}} = \Phi _{1}\,\Phi _{2} + \overline{\Phi ^{'}_{1}\,\Phi ^{'}_{2}}\,\,.$$Such product terms appear in the convective terms \( {\partial (\rho _{m}\,u_{j}\, u_{i})} / {\partial x_{j}} \) of the NS equations. They give rise to what are called turbulent stresses \(\tau ^{t}_{ij}\), in analogy with the laminar stresses \(\tau ^{l}_{ij}\).
- 3.
Body forces \(B_{i}\) are to be specified externally when they are present.
- 4.
The mass diffusivity is defined only for a binary mixture of two fluids a and b as \(D_{ab}\). Values of \(D_{ab}\) are given in Appendix F. In multicomponent gaseous mixtures, however, diffusivities for pairs of species are taken to be nearly equal and a single symbol D suffices for all species. Incidentally, in turbulent flows, this assumption of equal (effective) diffusivities holds even greater validity.
- 5.
It is also called the neighboring phase.
- 6.
In the chapters to follow, we shall encounter different conserved properties. A conserved property is one whose transport equation has a zero source. In combustion problems, \(\Psi \) is formed from the combinations of \(\omega _{k}\) and/or \(h_{m}\). However, \(\eta _{\alpha }\) is always a conserved property. Similarly, in an inert mixture (air + water vapor, for example), \(\omega _{vap}\) and \(\omega _{air}\) will be conserved properties .
- 7.
In mass transfer literature, B is also called a dimensionless driving force.
- 8.
To a first approximation, thickness \(\delta _{\Phi }\) may be taken to be the value of y where \(( \Phi - \Phi _{w}) / (\Phi _{\infty } - \Phi _{w}) \simeq 0.99\).
- 9.
Note that \(\text{ ln }\,(1 + B) \rightarrow B\) as \(B \rightarrow 0\), irrespective of the sign of B.
- 10.
Body forces \(B_{i}\) are ignored here.
- 11.
The transport equation is derived as follows: First, Eq. 6.83 is multiplied by \(\hat{u}_{i}\). The resulting equation is then time-averaged to yield an equation with dependent variable (E + k), where mean kinetic energy E \(= V^{2}\)/2. Next, Eq. 6.2 is multiplied by the time-averaged mean velocity \(u_{i}\) to yield an equation with dependent variable E. Subtracting this equation from the first equation yields an equation for k. This equation contains terms that need modeling. Equation 6.92 represents this modeled form, with \(\rho _{m}\,\epsilon \) representing the turbulent counterpart of \(\mu \,\Phi _{v}\).
- 12.
In modeling turbulent flows, the main governing idea is that turbulent kinetic energy generation (G) is maximum at large integral length scales denoted by l \(= l_{int}\). This energy generation breaks the turbulent eddies to smaller length scales, while the energy is simultaneously convected and diffused. The process continues down to ever smaller scales, until the smallest scale is reached. This smallest scale is called the Kolmogorov length scale l \(= l_{\epsilon }\). In this range of smallest scales, through the action of fluid viscosity, all fluctuations are essentially killed, resulting in a homogeneous and isotropic turbulence structure. This annihilation is associated with turbulent kinetic energy dissipation to heat. Thus, in turbulent flows, energy generation is directly proportional to \(\mu _{t}\) and hence, to l, whereas energy dissipation is associated with fluid viscosity \(\mu \) and is inversely proportional to l.
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Date, A.W. (2020). Derivation of Transport Equations. In: Analytic Combustion. Springer, Singapore. https://doi.org/10.1007/978-981-15-1853-9_6
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DOI: https://doi.org/10.1007/978-981-15-1853-9_6
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