Generalized Governing Equations

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The governing equations for natural convection are special cases of the generalized governing equations that were discussed in Chapter 2. Consider a multicomponent system with N components, the governing equations for a stationary reference frame can be expressed as


\frac{D\rho }{Dt}+\rho \nabla \cdot \mathbf{V}=0 \qquad \qquad(1)


\rho \frac{D\mathbf{V}}{Dt}=\rho \mathbf{g}-\nabla p+\nabla \cdot \mathbf{\tau } \qquad \qquad(2)


where the viscous stress tensor, τ, can be determined by using Newton’s law of viscosity [see eq. (1.53)]:


\mathbf{\tau }=2\mu \mathbf{D}-\frac{2}{3}\mu (\nabla \cdot \mathbf{V})\mathbf{I} \qquad \qquad(3)


where D is the rate of strain tensor:


\mathbf{D}=\frac{1}{2}\left[ \nabla \mathbf{V}+{{\left( \nabla \mathbf{V} \right)}^{T}} \right] \qquad \qquad(4)


For natural convection problems, it is often assumed that the fluid is incompressible, except in the first term on the right-hand side of eq. (2); this is referred to as the Boussinesq assumption. Under this assumption, the continuity equation (1) becomes:


\nabla \cdot \mathbf{V}=0 \qquad \qquad(5)


According to eq. (5), the second term on the right-hand side of eq. (3) will be zero. The momentum equation (2) then becomes:


\rho \frac{D\mathbf{V}}{Dt}=\rho \mathbf{g}-\nabla p+\nabla \cdot (\mu \nabla \mathbf{V}) \qquad \qquad(6)


where the left-hand side is the inertial term; the three terms on the right-hand side represent body force per unit volume, pressure force per unit volume, and viscous force per unit volume, respectively.

The density of a mixture is a function of its temperature and the mass fractions of its species. It can be expanded using a Taylor’s series near the vicinity of a reference point ({{T}_{\infty }},\text{ }{{\omega }_{1,\infty }},\text{ }{{\omega }_{2,\infty }},\text{ }\cdots {{\omega }_{N,\infty }}):


\rho ={{\rho }_{\infty }}+\frac{\partial \rho }{\partial T}(T-{{T}_{\infty }})+\sum\limits_{i=1}^{N}{\frac{\partial \rho }{\partial {{\omega }_{i}}}({{\omega }_{i}}-}{{\omega }_{i,\infty }})+\cdots


where {{\rho }_{\infty }} is density at the reference point. By defining the coefficient of thermal expansion, β, and composition coefficient of volume expansion, βm,i, as follows:


\beta \text{ }=-\frac{1}{{{\rho }_{\infty }}}{{\left( \frac{\partial \rho }{\partial T} \right)}_{p}} \qquad \qquad(7)
{{\beta }_{m,i}}\text{ }=-\frac{1}{{{\rho }_{\infty }}}{{\left( \frac{\partial \rho }{\partial {{\omega }_{i}}} \right)}_{p}} \qquad \qquad(8)


and neglecting the higher order terms in the Taylor’s series expansion, one obtains:


\rho \approx {{\rho }_{\infty }}-{{\rho }_{\infty }}\beta (T-{{T}_{\infty }})-{{\rho }_{\infty }}\sum\limits_{i=1}^{N}{{{\beta }_{m}}({{\omega }_{i}}-}{{\omega }_{i,\infty }}) \qquad \qquad(9)


which is valid only if \beta (T-{{T}_{\infty }})\ll 1\text{ and }{{\beta }_{m}}({{\omega }_{i}}-{{\omega }_{i,\infty }})\ll 1. Substituting eq. (9) into eq. (6), the momentum equation for natural convection is obtained:


\rho \frac{D\mathbf{V}}{Dt}=\left( -\nabla p+{{\rho }_{\infty }}\mathbf{g} \right)-{{\rho }_{\infty }}\mathbf{g}\beta (T-{{T}_{\infty }})


-{{\rho }_{\infty }}\mathbf{g}\sum\limits_{i=1}^{N}{{{\beta }_{m,i}}({{\omega }_{i}}-}{{\omega }_{i,\infty }})+\nabla \cdot (\mu \nabla \mathbf{V}) \qquad \qquad(10)


which is a generalized momentum equation because the effects of buoyancy forces due to both temperature and composition variations are considered. If it is assumed that the Dufour effect is negligible and the fluid is incompressible, the energy equation is:


\rho {{c}_{p}}\frac{DT}{Dt}=\nabla \cdot (k\nabla T)+{q}'''+\nabla \mathbf{V}:\mathbf{\tau } \qquad \qquad(11)

The conservation of species mass in terms of mass fraction for the ith species can be expressed as


\rho \frac{D{{\omega }_{i}}}{Dt}=-\nabla \cdot {{\mathbf{J}}_{i}}+{{{\dot{m}}'''}_{i}} \qquad \qquad(12)

For a binary system of A and B, one can apply Fick’s law to eq. (8) to obtain:


\rho \frac{D{{\omega }_{A}}}{Dt}=-\rho \nabla \cdot ({{D}_{AB}}\nabla {{\omega }_{A}})+{{{\dot{m}}'''}_{A}} \qquad \qquad(13)

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