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Friday, June 19, 2015

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})


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