Figure 7 the net heat transfer in a control volume [5]
The net heat transfer in the x – direction is equal to
\(\left[\left(q_{x}-\frac{\partial q_{x}}{\partial x}\frac{\text{δx}}{2}\right)-\left(q_{x}+\frac{\partial q_{x}}{\partial x}\frac{\text{δx}}{2}\right)\right]\text{δyδz}=-\frac{\partial q_{x}}{\partial x}\text{δxδyδz}\)
The net heat transfer in the y – direction is equal to
\(\left[\left(q_{y}-\frac{\partial q_{y}}{\partial y}\frac{\text{δy}}{2}\right)-\left(q_{y}+\frac{\partial q_{y}}{\partial y}\frac{\text{δy}}{2}\right)\right]\text{δxδz}=-\frac{\partial q_{y}}{\partial x}\text{δxδyδz}\)
The net heat transfer in the z – direction is equal to
\(\left[\left(q_{z}-\frac{\partial q_{z}}{\partial z}\frac{\text{δz}}{2}\right)-\left(q_{z}+\frac{\partial q_{z}}{\partial z}\frac{\text{δz}}{2}\right)\right]\text{δxδy}=-\frac{\partial q_{z}}{\partial z}\text{δxδyδz}\)
Then; the total heat rate per unit volume is the sum of all of the heat flow across the boundaries divided by δxδyδz
\(-\frac{\partial q_{x}}{\partial x}=-\frac{\partial q_{y}}{\partial y}=-\frac{\partial q_{z}}{\partial z}=-div\left(\mathbf{q}\right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.45\right)\)
The heat flux and the temperature gradient can related by Fourier’s law;
\(q_{x}=-k\frac{\partial T}{\partial x}\)\(q_{y}=-k\frac{\partial T}{\partial y}\)\(q_{z}=-k\frac{\partial T}{\partial z}\)
\(\mathbf{q}=-k\ \ grad\ T\)
\(-div\left(\mathbf{q}\right)=div\left(\text{k\ \ grad\ T}\ \right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.46\right)\)
In this way, Energy Equation can be written as below;
\(\rho\frac{\text{DE}}{\text{Dt}}==-div\left(P\mathbf{u}\right)+\left[\frac{\partial\left(u\sigma_{\text{xx}}\right)}{\partial x}+\frac{\left(\text{uτ}_{\text{yx}}\right)}{\partial y}+\frac{\partial\left(\text{uτ}_{\text{zx}}\right)}{\partial z}+\frac{\partial\left(v\sigma_{\text{yy}}\right)}{\partial x}+\frac{\left(\text{vτ}_{\text{xy}}\right)}{\partial y}+\frac{\partial\left(\text{vτ}_{\text{zy}}\right)}{\partial z}+\frac{\partial\left(w\sigma_{\text{zz}}\right)}{\partial x}+\frac{\left(\text{wτ}_{\text{yz}}\right)}{\partial y}+\frac{\partial\left(\text{wτ}_{\text{xz}}\right)}{\partial z}\right]+div\left(\text{k\ \ grad\ T}\ \right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.47\right)\)
  1. Turbulence
  2. What is Turbulence?
Turbulence is a the top level and a leading subject of fluid flow researches and during the last century some of the famous mathematician worked in this specific area like Reynolds, Taylor, Von – Karman, Parbdtl and his PhD student Blasius. Turbulence may be defined as a random, irregular, unpredictable motion in which each quantity of fluid flow properties fluctuates continuously with respect to the time and space [5]. Turbulence leads to increases drag, mixing, energy dissipation and heat transfer beside that it is a 3 – D flow [7]. For example, Figure 8 displays the water jet image visualized using laser-induced fluorescence technique under turbulent flow. It can be seen how the turbulence effect is high on the irregularity of the water distribution. Also, the turbulence is a recommended technique to increases the flame speed which enhance the heat release as illustrated in Figure 9 which is tangential swirl burner. Based on Figure 10 it can be seen that the instantaneous velocity fluctuate about its average value and can be written as indicated below;
\(U=\overset{\overline{}}{u}+\overset{\acute{}}{u\left(t\right)}\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(4.1\right)\)
Where \(\overset{\acute{}}{u\left(t\right)}\) it is the fluctuating velocity
\(\overset{\overline{}}{u}\) it is the time average velocity and can be calculated from
\(\overset{\overline{}}{u}=\frac{1}{t}\int_{0}^{t}\text{U\ dt}\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(4.2\right)\)