Figure 12 Control volume of Von-Karman Integral Momentum Equation [10]
Mass flow rate entering the c.v. upstream (ab):
\begin{equation} {\dot{m}}_{1}=\int_{0}^{\delta}\text{ρudy}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(7.1)\nonumber \\ \end{equation}
Mass flow rate leaving the c.v. downstream (dc):
\begin{equation} {\dot{m}}_{2}=\int_{0}^{\delta}\text{ρudy}+\frac{d}{\text{dx}}\left[\int_{0}^{\delta}\text{ρudy}\right]\text{dx\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ (7.2)\nonumber \\ \end{equation}
The net mass flow rate is
\begin{equation} {\dot{m}}_{2}-{\dot{m}}_{1}=\int_{0}^{\delta}\text{ρudy}+\frac{d}{\text{dx}}\left[\int_{0}^{\delta}\text{ρudy}\right]dx-\int_{0}^{\delta}\text{ρudy}\nonumber \\ \end{equation}\begin{equation} {\dot{m}}_{2}-{\dot{m}}_{1}=\frac{d}{\text{dx}}\left[\int_{0}^{\delta}\text{ρudy}\right]\text{dx\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(7.3)\nonumber \\ \end{equation}
Momentum flux entering ab =
\(\int_{0}^{\delta}{\rho u^{2}\text{dy}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (7.4)\)
Momentum flux entering cd =
\(\int_{0}^{\delta}{\rho u^{2}\text{dy}}+\frac{d}{\text{dx}}\left[\int_{0}^{\delta}{\rho u^{2}\text{dy}}\right]\text{dx\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.5\right)\)
Momentum flux entering through bc is given by
\(U_{\infty}*\left({\dot{m}}_{2}-{\dot{m}}_{1}\right)=U_{\infty}\frac{d}{\text{dx}}\left[\int_{0}^{\delta}\text{ρudy}\right]\text{dx\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.6\right)\)
Now the total drag force must be equal to the rate of change of momentum in flow and out flow;
\begin{equation} -F_{D}=\left.\ \text{Momentum\ Flux}\right\rceil_{\text{out}}-\left.\ \text{Momentum\ Flux}\right\rceil_{\text{in}}\nonumber \\ \end{equation}
\(-F_{D}=\int_{0}^{\delta}{\rho u^{2}\text{dy}}+\frac{d}{\text{dx}}\left[\int_{0}^{\delta}{\rho u^{2}\text{dy}}\right]\text{dx}-\int_{0}^{\delta}{\rho u^{2}\text{dy}}-U_{\infty}\frac{d}{\text{dx}}\left[\int_{0}^{\delta}\text{ρudy}\right]\text{dx}\)
\(-F_{D}=\frac{d}{\text{dx}}\left[\int_{0}^{\delta}{\rho u^{2}\text{dy}}\right]dx-U_{\infty}\frac{d}{\text{dx}}\left[\int_{0}^{\delta}\text{ρudy}\right]\text{dx}\)
\(-F_{D}=\rho\frac{d}{\text{dx}}\left[\int_{0}^{\delta}{[u}^{2}-uU_{\infty}]dy\right]\text{dx}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.7\right)\)
Let us multiplying and divided Eq. (7.7) by\(\frac{{U_{\infty}}^{2}}{{U_{\infty}}^{2}}\)
\(-\tau_{w}dx=\rho\frac{d}{\text{dx}}\left[\int_{0}^{\delta}{\frac{{U_{\infty}}^{2}}{{U_{\infty}}^{2}}[u}^{2}-uU_{\infty}]dy\right]\text{dx}\)
\(\tau_{w}=\rho{U_{\infty}}^{2}\frac{d}{\text{dx}}\left[\int_{0}^{\delta}{\frac{u}{U_{\infty}}\left(1-\frac{u}{U_{\infty}}\right)}\text{dy}\right]\text{\ \ }\)
\(\tau_{w}=\rho{U_{\infty}}^{2}\frac{d}{\text{dx}}\left[\delta\int_{0}^{1}{\overset{\overline{}}{U}\left(1-\overset{\overline{}}{U}\right)}d\overset{\overline{}}{y}\right]\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.8\right)\)
The above expression is the Von-Karman M.I.E. valid for laminar and turbulent shear layer. It has the following form with some manipulation;
Let\(I=\frac{\theta}{\delta}=\int_{0}^{1}{\overset{\overline{}}{U}\left(1-\overset{\overline{}}{U}\right)d\overset{\overline{}}{y}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.9\right)\)
\(\tau_{w}=\rho{U_{\infty}}^{2}\frac{d}{\text{dx}}\left[\delta\int_{0}^{1}{\overset{\overline{}}{U}\left(1-\overset{\overline{}}{U}\right)}d\overset{\overline{}}{y}\right]\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.10\right)\)
\(\tau_{w}=\rho{U_{\infty}}^{2}\frac{d\theta}{\text{dx}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.11\right)\)
\(\tau_{w}=\rho{U_{\infty}}^{2}I\frac{\text{dδ}}{\text{dx}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.12\right)\)
Laminar Boundary Layer
There are many Laminar velocity Profiles like the inserted below;
\(\overset{\overline{}}{U}=\overset{\overline{}}{y}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.13\right)\)
\(\overset{\overline{}}{U}=\frac{3}{2}\overset{\overline{}}{y}-\frac{1}{2}{\overset{\overline{}}{y}}^{3}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.14\right)\)
\(\overset{\overline{}}{U}=2\overset{\overline{}}{y}-{\overset{\overline{}}{y}}^{2}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.15\right)\)
\(\overset{\overline{}}{U}=\sin{\frac{\pi}{2}\overset{\overline{}}{y}\ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.16\right)\)
skin friction coefficient
Firstly, an expression of skin friction coefficient will be developed
\(C_{f}=\frac{\tau_{w}}{\frac{1}{2}\rho{U_{\infty}}^{2}}\rightarrow\tau_{w}=\frac{1}{2}\rho{U_{\infty}}^{2}*C_{f}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.17\right)\)
From Von-Karman IME:\(\tau_{w}=\rho{U_{\infty}}^{2}I\frac{\text{dδ}}{\text{dx}}\text{\ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.12\right)\)
Equate the above Eq. \(\left(7.12\right)\) with Eq.\(\left(7.17\right)\)two expression of the shear stress, we find out;
\(\frac{1}{2}\rho{U_{\infty}}^{2}*C_{f}=\rho{U_{\infty}}^{2}I\frac{\text{dδ}}{\text{dx}}\)
\(C_{f}=2*I*\frac{\text{dδ}}{\text{dx}}=2\frac{\text{dθ}}{\text{dx}}\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.13\right)\)
Rate of growth of L.B.L. over a flat plate
Now we will develop an expression of the rate of growth on flat plate
\(C_{f}=\frac{\tau_{w}}{\frac{1}{2}\rho{U_{\infty}}^{2}}\text{\ \ \ \ \ }\left(7.17\right)\)Also, From Newton’s law:\(\tau_{w}=\mu\frac{\partial u}{\partial y}\text{\ \ \ \ \ }\left(1.1\right)\)
\(C_{f}=\frac{\mu\left.\ \frac{\text{du}}{\text{dy}}\right\rceil_{w}}{\frac{1}{2}\rho{U_{\infty}}^{2}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ }\left(7.18\right)\)
\(C_{f}=\frac{2\mu}{\rho{U_{\infty}}^{2}}\left.\ \frac{\text{du}}{\text{dy}}\right\rceil_{y=0}\)
Since\(\overset{\overline{}}{y}=\frac{y}{\delta}\text{\ \ \ \ \ and\ \ \ \ }\overset{\overline{}}{U}=\frac{u}{U_{\infty}}\)we will get;
\(\left.\ \frac{\text{du}}{\text{dy}}\right\rceil_{y=0}=\left.\ \frac{d}{\text{dδ}\overset{\overline{}}{y}}\left(U_{\infty}\overset{\overline{}}{U}\right)\right\rceil_{y=0}=\frac{U_{\infty}}{\delta}\left.\ \frac{d\overset{\overline{}}{U}}{d\overset{\overline{}}{y}}\right\rceil_{y=0}\)
\(C_{f}=\frac{2\mu}{\rho{U_{\infty}}^{2}}*\frac{U_{\infty}}{\delta}\left.\ \frac{d\overset{\overline{}}{U}}{d\overset{\overline{}}{y}}\right\rceil_{y=0}=\frac{2\mu}{\rho\text{δU}_{\infty}}\left.\ \frac{d\overset{\overline{}}{U}}{d\overset{\overline{}}{y}}\right\rceil_{y=0}\)
Let us assume dimensionless parameter of velocity profile\(\left.\ D_{o}=\frac{d\overset{\overline{}}{U}}{d\overset{\overline{}}{y}}\right\rceil_{y=0}\)
\(C_{f}=\frac{2\mu D_{o}}{\rho\text{δU}_{\infty}}\text{\ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ }\left(7.18\right)\)
Now let us equalize the Eq. \(\left(7.13\right)\) with Eq.\(\left(7.18\right)\) ;\(\frac{2\mu D_{o}}{\rho\text{δU}_{\infty}}=2*I*\frac{\text{dδ}}{\text{dx}}\)
This is 1st ODE can be solved simply using the separation of variable method
\(\frac{\text{dδ}}{\text{dx}}=\frac{\mu D_{o}}{\rho\text{δU}_{\infty}I}\)
\(\int{\delta\text{dδ}}=\int\frac{\mu D_{o}}{\rho U_{\infty}I}\text{dx}\rightarrow\frac{\delta^{2}}{2}=\frac{\mu D_{o}}{\rho U_{\infty}I}x+c\)
\(at\ x=0,\ \delta=0\rightarrow c=0\)
\(\frac{\delta^{2}}{2}=\frac{\mu D_{o}}{\rho U_{\infty}I}x\) but\(Re=\frac{\rho U_{\infty}x}{\mu}\rightarrow\frac{\rho U_{\infty}}{\mu}=\frac{\text{Re}}{x}\)
\(\frac{\delta^{2}}{2}=\frac{x*D_{o}}{I*\text{Re}}x\rightarrow\delta=\sqrt{\frac{2D_{o}}{I}}\frac{x}{\sqrt{\text{Re}_{x}}}\)
\(\delta=\sqrt{\frac{2D_{o}}{I}}\frac{x}{\sqrt{\text{Re}_{x}}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.19\right)\)
Drag coefficient for flat plate
The drag force is the component of force on a body acting parallel to the direction of motion.
\(F_{D}=w\int_{0}^{L}{\tau_{w}\text{dx}}\) From one surface
For two upper and lower surface;
\(F_{D}=2*\left[w\int_{0}^{L}{\tau_{w}\text{dx}}\right]\)
\(F_{D}=2wL\tau_{w}\)
The drag coefficient is\(C_{D}=\frac{F_{D}}{\frac{1}{2}\rho{U_{\infty}}^{2}A}=\frac{2F_{D}}{\rho{U_{\infty}}^{2}A}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.20\right)\)
\(C_{f}=\frac{2\mu D_{o}}{\rho\text{δU}_{\infty}}\text{\ \ \ \ \ \ }\text{\ \ }\left(7.18\right)\)but \(\delta=\sqrt{\frac{2D_{o}}{I}}\frac{x}{\sqrt{\text{Re}_{x}}}\)
\(C_{f}=\frac{2\mu D_{o}}{\rho U_{\infty}*\sqrt{\frac{2D_{o}}{I}}\text{\ \ }\frac{x}{\sqrt{\text{Re}_{x}}}}=\frac{2\mu D_{o}}{\rho U_{\infty}x*\sqrt{\frac{2D_{o}}{I}}\text{\ \ }\frac{1}{\sqrt{\text{Re}_{x}}}}\)
\(C_{f}=\sqrt{\frac{{2D}_{o}I}{\text{Re}_{x}}}\text{\ \ \ \ \ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ \ }\left(7.21\right)\)
\(\overset{\overline{}}{C_{f}}=\frac{1}{L}\ \int_{0}^{L}{{C_{f}}_{x}\text{dx}}=\overset{\overline{}}{C_{f}}=\frac{1}{L}\ \int_{0}^{L}{\sqrt{\frac{{2D}_{o}I}{\text{Re}_{x}}}\text{dx}}=\frac{1}{L}\sqrt{\frac{{2D}_{o}\text{Iμ}}{\rho U_{\infty}}\ }\int_{0}^{L}\frac{\text{dx}}{\sqrt{x}}=\frac{2}{L}\sqrt{\frac{{2D}_{o}\text{Iμ}}{\rho U_{\infty}}\ }\sqrt{L}\)
\(C_{D}=2\sqrt{\frac{{2D}_{o}\text{Iμ}}{\rho U_{\infty}L}\ }=\sqrt{\frac{{8D}_{o}I}{\text{Re}_{L}}}=2\sqrt{2}\ \sqrt{\frac{D_{o}I}{\text{Re}_{L}}}\)
\(C_{D}=2\ \sqrt{\frac{{2D}_{o}I}{\text{Re}_{L}}}\text{\ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ \ }\left(7.22\right)\)
\({\therefore C}_{D}=2C_{f}\)