## Method 1. Write out each relevant parameters in terms of $MLT\theta Q$, corresponding to mass, length, time, temperature, and charge, respectively 2. Choose $k$ repeaters where $k$ is the number of fundamental parameters 3. Show that they cannot form a $\pi$ group by showing that $a=b=c=0$ in an equation similar to the following $ \begin{bmatrix} g^{a} & d_{e}^{b} & \rho^{c} \end{bmatrix} = [1] $ $ \left( \frac{L}{T^{2}} \right)^{a}(L)^{b}\left( \frac{M}{L^{3}} \right)^{c} = M^{0}L^{0}T^{0} $ 4. Then solve an equivalent equation except including each non-repeater parameter to form a $\pi$ group $ \begin{bmatrix} \gamma & \rho^{a} & g^{b} & d_{e}^{c} \end{bmatrix} $ $ \frac{M}{T^{2}} \left( \frac{M}{L^{3}} \right)^{a}\left( \frac{L}{T^{2}} \right)^{b}(L)^{c} = M^{0}L^{0}T^{0}\theta^{0} $ 5. Check that the $\pi$ group is dimensionless $ \pi_{1} = \frac{\gamma}{\rho gd_{e}^{2}} $ ## Matrix Version You can automate this process with matrices. Suppose we have the following parameters: | $h$ | $U$ | ${} \rho {}$ | $\mu$ | $D$ | ${} k_{f} {}$ | ${} c_{p} {}$ | ${} k_{s} {}$ | | ----------------------------- | ------------------- | ----------------------- | -------------------- | --- | ------------------------------ | --------------------------------- | ------------------------------ | | ${} \frac{M}{T^{3}\theta} {}$ | ${} \frac{L}{T} {}$ | ${} \frac{M}{L^{3}} {}$ | ${} \frac{M}{LT} {}$ | $L$ | ${} \frac{ML}{T^{3}\theta} {}$ | ${} \frac{L^{2}}{T^{2}\theta} {}$ | $\frac{ML}{T^{3}\theta}$ | We construct a dimensional matrix by forming column vectors for each repeater according to the exponents of fundamental units: $\begin{bmatrix}M \\ L \\ T \\ \theta\end{bmatrix}$ $\rho = \left[ \frac{M}{L^{3}} \right] = \begin{bmatrix}1 \\ -3 \\ 0 \\ 0\end{bmatrix}$ $D = \begin{bmatrix}L\end{bmatrix} = \begin{bmatrix}0 \\ 1 \\ 0 \\ 0\end{bmatrix}$ $c_{p} = \left[ \frac{L^{2}}{T^{2}\theta} \right] = \begin{bmatrix}0 \\ 2 \\ -2 \\ -1\end{bmatrix}$ $k_{s}= \left[ \frac{ML}{T^{3}\theta} \right] = \begin{bmatrix}1 \\ 1 \\ -3 \\ -1\end{bmatrix}$ $A = \begin{bmatrix}1 & 0 & 0 & 1 \\ -3 & 1 & 2 & 1 \\ 0 & 0 & -2 & -3 \\ 0 & 0 & -1 & -1\end{bmatrix}$ If the determinant of A is nonzero, then there is no nonzero solution and therefore the parameters cannot form a $\pi$ group and are viable repeaters. ``` % rho, D, cp, ks repeaters A = [1 0 0 1;… -3 1 2 1;… 0 0 -2 -3;… 0 0 -1 -1;] fprintf("Determinant of A = %.f\n", det(A)) ``` Now we follow the same process to make column vectors for the remaining parameters that we would like to nondimensionalize: ``` h = [1; 0; -3; -1]; u = [0; 1; -1; 0]; mu = [1; -1; -1; 0]; kf = [1; 1; -3; -1]; ``` And we can solve the system of linear equations to get our final nondimensionalized parameters: ``` H = linsolve(A, -h) % = [0; 1; 0; -1] U = linsolve(A, -u) % = [1; 1; 1; -1] Mu = linsolve(A, -mu) % = [0; 0; 1; -1] Kf = linsolve(A, -kf) % = [0; 0; 0; -1] ``` $\pi_{1} = h' = \frac{hD}{k_{s}}$ $\pi_{2} = u'= \frac{u\rho Dc_{p}}{k_{s}}$ $\pi_{3} = \mu'=\frac{\mu c_{p}}{k_{s}}$ $\pi_{4}=k_{f}'=\frac{k_{f}}{k_{s}}$ Relationships between dimensionless parameters are universal! ## Units | Name | Symbol | Equation | SI Units | Dimension | | ---------------------------------------------- | -------------- | ---------------------------------------------------------------------------------------------------------- | --------------------------------- | --------------------------------------- | | Velocity | $v$ | $v =\dot{x}$ | ${} \mathrm{m / s} {}$ | ${} L / T {}$ | | Acceleration | $g$ | ${} a = \dot{v} {}$ | ${} \mathrm{m / s^{2}} {}$ | ${} \mathrm{L / T^{2}} {}$ | | Force | $F$ | ${} F = ma {}$ | ${} \mathrm{N} {}$ | ${} \mathrm{ML / T^{2}} {}$ | | Energy / Work | $W$ | ${} W = \int Fdx {}$ | ${} \mathrm{J} {}$ | $\mathrm{ML^{2} / T^{2}}$ | | Power | $P$ | ${} P = \dot{W} {}$ | ${} \mathrm{W} {}$ | ${} \mathrm{ML^{2} / T^{3}} {}$ | | Density | ${} \rho {}$ | ${} \rho = m /V {}$ | ${} \text{kg} / \text{m}^{3} {}$ | ${} \mathrm{M / L^{3}} {}$ | | Viscosity | $\mu$ | ${} \tau_{xy}=\mu \left( \frac{ \partial u }{ \partial y } + \frac{ \partial v }{ \partial x } \right) {}$ | ${} \mathrm{Pa \cdot s} {}$ | ${} \mathrm{\frac{M}{LT}} {}$ | | Kinematic Viscosity / <br>Momentum diffusivity | $\nu$ | ${} \nu = \mu / \rho {}$ | ${} \mathrm{m^{2} / s} {}$ | $\mathrm{L^{2} / T}$ | | Thermal Diffusivity | ${} \alpha {}$ | ${} \alpha = k / \rho {}$ | ${} \mathrm{m^{2} / s} {}$ | $\mathrm{L^{2} / T}$ | | Heat Transfer Coefficient | $h$ | ${} R = \frac{1}{hA} {}$ | ${} \mathrm{\frac{W}{m^{2}K}} {}$ | ${} \mathrm{\frac{M}{T^{3}\theta}} {}$ | | Thermal Conductivity | $k$ | ${} R = \frac{L}{kA} {}$ | ${} \mathrm{\frac{W}{mK}} {}$ | ${} \mathrm{\frac{ML}{T^{3}\theta}} {}$ | | Specific Heat Capacity | $c$ | ${} \Delta U = mc\Delta T {}$ | ${} \mathrm{\frac{J}{kg~K}} {}$ | $\mathrm{\frac{L^{2}}{T^{2}\theta}}$ | ## Important Dimensionless Numbers | Parameter | Definition | Qualitative ratio | Importance | | :------------------: | :------------------------------------------------------: | :---------------------------------------------------------: | :--------------------: | | Reynolds number | $\mathrm{Re} = \frac{\rho U L}{\mu}$ | $\frac{\text{Inertia}}{\text{Viscosity}}$ | Always | | Nusselt Number | ${} \mathrm{Nu} = \frac{hD}{k} {}$ | ${} \frac{ \text{Convection}}{\text{Conduction}} {}$ | Convection | | Prandtl number | $\mathrm{Pr} = \frac{C_p \mu}{k}$ | $\frac{\text{Dissipation}}{\text{Conduction}}$ | Heat convection | | Rayleigh Number | $\mathrm{Ra} = \mathrm{Gr} \times \mathrm{\mathrm{Pr}}$ | $\frac{\text{diffusion time}}{\text{convection time}}$ | Natural Convection | | Grashof number | ${} \mathrm{Gr} = \frac{g\beta \Delta T L^3 }{\nu^2} {}$ | $\frac{\text{Buoyancy}}{\text{Viscosity}}$ | Natural convection | | Mach number | $\mathrm{Ma} = \frac{U}{A}$ | $\frac{\text{Flow speed}}{\text{Sound speed}}$ | Compressible flow | | Froude number | $\mathrm{Fr} = \frac{U^2}{gL}$ | $\frac{\text{Inertia}}{\text{Gravity}}$ | Free-surface flow | | Weber number | $\mathrm{We} = \frac{\rho U^2 L}{\gamma}$ | $\frac{\text{Inertia}}{\text{Surface tension}}$ | Free-surface flow | | Cavitation number | $\mathrm{Ca} = \frac{p - p_v}{\rho U^2}$ | $\frac{\text{Pressure}}{\text{Inertia}}$ | Cavitation | | Eckert number | $\mathrm{Ec} = \frac{U^2}{C_p T_o}$ | $\frac{\text{Kinetic energy}}{\text{Enthalpy}}$ | Dissipation | | Specific-heat ratio | $\gamma = \frac{C_p}{C_v}$ | $\frac{\text{Enthalpy}}{\text{Internal energy}}$ | Compressible flow | | Strouhal number | $\mathrm{St} = \frac{\omega L}{U}$ | $\frac{\text{Oscillation}}{\text{Mean speed}}$ | Oscillating flow | | Roughness ratio | $\frac{\varepsilon}{L}$ | $\frac{\text{Wall roughness}}{\text{Body length}}$ | Turbulent, rough walls | | Temperature ratio | $\frac{T_w}{T_o}$ | $\frac{\text{Wall temperature}}{\text{Stream temperature}}$ | Heat transfer | | Pressure coefficient | $C_p = \frac{p - p_\infty}{1/2 \rho U^2}$ | $\frac{\text{Static pressure}}{\text{Dynamic pressure}}$ | Fluid dynamics | | Lift coefficient | $C_L = \frac{L}{1/2 \rho U^2 A}$ | $\frac{\text{Lift force}}{\text{Dynamic force}}$ | Fluid dynamics | | Drag coefficient | $C_D = \frac{D}{1/2 \rho U^2 A}$ | $\frac{\text{Lift force}}{\text{Dynamic force}}$ | Fluid dynamics |