LearnChemE

Modeling and Similitude: Screencast

Determines Pi groups using Buckingham Pi theorem to scale force acting on a model to a prototype.

We suggest that after watching this screencast, you list the important points as a way to increase retention.

Important Equations:

$\Pi_1 = f(\Pi_2, \Pi_3, …, \Pi_n)$

This equation is know as the theory of models. It says that each  dimensionless group ($$\Pi$$-term) is a function of the other dimensionless groups. For example, the coefficient of drag of a golf ball is known to be a function of the Reynolds number and the relative roughness of the ball:

$C_D = f\left(Re,\frac{\epsilon}{D}\right)$In this case there are three $$Pi$$-terms: the coefficient of drag, $$C_D$$, the Reynolds number, $$Re$$, and the relative roughness, $$\frac{\epsilon}{D}$$. If we specify the Reynolds number and the relative roughness, then the coefficient of drag is uniquely defined.

The subscripts $$m$$ and $$p$$ represent “model” and “prototype”. For example, to maintain similitude, it is often important to keep the Reynolds number the same between the model and the prototype:

$Re_m = Re_p$ This means

$\frac{\rho_m\,V_m\,L_m}{\mu_m}\,=\,\frac{\rho_p\,V_p\,L_p}{\mu_p}$

Suppose the model is half as long as the prototype:

$L_m\,=\,\frac{1}{2}L_p$

To maintain similitude, the speed of the fluid over the model must be twice as fast:

$V_m\,=\,2V_p$