The medium residence time can be described as a quotient of the free channel volume and the volume throughput.

$$\bar{t} = \frac{V_{free}}{\dot{V}} \tag{1}$$

The substitution of both values by

$$\dot{V} \sim b h D N \text{ and } V_{free} \sim b h L \tag{2}$$

and the formation of the quotient for the model design and for the main design leads to Equation 3.

$$\frac{\bar{t}}{\bar{t}_0} = \frac{L D_0 N_0}{L_0 D N} = \left(\frac{D}{D_0}\right)^{\psi+\chi} \tag{3}$$

To be able to comply with the requirement of a constant residence time of the material in the model design as well as in the main design, the equation

$$\chi = -\omega \tag{4}$$

has to apply. This means that at a constant screw length to diameter ratio (ω = 0) both machines are operated with an identical screw speed.