Question 10.12: Find the minor loss due to a medical stopcock.

Again, we must design an appropriate experiment. Clearly, because the pipe-flow equation is but one scalar equation, we can solve for only one unknown, the K_{sc} (stopcock) of interest. The experiment should thus enable us to measure all other quantities and, in particular, to simplify measurement and interpretation. Recalling the general formula

let us consider a setup similar to that in Fig. 10.16, except with a stopcock in place of the pump. Hence, ensuring a laminar flow (\alpha =2), constant cross-sectional areas and thus \overline{\nu }_{1}=\overline{\nu }_{2}=\overline{\nu }, and a horizontal system, we are left with

              \frac{p_{1}}{\rho }-\frac{p_{2}}{\rho }=f\left(Re,\frac{e}{D} \right)\left(\frac{L_{1}}{D} \right)\left(\frac{\overline{\nu }^{2} }{2} \right)+f\left(Re,\frac{e}{D} \right)\left(\frac{L_{2}}{D} \right)\left(\frac{\overline{\nu }^{2} }{2} \right)+h_{m},
or
                            h_{m}=\frac{p_{1}-p_{2}}{\rho }-\frac{64}{Re}\left(\frac{L_{1}+L_{2}}{D} \right)\left(\frac{\overline{\nu }^{2} }{2} \right),
with
                                                      Re=\frac{\rho \overline{\nu }D }{\mu }.

Hence, to determine the loss coefficient, we must know \mu and \rho for our fluid, determine L_{1}, L_{2}, and D for the tubing, and measure p_{1} and p_{2}. Most impor-tantly, however, we must ensure that all assumptions are satisfied by the actual flow, as, for example, that it is indeed steady, incompressible, and laminar.