The Equation of Continuity — Question 10

Question 10

A moving control surface counts relative crossing speed

A uniform incompressible stream has density $\rho$, area $A$ and constant laboratory speed $v$ to the right. A control region extends from a fixed left plane to a right plane moving right at speed $u$, with $0<u<v$. There is no flux through its side boundary.

1. Find the mass inflow through the fixed plane, outflow through the moving plane, and mass accumulation inside the region.

2. Evaluate for $\rho=1000\,\mathrm{kg/m^3}$, $A=0.0020\,\mathrm{m^2}$, $v=3.0\,\mathrm{m/s}$ and $u=1.0\,\mathrm{m/s}$. What error results from using laboratory speed at both planes?

Question 10 - Solution

A moving control surface counts relative crossing speed

(a) Derivation and calculation

Flux is counted relative to each control surface:$$\boxed{\dot m_{\mathrm{in}}=\rho Av},\qquad\boxed{\dot m_{\mathrm{out}}=\rho A(v-u)}.$$$$\boxed{\frac{dM_{\mathrm{CV}}}{dt}=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}}=\rho Au}.$$$$\frac{dV_{\mathrm{CV}}}{dt}=Au\Rightarrow\frac{d(\rho V_{\mathrm{CV}})}{dt}=\rho Au.$$

(b) Result and interpretation

$$\boxed{\dot m_{\mathrm{in}}=1000(.00200)(3.0)=6.00\,\mathrm{kg/s}}.$$$$\boxed{\dot m_{\mathrm{out}}=1000(.00200)(3.0-1.0)=4.00\,\mathrm{kg/s}}.$$$$\boxed{dM_{\mathrm{CV}}/dt=2.00\,\mathrm{kg/s}}.$$Using laboratory speed at both planes falsely predicts zero accumulation despite the growing control-region volume.

Physical check

When $u\to0$, the region becomes fixed and the two mass fluxes become equal.