Flow Analysis of Water In a Control Volume Travis King BIEN 301 January 4, 2007

Problem P3.24 Given: Given: Water enters the bottom of the cone in Fig. P3.24 at a uniformly increasing average velocity V=Kt. Required: If d is very small, derive an analytic formula for the water surface rise h(t) for the condition h=0 at t=0. Assumptions: (1) Incompressible flow. (2) Since flow is incompressible, density does not change.

Sketch: Sketch: V=Kt h(t)Diameter d Cone θθ r h

Solution: Since we have conservation of mass, we can use Equation 3.20 (p.147) –If mass is constant then (dm/dt) system = 0 –Also, we have a one-dimensional inlet. If the inlet were not one- dimensional, we would have to integrate over the section to find mass flowing in. –However, since the inlet is one-dimensional: reduces to

–For this problem, our conservation of mass equation becomes: –Since no mass is leaving the system, the first summation term disappears leaving us with –We move the negative term over to yield –Simplify the integral and plug in the formula for the area of a circle to yield

–Volume of a cone: –Using our knowledge of right triangles, we know that we can find r by multiplying the height by the tangent of the angle. –Our volume equation then becomes: and our mass equation becomes:

–We then integrate our equation with respect to time to yield –Solving for our h term gives us: –Final Answer:

Application To BME Breathing Machines/Respirators –Must know the mass flow rate of oxygen that is being pumped into the lungs so that the machine does not pump in more oxygen than is being exhaled by the patient.

Questions?