Jewett, Walker, Reese, McMullan Problem 6.54* Jewett, Walker, Reese, McMullan

Given A swimming pool W by Y by h deep is to be emptied by gravity through a long pipe. Figure 1

Given Table 1 Constant Name Constant Value Coefficient of friction f Height h Area of Bottom W by Y Length of Pipe L Diameter of pipe D Roughness Value ε Table 1

Problem Derive a formula for the time to empty the tank from an initial level h0.

Assumptions Constant average friction factor Liquid Steady Incompressible Neglect Minor Losses Gage pressure. The area of the tank is much greater than the area of the pipe. Horizontal axis is at the bottom of the tank.

Solution Start with the energy equation: Therefore:

Solution Assume: So the energy equation becomes: Assume:

Solution Simplify equation 1.5: Flow in equals flow out: Substitute into equation 1.7:

Solution Factor out common terms: Isolate V1: Multiply equation 1.11 by ho2:

Solution Take the square root to get the final answer:

BME Application An application of this problem in Biofluids is an IV drip because IVs are emptied using gravity. It is very important for doctors to know how much time it will take to empty an IV bag. The tubing that the IV is emptied into will have certain characteristics that will affect the flow. All of these considerations need to be taken into account in order to administer proper dosages.