Mass Conservation Supplement

Mass Conservation Conservation of mass means mass is always conserved, it does not appear or disappear. If we look at a pump from a qualitative point of view, mass conservation says that: Rate mass goes in = Rate mass goes out + Rate mass accumulates in pump Quantitatively, this is expressed as: mFluid In Pump mIn = mOut + Equation ( 1 ) t Rate at Which Mass Flows In Rate of Increase of Mass Inside of The Pump Rate at Which Mass Flows Out If we can assume: 1. Rigid Pump: The pump body can’t change shape, meaning VolumePUMP is constant. 2. Incompressible: The fluid in the pump can not be compressed into a smaller volume so the amount of fluid per unit volume (density = ρFLUID) is constant. 3. Full Pump: The pump remains full of fluid (not draining or filling) during operation. Then we can say that the mass inside of the pump remains constant, or: mFluid In Pump t =0 To help you see this, think of a pump as a full cup of water. If the cup does not change shape (i.e. assumption 1), the water is not compressible (i.e. assumption 2) and the cup was initially full (i.e. assumption 3), then the water you pour into the cup must equal the water that comes out (over flows from) the cup. So we can say that the rate at which mass in the pump/motor changes is negligible. mIn = mOut + mFluid In Pump t Rate at Which Mass Flows In [Pour water into cup] Rate at Which Mass Flows Out [Water overflows] ~0 Rate at Which Mass INSIDE Changes [Mass inside of cup stays the same] Mass Conservation Cont. This leaves us with; mIn = mOut Equation ( 2 ) Rate at Which Mass Flows Out Rate at Which Mass Flows In Intuitively, this makes sense because if you can not add to, or subtract from something, then what goes in must equal what comes out. Now we need to put this in terms which are important to pumps and motors. FLOW IN: Shaft AIn VIn ρIn Pump or Motor FLOW OUT: WHERE: AOut Ai = Cross Sectional Area of Port i VOut ρOut Vi = Average Velocity at Port i ρι = Fluid Density at Port i Important Flow Variables In Hydraulic Pumps and Motors The mass flow in a pipe or pump/motor entry or exit (the inlet or outlet ports) is: mk = ρk • Ak • Vk Equation ( 3 ) Where k characterizes the mass flow as in or out of the pump. If we substitute equation 3 into equation 2 we obtain: ρIn • AIn • VIn = ρOut • AOut • VOut Equation ( 4 ) Since we are assuming the fluid is incompressible, the density of the fluid will remain constant and ρIn = ρOut. In equation 4, the densities cancel out leaving: AIn • VIn = AOut • VOut Equation ( 5 ) The quantity A x V is also know as the volume flow rate, Q. Therefore equation 5 can be written as: QIn = QOut Equation ( 6 ) Equation 6 is the result of mass conservation which is useful to hydraulic systems analysis