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Unformatted text preview: Surface of a rotating fluid P(r,z) dP = δP = ρrω 2 δr δP = − ρg δz 0 = ρrω 2 dr − ρgdz ρg ∫ dz = ρω 2 ∫ rdr
z0 0 z r δr δz dr + dz δP δP z= ω2 2 r + z0 g [Accumulation]=[input]-[output]+[source] example, a liquid of 25kg/m3 density enters into a tank, while a outlet of unknown density exits. If the inlet speed is .03m3/s and the outlet is .01 m3/s, what is the density of the liquid exiting the tank when there is 4m3 in the tank? Note that the initial volume of the tank is 2m3 d dt ( ρ out V ) = ρ inVin − ρ out Vout ( ρ outV ) = (25)(.02) − ( ρ out )(.01) dV = V −V
d dt dt in out dV dt = (.02) − (.01) = .01 V = .01t + 2 V = 4, t = 200 ρ out dV dt +V dρ out dt = .5 − .01ρ out ρ out (.01) + (.01t + 2) dρout = .5 − .01ρ out dt
solving ρ out = 25 − 25 (1 + .005t ) 2 t = 200, ρ out = 18.75 Kg m3 Vector notation uu MassFlowRate = ρ (V • n )dA = ρV cos(α )dA [accumulation] = [input ] − [output ] uu d dt ( ∫ ρdV ) = − ∫ ρ (V • n ) dA if α ≤ 90 ⇒ cos(α ) ≥ 0 uu ρ (V • n ) dA = Output − input ∫ ...
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This document was uploaded on 09/26/2010.
- Winter '09