Finite Control Volume –
Deformable CV Application
|
It's
time for your influenza vaccination.
The nurse decides to give you an injection using a syringe as
shown in the figure below. The inside
diameter of the syringe is 5 cm and the inside diameter of the
needle is 0.4 cm. Suppose the nurse
advances the plunger at a rate of 10 cm/sec.
Calculate the velocity
of the vaccine exiting the needle. Strategy: Construct the
control surface and control volume as shown above. Then apply conservation of
mass with a deformable control volume as given below. ∂/∂t ∫ ρ dV +
∫ ρ V . n dS =
0 note:
dV = element of volume cv cs ∂/∂t ∫ ρ dV =
ρ dV/dt
= ˗ ρ V1 A1 =
˗ ρ V1 [π (D1)2 ] /4 and ∫ ρ V . n
dS =
ρ V2 . n2 A2
= ρ V2 A2 = ρ V2[ π (D2)2
] /4 cs Note: V1 refers to the speed of the plunger
coincident with the left side of the deformable
control volume where V1 =
10 cm/sec, D1 = 5
cm, D2 =
0.4 cm, So ˗ ρ V1
[π (D1)2 ] /4 + ρ V2[ π (D2)2
] /4 =
0 V2 = V1 ( D1
/ D2 )2 = 125 cm/sec (result) |
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