Problem 1.53 [Difficulty: 5]
Given: Syring e pump to de liver 100 mL /min δV 0.001 in
min
⋅= δD 0.0005 in⋅=
Find: (a) Plot uncertainty in flow rate as a function of bore.
(b) Find combination of piston speed and bore resulting in minimum uncertainty in flow rate.
Solution: We will apply uncertainty concepts.
Governing Equations: Qπ
4D2
⋅V⋅= (Flow rate in syringe p ump)
2
1
2
1
1
1⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
±= L
xR u
x
R
R
x
u(Propagation of Uncertainties)
Now solving for the piston speed in terms of the bore: VD() 4Q⋅
πD2
⋅
=
2
1
22
2
1
22 2
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
±=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
±= VDVDQ u
V
Q
Q
V
u
D
Q
Q
D
u
V
Q
Q
V
u
D
Q
Q
D
u
So the uncertainty in the flow rate is:
()()
[]
2
1
22
2VDQ uuu +±= 0=
∂
∂
D
uQ
where uDδD
D
=uvδV
V
=The uncer tainty is minimized when
Subs titutin g expr essi ons in terms of bore we ge t: Dopt 32
π2
δDQ⋅
δV
⎛
⎜
⎝
⎞
⎠
2
⋅
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
1
6
=
Substituting all known values yields Dopt 1.76 in⋅=
Plugging this into the expression for the piston speed yields Vopt 2.50 in
min
⋅= and the uncertainty is uopt 0.0694 %⋅=
Graphs of the piston speed and the uncertainty in the flowrate as a function of the bore are shown on the following page.
