Description
In this , you are to model the flow of a liquid through a round based, conical funnel.
The funnel is of radius b = 275 mm and with height H to the (imaginary) apex of 450 mm.
The outlet is of radius a = 4 mm.
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Explanation & Answer
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Liquid Flow through a Funnel
Prework
Given Data
Dimension
Radius of the top of the funnel
Height of the funnel
Radius of the orifice at the bottom
Radius of the top of liquid at time t
Height of the liquid from orifice
Notation
b
h0
a
r(t)
h(t)
Measurement (mm)
275
450
4
?
?
Since, b is much larger than a and h0 is much larger than a. Hence, we can say that except
for the very small values of r(t) and h(t) both these quantities will be much larger than a and
Considering triangles as the figure below (Figure 1). Considering, triangles ΔPQV and ΔRSV.
Q
These two triangles are congruent, hence
P
b = 275
𝑄𝑆 𝑄𝑃
=
QR QV
r(t)
A R
S
h0 = 450
𝑟(𝑡)
𝑏
= h0
h(t)
𝑏
𝑟(𝑡) = ℎ0 ℎ(𝑡) ….. (1)
h(t)
Hence, at a time t, the volume of the fluid in
the funnel can be given by –
𝜋 2
𝑉(𝑡) =
𝑟 (𝑡)ℎ(𝑡)
3
𝜋 𝑏
𝑉(𝑡) = 3 (ℎ0)2 ℎ3 (𝑡)……. (2)
V
a=4
Now, rewriting the constant part of the above equation as K (Which is a constant based on
funnels shape and size) we can write –
𝑉(𝑡) =
1
𝑏
𝐾ℎ3 (𝑡)……. (2A), where 𝐾 = 𝜋(ℎ0)2
3
Part A
Liquid Delivery and Height
From the equat...