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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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 In this assignment, 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. This assignment is in three parts. Part A (2 Marks) Create a mathematical model of the liquid delivery through a round based conical funnel. Give height in terms of time (t). Part B (1 Mark) Calculate the time taken to drain the funnel from the full state. State all of the assumptions that you are using. Part C (2 Marks) Create a working MATLAB program which deploys your mathematical model above. For Your Submission It is important to show the steps that you have chosen to create your mathematical model of this system. Justifications are required for why you have made the decisions that you have. In your submission, you must provide a response for each of the three sections, including a copy of your working MATLAB program. Hint Tabulate all of the information that you have been provided. Derive other values, from the basic system geometry. List the relevant kinematics formulas to see what might suggest itself as a way forward. The system is changing dynamically, you will need to model the system in terms of the changing variables. What is changing? In what respect is it changing?
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Explanation & Answer

Here is the final answer with the MATLAB code and explanations

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...


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