Introduction

The Purpose of this lab is to study the

phenomena of tank draining and figuring out a mathematical model to accurately

describe what happened in the experiment. This lab experiment is useful to

understand on a small-scale, how tank draining works on theoretical and

technical aspect. This is important topic since it covers different areas of

Chemical Engineering.

One

area that tank draining is important is the instance that you are going

implement a nozzle or pipe into a storage tank. Before you can implement it,

you would need to know how fast will the water draining from the tank to with

an exit velocity. Another application that values this project is if there is a

tank rupture, it is important to understand all the factors that was used in this

lab in order to perform investigations as well as conduct Process Hazard Analysis

and improve on Process Safety Management.

Experimental Equipment and Procedures

Figure 1: Stainless Tank

A

tank that has a cylindrical top with a conical bottom shape (Figure 1) was used

for this experiment. The bottom of the tank is connected to the to a schedule

40, 1.25″ tube. Which is connected to a tee reducer which is then have a tube

connected to six different tube sizes and Diameters.

To

run the experiment, water will be filled up to the initial 30″ water height

above the tube. After the water has been filled to the set height, one of the

six tubes will be connected to the bottom of the pipe. To record data the, tee

that was connected to the pipe was opened and data was recorded at every inch

till a set water height was reached.

To

get a estimate theoretical model for the data for the tank draining, Several

assumptions were made here were the following assumptions:

1.

The fluid is incompressible

2.

The fluid is also inviscid

3.

The friction in the cylinder section and the conical

section, as well as the pipe and reducer are negligible.

4.

It is at constant density.

With these starting assumptions, the

model was begun by using the mass balance which is:.

Since there is no mass flow in the mass balance simplifies to . The Bernoulli Equation used for this

model was the head form of the Bernoulli Equation is the head form the

Bernoulli which is

(EQ.2).

Several terms can be canceled out with

the following reasons:

·

Both pressure terms can be canceled out since both are

held at atmospheric pressure

·

The work term can also be canceled out since there is

no work being done on the system

·

Since the water has an initial height an assumption was

made that one of the height is zero.

The

simplified equation will end up being . The assumption can also be made that

the velocity at the free surface (v2) is smaller than the velocity at the tube

(v1) so that term can go away so all is left to solve for is the velocity which

comes out to be . Since this a constant density draining

and has continuity, a valid equation to use is and set as since the water height is decreasing and

can solve for with the equation: . From this equation two separate models

were made for the cylindrical section and the conical section. is the area of the tube and is the area of the cylinder section/conical

section. One term that needed to be defined is . To get this term, a velocity term

needs to be defined. The velocity term was determined by taking the derivative

of the best fit equation from the experimental plot of water height with

respect to time and plugging either the time used with the experimental data

(the time to drain the cylindrical section/conical section can also be used).

Once a velocity is found, that velocity can be used to find the Reynolds number

from the equation (EQ.7) with D being the diameter of the tube and µ being the viscosity of the

water. The calculated Reynolds number used to find the friction factor. From the calculated for the Reynold

number in both the cylindrical section and conical section was calculated to be

laminar flow with each pipe which then the equation: 16/NRE to find the friction factor to find the head friction of the

tube which was calculated from the equation . For the first 8 initial fluid heights

which is in the cylindrical section of the tank, the area of the tubes is set

to be and the areas of both the cylinder section and

the conical section are both . For this lab, height was used to solve

for the time taken to drain the water. For the cylindrical section, EQ.9 and

EQ.10 was plugged into EQ.6 to get . This equation is a

separable differential equation and was solved for t to get However, got the friction from the tube

other than Tube No.1 (1/8″ ID, 24″ length) had very miniscule friction so it

could be safely assumed from the data that the term for the rest of the terms were

negligible. For the conical section it was assumed that the conical section was

shaped like a true cone. This is useful for the fact that you can find the

tangential angel of the cone in order to find the height of the fluid at any

given radius. The tangential angle was solved from the equation . The angle was found to be 31.34. This

was plugged in to find the radius of the cone in . The squared tangential angle turned

out to be .36. EQ.12 was plugged into EQ.6 to get . This is also a separable differential

equation and was solved for t. Since the friction can not be assumed negligible

in the first tube, the equation to solve for t was evaluated to be

t

. For the other tubes the equation

becomes

For

each tube, the cylindrical derived equation will be used for the first 8 inches

of fluid height until there is no water left in the cylinder section. After

total drainage in the cylinder section, the equation for the conical section of

tank will be used for the rest of the water heights.

Experimental Results and Discussion

(Tube No.1)

This

is one of the scatter plot of the Tube (No.1) that is graphed as water height with

respect to time in height. The experimental data was compared to the

theoretical model that was generated from the mathematical model. The following

graphs were formatted the exact way as the graph above. (Tube No.2 D data

with repsect ot time

From

the plotted data given above, one observation can be made in that the tank took

a substantial amount of time to drain rather than the experimental data points.

Another conclusion that can be made based off these graphs is that the length

of the tube seems to play a role in the accuracy between the theoretical and

the experimental. The most accurate that the theoretical model was Tube No.3

Relationship between theory and experimental results

As observed earlier, the theoretical

model data has some significant variation for each tube. However, it would be

safe to assume that the model closely models the draining of the water in the

cylindrical section than the conical section. This model is on the basis that

this is an unsteady state model for the fact that the cross-section area of the

tank is decreasing as the fluid height decreases. With the cylindrical section

of the tank it free surface and had constant cross-sectional area throughout

its section. On the other hand, the equation becomes unsteady when the water

height reaches the conical part of the section since the radius decreases as

height decreases.

One Equation/Law used for tank draining situations is Torricelli’s

Law. It theorizes that the velocity of the fluid is the same as the velocity at

rest (Nevers, 2005). It can be defined

from the Bernoulli as (Nevers, 2005). This equation does

not work with this model because this tank has two different shaped sections as

well as there is friction. The model uses a modified version of the Torricelli’s

Equation that factors in friction.

The Graphs also indicates that the velocity of the water exiting

the pipe increases as the height of the fluid decreases. This phenomenon can be

described for the Coriolis effect. The Coriolis effect describes the motion of

the fluid in the x,y,z gradient as well as the radial direction in cylindrical

coordinates (McCabe, Smith, & Harriot, 2005). As the fluid flows

down the tube the velocity in cylindrical coordinate directions which causes a Coriolis

effect to occur in the conical section as the channel for the streamline of the

water decreases. This causes interaction in the boundaries of the conical section

and momentum to occur in the tank results in an increase in turbulence (Moore, 1967).

Conclusion and Recommendations

In

Conclusion, a general mathematical model for tank draining of water was created.

The results from the theoretical data varied with the experimental data given.

Some recommendations would be to analyze the conical section more in depth as

well as studying the effect of having an initial flow rate and investigating the

conical section since the theoretical time was relatively long compared to the time

in the experimental data

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