Conservation of Mass

Fluid Mechanics ME 332Dominic Waldorf

Section 007 Tuesday 7:00 PMDr. A. M. Naguib and TA Yifeng Tian

Submission Date: March 11th, 2014

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Abstract

Conservation of mass is a very important concept in the studies of fluid

mechanics. The applications of this concept are very useful in solving real life fluid

mechanic problems. The purpose of this lap is to reinforce the control volume

formulation of mass conservation. In addition, the concept of the “discharge coefficient”

is introduced and its usage demonstrated. With the use of the Pressurized Flow System

(PFS), the discharge coefficient can be shown. The PFS consists of a large plenum,

which is pressurized by a fan that draws air from the ambient in the laboratory. The

system has one main inlet and two main outlets. The two major outlets are the stack and

the slit-jet. However, the system is not completely sealed so leakage has to be accounted

for when performing the experiment.

The experiment is broken up into two parts. The first part of the experiment is to

calculate the leakage of the system. To begin, the stack is completely closed which

leaves only one outlet in this part of the experiment. The Pitot tube is placed in the inlet

pipe. The Pitot tube is used to measure the pressure inside the flow of the fluid. The

pressure is then measured at different radiuses of the inlet tube. From the data collected

the inlet flow can be approximated. The outlet flow can also be calculated using the

single pressure differential across the slit jet. Because of mass conservation, the inlet

flow in theory is supposed to equal the outlet flow. However, since there is leakage in

the system, they do not equal. So the difference between the inlet flow and the outlet

flow should give the correct value for the leakage. This part of the experiment is

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essential because the volume flow rate and the discharge coefficient of the stack will be

flawed if the leakage is not included.

The second part of the experiment is to find the volume flow rate and the

discharge coefficient of the stack as a function of the ratio between the diameter of the

“cap” hole and the diameter of the stack. The stack opening is then changed with each

different “cap” in order to find the different pressures at the inlet of the system and the

different pressures on the inside of the Pressurized Flow System with respect to each

different “cap”. The “cap” in the experiment is the piece of circular metal that is placed

on top of the stack to vary the stack’s flow rate. The “cap” looks like a 2-dimensional

donut. There are seven different caps used in this part of the experiment, with each “cap”

having a different inner diameter. Once the different pressures are collected from

experiment, the inlet flow rate, and the slip-jet flow rate can be calculated using the

derived equations. Then using the calculated values of the inlet flow rate, the jet flow

rate, and leakage flow rate from part one; the flow rate of the stack can now also be

calculated at each different stack diameter. Conclusively, the discharge coefficient is also

calculated once the flow rate of the stack with each different “cap” is determined.

After the experiment is completed, graphs can be made to demonstrate the

relationship between the stack discharge coefficient and the ratio of the outer and inner

diameter of the “cap” placed on the top of the stack. The different trends of the data can

be determined as the diameter ratio approaches 1 and 0. Not only can the trends of the

data be determined from the results, but another key aspect of the lab can also be

demonstrated with the data collected. That is, the relationship of the discharge

coefficient and the ratio of the stack’s inner and outer diameter without the leakage flow

4

rate in consideration. Problems can then be observed with the different discharge

coefficient values, which show why the leakage flow rate needs to be included in the final

calculations of the experiment.

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Table of Contents

Introduction 6

Conservation of Mass…………………………………………………….….6

Discharge Coefficient………………………………………………………..9

Volume Flow Rate and Discharge Coefficient of PFS Inlet………………..11

Volume Flow Rate and Discharge Coefficient of PFS Stack……………....13

Experimental Setup 14

Volume Flow Rate and Discharge Coefficient of PFS Experiment 1…...…14

Volume Flow Rate and Discharge Coefficient of Stack Experiment 2.........16

Results and Discussion 18

Volume Flow Rate and Discharge Coefficient of PFS Results……………....18

Volume Flow Rate and Discharge Coefficient of Stack Results……………..20

Graph of Stack Results with Leakage Flow Rate………..…………………....22

Graph of Stack Results without Leakage Flow Rate……………………….....23

Conclusion 24

References 25

Appendices 26

6

Introduction

Conservation of Mass

Conservation of mass is an important principle when dealing with fluid

mechanics. This is mainly because fluid mechanics assumes that every fluid obeys

conservation of mass. When assuming the fluid obeys conservation of mass, many other

unknown variables can be derived. Since this assumption is made quite often in theory,

the principle of mass conservation should be proven using a real life model. If the model

produces plausible values for the calculated variables like the discharge coefficient and

volume flow rates, then conservation of mass has been proven to be correct in this case.

Conservation of mass states that for any system closed, the mass of the system is

conserved and does not change. The principle implies that mass can neither be created

nor destroyed, although it may be rearranged in space, or the entities associated with it

may be changed in form. Conservation of mass is demonstrated in this experiment by

allowing the inlet flow rate equal the summation of the outlet flow rates.

The idea of this lab experiment is to reinforce the control volume formulation of

mass conservation. Along with that, also demonstrate the concept of the discharge

coefficient, which is the ratio of the actual discharge to the theoretical discharge, and its

usage.

In order to demonstrate conservation of mass in this experiment, some equations

need to be derived. The general formula for mass conservation is represented by Equation

(1) where ∀ is the volume and Vr is the velocity vector.

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0=(∫CV

❑ ∂ ρ∂ t

d ∀)+∫CS

❑

ρ (V r ∙ n ) dA (1)

Now, if the flow is steady ( ∂

∂ x=0) or density if a constant, the first term on the left hand

side vanishes, and we simply get that the net mass flux through the control surface is

zero, or

∫CS

❑

ρ (V ∙ n ) dA=0 (2)

Equation (2) states that in steady flow the mass flows entering and exiting the control

volume must be equal. If the inlets and outlets are one-dimensional, for steady flow the

equation produced is

∑ ( ρi A iV i )¿=∑ ( ρ i Ai V i )out (3)

Using the definition of mass flow rate m in equation (4), equation (3) can be summed

down to equation (5)

m=∫CS

❑

ρ (V ∙n )dA (4)

∑ ( mi )¿=∑ ( mi)out=m (5)

Using a fixed control volume, for an incompressible flow, the term ∂ ρ∂t

is so small that

the term can be neglected. Density is also constant and can be removed from the equation

giving us:

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∫CS

❑

(V ∙ n )dA=Q (6)

For the case where density doesn’t change across the area, the mass flow = density x Q.

If the inlets and outlets are one-dimensional, we have

ρ 2∗V 2∗A 2=ρ 1∗V 1∗A 1 (7)

or Qin=Qout (8)

Equation 8 is possible because of the volume flow rate can be given as:

Q=∫CS

❑

(V ∙n ) dA (9)

Equation (9) allows us to define an average velocity Vavg that, when multiplied by the

section area, gives the correct volume flow:

V avg=QA

= 1A∫ (V ∙n ) dA (10)

The extension of these results to the case with multiple inlets and outlets is

straightforward. For steady flow through a flow system with N inlets and M outlets, the

mass conservation requires that:

9

∑i=1

M

(Qout )i=∑i=1

N

¿¿ (11)

Discharge Coefficient

The discharge coefficient Cd is dimensionless and only changes moderately with

value size. It is the ratio of the actual discharge to the theoretical discharge.

Consider a flow through a orifice of area A. Under inviscid flow conditions, meaning

frictionless flow, the exit velocity is uniform and the velocity is given by the Bernoulli

equation as

V inviscid=√ 2 ( ppl−prec )ρ

(12)

The ideal volume flow rate is then given by equation (13).

Qideal=V inviscid A (13)

The actual volume flow rate, Qactual, is less than the Qideal because the real flow is not

frictionless and some losses are always going to be present. Compared to the ideal

conditions, the actual velocity at the exit is expected to be non-uniform. However the

velocity equals the inviscid velocity only in the middle portion of the exit where the flow

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is not affected by viscous forces. Since the discharge coefficient is given as the ratio of

the actual discharge to the theoretical discharge the equation is

cd=Q actual

Qideal (14)

From the ratio and equation (14) it is clear that 0 < cd < 1. Discharge is not only the ratio

of volume flow rate, but also thought of as the ratio of the average velocity to the inviscid

velocity,

cd=V

V inviscid (15)

The discharge coefficient for a give flow passage depends on the geometry and the

Reynolds number. Once cd is determined, the volume flow rate can be determined.

Q=cd A Vinviscid=cd A √¿¿¿ (16)

Since cd has been determined to be .61, cd can substituted into the equation.

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Volume Flow Rate and Discharge Coefficient of PFS inlet

Equation (9) can be used to derive the velocity profile of the flow:

U (r )=√ 2∗[ pT (r )−pi ]ρ

(17)

From making the assumption that the flow is axisymmetric, the inlet flow rate can be

derived:

Qi=2π∫o

Ro

U (r )r dr (18)

Now using equation (12), Uo can be determined because the velocity field is suppose to

be uniform at Uo

Uo=√ 2∗[ patm−pi ]ρ

(19)

Since the ideal flow rate is known, the equation for the inlet discharge becomes

[ cd ] i=Qi/(πR o2Uo) (20)

And when the drag coefficient is substituted into equation (16), the slit-jet flow rate is

equal to

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Qj=.61 Aj √ 2∗[ ppl−patm ]ρ

(21)

To find the leakage, conservation of mass is used which derives the equation:

Qi=Qj+Qleak (22)

And the results from a past experiment yield the equation y = 1 – e^(a +bx +cx^2+dx^3)

with a=-1.05969 b=-2.47053 c=2.04658 and d=-.892334 which is used to find the

discharge coefficient at the inlet by using different values of [patm – pi] (in. H2O).

Therefore the actual inlet value can be described as

Qi=[ cd ] i(πR o2)√ 2 [ patm−pi ]ρ

(23)

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Volume Flow Rate and Discharge Coefficient of Stack

Now in order to find the discharge coefficient of the stack, conservation of mass needs to

be applied to the system once more

Qi=Qs+Qj+Qleak (24)

In order to find the stack discharge coefficient, equations (23) and (21) and the Qleak, the

discharge coefficient can be formulated into

[ cd ] s=Qs/¿ (25)

As is the area of the stack which is As=(π/4)*(Ds^2). Ds is fully opened and interchanges

with the other diameters.

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Experimental Setup

Experiment 1: Volume Flow Rate and Discharge Coefficient of PFS Inlet

In this part of the lab, the Pressurized Flow System (PFS) will be used. The

objective for this part of the lab is to find the leakage flow rate of the PFS. In order to

calculate this value, many steps need to be completed.

A. Zero the two pressure transducers and record their mp value.

B. Connect the first transducer’s (+) port to the Pitot tube and the (-) port to the static

probe. Connect the second transducer’s (+) port to the Pitot tube using a tee and

the (-) port to patm.

C. Make sure the stack is fully closed and start the PFS.

D. Run the Transverse VI to simultaneously acquire [pT(r) – pi] and [pT(r) – patm]

data. Use the corresponding mp values as the normalization constants so that the

final reading is in unit of in. H2O. Set the sample rate to 100 Hz for 500 samples.

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E. Carry out the Pitot tube survey in the following manner. Set the Pitot probe at the

extreme end of its travel; this defines a radial coordinate about 1 mm away from

the pipe wall. Therefore, the first point in the survey has a coordinate r = Ro – 1,

where Ro = 82.25 mm is the radius of the inlet pipe. In non-dimensional form,

this corresponds to r/Ro = .988. Perform the measurements at the following radial

locations:

point r (mm) r/Ro point r (mm) r/Ro1 81.25 0.988 8 62.25 0.7572 80.25 0.976 9 52.25 0.6353 79.25 0.964 10 42.25 0.5144 77.25 0.939 11 32.25 0.3925 75.25 0.915 12 22.25 0.2716 72.25 0.878 13 12.25 0.1497 68.25 0.83 14 2.25 0.027

F. Before you turn the PFS off, connect the (+) port of the second transducer to ppl

and record the DMM voltage E corresponding to the pressure differential

[ppl – patm] across the slit-jet. Also record the slit-jet width W and span L.

G. Note that [pT(r) – pi] gives the velocity U(r) according to equation (17). Your

survey will show that the profile of [pT(r) – pi], and therefore U(r), remains

practically uniform over the range 0 < r < ro and shows the presence of the thin

boundary layer over the range ro < r < Ro. The measured uniform velocity in the

central portion of the pipe will be referred to as Uc. Recall that the inviscid flow

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velocity Uo is determined from equation (19). Calculate the values Uc (ft/s), Uo

(ft/s), ro (mm), and ro/Ro from the data collected.

H. The inlet flow rate Qi can be computed by integrating the measured velocity

profile U(r) according to equation (18). The expression for Qi based on an

approximate form of the inlet velocity profile can be expressed by equation

Qi = (πUc/3)*(Ro^2 +Roro +ro^2).

I. Compute the value of the inlet flow rate Qi and the discharge coefficient [cd]i

J. Compute flow rate Qj out of the slit-jet using your measurement in part F and

equation (21), and determine the leakage flow rate Qleak.

Experiment 2: Volume Flow Rate and Discharge Coefficient of Stack

In this part of the lab, the Pressurized Flow System (PFS) will be used again. The

PFS will be used to determine flow rate Qs and the discharge coefficient [Cd]s as a

function of the stack opening d/Ds. Two pressure transducers will be used to measure

[patm – pi] and [ppl – patm] for each stack opening. The values of the “cap” diameters are

2.656’’, 3.0’’, 3.5’’, 4.0’’, 4.5’’, 5.312’’, and 6.687’’.

A. Zero the two pressure transducers and record their mp value.

B. Connect the first transducer’s (+) to patm and the (-) port to the inlet pipe static

probe. Connect the (+) port of the second transducer to ppl and its (-) port to

patm.

C. For each value of d/Ds, record the DMM voltages E1 and E2 corresponding to

[patm – pi] and [ppl – patm], respectively.

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D. Using the measured [patm – pi], equation (23), and the inlet discharge

coefficient, calculate the inlet flow rate Qi for each case.

E. Using the measured [ppl – patm] and equation (21), calculate the slit-jet flow

rate Qj for each case.

F. Compute the volume flow rate Qs out of the stack for each case, taking the

leakage flow rate Qleak into account. Use equation (25) to compute the stack

discharge coefficient [cd]s.

G. Plot the stack discharge coefficient [cd]s versus d/Ds.

H. Compute the volume flow rate Qs out of the stack for each case, taking the

leakage flow rate Qleak out of the equation. Use equation (25) to compute the

stack discharge coefficient [cd]s.

I. Plot the stack discharge coefficient [cd]s versus d/Ds without Qleak in

account.

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Results and Discussion

Experiment 1: Volume Flow Rate and Discharge Coefficient of PFS Inlet

The mp value for the two transducers is as follows:

(mp)1 = .5533 in. H2O/volt (mp)2 = .5533 in. H2O/volt

The DMM voltage, width W, and span L values recorded from the experiment were:

E = 4.7 volts W = ¾ inch L = 43.25 inch

From the data collected, the values of Uc, Uo, ro, and ro/Ro were calculated. Uo was

found from the graph, using 1.044 as [patm –pi]. 1.044 is where the center of where the

flow velocity is not uniform. The graph used can be found in the appendix. Uc was

found from the graph too, using 1.18 as [patm –pi]. 1.18 is where the values of the

velocity are uniform.

Uc (ft/s) Uo (ft/s) ro (mm) ro/Ro

71.86 67.59 75.25 0.915

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Recall that the region where [pT (r) – patm] is non-zero corresponds to streamlines that

have been affected by viscous shear in the boundary layer. According to [pT (r) – patm]

the boundary layer begins at r = 75.25 mm. The graph shows the velocity decreasing at

ro = 79.250 mm.

From the approximate integration:

Qi = 15.8198 ft^3/s [cd]i = 1.0231

The actual values:

Qi = 14.0924 ft^3/s [cd]i = .9114

Volume flow rate at slit-jet:

[ppl – patm] = 2.60051 slit-jet area Aj = WL = .22656 ft^2

Qj = 14.74 ft^3/s Qj/Qi = .9317

Qleak = 1.0798 ft^3/s Qleak/Qi = .06826

The values calculated from the equations derived seem to be plausible values that support

the hypothesis. Now the Qleak has been solved, the discharge coefficient for the stack

can be calculated.

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Experiment 2: Volume Flow Rate and Discharge Coefficient of Stack

The mp value for the two transducers is as follows:

(mp)1 = .5533 in. H2O/volt (mp)2 = .5533 in. H2O/volt

d inch E1 (volts) E2 (volts)fully-open 3.85 0.9

6.687 3.5 1.75.312 3.15 2.554.5 2.92 34 2.75 3.4

3.5 2.63 3.653 2.51 3.94

2.656 2.45 4.05

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From the data collected above, the rest of the unknown values could be calculated.

Table 1: Calculated Values with Leakage Flow Rate

The values calculated in Table 1 support the hypothesis. The discharge coefficients are reasonable numbers between 0 and 1.

d inch d/Ds

[patm - pi] (in. H2O)

[ppl - patm] (in.

H2O) Qi (ft^3/s)Qj

(ft^3/s)Qs

(ft^3/s) [cd]s

fully-open 1 2.1302 0.49797 22.0086 6.450915.557

7 0.969328162

6.687 0.8422 1.93655 0.94061 20.856 8.865911.990

1 0.7663422455.312 0.669 1.7489 1.410915 19.604 10.858 8.746 0.7232864334.5 0.5668 1.615636 1.704164 18.6367 11.934 6.7027 0.702807625

4 0.5038 1.521575 1.88122 17.92295 12.5385.3849

5 0.6801572053.5 0.4408 1.455179 2.019545 17.40707 12.991 4.4160 0.703137883

d inch d/Ds

[patm - pi] (in. H2O)

[ppl - patm] (in. H2O) Qi (ft^3/s)

Qj

(ft^3/s)Qs

(ft^3/s) [cd]s

fully-open 1 2.1302 0.49797 22.0086 6.4509 14.4779 0.902050836.687 0.8422 1.93655 0.94061 20.856 8.8659 10.9103 0.6973272785.312 0.669 1.7489 1.410915 19.604 10.858 7.6662 0.6339879324.5 0.5668 1.615636 1.704164 18.6367 11.934 5.6229 0.5895858384 0.5038 1.521575 1.88122 17.92295 12.538 4.30515 0.543770841

3.5 0.4408 1.455179 2.019545 17.40707 12.991 3.33627 0.5312093843 0.3778 1.388783 2.18002 16.8839 13.497 2.3071 0.481239287

2.656 0.3345 1.355585 2.240865 16.619897 13.684 1.856097 0.487196416

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73 0.3778 1.388783 2.18002 16.8839 13.497 3.3869 0.706475377

2.656 0.3345 1.355585 2.240865 16.619897 13.6842.9358

9 0.770627018

Table 2: Calculated Values without Leakage Flow Rate

The values in Table 2 are different because the leakage flow rate was not included when calculating the stack flow rate. This gave a larger value of the stack flow rate.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Stack [cd]s vs. d/Ds with Leakage Flow

d/Ds

Dis

char

ge C

oeff

icie

nt

[cd

]s

Figure 1: Stack Discharge Coefficient vs. d/Ds with Leakage Flow

The data from the experiment is quite accurate. As d/Ds approaches 0, the discharge coefficient is beginning to tend towards the discharge coefficient of the slit-jet, which is .61. Although the value is a little off, it is still fairly accurate being that fact that there is error when the experiment is done. Also as d/Ds approaches 1, the discharge coefficient is approaching unity.

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0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.2

0.4

0.6

0.8

1

1.2

Stack [cd]s vs. d/Ds without Leakage Flow

d/Ds

Dis

char

ge C

oeff

icie

nt [c

d]s

Figure 2: Stack Discharge Coefficient vs. d/Ds without Leakage Flow

The data from Figure 2 is taken from the discharge coefficient table that does not have the leakage flow rate in the calculations. When comparing the graph to the graph with the leakage include, there are many similarities and differences. The reason the graphs are reasonably similar is because the leakage flow rate is a relatively small value. However, the graph reaches a minimum before the other graph. As d/Ds approaches 0 in this graph, it begins to converge towards .8 instead of .61. As d/Ds approaches 1 in this graph, it approaches a higher value than the other graph.

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Conclusion

Overall the experiment performed was a success. After using the principle of

mass conservation, unknown discharge coefficients and flow rates were solved using the

equations derived in the lab. The most difficult part of the lab was the unit conversions.

If this lab is ever revised, I highly suggest that the author uses either metric or English

units, not both. There are many sources of error that could affect the data retrieved.

First, multiple groups calculated the different values of each stack cap. It is hard to tell

whether or not the other groups might have made an undetected mistake. Also the

devices used to calculate the different values in the experiment might not be up to par.

Odds are the equipment was not made by NASA, so there is room for manufacturing

error. The inlet tube could have also been blocked or had something disturb the flow

which would change the pressure retrieved from the Pitot tube. Even though there were

many areas where error could occur, the results collected were fairly accurate. For

example, the discharge coefficient for the stack, as d/Ds approached 0, converged

near .61, which is the discharge coefficient of the slit-jet. That relationship demonstrated

the importance of the leakage flow rate. When the discharge of the stack was calculated

without the leakage, the value was much farther away from .61 then when the leakage

was accounted for. This lab just demonstrated one method of applying conservation of

mass. Just image how many other real life applications this principle can be used for. To

conclude, the principles demonstrated in this lab are very useful and can be very

beneficial to society.

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References

White, Frank M., Fluid Mechanics, Seventh Edition

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Appendices

d inch [cd]i As (in^2)fully-open 0.99652 0.343849

6.687 0.990430.24388777

7

5.312 0.979640.15390173

7

4.5 0.968950.11044661

7

4 0.960210.08726646

3

3.5 0.953610.06681338

5

3 0.94680.04908738

5

2.656 0.943340.03847543

4

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