fluids case study on head loss

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ME 320 Case Study:

Pressure Losses Across Evaporators

Abstract

The question of whether or not the entrance length of a flow through a pipe

will be answered using experimental data taken from fluid flow through an

evaporator. Flow through a pipe is known to have two regions, an entrance region

and a fully developed region. Through the two parts of this report, the significance

of the entrance region will be put into question and it’ll be proven whether not a

fully developed approximation can be made. This will be done by tabulating

pressure losses in the different regions of the experimental flow and compared to

one another. An important result of this experiment is the predicted values of the

pressure loss in the different regions of the experimented flows and the comparison

of the approximations made during the analysis. At the conclusion of this report, it

will be proven that for the flows done in this experiment, the error made while

making a fully developed approximation is not significant enough to warrant a

reason for the analysis to be redone to include the entrance length, especially in

laminar flows.

Introduction

An evaporator is a heat exchanger in which a refrigerant liquid enters at low

pressure and temperature (relative to atmospheric) and leaves as a vapor. During

the vaporization process, the refrigerant "boils," absorbing energy from the

refrigerated space surrounding the evaporator, and everything within. The fluid

within the refrigerated space, typically air or water, is forced over the exterior sides

of the lateral tubes of the heat exchanger containing a volatile refrigerant (e.g.,

R134a is used in automobile cooling systems). Heat energy from the air/water flow

enters the lateral tubes of the heat exchanger by convection, and then is conducted

through the tube walls and into the refrigerant (again by convection.) The

schematic diagram in Fig 1 shows the geometry and the flow directions of the two

fluids in the evaporator under analysis.



If the refrigerant is in a saturated state and sufficient latent heat enters the

refrigerant, it changes phase (i.e., it “evaporates”). The “top dividing header” in an

evaporator distributes the liquid refrigerant to the lateral tubes and the “bottom

combining header” collects the vaporized refrigerant, ultimately to be condensed

again in a condenser later in the cycle. Each lateral tube has 11 channels that the

liquid flows through. In this case study you shall analyze the pressure drop from the

inlet to the outlet of a single lateral tube due to a liquid water flow that does not

change phase. The data were collected by Habte [ref. 4] as a preliminary analysis of

two-phase pressure drop in the full evaporator [5].



Setup, Data, and Methods of Analysis

The overall pressure drop across the evaporator is important to the overall

performance of the heat exchanger and will affect the choice of other components

in the system, such as the compressor. Although the overall pressure drop is the

sum of that within the two headers plus that within the lateral tubes, the greatest

contribution is in the pressure drop across each lateral tube in the evaporator. In

the experiment the pressure drop across a single lateral tube measured due to the

flow of liquid water at room temperature will be analyzed. The tests were conducted

with the procedure below. The tests were performed on a single aluminum flat tube

with 11 channels. The channel length was .6096 m, and because the tubes were

new and smooth when the tests were performed, roughness effects were negligible.

All data was collected at room temperature, about 20C. The two ends of the flat

tube were connected to plenum chambers each of which was connected to a

flexible hose. The two plenum chambers were connected to different sides of a

differential pressure gage that measures the pressure difference between the inlet

and outlet of the flat tube, P1 - P2. A variable-area flow meter was used to measure

the flow rate of water Q into the lateral tube. The flow rate of water through the

tube was set by adjusting the valve at the inlet to the test section. The variable area

flow meter (commonly known as a "rotameter") was calibrated before the test was

conducted to minimize error. The pressure drop measured with the differential

pressure gage was recorded for each fixed flow rate and the measurements were

repeated for different flow rates with the same tube geometry. Several

measurements were made at each flow rate and the pressure drops averaged to

improve accuracy.

The analysis will be in two parts. In the first part the head loss will be

evaluated experimentally and compare with standard engineering correlations by

approximating the flow as fully developed throughout the lateral tube. Recognizing

that there exists an entrance length, in the second part an analysis to estimate the

error introduced by the fully developed approximation made in part one will be

done and explained [5].



Analysis and Results

Part I: Analysis of the Pressure Drop Using the Full-Developed

Approximation

Part I (a)

The “effective diameter” for the noncircular pipes that reduce to the circular

pipe diameter when the cross-section is circular needs to be determined so that the

correlations from data generated for circular pipes can be used. The given geometry

given for the channels in the flat tubes of the evaporator is the width, a, is .0018 m

and the height, b, is .001 m. The “effective diameter,” can be interpreted as the

hydraulic diameter, DH. The equation for the hydraulic diameter is

DH=4Acp 1

where Ac is the cross-sectional area and p is the wetted perimeter. When the

dimensions of the channels are put into equation (1), the hydraulic diameter is

found to be .001286 m.

The Reynolds number is defined as the ratio of inertial forces to viscous

forces in the fluid. The equation to tabulate Reynolds number for a non-circular pipe

is

Re=ρVDHμ (2)

where ρ is the fluid density, µ is the fluid viscosity and Vavg is the average flow

velocity. For equation (2), ρ and µ was found for water at room temperature, 998.3

kg/m3 and .001003 kg/m*s respectively. The average velocity for each flow in the

data was found by using the equation

V=QAc (3)

where Q is the experimental flow rate and Ac is the cross sectional area of a

channel. The cross sectional area used in equation (3) was found to be .0000018

m2. The flow rate was divided by 11 to accommodate the 11 channels per tube that

the flow was going through. The tabulated Reynolds number is shown in Table 1.



Q volume flow rate[gal/hr]

P1 - P2 Pressure drop[psi] Q (m3/s)

Velocity(m/s) Re

1.0 0.075 9.63365E-08 0.053520295 68.50458

2.0 0.08 1.92597E-07 0.106998265 136.955

3.0 0.15 2.88857E-07 0.160476234 205.4054

4.0 0.325 3.85118E-07 0.213954203 273.8558

5.0 0.4 4.81378E-07 0.267432173 342.3062

6.0 0.525 5.77638E-07 0.320910142 410.75667.0 0.625 6.73899E-07 0.374388112 479.207

9.5 0.85 9.06778E-07 0.503765659 644.8069

12.5 1.23 1.19671E-06 0.664836842 850.9738

15.6 1.675 1.48663E-06 0.825908025 1057.141

18.6 1.8 1.77656E-06 0.986979208 1263.308

21.6 2.15 2.06649E-06 1.148050391 1469.475

24.7 2.6 2.35642E-06 1.309121574 1675.641

26.2 2.9 2.50138E-06 1.389657165 1778.725

27.7 2.95 2.64635E-06 1.470192757 1881.808

30.7 3.6 2.93628E-06 1.63126394 2087.975

33.8 4.2 3.2262E-06 1.792335123 2294.142

35.3 4.65 3.37117E-06 1.872870715 2397.22636.8 5.1 3.51613E-06 1.953406306 2500.309

Table 1. Given data with calculated average velocity and Reynolds number

Part I (b)

When a fluid is flowing through a pipe, there will be a few pressure losses

that the flow will induce across the length of the pipe. There will be both major and

minor head losses. The major head loss will either be laminar, transitional, or

turbulent depending on the Reynolds number and the critical Reynolds number for

the given pipe. The critical Reynolds number for the noncircular pipe is still

unknown in this experiment, so the laminar and turbulent head losses will be

tabulated for all flows. The equation for major head loss is

hf,maj=fLV22gDH 4



where f is the friction coefficient and L is the length of the pipe. For equation (4) the

length of the pipe is given as .6096 m. The friction coefficients for laminar and

turbulent flows are different. For laminar flow, the friction coefficient equation is

f=61.144Re (5)

The friction coefficient for turbulent flows can be found using the explicit form of

the Colebrook equation, which is

1f=-1.8log6.9Re+εDH3.71.11 6

where ε is the roughness factor. For equation (5) the roughness factor is zero

because the pipes are assumed to be smooth. When the friction coefficients are

found, they can be plugged back into equation (4) to find the major head losses for

each flow. The equation for the minor head loss in the flow is

hf,min=KLV22g 7

where K L is the loss coefficient. The loss coefficient for laminar and turbulent flows

can be tabulated by using Table 8-4 in Cengel and Cimbala [1]. The loss coefficient

is the sum of the coefficients acting on the flow. In our case the loss coefficient for

the entrance is 0.5 and for the exit it’s 1 per the write-up, so our total loss

coefficient is 1.5. Once we have the head losses tabulated, the total pressure loss

can be predicted. The equation for total pressure loss is

∆PL=ρghf 8

where ΔPL is the pressure loss. Major and minor total pressure losses can be

calculated by plugging the respected head loss into equation (8). Table 2 shows the

calculated major head losses (laminar and turbulent) and the total pressure losses

(laminar and turbulent). Also table 2 shows the calculated friction coefficients for

each flow. Table 3 shows the calculated minor losses for each flow. Since critical

Reynolds number is still unknown at this point, both laminar and turbulent losses

are found for each flow.



f lam hf,lam ΔPlam (Pa) f turb hf,turb ΔPturb (Pa)

0.892553 0.061769804 604.9316461 0.310583 0.021494 210.4987649

0.446453 0.123490759 1209.384892 0.183268 0.050693 496.4501777

0.297675 0.185211714 1813.838137 0.142102 0.088415 865.8773078

0.223271 0.246932669 2418.291383 0.120764 0.133562 1308.01435

0.178624 0.308653623 3022.744629 0.107356 0.185506 1816.720563

0.148857 0.370374578 3627.197874 0.097991 0.243815 2387.754588

0.127594 0.432095533 4231.65112 0.090997 0.308162 3017.926161

0.094825 0.581415071 5693.985589 0.079482 0.487336 4772.641107

0.071852 0.767313438 7514.548344 0.070586 0.753797 7382.173957

0.057839 0.953211806 9335.111099 0.064631 1.065142 10431.28409

0.0484 1.139110173 11155.67385 0.060286 1.418853 13895.28492

0.041609 1.32500854 12976.23661 0.056934 1.813006 17755.35021

0.03649 1.510906907 14796.79936 0.054245 2.246065 21996.43743

0.034375 1.603856091 15707.08074 0.053084 2.47678 24255.90184

0.032492 1.696805274 16617.36212 0.052023 2.716762 26606.126

0.029284 1.882703641 18437.92487 0.050147 3.224024 31573.90684

0.026652 2.068602009 20258.48763 0.048534 3.766926 36890.71851

0.025506 2.161551192 21168.76901 0.047807 4.051484 39677.49479

0.024455 2.254500376 22079.05038 0.047127 4.344657 42548.62932

Table 2. Major Pressure Losses

ΔPminor 2.144664

48.571874

519.28163

234.27393

653.54878

777.10618

6104.9461

3190.0113

1330.9424

6510.7233

4729.3539

5



986.83431283.164

41445.898

11618.344

21992.3737

2405.2532626.261

3

2856.982

Table 3. Minor Pressure Losses

Part I (c)

Figure 3. Pressure Losses vs. Flow Rate

Figure 3 shows the predicted the pressure losses (laminar and turbulent)

versus the flow rate for each flow. Also in this plot, the real data is plotted against

the predictions. The flow in the pipe is being considered as fully developed for this

portion, so error is seen within the graph.

Figure 4. Pressure Losses vs. Reynolds Number

In figure 4, the real data follows the predicted data fairly well given the fact

the entrance length of the flow has been ignored. Figure 2 shows that the major

pressure losses increase with the Reynolds Number. This is due to the fact that the

major pressure losses are dependent of Reynolds number. As the velocity increases,

the inertial forces increase, leading to an increasing Reynolds number, thus leading

to higher major pressure losses. From figure 2, the critical Reynolds number for the

noncircular pipes used in this experiment can be estimated. With this given plot, a

critical Reynolds number is estimated to be approximately between 1000 and 1500.

From this plot, it’s not fully clear where the flow laminar or turbulent, so this range

is the best estimate that can be made. For flows under this critical number, the flow

will be seen as fully laminar. For flows over the critical Reynolds number, the flow



will be fully turbulent. For flows in between these two Reynolds number, the flow

will be seen as transitional. The predicted values for the critical Reynolds numbers

are far less than that of circular pipes, which has a critical Reynolds number of

about 2300. This can be explained with the difference in the geometry of the pipes.

The fluid flowing through a noncircular pipe will have more surface area to “stick to”

(viscous forces) versus when it’s flowing through a circular pipe. The corners of the

noncircular pipe also add to the viscous affects because the flow cannot flow

smoothly through the pipe as to when there are no corners. In the fully developed

region of a flow, the velocity profile is assumed to be nearly parabolic. When in a

pipe with corners, the flow regions near the walls of the pipe are affected by the

corners, which explains the lower critical Reynolds number.

Part I (d)

Figure 5. Minor Pressure Losses Divided by Total Pressure Losses vs. Flow

Rate

Figure 5 shows the relative contribution of the theoretical minor loss

contribution to total pressure drop versus the flow rate. This ratio is found for both

the laminar and turbulent theoretical pressure losses. This plot shows that the ratio

of minor loss to total pressure loss in laminar flows increase more with flow rate

versus the ratio with turbulent pressure loss. This is because the velocity in the

turbulent flows is higher than that in laminar flows. The minor loss is low, but should

not be neglected in analysis because it does contribute to the pressure loss of the

flow. From this plot our estimate of critical Reynolds number made in part (c) is the

same.

Part I (e)

Once we have calculated the minor pressure losses for the flows, the

experimental pressure loss can be separated into major and minor pressure losses.

Once we have the experimental major pressure loss, the experimental friction

coefficient can be calculated.



fexp=∆Pmaj2DH3ρLμ2Re2 9

ΔPmaj f exp

514.9621324

0.7598069

543.008708

7

0.200455

61014.93196

20.166563

72206.52218

3 0.2037192704.35412

9 0.1598093542.64139

10.145386

94204.27717

50.126768

75670.53238

70.094434

7

8149.609007 0.077924111037.9951

20.068389

911681.2091

7 0.0506813836.8938

70.044369

216643.2045

80.041043

318548.8980

80.040594

518721.1898

20.036605

822828.7525

20.036257

626552.7276

2 0.03493329434.3601

10.035465

332306.2801

70.035782

2

Table 4. Experimental Major Pressure Loss and Friction Coefficient

Table 4 shows the tabulated experimental pressure losses and the experimental

friction coefficients using equation (9).

Figure 6. Friction Coefficient versus Reynolds Number (linear-linear)



Figure 7. Friction Coefficient versus Reynolds Number (log-log)

Figures 6 and 7 show the different friction coefficients (laminar, turbulent, and

experimental) versus the Reynolds number. The data points follow the same trend

as the Moody curves with fluctuations. The fluctuations in the data curve show uswhere the flow is laminar, transitional, and turbulent. With the help of these two

plots it’s easier to estimate Reynolds number from the data against the Moody

curves. From these plots, a critical Reynolds number is estimated to be 1300, which

is in our range of 1000-1500 made in part (c).

Part II: Estimating Accuracy of the “Fully Developed

Approximation.”

Part II (a)

In actual pipe flow, the flow is not fully developed for the whole length of the

pipe. There is an entrance length from the entrance to a certain length where the

flow has an increasing boundary layer which merges at the end of the entrance

length. The equation for the entrance length is

Le=.05ReDH (10)

Le,lam Le/L0.004404844

0.00722579

0.008806205

0.01444588

0.013207566

0.02166596

0.017608927

0.02888604

0.022010288

0.03610612

0.026411649 0.0433262

0.030813010.0505462

80.04146108

30.0680135

90.05471761

50.0897598

70.06797414

70.1115061

50.08123067

90.1332524

3



Table 5. Entrance Length for Laminar Flows

Table 5 shows the calculated entrance lengths and the ratio of the entrance

length to the whole length of the pipe for the laminar flows. This table shows that

the entrance length is much smaller compared to the entire length of the part, but

no real conclusions can be made yet about whether or not the entrance length can

be neglected or not.

Part II (b)

The flow at the inlet of the pipe is nearly uniform, so it is reasonable to say

that the peak velocity is approximately equal to the average velocity. This

approximation cannot be made at all other locations in the pipe because the peak

velocity exceeds the average velocity.

Velocity profile is given by:

ur= -R24µdPdx1- r2R2 (11)

The average velocity is given by:

V= -R28µdPdx (12)

Combining equation (11) and (12) gives:

ur=2V1- r2R2 (13)

For u (0), we have peak velocity. So,

Vpeak=2V (14)

Therefore, from equation (14) we can say that the average velocity in fully

developed laminar pipe flow is one half of the peak velocity. The average velocity is

the same at all axial locations x in the channel because the velocity profile no

longer depends on x in the fully developed region. When the boundary layers merge

at the end of the entrance region, the velocity profile for the rest of the flow is

parabolic to the outlet. In the fully developed region, the velocity is dependent of r

and x, r being the distance from the center of the pipe to the boundary layer. In the

fully developed region, the boundary no longer exists or in other words the walls of



the pipe become the boundary layer, thus the flow is constant in the fully developed

region.

Part II (c)

The Bernoulli equation is applicable on the streamline between the inlet and

the end of the entrance length because in this region the net viscous or frictional

forces are negligible compared to other forces acting on the fluid particles.

Pρ+V22+gz=constant 15

Equation (15) is the derived form of the Bernoulli equation. Since the Bernoulli

equation is applicable to the entrance length region of the flow, this equation can

be used to estimate ΔP1, which is defined as ΔP1=P1-Pe, as a function of the average

velocity. Pe is the pressure at the end of the entrance region, which will be called

point e and point 1 is the inlet of the pipe. The Bernoulli equation can be used to

estimate ΔP by setting the equation (15) for point 1 and point e equal to each other.

∆P1=ρ2Ve2-V12+ρgze-z1 (16)

After some manipulation, equation (16) will emerge. In this equation, both z terms

will be zero because we are using fluid particles along the streamline in the center

of the flow, so ze-z1 will equal zero. Using the approximations that V1 can be

approximated as the average velocity and that Ve=2Vavg, plugging these into

equation (16) would yield

∆P1=32ρV2 17

Equation (17) can be used to solve for the change in pressure in the entrance

region. Table 6 shows the tabulated changes in pressure in the entrance region for

the laminar flows.

ΔP1

4.289329

17.14375

38.56326

68.54787



107.0976

154.2124

209.8923

380.0226

661.8849

1021.447

1458.708

Table 6. Change in Pressure in Entrance Region

Part II (d)

The pressure loss for the fully developed region, ΔP2 is defined as Pe-P2, where

P2 is the pressure at the outlet. For the change in pressure in the fully developed

region, the Bernoulli equation is not applicable because the viscous effects of the

flow are not negligible.

∆P2=fLFDDHρV22 18

In equation (18), f is the experimental friction coefficient found in Part I (e) and LFD is

the total length of the pipe minus the entrance length. The change in pressure when

the entrance is ignored and the flow is assumed to be fully developed, ΔPFD, is thechange in pressure found using equations (4), (5), and (6) (laminar flows).

∆PFD=fLDHρV22 (19)

In equation (19) the f is the friction coefficient found using equation (5) for laminar

flows and L is the total length of the pipe.

L-L(entr) ΔP2 ΔPFD0.60519

5600.560

5604.931

60.60079

41191.91

41209.38

50.59639

2 1774.541813.83

8



0.591991

2348.437

2418.291

0.587592913.60

53022.74

50.58318

83470.04

53627.19

80.57878

74017.75

74231.65

10.56813

95306.71

75693.98

60.55488

26840.04

37514.54

80.54162

68294.18

99335.11

10.52836

99669.15

311155.6

7

Table 7. Length and Change in Pressure of Fully Developed Region after entranceregion and Change of Pressure if Whole Flow is Considered Fully Developed.

Table 7 shows the tabulated values of the length of the fully developed region after

the entrance length. This length is used to tabulate the change in pressure of the

fully developed region after the entrance length. Table 6 also shows the change in

pressure if the whole flow is considered fully developed, which is the change in

pressure for the fully developed region for laminar flows found in part I (b).

Part II (e)

With all the changes of pressure tabulated the percent error from part I can

be calculated using

ε=∆PFD∆P1+∆P2-1 20

Equation (20) can be manipulated to show the mathematical relationship between

the percent error and Reynolds number.

ε=fLDH3+fL-LeDH-1 21

In equation (21) the friction coefficient is a function of Reynolds number, which

shows the relationship of the error and Reynolds number. For equation (21) thefriction coefficient is the one predicted for laminar flow. This equation shows that

error will increase linearly with Reynolds number.



% Error 0.14873

4

0.297390.44608

6

0.594823

0.74360.89241

71.04127

41.40156

61.85044

92.29969

72.74931

2

Table 8. Percent Errors

Equation (21) calculates the error that is made by assuming that the flow through a

pipe is fully developed through the whole flow. The error calculated in equation (21)

is the error in one channel of the flow, so the error is multiplied by 11 to calculate

the error in the whole pipe. Table 8 shows the tabulated percent errors from the

fully developed approximation of part I. The percent error is found by multiplying

equation (21) by 100.

Figure 8. % Error of Making the Fully Developed Approximation versus the

Reynolds Number

Figure 8 shows the error from the fully developed approximation plotted against the

Reynolds number. The plot shows that percent error increases linearly with

Reynolds number, which is expected from equation (17). The calculated error shows

that the error made from the fully developed approximation is laminar flow is

insignificant. The analysis made in part I will not have to be redone when the

Reynolds number is below the critical value because the error is minimal.



Part II (f)

The relative error for a flow that is turbulent will be less compared to that of

laminar flow. In a fully turbulent flow, the boundary layers in the entrance region

will merge quicker than that of a laminar flow. The shorter entrance region will leadto the flow to more like a fully developed region, so making a fully developed

approximation will result in less of an error in turbulent flows.

Discussion and Summary

Things learned through this analysis:

• The flow through a noncircular pipe and a circular pipe is affected by the

different geometries. Also the circular pipe correlations can be used for

noncircular pipe flow after making a few approximations.

• The critical Reynolds number of a noncircular pipe was found to be about half

of that of a circular pipe. This was concluded after determining the predicted

values of major and minor pressure losses through the flow of the noncircular

pipes used in the experiment. This was explained by the fact that the flow

cannot flow smoothly through the noncircular pipes because of the geometry,

namely the corners.

• Making a fully developed approximation in laminar flows result in an

insignificant error when compared to the analysis done with the inclusion of

the entrance region. In turbulent flows, this error becomes even more

insignificant because the rate at which the boundary layers in the entrance

region will merge.

To reduce pressure loss while maximizing the aspect ratio of each lateral

tube, the length of the pipe or the flow rate can be reduced. The major pressure

loss is dependent on both of these things, so reducing them will reduce the overall

pressure loss. Also rounding the edges at the inlet and outlet of the pipe would

reduce the minor pressure loss of the flow, further reducing the pressure loss. Also

if the number of channels is reduced, this would lead to an increase in the effective

diameter, which would decrease the pressure loss.



References

1. Cengel, Y & Cimbala, J.M. 2006 Fluid Mechanics, McGraw-Hill, New York, NY

2. Cengel, Y.A., Boles, M.A. (year) Thermodynamics, An Engineering Approach (any

edition),

McGraw-Hill, New York, NY.

3. Incropera, F. P. and DeWitt, D.P. (year) Introduction to Heat Transfer (any

edition), John

Wiley and Sons, New York (or another undergraduate text in heat transfer).

4. Habte, M. 2003 Two-Phase Flow Mal-distribution in a Brazed Aluminum

Evaporator. M.S.

Thesis, The Pennsylvania State University, Department of Mechanical Engineering.

5. Write-Up

fluids case study on head loss

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