Pressure Losses Across
Evaporators
Chelsea Buxton
Abstract:
This case study analyzes the evaporator section of a refrigerant system. Fluid dynamics will be
used to address the effects of friction on pressure drop through the tubes. The flows within the tubes
transition from laminar to turbulent and move through the entrance length and fully developed region.
The main question being addressed is how much this friction affects the overall pressure drop and if it
needs to be taken into account when doing calculations to determine this pressure drop. Bernoulli’s
equation will be used along with the head loss factor. The data collected was the pressure change over
the tube length of the evaporator through the use of a differential pressure gage and the volume flow
rate from a variable-area flow meter. These two pieces of data will be compared with the experimental
data to find the error created from the use and negligence of friction/head loss. The most important
results taken from this analysis are that neglecting friction in laminar flows does not result in a large
error and the critical Reynolds number in a rectangular tube is slightly lower than the accepted critical
Reynolds number for circular tubes. The overall conclusion that can be made from this case study is that
friction is important but it can be neglected in the laminar flow region and result in small percent errors
but in the turbulent region the friction should be taken into account because the error will get larger
and larger at a linear rate.
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.) If the refrigerant is in a saturated state and sufficient latent heat enters the
refrigerant, it changes phase (i.e., it “evaporates”).
Figure 1 shows the overall geometry and flow directions of the fluids in the evaporator that is
the subject of this analysis. The liquid enters through the “top diving header”, flows through the lateral
tubes and collects in the “bottom combining header” in its vapor state. Figure 2 shows a cross sectional
view of the lateral tubes in which the liquid undergoes vaporization. Each individual lateral tube has 11
channels, also pictured in figure 2 along with their dimensions. If liquid water was placed into the
evaporator, no phase change would occur. This would result in a pressure drop between the inlet and
outlet of each lateral tube that is dependent upon the Reynolds number of the fluid. This pressure drop
is the same as the pressure drop from the inlet of a single tube channel to its exit. Data was collected by
Habte [ref 2] as a preliminary analysis of two-phase pressure drop in the full evaporator.
Flow transitions from laminar to turbulent when the Reynolds number exceeds a critical value
which is quoted to be approximately 2300 in a circular tube. Recrit can vary with the cross-sectional
shape of the channel, roughness, flow conditions, and other factors. Pressure drop is affected by this
Reynolds number along with inlet and outlet geometries, entrance length, and tube roughness.
Pressure drop due to frictional head loss in real tubes of non-circular cross section (in this case,
an individual flow channel of the lateral tubes of the evaporator) vary with flow rate. This overall
pressure drop is extremely important to the performance of the heat exchanger and has great impact
upon the other components that are chosen for the system, such as the compressor.
The first part of this case study will focus upon the experimental evaluation of head loss and
compare it was standard engineering correlations by approximating the flow as fully developed
throughout the lateral tube. The second part will recognize that there exists an entrance length causing
for the fully developed flow to be limited to length of the tube after this entrance length. An analysis
was carried out to estimate the error introduced by the fully developed approximation that were made
in the first part.
Setup, Data, and Methods of Analysis:
The tests were performed on a single aluminum flat tube with 11 channels as shown in Fig. 2.
The channel length was 24 inches, and because the tubes were new and smooth when the tests were
performed, roughness effects were negligible. All data were collected at room temperature, about 20C.
Test Setup
As illustrated in Fig. 3, the two ends of the flat tube were connected to plenum chambers each
of which was connected to a flexible hose. The cross section of the plenum is shown in Fig 4. 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 data that was collected from the
differential pressure gage and the variable-area flow meter are shown in table1.
Table1. Data collected by Hatbe in his experiment for two-phase flow mal-distribution in a
brazed aluminum evaporator.
Q volume flow rate [gal/hr]
1.0
2.0
3.0
4.0
5.0
6.0
7.0
9.5
12.5
15.6
18.6
21.6
24.7
P1 - P2 Pressure drop [psi]
0.075
0.08
0.15
0.325
0.4
0.525
0.625
0.85
1.23
1.675
1.8
2.15
2.6
26.2
27.7
30.7
33.8
35.3
36.8
2.9
2.95
3.6
4.2
4.65
5.1
The first objective carried out in this section of the case study was to see how well predicted
estimates for pressure drop vs. flow rate and Reynolds number using correlations for major loss (friction
factor) and minor loss (pressure loss coefficient) from a circular pipe compare with experimentally
measured pressure drops in the channels of rectangular cross section. These data and predictions were
then used to determine the Reynolds number below which the flow in the channels in laminar and
above which it is turbulent. Finally it was observed that real data of flows are more complex than the
basic flow studies displayed in test books. It was assumed in this section that the entire length of the
tube exhibits fully developed flow.
The data displayed in table 1 was converted into standards units of m3/s for volume flow rate
and Pa for pressure drop in order to keep the same units throughout the analysis. The evaporator has a
rectangular cross section rather than a circular one so an “effective diameter” needed to be calculated
for the analysis. The “effective diameter” that was used is the same as the hydraulic diameter, DH. The
calculation of this diameter then permitted the calculation of the Reynolds number for each flow rate Q
into the flat tube.
Through the approximation of the flow throughout the channels to be fully developed and
applying the friction factors for both laminar and turbulent flows, two predictions were made for
pressure drop as a function of flow rate Q and Reynolds number Re. From this calculated data two
curves were created; one for laminar flow and the other for turbulent flow permitting for the estimation
of the critical Reynolds number. The overall pressure drop is a function of pressure losses created by
both major and minor losses.
After the overall pressure change, the minor pressure changes, the major turbulent flow
pressure changes, and the major laminar pressure changes were calculated and plotted, the relative
contribution of the theoretical minor loss contribution to the total pressure drop was plotted against the
volume flow rate. This was then used to allow for another estimation of the critical Reynolds number.
Due to the fact that ΔPminor << ΔP, we were able to neglect the minor loss contribution to the
pressure change and just use the pressure change due to the major losses. We used these pressure
changes to calculate the friction factor for the given data, turbulent flow, and laminar flow. These were
then plotted against Reynolds number on the same graph (in both linear scale and log-log) to allow for
one final update of the critical Reynolds number.
By assuming in part 1 that the flow in the entire length of the tubes was fully developed, error
was created in the measurement of the friction factor. In reality, a boundary layer grows from the inlet
of the tube and grows until it has completely filled the tube. The length over which this occurs is called
the entrance length and the remaining portion of the tube contains the fully developed flow as show in
figure 5.
The objective of the second part of this case study was to estimate the percent error associated
with treating the flow as fully developed in the entire tube, including the entrance length. The first
pressure change occurs across Le and the second occurs across LFD. These two pressure differences were
added together to create the overall pressure change between the inlet and outlet of the tube. In this
section of the study we modeled the flow in the rectangular channels with the flow through a circular
tube with the effective diameter because there is no analytical solution for laminar flow in a rectangular
duct.
Flow is laminar where Re is less than Recrit and these are the flow rates upon which we focused
for this section of the study. The ratio of the entrance length to the overall length was calculated for
each flow rate where flow was laminar. Entrance length is a factor of both Reynolds number and
effective diameter.
Due to the fact that the flow at the inlet is nearly uniform, it can be assumed that the peak
velocity at the entrance can be approximated with the average velocity at the entrance. This cannot be
assumed at any other location because the peak velocity will exceed the average velocity. A relationship
was worked out between the peak velocity and the average velocity in the fully developed region using
the integration of the equation for the velocity profile over the radius of the tube. This was then used to
prove that average velocity is the same at all axial locations in each channel.
The Bernoulli equation is an approximate relation between pressure, velocity, and elevation and
is only valid in regions of steady, incompressible flow where net frictional forces are negligible [ref 1].
This means that it cannot be used in regions where friction occurs, or fully developed regions. This
allowed for the equation to be used to mathematically estimate the change in pressure over the
entrance length as a function of the average velocity.
Bernoulli’s equation was also used to estimate the pressure change over the fully developed
region and the pressure change if the whole length was fully developed as functions of average velocity
also, but in order to do this head loss had to be taken into account. The head loss adds the friction
factor into the equation.
The percent error between the pressure change of the tube including the entrance length and
the fully developed length was then calculated to demonstrate why it cannot be assumed that flow in
the tube is always fully developed. The result was plotted linearly vs. the Reynolds number. The
percent error was not found to be the same for any two Reynolds numbers.
To finalize the second part of the case study, the relative error was considered when flow was
turbulent and compared to the relative error that was calculated when the flow was laminar. These two
flows will result in different amounts of error.
Analysis and Results:
-Part1-
(a). The tubes that are in the evaporator that was used for the experiment were rectangular, not
circular, so an “effective diameter”(1) was needed for the calculations. This diameter is the same as the
hydraulic diameter:
2𝑎𝑏
𝐷𝐻 =
(1)
𝑎+𝑏
This is calculated where a and b are the cross sectional dimensions of each flow channel. This gave a
result of a hydraulic diameter of 0.001286 mm. This diameter, along with the volume flow rate was used
to calculate the Reynolds (2) number for each individual volume flow rate in the flat tube as shown in
table 2. The relationship between flow rate and average velocity was provided to simplify the equations
throughout the analysis (3). The volume flow rates that were provided were changed to the correct
units (m3/s) and divided by 11 due to the fact that there are 11 channels but for this experiment the
focus was only upon 1.
𝑅𝑒 =
̅ 𝐷𝐻
𝜌𝑉
(2)
𝜇
𝑄
𝑉̅ = 𝐴
(3)
Table 2. The volume flow rate in the standard units per individual tube along with the
number for each volume flow rate.
Q (m^3/s) Per Vent
9.620E-08
1.923E-07
2.884E-07
3.846E-07
4.807E-07
5.768E-07
6.729E-07
9.055E-07
1.195E-06
1.485E-06
1.774E-06
2.064E-06
2.353E-06
2.498E-06
2.643E-06
2.932E-06
3.222E-06
3.366E-06
3.511E-06
calculated Reynolds
Reynolds Number
68.28490432
136.5158066
204.7467089
272.9776112
341.2085135
409.4394158
477.6703181
642.7391651
848.2449513
1053.750737
1259.256524
1464.76231
1670.268096
1773.020989
1875.773882
2081.279668
2286.785455
2389.538348
2492.291241
(b). Pressure drop can be calculated as a function of the friction factor, density, and average velocity.
Major frictional losses are calculated using different equations for turbulent (5,6) and laminar (4) flow
and were found in the text [ref 1]. The equation for the frictional factors due to turbulent flow needed
to be restructured to be equivalent to the frictional factor, f:
LAMINAR:
𝑒
64
𝑓 (𝑅𝑒 , 𝐷) = 𝑅𝑒
(4)
TURBULENT:
1
√𝑓
= −1.8 log [
6.9
𝑅𝑒
+(
𝜖
𝐷
3.7
1.11
)
]
(5)
2
𝑓=(
−1
1.8 log(
6.9
)
𝑅𝑒
)
(6)
These frictional factors are the main driving force in the major pressure change that occurs between the
inlet and outlet of the tube. Equations (7,8) were obtained to directly calculate both the pressure
changes due to major and minor losses. It was taken into account that the roughness of the walls of the
tube was negligible:
1
𝐿
𝑒
∆𝑃𝑚𝑎𝑗 = 2 𝜌𝑉̅ 2 𝐷 𝑓 (𝑅𝑒 , 𝐷)
𝐻
1
∆𝑃𝑚𝑖𝑛 = 2 𝜌𝑉̅ 2 ∑ 𝐾
(7)
(8)
From the text book [ref 1] it was determined that the loss coefficient, K, for the inlet of a straight sharpedged tube is 0.5 and α for the outlet. It can be determined that the correction factor, α, is equal to 1.
These loss coefficients are the same for both laminar and turbulent flow which means that the same
equation was used for the pressure changes due to minor losses for both laminar and turbulent flow.
When the frictional losses were added into the equation for pressure changes (7,8) equations for the
major (9,10) pressure changes as functions of flow rate Q, and Reynolds number were found. The loss
coefficients were used to create an equation (11) for the minor losses that was independent of the
Reynolds number. These calculations values are displayed in table 3.
LAMINAR:
1
𝐿 64
∆𝑃𝑚𝑎𝑗 = 2 𝜌𝑉̅ 2 (𝐷 ) 𝑅𝑒
𝐻
(9)
TURBULENT:
2
∆𝑃𝑚𝑎𝑗
1
𝐿
= 2 𝜌𝑉̅ 2 (𝐷 ) (
𝐻
−1
1.8 log(
6.9
)
𝑅𝑒
)
(10)
LAMINAR + TURBULENT:
1
∆𝑃𝑚𝑖𝑛 = 2 𝜌𝑉̅ 2 (1.5)
(11)
The pressure changes due to the minor losses were added to the pressure changes due to the major
losses to calculate the overall pressure changes vs. flow rate Q and Reynolds number Re. These
calculated values are displayed in table 3.
Table 3. Calculated values for the change in pressure due to minor and major losses along with total pressure
change for both laminar and turbulent flows as functions of Q and Re.
[ΔPmajor]lam (Pa)
636.3150092
1272.126799
1907.938589
2543.750379
3179.562168
3815.373958
4451.185748
5989.384943
7904.397017
9819.409091
11734.42116
13649.43324
15564.44531
16521.95135
17479.45739
19394.46946
21309.48153
22266.98757
23224.49361
[ΔPmajor]turb (Pa)
211.4514349
498.373148
869.0078354
1312.549073
1822.836968
2395.617527
3027.691146
4787.608966
7404.721309
10462.5421
13936.31422
17807.15784
22059.98772
24325.67389
26682.3482
31663.70051
36994.95774
39789.27751
42668.16593
ΔPminor (Pa)
2.138322319
8.546526382
19.22461353
34.17258375
53.39043706
76.87817345
104.6357929
189.4494231
329.963822
509.2130694
727.1971655
983.9161102
1279.369903
1441.622368
1613.558545
1986.482036
2398.140375
2618.495113
2848.533562
ΔPlam (Pa)
638.4533
1280.673
1927.163
2577.923
3232.953
3892.252
4555.822
6178.834
8234.361
10328.62
12461.62
14633.35
16843.82
17963.57
19093.02
21380.95
23707.62
24885.48
26073.03
ΔPturb (Pa)
213.5898
506.9197
888.2324
1346.722
1876.227
2472.496
3132.327
4977.058
7734.685
10971.76
14663.51
18791.07
23339.36
25767.3
28295.91
33650.18
39393.1
42407.77
45516.7
(c). These calculations in table 3 were plotted vs. Q and Re to then determine what the critical Reynolds
number is. The turbulent and laminar flows were plotted on separate figures along with the data
provided from the experiment. By evaluating these plots the critical Reynolds number can be
determined from the point on each graph at which the data seems to follow the curve for each type of
flow. Figures 6 and 7 pertain to laminar flow, while 8 and 9 pertain to turbulent flow.
Pressure [Pa]
Pressure Change vs Q Laminar
40000
35000
30000
25000
20000
15000
10000
5000
0
0.000E+001.000E-062.000E-063.000E-064.000E-06
Laminar
data
Flow Rate (m3/s)
Figure 6. Graph of pressure change for laminar flow versus the provided volume flow rates.
Pressure [Pa]
Pressure Change vs Re Laminar
40000
35000
30000
25000
20000
15000
10000
5000
0
Laminar
data
0
500
1000
1500
2000
2500
3000
Reynold's Number
Figure 7. Graph of pressure change for laminar flow versus Reynolds number calculated from effective diameter
and the provided volume flow rates.
Pressure Change vs Q Turbulent
50000
Pressure [Pa]
40000
30000
Turbulent
20000
data
10000
0
0.000E+001.000E-06 2.000E-06 3.000E-06 4.000E-06
Flow Rate (m3/s)
Figure 8. Graph of pressure change for turbulent flow versus the provided volume flow rates.
Pressure Change vs Re Turbulent
50000
Pressure [Pa]
40000
30000
Turbulent
20000
data
10000
0
0
1000
2000
3000
Reynold's Number
Figure 9. Graph of pressure change for turbulent flow versus Reynolds number calculated from effective diameter
and the provided volume flow rates.
Through analysis of figures 6-9 it can be estimated that Recrit is about 2100. Recrit is the Reynolds
number at which the flow transitions from laminar to turbulent. Figure 7 shows that the data points
begin to stray from the laminar curve at a Reynolds number of about 2100. Figure 9 can be used to
show that at about 2100 the data points begin to follow the same slope as the turbulent curve. Recrit is
fairly obvious when using figure 7 because up until Re of 2100 the data points almost perfectly follow
the laminar curve, but in figure 9, it is not as obvious because the data points do not lay on the curve
they just follow the slope of it. This value of 2100 is fairly close to the quoted value of 2300 this
difference is most likely due to the fact that 2300 is the accepted value for circular tubes. This would
lead to the assumption that corners lead to the flow becoming turbulent at a lower Reynolds number
and volume flow rate. Logically this would make sense because corners create a surface that is much
less uniform than a circular tube. This would account for the difference in the critical Reynolds number.
(d). The relative contribution of the theoretical minor loss contribution to the total pressure drop can be
defined as ΔPminor/ΔP and when plotted vs. the volume flow rate from the theoretical estimates for both
laminar and turbulent correlations can be used to estimate the critical Reynolds number. This was the
next part of the analysis and is displayed in figure 10.
Minor Pressure change/Total Pressure Change
ΔPminor/ΔP vs Q
0.12
0.1
0.08
data
0.06
Laminar
0.04
Turbulent
0.02
0
0.000E+005.000E-071.000E-061.500E-062.000E-062.500E-063.000E-063.500E-064.000E-06
Flow Rate (m3/s)
Figure 10. The relative contribution of the theoretical minor loss contribution to the total pressure drop can be
defined vs. the volume flow rate for the data, laminar flow, and turbulent flow.
From figure 10 Recrit can be estimated to occur at a volume flow rate of approximately 3E-6 m3/s. This is
the point where the data curve seems to change from follow the laminar curve to the turbulent curve.
From looking at table 2 it can be determined that Recrit occurs at about 2100. This is the same
estimation that was obtained from the plots used in (c). The laminar curve is linear in all of the plots for
(c) and (d). The turbulent curves for sections (c) and (d) are both parabolic. The data curve changes
from linear to parabolic showing its transition from laminar to turbulent.
(e). ΔP was obtained from the experiment and ΔPminor was obtained through an equation (11) and is
much smaller than ΔP. This means that the data can be used to estimate the true friction factors as a
function of Reynolds number for this rectangular channel flow. The major losses were used to calculate
a measured friction factor for each flow rate using equations obtained from the text [ref 1]. The
pressure change for the major losses for the data can be calculated from the difference between the
pressure change obtained from the experiment and the calculated change in pressure due to minor
losses (12). This is then used to find the frictional factor for the data:
∆𝑃𝑚𝑎𝑗(𝑑𝑎𝑡𝑎) = ∆𝑃𝑑𝑎𝑡𝑎 − ∆𝑃𝑚𝑖𝑛𝑜𝑟
𝑓𝑑𝑎𝑡𝑎 = (
(12)
𝐷𝐻 ∆𝑃𝑚𝑎𝑗(𝑑𝑎𝑡𝑎)
) 1 ̅2
𝐿
𝜌𝑉
(13)
2
The frictional factors for the laminar flow (4), turbulent flow (6), and data (13) were calculated and then
plotted in two forms, linear and log-log versus the Reynolds number to be compared with the accepted
curve for fully developed circular pipe flow (moody curves). The curves drawn are figures 11 and 12.
Friction Factor
f vs Re Linear Scale
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
data
Laminar
Turbulent
0
500
1000
1500
2000
2500
3000
Reynold's Number
Figure 11. Linear plot of the friction factors vs. Reynolds number for laminar flow, turbulent flow, and the data.
f vs Re log-log scale
1
Friction Factor
1
10
100
1000
10000
data
0.1
Laminar
Turbulent
0.01
Reynold's Number
Figure 12. Log-log plot of the friction factors vs. Reynolds number for laminar flow, turbulent flow, and the data.
From evaluating figures 11 and 12 it seems to be a little bit more difficult to estimate Recrit. Most people
have a difficult time reading log-log plots, and the linear plot in figure 11 has lines that are extremely
close together and difficult to read. From these two figures Recrit can be estimated to remain around
2100. The fact that each plot has shown the critical Reynolds number to be 2100 leads to the
assumption that this is most likely a value that is fairly close to the true value.
Part II
(a). Due to the fact that the flow through the entire tube cannot be considered fully developed, there is
a section of the tube that is called the entrance length. The entrance length was determined through an
equation provided by the text [ref 1] (14):
𝐿𝑒 = 0.05𝑅𝑒𝐷𝐻
(14)
This entrance length is the region of the tube in which the boundary layer grows and then when the
boundary layer meets the fully developed flow begins. Le/L was tabulated for each flow rate where the
flow is laminar (Re<2100) and can be seen in table4.
Table 4. Le/L for each volume flow rate that occurs where the flow is laminar.
Le/L
0.007169
0.014332
0.021496
0.028659
0.035822
0.042986
0.050149
0.067479
0.089055
0.11063
0.132205
0.153781
0.175356
0.186144
0.196932
0.218507
(b). The flow at the inlet is nearly uniform leading to the assumption that the peak velocity at the
entrance is equal to the average velocity at the inlet. Using an equation for the velocity profile (15) from
the text [ref 1], a relationship (19) between the peak velocity and the average velocity in the fully
developed flow can be derived. Equations (16), (17) ,and (18) show steps of the derivation in order to
demonstrate the validity of the final relationship (19):
2
𝑟
𝑉𝑚𝑎𝑥 (1 − 𝑅2 ) = 𝑉̅
𝑅
𝑟2
∫0 𝑉𝑚𝑎𝑥 (1 − 𝑅2 ) 𝑑𝐴 = 𝑄
𝑑𝐴 = 2𝜋𝑟𝑑𝑟
𝑅
𝑟2
∫0 𝑉𝑚𝑎𝑥 (1 − 𝑅2 ) 2𝜋𝑟𝑑𝑟 = 𝑄
𝑉𝑒 = 2𝑉̅𝑒
(15)
(16)
(17)
(18)
(19)
Due to the fact that the velocity profile is the same at all points in the fully developed region and there is
a direction relationship between the peak velocity and the average velocity it can be assumed that the
average velocity is the same at all axial locations in the channels. The average velocities for all volume
flow rates in laminar flow can be seen in table 5.
Table 5. The average velocities for volume flow rates in laminar flow region.
Vavg
0.026722
0.053423
0.080123
0.106824
0.133525
0.160226
0.186926
0.251523
0.331943
0.412364
0.492784
0.573205
0.653625
0.693835
0.734045
0.814466
(c). By definition, the Bernoulli equation can only be used when frictional forces are negligible. The
frictional forces in the tube do not occur until it has developed into a fully developed flow so head loss
can be ignored because in the entrance length the flow is not fully developed. This entrance length
occurs between point 1 and point e. The Bernoulli equation (21) for this region can be used to estimate
ΔP1(20) as a function of average velocity (22):
∆𝑃1 = 𝑃1 − 𝑃𝑒
𝑃1
𝜌
1
+ 2 𝑉̅ 2 + 𝑔𝑧1 =
𝑃𝑒
𝜌
(20)
1
+ 2 𝑉𝑒2 + 𝑔𝑧𝑒
1
∆𝑃1 = 2 𝜌(𝑉𝑒2 − 𝑉̅ 2 )
(21)
(22)
The z terms in the equation (21) drop out due to the fact that they are both the same leaving just the
velocity and pressure terms to be used to calculate the final equation for the change in pressure.
(d). ΔP2 (23) and [ΔP]FD (24) can be calculated in much the same manner, but due to the fact that they
occur in the fully developed region this means that friction much be taken into account in the form of
head loss (26). Taking Bernoulli’s equation and calculating it over the fully developed region can be used
to find ΔP2 (25):
𝑃𝑒
𝜌
∆𝑃2 = 𝑃𝑒 − 𝑃2
(23)
∆𝑃𝐹𝐷 = ∆𝑃1 + ∆𝑃2
(24)
1
+ 2 𝑉𝑒2 + 𝑔𝑧𝑒 =
ℎ𝑓 = 𝑓
𝑃2
𝜌
̅2
𝐿𝐹𝐷 𝑉
𝐷𝐻 2
1
+ 2 𝑉𝑒2 + 𝑔𝑧2 + 𝜌𝑔ℎ𝑓
(25)
(26)
The head loss equation (26) can be taken directly from its definition and can be found in the text [ref 1].
The addition of this head loss factor to Bernoulli’s equation takes friction into account and with the
addition of the friction factor equation for laminar flow (4) can be used to find the pressure changes (27)
(28) when placed into (25) and (26). The length used to find the second pressure change is the length of
the fully developed region because this is the region across which the pressure change is being
calculated. When it is assumed that the entire region is fully developed ([ΔP]FD ) for the whole length is
used. All of the pressure changes are displayed in table6.
̅2
64 𝐿𝐹𝐷 𝑉
∆𝑃2 = 𝜌 𝑅𝑒
(27)
𝐷𝐻 2
̅2
64 𝐿 𝑉
∆𝑃𝐹𝐷 = 𝜌 𝑅𝑒 𝐷
𝐻
(28)
2
Table 6. The pressure changes across the entrance length, the fully developed length, and the entire length when
it is assumed that the whole tube is fully developed.
delP1
1.069161
4.273263
9.612307
17.08629
26.69522
38.43909
52.3179
94.72471
164.9819
254.6065
363.5986
491.9581
639.685
720.8112
806.7793
993.241
delP2
157.9383
313.4736
466.7315
617.7122
766.4156
912.8418
1056.991
1396.307
1800.119
2183.272
2545.767
2887.603
3208.781
3361.623
3509.3
3789.16
delPfd
159.0788
318.0317
476.9846
635.9376
794.8905
953.8435
1112.796
1497.346
1976.099
2454.852
2933.605
3412.358
3891.111
4130.488
4369.864
4848.617
(e). An equation was provided to calculate percent error (29) in the write up for this analysis:
% 𝑒𝑟𝑟𝑜𝑟 = 𝜀 =
∆𝑃𝐹𝐷
∆𝑃
−1
(29)
The ΔP can be calculated through the addition of ΔP1 and ΔP2 (30):
̅2
1
64 𝐿
𝑉
∆𝑃 = 2 𝜌(𝑉𝑒2 − 𝑉̅ 2 ) + 𝜌 𝑅𝑒 𝐷𝐹𝐷 2
𝐻
(30)
This can then be simplified to (31):
% 𝑒𝑟𝑟𝑜𝑟 =
̅
0.1𝑉
32𝜇𝐿
̅
−0.1𝑉
𝐷2
𝐻
(100)
(31)
The percent error calculated for each pressure difference in the laminar flow region was calculated and
can be seen in table 7. They were then graphed vs Reynolds number in figure 13.
Table 7. The overall pressure difference and percent error calculated for each of these pressure difference in the
laminar flow region.
delP
159.0075
317.7468
476.3438
634.7985
793.1109
951.2809
1109.309
1491.031
1965.1
2437.879
2909.365
3379.561
3848.466
4082.434
4316.079
4782.401
% error
0.044826
0.089658
0.134529
0.179441
0.224392
0.269385
0.314417
0.423531
0.559706
0.696252
0.833168
0.970457
1.108121
1.177094
1.246161
1.384578
Percent Error vs Re
1.6
Percent Error
1.4
1.2
1
0.8
0.6
Percent Error vs Re
0.4
0.2
0
0
500
1000
1500
2000
2500
Reynold's Number
Figure 13. Plot of the Percent Error vs the Reynolds number for each pressure change in the laminar flow region.
The percent errors are so small that it does not seem to warrant redoing analysis in part 1 to take
entrance length into account. The percents start out extremely small but then get larger as the
Reynolds number reaches the critical Reynolds number. The largest percent error calculated is 1.38%
which is very small. The percents also grow at a linear rate with Reynolds number.
(f). Due to the fact that the error grows linearly with Reynolds number and turbulent flows occur at
higher Reynolds numbers, it is to be assumed that the relative error will be larger when the flow is
turbulent. Physically, this would be logical due to the fact that turbulent flow is characterized by
velocity fluctuations and highly disordered motion [ref 1] creating greater friction between the fluid
particles and increasing the head loss. This means that the pressure change calculated from the
Bernoulli equation would lead to a higher error because of this larger head loss.
Discussion and Summary:
The analysis of the data and calculations of this case study created a deeper understanding of head
loss due to laminar and turbulent flow through pipes. A multitude of new knowledge was produced:

The critical Reynolds number for flow through a rectangular tube is similar to, but a little lower
than the critical Reynolds number that is accepted for flow through a circular tube.

The critical Reynolds number can be easily computed through the interpretation of graphs for
laminar and turbulent flows.

The effective diameter used in a rectangular tube is the same as the hydraulic diameter.

The minor losses for laminar and turbulent flow are the same.

The error that is calculated for pressure change when friction is taken into account and when it
is not is greater with regard to turbulent flow than laminar flow.
All of these new pieces of knowledge can be used to greater develop ones understanding of fluid
dynamics. Relating the equations and assumptions to real life scenarios make them much more
concrete and easier to understand.
Due to the fact that higher pressure drops lead to more expenses in the overall development of
evaporators methods to reduce the pressure drop while maximizing the aspect ratio need to be
implemented. Some possible methods of doing this are:

Through increasing the overall size of the tubes, this would increase the effective diameter
and decrease the pressure drop. To increase the aspect ratio, the width needs to be
increased more than the length.

The width of the tubes can be increased while the length remains the same to increase the
aspect ratio and slightly decrease the pressure drop. The pressure drop can be decreased
through the use of shorter tubes.
Overall, this case study led to the deepening of knowledge on the subject matter. Fluid
dynamics is evident all over the world and further understanding of this can be implemented into
different mechanisms, such as evaporators, to create products that work much more efficiently.
References:
1. Cengel, Y & Cimbala, J.M. 2006 Fluid Mechanics, McGraw-Hill, New York, NY
2. Habte, M. 2003 Two-Phase Flow Mal-distribution in a Brazed Aluminum Evaporator. M.S. Thesis,
The Pennsylvania State University, Department of Mechanical Engineering.