PRESSURE DROP IN PLATE HEAT EXCHANGERS FOR SINGLE-PHASE CONVECTION IN TURBULENT FLOW REGIME: EXPERIMENT AND THEORY PART 2







The characteristic velocity of the channel  is used for all three zones. This makes the data handling easier, and comparability of results between distribution zones and corrugated field is improved. A main disadvantage is that velocity  is not the physical velocity in distribution zones. The cross section in distribution zones differs from the one of the corrugated field and therefore velocity, too. As a result, friction factor finlet has only computational meaning.
From equations (10)–(13), we get
For PHEs P1–P4, four functions are received from experimental data:
Characteristic values for Reynolds number in the range Re = 1000,…, 10000 are chosen and called . Values  are according to equations (14) and (15), linear depending from . Using known mathematical procedure, we receive linear approximation for four points  for all characteristic values of . The resulting y-intercept of this linear function is  while  is the slope. The obtained values are presented in Table 2.

By approximating the data from Table 2, we obtain
The deviation of equations (16) and (17) is about 5% from data of Table 2. With equation (14), dimensionless pressure drop Fch becomes the form
The dimensionless pressure drop can be rearranged to Fanning friction factor of the channel (equations (6) and (14)). We get
According to equation (19), Fanning factor of the channel fch is related to characteristic length Lcorr. Earlier in Figures 6–8, the Fanning factor was based on characteristic length Lport. By using equations (16), (17), and (19) for plates P1–P4 with LL corrugated field, we receive
Calculated values from equation (20) are drawn as lines in Figure 9, and they are labeled with “Theory.” As can be seen, calculated data agree well with experimental data. The deviation is within ±5%. It must be emphasized that both equations (16) and (17) are only valid for specific geometry of distribution zone and corrugated field of plates P1–P4.

Distribution zones are equal for all patterns of corrugated fields. Due to this fact, equation (16) is valid for HH and HL channels, too. Based on experimental data for pressure drop ∆pch of HH and HL channels, the Fanning factor can be calculated from equations (11)–(13) for corresponding channel type.
To receive fcorr (HH), at first, dimensionless pressure drop is calculated for the HH corrugated field:
This allows us to calculate the Fanning friction factor of the channel for characteristic length Lcorr:
With known  from equation (16), we get
Values calculated from equation (23) result in a function depending on the Reynolds number. Following approximation is obtained from experimental data of the HH channel type:
The Fanning factor of the channel from equations (16) and (24) is
Calculated values of  from equation (25)—labeled as “Theory”—and experimental data according to equation (22) are compared in Figure 10.

Figure 10: Fanning friction factor  for plates P1–P4 with HH corrugated field.
For the HH channel type, data show only a slight dependency from the length of the corrugated field. This is due to a relatively low hydraulic resistance of distribution zones compared to HH corrugated field.
All calculations according to equations (21)–(25) are performed for HL channel type, too. We get
Calculated values of  from equation (27)—labeled as “Theory”—and experimental data are compared in Figure 11.


Figure 11: Fanning friction factor  for plates P1–P4 with HL corrugated field.
Experimental data agree with theoretical values within ±10%, and these are compatible with the uncertainty of the measurements.
The three-component model gives the opportunity to use data sets for twelve configurations (four plate lengths with three corrugation angles each) and describes with only four correlations for friction factors: , , , and . It is important to note that these correlations are length independent.
In Figure 12, all experimental data are compared with theoretical values for  of the three-component model. For this purpose, experimental data for all channels  were recalculated to experimental data for the corrugated field according to the following equation:













Figure 12: Fanning friction factor  vs. Reynolds number for plates P1–P4.
It can be seen that data from experiments of all four plates correspond to the theoretical calculations. Experimentally is confirmed that the factor  does not depend on the length of the corrugated field .
5.1. Dependence of Fanning Friction Factor  from Plate Length
Based on the proposed three-component model, a simple physical explanation of observed opposite behavior of Fanning friction factor  for HH and LL channels can be obtained. It is assumed that friction factor  is linear dependent from distribution zone length:where  is a unique relative pressure drop coefficient for each series of plate with equal geometry of distribution zone. It describes the specific hydraulic resistance of this zone. We derive from equation (19)
We introduce the relative length of the corrugated fieldand get
By increase in plate length Apl, both Lport and Lcorr increase; however, length of distribution zone remains unchanged:
With equation (31), we get,
As a result, if Lport increases, X approaches one (X ⟶ 1) and from equation (32),
For plates with very short corrugated field, X approaches zero (X ⟶ 0) and
The characteristic dependence of  from relative length of corrugated field X is shown in Figure 13. As a general rule, the Fanning friction factor of HH corrugated field is bigger than relative pressure drop of the inlet , while the friction factor of LL corrugated fields is smaller:







Figure 13: Characteristic dependence of friction factor  from relative length of 
corrugated field X.
For example, for PHE P1–P4 at Re = 1000, the friction factor is calculated as follows:
According to equation (35), an increase in plate length for the HH channel type should result in an increase in friction factor  that approaches . LL corrugated field has a relatively low pressure drop according to equation (37); therefore, with increase in plate length,  should decrease and approach . Our experimental data confirm this behavior.
5.2. Pressure Drop Fraction in Inlet and Outlet Distribution Zones
It is important to know what part of overall channel pressure drop corresponds to distribution zone. We introduce the inlet pressure drop ratio :
With equations (10) and (12),  can be written as
Equation (41) shows that always . The higher  is, the higher the portion of pressure drop in distribution zones is. For plates P1–P4, Figures 14–16 show calculated values for  for all three channel types.







Figure 14: Inlet pressure drop ratio  for plates P1–P4 for LL corrugated field.








Figure 15: Inlet pressure drop ratio  for plates P1–P4 for HL corrugated field.








Figure 16: Inlet pressure drop ratio  for plates P1–P4 for HH corrugated field.
The inlet pressure drop ratio  is highest in shortest plate P1 with LL corrugation angle. It is between 66% and 77%. Plates P2–P4 show values from 39% up to 61% as it can be seen in Figure 14. Plates with HH corrugated field shows values from 12% to 17% for shortest plate P1 (Figure 16). Longer plates have an inlet pressure drop ratio  from 4% up to 11%. The results of intermediate corrugation angle HL lay in-between HH and LL channels. Values for  from 28%–40% for plate P1 and 11%–29% for plates P2–P4 are calculated.
The calculated values of inlet pressure drop ratio show that the distribution zones play an important role in channel pressure drop for chevron angles of 30°–45° and lower.
5.3. Comparison of the Three-Component Model to Existing Data
Our approximation for the pressure drop factor of distribution zone  (equation (16)) is received by investigation of plates with particular distribution zone. Arsenyeva et al. [23] show the following equation:with friction coefficient fDZ calculated by
This equation is also obtained for a particular plate design. Therefore, a full quantitative match with our data can not be expected. Nevertheless, a comparison is of interest.
By comparing equations (12) and (42), it has to be noted that
For Re = 2700, we get from equation (43), fDZ = 38 and according to equation (44), finlet = 38/4 = 9.5 and from equation (16), finlet = 5.525 + 7500/2700 = 8.3.
The difference between equations (43) and (16) is about 13%; this is within the range of experimental uncertainties. Figure 17 shows calculated values of equations (43), (44), and (16). In the range Re = 1000, …, 40000, values from both equations agree well.




Figure 17: Pressure drop factor finlet for distribution zone from (i) equation (43) [23] and (ii) equation (16) the of three-component model.

6. Conclusions

(i)Pilot study of pressure losses of plate heat exchangers is conducted. The calculated Fanning friction factor is consistent with data from open literature.(ii)Experimentally, the dependence of Fanning friction factor on distance between ports Lport is determined. In case of corrugation pattern HH (chevron angle φ > 60°), Fanning friction factor  increases with increase in length Lport; in case of corrugation pattern LL (φ < 30°) in contrast, friction factor decreases with increase in length Lport. This dependence is confirmed by the data published in open literature.(iii)For pressure losses in PHE, the enhanced three-component model is offered. This model accounts for the hydraulic resistance of distribution zones and the correlations for corrugated field. It corresponds to experimental data with an error up to 10%.(iv)On the basis of three-component model is explained why at chevron angles φ > 60° Fanning friction factor  increases with increase in length Lport, and in case of φ < 30° in contrast, friction factor decreases with increase in length Lport.(v)The portion of hydraulic resistance of distribution zone in general resistance of channel is calculated. This portion can be considerably (up to 70%) in case of corrugation angles φ < 30°. For corrugation angles of φ > 60°, it can reach 10–15%.(vi)Comparison of the received dependence of hydraulic resistance of a distribution zone on the Reynolds number with the known data from literature is carried out. Good correspondence is obtained.

International Journal of Chemical Engineering Volume 2019, Article ID 3693657, 11 pages       https://doi.org/10.1155/2019/3693657

Nomenclature

A:Area of PHE (m2)
B:Width (m)
D:Port diameter (m)
De:Equivalent diameter = 2t (m)
Dh:Hydraulic diameter = 2t/FK (m)
F:Fanning friction factor, dimensionless
FK:Surface enlargement factor, dimensionless
L:Length (m)
Re:Reynolds number, , dimensionless
t:Wave amplitude (m)
T:Temperature, °C
IR:Inlet pressure drop ratio, dimensionless
w:Velocity (m/s)
X:Relative length of corrugated field, dimensionless
Δp:Pressure losses, Pa
Δwall:Wall thickness (m)
Λ:Wavelength of corrugation (m)
λ:Heat conduction coefficient (W/mK)
µ:Dynamic viscosity (kg/(m·s))
ρ:Density (kg/m³)
φ:Chevron corrugation angle relative to flow direction, degree
corr:Corrugation
ch:Channel
inlet:Inlet
mean:Mean
outlet:Outlet
pipe:Pipeline
pl:Plate
port:Port distance
proj:Projection
wall:At wall temperature.

Data Availability

The measurement data used to support the findings of this study have not been made available because these data are business secret of FUNKE Wärmeaustauscher Apparatebau GmbH.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors would like to thank our employer FUNKE Wärmeaustauscher Apparatebau GmbH for providing the opportunity to publish this article.

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