Characteristics of flow distribution in compact parallel flow heat exchangers, part I: Typical inlet header

Characteristics of

flow distribution in compact parallel flow heat exchangers,

part I: Typical inlet header

Chi-Chuan Wang

a

, Kai-Shing Yang

b

, Jhong-Syuan Tsai

c

, Ing Youn Chen

c,* a_{Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, ROC}

b_{Green Energy & Environment Research Labs, Industrial Technology Research Institute, Hsinchu 310, Taiwan, ROC} c_{Department of Mechanical Engineering, National Yunlin University of Science and Technology, Yunlin 640, Taiwan, ROC}

a r t i c l e i n f o

Article history:

Received 23 February 2011 Accepted 2 June 2011 Available online 12 June 2011 Keywords:

Flow distribution Header

Parallelflow heat exchangers

a b s t r a c t

This study experimentally and numerically investigates the single-phaseflow into parallel flow heat exchangers with inlet and outlet rectangular headers having square cross section and 9 circular tubes. The effects of inletflow condition, tube diameter, header size, area ratio, flow directions (Z and U-type), as well as the gravity are investigated. The experimental results indicate thatflow distribution for U-type flow is more uniform than Z-type flow. Depending on the inlet volumetric flow rate, the flow ratio at the first several tubes can be lower than 50% of the last tube for Z-type arrangement, and this phenomenon becomes more and more pronounced with the rising velocity at the intake conduit. The mal-distribution can be eased via reducing the branching tube size or increasing the entrance settling distance at the intake conduit. It is found that the influence of gravity on mal-distribution is negligible and the mal-distribution is associated with the jetflow pattern.

Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Flow distribution from a header into parallel channels appli-cable to heat exchangers is frequently encountered in heat transfer equipments, such as condensers, evaporators, boilers, solar energy flat-plate collectors, and cooling system of nuclear reactor. Appar-ently, the _{flow rates of single-phase distribution through the} parallel channels are often not uniform which could greatly affect the heat transfer performance, these heat transfer devices may suffer from significant performance drop subject to mal-distribution. Therefore, the issue of uniformflow distribution has recently received growing attentions for the heat exchanger design. The uneven distribution in parallel channels could be related to the stream velocity in the header (or manifold), size of the header, diameter of the tube, location and size of inlet port to the header,flow direction, orientation of the channels and the like. Thus, it is very imperative to understand the _{flow distribution} phenomena in the header and parallel channels as far as perfor-mance is concerned.

The single-phase distribution in a parallel_{flow heat exchanger} had been studied by several investigators. The single-phase distri-bution mechanism and calculation procedure are normally better

understood than two-phaseflow[1]. However, until now, there still lacks of general methodology for improving theflow distribution at header-tube junctions due to the complex interactions amid geometrical con_{figurations and inlet flow conditions at the inlet} header.

1.1. Parameters for counting theflow distribution

For counting theflow distribution among the parallel tubes, the dimensionless parameters,

b

i,

b

and

F

are used for evaluating the

flow distribution. Their definitions are given as following:

b

i ¼ Qi=Q (1)

Where

b

idenotes theflow ratio for ith tube, Qirepresents volume

flow rate for ith tube (m3_{/s), and Q is total volumeflow rate (m}3_/s).

To characterize the influence, Chiou[2]had used the concept of standard deviation to define the non-uniformity,

F

as:

F

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN i¼ 1ð

b

i

b

Þ2 N s (2) Where N is the number of total tubes in the parallel flow heat exchanger and

b

is the averageflow ratio for the total tubes which is defined as

b

¼ ðPN

i¼ 1

b

iÞ=N: The larger value of

F

indicates the

higher non-uniformity.

* Corresponding author. Tel.: þ886 5 5342601; fax: þ886 5 5312062. E-mail address:cheniy@yuntech.edu.tw(I.Y. Chen).

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Applied Thermal Engineering

j o u r n a l h o me p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a p t h e r m e n g

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1.2. Factors for affecting theflow distribution

From the previously reported results of the cited references, the trends for theflow distribution affected by several operating factors are briefly discussed in the following.

1.2.1. Flow area ratio

For evaluating the effect of the header size, the header-tubeflow area ratio (AR) had been defined as the ratio of the header cross-sectional area (A) to the total cross-sectional area of all branch tubes (AR¼ (N

p

D2/4)/A). Datta and Majumdar[3]had utilized the numerical method to calculate theflow distribution for two different outlet locations from the downstream header. Their results showed that theflow non-uniformity is increased as area ratio is increased.

Bajura and Jones[4]had theoretically analyzed theflow distri-bution and proposed a model for the prediction of_flow distribu-tion. The predicted values were compared with the measured data. For obtaining a more uniformflow distribution, AR < 1 is suggested for design.

Choi and Cho[5]had numerically investigated theflow distri-bution for electronic cooling with different AR values (4, 8, and 16) for a Reynolds number of 50, the case of AR¼ 4 produced the best coolant distribution. When the flow enters the tubes from the header, the eddyflow appeared at the inlet section for each tube, yet the engendered eddyflow becomes more pronounced with the rise of area ratio, leading to conspicuous mal-distribution.

Tong et al.[6] numerically examined the influences of cross section area of the header. The simplest way for attaining the outflow uniformity is to enlarge the headers for increasing their cross section area or reducing theflow area ratio.

1.2.2. Flow direction

The inlet and the exitflows may be located at the same side or the opposite side of the heat exchanger as called Z-type (parallel flow), or U-type (reverse flow) as shown inFig. 1. U-type and Z-type flow directions are the most commonly used configurations for parallelflow heat exchanger and normally the flow distribution of U-type is more uniform than Z-type at the sameflow area ratio[3].

This is because the pressure differences between the inlet header and outlet header for all the tubes are more evenly distributed for U-type than that of Z-type[7].

1.2.3. Inlet velocity to the header or entrance effect

In general, the higher inlet velocity to the header leads to a higher pressure drop at the header entrance. This is primarily associated with appreciable pressure drop with the jetflow into the header, thereby reducing theflow rate into branching tubes near the header entrance. The inlet velocity is affected by the totalflow rate and the cross section area at the entrance. Choi and Cho[5] indicated that theflow rate for the downstream branching tubes is significantly increased with the rise of inlet total flow rate to the header, and consequently the _{flow rate to the tubes near the} entrance is decreased.

Recently, Tong et al.[6]utilized the numerical simulation and indicated the larger size for the inlet tube having a better distribu-tion, while the smaller inlet size could induce jetflow at the entrance and cause reverse flow in the branching tubes at the upstream. The jetflow phenomenon becomes more significant with a higher inletflow rate or a smaller tube size at the entrance to the header. 1.2.4. Flow resistance

In a dividing header, the main_{fluid stream is decelerated due to} the loss of bothfluid and momentum through parallel tubes, thus causing a rise in pressure in the direction offlow. However, the frictional effect would cause a decrease of pressure in the _flow direction of header and tubes.

Bajura and Jones[4]inserted a small orifice inside the parallel tubes for increasing theflow resistance. The pressures at the inlet and outlet of the parallel tubes were measured, and their results indicated that the distribution of the pressure difference for the tubes is more uniform with respect to higherflow resistances.

Kubo and Ueda[8]also inserted various sizes of orifice (2, 4, 6, and 8 mm) into the parallel tubes for changing theflow resistance. The higher resistance with smaller orifice in the parallel tubes gives betterflow distribution.

1.2.5. Gravity effect

The gravity had been reported to cast a signi_{ficant effect on} the two-phase distribution in horizontal manifolds with vertical parallel tubes[9]due to uneven gas-liquid distribution in tubes with very high density difference between both phases. The effect of gravity to single-phaseflow distribution could be much less than two-phase flow due to the parallel tubes all filled with liquid. However, it is well known that the vertical upflow decreases the local pressure along theflow path, while the vertical down flow increases the local pressure. The gravity still has a minor effect on single-phase distribution in horizontal, vertical up and vertical downflows in the parallel tubes at various flow conditions. 2. Objective of the study

The literature survey reveals that the flow distribution in a configuration of header e parallel channels is very complex. Distribution of the _{flow into each parallel channel is not only} a function of theflow conditions in the header, but is also affected by the geometric size of the header and the parallel tubes, as well as theflow direction. In addition, the flow distribution problem may become even more critical in modern compact/micro heat exchangers. Severeflow mal-distribution could lead to overheating and cause entire failure of the thermal system. In this regard, the purpose of this study is to investigate theflow distribution subject to small and parallel tubes applicable for compact/micro heat exchangers. Investigations include the influences of Z-type and Nomenclature

A cross-sectionalflow area (m2

)

AR header-tubeflow area ratio ((N

p

D2/4)/A)

D inner tube diameter (m)

f fanning friction factor g gravity (m/s2)

G massflux (kg/m2_s)

h vertical height (m)

N number of total tubes

Qi volumeflow rate for ith tube (m3/s)

Q total volumeflow rate (m3_/s)

Greek symbols

b

i flow ratio for ith tube defined in Eq.(1)

b

averageflow ratio for the total tubes

D

L length for

D

P measurement

D

P measured pressure drop (N/m2₎

r

density (kg/m3)

F

non-uniformity defined in Eq.(2) Subscript

f friction

i ith

U-typeflow directions at horizontal, vertical up and vertical down orientations with typical headers. The effects of inletflow condi-tions, channel and header sizes are also investigated with various area ratios for optimizing theflow distribution. For comparison and better into physical insight of theflow distribution, a commercial software is also utilized to simulate the pressure andflow fields. 3. Experimental apparatus

3.1. Test rig

The test rig is primarily consisted of the water circulation and test section, as well as the measurement devices depicted inFig. 1.

Water is heated by a thermostat where water is maintained at 25C. The waterflow loop includes a variable speed gear pump. Part of the waterflow is circulated through the three way valve for flow rate control. The filter is designated to remove the impurities from the stream. A very accurate Yokogawa magneticflow meter (AXF005G) is installed at the downstream of the gear pump for measuring the waterflow rate. The accuracy of water flow meter is within0.045 L/min of the test span. Leaving the flow meter, the water temperature is measured by a resistance temperature device (Pt100

U

) having a calibrated accuracy of 0.1 K. The water temperature is near 25C during the tests. The pressure entering the test section is measured by a YOKOGAWA EJX pressure trans-ducer with an accuracy of 0.025%. Also, a YOKOGAWA EJ110

differential pressure transducer having an adjustable span of 1300e13 000 Pa installed across the inlet of the upstream header and outlet of the downstream header with a resolution of 0.3% of the measurements.

The test section includes an inlet and outlet headers along with 9 parallel tubes as shown inFig. 1. The foremost tube to the inlet location of header is termed 1st tube, and the aftermost one is termed 9th tube. The pressure drop for each tube at the location with 50 mm from the outlet header is measured by a YOKOGAWA EJ110 differential pressure transducer. Also, a 100 diameter calming tube length from the inlet header is used to ensure fully developed flow. The pressure taps are vertically drilled with a diameter of 0.5 mm. Leaving the test section, the water flows back to the thermal tank for recirculation.

The measurement of pressure drop in each tube is used to calculate theflow rates among the tubes. The total pressure drop includes gravitational dropð

D

Pg ¼

r

ghÞ and frictional drop

D

Pf, i.e.,

D

PT ¼

D

Pfþ

D

Pg (3)

D

P_f ¼ 4f

D

L D

G2

2

r

(4)

Where D: inner tube diameter,

D

L: tube length for

D

P measurement (50 mm), f: Fanning friction factor,

r

: density and G: massflux. With the measured

D

P for each tube, the mass_{flux (G}i) and volumeflow

rate (Qi¼ GiA/

r

) could be calculated. The uncertainty of the

calcu-lated volumeflow rate (Qi) from the pressure drop is estimated

to be 1.34%. The derived uncertainty of the flow ratio

b

and non-uniformity

F

is 2% and 3.88%, respectively. Detailed calculation of the uncertainty is shown in theAppendix.

3.2. Test sections

The test section consists of a distribution inlet header, 9 parallel tubes and an outlet header as shown in Fig. 1. The tubes and headers are made by transparent glass and acrylic fabric, respec-tively, forflow visualization. The geometric sizes of the test sections with a simplified schematic diagram are given in Table 1. The lengths of the parallel tubes are 300 mm and 400 mm, respectively, for 2 and 3 mm inner diameters with a pitch of 10 mm. The tube thickness for the 2 and 3 mm tubes is 1 mm. Three header sizes with square cross section of 7 mm 7 mm, 9 mm  9 mm and 12 mm 12 mm were tested. The inlet flow tube to the header has an inner diameter of 4 mm. The inlet tube is located near the center position of the cross section area of the inlet header, but the vertical distance between the center position of the inlet tube to the header entrance and the parallel tubes has a vertical distance,“b”, which is equal or greater than H/2 as shown inFig. 1, where H is the height of the header. Since b H/2, it indicates that the inlet tube is located at the center of the header or below it from 0 to 1.5 mm. To study the effect of velocity distribution in the header on theflow distribution,

several blocks with a length of 5 mm with identical cross section area of header were used to change the distance,“t”, between the inlet surface of the header entrance and thefirst tube, as well as the bottom surface to the last tube in the inlet headerFig. 1. The original length (L) of theflow path in the headers is 120 mm. The blocks have a hole exactly match the 4 mm hole at the entrance to the header from the inlet tube. Three different values of distance (t) are 18.5, 13.5 and 3.5 mm for the test sections with the header size of 9 mm 9 mm and the tube size of 3 mm as shown inTable 1. Also, there are 6 values of area ratio (AR) for the combination of different sizes of tube and header for testing.

4. Results and discussion

Theflow ratios (

b

) of the 9 parallel tubes for the vertical upflow are given inFig. 2(a)e(f) with header sizes being 7  7, 9  9 and 12 12 mm for Z-type and U-type flow arrangements with the original header length (L¼ 120 mm). Each figure also shows flow ratio for the volumetricflow rates of 0.5, 1 and 2 L/min having 2 and 3 mm inner diameter for the parallel tubes. Apparently, theflow distribution for U-typeflow is comparatively uniform than Z-type due to a more uniform pressure difference distribution across the parallel tubes between the inlet and outlet headers. In the mean-time, the Z-typeflow has less flow ratio in the first parallel tubes near the header entrance and a higherflow ratio distributed to the last tubes of header. The ratio between the least and the highest flow ratios for Z-type arrangement can be as high as 200% as shown inFig. 2(d). It should be mentioned that these two configurations (Z & U) had a very dramatic difference inflow distribution. The flow ratio is increased monotonically from thefirst tube to the last tube while a plateau offlow distribution is seen somewhere in-between thefirst and last tube. The results are in line with the theoretical calculations by Bassiouny and Martin[10,11]who investigated the flow distribution and pressure drop for U-type and Z-type plate heat exchangers. For Z-type plate heat exchanger, their calculation indicated that the pressure drop amid the intake conduit and exhaust conduit is always increasing from thefirst tube, suggesting the rise of velocity from thefirst to the last tube. For the U-type configuration, the pressure difference may be approximately fixed, monotonically increased or monotonically decreased depending on characteristics parameter of the plate heat exchanger.

Notice that the 2 mm tube has a betterflow distribution than that of 3 mm tube due to higherflow resistance. The results had been confirmed from some measurements and theoretical calcu-lations (e.g. Osakabe et al. [1], Lu and Wang[12]). On the other hand, the flow distribution for 0.5 L/min is more uniform than those of 1 and 2 L/min, respectively. One of the possible explana-tions for this phenomenon is attributed to the entryflow pattern into the header. When theflow enters a sudden enlargement cross section at the intake conduit, single jetflow may form when the velocity is sufficient high. The momentum of the jet flow lowers its static pressure at the upper stream, thereby reducing the effective pressure difference amid the inlet and outlet headers. As a conse-quence, theflow ratio is decreased for the branching tubes nearby the entry.

The non-uniformities (

F

) for Z-type and U-type_{flow directions} verse the variation offlow rate are respectively given inFig. 3. The effects of header size and tube diameter are also included inFig. 3, but the header length isfixed at L ¼ 120 mm. As shown in the figure,

F

increases monotonically with the rise of flow rate for 3 mm branching tube. By contrast,

F

is marginally increased asflow rate increased from 0.5 to 1 L/min, followed by a slight decrease with a further increase of volumetricflow rate to 2 L/min. Roughly speaking,

F

is relatively insensitive to the change of volumetric flow rate when the tube diameter is reduced to 2 mm. The major

Table 1

The geometric sizes of the test sections.

D mm H (mm) W (mm) AR t (mm) b (mm) L (mm) 3 12 12 0.442 18.5 6 120 9 9 0.785 18.5 5 120 13.5 110 3.5 90 7 7 1.298 18.5 5 120 2 12 12 0.196 19 6 120 9 9 0.349 19 5 120 7 7 0.557 19 5 120

explanation for this phenomenon is associated with the increase of viscous friction for smaller diameter tube as explained in preceding discussion. Additional reason is that theflow pattern in the parallel tubes had been changed from laminar flow to turbulent flow as totalflow rate reached 2 L/mm since the value of friction factor is momentously increased at theflow transition. The flow resistance in these tubes at turbulentflow (higher flow rate) is higher than the tubes at laminarflow (smaller flow rate). Thus, more fluid would distribute to the tubes with smaller resistance, and thus theflow distribution is improved with a smaller

F

. The increase of flow

resistance in parallel tubes would improve the_{flow distribution.} For the sameflow rate, the flow resistance of 2 mm tube is about 5 times of 3 mm tube, thus, the value of

F

for 2 mm tube at 2 L/min is much less than that of 3 mm as shown inFig. 3which indicates the betterflow distribution for 2 mm tubes, and the results also suggest that laminar flow may have a more severe mal-distribution problem. For both U-type and Z-type configurations, the effect of the tube size on theflow distribution is analogous but is different in magnitude as shownFig. 3. The betterflow distribution in U-type flow with smaller non-uniformity (

F

) is due to the uniform

pressure change distribution between the inlet and outlet of the parallel tubes.

For verifying the entrance effect in the header, the length of flowing path for the header was modified by inserting blocks with a dimension of 5 9  9 mm into the 9  9 mm header. Thus, the distance (t) between the inlet location and thefirst tube had been reduced from 18.5 mm to 13.5 mm and 3.5 mm, respectively while the header length (L) was changed from 120 mm to 110 mm and 90 mm, respectively. The lengths of L and t are also given inTable 1. The results offlow ratio of the 9 parallel tubes subject to the effect of entrance length (t) for Z-typeflow in 3 mm tubes are given in Fig. 4(a). By increasing the entrance length (t), theflow distribution is generally improved. The shorter entrance is prone to mal-distribution, yet this phenomenon becomes more severe when the inlet volumetricflow rate is increased. Normally the flow rate (or flow ratio) is the lowest in the first tube, and is gradually increased as the tube number increased. The results are explained in the foregoing discussion where the jet flow phenomenon

is more conspicuous with the inlet volumetric flow rate. The non-uniformity (

F

) verse the effect of entrance length (t) for Z-typeflow is also shown inFig. 4(b). For the same volumetricflow rate, the values of

F

for L¼ 90 mm are much higher than those of L¼ 110 and 120 mm, indicating a much severe mal-distribution problem associated with L ¼ 90 mm. With a shorter entrance length, the developed jetflow may considerably influence the flow distribution of thefirst several tubes. Conversely, with the rise of entrance length, the effect of jet flow is eased appreciably. The relation of non-uniformity (

F

) against the inlet volumetricflow is proportional to Q0.669.

The effect of gravity on theflow distribution had been studied for Z-type flow with the present 9  9 mm header and 3 mm parallel tubes. Tests are conducted for vertical up, vertical down and horizontal arrangements and results are depicted in Fig. 5. As seen in thefigure, the flow ratio is quite insensitive to change of orientation. For an easier comparison, the values of non-uniformity (

F

) for various orientations at different volume flow rates are plotted inFig. 6. The values of non-uniformity (

F

) for the vertical down orientation are marginally higher than the other two

Fig. 3. Non-uniformity Fvs. volumetricflow rate subject to Z-type and U-type arrangements.

Fig. 4. Effect of entrance length on theflow distribution of Z-type arrangement. C.-C. Wang et al. / Applied Thermal Engineering 31 (2011) 3226e3234 3231

orientations for Q¼ 1 and 2 L/min. At Q ¼ 3 L/min, non-uniformity is nearly the same amid the three orientations. Most of the tubes are in the laminarflow regime for Q ¼ 1 and 2 L/min where gravity still could impose a very small effect onflow distribution, espe-cially, for the vertical down orientation. For Q¼ 3 L/min, the flow regime moves toward transition/turbulentflow where the friction resistance takes control, thereby the influence of gravity becomes insignificant.

For a better understanding of theflow distribution character-istics in compact parallel_{flow heat exchangers at different flow} conditions, test results are compared with simulation from the EFD.lab commercial software. The simulation is based on the 9_{ 9 mm headers having L ¼ 90 mm with vertical up flow in} U-type and Z-typeflow directions. The parallel tubes have a diam-eter of 3 mm.Fig. 7shows the simulation vs. the experimental results for Q¼ 1 and 2 L/min in Z-type flow direction. The calcu-lated results are in good agreement with the experimental results, and the calculated values of non-uniformity (

F

) from the simula-tion and the experiment are 0.0239 and 0.0245, respectively, for Q ¼ 1 L/min Figs. 8, 9 and 10depict the velocity and pressure contours, as well as the flow velocity lines, respectively, with Q¼ 2 L/min in 90 mm header and 3 mm parallel tubes in U-type flow direction. As shown inFig. 8, a clear jetflow pattern presents at the entrance, yet the higher velocity at the header inlet is then gradually decreased along the conduit. Also, thefirst several tubes have the smallest lateral_{flow velocity, while the last one has the}

Fig. 5. Flow distribution with Z-typeflow for various orientations.

Fig. 6. Non-uniformity (F) for various orientations.

highestflow velocity. Due to the jet effect at the entrance, the pressure is smallest at the entrance and is largest at the end of the header as shown inFig. 9. Thus, thefirst tube near the entrance has the lowest pressure and the tube pressure is increased with the increase of tube number.Fig. 10shows theflow velocity lines in the front four and the header inlet. The result shows the jetflow is induced at the header inlet with a clear vortex circulated at the sides of the expanded jetflow. Also, a small eddy flow is observed

near the inlet of thefirst tube due to the circulation of the vortex flow. This vortex flow would reduce the flow rate of the first tube. Similar calculations were also reported by Fu et al.[13]who showed a secondaryflow is generated at the inlet of the first tube.

5. Conclusion

The single-phaseflow distribution in compact parallel flow heat exchanger is investigated by experiments and numerical simula-tions subject to various operating condisimula-tions. Theflow rates for all 9 parallel tubes are calculated by the measured pressure drops, the results are summarized as the following:

1. As total flow rate increases, the jet flow phenomenon becomes more pronounced, yielding smaller pressure differ-ence for thefirst several branching tubes between the intake and exhaust conduit. Therefore, a substantial reduction of flow rate in the first several tubes is observed. The mal-distribution caused by jet flow at the sudden expansion of the header is considerably eased by exploitation of a smaller diameter branching tube. Alternatively, the mal-distribution at the _{first several tubes can be significantly improved by} using a settling entrance length.

2. The gravity casts a very small effect on theflow distribution. The non-uniformity is slightly higher for vertical down flow with smallerflow rate. As the flow rate increases, the tube flow resistance also increases, thus, the gravity effect becomes insignificant.

3. The flow distribution for U-type flow is more uniform than Z-typeflow.

4. The trends of the localflow ratios between the simulations and experiments are very similar but with a small deviation of non-uniformity.

5. The numerical results clearly indicate a jetflow is generated with circulated vortex _{flow at the header inlet, as well as} a small eddyflow at the inlet of the first tube that would reduce theflow rate to the first tube.

Acknowledgements

The grant from National Science Committee (NSC 98-2221-E-224-049) of Taiwan is appreciated for supporting this study. Also, the authors are indebted tofinancial support from the Bureau of Energy of the Ministry of Economic Affairs, Taiwan.

Appendix. Detailed calculation of the uncertainty

From the pressure drop formula and relation between volu-metricflow rate and mass flux,

D

P ¼ 4fL D G2 2

r

(A1) G ¼ Q

r

A (A2) Hence Q ¼

D

PDA2 2fL

r

!1 2 (A3) Therefore the uncertainty of Q is obtained from the following

Fig. 8. Velocity simulation in 9 9 mm header with 90 mm length and D ¼ 3 mm tubes.

Fig. 9. Pressure simulation in 9 9 mm header with 90 mm length and D ¼ 3 mm tubes.

Fig. 10. Simulation offlow velocity lines in the header entrance and the front tubes.

d

Q ¼ " vQ v

D

P

dD

P 2 þ  vQ vD

d

D 2 þ  vQ vA

d

A 2 þ  vQ vL

d

L 2#12 ¼ 2 6 6 6 4 0 B B @1₂

D

PDA 2 2fL

r

!1 2 DA2 2fL

r



d

P 1 C C A 2 þ 0 B B @1₂

D

PDA 2 2fL

r

!1 2 

D

_2fLPA2

r



d

D 1 C C A 2 þ 0 B B @1₂

D

PDA 2 2fL

r

!1 2 2D_2fL

D

PA

r



d

A 1 C C A 2 þ 0 B B @1₂

D

PDA 2 2fL

r

!1 2 

D

PDA2 2fL2

_r

 ð1Þ 

d

D 1 C C A 23 7 7 5 1 2 ðA4Þ Employing the measured accuracies of the primary measurements (pressure drop, Employing the measured accuracies of the primary measurements (pressure drop, diameter, tube length, area.) to the derived parameter Q, one can estimate the associate uncertainty as

d

Qz0:000653

UQ ¼

d

_QQz0:0133z1:33%

As a result, estimation of

b

iis conducted with similar procedures:

b

i ¼ Qi Q_total (A5)

db

i ¼ " v

b

vQi

d

Qi 2 þ  v

b

i vQtotal

d

Q_total 2#12 ¼ "

d

Q_i Qtotal 2 þ Qi Q2 total

d

Qtotal !2#1₂ ðA6Þ Therefore,

db

iz9:9  104 U_bi ¼

db

i

b

iz0:02z2%

Similarly, the uncertainty of

F

is given in the following:

F

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN ið

b

i

b

Þ2 q N (A7)

b

¼ PN i

b

i N ¼

b

1þ .

b

9 9 (A8)

db

¼ " v

b

v

b

1

db

i 2 þ.  v

b

v

b

9

db

i 2#12 (A9)

db

z3:3  104

dF

¼ " v

F

v

b

i

db

i 2 þ  v

F

v

b

db

2#12 ¼ 2 6 6 4 0 B B @1₂  ð

db

i

b

Þ N 1 2 2ð

db

i

b

Þ N 

db

i 1 C C A 2 þ 0 B B @1₂  ð

db

i

b

Þ N 1 2 2ð

db



b

iÞ N 

db

1 C C A 23 7 7 5 1 2 ðA10Þ Thus,

dF

z0:00104 U_F ¼

dF

F

z0:0388z3:88% References

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