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Predicting the flow distribution in compact parallel flow heat exchangers
Applied Thermal Engineering 90 (2015) 551e558
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research paper
Predicting the flow distribution in compact parallel flow heat exchangers R. Camilleri*, D.A. Howey, M.D. McCulloch Energy and Power Group, Department of Engineering Science, University of Oxford, United Kingdom
h i g h l i g h t s A flow network can accurately simulate the flow in compact heat exchangers. The tube to header area ratio (AR) is a good indicator for flow mal-distribution. As AR increases, flow mal-distribution (FMD) becomes sensitive to increasing Re. As AR increases, FMD becomes sensitive to decreasing pipe length. Friction dominates FMD, but as AR increases inlet losses affect early channels.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 31 October 2014 Accepted 1 July 2015 Available online 14 July 2015
This paper presents a computationally efficient flow network model to predict the flow distribution in compact multi-channel parallel flow heat exchangers. Compact U-type and Z-type heat exchangers with nine parallel channels were used as test case on which the model was validated to within 4e8% in terms of non-dimensional flow distribution ratio. The model was used to perform a sensitivity analysis of flow mal-distribution with changes in operational and geometric boundary conditions. The results show that the tube to header area ratio is an important global parameter for controlling mal-distribution. As the area ratio increases, flow mal-distribution becomes more pronounced and more sensitive to increased Reynolds number and decreased parallel pipe length. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Compact heat exchanger Flow distribution Flow network model Heat exchanger design
1. Introduction This paper presents a flow network model to simulate flow distribution in compact parallel flow heat exchangers. The model is fast to solve and equips the designer with a useful tool to investigate the sensitivity of the head losses and flow mal-distribution from the components that make up the heat exchangers. Parallel flow heat exchangers have multiple applications such as evaporators and condensers in refrigeration units, solar energy collectors and cooling systems for electronics and nuclear reactors amongst others [1,2]. Flow mal-distribution in a heat exchanger can significantly reduce performance [3] and is therefore of interest for research. A number of papers focus on measuring techniques [1e5] or numerical simulation such as Computational Fluid Dynamics (CFD) [6]. Flow mal-distribution can be caused by a combination of
* Corresponding author. Tel. þ44 7582697457. E-mail addresses: [email protected] (R. Camilleri), david.howey@ eng.ox.ac.uk (D.A. Howey), [email protected] (M.D. McCulloch). http://dx.doi.org/10.1016/j.applthermaleng.2015.07.002 1359-4311/© 2015 Elsevier Ltd. All rights reserved.
heat exchanger manifold geometry, design features and operating conditions [7]. Therefore, despite these approaches providing detailed information and insight into the fluid behaviour, iterative improvement of the heat exchanger design is time consuming and costly. A design method that is fast and can accurately predict the flow distribution in compact heat exchangers is required, and is therefore the subject of this paper. Flow network models to predict the flow distribution in pipe networks are well established [8]. However compact heat exchangers often involve developing flows and therefore applying correlations from fully developed flows would result in significant errors. The pipe network approach was adapted to include appropriate minor losses and entrance effect friction factor correlations, enabling the flow network model to become an accurate design tool for compact heat exchangers. This paper is therefore organised in the following manner: A description of the heat exchanger geometry considered as a case study is provided in Section 2, thus setting the geometrical boundary conditions for this problem. Section 3 describes the iterative network model and
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R. Camilleri et al. / Applied Thermal Engineering 90 (2015) 551e558
Le N
Nomenclature A AR
b b CF D Dh DR
D dQ f fapp g hf K L
n
area (m2) tube/header area ratio flow ratio average flow ratio correction factor diameter of parallel pipes hydraulic diameter (m) diameter ratio head/flow ratio change in flow rate (m3/s) Darcy friction factor apparent friction factor acceleration due to gravity (m/s2) head loss (m) loss coefficient pipe length (m)
P Q Re s V
Subscripts i channel designation in inlet pipe j loop designation k time step iteration m manifold SE sudden enlargement PR pressure recovery
Section 4 describes the head loss coefficients for the various components. Finally Section 5 validates the model by comparing simulated results with experimental results found in the literature [1]. Finally Section 6 investigates the sensitivity of the flow distribution due to the various components that make up the compact heat exchanger.
2. Parallel flow heat exchangers Parallel flow heat exchangers consist of an inlet and an outlet manifold joined by a series of parallel branches. The heat exchanger is typically referred to as an U-type or a Z-type, depending on the flow direction. The heat exchangers investigated here each consist of two square manifolds joined by 9 parallel circular pipes as shown in Fig. 1a and b. The heat exchangers had a tube to header flow area ratio varying between 0.022 and 0.144, which was defined as:
AR ¼
NpD2 4Am
entry length (m) number of parallel pipes kinematic viscosity (m2/s) wet perimeter (m) flow rate (m3/s) Reynolds number non uniformity factor velocity (m/s)
(1)
3. Numerical modelling 3.1. The iterative flow network model The head loss hf along a pipe varies with flow rate Q as described by the DarcyeWeisbach equation and can be computed directly using:
hf ¼
fLV 2 fLQ 2 ¼ 2gD 2gDA2
(3)
In a flow network such as that in a parallel flow heat exchanger, the total head loss between inlet and outlet is equivalent to the sum of the pressure drops along any path connecting the inlet and the outlet. This is the same whatever path is chosen. The head loss is therefore dependent on the flow distribution. The flow along any one branch is determined by its resistance relative to neighbouring branches and therefore the flow distribution in the network must solved iteratively.
In which N is the number of parallel pipes and D the diameter of the parallel channels. Am is the manifold area defined as:
Am ¼ H2
(2)
The dimensions of the heat exchangers are given in Fig. 2 and Table 1.
Fig. 2. Geometrical dimensions of the heat exchanger investigated, to match Ref. [1].
Table 1 Geometrical dimensions of the simulated heat exchangers.
Fig. 1. Schematic showing the heat exchangers with a) U-type and b) Z-type flow directions.
Test case
D [mm]
H [mm]
L [mm]
Flow [lpm]
Area ratio
1 2 3 4
2 3 2 3
7 7 12 12
300 400 300 400
2 2 2 2
0.064 0.144 0.022 0.049
R. Camilleri et al. / Applied Thermal Engineering 90 (2015) 551e558
An iterative technique to determine the flow distribution in civil water infrastructures was demonstrated in Ref. [8]. The technique is based on the conservation equations as adapted from Kirchoff's laws for electrical circuits in which: 1. The sum of the current flow i at any junction n is zero.
X
in ¼ 0
(4)
In a flow network the sum of the flow rate into and out of any junction is kept equal to zero throughout the analysis by successive correction of the heads around the circuit. This method is hence referred to as the method of balancing heads. This method is directly applicable when the flow rate in and out of the pipe network is known [8]. 2. In the circuit shown, the sum of the voltage V at any closed loop m is zero.
X
Vm ¼ 0
Qj¼1 ¼ ½ Qab Qbc Qj¼2 ¼ Qbe Qef « Qj¼8 ¼ Qoq Qqr
Qcd Qda Qfc Qcb Qrp
Qpo
(6)
In the first iteration, a uniform flow distribution through the number of channels N is guessed such that:
Qad ¼ Qbc ¼ Qef ¼
The flow in the manifold is estimated at the first step by applying conservation of mass. Therefore:
Qab ¼ Qin Qad Qcd ¼ Qbc þ Qfc Qbe ¼ Qab Qbc « Qoq ¼ Qmo Qop Qpn ¼ Qop þ Qrp
Qin N
(7)
(8)
In the following iterations, an incremental correction for the flow in each loop is calculated using:
dQj ¼
hf ; j 2Dj
(9)
For which hf,j is the head loss in any single loop j and is estimated using:
hf;j ¼
4 X
hf ;i
(10)
i¼1
(5)
In a flow network the sum of the pressure drop around any closed loop is kept equal to zero throughout the analysis by successive correction of the flows into and out of the junctions. This method is therefore referred to as the method of balancing flows. This method is more convenient when the head at the inlet in and outlet of the pipe network is known [8]. In this work the method of balancing heads is employed. To demonstrate the application of the method, the heat exchanger shown in Fig. 1a is converted into a pipe network, shown in Fig. 3. Using an anti-clockwise positive flow convention, the flow vector for each loop Qj are defined by the individual flows in the pipes making the loop, so that:
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where i ¼ 1e4 are the individual pipe segments around any single loop. Likewise the Dj is the head/flow ratio for any loop and is calculated using:
Dj ¼
4 hf;i X i¼1
jQi j
(11)
The correction in the flow for each pipe is done algebraically using the adopted flow convention for each loop. For example for the first loop:
Qab;kþ1 ¼ Qab;k þ dQj¼1 Qbc;kþ1 ¼ Qbc;k þ dQj¼1 dQj¼2 Qcd;kþ1 ¼ Qcd;k þ dQj¼1 Qda;kþ1 ¼ Qda;k þ dQj¼1
(12)
The process is repeated for each loop and for a number of iterations until the incremental corrections in flow become very small. The model was solved using MATLAB. Special attention was taken to preserve the flow and head loss vectors, based on the flow convention taken in each loop. The iterative procedure shown in Fig. 4 continues until the maximum change in the branches' flows is less than 0.1%. For the heat exchangers shown in Fig. 1a and b the model is solved within 150e250 iterations and takes approximately 10 s. A typical residual convergence history chart is shown in Fig. 5. 3.2. The loss coefficients The flow conditions and geometry of small channels in compact heat exchangers often result in a developing laminar flow along the majority of the pipe length. Correlations in the model must therefore account for entrance effects. Similarly, losses associated with expansion, contraction or T-junctions are often looked upon as “minor losses” and ignored in pipe network models. However, in compact heat exchangers these may be significant and must be taken into consideration. The following sections investigate the loss coefficients applied to the various components in the model.
Fig. 3. Schematic of a simplified U-type flow network.
3.2.1. The friction factor in developing flows At the entry of a fluid into a pipe, the no slip condition at the wall dictates that the fluid velocity at the wall is zero, and a boundary layer along the pipe wall grows. The fluid at the centre of the flow accelerates to preserve continuity of mass. This is referred to as a
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R. Camilleri et al. / Applied Thermal Engineering 90 (2015) 551e558
Fig. 5. Typical convergence history chart for the iterative flow model.
In which A is the pipe flow area and P its wetted perimeter. The combined effects of shear stress at the wall and changes in momentum flux associated with changes to the velocity profile produce a higher friction factor [9]. The head loss equation (Eq. (2)) can be modified to take the entrance effects into consideration as follows:
hf ¼
4fapp Le V 2 2gD
¼
4fapp Le Q 2 2gDA2
(16)
The modified factor fapp incorporates the combined effects of shear stress at the wall surface and changes in momentum flux associated with changes in velocity profile. A correlation for fapp has been developed in Ref. [10]:
4fapp ðLe =Dh Þ ¼ 29:979
Le =Dh Re
0:6189 (17)
Shapiro [11] shows that the friction factor in the entry region can be up to 10 times larger than the Darcy friction factor of the developed flow, which is defined as:
f ¼ Fig. 4. Flow chart showing the process to solve the iterative flow model.
developing flow and occurs along an entry length Le until the velocity profile becomes parabolic and does not change profile along the further length of the pipe. In laminar flow, the entry length is approximately 20 diameters long and is defined in Ref. [9] as:
Le ¼ 0:05Re$Dh
(13)
Re is the Reynolds number defined as:
Re ¼
VDh n
64 Re
Fig. 6 compares the theoretical head loss along the nondimensional channel length for a fully developed flow with an assumed parabolic velocity profile, and the developing flow accounting for a changing velocity by Eq. (14) and (15). The difference in head loss between the developing and developed flows is seen to gradually and asymptotically approach each other as the inlet effects become relatively less significant. The entrance length in turbulent flow is very short and so when the flow is turbulent particularly in the manifold, the Blasius equation is used:
f ¼ 0:079Re0:25 (14)
4A P
(15)
(19)
After estimating the entry length, the total friction factor for each of the parallel channels may be calculated using:
The pipe hydraulic diameter Dh is defined as:
Dh ¼
(18)
hf ¼
V 2 4fapp Le þ f ðL Le Þ 2gD
(20)
R. Camilleri et al. / Applied Thermal Engineering 90 (2015) 551e558
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separation from the walls and becomes more pronounced with decreasing velocity. The head losses across each of the combining and dividing branches in T-junctions can be defined using:
hf ¼ K
Vi2 2g
(25)
Miller [13] shows how the loss coefficient K varies with flow rate, area ratio and the branches in a T-junction. The coefficient K therefore takes values of K31, K32 for dividing flows and, K13, K23 for combining flows as referenced to Fig. 7. A MATLAB table look up and interpolate function was used for each of the nine dividing and combining branches respectively. 4. Model validation The flow distribution within the parallel channels was analysed using a non-dimensional flow distribution ratio defined in Ref. [1] as: Fig. 6. Comparison of the theoretical head loss in laminar entry region for a developing and developed flow.
3.2.2. Entry losses The design of the inlet pipe into the manifold produces losses associated with a sudden enlargement. The effects of a sudden enlargement are typically correlated by:
hf ¼ KSE
2 Vin 2g
(21)
The variation of KSE with the inlet velocity and the diameter ratio DR was investigated in Ref. [12]. Miller [13] showed how the effects of sudden enlargement and sudden contraction are also experienced downstream of the inlet pipe i.e. pressure is gradually recovered along the length of the manifold. The author showed how the pressure recovery KPR varies with inlet velocity and inlet pipe/manifold area ratio for circular pipes. A MATLAB table look up and interpolate function was used so that KSE and KPR,i were determined at each location of the parallel branches and the head loss added to each of the pipes. As the fluid growth is a highly 3dimensional process and the data given in Ref. [13] is only for circular pipes, a correction factor was required to account for the square manifold. For the geometry considered, the following correction factors for the first three channels were empirically determined:
CC;1 ¼ 2:68DR þ 8:29 CC;2 ¼ 9:4DR þ 28:45 CC;3 ¼ 47:8DR þ 143:65
Dh >3 Din
Qi Qin
Figs. 8 and 9 compare the predicted flow distribution with the experimental results from Ref. [1] for an U type heat exchanger. Figs. 10 and 11 compare the flow distribution for a Z type heat exchanger. All the charts are shown in relation to geometrical detail shown in Table 1. The model is able to predict the flow distribution in heat exchangers to within 4e8% of measured values. 5. A sensitivity analysis Having validated the model with experimental results found in the literature, this section investigates the sensitivity of flow maldistribution to heat exchanger design parameters. As the number of parameters that can be tested are infinite, two test cases e Test Case 2 and Test Case 3 in an U-type heat exchanger are used as an example. Test Case 2 was chosen because it has the largest area ratio while Test Case 3 was chosen because it has the smallest area ratio. For consistency, these test cases are used during the assessment of the components making up the heat exchanger, Reynolds number and heat exchanger geometry. Similar trends were also noted on Z-type heat exchangers.
(22)
(23)
The effects of the flow rate (and the jet velocity as the fluid enters the manifold) on the flow distribution, the simulation was repeated for a number of inlet flow rates. The Reynolds number was defined at the inlet pipe diameter. Figs. 12 and 13 show the variation in flow distribution with flow rates for an U-type heat exchanger with area ratios of 0.144 and 0.022 respectively.
Hence in this work the head loss due to sudden enlargement along the length of the manifold was calculated using:
hf;i ¼
1 K K C V2 2g SE PR;i C;i i
(26)
5.1. Changes in the Reynolds number
For values of
1 > DR ¼
bi ¼
(24)
3.2.3. Losses from T-junctions and bends Combining and dividing manifolds are typically considered as a minor source of head losses. Yet in small systems their contribution may become significant. This phenomenon is governed by flow
Fig. 7. Definition of a) a dividing T-junction, b) a combining T-junction.
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R. Camilleri et al. / Applied Thermal Engineering 90 (2015) 551e558
Fig. 8. Comparison of the predicted and measured flow distribution for test cases 1 and 2: U-type heat exchanger, manifold dimensions 7 mm 7 mm, pipe diameters of 2 mm and 3 mm respectively.
Fig. 10. Comparison of the predicted and measured flow distribution for test cases 1 and 2: Z-type heat exchanger, manifold dimensions 7 mm 7 mm, pipe diameter of 2 mm and 3 mm respectively.
Fig. 9. Comparison of the predicted and measured flow distribution for test cases 3 and 4: U-type heat exchanger, manifold dimensions 12 mm 12 mm, pipe diameter of 2 mm and 3 mm respectively.
Fig. 11. Comparison of the predicted and measured flow distribution for test cases 3 and 4: Z-type heat exchanger, manifold dimensions 12 mm 12 mm, pipe diameter of 2 mm and 3 mm respectively.
It can be seen that the flow distribution in heat exchangers with larger area ratios is greatly affected by the variation of flow rate and inlet Reynolds number. Conversely the effect on heat exchangers with smaller area ratios is negligible. In the former case, as the flow rate increases so does the jet velocity at the entry of the manifold. With an increase in jet velocity, the sudden pressure losses at the inlet increases. Therefore the pipes closest to the inlet suffer from flow starvation.
The friction losses are seen to dominate the effects of the flow distribution. However, Fig. 14 shows that there is a clear effect of the inlet jet flow into the manifold, which dominates the head loss in the early pipes. The simulation was repeated for different flow rates and it was observed that this effect increases as the inlet flow rate and therefore the jet velocity increases. It is deduced that the flow distribution in the heat exchanger becomes more sensitive to inlet losses as the area ratio increases.
5.2. Contribution of the components
5.3. Changes in heat exchanger geometry
To investigate the contribution of each component to the flow mal-distribution, the pressure loss for each component was found as a percentage of the losses in the path to which it belonged. Figs. 14 and 15 show the component losses for U-type heat exchangers with area ratios of 0.144 and 0.022 respectively and an inlet flow rate of 2 lpm.
Finally, the effect of changing the heat exchanger geometry on its flow distribution is investigated. Increasing the number of parallel pipes N and the pipe diameter D automatically increases the area ratio. It is therefore expected that as the AR increases, flow mal-distribution increases. Conversely as the manifold area decreases the flow mal-distribution is expected to increase.
R. Camilleri et al. / Applied Thermal Engineering 90 (2015) 551e558
Fig. 12. The effects of the flow rate on flow distribution for an U-type geometry described in Test Case 2, AR ¼ 0.144.
Fig. 13. The effects of the flow rate on flow distribution for an U-type geometry described in Test Case 3, AR ¼ 0.022.
This section focuses on the effects of altering the overall heat exchanger height by increasing the length of the parallel pipes. The simulation was repeated for a range of pipe lengths between 125 mm and 1000 mm. Figs. 16 and 17 show the variation in flow distribution with flow rates for an U-type heat exchanger with area ratios of 0.144 and 0.022 respectively. The flow distribution in heat exchangers with a large AR is seen to be highly sensitive to the length of the parallel pipes. As the pipe length decreases the relative resistance of the manifold becomes more pronounced and therefore the flow mal-distribution increases. As the AR decreases, the heat exchanger becomes less sensitive to changes in parallel pipe length.
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Fig. 14. The contribution of the individual components to the head loss along any single path in an U-type flow configuration and geometry shown in Test Case 2, AR ¼ 0.144.
Fig. 15. The contribution of the individual components to the head loss along any single path in an U-type flow configuration and geometry shown in Test Case 3, AR ¼ 0.022.
early design stage and investigate the effects of various boundary, operational, and geometric conditions. The thermal designer can therefore run numerous simulations in a short time, identify major causes of flow mal-distribution and investigate trade-offs in heat exchanger parameters and performance. This allows him to make an informed judgement during the early design stages when important decisions are taken. Consequently an even flow distribution and an improved thermal performance of compact heat exchangers are achieved. The model was used to draw general observations and physical insight to heat exchanger design. In general it was noted that:
6. Discussion and conclusion This paper presents a fast and accurate numerical model to simulate the flow distribution in compact parallel flow heat exchangers. The model provides the thermal designer with the opportunity to simulate the flow distribution in heat exchangers at an
1. The perception that Z-type heat exchangers produce a more even-flow distribution is inaccurate. Compact heat exchangers should be investigated on a case-by-case basis. 2. The area ratio AR is an important overall performance parameter that may indicate the degree of flow mal-distribution. This
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R. Camilleri et al. / Applied Thermal Engineering 90 (2015) 551e558
ratio increases, the flow mal-distribution becomes more sensitive to sudden enlargement from the inlet pipe into the manifold. This provides a downstream effect along the length of the manifold and diminishes the flow in the first few pipes. The effect of the inlet jet velocity can be diminished by replacing the inlet pipe by a diffuser which gradually slows the inlet flow prior to entering the manifold and reduces sudden entry losses. Alternatively a diffusing material can be applied at the manifold inlet to dissipate the kinetic energy of the flow at the expense of an increase in heat exchanger pressure. 4. The flow mal-distribution is dependent on the relative resistances of each flow path. The influence of the manifolds and T-junctions is found to decrease as the resistance of the parallel pipes increases. This can be achieved for instance by increasing the length or decreasing the diameter of the parallel pipes. While flow mal-distribution decreases, this also comes at the expense of an overall increase in heat exchanger pressure. Acknowledgements Fig. 16. The effects of the variation in pipe length for an U-type flow configuration and geometry shown in Test Case 2, AR ¼ 0.144.
The authors would like to thank Seifert Systems Malta Ltd. for their support and generous funding. References
Fig. 17. The effects of the variation in pipe length for an U-type flow configuration and geometry shown in Test Case 3, AR ¼ 0.022.
parameter groups the relative effects of the manifold with respect to flow channels. 3. While flow mal-distribution was found to be heavily influenced by the friction factor and the design of the manifold, as the area
[1] C.C. Wang, K.S. Yang, J.S. Tsai, I.Y. Chen, Characteristics of flow distribution in compact parallel flow heat exchangers, part I: typical inlet header, Appl. Therm. Eng. 31 (2011) 3226e3235. [2] C.C. Wang, K.S. Yang, J.S. Tsai, I.Y. Chen, Characteristics of flow distribution in compact parallel flow heat exchangers, part II: modified inlet header, Appl. Therm. Eng. 31 (2011) 3235e3242. [3] B.P. Rao, P.K. Kumar, S.K. Das, Effect of flow distribution to the channels on the thermal performance of a plate heat exchanger, Chem. Eng. Process. 41 (2002) 49e58. [4] A. Jiao, Y. Li, C. Chen, Experimental Investigation on fluid flow maldistribution in plate-fin heat exchangers, Heat. Transf. Eng. 24 (2003) 25e31. [5] B.J. Jones, P.S. Lee, S.V. Garimella, Infrared micro-particle image velocimetry measurements and predictions of flow distribution in a microchannel heat sink, Int. J. Heat Mass Transf. 51 (2008) 1877e1887. [6] M.C. Lu, B.C. Yang, C.C. Wang, Numerical study of flow mal-distribution on the flow and heat transfer for multi-channel cold-plates, in: Proceedings of the 20th IEEE SEMI-THERM Symposium, San Jose, CA. [7] K.R. Shah, D.P. Sekulic, Chapter 12: Flow Maldistribution and Header Design, Fundamentals of Heat Exchanger Design, John Wiley and Sons, USA, 2003, pp. 809e863. [8] H. Cross, Analysis of Flow in Networks of Conduits or Conductors, 286, University of Illinois Eng. Experiment Station, 1936. [9] W. Kays, M. Crawford, B. Weigand, Convective Heat and Mass Transfer, McGraw Hill, 2004. [10] H.L. Langhaar, Steady flow in the transition of a straight tube, J. Appl. Mech. 64 (1942) A55eA58. [11] A.H. Shapiro, R.D. Smith, NACA, Friction Coefficients in the Inlet Length of Smooth Round Tubes, 1948. [12] E.F. Brater, H.W. King, Pipes, in: Handbook of Hydraulics, sixth ed., McGrawHill Book Company, New York, 1976, pp. 6e60. [13] D.S. Miller, Internal flows, in: Internal Flow Systems, second ed., Cranfield University, BHRA, Bedford, UK, 1990, pp. 96e97.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research paper
Predicting the flow distribution in compact parallel flow heat exchangers R. Camilleri*, D.A. Howey, M.D. McCulloch Energy and Power Group, Department of Engineering Science, University of Oxford, United Kingdom
h i g h l i g h t s A flow network can accurately simulate the flow in compact heat exchangers. The tube to header area ratio (AR) is a good indicator for flow mal-distribution. As AR increases, flow mal-distribution (FMD) becomes sensitive to increasing Re. As AR increases, FMD becomes sensitive to decreasing pipe length. Friction dominates FMD, but as AR increases inlet losses affect early channels.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 31 October 2014 Accepted 1 July 2015 Available online 14 July 2015
This paper presents a computationally efficient flow network model to predict the flow distribution in compact multi-channel parallel flow heat exchangers. Compact U-type and Z-type heat exchangers with nine parallel channels were used as test case on which the model was validated to within 4e8% in terms of non-dimensional flow distribution ratio. The model was used to perform a sensitivity analysis of flow mal-distribution with changes in operational and geometric boundary conditions. The results show that the tube to header area ratio is an important global parameter for controlling mal-distribution. As the area ratio increases, flow mal-distribution becomes more pronounced and more sensitive to increased Reynolds number and decreased parallel pipe length. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Compact heat exchanger Flow distribution Flow network model Heat exchanger design
1. Introduction This paper presents a flow network model to simulate flow distribution in compact parallel flow heat exchangers. The model is fast to solve and equips the designer with a useful tool to investigate the sensitivity of the head losses and flow mal-distribution from the components that make up the heat exchangers. Parallel flow heat exchangers have multiple applications such as evaporators and condensers in refrigeration units, solar energy collectors and cooling systems for electronics and nuclear reactors amongst others [1,2]. Flow mal-distribution in a heat exchanger can significantly reduce performance [3] and is therefore of interest for research. A number of papers focus on measuring techniques [1e5] or numerical simulation such as Computational Fluid Dynamics (CFD) [6]. Flow mal-distribution can be caused by a combination of
* Corresponding author. Tel. þ44 7582697457. E-mail addresses: [email protected] (R. Camilleri), david.howey@ eng.ox.ac.uk (D.A. Howey), [email protected] (M.D. McCulloch). http://dx.doi.org/10.1016/j.applthermaleng.2015.07.002 1359-4311/© 2015 Elsevier Ltd. All rights reserved.
heat exchanger manifold geometry, design features and operating conditions [7]. Therefore, despite these approaches providing detailed information and insight into the fluid behaviour, iterative improvement of the heat exchanger design is time consuming and costly. A design method that is fast and can accurately predict the flow distribution in compact heat exchangers is required, and is therefore the subject of this paper. Flow network models to predict the flow distribution in pipe networks are well established [8]. However compact heat exchangers often involve developing flows and therefore applying correlations from fully developed flows would result in significant errors. The pipe network approach was adapted to include appropriate minor losses and entrance effect friction factor correlations, enabling the flow network model to become an accurate design tool for compact heat exchangers. This paper is therefore organised in the following manner: A description of the heat exchanger geometry considered as a case study is provided in Section 2, thus setting the geometrical boundary conditions for this problem. Section 3 describes the iterative network model and
552
R. Camilleri et al. / Applied Thermal Engineering 90 (2015) 551e558
Le N
Nomenclature A AR
b b CF D Dh DR
D dQ f fapp g hf K L
n
area (m2) tube/header area ratio flow ratio average flow ratio correction factor diameter of parallel pipes hydraulic diameter (m) diameter ratio head/flow ratio change in flow rate (m3/s) Darcy friction factor apparent friction factor acceleration due to gravity (m/s2) head loss (m) loss coefficient pipe length (m)
P Q Re s V
Subscripts i channel designation in inlet pipe j loop designation k time step iteration m manifold SE sudden enlargement PR pressure recovery
Section 4 describes the head loss coefficients for the various components. Finally Section 5 validates the model by comparing simulated results with experimental results found in the literature [1]. Finally Section 6 investigates the sensitivity of the flow distribution due to the various components that make up the compact heat exchanger.
2. Parallel flow heat exchangers Parallel flow heat exchangers consist of an inlet and an outlet manifold joined by a series of parallel branches. The heat exchanger is typically referred to as an U-type or a Z-type, depending on the flow direction. The heat exchangers investigated here each consist of two square manifolds joined by 9 parallel circular pipes as shown in Fig. 1a and b. The heat exchangers had a tube to header flow area ratio varying between 0.022 and 0.144, which was defined as:
AR ¼
NpD2 4Am
entry length (m) number of parallel pipes kinematic viscosity (m2/s) wet perimeter (m) flow rate (m3/s) Reynolds number non uniformity factor velocity (m/s)
(1)
3. Numerical modelling 3.1. The iterative flow network model The head loss hf along a pipe varies with flow rate Q as described by the DarcyeWeisbach equation and can be computed directly using:
hf ¼
fLV 2 fLQ 2 ¼ 2gD 2gDA2
(3)
In a flow network such as that in a parallel flow heat exchanger, the total head loss between inlet and outlet is equivalent to the sum of the pressure drops along any path connecting the inlet and the outlet. This is the same whatever path is chosen. The head loss is therefore dependent on the flow distribution. The flow along any one branch is determined by its resistance relative to neighbouring branches and therefore the flow distribution in the network must solved iteratively.
In which N is the number of parallel pipes and D the diameter of the parallel channels. Am is the manifold area defined as:
Am ¼ H2
(2)
The dimensions of the heat exchangers are given in Fig. 2 and Table 1.
Fig. 2. Geometrical dimensions of the heat exchanger investigated, to match Ref. [1].
Table 1 Geometrical dimensions of the simulated heat exchangers.
Fig. 1. Schematic showing the heat exchangers with a) U-type and b) Z-type flow directions.
Test case
D [mm]
H [mm]
L [mm]
Flow [lpm]
Area ratio
1 2 3 4
2 3 2 3
7 7 12 12
300 400 300 400
2 2 2 2
0.064 0.144 0.022 0.049
R. Camilleri et al. / Applied Thermal Engineering 90 (2015) 551e558
An iterative technique to determine the flow distribution in civil water infrastructures was demonstrated in Ref. [8]. The technique is based on the conservation equations as adapted from Kirchoff's laws for electrical circuits in which: 1. The sum of the current flow i at any junction n is zero.
X
in ¼ 0
(4)
In a flow network the sum of the flow rate into and out of any junction is kept equal to zero throughout the analysis by successive correction of the heads around the circuit. This method is hence referred to as the method of balancing heads. This method is directly applicable when the flow rate in and out of the pipe network is known [8]. 2. In the circuit shown, the sum of the voltage V at any closed loop m is zero.
X
Vm ¼ 0
Qj¼1 ¼ ½ Qab Qbc Qj¼2 ¼ Qbe Qef « Qj¼8 ¼ Qoq Qqr
Qcd Qda Qfc Qcb Qrp
Qpo
(6)
In the first iteration, a uniform flow distribution through the number of channels N is guessed such that:
Qad ¼ Qbc ¼ Qef ¼
The flow in the manifold is estimated at the first step by applying conservation of mass. Therefore:
Qab ¼ Qin Qad Qcd ¼ Qbc þ Qfc Qbe ¼ Qab Qbc « Qoq ¼ Qmo Qop Qpn ¼ Qop þ Qrp
Qin N
(7)
(8)
In the following iterations, an incremental correction for the flow in each loop is calculated using:
dQj ¼
hf ; j 2Dj
(9)
For which hf,j is the head loss in any single loop j and is estimated using:
hf;j ¼
4 X
hf ;i
(10)
i¼1
(5)
In a flow network the sum of the pressure drop around any closed loop is kept equal to zero throughout the analysis by successive correction of the flows into and out of the junctions. This method is therefore referred to as the method of balancing flows. This method is more convenient when the head at the inlet in and outlet of the pipe network is known [8]. In this work the method of balancing heads is employed. To demonstrate the application of the method, the heat exchanger shown in Fig. 1a is converted into a pipe network, shown in Fig. 3. Using an anti-clockwise positive flow convention, the flow vector for each loop Qj are defined by the individual flows in the pipes making the loop, so that:
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where i ¼ 1e4 are the individual pipe segments around any single loop. Likewise the Dj is the head/flow ratio for any loop and is calculated using:
Dj ¼
4 hf;i X i¼1
jQi j
(11)
The correction in the flow for each pipe is done algebraically using the adopted flow convention for each loop. For example for the first loop:
Qab;kþ1 ¼ Qab;k þ dQj¼1 Qbc;kþ1 ¼ Qbc;k þ dQj¼1 dQj¼2 Qcd;kþ1 ¼ Qcd;k þ dQj¼1 Qda;kþ1 ¼ Qda;k þ dQj¼1
(12)
The process is repeated for each loop and for a number of iterations until the incremental corrections in flow become very small. The model was solved using MATLAB. Special attention was taken to preserve the flow and head loss vectors, based on the flow convention taken in each loop. The iterative procedure shown in Fig. 4 continues until the maximum change in the branches' flows is less than 0.1%. For the heat exchangers shown in Fig. 1a and b the model is solved within 150e250 iterations and takes approximately 10 s. A typical residual convergence history chart is shown in Fig. 5. 3.2. The loss coefficients The flow conditions and geometry of small channels in compact heat exchangers often result in a developing laminar flow along the majority of the pipe length. Correlations in the model must therefore account for entrance effects. Similarly, losses associated with expansion, contraction or T-junctions are often looked upon as “minor losses” and ignored in pipe network models. However, in compact heat exchangers these may be significant and must be taken into consideration. The following sections investigate the loss coefficients applied to the various components in the model.
Fig. 3. Schematic of a simplified U-type flow network.
3.2.1. The friction factor in developing flows At the entry of a fluid into a pipe, the no slip condition at the wall dictates that the fluid velocity at the wall is zero, and a boundary layer along the pipe wall grows. The fluid at the centre of the flow accelerates to preserve continuity of mass. This is referred to as a
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Fig. 5. Typical convergence history chart for the iterative flow model.
In which A is the pipe flow area and P its wetted perimeter. The combined effects of shear stress at the wall and changes in momentum flux associated with changes to the velocity profile produce a higher friction factor [9]. The head loss equation (Eq. (2)) can be modified to take the entrance effects into consideration as follows:
hf ¼
4fapp Le V 2 2gD
¼
4fapp Le Q 2 2gDA2
(16)
The modified factor fapp incorporates the combined effects of shear stress at the wall surface and changes in momentum flux associated with changes in velocity profile. A correlation for fapp has been developed in Ref. [10]:
4fapp ðLe =Dh Þ ¼ 29:979
Le =Dh Re
0:6189 (17)
Shapiro [11] shows that the friction factor in the entry region can be up to 10 times larger than the Darcy friction factor of the developed flow, which is defined as:
f ¼ Fig. 4. Flow chart showing the process to solve the iterative flow model.
developing flow and occurs along an entry length Le until the velocity profile becomes parabolic and does not change profile along the further length of the pipe. In laminar flow, the entry length is approximately 20 diameters long and is defined in Ref. [9] as:
Le ¼ 0:05Re$Dh
(13)
Re is the Reynolds number defined as:
Re ¼
VDh n
64 Re
Fig. 6 compares the theoretical head loss along the nondimensional channel length for a fully developed flow with an assumed parabolic velocity profile, and the developing flow accounting for a changing velocity by Eq. (14) and (15). The difference in head loss between the developing and developed flows is seen to gradually and asymptotically approach each other as the inlet effects become relatively less significant. The entrance length in turbulent flow is very short and so when the flow is turbulent particularly in the manifold, the Blasius equation is used:
f ¼ 0:079Re0:25 (14)
4A P
(15)
(19)
After estimating the entry length, the total friction factor for each of the parallel channels may be calculated using:
The pipe hydraulic diameter Dh is defined as:
Dh ¼
(18)
hf ¼
V 2 4fapp Le þ f ðL Le Þ 2gD
(20)
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separation from the walls and becomes more pronounced with decreasing velocity. The head losses across each of the combining and dividing branches in T-junctions can be defined using:
hf ¼ K
Vi2 2g
(25)
Miller [13] shows how the loss coefficient K varies with flow rate, area ratio and the branches in a T-junction. The coefficient K therefore takes values of K31, K32 for dividing flows and, K13, K23 for combining flows as referenced to Fig. 7. A MATLAB table look up and interpolate function was used for each of the nine dividing and combining branches respectively. 4. Model validation The flow distribution within the parallel channels was analysed using a non-dimensional flow distribution ratio defined in Ref. [1] as: Fig. 6. Comparison of the theoretical head loss in laminar entry region for a developing and developed flow.
3.2.2. Entry losses The design of the inlet pipe into the manifold produces losses associated with a sudden enlargement. The effects of a sudden enlargement are typically correlated by:
hf ¼ KSE
2 Vin 2g
(21)
The variation of KSE with the inlet velocity and the diameter ratio DR was investigated in Ref. [12]. Miller [13] showed how the effects of sudden enlargement and sudden contraction are also experienced downstream of the inlet pipe i.e. pressure is gradually recovered along the length of the manifold. The author showed how the pressure recovery KPR varies with inlet velocity and inlet pipe/manifold area ratio for circular pipes. A MATLAB table look up and interpolate function was used so that KSE and KPR,i were determined at each location of the parallel branches and the head loss added to each of the pipes. As the fluid growth is a highly 3dimensional process and the data given in Ref. [13] is only for circular pipes, a correction factor was required to account for the square manifold. For the geometry considered, the following correction factors for the first three channels were empirically determined:
CC;1 ¼ 2:68DR þ 8:29 CC;2 ¼ 9:4DR þ 28:45 CC;3 ¼ 47:8DR þ 143:65
Dh >3 Din
Qi Qin
Figs. 8 and 9 compare the predicted flow distribution with the experimental results from Ref. [1] for an U type heat exchanger. Figs. 10 and 11 compare the flow distribution for a Z type heat exchanger. All the charts are shown in relation to geometrical detail shown in Table 1. The model is able to predict the flow distribution in heat exchangers to within 4e8% of measured values. 5. A sensitivity analysis Having validated the model with experimental results found in the literature, this section investigates the sensitivity of flow maldistribution to heat exchanger design parameters. As the number of parameters that can be tested are infinite, two test cases e Test Case 2 and Test Case 3 in an U-type heat exchanger are used as an example. Test Case 2 was chosen because it has the largest area ratio while Test Case 3 was chosen because it has the smallest area ratio. For consistency, these test cases are used during the assessment of the components making up the heat exchanger, Reynolds number and heat exchanger geometry. Similar trends were also noted on Z-type heat exchangers.
(22)
(23)
The effects of the flow rate (and the jet velocity as the fluid enters the manifold) on the flow distribution, the simulation was repeated for a number of inlet flow rates. The Reynolds number was defined at the inlet pipe diameter. Figs. 12 and 13 show the variation in flow distribution with flow rates for an U-type heat exchanger with area ratios of 0.144 and 0.022 respectively.
Hence in this work the head loss due to sudden enlargement along the length of the manifold was calculated using:
hf;i ¼
1 K K C V2 2g SE PR;i C;i i
(26)
5.1. Changes in the Reynolds number
For values of
1 > DR ¼
bi ¼
(24)
3.2.3. Losses from T-junctions and bends Combining and dividing manifolds are typically considered as a minor source of head losses. Yet in small systems their contribution may become significant. This phenomenon is governed by flow
Fig. 7. Definition of a) a dividing T-junction, b) a combining T-junction.
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Fig. 8. Comparison of the predicted and measured flow distribution for test cases 1 and 2: U-type heat exchanger, manifold dimensions 7 mm 7 mm, pipe diameters of 2 mm and 3 mm respectively.
Fig. 10. Comparison of the predicted and measured flow distribution for test cases 1 and 2: Z-type heat exchanger, manifold dimensions 7 mm 7 mm, pipe diameter of 2 mm and 3 mm respectively.
Fig. 9. Comparison of the predicted and measured flow distribution for test cases 3 and 4: U-type heat exchanger, manifold dimensions 12 mm 12 mm, pipe diameter of 2 mm and 3 mm respectively.
Fig. 11. Comparison of the predicted and measured flow distribution for test cases 3 and 4: Z-type heat exchanger, manifold dimensions 12 mm 12 mm, pipe diameter of 2 mm and 3 mm respectively.
It can be seen that the flow distribution in heat exchangers with larger area ratios is greatly affected by the variation of flow rate and inlet Reynolds number. Conversely the effect on heat exchangers with smaller area ratios is negligible. In the former case, as the flow rate increases so does the jet velocity at the entry of the manifold. With an increase in jet velocity, the sudden pressure losses at the inlet increases. Therefore the pipes closest to the inlet suffer from flow starvation.
The friction losses are seen to dominate the effects of the flow distribution. However, Fig. 14 shows that there is a clear effect of the inlet jet flow into the manifold, which dominates the head loss in the early pipes. The simulation was repeated for different flow rates and it was observed that this effect increases as the inlet flow rate and therefore the jet velocity increases. It is deduced that the flow distribution in the heat exchanger becomes more sensitive to inlet losses as the area ratio increases.
5.2. Contribution of the components
5.3. Changes in heat exchanger geometry
To investigate the contribution of each component to the flow mal-distribution, the pressure loss for each component was found as a percentage of the losses in the path to which it belonged. Figs. 14 and 15 show the component losses for U-type heat exchangers with area ratios of 0.144 and 0.022 respectively and an inlet flow rate of 2 lpm.
Finally, the effect of changing the heat exchanger geometry on its flow distribution is investigated. Increasing the number of parallel pipes N and the pipe diameter D automatically increases the area ratio. It is therefore expected that as the AR increases, flow mal-distribution increases. Conversely as the manifold area decreases the flow mal-distribution is expected to increase.
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Fig. 12. The effects of the flow rate on flow distribution for an U-type geometry described in Test Case 2, AR ¼ 0.144.
Fig. 13. The effects of the flow rate on flow distribution for an U-type geometry described in Test Case 3, AR ¼ 0.022.
This section focuses on the effects of altering the overall heat exchanger height by increasing the length of the parallel pipes. The simulation was repeated for a range of pipe lengths between 125 mm and 1000 mm. Figs. 16 and 17 show the variation in flow distribution with flow rates for an U-type heat exchanger with area ratios of 0.144 and 0.022 respectively. The flow distribution in heat exchangers with a large AR is seen to be highly sensitive to the length of the parallel pipes. As the pipe length decreases the relative resistance of the manifold becomes more pronounced and therefore the flow mal-distribution increases. As the AR decreases, the heat exchanger becomes less sensitive to changes in parallel pipe length.
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Fig. 14. The contribution of the individual components to the head loss along any single path in an U-type flow configuration and geometry shown in Test Case 2, AR ¼ 0.144.
Fig. 15. The contribution of the individual components to the head loss along any single path in an U-type flow configuration and geometry shown in Test Case 3, AR ¼ 0.022.
early design stage and investigate the effects of various boundary, operational, and geometric conditions. The thermal designer can therefore run numerous simulations in a short time, identify major causes of flow mal-distribution and investigate trade-offs in heat exchanger parameters and performance. This allows him to make an informed judgement during the early design stages when important decisions are taken. Consequently an even flow distribution and an improved thermal performance of compact heat exchangers are achieved. The model was used to draw general observations and physical insight to heat exchanger design. In general it was noted that:
6. Discussion and conclusion This paper presents a fast and accurate numerical model to simulate the flow distribution in compact parallel flow heat exchangers. The model provides the thermal designer with the opportunity to simulate the flow distribution in heat exchangers at an
1. The perception that Z-type heat exchangers produce a more even-flow distribution is inaccurate. Compact heat exchangers should be investigated on a case-by-case basis. 2. The area ratio AR is an important overall performance parameter that may indicate the degree of flow mal-distribution. This
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ratio increases, the flow mal-distribution becomes more sensitive to sudden enlargement from the inlet pipe into the manifold. This provides a downstream effect along the length of the manifold and diminishes the flow in the first few pipes. The effect of the inlet jet velocity can be diminished by replacing the inlet pipe by a diffuser which gradually slows the inlet flow prior to entering the manifold and reduces sudden entry losses. Alternatively a diffusing material can be applied at the manifold inlet to dissipate the kinetic energy of the flow at the expense of an increase in heat exchanger pressure. 4. The flow mal-distribution is dependent on the relative resistances of each flow path. The influence of the manifolds and T-junctions is found to decrease as the resistance of the parallel pipes increases. This can be achieved for instance by increasing the length or decreasing the diameter of the parallel pipes. While flow mal-distribution decreases, this also comes at the expense of an overall increase in heat exchanger pressure. Acknowledgements Fig. 16. The effects of the variation in pipe length for an U-type flow configuration and geometry shown in Test Case 2, AR ¼ 0.144.
The authors would like to thank Seifert Systems Malta Ltd. for their support and generous funding. References
Fig. 17. The effects of the variation in pipe length for an U-type flow configuration and geometry shown in Test Case 3, AR ¼ 0.022.
parameter groups the relative effects of the manifold with respect to flow channels. 3. While flow mal-distribution was found to be heavily influenced by the friction factor and the design of the manifold, as the area
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