Oblique fluid flow and convective heat transfer across a tube bank under uniform wall heat flux boundary conditions

International Journal of Heat and Mass Transfer 91 (2015) 1259–1272

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Oblique fluid flow and convective heat transfer across a tube bank under uniform wall heat flux boundary conditions Li-Zhi Zhang ⇑, Yu-wen Ouyang, Zheng-Guo Zhang, Shuang-Feng Wang Key Laboratory of Enhanced Heat Transfer and Energy Conservation of Education Ministry, School of Chemistry and Chemical Engineering, South China University of Technology, Guangzhou 510640, China

a r t i c l e

i n f o

Article history: Received 3 April 2015 Received in revised form 20 August 2015 Accepted 20 August 2015 Available online 5 September 2015 Keywords: Tube bank Oblique flow Uniform wall heat flux Convective heat transfer Hollow fiber membrane bundle

a b s t r a c t Fluid flow and convective heat transfer in an oblique-flow tube bank under uniform wall heat flux boundary conditions are investigated. Most previous studies on heat transfer across tube banks have concerned only on two extreme cases, i.e., pure parallel flow or pure cross flow. For more general and practical conditions in engineering, fluid flow is in an intermediate state and it often impinges the tubes with an oblique angle as a result from duct or heat mass exchanger structures. In this research, fluid flow and convective heat transfer is investigated for an oblique-flow tube bank. Uniform wall heat flux boundary conditions are considered. Numerical solution has been performed to obtain the shell-side fluid field and convective heat transfer rates with various impinging angles. Experimental work is conducted to validate the solution. The analysis reveals that the convective heat transfer rates and friction factors rise with an increase in the oblique angles from 0° to 90° to the tube axes. Correlations are proposed for the prediction of convective heat transfer coefficients and friction factors at different oblique angles with various geometrical parameters. The results can be extended to mass transfer in hollow fiber membrane bundles with Chilton–Colburn analogy. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Heat mass exchangers are being increasingly used in energy and environmental technologies, for instance, total heat exchangers [1–3], membrane humidifiers/dehumidifiers [4–6], etc. Hollow fiber membrane banks (bundles) are the key components in these technologies to realize heat and mass exchange. Tube banks in heat/mass exchangers are usually arranged with two regular layouts, inline and staggered arrangements, which are characterized by the transverse, longitudinal and diagonal pitches [7–10]. Typically one fluid flows inside the fibers/tubes, while the other moves across the tubes, in a parallel-flow or cross-flow manner [11–14]. Analysis of pressure drop and heat/mass transfer in flow across the tube banks in shell side is of great importance in the design of heat/mass exchangers Friction factors and convective heat transfer rates are the most important parameters for system design. When the convective heat transfer coefficients are known, mass transfer coefficients can be estimated by heat mass analogy [15]. Therefore the fluid flow and convective heat transfer across tube banks are the focus of the study.

⇑ Corresponding author. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.08.062 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

The hollow fiber membrane bundles are very similar to conventional metal circular tube banks of shell-and-tube heat exchangers in structure. The difference is that the hollow fiber membranes have selective permeability for some gases and mass transfer may occur through the membranes [16,17]. In some cases, heat transfer data in tube banks can be applied to those in a hollow fiber membrane bundle when the effect of mass transfer on the fluid flow and heat transfer in a fiber bundle is negligible. For pure heat transfer over tube banks in heat exchangers, there have already been many detailed and classical works [18–21]. Uniform wall temperature [22,23] and uniform heat flux boundary conditions [24–26] are the two categories of classification for operating conditions. Under these two boundary conditions, fluid flow and heat transfer across tube banks for parallel flow and cross flow are investigated. The transport data are well accumulated. It is noteworthy that both parallel flow and cross flow are ideal conditions of shell side fluid flow. In practical applications, the inlet or outlet vents are usually smaller or bigger than the heat exchanger cores and they are projected with some attack angles, resulting in flow impinging the tube bundle with skewed angles between 0° and 90° to the tube axes. The fluid flow and heat transfer in tube banks would be inevitably affected by the impinging angles. Most previous studies [27–31] concerning the effect of oblique flow considered only several tubes, by neglecting the interactions between

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Nomenclature A A Atot b cp C d dh f Hp j Leff L Le m ma n Nu N p Pr q ro Re Sc SD Sh SL ST T u ua x, y, z

constant in correlation cross-sectional area normal to the main flow (m2) total area of convective heat transfer surface (m2) constant in correlation specific heat of air (J kg
1 K
1) constant in correlation fiber outer diameter (m) hydraulic diameter (m) friction factor; function of the independent variables heat transferred by the air stream (W) Colburn j factor for heat transfer effect tube length of convective heat transfer (m) rod length (m) Lewis number constant in correlation mass flow rate (kg/s) normal direction Nusselt number number of rows along the flow direction pressure (Pa) Prandtl number uniform heat flux transferred from the tube wall to the air stream (W/m2) outer radius of the heating tube (m) Reynolds number Schmidt number diagonal pitch (m) Sherwood number longitudinal pitch (m) transverse pitch (m) temperature (K) velocity of air (m/s) velocity of the air stream approaching the fiber bundle (m/s) coordinates in physical plane (m)

neighboring fibers in the tube bundle. A recent study of oblique flow and heat transfer across tube banks by the current research group has considered the interactions between neighboring fibers [32]. However, only a uniform temperature boundary condition is studied. The other more common and useful operating condition, uniform heat flux boundary condition, is still not addressed. This condition is more popular in heat exchangers like condensers and evaporators. To solve this problem, oblique flow over a circular tube bundle under uniform wall heat flux boundary condition will be elaborated here. This is a step forward. A finite volume based numerical solution in combination with experimental validation is conducted to obtain the shell-side fluid flow and convective heat transfer properties under various attacking angles. 2. Mathematical model 2.1. Governing equations For a more general purpose, convective heat transfer across a tube bank is considered. Metal tubes are used to substitute the hollow fiber membrane bundle, so only heat transfer is concerned and the boundary conditions can be well controlled by electric heating on surfaces. It is like a shell-and-tube heat exchanger. Air stream impinges the bundle with an impinging angle. Both inline and

Greek letters oblique angle (deg) density (kg/m3) dynamic viscosity (Pa  s) variable k heat conductivity (W/m/K) h dimensionless temperature of air f dimensionless temperature employed for the periodic boundary condition

a q l w

Superscripts 0 rear coordinate * dimensionless Subscripts a air b bulk i inlet; inner lm logarithmic mean m mean w wall mean max maximum x x-axis direction y y-axis direction z z-axis direction Abbreviations IA inline array SA staggered array H uniform heat flux boundary condition T uniform temperature boundary condition

staggered arrangements are considered for the bundle, as plotted in Fig. 1. Ambient air impinges the tube bundle with an oblique angle a between 0° and 90° to the axis of tube bank in the shell side. A uniform wall heat flux q is imposed on the tube surfaces. The air stream flows across the fiber bundle and is heated by the tube wall. For mass transfer problems encountered in membrane bundles, the boundary condition is in analogy to uniform wall mass flux on membrane surfaces. Resistance and convective heat transfer data are required for optimizing the design of heat exchangers. Commonly a commercial heat exchanger is tightly packed and may have hundreds of tubes in a tube bundle. Due to the numerous tubes in a module, direct modeling of the whole module will exceed laboratory-scale calculating capacity. To solve this problem, two periodic cells along the flow direction are selected to simplify the problem [16,17], as depicted in Fig. 2 for the two typical arrays. The periodic cells contain two or three fibers for each array. They consider the interactions between neighboring fibers. The fluid flow is assumed to be laminar and incompressible, Newtonian with constant thermal properties. Inlet and outlet temperature difference is less than 20 °C. The effects of temperature on thermal properties are less than 2%. The pressure drop is the pressure change of air during the process. Because the pressure change is less than 100 Pa, which is negligible compared to

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Air out

Air in

ST Uniform heat flux q

SL

(a) Air out

Air in ST

SD Uniform heat flux q

SL

(b) Fig. 1. Schematic of oblique flows across the tube banks: (a) inline layout; (b) staggered layout.

the total pressure of air of 101,325 Pa. So the influence of pressure change on thermal properties (determined by total pressure) is neglected. Shell-side fluid flow and heat transfer are described by the normalized governing equations in three-dimensional Cartesian coordinates [33] as below: Conservation equation of mass is

where dh is hydraulic diameter for the air stream and is equal to the tube outer diameter d, SL is longitudinal pitch of the tube bank, p refers to pressure and Re is Reynolds number. Conservation equation of energy is

@ux @uy @uz þ þ ¼0 @x @y @z

where Pr is Prandtl number and h is dimensionless temperature of air. Dimensionless coordinates are defined by

ð1Þ

where x, y and z are transverse, longitudinal and axial coordinates for air flow, respectively; subscript ‘‘x”, ‘‘y” and ‘‘z” refer to x-axis, y-axis and z-axis, respectively; u is velocity of air flow; superscript ‘‘*” represents dimensionless form. Conservation equations of momentum are

@u ux x @x

þ

@u uy x @y

þ

@u uz x @z

¼




1 @p dh 1 þ   2 @x SL Re

@ 2 ux @x2

þ

@ 2 ux @y2

þ

@ 2 ux @z2

!

ð2Þ @uy ux  @x

þ

@uy uy  @y

þ

@uy uz  @z

¼


1 @p dh 1 þ   2 @y SL Re

@ 2 uy @x2

þ

@ 2 uy @y2

þ

@ 2 uy @z2

!

ux

@h @h @h dh 1 @2h @2h @2h  þ uy  þ uz  ¼  þ 2 þ 2  2 @x @y @z @x @y @z SL Re Pr

@uz @u @u 1 @p dh 1 @ 2 uz @ 2 uz @ 2 uz þ uy z þ uz z ¼
þ   þ þ   2 @z @x @y @z SL Re @x2 @y2 @z2

!

ð4Þ

ð5Þ

x ¼

x SL

ð6Þ

y ¼

y SL

ð7Þ

z ¼

z SL

ð8Þ

Dimensionless velocities for the air stream in the three coordinates are given by

ux ¼

ux ua

ð9Þ

uy ¼

uy ua

ð10Þ

uz ¼

uz ua

ð11Þ

ð3Þ ux

!

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Dimensionless bulk temperature of air is calculated by

ZZ

u hdA

hb ¼ ZZ

Air in

C'

A B

y

E'

F D' E D

Num ¼

C

z

(a)

Dh ¼

Air in

A H B

y

E'

D' F G

E

C

um ¼

Fig. 2. Schematic of the periodic cells: (a) inline arrangement; (b) staggered arrangement.

where ua is velocity of the air stream approaching the fiber bundle, given by the Reynolds number for the air stream

qua dh l

ð12Þ

where l is dynamic viscosity and q is density of air. Dimensionless pressure for the air stream is defined by

p qu2a 2

p ¼ 1

ð13Þ

Dimensionless temperature is formulated by [34]

ð14Þ

where h is dimensionless temperature of air stream, ro is fiber outer radius, Ta is ambient temperature of air, q is heat flux imposed on the air-side tube surface and k is heat conductivity of air. The oblique angle a is defined as

a ¼ arctanðux =uz Þ

   dh ðdp=dzÞ qum dh 2 ¼ 
um qu2m =2 l

ZZ h

(b)

T
Ta qr o =k

ð18Þ

ð19Þ

where u*m is the mean dimensionless velocity and is given by

x

h¼

ðhw
hb ÞEFF0 E0
ðhw
hb ÞABB0 A0   ln ðhw
hb ÞEFF0 E0 =ðhw
hb ÞABB0 A0

ðfReÞ ¼

D

z

Re ¼

ð17Þ

where subscripts ‘‘w” and ‘‘b” represent ‘‘wall mean” and ‘‘bulk”, respectively; subscripts ‘‘ABB0 A0 ” and ‘‘EFF0 E0 ” represent the inlet and outlet planes of the periodic cells, respectively. For parallel flow, the term (ST/2Lsin a) in Eq. (17) is substituted by the cross sectional area normal to the inlet flow in the cell. For the special case of parallel flow, product of friction factor and Reynolds number (fRe) is applied to represent the characteristics of fluid flow in the shell side [35]

F' G'

H'

ST L sin a ðhb ÞEFF0 E0
ðhb ÞABB0 A0 Re Pr 2Am Dh

The equation is applicable to 0° < a 6 90°. In the equation, ST is transverse pitch of the tube bank, Am represents the tube surface area in the air side, which is proL/sin a and 2proL/sin a for the inline and the staggered arrays, respectively; Dh is the logarithmic mean temperature differences between tube surface and the air stream and it equals

x

A' B' C'

ð16Þ

where u* is dimensionless velocity of air and A means the crosssectional area normal to the flow direction. The heat flux released from tube surfaces is absorbed by the air stream when air flows in the periodic cell, resulting in an increase in the bulk temperature of air stream. The mean Nusselt number for the air stream across a periodic cell can be calculated by an energy balance, analyzed by

F' A' B'

u dA

ð15Þ

i 2
ðlum =dh ðdp=dzÞÞ dA ZZ dA

ð20Þ

For cross flow, mean friction factor in a periodic cell is useful to predict the characteristics of air stream across a tube bundle, given by [16]:

fm ¼

ðpÞEFF0 E0
ðpÞABB0 A0 0:5N qu2a;max

ð21Þ

where N is number of rows along the flow direction and maximum velocity ua,max is different at various geometries. For the aligned array, ua,max occurs at the transverse plane and is determined by the mass conservation for an incompressible fluid [36]

ua;max ¼

ST ua ST
d

ð22Þ

For the staggered arrangement, ua,max may occur at either the transverse plane or the diagonal plane, as shown in Fig. 1(b). It will occur at the diagonal plane when the geometrical parameters satisfy

" SD ¼

S2L

 2 #1=2 ST ST þ d þ < 2 2

ð23Þ

in which case ua,max is defined by

ua;max ¼

ST ua 2ðSD
dÞ

ð24Þ

L.-Z. Zhang et al. / International Journal of Heat and Mass Transfer 91 (2015) 1259–1272

or ua,max will occur at the transverse plane and take the same form as aligned array when the geometries do not satisfy Eq. (23). The critical Reynolds number deviating from laminar flow across tube banks is 2  105 [36]. In this study, low Reynolds numbers are investigated, because in heat and mass exchangers, the flow rates are usually low and fibers are fine. The Reynolds number of air stream is varied from 50 to 300 and thus the air stream is considered to be laminar. The following boundary conditions are specified for the periodic cells shown in Fig. 2. On the tube outer surface for the air stream, at planes BCC0 B0 , DEE0 D0 and HGG0 H0 , no-slip velocity and uniform wall heat flux boundary conditions are applied:

u ¼ 0 and q ¼
k

 @T  @nsurface

ð25Þ

Due to the periodicities of the fluid flow across the fiber bundle, at inlet and outlet surface planes ABB0 A0 and EFF0 E0 , the periodic boundary conditions of velocity for the air stream are defined by

ðux ÞABB0 A0 ¼ ðux ÞEFF0 E0

and ðuy ÞABB0 A0 ¼ ðuy ÞEFF0 E0

ð26Þ

The periodic boundary conditions of temperature can be described as

ðfÞABB0 A0 ¼ ðfÞEFF0 E0

ð27Þ

where f is dimensionless temperature employed for the periodic boundary condition, defined by [36–38]:

f ¼ ðh
hw Þ=ðhb
hw Þ 0

0

ð28Þ 0

0

0

0

0

0

0

0

0

0 0 0

At plane AFF A , CDD C , AHH A , GFF G , ABCDEF, A B C D E F , ABCDEFGHA and A0 B0 C0 D0 E0 F0 G0 H0 A0 on the periodic cells, symmetric boundary conditions are proposed:

@w ¼0 @n

ð29Þ

where w represents the velocity, pressure and temperature, and n refers to the normal directions. For such ducts with many tubes, fiber-to-fiber interaction will determine the flow finally. This is different to a single tube free flow. Periodicity will be finally developed after 4–5 cycles, as described in the classical book of [36]. At last the flow and heat mass transfer will become stable. Therefore a periodicity assumption for boundary conditions is valid. This boundary condition has been practiced by many other authors like [39]. In future applications of mass transfer for membrane bundles, convective mass transfer coefficients can be obtained from heat transfer coefficients by analogy. The Colburn j factor for heat transfer can be calculated by Ref. [40]

j¼

Nu RePr1=3

ð30Þ

In this study, only the periodic mean Nusselt number for oblique flow is calculated. Mass exchange in a membrane contactor may occur due to the permeability of hollow fiber membrane. In this case, the Colburn j factor for mass transfer is j = Sh/(Re Sc1/3). Mass transfer coefficients can be obtained from the Chilton–Colburn analogy [41] and given by

Sh ¼

Nu Le1=3

ð31Þ

where Le is Lewis number and calculated by

Le ¼

Pr Sc

ð32Þ

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2.2. Solution procedure The purpose is to develop a software package for calculations of fluid flow and heat transfer in periodic cells. The partial differential equations for momentum and energy are discretized by means of a finite volume method [42]. Since the velocity field is related to the temperature field, ADI techniques are applied to solve these equations. The periodic cells are in irregular shapes so a body-fitted coordinate system is applied to generate structured meshes. Grids with 61  21 for the inline array and 81  21 for the staggered array and 40 grids in z-axis are applied. It is noteworthy that normal outlet velocity is revised by satisfying the overall mass conservation during each iteration. The solution procedure is: (a) Based on Eqs. (25), (26), (29), solve the mass and momentum equations (1)–(4) for the air stream. These Navier–Stokes equations are solved using the SIMPLE algorithm [43,44]. Then the velocity fields and friction factors are obtained. (b) Assume the initial temperature for the whole domain. (c) Considering Eqs. (25) and (29), solve the energy equation (5) for the air stream. Get the temperature fields. (d) Based on the periodic boundary conditions of Eqs. (27) and (28), update the inlet values of dimensionless temperature. Give the new values as default values and return to step (c), until the old and the new values are converged. After these procedures, all the governing equations are satisfied simultaneously. The friction factors and convective heat transfer rates are obtained. To assure the accuracy of the results calculated, grid independence test is conducted. It is found that a finer mesh (grids with 61  31 for the inline array and 81  31 for the staggered array and 50 grids in z-axis) will have little merit (less than 1.0%) in further increasing the accuracy of calculation. Therefore the current mesh number is considered to be enough. For reasons of symmetry and simplicity in calculation and to disclose the fluid flow, surface planes ABCDEFA and ABCDEFGHA on the periodic cells in Fig. 2 are selected as the representative domains. The mesh structure for the surface planes on the periodic cells is schematically plotted in Fig. 3.

2.3. Experimental work An experiment is designed to study the oblique fluid flow and heat transfer across a tube bank. The experimental setup is depicted in Fig. 4. As seen, the system comprises of four main components: a blower, a front section, a test section and a rear section. Pre-conditioned air is supplied into the duct by a blower. The volumetric air flow rates can be regulated by the blower, resulting in different Reynolds numbers. In the front domain and rear domain, wind screens are installed in the duct to ensure a steady and uniform air stream to and from the tube bank. The test section is located between the front domain and the rear domain. Different sections are connected by custom-designed flanges. The whole ducting work is thermally insulated from the ambient environment by a layer of 20 mm thick insulation cotton. The test section is assembled with four smooth acrylic-glass plates of 5 mm in thickness, as schematically shown in Fig. 5. Take the test section with inline array as an example. 250 small holes are drilled evenly on the left and right organic glass plate surfaces, respectively. The heating tubes (rods) are inserted through these holes and stationed by insulation glue. They are 10 mm in outer diameter, 160 mm in effective length. They are made with

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(a)

(b) Fig. 3. Surface mesh structures on the periodic cells along the flow direction: (a) inline array; (b) staggered array.

Fig. 4. Experimental set-up for oblique flow heat transfer from a tube bank under constant heat flux boundary conditions. Heating rods are inserted across the duct with an oblique angle. Circled T and P represent thermometers and pressure drop gauges, respectively.

Nickel–Chromium alloy electric resistance wires, wrapped by stainless steel shells. The voids between the two layers are filled with magnesia powders. Thermocouples, type K and 3 mm in diameter, are inserted through the first and last rows of holes and are glued by conductive glue to the first and last rows of heating tubes. They are positioned evenly on the tube surfaces to measure the surface temperature. These measured values are considered as the tube surface temperatures and then averaged transversely to obtain the transversely mean temperatures. The heating tubes are designed to be in inline or staggered arrays. Oblique angles between the heating rods and the flow direction are pre-designed before the construction of the test section. The bundle of heating rods is heated by a power source, with power control. Two thermometers are stationed to and from the test section to measure the air inlet and outlet temperatures. The picture of the test section is shown in Fig. 6. The geometries of the test section are listed in Table 1. There are 50 rows along the flow, so the flow can be regarded as fully developed and the entry effects can be neglected. After the test section is assembled, the blower is switched on and the desired flow rate is set using the anemometer. When the flow rate and inlet fluid temperature is stabilized, the power sup-

Flange Flange

Air in

Holes for tubes

Fig. 5. Schematic of the test section. The circles represent the holes for inserting heating rods. The circles with dots are the holes for temperature measurements.

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The electric power supplied can be converted to heating power and monitored by a multimeter. Heat balance is calculated by comparing the heat transferred by air stream and the electric power supplied. A heat balance is checked to ensure the heat released by the heating rods is within 1% deviation from the heat transferred to the air stream. Pressure drop across the tube bundle is measured by a digital pressure drop gauge (DP-1000III). Anemometers (Testo-425) are used to measure the mean velocities to and from the test section. The differences between the inlet and outlet velocities are controlled to within 1%. Heat transferred by the air flow can be calculated by

Hp ¼ ma cp ðT fo
T fi Þ

ð33Þ

where Hp is heating power absorbed by the air stream, ma is air flow rate, cp is specific heat of air, Tfi and Tfo are inlet and outlet bulk temperatures of the air stream, respectively. Mean convective heat transfer coefficients are calculated by

hm ¼ Fig. 6. Picture of the test section with the heating rods. The holes for temperature measurements are covered by insulating cotton.

ply to the heating tubes is set to the desired value. Steady state is reached after about 40 min in each test when all the temperature readings (air and tube surface) are within 0.1 °C for about 20 min. Steady state readings from the thermocouples, pressure drop gauges and flow meters are recorded. The experimental procedure is repeated and averaged.

Hp Atot DT lm

ð34Þ

where Atot is total area of convective heat transfer surface, DTlm is the logarithmic mean temperature difference between the wall and air stream and defined by

DT lm ¼

ðT wi
T fi Þ
ðT wo
T fo Þ ln ½ðT wi
T fi Þ=ðT wo
T fo Þ

ð35Þ

where Twi and Two are mean wall temperatures of first row and last row of tubes, respectively. The average Nusselt number in the test section is calculated by

Table 1 Characteristic pitches of bundle and some transport properties. Name of properties Array pattern Impinging angles Number of lines Number of rows Transverse pitch Longitudinal pitch Effective length Tube outer diameter Tube outer radius Ambient temperature Velocity of air Density of air Dynamic viscosity of air Heat conductivity of air Specific heat of air

Symbol

Unit

a

deg

ST SL Leff d ro Ta ua

mm mm mm mm mm K m/s kg/m3 Pa  s W/m/K J/kg/K

q l

k cp

Array 1

Array 2

Inline 15, 30, 45, 60, 75 5 50 17.5 17.5 160.0 10.0 5.0 293.0 0.073–0.438 1.225 1.789  10
5 0.0263 1006.43

Staggered 15, 30, 45, 60, 75 5 51 17.5 17.5 160.0 10.0 5.0 293.0 0.073–0.438 1.225 1.789  10
5 0.0263 1006.43

Table 2 Comparisons of periodic mean Nusselt numbers under uniform heat flux boundary conditions (Num) and friction factors (fm) for cross flow, with inline arrangement and ST = SL. SL/d

Num (This study)

fm Refs. [36–38]

(This study)

Num Refs. [36–38]

Re = 50 1.25 1.5 1.75 2.0 2.5

19.41 9.05 6.03 4.67 3.36

21.13 9.59 6.88 5.47 3.75

fm Refs. [36–38]

(This study)

Refs. [36–38]

20.25 9.29 6.46 5.09 3.62

0.514 0.316 0.2336 0.1812 0.1142

0.517 0.318 0.232 0.180 0.115

22.58 10.40 7.53 5.98 3.83

0.2919 0.1822 0.1394 0.1113 0.0492

0.293 0.183 0.140 0.111 0.049

Re = 100 19.56 9.11 6.07 4.71 3.39

0.875 0.551 0.409 0.325 0.228

0.880 0.554 0.411 0.327 0.229

21.32 9.68 6.94 5.52 3.78

0.347 0.214 0.156 0.1216 0.0627

0.346 0.213 0.157 0.121 0.063

Re = 200 1.25 1.5 1.75 2.0 2.5

(This study)

20.09 9.22 6.40 5.04 3.59 Re = 300 22.37 10.31 7.56 6.03 3.86

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Table 3 Comparisons of periodic mean Nusselt numbers under uniform heat flux boundary conditions (Num) and friction factors (fm) for cross flow, with staggered arrangement and ST = SL. SL/d

Num

fm

(This study)

Refs. [36–38]

Num

(This study)

Refs. [36–38]

Re = 50 1.25 1.5 1.75 2.0 2.5

18.19 12.53 10.50 9.42 8.25

1.337 1.055 0.899 0.774 0.611

1.343 1.059 0.903 0.777 0.614

27.17 18.79 15.63 14.15 12.71

0.770 0.613 0.531 0.458 0.3424

0.773 0.615 0.533 0.460 0.344

21.25 15.34 12.76 11.45 10.13

Re = 200

Num ¼

(This study)

Refs. [36–38]

21.37 15.23 12.66 11.37 10.05

1.001 0.782 0.667 0.575 0.454

1.004 0.786 0.670 0.577 0.456

31.62 21.71 18.16 16.54 15.02

0.656 0.540 0.475 0.419 0.3186

0.653 0.538 0.473 0.421 0.320

Re = 300

27.28 18.94 15.75 14.27 12.80

hm dh k

31.88 21.87 18.27 16.65 15.14

ð36Þ

As seen, the mean heat transfer rates are calculated by the temperature differences across the whole duct. Local heat transfer coefficients are not measured because the adding of temperature sensors inside the bundle voids would disturb the flow. Further, temperature differences between two neighboring rows are small, which would lead to large errors. The average friction factor in the test section is formulated by

13 12 11 10

0.6 0.5

9 Nu

Dp ¼1 N q u2a;max 2

0.7 Num (IA) Num (SA) fm (IA) fm (SA)

ð37Þ

0.4 f

fm

Refs. [36–38]

Re = 100

18.04 12.44 10.45 9.36 8.19

1.25 1.5 1.75 2.0 2.5

fm

(This study)

8 0.3

7

where Dp is pressure drop between the inlet and outlet of the test section. The uncertainties for measurements are: temperature ±0.1 °C; volumetric flow rate ±0.5%; pressure drop ±1 Pa. The relative error of convective heat transfer and friction factor of air stream may be calculated by [45]:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n " 2  2 # X @f Dy u Dxi ¼t y @x y i i¼1

ð38Þ

where f is the function of the independent variables such as Nusselt number and friction factor, xi and 4xi are variables of the functions and absolute errors associated with the variables, respectively. The calculated final uncertainties are 8.9% for Nusselt number and 7.7% for friction factor.

6

0.2

5 0.1

4 3 10

20

30

40 50 60 Oblique angle α (deg)

70

0.0 80

Fig. 7. Shell-side mean Nusselt number (Num) and friction factor (fm) for the duct, SL/d = ST/d = 1.75 and Re = 100. The solid lines represent the numerical data and the discrete points stand for the experimental results. IA – inline array; SA – staggered array.

Num (IA) Num (SA) fm (IA) fm (SA)

14 3. Results and discussion

12

3.1. Model validation

0.7 0.6 0.5

Table 4 Comparisons of the fully developed (fRe) and Nusselt numbers (Nu) under uniform heat flux boundary conditions for parallel flow, ST = SL. SL/d

1.25 1.5 1.75 2.0 2.5

In-line arrangement

Staggered arrangement

Nu

Nu

(fRe)

10 Nu

0.4 f

It is better first to compare the numerical results with the available data for benchmark heat transfer problems. Though it is difficult to find transport data for oblique flow from available

0.3

8

0.2 6 0.1

(fRe)

(This study)

Ref. [20]

(This study)

Ref. [21]

(This study)

Ref. [20]

(This study)

Ref. [21]

4.76 4.83 4.14 3.51 2.70

4.72 4.87 4.17 3.54 2.72

22.49 15.85 11.50 9.90 7.44

22.60 15.92 11.55 9.94 7.48

10.77 7.68 5.74 4.60 3.34

10.86 7.75 5.79 4.64 3.37

27.42 21.15 14.98 12.30 8.19

27.55 21.23 15.05 12.36 8.23

4 25

50

0.0 75 100 125 150 175 200 225 250 275 300 325 Re

Fig. 8. Effects of different flow rates on the shell-side mean Nusselt number (Num) and friction factor (fm) for the duct, SL/d = ST/d = 1.75 and a = 45°. The solid lines are the numerical data and the discrete points represent the experimental results. IA – inline array; SA – staggered array.

L.-Z. Zhang et al. / International Journal of Heat and Mass Transfer 91 (2015) 1259–1272

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Fig. 9. Contours of velocity (m/s) on the surface ABCDEFA of the periodic cell in Fig. 2 with various oblique angles. Inline arrangement and Re = 100, SL/d = ST/d = 2. Viewed from the normal direction.

Fig. 10. Contour of velocity (m/s) on the surface ABCDEFGHA of the periodic cell in Fig. 2 with various oblique angles. Staggered arrangement and Re = 100, SL/d = ST/d = 2. Viewed from the normal direction.

literature, the data for pure parallel and cross flow tube banks are well-established. They can be regarded as two extreme cases for oblique flow. The corresponding skewed angles are 0° and 90° respectively. The periodic mean Nusselt number Num and friction factor fm (fRe for parallel flow) under uniform wall heat flux bound-

ary conditions are constants for hydro and thermally developed flows. The comparisons between the simulated data for the two extreme cases and those in Refs. [36–38] are listed in Tables 2 and 3 for cross flow with inline and staggered arrays, respectively. Table 4 lists the comparison for the parallel flow [20,21]. As seen,

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0.8

SL/d=2.0 (IA, H) SL/d=1.75 (IA, H)

0.7

SL/d=1.5 (IA, H) 0.6

SL/d=2.0 (SA, H) SL/d=1.75 (SA, H)

0.5

f

SL/d=1.5 (SA, H)

0.4 0.3 0.3 0.2 0.1 0

15

30 45 60 Oblique angle α (deg)

75

90

Fig. 11. Mean friction factors (fm) in a periodic cell with various oblique angles, ST = SL and Re = 100.

the maximum difference of the validation is less than 1.0% for Nu and 0.7% for f, which means that this method is successful in predicting the flows. For oblique flow, there is no available transport data. So the experimental data from Section 2.3 is used to validate the numerical results. The operating and geometrical conditions have also been listed in Table 1. The experimental results of the average Nusselt numbers and friction factors with various arrangements are calculated by Eqs. (33)–(37).

Totally five bundles with different angles are assembled. For each angle, six Reynolds numbers are tested. Comparisons are made between the calculated and experimentally obtained shellside average Nusselt numbers and friction factors for the duct, as depicted in Figs. 7 and 8. As seen, the deviations between the experimental results and numerical data are lower than 6.7% for heat transfer and 5.8% for friction factors. Overall this approach is effective in predicting the shell-side convective heat transfer and pressure drop in the tube bundle.

Fig. 12. Contours of dimensionless temperature on the surface ABCDEFA of the periodic cell in Fig. 2 with various oblique angles. Inline arrangement and Re = 100, SL/d = ST/d = 2. Viewed from the normal direction.

L.-Z. Zhang et al. / International Journal of Heat and Mass Transfer 91 (2015) 1259–1272

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Fig. 13. Contours of dimensionless temperature on the surface ABCDEFGHA of the periodic cell in Fig. 2 with various oblique angles. Staggered arrangement and Re = 100, SL/d = ST/d = 2. Viewed from the normal direction.

3.2. Effects of oblique angle on pressure drop After model validation, numerical work is performed to study the effects of oblique angle on flow field. The flow in a periodic cell is hydrodynamic developed. The periodic mean friction factors at different oblique angles are constants. Surfaces ABCDEFA and ABCDEFGHA on the periodic cells, as depicted in Fig. 2, are selected to disclose the fluid fields with inline and staggered array patterns, respectively. The operating conditions are: Re = 100, ST/d = SL/d = 2. Fluid fields at three typical oblique angles are plotted in Figs. 9 and 10 for the two arrays respectively. It is obvious that the mean velocities on the surfaces ABCDEFA or ABCDEFGHA at higher oblique angles exceed those at lower oblique angles. The velocity isolines near the symmetric boundary CD reveal that swirls are generated between the tube walls, due to the fact that the fluid turns abruptly when facing the rear tubes. Fig. 9 shows that the areas circled by velocity isolines and symmetric boundaries CD, which indicate strong swirls, increase as the oblique angle goes up. Fig. 10 demonstrates the phenomenon of recirculation is

strengthened with the rise in skewed angles. The momentum transfer is intensified by these swirls. The flow channel lengths ( perpendicular to view direction) between the tubes in periodic cells are larger with the decrease in oblique angle. The recirculating zones seem longer because the flow channels are longer. In contrary, quantitatively, the mean velocities on the surfaces of the periodic cells are lower at lower oblique angles. The curves of periodic mean friction factors in a periodic cell versus oblique angle are shown in Fig. 11 for both two arrays. The cases of pure cross flow (90°) and pure parallel flow (0°) have been extensively studied previously. So the experiments concerning cross flow and parallel flow are not repeated. Labels in Figs. 11, 14 and 15 represent the numerically-obtained results from 0° to 90°. The impinging angle in the experiments varies from 15° to 75°, which is applied to validate the numerical data from 15° to 75°. As seen, the friction factors rise with an increase in oblique angles. Also the friction factors increase with decreasing pitch-todiameter ratios (SL/d). The reason is that higher packing fraction intensifies the disturbance between the neighboring tubes. The

Fig. 14. Periodically mean Nusselt numbers (Num) with various oblique angles, ST = SL and Re = 100. The dashed lines are under uniform wall temperature boundary conditions.

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L.-Z. Zhang et al. / International Journal of Heat and Mass Transfer 91 (2015) 1259–1272

Fig. 15. j/f factor with various oblique angles and different flow rates, SL/d = ST/d = 2.

friction factors for staggered arrays are greater than those for aligned arrays at same pitch-to-diameter ratios due to the more complicated flow structure.

Constant

3.3. Effects of oblique angle on convective heat transfer The flow in a periodic cell is also thermally developed. It means that fully-developed cyclic mean Nusselt numbers are also constants. The oblique angle has great impacts on the convective heat transfer. Figs. 12 and 13 show the contours of dimensionless temperature on the surfaces ABCDEFA and ABCDEFGHA, for the inline and staggered arrangement respectively. Similarly three typical angles are chosen to figure out the differences. As seen, the mean values of isotherms rise with an increase in oblique angle, implicating

Table 5 Constants in correlations for calculation of the periodic mean Nusselt numbers and friction factors under uniform heat flux boundary conditions, inline arrangement and ST = SL. Constant

Table 6 Constants in correlations for calculation of the periodic mean Nusselt number and friction factors under uniform heat flux boundary conditions, staggered arrangement and ST = SL.

Angle 15

30

45

60

75

90

SL /d = 1.25 m C b a

0.230 3.102
0.660 6.508

0.154 7.264
0.614 7.149

0.113 10.496
0.592 7.078

0.080 13.977
0.618 8.608

0.066 16.121
0.626 9.505

0.068 16.676
0.613 9.235

SL /d = 1.5 m C b a

0.129 3.508
0.782 7.092

0.130 4.191
0.725 6.679

0.111 5.286
0.696 6.828

0.093 6.301
0.676 6.891

0.083 6.945
0.654 6.574

0.067 7.667
0.618 5.877

SL /d = 1.75 m C b a

0.117 3.111
0.813 3.261

0.128 3.142
0.687 3.994

0.161 2.950
0.605 3.164

0.147 3.442
0.617 3.729

0.136 3.774
0.619 4.053

0.120 4.162
0.607 4.121

SL /d = 2.0 m C b a

0.082 2.977
0.870 6.436

0.137 2.440
0.763 4.441

0.186 2.108
0.641 2.912

0.175 2.396
0.617 2.978

0.155 2.719
0.608 3.085

0.137 3.031
0.607 3.254

SL /d = 2.5 m C b a

0.096 2.136
0.859 4.504

0.113 2.058
0.846 4.802

0.107 2.210
0.878 6.037

0.088 2.506
0.874 6.176

0.095 2.495
0.839 5.450

0.075 2.815
0.862 6.347

Angle 15

30

45

60

75

90

SL /d = 1.25 m C b a

0.235 4.600
0.782 12.516

0.299 3.796
0.602 7.199

0.341 3.830
0.462 5.068

0.342 4.446
0.336 3.710

0.331 5.171
0.34 4.509

0.320 5.646
0.395 6.246

SL /d = 1.5 m C b a

0.169 4.428
0.760 8.980

0.276 3.040
0.549 4.609

0.371 2.280
0.407 3.261

0.347 3.031
0.363 3.411

0.328 3.669
0.386 4.424

0.312 4.088
0.373 4.468

SL /d = 1.75 m C b a

0.153 3.550
0.757 6.322

0.299 2.195
0.703 7.043

0.357 2.018
0.476 3.636

0.337 2.646
0.380 3.164

0.326 3.108
0.354 3.284

0.309 3.466
0.355 3.539

SL /d = 2.0 m C b a

0.219 2.230
0.719 4.518

0.329 1.613
0.576 3.077

0.350 1.861
0.471 2.793

0.338 2.375
0.410 2.930

0.333 2.721
0.36 2.872

0.319 2.980
0.344 2.909

SL /d = 2.5 m C b a

0.352 0.993
0.635 2.149

0.366 1.158
0.607 2.570

0.352 1.622
0.521 2.596

0.350 1.986
0.502 3.394

0.354 2.192
0.442 3.219

0.340 2.399
0.373 2.583

higher mean temperature in a periodic cell. The reason is that stronger swirls are generated at higher skewed angles, which would promote the mixture of cold bulk flow and hot fluid near the tube wall and decrease the thermal boundary layer thickness. Consequently the heat transfer performance is intensified. The periodic mean Nusselt numbers versus oblique angles with various longitudinal pitch-to-diameter ratios are plotted in Fig. 14 for the two layouts. The curves show that the Nusselt numbers rise with increasing oblique angles and they become stable when the angles approach 90°. Similarly the periodic mean Nusselt numbers increase with decreasing pitch-to-diameter ratios. At the same pitch-to-diameter ratios, the Nusselt numbers for the staggered arrays are much higher than those for the inline arrays. The reason is that in a staggered bank more tube surface touches the main stream.

L.-Z. Zhang et al. / International Journal of Heat and Mass Transfer 91 (2015) 1259–1272

3.4. Effects of oblique angle on the integrated performance In practical application, both heat transfer and pressure drop are important parameters for heat exchanger design. High heat transfer at low pressure drop is a desirable case. The integrated performance of heat exchanger is often estimated by the convective heat transfer to pressure drop ratio, j/f factor [46]. Fig. 15 shows the curves of j/f factor versus oblique angle under various flow rates. It is obvious that j/f factors for the inline arrays are greater than those for the staggered arrays, under large oblique angles at the same Reynolds numbers. The reason is that for the staggered arrays, pressure drop penalty exceeds the rise in heat transfer when compared with inline arrays. For both two arrays, j/f factors decrease with an increase in skewed angle, which indicates that integrated performance for parallel flow is relatively better than that for cross flow. The j/f factors for two typical arrays decrease with increasing Reynolds numbers. The reason is that larger flow rates result in higher pressure losses and decreased gains in convective heat transfer. The pressure drop plays a major role. 3.5. Correlations for performance evaluation Correlations for the estimation of pressure drop and convective transfer coefficient are useful in practical applications. To simplify equipment design, the numerical data are regressed to obtain the periodic mean Nusselt numbers and friction factors with various operating and geometrical parameters. These correlations take the following form [36]:

Nu ¼ CRem Pr1=3

ð39Þ

f ¼ aReb

ð40Þ

where the coefficients C, m, a, b are constants depending on the geometrical and operating configurations. The constants in the correlations for inline and staggered arrays are shown in Tables 5 and 6, respectively. The application range is 50 6 Re 6 300 in this study. The maximum errors of these correlations are below 8.0%. In Ref. [30], we have given the transport data for oblique flow under uniform temperature boundary conditions. For these two typical boundary conditions, the impacts of oblique angles on the transport phenomena are similar. The Nusselt numbers under uniform wall temperature conditions are plotted as dashed lines in Fig. 14. Generally most of the periodic mean Nusselt numbers under uniform wall heat flux boundary condition are larger than those under uniform temperature boundary condition at the same operating conditions, as shown in Fig. 14. This is in accordance to heat transfer in common pipes. For mass transfer problems encountered in membrane bundles, the uniform heat flux boundary condition is in analogy to uniform mass flux on membrane surfaces. With the Chilton–Colburn analogy [2,17], the Sherwood number can be estimated from Nusselt number.

4. Conclusions Oblique fluid flow and heat transfer across a tube bundle under uniform wall heat flux boundary conditions are investigated. The effects of the skewed angle on the fluid flow and heat transfer are numerically studied. A heat transfer experiment is conducted to validate the simulated data under skewed flows. Correlations are obtained to estimate the periodic mean friction factor and Nusselt number. It is concluded that:

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