better thermal calculations using modified generalized leveque equations for chevron plate heat...

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Better Thermal Calculations UsingModified Generalized Leveque Equationsfor Chevron Plate Heat ExchangersMazen M. Abu-Khader aa Department of Chemical Engineering, Faculty of EngineeringTechnology , Al‐Balqa Applied University , Amman , JordanPublished online: 01 Aug 2007.

To cite this article: Mazen M. Abu-Khader (2007) Better Thermal Calculations Using ModifiedGeneralized Leveque Equations for Chevron Plate Heat Exchangers, International Journal of GreenEnergy, 4:4, 351-366, DOI: 10.1080/15435070701193068

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351

BETTER THERMAL CALCULATIONS USING MODIFIED GENERALIZED LEVEQUE EQUATIONS FOR CHEVRON PLATE HEAT EXCHANGERS

Mazen M. Abu-KhaderDepartment of Chemical Engineering, Faculty of Engineering Technology,Al-Balqa Applied University, Amman, Jordan

Plate heat exchangers are becoming vital process energy exchange equipment in chemicalindustries. The present work looks into the applicability of the Generalized Leveque Equa-tion (GLE), which is used to generate thermal performance calculations for chevron-typeplates utilized in plate and frame heat exchangers. New Modified Generalized LevequeEquations are proposed to improve calculation accuracy, which will enhance energy utili-zation and cost reduction in the design stage. A case study was selected to examine the GLEapplicability when implemented to pure and mixed chevron plates. The GLE and othermodified equations are compared with known classical correlations for different corruga-tion angles of 30°, 45°, 50°, 60° and 65° as pure chevron plates, and 45°/90°, 23°/90°, 45°/45°, 23°/45°, and 23°/23° as mixed chevron plates. Both the GLE and single Modified Gen-eralized Leveque Equation (MGLE) gave good results only for 30° and 50° pure chevronangles and failed to give acceptable results in 45°, 60° and 65° chevron angles; both, how-ever, gave acceptable results with all tested mixed angles. Also, the GLE has the tendency tobe more applicable in the laminar region where Re < 2100, but the inaccuracy increaseswhen applied in the turbulent region, while MGLE gave slightly lower relative error at highRe. Based on the classification of soft and hard chevron plates, two new Modified General-ized Leveque Equations, MGLE(1) and MGLE(2), were developed from the set of Kumar’sdata for pure chevron plates. The proposed system of equations are capable of providingexcellent results for both pure and mixed chevron angles when compared with a large set ofclassical correlations available in the literature.

Keywords: Plate heat exchanger; Generalized leveque equation; Chevron plates; Frictionfactor; Pure angles; Mixed angles

INTRODUCTION

There is an increased interest in the development of Plate Heat Exchangers (PHE) aseffective and compact heat transfer equipment. During the last decade, great advanceshave been achieved in the fabrication technology and in the materials used. These defi-nitely have improved some of its previous performance limits. One of the major problemsfacing researchers when dealing with plate heat exchangers is that there are no codes orstandards that apply to these types of exchangers. Therefore, each manufacturer provides

Address correspondence to Mazen M. Abu-Khader, Department of Chemical Engineering, Faculty ofEngineering Technology, Al-Balqa Applied University, P.O. Box 15008, Marka (11134), Amman, Jordan.E-mail: [email protected]

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352 ABU-KHADER

its own range of different plate types. Each plate type has its own unique thermal andhydraulic correlations that remain confidential to manufacturers. So, despite the age of theplate heat exchanger, there is very little information in the open literature concerning itsperformance. Therefore, there is some suspicion over both the design and the integrity ofthese exchangers (Hills, 1997).

Plate heat exchangers (PHE) exhibit excellent heat transfer characteristics, whichallow more compact designs than that achieved with conventional shell and tube heatexchangers. They have a very large surface area in a small volume and can be modified fordifferent requirements simply by increasing or decreasing the number of plates needed.With these advantages, along with advances in material technology in the form of newtemperature- and pressure-resistant materials for gasket or graphite plates, it is now possi-ble to use this class of heat exchangers appropriately for power and chemical processes.The basic theoretical principles of the plate and frame heat exchangers and their industrialapplications are now well established in many text references (Saunders, 1988; Hewitt,1998). Also, there are several comprehensively compiled review materials on variousdesign aspects in the open literature (Shah & Focke, 1988; Thonon et al., 1995). The mainadvantages and benefits offered by the gasket-plate heat exchangers are thoroughly dis-cussed by Kakac and Liu (2002).

Due to the wide variety of chevron plates, the present research direction is todevelop a single general expression to replace several classical correlations based on dif-ferent angles and Re ranges available in the literature for pure and mixed angles, which areused to carry out thermal performance and design calculations for plate heat exchangers.Also these expressions have the capability to accommodate any proposed new mixedangles. Recently developed general expressions, like the Generalized Leveque Equation(GLE) presented by Martin (1996), and a complex one presented by Muley and Manglik(1999), are mainly a function of the chevron angle b, Reynolds number Re and frictionfactor f.

It is important to keep in mind that when trying to develop a single and generalexpression, usually the applicability and accuracy become questionable. The objec-tives of this work are to verify applicability of the Generalized Leveque Equation(GLE) to pure and mixed chevron plates at different inclination angles and to examinethe possibility of proposing another single Modified Leveque Eqution (MGLE) basedon pure classical correlations (Kumar, 1984), and another two new modified expres-sions based on the idea of being able to divide the different chevron plates into twoplate categories — Soft Plate and Hard Plate types—without sacrificing the accuracy.It is essential to understand certain basics and related issues on plate heat exchangerand corrugated chevron plates, which represent the key element in the development ofthe thermal design correlations.

CHEVRON PLATES CHARACTERISTICS

A popular design in the chemical industry is the herringbone or chevron corrugatedplate, illustrated in Figure 1. Previous work on corrugations of this type showed that thecorrugation angle has a major influence on the thermal and hydraulic performance of theexchanger (Focke et al., 1985; Okada et al., 1972). The angle of the chevron corrugationhas vital importance as a design variable (Heavner et al., 1993). In this work, the inclina-tion angle is measured from the horizontal axis termed β, whereas in some of the literatureit is measured from the vertical axis and termed f, like in Martin (1996). A plate with a

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THERMAL CALCULATIONS USING MODIFIED GLEs 353

low chevron angle β offers a high heat transfer coefficient and high pressure drop, whereas aplate with a high chevron angle has lower heat transfer and lower pressure drop. The low-and high-chevron angle plates can also be referred to as hard and soft plates, respectively,reflecting the resistance that they present to a flowing fluid. A high chevron angle, soft plate,can be considered greater than 50°, while a low chevron angle, hard plate, is ≤50°.

Manufacturers produce plates with a limited range of different corrugation angles.This causes a problem in the design of the exchanger because it means that it is not possibleto exactly match any combination of duty and pressure drop. To improve the range of possi-ble corrugation angles, manufacturers produce channels that are composed of two differentcorrugation angles (a mixed-angle channel). For example, a channel may be composed of a60° plate adjacent to a 30° plate. Information regarding the performance of a mixed anglechannel is very scarce. It has been suggested that the performance of a mixed angle channelcan be described by the correlation for a pure angle channel having the same average inclina-tion angle (HTFS, 1984). The overall performance of the 60°/30° channel and the 90°/0°channel was very similar to the pure 45° channel. In the 60°/30° case, the two sides of thechannel displayed very similar performances. In contrast, the 90°/0° channel displayed twoquite different performance curves, but the average performance of the two plates was stillvery similar to the pure 45° channel. In addition, the two mixed channels displayed almostidentical pressure drop characteristics to the pure 45° channel (Heggs & Walton, 1999).

The corrugations increase the surface area of the plate as compared to the originalflat area. To express the increase of the developed length in relation to projected length, asurface enlargement factor f is then defined as the ratio of the developed length to the flator projected length:

Figure 1 Chevron plate used in plate heat exchangers.

LwLvLp

Lh

Dp

ϕβ

f = =developed length

projected length

A

Ae

p

(1)

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354 ABU-KHADER

The value of f is a function of the corrugation pitch and the corrugation depth, and itvaries between 1.15 and 1.25. The value of 1.17 is assumed to be a typical average,(Kumar, 1984). The value of f is used to calculate the effective flow path.

Flow channel is a conduit formed by two adjacent plates between the gaskets. The crosssection of a corrugated surface being very complex, the mean channel spacing, b, is defined:

Where p is the plate pitch or the outside depth of the corrugated plate, t is the plate thick-ness and b is also the thickness of a fully compressed gasket, as the plate corrugations arein metallic contact. Channel spacing b is required for calculation of mass velocity andReynolds number. The hydraulic diameter of the channel dh is defined as:

with the approximation that b << effective plate width Lw.The correlations governing the heat transfer and pressure loss calculations in pure

chevron type plate heat exchangers have been reported by Kumar (1984), dependingmainly on chevron angle and Reynolds number Re as shown in Table 1. If the characteris-tic dimension of a chevron plate is taken as the hydraulic diameter dh, the plate length is Lp

b p t= − (2)

dA

Phc

w

=×

=4 4Channel Flow Area

Wetted Surface (3)

db L

b L

bh

w

w

=+

4

2

2( )( )

( )f»f

(4)

Table 1 Constants for single – phase heat transfer and pressure loss calculation in Chevron Plate Heat Exchang-ers (Kumar, 1984)

Chevron Angle (degree) Heat Transfer Pressure Loss

Reynolds Number

cn m Reynolds Number

K n

≤ 30 ≤ 10 0.718 0.349 <10 50.000 1.000>10 0.348 0.663 10–100 19.400 0.589

> 100 2.990 0.18345 < 10 0.718 0.349 < 15 47.000 1.000

10–100 0.400 0.598 15–300 18.290 0.652> 100 0.300 0.663 > 300 1.441 0.206

50 < 20 0.630 0.333 < 20 34.000 1.00020–200 0.291 0.591 20–300 11.250 0.631> 300 0.130 0.732 > 300 0.772 0.161

60 < 20 0.562 0.326 < 40 24.000 1.00020–400 0.306 0.529 40–400 3.240 0.457> 400 0.108 0.703 > 40 0.760 0.215

≥ 65 < 20 0.562 0.326 50 24.000 1.00020–500 0.331 0.503 50–500 2.800 0.451> 500 0.087 0.718 > 500 0.639 0.213

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and plate pressure drop, then friction factor (z = 4f ) becomes a function of Reynolds num-ber and plate type as follows:

Generally, each plate type has the following form of equation used to correlate its data:

where f is the friction fanning factor and the K and n depend on the flow region (laminar,transition and turbulent). A correlation in the form of Eq. (7) has been proposed by Kumarand the values of constants Cn and m are also given in Table (1).

The Reynolds number is based on channel mass velocity and hydraulic diameter dhof the channel. The Nusselt numbers Nu is also based on hydraulic diameter of the chan-nel. The Prandlt number Pr equals (cp m / k), while (m / mw) is the viscosity ratio, which isequal to unity in the turbulent flow.

GENERAL LEVEQUE EQUATION

The Generalized Leveque Equation (GLE) characterizes a new analogy between frac-tional pressure drop and heat transfer that may be used in tube bundles, in cross-rod matrices,and many other periodic arrangements of solids in a fluid flow. Martin (1996, 2002) presentedthe GLE and illustrated its applicability on external and internal flow situations, and in predict-ing packed bed heat or mass transfer from pressure drops. The classical basic form of the GLEwas formulated for the first time by Andre Leveque’s thesis in 1928. The Leveque modelassumes a linear velocity profile inside the concentration boundary layer (Focke et al., 1985):

Where is the mass transfer coefficient, D is the diffusion coefficient, L is the length andfinally S is the wall average velocity gradient. Although derived for planar flow with aconstant wall velocity gradient, the model can be applied to a duct geometry provided thatthe temperature (concentration) boundary-layer thickness is very small compared with theradius of the wall curvature. This will be the case for short transfer lengths and highPrandtl (Schmidt) numbers. The average wall velocity gradient S can be presented interms of the geometric constant K in the following expression:

zr

= =2

2

Δpd

u Lfn Plate typeh

P

(Re, ) (5)

fK

fnn

= = =z

b4 Re

(Re, ) (6)

Nu Cnm

w=

⎛⎝⎜

⎞⎠⎟

Re Pr //

1 31 6

mm (7)

k D S L= 0 8075 2 1 3. ( / ) / (8)

k

SK

dh

= ⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜⎞

⎠⎟Re

8 2

u (9)

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356 ABU-KHADER

where u is the kinematic viscosity and the geometric constant (K = z Re). By substitutingthe geometric constant in equation (9), the average wall velocity gradient can be rewrittenas follows;

If the average wall velocity gradient expression above is substituted into equation(8), a fully mass transfer coefficient expression can be developed:

Multiplying the previous equation (11) by (dh./D), the following equation isobtained:

This final form of equation (12) is called the General Leveque Equation (GLE). Asthe GLE has an analogy between mass and heat transfer, it can be written as follows(Martin, 1996);

This equation consists of four dimensionless groups (Nu, Pr, Sh, Sc), hydraulicdiameter to channel length ratio (dh /L) and the friction factor z. In the gasketed plate heatexchanger, L = Lv which is the distance between two port centers;

where Pc is the corrugation pitch. Therefore, the GLE is:

This equation may be easily adapted to correlate experimental data for plate heatexchangers by replacing the theoretical constant including the geometrical parameter (dh/Pc)and the theoretical exponent 1/3, if necessary, by appropriate values obtained from experi-mental results.

A correlation equation was also introduced by Martin (1996), which is used toevaluate heat transfer coefficient a in technical plates based on the Leveque analogy,which is based on heat transfer data for mixed chevron plate heat exchanger (Heavneret al., 1993), together with their data on pressure drop. Using the definition of the

Sdh

=⎛

⎝⎜⎞

⎠⎟⎛

⎝⎜

⎞

⎠⎟z

uRe2

28(10)

k Dd L

h

=⎛

⎝⎜

⎞

⎠⎟0 8075

8

122

2

1 3

.Re

/

zu

(11)

Shk d

D

d

Lh h= = ⎛

⎝⎜⎞⎠⎟

Sc 0 40375 2

1 3

. Re/

z (12)

Sh Nu d

Lh

Sc

1 3 1 32

1 3

0 40375/ /

/

Pr. Re= = ⎛

⎝⎜⎞⎠⎟

z (13)

L Pd

L

d

Pv ch

v

h

c

= =/ sin( ), sin( )2 2j j (14)

Sh Nu d

Ph

cSc

1 3 1 32

1 3

0 40375 2/ /

/

Pr. Re sin( )= =

⎛⎝⎜

⎞⎠⎟

z j (15)

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friction factor, a general expression for Reynolds number Re can be developed asshown in equation (16):

The general expression for Nusselt number can be presented as follows:

where; By inserting equation (16) into equation (17), the following expression is obtained:

and rearranging equation (18):

the correlation equation can be developed in the following form;

With the adopted exponent q = (m / [2–n]) and . Table 2 represents theconstants for the correlation equation as illustrated by Martin (1996). Taking the arith-metic mean of the five values of q and the geometric mean of the five values of Cq (which

z j jRe sin( )Re

Re sin( )2 224

2= =XK

n

X K n= −4 22Re sin( )j

X C= XXnRe2−

Re ( / ) /( )= −X CXn1 2 (16)

Nu Cnm* .Re= (17)

Nu Nu w* / [Pr ( / ) ]/ /= 1 3 1 6m m

Nu CX

CnX

m

n* =

⎛⎝⎜

⎞⎠⎟

−2(18)

NuC

Cn

Xq

q* [ Re sin( )]= z j 2 2 (19)

Nu Cqq* [ Re sin( )]= z j 2 2 (20)

C C Cq n Xq( / )

Table 2 Constants for the semi-empirical equation (Martin, 1996).

f (°) K n Cn m q CX Cq

(45–90)67.5 1.458 0.084 0.278 0.683 0.356 4.139 0.168(23–90)56.5 1.441 0.135 0.308 0.667 0.358 5.168 0.171(45–45)45 0.687 0.141 0.195 0.692 0.372 2.765 0.134(23–45)34 0.545 0.156 0.118 0.720 0.390 2.054 0.089(23–23)23 0.49 0.181 0.089 0.718 0.395 1.402 0.078

0.374a 0.122b

aArithmetic average.bGeometric average.

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358 ABU-KHADER

is appropriate averaging for a set of power laws) the following correlation equation wasobtained for heat transfer in technical plates based on the Leveque analogy:

As shown in Figure 1: f is the chevron angle with respect to vertical-axis, this meansf = 90–b and therefore, the equality ⏐sin (2f)⏐ = ⏐sin (2b)⏐ can be applied. Equation (21)can be rewritten with respect to horizontal-axis as follows:

The theoretical background and the required mathematical derivations of the abovecorrelation equation are fully addressed. Now it is possible to proceed in developing otherModified Generalized Leveque Equations (MGLEs).

NEW MODIFIED GENERALIZED LEVEQUE EQUATIONS

In many cases, data from experimental tests are not linear, and the most popularforms are the power and exponential forms y = axb or y = aebx. The normal equations forthose analogous to the preceding development for a least-square line can be developed bysetting the partial derivatives equal to zero. Such nonlinear simultaneous equations aremuch more difficult to solve than linear equations. Thus, the exponential forms are usuallylinearized by taking logarithms before determining the parameters.

Equation (17), which represents the general form of the classical correlations andequation (20), which represents the general form of the General Leveque equation give thefollowing form:

For the equation (23) to be in a linear form, the natural logarithms are taken for both sides:

A linear function of Y = a x + b can be used to fit the above equation through the use ofnormal equations. Where Y = ln Cn Rem, x = In [4 f Re2 sin(2b)], a = q, and b = ln Cq. Based onthe heat transfer and pressure loss data for pure chevron plates reported by Kumar (1984), acomputer program was used to evaluate new values for Cq and q for each chevron plate angle.Table 3 lists the new values of Cq and q for each pure chevron plate angle and the arithmeticmean of the five values of q and the geometric mean of the five values of Cq. Finally, anothernew single correlation equation is obtained which can be called the Modified General LevequeEquation [MGLE] for heat transfer in technical plates, shown in equation (25):

Nuw

=⎛⎝⎜

⎞⎠⎟

⎡⎣ ⎤⎦0 122 21 31 6

2 0 374. Pr Re sin( )/

/.m

mz j (21)

Nuw

=⎛⎝⎜

⎞⎠⎟

⎡⎣ ⎤⎦0 122 21 31 6

2 0 374. Pr Re sin( )/

/.m

mz b

(22)

C C fnm

qqRe [ Re sin( )]= 4 22 b (23)

ln Re ln ln [ Re sin( )]C C q fnm

q= + 4 22 b (24)

Nuw

=⎛⎝⎜

⎞⎠⎟

⎡⎣ ⎤⎦0 099 21 31 6

2 0 385. Pr Re sin( )/

/.m

mz b (25)

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The single MGLE can be divided into two equations; one equation, MGLE(1),for hard chevron plates (b ≤ 50) and a second equation, MGLE(2), for soft chevronplates (b > 50). Taking the arithmetic mean and the geometric mean of (q) and (Cq)respectively for hard chevron plates, as shown in Table 4, the MGLE(1) can be pre-sented in the following equation (26):

Likewise, taking the arithmetic mean and the geometric mean of (q) and (Cq)respectively for soft chevron angles as shown in Table 5, the MGLE(2) can be presentedin the following equation (27):

Now the applicability of GLE, MGLE, MGLE(1) and MGLE(2) can beexamined against both pure and mixed classical correlations at different chevronangles and different Re ranges using the following selected case study from theopen literature.

Table 3 New values of Cq and q for each pure chevron plateangles.

b (degree) Cq q

30 0.148 0.36545 0.157 0.37050 0.083 0.39860 0.074 0.39465 0.068 0.400

0.099b 0.385a


Table 4 Values of Cq and q for pure chevron plate angles (b ≤ 50).

β (degree) Cq q

30 0.148 0.36545 0.157 0.37050 0.083 0.398

0.124b 0.378a


Nuw

=⎛⎝⎜

⎞⎠⎟

⎡⎣ ⎤⎦0 124 21 31 6

2 0 378. Pr Re sin( )/

/.m

mz b (26)

Nuw

=⎛⎝⎜

⎞⎠⎟

⎡⎣ ⎤⎦0 071 21 31 6

2 0 397. Pr Re sin( )/

/.m

mz b (27)

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360 ABU-KHADER

CASE STUDY: RATING CALCULATIONS

An example for check calculations was selected from the literature (HTFS,1984).The example stated that cold water was heated by a waste water stream in counter-currentflow. The cold water, with a flow rate of 2.5 Kg/s, enters the gasketed plate heatexchanger at 10°C and is heated to 45°C. The waste water has the same flow rate andenters at 70°C, leaving at 35°C. The maximum permissible pressure drop for waste water(process stream) is 14 Kpa and for cold water (service stream) is 15 Kpa. This case studywas tested using different mass flow rates for each selected pure and mixed inclinationangles.

The first part is the use of pure chevron type plates with different corrugation anglesof 30°, 45°, 50°, 60° and 65°, and the second part is the use of mixed chevron angles witharithmetic average of 22.5°, 33.5°, 45°, 56° and 67°.

Figures 2 to 6 show that the GLE and the single MGLE equations demonstrate thesame behavior and give an excellent match, with chevron angles of 30° and 50°, but thereis a clear deviation when using chevron angles of 45°, 60° and 65° when compared withclassical correlations for pure angles. On the other hand, both the GLE and the singleMGLE also demonstrate the same behaviour with mixed angles and give an excellentmatch with all tested mixed angle correlations.

Table 5 Values of Cq and q for pure chevron plate angles (b > 50).

β (degree) Cq q

60 0.074 0.39465 0.068 0.400

0.071b 0.397a


Figure 2 Applicability of GLE and single MGLE to plate heat exchanger with pure chevron angle 30°.

Chevron Angle 30

0

50

100

150

200

250

14000120001000080006000400020000Reynolds No. (Re)

Nes

sult

No.

(Nu)

ClassicalCorrelationGLE

MGLE

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Nes

sult

No.

(Nu)

Chevron Angle 45

Reynolds No. (Re)14000120001000080006000400020000

0

50

100

150

200


MGLE


Nes

sult

No.

(Nu)

Chevron Angle 50

Reynolds No. (Re)14000120001000080006000400020000

0

50

100

150

200


MGLE


Chevron Angle 60

14000120001000080006000400020000Reynolds No. (Re)

0

50

100

150

Nes

sult

No.

(Nu)


MGLE

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362 ABU-KHADER

Both the GLE and the single MGLE have the same mathematical behaviour, whichmeans that the arithmetic average is a valid assumption for mixed chevron plates. Thisagrees with the experimental work conducted by Heggs and Walton (1999). The GLE orthe single MGLE can be used for design calculations when only mixed chevron plates areinvolved. The relative error in the results obtained from the GLE increases with theincrease of Re in the turbulent region. The results from single MGLE have slightly lowerrelative error than the ones obtained from the GLE when soft plates are in use.

Figures 7 to 11 show that both the MGLE(1) and MGLE(2) give better overall per-formance and accuracy than the GLE, especially as the applicability range goes furtherinto the turbulent region. This concludes that both the MGLE(1) and MGLE(2) for hardand soft plates respectively are capable of producing better results than the GLE in repre-senting the correlations for pure angle plates because both of these equations are actuallyderived based on various correlations at different Re ranges for each angle. Table 6 repre-sents the final proposed system of equations that are effective and more accurate in bothpure and mixed angle plates with a wide applicability range from laminar up to turbulentflows.


Chevron Angle 65

14000120001000080006000400020000Reynolds No. (Re)

0

50

100

150N

essu

lt N

o.(N

u) ClassicalCorrelationGLE

MGLE

Figure 7 Applicability of GLE and proposed MGLE (1) to plate heat exchanger with pure chevron Angle 30°.

Chevron Angle 30

14000120001000080006000400020000Reynolds No. (Re)

0

100

200

300

Nes

sult

No.

(Nu)

ClassicalCorrelation

GLE

MGLE(1)

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Chevron Angle 45

14000120001000080006000400020000Reynolds No. (Re)

0

80

40

160

120

200N

essu

lt N

o.(N

u)ClassicalCorrelation

GLE

MGLE(1)


Chevron Angle 50

14000120001000080006000400020000Reynolds No. (Re)

0

50

100

150

200

Nes

sult

No.

(Nu) Classical

Correlation

GLE

MGLE(1)


Chevron Angle 60

14000120001000080006000400020000Reynolds No. (Re)

0

40

120

80

160

Nes

sult

No.

(Nu) Classical

Correlation

GLE

MGLE(2)

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364 ABU-KHADER

CONCLUSION

The GLE presented by Martin (1996) is only applicable for a good range of mixedchevron plates at acceptable relative error. Both the GLE and single MGLE followed thesame behaviour when pure and mixed angles were used. Also they gave an excellentmatch with pure chevron angles of 30° and 50°, but failed to give an acceptable accuracywith pure chevron angles of 45°, 60° and 65° when compared with classical correlationsfor pure chevron plates. This concludes that taking the arithmetic average angle for themixed chevron plates is an acceptable assumption.

It was noticed that the single MGLE gave slightly better presentation at highReynolds number (Re) in the turbulent region and less relative deviation error with chev-ron angles greater than 50° in mixed angles than the GLE. The derived two equations of

Figure 11 Applicability of GLE and proposed MGLE(2) to plate heat exchanger with pure chevron Angle 65°.

Chevron Angle 65

14000120001000080006000400020000Reynolds No. (Re)

0

40

80

120N

essu

lt N

o.(N

u)ClassicalCorrelation

GLE

MGLE(2)

Table 6 Proposed system of equations to be used for chevron plate heat exchangers.

Plate Type Pure angle plates

Hard

Soft

Mixed angle plates

Hard

Soft

Nuw

=⎛⎝⎜

⎞⎠⎟

⎡⎣ ⎤⎦0 124 21 31 6

2 0 378. Pr Re sin ( )/

/.m

mz b

Nuw

=⎛⎝⎜

⎞⎠⎟

⎡⎣ ⎤⎦0 071 21 31 6

2 0 397. Pr Re sin ( )/

/.m

mz b

Nuw

=⎛⎝⎜

⎞⎠⎟

⎡⎣ ⎤⎦0 122 21 31 6

2 0 374. Pr Re sin ( )/

/.

mm

z b

Nuw

=⎛⎝⎜

⎞⎠⎟

⎡⎣ ⎤⎦0 099 21 31 6

2 0 385. Pr Re sin ( )/

/.

mm

z b

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the MGLE, based on hard plate MGLE(1), and soft plate MGLE(2), can easily replace theKumar’s correlations for pure chevron angles and any proposed new mixed angles at widerange of Reynolds numbers from laminar to high turbulent regions. These new proposedequations will help to achieve better plate heat exchanger design and more accurate ther-mal calculations at wide range of Reynolds numbers. This will affect directly the overallcost and lead to more efficient systems for energy saving.

NOMENCLATURE

Ac Channel flow area (m2)Ae Effective area (m2)Ap Single plate projected area (m2)b Mean gap between adjacent plate (m)Cn, Cx, Cq Constantscp Specific heat (J/ Kg.K)d Diameter (m)dh Hydraulic diameter (m)D Diffusion coefficient (m2 s−1)Dp Port diameter (m)f Fanning friction factork Thermal conductivity (W/m.K)

Mass transfer coefficient (m s−1)K Constant in equation (9)L Length (m)Lc Compressed plate pack length (m)Lh Horizontal port distance (m)Lp Plate length (m)Lv Vertical port distance (m)Lw Plate width (effective plate width) (m)m Constant in equation (10)n Constant in equation (9)Nt Total number of plates.Nu Nessult Number (dimensionless)Δp Total pressure drop (Kpa)p Plate pitch (m)Pc Corrugation pitch (m)Pr Prandtl Number (dimensionless)Pw Wetted perimeter (m)Re Reynolds Number (dimensionless)S Average velocity gradiant at wall (s−1)Sc Schemidt Number (dimensionless)Sh Sherwood Number (dimensionless)t Plate thickness (m)u Velocity (m/s)

Greek symbolsx Friction factor ( four times the fanning friction factor)b Chevron angle with respect to horizontal-axis (rad)j Chevron angle with respect to vertical-axis (rad)r Fluid density (Kg/m3)f Enlargement factoru Kinematic viscosity (m2 s−1)

k

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366 ABU-KHADER

m Dynamic viscosity at average inlet temperature (Pa.s)mw Dynamic viscosity at wall temperature (Pa.s)

Subscript1 First stream side2 Second stream side.

REFERENCES

Focke, W.W., Zachariades, J., Olivier, I. (1985). The effect of the corrugation inclination angle on thethermohydraulic performance of plate heat exchangers. Int. J. Heat Mass Transfer 28: 1469–1479.

Heavner Rl, Kumar, H, Wanniarachchi, A. (1993). Performance of an industrial plate heat-exchanger – Effect of Chevron angle. Proceedings of the 29th National Heat Transfer Confer-ence (89: 262–267), Atlanta, GA.

Heggs, P.J, Walton, C. (1999). Local transfer coefficients in corrugated plate heat exchanger chan-nels with mixed inclination angles. 6th U.K. Heat Transfer Conference, Edinburgh-UK.

Hewitt, G.F. (1998). Handbook of Heat Exchanger Design. New York: Begell House.Hills, P.D. (1997). So what’s wrong with a shell and tube exchanger? Obstacles to the use of com-

pact heat exchangers in the process industries. Compact Heat Exchangers for the Process Indus-tries (pp. 69–78). New York: Begell House.

HTFS. (1984). Plate heat exchanger design report. In M.F. Edwards, W.L. Wilkinson, andH. Kumar, Eds. University of Bradford.

Kakac, S., Liu, H. (2002). Heat Exchangers: Selection, Rating and Thermal Design, Second Edition.London, UK: CRC Press.

Kumar, H. (1984). The plate heat exchanger: construction and design. 1st U.K. National Heat Trans-fer Conference, IChemE Symposium Series No: 86, pp. 1275–1288.

Leveque, A. (1929). Les lois de la transmission de chaleur par convection. Ann. Mines vol.13.Martin, H. (1996). A theoretical approach to predict the performance of chevron-type plate heat

exchangers. Chemical Engineering and Processing 35: 301–310.Martin, H. (2002). The gerneralized Leveque Eguation (GLE) and its use to predict heat and mass

transfer from fluid friction. Twelfth International Heat Transfer Conference, Grenoble, France.Muley, A., Manglik, R.M. (1998). Experimental study of turbulent flow heat transfer and pressure

drop in a plate heat exchanger with chevron plates. Trans of the ASME, Journal of Heat Transfer121: 110–117.

Okada, K., Ono, M., Tomimara, T., Okuma, T., Konno, H., Ohtani, S. (1972). Design and heat trans-fer characteristics of new plate heat exchanger. Heat Trans – Jap Res. 1: 90–95.

Saunders, E.A.D. (1988). Heat Exchangers Selection, Design and Construction. New York: Longman.Shah, R.K., Focke, W.W. (1988). Plate Heat Exchangers and Their Design Theory. In Heat Transfer

Equipment Design, RK Shah, EC Subbarao, and RA Mashelkar, Eds., Washington DC: HemispherePublishing.

Thonon, B., Vidil, R., Marvillet, C. (1995). Recent research and developments in plate heat-exchangers. J. Enhanced Heat Transfer 2(1–2): 149–155.

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better thermal calculations using modified generalized leveque equations for chevron plate heat...

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