Heat transfer engineering an international journal, tập 31, số 13, 2010

It is desirable to express the two-phase frictional pressure gra-dient, dp/dzf, versus the total mass flux G in a dimensionless heat transfer engineering vol... CopyrightTaylor and Franc

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C Taylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457631003639059

Two-Phase Flow Modeling in

Microchannels and Minichannels

M M AWAD and Y S MUZYCHKA

Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St John’s, Newfoundland, Canada

In this article, three different methods for two-phase flow modeling in microchannels and minichannels are presented They

are effective property models for homogeneous two-phase flows, an asymptotic modeling approach for separated two-phase

flow, and bounds on two-phase frictional pressure gradient In the first method, new definitions for two-phase viscosity are

proposed using a one-dimensional transport analogy between thermal conductivity of porous media and viscosity in

two-phase flow These new definitions can be used to compute the two-two-phase frictional pressure gradient using the homogeneous

modeling approach In the second method, a simple semitheoretical method for calculating two-phase frictional pressure

gradient using asymptotic analysis is presented Two-phase frictional pressure gradient is expressed in terms of the asymptotic

single-phase frictional pressure gradients for liquid and gas flowing alone In the final method, simple rules are developed

for obtaining rational bounds for two-phase frictional pressure gradient in minichannels and microchannels In all cases,

the proposed modeling approaches are validated using the published experimental data.

INTRODUCTION

The pressure drop in two-phase flow through microchannels

and minichannels constitutes an important parameter because

pumping costs could be a significant portion of the total

op-erating cost As a result, expressions are needed to predict the

pressure drop in two-phase flow through microchannels and

minichannels accurately

Total pressure drop for two-phase flow in microchannels and

minichannels has three different components They are

fric-tional, acceleration, and gravitational components It is

neces-sary to know the void fraction (the ratio of gas flow area to total

flow area) to compute the acceleration and gravitational

com-ponents To compute the frictional component of pressure drop,

either the two-phase friction factor or the two-phase frictional

multiplier must be known [1]

There are two principal types of frictional pressure drop

mod-els in two-phase flow: the homogeneous model and the separated

flow model In the first, both liquid and vapor phases move at the

same velocity (slip ratio= 1) Consequently, the homogeneous

model has also been called the zero-slip model The

homoge-neous model considers the two-phase flow as a single-phase

flow having average fluid properties depending on mass quality

Thus, the frictional pressure drop is calculated by assuming a

The authors acknowledge the financial support of the Natural Sciences and

Engineering Research Council of Canada (NSERC).

Address correspondence to Dr M M Awad, Faculty of Engineering and

Applied Science, Memorial University of Newfoundland, St John’s,

Newfound-land, Canada, A1B 3X5 E-mail: awad@engr.mun.ca

constant friction coefficient between the inlet and outlet tions The prediction of the frictional pressure drop using thehomogeneous model is reasonably accurate only for bubble andmist flows since the entrained phase travels at nearly the samevelocity as the continuous phase In the second, two-phase flow

sec-is considered to be divided into liquid and gas streams Hence,the separated flow model has been referred to as the slip flowmodel The separated model is popular in the power plant indus-try Also, the separated model is relevant for the prediction ofpressure drop in heat pump systems and evaporators in refrig-eration The success of the separated model is due to the basicassumptions in the model, which are closely met by the flowpatterns observed in the major portion of the evaporators

In this study, three different methods for two-phase flow eling in microchannels and minichannels are presented They areeffective property models for homogeneous two-phase flows,

mod-an asymptotic modeling approach for separated two-phase flow,and bounds on two-phase frictional pressure gradient

The literature review on two-phase flow modeling in crochannels and minichannels can be found in tabular form in anumber of textbooks [2–6]

mi-PROPOSED METHODOLOGIES Homogeneous Property Modeling

In the first method, new definitions for two-phase ity are proposed [7] using a one-dimensional transport analogy

viscos-1023

between thermal conductivity of porous media [8] and viscosity

in two-phase flow The series and parallel combination rules for

thermal conductivity of porous media are analogous to existing

rules proposed by McAdams et al [9], and Cicchitti et al [10]

McAdams et al [9], introduced the definition of two-phase

vis-cosity (µm) based on the mass averaged value of reciprocals as

They proposed their viscosity expression by analogy to the

expression for the homogeneous density (ρm) Equation (1) leads

to the homogeneous Reynolds number (Rem) being equal to the

sum of the liquid Reynolds number (Rel) and the gas Reynolds

number (Reg) In the realm of two-phase flow viscosity models,

Collier and Thome [11] mentioned that the definition of µm

proposed by McAdams et al [9], Eq (1), is the most common

definition ofµm Cicchitti et al [10] introduced the definition

of two-phase viscosity (µm) based on the mass averaged value

as follows:

They used the preceding definition ofµm in place of the

definition proposed by McAdams et al [9] The only reason for

doing this, in addition to simplicity, was a reasonable agreement

with experimental data

Definitions for two-phase viscosity were generated by

analogy to the effective thermal conductivity using the

Maxwell–Eucken I and II models [12] Maxwell–Eucken I [12]

is suitable for materials in which the thermal conductivity of the

continuous phase is higher than the thermal conductivity of the

dispersed phase (kcont > k disp), like foam or sponge In this case,

the heat flow essentially avoids the dispersed phase In the case

of momentum transport, this is akin to a bubbly flow, where

the dominant phase is the liquid This definition for two-phase

viscosity is:

µm= µl2µl2µl+ µg+ µg− 2(µl+ (µl− µg)x− µg)x (3)

Maxwell–Eucken II [12] is suitable for materials in which

the thermal conductivity of the continuous phase is lower than

the thermal conductivity of the dispersed phase (kcont < k disp),

like particulate materials surrounded by a lower conductivity

phase In this case, the heat flow involves the dispersed phase

as much as possible In the case of momentum transport, this is

akin to droplet flow, where the dominant phase is the gas This

definition for two-phase viscosity is:

µm = µg2µg2µg+ µl+ µl− 2(µg+ (µg− µl)(1− µl)(1− x) − x) (4)

A definition for two-phase viscosity was also generated by

analogy to the effective thermal conductivity using the effective

medium theory (EMT [13, 14]) The effective medium theory

(EMT [13, 14]) is suitable for the structure that represents a

heterogeneous material in which the two components are tributed randomly, with neither phase being necessarily con-tinuous or dispersed In the case of momentum transport, thisaveraging scheme seems reasonable given the unstable and ran-dom distribution of phases in a liquid/gas flow This definitionfor two-phase viscosity is:

II models [12] This is proposed here as a simple alternative tothe effective medium theory:

µm= 12



µl2µl+ µg− 2(µl− µg)x2µl+ µg+ (µl− µg)x+ µg2µg2µg+ µl+ µl− 2(µg+ (µg− µl)(1− µl)(1− x) − x)

(7)

It is clear that these new definitions satisfy the followingtwo conditions: namely, (i) the two-phase viscosity is equal tothe liquid viscosity at mass quality = 0% and (ii) the two-phase viscosity is equal to the gas viscosity at mass quality=100% These new definitions overcome the disadvantages ofsome definitions of two-phase viscosity such as the Davidson

et al definition [15], Owens’s definition [16], and the Garc´ıa

et al definition [17, 18] that do not satisfy the condition at

x = 1, µm = µg For example, Garc´ıa et al [17, 18] fined the Reynolds number of two-phase gas–liquid flow usingthe kinematic viscosity of liquid flow (νl) instead of the kine-matic viscosity of two-phase gas–liquid flow (νm) They usedthis definition because the frictional resistance of the mixturewas due mainly to the liquid This was equivalent to defining

de-µmas

µm= µl

ρmρl

These new definitions of two-phase viscosity can be used

to compute the two-phase frictional pressure gradient using thehomogeneous modeling approach

It is desirable to express the two-phase frictional pressure

gra-dient, (dp/dz)f, versus the total mass flux (G) in a dimensionless

heat transfer engineering vol 31 no 13 2010

M M AWAD AND Y S MUZYCHKA 1025

form like the Fanning friction factor (fm) versus the Reynolds

number (Rem) The Fanning friction factor (fm) based on the

homogeneous model (fm) can be expressed as follows:

The Reynolds number based on the homogeneous model

(Rem) can be expressed as follows:

Equations (10) and (11) represent the two-phase density

based on the homogeneous model (ρm) and Reynolds number

based on the homogeneous model (Rem).

To satisfy a good agreement between the experimental data

and well-known friction factor models, assessment of the best

definition of two-phase viscosity among the different definitions

(old and new) is based on the definition that corresponds to the

minimum root mean square (RMS) error

The fractional error (e) in applying the model to each

avail-able data point is defined as:

e=

Predicted Available − Available (12)

For groups of data, the root mean square error, eRMS, is defined

as:

e R M S =

1

N N

K=1

e2K

1/2

(13)

For the case of microchannels and minichannels, the friction

factor is calculated using the Churchill model [19], which allows

for prediction over the full range of laminar–transition–turbulent

regions The Fanning friction factor (fm) can be predicted using

the Churchill model [19] as follows:

f m= 2

8

16

(15)

b m=

37530

Re m

16

(16)

The Churchill model [19] is preferable since it encompasses

all Reynolds numbers and includes roughness effects in the

The asymptotic analysis method was first introduced byChurchill and Usagi [21] in 1972 After this time, this method

of combining asymptotic solutions proved quite successful indeveloping models in many applications [24] Recently, it hasbeen applied to two-phase flow in circular pipes, minichannels,and microchannels [20] Moreover, Awad and Butt have shownthat the asymptotic method works well for petroleum industryapplications for flows through porous media [25], liquid–liquidflows [26], and flows through fractured media [27]

The main advantage of the asymptotic modeling method intwo-phase flow is taking into account the important frictionalinteractions that occur at the interface between liquid and gasbecause the liquid and gas phases are assumed to flow in thesame channel This overcomes the main disadvantage of theseparate cylinders model [28] for two-phase flow

Using the asymptotic analysis method, two-phase frictional

pressure gradient (dp/dz)fcan be expressed in terms of phase frictional pressure gradient for liquid flowing alone

single-(dp/dz)f,l and single-phase frictional pressure gradient for gas

flowing alone (dp/dz)f,gas follows:

X2

pl/p

(19)

On the other hand, if the two-phase frictional pressure

gradi-ent (dp/dz)fis presented in terms of the single-phase frictionalheat transfer engineering vol 31 no 13 2010

pressure gradient for gas flowing alone (dp/dz)f,g, then the model

can be expressed using the Lockhart–Martinelli parameter (X)

Equation (20) can be expressed in terms of a two-phase

fric-tional multiplier for gas flowing alone (φ2

g) as follows:

φ2

In this method, p is chosen as the value, which minimizes the

root mean square (RMS) error, eRMS (Eq (13)), between the

model predictions and the available data

Bounds

In the third method, simple rules were developed for

obtain-ing rational bounds for two-phase frictional pressure gradient

in minichannels and microchannels [29] This approach is very

useful in design and analysis, as engineers can then use the

re-sulting average and bounding values in predictions of system

performance The approach is also useful when conducting new

experiments, since it provides a reasonable envelope for the data

to fall within The bounds are intended to provide the most

re-alistic range of data and not firm absolute limits Statistically,

this is unreasonable as the upper and lower bounds would be

far apart The bounds are not fit to capture all data but rather a

majority of data points, as some outlying points are due to

ex-perimental error If a vast majority of data is within the bounds,

then a reasonable expectation is realistically assured

These bounds may be used to determine the maximum and

minimum values that may reasonably be expected in a two-phase

flow Further, by averaging these limiting values an acceptable

prediction for the pressure gradient is obtained, which is then

bracketed by the bounding values:

The bounds model can be in the form of two-phase frictional

pressure gradient versus mass flux at constant mass quality;

they may also be presented in the form of a two-phase frictional

multiplier, which is often useful for calculation and comparison

needs For this reason, development of lower and upper bounds

in terms of a two-phase frictional multiplier (φlandφg) versus

the Lockhart–Martinelli parameter (X) will also be presented.

Awad and Muzychka [30, 31] applied the bounds method for

the case of turbulent/turbulent flow in large circular pipes

be-cause, in practice, both Rel and Regare most often greater than

2,000 Faghri and Zhang [32] further commented that the use of

bounds alleviates the uncertainty in the separated flow models

In the present study, the method is applied for the case of

lam-inar/laminar flow in minichannels and microchannels because,

in practice, both Rel and Regare most often lower than 2,000

The lower bound is based on the Ali et al correlation [33] forlaminar–laminar flow This correlation is based on modification

of a simplified stratified flow model derived from the theoreticalapproach of Taitel and Dukler [34] for the case of two-phaseflow in a narrow channel The equations of the lower bound are


µgµl

(25)The equations of the lower bound are equivalent to the

Chisholm correlation [35] with C = 0 The physical

mean-ing of the lower bound (C= 0) is that the two-phase frictionalpressure gradient is the sum of the frictional pressure of liquidphase alone and the frictional pressure of gas phase alone Thismeans no pressure gradient caused by the phase interaction.Although the data points are in laminar–laminar flow, theycover different flow patterns such as bubble, stratified, and annu-lar As the mass flow rate of the gas in two-phase flow increases,the flow pattern changes from bubble until it reaches annular

at a high mass flow rate of gas As mentioned in the literature,the Chisholm correlation [35] has a good accuracy for annularflow pattern This is why the upper bound is based on Chisholmcorrelation [35] for laminar–laminar flow The equations of theupper bound are

0.5
µgµl


µgµl

M M AWAD AND Y S MUZYCHKA 1027

0.5
µgµl


µgµl



(31)

The equations of the mean model are equivalent to the Chisholm

correlation [35] with C= 2.5

This model can be applied for circular shapes using tube

diameter d, as well as using hydraulic diameter dh for

non-circular shapes For nonnon-circular shapes, the Hagen–Poiseuille

constant (fRe)= 16 will be changed For example, for a

rectan-gular channel with the aspect ratio of 0, the Hagen–Poiseuille

constant (fRe)= 24, while for a rectangular channel with the

as-pect ratio of 1 (square channel), the Hagen–Poiseuille constant

(fRe)= 14.23

RESULTS AND DISCUSSION

Comparisons of the two-phase frictional pressure

gradi-ent versus mass flux from published experimgradi-ental studies in

minichannels and microchannels are undertaken using the old

and new definitions of two-phase viscosity, after expressing

the data in dimensionless form as Fanning friction factor

ver-sus Reynolds number The published data include different

working fluids such as R717, R134a, R410A, and propane

(R290) at different diameters and different saturation

tempera-tures Also, examples of two-phase frictional multiplier (φland

φg) versus Lockhart–Martinelli parameter (X) using published

data of different working fluids, such as air–water mixture and

nitrogen–water mixture in laminar–laminar flow, from other

ex-perimental work are presented to validate the asymptotic model

and the bounds model in dimensionless form

Figures 1 and 2 show the Fanning friction factor (fm)

ver-sus Reynolds number (Rem) in minichannels and microchannels

using one of the old definitions (McAdams et al [9]) and one

of the new definitions (Maxwell–Eucken II [12]) of two-phase

viscosity on log-log scale The sample of the published data

includes Ungar and Cornwell’s data [36] for R 717 flow at Ts≈

74◦F (165.2◦C) in a smooth horizontal tube at d= 0.1017 inches

(2.583 mm), the Tran et al data [37] for R134a flow at saturation

pressure of 365 kPa and x≈ 0.73 in a smooth horizontal pipe

at d= 2.46 mm, the Cavallini et al data [38] for 410A flow at

T s= 40◦C and x= 0.74 in smooth multi-port minichannels at

hydraulic diameter of 1.4 mm, and Field and Hrnjak data [39]

Figure 1 f m versus Re min microchannels and minichannels using McAdams

et al [9] definition.

for propane (R 290) flow at reduced pressure of 0.23 and G≈

330 kg/m2-s in a smooth horizontal pipe at hydraulic diameter of0.148 mm The literature data represented a wide range of fluidproperties, across R717, R134a, R410A, and propane (R290).Equation (9) defines the measured Fanning friction factor, whileEqs (14)–(16) represent the predicted Fanning friction factor

Table 1 presents eRMS % values based on measured Fanning

fric-tion factor and predicted Fanning fricfric-tion factor using the sixdifferent definitions of two-phase viscosity for this sample ofthe published data It can be seen that two-phase viscosity based

on the Maxwell–Eucken II model [12] gives the best agreementbetween the published data and the Churchill model [19] with a

root mean square error (eRMS) of 16.47%.

In Figure 2, it is interesting to observe that the fluids with thehigher vapor–liquid density ratios, which were supposed to bemore appropriate for the Maxwell–Eucken II model of homoge-neous viscosity definition [12], might be thought to have betteragreement It can be seen from Figure 2 and Table 1 that thedefinition of effective viscosity based on the Maxwell–Eucken

II model [12] appears to be more appropriate for defining phase flow viscosity in microchannels and minichannels On

two-Figure 2 f m versus Re m in microchannels and minichannels using Maxwell–Eucken II [12] definition.

heat transfer engineering vol 31 no 13 2010

Table 1 e RMS % Values based on measured Fanning friction factor and

predicted Fanning friction factor in microchannels and minichannels using

different definitions of two-phase viscosity

Effective medium theory (EMT [13,14]) 23.60%

Arithmetic mean of Maxwell–Eucken I and II [12] 17.98%

the basis of the data considered, a nominal 5–6% gain in

accu-racy can be achieved using the homogeneous flow modeling

ap-proach When one considers the nature of the Maxwell–Eucken

II definition, whereby the dominant phase is the lower

viscos-ity phase, i.e., the gas, it is clear that this definition is most

appropriate for liquid/gas mixtures that have very high

den-sity ratios Thus, even for small mixture qualities, a significant

portion of the flow volume is occupied by gas, making the

Maxwell–Eucken II definition most appropriate

Figure 3 showsφlversus Lockhart–Martinelli parameter (X)

for laminar–laminar flow for different working fluids in smooth

microchannels and minichannels of different diameters at

dif-ferent conditions using the present asymptotic model and the

bounds model with the first three data sets in Table 2 Equation

(18) represents the present asymptotic model with different

val-ues of p as shown in Table 2 Equation (23) represents the lower

bound and Eq (26) represents the upper bound, while Eq (29)

represents the average

Figure 4 showsφgversus Lockhart–Martinelli parameter (X)

for laminar–laminar flow for different working fluids in smooth

microchannels and minichannels at different conditions using

the present asymptotic model and the bounds model with the

last two data sets in Table 2 Equation (21) represents the present

asymptotic model with different values of p as shown in Table 2.

Equation (24) represents the lower bound and Eq (27) represents

the upper bound, while Eq (30) represents the average

Figure 3 φl versus X for different sets of data.

Table 2 Values of the asymptotic parameter (p) in microchannels and

minichannels at different conditions

Lee and Lee [40] 0.78 ∗ 1/1.75 11.7% 14.07%

Chung and Kawaji [41] 0.1 1/1.7 13.44% 16.09% Kawaji et al [42] + 0.1 1/2.15 10.39% 11.34%

Kawaji et al [42] ++ 0.1 1/2.55 11.65% 17.36%

Ohtake et al [43] 0.32 ∗ 1/1.55 19.56% 24.16%

0.42 ∗ 16.08%∗∗ 18.24%∗∗ 0.49 ∗

∗Hydraulic diameter.

∗∗The two lower points are not taken into account.

+Gas in the main channel and liquid in the branch.

++Liquid in the main channel and gas in the branch.

To have a robust model, one value of the fitting parameter (p)

is chosen as p = 1/2 Choosing p = 1/2 is physically meaningful.

In fact, p = 1/2 in the asymptotic model corresponds to C = 2

in the bound model When p= 1/2, the root mean square (RMS)

error eRMS is 17.14%, or 15.69% if the two lower points ofOhtake et al data [43] are not taken into account Figure 3showsφlversus X for the first three data sets in Table 2, while

Figure 4 showsφgversus X for the last two data sets in Table

2 with p= 1/2 On the basis of the experimental data shown inFigures 3 and 4, it is clear that the experimental points set in a

form, when X→ 0, φl→ ∞, and φg → 1 and when X → ∞,

φl→ 1, and φg→ ∞ in line with the expected asymptotic havior of the Lockhart-Martinelli correlation [44] It can be seenthat there is a good agreement between the present asymptoticmodel and the different data sets in Figures 3 and 4

be-The mean model predicts the first three data sets in Table 2

ofφlwith the root mean square (RMS) error of 17.91%, 19.29%,and 10.49%, respectively while the asymptotic model gives theroot mean square (RMS) error of 14.07%, 16.09%, and 11.34%,respectively The mean model predicts the last two data sets

in Table 2 of φg with the root mean square (RMS) error of14.87%, and 28.04%, respectively while the asymptotic modelgives the root mean square (RMS) error of 17.36% and 24.16%,

Figure 4 φg versus X for different sets of data.

heat transfer engineering vol 31 no 13 2010

M M AWAD AND Y S MUZYCHKA 1029

Figure 5 φl versus X for Saisorn and Wongwises’s data [45] with various mass

flux values at d= 0.53 mm.

respectively In Figure 4, if the the two lower points of Ohtake

et al data [43] are not considered, the root mean square (RMS)

error will be 21.77% instead of 28.04%, while the asymptotic

model gives the root mean square (RMS) error of 18.24%

in-stead of 24.16% These outlying points are likely affected by

experimental error

The second method (the asymptotic model (p= 1/2 or C = 2))

and the third method (the bounds model (C = 2.5)) are also

validated against the recent data sets of Saisorn and

Wong-wises [45–47] that were published in 2008 and 2009 for

two-phase air–water flow in circular microchannels of d = 0.53,

0.15, and 0.22 mm, respectivley Figures 5–7 showφl versus

Lockhart–Martinelli parameter (X) for Saisorn and Wongwises’s

data [45–47] with various mass flux values for laminar–laminar

two-phase air-water flow in circular microchannels of d= 0.53,

0.15, and 0.22 mm, respectivley Figures 8–10 showφlversus

Lockhart–Martinelli parameter (X) for Saisorn and Wongwises’s

data [45–47] with various flow patterns for laminar–laminar

two-phase air–water flow in circular microchannels of d= 0.53,

0.15, and 0.22 mm, respectivley The observed flow patterns in

Figure 6 φl versus X for Saisorn and Wongwises’s data [46] with various mass

a 0.22-mm-diameter channel include throat-annular flow, lar flow, and annular–rivulet flow Equation (18) represents the

annu-present asymptotic model with p= 1/2 Equation (23) representsthe lower bound and Eq (26) represents the upper bound, while

Eq (29) represents the average In Figures 5–10, Saisorn andWongwises’s data [45–47] are also compared with the Mishima

and Hibiki correlation [48] (C = 21(1 − e−319d)) and the

English and Kandlikar correlation [49] (C = 5(1 − e−319d)).

It should be noted that the Saisorn and Wongwises tions (φ2

correla-i = 1 + (6.627/X0.761)) [45] for d = 0.53 mm and(φ2

i = 1 + (2.844/X1.666)) [46] for d= 0.15 mm neglect the

1/X2term, which represents the limit of primarily gas flow inthe Lockhart–Martinelli [44] formulation Neglecting this termignores this important limiting case, which is an essential contri-

bution As a result, at low values of X, the proposed correlations

Figure 8 φl versus X for Saisorn and Wongwises’s data [45] with various flow patterns at d= 0.53 mm.

heat transfer engineering vol 31 no 13 2010

Figure 9 φl versus X for Saisorn and Wongwises’s data [46] with various flow

patterns at d= 0.15 mm.

undershoot the trend of the data, limiting their use in the low X

range [50]

From Figures 3–10, it clear that the greatest departure of

bounds from the mean occurs at X= 1 From the relation that the

Lockhart–Martinelli parameter (X) for laminar–laminar flow is

equal to ((1−x)/x)0.5(ρg/ρl)0.5(µl/µg)0.5, this greatest departure

of bounds from the mean corresponds to x = 6.28% for the

air–water mixture at the atmospheric pressure while it

corre-sponds to x= 50% at the critical state (ρg= ρlandµg= µl) for

any working fluid

Figure 11 shows C parameter versus the channel

diame-ter (dh) for laminar–laminar flow using the asymptotic model

(p= 1/2 or C = 2), the bounds model (C = 2.5), the Chisholm

correlation [35] (C = 5), the Mishima and Hibiki correlation

[48] (C= 21(1 − e−319dh)), and the English and Kandlikar

cor-relation [49] (C= 5(1 − e−319dh)) It is found that the asymptotic

model (p = 1/2 or C = 2) is equivalent to the Mishima and Hibiki

correlation [48] at dh= 0.314 mm and equivalent to the English

and Kandlikar correlation [49] for laminar–laminar flow at dh=

1.601 mm Moreover, the bounds model (C= 2.5) is equivalent

Figure 10 φl versus X for Saisorn and Wongwises’s data [47] with various

flow patterns at d= 0.22 mm.

Figure 11 C parameter versus the channel diameter (d h).

to the Mishima and Hibiki correlation [48] at dh = 0.397 mmand equivalent to the English and Kandlikar correlation [49] for

laminar–laminar flow at dh= 2.173 mm

SUMMARY AND CONCLUSIONS

First, using a one-dimensional transport analogy betweenthermal conductivity in porous media and viscosity in two-phase flow, new definitions for two-phase viscosity are exam-ined These new definitions for two-phase viscosity satisfy thefollowing two conditions: (i)µm= µl at x= 0 and (ii) µm=

µgat x= 1 These new definitions of two-phase viscosity can beused to compute the two-phase frictional pressure gradient using

a homogeneous modeling approach Expressing two-phase tional pressure gradient in dimensionless form as Fanning fric-tion factor versus Reynolds number is also desirable in many ap-plications The models are verified using published experimentaldata for two-phase frictional pressure gradient in microchan-nels and minichannels after expressing these in a dimensionlessform as Fanning friction factor versus Reynolds number Thepublished data include different working fluids such as R717,R134a, R410A, and propane (R290) at different diameters anddifferent saturation temperatures To provide good agreementbetween the experimental data and well-known friction factormodels such as the Churchill model [19], selection of the bestdefinition of two-phase viscosity is based on the definition thatcorresponds to the minimization of the root mean square error

fric-(eRMS) From eRMS % values based on measured Fanning friction

factor and predicted Fanning friction factor using the six ent definitions of two-phase viscosity, it is shown that one ofthe new definitions of two-phase viscosity (Maxwell–Eucken II[12]) gives the best agreement between the experimental dataand well-known friction factor models in microchannels andminichannels These new definitions of two-phase viscosity can

differ-be used to analyze the experimental data of two-phase frictionalpressure gradient in microchannels and minichannels using thehomogeneous model

heat transfer engineering vol 31 no 13 2010

M M AWAD AND Y S MUZYCHKA 1031

Second, new two-phase flow modeling in microchannels and

minichannels is proposed, based upon an asymptotic

model-ing method The main advantage of the asymptotic modelmodel-ing

method in two-phase flow is taking into account the important

frictional interactions that occur at the interface between liquid

and gas, because the liquid and gas phases are assumed to flow in

the same channel This overcomes the main disadvantage of the

separate cylinders model for two-phase flow The only unknown

parameter in the asymptotic modeling method in two-phase flow

is the fitting parameter (p) The value of the fitting parameter

(p) is chosen to correspond to the minimum root mean square

(RMS) error eRMSfor any data set To have a robust model, one

value of the fitting parameter (p) is chosen as p= 1/2

Third, simple expressions are presented for obtaining bounds

for two-phase frictional pressure gradient in minichannels and

microchannels The lower bound is based on the Ali et al

cor-relation [33] for laminar–laminar flow This corcor-relation is based

on modification of a simplified stratified flow model derived

from the theoretical approach of Taitel and Dukler [34] for the

case of two-phase flow in a narrow channel The upper bound is

based on Chisholm correlation [35] for laminar–laminar flow

The mean model is based on the arithmetic mean of lower bound

and upper bound The model is verified using published

experi-mental data of two-phase frictional pressure gradient in circular

and noncircular shapes The bounds models are presented in a

di-mensionless form as two-phase frictional multiplier (φlandφg)

versus Lockhart–Martinelli parameter (X) for different working

fluids such as the air–water mixture and nitrogen–water mixture

The present model is very successful in bounding two-phase

frictional multiplier (φlandφg) versus Lockhart–Martinelli

pa-rameter (X) well for different working fluids over a wide range

of mass fluxes, mass qualities, and diameters The proposed

mean model provides a simple prediction of two-phase flow

parameters

Finally, the first method (homogeneous property modeling)

is recommended in predicting the two-phase frictional pressure

drop in microchannels and minichannels if we use the

homo-geneous model It is reasonably accurate only for bubble and

mist flows since the entrained phase travels at nearly the same

velocity as the continuous phase The second and third methods

(asymptotic modeling and bounds) are recommended in

pre-dicting the two-phase frictional pressure drop in microchannels

and minichannels if we use the separated model that originated

from the classical work of Lockhart and Martinelli [44]

N number of data points

dp/dz pressure gradient, Pa/m

cont continuous phase

disp dispersed (discontinuous) phase

Fundamen-[3] Kandlikar, S G., Garimella, S., Li, D., Colin, S., and King,

M R., Heat Transfer and Fluid Flow in Minichannels and Microchannels, Elsevier, Oxford, UK, 2006.

[4] Crowe, C T., Multiphase Flow Handbook, CRC/Taylor &

Francis, Boca Raton, FL, 2006

[5] Ghiaasiaan, S M., Two-Phase Flow, Boiling and sation in Conventional and Miniature Systems, Cambridge

Conden-University Press, New York, 2008

heat transfer engineering vol 31 no 13 2010

[6] Yarin, L P., Mosyak, A., and Hetsroni, G., Fluid Flow,

Heat Transfer and Boiling in Micro-Channels, Springer,

Berlin, 2009

[7] Awad, M M., and Muzychka, Y S., Effective Property

Models for Homogeneous Two-Phase Flows,

Experimen-tal Thermal and Fluid Science, vol 33, no 1, pp 106–113,

2008

[8] Carson, J K., Lovatt, S J., Tanner, D J., and Cleland,

A C., Thermal Conductivity Bounds for Isotropic, Porous

Materials, International Journal of Heat Mass Transfer,

vol 48, no 11, pp 2150–2158, 2005

[9] McAdams, W H., Woods, W K and Heroman, L C.,

Vaporization inside Horizontal Tubes II -Benzene-Oil

Mixtures, Trans ASME, vol 64, no 3, pp 193–200,

1942

[10] Cicchitti, A., Lombaradi, C., Silversti, M.,

Sol-daini, G., and Zavattarlli, R., Two-Phase Cooling

Experiments—Pressure Drop, Heat Transfer, and Burnout

Measurements, Energia Nucleare, vol 7, no 6, pp 407–

425, 1960

[11] Collier, J G., and Thome, J R., The Basic Models, in

Convective Boiling and Condensation, 3rd ed., chap 2,

p 45, Clarendon Press, Oxford, UK, 1994

[12] Hashin, Z., and Shtrikman, S., A Variational Approach

to the Theory of the Effective Magnetic Permeability of

Multiphase Materials, Journal of Applied Physics, vol 33,

pp 3125–3131, 1962

[13] Landauer, R., The Electrical Resistance of Binary Metallic

Mixtures, Journal of Applied Physics, vol 23, pp 779–784,

1952

[14] Kirkpatrick, S., Percolation and Conduction, Review of

Modern Physics, vol 45, pp 574–588, 1973.

[15] Davidson, W F., Hardie, P H., Humphreys, C G R.,

Markson, A A., Mumford, A R., and Ravese, T.,

Stud-ies of Heat Transmission Through Boiler Tubing at

Pres-sures from 500–3300 Lbs, Trans ASME, vol 65, no 6,

pp 553–591, 1943

[16] Owens, W L., Two-Phase Pressure Gradient, ASME

International Developments in Heat Transfer, Part II,

pp 363–368, 1961

[17] Garc´ıa, F., Garc´ıa, R., Padrino, J C., Mata, C., Trallero,

J L., and Joseph, D D., Power Law and Composite Power

Law Friction Factor Correlations for Laminar and

Tur-bulent Gas–Liquid Flow in Horizontal Pipelines,

Inter-national Journal of Multiphase Flow, vol 29, no 10,

pp 1605–1624, 2003

[18] Garc´ıa, F., Garc´ıa, J M., Garc´ıa, R., and Joseph, D D.,

Friction Factor Improved Correlations for Laminar and

Turbulent Gas-Liquid Flow in Horizontal Pipelines,

In-ternational Journal of Multiphase Flow, vol 33, no 12,

pp 1320–1336, 2007

[19] Churchill, S W., Friction Factor Equation Spans All Fluid

Flow Regimes, Chemical Engineering, vol 84, no 24,

pp 91–92, 1977

[20] Awad, M M., Two-Phase Flow Modeling in Circular Pipes,Ph.D Thesis, Memorial University of Newfoundland, St.John’s, Newfoundland, 2007

[21] Churchill, S W., and Usagi, R., A General Expression forthe Correlation of Rates of Transfer and Other Phenomena,

AIChE Journal, vol 18, no 6, pp 1121–1128, 1972 [22] Churchill, S W., Viscous Flows: The Practical Use of Theory, Butterworths, Boston, 1988.

[23] Kraus, A D., and Bar-Cohen, A., Thermal Analysis and Control of Electronic Equipment, Hemisphere, New York,

Measure-[25] Awad, M M., and Butt, S D., A Robust AsymptoticallyBased Modeling Approach for Two-Phase Flow in Porous

Media, ASME Journal of Heat Transfer, vol 131, no 10,

Con-Flow I, OMAE2009-79072, Honolulu, HI, May 31–June

[29] Awad, M M., and Muzychka, Y S., Bounds on Phase Frictional Pressure Gradient in Minichannels and

Two-Microchannels, Heat Transfer Engineering, vol 28, no.

Engi-IMECE2005-81493, 2005

[31] Awad, M M., and Muzychka, Y S., Bounds on Two-Phase

Flow Part II—Void Fraction in Circular Pipes, Proceedings

of IMECE 2005, Fluids Engineering, Forum on Recent Developments in Multiphase Flow, FE-11 B Numerical Simulations and Theoretical Developments for Multiphase Flows, Orlando, FL, IMECE2005-81543, 2005.

[32] Faghri, A., and Zhang, Y., Two-Phase Flow and Heat

Trans-fer, in Transport Phenomena in Multiphase Systems, Chap.

11, p 874, Elsevier, Burlington, MA, 2006

heat transfer engineering vol 31 no 13 2010

M M AWAD AND Y S MUZYCHKA 1033

[33] Ali, M., Sadatomi, M., and Kawaji, M., Adiabatic

Two-Phase Flow in Narrow Channels Between Two Flat Plates,

Canadian Journal of Chemical Engineering, vol 71, no.

5, pp 657–666, 1993

[34] Taitel, Y., and Dukler, A E., A Model for Predicting Flow

Regime Transitions in Horizontal and Near Horizontal

Gas–Liquid Flow, AIChE J., vol 22, no 1, pp 47–55,

1976

Lockhart–Martinelli Correlation for Two-Phase Flow,

International Journal of Heat Mass Transfer, vol 10, no.

12, pp 1767–1778, 1967

[36] Ungar, E K., and Cornwell, J D., Two-Phase

Pres-sure Drop of Ammonia in Small Diameter Horizontal

Tubes, 17th AIAA Aerospace Ground Testing Conference,

Nashville, TN, AIAA 92-3891, 1992

[37] Tran, T N., Chyu, M.-C., Wambsganss, M W., and France,

D M., Two-Phase Pressure Drop of Refrigerants During

Flow Boiling in Small Channels: An Experimental

Inves-tigation and Correlation Development, International

Jour-nal of Multiphase Flow, vol 26, no 11, pp 1739–1754,

2000

[38] Cavallini, A., Del Col, D., Doretti, L., Matkovic, M.,

Rossetto L., and Zilio, C., Two-Phase Frictional Pressure

Gradient of R236ea, R134a and R410A inside Multi-Port

Mini-Channels, Experimental Thermal and Fluid Science,

vol 29, no 7, pp 861–870, 2005

[39] Field, B S., and Hrnjak, P., Adiabatic Two-Phase Pressure

Drop of Refrigerants in Small Channels, Heat Transfer

Engineering, vol 28, no 8–9, pp 704–712, 2007.

[40] Lee, H., J., and Lee, S., Y., Pressure Drop Correlations for

Two-Phase Flow Within Horizontal Rectangular Channels

with Small Heights, International Journal of Multiphase

Flow, vol 27, no 5, pp 783–796, 2001.

[41] Chung, P M.-Y., and Kawaji, M., The Effect of Channel

Diameter on Adiabatic Two-Phase Flow Characteristics in

Microchannels, International Journal of Multiphase Flow,

vol 30, no 7–8, pp 735–761, 2004

[42] Kawaji, M., Mori, K., and Bolintineanu, D., The Effects

of Inlet Geometry on Gas–Liquid Two-Phase Flow in

Mi-crochannels, Proceedings of ICMM2005, 3rd International

Conference on Microchannels and Minichannels, Toronto,

Ontario, ICMM2005-75087, 2005

[43] Ohtake, H., Koizumi, Y., and Takahashi, H., Frictional

Pressure Drops of Single-Phase and Gas-Liquid

Two-Phase Flows in Circular and Rectangular Microchannels,

Proceedings of ICMM2005, 3rd International Conference

on Microchannels and Minichannels, Toronto, Ontario,

ICMM2005-75147, 2005

[44] Lockhart, R W., and Martinelli, R C., Proposed

Correla-tion of Data for Isothermal Two-Phase, Two-Component

Flow in Pipes, Chemical Engineering Progress Symposium

Series, vol 45, no 1, pp 39–48, 1949.

[45] Saisorn, S., and Wongwises, S., Flow Pattern, Void tion and Pressure Drop of Two-Phase Air–Water Flow in

Frac-a HorizontFrac-al CirculFrac-ar Micro-ChFrac-annel, ExperimentFrac-al mal and Fluid Science, vol 32, no 3, pp 748–760, 2008.

Ther-[46] Saisorn, S., and Wongwises, S., An Experimental vestigation of Two-Phase Air–Water Flow Through a

In-Horizontal Circular Micro-Channel, Experimental mal and Fluid Science, vol 33, no 2, pp 306–315,

Ther-2009

[47] Saisorn, S., and Wongwises, S., The Effects of ChannelDiameter on Flow Pattern, Void Fraction and Pressure Drop

of Two-Phase Air–Water Flow in a Horizontal Circular

Micro-Channel, Experimental Thermal and Fluid Science,

M M Awad is a postdoctoral fellow in mechanical

engineering in the Faculty of Engineering and plied Science, Memorial University of Newfound- land, Canada He received his Ph.D from Memo- rial University of Newfoundland in 2007 and his undergraduate degree and his master’s degree from Mansoura University, Egypt, in 1996 and 2000, re- spectively His research is related to two-phase flow modeling He was also a recipient of the ASME International Petroleum Technology Institute (IPTI) Award in 2005 and 2006.

Ap-Y S Muzychka is a professor of mechanical

en-gineering at Memorial University of Newfoundland, Canada His research focus is on the development of robust models for characterizing transport phenom- ena using fundamental theory These models are val- idated using experimental and/or numerical results.

He has published more than 80 papers in refereed journals and conference proceedings in these areas Presently, his research is focused on the modeling

of complex fluid dynamics and heat transfer lems in internal flows These include transport in porous media, compact heat exchangers, two-phase flow in oil and gas operations, non-continuum flow, mi- crochannel flows, and non-Newtonian flows He has also undertaken research

prob-in electronics packagprob-ing, contact heat transfer, and thermal design/optimization

of energy systems He is a registered professional engineer and a member of the American Institute for Aeronautics and Astronautics (AIAA) and the American Society of Mechanical Engineers (ASME).

heat transfer engineering vol 31 no 13 2010

Copyright
Taylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457631003639067

Heat Transfer Analysis for a

Concentric Tube Heat Exchanger

Including the Wall Axial Conduction

MEHMET EMIN ARICI

Department of Mechanical Engineering, Karadeniz Technical University, Trabzon, Turkey

The effect of wall axial conduction on the heat transfer in a concentric tube heat exchanger is examined for the inner flow

laminar flow regime The procedure used for the current analysis combines the analytical solution for the inner fluid with

a numerical approximation for the wall conduction and has the capability of handling the temperature variation for the

outer fluid Both parallel and counterflow cases are evaluated for the analysis, and results are presented in terms of the

axial variations of fluids and wall temperatures Effects of the heat capacity rate ratio of the fluids on the temperature

variations and on the mean heat flux are also pointed out The effect of the exchanger length is included for the analysis It is

concluded that the total heat transfer between the fluids is greatly influenced by the wall axial conduction for the counterflow

arrangement and is not ignorable when the heat capacity rate ratio of fluids are smaller than unity.

INTRODUCTION

One of the assumptions for the elementary heat exchanger

analysis is to neglect the axial conduction along the solid wall

[1] As a practical consideration, it is a reasonable approach to

prefer a highly conductive exchanger wall to reduce the total

thermal resistance between the fluids in radial direction On the

contrary, wall axial conduction in the direction of falling

tem-perature occurs and may show a significant effect The problem

of convection heat transfer for hot and cold fluids may need to

be treated together with the problem of axial conduction of the

wall separating the fluids

Conduction heat transfer in a solid wall interacting with fluid

flow and the convection heat transfer at the solid boundary is

an area of increasing research interest and is usually referred to

as the conjugate heat transfer problem This kind of problem is

an important aspect in the design and analysis of heat transfer

devices as well as in the interpretation of experimental data

Some basic parameters that lead to the heat transfer between a

solid wall and flowing fluid near the wall usually cannot be fixed

in advance In order to determine these parameters, such as the

Nusselt number, the interfacial temperature between the solid

Address correspondence to Professor Mehmet Emın Arıcı, Department

of Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon,

Turkey E-mail: arici@ktu.edu.tr

and fluid, and the temperature variation of both the solid walland the flowing fluid, the governing equations of the problemneed to be solved As the heat transfer of solid and fluid regions

is usually governed by two different conservation equationsand as some related variables couple the equations, a specialtreatment of the problem is required Current approaches to thetreatment of such a problem can be basically classified as nu-merical modeling, fully analytical solution, and experimentalinvestigation For a numerical approach, simultaneous solution

of the energy equation for the flowing fluid and the conductionequation for the solid region often needs to be obtained Most

of the numerical modeling approaches are capable of predictingthe temperature variation in radial and circumferential direc-tions as well as the flow direction and are capable of treating theinterfacial physical properties between the solid and the fluid re-gions Compared with multidimensional numerical methods, thecapability of fully analytical solutions is limited in terms of pa-rameters included and evaluation of dependent variables Fullyanalytical approaches can usually solve the one-dimensionalconduction equation and the fully developed two-dimensionalenergy equation of the flowing fluid in absence of axial conduc-tion Beyond this, other type of approaches, such as a combi-nation of any two methods, can be considered for the presentproblem

Numerous studies on the conjugate heat transfer problem can

be found in the literature The conjugate heat transfer studiesdiscussed here can be categorized mainly in two parts The first

1034

M E ARICI 1035

part of these studies deals with forced convection heat transfer

including the wall axial conduction effects for various

bound-ary conditions at the outer surface of the wall A fully numerical

study by Campo and Schuler [2] provides a finite-difference

procedure and examines the influence of a finite heated length

on the heat transfer characteristics of laminar flow through a

thick-walled circular pipe A one-dimensional domain

approxi-mation of the conduction equation for the wall is also included

in their work In their findings, there is no substantial

differ-ence in the interfacial temperature distribution yielded by the

one- and two-dimensional models A method to solve conjugate

heat transfer problems for the case of fully developed

lami-nar flow in a pipe, proposed by Barozzi and Pagliarini [3],

combines the superposition principle with the finite-element

method and expresses relative importance of the parameters

such as the Peclet number, wall-to-fluid conductivity ratio, and

dimensionless thickness In another work by the same authors

[4], wall heat conduction effects on laminar flow heat transfer

are experimentally investigated At the same time, the influence

of conduction along the wall on experimental results is

esti-mated by performing a numerical simulation of the test section

An analytical solution by Wijeysundera [5] is given for

lami-nar forced convection in circular and flat ducts with axial wall

conduction and external convection of the outer surface of the

wall By means of Duhamel’s superposition technique, he used

the expression for temperature distribution of fully developed

case and constant wall temperature boundary condition and

ob-tained the solution for the set of wall and flow region energy

equations

Faghri and Sparrow [6] analyzed the effect of the step change

in the external boundary condition They included externally

in-sulated upstream and uniformly heated downstream portions of

pipe wall for a Peclet number range of 5 to 50 in the analysis

The effect of axial conduction for both the fluid and the solid

region was taken into account and it was concluded that the

effect of wall axial conduction can readily overwhelm the

ef-fect of fluid axial conduction The conjugate problem of pipes

exposed to step change in the temperature was studied more

recently by Zueco et al [7] under developing flow condition for

both transient and steady-state cases They concluded that for

large values of Peclet number, the fluid axial conduction will be

negligible and results mainly depend on the wall characteristics

rather than on flow conditions As reported by Kays and

Craw-ford [8], investigations have also indicated as a general rule that

the axial conduction of a conventional pipe flow is negligible for

Peclet numbers higher than 100 Above this limit, the axial wall

conduction effect is examined under a step change in boundary

conditions and wall conductivity [9] Weigand and Gassner [10]

report that axial heat conduction effects in the fluid are

impor-tant for low Peclet numbers However, if short heating sections

are considered, these effects might drastically influence the heat

transfer behavior even for higher Peclet numbers

The second part of the studies concerns the wall axial

conduction effect of heat exchanger analysis In a work by

Ranganayakulu and Seetharamu [11], the combined effect of

wall axial conduction and nonuniformity of flow and ature at inlet port for a cross-flow plate heat exchanger is an-alyzed Another work on the effect of wall axial conduction

temper-in the cross-flow heat exchanger performed by Yuan and Kou[12] includes three fluid streams with three different arrange-ments Their conclusion indicates that the overall effects of thewall axial conduction seriously deteriorate the performance ofheat exchanger for each arrangement The effect of the axialheat conduction in the exchanger wall of a compact-plate cross-flow indirect evaporative cooler was investigated by Madhawa

et al [13] They found that performance deterioration for theevaporative cooler application is higher than the conservativeconditions Local effects of axial heat conduction in plate heatexchangers were examined by Ciofalo [14] with an analyticalmethod and tested by a CFD predictive tool It is stated in thestudy that axial heat conduction along the dividing walls of-ten plays some role and cannot be neglected Malinowski andBielski [15] studied the problem of transient and steady-state be-havior of a parallel-flow three-fluids heat exchanger with regard

to the transverse and axial wall conduction They showed thatgood thermal conductivity of the wall increases the influence ofaxial heat conduction and at the same time decreases the effect ofresistance in transverse direction In the book by Smith [16], ananalysis is performed for a counterflow plate fin heat exchangerwhen axial conduction effect is present It is stated in the analysisthat the equal heat capacity rate of fluids with wall axial con-duction gives the maximum reduction in the mean temperaturedifferences

The problem considered for the present work deals with theeffect of wall axial conduction for a concentric tube heat ex-changer An existing solution procedure [17] is used for theanalysis and is extended to include the temperature variation ofthe outer fluid of the exchanger The inner flow Peclet numberrange is chosen comparatively higher than that of reference [7],and only wall axial conduction is taken into consideration Sincethe present procedure is concerned only with the axial variation

of wall temperature, the competitive effect between the reducedthermal resistance across the wall and the axial conduction alongthe wall is excluded for the analysis It is understood from theprevious works that the axial heat transfer through the wall re-sults in altering the conventional heat transfer characteristics

of the exchanger This study aims to show the effect of thisalteration while assuming the conduction along the solid wall

is not neglected As laminar heating and cooling in small-size,modern instrumentation is gaining interest in increasing variety,such as micro-electromechanical systems and many compactheat exchangers, this kind of configuration becomes important

to researchers

ANALYSIS

The problem under consideration deals with two fluids,namely, the inner and outer fluids, and the wall that separates thefluids Since the conduction in the wall interacts with the fluidheat transfer engineering vol 31 no 13 2010

Figure 1 Schematic diagram of the problem: (a) longitudinal view for the

exchanger, and (b) the cross-sectional view for a differential wall element.

flow and the convection heat transfer, the problem is considered

as a conjugate heat transfer problem The exchanger geometry

and detailed wall cross-sectional view of the wall for energy

balance are shown in Figures 1a and b, respectively Inner flow

condition at x= 0 is hydrodynamically fully developed and the

pipe wall boundary conditions at x = 0 and x = L are insulated

boundary conditions

Applying the energy conservation principle to the pipe wall

element as demonstrated in Figure 1b, without energy

genera-tion and ignoring temperature variagenera-tion across the radial

direc-tion, a one-dimensional energy equation is obtained as

are the wall outer and inner heat fluxes, respectively

Using the definition of the inner flow local Nusselt number,

Nu = 2ri h i /k i, and Biot number Bi = ho r o /k i, together with

nondimensional variables as given in the Nomenclature section,

the energy equation in nondimensional form for the pipe wall

takes the final form

finite-control volume discretization to Eq (2), as in reference

[18], the discretization equation, which includes an unknown

coefficient, is obtained and is ready to be implicitly solved The

axial temperature distribution of the pipe wall is predicted by

the numerical solution of the above mentioned discretization

equation The wall outer surface boundary condition is

repre-sented by the second term of the equation and can be replaced

with any type of boundary condition The unknown heat transfer

characteristic is already in the last term of the equation Thus,

it is possible to start this solution procedure by setting a

re-quested boundary condition at r = ro and providing a guessed

local Nusselt number distribution at r = ri Because of the

iter-ative nature of the solution procedure, the correct temperature

distribution is obtained once the differences between the localNusselt numbers of the last two iteration steps are relativelysmall

On the other hand, the energy equation for the inner fluid iswritten for hydrodynamically fully developed laminar flow inthe absence of fluid axial conduction as follows:

ρucP ,i ∂T f

∂ x =

k i r

Solution of this equation for the constant temperature boundary

condition at r = riis given as (in reference [8])

whereλn are the eigenvalues, Rn are the corresponding

eigen-functions, and Cnare constants The local heat flux at the wallsurface can be readily evaluated from the derivative at the wallas

The local mean inner fluid temperature Tican be evaluated after

integration of the last equation from x = 0 to x+in

Since the constant wall temperature boundary condition at

r = ri cannot be sustained for the present problem, specialtreatment for obtaining the inner heat flux and the local meanfluid temperature is required Following the practice reported inreference [17] for the variable wall temperature case, the innerheat flux is given as follows:

T *

Figure 2 Comparative results of the present prediction for constant outer fluid

temperature case and the results of reference [5] for wall and inner fluid local

mean fluid temperatures.

constant and the local mean temperature of outer fluid is

calcu-lated via the conservation of energy principle for the pipe wall

section from x = 0 to x+ Using nondimensional variables and

after some manipulations, the outer flow local mean temperature

In order to close the solution procedure, Eqs (8), (9), and

(10), together with the definition of the Nusselt number and

the expression for the inner heat flux, should be added to thediscretized form of Eq (2) where unknown heat flux and localmean fluid temperature already exist

RESULTS AND DISCUSSION

This study has the objective of analyzing the effect of wallaxial conduction on heat transfer characteristics for a concentrictube heat exchanger A grid-independent study for the procedurewas performed in the reference [17] and is not repeated here.The grid-independent test showed that using a total of 200 con-trol volumes was accurate in terms of variation in wall and innerfluid local mean fluid temperatures To assess the validity ofthe solution procedure, a typical set of results in terms of ax-ial variation in nondimensional wall and inner fluid local meanfluid temperature for a constant outer fluid temperature case

is presented in Figure 2 The comparison is performed for thesame value of the parameterφ of the analytical solution given

by reference [5] The parameterφ is defined in the ture and its value used in Figure 2 is 5× 10−3 The predicted

nomencla-wall temperature is in good agreement with the analytical result

at all locations along the pipe, while the obtained bulk meanfluid temperature shows very small deviations near to the exit

of the pipe These deviations are assumed to be acceptable innumerical point of view By using the procedure described inthe previous section, the mean heat flux between the wall andthe inner fluid and the axial variations in wall and fluid tem-peratures are predicted for different wall to inner fluid thermalconductivity ratios, and for different outer to inner fluid heatcapacity rate ratios The effects of inner flow mean velocity andthe length of exchanger are also examined

Variations of the mean heat flux with the mean inflow locity are presented for various values of the heat capacity rateand the thermal conductivity ratios in Figure 3 and Figure 4,respectively The effect of heat capacity rate ratio on the mean

V+

0.4 0.6 0.8 1.0 1.2 1.4

C+ = 0.5

C+ = 1.0 C+ = 2.0 Counter

Figure 3 Variation of q+with V+for various values of C+.

heat transfer engineering vol 31 no 13 2010

0.4 0.6 0.8 1.0 1.2 1.4

k+ = 10 k+ = 500 k+ = 10000 Counter

Figure 4 Variation of q+with V+for various values of k+.

heat flux for the parallel flow is more precise than that of the

counterflow On the contrary, the effect of thermal conductivity

ratio on the mean heat flux for counterflow seems more

impor-tant than that of the parallel-flow case Variations of the mean

heat flux with the thermal conductivity ratio are presented for

various values of the heat capacity rate ratios and the

nondimen-sional exchanger length in Figures 5 and 6, respectively It is

concluded from these plots that the effect of the heat capacity

rate ratio and the effect of the exchanger length on the mean

heat flux are almost independent from the conductivity ratio for

the parallel flow case Dependence of the effect of the same

parameters on the mean heat flux is more pronounceable for the

counterflow case

The axial variations of wall and local mean fluid

temper-atures of the fluids are presented for three different values of

the heat capacity rate ratio in Figures 7–9 From an overall

ex-amination of the curves, the increase of thermal conductivity

ratio seems to have a noticeable effect only on variation of the

wall temperature The wall temperature flattens in profile and

increases in magnitude for high value of k+ However, only thecounterflow cases of the fluid temperature profiles are signif-icantly influenced by the flattened temperature profile These

influences are comparatively higher for small value of C+ For

the case of C+= 0.5, for instance, in Figure 7, discrepancies

in variation of temperature are quite significant for the flow The wall temperature exceeds the outlet temperature of the

counter-hot fluid temperature at X+= 0 for highly conductive wall Thiscauses transferring some part of the heat back to the hot fluidand results in less heat gain by the cold fluid This fact is alsosupported by Figure 4, which shows that the heat flux between

wall and inner fluid decreases with increasing k+.For all predicted results except the prediction for Figure 2,

values for r+ and Bi are kept constant as 1.2 and 12,

respec-tively The predictions are performed for k+= 100 and L+= 10

in Figure 3, for C+= 1 and L+= 10 in Figure 4, for V+= 6.6 ×

10−5 and L+ = 10 in Figure 5, and for C+ = 1 and

k+

0.4 0.6 0.8 1.0 1.2

1.4

C+ = 0.5 C+ = 1.0 C+ = 2.0 Counter

Figure 5 Variation of q+with k+for various values of C+.

heat transfer engineering vol 31 no 13 2010

0.4 0.6 0.8 1.0 1.2

1.4

L+ = 5 L+ = 10 L+ = 20 Counter

Figure 6 Variation of q+with k+for various values of L+.

V+= 6.6 × 10−5in Figure 6 The presented temperature

varia-tions in Figures 7 to 9 are performed for the fixed values of V+

= 6.6 × 10−5, which corresponds to Pe = 200, and of L+= 10

These figures indicate that the axial conduction effect should

be taken into account only for the counterflow heat exchanger

case especially for the small values of C+ As a

recommenda-tion in a practical point of view, this fact needs to be taken into

consideration for the designer

CONCLUSIONS

A combined numerical and analytical solution procedure is

used to examine the conjugate heat transfer for laminar flow

T + w

T + i

T + w

T + w

T + i

T + w

In general, the wall axial conduction effect is worth takinginto consideration for the counterflow case of the concentrictube heat exchangers Specifically, this effect becomes morepronounceable for the smaller values of the heat capacity rateratio The effect of the heat capacity rate ratio on the mean heatflux for the parallel flow case is more precise than that of thecounterflow counterpart Regardless of the flow configurations,

on the other hand, the length of the exchanger has a noticeableeffect on the mean heat flux

X+

0.0 0.2 0.4 0.6 0.8 1.0

T + i

T + w

T + i

T + w

T + i

T + w

T + i

T + w

T + w

T + i

T + w

T + w

T + i

T + i

T +

a) b)

+ = 10

Bi Biot number, ho r o/ki

c p,i specific heat of inner fluid

c p,o specific heat of outer fluid

C i heat capacity rate for inner fluid, ˙m i c p,i

C o heat capacity rate for outer fluid, ˙m o c p,o

r radial coordinate, radius

r+ nondimensional radial coordinate, r/ri

r∗ nondimensional radial coordinate, r/ri Pe

T∗ nondimensional temperature, [T −Tw]/[Ti(0)−Tw]

T i finite wall temperature differences between two

con-trol volumes

u inner flow fully developed laminar velocity

u m inner flow mean velocity

V+ nondimensional mean velocity, u2

m /(c p ,i Tmax)

x+ nondimensional axial coordinate, x/r

x∗ nondimensional axial coordinate, x/ri

Greek Symbols

φ dimensionless parameter used in Figure 2, kw[(ro/ri)2 – 1]/(2ki Pe 2)

λn eigenvalues

ξ dummy length variable in axial distance

ρ density of inner fluid

Subscripts

c cross-sectional

e inner fluid entrance

f inner fluid

i inner surface of the wall, local mean for inner fluid

o outer surface of the wall, local mean for outer fluid

M E ARICI 1041

[8] Kays, W M., and Crawford, M E., Convective Heat and

Mass Transfer, 2nd ed., McGraw-Hill, New York, 1987.

[9] Arıcı, M E., and Kaya, M E., Conjugate Heat

Trans-fer Analysis for Laminar Flow in Pipes Having a Step

Change in Boundary Conditions and Wall Conductivity,

Proc Institution of Mechanical Engineers, Part C,

Jour-nal of Mechanical Engineering Science, vol 221, no 8,

pp 917–925, 2007

[10] Weigand, B., and Gassner, G., The Effect of Wall

Con-duction for the Extended Graetz Problem for Laminar and

Turbulent Channel Flows, International Journal of Heat

and Mass Transfer, vol 50, pp 1097–1105, 2007.

[11] Ranganayakulu, C., and Seetharamu, K N., The

Com-bined Effects of Longitudinal Heat Conduction, Flow

Non-Uniformity and Temperature Non-Uniformity in

Cross-Flow Plate-Fin Heat Exchangers, International

Communications in Heat and Mass Transfer, vol 26, no.

5, pp 669–678, 1999

[12] Yuan, P., and Kou, H.-S., The Comparison of

Longitu-dinal Wall Conduction Effect on the Cross-Flow Heat

Exchanger Including Three Fluid Streams with Different

Arrangement, Applied Thermal Engineering, vol 21, pp.

1891–2007, 2001

[13] Madhawa, H D., Golubovic, H M., and Worek, W M,

The Effect of Longitudinal Heat Conduction in Cross Flow

Indirect Evaporative Air Coolers, Applied Thermal

Engi-neering, vol 27, pp 1841–1848, 2007.

[14] Ciofalo, M., Local Effects of Longitudinal Heat

Con-duction in Plate Heat Exchangers, International nal of Heat and Mass Transfer, vol 50, pp 3019–3025,

Jour-2007

[15] Malinowski, L., and Bielski, B., Transient TemperatureField in a Parallel-Flow Three-Fluid Heat Exchanger Withthe Thermal Capacitance of the Walls and the Longitudinal

Wall Conduction, Applied Thermal Engineering, vol 29,

no 5–6, pp 877–883, 2009

[16] Smith, E M., Thermal Design of Heat Exchangers, 3rd

ed., John Wiley & Sons, New York, 1990

[17] Arıcı, M E., Analysis of the Conjugate Effect of Wall and

Flow Parameters on Pipe Flow Heat Transfer, Proc tion of Mechanical Engineers, Part C, Journal of Mechan- ical Engineering Science, vol 215, no 3, pp 307–313,

Institu-2001

[18] Patankar, S V., Numerical Heat Transfer and Fluid Flow,

McGraw-Hill, New York, 1980

Mehmet Emın Arıcı is a professor in the Mechanical

Engineering Department of the Engineering Faculty

at Karadeniz Technical University, Trabzon, Turkey.

He received his Ph.D from Texas Tech University at Lubbock, TX, in 1993 His research interests include heat transfer enhancement techniques, computational fluid dynamics and heat transfer, and conjugate heat transfer.

heat transfer engineering vol 31 no 13 2010

Copyright
Taylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457631003639075

U-Tube Assembly Heat Exchanger

Performance Analysis Using Cyclic

Iteration

COREY CANTRELL and STEPHEN IDEM

Department of Mechanical Engineering, Tennessee Tech University, Cookeville, Tennessee, USA

This article outlines a technique to predict the performance of the convection passes of pulverized coal boilers, where the

heat exchangers are typically comprised of assemblies of U-tube elements Therein a heat exchanger comprised of n U-tubes

per assembly is modeled as 2n rows in the product gas flow direction, and the portions of the straight tubes making up each

assembly are considered to be distinct passes in crossflow It is assumed that in each pass the tube-side fluid is unmixed and

the gas-side fluid is mixed, and likewise the gas-side fluid is mixed between passes The model employs a straightforward

approach to predict the performance of a single pass as a building block to predict the performance of the overall assembly.

INTRODUCTION

This article describes a thermal performance model of the

convection passes commonly employed in a pulverized coal

boiler (PCB) These heat exchangers typically consist of either

pendant or cantilevered U-tubes, where water/steam flow

oc-curs inside the tubes The U-tubes are aligned to form tube bank

assemblies Hot combustion gases flow externally to the tubes,

and that flow may be normal to the tube banks or may occur

at an oblique angle The tube banks can be plumbed such that

the combustion gases and steam can variously flow in an

over-all parover-allel arrangement, an overover-all counterflow arrangement, or

a combination of such arrangements Refer to Figures 1–3 for

schematic diagrams of convection pass flow circuiting typically

encountered in a PCB power plant The ability to accomplish

performance predictions for such convection passes is

particu-larly important in the development of on-line fouling models;

refer to Cantrell [1]

Heat exchangers are devices that provide a means to

trans-fer energy between two or more fluids at diftrans-ferent temperatures

Heat transfer between these fluids takes place through a

separat-ing wall, which does not allow the fluids to mix Heat exchangers

are widely used in power production, automotive applications,

air conditioning, refrigeration, heat recovery, and

manufactur-ing Although there are many types of heat exchangers, this

Address correspondence to Professor Stephen Idem, Department of

Me-chanical Engineering, Tennessee Tech University, Cookeville, TN 38505-0001,

USA E-mail: sidem@tntech.edu

study will specifically address the performance of crossflowheat exchangers, which have the fluids flowing at right anglesthrough any pass A pass through a heat exchanger is defined as

a region that is exposed to identical thermofluid conditions, i.e.,temperatures, flow rates, etc., on both the tube and gas sides.Multipass crossflow heat exchangers may be described asseveral crossflow heat exchangers combined in series Mixingbetween the passes also has a strong impact on multipass heatexchanger performance If guide vanes, baffles, plate fins, etc.separate the flow such that no mixing can occur, then the flow

is assumed to be unmixed and the fluid temperature is dimensional in nature If there are no channel separators, thenthe fluid flow is termed mixed and there is a one-dimensionalfluid temperature variation in the flow path The direction of theflow is also important when considering multipass crossflowheat exchanger performance The fluids are considered to bemoving in parallel if the overall flow direction of both fluids isthe same In this configuration the outlet temperatures can becalculated without resorting to iteration The fluids are consid-ered to be moving in counterflow if both fluids are flowing inopposite directions When the flow between rows is mixed, thegas-side mixed mean temperature is used as the inlet value forthe adjacent row If the gas-side fluid remains unmixed betweenpasses, that fluid will have a nonuniform temperature profile as

two-it enters the adjacent pass This configuration requires an two-ative process in order to determine the outlet temperatures ofboth fluids

iter-Initial research on sensible heat exchanger performanceinvolved shell-and-tube heat exchangers The methods and

1042

C CANTRELL AND S IDEM 1043

t

II 1,2

t

II 1,

t

I n 2,

t

II 2,2

t

II 2,

t

I n 1,

T

I n 1,

1,1 II 2,2 T

II 2,1 T

T =

II 1,2

T

Figure 1 Overall parallel flow model.

principles applied to shell-and-tube heat exchangers were then

employed to model single-row and simple multirow crossflow

heat exchangers The first comprehensive analysis of heat

exchangers came when Nagle [2] presented correction factors

so that the logarithmic mean temperature difference for

counter-flow heat exchangers could be used to give the mean temperature

difference for multipass heat exchangers Nagle [2] claimed that

the correction factors depended only on the number of passes in

the exchanger and dimensionless ratios involving the inlet and

outlet temperatures of the two fluids However, the final equation

was solved by a graphical integration based on a trial and error

solution Smith [3] was one of the first researchers to present

the various flow arrangements in graphic form Bowman [4]

verified the performance formulas given by Underwood [5] for

a single pass shell with multipass tubes Bowman also extended

the results of Nagle [2] to exchangers with any number of shell

passes Fischer [6] derived an equation showing the correct

average temperature difference that should be used in multipass

heat exchangers, and Ten Broeck [7] used a graphical approach

to eliminate the need to calculate the mean temperature

difference in multipass shell-and-tube heat exchangers

The essential concept of the effectiveness–number of

trans-fer units (NTU) method of analyzing crossflow heat exchanger

performance was first presented by Nusselt [8] Bowman et al

t

II 1,2

t

II 1,

t

I n 2,

t

II 2,2

t

II 2,

t

I n 1,

T

I n 1,

1,1 II 2,2 T

II 2,1 T

T =

II 1,2

Figure 3 Combination flow model.

[9] were the first to present an analysis of the mean ture difference associated with both shell-and-tube and simplecrossflow heat exchangers They developed a correction factor

tempera-to the log-mean temperature difference for countercurrent flow,applicable to different types of flow arrangements for simplemultipass heat exchangers

In a crossflow arrangement, mixing of either fluid may or maynot occur A fluid is considered unmixed when it passes throughindividual flow channels or tubes with no fluid mixing betweenadjacent flow channels The concept of a mixed or unmixed flow

is an idealization that may not be observed in practice, as mostheat exchangers exhibit partial degrees of mixing DiGiovanniand Webb [10] developed algebraic expressions for ε-NTU ef-fectiveness for crossflow heat exchangers with arbitrary values

of partial mixing

Early work by researchers had presented the solution of thecrossflow heat transfer equations using tabular and graphicalforms Mason [11] solved the set of equations governing sensi-ble energy transfer in a simple crossflow heat exchanger usingLaplace transforms, where the solutions were represented inthe form of an infinite series Baclic [12] presented a seriessolution for simple crossflow heat exchanger effectiveness Itwas shown that the resulting formula provided results that wereequivalent to that obtained by Mason [11] A simple analyticalexpression for the effectiveness was presented for the case of asingle-pass crossflow heat exchanger with a capacity rate ratio

of unity The classic work of Kays and London [13] providesmany effectiveness–NTU relations for a wide variety of simpleheat exchanger geometries, and likewise includes an extensivecompilation of heat transfer and friction factor correlations formany different compact heat exchanger core designs Simonson[14] and Ribando et al [15] solved the differential equationsgoverning energy transfer in crossflow heat exchangers usingfinite-difference methods, thereby obtaining the fluid tempera-ture distribution throughout the exchanger

heat transfer engineering vol 31 no 13 2010

Stevens et al [16] reviewed the mean temperature difference

equations of all previously solved cases of simple crossflow

heat exchangers where there was one pass per row Using a

numerical integration process first introduced by Korst [17], data

were obtained for the single-pass case with both fluids unmixed

The cyclical reiteration method was used to solve all previously

unsolved cases of two- and three-pass crossflow heat exchangers

for counter-crossflow cases The correction-factor curves for

counter-crossflow exchangers revealed that the two-pass graphs

have areas where the effectiveness is lower than that of a true

parallel flow exchanger with the same NTU and thermal capacity

ratios, and is considerably lower than the corresponding

single-pass crossflow exchanger It was suggested that the results were

due to the fact that in some areas of the exchanger the coolant was

actually at a higher temperature than the hot-side fluid, resulting

in a reversed heat flow and a decreased effectiveness for both

fluids Moreover, it was found that this effect was present in the

three-pass case but was not as pronounced because of the lower

NTU per pass It was also found that for four or more

counter-crossflow passes the effectiveness relation for true counterflow

may be used with very little error Furthermore, for four or more

co-crossflow passes, the true parallel-flow effectiveness relation

could be used

Baclic and Gvozdenac [18] solved all previously unsolved

cases of two- and three-pass simple crossflow heat

exchang-ers with the passes coupled in identical order The results were

expressed in the form of an explicit formula of effectiveness

as a function of NTU, the heat capacity ratio, and the number

of passes Sekuli et al [19] presented a comprehensive review

of solution methods for obtaining effectiveness–NTU

relation-ships for numerous simple heat exchangers and heat exchanger

assemblies

Domingos [20] presented a method for analyzing the

per-formance of assemblies of heat exchangers This approach was

further expanded by Pignotti and his coworkers [21–25], and

Chen and Hsieh [26], to address a variety of fluid mixing

con-ditions and pass arrangements Their work provides a technique

for predicting the effectiveness of multistage heat exchangers

from the effectiveness of the individual stages In this context,

“multistage” refers to a heat exchanger that could be viewed

as two (or more) separate heat exchanger geometries combined

in series In many heat exchanger applications, improved heat

transfer and pressure drop performance is effectively pursued

with multistage designs This is particularly true in condensing

heat exchangers For example, refer to Jacobi et al [27], where

heat exchanger performance was described for an application

where one stage was designed for improved sensible heat

trans-fer and the other stage was optimized for latent heat transtrans-fer

CYCLIC ITERATION

In Incropera and DeWitt [28] it is demonstrated that the

steady performance of a heat exchanger can be expressed solely

as a function of the number of transfer units (NTU), the capacityrate ratio Cr, and the geometry/flow circuiting of the device Inthis paper performance predictions of U-tube convection passesare obtained using a variation of the cyclic iteration method firstdescribed in Stevens et al [16]; the approach is herein outlined.The present study will assume that the tube-side fluid is alwaysunmixed through each pass The gas-side fluid is assumed to becompletely mixed within each pass and between passes Phasechanges on the tube and gas side are disregarded, and constantthermal properties are assumed Furthermore, it is supposedthat the overall steam flow exhibits the maximum capacity rate,whereas the combustion gases possess the minimum capacityrate In every instance the tube- and gas-side flows are taken to

be steady

For any convection pass there are N U-tube assemblies across

the depth of the boiler cross section, where each assembly is

comprised of n U-tubes It is assumed that gas and steam flow

rates do not vary transverse to the gas-side flow Implicitly, theheat transfer surface area of the tube bends is disregarded in thisanalysis Hence the straight tubes making up each assembly are

modeled as 2n passes in the product gas flow direction It is

supposed that Uo is constant throughout the entire convectionpass, such that the NTUs are evenly distributed among the passes

of the heat exchanger In that case the NTUs per pass is givenby

Cr =m˙Acp,A

˙

However, when the steam flow through a convection pass is

evenly apportioned among the n tubes of each U-tube assembly,

under certain circumstances the capacity rate of fluid B per passcan be less than that of fluid A within that pass That occurswhen

C CANTRELL AND S IDEM 1045

remains as follows:

Cr= n ·m˙Acp,A

˙

Similarly, NTUis also redefined for the case stated in Eq (5)

since the tube-side fluid is now the minimum capacity fluid

within a pass It can be shown that

NTU=UoAo

Cmin = UoAo

2n

In this study it is assumed the tube-side flow is unmixed,

whereas the gas-side flow is taken to be mixed For steady-state

conditions and Cr = 0 it is shown in Incropera and DeWitt [28]

that the effectiveness per pass of a single-row crossflow heat

exchanger where the maximum capacity rate fluid is mixed and

the minimum capacity rate fluid is unmixed can be expressed as

ε= 1 − exp−C−1

r { 1 − exp[−C

Similarly per [28], for a single-row crossflow heat exchanger

where the maximum capacity rate fluid is unmixed and the

minimum capacity rate fluid is mixed, the effectiveness per pass

is expressed as

ε=C−1r 

(1− exp { −C

r[1− exp(−NTU)]}). (9)The performance of a single-row simple crossflow heat ex-

changer where the tube side has the maximum capacity rate

is calculated as follows Referring to Figure 4, performing an

energy balance on the heat exchanger implies

In this instance T1, t1,Cr, and ε are the known variables,

whereas T2 and t2 are unknown Solving Eqs (10) and (11)

simultaneously yields the outlet temperature of the minimum

Figure 4 Single-row simple crossflow heat exchanger.

capacity rate fluid as

is calculated as follows Referring to Figure 4, performing anenergy balance on the heat exchanger implies

of the overall capacity rate ratio, such that fluid B becomesthe minimum capacity rate fluid for a given pass Employingthe terminology implied by Figure 2, the convection pass inlettemperatures on both the steam and gas sides, i.e., t1 and T1,are presumed to be known For the first iteration the tempera-tures of fluid B entering each tube of section I, i.e., the tI

1,i, areguessed This permits the performance of each pass of section

I to be determined by means of Eqs (15) and (16), therebyyielding temperatures TI2,iand tI2,ileaving each pass Likewisefor section II, the performance of each pass is calculated sim-ilarly, which yields outlet temperatures TII2,i and tII2,i In everyinstance the gas-side outlet temperature for an upstream passserves as the inlet temperature for the subsequent downstreampass Thereafter the calculated temperatures tII2,i are compared

to the guessed values tI

1,i If these temperatures agree to within

a specified tolerance the computations are complete; otherwise,the calculations are repeated with the most recent values of tII2,iserving as the next assumed values for tI

1,i Fluid B is unmixedbetween the sections of the heat exchanger Hence in order toheat transfer engineering vol 31 no 13 2010

account for that, note the following:

tI1,i= tII2,1+n− i; i = 1, n (18)Upon convergence of the calculations, the average heat ex-

changer outlet temperature for fluid A is given by TII2,n

Further-more the average heat exchanger outlet temperature for fluid B

is given by

t2 =1n

n



i=1

If the heat exchanger is operated in overall parallel flow,

iter-ation is not required In that case with t1and T1assumed known,

the performances of sections I and II are likewise calculated

us-ing Eqs (16) and (17), where the gas-side outlet temperature

for an upstream pass serves as the inlet temperature for the next

downstream pass However since fluid B is unmixed between

the sections of the heat exchanger, this implies

n



i=1

In several instances the convection passes can be construed

as combinations of the fundamental parallel flow or counterflow

cases considered previously Therein the elementary models are

employed as “building blocks” to predict performance of the

more complex flow circuiting cases

It is informative to compare the performance of the heat

ex-changers found in a typical PCB power plant to the effectiveness

of a variety of elementary heat exchanger geometries As noted

in Incropera and DeWitt [28], the limit of zero capacity rate

ra-tio, i.e., Cr = 0, is obtained when one fluid in a heat exchanger

remains isothermal In that case for any heat exchanger pass

arrangement the effectiveness is given by

Figure 5 Convection pass effectiveness for minimum capacity rate fluid

mixed and maximum capacity rate fluid unmixed with n U-tube elements,

tube-side fluid unmixed between sections (overall parallel flow), C r = 0.0.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 6 Convection pass effectiveness for minimum capacity rate fluid

mixed and maximum capacity rate fluid unmixed with n U-tube elements,

tube-side fluid unmixed between sections (overall parallel flow), C r = 0.25.

For a concentric tube heat exchanger operated in pure parallelflow, the effectiveness is given by the following well-knownrelation:

ε=1− exp [−NTU(1 + Cr)]

1+ Cr

(23)Similarly, for a concentric tube heat exchanger operated in purecounterflow, the effectiveness is characterized as

ε= 1− exp [−NTU(1 − Cr)]

1− Crexp [−NTU(1 − Cr)]; Cr <1 (24)Likewise, for pure counterflow of a concentric tube heat ex-changer, it can readily be shown that

Figure 7 Convection pass effectiveness for minimum capacity rate fluid

mixed and maximum capacity rate fluid unmixed with n U-tube elements,

tube-side fluid unmixed between sections (overall parallel flow), C r = 0 5.heat transfer engineering vol 31 no 13 2010

C CANTRELL AND S IDEM 1047

Figure 8 Convection pass effectiveness for minimum capacity rate fluid

mixed and maximum capacity rate fluid unmixed with n U-tube elements,

tube-side fluid unmixed between sections (overall parallel flow), C r = 0.75.

Figure 9 Convection pass effectiveness for minimum capacity rate fluid

mixed and maximum capacity rate fluid unmixed with n U-tube elements,

tube-side fluid unmixed between sections (overall parallel flow), C r = 1.0.

exhibits lower effectiveness than a true parallel-flow heat

ex-changer having similar NTU and Cr values The effectiveness

is observed to decrease in proportion with decreasing values of

Cr For NTU≤ 3 and all values of Cr, effectiveness values for

all n values correspond closely to those for true parallel flow.

However, for NTU≥ 3 and all values of Cr the effectiveness

actually diminishes with increasing NTU

Figures 10 to 14 portray heat exchanger effectiveness

cal-culated by means of the cyclic iteration method for an overall

Figure 10 Convection pass effectiveness for minimum capacity rate fluid

mixed and maximum capacity rate fluid unmixed with n U-tube elements,

tube-side fluid unmixed between sections (overall counterflow), C r = 0.0.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 11 Convection pass effectiveness for minimum capacity rate fluid

mixed and maximum capacity rate fluid unmixed with n U-tube elements,

tube-side fluid unmixed between sections (overall counterflow), C r = 0.25.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 12 Convection pass effectiveness for minimum capacity rate fluid

mixed and maximum capacity rate fluid unmixed with n U-tube elements,

tube-side fluid unmixed between sections (overall counterflow), C r = 0.5.

counterflow U-tube configuration, as illustrated in Figure 2 A

counterflow U-tube heat exchanger having any number n of

U-tube elements exhibits lower effectiveness than a true terflow heat exchanger having equal values of NTU and Cr Theperformance of a U-tube heat exchanger described by any value

coun-of n elements closely matches that coun-of a true counterflow heat

exchanger for all values of Cr and NTU ≤ 1 For NTU ≥ 1this deviation is more apparent for larger values of Crand anynumber of U-tube assemblies For overall counterflow U-tube

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 13 Convection pass effectiveness for minimum capacity rate fluid

mixed and maximum capacity rate fluid unmixed with n U-tube elements,

tube-side fluid unmixed between sections (overall counterflow), C r = 0.75.heat transfer engineering vol 31 no 13 2010

Figure 14 Convection pass effectiveness for minimum capacity rate fluid

mixed and maximum capacity rate fluid unmixed with n U-tube elements,

tube-side fluid unmixed between sections (overall counterflow), C r = 1.0.

heat exchangers it is evident that effectiveness is inversely

pro-portional to Cr, whereas effectiveness increases monotonically

with NTU in that case

Referring to Figures 5 through 14, the effectiveness values

calculated for all numbers of U-tube assemblies with n≥ 2 are

essentially indistinguishable on the scale of the graphs, at any

particular combination of NTU and Cr This implies there is no

effectiveness penalty associated with adding additional U-tube

assemblies to a heat exchanger; however, the consequences in

terms of additional pressure loss were not examined in this study

In every instance a parallel-flow U-tube heat exchanger exhibits

lower effectiveness than a counterflow U-tube heat exchanger

having the same NTU and Crvalues

CONCLUSIONS

Performance predictions of U-tube convection passes, which

are typically employed in PCB power plants, were obtained

using a variation of the cyclic iteration method This general

method was first proposed by Stevens et al [16], subject to

fewer flow mixing and convection pass circuiting restrictions

than were employed in the present analysis However, the

sys-tematic approach outlined herein is particularly useful for the

de-velopment of on-line fouling models in coal-fired power plants

As discussed further in Cantrell [1], the resulting heat transfer

predictions represent conditions for each convection pass under

“clean” conditions These in turn can be compared to measured

steam-side heat transfer under actual (i.e., “dirty”) conditions

where fouling is present, thereby permitting the determination

of localized cleanliness factors within boiler Convection pass

performance predictions based on the method outlined in the

present paper are presented in Cantrell [1] for a typical PCB

power plant as functions of heat transfer surface area, and fluid

flow rates and convection heat transfer coefficients, which

ulti-mately depend on plant load

In this study the convection passes comprised of n U-tubes

per assembly were modeled as 2n passes in the product gas flow

direction, such that the straight tube sections making up each

assembly were considered to be distinct passes in crossflow.This is a reasonable assumption, in that the heat transfer surfacearea of the tube bends is typically less than 5% of the totalsurface area of a tube bank in a PCB power plant Generally

in a PCB power plant the overall capacity rate of the side fluid (steam) exceeds that of the gas-side fluid (combustionproducts) However, when the tube-side flow is divided among

tube-the n tubes of a U-tube assembly, tube-the gas-side capacity rate for

each pass may be greater than that of the tube-side fluid withinthat particular pass This occurs because the tube-side capacityrate decreases as the number of tubes increases, although theoverall tube-side capacity rate is constant As noted in Cantrell[1], this is particularly important when developing general PCBpower plant fouling models, in that the various convection passesencountered in practice can be comprised of greatly differentnumbers of U-tubes per assembly

The assumption employed in this model that the tube-sidefluid is unmixed is appropriate, since the fluid axial tempera-ture variation differs for each tube in a convection pass, andthe steam does not mix between successive passes The sup-position that the gas-side fluid was mixed within a pass wouldimply that its temperature was invariant perpendicular to thegas flow direction, i.e., in the tube longitudinal direction Whilenot strictly true, the product temperature change in the gas-sideflow direction far exceeds the temperature variation in the trans-verse direction, thereby closely approaching one-dimensionalbehavior associated with perfect mixing

As outlined in Cantrell [1], for a typical PCB power plantNTU values usually do not exceed 2.0 in the economizer, and0.8 in the other convection passes However, these values areaffected considerably by the plant load Likewise, capacity rateratios usually vary over the range 0.2≤ Cr≤ 1.0 In almost allpractical instances the numbers of U-tube assemblies compris-ing the convection passes of a PCB power plant correspond to

n≥ 2, and are of the configurations shown in Figures 1 through

3 Therefore, it can be concluded that use of a true flow heat exchanger performance model for such convectionpasses as the pendant/finishing superheaters and reheaters etc.may introduce minimal error However, application of the cycliciteration approach for the remaining convection passes, i.e., thelow-temperature superheater and economizer, may be justified.Over the range of Cr and NTU values encountered in a PCBpower plant, effectiveness values determined by the cyclic it-eration method may differ from those calculated using a truecounterflow model by as much as 6%

parallel-NOMENCLATURE

Ao total external surface area, m2(ft2)

Ao total external surface area per pass, m2(ft2)

Cmin minimum capacitance, kW/K (Btu/h-◦R)

Cmin minimum capacitance per pass, kW/K (Btu/h-◦R)heat transfer engineering vol 31 no 13 2010

C CANTRELL AND S IDEM 1049

cp,A specific heat of fluid A, kJ/kg-K (Btu/lb-m-◦R)

cp,B specific heat of fluid B, kJ/kg-K (Btu/lb-m-◦R)

Cr capacity rate ratio

Cr capacity rate ratio per pass

˙

mA mass flow rate of fluid A, kg/h (lb-m/h)

˙

mB mass flow rate of fluid B, kg/h (lb-m/h)

n number of U-tubes per assembly

N number of tube assemblies across the depth of the

boiler cross section

NTU number of transfer units

NTU number of transfer units per pass

PCB pulverized coal boiler

t1 temperature of tube-side fluid entering the

[1] Cantrell, C L., Performance Modeling of a Pulverized

Coal Boiler, Ph.D Dissertation, Tennessee Tech University,

Cookeville, TN, 2007

[2] Nagle, W M., Mean Temperature Difference in Multipass Heat

Exchangers, Industrial and Engineering Chemistry, vol 25, no.

6, pp 604–609, 1933

[3] Smith, D M., Mean Temperature Difference in Cross-Flow,

En-gineering, vol 138 pp 479-481, and vol 138, pp 606–607, 1934.

[4] Bowman, R A., Mean Temperature Difference Correction in

Mul-tipass Exchangers, Industrial and Engineering Chemistry, vol 28,

no 5, pp 541–544, 1936

[5] Underwood, A J V., The Calculation of the Mean Temperature

Difference in Multipass Heat Exchangers, Journal of the Institute

of Petroleum Technology, vol 20, pp 145–158, 1934.

[6] Fischer, F K., Mean Temperature Difference Correction in

Multi-pass Exchangers, Industrial and Engineering Chemistry, vol 30,

no 4, pp 377– 383, 1938

[7] Ten Broeck, H., Multipass Exchanger Calculations, Industrial and

Engineering Chemistry, vol 30, no 9, pp 1041–1042, 1938.

[8] Nusselt, N., A New Heat Transfer Formula for Cross-Flow,

Tech-nische Mechanik und Thermodynamik, vol 12, pp 417–422,

1930

[9] Bowman, R A., Mueller, A C., and Nagle, W M., Mean

Temper-ature Difference in Design, Trans ASME, vol 62, pp 283–294,

1940

[10] DiGiovanni, M A and Webb, R L., Uncertainty in

Effectiveness-NTU Calculations for Crossflow Heat Exchangers, Heat Transfer

Engineering, vol 10, no 3, pp 61–70, 1989.

[11] Mason, J L., Heat Transfer in Crossflow, Proceedings of the

Second U.S National Congress of Applied Mechanics, A.S.M.E.,

New York, pp 801-803, 1955

[12] Baclic, B S., A Simplified Formula for Crossflow Heat Exchanger

Effectiveness, Trans ASME, vol 100, pp 746–747, 1978 [13] Kays, W M., and London, A L., Compact Heat Exchangers, 3rd

ed., McGraw-Hill, New York, 1984

[14] Simonson, J R., Transient and Steady-State Analysis of

Cross-Flow Heat Exchangers by Programs in FORTRAN, Trans

Insti-tute of Chemical Engineers, vol 55, pp 53–58, 1977.

[15] Ribando, R J., O’Leary, G W., and Carlson-Skalak, S., GeneralNumerical Scheme for Heat Exchanger Thermal Analysis and

Design, Computer Applications in Engineering Education, vol 5,

pp 231–242, 1997

[16] Stevens, R A., Fernandez, J., and Woolf, J R Mean-TemperatureDifference in One, Two, and Three-Pass Crossflow Heat Exchang-

ers, ASME Trans., vol 79, pp 287–297, 1957.

[17] Korst, H H., Mean Temperature Difference in Multi-Pass

Cross-flow Heat Exchangers, Proceedings of the First U.S National

Congress of Applied Mechanics, A.S.M.E., New York, pp 949–

955, 1952

[18] Baclic, B S., and Gvozdenac, D D., Exact Explicit Equations forSome Two- and Three-Pass Crossflow Heat Exchangers Effec-

tiveness, in Heat Exchangers: Thermal-Hydraulic Fundamentals

and Design, eds S Kakac A E Bergles, and F Mayinger, pp.

481–494, Hemisphere, Washington, DC, 1981

[19] Sekuli, D P., Shah, R K., and Pignotti, A., A Review of tion Methods for Determining Effectiveness-NTU Relationships

Solu-for Heat Exchangers With Complex Flow Arrangements, Applied

Mechanics Review, vol 52, no 3, pp 97–117, 1999.

[20] Domingos, J D., Analysis of Complex Assemblies of Heat

Ex-changers, International Journal of Heat and Mass Transfer, vol.

12, pp 537–548, 1969

[21] Pignotti, A., and Cordero, G O., Mean Temperature Difference

in Multipass Crossflow, Journal of Heat Transfer, vol 105, pp.

584–591, 1983

[22] Pignotti, A and Cordero, G O., Mean Temperature Difference

Charts for Air Coolers, Journal of Heat Transfer, vol 105, pp.

592–597, 1983

[23] Pignotti, A., Effectiveness of Series Assemblies of Divided-Flow

Heat Exchangers, Journal of Heat Transfer, vol 108, pp 141–146,

[25] Shah, R K., and Pignotti, A., Thermal Analysis of Complex

Crossflow Exchangers in Terms of Standard Configurations,

Jour-nal of Heat Transfer, vol 115, pp 353–359, 1993.

heat transfer engineering vol 31 no 13 2010

[26] Chen, J D., and Hsieh, S H., General Procedure for

Effective-ness of Complex Assemblies of Heat Exchangers, International

Journal of Heat and Mass Transfer, vol 33, no 8, pp 1667–1674,

1990

[27] Jacobi, A M., Idem, S A., and Goldschmidt, V W., Predicting

the Performance of Multistage Heat Exchangers, Heat Transfer

Engineering, vol 14, no 1, pp 62–70, 1993.

[28] Incropera, F P., and DeWitt, D P., Fundamentals of Heat and

Mass Transfer, McGraw Hill, New York, 2007.

Corey Cantrell is an associate heat transfer engineer

for McQuay International, a member of the Daikin group, at its headquarters in Minneapolis, MN He received his Ph.D in mechanical engineering from Tennessee Technological University, Cookeville, TN.

His duties at McQuay include heat transfer and fluid flow analysis and validation for the Coil Engineering Department in the Applied Air Handling Unit.

Stephen Idem is a professor in the Department of

Mechanical Engineering at Tennessee Tech sity, Cookeville, TN He received his Ph.D in me- chanical engineering from Purdue University He has more than 22 years of experience in the areas of scale model testing, fluid flow measurement, and thermal modeling He is currently engaged in projects to ex- perimentally measure air and sound performance of propeller fans with systematic variation of inlet flow components, and is testing flat oval duct fittings to determine their total pressure loss coefficients He is also developing on-line thermal models of coal-fired power plants so as to optimize soot-blowing inter- vals.

Univer-heat transfer engineering vol 31 no 13 2010

Copyright
C Taylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457631003640313

Heat Exchangers Design Under

Variable Overall Heat Transfer

Coefficient: Improved Analytical and Numerical Approaches

MOSTAFA H SHARQAWY1and SYED M ZUBAIR2

1Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

2Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

In typical heat exchanger design methods it is generally assumed that the overall heat transfer coefficient is constant

and uniform; however, the heat transfer coefficients on the hot and cold sides of the heat exchanger may vary with flow

Reynolds number, surface geometries, fluid thermophysical properties, and other factors In this article we present simple

analytical and numerical methods for calculating heat transfer area for data sets introduced earlier in the literature For

the analytical methods presented in the article, the variation in the overall heat transfer coefficient with the local hot and

cold fluid temperature difference is expressed as a power-law model and as a general polynomial model The procedure for

calculating the heat transfer area with the power-law model is explained with respect to a simple closed-form solution, while

the polynomial model can also provide an analytical solution that seems to be quite accurate for the data sets examined It

is also shown that a Chebyshev numerical integration scheme that requires four points compared to the Simpson method of

three points is quite accurate (within 1% of the exact value).

INTRODUCTION

In a standard effectiveness–number of transfer units (ε-NTU)

and logarithmic mean temperature difference (LMTD) method

of heat exchanger analysis, it is assumed that the overall heat

transfer coefficient U is constant and uniform throughout the

exchanger and invariant with time However, in practical

ap-plications, the overall heat transfer coefficient varies along the

heat exchanger and depends on the heat transfer coefficients on

both hot and cold fluid sides as well as the wall and fouling

re-sistances The heat transfer coefficients are strongly dependent

on the flow Reynolds number, heat transfer surface geometry,

fluid thermophysical properties, entrance length effect due to

de-veloping thermal boundary layers, and other thermal-hydraulic

and geometric factors For example, as indicated by Shah and

Sekulic [1, 2], in a viscous liquid heat exchanger, a 10-fold

vari-ation in the heat transfer coefficient is possible when the flow

The authors acknowledge the support provided by King Fahd University of

Petroleum and Minerals for this research project.

Address correspondence to Professor Syed M Zubair, Department of

Mechanical Engineering, King Fahd University of Petroleum and Minerals,

Dhahran 31261, Saudi Arabia E-mail: smzubair@kfupm.edu.sa

pattern encompasses laminar, transition, and turbulent regions

on one side Thus, if the individual h values vary across the changer surface area, it is highly expected that U will not remainconstant and uniform in the exchanger

ex-The major variation of heat transfer coefficients of hot andcold fluids can be attributed to fluid property variations (or radi-ation) It consists of two components (Roetzel [3]): (a) distortion

of velocity and temperature profiles at a given free flow crosssection due to fluid property variations, and (b) variations in thefluid temperature along axial and transverse directions in theexchanger, depending on the flow arrangement; this effect is re-ferred to as a temperature effect Shah and Skeulic [2] indicatedthat the resulting axial changes in the overall mean heat transfercoefficient can be significant, and the variations in the local Ucould be nonlinear, depending on the type of fluid

The effect of variable overall heat transfer coefficient on thedesign of heat exchangers has been investigated by many re-searchers The early treatment in the literature is due to Colburn[4], where a linear variation of the overall heat transfer coeffi-cient with the temperature difference in the heat exchanger isconsidered and an analytical solution is given Different methods

of dealing with variable overall heat transfer coefficient havebeen proposed in classical heat transfer texts (Kern [5], Spalding

1051

[6], and Schlunder, [7]) In these methods, the exchanger is

di-vided into segments and the analysis is best performed by a

numerical or a finite-difference method

Some averaging procedures for determining the mean

over-all heat transfer coefficient values have been suggested by many

authors such as Peters [8], Roetzel [3], Hausen [9], and Roetzel

and Spang [10] However, Shah and Sekulic [1] have

reconsid-ered the applicability of Roetzel’s method in situations where

a highly nonlinear overall heat transfer coefficient is present

They concluded that none of the approximate methods will

ac-curately predict the exchanger surface area requirement when

the variation of the heat transfer coefficient on one of the two

fluid sides is highly nonlinear and the corresponding thermal

resistance is controlling the overall heat transfer coefficient

The objective of this article is to demonstrate simple

ap-proaches to determine the heat transfer area for the case of

variable overall heat transfer coefficient These approaches are

both numerical and analytical

HEAT EXCHANGER ANALYSIS

To design or evaluate a heat exchanger, it is important to relate

total heat transfer rates to quantities such as the inlet and outlet

fluid temperatures, the overall heat transfer coefficient, the fluid

capacitance rates, and the total surface area of the exchanger

The energy balance and the subsequent analysis are subject to

the following assumptions:

• The temperature is uniform across the cross section of the

fluid streams

• The overall heat transfer coefficient, U , is not constant

How-ever, it is varying along the heat exchanger

• There is a negligible heat loss from the exchanger

• There is a negligible longitudinal heat transfer rate through

the wall of the exchanger

For any given heat exchanger, an energy balance on an

ele-mental volume gives

d ˙ Q = − ˙Ch dT h = − ˙Cc dT c = (U
T ) dA (1)

where
T = Th−Tcis the local temperature difference between

the hot and cold fluids Integrating Eq (1) to calculate the total

heat transfer surface area gives

T hcurve or the area under the ( − ˙C c

U
T ) versus Tccurve will givethe exact surface area of the heat exchanger This area will take

into consideration the changes in the specific heat of both hot and

cold fluids as well as the local overall heat transfer coefficient

Integration of Eq (2) requires information about the hot or

cold fluid temperature distribution as well as the variation of the

heat transfer coefficient and specific heats with temperature Thebest way to integrate Eq (2) is by applying a numerical integra-tion method The Chebyshev’s numerical integration techniquethat requires four points only to perform the integration is exam-ined in the next section with respect to the complete numericalapproach It generally gives a higher accuracy as compared withother numerical integration techniques that use the same num-ber of intervals (like the Simpson and trapezoidal method) It isimportant to mention here that the method provided by Roetzeland Spang [10] and then modified by Shah and Sekulic [1] isbasically based on a Simpson integration method using threepoints The three points are: the inlet, exit, and a suggested in-termediate point that satisfies the geometric mean temperature

difference,
T 1/2 = (
T1×
T2)1/2 Their method requiredlengthy procedure and it is not accurate as concluded by Shahand Sekulic [1] The four-point Chebyshev integration technique

is recommended by the British Standard [11] and the CoolingTower Institute [12] to be employed in the cooling tower perfor-mance and analysis methods A discussion on the Chebyshevintegration technique usage for cooling towers can be found inMohiuddin and Kant [13]

Another approach that can be used to integrate Eq (2) isthe analytical integration, which can be performed if we knowthe relationship between the overall heat transfer coefficientand the temperature difference An analytical solution will be

presented for a polynomial relationship with n degrees as well

as a power-law correlation It is important to note that Colburn

[4] assumed a linear relationship between U and
T, which is a

special case of the polynomial method introduced in the present

article (i.e., polynomial of n= 1)

NUMERICAL METHOD

To integrate Eq (2) numerically, Chebyshev integrationmethod can be used where four points are required along the hot

or cold fluid If the four points of Chebyshev integral is applied

to this relation, the integral of Eq (2) can be expressed as

where the ch/U
T values in Eq (4) are dependent on the

fol-lowing intermediate temperatures of the hot fluid:

T h 1 = Th,i − 0.1 (Th,i − Th,o) (6a)heat transfer engineering vol 31 no 13 2010

M H SHARQAWY AND S M ZUBAIR 1053

T h 2 = T h,i − 0.4 (Th,i − Th,o) (6b)

T h 3 = Th,i − 0.6 (Th,i − Th,o) (6c)

T h 4 = T h,i − 0.9 (Th,i − Th,o) (6d)

Also, the cc/U
T values in Eq (5) are dependent on the

following intermediate temperatures of the cold fluid:

T c 1 = T c,i + 0.1 (Tc,o − Tc,i) (7a)

T c 2 = T c,i + 0.4 (Tc,o − Tc,i) (7b)

T c 3 = Tc,i + 0.6 (Tc,o − Tc,i) (7c)

T c 4 = T c,i + 0.9 (Tc,o − Tc,i) (7d)

To calculate the (c/U
T) values of Eqs (4) and (5) at these

intermediate temperatures, we can apply an energy balance

be-tween the inlet and intermediate fluid condition Consequently,

the temperature of the fluid at this intermediate location can be

calculated Hence, the specific heat (c), the overall heat transfer

coefficient (U ), and the local temperature difference (
T) at this

intermediate location can be determined This energy balance

can be applied for counter- and parallel-flow heat exchangers,

where j refers to the intermediate location for the hot or cold

fluid temperature and ¯c is the average specific heat between the

terminal and intermediate temperatures It is important to note

here that the effect of variable specific heat and heat transfer

coefficient with temperature is considered in the Chebyshev

numerical integration as the local value of (c/U
T) is calculated

at each intermediate temperature

ANALYTICAL METHODS

From the energy balance of Eq (1), we can write the variation

in the local temperature difference as

d (
T ) = dTh − dTc = d ˙Q

1

the fluid capacitance rates can be written as

where for a counterflow heat exchanger

T1= Th,i − Tc,o and
T2 = Th,o − Tc,i (13)Separating the variables and integrating along the completelength of the heat exchanger, results in

To integrate the right-hand side, we should know the relation

between the overall heat transfer coefficient, U , and the ature difference
T In this regard, we consider two models that

temper-may be applicable based on a best fit correlation for the givenheat exchanger problem

POWER-LAW MODEL

If the power-law correlation U = a
T nis the best fit

corre-lation for the U–
T data provided in a heat exchanger problem,

then the integration of Eq (14) can be easily handled as:

where the dimensionless temperature ratio is given by, τ =

T2/
T1 Considering the factor F as

Figure 1 gives the factor F for different values of τ and

n The figure is drawn for both cases when
T2 >
T1 or

T2 <
T1 It is important to note that for
T2 >
T1, weconsider τ =
T2/
T1 and ˙Q = F A U1
T1, while for the

case
T2 <
T1, it is defined as τ =
T1/
T2 and ˙Q =heat transfer engineering vol 31 no 13 2010

Figure 1 F factor plot for the power-law overall heat transfer coefficient

model; the effect of the power-law coefficient, n.

F A U2
T2 In the limit when n = 0, this gives U = a, which

is a constant value of the overall coefficient This gives

Suppose that the U –
T data of a given heat exchanger are

best represented with a polynomial correlation of the form:

By using the partial fraction expansion method [14], the

integration of Eq (23) can be simplified to give

where the R terms are the residues and the P terms are the poles

of Eq (23) Hence the solution of the above equation can be

written in the form

It is important to note here that if the overall heat transfercoefficient value is constant along the heat exchanger, there will

be only one pole and one residue for Eq (24) That is, P = 0

and R = 1/U Hence the surface area can be simplified to give

Also, if the U value is a linear function of the
T, as assumed

by Colburn [4], the surface area will be given as

RESULTS AND DISCUSSION

In order to use the analytical and numerical approaches tothe present problem, an example is solved using different meth-ods to calculate the surface area of a heat exchanger when theoverall heat transfer coefficient is not constant These methodsare as follows: (1) the exact numerical solution (using Simp-son integration with 200 points), (2) the Roetzel and Spang [10]method (i.e., Simpson numerical integration using three selectedpoints), (3) the Colburn [4] method (i.e., the logarithmic mean

difference of the U–
T product), (4) the Chebyshev integration

method, (5) the polynomial analytical method using a mial of a second degree, (6) the arithmetic mean of the terminal

polyno-U values, and (7) the U value determined at the arithmetic mean

of inlet and outlet temperatures of the Cminfluid stream.For comparison purposes, the same five sets of h-T dataprovided by Shah and Sekulic [1] are used These five setswere generated based on the data provided in Example B of theColburn’s original paper [4] The five sets of data are given inFigure 2 and the example is described as follows

0 50 100 150 200 250 300 350 400 450

Figure 2 Heat transfer coefficient on the hot liquid side versus local ature (Shah and Skeulic [1]).

temper-heat transfer engineering vol 31 no 13 2010

M H SHARQAWY AND S M ZUBAIR 1055

Table 1 Comparison of various averaging methods for the original Colburn

Chebyshev integral (present work) 18.86 +0.46

Polynomial analytical (present work) 19.18 −0.08

U at the arithmetic mean temperature 18.64 −0.69

Example

Straw oil is to be heated from 26.6◦C to 93.3◦C, flowing

at an average velocity of 0.914 m/s through a horizontal pipe

using steam at 108.3◦C The oil has a mean specific heat of

1.968 kJ/kg-K, a mean specific gravity of 0.85, and a mean

thermal conductivity of 0.135 W/m-K The viscosities of the oil

are 0.018 and 0.004 kg/m-s at 26.6◦C and 93.3◦C, respectively,

and a plot of viscosity versus temperature on logarithmic paper

may be considered as a straight line The outside and inside

diameters of the pipe are 6 and 5.3 cm, respectively; the thermal

conductivity of the pipe is 60.58 W/m-K Since the oil side heat

transfer resistance is controlling, the heat transfer coefficient on

the steam side is assumed to be constant at 12,208 W/m2-K

Using the data of this problem, Colburn [4] showed that the

local heat transfer coefficient on the oil side follows the

tempera-ture change nonlinearly (Figure 2, data set 1) This dependence,

in turn, causes a nonlinear change in the overall heat transfer

coefficient The other four sets were generated by Shah and

Skeulic [1] from the first one under the following assumptions:

For the second set of data, it is assumed that a hypothetical fluid

exchanges heat under the conditions of even more pronounced

nonlinearity with respect to local temperatures, as given by the

data set 2 in Figure 2 (all other conditions continues to be the

same) The third set of data assumes a linear dependence of the

local heat transfer coefficient on the oil side but stays at the same

terminal values as that of data set 1 The fourth set of data (data

set 4, Figure 2) corresponds to the mirror image of data set 2.Finally, the fifth set of data (data set 5) assumes an “oscillatory”behavior to find the impact of an oscillatory behavior on thesurface area calculation

The results of the comparison are presented in Tables 1 and

2 Table 1 summarizes the calculated surface areas and errorsassociated by using various methods for the original Colburnproblem (data set 1) The exact value is determined by using theSimpson integration method with 200 intervals Table 1 showsthat the Roetzel and Colburn methods cause an overpredic-tion, while the use of the arithmetic mean value of the terminal

Uvalues provides the largest error It can also be seen from the

table that the evaluation of U at various reference temperatures

as a general rule underestimates the correct result The goodestimation obtained by using the arithmetic mean temperature

as a reference temperature is an accident, rather than the rule.The numerical integration technique of Chebyshev gives theminimum error compared to other methods

Table 2 provides results for all five data sets and seven parison methods All positive numbers indicate an overestima-tion of the required surface areas compared to that by the exactnumerical values, and all negative numbers indicate underesti-mation It is clear from this table that the Chebyshev numericalintegration and the polynomial analytical methods proposed inthis article are the most accurate methods for the all sets ofdata even under the conditions of severe nonlinearities If thelocal heat transfer coefficient varies linearly (data set 3), theColburn model gives relatively good results compared with theChebyshev numerical integration However, if the nonlineari-ties are pronounced (data sets 2, 4, and 5), both the Chebyshevnumerical integration and the polynomial analytical solutiongive reasonably accurate results (within 1% for the numericalapproach and±2.5% for the analytical procedure)

com-CONCLUDING REMARKS

A simple and straightforward analytical and numerical tegration procedure for calculating the heat exchanger areafor the five data sets introduced in the literature by Shah andSkeulic [1] is introduced This can be summarized for the presentinvestigation:

in-Table 2 Comparison of various methods for averaging U , with the numerical values representing (A − Aexact)/Aexact × 100%

∗Based on second-order polynomial; if third-order polynomial is used as a better fit, the error is –0.25%.

heat transfer engineering vol 31 no 13 2010

(a) The closed-form analytical approach considers both the

power-law and polynomial models for the overall heat

trans-fer coefficient as a function of the local hot and cold fluid

temperature difference The famous Colburn linear variation

of the U value is shown as a special case of these models

(b) It is demonstrated that Chebyshev’s numerical integration

scheme that requires four points is accurate (within 1%) of

the exact numerical value

NOMENCLATURE

a constant, coefficient in a polynomial equation

A heat transfer area, m2

¯c average specific heat at constant pressure, J/kg-K

c specific heat at constant pressure, J/kg-K

˙

C fluid capacitance rate, W/K

F dimensionless factor introduced in Eq (19)

h heat transfer coefficient, W/m2-K

LMTD logarithmic mean temperature difference

˙

m mass flow rate, kg/s

n exponent of power-law model

NTU number of transfer units

[1] Shah, R K., and Sekulic, D P., Nonuniform Heat Transfer

Coef-ficients in Conventional Heat Exchanger Design Theory-Revised,

ASME Journal of Heat Transfer, vol 120, pp 520–525, 1998.

[2] Shah, R K., and Sekulic, D P., Fundamentals of Heat Exchanger

Design, John Wiley & Sons, Hoboken, NJ, 2003.

[3] Roetzel, W., Heat Exchanger Design With Variable Transfer

Co-efficients for Crossflow and Mixed Arrangements, International

Journal of Heat and Mass Transfer, vol 17, pp 1037–1049, 1974.

[4] Colburn, A P., Mean Temperature Difference and Heat Transfer

Coefficient in Liquid Heat Exchangers, Industrial & Engineering

Chemistry, vol 25, no 8, pp 873–877, 1933.

[5] Kern, D Q., Process Heat Transfer, McGraw-Hill, New York,

1950

[6] Spalding, D B., Heat Exchanger Theory, in Heat Exchanger

De-sign Handbook Series, Hemisphere, Washington, DC, 1983

[7] Schlunder, E U., Heat Exchanger Design Handbook,

Hemi-sphere, New York, 1989

[8] Peters, D L., Heat Exchanger Design With Transfer Coefficients

Varying With Temperature or Length of Flow Path, Warme- und

Stoffubertragung, vol 3, pp 220–226, 1970.

[9] Hausen, H., Heat Transfer in Counterflow, Parallel Flow, and

Cross Flow, McGraw-Hill, New York, 1983.

[10] Roetzel, W., and Spang, B., Design of Heat Exchangers, tion Ca; Heat Transfer, Section Cb; Computed Heat Transfer Co-

Sec-efficients, Section Cc, in VDI-Warmeatlas, 6th ed., VDI-Verlag

Specifica-[13] Mohiuddin, A K and Kant, K., Knowledge Base for the atic Design of Wet Cooling Towers Part I: Selection and TowerCharacteristics, International Journal of Refrigeration, vol 19, no

System-1, pp 43–5System-1, 1996

[14] Karris, S., Numerical Analysis Using MATLAB and Excel, 3rd ed.,

Orchard Publications, Norwood, MA, 2007

Mostafa H Sharqawy is a postdoctoral associate

in the Mechanical Engineering Department at sachusetts Institute of Technology, United States.

Mas-He earned his Ph.D from King Fahd University of Petroleum and Minerals, Saudi Arabia, in 2008 Cur- rently he is working on seawater desalination using thermal, solar and hybrid systems.

Syed M Zubair is a Distinguished Professor in the

Mechanical Engineering Department at King Fahd University of Petroleum & Minerals (KFUPM) He earned his Ph.D degree from Georgia Institute of Technology, Atlanta, Georgia, USA, in 1985 He has participated in several externally and internally funded research projects at KFUPM, and has pub- lished more than 120 research papers in internation- ally referred journals Due to his various activities in teaching and research, he was recognized with a Dis- tinguished Researcher award by the university in academic years 1993–1994, 1997–1998, and 2005–2006 as well as a Distinguished Teacher award in aca- demic years 1992–1993 and 2002–2003 In addition, he received a Best Applied Research award for “Electrical and Physical Properties of Soils in Saudi Arabia” from the GCC-CIGRE group in 1993.

heat transfer engineering vol 31 no 13 2010

Copyright
C Taylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457631003640321

Temperature and Heat Flux Behavior

of Complex Flows in Car Underhood Compartment

MAHMOUD KHALED1,2, FABIEN HARAMBAT2,

and HASSAN PEERHOSSAINI1

1Thermofluids, Complex Flows and Energy Research Group, Ecole polytechnique de l’universit´e de Nantes, Nantes, France

2PSA Peugeot Citro¨en, V´elizy Villacoublay, France

In this work heat transfer and temperature behavior of complex flows encountered in the vehicle underhood compartment is

experimentally studied and described with simple models Underhood thermal measurements made on a passenger vehicle

in a large-scale wind tunnel are reported here The underhood is instrumented by 80 surface and air thermocouples and 20

fluxmeters Measurements are carried out at three thermal functioning points, in all of which the engine is in operation and

the front wheels are positioned on the test facility with power-absorption-controlled rollers Models are proposed to predict

the maximum temperatures and time constants of the underhood components as functions of the car speed and car engine

power The relative errors of the models are 3.6% and 3.7%, respectively The maximum temperature and the time constant

are crucial in the design and optimization of the underhood aerothermal management system The results obtained in the

present work also provide a large database for validation of numerical codes dealing with underhood cooling management.

INTRODUCTION

In recent years, the automotive market has moved toward

high-performance engines and controlled climate systems At

the same time, geometrical restrictions related to style

con-straints must be satisfied More and more components must

be implemented in the underhood compartment, and the

cool-ing systems are already so large and complex that they can no

longer accommodate additional cooling requirements In

ad-dition, noise reduction measures have increased the insulation

required around the underhood compartment

All these facts constrain airflow passages in the confined

underhood space and thus give rise to complex aerothermal

phenomena consisting essentially of convective heat transfer,

radiative heat transfer, and internal flow topologies The

con-vective heat transfer is essentially achieved by the external air,

which at a given vehicle speed enters the underhood

compart-ment via the air-inlet openings in the car front end and exits via

Address correspondence to Professor Hassan Peerhossaini, Thermofluids,

Complex Flows and Energy research group–LTN, CNRS-UMR 6607, Ecole

polytechnique de l’universit´e de Nantes, rue C.Pauc, BP 50609, 44306 Nantes

cedex 3–France E-mail: hassan.peerhossaini@univ-nantes.fr

its air-outlet openings (which are classically in the wheel archesand enclosing the exhaust tunnel)

Flows in the vehicle underhood are thus complex flows inwhich the fluid (air) faces complex geometries and different hotsurfaces (components) that themselves exchange heat by radi-ation with their environment (Figure 1) Complex heat-transferphenomena (convection and radiation) therefore arise in the con-fined underhood geometry Experimental studies [1–4] of thesephenomena are rare and the literature often refers to numericalsimulations [5–7], which often do not have the accuracy required

to estimate temperatures and heat flux in the underhood more, numerical simulations must be validated experimentallyand necessitate experimental boundary conditions [8–12].The study presented here concerns temperature and heat fluxmeasurements carried out on a vehicle under the constrained-flow conditions of a wind tunnel in order to explore the aerother-mal conditions in the underhood Heat flux measured duringtests is overall (global) heat flux, which corresponds to the sum

Further-of the convective and radiative heat flux, since these two transfer modes always occur simultaneously in the underhoodregion The rest of this article is organized as follows: The exper-imental setup is described, particularly the instrumentation, con-trol parameters, and configuration details, then the experimental

heat-1057

Figure 1 Schematic of complex internal flows in the underhood control

vol-ume.

results are presented and the thermal situation in the underhood

is analyzed, and finally concluding remarks are presented

EXPERIMENTAL SETUP AND METHODS

Experimental setup and methods are described in detail in

[13] Here a brief account is presented

Test Facilities and Experimental Configurations

Measurements were carried out on a passenger car in wind

tunnel S4 of Saint-Cyr, France This is a 1/1-scale wind

tun-nel of 5× 4 m2 cross section and 0.12 blockage ratio During

aerothermal experiments, the car engine is in operation and the

front wheels entrain the test-facility rollers that permit control

and adjustment of the predefined wheel power and rotational

speed In each test, four parameters are required to establish

the thermal functioning point of the vehicle: wheel speed, wind

speed, gear ratio, and engine speed (rpm) Experiments are

car-ried out for the three different thermal functioning points

de-scribed in Table 1, which represent extreme thermal conditions

that a vehicle can encounter in actual use

The car computer is related to an external electronic device

that controls the cooling fan rotation speed The fan has two

rotational speeds The external device forces the fan to rotate

at its high speed during the constant-speed driving phase At

engine extinction (beginning of thermal soak), the electronic

device stops controlling the car calculator and the fan now runs

at its low speed during the minutes before it stops completely

The underhood of the car used in these experiments

is instrumented by surface and air thermocouples of types

T (copper/constantan, for easy implementation) and K

(chromel/alumel, for high-temperature locations) that permit

Table 1 Parameters defining the three thermal functioning points

Experimental Methods

For each experimental run (fixed thermal functioning point),the data (temperature and heat flux) recording covers threesuccessive phases These phases, constant-speed driving, slow-down, and thermal soak, represent the actual driving situationwith which automobiles are confronted

1 The constant-speed driving phase corresponds to driving avehicle at a specified vehicle speed and wind speed, for agiven gear ratio and engine rpm

2 The slowdown phase starts with the stopping of the tunnel airflow, going into the idle driving position This is

wind-an intermediate phase that relates the constwind-ant-speed drivingphase to the thermal soak phase

3 The thermal soak phase starts with engine extinction whenthe wind-speed airflow stops completely This phase corre-sponds to stopping of the vehicle after a high thermal charge.Components in the underhood compartment maintain hightemperatures even though the engine is off In this case, withthe car being stopped, the underhood cooling is assured only

by the air circulation provided by the fan at its low rotationalspeed It rotates for a short period (1 to 3 min) before itstops completely Therefore, in the absence of forced con-vection, natural convection is the only agent of underhoodcooling

Figure 2 provides a descriptive schematic of the different phases

of each test and their starting points

Constant-speed driving phase

Slow-down phase

Thermal soak phase

Slow-down phase

Thermal soak phase

Each experimental run starts by attaining the different

param-eters establishing the desired thermal functioning point (TFP-1,

TFP-2, or TFP-3) Then, once the predefined engine rpm is

reached, we start data acquisition and save the data for the three

successive phases just described The passage from the

constant-speed driving phase to the slowdown is made after temperature

and heat flux stabilization and is assured by the driver, who

re-leases the accelerator and stops the wind The instant of engine

extinction corresponds to the beginning of the thermal soak

phase In recording the data, the beginning of each phase is

clearly marked

RESULTS AND DISCUSSION

Temperature and Heat Flux Tendencies

In the constant-speed driving phase, typical temperature

trends of exponential forms are observed for almost all

com-ponents, air zones, and engine parameters Figure 3 shows some

of these typical exponential tendencies for three cases tested in

constant-speed driving for TFP-3 Each typical curve in Figure 3

(which starts fromT0) is now considered part of a curve having

the same tendency that starts from temperature T0at the initial

time t= 0 [13] Then in the coordinate system (t, T), the

ex-ponential temperature tendency corresponding to the

constant-speed driving phase starts from T0at time t0 By assuming that



(1)

Then, applying the translation in the coordinate system (t, T )

one can write:

Figure 4 Examples of typical heat flux density tendencies in constant-speed driving phase for TFP-3 Data corresponding to the air filter and the CAC inlet duct are nondimensionalized by their infinite heat fluxes and that of the cold box side by its initial heat flux.

On the other hand, in coordinates (t; T):



(4)Variation of the global heat flux densities exchanged at thesurface of different components in constant-speed driving alsoshows typical exponential behavior Figure 4 shows typicalglobal heat flux density variations at the surface of some com-ponents

The typical exponential forms followed by the global heatflux density at different positions in the constant-speed drivingphase are given by the following two equations:

400 exp 1 2 38 3

t T

=

400 exp 1 2 38 3

t T

t

ϕ

Figure 5 Comparisons of theoretical results with experimental data for

tem-perature and global heat flux time variation at the CAC inlet duct in TFP-3.

the relative error between experimental and theoretical values

are observed for all the experiments and all thermal

function-ing points, confirmfunction-ing the typical exponential tendencies in the

constant-speed driving phase for the temperature and the heat

flux

From a thermal point of view, components in the underhood

fall into two categories:

• High-temperature components, for which the temperature

varies between 600◦C and 900◦C Among these components

are the exhaust manifold, the elbows for the exhaust

manifold-turbine connections, the manifold-turbine, and the pre-catalysor;

• Low-temperature components, for which the temperature

varies between 30 and 300◦C

Among these components are the alternator, the cold box (zone

covering the computer and the battery), water outlet plenum,

cylinder head cover, electronic-assisted steering, starter,

ther-mal screens, inlet air distributor, steering junction, apron, and

triangle

Figures 6a and b show temperature variation in the

constant-speed driving mode for high-temperature and low-temperature

components, respectively As an example, at the end of the

constant-speed driving phase, the temperature is 830◦C at the

(a)

(b)

20 40 60 80 100 120 140 160 180 200 220

250 350 450 550 650 750 850

(a)

(b)

20 40 60 80 100 120 140 160 180 200 220

250 350 450 550 650 750 850

Time (s)

Exhaust manifoldConnection elbowTurbine

Figure 6 Temperature variation in constant-speed driving phase for TFP-2: (a) high-temperature components and (b) low-temperature components.

exhaust manifold and turbine and 615◦C at the elbows for theexhaust manifold–turbine connections For the low-temperaturecomponents, the temperature is 99◦C at the apron, 195◦C at theturbine screen, and 135◦C at the compressor oil duct

Engine parameters (control fluid temperatures) are also oftwo types:

• Hot underhood fluids, for which the temperature varies from

800◦C to 950◦C These engine parameters are the gas peratures at the turbo-compressor inlet and outlet and at thecatalysor inlet and outlet

tem-• Cool underhood fluids, for which the temperature varies from

50◦C to 200◦C Among these engine parameters are the engineoil temperature, water temperatures at the radiator inlet andoutlet, air temperature at the filter inlet, air temperatures atthe compressor inlet and outlet, air temperatures at the chargeair cooler inlet and outlet, and air temperature at the engineentry

Figures 7a and b show temperature variations in constantspeed-driving for high-temperature and low-temperature fluids,respectively At the end of the constant-speed driving phase, theorders of magnitude of the temperatures of the high-temperaturefluids are about 920◦C for the turbine inlet gas, 870◦C at theturbine outlet gas, and 845◦C at the catalysor outlet gas For theheat transfer engineering vol 31 no 13 2010

Figure 7 Temperature variation in constant-speed driving phase for TFP-2:

(a) high-temperature fluids and (b) low-temperature fluids.

low-temperature fluids, the air temperature at the compressor

outlet is 156◦C, the water temperature at the radiator outlet is

90◦, and the air temperature at the CAC outlet is 64◦C

For the different thermal functioning points and air zones

tested, the air temperature does not exceed 150◦C Figure 8

shows temperature variation in constant-speed driving at three

tested air zones

From the point of view of global heat flux variation at the

component surfaces, underhood components fall into two

Figure 8 Temperature variations at three air zones in constant-speed driving

2 Components for which the overall heat flux variation followsthe general form of Eq (6) These components absorb orrelease an overall heat flux, decreasing in absolute value withtime In other terms, the temperatures of these componentsincrease more rapidly than their thermal environment whenthey absorb heat, and their temperatures increase more slowlythan their thermal environment when they release heat.Tables 2 and 3 show respectively typical temperature andheat flux values for some underhood components, engine pa-rameters, and air zones (of different component categories justdescribed) for the three thermal functioning points From thesetables it is observed that the temperatures of engine parame-ters stabilize more rapidly than those of the components and airzones For example, for the positions shown in Table 2, the meantime constant for engine parameters is 42 s versus 132 s for thecomponents and 205 s for the air zones It is also observed thattemperature stabilization of components and engine parameters

at high temperatures is much faster than that of componentsand engine parameters at low temperatures For example, at thepositions shown in Table 2, the mean time constants for high-temperature components and engine parameters are respectively

44 s and 18 s, while for the low-temperature components andengine parameters mean time constants are 213 s and 65s Theoverall heat flux stabilization (Table 2) is also faster than tem-perature stabilization: At the CAC inlet duct, the mean timeconstant for temperature stabilization is about 360 s versus 50 sfor the overall heat flux stabilization

In general, time constants of both temperature and heat fluxstabilization vary over the three thermal functioning points Inother terms, time constants depend on the engine power and theupstream airflow velocity

Temperature–Heat Flux Coupling

This section examines the relation between overall heat fluxdensities and surface temperatures In the constant-speed driv-ing phase and for the three thermal functioning points, the heatabsorbed at some component surfaces is monitored Absorbedoverall heat fluxes are essentially correlated with absorbed con-vective heat fluxes when they absorb or release heat by radiation.For these cases, two types of variations can be distinguished:increasing convective heat flux, which imposes increasing over-all heat flux, and decreasing convective flux, which imposesdecreasing overall heat flux A typical example of the first type

is the air filter Figure 9 shows heat flux (overall, convective,and radiative) and temperature variations at the air filter surfaceheat transfer engineering vol 31 no 13 2010