Experimental Thermal and Fluid Science 45 (2013) 1–7

Contents lists available at SciVerse ScienceDirect

Experimental Thermal and Fluid Science

journal homepage: www.elsevier .com/locate /et fs

Convective heat transfer of a liquid dispersion system flowing in a pipe

Jerzy Hapanowicz, Patrycja Polaczek ⇑Opole University of Technology, Department of Chemical and Process Engineering, Opole, Mikołajczyka 5, Poland

a r t i c l e i n f o

Article history:Received 29 November 2011Received in revised form 8 November 2012Accepted 10 November 2012Available online 29 November 2012

Keywords:Heat transferTwo-phase flowFlow pattern

0894-1777/$ - see front matter � 2012 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.expthermflusci.2012.11.00

⇑ Corresponding author. Tel.: +48 77 40 06 326; faxE-mail address: p.polaczek@doktorant.po.edu.pl (P

a b s t r a c t

This report relates to the descriptive methods which are applicable to the heat transfer process betweenthe pipe wall and the unstable two-phase dispersion system which flows in the pipe, with the dispersionsystem formed by two immiscible liquids. The specific character of the flow was presented for a liquid–liquid system against the flow of other substances, and the resulting problems in the description of thethermal processes which take place in that system were addressed. Adaptations of the equations pro-vided in literature for convective heat transfer were suggested so that they could be used for liquid dis-persions. Based on the results of authors’ experiments, trueness of the presented relations was verified,and probable reasons were suggested for the discrepancies in relation to the measured test values. It wasconcluded that there is no method available now which could be used to describe accurately the convec-tive heat transfer for the flow of liquid dispersions in a horizontal pipe. Some of the suggested calcula-tions make it possible to provide estimate calculations only.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Heat transfer in the medium created by two liquid phases whichare insoluble in each other is practised commercially in some petro-chemical and carbo-chemical processes, in extraction processes, orin thermal concentration of substances. Those processes are usuallycarried out in membrane-type process equipment: in continuousflow tubular reactors with heating or cooling systems, or in shell-and-tube recuperative heat exchangers. Heat transfer betweenthe pipe wall and the liquid–liquid system which flows inside thepipe makes a typical example of the convective heat transfer phe-nomenon. However, it occurs under two-phase flow conditionssince no stable emulsions but rather unstable liquid dispersionsare formed by the liquid–liquid systems in the process equipment.

Which is specific for a liquid dispersion is that the state ofaggregation is the same for its components. The components, how-ever, have different chemical compositions and physical proper-ties, and they form separate phases. Each component mayperform the function of the dispersion medium under flow condi-tions since each of them is a fluid. In a gas–liquid system, consid-erable gas flow velocity is required for liquid-in-gas dispergation.In a liquid–liquid system, however, the flow rates of both phasesare not so critical; it is rather fractions of those phases to be deci-sive for the question which phase would form the continuous one.The type of the liquid–liquid dispersion which flows in a pipe indi-cates whether it was the oil component to be dispersed in water

ll rights reserved.7

: +48 77 40 06 381.. Polaczek).

(O/W system) or it was water to be dispersed in oil (W/O system).That issue is essential for the description of the convective heattransfer process since the pipe wall will contact the continuousphase to a greater degree than the dispersed phase. Hence, thermalresistance in the heat transfer between the wall and the liquid dis-persion should be controlled predominantly by the properties ofthe continuous phase. It is that phase to take majority of heat awayfrom the wall, and heat would then have to penetrate through thatphase to the core and to the dispersed phase elements.

As compared to the flow of emulsion, there is a specific featureof a dispersion – the shapes of the dispersed elements are not sta-ble under flow conditions and their sizes change over time. Thoseparameters are dependent on the flow conditions and they definethe type of the two-phase flow pattern in a pipe. Formation of flowpatterns can also be observed in other two-phase systems, e.g. gas–liquid. Yet, the patterns are specific in liquid–liquid systems whichresults from numerous possible relations between physical proper-ties of the component liquids.

The work on identification, nomenclature and the number ofavailable liquid–liquid two-phase flow patterns gave stronglydiversified results. A possible option for general systematics withinflow patterns, their photographs and detailed description was sug-gested by [1]. That systematic approach was based on the litera-ture review [2–6] and the authors’ research, and it comprises fivespecific groups of flow patterns for the liquid–liquid two-phaseflow in a horizontal pipe:

� drops (Dr) – dispersed phase forms spherical or ellipsoidal ele-ments. Those structures may have different sizes, but they donot exceed the pipe diameter;

Nomenclature

c specific heat, J/(kg K)d diameter of pipe, mg mass velocity, kg/(m2 s)x mass frictionA cross-sectional area, m2

F heat transfer surface, m2

G mass flow rate, kg/m3

L length of pipe, mNu Nusselt number (=ad/k)Pe Peclet number (=RePr)Pr Prandtl number, (=cg/k)Q heat flux, WR real friction phase in flowing system (void friction)Re Reynolds number (=tdq/g)T temperature, KV volumetric flow rate, m3/sa heat transfer coefficient, W/(m2 K)d statistical parameter, %g dynamic viscosity, Pa st average velocity, m/sk thermal conductivity, W/(m K)q density, kg/m3

U volume friction

Subscriptscal calculated valuecp continuous phasedp dispersed phaseexp experimental valueol oil phasem liquid–liquid mixtures superficial velocityw oil phasewall at wall temperatureTP two-phase system

Abbreviations for flow patternsAD annulus and dispersionD dispersionDr dropsDrP drops and plugsS stratification–W of water–O of oil

2 J. Hapanowicz, P. Polaczek / Experimental Thermal and Fluid Science 45 (2013) 1–7

� drops-plugs (DrP) – dispersed phase forms drops and plugs, andthe length of plugs is much higher than the pipe diameter;� dispersion (D) – dispersed phase forms small elements which

are carried by the continuous phase and which are uniformlyarranged over the pipe cross-section

Each of the above group of flow patterns may be applicable totwo-phase W/O and O/W systems. That makes an essential featureof liquid–liquid systems, as for example no liquid plugs will beformed in gas in case gas–liquid systems.

The systematic approach in question covers also W + O flow pat-terns. In that case, both oil and aqueous phases are continuous atthe same time, which in particular may be represented by:

� stratification (S) – both flowing liquids form separate layers. Thethickness of a layer changes to follow the fraction of that phasein a system. The liquids may sometimes undergo dispergationin each other at the interphase region;� annular-dispersion (AD) – a part of one liquid flows in a layer

along the pipe wall surface. The balance of that liquid formssmall drops which are carried closer to the pipe centre line bythe other liquid.

As compared to a gas–liquid system, the location of phasesagainst each other in a W + O liquid–liquid system may be morediversified. Depending on the relation of liquid density values,the upper layer in the (S) pattern may be formed by oil or by water.As regards the (AD) flow pattern, each liquid may flow as a layeralong the pipe wall, which is impossible for gas–liquid systems.

As can be concluded from that description of two-phase envi-ronments, the convective heat transfer mode for liquid dispersionsmay be different than for the liquid single-phase flow or for thegas–liquid two-phase flow. That is important inasmuch as thosetwo cases have so far been best researched and described. Hence,we may ask a question: To what extent the existing convectiveheat transfer description methods are true for the flow conditionsof unstable dispersions?

There are no reports available in the professional literaturewhich would present the accurate calculation method(s) for theconvective heat transfer coefficient, applicable for a liquid–liquid

two-phase system flowing in a pipe. And what is more, the calcu-lation models for the convective heat transfer process do not makeallowance in many instances for the complexity of the phenomenawhich may take place in liquid dispersions. This report attempts tocover the liquid–liquid systems by a few typical description meth-ods which are applicable to the convective heat transfer processtaking place when a liquid substance flows in a horizontal pipe.The results of authors’ experiments were used to verify thoseconsiderations.

2. Description methods for convective heat transfer conditions

As is known from the theory of heat transfer, conditions of thatprocess (represented by the Nusselt number) are controlled by theturbulent flow of a liquid (represented by the Reynolds number),by thermal properties of that liquid (represented by the Prandtlnumber), and by the relative length of the pipe along which thetemperature profile is formed in the liquid. Thus, a general relationfor the convective heat transfer conditions will have the form:

Nu ¼ f ðRe;Pr;d=LÞ ð1Þ

The Hausen equation is one of the basic equations applicable forthe laminar flow regime in a pipe. [7] made use of it for liquid–liquid systems. For the liquid–liquid two-phase system, it can bepresented as:

NuTP ¼ 3:66þ 0:0668PeTPd=L

1þ 0:04ðPeTPd=LÞ2=3 : ð2Þ

The modules in Eq. (2) may be defined as:

NuTP ¼aTPdkTP

; ð3Þ

PeTP ¼ ReTPPrTP ¼gTPdcTP

kTP: ð4Þ

The basic problem in application of Eq. (2) results from the needof defining equivalent properties of a liquid–liquid system. Whenthe system is assumed to be homogeneous, its properties are de-fined by the fractions of its components:

J. Hapanowicz, P. Polaczek / Experimental Thermal and Fluid Science 45 (2013) 1–7 3

xcp ¼Gcp

GTP¼ 1� xdp ð5aÞ

or

Ucp ¼Vcp

VTP¼

xcpqdp

xcpðqdp � qcpÞ þ qcp¼ 1�Udp: ð5bÞ

Having considered Eq. (5), one may put down as follows:

qTP ¼ ðxcp=qcp þ xdp=qdpÞ�1; ð6Þ

cTP ¼ xcpccp þ xdpcdp; ð7Þ

kTP ¼ xcpkcp þ xdpkdp: ð8Þ

The practical use of Eq. (2) is facilitated by the notation:

gTP ¼ tTPqTP ¼GTP

A¼ 4ðGcp þ GdpÞ

pd2 ð9Þ

and

PeTPd=L ¼ 4ðGcp þ GdpÞcTP

pdkTPL: ð10Þ

The use of Eq. (2) is limited since it is applicable for laminar flowconditions only. One should remember, however, that the viscosityof a liquid dispersion is high and it may be higher in many casesthan that of the continuous phase [1]. Because of that high viscos-ity, the flow velocity of a liquid–liquid system is not expected to behigher than 0.5 m/s in practice. The liquid dispersions are heatedup or cooled down in the process equipment where the pipe diam-eters are not very big. With all those conditions put together, itturns out that the laminar flow regime for a liquid–liquid systemmakes rather a typical case than a non-typical one.

Unfortunately, Eq. (2) gives no consideration to the viscosityspecification of the substance which is flowing in a pipe, which isshown in Eq. (4), whereas viscosity of a liquid dispersion is not aconstant parameter and it is dependent on the flow conditions[1]. Hence, it would be more advantageous to use an equation inwhich the viscosity parameter has been involved. Since the viscos-ity of a liquid dispersion is considerably high, one may additionallyexpect that it is essential to show the difference in the viscosity ofthe liquid core and that of the liquid boundary layer in the descrip-tion of the convective heat transfer conditions. That possibility isoffered by the Sieder-Tate correlations which are commonlyknown and referred to, e.g. by [8]. Each of those equations is validfor a different range of liquid flow conditions. The practical appli-cability of the heat transfer description method is extended in thisway to cover various flow conditions for liquid dispersions. Usingtwo phase subscripts, the Sieder-Tate equation can be presentedas:

� for laminar flow regime:

NuTP ¼ 1:86g

gwall

� �0:14

TP

�ReTPPrTP

dL

�1=3

; ð11aÞ

� for non-laminar flow regime:

NuTP ¼ 0:027g

gwall

� �0:14

TP

Re0:8TP Pr0:33

TP : ð11bÞ

The dimensionless numbers are defined as:

PrTP ¼cTPgTP

kTP; ð12Þ

ReTP ¼gTPdgTP¼ 4ðGcp þ GdpÞ

pdgTP: ð13Þ

ReTP < 2100 should be taken as the boundary condition for thelaminar flow. A problem in practical use of Eqs. (11a) and(11b)may come from the need to find the viscosity for a liquid–liquidsystem. That system is not a liquid solution but a mixture of twophases. Hence, its equivalent viscosity may not be calculated inaccordance with the additivity rule. It is correct, however, to makeuse of Eqs. (6)–(8) to establish other properties of a system. A num-ber of equations can be found in the professional literature for theviscosity of a liquid–liquid system. They were reviewed and ana-lysed by [1], with the conclusion that the viscosity of an unstableliquid dispersion, flowing in a pipe, is controlled by the flow condi-tions. To be more precise, it is defined by the amount of energywhich is needed to maintain a given phase dispergation level. Thatlevel is connected with the mixing rate under flow conditions, andthe amount of energy which is needed to force the flow is repre-sented by the pressure drop. For the purpose of this paper, how-ever, i.e. for the assessment of various methods employed topresent the convective heat transfer conditions, the viscosity of aliquid–liquid system may be found by means of the known Taylorequation, cited by [9]:

gTP

gcp¼ 1þ 5jþ 2

2jþ 2; j ¼

gdp

gcp: ð14Þ

Unfortunately, the two description methods for convective heattransfer conditions as suggested above, applicable to liquid disper-sion systems, do not take a phenomenon into consideration whichis essential for two-phase flows: the type of two-phase flow pat-tern which is formed under given flow conditions. There are noequations suggested in reports for convective heat transfer underliquid–liquid two-phase flow conditions, which would makeallowance for a specific flow pattern or at least a group of flow pat-terns. The equations for the gas–liquid two-phase flow are avail-able only. It should also be pointed out that some of thoseequations are for the vapour–liquid systems and they are true forthe boiling condition of the flowing liquid, hence they describethe process which is somewhat different.

The review and analysis of the equations which describe theconvective heat transfer under the gas–liquid two-phase condi-tions were presented by [10]. Some of them could be selectedwhich might be effective for convective heat transfer in the li-quid–liquid systems. Three criteria were taken into account: hori-zontal pipe arrangement, flow pattern which is close to that for aliquid–liquid system, and high viscosity of the liquid phase.

The equations which describe the convective heat transferunder gas–liquid two-phase conditions are based in most caseson the two-phase separated model. Each phase is assumed toflow independently along the pipe, and it controls only those phe-nomena which take place in the pipe volume actually occupied bythat phase. With that assumption, the convective heat transfercoefficient for the two-phase flow is referred to that for onephase. That phase is a liquid phase in a gas–liquid system, andit is understood to make the continuous phase at the same time.That approach is usually true for a gas–liquid system, except forfew cases like flow of dispersed liquid drops in a gas stream, orflow of mist.

Ref. [10] collected a few correlations in their report, one of thembeing the Shah equation. It was developed on the basis of the hor-izontal pipe tests. The tests covered the two-phase flow: air + a fewhigh-viscosity liquids, and the arrangement of phases in the pipewas much similar to flow patterns of the liquid–liquid two-phaseflow. After the subscripts are changed, the Shah model in questionmay be presented as below:

aTP

acp¼ 1þ tdp;s

tcp;s

� �0:25

: ð15Þ

4 J. Hapanowicz, P. Polaczek / Experimental Thermal and Fluid Science 45 (2013) 1–7

The structure of Eq. (15) does not impose the types of the liq-uids, i.e. of continuous and dispersed phases. The roles of liquidsare dependent on the type of the liquid dispersion which flowsin the pipe and exchanges heat with its wall. In accordance withEq. (15), the convective heat transfer under liquid–liquid two-phase conditions is referred to heat transfer conditions in the con-tinuous phase. That assumption is fairly true as it is predominantlythe continuous phase to contact the pipe wall. The effect of the dis-persed phase is allowed for by means of the correction for the ratioof superficial velocities of both phases, to be calculated as:

tdp;s ¼4Vdp

pd2 ; tcp;s ¼4Vcp

pd2 : ð16Þ

When Eqs. (16), (5a), and (5b) are used in (15), the model lookslike that:

aTP

acp¼ 1þ xdp

1� xcp

qcp

qdp

!0;25

; ð17aÞ

aTP

acp¼ 1þ Udp

1�Ucp

� �0;25

: ð17bÞ

Pursuant to the Shah’s suggestion, convective heat transfer inthe continuous phase may be described by the followingequations:

� for laminar flow regime:

Nucp ¼acpkcp

d¼ 1:86

ggwall

� �0:14

cp

Recp;sPrcpdL

� �1=3

ð18aÞ

� for non-laminar flow regime:

Nucp ¼acpkcp

d¼ 0:023

ggwall

� �0:14

cp

Re0:8cp;sPr0:4

cp ð18bÞ

The character of the flow is represented by the Reynoldsnumber:

Recp;s ¼tcp;sdqcp

gcp¼ 4Gcp

pdgcp: ð19Þ

If Recp,s < 2100, the flow is laminar.Hence, when a model for a separated two-phase system is uti-

lised, there is no need to calculate the equivalent viscosity value,which is an advantage beyond any doubt.

The two-phase flow is typically accompanied by the interfacialslip. It is shown by the difference in actual phase velocities whichmakes it difficult to find the actual turbulence level for the contin-uous phase. In turn, the turbulent conditions are closely related tothe convective heat transfer process within a liquid.

The slip is responsible for the apparent volume fractions ofphases as found at the pipe inlet from Eq. (5b), which are differentfrom the actual volumes of phases charged to the pipe. The actualpipe filling level is called, void fraction’’ in a gas–liquid system,which is not applicable to a liquid–liquid system since there areno void gas spaces in the continuous phase in that case. For thatreason, the term of ‘‘actual fraction’’ of a given phase will be usedin the further part of this report. That fraction should be under-stood as:

Rcp ¼Acp

A¼ Acp

Acp þ Adp¼ 1� Rdp ð20Þ

As R is significant for the description of hydrodynamics oftwo-phase flows, one may assume that its value should also beincorporated into the description of the convective heat transferconditions. That is possible owing to the Oliver–Wright

equation; its form for a gas–liquid system was presented anddiscussed by [10]. Its modified form, which makes allowancefor the type of a two-phase system, i.e. for a liquid dispersion,is as below:

NuTP ¼ Nucp1:2

R0:36cp

� 0:2Rcp

!: ð21Þ

The convective heat transfer process in the continuous phase isdescribed in that case by the following equation:

Nucp ¼ 1:615g

gwall

� �0:14

cp

Re�cpPrcpdL

� �1=3

: ð22Þ

The Reynolds number value for the continuous phase should becalculated at the assumption that the phase velocity results fromthe volume velocity of both phase streams, i.e.:

Re�cp ¼ðtcp;s þ tdp;sÞdqcp

gcp: ð23Þ

This approach to that velocity gives a value which is closer toactual conditions than the value obtained from Eq. (16).

Another problem in practical use of Eq. (20) is the calculation ofthe actual fraction of the continuous phase in the liquid dispersionwhich flows along a horizontal pipe. The method should not beemployed here which shows the value of Re for a gas–liquid two-phase system. That limitation comes directly from the relations be-tween physical properties of gas and liquid – those relations aredifferent for the properties of oil and water.

The calculation method for the actual phase fractions in a liquiddispersion flowing along a horizontal pipe was given by [11]. Thatmethod is based on a single equation which is applicable both toW/O and O/W systems. Its form is as follows:

Rcp ¼tcp;s

1:004ðtcp;s þ tdp;sÞ þ 0:0248ð24Þ

The four suggested calculation methods for convective heattransfer conditions, as presented above, are different first of all intheir approach to the phenomena which should affect the heattransfer between the pipe wall and a two-phase substance whichflows in the pipe. Eq. (2) has the simplest structure and theassumption is required that the liquid dispersion is homogeneous,while its temperature distribution profile is variable, and its viscos-ity has no effect on the convective heat transfer. The viscosityeffects were considered in Eqs. (11a) and (11b). Additionally, eachof those equations is applicable to a different type of the liquid dis-persion, which is still understood to make a homogeneous system.No homogeneity is one of the basic assumptions for Eq. (15). Itsconstruction complies with the assumptions of the model for theseparated system. In accordance with Eq. (15), convective heattransfer conditions in the continuous phase are of special impor-tance for the value of aTP, and the description of those conditionsis conditional upon the nature of the flow of that phase. Eq. (20)is also based on the model for a separated system, and additionallythe actual phase fractions were considered. The actual character ofthe flow of the continuous phase is better reflected in the methodadopted for the calculation of Recp here.

3. Experimental investigations

Experiments were needed to verify the calculation models aspresented above. Research on convective heat transfer under flowconditions of liquid dispersions in a horizontal pipe was conductedat the test stand in the laboratory of the Department of ProcessEngineering (Technical University of Opole). Research includedthe use of the equation:

J. Hapanowicz, P. Polaczek / Experimental Thermal and Fluid Science 45 (2013) 1–7 5

aTP ¼Q TP

FDTm¼ Q TP

FðTwall � TTPÞmð25Þ

Heat transfer flux based on the heat balance of the liquid- liquidmixture at the inlet and outlet pipe. Heat transfer surface area cor-responded to an electrically heated pipe with a length of 1 m andinternal diameter of 10 mm. The average temperature differencewas determined by measuring the temperature of the pipe walland flowing in the liquid–liquid mixture. The detailed descriptionof that stand, the merits of the test measurements and interpreta-tion of the results, as well as the assessment the their accuracy,were described in more details by [12,13].

Water and machine oil were used in the tests. The flow rateswere adjusted to produce both W/O and O/W dispersions in the testpipe. Attempts were made at the same time to obtain homoge-neous two-phase systems as far as possible, which was a pre-con-dition for some of the equations to be verified. Uniform systemswere provided by producing dispersion and drop flow patterns ofwater and oil in the pipe. Positions of experimental points wereshown in Fig. 1 which was plotted with the use of the map of flowpatterns for liquid–liquid two-phase flows as per [14]. Symbols –Wand –O in the nomenclature of structures represent respectivelythe water and oil.

The ranges of velocities for both phases and their volume frac-tions were presented in Table 1. Table 1 also contains the Reynoldsnumber values, as calculated from (13) and (22), and the Prandtlnumber values as per (12).

The values of U and g were specified in Table 1 for the pipe inletconditions, while Re and Pr are meant for the average temperaturein the heated liquid–liquid system, between the pipe inlet and pipeoutlet. The convective heat transfer from the pipe wall to the liquiddispersion was studied in the test pipe with the internal diameterof 10 mm and L/d = 100.

A set of a few tens of experimental findings was initiallyobtained. The items were removed form that set for which the

Fig. 1. Position of experimental points on background of map flow patterns forliquid–liquid system.

Table 1Parameters of tested liquid–liquid systems.

W/O systems O/W systems

Udp = 0.072–0.122 Udp = 0.162–0.226gw,s (kg/(m2 s)) 44–51 gw,s (kg/(m2 s)) 311–472gol,s (kg/(m2 s)) 340–537 gol,s (kg/(m2 s)) 67–78ReTP 98–175 ReTP 2540–4150PrTP 329–402 PrTP 9.6–11.4Re�cp 108–187 Re�cp 4084–5927

uncertainty error in the value of aTP exceeded 10%. The details ofthe uncertainty assessment, are given in [12]. The reliable resultsformed the authors’ internal database. It covered 26 cases of con-vective heat transfer conditions for W/O liquid dispersions flowingin a pipe, and 14 cases for O/W liquid dispersions. The informationon physical properties of phases at their average temperatures wasadded for each series of data in the base, so was the information onthe flow pattern which was formed in the pipe under given flowconditions. The type of flow pattern was established in accordancewith the procedure which had been described by [1].

4. Equations – assessment of correctness

The database information was used to verify correctness of thediscussed calculation models. Compatibility of the measured val-ues of the convective heat transfer coefficient and those calculatedwas preliminarily analysed on the basis of two simple statisticalparameters:

� mean error of calculated values against the measured ones.

TablValuliqui

E

(2(1(1(2

da ¼1n

Xn

i¼1

aexp;i � acal;i

aexp;i100%; ð26Þ

� mean dispersion of calculated values against the measuredones.

d�a ¼1n

Xn

i¼1

jaexp;i � acal;ijaexp;i

100%: ð27Þ

Table 2 presents the values of those parameters; they werecalculated separately for each equation, and separately for W/Oand O/W liquid dispersions.

The tabulated data show that the mean dispersion of the valuesof aTP as calculated for W/O systems with the use of those equa-tions is considerable. However, similar values of that dispersionmean that the correctness of those equations is comparable.

Hence, the assumptions made for adaptation of those equationsto a liquid–liquid system turned out to yield similar results of cal-culations. Accuracy of the relatively simple Eq. (2) was a bit betterfor W/O systems than that of Eq. (20) which is much more complexand in which allowance is made for inhomogeneity of dispersions.The value of error da is close to zero for Eqs. (11a) and (11b) whichmeans that the overrating and underrating levels for the value ofaTP,al were nearly equal for that equations. Applicability of individ-ual equations to liquid W/O dispersions can be compared betteragainst each other in Fig. 2.

As can be seen, the calculated values of aTP,cal do not changevery much for any equation used, while the measured valueschange considerably. Thus, the convective heat transfer conditionsin W/O dispersions are controlled by the parameters which werenot included into individual equations, or their incorporation isnot satisfactory.

It was revealed from a more precise analysis of the experimen-tal data that the values of aTP,cal were overrated for W/O dispersionsat higher dispersion velocities. Under experimental conditions

e 2es of da and d�a calculated from different equations and for different types ofd dispersions.

quation W/O system O/W system

da (%) d�a (%) da,(%) d�a (%)

) – for W/O and O/W 31.7 ±34.9 63.5 ±63.51a) – for W/O (11b) – for O/W 0.6 ±37.6 �17.4 ±24.55) – for W/O and O/W 25.1 ±39.1 �48.3 ±48.30) – for W/O and O/W 5.1 ±36.1 61.2 ±61.2

Fig. 2. Measured values of a versus calculated ones for W/O liquid dispersions.Fig. 4. Measured values of a versus calculated ones from Eq. (11b) for O/W liquiddispersions.

6 J. Hapanowicz, P. Polaczek / Experimental Thermal and Fluid Science 45 (2013) 1–7

(Vw = const.), that was equivalent to the situation in which the oilvolume fraction was high and the water volume fraction waslow. Fig. 1 shows that the flow pattern could be defined as (D–W) then.

Within the drop flow patterns (Dr–W), and in particular for thelowest oil volume fractions, the calculated values of aTP wereclearly underrated. It appears at the present stage of our researchthat the differences as mentioned above result from the changesin dispergation of water drops in the oil phase, i.e. from thechanges in water/wall interactions.

When the equations were reviewed for correctness, the findingswere not so unambiguous for O/W liquid dispersions as they werefor W/O dispersions. Table 2 demonstrates that the Eqs. (2) and(20) gave the values of aTP,cal which were far away from the mea-sured values. Moreover, all the calculated values were too lowfor both those equations. On the other hand, all the calculation re-sults of Eq. (15) are overpredicted. It is worth emphasising thatvarious attempts to describe the same phenomenon may go toopposite extremes sometimes. That is well illustrated in Fig. 3.The errors from Eqs. (2) and (20) are considerable and that is surelycaused by the fact that the convective heat transfer calculationmethod adopted in those equations gives absolutely no consider-ation to the liquid dispersion flow conditions. Table 1 shows thatthe flow was not laminar for O/W systems, while the form of Eq.

Fig. 3. Measured values of a versus calculated ones for O/W liquid dispersions.

(2) results from the assumption of a laminar flow regime, whichis correct and theoretically justifiable.

There are no limitations, either, for Eq. (20) as regards the flowregime of a liquid dispersion, and strictly speaking its continuousphase. The effect is analogous to those of Eq. (2), i.e. that equationis also incorrect.

In turn, the reason for overrated results from Eq. (15) can beperceived in the way of defining the right-hand side of that equa-tion. In its original form, Eq. (15) is applicable to the flow of a gas–liquid system and its right-hand side represents unequivocally theflow rate of gas in relation to the flow rate of liquid. However, theratio of phase flow velocities is of completely different order inliquid dispersions than in gas–liquid systems. Both phases are liq-uids anyway, and each phase can additionally make the continuousphase. Hence, the specific form of Eq. (15) should be different for aliquid dispersion.

Eq. (11b) turned out most effective for O/W dispersions. It holdsgood in the non-laminar flow regime of the total dispersion withthe conventional equivalent properties. The mean relative errorfor Eq. (11b) and for the O/W system, as specified in Table 2, wasnot small. However, when the experimental data were subjectedto closer examination, that error was found to be much lower forsome measurement points. That was illustrated in Fig. 4, wheresuccessive measurement points were shown together with theapplicable symbols for the flow patterns which were specific atthose points.

Fig. 4 demonstrates clearly that Eq. (11b) is less successful forthe points representing the flow of tiny oil elements in water (D–O). In the case of drop patterns (Dr–O), the measured values stayfairly well in line with the calculated ones. The statistical parame-ters, calculated from Eqs. (25) and (26) for the points within (Dr–O)flow patterns only, are as follows: da = 4.7% and d�a = 6.5%. These issatisfactory values, especially when the dynamic and in manycases stochastic course is considered for the processes whichaccompany all two-phase flows.

5. Conclusions

The review of literature reports shows that the descriptionmethods for the convective heat transfer under flow conditionsof a two-phase liquid dispersion system is unsatisfactory at pres-ent; one may even declare that just no good method is availablenow. When attempting to develop such a method, one has toremember the specific character of his liquid–liquid system, both

J. Hapanowicz, P. Polaczek / Experimental Thermal and Fluid Science 45 (2013) 1–7 7

against single-phase liquids and against other two-phase systems.The course of convective heat transfer under two-phase flow con-ditions is controlled by hydro-dynamic processes which take placein a pipe. The most critical of those factors are: changing degree ofphase dispergation, phase slip and actual phase fractions in theflowing liquid, and type flow pattern. One may thus declare thatthe convective heat transfer description methods which are appli-cable to single-phase liquids or gas–liquid two-phase systemsshould not be simply transferred to liquid–liquid systems. Theydo not address sufficiently the complex phenomena which takeplace under flow conditions of a liquid dispersion in a pipe.

Experimental verification of the values of aTP as calculated fromthe equations suggested in this report supports the statement thatthose equations should not be employed in situations when highaccuracy is required. Moreover, some of them should be avoidedeven for assessments, and especially for flowing O/W liquiddispersions.

As regards description of convective heat transfer conditions inW/O dispersions, all those equations turned out comparably accu-rate. For the needs of assessment calculations, it is not very impor-tant which equation to use. However, one has to go through anumber of intermediate calculations in all of them. From that pointof view, the easiest method is to use Eq. (2) which is pretty simple.

Only Eq. (11b) turned out to perform sufficiently in the case ofO/W liquid dispersions. One has to keep in mind, however, that thecalculation results obtained from this equation are correct only un-der oil drop flow pattern (Dr–O).

It is necessary to have accurate calculation equations at one’sdisposal for convective heat transfer conditions which are applica-ble to the flow of liquid dispersions in pipes. Such equations areindispensable to provide correct design engineering for some itemsof process equipment. But one would have to collect a much more

extensive set to diversified experimental data than is available nowwithin that field in order to develop the applicable equations. Onlywith the use of such comprehensive data it will be possible to findefficient empirical relations. Alternatively, the data may be em-ployed to verify the calculation models which have been developedwith the use of numerical simulation techniques of heat exchangeprocesses.

Acknowledgement

This study was made within the research project No 3611/B/T02/2008/34 financed from the funds intended to be spent on sci-ence in 2008-2011.

References

[1] J. Hapanowicz, The flow of liquid dispersion systems, Studia i monografie z.204, Opole University of Technology, 2007.

[2] Arirachakaran S., et al., An analysis of oil/water flow phenomena in horizontalpipe. in: SPE Paper 18836, SPE Prof. Prod. Operating Symp. 1989.

[3] M.E. Charles, G.W. Govier, G.W. Hodgson, Can. J. Chem. Eng. (2) (1961) 27.[4] T. Fujii et al., The flow characteristics of a horizontal immiscible equal-density-

liquids two phase folw, Int. Conf. Multiphase Flows (1991) 195.[5] J. Hapanowicz, L. Troniewski, Chem. Proc. Eng. (18) (1997) 575.[6] M. Nadler, D. Mewes, Int. J. Multiphase Flow 23 (1) (1997) 55.[7] S. Gat, N. Brauner, A. Ullman, Int, J. Heat Mass Transfer 52 (2009) 1385–1399.[8] T. Hobler, Movement and Heat Exchangers, WNT Warszawa, 1986.[9] R. Pal, J. Colloid Int. Sci. 231 (2000) 168–175.

[10] D. Kim, A.J. Ghajar, R.L. Dougherty, V.K. Ryali, Heat Transfer Eng. 20 (1) (1999)15–39.

[11] J. Hapanowicz, Int. J. of Multiphase Flow 34 (2008) 559–566.[12] B. Dybek, J. Hapanowicz, S. Witczak, Chem. Proc. Eng. (2) (2010) 31–32.[13] B. Dybek, The technical report of the research project No 3611/B/T02/2008/34

financed from the funds intended to be spent on science in 2008–2011,Influence of Hydrodynamic Phenomena in the Intensity of Heat TransferDuring the Flow of Liquid–Liquid in a Horizontal Pipe.

[14] J. Hapanowicz, Flow Meas. Instrum. 21 (3) (2010) 284–291.