immersed coil heat exchangers

Immersed Coil Heat Exchangers

William Logie

Institut für Solartechnik SPF ∗

October 22, 2007

Abstract

A large portion of presently available thermal storage tanks em-ploy heat exchangers between the collector subsystem and the stor-age or energy delivery subsystems. This trend is driven moreso incolder climates where protection of the collectors against frost andsnow was solved by circulating an anti-freeze mixture (commonly wa-ter and propylene glycol) between the collector and storage or load.The use of a heat exchanger with a non-freezing transfer fluid in a so-lar hot water system reduces the overall efficiency of the system. [2] [7]

Due to the coupling of the heat exchanger’s Heat Transfer Coef-ficient to convective and buoyant flows within the storage tank, anaccurate understanding of the penalty any heat exchanger brings to itssystem leads - even with simplification - to a non-trivial set of heat-momentum equations. [5]

Quantifying these characteristics through accurate visualisation maylead to better assumptions that will help bound the unclosable Navier-Stokes equation.

∗Oberseestraße 10, CH-8640 Rapperswil

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Institut für Solartechnik SPF

1 IntroductionThe last thirty years experience in solar research has gathered much calorificunderstanding in the behaviour of heat exchange taken from and delivered tothermal storage tanks. This report gathers together some of this knowledgeto identify further work involving contemporary experimental (and even-tually numerical) resources. It serves as an introduction to the principlesof heat transfer where immersed heat exchangers are implemented, concen-trates on well known examples used today and tentatively suggests what onemight learn through experimental observation of natural convection utilisingParticle Image Velocimetry (PIV) and Laser Induced Fluorescence (LIF).

As the title suggests, the character of heat exchangers mentioned in thisreport are specifically those which are immersed in solar thermal storage.The type of heat exchangers immersed depends on the type of tank andarrangement in which the heat transfer shall take place, which are many.We are foremost interested in the combination of vertical cylindrical storagetanks with varying immersed coil heat exchangers, which are most widelydispersed among existing solar thermal storage stock.

2 Theory - The problem of conventionDetailed study of immersed heat exchangers requires consideration of thenatural convection - both internally and externally - occuring near the im-mersed heat exchangers surfaces. This is the phenomenen where the fluidheated from the exchanger decreases in density and thus induces convectionaround or along the surface of heat exchange against gravity. Just as theflow past the exchanger is dependent on the heat transfer from it, so too isthe heat transfer coefficient of the exchanger dependent on the fluid veloc-ity past it. Much contemporary heat exchanger design accounts for naturalconvective heat transfer with the approximation (averaging) of laminar flowaround an infinitely long horizontal tube. Considering the convection inthree dimensions, as alluded to in the abstract of this paper, increases thecomplexity of calculation significantly.

Understanding the local flux or the total transfer rate in any convectionproblem depends on knowledge of the local and average surface friction,convective heat transfer and the convective mass transfer boundary layers.Determination of the coefficients for each, specifically the coupling betweenthem, is viewed as the problem of convection because, in addition to depend-ing on numerous fluid properties such as density (ρ), viscosity (µ), thermalconductivity (λ) and specific heat (Cp), the coefficients depend on the sur-face geometry and the flow conditions.

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2.1 Surface geometry

Considered are all spiral coils with variation in the following geometry:

• Internal and external radii of tube (diameters)

• Pitch of helical growth (in the z direction)

• Radius/diameter of helix

• Overall height of helix

• Entrance and exit effects to and from the heat exchanger

2.1.1 To fin or not to fin

Fins are used to increase the heat transfer from a surface by increasingthe effective surface area. However, the fin itself represents a conductionresistance to heat transfer from the original surface. For this reason (andothers, like that of water bubbles cavitating between the fins) there is noassurance that the heat transfer rate will be increased through the use offins.

Fin efficiency (ηf ) can be calculated from equations found in any tableof common fin efficiencies (e.g. Table 3.5, pp152 from [1]).

2.2 Flow conditions

The condition of flow within the fluids finding themselves on either side ofthe heat exchangers surfaces can influence greatly its ability to transfer heat.Both forced and bouyancy generated flows along the exchangers surfaceschange the gradients within the boundary layers and thus the rate at whichheat is able to travel along them. The greatest change in gradient comeswith the transition from laminar to turbulent flow.

2.2.1 Reynolds number

For dynamic similitude we measure the likelyhood of finding laminar orturbulent flow with the dimensionless Reynolds number, which is a balancebetween inertial and viscous forces.

Re = ρu∞L

µ(1)

for a given fluid (free flow) velocity u∞ travelling over a characteristic lengthL.

It is assumed that the global flow inside a solar storage tank with im-mersed heat exchangers remains laminar but it is very possible, dependingon the power with which the heat exchanger is charged, that local flows

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around the heat exchanger (within the convective boundary layers) can beturbulent.

Flow within the heat exchanger tube can be either laminar or turbu-lent depending on the flow rate of the pumped supply-side and the tubedimensions.

2.2.2 Rayleigh number

Something that will not be discussed in detail within this report but is validto the evaluation of information presented is the Rayleigh number, given asfollows;

RaL = GrLPr = gβ

µα(Ts − T∞))L3 (2)

Where:

Gr = Grashof number (ratio of bouyancy to viscosity)Pr = Prandtl number (ratio of momentum to thermal diffusivity)g = Acceleration due to gravityTs = Surface temperatureT∞ = Ambient temperatureµ = Kinematic voscosityα = Thermal diffusivityβ = Thermal expansion coefficient

2.2.3 External flow conditions

Before introducing the relevant theory relating to the convective heat transferboundary layer it needs mentioned that we interest ourselves primarily fornatural convection locally in the hope that accurate assessment on the localscale will give reliable global averages. We seek the local coefficients (as afunction of varying temperature, velocity and concentration gradients) tothen average them over the whole heat exchanger.

Evaluation of the tube cylindrical section as a combination of two-dimensionalradial conduction (with resistance Rw defined from an outer radius r2 andinner radius r1 over a characteristic length L) equation:

Rw =ln( r2

r1)

2πLλ. (3)

We can then include convection to observe that

1UA

= 1(ηohext2πr2L) ext

+R′′f,ext

(ηo2πr2L) ext+Rw

+R′′f,int

(ηo2πr1L) int+ 1

(ηohint2πr1L) int, (4)

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Figure 1: As shown in this 3D mesh of a single helical coil, Th, inominally finds itself at the top of the exchanger tube, such thatthe more effective counterflow heat exchange (where storage isstratified) takes place as the heating fluid moves downwards

which gives us the overall heat transfer area coefficent - UA. Subscriptsint and ext refer to internal and external surfaces of the heat exchangerrespectively, R′′f to the fouling factors on either side, h to the heat transferconvection coefficient and ηo to the overall surface efficiency; by way of (5),

ηo = 1− AfA

(1− ηf ). (5)

Our objective then becomes the development of the thermal field appli-cable to our curved surface - those gradient properties found within reachof the surface of the heat exchanger affecting the heat transfer convectioncoefficient.

Bending the conception from Figure (2) and looking closer at a crosssection (in this case taken from the very bottom of the heat exchangers -for simplicity - where no convective plumes from below can be expected) wecan seek definition of the convective heat transfer coefficient for each pointon the exchanger, seen as a horizontal cylinder.

Following the notation given in Figure (3) where the lower stagnationpoint is taken at x = 0, the local heat transfer coefficent is obtained as afunction of φ in degrees, where φ = 360

πxD for given diameters (D = 2r2) of

the tube [8].Using Fourier’s Law to determine the heat flux at the tube surface into

the convective boundary requires us to evaluate the integral from the tubessurface to the edge of its convection induced stream. For this local pointof interogation h is then obtained through Newton’s law of cooling and nor-

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Figure 2: Illustrates the Boundary Layer conditions relevant forconvection transport over an arbitrary surface (shown here underforced flow u∞) [1].

&%'$"!# r2

QQQs y

* x

66

qφ

Figure 3: Simple illustration of the natural convection boundarylayer over a heat exchanger tube section

malised for the temperature difference between free fluid and the tubes sur-face with Equation 6.

h = −λL

T∞ − TsTs − T∞

δT ∗

δy∗

∣∣∣∣y∗=0

(6)

The * implies distances normalised for characteristic length L and tem-peratures normalised for Newton’s law.

2.2.4 Nusselt number

The Nusselt number is another one of our dynamic similitude measures usedto describe the ratio of convective to conductive heat transfer for relation to’real’ situations.

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The dimensionless Nusselt expression is defined as follows:

Nu = hL

λ. (7)

An indicative graph of Nusselt number as a function of φ is found inFigure (4).

Figure 4: Nusselt correlation of tube section showing the perfor-mance of the local convective heat transfer coefficient along theboundary layer development [1]

Looking at how the Nusselt number changes when tubes are stacked ontop of one another (Figure 5) shows how the convective boundary layer caneffect heat transfer.

2.2.5 Internal flow conditions

Where flow is dominated by laminar velocity profiles, bouyancy forces dueto gravity become overwhelmed by the so called Dean flow - secondary flowresulting from coriolis forces, shown in Figure 6.

Of interest to note is a study made by Chagny (2001) in which internalconvective heat transfer of a standard and a chaotic helical coil configurationwere compared, shown in Figure 7. It was found that the chaotic config-uration achieved a higher convective heat transfer due to disruption of theDean flow for Laminar flows. Where transition to turbulence had occured,this advantage was insignificant.

3 Empirical correlations - TestingAn indiciative study on the conventional heat transfer performance of im-mersed heat exchangers is shown by Farrington and Bingham (1986) [5] in

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Figure 5: Nusselt numbers shown for a vertical array of 5 hori-zontal tubes (in an in-line bundle) [4]

Figure 6: Natural convection reacts not only to acceleration dueto gravity. Here we see the coriolis effect found in the form ofsecondary flow inside a helical coil: Dean flow [1]

the testing of four immersed heat exchangers. Following any good heat andmass transfer text [1], the performance equations fitting the load side of a so-lar thermal storage were constructed using log mean temperature differencecorrelation.

The effectiveness of a heat exchanger with regards to the system, mea-sured using equation (8). Here the temperature of the fluid entering theheat exchanger is written as Th,i (h identifying the heat source), the tem-perature of fluid exiting the heat exchanger with Th,o and the temperature

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Figure 7: Chagny (2001) LIF measurements of standard (a) andchaotic helical (b) configurations for varying Reynolds number [3]

of the storage with TS .ε =

Th,i − Th,oTh,i − TS

(8)

The heat transfer rate is measured by way of this same temperaturedifference acting on the specific capacitance CP of the fluid and its rate offlow m through the heat exchanger, written in equation (9).

q = mCP (Th,i − Th,o) (9)

The logarithmic mean temperature difference, used to relate the overallheat transfer area coefficient to the inlet and outlet temperatures, is givenin equation (10).

LMTD =Th,i − Th,oln( Th,i−TSTh,o−TS )

(10)

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This now allows us to formulate the overall heat transfer area coefficientin equation (11).

UA = q

LMTD(11)

A Nusselt correlation of:

NuD = CRaD (12)

was used for varying diameters D but made no account for varying pitch-to-diameter (P/D) ratios or different geometries.

Furbo (1984) [6] investigated many compact immersed coil heat exchang-ers (all with a P/D ratio of 1) and created a correlation function for deter-mining heat transfer from regression of empirical studies:

H = UA = A+B.TS (13)

Where:

A = C1 + C2. ln(Th,i − TS) andB = C3 + C4. ln(Th,i − TS)

The constants C1 through C4 depend on the design of the spiral exchanger.Nusselt correlation for hh,int.:

Nu = 0.016.P r0.34m .Re0.82.(Prm

Prw)0.25 (14)

Nusselt correlation for hh,ext.:

Nu = 0.60 + 0.387.Gr0.192

(1 + (0.559Prf

)9/16)8/27 (15)

And they can both be related back to H through Equation 4 for local orglobal areas of interest (interrogation areas).

4 ConclusionBoth the LMTD and the less iteratively intensive number of transfer units(NTU [1]) method - which requires only inlet temperatures to be known -is valid for any heat exchanger immersed in a storage medium that does nothave an appreciable change in temperature compared to the temperaturerise on the forced flow side of the heat exchanger. Seeing as immersed heatexchangers are well known for mixing the storage tank completely, wheneither in charge or discharge operation, this makes the modelling somewhateasier.

Due to the overwhelming use of immersed coil heat exchangers, thereis significant need for improvement in the knowledge of immersed heat ex-changers, namely:

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• Mixing caused from the heating or cooling (draw-off) of solar storagetanks with immersed spirals,

• Heat exchanger capacity rates for tall spirals (occupying different levelsof stratified tanks),

• Downward heat transfer caused by heat exchanger spirals and pipes,

• Natural convection caused by heat loss for tanks with low height todiameter ratios, and

• Heat exchangers of varying P/D ratio.

The greatest solar system effectiveness can be expected where the outlettemperature from the heat exchanger approaches the temperature of thefluid in the store without disturbing stratification (highest solar fraction).

Note: Further Literature from the Institute for Thermodynamics andHeating Technology (ITW - Stuttgart) will be looked into over the comingweeks.

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References[1] Frank P Incropera, David P Dewitt, Theodore L Bergman, Adrienne S

Lavine: “Fundamentals of Heat and Mass Transfer”, 6th Edition, JohnWiley & Sons, (2006)

[2] S Arora, J H Davidson, J Burch and S Mantell: “Thermal Penalty ofan Immersed Heat Exchanger in Integral Storage Systems”, ASME J.Sol. Energy Eng., Vol. 123, Issue 3, pp 180-186, (2001)

[3] C. Chagny, C. Castelain and H. Peerhossaini: “Chaotic heat transfer forheat exchanger design and comparison with a regular regime for a largerange of Reynolds numbers”, Journal of Applied Thermal Engineering,Vol. 20, (2000)

[4] Y. T. Krishne Gowda, P. A. Aswatha Narayana und K. N. Seetharamu:“Numerical investigation of mixed convection heat transfer past an in-line bundle of cylinders”, Journal of Heat and Mass Transfer, Vol 41,No. 11, (1997)

[5] Robert B Farrington, Carl E Bingham: “Testing and Analysis of Im-mersed Heat Exchangers”, Solar Energy Research Institute, (1986)

[6] S Furbo (Ph.D.): “Varmelargring til Solvarmeanlæg”, Laboratoriet forVarmeisoliering, Technical University of Denmark - DTU, (September1984)

[7] A Mertol, W Place and T Webster: “Detailed Loop Model (DLM)Analysis of Liquid Solar Thermosiphons with Heat Exchangers”, ISESSolar Energy, Vol. 27, No. 5, pp 367-386 (1981)

[8] Yogesh Jaluria: “Natural Convection Heat and Mass Transfer”, Perg-amon Press, (1975)

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