Heat Transfer Theoretical Analysis Experimental Investigations Systems Part 12 pot

The experimental results indicated that when the fuel cell orientation is vertical, two-phase flow pattern in anode channels can evolve from bubbly flow in normal gravity into slug flow

was made of transparent polycarbonate (PC) A high-speed video camera (VITcam CTC) with a CCTV C-mount lens (SE2514, AVENIR) was employed to capture two-phase flow images in the anode flow field A shutter speed of 3996 μs, a recording speed of 250 frames/s and a resolution of 1280×1024 pixels were set to visualize and record two-phase flow in the anode flow field

Oxygen gas with purity of 99.999%, without humidification, was used as oxidant reactant The oxygen gas flow rate was controlled by a mass flow controller (Cole Parmer, CZ-32907-67) at constant flow rate of 400 mL/min The prepared methanol solution was stored in a storage bag and was driven by a peristaltic pump and sent to a liquid flow meter (Cole Parmer, CZ-32908-43) The oxygen gas and the methanol solution were heated up before flowing into the anode channels The produced mixtures from DMFC were sent to two separate containers

Fig 13 Influence of gravity on the power performance of DMFC (Wan et al., 2006)

Fig 13 shows the preliminary results using gold-plating stainless steel as both the anode and the cathode bipolar plates (Wan et al., 2006) In despite of the deterioration of performance of the fuel cell, it is very evident that the cell performance falls more strongly with the degree of concentration polarization deepening

After re-design of the anode and the cathode bipolar plates, an in-situ visualization of phase flow inside anode flow bed of a small liquid fed direct methanol fuel cells in normal and reduced gravity has been conducted in a drop tower Beijing The experimental results indicated that when the fuel cell orientation is vertical, two-phase flow pattern in anode channels can evolve from bubbly flow in normal gravity into slug flow in microgravity (Fig 14) In normal gravity environment, the gravitational buoyancy is the principal detaching force The carbon dioxide bubbles were produced uniformly with tiny shape in normal gravity before the release of the drop tower The diameter of most bubbles, which were detached from the MEA surface, ranged from 0.05 to 0.3 mm in our experiments (Fig 14a, corresponding to 40 ms before the release) After detaching from the MEA surface, the carbon dioxide bubbles moved fast at a speed above 100 mm/s Considering that the mean velocity of liquid at the entry of channel was 3.03 mm/s, which was calculated from the

two-431 inlet flow rate of methanol solution, the speed of bubbles removal was quite fast because of buoyant lift force Big bubbles with fast velocity would push the small ones anterior When the bubbles collided with each other, coalescence took place and it was a dominative way of bubbles growth The typical flow pattern in the anode flow channels in normal gravity was bubbly flow

Fig 14 Gas-liquid two-phase flow pattern in vertical parallel channels of anode bipolar plate of DMFC in different gravity (Guo et al., 2009)

In microgravity, the carbon dioxide bubbles could not get away from the MEA surface in time (Fig 14b~d, corresponding to 1, 2 and 3 s after the release, respectively) At the beginning, the bubbles accreted on the wall of carbon cloth surface Then, the bubbles on the surface grew gradually because of producing carbon dioxide by anode electrochemical reaction The longer the time was, the bigger the bubble was Furthermore, the gravity affects not only detaching diameter, but also bubbles rising velocity The in-situ observation showed that once the capsule was released, the bubbles move was slowed down immediately Bubbles, which were detached from MEA, almost suspended in methanol solution The average rising velocities of bubbles in channels are near to the mean velocity of liquid, which was obviously slower than those in normal gravity because buoyancy lift was very weak and the bubbles removal was governed by viscous drag of fluid in the reduced gravity Some bubbles coalesced with each other and formed larger bubbles Those large bubbles decreased the effective area of fuel mass transfer and hence the DMFC performance deterioration took place The gravitational effect on power performance of DMFC is considerable when the concentration polarization is dominant in fuel cells operation The

higher the current density is, the bigger the effect of gravity is Increasing methanol feeding molarity is conducive to weaken the effect of gravity on performance of liquid fed direct methanol fuel cells Increasing feeding flow rate of methanol solution from 6 to 15 ml/min could reduce the size of carbon dioxide bubbles But the influence of gravity still exists (Ye

A compact transparent proton exchange membrane fuel cell (PEMFC) with a single serpentine channel in graphite cathode flow field, which had a square cross section of 2.0×2.0 mm2 and a rib width of 2.0 mm, was also designed and tested in short-term microgravity environment in the drop tower Beijing Hydrogen and oxygen gases with purity of 99.999%, without humidification, were used as fuel and oxidant reactant, respectively The experimental facility was similar with that for DMFC Its detail can be found in Liu (2008)

433

It was found that the accumulated liquid water in the vertical parts of flow channel for the vertical orientation configuration can be removed easily by the reactant gas in microgravity environment comparing with in normal gravity The PEMFC performance was then enhanced dramatically in microgravity because of the flooded areas in the flow channel before the release of the drop capsule was exposed to the reactant gas again However, for the horizontal orientation configuration with the lower outlet, liquid water produced in flow channel can move along the bottom of the channel in normal gravity and then flow freely out off the channel Then little liquid water was found and water columns to pinch off the flow channel were difficult to be formed in normal gravity On the contrary, the liquid water formed in microgravity was prone to stay in the flow channel, and the departure diameter of water droplets increased Therefore, the PEMFC performance was deteriorated due to liquid water flooding in the flow channel The influence of gravity on the characteristics of phase distribution and performance of PEMFC increases with the increase of the current, and/or increase with the decrease of the cell temperature

4 Further researches on two-phase flow in microgravity in china

Several new projects for two-phase flow in microgravity have been proposed to study pressure drop in in-tube condensation, flow boiling heat transfer enhancement of micro-pin-finned surface, membrane separation of two-phase air-water mixture, two-phase flows inside fuel cells and electrolysis cells, and so on These projects will be helpful for the development of space systems involving two-phase flow phenomena, as well as for the improvement of understanding of such phenomena themselves

5 Conclusion

Two-phase gas-liquid systems have wide applications both on Earth and in space Gravity strongly affects many phenomena of two-phase gas-liquid systems It can significantly alter the flow patterns, and hence the pressure drops and heat transfer rates associated the flow Advances in the understanding of two-phase flow and heat transfer have been greatly hindered by masking effect of gravity on the flow Therefore, the microgravity researches will be conductive to revealing of the mechanism underlying the phenomena, and then developing of more mechanistic models for the two-phase flow and heat transfer both on Earth and in space

The present chapter summarizes a series of microgravity researches on two-phase gas-liquid flow in microgravity conducted in the National Microgravity Laboratory/CAS (NMLC) since the middle of 1990’s, which included ground-based tests, flight experiments, and theoretical analyses In the present chapter, the major results obtained in these researches will be presented and analyzed

Up to now, the sole flow pattern map of two-phase gas-liquid flow in long-term, steady microgravity was obtained in the experiments aboard the Russian space station Mir, which

is intended to become a powerful aid for further investigation and development of phase systems for space applications Flow pattern map of two-phase air-water flow through a square channel in reduced gravity was obtained in the experiments aboard IL-76 parabolic airplane, too Mini-scale modeling was also used to simulate the behavior of microgravity two-phase flow on the ground The criteria of gravity-independence of two-phase gas-liquid flow were proposed based on experimental observations and theoretical

two-analyses A semi-theoretical Weber number model was proposed to predict the annular flow transition of two-phase gas-liquid flows in microgravity, while the influence of the initial bubble size on the bubble-to-slug flow transition was investigated numerically using the Monte Carlo method

slug-to-Pressure drops of two-phase flow through a square channel in reduced gravity were also measured experimentally, which were used to validate the common used correlations for microgravity applications It was found that much large differences exist between the experimental data and the predictions Among these models, the Friedel model provided a relative good agreement with the experimental data A new correlation for bubbly flow in microgravity was proposed successfully based on its characteristics, which indicates that there may exist a transition of flow structure in the range of two-phase Reynolds number from 3000 to 4000, which is similar to the laminar-to-turbulent transition in single-phase pipe flow

In-situ visualizations of two-phase gas-liquid flow inside fuel cells (DMFC and PEMFC) in different gravity conditions have carried out utilizing the drop tower Beijing The gravity influence of the cells performance, namely deterioration or enhancement, depends upon the operation conditions It also infers form the short-term microgravity experiments utilizing the drop tower Beijing that space experiments with long-term microgravity environment are needed

6 Acknowledgement

The studies presented here were supported financially by the National Natural Science Foundation of China (19789201, 10202025, 10432060, 50406010, 50976006), the Ministry of Science and Technology of China (95-Yu-34), the Chinese Academy of Sciences (KJCX2-SW-L05), and the Chinese National Space Agency The author really appreciates Prof W R Hu, Prof J C Xie, Mr S X Wan, Mr M G Wei, and all research fellows who have contributed

to the success of these studies The author also wishes to acknowledge the fruitful discussion and collaboration with Prof K S Gabriel (UOIT, Canada), and Profs H Guo and C.F Ma (Beijing University of Technology, China)

7 References

Bousman, W.S., 1995 Studies of two-phase gas-liquid flow in microgravity Ph.D thesis,

Univ of Houston, TX

Carron, I., Best, F., 1996 Microgravity gas/liquid flow regime maps: can we compute them

from first principles In: AIChE Heat Transfer Symp., Nat Heat Transfer Conf., August, Houston, TX

Chen, I., Downing, R., Keshock, E., Al-Sharif, M., 1991 Measurements and correlation of

two-phase pressure drop under microgravity conditions J Thermophy., 5, 514–523 Cheng, H., Hills, J.H., Azzopardi, B.J., 2002 Effects of initial bubble size on flow pattern

transition in a 28.9 mm diameter column Int J Multiphase Flow, 28(7), 1047–1062 Colin, C., 1990 Ecoulements diphasiques à bubbles et à poches en micropesanteur Thesis,

Institut de Mécanique des Fluides de Toulouse

Colin, C., Fabre J., Dukler A.E., 1991 Gas-liquid flow at microgravity conditions-I

Dispersed bubble and slug flow Int J Multiphase Flow, 17(4), 533–544

435 Colin, C., Fabre, J., McQuillen, J., 1996 Bubble and slug flow at microgravity conditions:

state of knowledge and open questions Chem Eng Comm., 141/142, 155–173 Dukler, A.E., Fabre, J.A., McQuillen, J.B., Vernon, R., 1988 Gas-liquid flow at microgravity

conditions: flow patterns and their transitions Int J Multiphase Flow, 14(4), 389–

400

Dukler, A.E., 1989 Response Int J Multiphase Flow, 15(4), 677

Gabriel, K.S., 2007 Microgravity Two-phase Flow and Heat Transfer Springer

Guo, H., Zhao, J.F., Ye, F., Wu, F., Lv, C.P., Ma, C.F., 2008 Two-phase flow and performance

of fuel cell in short-term microgravity condition Microgravity Sci Tech., 20(3-4): 265-270

Guo, H., Wu, F., Ye, F., Zhao, J.F., Wan, S.X., Lv, C.P., Ma, C.F., 2009 Two-phase flow in

anode flow field of a small direct methanol fuel cell in different gravities Sci China E-Tech Sci, 52(6): 1576 – 1582

Hewitt, G.F., 1996 Multiphase flow: the gravity of the situation In: 3rd Microgravity Fluid

Physics Conf., July 13–15, Cleveland, Ohio, USA

Jayawardena, S.S., Balakotaiah, V., Witte, L.C., 1997 Flow pattern transition maps for

microgravity two-phase flows AIChE J., 43(6), 1637–1640

Lee, J., 1993 Scaling analysis of gas-liquid two-phase flow pattern in microgravity In: 31st

Aerospace Sci Meeting Exhibit, Jan 11–14, Reno, NV

Liu, X., 2008 Two-phase flow dynamic characteristics in flow field of proton exchange

membrane fuel cells under micro-gravity conditions Ph.D thesis, Beijing University of Technology

Lowe, D.C., Rezkallah, K.S., 1999 Flow regime identification in microgravity two-phase

flows using void fraction signals Int J Multiphase Flow, 25, 433–457

McQuillen, J., Colin, C., Fabre, J., 1998 Ground-based gas-liquid flow research in

microgravity conditions: state of knowledge Space Forum, 3, 165–457

Reinarts, T.R., 1993 Adiabatic two phase flow regime data and modeling for zero and

reduced (horizontal flow) acceleration fields Ph.D thesis, Texas A&M Univ., TX Reinarts, T.R., 1995 Slug to annular flow regime transition modeling for two-phase flow in a

zero gravity environment In: Proc 30th Int Energy Conversion Eng Conf., July 30–August 4, Orlando, FL

Song, C.H., No, H.C., Chung, M.K., 1995 Investigation of bubble flow developments and its

transition based on the instability of void fraction waves Int J multiphase Flow, 21(3), 381–404

Wallis, G.B., 1969 One-dimensional Two-phase Flow McGraw-Hill Book Company, New

York

Wan, S.X., Zhao, J.F., Wei, M.G., Guo, H., Lv, C.P., Wu, F., Ye, F., Ma, C.F., 2006 Two-phase

flow and power performance of DMFC in variable gravity 3rd Germany-China Workshop on Microgravity & Space Life Sciences, October 8 - 11, 2006, Berlin, Germany

Ye, F., Wu, F., Zhao, J.F., Guo, H., Wan, S.X., Lv, C.P., Ma, C.F., 2010 Experimental

Investigation of Performance of a Miniature Direct Methanol Fuel Cell in Term Microgravity Microgravity Sci Tech., 22(3): 347-352

Short-Zhao J.F., 1999 A review of two-phase gas-liquid flow patterns under microgravity

conditions Adv Mech., 29(3): 369-382

Zhao J.F., 2000 On the void fraction matched model for the slug-to-annular transition at

microgravity J Basic Sci Eng., 8(4): 394–397

Zhao, J.F., 2005 Influence of bubble initial size on bubble-to-slug transition J Eng

Zhao, J.F., Hu W.R., 2000 Slug to annular flow transition of microgravity two-phase flow

Int J Multiphase Flow, 26(8), 1295–1304

Zhao, J.F., Xie, J.C., Lin, H., Hu, W.R., 2001a Experimental study on two-phase gas-liquid

flow patterns at normal and reduced gravity conditions Sci China E, 44(5), 553–

560

Zhao, J.F., Xie, J.C., Lin, H., Hu, W.R., Ivanov, A.V., Belyeav, A.Yu., 2001b Microgravity

experiments of two-phase flow patterns aboard Mir space station Acta Mech Sinica, 17(2), 151–159

Zhao, J.F., Xie, J.C., Lin, H., Hu, W.R., Ivanov, A.V., Belyeav, A.Yu., 2001c Experimental

studies on two-phase flow patterns aboard the Mir space station Int J Multiphase Flow, 27, 1931–1944

Zhao, J.F., Xie, J.C., Lin, H., Hu, W.R., Lv, C.M., Zhang, Y.H., 2001d Experimental study on

pressure drop of two-phase gas-liquid flow at microgravity conditions J Basic Sci Eng., 9(4), 373–380

Zhao, J.F., Xie, J.C., Lin, H., Hu, W.R., 2002 Pressure drop of bubbly two-phase flow

through a square channel at reduced gravity Adv Space Res., 29(4), 681–686 Zhao, J.F., Liu, G., Li, B., 2004a Two-phase flow patterns in a square micro-channel J

Thermal Sci., 13(2), 174–178

Zhao, J.F., Xie, J.C., Lin, H., Hu, W.R., Ivanov, A.V., Belyeav, A.Yu., 2004b Study on

two-phase gas-liquid flow patterns at partial gravity conditions J Eng Thermophy., 25(1), 85–87

Zhao, L., Rezkallah, K.S., 1995 Pressure drop in gas-liquid flow at microgravity conditions

Int J Multiphase Flow, 21(5), 837–849

Heinz Herwig and Tammo Wenterodt

Hamburg University of Technology

Germany

1 Introduction

Somebody, interested in heat transfer and therefore reading one of the many books about this

subject might be confronted with the following statement after the heat transfer coefficient h=

˙qw/ΔT has been introduced:

“The heat transfer coefficient h is a measure for the quality of the transfer process.”

That sounds reasonable to our somebody though for somebody else (the authors of this chapter)

there are two minor and one major concerns about this statement They are:

1 Heat cannot be transferred since it is a process quantity

2 The coefficient h is not a nondimensional quantity what it better should be.

3 What is the meaning of “quality”?

The major concern actually is the last one and it will be the crucial question that is raised andanswered in the following Around that question there are, however, some further aspectsthat should be discussed Two of them are the first two in the above list of concerns

The heat transfer coefficient h is typically used in single phase convective heat transfer

problems This is a wide and important field of heat transfer in general That is why theproblem of heat transfer assessment will be discussed for this kind of “conduction based heattransfer” in the following sections 2 to 4 In section 5 extensions to the overall heat transferthrough a wall, heat transfer with phase change and the fundamentally different “radiationbased heat transfer” will be discussed

2 The “quality” of heat transfer

2.1 Preliminary considerations

What is commonly named heat transfer is a process by which energy is transferred across

a certain system boundary in a particular way This special kind of energy transfer ischaracterised by two crucial aspects:

– The transfer process is initiated and determined by temperature gradients in the vicinity ofthe system boundary

– As a consequence of this transfer process there is a change of entropy on both sides of thesystem boundary That change can be interpreted as a transfer of entropy linked to theenergy transfer It thus is always in the same direction The strengths of both transferprocesses are not in a fixed proportion, but depend on the temperature level

According to these considerations the phrase “heat transfer” actually should be replaced by

“energy transfer in the form of heat” Since, however, “heat transfer” is established worldwide

Chapter Number Heat Transfer and Its Assessment

17

in the community we also use this phrase, but as a substitute for “energy transfer in the form

of heat”

It is worth noting that from a thermodynamics point of view heat is one of only two ways in which energy can be transferred across a system boundary The alternative way is work This

other kind of energy transfer is not caused by temperature gradients and is not accompanied

by entropy changes For further details see Moran & Shapiro (2003); Baehr & Kabelac (2009);Herwig & Kautz (2007), for example

The energy transferred in the form of heat from a thermodynamics point of view is internal

energy stored in the material by various mechanisms on the molecular level (translation,

vibration, rotation of molecules) The macroscopic view on this internal energy can not onlyidentify its amount (in Joule) but also its “usefulness” The thermodynamic term for that

is its amount of exergy Here exergy, also called available work, is the maximum theoretical

work obtainable from the energy (here: internal energy) interacting with the environment toequilibrium

According to this concept energy can be subdivided in two parts: exergy and anergy Hereanergy is energy which is not exergy (and thus “not useful”) If, however, energy has acertain value (its amount of exergy) a crucial question with respect to a transfer of energy

(heat transfer) is that about the devaluation of the energy in the transfer process.

2.2 Energy devaluation in a heat transfer process

What happens to the energy in a heat transfer process can best be analysed on the background

of the second law of thermodynamics This kind of analysis which considers the entropy,its transfer as well as its generation is called second law analysis (SLA) In a heat transfer

situation the entropy S is involved twofold:

– Entropy is transferred over the system boundary together with the transferred energy

In a thermodynamically reversible process entropy is transferred only and no entropy isgenerated This infinitesimal transfer rate is

generation rate per volume () is, see Bejan (1982); Herwig & Kock (2007),

Entropy generation always means loss of exergy According to the so-called Gouy–Stodola

theorem, see Bejan (1982), the exergy loss rate per volume due to heat conduction is

with ˙SC from (2) and T0as the temperature of the environment

The devaluation of energy that is transferred in the form of heat thus can be determined byintegrating the local exergy loss rate (3) over the volume of the system under consideration,

as will be shown in sec 4.2

2.3 Energetic and exergetic quality of heat transfer

Since heat transfer is caused by temperature gradients, or (integrated over a finite distance) bytemperature differencesΔT there are two questions about the “quality” of a transfer process:

– How much energy can be transferred in a certain situation with ΔT as the operating temperature difference? This is called the energetic quality of the transfer process.

– How much is the energy devaluated in a certain transfer situation withΔT as the operating temperature difference? This is called the exergetic quality of the transfer process.

Obviously two parameters are needed to answer both questions If, however, there is only the

heat transfer coefficient h or its nondimensional counterpart, the Nußelt number Nu, not all

information about the heat transfer process is available as will be demonstrated hereafter

3 Heat Transfer Assessment

3.1 The energetic quality of heat transfer

The energetic quality was introduced (sec 2.3) as the answer to the question “How muchenergy can be transferred in a certain situation with ΔT as the operating temperature

difference?” This amount of energy is finite (and not infinite) only because it occurs in a

real process subject to losses Finite values of the heat transfer coefficient h nevertheless are

not a direct measure for these losses

depend onΔT There are, however, situations in which this is not the case, like for natural

convection in general and for radiative heat transfer with large values ofΔT, see Herwig

(1997) for details

– Instead of h, its reciprocal 1/h=ΔT/ ˙qwwould be more appropriate Then finite values

(and not zero) for h−1would be due to a resistance which a heat flux ˙ Qw= ˙qwA encounters

on the heat transfer area A This is in analogy to the resistance R=ΔU/I that an electrical current I (I=iA; i: current density, A: cross section) encounters with a voltage ΔU By this analogy U corresponds to T and I to ˙ Qw

– Since h is part of the Nußelt number

size L and the thermal conductivity k of the fluid involved.

with a transfer area A

Fig 1 One dimensional heat transfer at a system boundary ( ˙qw=const)

Whenever general statements about a certain heat transfer situation are the objective theNußelt number is the preferred parameter

3.2 The exergetic quality of heat transfer

The exergetic quality was introduced (sec 2.3) as the answer to the question “how much isthe energy devaluated in a certain transfer situation withΔT as the operating temperature

difference?” This devaluation is directly measured by the exergy loss rate based on therelation (3)

For a heat transfer situation as sketched in Fig 1 which is characterised by Nu with respect

to its energetic behavior a second parameter is now introduced and named exergy loss number.

Its definition is

NE≡ ˙ELC

˙E = T0 ˙SC

with ˙ELCand ˙SCas integration of the local rates ˙ELCand ˙SCand ˙E as the exergy fraction of

the heat flux ˙Qw This exergy fraction corresponds to the transferred heat flux ˙Qwmultiplied

by the Carnot factor

ηC=1− T0

The crucial quantity in NEaccording to (6) is the overall entropy generation rate due to heat

conduction, ˙SC In a complex temperature field it emerges through a field integration of ˙SC,see section 4.2 below If, however, there is a one-dimensional heat transfer perpendicular tothe system boundary as sketched in Fig 1, the situation is different

Then only the two temperature levels Twand T∞count, and ˙SCis the difference of the transferrates ˙Qw/Twand ˙Qw/T∞, cf (1) for incremental parts, i.e

˙SC=Q˙w

1

Table 1 Heat transfer with Nu=100 in two different power cycles

performance is a Nußelt number correlation Nu=Nu(Re) This, however, is only theenergetic part of the performance and NEaccording to (9) should also be considered

To be specific, it is assumed that a heat transfer situation with Nu=100, ˙qw=103W/m2

and L=0.1 m occurs in two different power cycles One is a steam power cycle (SPC)

with water as the working fluid and a temperature level for heat transfer Tw=900 K.The alternative is an ORC cycle with ammonia (NH3) as working fluid and a temperature

level Tw=400 K When in both cycles Nu=100 with the same values for ˙qwand L holds

the temperature differenceΔT is larger by a factor 2.6 for ammonia compared to water This

is due to the different values of thermal conductivity of water (at T=900 K and p=250 bar)

and ammonia (at T=400 K and p=25 bar), assuming typical values for the temperature andpressure levels in both cycles

Table 1 collects all data of this example including the exergy loss number according to (9).For the ORC-cycle this number is 0.3 and this is 50 times that for the SPC-cycle Note, that

an amount of 0.6 % and 30 % less exergy after the heat transfer in a power cycle means: thatamount of available work is lost for a conversion into mechanical energy at the turbine of thecycle

Here the devaluation of the transferred energy obviously is an important aspect of the process.This devaluation cannot be quantified by the Nußelt number though its finite (and not infinite)value is due to the fact that losses occur in a real transfer process The exact amount of thelosses is given by the exergy loss number NE

4 Complex convective heat transfer problems

Heat transfer often occurs as convective heat transfer, i.e influenced and supported by a fluid

flow Especially when the flow is turbulent there is a strong impact on the heat transferperformance This is due to the strong effect turbulent fluctuations have on the transport ofinternal energy As a modeling strategy a so-called turbulent or effective thermal conductivity

can be defined which often is a magnitude larger than the molecular conductivity k, see

Munson et al (2009); Herwig (2006) for details

Increasing the flow rate almost always increases the heat transfer intensity This is reflected

by the increasing Nußelt numbers Nu(Re) There is, however, a prize to pay: it turns out that

not only the exergy losses due to the conduction of heat, ˙SCin (6), have to be accounted for,but that also the exergy losses due to the dissipation of mechanical energy in the fluid flowmust be considered Only when both losses are examined and accounted for together, one cananswer the question whether an increase in the flow rate is beneficial for the transfer process

as a whole

4.1 Fluid flow assessment

Before the heat transfer process as a whole is considered we want to address the losses in

a flow field Again these losses are exergy losses accompanied by entropy generation The

common way to characterise a flow with respect to its losses is to define a friction factor for a flow in pipes and channels, for example For external flows it would be a drag coefficient Both

parameters are finite values (and not zero) due to the fact that losses occur Again the questionarises, whether these parameters are an immediate measure of the exergy losses

It turns out, that the friction factor f , which will be considered in the following, strictly speaking is a parameter that assesses the energetic quality of the flow It is the answer to the

question “How much mechanical energy (measured by the total head of the flow) can betransferred along the pipe or channel in a certain situation withΔp as the operating pressure

difference?”

In analogy to the heat transfer process the second question is “How much is the mechanicalenergy (measured by the total head) devaluated in a certain situation withΔp as the operating pressure difference?” This is about the exergetic quality of the flow

Again in analogy to the heat transfer process we define two assessment parameters:

– Head loss coefficient K

K=f L

Dh ≡ 2ϕ

u2 m

(10)with ϕ as specific dissipation of mechanical energy and um as the mean velocity in thecross section of the pipe or channel This parameter is frequently used in fluid mechanics

By introducing K instead of f alone, a pipe or channel (of length L and with a hydraulic diameter Dh) is treated as a conduit component like bends, trijunction, diffusers etc.– Exergy loss coefficient KE

with T0 as the temperature of the environment and T∞ as that temperature level on

which the flow occurs The exergy loss rate due to dissipation ˙ELDis the integrated local

value ˙ELD=T0 ˙S

D with ˙SD defined later, cf (3) for the heat transfer counter parts

The background of both definitions can again be analysed by looking at the entropy and itsgeneration in the flow field Here the specific dissipationϕ and the entropy generation due

to dissipation, ˙SD, are closely related, though not the same The relation is, see Herwig et al.(2010), for example

˙

m ϕ=T∞˙SD (12)

Note, that T0 ˙SDand not T∞˙SDis the exergy loss rate according to the Gouy–Stodola theorem,

cf (3) That is why K is the energetic, but not the exergetic assessment parameter

For simple flow geometries such as straight pipes, bends, etc (10) and (12) can be combinedwith

and thus the dissipation rate ˙m ϕ according to (12) can be determined by accounting for the

local entropy generation rate per volume ()



∂v

∂y

2+



∂u

∂z +∂w ∂x

2+



∂v

∂z+∂w ∂y

2(15)

For details of the derivation see Herwig & Kock (2007); Herwig et al (2010) When the flow is

non-isothermal, T in (15) is different at different locations As long as temperature variations are small compared to T this effect is also small and can be neglected as a first approximation Then a unique temperature T∞appears in (12)

Equation (15) can be immediately used for the determination of ˙SD when the flow is laminar

or for a turbulent flow in a DNS-approach In a RANS approach ˙SD is split into ˙SD and ˙SD

like all other variables with



∂v

∂y

2+



∂u

∂z +∂w ∂x

2+



∂v

∂z+∂w ∂y

2(17)



∂v

∂y

2+



∂u

∂z +∂w ∂x

2+



∂v

∂z +∂w ∂y

2(18)

Only ˙SD can be determined directly once a CFD-solution of the flow field exists The

fluctuating part ˙SD must be subject to turbulence modeling For example, ˙SD can be linked

to the turbulent dissipation rateε which is known when a k–ε-model is used (but which is also

part of almost all other models) by

˙SD=ε

For details, again see Herwig & Kock (2007)

This approach can for example be used to determine the friction factor of a pipe with a specialroughness type, called Loewenherz-thread roughness, by integrating the entropy generationrate as shown in Fig 2, see Herwig et al (2008) for details The dark lines are numerical resultsbased on (17)-(19) They compare very well with experimental results from Schiller (1923) andshow that the classical Moody chart is not even a moderately good approximation for thiskind of rough pipes

Once the head loss coefficient K is known it is an easy though important step to find KEaccording to (11) Only KEis a direct measure for the devaluation of the transferred energy interms of lost available work (or exergy)

As an example, a pipe with a certain head loss coefficient K is part of a power cycle.

It is operated on different temperature levels when it is in a steam power cycle (SPC,

temperature T∞=900 K) or in an organic Rankine cycle (ORC, temperature T∞=400 K)

4.2 Convective heat transfer assessment

For a complete assessment of a convective heat transfer situation the energetic part is accounted for by the Nußelt number (with no need to look at the head loss coefficient) The exergetic part,

however, requires the combined consideration of losses in the temperature and the flow field

in order to determine the overall reduction of available work caused by the (convective) heattransfer

This, however, is straight forward within the SLA-analysis An overall exergy loss number NE

Since a device with a small number NEobviously is more efficient than one with a larger NE

we introduce an overall efficiency factor as

ηE≡1−NE= ˙E− (˙ELC+ ˙ELD)

WhenηEwith NEshould be applied to a complex convective heat transfer situation, ˙SCcannot

be determined like in (9) which only holds for a one-dimensional case Then ˙SCis found from



∂T

∂y

2+



∂T

∂y

2+



∂T

∂y

2+

4.3 An example

As an example a counter flow plate heat exchanger is analysed with respect to its heat transferperformance For that purpose a special 2D-geometry is chosen which corresponds to thegeometric situation in a stack of sinusodially formed plates shown in Fig 3 One element

of the cold part is “cut out” as the numerical solution domain, assuming periodic boundaryconditions (Fig 3 b) The boundary conditions are non-slip for the flow and temperature

boundary conditions for the heat transfer Basically a temperature rise T2−T1is set which isassumed to be linearly distributed between the cross sections and1  With T2 2−T1set, theoverall heat flux into the solution domain is also prescribed The questions to be answerednow are

– What is the Nußelt number Nu=Nu(Re)

– What is the overall exergy loss rate ˙EL= ˙ELD+ ˙ELCand thus the efficiency factorηE?– Is there a maximum ofηEwith respect to the Reynolds number?

2

2



Tw

H

Fig 3 2D-sinusodial plate arrangement in a plate heat exchanger

The flow part of the entropy generation ˙ELD/ ˙E will increase when the Reynolds number gets higher whereas the heat transfer part ˙ELC/ ˙E will decrease due to the favorable influence

of a stronger convection The overall effect ˙EL/ ˙E is expected to show a minimum which

corresponds to a maximum inηE

Details of the numerical approach, using the k– ω-SST turbulence model will not be given here

(since we want to concentrate on the assessment strategy), they can be found in Redecker(2010) Figure 4 shows the qualitative distribution of the time averaged velocityv=u, v,

temperature T and the local entropy generation rate ˙S= ˙S

D+ ˙S

C.The overall heat transfer performance is shown in Fig 5 The energetic part in terms of Nu(Re)

is shown in Fig 5(a), the exergetic part in terms ofηEin Fig 5(b) The Reynolds number Re=

1995, see Fig 5(b), turns out to be the optimal Reynolds number with respect to the loss ofavailable work, sinceηEhas its maximum for this parameter value

When this heat exchanger element is analysed in the “conventional way” without recourse

to the second law of thermodynamics, one would for example apply an often used

thermo-hydraulic performance parameter

η≡

St

introduced in Gee & Webb (1980) with St=Nu/RePr and f=K·Dh/L This parameter η is

used to compare a certain convective heat transfer device (with St0, f0) to modified versions

(with St, f ) and then to decide which of the modifications are beneficial There is, however,

no clear physical interpretation forη, except for the presumption that η>1 corresponds to animproved situation

Applied to the present example,η according to Fig 6 results without an indication of an

optimum within the Reynolds number range shown here With the combination of St and f

as two very different parameters in the definition ofη there is no clear physical meaning to be

recognised

We therefore suggest to useηE(or NE) in addition to Nu in order to get a complete assessment

of convective heat transfer situations

(a) Energetic assessment by Nu(Re)

1.0

ηE0.8

Fig 6 Heat transfer performance of a plate heat exchanger element, see Fig 5 for

comparison;η: thermo-hydraulic performance parameter (28)