Heat Mass Transfer (2007) 43: 207–215 DOI 10.1007/s00231-006-0086-x

O R I GI N A L

H. Herwig Æ F. Kock

Direct and indirect methods of calculating entropy generation rates in turbulent convective heat transfer problems

Received: 19 May 2004 / Accepted: 19 January 2005 / Published online: 18 March 2006  Springer-Verlag 2006

Abstract Computational fluid dynamics (CFD) solutions of turbulent convective heat transfer problems based on the mass, momentum and energy conservation principle provide all information to calculate the entropy production rate in such a transfer process. It can be determined in the post processing phase of a CFD calculation. Two methods are discussed in detail which can provide the information about the entropy production with different degrees of accuracy.

e g k U UH . SPRO, i

Turbulent dissipation rate (m2/s3) Viscosity (kg/ms) Thermal conductivity (W/mK) Dissipation of mech. energy (W/m3) Loss of mech. energy (WK/m3) Density (kg/m3) Entropy production rate (W/K m3)

1 Introduction List of A cf cp D L m_ Nu Pr q Re R s T t u, v, w V x, y, z

symbols Area (m2) Skin friction coefficient Specific heat (J/kg K) Diameter (m) Length (m) Mass flux (kg/s) Nusselt number Prandtl number Heat flux density (W/m2) Reynolds number Gas constant (m2/s2 K) Specific entropy (J/kg K) Temperature (K) Time (s) Velocity components (m/s) Volume (m3) Cartesian coordinates

Greek symbols a Thermal diffusivity (m2/s) at Turbulent thermal diffusivity (m2/s) H. Herwig (&) Æ F. Kock Technische Thermodynamik, TU Hamburg-Harburg, 21073, Hamburg, Germany E-mail: [email protected]

Computational fluid dynamics (CFD) over the past two decades has become state of the art in thermal engineering like in heat exchanger design. Pressure drop and heat transfer predictions often are accurate even in complex geometries, though a critical interpretation of the results is always mandatory, see for example Casey and Wintergerste (2000) as well as Ho¨lling and Herwig (2004). In standard versions commercial CFD-codes only take into account the first law of thermodynamics which is the basis for all types of energy equations incorporated in the programs. Predicting an efficient use of energy in thermal systems like compact heat exchangers or whole power plants, however, can only be achieved if also the second law of thermodynamics is accounted for, since the amount of available work (also called exergy) depends on the amount of entropy produced, see Bejan (1996). Therefore, a thermal apparatus producing less entropy by irreversibility destructs less available work (producing less anergy). This increases the total efficiency of a thermal system. The amount of entropy generated can be directly used as an efficiency parameter of the system, see for example Bejan (1978). For example, a heat transfer apparatus with small cross sections often encounters small temperature gradients and therefore small entropy production by heat transfer (For a fixed Nusselt number Nu=qw D/kDT characteristic temperature differences are DT  D!).

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However, due to the large pressure drop of this configuration there will be a large entropy production rate by dissipation. Since, however, both effects (heat transfer and pressure drop) have been linked to one single quantity (entropy production), the overall performance can be estimated by the total entropy production rate of the apparatus which should be as small as possible. If we had not this single quantity, two completely different parameters would serve to find out, for example, if an increase of heat transfer accompanied by an increase of pressure drop is an increase with respect to the overall performance of the apparatus. This, however, would be like comparing apples and pears. Therefore it would be welcome by many users of commercial CFD software if these codes could also determine the entropy production in momentum and heat transfer problems of technical interest, i.e. for complex turbulent flows. Before this can be achieved one should develop a general strategy how to get the information about the total entropy production ‘‘as cheap’’ as possible but also as precise as necessary. Two different strategies are the scope of our study.

ergy conservation which are incorporated in CFD-codes anyway. However, there is no need to solve (1) in order to determine s in the field, since s for single phase fluids is a function s(T, p) of temperature and pressure only. They both are known in a flow field once a conventional solution (based on mass, momentum and energy conservation) is found! In that sense the entropy s can be looked upon as a post-processing quantity that can be determined, once the flow- and temperature fields are known! But, does it mean that we also find the entropy production as a post-processing quantity? The answer is ‘‘yes’’, the procedure, however, is not quite trivial since we are dealing with turbulent flows. Equation (1) has to be time-averaged in order to find an equation for the time mean entropy s: According to the Reynolds averaged Navier Stokes (RANS) approach for turbulent flows, prior to time-averaging the equation all quantities are split into time-mean and fluctuating parts, i.e., s ¼ s þ s0 ; u ¼ u þ u0 ; . . . and inserted into (1). The time-averaged equation then reads

2 A short remark on entropy The starting point for all kinds of entropy analysis is the second law of thermodynamics which defines and mathematically describes the fundamentally important quantity called entropy. This quantity, despite its importance is not very popular. The main reason is that people are confronted with it in a rather late stage of their individual learning process. They then often oversee the fact that a question like ‘‘what is entropy?’’ naively assumes it could be answered by just linking it to one already known other quantity. Instead, ‘‘entropy’’ means much more: an own ‘‘entity’’ with complex relations to other physical quantities/aspects, like temperature and different forms of energy as well as their conversion, see for example Herwig (2000). Having that in mind, determining entropy generation in complex flows is nevertheless straightforward.

3 How to determine entropy in a flow field? The entropy in a flow field is a state variable that in its specific form s, i.e. entropy per unit mass (J/kgK), has a balance equation which for a single-phase incompressible flow reads, see for example Spurk (1989):

Here the two entropy production terms are marked by grey shaded boxes with the two functions U and UH given later. The unknown in this equation is s, provided the velocity and temperature field can be determined from the basic equations of mass, momentum and en-

After time-averaging of the equation new unknown terms emerge (closure problem). On the right hand side of (2) a group of terms is shown explicitly (coming from the left hand side and therefore has the minus sign). More terms will emerge when the time-averaging of the other terms on the right hand side will be performed explicitly (so far it is only formally indicated by the bars over these terms).

4 How to determine entropy production in a flow field? There are basically two ways how entropy production can be determined. For laminar flows, for which (1) holds, either • The terms U/T and UH/T2 are determined in detail and are then calculated directly, or • The entropy production [grey boxes in (1)] is calculated by equating it to the rest of the equation and then that rest (from the known temperature and pressure field) is calculated. In what follows, the first option will be called the direct method the second one the indirect method of determining entropy production in a flow field. For turbulent flows the situation is slightly different since additional turbulent terms appear in the balance equation, which now is (2), due to the time-averaging process. In the direct method U=T and UH =T 2 have to be calculated, which is complicated due to additional terms

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that appear after the time-averaging process is carried out as shown next. The indirect method is applicable only when either all additional turbulent terms in (2) outside the grey marked production terms are known or when they can be neglected. In all cases that can be found in the open literature they are neglected when the indirect method is used. The first term to be calculated in the direct method, ðU=T Þ; describes entropy production by dissipation which in a turbulent flow has two parts: viscous dissipation and turbulent dissipation (dissipation due to turbulent fluctuations of the velocity). With ðU=T Þ ¼ SPRO;D þ SPRO;D0 they are: " (      )  2 l @ u 2 @v 2 @w SPRO; D ¼ þ þ 2 @x @y @z T  2  2  @u @v @ u @w ð3Þ þ þ þ þ @y @x @z @x   #  2 @v @ w þ ; þ @z @y " (   0 2  0 2 ) l @u0 2 @v @w SPRO; D0 ¼ þ þ 2 @x @y @z T  0   0  @u @v0 2 @u @w0 2 þ þ þ þ @y @x @z @x #  0  @v @w0 2 þ : ð4Þ þ @z @y Here T¢ in the denominator appears only in higher order terms when expanded into a series and therefore is neglected. The first group represents entropy production by dissipation in the mean flow field, often referred to as direct dissipation. The second group of terms then is the so-called indirect or turbulent dissipation. The second term, UH =T 2 ; describes entropy production by heat transfer and for turbulent flows again has two parts: entropy production due to time mean temperature gradients and production due to gradients of the temperature fluctuations. With UH =T 2 ¼ SPRO;C þ SPRO;C 0 they are " 2  2  2 # k @T @T @T þ þ SPRO;C ¼ 2 ; ð5Þ @x @y @z T "   0 2  0 2 # k @T 0 2 @T @T þ þ SPRO;C 0 ¼ 2 : ð6Þ @x @y @z T Thus, for turbulent flows four groups of entropy production terms appear in the time-averaged entropy balance equation: 1. SPRO;D : entropy production rate by direct dissipation, 2. SPRO;D0 : entropy production rate by indirect (turbulent) dissipation,

3. SPRO;C : entropy production rate by heat conduction with mean temperature gradients, 4. SPRO;C0 : entropy production rate by heat conduction with fluctuating temperature gradients. In a rational approach the additional turbulent terms SPRO;D0 and SPRO;C0 cannot be neglected but should be subject to turbulence modelling.

5 The direct method In the direct method the entropy production terms are calculated in the post-processing phase of a CFD calculation. That means, they are determined by using the known field quantities (time-mean) velocity and temperature. Two of the four terms, SPRO;D and SPRO;C ; according to (3) and (5), respectively, can be determined immediately. The other two, SPRO;D0 and SPRO;C 0 ; first have to be related to the known field quantities through turbulence modelling. In Kock (2003) and Kock and Herwig (2004a, b) turbulence models are proposed that relate these terms to the turbulent dissipation rate e and the mean temperature T : Since in almost all turbulence models (ke, kx, Reynolds stress, ...) the dissipation rate e is included, this kind of modelling turbulent entropy generation is generally applicable. The two models are .e SPRO;D0 ¼ ; ð7Þ T at SPRO;C 0 ¼ SPRO;C : ð8Þ a Since now all four entropy production terms can be calculated, applying the direct method should be straight forward. I turns out, however, that numerical problems arise. In Fig. 1 the distribution of SPRO;C in the cross section of a heated pipe is shown for a laminar and a turbulent flow. In the turbulent case, due to the steep gradients of T close to the wall, entropy production accumulates in this part of the flow field, c.f. (5) for SPRO;C : In order to resolve it the numerical grid would have to be refined unacceptably in this region. However, this problem can be circumvented by finding wall functions for the entropy production terms. These wall functions are analytical expressions for SPRO;i ði ¼ D; D0 ; C; C 0 Þ in the immediate vicinity of the wall that can be found from asymptotic considerations (Re fi ¥, r fi 1). In these functions the universal character of turbulent flows close to a wall is expressed, like in the famous law of the wall that states a logarithmic velocity profile in the near wall region of a forced convection flow field. These wall functions are given in the Appendix, details of their derivation can be found in Kock (2003). With these functions the numerical grid needs no extreme refinement close to the wall.

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Fig. 1 Numerical calculation of SPRO;C for fully developed laminar and turbulent heated pipe flow with qw=const. Details in Kock (2003)

Next, two examples should illustrate the benefits of calculating entropy production in heat transfer design problems.

tropy production is an order of magnitude smaller than for example at Re=2.105. 5.2 Example 2: optimum turbulence promoters

5.1 Example 1: optimum diameter This example is adopted from Bejan (1996). His solution of the problem will be given in the next section where the indirect method is described. Figure 2 shows the problem: A fully developed turbulent pipe flow of water is heated over a length L by a constant wall heat flux density until its caloric mean temperature is raised from 300 to 310 K. This, however, can be performed with various values of D and L, provided the heat transfer surface pLD remains the same (in this example 0.42 m2). The question arises, whether there is an optimum value of D which corresponds to a minimum entropy production. This can be expected since for D fi 0 the pressure drop and therefore SPRO,D increases ad infinitum and for D fi ¥ this happens to the temperature differences in a cross section and thus for SPRO,C. Figure 3 shows numerical results of the direct method applied to seven different values of D. Instead of D the _ Reynolds number Re ¼ 4m=pgD is used in the diagram. _ Since 4m=pg is constant in this example we have D  Re1 so that small Reynolds numbers correspond to large diameters. There appears a definite minimum of entropy production at about Re=9.104. Here, the enFig. 2 Fully developed pipe flow of water (cp=4.2 kJ/kgK) with a heating section of length L T1 T2: caloric mean temperatures

A typical example of a complex convective heat transfer situation is given in Fig. 4, which shows a pipe of diameter D=25.4 mm and a length of 27.5 D. The turbulent flow of air is heated in the midsection of length L=15.5 D by imposing a constant wall heat flux density qw ¼ 8; 200 W=m2 : In order to increase the heat transfer performance of the pipe, a twisted tape (turbulence promoter) is inserted in the heating section. This leads to a considerable increase of the Nusselt number (Nu) but also increases the pressure loss coefficient cf of the whole device. This tape cuts the circular cross section into two half circles which spiral along the axis of the pipe. The slope of this spiralling is measured in terms of D/LT, where LT is the length over which the azimutal angle of the tape changes by 360. The wanted increase in heat transfer and the unwanted increase in pressure loss are shown in Fig. 4 for different slopes of the twisted tape, i.e. for different values of D/LT. Experimental results from Zhang et al. (1997) compare quite well with numerical calculations performed with the finite volume code CFX and a standard high Reynolds number ke turbulence model. Since both dimensionless parameters, Nu and cf, in-

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Nusselt Number is increased by 40%, but at the expense of an 400% increase in cf! Both effects, the wanted and the unwanted, can simultaneously be evaluated by calculating the overall entropy production of the system. If SPRO increases through a certain change in the system this change is counterproductive, otherwise it is beneficial in the sense that less entropy production means a reduced loss of available work. Figure 5 now can give a clear answer whether it makes sense to insert the tape or not and which value of D/LT is the best: • Yes, it makes sense since the overall entropy production is reduced for a certain parameter range of D/LT. • D/LT should be  0.18 since then the decrease in entropy production is the strongest, being nearly 8%, i.e. saves that percentage of exergy (available work). Fig. 3 Direct method: Numerical integration of (3), (5), (7) and (8) over the control volume (length L, diameter D)

crease monotonically for increasing D/LT, there is no optimum discernable. And, a more fundamental question arises (without answer, so far): Should one insert the twisted tape, since for example for D/LT=0.2 the

6 The indirect method Entropy production in turbulent flow with heat transfer according to (2) is the sum of U=T and UH =T 2 ; i.e.:

Fig. 4 Heat transfer augmentation by an inserted twisted tape; Re=um D/m=5.1·104; Nu0, cf0: no tape inserted. Experimental data from Zhang et al. (1997)

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Since one is interested in the total entropy production of the flow field, (9) must be integrated over the entire flow domain. This corresponds to the global balance shown in Fig. 6 which can be cast into the  : time mean velocity perpendicular following form ( w to a surface A):

Fig. 5 Total entropy production for various slopes of the twisted tape in a heated pipe flow; SPRO,0: no tape inserted

  @s  . SPRO ¼ U=T þ UH =T 2 ¼ |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} @t |fflfflfflfflfflffl{zfflfflfflfflfflffl} PRODUCTION CONVECTION  div! q =T |fflfflfflffl{zfflfflfflffl} MOLECULAR FLUX  0 0  @u s :   þ . @x |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} TURBULENT FLUX

ð9Þ

On the right hand side the last term represents a turbulent entropy flux analogous to the turbulent heat flux in the energy equation (which often is modelled by introducing a turbulent or eddy thermal diffusivity). Indeed, this is the only important additional turbulent extra term on the right hand side since time-averaging of the molecular flux term div! q =T does not lead to further terms on our level of approximation. Therefore, the local rate of entropy production [left hand side of (9)] can be calculated by determining the right hand side when there is a turbulence model for the turbulent entropy flux or when it is neglected.

Here, the turbulent flux is written in terms of a turbulent entropy flux vector ð! q =T Þt which could be modelled if necessary, i.e.

 0 0  @u s þ    ¼ div ð! . q =T Þt : @x

ð11Þ

Again, entropy production [left hand side of (10)] can be determined by calculating the right hand side of (10). This can be done whenever the turbulent flux term can be determined or neglected. In all case studies that can be found in the literature it has been neglected. This, however, is only justified in special situations as will be demonstrated by comparing such approximate solutions of the indirect method to the full solutions by the direct method given in the previous section. 6.1 Example 1, resumed Bejan (1996) solved this problem by the indirect method, neglecting the turbulent flux ˆ (which appears at the inlet A1 and outlet A2) completely and accounting for the molecular flux ´ only at the wall (i.e. at A0 in Fig. 6 but not at A1 and A2 where these terms are also nonzero). With these approximations it follows for the total entropy generation rate (in W/K), details in Kock (2003):

Fig. 6 Flow domain with volume V, impermeable wall A0, inlet A1 and outlet A2. The numbers refer to the terms in (10)

A0 A2 V 0

1

3

4

A1 3

2

3

4

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neglected by assuming a constant Nusselt number, may become a strong effect (covered by the direct method). Figure 7 shows exactly this trend. On the other hand this example for Re fi ¥ may serve as a test case for the solution by the direct method described in the previous section. 6.2 Example 2, resumed With the indirect method we make the same assumptions as in Example 1: the turbulent as well as the molecular flux over the entrance and exit cross sections are neglected, the entropy flux associated with the heat flux qw across the wall is accounted for. The convection terms  in (10) can be evaluated when the entropy s of air is assumed to be that of an ideal gas, i.e. Fig. 7 Comparing the indirect method Bejan (1996) to the results of the direct method. The two asymptotes in the double-log plot have the gradient +5.75 and 1.8 according to (13)

Z Z Z

4

SPRO dV ¼ 

q2w D2 pL

8m_ cp lnðT2 =T1 Þ cf þ Nu1 : p3 .2 D6 q2w kT1 T2 ð12Þ

which, with the empirical relations cf=0.316 Re1/5 and Nu=0.023 Re0.8 Pr0.4 for fully developed turbulent pipe flow, can be cast into the general form Z Z Z SPRO dV ¼ C1 Re5:75 þ C2 Re1:8 ; ð13Þ V

where C1, C2 follow from (12). Since the neglected terms in the cross sections A1 and A2 according! to (10) are proportional to the!axial heat flux density q and according to Fouriers law q  dT =dx holds with dT/dx  L1  D  Re1 we finally get ! q  Re1 : Therefore, the approximations (neglecting certain terms) are justified for large Reynolds numbers (if at all). Deviations from the full solution of the direct method thus can be expected for small Reynolds numbers, even more since small Reynolds numbers mean short pipes in which the thermal entrance effect,

Fig. 8 Comparing the indirect method, (15), to the results of the direct method

s2  s1 ¼ cp ln

T2 p2  R ln : T1 p1

ð14Þ

With these assumptions the balance according to (10) is, when A0 is the heat transfer surface Z Z Z Z p2 T2 qw _ ln þ mc _ p ln  SPRO dV ¼ mR dA : p T T 1 1 V A0 w |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} dissipation

conduction

ð15Þ Assuming that the temperature change due to viscous heating and gas expansion is small compared to the temperature increase by heat transfer we can separate the entropy production rate into the a part due to SPRO,D (dissipation) and one due to heat conduction SPRO,C (conduction) as indicated in (15). A further partitioning into effects by mean and fluctuating parts, however, is not possible. In Fig. 8 both parts of the entropy production rate are shown separately for the eleven parameter values of D/LT that were calculated by the direct method in the previous section. Whereas the dissipation part of entropy production deviates by about 15% the heat transfer entropy production is off by almost 50%. Obviously in this highly turbulent situation and especially with flow and temperature fields that are

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very different in the vicinity of the entrance compared to the exit, the flux terms ´ and ˆ in the general balance equation (10) are important and must not be neglected.

Table 1 Constants in the wall functions for pffiffiffiffiffiffiffiffiffi SPRO;C : Ts ¼ qw =.cp u2s ; Ecs ¼ u2s =cp Ts ; us ¼ sw /.

SPRO;D and

7 Discussion Two methods of accounting for entropy production in the post processing phase of a CFD solution for turbulent convective heat transfer problems have been introduced. • The direct method, based on (3), (5), (7) and (8). With these equations the four local parts of entropy production can be determined. Integration of these field quantities over the whole flow domain results in the overall entropy production rate in W/K. Approximations are involved in the turbulence modelling of SPRO;D0 and SPRO;C 0 ; in finding the wall functions and in integration by finite volumes. • The indirect method, based on the volume integrated time-averaged entropy balance equation (10). Though this equation is exact, its application for practical problems involves several approximations. Almost always the entropy flux terms across the inlet and outlet boundaries are neglected (since unknown). Further approximations are involved when inlet and outlet quantities are assumed to be mean values in the cross sections, neglecting their actual profiles. Obviously the direct method is superior and should be applied in complex flow situations. And, there is one more important advantage of this method: From the direct method we get the information how the overall entropy production is distributed between the four single  C 0 Þ; an information which the  D0 ; C; mechanisms ðD; indirect method cannot provide. It may, however, help to understand the physics of the complex process and be important in finding ways to reduce the overall entropy production in a technical device. One should never forget: the entropy production rate times ambient temperature is the rate at which exergy is destroyed. It is measured in Watt and has to be paid for.

þ þ The wall distances yln  and yln C  correspond to the D intersection of the asymptotic representation of the velocity and temperature profiles, respectively, for y+ fi þ þ 0 and y+ fi ¥. They are yln  ¼ 11:6 and yln C  ¼ 12:1 for D þ Pr = 0.71 and yln C ¼ 7:3 for Pr = 5, respectively. In order to find a representative value of the entropy production in the wall adjacent volume of finite volume approach, (16) is integrated over this volume. At a distance y+ mp (mp: midpoint, centre of the volume) we thus get:   rffiffiffiffiffiffi  pffiffiffiffiffiffi pffiffiffiffiffiffi AD p 1 þ þ ¼  b SPRO; a 2y  erf b   D mp D D Dmp þ 2ymp bD 2   pffiffiffiffiffiffi  erf  bD aD ;

þ SPRO; ¼  Cmp

8 Appendix

1 þ 2ymp



ð17Þ  rffiffiffiffiffiffi  pffiffiffiffiffiffi q ffiffiffiffiffi ffi AC p þ  erf bC 2ymp  bC aC 2 bC  qffiffiffiffiffiffi   erf  bC aC :

þ þ 8.1 Wall functions for SPRO;  and SPRO;C  D

The general form we assume for these two wall functions with respect to the mean profiles is h i 2 þ   C: SPRO;i ¼ Ai exp bi ðy þ  ai Þ ; i ¼ D; ð16Þ In Kock (2003) the constants are determined from asymptotic considerations (y+ fi 0) and DNS data. They are listed Table 1.

ð18Þ

þ þ and SPRO; 8.2 Wall functions for SPRO; D0 C0

These wall functions are found by patching the asymptotic representations at their intersection points þ þ yln  and yln C  ; respectively. D

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After an integration over the wall adjacent finite volume the midpoint values for the entropy production are  1 Tw þ þ SPRO;D0 mp ¼ þ 0:15Ecs yln 2ymp Ts D 

Tw2 1 Ts  þ logð2ymp Þ þ CDþ þEcs 2  log 1 þ T j Tw s   Ts þ þ logðyln D Þ þ CD  log 1 þ ; ð19Þ Tw 2 1 4 þ þ 0:15 Pryln SPRO;C 0 mp ¼ þ C 2ymp þ

1  þ þ CCþ Ts =Tw þ log yln C



1 

Ts =Tw þ log

þ 2ymp



þ

CCþ

3 5:

ð20Þ

with CDþ ¼ 5:0 and CCþ ¼ 13:7 Pr2=3  7:5:

References Bejan A (1978) General criterion for rating heat-exchanger performance. Int J Heat Mass Transfer 21:655–658

Bejan A (1996) Entropy generation minimization. CRC Press, Boca Raton Casey M, Wintergerste T (eds) (2000) ERCOFTAC Special Interest Group on Quality and Trust in Industrial CFD-Best Practice Guidelines, ERCOFTAC Publications (http://www.ercoftac.org) Herwig H (2000) Was ist Entropie? Eine Frage–Zehn Antworten. Forschung im Ingenieurwesen 66:74–78 Ho¨lling M, Herwig H (2004) CFD-today: Anmerkungen zum kritischen Umgang mit kommerziellen Software-Programmpaketen. Forschung im Ingenieurwesen 68:150–154 Kock F (2003) Bestimmung der lokalen Entropieproduktion in turbulenten Stro¨mungen und deren Nutzung zur Bewertung konvektiver Transportprozesse. Dissertation, TU HamburgHarburg Kock F, Herwig H (2004a) Local entropy production in turbulent shear flows: a high-Reynolds number model with wall functions. Int J Heat Mass Transfer 47:2205–2215 Kock F, Herwig H (2004b) Entropy production for turbulent shearflows and their implementation in CFD codes. In: Proceedings of the international symposium on advances in comp heat transfer, Norway April 19–24, paper CHT-04-112 Spurk JH (1989) Stro¨mungslehre. Springer, Berlin Heidelberg New York Zhang YM, Hang J, Lee C (1997) Heat transfer and friction characteristics of turbulent flow in circular tubes with twistedtape inserts and axial interrupted ribs. Enhanced Heat Transfer 4:297–308