Andreas Abdon and Bengt Sunden Division of Heat Transfer ... · PDF fileDivision of Heat...

A numerical method to calculate the coolingperformance of a rib-roughened duct

Andreas Abdon and Bengt Sunden

Division of Heat Transfer, Lund Institute of Technology, Box

Abstract

The trend in gas turbine combustor design is to use forced convection coolingwith enhanced surfaces instead of film cooling. Therefore there is a need todevelop design tools for the combustor cooling channels. This paper presents anumerical method to predict heat transfer and pressure drop in a ribbedrectangular channel by using the k-£ model with wall functions. The wallfunctions are based on experimental investigations of rib-roughened surfaces. Inthe optimization of the cooling channel the duct height, rib height, rib pitch andrib configuration are the variables while the mass flow rate and pressure drop arefixed. Besides the optimization study, a test case is examined to evaluate theperformance of the numerical method.

1 Introduction

Numerical simulations have been used to predict flow and heat transferin so-called enhanced cooling channels for some years. Unfortunately,the gas turbine engineers are still left without a powerful design toolapplicable for the cooling problems of gas turbine blades and combustionchambers.

The complex flow and heat transfer associated with enhancedgeometries for combustor wall cooling are indeed very difficult tohandle, as there are separated flow and turbulence present. Manyinvestigators have examined ribbed surfaces, which are commonly usedto increase The heat transfer of a plane surface. A large number of thenAestigators use numerical simulations in which the ribs are modelled injjiaJ. i e, me modelled geometry usually includes u rib pitch and the rib,_2:^ f"_< example Annan and Raba^ [1]. laco\ides 3rd Raisee [2]%L , * jru &iLJh/ iuC fl,'W amunu t%,. i cs ia fc cc'^iniex thai ;t cannot br

Transactions on Engineering Sciences vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533

382 Advanced Computational Methods in Heat Transfer

predicted accurately without very large computer resources in manycases. This makes the detailed models unattractive in an optimization

study where many different flow-situations have to be considered.

An alternative way to approach this problem is to consider theenhanced surface as a rough one. The detailed modelling of theenhancement promoters, for example ribs, is then avoided. Thismodification can be implemented through the use of wall functions of arough wall, which may be based on experimental correlations.Simulations that use wall functions of a rough wall have been presentedby Cunha [3] and Youn et al [4].

The main objective of this work is to apply and demonstrate anumerical calculation method which is applicable as an optimization-toolof enhanced cooling channels.

2 Problem formulation

This work focuses on prediction of heat transfer and pressure drop incooling channels attached to the hot wall of a gas turbine combustor. Thechannels are rectangular (the height and width of the channels are keptconstant in the flow direction) with attached ribs on one of the broaderwalls (the hot wall of the combustor). The ribs serve as turbulence andheat transfer promotors.

When designing a combustor cooling channel the mass-flow rate isgiven and there is a maximum pressure drop allowed for the cooling air.This means that the best design of the channels is the one which gives thehighest heat transfer for a fixed mass-flow rate and a fixed pressure drop.The cooling channel design parameters are as follows: duct height, ribheight, rib pitch and rib configuration (perpendicular, inclined, V-shapedor broken ribs).

To simulate the flow and heat transfer of the cooling channelsnumerically, the medium is considered as incompressible air withnegligible viscous dissipation. No buoyancy effects are present and thefluid properties are assumed independent of temperature. The flow in theducts is regarded as steady and periodically fully developed.

3 Governing equations

The governing equations to be solved are the momentum, continuity andenergy equations, plus the transport equations for the turbulent kineticenergy and dissipation added through the turbulence model. If thepressure and temperature are splitted up in one periodic part and onelinearly decreasing/increasing part, periodic boundary conditions may beapplied for all the variables:


Advanced Computational Methods in Heat Transfer 383

,y) = P*(x,y)-(3-x (1)

In the expressions above (3 represents the pressure gradient in the main

flow direction (x-axis) and a represents the temperature gradient in the

same direction. The temperature gradient in the main flow direction isdefined as:

9Ty /9x = q A /(me p) (3)

If the expressions above are inserted in the governing equations oneobtains:

=0

i a

?toi:L.d9X: 3X:

y-+*«- T** o.J

v -. •'- + , -uu3x, 3x,

_v+Y,

Pr 0,

3k -,_, au,- - u u - - - i -

(4)

(5)

(6)

(7), (8)

(9)

3e(10)

In the equations above the constants are set to: Ok=1.0, <3=\.2ll, a,=0.9,

q=0.09,Ci=1.44, C2=1.92.

4 Numerical solution method

A finite volume technique with a non-staggered body fitted grid isemployed for solving the governing equations. The hybrid-scheme(Versteeg and Malalasekera [5]) is used for discretization of theconvection terms and the pressure-velocity coupling is handled by theSFMPLEC algorithm (Versteeg and Malalasekera [5]). The equations aresolved in an iterative fashion and the code CFX 4.1b (AEA IndustrialTechnology [6]) is used.



The wall functions are correlated through experiments of ribbedrectangular channels. Several investigations by Han et al [7,8,9] haveprovided the data for the different configurations used here. In thesestudies, the dimensionless velocity and temperature fields are defined by(if the Prandtl number is constant):

i + R(e+) (11)

) + G(e+) (12)

The validity of these functions are limited to flow with the characteristicsknown as "fully rough" flow. This is attained when the condition e >50is satisfied, and this condition is fulfilled for all the calculations below.In Table 1, the parameters R and G are displayed for the differentconfigurations used in the calculations presented below.

Table 1. R and G for different rib configurations. R(e+) = a •

G(e+) = c • (e+y . The values are correlated for Pr~0.7.

and

Reference

Han [7]

Han [7]Han [7]Han et al [8]

Han et al [8]

Han et al [8]

Han & Zhang [9]

Han & Zhang [9]

Configuration

p/e=IO

p/e=15p/e=2060V

60 //

90 II

60 Vb

90 II

a

3.200

3.6884.0791.720

2.050

3.180

0.923

3.180

b

0.000

0.0000.000

0.000

0.000

0.000

0137

0.000

c

3.700

3.7003.700

1.350

1.520

3.970

1.348

2.660

d

0.280

0.2800.280

0.420

0.410

0.280

0.405

0.347

When applying the wall functions, there is a need to calculate the

dimensionless roughness height (e*). This is evaluated as:

/v 03)

The wall functions above are supposed to be independent of the overallgeometry (shape of duct) and only dependent of the roughness present onthe surface, but this has been questioned by Lee et al [10]. Therefore theyshould be used with some care.

The convergence criteria to be fulfilled in the simulations are a scaled

mass flow residual less than 210' and that the changes in friction factor

and mean Nusselt number are very small for a couple of hundrediterations. Underrelaxation is used and the URF's in the simulations are



set to 0.5 for the momentum- and 0.2 for the k- and E-equations. No

underrelaxation of the energy equation is necessary in these simulations.

The constant pressure gradient (3 is specified to create a flow field.

Through the rate of mass flow, the Reynolds number is coupled to p.

Thus various values of (3 correspond to various Reynolds numbers. For

the optimization study in this paper (where the mass flow rate andpressure drop are given) a special procedure is necessary, as theoptimization variables include both the channel height and rib size. Thechannel height is then fixed for each computed value, and the rib size isvaried in an iterative fashion in order to achieve the correct rate of massflow.

A DEC-Alpha 3000 computer is used in all the calculations below,and converged solutions are obtained within 3000 iterations. Thiscorresponds to a running time of about 15 minutes.

4.1 Definition of variables

The hydraulic diameter of the channel is defined as:

0% = 4A/P = 2WH/(W + H) (14)

The Reynolds number is then given as:

where the mean velocity is calculated through:

U = Ju-dA/JdA (16)

for a cross-sectional plane perpendicular to the main flow direction.The friction factor is calculated from:

The average Nusselt number (evaluated for all the heated surfaces)is defined as:

-TJ) (18)

"J-dA/jdA (19)

I dA/J|U| dA (20)

5 Results and discussion

5.1 Test case

The test case geometry is a rectangular duct (W=43.7 mm, H=20.4 mm)with all four walls heated by a constant heat flux. Two of the walls (the



broader ones) are equipped with square ribs (perpendicular to the main

flow direction). The fluid pumped through the duct is air at atmospheric

pressure with constant fluid properties (Tref=273K). This gives a Prandtlnumber equal to 0.714, and the Reynolds number is fixed to 20000. Inthe calculations, periodic boundary conditions are applied for the ductinlet and outlet to simulate fully developed flow and temperature fields.The grid consists of nonuniform rectangular cells and is defined byNX=3, NY=14, NZ=20. Increasing the number of cells in the cross-sectional plane of the duct alters the friction factor and heat transfercoefficient less than 2%. To simulate the effects of the ribs on the flowand heat transfer, the "rough" wall functions of Han [7] are applied forthe two ribbed walls. These are valid for the following flow conditions:rectangular channel (1<W/H<4) with two ribbed walls,

0.021<e/Dh<0.078, 10<p/e<20, 8000<Re<80000. Pr=0.704.The results of the test-calculations are compared to the correlations

developed by Han [7], who examined similar conditions as above byexperimental measurements. The size of the ribs is 0.584 mm, 1.0 mmand 2.2 mm in three different calculations (p/e=10 in all cases). Thiscorresponds to e/Dh=0.021, e/Dh=0.036 and e/D|,=0.079, respectively,and the results are displayed in Table 2.

Table 2. Results of test case simulations compared to Han [7].

e/Dh

0.021

0.036

0.079

0.036

0.036

p/e

10

10

1015

20

flQf

2.24

3.02

5.78

2.68

2.45

f Han 10'

2.41

3.42

6.83

2.99

2.70

Af (%)

-7.1

-11.7-15.4

-103

-93

Nu

74.5

81.7104.2

78.6

763

NUHan

91.4

102.3124.1

96.8

92.9

ANu(%)

-18.5

-201-16.0

-18.8

-1.7.9

In order to investigate the influence of the ratio p/e on the predictiveability, another three calculations are carried out. These are for a fixedrib height of 1 mm (e/Dh=0.036) and the results are also given in Table 2.

The test calculations above indicate that predictive ability of thenumerical method varies with relative rib size (e/Dh) and pitch (p/e).However, these variations are assumed to be sufficiently small, so thatthe method may be used to indicate the performance of differentconfigurations.

5.2 Investigation of grid sensivity

When using wall functions, one must always be careful to place the firstgrid-points at a distance (yp) from the wall, so that the functions are valid



in that point. According to Webb et al [11], the wall functions of the

rough walls should only be applicable for distances greater than theroughness height. This means that the grid-points next to the rough wall

should be arranged to have y?>e in the simulations. For the near wall

cells of the smooth walls, the value of y* should not be less than 30 toensure validity of the law of the wall. An investigation of the effect of y?on the friction factor and the heat transfer coefficient is displayed in Figs.

1 and 2 (values are scaled with the value at yp/e=l and labelled f and a*,

respectively). The calculations are carried out for a rectangular channel(W=43.7 mm, H=20.4 mm) with one ribbed wall (wall functions of Han[7] applied) heated by a constant heat flux. Air flows through the duct atRe= 16522 with constant fluid properties taken at Tref=720 K. Periodicboundary conditions are applied to achieve fully developed flowconditions. The grids used are nonuniform and defined by NX=3, NZ=20and NY=9-14 depending on the value of y?. Increasing the number ofcells in the cross-sectional plane of the duct alters the friction factor andheat transfer coefficient less than 1%.

1.1Z

1.1

1.08

1.06

1.04

1.02i -

r-».. f+

•••&•• a*

B*

Q / . *Q\ X .0--"fc. -° ••""B-nr'.*Qr • i

1.U4

1.02

1

0.98

n OA

...o-f*

-*-- a*

,X**

®'"' y

Ef"

0 1 2 3 4

yp/e

Figure 1. Influence of y? on f

and a when e=l .0 mm.

0.6 0.8 1 1.2

yp/e

Figure 2. Influence of y? on f

and a when e=2.5 mm.

1.4

Fig. 1 shows that f and a are indeed depending on the value of y?, and

there is a rather large deviation for values of y?/e>2. This deviation mayhowever be due to the size of the cells next to the walls. The size of thenear wall cells is determined by the value of y?, as a large y? value canonly be achieved by a large cell. In the simulations, large cells imply badresolution of gradients in the flow, and therefore loss of accuracy.

For large ribs, the condition of yp/e>l always means that large cells

have to be applied. This may lead to overpredictions in the friction factorand heat transfer coefficents as indicated in Fig. 1. To investigate thesensivity to changes in y? in these cases, a channel equipped with big ribsis simulated and the results are displayed in Fig. 2. In Fig. 2 it is shown



that a change in y?/e gives a larger change in f and a compared to the

small rib (Fig. 1) for the range 0.7<yp/e<1.4. Thus the results are muchmore sensitive to the value ofy? when large ribs are simulated.

The indication that large cells implies overpredictions and that the

results show great sensitivity to the value of yp for large ribs are twoimportant facts to be aware of when using this calculation method. Inorder to keep the size of the near wall cells as small as possible and

fulfill the demand of yp/e>l, the value of yp/e is 1.00±0.03 in all the

simulations in this paper.

5.3 Optimization study of a test combustor cooling channel

A rectangular duct with the same width and length as in the test caseabove is applied as a cooling channel of a gas turbine test combustor.One of the broader walls of the channel serves as the hot wall of thecombustor that is to be cooled, and this wall is equipped with ribs. Theheat load from the combustor is simulated as a constant heat flux on thesame wall. The mass-flow rate and the pressure drop of the cooling air isfixed, and the values of these parameters (rh = 0.018kg/s, dp/dx - 895Pa/m) are taken from a test combustor (at atmospheric pressure). Periodicboundary conditions are imposed on the inlet and outlet of the duct tosimulate fully developed flow and temperature fields. The cooling air isconsidered to have constant fluid properties (Tret-720K) and the Prandtlnumber is 0.69.

The aim of the optimization study is to maximize the heat transfer forthe given mass-flow rate and pressure drop. This may be done by alteringthe geometry which is affected by the following variables: coolingchannel height (H), rib pitch (p), rib size (e) and rib configuration (ribangle of attack, etc.). The results are displayed below for differentgeometries. In the calculations the grid is varied as the channel height ischanged (i.e. NY is altered), but the number of cells in the x- and z-directions are held constant (NX=3, NZ=20). Increasing the number ofcells in the cross-sectional plane of the duct alters the friction factor andheat transfer coefficient less than 2%.

To calculate the rib-roughened channel data in Fig. 3, the wallfunctions correlated by Han [7] are utilized. These are valid for theconditions outlined in section 5.1. As indicated in Fig. 3, the coolingchannel equipped with ribs clearly gives a higher heat transfer comparedto a smooth one (e=0). The smooth channel data is calculated by usingthe standard wall function applied on the hot wall. Unfortunately, thevalidity range of the wall functions of the rough wall covers only fullyrough flow (e*>50), which explains the lack of data for values of e/D%



below 0.02 for the ribbed ducts. A small value of e/Dh corresponds to

small ribs and also a small channel height. The results for the channels of

heights larger than H=23 mm (gives e/Dh=0.079) should be used with

care, as the "rough" wall functions are not correlated for e/Dh greaterthan 0.078. The Reynolds number for the data in Fig. 3 varies withchannel height and ranges from 15400 to 18100. As shown in the figure,there is a very small influence of relative rib size (e/Dh) and pitch (p/e)on the mean heat transfer coefficient.

130

125

120

115

110

105

100

00

O p/e=IUOp/e=15A p/e=20• e=0

0.02 0.04 0.06 0.08 012

Figure 3. Heat transfer coefficient as a function of e/Dh and p/e. Smoothduct is denoted e=0.

170

160

150

140

130

120

110

100

<>60Vn60//A90II• e=0

0.02 0.04 0.06 0.08 0.12

Figure 4. Heat transfer coefficient as a function of e/Dh for V-shaped

(60"), inclined (60") and perpendicular ribs (90°). Smooth duct isdenoted e=0.

To calculate the rib-roughened channel data in Fig. 4, the wall functionscorrelated by Han et al [8] are utilized. These are valid for the following



conditions: square channel (W/H=1) with two ribbed walls, e/Dh=0.0625,p/e=10, and 15000<Re<90000. This means that the wall functions areextrapolated to be valid for different channel aspect ratios and values ofe/Dh than the experiments. The smooth duct is inserted into Fig. 4 (samevalue as in Fig. 3) as a reference value. The Reynolds number variesfrom 15400 to 18100 in the data of Fig. 4.

The trend in Fig. 4 is that inclined ribs increase the heat transfercompared to the ones perpendicular to the main flow direction, and themean improvement of the heat transfer coefficient is about 30% for the60 V configuration.

160

150

140

130

120

110

100 1 1

o o

A A

0 0

/

0

* A A

1 L_

o60

A 90

• e=(

Vb

II

)

0.02 0.04 0.06 0.08 0.1 0.12

Figure 5. Heat transfer coefficient as a function of e/Dh for broken V-

shaped (60°) and perpendicular ribs (90°). Smooth duct is denoted e-0.

In Fig. 5, the data is calculated as before but the wall functions used arefrom Han & Zhang [9]. These are valid for: square channel (W/H=1)with two ribbed walls, e/Dh=0.0625, p/e=10, and 15000<Re<90000. Thebroken and V-shaped ribs provide approximately 30% increase in heattransfer coefficient compared to the perpendicular configuration. TheReynolds number varies from 15400 to 18100 in Fig. 5.

7 Conclusions

A numerical method for calculation of the cooling performance ofvarious types of rib-roughened channels in turbulent flow is presented.The method utilizes wall functions that imply fast results. It is applied inthe optimization of a gas turbine combustor cooling channel, with theaim to maximize the heat transfer for a given pressure drop and fixedmass-flow rate. The results of such an optimization study for arectangular duct with one ribbed wall of different rib size, pitch andconfiguration indicate the following; the heat transfer can be augmented



10-50% by using ribs compared to a smooth duct, and V-shaped or

broken V-shaped ribs may increase the heat transfer 30% compared toperpendicular ribs.

Unfortunately, the use of the calculation method is strongly limited bythe validity range of the wall functions. Also, the results are shown to besensitive to the grid structure.

8 References

[1] Arman, B., Rabas, T.J., "Two-layer-model predictions of heattransfer inside enhanced tubes", Numerical Heat Transfer, Part A,Vol.25, pp. 721-724, 1994.

[2] lacovides, H., Raisee, M., "Computation of flow and heat transferin 2-D rib roughened passages", Addendum 2nd Int. Symposium onTurbulence, Heat and Mass Transfer, Hanjalic, K., Peeters, T.W.J.(Editors), 1997.

[3] Cunha, F.J., "Turbulent flow and heat transfer in gas turbine bladecooling passages", ASME-Paper No. 92-GT-239, 1992.

[4] Youn, B., Yuen, C, Mills, A.F., "Friction factor for flow inrectangular ducts with one side rib-roughened", ASME J. FluidsEngineering, Vol. 116, pp. 488-493, 1994.

[5] Versteeg, H.K., Malalasekera, W., "An introduction tocomputational fluid dynamics the finite volume method", LongmanScientific & Technical, Essex, England. 1995.

[61 AEA Industrial Technology, "FLOW3D User Guide", CFDDepartment, Harwell Laboratory, U.K., 1992.

[7] Han, J.C., "Heat transfer and friction characteristics in rectangularchannels with rib turbulators", ASME J. Heat Transfer, Vol. 110,No2, pp. 321-328, 1988.

[8] Han, J.C., Zhang, Y.M., Lee, C.P., "Augmented heat transfer insquare channels with parallel!, crossed, and V-shaped angled ribs",ASME J. Heat Transfer, Vol. 113, No 3, pp. 590-596, 1991.

[9] Han, J.C., Zhang, Y.M., "High performance heat transfer ductswith parallell broken and V-shaped broken ribs", Int. J. Heat MassTransfer, Vol. 35, No 2, pp.513-523, 1992.

[101 Lee, B.K., Cho, N.H., Choi, Y.D., "Analysis of periodically fully

developed turbulent flow and heat transfer by k-e equation modelin artificially roughened annulus", Int. J. Heat Mass Transfer, Vol.31, No. 9, pp. 1797-1806, 1988.

[11] Webb, R.L., Eckert E.R.G., Goldstein, R.J., "Heat transfer frictionin tubes with repeated rib roughness", Int. J. Heat Mass Transfer,Vol. 14, No 4, pp. 601-618, 1971.


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