SkinFriction and Forced Convection from an Isothermal...

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SkinFriction and Forced Convection from an Isothermal Rough PlateAubrey G. Jaffer
Digilant2 Oliver Street, Suite 901
Boston MA 02109 [email protected]
Abstract
Developed are new correlations for skinfriction and forced convection from an isothermalplate having rootmeansquared heightofroughness εq. Corresponding to Prandtl and Schlichting’s“fullydeveloped roughness” region, the skinfriction correlation is independent of Reynolds number;the convection correlation is a linear function of Reynolds number (versus an exponent of 4/5 forturbulent convection from a smooth plate).
Measurements taken of a plate having precisely 3 mm of roughness match the convectioncorrelation within 3% at Reynolds numbers from 5000 to 50000. A model addressing the transitionfrom fullyrough turbulent to smooth turbulent convection for this bilevel test plate matches thosemeasurements within 2% from Re=5000 to 80000.
Convection measurements of a plate having 1.03 mm of roughness are within the expectedmeasurement uncertainties of the transition model.
An analysis of sandroughness finds that its relation to rootmeansquared heightofroughnessis nonlinear. The resulting scale error explains discrepancies between the Prandtl and Schlichtingformula, two recent papers, and the results presented here.
These new correlations and measurements provide a threefold tighter upper bound for theheight of admissibleroughness than is given in Schlichting’s book “Boundarylayer theory.”
Keywords: skinfriction; forced convection; roughness; sandroughness; admissibleroughness
Table of Contents
Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2PipePlate Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Rough to Smooth Turbulence Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Sand Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Admissible Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Measurement Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Measurement Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Forced Convection Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Directions for Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1. Scope
Convection from a rough surface is important in modeling weather and the thermal behavior of buildings.The related phenomenon of skinfriction has wide application to the fluid dynamics of vehicles and turbines.
The scope of this work is the total (not local) skinfriction and the total turbulent forced convection fromrectangular isothermal rough plates. There are selfsimilar roughness profiles for which the total skinfrictionand convection are welldefined and the local skinfriction and convection are not.
All forced convection measured by the apparatus was turbulent.What are the practical implications of not treating laminar forced convection? Reynolds numbers scale
with characteristiclength. At architectural and meteorological scales, laminar forced convection will rarelyoccur. Laminar forced flows occur frequently at scales much smaller than the 305 mm test plate. But atthose scales, isolated flat rough plates are of less interest than pipes, ducts, spheres, and cylinders.
1
2. Prior Work
In 1934 Prandtl and Schlichting published “Das Widerstandsgesetz rauher Platten” (The Resistance Law forRough Plates[1]) which brilliantly infers the relation for skinfriction resistance for rough plates from theiranalysis of Nikuradse’s measurements of sand glued inside pipes. In “Boundarylayer theory”[2] Prandtl andSchlichting give a formula for fully rough (large Re) total skinfriction coefficient for a rough flat plate:
cf =
(1.89 + 1.62 log10
l
kS
)−2.5102 < L/kS < 106 (1)
In “On the Skin Friction Coefficient for a Fully Rough Flat Plate”[3], Mills and Hang present a formulathey claim to be more accurate on the local skinfriction measurements done by Pimenta, Moffat, andKays[4]. Their total skinfriction formula is:
CD =
(2.635 + 0.618 ln
L
kS
)−2.57(2)
Putting these formulas into similar form:
cf = 0.299
(log10
14.68 · lks
)−2.5CD = 0.404
(log10
71.1 · LkS
)−2.57The roughness in both papers is reported in Nikuradse’s sandroughness metric kS , the height of “coarse
and tightly placed roughness elements such as for example coarse sand grains glued on the surface”. Prandtland Schlichting[1] gives the parameters of Nikuradse’s sand coatings:
kS grains/cm2 k−2S
.08 cm 150 156.25/cm2
.04 cm 590 625/cm2
.02 cm 1130 2500/cm2
.01 cm 4600 10000/cm2
Table 1 Nikuradse’s sand coatings
A correlation retaining its validity when all lengths are scaled by the same factor is a central tenet offluiddynamics theory. Because their packing densities scale as the inverse square of grain size, the .08 cmand .04 cm sand coatings should scale correctly as long as the glue thickness is proportional to grain size.Similarly the .02 cm and .01 cm sand coatings should scale.
The two roughest coatings (largest kS values) had packing densities slightly less than a square packing.The other two had packing densities less than half that of a square packing. But comparing the .02 cmand .04 cm sand coatings, their packing density ratio is nearly 1:2, not 1:4. Thus formulas (1) and (2) willhave incorporated a scale error (between measurements of large and small kS) and should not be applied toL/kS values much larger than 1000 (507 is the largest rough pipe R/kS value in Schlichting[2] Fig. 20.18).Section 5 compares these formulas with the present work.
2
3. PipePlate Analogy
The ratio of the mean heightofroughness in a pipe to the pipe diameter (its characteristic length) affectsthe friction factor used to compute pressure drops in flows through pipes. The Chilton and Colburn Jfactoranalogy (3) relates friction factors to turbulent forced convective heat transfer.
Nu = (f/2)Re Pr1/3 (3)
ASHRAE[5] and Lienhard[6] citing Dittus and Boelter (1930) give correlation (4) for convection in asmooth pipe:
Nu = 0.023 Re4/5 Pr0.3 (4)
Combining (3) and (4) (treating 0.3 ≈ 1/3) and solving for the Darcy friction factor fD = 4f yields (5):
fD = 8 · 0.023 Re−1/5 (5)
For turbulent flow, Re > 2300, fD in a pipe is governed by the ColebrookWhite equation (6):
1√fD
= 1.74− 2 log10
(kSR
+18.7
Re√fD
)= −2 log10
(kS
7.41R+
2.52
Re√fD
) (6)
Figure 1 is a Moody diagram relating fD to Re for rough pipes. It shows that 8 · 0.023 Re−1/5 (5)approximates the solution of equation (6) with kS = 0 over a wide range.
0.01
10000 100000 1x106
1x107
f D
Re
Darcy Friction Factor
fullyrough8*.664/Re
1/2
8*.037/Re1/5
kS/R = 102
kS/R = 103
kS/R = 104
kS/R = 0
8*.023/Re1/5
Figure 1When the sandroughness kS > 0, fD(Re) is asymptotically constant to the right of the fullyrough
trace. Thus Nu in (3) is proportional to Re in rough pipes (versus Re4/5 in smooth pipes).The correlations for laminar and turbulent convection from a smooth flat plate are:
Nu = 0.664 Re1/2 Pr1/3 (7)
Nu = 0.037 Re4/5 Pr1/3 (8)
Like the smooth pipe, turbulent forced convection from a smooth isothermal plate is proportional toRe4/5.
3
According to Lienhard and Lienhard[6], a very thin viscous sublayer “is responsible for a major fractionof the thermal resistance of a turbulent boundary layer.”
At the scale of the viscous sublayer, the inner radius curvature (being perpendicular to the flow) in alarge diameter pipe should be indistinguishable from a flat plate. Because most of the convective thermalresistance occurs close to the surface of a pipe or plate, it is reasonable to expect the turbulent correlationsfor pipe and plate to be related. Combining correlations (7) and (8) with the Chilton and Colburn Jfactoranalogy (3) yields:
fD = 8 · 0.664 Re−1/2 (9)
fD = 8 · 0.037 Re−1/5 (10)
The top two straight traces in Figure 1 are correlations (9) and (10); they do not lie near the curve fora smooth surface. But the trace for (10) is parallel to the trace for (5), both correlations being proportional
to Re−1/5.Although it must have the correct relation to dimensions and parameters, the scale of characteristic
length is otherwise a matter of convention. The forcedconvection correlations are usually monomials in Reand scale easily. With the characteristiclength of a pipe being perpendicular to the direction of flow andthe characteristiclength of a plate being in the direction of flow, it is not unreasonable that scaling may benecessary in order to make correspondences between them.
Scaling the characteristic length R→ L/9.5 and Re→ ReF /9.5 in the ColebrookWhite equation causescorrelations (9) and (10) to approximate the smooth convection curve in Figure 2. If the mean heightofroughness ε, which is perpendicular to the characteristiclength for both pipes and plates, can be correctlyscaled, the result may predict convection for rough plates.
Doubts about how to scale the roughness in the ColebrookWhite equation motivated the constructionof an apparatus designed to measure rough plate convection. The experiments described later in this paperfound that scaling the heightofroughness kS → ε 7.41/9.5 (such that the first term of the argument to log10
becomes ε/L) matched measurements well. The scaled ColebrookWhite equation is (11):
1√fD
= −2 log10
(ε
L+
9.5 · 2.51
ReF√fD
)ReF > 21850 (11)
Its asymptote as ReF →∞ is:
1√fD
= −2 log10
ε
LfD =
1
4[log10
εL
]2 fc =1
16[log10
εL
]2 (12)
0.01
0.1
10000 100000 1x106
1x107
f D
Re
Darcy Friction Factor withScaled Colburn Analogy to Flat Plate Convection
fullyroughε/L = 10
2
ε/L = 103
ε/L = 104
ε/L = 0 8*.664/Re
1/2
8*.037/Re1/5
SchultzGrunow
Figure 2
4
Figure 21.2 of Schlichting[2] shows that the SchultzGrunow formula (13) for turbulent skinfriction ofa smooth plate is well matched with experiment:
fc = 0.427(log10(Re)− 0.407)−2.64 (13)
The close match of this formula to ε/L = 0 in Figure 2 shows that the scaling by 9.5 works for thesmooth plate.
The scaled Colburn analogy asymptote (14) is the result of combining the scaled ColebrookWhiteequation asymptote (12) with the Chilton and Colburn Jfactor analogy (3):
NuF =ReF Pr1/3
32[log10
εLF
]2 (14)
Measurements displayed in Figure 3 show that rough surface convection is a linear function of ReFdown to ReF = 5490. Figure 3 shows that scaled Colburn analogy convection (top trace) is not linear. Itsasymptote (14) is a much better fit to the data.1
10
100
1000
1000 10000 100000
Nu
/Pr1
/3
Re
Scaled Colburn Analogy Convection
SCA ε/L = .01asymptote ε/L = .01
MeasuredSCA ε/L = 0
smooth turbulentsmooth laminar
Figure 3
The deviation of the measured data from correlation (14) at Re < 5000 can be accounted for by themixing of natural and forced convection, although the experimental uncertainty of the mixed model is largerthan 4% in that range. The deviation at Re > 50000 is addressed in Section 4.
Correlation (14) corresponds to the “fullydeveloped roughness” region of Prandtl and Schlichting,but extends to much lower Reynolds numbers. Natural convection prevents investigation of the forcedturbulentlaminar transition in the Convection Machine; measurement of skinfriction drag would be a bettermethodology for those other regions.
1 This may be related to the ColebrookWhite equation showing more gradual transitions between roughand smooth turbulent flows than Nikuradse’s rough pipe measurements.
5
4. Rough to Smooth Turbulence Transition
The roughness of the bilevel plate is not selfsimilar; it exhibits different convection behaviors at differentscales. The slope of the upper end of both curves in Figure 6 is 4/5, but with a magnitude possible only ifoperating with a much shorter characteristic length than the plate’s LF = 0.305 m.
At the leading edge of the plate, the smoothturbulent boundarylayer depth would be much smallerthan the heightofroughness; so fully rough convection (14) applies. The boundarylayer depth increaseswith distance x from the leading edge:
δtur(x) =0.37x
Re1/5x
= 0.37x4/5(L
Re
)1/5
= 0.37LRe4/5x
Re(15)
Solving for Rex where δtur = εpv, the peaktovalley height:
Rex =
(εpvL
Re
0.37
)5/4
Around
Rex >εpvLS
(εpvL
121× 103
0.37
)5/4
(16)
the flow transitions to a regime consisting of smooth turbulent convection (8) from the channels betweenposts at scale LF and smooth turbulent convection (8) from the post at scale LS = 8.3 mm, the length ofthe post in the direction of flow.
Where SS = L/26 is the post center spacing, the longitudinal (parallel to flow) channels between theposts have convection:
NuC/Pr1/3 =SS − LS
SS· 0.037 · Re4/5 (17)
The top of the posts occupy roughly half of the plate area; their convection is:
NuT /Pr1/3 =L2S
S2S
(LF
LS
)1/5
· 0.037 · Re4/5 (18)
The (LF /LS)1/5 factor converts the characteristic length from LS to LF .
Both post sides parallel to the flow also experience smooth turbulent convection at scale LS = 8.3 mm,but because the sides are within the channel’s boundary layer, TF , V and Re are not constant with elevation.The channel velocity profile provides velocity as a function of elevation y, boundary layer depth δ, and freestream velocity V :
v
V=
{(y/δ)
1/7when y ≤ δ;
1, otherwise.
By analogy with the derivation of the thermal profile for laminar flow from Lienhard and Lienhard[6],the turbulent thermal profile would be:
T − TwT∞ − Tw
=
{(y/δt)
1/7when y ≤ δt;
1, otherwise.≈
{(y/(δ Pr−1/3
))1/7when y ≤ δ;
1, otherwise.
The temperature profile will affect side convection by a factor of approximately Pr1/21 (y/δ)1/7
wheny ≤ δ; and 1 otherwise. In air, Pr1/21 is a reduction by 1.5% (for the post sides convection only).
6
0
50
100
150
200
250
300
1000 10000 100000fl
ow
x, m
m
ReF
Convection Mode Regions Versus ReF
ReM
at εpv=2mm
δ = 2mm
Figure 4Each vertical slice of Figure 4 represents the regions of convection on the surface of the 1 mm roughness
plate at the indicated value of ReF . The entire surface is in rough turbulent convection of correlation(14) below the ReM trace where (16) is not satisfied. Above the ReM trace, the convection is modeledas a combination of channel (17), posttop (18), and postside (19) convection. When ReF is larger than40000, there is a small region where the channel boundarylayer height is smaller than the post side heightεpv = 2 mm. Post side convection from this region was modeled, but its effect on total convection wasnegligible. For simplicity, the post side model uses formula (19) for all side convection.
The convection for a vertical slice of the post side is:
χ(x) =0.037
εpv
∫ εpv
0
(y
δF (x)Pr−1/3
)1/7(Rex
(y
δF (x)
)1/7)4/5
dy
=0.037 Pr1/21
εpv
∫ εpv
0
(y
0.37x4/5
(Re
L
)1/5)1/7
Re x
L
(y
0.37x4/5
(Re
L
)1/5)1/7
4/5
dy
= 0.038Re149/175 Pr1/21 x113/175 ε
9/35pv
L158/175
The convection from a row of post sides in line with the flow is:
NuS/Pr1/3 = 2εpvLS
S2S
(LF
LS
)1/51
LF
∫ LF
0
χ(εpv)dx
= 2εpvLS
S2S
(LF
LS
)1/5
0.0231 Pr1/21(εpvLF
)9/35
Re149/175(19)
Integrating the local convections for correlations (14), (17), (18), and (19) over distance:
ReM = min
(ReF ,
εpvLS
(εpvL
Ret0.37
)5/4)
Ret = 121× 103
NuF /Pr1/3 =ReM
32[log10
εLF
]2 +
[SS − LS
SS+L2S
S2S
(LF
LS
)1/5]
0.037(
Re4/5F − Re
4/5M
)
+ 2εpvLS
S2S
(LF
LS
)1/5
0.0231 Pr1/21(εpvLF
)9/35 (Re
149/175F − Re
149/175M
) (20)
7
4
2
0
2
4
1000 10000 100000
%
Re
Measurement versus Theory of 3.0 mm Roughness DownwardFacing Plate
Expected UncertaintyRough Turbulent Asymptote
RoughSmooth TransitionL
4 Mixed ∆T=11K
Figure 5
The constant Ret adds a degree of freedom to the plate model; but Figure 17 shows that 121000 fitsboth 3 mm roughness and 1 mm roughness datasets within their estimated uncertainties below Re=80000.Figure 5 shows that the percentage discrepancy from the mixed model is within ±2% below Re=80000.
10
100
1000
1000 10000 100000
Nu
/Pr1
/3
Re
Convection from Rough Plates
L3 Mixed ∆T=11K; 3.0 mm
L4 Mixed ∆T=11K; 3.0 mm
Rough Turbulent Asymptote 3.0 mm
Smooth Turbulent Asymptote
3.0 mm ∆T=11K Measured
L4 Mixed ∆T=11K; 1.03 mm
Rough Turbulent Asymptote
1.03 mm ∆T=11K Measured
Figure 6
While the L4 Mixed and Rough Turbulent Asymptote traces for 3 mm roughness are close in Figure 6,those traces for 1 mm roughness are not. This is because when ε of the plate was reduced from 3 mm to
8
1 mm, LS was not similarly scaled.
5. Sand Roughness
Modeling sand grains as unit radius spheres glued to the plate surface in a square array and modelingthe space not covered by a sphere as glue of height 0 ≤ g ≤ 1, the mean height of the enclosing square cellof area A containing a unit sphere is:
y =
(∫ 1
0
2πx(√
1− x2 + 1)dx+ (A− π) g
)/A =
(5
3π + (A− π) g
)/A
The arithmeticmean heightofroughness Ra and the rootmeansquared heightofroughness Rq are:
2Ra =
(∫ 1
0
2πx∣∣∣√1− x2 + 1− y
∣∣∣ dx+ (A− π) y − g)/
A
2Rq =
√(∫ 1
0
2πx(√
1− x2 + 1− y)2dx+ (A− π) (y − g)
2
)/A
For glue level 1 < g < 2, the grains are more than 50% submerged. Let xg =√
(1− (g − 1)2) =√2g − g2 be the radius of the glue opening through which the sphere is exposed. The mean height of the
enclosing square cell of area A containing a unit sphere is then:
y =
(∫ xg
0
2πx(√
1− x2 + 1)dx+
(A− πx2g
)g
)/A =
(4 + 3g2 − 2g3
3π +
(A− πx2g
)g
)/A
The arithmeticmean heightofroughness Ra and the rootmeansquared heightofroughness Rq are:
2Ra =
(∫ xg
0
2πx∣∣∣√1− x2 + 1− y
∣∣∣ dx+(A− πx2g
)y − g
)/A
2Rq =
√(∫ xg
0
2πx(√
1− x2 + 1− y)2dx+
(A− πx2g
)(y − g)
2
)/A
Figure 7 shows the conversion factor from ε to kS as a function of relative glue thickness.
1
10
100
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
ks/
ε
Glue Thickness (sand grain radius = 1)
SandRoughness ks per HeightofRoughness ε vs. Glue Thickness and Cell Area A
RootMeanSquared εq; A=4.20
ArithmeticMean εa; A=4.20
RootMeanSquared εq; A=8.77
ArithmeticMean εa; A=8.77
Figure 7In order to compare the Prandtl and Schlichting[1] formula (1) with formula (12) of present work,
different kS/ε conversion factors (and relative glue thicknesses) must be used at large and small kS values.
9
The unit cell area for the larger grains is 4.2. With L/ε = 150, compare 4cf for L/kS = RaL/ε in Table 2and L/kS = RqL/ε in Table 3 with fD = 0.0528 calculated from the scaled ColebrookWhite asymptote(12):
g y 1/Ra L/kS 4cf0.00 1.25 3.14 47.7 0.0880.70 1.42 4.97 30.2 0.1051.00 1.50 6.38 23.5 0.117
Table 2 Average roughness Ra versus largegrain sandroughness kS
g y 1/Rq L/kS 4cf0.00 1.25 2.66 56.4 0.0820.87 1.47 4.98 30.1 0.1051.00 1.50 5.64 26.6 0.111
Table 3 RMS roughness Rq versus largegrain sandroughness kS
fD = 0.0528 is out of the range of possible 4cf values for arithmeticmean or RMS roughness with g ≥ 0.However, there are values of 4cf/2 which are within range for both RMS and arithmeticmean roughness.
A factor of 2 is introduced in the last step of Prandtl and Schlichting’s paper[1]. While it is justifiedfor the local coefficient of resistance, the reason for its presence in the total coefficient of resistance is notobvious. In their example of the use of their formula, Mills and Hang[3] divide CD by 2.
Half the 4cf value from the g = 0.70 row in Table 2 matches fD = 0.0528, as does the g = 0.87 row inTable 3, for arithmeticmean and RMS heightofroughness respectively.
The unit cell area for the smaller grains is 8.77. The largest pipe which Nikuradse tested had a diameter of 0.101 m. With a coating made from the smallest sand grain size (.01 cm), L/kS = 1014 and thecorresponding fD = 0.0277. As before, there is no glue level g ≥ 0 whose 4cf matches fD, but half of 4cfdoes match at g = 0.90 and g = 0.99 in Tables 4 and 5.
g y 1/Ra L/kS 4cf0.00 0.60 2.61 388.6 0.0440.90 1.17 5.65 179.4 0.0551.50 1.57 17.98 56.4 0.082
Table 4 Average roughness Ra versus smallgrain sandroughness kS
g y 1/Rq L/kS 4cf0.00 0.60 2.46 411.4 0.0430.99 1.23 5.65 179.4 0.0551.50 1.57 13.95 72.7 0.075
Table 5 RMS roughness Rq versus smallgrain sandroughness kS
Afzal et al in “Turbulent flow in a machine honed rough pipe for large Reynolds numbers: General roughness scaling laws”[7] fit 5.333 for the RMS to sandroughness conversion factor and 6.45 for the arithmeticmean to sandroughness conversion factor. The mean of the 1/Rq values 4.98 and 5.65 is within 1% of 5.333(and not close to 6.45). This is strong evidence that RMS heightofroughness is the true metric for surfaceroughness in the scaled ColebrookWhite asymptote (12).
In “Skinfriction behavior in the transitionallyrough regime”[8], Flack et al measure skinfriction fromgritblasted surfaces. They find “The rootmeansquare roughness height (krms) is shown to be most stronglycorrelated with the equivalent sand roughness height (kS) for the gritblasted surfaces.”
Figure 8 shows that the MillsHang formula (2) is nearly coincident with the ColebrookWhite asymptote(12) over the range 100 < L/ε < 1000; so it is an improvement over Prandtl and Schlichting’s correlation(1). At L/ε > 1000, the inset in Figure 8 shows that both suffer from the scale error discussed in Section 2,growing to −13% and −18% relative to the scaled ColebrookWhite asymptote (12) at L/ε = 106.
10
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
0.065
100 1000
f D
L/ε
Fully Rough (Turbulent) Darcy Friction Factor
.5 Schlichting (kS = 5.65 εq) .99 glue
.5 MillsHuang (kS = 13.95 εq) 1.5 glue
Scaled ColebrookWhite Asymptote
.5 Schlichting (kS = 4.98 εq) .87 glue
0.01
1000 10000 100000 1x106
Figure 8
6. Admissible Roughness
“The amount of roughness which is considered “admissible” in engineering applications is that maximumheight of individual roughness elements which causes no increase in drag compared with a smooth wall.”2
Schlichting[2] develops a formula for kadm from its Fig. 21.9. (Resistance formula of sandroughened plate;coefficient of total skin friction):
U∞kadmν
= 100 (21)
A footnote states “The estimates performed in this section make no distinction between the equivalentsand height, kS , and the actual height, k, of a protuberance.” In the Section 5 model of sandroughness,the protuberance height is (1 − g/2)kS . For Nikuradse’s large grains that is 0.57 kS ; for small grains it is0.51 kS . So equation (21) overestimates the admissibleroughness height.
The admissibleroughness threshold does not occur in the “fullydeveloped roughness” region (Prandtland Schlichting[1]) which contains all the Convection Machine measurements. However, these measurementscan be used to develop an upper bound for admissibleroughness.
For a given RMS roughness Rq, the equalarea bilevel plate has the smallest possible k = 2Rq; whilelargegrain sandroughness would have k = 0.57·5.333Rq = 3.0Rq and smallgrain k = 0.51·5.333Rq = 2.7Rq.
The basis for equation (21) is that U∞kS/ν = Re kS/L is nearly invariant at the intercepts of theconstant kS/L and kS = 0 curves at Re from 105 to 108. For the 1.03 mm trace in Figure 6, this would bethe point below which the slope of measured Nu/Pr1/3 is 4/5. Natural convection obscures the slope belowRe=4419; so 4419 is an upper bound.
2 This definition does not imply that roughness always causes an increase in drag; for example in Figure 6the 1 mm roughness curve is less than the smoothturbulent asymptote from 3000 < Re < 7000.
11
RqU∞ν
= RqRe
L< 15 k
U∞ν
= kRe
L< 30 k < 30
L
Re(22)
This upper bound for k is a significant reduction from equation (21).
7. Apparatus
Figure 9 Figure 10Figure 9 shows the rough surface of the 30.5 cm× 30.5 cm test plate. Machined from MIC6 aluminum
to keep its emissivity low, the reflective surface is composed of (676) 8.28 mm × 8.28 mm × 6 mm postsspaced on 11.7 mm centers. The area of the top of each post is 0.686 cm2, half of the 1.37 cm2 cell. Thearithmeticmean (also the rootmeansquared) heightofroughness is thus 3 mm.
The back surface of the plate has 9 precision electronic resistors embedded as heating elements and atemperature sensor. There is 2.54 cm thick thermal foam insulation between the back of the plate and a0.32 mm thick sheet of aluminum. The aluminum sheet also has a temperature sensor at its center.
Figure 10 is a crosssection drawing of the plate assembly.
Figure 11The Convection Machine windtunnel (Figure 11) has a 61 cm × 35.6 cm crosssection and a 61 cm
depth. It allows the plate assembly to be centered with 15 cm of space on all sides. The fan pulling air
12
through the test chamber produces a maximum windspeed of 4.5 m/s (Re = 9× 104 for the 0.305 m squareplate). Its minimum nonzero windspeed is 0.12 m/s (Re = 2300).
The 99% boundarylayer depths [6] for the windtunnel versus the distance x from its front edge are(laminar and turbulent):
δlam = 4.92
√xν
Vδtur = 0.37x4/5
( νV
)1/5The boundary layer depths from the plate are similar, although the Reynolds numbers are lower with
its shorter characteristiclength. Figure 12 shows that 15 cm clearance between the plate and windtunnelwalls keeps the boundary layers from interacting at airspeeds within its range.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
δ,
99
% D
epth
in
m
Distance from front edge in m
WindTunnel Boundary Layer Thickness; DH = 0.449 m
laminar V=0.12 m/s; Re=3432laminar V=0.24 m/s; Re=6864
turbulent V=2.0 m/s; Re=57204turbulent V=4.5 m/s; Re=128708
Figure 12
The rotation rate of the fan is sensed by a phototransistor receiving the light from a LED beinginterrupted by the passing fan blades. Software controls a solidstate relay (supplying power to the fan) tomaintain the fan rotation rate dialed into switches. At rotation rates below 400 r/min, the software pulsespower to the fan according to a phase comparator. Above 400 r/min it servos the duty cycle of a 10 Hzsquarewave supplying power. It operates over a range of 32 to 1500 r/min.
The correspondence between fan rotations per second and windspeed is made using an anemometer.The ABM200 Airflow & Environmental Meter specifies an accuracy of ±0.5% of reading from 0.22 m/s to62.5 m/s. At speeds below 5 m/s, 0.5% is much better than can be achieved with hotwire anemometers,which specify accuracy as a percentage of 10 m/s fullscale.
Figure 13 shows the calibration curve for the empty windtunnel. These measurements make clear thatthe 0.22 m/s lower limit must be a misprint; the smallest nonzero reading my unit was able to report was0.66 m/s. Even at 1.25 m/s, the slope is on a trajectory to reach 0 when the fan is still moving air.
For any fan, there must be some rotation rate and windspeed Vs above which airflow does not increase.That flattening will appear gradually from slower rotation rates. Modeling the stall speed as Vs = 7.5 m/s,equation (23) is the solid line in Figure 13. From 500 r/min to 1500 r/min, it matches measurements witha standard deviation of 0.0075 m/s.
V =[(rS)−2 + V −2s
]−1/2S =
V1500
1500 r/min√
1− (V1500/Vs)2V1500 = 4.17195 m/s
(23)
13
1/sqrt((298.8/r)^2+1/7.50^2)Airspeed vs Fan Rotation 2017110418.4C58.5%100710Pa+90, 90
Vel
oci
ty,
m/s
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Fan Rotation, r/min0 200 400 600 800 1000 1200 1400 1600
Figure 13The plate assembly is suspended from six lengths of 0.38 mmdiameter steel piano wire terminated at
twelve zither tuning pins in wooden blocks above and around the test chamber. The wire is sheathed by0.95 mm Teflon tubing where it would otherwise contact the plate.
With the plate assembly in the windtunnel, the airspeed is increased in proportion to the reduction ofwindtunnel aperture by the plate’s crosssectional area:
Vp = VAwt
Awt − Cp(24)
The increase is 7.7% for the 3 mm roughness plate and 7.3% for the 1 mm roughness plate.
The ambient sensor board can be seen at the lower edge of the tunnel in Figure 11. This small boardmeasures the pressure, humidity, and temperature of the air at the windtunnel intake.3 The LM35 temperature sensor projects into the windtunnel test chamber; it is wrapped in aluminum tape so that radiativeheat transfer is minimized. The temperature sensor is powered only while it is being read so that selfheatingdoesn’t affect it; selfheating was a problem before it was switched.
Logging of the measurements is performed by a STMicroelectronics STM32F3 Discovery 32Bit ARMM4 72MHz development board. The custom electronics board which the STM32F3 plugs into, containspower supplies, heater control and drive, and signal conditioning.
A program was written for the microprocessor which takes measurements and writes them to the microprocessor’s nonvolatile RAM, controls the heater, servos the fan speed, and (later) uploads captured datato a computer over a USB serial interface. Every second for 108 minutes the program calibrates and readseach onchip 12 bit analogtodigital converter 16 times and sums its readings to create a 16 bit value.
3 It has since been moved to the chamber beneath the windtunnel where, although slower to respond toambient changes, it is unaffected by convection from the plate.
14
8. Measurement Methodology
Previous laboratory measurements (for example [4]) of forced convection have been performed by startingthe fluid flow and plate heater, waiting for the system to reach equilibrium (as indicated by a stable platetemperature), then recording the measurements of the physical quantities.
The expectation for this windtunnel was that it would support windspeeds up to 10 m/s. The test platewas made massive in order to maintain uniform temperature at high rates of convection. As a consequencethe plate’s temperature settling times range from minutes in the best case to hours at low airflow rates.
Ubiquitous electronic sensors and digital processing make possible an entirely different measurementmethodology. Rather than waiting for the plate to reach equilibrium before each measurement, the state ofall sensors is captured every second for 108 minutes and compared with simulations of the system, which israrely in equilibrium.
The physical parameters derived from measurements and specifications of both plate versions are:
symbol value value descriptionεq 3.0 mm 1.03 mm RMS heightofroughnessεa 3.0 mm 1.01 mm Arithmeticmean heightofroughnessAp 0.093 m2 0.093 m2 Plate Test AreaC 4690 J/K 4242 J/K Plate Thermal CapacityCb 146 J/K 146 J/K Back Thermal CapacityUI 0.075 W/K 0.075 W/K Insulation Thermal Conductanceεp 0.07 0.07 Plate Surface Emissivityεwt 0.90 0.90 WindTunnel Emissivity
Table 6 Physical Parameters
The effective windtunnel emissivity εwt may differ from the emissivity of mediumdensityfiberboard(0.90 [9]) because the temperatures of the (internal) tunnel surfaces may not be uniform, and the plateexchanges thermal radiation with objects in the room which may have different emissivities and temperaturesthan the tunnel and air.
With its low emissivity (εp = 0.07), the radiative heat loss from the rough face of the plate is only35 mW/K, about 3% of the 1.08 W/K expected for V = 1m/s convection. Even if the εwt value of 0.9 iswrong by 20%, forced convection measurements will be affected by less than 1%.
The measured dynamic physical quantities are:
symbol units descriptionΠH W Heater PowerV m/s Fluid Velocity (from fan rotation rate)TP K Plate TemperatureTF K Fluid (Air) TemperatureTB K Back Surface Temperature
Table 7 Dynamic Quantities
The calculated and simulated quantities are:
symbol units descriptionUS(V ) W/K Side Thermal ConductancehR W/(m2K) Radiative Surface Conductanceh(V ) W/(m2K) Convective Surface ConductanceUP (t) W/K Convective Thermal ConductanceTP (t) K Plate Surface Temperature
Table 8 Calculated and Simulated Quantities
The forced convective thermal conductance of the four sides US(V ) is computed by integrating the localforced convective surface conductance in series with the insulation thermal conductance over the area of thesides.
The short side is not an isothermal surface; it has a 4 mm metal strip and a wedge of extruded polystyrene(XPS) insulation. Each point’s effective surface conductance will depend on the temperature profile alongthe side.
15
A point on the side which is near to metal will have a large conductance. Summing the conductivitydivided by the shortest distance to metal at each angle will be roughly proportional to the local conductance(see Figure 14). The conductance through a slab with parallel isothermal faces is well known. The ratio ofthe slab conductance to the slab sum is the factor 0.637, which is used to normalize conductivities calculatedthis way.
Figure 14
With forced air the four sides have three distinct behaviors. The forced air flows parallel to the longdimension on two sides but flows into the windward side and away from the leeward side.
The forced convective component for the two parallel sides averages the local forced convective surfaceconductance in series with that point’s insulation conductance as computed above.
For the windward side, a similar calculation is done, but the wind speed grows linearly from 0 m/s atthe midline to V along each long edge. For the leeward side the wind speed grows linearly from 0 m/s alongeach long edge to V at the midline, but with the flow always turbulent. The leeward surface conductanceturned out to be within 1% of the windward surface conductance, so the windward value is used for both.
Convection for the sides uses the smooth turbulent correlation (8), but does not visibly differ from useof the scaled Colburn analogy asymptote (14) with 0.25 mm roughness. The expected convection throughthe sides is 9% of the convection from the rough surface at 4 m/s windspeed.
Collecting terms having a factor of TP − TF into UM (V ), the heat balance equation for the plate inforced convection is:
UM (V ) = US(V ) + h(V )Ap + εpεwthRAp
ΠH = UM (V ) [TP − TF ] + UI [TP − TB ] + CdTPdt
(25)
To track temperature measurements as functions of time requires characterization of the temperaturetime responses of the thermal masses and insulation (a combination of polyisocyanurate (PIR) and XPSfoams). Figure 15 shows the temperaturetime response of the downwardfacing plate to a 57.5 W step inheat drive applied starting at second 151. The dashed line is the temperature expected with a 15 s delay.Figure 15 and the heating phase of experiments show that a 15 s delay models the system well.
16
20160813T155911Z − 57.5W step response − Orientation=+90.00°
Temperature, C
22.8
23
23.2
23.4
23.6
23.8
24.0
24.2
24.4
Time, s150 162.5 175 187.5 200 212.5 225 237.5 250 262.5 275 287.5 300
Figure 15
The temperaturetime response of the insulation is best ascertained with the plate facing up so thatthe back is facing down for its minimum convection. Not a step response, this is the response to the plate’stemperature ramp. The blue trace showing back temperature in Figure 16 is superimposed on a red tracewhich models the insulation as thermal conductance UI = 75 mW/K with a 110 s delay between the plateand the back.
17
Ra
0
200000
400000
Time, s0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
k=0.026, Ra_U=305278, h=4.71W/(K.m^2), U=0.438W/K, Nu=13.78, Pr=0.73020160812T010335Z − upward Convection − Roughness=3.00mm; T=22.4+07.1°C; −90.00°
Temperature, C
23
24
25
26
27
28
29
30
31
32
Figure 16Including the delays for the insulation and plate thermal masses, and isolating TP (t):
TP (t) =ΠH(t− 14) + UM (V )TF (t) + UITB(t− 110)− CdTP (t)/dt
UM (V ) + UI(26)
Solving (26) as a finitedifference equation where dt = t− t′:
H(V, t) = ΠH(t− 14) + UM (V )TF (t) + UITB(t− 110)
TP (t) =H(V, t)[t− t′] + CTP (t′)
[UM (V ) + UI ] [t− t′] + C(27)
In equation (27), TP (t′) is the previous simulated value (not the measured value).Forced convection measurements are usually presented as Nu versus Re. In order to obtain Nu, the heat
balance equation is solved for UP = h(V )Ap:
η(V, t) = ΠH(t− 14)− UI [TP (t)− TB(t− 110)]
UP (t) =η(V, t)− C [TP (t)− TP (t′)] /[t− t′]
TP (t)− TF (t)−ApεpεwthR − US(V ) (28)
In the denominator of equation (28), TP (t) and TF (t) are the 11element cosine averages of plate andfluid temperature (centered at time t). Averaging is needed so that the derivative doesn’t correlate with thedenominator, causing bias.
18
9. Measurement Uncertainties
The Convection Machine measures heatflow into the plate from electrical heater dissipation and convective heat flow from the rate of change in plate temperature. For the 3 mm roughness plate, Table 9 computesthe sensitivity of heat flow to each parameter and the product of each sensitivity with the estimated parameter bias to yield the component uncertainty. In accordance with “ASME Measurement Uncertainty”[10], therootsumsquared (RSS) of these uncorrelated uncertainties is 2.0%. Reducing the roughness εq to 1.03 mm(εpv = 2 mm) raises the combined uncertainty to 2.9% in Table 10.
With a windspeed of 3 m/s or greater, emissivity and natural convection parameters don’t have asignificant effect on the combined uncertainty and aren’t included in Tables 9 and 10.
Symbol Nominal Sensitivity Bias Uncertainty ComponentT 19.5◦C 0.163%/◦C 1.0◦C 0.163% LM35C Temperature Sensor
∆T 10.5◦C 5.759%/◦C 0.15◦C 0.864% LM35C DifferentialP 101kPa 0.001%/Pa 1.5kPa 0.820% MPXH6115A6U Air PressureV 3.00m/s 18.454%/m
s 15mm/s 0.277% Airspeedb 10.000 4.304% 0.25 1.076% Logarithmic Base
Ret 121000 0.00006% 2000 0.124% RoughSmooth ThresholdLc 0.305m 11.560%/m 500um 0.006% Characteristic LengthLw 0.305m 213.753%/m 500um 0.107% Plate WidthLs 8.29mm 1236.477%/m 250um 0.309% Block Lengthεpv 6.00mm 7065.258%/m 150um 1.060% P2V HeightofRoughnesskpir 23.0mW
K·m 0.052%/mWK·m 1.2mW
K·m 0.060% PIR Thermal Conductivity
kxps 34.0mWK·m 0.064%/mW
K·m 1.7mWK·m 0.109% XPS Thermal Conductivity
Symbol Nominal Sensitivity Variability Uncertainty ComponentV 3.00m/s 18.454%/m
s 7.5mm/s 0.138% Airspeed1.99% RSS Combined Uncertainty
Table 9 Uncertainties for Plate with 3mm of Roughness
Symbol Nominal Sensitivity Bias Uncertainty ComponentT 19.5◦C 0.210%/◦C 1.0◦C 0.210% LM35C Temperature Sensor
∆T 10.5◦C 7.501%/◦C 0.15◦C 1.125% LM35C DifferentialP 101kPa 0.001%/Pa 1.5kPa 1.074% MPXH6115A6U Air PressureV 3.00m/s 24.165%/m
s 15mm/s 0.362% Airspeedb 10.000 0.446% 0.25 0.112% Logarithmic Base
Ret 121000 0.00003% 2000 0.066% RoughSmooth ThresholdLc 0.305m 32.377%/m 500um 0.016% Characteristic LengthLw 0.305m 281.685%/m 500um 0.141% Plate WidthLs 8.29mm 9347.066%/m 250um 2.337% Block Lengthεpv 2.00mm 2781.195%/m 150um 0.417% P2V HeightofRoughnesskpir 23.0mW
K·m 0.089%/mWK·m 1.2mW
K·m 0.102% PIR Thermal Conductivity
kxps 34.0mWK·m 0.112%/mW
K·m 1.7mWK·m 0.190% XPS Thermal Conductivity
Symbol Nominal Sensitivity Variability Uncertainty ComponentV 3.00m/s 24.165%/m
s 7.5mm/s 0.181% Airspeed2.89% RSS Combined Uncertainty
Table 10 Uncertainties for Plate with 1mm of Roughness
Convection measurements are 35 times more sensitive to ∆T = TP − TF than to T (the average of TFand TP ). The offsets in the measurement software for the plate, back, and ambient sensors were matchedto about ±0.15 K. In 300 consecutive readings at constant temperature, the variability for the ambient andplate sensors was 0.03 K, and 0.02 K for the back sensor at T = 19◦C.
The vertical dimensions of the aluminum plate’s rough surface are correct as far as can be determinedwith spot checks using a caliper depth gauge. The only accuracy specification for the ShopBot PRSstandardCNC (used to machine the rough surface of the plates) is a “Positional Repeatability” of 0.05 mm. Threetimes this value (150µm) is used for estimated bias of εpv in Tables 9 and 10.
19
10. Forced Convection Results
Figure 17 compares the measurements for the downwardfacing plates with equations (20) and (14) and theirpredicted uncertainties calculated for each measurement. Below Re=5000 the curves for equation (20) arecorrected for mixed convection, the subject of a future paper.
10
100
1000
1000 10000 100000
Nu
/Pr1
/3
Re
Convection and Measurement Uncertainties from Rough Plates
Expected Uncertainty 3.0 mm
L4 Mixed ∆T=11K; 3.0 mm
Rough Turbulent Asymptote 3.0 mm
3.0 mm ∆T=11K Measured
Expected Uncertainty 1.03 mm
L4 Mixed ∆T=11K; 1.03 mm
Rough Turbulent Asymptote
1.03 mm ∆T=11K Measured
Figure 17
Figure 18 shows a forced convection experiment (of 6144 s duration) with simulation. In the (upper)temperature graph, the solid trace is the plate temperature, while the dashed trace is the simulation of theplate temperature (taking the back and ambient temperatures as inputs).
Diamonds on the simulated temperature trace mark the beginning and end of the measurement period.The ending temperature difference with ambient is at most half of the peak temperature difference withambient. The simulated plate temperature is reset to the measured plate temperature at the start diamond.
Underneath the temperature graph is a graph of the air velocity versus time for mixed convection runs(and Raleigh number versus time for natural convection runs).
Each round dot in Figure 17 represents the convection measured over the time interval between thediamonds in its experiment. The dots at the left edge are the natural convections (where Re = 0). A graphof each dot’s run is provided in the supplementary files.
20
Velocity, m/s
0
1
2
Time, s0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
640±4r/min, V=2.218m/s, Re=42505, Ra_D=3301678, h=25.9W/(K.m^2), U=2.41W/K, Nu=301.920160807T144333Z − mixed Convection − Roughness=3.00mm; T=22.7+09.9°C; +90.00°
Temperature, C
24
26
28
30
32
34
36
Figure 18
10
100
1000
1000 10000 100000
Mixed Convection from Downward Facing Rough Plate
L4 Mixed; ∆T=4K; εq=0.75 mm RMS
L4 Mixed; ∆T=4K; εa=0.57 mm AVG
Rough Turbulent Asymptote; εq=0.75 mm RMS
Rough Turbulent Asymptote; εa=0.57 mm AVG
Figure 19
21
11. Directions for Further Work
Having equal areas of the plate at low and high elevations created a roughness whose height was the samewhether calculated using rootmeansquared or the arithmeticmean of absolutevalues. While Section 5finds evidence that ε should be calculated rootmeansquared, a definitive test would be to alter the plate sothat the two metrics yield significantly different ε values.
The plate will be machined to reduce the high area from 50% to 17.2% of the total plate area. Thenthe arithmeticmean heightofroughness εa = 0.57 mm and the RMS heightofroughness εq = 0.75 mm.Figure 19 shows the convection expected by formula (20) for the two roughness metrics. At Re=10000 therough turbulent convection with RMS εq is 9% larger than convection with arithmeticmean εa. This shouldbe sufficient to determine which is the true roughness metric.
LS is an input to inequality (16), but had the same value, 8.29 mm, for both plates. This modificationwill reduce LS to 2.43 mm, allowing its true effect on ReM and the transition model (20) to be discovered.
It is not difficult to devise a formula for offaxis forced convection; but with only 15 cm space betweenthe plate and the tunnel walls, offaxis flow would not be uniform in the Convection Machine. A significantlylarger windtunnel would be required to test it. Also, there is no guarantee that a tractable model could bedevised for the sides with nonorthogonal angles of wind incidence.
Acknowledgments
Thanks to John Cox and Doug Ruuska for machining the plate. Thanks to Martin Jaffer and Roberta Jafferfor their assistance and problemsolving suggestions.
12. References
[1] L. Prandtl and H. Schlichting. The Resistance Law for Rough Plates. Translation (David W. TaylorModel Basin). Navy Department, the David W. Taylor Model Basin, 1934. Translated 1955 by P.S.Granville.
[2] Hermann Schlichting, Klaus Gersten, Egon Collaborateur. Krause, Herbert Collaborateur. Oertel, andKatherine Mayes. Boundarylayer theory. Springer, Berlin, Heidelberg, Paris, 2000. Corrected printing2003.
[3] A. F. Mills and Xu Hang. On the skin friction coefficient for a fully rough flat plate. J. Fluids Eng,105(3):364–365, 1983.
[4] M. M. Pimenta, R. J. Moffat, and W. M. Kays. The Turbulent Boundary Layer: An ExperimentalStudy of the Transport of Momentum and Heat with the Effect of Roughness. Department of MechanicalEngineering, Stanford University, 1975.
[5] Mark S. Owen, editor. Handbook, ASHRAE Fundamentals (SI). American society of heating, refrigerating and airconditioning engineers, 2009.
[6] J.H. Lienhard IV and J.H. Lienhard V. A Heat Transfer Textbook. Phlogiston Press, Cambridge, MA,4th edition, 2016. Version 2.04.
[7] Noor Afzal, Abu Seena, and A. Bushra. Turbulent flow in a machine honed rough pipe for large reynoldsnumbers: General roughness scaling laws. Journal of Hydroenvironment Research, 7(1):81–90, 2013.
[8] Karen A Flack, Michael P Schultz, Julio M Barros, and Yechan C Kim. Skinfriction behavior in thetransitionallyrough regime. International Journal of Heat and Fluid Flow, 61:21–30, 2016.
[9] Rice R.W. Emittance factors for infrared thermometers used for wood products. Wood and FiberScience, 36:520–526, 2004.
[10] R.B. Abernethy, R.P. Benedict, and R.B. Dowdell. Asme measurement uncertainty. ASME. J. FluidsEng., 107(2):161–164, 1985.
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