Skin-Friction and Forced Convection from an Isothermal Rough PlateAubrey G. Jaffer

Digilant2 Oliver Street, Suite 901

Boston MA 02109 USagj@alum.mit.edu

Abstract

Developed are new correlations for skin-friction and forced convection from an isothermalplate having root-mean-squared height-of-roughness εq. Corresponding to Prandtl and Schlichting’s“fully-developed roughness” region, the skin-friction 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 fully-rough turbulent to smooth turbulent convection for this bi-level 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 sand-roughness finds that its relation to root-mean-squared height-of-roughnessis non-linear. 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 admissible-roughness than is given in Schlichting’s book “Boundary-layer theory.”

Keywords: skin-friction; forced convection; roughness; sand-roughness; admissible-roughness

Table of Contents

Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Pipe-Plate 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 skin-friction has wide application to the fluid dynamics of vehicles and turbines.

The scope of this work is the total (not local) skin-friction and the total turbulent forced convection fromrectangular isothermal rough plates. There are self-similar roughness profiles for which the total skin-frictionand convection are well-defined and the local skin-friction 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 characteristic-length. 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 skin-friction resistance for rough plates from theiranalysis of Nikuradse’s measurements of sand glued inside pipes. In “Boundary-layer theory”[2] Prandtl andSchlichting give a formula for fully rough (large Re) total skin-friction 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 skin-friction measurements done by Pimenta, Moffat, andKays[4]. Their total skin-friction 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 sand-roughness 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 offluid-dynamics 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. Pipe-Plate Analogy

The ratio of the mean height-of-roughness 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 J-factoranalogy (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 Colebrook-White 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

fully-rough8*.664/Re

1/2

8*.037/Re1/5

kS/R = 10-2

kS/R = 10-3

kS/R = 10-4

kS/R = 0

8*.023/Re1/5

Figure 1When the sand-roughness kS > 0, fD(Re) is asymptotically constant to the right of the fully-rough

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 J-factoranalogy (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 forced-convection correlations are usually monomials in Reand scale easily. With the characteristic-length of a pipe being perpendicular to the direction of flow andthe characteristic-length 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 Colebrook-White equation causescorrelations (9) and (10) to approximate the smooth convection curve in Figure 2. If the mean height-of-roughness ε, which is perpendicular to the characteristic-length 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 Colebrook-White equation motivated the constructionof an apparatus designed to measure rough plate convection. The experiments described later in this paperfound that scaling the height-of-roughness kS → ε 7.41/9.5 (such that the first term of the argument to log10

becomes ε/L) matched measurements well. The scaled Colebrook-White 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

fully-roughε/L = 10

-2

ε/L = 10-3

ε/L = 10-4

ε/L = 0 8*.664/Re

1/2

8*.037/Re1/5

Schultz-Grunow

Figure 2

4

Figure 21.2 of Schlichting[2] shows that the Schultz-Grunow formula (13) for turbulent skin-friction 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 Colebrook-Whiteequation asymptote (12) with the Chilton and Colburn J-factor 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 “fully-developed roughness” region of Prandtl and Schlichting,but extends to much lower Reynolds numbers. Natural convection prevents investigation of the forcedturbulent-laminar transition in the Convection Machine; measurement of skin-friction drag would be a bettermethodology for those other regions.

1 This may be related to the Colebrook-White 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 bi-level plate is not self-similar; 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 smooth-turbulent boundary-layer depth would be much smallerthan the height-of-roughness; so fully rough convection (14) applies. The boundary-layer 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 peak-to-valley 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), post-top (18), and post-side (19) convection. When ReF is larger than40000, there is a small region where the channel boundary-layer 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 Downward-Facing Plate

Expected UncertaintyRough Turbulent Asymptote

Rough-Smooth 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 data-sets 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 arithmetic-mean height-of-roughness Ra and the root-mean-squared height-of-roughness 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 arithmetic-mean height-of-roughness Ra and the root-mean-squared height-of-roughness 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)

Sand-Roughness ks per Height-of-Roughness ε vs. Glue Thickness and Cell Area A

Root-Mean-Squared εq; A=4.20

Arithmetic-Mean εa; A=4.20

Root-Mean-Squared εq; A=8.77

Arithmetic-Mean 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 Colebrook-White 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 large-grain sand-roughness 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 large-grain sand-roughness kS

fD = 0.0528 is out of the range of possible 4cf values for arithmetic-mean or RMS roughness with g ≥ 0.However, there are values of 4cf/2 which are within range for both RMS and arithmetic-mean 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 arithmetic-mean and RMS height-of-roughness respectively.

The unit cell area for the smaller grains is 8.77. The largest pipe which Nikuradse tested had a diam-eter 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 small-grain sand-roughness 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 small-grain sand-roughness kS

Afzal et al in “Turbulent flow in a machine honed rough pipe for large Reynolds numbers: General rough-ness scaling laws”[7] fit 5.333 for the RMS to sand-roughness conversion factor and 6.45 for the arithmetic-mean to sand-roughness 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 height-of-roughness is the true metric for surfaceroughness in the scaled Colebrook-White asymptote (12).

In “Skin-friction behavior in the transitionally-rough regime”[8], Flack et al measure skin-friction fromgrit-blasted surfaces. They find “The root-mean-square roughness height (krms) is shown to be most stronglycorrelated with the equivalent sand roughness height (kS) for the grit-blasted surfaces.”

Figure 8 shows that the Mills-Hang formula (2) is nearly coincident with the Colebrook-White 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 Colebrook-White 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 Mills-Huang (kS = 13.95 εq) 1.5 glue

Scaled Colebrook-White 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 sand-roughened 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 sand-roughness,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 admissible-roughness height.

The admissible-roughness threshold does not occur in the “fully-developed roughness” region (Prandtland Schlichting[1]) which contains all the Convection Machine measurements. However, these measurementscan be used to develop an upper bound for admissible-roughness.

For a given RMS roughness Rq, the equal-area bi-level plate has the smallest possible k = 2Rq; whilelarge-grain sand-roughness would have k = 0.57·5.333Rq = 3.0Rq and small-grain 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 smooth-turbulent 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 MIC-6 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. Thearithmetic-mean (also the root-mean-squared) height-of-roughness 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 cross-section drawing of the plate assembly.

Figure 11The Convection Machine wind-tunnel (Figure 11) has a 61 cm × 35.6 cm cross-section 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 wind-speed of 4.5 m/s (Re = 9× 104 for the 0.305 m squareplate). Its minimum nonzero wind-speed is 0.12 m/s (Re = 2300).

The 99% boundary-layer depths [6] for the wind-tunnel 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 characteristic-length. Figure 12 shows that 15 cm clearance between the plate and wind-tunnelwalls 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

Wind-Tunnel 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 photo-transistor receiving the light from a LED beinginterrupted by the passing fan blades. Software controls a solid-state 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 Hzsquare-wave supplying power. It operates over a range of 32 to 1500 r/min.

The correspondence between fan rotations per second and wind-speed is made using an anemometer.The ABM-200 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 hot-wire anemometers,which specify accuracy as a percentage of 10 m/s full-scale.

Figure 13 shows the calibration curve for the empty wind-tunnel. 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 wind-speed 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 mm-diameter 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 wind-tunnel, the airspeed is increased in proportion to the reduction ofwind-tunnel aperture by the plate’s cross-sectional 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 wind-tunnel intake.3 The LM35 temper-ature sensor projects into the wind-tunnel 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 self-heatingdoesn’t affect it; self-heating was a problem before it was switched.

Logging of the measurements is performed by a STMicroelectronics STM32F3 Discovery 32-Bit 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 micro-processor’s non-volatile 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 on-chip 12 bit analog-to-digital converter 16 times and sums its readings to create a 16 bit value.

3 It has since been moved to the chamber beneath the wind-tunnel 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 wind-tunnel was that it would support wind-speeds 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 height-of-roughnessεa 3.0 mm 1.01 mm Arithmetic-mean height-of-roughnessAp 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 Wind-Tunnel Emissivity

Table 6 Physical Parameters

The effective wind-tunnel emissivity εwt may differ from the emissivity of medium-density-fiberboard(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 mid-line 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 mid-line, 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 wind-speed.

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 temperature-time responses of the thermal masses and insulation (a combination of polyisocyanurate (PIR) and XPSfoams). Figure 15 shows the temperature-time response of the downward-facing 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 temperature-time 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 finite-difference 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 11-element 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 heat-flow into the plate from electrical heater dissipation and convec-tive 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 parame-ter bias to yield the component uncertainty. In accordance with “ASME Measurement Uncertainty”[10], theroot-sum-squared (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 wind-speed 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% Rough-Smooth 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 Height-of-Roughnesskpir 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% Rough-Smooth 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 Height-of-Roughnesskpir 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 downward-facing 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 root-mean-squared or the arithmetic-mean of absolute-values. While Section 5finds evidence that ε should be calculated root-mean-squared, 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 arithmetic-mean height-of-roughness εa = 0.57 mm and the RMS height-of-roughness 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 arithmetic-mean 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 off-axis forced convection; but with only 15 cm space betweenthe plate and the tunnel walls, off-axis flow would not be uniform in the Convection Machine. A significantlylarger wind-tunnel would be required to test it. Also, there is no guarantee that a tractable model could bedevised for the sides with non-orthogonal 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 problem-solving 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. Boundary-layer 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, refriger-ating and air-conditioning 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 Hydro-environment Research, 7(1):81–90, 2013.

[8] Karen A Flack, Michael P Schultz, Julio M Barros, and Yechan C Kim. Skin-friction behavior in thetransitionally-rough 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.

22