Measurement Uncertainty Estimation for Laser Doppler Anemometer

energies

Communication Measurement Uncertainty Estimation for Laser Doppler Anemometer

Karolina Weremijewicz 1 and Andrzej Gajewski 2,*

1 Students’ Scientiﬁc Society “Heat Engineer”, Faculty of Civil Engineering and Environmental Sciences, Bialystok University of Technology, Wiejska Street 45 A, 15-351 Białystok, Poland; [email protected] 2 Department of HVAC Engineering, Faculty of Civil Engineering and Environmental Sciences, Bialystok University of Technology, Wiejska Street 45 A, 15-351 Białystok, Poland * Correspondence: [email protected]; Tel.: +48-797-995-923

Abstract: Twenty percent of global electricity supplied to the buildings is used for preventing air temperature increase; its consumption for this prevention will triple by 2050 up to China’s present needs. Heat removed from the thermal power plants may drive cold generation in the absorption devices where mass and heat transfer are two-phase phenomena; hence liquid film break-up into the rivulets is extensively investigated, which needs knowledge of the velocity profiles. Laminar flow in a pipe is used in the preliminary study, velocity profile of developed flow is used as a benchmark. The study account writes the applied apparatus with their calibration procedure, and the uncertainty estimation algorithm. The calibration regression line with the slope close to one and a high Pearson’s coefficient value is the final outcome. Therefore, the apparatus may be applied in the principal research. Keywords: liquid film break down; rivulet; velocity profile; laminar flow; surface tension

Citation: Weremijewicz, K.; Gajewski, A. Measurement Uncertainty Estimation for Laser Doppler Anemometer. Energies 2021, 1. Introduction 14, 3847. https://doi.org/10.3390/ Crisis in cold generation approaches the world’s economy, for the air conditioners and en14133847 electric fans working against air temperature increase need nearly 20% of the total electrical energy used in buildings in the world. Electricity demands for these purposes is forecasted Academic Editor: Adrian Ilinca to triple by 2050, which would equal China’s present usage [1]. These demands reach the winter and summer peaks. Figure1 shows the former in December and January and the Received: 26 May 2021 latter in July and August; these seasonal peaks concern total energy production and its part Accepted: 23 June 2021 generated in thermal power plants, from fossil fuels combustion, or radioactive decay [2]. Published: 25 June 2021 Thermal power plants, in accordance with the second law of thermodynamics, need to remove heat to a heat sink which can be either surface water or a district heating system. Publisher’s Note: MDPI stays neutral However, in summer time, these systems supply heat only for generating domestic hot with regard to jurisdictional claims in water, which may be insufficient; this insufficiency might be overcome with cold produc- published maps and institutional affil- iations. tion, in the absorption refrigeration systems, from the removed heat temperatures of at least 80 ◦C[3]. In other words, the absorption refrigeration system might utilize heat that is removed from the thermal power plant during electricity generation. Absorption refrigeration systems might alternate from the traditional ones if they are equipped with compact and high-performance heat and mass exchangers [4]. A low usage of the ab- Copyright: © 2021 by the authors. sorption refrigeration systems stems from high ratio between the volume and the cooling Licensee MDPI, Basel, Switzerland. capacity [5]. For this reason, they are also experimentally studied [6,7]. An absorption This article is an open access article refrigerator uses two fluids, which change phases, and concentration flowing through the distributed under the terms and conditions of the Creative Commons absorber, generator, evaporator, and condenser [8,9]. Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Energies 2021, 14, 3847. https://doi.org/10.3390/en14133847 https://www.mdpi.com/journal/energies Energies 2021, 14, 3847 2 of 11 Energies 2021, 14, 3847 2 of 11

1800

1600

1400

1200

1000

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600

400 Electrical energy production[TWh] 200

Thermal power plants Total production

Figure 1. A report of International Energy Agency [2] into monthly electricity production. Figure 1. A report of International Energy Agency [2] into monthly electricity production. The change of phase or concentration occurs due to heat and mass transfer in thin liquidThe layers, change which of phase flow or down concentration a solid surface occurs (cf. due Figure to heat 3 in and [4]). mass Firstly, transfer these in layers thin liquidwere investigatedlayers, which because flow down of the a burnoutsolid surface in the (cf. steam Figure boilers 3 in or[4]). distillatory Firstly, these equipment. layers wereHartley investigated and Murgatroyd because [10of ]the investigated burnout in liquid the steam film breakboilers away or distillatory into the rivulets equipment. using Hartleytwo approaches: and Murgatroyd the first [10] one investigated is mechanical liquid equilibrium film break in away the highest into the point rivulets of the using dry twopatch, approaches: which results the first in the one force is mechanical balance in thisequilibrium point named in the stagnation highest point point; of the the other dry patch,method which employs results energy in the minimization, force balance assuming in this point the sum named of kinetic stagnation energy point; across the a other plane methodand the employs surface energy energy approaches minimization, minimum. assuming Bankoff the sum [11] balancedof kineticflow energy rates across and energiesa plane andof the the film surface and energy rivulets approaches at the critical minimum. film thickness; Bankoff it[11] was balanced assumed flow that rates a segment and ener- of giescircle of asthe a film cross-section and rivulets of theat the rivulet, critical and film the thickness; velocity it profiles was assumed in film, that and a in segment each slice of circleof the as rivulet, a cross-section are the same. of the Mikielewicz rivulet, and andthe velocity Moszynski profiles [12,13 in] appliedfilm, and minimization in each slice of of theenergy rivulet, extended are the bysame. surface Mikielewicz energy of and the Mo solid-gasszynski interface.[12,13] applied A common minimization feature of of en- the ergyinvestigations extended by presented surface energy above, inof the this soli paragraph,d-gas interface. is assumed A common laminar feature velocity of the profile in- vestigationsderived by Nusseltpresented (1916) above, (cf. in Madejski this paragraph, [14]); additionally, is assumed Hartley laminar and velocity Murgatroyd profile [de-10] rivedassumed by Nusselt a linear (1916) laminar (cf. profile Madejski for film, [14]); motivated additionally, by surface Hartley shear and only Murgatroyd or a von Karman[10] as- sumeduniversal a linear velocity laminar profile profile for turbulent for film, flow.motivated by surface shear only or a von Karman universalEl-Genk velocity and Saberprofile [15 for] continued turbulent researchflow. into film breakdown, applying the energetic approach;El-Genk their and analysis Saber [15] includes continued a two-dimensional research into film velocity breakdown, profile applying solved using the ener- Ritz geticmethod. approach; Perazzo their and analysis Gratton [includes16] derived a two-dimensional another two-dimensional velocity velocityprofile profile,solved whichusing Ritzis solved method. in closed Perazzo form, and forGratton both the[16] rivulet derived and another film, flowingtwo-dimensional along a vertical velocity plate; profile, the whichcontact is anglesolved was in closed assumed form, to be for either both perpendicularthe rivulet and to film, the flowing plate for along the rivulet a vertical or parallel plate; thefor contact the film. angle While was Tanasijczuk assumed etto al.be [ 17either] derived perpendicular a two-dimensional to the plate velocity for the profile rivulet for or a parallelrivulet pendingfor the film. from While a horizontal Tanasijczuk plate. et al. [17] derived a two-dimensional velocity profile forAtaki a rivulet and pending Bart [18] from applied a horizontal a velocity plate. profile for a gravity driven rivulet along an inclined plate in the analysis of their experiments. Another assumption was the static contact angle value 56◦.

Energies 2021, 14, 3847 3 of 11

Charogiannis et al. [19] experimentally determined laminar velocity proﬁle for a gravity driven rivulet along an inclined plate for a Reynolds number less than 5.9 and Kapitza number equal to 1.4 following its deﬁnition: σ Ka = (1) ρ(g sin αν4)1/3

where σ is surface tension, g is gravitational acceleration, α is an inclination of film to the horizon, and ν is kinematic viscosity; they also obtain a profile for wavy flow forced with frequencies 2 and 6 Hz, when the Reynolds number was about 5.2. The very familiar configurations of the absorbers, applied in the absorption cooling systems, consists of the horizontal tubes, which are covered with falling liquid film; this film is an absorbent solution that, flowing across the tubes, absorbs the vapor of a refrigerant [4]. The flow divides into three regimes: falling-film flowing around the tubes, droplet-formation flow that occurs on, and in, the vicinity of the lowest tube generatrix, and the droplet-fall flow that appears between two tubes; these regimes exist for LiBr/water [20] and aqueous alkaline nitrate solution [4]. When the absorbent solution is overflowing the tubes, two phenomena are occur- ring simultaneously: heat and mass exchange, which are mathematically modeled using two different algorithms; modelling heat exchange needs a knowledge of film thickness, which is computed from a velocity profile. Despite different geometry, both Alvarez and Bourouis [4] and Jeong and Garimella [20] apply, in their models, a laminar velocity profile derived by Nusselt, mentioned above, derived for a flow along the vertical plate. Accordingly, the film thickness δ is obtained from formula

3µΓ 1/3 δ(α) = . (2) ρ2g sin α

where µ is dynamic viscosity, Γ is mass flowrate per unit of wetted tube, ρ is solution density. Reynolds number for flowing film is defined by Equation [21]

4Γ Re = (3) µ

Combining Equations (2) and (3) we obtain a formula for film thickness as a function of the Reynolds number: 1/3 3µ2Re δ(α) = . (4) 4ρ2g sin α Figure2 shows a solution of Equation (4) for two Reynolds numbers: 100 and 1600; the latter is thought as the critical Reynolds number for films [21] which, under laminar flow, seems to be very thin, e.g., 0.5 mm or less under vertical flow; for the thicker films, a flow should not be laminar. Therefore, an in-depth research into the velocity profiles of thin liquid layers would be helpful in modelling absorbers; it should indicate whether the velocity profile, beyond laminar flow, is two dimensional, turbulent, and whose formula models best model the flows. The experimental investigations of the velocity profiles in the rivulets and liquid films, for a Reynolds number in range from 400 to 3000 and Kapitza number 3000–6000, determined with laser Doppler anemometer (LDA) usage, is the main area of the investigation, which will be done for water flows along copper, aluminum, brass, and stainless steel plates. The particular goal of the published preliminary studies is measurement uncertainty estimation and calibration of an applied LDA system. EnergiesEnergies2021 2021, 14, ,14 3847, 3847 4 of4 of 11 11

2.0

1.5

1.0 [mm] δ

Re=1600 0.5

Re=100

0.0 0 102030405060708090α [°] FigureFigure 2. 2.Film Film thickness thicknessδ asδ as a functiona function of of surface surface inclination inclination towards towards the the horizon. horizon.

2. MaterialsTherefore, and Methodsan in-depth research into the velocity profiles of thin liquid layers would be helpfulThe presented in modelling experimental absorbers; investigations it should indicate are carried whether out with the an velocity existing profile, pipeline beyond [22] thatlaminar is shown flow, as is an two internal dimensional, loop in Figureturbulent,3. A and researcher whose formula plugs in models a pump best (10) model supply- the ingflows. water, The through experimental a discharge investigations pipe (11), of an the upper velocity tank profiles (12), and in the an inletrivulets pipe and (13), liquid to afilms, transparent for a Reynolds pipe (15). number When in the range upper from tank 400 (12) to is3000 full and and Kapitza water is number overflowing 3000–6000, the brimdetermined of a pipe with (22), laser and returningDoppler anemometer to a bottom tank(LDA (21),) usage, a valve is the (16) main is opening. area of the Then investi- the assumedgation, which flow is will fixed be usingdone fo ther water needle flows valve along (19) copper, and the aluminum, flow meter brass, (20). Afterwardsand stainless thesteel researcher plates. The runs particular a computer goal program, of the publ whichished sends preliminary out the properstudies signals is measurement to the Burst un- Spectrumcertainty Analyzer estimation (BSA) and (5)calibration and the stepperof an applied motor LDA controller system. (6). BSA (5) sends, through an optical fiber (3), two coherent laser beams to the flow explorer (2). These beams intersect in2. the Materials assumed and point Methods after crossing the lens system in the front part of the flow explorer (2). PositioningThe presented the intersecting experimental beam in investigations the assumed pointsare carried is provided out with by an the existing stepper pipeline motor controller[22] that (6)is shown sending as the an signals,internal through loop in theFigure wires 3. (7),A researcher to the spindle plugs drives; in a pump these (10) drivers sup- shiftplying assembled water, through elements a of discharge linear stroke pipe (8) (11), along an x,upper y, and tank z axis. (12), It and turned an inlet out that pipe milk (13), is to thea transparent most effective pipe seeding (15). When substance. the upper tank (12) is full and water is overflowing the brim of aForasmuch pipe (22), and as the returning velocity to profile a bottom of the tank laminar (21), a flowvalve is (16) derived is opening. theoretically, Then the and as- provedsumed experimentally, flow is fixed using it is used the asneedle a benchmark valve (19) for and the the fluids flow flow meter velocity (20). measurements; Afterwards the aresearcher velocity profile runs isa determinedcomputer program, with a Hagen-Poiseuille which sends out equation the proper signals to the Burst Spectrum Analyzer (BSA) (5) and the stepper motor controller (6). BSA (5) sends, through 1 ∂p an optical fiber (3), two coherentv =laser beams tor2 the− R flow2 , explorer (2). These beams inter- (5) b 4µ ∂x sect in the assumed point after crossing the lens system in the front part of the flow explorer (2).p Positioning the intersecting beam in the assumed points is provided by the where ∂ means pressure gradient, r is radial distance, and R is the internal tube radius. stepper∂ xmotor controller (6) sending the signals, through the wires (7), to the spindle drives; these drivers shift assembled elements of linear stroke (8) along x, y, and z axis. It turned out that milk is the most effective seeding substance.

Energies 2021, 14, 3847 5 of 11 Energies 2021, 14, 3847 5 of 11

11 12

9 22 13 10 21 20 14 19 8 2 5 1 18 Z 15 X 6 16 Y 14 17 3 7 4

FigureFigure 3. 3.TheThe applied applied setup: setup: 1—intercrossing 1—intercrossing laser beam lasers, beams, 2—flow 2—flow explorer explorer with optics with system, optics 3— system, optical3—optical fiber, fiber, 4—signal 4—signal wire wireto photo-multiplie to photo-multiplier,r, 5—The 5—The Burst BurstSpectrum Spectrum Analyzer, Analyzer, 6—the 6—the stepper stepper motor controller, 7—x,y,z movement signal wires, 8—the elements of linear stroke along x,y,z axis Energies 2021, 14, 3847 motor controller, 7—x,y,z movement signal wires, 8—the elements of linear stroke along x,y,z6 of axis 11 with spindle drives, 9—personal computer, 10—pump, 11—discharge pipe, 12—upper tank, 13— with spindle drives, 9—personal computer, 10—pump, 11—discharge pipe, 12—upper tank, 13—inlet inlet pipe, 14—straight pipe coupler, 15—transparent tube, 16—valve, 17—temperature meter, 18— outletpipe, pipe, 14—straight 19—needle pipe valve, coupler, 20—float 15—transparent meter, 21—bottom tube, 16—valve, tank, 22—overflow 17—temperature pipe. meter, 18—outlet pipe, 19—needle valve, 20—float meter, 21—bottom tank, 22—overflow pipe. =arcsin sin , (10) 2.1.Forasmuch The Optics Systemas the velocity Adjustment profile of the laminar flow is derived theoretically, and proved experimentally, it is used as a benchmark for the fluids flow velocity measure- After a connection of the system =arcsin of apparatus sin the laser. beams intersect in any point; ments; a velocity profile is determined with a Hagen-Poiseuille equation (11) their intersection in the desired points is obtained as follows. Figure4 shows the laser beams refract four times, from right to1 left, passing three isotropic media: air, Plexiglas, Figure 4 shows every beam to =intersect must − cover, a fixed length d/2 along x (5)axis and water, so the basic equations system4 results from the law of refraction: = + + + + , (12) where means pressure gradient, r is radial distance, θ and R is the internal tube radius. θa p na sin = np sin , (6) where di = 30 mm is the internal tube diameter,2 and w2 = 5 mm is the tube wall thickness. 2.1. TheSubstituting Optics System Equations Adjustment (8)–(12) we obtain θp θw After a connection of the systemn pofsin apparatus= nw thesin laser, beams intersect in any point; (7) 2 2 their intersection in= the desired+ points + is obtained + as follows. + −2−Figure 4 shows the, laser (13) beamswhere refractna, np ,fournw are times, the from refractive right indexesto left, passing in air, Plexiglas,three isotropic and water,media: respectively;air, Plexiglas,θ a, θp, andwhereand water,θw a, = are 116.11so thethe angles basicmm isequations between measured system the with laser resultscalipers. beams from in thesethe law media. of refraction:

p sin = sin , a (6) w n =1.003 n =1.33 sin = sin , (7) 1 a 1 w 2 where na, np, nw are the refractive indexes2 in air, Plexiglas,1 p and water, respectively; θa, θp, 2 and θw, are the angles between the laser beams in these media. The angle between the laser beams in air θa is obtained from the equation 1 y d = = , (8)

where d is a distance between the laser beams in the optics, f is the focal length of the optics in air. After solving the Equations (6)–(8) we obtain the half angles of inclination in each medium: i n a =arctan , y d w (9) a

FigureFigure 4. 4. AA fragment fragment of of the the horizontal horizontal cross-section cross-section in in the the experimental experimental space. space.

Since d, f, w, di are known from the manufacturers’ data, a is measured, and the half angles are computed earlier Equation (13) is solved for ya.

= − 2 ∙ ∙ + −−2− . (14)

= − −. (15)

Figure 5 shows the course of the beams that intersect in an assumed measurement point in water at the distance yw from the internal wall, so every beam has to cover the fixed length:

= + + . (16)

Energies 2021, 14, 3847 6 of 11

The angle between the laser beams in air θa is obtained from the equation

θ 1 d d tan a = 2 = , (8) 2 f 2 f

where d is a distance between the laser beams in the optics, f is the focal length of the optics in air. After solving the Equations (6)–(8) we obtain the half angles of inclination in each medium:

θ d a = arctan , (9) 2 2 f

θ n θ w = arcsin a sin a , (10) 2 nw 2 θp n θ = arcsin a sin a . (11) 2 np 2 Figure4 shows every beam to intersect must cover a ﬁxed length d/2 along x axis

d θ θp θ θp θ = y tan a + wtan + d tan w + wtan + atan a , (12) 2 a 2 2 in 2 2 2

where di = 30 mm is the internal tube diameter, and w = 5 mm is the tube wall thickness. Substituting Equations (8)–(12) we obtain

d d θp θ θp d = y + wtan + d tan w + wtan + (a − 2w − d ) , (13) 2 a 2 f 2 in 2 2 in 2 f

where a = 116.11 mm is measured with calipers. Since d, f, w, di are known from the manufacturers’ data, a is measured, and the half angles are computed earlier Equation (13) is solved for ya. 2 f θp θ d y = f − 2·w·tan + d tan w − (a − 2w − d ) . (14) a d 2 in 2 in 2 f

Then the ordinate yl0 is read on the linear stroke element (8) shown in Figure3. The beams intersect in the point y1 when a distance from the optics (2) is equal to the focal length f. Thus, a base ordinate yl1 on the linear stroke element (8) is calculated from the formula yl1 = yl0 − ( f − ya). (15) Figure5 shows the course of the beams that intersect in an assumed measurement point in water at the distance yw from the internal wall, so every beam has to cover the ﬁxed length: d θ θp θ = y tan a + wtan + y tan w . (16) 2 a 2 2 w 2

Equation (16) is solved for another ya: 2 f θp θ y = f − w·tan + y tan w . (17) a d 2 w 2

Then the corresponding ordinate yl on the linear stroke element (8) is computed from the formula yl = yl1 + ( f − ya). (18) Energies 2021, 14, 3847 77 of of 11 11

p w n =1.48 a n =1.33 n =1.003

1 a 2 1 w 2 1 p 2 v d

w a y w y

FigureFigure 5. 5. AA course course of of the the laser laser rays rays from from th thee optics optics to to a measurement point. 2.2. Uncerainty Estimation Equation (16) is solved for another ya: Overall uncertainty estimation follows the Moffat [23] approach. A ﬁxed error Binst is equal to a difference of a benchmark = − laminar ∙ velocity + vb (yi). and a mean of 30 velocity(17) measurements vm(yi) in each point of an ordinate yi along the tube diameter:

Then the corresponding ordinate yl on the linear stroke element (8) is computed from B (y ) = v (y ) − v (y ). (19) the formula inst i b i m i

Standard deviation σ for the arithmetic mean vm(yi) is deﬁned by an equation = + −. (18)

30 !1/2 1 2 2.2. Uncerainty Estimation σ(yi) = ∑ vm(yi) − vj(yi) . (20) 30 j=1 Overall uncertainty estimation follows the Moffat [23] approach. A fixed error Binst is

equalAn to overalla difference uncertainty of a benchmark value δv ( ylaminari) is the velocity square root vb (y ofi) outsummedand a mean squaresof 30 velocity of the () i measurementsﬁxed error and doubledvymi in standard each point deviation of an ordinate because y 95% along conﬁdence the tube leveldiameter: is assumed. h i1/2 δ (y ) =By(B()(=−y vyvy)) ()2 + (2σ(y()))2 . (21) v i instinst ii b i mi i . (19)

The inverse of the squared the overall uncertainty() value is a weight wi Standard deviation σ for the arithmetic mean vymi is defined by an equation 1 wi = .12 (22) 1 30 ( ( ))2 2 σ=()yvyvy()δv ()yi − () imiji (20) 30 j=1 Since, in the lack of a ﬂow, both velocities equal zero, a. weighted regression line must cross origin; for that reason, the intercept b = 0 and the slope a is obtained with a An overall uncertainty value δv (yi) is the square root of outsummed squares of the formula [24]: fixed error and doubled standard deviationw v because(y )v ( y95%) confidence level is assumed. = ∑ i m i b i a 2 , (23) 12 ∑ wi[vm22(yi)] δ=()yBy() () +σ()2 () y vi insti i (21) and Pearson’s correlation coefﬁcient ris determined using an equation. [24]:

∑ w vm(y )v (y ) The inverse of the squaredr = the overalli uncertaintyi b i value is. a weight wi (24) n 2 2o1/2 ∑ w [vm(y )] ∑ w [v (y )] i i 1 i b i w = . i ()δ ()2 (22) 3. Results and Discussion viy The researcher adjusts the linear stroke element (8) at the location computed from EquationSince, (18) in the for lack each of of a 31flow, measurement both velocities points equal along zero, the a weighted diameter regression of the tube line (15), must and crosscarries origin; out the for velocitythat reason, measurements the intercept in b these= 0 and points; the slope then, a is the obtained researcher with repeats a formula the [24]:

Energies 2021, 14, 3847 8 of 11

()()  wvim y i v b y i a = , 2 (23) ()  wvimi y

and Pearson’s correlation coefficient r is determined using an equation [24]: ()()  wvim y i v b y i = . r 12 22 (24) {}wv() y wvy () imi ibi

3. Results and Discussion Energies 2021, 14, 3847 The researcher adjusts the linear stroke element (8) at the location computed 8from of 11 Equation (18) for each of 31 measurement points along the diameter of the tube (15), and carries out the velocity measurements in these points; then, the researcher repeats the measurementmeasurement procedure procedure 29 29 times. times. Figure Figure 6 showsshows thethe velocityvelocity proﬁleprofile fromfrom thethe averagedaveraged experimentalexperimental results results compared compared with with the benchmark laminar flow ﬂow for Re = 2137. 2137.

1.00

0.80

0.60

0.40

0.20

0.00

0.20 Laminar profile 0.40

relative radius r/R [-] r/R radius relative Experiment 0.60

0.80

1.00 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 v [m/s]

Figure 6. 6. A Acomparison comparison between between laminar laminar velocity velocity profile proﬁle Equation Equation (5) and (5) the and experimental the experimental results with results marked with velocity marked uncertainty.velocity uncertainty. Energies 2021, 14, 3847 9 of 11 To assess quality of the experiments the regression line computed from Equation (23) To assess quality of the experiments the regression line computed from Equation (23) is plotted in Figure7 with the Pearson’s coefﬁcient value computed from Equation (24). is plotted in Figure 7 with the Pearson’s coefficient value computed from Equation (24).

0.10

0.09

0.08 vb=0.9981vm 0.07 r2=0.9998

0.06

0.05 [m/s] b v 0.04 Regression line 0.03 Experiment 0.02

0.01

0.00 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 v [m/s] m FigureFigure 7. 7. TheThe plot plot of of the the experimental experimental result resultss and and regression regression line line Equation Equation (23). (23).

TheThe great great discrepancies discrepancies occur occur in in a a close close vicini vicinityty to to one one tube tube wall wall where where the the velocities velocities in laminar ﬂow v are close to zero, and the measured velocities vm are about 30 times in laminar flow vb bare close to zero, and the measured velocities vm are about 30 times higherhigher (cf. (cf. Figure Figure 7);7); it it may may result result from from the the thinner thinner wall wall in in this this place, place, which, which, unfortunately, unfortunately, may not be checked with the simple measurements because of the long distance from the tube ends. Figure 8 shows an impact of radial distance determination on the velocity profile measurements; there are plotted vertical errors bars that are the differences between radial distance of the measured velocity and the coordinate of laminar velocity profile. The differences in the first half of the diameter that is closer to the LDA system, which corresponds to the lower part of Figure 8, are significantly less than in the second half beyond the tube axis; the one exception is the first point that was discussed above. The bigger discrepancies arise when the measurement volume approaches the axis, and the maximum is reached 6 mm beyond the tube axis (r/R = 0.4). The errors in radial distance determination may be caused by other tube thickness in the measurement cross section, such as a non-circular cross section caused by several years of usage in the set-up, inappropriate adjustment of the element of linear stroke, different value of the refractive index, which might have been changed by the seeding particles.

Energies 2021, 14, 3847 9 of 11

may not be checked with the simple measurements because of the long distance from the tube ends. Figure8 shows an impact of radial distance determination on the velocity profile measurements; there are plotted vertical errors bars that are the differences between radial distance of the measured velocity and the coordinate of laminar velocity profile. The differences in the first half of the diameter that is closer to the LDA system, which corresponds to the lower part of Figure8, are significantly less than in the second half beyond the tube axis; the one exception is the first point that was discussed above. The Energies 2021, 14, 3847 10 of 11 bigger discrepancies arise when the measurement volume approaches the axis, and the maximum is reached 6 mm beyond the tube axis (r/R = 0.4).

1.00

0.80

0.60

0.40

0.20

0.00

0.20 Laminar profile 0.40

relative radius r/R [-] r/R radius relative Experiment 0.60

0.80

1.00 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 v [m/s]

FigureFigure 8. 8. AA comparison betweenbetween laminarlaminar velocity velocity proﬁle profile Equation Equation (5) (5) and and the the experimental experimental results results with with marked marked differences differ- encesbetween between radial radial distances. distances.

SummarizingThe errors in radialthe discrepancies, distance determination between the may benchmark be caused laminar by other flow tube profile thickness and the in measuredthe measurement velocity crossprofile, section, are an such uncertain as a non-circular combination cross of the section velocity caused and by radial several distance years determination.of usage in the set-up, inappropriate adjustment of the element of linear stroke, different value of the refractive index, which might have been changed by the seeding particles. 4. ConclusionsSummarizing the discrepancies, between the benchmark laminar flow profile and the measuredFor as much velocity as the profile,gradient are of anthe uncertain regression combination line is almost of theone, velocity and the andPearson’s radial distance determination. correlation coefficient is greater than minimal value, which is equal to 0.3557 for 29 de- grees4. Conclusions of freedom at 95% confidence level, it might be said, in conclusion, that the preliminary experiments are performed in a proper way, and the high quality of the further prin- For as much as the gradient of the regression line is almost one, and the Pearson’s cipal research may be maintained. correlation coefficient is greater than minimal value, which is equal to 0.3557 for 29 degrees Authorof freedom Contributions: at 95% confidence Conceptualization, level, it A.G.; might methodology, be said, in K.W. conclusion, and A.G.; that vali thedation, preliminary K.W. and A.G.;experiments formal analysis, are performed K.W. and A. inG.; a proper investigation, way, and K.W.; the resources, high quality A.G.; data of the curation, further K.W.; principal writing—originalresearch may draft be maintained.preparation, K.W. and A.G.; writing—review and editing, A.G.; visualization, K.W.; supervision, A.G.; project administration, A.G.; funding acquisition, A.G. and K.W. All au- thorsAuthor have Contributions: read and agreedConceptualization, to the published A.G.; version methodology, of the manuscript. K.W. and A.G.; validation, K.W. and A.G.; formal analysis, K.W. and A.G.; investigation, K.W.; resources, A.G.; data curation, K.W.; Funding:writing—original The paper draft was preparation, prepared at K.W. Students’ and A.G.; Scientif writing—reviewic Society “Heat and Engineer” editing, A.G.; at Bialystok visualization, Uni- versityK.W.; supervision, of Technology A.G.; and project were administration, financed by this A.G.; university. funding acquisition,The research A.G. was and carried K.W. Allout authors at the Bialystok University of Technology at the Department of HVAC Engineering as the project WZ/WB- have read and agreed to the published version of the manuscript. IIŚ/4/2019 and was subsidised by the Ministry of Science and Higher Education of the Republic of Poland from funding for statutory R&D activities. The research was conducted using equipment which was purchased thanks to either “INNO—EKO—TECH” Innovative research and didactic centre for alternative energy sources, energy efficient construction and environmental protection— project implemented by the Technical University of Bialystok (PB), co-funded by the European Un- ion through the European Regional Development Fund under the Programme Infrastructure and Environment or “Research on the efficacy of active and passive methods of improving the energy efficiency of the infrastructure with the use of renewable energy sources”—project was co-financed by the European Regional Development Fund under the Regional Operational Programme of the Podlaskie Voivodship for the years 2007–2013. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable.

Energies 2021, 14, 3847 10 of 11

Funding: The paper was prepared at Students’ Scientific Society “Heat Engineer” at Bialystok University of Technology and were financed by this university. The research was carried out at the Bialystok University of Technology at the Department of HVAC Engineering as the project WZ/WB- IIS/4/2019´ and was subsidised by the Ministry of Science and Higher Education of the Republic of Poland from funding for statutory R&D activities. The research was conducted using equipment which was purchased thanks to either “INNO—EKO—TECH” Innovative research and didactic centre for alternative energy sources, energy efficient construction and environmental protection— project implemented by the Technical University of Bialystok (PB), co-funded by the European Union through the European Regional Development Fund under the Programme Infrastructure and Environment or “Research on the efficacy of active and passive methods of improving the energy efficiency of the infrastructure with the use of renewable energy sources”—project was co-financed by the European Regional Development Fund under the Regional Operational Programme of the Podlaskie Voivodship for the years 2007–2013. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

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