APL390 Expt 4 Hot Wire Anemometry (HWA) Group- 4 Ashutosh Sharma 2020AM10643 Bhavesh Bhati 2020AM10645 Hridey Gupta 2020AM10650 Jai Shankar 2020AM10651 Parveen 2020AM10659 Raunak Choudhary 2020AM10664 Shubham Varshneya 2020AM10771 Aim: To obtain the King’s Law constants (A, B and n) and analyze submerged-jet profile variation Description of Experiment Technique (HWA)Hot-wire anemometry is a flow velocity measurement technique that relates the cooling characteristics of a flow to the velocity at that location. There are two configurations possible for an HWA setup1. Constant temperature mode 2. Constant current mode The latter is affected by thermal lag and prone to burning at low fluid velocities. Therefore, CTA is mostly used. In CTA, the voltage required to run a current to maintain a constant wire temperature is related to the local flow velocity through the King’s Law. 2 𝑛 𝐸 = 𝐴 + 𝐵𝑢 In practice, though, utilizing the temperature dependence of hot-wire resistance, the resistance is maintained constant to the value obtained from the over-heat ratio. ApparatusThe apparatus consists of1. A setup under a flat table top to generate a submerged jet in air. 2. A hot wire connected across a pair of prongs. 3. A boundary layer probe 4. A setup to mount the probe body and move it by desired amount in two directions. 5. BNC cables 6. Anemometer box 7. A manometer 8. A motor controller to control blower speed Experimental Procedure Resistance of the hot-wire probe was determined using a shorting probe (to get the resistance of leads and cable). Overheat ratio was applied on the hot-wire resistance. Calibration data was obtained by positioning the probe close to the nozzle outlet and running the blower at different speeds (corresponding to different jet exit velocities). The voltage across the hot wire leg of wheatstone was measured and displayed in the anemometer box. The pressure difference between the inlet (equal to atmospheric pressure) and outlet of the nozzle was measured using a manometer which was later related to the flow velocity using the Bernoulli’s equation. The calibration was again repeated at the end of the measurement to ensure consistency. The blower speed during measurement was chosen such that the resulting flow speed from the nozzle outlet would lead to a Reynolds number equal to that of the submerged water jet in the first PIV experiment, that is, 20000. 𝑅𝑒 = 20000 = Where, 𝐷𝑛𝑜𝑧𝑧𝑙𝑒 = 30𝑚𝑚, µ𝑎𝑖𝑟 ρ𝑎𝑖𝑟 ρ𝑎𝑖𝑟𝑣𝑛𝑜𝑧𝑧𝑙𝑒𝐷𝑛𝑜𝑧𝑧𝑙𝑒 µ𝑎𝑖𝑟 −5 = 1. 57 × 10 . Substituting in the above equation to obtain 𝑣𝑛𝑜𝑧𝑧𝑙𝑒. −5 𝑣𝑛𝑜𝑧𝑧𝑙𝑒 = 20000 ×(1.57 ×10 30 𝑚𝑚 2 𝑚 /𝑠) = 7. 85 𝑚/𝑠 Measurement involved taking voltage measurements at 13 radial locations at 7 different axial distances. The extent of radial locations at each axial position was decided by assuming a jet spread angle 𝑜 of 6. 5 . The following relation is obtained from the geometry of the setup𝐷 ( 2tan(θ/2) + 𝑥) tan θ = 𝑟 => 𝑟 = 𝐷/2 + 𝑥 tan θ 2 𝐷 = 30𝑚𝑚 𝑜 θ = 6. 5 Thus, different x values will yield different radial extents, increasing with increasing x. The first 12 radial readings at a given axial location were equally spaced, the last one was a little farther away to ensure that the velocity had entirely died down (by ensuring variation of voltage signal was negligible between the last two readings). The constants obtained from calibration and results from the measurement are summarized below. Results Calibration Plot between nozzle exit velocity and RPM 2 The results of the biquadratic fit (trendline, equation, and 𝑅 value) for calibration data before and after the experiment are shown below. Though the data doesn’t visibly change much (note the axes of both plots are the same so that the plots can be visually compared), the trendline equations are vastly different. The results of King’s law fit using the “Shooting” method is shown below with rms of normalized residuals mentioned below each plot. Again, though the data points for calibration before and after the experiment are visually identical, the regression leads to vastly different A, B and n in the King’s law relation, suggesting that the coefficients are highly sensitive to the data points. A third calibration (titled “together”) was done taking both the sets of data points (before and after the experiment) to get a better estimate of “average” calibration over the measurement. This was used for finding the velocity field from measurement data. NOTE: Though the regression results for calibration before and after are very different, the coefficients from any calibration will give velocity values very close to those from others. Therefore, the fitted King’s Law is given by2 5 6 0.21 𝐸 =− (7. 25 × 10 ) + (4. 18 × 10 ) 𝑈 Measurement- The velocity profiles at different axial distances are plotted against the radial coordinate. The measurement was done starting from origin and going along one direction, the velocity values were mirrored and plotted to show the full sectional profile. The initial few velocity profiles (starting from x = 1.1D upto 3D) can be identified as the top-hat velocity profile, it is only after x = 9D that it sufficiently spreads out and starts to take the usual Gaussian-like form. Velocity was normalized with the centreline velocity and position with axial position. The following plot showing self-similarity of the jet was obtained. The first two profiles, close to top-hat profiles don’t lie on the self-similarity curves. Error analysisThough the data on the error in Voltage signal (which was being noted manually) was not collected, the percentage (or fractional) error in velocity due to the same can be formulated as follows2 𝑛 𝐸 = 𝐴 + 𝐵𝑈 Taking differential of both sides- 𝑛−1 2𝐸 δ𝐸 = 𝐵𝑛𝑈 2 δ𝑈 𝑛 (2𝐸 ) δ𝐸/𝐸 = (𝑛𝐵𝑈 ) δ𝑈/𝑈 δ𝑈/𝑈 = 2(δ𝐸/𝐸) 2 𝑛(1−𝐴/𝐸 ) For a more accurate analysis, the errors in the (effectively) linear regression coefficients A and B can also be included using the expressions for standard error in Linear regression coefficients2 𝑆𝐸(β1 ) = σ (𝑛 − 2) · 𝑆𝑥𝑥 𝑆𝐸(β0) = 2 σ 2 · (1/𝑛 + 𝑥̄ /𝑆𝑥𝑥) Although, the process of calibration seemed far more accurate than the measurement. Discussion ● The reference velocities in calibration were obtained from manometer readings. The two limbs of the manometer were connected to inlet and outlet (open to atmosphere) of the nozzle having an area ratio of 5:1. The pressure difference measured as the difference in levels of the two limbs of the manometer was related to the flow velocity through the Bernoulli’s equation, ∆𝑝 = ρ𝑙𝑖𝑞𝑔∆ℎ = 1 2 2 ρ𝑎𝑖𝑟𝑣 𝑜𝑢𝑡𝑙𝑒𝑡 ⇒ 𝑣𝑜𝑢𝑡𝑙𝑒𝑡 = − 1 2 2 ρ𝑎𝑖𝑟𝑣 𝑖𝑛𝑙𝑒𝑡 = 12 25 2 ρ𝑎𝑖𝑟𝑣 𝑜𝑢𝑡𝑙𝑒𝑡 25ρ𝑙𝑖𝑞𝑔 ∆ℎ 12ρ𝑎𝑖𝑟 Where ∆ℎ is measured from the manometer. And the error in which is governed primarily by the least count of the scale on the manometer. One way to reduce this error is to 𝑜 incline the manometer at an angle (21. 6 for the manometer used) to increase the sensitivity of the manometer keeping the least count error same, thereby, reducing the relative error in ∆ℎ measurement. ● Although the hot-wire is most sensitive to the flow in the binormal direction, it is used with the flow in normal direction with slightly less sensitivity as it disturbs the flow much less in the latter configuration. The hot-wire probe is only translated axially and radially in the submerged jet experiment so its orientation relative to the flow direction is not expected to change. A change in relative flow direction would lead to a change in sensitivity and an error thereby. If the probe moves (for example, vibrates), it would add a flow velocity component relative to the probe which would appear as a noise in the signal. ● Axi-symmetry of the jet enables us to know the velocity profile at all angles by measuring it from centreline outwards along any one radial direction. ● Close to the edges, HWA is expected to have a positive bias error. The mean velocity of flow is close to zero at the edges, meaning that turbulent velocity fluctuations are dominant. Since only the magnitude of flow velocity is measured in HWA, all the fluctuations will be measured as positive, imparting a small positive mean proportional to the magnitude of turbulent fluctuations. ● Comparison of HWA and PIV: ○ PIV allows one to sample velocity at multiple points at the same time instant without disturbing the flow. The same, if attempted with HWA, would add to the complexity and bulkiness of the experimental apparatus linearly in proportion to the desired spatial resolution requirement. In addition, since the probe occupies the physical space where flow occurs, it disturbs the flow to whatever little extent depending on the kind of flow. ○ ○ PIV measures all components of the flow velocity as opposed to HWA which can measure only the resultant velocity at the probe's location. HWA is much less sophisticated as it requires only the probe to be placed in the flow. PIV, on the other hand, requires seeding of the flow with tracer particles. It also needs setting up of a camera, laser, and lens. The calibration resulted in an equation with n = 0.21, significantly lower than the typical range of n in King’s Law (0.4 to 0.5). A strong sensitivity of King’s law coefficients (A and B) on the data was observed. The calibration data (before and after the experiment) was visibly very similar, but the calibration relations differed. The Velocity profile obtained from the measurements well matched the theory. Close to the jet outlet, it started off as a top-hat velocity profile and became smooth and spread out for higher axial distances. Also, it was the profile of magnitude of velocity that was measured and not individual components, as opposed to PIV. The Voltage signal fluctuated a lot during measurement, making manual measurement very difficult. This also means that the electronic drift was the main contributor to experimental error as calibration looked comparatively much more precise. Logging the data as several time-series signals (one at each x and r location) would be much more practical. Since the flow is statistically stationary, the signal can be time-averaged (which should converge to the ensemble average) to obtain a much better estimate of flow velocity by reducing the random errors significantly.
