6. Friction and viscosity in gasses

Transcription

1 IR2 6. Friction an viscosity in gasses 6.1 Introuction Similar to fluis, also for laminar flowing gases Newtons s friction law hols true (see experiment IR1). Using Newton s law the viscosity of air uner normal pressure will be measure in the first part of the experiment. The viscosity will be etermine by the friction force of two counter spinning isks between which a layer of air is enclose. In the secon part of the experiment a vacuum pump will be use to reuce the air pressure between the two plates such that the mean free path of the air molecules is of the orer of magnitue or larger than the istance between the two plates. It will be shown that in this case the moel of air treate as a meia with viscosity fails an that it is necessary to take into account the properties of the single molecules. 6.2 Theory The experimental setup is shown in Fig Two flat, roun isks I an II having the same raius R are mounte on a common rotation axis separate by a istance. Disk II can be turne with negligible friction aroun this axis. On isk II a weight G is mounte at a istance l from the axis of rotation. Disk I will be riven by an electric motor to rotate with constant angular velocity ω. The air between the two isks will be set in motion by the rotating isk I ue to the resulting friction forces. This motion results in an torsional moment M R onto isk II. The isk II will turn until the torsional moment M G of the weight G compensates M R. For a specific eviation angle α, M G is given by: M G = G l sin α (6.1) The torque M R can be calculate from Newton s friction law. Since the istance between the two isk is small compare to the Raius R, the velocity profile between the two isks can be assume to be linear (see Fig. 1 in experiment IR1). Hence, the force applie to isk II is given by: F R = η A v, (6.2) 1

2 2 6. Friction an viscosity in gasses I II R l α motor ω G G Abbilung 6.1: Experimental setup for the etermination of the viscosity. where the velocity v an hence F R epen on the istance r from the axis of rotation as v(r) = ω r. (6.3) Thus the torque exerte in a circular ring with raius r an thickness r can be calculate as M R = r F = r η A v(r) = η 2 π ω r 3 r. (6.4) The total torque M R is obtaine by integrating of the raius an yiels: M R = R 0 M R = η 2 π ω R 0 r 3 r = η π ω R4. (6.5) 2 In the equilibrium state the two torsional momenta compensate mutually M R = M G. Equating Eq. 6.1 an Eq. 6.5 an solving for η yiels η = 2G l πr 4 sinα ω If R,, G, an l are known, η can be obtaine by measuring α as a function of ω. (6.6)

3 6.3. EXPERIMENTAL PART Experimental Part a) Viscosity of air at ambient pressure In the first part the viscosity of air in ambient conitions is measure. The experimental setup is alreay mounte an ajuste. The glass cover thus shoul not be lifte from the base plate. The imensions R,, G an l of the apparatus are liste at the experimental setup. Die angular velocity ω of isk I is measure by means of a stroboscope. This evice is similar to an electric flashlight for which the single light flashes repeately with a tunable frequency. Hint: Always write own the chosen electrical voltage for the riving motor uring the experiment, this makes it more convenient to reprouce the measurements. On the back sie of isk I a black fiel an a mirror are mounte. The rotating isk is illuminate by the stroboscope. If the rotation frequency of the isk is the same as the repetition rate of the light flashes, the isk will be illuminate always at the same position an seems to stop. This is calle a static picture of the isk. It is most easily observable by the position of black fiel or the mirror. In orer to measure ω, change the frequency of the light flashes until you observe a static picture of the isc. You can rea off the rotational frequency of the isc in units of Pulses per Minute irectly from the calibrate scale of the stroboscope. What o you observe if the flash frequency is an integer multiple of the rotation frequency or vice versa? When measuring the angular velocity o you thus have to scan through the frequencies of the stroboscope from low to high or from high to low values? Measure at a pressure of 1000 mbar (atmospheric pressure or about 730 Torr) the angle α of isk II as a function of the angular frequency ω of isk I. Repeat these measurements with increasing or ecreasing angular velocity to avoi hysteresis effects an take the average over these measurements. The resulting curve is shown in Fig For low velocities α increases linearly as expecte accoring to Eq. 6.6 with increasing ω (for small angles α the approximation sin α α can be mae). Above the critical velocity v krit the gas flow becomes turbulent, hence the friction forces, an the torsion angle α increases faster than linearly with increasing velocity. Determine the best line fit for at least five ata points below the critical velocity v krit an etermine the viscosity η of air from the slope of the line. Estimate the error on η. Estimate from the progression of the curve the critical velocity v krit an from this calculate the critical Reynols number Re krit (see experiment IR1).

4 4 6. Friction an viscosity in gasses α Re < Rekrit. α(υ) at p min v krit. v ~ ω Abbilung 6.2: Depenency of α on the rotational spee v as expecte accoring to Eq b) Investigation of the viscosity of air at low pressure Set the rotation frequency ω of isk I to a value below the critical velocity v krit. Turn on the vacuum pump an keep track of the torsion angle of isk II while evacuating the vacuum vessel. You will observe that the viscosity of air oes not epen on the pressure own to very low pressures. Use the iscussions in the Appenix to estimate at which pressure a rop in viscosity woul be observe for the experimental setup. At p = 10 6 bar an T = 20 C the mean free path of N 2 an O 2 molecules is approximately 7 cm. After the minimal pressure is reache, increase the rotation frequency ω to a value above the value foun for the critical velocity v krit at ambient pressure. Mark the measure torsion angle α in the α(v)-iagram from the first part of the experiment. Extrapolate the best fit line from the first part an explain why the measurement at minimal pressure lies on this line (see Appenix).

5 6.4. APPENDIX Appenix Pressure epenency of the gas viscosity In a gas the molecules move with a high velocity an collie with each other an the walls of the container. This motion is chaotic an has not preferre irection. Internal friction only results from a position epenent net mean rift velocity v superimpose on the chaotic movement. As an example consier a gas enclose by two parallel plates which move with a velocity v with respect to each other (see Fig. 1, experiment IR1). If the pressure of the gas is small such that the mean free path l between two collisions of the molecules is larger than the istance of the two plates, the molecules from the non-moving plate (mean velocity v = 0) reach the moving plate without colliing with other molecules (see Fig. 6.3). They will stick to the wall for a short moment an get a small amount of momentum. Eventually, on the plate a shear τ = F/A is exerte, which correspons to the transferre momentum per unit of time an area. τ Z m v (6.7) Here m is the molecule mass an Z the number of molecules which hit the plate per secon an unit area. Z is proportional to the molecule ensity which is proportional to the pressure p. Thus, τ p an since η τ the relation hols: η p. The mean free path l of the gas molecules is inversely proportional to the particle ensity an to the pressure p. If the pressure is increase such that l, the molecules reaching the moving plate o not irectly come from the non-moving plate but from a gas-layer at an average istance l from the wan in which they encountere collisions an acquire the mean velocity at this position: ( v(z = l) = v 1 l ) (6.8) υ υ z υ() l l» l «0 υ(z) Abbilung 6.3: Motion of the molecules between two plates for l (left panel) an l (right panel).

6 6 6. Friction an viscosity in gasses η p η~p for l» η inepenent of p for l«abbilung 6.4: Pressure epenency of the viscosity. The momentum transferre to the moving plate yiels: ( ) m v v(z = l) = m v l. (6.9) Using Eq. 6.7 we obtain τ Z m v l for l (6.10) Since Z p an l 1/p, the shear on the plate an thus the viscosity of the gas o not epen on the pressure at low particle ensity or pressure. Summarizing, a pressure epenence of the viscosity as shown in Fig. 6.4 is expecte.