1) The Millikan-Experiment
The Millikan experiment served in 1910 to determine the electrical elementary charge. In this process, atomized oil is added to the smallest droplet between the plates of a condenser oriented parallel to the gravitational force and is observed by a microscope. In the sputtering process, small amounts of electrical charge are torn from the oil droplets whose size is to be determined. Since the oil droplets in the microscope are only recognizable as scattering centers in the case of darkfield illumination, their diameter can only be determined indirectly. For the purpose of determining the electric charge, one proceeds therefore in two steps:
1. First, in a stress-free case, a drop-out oil droplet is observed and its sink rate c = x / t is determined by measuring the distance x traversed in a time t by means of a length measure visible in the microscope. The droplet diameter 2r can then be determined from the force balance.
2. Now apply an electric voltage U to the condenser, which is adjusted so that the falling oil droplet is brought to a standstill and into a suspended state. The force balance for this case allows the determination of the electric charge q.
In step 1 one can start from a creeping airflow around the oil droplet. In this case, for the resistance force the stokessche formula Fw = 6πηrc (η dynamic viscosity of the air) applies. In step 2 the electric force Fe = qE acts on the electrically charged oil droplet. The electric field strength E in the capacitor is related to the capacitor voltage U and the plate spacing d via the relationship E = U / d.
An Millikan-Experiment has made the following data for the sink rate c and the hold voltage U:
|
c[μm/s]
|
U[V]
|
2r[μm]
|
Re
|
q/e
|
|
3, 8
|
25, 2
|
|
|
|
|
5, 3
|
20, 1
|
|
|
|
|
3, 9
|
8, 6
|
|
|
|
|
3, 2
|
18, 7
|
|
|
|
|
4, 3
|
15, 4
|
|
|
|
Further were the boundary conditions and the substance data of the Experiment:
Density of the oil - Dichte des Ols ρδl = 0, 85 g/cm3
Pressure - Druck p = 1 bar
Temperature - Temperature 20oC
Kinematic Viscosity - Kinematische Viskositat der Luft v = 0, 1 cm2/s
Acceleration of Gravity - Erdbeschleunigubng g = 9, 81 m/s2
Plate Seperation - Plattenabstand d = 3 cm
Complete the above table:
A) Set the balance of forces for the test step 1 and determine the associated droplet diameters 2r for the table values c. Assess whether it is necessary to consider the buoyancy force for the oil droplet.
B) In all cases, calculate the Reynolds number Re for the air flow around the oil droplet and check whether the crit Re < 1 for creeping flows is fulfilled.
C) Calculate the respective electrical charge q for the test step 2. Show that it is a good approximation for integer multiples of the elementary charge e = 1.6 · 10^(-19)C (1 C = 1 Coulomb = 1 As = 1 Ampèresecond).
2) Power of a turbine in a water storage power station
On the inside of the reservoir of a water tank is the entrance to the downpipe 20 meters below the water level. The turbine house is 500 meters below. The pipeline has a diameter of 1 m and a total length of 4.0 km up to the turbine. The line exit to the surroundings behind the turbine also had the diameter 1 m. In the following, it is assumed that the pipeline is completely filled with water.
Task:
A) Calculate the power P delivered by a turbine operating at 95% efficiency as a function of the flow velocity c and sketch the function P (c) graphically. At which flow velocity copt is the maximum turbine output achieved? What is the maximum turbine output? At which speed cmax> 0 is the turbine output zero (no power draw from the fluid flow)? Compare this value with the one according to the torricellic outflow formula.
For the sake of simplicity, use the formula of Nikuradse for the pipe friction coefficient λ to calculate the loss number (value please specify!):
1/√λ = 2log10(3, 71 d/k)
It describes the asymptotic, roughness-dominated behavior for large Reynolds numbers. Check by comparing to the Colebrook diagram that the Nikuradze formula is applicable in the environment of c = copt, and make a judgment.
B) Using the assumption of the Nikuradze formula, sketch the pressure profile between a point in the tube inlet area and a point immediately before the turbine inlet as a function of the height coordinate z for constant slope (the pipe profile follows the oblique plane). In cases c = copt or c = cmax, explicitly specify the pressures at the pipe endpoints.
Roughness height of the tube wall k = 2 mm
Material data: Density of water ρw = 1000 kg/m3, viscosity vw = 1 mm2/s.
3) Part A: Ammonia synthesis
A gas mixture of molecular nitrogen N2 (species 1) and molecular hydrogen H2 (species 2) has been compressed to a pressure of p = 300 bar as a result of a rapid compression process. The temperature was 450oC. with a final volume of 4 liters. The mass ratio is
n1: n2 = 1: 3.
Under isothermal conditions, the ammonia synthesis reaction proceeds slowly in the material-compacted system
N2 + 3H2 → 2NH3.
Task:
A) Calculate the involved masses n1, n2 and the associated partial pressures p1, p2 at the beginning of the reaction.
B) What happens to the pressure p during the formation of ammonia and why?
C) At what value can the pressure of the gas mixture in the hypothetical case of a complete chemical reaction drop at most?
3) Part B: Dew point of exhaust gases
The exhaust gas of an engine or a combustion plant consists of dry mixed gas and water vapor. The analysis of an exhaust gas sample had the following composition: m (H2O) = 0.5 kg, m (CO2) = 3.0 kg, m (SO2) = 0.03 kg, m (N2) = 12.5 kg, m (O2) = 1.5 kg. During operation, the exhaust gas is released into an environment with p = 1 bar air pressure. At which temperature is the dew point at which the water vapor condenses for this gas mixture?
Task:
A) Calculate the water content x of the exhaust gas (eg in g of water vapor per kg of dry exhaust gas) and the corresponding partial pressure pD of the water vapor. At the dew point, this is equal to the saturation pressure pS. The associated temperature t can then be taken from the Mollier-h, x-diagram for moist air or also from a tabulated steam pressure curve.
B) The required atomic weights are M (H) = 1 g / mol, M (O) = 16 g / mol, M(C) = 12 g/mol, M (S) = 32 g / mol, M(N) = 14 g/mol
The general gas constant has the value Rm = 8.314 J/molK.
4. Thrust of a jet engine
A four-jet commercial aircraft flies at a constant cruising speed U = 950 km / h at 11 km altitude. From the engines, the air sucked in at the entrance speed U1 = U∞ = U (Fitting diameter 2.95 m, effective inflow cross section = 0.7 x fan cross section, U∞ flow velocity of the undisturbed air far away from the aircraft in its resting system) 65 times the value of the sound velocity in the surrounding air. The outlet nozzle is designed as a Laval nozzle so that the pressure in the emerging gas jet is equal to the ambient pressure.
A) Use the pulse set to calculate the thrust to be expected per engine. Calculate with the ICAO standard atmosphere and with R = 287 J/kgK and ? = 1.4 for air.
B) What is the size F / Mg of g = 9.81 m / s2 for an aircraft mass of M = 570 tonnes (Airbus A380) if F is the total thrust force? What is the effective resistance area cwA? What is the coefficient of drag cw when, as is usual in aeronautical engineering, the wing surface is used as the reference surface A? It is 846 m² for the Airbus A380 A. For the comparison, calculate the drag coefficient as the reference surface by using the end face, which is customary in vehicle dynamics and projected in the direction of movement. For this, estimate the latter by means of the following information: The Airbus 380 has an elliptical hull with an average diameter of approx. 7.7 m and four engines of the above-mentioned Dimensions.
C) How much is the thrust reduced if a conventional, convergent nozzle is used instead of the Laval nozzle with the same mass flow? The gas state at the inlet is characterized by p0 = 1.9 bar and t0 = 393oC.