fluid dynamics

Aims of Assessment
This task consists of problems to be solved by means of calculations or analytical evaluations. The exercises are designed to test your abilities to use techniques and approaches explained in the tutorial sessions.

Learning Outcomes
This piece of assessment will test you ability to meet learning outcomes 1-4 as described in your module booklet: –

1. Describe fundamental principles developed in fluid dynamics and heat and mass transfer, including dimension analysis and the meaning of dimensionless groups, and implement these principles to fires and explosions.
2. Determine fluid flow rates, pressure drops, describe flow in pipes, ducts, corridors and simple atria, and understand the governing parameters for different types of flow.
3. Describe heat, mass and momentum transfer in fluids and calculate transfer intensity using dimensionless criteria.
4. Examine the main characteristics of jet and buoyant flames, fire plumes and flows encountered in fire environments; calculate flame height and temperature above a fire.

Assignment Details

Answer ALL questions. The marks for each question are shown in square brackets [ ] next to the question.

The report should not exceed 2,500 words or equivalent in total.

Submission Details

The assignment should be prepared electronically and submitted via eLearn. The electronic submission should consist of a single PDF or DOC(x) file. Any further files will be disregarded.

Should any part of the assignment not provide sufficient information, the student should select her/his own data and explain the choice.

Return Date
Every attempt will be made to ensure that the work will be marked and available for collection by 4th May .

1. Classical Mechanics of Fluids (25 marks)

1.1. A rigid frame Helium balloon of volume 1600 m3 and total mass 40 kg is rising up in the Earth atmosphere. What is the altitude at which the balloon will settle if the atmosphere density is reducing with the altitude h [m] according to the formula , where = 1.2 kg/m3 and ï¡ » 5 x 10-5 kg/m4

[5 marks]

1.2. A pressure meter is calibrated using a Venturi meter attached to a sma11 orifice at the bottom of a water tank. The cross-section areas of the wide and narrow parts of the Venturi meter are of 2.7 cm2 and 0.7 cm2 correspondingly. What should be the pressure drop between the wide and narrow parts of the Venturi meter, when the water level in the tank is 2.1 m?
[5 marks]

1.3. Kolmogorov scale of velocity in homogeneous turbulence depends on the kinematic viscosity coefficient v [m2/s], specific dissipation rate [J/(kg s)] and, maybe, of fluid density [kg/m3]. Obtain the formula for this dependence using the inverse dimensional analysis.
[10 marks]

1.4. An aircraft of the cross-section area A = 9 m2 has the Newton’s aerodynamic drag coefficient Cd = 0.35. What should be the power of the aircraft’s engine to propel the aircraft with the cruising speed of 820 km/h?

[5 marks]

2. Heat Transfer and Thermochemistry (25 marks)

2.1. A solid rod of length l was heated non-uniformly and the difference between its maximal and minimal temperatures is . It was left thermally insulated and after time t its temperature became uniform through all its length. Estimate the thermal diffusivity [m2/s] of the rod. Use the inverse dimensional analysis.

[10 marks]

2.2. The non-dimensional Prandtl number Pr is used in combustion as the ratio of scales of momentum to thermal diffusivities. What formulas of the following ones are wrong to express this number? Justify your choice using the direct dimensional analysis for each of these formulas. Dimensions of the right-hand sides of all 5 formulas should be obtained and presented. Notations are usual: ï¬ – thermal conductivity, ï«- thermal diffusivity, D -material diffusivity, ï®- kinematic viscosity, ï­- dynamic viscosity, ï²- density.

a) , b) , c) , d) , e)
[10 marks]

2.3. Stoichiometric fuel-air ratio of gasoline is 1/14.7. What is the equivalence ratio Ï• of its lean mixture with the fuel-air ratio 1/16.9? What is the mass fraction of gasoline in this mixture?
[5 marks]

3. Fluid Dynamics of Combustion (25 marks)

3.1. Laminar burning speed [m/s] depends on thermal diffusivity [m2/s], laminar flame thickness [m], and, maybe, on the adiabatic burning temperature [K]. Obtain the dependence using the inverse dimensional analysis.
[10 marks]

3.2. Estimate the normal flame velocity in a methanol-air mixture with the equivalence ratio 0.75 and mass fraction of diluent 0.12 at temperature l9ÂºC and 0.75 of the normal atmospheric pressure. Use the Metghalchi-Keck correlation and data from the handouts.
[5 marks]

3.3. For the average amplitude of flow velocity fluctuations = 75 cm/s, compare the turbulent burning velocities , of wrinkled flames calculated using the original DamkÓ- hler formula and correlations by Schelkin, Klimov and Clavin & Williams as per handouts. Use calculated in the problem 3.2.
[5 marks]

3.4. What should be the fuel volumetric flow rate in order to produce a 45 cm high diffusive flame. Use the Roper’s model of the laminar diffusion flame at temperature = 2292 K, the diffusivity coefficient = 1.41 x 10-3 m2 /s, and the stoichiometric molar oxidizer-fuel ratio S = 1.8. Ambient temperature is 33ÂºC

[5 marks]

4. Shock and Detonation Waves (25 marks)

4.1. What is the Mach number of a shock wave if pressure jumps 11 times inside it? How many times temperature increases within this shock wave?
[5 marks]

4.2. Thickness of a shock wave depends on kinematic viscosity coefficient [m2/s], fluid velocity [m/s] across the shock and, maybe, on fluid density [kg/m3]. Obtain the dependence using the inverse dimensional analysis.
[10 marks]

4.3. Critically compare physical mechanisms which arc responsible for formation of premixed flame fronts and detonation waves. Provide a diagram. Copying lecture materials is not acceptable.
[5 marks]

4.4. A building structure may survive an overpressure shock up to 24 times of normal atmospheric pressure. Whether is it able to withstand the detonation wave in a detonating gas mixture with the heat release of 2.6 MJ/kg. The speed of sound is 331 m/s and ratio of specific heats of the product gases is 1.2.
[5 marks]

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