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Classical Mechanics of Fluids.
1.1 (a) To calculate the force that should be applied to the smaller cylinder in order to lift the challenger tank of 62.5 tonnes.
By the principle of Pascal, there is equal transmission of pressure in the fluid and hence pressure in both cylinders at equilibrium is equal. By definition, pressure is P = F/A where F represents force and A represents cross-section area. Hence
where A, R and F denote area, radius and force respectively.
(b) To use dimensional analysis to determine the hydrostatic law of pressure.
The dimensions of the given quantities are
where M, T and L denote units of mass, length and time respectively. By the basic law of dimensional analysis we take that
which implies the following relationship between the dimensions of the parameters
Inserting the symbolic dimensions into the above formula gives
The right and left hand side of the above relation gives the following system of linear equations
Î± = 1, -3Î± + Î² + Î³ = -1, -Î² = -1
Solving this linear system gives Î± = Î² = Î³ = 1. Hence
1.2 (a) ................(missing question)..................
(b) Explaining the power law of Eddies dissipation in homogeneous turbulence.
Turbulence causes the formation of eddies of varying length scales. Much of the kinetic energy of the turbulent motion is found in the large scale structures. The energy dissipates from these large scale structures to smaller structures which produces a hierarchy of eddies. Eventually structures are created that are small enough to allow molecular diffusion to take place and viscous dissipation of energy occurs.
Transfer of Heat 2.1. a) -----(missing question)---------------- b) To evaluate the formula for the Nusselt number through the estimation of the heat fluxes involved by their orders of magnitude.
The Nusselt number is the ratio of orders of magnitudes of convective and conductive heat fluxes. By the definition of the convective heat flux, its order of magnitude q can be estimated as
where h, Tw, and Tâˆž denote the convective heat transfer coefficient , wall temperature and flow temperatures respectively.
Similarly the conductive heat flux is given by
where Î» is the coefficient of conductive heat transfer and L is the length scale.
Getting the ratio of these two flexes yields the Nusselt number;
2.2. a) To explain the radiation heat flux formula.
The Stefan-Boltzmann law gives the radiation heat JR emitted by a unit surface area of a body at temperature T per unit time. That is
where Ïƒ is the Stefan-Boltzmann number and Îµ is the emissivity of the body surface . The emissivity Îµ lies between zero and unity. Only an ideal radiator called black body can emit radiation heat flux with
Îµ = 1. For any real body, the emissivity is always less than one.
(b) To calculate emissivity coefficient of a flat steel plate of total area 1.5^2 at temperature 600 CO radiates 30 KW/s.
The Stefan-Boltzmann law as stated above provides the heat radiation JR emitted by a unit surface at temperature T per unit time. The radiating power Q = 30 KW is calculated by getting the product of the radiation heat density JR and the surface area of the plate. This implies that JR = Q/A . Inserting the equation of the Stefan-Boltzmann law and using JR in terms of Q yields
3. Fluid Dynamics of Combustion 3.1. a) To explain what is the normal flame speed.
When a planar flame propagates through a pre-mixed combustible gas mixture in a tube, it creates a small pressure rise in its front. This pressure gradient acts like a piston pushing unburned gas away from the flame with the speed uu. Normal flame velocity or burning velocity is the planar flame front speed relative to the unburned gases:
SL = uv - uu.
Where uv is the visible flame speed as observed in the laboratory coordinate system.
b) To estimate the normal flame velocity in a pure stoichiometric mixture of propane and oxygen at temperature O 107C and normal atmospheric pressure using formulas and data from the handouts.
The normal flame velocity SL is
Where SLref = BM + B2 (... + ...M), Tu,ref = 298 K, Pref= 1 atm, Î³ = 218 - 0.8(... - 1).
Î² = -0.16 + 0.22 (... - 1), ..., YÇ½ and Tu is the equivalence ratio, diluents mass fraction and temperature of the unburned mixture respectively and P is the atmospheric pressure.
In this case ... = 1, because this is the stoichiometric mixture. Again, YÇ½ = 0 because the mixture is a pure one and P = Pref = 1 atm in virtue of the conditions given in the question. Values of
BM = 34.22, B2 = -138.65 and ...M= 1.08 are as known. With all these parameters in the Metghalchi and Keck correlation, we get
Simplifying this expression gives the required solution,
3.2 (a) To explain the turbulent burning velocity.
A definition of the turbulent burning velocity ST similar to the laminar burning velocity is impossible because the turbulent flames are severely wrinkled and corrugated. Therefore an averaged parameter, based on the continuity equation is a better way to define the turbulent burning velocity. If premixed combustible mixture of density Ï is consumed in a tube of cross-section area A by a turbulent flame with mass rate m, then the turbulent burning velocity of this flame is defined as
The locally unburned mixture is consumed at the laminar burning rate SL, then
and its substitution in the above formula gives
where A is the instantaneous surface area of the turbulent flame.
(b) For the average amplitude of flow velocity fluctuations u = 40 cm/s, to compare the turbulent burning velocities T S of wrinkled flames calculated using the original Damköhler formula and correlations by Schelkin, Klimov and Clavin & Williams. Use L S calculated in the problem 3.1.b.
The Damköhler formula:
Clavin and Williams yields:
- Diffusion flames 4.1 (a) To explain the Richardson and Froude numbers.
The Richardson number is the ratio of orders of magnitude of access of buoyancy and inertia. That is
where g is acceleration due to gravity, u and L are the integral velocity and length scale of the flow respectively. The Froude number Fr is the ratio of the velocity scales of the flow u and of the gravity waves (gL)1/2 and is given by
The Richardson and the Froude numbers are related by
(b) To evaluate the formula for the Richardson number through the estimation of the forces involved by their orders of magnitude.
The Richardson number Ri as defined above is the ratio of orders of magnitude of forces of buoyancy and inertia. The buoyancy force is of order Fb Î± ÏgV where Ï is the density of the fluid, g is the acceleration due to gravity and V is the volume of the fluid element. If the acceleration of the fluid element is a, then the order of force of inertia is F â‰ˆ ÏaV. The acceleration a is the flow moving at speeds of order u in times of order t is estimated as a âˆž u / t, where the time scale of fluid motion over distances of order of integral flow length scale L, in its sum is of order
These estimations of orders of magnitudes of acceleration and time, specific to the fluid element under consideration, yields
Finally , the Richardson number is given as:
- (a) The laminar diffusion flame height f L depends on the volumetric fuel flow rate Q & and diffusion coefficient D . o obtain a formula for f L using dimensional analysis.
The dimensions of the given quantities in symbolic form are as follows:
where L and T denotes units of length and time respectively. By dimensional analysis, suppose that
which implies that the dimensions of the parameters are related by
Inserting the symbolic dimensions in the right and left hand sine of the above formula gives
The above relationship gives
3Î± + 2Î² = 1
-Î± - Î² = 0
Solving this system of linear equation gives Î± = 1 and Î² = -1. Thus
(b) To calculate the height of the laminar diffusion at temperature 2200 fTK for the diffusivity coefficient 3 10âˆ’ D = m2 /s, volumetric flow rate 01 . 0 = Q & m3 /s and molar oxidizer-fuel ratio S = 2 . Fuel and oxidizer temperatures are equal to 40 CO .
Roper and Roper model gives the following diffusive flame height formula
where TF and T0 are the fuel and oxidizer temperatures respectively.
With the values of the given parameters into the formula we get the value of the height as 0.542m
- Giovangigli , Smooke,. Numerical modeling of symmetric laminar diffusion flames. 1992
- Charles (Editor), Xianming J.L (Editor), Vladimir, G.. Computational Fluid Dynamics in
Industrial Combustion. 2001 3. Oppenheim, A. K. 1st ed., Dynamics of Combustion Systems. 2002.