Classical Mechanics of Fluids - Assignment Example

1. Classical Mechanics of Fluids (35 marks) 1.1. The Navier-Stokes equations govern fluid flow in fires and fire protection systems. It is the foundation for water flows, gas flows, and fire simulations and modelling. Please list the Navier-Stokes equations together with the equations of energy conservation (not just the names but also the mathematical expressions) [4 marks]. The Navier-Stokes equations give description of motion of fluid using a set of partial differential equations. The equations are used in the describing mass conservation, momentum and energy flow in a fluid. Mass conservation equation is given by Momentum conservation is given by Energy conservation is given by Explain the physical meanings of each term in the momentum conservation equations [3 marks]. In the equation  is change pressure in the fluid,  a small change in distance,  is a small volume change and u is velocity of fluid. In conservation of momentum equation t is time  is a constant. Also in conservation of energy  is a constant,  is emissivity. Indicate what terms in the equation need turbulence modelling [2 marks]. There is need to have turbulence flow because the equation assume particles moving from outer layers to inner layers. In laminar flow this is not possible. Discuss the reasons why turbulence models are necessary by analyzing the scale of vortices in fluid flow and the required capacity of computers? [5 marks] Use of eddies at small scale presents the advantage of reduced need of calculation and thus translating to overall improvement. Obtaining transit solution does not present any difficult for LES modelling as the need of averaged parameters is eliminated (Bullen, 1978; Wakatsuki, 2001). Give an example of the source term in the equation of energy conservation [1 mark]. Pressure is a source term 1.2. A rise of 100mm in diameter is provided to a building to facilitate firefighting. The inlet of the riser is 1m above ground level. The outlet of the riser is 21m above the ground level. A hose of 45m in length and 80mm in diameter is connected to the outlet of the riser and horizontally laid on the floor. The hose has the same height as the riser outlet and its nozzle is open. Assuming the a fire engine is connected to the inlet of the riser and maintains a constant pressure of 12 bar at the inlet of the riser, please perform the following: (1) Draw a diagram to show the system; [5 marks] (2) assuming energy loss along the pipes can be omitted, calculate the pressure at the outlet of the riser where the diameter is 100mm using Bernoulli’s equation and the equation of continuity; [10 marks] Using Bernoulli equation P1 + += P2 + + (1) From Q = A1V1 = A2V2 and since cross section act the inlet and at outlet remain the same ie A1 = A2 it is implies that V1 =V2 This means that the velocity pressure cancels out P1 – P2 = (2) Rearranging P2 = (3) ρ = 1000 g=9.81 Substituting in eqn 3 P2= 1200000-1000*9.81*1-1000*9.81*21 P2 = 1200000- 9810-206010=1396200Pa P2= 9.84bar (3) assuming the pipe surface roughness is ε=0.1mm, density of water is 1000kg/m3, and viscosity μ=0.9×10-3 kg/(m∙s), calculate the flow velocity and volume flow rate at the outlet of the hose use the Swamee-Jain equation for the friction factor; [5 marks] From  Rearranging  2Dimensional analysis (15 marks) 2.1. Find the dimensions of the following term. 𝜌𝑣𝑎𝑏𝜇2 Where ρ is the density, v is the velocity, a and b are lengths, and µ is dynamic viscosity of the fluid. [5 marks] v = LT-1 ρ= ML-3 µ=ML-1T-1 a=L b=L 𝜌𝑣𝑎𝑏𝜇2 =ML-3 LT-1LL ML-1T-1 ML-1T-1 = M3L-1T-3 UCLAN FIRE www.uclan.ac.uk/fire 2 2.2. 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] The equation in indicial form is  =  (1) The basic dimensions of the elements are given as  = LT-1 = MLT-2M-1T-1 = LT-3 v = L2T-1 Substituting dimensions in place of variables LT-1 = (L2T-1)a (LT-3) b This can be simplified to L1 T-1 = L2a Lb T-1a T-3b Equating powers T-1= T-1a T-3b -1= -1a-3b Or a = 1-3b L1 = L2aLb 1 = 2a +b 1 = 2(1-3b)+b 2b = 1 b= ½ 1 = 2a +1/2 a= 1/4 Substituting for a=1/2 b = 1/4 substituting for a and b in equation  =  3. Heat Transfer, Thermochemistry and Fluid Dynamics of Combustion (25 marks) 3.1. Explain the process of the burning of wood (pyrolysis process, charring, flaming, and heat transfer process should be discussed). A timber beam is ignited in a compartment fire. After the beam burns for a period of time, a layer of char is formed on the surface of the beam, using the principle of heat transfer, explain how this layer of char affect the burning of the beam. [15 marks] Usually materials like wood are said to burn indirectly whereby the materials to not actually burn but combustion will tend to take place when there is reaction between oxygen and the gases that are released from the wood. When wood is subjected to high level of heat wood produces substances that are highly reactive with oxygen and this leads to this result high propensity of wood igniting and burning. Usually the ignition and combustion of wood comes as a result of pyrolysis of cellulose (the major component of wood) and the reaction of the products of this process with oxygen and other gases in the air. After the pyrolysis of cellulose as a result the high temperature exposure, the decomposition products may stay inside the wood or may be released as gaseous materials. The gaseous materials reacts among themselves and with oxygen a result of which is the release of heat which in turn result to further pyrolysis and more combustion reaction. There are two pathways that can be taken in the pyrolysis of wood this being dependant on environmental conditions such as the level of oxygen concentration, temperature, moisture, pH level and fire retardants. One of the pathway is the tar forming pathway, which is experienced when the temperature is in the tune of 300 °C which is said to be related to what is referred to as normal burning of wood (Bhattacharjee,2003). Here we have the pyrolysis process producing a lot of tar which includes levoglucosan which can easily be decomposed to burning gases when subjected to heat. The other is the char forming pathway, which involves transformation of cellulose to unstable active cellulose which undergoes further decomposition so that carbon dioxide and water are the main reaction products and the backbone of cellulose that would be compost mainly of carbon. The external factors that include the manner in which the wood is being heated, the rate of warming-up the materials affect pyrolysis of wood. External factors also play a major role in determination of rate of heat release. This makes it impossible for exact to be given. In the burning of the wood products at a constant rate of heat release per area, the boundary which we have between the pyrolysed material and the wood that is still intact would precede depth wise in the wood. Since any wood is susceptible to charring, the rate at which the wood will char β, will tally with the rate of propagation of the pyrolysis front. The charring rate is a very important quantity in determination the extent to which wooden can resist fire bearing in mind that the wood under the char is well protected from the fire and it preserves its original properties. 3.2. Define the Reaction Rate of a fire, then, discuss the factors that affect the reaction rate in a general secondary-order A + B → C + D chemical reaction. Give an example of such a chemical reaction of burning a gaseous fuel. [10 marks] Reaction rate may be defined as being change of concentration of a substance over a time interval in which the changes are being experienced. Temperature is one of the factors that have effect on the rate of reaction. Increasing the temperature of reactants will result to an increase in the rate of reaction beyond that experienced at a much lower temperature. Inclusion of a catalyst may also have a considerable effect on the reaction rate where by the catalyst will serve to increase the rate of reaction. The level of concentration of the reactants is another important factor in determination of the rate at which the reaction will take place with low concentration of reactants resulting to low reaction rate while high concentration of reactants could mean high rate of reaction and more “violent”. Combustion of PMMA in the air is a good example a general secondary order reaction. 4. Characteristics of Flames & Fire Plumes (25 marks) 4.1. Fire plumes are important in fire dynamics. Using a liquid fuel pool fire at the centre of a compartment as an example, explain the characteristics of a fire plume and generalize the axisymmetric plume model for calculating the smoke production rate and temperature along the axis of the fire plume. [15 marks] When it come combusting plume, there is enough supply of oxygen for complete burning but there is insufficient mixing. Because of the insufficient mixing 10x stoichiometric oxygen may be required for complete reaction to take place. Normally fire plumes will exhibit a centerline temperature as shown in Figure 1. Figure SOURCE: Quintiere. J.G (2006). Fire plumes and jets. Hydrogen safety. Belfast. Large fire flame temperatures will be described as in figure 1. The temperature will increase as the radiation fraction Xf decreases, will increase with D and is not significant function of fuel. Figure 3 SOURCE: Quintiere. J.G (2006). Fire plumes and jets. Hydrogen safety. Belfast. 4.2 Diffusion flames are common in compartment fires. Assuming a solid fuel is ignited, discuss and analyze factors that affect the spread of the flame on the solid fuel surface. If the fuel is a gas or a liquid, how the flame will spread? [10 marks] Studies of fire spread performed by Lefevrre et al (2004) has indicated that with high density the spread is likely to be lower with a longer melting step of foam. Experiment has also revealed that ignition delay time will be decreased when oxygen concentration pressure with an increase. Experiment done to investigate how temperature of opposed flow affect the behaviour of flames has revealed that the delay time will be decreased while the rate at which the flame is spreading will increase in response to increased temperature in the opposed flow. For the case of gaseous and liquid fuel upon the ignition of fire the spread of the fire will be dependent on whether there was sufficient air to support the gaseous fuel or not. References Bhattacharjee, S.et al (2003). “Predicting of critical fuel thickness of flame extinction in a quiescent microgravity environment,” Combustion and flame, Vol. 132, pp. 523-532, 2003. Lefebvre, J., et al (2004). “Flame spread of flexible polyurethane foam: comprehensive study,” Polymer Testing, Vol. 3, pp. 281-290. Fujita, O., Takahashi, J., Ito, K.(2000). Proceeding of the Combustion Institute, Vol. 28, Pittsburgh, p. 2761-2767. Niioka, T. et al (1981). “Gas –phase ignition of a splid fuel in a hot stagnation point flow,” 18thSymposium (International) on Combustion, pp. 741-747. Magee, R. S., Mcalevy, R. F., “The mechanism of flame spread,” J. of Fire and Flammability, Vol. 2, pp. 271-282, 1971 Quintiere. J.G (2006). Fire plumes and jets. Hydrogen safety. Belfast. Read More

After the pyrolysis of cellulose as a result the high temperature exposure, the decomposition products may stay inside the wood or may be released as gaseous materials. The gaseous materials reacts among themselves and with oxygen a result of which is the release of heat which in turn result to further pyrolysis and more combustion reaction. There are two pathways that can be taken in the pyrolysis of wood this being dependant on environmental conditions such as the level of oxygen concentration, temperature, moisture, pH level and fire retardants.

One of the pathway is the tar forming pathway, which is experienced when the temperature is in the tune of 300 °C which is said to be related to what is referred to as normal burning of wood (Bhattacharjee,2003). Here we have the pyrolysis process producing a lot of tar which includes levoglucosan which can easily be decomposed to burning gases when subjected to heat. The other is the char forming pathway, which involves transformation of cellulose to unstable active cellulose which undergoes further decomposition so that carbon dioxide and water are the main reaction products and the backbone of cellulose that would be compost mainly of carbon.

The external factors that include the manner in which the wood is being heated, the rate of warming-up the materials affect pyrolysis of wood. External factors also play a major role in determination of rate of heat release. This makes it impossible for exact to be given. In the burning of the wood products at a constant rate of heat release per area, the boundary which we have between the pyrolysed material and the wood that is still intact would precede depth wise in the wood. Since any wood is susceptible to charring, the rate at which the wood will char β, will tally with the rate of propagation of the pyrolysis front.

The charring rate is a very important quantity in determination the extent to which wooden can resist fire bearing in mind that the wood under the char is well protected from the fire and it preserves its original properties. 3.2. Define the Reaction Rate of a fire, then, discuss the factors that affect the reaction rate in a general secondary-order A + B → C + D chemical reaction. Give an example of such a chemical reaction of burning a gaseous fuel. [10 marks] Reaction rate may be defined as being change of concentration of a substance over a time interval in which the changes are being experienced.

Temperature is one of the factors that have effect on the rate of reaction. Increasing the temperature of reactants will result to an increase in the rate of reaction beyond that experienced at a much lower temperature. Inclusion of a catalyst may also have a considerable effect on the reaction rate where by the catalyst will serve to increase the rate of reaction. The level of concentration of the reactants is another important factor in determination of the rate at which the reaction will take place with low concentration of reactants resulting to low reaction rate while high concentration of reactants could mean high rate of reaction and more “violent”.

Combustion of PMMA in the air is a good example a general secondary order reaction. 4. Characteristics of Flames & Fire Plumes (25 marks) 4.1. Fire plumes are important in fire dynamics. Using a liquid fuel pool fire at the centre of a compartment as an example, explain the characteristics of a fire plume and generalize the axisymmetric plume model for calculating the smoke production rate and temperature along the axis of the fire plume. [15 marks] When it come combusting plume, there is enough supply of oxygen for complete burning but there is insufficient mixing.

Because of the insufficient mixing 10x stoichiometric oxygen may be required for complete reaction to take place. Normally fire plumes will exhibit a centerline temperature as shown in Figure 1. Figure SOURCE: Quintiere. J.G (2006). Fire plumes and jets. Hydrogen safety. Belfast.

Read More