The shaft shown in Figure P11-4 was designed in Problem 10-19. For the data in row (a) of Table P11-1, and the corresponding diameter of shaft found in Problem 10-19,….

## 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. Explain the physical meanings of each term in the momentum conservation equations. Indicate what terms in the equation need turbulence modelling and give the reasons why turbulence models are necessary? Give an example of the source term in the equation of energy conservation

** ****Assignment Details**

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

1. **Classical Mechanics of Fluids (25 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. Explain the physical meanings of each term in the momentum conservation equations. Indicate what terms in the equation need turbulence modelling and give the reasons why turbulence models are necessary? Give an example of the source term in the equation of energy conservation.

[15 marks]

1.2. A pressure meter is calibrated using a Venturi meter attached to a small orifice at the bottom of a water tank. The cross-section areas of the wide and narrow parts of the Venturi meter are of 6 cm^{2} and 2 cm^{2} 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 8.2 m? Please assume the Venturi meter is installed with pipes of cross-sectional area of 6 cm^{2} and the pipe finally discharges to open air at the outlet.

[10 marks]

**Dimensional analysis (25 marks)**

2.1. Find the dimensions of the following terms.

What term(s) are/is non-dimensional?

[12 marks]

2.2. Kolmogorov scale of velocity in homogeneous turbulence depends on the kinematic viscosity coefficient *v* [m^{2}/s], specific dissipation rate [J/(kg s)] and, maybe, of fluid density [kg/m^{3}]. Obtain the formula for this dependence using the **dimensional analysis**.

[13 marks]

3. **Heat Transfer, Thermochemistry and Fluid Dynamics of Combustion (25 marks)**

** **3.1. Burning of polymethylmethacrylate (PMMA) can be described by the following chemical reaction formula.

Explain the process of the burning of the PMMA (pyrolysis and reaction with oxygen) and calculate the stoichiometric fuel-air ratio. If 2.5kg of PMM is burnt, how much heat is produced?

[15 marks]

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]

4. **Characteristics of Flames & Fire Plumes (25 marks)**

** **4.1. Fire plumes are important in fire dynamics. Using a solid fuel fire at the centre of a compartment as an example, explain the characteristics of a fire plume and generalise the axisymmetric plume model for calculating the smoke production rate and temperature along the axis of the fire plume.

[15 marks]

4.2 Diffusion flames are common in compartment fires. Assuming a solid fuel is ignited, discuss and analyse 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]

Max total mark: 100.

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

The word limit is 1,500 words (+/-10%). This excludes footnotes but includes quotations.

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