Applied Analysis of the Navier-Stokes Equations - Assignment Example

Classical Mechanics of Fluids

Question One

The Navier-Stokes equations are partial differential equations that describe how the pressure, velocity, density and temperature of a viscous fluid in motion are related. The equations consists of a continuity equation for mass conservation, momentum conservation equations and equation of energy conservation. The three equations are expressed as below:

• Continuity equation /equation of mass conservation:

) = 0

• Momentum conservation equation:

+ .u u.u) + g

• Energy conservation equation:

+ . u = 0

• Equation of fluid motion:

= + F + u

Terms/variables in the Navier-Stokes equations

– Pressure

– Temperature

u - Velocity vector

- Fluid density

- External force per unit mass

- Thermodynamic energy

- Dynamic viscosity

- Gravitational acceleration

- Coefficient of heat conduction

Velocity and pressure are the terms that vary with conditions of flow and therefore, require turbulence modeling (Doering & Gibbon, 1995).

Modelling of turbulence flows is necessary because most of the flows encountered in engineering applications are turbulent in nature and predicting these flows is very challenging since there are no analytical solutions for the non-linear fluid flow equations. Capturing all the fluctuations or every single vortex in fluid flow may be impractical for turbulent flows. Capturing every single scale of scale of flow direct prediction would require very powerful computers that work very fast. These computers may not be available or may be very expensive. Hence, the need for turbulence modelling.

An example of a source term in the energy conservation equation is velocity.

Question 2

[A]

Figure 1: Diagram of the firefighting system

[B]

Assuming that the fluid is ideal fluid with zero viscosity,

Pressure at the riser inlet = 12 bar

Pressure at the riser outlet = unknown

From the equation A1V1 = A2V2, A1=A2, and therefore, V1=V2

The continuity equation states:

+ = +

P2 = P1 + -

= (1.2 ) + 1000 9.8 (1 20)

= (1.2 )196000

Pressure at the riser outlet = 1.004 Pascal or 10.4 bar

[C]

Surface roughness, ε = 0.1mm

= 1000kg/m3

μ = 0.9×10-3 kg/(m∙s)

Swamee-Jain equation for friction factor states:

=

Where:

ε – pipe surface roughness

= friction factor

D – Diameter of the pipe

RE – Reynolds number

Velocity is constant through the riser, and is given by:

Using the equation of continuity,

v2 = = = 30.938 m/s

Flow rate =(

RE = = =

= = 0.020955

Dimensional Analysis

Question 1

=

= = = s

Question 2

There are three variables for The Kolmogorov scale of velocity: fluid density, specific dissipation and kinematic viscosity. Their relation is:

=

In dimensional terms, this can be written as,

[L/T)= [L2/T] a [L2/T3] b [M/L3] c

Where a, b and c are unknown constants. We then compare the terms on both sides:

L

T

M

1=2a +2b -3c

-1 = -a-3b

0 = c

Since c=0,

1=2a +2b ……….. (i)

-1 = -a-3b ……….. (ii)

Solving equations (i) and (ii) simultaneously:

a = b = ¼

Therefore,

Kolmogorov scale of velocity is given by:(Szirtes, 2007).

Heat Transfer, Thermochemistry and Fluid Dynamics of Combustion

Question 1

Pyrolysis is the thermal decomposition of wood. The process is based on the decomposition of cellulose and the reactions between the products of pyrolysis and gases in the air. At about 160oC dehydration of wood is complete. As temperature rises to between 200oc and 280oC, the hemicellulose decomposes to yield mostly, products such as CO, CO2 and condensable vapors. Between 280oC and 320oC, cellulose begins to decompose yielding hydrocarbon volatile products. These volatile products contain 30-50% of the potential energy stored in the wood material. Chemically, wood is made of two components namely, cellulose and lignin. During the process of combustion, cellulose, which is softer and more reactive component, vaporizes and quickly gets decomposed. Lignin is harder and takes a lot more time to burn since it needs a lot of heat to reach ignition. Beyond 320oC, lignin is decomposed, accompanied by a relative increase in the content of carbon in the residual material. In all these stages of wood combustion of wood, gases given off from the decomposing wood must mix well with oxygen. Pyrolysis process of wood is coupled with mass and heat transfer processes.

Charring is a reaction process that occurs when wood undergoes incomplete combustion. The solid residue resulting from charring is known as char. Char is formed as the wood material continues to decompose, creating two layers of decomposed wood and intact wood. The action of thermal decomposition gets rid of hydrogen and oxygen from the wood material, so that charred residue is mainly composed of carbon.

When a mixture of the gases and solids emitted from the combusting wood material react altogether, a flame is produced. The nature of the flame produced depends on the chemical composition of the material and other products of combustion. In burning of wood, solid carbon particles produce a red-orange flame. The type of fuel in the combustion process determines the temperature and color of the flame produced. The maximum flame temperature of combusted wood material is 1027oC.

Heat transfer processes:

During the pyrolysis process of wood, heat is produced and transferred to the neighboring environment through two main processes; radiation and convection.

Heat transfer by radiation – This is a process by which heat is transferred through electromagnetic waves. The heat travels at a speed similar to that of light. The heat of radiation is higher closer to the heat source then further away from the radiation source.

Heat transfer by convection – In this process, hot gases and particles emitted from the burning wood travel to outer environments, transferring heat with them. The air around where combustion takes place is heated and becomes less dense, making it to rise as cold dense air moves down. This forms convectional currents that facilitate heat transfer to colder regions. The rate of heat transfer by convection is largely affected by the temperature difference between the heat source and the other regions.

The charred region forms a boundary layer between the decomposed material and the material that has not yet started burning. This boundary layer is known as the pyrolysis front. The pyrolysis front acts as an insulator, preventing heat transfer into the intact wood. It is known that insulating materials have poor thermal conduction properties. This slows down the pyrolysis process of the wood as the char layer provides insulation to stop further charring. Predictable charring in timber can be used in the determination of fire rating of the structural timbers, which is an important aspect of fire protection engineering.

Question 2

The reaction rate of fire is defined as the rate at which the fire combusts a material into the final products.

A + B → C + D

Temperature – Higher temperature will increase the kinetic energy of reacting molecules, leading to increased collisions. The colliding particles possess high activation energy, enabling them to have more collisions. The overall effect is increase in the rate of reaction.

Concentration of the reactants – Increasing the concentration of the reactants increases the number and frequency of collisions per unit time, therefore, increasing the rate of reaction.

Pressure – The concentration of a gas is directly related to the pressure of the gas. Increase in pressure increases the concentration, thus, increasing the rate of reaction. The reaction proceeds in the direction with fewer moles.

The nature and order of reaction – Some reactions occur faster than others depending on the nature of species involved, their physical state and the complexity of the reaction process. The order of reaction determines how pressure and concentration will affect the rate of reaction.

Surface area – Reactants with a larger surface area exposed to the reaction conditions will react faster than intact reaction species.

Catalysts – The presence of a catalyst will speed up the rate of reaction, as opposed to when the catalyst is absent.

Example of a chemical reaction when methane is burned:

CH4 (g) + 2O2 (g) CO2 (g) + 2H2O (l)

Characteristics of Flames & Fire Plumes

Question 1

Characteristics of a fire plume

Plume height – This is the mean vertical height of a plume rise. This height is expressed in relation to the height of a known object.

Plume velocity – This is the speed at which a plume is moving away from the point source.

Temperature – The temperature measures the thermal characteristics of a plume.

Plume turbulence – Unstable conditions existing in the atmosphere interfere with plume behavior and may alter it from laminar to turbulent flow. The nature of flow is determined by calculating the Froude’s number for a specific plume rise.

Air entrainment - This is the mount of air in a plume. Higher levels of entrained air increases the mass of the plume, and hence its travel velocity.

Plume mass – This refers to the mass flow of a given plume obtained from the speed and temperature characteristics (Quintiere, 1998).

Axisymmetric plume model is based on the assumption that a symmetrical axis line along the centerline of a plume forms a region of maximum temperature, and as you move towards the edges of the plume, temperatures decrease steadily (Yao, et al., 2007).

Question 2

Factors that affect the spread of the flame on the solid fuel surface

Availability of oxygen – If oxygen is available in the compartment, more oxygen will react with the fuel and the rate of burning will increase. Provided burning continues, fire will spread to other parts.

The size and surface area of the fuel – A larger size and surface area of a fuel will burn more efficiently than a smaller size fuel with smaller surface area. Increasing the size and surface area of fuel increases the interactions between the oxygen and fuel molecules, thus, favoring the spread of fire.

Density of the fuel – Lighter materials ignite faster and spread fire more quickly compared to heavier materials. High density materials insulating properties that makes heat to stay on the surface of the fuel and prevent spreading.

Moisture content of the fuel – Fuels with higher moisture content burn slower than those that have been dehydrated. Dehydrated fuels will burn faster and favor the spread of fire.

Amount of energy produced – Fuels which contain high thermal energy will burn at high temperatures. High temperatures in a compartment will make other fuel materials to start burning, and hence, spread fire to other parts.

A flame spreads faster for a gas or liquid because of increased surface area of the fuel. Liquids and gases are also mobile and will transfer the burning fuel to other parts.