Solution of Fluid Problems - Assignment Example

SOLUTION OF FLUID PROBLEMS Student Course Date Question 1 1.1 Classical mechanics of fluids The Navier-Stokes equations can be divided into the following subdivisions; i. The Cauchy momentum equation This is an equation expressing the conservation form of momentum and it takes the following form; Where; ii. Stokes’ stress constitutive equation It is an expression that is used for solids that are elastic and at the same time incompressible and it takes the following form (Arya, 1990); Where, For fluids that are viscous and incompressible Stokes’ stress constitutive equation is given as: In the convective form, the stokes’ equation for incompressible conditions is given as; In this equation; For incompressible flow, various equations are used. For elastic solids, an expression that considers linear stress, known as the constitutive equation is applied (Arya, 1990). This equation is expressed as; Where; Therefore, the constitutive equation for elastic solids becomes; In thermal hydraulics, an expression referred to as the constitutive equations for linear stress is used in fluids. The expression is given as; In the given equation; It is important to note that the dynamic viscosity µ and the bulk viscosity should remain constant in the equation. These two parameters are dependent on the density of the fluid. The most common and most important Navier-Stokes equation is the momentum equation. In compressible fluids, the convective form of the momentum equation is expressed as; In this equation, the bulk density is also considered to be constant throughout with parameter P which is given as; In most cases, the value of is given as zero. Momentum equation The moment equation is obtained from the mass conservation relation. This is done by getting the product of mass and the velocity (Arya, 1990). Therefore, by substitution in the mass conservation equation, the expression below is obtained. Where; This equation refers to the second rank tensors. Also, Then the equation below is obtained; In the above equation, the left expression refers to the mass continuity and is always equated to zero. The convective derivative then remains in the expression resulting to the given expression. By applying the derivative operator of the material, the expression can also be expressed as; The above expression majorly refers to the Newton’s second law of motion expressed as; Each term used in the Navier-Stokes equation is considered as a body force. In the equation for momentum conservation, the following terms need turbulence modeling. i. The flow expansion rate () ii. The fluid density (ρ) iii. The velocity of the fluid flow () iv. The tensor product ( Turbulence models Turbulent models are vital in calculating the mean of glow directly without following the normal procedure of first considering the flow field that has full time-dependent properties. It is important to note that a turbulent flow in any given fluid has many varied features on length scales that differ but ultimately interact. These turbulent models help in predicting the turbulence effects in a fluid flow (Arya, 1990). In turbulent flow, the vorticity as well as the velocity distribution are the major components used to determine the vortices characteristics. The energy equation The energy equation is given as; Where; 1.2 Firefighting equation Figure 1: diagram showing the system Bernoulli’s equation: From the given parameters; Then substituting in the equation given; Continuity equation: Therefore; Then; The Swamee-Jain equation for the friction factor is given as; Where; Assuming Then; Volume flow rate= velocity times pipe diameter Question 2 2.1 Dimensional analysis Dimension for 2.2 Kolmogorov scale of velocity Using dimensional analysis, the relation among the given parameters is given as; Thus, Question 3 3.1 Process of the burning of wood In wood products, the combustibility and ignitability is based on various physical and chemical properties that affect the various stages of burning. When wood is exposed to heat, it produces substances that easily react with oxygen resulting to the high tendency of the wood to ultimately ignite and consequently burn. For wood’s ignition and combustion major processes are involved; the cellulose thermal decomposition referred to as pyrolysis, various reactions among the products obtained as a result of pyrolysis and lastly the reaction of oxygen with the pyrolysis products (Drysdale, 1999). Cellulose is sensitive to temperature changes, at high temperatures; it undergoes pyrolysis resulting to products due to decomposition or gases. The gases are released to the atmosphere where they react with oxygen and in some cases; they also react with each other. Wood pyrolysis majorly depends on temperature, pH, fire retardants, moisture and the concentration of oxygen. Depending on temperature, two pathways can be formed, namely; the tar and char forming pathways at high and low temperatures respectively (Drysdale, 1999). Combustion releases heat which results to the production of fire. The heat released determines the rate of fire spread, in that the larger the amount of heat released the faster the fire spread and vice versa. The heat release rate is dependent on the material properties and internal structure as well as the external conditions. The properties of the material include; the combustion heat, gasification heat and specific heat capacity. On the other hand the external factors majorly include the net heat flux and the ambient oxygen concentration (Drysdale, 1999). When wood is exposed to heat released at a constant rate per unit area and burns, the front of pyrolysis continues to the given wood in the direction of depth. Due to the fact that all wood the undergoes pyrolysis is always considered to char, it is necessary to note that the charring rate is similar to the transmission rate of the front of pyrolysis. Any wood underneath the layer of char maintains its original properties therefore; the charring rate is mostly used to determine the resistance of wooden structures to fire. Density and the external flux of heat as well as the content of moisture influence the charring rate. Thus, when a layer of char forms on the beam’s surface, the beam’s resistance to fire is increased resulting to a reduction in fire spread (Glassman, Yetter, & Glumac, 2014). 3.2 Reaction rate of fire Reaction rate of fire refers to reaction speed for the fire reactants in certain conditions to give the expected products. The major factors affecting the rate of reaction include; i. Reaction nature; depending on the physical properties of the reactants, the reaction rate varies, in that it is faster in solids than in gases. ii. Concentration; the reaction rate depends on the reactant concentration. At a high concentration, the reaction rate is high owing to the fact that the collision rate increases. iii. Temperature; the reaction rate is high at high temperature and low at low temperature. This is because, at high temperatures, the colliding particles will have an increased activation energy resulting to more collisions thus increased reaction rate. For example, methane gas reacting with oxygen in the atmosphere QUESTION 4 4.1 Fire plume A fire plume consists of three main regions as shown in figure 2 below, Figure 2: regions of a fire plume The continuous flame is a region near the source of fuel. The region where flame flickers are experienced as a result of the entrainment of air is known as the intermittent flame. The buoyant flame is a region where there is no flame but rising hot gases are experienced. When experiencing stable conditions, the resulting fire plume’s symmetry will be about the fire axis. This condition is then known as an axisymmetric plume. Considering a liquid fuel pool fire at the centre of a compartment, the plume flow from its centerline will be outward in a radical manner. The plume can then be idealized as an inverted cone as shown in the figure 3 below. Figure 3: Idealized plume In a compartment where the liquid fuel pool fire is at the centre, the behavior of the flame in relation to the ceiling and the floor is shown in the figure below. Figure 4: liquid fuel pool fire behavior in a compartment The flame height (L) can then be calculated using the following relation; Where; Above the physical flame height, it is common to experience hot gases. The equation below can be used to calculate the temperature along the plume; : Where; To calculate the smoke produced the equation below is used; Where; 4.2 Diffusion flame Oxygen, fuel and source of heat are the basic components for a diffusion flame. For the diffusion flame to be achieved, key elements such as fuel, heat, oxygen, element proportioning, element mixing and the continuity of ignition have to be considered (Truesdell, & Rajagopal, 2009). Therefore, the factors that affect the spread of the flame are as discussed below; i. Heat: in most cases, solids do not burn. The input heat should therefore be sufficient enough to produce vapors as a result of combustion that ultimately burn. ii. Fuel: appropriate fuels that need little heat energy to initiate burning should be used iii. Oxygen: oxygen is required to facilitate the flame spread. The main source of oxygen in this case is the atmosphere. iv. Mixing and proportioning: to ensure continuous flame spread, the proportioning and mixing exercises must be sufficient. The major components to be mixed and appropriately proportioned are the vapors obtained from fuel and oxygen. v. Ignition continuity: heat transfer by conduction will ensure that the level of ignition is continuous at all times. Works cited Arya, A. P. 1990. Introduction to classical mechanics. Boston, Allyn and Bacon. Drysdale, D. 1999. An introduction to fire dynamics. Chichester, Wiley. Glassman, I., Yetter, R. A., & Glumac, N. 2014. Combustion. Amsterdam, Academic Press. Truesdell, C., & Rajagopal, K. R. 2009. An introduction to the mechanics of fluids. Boston, Birkhäuser. Read More