Fluid dynamics

a. Calculate density of the air at the building bottom.

P (pressure) = ρgh

ρ (density) = P/gh = 101.325/9.8*0.02 = 516.964285714 kg/m³

Density = 516.96 kg/m³

b. Assuming that density of the air is constant along the building height, calculate respectively pressure and temperature of the air at the top of the building.

P = ρgh = 516.96 * 9.8 * 0.02 = 101.325 kPa

P = ρRT

T = P/ρR = 101.325 / 516.96 * 8.31 = 0.0325 K

Temperature = 0.0325 Kelvin

c. If temperature is constant along the building height, calculate pressure and density of the air at the building top.

Pressure, P = ρgh = 516.96 * 9.8 * 0.02 = 101.32416 kPa

Density, ρ = P/gh = 101.32416/9.8*0.02 = 516.96 kg/m³

2. Dimensional analysis

Find the dimensions of Reynolds number, Re = ρul/μ, where ρ is the density, u is the velocity, l is a characteristic lengths and μ is the dynamic viscosity of fluid.

Dimensional formula is given by M⁰L⁰T⁰ where,

M = mass

L = length

T = time

Reynolds number, Re = ρul/μ = density * velocity * length * [dynamic viscosity]^-1

=> Density, ρ = mass/volume = mass * [volume]^-1

Dimensional formula of density = [M¹ L^-3 T⁰]

=> Velocity, u = displacement/time = displacement * [time]^-1

Dimensional formula of velocity = [M⁰L¹ T^-1 ]

=> Lentgh, l = [L]

=> Dynamic viscosity, μ = distance between layers * force * [area * velocity]^-1

Dimensional formula of dynamic viscosity = [L] * [M¹L¹T^-2] * [L * L¹T^-1]^-1 = [M¹L^-1T^-1]

After substituting these equation, we get,

Re = [M¹ L^-3 T⁰] * [M⁰L¹ T^-1 ] * [L] * [M¹L^-1T^-1]^-1 = [M⁰L⁰T⁰]

Therefore, the dimensional formula of Reynolds number is [M⁰L⁰T⁰].

Using the approach of dimensional analysis, derive the friction coefficient of pipe flow.

By using approach of dimensional analysis, the friction coefficient is given by [M⁰L⁰T⁰],

Coefficient of friction, f = Frictional force * [normal force]^-1

Force, F = mass * acceleration = mass * velocity * [time]^-1

Velocity, v = displacement * [time]^-1

Therefore, dimensions of force = [M] * [LT^-1] * [T]^-1 = [M¹L¹T^-2]

Since, coefficient of friction = frictional force * [normal force]^-1

f = [M¹L¹T^-2] * [M¹L¹T^-2]^-1 = [M⁰L⁰T⁰]

Coefficient of friction, f = [M⁰L⁰T⁰]

3. Shock wave and detonation

Shock wave tube is often used to study detonation.

A. Devise a shock wave tube for direct initiation of detonation and explain each section of the shock wave tube.

Detonation procedure can be described as a propulsion system with continuous burning and impulse of combustible mixture. It incudes the increased interest of using it for the detonation process in different technological devices including the concept of detonation engine (DE). However, the initiation of detonation can be done through number of methods like initiation at reflection of shock wave from a focusing surface (Tu, Yeoh and Liu, 2018). It also consist the way of spatial initiation with use of multi-charge schema through a variation of an amount of charges along with their spatial disposition from each other. Moreover, it consist basic schemes of detonation engines like 'weapon' schema of pulsing detonation engine (PDE), schema of supersonic pulsing detonation ramjet engine (PDRJE) and schema of a mixture burning with the help of steady rotating detonation wave.

B. Give the formulae to calculate the speed of the shock wave at the time t = 0, i.e., just when the diaphragm is broken down.

When time, T = 0, the velocity of shock waves is calculatde by the formula given below,

Velocity, v = u . kˆ + c_skˆ

where, u is velocity of fluid and k is the wave vector, in one dimension,

v = u . c_s

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Deflagration and detonation are two forms of explosion.

A. Given the definitions of deflagration and detonation and discuss their difference.

Deflagration: The terms deflagration refers to explosive procedure develop shock wave motive at less speed as compared to speed of sound in unreacted medium. It can be consider as rapid auto combustion of particles of explosive as a surface phenomenon (Johnson, 2016).

Detonation: This can be described as a process of explosion which results into shock wave moving at a speed more than the speed of sound in unreacted medium. It is known as an extreme rapid and self-decomposition of an explosive accompanies through high pressure temperature wave.

b. Illustrate the structure of a detonation and explain each section of the structure.

The synthesise of diamonds from carbon powder procedure consist use of explosives in which high pressure shock wave developed which move into explosive, initiating reaction. The detonation wave generated is given below.

(Source: A New Dynamic Powder Consolidation Technique Using Shock Waves, 2017)

The given figure consist two sections such as (a) structure of detonation shock wave and (b) pressure as the function of time. It includes the interface among chemical reaction region and denotation products whereas synthesis take place that is known as Chapman-Jouguet point (C-J). it is utilised for featuring explosive including pressure (P_CJ) and particle velocity(U_P)_CJ. The relaxation wave is caked as Taylor wave that is responsible for propagating in or product quickly behind the detonation wave. Conservative relationships derived from Rankine-Hugoniont.

c. Explain the direct and indirect initiations of detonation.

Direct initiation of denotation is transition from deflagration to denotion which refers to instantaneous formation of denotion in terms of asymptotic of strong blast wave from a powerful ignition source. It apply where powerful source facilitate a blast waver into an explosive gaseous mixture in order to develop Chapman–Jouguet (CJ) detonation.

Indirect initiation detonation refers to a denotation that starts with deflagration in explosive gaseous mixture which is a subsonic combustion wave which is ignited through weaker ignition source (Kajishima and Taira, 2017).

4. Fire Plumes

Formulate the ideal fire plume

A. List the main assumptions for setup of the ideal fire plume model.

Assumptions of ideal fire plume model for FARSITE:

• Fire spread is elliptical.
• Fire spread in any direction is independent of fire front.
• Fire acceleration is independent of fire behaviour and it is fuel dependent.
• Fires will instantly attain the expected elliptical shape due to change in burning situations.
• The elliptical shapes are independent of fuel and only identified through resultant wind-slope vector.
• Variation in direction and wind speed at high frequency do not affect shape of elliptical fire (Antar, 2019).
• Origin of an elliptical fire is located at the rear focus on ellipse.
• The spread of a continuous fire front can be around utilising finite number of points.
•  FARSITE model is not designed for determining if a fire will spread or not.

B. Write down and explain the basic equations for the ideal fire plume.

Equations for ideal fire plume is given here.

• Simple smoke filling equation,

d (ρV) = m_p

dt

where, ρ is density, V is volume and m_p is flow rate of fire plume.

• The volume of smoke layer, V = A_r (H_r - z)

where, A_r is floor area of room, H_r is ceiling area and z is smoke layer interface height.

- ρA, dz = m_p

dt

Assumed plume mass flow rate formula,

m_p = C_mQ_f^1/3z^5/3

where, C_m is plume coefficient for which Zukoski proposed C_m= 0.076

By combining equations,

-dz/z^5/3  = C_mQ_f^1/3 /ρA_{r .} dt

c. Derive the mass flow rate in the ideal fire plume.

The mass flow rate, M = mass/ time

If the fluid passes through an area (a) and velocity (v), then volume will be,

v = A * V * t

The mass (m) is simply density r times of the volume,

m = r * A * V * t

Through diving mass by time, we get

M = r * A * V

Therefore, mass flow rate is derived as, M = r * A * V

d. Derive the temperature increment in the ideal fire plume.

The derived formula of temperature is,

T_{p(centerline)} – t_a = 9.1 (T_a/g c_a² ρ_a²)^1/3 Q_c^2/3 (z-z₀)^-5/3

Through literature search, find out other formulae for the fire plume and compare them with the formulae of the ideal fire plume and further give your comments on them.

The fire formulae of fire plume are based on the concept of this model that includes the motion developed via source of buoyancy which exists through virtue of combustion and may incorporate an external source of momentum (Cockburn, 2018). In comparison, an ideal fire plume formulae are base on ideas situation of this model that has goal to derive simple algebraic equations for properties in plume and consider an assumption of top hat profile.

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