Mathematical Concepts for Mechanical Engineering Design

Mathematical Concepts for Mechanical Engineering Design
اسم المؤلف
Kaveh Hariri Asli, Hossein Sahleh, and Soltan Ali Ogli Aliyev
التاريخ
3 سبتمبر 2018
المشاهدات
48
التقييم
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Mathematical Concepts for Mechanical Engineering Design
Kaveh Hariri Asli, PhD, Hossein Sahleh, PhD, and
Soltan Ali Ogli Aliyev, PhD
CONTENTS
List of Abbreviations
List of Symbols
Preface
1. Heat Flow—From Theory to Practice
2. Dispersed Fluid and Ideal Fluid Mechanics
3. Modeling for Pressure Wave into Water Pipeline
4. Heat Transfer and Vapor Bubble
5. Mathematical Concepts and Computational Approach on
Hydrodynamics Instability
6. Mathematical Concepts and Dynamic Modeling
7. Modeling for Predictions of Air Entrance into Water Pipeline
Index
LIST OF ABBREVIATIONS
FD Finite differences
FE Finite elements
FV Finite volume
FVM Finite volume method
MOC Method of characteristics
PLC Program logic control
RTC Real-time control
WCM Wave characteristic method
LIST OF ABBREVIATIONS
FD Finite differences
FE Finite elements
FV Finite volume
FVM Finite volume method
MOC Method of characteristics
PLC Program logic control
RTC Real-time control
WCM Wave characteristic method
LIST OF SYMBOLS
V = water flow or discharge(m3 s ), ( ) lit s
C = the wave velocity (m s )
Eæ = modulus of elasticity of the liquid (water), MR a bt ct = + + 2 ,       kg m2
E = modulus of elasticity for pipeline material Steel, D D EFF F AV = α
d = outer diameter of the pipe (m)
δ = wall thickness (m)
V0
= liquid with an average speed ( ) ms
T = time( ) S
h
0= ordinate denotes the free surface of the liquid (m)
u = fluid velocity ( ) ms
l = wavelength
( ) hu x = amplitude a
h
dx
t
∂∂
= changingthe volume of fluid between planes in a unit time
h0 = phase velocity ( ) ms
vΦ = expressed in terms of frequency
f = angular frequency
ω = wave number
Φ = a function of frequency and wave vector
v k Φ ( ) = phase velocity or the velocity of phase fluctuations ( ) m s
l( ) k = wavelength
k = waves with a uniform length, but a time-varying amplitude
k**(ω ) = damping vibrations in length
ω = waves with stationary in time but varying in length amplitudes
psi0= saturated vapor pressure of the components of the mixture at an initial temperature of the mixture T0,(pa)
µ 2 , µ1 = molecular weight of the liquid components of the mixture
B = universal gas constant
pi = the vapor pressure inside the bubble (pa)
T
ki =temperature evaporating the liquid components (°C)
li = specific heat of vaporization
D = diffusion coefficient volatility of the components
N k0 , N c0 = molar concentration of 1-th component in the liquid and steam
cl =the specific heats of liquid
al = vapor at constant pressure
al = thermal diffusivity
ρ v = vapor density       kg m3
R = r = R(t)= radius of the bubble (m)
ll = coefficient of thermal conductivity
∆T = overheating of the liquid (°C)
b = is positive and has a pronounced maximum at k 0 = 0,02
p1 and p2 = the pressure component vapor in the bubble (pa)
p∞ = the pressure of the liquid away from the bubble (pa)
σ = surface tension coefficient of the liquid
n 1= kinematic viscosity of the liquid
k R = the concentration of the first component at the interface
ni = the number of molesList of Symbols xiii
V = volume(m3)
B = gas constant
Tv = the temperature of steam (°C)
/
ρi = the density of the mixture components in the vapor bubble kg 3
m
 
   
µi = molecular weight
psi = saturation pressure (pa)
li = specific heat of vaporization
k = the concentration of dissolved gas in liquid
v
Φ= speed of long waves
h = liquid level is above the bottom of the channel
ξ = difference of free surface of the liquid and the liquid level is above the bottom
of the channel (a deviation from the level of the liquid free surface)
u = fluid velocity (m s )
τ = time period
a = distance of the order of the amplitude
k = wave number
v
Φ (k)= phase velocity or the velocity of phase fluctuations
l(k) = wave length
ω**(k)= damping the oscillations in time
l = coefficient of combination
q = flow rate (m3 s )
µ = fluid dynamic viscosity
.
kg
m s
 
   
γ = specific weight
( 3 )
m
N
j = junction point (m)
y = surgetank and reservoir elevation difference (m)
k = volumetriccoefficient(GN m2)
T = period of motionxiv List of Symbols
A = pipe cross-sectional area (m 2 )
dp = static pressure rise (m)
hp = head gain from a pump (m)
hL = combined head loss (m)
Eυ = bulk modulus of elasticity (pa), 2
kg
m
 
   
α = kinetic energy correction factor
P = surge pressure (pa)
g = acceleration of gravity (m s2)
K = wave number
TP = pipe thickness (m)
EP = pipe module of elasticity, (pa) 2
kg
m
 
   
EW = module of elasticity of water (pa),
2
kg
m
 
   
C1 = pipe support coefficient
Y max = Max. Fluctuation
R0 = radiuses of a bubble (m)
D = diffusion factor
b = cardinal influence of componential structure of a mixture
N k0 , N c0= mole concentration of 1-th component in a liquid and steam
γ = Adiabatic curve indicator
c
l, cpv= specific thermal capacities of a liquid at constant pressure
a
l = thermal conductivity factor
ρ v = steam density 3
kg
m
 
   
R = vial radius (m)
ll = heat conductivity factor
k0 = values of concentration, therefore
wl = velocity of a liquid on a bubble surface (m s )
p1and p2 = pressure steam component in a bubble (pa)List of Symbols xv
p∞ = pressure of a liquid far from a bubble (pa)
σ and
n 1 = factor of a superficial tension of kinematics viscosity of a liquid
B = gas constant
Tv = temperature of a mixture (°C)
/
ρi = density a component of a mix of steam in a bubble       kg m3
µi = molecular weight
ji = the stream weight
i = components from an (i = 1,2) inter-phase surface in r = R(t)
wi = diffusion speeds of a component on a bubble surface (m s )
li = specific warmth of steam formation
k R = concentration 1-th components on an interface of phases
T0
, Tki = liquid components boiling temperatures of a binary mixture at initial
pressure p0, (°C)
D = diffusion factor
ll = heat conductivity factor
Nu
l= parameter of Nusselt
al = thermal conductivity of liquids
cl = factor of a specific thermal capacity
pel = Number of Pekle
Sh = parameter of Shervud
ped = diffusion number the Pekle
ρ = density of the binary mix       kg m3
t = time (s)
l0 = unitof length
V = velocity(m s )
S = length (m)
D = diameterof each pipe (m)xvi List of Symbols
R = piperadius (m)
v = fluiddynamic viscosity .
kg
m s
 
   
hp = head gain from a pump (m)
hL = combinedhead loss (m)
C = velocityof surge wave (m s )
γ
P = pressurehead (m)
Z = elevationhead (m)
g
V
2
2
= velocityhead (m)
γ = specific weight
( 3 )
m
N
Z = elevation(m)
H P = surgewave head at intersection points of characteristic lines (m)
VP = surgewave velocity at pipeline points- intersection points of characteristic
lines (m s )
V
ri = surgewave velocity at right hand side of intersection points of characteristic
lines (m s )
H
ri= surgewave head at right hand side of intersection points of characteristic
lines (m)
V
le = surgewave velocity at left hand side of intersection points of characteristic
lines (m s )
H
le = surgewave head at left hand side of intersection points of characteristic lines
(m)
p = pressure(bar),( 2)
m
N
dv = incrementalchange in liquid volume with respect to initial volume
  dρ ρ    = incremental change in liquid density with respect to initial densityList of Symbols xvii
SUPERSCRIPTS
C − = characteristic lines with negative slope
C + = characteristic lines with positive slope
SUBSCRIPTS
Min. = Minimum
Max.= Maximum
Lab. = Laboratory
MOC = Method of Characteristic
PLC = Program Logic Control
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