TRAINING AND DEVELOPMENT SERVICES
ISO 9001:2000 CERTIFICATED
FORMULAE
HANDOUT FOR
CERTIFICATE IN MEC
UPDATED AUGUST’ 06
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INDEX
PAGE
SECTION 1
AIRFLOW
1
PRESSURE SURVEYS
7
AIRFLOW MEASUREMENT
10
FANS
12
COMPRESSED AIR
14
SECTION 2
HEAT
15
PSYCHROMETRY
18
REFRIGERATION
22
GASES
27
RADIATION
28
SECTION 3
FIRES
30
DUST
31
NOISE
34
ILLUMINATION
41
MINE WATER
42
ECONOMICS
51
STATISTICS
54
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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AIRFLOW
Natural Ventilation Pressure [NVP]
a.
b.
Density Formula Method [when there are no fans in the circuit]
NVP
=
[wD - wU] x H x 9.79
Where NVP
=
natural ventilation pressure [Pa]
wD
=
mean density of downcast air [kg/m3]
wU
=
mean density of upcast air [kg/m3]
H
=
vertical distance from the top to the bottom of the circuit [m]
9.79
=
constant for gravitational acceleration [m/s2]
P-V Diagram Method [with or without fans in the circuit] and
NVE
=
Pv
NVP
=
NVE
v
Where NVE
=
natural ventilation energy [kJ/kg]
NVP
=
natural ventilation pressure [kPa]
P
=
barometric pressure [kPa]
v
=
specific volume [m3/kg]
REYNOLDS NUMBER
Re
=
wVD
μ
Where Re
w
V
D
=
=
=
=
Reynolds number [dimensionless]
density [kg/m3]
velocity [m/s]
diameter [m]
=
dynamic viscosity [Ns/m2]
μ
CONSERVATION OF ENERGY
u + Pv +
V2
+ Zg
2
Where u
=
internal energy [J/kg]
P
=
pressure [Pa]
v
=
specific volume [m3/kg]
V
=
velocity [m/s]
Z
=
elevation [m]
g
=
gravitational acceleration [9.79 m/s2]
=
Constant
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RESISTANCE
R
=
KCL w
x
A 3 1,2
Where R
=
resistance [Ns2/m8]
K
=
friction factor [Ns2/m4]
C
=
circumference [h + w]2 = hlge / πD = pipes [m]
L
=
length [m]
A
=
⎡D 2 ⎤
area [h x w] hlge / π ⎢ ⎥ = pipes [m2]
⎣ 4 ⎦
w
=
air density [kg/m3]
ATKINSON’S FORMULA
P
=
KCLQ 2 w
x
1.2
A3
Or
P
Where P
=
KCLV 2 w
x
A
1 .2
= pressure loss due to friction [Pa]
K
= friction factor [Ns2/m4]
C
= circumference [m]
L
= length [m]
Q
= air quantity [m3/s]
V
= air velocity [m/s]
A
= area [m2]
w
= air density [kg/m3]
PRESSURE REQUIRED TO OVERCOME FRICTIONAL RESISTANCE
P
Where P
= RQ2
= pressure required [Pa]
R
= resistance [Ns/m8]
Q
= air quantity [m3/s]
VELOCITY PRESSURE
VP =
V 2w
2
Where VP = velocity pressure [Pa]
V
= air velocity [m/s]
w
= air density [kg/m3]
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DARCY-WEISBACH EQUATION
p
Where p
=
λLwV ²
2D
= pressure [Pa]
λ
= Darcy Weisbach friction factor
L
= length [m]
w
= density [kg/m3]
V
= velocity [m/s]
D
= diameter [m]
λ
= 6.67K when ws = 1.2 kg/m3
AIR POWER
Wa =
pxQ
1000
OR
Wa =
RQ3
1000
Where Wa = air power [kW]
p
= pressure [Pa]
Q
= air quantity [m3/s]
R
= Resistance (Ns2/m8)
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TRIGONOMETRY
A
Y
X
B
C
To calculate length AC
Sin x =
opposite
⎛ AC ⎞
⎜
⎟
hypotenuse ⎝ AB ⎠
To calculate length AB
Cos x =
adjacent ⎛ BC ⎞
⎜
⎟
hypotenuse ⎝ AB ⎠
To calculate length BC
Tan x =
opposite ⎛ AC ⎞
⎜
⎟
adjacent ⎝ BC ⎠
EVASEÉS
Theoretical pressure regain = VPi - VPo
Where VPi
= velocity pressure at evaseé inlet [Pa]
VPo = velocity pressure at evaseé outlet [Pa]
Actual pressure regain can only be measured or theoretical pressure regain multiplied by evaseè
efficiency:Evasee efficiency =
Actual pressure regain
x 100
theoretical pressure regain
LEAKAGE
Duct efficiency =
power required for leakless column
x 100
power required for actual column
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SYSTEM RESISTANCE CURVES
These are calculated from a square law relationship derived from Atkinson’s formula for a constant
resistance
P ∝ Q2
Or
p1
p2
=
2
Q1
Q2 2
Where p
Q
= pressure [Pa]
= quantity [m3/s]
AIRWAYS IN SERIES
QT = Q1 = Q2
PT = P1 + P2
RT = R1 + R2
Where Suffix ‘T’ indicates total system conditions;
Suffix ‘1’ indicates conditions in airway 1;
Suffix ‘2’ indicates conditions in airway 2;
P
= pressure [Pa]
Q
= quantity [m3/s]
R
= resistance [Ns2/m8]
AIRWAYS IN PARALLEL
QT = Q1 + Q2
PT = P1 = P2
1
RT
=
1
R1
+
1
R2
REGULATORS
Ar = 1.2Q
w
p
Where Ar = regulator area [m2]
Q
= air quantity through regulator [m3/s]
p
= pressure used up by regulator [Pa]
w
= air density [kg/m3]
Or, when the air density is 1.2 kg/m3
Ar =
1.31Q
p
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BERNOULLI’S THEOREM [for frictionless flow]
TP1 = TP2
or
VP1 + SP1 = VP2 + SP2
Because
TP
Where TP
= SP + VP
= total pressure
SP
= static pressure
VP
= velocity pressure
BAROMETRIC PRESSURE INCREASE OR DECEASE
The approximate barometric pressure increase or decease in a vertical shaft = 1 kPa/100m
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PRESSURE SURVEYS
FULL VOLUME – REDUCED VOLUME METHOD [density effects ignored]
R
=
Where R
=
ΔB1 - ΔB2
QF² − QR²
resistance [Ns2/m8]
∆B1 =
difference in the barometric pressures at point [1] when the fans are running and
stopped [Pa]
∆B2 =
difference in the barometric pressures at point [2] when the fans are running and
stopped [Pa]
QF =
full volume flow [m3/s]
QR =
reduced volume flow [m3/s]
FULL VOLUME – REDUCED VOLUME METHOD [density effects included]
pf
=
ΔB ± 9.79 H [wmf − wmr ]
wmr [Qr² ]
1−
wmf [Qf² ]
Where pf = pressure loss for full volume flow [Pa]
∆B1
= difference in the barometric pressures with full and reduced volume flow [Pa]
Referring to the definitions in the previous formula ∆B = [∆B1 - ∆B2]
H
= difference in elevation [m]
Wmf = mean density at full volume flow [kg/m3]
Wmr = mean density at reduced volume flow [kg/m3]
NB
Qf
= full volume flow [m3/s]
Qr
= reduced volume flow [m3/s]
±
= Use the ‘+’ sign when depth increases from station [1] to station [2]
Use the ‘-‘ sign when depth decreases from station [1] to station [2]
DENSITY METHOD
The pressure loss:
•
the difference between the theoretical pressure increase or decrease and the actual pressure
increase or decrease
The theoretical pressure increase or decrease:[9.79 x H x wm]
where
9.79 - Constant for gravitational acceleration [m/s2]
H
- Difference in elevation [m]
wm
- Mean density [kg/m3]
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CORRECTION DUE TO BAROMETRIC PRESSURE VARIATIONS
ΔPth =
Where Ptb
Pcb
Pth
x ΔPcb
Pcb
= traverse barometer reading
= control barometer reading
∆Pcb = change in the control barometer reading
∆Ptb = corresponding change in the traverse barometer reading
PRESSURE / DENSITY RELATIONSHIP
Air pressure varies directly as an air density change:p1
w1
Where p
w
=
p2
w2
= pressure [Pa]
= density [kg/m3]
Suffix ‘1’ indicates conditions at one point in the system;
Suffix ‘2’ indicates conditions at another point in the same system.
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AIRFLOW MEASUREMENT
PITOT TUBE POSITIONS IN A CIRCULAR DUCT
2n − 1
4N
Rn = d
Where n
= the nth reading from the centre
Rn = radius of the reading [mm]
d
= duct diameter [mm]
N = number of readings across a diameter
ORIFICE PLATE
Q =
1.2Q
P
Where Q = air volume [m3/s]
P = differential pressure [Pa]
CONICAL INLET
Q = 1.11 D² C
Δp
w
Where Q = air density [m3/s]
D = duct diameter [m]
C = coefficient of discharge [from graphs]
∆p = measured pressure difference [Pa]
w = air density [kg/m2]
VENTURI METER
Q = 1.11 d² Cd E z
Δp
w
⎛
⎞
⎜
⎟
⎜
⎟
1
⎟
E= ⎜
⎜ ⎛ d 4 ⎞ 0. 5 ⎟
⎜ 1 − ⎜⎜ 4 ⎟⎟ ⎟
⎜
⎟
⎝ ⎝D ⎠ ⎠
Where Q = air quantity [m3/s]
d
= diameter of throat [m] (Venturi)
D = Column diameter (m)
Cd = coefficient of discharge [from graphs]
E = velocity of approach factor [from graphs]
z
= combination of factors for size, expansion and Reynolds number
∆p = measured pressure difference [Pa]
w = air density [kg/m3]
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GAS TRACER METHOD
a.
Tracer gas NOT in normal air:-
Q =
q x 10 6
C
Where Q = air volume or mass flow rate [m3/s of kg/s]
q = rate of tracer gas release [m3/s or kg/s]
C = concentration of tracer gas in air after mixing [part per million by volume or mass]
b.
Tracer gas IN normal air:-
Q =
q x 10 6
C1 − C2
Where Q = air volume or mass flow rate [m3/s of kg/s]
q = rate of tracer gas release [m3/s or kg/s]
C1 = concentration of tracer gas in air after mixing [part per million by volume or mass]
C2 = concentration of tracer gas found in normal air before mixing [part per million by
volume or mass]
c.
Volume of tracer gas:-
Vg =
Q =
m
a
= Q ∫o
Wg
Cdt
=
QA
m
w gA
Where Vg = volume of tracer gas [m3]
m = mass of tracer gas [kg]
wg = densities of tracer gas [kg/m3]
Q = airflow rate [m3/s]
C = tracer gas concentration by volume, part per unit
A = area under curve [∫o a Cdts]
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FANS
Efficiency
=
work output
x 100%
work input
Motor Efficiency
=
motor output power
x 100%
motor input power
Fan Efficiency
=
air power
x 100%
fan input power
Drive Efficiency
=
fan input power
x 100%
motor output power
Overall Efficiency =
air power
x 100%
motor input power
FAN LAWS
Air Density Change
When the air density changes from w1 to w2:1.
Q remains constant, i.e.: Q1 = Q2
2.
pαw
p1
w1
3.
=
p2
w2
Power α w
power 1
power 2
=
w1
w2
4.
Efficiency remains constant
Eff1 = Eff2
Fan Speed Change
When the fan speed changes from speed1 to speed2:1.
Q α speed
Q1
speed1
2.
=
Q2
speed2
Or
Q2 =
Q1 x speed2
speed1
p α speed2
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p1
speed1
3.
=
=
Or
p2
or
power2 =
p1 x [speed2]²
[speed1]²
power α speed3
power 1
[speed1]³
4.
p2
speed2
=
power 2
[speed2]³
power 1 x [speed2]³
[speed1]³
Efficiency remains constant
Eff1 = Eff2
Where
Q = fan air quantity [m3/s]
p = fan pressure [Pa]
power = fan power [kW]
w = air density [kg/m3]
speed = fan speed [r/s]
PULLEY SIZE CHANGES
1.
Fan pulley size change with a speed increase or decrease:New pulley size = Old pulley size x
2.
old speed
new speed
Motor pulley size change with a speed increase or decrease:New pulley size = Old pulley size x
new speed
old speed
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COMPRESSED AIR
Effect of auto compression
The increase in pressure due to auto compression can be derived from the equation:Pe =
⎡ gH ⎤
Ps exp ⎢
⎥
⎣ RT ⎦
Where Pe =
absolute pressure at end of column [kPa]
Ps =
absolute pressure at start of column [kPa]
G
=
gravitational acceleration m/s2 [9.79 m/s2]
H
=
vertical depth metres [m]
R
=
gas constant J/kgK [287 J/kgK]
T
=
absolute temperature [K]
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HEAT
AUTO-COMPRESSION OR DE-COMPRESSION
Heat increase or decrease: 0.979 kJ/kg/100m or 9,79 kJ/kg / 1000m of vertical depth
H =
gΔz
1000
H = heat increase
g = 0.979 kJ/kg/100m
Z = vertical depth
VIRGIN ROCK TEMPERATURE
V.R.T [approximate]:-
Gauteng
= 18 + (9 x depth In kilometre)
Free State
= 20+ (14.6 x depth in kilometre)
Klerksdorp
= 22 + (10,5 x depth in kilometres)
V.R.T. [accurate]:-
Gauteng
Where
D
=
[18.3 + 6D + 1.1 D2] °C
=
thickness of overlying strata [km]
Free State =
Where
[20 + 25.5 D1 + 14.2 D2 + 8.2 D3] °C
D1 =
thickness of Karoo diabase [km]
D2 =
thickness of lava [km]
D3 =
thickness of quartzite [km]
WET KATA FORMULA
H =
Where H =
θ
=
V =
0.7θ + θ v
wet kata reading
36.5 - wet bulb temperature [°C]
air velocity [m/s]
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Amount of Heat transferred
a)
Conduction
q =
KA [t1 − t 2]
b
Where q = Conductive heat transfer ratio [W]
K = Thermal conductivity of material [W/m°C]
A = Cross-sectional area [m²]
t1 - t2
= temperature difference of sources [°C]
B = Thickness [m]
b)
Convection
q = hcA (t1 - t2)
Where q = Convective heat transfer rate [w]
hc = Convection heat transfer co-efficient [W/m2]
(t1 - t2) = Temperature difference of sources [oC]
A = Cross-sectional area [m2]
c)
Radiation
q = 5,67 x 10-8 A1Fev(T14 - T24)
Where q = Radiative heat transfer [W]
5,67 x 10-8 = Stefan-Boltzmann constant
A1 = Smaller area of the two surfaces [m2]
Fev = Emissivity and view factor
(T14 - T24) = Absolute temperatures (K)
And
Fev =
1
⎞
1 A1 ⎛ 1
⎜⎜ − 1⎟⎟
+
∈1 A 2 ⎝ ∈2
⎠
OR
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q
⎡⎛ T ⎞ 4 ⎛ T ⎞ 4 ⎤
= 5,67 ⎢⎜ 1 ⎟ − ⎜ 2 ⎟ ⎥ x A1 x Fev
⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦
HEAT EQUATIONS
Static Heat Equation [no movement]
W =
Where W =
M Cp Δt
heat transferred [kJ]
M =
mass flow rate of substance [kg]
Cp =
thermal capacity of substance [kJ/kg °C]
Δt =
temperature difference [°C]
Flow Heat Equation [with movement]
q =
Where q =
M Cp Δt
heat transfer rate [kJ/s or kW]
M =
mass flow rate of substance [kg]
Cp =
thermal capacity of substance [kJ/kg °C]
Δt =
temperature difference [°C]
Wind Chill Equivalent Temperature:
WCET =
⎛ (10,45 + 10 v - v) x (33 - T) ⎞
⎟
33 - ⎜
⎜
⎟
22.04
⎝
⎠
WCET =
Wind Chill Equivalent Temperature
v =
Wind speed (m/s)
T =
air temperature, dry bulb (oC)
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PSYCHROMETRY
BOYLE’S LAW
P1V1 = P2V2
CHARLE’S LAW [T = absolute temp °C + K ] K - 273
V1 V 2
=
T1 T 2
UNIVESAL GAS LAW
P1V1 P2 V 2
=
T1
T2
Where P = absolute pressure [kPa]
V = volume [m3], volume flow rate [m3/s], Specific volume [m3/kg]
T = absolute temperature [K]
The Universal Gas Law can also be written as:Pv
= R
T
Where P = absolute pressure [kPa]
v = specific volume [m3/kg]
T = absolute temperature [K]
R = gas constant [kJ/kg
And the gas constant [R] for dry air = 0.2871 kJ/kg K
MASS FLOW OF AIR
M = Qxw
or
M =
Q
v
Where M = mass flow of air [kg/s]
Q = air quantity [m3/s]
w = air density [kg/m3]
v = air specific volume [m3/kg]
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For a constant mass flow of air:M1 = M2
Thus:
Q1w1 = Q2w2
Where M
= mass flow of air [kg/s]
Q
= air quantity [m3/s]
w
= air density [kg/m3]
Suffix ‘1’ indicates conditions at one point in a system;
Suffix ‘2’ indicates conditions at a second point in the system
CALCULATION OF PSYCHOMETRIC PROPERTIES
1.
2.
Vapour Pressure [Pw]
Pw
=
P' s − AP[tdb − twb ] kPa
Where P’s
=
0.6105 exp [17.27 twb/ [237.3 + twb]] kPa
A
=
0.000644 °C-1
P
=
pressure [kPa]
Moisture content [r] [kg/kg]
v
3.
4.
0.622 x Pw
P − Pw
Specific Volume [v]
v
=
0.287 x T
m³ / kg
P − Pw
Where T
=
273.15 + tdb K
=
1+ r
kg / m³
v
=
Ha + rH’w kJ/kg
=
1.005 tdb kJ/kg
Density [w]
w
5.
=
Enthalpy [H]
H
Where Ha
H’w =
1.8 tdb + 2501 kJ/kg
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6.
Sigma Heat [S]
S
=
Where H’w1 =
7.
4.18 twb kJ/kg
Relative Humidity [Φ]
φ
Where P’s
8.
H - rH’w1 kJ/kg
=
Pw
x 100%
P' s
=
0.6105 exp [17.27 tdb / [237.3 + tdb]] kPa
Dew point Temperature [tdp]
tdp
Where x
=
237.3 x
°C
17.27 − x
=
⎡ Pw ⎤
In⎢
⎥
⎣ 0.6105 ⎦
Heat removed for air :-
Q
=
M x ΔS
Where q
=
heat transfer rate [kW]
M
=
mass flow of dry air [kg/s]
∆S
=
change in sigma heat content [kJ/kg]
Amount of water evaporated/ condensed:-
R
=
M x Δr
1000
Where R
=
amount of water condensed [l/s]
M
=
mass flow of dry air [kg/s]
∆r
=
change in moisture content [g/kg]
Mixing of Airstreams:Sigma Heat Content:
Sc
=
[MA x SA ] + [MB x SB]
[MA + MB]
SC
=
sigma heat content of the mixture [kJ/kg]
[MA x SA]
=
the total kW of heat from air stream A
[MB x SB]
=
the total kW of heat from air stream B
[MA x SA] + [MB x SB]
=
the total kW in air stream C
Where:
[MA + MB] =
the total mass flow of air stream C
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Moisture Content:
rc
=
[MA x rA ] + [MB x rB]
[MA + MB]
rC
=
the moisture content of the mixture [g/kg]
[MA x rA]
=
the total moisture [g/s] from air stream A
[MB x SB]
=
the total moisture [g/s] from air stream B
[MA x SA] + [MB x SB]
=
the moisture [g/s] in air stream C
Where:
[MA + MB] =
the total mass flow of air stream C
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REFRIGERATION
In the formulae below, C.O.P. denotes Coefficient of Performance
a.
Heat balance
Condenser duty = evaporator duty + total input power to compressor
b.
Carnot C.O.P
Carnot COP =
T1
T 2 − T1
Where T1 = evaporating temperature [K]
T2 = condensing temperature [K]
c.
Overall compressor C.O.P:-
cooling at evaporator
kW
compressor motor input power
d.
Actual or nett compressor C.O.P.:-
cooling at evaporator
kW
compressor motor output power
e.
Overall plant C.O.P:-
cooling at coils
kW
total electric input power
f.
Overall compressor power/cooling ratio:-
compressor motor input power
kW
cooling at evaporator
g.
Actual or nett compressor power/ cooling ratio:-
compressor motor output power
kW
cooling at evaporator
h.
Overall plant power / cooling ratio:-
total electric input power
kW
cooling at coils
i.
Overall cycle efficiency
overall compressor C.O.P.
x 100%
Carnot L C.O.P.
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j.
Actual or nett cycle efficiency:-
actual or nett compressor C.O.P.
x 100%
Carnot C.O.P.
k.
Plant positional efficiency:-
cooling at coils [kW ]
x 100%
cooling at evaporator [kW ]
l.
Compressor motor input power:-
W = E Ipf n
Where W = electric power [kW]
E
= voltage [kV]
I
= current [amperes]
pf
= power factor [normally approximately 0.9]
n
= number of phases [normally 3]
m. Cooling tower efficiency [water]
Nw =
Where twi
twi − two
x 100%
twi − twbi
= temperature of water entering tower [°C]
two = temperature of water leaving tower [°C]
twbi = wet bulb temperature of air entering tower [°C]
n.
Cooling tower efficiency [air]
Na =
Sao − Sai
x 100%
Swi − Sai
Where Sao = sigma heat content, air leaving tower [kJ/kg]
Sai = sigma heat content, air entering tower [kJ/kg]
Swi = sigma heat content, water entering tower [kJ/kg]
o.
Cooling tower factor of merit
F
=
1
when R>1 and E = Na (Air Efficiency)
⎡
⎡1
⎤⎤
⎢1+ R ⎢ − 1⎥ ⎥
⎣E
⎦⎦
⎣
Or
F
=
1
when R<1 and E = Nw (Water Efficiency)
⎡ 1 ⎡1
⎤⎤
⎢1+ ⎢ − 1⎥ ⎥
⎦⎦
⎣ R ⎣E
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And
R
=
Mw x Cpw
Ma x C' a
Where:
C’a
=
Swi − Sai
twi − tai
PH Diagram
P = Pressure (kPa)
E
Condenser
A
D
Compressor
Evaporator
F
C
B
E on constant entropy
Line through C
H = Enthalpy (kJ/kg)
1.
Evaporator heat exchange
-
C - B [kJ/kg]
2.
Condenser heat exchange
-
D - A [kJ/kg]
3.
Heat of compression [actual] -
D - C [kJ/kg]
4.
Heat of compression [ideal]
E - C [kJ/kg]
-
Heat balance [on cycle]
Condenser heat exchange = Evaporator heat exchange + heat of compression
[D - A] = [C - B] + [D - C]
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5.
Carnot C.O.P.
T1
T 2 − T1
Where T1 - Absolute evaporating temperature [K]
T2 - Absolute condensing temperature [K]
6.
Actual compressor C.O.P.:-
evaporator heat exchange
actual work of compressio n
C−B
D−C
7.
Cycle efficiency:-
actual compresor C.O.P.
x 100%
Carnot C.O.P.
8.
Compressor efficiency:-
ideal work of compressio n
x 100%
actual work of compressio n
E−C
x 100%
D−C
9.
Percentage flash gas:-
B−F
x 100%
C−F
10.
Mass flow of refrigerant:-
total heat exchange at condenser
total heat exchange at Evaporator
OR
unit heat exchange of condenser
unit heat exchange of Evaporator
11.
Power consumed by - actual work of compression x compressor mass flow of refrigerant
- [D - C] x M
12.
Plant duty
- evaporator heat exchange x mass flow of refrigerant
- [C - B] x M
13.
Volume flow rate of refrigerant entering compressor
Q = M x vin
Where vin - constant volume at entrance of the compressor
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14.
Volume flow rate of refrigerant leaving compressor
Q = M x vout
Where vout - constant volume at exit of the compressor
15.
Percentage error:-
[a − c ] − b x 100
a
Where a
- condenser duty
b
- Compressor duty
c
- Evaporator duty
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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GASES
GAS DILUTION
where
Q1 x 106
- Q1
MAC − N
Q
=
Q
- fresh air volume or mass flow rate required for dilution [m3/s or kg/s]
Q1
- volume or mass flow rate of gas emission [m3/s or kg/s]
MAC
- maximum allowable gas concentration [after mixing] in parts Per million by
volume or mass
N
- gas concentration in normal air in parts per million by volume or mass
GAS MIXING
Percentage gas % =
Where:
Total Quantity gas
x100
Total quantity air + quantity gas
Total quantity gas = m3/s
Total quantity air + quantity gas = m3/s
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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RADIATION
1.
Working Level Month Per Annum [WML]
WML / Annum =
2.
Working Level Month Exposure
WLM
3.
number of hours worked per month x exposure rate
Maximum hours allowed per month
=
Time weighted exposure
WLM
4.
([number of weeks worked / year x number of hours / week] x
Mean Working Level
Maximum allowable monthly hours worked
=
[number of hours worked x exposure rate] + [number of hours
Worked x exposure rate
Maximum hours allowed per month
Residence Time [T]
T
Where T
V
⎡ 86.4 x 10 6 V ⎤
= ⎢
⎥
Et
⎦
⎣
1
1.85
NB; in brackets to the power of 1/1.85
- Residence time [s]
- volume of the tunnel or workings [m³]
Et - Radon production [p Ci/s]
5.
Radon Dilution
Q2 = Q1
Rn1
Rn2
Where Q1 - Air quantity prevailing
Q2 - Air quantity required for Rn2
Rn1 - Rn concentration prevailing
Rn2 - Rn concentration to be determined
6.
WL2 =
⎡Q ⎤
WL1 ⎢ 1 ⎥
⎣ Q2 ⎦
1.85
Where WL1 - prevailing condition
WL2 - desired condition
Q1
- quantity flowing [m³/s]
Q2
- quantity required for dilution [m³/s]
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Summation of an individual radiation dose :HID IRnD IThD IU ITh IUc ITc
+
+
+ +
+
+
≤1
HIDL
a
b
c
d
e
f
Where: HID
- is the deep dose equivalent index received in the year
IRnD
- is the annual exposure to radon daughter products
IThD
- is the annual exposure to thoron daughter products
IU
- is the annual intake of uranium ore dust
ITh
- is the annual intake of thorium ore dust
IUc
- is the annual intake of uranium concentrate
ITc
- is the annual intake of thorium concentrate
IIDL
- is the deep dose equivalent index limit
a
- is the annual limit of exposure to radon daughter products
b
- is the annual limit of exposure to thoron daughter products
c
- is the annual limit of intake of uranium ore dust
d
- is the annual limit of intake of thorium ore dust
e
- is the annual limit of intake of uranium concentrate
f
- is the annual limit of intake of thorium concentrate
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FIRES
EXPLOSIBILITY DIAGRAMS – US BUREAU OF MINES AND COWARD’S TRIANGLE
1.
Excess N2
- N2 - 3.7778 O2
2.
O2 deficiency
- 0.2647 N2 – O2
3.
Total combustibles, D
- CH4 + H2 + CO
4.
‘R’ values on USBM diagram
-
CH4
D
5.
CO/O2 deficiency ratio
-
CO
x 100%
O2 deficiency
6.
Young’s ratio
-
CO2
x 100%
O2 deficiency
7.
Willet’s ratio
-
CO2
x 100%
Excess N2 + total combustibl es, D + CO2
8.
x co-ordinate [USBM diagram]
- Excess N2 + 1.5 CO2
9.
y co-ordinate [USBM diagram]
- CH4 + 1.25H2 + 0.4 CO
10.
COWARD’S TRIANGLE and all above
[Graham Ratio]
Please refer to the “Environmental Engineering in South African Mines” Page 814 - 817
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DUST
DUST FILTRATION
Surface area of one filter bag:-
[m²] = [π D x L] + ⎡⎢ πD² ⎤⎥
⎣ 4 ⎦
Where D
L
- bag diameter [m]
- bag length [m]
DUST DILUTION
[Q1D1] + [Q2D2] =[Q1 + Q2]D3
Where Q1 - air volume of stream ‘1’ [m³/s]
Q2 - air volume of stream ‘2’ [m³/s]
D1 - dust content of stream ‘1’ [p/ml]
D2 - dust content of stream ‘2’ [p/ml]
D3 - dust content of mixture [p/ml]
Percentage particles stated size
100
80
60
40
20
0
0
1
2
3
4
5
6
7
8
Particle size [um]
Respirable sampling curve defined at the International Pneumoconiosis Conference in
Johannesburg, 1959
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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PERSONAL GRAVIMETRIC DUST SAMPLING
Calculation of the Time Weighted Average Concentration (TWA - CONC)
Calculation of Results
Step
Example
1. Note the average flow rate and sample time.
Obtain the pump flow rate.
2,2 Litres per minute
Determine the total sample time
8 hours 20 minutes
Convert total sample time to minutes
∴ Minutes = (8 x 60) + 20
= 500 minutes
2. Determine the sample volume
Results must be expressed in mg/m3
∴ Volume of air through pump = Flow rate
x time
Convert litres to m3
(1000 litres of air = 1 m3)
Volume = Flow rate (l/m) x time
= 2,2 x 500
= 1 100 litres of air
=
1100
1000
= 1,1 m3 sucked through
3. Determine the correction filter mass (Correction Factor)
Determine the average of pre and post
weighed control (blank) filters by:
•
•
weighing pre weighed control filter 3
x consecutively when weighing
sample filters
weighing post weighed control filter 3
x consecutively when weighing
exposed sample filters
Add together and divide by 3
Determine the correction factor by:
Subtract the pre weighed blank filter mass
from post weighed blank filter mass.
Post Filter mass
(mg)
Pre filter mass
(mg)
20,16
20,1
20,17
20,09
20,18
20,11
60,15
60,3
3
3
= 20,17 mg
= 20,10 mg
Correction factor = Post filter mass –
Pre filter mass
= 20,17 - 20,10
= 0,07 (Heavier, pickedup moisture)
If this mass is + subtract as a correction
factor.
As this 0, 07 mg is positive, it must be
subtracted from the sample filter mass.
If this mass is - add as a correction factor
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Step
Example
4. Determine the sample mass (mg)
Subtract the pre weighed sample mass
from the post weighed sample mass
Post weighed sample mass - Pre weighed
sample mass
Also weigh in the manner described in
step 3.
20,78 - 20,66
= 0,12 mg
5. Determine the correct sample mass (mg)
Subtract the correction factor (because it
is +) from the sample mass
Corrected Sample mass = Sample mass correction factor
Add correction factor if mass is -
= 0,12 - 0,07
= 0,05 mg
6. Determine the concentration (mg/m3)
Divide the corrected sample mass by the
volume of air sampled. (step 2 answer)
Concentration =
=
Mass ⎛ mg ⎞
⎜
⎟
Volume ⎝ m3 ⎠
0,05
1,1
= 0,046 mg/m3
7. Determine the TWA-CONC as applicable
Determine the time correction factor (i.e.
to convert actual sample time to an 8
hour (480 minutes) shift.
Multiply the concentration with the time
correction factor to obtain TWA - CONC
Time correction factor =
Actual Sample time
480
TWA CONC= Conc x time correction factor
= 0,046 x
= 0,046 x
Actual Sample time
480
500
480
= 0,048 mg/m3
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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NOISE
Background Noise
•
Equation:
LB =
10 log10 (10
Where LB =
•
L M 10
- 10 L A
10
)
Noise level for noise source alone – dB (A)
LM =
Measured noise level – dB (A)
LA =
Background noise level – dB (A)
Use attached table or graph:
Table for subtracting decibel values (correction for background noise).
Decibel value that must be subtracted
from the measured noise level
3
3
4-5
2
6-9
1
Decibel values which must be
subtracted from measured noise level
Difference between measured and
background noise level dB(A)
Difference between Measured Noise Level and
Background Noise dB (A)
Note:
A calculation having a difference of more than 10 dB (A) will indicate that the decibel
value to be subtracted is less than half a decibel and background correction can thus be
ignored.
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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Wavelength
The relationship between wavelength λ [m], speed c [m/s] and frequency f [Hz] is given in the
following formula: λ
=
c
[units as above]
f
Sound Intensity
I
Where: I
=
p²
wc
- Intensity [W/m²]
p
- Sound pressure [Pa]
w
- Density [kg/m³]
c
- Velocity of sound [m/s]
Sound Power Level
⎡ sound power ⎤
SWL = 10Log10 ⎢
⎥ dB
⎣ reference power ⎦
Where the reference power is 10-12 watt
Sound Pressure Level
SPL = 10 Log10
[sound pressure]²
[reference pressure]²
dB
Or
SPL = 20 Log10
[sound pressure]
[reference pressure]
dB
Where the reference pressure is the sound pressure at the threshold of hearing i.e. 2 x 10-5 Pa
Leq for Steady Noise Level
Leq = LA + C1
Where Leq - Equivalent noise level, dB [A]
LA - Measured level of steady noise, dB [A]
C1 - Impulse correction factor which is +10 dB where the noise is of
A repetitive nature [e.g. riveting or hammering] or where it occurred in single bursts
e.g. a drop forge hammer], and 0 dB in all other cases
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Leq For a fluctuating Noise Level
Leq = LA [av] + a C1
Where: Leq and Ci are as before
⎡ 1
LA[av] = 10 Log10 ⎢
⎣100
∑f
1 10
LAV 10 ⎤
⎥
⎦
Where: LAi - Noise level at the mid-point of the i-class dB [A]
F1 - Duration of the i-class sound level exposure expressed as a percentage of the total
analysis time [normalised to a 40 hour total period]
Other formulae (Logarithmic Mathematical Methods):-
Leq
Where: f1 to fn
=
L
L
L ⎤
⎡
10 Log10 ⎢f1anti log 1 + f2antilog 2 + fn anti log n ⎥ + C1
10
10
10 ⎦
⎣
- the ratios in relation to 40 hours of the duration of exposure to the sound
levels L1 to Ln
L1 to Ln - the sound levels of dB [A] of the exposures for the duration ratios f1 to fn
C1
- impulse correction factor which is +10 dB where the noise is of a repetitive
nature [e.g. riveting or hammering] or where it occurred in single bursts e.g. a
drop forge hammer], and 0 dB in all other cases.
OR
Leq
= 10 log F+90
Where F = Σf, where fn =
Cn
antilog [0,1 x (Lnoise – 90)]
40
and Cn = actual time of exposure at noise level (hours)
Exposure Factor [D]
D=
C1 C2
Cn
+
+ ....
T1 T 2
Tn
Where: C1 to Cn - Actual time of exposure at noise levels L1 to Ln
T1 to Tn - Permitted time of exposure at noise levels L1 to Ln
Average noise level
⎡T⎤
Lav = 85 - 10 Log10 ⎢ ⎥
⎣ 40 ⎦
⎡101 - L av ⎤
Antilog ⎢
⎥
⎣ 10 ⎦
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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When the Exposure Factor D has been obtained, the graph below is used to determine the
equivalent noise level, Leq
ADDITION OF SOUND LEVELS
Difference between the
two levels dB
Amount to be added to the
higher level dB
0
3
1
2,5
2
2
3
2
4
1,5
5
1
6
1
7
1
8
0,5
9
0,5
10 or more
0
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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Spheres
To calculate area:
Sphere
= 4πr2
Hemisphere = 2πr2
¼ Sphere
= πr2
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D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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EQUIPMENT TESTING [IN-DUCT METHOD FOR FANS AND SILENCERS]
SWL = SPL + 10 log A
For a 760 mm diameter duct the value of 10 log A is –3.5 dB
Control of Noise
a.
Fans
In the design stage the acoustical characteristic of a fan is not usually available and it is
often necessary to make an estimate. Three formulae often used are given below:i.
SWL = 97 + 10 log kW + 10 log P dB
ii.
SWL = 100 + 10 log Q + 20 log P dB
iii.
SWL = 95 + 20 log kW – 10 Q dB
Where SWL is the overall sound power level in the octave frequency bands 31.5 to 8 000 Hz
kW -
Rated motor power
P
-
Fan static pressure [kPa]
Q
-
Fan delivery quantity [m³/s]
Auxiliary In Line Axial Flow Fans
SWL = 100 + 10 log [QP²] dB
Where:
b.
P = Fan total pressure [kPa]
Rock drill
SWL = 140 + 10 log Q dB
Where:
c.
Q - Free air consumption [m³/s]
Diesel Equipment
i.
Exhaust Noise
SWL = 110 + 10 log kW dB
Where: kW - Rated power of the diesel
ii.
Engine Noise [below 300 kW]
SWL = 100 + 8 log kW dB
Where: kW - Rated power of the diesel
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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ILLUMINATION
1.
The relationship between wavelength [λ] speed [c] and frequency [f] is given in the following
formulae:λ =
c
[units as above]
f
Where: λ - wavelength [m]
c - Velocity [m/s]
f
2.
- Frequency [Hz]
Inverse Square Law
The inverse square law states that the illumination at any point on a surface varies directly
with a luminous intensity of the source and inversely as the square of the distance between
the source and the point. If the source is normal to the direction of the incident light, the law
may be expressed as:E =
l
d2
Where: E - Illumination [Lux] = 1 lm/m2
I
- Luminous intensity [cd]
d - Distance [m]
One solid steradian angle = 4π = 12.566
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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MINE WATER
Deep Cell Dust Concentration
The concentration in millions of particles per millilitre is given by:N x DF
CF
Where: N
- Total number of particles counted in 10 sections
DF - Dilution factor
CF - Cell factor
And cell factor: -
[CF] =
Where: x
L
xL²
105
- Depth of cell in micrometers
- Length of one side of counting section in micrometers [assuming the
counting section is a square]
If a 100 ml measuring flask is used, [assuming the distilled water and acid used both have
counts of zero
Dilution Factor [DF]
=
100
sample volume [ml]
Difference in Pipe Size
H
=
V2
2g
H
=
V 2 2 V 21
−
2g
2g
Where: V
-
Mean velocity
g
-
Gravitational acceleration [9.79 m/s²]
The friction co-efficient [λ]
Δp
wV ²
[L / Dh ]
2
λ
=
Where: w
-
Density of the substance [water = 1000 kg/m³]
L
-
Length (m)
Dh -
Equivalent hydraulic diameter (m) = 4A/C (A = Area, C = perimeter)
V
Mean Velocity (m/s)
-
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Where the Reynolds number is between 2500 and 1000 000
λ
=
0.316
R e 0.25
PUMP CHARACTERISTICS
Total [Manometric] head HT
HT = Hs + Hf
Where: HT - Total head [m] [liquid]
Hs - Static head [m] [liquid
Hf
- Head losses due to friction [m] [liquid]
To convert meters head to kPa: p
Where: p
=
Hwg
1000
- Pressure [kPa]
g
- Gravitational acceleration [m/s²]
H
- Head [m]
W - Density of liquid [kg/m³]
[the density of water is 1000 kg/m³]
Pump Power Requirements
Power
=
Where: Power
Q
Q x HT x g x SG
kW
1000η
- Kilowatts [kW]
- Volume flow rate [ℓ/s]
HT - Total head [m]
g
- Gravitational acceleration [m/s²]
SG - Specific gravity [water = 1]
η
- Pump efficiency expressed as a fraction of
1
1 [i.e. %]
100
Pump Efficiency
Efficiency =
Power output
x 100
Power input
Power output is determined by the following equation:H x Q x g x SG
kW
1000
Power [output]
=
Where: Power
- Kilowatts [kW]
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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Q
- Volume flow rate [ℓ/s]
HT - Total head [m]
g
- Gravitational acceleration [m/s²]
SG - Specific gravity [water = 1]
Energy Recovery System
Energy recovered
Where:
g
=
available head
x g x water flow x efficiency
1000
= Gravitational acceleration [9.79 m/s²]
Efficiency = Turbine efficiency [%]
Available Head = ΔH - Hf (m)
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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This pipe friction chart is used to read off directly the Darcy Weisbach’s information, without having
to perform long calculations and applies for both vertical and horizontal pipes.
To read the chart you need to know two of the following factors:
i.
Water flow rate (l /s)
ii.
Pipe diameter (mm)
iii.
Water velocity (m/s)
iv.
Head loss (m/100m)
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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Darcy Weisbach Formula
Hf
Where: Hf
=
λLV 2
2gD
= Head loss due to friction (m)
λ
= Pipe Friction Coefficient
L
= Pipe length (m)
V
= Mean velocity (m/s)
g
= Gravitational acceleration (9,79 m/s2)
D
= Pipe diameter (m) - is replaced by Dh for non-circular pipes (Dh =
Remember the Darcy Weisbach equation calculate head loss due to friction.
The Stanton Nikuradse Diagram
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4A
)
C
- 47 -
Stroh’s Equation
Hf = 2,04 x 10-9 x
M 1,92
xL
d 5,13
Where: Hf = Total head loss due to friction (m)
m = Water flow rate (l /s)
d = Pipe inside diameter (m)
EQUIVALENT LENGTHS OF STRAIGHT PIPES FOR VARIOUS FITTINGS
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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Pipe Inlet Losses
•
Pipe Inlet (Tapered and applicable to short pipe lengths)
V2
H =
2g
Where: H = Head required to accelerate water to design velocity.
(e.g. Pipe diameter reduces from 203mm to 153mm)
Reducers
2
H =
2
V2
V
− 1
2g
2g
Where subscript 1 is original diameter and subscript 2 is the reduced diameter.
Recommended Age Factor For Mine Water Piping
Age in Years
10
15
20
30
Age Factor
1,3
1,45
1,6
2,0
New equivalent pipe length = Given new pipe length x age factor.
Conversion Of Metres Head To Pressure And Vice Versa:
•
Metres head to pressure
ΔP = (H - Hf) x 9,79 (kPa)
•
Pressure To Metres head
Metres head =
ΔP
9,79
(Where ΔP = ΔH - Hf)
Temperature Increase In Pipes
•
Vertical pipes (Stroh’s Equation)
Δt
•
=
9,79
per 1000 m = 2,34 oC per 1000 m
4,187
Horizontal pipes (Joule Thompson effect) due to friction.
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ΔT = μ x ΔP
Where: ΔT = Temperature increase due to friction (K)
= 2,4 x 10-7
μ
ΔP = Pressure drop due to friction (Pa) (i.e.: Hf x 9,79 x 1000)
Energy Recovery
ΔH - Hf
xgxmxη
1000
Energy recovered =
Where: g = 9,79 m/s2
m = Water flow rate (l /s)
η = Turbine efficiency expressed as a fraction of 1.
Pump Power Requirements
•
Where Q = l /s
Power(in) =
•
Q x HT x g
1000 x η
Where Q = m3/s
Power(in) =
Q x HT x g
η
Important Notes:
•
1 m3/s of water flow rate = 1000 l /s of water.
•
Frictional head loss in pipes down a shaft above a turbine should not exceed 2,5m /
100m
•
Water terminal velocity is where the head loss is equal to 100 m / 100 m.
•
Water pressure increase due to elevation = 9,79 kPa / m
•
Water power operating on a turbine = P x Q (Kw)
Where: P = (ΔH - Hf) x 9,79
Q = Water flow rate in m3/s (i.e.
•
l /s
)
1000
Pump Total Head (HT) = Static Head (Hs) + Head loss due to friction (Hf)
∴ Ht = Hs + Hf
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•
Chilled water most economical pipe water velocity = ± 2 m/s
•
Recommended head loss in vertical pipes = less than 2,5 / 100 m
(I.e. not more than 2,5 % / 100m)
•
Recommended station water pressure = ± 1000 kPa
•
Minimum water pressure at coolers should be ± 100 kPa
•
Minimum water pressure at mining operations ± 300 kPa
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ECONOMICS
Simple Interest
I
Where: I
= pxnx
i
100
- interest to be paid [Rand]
p - principal invested [Rand]
n - time that the principal is invested [Years]
i
- interest rate [%]
Compound Interest
i ⎤
⎡
S = p ⎢1 +
100 ⎥⎦
⎣
n
Where: S - total sum of money at the end of the investment period [Rands]
p - principal invested [Rand]
i
- interest rate [%]
n - time that the principal is invested [Years]
Total Owing Cost
Value of capital cost plus present value of annual running cost
Present Value
Present value of 1 = v n =
1
[1 + I]n
The present value of 1 per year for n years at an interest rate of I
An =
Where: vn
1 − vn
i
- the present value which one unit of money in n years would have at the present time
n
- years
i
- interest rate [%]
NB: Please refer to the “Environmental Engineering In South Africa” hand book page 848 - 860
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D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006
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STATISTICS
MEAN
This is the arithmetical average of a set of values;
Σx
n
x=
Where: x =
means the average value
Σ=
The Greek letter sigma means the sum of all individual values of x.
n=
is the number of observations.
Example:
Mean of (10, 15, 29, 22, 16, 20) = (112/6) = 18.7
GEOMETRIC MEAN (GM)
GM = n Y1xY2 xY3 xY4 ....Yn
n = number of values y
Example:
Geometric mean of 2, 4, 6, 3, 5
GM =
5
2x4x6x3x5
= 3.7
MEDIAN (Me):
Is the middle value when all the observations are arranged in ascending order.
Me =
n +1
2
Example:
Median of (1, 3, 2, 5, 4, 6, 9, 8 and 7) = (1, 2, 3, 4, 5, 6, 7, 8, 9) equals 5 (i.e. 4 values below 5
and 4 above – applicable for odd or uneven number of
values only)
Median of (2, 4, 3, 1, 5, 8, 7, and 6) = (1, 2, 3, 4, 5, 6, 7, 8) equals 4.5 (the average of 4+5 or
n +1
)
using Me =
2
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STANDARD DEVIATION:
Is to determine by how much the individual observation vary from the mean.
To calculate the "Standard Deviation" from a set of observations the following formula is used:
(S) =
Σ ( x - x)2
n -1
Where: S = Standard deviation
Σ = The Greek letter sigma means the sum of all individual values of x.
n = the number of observations.
PERCENTILES:
Are the values in a set which divide the set into 100 equal parts.
QUARTILES:
The values in a set which divide the set into 4 equal parts.
RANKING:
(Array) to arrange numbers in ascending or descending order.
CONFIDENCE LIMITS on calculated parameter:
The statistician must decide if the mean of two sets of observations which show a difference from
each other do in fact represent a genuine difference in condition.
This is the Confidence interval for a population’s mean:
X
=
∂
± 1.95( n )
Where: ∂
=
Standard deviation
n
=
number of values
1.95 =
95% confident
D:\my data\Websites\COM MiningCertification\Documents\MEC Formulae Handout Updated - New.doc / June 2006