Equation Sheet for M E 433 Quizzes and Exams
Author: John M. Cimbala, Penn State University. Latest revision, 05 January 2015
Note: This equation sheet will be provided in electronic form as a pop-up window during the quizzes taken at the Testing
Center. Do not bring a printout of this equation sheet to the quiz. This equation sheet has information for the entire
semester in order of presentation; for the earlier quizzes, ignore the later pages. This sheet is also useful for homework, inclass poll questions, and the open-book portion of the final exam.
m 0.3048 m
1 mile
1 Btu
1 kPa  m 2 1 kN  m 1 kW  s
,
,
,
,
,
,
,
2
1 kN
1 ft
1.6093 m
1 kJ
1 kJ
1.055056 kJ
s
3
3
1 kg
1 ton
1m
4
1
1 tonne (metric ton)
1g
1m
,
,
, 6
, 6
, 9
, Vsphere    R p     D p  .
2.205 lbm 2000 lbm
3
6
1000 kg
10 μg 10 μm 10 nm
General and conversions: g  9.807
Molecular weights and mols: m  nM , M air  28.97 g/mol , M water  18.02 g/mol , Avagadro’s number: 6.0225  1023 .
P TSTP PSTP  101.325 kPa  760 mm Hg
Volume and mass flow rate: Q  V  UAc , m   Q  V , QSTP  Qactual
,
.
PSTP T
TSTP  25o C  298.15 K
Ideal gas: PV  nRuT , R  Ru / M , PV  mRT , P   RT , Ru  8.314
kJ
kJ
, Rair  0.287
.
kmol  K
kg  K
J
J
J
J
mj
j 1
j 1
j 1
j 1
mt
Ideal gas mixture: mt   m j , nt   n j , P   Pj , V  V j , f j 
J
PV  nt RuT , PjV  n j RuT , PV j  n j RuT , M t    y j M j  , c j 
j 1
Relative humidity and vapor pressure: RH 
PH2O
Pv ,H2O
Emission factors (EPA AP-42 and CHIEF): EF 
 100% 
mj
V
PH2O
Psat,H2O
 yj
Mj
Mt
, cmolar, j 
nj
V
, yj 

 100% , yH2O 
mpollutant
mraw material or product produced
,
cj
Mj
PH2O
Patm
nj
nt

Pj
P
, cj  yj

Vj
V
, Mj 
mj
nj
Mj P
, m j  c j Q .
Ru T
, Pv  Psat for any VOC .
m discharged  1  E  m generated
E = APCS removal efficiency
.
Combustion of hydrocarbons: Stoichiometric equation, C x H y  astoich  O 2  3.76N 2   bCO 2  cH 2 O  dN 2 ,
Simple dry air = 21% O2 , 79% N2 ,  
Flux chamber:
dm j
dt
V
Tank filling: m j , displaced 
Lapse rate:   
dc j
dt
 F / O2 n
 F / O2 n, stoichiometric

astoich
, where a = molar coeffient of oxidizer O2 ,
a
 c j ,a Qa  S j  c j Qa , m j , generated  S j   c j ,ss  c j ,a  Qa , EF 
M j Pj
RuT
m j , generated
some appropriate denom.
.

 
Qliquid in . Coriolis force: Fc  2m   U .


o
o
dT
C
C
, Normal lapse rate = 6.5
, Dry adiabatic lapse rate = 9.8
.
km
km
dz
Gradient diffusion: J A  b
dc j
da
A
, where a 
and A = mass, energy, momentum, … For mass, M j J j   Daj
.
dz
dz
V
d
Gaussian plume model:  y  ax b ,  z  cx  f , with x in units of km and y and z in units of m.
2
2 
 
 1  y   z  H   
exp     
 Absorbing ground: c j 
  , where H  hs   h = effective stack height.
 
2 U  y z
 2   y    z   
2 
2 

   2 
   2 
m j , s


1
y
z
H
z  H   


 1 y



exp     
 Reflecting ground: c j 
   exp     
  .
2 U  y z 
2   y    z   
2   y    z    




 
 
 
m j , s
,
 1  y 2  
 1  z   2 H  H  2  
 1  z  H 2 

T
exp      exp   
 Absorbing ground w/ inversion: c j 
  ,
   exp   
2 U  y z
2
2


 2   y   



z
z



  





where H = effective stack height and HT is the elevation of the reflecting part of the inversion.
 1  y 2 
m j , s
m j , s
exp      ; at y = 0, c j , F 
 Fumigating, reflecting ground, far downwind: c j , F 
.


 2y  
2 U  y H T
2 U  y H T


m j , s
Gaussian puff diffusion model:  xi   yi  ax ,  zi  cx , but x in units of m, not km, and yi and zi in units of m.
b
d
Use these empirical values for the instantaneous diffusion coefficients, depending on atmospheric stability conditions:
Stability condition
a
b
c
d
Unstable
Neutral
Very stable
0.14
0.06
0.02
0.92
0.92
0.89
0.53
0.15
0.05
0.73
0.70
0.61
Adapted from Slade (1968), as found in Heinsohn and Kabel (1999).
2
 
 1  x  Ut   y
exp  
 Absorbing ground: c j  x, y, z, t  
  
 2 xi yi zi
 2   xi    yi
2
2
  z  H   


   zi    .
  
 
 
y
 1
exp  
 Ground level dose, absorbing ground: D j  x, y,0  
2  
 U  yi zi
  yi
2
2
  H   


   zi    , (double for reflecting).
  
 
mj
mj

3
1
 3 
 CD
 
 Dp 2 vr vr .
, m p ,mean   p   D p ,am  mass   , Fgravity    p    D p g , Fdrag  
m p ,mean
6
6
8 C


 

Relative particle velocity = vr  v  U , where v is the particle velocity and U is the air velocity.

dv  p    3  CD 1  
g
vr vr , where C is the Cunningham correction factor,

Equation of particle motion:
dt
4  p C Dp
p

 vr D p




 0.55  
Kn 
, 
, C  1  Kn  2.514  0.80exp  
,
  , and CD = CD(Re), where Re 
0.499 8 P

Dp
 Kn  

Particles: cnumber, j 
CD 
cj


2
24
24
24
for Re < 0.1 , CD 
1  0.0916 Re  for 0.1 < Re < 5 , CD  1  0.158Re 3 for 5 < Re < 1000 .
Re
Re
Re
3
Air at STP (25oC): PSTP  101.325 kPa  760 mm Hg ,   1.184 kg/m ,   1.849  105 kg/  m s  ,   0.06704 μm .
Vt D p
p   2 C
4 p  
C
gD p
Dp g .
, Re 
. For Stokes flow (Re < 0.1), Vt 
3 
CD

18

Terminal settling velocity: Vt 
Gravimetric settling in ducts: Lc 
Laminar settling: E  D p  
cj
HU
= where Vt = function of Dp. Grade efficiency: E  D p   1 
:
Vt
c j (in)
 L
L
if L  Lc ; E  D p   1 if L  Lc . Well-mixed settling: E  D p   1  exp    .
Lc
 Lc 
 
 
m j , s
 1  y
exp  
Gaussian Plume with Particle Settling: c j 

2 U  y z
 2   y

 
x


2
z   H 0  Vt  
 
U


 
z






2


  , where H = H at x =0.
0



 
Inertial Separation Devices:
 vr D p
WU 
L
4  p   U 2
C
Dp
, Re 
, where rm = mean radius, Lc 
, c  c ,

3 
rm
CD
vr
rm
Inertial “settling” velocity: vr 
Laminar settling: E  D p   1 
cj
c j (in)
Standard Lapple cyclone: D p ,cut

cj
 x 
x
 1  exp    , where x  rm .
. Well-mixed settling: E  D p   1 
Lc
c j (in)
 Lc 
3 D23

, E  Dp  
128 Q   p   
2
1
 D
1  p
D
 p ,cut



2
QP
 Q 

, P  40.96  
.
 , Wblower 
blower
 WH 
Air Cleaners in Series and Parallel:
m
Qj
Parallel: E  D p overall  1   f j 1  E  D p  j  , where fj = volume fraction through cleaner j, f j 
.


Qtotal
j 1
m
Series: E  D p overall  1   1  E  D p  j  , where the volume flow rate of air through each cleaner is the same.


j 1
Rain, Spray Chambers, and Wet Scrubbers as Air Pollution Control Systems:
2
 p    Dp 2 U 0  Vt , p 

 r1  
Stk

Single-drop collection grade efficiency: Ed  D p      
.
 , where Stk 
18 Dc
 Rc   Stk  0.35 
2
 L
Dc
2 Qa Vc
Overall collection grade efficiency: E  D p   1  exp    , where Lc 
, Vc  Vt ,c  U a for a spray
3 Qs Vt ,c Ed  D p 
 Lc 
chamber. Lc must be estimated or calculated for a wet scrubber – depends on size and shape of the packing material.
Air Filters: ( = porosity, U0 = air speed, L = filter thickness, Ef(Dp) = single-fiber collection efficiency)
Stk 

p
   D p 2 U 0 /  
18 D f
Df
 L
 
Stk


, E f  Dp   
, E  D p   1  exp  
 , Lc  4 1  
E f  Dp 
 Stk  0.425 
 Lc
2

.

Electrostatic Precipitators: (ESPs)
 L
E  D p   1  exp    , where Lc is dependent on voltages, particle composition, air speed, gap widths, and many other
 Lc 
parameters. On quizzes and exams, Lc would either be given, or would be the variable to be calculate.
Polydisperse Aerosol Particle Statistics:
 j
= class (bin) number with range Dp,min, j < Dp  Dp,max, j, width Dp, j, and mid value Dp, j for j = 1 to J.
 nj
= number of particles in bin j, and nt = total number of particles in the sample, nt = nj .
 f (Dp, j) = fraction of particles per bin width = nj /(Dp, j nt).
Probability of particle in bin j: Prob  f  D p , j   D p , j 
a
nj
, Cumulative number distribution: F (a)   f  D p , j  dD p .
nt
0
Median diameter: F ( Dp ,50 )  0.50 . For number distribution, use Dp,50 (number); for mass use Dp,50 (mass).
Arithmetic mean diameter: D p ,am  0 D p f  D p dD p 


Geometric mean diameter: D p , gm  D
n1
p ,1
n2
J
1
nt
n D  .
j
j 1
nj
D p ,2 ...D p , j ...D p , J
nJ

p, j
1
nt
1
 exp 
 nt

  n ln  D   ,
J
j 1
j
p, j

Dp , gm  Dp ,50  Dp ,median .
J
2
 

 n j ln  Dp , j  
n j ln  D p , j   ln  D p ,am  




ln  g 

j 1
j 1 
Geometric standard deviation:  g  e
, ln  g  
, ln  D p ,am 
.
nt
nt  1
J
Or,  g 
D p ,50
D p ,15.9

D p ,84.1
D p ,50

D p ,84.1
D p ,15.9
, and g is the same whether based on the number or the mass distribution. So, we can
use either the number or mass values of Dp,50, Dp,15.9, and Dp,84.1, i.e.,  g 
D p ,50 (number)
D p ,15.9 (number)

D p ,50 (mass)
D p ,15.9 (mass)
, etc., where
D p , gm  number   D p ,50  number  , and D p , gm  mass   D p ,50  mass 
Conversion from number distribution to mass distribution: m j  n j  p
we plot histograms as
g  Dp, j 
D p , j

D 
6
p, j
3
. Mass fraction: g  D p , j  
mj
mt
a
for nonequal bin widths. Cumulative mass distribution: G (a)   g  Dp , j  dD p .
0
  ln D (mass)  ln D (number)  
  p,50
 
   p ,50
Since ln  D p ,50 (mass)   ln  D p ,50 (number)   3 ln   g   ,  g  exp 
.
3




Overall particle removal efficiency:
J
mj  J 
cj 

Eoverall    E  D p , j      E  D p , j 
.
mt  j 1 
coverall 
j 1 
2
, but