Transcript The overall height of a transfer unit

Chapter 8
Gas Absorption
1
Definition of Absorption:
•Absorption is a (column-type) process that
separates components of a gas mixture by their
solubility differences in the added solvent.
•In absorption, solute(s) is transferred from the
gas phase to the liquid phase.
•Desorption/stripping: the reverse of gas
absorption: recover valuable solute from the
absorbing solution and regenerate the solvents.In
stripping, the transfer is from the liquid to the gas
phase.
2
Illustrative figure
of packed column
Gas Gas
Gas
Liquid
Liquid
Gas
Liquid
Mist
catcher
Mist
catcher
Liquid
Mist
catcher
Clamp
plate
Clamp
plate
Mist catcher
Packing restrainer
Clamp plateColumnar
Columnar shell
shell she
Columnar
Columnar
shell
Columnar
shell
Gas
Packings
Packings
Packings
Packings
Packings
Liquid Support
Support
plate Suppo
Packing
Packing
Support plate
plate
Support
Mist
catcher
Liquidredistribu
redistribu
torto
Liquid
Liquid
redistri
Liquid
redistribu
tor
Liquid redistribu
tor
Columnar
Columnar
shell
she
Packing restrainer
4VS
Packings
Z  Packings
Packing
u
Packing
Support
Suppo
Gas
u  (0.5Liquid
~ 04.V
85
)u F redistribu
Liquid
redistri
to
S
DT 
Liquid

u
1)U  U min ;2) DT
 8(避免壁流现
d 填料
Mist
catcher
Gas
u  (0.5 ~ 0.85)u F
3
Clamp
plate
Liquid
[1.PACKINGS AND PACKED
TOWER DESIGN]
[Refer to Chapter 4 4.2 Packed Columns in
Chinese textbook]
Reading materials: pp. 546~557
4
2.PRINCIPLES OF ABSORPTION
Column diameter:
4VS
DT 
u
u  (0.5 ~ 0.85)u F
VS-gas volume flow rate, m3/s, uF-velocity
DT of liquid
1)U  U min ;2)
4VSS
flooding
D 
d 填料
 8(避免壁

u适宜
适宜(1) D should be rounded up;
•Attention:
T
TT
u(2)Checking:
85))uuFF
适宜
适宜  (0.5 ~ 0.85
DTT
1)U  U min
;
2
)
min
(Avoiding liquid

8(8避免壁流现象
) flow
dd packing
填料
along wall)
5
Tower height ZT
(1)Calculation from HETP: ZT=HETP*NT.
(2)Calculation from the number of transfer units:
ZT=HOy*NOy; etc.
•The height of the tower, and hence the total
volume of packing, depends on the magnitude of
the desired concentration changes and on the rate
of mass transfer per unit of packed volume.
•Calculations of the tower height, therefore, rest
on material balances, enthalpy balances, and
estimates of driving force and mass-transfer
coefficients, and equilibrium relationship.
6
Va y a
V
(1)Material balances
Vaa
y a La
V
a
•The types of equipment and designyyconcepts
xa
Laaused
a
in distillation are similar to those used
for
V
L
V
y
a
a
a
L
a
L
x
a
absorption and stripping. Here we choose
packed
a
Va Vya Ly a
x
column(differential-contact plant Vxaaa
)a asL the
aV
V
y
y
example.
a La xaLa
L
yLa
a Vax
Lower conc. end L LV y a
a xa a Lx a
V
LVa
a x y
稀端
L
y
x
x
a
a
a VaL
x
L
xxa
Vb
y
•Fig. 8.1-2 Material-balance
x
L
L
L
a
a
V
y
V
x
Ly
diagram for packed column.
VLb yb
V Vxxa yx
V
VVbb
Lb
y
b
Higher conc. end x xL V y
yyb y Vb x
x
浓端
L
b
b b 7
x
V
Ly y V
连续微分接触设备
Va y a
Material balances for the portion V
Vofaa the y a La
column above an arbitrary
section
yy ax-y Va x,
a
a) a
Total material: La  V  L  Va
(18L
.2V
L
Laa Va xyaa aL
(18.3)
Component A: La xa  Vy  Lx  VaVya V
Vxxaaaa yaa LLyaVa
V=molar flow rate of the gas
V
y
y
a La xaLa
L
yLa
a Vax
phase at x-y section
y
V
L
L
a
a xa Lax a
V
LVa
a x y
L= molar flow rate of the liquid
L
y
x
x
a
phase at x-y section
a
a VaL
x
L
xxa
yx Vb
L LVLa xV
a
y
Ly
VLb yb
V Vxxa yx
V
VVbb
Lb
y
b
V
L
x
x
xyybb y LVyb xb
8
b
x
V
Ly y V
(部分)
(任意的)
（ x-y 横截面）
Va y a
Overall material-balance equations, V
Vaa
y a La
yy a (18.4)Va x
Total material: La  Vb  Lb  Va
La a
a
V
y
(
18
.
5
)
Component A: La xa  Vb yb  Lb xb  VL
V
y
a
a
a
a
a
La
xaa L
Va Vya Ly a
Operating-line equation
x
Vxaaa
a L aV
V
y
y
Va ya  La xa
L
a La xaLa
L
y  x
(18.6) yLa
a Vax
y
V
V
V
L
L
a
a xa Lax a
V
LVa
a x y
L
y
x
x
a
a
a VaL
x
L
xxa
yx Vb
L LVLa xV
a
y
Ly
VLb yb
V Vxxa yx
V
VVbb
Lb
y
b
V
L
x
x
xyybb y LVyb xb
9
b
x
V
Ly y V
Operating-line equation:
axa xxb b
cu
r
ium

y
bb yya a
yba aa
ay a xxa a
ay
xyaa
ibr
tin
bb
a

b
uil
ine
gl
yb bb
byb bb 
bb yy 
b
yb
b
yb

b
b yb

b
b y a
ay b
yb
ya
x
ya a
a
x
a b
a
x
xaxb
b
byb
byb
by
Eq
•x and y represent the bulk
compositions of the liquid
and gas, respectively, in
contact with each other at
any given section through
the column.
yyb b
era
The operating-line must
lies above the equilibrium
line in order for absorption

to take place, i.e.,
y y 0
(18.6)
Op
Va ya  La xa
L
y  x
V
V
yb
ve
yb
axb
xxa
b
x10b
中溶质浓度较低时，气液流量的变化可以
忽略）.
axa xxb b
ve
cu
r
ium
ay
xyaa
ibr

y
bb yya a
yba aa
ay a xxa a
a

b
uil
tin
bb
b
yb
b
yb

b
b yb

b
b y a
ay b
yb
ya
x
ya a
a
x
a b
a
x
xaxb
b
byb
byb
by
Eq
gl
yb bb
byb bb 
bb yy 
ine
yyb b
era
•For dilute mixtures,
containing less than 10% of
soluble gas, the effect of
changes in total flow is
usually ignored and the
design is based on the
average flow rates（当混合气体
yb
Op
•Because of absorption, V
decreases from bottom to
top, and L increases from
top to bottom. These
changes make the operating
line slightly curved.
yb
axb
xxa
b
x11b
Absorption Driving forces at an arbitrary
section x-y:
A
Ay
Driving
y
force y
yx
x
x

x
A
A
yy
yy
xx
xx
A
Equilibrium
y curve
y

 Driving



force x
x

Absorption Driving forces: ( y 

y )

( x  x)
12
(2)Limiting (min) gas-liquid ratio
•1)Influence of liquid flow rate L
yb
Assume that V, xa, ya, yb are held constant and
yb L
decreases.
ibr
uil
Eq
Op
era
tin
ium
gl
cu
ine
rv
e
b
b
y
y
y
b 
b
b
b
L  ( L / V )  Driving force  Operating Cost
yb

y
b
b
b
b
b
y
But, the column height
yb( L / V )minb
b
byb bb 

b
y
y
b
b
a
•When L=Lmin(point b’),



b

b
absorption driving force=0, b yyb b
ayb
b y a

the column height=.
ay b
byb yya a
xyaa
yb
ya
yba aa
x
axb
ya a
a

ay a xxa a
x
x
13
xa
a b
b
•2)Minimum liquid-gas ratio (L/V)min
(L/V)min corresponding to L=Lmin(point b’) with
absorption driving force=0 and the column height=.
•In any actual tower the liquid rate L must be greater
than Lmin to achieve the specified change in gas
composition.
•3)Optimum liquid-gas ratio (L/V)opt
L  ( L / V )  Driving force  Operating/ e
e  Operating/ energy Cost  But, the column height
•By balancing the operating costs against the fixed
costs of the equipment, ( L / V )opt  (1.1 ~ 1.5)(L / V )min
14
yb
bb
by yya a
yba aa
ay a xxa a
axa xxb b
xba xx 
bb
e
ine
cu
rv
ium

b
tin
gl
yb  ya
L
   
 V  min xb  xa
yb bb
byb bb 
bb yy 
Op
era
•For normal curve,
yyb b
Eq
u il
ibr
•Calculations of (L/V)min
b
b
byb
yb

b
b
yb( L / V )minyb

b

b
y
y
b b
a


b
ayb
b y a
ay b
xyaa
yb
ya
x
axb
ya a
a

x
x
xba
a b
a
x
xb
xaxb
xb

15
x
xb
b
•For abnormal curve, (L/V)min can be computed
yb
from the slope（倾斜） of the operating
line ab’ that is
yb
tangent to the equilibrium
curve.
b
Op
era
tin
gl
ine
yybbL 
byb  ya by
  
b
yb bbV  min bxb  xa b 
yb
yb
(L / V ) min
b bb
y
b
bya
b
yb 
b yyb
 ve
y
r

a
b
b
u
a
y
c
b
b
m
u
i
r
b
i
l
yb yyaa

i
a
u
q
xyaa
y
E

b
b
y a aa
xya
axb
yb
a
a xxaa
xab
xba
ya

16
(3)Rate of absorption
•Volumetric mass transfer coefficients (Kya, etc.) are
used for most calculations, because it is more difficult
to determine the coefficients per unit area and because
the purpose of the design calculation is generally to
determine the total absorber volume.
•Kya=overall volumetric mass-transfer coefficient,
kmol/(m3·h·unit mole fraction).
•a=effective area of interface per unit packed volume,
m2/m3
17
•Simplicity Treatment
•The following treatment applies to lean gases低浓度气
体(up to 10% solute)：
•(a) Correction factors修正因子 for one-way diffusion
are omitted for simplicity.
•(b) Changes in gas and liquid flow rates (V and L) are
neglected.
•(c) kxa, kya, Kya, Kxa can be considered as constants.
18
•In chapters 6,7, the rate of mass transfer:
2·h·unit mole fraction)]
r=N
[kgmol/(m
r ()A
rN(
A ) k y ( y A  y Ai )  k x ( x Ai  x A )

N

K
(
y

y
r (
A)
y
A
A)

A
N A  K x ( x  xA )
19
•Let r =rate of absorption per unit volume,
kgmol/(m3·h)
r  k y a( y  yi )
(18.7)
r  k x a( xi  x)
(18.8)

r  K y a( y  y )

r  K x a ( x  x)
(18.9)
(18.10)
•It is hard to measure or to predict a, but in most cases
it is not necessary to know its actual value since design
calculations can be based on the volumetric
coefficients. α很难直接测定，一般不需要知道α具体值
20
•Determining the interface composition (yi, xi)
•(yi, xi) is also hard to measure, but it can be obtained
from the operating-line diagram using Eqs.(8.1-21)
and (8.1-22):
( y  yi )
kxa

( x  xi )
kya
(18.11)
•Thus a line drawn from the operating line with a
slope –kxa/kya will intersect the equilibrium line at (yi,
xi).
21
( y  yi )
kxa

( x  xi )
kya
kxa
(
18
.
11
)
Slope  
kya
y
x curve
k x a Equilibrium
SlopeOperating
 kx a line
kxa
y
a
i
Slope   kkyxaSlope

Slope ky a
kya
x
k
a
i
y
y
y

y
y
xy
kxa 
x
x
yxi Slope   k a x
y
yi
y
kxa
i
xyi i
Slope


y
xi 
xi
kya
x
yi x


y 
y
y
y
Driving force :
( y  yi )
( xi  x)

(y  y )

( x  x)
22
k y ( y A  y Ai )
Ky
k x ( x Ai  x A )
•Determining the overall coefficients: Using the local

slope

y Aiofthe
m xequilibrium
xA , m, we have
Ai , y A  m curve

A
y

y
y

y
11
A
Ai
Ai
1 m


K k ( y  y ) k ((x17.57x) )
y
yk A k Ai
x
Ai
A
K
y
y
x
 y Ai  m xAi , y
Therefore,
(17.56)

A
 m xA ,

y A  y Ai
y Ai  y A
1
1
1
m


K y k y( y A y Ai ) k x ( x Ai (18
x A.)12)
K ya k ya kxa


y Ai  m xAi , y A  m xA ,
Similarly,
1
1
1



K x a k x a m ky a
(17.56)
(18.13)
23
 y Ai  m xAi , y A  m xA ,
1
1
m



K ya k ya kxa
(18.12)
1
=overall resistance to mass transfer
K yy a
1
k yy a = resistance to mass transfer in the gas film
m
=resistance to mass transfer in the liquid film
k xx a
24
y
y
A
Ai
x
Ai
A

“controls”
Ai
A

y Aifilm
 m x , y  mand
xA , Liquid film “controls”
•Gas
1
1
m



(18.12)
K ya k ya kxa
1
m
1
1

,

 Gas film " controls"
When
k y a kxa K y a k y a
•Or, When the coefficients kya and kxa are of the same
order of magnitude, and m is very much greater than
1.0, the liquid film resistance is said to be controlling.
That is,
1
m
1
m

,

 Liquid film " controls"
kya
kxa K y a kxa
25
•Liquid film controlling means that any change in kxa
has a nearly proportional effect on both Kxa and Kya
and on the rate of absorption, whereas a change in kya
has little effect.
•Examples of Liquid film controls: Absorption of CO2,
H2,O2,Cl2 in water;
•When the solubility of the gas is very high, m is very
small and the gas-film resistance controls the rate of
absorption.
•Examples of gas-film controls: Absorption of HCl,
NH3 in water; NH3 in acid solution; SO2, H2S in basic
solvent.
26
•With gases of intermediate solubility both resistances
are important, but the term controlling resistance is
sometimes used for the larger resistance.
•The absorption of NH3 in water is often cited as an
example of gas-film control, since the gas film has
about 80 to 90 percent of the total resistance.
27
•(4)Calculation of tower height
•The following treatment applies to lean gases (up to
10% solute：
(a) Correction factors for one-way diffusion are
omitted for simplicity.
(b) Changes in gas and liquid flow rates (V and L) are
neglected.
(c) kxa, kya, Kya, Kxa can be considered as constants.
28
VL, ,yxLa , xa
V , ya
a
L
,
x
, ,yxa a Z V , y
a VL
S=cross sectional area;
V , yLa, xaaZ
V , y a ZV , y dZ Z
SdZ=differential volume in
L, xa
a
V
,
y
Z
L
,
x
dZ
a
adZ
height dZ.
Z
ydZdZ
VL ,,xy a Z
Z y
a, y
V
a
dZ ydZ x y
•The amount absorbed in
VZ, y a
y dZ x
section dZ is Vdy.
Z
x
y
y
Z
ZxT xy
dZ
dZ Z T
Vdy  rSdZ
ZT
x
dZ
x V , yZb T
y

Z T x V , yb
y
Vdy  K y a( y  y ) SdZ
y

dy
yZ T
ZT x  V
, yb
dx
x
y
Z
Z
V
,
y
x
b T L , xb
L
,
x
x
V
dy
V , yb V , by
L
,
x
b
0 dZ  ZT  K y aS y ( y  y ) ZZLTT,ZxT
L , xVb , ybb
b
L
,
x
V
,
y
b
V ,V
ybb, y
L , xb
b
•Fig. Diagram of packed
L,,xxbb
L
L , xb
absorption tower
29
a
T
b
a
•(5)Number of transfer units传质单元数
•The equation for column height can be written as
follows:
V /S
ZT 
K ya
yb
yb
dy
y ( y  y  )
a
dy
N oy  

(y  y )
ya
H oy
V /S

K ya
(18 .16 )
=overall number of transfer units
[NTU], based on gas phase.传质单
元数 []
=overall height of a transfer unit
[HTU], based on gas phase,传质单元高
30
度 [m].
ZT  Hoy Noy
(18.17)
(1)If the operating line and
yb yb yb
dydydy
equilibrium line are straight and
yb N NN


oy
oy
oy
  
parallel,
( y (y( y)y y) )
yb
yb
y3
yyb3
Operating line
yyyb32 yb
yyy3 y3y
y2
2a
1
yyx2aa y 2y1
 y  y
xyxaa1  ay1 1
xx xa
y a  y1
xa
Equilibrium
line x1
x2
x3  xb
y3
 
ya ya ya
yb ybyybayaya
y2  
(1.
NN
N

(18(18
.18
oy oy 
oy
  


y
y

y
y

y
yb  y1
y

y
y

y
y

y
b
a
b
a
b
a
xa N oyNN



oy oy
ya yayyaayaya
x1
yb ybyybayaya
x2 N oyNN

oy oy 
yb ybyybbybyb
x3  xb
31
yb
dy
N oyoperating

•If the
line
and equilibrium line are

(y  y )
y a parallel,
straight and
yb  y a
 N oy 
 NTP (18.18)

y y
y
yb
dy
dy
y

y
Nplates
a
NTP=Number
oy  [结合第7章比较]

NNoyoy   b of theoretical
(
y

y
)
 
y
(
y

y
)

y
Similarly,
a
a
yy
a
b
a
xb  xa
NN ox ybNoyya NTP N ox  x   x  NTP
yb oy

y

y
dy
b
by
yb  y a line (not
straight
and
equilibrium
yb operating
N oy•(2)
 For
a
N oy 


N

oy
(
y

y
)
y

y

parallel),
a
a
y

a
ya  ya


ybyy
(
y
)

(
y

y
)
a
b
b
a
a
yb  yay  y
N

oy

 N oy  N  b (18a.19) y L 

y

y
b
b
oy
y

y

b
y L yb  yb
ln b

y

y
a
a
yb  y a
32
N oy 

•Derivation of Eq.(18.19) :
•Assume
y   m xy b m x  b
yy y yy(mxyb()m x  b)
y  y 
V ( y yVb )( y L(yxb )xbL)( x  xb )
V

V

ym y 
 y  
y
( ym yb )( y xbyb) b xb   b
L

L

y ~ straight
y is straight
line relationsh
ip
y ~ y is
line relationsh
ip
yayb  ya
d (y ) d (yyb) 



dyyb  ya yb  ya
dy
dy1 ( y 1 ya()ydb (y )a )d (y )
dy

N oy  N oy       
33
y  y y  yy
yb  
yayb  ya
yb
yb
yb
yb
b
y ~ y is straight line relationship
d (y ) yb  ya


dy
yb  y a
N oy 
yb
yb

ya
dy
1 ( yb  ya )d (y )



y y
y
yb  ya
ya
yb
dy ( y  y ) y
N oy    N oy  b a ln b
( y  y ) yb  ya ya
ya
yb  y a
 N oy 
y L
yb  y a
N oy 
ya  ya
N 
yb  y a
(18.19)

b

a
(18.19) y  ( yb  y )  ( ya  y )
L

yb  yb
ln
ya  ya
34
yb
dy
•(3) When the
line
is straight but steeper
N oyoperating


yb than the equilibrium
(y  y )
y a line, as in Fig.18.13b,
xb
y

a
 N oy  NTP
NTP=Number of theoretical plates

b
yb  ya
N oy 
ya  ya
ya •Why?
y
yb  ya  yb  yb  ya  yya  y
a
N oy  b

 yb  ya  y L
y
y  yb
b
Operating line
b
y
yb  y a
x
b
b
 N oy  NTU 
 1  NTP
y L
ya xybb


y
y  yxba
a
b

a
aa
Equilibrium
line
yb  yb
yb  yy yyb b yb  ya  y35a
xa xb

•Similarly, for straight operating and equilibrium
yb
line (not parallel),
dy
N oy  

(
y

y
)

H
Nox
yZ
T
ox
a
xb  xa
 N ox 
(18.20)
x L
y
b  y a

( xb  xb)  ( xa  xa )
N

oy
x L  y yby 
a
ax dy
 xb
b
N oyx ln

xa(xy yx )
b
N oy 
ya
a
 a
yb  yb
L/S
H ox 
[ m]
K xa
36
•The overall height of a transfer unit [Hoy OR Hox]
can be defined
yb as the height of a packed section
dy
required
to
accomplish
a change in concentration
N oy 

( y  y )driving force in that section.
equal to theyaaverage

yb  ya  ( y  y  ) m  Z T  H oy
yb  y a
N oy 
ya  ya
•Common equations for calculations of height of
yb  y a
packedNsection:

oy
yb  yb
height of packed section=height of a transfer unit 
number of transfer units
•There are four kinds of transfer units:
37
•Four kinds of transfer units:
ZT  H y N y  H x N x  H oy N oy  H ox N ox
Gas film: H y 
V /S
kya
L/S
Liquid film: H x 
kxa
V /S
Overall gas: H oy 
K ya
Overall liquid: H ox 
L/S
K ya
yb
dy
y  yi
(18.21)
dx
Nx  
x x
xa i
(18.22)
Ny 

ya
xb
yb
dy
y  y
(18.23)
dx
N ox   
x x
xa
(18.24)
N oy 

ya
xb
38
(6)Alternate forms of transfer coefficients
The gas-film coefficients reported in the literature are
often based on a partial –pressure driving force instead of
a mole-fraction differencedy
and are written as kga or Kga.
J A  N A   Dv  M
A
•Similarly liquid-film coefficients
may be given as kLa or
db
BT the driving force
yA
KLa, where
dyA is a volumetric concentration
JNAA  db
N LA=k
Dv vMM dy
difference.
[k
defined
byAEq.(17.36)]
cD

0
BT
 db
y Ai
yA
k yaD 
K ya
N
db

dy
v M
A
 kA ga 
,K

ga 
0
y Ai
P
P
k x aDv
K xa
N kA La JA  , K(LcaAi c A )

 M BT
M
(17.20)
39
Gy
V Gx L
 ,
  LM
M S M S
 M M   x
Let GM 
GM
GM
Hy 
and H oy 
k g aP
K g aP
(18.25)
Gx /  x
Gx /  x
Hx 
and H ox 
(18.26)
kLa
KLa
G y V Gx L
Where, GM =molal

 mass
, velocity,

kgmol/m2 h
M S M S
=mass
velocity
of gas stream based on
G
G
V
L
y 
x

M


M
x
GM 

,
 cross section, kg/m2 h
total tower
M S GM S
Gstream
G y V Gx =mass
L
based
on
M velocity of liquid
M

H

and
H

(
18
.
25
)
M,M  y x
oy
M 
2
aP cross section,
K g aP
kg/m h
M S M total
Sk gtower
GM
GM
40
M M H

and
H

(
18
.
25
)
Gx /  x
Gx /  x
x y
oy
•The terms HG, HL, NG AND NL often appear in the
literature instead of Hy, Hx, Ny AND Nx, as well as the
corresponding terms for overall values,
but
here
the

y A  y Ai do not ysignify
1
different
subscripts
difference in
Ai  yany
A


(
17
.
56
)
either
K units
k ( yor magnitude.
y ) k (x  x )
y
y
A
Ai
x
Ai
A
•Relationships among
different kinds

 of height of a

my AxAi, y AiA  m xA ,y Ai  y A
1y Ai  unit:
transfer


K y 1 k y ( y1A  y Aim) k x ( x Ai  x A )



(18.12)
k a
K
a
k
a
 y y  m xy , y x m x ,
Ai
Ai
A
(17.56)
A
GM GM m GM LM



(18.27)
K y a k y a k x a LM
41
Ai
Ai
db
A
A
GM GM y m GM LM

(
18
.
27
)
N A  db  Dv  M 
dy
A

K
a
k
a
k x a LM
y
y y
0
BT
A
G yG y V VGxG Ai L L
GMGM  k a , , x K a
y S M
y
M
S
M
S
M
S
 kg a 
, Kga 
P

MMM M P
x x
Dv
G
GMGM (17.20)
G
M
M
N

J

(
c

c
)
1
H
and
H
.
25
)
y

y
A yH A
Ai
A   y Ai  y A(18

and
H
(
18
.
25
)
oy
A
Ai
y
oy
B
 k g aP
 K gKaP
(17.56)
T
k g aP
g aP
K y G k/y( y A  y Ai ) kGx (/xAi  x A )
H xHx  xLM x and
andHHoxoxGx x/  x x
(18
.26
(18
.26
))
aa
 y kkLxmx
, y  mx KK
,L aL a
Ai
Ai
A
A
mGM
 H oy  H y 
H x (18.27)
LM
42

A
1y Ai  myfrom
Similarly,
xAi, y  m xA ,y Ai  y
A


K y 1 k y ( y1A  y Ai 1) k x ( x Ai  x A )



(18.13)
m k a

K
a
k
a
1 y Aix  m
y xA ,y Ai  y A
y AxxAi, yyAiA  m


K y LkMy ( y ALM y Ai ) LMk x (GxMAi  x A )



 mk a G
K
a
k
a
x
x
 y  mx , y  mxy , M

Ai
A
Ai
Ai
A
(17.56)
(17.56)
A
LM
 H ox  H x 
H y (18.29)
mGM
EXAMPLE 18.3.
43