2 2 6
Chapter 6
1
dR,
k
0
-
X
a
1
■ K b
dx
(
v° 1
dr
(
(
v° ^
2
ds
'
(
(
v0 \V
1
+ 6
1 + 6 1 + '
1+6 1 + ^'-
l
a )
v
l
-
f
l
«JJ
dR2
_
c
dy
(y 0
+ c ) 2
Evaluation o f the derivatives at .x" = 0.5 and .s ' = 1
yields:
(13)
R t =
0.3 101 + 0.6202
dx
-
0.09616 (0.59175
dx
+ 0.11237
ds")
+ 0.11778
ds'
= 0.3 101 + 0.5633
dx
+ 0.10697
ds'
( 1 4 )
Rz
=
0 . 3 1 0 1
+ 0 . 9 5 1 9 ( 0 . 5 9 1 7 5
dx
+
0 . 1 1 2 3 7
ds')
=
0 . 3 1 0 1
+ 0 . 5 6 3 3
dx
+
0 . 1 0 6 9 7
ds’
( 1 5 )
Equations (14) and (15) can be used for an approxim ate calculation o f the steady state flux
T(x,s
3 at any
point
Q
=
(x.s
3.
The flux control coefficient CY (,t°, 1) is im m ediately obtained from (14) or (15) :
CY ( 0 . 5 , 1 ) = ^
dx
— = 0 .5 6 3 3 -
R?
0.5
0 .3 1 0 1
: 0 .9 0 8 2
( 1 6 )
T his value is o f course exact since (16) is exact for dx
0 and
d r ’
0.
E xtrapolation o f (15) to
x =
1
(i.e dx = 0.5) for constant 5 ’ gives ■/’(1,1) = 0.59175. This value is too
high since T’(.x) is dow nw ards concave, and extrapolation along the tangent o f the curve w ill predict too
high values o f J \ T he true value o f J ’( 1,1) is found by solution o f (3) for y and insertion o f y into (4 ).
J \ x a c , ( U ) = 0 . 5 5 2 7 8 .
Sim ilarly the control coefficients at
(1,1) can be estim ated b y the m ethod o f exam ple (6.3).
The
elasticities are: