This document discusses sediment transport and load in streams and rivers. It defines different types of sediment loads, including wash load (fine sediments) and bed material load (coarser sediments). It describes Einstein's method from 1942 for modeling sediment transport based on the probability that lift forces on sediment particles exceed submerged weight. Einstein's equation relates the fraction of particles moving in the bed load to a shear intensity parameter that depends on grain size, fluid properties, and channel geometry. The document provides the equations and approach used in Einstein's sediment transport modeling method.

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o f s e d im e n t in th c w a te r s h e d A n d th e tr a n s p o r t c a p a c
ity o f th e s tr e a m T ilì jc a ııy th e lin e r n ra te r ia ıw h ic h is
e a s iıv c a r r ie d in ıa r g e a m o u n ts h y ı1o w H a s ıim ite d
avaiıabiıi in th e w a te r s h e d T h is m a te r ia ı is a ls o
g e n c r a ıl c o n s id e r e d a s w a s h lo a d (s e e d e fin itio n a t th e
b e g in n in g o f th is s e c tio n ) T h e c o a r s e m a te r ia ıis m u c h
m o r e d iffic u ıt to m o v e h y I1o w s s o its r a te o f m o v e m e n t
is ıım ite d b y th e tr a n s p o r t c a p a c ity o f th e flo w T h is
m a te n a l is d e s ig n a te d a s b e d m a te r ia l ıo a d a n d in
c ıu d e s b o th b e d ıo a d a n d s u s p e n d e d lo a d A lth o u g h
th e r e is n o s h a r p d iv is io n b e tw e e n b e d m a te r ia l lo a d
a n d w a s h lo a d w a s h ıo a d is o fte n c o n s id e r e d to b e s ilt
a n d c la y m a te r ia ıs T h is is n o t a lw a y s a p p r o p r ia te b e
c a u s e in s o m e c h a n n e ıs h a v in g e x tr e m e ly s lo w flo w s
th e b e d m a te r ia l c o n s is ts o f fin e s iıts T h u s S im o n s a n d
S e n tu r k ( 19 7 7 19 9 2 ) d e lin e a te d a s e c o n d c r ite r io n th a t
u s e d th e D o f th e b e d m a te r ia l a s th e d iv id in g s iz e
b e tw e e n e
:m a te r ia l lo a d a n d w a s h lo a d W a s h lo a d is
o fte n a s s u m e d to tr a v e ıth r o u g h a s y s te m by s tr e a m flo w
w ith v e ņ ' little d e p o s itio n H e n c e ,
v e ry littıe o f th is
s m a ıl m a te r ia ıis d e p o s ite d
T o ta l s e d im e n t d is c h a r g e (a ıs o c a lle d to ta l s e d im e n t
lo a d ) c o n s is ts o f b o th b e d m a te r ia l lo a d a n d w a s h lo a d
G e n e r a lly to ta ıs e d im e n t lo a d c a n o n ly b e e s tim a te d if
th e w a s h lo a d is e s tim a te d b y m e a s u r e m e n t,
e x p e r i
m e n t o r u p la n d s e d im e n t y ie ld e q u a tio n s b e c a u s e m o s t
se d im e n t tr a n s p o r t m e th o d s c a n o n ıy d e te r m in e b e d
m a te r ia l ıo a d
E in s te in c o n c lu d e d th a t th e r e is a c lo s e r e la tio n s h ip
b e tw e e n b e d m a te r ia l lo a d a n d b e d lo a d b e c a u s e th e r e
is a c o n tin u o u s e x c h a n g e o f p a r tic le s b e tw e e n th e b e d
m a te r ia l a n d b e d lo a d In a d d itio n , p a r tic le s a r e tr a n s
p o r te d a ıo n g th e c h a n n e l b e d in a s e q u e n c e o f s te p s
w ith th e a v e r a g e d is ta n c e o f e a c h s te p p r o p o r tio n a l to
th e p a r tic le s iz e It w a s a ls o n o te d th a t a p a r tic u la r
p a r tic ıe is n o t a lw a y s m o v in g ,
b u t r a th e r ,
d e p o s its o n
th e b e d a fte r a fe w s te p s F in s re in fo u n d th a t th e
d e p o s itio n r a te p e r u n it a r e a d e p e n d s o n b o th th e
tr a n s p o r t r a te p a s t a p a r tic u la r p o in t a n d th e p r o b a b il
ity th a t th e fo r c e s o n th e p a r tic le a r e s u c h th a t th e
p a r tic ıe w ill d e p o s it
E in s ıe iıı
'
6 M e th o d
B e d ıııa d E in s te in (19 4 2 ) d e v e lo p e d a r e la tio n şłh ip
fo r i m e n t tr a n s p o r t b a s e d o n p ro b a b ility th e o ry
Ig n o r in g d r a g fo rc e e fre c ts o n a p a r tic le ,
E in s te in a 8
s u m e d th a t a p a rtic ıe w o u ıd b e d is ıo d ge d fr o m th e
c h a n n e ıb e d if the lift fo rc e e x c e e d e d th e s u b m e rg e d
w e ig h t d u r in g a n e x c h a n ge tim e T he p a r tic ıe w o u ld
m o v e a d is ta n c e o f A d b e fo re s tr ik in g th e s u r fa c e
a g a in ıf th e fo r c e ıa t th e p o in t o f im p a c t a r e s u c H th a t
ıift e x c e e d s s u bm e rg e d w e ig h t,
th e p a r tic ıe w ilıta k e
a n o th e r h o p o f d is ta n c e M b e fo re s tr ik in g the su r fa c e
7 Se d ım e n t P ro p e rtıe ıo n d Tre n ı
a g a in T h is c o n tin u e s u n til th e p a r tic le c o m e s to
p o in t w h e r e th e lift is le s s th a n th e s u b m e r ge d w e ight
T h e a c tu a l d is ta n c e a p a r tic le m o v e s is th u s a n do 1ï1
b u t th e a v e r a g e d is ta n c e th a t a p a r tic le m o y e s w aa
d e te r m in e d b y E in s te in to b e A ,
d o r
À d
A ı
d =
P '
(7 35)
w h e r e P is th e p r o b a b ility th a t th e lift fo rc e s , L , are
g r e a te r th a n th e s u b m e r g e d w e ig h t,
W ,
o r
p - P r o b (L W ) (7 36)
u n d e r th e c o n d itio n s w h e r e th e n u m b e r o f p a rtic le s o f
s iz e d b e in g e r o d e d is e q u a l to th e n u m be r be ing
d e p o s ite d ,
E in s te in s h o w e d th a t
w h e r e A is a c o n s ta n t,
is ,
is th e fr a c tio n o f p a rtic ıe s of
s iz e d m o v in g in th e b e d lo a d ,
ib
is th e fr a c tio n of
p a r tic le s o f s iz e d in th e b e d m a te r ia l,
a n d ￠is giv en
b y
w h e r e g .
Is s e d im e n t d is c h a r g e p e r u n it w idth o f size
d , 7 .
Is s p e c ific w e ig h t o f s e d im e n t,
is sp e c ińc w e ight
o f w a te r ,
a n d F is a fa c to r fo r s e ttlin g v e lo c ity giv e n by
(7 39)
w h e r e u is flu id v is c o s ity a n d s G is s e d im e n t speci{ic
g r a v ity T h e p r o b a b ility th a t lift is g r e a te r tha n su
m e r g e d w e ig h t is b a s e d o n flu id d y n a m ic s T he ldt
fo r c e w a s a s s u m e d to b e d e fin e d b y th e v e lo c ity a t the
to p o f a la m in a r s u b la y e r fo r m e d a s a r e s u lt o f drag
d u e to g r a in r o u g h n e s s (s e e c h a p te r 10 ),
o r
L C L p A ,ift
-
C . ;P K ,
d 2
(1 1 6 )
2
g R
'
S ,
(7 40)
w h e r e s is slo p e a n d u b
is th e v e lo c ity a t th e to p o fthe
b e d la y e r , g iv e n b y (s e e c h a p te r 10 )
2
U
2
P
w he r e R '
is łtae hy d r a u ıic r a d iu sK , is a c o e M e n t r e la tin g th e c r o s

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pa rtic ıe to th e s q u a r e o f its d ia m e te r d a n d C L
is th e
co e ffic ie n t o f lift M e th o d s fo r d e te r m in in g R '
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,
E in ste in p ro p o s e d th a t
p -
f( ), (7 4 2 )
w he re ıp is r e fe r r e d to a s th e s h e a r in te n s ity p a r a m e te r
giv e n b y
P . P d
a n d B is a c o n s ta n t
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th a t
=
f(tþ),
(7 4 4 )
w h e r e th e fu n c tio n a l r e la tio n s h ip m u s t b e d e ĥ n e d e x
p e r im e n ta ıly ,
s im ila r ly to F ig 7 2 4
B a s e d o n d a ta ta k e n in a r e s e a r c h n u m e ,
E in s te in
p r o p o s e d th a t
0 4 6 5 e
0 3 9 lılr
(7 4 5 )
In o r d e r to u s e E q (7 4 5 ),
it is n e c e s s a ry to d e fin e R '
v a lu e s fo r a g iv e n w a te r d e p th P r o c e d u r e s fo r m a k in g
th e s e c o m p u ta tio n s a r e g iv e n in C h a p te r 10
2 0 0
1 0 0 .
5 0 17
5 4 3 2
h î i
Ħg u re 7 2 1 H id in g fa c to r fo r u s e in E in s te in
'
s e q u a tio n
E in s te in 19 50 )
d i
X
d
5