Jökull - 01.12.1968, Page 28

X
T# ~
D
V.
D ’
T
T?'
D
where vm is the mean velocity and T0 is a re-
presentative temperature. With this equation
(9) becomes:
3T* VI
3t* F
1 / D \ \ 3T*
-in í--z* 1+8.48) —
dz*
x \/ -
1 1
P R + P,F
■ z* (1 - z*)
3T*
3z*
(11)
Here the following dimensionless groups have
been introduced:
The Froude number: F =
The Prandtl number: P :
V gD
v
k/yc
where v is the kinematic viscosity of the water.
— The Prandtl number depends only on the
properties of the water.
The Reynolds number: R:
D ■ v„
The boundary condition at the surface be-
comes with dimensionless quantities:
3Tj
3z*
S D
~k T~
- = N.
As equation (9), (or (11)), is rather complic-
ated we seek a more simple expression for
practical applications. Upon integrating equa-
tion (9) with respect to z, from z = 0 to z = D,
we obtain:
ru 3 i r
i, *ál+f
r k 3T 1 Z = D
L yc 3z J z=o
D 3T
v(z) ■ —■— dz
ÖX
(13)
In swift rivers it is known from measure-
ments that the variation of T with z is very
small (except in a thin layer close to the sur-
face if the heat exchange with the air is great).
In this case we can then assume 3T/3t and
3T/3x to be constant over the depth. With
this assumption and neglecting the heat ex-
change with the bottom, equation (13) becomes:
3T
3t
3T
3x
ycD
(14)
With dimensionless quantities we obtain for
(14):
3T*
3t*
3T*
3x*
= N.
1
r/
(15
Here the numbers Pt and F are not present
as we did not take the velocity distribution
into account.
For constant. S the general solution to (14) is:
(16)
Ns is the dimensionless Nusselt number at the
surface. The boundary conditions at the bottom
will be corresponding, we will denote these by
Nb.
From this we see that with our assumptions
the water temperature is a function of several
dimensionless groups:
T = f (•/, I, D/z0, F, Pt, P, R, Ns, Nb).
<f) (x — vt, T +---------) -- 0
V ycT)J
where <J> is an arbitrary function.
The solution can also by written:
T (x, t) = cp (x - vt) - —^ (17)
which for the characteristics x — vt = a = con-
stant gives:
a<o; T (x,t) = T (o, — a/v) — St/ycD. (18)
a > o; T (x,t) = T (a, o) — St/ycD. (18a)
In the stationary case 3T/3t = 0 and
Sx
T(x) = T(o) -
7cDvn
(19)
362 JÖKULL 18. ÁR