Jökull - 01.12.1973, Page 28

M = TAs
(35)
where As is the change o£ entropy of the transi-
tion. For most materials the slope of the Clap-
eyron curve is positive implying that heat is
absorbed when material transforms from the
more dense to the less dense phase and is re-
leased when the transition is in the opposite
direction.
dTi dTn _ qM
dx dx K
Tn = Tiii = Tc (P)
at x =d2 (37)
dTn dTm _ — qM
dx dx K
///////////////////////////////////////////////////^
T=0| />,A,
Pz ^2
PI Al
T= 0

x = 0 x = d( x = L/2 x = d2
x=L
Fig. 6. Strip model for an internally heated
two-phase fluid.
The strip model for a two phase fluid is
shown in Fig. 6. The phase boundaries are
located at di and d2 in the sinking and rising
branches of the cell respectively. The strip is
then divided into three regions. Phase 1 exists
in the section Oíx í di and d2 x :£ L;
whereas phase 2 exists to the section digx
-<d2.
For simplification, we assume that there is
uniform mass flow down the strip such that
Pl u^ = p2u2 = q, and that the heat sources vary
such that Pl = P2‘^2 — p\' Lastly, the
thermal conductivity, specific heat, thermal ex-
pansivity, and kinematic viscosity of the two
phases are assumed equal. Thus
Ki = K2 = K; ci = c2 = c;
oq = a2 = a; v^ = v2 = v-
The heat transport equation (24) must be
solved separately in each section of the strip.
In addition to the boundary conditions at the
ends of the strip, joining conditions are re-
quired at the phase boundaries. They are;
Ti= Tn = Tc(P)
at x = di (36)
where Tc (P) is the transition temperature. The
latent heat M is assumed to be constant and is
defined so that in (36) and (37) M is positive
for a normal phase transition.
The equation of state is modified slightly
because of the phase transition. We have
p = Pl( 1-«T); T=§TC(P)
(38)
p=(Pl +Ap)(l-aT); T^TC(P)
This modification is reflected in the equation
for the total head. In the present case, the
driving head is given by equation (39) where
L — d2 — di = AD is the difference in the
depths at which the phase change occurs in the
ascending and descending limbs of the convec-
tion cell. Equation (39) thus contains an addi-
tional pressure head due to the phase level
difference AD. This head helps drive the con-
vection and thus tends to counteract the nega-
tive effect of the latent heat.
The temperature distribution which satisfies
(24), (36) and (37) is given for segments I, II,
and III by equations (40), (41) and (42), respec-
tively, where T°(x) given by (25) represents the
temperature in a fluid without a phase transi-
tion. By substituting (40), (41), (42) into (39),
the driving head can be determined. Since the
relative displacement of the phase boundaries
will be small, the temperature and density
effects can be treated independently. Hence,
the phase levels will be assumed equal in com-
puting the head H due to the temperature
differences in the fluid. Moreover terms in-
volving the product aAp will be neglected in
(39). The total head is then given by (43) where
now b = cqL/K and b' = cqdi/K.
Inserting (43) into the flow equation (8) and
multiplying both sides by cL/K yields equation
(44). By defining a temperature Ti = pA8L2/K,
26 JÖKULL 23. ÁR