The models developed in Chapter VId to analyze the primary and the secondary sides of a PWR are based on the thermodynamic equilibrium assumption, except for the pressurizer and the secondary side of the steam generator, which were analyzed based on the thermodynamic non-equilibrium model. In the lumped parameter approach, the perfect mixing assumption is used and only one temperature is allocated to a node. Thus, a multi-node representation was required for regions such as the core and the steam generator primary side in which large temperature gradients exist (Figures VId.2.1, VId.3.2, and VId.7.1).
Another approach, originally developed by Myers and employed by Kao, allocates only one node to a region even if there is a large temperature gradient in the region. For example, one node is used to represent the PWR core despite the large temperature rise over the core. Similarly, the tube bundle region of the steam generator with large density gradient is modeled by only one control volume. This is possible by the introduction of the linear enthalpy profile model. In this model a volume-averaged mixture density,
ρ
* is defined as:
Similarly, a volume-averaged mixture enthalpy
h* is defined as:
where subscript m stands for mixture. If the volume-averaged mixture density and enthalpy are known, then the mass and energy of a node can be found from
m
=
ρ*V and
u
=
h*V –
PV, respectively. To find the volume-averaged mixture density and enthalpy in closed form, the mixture density profile in terms of pressure and enthalpy is needed to develop the above integrals.
a) To find such profile, show that at a given pressure, density of saubcooled water decreases almost linearly with increasing enthalpy. Also show that the specific volume of a two-phase mixture and of superheated steam increases linearly with enthalpy.
b) Now consider control volume i, connected to the control volumes
i
– 1 and
I
+ 1. Show that by a linear transformation, the volume-averaged density and enthalpy become functions of pressure and the inlet and exit mixture enthalpies, given by:
c) Using the linear enthalpy profile assumption show that in the single-phase region:
where in this region,
ρm,i
/
hm,i
is a constant. Also show that in the two-phase region:
where in this region, v
m,i
/
hm,i
is a constant.
d) Substitute these profiles in the above integrals to obtain expressions for ρ
i
*
and h
i
*.