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Problem 1 (30%) – Hydraulic analysis of the PWR primary system at cold zero-
power conditions
A greatly-simplified schematic of the PWR primary system is shown in Figure 1. The
core and steam generators are represented by two form losses of coefficients 7 and 4,
respectively. The loop can be modeled as a series of four identical round tubes of 1.45 m
ID and 10 m length. The flow within the loop is driven by a pump that delivers a
constant head, ΔPpump=200 kPa, regardless of the flow.
Figure 1. Simplified schematic of the PWR primary system

You are to evaluate the hydraulic behavior of the system at cold zero-power conditions. In
this situation the fluid can be considered isothermal at 20°C and atmospheric pressure.
The properties of water at this temperature and pressure are reported in Table 1.
i) Calculate the steady-state mass flow rate in the system. Clearly state all your
assumptions. (10%)
ii) Now consider flow start-up from stagnant conditions. At t=0 the pump is turned on
and the flow is established. Calculate the time it takes for the mass flow rate to reach
50% of its steady-state value. (15%) (Hint: use the following integral
iii) A nuclear engineer wishes to simulate the PWR primary system by means of an
experimental flow loop with the same form coefficients and geometrically similar, but
of 1/10 scale (the pump head is also scaled down to 1/10). Would such loop have the
same time constant of the PWR primary system? (5%)
Assumptions:
- Neglect the acceleration and friction terms (Facc and Ffric, respectively) in the
momentum equation.
Table 1. Water properties at 20°C.

Problem 2 (25%) – Surface tension effects in borated water draining from a BWR
Standby Liquid Control Tank.
BWRs have a Standby Liquid Control Tank (SLCT) containing highly-borated water at
room temperature that can be injected into the core, should the control rods fail to
shutdown the reactor during an accident. Over a long period of time, borated water
corrosion has created a small round hole of 0.5 mm diameter on the bottom of the
SLCT (Figure 2a). The contact angle between borated water and the SLCT material is θ
= 120°. The surface tension of borated water at room temperature is 0.07 N/m, and its
density is about 1,000 kg/m3 . The initial liquid level in the SLCT is 1 m.

i) Assuming that the SLCT top is open to the atmosphere, would you expect the
borated water to completely drain from the hole? (10%)
If so, explain why.
If not, calculate the level at which draining would stop.
ii) Now assume that the contact angle is 60°. Does the tank drain completely? Explain.
(5%)

iii) To prevent draining, a fellow MIT nuclear engineering student suggests sealing the
tank top and put a cover gas (Figure 2b). Would this in fact prevent draining? Does the
contact angle affect your answer? (10%)
Problem 3 (25%) – Flow split between a heated and an adiabatic channel.
Consider the two parallel channels shown in Figure 3. They are connected only at the
inlet and outlet plena, and both have flow area A, equivalent diameter De and length L.
Channel 1 is heated ( is the total heat rate), while channel 2 is adiabatic. Channel 1 has
an orifice at the inlet (of form loss coefficient K). The boundary conditions are as
follows:
- The inlet plenum temperature is To
-The total mass flow rate is mtot &
- The outlet plenum pressure is PL
The fluid specific heat and thermal expansion coefficient are cp and β, respectively. The
density of the fluid can be calculated by means of the Boussinesq approximation with To
and ρo as the reference temperature and density, respectively.

Figure 3. Parallel channels connected at plena.
i) Find an expression for the mass flow rate in channel 1 in terms of the heat rate,
geometry and properties only. (15%) (Hint: assume steady-state upflow in both
channels)
ii) Find an expression for at which the mass flow rate in channel 2 becomes zero. (5%)
Q&
iii) What happens to the flow in channel 2, if the heat rate in channel 1 is increased
beyond the threshold calculated in “ii”? (5%) (Note: provide only a qualitative
answer)

Assumptions:
- Heating in channel 1 is axially uniform.
- Assume single-phase flow in the system.
- Neglect acceleration and friction terms in both channels.
- All thermophysical properties (except density) can be considered independent of
temperature.
Problem 4 (20%) – Quenching experiments to simulate boiling heat transfer
during a LB-LOCA.
To simulate boiling heat transfer on the surface of the fuel pins during a Large-Break
Loss Of Coolant Accident (LB-LOCA) in a PWR, a nuclear engineer has designed a
very simple quenching experiment, in which a small copper sphere (∼1 cm diameter) is
heated up to very high temperatures (∼1,000°C), and then dropped in a large pool of
water at atmospheric pressure.
i) What are the differences between the experiment and the actual reactor situation
that are likely to have an effect on boiling heat transfer? (5%)
ii) Write the energy conservation equation describing the temperature history (T vs. t) of
the copper sphere during a quenching experiment? (5%) (Hint: neglect the temperature
gradient within the sphere, describe boiling heat transfer at the surface of the sphere by
means of a heat transfer coefficient, and assume that the water bulk is saturated)

iii) The boiling curve for the experimental conditions is shown in Figure 4. Provide a
qualitative sketch of the sphere temperature history for an initial temperature of
1,500°C. (10%)
Figure 4. Boiling curve for a sphere in saturated water at 1 atm.

Problem 1 (30%) – Hydraulic analysis of the PWR primary system at cold zero-
power conditions
i) The momentum equation for the loop is:
where m is the mass flow rate, L=40 m is the total length of the loop, A=1.65 m2 is the
flow area, Kcore=7 and Ksg=4 are the form loss coefficients for the core and steam
generator, respectively. The acceleration and friction terms were neglected in Equation 1,
as per the problem statement. Moreover the gravity term is zero because the fluid is
isothermal.
At steady-state and Equation 1 can be easily solved for the steady-state mass
flow rate, mss:
ii) Equation 1 can be re-written as follows:

Equation 3 can be integrated to find m(t) during start-up. Separating the variables, making
use of the hint in the problem statement, and setting the initial condition m(0)=0, one gets:
where the time constant, τ, is defined as follows:
Equation 4 is plotted in Figure 1. The time it takes to reach 50% of the steady-state value
can be calculated by setting m=0.5⋅mss in Equation 4, and solving for t.

Figure 1. PWR primary system mass flow rate during cold zero-power start-up.
iii) Equation 5 indicates that the time constant is proportional to the loop length and
inversely proportional to the square root of the pump head. Thus, it can be concluded that
the time constant for the scaled-down loop will be lower than for the actual PWR primary
system by a factor of √10≈3.16.
Problem 2 (25%) – Surface tension effects in borated water draining from a BWR
Standby Liquid Control Tank.

i) The water pressure at the bottom of the tank, Pℓ, can be calculated as follows:
where Patm is the atmospheric pressure, ρℓ is the borated water density and L is the level
in the tank. Let us now focus on the liquid/air interface at the hole. For a contact angle
>90°, the effect of surface tension is to oppose draining. The condition for static
equilibrium (i.e., no draining) is:
where σ is the surface tension and r is the radius of curvature, which can be derived from
simple geometric considerations:
with dH=0.5 mm. Combining Equations 7 and 8, one gets the maximum level of borated
water that can be held up by the surface tension in the hole, Lmax:
Since the initial level is higher than Lmax, the borated water will drain until L=Lmax.

ii) If the contact angle is <90°, the tank will drain completely because surface tension no
longer opposes draining.
iii) If the tank top is sealed and there is a cover gas, the borated water will drain until the
cover gas pressure, Pcg, becomes sufficiently low. The condition for static equilibrium
is:
where the positive sign on the right-hand side applies to contact angles >90° and the
negative sign to contact angles <90°. Thus, the contact angle will affect the equilibrium
pressure of the cover gas, but at a certain point draining will stop regardless of the value
of the contact angle.
Problem 3 (25%) – Flow split between a heated and an adiabatic channel.
i) The mass equation for the system is:
where m1 and m2 are the mass flow rate in channels 1 and 2, respectively. The energy
equations are:

where T1L and T2L are the temperature at the outlet of channels 1 and 2, respectively.
The momentum equations are:
where Po is the inlet plenum pressure. Equations 12 through 16 are 5 equations in the 5
unknowns m& 1 , m& 2 , T1L, T2L and Po. Substituting Equation 13 into Equation 15,
eliminating Po from Equations 15 and 16, and solving for , one gets:

iii) If the heat rate in channel 1 is increased beyond , the flow in channel 2 actually
reverses. Explanation: in this system the column weight in channel 2 sets the pressure drop
for both channels (see Equation 16). Focus now on channel 1. Because of the heating, the
column weight in channel 1 is lower than the total pressure drop (Equation 15). So in
general, channel 1 will have higher flow rate than channel 2. When the heating is so high
that the flow rate in channel 1 is higher than the total flow rate m Qo & tot, the flow in
channel 2 has to reverse to satisfy continuity (Equation 12).
Problem 4 (20%) – Quenching experiments to simulate boiling heat transfer during a
LBLOCA.
i) The main differences are geometry (spherical vs. cylindrical) and materials (copper vs.
zirconium). Geometry differences will have an effect mostly on film boiling and DNB.
Materials differences will have an effect mostly on nucleate boiling. Because of
geometry, size and materials differences, the experiment and reactor situation will also
have different thermal capacities, and thus different time scales.
ii) The energy balance for the sphere is:

where ρ and Cp are the copper density and specific heat, respectively, T, V and S are the
sphere temperature, volume and surface, respectively, q" is the heat flux at the surface, h
is the heat transfer coefficient, and Tsat is the saturation temperature of water.
iii) The qualitative sketch of the sphere temperature history for an initial temperature of
1,500°C is shown in Figure 2. The sphere goes through all heat transfer regimes,
including transition boiling, because the situation is temperature controlled, not heat-flux
controlled. Note that the film boiling region has the longest duration because of its large
temperature width. The concavity of the T-t curve can be determined by differentiating
Equation 19:
Since the term is obviously positive, the concavity depends only on the
derivative of the heat flux with respect to temperature. Thus, the concavity is positive
for film boiling, nucleate boiling and natural convection, but is negative for transition
boiling.

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