Thermal Hydraulics & Thermodynamics · C-TH-003

Natural Circulation — How a Thermal-Center Height Difference Replaces the Pump

Buoyancy between a low heat source and a high heat sink keeps decay heat moving when the reactor coolant pumps stop.

Concept C-TH-003

Definition

Natural circulation is flow around a loop driven, without any pump, by buoyancy: the density difference between coolant heated at the heat source (the core) and coolant cooled at the heat sink (the steam generator) acting in the gravity field. It is sustained only when the thermal center of the heat sink lies above the thermal center of the heat source.

In a PWR it is the basic means of carrying decay heat to the steam generators (SGs) whenever the reactor coolant pumps (RCPs) are stopped: loss of offsite power, station blackout (SBO) and post-accident cooling.

Principle and mechanism

Driving head and thermal centers

Integrating the hydrostatic pressure once around the loop gives the buoyancy driving pressure ΔPb = −g∮ρ dz. If the riser side (core outlet to SG inlet) and the downcomer side (SG outlet to downcomer) are each reduced to a column of effective mean density, this becomes:

ΔPb = (ρc − ρh) · g · ΔH
ρc: effective mean density on the downcomer (cold) side; ρh: effective mean density on the riser (hot) side
ΔH: elevation difference between the SG thermal center and the core thermal center

The thermal center is not the geometric center of a component but the equivalent height at which heat is effectively added or removed, defined as the heat-weighted mean elevation z_tc = ∫z dQ / ∫dQ. For a symmetric cosine axial power shape, the core thermal center is at the mid-height of the effective core. The SG thermal center lies within the U-tube bundle above the tubesheet, shifted toward the region that removes the most heat, and it moves with operating conditions. A driving force exists only if the heat sink is higher than the heat source (ΔH > 0), which is why a PWR places the vessel nozzles above the top of the core and the SG tubesheet above the nozzles.

PWR primary loop elevation layout showing the reactor vessel with the core thermal center below and the steam generator thermal center above, the height difference ΔH between them, riser-side lighter density ρh and downcomer-side heavier density ρc, the stopped RCP and crossover leg, and equation cards for the driving head ΔPb = (ρc − ρh)·g·ΔH, loop friction ΔPf = R·ṁ²/ρ, energy Q = ṁ·cp·ΔT, the result ṁ³ = ρ²βgΔH·Q/(cp·R), and the 20 % ΔH reduction giving ṁ × 0.928 and ΔT × 1.077 — Natural circulation is driven by the height difference ΔH between the core and SG thermal centers. Steady single-phase flow, mean-density columns and constant properties and loss coefficients are assumed; the core thermal center is for a symmetric power shape; component proportions are schematic.

Flow rate versus height difference

At steady state the driving head equals the loop friction. Assume steady single-phase flow with constant properties and loss coefficients.

Friction: ΔPf = R · ṁ²/ρ, with R = Σ(Ki + fi·Li/Di)/(2Ai²) (f is the Darcy friction factor)
Energy: Q = ṁ · cp · ΔT (Q is the heat carried by this loop)
Density difference: ρc − ρh ≈ ρβΔT (β is the volumetric expansion coefficient)

Solving the three together gives ṁ³ = ρ²βgΔH·Q / (cp·R), so:

ṁ ∝ (Q·ΔH)^1/3 and ΔT ∝ Q^2/3 · ΔH^−1/3
If friction is proportional to the flow rate (laminar flow), the exponent becomes 1/2. Plant loops are usually turbulent, so 1/3 is the default; the assumption should always be stated.

Worked example: shorter thermal-center distance. If ΔH falls by 20 % with Q and R unchanged:

ΔT'/ΔT = 0.8^−1/3 = 1.077 → temperature difference rises by about 7.7 %
ṁ'/ṁ = 0.8^1/3 = 0.928 → flow falls by about 7.2 %
Check: 0.928 × 1.077 = 1.000 (same heat removed)
With the laminar assumption: 0.8^−1/2 = 1.118 (about 11.8 % higher)

An example for a sense of scale: assume ΔH 10 m, a loop pressure loss of 0.7 MPa at rated flow, a rated core ΔT of 33 °C, ρβ ≈ 2.4 kg/m³K and decay heat of 2 % (assumed values, not design data). The flow ratio is then f³ = ρβgΔH·q·ΔT₀/ΔP₀ ≈ 2.2×10⁻⁴, giving a flow of about 6 % of rated and a core ΔT of about 11 °C. The actual flow may be lower because of the resistance of the stopped RCP impellers and the higher friction at low Reynolds number.

Three modes as inventory falls

As the coolant inventory decreases, natural circulation changes character.

Single-phase natural circulation: circulation driven by liquid density differences alone.
Two-phase natural circulation: bubbles in the riser make it lighter, so the driving head and the flow actually increase and pass through a peak.
Reflux condensation: with still less inventory, continuous liquid circulation weakens. Steam from the core rises through the hot leg into the U-tubes and condenses; the condensate returns to the core along the hot leg in the opposite direction to the steam (counter-current flow). In B&W plants this is called the boiler-condenser mode. The net loop circulation is nearly zero, but heat is still removed by condensation. The limits are counter-current flow limitation (CCFL) in the hot leg and SG inlet, and accumulation of non-condensable gas in the U-tubes.

In the Semiscale Mod-2A test S-NC-2 (core power 60 kW = 6 % of the facility rating), the loop flow was 0.36 kg/s at 100 % inventory. It peaked at 0.77 kg/s at 93.1 % (about 2.1 times the single-phase value), fell to 0.08 kg/s at 74.4 %, and was zero at 66.3 % and below.

Graph of loop mass flow rate from 0 to 0.8 kg/s against primary coolant inventory from 100 to 60 percent for Semiscale test S-NC-2, rising from 0.36 kg/s at 100 percent to a peak of 0.77 kg/s at 93.1 percent, then falling to 0.08 kg/s at 74.4 percent and zero at 66.3 percent, with bands for single-phase, two-phase and reflux condensation and small sketches of each mode in the hot leg and U-tube — Loop flow versus inventory in the Semiscale Mod-2A test S-NC-2 (core power 60 kW, 6 % of the facility rating; NENE 2021 paper 609, Table 1). The mode bands are illustrative, and inventory values depend on the facility and power.

What it means in a plant

Conditions for establishing and maintaining it, and what defeats it

Condition to establish and maintain	What breaks it (impeding factors)
Heat-sink thermal center above heat-source thermal center	Layout problem (insufficient ΔH)
SG heat sink: secondary-side level (auxiliary feedwater) + steam release (ADV)	SG dryout, loss of feedwater, stagnation in an isolated SG loop
Continuous flow path: no voids in the hot legs or at the top of the U-tubes	Steam voids, accumulation of non-condensable gas
Primary subcooling maintained (pressurizer pressure and level)	Saturation from excessive depressurization, loss of coolant
Even cooling among loops	Loop stagnation or partial U-tube reverse flow from asymmetric cooling
Boron and temperature mixing	Stratification, stagnant hot water in the upper head

Sources of non-condensable gas include dissolved hydrogen in the coolant, nitrogen from the safety injection tanks (SITs, entering after the tanks empty), hydrogen from the zirconium–water reaction, radiolysis, helium from B-10(n,α), hydrazine decomposition (N₂, NH₃), dissolved air in injected water, fission gas and helium from failed fuel, and air left after maintenance.

Confirming indications and operator actions

Natural circulation is judged established from the following indications (a concept shared by the natural-circulation verification attachment common to Westinghouse ERG procedures such as ES-0.1, ES-0.2 and ES-1.2; numbers are plant-specific):

RCS subcooling margin at or above the procedural criterion
SG pressure stable or decreasing
Hot-leg temperature (Th) and core exit temperature (CET) stable or decreasing
Cold-leg temperature (Tc) near the saturation temperature for SG pressure (primary–secondary coupling)
Core ΔT smaller than the full-power ΔT and stable. If ΔT keeps rising, check auxiliary feedwater and steam release first.

Operator actions: maintain SG narrow-range level with auxiliary feedwater and control the cooldown rate with the ADVs; maintain subcooling with pressurizer pressure and level; run the CRDM cooling fans and limit the cooldown and depressurization rates to prevent an upper-head bubble; before cooldown, raise boron to the cold-shutdown concentration, allowing for the slow mixing time under natural circulation.

Upper-head bubbles and how to detect them

In 1980, at St. Lucie Unit 1, depressurization during natural-circulation cooling flashed stagnant hot water in the upper head, and the pressurizer level rose abnormally fast. The event led to NRC GL 81-21. Means of detecting a bubble:

Reactor vessel level monitoring system (RVLMS; in CE-type plants, heated-junction thermocouples, HJTC, part of the ICCMS)
A rapid pressurizer level rise during depressurization that charging flow cannot explain
Comparison of CET and upper-head temperatures with the saturation temperature at RCS pressure
A mismatch in the charging–letdown balance

LOCA, SBO and natural circulation

LBLOCA: the RCS depressurizes quickly to containment pressure, and the core is cooled by ECCS injection and steam venting through the break. The SG secondary side is actually hotter and cannot act as a heat sink.
SBLOCA: RCS pressure stays near SG secondary pressure for a long time, so the SGs are the main heat sink. Decay heat is removed in the sequence single-phase → two-phase → reflux, so natural circulation is decisive.
Saturated coolant with an upper-head bubble: the upper head is outside the main circulation path, so natural circulation is still possible. If the bubble spreads into the hot legs and U-tubes, however, the plant moves into reflux and pressure control becomes difficult, so subcooling must be restored.
SBO: in a typical PWR, the turbine-driven auxiliary feedwater pump supplies water from a feedwater storage tank to the SGs, and steam is released through the main steam atmospheric dump valves (MSADVs).

Passive design examples

AP1000 uses primary-side natural circulation through the PRHR heat exchanger in the IRWST. Secondary-side passive systems, such as the PRHRS of the SMART integral reactor design, use SG secondary-side natural circulation to reject heat to condensation tanks. NuScale relies on natural circulation for normal operation itself.

Regulatory and design criteria

For example, in the United States the following criteria apply:

Basis	Key point
10 CFR 50 App. A, GDC 34	Requirements for the residual heat removal system
SRP 5.4.7, BTP 5-4 (formerly RSB 5-1)	Cold shutdown using only safety-grade systems; for a PWR, boron mixing and cooldown under natural circulation are tested or compared with tests of a similar design
RG 1.68 App. A (A-4.t)	Low-power natural-circulation test (heat removal, boron mixing, cooldown, depressurization) or comparison with prototype-plant hydraulic data
NUREG-0737 I.G.1	Post-TMI: natural-circulation training during low-power testing
NRC GL 81-21 (1981)	Evaluation of natural-circulation cooldown procedures and training; demonstration of cooldown without vessel voiding
APR1400 DCD 14.2.12.4.22	Natural-circulation test (responding to RG 1.68)

A first-of-a-kind (FOAK) plant performs the natural-circulation test; follow-on units may substitute earlier test results plus a comparison of RCS hydraulic loss coefficients. The framework of GL 81-21 and BTP 5-4 remains unchanged; in advanced reactors, passive natural-circulation systems (such as PRHR and PRHRS) have in addition become design requirements.

Easily confused pairs

	Single-phase natural circulation	Two-phase natural circulation	Reflux condensation	Forced circulation
Driver	Liquid density difference	Density difference enlarged by bubbles	Steam rising, condensate falling	RCP head
Loop flow	A few % of rated	Larger than single phase (peak)	Net circulation ≈ 0	100 %
Typical situation	LOOP or SBO, normal inventory	Early SBLOCA	SBLOCA with reduced inventory	Normal operation
Weak point	Voids, non-condensable gas	Further inventory loss	CCFL, non-condensable gas	Loss of power

Key points at a glance

Definition: buoyancy-driven circulation, without pumps, produced by the source–sink density difference and gravity.
Driving head: ΔPb = (ρc − ρh)·g·ΔH, with ΔH = SG thermal center − core thermal center > 0.
Thermal center: heat-weighted mean elevation z = ∫z dQ / ∫dQ.
Flow relation: ṁ³ = ρ²βgΔH·Q/(cp·R) → ṁ ∝ (QΔH)^1/3, ΔT ∝ Q^2/3·ΔH^−1/3 (turbulent friction; the exponent becomes 1/2 for laminar friction).
ΔH reduced by 20 %: ΔT up about 7.7 % (not 20 %), flow down about 7 %.
Modes: single phase → two phase (flow peak) → reflux condensation (limited by CCFL).
Conditions: ΔH > 0, SG heat sink (auxiliary feedwater, ADV), no voids or non-condensable gas, subcooling maintained.
Confirmation: subcooling margin; SG pressure, Th and CET stable or decreasing; Tc ≈ SG saturation temperature.
Basis: GDC 34, BTP 5-4, RG 1.68 natural-circulation test, GL 81-21.

Sketch of the reactor vessel and core at left and the steam generator with U-tubes at right, connected by the hot leg with density ρh and the cold leg with density ρc through a stopped RCP and loop seal, with dashed lines at the core and SG thermal centers and the height difference ΔH, and the equations ΔPb = (ρc − ρh)·g·ΔH and ṁ ∝ (Q·ΔH)^1/3 — Sketch: loop elevation layout and the thermal-center height difference ΔH.