##### Document Text Contents

Page 1

CHALMERS

NOISE APPLICATIONS IN LIGHT WATER REACTORS

WITH TRAVELING PERTURBATIONS

VICTOR DYKIN

Division of Nuclear Engineering

Department of Applied Physics

CHALMERS UNIVERSITY OF TECHNOLOGY

Göteborg, Sweden 2012

Page 41

3.1. Modeling of global and regional instabilities

application of the variational principle, the following reduced differential equations for

Ti(t), describing the fuel rod conduction dynamics, are obtained:

dT1,l,jϕ(t)

dt

= p11,jϕT1,l,jϕ(t) + p21,jϕT2,l,jϕ(t) + p31,jϕcq

2∑

i=0

ξi(Pi(t)− P̃i), (3.12)

dT2,l,jϕ(t)

dt

= p12,jϕT1,l,jϕ(t) + p22,jϕT2,l,jϕ(t) + p32,jϕcq

2∑

i=0

ξi(Pi(t)− P̃i), (3.13)

where pij are complicated coefficients which depend on the design and operational pa-

rameters, jϕ stands for the single- (1ϕ) or two-phase (2ϕ) regions, l is the channel number

between 1 and 4, and P̃0 is the steady state value of the fundamental mode.

3.1.3 Thermo-hydraulic model

In this Section the description of the thermal-hydraulic model for our ROM is given.

Since there are two axial coolant regions assumed in the channel, namely single-phase

and two-phase regions, with a constant flow cross section, the description is performed

in two separate sections, respectively. Within the scope of this Section, the procedure

to transform the PDEs, describing thermal-hydraulic processes, into simplified ODEs,

applying the variational method, is demonstrated.

Single-phase region

One starts with three local conservation equations written for mass, momentum and en-

ergy, respectively, as [20, 21]:

∂ρ∗(r̄∗, t∗)

∂t∗

+ ∇̄∗ · (ρ∗v̄∗)(r̄∗, t∗) = 0, (3.14)

∂(ρ∗v̄∗)

∂t∗

(r̄∗, t∗) + ∇̄∗ · (ρ∗v̄∗ ⊗ v̄∗)(r̄∗, t∗) =

∇̄∗ · ¯̄τ∗(r̄∗, t∗)− ∇̄∗ · (P ∗(r̄∗, t∗) ¯̄I) + ρ∗(r̄∗, t∗)ḡ∗, (3.15)

∂(ρ∗e∗)(r̄∗, t∗)

∂t∗

+ ∇̄∗ · (ρ∗e∗v̄∗)(r̄∗, t∗) = −∇̄∗ · q̄∗

′′

(r̄∗, t∗)

+q̄∗

′′′

(r̄∗, t∗) + ∇̄∗ · (¯̄τ∗ · v̄∗)(r̄∗, t∗)− ∇̄∗ · (P ∗v̄∗)(r̄∗, t∗) + (ρ∗ḡ∗ · v̄∗)(r̄∗, t∗), (3.16)

where ⊗ stands for the tensor multiplication and ¯̄I is the unit tensor.

Further, assuming the coolant flow mainly in the axial direction (i.e. neglecting the

radial flow), the time-dependent single-phase enthalpy h(z, t) can be expressed with a

second order polynomial as:

h(z, t) ≈ h2(z, t) = h(0, t) +

2∑

i=1

pi(t)z

i. (3.17)

Then, rewriting the energy balance equation in terms of enthalpy, after cross-section

averaging, the following dimensionless ODEs can be derived for the corresponding en-

thalpy time-dependent expansion coefficients pi(t) for each of four heated channels:

dp1,l(t)

dt

=

6

µl(t)

[NρNrNpch,1ϕ,l(t)− vinlet,l(t)p1,n(t)]− 2vinlet,l(t)p2,l(t), (3.18)

29

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Chapter 3. METHODS FOR MODELING THE EFFECT OF PROPAGATING

PERTURBATION IN CLOSED-LOOP SYSTEMS

dp2,l(t)

dt

= �

6

µ2l (t)

[NρNrNpch,1ϕ,l(t) � vinlet,l(t)p1,l(t)], (3.19)

where Nr and Nρ are dimensionless numbers, Fr is the Froude number and Npch,1ϕ(t) is

the so-called time dependent phase change number in the single-phase region which is

proportional to the wall heat flux q∗

′′

1ϕ and µ(t) is the boiling boundary (the axial elevation

in the reactor core where boiling starts).

Two-phase region

Following the procedure as in the single-phase region, one starts with the three local

conservation equations written for mass, momentum and energy for each coolant phase

region, respectively, as [20, 21]:

∂ρ∗k(r̄

∗, t∗)

∂t∗

+ r̄ ∗ �(ρ∗kv̄

∗

k)(r̄

∗, t∗) = 0, (3.20)

∂(ρ∗kv̄

∗

k)

∂t∗

(r̄∗, t∗) + r̄ ∗ �(ρ∗kv̄

∗

k

v̄

∗

k)(r̄

∗, t∗) =

r̄ ∗ �̄̄τ∗(r̄∗, t∗) � r̄ ∗ �(P ∗k (r̄

∗, t∗) ¯̄I) + ρ∗k(r̄

∗, t∗)ḡ∗, (3.21)

ρ∗(r̄∗, t∗)

∂h∗(r̄∗, t∗)

∂t∗

+ (ρ∗v̄∗)(r̄∗, t∗) �r̄ ∗ �h∗(r̄∗, t∗) =

�r̄ ∗ �̄q∗

′′

(r̄∗, t∗) + q∗

′′′

(r̄∗, t∗) + ¯̄τ∗(r̄∗, t∗) : [r̄ ∗

v̄∗(r̄∗, t∗)]

∂P ∗(r̄∗, t∗)

∂t∗

+

+v̄∗(r̄∗, t∗) �r̄ ∗P ∗(r̄∗, t∗), (3.22)

Here, k = l, v stands for the coolant phase: l for the liquid phase and v for the vapor

phase. Further, performing a radial space-averaging on the entire cross-sectional flow

area, assuming that both phases are in thermal equilibrium (homogeneous equilibrium

model), applying the mixture model and replacing the time-dependent flow quality with

the following second order polynomial profile:

x(z, t) � x2(z, t) = NρNr(d1(t)(z � µ(t)) + d2(t)(z � µ(t))2). (3.23)

and implementing the variational method to the resulting equations, after some rear-

rangements one gets the following dimensionless ODEs for the corresponding quality

time-dependent expansion coefficients di(t) for each of the four channels:

dd1,l(t)

dt

=

1

f2,l(t)

(f3,l(t)f1,l(t) + f4,l(t)), (3.24)

dd2,l(t)

dt

=

1

f5,l(t)

(f3,l(t)f1,l(t) + f6,l(t)). (3.25)

In the above, fi(t), i = 1, ..., 6 are complicated functions of time, depending on the design

and operational parameters, as well as phase variables, i.e. inlet velocity vinlet(t), pellet

temperature time-dependent coefficients Ti(t) , i = 1, 2, phase change number Npch,2ϕ,

mixture density ρm(z, t) and boiling boundaries µ(t).

The remaining two equations (3.15) and (3.21), written for the single- and two-phase

pressure drops, are used to derive the ODEs for the inlet velocity, using the pressure drop

balance and are not given here.

30

Page 81

Papers I-VIII

69

CHALMERS

NOISE APPLICATIONS IN LIGHT WATER REACTORS

WITH TRAVELING PERTURBATIONS

VICTOR DYKIN

Division of Nuclear Engineering

Department of Applied Physics

CHALMERS UNIVERSITY OF TECHNOLOGY

Göteborg, Sweden 2012

Page 41

3.1. Modeling of global and regional instabilities

application of the variational principle, the following reduced differential equations for

Ti(t), describing the fuel rod conduction dynamics, are obtained:

dT1,l,jϕ(t)

dt

= p11,jϕT1,l,jϕ(t) + p21,jϕT2,l,jϕ(t) + p31,jϕcq

2∑

i=0

ξi(Pi(t)− P̃i), (3.12)

dT2,l,jϕ(t)

dt

= p12,jϕT1,l,jϕ(t) + p22,jϕT2,l,jϕ(t) + p32,jϕcq

2∑

i=0

ξi(Pi(t)− P̃i), (3.13)

where pij are complicated coefficients which depend on the design and operational pa-

rameters, jϕ stands for the single- (1ϕ) or two-phase (2ϕ) regions, l is the channel number

between 1 and 4, and P̃0 is the steady state value of the fundamental mode.

3.1.3 Thermo-hydraulic model

In this Section the description of the thermal-hydraulic model for our ROM is given.

Since there are two axial coolant regions assumed in the channel, namely single-phase

and two-phase regions, with a constant flow cross section, the description is performed

in two separate sections, respectively. Within the scope of this Section, the procedure

to transform the PDEs, describing thermal-hydraulic processes, into simplified ODEs,

applying the variational method, is demonstrated.

Single-phase region

One starts with three local conservation equations written for mass, momentum and en-

ergy, respectively, as [20, 21]:

∂ρ∗(r̄∗, t∗)

∂t∗

+ ∇̄∗ · (ρ∗v̄∗)(r̄∗, t∗) = 0, (3.14)

∂(ρ∗v̄∗)

∂t∗

(r̄∗, t∗) + ∇̄∗ · (ρ∗v̄∗ ⊗ v̄∗)(r̄∗, t∗) =

∇̄∗ · ¯̄τ∗(r̄∗, t∗)− ∇̄∗ · (P ∗(r̄∗, t∗) ¯̄I) + ρ∗(r̄∗, t∗)ḡ∗, (3.15)

∂(ρ∗e∗)(r̄∗, t∗)

∂t∗

+ ∇̄∗ · (ρ∗e∗v̄∗)(r̄∗, t∗) = −∇̄∗ · q̄∗

′′

(r̄∗, t∗)

+q̄∗

′′′

(r̄∗, t∗) + ∇̄∗ · (¯̄τ∗ · v̄∗)(r̄∗, t∗)− ∇̄∗ · (P ∗v̄∗)(r̄∗, t∗) + (ρ∗ḡ∗ · v̄∗)(r̄∗, t∗), (3.16)

where ⊗ stands for the tensor multiplication and ¯̄I is the unit tensor.

Further, assuming the coolant flow mainly in the axial direction (i.e. neglecting the

radial flow), the time-dependent single-phase enthalpy h(z, t) can be expressed with a

second order polynomial as:

h(z, t) ≈ h2(z, t) = h(0, t) +

2∑

i=1

pi(t)z

i. (3.17)

Then, rewriting the energy balance equation in terms of enthalpy, after cross-section

averaging, the following dimensionless ODEs can be derived for the corresponding en-

thalpy time-dependent expansion coefficients pi(t) for each of four heated channels:

dp1,l(t)

dt

=

6

µl(t)

[NρNrNpch,1ϕ,l(t)− vinlet,l(t)p1,n(t)]− 2vinlet,l(t)p2,l(t), (3.18)

29

Page 42

Chapter 3. METHODS FOR MODELING THE EFFECT OF PROPAGATING

PERTURBATION IN CLOSED-LOOP SYSTEMS

dp2,l(t)

dt

= �

6

µ2l (t)

[NρNrNpch,1ϕ,l(t) � vinlet,l(t)p1,l(t)], (3.19)

where Nr and Nρ are dimensionless numbers, Fr is the Froude number and Npch,1ϕ(t) is

the so-called time dependent phase change number in the single-phase region which is

proportional to the wall heat flux q∗

′′

1ϕ and µ(t) is the boiling boundary (the axial elevation

in the reactor core where boiling starts).

Two-phase region

Following the procedure as in the single-phase region, one starts with the three local

conservation equations written for mass, momentum and energy for each coolant phase

region, respectively, as [20, 21]:

∂ρ∗k(r̄

∗, t∗)

∂t∗

+ r̄ ∗ �(ρ∗kv̄

∗

k)(r̄

∗, t∗) = 0, (3.20)

∂(ρ∗kv̄

∗

k)

∂t∗

(r̄∗, t∗) + r̄ ∗ �(ρ∗kv̄

∗

k

v̄

∗

k)(r̄

∗, t∗) =

r̄ ∗ �̄̄τ∗(r̄∗, t∗) � r̄ ∗ �(P ∗k (r̄

∗, t∗) ¯̄I) + ρ∗k(r̄

∗, t∗)ḡ∗, (3.21)

ρ∗(r̄∗, t∗)

∂h∗(r̄∗, t∗)

∂t∗

+ (ρ∗v̄∗)(r̄∗, t∗) �r̄ ∗ �h∗(r̄∗, t∗) =

�r̄ ∗ �̄q∗

′′

(r̄∗, t∗) + q∗

′′′

(r̄∗, t∗) + ¯̄τ∗(r̄∗, t∗) : [r̄ ∗

v̄∗(r̄∗, t∗)]

∂P ∗(r̄∗, t∗)

∂t∗

+

+v̄∗(r̄∗, t∗) �r̄ ∗P ∗(r̄∗, t∗), (3.22)

Here, k = l, v stands for the coolant phase: l for the liquid phase and v for the vapor

phase. Further, performing a radial space-averaging on the entire cross-sectional flow

area, assuming that both phases are in thermal equilibrium (homogeneous equilibrium

model), applying the mixture model and replacing the time-dependent flow quality with

the following second order polynomial profile:

x(z, t) � x2(z, t) = NρNr(d1(t)(z � µ(t)) + d2(t)(z � µ(t))2). (3.23)

and implementing the variational method to the resulting equations, after some rear-

rangements one gets the following dimensionless ODEs for the corresponding quality

time-dependent expansion coefficients di(t) for each of the four channels:

dd1,l(t)

dt

=

1

f2,l(t)

(f3,l(t)f1,l(t) + f4,l(t)), (3.24)

dd2,l(t)

dt

=

1

f5,l(t)

(f3,l(t)f1,l(t) + f6,l(t)). (3.25)

In the above, fi(t), i = 1, ..., 6 are complicated functions of time, depending on the design

and operational parameters, as well as phase variables, i.e. inlet velocity vinlet(t), pellet

temperature time-dependent coefficients Ti(t) , i = 1, 2, phase change number Npch,2ϕ,

mixture density ρm(z, t) and boiling boundaries µ(t).

The remaining two equations (3.15) and (3.21), written for the single- and two-phase

pressure drops, are used to derive the ODEs for the inlet velocity, using the pressure drop

balance and are not given here.

30

Page 81

Papers I-VIII

69