Research Article A Computing Approach with the Heat-Loss ...downloads.hindawi.com/journals/stni/2014/769346.pdf · e Nusselt number for natural convection heat transfer from a horizontal

Research ArticleA Computing Approach with the Heat-Loss Model for theTransient Analysis of Liquid Metal Natural Circulation Loop

Daogang Lu, Xun Zhang, and Chao Guo

Nuclear Science and Engineering School, North China Electric Power University, Changping District, Beijing 102206, China

Correspondence should be addressed to Xun Zhang; [email protected]

Received 10 May 2014; Accepted 10 August 2014; Published 28 August 2014

Academic Editor: Eugenijus Uspuras

Copyright © 2014 Daogang Lu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The transient behaviors of natural circulation loop (NCL) are important for the system reliability under postulated accidents.The heat loss and structure thermal inertia may influence the transient behaviors of NCL greatly, so a transient analysis modelwith consideration of heat loss was developed based on the MATLAB/Simulink to predict the thermal-hydraulic characteristic ofliquid metal NCL. The transient processes including the start-up, the loss of pump, and the shutdown of thermal-hydraulic ADSlead bismuth loop (TALL) experimental facility were simulated by using the model. A good agreement is obtained to validate thetransient model. The appended structure would provide significant thermal inertia and flatten the temperature distribution in thetransients. The oscillations of temperature and flow rate are also weakened. The temperature difference between hot leg and coldleg would increase with the decrease of heat loss, so the flow rate increases as well. However, a significant increase of hot sectiontemperature may cause a failure of facility integrity due to the decrease of heat loss. Hence, the full power of the core tank may alsobe limited.

1. Introduction

Natural circulation is physical phenomena that occur in agravity environment when a geometrically distinct heat sinkand heat source are connected through a flow path. In NCLthe fluid motion is generated by density differences in thefluid due to the temperature gradients or the phase change,which also means that no external sources of mechanicalenergy for the fluid motion are involved.Therefore, the natu-ral circulation principles are very attractive to the designersof nuclear power plant systems and components; not onlycould the system cost be reduced but also the reliability maybe enhanced.

Liquid metals such as lead (Pb), lead bismuth eutectics(LBE), sodium (Na), and sodium-potassium alloy (NaK)are candidate coolants for the next generation liquid metalreactors [1–5]. In general, a high mass flow rate could beprovided in liquid metals natural circulation loops due totheir significant thermal expansion coefficient which makeit possible to use natural circulation based primary coolantsystems and natural circulation based residual heat removalsystems in the design of nuclear power plants. We also

proposed a conceptual design of a small modular natu-ral circulation liquid metal fast reactor with alkali-metalthermal-to-electric conversion (AMTEC) units and provideda straightforward comparison of steady state natural circu-lation characteristics for different liquid metals in previousstudies [6]. On the other hand, the transient behaviors ofNCL would be more important because it can influencethe system performance under postulated accidents directly.Unlike the forced circulation, the operation of NCL may notbe easily controlled. The transient behaviors of NCL stronglydepend on the interaction between buoyancy and frictionalforces, so the performance of NCL may be influenced moresignificantly by other factors such as flow resistance, thermalinertia, and environmental heat loss.

Most of the experimental investigations and theoreticalanalysis methods have been carried out for symmetricaland asymmetric (determined by the relative position ofheater and cooler) water NCL. Vijayan et al. investigatedthe steady state and stability characteristics of single-phasenatural circulation in a rectangular loop with different heaterand cooler orientations experimentally and theoretically [7].Basu et al. theoretically investigated the dynamic response

Hindawi Publishing CorporationScience and Technology of Nuclear InstallationsVolume 2014, Article ID 769346, 12 pageshttp://dx.doi.org/10.1155/2014/769346

2 Science and Technology of Nuclear Installations

of a single-phase rectangular natural circulation loop todifferent excitations of input power [8]. Misale and Frogheristudied the influence of pressure drop on the transient behav-ior and stability of water experimental NCL by changing thediameter of orifices in the loop [9]. Besides, some computercodes, such as RELAP5 system code, could be applied tosimulate the transient behavior of the water NCL, and thecalculations could provide results which agree well with theexperimental data [10–12].

Some investigations are also carried out for the liquidmetal NCL. Ma et al. experimentally investigated the capa-bility and stability of TALL experimental facility and usedRELAP5 to estimate the natural circulation flow rate [13].Yiqiong used TRAC/AAA code to build a more detailedmodel to simulate the transient behavior of the loop. A goodagreement between simulation results and experimental datahad been obtained [14]. Abanades and Pena used compu-tational fluid dynamic codes to analyze the free convectioncoolingmode of the reference liquidmetal-cooled acceleratordriven system [15]. Wu et al. employed the linear analysismethod to study the stability of Argonne LBE NCL andgive the steady state fluid velocity and temperature differencebetween hot leg and cold leg under different heating power[16, 17]. Although there are many researches of naturalcirculation of liquidmetals, the study of transient behavior forliquid metal NCL is still limited. Furthermore, the influenceof structure thermal inertia and heat loss to ambient onthe performance of liquid metal NCL has not been paidmuch attention to in some of the previous studies. Therefore,more research is needed to analyze factors that influence thetransient behaviors and steady state conditions of the liquidmetal NCL.

In order to analyze the loop transient behavior andevaluate the natural circulation performance during varioustransient processes, a liquidmetalNCLmodelwas developed.The loop model including core tank model, pipe model, andintermediate heat exchangers (IHXs) model is built basedon the MATLAB/Simulink platform. In order to evaluatethe effectiveness of the loop model, the transient processesincluding the start-up, the loss of pump, and the shutdownof TALL experimental facility were simulated. Moreover, theinfluence of appended structure thermal inertia and heat lossto ambient on the performance ofNCL is evaluated in presentwork.

2. Calculation Model and Method

A general liquid metal NCL (TALL test facility) is shownin Figure 1. The primary loop of the natural circulationsystem mainly consists of core tank (heat source), IHX(heat sink), pipes (fluid flow path), electromagnetic (EM)pump (optional), and so forth. During operation, the coolantinitially at rest may be accelerated by the EM pump. Atthe same time, it would be heated by heaters to a specifictemperature when it flows through the core tank. Then, thecoolant flows from the outlet of core tank to the expansiontank through a long vertical pipe.The temperature of coolantwould be decreased to a specific value after it crosses the IHX.

Finally, the coolant flows down through a vertical pipe andreturns to the inlet of core tank to complete the circuit.

It can be seen that once the temperature difference isestablished, the coolant could be driven by the buoyancyforce without the operation of EM pump. So the naturalcirculation transient behaviors and steady state conditions aremainly determined by the interaction between buoyancy andfrictional forces. The temperature and flow rate oscillationswould always exist before the loop reaches steady state.Moreover, a fluctuation of temperature may cause a flowrate oscillation again even when the loop is initially at asteady state and vice versa. Therefore, the transient processand steady state could be predicted very well only whenthe temperature distribution and friction losses are exactlyevaluated.

2.1. Governing Equations and Correlations for Different Mod-els. As can be seen in Figure 1, the core tank holds theimmersion heater which simulates the fuel rod and providesspecific heating power to heat the coolant directly. Theheater consists of NiCr, filler of MgO, and stainless steel 316cladding.The transient heat conduction equation of structureneglecting axial heat conduction is expressed as follows:

𝜕𝑇𝑠

𝜕𝑡=

𝑘𝑠

𝜌𝑠𝐶𝑝,𝑠

1

𝑟

𝜕

𝜕𝑟(𝑟

𝜕𝑇𝑠

𝜕𝑟) +

𝑞V

𝜌𝑠𝐶𝑝,𝑠

. (1)

The coolant in the annular channel would be heated bythe heater. It should be noted that in the TALL facility someguide heaters and insulation are placed on different sectionsof the primary loop in order to reduce the environmentalheat loss. It means that the pipe wall, structure components,and insulation are also heated by the coolant. The energyequation with consideration of the boundary conditions forthe incompressible coolant in the core tank would be

𝜕𝑇𝑓

𝜕𝑡= −

𝑊

𝜌𝑓𝐴𝑓

𝜕𝑇𝑓

𝜕𝑠+

1

𝜌𝑓𝐶𝑝,𝑓

𝐴ℎ,rodℎrod (𝑇rod − 𝑇𝑓)

𝑉𝑓

+1


𝐴ℎ,wallℎwall (𝑇wall − 𝑇𝑓)

𝑉𝑓

,

(2)

where the heat transfer coefficient could be obtained byestimating the Nusselt number [18]

Nupri = 7 + 0.025(RepriPrpri)0.8

. (3)

As for the pipe wall, structure components, and insu-lation, the energy equation is the same as (1) except that𝑞V = 0 for wall and insulation. The outer surface of the pipeinsulation is assumed to be cooled by the air in room temper-ature (∼300K). The Nusselt number for natural convectionheat transfer from a horizontal and vertical cylinder can beexpressed in the following general form:

Nuair = 𝐶1(Gr Pr)𝐶2 , (4)

where the coefficients 𝐶1and 𝐶

1depending on Grashof

number Gr could be found in [19].

Science and Technology of Nuclear Installations 3

Core tankand heater

EM pump

IHX

Expansion tank

Heated part

To secondary loop

From secondary

loop

To sump tank for

LBE

Unheated part

Unheated part

0.03 cm

0.87m

0.6m 6m

0.75m

2.1m

6.8m

1.0m

6.5m

5m0.16m

1m

1.375

m

Figure 1: Schematic of the TALL test facility.

In the pipe model, the energy equation for the coolant is

𝜕𝑇𝑓

𝜕𝑡= −

𝑊

𝜌𝑓𝐴𝑝

𝜕𝑇𝑓

𝜕𝑠+

1


𝐴ℎℎwall (𝑇wall − 𝑇𝑓)

𝑉𝑓

. (5)

Theheat transfer through the pipe, structure components,and insulation could also be described by using (1).

In the TALL facility, the IHX is a single tube LBE-Glycerolheat exchangerwith the primary coolant in the inner tube andthe secondary coolant in the annulus. In the IHX model theenergy equation for primary side and secondary side wouldbe

𝜕𝑇𝑓,pri

𝜕𝑡= −

𝑊pri

𝜌𝑓,pri𝐴𝑓,pri

𝜕𝑇𝑓,pri

𝜕𝑠

+1

𝜌𝑓,pri𝐶𝑝,𝑓,pri

𝐴ℎ,priℎpri (𝑇wall − 𝑇𝑓,pri)

𝑉𝑓,pri

𝜕𝑇𝑓,sec

𝜕𝑡= −

𝑊sec𝜌𝑓,sec𝐴𝑓,sec

𝜕𝑇𝑓,sec

𝜕𝑠

+1

𝜌𝑓,sec𝐶𝑝,𝑓,sec

𝐴ℎ,secℎsec (𝑇wall − 𝑇𝑓,sec)

𝑉𝑓,sec

,

(6)

where the heat transfer coefficient in the secondary side ofIHX could be estimated by using the following correlations[20]:

Nusec =(𝑓/8) (Resec − 1000)Prsec1 + 12.7√𝑓/8 (Pr2/3sec − 1)

, (7)

where 𝑓 = (1.82𝑙𝑔Resec − 1.64)−2. As for the tube wall the

governing equation would be the same as (1) with 𝑞V = 0.

In the simulation the coolant is assumed to be incom-pressible so the mass flow rate is only a function of time andindependent of space coordinate; namely,

𝜕𝑊

𝜕𝑠= 0. (8)

On the other hand, the momentum equation for one-dimensional incompressible flow is

𝑑𝑊

𝑑𝑡= −

𝜕 (𝑊𝑢)

𝜕𝑠− 𝐴

𝜕𝑃

𝜕𝑠− 𝐴𝜌𝑔 cos 𝜃 − 𝑊

2

2𝜌

𝑓

𝐴𝐷. (9)

When Boussinesq approximation is employed, the density istaken to be constant except in the body force term. Thenintegrating on both sides of (9) along the loop we couldobtain

𝑁

∑

𝑖=1

𝐿𝑖

𝐴𝑖

𝑑𝑊

𝑑𝑡= Δ𝑃pump + 𝜌0𝑔𝛽∮𝑇𝑑𝑠

−𝑊2

2𝜌0

𝑁

∑

𝑖=1

[1

𝐴2𝑖

(𝑓𝑖𝐿𝑖

𝐷𝑖

+ 𝐾𝑖)] ,

(10)

where 𝜌 = 𝜌0[1−𝛽(𝑇−𝑇

0)]. For laminar flow𝑓 = 64/Re and

for turbulent flow 𝑓 = 0.316/Re0.25. Δ𝑃pump is the differencein pressure between the outlet and inlet of EMpump. It wouldbe equal to zerowhen the pump is out of work and the coolantis completely driven by the buoyancy.

In addition, the liquid metal NCL model also includessome other submodels. The junction model is mainly usedto estimate the coefficient of local resistance. For suddenexpansion and contraction of a stream with uniform velocitydistribution, the coefficient would be

𝐾exp = (1 −𝐴 in𝐴out

)

2

, 𝐾con = 0.5 (1 −𝐴out𝐴 in

) . (11)


The resistance coefficient of the bend is

𝐾bend = 0.0175𝑓𝑅

𝐷𝛾 + 𝐴

1𝐵1,

𝐴1=

{{{

{{{

{

0.9 sin 𝛾, 𝛾 ≤ 70∘

,

1.0, 𝛾 = 90∘

,

0.7 +0.35𝛾

90∘, 𝛾 ≥ 100

∘

,

𝐵1=

{{{

{{{

{

0.21

(𝑅/𝐷)2.5,

𝑅

𝐷< 1.0,

0.21

√𝑅/𝐷,

𝑅

𝐷> 1.0.

(12)

As for the multihole plates the resistance coefficientscould be obtained by the following correlations:

𝐾grid =

{{{{

{{{{

{

𝜁𝜑+ 𝛼0(1 + 0.707√1 − 𝑓 − 𝑓)

21

𝑓2, Re < 10

5

,

(1 + 0.707√1 − 𝑓 − 𝑓)

21

𝑓2, Re > 10

5

,

(13)

where the cross-section coefficient 𝑓 is equal to the ratio offlow area of the obstruction cross-section to the area of theobstruction front. The coefficients 𝜁

𝜑and 𝛼

0are a function

of Re, respectively, and could be determined on the graph of[21].

2.2. Discretization of the Energy Equation. The liquid metalNCL model is mainly composed of core tank model, IHXmodel, and pipe model. Figure 2 shows the calculation cellsfor these models.

Figure 2(a) shows the radial nodding in an axial sliceof the core tank. The NiCr part of the heater is dividedinto three (user input) nodes; the filler, clad, coolant, pipewall, and appended structure have one node each. Theinsulation is divided into three (user input) nodes. Becausethe SIMULINK is a time-based simulation platform, thediscretization of the partial derivative with respect to timethat appears in the governing equations is not necessary [22].By using the finite volume method the general form of thediscretization of the energy equation for structure could beexpressed as

𝑑𝑇𝑗

𝑑𝑡=

1

𝜌𝑗𝐶𝑝,𝑗𝑉𝑗

× [𝑎𝑗−1

𝑇𝑗−1

− (𝑎𝑗−1

+ 𝑎𝑗+1

) 𝑇𝑗+ 𝑎𝑗+1

𝑇𝑗+1

]

+𝑞V

𝜌𝑗𝐶𝑝,𝑗

,

(14)

where 𝑎 = 𝑘𝐴/𝛿 for heat conduction and 𝑎 = ℎ𝐴

for convective heat transfer. It should be noted that thethermal conductivity 𝑘 of the adjacent structures may bevery different. So the numerical treatment of coefficient 𝑎 atinterface could be expressed as follows [23].

According to Fourier’s law the heat flux density across theinterface (Figure 3) would be

𝑞 =𝑇𝑗,𝑗+1

− 𝑇𝑗

𝛿−/𝑘𝑗

=𝑇𝑗+1

− 𝑇𝑗,𝑗+1

𝛿+/𝑘𝑗+1

=𝑇𝑗+1

− 𝑇𝑗

(𝛿−/𝑘𝑗) + (𝛿+/𝑘

𝑗+1).

(15)

So the right side coefficient 𝑎 would be 𝑎 = 𝑘𝐴/𝛿 =

𝐴/(𝛿−

/𝑘𝑗+ 𝛿+

/𝑘𝑗+1

). It can also be seen that the temperatureat the interface would be

𝑇𝑗,𝑗+1

=(𝑇𝑗+1

𝛿−

/𝑘𝑗+ 𝑇𝑗𝛿+

/𝑘𝑗+1

)

(𝛿−/𝑘𝑗+ 𝛿+/𝑘

𝑗+1)

. (16)

Obviously, the interface temperature is determined by theconductivity of materials (also a function of temperature)and the node positions. As for the coolant in the channel(Figure 2(a)) the discretization of (2) would be

𝑑𝑇𝑖

𝑑𝑡=

𝑊

𝜌𝑉𝑖−1,𝑖

(𝑇𝑖−1

− 𝑇𝑖) +

1

𝜌𝐶𝑝𝑉𝑖−1,𝑖

× [𝑎clad𝑇clad,𝑖 − (𝑎clad + 𝑎wall) 𝑇𝑖−1,𝑖 + 𝑎wall𝑇wall,𝑖] ,

(17)

where the average coolant temperature in the control volume𝑇𝑖−1,𝑖

= (𝑇𝑖−1

+ 𝑇𝑖)/2. In addition, the boundary conditions

need to be specified to solve the discretized equations, sothe convective heat transfer coefficient of the exterior ofinsulation and the air temperature are specified. As can beseen in Figure 4 that five nodes are used to simulate theheaters of core tank, three nodes are used to simulate theupper part of core tank and junction 001 would be used toestimate the pressure drop coefficients when the coolant flowsthrough the multihole plates.

In the radial direction of IHX model (Figure 2(b)) twonodes are used to represent the primary coolant and sec-ondary coolant, respectively, and one node is used for the tubewall. The same discretization method is employed to obtaindiscretized equations in (6) which are similar to (17) exceptthat there is only one convection heat transfer surface. It canbe noted that the whole secondary loop is not simulated inthis loop model because the main function of the secondaryloop is to provide the steady coolant for the LBE-Glycerolheat exchanger. On the other hand, the heat loss throughthe insulation of the secondary side of IHX is neglecteddue to a relatively lower secondary coolant temperature andsmaller overall heat transfer coefficient compared to primaryloop.

In Figure 2(c) the pipe model is represented by six meshintervals: coolant, pipe wall, and appended heat structure(represent the guider heaters, valve, thermocouples, etc.)have one node each and another three are put for modelinginsulation. The discretized equations for pipe model couldalso be obtained by using the same method that is employedin core tank model. In Figure 4 the pipes are connected bymany junctions which are used to estimate the coefficient oflocal resistance.


Environment T1 Tpin Tfiller Tclad

Ti,fluid

i

i + 1

i − 1

Ti−1,fluid

Twall Tstruc Tinsulation Ti,N

Axis

(a)

Primary flow

Secondary flow

Tube wall

Ti−1,p Ti−1,s

Ti,sTi,p

Ti,w

(b)

Environment Ti,wallTstruc Tinsulation Ti,N

Ti+1,p

Ti,p

(c)

Figure 2: Calculation cells for (a) core tank model, (b) IHX model, and (c) pipe model.

2.3. Solution Procedures. The transient temperature distri-bution at the next time step can be numerically obtainedby solving the discretized energy equations for given initialconditions and boundary conditions. Then, the buoyancyterm in (10) can be solved by integrating the temperaturedistribution along the entire length of the loop. At the sametime, themass flow rate at the next time step through the loopcan be obtained by solving (10). Finally, the mass flow rate issubstituted into the energy equations for the coolant in thedifferent sections of the loop and the next iteration is started.

It can be seen from (14) and (17) that the discretizedenergy equations for structure and coolant could be rewrittenin the form of matrix multiplication expressed as follows:

𝑑𝑇

𝑑𝑡= 𝑎𝑇 + 𝑆, (18)

where 𝑇 = [𝑇1⋅ ⋅ ⋅ 𝑇𝑁]𝑇; 𝑎 is a𝑁 × 𝑁matrix and the general

source term 𝑆 = [𝑆1⋅ ⋅ ⋅ 𝑆𝑁]𝑇. Figure 5(a) shows the simplified

solution diagram for the heated part in core tank model byusing MATLAB/Simulink.

In core tank model the coefficient matrix 𝑎 and generalsource term 𝑆 are solved, respectively. The heat source andthe boundary conditions including the inlet temperature ofcore tank and the environment temperature are used to cal-culate the general source term. The temperature distributionwould be used to determine the material properties and thebuoyancy term in the momentum equation. Meanwhile, theoutlet temperature of the model would be considered as theinlet temperature of the next adjacent model. As a result, theNCL is composed of these connectedmodels which representthe transient temperature distribution in different sections ofthe loop.


Interface

𝛿

Tj,j+1

Tj Tj+1

𝛿− 𝛿+

Figure 3: Numerical treatment of coefficient 𝑎 at interface.

Volume

Pipe 002

Jun 003Jun 004

Jun 005Pipe 003

IHXmodel

Jun 006

Pipe 004

Jun 007

Jun 008

Pumpmodel

Pipe 005

Jun 009

Pipe 006Jun 010-011

Pipe 007

Jun 001Core tank

model

Jun 002

Pipe 001

Figure 4: Nodalization of the NCL.

Figure 5(b) shows the Simulink model for momentumequation solver. The driving force is mainly provided bythe pump and the buoyancy caused by the temperaturedifference between the hot leg and cold leg. The buoyancywould be determined by the temperature distribution in thecore tank, IHX, and pipes. At the same time, the flow lossestake place either as friction loss due to wall friction or aslocal flow loss depending on channel geometry. The frictionloss in each section of the loop has been calculated in thecorresponding model. The local flow losses are calculated inthe junction models.Then the transient mass flow rate wouldbe determined by the interaction between driving forces and

resistance. Finally, the mass flow rate at the next time stepis transferred to other models to obtain the temperaturedistribution in each section of the loop. Consequently, thetransient behaviors of the liquid metal NCL can be predictedby using the Simulink model for the loop. On the otherhand, because the problem is supposed to be stiff, an implicitcontinuous variable-step solver (ode15s) is chosen for thesimulation. Some other parameters such as loop geometryand material properties (the appended structure is assumedto be composed of stainless steel 316) could be found in[14, 24–26].

3. Results and Discussion

Some transient experiments had been carried out on theTALL facility to study the steady state and transient thermal-hydraulics performance of LBE-cooled reactors [14]. Mean-while, the TRACE code was used to simulate the transientbehavior of TALL facility without consideration of thermalinertia caused by appended structure such as rope heaters,valves, and sensors. In the present study the start-up, lossof primary pump, and shutdown transient experiments aresimulated with appended structure to test the liquid metalNCL model and study the influence of thermal inertia on theperformance of NCL.

3.1. Start-Up Experiment. In the start-up experiment theloop would be started by forced circulation. The initial LBEtemperature in the primary loop is about 200∘C. The initialmass flow rate for primary and secondary loop coolantis 0 kg/s. Meanwhile, the heater in the core tank was outof work at the initial state. At about 140 s the transientexperiment was started. The primary and secondary looppumps were switched on to work, whereas the power wassupplied to the core tank heater. The mass flow rate of LBEand Glycerol would immediately reach their final value of0.91 kg/s and 0.69 kg/s, respectively. A constant power of8.5 kW is supplied to the heater at the same time. Figure 6illustrates the experiment and simulation results of the start-up forced circulation.

It can be seen from Figure 6 that at the beginning of theexperiment the outlet temperature of IHX would decreasedue to the cooling by the secondary coolant flow. Aftera few seconds, the cold coolant reaches the inlet of coretank. It should be noted that the inlet temperature of coretank does not decrease sharply as the outlet temperature ofIHX. It seems that the structure of the cold leg plays animportant part in flattening the temperature distribution dueto the thermal inertia. On the other hand, the minimum inlettemperature of core tank is slightly higher than that of IHX.This is because of the heat transfer from the cold leg structureto the coolant due to a higher temperature of appendedstructure.The outlet temperature of core tank would increasedue to the heat supply. However, because of the decreaseof the inlet temperature, the core tank outlet temperaturewould stop increasing in a short period of time and thenkeep increasing. The inlet temperature of IHX also increaseswith the outlet temperature of core tank. Finally, a steady


Q(t)

MATLABfunctionFrom outlet of

pipe 007Tinlet

TenvironmentConstant

Materialproperties

MATLABfunctionGeometry

From fileand mesh

MATLABfunction

From momentumequation solver

W

f

a

MATLABfunction

To momentumequation solver

S

aTMatrix

Product

multiply

+

Add

From file+

Initialtemperature

dT

dt1/s

T0Integrator

TMATLABfunction

Tout letTo junction 001

To momentumequation solver

(a)

Pumpmodel

From core tankmodel, IHX model,

and pipe models

From core tankmodel, IHX model,

and pipe models

From junctionmodels

MATLABfunction

MATLABfunction

MATLABfunction

MATLABfunction

To core tank model,IHX model, and

pipe models

Initial massflow rate

From file

Integrator

W1/s

W0

dW

dtTi

fi

Ki W2

2𝜌0

N∑i=1

[ 1

A2i

(fiLi

Di

+ Ki)]

ΔPpump (t)

𝜌0g𝛽∮Tds

(b)

Figure 5: Simulink model for (a) heated part in core tank model and (b) momentum equation solver.

350

300

250

200

150

Tem

pera

ture

(∘C)

0 1000 2000 3000 4000 5000 6000

Time (s)

Outlet temperatureInlet temperatureSimulation

ExperimentRef. [14]

(a)

350

300

250

200

150

Tem

pera

ture

(∘C)

0 1000 2000 3000 4000 5000 6000

Time (s)


ExperimentRef. [14]

(b)

Figure 6: Inlet and outlet temperature variation of (a) core tank and (b) IHX.


state loop temperature distributionwould be established afterabout 4000 s. By comparing Figures 6(a) and 6(b) we cansee that the outlet temperature of core tank/IHX is higherthan the inlet temperature of IHX/core tank at steady state.The heat loss to ambient would decrease the temperature ofcoolant when it flows through the pipes.

It can be seen from Figure 6 that both of the simulationresults show good agreement with the experimental data.Moreover, both results indicate that the outlet temperatureof core tank/IHX is higher than the inlet temperature ofIHX/core tank at steady state because the heat loss to ambientis considered in present simulations and [14]. However, thesteady state temperatures are still somewhat different whichmay be caused by the different calculation methods andassumptions in the simulations. Furthermore, the influenceof appended structure thermal inertia was also consideredin present study which, however, was not included in [14].Therefore, the operating temperature of the loop in the tran-sient would rise more slowly with consideration of appendedstructure which is closer to the experimental process. It isclear that in the start-up transient not only the coolant in theloop but also all the structure surrounding the loop wouldbe heated. Therefore, the thermal inertia of the structure willextend the time required to reach steady state. It should benoted that a completely stable initial condition is not easy tobe established because of a small natural circulation residualflow and the use of rope heaters in the experiment. On theother hand, it is difficult to predict the structure surroundingthe loop and the environmental heat loss very exactly; thus,a longer time is taken to reach the steady state in theexperiment than in both of the simulation results. In addition,it seems that there is more heat loss in the experimentthan in the prediction. As a result, a good simulation ofthermal inertia and heat loss is needed to obtain an accurateprediction of the transient behavior of the NCL.

3.2. Loss of Primary Pump Experiment. When the primarypump is out of work, the coolant would be completely drivenby the buoyancy. It means that the power of the heaterwould be removed by natural circulation and the loss offlow accidents can be well prevented. Therefore, a naturalcirculation based primary circuit is often preferred in somedesigns of advanced nuclear power plants.

In the loss of primary pump experiment the loop wouldinitially work at steady state forced circulation with a heaterpower of 12.4 kW. At about 550 s the primary pump wasswitched off, and then the rotational speed of primary pumpwould reduce to 0 in about 2 s. Figure 7 illustrates thevariation of temperature and velocity.

It can be seen that the velocity of coolant would decreasedramatically when the pump stops working. Therefore, theoutlet temperature of core tank increases when the heaterpower is kept constant, while the outlet temperature of IHXdecreases suddenly because the primary coolant is furthercooled by the secondary coolant when the velocity of primarycoolant decreases. It should be noted that Glycerol waschosen as the intermediate fluid in the secondary loop. Itsboiling point (∼290∘C) is higher than the melting point of

LBE (∼125∘C). It means that theminimum temperature in theloop could be much greater than the melting point of LBEso that the solidification of LBE in the heat exchanger can beavoided. It is clear that the structure again causes a thermalinertia in the transient experiment. The heat supplying andabsorbing of the structure would influence the transienttemperature distribution in cold leg and hot leg. Meanwhile,the temperature difference between the inlet and outlet ofcore tank/IHX at steady state would increase greatly whenthe natural circulation flow rate is only about half of theflow rate provided by forced circulation. It can also be notedthat the inlet temperature of IHX is obviously lower thanthe outlet temperature of core tank under natural circulationcondition.

A good agreement between the simulation results andthe experimental data can be obtained. In addition, thesimulation results show that the oscillations of temperatureand flow rate would be more obvious (amplitude and fre-quency) when lower thermal inertia is employed. As we knowthat the structure including the pipe wall was not generallyconsidered in most of previous studies on the stability ofNCL [7, 8, 10, 16, 17]. It seems that the thermal inertiawould play an important part in the stability of NCL. Thismay also explain why the instability of NCL is sometimesoverpredicted in the simulation compared to the experiment.From Figure 7 we can also see that the simulation resultsreach the stable state earlier than the experimental data. Thismay be because the thermal and coolant inertia caused by theauxiliary equipment (e.g., expansion tank) are not accuratelycalculated in the simulation.On the other hand, the appendedstructure is assumed to be uniformly distributed on the outersurface of pipe in the simulation which is different from theactual conditions.

3.3. Shutdown Experiment. The shutdown procedure of theTALL facility is also simulated in present work. The initialpower is 17 kW when the mass flow rate is about 0.91 kg/s. Atthe beginning of the transient experiment, the power wouldbe decreased to zero at about 280 seconds while the forcednatural circulation working condition was kept. Figure 8presents the inlet and outlet temperature variation of coretank and IHX.

It can be seen from Figure 8 that the outlet temperature ofcore tank would decrease dramatically due to the shutdownof heating power while the rate of decrease of IHX inlettemperature is a little slower due to the thermal inertia. Onthe other hand, it can be observed that the cold coolant atthe outlet of IHX would be somewhat reheated by the pipeand structure when it flows through the cold leg. Hence, itstemperature rises when the coolant reaches the inlet of thecore tank. An opposite feature is observed in the start-upexperiment.

Generally, both of the simulation results show goodagreement with the experimental data in Figure 8. However,the operating temperature of the loop would be decreasedmore quickly with lower thermal inertia. The simulationresults would be closer to the experimental data when theappended structure is considered.


Tem

pera

ture

(∘C)

0 1000 2000 3000 4000

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600

500

400

300

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(a)

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(c)

Figure 7: Variation of (a) core tank inlet/outlet temperature, (b) IHX inlet/outlet temperature, and (c) velocity of coolant in IHX.

4. Influence of Heat Loss onthe Performance of NCL

In most of previous studies the heat loss to ambient in theNCL was not considered. Generally, the hot leg and coldleg are assumed to be perfectly adiabatic in the theoreticalanalysis. It had been proved to be reasonable as the calcu-lation results show good agreement with the experimentaldata. It seems that the heat loss to ambient is limited when theloop operating temperature is not much higher than ambienttemperature or the experimental loop size is not very large.Hence, the influence of heat loss on the performance of NCLwould depend on the specific case.

In the loss of primary pump transient of TALL facility itcan be seen that the inlet temperature is significantly lower

than the outlet temperature of core tank especially whena stable natural circulation is established in primary loop. Inorder to evaluate the effect of heat loss on the performanceof NCL, the thermal conductivity of the insulation is set tobe a minimum value (e.g., 10−4W/m/K) to simulate the loopperformance without heat loss. Table 1 shows the thermalconductivity of insulation material (asbestos) at differenttemperatures. Figure 9 illustrates the calculation results withand without consideration of heat loss.

It is clear that the heat loss plays an important part inthe performance of the loop, no matter what the loop isat the condition of forced circulation or natural circulation.Obviously, the average operating temperature of the NCLincreases a lot (∼30∘C) when the hot leg and cold leg areassumed to be adiabatic. Under this condition the total heat


Tem

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(∘C)

0 500 1000 1500 2000

Time (s)


Experiment

500

400

300

200

Ref. [14]

(a)

Tem

pera

ture

(∘C)

0 500 1000 1500 2000

Time (s)


Experiment

500

400

300

200

Ref. [14]

(b)

Figure 8: Inlet and outlet temperature variation of (a) core tank and (b) IHX.

Table 1: Thermal conductivity of insulation material.

Temperature (∘C) 10 50 100 150 200 300 400 500 800Thermal conductivity (W/m/K) 0.034 0.037 0.042 0.049 0.059 0.083 0.116 0.157 0.157

is completely removed by the IHX, so the primary coolanttemperature would increase in order to extract more heatto the secondary loop. On the other hand, the average hotleg temperature would increase ∼40∘C while the cold legtemperature increases ∼25∘C under natural circulation. It canbe concluded thatmore heat is dissipated from the hot sectiondue to its higher temperature. In addition, fromTable 1we cansee that the thermal conductivity of insulation material willincrease with temperature which may also cause an increaseof heat loss through the hot leg.

The capability of the natural circulation of the loop willalso be influenced by the heat loss to ambient. It can be seenfrom Figure 9(c) that the temperature difference betweenhot leg and cold leg would be increased by decreasingthe heat loss through pipes. It means that the flow rateor the capability of the natural circulation could also beincreased due to the increase of buoyancy (Figure 9(d)).It is preferred to have as high natural circulation flow aspossible in a reactor design. However, reducing the heat lossmay also bring some undesirable results. The corrosion ratemay also be increased with the operating temperature whichundermines the integrity of the facilities. For example, theoutlet temperature of the core tank is restricted to ∼460∘C inthe TALL facility. It also means that the total core tank powerwill therefore be limited.

5. Conclusions

A liquid metal NCL computing approach with the heat-loss model, including core tank model, IHX model, and

pipe model, is developed based on the MATLAB/Simulinkplatform to investigate the transient behaviors and steadystate characteristics of the loop. Three types of transientexperiments (the start-up, the loss of pump, and the shut-down) on the TALL facility are simulated to validate themodel and evaluate the influence of appended structurethermal inertia and heat loss on the performance of NCL.

A good agreement between the simulation results and theexperimental data of the transient behaviors for the primaryLBE coolant is obtained, which provides good validation forthe MATLAB/Simulink models. It is found that the thermalinertia greatly influence the temperature variation duringthe transient processes.The coolant temperature distributionin the pipe would be flattened due to heat exchange withstructure. It seems that the simulation results reach thesteady state earlier than the experimental data. This maybe because the thermal and coolant inertia caused by theauxiliary equipment are not accurately considered in thesimulation. On the other hand, the structure thermal inertiaoften neglected in the previous analysis of the stability ofNCL may play an important role in the stability behaviorof NCL. The oscillations of temperature and flow rate inthe experiment may be not obvious which is unlike thesimulation results.Thismay explainwhy theNCL is supposedto be unstable but the instability behavior is not observed inthe experiment.

The influence of heat loss through the pipes on theperformance of NCL is also evaluated. Obviously, the averageoperating temperature of the NCL would increase withthe decrease of heat loss. Furthermore, the temperature ofhot section would increase more significantly compared to


Tem

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ture

(∘C)

0 1000 2000 3000 4000

Time (s)

With heat lossesAdiabatic

Inlet temperatureOutlet temperature

600

500

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300

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100

(a)

Tem

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Inlet temperatureOutlet temperature

600

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(b)

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0.7

0.6

0.5

0.4

0.3

Velo

city

(m s−

1)

0.9

1.0

1.1

(d)

Figure 9: Influence of the heat loss on the performance of NCL. (a) Inlet and outlet temperature of core tank; (b) inlet and outlet temperatureof IHX; (c) temperature difference between average temperatures of hot leg and cold leg; (d) coolant velocity in IHX.

the cold section temperature. It may even exceed the uppertemperature limit and so that undermine the integrity of thefacilities, so the heat power is also limited at the same time.However, the capability of the NCL could be increased due tothe decrease of heat loss.

Nomenclature

𝐴 : Area (m2)𝐶𝑝: Specific heat capacity (J⋅kg−1⋅K−1)

𝐷: Loop diameter (m)𝑓: Friction coefficient𝑔: Gravity (m⋅s−2)Gr: Grashof numberℎ: Heat transfer coefficient (W⋅m−2⋅K−1)𝑘: Thermal conductivity (W⋅m−1⋅K−1)

𝐾: Local resistance coefficient𝐿: Length (m)Nu: Nusselt number𝑃: Pressure (Pa)Pr: Prandtl number𝑞: Heat flux density (W⋅m−2)𝑞V: Volumetric rate of heat generation (W⋅m−3)𝑟: Radius (m)𝑅: Mean radius of curvature of the bend (m)Re: Reynolds number𝑠: Space coordinate (m)𝑡: Time (s)𝑇: Temperature (∘C)𝑢: Velocity (m⋅s−1)𝑉: Volume (m3)𝑊: Mass flow rate (kg⋅s−1).


Subscripts𝑠: Structure𝑓: Fluidℎ: Heat transferpri: Primary loopsec: Secondary loop0: Reference value.

Greek Symbols

𝛽: Thermal expansion coefficient (K−1)𝛾: Angle of bend of the curved channel (∘)𝜃: Inclination angle (∘)𝜌: Fluid density (kg⋅m−3).

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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Research Article A Computing Approach with the Heat-Loss ...downloads.hindawi.com/journals/stni/2014/769346.pdf · e Nusselt number for natural convection heat transfer from a horizontal