A REPORT

A REPORT

ON

DEVELOPING A NUMERICAL MODEL

FOR SIMULATING COOLANT BOILING

IN REACTOR CORE

BY

Name(s) of the Student(s)

ID.No.(s)

Discipline(s)

A. SAI DARSHAN 2012A1PS487H B.E.(HONS) CHEMICAL

ENGINEERING

Prepared in partial fulfillment of the

Practice School I Course Nos

BITS C221/BITS C231/BITS C241

At

INDIRA GANDHI CENTRE

FOR ATOMIC RESEARCH

A Practice School-I Station of

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI

(JULY 2014)

Format of an Abstract Sheet

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE

PILANI (RAJASTHAN)

Practice School Division

Station: Indira Gandhi Centre for Atomic Research Centre: -------------------------------------

Duration: 8 weeks Date of Start: 23rd May 2014

Date of Submission:------------------------------

Title of the Project: DEVELOPING A NUMERICAL MODEL FOR SIMULATING

COOLANT BOILING WITHIN REACTOR COREID No./

2012A1PS487H

Name(s)/

A. SAI DARSHANDiscipline(s)/of the student(s)

B.E. (HONS) CHEMICAL ENGINEERINGName and Designation Of the expert:K. Natesan, Scientific Officer G, Mechanics and Hydraulics Division, Reactor Design Group

Name of the

PS Faculty: ETHIRAJ SENTHAMARAI KANNAN

Key Words: PFBR, Coolant, Heat Transfer, Boiling.Project Areas: Coolant in Prototype Fast Breeder Type ReactorAbstract:

Coolant flow is circulated through reactor core of nuclear reactor to remove the fission heat generated. Under an unlikely event of loss in pumping system, automatic shutdown of the reactor takes place to protect the reactor from serious consequences. Multiple and redundant shutdown systems are provided in the reactor design to achieve this. Under a very unlikely event of loss of coolant pumping system along with failure of reactor shutdown system, large imbalance between the heat generated in the core and the heat removed by the coolant would arise. This leads to sharp rise in temperature of reactor core. The sodium coolant may begin to boil under this situation leading to voiding of the core. In a fast reactor, coolant voiding results in positive reactivity effects and power produced in the core increases. This effect leads to severe thermal consequences. Under this scenario, the knowledge of the temperature profile in the reactor core will be pivotal in handling of such a situation in the plant.

The report starts off with a brief introduction of PFBR, following which the concepts used in tackling the problem are described. The simplest and ideal case of steady state heat transfer is analyzed followed by a transient formulation for fuel rod using finite volume and finite difference approaches. The discretized equations have been solved using both implicit and explicit methods. Then using the best technique and algorithm, the analysis of typical loss of coolant event along with failure of reactor protection system has been analysed. A parametric study has been carried out by varying the coast down time of pump and the sensitivity of rate of propagation of boiling front in the core has been established. It is observed from the analysis that there is possibility of fuel melting in the central region of fuel during the transient.

The present work could be extended by incorporating melting model for fuel and reactor physics associated power changes in core. The tool developed could be interfaced with thermal hydraulic models for rest of the plant for predicting the integrated behavior of whole plant during transients.Signature(s) of Student(s) Signature of PS FacultyAcknowledgements

I would like to thank my guide, Mr. Natesan, Thermal hydraulics Section, Reactor Design Group, Indira Gandhi Centre for Atomic Research Kalpakkam for the unending support he gave me during the last 8 weeks, always being ready help me and clear my doubts.

I would also like to express my gratitude to Mr. Sai Baba and Mr. Kabali who went out of their way to help me get a very good project.

I would like to express my whole-hearted thanks to my friends and colleagues for sharing their knowledge and experience with me and bringing the project work in a good shape.

I register my special thanks to all my friends to make my stay at Kalpakkam enjoyful and homely.At last but not least, I express my sincere thanks to my parents, for their support and constant encouragement to carry out this work.

Table of ContentsChapterTitlePage number

1Introduction8

1.1Introduction to Indian nuclear power program8

1.2Prototype Fast Breeder Reactor8

1.2.1Reactor Core10

1.2.2Fuel sub- assembly10

1.2.3Reactor assembly10

1.2.4Heat transport system15

1.3Motive for Project16

2Concepts Involved18

2.1Steady State Heat Transfer18

2.2Transient Heat Transfer20

2.2.1Eulers Explicit Method21

2.2.2Eulers Implicit Method23

2.2.3Finite Volume Method24

2.2.4Finite Difference Method 25

2.3Boiling27

2.4Influence of Pump Inertia30

2.5Input Data31

3Simulation Algorithms32

3.1Steady State Formulation32

3.2Transient Formulation, excluding temperature variation for coolant35

3.2.1Finite Volume Method with Eulers Explicit Method35

3.2.2Finite Difference Method with Eulers Explicit Method37

3.2.3Finite Volume Method with Eulers Implicit Method38

3.2.4Finite Difference Method with Eulers Implicit Method41

3.3Transient Formulation, including temperature variation for coolant (no boiling)44

3.3.1Temperature Formulation for coolant44

3.3.2Enthalpy Formulation for Coolant48

3.4Transient Formulation for boiling coolant50

3.4.1Coolant Enthalpy formulation 50

3.4.2Boiling Front vs Time54

4Results and Discussions56

4.1Steady State Formulation56

4.1.1Rod of unit length with constant heat generation56

4.1.2Rod of length 1.6 metres with varying volumetric heat generation57

4.2Transient Formulation without temperature variation for coolant61

4.2.1Finite Volume Method with Eulers Explicit Technique for a rod of unit length61

4.2.2Finite Difference Method with Eulers Explicit Technique for a rod of unit length62

4.2.3Finite Volume Method with Eulers Implicit Technique for a rod of unit length64

4.2.4Finite Difference Method with Eulers Implicit Technique for a rod of unit length66

4.2.5Finite Difference Method with Eulers Implicit Technique for an actual fuel rod67

4.3Transient Formulation including Temperature variation for coolant72

4.3.1Temperature formulation for coolant72

4.3.2Enthalpy formulation for coolant73

4.3.3Enthalpy and Temperature formulations for coolant for a 10 percent increase in volumetric heat generation75

4.3.4Enthalpy and Temperature formulations for coolant for a 10 percent decrease in mass flow rate76

4.4Boiling Scenario78

4.4.1Temperature and enthalpy formulations due to boiling coolant78

4.4.2Boiling Front for various Flow Half Times86

5Conclusions87

6Future Work to be done88

7References89

Chapter 1

Introduction

The present chapter gives a brief introduction to the nuclear power programme in India and the three stages of Indian nuclear power programme. Among the 3 stages the second stage is the development of fast breeder reactors. An introduction to the Indian fast breeder reactor, prototype fast breeder reactor (PFBR) and the technical details of the reactor is given in this chapter. The boiling of liquid sodium in the reactor is taken as the current area of study with focus on finding the temperature distribution in the boiling coolant. The ill effects of such an event on the core of the reactor are also discussed. Finally the objectives of the present work are listed.

1.1 Introduction to Indian nuclear power program

The Indian nuclear power program has been envisaged to have three stages for effective utilization of limited uranium and vast thorium resources. In the first stage, Pressurized Heavy Water Reactors are deployed which use natural uranium as fuel and generate plutonium as a byproduct, which is fissile. Only a very small percentage (0.7%) of naturally occurring uranium which is fissile is used in the PHWR, essentially all of the rest is fertile. In order to utilize the rest of uranium, the second stage is envisaged. In the second stage, Liquid Metal cooled Fast Breeder Reactors (LMFBR) is deployed. In FBR, the plutonium generated in the first stage of reactors is used as fuel for fission energy. The core of FBR is surrounded by natural or depleted uranium as blanket material which in due course gets converted to plutonium. The plutonium thus generated in an FBR is more than that is used for fission, leading to breeding ratio >1[1]. The third stage reactors are also breeders working on thorium - uranium cycle.

The Indira Gandhi Centre for Atomic Research (IGCAR) is entrusted with the responsibility of developing fast breeder reactor technology for India. A loop type Fast Breeder Test Reactor (FBTR) of 40 MW thermal energy is under operation at this centre. As a natural extension of FBTR, a larger capacity (500 MWe) pool type Prototype Fast Breeder Reactor (PFBR) is under construction at Kalpakkam [2].1.2 Prototype Fast Breeder ReactorThe PFBR is a 500 MWe, sodium cooled, pool type, mixed oxide (MOX) fuelled reactor having two secondary loops [2]. The main characteristics of the PFBR are given in table 1.1. The PFBR uses a proven fuel cycle MOX based in its proven capability of safe operation to high burn up, ease of fabrication and proven reprocessing. A pool type design is used for primary circuit layout because of the safety features of pool type, i.e., main vessel with no nozzles leads to high integrity of the vessel, relatively large thermal inertia leads to ease in design of decay heat removal, large diameter of the main vessel with internals leads to significant lowering the strain in the main vessel in case of core disruptive accident.

Liquid sodium, having a very high heat transfer coefficient (~25,000 W/m2-K in turbulent forced convection regime) is the natural choice as heat transport medium. This is essential to remove very high heat flux values (~1.5 MW/m2) encountered in the core. Another favorable point for the selection of sodium is its high boiling point (940(C), so that the systems need not be pressurized to achieve high temperature.

The main drawback of sodium is its violent chemical reaction with air and water. Hence, in all the sodium systems inert argon gas is maintained above the sodium free surfaces to avoid sodium-air contact.Table 1.1: Main characteristics of PFBR

Thermal power (MWt)1250

Electric output (MWe)500

Core height (mm)1000

Core diameter (mm)1900

FuelPuO2-UO2

Fuel pin outer diameter (mm)6.6

Pins per fuel subassembly217

Fuel clad material20% CW D9

Diameter of main vessel (mm)12900

Primary circuit layoutPool

Primary inlet/outlet temp (c)397/547

Steam temperature (c)490

Steam pressure (MPa)16.6

Reactor containmentRectangular

Plant life (years)40

No. of shutdown systems2

No. of decay heat removal systems2

1.2.1 Reactor Core

The reactor core (Refer figure 1.1) is the source of heat from nuclear fission. A homogeneous core concept with two fissile enrichment zones of 21/28% of PuO2 is adopted for power flattening. The active core where most of the heat generated consists of 181 fuel subassemblies (SA). Each SA contains 217 helium bonded pins of 6.6 mm diameter. Each pin has a 1000 mm column of annular MOX fuel pellets and 300 mm each of upper and lower blanket columns. The clad material used is 20%CW 15Ni-14Cr-2Mo+Si+Ti (D9). Spent subassemblies are stored for one campaign in internal storage with one third of the active core being replaced during every fuel handling campaign. Two rows of blanket subassemblies are provided surrounding the outer fuel region. Twelve absorber rods, i.e., nine control and safety rods (CSR) and three diverse safety rods (DSR) are arranged in two rings. In addition, axial shielding is provided within the subassemblies and radial shielding subassemblies are provided within the core.1.2.2 Fuel sub- assembly

There are 181 fuel subassemblies in 8 rings (Refer figure 1.2). About 90% of reactor power is generated in the fuel SA. The fuel SA is vertically held on the grid plate. Each fuel SA consists of a foot and a handling head welded to the central hexagonal sheath, which houses 217 fuel pins arranged in triangular pitch. The coolant enters through multiple slots provided at 2 levels in the foot .The handling head of a fuel SA consists of an adapter to provide alternate path for flow in the event of blockage at the top and it has an internal groove to facilitate handling by transfer arm. Fuel pellets make up the active core region, and blanket pellets provide the axial boundaries. The fission gas plenum is located both above the upper blanket region, and lower gas plenum region (below the lower blanket).

The fuel SA is designed to contact at buttons formed on the 6 faces of the hexagon at the level of the middle of top axial blanket. Buttons are provided on the SA to act as restraining pads against bowing. They help to localize the restraint forces so that removal of SA from the core does not become complicated. They are located such at the core reactivity change due to bowing from normal operating state to fuel handling state is negative and minimum.

1.2.3 Reactor assembly

In this reactor the core, primary pumps, intermediate heat exchanger and primary pipe connecting the pumps and the grid plate are all housed in a large diameter vessel called the main vessel (figure 1.4). The main vessel is cooled by cold sodium to enhance its structural integrity. The core subassemblies and the grid plate are supported by core support structure which rests on the main vessel. A core catcher is provided at the bottom to take care of the meltdown of the subassemblies and prevents the debris from coming in contact with the main vessel. The main vessel is surrounded by the safety vessel which helps to keep the sodium level above the inlet windows of the intermediate heat exchanger ensuring continued cooling of the core in case of leak of main vessel. The sodium pool is divided into hot pool and cold pool by a thin structure known as inner vessel. The main vessel is closed at its top by top shield, which includes roof slab, large and small rotatable plugs and control plug. A concrete vault is constructed as a biological shielding in the radial and bottom axial direction outside the main vessel.

Figure 1.1: Reactor Core

Figure 1.2: Fuel subassembly

Figure 1.3: Reactor assembly

Figure 1.3: A schematic of a fuel pin1.2.4 Heat transport systemThe liquid sodium enters the core at 397 C and leaves at 547 C. The hot primary sodium is radioactive and therefore heat is transferred to the secondary sodium from the primary sodium through four heat exchangers. The non radioactive secondary sodium is circulated through two independent secondary loops, each having a sodium pump, two heat exchangers and four steam generators. The decay heat is removed using the operation grade decay heat removal system of maximum 20 MWt capacity in the steam water system under normal conditions. In case of failure of the steam water system, the decay heat is removed by a passive safety grade decay heat removal circuit consisting of 4 independent loops of 8 MWt capacities each. Figure 1.4 shows the flow sheet of main heat transport system.

Figure 1.4: Flow sheet of prototype fast breeder reactor.

1.3 Motive for the Project

The function of a coolant in a nuclear reactor is to remove heat from thenuclear reactor core and transfer it to electrical generators and the environment.The malfunctioning of any component related to the coolant flow can cause severe ramifications for the reactor and eventually lead to a disaster. Thus the coolant plays a critical role in the effective functioning of a reactor.

The situation envisaged is one where the shut down system has failed, and the coolant pump has tripped, thereby reducing the mass flow rate of the liquid sodium. With the same amount of power being generated in the core, and sodium flowing at a reduced rate, coolant temperature increases. This venture concentrates on the case of temperature increasing to the extent that the liquid sodium begins to boil.

The properties of vapor, being highly different from that of a liquid, will tremendously affect the rate and the manner of heat transfer from the reactor core. A vapor heat transfer coefficient is much lower than that of a liquid, owing to its vastly lower thermal conductivity, thereby massively reducing the heat it gains from the fission process. Thus the heat produced in the reactor core is not carried away effectively. This energy instead is accumulated in the cladding of the fuel rod.

The implications are that overheating of the fuel rod can arise, eventually causing melting of the cladding and the core. Further, voiding of the core occurs leading to reactivity changes resulting in increase in power produced causing severe thermal consequences.

In case of such a development, possessing the knowledge of the temperature of the boiling coolant, at various radial and axial sections of a fuel pin will be pivotal in taking appropriate measures to avert a disaster. The aim of this project is thus the development of a numerical model to simulate the temperature distribution in the boiling liquid sodium.

Chapter 2Concepts Involved

The fuel pin being cylindrical in shape, the general heat conduction equation in cylindrical coordinates will be employed [4]

(2.1)It is often conjectured that heat transfer in angular and axial directions are negligible compared to that in the radial direction. So the equation is simplified to ignore the terms dependant on angular positions and height, resulting in

(2.2) Taking k / ((Cp) = ( (thermal diffusivity)

(2.3) The heat generation term here, q, will be zero if there are no heat generation sources within the material.

Heat Transfer is broadly classified into Steady State Heat Transfer and Transient Heat Transfer depending on whether the properties involved are independent of time or not, respectively. These concept are described below

2.1 Steady State Heat Transfer

Steady state essentially implies that all properties are independent of time. In case of heat transfer, a Steady State condition means that temperature of all positions will not change with time, irrespective of their variation with space coordinates. Further no heat will be accumulated by any material, with all the heat transferred in being of the same magnitude as the heat leaving the material. This method is applied to the study pursued below.

In the general heat conduction equation adopted, the term ((T/(t) now becomes zero, and it simplifies to

(2.4)

On solving this differential equation, a generic equation for temperature of the MOX pellet as a function of radius is obtained :

(2.5)Here k thermal conductivity of the material

And c1 and c2 are constants, whose values become known by applying suitable boundary conditions.

For the regions where no heat is accumulated and the heat flow remains a constant value, found using the relation [4]

Heat Transferred = Temperature difference / Thermal Resistance (2.6).The Thermal Resistance in the above equation depends on whether the mode of heat conduction is conduction or convection.

If it is radial conduction, Thermal resistance is

(2.7)

In case of convection, Thermal resistance is

(2.8)

The Heat Transfer coefficient is often found using a relation for the Nusselt number

(2.9)h heat transfer coefficient

Nu Nusselt numberK thermal conductivity of medium

D Diameter of area of flow

The Nusselt number calculation varies depending on the type of medium to which heat is being convected. In case of liquid metals, the correlation used is [4] -

Nu = 5 + 0.025(Re * Pr)0.8 (2.10)

Where Re Reynolds number and Pr Prandtl number of medium

To find Re, the equation [4]

(2.11)

Where u velocity of flow, d - Diameter, Kinematic Viscosity

In case of a non circular pipe, the Hydraulic Diameter is used to find the Reynolds number.

This is calculated with the relation [4]

(2.12)2.2 Transient Heat Transfer

Up till now, the concept of steady state heat transfer has been used, which is evidently not realistic. No material can have zero accumulation of heat. Some of the energy is absorbed by the substance and thus causes its temperature to vary with time.

In the equation being used here -:

The formulation of transient heat conduction problems differs from that of steady ones in that the transient problems involve an additional term representing the change in the energy content of the medium with time. This additional term appears as a first derivative of temperature with respect to time in the differential equation, and as a change in the internal energy content during in the energy balance formulation. The nodes and the volume elements in transient problems assuming all heat transfer is into the element for convenience, the energy balance on a volume element during a time interval can be expressed as [5]-

where the heat transfer rate normally consists of conduction terms for interior nodes, but may involve convection, heat flux, and radiation for boundary nodes.Noting that E element = mCT = V element CT , where r is the density and C is the specific heat of the element, dividing the relation above by t gives [5]

or, for any node m in the medium and its volume element,

whereandare the temperatures of nodemat timesti= iandti+1=(i+1),respectively, andrepresents the temperature change of the node during the time intervalbetween the time stepsi andi+1.Note that the ratio , is simply the finite difference approximation of the partial derivative ((T/(t) that appears in the differential equations of transient problems. Therefore, we would obtain the same result for the finite difference formulation if we followed a strict mathematical approach instead of the energy balance approach used above. Also note that the finite difference formulations of steady and transient problems differ by the single term on the right side of the equal sign, and the format of that term remains the same in all coordinate systems regardless of whether heat transfer is one, two or three dimensional. The nodal temperatures in transient problems normally change during each time step, and it is often contemplated whether to use temperatures at the previous time step i or the new time step i+1 for the terms. Both are reasonable approaches and are used in practice. The finite difference approach is called the explicit method in the first case and the implicit method in the second case.

2.2.1 Eulers Explicit Method

The time derivative is expressed in forward difference form in the explicit case. Note that the formulation is an expression between the nodal temperatures before and after a time interval and is based on determining the new temperatures using the previous temperatures

Explicit Method -: all sides Qi + Qigen = Velement C (Ti+1 m - Ti m ) / t (2.13)The explicit given above [5] is quite general and can be used in any coordinate system regardless of the dimension of heat transfer. The volume elements in multidimensional cases simply have more surfaces and thus involve more terms in the summation.

The explicit method is easy to use, but is suffers from an undesirable feature that severely restricts utility [5]: the explicit method is not unconditionally stable, and the largest permissible value of the time step t is limited by the stability criterion. If the time step t is not sufficiently small, the solutions obtained by the explicit method may oscillate wildly and diverge from the actual solution. To avoid such divergent oscillations in nodal temperatures, the value of t must be maintained below a certain upper limit established by stability criterion. It can be shown mathematically or by a physical argument based on the second law of thermodynamics that thestability criterion is satisfied if the coefficients of all Ti m and Ti+1 m in the expressions (called the primary coefficients) are greater than or equal to zero for all nodes m. Of course, all the terms involving for a particular node must be grouped together before this criterion is applied.

Different equations for different nodes may result in different restrictions on the size of the time step t, and the criterion that is most restrictive should be used in the solution of the problem. A practical approach is to identify the equation with the smallest primary coefficient since it is the most restrictive and to determine the allowable values of t by applying the stability criterion to that equation only. A t value obtained this way will satisfy the stability criterion for all other equations in the system.The condition that will be used here will be [5] -

(2.14)When the material of the medium and thus its thermal diffusivity ( is known and the value of the mesh size is specified, the largest allowable value of the time step t can be determined from the above relation. For example, in the case of a brick wall ( = 0.45 x 10-6m2/s) with a mesh size of x = 0.01m , the upper limit of the time step is

The boundary nodes involving convection and/or radiation are more restrictive than the interior nodes and thus require smaller time steps. Therefore, the most restrictive boundary node should be used in the determination of the maximum allowable time step when a transient problem is solved with the explicit method.2.2.2 Eulers Implicit Method

The time derivative is expressed in backward difference form in the implicit case.

Implicit Method -: all sides Qi + Qigen = Velement C(Ti m - Ti-1 m) / t (2.15)The implicit given above [5] is quite general and can be used in any coordinate system regardless of the dimension of heat transfer. The volume elements in multidimensional cases simply have more surfaces and thus involve more terms in the summation.

The disadvantage of the implicit method is that it results in a set of equations that must be solved simultaneously for each time step. The set of equations when expressed as a matrix form a tridiagonal matrix. These matrices in general are more complicated to solve than the simple Marching Method of the Explicit Technique and hence this method is used only when the Explicit formulation of the same problem requires an absurdly small and a non feasible time step.

There are many technique used to solve equations involving tridiagonal matrix. A popular method used is the Thomas Algorithm [6] which is described below

a A vector whose elements are the diagonal elements of the tridiagonal matrix

b - A vector whose elements are the lower diagonal elements of the tridiagonal matrixc - A vector whose elements are the upper diagonal elements of the tridiagonal matrixf - A vector whose elements are constant valuesIf n = length of fv A vector of n elements whose initial values are indeterminate. This will help the calculationsy A vector consisting of the unknowns

If a(i), b(i), c(i), represent the ith elements of the a, b and c vectors, the code to solve for unknowns in y is -

y = v;w = a(1);y(1) = f(1) / w ;for i = 2 : n v(i-1) = c(i-1) / w ; w = a(i) - (b(i) * v(i-1)); y(i) = (f(i) - b(i)*y(i-1)) / w ;endfor j = (n-1) : -1 : 1 y(j) = y(j) - (v(j) * y(j+1));end2.2.3 Finite Volume Method

The finite volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations. Values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh.. For each control volume the basic heat transfer equation is applied-

Heat transferred in - Heat transferred out + Heat generated within the Control volume =

Rate of change in internal energy of the volume

This finite volume technique is applied to the terms on the LHS of (2.21) while the terms dependant on time will have the finite difference method applied.

For all control volumes except those at the ends, the full control volumes are considered for analysis. The control volumes at the ends are analyzed by considering only half their volumes, in a method called half control volume analysis.

So the control volumes in the middle, apart from those at the ends will have similar equations, while those at the ends will have a unique equation.

For all the control volumes in the fuel region -

Heat transferred in/out = change in temperature / thermal resistance

Heat generated = Heat generated per unit volume * Volume

Rate of Change of internal energy = mass * Specific heat * Rate of change in temperature

(2.16)

For eq 2.16, the mass of a CV is needed. This is estimated by -

Mass = density * Volume

.Volume of a CV is found by using

V (i) = * (r(i)2) - r(i-1)2)* l (2.17)

Where r(i) and r(i-1) represent radial distance of the outer edges of the ith and (i-1)th control volumes from the centre and

l is the length of the material

The Finite Volume Method can be used with Eulers Explicit and Implicit Method depending on whether the Derivative of Temperature with respect to time is taken as a forward or a backward derivative. Both techniques are discussed in Chapter 3.2.2.4 Finite Difference Method

Finite-difference method is numerical method for approximating the solutions to differential equations using finite difference equations to approximate derivatives. It involves discretizing the problem in the space variables and solving for temperatures at discrete points called the nodes.The general heat conduction equation in cylindrical coordinates is -

When the finite difference method is applied the above equation becomes

(2.18)Where

T(i,j) Temperature of the ith node at the jth time step T(i+1,j) Temperature of the (i+1)th node at the jth time step T(i-1,j) - Temperature of the (i-1)th node at the jth time stepr Radial distance of the node from the centre

r distance between 2 nodes

t time step

q heat generated per unit volume

k thermal conductivity of the material

thermal diffusivity of the material

T(i,j+1) Temperature of the ith node at the next time

The above equation is written assuming explicit technique is used. However even the Implicit Technique can be used, wherein the RHS of the equation becomes

The above equation though is applied only if the mode of heat transfer is only conduction.

In the first and the last nodes, there is a boundary/interface where convection heat transfer occurs and so the above equation cannot be applied

In the first and the last nodes, the finite control volume's half control volume approach is applied with the basic equation being -

Heat transferred in - Heat transferred out + Heat generated within the Control volume =

Rate of change in internal energy of the volume

So mathematically this becomes

m*C*(Change in temp) / time step = qin - qout + q(generated)

If the heat transferred either in or out at a boundary is through conduction, in the finite difference method it is evaluated as

K* Am * T / r (2.19)Where Am = (Ai + Ao) / 2

Ie the mean area of cross section between 2 nodes

If the heat transferred in or out is through convection, it is evaluated as

hAT (2.20)

2.3 Boiling

Boiling is the rapid vaporization of a liquid, which typically occurs when a liquid is heated to a temperature such that its vapor pressure is above that of the surroundings, such as air pressure.

When boiling occurs, interesting consequences occur for the mediums temperature as well as enthalpy.

When a liquid medium receives energy, its temperature increases continuously as long as boiling does not occur. When boiling begins however, the temperature remains constant at the value it had when boiling first began. During the course of boiling, the entire mass of liquid is converted to gaseous state, and the thermal reading retains the same magnitude till the last drop of liquid is vapourized. This phenomenon is explained by the fact that the energy goes into breaking the intermolecular bonds, but the average kinetic energy stays invariable and so does the temperature until all of the bonds are broken and the substance is in the vapor state.

The enthalpy of the medium however does not remain constant during the boiling. The enthalpy is the measure of the total energy of the material and since heat is continuously supplied during the process, it cannot remain constant. The question often arises as to how to calculate the enthalpy of a mixture of liquid and vapour during the process of boiling, ie- when the entire liquid has not completely become vapour. The enthalpies of liquid and vapour media of various substances have been tabulated but there is never info available directly for a liquid vapour mixture.

The enthalpy of a boiling mixture is determined using the notions of Enthalpy of Vapourisation and Quality of a liquid-vapour mixture.

The Enthalpy of Vapourization, (symbol Hlv) also known as the (latent) heat of vaporization or heat of evaporation, is the enthalpy change required to transform a given quantity of a substance from a liquid into a gas at a given pressure (often atmospheric pressure, as in STP). The heat of vaporization is temperature-dependent, though a constant heat of vaporization can be assumed for small temperature ranges and for reduced temperature Tr