Solving of differential equation of heat conduction with differnt numerical methods is a very useful to estimate th heating/cooling curves of a specimen

Differential equation of heat conduction:

Differential equation in 1D (plate-like specimen):

λ: heat conduction coefficient(W/(mK)), ρ : density(kg/m³),

c specific heat capacity (J/(kg°C)),

Divide the plate N equal part:

T₀ T_L

X_i: position coordinate

Solving of differential equation of heat conduction with differnt numerical methods is a very useful to estimate th heating/cooling curves of a specimen.

In the course of explicite differences method we divide the plate to N qual part. At the boundary of parts can we calculate the temperature. The division may be 3 dimensional depend on the specimen geometry. Most of cases the 1 or 2 dimensional division give a good approximation. For this calculation necessary to known the material properies like heat conduction coefficient or density.

t

Explicite finite differences method

x₀ x₁ x₂ X_N-1X_N

T(x,t): the tremperature inside the plate function of time and position

Temperature at discrete time t_nand position j:

x

Approximation of first derivative with respect to time:

Approximation of first derivative with respect to position:

Th temperature of plate at time t and position x: T(x,t). The discrete value of this function at discrete time n and at discrete position j: Tjn

. The first and second derivatives in the differential equation can we approximate with finite differences. The approximation is better if the time interval and distance of discrete positions are smaller.



Approximation of scond derivative:

Th approximation of differential equation of heat conduction with finite differences:

Temperature at time n+1:

  ^{x a}

It is possible to calculate the temperature field inside the plate at time n+1 from temperature field at time n.

Explicite finite differences method

Approximate the second derivative of temperature respect to place, and substitute the first and second derivatives in the differential equation. It is noticeable that the temperature may be calculated at time n+1 inside the plate, if we known the temperature field at time n. The temperature at position j can we calculate from position j-1 and j+1. Now there is a problem with calculating the temperature at the boundary of plate. Then we must to use boundary conditions to calculate this temperatures.

Boundary conditions

Of first kind: the temperature is known on the sheet boundary

Then known: ^{n 1}

The most widely used boundary condition of the third kind:

Heat transfer equations:

After substitution the temperature on the plate boundary:

  _

With similar methods can generate an equation for the left side as well.

Explicite finite differences method

To boundary condition stand-up we must to known the properties of real heat transfer process. We can use the boundary condition of first kind if we know the temperature on the boundary of plate. In the reality this is very rare. Heating or cooling in a furnace we must to calculate with heat transfer equations. This is possible with boundary condition of third kind. Now we must to assume two new coordinates. One on the position -1 and one on the position N+1. At this positions the temperature is equal to the furnace temperature. If we use this simplification then possible to calculate the temperature on the boundary of plate.

Initial condition:

  x f T

_j⁰



The initial condition is the temperature field at time t=0, it must known to start the algorithm.

There are several condition wich must to use for right computation, the most important:

The value of r in most cases is between 0.2..0.3.

From this the maximum time interval for givenΔx and a :

 

The method of explicite finite differences give a very good approximation for practice. The 2 or 3 dimensional equation may be expressed simply on the basics of 1D case.

Explicite finite differences method

The next step is to give the initial condition. The initial condition is the temperature field at time 0.

This means, that at time 1 (after time interval Δt) it is possible to calculate the temperature field.

Very important to select the right value of time step interval, wich limited with a and Δx.

Differences between boundary conditions of first and of third kind.

Explicite finite differences method

The left video presents the temperature field of plate heated in a furnace with firs-order boundary condition, while the right video with three order boundary condition. The first order boundary condition give a bad result, but the three order boundary condition give a good result. The first order boundary condition say that the temperature on the boundary of plate is equal to the temperature of the furnace. This is not true, because the temperature decrease near the surface of plate. There is heat transfer witr estimated coeffitients. The left side of charts is the left side of plate, and the right side of chart presents the right side of plate.

Modeling of transformations in materials

Modeling of transformations in metals and alloys is possible with time-temperature parameters or with kinetic functions.



t,T,x



0

F 

If the temperature is constans, then thn we assume isothermal kinetic function:

Pearlitic transformation, isothermal case:

T (°C)

t (s)

Beginning of transformation: 1% transformed volume fraction

Transformation finish curve: transformed volume fraction is 99%

Models for transformation processes are empirical equations, time-temperatur parameters or kinetic functions.The amount of transformed fraction can expressed in the form: X=Vtransformed/Vtotal where Vtransformed and Vtotal is the transformed volume and total specimen volume, respectively.The beginning of transformation connected to 1% transformed fraction (because measurement reasons), while the finish of transformation connected to 99%transformed fraction. The broken line presents the maximum temperature of bainitic transformation, between this temperature (~500-550°C) and Ms temperature austenite transformed to bainet and not to pearlite. The bainitic transformation can be described as the pearlitic.

Modeling of pearlitic transformation, isothermal case T (°C)

t (s) X (-)

t (s) 100%

99%

1%

Isotherm TTT curves

Estimate the isothermal kintic function

Isotherm transformation curves

Time required for 1%

and 99% transformed fraction

Modeling of transformation

t_k t_v

T₁

T₂

Modeling of transformations in materials

Isothermal TTT-curves give two points of isothermal kinetic funcion. If we know this two point, we can estimate the parameters of kinetic function. The parameters of kinetic function depend strongly on temperature. Now we must slect a kinetic function, and evaluate its parameters.

Modeling of pearlitic transformation, isothermal case:

The kinetic function is now the well known Avrami-equation:



Th time required for 1% and 99% transormation: t_kand t_v

Then 1% an 99% transformed fraction can be expressed in the form:

and Solve this equation for n and k:

 and in another form:

Modeling of transformations in materials

Thermally activated transformation processes often described with Avrami equations. This is a two-parameter kinetic function. Tke first two-parameter called transformation rate constans (k) most of cases depend on the temperature according to Arrhenius equation. The second parameter, the Avrami exponent has a geometrical meaning, but most of expriments show that the value of these exponent is not corresponding with its theoretical value. The Avrami exponent in most of cases have temperature and volume fraction dependence.

Example for modeling of pearlitic transformation

T (°C)

t (s) 50CrMo4 isothermal transformation diagram, choose T=580°C

t

_k

t

_v

Modeling of transformations in materials

Tis figure shows the isothermal transformation curves for steel 50CrMo4. Choose from the chart a transformation temperature, for example 580°C. Then estimate the time for 1% and 99%

transformed fraction. These values depend very strongly on the austenitising temperature and time (austenite grain size), chemical composition, initial properties of specimen.This means that the constans of kinetic function depend strongly on this parameters.



Time required for 1% and 99% transformed fraction: t_k=395s and t_v=16000s From this the value of n and k:

1.00E+02 1.00E+03 1.00E+04 1.00E+05

t (s)

Modeling of transformations in materials

On the previous slide we were estimated the time required 1% and 99% transformed fraction. From these times now we can evaluate the Avrami parameters n and k. Then we plotted the transformed fraction of pearlitic transformation as a function of time. This simulation is true only at temperature 580°C. At another temperatures the transformation can be calculated as this example. In fact, the isothermal transformation is very rare. It is important to choose the right kinetic function. For example for reversible and irreversible processes or diffusional or martensitic transformation we must choose different kinetic functions.

Modeling of transformation processes, non-isothermal case:

Start from the isothermal kinetic function: F



t,T,x



0 Take the derivative respect to time: ^f



^t,T,x



dt dx  Most cases the isothermal kinetic function can written as:

Take the derivative respect to time:

 

x K

 

T,t 0

The transformation rate depend on the tmperature and transformed fraction. If the temperature change according to function T=T(t), then:

      _

¹

   

From this the general kinetic funcion:

     

Modeling of transformations in materials

The most of heat treatments becomes not in isothermal conditions. This means, that the isothermal kinetic function is not true. But the rate of process can expressed at arbitrary temperature and transformed fraction. By most of cases we can separate the variables, then the sum of two funcions give the kinetic function. F(x) is a function of transformed fraction, and K(T) is a function of temperature. If we know the transformed fraction and temperature at time t, then we can evaluate the transformed fraction at time t+dt, dt is infinitesimal. The series of numerical values of temperature can we calculate the transformed fraction.

Modeling of transformations, non-isothermal case

The transformation rate can often expressed with this differential equation:

 

₀ ^a1

 

^{a2 a3}

No. Differential equation Solution with initial condition

x(0)=0 Parameters

Modeling of transformations in materials

The table content kinetic functions for modeling thermally activated processes. No. 1 is the classical Avrami kinetic function, this is useful for modeling transformations in steels, recrystallization, precipitation. The second is the well known Austin-Rickett kinetic function, used for modeling of precipitation. The No. 3 is the Beck’s kinetic function, applicable to describe the grain coarsening.

Modeling of grain coarsening by austenitising Evaluating of Ac3 temperature

3 3

• c=3…25, depend on the initial microstructure,

• β heating rate

Size of austenite grain size in isothermal case:

Kinetic differential equation:

from this the general kinetic function:

 

ⁿ

Modeling of transformations in materials

The amount of constituent after transformation of austenite depend strongly on austenite grain size.

The austenite grain size increase with a power function. The non-isothermal kinetic function can expressed if we take the derivative respect to time of isothermal kinetic function, and after that subtituting the time with a funcion of grain size. If we solving this differential equation, then the grain size may calculate in non-isothermal case as well.

Modeling of phase transformations of steels

Isothermal case: b_i: maximal amount of

constituent i

j: number of constituent Y: total transformed fraction

 

^1,i

 

^{2,i a3,i}

Eutectoid steel phase transformation during continous cooling:

Modeling of transformations in materials

The total volume fraction is the sum of folume fractions of constituent. The amount of constituent can estimate to solve the diffrential quation of individual constituent. Charts show the transformation of an eutectoid steel after austenitisation. The simulation calculated with linear continous cooling curve with cooling rate β. The chart on the right side show the amount of constituents during transformation.The first time starts pearlite transformation after this bainite and martensite.

x_martensite

Kinetic treatment of martensitic transformation:



^{1 x x x}

 ^

^{1 exp}

^

^0.011

^

^{M T}

^{  }

x_martensite  _ferrite _pearlite _bainite    _s

This is an approximation for M_stemperature:

Mo 21 Ni 17 Cr 17 Mn 33 C 474 561

M

_s

      

ΔT= M_s-T (°C) Rest austenite

where Ms the martensite start temperature, T is the temperature of cooling below Ms.

Modeling of phase transformations of steels

Modeling of transformations in materials

The amount of martnsite depend mainly on the cooling temperature below Ms. The amount of martensite can approximate with equation Koistinen-Marburger. The temperature Ms approximated from chemical composition. The amount of martensite is maximal, if the temprature of specimen decrease until Mf temperature.

Estimation of Hardness after continous transformation of autenite

isothermal transformation the hardness of constituent depend on temperature, that’s called isothermal hardness.

The final hardness can estimated with th series of isothermal hardness. Hardness component for given Tk temperature:

mart

bainite and martensite at given temperature, respectively.

mart mart HV x

HV

HV



   and the final hardness of specimen:

Modeling of transformations in materials

The hardness of constituent depend on the chemical composition and the temperature of formation.

The hardness of a constituent at given temperature called isothermal hardness.The final hardness of specimen can approximated with a sum of isothermal hardness values.

Modeling of tempering

Modeling of tempering using time-temperature parameters

Tempering model

Time of tempering Temperature of tempering

Chemical composition, austenitising parameters, etc.

Effect of time and temperature

Time- temperature parameter (Hollomon-Jaffe):

 

 lg t k 

T

P   

where T temperature of tempering, t time of tempering, k constant.

• Parameters of this type have not physical meaning, but approximate very good the experiments.

• Connect a time-temperature parameter and a mechanical properties.:

c

where HV is the Vickers hardness, a,b,c are constans.

Modeling of transformations in materials

The mechanical properties after tempering may be exactly adjusted using time temperature parameters. The two main parameter, wich affected tempering process is the temperature and time.

Hollomon and Jaffe disovered a time-temperature parameter, wich consider the effect of tempering temperature and time as well. With this complex parameter it can be predicted the material properties during tempering. The time-temperature parameter connected to a material properties with a polinomial or power function.

Modeling of tempering

Time–temperature parameters

No. Time temperature parameter Calculating of hardness using time temperature parameter the hardness of tempered specimen.

Modeling of transformations in materials

The table contents several time-temperature parameters. Time-temperature parameters have not physical meaning. The temperature hat a greater effect to tempering process. Thiss appears in the equations as well: the parameter depend with an exponential function of temperature, or with logaritmic function of time. These parameters work in isothermal case.

Modeling of tempering during temperature change

Several time-temperature parameter may be usded for modeling tempering during continous heating/cooling.

No. Time temperature parameter Equation for hardness calculation

1.

The theoretical basics are clarified korábban.

There are several time-temperature parameter, and an application for hardness esttiumation:

Modeling of transformations in materials

Time-temperature parameters may be used for non-isothermal tempering proceses as well. In this case, the non-isothermal equation can be expressed from the isothermal equation.

Tempering charts

P, time-temperature parameter HV hardness

t, time of tempering, (h) T, temperature of tempering (°C)

Modeling of transformations in materials

In the practice two types of tempering charts used. On the left side on the x-axis is the time-temprature parameter, and on the y-axis is the hardness. The parameter P may be calculated from time-temperature history.

Modeling of recrystallization

n,k=f(x) Local Avrami exponent and rate konstans

Apaptation of parameters by isothermal case:

The rate of process is depend on:

• the temperature

• the amount of deformation

• the deformation rate

• the initial grain size

• Stb.

Effect of this parameters appears in functions.:



After repeating on several temperatures:

Modeling of transformations in materials

Recrystallization is a softening process, wich take place cold formed materials during heating. The kinetic differential equation depend on many parameters. The rate of process is very sensitive of the temperature change. A few grade change of temperature occurs very large differences in transformation rate. For this reason, the calculation of parameters need special experiments and mathematical procedures.

 

₀ ^a1

 

^{a2 a3} can be predicted from isothermal or from non-isothermal temperature experiments

Modeling of recrystallization

Modeling of transformations in materials

Recrystallization process during temperature change can be modelled with several methods. The chart shows the temperature of a specimen and the transformed fraction during this temperature change. The transformed fraction can be connected with several material properties for example hardness or conductivity. Material properties can be adjusted during heat treating with exact models.

Modeling of welding

Initial parameters:

• Material properties

• Initial temperature

• Welding process parameters:

• welding direction

• welding velocity

• welding power

• dimensions of ellipsoid

• absorption coefficient

Fem grid of gap with lack of root fusion:

Figures show sveral input paramters for welding simulation. The most important parameter is the density of nodes. For validation of models necessary take experiments. The input parameters are matrial properties, initial and boundary conditions, welding parameters.

In document Materials technology (Pldal 88-112)