Module 06: Transient Thermal Analysis

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1 © 2019 ANSYS, Inc.

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Module 06: Transient Thermal Analysis

ANSYS Mechanical Heat Transfer

Release 2019 R3

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Module 06 Topics

1. Transient Theory 2. Time Stepping 3. Transient Loading

4. Transient Postprocessing 5. Phase Change

6. Workshop 06.1 – Soldering Iron

7. Workshop 06.2 – Phase Change

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06.01 Transient Theory

Like steady-state analyses, transient analyses may be linear or nonlinear.

If nonlinear, the same preprocessing considerations apply as with steady state nonlinear analysis.

The most significant difference between steady-state and transient analyses lies in the Loading and Solution procedures.

We will focus on these procedures after a brief presentation of the numerical

methods employed during transient thermal analysis.

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Recall the governing equation for thermal analysis of a linear system written in matrix form. The inclusion of the heat storage term differentiates

transient systems from steady-state systems:

In a transient analysis, loads may vary with time . . .

. . . or, in the case of a nonlinear transient analysis, time AND temperature:

Heat Storage Term = (Specific Heat Matrix) x (Time Derivative of Temperature)

06.01 Transient Theory

𝑪 ሶ𝑻 + 𝑲 𝑻 = {𝑸}

𝑪 ሶ𝑻 + 𝑲 𝑻 = {𝑸(𝒕)}

𝑪(𝑻) ሶ𝑻 + 𝑲(𝑻) 𝑻 = {𝑸(𝑻, 𝒕)}

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Time-Varying Loads Time-Varying Response

When the response of a system over time is required (due to time varying loads and/or boundary conditions in conjunction with thermal mass effects), a Transient Analysis is performed.

Thermal energy storage effects are included in a transient solution.

06.01 Transient Theory

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Time has a physical meaning

• For steady-state, time is used to track loading history

• For transient, thermal mass, thermal inertia and rate-dependence are active

You can turn off thermal inertia, or time integration effects on a load step basis, in the Analysis Settings Details

• Useful for introducing steady state solutions into the loading history

• For example, to initialize temperatures to a steady-state solution

06.01 Transient Theory

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Material Property Considerations for Transient Analyses:

In addition to thermal conductivity (k), density (r) and specific heat (c ) material properties must be specified for entities which can conduct and store thermal energy.

These material properties are used to calculate the heat storage characteristics of each element which are then combined in the Specific Heat Matrix [C].

06.01 Transient Theory

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• Nonlinear solutions in ANSYS Mechanical are

fundamentally based on the full Newton-Raphson iteration procedure.

• For transient thermal analysis cases where conductivity nonlinearities are mild, a Quasi Newton-Raphson

algorithm, the Fast-Thermal Transient Solver, is also offered.

Mechanical will use this solution option by default. This Corresponds to the THOPT APDL command

The setting can be overridden in the “Analysis Settings Details”

06.01 Transient Theory

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QUASI Solver

• Speeds up solution time by avoiding the reformulation of the systems conductivity matrix for each time step or iteration.

• Certain physics features require full Newton-Raphson.

• Highly nonlinear solutions may be more efficient with full Newton Raphson.

• Two flavors:

Multipass Iterative

06.01 Transient Theory

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The temperature of a transient thermal system changes continuously from instant to instant:

When performing a thermal transient analysis, a time integration procedure is used to obtain solutions to the system equations at discrete points in time. The change in time between solutions is called the integration time step (ITS).

T

t

T

t

D t

t

n

t

n+1

t

n+2

Generally, the smaller the ITS, the more accurate the solution becomes.

06.02 Time Stepping

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The time integration operator is modifiable and is based on generalized trapezoidal rule:

• θ (THETHA) or Euler parameter

• 1.0 -- Backward Euler

• 0.5 -- Midpoint or Crank-Nicholson

• Program selected and defaults to 1.0 for most ANSYS Mechanical analyses

• To guarantee stability, 𝜽 ≥ 0.5

• θ is changed with the APDL command TINTP via a command object

OSLM, the oscillation limit will be discussed in later slides

06.02 Time Stepping

𝑻

𝒏+𝟏

= 𝑻

𝒏

+ 𝟏 − 𝜽 𝚫𝒕 ሶ𝑻

𝒏

+ 𝜽𝚫𝒕 ሶ𝑻

𝒏+𝟏

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• Selection of a reasonable time step size is important because of its impact on solution accuracy and stability:

If the time step size is too small, then solution oscillations may occur which could result in temperatures which are not physically meaningful (e.g. thermal undershoot).

If the time step is too large, then temperature gradients will not be adequately captured.

• One approach is to specify a relatively conservative initial time step and allow Automatic Time Stepping to increase the time step as needed.

• The guidelines on the following slides are presented as a way to

approximate a reasonable initial time step size for use with Automatic Time Stepping.

06.02 Time Stepping

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• Approximate a reasonable time step size for thermal transient ANSYS use the Biot and Fourier numbers.

• The Biot Number is the dimensionless ratio of convective and conductive thermal resistances, where D x is the mean element width, h is the average film coefficient, and K is an averaged conductivity.

• The Fourier Number is a dimensionless time ( D t/t ) which quantifies the relative rates of heat conduction vs. heat storage for an element of width D x: Where r and c are averaged density and specific heat, respectively.

)

2

( 4

x C

t Fo K

D

= D r

K x Bi h D

=

If B

i

< 1, then we use the F

0

to calculate D t. Otherwise, we use B

i

∙F

0

to calculate D t.

06.02 Time Stepping

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For example reasonable time step size for thermal transient analyses dominated by conduction can be approximated using the “Fourier number”:

Where:

‐ Δt is ITS time step (Initial Time Step)

‐ x is the average element length

‐ K is the average thermal conductivity

‐ ρ is average density

‐ C is average specific heat

A suggested minimum initial time step (ITS):

If Δt is 100 times the ITS suggestion, ANSYS issues a warning.

)

2

( 4

x C

t Fo K

D

= D r

K c t x

4

2

r

= D D

06.02 Time Stepping

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To help evaluate the accuracy of the time integration algorithm, ANSYS computes and reports some helpful quantities after every solution:

The Response Eigenvalue represents the dominant system eigenvalue for the most recent time step solution (reported in Solution information). Can be viewed as a Fourier Number for the discretized system.

The Oscillation Limit is a dimensionless quantity that is simply the product of the Response Eigenvalue and the current time step size (reported in Solution information).

     

  T T T   C K   T T

T

r D D

D

= D

r

t n

f = D 

It is typically desirable to maintain the oscillation limit below 0.5 to ensure that the transient response of the system is being adequately characterized.

06.02 Time Stepping

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By default, the Automatic Time Stepping (ATS) feature bases time step prediction on the Oscillation Limit. ATS seeks to maintain the Oscillation Limit below 0.5 within a tolerance, and will adjust the ITS to satisfy this criterion.

Notice how ATS gradually reduces the ITS based on the Oscillation Limit. This sample was taken from the ANSYS Output Window

during a nonlinear transient analysis.

Time step metrics can be viewed in the Solution Information.

06.02 Time Stepping

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Factors that influence the automatic time stepping algorithm:

• Rate of convergence.

• Limits on time step size set by user.

• Minimum recommended time step.

• Oscillation limit (eigenvalue) calculation.

Typically, an analyst will have good experience regarding the appropriate time step size for the problem class.

• If not, a transient time step size convergence study may prove useful.

• Analogous to mesh convergence studies that are used to determine when spatial discretization is sufficiently accurate.

06.02 Time Stepping

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While loads can be applied as constants in transient analyses, often they vary with time.

In Mechanical, thermal loads can be defined as constants, tables or functions.

Here we will illustrate using specific examples.

Table Loads Function Loads

06.03 Transient Loading

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Example 1: the heating coil experiences joule heating as power is cycled on and off at 1 second intervals:

‐ Notice in the table a small-time increment is used to ramp the load on and off quickly, simulating a step function.

‐ Each new time point must increase in value.

06.03 Transient Loading

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Example 2: the same heating coil undergoes sinusoidal loading according to the function (0.1+(0.1*sin(180*time))):

Notice the table is populated by evaluating the function at 200 equally spaced time points.

06.03 Transient Loading

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• In addition to time varying loads and thermal boundary

conditions, transient problems always have an initial condition.

• The simplest initial conditions is a homogenous /uniform temperature field.

• It is also possible to map an imported (or self-imported solution) spatially varying temperature field to an initial condition or time varying temperature constraint.

• As mentioned in an earlier slide, you can turn off time

integration effects in order to use a steady-state load step to provide an initial temperature field.

External Data Application

06.03 Transient Loading

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Post processing transient results is done by requesting results from particular time points:

‐ RMB on the graph or table at the desired time point and choose “Retrieve This Result”.

OR

‐ Enter the desired time in the details for a result and RMB “Retrieve This Result”.

06.04 Transient Postprocessing

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Often the desired quantity is the result

variation over time at a point rather than a contour of the overall model.

A graph is useful in displaying results vs.

time.

Here a temperature probe is scoped to a

local coordinate system and the temperature variation is plotted in the graph area.

06.04 Transient Postprocessing

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Phase Change - A change of energy to a system (either added or taken away) causes a substance to change phase.

‐ The Common phase change processes are called freezing, melting, vaporization, or condensation.

Phase - A distinct molecular structure of a substance, homogeneous throughout

‐ There are three principal phases:

ANSYS Analyses

Liquid Gas Solid

06.05 Phase Change

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Latent Heat:

‐ When a substance changes phase, the temperature remains constant or nearly constant throughout the change.

‐ For example, solid ice at 0 C is ready to melt:

➢ Heat is added to the ice and it becomes liquid water.

➢ When the ice has just become completely liquid, it is still 0 C.

‐ Where did the heat energy go, if there was no temperature change?

➢ The heat energy is absorbed by changes in the molecular structure of the substance.

➢ The energy required for the substance to change phase is called its latent heat.

‐ A phase change analysis must account for the latent heat of the material.

‐ Latent heat is related using the enthalpy property which varies with temperature. Therefore, a thermal phase change analysis is non-linear.

cdT H

T

c H

= r

r

: to according )

( re temperatu and

), ( heat specific

), ( density to

related is

, Enthalpy,

06.05 Phase Change

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During phase change, a small temperature range exists where both the solid and liquid phases exist together.

The temperature at which the substance is completely liquid (the liquidus temperature) is T

L

.

The temperature at which the substance is completely solid (the solidus temperature) is T

S

.

ΔH, Latent Heat

T L T S

H A Change of Phase is Indicated by a Rapid

Variation in Enthalpy with Respect to Temperature.

T

T

S

= Solidus Temperature T

L

= Liquidus Temperature

Note: In this diagram, T

L

-T

S

is small. For a pure material, T

L

-T

S

would be zero.

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Applications involving phase change which can be approached using ANSYS Mechanical products are:

‐ The freezing (or solidification) of a liquid.

‐ The melting of a solid.

A phase change analysis must be solved as a thermal transient analysis.

Phase change analysis setup:

‐ Transient analysis type.

‐ A small initial and minimum time step sizes.

‐ Use automatic time stepping.

‐ Generally the “Line Search” solution option is preferred.

‐ ANSYS enthalpy data (material property) must be specified in units of energy/volume.

‐ Full Newton-Raphson solution algorithm.

‐ Set time integration parameter, θ, to 0.5

06.05 Phase Change

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Enthalpy Definitions/Calculations (reference):

Equations 1 through 7 can be used to calculate enthalpy values to enter as material properties 1. C

avg

= (C

S

+ C

L

)/2 : Average specific heat

2. C* = C

avg

+ (L / (T

L

– T

S

)) : Specific heat for transition

3. H

-

= p*C (T – T

0

) : Enthalpy below solid temperature 4. H

S

= p C

S

(T

S

– T

0

) : Enthalpy at solid temperature

5. H

TR

= H

S

+ pC (T

L

– T

S

) : Enthalpy between solid/liquid temperatures 6. H

L

= H

S

+ pC* (T

L

– T

S

) : Enthalpy at liquid temperature

7. H

+

= H

L

+ pC

L

(T – T

L

) : Enthalpy above liquid temperatureC

S

: specific heat of solid

C

L

: specific heat of liquidP: density

T

S

: solidus temperatureT

L

: liquidus temperatureL: latent heat

06.05 Phase Change

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Essential Steps for phase change analysis with ANSYS Mechanical

1. Define Enthalpy Curve in Engineering Data If enthalpy is defined, then specific heat properties are ignored during solution.

2. Activate Full Newton-Raphson as transient thermal solution option.

If transient thermal solver option defaults to the

“fast thermal transient” option - the QUASI algorithm - then enthalpy will be ignored.

06.05 Phase Change

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Please refer to your Workshop Supplement for instructions on:

Workshop 06.1 – Soldering Iron

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Please refer to your Workshop Supplement for instructions on:

Workshop 06.2 – Phase Change

06.07 Workshop 06.2 – Phase Change

Figure

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