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**Module 06: Transient Thermal ** **Analysis**

### ANSYS Mechanical Heat Transfer

### Release 2019 R3

**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.**

**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.**

**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)**

**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**

**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.**

**a Transient Analysis is performed.**

**Thermal energy storage effects are included in a transient solution.**

**06.01 Transient Theory**

**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**

**You can turn off thermal inertia, or time integration**

**• 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.**

**In addition to thermal conductivity (k), density**

### ‐ **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**

**• 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**

**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.**

**Generally, the smaller the**

**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**

**OSLM, the oscillation limit will**

**06.02 Time Stepping**

### 𝑻

_{𝒏+𝟏}

### = 𝑻

_{𝒏}

### + 𝟏 − 𝜽 𝚫𝒕 ሶ𝑻

_{𝒏}

### + 𝜽𝚫𝒕 ሶ𝑻

_{𝒏+𝟏}

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

**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 size is too small, then solution oscillations may occur which could result in**

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

**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.**

**• One approach is to specify a relatively conservative initial time step and**

**• 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.**

**approximate a reasonable initial time step size for use with Automatic**

**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.**

**is the dimensionless ratio of convective and conductive thermal**

**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. **

**is a dimensionless time (**

**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**

**t. Otherwise, we use B**

_{i}**∙F**

_{0 }**to calculate ** D *t.*

**06.02 Time Stepping**

**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**

**‐ x**

**‐ K** **is the average thermal conductivity**

**‐ K**

**‐ ρ** **is average density**

**‐ C** **is average specific heat **

**‐ C**

**A suggested minimum initial time step (ITS):**

**A suggested minimum initial time step (ITS):**

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

**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}

_{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**

**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.**

**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 **

**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.**

**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**

**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**

**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.**

**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**

**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**

**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**

**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?**

**‐ 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**

**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 **

**Indicated by a Rapid**

**Variation in Enthalpy with ** **Respect to Temperature.**

**Variation in Enthalpy with**

### T

**T**

**T**

_{S}**= Solidus Temperature** **T**

**= Solidus Temperature**

**T**

_{L}**= Liquidus Temperature**

**= Liquidus Temperature**

**Note:** **In this diagram, T**

**Note:**

**In this diagram, T**

_{L}**-T**

**-T**

_{S}**is small. For a ** **pure material, T**

**is small. For a**

**pure material, T**

_{L}**-T**

**-T**

_{S}**would be zero.**

**would be zero.**

**06.05 Phase Change**

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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**

**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 temperature** – **C**

_{S}**: specific heat of solid**

### – **C**

_{L}**: specific heat of liquid** – **P: density**

### – **T**

_{S}**: solidus temperature** – **T**

_{L}**: liquidus temperature** – **L: 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.**

**“fast thermal transient” option - the QUASI**

**06.05 Phase Change**

**Please refer to your Workshop Supplement for instructions on:**

**Please refer to your Workshop Supplement for instructions on:**

**Workshop 06.1 –** **Soldering Iron**

**Workshop 06.1 –**

**Soldering Iron**

**06.06 Workshop 06.1 – Soldering Iron**

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**Please refer to your Workshop Supplement for instructions on:**

**Please refer to your Workshop Supplement for instructions on:**

**Workshop 06.2 –** **Phase Change**

**Workshop 06.2 –**

**Phase Change**