**ANSYS Methodology****:**

This analysis was carried out by using Transient Thermal module in Ansys Workbench. A Transient Thermal analysis is one in which thermal boundary conditions depends on time. It follows basically the same procedures as a Steady-State Thermal analysis. The main difference is that most applied loads in a transient analysis are functions of time.

A transient thermal analysis can be either linear or nonlinear. Temperature dependent material properties (thermal conductivity, specific heat or density), or temperature dependent convection coefficients or radiation effects can result in nonlinear analyses that require an iterative procedure to achieve accurate solutions. The thermal properties of most materials do vary with temperature, so the analysis usually is nonlinear.

Moreover, to better understand the Transient Thermal analysis, there is a need to understand few dimensionless numbers which are discussed below:

**Fourier Number Fo:**

It signifies the degree of penetration of the heating or cooling effect through solid. For small values of α/S2, for instance a large τ is required to obtain a significant temperature change.

**Biot Number Bi:**

It signifies the ratio of the resistance to heat transfer at the surface of a solid to that within the internal body to that with in the internal body. It is somewhat similar to Nusselt number used in convective heat transfer.

**Dimensionless length parameter x/S:**

It indicates the distance of any location from an arbitrary origin as compared to the plate thickness. In case of solids having circular cross section this parameter will have the value (r/R).

Now, let us discuss a Transient Thermal problem and try to solve it through analytical procedure as well as simulate it in ANSYS to validate the results.

**Problem Statement****: **

A bar 12.5 cm diameter is heated in a furnace to a uniform temperature of 200°C. It is then left to cool in a forced air stream at 40°C, h = 150 W/mK.

- How long will it take for the center of the bar to cool down to 50°C?
- What will be the temperature of surface and at mid-radius at this instant? {K = 41.5 W/mK,
= 0.0725 m*α*^{2}/hr.}

**Analytical Solution****:**

At first the problem was solved analytically and then simulated in ANSYS Transient Thermal module for result verification.

- To calculate the time required by the Bar to cool down to 50°C first we need to find out dimensionless temperature distribution in solid by the given mentioned formula:

After finding the value of dimensionless temperature distribution in solid we need to find the Biot number from the below mentioned formula:

For the above value of 1/B_{i }the value of “F_{o}” from “Heisler chart for central temperature history in cylinder” is:

Since, F_{o }= *ατ*/S^{2}

Therefore,

- Now, for the value of 1/B
_{i }= 4.43 and r/S = 0.5, the value of correction factor from “Heisler Position correction factor chart for temperature history in cylinder” would be:

Correction Factor = 0.96

So the temperature distribution is given by:

t = 49.6°C.

Therefore, the temperature at mid-radius is given by:

Similarly the correction factor for r/S = 1 is found to be 0.89 and therefore the surface temperature is given by:

The problem is solved analytically and now the same problem is simulated in ANSYS Workbench for result verification.

**Geometry and Details: **

**Meshed Model****:**

**Boundary Conditions****:**

After setting up the boundary conditions the problem was solved to find out the Temperature distribution in the Bar in 1183 seconds.

**Result****:**

**Temperature Distribution****:**

**Temperature drop with respect to time****:**

**Conclusion****: **

After analyzing the result, it is observed that the temperature at mid-radius of the Bar was found to be 50.59°C and at surface it is 49.89°C.

**Final Deduction****: **

Transient Thermal analysis for the Cylindrical Bar was carried out using ANSYS, and it is observed that the analytical result matches with the simulated results.

Results obtained from analytical solution and Simulation is presented below: