FINITE ELEMENT MODELLING & ANALYSIS OF HEAT CONDUCTION IN A DUCTED CYLINDER USING THE FE

PACKAGE-ANSYS

6.1 Temperature distribution in the cylinder

Figure 1-Unrefined mesh Figure 2-Contour of nodal temperatures for unrefined mesh

Figure 3-Mesh with no ducts Figure 4-Contour of nodal temperatures for mesh with no ducts

Figure 5-Refined mesh Figure 6-Contour of nodal temperatures for mesh

Figure 1 above shows the set up for investigating the heat transfer problem. The inner radius on the left side is kept constant at a temperature of 200o while the outer radius on the right side is kept at a constant temperature of 15o. The upper line and the two lower straight lines left and right of the duct radius have no heat flux passing through them. The duct radius acts as a cooling surface with a constant heat flux of 1000W/m2 leaving the cylinder body. The mesh is initially unrefined and presented a warning related to three of the cells.

Figure 2 shows the nodal temperature distribution within the plate. It is observable that the separation lines between two colours (which are essentially lines of constant temperature) tend to bend towards the radius of the cooling duct and not remain parallel to the two inner and outer radii (as one would expect in the absence of the cooling duct and as it can be observed in Figure 3, where the cylinder with no cooling ducts is investigated). The cooling duct is effective in reducing the overall temperature throughout the element, as temperature below the outer constant of 15o are encountered (namely, values as low as 12.14o). These temperatures below 15o most likely occur in the vicinity of the cooling duct or on the boundary of the cooling duct. Cold temperatures are distributed across a larger surface, as compared to the case with no ducts in Figure 4.

A refined mesh is also investigated to ensure reliability of the proposed solution. The two sets of results are sufficiently close, however differences exist. Looking at Figures 2 and 6, constant temperature lines seem to be slightly different, with the ones in Figure 6 being more continuous. More importantly, the minimum temperature found from the refined investigation is 11.5162o, as compared to 12.14o from the initial unrefined simulation. Thus, one can conclude that differences exist and that accuracy was improved by employing a refined mesh.

Figure 7-Temperature distribution along radial path for unrefined

mesh

Figure 8-Temperature distribution along radial path

for refined mesh

Figure 9-Temperature distribution along radial path

for mesh with no ducts

Figures 7, 8 and 9 show the temperature distribution along the radial path. As observable, they are almost identical, which leads to the conclusion that mesh refinement improved results accuracy near the cooling duct but did not produce significant differences far away from it. In other words, mesh refinements were needed only in the vicinity of the cooling duct. More importantly, since Figure 9 represents the same radial temperature distribution with no ducts whatsoever, one can conclude that radial temperature distribution 22.5o away from the centre of the cooling duct is not affected at all by the existence of the latter. This is due to the symmetry of the problem-the line that is investigated is positioned halfway radially between two identical cooling ducts, the effects of which cancel out. This conclusion is also supported by inspecting the upper line in Figures 2, 4 and 6: each colour has roughly the same proportion of the whole radius, with differences being caused by variations in colour scale.

The formula for the theoretical temperature distribution in radial direction is given below:

(1)

Where: T=temperature@r T1=temperature@r1 and T2=temperature@r2

To relate with the present case under investigation, r2 is the outer radius that is kept at a constant temperature T2 of 15o C, r1 is the inner radius that is kept at a constant temperature T1 of 200o C, and r is the variable radius with r1<r< r2, T1>T(r)> T2.

The results from the theoretical solution are found in Excel Chart 1. As observable, differences exist, and can be better highlighted if the two charts are placed one on top of the other (Excel Chart 2).

The theoretical distribution is much more linear than the experimental one (obtained in Ansys). This is due to the inherent limitations of using a finite element approach. Probably, if an infinitely refined mesh was employed (i.e. an infinite number of cells), the experimental solution would match the theoretical one. However, the computational time would, under the previously mentioned circumstances, render the case as impractical.

15

33.5

52

70.5

89

107.5

126

144.5

163

181.5

200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TEMPERATURE

DISTANCE

Excel Chart 1-Temperature distribution in radial direction

Excel Chart 2-Temperature distribution in radial direction, red=theoretical, turquoise=experimental

Figures 10, 12 and 14 show vectors of thermal flux for the three cases considered-unrefined mesh, refined mesh and no ducts mesh respectively. Figures 11, 13 and 15 show contour plots of thermal flux across the previously mentioned three meshes. It is observable that in the case of no ducts (Figures 14 and 15) the heat flux is radial and decreasing from the inner towards the outer surface. This behaviour is somewhat observable in Figures 10 and 11, however the presence of cooling ducts brings some

Figure 10-Vectors of thermal flux for unrefined mesh

Figure 11-Contour of thermal flux for unrefined mesh

Figure 14-Vectors of thermal flux for mesh with no ducts

Figure 15-Contour of thermal flux for mesh with no ducts

Figure 12-Vectors of thermal flux for refined mesh

Figure 13-Contour of thermal flux for refined mesh

changes. Firstly, there is a region of high thermal near the cooling duct radius in the middle. In this region, thermal flux is more or lass tangential to the cooling duct radius. heat flows from the cylinder to the cooling liquid trough the left and right boundaries of the duct radius. One important observation is that the presence of the cooling duct bends the direction of thermal flux, as vectors of thermal flux tend to become parallel to the radius of the duct in its vicinity. However, further away from the cooling duct, in the upper region of the plots in Figures 10 and 12, the cooling ducts seem to have no effect on the direction of the thermal flux vectors. This is most likely due to the symmetry of the problem-the line I mentioned is positioned halfway between two cooling ducts and their effects cancel out. This is consistent with the previous observation made after investigating plots in Figures 7, 8, 9.

By looking at Figure 15, one can observe that the arcs of constant heat flux (arcs that delimitate two different colours) are parallel to the inner and outer radii of the cylinder. However, in Figures 11 and 13 these seem to bend towards the centre of the cooling duct, which is most likely cause by the presence of the latter.

Also, significant differences can be observed between the refined (Figures 12, 13) and unrefined (Figures 10, 11) meshes, especially in the vicinity of the cooling duct radius. Maximum heat transfer in Figure 12 is 3231.61 W/(unit area) whereas maximum heat transfer in Figure 10 is 3006.42 W/(unit area). It is thus apparent that mesh refinement resulted in improved accuracy. More generally, mesh refinement allowed for a better visualisation of results (as seen by comparing Figures 11 and 13) through improved results resolution.

To get a better understanding of the heat flux across this section, contour plots of thermal flux in X and Y directions were also generated.

Figure 16-Contour of X thermal flux for unrefined mesh

Figure 17-Contour of Y thermal flux for unrefined mesh

In the case of no ducts (Figures 20, 21) all heat transfer takes place in positive X and Y directions, while the amount of flux going in the X direction is greater than the one going in the Y direction. The presence of ducts clearly changes this pattern, as observed in Figures 18, 19. First of all, thermal flux goes in both positive and negative directions and the predominance of X-direction heat transfer is no longer present. These plots are useful to identify the regions where cooling takes place intensely-as observed before, there are two regions on the bottom line of the plot, left and right of the cooling duct radius, where heat transfer takes place mostly in the X direction (Figure 18). However, there is another region of high heat transfer on the upper left side of the radius where intense heat transfer takes place, away from the cylinder, in the negative Y direction (Figure 19). Also, the red region of positive Y transfer in the vicinity of the cylinder in the upper right side of the cooling radius supports the previous statement that heat flux vectors bend around the duct (Figure 19). Again, the refined mesh provides improved results resolution in both the X and Y directions where differences between results seem to be significant (as seen by comparing Figures 16 and 18, Figures 17 and 19).

Figure 20-Contour of X thermal flux for mesh with no ducts

Figure 21-Contour of Y thermal flux for mesh with no ducts

Finally, one more set of plots was generated to look at thermal gradients and their variations along the refined, unrefined and no ducts meshes.

Firstly, looking at Figure 24, the results seem very much in line with the expectations: as the outer radius is kept at a constant temperature of 15o C and the inner radius is kept at a constant temperature of 200o C, the temperature gradients should point towards the cylinder centre radially and should also be higher near the inner radius, where temperatures are higher (since the temperature distribution is logharitmic radially(2)). Figure 23 shows temperature gradients away from the cooling duct radius, with the highest magnitude in the middle-right region. This is definitely an indication of heat extraction through the cooling duct, as the duct radius is at a lower temperature compared to the surroundings. the lower-right region of the cooling duct radius does not provide such a steep gradient despite being closer to the inner radius, as the thermal flux vectors bend and the steady-state solution gives relatively constant temperatures in the region (as seen before from Figures 4, 12).

The refined mesh provides different results from the unrefined mesh, as observed from the minimum and maximum temperature gradients (Figures 22, 23) and also better results resolution.

What is more, Figure 24 supports the statement that cylindrical or spherical components only experience temperature gradients in the radial direction and can thus be treated as one-dimensional(3). This one-directional behaviour is also partially observed in Figures 22 and 23 in the upper part and is caused by the symmetry of the problem.

Figure 22-Vectors of thermal gradients for unrefined mesh

Figure 23-Vectors of thermal gradients for refined mesh

Figure 24-Vectors of thermal gradients for unrefined mesh

6.2 Analysis of the 5.2 problem

To solve the 5.2 problem, an iterative approach was employed. The refined mesh was used for a higher level of accuracy and the loads were modified as such: a constant heat flux of 5000 W/(unit area) was applied onto the inner radius, the outer radius was kept at a constant temperature of 150 C and the upper and lower straight lines were kept at the previous conditions (i.e. 0 heat flux) to maintain the case symmetry. The heat flux rate through the cooling duct arc was negatively incremented until the nodal temperature contour plot showed no temperatures above 250o C in the cylinder.

Figure 25-Contour of nodal temperatures for mesh at a heat flux rate of -1000 W/(unit

area) through the duct arc

Figure 26-Contour of nodal temperatures for mesh at a heat flux rate of -2000 W/(unit

area) through the duct arc

Figure 27-Contour of nodal temperatures for mesh at a heat flux rate of -3000 W/(unit

area) through the duct arc

Figure 28-Contour of nodal temperatures for mesh at a heat flux rate of -3350 W/(unit

area) through the duct arc

At a heat flux rate of -1000 W/(unit area) through the duct arc, the maximum temperature was 386.321o C (Figure 25). At a heat flux rate of -2000 W/(unit area) through the duct arc, the maximum temperature was 326.994o C (Figure 26). At a heat flux rate of -3000 W/(unit area) through the duct arc, the maximum temperature was 268.579o C. At a heat flux rate of -3350 W/(unit area) through the duct arc, the maximum temperature was 249.422o C, below the required threshold of 250o C. Thus the answer to the question is that a heat flux rate through the duct arc of -3350 W/(unit area) is required to keep the temperature in the cylinder below 250o C, although a value of 3400 W/(unit area) would be safer, to have same margin for ANSYS computational errors.

References

(1) Holman, J.P., Heat Transfer, Tenth Edition, McGraw-Hill, New York, 2010. (2) Mills, A.F., Heat Transfer, Second Edition, Prentice-Hall, New Jersey, 1999. (3) Incropera, F. P., Dewitt, D.P., Bergman, T.L. and Lavine, A.S., Fundamentals of Heat

and Mass Transfer, 6th edition, John Wiley & Sons, 2007.

Andrei Traian Tudor Mechanical Engineering MEng Student ID 8582294