For contour plots and prole plots

MAE 598 - Applied Computational Fluid Dynamics

Project-I

MECHANICAL ENGINEERING DEPARTMENT

Project 1

condition. The bottom wall of the heater is assigned Dirichlet temperature boundary condition as T = 55oC. Standard k-epsilon turbulence model is used in the present simulation with full buoyancy eect. The pressure based solver is to be used in the present simulation and steady state solution is desired. The gravitational acceleration is taken as g = −9.81m s2.

The density is set as Boussinesq for water in order to allow density variation with tem-perature. The operating density is evaluated from below Eq. 1 from the given reference

ρ = 999.85308+6.3269×10−2T−8.523829×10−3T2+6.943248×10−5T3−3.821216×10−7T4 (1)

Part b
To simulate the above given problem with the reduced gravity g = −1.62m s2.

layers having growth rate of 1.2 at the wall. In the present simulation, z is the direction parallel to the gravity vector. g = −9.81m s2. Although, symmetry can be invoked, the present simulation is performed on full geometry.

Z Z

Figure 2: (a) Velocity contour at the plane of symmetry (here z-velocity) and (b) Tem-perature contour at the plane of symmetry.

The convergence criteria is achieved when the temperature between two iterations with interval of 100 is less than 0.05 K. The same is achieved between iterations 1900 and 2000 as observed in the Fig. 3. The temperature dierence is observed as 0.045 K over the given interval of iterations.

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Figure 3: Line plot of the outlet temperature as a function of the number of iterations.

Task 1(b)
Deliverable (1 & 2)
The velocity and the temperature contour plot is shown in the Fig. 4 when g = −1.62m s2.

The outlet temperature at the steady state is T = 287.82 K which is calculated using Eq. 3. The convergence criteria is achieved when the temperature between two iterations with interval of 100 is less than 0.05 K. The same is achieved between iterations 2000 and 2100 as observed in the Fig. 5. The temperature dierence is observed as 0.0404 K over the given interval of iterations. The nature of temperature plot is somewhat opposite to that expected which is mainly due to some singularity occurring at a signicantly nearer grid points causing an initial value of temp. However, the steady state is still achieved.

Figure 5: Line plot of the outlet temperature as a function of the number of iterations.

Solution

Figure 6 (a) shows the computational domain of the coiled water heater and Fig. 6 (b) shows the grid used in the present simulation and (c) shows the grid at inlet and outlet surface of the helical pipe. The element size for the triangular mesh is taken as 1 cm. The ination is led by 'Program Controlled' with 5 layers having growth rate of 1.2 at the wall.

5

Figure 7: Plot for ∆TNumerical = Tout − Tin v/s the inlet velocity Vin
As observed from the Eq. 4 of ∆TAnalytical, there is an inverse relation between the temper-ature dierence ∆T and the inlet velocity Vin. The similar relationship is also discernible from the numerical results in Table. 1 and Fig. 7 where the value of ∆TNumerical is reducing with an increase in the inlet velocity Vin. This is mainly due to the fact that as the water moves at higher velocity inside the helical pipe, it will have a lesser residence time, which in fact, causes it to gain less amount of heat as compared to the water which moves at slower velocity inside helical pipe of same length and with constant heat ux at the wall.

Figure 8: (a) Velocity magnitude at the outlet surface and (b) Temperature contour at the outlet surface.

As observed from the Fig. 8 (a), the velocity of water at the outer edge of the pipe is higher as compared to the velocity of water at the inner edge. This is mainly due to the fact that water particles nearer to the outer edge has to travel more distance in same interval of time as compared to those which travel nearer to the inner edge. Due to this phenomena, the heat gained by water nearer to the outer edge will be less as compared to heat gained by water nearer to inner edge. As a result, the temperature of water nearer to the outer edge will less as compared to the temperature of water nearer to the inner edge. The same is discernible from Fig. 8 (b).

model is to be used in the present simulation. The density based solver is to be used in the present simulation and transient simulation is to be performed. The density to be set as 'Ideal Gas' for air.

Figure 9 (a) shows the computational domain of chamber and Fig. 9 (b) shows the grid used at the plane of symmetry in the present simulation. In the present simulation the element size for grid generation is taken as 1 cm. Programmed controlled ination of 5 layers with 1.2 growth rate is allowed at boundary of the chamber. Moreover, an additional renement of the grid is provided nearer to the top and bottom wall of the chamber as observed from the Fig. 9 (b).

Z Z
Y

Deliverable (1)
The line plot for Vout v/s time from t = 0 s to t = 10 s is shown in the Fig. 10. As the temperature of the air increase in the chamber it expands and as a result a continuous increase in the velocity (area weighted average) of the air, normal to the outlet, is observed. In the present simulation, the time step size is chosen as ∆t = 0.1 s. The number of time steps is kept as 100 to achieves the solution till t = 10 s. The maximum iterations per time step is chosen as 20. The value of Vout is 0.00113m safter t = 10 s. A detail investigation was carried out by further renement of the time step size (∆t = 0.05, 0.01 and 0.005 s), however, the nature of curve for Vout was found to be oscillating, which in fact suggested that a further renement in the grid was required.

Figure 10: Line plot for Vout v/s ow time t from t = 0 s to t = 10 s.

Figure 11: (a) Pressure Contour, (b) Velocity magnitude contour and (c) Temperature

Solution

Deliverable (3)
Figure 12 (a) shows the computational domain of the quarter chamber and Fig. 12 (b) shows the grid used in the present simulation. In the present simulation the element size for grid generation is taken as 1 cm. Programmed controlled ination of 5 layers with 1.2 growth rate is allowed at boundary of the chamber. Moreover, an additional renement of the grid is provided nearer to the bottom wall of the chamber as observed from the Fig. 12 (b).

Figure 13: Line plot for Vout v/s ow time t from t = 0 s to t = 10 s.

Deliverable (2)
The pressure contour, velocity magnitude contour and the temperature contour are shown at the plane of the symmetry at t = 10 s, in Fig. 14 (a), (b) and (c) respectively. It is discernible that the results obtained in the present simulation over a quarter domain do corroborate with that obtained in task 3. However, there are very minor discrepancies in the result which are caused mainly due to dierence between the grid generated in original computational domain and the grid generated in the quarter domain, and in addition to that, also due to some numerical error.

Figure 14: (a) Pressure Contour, (b) Velocity magnitude contour and (c) Temperature

contour at the plane of symmetry.