**3. Results**

As the results of the numerical simulation, fluids behave as expected according to the theory; for the case of water, as has flow within pipes, it is expected that the hydrodynamic and thermal layers are fully formed and have a length region input and a fully developed area.

As can be seen from **Figure 6**, tubes have an overall length sufficient for the water flow developing hydrodynamically and with a predominance of the fully developed flow as established in the calculations in Section 1.3.

**1.** The partial differential equations are integrated into all generated control volumes; the Navier-Stokes and models of thermal energy and turbulence are assigned to each node of

**2.** These integral equations become a global system of equations because each control

**3.** The global system of equations is solved iteratively; this is necessary because of the nonlinear nature of the equations, and then the exact solution is approached for the solution. That is, the numerical solution obtained by each iteration should converge

**External domain (turbulent air) Magnitude Temperature**

Air inlet Inlet Mass flow m˙ =0.1284 kg / s 336.3100 K Air outlet Outlet Pressure 0 Pa 306.0400 K Outer tube wall Wall Convection No slip h1 =77.6200W / <sup>m</sup><sup>2</sup> <sup>K</sup> 302.1500 K **Internal domain (laminar water) Condition Temperature** Water inlet Inlet Mass flow m˙ = 0.1987 kg/s 291.9500 K Water outlet Outlet Pressure 0 Pa 293.3100 K Inner tube wall Wall Convection no slip h2 = 184.1873 W / <sup>m</sup><sup>2</sup> <sup>K</sup> 302.0400 K

Postprocessing is the final stage of CFD simulation, which allows visualizing the results of flow behavior inside the domain. There are several options to observe it, for example, contours, streamlines, and vectors. In the next section, the examples of post-processing are presented.

As the results of the numerical simulation, fluids behave as expected according to the theory; for the case of water, as has flow within pipes, it is expected that the hydrodynamic and thermal

As can be seen from **Figure 6**, tubes have an overall length sufficient for the water flow developing hydrodynamically and with a predominance of the fully developed flow as

layers are fully formed and have a length region input and a fully developed area.

Case Wall Adiabatic no slip – –

each element or subdomain.

200 Numerical Simulation - From Brain Imaging to Turbulent Flows

volume is connected to the adjacent nodes.

toward and approach a condition previously considered.

**Zone Boundary type Condition Variables**

**Table 4.** Boundary conditions for numerical simulation.

established in the calculations in Section 1.3.

**2.3. Postprocessing**

**3. Results**

**Figure 6.** Formation and development of the hydrodynamic boundary layer, velocity vectors in the inlet region, and development of the velocity boundary layer in the total length tubes.

The temperature of tubes wall is not constant for the thermal boundary layer, and, therefore, the thermal boundary layer has different temperatures at the interface of the tube wall and the fluid; as can be observed, it behaves correctly. **Figure 7** shows that the fluid cooler is located at the center of the tube, and the fluid is at a higher temperature at the boundary.

Theoretically, the air velocity should be higher where the circulating cross section (**Figure 8**) is reduced; wake vortices should also be formed, generated by the effect of separation of the hydrodynamic boundary layer in the tube wall (**Figure 9**).

**Figure 7.** Development of thermal boundary layer: middle plane of tubes and transversal plane at different heights.

Analysis of Heat Transfer in an Experimental Heat Exchanger Using Numerical Simulation http://dx.doi.org/10.5772/63957 203

**Figure 8.** Air velocity contour in a cross section cut of 24 cm of the water inlet to the arrangement of pipes.

**Figure 9.** Close-up view of streamlines of airflow at the transversal plane.

The temperature of tubes wall is not constant for the thermal boundary layer, and, therefore, the thermal boundary layer has different temperatures at the interface of the tube wall and the fluid; as can be observed, it behaves correctly. **Figure 7** shows that the fluid cooler is located

Theoretically, the air velocity should be higher where the circulating cross section (**Figure 8**) is reduced; wake vortices should also be formed, generated by the effect of separation of the

**Figure 7.** Development of thermal boundary layer: middle plane of tubes and transversal plane at different heights.

at the center of the tube, and the fluid is at a higher temperature at the boundary.

hydrodynamic boundary layer in the tube wall (**Figure 9**).

202 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Figure 10.** Contour static gauge pressure in the matrix of tubes.

**Figure 11.** Outline temperature air flowing through the tube arrangement.

The maximum speed calculated theoretically was 1.711 m/s and from simulation it is 1.565 m/ s, which represents a percentage error of 8.5% that can be considered small due to the com‐ plexity of the phenomenon. In the simulation results, it is possible to observe the formation of high-pressure areas where air directly enters the wall of the tube and low-pressure zones where the hydrodynamic wake zone (**Figure 10**) is generated.

For air temperatures, in a manner analogous to the water, the lower temperature is found at the boundary between the outer tube wall and the fluid as shown in **Figure 11**.

The average temperature difference in the tubes is approximately 4°C between numerical simulation and experimentally obtained (**Figure 12**) whereas the average temperature difference at the air outlet is about 6°C (**Figure 13**).

**Figure 12.** Temperatures measured vs. numerically simulated in the outer wall of tubes from the heat exchanger.

**Figure 13.** Temperatures measured vs. numerically simulated in the air outlet of the heat exchanger.

**Figure 11.** Outline temperature air flowing through the tube arrangement.

204 Numerical Simulation - From Brain Imaging to Turbulent Flows

the hydrodynamic wake zone (**Figure 10**) is generated.

difference at the air outlet is about 6°C (**Figure 13**).

The maximum speed calculated theoretically was 1.711 m/s and from simulation it is 1.565 m/ s, which represents a percentage error of 8.5% that can be considered small due to the com‐ plexity of the phenomenon. In the simulation results, it is possible to observe the formation of high-pressure areas where air directly enters the wall of the tube and low-pressure zones where

For air temperatures, in a manner analogous to the water, the lower temperature is found at

The average temperature difference in the tubes is approximately 4°C between numerical simulation and experimentally obtained (**Figure 12**) whereas the average temperature

**Figure 12.** Temperatures measured vs. numerically simulated in the outer wall of tubes from the heat exchanger.

the boundary between the outer tube wall and the fluid as shown in **Figure 11**.

Although temperatures are not exactly the same, the differences are relatively small as well as the behavior is expected theoretically; it should also be noted that when working with heat transfer convection errors in physical quantities will usually be larger than with phenomena such as conduction, which is a complex phenomenon, as there are a greater number of variables involved.

**Table 5** shows the results of calculus of heat transfer rate from both working fluids that are obtained by energy balances. In addition, the Nusselt number was calculated to subsequently obtain the convective coefficients of both fluids.


**Table 5.** Comparison of the magnitudes of the rate of heat transfer test analyzed.

In order to represent the efficiency of numerical simulations, the following ratio was calculated:

$$\eta = \frac{Q\_{\text{Re all}}}{Q\_{\text{Silumination}}} = \frac{1131.19}{1333.17} = 0.85$$

which is quite acceptable for such heat transfer problems.
