**1.2. Instrumentation**

certain areas or arrangements to which more heat is transferred so that the more information about the process is known; it will be possible to have a device with lower maintenance costs or energysupply.Currently, computationalfluiddynamics (CFD)hasbeenemergedasapowerful tool for the analysis of various processes that deal with fluids, and in the case of heat exchang‐ ers it is widespread and has proven to be a reliable tool for analysis. An example is the study by Z.C. Liu and W. Liu. about the geometry of the pipes used in heat exchangers of shell and tube type,whichhas beenshowntomodifyingthegeometryofthe tube andmodifyingthe fluidflow patterns, which results in variations in heat transfer as well as the various flow regimes, and that as the fluid velocity increases thereby increasing the turbulence the heat transfer also

The study of the loss of efficiency in heat exchangers because of corrosion or malfunction of the system is also important, as presented in the work of Torres-Tamayo et al., which explains how it affects resource consumption; in this work the coefficients of heat transfer by convection are determined as well as the impact of this corrosion decreased the system efficiency because it showed a decrease in the thermal efficiency of 70% in heat exchangers due to such problems

The experimental heat exchanger is of tubular type, cross flow occurs at 90°, it comprises vertical tubes placed in a 10 × 4 matrix in an arrangement of square type [4] as shown in

**Figure 1.** Frontal, lateral, and isometric view from heat exchanger (dimension in millimeters).

The heat exchanger is provided with an acrylic duct that is used to conduct the air to the tube arrangement (**Figure 2**). Water flows through 40 tubes vertically and constantly by means of a recirculation circuit which comprises a pump and a reservoir, and the air is supplied by a

increases [1].

[2, 3].

**Figure 1**.

vertical fan.

**1.1. Experimental heat exchanger**

190 Numerical Simulation - From Brain Imaging to Turbulent Flows

The experimental heat exchanger is implemented with 32 fixed thermocouples, a flow meter is used to measure the flow of water, and readings of air velocity are taken during the test with an anemometer. Thirty-two thermocouples are connected to a data acquisition system in which the analogic signals received are changed to digital signals and then processed and captured in a computer using the LAB-VIEW software. In **Figure 3**, a diagram of the device with basic instrumentation installed therein to monitor the variables of interest is shown.

**Figure 3.** Diagram of instrumentation.

An anemometer was used to measure the velocity of the air entering to feed channel to heat exchanger (**Figure 4**).

**Figure 4.** Hot-wired anemometer.

A flowmeter is installed in the water supply pipe before entering into the arrangement of heat exchanger tubes (**Figure 5**).

**Figure 5.** Turbine flow meter installed in the water supply pipe.

The procedure for data collection is as follows:


**3.** Resistances are turned on for heating the air.

An anemometer was used to measure the velocity of the air entering to feed channel to heat

A flowmeter is installed in the water supply pipe before entering into the arrangement of heat

exchanger (**Figure 4**).

192 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Figure 4.** Hot-wired anemometer.

exchanger tubes (**Figure 5**).

**Figure 5.** Turbine flow meter installed in the water supply pipe.

The procedure for data collection is as follows:

**2.** Ingress of water (just to fill pipes).

**1.** Ingress of air.



**Table 1.** Averaged variables from experimental measurements.


**Table 2.** Averaged properties of fluids.

On the completion of the measurements and obtaining the data needed, these are organized and averages are computed, which are used for numerical simulation. The averages of the main variables measured are presented in **Table 1**. With the acquired data and obtaining the properties of both fluids, it is possible to calculate the rate of heat transfer and convective coefficients, and then they are used to perform numerical simulation. It is also necessary to calculate the average of fluid properties (**Table 2**).

### **1.3. Theoretical approach**

The aim of the presented calculations is to obtain heat transfer and convective coefficients for both fluids. In reality, the convection phenomenon is complex and a constant numerical value for the heat convective coefficient is obtained; while stationary the heat transfer phenomenon is impossible; however, it is possible to obtain a representative average performing theoretical assumptions that are valid; in this case, the variables will be obtained with a theoretical model in which the same amount of heat is transferred with constant convective coefficients per unit average value area along the entire contact surface for both fluids [5]. The nomenclature of variables used on analytical calculation can be found in **Table 3**.



**Table 3.** Nomenclature.

On the completion of the measurements and obtaining the data needed, these are organized and averages are computed, which are used for numerical simulation. The averages of the main variables measured are presented in **Table 1**. With the acquired data and obtaining the properties of both fluids, it is possible to calculate the rate of heat transfer and convective coefficients, and then they are used to perform numerical simulation. It is also necessary to

The aim of the presented calculations is to obtain heat transfer and convective coefficients for both fluids. In reality, the convection phenomenon is complex and a constant numerical value for the heat convective coefficient is obtained; while stationary the heat transfer phenomenon is impossible; however, it is possible to obtain a representative average performing theoretical assumptions that are valid; in this case, the variables will be obtained with a theoretical model in which the same amount of heat is transferred with constant convective coefficients per unit average value area along the entire contact surface for both fluids [5]. The nomenclature of

calculate the average of fluid properties (**Table 2**).

194 Numerical Simulation - From Brain Imaging to Turbulent Flows

variables used on analytical calculation can be found in **Table 3**.

**Variable Symbol Units** Density *ρ* kg/m3 Specific heat *Cp* J/kg K Thermal conductivity coefficient *k* W/m K Dynamic viscosity μ kg/m s

Prandtl number *Pr* – Temperature *T* K Heat transfer *Q* W Velocity *v* m/s Volumetric flow *ύ* m3

Mass flow m˙ kg/s Temperature gradient *ΔT* K Convective coefficient *h* W/m2

Nusselt number *Nu* – Reynolds number *Re* – Transversal section area *AST* m2 Heat transfer area *ATC* m2 Inner diameter *Di* <sup>m</sup>

/s

K

**1.3. Theoretical approach**

Subindex 1: Property or variable belonging to air.

Subindex 2: Property or variable belonging to water.

From equation for energy balance:

$$Q = \dot{m}\_2 C p\_2 \Delta T\_{\text{2\text{-balance}}} \tag{1}$$

The total water mass flow:

$$
\dot{m}\_2 = \dot{o}\_2 \,\rho\_2 \tag{2}
$$

The temperature gradient for the energy balance is:

D =- *T TT* 2 2 2 *balance outlet inlet* (3)

As the heat transfer rate is known, the heat transfer coefficient can be obtained from:

$$Q\_{2\text{ conv}} = h\_2 A\_{\tau 2} \left(\Delta T\_{C2}\right) \tag{4}$$

*ΔT*C2 can be obtained from:

$$
\Delta T\_{C2} = T\_{\text{nonor wall}} - T\_{\text{natur average}}
$$

where

$$T\_{\text{waar anvuga}} = \frac{T\_{2\text{ oute}} + T\_{2\text{ inlet}}}{2}$$

the temperature of the inner wall is not known; however, the temperature of the outer wall, the tube thickness 1 mm, and thermal conductivity of copper tube, which is 380 W/m K and also has 40 tubes, are known; and as *Qcond* =*Qconvection* =*Q*2 it follows that:

$$T\_{inormal} = T\_{oantrow} - \frac{\left(\bigvee\_{40} (Q\_{condraction}) \ln\left(\frac{r\_{oantrow}}{r\_{invar}}\right)}{2\pi Lk} \tag{5}$$

As the temperature of the inner wall and the average water temperature are known, it is possible to obtain the heat transfer coefficient, which from Eq. (4) yields

$$h\_2 = \frac{Q\_{2\text{ communication}}}{A\_{r2} \left(\Delta T\_{zc}\right)}$$

Now we can calculate the energy balance for air. First, the volumetric flow must be obtained:

$$
\dot{\upsilon}\_1 = \upsilon\_1 A\_{s1} \tag{6}
$$

The air mass flow is calculated with the following equation:

$$
\dot{m}\_1 = \dot{\upsilon}\_1 \rho\_1 \tag{7}
$$

Then, *Q*2 = *Q*1 and it is possible to obtain the heat transfer coefficient of air, using *h* from Eq. (4):

$$h\_1 = \frac{Q\_2}{A\_{1T} \left(\Delta T\_{1C}\right)}$$

The area for convection transfer to the air is

$$A\_{T1} = 40\pi D\_o L$$

And the temperature gradient *ΔT*1*<sup>C</sup>* would be

$$\Delta T\_{1C} = T\_{\text{амичақат}} - T\_{\text{ошичала}}$$

As the temperature of the outer wall is known, the average temperature is obtained as follows:

$$T\_{\left(^{\text{anwarga }abr}\right)} = \frac{\left(T\_{\left(^{\text{anwarc}}\right)} + T\_{\left(^{\text{ouzlarc}}\right)}\right)}{2}$$

All the above calculations are based on energy balances, but it is possible based on Nusselt numbers for comparison calculations. To calculate the Nusselt number for water, first obtain the Reynolds number to determine the conditions of the fluid:

$$Re\_2 = \frac{\rho\_2 \mathbf{v}\_2 D\_i}{\mu\_2} \tag{8}$$

The water velocity is given by

2 outlet 2 inlet

2 *water average T T <sup>T</sup>* <sup>+</sup> <sup>=</sup>

also has 40 tubes, are known; and as *Qcond* =*Qconvection* =*Q*2 it follows that:

*outt*

possible to obtain the heat transfer coefficient, which from Eq. (4) yields

*innerwal e a l rw ll*

196 Numerical Simulation - From Brain Imaging to Turbulent Flows

The air mass flow is calculated with the following equation:

The area for convection transfer to the air is

And the temperature gradient *ΔT*1*<sup>C</sup>* would be

the temperature of the inner wall is not known; however, the temperature of the outer wall, the tube thickness 1 mm, and thermal conductivity of copper tube, which is 380 W/m K and

( )( ) <sup>1</sup> ln <sup>40</sup>

As the temperature of the inner wall and the average water temperature are known, it is

*<sup>r</sup> <sup>Q</sup> <sup>r</sup> T T*

2

& =

1

*<sup>Q</sup> <sup>h</sup> A T* <sup>=</sup> <sup>D</sup>

*<sup>Q</sup> <sup>h</sup> A T* <sup>=</sup> <sup>D</sup>

2 *conduction*

( ) <sup>2</sup>

2 2 *convection T C*

Now we can calculate the energy balance for air. First, the volumetric flow must be obtained:

r

Then, *Q*2 = *Q*1 and it is possible to obtain the heat transfer coefficient of air, using *h* from Eq. (4):

( ) <sup>2</sup>

1 1 *T C*

<sup>1</sup> 40 *A DL* = p*oT*

p*Lk*

*inn outter*

æ ö ç ÷ è ø = - (5)

*er*

1 11 = *st ύ A v* (6)

*m ύ* 1 11 (7)

$$\nu\_2 = \frac{\not b\_2}{A\_{\text{st2}}} \tag{9}$$

The area *AST* 1 of the cross section for the flow of water is

$$A\_{\text{sr2}} = 40 \left( \frac{\pi D\_o^{\cdot^2}}{4} \right)^2$$

The Reynolds number describes the flow rate of fluids according to their size, and flow in a tube is considered laminar if it holds that *Re* < 2300. It is considered in transition if 2300 <*Re* <4000. And it is considered turbulent if *Re* > 4000 [5, 6].

The Nusselt number for a constant flow of heat per unit area for a fully developed flow and laminar flow along a circular pipe is 4.36. In this case, the water has a laminar flow and therefore the Nusselt number for this fluid is already known.

The heat transfer coefficient would be described by the following equation:

$$h\_2 = \frac{Nu\_2k\_2}{D\_i} \tag{10}$$

And the heat transfer is obtained from Eq. (4):

$$Q\_{2\text{-conv}} = h\_2 A\_{T2} \left(\Delta T\_{C2}\right)$$

where

$$\mathcal{A}\_{r\_2} = 40\pi D\_i L$$

Now the mass coefficient is calculated based on the Nusselt number for air leaving the heat transferred by air which is absorbed by water *Q*<sup>2</sup> =*Q*1.

It is necessary to know the hydraulic diameter of the cross section that crosses in air:

$$D\_h = L\_c = \frac{4A\_c}{P} \tag{11}$$

As the air flows through the matrix or tube bank, the speed will remain unchanged because the volume of air flow will be lower in the area where this is in contact with the tubes, and to maintain flow mass, the speed increases accordingly. Then, it is interesting to know the maximum speed reached by the air; this is the type of arrangement of the tube bank, and this case is rectangular arrangement type and the equation for the maximum speed is defined as

$$\nu\_{\text{max}} = \nu \frac{S\_{\text{\tiny}}}{S\_{\text{\tiny}} - D\_o} \tag{12}$$

And the Reynolds number according to Eq. (8) is

$$Re\_2 = \frac{\rho \nu\_{\text{max}} L\_\text{g}}{\mu} \tag{13}$$

If the Reynolds number is the turbulent regime as in this case, the equation determined by Churchill and Bernstein can be applied, which is valid for all Reynolds numbers [6]:

$$Nu\_1 = \frac{hD\_o}{k} = 0.3 + \frac{0.62 Re^{1/2} Pr^{1/3}}{\left(1 + \left(\frac{0.4}{Pr}\right)^{2/3}\right)^{1/4}} \left(1 + \left(\frac{Re}{282,000}\right)^{3/8}\right)^{4/5} \tag{14}$$

As the thermal conductivity of air is known from Eq. (10), it is possible to know the mass coefficient:

$$h\_1 = \frac{Nu\_1 k\_1}{D\_\odot}$$
