**2.2. Tubular flow of a scalar quantity and heat conduction in the radial direction**

A physical problem used in this study for assessing the numerical schemes is the numerical prediction of the energy transfer in a triple tube heat exchanger [17]. The heat exchanger has a three-dimensional curvilinear geometry similar to a circular cylinder. Three fluids are being considered to flow inside the domain that is separated by solid. Chilled water (10°C) flows in the inner tube, hot water (70°C) flows in the inner annulus and normal tap water (18°C) flows in the outer annulus. The three fluids follow a co-current flow. The numerical model is first validated and then used to compare the performance of various numerical schemes.

The boundary conditions applied for solving the above physical problem are the following:

Water with uniform properties (temperature, density, thermal capacity, kinematic viscosity) enters the three different inlets of the heat exchanger. The heat exchanger is separated to three components that are filled with water of different properties and the solid parts that prevent the water vertical flow. Heat is exchanged between the fluids through conduction. The solid is steel at 27°C of uniform properties (density, viscosity, specific heat, thermal conductivity, thermal expansion coefficient, compressibility). At the three outlets of the heat exchanger the boundary condition of zero mass flow is applied.

**Figure 3.** Local temperature variation of the co-current three fluid streams along the length of the heat exchanger.

In **Figure 3** the numerical prediction of the temperature distribution along the heat exchanger is validated against experimental data. The average% relative error of the numerical prediction does not exceed the value of 20%.

flow direction than the conventional numerical scheme (UPWIND) and the non-linear numerical scheme (van LEER). It is concluded that the numerical diffusion errors are minimized when the discretization schemes (SUCCA, SUPER) are applied to predict the transport of the scalar quantity C1 when the air-flow direction appears the major inclination angle 45° to the grid lines. The performance of the van LEER numerical scheme is also satisfactory in predicting the scalar concentration distribution. The conventional UPWIND numerical scheme that does not

A physical problem used in this study for assessing the numerical schemes is the numerical prediction of the energy transfer in a triple tube heat exchanger [17]. The heat exchanger has a three-dimensional curvilinear geometry similar to a circular cylinder. Three fluids are being considered to flow inside the domain that is separated by solid. Chilled water (10°C) flows in the inner tube, hot water (70°C) flows in the inner annulus and normal tap water (18°C) flows in the outer annulus. The three fluids follow a co-current flow. The numerical model is first

take into account the phenomenon of false-diffusion presents the poorest accuracy.

**2.2. Tubular flow of a scalar quantity and heat conduction in the radial direction**

validated and then used to compare the performance of various numerical schemes.

boundary condition of zero mass flow is applied.

70 Numerical Simulations in Engineering and Science

The boundary conditions applied for solving the above physical problem are the following: Water with uniform properties (temperature, density, thermal capacity, kinematic viscosity) enters the three different inlets of the heat exchanger. The heat exchanger is separated to three components that are filled with water of different properties and the solid parts that prevent the water vertical flow. Heat is exchanged between the fluids through conduction. The solid is steel at 27°C of uniform properties (density, viscosity, specific heat, thermal conductivity, thermal expansion coefficient, compressibility). At the three outlets of the heat exchanger the

**Figure 3.** Local temperature variation of the co-current three fluid streams along the length of the heat exchanger.

In **Figure 4** the numerical results of the enthalpy distribution at the lateral plane in the middle of the heat exchanger applying three different numerical schemes (HYBRID, van LEER, SUPER) [6, 8, 12] is presented.

Comparing the performance of the numerical schemes in predicting the transfer of the scalar quantity H1 it is concluded that the numerical prediction is equally satisfactory for the three discretization schemes.

**Figure 4.** Enthalpy distribution at the lateral plane of the heat exchanger applying: (a) the HYBRID, (b) the van Leer, (c) the SUPER numerical scheme.

### **2.3. Airflow in a backward facing step**

A physical problem appropriate for assessing the accuracy of various numerical schemes is the prediction of the airflow velocity vectors distribution in a backward facing step.

According to the numerical results that agree well with the experimental data the performance of the numerical schemes (UPWIND, HYBRID, van LEER, SUPER) is equally satisfactory. The

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A physical problem that has been used for assessing the performance of various numerical schemes is the dispersed air-particles flow in a model room geometry (**Figure 7**) [19]. A two-phase Euler-Euler mathematical model is applied to calculate the oblique inflow in the interior of the internal space. The accuracy of the four numerical schemes: (a) the conventional first-order UPWIND numerical scheme, (b) the second-order HYBRID scheme, (c) the nonlinear van LEER and (d) the flow-oriented SUPER scheme [6, 8, 12] are compared in the case

At the inlet the mass flow rate of each phase is multiplied by its volume fraction. The dispersed turbulent air-particles flow enters the room with uniform velocity (0.225 m/s) in an inclined direction. Turbulence is modeled by the RNG k-ε model [20]. Particles are assumed

The turbulence kinetic energy of the air-phase that is applied at the inlet is defined as [13, 21]

At the outlet both air phases are supposed to exhaust at an environment of fixed uniform pres-

sure. At the walls the no-slip and no-penetration condition is applied for both phases.

<sup>3</sup>/<sup>4</sup> *k* <sup>3</sup>/<sup>2</sup> \_\_\_

the turbulence intensity, considered

<sup>ℓ</sup> , where ℓ the turbulence length scale is taken as

= 0.0845 an empirical constant of the turbulence model. The

velocity distribution at the longitudinal plane of the domain

to be transported and dispersed due to turbulence of the carrier fluid (air).

where *<sup>U</sup> avg*. The mean air inlet velocity and *<sup>T</sup> <sup>i</sup>*

mathematical model uses the logarithmic "wall functions" near the solid surfaces.

same conclusion is derived for the case of air-particles flow in a backward facing step.

**2.4. Inclined air-particles flow**

of inclined inflow (θ = 45°).

*2.4.1. Boundary conditions*

**Figure 7.** Geometry of the model room.

*<sup>k</sup> in* <sup>=</sup> <sup>3</sup> \_\_

<sup>ℓ</sup> = 0.07 *<sup>d</sup> <sup>h</sup>*

*2.4.2. Results*

2 ( *<sup>U</sup> avg <sup>T</sup> <sup>i</sup>*) 2

as 6%. The dissipation rate is given by *<sup>ε</sup>* <sup>=</sup> *<sup>C</sup> <sup>μ</sup>*

**Figure 8a**–**c** present the vertical *w*<sup>1</sup>

(*<sup>d</sup> <sup>h</sup>* hydraulic diameter of the duct) and *<sup>C</sup> <sup>μ</sup>*

at distance 0.2, 0.4, 0.6 m from the supply inlet.

The separation of the airflow, the recirculation zone formed forward the step and the reattachment length are the main characteristics of the flow in this geometry. The numerical solution of the airflow field in a backward facing step is of major concern due to the difficulty in accurate prediction of these complex phenomena. The flow field may become more complex in the presence of particles.

In this study the two-dimensional backward facing step of Benavides and Wachem [18] is investigated.

In **Figures 5** and **6** the velocity flow field is presented applying various numerical schemes (UPWIND, HYBRID, van LEER, SUPER) [6, 8, 12] and validated against experimental data [18].

**Figure 5.** Velocity flow field applying various numerical schemes (upwind, hybrid, van Leer, super).

**Figure 6.** Vertical velocity distribution applying various numerical schemes (UPWIND, HYBRID, van LEER, SUPER) and comparison with the experimental data.

According to the numerical results that agree well with the experimental data the performance of the numerical schemes (UPWIND, HYBRID, van LEER, SUPER) is equally satisfactory. The same conclusion is derived for the case of air-particles flow in a backward facing step.

## **2.4. Inclined air-particles flow**

**2.3. Airflow in a backward facing step**

72 Numerical Simulations in Engineering and Science

in the presence of particles.

and comparison with the experimental data.

investigated.

A physical problem appropriate for assessing the accuracy of various numerical schemes is

The separation of the airflow, the recirculation zone formed forward the step and the reattachment length are the main characteristics of the flow in this geometry. The numerical solution of the airflow field in a backward facing step is of major concern due to the difficulty in accurate prediction of these complex phenomena. The flow field may become more complex

In this study the two-dimensional backward facing step of Benavides and Wachem [18] is

In **Figures 5** and **6** the velocity flow field is presented applying various numerical schemes (UPWIND, HYBRID, van LEER, SUPER) [6, 8, 12] and validated against experimental data [18].

**Figure 5.** Velocity flow field applying various numerical schemes (upwind, hybrid, van Leer, super).

**Figure 6.** Vertical velocity distribution applying various numerical schemes (UPWIND, HYBRID, van LEER, SUPER)

the prediction of the airflow velocity vectors distribution in a backward facing step.

A physical problem that has been used for assessing the performance of various numerical schemes is the dispersed air-particles flow in a model room geometry (**Figure 7**) [19]. A two-phase Euler-Euler mathematical model is applied to calculate the oblique inflow in the interior of the internal space. The accuracy of the four numerical schemes: (a) the conventional first-order UPWIND numerical scheme, (b) the second-order HYBRID scheme, (c) the nonlinear van LEER and (d) the flow-oriented SUPER scheme [6, 8, 12] are compared in the case of inclined inflow (θ = 45°).

**Figure 7.** Geometry of the model room.

### *2.4.1. Boundary conditions*

At the inlet the mass flow rate of each phase is multiplied by its volume fraction. The dispersed turbulent air-particles flow enters the room with uniform velocity (0.225 m/s) in an inclined direction. Turbulence is modeled by the RNG k-ε model [20]. Particles are assumed to be transported and dispersed due to turbulence of the carrier fluid (air).

The turbulence kinetic energy of the air-phase that is applied at the inlet is defined as [13, 21] *<sup>k</sup> in* <sup>=</sup> <sup>3</sup> \_\_ 2 ( *<sup>U</sup> avg <sup>T</sup> <sup>i</sup>*) 2 where *<sup>U</sup> avg*. The mean air inlet velocity and *<sup>T</sup> <sup>i</sup>* the turbulence intensity, considered as 6%. The dissipation rate is given by *<sup>ε</sup>* <sup>=</sup> *<sup>C</sup> <sup>μ</sup>* <sup>3</sup>/<sup>4</sup> *k* <sup>3</sup>/<sup>2</sup> \_\_\_ <sup>ℓ</sup> , where ℓ the turbulence length scale is taken as <sup>ℓ</sup> = 0.07 *<sup>d</sup> <sup>h</sup>* (*<sup>d</sup> <sup>h</sup>* hydraulic diameter of the duct) and *<sup>C</sup> <sup>μ</sup>* = 0.0845 an empirical constant of the turbulence model. The mathematical model uses the logarithmic "wall functions" near the solid surfaces.

At the outlet both air phases are supposed to exhaust at an environment of fixed uniform pressure. At the walls the no-slip and no-penetration condition is applied for both phases.

### *2.4.2. Results*

**Figure 8a**–**c** present the vertical *w*<sup>1</sup> velocity distribution at the longitudinal plane of the domain at distance 0.2, 0.4, 0.6 m from the supply inlet.

domain is equal to the atmospheric pressure. The initial relative humidity condition tested is 90%. The overall water vapor mass of humid air at the initial temperature 303 K is taken from

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In **Figure 9** the vertical temperature distribution in the middle of the domain at the height

The temperature of humid air near the floor is below the dew point (301.2 K) and water phase humidity covers the whole surface. The hot air close to the floor that comes into contact with the cold surface, reduces its temperature and flows down due to the gravity. The remained

In **Figure 10** the vertical temperature distribution in the region close to the floor calculated by three different numerical schemes (HYBRID, van LEER, SUPER) [6, 8, 12] is presented.

Temperature profile in the region of major gradient near the floor surface is predicted more

In **Figure 11** the vertical absolute humidity ratio (kg H2O/kg of dry air) predicted by the three

Heat convection is accompanied by mass transfer and phase change of humid air. The larger gradient of temperature profile predicted by the SUPER scheme [10] leads to the formation of larger amount of water phase. Comparing the performance of the discretization schemes a more accurate solution of the condensation procedure is observed when applying the SUPER

different numerical schemes (HYBRID, van LEER, SUPER) is presented at time 360 s.

**Figure 9.** Vertical temperature distribution at time 360 s for initial humidity condition 90%.

the psychrometric chart [23].

*2.5.2. Results*

scheme.

(0–4 m) is presented.

hot air flows up to the roof.

abrupt by the SUPER scheme.

**Figure 8.** Vertical w<sup>1</sup> velocity distribution at the longitudinal plane of the domain and (a) 0.2 m from the inlet; (b) 0.4 m from the inlet; and (c) 0.6 m from the inlet

As far as the accuracy of the four numerical schemes in the case of inclined inflow (θ = 45°) is concerned the higher-order and non-linear schemes (hybrid and van Leer numerical scheme) present a similar performance with the first-order upwind numerical scheme and a different performance than the flow-oriented scheme (SUPER scheme). The vertical w<sup>1</sup> velocity distribution predicted by the SUPER scheme presents a more abrupt and accurate profile due to the successful minimizing of the false-diffusion errors.

### **2.5. Heat and mass transfer**

A physical problem that has been used to evaluate the performance of the numerical accuracy is the water-vapor condensation of humid air in the three dimensional geometry of a real-scale indoor space. A two-phase flow Euler-Euler mathematical model has been developed, wherein the humid air and water droplets are being treated as separate phases. The two phases exchange momentum and energy and, as the temperature drops below the dew point of humid air, mass transfer and phase change of water vapor to liquid takes place. The flow of humid air inside the room is buoyancy driven in the temperature range of 290–303 K. The properties of humid air (enthalpy, relative humidity, concentration of water vapor, saturation vapor pressure) vary with the temperature [22]. The dimensions of the domain are: width (X) × height (Y) × length (Z) = 4.0 × 4.0 × 8.0 m.
