**4. Results and discussion**

The condensing behavior and evaporating dynamics of refrigerant R404a have been simulated using the VOF approach. Due to the temperature difference, and heat flux, the refrigerant condenses throughout the spiral tubing array, whereas sporadic evaporation occurs given the inertial and body forces present in the flow. The simulation was initialized by filling the total volume of the condenser section with both, the primary fluid (liquid water) and the secondary fluid domain (refrigerant) in a vapor state.

**Figure 4** shows the domain initialization at t = 0.0 s. After this instant, the water starts to flow while the refrigerant is still in a full vapor state. The magnitude and direction of the water velocity is presented by the streak line in the internal flow section, and the quality of the secondary flow is indicated by the contour interface.

At t = 0.15 s, the refrigerant leaving the condenser has fully condensed near the outlet section based on the sub-cooled temperature boundary condition. Upstream the copper tube, the refrigerant begins a phase-transitioning throughout the lower section of the spiral geometry (shown in **Figure 5**) induced by the fluid temperature distribution.

The two-phase flow in the copper tube shows complicated details, with various phase-transition patterns and irregular oscillation manifesting at the periphery of the geometry due to the centrifugal [42] and inertial forces. The mean vapor quality found in the refrigerant region is 60%.

The water around the copper elbow carrying vapor refrigerant near the inlet develops a high velocity zone due to the reduction of area enhancing the heat flux in

the system [43]. On the other hand, the vapor refrigerant develops a vortex pair at the beginning of the first spiral. This is probably induced by the tube curvature close to the inlet elbow. Interestingly, no other vortices develop in subsequent curved regions which suggests that the vortex develops due to the thermal and inertial characteristics of the vapor refrigerant entering the tubular array and the

*Numerical Analysis of a Water-Cooled Condenser at Startup Conditions for Refrigeration…*

*DOI: http://dx.doi.org/10.5772/intechopen.97203*

*Phase-transition and velocity field development at the outlet of the condenser at t = 0.15 s.*

**Figure 6** shows the refrigerant phase transition details at 0.5 s. At this instant, the

refrigerant has completely transitioned to liquid at the last quarter of the spiral leading to the outlet of the condenser. In the middle section of the condenser, the saturated vapor condenses and evaporates along the outer wall in repeated patters of saturated vapor and multi-phase flow swirling due to gravity and rotational speed. Since the wall temperature across the copper wall was assumed constant, the phase-transition region extends across a large region of the tubular array. This boundary condition prevented a flash point transition between phases [44]. Furthermore, the outlet temperature was specified to remain in a sub-cool state allowing the saturated liquid to develop further into the inner regions of the

proximity to the 90° elbow located at the inlet.

condenser.

**19**

**Figure 5.**

*Numerical Analysis of a Water-Cooled Condenser at Startup Conditions for Refrigeration… DOI: http://dx.doi.org/10.5772/intechopen.97203*

**Figure 5.**

**4. Results and discussion**

*Heat Transfer - Design, Experimentation and Applications*

erant) in a vapor state.

found in the refrigerant region is 60%.

distribution.

**Figure 4.**

**18**

The condensing behavior and evaporating dynamics of refrigerant R404a have been simulated using the VOF approach. Due to the temperature difference, and heat flux, the refrigerant condenses throughout the spiral tubing array, whereas sporadic evaporation occurs given the inertial and body forces present in the flow. The simulation was initialized by filling the total volume of the condenser section with both, the primary fluid (liquid water) and the secondary fluid domain (refrig-

**Figure 4** shows the domain initialization at t = 0.0 s. After this instant, the water starts to flow while the refrigerant is still in a full vapor state. The magnitude and direction of the water velocity is presented by the streak line in the internal flow section, and the quality of the secondary flow is indicated by the contour interface. At t = 0.15 s, the refrigerant leaving the condenser has fully condensed near the outlet section based on the sub-cooled temperature boundary condition. Upstream the copper tube, the refrigerant begins a phase-transitioning throughout the lower section of the spiral geometry (shown in **Figure 5**) induced by the fluid temperature

The two-phase flow in the copper tube shows complicated details, with various phase-transition patterns and irregular oscillation manifesting at the periphery of the geometry due to the centrifugal [42] and inertial forces. The mean vapor quality

The water around the copper elbow carrying vapor refrigerant near the inlet develops a high velocity zone due to the reduction of area enhancing the heat flux in

*Numerical solution of the phase transition of R404a in the water-condenser at t = 0.0 s.*

*Phase-transition and velocity field development at the outlet of the condenser at t = 0.15 s.*

the system [43]. On the other hand, the vapor refrigerant develops a vortex pair at the beginning of the first spiral. This is probably induced by the tube curvature close to the inlet elbow. Interestingly, no other vortices develop in subsequent curved regions which suggests that the vortex develops due to the thermal and inertial characteristics of the vapor refrigerant entering the tubular array and the proximity to the 90° elbow located at the inlet.

**Figure 6** shows the refrigerant phase transition details at 0.5 s. At this instant, the refrigerant has completely transitioned to liquid at the last quarter of the spiral leading to the outlet of the condenser. In the middle section of the condenser, the saturated vapor condenses and evaporates along the outer wall in repeated patters of saturated vapor and multi-phase flow swirling due to gravity and rotational speed.

Since the wall temperature across the copper wall was assumed constant, the phase-transition region extends across a large region of the tubular array. This boundary condition prevented a flash point transition between phases [44]. Furthermore, the outlet temperature was specified to remain in a sub-cool state allowing the saturated liquid to develop further into the inner regions of the condenser.

**Figure 6.** *Numerical solution of the phase transition of R404a in the water-condenser at t = 0.5 s.*

**Figure 7** shows the condensing and evaporating characteristics of R404a at t = 1.25 s. At the tube periphery and after the first revolution, the saturated vapor begins to condense, transitioning into a two-phase flow with a 90% quality and further decreasing into a quality of 82% while displaying a parabolic profile along the outer surface wall. Then, the mixture evaporates back to a saturated vapor state before it reaches its full condensation state after the third revolution.

Since the heat transfer can be enhanced or penalized by geometrical factors [45], the widely used Dittus-Boelter (Eq. (17)) was employed to find the Nusselt number in the internal tubing section for the vapor refrigerant flow. Based on the fundamental restrictions of the geometry, parametric modifications to the equation determining the Nusselt number (Eq. (18)) need to be considered based on constant wall temperature assumptions [46], curvature ration [47], and fluid regime [48] such that:

$$Nu\_v = \frac{\left(\frac{\zeta}{8}\right) Re\_v Pr\_v}{1.07 + 12.7\left(\frac{\zeta}{8}\right)^{0.5} \left(Pr\_v \, ^{0.66} - 1\right)} \, 4\varepsilon 3 < Re < 5\epsilon 6, 0.5 < Pr < 2\epsilon 3 \tag{17}$$

and

$$Nu\_{v-HLX} = Nu\_v \left( \mathbf{1} + \mathbf{0}.\mathbf{35} \left( \frac{D\_{\Omega\_{\rm b-w}}}{D\_{\rm HLx}} \right) \right). \tag{18}$$

where the friction factor is given by:

*DOI: http://dx.doi.org/10.5772/intechopen.97203*

and

**21**

**Figure 7.**

given that the Prandtl and Reynolds numbers are:

*Numerical solution of the phase transition of R404a in the water-condenser at t = 1.25 s.*

*Numerical Analysis of a Water-Cooled Condenser at Startup Conditions for Refrigeration…*

the refrigerant line and reducing the number of spirals.

*Prv* <sup>¼</sup> *<sup>μ</sup>vcp*�*<sup>v</sup> Kv*

> *\* vD*<sup>Ω</sup><sup>3</sup> *μv*

**Figure 8** shows the Nusselt number with respect to vapor refrigerant mass flow

*Re* <sup>¼</sup> *<sup>ρ</sup>vu*

as a function different tube diameter. For *D*<sup>Ω</sup><sup>3</sup> ¼ 0*:*0063*m*, the Nusselt number calculated ranges between 900 and 1500 at different vapor mass flows. On the other hand, the Nusselt number at *D*<sup>Ω</sup><sup>3</sup> ¼ 0*:*0191*m*, ranges between 2250 and 4200, suggesting that the heat transfer enhancement occurs by increasing the diameter of

*<sup>f</sup>* <sup>¼</sup> ð Þ <sup>1</sup>*:*82 ln *Re* � <sup>1</sup>*:*<sup>62</sup> �<sup>2</sup> (19)

*:* (21)

(20)

*Numerical Analysis of a Water-Cooled Condenser at Startup Conditions for Refrigeration… DOI: http://dx.doi.org/10.5772/intechopen.97203*

**Figure 7.** *Numerical solution of the phase transition of R404a in the water-condenser at t = 1.25 s.*

where the friction factor is given by:

$$f = \left(\mathbf{1.82} \ln \text{Re} - \mathbf{1.62}\right)^{-2} \tag{19}$$

given that the Prandtl and Reynolds numbers are:

$$Pr\_v = \frac{\mu\_v c\_{p-v}}{K\_v} \tag{20}$$

and

**Figure 7** shows the condensing and evaporating characteristics of R404a at t = 1.25 s. At the tube periphery and after the first revolution, the saturated vapor begins to condense, transitioning into a two-phase flow with a 90% quality and further decreasing into a quality of 82% while displaying a parabolic profile along the outer surface wall. Then, the mixture evaporates back to a saturated vapor state

Since the heat transfer can be enhanced or penalized by geometrical factors [45], the widely used Dittus-Boelter (Eq. (17)) was employed to find the Nusselt number in the internal tubing section for the vapor refrigerant flow. Based on the fundamental restrictions of the geometry, parametric modifications to the equation determining the Nusselt number (Eq. (18)) need to be considered based on constant wall temperature assumptions [46], curvature ration [47], and fluid regime [48]

4*e*3< *Re* <5*e*6, 0*:*5<*Pr* <2*e*3 (17)

*:* (18)

*D*<sup>Ω</sup>3�<sup>∞</sup> *D*HLx

before it reaches its full condensation state after the third revolution.

*Numerical solution of the phase transition of R404a in the water-condenser at t = 0.5 s.*

*Heat Transfer - Design, Experimentation and Applications*

*f* 8 

> 8 <sup>0</sup>*:*<sup>5</sup>

<sup>1</sup>*:*<sup>07</sup> <sup>þ</sup> <sup>12</sup>*:*<sup>7</sup> *<sup>f</sup>*

*Re vPrv*

*Prv* <sup>0</sup>*:*<sup>66</sup> � <sup>1</sup>

*Nuv*�*HLX* ¼ *Nuv* 1 þ 0*:*35

such that:

**Figure 6.**

and

**20**

*Nuv* ¼

$$Re = \frac{\rho\_v \overrightarrow{\boldsymbol{\mu}}\_v D\_{\Omega\_3}}{\mu\_v}. \tag{21}$$

**Figure 8** shows the Nusselt number with respect to vapor refrigerant mass flow as a function different tube diameter. For *D*<sup>Ω</sup><sup>3</sup> ¼ 0*:*0063*m*, the Nusselt number calculated ranges between 900 and 1500 at different vapor mass flows. On the other hand, the Nusselt number at *D*<sup>Ω</sup><sup>3</sup> ¼ 0*:*0191*m*, ranges between 2250 and 4200, suggesting that the heat transfer enhancement occurs by increasing the diameter of the refrigerant line and reducing the number of spirals.

**Acknowledgements**

*DOI: http://dx.doi.org/10.5772/intechopen.97203*

experimental analysis.

**Conflict of interest**

**Nomenclature**

*u*

**Subscripts**

**Greek letters**

*τ*

**23**

The author declares no conflict of interest.

*kg* h i

*s* � �

> *m* � �

*s* � �

*As* Surface area *<sup>m</sup>*<sup>2</sup> ½ � *D* Diameter½ � *m* h Enthalpy *kJ*

T temperature ½ � *K*

*α* Vapor volume fraction

*ς* Mass transfer intensity <sup>1</sup>

*ξ* Surface tension *<sup>N</sup>*

<sup>¼</sup> Stress tensor ½ � *Pa*

*S* Source

t Time ½ �*s V* Volume

x Quality

cr Critical cu Copper eff Effective l Liquid o Reference Ref Refrigerant sat Saturation v Vapor

*ρ* Density

*μ* Viscosity Ω 3*D* domain

*\** Velocity *<sup>m</sup>*

The author is grateful to FBD L.P. for the financial support to perform the

*Numerical Analysis of a Water-Cooled Condenser at Startup Conditions for Refrigeration…*

**Figure 8.** *Predicted Nusselt number as a function vapor refrigerant mass flow as a function of inlet diameter.*
