The chosen configuration (
Figure 4) is the “circular tube fin”. It has several advantages such as reduced weight, better temperature control and easier transport. The type of tube is often chosen to reduce losses [
13,
14,
15]. The inner diameter of the tube is 8 mm, the fin length is 3 mm, the tube thickness is 1 mm and the distance between two consecutive fins is 2 mm. The staggered arrangement is triangular, to avoid interference problems, and the distance between each arrangement center is 17 mm (the distance will be called p
t or p
l if it is from tube to tube or from tube to the outer shell).
After the number of tubes is set, the width, thickness, and length are defined. At this point, we have to distinguish what happens in the mono-phase or the two-phase condensation. The analysis will be conducted for both the working fluid (inside) and the water (outside).
4.1. Monophasic Condensation
Working fluid side
The fluid characteristics (viscosity coefficient, thermal conductivity, etc.), were derived from the Coolprop library. First, the enthalpy difference is calculated to compute the thermal exchange. Then, the LMTD is derived with the method previously described. It is now necessary to compute the overall heat coefficient U in order to find the required exchange area. To compute the coefficient U we have to introduce the basic dimensionless numbers: Reynolds, Nusselt, Prandtl, and Froude. The
Reynolds number (Re) is defined as:
where ρ is the fluid density, u
m is the fluid velocity, μ is the dynamic viscosity, and D
h is the hydraulic diameter (it will be different in the working fluid or cooling fluid case).
The
Nusselt number (Nu) is a dimensionless number expressed by the ratio convective to conductive heat transfer. A Nusselt number close to one is characteristic of “slug flow“ or laminar flow. A larger Nusselt number corresponds to more active convection, with turbulent flow typically in the 100–1000 range.
The Dittus and Boelter equation, which can be applied to the inlet region (where turbulent flow is not developed), will be used to compute the Nusselt number:
This equation is only used for single-phase operating.
The
Prandtl number (Pr) is a dimensionless number, defined as the ratio of momentum diffusivity to thermal diffusivity. That is, the Prandtl number is given by:
where μ is the dynamic viscosity, c
p is the specific heat capacity and λ is the thermal conductivity. For gases, the Prandtl number varies from 0.2 to 1; for water or liquid, it varies from 1 to 10.
The
Froude number (Fr) is a dimensionless number defined as the ratio of the flow inertia over the external field forces (the latter in, many applications, is simply due to gravity). The Froude number is based on the speed–length ratio, and it is defined as:
where
u is a characteristic flow velocity,
g is the gravity and
l is the characteristic length.
Once all quantities have been defined, the following procedure will be used.
Firstly, the Prandtl number is computed, and secondly, the calculation of the fluid velocity inside the pipes is evaluated, permitting the evaluation of the Reynolds number, which leads to the Nusselt numbers with Equation (13).
After computing the Nusselt numbers, it is then possible to estimate the hi that is the heat transfer coefficient by Equation (12).
At this point, it is necessary to introduce the areas. When the numbers of tubes are known, where N
s is the number of pipes where the mass flow condensate, N
r is the numbers of transits of the same bundle of tubes, and the total number N
t, the geometrical properties can be calculated. The inner area of the pipes is:
and the minimum free flow area is:
Cooling fluid side
As previously mentioned, the characteristics of the fluids are derived from the Coolprop library. The main areas (external pipes side) are the A
p, the primary area, the fin area A
f and the heat transfer surface area A
o. The fin area is defined as:
The primary area is the difference between the pipe surface area and the area blocked by the fins.
The
Ao is the total heat transfer area, computed by the sum of the primary area and the fin area.
Another important parameter to determine is the minimum areas among the pipelines, where the water flows. If a triangular configuration is chosen [
10], it is possible to consider that surface, as a flat surface. In this way, the following simplified formula can be used, without significant errors.
The minimum area is represented in
Figure 5.
The hydraulic diameter is calculated as follows:
The Reynolds number for the cooling fluid is finally calculated with the hydraulic diameter. The velocity in the Re number is expressed by:
Finally, in this case, the Nusselt number changes and it is defined by the equation:
Collecting all obtained data, the overall heat transfer coefficient U can be computed:
Using the LMTD approach, the surface required for the heat exchange is defined by:
where Q is the power exchanged, U the overall heat transfer coefficient, and
is the LMTD.
Pressure drop
In this paper, the pressure drop equation, proposed by Fanning has been used, with the Fanning friction factor “f” proposed by Taitel and Dukler:
In Equation (28), L
e is the equivalent length of the pipe, ρ is the inner fluid density, u is the pipe fluid velocity and d
i is the internal pipe diameter. The f factor is a function of the Re and the roughness of the pipe. For smooth pipes, the friction factor, in a range of 3000 < Re < 10
5, can be approximated by:
If Re > 10
5, the following equation is more accurate:
4.2. Biphasic Condensation
As the condensation process proceeds along the pipes, the working fluid velocity decreases. At first, the condensation will occur on the wall of the pipes, then layer by layer, the liquid phase will increase. When the fluid is condensing, its thermodynamic characteristics change. The same procedure for the mono-phase has been adopted, introducing some necessary changes [
22,
23,
24]. The evolution of the working fluid, from quality “0” to quality “1”, has been divided into four-parts.
Part one, when the quality x is within 0 ÷ 0.75.
Part two, when x = 0.75 ÷ 0.5.
Part three, when x = 0.5 ÷ 0.25.
Part four, when x = 0.25 ÷ 0.
Similarly, for the cooling fluid side, there will be four corresponding stages. The four stages of the cooling water have been computed, assuming that every property is changing linearly. From the working fluid side, the Martinelli parameter is introduced to compute the Nusselt number. The formulae used for the R245fa is:
Re is the equivalent Reynolds number defined as:
where G
eq is the liquid equivalent mass flow rate, expressed by the following equation:
x is the quality, and the ρl and ρg are, respectively, the liquid and the gas density.
Pressure drop
The approach to predict the pressure drop is the “homogeneous model”, based on the experimentally derived theory. The pressure drop equation used is based on Kedzierski and Goncalves’ theory, modified by Pierre, for the refrigerant fluid approach:
where ν is the specific volume of the two-phases fluid, L
e is the equivalent length and f
n is the new friction factor:
This f
n is based on the Reynolds and K
f number, when the fluid is all liquid, so Re is defined as:
The K
f number is expressed by:
where α is the latent heat, provided by the Fluid-pro database,
is the quality of the gas and g is the gravity.