1 INTRODUCTION
The pulsating (or oscillating) heat pipe (PHP) is a simple capillary tube bent into branches meandering between heat source and sink and partially filled with a two-phase, usually single component, working fluid. A pattern of multiple vapor bubbles separated by liquid plugs forms inside the tube. When heat power is applied, the whole train of bubbles and plugs begins to oscillate spontaneously due to evaporation and condensation occurring in the hot evaporator part and a cooler condenser, respectively. A combination of the latent heat transfer produced by the phase change and the convective boiling and condensation leads to a high performance of the PHP. Combined with its simplicity (thus high reliability and low cost), it makes PHP of high industrial potential. Last decades, researchers have extensively studied PHP [1] with dozens of research articles published each year in leading scientific journals. However, the PHP functioning is intrinsically non-stationary, so very complex and not completely understood. Simple PHP design tools are still absent, and the only way to predict their functioning is numerical simulation [2].
One of the most critical issue for the PHP modeling is the vapor phase thermodynamic state. In most two-phase heat and mass exchange applications (like boiling [3]), the vapor is mostly at saturation, so its compressibility can be neglected. The PHP case is different. Because of its non-stationary functioning, the vapor state continuously varies, so both the current state and its change should be understood. The vapor at equilibrium with the liquid (at saturation) condenses at compression without giving an elastic response necessary for the oscillations.
The vapor state was studied experimentally by several research teams [4,5,6,7] by inserting the thermocouples into the PHP channel. Such studies are extremely delicate because of the difficulties in measuring the vapor temperature during the oscillations. The convective heat exchange of the vapor with a thermocouple is weak, and the measurement frequency is limited by the thermocouple thermal mass; the parasitic radiative heat exchange with the internal surfaces of the liquid and dry channel needs to be carefully compared to the convective heat exchange. While such an evaluation has been done in some works [4,7], it was neglected in others, thus reducing their data reliability. Gully et al. [4] have shown that the vapor is nearly always superheated in the evaporator and rarely approaches the coexistence curve; the vapor superheating ~ 10 K (with respect to \(T_{sat}(p)\)) was measured. Another team [5,6] observed that, while the vapor is superheated in the evaporator part that remains dry during the oscillations, the measured \(T_v\) approaches \(T_{sat}\) in the areas where the liquid film covers the PHP channel walls in the evaporator and adiabatic sections. However, the above thermocouple heat transfer analysis was not performed in these works. In addition, the cooling of the thermocouple by evaporation of liquid film left on it after the receding of the liquid plug cannot be excluded. Recently, Mameli et al. [7] evaluated the vapor state in the condenser. Their data suggest that most of the time, \(T_v<T_{sat}\) (i.e., the vapor is subcooled and thus metastable). It is difficult to understand from the physical viewpoint as the liquid films should always surround the vapor in the condenser and the metastability does not occur for condensation to the liquid surface. Mameli et al. expressed themselves reserves about these data, accounting for the limitations on the measurement method. The question of the vapor thermodynamic state in the PHP thus remains open.
In the numerical modeling approaches, the state of a vapor bubble situated inside the PHP tube is usually described in terms of a single temperature \(T_v\). Such a description is precise during the fast bubble motion, where the thickness of the thermal boundary layer developed inside the vapor is much smaller than the tube diameter due to the weakness of the vapor thermal conductivity [2]. In other cases, \(T_v\) corresponds to the spatially averaged vapor temperature.
Two PHP modeling approaches can be distinguished. In their pioneering article on the PHP modeling, Shafii et al. [8] assumed that the vapor behavior may be assimilated to that of the ideal gas that is described by the equation of state (EOS):
\begin{equation}
p = \frac{m R_v T_v}{V}
\tag{1}
\end{equation}
Such an assumption was adopted in a large majority of modeling approaches.
Instead of using EOS, Eq. (1), Rouaze et al. [9] assumed the vapor to always remain at saturation, evolving along the gas branch of the coexistence curve (right part of the red curve in Fig. 1b), where
\begin{equation}\label {psat} p=p_{sat}(T_v). \tag{2} \end{equation}
During the volume variation (vapor compression/dilatation) due to oscillations, vapor density \(\rho _{sat}\) changes. They find the new vapor temperature by inverting the \(\rho _{sat}(T)\) curve (shown in Fig. 2). We note that the saturated vapor model can also create a kind of elastic response: when the vapor volume reduces at constant mass, \(\rho _{sat}\) increases (i.e. \(\rho _{sat}^{-1}\) decreases) and the pressure rises according to Fig. 1b together with the vapor temperature. Note that such an assumption is hardly justified because it is not clear why the fluid cannot deviate from this particular trajectory (i.e., the coexistence curve) in the thermodynamic space. In addition, a change in vapor enthalpy appearing when the vapor evolves along the coexistence curve is not balanced; the energy conservation is not satisfied.
Fig. 1. Fluid phase diagram: a) in the p-T coordinates, and b) in the p-V coordinates; the isotherms are in green and the coexistence curve is in red
Many modeling approaches [10,11,12,13,14,15] try to reconcile these two models. On one hand, Eq. (1) describes the vapor state within them. On the other, they assume the channel wall-fluid heat exchange rate to be proportional to the difference \((T_w-T_v)\) of the wall and vapor temperatures. Note that in reality, the wall heat flux is spent mainly for the phase change, the rate of which is well known to be proportional to \((T_{w}-T_{sat})\) [3]. They justify the above assumption by the hypothesis that the vapor is saturated (\(T_v=T_{sat}\)).
The assumptions of superheated vapor and saturated vapor are, however, mutually exclusive if applied in dynamics. Indeed, the pressure of the saturated vapor is entirely defined by \(T_v\) (cf. Eq. (2)) rather than by \(T_v\), \(m\), and \(V\) in the EOS, Eq. (1), of superheated vapor.
In this article, we describe a modeling approach in which the vapor is allowed to change its state from superheated to saturated. This approach is used in the CASCO code (advanced PHP simulation code, the abbreviation of its name in French: Code Avancé de Simulation de Caloduc Oscillant) [16,17]. In the second part of this communication, we discuss the fluid merit number that appears to be linked to the vapor thermodynamic state evolution in PHPs.
2 VAPOR THERMODYNAMIC STATE
The phase diagram for common fluids is sketched in Fig. 1. The fluid equation of state (EOS, i.e., the set of thermodynamic states that can be occupied by the fluid) is a surface in the coordinates \((p,V,T)\). The two phase (liquid-gas) coexistence region situates inside the red bell-shaped curve called coexistence curve as seen in the projection of this surface to the \(p-V\) plane (Fig. 1b). Its projection to the \(p-T\) plane is just a line called saturation curve shown in Fig. 1a. A part below the saturation curve (where the point D situates) corresponds to the gas phase. The point A (saturated vapor) is situated at the coexistence curve. The gas state D is often referred to as superheated vapor because, at equal pressures, its temperature is higher than that of the saturated vapor, \(T_\mathrm {D}>T_{sat}(p_\mathrm {D})\), cf. Fig. 1a.
The above assumption about the vapor behavior similar to the ideal gas can be easily checked by comparing the density \(\rho _{sat}\) of saturated vapor calculated from Eqs. (1,2)
\begin{equation}\label {rsat} \rho _{sat}=\frac {p_{sat}(T_v)}{R_vT_v}, \tag{3} \end{equation}
to the experiment. Fig. 2 shows, for the water example, that the saturated vapor can be approximated with good accuracy by the ideal gas. For this reason, there is no need to use more sophisticated EOS forms.
Fig. 2. Density of the water saturated vapor depending on its temperature.
When the superheated vapor is compressed or dilated (i.e., its volume changes), the vapor state follows EOS, Eq. (1), and the pressure responds, which creates a restoring force for the oscillation. However, to manage such events in the model, a solution proposed by Shafii et al. [8] can be used. The vapor state obeys Eq. (1) as long as \(p<p_{sat}(T_v)\) (point D in Fig. 1). Once the pressure obtained with Eq. (1) becomes larger than \(p_{sat}(T_v)\), the vapor pressure is imposed to be equal to \(p_{sat}\) and the vapor mass is made tobe equal to \(\rho _{sat} V\). The extra vapor mass condenses and is added to the liquid plug for mass conservation. In other words, the point B in Fig. 1 is shifted to the point C. This means that the subcooled vapor (which is a metastable or even thermodynamically unstable state) is not allowed in our model. This is justified from the physical point of view as discussed above.
The vapor temperature is not entirely homogeneous during vapor compression or expansion. Thermal boundary layers are developed near both the tube walls at temperature \(T_{w}\) and the liquid films at temperature \(T_{sat}\) covering the walls. The thickness of these layers is a complex yet unsolved issue. Direct numerical simulations [4] of the gas flow during oscillatory compression performed with ANSYS CFX software for oxygen at about 70 K showed a thickness of \(\sim \) 0.25 mm in the tube of \(2\) \(\mathrm {mm}\) diameter. The vapor flow remains mostly laminar in the thermal boundary layers; turbulent mixing is absent near the wall. For this reason, the vapor description with a unique temperature introduced above is reasonable. Because of a large temperature difference, the vapor heat exchange is mainly that with the dry wall area of the length \(L_{d}\). The heat exchange coefficient is \(U_{v}= Nu_{v}\lambda _{v}/d\) and the heat exchange rate is:
\begin{equation}\label {Qv} \dot Q_{v}=U_{v}\pi dL_{d}(T_{w}-T_v). \tag{4} \end{equation}
The above numerical calculation resulted in \(Nu_{v}\simeq 6\). The heat exchange with the wetted wall area is usually smaller because of the smallness of the temperature difference \((T_{sat}-T_v)\); it is neglected here.
As mentioned above, while the superheated vapor behavior is entirely coherent with the experimental measurements and theoretical model presented by Gully et al. [4], the saturated state was observed by Kim’s group [5,6].
3 VAPOR STATE CHANGE
The vapor energy equation is different for superheated vapor in different thermodynamic states and should be derived from the first law of thermodynamics.
3.1 State Change in the Vapor Domain
When the vapor is superheated and obeys the ideal gas law, Eq. (1), Shafii et al. [8] showed that it can be described by the equation (see [18] for its detailed derivation):
\begin{equation}\label {ensup} mc_{vv}\dot T_v=R_{v}T_v \dot m-p\dot V+\dot Q_{v} \tag{5} \end{equation}
where dot means the time derivative. This equation can be reduced by using both EOS, Eq. (1) and Mayer’s relation \(c_{pv}=c_{vv}+R_{v}\) valid for the ideal gas:
\begin{equation}\label {dotp} \dot p=\frac {p}{T_v} \frac {\gamma }{\gamma -1}\dot T_v-\frac {\dot Q_{v}}{V}, \tag{6} \end{equation}
where \(\gamma =c_{pv}/c_{vv}\) is the adiabatic index.
If one neglects the heat transfer \(\dot Q_v\) to the vapor, which is generally smaller than the effect of adiabatic compression, Eq. (6) can be integrated directly resulting in the adiabatic formula with no explicit dependence on time,
\begin{equation}\label {dotp1} p=CT_v^\frac {\gamma }{\gamma -1}, \tag{7} \end{equation}
where the integration constant \(C\) is determined at the initial time moment. This approximation is convenient because it reduces the number of differential equations and, for this reason was used in many modeling approaches [19]. Such an approximation can be justified by the smallness of \(\dot Q_v\) to the vapor (and can indeed be neglected) when calculating the overall heat transfer in the stable oscillation regime. However, in the vapor energy balance \(\dot Q_v\) is important. In particular, \(\dot Q_v\) term is essential for the start-up threshold determination [20,21], so a more general form, Eq. (5), is kept in CASCO.
3.2 Saturated Vapor State Change
Besides the physical reasons discussed above, there is a practical reason requiring the introduction of the saturated vapor state. If one assumes that the vapor always satisfies the vapor domain EOS, Eq. (1), and never reaches the saturation state, the numerical simulation code can become unstable at large heat powers or large simulation times: \(T_v\) exhibits increasingly large temporal oscillation around more slowly changing \(T_w\). A \(T_v\) rise causes a \(p\) rise through EOS, Eq. (1), which leads to \(T_{sat}\) increase. The vapor mass change rate is defined by the wall fluid heat flux so the vapor bubble mass change rate expression:
\begin{equation}\label {mdotf} \dot m\sim (T_w-T_{sat}), \tag{8} \end{equation}
as explained above and can thus become negative. Because of Eq. (5) where \(\dot m\) is the largest term, \(T_v\) decreases. However its decrease is much stronger than its previous increase. This effect can cause artificial oscillations that lead to huge vapor temperature (either positive or negative), and eventually the code crashes. To prevent it, the saturated vapor state needs to be introduced.
When the vapor is at saturation, it obeys the law as in Eq. (2). Most often, the saturation is attained in the condenser where there is no dry channel wall; the vapor sensible heat exchange \(\dot Q_v\) with the environment is negligibly small so one can assume
\begin{equation}\label {ensat} \dot {T}_v=0, \tag{9} \end{equation}
and the vapor pressure does not change either. All heat exchange leads to the phase change at the vapor-liquid interface, which maintains a constant vapor density \(\rho _{sat}(T_v)\) and the vapor mass evolution obeys
\begin{equation}\label {mvsat} \dot {m}=\dot {m}_{sat}\equiv \rho _{sat}\dot V, \tag{10} \end{equation}
instead of Eq. (8). In other words, the vapor condenses when compressed and vice-versa, evaporation occurs when the vapor is dilated. The condensed (or evaporated) liquid mass should be added to (or removed from) the liquid domain to conserve the total fluid mass.
The \(\dot {m}\) definition in Eq. (10) cannot always be applied. Indeed, this would lead to impossibility for a bubble to leave the saturation state even during a strong film condensation (i.e. vapor mass reduction) or the bubble expansion. Within an adequate thermodynamic model, both these changes should lead to a pressure reduction so the vapor may possibly become superheated, cf. Fig. 1b). For this, an additional criterion based on the pressure change needs to be introduced; it is described hereafter.
Suppose the vapor is at saturation (point A in Fig. 1). While the vapor remains saturated, its evolution is governed by Eqs. (9,10), which means that the vapor state does not move from the point A. To determine if the vapor quits this state, one needs to calculate the pressure change \(\textrm {d}p_{gas}=\dot {p}\textrm {d}t\) under the hypothesis that the vapor is superheated, where \(\textrm {d}t\) is the time step and \(\dot {p}\) is obtained from the EOS (with Eq. (6) for the ideal gas EOS). Since the point A situates at the coexistence curve, all the parameters in Eq. (6) have to be taken at the coexistence. In Fig. 1, the pressure change \(\textrm {d}p_{gas}\) corresponds to the transitions A\(\to \)B or A\(\to \)D. The corresponding \(T_v\) change is \(\textrm {d}T=\dot {T}_v\textrm {d}t\) with \(\dot {T}_v\) given by Eq. (5).
Two situations are possible. In the first case, the new vapor state would fall inside the coexistence curve (point B in Fig. 1). Evidently, in this case the vapor should remain saturated. In the second case (point D in Fig. 1), the vapor should quit the saturation state and become gas; the mass and temperature derivatives should now be defined with Eqs. (5, 8). Since the isotherm is a decreasing function (see Fig. 1), to distinguish two these cases, one needs to compare \(\textrm {d}p_{gas}\) with the change
\begin{equation}\label {dpsat} \textrm {d}p_{sat}=\left .\frac {\textrm {d}p}{\textrm {d}T}\right |_{sat}\textrm {d}T, \tag{11} \end{equation}
that correspond to the transition A\(\to \,\)C in Fig. 1. The vapor stays at saturation while \(\textrm {d}p_{gas}\geq \textrm {d}p_{sat}\). The vapor leaves the saturation state when \(\textrm {d}p_{gas}<\textrm {d}p_{sat}\). Note that Fig. 1 apply to the case \(\textrm {d}T>0\). One can easily check that the same criterion is valid for the case \(\textrm {d}T<0\).
4 FLUID MERIT NUMBER
The above vapor state change criterion (i.e. a criterion of exiting the saturation state) reduces to
\begin{equation}\label {crit} \frac {1}{N}-\frac {\dot Q_{v}}{\dot T_vV}\left .\frac {\textrm {d}T}{\textrm {d}p}\right |_{sat}<1, \tag{12} \end{equation}
where
\begin{equation}\label {N} N=\frac {(\gamma -1)T}{\gamma p_{sat}(T)}\left . \frac {\text {d}p}{\text {d}T} \right |_{sat}. \tag{13} \end{equation}
For a given fluid, the number \(N\) depends only on the temperature (Fig. 3) and exhibits a weak temperature variation. For some fluids like FC-72, \(N<1\), while for others (ethanol or water), \(N>1\) over all the temperature range of interest.
Fig. 3. Fluid merit number N related to the vapor properties
The objective is to analyze the fluid merit for the PHP startup and stable functioning. This analysis is based on the PHP simulation results obtained with the numerical code CASCO [22], showing the importance of bubble nucleation and growth in the evaporator section. In particular, the stable oscillations start when multiple vapor bubbles are generated in the evaporator. Consider a bubble freshly generated at a nucleation (hot) spot at the wall. The second bubble can be generated at the same hot spot while the first bubble is advected because of the liquid plug displacement during oscillating motion. Two bubbles become neighbors. The oscillations are amplified when both bubbles expand, so a large pressure difference is created. However, when the growth of the second bubble leads to retraction and disappearance of the first, the oscillating motion is weak, and the PHP performance is poor. The growth of the second bubble inevitably leads to the compression of the first. At its birth, the vapor in the bubble is saturated. If the vapor remains saturated in the first bubble after generation of the second, the growth of the second bubble will lead to the contraction (because of condensation) of the first and its eventual disappearance. If, on the contrary, the vapor in the first bubble becomes superheated (i.e., elastic), it can resist the compression and expand if evaporation into it occurs. For this reason, the ability of the vapor to change its state (saturated\(\to \)superheated) is the major factor for the sustainment of oscillations.
In the vapor state change criterion, Eq. (12), the first (\(N^{-1}\)) term is more important than the second (containing \(\dot Q_{v}\)). The mean value of the latter is generally small (because of the weak vapor heat exchange) and positive: the tube walls are hotter than the vapor in the evaporator. One can see that if \(N<1\), the bubbles can rarely exit the saturation state. The only effect that can drive them out of saturation is the vapor heat exchange. On the contrary, the bubbles of the fluids with \(N>1\) leave the saturation state, helping to establish the stable oscillations. For this reason, \(N\) characterizes the fluid merit for usage in PHPs.
Such a fluid merit definition is coherent with the oscillation start-up threshold found analytically for the single-branch PHP [21,2]. It was found that the oscillation start-up threshold decreases (which is beneficial) with \(\text {d}p/\text {d}T |_{sat}\).
Fig. 4. The PHP geometry and the phase distribution pictured by the CASCO postprocessor
5 SIMULATION RESULTS
The simulation is performed with the in-house CASCO software (French abbreviation for Code Avancé de Simulation du Caloduc Oscillant: advanced PHP simulation code) version 4 [23,22,16] for three different fluids (water, ethanol and FC-72) for the conventional PHP geometry (Fig. 4). Thin liquid films (in violet) cover the internal tube walls inside the vapor bubbles, except of the dry areas (in gray); the liquid plugs are shown in blue. The wall temperature is shown with the color varying from blue to red; its scale is indicated at the top. The round turns (that are accounted for with the appropriate liquid pressure losses) are not pictured for simplicity. The black lines connect equivalent points that correspond to the extreme points of each turn. An example of FC-72 in horizontal orientation for \(P_e=\) 20W at \(t=\) 140.198 s is displayed. The total power \(P_e\) injected into the evaporator and the temperature \(T_c\) at the internal tube walls are imposed in the simulation. Some portions of the tube belong to the evaporator. The length of one portion is denoted \(L_e\), and their number is referred to as \(N_{turn}\). To study the local fluid-tube thermal interaction, each of them is assumed to be heated independently and homogeneously, with the power \(P_e/N_{turn}\). The PHP parameters are shown in Table 1.
Table 1. Parameters used for the simulation
| Parameters | |
|---|---|
| Length of evaporator | \(L_e=\) 126 mm |
| Length of adiabatic section | \(L_a=\) 126 mm |
| Length of condenser | \(L_c=\) 110 mm |
| Number of turns | \(N_{turn}=\) 10 |
| Feedback section length | \(L_{fb}=\) 130 mm |
| Filling ratio | \(\phi =\) 0.5 |
| Inner diameter | \(d=\) 1.4 mm |
| Outer diameter | 3.2 mm |
| Condenser temperature | \(T_c=22^{\circ }\)C |
| Tube material | Copper |
| Time step | \(\Delta t= \) 10 µs |
| Tube mesh size | 2 mm |
| Bubble deletion threshold | \(L_{thr}=\) 10 m |
| Nucleated bubble length | \(L_{nucl}=\) 500 µm |
| Nucleation barrier | \(\Delta T_{nucl}=15^{\circ }\)C |
The liquid film thickness \(\delta \) is a fixed parameter in CASCO. We calculated it a posteriori from the average menisci velocity observed in simulations using the Aussillous & Quéré formula [24,20]. The thickness is not very sensitive to the velocity and can be fixed for each fluid. In present simulations, it is \(40\) \(\mathrm {\text {µ}m}\) for water, \(80\) \(\mathrm {\text {µ}m}\) for ethanol and \(60\) \(\mathrm {\text {µ}m}\) for FC-72. Fig. 5 illustrates the performance of water and ethanol.
Fig. 5. Time evolution of the averaged over evaporator tube sections temperature Te for water and ethanol in horizontal (0∘) and vertical favorable (90∘), evaporator under condenser) orientations for Pe = 100 W
One can see that in this particular situation, both water and ethanol PHP show stable chaotic oscillations after an initial start-up period. In agreement with the above theory, the start-up of ethanol PHP in horizontal orientation is more difficult than that of water PHP. In particular, an initial evaporator overheating above the stable functioning temperature is needed to start-up the oscillations. In the beginning of evolution, most bubbles that existed initially disappear (the liquid plugs coalesce) when their length becomes smaller than \(L_{thr}\). The system starts up by the generation of multiple bubbles [22].
Another manifestation of the difference between these fluids concerns different orientations. One can see that for water the performances in horizontal and vertical orientations are similar. It means that the pressure difference in the neighboring bubbles created during oscillation is much larger than the hydrostatic pressure (which is beneficial). This is not the case for ethanol, where the difference between horizontal and vertical orientations is striking.
The case of FC-72 is very different: for the horizontal orientation, stable oscillations appear only for a few particular choices of parameters. Usually the system dries out without attaining a stable oscillation regime. The initial stage dynamics is different from water or ethanol cases. After disappearance of the initial small bubbles and heating of the tube walls in the evaporator section, bubble generation starts in long liquid plugs. At each time step, several bubbles of the size \(L_{nucl}\) are allowed to be nucleated sequentially once the difference between the wall temperature and the local saturation temperature exceeds the nucleation barrier (also called the superheating of the onset of nucleate boiling) [16]. As the vapor density is smaller, the ends of the original plug displace slightly during each nucleation event to accommodate the newly generated bubble volume. Therefore, each subsequent bubble nucleation leads to a compression of the previously nucleated bubbles. Because of a small \(N<1\) for FC-72, in many cases, a new nucleation leads to recondensation of the bubbles that were nucleated at the previous time step (Fig. 6). Instead of expanding (as the newly generated bubbles do in the water case [22]), they recondense. This process does not create oscillations, so the PHP eventually dries out. In Fig. 6, we show four consecutive snapshots separated by \(1\) \(\mathrm {ms}\) (\(100\Delta t\)). The evaporator part of the 16th branch (counted from the bottom) is presented. Most newly nucleated bubbles disappear between snapshots. The only bubble that survives two consecutive snapshots is indicated with an arrow.
The fluid hierarchy provided by the number \(N\) (Fig. 3) agrees with the experimental study of Thapa et al. [25], who used several fluids in the same PHP prototype. Its geometry was close to ours. They showed that water exhibited the best performance (i.e., the smallest PHP thermal resistance). Ethanol was slightly worse, while FC-72 was the worst.
Fig. 6. Bubble generation and disappearance in the evaporator for the FC-72 case, the same as in Fig. 4, with the same Tw scale
The number \(N\) characterizes the capacity of a given fluid to provide the oscillatory behavior. However, it is evident that the fluid merit is not defined solely by \(N\). The liquid phase properties are also important. A ‘good’ fluid should have a small viscosity (for small viscous losses). High values of both latent heat and heat capacity of liquid are required to provide an efficient heat exchange.
Several attempts were undertaken in the past to propose an empirically-based fluid merit number accounting for both liquid and vapor properties. Probably the most often cited is that of Kim & Kim [26]:
\begin{equation}\label {MKim} M_{Kim} = \frac {c_{pl}R_v T_{sat}}{\nu _l h_{lv}}\left .\frac {\text {d}p}{\text {d}T}\right |_{sat}. \tag{14} \end{equation}
Note that this quantity is dimensional and thus does not result from a physical analysis. According to the graph [26] based on Eq. (14), water shows the smallest (i.e., the worst) \(M_{Kim}\), while FC-72, the largest. Although such an evaluation agrees with the experimental data of Kim & Kim [26], it is contrary to those of Thapa et al. [25]. Note that the geometries of their prototypes were entirely different. Such a comparison shows a necessity to search for a physically based merit number (possibly including the PHP geometrical parameters), which is a task for future research.
6 CONCLUSIONS
In conclusion, we propose here a model of the vapor behavior in the fast transients that occur in the PHP and derive the fluid merit factor. One notes that this factor corresponds only to the vapor phase properties and not to the liquid properties responsible for many other phenomena impacting the fluid merit, like liquid viscosity and wetting. Such a merit factor allows us to explain the numerical simulation results obtained for three different fluids with the code CASCO. We show that, in agreement with the merit factor expression, FC-72 is generally worse for PHP applications than ethanol or water, which is confirmed by experimental studies of other researchers.
Nomenclature
| \(c_{p\varphi }\) | specific heat at constant pressure of the phase \(\varphi =(l,v)\) [\(\frac {\mathrm {J}}{\mathrm {kg}\,\mathrm {K}}\)] |
| \(c_{vv}\) | specific heat at constant volume of vapor [\(\frac {\mathrm {J}}{\mathrm {kg}\,\mathrm {K}}\)] |
| \(d\) | PHP channel inner diameter [\(\mathrm {m}\)] |
| \(h_{lv}\) | latent heat [\(\frac {\mathrm {J}}{\mathrm {kg}}\)] |
| \(L\) | length [\(\mathrm {m}\)] |
| \(m\) | mass of vapor [\(\mathrm {kg}\)] |
| \(N\) | fluid merit number |
| \(N_{turn}\) | number of PHP turns |
| \(Nu\) | Nusselt number |
| \(p\) | pressure [\(\mathrm {Pa}\)] |
| \(P, \dot Q\) | power [\(\mathrm {W}\)] |
| \(R\) | specific gas constant [\(\frac {\mathrm {J}}{\mathrm {kg}\,\mathrm {K}}\)] |
| \(S\) | cross-section area [\(\mathrm {m}^{2}\)] |
| \(T\) | temperature [\(\mathrm {K}\)] |
| \(t\) | time [\(\mathrm {s}\)] |
| \(U\) | heat transfer coefficient [\(\frac {\mathrm {W}}{\mathrm {m}^{2}\,\mathrm {K}}\)] |
| \(V\) | vapor bubble volume [\(\mathrm {m}^{3}\)] |
| \(x\) | coordinate along the PHP tube [\(\mathrm {m}\)] |
| \(\gamma \) | vapor adiabatic index |
| \(\Delta \) | difference |
| \(\delta \) | thickness [\(\mathrm {m}\)] |
| \(\lambda \) | heat conductivity [\(\frac {\mathrm {W}}{\mathrm {m}\,\mathrm {K}}\)] |
| \(\nu \) | liquid kinematic viscosity [\(\frac {\mathrm {m}^{2}}{\mathrm {s}}\)] |
| \(\rho \) | density [\(\frac {\mathrm {kg}}{\mathrm {m}^{3}}\)] |
| \(\phi \) | volume fraction of liquid in PHP |
| \(a\) | adiabatic |
| \(c\) | condenser |
| \(d\) | dry |
| \(e\) | evaporator |
| \(f\) | liquid film |
| \(fb\) | feedback section (vertical in Fig. 4) |
| \(l\) | liquid |
| \(nucl\) | nucleation |
| \(sat\) | at saturation |
| \(thr\) | threshold |
| \(v\) | vapor |
| \(w\) | internal tube wall |
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