## Abstract

Thermal ground planes (TGPs) are flat, thin (external thickness of 2 mm) heat pipes which utilize two-phase cooling. The goal is to utilize TGPs as thermal spreaders in a variety of microelectronic cooling applications. In addition to TGPs and flat heat pipes, some investigators refer to similar devices as vapor chambers. TGPs are novel high-performance, integrated systems able to operate at a high power density with a reduced weight and temperature gradient. In addition to being able to dissipate large amounts of heat, they have very high effective axial thermal conductivities and (because of nanoporous wicks) can operate in high adverse gravitational fields. A three-dimensional (3D) finite element model is used to predict the thermal performance of the TGP. The 3D thermal model predicts the temperature field in the TGP, the effective axial thermal conductivity, and the evaporation and the condensation rates. A key feature of this model is that it relies on empirical interfacial heat transfer coefficient data to very accurately model the interfacial energy balance at the vapor–liquid saturated wick interface. Wick samples for a TGP are tested in an experimental setup to measure the interfacial heat transfer coefficient. Then the experimental heat transfer coefficient data are used for the interfacial energy balance. Another key feature of this model is that it demonstrates that for the Jakob numbers of interest, the thermal and flow fields can be decoupled except at the vapor–liquid saturated wick interface. This model can be used to predict the performance of a TGP for different geometries and implementation structures. This paper will describe the model and how it incorporates empirical interfacial heat transfer coefficient data. It will then show theoretical predictions for the thermal performance of TGP's, and compare with experimental results.

## Introduction

Heat pipes operate in a wide variety of heat transfer related applications where low temperature drops and high power are required. Flat heat pipes are an efficient technology for spacecraft cooling and electronic cooling applications due to their high thermal conductivity, low weight, and reliability [1].

In order for a heat pipe to work properly, the net capillary pressure difference between wet and dry pressure points must be greater than the sum of pressure losses in the system. Heat pipe performance is typically constrained by the flow resistance of the wick, which limits mass flow and the total heat load the system is able to transport. If the heat load on a heat pipe is increased, the mass flow inside the device increases. As the axial pressure gradient of the liquid within the wick structure increases, a point is reached where the maximum permissible capillary pressure difference across the vapor–liquid interface in the evaporator equals the total pressure losses in the system. The maximum heat transport of the device is reached at this point. So if the heat load exceeds this point, the wick will dry-out in the evaporator region and the heat pipe will not work. This point is called the capillary limit [2]. In addition to the capillary limit, there are other operating limits for heat pipes. For example, viscous, entrainment, sonic, and boiling limits. Since the TGP is expected to operate in a 20 g environment, the viscous and hydrostatic pressure difference in the liquid saturated wick are often very large. Therefore, the capillary limit is expected to be the most critical operating limit and studied in this research.

Recently, high performance aircraft utilizes complex electronics cooling packages. Theses electronic packages contain state-of-the-art high density circuit boards. In order to keep these packages in an appropriate thermal environment, a thermal management device such as heat pipe, loop heat pipe, or two-phase pumped loop is built into the support structure. It is very important for the thermal designer to meet the thermal performance not only on the ground but also during flight and maneuvers too [3]. These aircraft packages may undergo acceleration levels of 5–7.3 g-forces. Understanding the heat pipe performance under the resulted external body forces which are produced from system acceleration and random vibration is a vital issue. Since these forces may easily deprime the wick structure and force restart scenarios [4]. A flexible copper/water heat pipe was successfully tested and performed on a spin table up to 10-g acceleration load under steady state condition in the transverse direction to the heat pipe axis by Rengasamy [4]. Richardson et al. [5] tested water/stainless steel sintered wick heat pipe with 0–12 g in the longitudinal direction and found that as the acceleration increases the heat transfer capacity decreases. Phillips et al. [6] designed, fabricated, and tested loop heat pipes for specific avionics cooling applications aboard to the McDonnell Douglas F/A-18 Hornet. One of these loop heat pipe was tested up to 10 -g at a spin table in Russia. De Bock et al. [7] presented an approach to identify the feasible design space for high-g operation. The experimental data showed an effective thermal conductivity of 225–436 W/m·K at 0–10 g forces. Chamarthy et al. [8] conducted flow visualization experiments to evaluate capillary and permeability characteristic of sintered copper wick which used in Ref. [7] from 0 to 10 g using an ultraviolet fluorescent flow visualization method. A spin table is used to identify the effect of g-force on the capability of the liquid wicking structure to maintain water. It was found that the sintered wick held water but the intensity of ultraviolet fluorescent decreases with acceleration and height.

The TGP is an advanced planar heat pipe designed for cooling microelectronics in high gravitational fields, as in common heat pipes, but with a high total heat transport capability. Major advantages, however, include the ability to integrate directly with the microelectronic substrate for a wide range of applications due to the substrate being made out of material that is coefficient of thermal expansion (CTE) matched with common semiconductor materials such as Si, SiGe, aluminum nitride (AlN), and SiC. Copper and AlN were selected to meet the CTE mismatch requirement of the TGP program and also because of structural integrity considerations. Experimental results from a TGP with a copper substrate are reported in this work.

Advantages of the TGP, as well as other heat pipes, include a very high effective axial thermal conductivity, reliability, no moving parts, and no need for external power. Unlike conventional heat pipes, the TGP can be used in applications where space is extremely limited. The proposed TGP as shown in Fig. 1 is a thin planar heat spreader that is capable of moving heat from multiple chips to a distant thermal sink. This TGP may have a thermal conductivity more than 100 times greater than common copper alloy backed substrates. In addition, a nanostructured wick will enable the TGP to operate in an adverse gravity environment of up to 20 g [2,10]. Since the TGP utilizes the transport of the latent energy from the evaporator to the condenser, the TGP has an extremely large effective thermal conductivity.

As shown in Fig. 1, the heat transferred to the evaporator section by an external source is conducted through the TGP wall and wick structure, and then vaporizes the working fluid in the wick. As vapor is formed, its pressure increases, which drives the vapor to the condenser, where the vapor releases its latent heat of vaporization to the heat sink in the condenser. The condensed fluid returns to the evaporator due to a pressure difference. By transporting the latent heat of vaporization, the TGP is able to transfer high heat loads with small temperature differences. This process will continue as long as there is sufficient capillary pressure to force the condensed liquid from the condenser to the evaporator.

Ababneh et al. [11,12] developed a thermal resistance model to predict the thermal performance of the TGP, including the effects of the presence of noncondensable gases (NCGs). Viscous laminar flow pressure losses are predicted to determine the maximum heat load when the capillary limit is reached.

Some of the experimental work was performed to measure wick performance as a function of evaporation rates for sintered copper wicks for simulating heat pipe. Wick testing apparatus is used to compare the performance of different sintered wick samples of different porosity [13].

Nam et al. [14] reported the heat transfer
performance of superhydrophilic Cu micropost wicks fabricated on thin silicon
substrates. Hanlon and Ma [15] introduced a
2D model to predict the overall heat transfer capability for a sintered wick and
conducted an experimental study to predict the effective parameters for evaporation
heat transfer from a sintered porous wick. The results showed that choosing a
suitable wick porosity, particle radius, effective thermal conductivity, and wick
structure thickness would enhance the evaporation heat transfer coefficient. The
selected wick is 500 *μ*m thick sintered copper wicks (with porosity *ε* ≈ 50%, average pore diameter of 11.30 *μ*m and
wick thermal conductivity *k*_{sintered copper} ≈ 170 W/m·K).

The MTE setup is used to measure the thermal resistance at the TGP. A simple finite element model is used to calculate the thermal resistances based on the experimental data. Then the experimental results are utilized to develop a 3D finite element model to predict the performance of the TGP. Since this model is based on empirical measurements, it can be used to predict the performance of TGP for different geometries and implementation configurations. The 3D thermal model predicts the temperature field in the TGP, the effective axial thermal conductivity, and the evaporation and the condensation rates for a given heat input. The manuscript will describe the model and how it incorporates empirical interfacial heat transfer coefficient data. It will then present theoretical predictions for the thermal performance of TGP's, and compare with experimental results.

## MTE

The mass transport setup simulates an “open functional TGP.” To characterize the
performance of the wick samples for a TGP, an experiment was designed to measure
mass transport in wicks. The test sample shown in Fig. 2 is 30 mm × 30 mm × 1 mm thick, with a 2 mm wide wall around the
perimeter. The heated region is 10 mm × 30 mm. The area of the wick exposed for
evaporation is 8 mm × 26 mm. To simulate the TGP, the system includes a vacuum
vessel having de-ionized (DI) water, which is evacuated to reach saturated
conditions. A wick in a vertical arrangement inside the vessel draws liquid from the
pool in the bottom of the vessel until saturated. Heat is applied to the top of the
wick and the temperature is measured with a thermocouple inserted in the heater
(*T*_{hot}). Liquid in the wick starts to evaporate and vapor escapes the wick
structure as input heat flux is increased (*Q*_{in}).

The natural convection thermal resistance is three orders of magnitude larger than
the evaporation thermal resistance. Hence, in this study, the heat loss due to
natural convection between *T*_{hot} and *T*_{sat} is not considered.

The thermal resistance pathway for the system is shown in Fig. 3. The heat supplied by the heater is split into two paths. Most of the heat is dissipated to the vapor through evaporation $(Qevap=((Thot-Tsat)/1/(hevap\u2003.\u2003A)))$, but some of it is conducted and convected (nonlinear resistance) to the liquid pool, substrate wall, and the surrounding. The heat losses by conduction and convection are given by $(Qcond=((Thot-Tcold)/1/(hcond\u2003.\u2003A)))$.

## Modeling Procedure

The modeling process involves three main steps to simulate and predict the performance of the TGP:

### Step 1: Determine the Q_{cond} to the Pool.

In the experiment, the conduction and nonlinear resistances are measured
experimentally by running the test at ambient conditions (baseline case),
effectively turning off evaporation heat transfer. In the model, the wick is
truncated at the wick–pool interface and a convection boundary condition is
applied at this surface. To find the value of the heat transfer coefficient *h*_{cond} at the surface of the wick, which is exposed to water, a finite
element model using workbench ANSYS [16] is developed to represent the baseline experiment. Figure 4 shows the boundary conditions for the MTE
baseline case. The boundary conditions are:

**Surface A**: (Heater–wick interface as shown on the bottom surface of
Fig. 4) A variable heat input in applied at
the back surface of the wick to represent the heater. A heat input of [0–10] W
is applied to match with the experimental conditions.

**Surface B**: (Evaporator surface as shown on the yellow top surface
of Fig. 4) A convection boundary condition
for the wick which is exposed to saturated vapor, with a temperature *T*_{sat} which corresponds with the heat flux at boundary condition A,
(*h*_{evap} ≈ 0 W/m^{2}·K) because we assume there is no
evaporation.

**Surface C:** (Wick–pool interface as shown on the yellow right
surface of Fig. 4) A convection boundary
condition is applied at the wick–pool interface. The experimentally measured
submerged substrate temperature, *T*_{cold}, is used as the ambient temperature. For each heat input, *h*_{cond} is varied until it matches with the experimentally measured *T*_{hot} value. All other surfaces are perfectly insulated.

The result for the *h*_{cond} as a function of the heat input is shown in Fig. 5. In the experiment, however, *h*_{evap} cannot be completely blocked as there will be natural convection
at that surface. Hence, the curve shown in Fig. 5 is extrapolated to *Q*_{in} = 0 W and that value is assumed to be the actual value for *h*_{cond}. To validate this assumption, the amount of energy transported
through evaporation is found to be more than 95% out of the total energy.
Because of the values of *Q*_{in}, *Q*_{evap}, and *Q*_{cond} were estimated, the amount of energy transported through
evaporation at every *Q*_{in} can be determined.

### Step 2: Determine the h_{evap} at the Evaporator Surface.

In order to estimate the effective heat transfer coefficient at the evaporator
surface, a transient finite element model using workbench ANSYS is
developed. The evaporation heat transfer coefficient (*h*_{evap}) at the wick surface is modeled as a function of
Δ*T* (Δ*T* = *T*_{average surface of wick} – *T*_{saturation}). Again in this step, the experimental results are used to
empirically determine *h*_{evap}. Figure 6 shows the
boundary conditions and the result for the MTE with evaporation. The boundary
conditions are:

**Surface A:** (Heater–wick interface as shown on the bottom surface of
Fig. 6) A variable heat input in applied at
the back surface of the wick to represent the heater. A heat input of [0–60] W
is applied to match with the experimental conditions.

**Surface B:** (Evaporator surface as shown on the yellow top surface
of Fig. 6) A convection boundary condition
for the wick with an experimentally determined temperature *T*_{sat} which corresponds with the heat flux at boundary condition
(surface A), *h*_{evap} is assumed to be uniform at the wick surface, the *h*_{evap} keeps changing to match with the thermocouple value
(*T*_{hot}). Notably, the thermocouples accuracy is evaluated as ±0.78 °C
[17].

**Surface C:** (Wick–pool interface as shown on the yellow front
surface of Fig. 6) A convection boundary
condition is applied at the wick–pool interface. The value of *h*_{cond} = 13,910 W/m^{2}·K is applied at this surface (from step
1).

For each heat input value *h*_{evap} is varied until the *T*_{hot} and *T*_{cold} match with the experimental value, which is then assumed to be
the effective evaporation heat transfer coefficient at that temperature. (For
example, for the case shown in Fig. 7, *h*_{evap} = 29,800 W/m^{2}·K to match *T*_{hot} = 76.8 °C, at a heat flux of 1.667 × 10^{5}^{ }W/m^{2}.)

The MTE is repeated for six wick samples for reliability and to show that our
results can be replicated. Figure 7 shows
the experimentally measured *h*_{evap} as a function of the input power for a uniform (an average curve
of one of the six samples) wick. Figure 8 shows the *h*_{evap} values estimated using the model as a function of
Δ*T*. The raw data from the experiment is used to obtain this
curve. This curve will serve as a boundary condition in the TGP model.

## Numerical Model for Temperature and Thermal Conductivity Predictions for the TGP

### Step 3: Thermal Model of the TGP.

The length of TGP was varied (3 cm, 9 cm, and 15 cm) where the
evaporator/condenser lengths are fixed at 1 cm. The width is fixed at 3 cm for
all lengths. It has total external thickness of 2 mm (bottom and top copper
substrate has thickness of 1 mm each) with an internal thickness of 0.5 mm for
both. 500 *μ*m thick sintered copper wicks (with *ε* ≈ 50%, pore diameter of 11.30 *μ*m and *k*_{sintered copper} ≈ 170 W/m·K using Laser Flash method) are assumed on
the bottom substrate, where the vapor space is assumed to be
500 *μ*m in the top substrate. For more details about the
related experiments, see Ref. [18].

As mentioned in the previous section, Step 2: Determine the h_{evap} at
the Evaporator Surface, the curve shown in Fig. 8 is used as an input for the evaporation heat transfer
coefficients. The evaporation heat transfer coefficient is determined
empirically based on the experimental data. For the condenser surface, the heat
transfer coefficient is assumed to be a constant value of
≈10,000 W/m^{2}·K.

Past investigators, such as Vadakkan et al. [19], solved for the energy transport in the liquid saturated wick
utilizing a Brinkman–Forchheimer extended Darcy model for the fluid flow and an
energy equation which uses the superficial liquid velocity in a convection term.
This approach is inappropriate for the current research. Utilizing a porous
media energy equation assumes that there is local thermodynamic equilibrium
between the solid and liquid phases. Since the ratio of solid to liquid thermal
conductivities (*k*_{sintered copper}/*k*_{water} ≈ 275) is very large, the local thermal equilibrium assumption
is not valid for the current study. For the TGPs investigated which utilize
water as the working fluid, Ja << 1, and convection in the liquid can be
neglected. Therefore, the energy transport with the fluid saturated wick is
purely by diffusion. Just as important as not having a convection term in the
energy equation (u dT/dx ≈ 0), is the assumption that *h*_{evap} is only a function of temperature. Taken together, these two
assumptions allow the temperature and velocity fields within the liquid
saturated wick to be decoupled. Conduction equations for the substrate and wick
can be solved utilizing a convection boundary condition for the evaporation at
the vapor–liquid saturated wick interface. The evaporation or condensation rate
at the interface determines the normal velocities within the vapor and liquid
(superficial velocity in the liquid). These velocities are then used as boundary
conditions to solve the momentum equations in the vapor and liquid saturated
wick. The thin film resistance is much larger than the vertical wick and
substrate thermal resistances where the energy transport within the substrate
and the vertical wick by conduction.

The vapor core in the TGP was treated as stagnate, with constant temperature
(*T*_{vap} = *T*_{sat}).

Notably, the evacuation and filling procedures for the TGP can be found in Ref. [20].

Figure 8 is used as a boundary condition for the TGP as shown in Fig. 9. The boundary conditions in this model are as follows:

**Surface A:** (Heater–wick interface as shown on the blue left bottom
surface of Fig. 9(i)) a
variable heat input in applied at the back surface of the wick to represent the
heater.

**Surface B:** (Evaporator surface as shown on the yellow top surface
of Fig. 9(ii)) a
convection boundary condition is applied at the evaporator surface with a
temperature *T*_{sat} for the vapor. The evaporation heat transfer coefficient
(*h*_{evap}) is determined from *h*_{evap} versus Δ*T* curve which is given in Fig. 8.

**Surface C:** (condenser surface as shown on the red right bottom
surface of Fig. 9(i)) a
constant temperature boundary condition is assumed at the condenser section.

The temperature distribution for (3, 9, and 15 cm length × 3 cm width × 2 mm
thickness) TGP is shown in Figs. 10, 11, and 12, respectively. It is clear from Figs. 11 and 12 that the
fraction of the heat transferred by evaporation increases as the length of the
TGP increases. Figure 10 shows (as opposed
to diffusion through the substrate) that for *L*_{TGP} = 3 cm, conduction through substrate is as important as *evaporation.*

Finally, the equivalent effective thermal conductivity for a copper substrate TGP
as predicted by the model is shown in Fig. 13. It is clear from the figure that as the TGP's length increases
the effective thermal conductivity increases. Based on the current uniform wick
and the assumptions made, the performance of the TGP will be ∼5000 W/m·K for a
15 cm long and *Q*_{in} = 30 W TGP. Notably, the resulting ANSYS model is unique
to this particular TGP configuration.

## Comparison Between the ansys Model With the Experimental Results for the TGP

To verify the results of the ANSYS model, several comparisons were made with
experiments on different TGP samples. Figures 14–19 show the comparison of temperature distributions between the ANSYS model
and the experimental results for the TGP which is shown in Fig. 9 with *L*_{TGP} = 9 and 15 cm, different input power levels (*Q*_{in} = 10 W, 15 W, and 20 W) and *T*_{condenser} = 60, 75, and 90 °C.

Figures 14–16 show the comparison between experimental data and the ANSYS model temperature distributions for the TGP with 15 cm long, *Q*_{in} = 10 W, 15 W, and 20 W, assuming the exterior surfaces exposed to
ambient air and natural convection coefficient
(*h* = 10 W/m^{2}·K) and constant condenser's
temperatures (*T*_{condenser} = 60, 75, and 90 °C), respectively. The ANSYS model
showed excellent agreement with the experimental results. For the three cases
(*L*_{TGP} = 15 cm) at *Q*_{in} = 10 W the model match the experimental results. As *Q*_{in} increases (*Q*_{in} = 15 W) the model still shows good agreement with the experiment over
most of the TGP length. Near the evaporator, however
(*x* = 0–0.01 m), the predicted temperatures are smaller than the
experimental values. One possible explanation for this is the onset of dry-out as
the power is increased. The effect is more noticeable as *Q*_{in} increases (*Q*_{in} = 20 W). The model can be used to demonstrate why the evaporator
temperature increases at the onset of dry-out. For total dry-out, there is no
evaporation at the vapor–liquid saturated wick interface. Therefore, the following
case was modeled: *L*_{TGP} = 15 cm, *Q*_{in} = 20 W, *T*_{condenser} = 75 °C and *h*_{evap} ≈ 0 W/m^{2}·K. The result shows that *T* = 170.50 °C at *x* = 0 m. While this is not
intended to be a dry-out model, it does demonstrate that a rise in evaporator
temperature is consistent with the onset of dry-out.

Figures 17–19 show the comparison
between experimental data and the ANSYS model temperature distributions for
the TGP with 9 cm long, *Q*_{in} = 10 W, 15 W, and 20 W, assuming the exterior surfaces exposed to
ambient cooling temperatures and natural convection coefficient
(*h* = 2 W/m^{2}·K) and constant condenser's temperatures
(*T*_{condenser} = 60, 75, and 90 °C), respectively. The ANSYS model
showed good agreement with the experimental results. As *Q*_{in} increases the model still shows good agreement with the experiment
over most of the TGP length. The predicted temperatures are smaller than the
experimental values near the evaporator section. One possible explanation for this
is the onset of dry-out as the power is increased. The effect is more obvious as *Q*_{in} increases (*Q*_{in} = 15 and 20 W). The model can be used to validate why the evaporator
temperature increases at the onset of dry-out as discussed before.

The quality of agreement with the model is in part attributed to the empirical
coefficients attained from experiments that were performed in this work. Since the *h*_{evap} was measured without a lid. So it seems like the evaporation heat
transfer changes when the vapor is confined, as opposed to being open. This could be
a part of proposed future work. Also, as the *T*_{condenser} increases the *T*_{vap} or *T*_{sat} increases so the maximum heat transport capability will be
increasing, so the model will be more accurate for higher *T*_{condenser}. Finally, temperature measurement uncertainty would
be ± 0.5 °C. That could explain slight deviation at the adiabatic section.

## Conclusions

The present paper describes a 3D finite element thermal model of a TGP which incorporates empirical interfacial heat transfer coefficient data.

The model is used to predict the thermal performance of the TGP and compare the results with the experimental data. The 3D thermal model predicts the temperature field in the TGP, the effective axial thermal conductivity, and the evaporation and the condensation rates.

A key feature of this model is that it depends on empirical interfacial heat transfer coefficient data to very precisely model the interfacial energy balance at the vapor–liquid saturated wick interface. Wick samples for a TGP are tested in an experimental setup to measure the interfacial heat transfer coefficient. Then the experimental heat transfer coefficient data are used for the interfacial energy balance.

An additional significant feature of this model is that it demonstrates that, for Jakob numbers of interest, the thermal and flow fields can be decoupled except at the vapor–liquid saturated wick interface. For the TGPs investigated which utilize water as the working fluid, Ja << 1, and convection in the liquid can be neglected (u dT/dx ≈ 0). Hence, the temperature field is decoupled from the velocity field.

The axial effective thermal conductivity of the TGP increases as the TGP's length increases because the thermal resistance of the system is constant while as the length of the TGP increases the amount of the spreading heat increases.

To verify the results of the ANSYS model several of the experimental work
was done for different TGP samples. The results show the comparison of temperature
distributions between the ANSYS model and the experimental results for the
TGP with *L*_{TGP} = 9, 15 cm, different input power levels (*Q*_{in} = 10 W, 15 W, and 20 W) and *T*_{condenser} = 60, 75, and 90 °C.

The ANSYS model showed excellent agreement with the experimental results.
The difference between the model and the experimental values for the evaporator
section is due to close to the dry-out condition. The quality of agreement with the
model is in part attributed to the empirical coefficients attained from experiments
that were performed in this work. Also, as the *T*_{condenser} increases the *T*_{vap} or *T*_{sat} increases so the maximum heat transport capability will be
increasing, so the model will be more accurate for higher *T*_{condenser}. Finally, temperature measurement uncertainty would be ±0.5 °C.
That could explain slight deviation at the adiabatic section.

## Acknowledgment

This paper is based upon work supported by DARPA under SSC SD Contract No. N66001-08-C-2008. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the SSC San Diego. Sincere appreciation is expressed to Professor Avram Bar-Cohen and Mr. James Schmidt for their valuable comments.

*A*=area (m

^{2})- CTE =
coefficient of thermal expansion

*g*=gravitational acceleration (m/s

^{2})*h*=heat transfer coefficient (W/m

^{2}·K)*h*_{cond}=conduction heat transfer coefficient (W/m

^{2}·K)*h*_{evap}=evaporation heat transfer coefficient (W/m

^{2}·K)- Ja =
Jakob number

*k*_{eff}=axial effective thermal conductivity (W/m·K)

*L*_{TGP}=length of the TGP (m)

- MTE =
mass transport experiment

*Q*_{cond}=heat conducted to the liquid pool (W)

*Q*_{in}=input power (W)

*t*=thickness (m)

*T*=temperature (°C)

*T*_{condenser}=condenser temperature (°C)

- TGPs =
thermal ground planes

*T*_{hot}=the measured temperature of the thermocouple which is inserted in the heater (°C)

*T*_{sat}=saturation temperature (°C)

*T*_{vap}=vapor temperature (°C)

*W*=Watt

- Δ
*T*= *T*_{average surface of wick}–*T*_{saturation}(°C)*ε*=porosity