**4.3 Two-phase heat exchanger design**

## *4.3.1 Design fundamentals*

*4.3.1.1 Carrying capacity*

The performance of a two-phase heatsink is measured by its carrying capacity, *Q*, defined below.

Definition

The *carrying capacity*, *Q*, of a two-phase heat exchanger is defined as the amount of heat the device can move out per unit time.

From its definition, *Q* is an *energy per unit time*, which is a *power* which should be expressed in *Watts*.

For a given two-phase heat exchanger, the value of *Q* depends on several parameters: the size of the vessel, the latent heat of the transfer liquid used, the thermal conductivity of the metal constituting the vapor chamber or the heat pipe, convections (inside and outside the vessel), etc. However, we should note that, in two-phase heat sinks, energy is essentially transferred through the vaporizationcondensation process, involving latent heat.

#### *4.3.1.2 Two-phase thermal budget*

#### Definition

The *thermal budget*, Δ*T*, of a two-phase heat exchanger is defined as the difference between the temperature of the electronic component (heat source),*Tc*, and the temperature of the ambiance where the component will operate,*Ta*:

$$
\Delta T = T\_c - T\_a \tag{22}
$$

The carrying capacity, *Q*, is proportional to the thermal budget Δ*T* (see Eq. (9)). Thus for low thermal budgets, it will be difficult to achieve high values of *Q*. In this case, two-phase heat sinks should be used, mainly when Δ*T* < 40°C.

#### *4.3.1.3 Effective thermal conductivity*

Heat exchange in two-phase sinks is actually complicated to model. It involves several heat transfer mechanisms: conduction through the vapor chamber or the heat pipe metal envelope, evaporation, condensation, convection inside and outside the vessel, etc. The exact representations of these transfers are interesting on the theoretical level of analysis. But, for sizing and design purposes, a more practical approach has been adopted by manufacturers and system designers. This approach is based on the thermal resistance model, in a way similar to that presented in Subsection 2.1 for single-phase heat sinks.

Let us recall that in conduction and in convection, thermal resistance models permit to express the power transferred *Q* in terms of *R*th and the thermal budget, Δ*T*, as follows [2, 47]:

*<sup>Q</sup>* <sup>¼</sup> <sup>Δ</sup>*<sup>T</sup> Rth* , where, *R*th is the thermal resistance, which is expressed differently for conduction and convection:

• For a convection heat transfer:

$$R\_{th}^{conv} = \frac{1}{hA} \tag{23}$$

Where *h* is the convective heat transfer coefficient and *A* the heat transfer area.

• For a conduction heat transfer:

$$R\_{th}^{cond} = \frac{L}{kA} \tag{24}$$

Where *L* is the material thickness, *A* the heat transfer area, and *k* is the *thermal conductivity*.

Similarly, two-phase heat exchanger energy transfers are represented using an *effective thermal resistance*, *Reff*, such that the carrying capacity, *Q*, is given by:

$$Q = \frac{\Delta T}{R\_{\mathcal{G}}} \tag{25}$$

With *Reff* <sup>¼</sup> *<sup>L</sup> A keff* , where *keff* is the *effective thermal conductivity* of the two-phase heat exchanger and *A* is its *heat transfer area of the heat sink*.

Definition

The *effective thermal conductivity*, *keff*, of a two-phase heat exchanger is defined as follows:

$$k\_{\ell\overline{\ell}} = \frac{QL}{A\Delta T} \tag{26}$$

Where *Q* is the *carrying capacity*, *A* is its *heat transfer area*, Δ*T* is the thermal budget (temperature difference between evaporator and condenser sections) and *L* is the *effective length* of the vessel which is the *distance from the midpoint of the evaporator to the midpoint of the condenser*.

It should be noted that, unlike the thermal conductivity of materials, the *two-phase heat exchanger thermal conductivity* varies with length. It is not a physical property of the material, but it is a modeling representation of heat transfer in two-phase heat sinks.

This modeling—using effective thermal conductivity, *keff*—makes it possible to represent *two-phase heatsink energy transfers* by *a simple thermal resistance model*, just the same way thermal resistance was used in single-phase heat sinks.

Effective thermal conductivities of two-phase heat exchangers range from 1500 to 50,000 W/m°C. For comparison, the thermal conductivities of the best energy conductors are around 400 W/m°C. Effective thermal conductivities of two-phase heat exchangers can reach 10–100 times the conductivity of the best thermal conductors like copper.

#### *4.3.2 Design steps*

Generally, manufacturers of two-phase heat exchangers supply, among other information on their product, the following parameters:


The following five-step procedure is proposed to design two-phase heat exchangers.

*Heat Exchangers for Electronic Equipment Cooling DOI: http://dx.doi.org/10.5772/intechopen.100732*

STEP 1: The datasheet of the electronic component to be cooled gives the power and operating temperature: *Q* (W) and *Top* (°C).

STEP 2: Knowing ambient temperature,*Ta*, calculate the thermal budget Δ*T* as follows:

$$
\Delta T = T\_{op} - T\_a \tag{27}
$$

STEP 3: Calculate the *design thermal budget* Δ*TD* as follows:

$$
\Delta T\_D = \Delta T - \dots \tag{28}
$$

STEP 4: Divide the *design thermal budget*, Δ*TD*, by the power, *Q*, of the electronic equipment to be cooled. This division will give the maximal thermal resistance to be assured by the two-phase heatsink: *RMax*

STEP 5: Select the two-phase heatsink having a thermal resistance *R* < *RMax*. Rules of thumb


#### **A. Appendix 1**

Determination of heat transfer coefficients for electronic components and circuit boards under natural convection

#### **A.1 Electronic components under natural convection in air**

$$\mathbf{h} = \mathbf{3.53} \left( \frac{\mathbf{T\_w} - \mathbf{T\_a}}{\mathbf{H}} \right)^{0.25} \tag{29}$$

Where:

*H* is the height of the small electronic component, *expressed in meters Tw* and *Ta* are, respectively, the wall temperature and the ambient temperature

#### **A.2 Circuit boards under natural convection in air**

$$\mathbf{h} = 2.44 \left( \frac{\mathbf{T\_w} - \mathbf{T\_a}}{\mathbf{H}} \right)^{0.25} \tag{30}$$

Where:

*H* is the height of the electronic board considered, *expressed in meters Tw* and *Ta* are, respectively, the board and the ambient temperatures. *Heat Exchangers*
