2. Design

Based on the problem specifications, the heat exchanger construction type, flow arrangement, surface or core geometry, and materials must be selected. The most common problems in heat exchanger design are rating and sizing. This chapter discusses the basic design methods for two fluid direct transfer heat exchangers. The rating problem is concerned with the determination of the heat transfer rate and the fluid outlet temperatures for prescribed fluid flow rates, inlet temperatures, and allowable pressure drop for an existing heat exchanger; hence, the heat transfer surface area and the flow passage dimensions are available. The sizing problem, on the other hand, involves determination of the dimensions of the heat exchanger, that is, selecting an appropriate heat exchanger type and determining the size to meet the requirements of specified hot and cold fluid inlet and outlet temperatures, flow rates, and pressure drops. Problem definition and design process passes through the following parameters choice:


• Flow channel geometry/shape is important for high performance microchannel HEs (see

• Additive manufacturing or improvements in conventional fabrication methods are needed

• Additional benefits can be obtained by incorporating compound enhancement methods. The features of the most commonly used heat exchangers in aviation are listed in Table 1. It shows that the materials most often used are aluminum, copper, and carbon steel, while the

Based on the problem specifications, the heat exchanger construction type, flow arrangement, surface or core geometry, and materials must be selected. The most common problems in heat exchanger design are rating and sizing. This chapter discusses the basic design methods for two fluid direct transfer heat exchangers. The rating problem is concerned with the determination of the heat transfer rate and the fluid outlet temperatures for prescribed fluid flow rates, inlet temperatures, and allowable pressure drop for an existing heat exchanger; hence, the heat transfer surface area and the flow passage dimensions are available. The sizing problem, on

Table 1. Materials and commonly used fluids and sizes in aviation heat exchangers.

Figure 3).

Materials of construction

Fluids commonly worked with air

Heat transfer capacity (typical)

Depths from 0.75 in. (19 mm) to 24 in. (61 cm) Heights from 1 in. (25 mm) to 52 in. (132 cm) Widths from 4 in. (100 mm) to 52 in. (132 cm)

Aluminum Carbon steel Copper Cupronickel Nickel alloys Stainless steel

Coolants

Petroleum products Refrigerants Water

200 W to 300 KW Typical unit size range

2. Design

to resolve current challenges.

154 Heat Exchangers– Advanced Features and Applications

typical sizes range between 100 mm and 132 cm.


See the heat exchanger design methodology in Ref. [9] and summarized in next sections (see Figure 4).

Figure 4. Design loop of a heat exchanger.

#### 2.1. Problem definition

Generally the heat transfer rate in a heat exchanger can be calculated by

$$
\Delta Q = \Delta T\_{lm} \mathcal{U} \beta V \tag{1}
$$

Therefore, improvements of heat transfer can be achieved by increasing exchanger volume V, area density β of the exchanger, logarithmic mean temperature difference ΔTlm, or overall heat transfer coefficient U, including the heat transfer coefficients and the conductivity of the wall. The convective heat transfer coefficient of gases usually is one or two orders of magnitude lower than that of liquids. For this reason, a high heat transfer area is necessary for realizing a high heat transfer rate, especially if one or more fluids are gaseous. This means the surface must be compact. It is defined that a heat exchanger is compact, if it incorporates at least one compact surface. On the other hand, heat exchangers with densities of 6600 m2 /m<sup>3</sup> are also used. The logarithmic mean temperature is calculated by the formula:

$$
\Delta T\_{lm} = \frac{\left(\Delta T\_H - \Delta T\_C\right)}{\ln \left(\frac{\Delta T\_H}{\Delta T\_C}\right)}\tag{2}
$$

Log-mean temperature difference (LMTD) is a good measure of the effectiveness of similar heat exchangers of different designs. Often, LMTD (counter flow) > LMTD (parallel flow). When there is insufficient information to calculate the log-mean temperature difference (LMTD), the so-called number of transfer units (NTU) method is used to calculate the rate of heat transfer in heat exchangers (especially countercurrent exchangers). In heat exchanger analysis, if the fluid inlet and outlet temperatures are specified or can be determined by simple energy balance, the LMTD method can be used, but when these temperatures are not available. The NTU or the effectiveness method is used. The maximum heat transfer rate, Qmax, is evaluated by

$$Q\_{\text{max}} = (mc)\_{\text{min}} (T\_{h,i} - T\_{c,i}) \tag{3}$$

The heat transfer rate of the heat exchanger, Q, is calculated by

$$Q\_{\mathcal{C}} = m\_{\mathcal{C}} \mathbf{c}\_{\mathcal{C}} (T\_{\mathcal{C},o} - T\_{\mathcal{C},i}) \tag{4}$$

The effectiveness (NTU method) ε is calculated by

$$
\varepsilon = \frac{Q}{Q\_{\text{max}}} \tag{5}
$$

For more details, the reader can refer to the works [12, 13]. Heat flux is calculated by

$$q = \frac{Q}{A} = \frac{m\_c c\_c (T\_{c,o} - T\_{c,i})}{n L\_c W\_c} \tag{6}$$

or

$$q = \mathcal{U}\Lambda T\_{lm} = \frac{\Delta T\_{lm}}{\sum R} \tag{7}$$

the overall thermal resistance P R is determined by

$$\sum R = R\_{\text{cond}} + R\_{\text{conv}} \tag{8}$$

where m\_ is the mass flow rate (subscripts h and c stand for the hot side and cold side, respectively), n is the number of channels, c is the specific heat, Th,i, Th,o, Tc,i, and Tc,o are inlet and outlet temperatures of the hot and cold side, respectively, q is the heat flux, A is the heat transfer area, k is the overall heat transfer coefficient, Rcond = δ/λ is the conductive thermal resistance, Rconv = 1/hh + 1/hc is the convective thermal resistance, hh and hc are the convective heat transfer coefficients of the hot side and the cold side, respectively, δ is the thickness, and λ is the thermal conductivity [14].

Another parameter that is really important during the design process of a cooling system is the total pressure drop, sum of a cold side pressure drop, and hot side pressure drop,

$$
\Delta p\_{\text{tot}} = \Delta p\_c + \Delta p\_h \tag{9}
$$

Manufactures of heat exchangers have then the possibility to choose a wide variety of materials, such as aluminum, stainless steel, copper, and cupronickel, chosen based upon heat transfer or environmental requirements

The Reynolds number Re is calculated by

compact surface. On the other hand, heat exchangers with densities of 6600 m2

<sup>Δ</sup>Tlm <sup>¼</sup> <sup>ð</sup>ΔTH � <sup>Δ</sup>TC<sup>Þ</sup> ln <sup>Δ</sup>TH ΔTC

Log-mean temperature difference (LMTD) is a good measure of the effectiveness of similar heat exchangers of different designs. Often, LMTD (counter flow) > LMTD (parallel flow). When there is insufficient information to calculate the log-mean temperature difference (LMTD), the so-called number of transfer units (NTU) method is used to calculate the rate of heat transfer in heat exchangers (especially countercurrent exchangers). In heat exchanger analysis, if the fluid inlet and outlet temperatures are specified or can be determined by simple energy balance, the LMTD method can be used, but when these temperatures are not available. The NTU or the effectiveness method is used. The maximum heat transfer rate, Qmax, is

> <sup>ε</sup> <sup>¼</sup> <sup>Q</sup> Qmax

<sup>A</sup> <sup>¼</sup> mcccðTc, <sup>o</sup> � Tc,i<sup>Þ</sup> nLcWc

<sup>q</sup> <sup>¼</sup> <sup>U</sup>ΔTlm <sup>¼</sup> <sup>Δ</sup>Tlm

where m\_ is the mass flow rate (subscripts h and c stand for the hot side and cold side, respectively), n is the number of channels, c is the specific heat, Th,i, Th,o, Tc,i, and Tc,o are inlet and outlet temperatures of the hot and cold side, respectively, q is the heat flux, A is the heat transfer area, k is the overall heat transfer coefficient, Rcond = δ/λ is the conductive thermal

For more details, the reader can refer to the works [12, 13]. Heat flux is calculated by

<sup>q</sup> <sup>¼</sup> <sup>Q</sup>

used. The logarithmic mean temperature is calculated by the formula:

156 Heat Exchangers– Advanced Features and Applications

The heat transfer rate of the heat exchanger, Q, is calculated by

The effectiveness (NTU method) ε is calculated by

the overall thermal resistance P R is determined by

evaluated by

or

/m<sup>3</sup> are also

(5)

(6)

� � (2)

Qmax ¼ ðmcÞminðTh,i–Tc,iÞ (3)

Qc ¼ mcccðTc, <sup>o</sup>–Tc,iÞ (4)

<sup>P</sup> <sup>R</sup> (7)

<sup>X</sup><sup>R</sup> <sup>¼</sup> <sup>R</sup>cond <sup>þ</sup> <sup>R</sup>conv (8)

$$Re = \frac{\rho w D\_h}{\mu} = \frac{2\dot{m}}{\mu(W\_c + D\_c)}\tag{10}$$

pressure drop due to friction is determined by [14, 15]:

$$
\Delta p = 2f\rho u^2 \frac{L}{D\_h} = 2f \text{Re} \frac{L}{D\_h^{-2}} w\mu \tag{11}
$$

where Dh = 4Ac/P is the hydraulic diameter, u is the flow velocity, μ is the dynamic viscosity, ρ is the density, Ac is the cross-sectional area, P is the wetted perimeter, L is the channel length, and f is the Fanning friction factor [15].

The performance index, ξ, is determined by [16–21]

$$\frac{\mathcal{Q}\_c}{\Delta p\_t} = \frac{m\_c c\_c (T\_{c,o} - T\_{c,i})}{\Delta p\_h + \Delta p\_c} \tag{12}$$

In aviation, the cold fluid is often represented by air, whereas the hot fluid is for example the oil that must reduce the engine temperatures. The real parameter of choice is the group hA based on the following simplified equation:

$$Q = hA(T\_w - T\_f) \tag{13}$$

where Tw and Tf are the hot wall temperature and the cooling air temperature, respectively. First of all, an aerospace engineer tries to reduce the size of a heat exchanger for a fixed heat duty. To contain the HE size, the surface geometry is designed in order to increase the heat transfer area, using the so-called compact heat exchangers, that can be of plate fin or tube fine typology, see Ref. [22]. A particular surface geometry affects both the heat transfer coefficient h and also the pressure drop Δp. These parameters are correlated with the flow regime of each fluid and so they depend on the value of Reynolds number Re. A useful correlation in this sense can be given by the colburn factor that can be calculated by

$$j = \frac{h}{\rho w c\_p} Pr^{2/3} \tag{14}$$

where cp is the air specific heat, h is the convective heat transfer coefficient, w is the air flow velocity, ρ is the density, and Pr = υ/α is the Prandtl number in the flow conditions under investigation, υ is the kinematic viscosity, and α is the thermal diffusivity.

#### 2.2. Computational methodologies

The numerical methods that can be used to investigate the deeper behavior of a heat exchanger are often based on the Navier-Stokes solution rather than on a balance of heat fluxes and mass flow rates.

#### 2.2.1. Governing equations

The domain under investigation must be discretized my means of a preprocessor—grid generator and on it the following equations are to be manipulated numerically in order to have the fluid and thermal fields of the case, as follows:

Mass

$$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x\_j} [\rho u\_j] = 0 \tag{15}$$

Momentum

$$\frac{\partial}{\partial t}(\rho u\_i) + \frac{\partial}{\partial x\_j}[\rho u\_i u\_j + p\delta\_{ij} - \tau\_{ji}] + S\_{F,i} = 0, \; i = 1, 2, 3 \tag{16}$$

Energy

$$\frac{\partial}{\partial t}(\rho e) + \frac{\partial}{\partial \mathbf{x}\_j}[\rho u\_\rangle e + u\_\rangle p - u\_i \tau\_{ij}] + \mathcal{S}\_E = 0 \tag{17}$$

Based on the typology of flow regime, other equations can be solved to take into account the turbulent fluctuations. It must be reminded that the flow regime is strictly related to the heat transfer that can be achieved in the heat exchanger. Often other heat sources can play a fundamental role in the solution of the aforementioned equations. In this case, other equations have to be added to the basic ones increasing the times to reach the solution, so before proceeding to solve the equations governing the fluid-dynamics inside and out of the cooling ducts, the right evaluations in terms of time and number of simulations must be done.

#### 2.2.2. Design approaches

Often the manufacturers offer characteristic curves of a heat exchanger on their catalogs so to make possible a preliminary selection of the right choice. Nevertheless, this is not sufficient and so the designer has to deep analysis using iterative procedures that pass through a number of numerical simulations or graphical assessments.

#### 2.2.3. Numerical tools

<sup>j</sup> <sup>¼</sup> <sup>h</sup> ρwcp

investigation, υ is the kinematic viscosity, and α is the thermal diffusivity.

2.2. Computational methodologies

158 Heat Exchangers– Advanced Features and Applications

fluid and thermal fields of the case, as follows:

∂ ∂t <sup>ð</sup>ρuiÞ þ <sup>∂</sup> ∂xj

> ∂ ∂t

of numerical simulations or graphical assessments.

<sup>ð</sup>ρeÞ þ <sup>∂</sup> ∂xj

flow rates.

Mass

Momentum

Energy

2.2.2. Design approaches

2.2.1. Governing equations

where cp is the air specific heat, h is the convective heat transfer coefficient, w is the air flow velocity, ρ is the density, and Pr = υ/α is the Prandtl number in the flow conditions under

The numerical methods that can be used to investigate the deeper behavior of a heat exchanger are often based on the Navier-Stokes solution rather than on a balance of heat fluxes and mass

The domain under investigation must be discretized my means of a preprocessor—grid generator and on it the following equations are to be manipulated numerically in order to have the

Based on the typology of flow regime, other equations can be solved to take into account the turbulent fluctuations. It must be reminded that the flow regime is strictly related to the heat transfer that can be achieved in the heat exchanger. Often other heat sources can play a fundamental role in the solution of the aforementioned equations. In this case, other equations have to be added to the basic ones increasing the times to reach the solution, so before proceeding to solve the equations governing the fluid-dynamics inside and out of the cooling

Often the manufacturers offer characteristic curves of a heat exchanger on their catalogs so to make possible a preliminary selection of the right choice. Nevertheless, this is not sufficient and so the designer has to deep analysis using iterative procedures that pass through a number

ducts, the right evaluations in terms of time and number of simulations must be done.

∂ρ ∂t þ ∂ ∂xj Pr<sup>2</sup>=<sup>3</sup> (14)

½ρuj� ¼ 0 (15)

½ρuiuj þ pδij � τji� þ SF,<sup>i</sup> ¼ 0, i ¼ 1, 2, 3 (16)

½ρuje þ ujp � uiτij� þ SE ¼ 0 (17)

There are a number of numerical tools capable of simulating a heat exchanger, some simplified and other with the possibility of considering more details. They are dedicated codes or numerical models inserted in more complex numerical packages. Often the design can require more software, from computer-aided design to computational fluid dynamics solvers. The most important parameters to monitor during an HE simulation are follows:


Heat exchangers are designed to maximize the surface area of the wall between two fluids, while minimizing resistance to fluid flow through the exchanger by means of thermal analyses, CFD, and FEA, to ensure efficient and effective optimized designs. Different commercial codes are present on the market:


They include packages able to simulate the flow fields inside the tubes and also specific numerical models capable of simulating special fluids like nanofluids or special structures like porous media [23–26].

#### 2.3. Economic evaluation

Once the heat exchanger size and characteristics are chosen, the designer has to proceed to an economical evaluation of this choice in terms of maintenance costs and, if the case, reinstallation costs. The components inside the heat exchangers have to be free from deposits and dirt built up during flying operations. This is vital because substandard cleaning could result in a loss of pressure in the heat exchanger, which is unacceptable. Therefore, they need to be cleaned at regular intervals. In the past years, airline companies needed to hire engineers who would conduct elaborate investigations into the dirt accumulation and physical/chemical surface analysis of the aluminum plates in the center. Now, there are agents for heat exchanger services who perform it by using a scanning electron microscope to recognize the different elements of the mount up dirt. Nevertheless, maintenance and reinstallation involve often higher costs that must be considered in the economic evaluation of the HE typology to choose already in the preliminary design phase.
