**2. Basic equation of heat transfer**

In most heat transfer problems, hot and cold fluids are divided by a solid wall. In this case, the mechanism of heat transfer from hot fluid to the cold fluid can be categorized into three steps:


**Figure 1** shows a schematic of heat transfer between two fluids. As it can be seen, thermal resistance (*R*) is present at each stage of the transfer. Thermal resistance is a thermal (physical) property that indicates the resistance of each material to heat transfer due to temperature differences that can be calculated from [3]:

$$\begin{aligned} R &= \frac{L}{KA} \quad , \text{ for conduction} \\ R &= \frac{1}{hA} \quad , \text{ for convection} \end{aligned} \tag{1}$$

Where *L* is the thickness of the wall, *A* is the cross-sectional area in which heat transfer occurs, and *K* and *h* are conduction and convection heat transfer coefficient, respectively.

Heat transfer in each stage can be calculated as follows [4]:

$$Q = \frac{T\_1 - T\_2}{R\_{c,H}} = \frac{T\_2 - T\_3}{R\_{f,H}} = \frac{T\_3 - T\_4}{R\_w} = \frac{T\_4 - T\_5}{R\_{f,C}} = \frac{T\_5 - T\_6}{R\_{c,C}}\tag{2}$$

Where:

*Rc,H* = thermal resistance for convection in the hot side.

**Figure 1.** *A schematic of heat transfer in heat exchangers.*

*Heat Exchanger Design and Optimization DOI: http://dx.doi.org/10.5772/intechopen.100450*

*Rf,H* = fouling resistance of hot side.

*Rw* = wall resistance.

*Rf,C* = fouling resistance of cold side.

*Rc,C* = thermal resistance for convection in cold side.

**Table 1** shows the fouling resistance of the most common fluids used in heat exchangers. The overall heat transfer coefficient can be obtained from Eq. (2) as follows [4]:

$$Q = \frac{T\_1 - T\_6}{R\_{\text{c,H}} + R\_{f,H} + R\_w + R\_{f,C} + R\_{\text{c,C}}} \tag{3}$$

or:

$$Q = \frac{T\_1 - T\_6}{\frac{1}{h\_H A\_H} + \frac{r\_{f,H}}{A\_H} + \frac{r\_w}{A\_H} + \frac{r\_{fC}}{A\_C} + \frac{1}{h\_C A\_C}}\tag{4}$$


#### **Table 1.**

*Fouling factors for different types of fluid [5, 6].*

Where:

*hH* and *hC* = convection heat transfer coefficient of the hot and cold sides, respectively.

*AH* and *AC* = and surface area of wall in the hot and cold side, respectively.

The *rw* can be calculated for flat wall and cylindrical walls using Eqs. (5) and (6), respectively.

$$r\_w = \frac{d\_w}{KA}, \text{ for flat wall} \tag{5}$$

$$r\_w = \frac{\ln\left(\frac{r\_o}{r\_i}\right)}{2\pi LK}, \text{ for cylindrical wall} \tag{6}$$

Where *dw* is the thickness of the wall, and *ro* and *ri* are the outside and inside diameter of the wall, respectively.

Total thermal resistance can be expressed as [7]:

$$R\_l = \frac{\mathbf{1}}{h\_H A\_H} + \frac{r\_{f,H}}{A\_H} + \frac{r\_w}{A\_H} + \frac{r\_{f,C}}{A\_C} + \frac{\mathbf{1}}{h\_C A\_C} \tag{7}$$

The rate of heat transfer (*Q*) can be determined from

$$Q = UA\Delta T\tag{8}$$

Where *U* is the overall heat transfer coefficient.

$$U = \frac{1}{\frac{1}{h\_H \frac{A\_H}{A\_{rf}}} + \frac{r\_{f,H}}{\frac{A\_H}{A\_{rf}}} + \frac{r\_w}{\frac{A\_H}{A\_{rf}}} + \frac{r\_{f,C}}{\frac{A\_C}{A\_{rf}}} + \frac{1}{h\_C \frac{A\_C}{A\_{rf}}}} \tag{9}$$

Where *Aref* is a reference area. If the heat transfer is carried out over a pipe, the inside and outside surface areas of the pipe are not equal. Hence, the *Aref* must be determined (The outer surface of the pipes is usually selected).

### **3. Thermal design of heat exchangers**

The thermal design of heat exchangers can be performed by several methods. The most commonly used methods are log-mean temperature difference (LMTD) and effectiveness-number of transfer units (*ε***-**NTU) [8]. The LMTD was used to calculate heat transfer when the inlet and outlet temperatures of fluids are specified. When more than one inlet and/or outlet temperature of the heat exchanger is unknown, LMTD may be calculated by trial and errors solution. In this case, the ε**-**NTU method is commonly used [3].

#### **3.1 The log-mean temperature difference (LMTD) method**

As mentioned earlier, by determining the temperature difference between hot and cold fluids, the amount of heat transfer can be calculated from Eq. (3). **Figure 2** shows temperature changes of hot and cold fluids along with a heat exchanger with different types of flow configuration. As it can be seen the temperature difference between hot and cold fluids can vary along with the heat exchanger. Terminal temperatures of hot and cold fluids (*TH,out* anf *TC,out*) are very effective factors in a heat exchanger design. If *TC,out* is lower than *TH,out* for countercurrent flow,

#### **Figure 2.**

*Temperature changes of hot and cold fluids along with a heat exchanger with different types of flow configuration.*

temperature approach occurs. In contrast, if *TC,out* is higher than the *TH,out* for countercurrent flow, temperature cross happens [9]. But if *TC,out* is equal to *TH,out*, temperature meet takes place. Based on the second law of thermodynamics, temperature cross can never take place for heat exchangers with co-current flow configuration [10]. In 1981, Wales [11] proposed a parameter G, which can be used to determine the temperature conditions in the heat exchangers. Eq. (10) defines the *G* parameters that can be between �1 and 1.

$$\begin{aligned} G &= \frac{T\_{H,out} - T\_{C,out}}{T\_{H,in} - T\_{C,in}}, \\ G &> 0 \rightarrow \text{ temperature approach} \\ G &= \mathbf{0} \rightarrow \text{temperature} \ \text{met} \\ G &< \mathbf{0} \rightarrow \text{temperature} \ \text{cross} \end{aligned} \tag{10}$$

Since the temperature difference between hot and cold streams varies along with the heat exchanger, the basic question is which temperature difference should be considered to calculate the heat transfer rate. To answer this question, consider **Figure 3** which shows heat transfer between two parallel and co-current fluids. Based on **Figure 3**, the heat transfer for the specified heat transfer area can be written in the form:

$$dQ = U\_{\perp} \Delta T\_{\perp} dA \tag{11}$$

**Figure 3.** *Heat transfer between two parallel and co-current fluids [3].*

It can be said that the amount of heat transferred is reduced from the hot fluid and added to the cold fluid. Therefore [3]:

$$dQ = -C\_{p,H} \, dT\_H \tag{12}$$

$$dQ = \mathbb{C}\_{p, \mathbb{C}} \, | \, dT\_{\mathbb{C}} \, \tag{13}$$

Where *Cp,H* and *Cp,C* are specific heat capacities of hot and cold fluids, respectively. The temperature difference can be written as below:

$$d(\Delta T) = dT\_H - dT\_C \tag{14}$$

By the combination of Eqs. (12) and (13) with Eq. (14):

$$d(\Delta T) = -\frac{dQ}{\mathcal{C}\_{p,H}} - \frac{dQ}{\mathcal{C}\_{p,C}}\tag{15}$$

By defining <sup>1</sup> *<sup>M</sup>* <sup>¼</sup> <sup>1</sup> *Cp*,*<sup>C</sup>* <sup>þ</sup> <sup>1</sup> *Cp*,*H*, Eq. (15) can be written as follow:

$$dQ = M \,\, d(\Delta T) \tag{16}$$

Assuming the *M* is constant along with the heat exchanger:

$$\int\_{0}^{Q} dQ = -M \int\_{\Delta T\_{in}}^{\Delta T\_{out}} d(\Delta T) \to Q = -M(\Delta T\_{out} - \Delta T\_{in}) \tag{17}$$

On the other hand, by placing Eq. (16) in Eq. (11):

$$-M\frac{d(\Delta T)}{\Delta T} = UdA = A\frac{UdA}{A} \tag{18}$$

$$A\int\_{0}^{A} \frac{U dA}{A} = -M \int\_{\Delta T\_{in}}^{\Delta T\_{out}} \frac{d(\Delta T)}{\Delta T} \rightarrow U\_m A = -M \ln \left(\frac{\Delta T\_{out}}{\Delta T\_{in}}\right) \tag{19}$$

Where *Um* is the mean overall heat transfer coefficient which can be defined as below:

$$U\_m = \bigcup\_{0}^{A} \frac{dA}{A} \tag{20}$$

The heat transfer rate can be written as flow:

$$Q = U\_m A \Delta T\_{LMTD} \tag{21}$$

Where Δ*TLMTD* is the logarithmic mean temperature difference (LMTD) which can be defined as below:

$$
\Delta T\_{LMTD} = \frac{\Delta T\_{out} - \Delta T\_{in}}{\ln \left(\frac{\Delta T\_{out}}{\Delta T\_{in}}\right)} \tag{22}
$$

*Heat Exchanger Design and Optimization DOI: http://dx.doi.org/10.5772/intechopen.100450*

The simplicity of the LMTD method has led to its use in the design of many heat exchangers by introducing a correction factor, *F*, according to Eq. (23) [12]. The F is generally expressed in terms of two non-dimensional parameters, thermal effectiveness (*P*), and heat capacity ratio (*R*). The *P* and *R* are defined as Eqs. (24) and (25), respectively [13]. **Figure 4** shows the correction factor for common shell and tube heat exchangers.

$$Q = UAF\Delta T\_{LMTD} \ , \ \ \ 0 < F < \mathbf{1} \tag{23}$$

$$P = \frac{T\_{C,out} - T\_{C,in}}{T\_{H,in} - T\_{C,in}} \tag{24}$$

$$R\_{cr} = \frac{T\_{H,in} - T\_{H,out}}{T\_{C,out} - T\_{C,in}} \tag{25}$$
