1. Introduction

Transient effectiveness methodology is a new analytical method which is developed for studying the dynamic performance of a heat exchanger. The concept was originally introduced by Cima and London in 1958 and used as a signature in representing the heat exchanger transient performance. The concept was then used for developing generalized transient effectiveness for plotting the transient response of a counter-flow heat exchanger [1]. In some of the recent

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

studies [2, 3], the transient effectiveness concept is used for developing a new methodology for dynamic characterization of cross-flow heat exchangers. In this chapter, a complete summary and review of the transient effectiveness method is provided, including the methodology development, transient effectiveness characterization, modeling validation, as well as the three major application and usefulness of the transient effectiveness. The heat exchanger configuration considered in most of the studies as well as in this chapter is an unmixed-unmixed cross flow one. It needs to be mentioned here that the majority of the work and results are summarized and published in different scientific journals by the same group of authors. This work provides a complete connection of all the existing research efforts and major results related to the transient effectiveness methodology. The readers can obtain a clear idea of this methodology and utilize it in the corresponding research and studies directly.

#### 2. Transient effectiveness

#### 2.1. Governing equations and numerical solution

Effectiveness which is defined as the ratio of actual heat transferred rate over the maximum heat transfer rate is introduced for characterizing heat exchanger steady-state performance. Cima and London [1] extended this concept to a time-dependent one in [1]. In their study, a generalized transient effectiveness was developed based on Eqs. (1a) and (1b), and then used as a means for representing the transient analog results for a counter-flow heat exchanger instead of using outlet temperatures.

$$\varepsilon\_{h}(t) = \frac{c\_{p\_{k}}[T\_{h,in}(t) - T\_{h,out}(t)]}{c\_{p\_{\min}}[T\_{h,in}(t) - T\_{c,in}(t)]} \tag{1a}$$

$$\varepsilon\_{\varepsilon}(t) = \frac{c\_{p\_{\varepsilon}}[T\_{c,out}(t) - T\_{c,in}(t)]}{c\_{p\_{\min}}[T\_{h,in}(t) - T\_{c,in}(t)]} \tag{1b}$$

The transient effectiveness concept and its governing equations were introduced and directly used for characterizing dynamic performance of a cross-flow heat exchanger in references [2, 3]. In these studies, the transient effectiveness governing equations are solved numerically by coupling them with thermal dynamic heat exchanger equations which are shown in Eqs. (2)–(4). These three sets of governing equations are widely used in most of the existing literature [4–9] for solving similar problems. A full numerical solution for these equations and a comprehensive heat exchanger transient behavior characterization using numerical modeling are conducted in [10–13]. Most of the variation scenarios were covered in these studies, including single fluid temperature variations, fluid mass flow rate variations, as well as multiple variation combinations.

$$\frac{\partial T\_{\text{wall}}}{\partial t} = r\_c^\emptyset \cdot T\_c + \mathcal{R} \cdot r\_h^\emptyset \cdot T\_h - (r\_c^\emptyset + \mathcal{R} \cdot r\_h^\emptyset) \cdot T\_{\text{wall}} \tag{2}$$

$$V\_c \frac{\partial T\_c}{\partial t} = r\_c^\circ \cdot T\_{wall} - r\_c^\circ \cdot T\_c - r\_c \frac{\partial T\_c}{\partial X} \tag{3}$$

Transient Effectiveness Methods for the Dynamic Characterization of Heat Exchangers http://dx.doi.org/10.5772/67334 195

$$\frac{dV\_h}{R}\frac{\partial T\_h}{\partial t} = r\_h^\circ \cdot T\_{\text{wall}} - r\_h^\circ \cdot T\_h - r\_h \frac{\partial T\_h}{\partial Y} \tag{4}$$

#### 2.2. Transient effectiveness method verification

studies [2, 3], the transient effectiveness concept is used for developing a new methodology for dynamic characterization of cross-flow heat exchangers. In this chapter, a complete summary and review of the transient effectiveness method is provided, including the methodology development, transient effectiveness characterization, modeling validation, as well as the three major application and usefulness of the transient effectiveness. The heat exchanger configuration considered in most of the studies as well as in this chapter is an unmixed-unmixed cross flow one. It needs to be mentioned here that the majority of the work and results are summarized and published in different scientific journals by the same group of authors. This work provides a complete connection of all the existing research efforts and major results related to the transient effectiveness methodology. The readers can obtain a clear idea of this methodol-

Effectiveness which is defined as the ratio of actual heat transferred rate over the maximum heat transfer rate is introduced for characterizing heat exchanger steady-state performance. Cima and London [1] extended this concept to a time-dependent one in [1]. In their study, a generalized transient effectiveness was developed based on Eqs. (1a) and (1b), and then used as a means for representing the transient analog results for a counter-flow heat exchanger

½Th,inðtÞ � Th, outðtÞ�

½Tc, outðtÞ � Tc,inðtÞ�

The transient effectiveness concept and its governing equations were introduced and directly used for characterizing dynamic performance of a cross-flow heat exchanger in references [2, 3]. In these studies, the transient effectiveness governing equations are solved numerically by coupling them with thermal dynamic heat exchanger equations which are shown in Eqs. (2)–(4). These three sets of governing equations are widely used in most of the existing literature [4–9] for solving similar problems. A full numerical solution for these equations and a comprehensive heat exchanger transient behavior characterization using numerical modeling are conducted in [10–13]. Most of the variation scenarios were covered in these studies, including single fluid temperature variations, fluid mass flow rate variations, as well as

cpmin <sup>½</sup>Th,inðtÞ � Tc,inðtÞ� (1a)

cpmin <sup>½</sup>Th,inðtÞ � Tc,inðtÞ� (1b)

<sup>h</sup>Þ � Twall (2)

<sup>∂</sup><sup>X</sup> (3)

ogy and utilize it in the corresponding research and studies directly.

<sup>ε</sup>hðtÞ ¼ cph

<sup>ε</sup>cðtÞ ¼ cpc

<sup>c</sup> � Tc þ R � r

<sup>c</sup> � Twall � r

β <sup>h</sup> � Th � ðr

β <sup>c</sup> � Tc � rc β <sup>c</sup> þ R � r β

∂Tc

2. Transient effectiveness

194 Heat Exchangers– Design, Experiment and Simulation

instead of using outlet temperatures.

multiple variation combinations.

∂Twall <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>r</sup> β

Vc ∂Tc <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>r</sup> β

2.1. Governing equations and numerical solution

The methodology and the numerical solution are verified by comparison with several published results in [1, 14, 15]. First, several published analytical solutions and analog solutions for the transient effectiveness of a 1D contour-flow heat exchanger are used [14]. The equivalent method was used and the same transient effectiveness equations were integrated into the numerical code and then compared to the results presented in a form as generalized transient effectiveness. Figure 1(a) shows a comparison of the numerical solutions and the analytical data points [14, 15]. This case represents a response of a heat exchanger under a fluid

Figure 1. (a) Comparison of the numerical solutions with the analytical results in [14] and analog results in [1, 15]; (b) Comparison of the numerical solutions with the analog results in [1] for NTU 1–1.5 and NTU 1.5–1.

inlet temperature step change. Figure 1(b) shows comparison between the analog solution [1] and numerical solution under a step change in the fluid mass flow rate. The mass flow rate step increase results in the NTU variation from 1 to 1.5, and the step decrease results in the NTU variation from 1.5 to 1, which is also mentioned in the figure caption. It can be seen from both figures that the results are in good agreement, and different scenarios including fluid mass flow rate change and inlet temperature change are validated.

#### 2.3. Parametric study

A detailed study of the characterizing transient effectiveness under different variation conditions including both inlet temperature change and mass flow rate change is presented in [2, 3]. It is found in these studies that the transient effectiveness can be used as a measure of the heat exchanger dynamic performance from one steady state to the new equilibrium state under certain inputs. In addition, the impact of modeling physical parameters, including NTU, E, R, and V, can be represented on the effectiveness curves. NTU results are chosen as an example to discuss in this section. More detailed parametric results are summarized in references [2, 3].

#### 2.3.1. Inlet temperature variation

The inlet temperature variation does not influence the final steady-state values of the effectiveness curve lines. This means that the effectiveness curve always returns to the initial value after a certain transient variation. Figure 2 shows the transient effectiveness of two fluids plotted versus nondimensional time for a wide range of NTU values for the step change. It can be seen from the figure that NTU governs both transient and steady-state variation of effectiveness. The larger the NTU value, the longer time is taken to reach the final steady state. When comparing the hot fluid transient effectiveness curve and the cold fluid transient effectiveness curve, a time lag is seen on the hot fluid curves. This time lag indicates that the corresponding fluid takes some time to begin to respond at the outlet, after the variation is applied at the inlet.

Figure 2. Effect of NTU on the transient effectiveness results with E = 1, Vh = Vc = 1, R = 1; step change to the hot fluid inlet temperature.

The larger the NTU value, the longer the time lag. A larger NTU value can be simply understood as a larger heat exchanger physical size. Therefore, the time lag is longer for a larger NTU value.

#### 2.3.2. Fluid mass flow rate variation

inlet temperature step change. Figure 1(b) shows comparison between the analog solution [1] and numerical solution under a step change in the fluid mass flow rate. The mass flow rate step increase results in the NTU variation from 1 to 1.5, and the step decrease results in the NTU variation from 1.5 to 1, which is also mentioned in the figure caption. It can be seen from both figures that the results are in good agreement, and different scenarios including fluid

A detailed study of the characterizing transient effectiveness under different variation conditions including both inlet temperature change and mass flow rate change is presented in [2, 3]. It is found in these studies that the transient effectiveness can be used as a measure of the heat exchanger dynamic performance from one steady state to the new equilibrium state under certain inputs. In addition, the impact of modeling physical parameters, including NTU, E, R, and V, can be represented on the effectiveness curves. NTU results are chosen as an example to discuss in this section. More detailed parametric results are summarized in references [2, 3].

The inlet temperature variation does not influence the final steady-state values of the effectiveness curve lines. This means that the effectiveness curve always returns to the initial value after a certain transient variation. Figure 2 shows the transient effectiveness of two fluids plotted versus nondimensional time for a wide range of NTU values for the step change. It can be seen from the figure that NTU governs both transient and steady-state variation of effectiveness. The larger the NTU value, the longer time is taken to reach the final steady state. When comparing the hot fluid transient effectiveness curve and the cold fluid transient effectiveness curve, a time lag is seen on the hot fluid curves. This time lag indicates that the corresponding fluid takes some time to begin to respond at the outlet, after the variation is applied at the inlet.

Figure 2. Effect of NTU on the transient effectiveness results with E = 1, Vh = Vc = 1, R = 1; step change to the hot fluid inlet

mass flow rate change and inlet temperature change are validated.

2.3. Parametric study

196 Heat Exchangers– Design, Experiment and Simulation

2.3.1. Inlet temperature variation

temperature.

The characteristics of the effectiveness under fluid mass flow rate change are discussed in this section. Figure 3 shows that the steady-state conditions of the effectiveness curve changes due to a change in fluid mass flow rate. The difference is clearly seen from the transient effectiveness between a cold fluid mass flow rate change and a hot fluid mass flow rate change.

Figure 3. Effect of NTU on the transient effectiveness results with E = 0.5, R = 1, V = 1; (a) step change to hot fluid mass flow rate; (b) step change to cold fluid mass flow rate.

A step change is seen in all the cold fluid curves before the curves move smoothly and reach a steady state. In Eq. (1b), Tc,out(t), and Cc/Cmin (Cmin is considered as a hot fluid capacity rate, which gives Cc/Cmin = E) govern the cold fluid effectiveness variations. The previously mentioned step change at the very beginning is due to the step change of Cc(E). In terms of a mass flow rate ramp variation scenario, the variation in transient effectiveness curves at the beginning is dominated by E and Tc,out(t). The transient effectiveness also illustrated the combination impact of the fluid mass flow rate variation and the physical parameters. As an example, Figure 3 shows the transient effectiveness results versus different NTU values. The impact of the NTU value on the steady-state and transient performance of the transient effectiveness can be clearly seen in the curves of the figure. As an example, the larger the NTU value the longer time the heat exchanger takes to reach steady state. The difference of the variation of the cold fluid and the hot fluid is clearly distinguished in the same figure.

It can be seen that the transient effectiveness curves are able to represent the transient response of heat exchangers under different variation conditions by comparing the curves shown in Figure 3(a) and (b). This means that a transient effectiveness curve represents more physical information than an outlet temperature curve, since the curves are distinguished clearly when different boundary conditions are applied. In addition, the transient effectiveness curves also reflect the influences of the physical parameters on the transient and steady-state responses of the heat exchangers.

#### 2.4. Experimental verification

Experimental measurements on a liquid to air cross-flow heat exchanger cores are presented in reference [16], in which the liquid mass flow rate or inlet temperature varied in time following controlled functional forms (step jump and ramp). The specific design enables the control of transient variations in the inlet temperature and mass flow rate on both the air and water flow streams supplied to the heat exchanger device. More details regarding the entire experimental setup and tests can be found in reference [16]. The experimental data were used to characterize and validate the transient effectiveness methodology and the transient numerical solution in reference [17], and the more comprehensive understanding of the characteristics of the transient effectiveness is obtained.

For modeling a specific heat exchanger, the modeling physical parameters (E, NTU, R, Vh, Vc) need to be extracted and calculated using the heat exchanger hardware data and one set of steady-state experimental data for modeling a specific heat exchanger and specific dynamic physical scenarios. The procedure can be interpreted as integrating the hardware data into the nondimensional mathematical model (Eqs. (2)–(4)) to model a specific heat exchanger device. One of the methodologies can be referenced to calculate the physical parameter and is presented in reference [18].

#### 2.4.1. Inlet temperature variation

Several functional forms are designed to vary the water inlet temperature and water flow rate. Ramp functions for water inlet temperature change and step functional forms of water flow rate change are selected to present here. The detailed information of each experimental case


designed is shown in Table 1. The physical parameters used in the numerical solution for each case are also summarized in Table 1.

Table 1. Test cases.

A step change is seen in all the cold fluid curves before the curves move smoothly and reach a steady state. In Eq. (1b), Tc,out(t), and Cc/Cmin (Cmin is considered as a hot fluid capacity rate, which gives Cc/Cmin = E) govern the cold fluid effectiveness variations. The previously mentioned step change at the very beginning is due to the step change of Cc(E). In terms of a mass flow rate ramp variation scenario, the variation in transient effectiveness curves at the beginning is dominated by E and Tc,out(t). The transient effectiveness also illustrated the combination impact of the fluid mass flow rate variation and the physical parameters. As an example, Figure 3 shows the transient effectiveness results versus different NTU values. The impact of the NTU value on the steady-state and transient performance of the transient effectiveness can be clearly seen in the curves of the figure. As an example, the larger the NTU value the longer time the heat exchanger takes to reach steady state. The difference of the variation of the cold

It can be seen that the transient effectiveness curves are able to represent the transient response of heat exchangers under different variation conditions by comparing the curves shown in Figure 3(a) and (b). This means that a transient effectiveness curve represents more physical information than an outlet temperature curve, since the curves are distinguished clearly when different boundary conditions are applied. In addition, the transient effectiveness curves also reflect the influences of the physical parameters on the transient and steady-state responses of

Experimental measurements on a liquid to air cross-flow heat exchanger cores are presented in reference [16], in which the liquid mass flow rate or inlet temperature varied in time following controlled functional forms (step jump and ramp). The specific design enables the control of transient variations in the inlet temperature and mass flow rate on both the air and water flow streams supplied to the heat exchanger device. More details regarding the entire experimental setup and tests can be found in reference [16]. The experimental data were used to characterize and validate the transient effectiveness methodology and the transient numerical solution in reference [17], and the more comprehensive understanding of the characteristics of the tran-

For modeling a specific heat exchanger, the modeling physical parameters (E, NTU, R, Vh, Vc) need to be extracted and calculated using the heat exchanger hardware data and one set of steady-state experimental data for modeling a specific heat exchanger and specific dynamic physical scenarios. The procedure can be interpreted as integrating the hardware data into the nondimensional mathematical model (Eqs. (2)–(4)) to model a specific heat exchanger device. One of the methodologies can be referenced to calculate the physical parameter and is

Several functional forms are designed to vary the water inlet temperature and water flow rate. Ramp functions for water inlet temperature change and step functional forms of water flow rate change are selected to present here. The detailed information of each experimental case

fluid and the hot fluid is clearly distinguished in the same figure.

the heat exchangers.

2.4. Experimental verification

198 Heat Exchangers– Design, Experiment and Simulation

sient effectiveness is obtained.

presented in reference [18].

2.4.1. Inlet temperature variation

Two important characteristics of the transient effectiveness are discussed in this section. In case 1, the water inlet temperature is lower than the air inlet temperature at the beginning, and then becomes the hot fluid after the variation. This is the scenario that the cold fluid becomes the hot fluid due to the temperature change. When plotting the transient effectiveness curves, a mathematical singularity point is seen. In Eqs. (1a) and (1b), the term Th,in Tc,in will vary from a positive value to a negative value. This performance is characterized by both simulation modeling and experimental testing. By comparing with the regular fluid temperature curve, the transient effectiveness curves can capture this special scenario. At the same time, they contain all the steady-state and transient characteristics. In case 2, since the water and air are at the same temperature, no singularity point is generated. By comparing the results of cases 1 and 2, the difference between the boundary conditions applied and the transient response is clearly reflected on the transient effectiveness curves. The initial conditions of case 2 can be considered as an idle condition. When plotting the transient effectiveness curve for this special case, there will be a sensitive region after the variation is applied at the time between 10 and 15 s, which is shown in Figure 5. Since the two fluid inlet temperatures are same, the numerator and denominator in Eqs. (1a) and (1b) equal 0. Then, even a very miner error in either temperature data may result in a major difference in the effectiveness value. In both Figures 4 and 5, the numerical results show a faster response than the experimental results. This is because axial dispersion and longitudinal conduction are neglected in the numerical modeling. When using the transient effectiveness curve plotting the fluid inlet temperature variation cases, the variation form and some of the corresponding characteristics in the fluid inlet temperature can be represented at the same time.

#### 2.4.2. Mass flow rate variation

In cases 3 and 4, variations are applied to the water fluid mass flow rate. Figures 6 and 7 show the transient effectiveness results of these the two cases, respectively. In terms of the steadystate results, the increase in the water (Cmax fluid) leads to an increase in the effectiveness value, which is shown in Figure 6. In case 4, a special scenario is considered in which the

Figure 4. Transient effectiveness results of case 1.

Figure 5. Transient effectiveness of case 2.

minimum capacity fluid (water) becomes the maximum capacity fluid, due to the change in the fluid mass flow rate. Then the air becomes the minimum capacity fluid due to the change. This scenario may be seen in an actual heat exchanger industrial application, especially in certain failure scenarios. It can be seen in Figure 7 that step changes are seen on both of the curves, before the curves move smoothly and reach the final steady state. By comparing the results shown in Figures 6 and 7, it can be found that the transient effectiveness results are clearly different when different fluid mass flow rate variation scenarios are applied. Again, this transient effectiveness methodology can present the heat exchanger dynamic performance in a more comprehensive manner in fluid mass flow rate variation scenarios. It contains the information of the two fluids dynamic responses, and the corresponding variations applied as well (both the fluid inlet temperature and the fluid mass flow rate).

Figure 6. Transient effectiveness results of case 3.

Figure 7. Transient effectiveness results of case 4.

#### 2.4.3. Summary

minimum capacity fluid (water) becomes the maximum capacity fluid, due to the change in the fluid mass flow rate. Then the air becomes the minimum capacity fluid due to the change. This scenario may be seen in an actual heat exchanger industrial application, especially in certain failure scenarios. It can be seen in Figure 7 that step changes are seen on both of the curves, before the curves move smoothly and reach the final steady state. By comparing the results shown in Figures 6 and 7, it can be found that the transient effectiveness results are clearly different when different fluid mass flow rate variation scenarios are applied. Again, this transient effectiveness methodology can present the heat exchanger dynamic performance in a more comprehensive manner in fluid mass flow rate variation scenarios. It contains the information of the two fluids dynamic responses, and the corresponding variations applied as

well (both the fluid inlet temperature and the fluid mass flow rate).

Figure 5. Transient effectiveness of case 2.

Figure 4. Transient effectiveness results of case 1.

200 Heat Exchangers– Design, Experiment and Simulation

This section provides several important characteristics of the transient effectiveness for dynamic characterization of a heat exchanger transient performance. Several experimental test cases are selected and analyzed. Two cases of fluid inlet temperature change and two cases of fluid mass flow rate change cases provide a more complete understanding of the transient effectiveness method in characterizing the dynamic performance of the heat exchanger. The transient effectiveness methodology can be used as an alternative for representing the dynamic performance of the heat exchanger. It is a more effective way than using the fluid temperature results, and it contains more information, including the variation condition applied to the heat exchanger, initial conditions and some special circumstances such as the cold fluid becoming as the hot fluid, Cmin fluid becoming Cmax fluid, and so on.
