4. CFD compact heat exchanger modeling

This section discusses another important application of the transient effectiveness concept and model, which can be used in developing computational fluid dynamics (CFD) compact transient heat exchanger modeling methodologies. There are two methods which are proposed in references [21, 22]. The methods can be used to model different types of heat exchangers, including a counter-flow heat exchanger and a cross-flow heat exchanger. In addition, the compact models developed can be used to model different variation scenarios, including fluid inlet temperature variation, fluid mass flow rate variation, and multiple combination variation scenarios.

#### 4.1. Compact modeling methodology I

#### 4.1.1. Modeling methodology development

It has been shown in previous studies that the transient effectiveness is able to characterize the dynamic response of heat exchangers. When studying heat exchanger dynamic response, the transient input can be either an inlet temperature variation or a mass flow rate variation. This case may become more complicated when considering multiple variation combination scenarios. Then the outlet temperature transient performance will be a complicated form, as shown in references [12, 23]. The transient effectiveness is correspondingly more complicated, due to the fact that the transient effectiveness is reflecting the variation in both the fluid inlet temperature and the outlet temperature. When comparing with the steady-state ε-NTU method, the first methodology was developed by extending the concept to a time-dependent effectiveness.

#### 4.1.2. CFD compact heat exchanger model

cooler heat exchanger. The temperature results are collected at different locations, capturing a detailed response sequence. However, since the heat exchanger units are connected to each other using the internal loop and external loop, it is very difficult to characterize the response time of certain heat exchanger by using any temperature result. The temperature results vary during the entire test run. The transient effectiveness method provides a way to observe individual component performance, even though it is in a closed coupled system, by fliting the influence of the neighbored components. The buffer unit transient effectiveness curves have reached steady-state conditions, while the temperatures are still varying. This illustrates that the buffer unit itself has reached a steady-state thermal-exchange condition during a transient event. This can be understood as a self "steady-state" condition in a transient environment. In this condition, even though the corresponding fluid temperatures vary with time, the heat

This section illustrates that the transient effectiveness can be used for characterizing the dynamic response of a closed coupled heat transfer loop, which has multiple heat exchanger units installed. It also represents the thermal capacitance impact of each component during different transient events. In addition, some detailed physical insights, which cannot be directly captured

This section discusses another important application of the transient effectiveness concept and model, which can be used in developing computational fluid dynamics (CFD) compact transient heat exchanger modeling methodologies. There are two methods which are proposed in references [21, 22]. The methods can be used to model different types of heat exchangers, including a counter-flow heat exchanger and a cross-flow heat exchanger. In addition, the compact models developed can be used to model different variation scenarios, including fluid inlet temperature

It has been shown in previous studies that the transient effectiveness is able to characterize the dynamic response of heat exchangers. When studying heat exchanger dynamic response, the transient input can be either an inlet temperature variation or a mass flow rate variation. This case may become more complicated when considering multiple variation combination scenarios. Then the outlet temperature transient performance will be a complicated form, as shown in references [12, 23]. The transient effectiveness is correspondingly more complicated, due to the fact that the transient effectiveness is reflecting the variation in both the fluid inlet temperature and the outlet temperature. When comparing with the steady-state ε-NTU method, the first methodology was developed by extending the concept to a time-dependent effectiveness.

variation, fluid mass flow rate variation, and multiple combination variation scenarios.

from temperature results, can be indicated by the transient effectiveness results.

exchanger has approached a steady-state condition.

206 Heat Exchangers– Design, Experiment and Simulation

4. CFD compact heat exchanger modeling

4.1. Compact modeling methodology I 4.1.1. Modeling methodology development

3.3. Summary

The ε-NTU heat exchanger modeling methodology has been widely used in heat exchanger steady-state studies. This method and theoretical equations have been incorporated into most of the commercial CFD codes with a heat exchanger modeling option. For heat exchanger steady-state modeling, effectiveness performance data of the heat exchangers are used to obtain the corresponding term εCmin, and then the ε-NTU equation is used to represent the corresponding heat exchanger unit under certain flow rate operating conditions. The transient effectiveness concept discussed in the previous section is extended here to develop a compact transient heat exchanger model. The modeling methodology uses the transient effectiveness in the standard ε-NTU heat exchanger equation to extend the ε-NTU model to a compact transient heat exchanger form, as shown in Eq. (8). This transient effectiveness is denoted as ε<sup>T</sup> <sup>0</sup> in the current study. By applying transient effectiveness to the equation, the transient compact model can be developed.

$$\mathbb{Q}' = \varepsilon' \mathbb{C}'\_{\text{min}} (T'\_{h,in} - T'\_{c,in}) \tag{8}$$

A CFD compact transient heat exchanger model is developed based on this transient methodology using the commercial code FloTherm [24]. The basic methodology correlates a negative linear source function as in Eq. (9) to Eq. (8) to represent the heat exchanger model. In FloTherm, the value and the coefficient can be set as transient variables for this linear heat source function. Therefore, Eq. (9) can be correlated to the transient compact heat exchanger model shown in Eq. (8). The detailed description of the FloTherm linear source model and the correlation method can be found in reference [24].

$$Q = \text{Coefficient} \times (T\_{c,in} - T\_{h,out}) = \text{Coefficient} \times (T\_{water,in} - T\_{air,out}) \tag{9}$$

#### 4.1.3. Verification with thermal dynamic model

Thermal dynamic modeling results and the experimental test results are used as the input for calculating the transient effectiveness, and the effectiveness is then integrated into the CFD model. Then outlet temperatures predicted by the CFD compact model are compared with the thermal dynamic modeling results and experimental results. The detailed validation study is presented in reference [21]. Here a multiple variation combination case is presented as an example. It can be seen in Figure 10 that the CFD compact modeling results are in good agreement with the thermal dynamic modeling results.

#### 4.1.4. Verification with experimental data

In this section, this CFD compact model is verified using experimental data. The experimental tests discussed in the previous section are used. The original data are summarized in reference [19]. The transient test results, including the fluid mass flow rate and temperature variations, are incorporated into Eqs. (1a) and (1b) to calculate the transient effectiveness for the heat exchanger unit under different scenarios. Then the transient effectiveness (ε<sup>T</sup> 0 ) is used in the CFD model. The dry cooler results are chosen as an example to discuss in this section. The detailed formulas for the transient effectiveness calculations are shown in Eq. (10).

Figure 10. Hot fluid outlet temperature results, case 1: cot fluid inlet temperature step change and cold fluid mass flow rate ramp change; case 2: hot fluid inlet temperature step change and cold fluid mass flow rate step change.

For an air to liquid cross-flow heat exchanger—dry cooler:

$$\varepsilon\_{T} = \frac{c\_{\text{external}}(\pi) \cdot \left[T\_{A\text{unbient air}}(\pi) - T\_{\text{Exhuast air}}(\pi)\right]}{c\_{\text{min}}(\pi) \cdot \left[T\_{A\text{unbient air}}(\pi) - T\_{\text{Postbufffer}}(\pi)\right]} \tag{10}$$

The test data discussed in Section 3 is used here for calculating the transient effectiveness of the dry cooler, and the three cases shown in Table 2 are plotted in Figure 11. The comparison results are shown in Figure 12, and the two sets of results are in good agreement.

Figure 11. Hot fluid transient effectiveness in the three test cases.

Figure 12. Comparison of the hot fluid outlet temperatures results (secondary fluid) in case 1 and case 2.

#### 4.2. Compact modeling methodology II

For an air to liquid cross-flow heat exchanger—dry cooler:

<sup>0</sup> <sup>¼</sup> cexternalðτÞ�½TAmbient airðτÞ � TExhaust airðτÞ�

Figure 10. Hot fluid outlet temperature results, case 1: cot fluid inlet temperature step change and cold fluid mass flow

rate ramp change; case 2: hot fluid inlet temperature step change and cold fluid mass flow rate step change.

The test data discussed in Section 3 is used here for calculating the transient effectiveness of the dry cooler, and the three cases shown in Table 2 are plotted in Figure 11. The comparison

results are shown in Figure 12, and the two sets of results are in good agreement.

<sup>c</sup>minðτÞ�½TAmbient airðτÞ � TPostbuf f erðτÞ� (10)

εT

208 Heat Exchangers– Design, Experiment and Simulation

Figure 11. Hot fluid transient effectiveness in the three test cases.

The limitation of modeling methodology I is that the transient effectiveness, which used as the input for the CFD compact model, is generated based on the existing solutions from either the thermal dynamic model or experimental tests. This means that a transient effectiveness curve only represents a specific case and can only be used for modeling one certain transient case. Then the CFD model can be only used for modeling the cases with the same boundary condition, due to the limitation of the transient effectiveness used in the code. In addition, as discussed in the previous section, the transient effectiveness variation can be very complex for certain scenarios. This limitation results in the fact that this compact model may not be applied to a system level modeling work. Therefore, a derivative transient effectiveness method is developed.

#### 4.2.1. Modeling methodology development

Eq. (11) is generated by adding the three governing partial differential equations and used as a simplified correlation in representing heat exchanger transient performance. By considering a single energy balance equation to represent the cross-flow heat exchanger using fluid inlet and out flow, Eq. (11) can be expressed in Eq. (12). The energy balance equation, together with the ε-NTU methodology, is shown in Eq. (13). Eq. (14) is then generated by substituting Eq. (13) into Eq. (12). Here, a new term (pVcp)heat exchanger is introduced, which is a lumped thermal capacitance of the heat exchanger, including the capacitance of the heat exchanger metal and the two fluids. By lumping the thermal capacitances together, Eq. (14) is then expressed as Eq. (15). The term Theat exchanger represents the heat exchanger temperature, which can be understood as an averaged value of the two fluids and the heat exchanger metal. The negative sign represents a negative heat source. Eq. (15) is the governing equation of the second methodology. Eq. (16) is the expression of the fluid outlet temperature, and the hot fluid is used as an example.

$$\text{MC}\frac{\partial T\_{\text{wall}}}{\partial t} + \text{C}\_{\text{c}}^{\prime}\frac{\partial T\_{\text{c}}}{\partial t} + \text{C}\_{h}^{\prime}\frac{\partial T\_{h}}{\partial t} + (m^{\prime}c)\_{h}\frac{\partial T\_{h}}{\partial (y/L\_{h})} + (m^{\prime}c)\_{c}\frac{\partial T\_{c}}{\partial (\mathbf{x}/L\_{c})} = \mathbf{0} \tag{11}$$

$$\text{MC}\frac{\partial T\_{\text{null}}}{\partial t} + \text{C}\_{c}^{\prime}\frac{\partial T\_{c}}{\partial t} + \text{C}\_{h}^{\prime}\frac{\partial T\_{h}}{\partial t} + (m^{\prime}c)\_{h} \cdot (T\_{h,out} - T\_{h,in}) + (m^{\prime}c)\_{c} \cdot (T\_{c,out} - T\_{c,in}) = 0 \tag{12}$$

$$\dot{Q} = (m'c)\_{\varepsilon} \cdot (T\_{c,out} - T\_{c,in}) = (m'c)\_{h} \cdot (T\_{h,out} - T\_{h,in}) = \varepsilon \cdot C\_{\min} \cdot (T\_{h,in} - T\_{c,in}) \tag{13}$$

$$\text{MC}\frac{\partial T\_{\text{wall}}}{\partial t} + \text{C}\_{c}^{\prime}\frac{\partial T\_{c}}{\partial t} + \text{C}\_{h}^{\prime}\frac{\partial T\_{h}}{\partial t} + (m^{\prime}c)\_{h} \cdot (T\_{h,out} - T\_{h,in}) + \varepsilon\text{C}\_{\text{min}} \cdot (T\_{h,in} - T\_{c,in}) = 0 \tag{14}$$

$$-\dot{Q} = -\varepsilon \mathbb{C}\_{\text{min}} \cdot (T\_{h,\text{in}} - T\_{c,\text{in}}) = \dot{m}c\_p (T\_{h,\text{out}} - T\_{h,\text{in}}) + (pVc\_p)\_{\text{heat exchange}} \times \frac{\partial T\_{\text{heat exchange}}}{\partial t} \tag{15}$$

$$T\_{h,\ out} = \frac{-\varepsilon \mathbb{C}\_{\min} \cdot (T\_{h,in} - T\_{c,in}) - (pVc\_p)\_{\text{heat exchanger}} \times \frac{d\Gamma\_{\text{heat exchanger}}}{dt}}{(\dot{m}c\_p)\_{h}} + T\_{h,\ in} \tag{16}$$

#### 4.2.2. CFD compact heat exchanger model

In reference [22], a CFD compact model is realized in the commercial CFD code FloTherm using methodology II. The detailed procedure regarding the model development is presented in reference [22]. The heat exchanger is modeled using a linear heat source module, as shown in Eq. (9), and server solid blocks, which are used to represent the thermal capacitance. Two heat source modules are used to represent the supply fluid inlet temperature and mass flow rate. The user is able to manipulate the parameters in the linear heat source module and material setting in the solid rods module to correlate it to the governing equation (Eq. (15)) of methodology II.

#### 4.2.3. Model verification

It was mentioned that the lumped capacitance term is dominated by the capacitance of the heat exchanger coil and the two fluids, as well as their corresponding weight. For modeling verification purposes, a method was used to adjust the estimated thermal properties initially considered in the model, instead of deriving the actual lumped capacitance value. A method for lumping the three capacitance terms is a comprehensive study, which requires developing a complex physical correlation. In addition, it may vary from case to case. Basically, when using a lumped capacitance value, it should have the same impact on the heat exchanger transient response. Therefore, the curve adjustment method was used. The detailed procedure for adjusting the curve is presented in reference [22].

#### 4.2.3.1. Inlet temperature variation scenario

The inlet temperature variation case is considered in this section, and the fluid mass flow rate is set to a constant value. For the cross-flow heat exchanger model considered in this work, the hot fluid is modeled as the supply fluid and as the Cmin fluid. Based on Eq. (17), the variation in the fluid inlet temperature either Th and/or Tc will impact the left side of the equation. The effectiveness and Cmin are constant values, since the fluid mass flow rate is set at a constant value.

MC <sup>∂</sup>Twall

210 Heat Exchangers– Design, Experiment and Simulation

MC <sup>∂</sup>Twall

MC <sup>∂</sup>Twall

methodology II.

4.2.3. Model verification

<sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>C</sup><sup>o</sup> c ∂Tc <sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>C</sup><sup>o</sup> h ∂Th <sup>∂</sup><sup>t</sup> þ ðm<sup>0</sup>

<sup>Q</sup>\_ ¼ ðm<sup>0</sup>

<sup>∂</sup><sup>t</sup> <sup>þ</sup> Co c ∂Tc <sup>∂</sup><sup>t</sup> <sup>þ</sup> Co h ∂Th <sup>∂</sup><sup>t</sup> þ ðm<sup>0</sup>

4.2.2. CFD compact heat exchanger model

adjusting the curve is presented in reference [22].

4.2.3.1. Inlet temperature variation scenario

<sup>∂</sup><sup>t</sup> <sup>þ</sup> Co c ∂Tc <sup>∂</sup><sup>t</sup> <sup>þ</sup> Co h ∂Th <sup>∂</sup><sup>t</sup> þ ðm<sup>0</sup>

cÞ<sup>c</sup> � ðTc, out � Tc,inÞ¼ðm<sup>0</sup>

�Q\_ ¼ �εCmin � ðTh,in � Tc,inÞ ¼ mc\_ <sup>p</sup>ðTh, out � Th,inÞþðpVcpÞheat exchanger ·

Th, out <sup>¼</sup> �εCmin � ðTh,in � Tc,inÞ�ðpVcpÞheat exchanger ·

ðmc\_ <sup>p</sup>Þ<sup>h</sup>

In reference [22], a CFD compact model is realized in the commercial CFD code FloTherm using methodology II. The detailed procedure regarding the model development is presented in reference [22]. The heat exchanger is modeled using a linear heat source module, as shown in Eq. (9), and server solid blocks, which are used to represent the thermal capacitance. Two heat source modules are used to represent the supply fluid inlet temperature and mass flow rate. The user is able to manipulate the parameters in the linear heat source module and material setting in the solid rods module to correlate it to the governing equation (Eq. (15)) of

It was mentioned that the lumped capacitance term is dominated by the capacitance of the heat exchanger coil and the two fluids, as well as their corresponding weight. For modeling verification purposes, a method was used to adjust the estimated thermal properties initially considered in the model, instead of deriving the actual lumped capacitance value. A method for lumping the three capacitance terms is a comprehensive study, which requires developing a complex physical correlation. In addition, it may vary from case to case. Basically, when using a lumped capacitance value, it should have the same impact on the heat exchanger transient response. Therefore, the curve adjustment method was used. The detailed procedure for

The inlet temperature variation case is considered in this section, and the fluid mass flow rate is set to a constant value. For the cross-flow heat exchanger model considered in this work, the hot fluid is modeled as the supply fluid and as the Cmin fluid. Based on Eq. (17), the variation in the

cÞh

cÞ<sup>h</sup> � ðTh, out � Th,inÞþðm<sup>0</sup>

∂Th ∂ðy=LhÞ

þ ðm<sup>0</sup> cÞc

∂Tc ∂ðx=LcÞ

cÞ<sup>h</sup> � ðTh, out � Th,inÞ ¼ εCmin � ðTh,in � Tc,inÞ (13)

∂Theat exchanger ∂t

cÞ<sup>h</sup> � ðTh, out � Th,inÞ þ εCmin � ðTh,in � Tc,inÞ ¼ 0 (14)

¼ 0 (11)

cÞ<sup>c</sup> � ðTc, out � Tc,inÞ ¼ 0 (12)

∂Theat exchanger

<sup>∂</sup><sup>t</sup> (15)

þ Th, in (16)

$$-\varepsilon \mathbb{C}\_{\text{min}} \cdot (T\_{h,in} - T\_{c,in}) = -\varepsilon \mathbb{C}\_{\text{min}} \cdot (T\_{a,in} - T\_{w,in}) = \dot{m}c\_p (T\_{a,out} - T\_{a,in}) + (pVc\_p)\_{\text{heat exchange}} \tag{17}$$

$$\times \frac{\partial T\_{\text{heat exchange}}}{\partial t} \tag{17}$$

The analytical and numerical solutions of the thermal dynamic model shown in Eqs. (2)–(4) are used to verify the compact model shown in Eq. (17). A hot fluid inlet temperature step change scenario is used as an example in this section. Figure 13 shows several sets of results, including the CFD modeling results, which are illustrated by solid lines, the analytical results presented in reference [8], which are indicated by discrete round black points, and the numerical results, which are plotted in dashed lines. The detailed information of each case is shown in the figure legend. It can be seen that the three sets of solutions are in good agreement for the case NTU = 1.5. It needs to be noted that axial dispersion is dismissed in both the numerical results and the analytical solution. It has been concluded in references [8, 25–27] that the primary fluid responds immediately, with no time delay to the sudden variation applied at the inlet. It also has been concluded that the axial dispersion has a clear impact on the fluid dynamic performance, when the NTU value is larger than 2. It can be seen in Figure 13 that the CFD model results are in good agreement with the numerical solution. By comparing the two NTU = 2 cases (with and without axial dispersion), it can be seen that both the steady-state and transient performances of the outlet temperatures are influenced by the axial dispersion. Even for the same modeling case (NTU = 2), since the numerical results are used to calculate the ε or the coefficient value used in the CFD model, the CFD modeling results are different. It is seen that the CFD curve responds rapidly at the early response for the two NTU = 2 cases. Similar performance has been presented in reference [28].

Figure 13. Comparison of the CFD modeling results with analytical and numerical solutions.

#### 4.2.4. Fluid mass flow rate variation scenario

#### 4.2.4.1. Mass flow rate variation-based transient effectiveness

It has been discussed in the previous section that modeling a case that involves fluid mass flow rate changes is more complicated than modeling a fluid inlet temperature variation, due to the changing in the heat transfer coefficient. Therefore, the impact of the fluid mass flow rate variation on the heat transfer coefficient should be considered. In this section, both the hot fluid and the cold fluid inlet temperatures are considered as constant. In Eq. (18), ε<sup>0</sup> <sup>m</sup> is defined as a time-dependent variable and it represents the effectiveness changes due to fluid mass flow rate variations. The term Cmin is also a variable in the cases that the Cmin fluid mass flow rate changes.

$$-\varepsilon\_m' \mathbb{C}\_{\text{min}} \cdot (T\_{h,in} - T\_{c,in}) = -\varepsilon\_m' \mathbb{C}\_{\text{min}} \cdot (T\_{a,in} - T\_{w,in}) = \dot{m}c\_p (T\_{a,out} - T\_{a,in}) + (pVc\_p)\_{\text{heat exchange}}$$

$$\times \frac{\partial T\_{\text{heat exchange}}}{\partial t}$$

It is important for modeling the heat exchanger transient response to correctly characterize the effectiveness due to variations in the fluid mass flow rate, and in the corresponding heat transfer coefficient. Based on the steady-state ε-NTU results, different steady-state mass flow rates and heat transfer coefficients govern the NTU values. Thus, the ε value changes due to the variation of NTU. This concept is extended to a "mass flow rate variation based" transient effectiveness. Due to the mass flow rate variation, the heat transfer coefficient changes are denoted by the NTU<sup>0</sup> value in Eq. (19). In addition, mass flow rate variations lead to changes in the heat capacity rate ratio (E<sup>0</sup> ), as in Eq. (20). The detailed mathematical procedure is presented in reference [22]. Then the "mass flow rate based transient effectiveness (ε<sup>m</sup> 0 )" concept is defined by extending the theoretical steady-state correlation of ε and NTU to the transient case. The theoretical steady-state correlations are shown in Eqs. (21) and (23) for a cross-flow heat exchanger and for a counter-flow heat exchanger, respectively. By integrating the NTU<sup>0</sup> and E<sup>0</sup> equations (Eqs. (19) and (20)), the mass flow rate variation-based transient effectiveness can be expressed as Eqs. (22) and (24). They are designated as the ε-NTU transient theoretical correlations. The transient theoretical correlations are used to calculate the corresponding mass flow rate based transient effectiveness under the corresponding mass flow rate variations for the CFD heat exchanger models.

$$NTUl' = \frac{(mc\_p)\_{\text{min}}}{(m'c\_p)\_{\text{min}}} \cdot r\_c^\beta \cdot \frac{(r\_h^\beta + R \cdot r\_h^\beta)}{r\_c^\beta + R \cdot r\_h^\beta} \cdot NTUl \tag{19}$$

(18)

$$E' = \frac{(m'c\_p)\_h}{(m'c\_p)\_c} = \frac{r\_h}{r\_c}E\tag{20}$$

For a unmixed-unmixed cross-flow heat exchanger

$$\varepsilon = 1 - \exp\left\{\frac{NTU^{0.22}}{E} \left[\exp\left(-E \cdot NTU^{0.78}\right) - 1\right]\right\} \tag{21}$$

#### Transient Effectiveness Methods for the Dynamic Characterization of Heat Exchangers http://dx.doi.org/10.5772/67334 213

$$\varepsilon\_{m}^{\prime} = 1 - \exp\left\{ \frac{NTU^{0.22}}{E} \left[ \exp\left( -E \cdot NTU^{0.78} \right) \right] \right\} \tag{22}$$

For a counter cross-flow heat exchanger

4.2.4. Fluid mass flow rate variation scenario

212 Heat Exchangers– Design, Experiment and Simulation

mCmin � ðTh,in � Tc,in޼�ε<sup>0</sup>

the heat capacity rate ratio (E<sup>0</sup>

rate variations for the CFD heat exchanger models.

For a unmixed-unmixed cross-flow heat exchanger

ε ¼ 1 � exp

NTU<sup>0</sup> <sup>¼</sup> <sup>ð</sup>mcpÞmin ðm<sup>0</sup> cpÞmin � r β c � ðr β <sup>h</sup> þ R � r β hÞ

> cpÞ<sup>h</sup> ðm<sup>0</sup> cpÞ<sup>c</sup>

¼ rh rc

NTU<sup>0</sup>:<sup>22</sup>

<sup>E</sup><sup>0</sup> <sup>¼</sup> <sup>ð</sup>m<sup>0</sup>

� ε<sup>0</sup>

4.2.4.1. Mass flow rate variation-based transient effectiveness

·

It has been discussed in the previous section that modeling a case that involves fluid mass flow rate changes is more complicated than modeling a fluid inlet temperature variation, due to the changing in the heat transfer coefficient. Therefore, the impact of the fluid mass flow rate variation on the heat transfer coefficient should be considered. In this section, both the hot fluid

time-dependent variable and it represents the effectiveness changes due to fluid mass flow rate variations. The term Cmin is also a variable in the cases that the Cmin fluid mass flow rate changes.

It is important for modeling the heat exchanger transient response to correctly characterize the effectiveness due to variations in the fluid mass flow rate, and in the corresponding heat transfer coefficient. Based on the steady-state ε-NTU results, different steady-state mass flow rates and heat transfer coefficients govern the NTU values. Thus, the ε value changes due to the variation of NTU. This concept is extended to a "mass flow rate variation based" transient effectiveness. Due to the mass flow rate variation, the heat transfer coefficient changes are denoted by the NTU<sup>0</sup> value in Eq. (19). In addition, mass flow rate variations lead to changes in

presented in reference [22]. Then the "mass flow rate based transient effectiveness (ε<sup>m</sup>

concept is defined by extending the theoretical steady-state correlation of ε and NTU to the transient case. The theoretical steady-state correlations are shown in Eqs. (21) and (23) for a cross-flow heat exchanger and for a counter-flow heat exchanger, respectively. By integrating the NTU<sup>0</sup> and E<sup>0</sup> equations (Eqs. (19) and (20)), the mass flow rate variation-based transient effectiveness can be expressed as Eqs. (22) and (24). They are designated as the ε-NTU transient theoretical correlations. The transient theoretical correlations are used to calculate the corresponding mass flow rate based transient effectiveness under the corresponding mass flow

> r β <sup>c</sup> þ R � r β h

<sup>E</sup> <sup>½</sup> exp ð�<sup>E</sup> � NTU<sup>0</sup>:<sup>78</sup>Þ � <sup>1</sup>� ( )

mCmin � ðTa,in � Tw,inÞ ¼ mc\_ <sup>p</sup>ðTa, out � Ta,inÞþðpVcpÞheat exchanger

), as in Eq. (20). The detailed mathematical procedure is

<sup>m</sup> is defined as a

(18)

0 )"

(21)

� NTU (19)

E (20)

and the cold fluid inlet temperatures are considered as constant. In Eq. (18), ε<sup>0</sup>

∂Theat exchanger ∂t

$$\varepsilon = \frac{1 - \exp\left[-NTU \cdot (1 - E)\right]}{1 - E' \cdot \exp\left[-NTU \cdot (1 - E)\right]} \tag{23}$$

$$\epsilon'\_m = \frac{1 - \exp\left[-NTU' \cdot (1 - E')\right]}{1 - E' \cdot \exp\left[-NTU' \cdot (1 - E')\right]} \tag{24}$$

Two methodologies have been developed based on the transient effectiveness methodology. The first transient effectiveness is the temperature-based transient effectiveness, or full transient effectiveness. The second transient effectiveness is denoted as the mass flow rate based transient effectiveness method, or partial transient effectiveness. The major difference between the two transient effectiveness models is that the partial transient effectiveness only considers the impact of the variations in the fluid mass flow rate and the corresponding heat transfer coefficient, and thermal capacitance effects are dismissed.

#### 4.2.4.2. Verification with numerical solution of thermal dynamic model

An example is selected here to perform the CFD compact model verification in modeling fluid mass flow rate changes. A set of numerical solutions for the thermal dynamic models are used. Two variation cases are considered: they are a ramp increase in the cold fluid mass flow rate, and a ramp increase in the hot fluid mass flow rate. To show the difference between the two modeling methodologies, both the full transient effectiveness and partial transient effectiveness are presented together. This difference can be seen clearly in Figure 14, between the two effectiveness models which are calculated using Eqs. (1a) and (22) for the same variation case. It is found that the hot fluid mass flow rate variation leads a larger difference between the two final steady states, which is not seen for the cold fluid mass flow rate variation case. One possible reason is that the hot fluid is modeled as the Cmin fluid. Therefore, Eq. (19) is used to calculate NTU<sup>0</sup> . Based on Eq. (19), rh may result in a larger impact on the NTU<sup>0</sup> value than on rc. The temperature results are plotted in Figure 15, and the hot fluid outlet temperatures are used to compare with the previously verified numerical solutions. It can be seen that the compact modeling results are in good agreement with the numerical solutions.

#### 4.2.4.3. Validation with transient experimental data

Several experimental data presented in reference [29] are used to validate the modeling methodology. It needs to be mentioned that the data shown in reference [29] is for a counter-flow heat exchanger. By considering the CFD model as a black box, the counter-flow heat exchanger is modeled using the same model as the cross-flow heat exchanger, with proper modification for the model dimensions. In terms of calculating the partial transient effectiveness, Eq. (24) is used.

Figure 14. Transient effectiveness of the hot fluid under mass flow rate ramp change.

Figure 15. Outlet temperature of the hot fluid.

The mass flow rate variation magnitude was considered as rh = 1.56/0.45 and applied to the hot fluid mass flow rate. The analytical solution presented in reference [29] is also plotted in the same figure. Therefore, Figure 16 shows the experimental data, the CFD modeling results, and the analytical results. In addition, the effect of the lumped specific heat used in the current compact model is studied. Eq. (25) represents the nondimensional Peclet number. This number is used to represent the ratio of the thermal energy transported to the other fluid through convection to the energy conducted within the fluid. A small PeL value represents a stronger conduction effect. A large PeL value indicates that the impact of axial conductance is minimal. In the current CFD model, when using a relatively large heat exchanger specific heat, the axial dispersion effect can be reduced significantly. Therefore, the set point of the specific heat value has a major impact on the conductance. It can be seen in Figure 16 that the solutions are in good agreement. When the axial dispersion impact is considered in the CFD model, the corresponding results are in good agreement with the experimental data. When the axial dispersion impact is neglected in the CFD model, the corresponding results are in good agreement with the analytical solution. The impact of axial dispersion can be seen clearly in delaying the transient response.

$$Pe\_L = \frac{\text{UL}}{k/\rho \text{C}\_p} = \frac{\text{UL} \cdot \dot{m} c\_p}{k} \tag{25}$$

Figure 16. Fluid outlet temperature under mass flow rate step change, case 1.

#### 4.3. Summary

In this section, the transient effectiveness concept is used to develop heat exchanger modeling methodologies. Detailed development procedures are provided. The first method is to extend the steady-state effectiveness concept to a transient concept, and the calculation of this transient effectiveness is based on the actual temperature results. This method can be used to integrate the numerical and analytical solutions and experimental data into the CFD model. The second modeling method is to extend the steady-state theoretical correlation ε-NTU to a transient correlation. This method is then used for developing CFD compact transient heat exchanger models for modeling the scenario that fluid mass flow rates change. This section provides a comprehensive summarization of the compact modeling methodology validation. Experimental data, analytical solutions, and numerical solutions are used to compare with the compact modeling results. The results show that the transient effectiveness-based CFD compact models are in good agreement with the experimental data and analytical solutions for different variation scenarios, including fluid inlet temperature changes, fluid mass flow rate changes, and combinations of multiple variations cases.

### 5. Conclusion

The mass flow rate variation magnitude was considered as rh = 1.56/0.45 and applied to the hot fluid mass flow rate. The analytical solution presented in reference [29] is also plotted in the same figure. Therefore, Figure 16 shows the experimental data, the CFD modeling results, and the analytical results. In addition, the effect of the lumped specific heat used in the current compact model is studied. Eq. (25) represents the nondimensional Peclet number. This number is used to represent the ratio of the thermal energy transported to the other fluid through convection to the energy conducted within the fluid. A small PeL value represents a stronger conduction effect. A large PeL value indicates that the impact of axial conductance is minimal. In the current CFD model, when using a relatively large heat exchanger specific heat, the axial dispersion effect can be reduced significantly. Therefore, the set point of the specific heat value has a major impact on the conductance. It can be seen in Figure 16 that the solutions are in good agreement. When the axial dispersion impact is considered in the CFD model, the corresponding results are in good

Figure 14. Transient effectiveness of the hot fluid under mass flow rate ramp change.

Figure 15. Outlet temperature of the hot fluid.

214 Heat Exchangers– Design, Experiment and Simulation

The aim of this chapter is to provide a comprehensive review of the transient effectiveness methodology for heat exchanger analysis. This chapter provides a thorough connection of all the transient effectiveness-related knowledge/work. Novel transient effectiveness methodologies for studying heat exchanger transient characterization are introduced, and a detailed analytical, numerical, and experimental study of these models is presented. Mathematical models, analytical and numerical analysis, experimental testing, and validating studies provide a better understanding of the transient effectiveness methodology. It is shown that the transient effectiveness methodology is very useful for thermal dynamic characterization of heat exchangers and the development of compact/CFD transient models. In addition, it is found that methodology is also useful for analyzing cooling system transient experimental results.

The transient effectiveness curves represent both the heat exchanger dynamic behavior and the corresponding boundary conditions on a single curve. It depicts the heat exchanger transient response in a more comprehensive manner, when compared with outlet temperature curves.

The transient effectiveness methodology is shown to be useful for characterizing the thermal capacitance effects of the entire system, as well as each component, during transient events. The transient effectiveness curves clearly capture the transient response and the impact of thermal capacitance on each heat exchanger unit.

Two CFD compact modeling methodologies are developed and validated, namely a full transient effectiveness methodology and a partial transient effectiveness methodology. These two compact models are accurate and fast, and can be integrated into large scale models, such as system/building level models.
