**3. Computational analyses**

#### **3.1 Mathematical model of the heat exchanger**

The mathematical model of the considered heat exchanger has been worked out taking into account the following simplifying assumptions (only most important):


It has been also assumed that the air inflow is non-uniform and the water inflow may be non-uniform. An influence of temperature on thermal properties of the agents has been taken into account too.

Fig. 8. Model heat exchanger and the recurrent fragment.

HE-1/1 1.556 4.5·10-4 50 49.8 43.9 11.03 HE-1/2 1.556 4.5·10-4 70 68.9 56.7 22.61 HE-1/3 1.556 4.5·10-4 90 86.4 67.9 34.08 HE-1/4 1.083 4.5·10-4 50 49.7 44.5 9.72 HE-1/5 1.083 4.5·10-4 70 69.2 58.7 19.42 HE-1/6 1.083 4.5·10-4 90 88.0 72.2 29.11 HE-2/1 2.04 4.5·10-4 50 48.2 42.8 10.07 HE-2/2 2.04 4.5·10-4 70 69.6 62.0 14.08 HE-2/3 2.04 4.5·10-4 90 90.2 79.5 19.58 HE-2/4 1.063 4.5·10-4 50 48.0 45.6 4.48 HE-2/5 1.074 4.5·10-4 70 68.5 62.0 12.04 HE-2/6 2.033 4.5·10-4 90 89.8 79.0 19.76 HE-3/1 1.876 4.5·10-4 50 49.3 42.7 12.39 HE-3/2 1.876 4.5·10-4 70 69.1 59.8 17.31 HE-3/3 1.877 4.5·10-4 90 87.8 74.6 24.08 HE-3/4 1.052 4.5·10-4 50 50.1 47.1 5.51 HE-3/5 1.052 4.5·10-4 70 69.6 61.6 14.81 HE-3/6 1.877 4.5·10-4 90 88.7 75.4 24.30

The results of the measurements for the three considered heat exchangers are summarized in Table 1. All the measurements have been repeated for three times in order to verify repeatability of results. Presented in the last column heat flow rates, of course, refer to the conditions of non-uniform air flow. In order to determine the impact of this inequality on the efficiency of considered heat exchangers in the next stage the computational analysis was carried out. The measured inlet media parameters were used as input for calculations.

The mathematical model of the considered heat exchanger has been worked out taking into

It has been also assumed that the air inflow is non-uniform and the water inflow may be non-uniform. An influence of temperature on thermal properties of the agents has been

account the following simplifying assumptions (only most important):

real rib is replaced with a round or a plate-elliptic rib of the same surface.

tB(1) ºC

tw,in ºC

tw,out ºC

*Qw* kW

*Vw* m3/s

Measurement

(1): the temperature set at the electric boiler outlet

Table 1. Results of measurements.

**3. Computational analyses** 

 steady state conditions, one-dimensional media flow, radiation is neglected, heat losses are neglected,

taken into account too.

heat flow is normal to a boundary,

**3.1 Mathematical model of the heat exchanger** 

*Va* m3/s

No.

The analysed real cross-flow heat exchanger has been replaced with a model rectangular heat exchanger. The model was then divided into elementary fragments (Fig. 8). Each fragment represents a recurrent element of the real heat exchanger - a single tube with the rib (Piątek, 2003).

The energy balance equations for each fragment constitute the mathematical basis of the model. Assuming that the water flows along the X axis and the air flows along the Y axis the energy balance for a recurrent fragment may be written as follow:

$$d\boldsymbol{d\hat{Q}} = -\dot{m}\_w c\_{pw} \frac{\partial T\_w}{\partial \mathbf{x}} \, d\boldsymbol{y}dz = \dot{m}\_a c\_{pa} \frac{\partial T\_a}{\partial \mathbf{y}} \, d\mathbf{x}dz = h\_a \left(T\_m - T\_a\right) dA \tag{3}$$

where ha is an average heat transfer coefficient on the gas side for all the ribbed surface and Tm is the average temperature of rib and pipe surface.

The inlet temperatures of the mediums are known so the following boundary conditions may be used:

$$T\_w(0, y, z) = T\_{w,in} \quad T\_a(\ge, 0, z) = T\_{a,in} \tag{4}$$

The mass flow rates of the fluids are described by the following formulas:

$$d\dot{m}\_w = \frac{g\_w \cdot \dot{m}\_w}{Y\_{\text{max}} Z\_{\text{max}}} dy dz \tag{5}$$

$$d\dot{m}\_a = \frac{\mathcal{g}\_a \cdot \dot{m}\_a}{X\_{\text{max}} Z\_{\text{max}}} d\mathbf{x} dz \tag{6}$$

The inequality factors gw and ga are defined as follows:

$$\mathcal{g}\_w = \frac{w\_w}{w\_{w,m}}\tag{7}$$

Impact of a Medium Flow Maldistribution on a Cross-Flow Heat Exchanger Performance 129

analytical solution for real cases and extensive measurements are necessary. A statistic analysis of the experimental results allows formulating an empirical correlation. A large number of such relationships have been worked so far. It should be however mentioned here that their application is limited to the heat exchangers of the same or very similar constructions to the experimental units. A review of available correlations allowed choosing those applicable for the heat exchangers under consideration. Six formulas have been

0.418

*m c <sup>D</sup> Nu*

Pr

0.681 0.333 0.134 Re Pr *s s Nu*

0.6 0.333 0

*<sup>A</sup> <sup>A</sup> Nu*

2

,

0.625 0.333 0.3 Re Pr *r p*

The relationships shown above have been used to calculate the heat transfer coefficient for the air velocity ranging from 2 to 20 m/s and for the air temperatures equal to 10ºC, 20ºC or

Figure 9 illustrates how big the discrepancy of the heat transfer coefficient is obtained depending on the choice of Nusselt number relationship. The use of different empirical correlations does not lead to conclusive results, but difficult to find criteria for selecting the

30ºC. The range of the air parameters has been established based on the experiments.

0.26 Re Pr

*tp*

*A s ds s d l*

*Nu*

*e tp tp A s s*

0.011 Re

2/3

,max

*l*

*a p h*

*a*

0.633 *Nu* 0.3375 Re (11)

(10)

*k*

0.2 0.1131

(12)

1/2 1/3 *Nu* 1.0 Re Pr (13)

<sup>0</sup> 2 1

*A lld A ds*

 

0.375

(14)

(15)

0.6 0.15

0

*e G*

*A A*

*G*

*p*

*A*

*A*

investigated (Kays and London, 1998; Welty et al., 2008):

Kays and London correlation:

Brigs and Young correlation:

Norris and Spofford correlation:

0

Berman correlation:

Paikert correlation:

Schmidt correlation:

where

$$\mathbf{g}\_a = \frac{w\_a}{w\_{a,m}} \tag{8}$$

The subscript m in relationships (7) and (8) means the average velocity of the medium. Information about the non-uniform flow of the air is put into the model on the basis of measurements. A non-uniform water inlet to the exchanger may be set arbitral by a function or on the basis of numerical simulations (Bury et al., 2007a).

The control volume method based model of heat transfer for the recurrent fragment of the heat exchanger has been worked out to calculate the average temperature of the ribs and tube outer surface. The detailed description of the model and equations can be found in (Piątek, 2003).

The parameters calculated with the model of the recurrent fragment are: outlet and average temperature of the water flowing in the pipe, average temperature of the air, average temperature of the rib and the pipe surface, average values of the heat transfer coefficients at the gas side and the heat flux transported in the recurrent fragment. The heat transfer coefficient from the hot water to the pipe has been computed from Colburn's formula (Welty et al., 2008):

$$Nu = 0.023 \cdot \text{Re}^{0.8} \cdot \text{Pr}^{1/3} \tag{9}$$

The heat transfer coefficient on the gas side may be determined on the way of the numerical simulations for a numerical model of the recurrent fragment of the considered heat exchanger (see subsection 3.2.2 and Bury and Składzień, 2006) or may be computed from one of available Nusselt number correlations.

The calculation procedure for the whole exchanger model is iterative and it is repeated for all the recurrent fragments of the considered heat exchanger. First, the air temperature increase in the analysed fragment is assumed. Next, the heat transfer coefficients for the water and the gas sides are calculated as well as the rib and pipe surface average temperature. The heat flux transported in the recurrent fragment is then computed and the accuracy criterion is checked. If the criterion is satisfied the procedure is realized for the next fragment. If the criterion is not fulfilled the described procedure is then repeated for the given recurrent fragment till the demanded accuracy is achieved.

The validation procedure was performed by means of comparison of the experimental and numerical results. The total heat flux transported in the heat exchanger is the main compared parameter and it is the basis for evaluation of the code. Significant differences have been recorded between experimental and numerical data after the initial validation of the model (Bury et al., 2008a). Minor changes have been put into the code and the validation procedure was then repeated with usage of the infra-red thermography measurements results also. The last stage of the research was the sensitivity analysis (Bury et al., 2008b). This analysis has shown that the heat transfer coefficient from ribbed surfaces to the gas may be the reason for recorded discrepancies between numerical and experimental results.

#### **3.2 Heat transfer coefficient on the gas side**

#### **3.2.1 Application of Nusselt number correlations**

A traditional analysis of the convective heat transfer for simple cases is based on the similarity theory and application of the dimension analysis. It is very difficult to find an analytical solution for real cases and extensive measurements are necessary. A statistic analysis of the experimental results allows formulating an empirical correlation. A large number of such relationships have been worked so far. It should be however mentioned here that their application is limited to the heat exchangers of the same or very similar constructions to the experimental units. A review of available correlations allowed choosing those applicable for the heat exchangers under consideration. Six formulas have been investigated (Kays and London, 1998; Welty et al., 2008):

Kays and London correlation:

$$Nu = \frac{0.011 \cdot \text{Re}^{-0.418} \cdot \dot{m}\_{a,\text{max}} \cdot c\_p}{\text{Pr}^{2/3}} \left(\frac{D\_h}{k\_a}\right) \tag{10}$$

Berman correlation:

128 Heat Exchangers – Basics Design Applications

The subscript m in relationships (7) and (8) means the average velocity of the medium. Information about the non-uniform flow of the air is put into the model on the basis of measurements. A non-uniform water inlet to the exchanger may be set arbitral by a function

The control volume method based model of heat transfer for the recurrent fragment of the heat exchanger has been worked out to calculate the average temperature of the ribs and tube outer surface. The detailed description of the model and equations can be found in (Piątek, 2003). The parameters calculated with the model of the recurrent fragment are: outlet and average temperature of the water flowing in the pipe, average temperature of the air, average temperature of the rib and the pipe surface, average values of the heat transfer coefficients at the gas side and the heat flux transported in the recurrent fragment. The heat transfer coefficient from the hot water to the pipe has been computed from Colburn's formula (Welty

The heat transfer coefficient on the gas side may be determined on the way of the numerical simulations for a numerical model of the recurrent fragment of the considered heat exchanger (see subsection 3.2.2 and Bury and Składzień, 2006) or may be computed from

The calculation procedure for the whole exchanger model is iterative and it is repeated for all the recurrent fragments of the considered heat exchanger. First, the air temperature increase in the analysed fragment is assumed. Next, the heat transfer coefficients for the water and the gas sides are calculated as well as the rib and pipe surface average temperature. The heat flux transported in the recurrent fragment is then computed and the accuracy criterion is checked. If the criterion is satisfied the procedure is realized for the next fragment. If the criterion is not fulfilled the described procedure is then repeated for the

The validation procedure was performed by means of comparison of the experimental and numerical results. The total heat flux transported in the heat exchanger is the main compared parameter and it is the basis for evaluation of the code. Significant differences have been recorded between experimental and numerical data after the initial validation of the model (Bury et al., 2008a). Minor changes have been put into the code and the validation procedure was then repeated with usage of the infra-red thermography measurements results also. The last stage of the research was the sensitivity analysis (Bury et al., 2008b). This analysis has shown that the heat transfer coefficient from ribbed surfaces to the gas may be the reason for recorded discrepancies between numerical and experimental results.

A traditional analysis of the convective heat transfer for simple cases is based on the similarity theory and application of the dimension analysis. It is very difficult to find an

*g*

or on the basis of numerical simulations (Bury et al., 2007a).

one of available Nusselt number correlations.

**3.2 Heat transfer coefficient on the gas side** 

**3.2.1 Application of Nusselt number correlations** 

given recurrent fragment till the demanded accuracy is achieved.

et al., 2008):

, *<sup>a</sup> <sup>a</sup> a m w*

*<sup>w</sup>* (8)

0.8 1/3 *Nu* 0.023 Re Pr (9)

$$\mathbf{N}\mu = 0.3375 \cdot \mathbf{Re}^{0.633} \tag{11}$$

Brigs and Young correlation:

$$Nu = 0.134 \cdot \text{Re}^{0.681} \cdot \text{Pr}^{0.333} \left(\frac{s}{l}\right)^{0.2} \left(\frac{s}{\delta}\right)^{0.1131} \tag{12}$$

Norris and Spofford correlation:

$$\mathbf{N}\mu = \mathbf{1}.0 \cdot \mathbf{Re}^{1/2} \cdot \mathbf{Pr}^{1/3} \tag{13}$$

Paikert correlation:

$$Nu = 0.26 \cdot \text{Re}^{0.6} \cdot \text{Pr}^{0.333} \left(\frac{A\_0}{A\_\varepsilon}\right)^{0.6} \left(\frac{A}{A\_{G0}}\right)^{-0.15} \tag{14}$$

where

$$\left(\frac{A\_0}{A\_c}\right) = \frac{s\_{tp}\left(s+\delta\right)}{\left(s\_{tp}-d\right)\cdot s + \left(s\_{tp}-d-2\cdot l\right)\cdot \delta}, \left(\frac{A}{A\_{G0}}\right) = 1 + \frac{2\cdot l\cdot \left(l+d+\delta\right)}{d\cdot \left(s+\delta\right)}$$

Schmidt correlation:

$$Nu = 0.3 \cdot \text{Re}^{0.625} \cdot \text{Pr}^{0.333} \left(\frac{A\_{r+p}}{A\_p}\right)^{-0.375} \tag{15}$$

The relationships shown above have been used to calculate the heat transfer coefficient for the air velocity ranging from 2 to 20 m/s and for the air temperatures equal to 10ºC, 20ºC or 30ºC. The range of the air parameters has been established based on the experiments.

Figure 9 illustrates how big the discrepancy of the heat transfer coefficient is obtained depending on the choice of Nusselt number relationship. The use of different empirical correlations does not lead to conclusive results, but difficult to find criteria for selecting the

Impact of a Medium Flow Maldistribution on a Cross-Flow Heat Exchanger Performance 131

model. These computations have been realized for the air inlet temperatures of 10ºC, 20ºC or 30ºC, and the velocity ranging from 2 m/s to 20 m/s. The water temperature has been assumed equal to 90ºC, and the heat transfer coefficient inside the pipes has been calculated

Fig. 10. The recurrent element (left) and the recurrent fragment (right) of the heat exchanger

The averaged value of the heat transfer coefficient at the air side has been calculated based on the known fields of temperature for the rib surface and the pipe surface as well as the average temperature of the air and the transferred heat flux – see (Bury and Składzień, 2006) for details. The results for the HE-1 exchanger obtained by using the recurrent element

Fig. 11. Heat transfer coefficient versus the air inlet velocity – HE-1 exchanger, recurrent

from the Colburn relationship.

model are presented in Fig. 11.

element model.

HE-1.

correct equation for the present case (range of Reynolds numbers and the equivalent diameter of the pipes are not sufficient criteria). The Kays and London correlations (presented for the specific geometry of the heat exchanger core) seem to be the most accurately determined according to empirical findings. But it is hard to tell what the impact of differences in the geometric parameters of the heat exchangers cores used in the study is.

Fig. 9. Comparison of results obtained by different Nusselt number correlations for HE-1 heat exchanger.
