**2. Measurement of mixing uniformity in DCHE**

#### **2.1. Experimental**

#### *2.1.1. Apparatus*

The schematic of the experiment employed in the present research is sketched in **Figure 1** [32]. There are two circulation loops in the test device for this experiment. The first loop, which consists of the DCHE (1), electric heater (2), heat transfer fluid (HTF), pump (7) and connecting inlet and outlet pipelines, is a continuous-phase circulation loop for fluid flow, and the other, which consists of the DCHE (1), centrifugal pump (4), plate condenser (5), centrifugal pump (6) and connecting inlet and outlet pipelines, is a dispersed-phase circulation loop for working medium flow. The temperature control device, gear oil pump (3), regulates the initial temperature difference arising from heat exchange. The frequency control cabinet, gas mass flow-meter (8), regulates the rates of flow of the HTF and working medium. The patterns were imaged by a high-speed shutter video camera, which was placed at the second viewing window. In the bubble evaporation process, we could observe the most active stage of the bubbling regime. HTF and the refrigerant R-245fa (1, 1, 1, 3, 3 pentafluoropropane) were used as the continuous phase and the dispersed phase in all runs, respectively.

#### *2.1.2. Experimental design*

The settings of the experimental plan affecting the heat transfer capacity of the tested DCHE are determined through the orthogonal array (OA) experimental design method.

As listed in **Table 1**, design parameters with four factors and three levels were selected to investigate the influence of heat transfer capacity. The L<sup>9</sup> (3<sup>4</sup> ) orthogonal array table was chosen for designing the experiment. The interaction between the design parameters was neglected in the present study. *H* is the height of HTF in the DCHE, *ΔT* is the initial heat transfer temperature difference, *Ug* is the refrigerant flow rate, and *U*<sup>0</sup> is the flow rate of the HTF.

**Figure 1.** Experimental equipment for direct-contact heat transfer.


**Table 1.** Design parameters and levels.

with the *L*<sup>2</sup>

**1.10. Chapter structure**

150 Heat Exchangers– Design, Experiment and Simulation

**2.1. Experimental**

*2.1.1. Apparatus*

*2.1.2. Experimental design*

fer temperature difference, *Ug*

norm, namely, the centred discrepancy (CD) and the wrap-around discrepancy

(WD). The centred discrepancy (CD) and the wrap-around discrepancy (WD) satisfy a Koksma-Hlawka type inequality according to Xu et al. [39]. According to the theory of UC-LD, if the image shows a low CD or WD that can be called homogeneous mixing of a set or bubble swarm pattern, and the use of UC-CD and UC-WD provides at least likely to get a good approximation of mixing bubble swarm of spatial distribution. Additionally, UC-CD and UC-WD exhibit some advantages including rotation invariance, reflection invariance and projection uniformity [39].

The chapter is organized as follows. In the next section, experiments and methodology are presented; the results and discussion are presented subsequently; the conclusion is briefly summarized in this section finally. Then the acknowledgements and references are presented in the end.

The schematic of the experiment employed in the present research is sketched in **Figure 1** [32]. There are two circulation loops in the test device for this experiment. The first loop, which consists of the DCHE (1), electric heater (2), heat transfer fluid (HTF), pump (7) and connecting inlet and outlet pipelines, is a continuous-phase circulation loop for fluid flow, and the other, which consists of the DCHE (1), centrifugal pump (4), plate condenser (5), centrifugal pump (6) and connecting inlet and outlet pipelines, is a dispersed-phase circulation loop for working medium flow. The temperature control device, gear oil pump (3), regulates the initial temperature difference arising from heat exchange. The frequency control cabinet, gas mass flow-meter (8), regulates the rates of flow of the HTF and working medium. The patterns were imaged by a high-speed shutter video camera, which was placed at the second viewing window. In the bubble evaporation process, we could observe the most active stage of the bubbling regime. HTF and the refrigerant R-245fa (1, 1, 1, 3, 3 pentafluoropropane) were used as the continuous phase and the dispersed phase in all runs, respectively.

The settings of the experimental plan affecting the heat transfer capacity of the tested DCHE

As listed in **Table 1**, design parameters with four factors and three levels were selected to

chosen for designing the experiment. The interaction between the design parameters was neglected in the present study. *H* is the height of HTF in the DCHE, *ΔT* is the initial heat trans-

is the refrigerant flow rate, and *U*<sup>0</sup>

(3<sup>4</sup>

) orthogonal array table was

is the flow rate of the HTF.

are determined through the orthogonal array (OA) experimental design method.

investigate the influence of heat transfer capacity. The L<sup>9</sup>

**2. Measurement of mixing uniformity in DCHE**

As shown in **Table 2**, the numbers E<sup>1</sup> –E<sup>9</sup> denote different experimental levels according to the orthogonal array table.

#### **2.2. Pattern acquisition and processing**

A high-speech video camera was employed to obtain the patterns, and the brand used was PRAKTICA from Germany, with resolution 4 million pixels with no LED light. The images,


**Table 2.** Design experiments according to four factors and three levels orthogonal table.

which were blurred in photographing, can be improved using some image processing techniques. It takes 8 minutes to shoot in each occasion of the orthogonal experiment. Because of difficulties in storing and calculating these images, we choose equal interval sampling from 6000 images, in total, 12,000 images are collected.

**Figure 2** is randomly obtained in the present image-processing process. In order to suppress the background of the original image, eliminate noise and enhance the image, grayscale transformation, top-hat transform is used here. The binaruzation operation was used to calculate the Betti numbers. With a dilation process, an erosion process named as an opening was executed. This process, aiming at removing tiny or isolate points at the finer locations, and smoothing the boundaries of larger points, could not change the size of the image significantly. In contrast, with a dilation erosion process, a dilation process named as an opening was executed. This operation, aiming at filling up tiny pores within the points, connecting nearby points, and smoothing the borders, could not alter the size of the image significantly. The opening is used here to remove small holes representing sile bubbles or small bubble swarms of the binarization images. Since the behavioural characteristics of bubble swarms could not be accurately portrayed by binarized images with noise, an opening operation must be executed to eliminate image noise by the appropriate thresholds selected.

Thus, the white area indicates the bubble swarm, and the black area refers to the continuous phase. As the experimental conditions, the captured image is relatively fuzzy; however, its quality can be improved by using the digital image processing techniques. The resultant image that could be used for the following analysis was identifiable.

#### **2.3. Methodology**

which were blurred in photographing, can be improved using some image processing techniques. It takes 8 minutes to shoot in each occasion of the orthogonal experiment. Because of difficulties in storing and calculating these images, we choose equal interval sampling from

**Figure 2** is randomly obtained in the present image-processing process. In order to suppress the background of the original image, eliminate noise and enhance the image, grayscale transformation, top-hat transform is used here. The binaruzation operation was used to calculate the Betti numbers. With a dilation process, an erosion process named as an opening was executed. This process, aiming at removing tiny or isolate points at the finer locations, and smoothing the boundaries of larger points, could not change the size of the image significantly. In contrast, with a dilation erosion process, a dilation process named as an opening was executed. This operation, aiming at filling up tiny pores within the points, connecting nearby points, and smoothing the borders, could not alter the size of the image significantly. The opening is used here to remove small holes representing sile bubbles or small bubble swarms of the binarization images. Since the behavioural characteristics of bubble swarms could not be accurately portrayed by binarized images with noise, an opening operation must be executed to eliminate image noise by the appropriate

Thus, the white area indicates the bubble swarm, and the black area refers to the continuous phase. As the experimental conditions, the captured image is relatively fuzzy; however, its quality can be improved by using the digital image processing techniques. The resultant

image that could be used for the following analysis was identifiable.

**Figure 2.** Treatment for one piece of bubble swarm patterns.

6000 images, in total, 12,000 images are collected.

152 Heat Exchangers– Design, Experiment and Simulation

thresholds selected.

#### *2.3.1. Performance evaluation of the DCHE model*

Owing to the complexity of the DCHE multiphase structure, heat exchange performance has often been expressed in terms of the volumetric heat transfer coefficient, *hV* , which is given by [43]:

$$h\_V = \frac{Q}{V \times LMTD} \tag{1}$$

where *V* is the volume of the continuous phase in the DCHE, and *Q* is the rate of heat transfer from the continuous phase to the dispersed phase, given by:

$$Q = \left. m \left( h\_{\rm do} - h\_{\rm do} \right) \right. \tag{2}$$

where *m* is mass flow-rate of the dispersed phase steam, and *h* is the enthalpy of the dispersed phase. The *LMTD* in Eq. (1) is the logarithmic mean temperature difference, which is defined as:

phase. The LMTD in Eq. (1) is the logarithmic mean temperature difference, which is defined as: 
$$\text{LMTD} = \frac{(T\_{\text{ai}} - T\_{\text{do}}) - (T\_{\text{co}} - T\_{\text{d}})}{\ln\frac{(T\_{\text{ai}} - T\_{\text{ao}})}{(T\_{\text{oo}} - T\_{\text{d}})}} \tag{3}$$

where *T* is temperature. In all the equations, the subscript *c* refers to the continuous phase, *d* refers to the dispersed phase, *i* refers to the inlet, and *o* refers to the outlet.

#### *2.3.2. Computational homology (Betti numbers)*

Box-counting with erosions method, which was developed by Le Coënt et al. [15], can be applied to quantify the mixture homogeneity; however, it is not available for quantifying the mixture non-homogeneity. As shown in the experiment, some agglomerates still exist in the vessel after stirring for quite a long time. With computational homology, an original analysis method aiming at getting the quantification of the mixture homogeneity and non-homogeneity was proposed.

As we all know that the zeroth Betti number and the first Betti number have the following information [18, 44]: *β*<sup>0</sup> equals the number of connected components that make up the space, and *β*<sup>1</sup> provides a measure of the number of tunnels in the structure. In a two-dimensional domain, tunnels are reduced to loops. Since an image is three-dimensional, it has three Betti numbers: *β*<sup>0</sup> , *β*<sup>1</sup> , and *β*<sup>2</sup> . *β*<sup>2</sup> measures the number of completely enclosed cavities, such as the interior of a sphere. *β*<sup>0</sup> indicates the number of pieces, and *β*<sup>1</sup> represents the number of the holes. In other words, the mixing effect will vary with the number of pieces in the glass vessel. So *β*<sup>0</sup> and *β*<sup>1</sup> are used to get such a characterization of the mixture homogeneity and the mixture non-homogeneity, respectively.

*β*0 , *β*<sup>1</sup> and their averages *β*¯ <sup>0</sup> , *β*¯ 1 of the binary images of the patterns can be obtained at different submerged lengths of the lance and flow rates of the gas. Also, we may obtain the value of time *T* (time unit: seconds) at which *β*<sup>0</sup> of the black/white image is equal to *β*¯ <sup>0</sup> . The time *T* can be employed to obtain the minimum mixing time.

Set

$$\chi^{\circ} \coloneqq \left\{ t \, \middle| \, \beta\_{\circ}(t) > \beta\_{\circ}, t > T \right\} \quad \chi^{\circ} \coloneqq \left\{ t \, \middle| \, \beta\_{\circ}(t) < \beta\_{\circ}, t > T \right\} \quad A \coloneqq \frac{1}{2} \left[ \frac{1}{m} \sum\_{t \in \bar{\chi}^{\circ}} \beta\_{\circ}(t) - \frac{1}{n} \sum\_{t \in \bar{\chi}^{\circ}} \beta\_{\circ}(t) \right] \tag{4}$$

where *β*<sup>0</sup> (*t*) denotes the zeroth Betti number of the binary image of the pattern, which is captured at the time *t*, and *m*, *n* are the numbers of elements in *χ*<sup>+</sup> , *χ*<sup>−</sup> . *A* is used to estimate the deviation amplitude of *β*<sup>0</sup> (t) from their average *β*¯ <sup>0</sup> .

In two-dimensional cases, *β*<sup>0</sup> is the number of connected components, such as black regions. The number of these holes, which is completely enclosed by cubes/pixels, is measured by *β*<sup>1</sup> , and *β*<sup>1</sup> represents the number of the holes in the domain. One can easily count these white regions. As shown in **Figure 2**, *β*<sup>0</sup> represents the number of continuous phases, whereas *β*<sup>1</sup> represents the number of bubble swarms.

The calculation of Betti number is difficult, and the methods are only in their early stages [44]. The free software package CHomP was used to calculate Betti numbers [44, 45]. We could compute *β*<sup>0</sup> and *β*<sup>1</sup> of the open operation images of the patterns at different experimental levels using the CHomP software package [45].

Subsequently, we obtained the value of time *t* (seconds) that can be used to estimate the pseudo-homogeneous time with *β*<sup>1</sup> representing the average of *β*<sup>1</sup> of the open operation image after the pseudo-homogeneous process. The bubble sizes were found to be almost the same by inspecting many test images. With the pseudo-homogeneous time, the entire visible area was covered by the bubbles [36]. Just as we all known, combining the evolution of Betti numbers, we can distinguish whether the distribution is uniform or not. In the beginning, the Betti number increases and then rapidly stabilizes after fluctuations.

As **Figure 3** shows, a conversion operation of open operation images was performed. The results showed that a black-and-white conversion directly leads to a switch between the corresponding objects of the zeroth and first Betti numbers [46]. To illustrate, *β*<sup>0</sup> and *β*<sup>1</sup> represent the number of the continuous phase in Figure 3a, and the opposite in Figure 3b, respectively. Since it is the white pores that most directly reflect the flow patterns of the bubble swarms, *β*<sup>1</sup> is still used to characterize the number of bubble swarms.

#### *2.3.3. Three-sigma method*

Let *X* be a normally *N*(*μ*, *σ*<sup>2</sup> ) distributed random variable. For any *k* > 0, *P*{|*X*- *μ*|< *kσ*} = 2Φ{*k*}−1, where Φ{·} is the distribution function of the standard normal law; whence, in particular, for *k* = 3 it follows that *P*{ *μ*−3*σ* < *X* < *μ*+3*σ*} = 0.9973. The latter equation means that *X* can differ from its expectation by a quantity exceeding 3σ on the average in not more than 3 times in a thousand trials [47]. This circumstance is sometimes used by an experimenter in certain problems, by assuming that {|*X*− *μ*|> 3*σ*} is practically impossible, and consequently, {|*X*−*μ*|< 3*σ*} is practically certain. The probability of exceeding the range of "*μ*±3*σ*" occurring twice is 7.29 × 10<sup>6</sup> . Indeed the experimental time series of Betti numbers approximate normal distribution, as shown in **Figure 4** [36].

**Figure 3.** Influence of the boundary on the Betti numbers.

time *T* (time unit: seconds) at which *β*<sup>0</sup>

154 Heat Exchangers– Design, Experiment and Simulation

(*t*) > *β*¯ 0

Set

*χ*<sup>+</sup> := {*t*

where *β*<sup>0</sup>

and *β*<sup>1</sup>

compute *β*<sup>0</sup>


deviation amplitude of *β*<sup>0</sup>

In two-dimensional cases, *β*<sup>0</sup>

and *β*<sup>1</sup>

*2.3.3. Three-sigma method*

Let *X* be a normally *N*(*μ*, *σ*<sup>2</sup>

regions. As shown in **Figure 2**, *β*<sup>0</sup>

represents the number of bubble swarms.

using the CHomP software package [45].

increases and then rapidly stabilizes after fluctuations.

is still used to characterize the number of bubble swarms.

pseudo-homogeneous time with *β*<sup>1</sup>

be employed to obtain the minimum mixing time.

, *t* > *T*} *χ*<sup>−</sup> := {*t*

tured at the time *t*, and *m*, *n* are the numbers of elements in *χ*<sup>+</sup>

(t) from their average *β*¯ <sup>0</sup>


(*t*) < *β*¯ 0

The number of these holes, which is completely enclosed by cubes/pixels, is measured by *β*<sup>1</sup>

The calculation of Betti number is difficult, and the methods are only in their early stages [44]. The free software package CHomP was used to calculate Betti numbers [44, 45]. We could

Subsequently, we obtained the value of time *t* (seconds) that can be used to estimate the

after the pseudo-homogeneous process. The bubble sizes were found to be almost the same by inspecting many test images. With the pseudo-homogeneous time, the entire visible area was covered by the bubbles [36]. Just as we all known, combining the evolution of Betti numbers, we can distinguish whether the distribution is uniform or not. In the beginning, the Betti number

As **Figure 3** shows, a conversion operation of open operation images was performed. The results showed that a black-and-white conversion directly leads to a switch between the cor-

the number of the continuous phase in Figure 3a, and the opposite in Figure 3b, respectively. Since it is the white pores that most directly reflect the flow patterns of the bubble swarms, *β*<sup>1</sup>

2Φ{*k*}−1, where Φ{·} is the distribution function of the standard normal law; whence, in particular, for *k* = 3 it follows that *P*{ *μ*−3*σ* < *X* < *μ*+3*σ*} = 0.9973. The latter equation means that *X* can differ from its expectation by a quantity exceeding 3σ on the average in not more than 3 times in a thousand trials [47]. This circumstance is sometimes used by an experimenter in certain problems, by assuming that {|*X*− *μ*|> 3*σ*} is practically impossible, and consequently, {|*X*−*μ*|< 3*σ*} is practically certain. The probability of exceeding the range of "*μ*±3*σ*" occurring

responding objects of the zeroth and first Betti numbers [46]. To illustrate, *β*<sup>0</sup>

representing the average of *β*<sup>1</sup>

(*t*) denotes the zeroth Betti number of the binary image of the pattern, which is cap-

.

represents the number of the holes in the domain. One can easily count these white

of the open operation images of the patterns at different experimental levels

of the black/white image is equal to *β*¯ <sup>0</sup>

, *<sup>t</sup>* <sup>&</sup>gt; *<sup>T</sup>*} *<sup>A</sup>*:<sup>=</sup> \_\_1

2[ \_1 *m* ∑ *t*∈*χ*<sup>+</sup> *β*0 (*t*) <sup>−</sup> \_1 *<sup>n</sup>* <sup>∑</sup>*<sup>t</sup>*∈*χ*<sup>−</sup> *β*0 (*t*) ] (4)

, *χ*<sup>−</sup>

is the number of connected components, such as black regions.

) distributed random variable. For any *k* > 0, *P*{|*X*- *μ*|< *kσ*} =

represents the number of continuous phases, whereas *β*<sup>1</sup>

. The time *T* can

,

. *A* is used to estimate the

of the open operation image

and *β*<sup>1</sup>

represent

**Figure 4.** Betti number histogram with a normal distribution fit.

Two consecutive points beyond the limits are viewed as exception criteria.

Step 1: Giving a time point *t* 0 > *t*, and *t* is the mixing time, a homogeneous trend was presented during the evolution of Betti number time series of bubble swarm.

Step 2: Calculating the mean *t* and standard deviation *σ* of Betti numbers time series after the time *t* 0 .

Step 3: Determining whether an event exceeds the range of "*μ*±3*σ*" occurring twice, as *t* 0 in reverse order. If so, then the moment is defined as mixing time, *t*. In order to quantify the macro-mixing efficiency using the Betti numbers, the data of the Betti numbers satisfying approximately normal distribution are collected from mixing homogeneity process, then let *μ* represents the estimated mean and *σ* represents the estimated standard deviation, and the mixing time is the time when the critical point exceeds the range of *μ*-3*σ* in reverse order twice.

The technique by itself is not limited to transparent tanks. It can be used in conjunction with electrical resistance tomography (ERT), position emission tomography (PET) and magnetic resonance imaging (MRI) [36].

#### *2.3.4. Measures of uniformity*

A popular figure of merit is the star discrepancy [48] and its generalization the *Lp* -star discrepancy. Let *Fu* (*x*)=*x*<sup>1</sup> *x*2 …*xs* be the uniform distribution function on *Cs* , where *x* = (*x*<sup>1</sup> , *x*<sup>2</sup> , …, *xs* ). Let *FP*(*x*) be the empirical distribution function of *P* = {*x*<sup>1</sup> , *x*<sup>2</sup> , …, *xn* }:

$$F\_p(\mathbf{x}) = \frac{1}{n} \sum\_{i=1}^{n} \mathbf{1}\_{[v\_i \ast \mathbf{s})}(\mathbf{x}) \tag{5}$$

where **1***A*(*x*) is the indicator function. Then the *Lp* -star discrepancy can be defined as the *Lp* norm of difference between uniform and empirical distribution function, and then the *Lp* discrepancy can be defined as:

$$D\_p(P) = \left[ \int\_{\mathbb{C}} \left| F\_u \{ \mathbf{x} \} - F\_p \{ \mathbf{x} \} \right|^p dx \right]^{\frac{1}{p}} \tag{6}$$

By taking *p* = ∞, *L*∞ discrepancy, which defined as the maximum deviation between these two distributions, is called the star discrepancy [48]. It is probably the most commonly used and can be expressed in another way as follows:

$$D\_p^\*(\mathcal{P}) = \sup\_{\mathbf{x} \in \mathbb{C}} \left| F\_u(\mathbf{x}) - F\_p(\mathbf{x}) \right| \tag{7}$$

With the discrepancy criterion in mind, we next discuss how to construct a uniformity coefficient. *x*=(*x*<sup>1</sup> , *x*<sup>2</sup> , …, *xs* )∈*Cs* , [**0**, *x*] = [0, *x*<sup>1</sup> ]×[0, *x*<sup>2</sup> ]×…×[0, *xs* ] is the rectangle determined by the origin O and x decided on *Cs* . *Vol*([**0**, *x*]) denotes the volume of the rectangular solid [**0**, *x*], where *Vol*([**0**, *x*])=*x*<sup>1</sup> *x*2 …*xs* =*Fu* (*x*). Let || be the gained number of points in a group. The function of |P∩ [**0**, *x*]/*n*| represents an empirical distribution, as shown below:

$$F\_p(\mathbf{x}) = \frac{1}{n} \sum\_{l=1}^{n} \mathbf{1}\_{\left[x\_l \approx l\right]}(\mathbf{x}) = \frac{\left|P \cap \left[\mathbf{0}, \mathbf{x}\right]\right|}{n} \tag{8}$$

Definition 2.1. The local discrepancy function is

$$dis\ c^\*(\mathbf{x}) = F\_\mu(\mathbf{x}) - F\_p(\mathbf{x}) = Vol(\mathbf{[0,x]}) - \frac{|P \cap \{0, \mathbf{x}\}|}{\mathbb{N}} \tag{9}$$

The difference between theory and empirical distribution can be used to measure the local discrepancy function with a rectangle [**0**, *x*]. It can be expressed in another way as follows:

$$dis\ c\_{i}(s,t) = \frac{vol\_{i}(s,t)}{vol\_{i}(\cdot,t)} - \frac{hol\_{i}(s,t)}{hol\_{i}(\cdot,t)}\tag{10}$$

where *i* denotes the four corners of an image, *i* = 1, 2, 3, 4, *vol* denotes the volume of the rectangular solid, *hol* is the number of bubbles.

Definition 2.2. The mean absolute discrepancy is often defined as follows:

Step 3: Determining whether an event exceeds the range of "*μ*±3*σ*" occurring twice, as *t*

A popular figure of merit is the star discrepancy [48] and its generalization the *Lp*

(*P*) <sup>=</sup> [ <sup>∫</sup> *Cs* |*Fu*

<sup>∗</sup>(*P*) = sup

be the uniform distribution function on *Cs*

(*x*) = \_\_1 *n* ∑ *i*=1 *n* 1[*xi* ,∞)

norm of difference between uniform and empirical distribution function, and then the *Lp*

**(x)** − *Fp* **(x)** | *p dx*] \_\_1 *p*

By taking *p* = ∞, *L*∞ discrepancy, which defined as the maximum deviation between these two distributions, is called the star discrepancy [48]. It is probably the most commonly used and

*<sup>x</sup>*∈*Cs*|*Fu*

With the discrepancy criterion in mind, we next discuss how to construct a uniformity coef-

The difference between theory and empirical distribution can be used to measure the local discrepancy function with a rectangle [**0**, *x*]. It can be expressed in another way as follows:

]×[0, *x*<sup>2</sup>

**(x)** − *Fp* **(x)**

]×…×[0, *xs*

. *Vol*([**0**, *x*]) denotes the volume of the rectangular solid [**0**, *x*],

(*x*). Let || be the gained number of points in a group. The func-

, *x*<sup>2</sup> , …, *xn* }:

resonance imaging (MRI) [36].

156 Heat Exchangers– Design, Experiment and Simulation

*2.3.4. Measures of uniformity*

(*x*)=*x*<sup>1</sup> *x*2 …*xs*

crepancy can be defined as:

*Dp*

*Dp*

, *x*<sup>2</sup> , …, *xs*

origin O and x decided on *Cs*

*FP*

*dis c*∗(*x*) = *Fu*

can be expressed in another way as follows:

)∈*Cs*

Definition 2.1. The local discrepancy function is

*x*2 …*xs* =*Fu*

, [**0**, *x*] = [0, *x*<sup>1</sup>

tion of |P∩ [**0**, *x*]/*n*| represents an empirical distribution, as shown below:

(*x*) = \_\_1 *n* ∑ *i*=1 *n* 1[*xi* ,∞)

(*x*) − *FP*

*FP*

*FP*(*x*) be the empirical distribution function of *P* = {*x*<sup>1</sup>

where **1***A*(*x*) is the indicator function. Then the *Lp*

ancy. Let *Fu*

ficient. *x*=(*x*<sup>1</sup>

where *Vol*([**0**, *x*])=*x*<sup>1</sup>

reverse order. If so, then the moment is defined as mixing time, *t*. In order to quantify the macro-mixing efficiency using the Betti numbers, the data of the Betti numbers satisfying approximately normal distribution are collected from mixing homogeneity process, then let *μ* represents the estimated mean and *σ* represents the estimated standard deviation, and the mixing time is the time when the critical point exceeds the range of *μ*-3*σ* in reverse order twice. The technique by itself is not limited to transparent tanks. It can be used in conjunction with electrical resistance tomography (ERT), position emission tomography (PET) and magnetic

0 in


). Let


dis-

(6)

, *x*<sup>2</sup> , …, *xs*

, where *x* = (*x*<sup>1</sup>

**(x)** (5)

<sup>|</sup> (7)

] is the rectangle determined by the

(*x*) <sup>=</sup> \_\_\_\_\_\_\_\_\_ <sup>|</sup>*<sup>P</sup>* <sup>∩</sup> **[0, x]**<sup>|</sup> *<sup>n</sup>* (8)

(*x*) <sup>=</sup> *Vol*(**[ 0, x]**) <sup>−</sup> \_\_\_\_\_\_\_\_\_ <sup>|</sup>*<sup>P</sup>* <sup>∩</sup> **[0, x]**<sup>|</sup> *<sup>n</sup>* (9)


$$MAD(\mathbf{s}, t) = \frac{1}{4} \sum\_{i=1}^{4} \left| dis \, c(\mathbf{s}, t) \right| \tag{11}$$

In **Figure 5**, the influence of iteration steps on the measurement is not pronounced. The MAD (mean absolute discrepancy) is conducted by the four corners of an image.

Definition 2.3. Uniformity coefficient (UC) at time *t* is often defined by

$$LIC(t) = 1 - Median\{MAD(s, t)\}\tag{12}$$

$$IL\,\mathbf{C}\_{\mathbf{r}}(t) = 1 - \sqrt{\frac{1}{S}\sum\_{s=1}^{S}MAD(\mathbf{s}, t)}\tag{13}$$

In every case, the degree of mixing uniform could be detected successfully by the uniformity coefficient method (**Figure 6**). After certain processing, the value range of UC is usually [0, 1]. We also denote that the measurement is not pronouncedly affected by the iterative steps.

In **Figure 7**, when the pixels sizes are reduced from 16:9 to 4:3, the influence of homogenization curve by the uniformity coefficient method is not reduced [40]. However, the trend of homogenization curve by Betti numbers method becomes unclear.

**Figure 5.** Effect of initial positions on uniformity coefficient.

**Figure 6. Effect of expressions (left) and iteration steps (right) on uniformity coefficient.**

**Figure 7.** Effect of different pixels (16:9 and 4:3) on uniformity coefficient.

The evolution of the UC and Betti numbers of binary images at different image sizes was clearly shown in **Figure 7**.

Quasi-Monte Carlo method is the most commonly used measure of uniformity in the literature, especially when *p* = ∞ and 2. When *p* = 2, Warnock [49] gave an analytic and simple formula for calculating *L*<sup>2</sup> -star discrepancy as follows:

$$D\_2^\*(P) = \left\{ \left(\frac{1}{3}\right)^s - \frac{1}{n} \sum\_{l=1}^n \prod\_{j=1}^s \frac{1 - \mathbf{x}\_{\vec{\eta}}^2}{2} + \frac{1}{n^2} \sum\_{l \neq 1}^n \prod\_{l=1}^s \left[1 - \max\left(\mathbf{x}\_{l'} \mathbf{x}\_{l\_l}\right)\right] \right\}^{\frac{1}{2}} \tag{14}$$

where *xk* =(*xk*<sup>1</sup> , *xk*<sup>2</sup> , … , *xks*). Unfortunately, the *L*<sup>2</sup> -star discrepancy exhibits some limitations, as pointed out by Heinrich and Hickernell [41, 42]. To overcome these disadvantages, other discrepancies were proposed. From the definition of discrepancy, its formula is calculated as follows if objective function takes the uniform distribution function on *X*:

$$D(P, \mathcal{K}) = \left\{ f\_{\mathcal{X}^\perp} \mathcal{K} \{ \mathbf{x}, y \} \text{d} \, F\_{\boldsymbol{\mu}}(\mathbf{x}) \text{d} \, F\_{\boldsymbol{\mu}}(y) - \frac{2}{n} \sum\_{l=1}^{n} \mathbb{I}\_{\boldsymbol{\lambda}^\perp} \mathcal{K} \{ \mathbf{x}, y \} \text{d} \, F\_{\boldsymbol{\mu}}(y) + \frac{1}{n^2} \sum\_{l=1}^{n} \mathcal{K} \{ \mathbf{x}\_{\boldsymbol{\nu}}, \mathbf{x}\_{\boldsymbol{\lambda}} \} \right\}^{\frac{1}{2}} \tag{15}$$

According to Fang et al. [50], the reproducing kernel functions are taken, respectively, as follows,

$$K^{\epsilon}(z,t) = 2^{-s} \prod\_{\beta=1}^{s} \left( 2 + \left| z\_{\beta} - \frac{1}{2} \right| + \left| t\_{\beta} - \frac{1}{2} \right| - \left| z\_{\beta} - t\_{\beta} \right| \right) \tag{16}$$

$$K^{\omega}(z,t) = \prod\_{\gamma=1}^{s} \left(\frac{3}{2} - \left|z\_{\gamma} - t\_{\gamma}\right| + \left|z\_{\gamma} - t\_{\gamma}\right|^2\right) \tag{17}$$

hence, the analytical expressions for centred discrepancy and wrap-around discrepancy are as follows:

$$\begin{aligned} \text{Min.} \text{ expresonsous on continue unset-punte y una wra-monuan tensor-punte y anu } \\ \text{CD}(\mathbf{f}) &= \left\{ \left( \frac{13}{12} \right)^{\circ} - \frac{2}{n} \sum\_{i=1}^{n} \prod\_{j=1}^{i} \left( 1 + \frac{1}{2} \left| \mathbf{x}\_{ij} - \mathbf{0.5} \right| - \frac{1}{2} \left| \mathbf{x}\_{ij} - \mathbf{0.5} \right|^{2} \right) \right. \\ &\left. + \frac{1}{n^{2}} \sum\_{i=1}^{n} \prod\_{j=1}^{i} \left( 1 + \frac{1}{2} \left| \mathbf{x}\_{ij} - \mathbf{0.5} \right| + \frac{1}{2} \left| \mathbf{x}\_{kj} - \mathbf{0.5} \right| - \frac{1}{2} \left| \mathbf{x}\_{ij} - \mathbf{x}\_{kj} \right| \right) \right\}^{\frac{1}{2}} \end{aligned} \tag{18}$$

$$\text{VD}\mathbf{D}(t) = \left\{ -\left(\frac{4}{3}\right)^s + \frac{1}{n}\left(\frac{3}{2}\right)^s + \frac{2}{n^2} \sum\_{l=1}^{n-1} \sum\_{k=1}^n \prod\_{j=1}^s \left(\frac{3}{2} - \left|\mathbf{x}\_{lj} - \mathbf{x}\_{kj}\right| + \left|\mathbf{x}\_{lj} - \mathbf{x}\_{kj}\right|^2 \right) \right\}^{\frac{1}{2}} \tag{19}$$

UC-CD and UC-WD related to time *t* are defined and denoted by

$$\text{UC}-\text{CD}(t) = 1-\text{CD}(t), \text{UC}-\text{WD}(t) = 1-\text{WD}(t) \tag{20}$$

where CD*(t*) and WD(*t*) denote the modified discrepancy.

Many bubble patterns are related to time *t* and one piece of patterns corresponds to one *t*. For the calculation of UC−CD and UC−WD, using the *x*-axis and *y*-axis values of a Cartesian coordinate system to determine the position of bubble swarm is necessary.

Step 1: Transform image matrix to 0–1 matrix.

The evolution of the UC and Betti numbers of binary images at different image sizes was

Quasi-Monte Carlo method is the most commonly used measure of uniformity in the literature, especially when *p* = ∞ and 2. When *p* = 2, Warnock [49] gave an analytic and simple

as pointed out by Heinrich and Hickernell [41, 42]. To overcome these disadvantages, other discrepancies were proposed. From the definition of discrepancy, its formula is calculated as

[1 − max(*xij*

, *xlj*)]}


\_\_1 2

(14)


**Figure 6. Effect of expressions (left) and iteration steps (right) on uniformity coefficient.**

158 Heat Exchangers– Design, Experiment and Simulation

clearly shown in **Figure 7**.

formula for calculating *L*<sup>2</sup>

<sup>∗</sup>(*P*) <sup>=</sup> {(

\_\_1 3) *s* − \_\_1 *n* ∑ *i*=1 *n* ∏ *j*=1 *<sup>s</sup>* 1 − *xij* 2 \_\_\_\_ <sup>2</sup> <sup>+</sup> \_\_1 *<sup>n</sup>*<sup>2</sup> ∑ *i*,*l*=1 *n* ∏ *j*=1 *s*

**Figure 7.** Effect of different pixels (16:9 and 4:3) on uniformity coefficient.

, … , *xks*). Unfortunately, the *L*<sup>2</sup>

follows if objective function takes the uniform distribution function on *X*:

*D*<sup>2</sup>

=(*xk*<sup>1</sup> , *xk*<sup>2</sup>

where *xk*

Step 2: Search the coordinates of one bubble located in top-left and bottom-right corners.

Step 3: Calculate the mean values of rows and columns of the two above coordinates.

In our work, *M* = 1280 and *N* = 720, then readjusting every piece of bubble patterns to unified pixel size of *M* × *N*. The image pixel matrix is transformed into coordinate within [0,1]

$$\mathbf{x} = \frac{\mathbf{x}\_0 - 1}{M - 1}, \quad \mathbf{y} = \frac{N - y\_0}{N - 1} \tag{21}$$

and where *x*<sup>0</sup> and *y*<sup>0</sup> denote the column and row of one image matrix, *x* and *y* denote the coordinates of horizontal axis and vertical axis in Cartesian coordinate system. Furthermore, in this work, *s* = 2, so (*x*, *y*) of one experimental point is equal to (*xi*<sup>1</sup> , *xi*<sup>2</sup> ) of Eqs. (7) and (8).

The origin of coordinates lies in the bottom-left (BL) corner of one piece of pattern. Certainly, other three groups of transformation ways are used to make origin of coordinates locate in bottom-right (BR), top-right (TR) and top-left (TL) coiners' of one piece of patterns, respectively. Detailed formulas as follows,

$$\mathbf{x} = \frac{M - \mathbf{x}\_0}{M - 1} \quad y = \frac{N - y\_0}{N - 1} \quad \mathbf{x} = \frac{M - \mathbf{x}\_0}{M - 1} \quad y = \frac{y\_0 - 1}{N - 1} \quad \mathbf{x} = \frac{\mathbf{x}\_0 - 1}{M - 1} \quad \mathbf{x} = \frac{y\_0 - 1}{N - 1} \tag{22}$$

More interesting, these transform methods are different but corresponding to the coordinates rotate operation for the rectangular plane coordinate system. Hence, we will talk about the rotational invariance and neglect the different transform methods in next section.

#### **2.4. Mixing quantification by Betti numbers**

#### *2.4.1. Multiphase mixing quantification*

Now this new method is used to study the influence of the flow rate and the submerged length on the degree of the mixing homogeneity and non-homogeneity of solid and liquid. The acquisition system was shown in **Figure 8**. The patterns were gained at the speed of 30 frames per second by a camera taking 10,000 images in each experiment.

**Figure 9** shows that an initial image was subtracted from each image.

#### *2.4.2. Characterization of heat transfer process*

**Figure 10** shows the evolution of *β*<sup>1</sup> in the open operation images produced in representative experimental cases. *β*<sup>1</sup> for each pattern is shown as a solid line, and the horizontal dotted line corresponds to the average of *β*1 [32]. The vertical dotted line corresponds to the pseudo-homogeneous time *t*.

Experiments indicate that, in **Figure 11**, volumetric heat transfer coefficient shows good correlation with average Betti number and pseudo-average-time value [32]. An interesting tendency is found in the better cases of L<sup>6</sup> and L<sup>2</sup> , in which the larger first Betti numbers averages and

**Figure 8.** Scheme of experimental equipment.

**Figure 9.** Binarization for one piece of images.

*<sup>x</sup>* <sup>=</sup> *<sup>M</sup>* <sup>−</sup> *<sup>x</sup>* \_\_\_\_\_0

*<sup>M</sup>* <sup>−</sup> <sup>1</sup> *<sup>y</sup>* <sup>=</sup> *<sup>N</sup>* <sup>−</sup> *<sup>y</sup>* \_\_\_\_\_0

**2.4. Mixing quantification by Betti numbers**

*2.4.2. Characterization of heat transfer process*

**Figure 10** shows the evolution of *β*<sup>1</sup>

tative experimental cases. *β*<sup>1</sup>

pseudo-homogeneous time *t*.

is found in the better cases of L<sup>6</sup>

**Figure 8.** Scheme of experimental equipment.

*2.4.1. Multiphase mixing quantification*

160 Heat Exchangers– Design, Experiment and Simulation

*<sup>N</sup>* <sup>−</sup> <sup>1</sup> *<sup>x</sup>* <sup>=</sup> *<sup>M</sup>* <sup>−</sup> *<sup>x</sup>* \_\_\_\_\_0

30 frames per second by a camera taking 10,000 images in each experiment.

**Figure 9** shows that an initial image was subtracted from each image.

and L<sup>2</sup>

rotational invariance and neglect the different transform methods in next section.

*<sup>M</sup>* <sup>−</sup> <sup>1</sup> *<sup>y</sup>* <sup>=</sup> *<sup>y</sup>*<sup>0</sup> <sup>−</sup> <sup>1</sup> \_\_\_\_

More interesting, these transform methods are different but corresponding to the coordinates rotate operation for the rectangular plane coordinate system. Hence, we will talk about the

Now this new method is used to study the influence of the flow rate and the submerged length on the degree of the mixing homogeneity and non-homogeneity of solid and liquid. The acquisition system was shown in **Figure 8**. The patterns were gained at the speed of

dotted line corresponds to the average of *β*1 [32]. The vertical dotted line corresponds to the

Experiments indicate that, in **Figure 11**, volumetric heat transfer coefficient shows good correlation with average Betti number and pseudo-average-time value [32]. An interesting tendency

*<sup>N</sup>* <sup>−</sup> <sup>1</sup> *<sup>x</sup>* <sup>=</sup> *<sup>x</sup>*<sup>0</sup> <sup>−</sup> <sup>1</sup> \_\_\_\_

in the open operation images produced in represen-

, in which the larger first Betti numbers averages and

for each pattern is shown as a solid line, and the horizontal

*<sup>M</sup>* <sup>−</sup> <sup>1</sup> *<sup>x</sup>* <sup>=</sup> *<sup>y</sup>*<sup>0</sup> <sup>−</sup> <sup>1</sup> \_\_\_\_

*<sup>N</sup>* <sup>−</sup> <sup>1</sup>. (22)

**Figure 10.** The evolution of Betti numbers at E<sup>4</sup> .

**Figure 11.** Fitting of *β<sup>t</sup>* and *h* ¯ *V*.

shorter pseudo-homogeneous times correspond to a higher volumetric heat transfer coefficient *hV* , while L<sup>4</sup> and L<sup>7</sup> are the worse cases. As shown in **Table 3**, *t* and *h*¯ *<sup>V</sup>* are nearly opposite in all cases except L<sup>5</sup> , L<sup>8</sup> and L<sup>9</sup> . Therefore, the parameters *t* and *β*¯ <sup>1</sup> , which can be used to characterize a bubbling flow pattern, are both related to *h*¯ *<sup>V</sup>* .


**Table 3.** The data of the *t*, *β* ¯ 1, *β<sup>t</sup>* , and *h* ¯ *V* for the entire orthogonal array table.

Let *β<sup>t</sup>* <sup>=</sup> ¯*β*<sup>1</sup> <sup>⋅</sup> *<sup>t</sup>* −1 , the tendency of *β<sup>t</sup>* is consistent with that of ¯ *hV* . According to our analysis, a linear relation between *β<sup>t</sup>* and ¯ *hV* seems to be the outcome. Let ¯ *hV* = *a* × *β<sup>t</sup>* + *b*. The least-squares fitting method was used to obtain the parameters *a* and *b* between *β<sup>t</sup>* and ¯ *hV* . In this work, *a* = 0.4241, *b* = 0.4547. The linear relationship is illustrated in **Figure 11**. The correlation coefficient is 0.95. In the end, we constructed a model on the parameters *β<sup>t</sup>* and ¯ *hV* , which points out the relationship between the flow pattern of a bubble swarm and the heat transfer performance of a DCHE.

#### *2.4.3. Accurate estimation of mixing time*

#### (1) Mixing time estimations by different methods

Based on the above, *β<sup>t</sup>* has been defined by the Betti number average as well as the mixing time, which is synergistic with ¯ *hV* . Correlation degree and correlation coefficient are used to investigate about the bubble swam patterns and heat transfer performance for the mixing time evaluation effectiveness. According to reference [36], the computing results show that the synergy by our 3σ method between *β<sup>t</sup>* and ¯ *hV* is much better than the other methods. In **Figure 12**, the plots show the evolution and determination of mixing time measured by different methods [36].

It is found that the correlation coefficient between *t* 3σ and mixing time estimated by these methods is −0.2304 (mean method), 0.9494 (slope method), 0.9265 (SD method) and 0.9731 (3σ method).

In **Table 4**, *μ* is the mean of Betti number time series after the time *t* 0 . *t* mean is the mixing time by mean method. *t* slope is the mixing time by slope method. *tsd* is the mixing time by SD method. *t* 3σ is the mixing time defined by 3σ method. *T*3σ is the inhomogeneous time by 3σ method. *δ<sup>t</sup>* is the difference between *t* 3σ and *T*3σ. *β*mean is calculated by mean method, and the others are defined similarly [40]. It can be quantified by time intervals *δ<sup>t</sup>* between inhomogeneous time and mixing time. From the view point of the time interval, the transitional state appears the following forms: sudden change case (*δ<sup>t</sup>* = 0); interval case (*δ<sup>t</sup>* > 0); overlapping case (*δ<sup>t</sup>* < 0).

#### (2) Simulation experiments

By real data analysis of the Bitti number data, we have compared the proposed method with mean method, slope method and SD method. In order to assess the effectiveness of the new

**Figure 12.** Evolution and determination of mixing time measured by two methods.

shorter pseudo-homogeneous times correspond to a higher volumetric heat transfer coefficient

are nearly opposite in all

, which can be used to characterize

. According to our analysis, a lin-

, which points out the relationship

+ *b*. The least-squares fitting

. In this work, *a* = 0.4241, *b*

mean is the mixing time by

< 0).

between inhomogeneous time

> 0); overlapping case (*δ<sup>t</sup>*

are the worse cases. As shown in **Table 3**, *t* and *h*¯ *<sup>V</sup>*

**Parameter E<sup>1</sup> E<sup>2</sup> E<sup>3</sup> E<sup>4</sup> E<sup>5</sup> E<sup>6</sup> E<sup>7</sup> E<sup>8</sup> E<sup>9</sup>** *T* 159 93 120 226 126 92 264 105 145

*β<sup>t</sup>* 1.26 2 1.23 0.79 1.61 2.25 0.71 1.50 1.52

.

197 186 208 177 197 189 187 176 196

0.96 1.21 0.86 0.83 1.20 1.44 0.75 1.11 1.19

is consistent with that of ¯

, and *h* ¯ *V* for the entire orthogonal array table.

= 0.4547. The linear relationship is illustrated in **Figure 11**. The correlation coefficient is 0.95. In

gate about the bubble swam patterns and heat transfer performance for the mixing time evaluation effectiveness. According to reference [36], the computing results show that the synergy by

show the evolution and determination of mixing time measured by different methods [36].

is −0.2304 (mean method), 0.9494 (slope method), 0.9265 (SD method) and 0.9731 (3σ method).

3σ is the mixing time defined by 3σ method. *T*3σ is the inhomogeneous time by 3σ method. *δ<sup>t</sup>*

and mixing time. From the view point of the time interval, the transitional state appears the

By real data analysis of the Bitti number data, we have compared the proposed method with mean method, slope method and SD method. In order to assess the effectiveness of the new

= 0); interval case (*δ<sup>t</sup>*

seems to be the outcome. Let ¯

between the flow pattern of a bubble swarm and the heat transfer performance of a DCHE.

*hV*

 and ¯ *hV*

has been defined by the Betti number average as well as the mixing time,

slope is the mixing time by slope method. *tsd* is the mixing time by SD method.

3σ and *T*3σ. *β*mean is calculated by mean method, and the others are

. Correlation degree and correlation coefficient are used to investi-

is much better than the other methods. In **Figure 12**, the plots

3σ and mixing time estimated by these methods

0 . *t*

*hV* = *a* × *β<sup>t</sup>*

 and ¯ *hV*

. Therefore, the parameters *t* and *β*¯ <sup>1</sup>

*hV*

*β* ¯ 1

*h* ¯ *V* , while L<sup>4</sup>

Let *β<sup>t</sup>* <sup>=</sup> ¯*β*<sup>1</sup> <sup>⋅</sup> *<sup>t</sup>*

−1

**Table 3.** The data of the *t*, *β* ¯ 1, *β<sup>t</sup>*

ear relation between *β<sup>t</sup>*

Based on the above, *β<sup>t</sup>*

which is synergistic with ¯

our 3σ method between *β<sup>t</sup>*

is the difference between *t*

(2) Simulation experiments

following forms: sudden change case (*δ<sup>t</sup>*

mean method. *t*

*t*

cases except L<sup>5</sup>

and L<sup>7</sup>

, L<sup>8</sup>

162 Heat Exchangers– Design, Experiment and Simulation

and L<sup>9</sup>

a bubbling flow pattern, are both related to *h*¯ *<sup>V</sup>*

, the tendency of *β<sup>t</sup>*

*2.4.3. Accurate estimation of mixing time*

 and ¯ *hV*

the end, we constructed a model on the parameters *β<sup>t</sup>*

(1) Mixing time estimations by different methods

It is found that the correlation coefficient between *t*

*hV*

 and ¯ *hV*

In **Table 4**, *μ* is the mean of Betti number time series after the time *t*

defined similarly [40]. It can be quantified by time intervals *δ<sup>t</sup>*

method was used to obtain the parameters *a* and *b* between *β<sup>t</sup>*

method and provide more evidences of good performance of this method, the mean absolute error (MAE) and the mean square error (MSE) are often used.

$$MAE = \frac{1}{n} \sum\_{i=1}^{n} \left| t\_i - \stackrel{\scriptstyle \Delta}{t}\_i \right| \quad MSE = \frac{1}{n} \sum\_{i=1}^{n} \left( t\_i - \stackrel{\scriptstyle \Delta}{t}\_i \right)^2 \tag{23}$$

where *t i* is the real mixing time, and *t* ^ *i* is the estimate of *ti* .


**Table 4.** Computing results of mixing performance by four methods.

From **Table 5**, we can see that proposed method has a distinct advantage [36]. **Figure 13** shows an example of 1000 simulation results.


**Table 5.** Comparison of computer simulation results by 1000 times.

**Figure 13. One of these simulation results by 1000 times.**

#### **2.5. Measuring bubbles uniformity by discrepancy**

#### *2.5.1. Quantifying mixing efficiency using L<sup>2</sup> -star discrepancy*

#### (1) Quantification of mixing efficiency

The variation of the uniformity coefficient with frames can be an effective method to determine the critical mixing time and mixing uniform.

In **Figure 14**, quantitative comparisons of the homogenization curve and mixing time predicted by the uniformity coefficient method are conducted with reported experimental data and other predictions by the Betti numbers method.

The comparisons show that good agreements of the mixing time obtained by Betti numbers method and uniformity coefficient method have also been achieved as given in **Table 6** [40].

(2) Recognition of local and global uniformity

**Numerical simulations.** Generated small sets were used with the same Betti number randomly to assess the performance of the UC implementation for approximating the discrepancy of a given set of points. By checking a large number of experimental images, we found the sizes

From **Table 5**, we can see that proposed method has a distinct advantage [36]. **Figure 13**

**Index Mean method Slope method SD method 3σ method** MAE 6.22 6.66 8.07 3.38 MSE 41.78 44.85 101.85 12.11

*-star discrepancy*

The variation of the uniformity coefficient with frames can be an effective method to determine

In **Figure 14**, quantitative comparisons of the homogenization curve and mixing time predicted by the uniformity coefficient method are conducted with reported experimental data and other

The comparisons show that good agreements of the mixing time obtained by Betti numbers method and uniformity coefficient method have also been achieved as given in **Table 6** [40].

**Numerical simulations.** Generated small sets were used with the same Betti number randomly to assess the performance of the UC implementation for approximating the discrepancy of a given set of points. By checking a large number of experimental images, we found the sizes

shows an example of 1000 simulation results.

164 Heat Exchangers– Design, Experiment and Simulation

**Table 5.** Comparison of computer simulation results by 1000 times.

**2.5. Measuring bubbles uniformity by discrepancy**

*2.5.1. Quantifying mixing efficiency using L<sup>2</sup>*

**Figure 13. One of these simulation results by 1000 times.**

the critical mixing time and mixing uniform.

predictions by the Betti numbers method.

(2) Recognition of local and global uniformity

(1) Quantification of mixing efficiency

**Figure 14.** Evolution of mixing efficiency by the uniformity coefficient method (denoted as U9) and Betti numbers method (denoted as B9).


**Table 6.** Computing results of mixing time by the Betti numbers method (*t*) and uniformity coefficient method (*t*\* ) at the whole orthogonal arrays table.

of the bubbles are nearly the same. Two hundred and thirty-four bubbles have the same small blank area of the radius of the circle. MATLAB software randomly selects the centre. Although 234 bubbles have the same size, they are spread in different places.

These plots of **Figure 15** are got by simulation with 720 lines and 1280 rows. Among them, the simulation 1 (**Figure 15a**) is the corresponding local uniform description, simulation 2 (**Figure 15b**) corresponds to the portrayal of global uniform. The lattice points generate the example of lattice uniform used in the demonstration of the algorithm. Under the guidance of the lattice points method, the simulation 3 (**Figure 15c**) performs for a lattice points set, which has 234 bubbles. It is can be seen that the lattice uniform is most accurate uniformly. One can take set of points or objects, which are generated by these experiments and simulations to check the algorithm.

**Figure 15.** Numerical simulations and MAD evolutions of different mixing efficiency.

**Experimental examples.** The plots in **Figure 16** are obtained by experiments E<sup>3</sup> and E<sup>6</sup> . The experiments are response to the worse and better cases. **Figure 16a**–**c** shows the difference of mixing uniformity in case 1 of E<sup>6</sup> with *β*1 = 160, whereas **Figure 16e**, **d**, and **f** shows the difference of mixing uniformity in case 2 of E<sup>3</sup> with *β*1 = 194. Comparison results show that the different experimental cases with the same Betti numbers can be identified by the MAD evolutions. The results in **Figure 16** indicate that the uniformity coefficient method might be enough to obtain a good estimation and quantification of multiphase mixing effects.

**Figure 16.** MAD evolutions of different experimental cases with the same Betti numbers.

#### *2.5.2. Modified L<sup>p</sup> -star discrepancy for measuring mixing uniformity*

(1) Video-frequency image sequence of experimental cases

In **Figure 17**, quantitative comparisons of the homogenization curve utilizing uniformity coefficient with modified discrepancy methods are conducted with reported experimental data and the other method is UC-LD. The variation of the UC-LD and UC-WD versus *i* can be effectively used to estimate the critical mixing time and mixing uniform because of the similar evolutionary trend and regularity of most experimental cases. Especially here there is a clearly and distinctively different at the early phase of mixing process of experimental case E<sup>2</sup> (see **Figure 17a**), and obvious differences in numerical performance have a deep significant role to assess the mixing efficiency. For another experimental case E<sup>7</sup> , UC-CD and UC-WD successfully measured the mixing process from no uniformity to uniformity clearly, just like UC-LD (see **Figure 17b**).

#### (2) Verification of properties

**Experimental examples.** The plots in **Figure 16** are obtained by experiments E<sup>3</sup>

**Figure 15.** Numerical simulations and MAD evolutions of different mixing efficiency.

**Figure 16.** MAD evolutions of different experimental cases with the same Betti numbers.

enough to obtain a good estimation and quantification of multiphase mixing effects.

of mixing uniformity in case 1 of E<sup>6</sup>

166 Heat Exchangers– Design, Experiment and Simulation

difference of mixing uniformity in case 2 of E<sup>3</sup>

experiments are response to the worse and better cases. **Figure 16a**–**c** shows the difference

the different experimental cases with the same Betti numbers can be identified by the MAD evolutions. The results in **Figure 16** indicate that the uniformity coefficient method might be

with *β*1 = 160, whereas **Figure 16e**, **d**, and **f** shows the

with *β*1 = 194. Comparison results show that

and E<sup>6</sup>

. The

Suppose coordinates of bubble swarms in **Figure 2** can be denoted as follows,

$$\mathbf{X}\_p = \begin{pmatrix} \mathbf{x}\_{11}\mathbf{x}\_{21} \\ \mathbf{x}\_{21}\mathbf{x}\_{22} \\ \vdots \\ \mathbf{x}\_{n1}\mathbf{x}\_{n2} \end{pmatrix} \tag{24}$$

where *n* is the number of bubble swarms or experimental points noted earlier. With one important note as mentioned above, the first rank elements *x*11, *x*21,…, *xn*<sup>1</sup> of *XP* correspond to the *x* values identified in Eq. (21) the second rank elements *x*12, *x*22, …, *xn*<sup>2</sup> of *XP* correspond to the *y* values identified in Eq. (21). UC-CD and UC-WD are formulated as the two methods of measures of uniformity. In addition, they are dimensionless, which vary from 0 to 1. And it is hard for it to reach the certain endpoint values.

**Invariance to permutation.** UC-CD and UC-WD are permanent for disrupted order of the experimental factors or the experimental points. **Table 7** shows verification data about invariance to permutation. Bubbles disordered in **Table 7** means that the operator randomly shuffles

**Figure 17.** Uniformity coefficients of E<sup>2</sup> and E<sup>7</sup> .

rows in Eq. (24). Coordinates disordered in **Table 7** means that the operator randomly shuffles columns in Eq. (24). It is quite clear found that the two parameters identically equal to 0.9751 and 0.9723 individually even though the disorder of rows or ranks happened.


**Table 7.** The data of verification of invariance to permutation.

**Invariance under reflection.** In theory, UC-CD(*P*) and UC-WD(*P*) are invariant if *xi*<sup>1</sup> and *xi*<sup>2</sup> are replaced individually by 1-*xi*<sup>1</sup> and 1-*xi*<sup>2</sup> , 1≤*i*≤*n*. The data of verification of invariance under reflection are depicted in **Table 8**. More in detail, *x* = 1/2 in **Table 8** means that the operator rotates the origin of coordinates system from bottom-left to bottom-right in a piece of patterns, *y* = 1/2 in **Table 8** means that the operator rotates the origin of coordinates system from bottom-left to top-left in a piece of patterns, *x* = 1/2 and *y* = 1/2 (both) in **Table 8** means that the manipulator spins the origin of coordinates system from bottom-left to top-left. It is obviously found that the two parameters individually equal to 0.9751 and 0.9723 even though the origin of coordinates system is rotated.

**Projection uniformity.** The projection uniformity over all sub-dimensions can be considered, and UC-CD(*P*) and UC-WD(*P*) are also invariant to it in theory. The data of verification of projection uniformity are depicted in **Table 9**. Projected to *x*-axis in **Table 9** refers to set *xi*2 = 0 in Eq. (24), projected to *y*-axis refers to *xi*1 = 0 and projected to origin refers to *xi*1 = 0 and *xi*2 = 0. It is quite clear that the projection uniformity over all sub-dimensions can be obtained. All this to say the consideration is not insignificant for high-dimensional cases.


**Table 8.** The data of verification of invariance under reflection.


**Table 9.** The data of verification of projection uniformity.

(3) Time complexity. The time complexity of different methods is shown in **Table 10**. Through experimental comparison, we may draw the conclusion that UC-CD and UC-WD can replace UC-LD and Betti numbers to some extent. Determination of the position of bubble swarms spends too much time, which leads to make the upper time complicated. But, other progressive technology can change this disadvantage.


**Table 10.** The data of time of different methods.

rows in Eq. (24). Coordinates disordered in **Table 7** means that the operator randomly shuffles columns in Eq. (24). It is quite clear found that the two parameters identically equal to 0.9751

**Bubble Coordinates Both**

and 0.9723 individually even though the disorder of rows or ranks happened.

**UC-CD** 0.9289 0.9289 0.9289 0.9289 UC-WD 0.9311 0.9311 0.9311 0.9311

**Invariance under reflection.** In theory, UC-CD(*P*) and UC-WD(*P*) are invariant if *xi*<sup>1</sup>

reflection are depicted in **Table 8**. More in detail, *x* = 1/2 in **Table 8** means that the operator rotates the origin of coordinates system from bottom-left to bottom-right in a piece of patterns, *y* = 1/2 in **Table 8** means that the operator rotates the origin of coordinates system from bottom-left to top-left in a piece of patterns, *x* = 1/2 and *y* = 1/2 (both) in **Table 8** means that the manipulator spins the origin of coordinates system from bottom-left to top-left. It is obviously found that the two parameters individually equal to 0.9751 and 0.9723 even though the origin

**Projection uniformity.** The projection uniformity over all sub-dimensions can be considered, and UC-CD(*P*) and UC-WD(*P*) are also invariant to it in theory. The data of verification of projection uniformity are depicted in **Table 9**. Projected to *x*-axis in **Table 9** refers to set *xi*2 = 0 in Eq. (24), projected to *y*-axis refers to *xi*1 = 0 and projected to origin refers to *xi*1 = 0 and *xi*2 = 0. It is quite clear that the projection uniformity over all sub-dimensions can be obtained. All this

(3) Time complexity. The time complexity of different methods is shown in **Table 10**. Through experimental comparison, we may draw the conclusion that UC-CD and UC-WD can replace

**Modified UC No projected** *y* **= 0** *x* **= 0 Projected to origin**

UC-CD 0.9289 0.3946 0.3936 0.0554 UC-WD 0.9311 0.5263 0.5253 0.3128

and 1-*xi*<sup>2</sup>

to say the consideration is not insignificant for high-dimensional cases.

**Modified UC No reflected** Reflected x = 1/2 y = 1/2 Both UC-CD 0.9289 0.9289 0.9289 0.9289 UC-WD 0.9311 0.9311 0.9311 0.9311

are replaced individually by 1-*xi*<sup>1</sup>

**Modified UC In proper order Disordered**

168 Heat Exchangers– Design, Experiment and Simulation

**Table 7.** The data of verification of invariance to permutation.

**Table 8.** The data of verification of invariance under reflection.

**Table 9.** The data of verification of projection uniformity.

of coordinates system is rotated.

and *xi*<sup>2</sup>

, 1≤*i*≤*n*. The data of verification of invariance under

(4) Numerical simulations and experimental examples. In order to assess the performance of UC-CD and UC-WD implementation for approximating the discrepancy of a given set of points, the three sets in **Figure 14** were used. **Table 11** shows that UC-LD of the three simulated images are affected by initial position, but UC-CD and UC-WD not. Comparing the modified UC of **Figure 14b** and c, the absolute difference |0.9751−0.9657| = 0.0094 is less than |0.9623−0.9464| = 0.0159 since **Figure 14b** and **c** seems to have the same degree of mixing uniformity. So it is concluded that UC-CD may outperform UC-WD and perform more sensitive for practical engineering application in some sense. The data in **Table 11** also show that the difference of mixing uniformity coefficients including UC-LD, UC-CD and UC-WD with the same Betti numbers in **Figure 16a**, b, d and e. Meanwhile, it is noticed that different initial positions are response to different UC-LDs, which bring unreasonable and bias measurement of uniformity in practice. In other words, UC-LD may result in multiple values, but UC-CD and UC-WD do not have this problem. Moreover, the absolute difference of UC-CDs is larger than that of UC-WDs. The comparison result shows that UC-CD performs more sensitive than UC-WD in identifying the different patterns with the same Betti numbers. Those are the major of our presented work in this part.

#### **2.6. Conclusions**

**1.**  Because a new technique based on algebraic topology was introduced for quantifying the efficiency of multiphase mixing, the mixture homogeneity and the non-homogeneity


**Table 11.** The data of numerical simulations and experimental examples.

of the mixture can be characterized by the Betti numbers for binary images of the patterns. The zeroth Betti numbers *β*<sup>0</sup> are used to estimate the numbers of pieces in the patterns, leading to a useful parameter to characterize the mixture homogeneity. The first Betti numbers *β*<sup>1</sup> are introduced to characterize the non-homogeneity of the mixture. This novel method may be applied for studying a variety of multiphase mixing problems in which multiphase components or tracers are visually distinguishable.


ing rotation invariance (reflection invariance), permutation invariance and the ability to measure projection uniformity. Analysing real experimental cases and simulating to evaluate the performance of the novel method. The experimental results show that UC-CD presents more sensitive performance than UC-WD so the UC-CD is more appropriate for industry.

In summary, we believe that on the basis of a large amount of previously published works, the complexity of the bubble swarm patterns can be reduced and their mechanisms clarified, and the heat transfer performance in a DCHE can be elucidated.
