Digital Twin of Heat Exchanger

Abstract

How to create proper digital twin of plate heat exchange. The chapter will investigate alternatives for modeling and propose the best choices in modeling to trade-off between accuracy of the calculation vs. speed. Lump sum vs. distributed model, how to make plate model discrete, temperature distribution models, and details in heat exchange wall modeling will be considered. Last part would focus on experimental evaluation of the simulation using IR temperature vision measurements. Conclusions would take concrete examples of dynamic responses of heat exchange units, comparing numeric simulation with experimental results.

1. Introduction

Heat exchangers are used for heat and cool transfer in different process industries, heating/cooling, and other applications. Plate heat exchangers are for majority of applications where pressure is not so high, the most cost-efficient solution. Brazed are used for smaller sizes, and more clean media, while gasketed are meant more for applications where occasional cleaning is needed.

The simplest control loop depicted in Figure 1 is to control the outlet temperature on secondary side of the heat exchanger by controlling the flow on primary at given conditions of inlet temperatures on primary and secondary sides.

Figure 1.

Control loop schematics.

The system consists of a heat exchanger as the central component of the control system. The control can be performed using an electronic controller (usually due to the speed of response of the proportional-integral controller system) and the valve with a drive. In the return control loop, the measuring sensor for the temperature represents the guiding control variable.

The speed of process, seen from dynamics point of view, is a major challenge, as the time constants of the measuring system for the temperature of the heat carrier or hot water need to be small due to a relatively small control volume. Time constants may be less than few seconds.

2. Literature overview

In the field of heat exchangers, we can highlight the contribution [1], which is an overview of the current literature on the state of heat exchangers. Aslam Bhutta et al. [2] and Zhang et al. [3] discuss the review of contributions dealing with the topic of numerical simulation of heat transmissions. In Ref. [4] the case of a numerical simulation of a plate heat exchanger is considered. Freund and Kabelac [5] also deals with modeling of dynamic responses of plate heat transmissions [6]. The topic of evaporation in heat exchangers is discussed. The paper [7] deals with the experimental determination of the operational factors of heat exchangers in district heating for heating and hot sanitary water. Sources [8, 9] present an experimental method of visualizing the flow in a flat heat exchanger with a microplate structure [10]. The authors are also involved in visualizing the flow and determining local heat transfer. The source [11] analyzes the local heat transfer conditions. While the source [12] addresses the impact of the final plates on the operation of the heat exchanger. The paper [13] focuses on a study of the sensitivity of the geometric properties of a heat exchanger to its efficiency. The paper [14] deals with the technical requirements and cost analysis of heat transmissions in low-temperature systems. A similar theme of low-temperature systems is also in the paper [14, 15], with the emphasis in publication [15] on the impact of low-temperature systems on heating and the preparation of hot water without accumulation and in the source [14], on the main changes in the structure of the exchangers and thus the dynamic properties of heat transfer for such systems. A comparison between the plate and the tube heat exchanger is made in Ref. [16]. The source [17] discusses experimental and simulation methods in the field of heat exchangers. In Ref. [5], a study is carried out on how to determine local heat transfer coefficients in heat exchangers [18]. It also addresses the sensitivity of the parameter analysis to the numerical model of the heat exchanger. The link between the flow rate and the heat transfer under the micro heat exchanger conditions is given in Ref. [19]. Laszczyk [20] provides a method of a simplified heat exchanger model. Improved heat transfer in heat exchangers deals with [21]. Heat exchangers shell and tube are treated in Refs. [22]. Sharifi et al. [23] present a dynamic simulation of the heat exchanger. General dimensioning and dynamic analysis can be found in Refs. [24, 25]. The more generic described heat transfer dynamics depend, of course, on the configuration and sample on the heat exchanger plate, the various examples described in Refs. [26, 27, 28]. Many studies use CFD to analyze current patterns, distribution of pressure field, and temperature field, such as Ref. [4]. Sharifi et al. [23] provide a general model of heat transfer. Semiempirical equations are important for determining the coefficient of heat transfer, such as in Refs. [29, 30, 31]. Carlomagno and Cardone [17] describe the method of noninvasive measurement of heat exchanger parameters. Aprea and Renno [32] describe the design of tests for the portable evaporator function, that is, similar functionality to a heat exchanger. The use of MATLAB Simulink is shown in Ref. [33]. Integrated thermo-fluid dynamics and modeling reliability are shown in Ref. [34]. The literature also lists several approaches to modeling heat exchangers for different purposes, for example in Ref. [35], a model mode that is object-oriented using the Modelica programming language and provides a model based on the control volume method. Using Laplace’s transformation, dynamic analysis of plate heat transfer is given in Ref. [36]. While in the Ref. [37] heat exchanger is described with the hyperbolic first order differential equations. Furthermore, in Ref. [38], a simplified mathematical model of a heat exchanger based on the final volume method is also used in the interpolation scheme. Similar is used in Ref. [39]. Further more modelling of the heat exchangers in the system and elements are the system are described in Refs. [40, 41, 42, 43, 44].

3. Basics of modeling

Seen from dynamic behavior point of view heat exchanger is a system with distributed parameters, meaning that the temperature continually changes throughout its length, thus traditional approach of direct Laplace transformation becomes too complex to solve, as we end up in double Laplace transformation, causing numeric problems with solution.

If we want to reduce the model to lump sum, then discretization is needed. Of course, the question is how many sections we should use to get good approximation with minimum numeric calculation load. The answer is in what kind of constitutional equation we use for temperature distribution in every cell. One has three options, one is if cells represent perfect mixing cell, and the assumption is that temperature immediately drops to exit temperature, and the second option is the opposite where the assumption is that mixing is imperfect, so the temperature is kept on inlet value until it leaves the cell. One can intuitively feel that those options are not realistic. The third option is to use distribution, it can be linear or to get closer to reality logarithmic. Experiments show that linear distribution is the best choice between number of cells, calculation effort, and minimized calculation error.

The second important question is how to model the thin metal wall between two media streams. One can model it as a conductive element of just ignoring the conduction. Given the ratio between heat convection resistance factor and heat conduction in a thin metal wall, it can be very well assumed that this is low resistance to heat compared to heat transfer for media to wall.

It is also important that proper material and turbulence models are used, thus in next chapter, we will investigate properties of water.

3.1 Basic properties

The material properties depend, of course, on different parameters. In the example shown, the impact of pressure and temperature is greatest, but we are particularly interested in variables:

• Density,

• Viscosity, and

• Specific heat.

The density gives us the dependence of the mass change per volume unit. In general, we can write down the dependence:

$$\mathit{d\rho }={\left(\frac{\partial \rho }{\partial T}\right)}_p\mathit{dT}+{\left(\frac{\partial \rho }{\partial p}\right)}_T\mathit{dT}$$E1

When the coefficient of temperature extension is applied, the following shall be obtained:

$$\frac{\mathit{d\rho }}{\rho }=-\mathit{\beta dT}+\frac{1}{\epsilon }\mathit{dp}$$E2

The eq. (2) specifies that the density depends on the temperature and the suitability of the substance. In this case, the heat carrier will be used as a heat carrier for district heating, and for this purpose, the constitution equation is by source [45]:

$$\rho =\left(1000+\mathrm{0,067}\left(1-e^{-\mathrm{0,016}T}\right)-\mathrm{0,001}T\right)\mathit{za}{0}^{°}C<T<{50}^{°}C$$E3

$$\rho =\left(1.067-e^{-\mathrm{0,023}\left(T-50\right)}-\mathrm{0,001}T\right)\mathit{za}{50}^{°}C<T<{200}^{°}C$$E4

Viscosity can be obtained from the viscous tensor as:

$$\tau =\eta \frac{d{v}_x}{\mathit{dy}}$$E5

The viscosity is therefore greater, the smaller the shear deformity. For water, the kinematic viscosity equation can be summarized as [45]:

$$\nu =1,75e^{-\mathrm{0,029}T}{10}^{-6}\text{for}{0}^{°}\mathrm{C}<\mathrm{T}<{20}^{°}\mathrm{C}$$E6

$$\nu =0,98e^{-0,02\left(T-20\right)}{10}^{-6}\text{for}{20}^{°}\mathrm{C}<\mathrm{T}<{50}^{°}\mathrm{C}$$E7

$$\nu =0,55{\left(\frac{50}{T}\right)}^{0,9}{10}^{-6}\text{for}{50}^{°}\mathrm{C}<\mathrm{T}<200^{°}\mathrm{C}$$E8

Specific heat is a greatness that determines how much energy we need to produce to heat the substance by 1 K.

$$c_p\left(T\right)=\frac{1}{m}\frac{\mathit{dh}}{\mathit{dT}}$$E9

or it can be written as an approximation equation as:

$$c_p=-7·{10}^{-7}{T}^3+8·{10}^{-5}{T}^2-0,0031T+4,2161\text{for}{0}^{°}\mathrm{C}<\mathrm{T}<{50}^{°}\mathrm{C}$$E10

$$\begin{array}{l}{c}_p=-6·{10}^{-11}{T}^5+2·{10}^{-8}{T}^4-3·{10}^{-6}{T}^3+0,0002T^2-0,\\ 0069T+\mathrm{4,269}\text{For}{51}^{°}\mathrm{C}<\mathrm{T}<100^{°}\mathrm{C}\end{array}$$E11

4. Heat exchanger

Heat exchanger is the heart of the thermal substation, as we control heat transfer with changing thermal and hydraulic parameters. Despite the different types of heat transferors, in the practice of district heating, plate heat exchangers have established themselves, namely, for smaller dimensions and relatively good media quality (clean and noncorrosive), dedicated laptops, and all other applications are covered by gasketed heat exchangers.

In general, all plate laptops have a sample on plates shaped like a fish bone, the only other ones on the market are Danfoss’s microplate heat exchangers, which have a point-shaped sample. The sample on the heat transfer plates is intended to increase the heat transfer area and increase turbulence in order to increase the heat transfer coefficient.

Modern heat exchangers also have an input distribution section and an output collection section to increase the efficiency of the panels, and the task of these two sections is to redistribute the medium to flow evenly throughout the plate and thus increase the efficiency of the surface.

There are quite a few dynamic models of heat transfers in the literature [1]. The simplest assumes a first-order system with a possible delay, but such a heat exchanger model is inappropriate because the validity of the mathematical model is too limited. More interesting are the models where the dual Laplace transformation is used, but they need to be linearized, which in turn is in too limited force range for our case. Therefore, the heat exchanger will be considered as a system with distributed parameters which means that the heat exchanger will be considered as several consecutive systems with no distributed parameters. The directions of the currents in the heat exchanger are schematically shown in Figure 2.

Figure 2.

Schematics of heat exchanger.

4.1 General assumptions and limitations of the mathematical model

When the heating medium enters the heat exchanger, the more distant plates (for example, 100 to 200), represent a greater hydraulic resistance against the water flow. Therefore, for the first time, water flows over nearby plates faster and the second time there is a certain delay due to the flow of water. As this will be negligible for our case (about 20 panels), we will not take this effect into account.

In general, the heating medium has a higher temperature than the surrounding area, so the heat transfer takes place not only from the warmer medium to the cooler medium but also in the surrounding area. Due to the relatively good thermal insulation, we will neglect these losers. The translation of heat over steel plates exists, but due to the thickness of these (about 0,2 mm) and the relatively high coefficient of heat translation, this will be ignored. We will also pre-bet that the heat exchanger is clean without the suspended limestone on the walls. The most unrealistic assumption will be the uniformity of the distribution of the heating medium by plate. Assuming that the mixing section is relatively short toward the entire surface, we can also take this assumption into account.

In the heat exchanger model, we considered that the heat exchanger is a nonlinear system with distributed parameters. Therefore, in the model, the heat exchanger will be divided into several smaller subunits, in which linearity will be assumed. The heat exchanger has different dynamic responses according to the input parameters, whether it is the temperature or the mass flow rate of the heat carrier. It is important to determine the dynamic properties of the heat exchanger, however, to respond to the temperature change.

4.2 Static characteristics

The static characteristic of the heat exchanger is the ratio of mass flow rate to the heat ousted by the energy equation:

$$P=\dot{m}{c}_p\mathrm{\Delta }T$$E12

If only this is considered and when the efficiency of the heat exchanger is introduced, which determines the ratio between actual thermal power and maximum thermal power:

$$\epsilon =\frac{P_{\mathit{dej}}}{P_{\mathit{\text{maks}}}}$$E13

and for primary side

$${\epsilon }_h=\frac{T_{11}-T_{12}}{T_{11}-T_{21}}$$E14

4.3 Dynamic characteristics

The mathematical model of the heat exchanger is basically the problem of switching heat from a hot medium over the wall to a cold medium. This can be described using the energy equation for a cold and hot medium:

$${\rho }_i{c}_{\mathit{pi}}\frac{{\partial }^2{T}_i}{\partial x\partial t}=\nabla ·\left({\lambda }_i\nabla T_i\right)-\underset{¯}{w_i}{\rho }_i{c}_{\mathit{pi}}\nabla T_i+r_i\left(-∆I_i\right)+{\upsilon }_{\mathit{Ti}}\left({\rho }_i{T}_i\frac{\partial p_i}{\partial t}+\underset{¯}{w_i}{T}_i\nabla p_i\right)$$E15

Just as in the case of a temperature sensor, there are no chemical reactions and generated heat due to the mechanical part, which is why the last two articles of the eq. (15) are off. However, since the temperature gradient and the translation of heat over the wall are ignored, the equation can be simplified by consideration in:

$$m_i{c}_{\mathit{pi}}\frac{d{T}_i}{\mathit{dt}}={\dot{m}}_i{c}_{\mathit{pi}}\frac{d{T}_i}{\mathit{dx}}\pm {\alpha }_i{A}_i\left(T_{\mathit{hi}}-T_{\mathit{ci}}\right)$$E16

Therefore, assuming one dimension model with complete mixing in the cell we obtain a model based on the energy balance of the heat exchanger:

• heat exchanger hot side:

  $$m_h{c}_{\mathit{ph}}\frac{d{T}_h}{\mathit{dt}}={\dot{m}}_h{c}_{\mathit{ph}}\left(T_{\mathit{hi}}-T_{\mathit{ho}}\right)-{\alpha }_h{A}_h\left(T_h-T_w\right)$$E17

• cool side of the heat exchanger:

  $$m_c{c}_{\mathit{pc}}\frac{d{T}_c}{\mathit{dt}}={\dot{m}}_c{c}_{\mathit{pc}}\left(T_{\mathit{ci}}-T_{\mathit{co}}\right)+{\alpha }_c{A}_c\left(T_w-T_c\right)$$E18

• and the wall of heat exchanger:

  $$m_w{c}_{\mathit{pw}}\frac{d{T}_w}{\mathit{dt}}={\alpha }_h{A}_h\left(T_h-T_w\right)-{\alpha }_c{A}_c\left(T_w-T_c\right)$$E19

However, since the heat exchanger system is a system with distributed parameters, the analytical solutions of the above model are impossible. In the trace of this, the heat exchanger will be divided into several units (Figure 3). For each unit, the balance sheet equations can be recorded:

Figure 3.

(a) Heat exchanger section, (b) one cell, (c) resistance scheme.

• heat exchanger hot side:

  $$m_h{c}_{\mathit{ph}}\frac{d{T}_{\mathit{hi}+1}}{\mathit{dt}}={\dot{m}}_h{c}_{\mathit{ph}}\left(T_{\mathit{hi}}-T_{\mathit{hi}+1}\right)-{\alpha }_h{A}_h\left(\frac{T_{\mathit{hi}}+T_{\mathit{hi}+1}}{2}-T_{\mathit{wi}}\right)$$E20

• cool side of the heat exchanger:

  $$m_c{c}_{\mathit{pc}}\frac{d{T}_{\mathit{ci}-1}}{\mathit{dt}}={\dot{m}}_c{c}_{\mathit{pc}}\left(T_{\mathit{ci}-1}-T_{\mathit{ci}}\right)+{\alpha }_c{A}_c\left(T_{\mathit{wi}}-\frac{T_{\mathit{ci}-1}+T_{\mathit{ci}}}{2}\right)$$E21

• and the wall of heat exchanger:

$$m_w{c}_{\mathit{pw}}\frac{d{T}_{\mathit{wi}}}{\mathit{dt}}={\alpha }_h{A}_h\left(\frac{T_{\mathit{hi}}+T_{\mathit{hi}+1}}{2}-T_{\mathit{wi}}\right)-{\alpha }_c{A}_c\left(T_w-\frac{T_{\mathit{ci}-1}+T_{\mathit{ci}}}{2}\right)$$E22

An important assumption is what the temperature distribution in each cell will be. In general, it can be assumed to jump (which is impossible in the heat exchanger, but if the sections are infinitesimally narrow, the assumption is good enough for the apportionment, while greatly simplifying the calculation), which means that the medium has an exit temperature from the cell immediately upon entering the cell. Such a model is very easy to use but the assumption is valid only if mixing is practically ideal. A more realistic model is where the temperature changes linearly along the cell. Thus, the model is numerically still sustainable, but to us, results that are close to real are why this model is used in the above equations. The third model considers the real logarithmic distribution of the temperature in the cell (Figure 4).

Figure 4.

Temperature distribution in the cell.

By rearranging eqs. (20)–(22) and inserting variables:

$${\tau }_{h/c_i}=\frac{m_{h/c_i}}{{\dot{m}}_{h/c_i}}$$E23

$${\delta }_{w_i}=\frac{m_{w_i}{c}_{p_{h/e_i}}}{A_i}$$E24

$${\gamma }_{h/c_i}=\frac{A_i{\alpha }_{h/c_i}}{m_{h/c_i}{c}_{p_{h/c_i}}}$$E25

One can get a matrix for the i-th cell:

$$\left[\begin{array}{c}\dots \\ {\dot{T}}_{h_{i-1}}\\ {\dot{T}}_{h_i}\\ \dots \\ {\dot{T}}_{w_i}\\ \dots \\ {\dot{T}}_{c_i}\\ {\dot{T}}_{c_{i-1}}\\ \dots \end{array}\right]=\begin{array}{l}\left[\begin{array}{cccccccccccccc}\dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & 0& \frac{1}{{\tau }_{h_i}}& -\frac{1+{\gamma }_{h_i}}{{\tau }_{h_i}}& 0& \dots & 0& \frac{{\gamma }_{h_i}}{{\tau }_{h_i}}& 0& \dots & \dots & \dots & \dots & 0\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & 0& \frac{{\gamma }_{h_i}}{{\delta }_{w_i}}& 0& \dots & 0& -\frac{{\gamma }_{h_i}+{\gamma }_{c_i}}{{\delta }_{w_i}}& 0& \dots & 0& \frac{{\gamma }_{c_i}}{{\delta }_{w_i}}& 0& \dots & 0\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & 0& -\frac{1}{{\tau }_{c_i}}& \frac{1+{\gamma }_{c_i}}{{\tau }_{c_i}}& 0& \dots & 0& -\frac{{\gamma }_{c_i}}{{\tau }_{c_i}}& 0& \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \end{array}\right]·\left[\begin{array}{c}\dots \\ T_{h_{i-1}}\\ T_{h_i}\\ \dots \\ T_{w_i}\\ \dots \\ T_{c_i}\\ T_{c_{i-1}}\\ \dots \end{array}\right]\\ +\left[\begin{array}{cc}\frac{1}{{\tau }_{h_1}}& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& \frac{1}{{\tau }_{c_1}}\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right]·\left[T_{h_{\mathit{\text{in}}}}{T}_{c_{\mathit{\text{in}}}}\right]\end{array}$$E26

or:

$${\underset{¯}{\dot{T}}}_{H/w/c}=\underset{¯}{A}\left({\tau }_{H/c},{\gamma }_{H/c},{\delta }_w\right)·{\underset{¯}{T}}_{H/w/c}+\underset{¯}{B}\left({\tau }_{H/c_1}\right)·{\underset{¯}{T}}_{H/c_{\mathit{\text{in}}}}$$E27

4.4 Laplace transformation

The above-mentioned equation system can be mapped to:

$$T_{\mathit{hi}+1}=\frac{\left(\frac{1}{{\tau }_h}-\frac{1}{{\delta }_h}\right)}{s+\left(\frac{1}{{\tau }_h}+\frac{1}{{\delta }_h}\right)}{T}_{\mathit{hi}}+\frac{\frac{2}{{\delta }_h}}{s+\left(\frac{1}{{\tau }_h}+\frac{1}{{\delta }_h}\right)}{T}_{\mathit{wi}}=G_{h1}{T}_{\mathit{hi}}+G_{h2}{T}_{\mathit{wi}}$$E28

$$T_{\mathit{ci}-1}=\frac{\left(\frac{1}{{\tau }_c}-\frac{1}{{\delta }_c}\right)}{s+\left(\frac{1}{{\tau }_c}+\frac{1}{{\delta }_c}\right)}{T}_{\mathit{ci}}+\frac{\frac{2}{{\delta }_c}}{s+\left(\frac{1}{{\tau }_c}+\frac{1}{{\delta }_c}\right)}{T}_{\mathit{wi}}=G_{c1}{T}_{\mathit{ci}}+G_{c2}{T}_{\mathit{wi}}$$E29

$$T_{\mathit{wi}}=\sum _{j=1}^n\frac{{\gamma }_h}{s+2\left({\gamma }_h+{\gamma }_c\right)}{T}_{\mathit{hj}}+\sum _{j=1}^n\frac{{\gamma }_c}{s+2\left({\gamma }_h+{\gamma }_c\right)}{T}_{\mathit{cj}}$$E30

to obtain a simplified record:

$$T_{\mathit{cn}}=\prod _{j=1}^n{G}_h{T}_{\mathit{hj}}+\prod _{j=1}^n{G}_c{T}_{\mathit{cj}}$$E31

From what we see, we get an nth-order system that depends on changes in input temperatures as well as material, geometric, and flow parameters.

4.5 Constitution equations

The material properties are generally described in the literature [45] and will not be further stated, but the determination of the heat transfer coefficient is more important. When using no dimensional numbers and links between them:

$$\mathit{Nu}=C{\mathit{Pr}}^m{\mathit{Re}}^n$$E32

And for our case, Reynolds’ number of

$$\mathit{Re}=\frac{\dot{m}}{\mathit{o\nu }}$$E33

Nusselt number:

$$\mathit{Nu}=\frac{\mathit{\alpha A}}{\mathit{o\lambda }}$$E34

The coefficients depend on the shape of the channel and the Reynolds number. For our example, the coefficients are:

$$\mathrm{C}=0,2;\mathrm{m}=0,67;\mathrm{n}=0,4$$E35

Valid for Reynolds numbers:

$$\mathit{Re}>100$$E36

5. Experimental model

Although most of the models of heat exchangers are on the market with a fish bone sample, it was still selected to test the heat exchanger with a microplate sample. The reason is that heat exchangers with a microplate are more efficient because the flow between the plates is more subdued and the distribution by plate is generally better, which gives us about a 20–30% lower pressure drop or a much better heat transfer. The comparison between the plates is shown in Figure 5.

Figure 5.

Comparison between (b) microplate and (a) fishbone heat exchanger.

The test was performed on a heat exchanger with three plates in different configurations. Four cases were tasted with different flow configurations and speeds as per Table 1. Although the test was carried out on only three channels, the multi plates’ behavior is similar. The geometry of the heat exchanger is shown in Figure 6.

Table 1.

Test design.

Figure 6.

Heat exchanger geometry.

For this purpose, a device was designed to allow for a rapid switch between hot and cold water. The heat exchanger was flushed with cold water and, at the moment of disturbance, a stream of hot water was flushed in. The switching was made with magnetic valves. Despite the fast-switching magnetic valves, the disturbance was not stepped but was approximately similar to the second-order response. To record this phenomenon, the entry signal was recorded, simulated and compared to the real state. Comparisons are shown in Figures 7–10.

Figure 7.

Disturbance A.

Figure 8.

Disturbance B.

Figure 9.

Disturbance C.

Figure 10.

Disturbance D.

In Figure 7−10, input disturbances are simulated, while in Figure 11, deviations are depicted.

Figure 11.

Disturbance error between approximation and real one.

The discrepancies between the approximated and the measured disturbance are shown in Figure 11.

A FLIR A600 IR thermal camera was used to measure the dynamic temperature change. Flow measurements were carried out through weighing. Since the thermal camera can take a large number of images per second of 4000 images per second, it was assumed that the images are static at 0.1 s, which virtually eliminates the dynamic error of measurement. The temperature measurement error is +/−0,4°C.

In addition to the temperatures on the outside of the plate, the thermal camera also visualized the temperature field. The change in the temperature field is shown in Figures 12–14 as a comparison between type A and other types.

Figure 12.

Case A and B.

Figure 13.

Case A and C.

Figure 14.

Case A and D.

The results show us that in all cases the distribution of temperature is uneven, which is probably due to the flow conditions on the panel itself. Despite the use of microchannel technology, the apparent distribution of water across the heat exchanger plate is still uneven, or not ideal for heat transfer.

The main observations are:

• In the case of hot water flow in the middle, the spread of hot temperature disturbance is more limited to the center of the heat exchanger and therefore better, the reason for this is the higher heat take- and the “filtered” view of the hot plate,

• The C and D responses are quite similar. The reason is simple and is based on 2. The law of thermodynamics, says that the heat currents are the primary and secondary sides in equilibrium and therefore, with the temperature regime unchanged, the main influential parameter may be a flow rate that is smaller relative to A, and therefore the response is slower,

• A similar reason for the slower response in case B is that we have more cold water than hot water, and therefore the overheating of cold water is slower.

The results of the dynamic time response tests of the heat exchanger are shown in Figure 15.

Figure 15.

Experimental results.

One can conclude from Figures 12 and 13, that case B is not relevant for temperature front propagation. Comparing then cases A, C, and D one can see that speed of water is main contributor to temperature front propagation. As in A-D cases hot is on outside border, then there is very little difference between A and C, while D with half speed, is noticeable slower than case A.

6. Numeric experiment

The solution to the eq. (27) considering the geometry of the heat exchanger in Figure 16 (Heat exchanger was divided into 10 segments — Table 2 indicates a shift from the starting position) and the specifics of the experiment (input disturbance does not step)) is shown in Figure 17−20. The equation system was solved with the MATLAB Simulink software package using the RK4 numerical method.

Figure 16.

Heat exchanger sections.

Segment	0	1	2	3	4	5	6	7	8	9	10
Distance; mm	0	47,9	95,8	143,7	191,6	239,5	287,4	335,3	383,2	431,1	479

Table 2.

Segments position.

Figure 17.

Simulation A.

Figure 18.

Simulation B.

Figure 19.

Simulation C.

Figure 20.

Simulation D.

Figure 17−20 show simulation results split into sections. One can see the development of temperature as a function of time. The last 10th curve represents result of heat exchanger outlet.

7. Comparison between numerical calculation and tests results obtained and discussion

A comparison between the measured and simulated results is shown in Figure 21−24.

Figure 21.

Comparison A.

Figure 22.

Comparison B.

Figure 23.

Comparison C.

Figure 24.

Comparison D.

The deviation between the measured and simulated results is relatively small and is less than 10%. The only difference is case D in the initial state because the heat exchanger was flushed with cold water and was not at ambient air temperature, as assumed in the simulation. Therefore, the initial situation is different. The deviations are shown in Figure 25.

Figure 25.

Error between simulation and measurements.

8. Conclusion

Obtaining digital twin from plate heat exchanger is relatively easy when behavior of the heat transfer coefficients is known. It is necessary to sectionize heat exchanger to get lump sum model, which is easy to resolve. A number of sections determine the accuracy, but also the distribution within the cell. The model can be further simplified and improved.

The mathematical model can be simplified by ignoring the intermediate wall, complete cell mixing, and independence of material properties from temperature.

However, it can be improved by increasing the number of cells to improve the approximation of the real state or by considering the logarithmic distribution of the temperature in the cell.

Conflict of interest

“The authors declare no conflict of interest.”

Nomenclature

A

${\mathrm{m}}^2$

Area

C

-

Constant

$c_p$

$\mathrm{J}/\left(\mathrm{kgK}\right)$

Specific heat

G

Transfer function

h

J

Enthalpy

I

$\mathrm{J}/\mathrm{kg}$

Reaction enthalpy

m

kg

Mass

$\dot{m}$

$\mathrm{kg}/\mathrm{s}$

Mass flow

Nu

-

Nusselt number

o

m

Circumference

P

W

Power

Pr

-

Prandtl number

p

Pa

Pressure

Re

-

Reynolds number

r

$\text{kmol}/\mathrm{s}$

Velocity of chemical reaction

s

Laplace operator

T

°C, K

Temperature

t

s

Time

${\mathit{v}}_{\mathit{x}}$

$\mathrm{m}/\mathrm{s}$

Shear velocity

$v_T$

${\mathrm{m}}^3/\mathrm{kg}$

Specific volume

w

$\mathrm{m}/\mathrm{s}$

Fluid speed

x

m

Coordinate

y

m

Coordinate

$\alpha$

$\mathrm{W}/\left({\mathrm{m}}^2\mathrm{K}\right)$

Heat convection coefficient

β

${\mathrm{m}}^3/\mathrm{K}$

Temperature dilatation coefficient

γ

$1/\mathrm{s}$

Heat exchanger coefficient

δ

$\mathrm{J}/\left({\mathrm{m}}^2\mathrm{K}\right)$

Heat exchanger coefficient

ε

$\mathrm{N}/{\mathrm{mm}}^2$

Compression Module

ε

-

Heat exchanger efficiency

η

Pa·s

Dynamic viscosity

$\rho$

$\mathrm{kg}/{\mathrm{m}}^3$

Density

$\lambda$

$\mathrm{W}/\left(\mathrm{mK}\right)$

Heat conduction coefficient

ν

$\mathrm{kg}/\left(\mathrm{ms}\right)$

Kinematic viscosity

τ

$\mathrm{N}/{\mathrm{m}}^2$

Shear stress

τ

s

Heat exchanger coefficient

1

Primary side

11

Primary side inlet

12

Primary side outlet

+1

Next cell

−1

Previous cell

2

Secondary side

21

Secondary side inlet

22

Secondary side outlet

c

Cold side

eff

Effective

dej

Real

h

Hot side

i

Inlet

i

i-th element

j

j-th element

m

Exponent

n

Exponent

o

Outlet

p

Pressure

w

Wall