“If you read the inscription “buffalo” on the cage of an elephant, do not believe your eyes” Kozma Prutkov
In the previous it was shown why an object model is needed, and it was proved that without this object model one can speak of model based design only as a marketing blizzard, meaningless and merciless. But when a model of an object appears, competent engineers always have a reasonable question: what evidence is there that the mathematical model of an object corresponds to a real object.
One example of an answer to this question is given in In this article, we will consider an example of creating a model for aircraft air conditioning systems, diluting the practice with some theoretical considerations of a general nature.
Creation of a reliable object model. Theory
In order not to pull rubber, I will immediately tell you about the algorithm for creating a model for model-based design. It has just three simple steps:
Step 1. Develop a system of algebraic-differential equations that describe the dynamic behavior of the simulated systems. It's easy if you know the physics of the process. Many scientists have already developed for us the basic physical laws of the name of Newton, Brenouli, Navier Stokes and other Compasses and Rabinovich Stangels.
Step 2. Select in the resulting system a set of empirical coefficients and characteristics of the modeling object, which can be obtained from tests.
Step 3. Test the object and adjust the model according to the results of field experiments so that it corresponds to reality, with the required degree of detail.
As you can see, just like two three.
Example of practical implementation
The air conditioning system (ACS) in an aircraft is linked to an automatic pressure maintenance system. The pressure in the aircraft must always be greater than the external pressure, while the rate of pressure change must be such that the pilots and passengers do not bleed from the nose and ears. Therefore, the air inflow and outflow control system is important for safety, and expensive test systems are put on the ground to develop it. They create temperatures and pressures of the flight altitude, reproduce the modes of takeoff and landing at airfields of different heights. And the issue of developing and debugging control systems for VCS rises to its full height. How long will we run the test bench to get a satisfactory control system? Obviously, if we tune the control model to the object model, then the cycle of work on the test bench can be significantly reduced.
The aircraft air conditioning system consists of the same heat exchangers as any other thermal system. Battery - it is also a battery in Africa, only air conditioning. But due to the limitations of the take-off weight and dimensions of aircraft, heat exchangers are made as compact as possible and as efficient as possible in order to transfer as much heat as possible with less weight. As a result, the geometry becomes quite bizarre. As in the case under consideration. Figure 1 shows a plate heat exchanger that uses a membrane between the plates to improve heat transfer. Hot and cold coolant alternate in the channels, while the direction of flow is transverse. One coolant is supplied to the front cut, the other to the side.
To solve the problem of SCR control, we need to know how much heat is transferred from one medium to another in such a heat exchanger per unit time. The rate of temperature change depends on this, which we regulate.
Figure 1. Scheme of an aircraft heat exchanger.
Problems of modeling. Hydraulic part
At first glance, the task is quite simple, it is necessary to calculate the mass flow through the channels of the heat exchanger and the heat flow between the channels.
The mass flow rate of the coolant in the channels is calculated using the Bernoulli formula:
where:
ΔP is the pressure difference between two points;
ξ is the friction coefficient of the coolant;
L is the channel length;
d is the hydraulic diameter of the channel;
ρ is the coolant density;
ω is the coolant velocity in the channel.
For a channel of arbitrary shape, the hydraulic diameter is calculated by the formula:
where:
F is the area of the passage section;
P is the wetted perimeter of the channel.
The coefficient of friction is calculated by empirical formulas and depends on the flow velocity and properties of the coolant. For different geometries, different dependencies are obtained, for example, the formula for turbulent flow in smooth pipes:
where:
Re is the Reynolds number.
For flow in flat channels, the following formula can be used:
From the Bernoulli formula, one can calculate the pressure drop for a given speed, or vice versa, calculate the coolant velocity in the channel, according to a given pressure drop.
Heat exchange
The heat flow between the coolant and the wall is calculated by the formula:
where:
α [W/(m2×deg)] – heat transfer coefficient;
F is the area of the flow section.
For the problems of the flow of heat carriers in pipes, a sufficient number of studies have been carried out and there are many calculation methods, and as a rule, everything comes down to empirical dependencies, for the heat transfer coefficient α [W/(m2×deg)]
where:
Nu is the Nusselt number,
λ is the thermal conductivity of the liquid [W/(m×deg)] d is the hydraulic (equivalent) diameter.
To calculate the Nusselt number (criterion), empirical criterion dependencies are used, for example, the formula for calculating the Nusselt number of a round pipe looks like this:
Here we already see the Reynolds number, the Prandtl number at the wall temperature and the liquid temperature, and the non-uniformity coefficient. ()
For corrugated plate heat exchangers, the formula is similar ( ):
where:
n = 0.73 m =0.43 for turbulent flow,
coefficient a - varies from 0,065 to 0.6 depending on the number of plates and the flow regime.
We take into account that this coefficient is calculated only for one point in the flow. For the next point, we have a different liquid temperature (it has warmed up or cooled down), a different wall temperature, and, accordingly, all Reynolds numbers, Prandtl numbers float.
At this point, any mathematician will say that it is impossible to calculate exactly a system in which the coefficient changes by 10 times, and he will be right.
Any practical engineer will say that each heat exchanger is different in manufacture and it is impossible to count the systems, and they will also be right.
What about Model-Based Design? Is it all gone?
Advanced sellers of Western software in this place will sell you Supercomputers and 3D-calculation systems, such as "no way without it." And you need to run the calculation for a day to get the temperature distribution within 1 minute.
It is clear that this is not our option, we need to debug the control system, if not in real time, then at least in the foreseeable future.
Solution by poking
A heat exchanger is manufactured, a series of tests are carried out, and a table of the efficiency of a steady temperature is set at given heat carrier flow rates. Simple, fast and reliable as the data comes from trials.
The disadvantage of this approach is that there are no dynamic characteristics of the object. Yes, we know what the steady-state heat flux will be, but we don’t know how long it will take to set when switching from one operating mode to another.
Therefore, having calculated the necessary characteristics, we set up the control system directly during the tests, which we initially would like to avoid.
Model-Based Approach
To create a dynamic heat exchanger model, it is necessary to use test data to eliminate uncertainties in empirical calculation formulas - Nusselt numbers and hydraulic resistance.
The solution is simple, like all ingenious. We take an empirical formula, conduct experiments and determine the value of the coefficient a, thereby eliminating the uncertainty in the formula.
As soon as we have a certain value of the heat transfer coefficient, all other parameters are determined by the basic physical laws of conservation. The temperature difference and the heat transfer coefficient determine the amount of energy transferred to the channel per unit time.
Knowing the energy flow, it is possible to solve the equations of conservation of energy mass and momentum for the coolant in the hydraulic channel. For example this:
For our case, the heat flux between the wall and the coolant, Qwall, remains undefined. You can see more
As well as the equation of the temperature derivative for the channel wall:
where:
∆Qwall is the difference between the incoming and outgoing flow to the channel wall;
M is the mass of the channel wall;
CCP is the heat capacity of the wall material.
Model Accuracy
As mentioned above, in the heat exchanger we have a temperature distribution over the surface of the plate. For a steady-state value, one can take the average over the plates and use it, presenting the entire heat exchanger as a single concentrated point, at which, at one temperature drop, heat is transferred through the entire surface of the heat exchanger. But for transient regimes, such an approximation may not work. The other extreme is to make several hundred thousand points and load the Supercomputer, which also does not suit us, since the task is to set up the control system in real time, and preferably faster.
The question arises, how many sections should the heat exchanger be divided into in order to obtain acceptable accuracy and speed of calculation?
As always, by chance, I had a model of an amine heat exchanger at hand. The heat exchanger is a tube, a heating medium flows in the pipes, and a heated medium flows between the sacks. To simplify the problem, the entire tube of the heat exchanger can be represented as one equivalent pipe, and the pipe itself can be represented as a set of discrete computational cells, in each of which a point heat transfer model is calculated. A diagram of the model of one cell is shown in Figure 2. The hot air channel and the cold air channel are connected through a wall, which ensures the transfer of heat flow between the channels.
Figure 2. Model of a heat exchanger cell.
The tubular heat exchanger model is easily customizable. You can change only one parameter - the number of sections along the length of the pipe and look at the calculation results for different partitions. Let's calculate several options, starting with splitting into 5 points along the length (Fig. 3) and up to 100 points along the length (Fig. 4).
Figure 3. Stationary temperature distribution of 5 calculated points.
Figure 4. Stationary temperature distribution of 100 calculated points.
As a result of calculations, it turned out that the steady-state temperature when divided into 100 points is 67,7 degrees. And when divided into 5 calculated points, the temperature is 72 degrees C.
The calculation speed relative to real time is also displayed in the lower part of the window.
Let's see how the steady temperature and calculation speed change depending on the number of calculation points. The difference in steady-state temperatures in calculations with a different number of calculation cells can be used to assess the accuracy of the result obtained.
Table 1. Dependence of temperature and calculation speed on the number of calculation points along the length of the heat exchanger.
| Number of calculated points | Steady-state temperature | Calculation speed |
| 5 | 72,66 | 426 |
| 10 | 70.19 | 194 |
| 25 | 68.56 | 124 |
| 50 | 67.99 | 66 |
| 100 | 67.8 | 32 |
Analyzing this table, we can draw the following conclusions:
- The calculation speed decreases in proportion to the number of calculation points in the heat exchanger model.
- The change in the accuracy of the calculation occurs exponentially. As the number of points increases, the refinement at each subsequent increase decreases.
In the case of a plate heat exchanger with a cross flow of the coolant, as in Figure 1, the creation of an equivalent model from elementary computational cells is slightly more complicated. We need to connect the cells in such a way as to organize cross-flow. For 4 cells, the diagram will look as shown in Figure 5.
The coolant flow is divided along the hot and cold branches into two channels, the channels are connected through thermal structures, so that when passing through the channel, the coolant exchanges heat with different channels. Simulating crossflow, the hot coolant flows from left to right (see Fig. 5) in each channel, sequentially exchanging heat with the channels of the cold coolant, which flows from bottom to top (see Fig. 5). The hottest point is in the upper left corner, since the hot coolant exchanges heat with the already heated coolant of the cold channel. And the coldest one is in the lower right, where the cold coolant exchanges heat with the hot coolant, which has already cooled down in the first section.
Figure 5. Cross flow model of 4 computational cells.
Such a model for a plate heat exchanger does not take into account the transfer of heat between cells due to thermal conductivity and does not take into account the mixing of the coolant, since each channel is isolated.
But in our case, the last limitation does not reduce the accuracy, since in the design of the heat exchanger, the corrugated membrane divides the flow into many isolated channels along the coolant (see Fig. 1). Let's see what happens to the calculation accuracy when simulating a plate heat exchanger with an increase in the number of calculation cells.
To analyze the accuracy, we use two options for dividing the heat exchanger into calculated cells:
- Each square cell contains two hydraulic (cold and hot flows) and one thermal element. (see figure 5)
- Each square cell contains six hydraulic elements (three sections each in hot and cold streams) and three thermal elements.
In the latter case, we use two types of connection:
- counter flow of cold and hot streams;
- co-current of cold and hot flow.
A counter flow increases efficiency compared to a cross flow, while a tail flow reduces it. With a large number of cells, the flow is averaged and everything becomes close to the real transverse flow (see Figure 6).
Figure 6. Four-cell cross-flow model with 3 elements.
Figure 7 shows the results of the steady-state stationary temperature distribution in the heat exchanger when air is supplied through the hot line with a temperature of 150 °C, and through the cold line - 21 °C, for various options for partitioning the model. The color and numbers on the cell reflect the average wall temperature in the calculation cell.
Figure 7. Steady-state temperatures for different design schemes.
Table 2 shows the steady-state temperature of the heated air after the heat exchanger, depending on the division of the heat exchanger model into cells.
Table 2. Dependence of temperature on the number of calculated cells in the heat exchanger.
| Model dimension | Steady-state temperature 1 item per cell |
Steady-state temperature 3 items per cell |
| 2x2 | 62,7 | 67.7 |
| 3 × 3 | 64.9 | 68.5 |
| 4x4 | 66.2 | 68.9 |
| 8x8 | 68.1 | 69.5 |
| 10 × 10 | 68.5 | 69.7 |
| 20 × 20 | 69.4 | 69.9 |
| 40 × 40 | 69.8 | 70.1 |
With an increase in the number of computational cells in the model, the final steady-state temperature increases. The difference between the steady-state temperature at different partitions can be considered as an indicator of the accuracy of the calculation. It can be seen that with an increase in the number of calculation cells, the temperature tends to the limit, and the increase in accuracy is not proportional to the number of calculation points.
The question arises, what accuracy of the model do we need?
The answer to this question depends on the purpose of our model. Since this article is about Model-Based Design, we create a model to configure the control system. This means that the accuracy of the model must be comparable to the accuracy of the sensors used in the system.
In our case, the temperature is measured by a thermocouple with an accuracy of ±2.5°C. Any higher accuracy for the purposes of setting up the control system is useless, our real control system will simply “not see” it. Thus, if we assume that the limiting temperature for an infinite number of partitions is 70 °C, then a model that gives us more than 67.5 °C will be of sufficient accuracy. All models with 3 points in the calculation cell and models larger than 5x5 with one point in the cell. (Highlighted in green in Table 2)
Dynamic modes of operation
To evaluate the dynamic mode, let us evaluate the process of temperature change at the hottest and coldest points of the heat exchanger wall for different variants of design schemes. (see fig. 8)
Figure 8. Warming up the heat exchanger. Models of 2x2 and 10x10 dimensions.
It can be seen that the time of the transient process and its very nature practically do not depend on the number of calculation cells, and are determined solely by the mass of the heated metal.
Thus, we conclude that for an honest simulation of a heat exchanger in modes from 20 to 150 ° C, with the accuracy required by the SCR control system, about 10 - 20 calculation points are sufficient.
Setting up a dynamic model by experiment
Having a mathematical model, as well as the data of the heat exchanger blowdown experiment, we need to make a simple correction, namely, to introduce the intensification factor into the model, such that the calculation coincides with the experimental results.
Moreover, using the graphical model creation environment, we will do this automatically. Figure 9 shows the algorithm for selecting heat transfer intensification coefficients. The data obtained from the experiment is fed to the input, the heat exchanger model is connected, and the necessary coefficients for each of the modes are obtained at the output.
Figure 9. Algorithm for selecting the intensification factor based on the results of the experiment.
Thus, we determine the same coefficient for a Nusselt number and eliminate the uncertainty in the calculation formulas. For different modes of operation and temperatures, the values of the correction factors may vary, however, for similar modes of operation (normal operation), they turn out to be very close. For example, for a given heat exchanger for various modes, the coefficient is from 0.492 to 0.655
If we apply a coefficient of 0.6, then in the investigated operating modes, the calculation error will be less than the thermocouple error, thus, for the control system, the mathematical model of the heat exchanger will be fully adequate to this model.
Results of tuning the heat exchanger model
To assess the quality of heat transfer, a special characteristic is used - efficiency:
where:
effhot is the efficiency of the heat exchanger for the hot coolant;
TMountainsin is the temperature at the inlet to the heat exchanger along the path of the hot coolant;
TMountainsout - temperature at the outlet of their heat exchanger along the path of the hot coolant;
Tthe hallin is the temperature at the inlet to the heat exchanger along the path of the cold heat carrier.
Table 3 shows the deviations of the efficiency of the heat exchanger model from the experimental one at various flow rates in the hot and cold lines.
Table 3. Errors in calculating the heat transfer efficiency in %
In our case, the selected coefficient can be used in all modes of operation we are interested in. In the event that at low flow rates, where the error is greater, the required accuracy is not achieved, we can use a variable intensification factor, which will depend on the current flow rate.
For example, in Figure 10, the intensification factor is calculated according to a given formula depending on the current flow rate in the channel cells.
Figure 10. Variable heat transfer enhancement factor.
Conclusions
- Knowledge of physical laws allows you to create dynamic object models for Model-Based Design.
- The model must be verified and adjusted according to the test data.
- Modeling tools should allow the developer to customize the model based on the test results of the object.
- Use the correct model-oriented approach and you will be happy!
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Source: habr.com