A method for overall identification of the isobaric specific heat capacity of a variable-property working fluid

Abstract

This invention provides a method for the overall identification of the isobaric specific heat capacity of a variable-property working fluid, comprising the following steps: conducting heat transfer tests on the variable-property working fluid under different inlet temperature conditions to obtain the inlet and outlet temperatures and the total heat transfer of the variable-property working fluid under various operating conditions; obtaining the maximum and minimum temperature values ​​of the inlet and outlet temperatures under all operating conditions, dividing the temperature range into intervals with fixed temperature intervals, and using the lower limit temperature of each temperature range as the qualitative temperature of the isobaric specific heat capacity of that temperature range; establishing an artificial neural network for each temperature range, using the inlet and outlet temperatures and the total heat transfer of the heat transfer process under different operating conditions as sample data, and training all artificial neural networks simultaneously to achieve the identification of the isobaric specific heat capacity of each temperature range. This invention effectively improves the identification accuracy by using the overall identification approach of "multiple operating conditions instead of multiple measurement points" to identify the isobaric specific heat capacity of a variable-property working fluid.

Description

Technical Field

[0001] This invention belongs to the field of thermal property identification technology, and specifically relates to a method for the overall identification of the specific heat capacity of a variable-property working fluid at constant pressure. Background Technology

[0002] In recent years, with the rapid development of high-tech fields, the application areas of heat exchange systems have gradually expanded, and the use of variable-property working fluids in heat exchange systems has become increasingly widespread. For example, in transcritical refrigeration systems, supercritical working fluids absorb and release heat, and in evaporators and condensers commonly used in daily life, the working fluid undergoes a variable-property heat exchange process in the above heat exchange systems. The physical properties of the working fluid change drastically with temperature changes, and the assumption of constant physical properties is not applicable. Further research is needed to explore the analysis and optimization calculation of heat exchangers.

[0003] In this type of variable property heat transfer process, the identification of traditional physical property parameters is often affected by the traditional local analysis approach. Multiple thermocouples are placed in the variable property heat transfer experimental section, and the local average physical properties of the working fluid in the small section are calculated and measured by segmentation.

[0004] Traditional methods for measuring the physical properties of heat transfer processes involving changes in physical properties have the following shortcomings:

[0005] Without considering flow, heat transfer and the relationship between upstream and downstream sections, each section is calculated separately. The physical property calculation results are greatly affected by the number and accuracy of the measuring points.

[0006] For the "peak" segment where the physical properties change most drastically, the changes in physical properties during the heat exchange process are very drastic, and it often only accounts for a small part of the heat exchange test section. In the limited space, it is difficult to arrange enough thermocouples, which leads to a large error in the identification of the physical properties of the "peak" segment. Summary of the Invention

[0007] The technical problem solved by this invention is to overcome the shortcomings of the prior art and provide a method for overall identification of the specific heat capacity of a working fluid under constant pressure, which avoids the limitation of the measurement results of physical properties being affected by the number of measurement points in the traditional method and improves the accuracy of identification of abrupt physical properties.

[0008] The technical solution of this invention is:

[0009] A method for the overall identification of the isobaric specific heat capacity of a variable-property working fluid includes the following steps:

[0010] (1) Heat exchange tests were conducted on the variable-property working fluid under different inlet temperature conditions to obtain the inlet and outlet temperatures of the variable-property working fluid under various different working conditions, and the total heat exchange in the heat exchange process under each working condition was measured.

[0011] (2) Obtain the maximum and minimum temperatures of the inlet and outlet temperatures under all operating conditions, and divide the temperature range formed by the minimum and maximum temperatures into fixed temperature intervals. Divide into N temperature ranges, and use the lower limit temperature of each temperature range as the qualitative temperature of the isobaric specific heat capacity of that temperature range;

[0012] (3) Under each operating condition, the state variables of each temperature range are marked according to their inlet and outlet temperatures. The value range of the state variables is [0, 1]. When the temperature range is entirely between the inlet and outlet temperatures, its state variable is marked as 1. When the temperature range is entirely outside the inlet and outlet temperatures, its state variable is marked as 0. When part of the temperature range is between the inlet and outlet temperatures, its state variable is marked as a value between (0, 1). Obtain the state variables of each temperature range under all operating conditions.

[0013] (4) Establish an artificial neural network for each temperature range. The input parameter of each artificial neural network is the state variable of the temperature range, and the output parameter is the isobaric specific heat capacity of the temperature range. Use the state variable of each temperature range under different working conditions and the total heat exchange of the heat exchange process as sample data to train N artificial neural networks at the same time. When the error function value meets the convergence requirement, obtain the isobaric specific heat capacity of the variable working fluid in each temperature range through the artificial neural network corresponding to each temperature range.

[0014] Preferably, the error function is calculated using the following expression:

[0015]

[0016] in, This represents the total heat transfer during a heat exchange process under a specific operating condition. Representing the The specific heat capacity at constant pressure under the corresponding operating condition is output by the neural network for each temperature range. Representing the The state quantities of a temperature range under this operating condition ; Represents mass flow rate.

[0017] Preferably, the convergence condition is as follows: .

[0018] Preferably, in step (3), when a portion of the temperature range falls between the inlet and outlet temperatures, its state variable is marked as a value between (0, 1), specifically:

[0019] If a certain working condition Next, the The lower limit temperature of a temperature range is lower than the outlet temperature of that operating condition, and the upper limit temperature is higher than the outlet temperature of that operating condition. Its state variables are: ,in, Representative working conditions The outlet temperature, Representing the The lower limit temperature of a temperature range; if a certain operating condition Next, the The lower limit temperature of a temperature range is lower than the inlet temperature of that operating condition, and the upper limit temperature is higher than the inlet temperature of that operating condition. The state variables are calculated using the following expression: ,in, Representative working conditions The inlet temperature, Representing the The upper limit temperature of a temperature range.

[0020] Preferably, the artificial neural network includes an input layer, a first hidden layer, a second hidden layer, a third hidden layer, and an output layer.

[0021] Preferably, the number of nodes in the first hidden layer is 8, the number of nodes in the second hidden layer is 16, and the number of nodes in the third hidden layer is 8.

[0022] Preferably, the activation function of the first hidden layer is ReLU, the activation function of the second hidden layer is ReLU, the activation function of the third hidden layer is ReLU, and the activation function of the output layer is sigmoid.

[0023] Preferably, the heat transfer test on the variable-property working fluid under different inlet temperature conditions to obtain the inlet and outlet temperatures of the variable-property working fluid under various different operating conditions specifically includes:

[0024] A counter-current heat exchanger with shell-and-tube design was used to conduct heat exchange tests on the variable-property working fluid. The variable-property working fluid was placed inside the inner tube of the heat exchanger and flowed in the outer tube. The pressure of the variable-property working fluid and the cooling water were kept constant. The inlet temperatures of the variable-property working fluid and the cooling water were changed. Multiple heat exchange tests were conducted and the outlet temperature of the variable-property working fluid after heat exchange was measured to obtain the inlet and outlet temperatures of the variable-property working fluid under various operating conditions.

[0025] Preferably, the fixed temperature interval The value range is 0.1K to 2K.

[0026] Preferably, the number of different working conditions ranges from 10,000 to 10,000.

[0027] The advantages of this invention compared to the prior art are:

[0028] The present invention provides a method for the overall identification of the isobaric specific heat capacity of variable-property working fluids. This method treats the heat exchange process of the variable-property working fluid as a whole for property identification. By measuring a large number of inlet and outlet temperatures under various operating conditions as sample data, a neural network model is established for each temperature range to identify the isobaric specific heat capacity of each temperature range. This avoids the limitation of traditional methods where the property measurement results are affected by the number of measurement points, and solves the problem that traditional methods cannot conduct local investigations of ranges with drastic property changes, effectively improving the accuracy of property identification for variable-property working fluids. Attached Figure Description

[0029] Figure 1 This is a schematic flowchart of the method for overall identification of the specific heat capacity of a variable-property working fluid under constant pressure according to the present invention.

[0030] Figure 2 This is a supercritical CO2 countercurrent heat exchanger according to an embodiment of the present invention;

[0031] Figure 3 This is a thermal circuit diagram of a supercritical CO2 countercurrent heat exchanger based on the inlet temperature difference, as described in an embodiment of the present invention.

[0032] Figure 4 This is a schematic diagram of "multiple working conditions replacing multiple measuring points" in an embodiment of the present invention;

[0033] Figure 5 This refers to a portion of the basic component units occupied at the outlet temperature of the supercritical CO2 heat exchange process in an embodiment of the present invention.

[0034] Figure 6 This is the overall identification approach for the supercritical CO2 property-modifying heat transfer process in the embodiments of the present invention;

[0035] Figure 7 This is an example of an artificial neural network structure in an embodiment of the present invention;

[0036] Figure 8 The error curve of the artificial neural network in the embodiment of the present invention;

[0037] Figure 9 This is a diagram showing the identification results of the specific heat capacity of supercritical CO2 at constant pressure in an embodiment of the present invention.

[0038] Figure 10 This is the error in the identification result of the supercritical CO2 isobaric specific heat capacity in the embodiments of the present invention. Detailed Implementation

[0039] The features and advantages of the present invention will become clearer and more explicit through the following detailed description.

[0040] This invention provides a method for the overall identification of the isobaric specific heat capacity of a variable-property working fluid, such as... Figure 1 As shown.

[0041] Taking the supercritical CO2 heat transfer process as an example, we will first conduct an overall analysis of the supercritical CO2 heat transfer process:

[0042] The structure of a shell-and-tube supercritical CO2 countercurrent heat exchanger is as follows: Figure 2 As shown, supercritical CO2 flows in the inner tube of the heat exchanger, while cooling water flows in the outer tube to cool the supercritical CO2, and the flow direction is opposite to that of the supercritical CO2 fluid. Dissipation from the outer surface of the heat exchanger and axial heat conduction are ignored. It is assumed that both hot and cold fluids undergo isobaric heat exchange, and that the cold-side fluid is a fluid with constant properties.

[0043] The supercritical CO2 countercurrent heat exchanger is uniformly divided into N segments by area, and each segment is denoted as i = 1, 2, …, N along the flow direction of the supercritical CO2 fluid. When the area of ​​each segment is sufficiently small, the isobaric specific heat capacity of supercritical CO2 during heat exchange in that segment can be approximated as a constant. Based on the heat flow method, the thermal path diagram of the supercritical CO2 countercurrent heat exchanger based on the inlet temperature difference can be drawn, as follows. Figure 3 As shown.

[0044] Based on the thermal circuit diagram of the supercritical CO2 countercurrent heat exchanger, the overall heat transport model for this variable property heat transfer process can be obtained:

[0045]

[0046] The inlet temperature of supercritical CO2 is denoted as T. h,in The inlet temperature of the cold water is denoted as T. c,in The heat exchange in each stage is denoted as Q. j The thermal resistance of each heat exchange segment is denoted as R. j ε represents the additional thermal kinetic potential. The subscript h represents the hot fluid, the subscript c represents the cold fluid, and the subscript in represents the inlet.

[0047]

[0048]

[0049]

[0050] in, To exchange heat, For quality flow, For isobaric specific heat capacity, The thermal conductivity of the heat exchanger.

[0051] By employing the solution method of the heat flow model under varying physical properties, numerical simulations of heat transfer processes under different operating conditions can be performed, thereby obtaining a large number of inlet and outlet parameters for the heat transfer process under various operating conditions.

[0052] The following is a comprehensive analysis of the heat transfer process involving changes in physical properties:

[0053] Based on the fundamental characteristics of the heat flow method, the heat circuit diagram establishes the relationship between system topological constraints and component characteristic parameters. From the heat circuit diagram, the additional thermodynamic potential of each segment can be analyzed as follows:

[0054]

[0055] The temperature change of a thermally variable fluid can be expressed as:

[0056]

[0057] Where T is temperature and Q is heat exchange. To add thermal kinetic potential, the subscript h represents the hot fluid, the subscript c represents the cold fluid, the subscript in represents the inlet, and the subscript out represents the outlet. G=m cp, m is the mass flow rate, and cp is the specific heat capacity at constant pressure.

[0058] Traditional methods for measuring the isobaric specific heat capacity of fluids with variable properties involve placing multiple thermocouples in the experimental section of the heat transfer process and calculating the local properties of the working fluid in each segment. This method neglects the interaction between flow, heat transfer, and preceding and following segments, calculating each segment separately and thus fragmenting the overall heat transfer process. Furthermore, the calculated property results are highly dependent on the number and accuracy of the measurement points. This invention proposes a holistic parameter identification approach that utilizes "multiple operating conditions instead of multiple measurement points." For example... Figure 4 As shown, the isobaric specific heat capacity of the hot-side fluid at each corresponding temperature segment is considered as a fundamental element characteristic, represented by a small square in the figure. Under different operating conditions with varying inlet and outlet temperatures of the hot-side fluid, each operating condition is composed of different small squares, containing different fundamental element characteristics. By training an artificial neural network under a large number of known operating conditions with different inlet and outlet temperatures, the characteristic of each fundamental element—that is, the supercritical CO2 isobaric specific heat capacity corresponding to the qualitative temperature of each segment—can be identified holistically.

[0059] Next, we will combine artificial neural networks to achieve overall identification of feature parameters:

[0060] like Figure 2 The basic parameters of the supercritical CO2 countercurrent heat exchanger are: length L = 20 m, inner and outer diameters of the inner tube and outer inner diameter r1 = 5 × 10⁻⁶ m. -3 m, r2 = 6.35 × 10 -3 m, r3 = 8.3 × 10 -3 m, the inner and outer tubes are made of stainless steel and an insulating material, respectively. The heat transfer coefficient of stainless steel is λ. w =16 W·m -1 ·K-1 Because the isobaric specific heat capacity of supercritical CO2 at the same temperature differs under different pressures, the heat transfer pressure of the fluid should be constant in numerical simulations under different operating conditions. The pressure p of supercritical CO2... CO2 Take 12 MPa, cooling water pressure p water =0.1 MPa.

[0061] Next, sufficient operating conditions with different inlet and outlet temperatures of supercritical CO2 need to be obtained for the training of the artificial neural network. In this embodiment, numerical simulation is used to obtain these conditions, but they can also be obtained through experiments.

[0062] To numerically simulate sufficient operating conditions with different supercritical CO2 inlet and outlet temperatures for training the artificial neural network, the supercritical CO2 inlet temperature is taken as T. CO2,in = 314 K …… T CO2,in = 388.5 K, with supercritical CO2 inlet temperatures set at intervals of 0.5 K. The cooling water inlet temperatures are respectively set to T. water,in = 283 K …… T water,in = 303K, with an inlet temperature for cooling water set every 0.5K, and the mass flow rates of both being taken as m. CO2 = m water = 7.5 × 10⁻² kg, …… m CO2 = m water = 11.5 × 10 -2 kg, every 1×10 -2 kg sets the mass flow rate of the fluid on both sides. There are a total of 5 × 41 × 150 = 30750 different operating conditions. The solution method of the heat flow model under the variable property operating conditions is used to numerically simulate the different operating conditions, and obtain the inlet and outlet temperatures of the supercritical CO2 variable property heat transfer process and the corresponding total heat transfer.

[0063] Because different operating conditions of the supercritical CO2 property-modifying heat transfer process contain different basic element units, the heat transfer process is segmented by temperature under each operating condition. Each segment has a fixed temperature difference, such as 0.5 K. Each basic element unit represents the isobaric specific heat capacity of supercritical CO2 at that specific temperature. The number and characteristics of the basic element units differ under operating conditions with different inlet and outlet temperature differences. If two operating conditions differ only by a single basic element unit, the characteristics of that basic element unit can be obtained. Given a large number of operating conditions with different inlet and outlet temperatures, an artificial neural network can be trained to identify the characteristics of each basic element unit.

[0064] After a comprehensive analysis of the supercritical CO2 heat transfer process with varying physical properties, the inlet and outlet temperatures of the supercritical CO2 fluid along the heat transfer process were identified based on numerical simulations. The data analysis revealed that the highest temperature reached by supercritical CO2 in different heat transfer processes reached T. h.max = 388.5K, the lowest temperature reached T h.min = 290K. Within this temperature range, taking a fixed temperature difference of ΔT = 0.5 K, we obtain N basic component units.

[0065]

[0066] In this embodiment, N = 198, and the basic element units are labeled 1, 2, ..., N according to their qualitative temperatures from low to high. The qualitative temperature Ti of each basic element unit is taken as the lower limit of the corresponding temperature range. For any given operating condition, only a portion of the basic element units will be used. A 1×N label vector is generated for each operating condition to mark which basic element units are used under that condition.

[0067] For a given operating condition k, if a basic component unit i is used in this operating condition, it meets the determination formula:

[0068]

[0069] Then, the i-th element of the marker vector is recorded as 1; otherwise, it is recorded as 0. Because the process is segmented according to a fixed temperature difference, the supercritical CO2 heat exchange process may only occupy a portion of the basic element units at the outlet temperature, not all of them. Figure 5 As shown, the shaded area represents the unoccupied portion of the basic element unit j.

[0070] At this point, the j-th element of the marker vector is denoted as , For each working condition, we can obtain its corresponding label vector m:

[0071]

[0072] The overall identification approach for supercritical CO2 property-modifying heat transfer processes is as follows: Figure 6 As shown. For each operating condition with different inlet and outlet temperatures, its label vector m is obtained. These vectors correspond one-to-one with N basic component units, and N identical artificial neural networks are constructed and trained simultaneously, as shown. Figure 6 As shown in the selected section, the input parameter of each artificial neural network is the element i corresponding to the label vector m for this working condition, and the output parameter is the supercritical CO2 isobaric specific heat capacity c identified at this qualitative temperature. p,i The structure of a neural network is as follows: Figure 7As shown, a 5-layer fully connected neural network is used, with the number of nodes being 1, 8, 16, 8, and 1 respectively. The activation functions for each layer are f1=relu, f2=relu, f3=relu, and f4=sigmoid, respectively. The test set accounts for 10% of the total data. For each training iteration, the neural network obtains the output parameters c. p,1 ...c p,N Then, calculate the heat transfer Qi within each small temperature range:

[0073]

[0074] The corresponding total heat exchange under the operating conditions is Error function of artificial neural networks The expression is:

[0075]

[0076] when When the convergence condition is met, the isobaric specific heat capacity of the variable working fluid in each temperature range can be obtained through the BP neural network corresponding to each temperature range.

[0077] Therefore, it is possible to identify the characteristics of each basic component unit based on a large number of different inlet and outlet temperatures using artificial neural networks. Only the hot-side inlet and outlet temperatures and the total heat transfer in the supercritical CO2 variable property heat transfer process are needed to identify the local isobaric specific heat capacity along the heat transfer process. This is an identification approach that uses the overall system to identify the local components, replacing multiple measurement points with multiple operating conditions.

[0078] The following are the overall recognition results based on artificial neural networks in this embodiment:

[0079] Based on the proposed overall identification approach for supercritical CO2 variable property heat transfer processes and numerical simulations, a large number of inlet and outlet temperatures and total heat transfer data for different supercritical CO2 variable property heat transfer processes are obtained. The error function of the artificial neural network increases with the number of training steps, as shown below. Figure 8 As shown, the training effect is good, and the error between the heat exchange obtained from the training and the total heat exchange is within 3%.

[0080] The output of the neural network, namely the supercritical CO2 isobaric specific heat capacity identified at different qualitative temperatures, was compared with the supercritical CO2 isobaric specific heat capacity in the property library. Since the numerical simulation used the property data of supercritical CO2 from the property library, the supercritical CO2 isobaric specific heat capacity in the numerical simulation experiment was consistent with the data in the property library. Figure 9 The figure shows a comparison between the isobaric specific heat capacity of supercritical CO2 in the property library and the isobaric specific heat capacity of supercritical CO2 identified by an artificial neural network. Figure 10The figure shows the error of the identification results. As can be seen from the figure, the error of most identification results is within 3%, especially in the peak segment where the physical properties change drastically, the relative error is small. The method provided by this invention can identify the thermophysical properties of heat transfer processes with changing physical properties with high accuracy.

[0081] The contents not described in detail in this specification are common knowledge to those skilled in the art.

Claims

1. A method for overall identification of the isobaric specific heat capacity of a variable-property working fluid, characterized in that, Includes the following steps: (1) Heat exchange tests were conducted on the variable-property working fluid under different inlet temperature conditions to obtain the inlet and outlet temperatures of the variable-property working fluid under various different working conditions, and the total heat exchange in the heat exchange process under each working condition was measured. (2) Obtain the maximum and minimum temperatures of the inlet and outlet temperatures under all operating conditions, and divide the temperature range formed by the minimum and maximum temperatures into fixed temperature intervals. Divide into N temperature ranges, and use the lower limit temperature of each temperature range as the qualitative temperature of the isobaric specific heat capacity of that temperature range; (3) Under each operating condition, the state variables of each temperature range are marked according to their inlet and outlet temperatures. The value range of the state variables is [0, 1]. When the temperature range is entirely between the inlet and outlet temperatures, its state variable is marked as 1. When the temperature range is entirely outside the inlet and outlet temperatures, its state variable is marked as 0. When part of the temperature range is between the inlet and outlet temperatures, its state variable is marked as a value between (0, 1). Obtain the state variables of each temperature range under all operating conditions. (4) Establish an artificial neural network for each temperature range. The input parameter of each artificial neural network is the state variable of the temperature range, and the output parameter is the isobaric specific heat capacity of the temperature range. Use the state variable of each temperature range under different working conditions and the total heat exchange of the heat exchange process as sample data to train N artificial neural networks at the same time. When the error function value meets the convergence requirement, obtain the isobaric specific heat capacity of the variable working fluid in each temperature range through the artificial neural network corresponding to each temperature range.

2. The method for overall identification of the isobaric specific heat capacity of a variable-property working fluid according to claim 1, characterized in that, The error function is calculated using the following expression: in, This represents the total heat transfer during a heat exchange process under a specific operating condition. Representing the The specific heat capacity at constant pressure under the corresponding operating condition is output by the neural network for each temperature range. Representing the The state quantities of a temperature range under this operating condition ; Represents mass flow rate.

3. The method for overall identification of the isobaric specific heat capacity of a variable-property working fluid according to claim 2, characterized in that, The specific convergence condition is as follows: .

4. The method for overall identification of the isobaric specific heat capacity of a variable-property working fluid according to claim 3, characterized in that, In step (3), when a portion of the temperature range falls between the inlet and outlet temperatures, its state variable is marked as a value between (0, 1), specifically: If a certain working condition Next, the The lower limit temperature of a temperature range is lower than the outlet temperature of that operating condition, and the upper limit temperature is higher than the outlet temperature of that operating condition. Its state variables are: ,in, Representative working conditions The outlet temperature, Representing the The lower limit temperature of a temperature range; if a certain operating condition Next, the The lower limit temperature of a temperature range is lower than the inlet temperature of that operating condition, and the upper limit temperature is higher than the inlet temperature of that operating condition. The state variables are calculated using the following expression: ,in, Representative working conditions The inlet temperature, Representing the The upper limit temperature of a temperature range.

5. The method for overall identification of the isobaric specific heat capacity of a variable-property working fluid according to claim 4, characterized in that, The artificial neural network includes an input layer, a first hidden layer, a second hidden layer, a third hidden layer, and an output layer.

6. The method for overall identification of the isobaric specific heat capacity of a variable-property working fluid according to claim 5, characterized in that, The first hidden layer has 8 nodes, the second hidden layer has 16 nodes, and the third hidden layer has 8 nodes.

7. The method for overall identification of the isobaric specific heat capacity of a variable-property working fluid according to claim 6, characterized in that, The activation function of the first hidden layer is ReLU, the activation function of the second hidden layer is ReLU, the activation function of the third hidden layer is ReLU, and the activation function of the output layer is sigmoid.

8. A method for overall identification of the isobaric specific heat capacity of a variable-property working fluid according to any one of claims 1 to 7, characterized in that, The heat transfer test of the variable-property working fluid under different inlet temperature conditions, to obtain the inlet and outlet temperatures of the variable-property working fluid under various operating conditions, specifically includes: A counter-current heat exchanger with shell-and-tube design was used to conduct heat exchange tests on the variable-property working fluid. The variable-property working fluid was placed inside the inner tube of the heat exchanger and flowed in the outer tube. The pressure of the variable-property working fluid and the cooling water were kept constant. The inlet temperatures of the variable-property working fluid and the cooling water were changed. Multiple heat exchange tests were conducted and the outlet temperature of the variable-property working fluid after heat exchange was measured to obtain the inlet and outlet temperatures of the variable-property working fluid under various operating conditions.

9. The method for overall identification of the isobaric specific heat capacity of a variable-property working fluid according to claim 8, characterized in that, The fixed temperature interval The value range is 0.1K to 2K.

10. The method for overall identification of the isobaric specific heat capacity of a variable-property working fluid according to claim 9, characterized in that, The number of different working conditions ranges from 10,000 to 10,000.

Images