2020 Higher Education Club National College Students Mathematical Modeling Contest Questions
(Please read the "National College Students Mathematical Modeling Contest Essay Format Specification") Question A
Furnace temperature curve In the production of electronic products such as integrated circuit boards, printed circuit boards with various electronic components need to be placed in a reflow furnace, and the electronic components are automatically soldered to the circuit board by heating. In this production process, keeping all parts of the reflow furnace at the temperature required by the process is essential to product quality. At present, many works in this area are controlled and adjusted through experimental tests. This question aims to conduct analysis and research through mechanism models. There are several small temperature zones inside the reflow furnace, which can be divided into 4 large temperature zones functionally: preheating zone, constant temperature zone, reflow zone, and cooling zone (as shown in Figure 1). The two sides of the circuit board are placed on the conveyor belt and enter the furnace at a constant speed for heating and welding. Figure 1 Schematic diagram of the cross-section of the reflow furnace. There are 11 small temperature zones in a reflow furnace, as well as the front and back areas of the furnace (see Figure 1). The length of each small temperature zone is 30.5 cm. With a gap of 5 cm, the length of the area before and after the furnace is 25 cm. After the reflow furnace is started, the air temperature in the furnace will stabilize in a short time, after which the reflow furnace can be used for welding. There is no special temperature control for the gap between the area in front of the furnace, the area behind the furnace and the small temperature zone. The temperature is related to the temperature of the adjacent temperature zone, and the temperature near the boundary of each temperature zone may also be affected by the temperature of the adjacent temperature zone. . In addition, the temperature of the production workshop is maintained at 25ºC. After setting the temperature of each temperature zone and the furnace passing speed of the conveyor belt, the temperature of the center of the welding area at some positions can be tested by the temperature sensor, which is called the furnace temperature curve (ie the temperature curve of the center of the welding area). The attachment is the data of the furnace temperature curve in a certain experiment. The set temperature of each temperature zone is 175ºC (small temperature zone 15), 195ºC (small temperature zone 6), 235ºC (small temperature zone 7), 255ºC (small temperature zone) Zone 8 9) and 25ºC (small temperature zone 10~11); the furnace passing speed of the conveyor belt is 70 cm/min; the thickness of the welding zone is 0.15 mm. The temperature sensor starts to work when the temperature in the center of the welding area reaches 30ºC, and the circuit board enters the reflow furnace to start timing. In actual production, the product quality can be controlled by adjusting the set temperature of each temperature zone and the passing speed of the conveyor belt. On the basis of the above experimental set temperature, the set temperature of each small temperature zone can be adjusted within the range of ºC. During the adjustment, the temperature in the small temperature zone 15 should be the same, the temperature in the small temperature zone 89 should be the same, and the temperature in the small temperature zone 10 should be the same.The temperature in 11 is maintained at 25ºC. The speed adjustment range of the conveyor belt is 65 100 cm/min. In the production of reflow furnace circuit board welding, the furnace temperature curve should meet certain requirements, which is called the process limit (see Table 1). Table 1 Process limit limit name Lowest value Highest value Unit temperature rising slope ºC/s Temperature falling slope ºC/s Time during temperature rise from 150ºC to 190ºC s Time when temperature is greater than 217ºC s Peak temperature ºC Please answer the following questions: Question 1 Please establish a mathematical model for the temperature change law of the welding area. Assuming that the conveyor belt passes through the furnace at 78 cm/min, the temperature settings of each temperature zone are 173ºC (small temperature zone 15), 198ºC (low temperature zone 6), 230ºC (small temperature zone 7) and 257ºC (low temperature zone). Zone 8 9), please give the temperature change in the center of the welding area, list the temperature of the middle point of the small temperature zone 3, 6, 7 and the end of the small temperature zone 8, draw the corresponding furnace temperature curve, and Store the temperature at the center of the welding area every 0.5 s in the provided result.csv. Question 2 Assume that the temperature settings of each temperature zone are 182ºC (small temperature zone 15), 203ºC (small temperature zone 6), 237ºC (small temperature zone 7), and 254ºC (small temperature zone 8).9) Please confirm the maximum allowable conveyor belt passing speed. Question 3 During the welding process, the time in which the temperature in the center of the welding area exceeds 217ºC should not be too long, and the peak temperature should not be too high. The ideal furnace temperature curve should minimize the area covered by the peak temperature above 217ºC (the shaded part in Figure 2). Please determine the optimal furnace temperature curve under this requirement, as well as the set temperature of each temperature zone and the furnace passing speed of the conveyor belt, and give the corresponding area. Figure 2 Schematic diagram of furnace temperature curve Problem 4 In the welding process, in addition to meeting the process limit, it is also hoped that the furnace temperature curve with peak temperature as the center line exceeding 217ºC should be as symmetric as possible (see Figure 2). Please combine with Question 3 to further provide the optimal furnace temperature curve, the set temperature of each temperature zone and the conveyor belt passing speed, and give the corresponding index value. 2020 Higher Education Club Cup National College Students Mathematical Modeling Contest Topics (please read the "National College Student Mathematical Modeling Contest Paper Format Specification" first) Question A: In the production of integrated circuit boards and other electronic products, it is necessary to install various electronics The printed circuit board of the component is placed in the reflow furnace, and the electronic component is automatically soldered to the circuit board by heating. In this production process, keeping all parts of the reflow furnace at the temperature required by the process is essential to product quality. At present, many works in this area are controlled and adjusted through experimental tests. This question aims to conduct analysis and research through mechanism models. There are several small temperature zones inside the reflow furnace, which can be divided into 4 large temperature zones functionally: preheating zone, constant temperature zone, reflow zone, and cooling zone (as shown in Figure 1). The two sides of the circuit board are placed on the conveyor belt and enter the furnace at a constant speed for heating and welding. Figure 1 Schematic diagram of the cross-section of the reflow furnace. There are 11 small temperature zones in a reflow furnace and the front and back areas of the furnace (see Figure 1), and the length of each small temperature zone is 30. 5 cm, there is a gap of 5 cm between adjacent small temperature zones, and the length of the area before and after the furnace is 25 cm. After the reflow furnace is started, the air temperature in the furnace will stabilize in a short time, after which the reflow furnace can be used for welding. There is no special temperature control for the gap between the area in front of the furnace, the area behind the furnace and the small temperature zone. The temperature is related to the temperature of the adjacent temperature zone, and the temperature near the boundary of each temperature zone may also be affected by the temperature of the adjacent temperature zone. . In addition, the temperature of the production workshop is maintained at 25ºC. After setting the temperature of each temperature zone and the furnace passing speed of the conveyor belt, the temperature of the center of the welding area at some positions can be tested by the temperature sensor, which is called the furnace temperature curve (ie the temperature curve of the center of the welding area). The attachment is the data of the furnace temperature curve in a certain experiment. The temperature set in each temperature zone is 175ºC (small temperature zone 15), 195ºC (small temperature zone 6), 235ºC (low temperature zone 7), 255ºC (low temperature zone 8 9) and 25ºC (low temperature zone 10~11); the furnace passing speed of the conveyor belt is 70 cm/min; welding The thickness of the area is 0.15 mm. The temperature sensor starts to work when the temperature in the center of the welding area reaches 30ºC, and the circuit board enters the reflow furnace to start timing. In actual production, the product quality can be controlled by adjusting the set temperature of each temperature zone and the passing speed of the conveyor belt. On the basis of the above experimental set temperature, the set temperature of each small temperature zone can be adjusted within the range of ºC. During the adjustment, the temperature in the small temperature zone 15 should be the same, the temperature in the small temperature zone 89 should be the same, and the temperature in the small temperature zone 10 11 should be kept at 25ºC. The speed adjustment range of the conveyor belt is 65 100 cm/min. In the production of reflow furnace circuit board welding, the furnace temperature curve should meet certain requirements, which is called the process limit (see Table 1). Table 1 Process limit limit name Lowest value Highest value Unit temperature rising slope ºC/s Temperature falling slope ºC/s Time during temperature rise from 150ºC to 190ºC s Time when temperature is greater than 217ºC s Peak temperature ºC Please answer the following questions: Question 1 Please establish a mathematical model for the temperature change law of the welding area. Assuming that the conveyor belt passes through the furnace at 78 cm/min, the temperature settings of each temperature zone are 173ºC (small temperature zone 15), 198ºC (low temperature zone 6), 230ºC (small temperature zone 7) and 257ºC (low temperature zone). Zone 8 9), please give the temperature change in the center of the welding area, list the temperature of the middle point of the small temperature zone 3, 6, 7 and the end of the small temperature zone 8, draw the corresponding furnace temperature curve, and Store the temperature at the center of the welding area every 0.5 s in the provided result.csv. Question 2 Assume that the temperature settings of each temperature zone are 182ºC (small temperature zone 15), 203ºC (small temperature zone 6), 237ºC (small temperature zone 7), and 254ºC (small temperature zone 8).9) Please confirm the maximum allowable conveyor belt passing speed. Question 3 During the welding process, the time in which the temperature in the center of the welding area exceeds 217ºC should not be too long, and the peak temperature should not be too high. The ideal furnace temperature curve should minimize the area covered by the peak temperature above 217ºC (the shaded part in Figure 2). Please determine the optimal furnace temperature curve under this requirement, as well as the set temperature of each temperature zone and the furnace passing speed of the conveyor belt, and give the corresponding area. Figure 2 Schematic diagram of furnace temperature curve Problem 4 In the welding process, in addition to meeting the process limit, it is also hoped that the furnace temperature curve with peak temperature as the center line exceeding 217ºC should be as symmetric as possible (see Figure 2). Please combine with Question 3 to further provide the optimal furnace temperature curve, the set temperature of each temperature zone and the conveyor belt passing speed, and give the corresponding index value.