Multi-channel synchronous control temperature field equalization control system and algorithm

Abstract

The invention discloses a multi-channel synchronous control temperature field balance control system and a temperature field air fluid balance control algorithm, the system comprises a computer, a high temperature test box and a PLC controller, the PLC controller is connected with and controls a servo cross-flow fan blade module through an EtherCAT communication technology, and the PLC controller is connected with the servo cross-flow fan blade module based on the temperature field air fluid balance control algorithm. And precise and balanced control of air fluid in the test box is realized. Compared with the prior art, the multi-channel synchronous control temperature field equalization control system and algorithm have the advantages that the uniformity of the temperature field can be effectively controlled, the control precision of the temperature field is high, and the response speed is high.

Description

Technical Field

[0001] The invention relates to the technical field of temperature field uniformity control of a high temperature test chamber in integrated circuit chip packaging, and in particular to a multi-channel synchronous control temperature field balance control system and algorithm. Background Art

[0002] At present, high temperature test chambers are widely used in industry and research, but there are still some shortcomings in terms of temperature field uniformity. The uneven temperature field will lead to substandard integrated circuit chip product quality and deviation in test data, affecting the reliability evaluation of the chip. Therefore, how to improve the uniformity of the temperature field inside the high temperature test chamber has become a technical problem that needs to be solved urgently.

[0003] Integrated circuit chips (ICs) play a vital role in modern electronic devices. With the continuous development of integrated circuit chip technology, the requirements for their packaging and testing are becoming higher and higher. High temperature test chambers are used to simulate the working state of ICs in high temperature environments and test their thermal reliability and durability. In addition, during the integrated circuit packaging process, the curing process also requires the use of high temperature test chambers. The curing process refers to curing the packaging material through high temperature to enhance the stability and reliability of the packaging structure. In order to ensure the accuracy and reliability of the packaging, the high temperature test chamber needs to have the characteristics of high temperature field uniformity, high control accuracy and fast response speed.

[0004] However, existing high-temperature test chambers still have many problems in these aspects and cannot meet high-standard testing requirements: insufficient uniformity of temperature field; low temperature control accuracy; and slow dynamic response speed. Summary of the invention

[0005] The technical problem to be solved by the present invention is to overcome the above technical defects and provide a multi-channel synchronous control temperature field balance control system and algorithm that can effectively control temperature uniformity while having high temperature control accuracy and fast response speed.

[0006] In order to solve the above technical problems, the technical solution provided by the present invention is: a multi-channel synchronous control temperature field balance control system, including a computer, a test box and a PLC controller, the PLC controller is connected to control the servo crossflow fan module, the test box and the servo crossflow fan module are provided with air supply outlets, and multiple groups of experimental modules are also provided in the test box, and temperature sensors are provided near the multiple experimental modules. The computer is connected to control the PLC controller, and the test box is also provided with a test box operation screen.

[0007] Preferably, the servo crossflow blade module includes a servo drive and a crossflow blade controlled by the servo drive, the power source of the crossflow blade is a servo motor, and the PLC controller is connected to control the servo drive.

[0008] Preferably, the temperature sensor model is PT100, and three groups are provided on each layer.

[0009] Preferably, it also includes a control module, a temperature monitoring module and a fan blade control module. The control module is used to collect data from the temperature monitoring module and provide feedback to control the fan blade control module. The temperature monitoring module is used to monitor the temperature of each layer in the test chamber. The fan blade control module controls the temperature uniformity of each layer in the test chamber.

[0010] On the other hand, the present invention discloses a multi-channel synchronous control temperature field balance control algorithm, including establishing an aerodynamic model based on a test box, using a temperature field air fluid balance control algorithm to adjust the temperature of each layer in the test box, and the temperature field air fluid balance control algorithm is constructed based on PID control and fuzzy logic control;

[0011] The temperature monitoring module monitors the temperature of each layer in real time, the control module calculates the temperature uniformity error, and controls the fan blade speed of the fan blade control module through the temperature field air fluid balance control algorithm, so that the temperature uniformity of each layer reaches the predetermined target.

[0012] Preferably, the method comprises the following steps:

[0013] S1: Temperature sensor readings: real-time collection of three temperature sensor data T for each layer ij (t);

[0014] S2: Calculate the average temperature: Calculate the average temperature of each layer Calculate temperature variance: Calculate the temperature variance of each layer Calculate the temperature uniformity error: Calculate the temperature uniformity error e of each layer i (t);

[0015] S3: Convective heat transfer calculation: Calculate the convection heat transfer Q based on the fan blade speed and temperature difference conv ;

[0016] S4: Thermal conductivity calculation: Calculate the thermal conductivity Q based on the temperature difference and material properties cond ;

[0017] S5: Radiation heat transfer calculation: Calculate the radiation heat transfer Q based on the temperature difference rad ;

[0018] S6: PID controller output: The PID controller calculates the initial adjustment signal u based on the error i (t);

[0019] S7: Fuzzy logic controller output: The fuzzy logic controller fine-tunes the PID output according to the error and the error change rate to obtain the adjustment signal Δu i (t);

[0020] S8: Adjusted control signal: Synthesize PID and fuzzy logic controller output to obtain the final control signal u′ i (t);

[0021] S9: Fan blade speed adjustment: adjust the fan blade speed V according to the final control signal i (t), ensuring uniform temperature in each layer.

[0022] Preferably, the mathematical model of the PID controller is:

[0023]

[0024] Where: u(t) is the control output, e(t) is the error, K p is the proportional gain, K i is the integral gain, K d is the differential gain;

[0025] Parameter optimization using critical ratio method:

[0026] K p =0.6K u ; K i =2K p / T u ; K d =K p T u / 8;

[0027] Where: K u is the critical gain, T u is the critical period;

[0028] The fuzzy logic controller includes a i (t) and error change rate Δe i (t) Fine-tune PID output;

[0029] The fuzzy logic controller outputs Δu i (t) is calculated as:

[0030] Δu i (t) = F(e i (t), Δe i (t));

[0031] Wherein, F represents the fuzzy logic control rule.

[0032] The adjusted control signal u i The calculation formula of ′(t) is:

[0033] u i ′(t)=u i (t)+Δui (t);

[0034] Fan blade speed V i (t) According to the control signal u′ i (t) Adjust to ensure uniform temperature on each layer:

[0035] V i (t) = f(u′ i (t));

[0036] Wherein, f is the mapping relationship between the fan blade speed and the control signal.

[0037] Preferably, the mathematical model of the aerodynamic model is:

[0038]

[0039] Where: u is the flow velocity, p is the pressure, ρ is the density, ν is the kinematic viscosity, and f is the external force.

[0040] Preferably, the average temperature formula when the temperature monitoring module monitors the temperature of each layer in real time is:

[0041]

[0042] in: is the average temperature of each layer, (i) is the number of layers, (t) is the time, T i1 (t), T i2 (t), T i3 (t) is the reading of multiple sensors;

[0043] Temperature variance of each layer The calculation formula is:

[0044]

[0045] Calculate the temperature uniformity error e i (t) is calculated as:

[0046] e i (t) = σ i，target -σ i (t);

[0047] Where: i，target is the uniformity of the target temperature.

[0048] Preferably, the method further includes calculating the convective heat transfer according to the motor speed and the temperature difference, and the formula is:

[0049] Q conv =h A(T surface -T fluid );

[0050] Where: h is the convective heat transfer coefficient, A is the heat transfer area, T surface is the surface temperature, T fluid is the fluid temperature;

[0051]

[0052] Where: k is the thermal conductivity, ΔT is the temperature difference, and d is the thickness;

[0053]

[0054] Where: ∈ is the surface emissivity, σ is the Stefan-Boltzmann constant, T surface and T surrounding are the surface temperature and the ambient temperature respectively.

[0055] The advantages of the present invention compared with the prior art are:

[0056] The control system of the present invention has a reasonable structure and uses PLC for centralized control, which reduces the complexity of the system, improves the reliability and stability of the overall system, and can operate stably for a long time to meet high-intensity testing requirements. By optimizing the fluid dynamics characteristics and structural design, energy consumption is reduced, the operating efficiency of the system is improved, and it helps to save energy and reduce operating costs. It has a wide temperature range (room temperature to 170°C), can adapt to different testing environments and requirements, has good versatility and adaptability, and is suitable for the packaging of various integrated circuit chips;

[0057] Through the temperature field air fluid balance control algorithm, each servo motor equipped with a cross-flow fan blade is precisely controlled to promote the movement of air fluid inside the high-temperature test chamber, make the temperature field distribution more uniform, ensure the temperature consistency of each chip package, and thus improve the reliability of the packaging results. The temperature uniformity control system based on the servo motor is adopted, and the temperature is collected by using a high-precision PT100 temperature sensor. Combined with the precise control algorithm of the PLC, the precise temperature control is achieved, and the temperature stability reaches ±0.5℃, and the uniform distribution reaches ±1℃, which meets the high requirements of the test standards;

[0058] With high-speed pulse commands and closed-loop control systems, the servo motor can respond quickly to temperature changes and quickly adjust the rotation speed of the fan blades to maintain a stable temperature field inside the test chamber, thereby improving the dynamic response speed and adjustment capabilities of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 It is a structural diagram of a multi-channel synchronous control temperature field balance control system.

[0060] Figure 2 It is a structural diagram of a multi-channel synchronous control temperature field balance control system.

[0061] Figure 3 This is the servo motor structure diagram.

[0062] Figure 4 It is a structural diagram of the PLC system.

[0063] Figure 5 It is a structural diagram of the test chamber.

[0064] Figure 6 It is a schematic diagram of the structure of the test chamber from the second perspective.

[0065] Figure 7 It is a schematic diagram of the temperature control point structure of an embodiment of the present invention.

[0066] Figure 8 This is the first layer fan blade speed control diagram.

[0067] Fig. 9 This is the second layer fan blade speed control diagram.

[0068] Fig.10 This is the third-layer fan blade speed control diagram.

[0069] Fig.11 This is the 4th layer fan blade speed control diagram.

[0070] Fig.12 This is a schematic diagram of temperature monitoring point A1.

[0071] Fig.13 This is a schematic diagram of temperature monitoring point A2.

[0072] Fig.14 This is a schematic diagram of temperature monitoring point A3.

[0073] Fig.15 This is a schematic diagram of temperature monitoring point B1.

[0074] Fig.16 This is a schematic diagram of temperature monitoring point B2.

[0075] Fig.17 This is a schematic diagram of temperature monitoring point B3.

[0076] Fig.18 This is a schematic diagram of temperature monitoring point C1.

[0077] Fig.19 This is a schematic diagram of temperature monitoring point C2.

[0078] Fig. 20 This is a schematic diagram of temperature monitoring point C3.

[0079] Fig.21 This is a schematic diagram of temperature monitoring point D1.

[0080] Fig. 22 This is a schematic diagram of temperature monitoring point D2.

[0081] Fig.23 This is a schematic diagram of temperature monitoring point D3.

[0082] As shown in the figure: 1. Computer, 2. Test box, 3. PLC controller, 4. Experimental module, 5. Test box operation screen, 6. Servo drive, 7. Crossflow fan blade, 8. Servo motor, 9. Temperature sensor. DETAILED DESCRIPTION

[0083] The present invention is further described in detail below in conjunction with the accompanying drawings.

[0084] Combined with Figure 1-23 As shown, the present invention proposes a multi-channel synchronous control temperature field balance control algorithm, which realizes the precise adjustment of the temperature of each layer in the test box by combining PID control and fuzzy logic control.

[0085] Create a fuzzy controller, set PID control parameters, and initialize the fan blade speed. The system monitors the temperature of each layer in real time, calculates the temperature uniformity error, and adjusts the fan blade speed through the temperature field air fluid balance control algorithm based on PID and fuzzy logic controller to make the temperature uniformity of each layer reach the predetermined target.

[0086] Several thermal resistance temperature sensors are arranged inside the high temperature test chamber, evenly distributed at various key positions, to collect temperature data in real time.

[0087] The temperature sensor is connected to the signal processing module through a signal line. The signal processing module converts the analog signal into a digital signal and transmits it to the PLC.

[0088] The present invention optimizes the temperature distribution by adjusting the rotation speed of the crossflow fan blades. The adjustment of the rotation speed of any one motor requires the synchronous adjustment of the rotation speeds of the other three motors to ensure the stability and uniformity of the overall temperature field.

[0089] First, the temperature data of each layer is collected in real time to calculate the temperature uniformity error. The PID controller generates a preliminary adjustment signal based on the error, and fine-tunes it through the fuzzy logic controller, and finally adjusts the fan blade speed to optimize air flow and heat transfer.

[0090] During use, the influence of wind speed on temperature distribution, as well as heat transfer methods such as convection, conduction and radiation, are taken into consideration. The Navier-Stokes equations are used to simulate the aerodynamic model to accurately describe the influence of air flow on temperature distribution, thereby improving the control effect of temperature uniformity and ensuring that the temperature difference between each layer remains within the ideal range.

[0091] When used specifically:

[0092] The PID controller is used to adjust the fan speed to control the temperature uniformity. Its mathematical model is:

[0093]

[0094] Among them, u(t) is the control output; e(t) is the error; K p is the proportional gain; K i is the integral gain, K d is the differential gain;

[0095] The fuzzy logic controller adjusts the fan speed according to the uniformity error and the error change rate. The fuzzy logic rules are optimized through experimental data and expert knowledge to ensure that the rules can accurately reflect the behavior rules of the system. Based on the definition of fuzzy sets, it is as follows:

[0096] IF UniformityError is Negative AND DeltaUniformityError is NegativeTHEN FanSpeedAdju

[0097] stment is Decrease

[0098] IF UniformityError is Zero AND DeltaUniformityError is Positive THENFanSpeedAdjustment isIncrease;

[0099] The Navier-Stokes equations are used to simulate the flow of air in the test chamber.

[0100]

[0101] Where u is the flow velocity, p is the pressure, ρ is the density, v is the kinematic viscosity, and f is the external force;

[0102] Temperature sensor readings: Let the three temperature sensor readings of layer (i) be, T i1 (t), T i2 (t), T i3 (t), where (i) represents the number of layers, ranging from (1, 2, 3, 4), and (t) is the time. Calculate the average temperature, the average temperature of each layer The calculation formula is:

[0103]

[0104] Temperature variance of each layer The calculation formula is:

[0105]

[0106] Calculate the temperature uniformity error e i (t) is calculated as:

[0107] e i (t) = σ i，targer -σ i (t)

[0108] Among them, σ i，target is the uniformity of the target temperature.

[0109] Convective heat transfer, conductive heat transfer and radiative heat transfer use a more detailed model. The convection heat transfer is calculated based on the motor speed and temperature difference. The heat transfer formulas for convection heat transfer, conductive heat transfer and radiative heat transfer are as follows:

[0110] Q conv =h A(T surface -T fluid )

[0111] h is the convective heat transfer coefficient, A is the heat transfer area, T surface is the surface temperature, T fluid is the fluid temperature.

[0112]

[0113] k is the thermal conductivity, ΔT is the temperature difference, and d is the thickness.

[0114]

[0115] ∈ is the surface emissivity, σ is the Stefan-Boltzmann constant, T surface and T surrounding are the surface temperature and the ambient temperature respectively.

[0116] Tuning of PID controller, parameter K of PID controller p , K i and K d Tune according to the system and use the critical proportionality method to optimize parameters.

[0117] K p =0.6K u ; K i =2K p / T u ; K d =K p T u / 8

[0118] Among them, K u is the critical gain, T u is the critical period.

[0119] The fuzzy logic controller is based on the error e i (t) and error change rate Δe i (t) Fine-tune the PID output. Fuzzy logic controller output Δu i (t) is calculated as:

[0120] Δu i (t) = F(e i (t), Δe i (t))

[0121] Wherein, F represents the fuzzy logic control rule.

[0122] The adjusted control signal u′ i (t) is calculated as:

[0123] u′ i (t) = u i (t)+Δu i (t)

[0124] Fan blade speed V i (t) According to the control signal u′ i (t) Adjust to ensure uniform temperature on each layer:

[0125] V i (t) = f(u i ′(t))

[0126] Wherein, f is the mapping relationship between the fan blade speed and the control signal.

[0127] When the present invention is implemented,

[0128] S1: Temperature sensor 9 reading: real-time collection of data T of three temperature sensors 9 on each layer ij (t);

[0129] S2: Calculate the average temperature: Calculate the average temperature of each layer Calculate temperature variance: Calculate the temperature variance of each layer Calculate the temperature uniformity error: Calculate the temperature uniformity error e of each layer i (t);

[0130] S3: Convective heat transfer calculation: Calculate the convection heat transfer Q based on the fan blade speed and temperature difference conv ;

[0131] S4: Thermal conductivity calculation: Calculate the thermal conductivity Q based on the temperature difference and material properties cond ;

[0132] S5: Radiation heat transfer calculation: Calculate the radiation heat transfer Q based on the temperature differencerad ;

[0133] S6: PID controller output: The PID controller calculates the initial adjustment signal u based on the error i (t);

[0134] S7: Fuzzy logic controller output: The fuzzy logic controller fine-tunes the PID output according to the error and the error change rate to obtain the adjustment signal Δu i (t);

[0135] S8: Adjusted control signal: Synthesize PID and fuzzy logic controller output to get the final control signal u i ′(t);

[0136] S9: Fan blade speed adjustment: adjust the fan blade speed V according to the final control signal i (t), ensuring uniform temperature in each layer.

[0137] During specific use, the temperature changes of each layer and the temperature change curves of each sensor can be displayed in real time through multiple graphic windows, thereby achieving precise control of the temperature in the test chamber.

[0138] The working principle of the present invention is:

[0139] Temperature data collection: Multiple thermal resistance temperature sensors are arranged inside the box to collect temperature data at different locations in real time.

[0140] Data processing and feedback: The data collected by the temperature sensor is transmitted to the PLC through signals, and the PLC performs data analysis and processing.

[0141] Control command sending: Based on temperature data and preset control algorithm, PLC sends high-speed pulse commands to the servo drive.

[0142] Servo motor control: The servo driver controls the rotation of the servo motor according to the PLC instructions. The servo motor drives the fan blades to rotate, disturbing the air flow inside the box.

[0143] Closed-loop feedback system: The rotation information of the servo motor is fed back to the servo driver through the encoder, forming a closed-loop control system to ensure high-speed and precise operation of the servo motor.

[0144] Temperature field regulation: The rotation of the fan blades disturbs the air flow, making the temperature field inside the box more uniform, thus achieving the goal of temperature control.

[0145] After receiving the temperature data, the PLC performs data analysis and processing according to the preset control system and generates corresponding control instructions. The instructions include adjusting the rotation speed of the fan blades to achieve uniform distribution of the temperature field.

[0146] The PLC sends the generated control instructions to the servo driver in the form of high-speed pulses. The servo driver controls the rotation of the servo motor according to the instructions. The servo motor feeds back the rotation information to the servo driver through the encoder, forming a closed-loop control system to ensure the high-speed and precise operation of the servo motor.

[0147] The servo motor drives the fan blades to rotate, disturbing the air flow inside the box to make the temperature field evenly distributed. The PLC continuously adjusts the rotation speed of each crossflow fan blade according to the real-time temperature data to ensure the accuracy and stability of temperature control.

[0148] The contents not described in detail in this specification belong to the prior art known to professional and technical personnel in this field.

[0149] The present invention and its embodiments are described above, and such description is not restrictive. The drawings show only one embodiment of the present invention, and the actual structure is not limited thereto. In short, if ordinary technicians in the field are inspired by it, without departing from the purpose of the invention, they can design a structure and embodiment similar to the technical solution without creativity, which should belong to the protection scope of the present invention.

Claims

1. A multi-channel synchronous control temperature field balance control system, characterized in that: The invention comprises a computer (1), a test box (2) and a PLC controller (3), wherein the PLC controller (3) is connected to control a servo crossflow fan module, an air supply port is provided in the test box (2) and the servo crossflow fan module, a plurality of groups of experimental modules (4) are also provided in the test box (2), temperature sensors (9) are provided near the plurality of experimental modules (4), the computer (1) is connected to control the PLC controller (3), and a test box operation screen (5) is also provided on the test box (2).

2. A multi-channel synchronous control temperature field balance control system according to claim 1, characterized in that: The servo crossflow fan module comprises a servo driver (6) and a crossflow fan (7) controlled by the servo driver (6); the power source of the crossflow fan (7) is a servo motor (8); and the PLC controller (3) is connected to control the servo driver (6).

3. A multi-channel synchronous control temperature field balance control system according to claim 1, characterized in that: The temperature sensor (9) is of PT100 type, and three sets are provided on each layer.

4. A multi-channel synchronous control temperature field balance control system according to claim 3, characterized in that: It also includes a control module, a temperature monitoring module and a fan blade control module. The control module is used to collect data from the temperature monitoring module and provide feedback to control the fan blade control module. The temperature monitoring module is used to monitor the temperature of each layer in the test box (2). The fan blade control module controls the temperature uniformity of each layer in the test box (2).

5. A multi-channel synchronous control temperature field balance control algorithm, applied to the temperature control system according to any one of claims 1 to 4, characterized in that: The method includes establishing an aerodynamic model based on the test box (2), and using a temperature field air fluid balance control algorithm to adjust the temperature of each layer in the test box (2), wherein the temperature field air fluid balance control algorithm is constructed based on PID control and fuzzy logic control; The temperature monitoring module monitors the temperature of each layer in real time, the control module calculates the temperature uniformity error, and controls the fan blade speed of the fan blade control module through the temperature field air fluid balance control algorithm, so that the temperature uniformity of each layer reaches the predetermined target.

6. A multi-channel synchronous control temperature field balance control algorithm according to claim 5, characterized in that: The following steps are involved: S1: Temperature sensor (9) readings: real-time collection of data from three temperature sensors (9) on each floor T ij (t); S2: Calculate the average temperature: Calculate the average temperature of each layer Calculate temperature variance: Calculate the temperature variance of each layer Calculate the temperature uniformity error: Calculate the temperature uniformity error e for each layer i (t); S3: Convective heat transfer calculation: Calculate the convection heat transfer Q based on the fan blade speed and temperature difference conv ; S4: Thermal conductivity calculation: Calculate the thermal conductivity Q based on the temperature difference and material properties cond ; S5: Radiation heat transfer calculation: Calculate the radiation heat transfer Q based on the temperature difference rad ; S6: PID controller output: The PID controller calculates the initial adjustment signal u based on the error i (t); S7: Fuzzy logic controller output: The fuzzy logic controller fine-tunes the PID output according to the error and the error change rate to obtain the adjustment signal Δu i (t); S8: Adjusted control signal: Synthesize PID and fuzzy logic controller output to obtain the final control signal u′ i (t); S9: Fan blade speed adjustment: adjust the fan blade speed V according to the final control signal i (t), ensuring uniform temperature in each layer.

7. A temperature control method for a high temperature test chamber according to claim 6, characterized in that: The mathematical model of the PID controller is: Where: u(t) is the control output, e(t) is the error, K p is the proportional gain, K i is the integral gain, K d is the differential gain; Parameter optimization using critical ratio method: K p =0.6K u ;K i =2K p / T u ;K d =K p T u / 8; Where: K u is the critical gain, T u is the critical period; The fuzzy logic controller includes a i (t) and error change rate Δe i (t) Fine-tune PID output; The fuzzy logic controller outputs Δu i (t) is calculated as: Δu i (t)=F(e i (t),Δe i (t)); Wherein, F represents the fuzzy logic control rule. The adjusted control signal u′ i (t) is calculated as: u' i (t)=u i (t)+Δu i (t); Fan blade speed V i (t) According to the control signal u′ i (t) Adjust to ensure uniform temperature on each layer: V i (t)=f(u′ i (t)); Wherein, f is the mapping relationship between the fan blade speed and the control signal.

8. The multi-channel synchronous control temperature field balance control algorithm according to claim 6, characterized in that: The mathematical model of the aerodynamic model is: Where: u is the flow velocity, p is the pressure, ρ is the density, ν is the kinematic viscosity, and f is the external force.

9. The multi-channel synchronous control temperature field balance control algorithm according to claim 6, characterized in that: The average temperature formula when the temperature monitoring module monitors the temperature of each layer in real time is: in: is the average temperature of each layer, (i) is the number of layers, (t) is the time, T i1 (t), T i2 (t), T i3 (t) is the reading of multiple sensors; Temperature variance of each layer The calculation formula is: Calculate the temperature uniformity error e i (t) is calculated as: e i (t)=σ i，target -s i (t); Where: i，target is the uniformity of the target temperature.

10. The multi-channel synchronous control temperature field balance control algorithm according to claim 6, characterized in that: It also includes the calculation of convective heat transfer based on motor speed and temperature difference, the formula is: Q conv =hA(T surface -T fluid ); Where: h is the convective heat transfer coefficient, A is the heat transfer area, T surface is the surface temperature, T fluid is the fluid temperature; Where: k is the thermal conductivity, ΔT is the temperature difference, and d is the thickness; Where: ∈ is the surface emissivity, σ is the Stefan-Boltzmann constant, T surface and T surrounding are the surface temperature and the ambient temperature respectively.

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