SPÉCIFICATION TECHNIQUE CEI IEC TECHNICAL SPECIFICATION TS 60695 11 40 Première édition First edition 2002 02 Essais relatifs aux risques du feu � Partie 11 40 Flammes d''''essai � Essais de confirmation[.]
Dynamique générale
Les flammes sont constituées de mélanges gazeux à réaction chimique exothermique d'oxydant (généralement un mélange d'oxygène ou d'air) et de réducteur ou combustible
Typically, a combustible gas or steam is used For standardized test flames, appropriate equipment with combustible material supplies delivers a continuous flow of gaseous reactants to sustain the desired standardized flame.
Brûleurs et flammes de diffusion
Diffusion flames are generated by a straightforward flow of combustible gas that ignites upon mixing with air at the outlet of the equipment The advantages of diffusion flames include the simplicity of the apparatus, often just a basic tube, and a more accurate representation of real flames that may be encountered in the equipment.
L'inconvénient est qu'elles sont moins stables d'un point de vue géométrique.
Flammes à prémélange et brûleurs à mélange préalable
Premixed flames are generated by mixing a portion of the combustion air with the combustion gas before reaching the combustion point at the burner tube's end The remaining combustion air is supplied similarly to diffusion flames The resulting flame features an inner cone that is typically a light blue color, formed by the mixture of premixed gas and air along with excess combustible gas, and an outer cone that appears a slightly darker blue as the additional air needed diffuses into the upper part of the flame The inner cone operates at a significantly lower temperature and is reducing, while the outer cone is much hotter and oxidizing.
Flame stabilizer assemblies are sometimes added around the outlet orifice to enhance flame stability and prevent it from escaping the upper part of the burner They provide a layer of premixed gas with reduced velocity at the interface between the main central flow and calm air, moderating the velocity variation between these flows Gases enter the stabilized flame through calibrated passages at the upper end of the burner tube Premixed flames offer advantages such as higher efficiency and flame temperature, along with the ability to regulate both combustion air and fuel gas However, they require more complex equipment compared to diffusion flames In premixed burners, combustion air can be either directly controlled or introduced through an adjustable shutter assembly that creates a Venturi effect.
4.3.1 Brûleurs à prémélange: air et gaz mesurés
These designs ensure the regulation of both the measured combustible gas and air Both elements can be introduced through specified orifices with defined flow rates and back pressures Typically, an additional ramp is utilized for the measured air, while a restrictive orifice for the combustible gas is essential to achieve a high gas flow rate for an optimal mixture The measurement of both air and combustible gas significantly enhances the control of the resulting standardized flames.
4.3.2 Brûleurs de type Venturi à prémélange d'air
In traditional designs, such as Bunsen and Tirrill burners, air is drawn in through an adjustable shutter that utilizes the Venturi effect of high-speed combustible gas from a restrictive orifice This orifice can be either fixed or adjustable with a conical needle The settings for gas flow, back pressure, and air shutter are particularly critical These burners, lacking an air distributor, have fewer components and are generally simpler and more compact than measured air designs.
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4 Test flames, burner types and dynamics
Flames are formed from exothermic chemical reactions between gaseous mixtures of oxidizers, such as oxygen or air, and fuels, typically in the form of gas or vapor To create standardized test flames, specialized equipment is used to provide a continuous supply of combustible materials, ensuring a consistent flow of the necessary gaseous reactants to sustain the desired flame characteristics.
Diffusion flames are generated when fuel gases flow and ignite upon mixing with air at the exit orifice of the equipment The primary benefits of diffusion flames include the simplicity of the hardware, often consisting of just a basic tube, and their ability to closely mimic real flames found in various applications However, a notable drawback is their geometric instability.
4.3 Pre-mixed flames and burners
Pre-mixed flames are created by combining a portion of combustion air with fuel gas before reaching the combustion point at the burner tube's end The remaining combustion air is introduced similarly to diffusion flames This results in a flame with a lighter blue inner/lower cone, which consists of the pre-mixed gas and air mixture with excess fuel gas, and a darker blue outer/upper cone where additional air diffuses The inner/lower cone operates at a lower temperature and is chemically reducing, while the outer/upper cone is hotter and more oxidizing.
Flame stabilizer assemblies are installed around the exit orifice to improve flame stability and prevent flames from escaping the burner top They create a layer of pre-mixed gases with reduced velocity, which helps moderate the velocity gradient between the high-velocity central flow and the surrounding still air Gases enter the stabilized flame through metering holes in the burner tube's upper end Pre-mixed flames offer higher efficiency and temperature, allowing for precise metering of combustion air and fuel gas, although they require more complex hardware compared to diffusion flames In pre-mixed burners, combustion air can be metered directly or supplied through an adjustable open shutter assembly utilizing the Venturi effect.
4.3.1 Metered air pre-mixed burners
These designs enable precise control of metered air and fuel gas, which are introduced through designated orifices with specific flow rates and back pressures Typically, an additional manifold assembly is utilized for the metered air, while a restrictive fuel gas orifice is essential to achieve a high gas stream flow rate for optimal mixing The combination of metered air and fuel gas significantly enhances the control of standardized flames.
4.3.2 Venturi air pre-mixed burners
Traditional burner designs, such as Bunsen and Tirrill types, utilize the Venturi effect to draw air through an adjustable shutter, driven by a high-speed fuel gas stream from a restrictive orifice This orifice can be fixed or adjustable with a tapered needle, making gas flow rate, back pressure, and air shutter adjustments crucial for optimal performance These burners are simpler and less bulky than metered air designs, as they do not include an air manifold and have fewer components.
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Their operation is simpler, and in the case of adjustable orifices, they allow the use of the same equipment to produce a range of flame sizes However, the flames are less consistent and not as well-suited to the requirements of testing.
Gaz combustibles
A unique chemical compound, such as methane, propane, or butane, is often required to have a purity of at least 98% for methane and propane, or at least 95% for butane However, equivalent performance mixtures may also be suitable It is important to specify the highest possible technical purity.
Généralités
Calorimetry tests on a copper block utilize a standardized setup where a mass is placed in the test flame, and the time/temperature characteristics are recorded to indicate the heat rate delivered to the copper sample The measured hourly rate of temperature increase is directly proportional to the rate of change of the net enthalpy of the copper block and inversely proportional to its heat capacity The hourly rate of change in the net enthalpy is influenced by convective, conductive, and radiative effects.
Procédure et disposition du matériel
The flames should be adjusted and optimized in a draft-free environment A copper block is positioned in the flame, and the time required for the temperature of the copper block to exceed a specified range is recorded.
Thermocouple
The thermocouple must be tailored to operate at the expected maximum temperature, ensuring it is sufficiently minimized to avoid affecting the thermal capacity of the copper block It should also possess adequate roughness to allow for the suspension of the copper block Additionally, the thermocouple needs to withstand the extremely high temperatures found just above the copper block in the upper flame zone during testing Its connection should be centrally located and inserted to a specified depth within the copper block through a drilled hole Due to potential challenges with other fastening methods, it is advisable to secure the wire by deforming the copper block material around the thermocouple while taking care to prevent damage.
Nature du bloc de cuivre
The nature of the copper block, identified as Cu-ETP UNS C11000, is characterized by its oxygen-containing electrolytic plating This electrical-grade copper exhibits several essential properties: a melting point significantly higher than the maximum testing temperatures, high thermal diffusivity, a well-defined chemical composition as an easily available elemental metal, and suitability for machining.
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Adjustable orifices simplify operation and enable the same equipment to generate various flame sizes However, the flames produced may lack consistency, making them less ideal for testing purposes.
A single chemical compound such as methane, propane or butane, of a purity not less than
For optimal performance, it is often recommended to use methane and propane with a purity of 98% or higher, and butane with at least 95% purity However, suitable mixtures that perform equivalently may also be acceptable It is essential to specify a technical laboratory grade with the highest possible purity.
Copper block calorimetry tests involve a standard setup where a mass is placed in the test flame, allowing for the recording of time and heating characteristics to assess the heat delivery rate to the copper block The rate of temperature increase is directly proportional to the copper block's net heat content increase and inversely proportional to its heat capacity This change in net heat content is influenced by convective, conductive, and radiative effects.
To ensure accurate results, flames must be finely tuned and evaluated in a draught-free setting A copper block is placed within the flame, and the duration needed for the temperature of the copper block to rise within a designated range is recorded.
The thermocouple shall be suitable for operation up to the maximum anticipated temperature.
The thermocouple must be small enough to avoid affecting the heat capacity of the copper block while being robust enough to support the block's suspension It should endure the high temperatures found just above the copper block in the flame's upper region during testing The junction of the thermocouple needs to be positioned centrally and at a specific depth within the copper block, achieved by inserting it into a drilled hole To mitigate potential issues with alternative attachment methods, it is advisable to secure the thermocouple wire by peening the surrounding copper block material, ensuring that no damage occurs.
The copper block material (electroplated tough pitch copper) is identified as Cu-ETP
UNS C11000 is an electrical grade copper known for its key characteristics, including a melting point significantly higher than expected test temperatures, high thermal diffusivity, a well-defined chemical composition as a readily available quality grade elemental metal, and excellent machinability.
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Masse du bloc de cuivre
The mass of the copper block directly affects the testing time range due to the material's thermal capacity, making it essential to select it as the primary parameter for determining suitable ranges.
Géométrie du bloc de cuivre
The geometry of the copper block should be as simple as possible, clearly defined, and suitable for practical production Its size and shape must be designed to minimize disruption to the flame, ensuring it is completely enveloped in the critical areas of the flame.
Positionnement du bloc de cuivre
It is recommended to suspend the copper block using the thermocouple wire and position it at the start of the test at the center and within the flame envelope, at a defined distance above the burner material, ensuring it is fully enveloped in the critical part of the flame For circular premixed flames, the optimal position is typically just above the inner cone and entirely within the outer cone Linear flames require special placement to ensure complete contact of the probe with the critical part of the flame (see Figure 1).
Plages de températures d'essai
The copper block experiences a rise in temperature due to a combination of convection and conduction, while it loses heat through thermal radiation.
The radiation effect is proportional to the fourth power of the absolute temperature of the copper block and significantly depends on its surface emissivity To minimize variability due to changes in surface emissivity, it is advisable to limit the maximum test temperature to levels where radiation is minimal, yet sufficiently high to achieve an adequate range for appropriate testing durations A temperature range between 100 °C and 700 °C is deemed suitable.
General dynamics
Flames are formed from exothermic chemical reactions between gaseous mixtures of oxidizers, such as oxygen or air, and reducers or fuels, typically in the form of fuel gas or vapor To create standardized test flames, specialized equipment is used to provide a continuous supply of combustible materials, ensuring a consistent flow of the necessary gaseous reactants to sustain the desired flame characteristics.
Diffusion flames and burners
Diffusion flames are generated when fuel gases flow and ignite upon mixing with air at the exit orifice of the equipment The primary benefits of diffusion flames include the simplicity of the hardware, often consisting of just a basic tube, and their ability to closely mimic real flames found in various applications However, a notable drawback is their geometric instability.
Pre-mixed flames and burners
Pre-mixed flames are created by combining a portion of combustion air with fuel gas before reaching the combustion point at the burner tube's end The remaining combustion air is introduced similarly to diffusion flames This results in a flame with a lighter blue inner/lower cone, which consists of the pre-mixed gas and air mixture with excess fuel gas, and a faint darker blue outer/upper cone where additional air diffuses The inner/lower cone operates at a lower temperature and is chemically reducing, while the outer/upper cone is hotter and more oxidizing.
Flame stabilizer assemblies are utilized around the exit orifice to improve flame stability and prevent flames from escaping the burner top They create a layer of pre-mixed gases with reduced velocity, which helps moderate the velocity gradient between the high-velocity central flow and the surrounding still air Gases enter the stabilized flame through metering holes at the burner tube's upper end Pre-mixed flames offer higher efficiency and temperature, allowing for precise metering of combustion air and fuel gas, although they require more complex hardware compared to diffusion flames In pre-mixed burners, combustion air can be metered directly or supplied through an adjustable open shutter assembly using the Venturi effect.
4.3.1 Metered air pre-mixed burners
These designs enable precise control of metered air and fuel gas through designated orifices with specific flow rates and back pressures Typically, an additional manifold assembly is utilized for metered air, while a restrictive fuel gas orifice is essential to achieve a high gas stream flow rate for optimal mixing The combination of metered air and fuel gas significantly enhances the control of standardized flames.
4.3.2 Venturi air pre-mixed burners
Traditional burner designs, such as Bunsen and Tirrill types, utilize the Venturi effect to draw air through an adjustable shutter, driven by a high-speed fuel gas stream from a restrictive orifice, which can be fixed or adjustable with a tapered needle The gas flow rate, back pressure, and air shutter adjustments are crucial for optimal performance These burners are simpler and less bulky than metered air designs, as they do not include an air manifold and have fewer components.
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Their operation is simpler, and in the case of adjustable orifices, they allow the use of the same equipment to produce a range of flame sizes However, the flames are less consistent and not as well-suited to the requirements of testing.
A unique chemical compound such as methane, propane, or butane is often required to have a purity of at least 98% for methane and propane, or at least 95% for butane However, equivalent performance mixtures may also be suitable It is important to specify the highest possible technical purity.
Calorimetry tests on a copper block utilize a standardized setup where a mass is placed in the test flame, and the time/temperature characteristics are recorded to indicate the heat rate delivered to the copper sample The measured hourly rate of temperature increase is directly proportional to the net enthalpy increase of the copper block and inversely proportional to its heat capacity The hourly rate of change in the net enthalpy of the copper block results from convective, conductive, and radiative effects.
5.2 Procédure et disposition du matériel
Flames should be adjusted and optimized in a draft-free environment A copper block is placed in the flame, and the time required for the temperature of the copper block to exceed a specified range is recorded.
The thermocouple must be tailored to operate at the expected maximum temperature, ensuring it is sufficiently small to avoid affecting the thermal capacity of the copper block It should also be rough enough to allow for the suspension of the copper block Additionally, the thermocouple must withstand the extremely high temperatures found just above the copper block in the upper flame zone during testing Its connection should be centrally located and inserted to a specified depth within the copper block through a drilled hole Due to potential challenges with other fastening methods, it is advisable to secure the wire by deforming the copper block material around the thermocouple while taking care to prevent damage.
5.4 Nature du bloc de cuivre
The nature of the copper block, identified as Cu-ETP UNS C11000, is characterized by its electroplated oxygen content This electrical-grade copper exhibits several essential properties: a melting point significantly higher than the maximum testing temperatures, high thermal diffusivity, a well-defined chemical composition as an easily available elemental metal, and suitability for machining.
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Adjustable orifices simplify operation and enable the same equipment to generate various flame sizes However, the flames produced may lack consistency, making them less ideal for testing purposes.
Fuel gases
A single chemical compound such as methane, propane or butane, of a purity not less than
For optimal performance, it is often recommended to use methane and propane with a purity of 98% or higher, and butane with at least 95% purity However, mixtures that perform equivalently may also be acceptable It is essential to specify a technical laboratory grade with the highest purity available.
General
Copper block calorimetry tests involve a standard setup where a mass is placed in the test flame, allowing for the recording of time and heating characteristics to assess the heat delivery rate to the copper block The rate of temperature increase is directly proportional to the net heat content increase of the copper block and inversely proportional to its heat capacity This change in net heat content is influenced by convective, conductive, and radiative effects.
Procedure and hardware arrangement
To ensure accurate results, flames must be finely tuned and evaluated in a draught-free setting A copper block is placed within the flame, and the duration needed for the temperature of the copper block to rise within a designated range is recorded.
The thermocouple shall be suitable for operation up to the maximum anticipated temperature.
The thermocouple must be small enough to avoid affecting the heat capacity of the copper block while being robust enough to support the block's suspension It should endure the high temperatures found just above the copper block in the flame's upper region during testing The thermocouple junction needs to be positioned centrally and at a specific depth within the copper block, achieved by inserting it into a drilled hole To mitigate potential issues with other attachment methods, it is advisable to secure the wire by peening the surrounding copper block material, ensuring that no damage occurs.
Copper block material
The copper block material (electroplated tough pitch copper) is identified as Cu-ETP
UNS C11000 is an electrical grade copper known for its key characteristics, including a melting point significantly higher than expected test temperatures, high thermal diffusivity, a well-defined chemical composition as a quality grade elemental metal, and excellent machinability.
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
5.5 Masse du bloc de cuivre
The mass of the copper block directly affects the range of testing times due to the material's thermal capacity, making it essential to select it as the primary parameter for determining suitable ranges.
5.6 Géométrie du bloc de cuivre
The geometry of the copper block should be as simple as possible, clearly defined, and suitable for practical production Its size and shape must be designed to minimize flame disturbance while ensuring it is completely enveloped in the critical areas of the flame.
5.7 Positionnement du bloc de cuivre
It is recommended to suspend the copper block using the thermocouple wire and position it at the start of the test at the center and within the flame envelope, at a defined distance above the burner material, ensuring it is fully enveloped in the critical part of the flame For circular premixed flames, the optimal position is typically just above the inner cone and entirely within the outer cone Linear flames require special placement to ensure complete contact of the probe with the critical part of the flame (see Figure 1).
The copper block experiences an increase in temperature due to a combination of convection and conduction, while it loses heat through thermal radiation.
The radiation effect is proportional to the fourth power of the absolute temperature of the copper block and significantly depends on its surface emissivity To minimize variability in surface emissivity changes, it is advisable to limit the maximum test temperature to levels where radiation is minimal, yet sufficiently high to achieve an adequate range for appropriate testing durations A temperature range between 100 °C and 700 °C is deemed suitable.
The range of test durations should ideally be reasonable to ensure meaningful recording This range is best determined by selecting the mass of the copper block and the maximum test temperature It is recommended that the test duration falls between 30 seconds and 90 seconds whenever possible.
7 Dynamique de calorimétrie du bloc de cuivre et théorie
The effectiveness of a test flame is directly related to its ability to provide heat to the test specimen Calorimetry tests on a copper block utilize a standardized setup where a mass is placed in the test flame, and the time/temperature characteristics are recorded to indicate the heat rate supplied to the copper block The measured hourly rate of temperature increase is directly proportional to the rate of increase in the net enthalpy of the copper block and inversely proportional to its heat capacity The hourly change in the net enthalpy of the copper block results from convective, conductive, and radiative effects.
Convective and conductive heat increases are directly proportional to the temperature difference between the flame and the copper block, while radiative loss is proportional to the fourth power of the absolute temperature of the copper block These factors lead to the following differential equation governing the absolute temperature (T) as a function of time (t):
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Copper block mass
The mass of the copper block significantly affects the duration of test times due to its material heat capacity, making it a crucial parameter for establishing appropriate ranges.
Copper block geometry
The geometry of the copper block must be straightforward and easily defined for practical manufacturing Its size and shape should minimize disruption to the flame while ensuring it is fully contained within the critical flame zones.
Copper block positioning
To achieve accurate testing, the copper block should be suspended by the thermocouple wire and centrally positioned at a specified distance above the burner hardware, fully enveloped within the critical area of the flame For pre-mixed flames with a circular cross-section, the ideal placement is just above the inner cone and entirely within the outer cone Special positioning is necessary for linear flames to ensure the probe makes complete contact with the critical part of the flame.
Test temperature ranges
The copper block absorbs heat through convection and conduction while losing it via thermal radiation, which is proportional to the fourth power of its absolute temperature and influenced by surface emissivity To minimize variability from changes in emissivity, it is advisable to restrict the maximum test temperature to levels that reduce radiation effects, yet remain sufficiently high for effective testing A temperature range of 100 °C to 700 °C is deemed appropriate for this purpose.
Test time ranges
For effective recording, it is essential to select a reasonable range of test times, ideally between 30 to 90 seconds This range can be optimized by considering the mass of the copper block and the maximum test temperature.
7 Copper block calorimetry dynamics and theory
The effectiveness of a test flame is crucial for delivering heat to the test specimen In copper block calorimetry tests, a standard mass is placed in the test flame, and the time and heating characteristics are recorded to indicate the heat delivery rate The rate of temperature increase is directly proportional to the copper block's net heat content and inversely proportional to its heat capacity This change in net heat content results from convective, conductive, and radiative effects, with convective and conductive heat increases being proportional to the temperature difference between the flame and the copper block In contrast, radiative loss is proportional to the fourth power of the copper block's absolute temperature These factors lead to a differential equation that describes the absolute temperature (T) as a function of time (t).
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The constants A, B, and C, along with the initial starting temperature (T₀), are essential for solving the equation Once these values are known or assumed, the solutions can be accurately determined using a numerical method, such as the Runge-Kutta method.
Le tableau suivant comprend des données types, bien que non idéales (t (100 − 700) ≈57,5s) pour la flamme de 500 W nominale, à base de méthane, possédant une hauteur totale de
125 mm avec un cône intérieur bleu de 40 mm.
Tableau 2 – Données types pour une flamme de 500 W nominale, à base de méthane
The initial temperature \( T_0 \) is crucial for the time-temperature characteristic solution, which is why data is sampled more frequently during the first 15 seconds of the test It is observed and theoretically anticipated that the test characteristic at lower temperature intervals should closely adhere to a parabolic shape.
The constants \( k_1 \) are determined using second-order polynomial regression to find the best parabolic fit for the first 15 seconds of data, represented as \( T(t) = 4.57 + 15.34t - 0.09384t^2 \) Consequently, \( T_0 \) is accurately identified as \( k_0 = 4.57^\circ C \) The accuracy of the parabolic fit during the initial 15 seconds is demonstrated by the data in column 3 of Table 2.
Determining the best-fit constants A, B, and C is a complex task; however, a reasonable initial approximation can be easily obtained by noting the temperatures and approximate slopes near the beginning, midpoint, and end of the characteristic These values can be substituted into the differential equation to create a system of three linear equations, which can then be solved using a method like Gauss-Jordan elimination.
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) (T A BT CT t d/d − − (1) where A, B and C are characteristic constants If A, B, C and the initial starting temperature
(T 0 ) are known or assumed, then solutions of the equation can be determined to arbitrary precision using a numerical method such as the Runge-Kutta method.
The following table includes typical, though not ideal (t (100 − 700) ≈ 57,5 s) data for the nominal
500 W methane flame having an overall height of 125 mm with a 40 mm inner blue cone.
Table 2 – Typical data for a nominal 500 W methane flame
The initial temperature \( T_0 \) plays a crucial role in the time-temperature characteristics of the solution, leading to more frequent data sampling during the first 15 seconds of the test Both practical experience and theoretical analysis suggest that the test characteristics at lower temperatures closely adhere to a parabolic form represented by \( T(t) = k_0 + k_1 t - k_2 t^2 \) To analyze the data, the constant \( k_1 \) is determined using second-order polynomial regression, ensuring the best parabolic fit for the initial 15 seconds of data.
T = + − T 0 is therefore precisely determined as k 0 4,57°C The closeness of the parabolic fit over the first 15 s is illustrated by the figures in column 3 of table 2.
Finding the optimal constants A, B, and C can be challenging; however, a reasonable initial estimate can be obtained by observing the temperatures and approximate slopes at the beginning, midpoint, and end of the characteristic By substituting these values into the differential equation, a system of three linear equations is formed, which can be solved using methods like Gauss-Jordan elimination.
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For any approximation of A, B, and C, a heating characteristic can be developed from a precisely determined T0 using the Runge-Kutta method, allowing for the calculation of the mean squared error for actual test data Starting with the initial Gauss-Jordan approximation of A, B, and C, increasingly finer three-dimensional grid searches can be conducted on A, B, and C to identify the optimal values, thereby minimizing the resulting mean squared errors to ultimately determine the characteristic of the best fit.
La figure 2 illustre le résultat de cette procédure sur les données précédentes de l'éprouvette d'essai en réduisant l'erreur quadratique moyenne à environ 2,9 °C seulement.
Les données d'essai sont représentées en cercles à des intervalles de 5 °C avec la courbe
(inférieure) du meilleur ajustement qui en résulte en utilisant les paramètres déterminés:
A = 18,75, B = 0,011 03 et C = 1,580 On note un intérêt particulier pour la courbe supérieure résultant du réglage C = 0 ou en éliminant mathématiquement l'effet du rayonnement thermique La ligne supérieure à traits interrompus est l'asymptote résultante
= A/B = 1 427 °C, qui est la température effective de la flamme L'asymptote caractéristique résultante est la dernière température d'essai, aisément obtenue comme la solution de
Parameters A, B, and C, along with the effective flame temperature and ultimate test temperature, are specific to a single heating characteristic This specificity allows for a more precise specification of test flames, moving beyond the simplistic time interval between two temperatures that is not unique to any particular flame or testing characteristic.
If the calibration parameter for confirmation is the time required to raise the temperature of the copper block over a specified range (for instance, from 100 °C to 700 °C as outlined in previous specifications), then the aforementioned parabolic fitting is highly beneficial for discrete data points It can be demonstrated that this parabolic fitting is very accurate below 800 °C.
If the radiation factor (C) is omitted from the differential equation, precise solutions can be obtained, such as \$T(t) = T_f - (T_f - T_0)e^{-Bt}\$, where \$T_f = \frac{A}{B}\$ represents the effective flame temperature and \$T_0 = T(0)\$ is the initial temperature By expanding the term \$e^{-Bt}\$ as an infinite series, further insights can be derived.
Below 800 °C, all terms with powers of (t) greater than 2 are small and can be disregarded Notably, the reduced error from ignoring radiation is positive, while the reduced error from neglecting higher powers of (t) is negative Fortunately, these errors tend to cancel each other out, resulting in a high accuracy of the parabolic fit for data below 800 °C In fact, the parabolic fit remains quite accurate even up to 900 °C, but it deviates significantly from actual data as temperatures approach higher levels.
1 000 °C, comme l'illustrent les figures 3, 4 et 5 suivant les graphiques des ajustements paraboliques pour les données en tableau, relevées respectivement jusqu'à 800 °C, 900 °C et
1 000 °C, en indiquant t (100 − 700) et les erreurs quadratiques moyennes.
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Using the Runge-Kutta method, a heating characteristic can be developed from the accurately determined T 0 for any approximations of A, B, and C By calculating the root mean square (r.m.s.) error against actual test data, we can refine our initial Gauss-Jordan approximations of A, B, and C This process involves conducting progressively tighter three-dimensional computer grid searches to identify improved values, ultimately minimizing the r.m.s errors and achieving the best fit characteristic.
Figure 2 illustrates the result of this procedure on the previous test specimen data with the r.m.s error reduced to about only 2,9 °C.
Test data are represented as circles at 5 °C intervals, with the best fit curve determined by parameters A = 18.75, B = 0.01103, and C = 1.580 Notably, the upper curve is derived by setting C = 0, effectively removing the influence of thermal radiation The upper dashed line indicates the asymptote, calculated as A/B = 1,427 °C, which reflects the effective flame temperature Additionally, the characteristic asymptote represents the ultimate probe temperature, determined by solving d/dt(T) = 0, resulting in T_p = 23 °C.