Pressure-type flow sensor, flow rate computation device, flow rate computation method, flow rate computation program, and fluid control device

Abstract

To enable accurate flow rate computation over a wide range from a small flow rate to a large flow rate.SOLUTION: A pressure-type flow sensor is provided, comprising a flow resistance element 33 provided in a flow path 2R, an upstream pressure sensor 31a for detecting upstream pressure p1 on the upstream side of the flow resistance element 33, a downstream pressure sensor 31b for detecting downstream pressure p2 on the downstream side of the flow resistance element 33, and a flow rate computation unit 4a for computing a flow rate Q based on the upstream pressure p1 and the downstream pressure p2, where the flow rate computation unit 4a is configured to compute the flow rate Q on the basis of viscous resistance, inertial resistance and a degree of dilution effect.SELECTED DRAWING: Figure 1

Description

Technical Field

[0001] The present invention relates to a pressure-type flow sensor, a flow rate calculation device, a flow rate calculation method, a flow rate calculation program, and a fluid control device.

Background Art

[0002] The meanings of the symbols shown in the following equations are as follows.

[0003]

Equation

[0004] A pressure-type flow sensor calculates the flow rate flowing through a flow path by using the upstream pressure of a fluid resistance element provided in the flow path and the downstream pressure of the fluid resistance element.

[0005] Specifically, the velocity distribution obtained from the theory of Poiseuille flow such as the Hagen-Poiseuille equation or the two-dimensional Poiseuille flow equation

[0006]

Equation

[0007]

Equation

[0008]

Equation

[0009]

Number

Prior Art Documents

Patent Documents

[0010]

Patent Document 1

Summary of the Invention

Problems to be Solved by the Invention

[0011] As seen in pressure-type mass flow controllers, the measurement range of a flow meter is limited by the appearance of nonlinearity in the differential pressure flow characteristics due to inertial resistance in the large flow rate range, and the significant influence of rarefied gas effects at low flow rate and low pressure conditions. The basic equation based on the theory of Poiseuille flow does not include the above physical effects, and even if corrections are made using polynomials, etc., the large error in the basic equation deteriorates the final accuracy. Therefore, problems arise in measuring accurately over a wide measurement range from small to large flow rates with a single fluid resistance element structure.

[0012] Therefore, the present invention has been made to solve the above-mentioned problems, and an object of the present invention is to calculate the flow rate with high accuracy in a wide range from a small flow rate region to a large flow rate region. [Means for solving the problem]

[0013] That is, the pressure type flow sensor according to the present invention comprises a fluid resistance element provided in a flow path, an upstream pressure sensor for detecting the upstream pressure of the fluid resistance element, a downstream pressure sensor for detecting the downstream pressure of the fluid resistance element, and a flow rate calculation unit for calculating a flow rate based on the upstream pressure and the downstream pressure, and the flow rate calculation unit calculates the flow rate based on the viscous resistance, the inertial resistance, and the degree of influence due to rarefaction. Note that rarefaction means that the pressure of the gas becomes very low or the system becomes minute, and the mean free path of the molecules becomes a size that cannot be ignored compared to the representative dimension of the system. At this time, the local equilibrium in the molecular motion of the gas is lost and the rarefied gas effect appears.

[0014] Such a pressure type flow sensor calculates the flow rate based on the influence of viscous resistance, inertial resistance, and dilution, and can therefore calculate the flow rate with high accuracy over a wide range from small flow rates to large flow rates. Specifically, the flow rate is calculated based on the pressure loss (viscous resistance) in the laminarized fluid resistance element, the pressure loss due to inertia (inertial resistance) at the inlet and outlet of the fluid resistance element, at branch junctions, etc., and the degree of influence of flow dilution. Therefore, the flow rate can be accurately calculated over a wide range from the small flow rate region to the large flow rate region. Here, the inertial resistance has a greater influence in the large flow rate region and a smaller influence in the small flow rate region. Also, the degree of influence due to flow dilution increases in the small flow rate region where the primary pressure approaches the secondary pressure, and decreases in the large flow rate region where the primary pressure is high. By incorporating these inertial resistance and the degree of influence due to dilution, the flow rate can be accurately calculated over a wide range from the small flow rate region to the large flow rate region.

[0015] As a specific embodiment of the flow rate calculation unit, it is desirable that the flow rate calculation unit calculates the flow rate based on the following relational expression. Relational expression: p1 2 -p2 2 =K1×Q+K2×Q 2 +K3×(p1-p2) Here, p1 is the upstream pressure, p2 is the downstream pressure, Q is the flow rate, and K1 to K3 are coefficients. Also, K1×Q is the term indicating viscous resistance, and K2×Q 2 is the term indicating inertial resistance, and K3×(p1-p2) is the term indicating the degree of influence of dilution (hereinafter referred to as the dilution correction term).

[0016] The above relational expression is the basic formula of a pressure type flow meter obtained by applying a dilution correction analogized from the theoretical formula of a dilute gas to the Forchheimer's formula, and corrects the errors in the large flow rate region and the small flow rate region of the conventional basic formula represented by Equation (7).

[0017] For the explanation of the relational expression, first consider the basic equation including the effect of inertia. When there is a partial expansion / contraction or a branching / merging point in the flow between the upstream pressure sensor and the downstream pressure sensor including the fluid resistance element, characteristics depending on the inertia of the flow appear, and non-linearity occurs in the differential pressure flow rate characteristics. There is the Forchheimer's equation (Equation (8)) as a model of the flow including the viscous resistance as appears in the Poiseuille flow and the effect of the above-mentioned inertial resistance. By obtaining the basic equation from this equation, the effect of inertia can be included. The first term on the right side of Equation (8) represents the viscous resistance, and the second term represents the inertial resistance, and they are characterized by the model coefficients a F and b F respectively. In the Forchheimer's equation, since a fine and complex flow path system such as a porous body is modeled as a one-dimensional system having space only in the flow direction, the apparent flow velocity u s is converted to the flow rate by introducing the total cross-sectional area A a of the flow path at a certain position. That is, since the relationship of Equation (9) holds between the apparent flow velocity u a and the mass flow rate q, the Forchheimer's equation can be expressed as Equation (10). Allowing the density change by the gas state equation and assuming that the system is homogeneous in the flow direction, Equation (11) is obtained by integrating in the flow direction. When converted to the standard volume flow rate, it becomes Equation (12).

[0018]

Number

[0019] Furthermore, in a certain gas species, under the condition that the temperature is constant and the change in the compressibility factor z is small, it can be simplified as Equation (13).

[0020]

Number

[0021] Since Equation (13) is a quadratic equation of flow rate, a function of pressure with respect to flow rate can be obtained from the formula for its solution, and by using this, the flow rate can be calculated from the pressure. Alternatively, the flow rate can also be obtained from numerical calculations such as iterative calculations.

[0022] The coefficients K1 and K2 are empirically obtained by fitting Equation (13) to the data representing the relationship between the flow rate and pressure of the fluid flowing through the flow path. Since Equation (13) is a quadratic polynomial with respect to the flow rate, the linear least squares method can be applied, and the fitting can be easily performed.

[0023] Assuming that the flow can be approximated as circular pipe flow or planar flow, K1 is

[0024]

Number

[0025] Equation (12) can also be used as a basic equation. In that case, the coefficients a F and b F of the Forchheimer equation become the coefficients of the basic equation. They are coefficients that vary depending on the flow path shape and can be obtained theoretically or empirically. a p can be theoretically given as in Equation (6) when it can be approximated as circular pipe flow or planar flow.

[0026] Even for an arbitrary cross-sectional shape, if it is isotropic, it can be approximated as circular pipe flow using the hydraulic diameter or hydraulic radius, and the coefficient a F can be theoretically obtained.

[0027] When there are multiple flow paths and they are located in a parallel or series relationship, the K 1i or a Fi of each flow path obtained theoretically can be used to give the K1 or a F of the entire flow path system as follows.

[0028] [Number]

[0029] Next, in addition to the inertial effect that satisfies the relational expression, a basic equation including the dilution effect will be described. In the actual measurement data of the mass flow controller, the basic equation (Equation (13)) based on the Forchheimer equation has a large fitting error due to the dilution effect in the low-pressure and low-flow rate regions. Therefore, it is necessary to consider the dilution effect to achieve wide-range and high-precision measurement. Note that the fitting error is the relative error with respect to the flow rate data of the flow rate predicted by the regression curve obtained by data fitting, and will be used in the same meaning hereinafter. In the field of rarefied gases, applying Maxwell's velocity slip, which is a velocity boundary condition expressing the dilution effect in the theory of fluid dynamics, the velocity distributions of internal flow in a circular pipe and planar flow including the dilution effect in the slip flow region at low Reynolds number and low Mach number are obtained. That is,

[0030] [Number] where α is the tangential momentum accommodation coefficient, which represents the degree of diffuse reflection in molecular reflection at the wall surface. Usually, 1 is given assuming complete diffuse reflection. λ is the mean free path. Multiplying both sides of the velocity distribution by the density, allowing for density changes according to the gas state equation, and then integrating over the cross-section, Equation (23) is obtained. c r is a coefficient that changes depending on the flow path shape, l is the representative length, and Kn is the Knudsen number.

[0031] [Number] On the other hand, in the case based on the Poiseuille flow equation, similarly, by integrating the velocity distributions (Equations (1) and (2)) over the cross-section,

[0032] [Number] can be expressed as. Comparing this with Equation (23), since there is a relationship between the mass flow rate of the continuous flow field and the mass flow rate including the dilution effect, this is interpreted as the dilution correction of the mass flow rate.

[0033] [Number] Since there is a relationship between them, this is interpreted as the dilution correction of the mass flow rate.

[0034] Note that c r becomes as shown in Equation (24) for internal flow in a circular pipe and planar flow, but can also be obtained theoretically or empirically for other channel shapes.

[0035] Integrating Equation (27) in the flow direction based on the mass conservation law of the flow and multiplying by the number of channels n in the case of multiple channels,

[0036] [Number] the mass flow rate is obtained. Comparing this with Equation (4) shown previously,

[0037] [Number] it is obtained. Therefore, the coefficient c r and the coefficient a p can estimate one from the other.

[0038] When analogizing the form in which the Forchheimer equation (Equation (10)) expressed in terms of the mass flow rate is corrected for dilution from Equation (28),

[0039] [Number] is obtained. Applying the gas state equation to this and integrating in the flow direction,

[0040] [Number] is obtained. Note that the mean free path λ is

[0041]

Number

[0042] Furthermore, when p1 is sufficiently larger than p2, considering the conditions of specific gas species and pressure ranges in the mass flow controller, it is approximated as in Equation (37). When this is converted to the standard volume flow rate, Equation (38) is obtained.

[0043]

Number

[0044]

Number

[0045] Since Equation (39) is a quadratic equation of the flow rate, a function of pressure with respect to the flow rate can be obtained from the solution formula, and by using this, the flow rate can be calculated from the pressure. Alternatively, the flow rate can also be obtained from numerical calculations such as iterative calculations.

[0046] The coefficients K1 to K3 are empirically obtained by fitting Equation (39) to the data representing the relationship between the flow rate and pressure of the fluid flowing through the flow path. Since Equation (39) is a quadratic polynomial with respect to the flow rate, the linear least squares method can be applied and the fitting can be easily performed.

[0047] K1 can also be given by using the shape coefficient a of the Poiseuille flow according to Equations (14) and (15), similar to the case of the basic formula (Equation (13)) based on the Forchheimer's equation. p a pis a coefficient that varies depending on the flow path shape and is given as in Equation (6) when it can be approximated as flow in a circular pipe or planar flow, but can also be determined theoretically or empirically for other flow path shapes.

[0048] Comparing Equation (38) and Equation (39), K3

[0049]

Number

[0050] Equation (38) can also be used as the basic equation. In that case, the coefficients a F , b F of the Forchheimer equation and the coefficient c r of the dilution correction become the coefficients of the basic equation. Each of them is a coefficient that varies depending on the flow path shape and can be determined theoretically or empirically. a p can be theoretically given as in Equation (6) when it can be approximated as flow in a circular pipe or planar flow, and c r can be given as in Equation (24) when it can be approximated as flow in a circular pipe or planar flow.

[0051] If it is isotropic in any cross-sectional shape, it may be approximated as flow in a circular pipe using the hydraulic diameter or hydraulic radius.

[0052] When there are multiple flow paths and they are located in a parallel or series relationship, the K 1i or a Fi of each flow path determined theoretically can be used to give K1 or a F of the entire flow path system as in Equations (16) to (19).

[0053] The approximation was made under the condition that p1 is sufficiently larger than p2. However, in the mass flow controller, in practice, ranges outside this are also included in the operating range. However, Equation (39) is useful because the fitting error with respect to the calibration dataset is improved compared to conventional ones based on the theory of Poiseuille flow.

[0054] It is also possible to use Equation (33) before the approximation is applied as the basic equation. However, obtaining the coefficient c r from fitting requires determining at least b F from fitting as well, which leads to a multivariable non-linear optimization problem and is difficult to solve stably, accurately, and quickly. If c r is given theoretically, a F and b F can be easily obtained from the linear least squares method. However, in this case, the trial results showed that there was not much difference from the case of the approximated Equation (39).

[0055] When applying to a flowmeter, as in the past, corrections for non-linearity of the flow rate-pressure characteristics, temperature, pressure, gas species, and individual differences can be added to the flow rate obtained by the above relational expression by a higher-order polynomial or the like.

[0056] Since the above relational expression corrects the effects of inertia and rarefaction based on theory, it can be expected that the overall measurement accuracy obtained will be improved in addition to the case of using the conventional basic equation when the degrees of freedom of the entire final flowmeter calculation formula with other corrections added are the same. Alternatively, it can be expected that the degrees of freedom of the corrections required to obtain the same measurement accuracy as in the past will be reduced.

[0057] As a specific implementation mode of fitting, it is conceivable that it is obtained from the weighted least squares method using the reciprocal of the flow rate as a weight.

[0058] When determining the coefficients K1 to K3 by fitting, in the ordinary least squares method, the fitting error becomes large in the low to medium flow rate range. Therefore, it is desirable to obtain the coefficients K1 to K3 from the least squares method weighted by the reciprocal of the flow rate as shown in the following equation. This weighting corresponds to emphasizing the low flow rate range.

[0059]

Number

[0060] The dilution correction term effective in the low flow rate range has degrees of freedom, and the above weighting emphasizes the low flow rate, thereby improving the degree of fitness. As a result, the flow rate can be calculated accurately especially in the low flow rate range. Also, in the above theoretical coefficient, the Kn number uses the flow path size, but due to dimensional errors, the fitting error may increase depending on the application target. Therefore, by making the coefficient K3 an empirical coefficient, the fitting error associated with dimensional errors can be reduced.

[0061] The data may be obtained from actual measurement or from simulation. Generally, in fitting for calibration of measuring instruments, actual measurement data is used, but for obtaining only some fitting coefficients separately by simulation, advantages can arise from the difficulty of actual measurement and the like.

[0062] Further, the flow rate calculation device according to the present invention is characterized in that it calculates the flow rate based on the upstream pressure of the fluid resistance element provided in the flow path, the downstream pressure of the fluid resistance element, viscous resistance, inertial resistance, and the degree of influence of dilution.

[0063] Further, the flow rate calculation method according to the present invention is a flow rate calculation method using an upstream pressure sensor that detects the upstream pressure of a fluid resistance element provided in a flow path and a downstream pressure sensor that detects the downstream pressure of the fluid resistance element, and is characterized in that it calculates the flow rate based on the upstream pressure, the downstream pressure, viscous resistance, inertial resistance, and the degree of influence of dilution.

[0064] Furthermore, the flow rate calculation program according to the present invention is a flow rate calculation program used in a pressure type flow sensor that measures the flow rate based on the upstream pressure of a fluid resistance element provided in a flow path and the downstream pressure of the fluid resistance element, and causes a computer to have a function of calculating the flow rate based on the upstream pressure, the downstream pressure, viscous resistance, inertial resistance, and the influence degree of thinning.

[0065] Note that the flow rate calculation program may be distributed electronically or may be recorded on a program recording medium such as a CD, DVD, or flash memory.

[0066] In addition, the fluid control device according to the present invention is characterized by including the above-described pressure type flow sensor and a fluid control valve provided on the upstream side or the downstream side of the pressure type flow sensor.

Effects of the Invention

[0067] Thus, according to the present invention, the flow rate can be accurately calculated in a wide range from a small flow rate region to a large flow rate region.

Brief Description of the Drawings

[0068]

Figure 1

Figure 2

Figure 3

Figure 4

Mode for Carrying Out the Invention

[0069] <One Embodiment of the Present Invention> Hereinafter, an embodiment of a fluid control device incorporating a pressure type flow sensor according to the present invention will be described with reference to the drawings. It should be noted that, for the sake of clarity, all the figures shown below are schematically drawn with appropriate omissions or exaggerations. The same components are denoted by the same reference numerals and the description thereof will be omitted as appropriate.

[0070] <Device Configuration> The fluid control device 100 of the present embodiment is, for example, used in a semiconductor manufacturing process, and is provided in one or a plurality of gas supply lines to control the flow rate of the process gas flowing through each gas supply line.

[0071] Specifically, the fluid control device 100 is a so-called differential pressure type mass flow controller (differential pressure type MFC), and as shown in FIG. 1, includes a flow path block 2 in which a plurality of internal flow paths 2R are formed, and a fluid control device 3 mounted on the flow path block 2.

[0072] The flow path block 2 is provided with an introduction port 21 for introducing fluid into the internal flow path 2R and a derivation port 22 for deriving fluid from the internal flow path 2R. An upstream side pipe H1 is connected to the introduction port 21, and an upstream side pneumatic valve V1 is provided in the upstream side pipe H1. Further, a downstream side pipe H2 is connected to the derivation port 22, and a downstream side pneumatic valve V2 is provided in the downstream side pipe H2.

[0073] The fluid control device 3 controls the fluid in the internal flow path 2R, and has a flow rate sensor 31 for measuring the flow rate of the fluid flowing through the internal flow path 2R, and a fluid control valve 32 provided upstream of the flow rate sensor 31. The valve opening degree of the fluid control valve 32 is feedback controlled by a control unit 4 described later.

[0074] The flow rate sensor 31 is a pressure type flow rate sensor, and includes an upstream pressure sensor 31a provided upstream of a fluid resistance element 33 such as a restrictor or an orifice provided in the internal flow path 2R, and a downstream pressure sensor 31b provided downstream of the fluid resistance element 33. Then, based on the upstream pressure p1 of the fluid resistance element 33 detected by the upstream pressure sensor 31a and the downstream pressure p2 of the fluid resistance element 33 detected by the downstream pressure sensor 31b, the flow rate Q flowing through the internal flow path 2R is calculated by the flow rate calculation unit 4a of the control unit 4 described later.

[0075] The fluid control valve 32 is provided upstream of the pressure type flow rate sensor 31. Specifically, the fluid control valve 32 controls the flow rate by moving a valve body forward and backward with respect to a valve seat by an actuator of, for example, a piezo actuator. Note that the fluid control valve 32 is controlled by the valve control unit 4b of the control unit 4.

[0076] The control unit 4 includes a flow rate calculation unit 4a that calculates the flow rate Q flowing through the internal flow path 2R based on the upstream pressure p1 and the downstream pressure p2, and a valve control unit 4b that controls the fluid control valve 32 based on the flow rate Q calculated by the flow rate calculation unit 4a. Note that the control unit 4 is a so-called computer including, for example, a CPU, a memory, an A / D·D / A converter, and input / output means, and by executing a fluid control program stored in the memory and causing various devices to cooperate, functions as the flow rate calculation unit 4a, the valve control unit 4b, etc.

[0077] However, the flow rate calculation unit 4a of the present embodiment calculates the flow rate Q based on the influence degrees of viscous resistance, inertial resistance, and thinning.

[0078] Specifically, the flow rate calculation unit 4a calculates the flow rate Q based on the following relational expression F. Relational expression F: p1 2 - p2 2 = K1 × Q + K2 × Q 2 + K3 × (p1 - p2)

[0079] Here, p1 is the upstream pressure, p2 is the downstream pressure, Q is the flow rate, and K1 to K3 are coefficients.

[0080] Also, K1×Q is the term representing viscous resistance, and K2×Q 2 is the term representing inertial resistance, and K3×(p1 - p2) is the term indicating the degree of dilution effect.

[0081] K1 to K3 are empirical coefficients obtained by fitting the relational expression F to the data representing the relationship between the measured value of the flow rate flowing through the flow path 2R and the pressure. Here, the linear least squares method can be used for fitting. Note that K1 to K3 may also be obtained by fitting the relational expression F to the relationship between the simulation value of the flow rate flowing through the flow path 2R and the pressure.

[0082] When obtaining the coefficients K1 to K3 by fitting, since the fitting error becomes large in the low to medium flow rate range with the ordinary least squares method, it is desirable to obtain the coefficients K1 to K3 from the weighted least squares method using the reciprocal of the flow rate as a weight as shown in the following formula. This weighting corresponds to emphasizing the low flow rate range.

[0083]

Equation

[0084] The dilution correction term effective in the low flow rate range has degrees of freedom, and the above weighting emphasizes the low flow rate, thereby improving the degree of fit. As a result, the flow rate can be calculated accurately particularly in the low flow rate range. Also, in the above theoretical coefficients, the Kn number uses the flow path size, but there may be a dimensional error, so the fitting error may increase depending on the application target. Therefore, by using the coefficient K3 as an empirical coefficient, the fitting error associated with the dimensional error can be reduced.

[0085] <Fitting results of experimental data and each formula> Figure 2 is a graph showing experimental data and basic equations based on the equation of Poiseuille flow with coefficients determined by fitting thereto, a basic equation based on Forchheimer's equation, and a basic equation based on the power law. The gas type in this experiment is nitrogen gas. Each equation is shown below, and the coefficients K1 and K2 in each equation are different from each other.

[0086]

Number

[0087] In the basic equation based on Poiseuille flow, the fitting results have large errors on both the low-flow rate side and the high-flow rate side. Also, in the basic equation based on Forchheimer's equation, the fitting is the best compared to other equations, especially excellent in the high-flow rate region, and has an error equivalent to that of the basic equation based on the power law in the low-flow rate region.

[0088] <Fitting results of experimental data to the relational expression of the present embodiment> Figure 3 is a graph showing (a) the fitting error by the basic equation based on Forchheimer's equation and (b) the fitting error by the relational expression (present formula) of the present embodiment. The gas type in this experiment is nitrogen gas. Each equation is shown below, and the coefficients K1 and K2 in each equation are different from each other.

[0089]

Number

[0090] Comparing (a) and (b) of Figure 3, it can be seen that by introducing a dilution correction term into Forchheimer's equation, the negative error in the low-flow rate region of Forchheimer's equation is improved. Also, in the high-flow rate region where the influence of inertia becomes large, the accuracy of fitting by introducing the dilution correction term is not significantly impaired.

[0091] <Effect of the present embodiment> Thus, according to the fluid control device 100 in the present embodiment, since the flow rate is calculated based on the viscous resistance, the inertial resistance, and the influence degree of thinning, the flow rate can be accurately calculated in a wide range from the small flow rate region to the large flow rate region. As a result, the flow rate can be accurately controlled in a wide range from the small flow rate region to the large flow rate region.

[0092] Specifically, the pressure loss (viscous resistance; K1×Q) in the laminarized fluid resistance element, and the pressure loss due to inertia (inertial resistance; K2×Q 2 ) at the inflow and outflow portions of the fluid resistance element and at branch junctions, etc., and the influence degree due to the thinning of the flow (influence degree of thinning; K3×(p1 - p2)) are used to calculate the flow rate, so the flow rate can be accurately calculated in a wide range from the small flow rate region to the large flow rate region. Here, the inertial resistance has a greater influence in the large flow rate region and a smaller influence in the small flow rate region. Also, the influence degree due to the thinning of the flow becomes greater in the small flow rate region where the primary pressure approaches the secondary pressure, and becomes smaller in the large flow rate region where the primary pressure is large. By incorporating these inertial resistance and the influence degree of thinning, the flow rate can be accurately calculated in a wide range from the small flow rate region to the large flow rate region.

[0093] Further, in the present embodiment, since the fluid resistance element 33 is provided on the downstream side of the fluid control valve 32, in the low flow rate region, the environment of the fluid resistance element 33 becomes a low pressure, and an increase in the flow rate due to the thinning of the flow occurs. Even in such a case, in the present embodiment, since the flow rate is calculated based on the influence due to the thinning of the flow (influence degree of thinning; K3×(p1 - p2)), the flow rate can be accurately calculated.

[0094] <Another Embodiment 1> In the above embodiment, K1 can also be the theoretical coefficient represented by Equation (14). Specifically, it becomes the following equation (re - listed). Here, a p is a coefficient depending on the flow path shape, and when the system can be approximated to an in - tube flow (Tube flow) or a plane flow (Plane flow), it becomes as shown in Equation (6). Also, As is the total cross-sectional area of the flow path, and when the flow path can be approximated by a circular pipe or an assembly flow path in which sufficiently flat rectangular pipes are connected in parallel, it is the cross-sectional area A of one flow path multiplied by the number of flow paths n.

[0095] In this embodiment, by replacing the theoretical coefficient with an empirical coefficient, the degree of freedom of data fitting can be reduced. Although the influence of dimensional errors caused by processing errors occurs in the theoretical coefficient, as long as the fitting accuracy is maintained, when the number of data points is small or the error with respect to the true value of the data is large, an increase in the error of the formula with respect to the true value due to overfitting can be prevented. That is, an improvement in the validity of the formula can be expected.

[0096]

Number

[0097] At this time, the flow rate calculation unit 4a calculates the flow rate Q based on the following formula determined by the empirical coefficients K2 and K3.

[0098]

Number

[0099] <Another Embodiment 2> In the above embodiment, K3 can also be a theoretical coefficient represented by Equation (40). Specifically, it becomes the following equation (reposted). Here, c r is a coefficient depending on the flow path shape, and l is a representative length. c r becomes the value shown in Equation (24) when the system can be approximated by tube flow or plane flow. At this time, for l, the length shown in Equation (25) is used, that is, the radius r of the circular pipe in tube flow pipe , and the flow path height h in plane flow is used.

[0100] In this embodiment, similar to the other embodiment 1, the degree of freedom of data fitting can be reduced by replacing the empirical coefficient with the theoretical coefficient. Although the dimensional error caused by the machining error occurs in the theoretical coefficient, as long as the fitting accuracy is maintained, when the number of data points is small or the error with respect to the true value of the data is large, an increase in the error of the formula with respect to the true value due to overfitting can be prevented. That is, an improvement in the validity of the formula can be expected. Note that since the term for replacing the empirical coefficient with the theoretical coefficient is different from the other embodiment 1, it is assumed that the effect will be different from the other embodiment 1 depending on the fluid resistance element, the range of physical conditions to be applied, and the characteristics of the data, and there is room for selection.

[0101]

Number

[0102] At this time, the flow rate calculation unit 4a calculates the flow rate Q based on the following formula determined by the empirical coefficients K1 and K2.

[0103]

Number

[0104] <Other Embodiment 3> In the above embodiment, in the same manner as in the other embodiment 1 and the other embodiment 2, both K1 and K3 can be set as the theoretical coefficients represented by Formula (14) and Formula (40).

[0105] In this embodiment, since more empirical coefficients are replaced with theoretical coefficients compared to the other embodiment 1 and the other embodiment 2, the degree of freedom of data fitting is further reduced. Therefore, although the influence of dimensional errors caused by machining errors in the theoretical coefficients becomes greater, as long as the fitting accuracy is maintained, when the number of data points is small or the error with respect to the true value of the data is large, it is possible to more highly prevent an increase in the error of the formula with respect to the true value due to overfitting. That is, a further improvement in the validity of the formula can be expected.

[0106]

Number

[0107] <Fitting error when applying the theoretical coefficient to coefficient K3> Figures 4(a), (b), and (c) show the fitting errors when applying the theoretical coefficient to coefficient K3. In FIG. 4, "K1K2" means the case where empirical coefficients are applied to coefficients K1 and K2 and the theoretical coefficient is applied to coefficient K3, and "K1K2K3" means the case where empirical coefficients are applied to all of coefficients K1 and K3, that is, the relational expression of the said embodiment. "fit" means fitting by the ordinary linear least squares method, and "fit weighted" means the case of fitting by the weighted linear least squares method that emphasizes the low flow rate range.

[0108] Looking at the case of nitrogen (N2) shown in Fig. 4(a), "K1K2K3 fit weighted" has the freedom to fit in the low flow rate range, and the fitting error is minimized by applying a weighting that emphasizes the low flow rate range. "K1K2 fit" has a small fitting error, but since it theoretically predicts K3, it has no freedom to fit in the low flow rate range, so the fitting error is larger compared to "K1K2K3 fit weighted". "K1K2 fit weighted" is improved compared to "K1K2 fit" by the weighting that emphasizes the low flow rate range, but since it has no freedom to fit in the low flow rate range, the fitting error is larger compared to "K1K2K3 fit weighted". Also, "K1K2K3 fit" is not utilized by the ordinary linear least squares method even if it has the freedom to fit in the low flow rate range.

[0109] In the cases of sulfur hexafluoride (SF6) and helium (He) shown in Figs. 4(b) and 4(c) as well, similar to nitrogen (N2), "K1K2K3 fit weighted" has the smallest fitting error, and the result is that "K1K2 fit" is inferior but functional.

[0110] Also, for each term or part or all of the coefficients of K1×Q + K2×Q 2 + K3×(p1 - p2) in the relational expression of the above embodiment, a database may be constructed in advance. In this case, the target can be obtained by simply referring to or applying an interpolation method to the database to calculate an approximate value of the data, and it can also be used for flow rate calculation. Here, as variables of the database, flow rate, pressure, temperature, gas species, or a combination thereof can be considered. Also, when the flow rate is included as a variable in the database, in flow rate calculation, first, a temporary value such as 0 is given to the target term or coefficient for flow rate calculation, and then based on the calculated flow rate, the target value is obtained from the database, and the flow rate can be obtained from numerical calculations such as iterative calculations where the flow rate calculation is performed again using that value. For example, it is conceivable to set the final flow rate when the change in the flow rate value obtained by iterative calculation becomes smaller than the allowable value, or when a predetermined number of iterative calculations are performed.

[0111] Also, in the above embodiment, the pressure type flow sensor 31 is incorporated into the fluid control device, but the pressure type flow sensor 31 alone may be used.

[0112] In the above embodiment, the flow rate is expressed in standard volume flow rate (SCCM), but it can also be converted into mass flow rate.

[0113] Also, in the above formulas of the present invention, it is assumed that p1>p2. However, when p1<p2, that is, in the case of reverse flow, the roles of p1 and p2 may be interchanged.

[0114] In addition to the combination of the upstream pressure p1 and the downstream pressure p2, the pressure type flow sensor 31 may measure the difference (differential pressure Δp) between the inlet pressure and the outlet pressure of the flow path using, for example, a differential pressure sensor, and calculate the flow rate from the differential pressure Δp and the upstream pressure p1, or from the differential pressure Δp and the downstream pressure p2.

[0115] The data used when performing fitting may be based on simulation or theoretical formulas in addition to those based on actual measurement.

[0116] In addition, various modifications and combinations of embodiments may be made as long as they do not depart from the spirit of the present invention.

Explanation of Reference Numerals

[0117] 100 ··· Fluid control device Q ··· Flow rate p1 ··· Upstream pressure p2 ··· Downstream pressure 2R ··· Flow path 31 ··· Pressure type flow sensor 31a ··· Upstream pressure sensor 31b ··· Downstream pressure sensor 32 ··· Fluid control valve 33 ··· Fluid resistance element 4a ··· Flow rate calculation unit

Claims

1. A fluid resistance element provided in a flow path, an upstream pressure sensor that detects the upstream pressure of the fluid resistance element, a downstream pressure sensor that detects the downstream pressure of the fluid resistance element, and a flow rate calculation unit that calculates a flow rate based on the upstream pressure and the downstream pressure, wherein the flow rate calculation unit calculates the flow rate based on viscous resistance, inertial resistance, and the degree of influence of thinning, and is a pressure type flow sensor.

2. The pressure type flow sensor according to claim 1, wherein the flow rate calculation unit calculates the flow rate based on the following relational expression. Relationship: p １ ２ - p ２ ２ = K １ × Q + K ２ × Q ２ + K ３ × (p １ - p ２ ) Here, p １ is the upstream pressure, p ２ is the downstream pressure, Q is the flow rate, and K １ to K ３ are coefficients. Also, K １ ×Q is the term indicating viscous resistance, and K ２ ×Q ２ is the term indicating inertial resistance, and K ３ ×(p １ −p ２ ) is the term indicating the degree of influence of thinning.

3. Coefficient K １ to K ３ is obtained by the weighted least squares method using the reciprocal of the flow rate as a weight in fitting the relational expression to the relationship between the measured value or simulation value of the flow rate of the fluid flowing through the flow path and the pressure. The pressure formula flow rate sensor according to claim 2.

4. The flow of the fluid resistance element can be approximated as tube flow or plane flow, and the coefficient K １ is the length L of the fluid resistance element, the radius r of the tube flow ｐｉｐｅ or the channel height h of the plane flow, the area A of the channel, the total cross-sectional area A of the channel ｓ , the number of channels n, the temperature T, the specific gas constant R, the viscosity coefficient μ, the compressibility factor z, the unit conversion coefficient φ from the SI unit system of the volumetric flow rate under standard conditions, the pressure p under standard conditions ｓｔｄ , the temperature T under standard conditions ｓｔｄ and the compressibility factor z under standard conditions ｓｔｄ by 【Number 1】 is given, and coefficient K ２ and coefficient K ３ is obtained from the weighted least squares method with the reciprocal of the flow rate as the weight in fitting the relational expression to the relationship between the measured value or simulation value of the flow rate of the fluid flowing through the flow path and the pressure. The pressure formula flow rate sensor according to claim 2.

5. The flow of the fluid resistance element can be approximated by tube flow or plane flow, and the coefficient K ３ is the radius r of the tube flow ｐｉｐｅ or the channel height h, temperature T, specific gas constant R, viscosity coefficient μ, and tangential momentum accommodation coefficient α of the plane flow 【Number 2】 is given, and the coefficient K １ and the coefficient K ２ is obtained from the weighted least squares method with the reciprocal of the flow rate as a weight in fitting the relational expression to the relationship between the measured value or simulation value of the flow rate of the fluid flowing through the flow path and the pressure. The pressure formula flow rate sensor according to claim 2.

6. The flow of the fluid resistance element can be approximated to tube flow or plane flow, and the coefficient K １ is as described in claim 4, and the coefficient K ２ is obtained from the weighted least squares method with the reciprocal of the flow rate as the weight in fitting the relational expression to the relationship between the measured value or simulation value of the flow rate of the fluid flowing through the flow path and the pressure. The coefficient K ３ is as described in claim 5, and the pressure formula flow rate sensor according to claim 2.

7. A flow rate calculation device that calculates a flow rate based on the upstream pressure of a fluid resistance element provided in a flow path, the downstream pressure of the fluid resistance element, viscous resistance, inertial resistance, and the degree of influence of thinning.

8. A flow rate calculation method using an upstream pressure sensor that detects the upstream pressure of a fluid resistance element provided in a flow path and a downstream pressure sensor that detects the downstream pressure of the fluid resistance element, wherein the flow rate is calculated based on the upstream pressure, the downstream pressure, viscous resistance, inertial resistance, and the degree of influence of the thinning.

9. A flow rate calculation program used for a pressure type flow sensor that measures a flow rate based on the upstream pressure and the downstream pressure of a fluid resistance element provided in a flow path, wherein the computer is provided with a function of calculating a flow rate based on the upstream pressure, the downstream pressure, viscous resistance, inertial resistance, and the degree of influence of the thinning.

10. A fluid control device comprising the pressure type flow sensor according to any one of claims 1 to 6, and a fluid control valve provided upstream or downstream of the pressure type flow sensor.

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