THERMAL MASS FLOW CONTROLLER SCALING RELATIONS

Chiun Wang, Ph.D. CareFusion, Inc.

22745 Savi Ranch Pkwy Yorba Linda, CA 92887

(714) 922-7595 ccwang_2000@yahoo.com

Abstract - Thermal mass flow controllers precisely control the timely delivery of a great variety of gases during the MEMS and semiconductor manufacturing processes. Many of these gases cannot be used for the calibration and the tuning of the controllers because they are corrosive and can easily contaminate the ultra-clean high-vacuum processes. Although a non-reactive surrogate gas such as nitrogen may be used for the calibration or tuning, the controllers are eventually required to be accurate for the actual process gases themselves. To deal with these mutually conflicting requirements, scaling relations that guarantee the accuracy for the process gases become extremely important for the fabrication of high-accuracy mass flow controllers. The three main components of the MFC, i.e. the thermal mass flow sensor, the laminar flow element, and the flow control valve, are each governed by a different set of thermal fluid dynamic relations, some of which are significantly nonlinear, making the scaling of especially multi-gas thermal mass flow controllers both interesting and challenging. In this paper the theoretical relations governing different components of the thermal mass flow controllers are reviewed. Examples are then given to show how accurate scaling laws may be developed for each MFC component to guide the design of the thermal mass flow controllers, and to optimize their calibration and tuning processes.

INTRODUCTION Thermal mass flow controllers (MFCs) play an important role in the fabrication of various micro-electromechanical (MEMS) and semiconductor devices. They precisely administer the timely delivery of hundreds of process gases utilized during the various stages of doping, etching, cleaning, and chemical vapor deposition, all of which occurring in the ultra-clean environment. Modern MFCs must meet high accuracy requirements sometimes exceeding 1% of reading over the entire (1% to 100% of full-scale) range of operation. It is not feasible for mass flow controller manufacturers to calibrate the MFCs directly with the process gases. For gases that are corrosive, corrosion will contaminate the MFC during calibration and render it unacceptable for the ultra-clean semi-conductor processes. Even when the process gases are non-corrosive, since the MFCs are nonlinear devices and the nonlinearity varies with both the gas and the range of the controllers, calibrating a multi-gas MFC by using the exotic process gases is a formidably expensive operation. The intriguing question is then without direct calibration how can one achieve the required process-gas accuracy? Historically the industry’s approach was to assume that the MFCs are linear, and to calibrate the MFCs with an appropriate inert surrogate gas, such as nitrogen, at flow rates equal to the process-gas flow-rate multiplied by a certain 'gas conversion factor' or 'gas correction factor'. This method often leads to miserable inaccuracy because the MFC output is a significantly nonlinear function of the flow rate. While the accuracy requirements at the dawn of the semi-conductor industry could be a few percent of the full-scale flow rate, the accuracy requirements for modern semi-conductor processes is typically better than 1% of the flow reading. Using a constant conversion factor for scaling simply would not deliver the high accuracy that is required for most modern MFC applications. To deal with these issues, at first sight it might seem that one could build a batch of identical MFCs, and test and calibrate each and all of them with both the process gas and the inert calibration gas to establish

their mutual conversion functional relationship, with the hope that these relations remain invariant and can therefore guarantee the process-gas accuracy of the production MFCs. Unfortunately due to the small dimensions of some of the MFC components and the unforgiving manufacturing tolerances involved, no two MFCs may be considered identical or 'close enough' to share the same performance characteristics. It is also not practical to develop physical models that can be adapted to variations in the device dimensions. Bear in mind that certain tolerance of the devices, such as the thickness of the insulation layer over the resistance heating wires, or the thermal contact resistance between the heating wire and the sensor tube, or the radius of curvature at the corner of the entrance to the tubular laminar flow elements, cannot even be quantified with or without taking the components completely apart. Thus the cost-effective calibration of high performance multi-gas MFCs has been a great technical challenge. The three main components of the MFC, i.e. the thermal mass flow sensor, the laminar flow element, and the flow control valve with the control electronics, as shown in Fig. 1, are each governed by a different set of thermal fluid dynamic relations. In this paper the governing equations for the different components of the thermal mass flow controllers are reviewed. Examples are then given to show how similarity analysis may be used to derive accurate scaling laws. These scaling laws are useful for the design of thermal mass flow controllers and the optimization of their calibration and tuning processes.

Fig. 1 Thermal mass flow controller, cross-sectional view.

THERMAL MASS FLOW SENSOR

The typical thermal mass flow sensor consists of a stainless steel sensor tube with its two ends welded to the sensing port of the MFC, as shown in Fig. 2. Two temperature-sensitive resistance elements are wound over the sensor tube as shown in Fig. 3, and are electrically connected in series to form one branch of a Wheatstone bridge. An electric current flowing through the two resistance elements raises

the wire temperature to approximately ~100 Co above the ambient. The two resistance elements also

function as the differential temperature sensor. When there is no fluid flow, the electric current heats up the sensor tube symmetrically and there is no differential voltage output. When there is flow, the fluid cools down the upstream resistance element more than the downstream element. The difference in temperature between the two resistance elements is indicative of the mass flow rate. Strictly speaking the convective heat transfer in the thermal mass flow sensor involves the conjugated heat transfer problem. Fortunately the sensors are made of thin-walled stainless steel tubing so that the thermal conductivity of the tube wall does not significantly diffuse the temperature gradient introduced by the flow. In fact the tube walls are often so thin, with the tube length to the wall thickness ratio exceeding several hundred, that the temperature over any axial cross section of the metal tube is essentially uniform under the steady-state conditions. For continuum flows the fluid adjacent to the tube inside wall is also at the same temperature as the wall. This tremendously simplifies the mathematical problem because now we only need to deal with the temperature distribution in the fluid.

Similarity Analysis

Assume the radial component of velocity n to be identically zero in the sensor tube and the laminar flow has a fully-developed velocity profile, as shown in Fig. 3. For gases with constant thermal conductivity k,

constant density r and constant specific heat cp, the energy equation for convective heat transfer in the cylindrical coordinates reads:

x

T

k

cu

x

T

r

Tr

rr

p

¶

¶=

¶

¶+÷÷

ø

öççè

æ

¶

¶

¶¶ r

2

21

(1)

where T stands for the gas temperature, r is the radial- and x the axial- coordinate along the length of the tube. Eq. (1) may be written in dimensionless form [1] as:

2 2

2 2 2

1 1

2 (Re Pr)D

u

r r r x x

q q q q+

+ + + + +

¶ ¶ ¶ ¶+ = -

¶ ¶ ¶ × ¶ (2)

where

0

00

( ), / ,

( )

/, / ,

Re Pr

Re ; Pr ; , Re Pr .

W

W

D

p pD D

T Tu u V

T T

x rx r r r

c V D cV Dand

k k

q

m rrm

+

+ +

-= =

-

= =

= = × =

In the above, r0 is the radius and D is the inside diameter of the tube. T0 is the temperature of the ambient. TW is the tube inside wall temperature and is a constant for the constant-temperature

Fig. 2 Thermal mass flow sensor.

Fig. 3 Steady-state convective heat transfer in the sensor tube. sensor under consideration. V is the reference flow velocity. V may be identified with the maximum in the sectional velocity profile. The heat conduction of the fluid in the axial direction is retained in the above formulation. Eq.(2) tells us that the thermal conduction in the axial direction is important when

PrRe ×D is small, occurring at low flow-rates in the thermal mass flow sensor.

For a sensor with two adjacent heating coils each with length L, Eq. (2) applies over the region:

( )PrRe

2~0

D

DL

x×

=+ (3)

Eqs.(2) and (3) suggest that the temperature distribution in the fluid is governed by the two non-

dimensional parameters: PrRe ×D and L/D. For a constant-temperature sensor, the sensor output is

determined by the amount of heat generated by the electric current to offset the effect of the gas flow so as to keep the resistors at the constant temperature. For a constant-temperature thermal mass-flow sensor with two resistance elements symmetrically located on the tube, the output voltage S is proportional to the difference between the heat input to the upstream and to the downstream element. The heat loss through the tube-ends as well as the heat loss to the ambient through the thermal insulation are both omitted from the consideration on the ground that, since the coil windings are maintained at constant temperature, these heat losses do not vary with the gas flow and therefore do not affect the sensor differential output.

With the temperature distribution in the fluid written as:

þýü

îíì ×= ++

D

Lrx D ;PrRe;;θθ (4)

, the heat-flux from the heating coil to the fluid per unit length of the tube wall is

1

00 )(22

0 =+

= +÷÷ø

öççè

æ

¶

¶--=÷÷

ø

öççè

æ

¶

¶=

r

W

rrr

TTkr

Tkr

dx

dq qpp (5)

Using Eqs.(3)-(5) the following expression for the sensor output is obtained:

( )ïþ

ïýü

ïî

ïíì

÷÷ø

öççè

æ-÷÷

ø

öççè

æ××-×-= ò ò

× ×

×++

+

=+

+

=+

PrRe

1

PrRe

2

PrRe

10

11

0

θθPr)(Re S DD

L

DD

L

DD

Ldx

dr

ddx

dr

dTTGkD

rr

DWp (6)

where G is the electronic amplifier gain. The quantity in the curly bracket is a function of the two non-

dimensional parameters PrRe ×D and L/D only. Except for the gas thermal conductivity k, the

quantities G, D, L/D , and (Tw – T0) are all fixed sensor design constants. For sensors of a given

design, clearly Eq. (6) suggests that the quantity S/k is a function of PrRe ×D , i.e., the Péclet number

of the flow.

Scaling Relation

Without explicitly solving the partial differential equation, we thus arrived at the following scaling relation for the constant-temperature sensor:

{ }þýü

îíì

=×µk

cDVWW

p

D

rPrRe

k

S (7)

where W represents the integral on the right-hand-side of Eq.(6). The similarity theory and sensor model was verified by data from the constant-temperature sensors with 0.0135 inches tube ID and 0.5 inches-long heated section [2]. Fig. 4 shows the sensor electric output signal plotted against the flow-rate for several process gases with widely varying nonlinear characteristics. Fig. 5 shows the same data plotted by using the non-dimensional quantities given in Eq. (7), i.e. S/k ~ ReLPr. In these

data, gas property constants [3] at an average temperature of 85 °C were used for calculating the dimensionless parameters. As shown in Fig. 5, the S/k ~ ReLPr curves from the 9 gases tested all collapsed into a narrow band, suggesting that the similarity theory works quite well.

Examination of the data from multiple sensor samples suggests that the residual error in Fig. 5 may be further reduced by adjusting the gas specific heat cp with a multiplier ec and the thermal conductivity k by a multiplier ek. It turns out even with all the manufacturing tolerances involved, the values of the correction constants ec and ek do not vary significantly among sensors of the same design. This suggests the constants ec and ek are not arbitrary fudge factors but are associated with either the approximations inherent in the model or the uncertainty in the gas property data. The optimal values of ec and ek were chosen by comparing against the live-gas test data.

The thermal mass-flow sensor similarity model has since been extensively verified against data obtained from a great number of process gases. In Fig. 6 the sensor output for over 30 different process gases are plotted along with the calibration gas argon, for which the gas properties are assumed to be exact (with ec and ek both equal to 1.0). The gas property correction constants ec and ek for all of the gases tested are listed in Table 1. The fact that these constants all fall pretty close to 1.0 suggests that the gas property corrections are moderate and the similarity theory is largely sound. Of particular interest in Fig. 6 are the data for xenon (Xe) recently obtained from the pressure rate-of-rise measurement. Xenon is known to be one of the gases exhibiting the worst nonlinearity in the thermal mass flow sensor output. The xenon data in Fig. 6 also comply with the similarity scaling relationship, providing further proof for the validity of the theory and the model in Eq. (7).

Explicit mathematical solutions for the thermal mass flow sensor may also be obtained from Eqs. (2-6) with the appropriate boundary conditions. For instance the solution the author obtained for the constant-temperature sensor satisfactorily predicts all the important features of the sensor nonlinear characteristics including the gas-to-gas scaling relationship presented above. Although these solutions enlighten the sensor design process, they do not exactly predict the output curves for any particular sensor, potentially due to the manufacturing tolerance issues mentioned earlier. The similarity scaling relation in Eq. (7), however, remains valid for any constant-temperature sensor that was tested. This makes it possible for the thermal mass flow controller manufacturers to build sensors that are accurate for the process gas even if they are calibrated only by using an inert gas.

Fig. 4 Raw sensor data.

0.0

0.5

1.0

1.5

0 5 10 15 20 25 30 35 40 45

Se

nso

r O

utp

ut

Actual Gas Flow (sccm)

Ar

CF4

SF6

CO2

CH4

CHF3

He

N2

Fig. 5 Sensor data plotted by using the similarity variables.

Fig. 6 Sensor data plotted by using the similarity variables after gas property correction.

0

20

40

60

80

100

0 1000 2000 3000 4000 5000 6000

(Sensor

Outp

ut)

/ k

Re Pr

Ar

CF4

SF6

CO2

CH4

CHF3

He

N2

0

20

40

60

80

100

0 1000 2000 3000 4000 5000 6000

(Se

nso

r O

ut)

/(k*E

k )

F = Re Pr *Ec

Sensor Output in Similarity VariablesN2

He

Ar

CF4

SF6

CO2

CH4

CHF3

HBr

Cl2

NH3

CH3F

C2H4

BCl3

SiCl4

C2F6

C4F8

CH2F2

C4F6

Xe

Table 1 Values of the ec and ek constants

Gas ec ek

N2 0.99 0.95

He 1.00 1.00

Ar 1.00 1.00

CH4 1.00 1.00

CO2 1.00 1.00

CF4 1.00 0.93

CHF3 0.98 1.02

SF6 1.00 0.93

HBr 0.98 1.03

Cl2 0.99 1.15

NH3 0.98 1.0

CH3F 0.96 0.96

C2H4 0.97 0.98

BCl3 0.98 1.04

SiCl4 0.97 1.11

C2F6 0.98 1.00

C4F8 0.97 1.00

CH2F2 0.99 1.00

C4F6 1.15 1.00

Xe 0.98 1.07

LAMINAR FLOW ELEMENTS

Laminar flow elements are frequently used in flow meters either as the flow-sensing element or as the flow-splitting device. When used as the flow-sensing element, the laminar flow element generates the pressure drop to deduce the flow rate. When used as a flow-splitting device, the laminar flow element divides the flow between the sensor flow path and the bypass flow paths. In both cases the pressure drop characteristics directly impacts the accuracy of the flow meter.

The pressure drop in the fully-developed laminar flow in round tubes is well-known and is governed by Hagen-Poiseuille’s relation. Unfortunately no laminar flow element delivers purely fully-developed flow. In fact any laminar flow element must have an entrance, where the boundary layer grows and the loss of the dynamic pressure in the potential core contributes significantly to the pressure drop. The entrance effect makes the pressure-flow characteristics for most laminar flow elements significantly nonlinear, with the nonlinearity depending on both the inlet geometry and the gas viscosity. The viscosity effect may be

characterized by the Reynolds number ReD =ruD/µ. The larger the Reynolds number, the more nonlinear the pressure drop characteristics.

The bluntness of the lip at the tube entrance affects the nonlinearity associated with the loss of the total

pressure, with the exact shape of the lip determined by the manufacturing tolerance. Consider laminar flow elements composed of bundles of round tubes for example. As a result of the electro-chemical polishing process the corner radius at the tube entrance is non-uniform and may range anywhere from 0.0005" to 0.001”. For tubes with 0.020” ID and 0.005” thick wall the corresponding corner to radius ratio can easily vary between 5% and 10%, enough to make a significant difference in the flow pattern and the pressure drop characteristics, as illustrated in Fig. 7. As a matter of fact the exact geometry of many practical laminar flow elements cannot even be accurately quantified except by dissection. Without knowing the exact geometry, any attempt to describe the pressure flow characteristics explicitly by using a closed-form formula is futile.

Similarity Analysis

To conduct a formal similarity analysis for the laminar flow in a tube with entrance effect one begins with the Navier-Stokes equations in cylindrical coordinates. After going through the standard procedures of simplification and non-dimensionalization, the following results are obtained:

( )10

ˆ

ur v

x r r

++ +

+ +

¶ ¶+ =

¶ ¶ (8)

ˆ 1

ˆ ˆ

u u d p uu u r

x d xr r r r

+ + ++ + +

+ + + +

æ ö¶ ¶ ¶ ¶+ =- + ç ÷

¶ ¶ ¶ ¶è ø (9)

where

0

0

; ;r u vr

r u vr V

rm

+ + += = =

and

0

2

2

1x̂ 4 4

Re

ˆ

D

x x x

VD D DVr

pp

V

m mrr

r

= = =

¢=

Eq. (8) and (9) suggest the solution for pressure distribution in the form ( )ˆ x̂p f= , with which the

pressure drop'

Lp at x = L becomes

2

1

Re

L

D

p Lf

DVr

¢ æ ö= ç ÷

è ø (10)

It is interesting to approach the same problem also from dimensional analysis. There are totally six (6)

physical quantities governing the problem of laminar flow in a tube with the entrance effects: the tube inside diameter D, the length L, the fluid viscosity µ, the fluid density ρ, the mean flow speed U, and the

overall pressure drop Dp. Buckingham’s Π-theorem dictates that there are three (6-3) dimensionless

parameters governing the problem. While these parameters may be chosen as D/L, ReD, and ½ru2/Dp,

Langharr's work [5] suggests ½ru2/Dp to be a function of ReD(D/L), as given in Eq. (10).

In the above analysis we deliberately omitted the variation in the entrance geometry as a controlling

parameter. This is permitted only based on the assumption that the similarity relationship to be obtained will only be applied to devices of exactly the same entrance geometry. Since in practice due to the manufacturing tolerance issue the exact geometry cannot be guaranteed, the above assumption means that the similarity relation must be applied to nothing but exactly the same device. Although this may sound restrictive, using the similarity relationship to scale the calibration data from the calibration gas to the process gas for individual MFC is exactly what we intend to achieve. Besides, since for accuracy reasons one must individually calibrate each nonlinear laminar flow element anyway, why not explore the similarity relation and take full advantage of the calibration data?

Scaling Relation

Extensive tests and measurements were carried out to verify the similarity relation by using a variety of

gases over a wide range of Reynolds numbers. In these tests each laminar flow element was calibrated and then scaled as a unique device with the intent that any distinctive features in the flow characteristics due to part tolerance are completely preserved. In order to avoid the denominator on both sides of the

Eq.(10) to vanish simultaneously at no flow, to correlate the experimental data, the similarity relation is written in the following alternative form:

2

Re½ ~ D

Vf

P

D

L

r æ öç ÷D è ø

. (11)

Fig. 8 shows the test data collected from 9 different gases using a laminar flow element consisting of

138 round tubes each 1.5" long, with 0.030"ID and 0.004" thick wall. The tubes were electro-chemically polished and packed in hexagonal patterns inside a housing with a hexagonal cutout. Typical tolerances were ±0.0005” for the wall thickness and ±0.001” for the ID, respectively. The corner radius at the tube inlet was not controlled and it falls somewhere between 10% and 30% of the wall thickness. As shown in Fig. 8, there was very little random error at either the low or the high flow-rates for any of the gases studied, suggesting that the similarity principle holds quite well.

To account for the temperature difference between the gas flowing through the sensor and through the

laminar flow element, a gas specific bias-function [4] was introduced to further reduce the systematic error in the model. With these the similarity principle was checked out satisfactorily among the data collected from different gases using any single laminar flow element. The similarity plots spread out somewhat more pervasively, however, when data from different laminar flow elements (albeit of the same design) are cross-compared, even after the bias functions are introduced. This seems to justify our speculation that device-to-device part variation impairs the similarity relationship. The flow rate data reported above were obtained by using the DryCal moving-piston provers made by BIOS International Corporation.

Fig.7 Effect of inlet geometry on the laminar boundary layer near the entrance of a tube.

Fig. 8 Pressure drop in a laminar flow element with 138 x 0.030 inches ID tubes that are 1.5 inches

long.

VALVE TUNING CONDITIONS

The flow control valve together with the valve-drive electronic circuit regulates the flow by way of a feed-back control loop. The required valve stroke depends on both the flow rate and the supply pressure available to the process gas of interest. Of course the dynamic performance of the solenoid valve, for example its current-displacement characteristics, also depends on the actuator's electro-mechanical design such as the size of the coil, the magnetic path, and the stiffness of the return spring.

When the MFCs are being built in the factory, the valve is typically tested by using an inert gas. After

the valve dynamics is optimized by using an inert tune gas at a certain pressure, at the same drive current it behaves differently when flowing the process gas that may have a different molecular weight, different thermal physical properties, and different supply pressure. Besides, the mass flow controllers normally operate under the choked conditions with its downstream exposed to the semi-conductor process vacuum. The MFC manufacturers, however, sometimes cut corners by tuning the MFC with the exhaust exposed to not the vacuum but the ambient atmospheric pressure, where the valve may or may not be choked depending on the supply pressure of the tune gas. All of these complicate the valve tuning process. Especially when the engineering solution must deal with hundreds of semi-conductor process gases, how to effectively choose the proper tuning gas and the tuning pressure is a question of practical importance.

Although the valve normally works under the choked conditions with the gas acting like a compressible fluid in the semiconductor vacuum processes, there is usually so much reserve in the valve stroke that incompressible flow relations may often be used as a rough estimate for calculating the stroke. For simplicity, we will assume incompressible flow here just to illustrate how the valve tuning conditions may be determined. When more accurate calculations are needed compressible flow relations must be used with some extra effort.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0 5 10 15

U2/2

DP

Re D/L

Ar

N2

CO2

CHF3

CF4

SF6

In order to tune a valve for ideal dynamic performance, the valve must be operated under the right valve opening and the right differential pressure. The equation for valve sizing is well known and needs no mentioning here. The method for scaling and choosing the proper tuning conditions will be discussed below. Consider the valve at an effective opening area A. Neglecting the compressibility effect, the flow rate is related to the pressure difference by

( )2

2

2

1 1

2 2

uAp u

A

rr

rD µ = (12)

, where ρ is the gas density and u is the gas velocity at the valve throat. Since we will only be dealing with pressures ratios in the following discussion, a valve orifice discharge coefficient of 1.0 is used here without loss of generality. The effect of Reynolds number on the discharge coefficient is also neglected.

With the quantity ρuA equated to the mass flow rate m , and after introducing the perfect gas law, Eq.(12)

becomes:

( )2ˆ

MPm p A

RT= D (13)

where M represents the molecular weight and R̂ the universal gas constant. Taking the ratio of the mass flow rate between the surrogates (subscript s) tune gas and the process gas, for the same valve opening area A we obtain:

s s s sm p P M

m p P M

D=

D (14)

With the mass flow rate expressed in moles per minute m nM= and n directly proportional to the

volumetric flow rate Q, Eq.(14) becomes

s s s s

s

Q n p P M

Q n p P M

D= =

D (15)

Assume the MFC output voltage e to be linearly related to the mass flow rate by a constant 'conversion

factor' CF, i.e., FQ n C eµ µ × . Finally the MFC output for the surrogate gas is related to that for the

process gas by:

,

s s s F

s F s

e p P M C

e p P M C

D=

D (16)

In the above, we have neglected the nonlinearity of both the sensor and the laminar flow elements to simplify the formulation as much as possible so that it does not obscure the main objective. Normally the MFC manufacturers tune the valve for the optimum transient response at a few selected set-points, for examples at 25%, 50%, 75% and 100% of the sensor full-scale output. These set-points correspond to the same percentages of sensor full-scale output voltage e for the process gas and for the

surrogate tune gas, respectively, i.e.

,

1s s sF

s F s

p P eM C

p P M C e

D= =

D (17)

Eq. (17) provides a method for choosing the correct tune gas and pressure for any process gas at any pressure of interest.

To give an example, provided that the MFC is built for a certain process gas where the discharge

pressure is 1 atm. Assume that the MFC is tuned with ambient discharge so that 14.7P p psia= D +

and 14.7s sP p psia= D + . Eq. (17) may now be written as

2,

2

( 14.7 )

( 14.7 )

s F ss s

F

M Cp p psia

p p psia M C

D D +=

D D + (18)

For a chosen tune gas, the factor on the right hand side of Eq. (18) is a constant. Represent this constant by Θ, we have

2 14.7 ( 14.7) 0s sp p p pD + D -Q× D × D + = (19)

, for which the only meaningful solution is:

( )20.5* 14.7 4 ( 14.7) 14.7sp p pD = + × Q × D × D + - (20)

This example illustrates how the tune pressure may be calculated for any process gas at any vapor pressure with any given tune gas. Valve tuning conditions are important because many process gases are associated with low vapor pressure or high molecular weight that the valve will perform poorly if the wrong tune gas or the wrong tune pressure is used.

CONCLUSIONS

Scaling relations that ensure the accuracy for the actual process gases are important for the fabrication

of high-performance mass flow controllers. The three main components of the MFC, i.e. the thermal mass flow sensor, the laminar flow element, and the flow control valve, are each governed by a different set of thermal fluid dynamic relations, some of which are significantly nonlinear, making the scaling of especially multi-gas thermal mass flow controllers challenging. In this paper the governing equations for the three MFC components were reviewed, and the scaling relations were presented.

In retrospect, it is rather unfortunate that the word scaling sometimes leaves people a false sense of

vagueness and imprecision. While some scaling laws may be approximate when the physical processes are modeled with approximate theories, those derived rigorously from precise theories using similarity analysis can be highly accurate. Proper understanding of the underlying physical process is critical for developing the accurate scaling laws. Proper application of the scaling laws is critical for the calibration of the multi-gas mass flow controllers. References [1] Wang, C. "A Similarity theory for Thermal Mass Flow Sensor and Its Gas Conversion Factors", Measurement Science Conference, Jan. 25, 2007, Long Beach, CA. [2] Wang, C. “Thermal Mass Flow Sensor Similarity Theory – Comparison with Experiment”, Measurement Science Conference, Mar. 13, 2008, Anaheim, CA. [3] "Le Gas Encyclopedia" L'air Liquide, 1976. [4] Wang, C. “Calibration of Nonlinear Laminar Flow Elements for Multigas Applications”, Measurement Science Conference, 2009 Pasadena, CA. [5] Langharr, H.L. “Steady Flow in the Transition Length of a Straight Tube”, J. Applied Mechanics, Vol. 9, No. 2, A55-58, 1942. [6] Kays, W. M., and Crawford, M. “Convective Heat and Mass Transfer”, 2

nd ed. 1980, and 3rd ed.

1993, McGraw-Hill.

(Published in: Measurement Science Conference, Anaheim, CA. March 2012)