Baker R.C. Flow Measurement Handbook: Industrial Designs, Operating Principles, Performance, and Applications

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15.A.3 GENERAL VECTOR EQUATION 387

and

(15.A.15)

with the addition of the final term, which relates to the local heat addition. If all

heat is added across boundaries with the flow, then the last term disappears. Hemp

uses Equations (15.A.14) and (15.A.15) to develop his weight function theory for

thermal diffusion flowmeters. This is an elegant theory, and the interested reader is

encouraged to refer to Hemp's papers. However, as he comments, the work at present

is probably mainly of theoretical interest.

We can write Equation

(15.A.

13)

for flow in a two-dimensional channel as

(

dz pc

\dx

(15.A.16)

We can simplify this for flow in a tube where we neglect temperature gradients

in any but the axial direction. Referring to Figure 15.A.2 and rewriting the gradient

and the Laplacian as finite difference expressions, we obtain

(15.A.17)

If we take the area of the duct to be A and the heat source to be at the center, which

has a temperature T

, we can rewrite the equation in terms of mass flow:

convection

term

2kA (Ti - 2T

conduction

term

) 2LA

+ -—^a

(15.A.18)

heat input

term

The first and last terms constitute the simplest approximation for the behavior of

the thermal gas flowmeter. Heat is convected away by the flow, and the difference

of temperature between inlet and outlet is related to the heat input, the specific

heat, and the mass flow. This assumes that the first term on the right-hand side

can be neglected. However, in the case of no flow, the heat added is conducted

away, and the central term cannot be neglected because it gives the size of the

conduction.

L L

7",

Figure 15.A.2. Diagram of tube to illustrate the use of finite differences.

388

THERMAL FLOWMETERS

15.A.4 HASTINGS FLOWMETER THEORY

Figure 15.A.3 shows the heat flows in the Hastings flowmeter based on an approx-

imate one-dimensional theory. From the heat flows in the diagrams, we can derive

the simple equation of the meter. We consider first the wall, and specifically an ele-

ment of length 8z. Here we have a heat addition as a result of the electric current /,

which flows down the tube wall of resistance R per unit length. Thus the heat input

is I

R8z. From this element, heat flows along the wall, of amount <5Qh, which can

be related to the temperature gradient in the wall

Qh = -kA^- (15.A.19)

where A is the cross-section of the wall, and k is the thermal conductivity of the wall

material. Heat also flows out as heat loss from the meter, fiT,

being a dimensional

constant to give the heat loss based on the wall temperature, and it flows into the

fluid, q±. From this we have an equation

(15.A.20)

ELECTRIC

CURRENT

ELECTRIC

CURRENT

| /

OUT

CENTER

FLOW

LINE

\iThz

ELECTRIC

CURRENT•

HEAT FLOWS

IN TUBE WALL

9±5z

(b)

HEAT FLOWS

IN FLUID

Figure 15.A.3. Geometry and heat flows for the Hastings flowmeter: (a) Schematic diagram;

(b) Heat flows within the meter.

15.A.5 WEIGHT VECTOR THEORY FOR THERMAL FLOWMETERS 389

8z then flows into the fluid, which is assumed to be at approximately the same

temperature as the wall at each point along its length. This heat causes a rise in the

temperature of the fluid. The rate of heat addition q

8z is equal to the product of

the temperature rise of the fluid, the specific heat of the fluid at constant pressure,

and the mass flow rate. Thus we can write

Finally, we can combine Equations (15.A.20) and (15.A.21) and use Equation

(15.A.19) to eliminate Qh

8z =

q± =

qvC

~dz

^ ^ fiT + IR = 0 (15.A.22)

This equation is quoted by Hemp (1995a), and he also quotes a solution due to

Blackett and Henry (1930), which for low flow rates can be written in a linear

form

-Ti=

(15.A.23)

Hemp suggests that the nonlinear complete solution is probably invalid outside

the linear range due to the assumptions about uniform temperature across any cross-

section of the flowmeter. For other solutions and more details, the reader is referred

to Hemp (1994a, 1995a), Blackett and Henry (1930), Komiya et al. (1988), Brown

and Kronberger (1947), and Widmer et al. (1982).

15.A.5 WEIGHT VECTOR THEORY FOR THERMAL FLOWMETERS

Hemp (1994a, 1995a) presented an extension of the weight vector theory to thermal

flowmeters. Again the ideal weight vector must satisfy

V x W = 0 (15.A.24)

For a typical thermal diffusion flowmeter, the signal

- T

f\-Wdv

(15.A.25)

and the weight vector is given by

W = pc

(15.A.26)

390 THERMAL FLOWMETERS

CAB

Figure 15.A.4. Diagram for Hemp (1994a) theory.

where T

is the temperature distribution resulting from unit heat flux injected into

(Figure 15.A.4), and T

is the temperature distribution when unit heat flux enters

at C and leaves at B. It appears that the usefulness of Hemp's theory is limited at

present by the nature of the cases for which it is valid and the flow rates that are low

enough to ensure linear conditions.

CHAPTER

Angular Momentum Devices

16.1 INTRODUCTION

I am indebted to Medlock (1989) for notes on three early devices that attempt to use

change of angular momentum to obtain mass flow.

Katys (1964) used an electric synchronous motor with a special rotor supported

internally in the pipe and a stator outside the pipe. Fluid passed through vanes in the

rotor and acquired angular momentum. From the power used or the torque needed

to drive the rotor, it was claimed that the mass flow could be deduced.

According to Medlock, the Bendix meter "measures the torque required to impart

angular momentum to the liquid. One end of a calibrated spring is driven at a

constant speed and the other end is connected to a freely rotatable turbine. The

torsion developed in the spring is a measure of mass flow rate."

The twin rotor turbine meter (Potter 1959) attempted to measure mass flow by

means of two in-line turbine rotors of different blade angles, which are joined by

a torsion spring. Despite discussions with Medlock about this meter, I am not con-

vinced that this is a true mass flow meter and suggest that the valid derivative of the

Bendix meter is the fuel flow transmitter described later.

An early design with many similarities to the current commercial designs was a

device due to Orlando and Jennings (1954) shown in Figure

16.1.

A constant speed

motor drives an impeller imparting swirl to the liquid.

The angular momentum, which is removed from this liquid per second as it

leaves the driven rotor and moves into the tethered rotor, is

coR

(16.1)

where

is the angular velocity of the rotor, and R is the radius of the annulus in

which the flow takes place. The force on the spring restraining the tethered rotor

will be equal to the loss in angular momentum per second and will be indicated by

the angular deflection of the tethered rotor. The mass flow will therefore be given by

sO/(coR

)

(16.2)

where 0 is the angular deflection, and s is the spring constant. It is important to

note the following.

i. The flow will not precisely follow the vanes unless sufficient length is allowed to

force the flow to be axial relative to the blades.

391

392

ANGULAR MOMENTUM DEVICES

Constant

speed

motor

Section

through

impellers

Driven

impeller

Restrained

impeller

Figure 16.1. Orlando and Jennings device.

ii.

The flow profile in the annulus will modify the effective value of R.

iii.

The motor speed must be known.

16.2 THE FUEL FLOW TRANSMITTER

Much of the material in this section comes from a lecture by Rowland (1989). The

main features of a particular commercial device are shown in Figure 16.2 and in

exploded view in Figure 16.3. This design eliminates the electrical drive by extract-

ing power from the liquid flow to drive the turbine and provides one example of

commercial meters of this type.

GUIDE VANES TURBINE STRAIGHTENING PICK OFF SPRING DRUM

VANES ' 1

MEASUREMENT ASSEMBLY

PICK OFF

Figure 16.2. Diagram of the fuel flow transmitter (reproduced with permission of GEC-Marconi

Avionics).

MEASUREMENT ASSEMBLY

Figure

16.3.

Exploded diagram of the fuel flow transmitter (reproduced with permission of GEC-Marconi Avionics).

394 ANGULAR MOMENTUM DEVICES

16.2.1 QUALITATIVE DESCRIPTION

OPERATION

As the flow enters the unit, a bypass assembly controls the amount of liquid passing

through the power turbine in order to avoid the assembly's running too fast or too

slow. The bypass mechanism consists of a spring-loaded bypass valve that contains

the speed of the unit to within the range 70-400 rpm. At the lowest flow rates, all

the liquid passes through the turbine, but as the flow rate increases, the build-up of

pressure across the turbine progressively opens the valve and allows an increasing

portion of the total liquid flow to bypass the turbine.

Depending on the opening of the bypass valve, a proportion of the flow passes

through a stator with guide vanes that create swirl in the flow. The flow then enters

the power turbine, which drives a shaft with the measurement assembly on the other

end. However, having extracted power from the flow in this way, it is essential that

the remaining swirl after the power turbine is completely removed from the liquid.

After the turbine and bypass valve, therefore, the flows combine to pass through

the straightening vanes, before passing into the driven measurement assembly. The

meter proper then follows and in this particular design, unlike the example described

earlier, measures the torque needed to impart the angular momentum to the liquid.

Within the rotating measurement assembly, a tethered impeller imparts to the liq-

uid the angular momentum. The ratio of blade length to mean space between the

impeller vanes is 8:1. Clearances are kept to a minimum to reduce leakage, and the

meter is insensitive to installation attitude.

In the process, the impeller experiences a torque that causes the restraining spring

made of NiSpan-C902, a material whose change of stiffness with temperature can

be adjusted by heat treatment, to tighten. To avoid overtightening the spring, the

bypass control of speed also reduces the necessary deflection of the spring. This can

be seen from the fact that, without the bypass, the angular velocity would be approx-

imately proportional to the mass flow rate and so the torque would be proportional

to the square of the angular velocity. By allowing some of the fluid to bypass the

drive turbine, the angular velocity is reduced, and the speed need not rise in propor-

tion to the mass flow rate. The relationship between flow rate and rotational speed

is shown in Figure 16.4.

The bearings are precision ball races. Friction errors and vibration errors can

result from spurious torques at the impeller bearings. The fuel acts as a lubricant and

must therefore be clean (filtered).

The deflection is measured by means of a pair of electromagnetic pickup coils

placed on the outside of the nonmagnetic meter body. A pair of small powerful

magnets displaced 180° apart are attached to the circumference of both drum and

impeller. Signals are therefore generated for each 180° of rotation of the assembly.

16.2.2 SIMPLE THEORY

We can obtain an expression for the mass flow rate. We write the spring constant

as s. The torque causes the impeller to be displaced by an angle 0 from the rest of

the measurement assembly. The displacement in the range 0-160° is measured by

recording the time of transit of a marker on the assembly drum and a marker on the

impeller. A knowledge of the time delay r and the angular velocity of the assembly

will then allow the angular displacement to be deduced. Maximum calibration

16.2 THE FUEL FLOW TRANSMITTER 395

1ST STAGE

SPRING

BOTH SPRINGS

OPERATING

FLOW RATE

Figure 16.4. Relation between flow rate and rotational speed for the fuel flow

transmitter (reproduced with permission of GEC-Marconi Avionics).

spring hysteresis is approximately 0.05% of

FSD.

Thus

= sO/(coR

)

where the effective radius of the flow annulus is taken as R. This is affected by the

velocity profile in the impeller and the shape of the channel between the vanes.

Hub-to-tip ratio is generally greater than 0.7.

The time difference between the markers is

r =

0/Q)

(16.3)

so that

= sr/R

(16.4)

and we, thus, note that the mass flow may be obtained as directly proportional to

r, provided s is constant, and our simple description of operation is adequate.

16.2.3 CALIBRATION ADJUSTMENT

The meter has three methods of calibration adjustment (Figure 16.5):

i. Zero adjustment achieved by moving the impeller pickup circumferentially out

of alignment with the drum coil so as to introduce an apparent zero flow adjust-

ment.

ii.

Rate adjustment achieved when the meter is partially disassembled by altering

the clamping point of the spring and hence its effective length. This will alter

the value of

in the preceding equations.

iii.

Swirl adjustment by means of

small rudder in the flow upstream of the measure-

ment assembly creates a small local flow deflection and affects the performance

throughout the characteristic of the meter.

396

ANGULAR MOMENTUM DEVICES

FLOW RATE

ZERO (CONSTANT SPEED)

Figure 16.5. Calibration adjustments (reproduced with permission of GEC-

Marconi Avionics).

16.2.4 METER PERFORMANCE AND RANGE

The meter is adjusted to rotate at 150 rpm at maximum flow and between 150 and

300 rpm at other flow rates. Figure 16.4 indicates an increase in rotational speed

at lower flows resulting from the action of the bypass valve to maintain spring

wind-up. The assembly is fitted with a stop to prevent, at low flow rates, a reverse

oscillation causing spurious readings. It has a flow range of about 115-4,500 kg/h.

The temperature range is -40 to 150°C, and the maximum line pressure is about

130 bar with a maximum pressure drop of about 0.5 bar.

Accuracy appears to be claimed as ±1% of rate over a 7:1 flow range.

It is desirable to install the meter with a straight upstream length of pipework

to avoid swirl and profile distortion effects. Because this is seldom possible on an

aircraft, the calibration should be done with a representative piece of inlet pipework.

Referring to Equation (16.4), we can review the likely sources of uncertainty

in this meter that will influence the claims for its accuracy. It is unlikely that the

impeller completely imparts the assumed angular velocity to all the fluid. In addi-

tion, there will be some leakage and drag due to the gap between rotor, tube, and

vanes,

although the shroud may reduce some of these. These may be approximately

proportional to the mass flow rate so error may be of second order. The spring may

introduce nonlinearity or hysteresis and may give problems if not correctly installed.

The time difference may be of order 100 ms, and so measurement will be to within

about 100 /xs. There are also the characteristic changes due to the means of zero

adjustment. These may be incorporated into a calibration characteristic if a flow

computer is available to adjust the flow signal accordingly.

The material of the body is high strength aluminum alloy.

16.2.5 APPLICATION

The meter appears to have primarily one application to date. This is for the mea-

surement of fuel flow in aircraft.