EXPERIMENTAL TECHNIQUE
A N A L Y S I S A N D C A L I B R A T I O N OF T U B U L A R
FLOW M E T E R S D O N A L D A. W I L L O U G H B Y A N D P A U L A.
K I T T L E
Redstone Research Laboratories, Rohm and Haas Go., Huntsville, Ala. Capillary flowmeters are capable of accurate measurement of gas flow rates when the flow rate is sufficiently low. However, for large flow rates entrance effects exist throughout a tube of reasonable length and must be accounted for in the meter calibration. Theoretical solutions for entrance-region tube flow were used in conjunction with the theory of rounded-entrance flowmeters to obtain a theoretical solution relating flow rate, downstream pressure, pressure drop, and temperature. A good measure of the flow rate was obtained directly from the theory, and excellent results were obtained when an experimentally determined correction factor was used.
and Katz (4)have shown that entrance effects on the pressure drop in tubes are negligible when ( L / d ) l/Red > 1, Under these conditions the pressure drop is given by the well known Poiseuille equation NUDSEN
AP =
8 pLU
dzsc
and the volumetric flow rate is given by
Q
=
Td4& --AP 32 pL
A meter operating under these conditions will be referred to as a capillary flowmeter to distinguish it from a tubular flowmeter in which entrance effects are important. Obviously, the volumetric flow rate a t constant temperature is a linear function of the pressure drop in a capillary meter. As the mass flow rate increases it becomes difficult to design a capillary flowmeter of reasonable length for gases, which will satisfy the requirement that ( L / d ) l/Red > 1. This difficulty is more clearly seen when the requirement is rearranged to the form 7rpL/4 tir > 1. However, in order to retain conveniently measurable pressure drops over a range of pressures and flow rates it is desirable to retain features of the capillary flowmeter rather than to use a n orifice or Venturi-type meter. T h e capillary meter AP is independent of pressure, whereas AP varies with pressure in a n orifice or Venturi meter. A tubular flowmeter of the type shown in Figure 1 has some of the features of the capillary meter and can be analyzed and calibrated as described below. Flow development in the entrance region of a circular tube has been studied by a number of investigators ( 7 , 3, 5, 6, 9 ) , who generally assumed that a uniform velocity profile existed a t the entrance of the tube; analytical solutions were obtained using a simplified form of the Navier-Stokes equations. T h e linearization procedure employed by Sparrow and Lin ( 9 ) is unique in that the pressure drop calculated from momentum considerations is identical to that calculated from mechanical energy considerations. This solution was used in this work. However, Sparrow’s results are practically identical to those of Langhaar (6). I n practice a uniform velocity profile does not exist a t the tube entrance unless a rounded or bell-shaped entrance is used. Rivas and Shapiro (7) have shown that the condition at the entrance of a rounded entrance flowmeter or tube is the 304
l&EC FUNDAMENTALS
same as that predicted by theory at a position a short distance downstream of the entrance. Results from Rivas and Shapiro (7) are shown graphically in Figure 2, where the ratio of a n equivalent straight tube length for the rounded entrance to the tube diameter is presented as a function of Reynolds number. A least-squares fit of these data to a five-term polynomial gave Equation 3, which is included on Figure 2.
r;)
= 2.208
- 1.032 [ln(Red)] + 0.1791[ln(Red)]2 -
+
0.01214[ln(Red)I3 0.0002803[ln(Rea)]4 (3) T h e effective tube length for the meter, L’, is then the sum of L and Leq.
I
I
1---L----1
T w DIRECTION 7 I
OF FLOW
I
I
Figure 1.
Tubular flowmeter 1. 2. 3. 4.
Inlet chamber Tube inlet Tube exit Exit chamber
I
0.40 0.35
- ---
,
3
4
8
- RECOMMENDED
0.30 - IN C71 FOR ROd < 800
Leq -
d
A
,
CURVE USE9 IN THIS
RECOMMENDED
-
0.25
Red > 1000
r
k - p
0.20
$ 1000 l0,OOO
0,15,00
Red Figure 2.
Equivalent length of entrance
T h e pressure drop between the inlet reservoir and the tube exit chamber is given (7, 9 ) by (4) where $ = n L ’ ~ / m or, equivalently, $ = (L’/d)/4 Red. T h e first term C J the right-hand side of Equation 4 accounts for the pressure drop between the reservoir and the tube inlet, the second term represents the pressure loss for fully developed laminar flow, and the third term represents the additional pressure loss associated with the tube inlet region due to combined considerations of momentum and friction. I t is reasonable to assume that the static pressure at the tube exit, Pa, and the exit chamber pressure, P4, are identical, since kinetic energy losses associated with a sudden expansion in area are severe (70). Therefore, Equation 4 can be considered to represent the total pressure drop between the two reservoirs. T h e inlet correction factor, K , approaches a value of 1.24 as the flow becomes fully developed. A plot of K as a function of $ is given by Sparrow and Lin ( 9 ) . A least square fit of these data in the range 0.005 < $ < 0.23 to a five-term polynomial gave
K
=
0.3701
+ 19.50 $ - 199.8 $’ + 956.9
$3
-
1693 fi4
(5)
For $ > 0.23 it is satisfactory to use the fully developed value of 7,-
A.
