Ind. Eng. Chem. Fundam. 1983,22, 163-166
163
INT = the integer of a reduced improper fraction k = thermal conductivity of tube metal, Btu/ft h OF (W/m
C)
m, n = integers qsv = average flux density over surface of tube, Btu/ft2 h (kW/m2) qe(qi) = external (internal) local surface flux density qm = maximum external surface flux density on tube (at 6 = 0) - I
Qe = ratio of ( local external flux density at B)/(maximum
Te - T B cn( -- Co(B+ In R ) + Cn qmre/k
n=l
n
external flux density) re (ri,r,) = external (internal,arithmeticmean) radius of tube, ft (m) R = ratio, re/ri S = distance between center lines of two parallel rows of tubes, ft (m) T(T,, Ti) = temperature of tube metal at arithmetic mean radius (at re, at ri), O F ("C) TB = bulk fluid temperature in tubes, O F ("C) = shorthand designation of exchange area A,F,, (= AY,,)
5
R2nHn - 1 ) cos n8 R2"Hn 1
+
Greek Letters = absorptance of tube surface e = emittance of tube surface p = reflectance of tube surface 0 = angular position on tube surface, measured in radians from plane normal to radiating plane CY
Nomenclature A (Ap, AT) = area (of plane, of tube) A , = a constant, defined before eq A1 B = Biot number, k / h r i c = constant in eq 20 C = center-to-center distance between tubes in a row C0,,4,n= coefficients in cosine series D = tube diameter D = determinant D' = the x-row, y-column signed minor or cofactor of D E = dimensionless parameter defined following eq 25 F (FW,Fp).= direct-viewfactor (fraction of radiation leaving P which is intercepted by T, or vice versa) pz,,= fraction of radiation emitted by x which impinges on y, directly and by aid from adiabatic reradiating surface 3,, = fraction of radiation emitted by x which is received by y, directly and by aid from reradiating adiabatic surfaces and by multiple reflection in whole system GB = fractional view, from a tube spot, of surroundings on the 6 = ?r side, or back side GF = fractional view, from a tube spot, of surroundings on the 0 = 0 side, or front side h = heat transfer coefficient, Btu/ft2 h O F (kW/m2 C)
Subscripts B = black (except on TB) e = external i = internal m = maximum P = plane PT = from plane to tube T = tube TP = from tube to plane Literature Cited Hopi, H. C. "Radiant Heat Transmission"; Chapter 4 in McAdams, W. H. Heat Transmlssion", 3rd ed.:McGraw-Hill: New York, 1954. Hottei, H. C.; Sarofim, A. F. "Radiative Transfer"; McGraw-Hill: New York, 1967; p 94. Hottel, H. C. "Heat Transfer in Fiames"; Afgan, N. H.; Beer, J. M., Ed.; Scripta (Wiley): New York, 1974; Chapter I.
Received for review April 14, 1982 Accepted January 28, 1983
Gas Flow through Rotameters Harry Levln* Jet Propulsion Laboratoty, California Institute of Technology, Pasadena, California 9 1 109
Monica M. Escorza Stanford Unlversity, Stanford, California 94305
From data available for small rotameters that use spherical floats in gas flow, a linear relationship is presented as a ood fit for variable volumetric flow Q , density p, and viscosity I.( at constant float height. The equation is (Qpl')-' = A ( ~ / p l ' -k ~ )8 , where A and 8 are constants for a given tube and height. Applying the equation at Q p becomes constant; on the other hand, at high Reynolds numbers (Re > 2000), low Reynolds numbers (Re < l), Qp1'2 becomes constant. The equation can be used to obtain an indirect calibration with any gas of known p and k. The constancy of Q B at low Q suggests the development of simple, inexpensive gas viscometers using __ rotameter technology;
Textbook and manufacturers' information suggesta that in relationships for flow through rotameters, the volumetric rate of flow Q, the fluid density p, and the viscosity p occur in combinations Qp1l2 and p 1 I 2 / p and, to a close approximation
QP'/' = ~ ( P ' / ~ / L L ) at constant float mass, float density, and height.
This paper examines how eq 1 applies to different regimes of gas flow. It is important to evaluate the function f of eq 1to obtain an insight into gas dynamics and also to reduce the work of calibration to a minimum in going from one gas to another. For these reasons, we sought to transform eq 1 to its most useful form. Using data available for small rotameters with spherical floats, we determined by trial and error that an explicit
(1)
0196-4313/03/1022-0163$01.50/0
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1983 American Chemical Society
164
Ind. Eng. Chern. Fundarn., Vol. 22, No. 2. 1983
, C 32
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102
Figure 1. In the series of rotameters, a straight line relationship of ( Q P ' / ~ )and ~ / p * is/ ~applied for seven Gases and five heights. Observe that the intercepts approach zero for the smallest tube; the slopes approach zero for the largest tube.
form for eq 1-and perhaps the best fit-at a given height is provided by the linear expression (Qp'/')-l = A ( p / p ' j 2 ) + B (2) where A and B are constants for a given tube, spherical float, float mass, and height. We were able to carry out the study because of the commercial availability of a series of six small, or purgetype, rotameters manufactured by the Brooks Instrument Division of the Emerson Electric Co. Each model, or size, of rotameter is provided by the manufacturer with general, rather than individual, calibration charts, H vs. Q (at
standard temperature and pressure) for each of a large number of gases. The six rotameters all use spherical floats. They encompass more than four orders of magnitude of flow rates, ranging from a few cubic centimeters per minute to more than 100 L/min. The rotameters are identified here as A, ..., F, in order of increasing flow rate. Their tubes, each with 150 mm of calibrated height, are right cylinders of glass. They have complex bores, each model having a bore somewhat different from the others. The bores are tapered for increasing diameter with height. Each bore has a symmetrical set of 3 or 6 length-running inner projections, which
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
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vary in shape as flats or ribs and which increase in thickness with height. The inner projections serve as guides for the float and hold it central to the flow at each height. The floats used in the study were made either of glass or stainless steel, milled to precise tolerances for interchangeability. We selected seven gases for the study which relied entirely on the Brooks Instrument data. For each tube and gas, there is available a Brooks calibration chart of H vs. Q at STP (21.1 OC and 1atm), as mentioned above. Table I lists these gases and their densities and viscosities in order of increasingp / p 1 l 2ratios. Aa is evident, they present for analysis widely differing densities and viscosities in a
10-fold variation of p / p 1 / 2 values. Figures lA, B, C, and D show the application of eq 2 to the series of rotameters. It is seen that as the flow becomes larger (e.g., as it increases in liters per minute, as in tube F), the slope A approaches zero. For a given height, the quantity Qp1/2then becomes constant. Constancy of Qp112 is known to apply to high-flow-rate rotameters. On the other hand, it is seen that as the flow becomes small (e.g., a few cubic centimeters per minute, as in tube A), the intercept B approaches zero. For a given height, the quantity Q p then becomes constant. Transition from the constancy of Q p at low flow rates to the constancy of Qp1J2at high flow rates is shown in
166
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
Table I. Table of Gases and Properties ~~~~
~~~~
~
~
gas
P , gicm' x 103
P , cP3 x 10
butane chlorine silane nitrogen argon hydrogen helium
2.49 2.99 1.34 1.16 1.66 0.0834 0.166
7.60 13.1 11.2 17.6 22.2 8.85 19.8
PIP
x
_____.__ _________._
i o 3 --
1.52 2.46 3.07 5.16 5.45 9.69 15.4
a Properties of gases are given at STP (i.e., at 21.1 "C and 1 atm), as listed o n Brooks' calibration sheets. @ / P I ' * is in units of ( g cm)l'z/s.
Table 11. Nitrogen Flow Past Float at H = 60 mm ~
~~
tube
float material
Q, cm3/
Q/S,
min
cmls
Ref
45.7 140 320
3.2 19.7 1430
__.___
A C F
glass glass stainless steel
14.5 91.8 15 300
Figures 2A, B, C, and D, where Qp is plotted vs. Qp1i2over the series of tubes. As the next step in investigation of gas flow through rotameters, tubes A, C, and F were cut into sections at various heights. Their cross-sectional dimensions were measured by microscope and shadowgraph. The floats too were measured, and the glass and stainless steel floats were found to have exactly the same diameter. For tubes A and C, Df = 0.1248 in.; for tube F, Df = 0.2498 in. For each of the above tubes, a Reynolds number Ref was computed for nitrogen flow at H = 60 mm, the height mark being taken coincident with the horizontal diameter at the float. For R e f , a quotient 4mQp/Sp was used at height H, where the cross-section area is S for flow between wall and float, and where m is the hydraulic radius defined as S divided by the wetted perimeter. Table I1 lists flow rates and Reynolds numbers for nitrogen at STP. The R e f for tube F (stainless steel float) in Table 11, together with the trend of Qp vs. Qp1i2for tube F (stainless steel float) in Figure 2D, suggest that Qp'12 becomes constant as the flow past the float becomes fully turbulent (perhaps at Ref = 2000). Constancy of Qp112then belongs to the regime of turbulent flow at the float, i.e., at the maximum flow constriction. (In that situation, viscosity has no influence on float height, just as in the hypothetical case of a nonviscous fluid, or at p/p112 = 0.) According to Newton's law for bodies in turbulent motion, the resistant force of a (float) body to motion is proportional to the quantity 0 , 2 V Z p . Hence, in turbulent flow past the float, the resistance force is proportional to the square of Qp1i2, which is constant at fixed H. At the lowest volumetric flow rates, as seen for tube A (glass float), the trend is clearly to a zero intercept ( B = 0) in Figure 1A and to a constant value of Qp in Figure 2A. Table I1 indicates that the threshold for constant Q p is at low R e f , perhaps as low as R e f = 1, which is surprisingly deep within the region of laminar flow. The apparent discrepancy, however, may belong to the less than satisfactory application of hydraulic radius to other-than-cylindrical geometries in laminar flow, in contrast to turbulent flow (as pointed out by Vennard, 1940). Thus, in the rotameter series A, ..., F, apparently there is a wide transition region, from Ref's = 1 to 2000, where
both p and p are important to flow meter calibration. Below Ref = 1, again there is a consistency with basic fluid dynamics, since the resistance force of a (float) body to motion is now proportional to DfVp, or to constant Qp at a given height in a rotameter. This resembles Stokes' law for a particle in a fluid in laminar flow. We refer to Dallavalle (1943) for a fundamental discussion of the forces on particles in laminar and turbulent flow. The situation of constancy of Qp at low Q suggests the design and construction of simple, convenient, and inexpensive gas rotameters as comparative gas viscometers at low Q. We propose to use eq 2 for indirect calibration of a working gas and a small rotameter with a spherical float. The procedure calls for separate Q vs. H calibrations of flow tube and float to be made with two gases other than the working gas. For example, the two gases may be nitrogen and helium, or perhaps a better choice may be argon and dichlorodifluoromethane (Freon 12), since the p / p ' i z values of most other gases lie between the values for the latter two gases. As in Figure 1, (Qp'l2)-' is plotted vs. p / p ' i 2 for each of the two gases at selected heights. A straight line is then drawn through the two points at each H. From the plot, H vs. Q values for the working gas or for any other gas of known p / p 1 I 2 can be picked (i.e., predicted). With careful calibration, accuracies greater than *2.0% appear to be obtainable for the tube, float, and working gas. The above indirect calibration would be made in circumstances where direct calibration with the working gas would be difficult, inconvenient, or costly because of reactivity, or hazard, or the influence of a second phase, or a combination of the above.
Nomenclature D - rotameter spherical float diameter, cm height of rise of float, mm Q = volumetric flow rate, cm3 min-' or L min-' Ref = Reynolds number at H, equal to 4mQplSp S = cross-sectional area for flow between rotameter wall and float at H, cm2 V = fluid (gas) velocity, cm s-' m = hydraulic radius, defined as S divided by the wetted perimeter at the float at H, cm p = fluid (gas) viscosity, CP p = fluid (gas) density, g cm-3 In application to tubes A, ...,F, values of Q, p, and p are taken at STP (21.1 OC and 1 atm) in this paper.
Hfz
Acknowledgment The study described in this paper was performed at the Jet Propulsion Laboratory, California Institute of Technology, and was sponsored by the Department of Energy through an agreement with the National Aeronautics and Space Administration.
Literature Cited "Callbration Data for 150 MM Variable Area Flowmeter Tubes": Brooks Instrument Division, Emerson Electric Co: New York, 1980. Dallavalle, J. M. "Mlcromerttlcs"; Pkman: New York, 1943; Chapter 2. Vennard, J. K. "Elementary Fluid Mechanics"; Wiley: New York, 1940; Chapter 6, p 164.
Received for review October 13,1981 Revised manuscript received November 29, 1982 Accepted January 19,1983
