ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Heat Transfer fro Maximum Flow Rates in Thermal Conductivity Measurements A.
C. PETERSON,
A. J. MADDEN, JR.,
AND
EDGAR L. P R E T
University o f Minnesofo, Minneapolis, M i n n
0
iYE of the most convenient methods for the continuous analysis of simple gas mixtures is based on the measurement
of the thermal conductivity as an index of the gas composition. One method of carrying out such an analysis involves bleeding off a fraction of the gas stream and pasEing it through a thermal conductivity cell. The latter consists essentially of a heated wire located a t the axis of a thermostated tube. While other arbitrary conditions might be used for specific problems, the heat loss from the wire preferably should depend solely upon the thermal conductivity of the gas mixt,ure and should be independent of the flow rate of the gas. Investigatore deeiring high precision in thermal conductivity measurements have been faced with t,he problem of determining the allowable flow rabes through the conductivity cells that do not affect the measured conducti.v.itg. Wynkoop and Wilhelm ( I O j , for example, found that, for the analysis of hydrogen-ethylene and hydrogen-ethylene-ethane mixtures in their studies on t.he hydrogenation of ethylene, t,he conductivity cells were free of effects due to total gas flow only a t rates less than 40 cc. per second. I n general, a need exists for predicting the maximum allowable gas flow rates as a function of gas properties and cell dimensions. Basically, the problem is one of determining the effect of comparat,ively low gas flow rates on the heat transfer from wires, located a t the axis of vertical tubes, to gases in parallel flow. Other investigators, such as RIueller ( 4 ) , studied the heat t.ransfer rates from wires t,o gases in parallel flow but only under conditions where the velocities were high and outside the range of interest for thermal conductivity cells. The results reported in this paper are based on a study with two sizes of wires and two tube diamet,ers. Wire temperatures were varied from 35" to 120" C. Air a t the ambient temperature a,nd pressure was used exclusively, but, the generalized results have been confirmed in this laboratory by Stover (8))n-ho also used argon and helium. The present investigation is a continuation of work on heat transfer studies with wires. Related work on heat t,ransfer from wires t o water a t IOTYvelocities and to various gases a t subatmospheric pressures has been reported (1-8, 6). Nusselt Number Is Part of Group Used for Analysis of Experimental Results
The steady-state heat loss by conduction from a heated wire through a gas to a concentric tube, maintained a t a lower temperature, may be pxpressed as Y =
2nLk(t,
in
- tf)
D _f
D,
A heat transfer coefficient for the wire may be defined as 2ic
from R-hich the Kusselt number for the wire is
2038
Dt Thus, for pure conduct'ion, Nu In -- = 2.0.
D,
As convect,ion sets in, the value for Nu In
D
becomes greater D 'il D than 2. The magnitude of the quantity, Xu In L! is thus a useful D w, criterion of the heat transfer mechanism and is used in the analysis of the experimental result,s. Measurements Are Made of Heat Transfer from Wire to Gas in Parallel Flow
The apparatus consisted essentially of a length of pure nickel m-ire located axially in a vertical glass t,ube through which the gas being invcstigat.ed was passed. The wire was hciited electrically along its entire length, but heat loss mca.surements Tere made only for the central, 6-inch test eect,ion. The wire flow tube assembly and auxiliary apparat,us are shown in Figure 1. The floir tube was thermostated in room air, which was substantially constant for a series of runs. A typical !Tire assembly is shown in Figure 2. It i3 similar to that used by Taylor and Johnston (9) in their potential lead type hot. wire cell for the precise measurement of the thermal conductivit,ies of gases. By studying the central portion of a longer heated !Tire, the effect of heat conduction from the ends of the mire is minimized. Two Lrires, 0.0028 and 0.01 inch in diameter, and tIro glass flow tubes, 0.438 and 0.871 inch in diameter, v-ere investigated in this work. Wire Assembly. The wire assembly was suspended from a male 29/12 standard taper ground-glass joint, A , by three tungsten electrodes, B , 0.03 inch in diameter, sealed in glass approximately 4 mm. apart. Each tungsten electrode had a small nickel plate, C, spot-velded to it to facilitate at,tachment of t'he wires by soft soldering. The n-ire to be heated was the central wire of the assembly. The two outer wires served as potential leads. The lower end of the central \Tire xvas soldered to a small nickel plat,e, C', spot-welded t,o the central electrode of an assembly of t,hree turigst,en electrodes, B ' , imbedded in a glass bead, E, 0.25 inch in diameter. The lower end of the central tungsten elcctrode w-as silver-soldered to a small cylindrical brass vieight,, P. The out,er tungsten wires were bent to form hooks. Att'ached to these Tvere small springs D, coiled from 0.005-inch piano wire, Tvhich were soldered a t their other end to outer potential lead v-ires G of the main three-Tvire assembly. A wire, J , from the bot'tom of the brass weight made cont'act with a cup of mercury, K . The potential lead wires, G, 0.0028 inch in diameter, made contact with the central heated wire through small nickel jumper wires H , 0.0018 inch in diameter. These !%-ereattached to t,he central wire and the appropriate lead wire by a minute quantity of soft solder. The soldering operation was carried out under a microscope, using a very small Boldering iron t,o minimize the size of the result'ant' joint. Flow Tube. Each mire assembly \vas studied in each of two
INDUSTRIAL AND ENGINEERING CHEMISTRY
voi. 46, N ~ 10 .
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT building and dried by passing through an adsorption train O F anhydrous calcium chloride and Dehydrite. Calibrated rot,ametera were used to measure the flow rate of the air. Experimental Method. For a fixed gas f l o ~rate and current flow through the wire, a steady state is rapidly attained whereby the wire reaches a conetant temperature. This is indicat,cd experimentally by a conetant resistance of the wire test, section. L-nder steady-state conditions the heat loss from the wire i p t,hr
12R loss, which, from Ohm's law, is equal to
EWE,
----, R,
where E, is
the voltage drop across the wire test section, E , is the voltage drop acrose the standard manganin resistance in series with wire, and E , is the resistance of t'he standard resistance. The resistance of the wire test section, R,, was determined from the same measurements since R, =
Figure 1.
Apparatus
borosilicate glaes flow tubes. The essential part of each flow tube was a section of straight tubing, approximately 12 inches long. The test section of the heat,ed wire was roughly centered u-ith respect to this length. The straight portion of the flow tube terminated in conical entrance and exit sections as ehoivn in Fiyure 1. The average inside diameters of the central 6-inch sections of the two flow t'ubes were found t,o be 0.438 and 0.871 inch, respectively. The average diameter was determined in each case by weighing the amount of water that could be drained from thr section and measuring the length of the water column drained out. The latter v a s determined with a cathetometer. From these data and the kno.ivn density of water, t,he average tube diameter was calculated. The average diameter of each wire was determined by weighing several measured lengths of wire taken from the same spool and calculating the diameter from these dat'a and the densit'y of nickel. Pertinent physical data for t'he wire assemblies and the location of the wire test eections relative t o the top of the flow tube are given in Table I.
Table I. Tire
Assembly ?io,
I I1
Heated Wire Length, Inches 15.75
W t . of $i?$t,
~f-a.,
2
Grams
Inches
4l/&
30
2'12
Spacing ~(Figure , , 2 ) , Inches X
Y
0 0028
51/*
6
2'/e 78 2l/2 a Distance from throat of constriction a t top of 8ow tu35 t o upper potential lead wire of test section. 13.75
0.0100
51,'s
Both upward and downward flow were investigated inthisstudy. Thesedatawere found to be identical, and most of the TO data were obtained with downivard flow. The experimental results are presented g r a p h i c a l l y i n F i g u r e 3, p l o t t e d a s D Nu In -t versus the ReyD, DtG . Since nolds number, -. M
$:
G
AK
TO D.C. POWER
The mire was heated electrically by means of direct current obtained from two 6-volt lead storage batteries. The current flow mas controlled by means of decade resistance boxes in series with the wire. A standard manganiii resist'or, also in series with the wire, served as a reference for the heat transfer measurement.;. The potential drops across the standard manganin resktor and the wire test section were measured v i t h a Leeds RS Korthrup Type Ii2 potentiometer. I n k t and exit air temperatures nere measured with calibrated 0.1' C. thermometers, located a8 shown in Figure 1. Air was taken from the compressed air lines of the October 1954
The temperature-resist,ance
At l o w Reynolds Numbers Rate of Heat Transfer Is Independent of Gas Flow Rate
z
Dimensions of Wire Assemblies Heated Kire ~ i ~ Inches
-E. R,. E,
relationship of each wire assembly had been previously det>ermined by calibration in a thermostated bath over the range 25" to 100" C. From the potential drop measurements of each run, the steady-state heat loss from the mire and the wire temperature were thus determined. For the conditions of this work the heat loss from the wires by radiation was generally negligible. Radiat,ion losses for the two wires used in this work had been previously determined ( 2 ) . The wires were suspended within a large glass bell jar and the syst'em was evacuated to approsimately 10-6 mm. of mercury. Radiation losses were measured for different wire temperatures. At the highest wire t,empprat,ures used in t,his work, t,he percentage of the total heat loss due to radiation was of the order of 0.7% for the small wire and 2.0% for the large wire.
Figure 2. A.
SOURCE
Wire Assembly
Ground-glass joint
E, 8'. Tungsten electrodes
c, C'.
Nickel plates
D. Springs E.
Glass b e a d
F. Brass weight G. Lead wires N, H'. Jumper wires 1.
I.
K.
Heated wire Wire Mercury
INDUSTRIAL AND ENGINEERING CHEMISTRY
the wire diameter is negligible in comparison with t,he tube diameter, t'hc lat,ter is essentially equal to the equivalent diameter, usually used in characterizing flow in annular spaces. In calculating the Xu+ selt and Reynolds numbers, the physical propcrties of t,he gas were evaluated a t the arithmetic mean of the wire tr.mperat,ure aiid the average gas temperature. The latter was taken as the average of i d e t aiid out1e t g a s t e m p e r a t u r e P, taken a t the locations shown in Figure 1. For evaluating the lieat transfer coefficient, the At vias 2039
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT range of thermal conductivities and densities. The thermal conductivities of helium, air, and argon are in the approximate ratio of 10:2:1, and their densities are in the ratio 1:7 :10. Under conditions of flow where the Reynolds
6 a
4
c
3
number, 2
4
20
7 10
40
70 100
200
1000 2000
400
9lJf2. Figure 3.
Correlation of Experimental Data
Wire Diam., Inch
Tube Diam., Inch
Gas
0.0028 0.0100 0.0028 0.0100 0.0028 0.0028 0.0028
0.438 0.438 0.871 0.871 0.301 0.301 0.301
Air Air Air Air Air Helium Argon
2040
D E G h
Stover (8) Stover ( 8 ) Stover (8)
Dt
o,,
the heat transfer coefficient is based on the temperature difference between the wire and the tube wall. In view of the small heat fluxes involved and the location of the thermometers for the gas temperature measurement (Figure l), the average gas temperature as measured should be very close to the tube wall temperature, For Reynolds numbers less than 100, inlet and outlet gas temperatures did not differ by more than about 1' C. Figure 3 shows that when the Reynolds number is less than 100, heat loss from the wire occurs by conduction through the gas to the tubewall. Thisplotis based on wire temperatures ranging from 3 5 O to 120° C. Also plotted on Figure 3 are the confirmatory data of Stover (8), who used an 0.0028-inch wire in a 0.301-inchdiameter tube. D a t a were obtained for helium, air, and argon near room temperature and a t the ambient pressure. In this case the tube was thermostated with a water jacket for the low velocity data, and the heat transfer coefficients were based on the temperature difference between the wire and the therniostated tube. Workers in the thermal diffusion field have measured the incidence of turbulence in gases in free convection between concentric vertical cylinders. Onsager and Watson (6) and Simon (7) found a change from laminar to turbulent flow occurring a t approximately the same Reynolds number-Le., 100-for temperature differences in the same range as for this investigation. The combination of the present work and the confirmatory experiments of Stover embraces systems of two wire diameters, three tube sizes, and three gases. The three gases cover a wide
END
-,P
is less than 100, the data of Figure 3
indicate that heat loss from a wire to a concentric tube occurs principally by conduction from the wire through the gas to the tube. Variations in flow rates below this criterion have no measurable effect on the heat loss. Hence, this establishes a desirable range for thermal conductivity cell operation. Nomenclature
taken as the difference between the wire temperature and the average gas temperature. In the derivation of the conduction value of 2.0 for Nu In
DtG
= diameter, ft. = potential drop, v. = mass velocity of gas flow, lb./(hr.)(sq. ft.)
= heat transfer coefficient, B.t.u./(hr.)(sq. ft.)(" F.) I = current flow, amp. k = thermal conductivity of gan, B.t.u./(hr.)(ft.)(" F.) L = length, ft. In = natural logarithm = viscosity of gas, lb./(ft.)(hr.) p Nu = Nusselt number = rate of heat loss (exclusive of radiation), B.t.u./hr. ?! = resistance, ohms t = temperature, O F.
Subscripts s indicates standard resistance t indicates tube w indicates wire test section
literature Cited (1) Kyte, J. R., Madden, A. J., and Piret, E. L., Chem. Eny.
P T O ~ T49, . , 653-62 (1953). (2) Madden, A. J., and Piret, E. L., "General Discussion on Heat Transfer," p. 328, Inst. Riech. Engrs. (London) and Am. SOC. hfech. Engrs., New York, 1951. (3) Middlebrook, G. B., and Piret, E. L., IND. EXG.CHEY., 42, 1511-13 (1950). (4) Mueller, A. C., Trans. Am. Inst. Chem. Engrs., 38, 613-29 (1942). (5) Onsager, L., and Watson, W.W., Phys. Rep., 56, 474 (1939).
(6) Piret, E. L., James, W., and Stacy, M. W., IND.ENG.CHEM., 39, 1098-1103 (1947).
(7) Simon, R., Phys. Rev., 69, 596 (1946). (8) Stover, R. W., M.S. thesis, University of blinnesota, 1951. (9) Taylor, W. J., and Johnston, H. L., J . Chem. Phys., 14, 219-33 (1946).
(10) Wynkoop, R., and Wilhelm, R. H., Chem. Eng. Progr., 46, 30010 (1950). RECEIVED for review December 18, 1953.
ACCEPTED J u n e 7, 1054.
OF ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT SECTION
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46, No. 10
