Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
K = a constant in eq 12 U = superficial fluidizing gas velocity U B = absolute bubble rise velocity UB- = rising velocity of an isolated bubble U,, = superficial critical fluidizing velocity UGB = superficial bubble gas velocity U I = superficial interstitial gas velocity Umf = superficial minimum fluidization velocity Greek Letters a = ratio of bubble velocity to interstitial gas velocity a t minimum fluidization CB = fraction of bed occupied by bubbles emf = voidage a t incipient fluidization
Literature Cited Bukur, D. B., Amundson, N. R., "Some Model Studies of Fluidized Bed Reactws", p IO, Chemical Engineering Department, University of Minnesota, 1974. Bukur. D., Caram, H. S.,Amundson, N. R., "Chemical Reactor Theory-A Review", L. Lapidus and N. R. Amundson, Ed., p 692, Prentice-Hall, Englewood Cliffs, N.J., 1977.
123
Chavarie, C., Grace, J. R., lnd. Eng. Chem. Fundam., 14, 79 (1975a). Chavarie, C., Grace, J. R., lnd. Eng. Chem. Fundam., 14, 86 (1975b). Davidson, J. F., Harrison, D., "Fluidized Particles", p 100, Cambridge University Press, 1963. Fryer, C., Potter, 0. E., lnd. Eng. Chem. Fundam., 11,338 (1972). Grace, J. R., A.l.Ch.E. Symp. Ser., 67, No. 116. 159 (1971). Kunii, D., Levenspiel, O., lnd. Eng. Chem. Fundam., 7, 446 (1968a). Kunii. D.. Levenspiel, O., lnd. Eng. Chem. Fundam., 7, 481 (1968b). Latham, R., Potter, 0. E., Chem. Eng. J., 1, 152 (1970). Murray, J. D., J. Fluid Mech., 21, 465 (1965). Partridge, 6. A., Rowe, P. N.. Trans. lnst. Chem. Eng., 44, T349, (1966a). Partridge, B. A., Rowe, P. N., Trans. lnst. Chem. Eng., 44, T335 (1966b). Potter, 0. E., "Fluidization", J. F. Davidsonand D. Harrison, Ed., p 332, Academic Press, London, 1971. Rowe, P. N., Partridge, B. A,, Trans. lnst. Chem. Eng., 43, 157 (1965). Stephens, G. K., Sinclair, R. J., Potter, 0. E., Powder Techno/., 1, 157 (1967). Toomey, R. D.,Johnstone, H. P., Chem. Eng. Prog., 48 (9, 220 (1952). Van Deemter, J. J., "Proceedings of the International Symposium on Fluidization", A. A. H. Drinkenburg, Ed., p 334, Netherlands University Press, Amsterdam, 1967.
Received for reoiew M a y 2, 1977 Accepted January 26, 1978
EXPERIMENTAL TECHNIQUES
A High-Frequency Rheometer Syamal K. Poddar, Paul Klelnsmlth, and Willlam C. Forsman* Department of Chemical and Biochemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19 174
A rheometer has been developed to study the high-frequency, linear viscoelastic behavior of liquids (solvents or
solutions) with viscosities varying over a wide range. This instrument could be of particular interest in the study of the viscoelastic effects associated with the presence of short-range interactions such as intramolecular hydrogen bonding or chain solvation in polymer solutions. The apparatus consists of a glass rod bonded to a quartz crystal of the same diameter which is driven in torsional oscillation by a pulse modulated sinusoidal emf. The wave packets are reflected by the free end of the rod, return up the rod, and are sensed by the quartz crystal. The primary data are the differences in phase shift and attenuation of the reflected waves in the rod assembly when the rod is partially immersed in a viscoelastic solution and when in air. The paper deals with the development of the equipment, its calibration (which also tests the basic assumptions underlining the operation of the equipment and its mathematical description), and testing the equipment with distilled water, glycerol, and a glycerol-water mixture. Equations have been developed for determining the complex dynamic modulus G* of liquids.
Introduction The instrument described here is a modification of a design introduced by McSkimin (1952) that utilizes a quartz crystal vibrating in a torsional mode, which in turn excites traveling torsional wave in a glass rod that serves as the viscometer probe. Recent modifications of this apparatus have been reported by Barlow and Erginsav (1972), Barlow and Lamb (1959), and Barlow et al. (1964, 1969), who focused on improving the stability and accuracy of the equipment and extending the frequency to the megahertz range. A similar device was described by Glover et al. (1968). An alternative design described by Mason (1947) which uses vibrations excited in a totally immersed crystal is potentially operated in the same frequency range and more accurate for determining the dynamic viscosity of relatively nonviscous fluids. Unfortunately, this instrument uses water'soluble crystals and is thus unsuitable for measurements on aqueous solutions. 0019-7874/78/1017-0123$01.00/0
Analysis of the operational characteristics of a rheometer employing traveling torsional waves was given in McSkimin's (1952) paper and discussed by Barlow (1959) and Mewis (1967). The analysis given here adopts a different point of view and offers a detailed method for processing of experimental data. In addition, it was our intent to explore the suitability of this instrument for measuring high-frequency viscoelastic behavior of solutions (or simple liquids) having dynamic viscosities in the centipoise range. Finally, we should like to add that the calibration technique we describe for this equipment is new and offers proof of the adequacy of the basic physical assumptions and the mathematical description (McSkimin, 1952; Barlow, 1959; Mewis, 1967).
Description A component diagram is given in Figure 1, and a photograph of the complete experimental setup with all the electrical
0 1978 American Chemical Society
124
Ind. Eng. Chem. Fundam.. VoI. 17, No. 2, 1978 BRIDGE
AI
1
M OSCILLOSCOPE
OEN.
Figure I. Component block diagram
Figure 2. Photograph of the complete apparatus: (1) and (2) constant voltage power supply units; (3) electronic counter for frequency measurement; (4) tuned amplifier; ( 5 ) phase generator; (6) frequency generator; (7) pulse generator; (8) oscilloscope; (9) attenuator; (10) solution chamber; (11)bridge; (12) gate; (14) buffer amplifier; (13) and (16) temperature-controlled bath far heating and cooling, respectively; (15) cathetometer; (17) thermocouple connected with millivolt meter.
connections and thermostating lines is shown in Figure 2. Detail of the solution chamber holding the glass rod-quartz crystal assembly and the thermostating arrangements is depicted in Figure 3. A General Radio 1309-A oscillator supplies a continuous sine wave signal to the Dytronics 410 phase generator, which in turn supplies two continuous sine wave signals of constant amplitude ratio, hut variable phase, to the two balance arms indicated in the Figure 1.One of these signals is pnlse-modulated by the gate circuit using a General Radio 1340 pulse generator. This pulse-modulated signal is transformer-coupled to the bridge circuit, supplying a balanced, pulsed sinusoidal emf across the terminals of a torsional piezoelectric crystal attached to a Pyrex rod of equal diameter. The tor sional wave packet generated by this emf then travels down the length of the rod and is reflected hack to the piezoelectric crystal, creating an unbalanced counter emf which is detected by the buffer amplifier. Each packet oscillates at about 80 kHZ for a duration of s. The interval between the packets is 10-2 s. Each packet thus is reflected up and down the rod enough times to he totally dissipated before the rod is excited by the succeeding packet. The signal generated hy the piezoelectric crystal in response to the reflected waves is then balanced with the second
phase-shifted conrinuuiis wave signal from thr phair generiittrr Iiy menns dil Varconi TF 2lfiOattenuator. Thenrtenuatim and phaie shift values required to provide this halanre, Iimh when the wd iSosrillating i n air and when it is partially immrrstd in it,lutiim or solvent, provide the data from which G' and G" for the fluid can he determined. The present apparnrus ditters from rhar of McSkimin primarily i n simplification nf adjurrmrnt filni.tiuns and in modificationi rivluirrd for wmpatihilitv with modern sulidstate auxiliary equipment. Gate circuit controls have been aimplifird t o improve balancinq, and an ampliiier has hern nildid for use with the General liadi,, generator. MrSkimin's wrinhle-gain phase shifter hns h r m replawd by a constnntgain phase genrrator, eliminating interaction with rhr artrniiatur in setting the final balance of the retlrrted e m f h m the rod. The present phase and attenuator devices pro\ ide a full t hrce decades of rontrol. imDrovine. ovrrnll simriliritv and arriir;icy of the method. Assumptionsand Primary Data We will imagine a torsional WRVP packet mwing down the elnstir rod,firsr throiigha lPngrh~,fI).uen,dundthcnintothat part of the rod immersed in a visvoclastic fluid. Ignoring The viicous drag of air, rhe attenuation of thv wave packet in the hare portim t f thi. r i d is due only to rhe internal friction of thi. rod. After the packet enters that parr of the rod immersed in rhi: viscoelaitic fluid, attenuation is dur hnth to internal friction i n the rod and the viscous drag at its surtare; in addition, rhrre IS n iihift in phase due to the visroelnstir resptinw of the fluid. The experimcnt mosr easily visualized is one in which a sen~iinfiniwrod sustlins a wntinuous wave train. Our intuition tells us thnr amplitudeand phase measurementsol the wave propapatiun phenomenon in the hare rod and wirh the roil partially immersed in rhe fluid wuuld hesufficient t o determine G'anil (;"of the fluid, when simple physiralnnil geometrir ronstants are known. In practice, of course, ir is impossihle r o do t h r iilii)ve experimenr, and if is even imprnctir;il to suhstitute a rud lung rnough to attenuate totally a rontinuoiis wavr train (which wnuld he amenahle M the same nnalysisj. This meani that a rod of finite lrngth must he used. along wirh pulsed trains, or
Ind. Eng.
Chem. Fundarn., Vol. 17, No. 2, 1978 125
PLANE OF REFLECTION
OUARTZ CRYSTAL
h
-
OUARTZ CRYSTAL (REFLECTED I
IMMERSION D E P T H
1. 2 h
Flrsl
ief:ecr:on
iecoid
?efle:tior
Third Reflection
Fourth Reflection
Fifth. Reilecfion
Sixth Reflection
. E F F E C T I V E P A T H L E N G T H OF TORSIONAL WAVE PASSING THROUGH F L U I D
Figure 4. Equivalent system for perfect reflection condition. wave packets. The first assumption, therefore, is that the pulse, or wave packet, contains a sufficient number of sinusoids to be adequately described by the mathematics of continuous wave trains (the analysis of which is given by McSkimin). If a finite rod is used, employing a single quartz crystal as driver and detector as described in the previous section, the mathematics is tractable and the results of the experiment interpretable only if there is perfect reflection of the wave packet from the free end of the rod. Since the torsional wave velocity is virtually independent of frequency, there is thus a discrete set of experimental frequencies that give perfect reflection. Although perfect reflection has been assumed (McSkimin, 1952; Barlow and Lamb, 1959; Barlow et al., 1964,1969; Barlow and Erginsav, 1972), there have been no reports of how well this assumption applies, or of the effect of deviations from perfect reflection. Even if the pulsed train is adequately described by a continuous sinusoidal input, and perfect reflection is achieved, one other assumption bears examination. McSkimin assumed that the behavior of the wave phenomenon could be adequately described by first-mode solutions to Bessel's equation. In this paper we propose testing these assumptions experimentally. Given the condition of perfect reflection, the equivalent physical system is shown in Figure 4.Each succeeding reflected wave packet sensed by the quartz crystal (acting as pick-up transducer) can thus be interpreted as a succession of torsional wave packets passing from point A to point B in Figure 4.I t is clear, therefore, that the phase shift between the first and second reflections should be the same as between the second and third reflections, etc. Also the attenuation per reflection should be the same. The experimental data are taken as follows. With signal two from the phase generator disconnected, the buffer ampiifier delivers a signal to the oscilloscope, which displays the input and reflected waves as depicted in Figure 5 . The oscilloscope can, however, also be adjusted to display any one of the reflections. Signal two from the phase generator is then passed through the attenuator and added (in the buffer amplifier) to the signal from the crystal. The phase and amplitude of this signal is then adjusted to nullify exactly the first reflected signal, giving the first reflection's attenuation (in decibels, defined in this paper as (20)-1 log [initial amplitude/final amplitude]) and phase angle relative to signal one from the phase generator. These quantities are designated Do(1)and ao(l)for bare rod readings and D ( l )and A(l) for readings with fluid in the instrument. The phase and attenuation of signal two is then readjusted to nullify the second reflected wave (or D @ )and A@)if fluid is present), the giving Do(2)and attenuation and phase angle of the second reflection relative to signal one. This process can be repeated for as many reflections as needed, generating a series of quantities Docn), Ao(n), and A(.). The analysis given in the next section shows that the fol-
Figure 5. Nature of scope trace showing various reflected waves without signal two.
lowing data are required in the computation of G' and G" of the fluid. For n > m Do(n)- Do(m) Do = n-m which is the attenuation per reflection of the bare rod using reflections n and m D ( n )- D(m) D= n-m which is the attenuation per reflection for the loaded rod using reflections n and m A,(.) - Ao(m) lo= n-m which is the phase shift per reflection for the bare rod using reflections n and m; identically equal to zero for perfect reflection since Ao(n) = a ~ for ( ~all)n and m , and ~ ( n-)
A=
A(m)
n-m which is the phase shift per reflection for the loaded rod using reflections n and m. Other physical quantities required in the analysis are a = radius of the rod; 1 = 2h; h = depth of immersion of the rod; L = total length of the rod; p , = density of the rod; p = density of the fluid; G,* = complex shear modulus of rod; G* = complex shear modulus for liquid; u = shear wave velocity of glass rod; f = frequency; f r = frequency for a perfect reflection; and, w = 2Tf. Mathematical Development (a) Modeling. The mathematical model of the traveling wave system described above was developed by McSkimin (1952). Using that development as a starting point, we developed a computer algorithm for data reduction. The differential equations describing the torsional wave displacement u, in the elastic rod and the accompanying shear wave displacement u in the viscoelastic loading medium (which is the fluid sample) can be written as (McSkimin, 1952)
and
126
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
along with the boundary conditions au au, =-
at
at
-
(r = a)
(E)+ Po2
P2 = k 2
(4)
We can relate the parameters (3 and PO to experiments through the following considerations. The bare-rod damping PO’ can be obtained from wave attenuation by the equation
(5)
00’= (2.303/4OL)Do
(13)
ETamination of eq 7b shows that the bare-rod wave velocity v is related to PO’’ by
and
u
+
0 as r
+
0)
(6)
= w/u
PO’’
Solutions satisfying these equations are u, = J l ( k r ) exp(iwt
- Pz)
u = H l @ ) ( k f r )exp(iwt - P z )
(74
GI* = -w2p,/(Po’
(8)
where J1 and H I @ )are first-order Bessel and Hankel functions of the second kind. The quantities w and are the angular frequency and ’ complex damping coefficient of both the traveling wave in the rod and the viscoelastic shear wave in the loading medium (fluid sample), k and kf are the wave numbers for those waves in the rod and viscoelastic medium. The complex quantity P = p’ t ipll includes a real part p’ which is the physical damping constant and an imaginary part pll which determines the phase of the torsional wave. . Substitution of eq 7 and 8 into eq 1 and 2 yields the dispersion relations for the rod and liquid as given below
(14)
In addtion, PO can be given further significance through applying eq 9b.
= w2pr/(p0”2
+ iPo”)2
- 00’2 + 2ipOpo”)
(15) (16)
Since PO’’ is three orders of magnitude greater than Po’ for the glass rod used in this instrument, we can write GI*
w2pr/Po”2 N
(17)
pv2
with negligible error. Attenuation of the rod immersed in the liquid is due to both loaded and unloaded surfaces or
+
D = [20(2L - 1)/2.303]/30’ (201/2.303)p’
(18)
P’ = (2.303/201)0 - [(2L - I)/1]P0’
(19)
or
The shear impedance per unit surface area for the fluid and that for the rod are equal at the rod-fluid interface. Thus
The experimental phase shift per reflection for the loaded rod relative to the bare rod occurs over the length of the rod immersed in the fluid. Consequently pll = Po”
+ (~/1801)A
(20)
In summary, experimentation gives all the terms in eq 9c, lob, and 12 except k, k , , and G * . These three equations can thus be solved giving G * , the dynamic modulus of the fluid. Taking the required derivatives of u and u, expressed by eq 7a and 8 and putting those in eq 11,we get McSkimin’s impedance matching relationship as shown below
For the bare rod, the shear stress at the surface must be zero. This gives k , = 0 which makes eq 7a invalid for this special case. The correct first mode solution to Bessel’s differential equation for k = 0 is then
u, = Cr exp(iwt - 6 0 2 )
(7b)
where C is a constant. The corresponding dispersion relationship is
Although the rod is considered to be very nearly purely elastic, it nevertheless demonstrates a small amount of internal friction. I t is therefore necessary to write G,* = G,’ iG,” with the imaginary part, G,”, determining the damping. Consequently, we must take PO as complex with a real part PO’ which is the experimental damping of the bare rod and an imaginary part Po’’ that determines its phase angle relative to some arbitrary reference point. It is thus convenient to write eq 9a and 10a as
+
- 002 t P2 = k I 2
(9c)
Computation For a glass rod excited a t the frequency used with this instrument 1 ka I
