Article pubs.acs.org/IECR
Temperature Rise in Large-Amplitude Oscillatory Shear Flow from Shear Stress Measurements A. J. Giacomin,†,‡,* R. B. Bird,†,§ and H. M. Baek†,‡ †
Rheology Research Center, ‡Mechanical Engineering Department, and §Chemical and Biological Engineering Department, University of WisconsinMadison, Wisconsin 53706, United States ABSTRACT: Recently, we derived the temperature rise in a viscoelastic fluid undergoing large-amplitude oscillatory shear flow (LAOS) using the corotational Maxwell model [Giacomin et al. Phys. Fluids, 2012, 24, 103101]. The results of this derivation are used to estimate the temperature rise a priori. In the present paper, we show how to calculate the temperature rise in any liquid af ter measuring the shear stress response to LAOS. Specifically, if the measured response is represented by a Fourier series of odd harmonics, we can then combine this Fourier series with the equation of energy, written in terms of temperature, to calculate the temperature rise. This yields both a time-averaged contribution to the temperature rise, and an oscillating part. We find both contributions to be significant. We estimate the oscillating part with an analytical solution for the worst case, LAOS between adiabatic plates. We can thus use this calculation to see if the ±0.5 °C temperature requirement of the current standard for oscillatory shear flow testing [BS ISO 6721-10:1999] is satisfied. Our work proceeds without the choice of a constitutive equation, and can thus be used to estimate the temperature rise a posteriori for any LAOS measurement on any fluid.
I. INTRODUCTION For elastic liquids, perhaps the most popular test of viscoelasticity is performed in oscillatory shear flow.1−3 Here, the liquid is subjected to a homogeneous cosinusoidal shear rate (corresponding to a linear velocity profile): γ(̇ t ) = γ 0̇ cos ωt
(1)
which is defined by two independently controlled variables, the shear rate amplitude, γ̇0, and the angular frequency, ω. When viscoelastic properties are measured in this way, we worry about the temperature rise in the sample due to its own viscous heating. Figure 1 illustrates this experiment, along with its concomitant temperature rise. Specifically, we aim to satisfy the 0.5 °C temperature requirement of the current standard for oscillatory shear flow testing, that is, BS ISO 6721-10:1999 (§ 5.3 of ref 4). When the shear rate amplitude is large relative to the angular frequency:
γ 0̇ >1 ω
Figure 1. Orthomorphic sketch of oscillating velocity profile (left) and nonoscillating part of temperature rise (right) between two adiabatic walls [eq 14]. The uniform temperature profile is increasing linearly with time, without bound. Shown using Cartesian coordinates with the origin on the midplane.
(2)
we call the oscillatory shear flow large-amplitude. When the amplitude of the oscillatory shear flow is not large, the shear stress response normally alternates at a single frequency, ω. In 1935, Gemant conceived the complex viscosity, η*(ω), for interpreting this shear stress response to small amplitude oscillatory shear flow.5,6 Today, the real and imaginary parts, η′(ω) and −η″(ω) of this complex function of the frequency are used commonly to describe the linear viscoelastic behavior of liquids. In 1957, The Society of Rheology adopted the complex viscosity into its official nomenclature and symbols.7−9 For large-amplitude oscillatory shear flow (LAOS), with few exceptions,10 the alternating shear stress responds at more than one odd multiple of the test frequency. Although graphical methods have been developed to analyze these complications, © 2013 American Chemical Society
including loops of shear stress versus shear rate,11−13 the most popular way of analyzing the measured shear stress is by fitting the constitutive equation that takes the form of a Fourier series. This equation is restricted to the shear stress component of the extra stress tensor in simple shear flow and, in practice, is just Received: Revised: Accepted: Published: 2008
October 12, 2012 December 24, 2012 December 27, 2012 January 25, 2013 dx.doi.org/10.1021/ie302786a | Ind. Eng. Chem. Res. 2013, 52, 2008−2017
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used for alternating shear flows. Specifically, we use the series of odd harmonics given by: γ 0̇ τyx(θ , γ ̇ , ω) ≡ − ω 0
Recently, we derived an analytical solution for the temperature rise in a corotational Maxwell fluid undergoing LAOS.15 We find that the oscillating part of the temperature rise can be as important as the nonoscillating part. The two worked examples in ref 15 show how to use the analytical solution to predict a priori the temperature rise for a fluid of known zero shear viscosity and of known relaxation time. In this paper, we turn our attention to the corresponding a posteriori calculation. We derive an expression for the temperature rise as a function of the set of measured values [Gn′(ω, γ̇0), Gn″(ω, γ̇0)]. In other words, this paper allows practitioners to estimate, immediately after an LAOS measurement, just how hot their sample got. We arrive at this expression by first incorporating the Fourier series in eq 3 into the time-averaged equation of energy written in terms of the temperature rise. We then get the oscillating part of the temperature rise by inserting eq 3 into the equation of energy written in terms of the temperature rise, and without any time averaging. In Table I, we classify the literature on analytical solutions for viscous heating in oscillatory shear flow. This paper attacks this problem without any specific constitutive hypothesis and can thus be applied to any fluid.
∞
∑ [Gn′(ω , γ 0̇ ) sin nθ n=1 odd
+ Gn″(ω , γ 0̇ ) cos nθ ]
(3)
where the set of pairs of Fourier coefficients [Gn′(ω, γ̇0), Gn″(ω, γ̇0)] is defined with reference to eq 1 (see ref 14). Of course, eq 3 applies only after alternance is achieved, that is, a few cycles after the LAOS experiment begins (we use eq 112 of ref 15 to estimate how many cycles are needed). Philippoff used analog computers to determine the components of the harmonics of the shear stress that are in-phase, γ̇0Gn″/ω, or outof-phase, −γ̇0Gn′/ω, with the shear rate [given by eq 1] (see Fig. 4 of ref 16). Krieger simplified this determination by using the discrete Fourier transform (DFT).17 Nowadays, those measuring LAOS behavior normally sample the shear stress digitally, and then use the discrete Fourier transform to analyze the measurement, and good software is now freely available for this purpose.18 This use of the DFT [or its popular approximation, the fast Fourier transform (FFT)] provides the experimentalist with the opportunity to improve precision by removing background noise objectively.19,20 The DFT can even be deduced, with a minimum of effort, from loops of the shear stress versus the shear rate.21 This paper shows how to use the Fourier coefficients, Gn′(ω, γ̇0) and Gn″(ω, γ̇0), to calculate the temperature rise through the liquid, T(y, ωt), that has undergone LAOS. In rheometry, the temperature rise introduces measurement error. Specifically, the Fourier coefficients for the first harmonic, G1′(ω, γ̇0) and G1″(ω, γ̇0), descend with temperature, and so, if the temperature rise is excessive, the measured values of G1′(ω, γ̇0) and G1″(ω, γ̇0) will be lower than they should be. The higher harmonics in oscillatory shear flow are also temperature sensitive (see Figs. 3 and 4 of ref 22; ref 23) so the temperature rise can also corrupt the measurement of the higher harmonics. In oscillatory shear flow measurements, the temperature rise is not excessive when:
II. ENERGY EQUATION: TIME-AVERAGED DISSIPATION We begin our analysis by taking the time-average of the equation of energy written in terms of temperature (for a fluid whose enthalpy is at most a function of temperature and pressure; see eq 18 of ref 15): ρCp̂
X̅ (y , t ) ≡
2γ 0̇ ⎛⎜ a ω ⎜⎝ λ
|dη0 /dT | ⎞ ⎟⎟ k ⎠
(6)
ω 2π
∫t
t + 2π / ω
X(y , t ′) dt ′
(7)
where y is position and is defined in Table II (see also Figures 1 or 2). We next define the time-averaged dissipation: Hv ≡ −
where the Rothstein number, Ro, is defined by: Ro ≡
∮ τyxγyẋ dt
By the time-averaged quantity, X(t), we mean an average over exactly one period, 2π/ω, of the imposed rate of deformation:
(4)
Ro ≪ 1
∂T̅ ∂ 2T̅ ω =k − ∂t 2π ∂y 2
ω 2π
∮ τyxγyẋ dt = − 21π ∫0
2π
τyxγyẋ dθ
(8)
where θ ≡ ωt.
III. VISCOUS DISSIPATION TERM FOR ANY FLUID Substituting eq 3 into eq 8 gives:
(5)
following eqs 1 and 2 of ref 24 and which depends on the temperature sensitivity of η0, that is, on dη0/dT. This Ro is the ratio of the square root of the Nahme−Griffith number for oscillatory shear flow, 2aWe(|dη0/dT|/k/λ)1/2, to the Deborah number.24 Since eq 5 is used for a priori estimation of the temperature rise of an expected value for the fluid relaxation time, λ, along with η0(T). The method outlined in this paper does not require estimates of the fluid relaxation time, λ, or η0(T), or any other supposition about the fluid undergoing LAOS. We thus call our method an a posteriori calculation of the temperature rise in a fluid in LAOS. In rheometry, we can tolerate only a small temperature rise in the sample. We thus limit our analysis to small dimensionless temperature rises (Θ ≪ 1) by neglecting the temperature dependences of the rheological properties.
Hv = −
(γ 0̇ )2 2πω
∫0
2π
∞
[− ∑ {Gn′(ω , γ 0̇ ) sin(nθ ) n=1 odd
+ Gn″(ω , γ 0̇ ) cos(nθ )}] cos θ dθ
(9)
Evaluating the integral in eq 9 yields just one nonzero term, thus: Hv =
(γ 0̇ )2 G1″(ω , γ 0̇ ) 2ω
(10)
If we substitute eq 10 into the definition from Table III for the corresponding dimensionless Fourier modulus, 1″, we get the identity: 1″ = 2 2009
(11) dx.doi.org/10.1021/ie302786a | Ind. Eng. Chem. Res. 2013, 52, 2008−2017
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Table I. Literature on Analytical Solutions for Viscous Heating in Oscillatory Shear Flowa constitutive excluded flow thermal rheological runaway model forces field BCs properties time T0a
II
f(T)
Schapery (1964)
LVE
Bird (1965); Ding et al. (1999); Bird et al. (2007)
N
Schapery and Cantey (1966) Pipkin (1972, 1986)
LVE LVE
II
Pipkin (1972, 1986)
LVE
II
Tanner (1985, 1988, 2000)
LVE
II
aa
f(T)
Giacomin (1987)
LVE
II
T0a
f(T)
Ding et al. (1999) Giacomin et al. (2012) Giacomin et al. (2012) Giacomin et al. (2012) Giacomin et al. (2012) Giacomin et al. (2012) Giacomin et al. (2012) this paper
LVE CM CM CM N LVE CM any fluid
het
het II II II II II II II
∞
T0T0
T0a
II
