1 Introduction

Glycerol is used in many applications within the scientific fields of fluid mechanics, chemistry, medicine, and biology. In daily life glycerol is commonly used in pharmaceutical and personal care products, as well as an anti-freeze and in food industry, because it is cheap and non-toxic (Ayoub and Abdullah 2012). In scientific work the exact determination of the density of aqueous glycerol solutions is important. By mixing glycerol with water, the density of the solution at room temperature can be adapted in the range from 1000 kg/m\(^{3}\) (pure water) to 1260 kg/m\(^{3}\) (pure glycerol). This possibility of adapting the liquid density is why glycerol is used in many scientific fields in particular in fluid mechanics and biotechnology. A specific example for the application of this technique in fluid mechanics is the matching of the liquid density to polymer particles, which allows them to follow the streamlines. This property can be used in measurement technologies to determine the structure of a flow field (Adrian and Westerweel 2011; Raffel et al. 2018). In biotechnology, glycerol density gradients allow to separate biological material by centrifugation (Hansen et al. 1987). Due to the various applications mentioned above, densities of aqueous glycerol solutions have been measured since more than a century. Whereas first measurements by Gerlach (1884) and Strohmer and Gerlach (1885) were accurate to four significant digits, several detailed investigations on the properties of aqueous glycerol solutions have been published later in the early 20th century by Washburn and West (1926) and Timmermans (1935), as well as by Bosart and Snoddy (1927) and Bosart and Snoddy (1928) and determined the density of glycerol and its aqueous solution with a precision up to five significant digits. A summary of tables on glycerol measurements can be found in a publication by the Glycerine Producers’ Association (1963). On the other hand, a precise formula to calculate the density based on temperature and glycerol content has not yet been presented. An attempt to summarize these measurements in a formula has been presented by Cheng (2008). However, in that work, the density of the solution is wrongly calculated using the mass fractions of the solutes instead of the volume fractions. Moreover the effect of volume contraction is not taken into account. In the work presented here, these aspects are corrected to develop a model for the the density of aqueous glycerol solutions that precisely fits with the measured data of Bosart and Snoddy (1928).

2 Density relation for aqueous glycerol solutions

Assuming an ideal solution, its density \(\rho _{\text {s}}\) is given by

$$\begin{aligned} \rho _{\text {s}}(T,\phi _i)=\sum _{i=1}^N\rho _i(T)\,\phi _i, \end{aligned}$$
(1)

where N is the number of components, T the temperature value in \(^\circ {\text {C}}\), \(\phi _i\) the volume fraction of the ith component, and \(\rho _i(T)\) its temperature-dependent density. However, Eq. (1) does not take into account the volume contraction, an effect that is typically small but occurs for most liquid mixtures and leads to a solution volume \(V_{{\text {s}}}\) that is smaller than the sum of the component volumes \(\sum _{i} V_{i}\) (Prigogine et al. 1957). To set up a model with high accuracy, this effect is also taken into account in this work. Thus, the volume contraction coefficient \(\kappa\) is introduced as the ratio:

$$\begin{aligned} \kappa =\frac{1}{V_{{\text {s}}}}\sum _{i} V_{i}. \end{aligned}$$
(2)

By analyzing measured data of aqueous glycerol solutions, it can be found that \(\kappa\) not only depends on the volume fraction of glycerol, but also on the temperature of the solution (Bosart and Snoddy 1928). With the volume contraction the density of an aqueous glycerol solution is given by

$$\begin{aligned} \rho _{\text {s}}(T,\phi _{\text {g}})=\kappa (T,\phi _{\text {g}})\,\left[ \rho _{\text {g}}(T)\phi _{\text {g}}+\rho _{\text {0}}(T)(1-\phi _{\text {g}})\right] , \end{aligned}$$
(3)

where \(\phi _{\text {0}}\) and \(\phi _{\text {g}}\) are the volume fractions of water and glycerol and \(\rho _{\text {0}}(T)\) and \(\rho _{\text {g}}(T)\) the densities, respectively. As the mass fraction \(w_i\) is more commonly used than the volume fraction \(\phi _i\) to describe the amount of the components in a solution, the conversion:

$$\begin{aligned} \phi _{\text {g}}=\left[ 1+\frac{\rho _{\text {g}}(T)}{\rho _{\text {0}}(T)}\left( \frac{1}{w_{\text {g}}}-1\right) \right] ^{-1} \end{aligned}$$
(4)

is used in the following. The resulting expression for the density of an aqueous glycerol solution is then

$$\begin{aligned} \rho _{\text {s}}(T,w_{\text {g}})=\kappa (T, w_{\text {g}})\,\left[ \rho _{\text {0}}(T)+\frac{\rho _{\text {g}}(T)-\rho _{\text {0}}(T)}{1+\frac{\rho _{\text {g}}(T)}{\rho _{\text {0}}(T)}\left( \frac{1}{w_{\text {g}}}-1\right) }\right] . \end{aligned}$$
(5)

3 Model for the density of aqueous glycerol solutions

The model presented in this work is based on experimental data which is shown in Table 1 of the supplementary material where the density of aqueous glycerol solutions \(\rho _{\text {s}}\) was measured by Bosart and Snoddy in a temperature range between 15 and 30 \(^{\circ }\)C for glycerol weight fractions between 0 and 100% with an uncertainty \(\Delta \rho _{\text {s}}\approx \,0.1\,{\text {kg}}/{\text {m}}^{3}\) (Bosart and Snoddy 1928). Based on this data, a model will be developed in this work to describe the density of aqueous glycerol solutions. The model uses the density of pure water \(\rho _{\text {0}}\) (Linstrom and Mallard 2005; Wagner and Pruß 2002) and pure glycerol \(\rho _{\text {g}}\) (Bosart and Snoddy 1928) to calculate \(\rho _{\text {s}}\). The uncertainties of the measured densities of the pure components are \(\Delta \rho _{\text {0}}\approx \,0.01\,{\text {kg}}/{\text {m}}^{3}\) and \(\Delta \rho _{\text {g}}\approx \,0.1\,{\text {kg}}/{\text {m}}^{3}\). With Eq. (2) and the given uncertainties, the uncertainty of the volume contraction coefficient is \(\Delta \kappa \approx 0.0002\) according to linear error calculation.

The densities of the pure liquids can be approximated with the empirical expressions:

$$\rho _{\text {0}}(T) = 1000\left( {1 - \left| {\frac{{T - 3.98}}{{615}}} \right|^{{1.71}} } \right) \frac{{{\text{kg}}}}{{{\text{m}}^{{\text{3}}} }}$$
(6)

for water and

$$\rho _{\text {g}}(T) =\left(1273-0.612\,T\right) \,\frac{\text{kg}}{{\text{m}^{3}}}$$
(7)
Fig. 1
figure 1

Coefficient \(\kappa\) of aqueous glycerol solutions as a function of glycerol mass fraction \(w_{\text {g}}\) and Temperature T. Data points where calculated from the data of Bosart and Snoddy (1928) (dotted lines). Model for contraction is given by Eq. (8) (solid lines). The maximum deviation of the model from the measured data is approximately 0.0004. A maximum of the volume contraction can be observed for glycerol mass fractions of about \(60\%\). With increasing temperature, \(\kappa\) is decreasing according to Eq. (9)

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for glycerol (adapted from Cheng (2008) to fit the data of Linstrom and Mallard (2005) and Bosart and Snoddy (1928), respectively). Equation 6 describes the temperature dependent density of water where the maximum deviation from the measured data is less than 0.3 kg/m\(^{3}\) (0.03%) in the temperature range between 0 and \(100\,^\circ {\text {C}}\). For glycerol, the temperature dependent density is given by Eq. (7) and the maximum deviation from the measured data is less than 0.4 kg/m\(^{3}\) (0.03%) in the temperature range between 15 and \(30\,^\circ {\text {C}}\). For pure liquids \(\kappa\) has to be exactly one. To guarantee this it can be approximated by the function

$$\begin{aligned} \kappa (T, w_{\text {g}})=1+A\,\sin (w_{\text {g}}^{1.31}\pi )^{0.81}, \end{aligned}$$
(8)

where A is the temperature dependent coefficient

$$\begin{aligned} A=1.78\times 10^{-6}\,T^2-1.82\times 10^{-4}\,T+1.41\times 10^{-2}. \end{aligned}$$
(9)
Fig. 2
figure 2

Comparison between density measurements and analytical density models of aqueous glycerol solutions with various glycerol contents by weight (w/w). Solid lines represent the model given in Eq. (5), dashed lines the model of Cheng (2008), dotted lines the model of Cristancho et al. (2011), and the cross-markers the measured data by Bosart and Snoddy (1928). Comparing the models with experimental data in the temperature range between 15 and \(30\,^\circ {\text {C}}\), the models of Cheng and Cristantcho et al. show a clear deviation of up to \(2\%\). In contrast the model presented in this work that takes into account the volume contraction has a maximum deviation from the measured data of less than \(0.07\%\) and is much closer to the measurement error of \(0.01\%\)

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Equations (8) and (9) where determined based on the measurements of Bosart and Snoddy (1928). Figure 1 shows a comparison between \(\kappa\) calculated by the model given in Eq. (8) and the same coefficient determined based on the measured data as a function of glycerol content and temperature. The maximum deviation of \(\kappa\) from the measured data is 0.0004 which is only by a factor of 2 higher than the calcuated uncertainty based on the measurements.

The density of aqueous glycerol solutions is calculated by substituting Eqs. (6)–(8) into Eq. (5). When comparing the results with the measurements which are given in Bosart and Snoddy (1928) in the temperature range between 15 and \(30\,^\circ {\text {C}}\), the maximum deviation between model and measured data is less than 0.7 kg/m\(^{3}\) (0.07%). In comparison, the commonly used model of Cheng (2008), which does not take into account volume contraction shows a maximum deviation from measured data of about \(2\%\). A more recent but less known work by Cristancho et al. (2011) is using the Jouyban–Acree model (Jouyban et al. 2004) to estimate the density of aqueous glycerol solutions and achieved results with a maximum deviation from measured data of about \(0.4\%\). The model presented in this work is, therefore, strongly improving other density models of aqueous glycerol solutions and is much closer to the measurement uncertainty of \(0.01\%\).

Figure 2 shows a comparison of the model presented in this work with the models of Cheng (2008) and Cristancho et al. (2011), respectively. The measured data of Bosart and Snoddy (1928) is also shown in the figure. To enable an easy implementation of the above results in other scientific work, a matlab function with the model is provided in the supplementary material. An online calculator for the density and viscosity of aqueous glycerol solutions is provided by Chris Westbrook (http://www.met.reading.ac.uk/~sws04cdw/viscosity_calc.html). The tables in the supplementary material show a comparison between the measured data (Supplementary Table 1) and the model (Supplementary Table 2) as well as calculated densities of aqueous glycerol solutions at typical lab temperatures between 15 and \(29\,^\circ {\text {C}}\) (Supplementary Tables 3, 4, 5).