Droplet Evaporation-Based Approach for Microliter Fuel Property Measurements

Abstract

Small-volume, high-throughput screening techniques are sought to enable downselection from a large candidate pool of bio-blendstocks to a select few, having physical properties consistent with requirements of downsized, turbo-boosted internal combustion engines. This work presents a droplet evaporation-based approach to predict heat of vaporization, vapor pressure, diffusion coefficient, and Lennard–Jones parameters for an unknown fuel. Two different schemes, considering the isothermal evaporation of a moving droplet in ambient air, are proposed, which combine droplet velocity and temperature measurements, with some known properties to predict unknown properties. The schemes utilize an inverse solution of a transient model of droplet evaporation solved in an iterative fashion. A baseline scheme, which only requires droplet size change measurements, is evaluated using test data for three liquid fuels, comprising of alkanes and alcohols, as obtained in a temperature-controlled chamber. Results yield temperature-dependent heat of vaporization and vapor pressure predictions within 10 % and 22 %, respectively, of reference values. The advanced scheme, which additionally requires droplet temperature measurement, is numerically evaluated in the current work and will be experimentally validated in future efforts. The advanced scheme is found to significantly improve prediction quality, with deviations less than 2 % and 1 % for heat of vaporization and vapor pressure, while also predicting diffusion coefficient and Lennard–Jones parameters within 5 % and 8 %, respectively. The combined set of approaches, which primarily track droplet evaporation, can be incorporated into a small-volume, high-throughput fuel screening process.

Appendices

Appendix A: Property Definition

$$\begin{aligned}&Sh^*=2+\dfrac{(Sh_0-2)}{F_M}; \quad Nu^*=2+\dfrac{(Nu_0-2)}{F_T}, \end{aligned}$$

(A1)

$$\begin{aligned}&Sh_0=1+(1+Re Sc)^{1/3} \Phi \left( Re \right) ; \quad Nu_0=1+(1+Re Pr)^{1/3} \Phi \left( Re \right), \end{aligned}$$

(A2)

$$\begin{aligned}&F_{(T,M)}=(1+B_{(T,M)})^{0.7} \dfrac{\text {ln} \left( 1+B_{(T,M)} \right) }{B_{(T,M)}}, \end{aligned}$$

(A3)

where $\Phi \left( Re \right) =1$ for $Re\le 1$ and $\Phi \left( Re \right) =Re^{0.077}$ when $1<Re\le 400$

The fuel vapor mass fraction of the film and its temperature are evaluated based on ’1/3 rule’ recommended in the work of Yuen and Chen  [44] given by,

$$\begin{aligned} \bar{T}_f=T_i+\dfrac{(T_{\infty }-T_i)}{3}; \quad \bar{Y}_F=Y_i+\dfrac{(Y_{\infty }-Y_i)}{3}. \end{aligned}$$

(A4)

For the simulations, which provide input to Scheme-II, the empirical diffusion coefficient Eqn.A5 defined in the work of Fuller et. al  [40] was employed.

$$\begin{aligned} \bar{\chi }_{fc}=\dfrac{0.00143 \bar{T}^{1.75}_f\times 10^{-4}}{P_{\infty }M^{0.5}_{fc}\left( \zeta ^{1/3}_{c}+\zeta ^{1/3}_{F} \right) ^2}, \end{aligned}$$

(A5)

where $\zeta$ is the diffusion volume, evaluated based on sum of element wise contributions. For the present study, the diffusion volume for air $\zeta _c=19.7$ and diffusion volume of $\textit{n}$-heptane vapor, $\zeta _F=148.2$.

Similarly, the temperature-dependent HOV of $\textit{n}$-heptane is defined based on the handbook of Yaws  [38], given by

$$\begin{aligned} H_{F}=(A_v(1-T_i/T_c)^{n_v})\times 10^6/MW_F; \end{aligned}$$

(A6)

where for $\textit{n}$-heptane, $A_v=49.73 , T_c=540.26,~ \& ~n_v=0.360$. The temperature-dependent vapor pressure is evaluated by Antoine equation given by,

$$\begin{aligned} \log _{10} \left( P_{vap} \right) =A-\dfrac{B}{(C+T_i-273)}, \end{aligned}$$

(A7)

where for $\textit{n}$-heptane, $A=4.0404, B=1263.909, C=216.432$.

The vapor phase thermal conductivity was defined based on the work of Chung et. al  [3], expressed as,

$$\begin{aligned}&k_F= \dfrac{\mu _F 3.75 \psi R_u}{MW_F}, \end{aligned}$$

(A8)

$$\begin{aligned}&\psi =1+\alpha \left[ \dfrac{\left( 0.215+0.28288 \alpha -1.061 \beta +0.2665 Z \right) }{\left( 0.6366+\beta Z+1.061 \alpha \beta \right) }\right], \end{aligned}$$

(A9)

where $\alpha =\dfrac{C_v}{R}-1.5$, $\beta =0.7862-0.7109 \omega + 1.3618 \omega ^2$ and $Z=2.0+10.5 \dfrac{T_f}{T_c}$. $T_c$ represents the critical temperature of the fuel and $\omega$ refers to the accentric factor of the fuel. For fuel vapor specific heat and viscosity, fourth order and third order polynomials of temperature are, respectively, employed, with the material specific polynomial coefficients listed in the work of Poling et. al  [3] and Yaws et. al [38]

Appendix B: Algorithm for Scheme-II

The pertinent steps of the Scheme-II algorithm are listed below:

• 1. Using the droplet temperature, experimental data and property inputs as listed in Table. 1, the governing equation for droplet velocity is utilized to solve for interface fuel vapor mass fraction $Y_{Fi}$. Accordingly, Eq. 2 is interpreted as an implicit function of $Y_{Fi}$ and $T_i$, as shown below:

  $$\begin{aligned} \frac{du}{dt}+\frac{3 C_D \bar{\rho }_{f}}{4D \rho _l}(u-u_{\infty })^2-g_0\left( 1-\frac{\bar{\rho _f}}{\rho _l}\right) =E(Y_{Fi},T_i)=0. \end{aligned}$$

  (B1)

  The above equation can be solved for unknown interface mass fraction $Y_{Fi}$, using any standard root finding technique.

• 2. The interface mass fraction ($Y_{Fi}$) is utilized to find the instantaneous vapor pressure of the droplet, which is defined as

  $$\begin{aligned} P_{vap}=P_\infty \left\{ \frac{Y_{Fi} MW_{c}}{MW_F+Y_{Fi}(MW_{c}-MW_F)} \right\}, \end{aligned}$$

  (B2)

  where MW is the molecular weight of carrier gas (subscript ’c’) and fuel (subscript ’F’).

• 3. Using the interface mass fraction and temperature, the mass balance equation Eq. 5 is solved for the unknown film diffusion coefficient $\bar{\chi }_{fc}$, which is expressed as:

  $$\begin{aligned} \bar{\chi }_{fc}=\frac{\dot{m}_l}{\pi \bar{\rho }_f{D} {\text {Sh}}^* ln(1+B_M)} .\end{aligned}$$

  (B3)

• 4. The above steps are repeated for experiments with different carrier gas temperature. The resulting equilibrium droplet temperature for each experiment and vapor pressure, evaluated from the Step.1 , are employed to find the Antoine constants (A, B, & C) for the tested liquid, by fitting Antoine equation to the experimental data, expressed as,

  $$\begin{aligned} \log _{10} \left( P_{vap} \right) =A-\dfrac{B}{(C+T_i-273)}. \end{aligned}$$

  (B4)

• 5. Using the Antoine constants in Antoine equation Eq. B4, the boiling point ($T_{boil}$) of the tested liquid can be determined. Also, HOV can be determined by using the differential form of Clausius–Clapeyron equation, given by Eqn. B5, instead of Eqn. 10, which can improve the accuracy of HOV predictions.

  $$\begin{aligned} \dfrac{dP_{vap}}{dT}=\dfrac{P_{vap} H_F MW_F}{T^2 R_u}. \end{aligned}$$

  (B5)

• 6. The predicted diffusion coefficients, for each experiment, is used to evaluate the Lennard–Jones parameters, using the following empirical definition for diffusion coefficient, based on the work of Wilke and Lee  [45].

  $$\begin{aligned} \bar{\chi }_{fc}=\frac{[3.03-(0.98/{MW}^{0.5}_{fc})] (10^{-7}) \bar{T}^{3/2}_{f}}{P_{\infty }({MW}^{0.5}_{fc})\sigma ^2_{fc}\Omega _D}, \end{aligned}$$

  (B6)

  where

  $$\begin{aligned} MW_{fc}&=2\left[ \frac{1}{MW_F}+\frac{1}{MW_c}\right] ^{-1}\\ \sigma _{fc}&=\frac{(\sigma _c+\sigma _F)}{2}.\\ \end{aligned}$$

  The subscript ’c’ represents the carrier gas and ’F’ represents the fuel vapor. $\Omega _d$ represents the collision integral, in terms of Lennard–Jones parameters ($\sigma$, $\epsilon$), given by:

  $$\begin{aligned} \Omega _d=\frac{A}{(T^*)^B}+\frac{C}{\text {exp}(DT^*)}+\frac{E}{\text {exp}(FT^*)}+\frac{G}{\text {exp}(HT^*)}, \end{aligned}$$

  (B7)

  where

  $$\begin{aligned} T^*=k_bT/\epsilon _{fc}&A=1.06036&B=0.15610 \\ C=0.19300&D=0.47635&E=1.03587 \\ F=1.52996&G=1.76474&H=3.89411 \\ \epsilon _{fc}=\left( \epsilon _c \epsilon _F \right) ^{0.5}&&\\ \end{aligned}$$

  The two Lennard–Jones potential ($\sigma , \epsilon$) parameters are evaluated by non-linear fitting of Eq. B6 on the diffusion coefficient predictions obtained from Step. 6 of the current algorithm. Using the Boiling point $T_{boil}$ evaluated in Step. 1 of the present algorithm, based on the work of Wilke and Lee  [45], the Lennard–Jones energy parameter can be defined as $\dfrac{\epsilon }{k_b}=1.15~T_{boil}$. Using this definition for $\dfrac{\epsilon }{k_b}$, the diffusion coefficient equation Eq. B6 can be used only to evaluate the Lennard–Jones length ($\sigma$), which is observed to improve the prediction accuracy.