Pasquill’s Influence: On the Evaporation from Various Liquids into the Atmosphere

C. H. Huang Westinghouse Hanford Company, Richland, Washington

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Abstract

A theory of evaporation from a plane, free-liquid surface was first introduced by O. G. Sutton. However, in recognizing the shortcomings of Sutton’s theory, F. Pasquill proposed a generalized theory—that is, to modify Sutton’s theory by replacing the air viscosity in the Sutton model with the molecular diffusivity. To verify his theory, Pasquill also carried out a series of experiments. In this study, the author further considers the modification of the Sutton–Pasquill theory by introducing a Schmidt number of −2/3 and provides some theoretical justifications. The results are in very good agreement with the experimental data. In addition, a two-layer evaporation theory that includes an interfacial sublayer is developed.

Corresponding author address: Dr. C. H. Huang, Fluor Daniel Northwest, Inc., P.O. Box 1050, Richland, WA 99352.

chin-hua_huang.@rl.gov

Abstract

A theory of evaporation from a plane, free-liquid surface was first introduced by O. G. Sutton. However, in recognizing the shortcomings of Sutton’s theory, F. Pasquill proposed a generalized theory—that is, to modify Sutton’s theory by replacing the air viscosity in the Sutton model with the molecular diffusivity. To verify his theory, Pasquill also carried out a series of experiments. In this study, the author further considers the modification of the Sutton–Pasquill theory by introducing a Schmidt number of −2/3 and provides some theoretical justifications. The results are in very good agreement with the experimental data. In addition, a two-layer evaporation theory that includes an interfacial sublayer is developed.

Corresponding author address: Dr. C. H. Huang, Fluor Daniel Northwest, Inc., P.O. Box 1050, Richland, WA 99352.

chin-hua_huang.@rl.gov

Introduction

Pasquill’s development of an evaporation model, as well as his experimental work on the subject, are important in view of the recent emphasis on toxic chemical releases to the environment. Pasquill’s contributions to the field of “atmospheric diffusion” are enormous and well known. The Pasquill stability classification (Pasquill 1961) enables us to apply the Gaussian diffusion model to solving air pollution problems in our daily life. The Gaussian diffusion model has been widely and routinely applied in industry to estimate the air concentration and to manage radioactive and hazardous wastes in recent years. Equally important as his contribution to the Gaussian diffusion model is Pasquill’s influence on the subject of evaporation from various liquid surfaces into the atmosphere.

An evaporation model taking into account the vertical variations of the mean wind speed and the eddy exchange coefficient was first introduced by Sutton (1934). The Sutton model made it possible to estimate toxic chemical release from a smooth liquid surface to the environment. However, in the Sutton model, the process of vapor transfer is based on the momentum exchange involving the air viscosity. Subsequently, Pasquill (1943) modified Sutton’s evaporation theory by introducing the molecular diffusivity. This replaced the viscosity in the Sutton model, an important missing parameter. The Sutton–Pasquill evaporation model has been widely applied in industry for half a century. However, from these two parameters (air viscosity and molecular diffusivity), a nondimensional parameter can be formed and used to modify the Pasquill evaporation model. Experimental data in laboratories and in the field indicate that the rate of evaporation from a liquid surface is dependent upon the Schmidt number. Thus, in this study, we will consider the modification of the Sutton–Pasquill model and provide a theoretical justification.

Since Sutton’s model is the basic building foundation for subsequent development in this study, it is necessary to give a brief description of his model.

Sutton’s evaporation model

A theory of the removal of water vapor from a free-liquid surface into the atmosphere was put forward by Sutton(1934). He assumed similarity between the vapor transfer and the momentum transfer. The basic equation Sutton considered is a two-dimensional diffusion equation, which may be written as
i1520-0450-36-8-1021-e1a
where
i1520-0450-36-8-1021-e1b

The evaporation problem considered by Sutton is a strip of a free-liquid surface with an infinite extent in the crosswind direction and a finite dimension in the downwind direction. The wet surface area is defined by 0 < x < x0 for z = 0, while for x < 0 or x > x0, the ground is assumed to be dry, as shown in Fig. 1. Figure 1 shows the development of an internal concentration boundary layer. In Eq. (1a), where x and z are the coordinates in the downwind and the vertical directions, respectively, u(z) is the horizontal mean wind speed, χ is the vapor concentration, and Kz is the eddy diffusivity. The mean wind speed and the eddy diffusivity are assumed to vary with height and with power indices of m and n, respectively, and where u1 in Eq. (1b) and K1 in Eq. (1c) are constant values.

With the wind profile and the eddy diffusivity specified, the solution of Eq. (1a) for the rate of evaporation is obtained as
i1520-0450-36-8-1021-e2
where A = s (KKz); u is the wind speed measured at a reference height of interest (for field observations, the standard height is usually taken at 10 m).

Thus, the total rate of evaporation per unit width in the crosswind direction over a fetch x0 from a smooth surface can be calculated from Eqs. (2) or (3), where χs is the saturated vapor concentration at the liquid surface. For practical applications, the values of K for various values of the atmospheric turbulence intensity index n were provided by Pasquill (1943). The equation has often been used to estimate the release of hazardous materials into the atmosphere.

Pasquill’s experimental work

Various researchers considered the evaporation from a plane liquid surface. However, insufficient attention has been paid studying the aerodynamic effects on evaporation. Thus, Pasquill (1943) set out to conduct his experiments and to investigate the effects of the airflow over a liquid surface on the rate of evaporation. He examined the results from an experiment conducted in a wind tunnel in the light of the evaporation theory in turbulent flow developed by Sutton (1934). Recognizing the shortcoming of Sutton’s theory—that is, that the process of evaporation from a free-liquid surface is controlled by the viscosity of air rather than the molecular diffusivity—he subsequently proposed a generalized theory of evaporation based on the modification of Sutton’s theory. However, Pasquill’s generalized theory is mainly dependent on general physical considerations. To justify his theory, Pasquill carried out his experimental work in a wind tunnel and used bromobenzene as a liquid for the experiment. In his experiment, he showed that Sutton’s original theory overestimated the rate of evaporation, which is also reproduced here and shownin Fig. 2 to facilitate the discussion of his work. Thus, to remedy the shortcoming of Sutton’s theory, Pasquill suggested replacing the viscosity in Sutton’s theory by the molecular diffusivity.

Comparison of experimental data with theory

In this section, the experimental work conducted by Pasquill (1943) will be compared with the theories of evaporation (Sutton 1934; Pasquill 1943).

To verify the evaporation model, Pasquill (1943) used bromobenzene to conduct a series of experiments. The results are shown in Fig. 2, which also shows the comparison of the experimental data with Sutton’s theory (Pasquill 1943), where E is the total rate of evaporation, u1 cm the wind velocity measured at a height of 1 cm, D the molecular diffusivity, and Ps the saturation vapor pressure at the liquid surface. In Fig. 2, the data of the rate of evaporation obtained from wind tunnel experiments are normalized by the theory for the wind speed of 5 m s−1. In general, the results are in good agreement with Sutton’s theory, although Sutton’s original theory somewhat overestimated the rate of evaporation. In Sutton’s theory, Pasquill argued that the diffusion of momentum, heat, and mass is controlled primarily by the movements of eddies or masses of fluid. Eventually, these eddies mix with the air in its surroundings and must ultimately depend on the molecular motion. In the case of mass transfer, the molecular diffusion must play a dominant role for the turbulent transfer process of a substance. Therefore, Pasquill considered the molecular diffusion an important parameter in controlling the process of mass transfer.

On the basis of the above analysis of the vapor exchange coefficient, Pasquill suggested modifying Sutton’s theory and calculated the rate of evaporation from a free-liquid by replacing the parameter of the kinematic viscosity of air (ν) by the molecular diffusivity (D). In his paper, Pasquill further demonstrated the superiority of the so-called generalized theory (Pasquill 1943) over Sutton’s theory of evaporation.

To date, his work has in some ways been widely adopted by many practitioners in the industries for the calculation of the toxic releases of volatile liquids into the environment.

Theoretical justification

In the laboratories and field experiments, it appears that the rate of evaporation from various liquids is dependent upon a nondimensional parameter, the Schmidt number, with a power index of −2/3 rule (see MacKay and Matsugu 1973). However, MacKay and Matsugu (1973) used the experimental data mainly to obtain an empirical correlation for the evaporation formula. Thus, a theoretical justification to confirm their formula is needed.

In the following section, it will be shown that the rate of evaporation is dependent on the −2/3 power of the Schmidt number. As a demonstration of the concept, a set of basic equations will be used to obtain an analytical solution for the rate of evaporation. The basic equation considered is a two-dimensional diffusion equation of the form given in Eq. (1a).

Evaporation in laminar and turbulent flows

The solution for Eq. (1a) can be obtained if the boundary conditions, the wind profile, and the molecular diffusivity are specified. The boundary conditions are χ = 0 for z> 0, x = 0; χ = 0 for z = ∞; and χ = χs for z = 0, x > 0. The wind profile u = u2z/ν, where u∗ is the friction velocity, and the molecular diffusivity Kz = D for z ≥ 0. For the flow over a smooth surface, in the interfacial sublayer, the diffusion process is presumably controlled by the molecular diffusivity, and there exists a linear wind profile in this sublayer. With the boundary conditions, the wind profile, and the molecular diffusivity specified, the solution of the diffusion equation, Eq. (1a), can readily be obtained for the vapor density and the vertical flux of a diffusion substance (see, e.g., Bird et al. 1966 for the method of solution for a falling film diffusion problem). Thus, the mass transfer coefficient [also see Eq. (16) for the solution] may be expressed as
i1520-0450-36-8-1021-e4
where A0 is a constant. The mass transfer coefficient in Eq. (4) is defined as km = E/χs, where E is the mass flux.
Since we are dealing with the diffusion process of an airflow over a smooth surface, the drag force on such a flat surface in Eq. (4) will also be considered. The shearing stress τ on a smooth, flat surface in the laboratory and in the field may be expressed as
i1520-0450-36-8-1021-e5
where a is a constant that is equal to 0.029 according to the laboratory experiments, U is an undisturbed mean wind speed, and Rex is the Reynolds number.
Substituting Eq. (5) into Eq. (4), we have the mass transfer coefficient as
kmA−2/3Re−2/5xU,
where A is a constant. In practice, the constant A should be determined from experimental data. Equation (6) shows that the mass transfer coefficient follows the −2/3 power of the Schmidt number (ν/D); it also depends on the Reynolds number.
Equation (6) can be put into the form of the so-called J factor (e.g., Bird et al. 1966) as
i1520-0450-36-8-1021-e7
Then, the rate of evaporation over a fetch x may be expressed as
ETB−2/3Re3/5xχs

The value of B is a constant. The above results demonstrate that the rate of evaporation is dependent upon the −2/3 power of the Schmidt number.

For a turbulent flow over a flat plate, the correlation of the mass transfer coefficient with the experimental data may be expressed by the J factor (e.g., Bird et al. 1966) as
JRe−1/5x
and hence the rate of evaporation over a fetch x may be written as
ET−3−2/3χsu0.8x0.8

Surface renewal theory

A somewhat different treatment of the diffusion process in the interfacial sublayer was given by Brutsaert (1975). He formulated the rate of evaporation from the air–water interface based on the surface renewal theory. Liu et al. (1979) also include an interfacial sublayer in their study of the air–sea exchange of water vapor. Brutsaert (1975) showed that for the flow over a smooth surface, the interfacial transfer coefficient or the rate of local evaporation may be expressed as a Dalton’s number:
i1520-0450-36-8-1021-e11
where Cs is a dimensionless constant and χh is the vapor density at the top of the interfacial sublayer.
Equation (11) further demonstrates that the rate of local evaporation of a substance from a free-liquid surface into the atmosphere is a function of the Schmidt number with a power index of −2/3. Furthermore, the mass transfer coefficient, according to Brutseart (1975), may also be expressed as
i1520-0450-36-8-1021-e12
where αe is the inverse of the turbulent Schmidt number and Cd is the drag coefficient.
As a first approximation, if we assume that the second term in Eq. (12) can be neglected and that by virtue of Eq. (5) the friction velocity u∗ in Eq. (12) can be eliminated, then Eq. (12) becomes
km−3−2/3u0.9x−0.1
Thus, the rate of evaporation may be written as
ET−3−2/3χsu0.9x0.9
Equation (14) results in an expression similar to Sutton and Pasquill’s formulas, but it contains a Schmidt number, Sc−2/3. The constants associated with the transfer coefficient in Eq. (13) and the rate of evaporation in Eq. (14) are quite close to those values obtained by MacKay and Matsugu (1973) and by the present author (see Tables 1 and 2). Thus, the Schmidt number can be incorporated into Sutton and Pasquill’s formulas for the parameterization of the vapor transfer from a liquid surface.

Again, Eq. (14) shows that the rate of evaporation is dependent upon the Schmidt number with the power index of −2/3.

A theory

Another theoretical treatment of the vapor transfer from a liquid surface into the atmosphere is considered here. It is assumed that the atmosphere consists of two layers: one is the interfacial sublayer, and the other is a layer above the sublayer. The rate of evaporation can be obtained for these two layers. Solving Eq. (1a) by the methodof the Laplace transform, one can obtain the solution of vapor concentration χ [=AfΓ(ν, η); see below for the symbols], with a free parameter Af for the upper layer of the atmosphere. In the interfacial sublayer, the rate of evaporation may be expressed by Eq. (11), where the molecular diffusion is a controlling factor for the process of vapor transfer. Furthermore, assuming the continuities of the rate of evaporation and the vapor concentration at the top of the interfacial sublayer (or any arbitrary lower atmospheric layer) and matching the variables of these two layers together at the top of the interfacial sublayer, we obtain the solution for the local rate of evaporation as
i1520-0450-36-8-1021-e15
where
i1520-0450-36-8-1021-eq1
and Γ(ν, η) is the incomplete gamma function. Equation (15) shows that in the atmospheric boundary layer, the rate of evaporation is dependent upon a Schmidt number of Sc−2/3.
For z → 0 and neglecting the term C1, Eq. (15) re-duces to
i1520-0450-36-8-1021-e16
For the conjugate power law of n = 1 − m, Eq. (16) is reduced to a solution that may be shown to be equivalent to the Sutton (1934), Pasquill (1943), and Calder (1949) solutions. Therefore, the term C1eη in Eq. (15) may be considered as the resistance to the evaporation due to the presence of the interfacial sublayer.

Results

Pasquill (1943) analyzed the experimental data obtained by the previous investigators. He compared his own experimental data and other data obtained by previous researchers with the generalized theory. The results are shown in Fig. 3. Figure 3 shows that the rate of evaporation has been normalized by the theoretical rate of evaporation for wind speed, u = 3 m s−1, measured at a height of 1 cm in wind tunnel experiments. The results show that Pasquill’s experimental data are in good agreement with the generalized theory.

But, as can be seen from Fig. 3, the experimental data obtained by other authors are not entirely in agreement with his theory. However, from these two parameters (ν and D), a nondimensional parameter, the Schmidt number, can be formed. Furthermore, the experimental data obtained in the laboratory and the field, and the theories described above in section 5 all suggest that the rate of evaporation is dependent on a Schmidt number, Sc−2/3. Thus, the rate of evaporation may be expressed as
ETc−2/3χsu0.78x0.89
and, hence, the mass transfer coefficient as
kmc−2/3u0.78x−0.11
where c is a constant to be determined from the experimental data. The value c is found to be approximately 5.5 × 10−3 based on the best fit of Pasquill’sexperimental data, corresponding to curve A in Fig. 3; it is equal to 3.28 × 10−3 Sc2/3 for Sc = 2.162 (see Table 2). The value c′ can be calculated from the relationship between ET and km (i.e., c′ = 0.89c). The calculated values of c′ and c are given in Tables 1 and 2, respectively. Equation (17) is also plotted in Fig. 3 for various liquids—bromobenzene, toluene, and water. Equation (17) shows much better agreement than the generalized theory with the experimental data for various liquids. The results indicate that the rate of evaporation depends on the −2/3 rule of the Schmidt number. It is evident that the rate of evaporation for various liquids also varies with the Schmidt number.

Similar to the plot shown in Fig. 3, the results for the plot of the product of the total rate of evaporation and the 2/3 rule of the Schmidt number are shown in Fig. 4. Figure 4 shows that the three curves generated from the proposed evaporation equation, Eq. (17), as displayed in Fig. 3, collapse into a single curve. The results indicate that the proposed equation of the rate of evaporation for various liquids is in very good agreement with the experimental data.

The J factor for turbulent flow with a fetch of 1 m is also plotted in Fig. 4 and is used in making the comparison with the theory. The figure shows that in general, the theory is in reasonably good agreement with the experimental data, but predicts a higher value for the evaporation rate than the J factor.

The various formulas for the mass transfer coefficient (transfer velocity) and the equations used to predict the total rate of evaporation by various authors are given in Tables 1 and 2, respectively. The formulas with the numbers appearing in Tables 1 and 2 are in the metric system.

As can be seen from Tables 1 and 2, the constants associated with the mass transfer coefficient and the rate of evaporation of bromobenzene or water vapor predicted by the present author are remarkably close to those predicted by MacKay and Matsugu (1973).

Knowledge of the mass transfer coefficient, such as given in Tables 1 and 2, is needed in various practical applications for the estimate of chemical releases into the environment. For example, the mass transfer coefficient proposed by MacKay and Matsugu (1973) has been applied to the accidental event of the Exxon Valdez oil spill by Hanna and Drivas (1993).

Similar to the evaporation formula for the rectangular area, for a circular area of radius r, the rate of evaporation from various liquid surfaces into the atmosphere may be written as
ETrc−2/3χsu(2−n)/(2+n)r(4+n)/(2+n)
In practical applications, we may set n = 0.25. Then, Eq. (19) becomes
ETrc−2/3χsu0.78r1.9
where c = 1.64 × 10−2.

Conclusions

Pasquill’s equation for predicting the rate of evaporation for bromobenzene is close to the equation developed by MacKay and Matsugu (1973) and the present author. Pasquill’s experimental data have been utilized to develop an equation to estimate the rate of evaporation for the airflow over a flee-liquid surface in the atmospheric boundary layer. The modified evaporation equation is based on the theoretical treatment of the evaporation process for airflow over a fetch. The predicted evaporation equation by the present author is remarkably close to that of MacKay and Matsugu (1973). It appears that the evaporation equation including a −2/3 of the Schmidt number is in good agreement with a wide range of experimental data. However, additional experimental work in the laboratory or the field is needed to verify the rate of evaporation for various liquids under a much better controlled condition.

In addition, theoretical justifications have been provided for the expression of the rate of evaporation over a fetch that includes a −2/3 Schmidt number in the formula. The rule of −2/3 Schmidt number in the evaporation formulation has also been confirmed by the observations obtained in laboratories and in the field. For a laminar flow, and with the assumptions of a linear wind profile and constant diffusivity, it is found that the mass transfer coefficient is dependent upon the −2/3 power of the Schmidt number and a Reynolds number. Furthermore, in this study, a two-layer evaporation model including an interfacial sublayer and an evaporation formula for a circular area that depends upon a −2/3 power of the Schmidt number are developed.

Pasquill’s experimental work on the evaporation of various liquids still remains one of the important viable sources that may be used to justify semiempirical or theoretical work. It is a new challenge to continue Pasquill’s efforts in the estimation of the rate of evaporation from various liquid surfaces into the atmosphere and, especially, to conduct the experiments under much more controlled conditions. It is up to us to carry the torch of experimental or theoretical work left to us by Pasquill—a man of great contributions to the fields of turbulence, diffusion, and evaporation.

REFERENCES

  • Bird, R. B., W. E. Stewart, and E. N. Lightfoot, 1966: Transport Phenomena. John Wiley & Sons, 780 pp.

  • Brutsaert, W., 1975: A theory for local evaporation (or heat transfer) from rough and smooth surfaces at ground level. Water Resour. Res.,11, 543–550.

  • Calder, K. L., 1949: Eddy diffusion and evaporation in flow over aerodynamically smooth and rough surface: A treatment based on laboratory laws of turbulent flow with special reference to conditions in the lower atmosphere. Quart. J. Mech. Appl. Math.,2, 153–176.

  • Hanna, S. R., and P. J. Drivas, 1993: Modeling VOC emissions and air concentration from the Exxon Valdez oil spill. J. Air Waste Manage. Assoc.,43, 298–309.

  • Liu, W. T., K. B. Katsaros, and J.A. Businger, 1979: Bulk parameterization of air–sea exchanges of heat and water vapor including the molecular constraints at the interface. J. Atmos. Sci.,36, 1722–1935.

  • MacKay, D., and R. S. Matsugu, 1973: Evaporation rate of hydrocarbon spills on water and land. Can. J. Chem. Eng.,5, 434–439.

  • Pasquill, F., 1943: Evaporation from a plane, free-liquid surface into a turbulent air stream. Proc. Roy. Soc. London, Ser. A,182, 75–94.

  • ——, 1961: The estimation of the dispersion of windborne material. Meteor. Mag.,90, 33–49.

  • Sutton, O. G., 1934: Wind structure and evaporation in a turbulent atmosphere. Proc. Roy. Soc. London, Ser. A,146, 701–722.

Fig. 1.
Fig. 1.

Evaporation from a liquid surface into the atmosphere and the growth of a concentration boundary layer.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1021:PSIOTE>2.0.CO;2

Fig. 2.
Fig. 2.

Comparison of experimental and theoretical values for the rate of evaporation of bromobenzene into a turbulent airstream.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1021:PSIOTE>2.0.CO;2

Fig. 3.
Fig. 3.

Rate of evaporation into a turbulent airstream; comparison of experimental and theoretical values. Theory: curve A is bromobenzene, B is toluene, and C is water.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1021:PSIOTE>2.0.CO;2

Fig. 4.
Fig. 4.

Comparison of the theory with the experimental data.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1021:PSIOTE>2.0.CO;2

Table 1.

Constants associated with various transfer coefficients (km) by various authors.

Table 1.
Table 2.

The rates of evaporation (ET) according to various authors.

Table 2.
Save
  • Bird, R. B., W. E. Stewart, and E. N. Lightfoot, 1966: Transport Phenomena. John Wiley & Sons, 780 pp.

  • Brutsaert, W., 1975: A theory for local evaporation (or heat transfer) from rough and smooth surfaces at ground level. Water Resour. Res.,11, 543–550.

  • Calder, K. L., 1949: Eddy diffusion and evaporation in flow over aerodynamically smooth and rough surface: A treatment based on laboratory laws of turbulent flow with special reference to conditions in the lower atmosphere. Quart. J. Mech. Appl. Math.,2, 153–176.

  • Hanna, S. R., and P. J. Drivas, 1993: Modeling VOC emissions and air concentration from the Exxon Valdez oil spill. J. Air Waste Manage. Assoc.,43, 298–309.

  • Liu, W. T., K. B. Katsaros, and J.A. Businger, 1979: Bulk parameterization of air–sea exchanges of heat and water vapor including the molecular constraints at the interface. J. Atmos. Sci.,36, 1722–1935.

  • MacKay, D., and R. S. Matsugu, 1973: Evaporation rate of hydrocarbon spills on water and land. Can. J. Chem. Eng.,5, 434–439.

  • Pasquill, F., 1943: Evaporation from a plane, free-liquid surface into a turbulent air stream. Proc. Roy. Soc. London, Ser. A,182, 75–94.

  • ——, 1961: The estimation of the dispersion of windborne material. Meteor. Mag.,90, 33–49.

  • Sutton, O. G., 1934: Wind structure and evaporation in a turbulent atmosphere. Proc. Roy. Soc. London, Ser. A,146, 701–722.

  • Fig. 1.

    Evaporation from a liquid surface into the atmosphere and the growth of a concentration boundary layer.

  • Fig. 2.

    Comparison of experimental and theoretical values for the rate of evaporation of bromobenzene into a turbulent airstream.

  • Fig. 3.

    Rate of evaporation into a turbulent airstream; comparison of experimental and theoretical values. Theory: curve A is bromobenzene, B is toluene, and C is water.

  • Fig. 4.

    Comparison of the theory with the experimental data.

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