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# Success of coastal wetlands restoration is driven by sediment availability

## Abstract

Shorelines and their ecosystems are endangered by sea-level rise. Nature-based coastal protection is becoming a global strategy to enhance coastal resilience through the cost-effective creation, restoration and sustainable use of coastal wetlands. However, the resilience to sea-level rise of coastal wetlands created under Nature-based Solution has been assessed largely on a regional scale. Here we assess, using a meta-analysis, the difference in accretion, elevation, and sediment deposition rates between natural and restored coastal wetlands across the world. Our results show that restored coastal wetlands can trap more sediment and that the effectiveness of these restoration projects is primarily driven by sediment availability, not by wetland elevation, tidal range, local rates of sea-level rise, and significant wave height. Our results suggest that Nature-based Solutions can mitigate coastal wetland vulnerability to sea-level rise, but are effective only in coastal locations where abundant sediment supply is available.

## Introduction

Coastal wetlands provide a wide range of ecosystem services (valued up to US\$194,000 ha−1 yr−1), including the support of commercial fisheries, carbon sequestration, natural coastal protection, and water quality improvement1,2. Historically, salt marshes and mangroves have been converted in urban development or agricultural fields at staggering rates; it is estimated that Europe has lost 50% of salt marshes to reclamation3. In Southeast Asia deforestation destroys 1.3% of mangrove forests every year4.

In recent decades, a leading cause of widespread coastal wetland loss is the inability to build vertically at rates comparable to relative sea-level rise (SLR)5,6, and global-scale assessments of coastal wetland dynamics indicate that SLR represents a critical threat to these ecosystems7,8. Wetland disappearance can accelerate due to the substantial increase in inundation and flooding triggered by accelerated SLR, storm surges, and land subsidence9,10. A reduction in sediment supply due to river damming can further compromise the resilience of coastal wetlands11,12. Recent results have suggested that the vulnerability to SLR might be lower than expected due to feedbacks between ecology and geomorphology13 and the availability of inland areas for marsh migration14. Even if a smaller fraction of wetland area will be affected by SLR, there is still the need to reestablish these important ecosystems where they were historically reclaimed. In fact, a growing world population has turned coastal wetlands into agricultural fields, urban and industrial developments, and intensive aquaculture. This conversion has resulted in a loss of habitat for many fish and birds, as well as loss of important ecosystem services such as erosion mitigation, water purification, carbon storage, and natural flood defense10,15.

In recent years, large investments have been made to protect the coast from SLR and storms, mostly by building traditional hard structures like seawalls and breakwaters. These solutions are expensive and non-sustainable in the long term16,17. Numerical and field studies suggest that coastal wetlands have a high potential to reduce flooding and coastal erosion18,19,20. Government agencies, nonprofit organizations, and businesses have developed an increasing interest in Nature-based Solutions, which are defined as actions to protect, sustainably manage, create, and restore natural or modified ecosystems for providing solutions to climate mitigation and adaptation challenges21,22,23. In coastal wetlands, Nature-based Solutions encompass the creation of living shorelines through vegetation planting, hydrological re-connection of reclaimed wetlands to the sea, managed retreat from the shore through removal of flood defenses, and thin-layer sediment placement that increases wetland elevation and enhances coastal resilience10,24,25,26,27. There is a growing body of studies showing that Nature-based Solutions are a long-term and cost-efficient strategy to help safeguard wetlands from SLR, thus protecting associated ecosystem services26,28. However, the effectiveness of Nature-based Solutions for SLR mitigation and adaptation has not been globally assessed, owing to the complex feedbacks of multiple environmental processes driven by elevation, vegetation, local relative SLR rate, tidal range, and sediment availability29. Recent modeling advances have provided the ability to conduct meta-analyses to fill research gaps at a global scale13,30.

The vulnerability of coastal wetlands to SLR depends on whether they are able to vertically build at rates equal or greater than relative SLR13,31,32. Here we provide the first global synthesis and meta-analysis of the contributions of Nature-based Solutions to vertical accretion and surface elevation gain. Our aims are: (1) to quantitatively assess the effects of Nature-based Solutions on accretion, elevation change, and sediment deposition in comparison to natural wetland sites; and (2) to examine the relationship between effect size and environmental factors, including suspended-matter concentration, elevation, tidal range, local relative SLR, and wave height.

## Results

In general, Nature-based Solutions significantly enhance resilience to SLR along the shorelines of the Atlantic, Pacific, and Indian Oceans. Hedges’ g* effect size of surface elevation change rate in the European Atlantic coast is significantly higher than that of the US Atlantic coast, while the US Pacific coast has the lowest Hedges’ g* value. Although the sample size in the Indo-Pacific region (China and Sri Lanka) is small, results show the enormous potential in enhancing vertical accretion rate in this region33,34 (Fig. 1). These solutions will prevent coastal wetland loss in the future, by storing sediment near the shore.

Nature-based Solutions significantly increased the accretion rate by 19.58 ± 6.66 mm yr−1 in salt marshes and by 2.91 ± 0.63 mm yr−1 in mangroves (Fig. 2c). The standardized effect size of the accretion rate was lower in salt marshes than in mangroves (mean Hedges’ g* are 1.66 and 4.09, respectively) (Fig. 2a). Rates of surface elevation change in restored sites were higher by 10.35 ± 1.56 mm yr−1 in salt marshes and by 2.55 ± 0.63 mm yr−1 in mangroves (Fig. 2c). The standardized effect size of surface elevation gain was not significantly different among salt marshes and mangroves (Fig. 2a). For rates of elevation change, salt marshes can keep pace vertically with current local relative SLR in restored sites, but not in natural reference sites. In pristine areas, some salt marsh ecosystems are slowly drowning because their mean accretion deficit (elevation change rate minus relative SLR rate) is greater than 0.5 mm yr−1 (Fig. 2d). Nature-based Solutions can shift elevation change from deficit to gain. In general, Nature-based Solutions in salt marshes could increase sediment trapping by on average 185.71 ± 56.35 ton ha−1 yr−1 with respect to adjacent natural wetlands (Fig. 2b).

On a global scale, rates of accretion and elevation change in Nature-based Solutions projects are significantly correlated to the concentration of total suspended matter (TSM) in the water column (Fig. 3 and Supplementary Fig. 2). These relationships link the supply of sediments to the maintenance of soil elevation in salt marshes and mangrove forests across the Atlantic, Pacific, and Indian Oceans. Other variables (elevation relative to mean sea level (MSL), significant wave height, tidal range, regional rate of SLR, and elevation difference between restored and natural sites) explain a smaller proportion of the variation in accretion and rates of elevation change in salt marshes (Figs. 3 and 4, and Supplementary Figs. 2, 3, and 5).

The effect of Nature-based Solutions significantly increases with increasing TSM; trends are generally similar in salt marshes and mangroves (Fig. 3a, b and Supplementary Fig. 2a, b). For the first time we prove with field data collected across the world that the success of restoration projects is primarily driven by sediment availability. Consistent with numerical models, the effect of Nature-based Solutions is always positive when suspended sediment concentrations are greater than 20 g m−3 in salt marshes35. Hedges’ g* of accretion and rates of elevation change in mangroves is negatively correlated with wetland elevation (Fig. 3d and Supplementary Fig. 2d). The effect size of accretion in salt marshes increases first, and then decreases with an increase in elevation (Fig. 3c), however, the effect size of elevation change rate in salt marshes does not display a significant change when elevation increases (Supplementary Fig. 2c). Hedges’ g* of accretion and rates of elevation change in salt marshes is negatively correlated with elevation difference between restored and natural sites (Supplementary Fig. 5a, c), however, the effect size for mangroves does not display a significant change when the difference in elevation between restored and natural sites increases (Supplementary Fig. 5b, d). The effect size of accretion rate for salt marshes is positively correlated with SLR and tidal range, however, Hedges’ g* is not correlated with significant wave height (Fig. 4). The effect size of rates of elevation change for salt marshes is also positively correlated with SLR, but it is not correlated to tidal range and significant wave height (Supplementary Fig. 3). For mangrove ecosystems, the effect size of accretion rate in areas with large waves is significantly higher than in areas with small waves (Supplementary Fig. 4), which is consistent with sediment resuspension during storms or wave set up facilitating the flux of inorganic sediments into low-lying mangroves36,37. The effect size of both accretion and elevation change is not correlated to the percentage of sediment trapped by dams in nearby large rivers (Supplementary Fig. 6).

In general, effect size of accretion is more correlated to the variables we considered than to the effect size of surface elevation change (Fig. 3 vs Supplementary Fig. 2, Fig. 4 vs Supplementary Fig. 3, and Supplementary Fig. 5a, b vs Supplementary Fig. 5c, d). The surface elevation table – marker horizon (SET–MH) method has been used to successfully measure sediment vertical accretion, changes in relative elevation, and shallow soil processes (subsidence and expansion due to root production) worldwide13,32,38. This method allows to separate the contribution to surface elevation change due to surface accretion processes from that due to subsurface processes such as shallow subsidence, water table fluctuations, and root accumulation39,40. Therefore, changes in surface elevation depend not only on vertical accretion, but also on shallow subsidence, sediment compaction, and root zone expansion41. Local shallow subsidence complicates the relationship between changes in surface elevation and the physical factors investigated here. In salt marshes, elevation change is much lower than accretion, because of the large fraction of organic matter that can be compacted or that decays in time (Fig. 2c)42. In mangroves, less organic material is accumulated in the soil and mangrove roots better resist the decay, as a result, elevation change is closer to accretion41,43.

## Discussion

Nature-based Solutions can mitigate coastal wetland vulnerability to SLR by leveraging on ecogeomorphic feedbacks between flooding, vegetation, organic matter accretion, and sediment deposition. Restored wetlands may be lower in the tidal frame than natural ones, because of erosion or soil compaction after land reclamation in pre-restoration sites44,45. Therefore, they experience higher hydroperiods, and more time for sediment deposition than natural ones (Supplementary Fig. 5). A lower elevation also increases tidal prism, and therefore the volume of water and sediments transported to the wetland. Over the course of time, if there is enough sediment supply, the elevation difference between restored and natural marsh will vanish, together with the accretion differential45,46,47. The difference in accretion rates was not significant in areas with low total-suspended-matter concentration45. Note that the correlation between effect size and TSM is much stronger than the correlation with elevation difference (Fig. 3a, b vs Supplementary Fig. 5), therefore sediment availability is more important for restoration success than longer hydroperiods and larger tidal prisms. This is because larger volumes of water flooding the wetlands also require high-suspended-sediment concentrations to trigger accretion. Therefore, the success of the Nature-based Solution is intrinsically linked to the location of the project and the local geomorphology, for example, to the elevation within the tidal frame and, more importantly, to sediment availability.

Restored wetlands can also have higher vegetation density, because of artificial planting or a position in the tidal frame more favorable to vegetation development. Higher vegetation biomass favors sediment trapping and accretion41,45,48. Salt marsh grass or mangrove planting, one of the traditional wetland restoration approaches, can reduce tidal current speeds, prevent erosion, trap more sediment, and promote belowground root production to facilitate accretion and sediment retention49,50,51. Hydrological restoration like managed realignment (MR) or controlled reduced tide (CRT) can increase tidal inundation compared to pre-restoration conditions by breaching artificial barriers and reestablishing tidal exchange between the restoration site and the adjacent estuary or sea, hence promoting more frequent and longer episodes of mineral sediment deposition, enhancing vegetation growth, and accelerating rates of mineral and organic matter accumulation10,52. Our results show that Hedges’ g* effect size of accretion and rate of elevation change is not significantly different between planting and hydrological restoration (Fig. 5). According to published data, however, the median cost for restoration of 1 hectare of salt marsh with planting is 10–20 times higher than the cost for hydrological restoration26. Therefore, hydrological restoration, in which low-lying inland areas are converted to coastal wetlands, is an economical and effective restoration strategy to facilitate accretion and vertical increase in elevation, when sufficient additional accommodation space is available10,14. In addition, thin-layer sediment placement (also known as thin-layer deposition, sediment augmentation, or sediment replenishment) and diversion of sediment-laden riverine discharge to wetlands can also increase sediment supply, promoting elevation gain and wetland expansion53,54.

In mangroves, an increase in elevation decreases the difference in sediment trapping capacity between restored and natural sites (Fig. 3d). This is expected because mangroves higher in the intertidal frame and subject to lower flooding depth have likely a lower hydroperiod. With a lower hydroperiod, suspended sediment has less time to deposit, reducing sediment accretion and elevation change55. In salt marshes, however, the effect of restoration is more complex, with the difference in sediment accretion first increasing and then decreasing as a function of elevation (Fig. 3c). We attribute this behavior to ecogeomorphic feedbacks between elevation and salt marsh vegetation. Biomass of salt marsh plants first increases with elevation and then decreases, with optimal conditions for vegetation growth occurring at intermediate elevations50,56. Biomass controls sediment accretion by trapping sediment on stems and leaves, increasing belowground production of organic matter, and slowing tidal currents, thus promoting deposition57. This feedback between elevation and vegetation controls the relationship between accretion and elevation (Fig. 3c). Restored mangroves along shorelines with high wave energy trap more sediments than natural ones (Supplementary Fig. 4). This result is in agreement with field measurements showing that waves are instrumental in resuspending bottom material and advecting it in the mangrove forest58. In salt marshes, this effect is subdue because the thick canopy promotes wave dissipation reducing transport to the marsh interior59,60. In salt marshes, tidal range and relative SLR rate explained a smaller proportion of the variation in the effect size of vertical accretion, however, these factors are positively correlated with the effect size (Fig. 4a, b). The trends are consistent with previous theory and results, showing that coastal ecosystems are likely to survive at sites with high tidal range and rates of SLR35,43,50,61. Moderate rates of SLR not only increase the frequency and duration of tidal inundation, but may also stimulate vegetation growth, accelerating the rate of accretion50,61. In fact, rates of mineral sediment deposition increase with the frequency and duration of tidal inundation, while stimulated vegetation growth also promote inorganic sediment trapping and in situ organic accretion57,61. Vegetation growth range expands with tidal range, so that vegetation surfaces in macrotidal and mesotidal marshes can more easily accommodate SLR than in microtidal marshes43,61.

The success of Nature-based Solutions is not related to the percentage of sediment trapped by dams in the nearest large river (Supplementary Fig. 6). We ascribe this result to the complex and local nature of sediment supply to salt marshes. In coastal and estuarine systems, sediment supply not only depends on upstream or seaward sediment inputs, but also on sediment re-distribution through riverine and tidal channels62,63. Many salt marshes are fed by sediments originating from tidal flats and the nearshore area64,65. River sediment load is rarely discharged directly on salt marshes; rather, the sediment is first delivered to coastal bays and the inner shelf and then reworked by marine processes, which can mediate and modulate the sediment flux to the marsh62,63. Moreover, small rivers located close to the marsh might have a stronger effect on restoration projects than the sediment discharged by distant large rivers, because sediment supply decreases with distance from the river mouth12,53. Several large rivers are also dammed, decreasing the flux of sediment to the coast11,66. For example, the sediment load of the Yellow River in China, one the world’s largest, has decreased by approximately 90% from 1950s to 2010s67. The Mississippi River sediment load has also been reduced by 50% after the construction of dams68. Therefore, restored sites should be located where suspended-matter concentration is high, like in estuaries, near river mouths, or along muddy shorelines. Sediment transport pathways and budgets should be integrated into the early phases of Nature-based Solutions planning.

In conclusion, Nature-based Solutions can be an effective strategy to trap sediment along the shore and mitigate coastal wetland vulnerability to SLR69. Plantation and hydrologic restoration (MR or CRT), the two most common Nature-based Solutions in salt marshes and mangroves ecosystems, can enhance vegetation growth, prevent erosion, and accelerate rates of mineral and organic matter accumulation. Furthermore, results from our synthesis indicate that the effectiveness of Nature-based Solutions for SLR mitigation and adaption is strongly linked to the local availability of suspended matter in coastal waters. A reliable sediment supply is needed for wetland accretion, and is more important than the local rate of SLR for restoration success. This is why the effect size along the North-European and Indo-Pacific coasts is higher than along the US coast (Fig. 1). The North-European and Indo-Pacific coasts have more sediment availability, while the US coast is sediment starved14 (Fig. 1). When sediment availability is scarce, restoration projects might fail, preventing the attainment of the surface elevation needed for normal wetland ecological functions70,71,72. Unfortunately, the sediment flux to the coast has been reduced in recent decades by 1.4 ± 0.3 billion metric tons per year, because of retention within dams and reservoirs11. As a result, the ability of Nature-based Solutions to trap sediment is diminishing68,73. Dam regulation, and targeted management of upstream watersheds are therefore vital for the coastal sediment budget and the survival of coastal wetlands.

## Methods

### Literature search

To build a comprehensive database of the impacts of Nature-based Solutions on the resilience of coastal wetlands to SLR we reviewed primary literature, reports, and other datasets. We carried out a systematic review in the ISI Web of Science database (www.isiwebofknowledge.com) on 22 April 2019 with no restriction on publication year and subject areas, using the following search terms: TS = (salt marsh* OR saltmarsh* OR tidal marsh* OR mangrove* OR mangal*) AND TS = (restor* OR rehab* OR recov* OR creat* OR reestab* OR reveget* OR afforest*) AND TS = (elevation* OR sediment* OR accretion* OR erosion* OR deposit*), and TS = (living shoreline* OR nature-based OR thin-layer placement OR dredge material OR managed realignment OR managed retreat*) AND TS = (elevation* OR sediment* OR accretion* OR erosion* OR deposit*). This resulted in 3516 publications. We also included 45 other published and unpublished papers or reports from references that were relevant.

We examined the title and abstract of each publication to assess their potential for meeting the selection criteria for inclusion in the review. In all, 268 studies were identified that potentially met the selection criterion. We only selected studies that: (1) examined the effects of Nature-based Solutions on accretion, elevation change, and sediment deposition in restoration projects or field experiments; (2) used surface elevation tables (SETs) and feldspar-marker horizons (MH) to measure soil surface elevation and vertical accretion rates, following the method of Cahoon et al.38, and used sediment traps to measure the deposition of sediments, as indicated by Reed74; and (3) reported sample sizes and some measure of variance (e.g., standard deviations/errors) for each measured variable in both Nature-based Solutions and natural reference systems. Reference natural wetlands used herein were typically adjacent to the nearby wetlands where Nature-based Solutions were applied. Restored and natural sites are similar in species composition and share the same tidal range, rate of SLR, sediment supply, and wave height. We calculated the difference in elevation between restored and natural sites, because the elevation of the two sites may be different44,45. The literature selection procedure is shown in Supplementary Fig. 1 as a PRISMA flow diagram. This methodology resulted in 225 experiments/observations reported in 52 published and unpublished studies, which formed the basis of the meta-analysis. An overview map of the worldwide locations of Nature-based Solutions projects is provided in Fig. 1.

### Data extraction and data source

For each retained publication, we extracted data at sites where Nature-based Solutions were applied and reference sites from the main text, tables, and figures of the articles. Data from plots and figures were extracted graphically using WebPlotDigitizer (available online). When accretion and/or elevation change rates were reported for multiple dates, we calculated the averaged rates across the entire measurement period to minimize the effect of restoration duration on rates of accretion and elevation change. When accretion/elevation change and sediment deposition were reported in mm or cm and g m−2, respectively, we divided the results by the measurement duration (in years) to obtain mm yr−1 and ton ha−1 yr−1.

In addition, we also recorded the following variables for each study: author(s), year, study location, latitude, longitude, habitat (salt marshes or mangroves), project duration (in years) and restoration method, tidal range, regional relative SLR rate, elevation relative to MSL, and the difference in elevation between restored and natural reference sites. Not all information required for the database was directly available in every publication, therefore, additional information was derived where possible. Latitude and longitude data were obtained by locating the study site on Google Earth. Tidal range and regional relative SLR data were obtained from the nearest Center for Operational Oceanographic Products and Services (CO-OPS) tide station (https://tidesandcurrents.noaa.gov/) and National Oceanic and Atmospheric Administration (NOAA) tide gauge (https://tidesandcurrents.noaa.gov/sltrends/sltrends.html), respectively. Elevation relative to MSL data was obtained from other relevant references. If unavailable in the publication, the elevation of study area in the USA was calculated using the USGS National Elevation Dataset (NED; http://nationalmap.gov/elevation.html), which is a set of ~3 m resolution, best-quality elevation data widely used in geomorphic studies75, and converted to elevation relative to local MSL.

The significant wave height was derived from the NOAA WAVEWATCH III (data freely available from https://polar.ncep.noaa.gov/waves/hindcasts/nopp-phase2.php), which contains monthly values of significant wave height in the Earth’s oceans between March 2005 and April 2019. The significant wave height was extracted with MATLAB, using the significant wave height value of the pixel containing, or closest to, the Nature-based Solution sites. We calculated the monthly averages during the measurement period to represent the local significant wave height. If no data were available during the measurement period, we used the monthly averages from March 2005 to April 2019. Sediment supply is critical for surface elevation gains. Therefore, we also explored the relationship between restoration success (e.g., effect size) and TSM. The TSM in coastal waters was derived from remotely sensed GlobColour MERIS and OLCIA imagery (data freely available from http://hermes.acri.fr/), which contains monthly values of TSM in the Earth’s oceans and lakes between 2002 and 2019 (data between April 2012 and March 2016 is missed). MERIS data have already been used to calculate local sediment availability in global estimates of wetland response to SLR14,43. TSM was extracted with MATLAB using the TSM value of the pixel containing, or closest to, the Nature-based Solution sites. We calculated the monthly averages during the measurement period to represent the local sediment availability. Monthly coverage of MERIS, however, is incomplete. If no data were available during the measurement period, we used the monthly averages from April 2002 to April 2019.

To investigate whether a reduction in sediment supply caused by dams in nearby fluvial watersheds affects the success of Nature-based Solutions, we relate accretion rates and changes in marsh elevation to a global dataset of anthropogenic sediment retention in large rivers76. To each salt marsh site, we assign the percentage of sediment trapped by dams in the nearest large river, and if large rivers are not present, we assign a value of zero.

### Meta-analysis

To standardize and compare data, we quantified the effect size by calculating the Hedges’ g*, which is a metric commonly used in meta-analysis and can quantify the unbiased, standardized mean difference in sediment deposition between restored and natural sites77. Positive g* values indicate that Nature-based Solutions increase coastal wetland resistance to SLR, by increasing the rates of accretion and elevation change, or sediment deposition, while negative values indicate Nature-based Solutions fail to increase the resistance.

$$g^ \ast = \frac{{X_{{\mathrm{NbS}}} - X_{{\mathrm{ref}}}}}{S} \times J,$$
(1)

where X and S denote the mean and standard deviation of the measured variable, respectively. The subscript NbS refers to Nature-based Solution wetlands (restored or newly created wetlands) and ref to reference wetlands, respectively. J is a correction factor for small sample bias, and S is the pooled standard deviation.

$$J = 1 - \frac{3}{{4df - 1}},$$
(2)
$$S = \sqrt {\frac{{\left( {N_{{\mathrm{NbS}}} - 1} \right)S_{{\mathrm{NbS}}}^2 + (N_{{\mathrm{ref}}} - 1)S_{{\mathrm{ref}}}^2}}{{N_{{\mathrm{NbS}}} + N_{{\mathrm{ref}}} - 2}}} ,$$
(3)

where N is the sample size, df is the degrees of freedom used to estimate S, for two independent groups is NNbS + Nref−2.

Using random-effects models77, we estimated mean effect sizes g* and 95% bootstrapped confidence intervals (95% CI) on accretion, elevation change, and sediment deposition rates for salt marshes and mangroves. Treatment effects were considered significant if the 95% CI did not overlap zero. All analyses were conducted using R 3.6.1 (R Core Team 2019) and its metafor package.

## Data availability

Data supporting the analyses and results of this study are available in the Zenodo repository, https://doi.org/10.5281/zenodo.4452745.

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## Acknowledgements

We thank Denise Reed for sharing data. We are grateful to the following for discussions and comments: Qiang He, Olivier Gourgue, Xiaohe Zhang, Huan Mi, Song Jin, Yepei Chen, and Xiaojun She. We also thank the China Scholarship Council. This work was supported financially by the Key Project of the National Natural Science Foundation of China (U1901212, 51639001) and the Fund for Innovative Research Group of the National Natural Science Foundation of China (51721093). S.F. was supported by NSF grants DEB-1832221 to the Virginia Coast Reserve Long-Term Ecological Research project and OCE-1637630 to the Plum Island Ecosystems Long-Term Ecological Research project.

## Author information

Authors

### Contributions

Z.Z.L., S.F., and B.S.C. designed the study; Z.Z.L. collected and analyzed the data; Z.Z.L. and S.F. created the figures; Z.Z.L., S.F., and B.S.C. discussed the results; all authors wrote and reviewed the manuscript.

### Corresponding author

Correspondence to Baoshan Cui.

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### Competing interests

The authors declare no competing interests.

Peer review information Primary handling editors: Adam Switzer, Heike Langenberg

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Reprints and Permissions

Liu, Z., Fagherazzi, S. & Cui, B. Success of coastal wetlands restoration is driven by sediment availability. Commun Earth Environ 2, 44 (2021). https://doi.org/10.1038/s43247-021-00117-7

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• Sérgio A. Netto

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text:
# Diffusivity of Water versus Sarin (Nerve Agent) in Air at 10 Degrees Celsius (50 Degrees Fahrenheit) and 1 Atmosphere

Diffusivity of Water versus Sarin in Air at 10 Degrees Celsius (50 Degrees Fahrenheit) and 1 Atmosphere[see bottom of post]

1991 Gulf War veterans are suffering from 1991 Gulf War Illness[3;References]. Scientific research suggests the combination of experimental medication, pyridostigmine bromide as an example, over use of pesticides, chemical weapon-sarin as an example-destruction at plants and football sized bunkers, oil fires, etc as the potential cause[6-9].

Dr. Robert Haley, MD, UT SouthWestern Medical Center, and Intelligence Analyst James Tuite have reported how 1991 Gulf War veterans might have been contaminated with chemical weapons prior to the ground war, “Desert Storm”[9]. In fact, their work provides data proving that sophisticated equipment detected chemical weapons in Saudi Arabia prior to the ground war[9a]. It is also hypothesized that the “toxic cocktail” has caused autonomic dysfunction, nerve death, and brain death[9-14].

As a 1991 Gulf War veteran, I have been affected. I am also a chemical engineer with a degree in biological sciences. Like most educated, I have lost much of my knowledge in chemical engineering and biological sciences, but I can, if I find a good example, still “plug and chug” by using “tested and trusted” equations, which is advised anyhow. 🙂 Here, I compare the diffusivity of sarin vapor and water vapor in air by using Chapman and Enskog equation with Brokaw relations for polar gases correction. I have shown that the equation can be used when considering the diffusivity of polar in a non-polar matrix[19]. After performing the latter calculation, I noticed that reference [1] also suggests Brokaw relations to be used for diffusivity of one polar gas molecule in a non-polar matrix[1].

I will be comparing the diffusivity of polar sarin = A in non-polar air = B at 10$\textdegree$C and 1 atmosphere. I chose 10$\textdegree$C because I discovered data, possibly experimental, that stated that 90% volume of 1 mm sarin drop on a non-absorbable surface at 10$\textdegree$C evaporated in 0.24 hours[17].

Equations

Chapman and Enskog Equation[1]. Reference [1] reports that this equation has a “Average absolute error” of 7.9% when used without Brokaw relations. The range is from 0% to 25%. The authors[1] did not provide an average for Browkaw relations but do provide specific absolute error values. When I averaged the Brokaw values[1], I obtained a 10.9% average absolute error with a range from 0% to 33%.

Chapman and Enskog Equation[1]

$D_{AB} = \frac{3}{16} \frac{(\frac{4 \pi \kappa T}{M_{AB}})^{1/2}}{n \pi \sigma_{AB}^2 \Omega_D} f_D$

Neufield, et al. Equation

$\Omega_D = \frac{A}{(T^*)^B} + \frac{C}{\exp{((D)(T^*))}} + \frac{E}{\exp{((F)(T^*))}} + \frac{G}{((H)(T^*))}$

Polar Gases: Brokaw Relations

$\Omega_D(Neufield) + \frac{0.19 \delta_{AB}^2}{T^*}$

$T^* = \frac{\kappa T}{\epsilon_{AB}}$

$\delta = \frac{1.94x10^3 \mu_p^2}{V_bT_b}$

$\mu_p = dipole \ moment, \ debyes$

$V_b = liquid \ molar \ volume \ at \ the \ normal \ boiling \ point, \ \frac{cm^3}{mol}$

$T_b = normal \ boiling \ point \ (1 \ atm), K$

$\frac{epsilon}{\kappa} = 1.18(1 + 1.3\delta^2)T_b$

$\sigma = (\frac{1.585V_b}{1 + 1.3 \delta^2})^{1/3}$

$\delta_{AB} = (\delta_A \delta_B)^{1/2}$

$\frac{\epsilon_{AB}}{\kappa} = (\frac{\epsilon_A}{\kappa} \frac{\epsilon_B}{\kappa})^{1/2}$

$\sigma_{AB} = (\sigma_A \sigma_B)^{1/2}$

When $f_D$ is chosen as unity and “n” is expressed by the ideal-gas law, the Chapman-Enskog Equation

$D_{AB} = \frac{0.00266 T^{3/2}}{PM_{AB}^{1/2} \sigma_{AB}^2 \Omega_D}$

Brokaw Diffusivity: Water in Air at 10$\textdegree$C and 1 Atmosphere

Molecular Weight

Water:

$M_A = M_{H_2O} = 2(MW_H) + 1(MW_O) = 2(1.008) + 1(16.00) = 18 \frac{g}{mol}$

Air: 1 mole basis

$21\% \frac{molO_2}{mol} \ O_2 \ and \ 79\% \frac{molN_2}{mol}\ N_2$

$Moles \ O_2 = 0.21 \frac{molO_2}{mole}(1 \ mol) = 0.21 \ molO_2; Moles \ N_2 = 0.79 \frac{molN_2}{mol} (1 \ mol) = 0.79 \ molN_2$

Grams oxygen:

$0.21 \ molO_2(MW_{O_2}) = 0.21 molO_2(\frac{32 \ grams \ O_2}{mol \ O_2}) = 6.72 \ grams \ O_2$

Grams nitrogen:

$0.79 \ molN_2(MW_{N_2}) = 0.21 molO_2(\frac{ 28 \ grams \ N_2}{mol \ O_2}) = 22.12 \ grams \ N_2$

Air: $M_B = M_{air} = \frac{(6.72 + 22.12)}{1mol} = 28.8 \frac{g}{mol}$

$M_{AB} = 2[\frac{1}{M_A} + \frac{1}{M_B}]^{-1} = 2[\frac{1}{18} + \frac{1}{28.8}]^{-1} = 22.2$

Need: $\sigma; \delta; \Omega_D$

Note: I will only be calculating a delta value for water because air is non-polar[1;19].

$\delta_A = \delta_{H_2O} = \frac{1.94x10^3 \mu_p^2}{V_bT_b}$

From [16]: $V_b \frac{cm^3}{mol} = 18.045 \frac{cm^3}{mol}$

From [20]: $\mu_{p_{H_2O}} = 1.855$

$T_b = 373 K$

$\delta_{A_{H_2O}} = \frac{1.94x10^3(1.855)^2}{(18.045)(373)} = \frac{6.68x10^3}{6.73x10^3} = 0.992$

$\frac{\epsilon_{A}}{\kappa} = 1.18(1 + 1.3 \delta_{A}^2)T_b = 1.18(1 + 1.3(0.992)^2)373 K = 1003 K$

$\sigma_A = (\frac{1.585V_b}{1 + 1.3\delta_A^2})^{1/3} = (\frac{1.585 (18.045)}{1 + 1.3(0.992)^2})^{1/3} = (12.55)^{1/3} = 2.32 \AA$

Need $T^*$ to calculate $\Omega_D$

$T^* = \frac{\kappa T}{\epsilon_{AB}}$

$\frac{\epsilon_{AB}}{\kappa} = (\frac{\epsilon_A}{\kappa} \frac{\epsilon_B}{\kappa})^{1/2}$

Water: $\frac{\epsilon_A}{\kappa} = 1003 K; Air[1, Appendix B]:78.6 K$

$\frac{\epsilon_{AB}}{\kappa} = \sqrt{\frac{\epsilon_A}{\kappa} \frac{\epsilon_B}{\kappa}} = \sqrt{(1003 K)(78.6 K)} = 280.8 K$

$T^* = \frac{\kappa T}{\epsilon_{AB}}$

$\frac{\epsilon_{AB}}{\kappa T} = \frac{280.8 K}{283 K} = 0.992$

$T^* = \frac{\kappa T}{\epsilon_{AB}} = \frac{1}{0.992} = 1.01$

Neufield, et al.:

$\Omega_D = \frac{A}{(T^*)^B} + \frac{C}{\exp{((D)(T^*))}} + \frac{E}{\exp{((F)(T^*))}} + \frac{G}{\exp((H)(T^*))} =$

$\Omega_D = \frac{1.06036}{1.01^{0.15610}} + \frac{0.19300}{\exp{((0.47635)(1.01))}} + \frac{1.03587}{\exp{((1.52996)(1.01))}} + \frac{1.76474}{\exp{((3.89411)(1.01))}} =$

$\Omega_D = 1.43$

$\Omega_D = \Omega_D(Neufield) + \frac{0.19 \delta_{AB}^2}{T^*}$ changed to $\Omega_D(Neufield) + \frac{0.19 \delta_A^2}{T^*}$

$\Omega_D = 1.43 + \frac{0.19(0.992)^2}{1.01} = 1.62$

Need $\sigma_{AB} = \sqrt{\sigma_A \sigma_B}$

Water: 2.32 $\AA$; Air (Appendix B[1]): 3.711 $\AA$

$\sigma_{AB} = \sqrt{\sigma_A \sigma_B} = \sqrt{(2.32)(3.711)} = 2.93 \AA$

Diffusivity: Polar water in non-polar air at 10$\textdegree$C and 1 atmosphere

$D_{AB} = \frac{0.00266 T^{3/2}}{PM_{AB}^{1/2} \sigma_{AB}^2 \Omega_D} = \frac{0.00266 (283)^{3/2}}{1 (22.2)^{1/2} (2.93)^2 (1.62)} = \frac{12.66}{65.53} =$

$D_{AB} = 0.193 \frac{cm^2}{s}$

Brokaw Diffusivity of Sarin in Air at 10$\textdegree$C and 1 Atmosphere

Molecular Weight

Sarin, $C_4H_{10}FO_2P$:

$M_A = M_{C_4H_{10}FO_2P} = 4(MW_C) + 10(MW_H) + 1(MW_F) + 2(MW_O) + 1(MW_P) =$

$M_{C_4H_{10}FO_2P} = 4(12.01) + 10(1.008) + 1(19.00) + 2(16.00) + 1(30.97) = 140.1 \frac{g}{mol}$

Air: 1 mole basis

$21\% \frac{molO_2}{mol} \ and \ 79\% \frac{molN_2}{mol}$

0.21 $\frac{molO_2}{mol}$(1 mol) = 0.21 mol oxygen gas; 0.79 $\frac{molN_2}{mol}$(1 mol) = 0.79 mol nitrogen gas

Grams oxygen:

$0.21 (molO_2)(32 \frac{gO_2}{molO_2}) = 6.72 grams \ O_2$

Grams nitrogen:

$0.79 (molN_2)(28 \frac{gN_2}{molN_2}) = 22.1 grams \ N_2$

Air: $M_B = M_{air} = \frac{(6.72 + 22.12)}{1 mol} = 28.8 \frac{g}{mol}$

$M_{AB} = 2[\frac{1}{140.1} + \frac{1}{29.0}]^{-1} = 48.1$

Need: $\delta; \sigma; \Omega_D$

Note: I will only be calculating the delta value for the polar gas sarin because air is non-polar[1;19].

$\Omega_D = \Omega_D(Neufield) + \frac{0.19 \delta_{AB}^2}{T^*}$ changed to $\Omega_D(Neufield) + \frac{0.19 \delta_A^2}{T^*}$

$T^* = \frac{\kappa T}{\epsilon_{AB}}$

$\frac{\epsilon_i}{\kappa} = 1.18(1 + 1.3\delta_i^2)T_b$

Sarin: $\frac{\epsilon_A}{\kappa} = 1.18(1 + 1.3(0.418)^2)(420) = 608.2 K$

$\frac{\epsilon_{AB}}{\kappa} = (\frac{\epsilon_A}{\kappa} \frac{\epsilon_B}{\kappa})^{1/2}$

Sarin: $\frac{\epsilon_A}{\kappa} = 608.2 K$

Air[Appendix B;1]: $\frac{\epsilon_B}{\kappa} = 78.6 K$

$\frac{\epsilon_{AB}}{\kappa} = \sqrt{\frac{\epsilon_A}{\kappa} \frac{\epsilon_B}{\kappa}} = \sqrt{(608.2)(78.6)} = 216.1 K$

$T^* = \frac{\kappa T}{\epsilon_{AB}}$

$\frac{\epsilon_{AB}}{\kappa T} = \frac{216.1}{283} = 0.764$

$T^* = \frac{\kappa T}{\epsilon_{AB}} = \frac{1}{0.764} =1.31$

$\delta_A = \frac{1.94x10^3 \mu_p^2}{V_bT_b}$

$\mu_p =$ dipole moment, debyes

$V_b =$ liquid molar volume at the normal boiling point, $\frac{cm^3}{mol}$

$T_b =$ normal boiling point (1 atm), K

Sarin[18;16a]: $\delta_A = \frac{1.94x10^3(3.44)^2}{(130.9)(420)} = 0.418$

$\sigma_i = (\frac{1.585V_b}{1 + 1.3\delta_i^2})^{1/3}$

Sarin[16a]: $\sigma_A = (\frac{1.585(130.9)}{1 + 1.3(0.418)^2})^{1/3} = 5.5 \AA$

$\sigma_{AB} = (\sigma_A \sigma_B)^{1/2}$

Sarin: $\sigma_A = 5.5 \AA$

Air[Appendix B;1}: $\sigma_B = 3.711 \AA$

$\sigma_{AB} = \sqrt{(\sigma_A)(\sigma_B)}= \sqrt{(5.5)(3.711)} = 4.52 \AA$

$\Omega_D = \frac{A}{(T^*)^B} + \frac{C}{\exp{((D)(T^*))}} + \frac{E}{\exp{((F)(T^*))}} + \frac{G}{\exp{((H)(T^*))}} =$

$\Omega_D = \frac{1.06036}{(1.31)^{0.15610}} + \frac{0.19300}{\exp{((0.47635)(1.31))}} + \frac{1.03587}{\exp{((1.52996)(1.31))}} + \frac{1.76474}{\exp{((3.89411)(1.31))}} =$

$\Omega_D = 1.24$

$\Omega_D = \Omega_D(Neufield) + \frac{0.19 \delta_A^2}{T^*} = 1.24 + \frac{0.19(0.418)^2}{1.31} = 1.27$

Chapman-Enskog equation after polar correction

Diffusivity of Sarin in Air:

$D_{AB} = \frac{0.00266T^{3/2}}{PM_{AB}^{1/2}\sigma_{AB}^2 \Omega_D} = \frac{0.00266(283)^{3/2}}{1(48.1)^{1/2}(4.52)^2(1.27)} = \frac{12.66}{180.0} = 0.070 \frac{cm^2}{s}$

Diffusivity Comparison in Air: Water Versus Sarin in Descending Order

Water: $D_{AB} = 0.193 \frac{cm^2}{sec}$

Sarin: $D_{AB} = 0.070 \frac{cm^2}{sec}$

Diffusivity Ratio: $\frac{Water}{Sarin} = \frac{0.193}{0.070} = 2.74$

References:

[1] Poling, Bruce E.; Prausnitz, John M.; O’Connell, John P. (2001) The Properties of Gases and Liquids, Fifth Edition. New York: Mcgraw-Hill.

[2] Welty, James R.; Wicks, Charles E.; Wilson, Robert E. (1984) Fundamentals of Momentum, Heat, and Mass Transfer, third edition. New York: John Wiley & Sons.

[3] Harding, Byron. 1991 Gulf War Illnesses and Differing Hypotheses: Nerve and Brain Death Versus Stress, December 2012. gather.com[online] 2012. Available from: http://www.gather.com/viewArticle.action?articleId=281474981824775

[4] Removed

[4a] Removed

[6] National Academies Press. Institute of Medicine. Committee on Gulf War and Health: Health Effects of Serving in the Gulf War, Update 2009. Board on Health of Select Populations. Gulf War and Health, Volume 8. nap.edu[online]. 2010. pp. 320. Available from: http://www.nap.edu/catalog.php?record_id=12835 ISBN-10: 0-309-14921-5; ISBN-13: 978-0-309-14921-1

[7] Research Advisory Committee on Gulf War Veterans’ Illnesses. Gulf War Illness and Health of Gulf War Veterans. Scientific Findings and Recommendations, 2008. va.gov[online]. 2012. Available from: http://www.va.gov/RAC-GWVI/docs/Committee_Documents/GWIandHealthofGWVeterans_RAC-GWVIReport_2008.pdf

[8] Research Advisory Committee on Gulf War Veterans’ Illnesses. Research Advisory Committee on Gulf War Veterans’ Illnesses Findings and Recommendation, June 2012. va.gov[online]. 2012. Available from: http://www.va.gov/RAC-GWVI/docs/Committee_Documents/CommitteeDocJune2012.pdf

[9] Kennedy, Kelly. Study: Wind blew deadly gas to U.S. troops in Gulf War, December 2012. ustoday.com[online]. 2012. Available from: http://www.usatoday.com/story/news/world/2012/12/13/sarin-gas-gulf-war-veterans/1766835/

[9a] Haley, Robert W.; Tuite, James J. Meteorological and Intelligence Evidence of Long-Distance Transit of Chemical Weapons Fallout from Bombing Early in the 1991 Persian Gulf War, December 2012. karger.com[online]. 2012. vol. 40. pp. 160-177. Available from: http://content.karger.com/ProdukteDB/produkte.asp?Aktion=ShowFulltext&ArtikelNr=345123&Ausgabe=257603&ProduktNr=224263 DOI: 10.1159/000345123

[9b] Haley, Robert W.; Tuite, James J. Epidemiologic Evidence of Health Effects from Long-Distance Transit of Chemical Weapons Fallout from Bombing Early in the 1991 Persian Gulf War, December 2012. karger.com[online]. vol. 40. pp. 178-189. Available from: http://content.karger.com/ProdukteDB/produkte.asp?Aktion=ShowFulltext&ArtikelNr=345124&Ausgabe=257603&ProduktNr=224263 DOI: 10.1159/000345124

[10] Oswal, DP; Garrett, TL; Morris, M; Kucot, JB. Low-dose sarin exposure produces long term changes in brain neurochemistry of mice. Neurochem Res[online]. 2013. vol. 1. pp. 108-116. Available from: http://www.ncbi.nlm.nih.gov/pubmed/23054072 doi: 10.1007/s11064-012-0896-9

[11] Shewale, SV.; Anstadt, MP; Horenziak, M; Izu, B.; Morgan, EE.; Lucot, JB.; Morris, M. Sarin causes autonomic imbalance and cardiomyopathy: an important issue for military and civilian health, July 2012. J. Cardiovasc Pharmacol.[online]. 2012. vol 60(1). pp. 76-87. Available from: http://www.ncbi.nlm.nih.gov/pubmed/22549449 doi: 10.1097/FJC.0b013e3182580b75

[12] DTIC. Online Information for the Defense Community.Chan, Victor T; Soto, Armando; Wagner, Jessica A; Watts, Brandy S.; Walters, Amy D.; Hill, Tiffany M. Mechanisms of Organophosphates (OP) Injury: Sarin-Induced Hippocampal Gene Expression Changes and Pathway Perturbation, Jan 2012. dtic.mil[online]. 2012. Available from: http://www.dtic.mil/docs/citations/ADA560343

[13] Medical News Today. Low-Level Exposure to Organophosphate Pesticides Damage Brain and Nervous System, December 2012. medicalnewstoday.com[online]. 2012. Available from: http://www.medicalnewstoday.com/releases/253534.php

[14] Fulco, Carolyn E; Liverman, Catharyn T.; Sox, Harold C. National Academy Press. Committee on Health Effects Associated with Exposures During the Gulf War. Gulf War and Health: Volume 1. Depleted Uranium, Sarin, Pysidostigmine Bromide, Vaccines, 2000. Effects of Long-Term Exposure to Organophosphate Pesticides in Humans. nap.edu[online]. 2012. Available from: http://www.nap.edu/openbook.php?record_id=9953&page=R1

[15] NCBI.PubChem. Sarin-Compound Summary (CID 7871). pubmed.ncbi.nlm.nih.gov[online]. 2012. Available from: http://pubchem.ncbi.nlm.nih.gov/summary/summary.cgi?cid=7871

[16] ChemSpider. The free chemical database. Water. chemspider.com[online]. 2013. Available from: http://www.chemspider.com/Chemical-Structure.937.html?rid=01a81689-c122-434f-a0a1-b4e6e3ca8109

[16a] ChemSpider. The free chemical database. Sarin (isopropyl methylphosphonofluoridate). chemspider.com[online]. 2013. Available from: http://www.chemspider.com/Chemical-Structure.7583.html?rid=8885b92c-43db-4dbf-a9fd-280d32df0450

[17] US National Library of Medicine. WISER: Wireless Information System for Emergencey Responders. Sarin, CAS RN: 107-44-8. Volatilization. webwiser.nlm.nih.gov[online]. 2012. Available from: http://webwiser.nlm.nih.gov/getSubstanceData.do;jsessionid=E6C28B95977867F872631D36CDD61D42?substanceID=151&displaySubstanceName=Sarin&UNNAID=&STCCID=&selectedDataMenuItemID=81

[18] Lee, Ming-Tsung; Vishnyakov, Aleksey; Gor, Gennady Yo.; Neimark, Alexander V. Interactions of Phosphororganic Agents with Water and Components of Polyelectrolyte Membranes, October 2011. J. Physical Chemistry[online]. 2012. Available from: http://www.princeton.edu/~ggor/Gor_JPCB_2011.pdf

[19] Harding, Byron. Chapman and Enskog Versus Hirschfelder Equation when Compared to Experimental Value at 25 Degree C and 1 Atm, and Non-Polar Versus Brokaw Polar Method, January 2013. Available from: https://chrisbharding.wordpress.com/2013/01/04/chapman-and-enskog-versus-hirschfelder-equation-and-compared-to-experimental-value-at-25c-and-1-atm/

[20] Gregory, J.K.; Clary, D.C.; Liu, K.; Brown, M.G.; Saykally, R.J. The Water Dipole Moment in Water Clusters, February 1997. science[online]. vol. 275. pp. 814. Available from: http://www.cchem.berkeley.edu/rjsgrp/publications/papers/1997/187_gregory_1997.pdf

# Temporary Divergence: Diffusivity: That Smell-Methyl Mercaptan (Methanethiol), Odorless Natural Gas, Odorless Propane, and Even Flatulence

Temporary Divergence: Diffusivity: That Smell-Methyl Mercaptan (Methanethiol), Odorless Natural Gas, Odorless Propane, and Even Flatulence

Lynyrd Skynyrd. That Smell. youtube.com[online]. 2013. Available from: http://youtu.be/ZDB-yswOrzc

Diffusivity of Methyl Mercaptan Versus Methane and Propane

Methyl mercaptan[3-6], “methanethiol”, is the byproduct of many natural processes. Flatulence is one example[7]. Because of its odor threshold, 1 ppb has been reported[4], methanethiol is also added to odorless natural gas, methane, and odorless propane for detection purposes. Apparently, it is used as a communication warning system in mining operations as well[4].

In this blog post, I will be comparing the diffusivity of the polar chemical methanethiol to the non-polar chemicals methane and propane in air. I have heard reports that the diffusivity of methanethiol is significantly greater than methane and propane. See bottom of post for diffusivities.Since reference[1] has tabular values for methane and propane in appendix B, I will use the tabular values and the Chapman-Enskog equation to calculate diffusivity values for methane and propane. For methanethiol, I will use Fuller, et al equation and tabular values for the atoms making up methanethiol, $CH_3-SH$

Chapman-Enskog Equation. From reference[1], the average absolute error of this “theoretical equation” is 7.9%

$D_{AB} = \frac{3}{16} \frac{(\frac{4 \pi \kappa T}{M_{AB}})^{1/2}}{n \pi \sigma_{AB}^2 \Omega_D} f_D$

If $f_D$ is chosen as unity and “n” expressed by ideal-gas law

$D_{AB} = \frac{0.00266 T^{3/2}}{P M_{AB}^{1/2} \sigma_{AB}^2 \Omega_D}$

For Non-polar gases: Methane and Propane

Methane $CH_4$ in air at 25$\textdegree$C and 1 atmosphere (atm)

$M_{AB} = 2[\frac{1}{M_A} + \frac{1}{M_B}]^{-1}$

$M_A = M_{CH_4} = 1(MW_C) + 4(MW_H) = 1(12.01) + 4(1.008) = 16.0 \frac{g}{mol}$

$M_B = M_{air} = 29.0 \frac{g}{mole}$

$M_{AB} = 2[\frac{1}{M_A} + \frac{1}{M_B}]^{-1} = 2[\frac{1}{16.0} + \frac{1}{29.0}]^{-1} = 20.6$

Need $\Omega_D$

Neufield, et al.: $\Omega_D = \frac{A}{(T^*)^B} + \frac{C}{\exp{((D)(T^*))}} + \frac{E}{\exp{((F)(T^*))}} + \frac{G}{\exp{((H)(T^*))}}$

$T^* = \frac{\kappa T}{\epsilon_{AB}}$

From appendix B[1]

Methane: $\sigma = 3.758 \AA; \frac{\epsilon_A}{\kappa} = 148.6 K$

Air: $\sigma = 3.711 \AA; \frac{\epsilon_B}{\kappa} = 78.6 K$

$\sigma_{AB} = \frac{\sigma_A + \sigma_B}{2} = \frac{3.758 + 3.711}{2} = 3.735 \AA$

$\frac{\epsilon_{AB}}{\kappa} = \sqrt{(148.6)(78.6)} = 108.1 K$

$\frac{\epsilon_{AB}}{\kappa T} = \frac{108.1 K}{298 K} = 0.363$

$T^* = \frac{\kappa T}{\epsilon_{AB}} = \frac{1}{0.363} = 2.76$

$\Omega_D = \frac{A}{(T^*)^B} + \frac{C}{\exp{((D)(T^*))}} + \frac{E}{\exp{((F)(T^*))}} + \frac{G}{\exp{((H)(T^*))}} =$

$\Omega_D =\frac{1.06036}{(2.76)^{0.15610}} + \frac{0.19300}{\exp{((0.47635)(2.76))}} + \frac{1.03587}{\exp{((1.52996)(2.76))}} + \frac{1.76474}{\exp{((3.89411)(2.76))}} =$

$\Omega_D = 0.972$

Diffusivity of Methane in Air:

$D_{AB} = \frac{0.00266 T^{3/2}}{P M_{AB}^{1/2} \sigma_{AB}^2 \Omega_D} = \frac{0.00266 (298)^{3/2}}{1(20.6)^{1/2} (3.735)^2 (0.972)} = \frac{13.68}{61.54} = 0.222 \frac{cm^2}{s}$

Propane $CH_3CH_2CH_3$ in air at 25$\textdegree$C and 1 atmosphere (atm)

$M_{AB} = 2[\frac{1}{M_A} + \frac{1}{M_B}]^{-1}$

$M_A = M_{CH_3CH_2CH_3} = 3(MW_C) + 8(MW_H) = 3(12.01) + 8(1.008) = 44.1 \frac{g}{mol}$

$M_B = M_{air} = 29 \frac{g}{mol}$

$M_{AB} = 2[\frac{1}{44.1} + \frac{1}{29}]^{-1} = 35.0$

Need $\Omega_D$

Neufield, et al.: $\Omega_D = \frac{A}{(T^*)^B} + \frac{C}{\exp{((D)(T^*))}} + \frac{E}{\exp{((F)(T^*))}} + \frac{G}{\exp{((H)(T^*))}}$

$T^* = \frac{\kappa T}{\epsilon_{AB}}$

From appendix B[1]

Propane: $\sigma_A = 5.118 \AA; \frac{\epsilon_A}{\kappa} = 237.1 K$

Air: $\sigma_B = 3.711 \AA; \frac{\epsilon_B}{\kappa} = 78.6 K$

$\sigma_{AB} = \frac{\sigma_A + \sigma_B}{2} = \frac{5.118 + 3.711}{2} = 4.42 \AA$

$\frac{\epsilon_{AB}}{\kappa} = \sqrt{(\frac{\epsilon_A}{\kappa})(\frac{\epsilon_B}{\kappa})} = \sqrt{(237.1)(78.6)} = 136.5 K$

$\frac{\epsilon_{AB}}{\kappa T} = \frac{136.5}{298} = 0.458$

$T^* = \frac{\kappa T}{\epsilon_{AB}} = \frac{1}{0.458} = 2.18$

$\Omega_D = \frac{A}{(T^*)^B} + \frac{C}{\exp{((D)(T^*))}} + \frac{E}{\exp{((F)(T^*))}} + \frac{G}{\exp{((H)(T^*))}} =$

$\Omega_D = \frac{1.06036}{(2.18)^{0.15610}} + \frac{0.19300}{\exp{((0.47635)(2.18))}} + \frac{1.03587}{\exp{((1.52996)(2.18))}} + \frac{1.76474}{\exp{((3.89411)(2.18))}} =$

$\Omega_D = 1.05$

Diffusivity of Propane in Air:

$D_{AB} = \frac{0.00266T^{3/2}}{PM_{AB}^{1/2} \sigma_{AB}^2 \Omega_D} = \frac{0.00266(298)^{3/2}}{1 (35.0)^{1/2} (4.42)^2 (1.05)} = \frac{13.68}{121.36} =$

$D_{AB} = 0.113 \frac{cm^2}{s}$

For polar molecule $CH_3-SH$, will use Fuller, et al. equation. From reference[1], the absolute relative error of this equation is 5.4%. Authors report an average absolute error of about 4% when using

Fuller, et al.

$D_{AB} = \frac{0.00143 T^{1.75}}{PM_{AB}^{1/2}[(\sum \nu)_A^{1/3} + (\sum \nu)_B^{1/3}]^2}$

T = 25$\textdegree$C = 298 K; P = 1 atm

$M_{AB} = 2[\frac{1}{M_A} + \frac{1}{M_B}]^{-1}$

$M_A = M_{CH_3SH} = 1(C) + 4(H) + 1(S) = 1(12.01) + 4(1.008) + 1(32.07) = 48.11 \frac{g}{mol}$

$M_B = M_{air} = 29 \frac{g}{mol}$

$M_{AB} = 2[\frac{1}{48.11} + \frac{1}{29}]^{-1} = 18.1$

Summation of “Atomic and Structural Diffusion Volume Increments from table 11-1[1]

$(\sum \nu)_A = (\sum \nu)_{CH_3SH} = 1 (C) + 4(H) + 1(S) = 1(15.9) + 4(2.31) + 1(22.9) = 40.04$

$(\sum \nu)_B = (\sum \nu)_{air} = 19.7$

$D_{AB} = \frac{0.00143T^{1.75}}{PM_{AB}^{1/2}[(\sum \nu)_A^{1/3} + (\sum \nu)_B^{1/3}]^2} = \frac{0.00143(298)^{1.75}}{1(18.1)^{1/2}[(40.0)_A^{1/3} + (19.7)_B^{1/3}]^2} =$

Diffusivity of methanethiol in air

$D_{AB} = \frac{30.56}{159.38} = 0.192 \frac{cm^2}{s}$

Diffusivities in air in decreasing order

Methane: $D_{AB} = 0.222 \frac{cm^2}{s}$

Methanethiol: $D_{AB} = 0.192 \frac{cm^2}{s}$

Propane: $D_{AB} = 0.113 \frac{cm^2}{s}$

At a detection threshold of 1 part per billion (ppb) and the above diffusivities, one might detect methanethiol prior to experiencing propane. In truth, there is an equation that takes “mixture” into account but I do not know the percent mixture of each component[2].

Equation for mixture

$D_{1-mixture} = \frac{1}{\frac{z_2}{D_{1-2}} + \frac{z_3}{D_{1-3}} + .... + \frac{z_n}{D_{1-n}}}$

$z_n$ is the mole fraction of component “n” in the gas mixture evaluated on a component-1-free basis

$z_2 = \frac{y_2}{y_2 + y_3 + ... + y_n}$

ihatemyhate. Friends Selection – Ross Flirts. youtube.com[online]. 2013. Available from: http://youtu.be/kH5JhYsfNMA

• Wait until 2nd attempt

References:

[1] Poling, Bruce E.; Prausnitz, John M.; O’Connell, John P. (2001) The Properties of Gases and Liquids, Fifth Edition. New York: Mcgraw-Hill.

[2] Welty, James R.; Wicks, Charles E.; Wilson, Robert E. (1984) Fundamentals of Momentum, Heat, and Mass Transfer, third edition. New York: John Wiley & Sons.

[3] ScienceBlogs. Molecule of the day. Methanethiol (They put that in, you know), March 2009. scienceblogs.com[online]. 2013. Available from: http://scienceblogs.com/moleculeoftheday/2009/03/18/methanethiol-they-put-that-in/

[4] Wikipedia. Methanethiol. Also known as methyl mercaptan. en.wikipedia.org[online]. 2013. Available from: http://en.wikipedia.org/wiki/Methanethiol

[5] NCBI.PubChem Substance. Methanethiol-Substance Summary (SID 3699). Also known as Methylmercaptan (CAS: 74-93-1). pubchem.ncbi.nlm.nih.gov[online]. 2013. Available from: http://pubchem.ncbi.nlm.nih.gov/summary/summary.cgi?sid=3699

[6] US National Institute of Standards and Technology. NIST. Methanethiol. webbook.nist.gov[online]. 2013. Available from: http://webbook.nist.gov/cgi/cbook.cgi?ID=74-93-1&Units=SI

[7] Wikipedia. Flatulence. en.wikipedia.org[online]. 2013. Available from: http://en.wikipedia.org/wiki/Flatulence

# Diffusivity: Chapman and Enskog Versus Hirschfelder Equation when Compared to Experimental Value at 25 Degree C and 1 Atm, and Non-polar Versus Brokaw Polar Method

Title: Diffusivity: Chapman and Enskog Versus Hirschfelder Equation when Compared to Experimental Value at 25$\textdegree$C and 1 Atm, and Non-polar Versus Brokaw Polar Method

Chapman and Enskog Equation

$D_{AB} = \frac{3}{16} \frac{(\frac{4 \pi \kappa T}{M_{AB}})^{1/2}}{n \pi \sigma_{AB}^{2} \Omega_D}f_D$

When $f_D$ is unity and n is expressed by the ideal gas law

$D_{AB} = \frac{0.0026 T^{3/2}}{PM_{AB}^{1/2} \sigma_{AB}^{2}\Omega_D}$

Hirschfelder, Bird, and Spotz Equation

$D_{AB} = \frac{0.001858 T^{3/2}[(\frac{1}{M_A}) + (\frac{1}{M_B})]^{1/2}}{P \sigma_{AB}^{2} \Omega_D}$

Non-polar Comparison

There are suggested correction factors for polar compounds. Since water is a polar compound, I will use these factors in a later comparison. I am doing a non-polar comparison because I was surprised with the closeness of the Hirschfelder equation previously when using non-polar factors.

Hirschfelder, Bird, and Spotz equation[2]

$D_{AB} = \frac{0.001858T^{3/2}[\frac{1}{M_A} + \frac{1}{M_B}]^{1/2}}{P \sigma_{AB}^{2} \Omega_D}$

$M_A = M_{H_2O} = 18 \frac{g}{mol}; M_B = M_{air} = 29 \frac{g}{mol}$

T = 298 K; P = 1 atm

Need $\sigma_{AB} \ and \ \Omega_D$. For comparative purposes, will use tabular values from [1].

Water: $\sigma_A = 2.641 \AA; \frac{\epsilon_A}{\kappa} = 809.1 K$

Air: $3.711 \AA; \frac{\epsilon_B}{\kappa} = 78.6 K$

$\sigma_{AB} = \frac{\sigma_A + \sigma_B}{2} = \frac{2.641 + 3.711}{2} = 3.18 \AA$

$\frac{\epsilon_{AB}}{\kappa} = \sqrt{\frac{\epsilon_A}{\kappa} \frac{\epsilon_B}{\kappa}} = \sqrt{(809.1)(78.6)} = 252.2 K$

Need $T^* = \frac{\kappa T}{\epsilon_{AB}}$ to calculate $\Omega_D$

Neufeld, et al.: $\Omega_D = \frac{A}{(T^*)^{B}} + \frac{C}{\exp{DT^*}} + \frac{E}{\exp{FT^*}} + \frac{G}{\exp{HT^*}}$

The constants

A;B;C;D;E;F;G;H will be placed in the Neufeld, et. al. equation

First calculate $T^*$

$\frac{\epsilon_{AB}}{\kappa T} = \frac{252.2 K}{298 K} = 0.846$

$T^{*} = \frac{\kappa T}{\epsilon_{AB}} = \frac{1}{0.846} = 1.18$

$\Omega_D = \frac{A}{(T*)^B} + \frac{C}{\exp{DT^*}} + \frac{E}{\exp{FT^*}} + \frac{G}{\exp{HT^*}} =$

$\frac{1.06036}{1.18^{0.15610}} + \frac{0.19300}{\exp{((0.47635)(1.18))}} + \frac{1.03587}{\exp{((1.52996)(1.18))}} + \frac{1.76474}{((3.89411)(1.18))} = \Omega_D$

$\Omega_D = 1.33$

Back to Hirschfelder equation

$D_{AB} = \frac{0.001858 T^{3/2}[\frac{1}{M_A} + \frac{1}{M_B}]^{1/2}}{P \sigma_{AB}^{2} \Omega_D} = \frac{0.001858 (298)^{3/2}[\frac{1}{18} + \frac{1}{29}]^{1/2}}{1 (3.18)^{2} 1.33} =$

$D_{AB} = \frac{2.87}{13.45} = 0.213 \frac{cm^2}{s}$

Compared to experimental value from [2] at 25$\textdegree$C and 1 atm

Diffusivity of water in air: $D_{AB} = 0.260 \frac{cm^2}{s}$

Chapman and Enskog equation

$D_{AB} = \frac{0.00266 T^{3/2}}{P M_{AB}^{1/2} \sigma_{AB}^2 \Omega_D}$

$M_{AB} = 2[\frac{1}{M_A} + \frac{1}{M_B}]^{-1} = 2[\frac{1}{18} + \frac{1}{29}]^{-1} = 22.21$

T = 298 K; P = 1 atm; $\sigma_{AB} = 3.18 \AA; \ \Omega_D = 1.33$

$D_{AB} = \frac{0.00266 (298)^{3/2}}{1 (22.21)^{1/2} (3.18)^2 (1.33)} = \frac{13.68}{63.38} =$

$D_{AB} = 0.216 \frac{cm^2}{s}$

Compared to Hirschfelder equation and experimental value

Hirschfelder[2]: $D_{AB} = 0.213 \frac{cm^2}{s}$

Percent Difference:

$\frac{Hirschfelder - Experimental}{Experimental} x 100 = \frac{0.213 - 0.260}{0.260} x 100 = -18.1\%$

Experimental: $D_{AB} = 0.260 \frac{cm^2}{s}$

Chapman[1]: $D_{AB} = 0.216 \frac{cm^2}{s}$

Percent Difference:

$\frac{Chapman - Experimental}{Experimental} x 100 = \frac{0.216 - 0.260}{0.260} x 100 = -16.9\%$

Polar Molecule Correction Comparison

Sadly, I have discovered that most empirical correlations lack sufficient data to estimate the diffusivity of many compounds. As an example, I, as a 1991 Gulf War veteran, desire to calculate the diffusivity of sarin in air. I have discovered that most empirical correlations do not take the phosphorus atom into consideration. Also, the Brokaw relationships for polar gases have correction equations that consider polar diffusing through polar.

In my analysis, I will first use the Brokaw method and only consider the polar molecule of water since air is non-polar. I will highlight the potential error of using this method by “error” when I use the correction equations. To be specific, $\Omega_{D_{H_2O}}$ will be calculated based on the new variable $\delta_{H_2O}$ instead of a $\delta_{AB}$ of two polar species. I hope to see if I can use the Brokaw method to calculate the diffusivity of sarin in air since the correction factors used in the Brokaw method include phosphorus and fluorine. I wish I could have found a phosphorus “diffusion volume increment”, $\nu$, but I could not find one. I did find a fluorine[1], Nitrogen, Sulfur, Iodine, etc and might use either Nitrogen or Sulfur in the Fuller, et al. equation as an estimate when I calculate the diffusivity of sarin in air.

Since both equations gave approximately the same value during the non-polar comparison, I will use the Chapman-Enskog equation. Once again, the temperature and pressure are 25$\textdegree$C and 1 atmosphere.

Chapman-Enskog equation[1]

$D_{AB} = \frac{0.00266 T^{3/2}}{PM_{AB}^{1/2}\sigma_{AB}^2 \Omega_D}$

T = 298 K; P = 1 atm

$M_{AB} = 2[\frac{1}{M_A} + \frac{1}{M_B}]^{-1} = 2[\frac{1}{18} + \frac{1}{29}]^{-1} = 22.21$

Brokaw Method

Neufield, et al. Relation:

$\Omega_D = \frac{A}{(T^*)^B} + \frac{C}{\exp{(DT^*)}} + \frac{E}{\exp{(FT^*)}} + \frac{G}{\exp{(HT^*)}}$

$T^* = \frac{\kappa T}{\epsilon_{AB}}$

$\Omega_D = \Omega_D(Neufield) + \frac{0.19 \delta_{AB}}{T^*}$

Possible error: Since air is non-polar, I will only be using the delta of water, $\delta_A$ in the above equation.

Changed equation: $\Omega_D = \Omega_D(Neufield) + \frac{0.19 \delta_A}{T^*}$

For Water (A)

$\frac{\epsilon_A}{\kappa} = 1.18(1 + 1.3\delta_A^2)T_b; T_b = normal \ boiling \ point (1 atm), K = 373 K$

$\delta_A = \frac{1.94 x 10^3 \mu_p^2}{V_b T_b}; V_b = Molar \ Volume \ at \ T_b; \mu_p = Dipole \ moment, D$

Calculation by Le Bas method[1]

$V_b = 2(H) + 1(O) = 2(3.7) + (7.4) = 14.8 \frac{cm^3}{mol}$

Percent Difference: $\frac{Calculated - Experimental}{Experimental} = \frac{14.8 - 18.8}{18.8} = -21.3\%$

Note: ChemSpider provides a “Molar Volume” of $18.045 cm^3$ for water. Although I assume the latter was calculated at normal boiling point, I am not certain. Still, the value is extremely close to the experimental value in reference [1], $18.8 \frac{cm^3}{mol}$. For this reason, I will use the Chemspider value for water. Why? ChemSpider also provides a Molar Volume value for Sarin. If ChemSpider responds by email that the value was not calculated at boiling point, I will reconsider. Still, the percent difference of ChemSpider for water Molar Volume when compared to experimental[1] is:

ChemSpider Percent Difference:

$\frac{ChemSpider - Experimental}{Experimental}x100 = \frac{18.045 - 18.8}{18.8}x 100 = -4.02\%$

Delta: $\delta_{A} = \frac{1.94x10^3 \mu_p}{V_bT_b}$

$\mu_p = dipole \ moment, \ debyes$

$V_b = liquid \ molar \ volume \ at \ normal \ boiling \ point, \ \frac{cm^3}{mole}$

$T_b = normal \ boiling \ point \ (1 atm), \ K$

$\delta_{A} = \frac{1.94 x 10^3(1.855)^2}{(18.045)(373)} = 0.992$

$\frac{\epsilon_A}{\kappa} = 1.18(1 + 1.3 \delta_A^2)T_b = 1.18(1 + 1.3(0.992)^2)373 = 1003 \ K$

$\sigma_A = (\frac{1.585V_b}{(1 + 1.3\delta_A^2)})^{1/3} = (\frac{1.585(18.045)}{(1 + 1.3(0.992)^2)})^{1/3} = 2.32 \AA$

Need: $T^* = \frac{\kappa T}{\epsilon_{AB}}$

Water: $\sigma_A = 2.32 \AA; \frac{\epsilon_A}{\kappa} = 1003 K$

From [1]: Air: $\sigma_B = 3.711 \AA; \frac{\epsilon_B}{\kappa} = 78.6 K$

$\frac{\epsilon_{AB}}{\kappa} = \sqrt{\frac{\epsilon_A}{\kappa}\frac{\epsilon_B}{\kappa}} = \sqrt{(1003.0)(78.6)} = 280.8$

$\frac{\epsilon_{AB}}{\kappa T} = \frac{280.8}{298.0} = 0.942$

$T^* = \frac{\kappa T}{\epsilon_{AB}} = \frac{1}{0.942} = 1.06$

Neufield, et al. Relation

$\Omega_D = \frac{A}{(T^*)^B} + \frac{C}{\exp{((D)(T^*))}} + \frac{E}{\exp{((F)(T^*))}} + \frac{G}{\exp{((H)(T^*))}} =$

$\Omega_D = \frac{1.06036}{(1.06)^{0.15610}} + \frac{0.19300}{\exp{((0.47635)(1.06))}} + \frac{1.03587}{\exp{((1.52996)(1.06))}} + \frac{1.76474}{\exp{((3.89411)(1.06))}} =$

$\Omega_D = 1.4$

Brokaw relation for polar molecules

$\Omega_D = \Omega_D(Neufield) + \frac{0.19 \delta_{AB}^2}{T^*}$

Changed for water only: $\Omega_D = \Omega_D (Neufield) + \frac{0.19 \delta_A^2}{T^*}$

$\Omega_D = 1.4 + \frac{(0.19)(0.992)^2}{1.06} = 1.57$

Check Chapman and Enskog equation

$D_{AB} = \frac{0.00266 T^{3/2}}{P M_{AB}^{1/2} \sigma_{AB}^2 \Omega_D}$

T = 298 K; P = 1 atm; $M_{AB} = 22.21 \ and \ \Omega_D = 1.57$

Need: $\sigma_{AB}^2$

From Brokaw relation

$\sigma_{AB} = \sqrt{\sigma_A \sigma_B} = \sqrt{(2.32)(3.711)} = 2.93 \AA$

Chapman and Enskog equation for diffusivity of polar water in non-polar air

$D_{AB} = \frac{0.00266 (298)^{3/2}}{1(22.21)(2.93)^2(1.57)} = \frac{13.68}{63.52} = 0.215 \frac{cm^2}{s}$

Non-polar versus polar comparisons

Using: $D_{AB} = \frac{0.00266 T^{3/2}}{PM_{AB}^{1/2} \sigma_{AB}^2 \Omega_D}$ and Brokaw relationships for polar corrections

Chapman and Enskog non-polar molecule[1]: $D_{AB} = 0.216 \frac{cm^2}{s}$

Chapman and Enskog with Brokaw relationships for polar molecules[1]: $D_{AB} = 0.215 \frac{cm^2}{s}$

Hirschfelder nonpolar[2]: $D_{AB} = 0.213 \frac{cm^2}{s}$

The experimental value[2]: $D_{AB} = 0.260 \frac{cm^2}{s}$

Note: All the ‘calculated” values are quite close. As such, I assume I can, when needed, use Chapman and Enskog and the Brokaw method to calculate the diffusivity of a polar molecule of sarin in non-polar air.

References

[1] Poling, Bruce E.; Prausnitz, John M.; O’Connell, John P. (2001) The Properties of Gases and Liquids, Fifth Edition. New York: Mcgraw-Hill.

[2] Welty, James R.; Wicks, Charles E.; Wilson, Robert E. (1984) Fundamentals of Momentum, Heat, and Mass Transfer, third edition. New York: John Wiley & Sons.

[3] Harding, Byron. Chapter 24: Fundamentals of Mass Transfer. Diffusivity of water in Air at 20 Degrees Celsius and 1 Atmosphere, December 2012. chrisbharding.wordpress.com[online]. 2012. Available from: https://chrisbharding.wordpress.com/2012/12/28/chapter-24-fundamentals-of-mass-transfer-diffusivity-of-water-in-air-at-25-degrees-celsius/

[4] ChemSpider. The free chemical database. chemspider.com[online]. 2013. Available from: http://www.chemspider.com/

[5] Gregory, J.K.; Clary, D.C.; Liu, K.; Brown, M.G.; Saykally, R.J. The Water Dipole Moment in Water Clusters, February 1997. science[online]. vol. 275. pp. 814. Available from: http://www.cchem.berkeley.edu/rjsgrp/publications/papers/1997/187_gregory_1997.pdf

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# Topological Manifolds: Definition and Properties

In this post, I would like to explore some basic properties of Topological manifolds.

As a pre-concept of Differential manifolds, the definition and properties of topological manifolds is quite important. In fact, we will emphasis on what kind of impact the locally Euclidean will make on the original topological space.
1. The Definition of Topological Manifolds Let’s first give the definition of Topological Manifolds.

1. $M$ is Haussdorff. I.e., for any points $p,q$, $p\neq q$, then there are neighbourhoods $U, V$ of $p$ and $q$ respectively, such that $U\cap V=\emptyset$.
2. For each $x\in M$, there is a neighbourhood $U$ of $x$ and an integer $n\geq0$, such that $U$ is homeomorphic to $\R^n$.
Then, we call that $M$ is a Topological Manifolds.

On the other hand, suppose that for each $x\in M$, there exists a neighborhood $U$ of $x$, then, by the definition of neighborhood, we must have an open set $V$, such that $x\in V\subset U$. Now, we need the easy verified facts: If $\phi$ is a homeomorphic between the topological space $M$ and $N$, and $U$ is a subset of $M$, then under the induced topology, $\phi$ is a homeomorphic between $U$ and $\phi(U)\subset N$. Apply this fact, Since $\phi$ is a homemorphic between $U$ and $\R^n$, and $V\subset U$ is open, we know that $\phi(V)$ is also open in $\R^n$. Noted that $\phi(x)\in\phi(V)$, we can find an open ball $W$, $\phi(x)\in W\subset \phi(V)$, use again that $\phi^{-1}$ is a homeomorphic between $\R^n$ and $U$, we know that $\phi^{-1}(W)$ is open in $U$. It is apparently that $\phi^{-1}(W)\subset V\subset U$, and $\phi$ is a homeomorphic between $\phi^{-1}(W)$ and $W$. Now by the well known fact that any open ball in $\R^n$ is homeomorphic to $\R^n$, from which we complete the proof.

\begin{align*}
A_+&=\set{(x,y)|x\geq0,y=1},\\
A_-&=\set{(x,y)|x\geq0,y=-1},\\
B&=\set{(x,y)|x< 0, y=0}.
\end{align*}
Define the topology as follows, for the neighborhood of points other than $(0,\pm1)$, we use the induced topology from $\R^2$; and added $N^\pm_\eps=\set{(x,\pm1)|0\leq x< \eps}\cup\set{(x,0)|-\eps\leq x< 0}$ as neighborhood of $(0,\pm1)$.

It is a directly verification that the above definition do defined a topology for $X$ and it is locally Euclidean. Note that $X$ can’t be Hausdorff, since $(0,\pm1)$ can’t be separated by two non-intersection open set.

2. The Properties of Topological Manifolds Now, we’ll turn to the properties of topological manifolds.

1. $M$ is locally connected, i.e., for every point $x\in M$, and any neighborhood $U$ of $x$, there is a open connected set $V$, such that $x\in V\subset U$;
2. Since $M$ is locally connected, then every connected component of $M$ is open;
3. If $M$ is connected, then it is connected by arc, i.e., for any two points $p,q\in M$, there exist an continuous map (called arc or path) $\phi:[0,1]\to M$, such that $\phi(0)=p, \phi(1)=q$;
4. $M$ is locally compact, i.e., for every point $x\in M$, and any neighborhood $U$ of $x$, there is a open set $V$ with $\overline V$ is compact, such that $x\in V\subset \overline V\subset U$;
5. Since every Hausdorff locally compact space is regular, so $M$ is regular. (Recall that a space is regular, if the point and closed set can be separated by their prospectively neighborhoods.)
The above property is not true for infinite dimensional space.
6. Manifold may not normal. (Recall that a space is normal if any two closed subset can be separated by their neighborhoods.)

Another important thing is about the second countability, we always need to assume that $M$ has a countable basis.

1. Each component of $M$ is $\sigma$-compact(also called compact at infinity), i.e., there exist a family of compact sets $\set{K_m}_{m=1}^\infty$, such that $K_{i-1}\subset K_i^\circ\subset K_{i+1}$ and $M=\cup_{m=1}^\infty K_m$;
2. Each component of $M$ is second-countable;
3. $M$ is metrizable;
4. $M$ is paracompact, i.e., each open covering has a locally finite refinement.
thus, any compact manifold is metrizable.


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# zbMATH — the first resource for mathematics

Optimal inequalities for generalized logarithmic, arithmetic, and geometric means. (English) Zbl 1187.26015
Summary: For $$p\in\mathbb R$$, the generalized logarithmic mean $$L_p(a,b)$$, arithmetic mean $$A(a,b)$$, and geometric mean $$G(a,b)$$ of two positive numbers $$a$$ and $$b$$ are defined by $$L_p(a,b)=a$$, for $$a=b$$, $$L_p(a,b)= [(b^{p+1}-a^{p+1})/((p+1)(b-a))]^{1/p}$$, for $$p\neq 0$$, $$p\neq-1$$, and $$a\neq b$$, $$L_p(a,b)= (1/e)(b^b/a^a)^{1/(b-a)}$$, for $$p=0$$, and $$a\neq b$$, $$L_p(a,b)= (b-a)/(\log b-\log a)$$, for $$p=-1$$, and $$a\neq b$$, $$A(a,b)= (a+b)/2$$, and $$G(a,b)= \sqrt{ab}$$, respectively. In this paper, we find the greatest value $$p$$ (or least value $$q$$, resp.) such that the inequality $$L_p(a,b)<\alpha A(a,b)+ (1-\alpha) G(a,b)$$ (or $$\alpha A(a,b)+ (1-\alpha)G(a,b)< L_q(a,b)$$, resp.) holds for $$\alpha\in(0,1/2)$$ (or $$\alpha\in(1/2,1)$$, resp.) and all $$a,b>0$$ with $$a\neq b$$.

##### MSC:
 2.6e+61 Means
Full Text:
##### References:
 [1] Alzer, H, Ungleichungen für mittelwerte, Archiv der Mathematik, 47, 422-426, (1986) · Zbl 0585.26014 [2] Alzer, H; Qiu, S-L, Inequalities for means in two variables, Archiv der Mathematik, 80, 201-215, (2003) · Zbl 1020.26011 [3] Burk, F, The geometric, logarithmic and arithmetic Mean inequality, The American Mathematical Monthly, 94, 527-528, (1987) · Zbl 0632.26008 [4] Janous, W, A note on generalized Heronian means, Mathematical Inequalities & Applications, 4, 369-375, (2001) · Zbl 1128.26302 [5] Leach, EB; Sholander, MC, Extended Mean values. II, Journal of Mathematical Analysis and Applications, 92, 207-223, (1983) · Zbl 0517.26007 [6] Sándor, J, On certain inequalities for means, Journal of Mathematical Analysis and Applications, 189, 602-606, (1995) · Zbl 0822.26014 [7] Sándor, J, On certain inequalities for means. II, Journal of Mathematical Analysis and Applications, 199, 629-635, (1996) · Zbl 0854.26013 [8] Sándor, J, On certain inequalities for means. III, Archiv der Mathematik, 76, 34-40, (2001) · Zbl 0976.26015 [9] Shi, M-Y; Chu, Y-M; Jiang, Y-P, Optimal inequalities among various means of two arguments, No. 2009, 10, (2009) · Zbl 1187.26017 [10] Carlson, BC, The logarithmic Mean, The American Mathematical Monthly, 79, 615-618, (1972) · Zbl 0241.33001 [11] Sándor, J, On the identric and logarithmic means, Aequationes Mathematicae, 40, 261-270, (1990) · Zbl 0717.26014 [12] Sándor, J, A note on some inequalities for means, Archiv der Mathematik, 56, 471-473, (1991) · Zbl 0693.26005 [13] Lin, TP, The power mean and the logarithmic Mean, The American Mathematical Monthly, 81, 879-883, (1974) · Zbl 0292.26015 [14] Pittenger, AO, Inequalities between arithmetic and logarithmic means, Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, 678-715, 15-18, (1981) · Zbl 0469.26009 [15] Imoru, CO, The power mean and the logarithmic Mean, International Journal of Mathematics and Mathematical Sciences, 5, 337-343, (1982) · Zbl 0483.26012 [16] Chen, Ch-P, The monotonicity of the ratio between generalized logarithmic means, Journal of Mathematical Analysis and Applications, 345, 86-89, (2008) · Zbl 1160.26012 [17] Li, X; Chen, Ch-P; Qi, F, Monotonicity result for generalized logarithmic means, Tamkang Journal of Mathematics, 38, 177-181, (2007) · Zbl 1132.26326 [18] Qi, F; Chen, Sh-X; Chen, Ch-P, Monotonicity of ratio between the generalized logarithmic means, Mathematical Inequalities & Applications, 10, 559-564, (2007) · Zbl 1127.26021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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## Airbus exploring hydrogen fuel cell propellor “pods” for aircraft propulsion

##### 03 January 2021

Airbus is conducting studies to determine how scalable a hydrogen fuel cell “pod” configuration, among others, could be to large commercial aircraft. The aviation industry has developed numerous configurations—twinjet, s-duct, winglets, contra-rotating propellers—over the last five decades that have enabled aircraft to fly higher, faster and longer. Now, Airbus engineers are unveiling a new configuration as part of the ZEROe program (earlier post) that could enable a passenger aircraft to fly farther than ever without emissions.

The innovative approach consists of six, eight-bladed hydrogen-fuel-cell-powered “pods” mounted beneath the aircraft wing.

The ‘pod’ configuration is essentially a distributed fuel cell propulsion system that delivers thrust to the aircraft via six propulsors arranged along the wing. Hydrogen fuel cells have very different design considerations, so we knew we had to come up with a unique approach.

—Matthieu Thomas, ZEROe Aircraft Lead Architect

Hydrogen fuel cell technology has yet to be scaled up to a passenger-size large commercial aircraft. Smaller experimental hydrogen aircraft, comprising up to 20 seats, can rely on a traditional fixed-wing configuration with two propellers. But more passenger capacity and longer range require another solution. This is why Airbus is studying a variety of configurations, including “pods,” to determine which option has the potential to scale up to larger aircraft.

This ‘pod’ configuration is a great starting point to nurture further inquiry into how we can scale up hydrogen technology to commercial aircraft. This is one option, but many more will be conceptualised before we make a final selection, a decision that is expected by 2025.

—Glenn Llewellyn, VP of Zero-Emission Aircraft

The pod configuration features six standalone turboprop motors based on a distributed hydrogen fuel cell propulsion system and removable fixtures for quick assembly and disassembly.

Each pod is essentially a stand-alone propeller propulsion system powered by hydrogen fuel cells. It consists of the following elements:

• A propeller
• Electric motors
• Fuel cells
• Power electronics
• LH2 tank
• A cooling system
• A set of auxiliary equipment

Another feature of the pod configuration is its removable fixtures. This means each pod can be disassembled and reassembled in record time. This approach could provide a practical and rapid solution for maintenance and potentially hydrogen refueling at airports.

The pod’s eight-bladed propellers, made of composite materials, are shaped to provide added thrust during the takeoff and climb-out phases of flight. The advanced airfoil design is expected to lead to improved efficiency and performance.

Although advanced in its design, the pod configuration still requires a lot of work to determine whether it could be a suitable solution. To date, it remains one of the technology options that Airbus engineers are considering as they work towards launching the ZEROe program.

A patent application for the pod configuration was published in December 2020, 18 months after its initial submission. This highlights that Airbus has been working on ZEROe since at least 2018. Several more patent applications are expected to be submitted over the coming months and years as R&D continues on the ZEROe program.

LH2 tank
correct

Scientists at MIT believe that to make commercial electric aircraft viable using batteries will take an energy density of about 1,500 Wh/kg; todays common energy density runs about 300Wh/kg which leaves the door wide open for H2 in this transportation segment.

The hydrogen-fuel-cell-powered “pod” configuration looks like a system that could work in the near term. Probably, based on the ATR 72-600 turboprop aircraft (instead of two PW127 engines @ 2,475 hp each, six 825 hp electric motors, and six 450 kW fuel cells - 78% cruise power). Liquid H2 stored in "slipper tanks similar to the Lockheed JetStar wing mounted fuel tanks.
So this could definitely work. Though Liquid H2 fuel costs and infrastructure are another challenge for practical air travel.

For a mid-size to big airliner, the gas turbine engine is efficient enough when burning LH2 and therefore Fuel Cell is not necessary. With LH2 that is very cold, it can be used to cool the air intake charge before entering the compressor, thus significantly reducing the compression work and hence increase thermodynamic efficiency. Mid-size airliner turbine engine can see an increase in thermal efficiency by 10-20%, with similar gain in power density, as the result of using LH2 to cool the air intake charge.

LH2 would best be placed in the rear of the aircraft in one big spherical tank in order to minimize the surface to volume ratio and hence reducing insulation weight and boil-off rate. The fuselage is simply lengthened enough accommodate the LH2 tank without reducing cargo and passenger space. An added advantage would be great reduction in post-crash fire hazard after impact-survivable crashes.

The wing fuel tanks for kerosene could be left intact in order to have a jet-fuel capability in case of operating at airports that do not carry LH2. LH2 may not be available in many airports at the beginning of the Hydrogen Economy. This is an added advantage of using existing aircraft designs, existing gas turbine engines modified for use with LH2, and the fuselage will simply need lengthening, without having to replace the whole aircraft. With modern fly-by-wire controls in newer generations of aircraft, lengthening of the fuselage would be easier, by simply using longer electric cables to transmit control commands from the cockpit. Hydraulic controls can also be accommodated by simply using longer hydraulic piping.

Lengthening the fuselage is not always an option; the aircraft has to rotate a certain amount for takeoff, and a longer fuse means you can't rotate as far without tail strikes.  Extending the landing gear has its own problems.

Surface/volume ratio issues on aircraft H2 tanks are probably not a big factor, as boiloff will be far slower than consumption in flight.  If you're concerned about it, run an APU to operate a cooler and top off the tanks right before flying.

When did liquefying and storing hydrogen become cheap and lightweight?

The ATR 72-600 jointly manufactured by Airbus and Leonardo is still in production and few modifications would be necessary. Probably just wing extensions like Boeing did on the 777X. What is interesting is that this concept shows a straightforward path to using Fuel Cell electric power.

The PW127 used in the ATR72-600 is 27% efficient at Max cruise (https://en.wikipedia.org/wiki/Brake-specific_fuel_consumption) and burns about 735 kg per hour (https://www.fliegerfaust.com/atr-72-sells-better-q400-2469153415.html). So an FC should burn about half that amount with each pod weighing less than1000 kg (Note: the ATR72 carries 5,000 kg of fuel).

Liquid H2 is too expensive for commercial aviation (even for space travel). However, H2 storage with the volumetric density of LH2 might solve the problem. Either "Blue" or "Green" Ammonia or the advanced solid state H2 storage being developed in Australia might work better.

The real potential for this concept would be in eVTOL. Look at the ASX Modi-One for example, which is a 4 engine tilt wing VTOL. The "Pod" concept could work well for this design.

@EP,
Great point regarding the potential of tail-strike with lengthening of the fuselage. There are several potential solution short of lengthening the landing gear strut.
1.. No modification to the aircraft length and reduce seat count on existing aircraft, while accepting heavier cargo payload in the front cargo section to make up for lower seat count, because it would cost quite a bit to lengthen the fuselage existing aircraft.
2.. On aircraft not-yet-built, the fuselage would be designed to be lengthened BOTH in front of the wings AND behind the wings in order to maintain optimum balance. This would minimize the fuselage lengthening requirement behind the landing gear.
3.. Furthermore, the use of LH2 while carrying the same payload would mean much reduced Maximum Take Off Weight (MTOW) and hence no need for high angle of attack like before.
4.. If higher payload in the form of heavier cargo load is taken on to offset the much-lower fuel weight, then the aircraft will not be allowed to be used on short runways, only on longer runways, or higher flap angle will be used to reduce rotation angle requirement on takeoff on shorter runways, because takeoff distance is always longer than landing distance.
5.. In practice, building some extra speed before rotation would allow for reduction in angle of rotation and would enhance safety if runway is long enough.

@EP
On second thought, BOTH LH2 tanks, in front and in the rear are needed due to problem with weight balance as the fuel is being consumed. The front LH2 tank can be placed below the passenger floor, in the front cargo area, while the rear tank can be placed in the tail section, thus taking up less passenger seat space, though cargo space will be reduced and will have to take on denser cargo to make up for the loss of some cargo space, unless the fuselage is being lengthened as discussed.

@gryf,
Yes, smaller-aircraft turboprop engines have poor efficiency and would be better off replaced with FC that can double the fuel efficiency. Additionally, due to the much lighter fuel weight of LH2, payload can increase significantly, thus allowing LH2 price per BTU to be as high as 3 times the price of jet kerosene fuel. So, if jet fuel now costs $2 per gallon, then LH2 cost of$6 per kg would be tolerable. Back in 2001, NASA paid $3.66 per kg of LH2 and$0.16 per kg of LO2. Adding inflation and the bulk price of LH2 today would still be affordable to replace kerosene jet fuel.

@Albert E Short
It didn't

@E-P
On the third thought, instead of lengthening the fuselage design in not-yet-build aircraft, the fuselage could be widened to accommodate larger volume instead of lengthening the aircraft. For example, the B 737 500 had fuselage width of 3.75 meter, and we want to widen it about 0.5 m to accommodate another passenger's row. With the length of the untapered fuselage being 26 m, the old volume is 287 m^3 vs the new volume to be 368 m^3 for a gain of 81 m^3. The plane carries about 15,000 kg of fuel, so with H2 weighing a third as much, will end up carrying ~5,000 kg of LH2. With a density of 71 kg per m^3, this would occupy a volume of 70 m^3, which is below the volume gained from widening the plane.

Of course, due to the higher efficiency of the engine running on LH2 and the lower Max takeoff wt, we will need quite a bit less LH2 to travel the same distance, so some of the extra volume can end up as polyurethane insulation foam that can also add immense structure strength to the fuel tank as well, and any left-over can be used to increase payload to compensate for a lot lower fuel weight.

With a fuselage width widened from 3.75 m to 4.25 m while keeping the length the same, there will be a 13% gain in skin surface area. Assuming that the skin surface area of the fuselage is only 40% of total skin surface area of the plane, then this gain = 40% x 13% = 5% increase in total skin surface area of the plane, so this is a negligible penalty in cruising drag that can be made up for by slightly decreasing cruising speed of about roughly 1%.

Albert E. Short asked: "When did liquefying and storing hydrogen become cheap and lightweight? "
Answer: Ask NASA, who paid $3.66 per kg of LH2 in 2001. With inflation between 2002 and today = 45%, then$3.66 x 1.45 = $5.3 per kg. In regional airliners with PT6 turboprops, the FC system can double the efficiency propeller to propeller, thus we can divide this cost in 1/2 =$2.65 per GGe of jetfuel. Even better, the very lightness of LH2 permit gain in payload weight, even doubling the payload weight that is especially applicable to heavy cargo, so we can divide this $2.65 by half again, to$1. 325.

In larger airliners with higher-efficiency gas-turbines, the efficiency gain using LH2 is not as dramatic as with smaller regional airliners, but at least the cost of LH2 can equal the cost of jetfuel per ton-mile of payload.

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