# Yet more on cross winds effects on trains

This post and its attachments continues a sporadic (and what some might call obsessional) series on the effects of cross winds on trains. It gives links to a downloadable report and a downloadable spreadsheet.

The spreadsheet gives a simple and straightforward way of calculating the CWCs using the methodology described in the report. It is made available on the basis that the coding has not been verified in any rigorous fashion, and that the user takes full responsibility for the output. That warning being given, I hope some will find it of interest. There are two worksheets. For both the user-defined parameters are highlighted in yellow. The first calculates the CWC from a user-specified value of the characteristic velocity The second calculates the value of characteristic velocity from the vehicle geometric, mass and aerodynamic parameters as in section 11 of the above report. It uses the same values for these parameters as used in the report, but these can be changed as required.

# More on Cross Wind Characteristics

Wind blows train off tracks in Xinjiang, Shanghai Daily 02-28-2007

In two previous blog posts I have discussed the method for calculating cross wind characteristics for train overturning in high winds that is set out in Baker et al (2019). In the first, I used the approach to look at what might be regarded as the “best” shape for trains in overturning terms, and in the second I looked at the methodology itself and tried to understand the quite complex form of the solution of the governing equations. In this post, I will consider the shape of the cross wind characteristics that are predicted by the method and consider how the characteristics change as the form of the lee rail aerodynamic rolling moment characteristic changes.

The method itself is straightforward and is given in the box below taken from a previous blog post. It assumes a simple three mass model of a train under the action of a wind gust and, through a suitable assumption for the form of the rolling moment characteristic allows reasonably simple formulae for the cross wind characteristic to be calculated. The method is considerably simpler than the methodology outlined in “Railway Applications – Aerodynamics, Part 6: Requirements and Test Procedures for Cross Wind Assessment. CEN EN 14067-6:2018” where a multi degree of freedom dynamic model of the train is required, and an artificial wind gust is imposed. I am strongly of the view that the complexity of the latter method is unjustified for two basic reasons. Firstly the use of a highly accurate multi-degree of freedom dynamic model is inappropriate when the input wind gust and aerodynamic characteristics have major uncertainties associated with them and the output is used in very approximate risk calculations; and secondly because the CEN method of specifying the wind gust is theoretically unsound and not representative of a real wind gust as I have argued elsewhere. In any case the methodology I use here has actually been compared against the CEN methodology and can be made to be in good agreement if properly calibrated. I would be the first to admit that a more detailed calibration of the method for a range of “real” effects such as track roughness, turbulence scale, suspension effects etc. is probably required, but its simplicity of use has much to commend it, particularly in helping to understand the physical processes involved.

The methodology of Baker et al (2019)

Those points being made, now let us turn to the matter in hand. The methodology starts from a curve fit of the measured or calculated lee rail rolling moment coefficients. The forms chosen are shown in Figure 1 below and effectively requires the specification of four parameters – the lee rail rolling moment coefficient at 30 and 90 degrees yaw, and the exponents of the curve fits n1 and n2, the first in the low yaw angle range, and the second in the high yaw angle range.

Figure 1. Curve fit formats to lee rail rolling moment characteristic

This curve fit then leads to the formulae for CWCs in the two yaw angle ranges given as equations A and B in the box above.. These give the values of normalized overturning wind speed against normalized vehicle speed, as a function of wind direction, the ratio of the lee rail moment coefficients at 90 degrees and 30 degrees and the two exponents. The normalization is through the characteristic velocity which is a function of the train and track characteristics, including the rolling moment coefficient at 30 degrees yaw.

Let us firstly consider the normalized CWCs calculated from this method. Figure 2 shows these for wind directions relative to the train direction of travel from 70 degrees to  110 degrees (where 90 degrees is the pure cross wind case). The lee rail rolling moment coefficients at 30 and 90 degrees are 4 and 6 respectively, and the exponents n1 and n2 are 1.5 and -1, all of which are typical values for a range of trains. From the figure it can be clearly seen that there are two parts of the cross wind characteristic – a low yaw angle range at the higher vehicle speeds, where the normalized overturning wind speed decreases slowly with increases in normalized vehicle speed; and a high yaw angle range for low vehicle speeds, where the normalized wind speed increases above the low yaw angle value, in some cases quite significantly. In general terms the low yaw angle curve is probably of more practical relevance as it corresponds to the normal train operating conditions, at least for high speed trains. Here there can be seen to be little variation of the characteristic with wind angle over the range from 70 to 90 degrees. The minimum value is usually at a wind angle of around 80 degrees, but the minimum is very flat and the values of normalised wind speed for a pure cross wind of 90 degrees are very close to the minimum values.

Figure 2 CWC variation with wind direction

Figure 4 CWC variation with high yaw angle exponent n2

Figure 3 CWC variation with low yaw angle exponent n1

Figure 5 CWC variation with ratio R of lee rail rolling moment coefficients at 90 and 30 degrees yaw

Figures 3 to 5 show the variation of the CWC at a wind direction of 90 degrees for a range of values of the two exponents n1 and n2 and the ratio R of the rolling moment coefficients at 90 and 30 degrees.  Firstly the low yaw angle exponent is allowed to vary between 1.0 and 2.0. Earlier work has shown that blunt nosed leading vehicle tend to have a value of n1 of around 1.1 to 1.3, and streamlined leading vehicles have values between from 1.4 and 1.7. There can be seen to be very considerable variation in the CWCs throughout the vehicle speed range as this parameter varies, with the lower values resulting in lower, and thus more critical CWCs (but remember that these are non-dimensional curves – we will deal with the dimensional case below). Variations in the high yaw angel exponent n2 and the ratio of the rolling moment coefficients have a somewhat smaller and more localized effect in the low vehicle speed range only. As to which are the most important parameters, that depends upon the type of train – for high speed trains, the low yaw angle range is critical, but for low speed trains, the yaw angles experienced in practice span the high and low yaw angle ranges so both are important.

To simplify things further, the figures suggest that if the CWCs for the low yaw angle range were used throughout the speed range, then this would be a conservative approach. Figure 6 shows such CWCs for the conditions of figure 3 for a wind direction of 90 degrees, which is very close to the minimum, critical, value, and a range of values of the exponent n1. Note that at zero normalised speed, the normalised wind speed is 1.0 in all cases. By setting the wind direction to 90 degrees, equation A in the box above takes on a very straightforward form, and values of normalised wind speed can be found for any value of normalised vehicle speed for any value of n1, although an iterative solution is required.

Figure 6. CWCs for all vehicle speeds using low yaw angle formulation only.

All the CWCs presented above have been in a dimensionless form. These can easily be converted to a dimensional form by multiplying the velocities on both axes by the characteristic velocity. This is know to vary between about 30m/s for conventional low speed trains to around 40m/s for high speed trains. The variation in the CWC for 90 degrees wind direction from Figure 2 for these two characteristic velocities is shown in figure 7. The value for 30 m/s lies well below the 40 m/s curve, with very much lower overturning wind speeds at any one vehicle speed. However whilst the high speed train with a characteristic velocity of 40 m/s has a top speed of above 300 km/h, the top speed for the low speed, conventional train with a value of 30 m/s will be around 160 km/h. So a direct comparison at the same speed is not entirely appropriate.

Figure 7 CWCs in dimesnional form

# Cross Wind Characteristics – a mathematical curiosity

Readers of this blog will know that one of the subjects that I have worked on over the last 40+ years has been the effect of cross winds on trains. By this time, one would have thought that I should have plumbed the depths of the topic, but it still has the ability to surprise. In this short (and very nerdy) post I want to describe a mathematical curiosity associated with this subject that I have recently become aware of.

Box 1, CWC Calculation methodology

In the book “Train Aerodynamics – fundamentals and applications” I set out a simple methodology for calculating Cross Wind Characteristics (CWCa) – essentially plots of overturing wind speed against train speed. This is based on a simple three mass model and the equations are set out in Box 1 above – equation (A) for the low yaw angle range, and equation (B) for the high yaw angle range. I won’t describe this in further detail here – the book only costs £132 on Amazon, so any interested readers can find a fuller description there and provide some minimal royalties to myself and the other authors.

Recently I have had occasion, as part of a consultancy project, to develop simple spreadsheet to enable CWCs to be calculated for a range of different types of rail vehicle. The method I chose was to solve equation (A) for low yaw angles below the critical yaw angle and equation (B) for high yaw angles above the critical angle, using the Newton Raphson iterative method. These equations give an explicit solution for the overturning wind speed at a train speed of zero. The value of train speed is then increased in small increments up to the vehicle maximum operating speed, with the first estimate in the iteration at any one wind speed being the converged value of wind speed from the previous calculation with a slightly lower train speed. Convergence is usually very rapid, usually just one or two iterations.

Figure 1 Calculated CWCs for n1=1.5, n2=0 for wind directions up to 90 degrees

Figure 2 Calculated CWCs for for n1=1.5, n2=0 for wind directions above 90 degrees

Figure 3 Calculated CWCs for for n1=1.5, n2=-0.5 for wind directions above 120 degrees

The methodology in general worked well, and some of the results for different wind directions relative to the train direction of travel are shown in Figures 1 and 2 (for lee rail rolling moment coefficients at 30 and 90 degrees of 2.2 and 3.5 respectively and parameters n1 and n2 of 1.5 and 0.0, i.e. a steadily increasing rolling moment coefficient up to the critical yaw angle, and a constant value above that angle). The two yaw angle ranges can be clearly seen, with the lower yaw angle range at the higher train speeds, and the higher yaw angle range at the lower train speeds. For the train aerodynamic characteristics shown here, the calculations are very stable up to a wind direction of 120 degrees. However, if the calculation is carried out for higher wind directions, then something odd happens and the iteration becomes unstable as can be seen for the 135 and 150 degree cases in figure 2. This effect is even more severe for different rolling moment characteristics. Figure 3 shows the CWCs for the same rolling moment coefficients and value of n1, but with a value of n2=-0.5 and thus with a peak at the critical yaw angle, which is typical of high-speed trains. Here we can see major instabilities for wind directions above 120 degrees. I was very puzzled as to why this was the case. Whilst in practical terms this is of no significance, as the overturning wind speeds for such wind directions are high and not close to the minimum critical value at any one vehicle speed, but nonetheless it would still be good to understand what was going on.

After playing around with the equations for a while, I found the best way to understand this was to regard equations (A) and (B) as quadratic equations in train speed and solve for train speed for a range of values of overturning wind speed. This is the wrong way round of course, as the vehicle speed is really the independent variable that can be specified, and the wind speed is the dependent variable that needs to be calculated but solving the equations in this way proved to be illustrative.

As the equations are quadratics, there are two solutions for train speed for each value of wind speed for each equation, and regions of the vehicle speed / wind speed plane where no solutions exist. There are thus four distinct solutions to the equations, two for the low yaw angle range and two for the high yaw angle range. These are shown for a range of different wind directions in Figure 4 for the same case as in figures 1 and 2. Here the solutions are shown for both positive and negative train speeds. The critical yaw angle condition is indicated by the short-dotted lines – between the lines the high yaw angle curves will form the CWC and outside them the CWC will be formed from the low yaw angle curves. The calculated CWCs (in the positive velocity quadrant) are shown by the long-dotted line.

a) Wind direction = 30 degrees

b) Wind direction = 60 degrees

c) Wind direction = 90 degrees

d) Wind direction = 120 degrees

e) Wind direction = 150 degrees

Figure 4 Complete solutions of equations A and B for for n1=1.5, n2=0

Consider first the 90 degrees yaw angle case (Figure 4c). Here the solutions are symmetric about the wind speed axis, and the CWC simply takes the positive high yaw angle solution at low vehicle speeds, and the low yaw angle solution at higher vehicle speeds. As the wind direction moves away from this case, the solutions become skewed, although there is still a degree of symmetry about the 90 degreecase, with the 30 degrees case being the image of the 150 degrees case, and the 60 degrees case being the mirror image of the 120 degrees case.

For the 30 degree case the CWC is formed entirely from a solution to  a low yaw angle equation. At 60 and 90 degrees the CWC is formed from one low yaw angle solution, and one high yaw angle solution. At 120 degrees, the CWC consists of one low yaw angle solution and two high yaw angle solutions, whilst at 150 degrees the CWC consists of two low yaw angle and two high yaw angle solutions. There is thus considerable complexity here that is not fully revealed by simply considering the direct calculation of the CWC.

But coming back to the reason for this study, a consideration of the 150 degrees case shows the reason for the instabilities in figures 2 and 3. One of the high yaw angle curves that comprise the CWC doubles back on itself – ie there are two values of normalized wind speed that have the same values of train speed. The iterative method is thus jumping from one value to another and not converging,

As I said, this is not a practical issue as the overturning wind speeds in the wind direction range above 120 degrees are significantly higher than the minimum values which tend to occur around a wind direction of 80 degrees. The iterative calculation method for wind speed at a particular vehicle speed should only be used with caution in this range, and if values are required, the rather more cumbersome solutions for vehicle speed at a particular value of wind speed should be used. In personal terms the graphs of the solutions of figure 4 are rather attractive and their symmetry and form satisfying, and it was fun trying to sort out the reason for the instabilities. Being retired one has the leisure for this sort of thing! Perhaps however it is no bad thing to appreciate a little more the complexities behind what is intended to be a simple calculation method for CWCs.

# Measurements of Carbon Dioxide concentrations in a church

The measurements reported in this post were made by colleagues of the School of Engineering at the University of Birmingham – Dr David Soper and Dr Mike Jesson – whose help is gratefully acknowledged.

## Introduction

Over the course of the Covid-19 pandemic, there has understandably been increased concern over ventilation within buildings and on buses and trains etc. This has been reflected in church circles where church ventilation has also been much discussed. Whilst more modern churches will have been specifically designed with ventilation in mind, with proper ventilation paths between windows and doors, the same cannot be said about older churches. For many such churches the only ventilation is offered by the opening of doors, and by leakage through windows and roofs. Because of the large vertical size of such buildings, this lack of ventilation is ameliorated by the ability of any pollutants of pathogens to diffuse throughout the large church space.

One such church is St. Michael on Greenhill in Lichfield (figure 1 below), which is essentially two large, connected boxes – a nave, and a chancel, with a main door in the north wall of the nave and a smaller door into the choir vestry on the south side, and internal doors between the vestry area, the nave and the chancel (figure 2). A though ventilation path is rarely established however as the external and internal doors are seldom open at the same time. There are plans to build new parish rooms to the south of the church, on the grassed area of the figure below.

This brief post outlines a short series of measurements to measure carbon dioxide (CO2) levels in St. Michael’s. CO2 is produced naturally by people during breathing and CO2 concentration levels are often taken to be an indication of pathogen levels when the population is infected. These measurements were made on Sunday May 15th 2022, when the service pattern was somewhat different from normal, with the normal 8.00 and 10.00 Holy Communion services supplemented by the Annual Parochial Church Meeting (APCM) at 11.15 and a 4.00 service at which a new Rector was Instituted by the Bishop and Archdeacon. As such it gave the opportunity to look at the effects of different congregation numbers (10 in the chancel for the 8.00 service, 50 for the 10.00 service and the APCM, and 150 for the Institution). A screen shot of a video of the Induction service is shown in figure 3 to give some idea of the density of the congregation.

## The measurements

Carbon Dioxide measurements were made with small transducers and data loggers at different points around the church. These were attached to pillars of left on suitable window ledges. These sampled automatically every minute and the results were transmitted wirelessly to a Raspberry Pi computer and from there to a University of Birmingham web site from where the data could be accessed in real time. These measurements were supplemented by measurements of temperature and pressure using further transducers with built in data loggers.

For the sake of simplicity only the results from two of the CO2 sensors will be shown, as the results from them all were very similar. The location of these are shown on the plan of Figure 2 – one on a pillar in the nave, and one on a window ledge in the chancel. The photographs of the instruments shown in figure 4 indicate that they are quite small and discrete and indeed were barely noticed by the congregation. The results will be presented from midnight on Saturday May 14th to midnight on Sunday May 15th.

## The results of the trials

The weather on May 15th was quite pleasant with early morning temperatures of 10°C rising to around 20°C in the late afternoon and evening. The external humidity varied from 20% to 100% throughout the day. Inside the church however there was far less variation with temperatures between 16 and 21°C and humidity between 55 and 70%. The was a light southerly wind in the morning, with a somewhat stronger easterly wind from mid-afternoon onwards.

The results of the CO2 measurements are shown on the graph of figure 5. These are shown in terms of parts per million (ppm) of carbon dioxide in the atmosphere by volume and are relative to a general background level of around 400 ppm.

The church was opened at around 7.30 am for the 8.00 Holy Communion service held in the chancel, which went on until till around 8.45. Around 10 people attended. There can be seen to be a small increase in CO2 levels in the chancel over the course of the service (A). Later in the morning there was a 10.00 Holy communion service in the nave with around 50 in the congregation, with a small choir of 4 or 5 in the chancel. This was followed immediately by the APCM from 11.15 to 11.45 in the nave with about the same number attending. During this period there can be seen to be a steady increase in CO2 levels both in the nave and the chancel (B). At 12.00 the church emptied and the doors were closed. This led to a steady decrease in concentrations (C) till about 2.00 when people started to arrive at the church to set up for the major service of the day – the Institution of the new Rector by the Bishop of Lichfield. At this point both the main door and the choir vestry door were opened (as Gazebos were being set up to the south of the church for refreshments after the service), and a ventilation path was opened through the church, with major CO2 concentration reductions (D). Around 3.00 the congregation for the 4.00 Induction service began to arrive and the church rapidly filled with around 150 attending, including a choir of around 20 in the chancel. There were significant increases in CO2 concentrations during the course off the service through till around 5.30 (E). When the service was over, both the main door and the choir vestry door were again opened, and there was a rapid drop in concentration levels till around 7.00 when the choir vestry door was closed (F). After some clearing up, the church emptied by around 8.00 and there was a gradual fall off in concentration levels (G).

Two main points emerge from these measurements. Firstly, and quite obviously, the levels of CO2 increase with the number of people in church and with the time they spend there – B and E on the above figure. Secondly it is clear that there are two different types of ventilation – the slow diffusion of CO2 throughout the building and leakage through the building envelope – roof, doors, windows etc. (C and G); and the rapid lowering of concentration levels when there is a direct ventilation path through the building between the two doors (D and F).

Now from the slope of the graph for the times when concentrations are falling, it is possible to get estimates of the time it takes for the concentrations to fall by 50%. For C and G these times are around 2.5 hours, whilst for D and F these times are between 10 and 30 minutes. Thus the through ventilation reduces the carbon dioxide levels much more quickly than simple diffusion and leakage.

## Implications

The results show firstly that the method that was used is a simple and viable way of assessing the main ventilation parameters in a church. Colleagues from the University of Birmingham recognise that there is still work to on improving the frequency response of the sensors but overall the method has much promise. Secondly there are some implications for St. Michael’s itself – that large congregations in the church for lengthy periods of time can result in significant CO2 concentrations (and thus pathogens in times of infection), and that through ventilation is much more effective in reducing these concentrations than simply relying on diffusion and leakage. In the Parish Rooms developments that are under consideration for the area adjoining the choir vestry, it may be worth investigating if it is possible to design through ventilation paths through the church and the new development.

# Pollutants, pathogens and public transport – ventilation, dispersion and dose

## Preamble

The ventilation of buses and trains has come to be of some significance to the travelling public in recent years for a number of reasons. On the one hand, such vehicles can travel through highly polluted environments, such as urban highways or railway tunnels, with high levels of the oxides of nitrogen, carbon monoxide, hydrocarbons and particulate matter that can be drawn into the passenger compartments with potentially both short- and long-term health effects on passengers. On the other, the covid-19 pandemic has raised very significant concerns about the aerosol spread of pathogens within the enclosed spaces of trains and buses. There is a basic dichotomy here – to minimise the intake of external pollutants into vehicles, the intake of external air needs to be kept low, whilst to keep pathogen risk low, then high levels of air exchange between the outside environment and the internal space are desirable. This post addresses this issue by developing a common analytical framework for pollutant and pathogen dispersion in public transport vehicles, and then utilises this framework to investigate specific scenarios, with a range of different ventilation strategies.

The full methodology is given in the pdf that can be accessed via the button opposite. This contains all the technical details and a full bibliography. Here we give an outline of the methodology and the results that have been obtained.

## Analysis

The basic method of analysis is to use the principle conservation of mass of pollutant or pathogen into and out of the cabin space. In words this can be written as follows.

Rate of change of mass of species inside the vehicle = inlet mass flow rate of species + mass generation rate of species within the vehicle – outlet mass flow rate of species– mass flow rate of species removed through cleaning, deposition on surfaces or decay.

This results in the equation shown in Box 1 below, which relates the concentration in the cabin to the external concentrations, the characteristics of the ventilation system and the characteristics of the pollutant or pathogen. The basic assumption that is made is of full mixing of the pollutant or pathogen in the cabin. The pdf gives full details of the derivation of this equation, and of analytical solutions for certain simple cases. It is sufficient to note here however that this is a very simple first order differential equation that can be easily solved for any time variation of external concentrations of pollutant generation by simple time stepping methods. For gaseous pollutants, the rate of deposition and the decay rate are both zero which leads to a degree of simplification.

Box 1. The concentration equation

The pdf also goes on to consider the pollutant or pathogen dose that passengers would be subjected to – essentially the integration of concentration of time history – and then uses this in a simple model of pathogen infection. This results in the infection equation shown in Box 2. Essentially it can be seen that the infection risk is proportional to the average concentration in the cabin and to journey length.

Box 2. Infection equation

The main issue with this infection model is that it assumes complete mixing of the pathogen throughout the cabin space and does not take account of the elevated concentrations around an infected individual. A possible way to deal with this is set out in the pdf. Further work is required in this area.

## Ventilation types

The concentration and infection equations in Boxes 1 and 2 do not differentiate between the nature of the ventilation system on public transport vehicles. Essentially there are five types of ventilation.

• Mechanical ventilation by HVAC systems
• Ventilation through open windows
• Ventilation through open doors
• Ventilation by a through flow from leakage at the front and back of the vehicle (for buses only)
• Ventilation due to internal and external pressure difference across the envelope.

Simple formulae for the air exchange rates per hour have been derived and are shown in Box 3 below. By substituting typical parameter values the air exchange rates are of the order of 5 to 10 air changes per hour for the first four ventilation types, but only 0.1 for the last. Thus ventilation due to envelope leakage will not be considered further here, although it is of importance when considering pressure transients experienced by passengers in trains.

Box 3. Ventilation types

## Scenario modelling

In what follows, we present the results of a simple scenario analysis that investigates the application of the above analysis for different types of vehicle with a range different ventilation systems, running through different transport environments. We consider the following vehicle and ventilation types.

• An air-conditioned diesel train, with controllable HVAC systems.
• A window and door ventilated diesel train.
• A bus ventilated by windows, doors, and externally pressure generated leakage.

Two journey environments are considered.

• For the trains, a one-hour commuter journey as shown in figure 1, beginning in an inner-city enclosed station, running through an urban area with two stations and two tunnels, and then through a rural area with three stations (figure 1).
• For buses, a one-hour commuter journey, with regular stops, through city centre, suburban and rural environments (figure 2).

Results are presented for the following scenarios.

• Scenario 1. Air-conditioned train on the rail route, with HVACs operating at full capacity throughout.
• Scenario 2. As scenario 1, but with the HVACs turned to low flow rates in tunnels and enclosed stations, where there are high levels of pollutants.
• Scenario 3. Window ventilated train on rail route with windows open throughout and doors opened at stations.
• Scenario 4. As scenario 3, but with windows closed.
• Scenario 5. Window, door and leakage ventilated bus on bus route with windows open throughout and doors opened at bus stops.
• Scenario 6. As scenario 5, but with windows closed.

Details of the different environments and scenarios are given in tables 1 and 2.  Realistic, if somewhat arbitrary levels of environmental and exhaust pollutants are specified for the different environments – high concentrations in cities and enclosed railway and bus stations and lower concentrations in rural areas. The air exchange rates from different mechanisms are also specified, with the values calculated from the equations in Box 3. Note that, in any development of this methodology, more detailed models of the exhaust emissions could be used that relate concentrations at the HVAC systems and window openings to concentrations at the stack, which would allow more complex speed profiles to be investigated, with acceleration and deceleration phases.

Figure 1. The rail route

Figure 2. The bus route

Table 1. The rail scenarios

Table 2. The bus scenarios

The results of the analysis are shown in figures 3 and 4 below for the train and bus scenarios respectively. Both figures show time histories of concentrations for NO2, PM2.5, CO2 and Covid-19, together with the external concentrations of the pollutants.

For Scenario 1, with constant air conditioning, all species tend to an equilibrium value that is the external value in the case of NO2 and PM2.5, slightly higher than the external value for CO2 due to the internal generation and a value fixed by the emission rate for Covid 19.

For Scenario 2, with low levels of ventilation in the enclosed station and in the tunnels, NO2 and PM2.5 values are lower than scenario 1 at the start of the journey where the lower ventilation rates are used, but CO2 and Covd-19 concentrations are considerably elevated. When the ventilation rates are increased in the second half of the journey all concentrations approach those of Scenario 1.

The concentration values for scenario 3, with open windows, match those of Scenario 1 quite closely as the specified ventilation rates are similar. However, for Scenario 4, with windows shut and only door ventilation at stations, such as might be the case in inclement weather, the situation is very different, with steadily falling levels of NO2 and PM2.5, but significantly higher values of CO2 and Covid-19. The latter clearly show the effect of door openings at stations.

Figure 3. The train scenario results

Now consider the bus scenarios in figure 4. For both Scenario 5 with open windows and doors, and Scenario 6 with closed windows and open doors, the NO2 and PM2.5 values tend towards the ambient concentrations and thus fall throughout the journey as the air becomes cleaner in rural areas. The internally generated CO2 and Covid-19 concentrations for CO2 and Covid-19 are however very much higher for Scenario 6 than for Scenario 5.

Figure 5. The bus scenarios

The average values of concentration for all the scenarios is given in Table 3. The dose and, for Covid-19, the infection probability, are proportional to these concentrations. For NO2 and PM10 the average concentrations reflect the average external concentrations, and, with the exception of Scenario 4, where there is low air exchange with the external environment for part of the journey. The average concentrations for CO2 and Covid-19 for the less ventilated Scenarios 4 and 6 are significantly higher than the other. For Covid-19, the effect of closing windows on window ventilated trains and buses raises the concentrations, and thus the infection probabilities, by 60% and 76% respectively.

Table 3. Average concentrations

The major strength of the methodology described above is its ability, in a simple and straightforward way, to model pollutant and pathogen concentrations for complete journeys, and to investigate the efficacy of various operational and design changes on these concentrations. It could thus be used, for example, to develop HVAC operational strategies for a range of different journey types. That being said, there is much more that needs to be done – for example linking the methodology with calculations of exhaust dispersion around vehicles, with models of particulate resuspension or with models of wind speed and direction variability. It has also been pointed out above that the main limitation of the infection model is the assumption of complete mixing. The full paper sets out a possible way forward that might overcome this. Nonetheless the model has the potential to be of some utility to public transport operators in their consideration of pollutant and pathogen concentrations and dispersion within their vehicles.

# The calculation of Covid-19 infection rates in churches

## Preamble

In a recent post, I looked at the risk of Covid infection on GB trains, based on the spreadsheet calculation methodology of Professor Jimenez and his team at the University of Colorado – Boulder. This method is based solely on aerosol transmission, which is now regarded as being of much more significance than transmission by surface contamination, and the risk of the latter can be easily reduced by normal hygiene precautions. In this post, I apply the same methodology specifically to the case of churches and include a downloadable EXCEL spreadsheet that might be of use to others. There is a level of self-interest of course, as I am a minister at an Anglican church which will shortly be faced with decisions concerning the nature of worship as the Covid restrictions are removed.  Essentially the spreadsheet gives a numerical value for the risk of Covid infection with specified amelioration methods in place (social distancing, masks, no singing etc.) and allows a rational assessment of safety to be made.

At the outset, it needs to be made clear that there are very many assumptions in the methodology of Jimenez, with some of the parameters not well specified, and the base values of risk that the model gives must be regarded as indicative only and it is best used in a comparative sense. In what follows, I first describe the input and output parameters of the spreadsheet, and then look at how it might be used to compare risk levels for different situations.

The spreadsheet is quite simple and straightforward, and requires no specific expertise to use. A screenshot is given above. The brown cells are input parameters, and the blue cells the output parameters The former are as follows.

• Length, width and height of worship area. The model effectively assumes that the worship area is a three-dimensional box. This is clearly not usually the case, and some degree of judgement will be required in assigning the length, width and height. All dimensions are in metres.
• Duration of worship is specified in hours.
• The ventilation with outside air is specified in air changes per hour. For most old churches that have been well maintained, this will be small and a value of 1.0 can be assumed. For particularly drafty churches, this could be rather higher (at say 3.0). For air-conditioned worship areas a value of 10.0 is appropriate.
• For the decay rate of the virus and the deposition to surfaces standard parameters are assumed. Normally the value for additional control measures will be zero unless there is filtering of recirculated air.
• The number in the choir and congregation are self-explanatory. Ministers should be included in the latter. Because of lack of reliable data on breathing rates and virus emission rates in children, no breakdown by age is required. This is probably a conservative assumption.
• The fractions of time that the choir sings and the fraction of time that the congregation sings are both values between 0 and 1.0. The choir fraction is when they are singing alone – it is assumed they will join with the congregation when the latter sing.
• The fraction of population that is immune is taken to be the proportion of the population that have received a full course of vaccinations, multiplied by 0.9 to allow for virus escape. At the time of writing in the UK, this parameter has a value of around 0.5.
• The parameter that allows for virus transmission enhancement due to variants has a base value of 1.0, a value of 1.5 for the alpha variant, and a value of 2.0 for the delta variant.
• A choice of values for masks efficiency for both breathing in and out are given.
• The fraction of the congregation with masks is a number between 0 and 1.0.
• The probability of being infective is taken from regional ONS data. For example, if the ONS figure of those infected is 1 in 500, then the probability will be 1/500 = 0.002.
• The hospitalization and death rates of those infected can also be taken from ONS data and have small values just above 0.0. At the time of writing the hospitalization rate is around 0.02 (2%) and the death rate is almost negligible and is taken as 0.001 (0.1%).

The next set of parameters in the spreadsheet are those that emerge from the calculation process and are not of direct interest to users. These lead on to the output parameters, which are as follows.

• The probabilities of covid infection, hospitalisation and death of a person attending the service of worship.
• These probabilities expressed as risk – for example a risk of 1 in 1000 of infection.
• The number of covid cases, hospitalisations and deaths arising from attending the service.

## Comparing risk

The absolute values of probability and risk must only be regarded as approximate. Indeed, Jimenez emphasises that there is a great deal of uncertainty around many of the assumed parameter and urges caution in the interpretation of the results. At best, the results will be accurate to within an order of magnitude. The main utility of the model would seem to be to assess changes in risk – for example, any particular congregation may be comfortable with a certain set of Covid amelioration methods (no singing, masks etc.) and the method can be used to see how this risk might change as these measures are relaxed.

As an example of this, let us consider a church (which is not dissimilar to the one where I am a minister), where the congregation is currently capped at 60, there is 100% marks wearing, and only the choir of 6 sings. For the current infection rate of 1 in 150, this gives a risk of infection of 1 in 18100 for a one-hour service. This level of risk would seem to be acceptable to the congregation. Indeed, for one person attending similar services each week for one year, the risk of covid infection is close to the UK risk of injury in a vehicle accident in a year.

Firstly, suppose that a capacity of 100 is allowed (i.e. social distancing regulations are abolished). This increases the risk of infection to 1 in 11800. Now suppose that in addition masks are no longer required. This leads to a risk of infection of 1 in 4100. Allowing congregational singing raises the risk further to 1 in 1600. As all these figures are dependent upon regional infection rate, they also allow for the congregation to decide at what infection level restrictions can be removed. Should the infection level fall to 1 in 1000, then the overall risk with no amelioration measures decreases from 1 in 1600 to 1 in 11300. Whilst these figures are themselves only approximate, they nonetheless give any congregation the information to make a rational choice of how to proceed as restrictions are eased.

## Closing comment

In order to make the spreadsheet as easy to use as possible, I have deliberately kept it simple and have not included too many options. However, if anyone has any suggestions for improvements / useful additions, then please contact me on c.j.baker@bham.ac.uk.

# Giovanni Solari 1953-2020

On April 19th 2021 an online memorial event was held to celebrate the life of Prof Giovanni Solari of the University of Genoa who died five months previously. His career is well described in a memorial article in the Journal of Wind Engineering that can be found here. I was one of over 20 friends and colleagues who spoke at the event. My short contribution is given below.

Giovanni Solari has left us a very considerable legacy, and I would like to briefly consider three aspects of this. The first is his legacy to the wind engineering community. He was the first President of the International Association of Wind Engineering and held that role from 2003 to 2007. But that role involved much more than a ceremonial aspect. He was instrumental in turning the IAWE from a very loose association that met for an extended supper every four years at the major conferences to a legally organised society with a properly formulated constitution, member organisations and a functioning secretariat. This involved much work with lawyers (an unenviable task) and much travelling and discussion. In a real way the existence of an international wind engineering community is one of Giovanni’s major legacies.

The second aspect I want to mention is his intellectual legacy. Giovanni had the gift of being able to take a complex physical or engineering problem, often in the field of structural dynamics, and to express this problem mathematically in such a way that he could obtain closed form solutions for the engineering parameters of interest. These were often complex but allowed a proper appreciation of the role of different material and loading properties to be understood and generalised. Giovanni was the master of the closed form solution. In these days, when it is so easy simply to throw computer power at a difficult problem through complex CFD of FE analysis, the need for such closed form solution becomes all the greater to inform calculations and to actually understand the issues in depth. Giovanni’s intellectual legacy, of doing the hard thinking and analysis before resorting to numerical calculation, is a very important one to keep hold of.

The third of the legacies I want to mention is a personal one. I believe I first met Giovanni at the first European Conference on Wind Engineering in the early 1990s. Certainly we began to correspond after that (and remember those were the days before the instant gratification of emails) and I paid a memorable visit to Genoa around that time where the highlight for me was the ability to spend some hours in the library, which was much better resourced in wind engineering terms than that of my own institution. I was received with courtesy and kindness and Giovanni spent time showing me around the city that he clearly loved. Over the years that same courtesy and kindness has been shown by Giovanni to numerous people – from research students at the very start of their careers to the more senior of us. And that is how many of us, myself included, who remember him – for his personal legacy as much as for his undoubted scholarship, organisational and intellectual legacies, as the kindest and most courteous of friends and colleagues. He will be very much missed by many in the community.

Giovanni – Requiescat in pace

# The Midland Tornado of 1545

Preamble

In the recently published book “Come wind, come weather” (Lichfield Press, 2021), Trevor James draws attention to the Midland Tornado of 1545 which caused very considerable damage along a very long storm track in Derbyshire. A description of the damage from that time is given in the Derbyshire edition of the Magna Britannia in 1817 and is reproduced below. In this short post, the nature of the storm will first be discussed and then we will make some estimates of the windspeeds that occurred, using modern damage scales. I will then address the question as to whether the event was actually a tornado, or some other type of wind storm.

The event

From the Magna Britannia.

” At Darbie the 25th daye of June 1545.

“Welbeloved sonne I recomend me unto you, gevyng you Godds blessyng & myne. Son this is to sertifie you of soche straunge newes, as that.hath of late chauns,ed in thes p’ties; that is to wytt, apon Satterday last past, being the 20th daye of this moneth, on Say’te Albons day, we had in thes p’tyes great tempest. … wether, about xi of the clok before none: & in the same tempest, The dev[ill] as we do suppose beganne in Nedewood, wch is ix myles from Da[rbie]; & there he caste downe a great substance of wood; & pulled up by the rotts: & from thens he came to Enwalle [Etwall] wher at one Mre Powret [Porte] dothe dwell, & he pulled downe ij great elmes, that there was a dossyn or xvj loode apon a piese of them; & went to the churche & pullyd up the leade, & flonge it apon a great elme that stondyth a payer of butt lenghthes from the churche, &. … it hangyd apon the bowys lyke stremars; & afte. …….. tourned. …… & the grounsells upwards & some layd bye apon. ….. heape &. ……. that was apon viij bayes long he set it a…….. gge & the. …… ro[ots] sett upwards; & he hathein the same towne lefte not past iiij or v housses hole. And from thence he came a myle a this syde, & there grewe opon Ix or iiijxx wyllowes, & apon xij or xvi he hathe brokyn in the mydds, & they were as great as a mans body: & so he lefte them lyke a yard and a half hye: And from thence he went to Langley, wch is lyke iiij myles from Darby, & there he hath pullyd downe a great p’te of the churche, & rowled up the leade & lefte it lyeing, & so went to Syr Wyllam Bassetts place in the same [towne] & all so rente it, & so pullyd a great parte of it downe wth his. …..& the wood that growethe abowte his place, & in his parke he pulled downe his pale & dryve out his deare, & pulled downe his woods, & so[me] broken in the mydds that was xvj or xx loode of wood of some one tre. And after that he went into the towne to Awstens housse of Potts & hath slayne his sonne & his ayer, & perused all the hole towne, that he hath left not past ij hole howsses in the same towne. And from thence he went to Wy’dley lane, & there a nourse satt wt ij chylderen uppon her lappe before the fyre, & there he flonge downe the sayde howse, & the woman fell forwards ap[on the] yongechyl[dren] afore the fyre, & a piese of ty’ber fell apon her. …… & so killed [her] but the chylderen were savyd, & no more hurte, [and none] of the house left standyng but the chymney, & there as the house stode, he flange a great tre, that there is viij or x lood of wood apon it. And from thence he went to Belyer [Belper] & there he hath pullyd & rent apon xl housses; & from thence he wente to Belyer [Belper] wood & he hathe pullyd downe a wonderous thyng of wood & kylled many bease; & from thens to Brege [Heage] & there hath he pulled downe the chappyl & the moste parte of the towne; & from thens to WynfeldmaJ that is the Erie of Shrowseberys [Wingfield Park], & in the parke he pulled him downe a lytell…… & from thens to Manfyld [Mansfield] in Shrewood & there I am sure he hath done [no] good, & as it is sayd he hathe donne moche hurte in Chesshire &….. shire. And as the noyse goeth of the people ther felle in some places hayle stons as great as a mans fyste, & some of them had prynts apon them lyke faces. This is trewe & no fables, there is moche more hurte done besyds, that were to moche to wryte, by the reporte of them that have sene it; and thus fare you well.”

James is persuaded that the account is genuine, not least by the mention of the damage to the chapel at Heage. The church at Heage was indeed officially a chapel (dependent upon another church) at the time and there are records elsewhere that indicate it was rebuilt after the storm. James quotes a further source (Warkworth’s Chronicle) which again suggests strong winds in Cheshire and Lancashire on that day.

The personification of the event as the “devil” is of interest and may reflect both the belief that such events were demonically rather than divinely inspired but might also refer to the name of such events – indeed even today small whirlwinds are referred to as dust-devils or something similar.

The description allows the track of the storm to be determined quite accurately, and this is shown on the map of figure below. In all the track where precise damage details are given seems to have been about 40 km long at least.

Wind speeds

On the assumption that we are dealing with an tornado here, rather than another type of windstorm (see below), is it possible to obtain estimates of what the windspeeds actually were? Tornado windspeeds are usually estimated by inspecting the damage that they cause, and then using a damage classification method to determine the broad range of wind speeds that would cause that damage. Two methods are commonly used – the Enhanced Fujita (EF) scale developed in the US, and the T scale developed by the Tornado Research Association TORRO. Extracts from the damage descriptors are given below.

Enhanced Fujita EF Scale

EF2         49–60m/s      Roofs torn off from well-constructed houses; foundations of frame homes shifted; mobile homes completely destroyed; large trees snapped or uprooted; light-object missiles generated; cars lifted off ground.

EF3         60-74m/s      Entire stories of well-constructed houses destroyed; severe damage to large buildings such as shopping malls; trains overturned; trees debarked; heavy cars lifted off the ground and thrown; structures with weak foundations are badly damaged.

EF4 74-89m/s Well-constructed and whole frame houses completely leveled; some frame homes may be swept away; cars and other large objects thrown and small missiles generated.

TORRO T scale

T4           52 – 61m/s           Motor cars levitated. Mobile homes airborne / destroyed; sheds airborne for considerable distances; entire roofs removed from some houses; roof timbers of stronger brick or stone houses completely exposed; gable ends torn away. Numerous trees uprooted or snapped.

T5          62- 72m/s            Heavy motor vehicles levitated; more serious building damage than for T4, yet house walls usually remaining; the oldest, weakest buildings may collapse completely.

T6           73 – 83m/s          Strongly built houses lose entire roofs and perhaps also a wall; windows broken on skyscrapers, more of the less-strong buildings collapse.

To give some context, the Birmingham Tornado of 2005 (pictured above), one of the strongest in recent years, was classified as a T5 event.

It is immediately clear that these descriptions are very subjective and the classification of damage into a particular class is not straightforward. It is perhaps less clear that each of these scales is to some extent culturally dependent. The EF scale reflects North American building and vehicle types, and the T scale reflects building and vehicle types from the UK in the 1970s when the scale was first produced. Neither reflects building practice in the 1540s and neither were there any cars to be lifted up in that period!  Nonetheless, the description of 1545 can be used within these classifications at least in an approximate way. I would thus classify the 1545 event as borderline EF2 / EF3 or borderline T4/T5. These classifications give a wind speed range around 135 mph or 60 m/s. However,  Neaden, in a study of tornado risk for the HSE, based on the above description, assigns a category of T6 to the event, giving a wind speed of at least 73m/s. This again illustrates the subjectivity of the classification.

The T scale also gives a classification based on path length

T4 2.2 to 4.6km

T5 4.7 to 9.9km

T6 10 to 21km

T7 22 to 46km

On the basis of path length Neaden again gives a T6 classification, although the length as shown in figure 1 suggests a T7 classification. My view would be that the event should properly be categorised as EF2/EF3 or T4/T5 with an unusually long path length but the subjectivity of this assessment must again be emphasised.

Was it a tornado?

The question that then needs to be addressed is whether or not the 1545 event was a tornado or some other storm type – presumably one of the usual extra-tropical cyclones that pass across the UK quite frequently. Even in such storms there are known to be smaller tracks of major damage. In the 1987 storm for example there was a swathe of extreme damage to trees in a arc a few miles wide across the south of England. This was attributed to a high-level jet of wind sweeping down to ground level – a phenomenon know as a “sting jet” because of the scorpion tail-like cloud formation with which such events are often associated.

The points that suggest the event was a tornado are firstly the existence of a coherent storm track, albeit significantly longer than would normally be the case, and secondly the fact that the event occurred in June, when extra-tropical cyclones are uncommon but tornadoes are. On the other hand, the point that suggest the damage was due to an extra-tropical cyclone is the reference in more than one source to concurrent strong winds in Cheshire and Lancashire. Indeed, wind speeds of 60 m/s have been measured in extra-tropical cyclones in the past – for the 1987 Burns Night storm for example the peak wind speed was somewhere around this value.

Referring again to the work of Neaden, his data indicates that between 1800 and 1985 there were around 10 tornadoes with a classification of T5 or higher in Derbyshire. This indicates that one would occur on average every 20 years or so. One would expect that most of these would have path lengths of a few kilometres and thus the effects would be localised – and in a rural county like Derbyshire not much damage might be recorded.

Now, one might expect that an extra-tropical cyclone with wind speeds of the order of 60m/s would only occur in lowland Britain once every 200 to 300 years – indeed the 1987 storm was assessed as having this return period. Thus they are very rare events indeed.

A comparison of the likelihoods of a T5 tornado and an extra-tropical storm with the same windspeeds thus suggests to me that it is most likely that the 1545 event was indeed a tornado with an EF2 / EF3 or T4/T5 classification, albeit with an unusually long storm track, but also that it was quite possibly embedded in a larger extra-tropical cyclone of some strength. However, as with any other historical phenomenon of this type, absolute certainty as to its cause is of course not possible.

# International Wind Engineering seminars 2020/21 – some reflections

A Japanese version of this post can be found here

Between October 2020 and March 2021, I organised a series of six International Wind Engineering Seminars, through the University of Birmingham, my employer before I retired. These were sponsored by the International Association of Wind Engineering (IAWE) and delivered via Zoom. On the web page for this seminar series, I give the justification for organising it as follows.

“Because of the Covid19 pandemic, opportunities for the international wind engineering community to meet physically have been very much restricted and are likely to remain so for at least the next year. To enable the community to continue to interact with each other, at least in a virtual way, the University of Birmingham is organizing a series of six seminars via Zoom from October 2020 to March 2021.”

In this post, I want to reflect on how these seminars were delivered and received, what lessons might be learnt, and ask some questions concerning the future.

Each seminar consisted of a main speaker, followed by either a panel discussion or between two and four shorter presentations. The dates and topics are given in table 1. As these seminars were set up in some haste in August / September 2020, I mainly called upon my circle of contacts to be the main speakers at the events, and they suggested other speakers or panel members. I am indebted to all the speakers for taking part and spending considerable time in preparation. The nature of the delivery and follow up evolved over the course of the series. After the first seminar it became clear that I could not both chair the sessions and organise the questions in Chat to put to the speakers. Thus, from seminars 2 to 6, I was assisted by Grace Yan from Missouri who collated all the questions that were put on Chat and forwarded them to me to put to the speakers. Her help was hugely appreciated. For seminars 3, 4 and 6 the presenters and panelists were also asked to provide written answers to questions, and these were posted on the web pages that were for each of the seminars. All the presentations (and for seminars 5 and 6 the questions and answers) were recorded using the Zoom Record function and these recordings were place on my YouTube site and linked to the appropriate page. These pages also included talk abstracts and speaker biographies. After the third seminar I realised that YouTube could not be accessed from all parts of the world, so a link to the Zoom cloud versions was also given. From seminar 4 onwards, these could also be downloaded as required. The time chosen for the seminars (after the first) was 12.00 UK time, this being the best compromise for most time zones, with the exception of the west coast of the America and Australasia. I tried to institute a separate Q and A session for these time zones a day or so after the seminar, but there was insufficient take up to make it worthwhile. Thus the whole process was a considerable learning experience for me.

Table 1 Seminar dates, titles and speakers

It must be mentioned at this point that the third seminar occurred shortly after the death of Prof Giovanni Solari, who was instrumental in the setting up of the IAWE, and the speaker, Prof Kareem, paid tribute to him in his talk.

Table 2 shows the bare statistics for the seminars. The size of the distribution list for publicity grew through the series from the original 688 of the mailing list for the abortive BBAA conference to 1525 for seminar 6.  By seminar 3 the size of the list became so large that my e mail account was temporarily stopped as it was thought it had been hacked and was sending out spam. Thereafter I sent the information around in smaller batches. The number of registrants varied between 279 and 616, although only around 50 to 70% of these actually connected. The number of video views was also encouraging although again one must interpret these numbers cautiously as only around 20 to 30% of the views were for more than a few minutes. Note that these statistics are up to March 14th 2021 only, and as the views continued for several months after each seminar, the number of video views for the 2021 seminars will not be the final values.

Table 2 Seminar statistics (up to March 14th 2021)

Table 3 shows a breakdown of the views of the seminar web pages by month (which includes links to the videos). As expected these peak just before and just after the seminar, but all the seminars attract a significant number of views for a number of months after the event, which suggest that the subject matter is of ongoing interest. Again, note that this date only extends to the middle of March 2021,and a significant number of views could be expected for the later seminars after this date.

Table 3 Views of seminar web pages (up to March 15th 2021)

Table 4 shows the location of those who registered, as far as could be judged from email addresses. The generic .com address contains registrants from a wide variety of countries, and this rather skews the results. Nonetheless, it can be seen that whilst those countries where wind engineering is well established are well represented, a very wide range of countries was represented overall.

Table 4 Locations of registrants

Thus the numbers suggest that there was a significant number of wind engineers around the world who appreciated the seminar series and found them useful, and indeed that is what has been suggested by the informal feedback I have received. Again, caution is required to avoid over interpretation – the level of engagement with online seminars is likely to be much less than with in person presentations – I for one tend to do things such as checking my e mail / cricket scores when attending such virtual events – but not when I am chairing of course! But broadly the seminar series seems to have met a need. But there are needs it hasn’t addressed, for example the inclusion of a social aspect for informal discussion and the inclusion of young researchers in a meaningful way etc. To address this sort of issue, other formats can be envisaged – for example I can think of the following.

• Specific discussion topics could be set, and potential attendees asked to submit short abstracts of a two minute, two slide talk, from which a balanced group of young and established researchers could be selected for a series of short presentations and a more relaxed discussion. These could be recorded and put on-line for all to see.
• Interviews (by me or others) of a range of wind engineers, talking about their careers, their successes and failures etc., which could again be recorded and put on-line.
• The use of a platform such as Gather Town, which seems to allow for multiple individual conversations within a group structure and could be used for, say, virtual poster sessions (but note I have never used this, although on the face of things it seems potentially useful.)

And there are no doubt other possibilities. The question then arises as to what should happen next. I don’t intend to organise any more such seminars till September at least – amongst other things I wish to watch a number of cricket matches rather than just checking the scores, and to re-acquaint myself with a number of heritage railways in Wales. So, I put the following questions to the wind engineering community.

• Should something similar be organised for next winter as I suspect international travel won’t resume in any real sense until Summer 2022 at best? Note that I am not necessarily implying that should something felt to be necessary, then I would be the one to organise it!
• If so, what should the format be – just one speaker, or more than one speaker, or something completely different?
• Are there any suggestions for topics and speakers?
• Are there any other suggestions for possible related activities, such as I mention above.

There is also a larger question of course about the future of the four year cycle of Wind Engineering conferences and whether such a cycle is still sustainable – see for example the initiative of Glasgow University which is urging academics to reduce overseas travel as part of the greening of its activities. But that is a discussion for others to have within the IAWE committee.

Please make any comments in the comment box attached to this post, or, if you wish, email me directly on c.j.baker@bham.ac.uk. Thanks in advance.

# Some thoughts on ventilation and pathogen concentration build up

## Introduction

Up till recently most attention had been focused on the spread of Covid-19 by near field transmission – being in close proximity to an infected person for a certain amount of time, and rather ad hoc social distancing rules have been imposed to attempt to reduce transmission. However, there is another aspect of transmission – the gradual build up of pathogen concentrations in the far field in enclosed spaces due to inadequate ventilation. The importance of this mode of transmission is beginning to be recognised – see for example a recent seminar hosted by the University of Birmingham. The main tool that seems to have been used for both near and far field dispersion is Computational Fluid Dynamics (CFD) – see the graphic above from the University of Minnesota for example. Now whilst such methods are powerful and can produce detailed information, they are very much situation specific and not always easy to generalise. This post therefore develops a simple (one could even say simplistic) method for looking at the far field build up of pathogens in an enclosed space, in a very general way, to try to obtain a basic understanding of the issues involved and arrive at very general conclusions.

## The model

We begin with equation (1) below. This is a simple differential equation that relates the rate of change of concentration of pathogen in an enclosed volume to the pathogen emitted from one or more individuals via respiration and the pathogen removed by a ventilation system. This assumes that the pathogen is well mixed in the volume and is a simple statement of conservation of volume.

From the point of view of an individual, the important parameter is the pathogen dose. This is given by equation (2) and is the volume of pathogen ingested over time through respiration. The respiration rate here is assumed to be the same as that of the infected individual.

Equations (1) and (2) can be expressed in the normalised form of equations (3) and (4) and simply solved to give equations (5) and (6).

Equations (5) and (6) are plotted in figures 1 and 2. Note that an increment of 1.0 in the normalised time in this figure corresponds to one complete air change in the enclosed volume. It can be seen that after around three complete air changes the concentration of pathogen reaches an equilibrium value and the dose increases linearly, whatever the starting concentration. To the level of approximation that we are considering here we can write the relationship between normalised dose and time in the form of equation (7), which results in the non-normalised form of equation (8).

Assuming that there is a critical dose, the critical time after which this occurs is then given by equation (9).

Equation (9), although almost trivial, is of some interest. It indicates that the time required for an individual to receive acritical dose of pathogen is proportional to the volume of the enclosure and the ventilation rate. This is very reasonable – the bigger the enclosure and the higher the ventilation, the longer the time required. The critical time is inversely proportional to the concentration of the emission, which is again reasonable, but inversely proportional to the square of the respiration rate. This is quite significant and a twofold increase in respiration rate (say when taking exercise or dancing) results in the time for a critical dose being reduced by a factor of 4, or alternatively the need for ventilation rate to increase by a factor of 4 to keep the critical time constant. Similarly if there are two rather than one infected individuals in the space, then the respiration rate will double, with a reduction in the critical time by a factor of four.

## Discussion

Now consider the implications of this equation for two specific circumstances that are of concern to me – travelling on public transport (and particularly trains) and attending church services. With regard to the former, perhaps the first thing to observe is that there is little evidence of Covid-19 transmission on trains, and calculated risks are low. In terms of the far field exposure considered here, respiration rates are likely to be low as passengers will in general be relaxed and sitting. This will increase the time to for a critical dose. On modern trains there will be an adequate ventilation system, and the time to reach a critical dose will be proportional to its performance. Nonetheless the likelihood of reaching the critical level increases with journey time – thus there is a prima facie need for better ventilation systems on trains that undergo longer journeys than those that are used for short journeys only. For trains without ventilation systems (such as for example the elderly Class 323 stock I use regularly on the Cross City line) has window ventilation only, and in the winter these are often shut. Thus ventilation rates will be low and the time to achieve a critical dose will be small.

Now consider the case of churches. Many church buildings are large and thus from equation (9) the critical times will be high. However most church buildings do not possess a ventilation system of any kind, and ventilation is via general leakage. Whilst for many churches this leakage this can be considerable (….the church was draughty to day vicar….), some are reasonable well sealed – this will thus, from equation (9) tend to reduce the critical time. In this case too the respiration rate is important. As noted above the critical time is proportional to the respiration rate squared. As the rate increases significantly when singing, this gives a justification for the singing bans that have been imposed.

The above analysis is a broad brush approach indeed, and in some ways merely states the obvious. However it does give something of a handle on how pathogen dose is dependent on a number of factors, that may help in the making of relevant decisions. To become really useful a critical dose and initial pathogen concentration need to be specified together with site specific values of enclosed volume, ventilation rate and expected respiration rates. This would give at least approximate values of the time taken to reach a critical dose in any specific circumstance.