From 6ed1282b27c9d923972996c8d06f8fe3fa41009d Mon Sep 17 00:00:00 2001 From: Luis Aleixo Date: Tue, 11 Oct 2022 12:07:23 +0200 Subject: [PATCH 1/7] Fixed typo in documentation --- caimira/docs/full_diameter_dependence.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/caimira/docs/full_diameter_dependence.rst b/caimira/docs/full_diameter_dependence.rst index 09461003..542c659d 100644 --- a/caimira/docs/full_diameter_dependence.rst +++ b/caimira/docs/full_diameter_dependence.rst @@ -144,7 +144,7 @@ The initial concentration of virions at the mouth/nose, :math:`C_{0, \mathrm{SR} :math:`C_{0, \mathrm{SR}}(D) = N_p(D) \cdot V_p(D) \cdot \mathrm{vl_{in}} \cdot 10^{-6}`, given by :meth:`caimira.models.Expiration.jet_origin_concentration`. It computes the same quantity as :meth:`caimira.models.Expiration.aerosols`, except for the mask inclusion. As previously mentioned, it is normalized by the **viral load**, which is a diameter-independent property. -Note, the :math:`10^{-6}` factor corresponds to the conversion from :math:`\mathrm{ΞΌm}^{3} \cdot \mathrm{cm}^{-3}` to :math:`\mathrm{mL} \cdot m^{3}`. +Note, the :math:`10^{-6}` factor corresponds to the conversion from :math:`\mathrm{ΞΌm}^{3} \cdot \mathrm{cm}^{-3}` to :math:`\mathrm{mL} \cdot m^{-3}`. Note that similarly to the `long-range` approach, the MC integral over the diameters is not calculated at this stage. From faf41bed4abe45a096de00c0681bddac0c4658ec Mon Sep 17 00:00:00 2001 From: Luis Aleixo Date: Thu, 20 Oct 2022 18:17:55 +0200 Subject: [PATCH 2/7] removed error from disclaimer --- caimira/apps/templates/common_text.md.j2 | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/caimira/apps/templates/common_text.md.j2 b/caimira/apps/templates/common_text.md.j2 index b35e0858..79648ed6 100644 --- a/caimira/apps/templates/common_text.md.j2 +++ b/caimira/apps/templates/common_text.md.j2 @@ -34,7 +34,7 @@ We wish to thank CERN’s HSE Unit, Beams Department, Experimental Physics Depar

The risk assessment tool simulates the airborne spread SARS-CoV-2 virus in a finite volume, assuming homogenous mixing for the long-range component and a two-stage jet model for short-range, and estimates the risk of COVID-19 airborne transmission therein. - The results DO NOT include short-range airborne exposure (where the physical distance is a significant factor) nor the other known modes of SARS-CoV-2 transmission. + The results DO NOT include other known modes of SARS-CoV-2 transmission, such as contact or fomite. Hence, the output from this model is only valid when the other recommended public health & safety instructions are observed, such as adequate physical distancing, good hand hygiene and other barrier measures.

From a32c4f1ae3f3d889b1563492f349214a258c858f Mon Sep 17 00:00:00 2001 From: Luis Aleixo Date: Fri, 28 Oct 2022 12:17:12 +0200 Subject: [PATCH 3/7] update to documentation on dilution factor --- caimira/docs/full_diameter_dependence.rst | 22 +++++++++++++++--- caimira/models.py | 28 +++++++++++------------ 2 files changed, 32 insertions(+), 18 deletions(-) diff --git a/caimira/docs/full_diameter_dependence.rst b/caimira/docs/full_diameter_dependence.rst index 542c659d..04db938d 100644 --- a/caimira/docs/full_diameter_dependence.rst +++ b/caimira/docs/full_diameter_dependence.rst @@ -139,8 +139,24 @@ Very similar to what we did with the **emission rate**, we need to calculate the During a given exposure time, multiple short-range interactions can be defined in the model. In addition, for each individual interaction, the expiration type may be different. -To calculate the short-range component, we first need to calculate what is the **concentration at the jet origin**, that depends on the diameter :math:`D`. -The initial concentration of virions at the mouth/nose, :math:`C_{0, \mathrm{SR}}(D)` is calculated as follows: +To calculate the short-range component, we first need to calculate what is the **dilution factor**, that depends on the distance :math:`x` as a random variable, from a log normal distribution in :meth:`caimira.monte_carlo.data.short_range_distances`. +This factor is calculated in a two-stage model, based in a transition point, defined as follows: + +:math:`\mathrm{xstar}=𝛽_{\mathrm{x1}} \cdot (\mathrm{BR} \cdot u_{0})^\frac{1}{4} \cdot (\mathrm{tstar} + t_{0})^\frac{1}{2} - x_{0}`, + +where :math:`\mathrm{BR}` is the expired flow rate during the expiration period, converted from :math:`m^{3} h^{-1}` to :math:`m^{3} s^{-1}`, :math:`u_{0}` is the expired jet speed (in :math:`m s^{-1}`) given by :math:`u_{0}=\frac{\mathrm{BR}}{A_{m}}`, being :math:`A_{m}` the area of the mouth assuming a perfect circle (average mouth diameter :math:`D` of `0.02m`). +The time of the transition point :math:`\mathrm{tstar}` is defined as 2s. The distance of virtual origin :math:`x_{0}` given by :math:`x_{0}=\frac{D}{2𝛽_{\mathrm{r1}}}` (in m), and the time of virtual origin on puff-like stage is given by :math:`t_{0}=(\frac{x_{0}}{𝛽_{\mathrm{x1}}})^2 \cdot (\mathrm{BR} \cdot u_{0})^-\frac{1}{2}` (in s). +Having the distance for the transition point, we can calculate the dilution factor at the transition point, defined as follows: + +:math:`\mathrm{Sxstar}=2𝛽_{\mathrm{r1}}\frac{(xstar + x_{0})}{D}`. + +The remaining dilution factors, either in the jet- or puff-like stages are calculated as follows: + +:math:`\mathrm{factors}(x)=\begin{cases}\hfil 2𝛽_{\mathrm{r1}}\frac{(x + x_{0})}{D} & \textrm{if } x < \mathrm{xstar},\\\hfil \mathrm{Sxstar} \cdot \biggl(1 + \frac{𝛽_{\mathrm{r2}}(x - xstar)}{𝛽_{\mathrm{r1}}(xstar + x_{0})}\biggl)^3 & \textrm{if } x > \mathrm{xstar}.\end{cases}` + +The variables :math:`𝛽_{\mathrm{r1}}`, :math:`𝛽_{\mathrm{r2}}` and :math:`𝛽_{\mathrm{x1}}` are defined as `0.18`, `0.2`, and `2.4` respectively. The dilution factor for each distance `x` is then stored in the :math:`\mathrm{factors}` array that is returned by the method. + +Having the dilution factors, the **initial concentration of virions at the mouth/nose**, :math:`C_{0, \mathrm{SR}}(D)`, is calculated as follows: :math:`C_{0, \mathrm{SR}}(D) = N_p(D) \cdot V_p(D) \cdot \mathrm{vl_{in}} \cdot 10^{-6}`, given by :meth:`caimira.models.Expiration.jet_origin_concentration`. It computes the same quantity as :meth:`caimira.models.Expiration.aerosols`, except for the mask inclusion. As previously mentioned, it is normalized by the **viral load**, which is a diameter-independent property. @@ -155,7 +171,7 @@ The former operation is given in method :meth:`caimira.models.ShortRangeModel._l one solution would be to recompute the values a second time using :math:`D_{\mathrm{max}} = 100\mathrm{ΞΌm}`; or perform a approximation using linear interpolation, which is possible and more effective in terms of performance. We decided to adopt the interpolation solution. The set of points with a known value are given by the default expiration particle diameters for long-range, i.e. from 0 to 30 :math:`\mathrm{ΞΌm}`. -The set of points we want the interpolated values are given by the short-range expiration particle diameters, i.e. from 0 to 100:math:`\mathrm{ΞΌm}`. +The set of points we want the interpolated values are given by the short-range expiration particle diameters, i.e. from 0 to 100 :math:`\mathrm{ΞΌm}`. To summarize, in the code, :math:`C_{\mathrm{SR}}(t, D)` is computed as follows: diff --git a/caimira/models.py b/caimira/models.py index 4a96be97..f2b104a3 100644 --- a/caimira/models.py +++ b/caimira/models.py @@ -1156,7 +1156,7 @@ class ShortRangeModel: ''' # Average mouth diameter D = 0.02 - # Convert Breathing rate from m3/h to m3/s + # The expired flow rate during the expiration period, converted from m3/h to m3/s BR = np.array(self.activity.exhalation_rate/3600.) # Area of the mouth assuming a perfect circle Am = np.pi*(D**2)/4 @@ -1164,30 +1164,28 @@ class ShortRangeModel: u0 = np.array(BR/Am) tstar = 2.0 - Cr1 = 0.18 - Cr2 = 0.2 - Cx1 = 2.4 + 𝛽r1 = 0.18 + 𝛽r2 = 0.2 + 𝛽x1 = 2.4 - # The expired flow rate during the expiration period, m^3/s - Q0 = u0 * np.pi/4*D**2 # Parameters in the jet-like stage - x01 = D/2/Cr1 + x0 = D/2/𝛽r1 # Time of virtual origin - t01 = (x01/Cx1)**2 * (Q0*u0)**(-0.5) + t0 = (x0/𝛽x1)**2 * (BR*u0)**(-0.5) # The transition point, m - xstar = np.array(Cx1*(Q0*u0)**0.25*(tstar + t01)**0.5 - x01) + xstar = np.array(𝛽x1*(BR*u0)**0.25*(tstar + t0)**0.5 - x0) # Dilution factor at the transition point xstar - Sxstar = np.array(2*Cr1*(xstar+x01)/D) + Sxstar = np.array(2*𝛽r1*(xstar+x0)/D) distances = np.array(self.distance) factors = np.empty(distances.shape, dtype=np.float64) - factors[distances < xstar] = 2*Cr1*(distances[distances < xstar] - + x01)/D + factors[distances < xstar] = 2*𝛽r1*(distances[distances < xstar] + + x0)/D factors[distances >= xstar] = Sxstar[distances >= xstar]*(1 + - Cr2*(distances[distances >= xstar] - - xstar[distances >= xstar])/Cr1/(xstar[distances >= xstar] - + x01))**3 + 𝛽r2*(distances[distances >= xstar] - + xstar[distances >= xstar])/𝛽r1/(xstar[distances >= xstar] + + x0))**3 return factors def _long_range_normed_concentration(self, concentration_model: ConcentrationModel, time: float) -> _VectorisedFloat: From a898e72bd54fc8ad93b63fe0a5fae5b41c0b6261 Mon Sep 17 00:00:00 2001 From: Luis Aleixo Date: Wed, 2 Nov 2022 15:40:40 +0100 Subject: [PATCH 4/7] =?UTF-8?q?Q0=CF=86=20with=20instead=20of=20BR=20direc?= =?UTF-8?q?tly?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- caimira/models.py | 18 ++++++++++-------- 1 file changed, 10 insertions(+), 8 deletions(-) diff --git a/caimira/models.py b/caimira/models.py index f2b104a3..1efde56d 100644 --- a/caimira/models.py +++ b/caimira/models.py @@ -1155,11 +1155,11 @@ class ShortRangeModel: The dilution factor for the respective expiratory activity type. ''' # Average mouth diameter - D = 0.02 - # The expired flow rate during the expiration period, converted from m3/h to m3/s + mouth_diameter = 0.02 + # Convert Breathing rate from m3/h to m3/s BR = np.array(self.activity.exhalation_rate/3600.) # Area of the mouth assuming a perfect circle - Am = np.pi*(D**2)/4 + Am = np.pi*(mouth_diameter**2)/4 # Initial velocity from the division of the Breathing rate with the area u0 = np.array(BR/Am) @@ -1168,20 +1168,22 @@ class ShortRangeModel: 𝛽r2 = 0.2 𝛽x1 = 2.4 + # The expired flow rate during the expiration period, m^3/s + Q0 = u0 * np.pi/4*mouth_diameter**2 # Parameters in the jet-like stage - x0 = D/2/𝛽r1 + x0 = mouth_diameter/2/𝛽r1 # Time of virtual origin - t0 = (x0/𝛽x1)**2 * (BR*u0)**(-0.5) + t0 = (x0/𝛽x1)**2 * (Q0*u0)**(-0.5) # The transition point, m - xstar = np.array(𝛽x1*(BR*u0)**0.25*(tstar + t0)**0.5 - x0) + xstar = np.array(𝛽x1*(Q0*u0)**0.25*(tstar + t0)**0.5 - x0) # Dilution factor at the transition point xstar - Sxstar = np.array(2*𝛽r1*(xstar+x0)/D) + Sxstar = np.array(2*𝛽r1*(xstar+x0)/mouth_diameter) distances = np.array(self.distance) factors = np.empty(distances.shape, dtype=np.float64) factors[distances < xstar] = 2*𝛽r1*(distances[distances < xstar] - + x0)/D + + x0)/mouth_diameter factors[distances >= xstar] = Sxstar[distances >= xstar]*(1 + 𝛽r2*(distances[distances >= xstar] - xstar[distances >= xstar])/𝛽r1/(xstar[distances >= xstar] From 4f091946ca8fa102c031719b35b755fa9f15cfcd Mon Sep 17 00:00:00 2001 From: Luis Aleixo Date: Wed, 2 Nov 2022 15:41:20 +0100 Subject: [PATCH 5/7] Updated documentation --- caimira/docs/full_diameter_dependence.rst | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/caimira/docs/full_diameter_dependence.rst b/caimira/docs/full_diameter_dependence.rst index 04db938d..3e30599d 100644 --- a/caimira/docs/full_diameter_dependence.rst +++ b/caimira/docs/full_diameter_dependence.rst @@ -142,19 +142,19 @@ In addition, for each individual interaction, the expiration type may be differe To calculate the short-range component, we first need to calculate what is the **dilution factor**, that depends on the distance :math:`x` as a random variable, from a log normal distribution in :meth:`caimira.monte_carlo.data.short_range_distances`. This factor is calculated in a two-stage model, based in a transition point, defined as follows: -:math:`\mathrm{xstar}=𝛽_{\mathrm{x1}} \cdot (\mathrm{BR} \cdot u_{0})^\frac{1}{4} \cdot (\mathrm{tstar} + t_{0})^\frac{1}{2} - x_{0}`, +:math:`\mathrm{xstar}=𝛽_{\mathrm{x1}} \cdot (Ο†Q_{0} \cdot u_{0})^\frac{1}{4} \cdot (\mathrm{tstar} + t_{0})^\frac{1}{2} - x_{0}`, -where :math:`\mathrm{BR}` is the expired flow rate during the expiration period, converted from :math:`m^{3} h^{-1}` to :math:`m^{3} s^{-1}`, :math:`u_{0}` is the expired jet speed (in :math:`m s^{-1}`) given by :math:`u_{0}=\frac{\mathrm{BR}}{A_{m}}`, being :math:`A_{m}` the area of the mouth assuming a perfect circle (average mouth diameter :math:`D` of `0.02m`). -The time of the transition point :math:`\mathrm{tstar}` is defined as 2s. The distance of virtual origin :math:`x_{0}` given by :math:`x_{0}=\frac{D}{2𝛽_{\mathrm{r1}}}` (in m), and the time of virtual origin on puff-like stage is given by :math:`t_{0}=(\frac{x_{0}}{𝛽_{\mathrm{x1}}})^2 \cdot (\mathrm{BR} \cdot u_{0})^-\frac{1}{2}` (in s). +where Ο† is a coefficient of 1, the :math:`Q_{0}` is the expired flow rate during the expiration period, converted from :math:`m^{3} h^{-1}` to :math:`m^{3} s^{-1}`, :math:`u_{0}` is the expired jet speed (in :math:`m s^{-1}`) given by :math:`u_{0}=\frac{Q_{0}}{A_{m}}`, :math:`A_{m}` being the area of the mouth assuming a perfect circle (average `mouth_diameter` of `0.02m`). +The time of the transition point :math:`\mathrm{tstar}` is defined as 2s. The distance of the virtual origin of the puff-like stage is defined by :math:`x_{0}=\frac{\textrm{mouth_diameter}}{2𝛽_{\mathrm{r1}}}` (in m), and the corresponding time is given by :math:`t_{0}=(\frac{x_{0}}{𝛽_{\mathrm{x1}}})^2 \cdot (Q_{0} \cdot u_{0})^{-\frac{1}{2}}` (in s). Having the distance for the transition point, we can calculate the dilution factor at the transition point, defined as follows: -:math:`\mathrm{Sxstar}=2𝛽_{\mathrm{r1}}\frac{(xstar + x_{0})}{D}`. +:math:`\mathrm{Sxstar}=2𝛽_{\mathrm{r1}}\frac{(xstar + x_{0})}{\textrm{mouth_diameter}}`. The remaining dilution factors, either in the jet- or puff-like stages are calculated as follows: -:math:`\mathrm{factors}(x)=\begin{cases}\hfil 2𝛽_{\mathrm{r1}}\frac{(x + x_{0})}{D} & \textrm{if } x < \mathrm{xstar},\\\hfil \mathrm{Sxstar} \cdot \biggl(1 + \frac{𝛽_{\mathrm{r2}}(x - xstar)}{𝛽_{\mathrm{r1}}(xstar + x_{0})}\biggl)^3 & \textrm{if } x > \mathrm{xstar}.\end{cases}` +:math:`\mathrm{factors}(x)=\begin{cases}\hfil 2𝛽_{\mathrm{r1}}\frac{(x + x_{0})}{\textrm{mouth_diameter}} & \textrm{if } x < \mathrm{xstar},\\\hfil \mathrm{Sxstar} \cdot \biggl(1 + \frac{𝛽_{\mathrm{r2}}(x - xstar)}{𝛽_{\mathrm{r1}}(xstar + x_{0})}\biggl)^3 & \textrm{if } x > \mathrm{xstar}.\end{cases}` -The variables :math:`𝛽_{\mathrm{r1}}`, :math:`𝛽_{\mathrm{r2}}` and :math:`𝛽_{\mathrm{x1}}` are defined as `0.18`, `0.2`, and `2.4` respectively. The dilution factor for each distance `x` is then stored in the :math:`\mathrm{factors}` array that is returned by the method. +The parameters :math:`𝛽_{\mathrm{r1}}`, :math:`𝛽_{\mathrm{r2}}` and :math:`𝛽_{\mathrm{x1}}` are defined as `0.18`, `0.2`, and `2.4` respectively. The dilution factor for each distance `x` is then stored in the :math:`\mathrm{factors}` array that is returned by the method. Having the dilution factors, the **initial concentration of virions at the mouth/nose**, :math:`C_{0, \mathrm{SR}}(D)`, is calculated as follows: From eb87574634584995a67d4d8524b0c5dc9575d1a0 Mon Sep 17 00:00:00 2001 From: Luis Aleixo Date: Wed, 2 Nov 2022 16:53:19 +0100 Subject: [PATCH 6/7] Updated docs text --- caimira/docs/full_diameter_dependence.rst | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/caimira/docs/full_diameter_dependence.rst b/caimira/docs/full_diameter_dependence.rst index 3e30599d..40dfdf84 100644 --- a/caimira/docs/full_diameter_dependence.rst +++ b/caimira/docs/full_diameter_dependence.rst @@ -140,12 +140,12 @@ During a given exposure time, multiple short-range interactions can be defined i In addition, for each individual interaction, the expiration type may be different. To calculate the short-range component, we first need to calculate what is the **dilution factor**, that depends on the distance :math:`x` as a random variable, from a log normal distribution in :meth:`caimira.monte_carlo.data.short_range_distances`. -This factor is calculated in a two-stage model, based in a transition point, defined as follows: +This factor is calculated in a two-stage expiratory jet model, with its transition point defined as follows: -:math:`\mathrm{xstar}=𝛽_{\mathrm{x1}} \cdot (Ο†Q_{0} \cdot u_{0})^\frac{1}{4} \cdot (\mathrm{tstar} + t_{0})^\frac{1}{2} - x_{0}`, +:math:`\mathrm{xstar}=𝛽_{\mathrm{x1}} (Q_{0} \cdot u_{0})^\frac{1}{4} \cdot (\mathrm{tstar} + t_{0})^\frac{1}{2} - x_{0}`, -where Ο† is a coefficient of 1, the :math:`Q_{0}` is the expired flow rate during the expiration period, converted from :math:`m^{3} h^{-1}` to :math:`m^{3} s^{-1}`, :math:`u_{0}` is the expired jet speed (in :math:`m s^{-1}`) given by :math:`u_{0}=\frac{Q_{0}}{A_{m}}`, :math:`A_{m}` being the area of the mouth assuming a perfect circle (average `mouth_diameter` of `0.02m`). -The time of the transition point :math:`\mathrm{tstar}` is defined as 2s. The distance of the virtual origin of the puff-like stage is defined by :math:`x_{0}=\frac{\textrm{mouth_diameter}}{2𝛽_{\mathrm{r1}}}` (in m), and the corresponding time is given by :math:`t_{0}=(\frac{x_{0}}{𝛽_{\mathrm{x1}}})^2 \cdot (Q_{0} \cdot u_{0})^{-\frac{1}{2}}` (in s). +where the :math:`Q_{0}` is the expired flow rate during the expiration period, in :math:`m^{3} s^{-1}`, :math:`u_{0}` is the expired jet speed (in :math:`m s^{-1}`) given by :math:`u_{0}=\frac{Q_{0}}{A_{m}}`, :math:`A_{m}` being the area of the mouth assuming a perfect circle (average `mouth_diameter` of `0.02m`). +The time of the transition point :math:`\mathrm{tstar}` is defined as `2s` and corresponds to the end of the exhalation period, i.e. when the jet is interrupted. The distance of the virtual origin of the puff-like stage is defined by :math:`x_{0}=\frac{\textrm{mouth_diameter}}{2𝛽_{\mathrm{r1}}}` (in m), and the corresponding time is given by :math:`t_{0}=(\frac{x_{0}}{𝛽_{\mathrm{x1}}})^2 \cdot (Q_{0} \cdot u_{0})^{-\frac{1}{2}}` (in s). Having the distance for the transition point, we can calculate the dilution factor at the transition point, defined as follows: :math:`\mathrm{Sxstar}=2𝛽_{\mathrm{r1}}\frac{(xstar + x_{0})}{\textrm{mouth_diameter}}`. @@ -154,7 +154,7 @@ The remaining dilution factors, either in the jet- or puff-like stages are calcu :math:`\mathrm{factors}(x)=\begin{cases}\hfil 2𝛽_{\mathrm{r1}}\frac{(x + x_{0})}{\textrm{mouth_diameter}} & \textrm{if } x < \mathrm{xstar},\\\hfil \mathrm{Sxstar} \cdot \biggl(1 + \frac{𝛽_{\mathrm{r2}}(x - xstar)}{𝛽_{\mathrm{r1}}(xstar + x_{0})}\biggl)^3 & \textrm{if } x > \mathrm{xstar}.\end{cases}` -The parameters :math:`𝛽_{\mathrm{r1}}`, :math:`𝛽_{\mathrm{r2}}` and :math:`𝛽_{\mathrm{x1}}` are defined as `0.18`, `0.2`, and `2.4` respectively. The dilution factor for each distance `x` is then stored in the :math:`\mathrm{factors}` array that is returned by the method. +The penetration coefficients in the jet-like stage :math:`𝛽_{\mathrm{r1}}`, :math:`𝛽_{\mathrm{r2}}` and :math:`𝛽_{\mathrm{x1}}` are defined by the following empirical values `0.18`, `0.2`, and `2.4` respectively. The dilution factor for each distance :math:`x` is then stored in the :math:`\mathrm{factors}` array that is returned by the method. Having the dilution factors, the **initial concentration of virions at the mouth/nose**, :math:`C_{0, \mathrm{SR}}(D)`, is calculated as follows: From cf330ab9af43d4e0a0b343dac20504978faa2a8a Mon Sep 17 00:00:00 2001 From: Luis Aleixo Date: Thu, 3 Nov 2022 09:15:05 +0100 Subject: [PATCH 7/7] updated t0 formula according to published paper --- caimira/docs/full_diameter_dependence.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/caimira/docs/full_diameter_dependence.rst b/caimira/docs/full_diameter_dependence.rst index 40dfdf84..d4a3b628 100644 --- a/caimira/docs/full_diameter_dependence.rst +++ b/caimira/docs/full_diameter_dependence.rst @@ -145,7 +145,7 @@ This factor is calculated in a two-stage expiratory jet model, with its transiti :math:`\mathrm{xstar}=𝛽_{\mathrm{x1}} (Q_{0} \cdot u_{0})^\frac{1}{4} \cdot (\mathrm{tstar} + t_{0})^\frac{1}{2} - x_{0}`, where the :math:`Q_{0}` is the expired flow rate during the expiration period, in :math:`m^{3} s^{-1}`, :math:`u_{0}` is the expired jet speed (in :math:`m s^{-1}`) given by :math:`u_{0}=\frac{Q_{0}}{A_{m}}`, :math:`A_{m}` being the area of the mouth assuming a perfect circle (average `mouth_diameter` of `0.02m`). -The time of the transition point :math:`\mathrm{tstar}` is defined as `2s` and corresponds to the end of the exhalation period, i.e. when the jet is interrupted. The distance of the virtual origin of the puff-like stage is defined by :math:`x_{0}=\frac{\textrm{mouth_diameter}}{2𝛽_{\mathrm{r1}}}` (in m), and the corresponding time is given by :math:`t_{0}=(\frac{x_{0}}{𝛽_{\mathrm{x1}}})^2 \cdot (Q_{0} \cdot u_{0})^{-\frac{1}{2}}` (in s). +The time of the transition point :math:`\mathrm{tstar}` is defined as `2s` and corresponds to the end of the exhalation period, i.e. when the jet is interrupted. The distance of the virtual origin of the puff-like stage is defined by :math:`x_{0}=\frac{\textrm{mouth_diameter}}{2𝛽_{\mathrm{r1}}}` (in m), and the corresponding time is given by :math:`t_{0} = \frac{\sqrt{\pi}D^3}{8𝛽_{\mathrm{r1}}^2𝛽_{\mathrm{x1}}^2Q_{0}}` (in s). Having the distance for the transition point, we can calculate the dilution factor at the transition point, defined as follows: :math:`\mathrm{Sxstar}=2𝛽_{\mathrm{r1}}\frac{(xstar + x_{0})}{\textrm{mouth_diameter}}`.