Here I present the DIY air sterilisation project which this is about.
I want to calculate the overall sterilization rate of the device under some simplifying assumptions. (UVC is the hard ultra violet (UV) light killing the germs.)
schema drawing
I have
- the effective UVC power $E$ of my germicidal lamp. The lamp is a Hg-"neon" tube along the axis of the cylindrical device.
- the length $L$ of the lamp
- the radius $R_l$ of the lamp
- the inner radius $R_c$ of the cylinder
- a radius-invariant constant airflow through the cylinder (we are way into turbulent flow) with the speed $v$, so that I can calculate a mean residence time of the virus traveling through the killzone for
- a time $t = L/v$
- the UVC irradiation intensity $I$ at a given radius $r$ in the cylinder $I(r) = \frac{E}{2\pi \, L \, r}$
- a irradiation dose $d(r) = I(r)*t$
- a virus-specific susceptibilty rate $k$ to calculate the specific killrate $P(r)=1-e^{-k \, d(r)}$
- a infinitisimal thin hollow cylinder (thickness $dr$) at radius $r$ around the axis of the cylinder with the volume $V(r) = 2 \pi L \, r \, dr$ , which is proportional to the amount of germs exposed to the specific irradiation dose $d(r)$.
Now I look for the overall kill rate for the volume between $R_l$ and $R_c$.
Would it be correct to calculate $\frac{1}{R_c - R_l}\cdot\int_{R_l}^{R_c} P(r) \cdot V(r)dr = \frac{1}{R_c - R_l}\cdot\int_{R_l}^{R_c}2 \pi L \, r \, (1-e^{-k \frac{E}{2\pi\,v \, r}}) dr$ ?
Is that the differential equation I need? What is a solution for that?
Here are some matlab lines which are unfortunatly not correct as the result is independent of most of the variables.
syms R_l R_c L r E k v V(r);
u = symunit;
assume(0<R_l <R_c);
t = L*u.m/(v*u.m/u.s);
I = E*u.W/(2*pi*L*u.m*r*u.m);
d = I * t;
P = 1-exp(-k*(u.cm^2/u.mJ)*d);
diff(V*u.m^3) == 2*pi* L*u.m * r*u.m *diff(r*u.m);
killrate = V*P
latex(killrate);
mean_killrate = int( killrate ,r, R_l, R_c)/(R_c -R_l);
mean_killrate_small = eval( subs(mean_killrate, {R_l R_c L E k v}, {0.016/2 0.125/2 0.113 0.95 0.41 3}))
mean_killrate_medium = eval( subs(mean_killrate, {R_l R_c L E k v}, {0.016/2 0.125/2 0.205 4 0.41 3}))
mean_killrate_large = eval( subs(mean_killrate, {R_l R_c L E k v}, {0.040/2 0.125/2 0.51 27 0.41 3}))
edit: updated the matlab code to use diff() for differential terms and measurement units (to find inconcistencies in the equations).