# How to write mass balance equation for the scenario when molecules are getting generated inside an sphere?

I am working on the physical understanding of diffusion through a nano machine(a shell with hollow interior in which molecules are produced) and have taken few assumptions:

1. There is rate $r_T$ with which molecules are produced inside this machine, and we are not concerned with the production process.
2. The machine is spherical shell and emitting molecules through pores on its body.
3. We are diffusing in an infinite radius sphere where the center of the nano machine coincides with the center of sphere and is stationary.
4. Let the radius of the spherical be $r+\delta r$ where r is the radius of inner side region where mass is getting produced with rate $r_T(t)$ and $\delta r$ the width of shell. $j_1$ is flux per unit area.
5. Assuming that there is a porous outer through which molecules are diffusing with Partition coefficient 'H'.

Now I am trying to write mass balance equation for it so as to get diffusion flux in terms of concentration.For this I apply this formula: input +generation-output-consumption=accumulation Taking its rate/differential form, I know that rate of accumulation in $\delta r$ will be zero.

so, by somewhat same approach as when we have a spherical solute dissolving sparingly in water I got: $$r_T(t)+(4\pi r^2j_1)_r - (4\pi r^2 j_1)_{r+\delta r}=0$$ and when I try to solve it taking $\delta r$ to be very small and dividing both sides by the volume of the shell, I got this equation: $$\frac {r_t(t)}{4\pi r^2 \delta r}=\frac{1}{r^2} \frac{d (r^2 j_1)}{dr}$$

which seems to be unbalanced, further applying it to Fick's first law doesn't make any sense. How should I proceed in my case? Is my mass balance equation wrong? 

• You're pretty close. Is $r_t(t)$ supposed to be the total rate of generation from r = 0 to r = R, or is it supposed to to be rate of generation per unit volume? – Chet Miller Mar 1 '17 at 21:05
• Thanks for your interest, $r_T(t)$ is the total rate of generation from r = 0 to r = R. – Userhanu Mar 2 '17 at 4:27
• Then the left hand side of the equation should read: $\frac{3r_t(t)}{4\pi R^3}$ – Chet Miller Mar 2 '17 at 11:41
• Incidentally, is this problem time-dependent, or are you assuming quasi steady state? – Chet Miller Mar 2 '17 at 15:12
• $r_T(t)$ is the rate of production of particles with respect to time. I think quasi steady state is an easier thing to do. You are asking that because in latter case we have equilibrium established?Then yes quasi steady state. – Userhanu Mar 5 '17 at 6:47