What is the Validity of the Rate Equation in Different Reactor Types?

[Rahulx084]

What is the definition of rate $(-r_a)=-\frac{dCa}{dt}$ or $(-r_a)=-1/V\frac{dNa}{dt}$ ? I think the general one is the second one and first one is for constant volume reaction system. Is the above written rate equation only valid to batch reactor? If not can we use this in a PFR or MFR? In PFR we know the rate comes out to be $(-r_a)=-\frac{dFa}{dV}$ ,can we equate $-1/V\frac{dNa}{dt} = -\frac{dFa}{dV}$ ? If not then why? Is the rate equation not valid everywhere?

 

[Chestermiller]

You are aware that, in a PFR reactor, the reactor is assumed to operate at steady state so the temperature at any fixed location in the reactor is not changing with time, correct? It is changing with spatial position (cumulative volume V) through the reactor. And, if we define V/Q as the cumulative residence time from the inlet to cumulative volume V (i.e., $t_r$), then $$\frac{dF_a}{dV}=Q\frac{dC_a}{dV}=r_a$$or$$\frac{dC_a}{dt_r}=r_a$$ So the concentration varies with the cumulative residence time in the PFR in exactly the same way it varies with the clock time for a batch reactor.

 

[Rahulx084]

okay, so does that mean we can't equate $-dCa/dt$=$\frac{Ca_o-Ca}{t_m}$ (equating design equation of cstr with rate equation), we can't do the above thing right?As here the concentration isn't dropping gradually instead its sudden drop and also there is uniformity in the cstr

 

[Chestermiller]

okay, so does that mean we can't equate $-dCa/dt$=$\frac{Ca_o-Ca}{t_m}$ (equating design equation of cstr with rate equation), we can't do the above thing right?As here the concentration isn't dropping gradually instead its sudden drop and also there is uniformity in the cstr

Yes. The equation should really read: $$\frac{(C_a-C_{a0})}{t_m}=r_a|_{C_a}$$
I've never liked the idea of writing $dC_a/dt=r_a$ because it only applies to batch reactors.

 

[Rahulx084]

Okay great , Thank you so much :)