Query regarding approach to solve a given differential equation.

Question

There's a equation

$$N(t) = N(t)\frac{P(t,z)}{B}-C\frac{d(P(t,z))}{dz}$$ $$N(t) = A\frac{dP(t,z)}{dt}$$

Constants:

B,

C=3.9878*10⁻⁷,

$A=0.11941$,

Variables:

N(t) is a function of t and is defined at a point along z, it's a vector quantity.

P(t) is a function of t as well as z

Background:

I encountered this equation when I was analysing a mass transfer problem. N is the flux of molecules across the surface of a liquid in contact with an air film of a certain thickness. P is the partial pressure at point z at time t. The situation is as follows: a volatile chemical spills on the floor of a poorly ventilated laboratory. We have to find out the time required for the complete evaporation of the chemical. Since the space is poorly ventilated, the vapour will accumulate in the air, reducing the ease with which the liquid can evaporate. This will also reduce N over time.

My Question: How should I solve the differential equation mentioned at the start of the question? I am looking for a method that can be done by hand and is not too complicated.

My attempt: I attempted to solve the question by using computational techniques. I plotted the moles of chemical in air against time and used curve fitting to obtain a second-degree polynomial that fit 100% and a linear equation that fit 99%. However, this method requires a computer, which is not available in the exam hall.

For anyone interested in actual question:

Problem: A reagent bottle containing 3 kg of isopropanol (C$_3$H$_8$O) breaks and spills on the floor of a 3 m x 4 m x 3 m room at 27°C and 1 atm. The liquid evaporates and mixes with the air in the room.

Objective: To determine the thickness of the air film through which the isopropanol vapor diffuses and the time required for complete evaporation under two scenarios: (a) with ventilation and (b) without ventilation.Make necessary assumptions and and data.

Answer 1

Hint.

Substituting the second equation into the first we have the quasi linear PDE

$$ AP_t-\frac AB P P_t + C P_z=0 $$

so we have

$$ \frac{dt}{A-\frac AB P}=\frac{dz}{C}=\frac {dP}{0} $$

and solving

$$ \cases{ P = k_1\\ \frac{t}{A-\frac AB k_1}-\frac zC = k_2 } $$

then a general solution $P$ obeys

$$ \phi\left(P,\frac{t}{A-\frac AB P}-\frac zC\right)=0 $$

Answer 2

$$A\frac{\partial P}{\partial t} = A\frac{\partial P}{\partial t}\frac{P}{B}-C\frac{\partial (P)}{\partial z}$$ $$A\left(\frac{P}{B}-1\right)\frac{\partial P}{\partial t}-C\frac{\partial P}{\partial z}=0 $$ This is a first order PDE. The Charpit-Lagrange characteristic ODEs are : $$\frac{dt}{A\left(\frac{P}{B}-1\right)}=\frac{dz}{-C}=\frac{dP}{0}$$ An obvious first characteristic equation comes from $dP=0$ : $$P=c_1$$ A second characteristic equation comes from solving $\frac{dt}{A\left(\frac{c_1}{B}-1\right)}=\frac{dz}{-C}$ : $$A\left(\frac{c_1}{B}-1\right)z+Ct=c_2$$ The general solution of the PDE on implicit form $c_2=F(c_1)$ is : $$\boxed{A\left(\frac{P}{B}-1\right)z+Ct=F(P)}$$ $F$ is an arbitrary function. They are infinity many solutions of the PDE.

If some valid initial condition is specified one can determine the function $F$.

In order to go further in the solving please add to the question the initial condition clearly expressed on mathematical form.