My question is the last part of the d) part of the exercise 1.17 in Mark Holms' Introduction to Applied Mathematics. The exercise is given below, where I have emphasized the part of it that is my question.
1.17. A thermodynamic model for the concentration $u$ and temperature $q$ of a mixture consists of the following equations( Gray and Scott [1994])
$\frac{\text{d}u}{\text{d}t}=k_1-k_2ue^{k_3q}$,
$\frac{\text{d}q}{\text{d}t}=k_4ue^{k_3q}-k_5q$.
The initial conditions are $u=0$ and $q=0$ at $t=0$.
(a) What are the dimensions of the $k_i$'s?
(b) Explain why the rule of thumb for scaling used in the projectile problem does not help here
(c) Find the steady-state solution, that is, the solution of the differential equations with $u'=0$ and $q'=0$
(d) Nondimensionalize the problem using the steady-state solution from (c) to scale $u$ and $q$. Make sure to explain how you selected the scaling for $t$
Please note that this is not homework, but part of a self-study.
Suggested solution (shown only for (d)):
Result from (c):
$q_{\text{eq}}=\frac{k_4k_1}{k_2k_5}$
$u_{\text{eq}}=\frac{k_1}{k_2}e^{-\frac{k_3k_4k_1}{k_2k_5}}$
To do the scaling I first need to make a change of variables:
$u=u_cu^*$
$q=q_cq*$
$t=t_ct^*$
By using the chain rule, we get that the system of the ODEs becomes:
$\frac{u_c}{t_c}\frac{\text{d}u*}{\text{d}t*}=k_1-k_2u_cu^*e^{k_3q_cq^*}$
$\frac{q_c}{t_c}\frac{\text{d}q*}{\text{d}t*}=k_4u_cu^*e^{k_3q_cq^*}-k_5q_cq^*$
Then we need to form dimensionless groups of the parameters in the ODEs. We get that:
$\frac{u_c}{k_1t_c}\frac{\text{d}u*}{\text{d}t*}=1-\frac{k_2}{k_1}u_cu^*e^{k_3q_cq^*}$
$\frac{1}{k_5t_c}\frac{\text{d}q*}{\text{d}t*}=\frac{k_4u_cu^*}{k_5q_c}e^{k_3q_cq^*}-q^*$
so that the dimensionless independent groups are:
$\Pi_1=\frac{1}{k_5t_c}$
$\Pi_2=\frac{k_4u_c}{k_5q_c}$
$\Pi_3=\frac{u_c}{k_1t_c}$
$\Pi_4=\frac{k_2}{k_1}u_c$
$\Pi_5=k_3q_c$
We know that we should choose $u_c=u_{\text{eq}}$ and $q_c=q_{\text{eq}}$. For $t_c$ I see 3) possible solutions:
1) Find the time it takes for the solution, starting with $q=0, u=0$ at $t=0$ to reach $u_{\text{eq}}$ and $q_{\text{eq}}$
2) Set $\Pi_1=1$
3) Set $\Pi_3=1$
To use 1), we need to solve the system of ODEs, so we do not use 1). We are then left with 2) and 3). Choosing between 2) and 3) is choosing which of the terms $\frac{\text{d}q*}{\text{d}t*}$ and $\frac{\text{d}u*}{\text{d}t*}$ is the most important. If for instance I set $\Pi_1=1$, then $\frac{\text{d}q*}{\text{d}t*}$ is the most important term of the two mentioned.
Is my argumentation correct?