Expanding exponential function

Question

$W((\exp⁡(UA/WC)-1)/\exp⁡(UA/WC)) = K$

How to obtain W? This is derived from an engineering equation (isothermal cooling jacket).

Answer 1

In order to avoid further confusion, let me rewrite the equation as $$x e^{-\frac{A U}{C x}} \left(e^{\frac{A U}{C x}}-1\right)=K$$ where $x$ is your $W$. Now, let $y=\frac{A U}{C x}$ to make the equation $$\frac{A U e^{-y} \left(e^y-1\right)}{C y}=K$$ that is to say $$\frac{1-e^{-y}}{y}=\alpha$$ where $\alpha=\frac{C K}{A U}$.

There is no elementary solution to this equation but it can express in terms of Lambert function and the solution is $$y=\frac{1}{\alpha }+W\left(-\frac{e^{-1/\alpha }}{\alpha }\right)$$ The Wikipedia page gives many examples of the manners things have to be manipulated to arrive to such expressions.

The Wikipedia page gives also series expansions for the evaluation of $W(.)$.

If you cannot use Lambert function, only numerical methods would solve the equation.

More generally, any equation which can write or rewrite $A+Bx+C\log(D+Ex)=0$ shows explict solution(s) in terms of Lambert function (i real and/or complex domains).