How to extract pure potassium carbonate from ash?
The composition of the wood ashes varies with the nature and the origin of the wood. In the average, it is a mixture of $40$ % - $70$ % $\ce{CaCO3}$, $5$% - $10$% $\ce{MgCO3}$, $5$% - $10$% $\ce{K2CO3}$, $5$% - $10$% $\ce{Na2CO3}$, $2$% - $5$% $\ce{SiO2}$, $2$% - $5$% $\ce{Ca3(PO4)2}$, $0.5$% -$2$% $\ce{Al2O3}$, &lt;$1$% $\ce{NaCl}$, &lt;$1$% $\ce{KCl}$, plus traces of manganese and iron. So it is difficult to purify, and/or to extract any of these compounds, as a lot of these compounds are insoluble in water: $\ce{CaCO3, MgCO3}$, $\ce{SiO2, Ca3(PO4)2, Al2O3}$. The only soluble compounds are $\ce{K2CO3}$ and $\ce{Na2CO3}$, plus traces of $\ce{NaCl}$ and $\ce{KCl}$, and they form less than $20$% of the total mass of the ashes. The solution obtained from washing the wood ashes is a mixture of sodium and potassium carbonate. The problem is how to separate those two carbonates. Fractional crystallization does not seem easy. I would rather transform them into nitrates by adding nitric acid : at $0°$C, $\ce{KNO3}$ is  not very soluble ($5$ g/$100$ mL), which is much less than $\ce{NaNO3}$ ($73$ g/$100$ mL). So obtaining $\ce{KNO3}$ from the mixture by cooling a hot solution should be relatively easy.
Ref.: J. Hartwig, Ann. der Chem. u. Pharm. XLVI, p. 97. A. Wurtz, Dictionnaire de chimie pure et appliquée, Vol. 1, p. 641 (1891).