Purpose

For a given charge , The change of burning surface area with meat thickness $A_{\mathrm{b}}\left(e\right)$ It is known. , It is necessary to calculate the relationship between combustion chamber pressure and time according to this curve $p_{\mathrm{c}}\left(t\right)$

Continuity equation

1 The flow rate of gas produced by charge combustion

$${\dot{m}}_1={\rho }_{\mathrm{p}}{A}_{\mathrm{b}}r=a{\rho }_{\mathrm{p}}{A}_{\mathrm{b}}{p}_{\mathrm{c}}^{\mathrm{n}}$$ here ${\rho }_{\mathrm{p}}$ Is the density of the charge ,$r$ Is the linear burning rate , You can use $a{p}_{\mathrm{c}}^{\mathrm{n}}$ Empirical formula calculation

2 The flow of gas from the nozzle

$${\dot{m}}_2=\frac{p_{\mathrm{c}}{A}_{\mathrm{t}}}{c^{\ast }}$$ In style $c^{\ast }$ Is the characteristic velocity of propellant ,$A_{\mathrm{t}}$ Is the throat area of the nozzle , It should be noted that , The premise of using this formula is that the nozzle is in the maximum flow state （ Throat speed ）, Therefore, it may not be applicable to the engine starting process and shutdown

3 The gas flow accumulated in the combustion chamber

$${\dot{m}}_3=\frac{\mathrm{d}\left({\rho }_{\mathrm{g}}{V}_{\mathrm{f}}\right)}{\mathrm{d}t}$$ In style $V_{\mathrm{f}}$ Is the volume of the combustion chamber cavity ,${\rho }_{\mathrm{g}}$ Is the density of the gas in the combustion chamber , Assume that the temperature and molecular weight of the gas remain unchanged , Bring the complete gas equation of state into , Yes
$${\dot{m}}_3=\frac{1}{RT}\frac{\mathrm{d}\left(p_{\mathrm{c}}{V}_{\mathrm{f}}\right)}{\mathrm{d}t}=\frac{1}{{\left(c^{\ast }\Gamma \right)}^2}\frac{\mathrm{d}\left(p_{\mathrm{c}}{V}_{\mathrm{f}}\right)}{\mathrm{d}t}$$ In style $R$ Is the complete gas constant ,$T$ It's the gas temperature . In order to avoid directly calculating the gas temperature （ Chemical equilibrium calculation is required , Not easy to use ）, So $\Gamma$ function , Use ${\left(c^{\ast }\Gamma \right)}^2$ Replace $RT$, It can be proved that the two are equal
$$\Gamma =\sqrt{\gamma }{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2\left(\gamma -1\right)}}$$ In style $\gamma$ Is the specific heat ratio , It can be simply taken as 1.2
So use the derivative formula of the product , Yes
$${\dot{m}}_3=\frac{1}{{\left(c^{\ast }\Gamma \right)}^2}\frac{\mathrm{d}\left(p_{\mathrm{c}}{V}_{\mathrm{f}}\right)}{\mathrm{d}t}=\frac{1}{{\left(c^{\ast }\Gamma \right)}^2}\left(p_{\mathrm{c}}\frac{\mathrm{d}{V}_{\mathrm{f}}}{\mathrm{d}t}+V_{\mathrm{f}}\frac{\mathrm{d}{p}_{\mathrm{c}}}{\mathrm{d}t}\right)$$

4 Flow balance

$${\dot{m}}_1={\dot{m}}_2+{\dot{m}}_3$$ There are
$$a{\rho }_{\mathrm{p}}{A}_{\mathrm{b}}{p}_{\mathrm{c}}^{\mathrm{n}}=\frac{p_{\mathrm{c}}{A}_{\mathrm{t}}}{c^{\ast }}+\frac{1}{{\left(c^{\ast }\Gamma \right)}^2}\left(p_{\mathrm{c}}\frac{\mathrm{d}{V}_{\mathrm{f}}}{\mathrm{d}t}+V_{\mathrm{f}}\frac{\mathrm{d}{p}_{\mathrm{c}}}{\mathrm{d}t}\right)\text{\hspace{1em}(1)}$$

Burning surface displacement equation

Burning will cause the burning surface to move , Yes
$$\frac{\mathrm{d}e}{\mathrm{d}t}=a{p}_{\mathrm{c}}^{\mathrm{n}}\text{\hspace{1em}(2)}$$ Combustion will also cause the volume of the combustion chamber cavity to expand , Yes
$$\frac{\mathrm{d}{V}_{\mathrm{f}}}{\mathrm{d}e}=A_{\mathrm{b}}$$ Combine these two formulas , It's not hard to get
$$\frac{\mathrm{d}{V}_{\mathrm{f}}}{\mathrm{d}t}=A_{\mathrm{b}}a{p}_{\mathrm{c}}^{\mathrm{n}}\text{\hspace{1em}(3)}$$

The final set of differential equations

Will type $(3)$ Plug in $(1)$ It can be reduced to
$$a{\rho }_{\mathrm{p}}{A}_{\mathrm{b}}{p}_{\mathrm{c}}^{\mathrm{n}}=\frac{p_{\mathrm{c}}{A}_{\mathrm{t}}}{c^{\ast }}+\frac{1}{{\left(c^{\ast }\Gamma \right)}^2}\left(A_{\mathrm{b}}a{p}_{\mathrm{c}}^{\mathrm{n}+1}+V_{\mathrm{f}}\frac{\mathrm{d}{p}_{\mathrm{c}}}{\mathrm{d}t}\right)\text{\hspace{1em}(4)}$$ Will type $(2)(3)(4)$ After finishing , Write them together to get the final set of differential equations
$$\frac{\mathrm{d}{p}_{\mathrm{c}}}{\mathrm{d}t}=\frac{{\left(c^{\ast }\Gamma \right)}^2}{V_{\mathrm{f}}}\left(a{\rho }_{\mathrm{p}}{A}_{\mathrm{b}}{p}_{\mathrm{c}}^{\mathrm{n}}-\frac{p_{\mathrm{c}}{A}_{\mathrm{t}}}{c^{\ast }}\right)-\frac{A_{\mathrm{b}}a{p}_{\mathrm{c}}^{\mathrm{n}+1}}{V_{\mathrm{f}}}$$$$\frac{\mathrm{d}{V}_{\mathrm{f}}}{\mathrm{d}t}=A_{\mathrm{b}}a{p}_{\mathrm{c}}^{\mathrm{n}}$$$$\frac{\mathrm{d}e}{\mathrm{d}t}=a{p}_{\mathrm{c}}^{\mathrm{n}}$$ Given Initial conditions after , Numerical calculation method can be used to solve