> For the complete documentation index, see [llms.txt](https://wiki.gtpropulsivelanders.org/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://wiki.gtpropulsivelanders.org/propulsive-landers-gt-docs/wiki/hybrid-governing-relations.md). # Hybrid Governing Relations {% embed url="" %} Team's favorite reference for these are: * Highly recommended for those who wish to understand in-depth how design choices affect hybrid performance and how changing one variable lead to change in another. * Contains all the equations up to the point which you will need to design some sort of solver to "optimize" for the length of your engine. In air quotes because there are different levels of optimization! ### Port Diameter Growth For constant oxidizer mass flow rate, the port diameter as a function of time is: $$ D\_p(t)=\left\[D\_{pi}^{2n+1}+\frac{(2n+1)2^{2n+1}a}{\pi^n}\dot{m}\_{ox}^{,n}t\right]^{\frac{1}{2n+1}} $$ ### Oxidizer Mass Flux The instantaneous oxidizer mass flux through the port is: $$ G\_{ox}(t)=\frac{4\dot{m}\_{ox}}{\pi D\_p^2} $$ ### Regression Rate The fuel regression rate is modeled as: $$ \dot{r}(t)=aG\_{ox}^{n} $$ ### Fuel Mass Flow Rate The instantaneous fuel mass flow rate is: $$ \dot{m}\_f(t)=\rho\_f \pi D\_p L \dot{r} $$ ### Oxidizer-to-Fuel Ratio The instantaneous mixture ratio is: $$ \frac{O}{F}=\frac{\dot{m}\_{ox}}{\dot{m}\_f} $$ ### Chamber Pressure Using total mass flow rate, nozzle throat area, discharge coefficient, and characteristic velocity: $$ P(t)=\frac{(\dot{m}*{ox}+\dot{m}*f)c^\**{theo}\eta\_c}{A*{nt}C\_d} $$ ### Thrust The thrust can be estimated from total mass flow rate, characteristic velocity, combustion efficiency, thrust coefficient, and nozzle efficiency: $$ T(t)=(\dot{m}*{ox}+\dot{m}*f)c^\**{theo}\eta\_c C*{F,theo}\eta\_n $$ ### Initial and Final Port Diameter Relation For constant oxidizer flow rate, the initial and final port diameters are related by: $$ \left(\frac{D\_{pf}}{D\_{pi}}\right)^{2n+1}-1= \frac{(2n+1)2^{2n+1}a}{D\_{pi}^{2n+1}\pi^n}\dot{m}\_{ox}^{,n}t\_b $$ ### Final Port Diameter From Total Oxidizer Mass The final port diameter can also be related to total oxidizer mass: $$ D\_{pf}= \left\[ \left( \frac{(2n+1)2^{2n+1}a}{\pi^n} \right) \frac{M\_{ox}^{n}t\_b^{1-n}} {1-\left(D\_{pi}/D\_{pf}\right)^{2n+1}} \right]^{\frac{1}{2n+1}} $$ ### Fuel Grain Length The fuel grain length can be related to total oxidizer mass, target O/F, fuel density, and port diameter change: $$ L= \frac{4M\_{ox}} {\pi \rho\_f (O/F)\left(D\_{pf}^2-D\_{pi}^2\right)} $$ --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter: ``` GET https://wiki.gtpropulsivelanders.org/propulsive-landers-gt-docs/wiki/hybrid-governing-relations.md?ask=&goal= ``` `ask` is the immediate question: it should be specific, self-contained, and written in natural language. `goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.