1. **Problem Statement:** Find the washout function for both Continuous Stirred Tank Reactor (CSTR) and pipe flow using Laplace transforms for laminar and turbulent flow conditions. 2. **Background:** The washout function describes the response of a system to an impulse input, often used in chemical engineering to characterize reactors and flow systems. 3. **CSTR Washout Function:** - The governing equation for concentration $C(t)$ in a CSTR with flow rate $Q$, volume $V$, and input concentration $C_0$ is: $$\frac{dC}{dt} = \frac{Q}{V}(C_0 - C)$$ - Taking Laplace transform (denote $\mathcal{L}\{C(t)\} = \hat{C}(s)$): $$s\hat{C}(s) - C(0) = \frac{Q}{V}(\frac{C_0}{s} - \hat{C}(s))$$ - Assuming zero initial concentration $C(0)=0$ and impulse input $C_0=1$: $$s\hat{C}(s) = \frac{Q}{V}\frac{1}{s} - \frac{Q}{V}\hat{C}(s)$$ - Rearranged: $$\hat{C}(s)\left(s + \frac{Q}{V}\right) = \frac{Q}{V}\frac{1}{s}$$ - So the Laplace domain washout function is: $$\hat{C}(s) = \frac{\frac{Q}{V}}{s\left(s + \frac{Q}{V}\right)}$$ - Inverse Laplace transform gives the time domain washout function: $$C(t) = 1 - e^{-\frac{Q}{V}t}$$ 4. **Pipe Flow Washout Function:** - For laminar flow, the velocity profile is parabolic, and the washout function can be modeled by the axial dispersion model: $$\frac{\partial C}{\partial t} + u \frac{\partial C}{\partial x} = D \frac{\partial^2 C}{\partial x^2}$$ where $u$ is average velocity, $D$ is dispersion coefficient. - Taking Laplace transform in time and solving with boundary conditions yields the washout function in Laplace domain: $$\hat{C}(s) = \exp\left(-\frac{x}{2D}(u - \sqrt{u^2 + 4Ds})\right)$$ - For turbulent flow, dispersion is much higher, and $D$ is replaced by an effective turbulent dispersion coefficient $D_t$ which is larger. - The washout function form remains similar but with $D_t$ instead of $D$: $$\hat{C}(s) = \exp\left(-\frac{x}{2D_t}(u - \sqrt{u^2 + 4D_ts})\right)$$ 5. **Summary:** - CSTR washout function in time domain: $$C(t) = 1 - e^{-\frac{Q}{V}t}$$ - Pipe flow washout function in Laplace domain: $$\hat{C}(s) = \exp\left(-\frac{x}{2D}(u - \sqrt{u^2 + 4Ds})\right)$$ for laminar flow. - Replace $D$ by $D_t$ for turbulent flow. 6. **Interpretation:** - The CSTR washout function shows exponential approach to steady state. - The pipe flow washout function accounts for convection and dispersion effects, with dispersion higher in turbulent flow. This detailed derivation uses Laplace transforms to find washout functions for both reactor and pipe flow under laminar and turbulent conditions.