From 6029a956986f4a02e4d5bd403b9e3de3503378f0 Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Wed, 2 Mar 2022 01:36:22 +0000 Subject: [PATCH] most of the notes on feb 22nd --- .../fluid_dynamics.md | 154 ++++++++++++++++++ ...ot-2022-03-01T22:47:31,932087932+00:00.png | Bin 0 -> 26605 bytes ...ot-2022-03-01T22:59:14,129582752+00:00.png | Bin 0 -> 9184 bytes 3 files changed, 154 insertions(+) create mode 100644 uni/mmme/1048_thermodynamics_and_fluid_mechanics/images/vimscrot-2022-03-01T22:47:31,932087932+00:00.png create mode 100644 uni/mmme/1048_thermodynamics_and_fluid_mechanics/images/vimscrot-2022-03-01T22:59:14,129582752+00:00.png diff --git a/uni/mmme/1048_thermodynamics_and_fluid_mechanics/fluid_dynamics.md b/uni/mmme/1048_thermodynamics_and_fluid_mechanics/fluid_dynamics.md index 259fe4c..b4e1f46 100755 --- a/uni/mmme/1048_thermodynamics_and_fluid_mechanics/fluid_dynamics.md +++ b/uni/mmme/1048_thermodynamics_and_fluid_mechanics/fluid_dynamics.md @@ -384,3 +384,157 @@ Head form: \frac{p_s}{\rho g} + z + \frac{v^2}{2g} &= \text{total head} \\ \frac{p_s}{\rho g} + z &= \text{piezometric head} \end{align*} + +# Steady Flow Energy Equation (SFEE) + +SFEE is a more general equation that can be applied to **any fluid** and also is also takes +**heat energy** into account. +This is useful in applications such as a fan heater, jet engines, ICEs, and steam turbines. + +The equation deals with 3 types of energy tranfer: + +1. Thermal energy transfer (e.g. heat tranfer from central heating to a room) +2. Work energy transfer (e.g. shaft from car engine that turns wheels) +3. Energy transfer in fluid flows (e.g. heat energy in a flow, potential energy in a flow, kinetic + energy in a flow) + +## Derivation of Steady Flow Energy Equation + +#### Consider a control volume with steady flows in and out and steady transfers of work and heat. + +The properties don't change with time at any any location and are considered uniform over inlet and +outlet areas $A_1$ and $A_2$. + +For steady flow, the mass, $m$, of the fluid **within the control volume** and the total energy, $E$, +must be constant. + +$E$ includes **all forms for energy** but we only consider internal, kinetic, and potential energy. + +#### Consider a small time interval $\delta t$. + +During $\delta t$, mass $\delta m_1$ enters the control volume and $\delta m_2$ leaves: + +![](./images/vimscrot-2022-03-01T22:47:31,932087932+00:00.png) + +The specific energy $e_1$ of fluid $\delta m_1$ is the sum of the specific internal energy, specific +kinetic energy, and specific potential energy: + +$$e_1 = u_1 + \frac{v_1^2}{2} gz_1$$ +$$e_2 = u_2 + \frac{v_2^2}{2} gz_2$$ + +Since the mass is constant in the control volume, $\delta m_1 = \delta m_2$. + +#### Applying the First Law of Thermodynamics + +The control volume is a system for which $\delta E_1$ is added and $\delta E_2$ is removed:: + +$$\delta E = \delta E_2 - \delta E_1$$ + +$E$ is constant so applying the +[first law of thermodynamics](thermodynamics.html#st-law-of-thermodynamics) +we know that: + +$$\delta Q + \delta W = \delta E$$ + +We can also say that: + +$$\delta E = \delta E_2 - \delta E_1 = \delta m(e_2 - e_1)$$ + +#### The Work Term + +The work term, $\delta W$, is mae up of shaft work **and the work necessary to deform the system** +(by adding $\delta m_1$ at the inlet and removing $\delta m_2$ at the outlet): + +$$\delta W = \delta W_s + \text{net flow work}$$ + +Work is done **on** the system by the mass entering and **by** the system on the mass leaving. + +For example, at the inlet: + +![](./images/vimscrot-2022-03-01T22:59:14,129582752+00:00.png) + +$$\text{work done on system} = \text{force} \times \text{distance} = p_1A_1\delta x = p_1\delta V_1$$ + +Knowing this, we can write: + +$$\delta W = \delta W_s + (p_1\delta V_1 - p_2\delta V_2)$$ + +#### Back to the First Law + +Substituting these equations: + +$$\delta E = \delta E_2 - \delta E_1 = \delta m(e_2 - e_1)$$ +$$\delta W = \delta W_s + (p_1\delta V_1 - p_2\delta V_2)$$ + +into: + +$$\delta Q + \delta W = \delta E$$ + +gives us: + +$$\delta Q + \left[ \delta W_s + (p_1\delta V_1 - p_2\delta V_2)\right] = \delta m (e_2-e_1)$$ + +Dividing everything by $\delta m$ and with a bit of rearranging we get: + +$$q + w_s = e_2-e_1 + \frac{p_2}{\rho_2} - \frac{p_1}{\rho_1}$$ + +#### Substiute Back for $e$ + +$$e = u + \frac{v^2}{2} + gz$$ + +This gives us: + +$$q + w_s + \left[ u_2 + \frac{p_2}{\rho_2} + gz_2 + \frac{v_2^2}{2} \right] - \left[ u_1 + \frac{p_1}{\rho_1} + gz_1 + \frac{v_1^2}{2} \right]$$ + +#### Rearrange and Substitute for Enthalpy + +By definition, enthalpy $h = u + pv = u + \frac p \rho$. +This gives us the equation: + +$$q + w_s = (h_2 - h_1) + g(z_2-z_1) + \frac{v_2^2-v_1^2}{2}$$ + +This equation is in specific energy form. + +Multiplying by mass flow rate will give you the power form. + +## Application of the Steady Flow Energy Equation + +#### Heat Transfer Devices + +Like heat exchangers, boilers, condensers, and furnaces. + +In this case, $\dot W = 0$, $\delta z ~ 0$, and $\delta v^2 ~ 0$ so the equation can be simplified +to just + +$$\dot Q = \dot m(h_2-h_1) = \dot m c_p(T_2-T_1)$$ + +#### Throttle Valve + +No heat and work transfer. +Often you can neglect potential and kinetic energy terms, giving you: + +$$0 = h_2-h_1)$$ + +#### Work Transfer Devices + +e.g. Turbines, Pumps, Fans, and Compressors + +For these there is often no heat transfer ($\dot Q = 0$) and we can neglect potential +($\delta z ~ 0$) and kinetic ($\delta v^2 ~ 0$) energy terms, giving us the equation + +$$\dot W = \dot m (h_2-h_1) = \dot m c_p(T_2-T_1)$$ + +#### Mixing Devices + +e.g. hot and cold water in a shower + +In these processes, work and heat transfers are not important and you can often +neglect potential and kinetic energy terms, giving us the same equation as for the throttle valve +earlier: + +$$0 = h_2-h_1$$ + +which you may want to write more usefully as: + +$$\sum \dot m h_{out} = \sum \dot m h_{in}$$ + diff --git a/uni/mmme/1048_thermodynamics_and_fluid_mechanics/images/vimscrot-2022-03-01T22:47:31,932087932+00:00.png b/uni/mmme/1048_thermodynamics_and_fluid_mechanics/images/vimscrot-2022-03-01T22:47:31,932087932+00:00.png new file mode 100644 index 0000000000000000000000000000000000000000..a2099545a9a8f16ede5809a91f932a356144e3d3 GIT binary patch literal 26605 zcmeFZi945T+ctb5rBGBPWynxbL{dtI_@yX`NSTLZo+_C#G?1}Dp+S_$l&Q=^WGqc) z$yiEeDIvnUpVqpc_j$JW`@TQm-M+SMt$W?^yMEVop2v9{`?2r)aRnYda)5Oi_cDs2 zSXB?wv?z)}jH2ilF)ha5tgd)@n4u`)xgNpKL?5v5XUFZFkF0F$FCE7z9KyQ_>F7uA!{ 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