From 02a98be18e12f50f873a6a5f693f0b620002e8fc Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Sun, 26 Dec 2021 22:00:00 +0000 Subject: [PATCH] finish notes on semester 1 thermodynamics --- .../thermodynamics.md | 159 +++++++++++++++++- 1 file changed, 158 insertions(+), 1 deletion(-) diff --git a/uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md b/uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md index c365e38..894bc36 100755 --- a/uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md +++ b/uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md @@ -40,7 +40,7 @@ The system is not in equilibrium if parts of the system are at different conditi #### Adiabatic -A process in which does not cross the system boundary +A process in which heat does not cross the system boundary ## Perfect (Ideal) Gasses @@ -204,6 +204,41 @@ Entropy is defined as the following, given that the process s reversible: $$S_2 - S_1 = \int\! \frac{\mathrm{d}Q}{T}$$ +#### Change of Entropy of a Perfect Gas + +Consider the 1st corollary of the 1st law: + +$$\mathrm dq + \mathrm dw = \mathrm du$$ + +and that the process is reversible: + +\begin{align*} +\mathrm ds &= \frac{\mathrm dq} T \bigg|_{rev} \\ +\mathrm dq = \mathrm ds \times T \\ +\mathrm dw &= -p\mathrm dv \\ +\end{align*} + +The application of the 1st corollary leads to: + +$$T\mathrm ds - p\mathrm dv = \mathrm du$$ + +Derive the change of entropy + +\begin{align*} +\mathrm ds &= \frac{\mathrm du}{T} + \frac{p \mathrm dv}{T} \\ +\\ +\mathrm du &= c_v \mathrm{d}T \\ +\frac p T &= \frac R v \\ +\\ +\mathrm ds &= \frac{c_v\mathrm{d}T}{T} \frac{R\mathrm dv}{v} \\ +s_2 - s_1 &= c_v\ln\left(\frac{T_2}{T_1}\right) + R\ln\left(\frac{v_2}{v_1}\right) +\end{align*} + +There are two other forms of the equation that can be derived: + +$$s_2 - s_1 = c_v\ln\left(\frac{p_2}{p_1}\right) + c_p\ln\left(\frac{v_2}{v_1}\right)$$ +$$s_2 - s_1 = c_p\ln\left(\frac{T_2}{T_1}\right) - R\ln\left(\frac{p_2}{p_1}\right)$$ + ### Heat Capacity and Specific Heat Capacity Heat capacity is quantity of heat required to raise the temperature of a system by a unit @@ -375,3 +410,125 @@ The 1st Law of Thermodynamics can be thought of as: > The internal energy of a closed system remains unchanged if it > [thermally isolated](#thermally-insulated-and-isolated-systems) from its surroundings + +# Polytropic Processes + +A polytropic process is one which obeys the polytropic law: + +$$pv^n = k \text{ or } p_1v_1^n = p_2v_2^n$$ + +where $n$ is a constant called the polytropic index, and $k$ is a constant too. + +A typical polytropic index is between 1 and 1.7. + +
+ + +#### Example 1 + +Derive + +$$\frac{p_2}{p_1} = \left(\frac{T_2}{T_1}\right)^{\frac{n}{n-1}}$$ + + + +\begin{align*} +p_1v_1^n &= p_2v_2^n \\ +pv &= RT \rightarrow v = R \frac{T}{p} \\ +\frac{p_2}{p_1} &= \left( \frac{v_1}{v_2} \right)^n \\ + &= \left(\frac{p_2T_1}{T_2p_1}\right)^n \\ + &= \left(\frac{p_2}{p_1}\right)^n \left(\frac{T_1}{T_2}\right)^n \\ +\left(\frac{p_2}{p_1}\right)^{1-n} &= \left(\frac{T_1}{T_2}\right)^n \\ +\frac{p_2}{p_1} &= \left(\frac{T_1}{T_2}\right)^{\frac{n}{1-n}} \\ + &= \left(\frac{T_2}{T_1}\right)^{\frac{n}{n-1}} \\ +\end{align*} + +
+ + +How did you do that last step? + + + +For any values of $x$ and $y$ + +\begin{align*} +\frac x y &= \left(\frac y x \right) ^{-1} \\ +\left(\frac x y \right)^n &= \left(\frac y x \right)^{-n} \\ +\left(\frac x y \right)^{\frac{n}{1-n}} &= \left(\frac y x \right)^{\frac{-n}{1-n}} \\ + &= \left(\frac y x \right)^{\frac{n}{n-1}} \\ +\end{align*} + +
+
+ +# Isentropic + +*Isentropic* means constant entropy: + +$$\Delta S = 0 \text{ or } s_1 = s_2 \text{ for a precess 1-2}$$ + +A process will be isentropic when: + +$$pv^\gamma = \text{constant}$$ + +This is basically the [polytropic law](#polytropic-processes) where the polytropic index, $n$, is +always equal to $\gamma$. + +
+ + +Derivation + + + +\begin{align*} +0 &= s_2 - s_1 = c_v \ln{\left(\frac{p_2}{p_1}\right)} + c_p \ln{\left( \frac{v_2}{v_1} \right)} \\ +0 &= \ln{\left(\frac{p_2}{p_1}\right)} + \frac{c_p}{c_v} \ln{\left( \frac{v_2}{v_1} \right)} \\ + &= \ln{\left(\frac{p_2}{p_1}\right)} + \gamma \ln{\left( \frac{v_2}{v_1} \right)} \\ + &= \ln{\left(\frac{p_2}{p_1}\right)} + \ln{\left( \frac{v_2}{v_1} \right)^\gamma} \\ + &= \ln{\left[\left(\frac{p_2}{p_1}\right)\left( \frac{v_2}{v_1} \right)^\gamma\right]} \\ + e^0 = 1 &= \left(\frac{p_2}{p_1}\right)\left( \frac{v_2}{v_1} \right)^\gamma \\ +&\therefore p_2v_2^\gamma = p_1v_1^\gamma \\ +\\ +pv^\gamma = \text{constant} +\end{align*} + +
+ +During isentropic processes, it is assumed that no heat is transferred into or out of the cylinder. +It is also assumed that friction is 0 between the piston and cylinder and that there are no energy +losses of any kind. + +This results in a reversible process in which the entropy of the system remains constant. + +An isentropic process is an idealization of an actual process, and serves as the limiting case for +real life processes. +They are often desired and often the processes on which device efficiencies are calculated. + +## Heat Transfer During Isentropic Processes + +Assume that the compression 1-2 follows a polytropic process with a polytropic index $n$. +The work transfer is: + +$$W = - \frac{1}{1-n} (p_2V_2 - p_1V_1) = \frac{mR}{n-1} (T_2-T_1)$$ + +Considering the 1st corollary of the 1st law, $Q + W = \Delta U$, and assuming the gas is an ideal +gas [we know that](#obey-the-perfect-gas-equation) $\Delta U = mc_v(T_2-T_1)$ we can deduce: + +\begin{align*} +Q &= \Delta U - W = mc_v(T_2-T_1) - \frac{mR}{n-1} (T_2-T_1) \\ + &= m \left(c_v - \frac R {n-1}\right)(T_2-T-1) +\end{align*} + +Now, if the process was *isentropic* and not *polytropic*, we can simply substitute $n$ for +$\gamma$ so now: + +$$Q = m \left(c_v - \frac R {\gamma-1}\right)(T_2-T-1)$$ + +But since [we know](#relationship-between-specific-gas-constant-and-specific-heats) $c_v = \frac R {\gamma - 1}$: + +$$Q = m (c_v-c_v)(T_2-T_1) = 0 $$ + +This proves that the isentropic version of the process adiabatic (no heat is transferred across the +boundary).