From b05441973e5adadbc72f050ddfb68712d12e5b3d Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Sun, 26 Dec 2021 22:58:19 +0000 Subject: [PATCH] attempt to better organise thermodynamics page --- .../thermodynamics.md | 426 +++++++++--------- 1 file changed, 213 insertions(+), 213 deletions(-) diff --git a/uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md b/uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md index 894bc36..524dff9 100755 --- a/uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md +++ b/uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md @@ -116,211 +116,6 @@ c_p &= \frac{c_p}{\gamma} + R \\ c_p &= \frac{\gamma}{\gamma -1} R \end{align*} - - - - -## Properties of State - -*State* is defined as the condition of a system as described by its properties. -The state may be identified by certain observable macroscopic properties. -These properties are the *properties of state* and they always have the same values for a given -state. - -A *property* can be defined as any quantity that depends on the *state* of the system and is -independant of the path by which the system arrived at the given state. -Properties determining the state of a thermodynamic system are referred to as *thermodynamic -properties* of the *state* of the system. - -Common properties of state are: - -- Temperature -- Pressure -- Mass -- Volume - -And these can be determined by simple measurements. -Other properties can be calculated: - -- Specific volume -- Density -- Internal energy -- Enthalpy -- Entropy - -### Intensive vs Extensive Properties - -In thermodynamics we distinguish between *intensive*, *extensive*, and *specific* properties: - -- Intensive --- properties which do not depend on mass (e.g. temperature) -- Extensive --- properties which do depend on the mass of substance in a system (e.g. volume) -- Specific (extensive) --- extensive properties which are reduced to unit mass of substance - (essentially an extensive property divided by mass) (e.g. specific volume) - -### Units - -
- -Property | Symbol | Units | Intensive | Extensive ---------------- | ------ | --------------- | --------- | --------- -Pressure | p | Pa | Yes | -Temperature | T | K | Yes | -Volume | V | m$^3$ | | Yes -Mass | m | kg | | Yes -Specific Volume | v | m$^3$ kg$^{-1}$ | Yes | -Density | $\rho$ | kg m$^{-3}$ | Yes | -Internal Energy | U | J | | Yes -Entropy | S | J K$^{-1}$ | | Yes -Enthalpy | H | J | | Yes - -
- -### Density - -For an ideal gas: - -$$\rho = \frac{p}{RT}$$ - -### Enthalpy and Specific Enthalpy - -Enthalpy does not have a general physical interpretation. -It is used because the combination $u + pv$ appears naturally in the analysis of many -thermodynamic problems. - -The heat transferred to a closed system undergoing a reversible constant pressure process is equal -to the change in enthalpy of the system. - -Enthalpy is defined as: - -$$H = U+pV$$ - -and Specific Enthalpy: - -$$h = u + pv$$ - -### Entropy and Specific Entropy - -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 -temperature: - -$$C = \frac{\mathrm{d}Q}{\mathrm{d}T}$$ - -Specific heat capacity is the quantity of heat required to raise the temperature of a unit mass -substance by a unit temperature: - -$$c = \frac{\mathrm{d}q}{\mathrm{d}T}$$ - -
- - -#### Heat Capacity in Closed Systems and Internal Energy - -The specific heat transfer to a closed system during a reversible constant **volume** process is -equal to the change in specific **internal energy** of the system: - -$$c_v = \frac{\mathrm{d}q}{\mathrm{d}T} = \frac{\mathrm{d}u}{\mathrm{d}T}$$ - - - -This is because if the change in volume, $\mathrm{d}v = 0$, then the work done, $\mathrm{d}w = 0$ -also. -So applying the (1st Corollary of the) 1st Law to an isochoric process: - -$$\mathrm{d}q + \mathrm{d}w = \mathrm{d}u \rightarrow \mathrm{d}q = \mathrm{d}u$$ - -since $\mathrm{d}w = 0$. - -
- -
- - -#### Heat Capacity in Closed Systems and Enthalpy - -The specific heat transfer to a closed system during a reversible constant **pressure** process is -equal to the change in specific **enthalpy** of the system: - -$$c_p = \frac{\mathrm{d}q}{\mathrm{d}T} = \frac{\mathrm{d}h}{\mathrm{d}T}$$ - - - -This is because given that pressure, $p$, is constant, work, $w$, can be expressed as: - -$$w = -\int^2_1\! p \,\mathrm{d}v = -p(v_2 - v_1)$$ - -Applying the (1st corollary of the) 1st law to the closed system: - -\begin{align*} - q + w &= u_2 - u_1 \rightarrow q = u_2 - u_1 + p(v_2 - v_1) \\ - q &= u_2 + pv_2 - (u_1 + pv_1) \\ - &= h_2 - h_1 = \mathrm{d}h \\ - \therefore \mathrm{d}q &= \mathrm{d}h -\end{align*} - -
- -
- - -#### Ratio of Specific Heats - -$c_p > c_v$ is always true. - - - -Heating a volume of fluid, $V$, at a constant volume requires specific heat $q_v$ where - -$$q_v = u_2 - u_1 \therefore c_v = \frac{q_v}{\Delta T}$$ - -Heating the same volume of fluid but under constant pressure requires a specific heat $q_p$ where - -$$q_p =u_2 - u_1 + p(v_2-v_1) \therefore c_p = \frac{q_p}{\Delta T}$$ - -Since $p(v_2-v_1) > 0$, $\frac{q_p}{q_v} > 1 \therefore q_p > q_v \therefore c_p > c_v$. - -The ratio $\frac{c_p}{c_v} = \gamma$ -
## Thermodynamic Processes and Cycles @@ -355,7 +150,7 @@ Constant volume process ## Heat and Work -Heat and Work are different forms of enery transfer. +Heat and Work are different forms of energy transfer. They are both transient phenomena and systems never possess heat or work. Both represent energy crossing boundaries when a system undergoes a change of state. @@ -385,10 +180,6 @@ In thermally insulated systems and isolated systems, heat transfer cannot take p In thermally isolated systems, work transfer cannot take place. -# Process and State Diagrams - -Reversible processes are represented by solid lines, and irreversible processes by dashed lines. - # 1st Law of Thermodynamics The 1st Law of Thermodynamics can be thought of as: @@ -411,7 +202,216 @@ 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 +# Properties of State + +*State* is defined as the condition of a system as described by its properties. +The state may be identified by certain observable macroscopic properties. +These properties are the *properties of state* and they always have the same values for a given +state. + +A *property* can be defined as any quantity that depends on the *state* of the system and is +independant of the path by which the system arrived at the given state. +Properties determining the state of a thermodynamic system are referred to as *thermodynamic +properties* of the *state* of the system. + +Common properties of state are: + +- Temperature +- Pressure +- Mass +- Volume + +And these can be determined by simple measurements. +Other properties can be calculated: + +- Specific volume +- Density +- Internal energy +- Enthalpy +- Entropy + +## Intensive vs Extensive Properties + +In thermodynamics we distinguish between *intensive*, *extensive*, and *specific* properties: + +- Intensive --- properties which do not depend on mass (e.g. temperature) +- Extensive --- properties which do depend on the mass of substance in a system (e.g. volume) +- Specific (extensive) --- extensive properties which are reduced to unit mass of substance + (essentially an extensive property divided by mass) (e.g. specific volume) + +## Units + +
+ +Property | Symbol | Units | Intensive | Extensive +--------------- | ------ | --------------- | --------- | --------- +Pressure | p | Pa | Yes | +Temperature | T | K | Yes | +Volume | V | m$^3$ | | Yes +Mass | m | kg | | Yes +Specific Volume | v | m$^3$ kg$^{-1}$ | Yes | +Density | $\rho$ | kg m$^{-3}$ | Yes | +Internal Energy | U | J | | Yes +Entropy | S | J K$^{-1}$ | | Yes +Enthalpy | H | J | | Yes + +
+ +## Density + +For an ideal gas: + +$$\rho = \frac{p}{RT}$$ + +## Enthalpy and Specific Enthalpy + +Enthalpy does not have a general physical interpretation. +It is used because the combination $u + pv$ appears naturally in the analysis of many +thermodynamic problems. + +The heat transferred to a closed system undergoing a reversible constant pressure process is equal +to the change in enthalpy of the system. + +Enthalpy is defined as: + +$$H = U+pV$$ + +and Specific Enthalpy: + +$$h = u + pv$$ + +## Entropy and Specific Entropy + +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 +temperature: + +$$C = \frac{\mathrm{d}Q}{\mathrm{d}T}$$ + +Specific heat capacity is the quantity of heat required to raise the temperature of a unit mass +substance by a unit temperature: + +$$c = \frac{\mathrm{d}q}{\mathrm{d}T}$$ + +
+ + +### Heat Capacity in Closed Systems and Internal Energy + +The specific heat transfer to a closed system during a reversible constant **volume** process is +equal to the change in specific **internal energy** of the system: + +$$c_v = \frac{\mathrm{d}q}{\mathrm{d}T} = \frac{\mathrm{d}u}{\mathrm{d}T}$$ + + + +This is because if the change in volume, $\mathrm{d}v = 0$, then the work done, $\mathrm{d}w = 0$ +also. +So applying the (1st Corollary of the) 1st Law to an isochoric process: + +$$\mathrm{d}q + \mathrm{d}w = \mathrm{d}u \rightarrow \mathrm{d}q = \mathrm{d}u$$ + +since $\mathrm{d}w = 0$. + +
+ +
+ + +### Heat Capacity in Closed Systems and Enthalpy + +The specific heat transfer to a closed system during a reversible constant **pressure** process is +equal to the change in specific **enthalpy** of the system: + +$$c_p = \frac{\mathrm{d}q}{\mathrm{d}T} = \frac{\mathrm{d}h}{\mathrm{d}T}$$ + + + +This is because given that pressure, $p$, is constant, work, $w$, can be expressed as: + +$$w = -\int^2_1\! p \,\mathrm{d}v = -p(v_2 - v_1)$$ + +Applying the (1st corollary of the) 1st law to the closed system: + +\begin{align*} + q + w &= u_2 - u_1 \rightarrow q = u_2 - u_1 + p(v_2 - v_1) \\ + q &= u_2 + pv_2 - (u_1 + pv_1) \\ + &= h_2 - h_1 = \mathrm{d}h \\ + \therefore \mathrm{d}q &= \mathrm{d}h +\end{align*} + +
+ +
+ + +### Ratio of Specific Heats + +$c_p > c_v$ is always true. + + + +Heating a volume of fluid, $V$, at a constant volume requires specific heat $q_v$ where + +$$q_v = u_2 - u_1 \therefore c_v = \frac{q_v}{\Delta T}$$ + +Heating the same volume of fluid but under constant pressure requires a specific heat $q_p$ where + +$$q_p =u_2 - u_1 + p(v_2-v_1) \therefore c_p = \frac{q_p}{\Delta T}$$ + +Since $p(v_2-v_1) > 0$, $\frac{q_p}{q_v} > 1 \therefore q_p > q_v \therefore c_p > c_v$. + +The ratio $\frac{c_p}{c_v} = \gamma$ + +
+ +# Process and State Diagrams + +Reversible processes are represented by solid lines, and irreversible processes by dashed lines. + +# Isentropic and Polytropic Processes + +## Polytropic Processes A polytropic process is one which obeys the polytropic law: @@ -462,7 +462,7 @@ For any values of $x$ and $y$ -# Isentropic +## Isentropic Processes *Isentropic* means constant entropy: @@ -506,7 +506,7 @@ An isentropic process is an idealization of an actual process, and serves as the real life processes. They are often desired and often the processes on which device efficiencies are calculated. -## Heat Transfer During Isentropic Processes +### Heat Transfer During Isentropic Processes Assume that the compression 1-2 follows a polytropic process with a polytropic index $n$. The work transfer is: