---
title: "ChEn 433 Steam"
author: David Lignell
date: Class 7
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# Rankine cycle
::: {.cols2}
::: {.col2-l}
* Key power generation cycle: coal, nuclear, natural gas, etc.
* Heat input from a heat source: combustion, nuclear, geothermal, solar.
* Heat rejection: river, ocean, lake, atmosphere (dry cooling).
    * [Water requirements](http://large.stanford.edu/courses/2017/ph241/styles2/) of power generation are substantial.
    * [Review of water consumption for electricity generation](https://iopscience.iop.org/article/10.1088/1748-9326/7/4/045802/pdf)
* Water is used as the working fluid.
    - available
    - non-toxic, non-hazardous, non-reactive
    - pump liquid, phase change gives $Q_{isoT}$
:::
::: {.col2-v}
<img src="https://ars.els-cdn.com/content/image/3-s2.0-B9780128095973001061-f0100106-28-9780128095973.jpg" title="https://upload.wikimedia.org/wikipedia/commons/b/be/Rankine_cycle_Ts.png" width=800 alt="image">
:::
:::

## Rankine cycle
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::: {.col2-l}
* 1-2: $W_{in}$, pump liquid from low P to high P; isentropic compression
* 2-3: $Q_{in}$, heat at constant pressure: boiling
* 3-4: $W_{out}$, expand fluid through turbine; isentropic expansion
* 4-1: $Q_{out}$, cool in condenser; vapor to liquid
:::
::: {.col2-v}
<img src="https://upload.wikimedia.org/wikipedia/commons/b/be/Rankine_cycle_Ts.png" title="https://upload.wikimedia.org/wikipedia/commons/b/be/Rankine_cycle_Ts.png" width=600 alt="image">
:::
:::


## Rankine cycle: issues

<img src="https://upload.wikimedia.org/wikipedia/commons/b/be/Rankine_cycle_Ts.png" title="https://upload.wikimedia.org/wikipedia/commons/b/be/Rankine_cycle_Ts.png" width=400 alt="image">

::: fragment
***Issues?***
:::
::: incremental
* Peak temperature $→$ efficiency $→$ superheat
* Turbine steam quality > 90%
* Pumping a liquid vs vapor vs mixture.
* [Fluid choice](https://en.wikipedia.org/wiki/Working_fluid_selection)
    * Temperature range. [Organic Rankine cycle (wiki)](https://en.wikipedia.org/wiki/Organic_Rankine_cycle)
        * [Organic Rankine Cycle article](https://www.sciencedirect.com/science/article/pii/B978008100510100017X)
:::

::: notes
* Cengel thermo book 3rd edition, page 556;
* Compare in context of Carnot: box below the dome, box above the dome.
:::

## Rankine cycle: TS
<img src="rankine1.png" width=1200 alt="image">

## Rankine cycle: PV
<p style="color:red">
***What does the PV diagram look like?***
</p>
::: fragment
<img src="rankine2.png" width=1200 alt="image">
:::

## Rankine cycle: nonidealities

<img src="rankine1.png" width=400 alt="image">

::: fragment
***Nonidealities?***
:::
::: incremental
* Pressure drop in the boiler
* Nonideal turbine
* Subcool the condensed liquid to avoid pump cavitation
* Heat losses to surroundings
:::

::: notes
* horizontal blue curve drops due to pressure drop
* red line isn't vertical
:::

## Rankine cycle: increase $η$

<img src="rankine1.png" width=400 alt="image">

::: fragment
***Increase efficiency?***
:::
::: incremental
* Decrease condenser pressure (hence temperature)
    * real condensers operate below atmospheric pressure
    * limited by the saturation pressure corresponding to $T_{cooling}$.
        * river at 20 $^oC$ $→$ P=2312 Pa = 0.023 atm.
        * Typically 2-7 kPa
    * but...quality of steam in turbine, and... air leakage
* Superheat steam to higher T, even supercritical
    * higher turbine exit steam quality
    * but...limited by material performance and cost: [620 $^oC$ max](https://www.sciencedirect.com/topics/engineering/steam-inlet-temperature).
:::

::: notes
* decreasing condenser pressure: extracting more work from the turbine
:::

## Rankine cycle: increase $η$

<img src="rankine1.png" width=400 alt="image">

::: fragment
***Increase efficiency?***
:::
::: incremental
* Increase boiler pressure ($P_c$=218 atm)
    * [Supercritical steam](https://en.wikipedia.org/wiki/Supercritical_steam_generator)
    * [Turk Plant](https://en.wikipedia.org/wiki/John_W._Turk_Jr._Coal_Plant) P > 310 bar, T > 600 $^o$C.
    * At a given $T_{turbine}$, higher P results in lower steam quality, so reheat
* Reheat
    * 4-5% higher efficiency. 
    * Usually limited to single reheat stage (except for supercritical plants)
        * A second reheat gives $Δ η$ about half the initial gain.
        * Avoid superheated turbine exhaust.
    * Optimal reheat pressure ~ 1/4 $P_{max}$
:::



## Rankine cycle: steam with reheat
<img src="https://miro.medium.com/max/1262/0*52Tld4UXbWrLFiJY.png" title="https://miro.medium.com/max/1262/0*52Tld4UXbWrLFiJY.png" width=800 alt="image">

* 1-2: $W_{in}$, pump liquid from low P to high P; isentropic compression
* 2-3: $Q_{in}$, heat at constant pressure: heat, boil, superheat
* 3-4: $W_{out}$, expand fluid through turbine; isentropic expansion
* 4-5: $Q_{in}$, re-heat at constant pressure: superheat
* 5-6: $W_{out}$, expand fluid through turbine; isentropic expansion
* 6-1: $Q_{out}$, cool in condenser; vapor to liquid

# Conversion efficiencies
<a href="https://en.wikipedia.org/wiki/Energy_conversion_efficiency">
<img src="efficiencies.png" title="https://miro.medium.com/max/1262/0*52Tld4UXbWrLFiJY.png" width=1300 alt="image">
</a>

# Steam properties: Cantera

<iframe width="1600px" height="900px" src="https://cantera.org/" ></iframe>

## Cantera Links

* [Cantera.org](https://cantera.org)
* [Python thermodynamic properties](https://cantera.org/documentation/docs-2.5/sphinx/html/cython/thermo.html)
* [Phase objects: water, R134a, heptane, etc.](https://cantera.org/documentation/dev/sphinx/html/cython/importing.html#pure-fluid-phases)
* [C++ reference](https://cantera.org/documentation/docs-2.5/doxygen/html/)
    - useful for detailed documentation, units, available properties.
* [General documentation](https://cantera.org/documentation/)
* [Installation with Conda](https://cantera.org/install/conda-install.html)

## Cantera example: steam tables

<iframe width="1600px" height="900px" src="https://nbviewer.jupyter.org/url/ignite.byu.edu/che433/lectures/lecture_07_steam/steam.ipynb" ></iframe>

## Cantera example: TS diagram

<iframe width="1600px" height="900px" src="https://nbviewer.jupyter.org/url/ignite.byu.edu/che433/lectures/lecture_07_steam/steam_TS_diagram.ipynb" ></iframe>

## Cantera example: Rankine cycle
<iframe width="1600px" height="900px" src="https://nbviewer.jupyter.org/url/ignite.byu.edu/che433/lectures/lecture_07_steam/rankine_.ipynb" ></iframe>

# Work potential

* For conversion of thermal energy to electricity or mechanical work, energy alone is not enough.
    * A 1 K temperature change in the oceans would supply 17000 CMO!
    * But at the average surface temperature, the thermal conversion efficiency would be ~0.003.
* We really want the *work potential* of an energy source.
    - Names: *exergy*, *availability*, *lost work*, *second law analysis*

How much work can be extracted in going from one state to another?

<p style="color:gray">
Or, how much work is required in going from one state to another?
</p>

## Work Potential: KE, PE

The wind blows at 10 m/s, what is the work potential per unit mass?

::: fragment
What is the work potential in a 5.1 meter high waterfall?
:::

## Work potential: heat transfer

We transfer 1 kW of power from a thermal energy source at 500 K.

What is the work potential?

::: fragment
Need to define the surrounding state, so, 300 $K$, say.
:::

::: fragment
::: {.red}
WP here is simply 1 kW times the Carnot efficiency:
$$W = 1\,\mbox{kW} ⋅ \underbrace{(1 - 300/500)}_{η_c} = 0.40\,\mbox{kW}$$
:::
:::

## Work potential: air, change state

**Air**

| quantity     | state 1 | state 2 |
|--------------|:-------:|:-------:|
| T (K)        |  573    | 300     |
| P (atm)      |  10     | 1       |
| S (J/kg$⋅$K) |  6892   | 6892    |
| H (kJ/kg)    |  282    | 1.91    |

*How much work can be extracted in moving from state 1 to state 2?*

::: fragment
<p style="color:red">
Isentropic
<br>
Work is just $Δ H$
<br>
That is, the maximum 1$^{st}$ law work can be extracted 
</p>
:::

## Work potential: steam, change state

**Water**

| quantity     | state 1 | state 2 |
|--------------|:-------:|:-------:|
| Phase        | supercritical   | liquid |
| T (K)        | 800     | 300     |
| P (atm)      | 10      | 1       |
| S (J/kg$⋅$K) | 11350   | 3913    |
| H (kJ/kg)    | -12434  | -15858  |

*How much work can be extracted in moving from state 1 to state 2?*

::: {.fragment .red}
* First law: $Δ H = W + Q$
* Max work for min Q
* Second law: $Q_{min}=T_{min}Δ S$
:::

## Lost work

* Consider an unsteady, open system
* **Energy balance**
\begin{align}
\frac{d(mu)_{sys}}{dt} &= \dot{W} + \dot{Q} + \sum_i\dot{m}_iu_i \\
&=\sum_i\dot{m}_iPv_i + \dot{W}_\phi + \sum_j\dot{Q}_j + \sum_i\dot{m}_iu_i
\end{align}

Here, $\dot{W}_\phi$ is "other" work not including injection work (which goes into enthalpy).

::: {.red}
$$\frac{d(mu)_{sys}}{dt}  = \sum_i\dot{m}_ih_i + \dot{W}_\phi + \sum_j\dot{Q}_j$$
:::

## Lost work

* **Entropy balance**
    * Here, we define the system so that it exchanges heat with the surrounding reservoirs $j$ at the respective reservoir temperature. In this way, all irreversibilities are within the system, and $\dot{S}_{sys} + \dot{S}_{surr} = \dot{S}_{irrev}$.
$$\frac{d(ms)_{sys}}{dt} = \sum_i\dot{m}s_i + \underbrace{\sum_j\dot{S}_{j,HT}}_{\sum_j\dot{Q}_j/T_j} + \dot{S}_{irrev}$$

::: {.red}
$$\frac{d(ms)_{sys}}{dt}  = \sum_i\dot{m}_is_i + \sum_j\dot{Q}_j/T_j + \dot{S}_{irrev}$$
:::

## Lost work
* The term $\dot{S}_{irrev}$ is identified as the *lost work* divided by the *surrounding* temperature $T_0$:
$$\dot{S}_{irrev} = \frac{W_L}{T_0}$$
* We multiply the entropy balance by $T_0$ and subtract the result from the energy balance. This gives a relation for the combined first and second laws:
$$\frac{d(m(u-T_0s))_{sys}}{dt} = \sum_i\dot{m}_i(h_i-T_0s_i) + \sum_jQ_j\left(1-\frac{T_0}{T_j}\right) + \dot{W}_\phi -\dot{W}_L$$
* Solve for $\dot{W}_\phi$:
$$\dot{W}_\phi = \left[\frac{d(m(u-T_0s))_{sys}}{dt} - \sum_i\dot{m}_i(h_i-T_0s_i) - \sum_jQ_j\left(1-\frac{T_0}{T_j}\right)\right] + \dot{W}_L$$

## Lost work
* Again:
$$\dot{W}_\phi = \frac{d(m(u-T_0s))_{sys}}{dt} - \sum_i\dot{m}_i(h_i-T_0s_i) - \sum_jQ_j\left(1-\frac{T_0}{T_j}\right) + \dot{W}_L$$
* If the lost work is zero, then we have a reversible process, by which:

::: {.red}
$$\dot{W}_{\phi,rev} = \frac{d(m(u-T_0s))_{sys}}{dt} - \sum_i\dot{m}_i(h_i-T_0s_i) - \sum_jQ_j\left(1-\frac{T_0}{T_j}\right)$$
:::

::: {.gray}
* $Q$ is positive into the system
* $\dot{m}$ is positive into the system
* $\dot{W}_{\phi,rev}$ is positive into the system
:::


<!--

## Exergy of internal energy

$$dU = -dW - dQ,$$
where dW and dQ are work and heat done **by** the system.

What is the **useable** work due to the work and heat transfer processes?

::: {.columns}
::: {.column}
**Work**
$$dW = PdV = \underbrace{(P-P_0)dV}_{dW^u_{PV}} + P_0dV$$
<p style="color:blue">
$$dW = dW^u_{PV} + P_0dV$$
</p>
:::
::: {.column}
**Heat**
\begin{align}
dW^u_{HT} &= dQη_{c} \\
          &= dQ(1-T_0/T) \\
          &= dQ - dQT_0/T \\
          &= dQ + TdST_0/T \\
          &= dQ + T_0dS
\end{align}
<p style="color:blue">
$$dQ = dW^u_{HT} - T_0dS$$
</p>
:::
:::
Insert in energy relation, then combine the useful work terms, and rearrange:
$$dU = -dW^u_{PV} - P_0dV -dW^u_{HT} + T_0dS$$
$$dU = -dW^u - P_0dV + T_0dS$$
$$dW^u = -dU - P_0dV + T_0dS$$

## Exergy of system

$$dW^u = -dU - P_0dV + T_0dS$$
Now, integrate from the initial state to the dead state $0$:
$$W^u = (U-U_0) + P_0(V-V_0) - T_0(S-S_0)$$
We can add in any contributions from KE and PE, and write per unit mass:
<p style="color:red">
$$w^u = (u-u_0) + P_0(v-v_0) - T_0(s-s_0) + \frac{\mathrm{v}^2}{2} + g(z-z_0)$$
</p>
For a **flow system**, we have the exergy of the flow work $=(P-P_0)V$, which, added per mass to the above gives (with $h=u+Pv$):
<p style="color:rgb(162,11,26)">
$$w^u = (h-h_0) - T_0(s-s_0) + \frac{\mathrm{v}^2}{2} + g(z-z_0)$$
</p>

-->