Home
Bookshelves
Introductory Engineering
Basic Engineering Science - A Systems, Accounting, and Modeling Approach (Richards)
9: Appendices
9.3: Appendix C- Summary of Conservation and Accounting Equations, Unit Conversions, Property Models, Thermophysical Property Data

9.3: Appendix C- Summary of Conservation and Accounting Equations, Unit Conversions, Property Models, Thermophysical Property Data

Last updated
Save as PDF

Page ID: 85361

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

C.1: Basic Conservation & Accounting Equations

Basic Conservation & Accounting Equations
Accounting Equation for Generic, Extensive Property \(B\): Extensive Property ----- \(\displaystyle B_{sys} (t) = \iiint\limits_{V_{sys}} b_{(x, y, z, t)} \rho_{(x, y, z, t)} dV\) Accounting Equation ----- \(\dfrac{dB_{sys} (t)}{dt} = \dot{B}_{in} - \dot{B}_{out} + \dot{B}_{gen} - \dot{B}_{cons}\) \[\underbrace{\frac{d B_{sys}}{d t}}_{\text {Accumulation}} = \underbrace{ \underbrace{\left\{\dot{B}_{in}-\dot{B}_{out}\right\}}_{\text {non-flow boundaries}} + \underbrace{\left\{\sum_{in} \dot{m}_{i} b_{i}-\sum_{out} \dot{m}_{e} b_{e}\right\}}_{\text {flow boundaries}} }_{\text {Transport}} + \underbrace{\left\{\dot{B}_{\text {gen}} - \dot{B}_{\text {cons}}\right\}}_{\text {Generation/Consumption}} \nonumber \]	Conservation of Linear Momentum: \(\displaystyle \quad \mathbf{P}_{sys} = \int\limits_{V_{sys}} \mathbf{V} \rho \ dV\) \[\frac{d \mathbf{P}_{sys}}{dt} = \sum_{j} \mathbf{F}_{\text{ext, } j} + \left\{\sum_{in} \dot{m}_{i} \mathbf{V}_{i} - \sum_{out} \dot{m}_{e} \mathbf{V}_{e} \right\} \nonumber \]
Conservation of Mass: \(\displaystyle \quad m_{sys} (t)=\iiint\limits_{V_{sys}} \rho_{(x, y, z, t)} \ d V\) \[\frac{d m_{sys}}{dt} = \sum_{in} \dot{m}_{i} - \sum_{out} \dot{m}_{e} \nonumber \] \[\text{where } \dot{m} = \int\limits_{A_C} \rho V_{n} \ dA_{C} = \underbrace{\rho A_{C} V_{\text{avg}}}_{\text{1-D Flow Assumption}} \quad \text{(the mass flow rate)} \nonumber \]	Conservation of Angular Momentum: \(\displaystyle \quad \mathbf{L}_{o, \ sys} = \int\limits_{V_{sys}} (\mathbf{r} \times \mathbf{V}) \rho \ dV\) \[\frac{d \mathbf{L}_{o, \ sys}}{dt} = \sum_{j} \mathbf{M}_{o, \ j} + \left\{\sum_{in} \dot{m}_{i} \left(\mathbf{r}_{i} \times \mathbf{V}_{i}\right) -\sum_{out} \dot{m}_{e} \left(\mathbf{r}_{e} \times \mathbf{V}_{e} \right)\right\} \nonumber \] \[\text{where } \mathbf{M}_{o, j} = \mathbf{r}_{j} \times \mathbf{F}_{j} \quad \text{or} \quad M_{\text{couple, } j} \nonumber \]
Chemical Species Accounting: \(\quad m_{j}=n_{j} M_{j}\) \[\text{mass} \quad \rightarrow \quad \frac{d m_{j, \ sys}}{d t} = \sum_{in} \dot{m}_{j, i} - \sum_{out} \dot{m}_{j, e} + \left(\dot{m}_{j, \ gen}-\dot{m}_{j, \ cons}\right) \nonumber \] \[ \text{molar} \quad \rightarrow \quad \frac{d n_{j, \ sys}}{dt} = \sum_{in} \dot{n}_{j, \ i} - \sum_{out} \dot{n}_{j, \ e} + \left( \dot{n}_{j, \ gen} - \dot{n}_{j, \ cons} \right) \nonumber \]	Conservation of Energy: \[\begin{gathered} E_{sys} = \int\limits_{V_{sys}} e \rho \ dV \quad \text { where } \quad e = u+\frac{V^{2}}{2}+gz+e_{\text {spring}} + \ldots \\ \frac{dE_{sys}}{dt} = \dot{Q}_{\text{net, in}} + \dot{W}_{\text {net, in}} + \left\{ \sum_{in} \dot{m}_{i} \left(h_{i} + \frac{V_{i}^{2}}{2} + gz_{i} \right) - \sum_{out} \dot{m}_{e} \left(h_{e} + \frac{V_{e}^{2}}{2} + gz_{e}\right) \right\} \end{gathered} \nonumber \]
Conservation of Charge: \(\displaystyle \quad q_{sys} = \int\limits_{V_{sys}} \tilde{q} \rho \ dV\) \[\frac{d q_{sys}}{dt} = \sum_{in} \dot{q}_{i} - \sum_{out} \dot{q}_{e} \nonumber \]	Entropy Accounting: \(\displaystyle \quad S_{sys} = \int\limits_{V_{sys}} s \rho \ dV \quad \text{and} \quad S_{gen} \geq 0\) \[\frac{d S_{sys}}{dt} = \sum_{j} \frac{\dot{Q}_{j}}{T_{b, \ j}} + \left\{ \sum_{in} \dot{m}_{i} s_{i} - \sum_{out} \dot{m}_{e} s_{e}\right\} + \dot{S}_{gen} \nonumber \]

Basic Conservation & Accounting Equations

Accounting Equation for Generic, Extensive Property \(B\):

Extensive Property ----- \(\displaystyle B_{sys} (t) = \iiint\limits_{V_{sys}} b_{(x, y, z, t)} \rho_{(x, y, z, t)} dV\)

Accounting Equation ----- \(\dfrac{dB_{sys} (t)}{dt} = \dot{B}_{in} - \dot{B}_{out} + \dot{B}_{gen} - \dot{B}_{cons}\)

\[\underbrace{\frac{d B_{sys}}{d t}}_{\text {Accumulation}} = \underbrace{ \underbrace{\left\{\dot{B}_{in}-\dot{B}_{out}\right\}}_{\text {non-flow boundaries}} + \underbrace{\left\{\sum_{in} \dot{m}_{i} b_{i}-\sum_{out} \dot{m}_{e} b_{e}\right\}}_{\text {flow boundaries}} }_{\text {Transport}} + \underbrace{\left\{\dot{B}_{\text {gen}} - \dot{B}_{\text {cons}}\right\}}_{\text {Generation/Consumption}} \nonumber \]

Conservation of Linear Momentum: \(\displaystyle \quad \mathbf{P}_{sys} = \int\limits_{V_{sys}} \mathbf{V} \rho \ dV\)

\[\frac{d \mathbf{P}_{sys}}{dt} = \sum_{j} \mathbf{F}_{\text{ext, } j} + \left\{\sum_{in} \dot{m}_{i} \mathbf{V}_{i} - \sum_{out} \dot{m}_{e} \mathbf{V}_{e} \right\} \nonumber \]

Conservation of Mass: \(\displaystyle \quad m_{sys} (t)=\iiint\limits_{V_{sys}} \rho_{(x, y, z, t)} \ d V\)

\[\frac{d m_{sys}}{dt} = \sum_{in} \dot{m}_{i} - \sum_{out} \dot{m}_{e} \nonumber \]

\[\text{where } \dot{m} = \int\limits_{A_C} \rho V_{n} \ dA_{C} = \underbrace{\rho A_{C} V_{\text{avg}}}_{\text{1-D Flow Assumption}} \quad \text{(the mass flow rate)} \nonumber \]

Conservation of Angular Momentum: \(\displaystyle \quad \mathbf{L}_{o, \ sys} = \int\limits_{V_{sys}} (\mathbf{r} \times \mathbf{V}) \rho \ dV\)

\[\frac{d \mathbf{L}_{o, \ sys}}{dt} = \sum_{j} \mathbf{M}_{o, \ j} + \left\{\sum_{in} \dot{m}_{i} \left(\mathbf{r}_{i} \times \mathbf{V}_{i}\right) -\sum_{out} \dot{m}_{e} \left(\mathbf{r}_{e} \times \mathbf{V}_{e} \right)\right\} \nonumber \]

\[\text{where } \mathbf{M}_{o, j} = \mathbf{r}_{j} \times \mathbf{F}_{j} \quad \text{or} \quad M_{\text{couple, } j} \nonumber \]

Chemical Species Accounting: \(\quad m_{j}=n_{j} M_{j}\)

\[\text{mass} \quad \rightarrow \quad \frac{d m_{j, \ sys}}{d t} = \sum_{in} \dot{m}_{j, i} - \sum_{out} \dot{m}_{j, e} + \left(\dot{m}_{j, \ gen}-\dot{m}_{j, \ cons}\right) \nonumber \]

\[ \text{molar} \quad \rightarrow \quad \frac{d n_{j, \ sys}}{dt} = \sum_{in} \dot{n}_{j, \ i} - \sum_{out} \dot{n}_{j, \ e} + \left( \dot{n}_{j, \ gen} - \dot{n}_{j, \ cons} \right) \nonumber \]

Conservation of Energy:

\[\begin{gathered} E_{sys} = \int\limits_{V_{sys}} e \rho \ dV \quad \text { where } \quad e = u+\frac{V^{2}}{2}+gz+e_{\text {spring}} + \ldots \\ \frac{dE_{sys}}{dt} = \dot{Q}_{\text{net, in}} + \dot{W}_{\text {net, in}} + \left\{ \sum_{in} \dot{m}_{i} \left(h_{i} + \frac{V_{i}^{2}}{2} + gz_{i} \right) - \sum_{out} \dot{m}_{e} \left(h_{e} + \frac{V_{e}^{2}}{2} + gz_{e}\right) \right\} \end{gathered} \nonumber \]

Conservation of Charge: \(\displaystyle \quad q_{sys} = \int\limits_{V_{sys}} \tilde{q} \rho \ dV\)

\[\frac{d q_{sys}}{dt} = \sum_{in} \dot{q}_{i} - \sum_{out} \dot{q}_{e} \nonumber \]

Entropy Accounting: \(\displaystyle \quad S_{sys} = \int\limits_{V_{sys}} s \rho \ dV \quad \text{and} \quad S_{gen} \geq 0\)

\[\frac{d S_{sys}}{dt} = \sum_{j} \frac{\dot{Q}_{j}}{T_{b, \ j}} + \left\{ \sum_{in} \dot{m}_{i} s_{i} - \sum_{out} \dot{m}_{e} s_{e}\right\} + \dot{S}_{gen} \nonumber \]

C.2 Unit Conversions

Unit Conversions
Length \[\begin{aligned} & 1 \mathrm{~ft} = 12 \mathrm{~in} = 0.3048 \mathrm{~m} = 1/3 \mathrm{~yd} \\ & 1 \mathrm{~m} = 100 \mathrm{~cm} = 1000 \mathrm{~mm} = 39.37 \mathrm{~in} = 3.2808 \mathrm{~ft} \\ & 1 \mathrm{~mile} = 5280 \mathrm{~ft} = 1609.3 \mathrm{~m} \end{aligned} \nonumber \]	Force \[\begin{aligned} & 1 \mathrm{~N} = 1 \mathrm{~kg} \cdot \mathrm{m}/\mathrm{s}^{2} = 0.22481 \mathrm{~lbf} \\ & 1 \mathrm{~lbf} = 1 \mathrm{~slug} \cdot \mathrm{ft}/\mathrm{s}^{2} = 32.174 \mathrm{~lbm} \cdot \mathrm{ft}/\mathrm{s}^{2} = 4.4482 \mathrm{~N} \end{aligned} \nonumber \]
Mass \[\begin{aligned} &1 \mathrm{~kg} = 1000 \mathrm{~g} = 2.2046 \mathrm{~lbm} \\ &1 \mathrm{~lbm} = 16 \mathrm{~oz} = 0.45359 \mathrm{~kg} \\ &1 \mathrm{~slug} = 32.174 \mathrm{~lbm} \end{aligned} \nonumber \]	Pressure \[\begin{aligned} & 1 \mathrm{~atm} = 101.325 \mathrm{~kPa} = 1.01325 \mathrm{~bar} = 14.696 \mathrm{~lbf}/\mathrm{in}^{2} \\ & 1 \mathrm{~bar} = 100 \mathrm{~kPa} = 10^{5} \mathrm{~Pa} \\ & 1 \mathrm{~Pa} = 1 \mathrm{~N}/\mathrm{m}^{2} = 10^{-3} \mathrm{~kPa} \\ & 1 \mathrm{~lbf}/\mathrm{in}^{2} = 6.8947 \mathrm{~kPa} = 6894.7 \mathrm{~N}/\mathrm{m}^{2} \\ & \quad \left[ \mathrm{lbf}/\mathrm{in}^{2} \text{ often abbreviated as “psi”} \right] \end{aligned} \nonumber \]
Temperature Values \[\begin{aligned} & (\mathrm{T} / \mathrm{K}) = \left(\mathrm{T} / { }^{\circ} \mathrm{R}\right) / 1.8 \\ & (\mathrm{T} / \mathrm{K}) = \left(\mathrm{T} / { }^{\circ} \mathrm{C} \right) + 273.15 \\ & \left(\mathrm{T} / { }^{\circ} \mathrm{C} \right) = \left[ \left( \mathrm{T} /{ }^{\circ} \mathrm{F} \right) - 32 \right]/1.8 \\ & \left(\mathrm{T} / \mathrm{R} \right) = 1.8 (\mathrm{T}/\mathrm{K}) \\ & \left( \mathrm{T}/{ }^{\circ} \mathrm{R}\right) = \left( \mathrm{T}/{ }^{\circ} \mathrm{F} \right) + 459.67 \\ & \left( \mathrm{T}/{ }^{\circ} \mathrm{F} \right) = 1.8 \left( \mathrm{T}/{ }^{\circ} \mathrm{C}\right) + 32 \end{aligned} \nonumber \]	Energy \[\begin{aligned} & 1 \mathrm{~J} = 1 \mathrm{~N} \cdot \mathrm{m} \\ & 1 \mathrm{~kJ} = 1000 \mathrm{~J} = 737.56 \mathrm{~ft} \cdot \mathrm{lbf} = 0.94782 \mathrm{~Btu} \\ & 1 \mathrm{~Btu} = 1.0551 \mathrm{~kJ} = 778.17 \mathrm{~ft} \cdot \mathrm{lbf} \\ & 1 \mathrm{~ft} \cdot \mathrm{lbf} = 1.3558 \mathrm{~J} \end{aligned} \nonumber \]
Temperature Differences \[\begin{aligned} & (\Delta \mathrm{T}/{ }^{\circ} \mathrm{R}) = 1.8 (\Delta \mathrm{T}/\mathrm{K}) \\ & (\Delta \mathrm{T}/{ }^{\circ} \mathrm{R}) = (\Delta \mathrm{T}/{ }^{\circ} \mathrm{F}) \\ & (\Delta \mathrm{T}/\mathrm{K}) = (\Delta \mathrm{T}/{ }^{\circ} \mathrm{C}) \end{aligned} \nonumber \]	Energy Transfer Rate \[\begin{aligned} & 1 \mathrm{~kW} = 1 \mathrm{~kJ}/\mathrm{s} = 737.56 \mathrm{~ft} \cdot \mathrm{lbf}/\mathrm{s} = 1.3410 \mathrm{~hp} = 0.94782 \mathrm{~Btu}/\mathrm{s} \\ & 1 \mathrm{~Btu}/\mathrm{s} = 1.0551 \mathrm{~kW} = 1.4149 \mathrm{~hp} = 778.17 \mathrm{~ft} \cdot \mathrm{lbf}/\mathrm{s} \\ & 1 \mathrm{~hp} = 550 \mathrm{~ft} \cdot \mathrm{lbf}/\mathrm{s} = 0.74571 \mathrm{~kW} = 0.70679 \mathrm{~Btu}/\mathrm{s} \end{aligned} \nonumber \]
Volume \[\begin{aligned} & 1 \mathrm{~m}^{3} = 1000 \mathrm{~L} = 10^{6} \mathrm{~cm}^{3} = 10^{6} \mathrm{~mL} = 35.315 \mathrm{~ft}^{3} = 264.17 \mathrm{~gal} \\ & 1 \mathrm{~ft}^{3} = 1728 \mathrm{~in}^{3} = 7.4805 \mathrm{~gal} = 0.028317 \mathrm{~m}^{3} \\ & 1 \mathrm{~gal} = 0.13368 \mathrm{~ft}^{3} = 0.0037854 \mathrm{~m}^{3} \end{aligned} \nonumber \]	Specific Energy \[\begin{aligned} & 1 \mathrm{~kJ}/\mathrm{kg} = 1000 \mathrm{~m}^{2}/\mathrm{s}^{2} \\ & 1 \mathrm{~Btu}/\mathrm{lbm} = 25037 \mathrm{~ft}^{2}/\mathrm{s}^{2} \\ & 1 \mathrm{~ft} \cdot \mathrm{lbf}/\mathrm{lbm} = 32.174 \mathrm{~ft}^{2}/\mathrm{s}^{2} \end{aligned} \nonumber \]
Volumetric Flow Rate \[\begin{aligned} & 1 \mathrm{~m}^{3}/\mathrm{s} = 35.315 \mathrm{~ft}^{3}/\mathrm{s} = 264.17 \mathrm{~gal}/\mathrm{s} \\ & 1 \mathrm{~ft}^{3}/\mathrm{s} = 1.6990 \mathrm{~m}^{3}/\mathrm{min} = 7.4805 \mathrm{~gal}/\mathrm{s} = 448.83 \mathrm{~gal}/\mathrm{~min} \end{aligned} \nonumber \]

Unit Conversions

Length

\[\begin{aligned} & 1 \mathrm{~ft} = 12 \mathrm{~in} = 0.3048 \mathrm{~m} = 1/3 \mathrm{~yd} \\ & 1 \mathrm{~m} = 100 \mathrm{~cm} = 1000 \mathrm{~mm} = 39.37 \mathrm{~in} = 3.2808 \mathrm{~ft} \\ & 1 \mathrm{~mile} = 5280 \mathrm{~ft} = 1609.3 \mathrm{~m} \end{aligned} \nonumber \]

Force

\[\begin{aligned} & 1 \mathrm{~N} = 1 \mathrm{~kg} \cdot \mathrm{m}/\mathrm{s}^{2} = 0.22481 \mathrm{~lbf} \\ & 1 \mathrm{~lbf} = 1 \mathrm{~slug} \cdot \mathrm{ft}/\mathrm{s}^{2} = 32.174 \mathrm{~lbm} \cdot \mathrm{ft}/\mathrm{s}^{2} = 4.4482 \mathrm{~N} \end{aligned} \nonumber \]

Mass

\[\begin{aligned} &1 \mathrm{~kg} = 1000 \mathrm{~g} = 2.2046 \mathrm{~lbm} \\ &1 \mathrm{~lbm} = 16 \mathrm{~oz} = 0.45359 \mathrm{~kg} \\ &1 \mathrm{~slug} = 32.174 \mathrm{~lbm} \end{aligned} \nonumber \]

Pressure

\[\begin{aligned} & 1 \mathrm{~atm} = 101.325 \mathrm{~kPa} = 1.01325 \mathrm{~bar} = 14.696 \mathrm{~lbf}/\mathrm{in}^{2} \\ & 1 \mathrm{~bar} = 100 \mathrm{~kPa} = 10^{5} \mathrm{~Pa} \\ & 1 \mathrm{~Pa} = 1 \mathrm{~N}/\mathrm{m}^{2} = 10^{-3} \mathrm{~kPa} \\ & 1 \mathrm{~lbf}/\mathrm{in}^{2} = 6.8947 \mathrm{~kPa} = 6894.7 \mathrm{~N}/\mathrm{m}^{2} \\ & \quad \left[ \mathrm{lbf}/\mathrm{in}^{2} \text{ often abbreviated as “psi”} \right] \end{aligned} \nonumber \]

Temperature Values

\[\begin{aligned} & (\mathrm{T} / \mathrm{K}) = \left(\mathrm{T} / { }^{\circ} \mathrm{R}\right) / 1.8 \\ & (\mathrm{T} / \mathrm{K}) = \left(\mathrm{T} / { }^{\circ} \mathrm{C} \right) + 273.15 \\ & \left(\mathrm{T} / { }^{\circ} \mathrm{C} \right) = \left[ \left( \mathrm{T} /{ }^{\circ} \mathrm{F} \right) - 32 \right]/1.8 \\ & \left(\mathrm{T} / \mathrm{R} \right) = 1.8 (\mathrm{T}/\mathrm{K}) \\ & \left( \mathrm{T}/{ }^{\circ} \mathrm{R}\right) = \left( \mathrm{T}/{ }^{\circ} \mathrm{F} \right) + 459.67 \\ & \left( \mathrm{T}/{ }^{\circ} \mathrm{F} \right) = 1.8 \left( \mathrm{T}/{ }^{\circ} \mathrm{C}\right) + 32 \end{aligned} \nonumber \]

Energy

\[\begin{aligned} & 1 \mathrm{~J} = 1 \mathrm{~N} \cdot \mathrm{m} \\ & 1 \mathrm{~kJ} = 1000 \mathrm{~J} = 737.56 \mathrm{~ft} \cdot \mathrm{lbf} = 0.94782 \mathrm{~Btu} \\ & 1 \mathrm{~Btu} = 1.0551 \mathrm{~kJ} = 778.17 \mathrm{~ft} \cdot \mathrm{lbf} \\ & 1 \mathrm{~ft} \cdot \mathrm{lbf} = 1.3558 \mathrm{~J} \end{aligned} \nonumber \]

Temperature Differences

\[\begin{aligned} & (\Delta \mathrm{T}/{ }^{\circ} \mathrm{R}) = 1.8 (\Delta \mathrm{T}/\mathrm{K}) \\ & (\Delta \mathrm{T}/{ }^{\circ} \mathrm{R}) = (\Delta \mathrm{T}/{ }^{\circ} \mathrm{F}) \\ & (\Delta \mathrm{T}/\mathrm{K}) = (\Delta \mathrm{T}/{ }^{\circ} \mathrm{C}) \end{aligned} \nonumber \]

Energy Transfer Rate

\[\begin{aligned} & 1 \mathrm{~kW} = 1 \mathrm{~kJ}/\mathrm{s} = 737.56 \mathrm{~ft} \cdot \mathrm{lbf}/\mathrm{s} = 1.3410 \mathrm{~hp} = 0.94782 \mathrm{~Btu}/\mathrm{s} \\ & 1 \mathrm{~Btu}/\mathrm{s} = 1.0551 \mathrm{~kW} = 1.4149 \mathrm{~hp} = 778.17 \mathrm{~ft} \cdot \mathrm{lbf}/\mathrm{s} \\ & 1 \mathrm{~hp} = 550 \mathrm{~ft} \cdot \mathrm{lbf}/\mathrm{s} = 0.74571 \mathrm{~kW} = 0.70679 \mathrm{~Btu}/\mathrm{s} \end{aligned} \nonumber \]

Volume

\[\begin{aligned} & 1 \mathrm{~m}^{3} = 1000 \mathrm{~L} = 10^{6} \mathrm{~cm}^{3} = 10^{6} \mathrm{~mL} = 35.315 \mathrm{~ft}^{3} = 264.17 \mathrm{~gal} \\ & 1 \mathrm{~ft}^{3} = 1728 \mathrm{~in}^{3} = 7.4805 \mathrm{~gal} = 0.028317 \mathrm{~m}^{3} \\ & 1 \mathrm{~gal} = 0.13368 \mathrm{~ft}^{3} = 0.0037854 \mathrm{~m}^{3} \end{aligned} \nonumber \]

Specific Energy

\[\begin{aligned} & 1 \mathrm{~kJ}/\mathrm{kg} = 1000 \mathrm{~m}^{2}/\mathrm{s}^{2} \\ & 1 \mathrm{~Btu}/\mathrm{lbm} = 25037 \mathrm{~ft}^{2}/\mathrm{s}^{2} \\ & 1 \mathrm{~ft} \cdot \mathrm{lbf}/\mathrm{lbm} = 32.174 \mathrm{~ft}^{2}/\mathrm{s}^{2} \end{aligned} \nonumber \]

Volumetric Flow Rate

\[\begin{aligned} & 1 \mathrm{~m}^{3}/\mathrm{s} = 35.315 \mathrm{~ft}^{3}/\mathrm{s} = 264.17 \mathrm{~gal}/\mathrm{s} \\ & 1 \mathrm{~ft}^{3}/\mathrm{s} = 1.6990 \mathrm{~m}^{3}/\mathrm{min} = 7.4805 \mathrm{~gal}/\mathrm{s} = 448.83 \mathrm{~gal}/\mathrm{~min} \end{aligned} \nonumber \]

C.3 Substance Models

Two Substance Models (Constitutive Relations)
	Equation of State
	Ideal Gas Model with room-temperature specific heats	Incompressible Substance Model with room temperature specific heats
Used to model behavior of	gases and vapors	liquids and solids
Basic Model Assumptions	Pressure, volume, and temperature obey the ideal gas relation: \[PV = NR_{u}T \nonumber \] Specific internal energy depends only on temperature, \(u = u(T)\). Molar mass of an ideal gas equals the molar mass of the real substance: \[M_{\text{ideal gas}} = M_{\text{real stuff}} \nonumber \] The specific heats are independent of temperature, i.e. they are constants.	The density of the substance is a constant. Specific internal energy depends only on temperature, \(u = u(T)\). Molar mass of an incompressible substance equals the molar mass of the real substance: \[M_{\text{incomp substance}} = M_{\text{real stuff}} \nonumber \] The specific heats are independent of temperature, i.e. they are constants.
\(P \ - \ T \ - \ \rho\) and \(P \ - \ T \ - \ \upsilon\) relations	\(P=\rho RT\) and \(P \upsilon = RT\) where \(R = R_{u}/M\)	\(\upsilon = 1/\rho = \text{constant}\) Evaluated at room temperature
Specific heat relations	\(c_{\text{p}} - c_{\text{v}} = R; \quad k = c_{\text{p}} / c_{\text{v}}\)	\(c_{\text{p}} = c_{\text{v}} = c, \text{ a constant}\)
\(c_{\text{p}}\) and \(c_{\text{v}}\) values	Evaluated at room temperature	Evaluated at room temperature
\(\Delta u\) — specific internal energy	\(\Delta u = u_{2}-u_{1} = c_{\text{v}} \left(T_{2}-T_{1}\right)\)	\(\Delta u = u_{2}-u_{1} = c \left(T_{2}-T_{1}\right)\)
\(\Delta h\) — specific enthalpy	\(\Delta h = h_{2}-h_{1} = c_{\text{p}} \left(T_{2} - T_{1}\right)\)	\[\begin{aligned} \Delta h &= h_{2}-h_{1} \\ &= \left(u_{2}+P_{2} \upsilon\right) - \left(u_{1}+P_{1} \upsilon\right) \\ &= \left(u_{2} - u_{1}\right) + \upsilon \left(P_{2}-P_{1}\right) \\ \text{thus}& \\ \Delta h &= \Delta u + \upsilon \Delta P = c \Delta T + \upsilon \Delta P \end{aligned} \nonumber \]
\(\Delta s\) — specific entropy Note: All temperatures are absolute values, i.e. \(\mathrm{K}\) or \({ }^{\circ} \mathrm{R}\), in the entropy relations	\[\begin{aligned} \Delta s &= s_{2}-s_{1} \\ &= c_{\text{p}} \ln \left(T_{2}-T_{1}\right) - R \ln \left(P_{2}-P_{1}\right) \\ &= c_{\text{v}} \ln \left(T_{2} / T_{1}\right) + R \ln \left(\upsilon_{2}-\upsilon_{1}\right) \end{aligned} \nonumber \]	\[\begin{aligned} \Delta s &= s_{2}-s_{1} \\ &= c \ln \left(T_{2}/T_{1}\right) \end{aligned} \nonumber \]

Ideal Gas Equation
Molar Basis	Mass Basis
\[\begin{array}{c} PV=nRT \\ P \bar{\upsilon} = R_{u} T \quad \text{and} \quad P = \bar{\rho} R_{u} T \end{array} \nonumber \] \[\begin{aligned} \text{where} & \\ P &= \text{absolute pressure of gas } \left[\text{kPa or lbf} / \mathrm{ft}^{2}\right] \\ V &= \text{volume of gas } \left[\mathrm{m}^{3} \text{ or } \mathrm{ft}^{3}\right] \\ n &= \text{number of moles of gas } \left[\text{kmol or lbmol}\right] \\ R_{u} &= \text{universal gas constant (the same for every gas)} \\ &\quad\quad \left[ \mathrm{kJ}/\left(\mathrm{kmol} \cdot \mathrm{K}\right) \text{ or } \left(\mathrm{ft} \cdot \mathrm{lbf}\right) / \left(\mathrm{lbmol} \cdot { }^{\circ} \mathrm{R}\right) \right] \\ T &= \text{absolute temperature of gas } \left[\mathrm{K} \text{ or } { }^{\circ} \mathrm{R}\right] \\ \bar{\rho} &= \text{molar density} = 1/\bar{\upsilon} \ \left[\mathrm{kmol}/\mathrm{m}^{3} \text{ or } \mathrm{lbmol}/\mathrm{ft}^{3} \right] \\ \bar{\upsilon} &= \text{molar specific volume } \left[\mathrm{m}^{3}/\mathrm{kmol} \text{ or } \mathrm{ft}^{3}/\mathrm{lbmol}\right] \end{aligned} \nonumber \]	\[\begin{array}{c} PV=mRT \\ P \upsilon = RT \quad \text{and} \quad P = \rho R T \end{array} \nonumber \] \[\begin{aligned} \text{where} & \\ P &= \text{absolute pressure of gas } \left[\text{kPa or lbf} / \mathrm{ft}^{2}\right] \\ V &= \text{volume of gas } \left[\mathrm{m}^{3} \text{ or } \mathrm{ft}^{3}\right] \\ m &= \text{mass of gas } \left[\mathrm{kg} \text{ or } \mathrm{lbm}\right] \\ R &= \text{specific gas constant (different for each gas)} \\ &\quad\quad \left[\mathrm{kJ} / \left(\mathrm{kg} \cdot \mathrm{K}\right) \text{ or } \left(\mathrm{ft} \cdot \mathrm{lbf}\right) / \left(\mathrm{lbmol} \cdot { }^{\circ} \mathrm{R}\right) \right] \\ T &= \text{absolute temperature of gas } \left[\mathrm{K} \text{ or } { }^{\circ} \mathrm{R}\right] \\ \rho &= \text{density} = 1/\upsilon \ \left[ \mathrm{kg}/\mathrm{m}^{3} \text{ or } \mathrm{lbm}/\mathrm{ft}^{3} \right] \\ \upsilon &= \text{specific volume } \left[ \mathrm{m}^{3}/\mathrm{kg} \text{ or } \mathrm{ft}^{3}/\mathrm{lbm} \right] \end{aligned} \nonumber \]
\(\text{and}\) \[\begin{aligned} R_{u} &= 8.314 \ \frac{\mathrm{kJ}}{\mathrm{kmol} \cdot \mathrm{K}} = 8.314 \ \frac{\mathrm{J}}{\mathrm{mol} \cdot \mathrm{K}} \\ &= 1545 \ \frac{\mathrm{ft} \cdot \mathrm{lbf}}{\mathrm{lbmol} \cdot { }^{\circ} \mathrm{R}} \end{aligned} \nonumber \]	\(\text{and}\) \[ R = \frac{R_{u}}{M} \nonumber \] \(\text{where}\) \[\quad M = \text{molecular weight (molar mass) of a specific gas} \nonumber \]

Ideal Gas Equation

Molar Basis

Mass Basis

\[\begin{array}{c} PV=nRT \\ P \bar{\upsilon} = R_{u} T \quad \text{and} \quad P = \bar{\rho} R_{u} T \end{array} \nonumber \]

\[\begin{aligned} \text{where} & \\ P &= \text{absolute pressure of gas } \left[\text{kPa or lbf} / \mathrm{ft}^{2}\right] \\ V &= \text{volume of gas } \left[\mathrm{m}^{3} \text{ or } \mathrm{ft}^{3}\right] \\ n &= \text{number of moles of gas } \left[\text{kmol or lbmol}\right] \\ R_{u} &= \text{universal gas constant (the same for every gas)} \\ &\quad\quad \left[ \mathrm{kJ}/\left(\mathrm{kmol} \cdot \mathrm{K}\right) \text{ or } \left(\mathrm{ft} \cdot \mathrm{lbf}\right) / \left(\mathrm{lbmol} \cdot { }^{\circ} \mathrm{R}\right) \right] \\ T &= \text{absolute temperature of gas } \left[\mathrm{K} \text{ or } { }^{\circ} \mathrm{R}\right] \\ \bar{\rho} &= \text{molar density} = 1/\bar{\upsilon} \ \left[\mathrm{kmol}/\mathrm{m}^{3} \text{ or } \mathrm{lbmol}/\mathrm{ft}^{3} \right] \\ \bar{\upsilon} &= \text{molar specific volume } \left[\mathrm{m}^{3}/\mathrm{kmol} \text{ or } \mathrm{ft}^{3}/\mathrm{lbmol}\right] \end{aligned} \nonumber \]

\[\begin{array}{c} PV=mRT \\ P \upsilon = RT \quad \text{and} \quad P = \rho R T \end{array} \nonumber \]

\[\begin{aligned} \text{where} & \\ P &= \text{absolute pressure of gas } \left[\text{kPa or lbf} / \mathrm{ft}^{2}\right] \\ V &= \text{volume of gas } \left[\mathrm{m}^{3} \text{ or } \mathrm{ft}^{3}\right] \\ m &= \text{mass of gas } \left[\mathrm{kg} \text{ or } \mathrm{lbm}\right] \\ R &= \text{specific gas constant (different for each gas)} \\ &\quad\quad \left[\mathrm{kJ} / \left(\mathrm{kg} \cdot \mathrm{K}\right) \text{ or } \left(\mathrm{ft} \cdot \mathrm{lbf}\right) / \left(\mathrm{lbmol} \cdot { }^{\circ} \mathrm{R}\right) \right] \\ T &= \text{absolute temperature of gas } \left[\mathrm{K} \text{ or } { }^{\circ} \mathrm{R}\right] \\ \rho &= \text{density} = 1/\upsilon \ \left[ \mathrm{kg}/\mathrm{m}^{3} \text{ or } \mathrm{lbm}/\mathrm{ft}^{3} \right] \\ \upsilon &= \text{specific volume } \left[ \mathrm{m}^{3}/\mathrm{kg} \text{ or } \mathrm{ft}^{3}/\mathrm{lbm} \right] \end{aligned} \nonumber \]

\(\text{and}\)

\[\begin{aligned} R_{u} &= 8.314 \ \frac{\mathrm{kJ}}{\mathrm{kmol} \cdot \mathrm{K}} = 8.314 \ \frac{\mathrm{J}}{\mathrm{mol} \cdot \mathrm{K}} \\ &= 1545 \ \frac{\mathrm{ft} \cdot \mathrm{lbf}}{\mathrm{lbmol} \cdot { }^{\circ} \mathrm{R}} \end{aligned} \nonumber \]

\(\text{and}\) \[ R = \frac{R_{u}}{M} \nonumber \]

\(\text{where}\) \[\quad M = \text{molecular weight (molar mass) of a specific gas} \nonumber \]

Thermophysical Property Data for Some Common Substances (SI Units)
Gases (at \(25^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\))
Substance		Molar Mass	\(\dfrac{R}{\left[ \dfrac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}} \right]}\)	\(\dfrac{c_{v}}{\left[ \dfrac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}} \right]}\)	\(\dfrac{c_{p}}{\left[ \dfrac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}} \right]}\)	\(k\)	\(\dfrac{T_{c}}{\mathrm{K}}\)	\(\dfrac{P_{c}}{\mathrm{bar}}\)
Acetylene	\(\mathrm{C}_{2} \mathrm{H}_{2}\)	\(26.04\)	\(0.3193\)	\(1.37\)	\(1.69\)	\(1.23\)	\(309\)	\(62.4\)
Air	--	\(28.97\)	\(0.2870\)	\(0.718\)	\(1.005\)	\(1.40\)	\(133\)	\(37.7\)
Ammonia	\(\mathrm{NH}_{3}\)	\(17.04\)	\(0.4879\)	\(1.66\)	\(2.15\)	\(1.30\)	\(406\)	\(112.8\)
Carbon dioxide	\(\mathrm{CO}_{2}\)	\(44.01\)	\(0.1889\)	\(0.657\)	\(0.846\)	\(1.29\)	\(304.2\)	\(73.9\)
Carbon monoxide	\(\mathrm{CO}\)	\(28.01\)	\(0.2968\)	\(0.744\)	\(1.04\)	\(1.40\)	\(133\)	\(35.0\)
Ethane	\(\mathrm{C}_{2} \mathrm{H}_{6}\)	\(30.07\)	\(0.2765\)	\(1.48\)	\(1.75\)	\(1.18\)	\(305.4\)	\(48.8\)
Ethylene	\(\mathrm{C}_{2} \mathrm{H}_{4}\)	\(28.05\)	\(0.2964\)	\(1.23\)	\(1.53\)	\(1.24\)	\(283\)	\(51.2\)
Helium	\(\mathrm{He}\)	\(4.003\)	\(2.077\)	\(3.12\)	\(5.19\)	\(1.67\)	\(5.2\)	\(2.3\)
Hydrogen	\(\mathrm{H}_{2}\)	\(2.016\)	\(4.124\)	\(10.2\)	\(14.3\)	\(1.40\)	\(33.2\)	\(13.0\)
Methane	\(\mathrm{CH}_{4}\)	\(16.04\)	\(0.5183\)	\(1.70\)	\(2.22\)	\(1.31\)	\(190.7\)	\(46.4\)
Nitrogen	\(\mathrm{N}_{2}\)	\(28.01\)	\(0.2968\)	\(0.743\)	\(1.04\)	\(1.40\)	\(126.2\)	\(33.9\)
Oxygen	\(\mathrm{O}_{2}\)	\(32.00\)	\(0.2598\)	\(0.658\)	\(0.918\)	\(1.40\)	\(154.4\	\(50.5\)
Propane	\(\mathrm{C}_{3} \mathrm{H}_{8}\)	\(44.09\)	\(0.1886\)	\(1.48\)	\(1.67\)	\(1.13\)	\(370\)	\(42.5\)
Refrigerant 134a	\(\mathrm{C}_{2} \mathrm{F}_{2} \mathrm{H}_{2}\)	\(102.03\)	\(0.08149\)	\(0.76\)	\(0.85\)	\(1.12\)	\(374.3\)	\(40.6\)
Water (Steam)	\(\mathrm{H}_{2} \mathrm{O}\)	\(18.02\)	\(0.4614\)	\(1.40\)	\(1.86\)	\(1.33\)	\(647.3\)	\(220.9\)

Liquids				Solids*
Substance	Temp. \(({ }^{\circ} \mathrm{C})\)	\(\dfrac{\rho}{\left[ \dfrac{\mathrm{kg}}{\mathrm{m}^{3}}\right]}\)	\(\dfrac{c_{p}}{\left[ \dfrac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}} \right]}\)	Substance	\(\dfrac{\rho}{\left[ \dfrac{\mathrm{kg}}{\mathrm{m}^{3}}\right]}\)	\(\dfrac{c_{p}}{\left[ \dfrac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}} \right]}\)
Ammonia	\(25\)	\(602\)	\(4.80\)	Aluminum	\(2,700\)	\(0.902\)
Benzene	\(20\)	\(879\)	\(1.72\)	Brass, yellow	\(8,310\)	\(0.400\)
Brine \((20 \% \mathrm{NaCl})\)	\(20\)	\(1,150\)	\(3.11\)	Brick (common)	\(1,922\)	\(0.79\)
Ethanol	\(25\)	\(783\)	\(2.46\)	Concrete	\(2,300\)	\(0.653\)
Ethyl Alcohol	\(20\)	\(789\)	\(2.84\)	Copper	\(8,900\)	\(0.386\)
Ethylene Glycol	\(20\)	\(1,109\)	\(2.84\)	Glass, window	\(2,700\)	\(0.800\)
Kerosene	\(20\)	\(820\)	\(2.00\)	Iron	\(7,840\)	\(0.45\)
Mercury	\(25\)	\(13,560\)	\(0.139\)	Lead	\(11,310\)	\(0.128\)
Oil (light)	\(25\)	\(910\)	\(1.80\)	Silver	\(10,470\)	\(0.235\)
Refrigerant 134a	\(25\)	\(1,206\)	\(1.42\)	Steel (mild)	\(7,830\)	\(0.500\)
Water	\(25\)	\(997\)	\(4.18\)	* Evaluated at room temperature.
Values adapted from K. Wark, Jr. and D. E. Richards, Thermodynamics, 6th ed. (McGraw-Hill, New York, 1999) and Y. A. Cengul and M. A. Boles, Thermodynamics, 4th ed. (McGraw-Hill, New York, 2002).

Thermophysical Property Data for Some Common Substances (USCS Units)
Gases (at \(77^{\circ} \mathrm{F}\) and \(1 \mathrm{~atm}\))
Substance		Molar Mass	\(\dfrac{R}{\left[ \dfrac{\mathrm{ft} \cdot \mathrm{lbf}}{\mathrm{lbm} \cdot { }^{\circ} \mathrm{R}} \right]}\)	\(\dfrac{c_{v}}{\left[ \dfrac{\mathrm{Btu}}{\mathrm{lbm} \cdot { }^{\circ} \mathrm{R}} \right]}\)	\(\dfrac{c_{p}}{\left[ \dfrac{\mathrm{Btu}}{\mathrm{lbm} \cdot { }^{\circ} \mathrm{R}} \right]}\)	\(k\)	\(\dfrac{T_{c}}{\text{ }^{\circ} \mathrm{R}}\)	\(\dfrac{P_{c}}{\mathrm{atm}}\)
Acetylene	\(\mathrm{C}_{2} \mathrm{H}_{2}\)	\(26.04\)	\(59.33\)	\(0.328\)	\(0.404\)	\(1.23\)	\(556\)	\(61.6\)
Air	--	\(28.97\)	\(53.33\)	\(0.171\)	\(0.240\)	\(1.40\)	\(239\)	\(37.2\)
Ammonia	\(\mathrm{NH}_{3}\)	\(17.04\)	\(90.67\)	\(0.397\)	\(0.514\)	\(1.29\)	\(730\)	\(111.3\)
Carbon dioxide	\(\mathrm{CO}_{2}\)	\(44.01\)	\(35.11\)	\(0.156\)	\(0.202\)	\(1.29\)	\(548\)	\(72.9\)
Carbon monoxide	\(\mathrm{CO}\)	\(28.01\)	\(55.16\)	\(0.178\)	\(0.249\)	\(1.40\)	\(239\)	\(34.5\)
Ethane	\(\mathrm{C}_{2} \mathrm{H}_{6}\)	\(30.07\)	\(51.38\)	\(0.353\)	\(0.419\)	\(1.19\)	\(549\)	\(48.2\)
Ethylene	\(\mathrm{C}_{2} \mathrm{H}_{4}\)	\(28.05\)	\(55.08\)	\(0.294\)	\(0.365\)	\(1.24\)	\(510\)	\(50.5\)
Helium	\(\mathrm{He}\)	\(4.003\)	\(386.0\)	\(0.744\)	\(1.24\)	\(1.67\)	\(9.3\)	\(2.26\)
Hydrogen	\(\mathrm{H}_{2}\)	\(2.016\)	\(766.4\)	\(2.43\)	\(3.42\)	\(1.40\)	\(59.8\)	\(12.8\)
Methane	\(\mathrm{CH}_{4}\)	\(16.04\)	\(96.32\)	\(0.407\)	\(0.531\)	\(1.30\)	\(344\)	\(45.8\)
Nitrogen	\(\mathrm{N}_{2}\)	\(28.01\)	\(55.16\)	\(0.178\)	\(0.248\)	\(1.39\)	\(227\)	\(33.5\)
Oxygen	\(\mathrm{O}_{2}\)	\(32.00\)	\(48.28\)	\(0.157\)	\(0.219\)	\(1.40\)	\(278\)	\(49.8\)
Propane	\(\mathrm{C}_{3} \mathrm{H}_{8}\)	\(44.09\)	\(35.04\)	\(0.355\)	\(0.400\)	\(1.13\)	\(666\)	\(42.1\)
Refrigerant 134a	\(\mathrm{C}_{2} \mathrm{F}_{4} \mathrm{H}_{2}\)	\(102.03\)	\(15.14\)	\(0.184\)	\(0.203\)	\(1.10\)	\(672.8\)	\(40.1\)
Water (Steam)	\(\mathrm{H}_{2} \mathrm{O}\)	\(18.02\)	\(87.74\)	\(0.335\)	\(0.445\)	\(1.33\)	\(1165\)	\(218.0\)

Liquids				Solids*
Substance	Temp \(({ }^{\circ} \mathrm{F})\)	\(\dfrac{\rho}{\left[ \dfrac{\mathrm{lbm}}{\mathrm{ft}^{3}} \right]}\)	\(\dfrac{c_{p}}{\left[ \dfrac{\mathrm{Btu}}{\mathrm{lbm} \cdot { }^{\circ} \mathrm{R}} \right]}\)	Substance	\(\dfrac{\rho}{\left[ \dfrac{\mathrm{lbm}}{\mathrm{ft}^{3}} \right]}\)	\(\dfrac{c_{p}}{\left[ \dfrac{\mathrm{Btu}}{\mathrm{lbm} \cdot { }^{\circ} \mathrm{R}} \right]}\)
Ammonia	\(80\)	\(37.5\)	\(1.135\)	Aluminum	\(170\)	\(0.215\)
Benzene	\(68\)	\(54.9\)	\(0.411\)	Brass, yellow	\(519\)	\(0.0955\)
Brine \((20 \% \mathrm{NaCl})\)	\(68\)	\(71.8\)	\(0.743\)	Brick (common)	\(120\)	\(0.189\)
Ethanol	\(77\)	\(48.9\)	\(0.588\)	Concrete	\(144\)	\(0.156\)
Ethyl Alcohol	\(68\)	\(49.3\)	\(0.678\)	Copper	\(555\)	\(0.0917\)
Ethylene Glycol	\(68\)	\(69.2\)	\(0.678\)	Glass, window	\(169\)	\(0.191\)
Kerosene	\(68\)	\(51.2\)	\(0.478\)	Iron	\(490\)	\(0.107\)
Mercury	\(77\)	\(847\)	\(0.033\)	Lead	\(705\)	\(0.030\)
Oil (light)	\(77\)	\(56.8\)	\(0.430\)	Silver	\(655\)	\(0.056\)
Refrigerant 134a	\(32\)	\(80.9\)	\(0.318\)	Steel (mild)	\(489\)	\(0.119\)
Water	\(68\)	\(62.2\)	\(1.00\)	* Evaluated at room temperature.
Values adapted from K. Wark, Jr. and D. E. Richards, Thermodynamics, 6th ed. (McGraw-Hill, New York, 1999) and Y. A. Cengul and M. A. Boles, Thermodynamics, 4th ed. (McGraw-Hill, New York, 2002).