5.5 Key equations

Claire Yu Yan

5. The First Law of Thermodynamics for a Control Volume

5.5 Key equations

Constant-pressure and constant-volume specific heats

Constant-pressure specific heat

C_{p} = {(\frac{\partial h}{\partial T})}_{p}

Constant-volume specific heat

C_{v} = {(\frac{\partial u}{\partial T})}_{v}

Relations between

C_{p}

and

C_{v}

for ideal gases

k = \frac{C_{p}}{C_{v}} C_{p} = C_{v} + R

$C_{v} = \frac{R}{k - 1} C_{p} = \frac{k R}{k - 1}$

Specific enthalpy

Change in specific enthalpy

Δ h = h_{2} - h_{1}

Change in specific enthalpy for ideal gases

Δ h = h_{2} - h_{1} = C_{p} (T_{2} - T_{1})

(assuming constant

C_{p}

in the temperature range)

Relation between

Δ h

and

Δ u

for solids and liquids

Δ h \approx Δ u \approx C_{p} (T_{2} - T_{1})

Mass conservation equations in a control volume

Volume flow rate

\dot{V} = \frac{d V}{d t} = V_{a v g, n} A = \dot{m} v

Mass flow rate

\dot{m} = \frac{d m}{d t} = ρ V_{a v g, n} A = ρ \dot{V}

Transient flow

\frac{d m_{C V}}{d t} = \sum {\dot{m}}_{i} - \sum {\dot{m}}_{e} \neq 0

Steady flow

\frac{d m_{C V}}{d t} = \sum {\dot{m}}_{i} - \sum {\dot{m}}_{e} = 0

Energy conservation equations in a control volume

Transient flow

\frac{d E_{C V}}{d t} \neq 0

$\begin{aligned} \frac{d E_{C V}}{d t} = {\dot{Q}}_{c v} - {\dot{W}}_{c v} & + \sum {\dot{m}}_{i} (h_{i} + \frac{1}{2} V_{i}^{2} + g z_{i}) \\ - \sum {\dot{m}}_{e} (h_{e} + \frac{1}{2} V_{e}^{2} + g z_{e}) \end{aligned}$

Steady flow

\frac{d E_{C V}}{d t} = 0

${\dot{Q}}_{c v} + \sum {\dot{m}}_{i} (h_{i} + \frac{1}{2} V_{i}^{2} + g z_{i}) = {\dot{W}}_{c v} + \sum {\dot{m}}_{e} (h_{e} + \frac{1}{2} V_{e}^{2} + g z_{e})$

Mass and energy conservation equations for steady-state, steady-flow (SSSF) devices

SSSF device

Assumptions

Mass conservation

Energy conservation

Expansion device

Adiabatic flow; Negligible work transfer with the surroundings; Negligible changes in kinetic and potential energies

{\dot{m}}_{i} = {\dot{m}}_{e}

h_{i} = h_{e}

Nozzle and diffuser

Adiabatic flow; Negligible work transfer with the surroundings; Negligible change in potential energy

{\dot{m}}_{i} = {\dot{m}}_{e}

h_{i} + \frac{1}{2} V_{i}^{2} = h_{e} + \frac{1}{2} V_{e}^{2}

Mixing chamber

Negligible work transfer with the surroundings; Negligible changes in kinetic and potential energies

\sum {\dot{m}}_{i} = \sum {\dot{m}}_{e}

{\dot{Q}}_{c v} + \sum {\dot{m}}_{i} h_{i} = \sum {\dot{m}}_{e} h_{e}

Heat exchanger

Negligible work transfer with the surroundings; Negligible changes in kinetic and potential energies

{\dot{m}}_{i} = {\dot{m}}_{e}

(for each of the hot and cold streams, separately)

{\dot{Q}}_{c v} + \sum {\dot{m}}_{i} h_{i} = \sum {\dot{m}}_{e} h_{e}

Turbine

Adiabatic flow; Negligible changes in kinetic and potential energies

{\dot{m}}_{i} = {\dot{m}}_{e} = \dot{m}

{\dot{W}}_{s h a f t} = \dot{m} (h_{i} - h_{e})

Compressor

Adiabatic flow; Negligible changes in kinetic and potential energies

{\dot{m}}_{i} = {\dot{m}}_{e} = \dot{m}

{\dot{W}}_{s h a f t} = \dot{m} (h_{e} - h_{i})

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Introduction to Engineering Thermodynamics Copyright © 2022 by Claire Yu Yan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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