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11.7.1: The Control Volume Analysis/Governing equations

  • Page ID
    813
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    Figure 11.17 describes the flow of gas from the left to the right. The heat transfer up stream (or down stream) is assumed to be negligible. Hence, the energy equation can be written as the following:

    \[ \dfrac{d\, Q }{ \dot{m} } = c_p dT + d \dfrac{U^2 }{ 2} = c_p dT_{0}
    \label{isothermal:eq:CV}
    \]

    The momentum equation is written as the following

    \[ -A\, dP - \tau_{w}\, dA_{\text{wetted area}} = \dot{m}\, dU
    \label{isothermal:eq:momentum}
    \]

    where \(A\) is the cross section area (it doesn't have to be a perfect circle; a close enough shape is sufficient.). The shear stress is the force per area that acts on the fluid by the tube wall. The \(A_{wetted\;\;area}\) is the area that shear stress acts on. The second law of thermodynamics reads

    \[ {s_2 - s_1 \over C_p} = \ln {T_2 \over T_1 } - {k -1 \over k}
    \ln {P_2 \over P_1}
    \label{isothermal:eq:2law}
    \]

    The mass conservation is reduced to

    \[ \dot {m} = \text{constant} = \rho\, U\, A
    \label{isothermal:eq:mass}
    \] Again it is assumed that the gas is a perfect gas and therefore, equation of state is expressed as the following:

    \[ P = \rho\, R\, T
    \label{isothermal:eq:state}
    \]

    Contributors and Attributions

    • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.


    This page titled 11.7.1: The Control Volume Analysis/Governing equations is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


    This page titled 11.7.1: The Control Volume Analysis/Governing equations is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Genick Bar-Meir via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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