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Engineering LibreTexts

11.9.1: Introduction

Consider a gas flowing through a conduit with a friction (see Figure 11.19). It is advantages to examine the simplest situation and yet without losing the core properties of the process. The mass (continuity equation) balance can be written as
\[
    \begin{array}{c}
    \dot{m} = \rho\, A \,U = constant\  \hookrightarrow
    \rho_1 \, U_1 = \rho_2\, U_2
    \end{array}
    \label{fanno:eq:mass} \tag{1}
\]
The energy conservation  (under the assumption that this model is adiabatic flow and the friction is not transformed into thermal energy) reads
\[
    \begin{array}{rl}
    {T_{0}}_1 &=  {T_{0}}_2     \ \hookrightarrow
    T_1 + \dfrac{ {U_1}^2 }{ 2\,c_p} &=   
    T_2 + \dfrac{ {U_2}^2 }{ 2\,c_p}  
    \end{array}
    \label{fanno:eq:energy} \tag{2}
\]
Or in a derivative from  
\[
    C_p\, dT +d \left( U^2 \over 2 \right)  =  0   
    \label{fanno:eq:energyDerivative} \tag{3}
\]
Again for simplicity, the perfect gas model is assumed is slower in reality. However, experiments from many starting with 1938 work by has shown that the error is not significant. Nevertheless, the comparison with reality shows that heat transfer cause changes to the flow and they need/should to be expected. These changes include the choking point at lower Mach number.

Contributors

  • 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.