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5.4: Reynolds Transport Theorem

It can be noticed that the same derivations carried for the density can be carried for other intensive properties such as specific entropy, specific enthalpy. Suppose that \(g\) is intensive property (which can be a scalar or a vector) undergoes change with time. The change of accumulative property will be then
\[
    \label{mass:eq:intensiveProperty}
    \dfrac{D}{Dt} \int_{sys} f\,\rho dV =  
    \dfrac{d}{dt} \int_{c.v.} f\,\rho dV + \int_{c.v} f\,\rho\,U_{rn} dA  \tag{25}
\]
This theorem named after Reynolds, Osborne, (1842-1912) which is actually a three dimensional generalization of Leibniz integral rule. To make the previous derivation clearer, the Reynolds Transport Theorem will be reproofed and discussed. The ideas are the similar but extended some what. Leibniz integral rule is an one dimensional and it is defined as
\[
    \label{mass:eq:Leibniz}
    \dfrac{d}{dy} \,\int_{x_1(y)}^{x_2(y)}f(x,y) \, dx =
        \int_{x_1(y)}^{x_2(y)}\dfrac {\partial f}{\partial y}\,dx
    +    f(x_2,y)\, \dfrac{dx_2}{dy} - f(x_1,y)\, \dfrac{dx_1}{dy}  \tag{26}
\]
Initially, a proof will be provided and the physical meaning will be explained. Assume that there is a function that satisfy the following
\[
    \label{mass:eq:aF}
    G(x,y)= \int^x f\left ( \alpha,\,y \right) \,d\alpha \tag{27}  
\]
Notice that lower boundary of the integral is missing and is only the upper limit of the function is present . For its derivative of equation (??) is
\[
    \label{mass:eq:daF}
    f(x,y) = \dfrac{\partial G}{\partial x}  \tag{28}
\]
differentiating (chain rule \(d\,uv = u\,dv+v\,du\)) by part of left hand side of the Leibniz integral rule (it can be shown which are identical) is
\[
    \label{mass:eq:daFm}
    \dfrac{d\, \left[G(x_2,y)-G(x_1,y) \right] }{dy}
    = \overbrace{\dfrac{\partial G}{\partial x_2} \dfrac{dx_2}{dy}}^{1}  +
      \overbrace{\dfrac{\partial G}{\partial y}(x_2,y)}^{2} -
      \overbrace{\dfrac{\partial G}{\partial x_1}\dfrac{dx_1}{dy}}^{3} -
      \overbrace{\dfrac{\partial G}{\partial y}(x_1,y)}^{4}  \tag{29}
\]
The terms 2 and 4 in equation (??) are actually (the \(x_2\) is treated as a different variable)
\[
    \label{mass:eq:2and4}
    \int_{x_1(y)}^{x_2(y)}\dfrac{\partial\,f(x,y) }{\partial y}\,dx  \tag{30}
\]
The first term (1) in equation (??) is
\[
    \label{mass:eq:term1}
    \dfrac{\partial G}{\partial x_2} \dfrac{dx_2}{dy} =  
        f(x_2,y)\, \dfrac{dx_2}{dy} \tag{31}  
\]
The same can be said for the third term (3). Thus this explanation is a proof the Leibniz rule. The above "proof'' is mathematical in nature and physical explanation is also provided. Suppose that a fluid is flowing in a conduit. The intensive property, \(f\) is investigated or the accumulative property, \(F\). The interesting information that commonly needed is the change of the accumulative property, \(F\), with time. The change with time is
\[
    \label{mass:eq:F}
    \dfrac{DF}{Dt} = \dfrac{D}{Dt} \int_{sys} \rho\, f\, dV    \tag{32}
\]
For one dimensional situation the change with time is
\[
    \label{mass:eq:F1D}
    \dfrac{DF}{Dt} = \dfrac{D}{Dt} \int_{sys} \rho\, f\, A(x) dx  \tag{33}  
\]
If two limiting points (for the one dimensional) are moving with a different coordinate system, the mass will be different and it will not be a system. This limiting condition is the control volume for which some of the mass will leave or enter. Since the change is very short (differential), the flow in (or out) will be the velocity of fluid minus the boundary at \(x_1\), \(U_{rn}=U_{1}−U_{b}\). The same can be said for the other side. The accumulative flow of the property in, \(F\), is then
\[
    \label{mass:eq:flowOut1D1}
    F_{in} = \overbrace{f_1\,\rho}^{F_1}\,\overbrace{ U_{rn}}^{\dfrac{dx_1}{dt}}   \tag{34}  
\]
The accumulative flow of the property out, \(F\), is then
\[
    \label{mass:eq:flowOut1D}
    F_{out} = \overbrace{f_2\,\rho}^{F_2}\,\overbrace{ U_{rn}}^{\dfrac{dx_2}{dt}}     \tag{35}
\]
The change with time of the accumulative property, \(F\), between the boundaries is
\[
    \label{mass:eq:cvD}
    \dfrac{d}{dt} \int_{c.v.} \rho(x) \,f\,A(x)\,dA   \tag{36}
\]
When put together it brings back the Leibniz integral rule. Since the time variable, \(t\), is arbitrary and it can be replaced by any letter. The above discussion is one of the physical meaning the Leibniz rule. Reynolds Transport theorem is a generalization of the Leibniz rule and thus the same arguments are used. The only difference is that the velocity has three components and only the perpendicular component enters into the calculations.

Reynolds Transport

\[
    \label{mass:eq:RT}
    \dfrac{D}{DT} \int _{sys} f\, \rho dV =  
        \dfrac{d}{dt} \int_{c.v} f\,\rho \, dV  + \int_{S_{c.v.}}  f\,\rho \, U_{rn}\, dA   \tag{37}
\]

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.