Dual-Phase Lag (DPL) model

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<center><math>{\mathbf{q''}}({\mathbf{r}},t + \tau ) =  - k\nabla T({\mathbf{r}},t)    \qquad \qquad(1)</math></center>
<center><math>{\mathbf{q''}}({\mathbf{r}},t + \tau ) =  - k\nabla T({\mathbf{r}},t)    \qquad \qquad(1)</math></center>
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which indicates that there is a delay between the heat flux vector and the temperature gradient. For the same point in the conduction medium, the temperature gradient is established at time <math>t</math>, but the heat flux vector will be established at a later time <math>t + \tau </math>, i.e., the relaxation, <math>\tau </math>, can be interpreted as the time delay from the onset of the temperature gradient to the heat flux vector. While the thermal wave model assumes that the temperature gradient always precedes the heat flux, [[#References|Tzou (1997)]] proposed a dual-phase lag model that allows either the temperature gradient (cause) to precede heat flux vector (effect) or the heat flux vector (cause) to precede the temperature gradient (effect), i.e.,
which indicates that there is a delay between the heat flux vector and the temperature gradient. For the same point in the conduction medium, the temperature gradient is established at time <math>t</math>, but the heat flux vector will be established at a later time <math>t + \tau </math>, i.e., the relaxation, <math>\tau </math>, can be interpreted as the time delay from the onset of the temperature gradient to the heat flux vector. While the thermal wave model assumes that the temperature gradient always precedes the heat flux, [[#References|Tzou (1997)]] proposed a dual-phase lag model that allows either the temperature gradient (cause) to precede heat flux vector (effect) or the heat flux vector (cause) to precede the temperature gradient (effect), i.e.,
<center><math>{\mathbf{q''}}({\mathbf{r}},t + {\tau _q}) =  - k\nabla T({\mathbf{r}},t + {\tau _t})    \qquad \qquad(2)</math></center>
<center><math>{\mathbf{q''}}({\mathbf{r}},t + {\tau _q}) =  - k\nabla T({\mathbf{r}},t + {\tau _t})    \qquad \qquad(2)</math></center>
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where <math>{\tau _q}</math> is the phase lag for the heat flux vector, while <math>{\tau _T}</math> is the phase lag for the temperature gradient. If <math>{\tau _q} > {\tau _T}</math>, the local heat flux vector is the result of the temperature gradient at the same location but an early time. On the other hand, if <math>{\tau _q} < {\tau _T}</math>, the temperature gradient is the result of the heat flux at an early time. The first order approximation of eq. (2) is:
where <math>{\tau _q}</math> is the phase lag for the heat flux vector, while <math>{\tau _T}</math> is the phase lag for the temperature gradient. If <math>{\tau _q} > {\tau _T}</math>, the local heat flux vector is the result of the temperature gradient at the same location but an early time. On the other hand, if <math>{\tau _q} < {\tau _T}</math>, the temperature gradient is the result of the heat flux at an early time. The first order approximation of eq. (2) is:
<center><math>{\mathbf{q''}} + {\tau _q}\frac{{\partial {\mathbf{q''}}}}{{\partial t}} =  - k\left[ {\nabla T + {\tau _T}\frac{\partial }{{\partial t}}(\nabla T)} \right]    \qquad \qquad(3)</math></center>
<center><math>{\mathbf{q''}} + {\tau _q}\frac{{\partial {\mathbf{q''}}}}{{\partial t}} =  - k\left[ {\nabla T + {\tau _T}\frac{\partial }{{\partial t}}(\nabla T)} \right]    \qquad \qquad(3)</math></center>
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Substituting eq. (3) into eq. (3.1), the energy equation based on the dual-phase lag model is
Substituting eq. (3) into eq. (3.1), the energy equation based on the dual-phase lag model is
 
 
<center><math>\frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} + \frac{{{\tau _q}}}{\alpha }\frac{{{\partial ^2}T}}{{\partial {t^2}}} = {\nabla ^2}T + {\tau _T}\frac{\partial }{{\partial t}}({\nabla ^2}T) + \frac{1}{k}\left( {q''' + {\tau _q}\frac{{\partial q'''}}{{\partial t}}} \right)    \qquad \qquad(4)</math></center>
<center><math>\frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} + \frac{{{\tau _q}}}{\alpha }\frac{{{\partial ^2}T}}{{\partial {t^2}}} = {\nabla ^2}T + {\tau _T}\frac{\partial }{{\partial t}}({\nabla ^2}T) + \frac{1}{k}\left( {q''' + {\tau _q}\frac{{\partial q'''}}{{\partial t}}} \right)    \qquad \qquad(4)</math></center>
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which reduces to the parabolic conduction equation <math>\frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} = {\nabla ^2}T + \frac{{q'''}}{k} </math> from [[Basics of heat conduction]] if both <math>{\tau _q}{\rm{ and }}{\tau _T}</math> are zero. In the absence of phase lag for temperature gradient (<math>{\tau _T} = 0</math>), eq. (4) is reduced to the hyperbolic conduction model, eq. <math>\frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} + \frac{\tau }{\alpha }\frac{{{\partial ^2}T}}{{\partial {t^2}}} = {\nabla ^2}T + \frac{1}{k}\left( {q''' + \tau \frac{{\partial q'''}}{{\partial t}}} \right) </math> from [[Hyperbolic model]].
which reduces to the parabolic conduction equation <math>\frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} = {\nabla ^2}T + \frac{{q'''}}{k} </math> from [[Basics of heat conduction]] if both <math>{\tau _q}{\rm{ and }}{\tau _T}</math> are zero. In the absence of phase lag for temperature gradient (<math>{\tau _T} = 0</math>), eq. (4) is reduced to the hyperbolic conduction model, eq. <math>\frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} + \frac{\tau }{\alpha }\frac{{{\partial ^2}T}}{{\partial {t^2}}} = {\nabla ^2}T + \frac{1}{k}\left( {q''' + \tau \frac{{\partial q'''}}{{\partial t}}} \right) </math> from [[Hyperbolic model]].

Revision as of 19:14, 21 December 2009

Equation {\mathbf{q''}} + \tau \frac{{\partial {\mathbf{q''}}}}{{\partial t}} =  - k\nabla T  from Hyperbolic model can be viewed as the first order approximation of the following equation,

{\mathbf{q''}}({\mathbf{r}},t + \tau ) =  - k\nabla T({\mathbf{r}},t)    \qquad \qquad(1)


which indicates that there is a delay between the heat flux vector and the temperature gradient. For the same point in the conduction medium, the temperature gradient is established at time t, but the heat flux vector will be established at a later time t + τ, i.e., the relaxation, τ, can be interpreted as the time delay from the onset of the temperature gradient to the heat flux vector. While the thermal wave model assumes that the temperature gradient always precedes the heat flux, Tzou (1997) proposed a dual-phase lag model that allows either the temperature gradient (cause) to precede heat flux vector (effect) or the heat flux vector (cause) to precede the temperature gradient (effect), i.e.,

{\mathbf{q''}}({\mathbf{r}},t + {\tau _q}) =  - k\nabla T({\mathbf{r}},t + {\tau _t})    \qquad \qquad(2)


where τq is the phase lag for the heat flux vector, while τT is the phase lag for the temperature gradient. If τq > τT, the local heat flux vector is the result of the temperature gradient at the same location but an early time. On the other hand, if τq < τT, the temperature gradient is the result of the heat flux at an early time. The first order approximation of eq. (2) is:

{\mathbf{q''}} + {\tau _q}\frac{{\partial {\mathbf{q''}}}}{{\partial t}} =  - k\left[ {\nabla T + {\tau _T}\frac{\partial }{{\partial t}}(\nabla T)} \right]    \qquad \qquad(3)


Substituting eq. (3) into eq. (3.1), the energy equation based on the dual-phase lag model is

\frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} + \frac{{{\tau _q}}}{\alpha }\frac{{{\partial ^2}T}}{{\partial {t^2}}} = {\nabla ^2}T + {\tau _T}\frac{\partial }{{\partial t}}({\nabla ^2}T) + \frac{1}{k}\left( {q''' + {\tau _q}\frac{{\partial q'''}}{{\partial t}}} \right)    \qquad \qquad(4)


which reduces to the parabolic conduction equation \frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} = {\nabla ^2}T + \frac{{q'''}}{k} from Basics of heat conduction if both τqandτT are zero. In the absence of phase lag for temperature gradient (τT = 0), eq. (4) is reduced to the hyperbolic conduction model, eq. \frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} + \frac{\tau }{\alpha }\frac{{{\partial ^2}T}}{{\partial {t^2}}} = {\nabla ^2}T + \frac{1}{k}\left( {q''' + \tau \frac{{\partial q'''}}{{\partial t}}} \right) from Hyperbolic model.

References

Further Reading

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