# Dual-Phase Lag (DPL) model

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- | + | The heat flux and temperature gradient in the [[hyperbolic model]] was related by | |

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+ | <center><math>{\mathbf{q''}} + \tau \frac{{\partial {\mathbf{q''}}}}{{\partial t}} = - k\nabla T </math></center> | ||

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+ | which can be viewed as the first order approximation of the following equation, | ||

<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]]. | ||

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==References== | ==References== | ||

+ | Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Elsevier, Burlington, MA. | ||

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+ | Faghri, A., Zhang, Y., and Howell, J. R., 2010, ''Advanced Heat and Mass Transfer'', Global Digital Press, Columbia, MO. | ||

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+ | Tzou, D. Y., 1997, ''Macro- to Microscale Heat Transfer'', Taylor and Francis, Washington, DC. | ||

==Further Reading== | ==Further Reading== | ||

==External Links== | ==External Links== |

## Current revision as of 01:41, 4 July 2010

The heat flux and temperature gradient in the hyperbolic model was related by

which can be viewed as the first order approximation of the following equation,

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.,

where τ_{q} is the phase lag for the heat flux vector, while τ_{T} is the phase lag for the temperature gradient. If τ_{q} > τ_{T}, 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 τ_{q} < τ_{T}, the temperature gradient is the result of the heat flux at an early time. The first order approximation of eq. (2) is:

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

which reduces to the parabolic conduction equation from Basics of heat conduction if both τ_{q}andτ_{T} are zero. In the absence of phase lag for temperature gradient (τ_{T} = 0), eq. (4) is reduced to the hyperbolic conduction model, eq. from Hyperbolic model.

## References

Faghri, A., and Zhang, Y., 2006, *Transport Phenomena in Multiphase Systems*, Elsevier, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, *Advanced Heat and Mass Transfer*, Global Digital Press, Columbia, MO.

Tzou, D. Y., 1997, *Macro- to Microscale Heat Transfer*, Taylor and Francis, Washington, DC.