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 Heat flow in unsteady-state heat conduction may be expressed in terms of which of the following two dimensionless numbers?


1. Nusselt and Fourier numbers
2. Prandtl and Schmidt numbers
3. Grashof and Graetz numbers
4. Biot and Fourier numbers

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Correct Answer - Option 4 : Biot and Fourier numbers

Explanation:

In case of unsteady state heat conduction, the temperature keeps changing with respect to time. so in this case, storage or accumulation of energy occurs. So in this case Fourier and Biot numbers are helpful to study the unsteady state.

Fourier number is the ratio of heat conduction through medium to the rate of thermal energy storage. It is a dimension less number and it is denoted as Fr. It is also defined as ratio of operating time to diffusion time. the main significance of this number is it decides the thermal response of body.

Biot number is dimensionless number, the ratio of internal conductive resistance to external convective resistance and it is denoted as Bi. the main significance of this number is, it measures the temperature gradient of body.

If Bi < 0.1, then it indicates Lumped body.

Schmidt number (Sc): Ratio of molecular diffusivity of momentum to molecular diffusivity of mass transfer.

\(Sc=\frac{\mu}{\rho D}=\frac{\nu}{ D}\)

Where v is kinematic viscosity and D is diffusion coefficient

Prandtl number:

Prandtl number is the ratio of momentum diffusivity to thermal diffusivity.

\(Pr = \frac{\nu }{\alpha } = \frac{\mu }{{\frac{{\rho k}}{{\rho {C_p}}}}} = \frac{{\mu {C_p}}}{k}\)

Where α is thermal diffusivity.

Now,

\(\frac{{{S_c}}}{{Pr}} = \frac{{\frac{v}{D}}}{{\frac{v}{\alpha }}} = \frac{\alpha }{D}\)   

Fourier number:   

Fourier number is used in transient heat conduction. It is the ratio of diffusive or conductive transport rate to the quantity storage rate.

\(\begin{array}{l} {F_0}\; = \;\frac{{Conduction\;rate}}{{Thermal\;energy\;storage\;rate}}\\ {F_0}\; = \;\frac{{\alpha t}}{{{L^2}}} \end{array}\)

where \(\alpha \; = \;\frac{k}{{{\rho \times{{C_p}}}}}\) is thermal diffusivity, t = characteristic time, L = Length through which conduction occurs

Grashof number:

The dimensionless parameter which represents the natural convection effects is called the Grashof number.

Grashof number, Gr, is the ratio between the buoyancy force and the viscous force:

\(Gr = \frac{{g\beta \left( {{T_s} - {T_\infty }} \right)L_c^3}}{{{\nu ^2}}}\)

Nusselt number is a function of the Grashof number and the Prandtl number alone. Nu = f (Gr, Pr)

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