HEAT TRANSFER

Just like any transport phenomenon, heat flow is possible in the presence of a driving force. In heat transfer, this driving force is the temperature gradient.

Modes of Heat Transfer
Conduction Through Material Medium Absence of Bulk Fluid Motion
Convection Presence of Bulk Fluid Motion on top of diffusion
Radiation No Material Medium

 

I. Conduction

A. Fourier’s Law

Conduction refers to the mode of transfer of heat by the collision of particles within the material.

Fourier’s Law of Conduction states that the heat flux across a material is proportional to the negative gradient in temperature.

\frac{q_{x} }{ A} = -k\frac{dT}{dx}
8th Perry’s Handbook, Eq. 5 -1, p. 5

q – rate of heat transfer (W)
k – thermal conductivity (W/m.K)
A – area perpendicular to hear flow (m2)
T – Temperature of the surface perpendicular to heat flow (K)
General Heat Equation: Cartesian Coordinate Form
 

    k\frac{ \partial ^{2}T }{ \partial x^{2}} + k\frac{ \partial ^{2}T }{ \partial y^{2}} + k\frac{ \partial ^{2}T }{ \partial z^{2}} + S = \rho c \frac{ \partial T }{ \partial t }
     8th Perry’s Handbook, Eq. 5 -16, p. 5
 
 
Special Cases:
    a. Steady-State
    k\frac{ \partial ^{2}T }{ \partial x^{2}} + k\frac{ \partial ^{2}T }{ \partial y^{2}} + k\frac{ \partial ^{2}T }{ \partial z^{2}} + S = 0
     8th Perry’s Handbook, Eq. 5 – 15, p. 5- 6
 

    b. Steady-State, No Heat Generation
    \frac{ \partial ^{2}T }{ \partial x^{2}} + \frac{ \partial ^{2}T }{ \partial y^{2}} + \frac{ \partial ^{2}T }{ \partial z^{2}} = 0
 

    c. 1D Heat Flow, Steady-State
    \frac{ \partial ^{2}T }{ \partial x^{2}} = 0
 

    d. 1D heat flow, Unsteady-State
    k\frac{ \partial ^{2}T }{ \partial x^{2}} = \rho c\frac{ \partial T }{ \partial t }
    \alpha \frac{ \partial ^{2}T }{ \partial x^{2}} = \frac{ \partial T }{ \partial t }
     8th Perry’s Handbook, Eq. 5 – 17, p. 5-6

k – thermal conductivity
\rho – density
c – heat capacity
S – heat generated by the material
\alpha = \frac{ k }{ c\rho } – thermal diffusivity. Ratio of thermal conductivity and product of density and heat capacity.
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