Stated that in the boundary layer conduction-convection. Determine convective heat transfer coefficient in terms of temperature gradient applying Fourier’s law of conduction.
The correct answer and explanation is:
Correct Answer:
To determine the convective heat transfer coefficient (h) in terms of the temperature gradient using Fourier’s law of conduction, consider a solid surface in contact with a fluid. At the boundary layer, the energy is transferred by conduction from the wall to the fluid and then by convection into the fluid bulk.
According to Fourier’s law of conduction: q′′=−k(∂T∂y)y=0q” = -k \left( \frac{\partial T}{\partial y} \right)_{y=0}
Where:
- q′′q” is the heat flux at the surface (W/m²),
- kk is the thermal conductivity of the fluid (W/m·K),
- (∂T∂y)y=0\left( \frac{\partial T}{\partial y} \right)_{y=0} is the temperature gradient at the wall in the direction normal to the surface.
Also, by Newton’s law of cooling (convective heat transfer): q′′=h(Ts−T∞)q” = h (T_s – T_\infty)
Where:
- hh is the convective heat transfer coefficient (W/m²·K),
- TsT_s is the surface temperature,
- T∞T_\infty is the free stream (bulk fluid) temperature.
Equating both expressions for heat flux:
h(Ts−T∞)=−k(∂T∂y)y=0h (T_s – T_\infty) = -k \left( \frac{\partial T}{\partial y} \right)_{y=0} h=−kTs−T∞(∂T∂y)y=0\boxed{h = -\frac{k}{T_s – T_\infty} \left( \frac{\partial T}{\partial y} \right)_{y=0}}
Explanation (Approx. 300 words):
In heat transfer, the boundary layer is a region close to the surface of a solid where temperature gradients are significant. At this interface, heat is first transferred by conduction due to molecular collisions and then moves into the fluid via convection due to bulk fluid motion.
To relate conduction to convection, we use Fourier’s law to describe the heat flux conducted from the solid surface into the fluid. This law tells us that the heat flux is proportional to the temperature gradient at the wall and the thermal conductivity of the fluid. On the other hand, Newton’s law of cooling describes the same flux from a convective perspective — as being proportional to the temperature difference between the surface and the fluid bulk, multiplied by the convective heat transfer coefficient hh.
By equating both expressions for heat flux, we bridge conduction at the wall with convection in the fluid. The resulting formula expresses hh, the convective heat transfer coefficient, as a function of the local temperature gradient at the wall and the temperature difference across the boundary layer.
This relationship is fundamental in heat transfer analysis, particularly in computational simulations and boundary layer theory. It shows that hh is not a fixed property but depends on the local thermal conditions and flow characteristics, especially the steepness of the temperature gradient at the surface. A steeper gradient indicates a higher hh, meaning more efficient heat transfer by convection.