Energy Equation
This discussion involving the energy equation is with respect to rotationally symmetric flow. This analysis will take into consideration the effect of gravitational acceleration on flow and heat transfer. The mathematical equations to follow apply both to rotational symmetric flow as well as two-dimensional planar flow.
- rotational direction - tangential velocity - planar direction - normal velocity - radius
This equation expands to the following with the addition of physical fluid properties:
In this equation the superscript n is to differentiate between rotationally symmetric flow from planar flow. The following characteristics of this equation hold true. - rotationally symmetric flow - planar, two-dimensional flow - gravitational acceleration
From here we can further simplify the momentum equation by setting the bulk fluid velocity to 0.
This relation shows that the pressure gradient is simply a product of the bulk fluid density and the gravitational acceleration. The next step is to plug in the pressure gradient into the momentum equation.
Further simplification of the momentum equation comes by substituting the volume expansion coefficient, density relationship found above into the momentum equation.
To find the Grashof Number from this point the preceding equation must be non-dimesionalized. This means that every variable in the equation should have no dimension. This is done by dividing each variable by corresponding constant quantities. Lengths are divided by a characteristic length . Velocities are divided by appropriate reference velocities which considering the Reynolds number gives Temperatures are divided by the appropriate temperature difference These dimensionless parameters look like the following:
, .
The asterisks represent dimensionless parameter. Combining these dimensionless equations with the momentum equations gives the following simplified equation.
- surface temperature - bulk fluid temperature - characteristic length
The dimensionless parameter enclosed in the brackets in the preceding equation is known as the Grashof Number
Read more about this topic: Grashof Number, Derivation of Grashof Number
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