Page 222 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
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7.8.1 Nusselt number
The Nusselt number is derived as follows. Let us assume that a hot surface is cooled
by a cold fluid stream. The heat from the hot surface, which is maintained at a constant
temperature, is diffused through a boundary layer and convected away by the cold stream.
This phenomenon is normally defined by Newton’s law of cooling per unit surface area as
∂T
h c (T w − T f ) =−k (7.175)
∂n
where h c is the heat transfer coefficient, k is an average thermal conductivity of the fluid,
T f is the free stream temperature of the fluid and n is the normal direction to the heat
transfer surface. The above equation can be rewritten as
h c L 1 ∂T
=− L (7.176)
k T w − T f ∂n
where L is any characteristic dimension. The quantity on the left-hand side of the above
equation is the Nusselt number. If we apply non-dimensional scales, as discussed in Section
3, we can rewrite the above equation as

∂T ∗
N u =− (7.177)
∂n ∗
where N u is the local Nusselt number. It should be observed that the local Nusselt number
is equal to the local, non-dimensional, normal temperature gradient. The above definition of
the Nusselt number is valid for any heat transfer problem as long as the surface temperature
is constant, or a reference wall temperature is known. However, for prescribed heat flux
conditions, a different approach is required to derive the Nusselt number. Let us assume a
surface subjected to a uniform heat flux q. We can write locally

∂T
q =−k = h c (T w − T f ) (7.178)
∂n
where T w is not a constant. The Nusselt number relation can be obtained by multiplying
the RHS of the previous equations by L/k,thatis,
h c L qL
(T w − T f ) = (7.179)
k k
Rearranging, we obtain
qL
k
N u = (7.180)
(T w − T f )
When a wall is subjected to heat flux boundary conditions, the temperature scale is
qL/k, which non-dimensionalizes the temperature. Therefore, the above equation can be
rewritten as
1
N u = (7.181)
T − T f ∗
∗
w