Page 184 - Mechanical Engineers' Handbook (Volume 4)

2 Convection Heat Transfer 173

these types of situations, the convection heat transfer can be described as q hA (T T ),
s
s
where T is the adiabatic surface temperature or recovery temperature, and is related to the
s
recovery factor by r (T s T )/(T T ). The value of the stagnation temperature, T ,

0

0
is related to the free stream static temperature, T , by the expression

T 0 1 2
T 1 2 M
where is the specific heat ratio of the fluid and M is the ratio of the free stream velocity
and the acoustic velocity. For the case where 0.6 Pr 15,
5
r Pr 1/2 for laminar flow (Re 5 10 )
x
5
r Pr 1/3 for turbulent flow (Re 5 10 )
x
Here, all of the fluid properties are evaluated at the reference temperature T T 0.5(T s
ref

T ) 0.22(T T ). Expressions for the local heat-transfer coefficients at a given distance
s

x from the leading edge are given as 2
Nu 0.332Re 0.5 Pr 1 / 3 for Re 5 10 5
x
x
x
5
Nu 0.0292Re 0.8 Pr 1 / 3 for 5 10 Re 10 7
x
x
x
7
Nu 0.185Re (logRe ) 2.584 for 10 Re 10 9
x
x
x
x
In the case of gaseous fluids flowing at very high free stream velocities, dissociation of the
gas may occur, and will cause large variations in the properties within the boundary layer.
For these cases, the heat-transfer coefficient must be defined in terms of the enthalpy dif-
ference, i.e., q hA(i i ), and the recovery factor will be given by r (i i )/(i
s
0
s
s
s
i ), where i represents the enthalpy at the adiabatic wall conditions. Similar expressions to

s
those shown above for Nu can be used by substituting the properties evaluated at a reference
x
enthalpy defined as i ref i 0.5(i i ) 0.22(i s i ).
s

High-Speed Gas Flow past Cones
For the case of high-speed gaseous flows over conical shaped objects the following expres-
sions can be used:
Nu 0.575Re 0.5 Pr 1 / 3 for Re 10 5
x
x
x
Nu 0.0292Re 0.8 Pr 1 / 3 for Re 10 5
x
x
x
where the fluid properties are evaluated at T as in the plate. 12
ref
Stagnation Point Heating for Gases
When the conditions are such that the flow can be assumed to behave as incompressible, the
13
Reynolds number is based on the free stream velocity and h is defined as q hA (T T ) .
s
Estimations of the Nusselt can be made using the following relationship
Nu CRe 0.5 Pr 0.4
D
D
where C 1.14 for cylinders and 1.32 for spheres, and the fluid properties are evaluated at
the mean film temperature. When the flow becomes supersonic, a bow shock wave will occur
just off the front of the body. In this situation, the fluid properties must be evaluated at the
stagnation state occurring behind the bow shock and the Nusselt number can be written as