Allen/Eggers Hypersonic Drag, 1957
In METRIC!
Symbols: |
||
$ T_r |
Kelvins |
Recovery temperature |
T_w |
Kelvins |
Wall temperature (relatively small, will be ignored) |
T |
Kelvins |
Temperature at altitude |
M |
unitless |
Mach number at altitude |
H |
J / m2 |
Heat transferred per unit area |
h |
J / m2 - K |
Heat transfer coefficient |
C_v |
J / kg K |
Specific heat capacity at constant volume |
C_p |
J / kg K |
Specific heat capacity at constant pressure |
C_f |
? |
Skin effect coefficient |
\gamma |
C_p / C_v |
Specific heat capacity ratio, typically 1.4 |
\sigma |
meters |
nose radius |
k_r |
? |
Thermal conductivity at the recovery temperature |
Nu_r |
unitless |
Nusselt number |
Re_{\sigma} |
unitless |
Reynolds number for nose cone radius \sigma |
Pr |
unitless |
Prandtl number = 1 |
\mu_r |
? |
coefficient of viscosity at the recovery temperature |
note 1: in the original document, many variables have subscript ._l indicating "local" or at altitude, a complication not needed here
note 2: \gamma can be higher for diatomic or ionized gasses.
Assuming that the Prandtl number is unity.
Eq 23: T_r = T \left( 1 + { { \gamma-1 } \over 2 } M^2 \right) \approx { { \gamma-1 } \over 2 } M^2 T
Eq 25: ( T_r - T_w) = V^2 / 2 C_p ... since V_{sound} = \sqrt{ ( \gamma - 1 ) C_p T } . at altitude.
Eq 26: h = { 1 \over 2 } ~ C_f ~ C_p ~ \rho ~ V Heat transfer coefficient (all subcripted ._l in the original) .
- .. much omitted ...
Page 17: (modified to ditch minus sign)
Eq 42a(?): \Large { { d H_s } \over { d t } } = { { N_{ur} k_r ( T_r - T_w ) } \over \sigma } \approx { { N_{ur} k_r T_r } \over \sigma } Heat transfer rate per unit area at the stagnation point
Page 18:
Eq 42b(?): Nu_r = 0.934 ~ Re_{\sigma}^{0.5} ~ Pr^{ 0.4 } Nusselt number at recovery temperature (unitless)
"note that???" Re_{\sigma} ~ = ~ \rho ~ V ~ \sigma / \mu_r