Differences between revisions 5 and 6

Allen/Eggers Hypersonic Drag, 1957

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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

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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

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-  ⇤ ← Revision 5 as of 2016-12-12 00:44:00 → 
  Size: 1473
  Editor: KeithLofstrom
  Comment:
+   ← Revision 6 as of 2016-12-12 02:17:16 → ⇥
  Size: 2927
  Editor: KeithLofstrom
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
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-||<-3>'''Symbols:'''                                                          ||
|| $ T_r     ||  Kelvins      || Recovery temperature                         ||
|| $ T_w   $ ||  Kelvins      || Wall temperature                             ||
|| $ T     $ ||  Kelvins      || Temperature at altitude                      ||
|| $ M     $ ||  -            || Mach number at altitude                      ||
|| $ H     $ ||  J / m^2^     || Heat transferred per unit area               ||
|| $ h_l   $ ||  J / m^2^ - K || Heat transfer coefficient                    ||
|| $ C_v   $ ||  J / kg K     || [[ https://en.wikipedia.org/wiki/Heat_capacity | Specific heat capacity ]] at constant volume    ||
|| $ C_p   $ ||  J / kg K     || Specific heat capacity at constant pressure  ||
|| $ gamma $ || $ C_p / C_v $ || Specific heat capacity ratio, typically 1.4  ||
+||<-3>'''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 / m^2^     || Heat transferred per unit area                               ||
|| $ h     $ ||  J / m^2^ - K || Heat transfer coefficient                                    ||
|| $ C_v   $ ||  J / kg K     || [[ https://en.wikipedia.org/wiki/Heat_capacity | 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         ||
-Line 17:
+Line 24:
- . note: $ gamma $ can be higher for diatomic or ionized gasses.
+ . 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.
-Line 23:
+Line 31:
-Eq 25: $ ( T_r - T_W) = V^2 / 2 C_p $ ... since $ V_{sound} = \sqrt{ ( \gamma - 1 ) C_p T } $ . at altitude
+Eq 25: $ ( T_r - T_w) = V^2 / 2 C_p $ ... since $ V_{sound} = \sqrt{ ( \gamma - 1 ) C_p T } $ . at altitude.
-Line 25:
+Line 33:
-Heat transfer coefficient h_l
+Eq 26: $ h = { 1 \over 2 } ~ C_f ~ C_p ~ \rho ~ V $ Heat transfer coefficient (all subcripted $._l$ in the original) .
-Line 27:
+Line 35:
- . note: $ gamma $ can be higher for diatomic or ionized gasses.
+ ... much omitted ...
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+Line 37:
+Page 17: (modified to ditch minus sign)
-Line 30:
+Line 39:
+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 $

Diff for "DragAllenEggers1957"

Allen/Eggers Hypersonic Drag, 1957

In METRIC!