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$                       RIGID FORMAT No. 3, Approach Heat,
$            Nonlinear Radiation and Conduction of a Cylinder (3-6-1)
$ 
$ A. Description
$ 
$ This problem illustrates the solution of a combined conduction and radiation
$ heat transfer analysis. The model is a two-dimensional representation of a
$ long cylinder subject to radiant heat from a distant source. The shell has
$ internal radiation exchange, external radiation loss, and conduction around
$ the circumference.
$ 
$ B. Input
$ 
$ The NASTRAN model uses ROD elements to represent the circumferential heat flow
$ and HBDY elements to represent the inside and outside surfaces. The radiation
$ exchange factors for the inside of the cylinder are defined on the RADMTX data
$ cards. The incoming vector flux is defined on the QVECT data card. The model
$ parameters are:
$ 
$ R  =  2.0 ft                      (Radius of shell)
$ 
$ t  =  .001 ft                     (Thickness)
$ 
$ l  =  20.306 ft                   (Axial length)
$ 
$ epsilon = alpha = 0.1             (Emissivity and absorptivity)
$ 
$                  2
$ q  =  425 BTU/(ft -hr)            (Source flux density)
$  v
$ 
$ k  =  94.5 BTU/(hr-ft-deg. F)      (Conductivity of shell)
$ 
$                  -8        2          4
$ sigma = .174 x 10   BTU/(ft -hr-deg. R ) (Stefan-Boltzmann radiation
$                                    constant)
$ 
$ C. Theory
$ 
$ A closed-form solution to this problem is not available. However, the solution
$ may be validated by checking the global net heat flow, the local net heat
$ exchange, and the estimated average temperature.
$ 
$ An estimate of the average temperature may be obtained from the equations:
$ 
$    Q   = alpha q  lR integral from -pi/2 to pi/2 of cos theta d theta =
$     in          v
$                                                                            (1)
$    2 alpha lRq
$               v
$ 
$ and
$ 
$                         _4
$    Q    = epsilon sigma T  (2 pi Rl)                                       (2)
$     out
$ 
$ where Q   is the total input from the source, Q    is the net flux radiated
$        in   _                                  out
$ outward and T is the average absolute temperature.
$ 
$ Since the net heat flow must be zero in a steady-state analysis, Equations (1)
$ and (2) are equated to obtain:
$ 
$           q
$    _4      v
$    T  = --------                                                           (3)
$         pi sigma
$ 
$ D. Results
$ 
$ The average value of temperature from the NASTRAN results shows 57.87 degrees
$ F. The estimated average temperature from Equation (3) above is 68 degrees.
$ The difference is due to the non-uniform radiation effects.
$ 
$ A second check is provided by computing the global net heat flow error in the
$ system. Summing the net flow into each element gives a net heat flow error
$ several orders of magnitude less than the total heat from the source. As a
$ further check, the local net heat flow error at grid point 2 was calculated by
$ summing the contributions from the connected elements. The heat flow terms, as
$ calculated by NASTRAN, were:
$ 
$    Q      = 59.420     (Flow through ROD #2 (flux - area))
$     2
$ 
$    Q      = 97.862     (Flow through ROD #3 (flux - area))
$     3
$ 
$    Q      = -133.564   (Inside radiation flow into HBDY #42)
$     r42
$ 
$    Q      = -85.352    (Inside radiation flow into HBDY #43)
$     r43
$ 
$    Q      = -305.418   (Outside radiation into HBDY #22)
$     r22
$ 
$    Q      = -257.930   (Outside radiation into HBDY #23)
$     r23
$ 
$    Q      = 481.157    (Vector flux input to HBDY #22)
$     v22
$ 
$    Q      = 381.848    (Vector flux input to HBDY #23)
$     v23
$ 
$ The net flow error into grid point 2 is:
$ 
$    _    1
$    Q  = - (Q    + Q    + Q    + Q    + Q    + Q   ) + Q  - Q  = 1.9 BTU    (4)
$     2   2   r22    r23    r42    r43    v22    v23     2    3
$ 
$ This error is less than 1% of the total heat flow input at the point.
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