Matrix or Vector | Geometry / Midside Nodes [1] | Shape Functions | Integration Points |
---|---|---|---|
Convection Surface Matrix and Load Vector; and Heat Generation Load Vector | Quad with midside nodes | Equation 11–91 | 3 x 3 |
Quad without midside nodes | Equation 11–79 | 2 x 2 | |
Triangle with midside nodes | Equation 11–63 | 6 | |
Triangle without midside nodes | Equation 11–106 | 3 |
Load Type | Distribution |
---|---|
All Loads | Same as shape functions |
When the extra node is not present, the logic is the same as given and as described in Derivation of Heat Flow Matrices. The discussion below relates to theory that uses the extra node.
The conductivity matrix is based on one-dimensional flow to and away from the surface. The form is conceptually the same as for LINK33 (Equation 13–27) except that the surface has four or eight nodes instead of only one node. Using the example of convection and no midside nodes are requested (KEYOPT(4) = 1) (resulting in a 5 x 5 matrix), the first four terms of the main diagonal are:
(13–240) |
where:
h_{u} = output argument for film coefficient of USRSURF116 |
{N} = vector of shape functions |
which represents the main diagonal of the upper-left corner of the conductivity matrix. The remaining terms of this corner are all zero. The last main diagonal term is simply the sum of all four terms of Equation 13–240 and the off-diagonal terms in the fifth column and row are the negative of the main diagonal of each row and column, respectively.
If midside nodes are present (KEYOPT(4) = 0) (resulting in a 9 x 9 matrix) Equation 13–240 is replaced by:
(13–241) |
which represents the upper-left corner of the conductivity matrix. The last main diagonal is simply the sum of all 64 terms of Equation 13–241 and the off-diagonal terms in the ninth column and row are the negative of the sum of each row and column respectively.
Radiation is handled similarly, except that the approach discussed for LINK31 in LINK31 - Radiation Link is used. A load vector is also generated. The area used is the area of the element. The form factor is discussed in a subsequent section.
An additional load vector is formed when using the extra node by:
(13–242) |
where:
{Q^{c}} = load vector to be formed |
[K^{tc}] = element conductivity matrix due to convection |
TEMVEL from USRSURF116 is the difference between the bulk temperature and the temperature of the extra node.
There is special logic that accesses FLUID116 information where FLUID116 has had KEYOPT(2) set equal to 1. This logic uses SURF151 or SURF152 with the extra node present (KEYOPT(5) = 1) and computes an adiabatic wall temperature (KEYOPT(6) = 1). For this case, T_{v}, as used above, is defined as:
(13–243) |
where:
F_{R} = recovery factor (see Equation 13–244) |
V_{abs} = absolute value of fluid velocity (input as VABS on R command) |
Ω = angular velocity of moving wall (input as OMEGA on R command) |
Ω_{ref} = reference angular velocity (input as (A_{n})_{I} and (A_{n})_{J} on R command of FLUID116) |
F_{s} = slip factor (input as SLIPFAI, SLIPFAJ on R command of FLUID116) |
V_{116} = velocity of fluid at extra node from FLUID116 |
g_{c} = gravitational constant used for units consistency (input as GC on R command) |
J_{c} = Joule constant used to convert work units to heat units (input as JC on R command) |
The recovery factor is computed as follows:
(13–244) |
where:
C_{n} = constant used for recovery factor calculation (input as NRF on R command) |
μ^{f} = viscosity of fluid (from FLUID116) |
ρ^{f} = density of fluid (from FLUID116) |
D = diameter of fluid pipe (from FLUID116) |
(13–245) |
where:
V = velocity used to compute Reynold's number |
The adiabatic wall temperature is reported as:
(13–246) |
where:
T_{aw} = adiabatic wall temperature |
T_{ex} = temperature of extra node |
KEYOPT(1) = 0 or 1 is ordinarily used for turbomachinry analysis, whereas KEYOPT(1) = 2 is ordinarily used for flow past stationary objects. For turbomachinery analyses T_{ex} is assumed to be the total temperature, but for flow past stationary objects T_{ex} is assumed to be the static temperature.
After the first coefficient has been determined, it is adjusted if KEYOPT(7) = 1:
(13–247) |
where:
h_{f} = unadjusted film coefficient |
T_{S} = surface temperature |
T_{B} = bulk temperature (T_{aw}, if defined) |
n = real constant (input as ENN on RMORE command) |
The form factor is computed as:
(13–248) |
also,
F = form factor (output as FORM FACTOR) |
Developing B further
α = angle between element z axis at integration point being processed and the line connecting the integration point and the extra node (see Figure 13.23: Form Factor Calculation) |
F is then used in the two-surface radiation equation:
(13–249) |
where:
σ = Stefan-Boltzmann constant (input as SBCONST on R command) |
ε = emissivity (input as EMIS on MP command) |
A = element area |
Note that this “form factor” does not have any distance affects. Thus, if distances are to be included, they must all be similar in size, as in an object on or near the earth being warmed by the sun. For this case, distance affects can be included by an adjusted value of σ.