Fix use of emissivity for cylindrical and spherical surfaces

modules/heat_transfer/include/materials/GapConductance.h

@@ -120,6 +120,8 @@ class GapConductance : public Material
   const VariableValue * const _gap_conductivity_function_variable;
 
   const Real _stefan_boltzmann;
+  const Real _emissivity_primary;
+  const Real _emissivity_secondary;
   Real _emissivity;
 
   const Real _min_gap;

modules/heat_transfer/src/materials/GapConductance.C

@@ -49,7 +49,7 @@ GapConductance::validParams()
                         "then required.");
   params.addParam<BoundaryName>("paired_boundary", "The boundary to be penetrated");
 
-  params.addParam<Real>("stefan_boltzmann", 5.669e-8, "The Stefan-Boltzmann constant");
+  params.addParam<Real>("stefan_boltzmann", 5.670374e-8, "The Stefan-Boltzmann constant");
 
   params.addParam<bool>("use_displaced_mesh",
                         true,
@@ -128,6 +128,8 @@ GapConductance::GapConductance(const InputParameters & parameters)
                                             ? &coupledValue("gap_conductivity_function_variable")
                                             : nullptr),
     _stefan_boltzmann(getParam<Real>("stefan_boltzmann")),
+    _emissivity_primary(getParam<Real>("emissivity_primary")),
+    _emissivity_secondary(getParam<Real>("emissivity_secondary")),
     _min_gap(getParam<Real>("min_gap")),
     _min_gap_order(getParam<unsigned int>("min_gap_order")),
     _max_gap(getParam<Real>("max_gap")),
@@ -139,13 +141,8 @@ GapConductance::GapConductance(const InputParameters & parameters)
     _p1(declareRestartableData<Point>("cylinder_axis_point_1", Point(0, 1, 0))),
     _p2(declareRestartableData<Point>("cylinder_axis_point_2", Point(0, 0, 0)))
 {
-  // Set emissivity but allow legacy naming; legacy names are used if they
-  // are present
-  const auto emissivity_primary = getParam<Real>("emissivity_primary");
-  const auto emissivity_secondary = getParam<Real>("emissivity_secondary");
-
-  _emissivity = emissivity_primary != 0.0 && emissivity_secondary != 0.0
-                    ? 1.0 / emissivity_primary + 1.0 / emissivity_secondary - 1
+  _emissivity = _emissivity_primary != 0.0 && _emissivity_secondary != 0.0
+                    ? 1.0 / _emissivity_primary + 1.0 / _emissivity_secondary - 1
                     : 0.0;
 
   if (_quadrature)
@@ -337,19 +334,35 @@ GapConductance::h_radiation()
   /*
    Gap conductance due to radiation is based on the diffusion approximation:
 
-      qr = sigma*Fe*(Tf^4 - Tc^4) ~ hr(Tf - Tc)
-         where sigma is the Stefan-Boltzmann constant, Fe is an emissivity function, Tf and Tc
-         are the fuel and clad absolute temperatures, respectively, and hr is the radiant gap
-         conductance. Solving for hr,
+      q12 = sigma*Fe*(T1^4 - T2^4) ~ hr(T1 - T2)
+            where sigma is the Stefan-Boltzmann constant, Fe is an emissivity
+            function, T1 and T2 are the temperatures of the two surfaces, and
+            hr is the radiant gap conductance. Solving for hr,
+
+      hr = sigma*Fe*(T1^4 - T2^4) / (T1 - T2)
+           which can be factored to give:
+
+      hr = sigma*Fe*(T1^2 + T2^2) * (T1 + T2)
+
+   Assuming the gap is between infinite parallel planes, the emissivity
+   function is given by:
+
+      Fe = 1 / (1/e1 + 1/e2 - 1)
+
+   For cylinders and spheres, see Fundamentals of Heat and Mass Transfer,
+      Sixth Edition, John Wiley & Sons, Table 13.3.
+
+   For cylinders:
+
+      Fe = 1 / (1/e1 + (1/e2 - 1) * (r1/r2))
 
-      hr = sigma*Fe*(Tf^4 - Tc^4) / (Tf - Tc)
-         which can be factored to give:
+      q21 = -q12 * (r1/r2)
 
-      hr = sigma*Fe*(Tf^2 + Tc^2) * (Tf + Tc)
+   For spheres:
 
-   Approximating the fuel-clad gap as infinite parallel planes, the emissivity function is given by:
+      Fe = 1 / (1/e1 + (1/e2 - 1) * (r1/r2)^2)
 
-      Fe = 1 / (1/ef + 1/ec - 1)
+      q21 = -q12 * (r1/r2)^2
   */
 
   if (_emissivity == 0.0)
@@ -360,9 +373,18 @@ GapConductance::h_radiation()
   Real surface_integration_factor = 1.0;
 
   if (_gap_geometry_type == GapConductance::CYLINDER)
-    surface_integration_factor = 0.5 * (_r1 + _r2) / _radius;
+  {
+    _emissivity = 1.0 / _emissivity_primary + (1.0 / _emissivity_secondary - 1) * _r1 / _r2;
+    if (_r2 == _radius)
+      surface_integration_factor = _r1 / _r2;
+  }
   else if (_gap_geometry_type == GapConductance::SPHERE)
-    surface_integration_factor = 0.25 * (_r1 + _r2) * (_r1 + _r2) / (_radius * _radius);
+  {
+    _emissivity =
+        1.0 / _emissivity_primary + (1.0 / _emissivity_secondary - 1) * _r1 * _r1 / (_r2 * _r2);
+    if (_r2 == _radius)
+      surface_integration_factor = _r1 * _r1 / (_r2 * _r2);
+  }
 
   const Real temp_func =
       (_temp[_qp] * _temp[_qp] + _gap_temp * _gap_temp) * (_temp[_qp] + _gap_temp);
@@ -379,9 +401,18 @@ GapConductance::dh_radiation()
   Real surface_integration_factor = 1.0;
 
   if (_gap_geometry_type == GapConductance::CYLINDER)
-    surface_integration_factor = 0.5 * (_r1 + _r2) / _radius;
+  {
+    _emissivity = 1.0 / _emissivity_primary + (1.0 / _emissivity_secondary - 1) * _r1 / _r2;
+    if (_r2 == _radius)
+      surface_integration_factor = _r1 / _r2;
+  }
   else if (_gap_geometry_type == GapConductance::SPHERE)
-    surface_integration_factor = 0.25 * (_r1 + _r2) * (_r1 + _r2) / (_radius * _radius);
+  {
+    _emissivity =
+        1.0 / _emissivity_primary + (1.0 / _emissivity_secondary - 1) * _r1 * _r1 / (_r2 * _r2);
+    if (_r2 == _radius)
+      surface_integration_factor = _r1 * _r1 / (_r2 * _r2);
+  }
 
   const Real temp_func = 3 * _temp[_qp] * _temp[_qp] + _gap_temp * (2 * _temp[_qp] + _gap_temp);
 

modules/heat_transfer/test/tests/gap_heat_transfer_balance/gold/large_gap_heat_transfer_test_cylinder_out.csv

@@ -1,3 +1,3 @@
 time,Tcore_avg,Tcore_max,Tcore_min,Trpv_avg,Trpv_max,Trpv_min,heat_balance,ptot,rpv_convective_out
 0,0,0,0,0,0,0,0,0,0
-1,5393.5775548207,6113.3891835742,4759.3589179598,4352.9526929928,4609.8983226228,4133.5700499916,7.4443014717573e-05,628318.53071796,628365.30464359
+1,5393.6644156926,6113.4760465477,4759.4462004543,4352.9605673544,4609.906406692,4133.5778074715,7.6386029698641e-05,628318.53071796,628366.52547591

modules/heat_transfer/test/tests/gap_heat_transfer_balance/gold/large_gap_heat_transfer_test_rz_cylinder_out.csv

@@ -1,3 +1,3 @@
 time,Tcore_avg,Tcore_max,Tcore_min,Trpv_avg,Trpv_max,Trpv_min,flux_from_core,flux_into_rpv,heat_balance,ptot,rpv_convective_out
 0,0,0,0,0,0,0,0,0,0,0,0
-1,5321.3794083679,6043.9329433687,4600.3618143312,4279.5038918932,4594.2083328607,4004.8357991759,627533.46021351,-627605.34282851,1.5600638240969e-13,628318.53071796,628318.53071806
+1,5321.4889932475,6044.0425282483,4600.4713992108,4279.503891892,4594.2083328594,4004.8357991749,627533.46021336,-627605.34282826,-1.3210516705238e-13,628318.53071796,628318.53071788

modules/heat_transfer/test/tests/gap_heat_transfer_balance/gold/large_gap_heat_transfer_test_sphere_out.csv

@@ -1,3 +1,3 @@
 time,flux_left,flux_right,heat_balance,ptot,sphere_convective_out,temp_left,temp_right
 0,0,0,0,0,0,0,0
-1,104721.76094827,-102359.96262201,1.4653101375897e-07,104718.22222357,104718.23756804,738.7371726696,565.45711490661
+1,104721.69029332,-102359.97052264,1.8263860794812e-08,104718.22222357,104718.22413613,835.27391832054,565.45708251443

modules/heat_transfer/test/tests/gap_heat_transfer_mortar/gold/bc_gap_heat_transfer_displaced_radiation_out_NodalTemperature_0001.csv

@@ -1,15 +1,15 @@
 id,temp,x,y,z
-0,70.621675442894,0.4,1,0
-8,70.621675442894,0.4,0,0
-10,70.621675442894,0.4,0.2,0
-11,70.621675442894,0.4,0.4,0
-12,70.621675442894,0.4,0.6,0
-13,70.621675442894,0.4,0.8,0
-19,58.75664910825,0.6,1,0
-22,58.75664910825,0.6,0,0
-23,58.75664910825,0.6,0.85714285714286,0
-24,58.75664910825,0.6,0.71428571428571,0
-25,58.75664910825,0.6,0.57142857142857,0
-26,58.75664910825,0.6,0.42857142857143,0
-27,58.75664910825,0.6,0.28571428571429,0
-28,58.75664910825,0.6,0.14285714285714,0
+0,70.620771417379,0.4,1,0
+8,70.62077141738,0.4,0,0
+10,70.62077141738,0.4,0.2,0
+11,70.62077141738,0.4,0.4,0
+12,70.62077141738,0.4,0.6,0
+13,70.62077141738,0.4,0.8,0
+19,58.758457159288,0.6,1,0
+22,58.758457159288,0.6,0,0
+23,58.758457159288,0.6,0.85714285714286,0
+24,58.758457159288,0.6,0.71428571428571,0
+25,58.758457159288,0.6,0.57142857142857,0
+26,58.758457159288,0.6,0.42857142857143,0
+27,58.758457159288,0.6,0.28571428571429,0
+28,58.758457159288,0.6,0.14285714285714,0

modules/heat_transfer/test/tests/gap_heat_transfer_mortar/gold/large_gap_heat_transfer_test_cylinder_out_NodalTemperature_0001.csv

@@ -1,81 +1,81 @@
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modules/heat_transfer/test/tests/gap_heat_transfer_mortar/gold/large_gap_heat_transfer_test_rz_cylinder_out_NodalTemperature_0001.csv

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