Understanding temperature coupling BCs

chriss85 · October 28, 2014, 07:44

I'm currently trying to understand the boundary conditions for temperature coupling between two regions, namely compressible::turbulentTemperatureCoupledBaffleMix ed and compressible::turbulentTemperatureRadCoupledMixed (source files located in turbulenceModels/compressible/turbulenceModel/derivedFvPatchFields/ ). In the first BC, radiation is neglected and the problem thus simplifies. In the .C file, one finds the following comment:

Quote:

// Both sides agree on
// - temperature : (myKDelta*fld + nbrKDelta*nbrFld)/(myKDelta+nbrKDelta)
// - gradient : (temperature-fld)*delta
// We've got a degree of freedom in how to implement this in a mixed bc.
// (what gradient, what fixedValue and mixing coefficient)
// Two reasonable choices:
// 1. specify above temperature on one side (preferentially the high side)
// and above gradient on the other. So this will switch between pure
// fixedvalue and pure fixedgradient
// 2. specify gradient and temperature such that the equations are the
// same on both sides. This leads to the choice of
// - refGradient = zero gradient
// - refValue = neighbour value
// - mixFraction = nbrKDelta / (nbrKDelta + myKDelta())

I completely agree to and understand the value of the temperature on both sides of the patch. This is just a weighted average of the temperatures in the cells weighted with the heat conductivities divided by the length from center of the cell to the patch face.
I then assume that the gradient which is written here is only meant for the first side, and that the neighbour side is meant to be (temperature-nbrFld)*nbrDelta (possibly negative when the direction of the surface normal vector is considered).

Now the two strategies below are meant to be used in iterative solvers I suppose. Does anyone know how the second one can be derived from the formulas? Or is it somewhat empirically determined to accelerate convergence maybe?

For the second BC, things become more difficult because there are additional radiative heat fluxes, and no explaining comment. The strategy which is used here appears to be a bit different. The formulas used are:
$refValue_o=\frac{K_{\Delta_n} \cdot T_n}{\alpha}=\frac{\frac{\kappa_o}{\Delta_o} \cdot T_n}{\frac{\kappa_n}{\Delta_n}-\frac{Qr_o+Qr_n}{T_o}} \newline refGrad=0\newline valueFraction=\frac{\alpha}{\alpha + K_{\Delta_o}}=\frac{\frac{\kappa_n}{\Delta_n}-\frac{Qr_o+Qr_n}{T_o}}{\frac{\kappa_n}{\Delta_n}-\frac{Qr_o+Qr_n}{T_o} + \frac{\kappa_o}{\Delta_o}}$
, where o=owner, n=neighbour, Delta=distance between cell center and patch face, kappa heat conductivity and Qr radiative heat flux.

Does anyone understand this or has seen a derivative? I think this could be interesting for different heat transfer applications at boundaries.

chriss85 · October 31, 2014, 08:31

I've spent some more time thinking about this. Using the equations above inserted into the mixed BC formula:
$T_{F_0}=\frac{\alpha}{\alpha + \frac{\kappa_1}{\Delta_1}} \frac{\frac{\kappa_1}{\Delta_1}T_{F_1}}{\alpha} + (1 - \frac{\alpha}{\alpha + \frac{\kappa_1}{\Delta_1}}) T_0 =\frac{\alpha}{\alpha + \frac{\kappa_1}{\Delta_1}} \frac{\frac{\kappa_1}{\Delta_1}T_{F_1}}{\alpha} + \frac{\frac{\kappa_0}{\Delta_0}}{\alpha + \frac{\kappa_1}{\Delta_1}} T_0 =\frac{\frac{\kappa_1}{\Delta_1}}{\alpha + \frac{\kappa_1}{\Delta_1}} T_{F_1} + \frac{\frac{\kappa_0}{\Delta_0}}{\alpha + \frac{\kappa_1}{\Delta_1}} T_0$ , here with 0: first region, 1: second region, so T0: temperature in first region cell, T_F_0: temperature in first region patch.
I'm wondering if the signs at the radiation fluxes are correct. For positive Q_r (meaning outgoing radiation) an iteration procedure using this formula is bound to diverge, because the sum of the two weighting factors in front of the temperatures is not 1???

As an example: Consider radiation coming out of a gas (0) being absorbed completely by a solid (1). In this case, Qr_0 > 0, Qr_1 = 0 and the sum of both factors is larger than 1. In my understanding this should result in an increasing temperature on each iteration.

I'm going to test this series one dimensionally with MATLAB to investigate the convergence properties. Does anyone have any better explanation for the reasoning behind these formulas or maybe a literature recommendation?

chriss85 · November 3, 2014, 11:06

See http://www.cfd-online.com/Forums/ope...tml#post517173 for further results.

atulkjoy · November 26, 2018, 00:20

Quote:

Originally Posted by chriss85

I've spent some more time thinking about this. Using the equations above inserted into the mixed BC formula:
$T_{F_0}=\frac{\alpha}{\alpha + \frac{\kappa_1}{\Delta_1}} \frac{\frac{\kappa_1}{\Delta_1}T_{F_1}}{\alpha} + (1 - \frac{\alpha}{\alpha + \frac{\kappa_1}{\Delta_1}}) T_0 =\frac{\alpha}{\alpha + \frac{\kappa_1}{\Delta_1}} \frac{\frac{\kappa_1}{\Delta_1}T_{F_1}}{\alpha} + \frac{\frac{\kappa_0}{\Delta_0}}{\alpha + \frac{\kappa_1}{\Delta_1}} T_0 =\frac{\frac{\kappa_1}{\Delta_1}}{\alpha + \frac{\kappa_1}{\Delta_1}} T_{F_1} + \frac{\frac{\kappa_0}{\Delta_0}}{\alpha + \frac{\kappa_1}{\Delta_1}} T_0$ , here with 0: first region, 1: second region, so T0: temperature in first region cell, T_F_0: temperature in first region patch.
I'm wondering if the signs at the radiation fluxes are correct. For positive Q_r (meaning outgoing radiation) an iteration procedure using this formula is bound to diverge, because the sum of the two weighting factors in front of the temperatures is not 1???

As an example: Consider radiation coming out of a gas (0) being absorbed completely by a solid (1). In this case, Qr_0 > 0, Qr_1 = 0 and the sum of both factors is larger than 1. In my understanding this should result in an increasing temperature on each iteration.

I'm going to test this series one dimensionally with MATLAB to investigate the convergence properties. Does anyone have any better explanation for the reasoning behind these formulas or maybe a literature recommendation?

HI Hi chriss

Did you find a way to couple radiation of solid and gas phase. ????

atulkjoy · November 26, 2018, 00:24

Quote:

Originally Posted by chriss85

I've spent some more time thinking about this. Using the equations above inserted into the mixed BC formula:
$T_{F_0}=\frac{\alpha}{\alpha + \frac{\kappa_1}{\Delta_1}} \frac{\frac{\kappa_1}{\Delta_1}T_{F_1}}{\alpha} + (1 - \frac{\alpha}{\alpha + \frac{\kappa_1}{\Delta_1}}) T_0 =\frac{\alpha}{\alpha + \frac{\kappa_1}{\Delta_1}} \frac{\frac{\kappa_1}{\Delta_1}T_{F_1}}{\alpha} + \frac{\frac{\kappa_0}{\Delta_0}}{\alpha + \frac{\kappa_1}{\Delta_1}} T_0 =\frac{\frac{\kappa_1}{\Delta_1}}{\alpha + \frac{\kappa_1}{\Delta_1}} T_{F_1} + \frac{\frac{\kappa_0}{\Delta_0}}{\alpha + \frac{\kappa_1}{\Delta_1}} T_0$ , here with 0: first region, 1: second region, so T0: temperature in first region cell, T_F_0: temperature in first region patch.
I'm wondering if the signs at the radiation fluxes are correct. For positive Q_r (meaning outgoing radiation) an iteration procedure using this formula is bound to diverge, because the sum of the two weighting factors in front of the temperatures is not 1???

As an example: Consider radiation coming out of a gas (0) being absorbed completely by a solid (1). In this case, Qr_0 > 0, Qr_1 = 0 and the sum of both factors is larger than 1. In my understanding this should result in an increasing temperature on each iteration.

I'm going to test this series one dimensionally with MATLAB to investigate the convergence properties. Does anyone have any better explanation for the reasoning behind these formulas or maybe a literature recommendation?

Quote:

Originally Posted by chriss85

I'm currently trying to understand the boundary conditions for temperature coupling between two regions, namely compressible::turbulentTemperatureCoupledBaffleMix ed and compressible::turbulentTemperatureRadCoupledMixed (source files located in turbulenceModels/compressible/turbulenceModel/derivedFvPatchFields/ ). In the first BC, radiation is neglected and the problem thus simplifies. In the .C file, one finds the following comment:

I completely agree to and understand the value of the temperature on both sides of the patch. This is just a weighted average of the temperatures in the cells weighted with the heat conductivities divided by the length from center of the cell to the patch face.
I then assume that the gradient which is written here is only meant for the first side, and that the neighbour side is meant to be (temperature-nbrFld)*nbrDelta (possibly negative when the direction of the surface normal vector is considered).

Now the two strategies below are meant to be used in iterative solvers I suppose. Does anyone know how the second one can be derived from the formulas? Or is it somewhat empirically determined to accelerate convergence maybe?

For the second BC, things become more difficult because there are additional radiative heat fluxes, and no explaining comment. The strategy which is used here appears to be a bit different. The formulas used are:
$refValue_o=\frac{K_{\Delta_n} \cdot T_n}{\alpha}=\frac{\frac{\kappa_o}{\Delta_o} \cdot T_n}{\frac{\kappa_n}{\Delta_n}-\frac{Qr_o+Qr_n}{T_o}} \newline refGrad=0\newline valueFraction=\frac{\alpha}{\alpha + K_{\Delta_o}}=\frac{\frac{\kappa_n}{\Delta_n}-\frac{Qr_o+Qr_n}{T_o}}{\frac{\kappa_n}{\Delta_n}-\frac{Qr_o+Qr_n}{T_o} + \frac{\kappa_o}{\Delta_o}}$
, where o=owner, n=neighbour, Delta=distance between cell center and patch face, kappa heat conductivity and Qr radiative heat flux.

Does anyone understand this or has seen a derivative? I think this could be interesting for different heat transfer applications at boundaries.

Nice work Chriss

Bloerb · November 26, 2018, 11:38

At the interface between fluid and solid (or solid and solid) the following is true:

and

This means, that the heat flux exiting one domain enters the other and that both regions agree on temperature.

The heat flux can be written as:

$\kappa_s \nabla T_s=Q_s = -Q_f = -\kappa_f \nabla T_f$

The gradient at the wall can be expressed as the difference between the value at the cell center and wall face devided by the distance between those. OpenFOAM uses the inverse of that however: $\Delta=\frac{1}{\delta}$ .
$\nabla T = \frac{T_c-T_f}{\delta}=\Delta(T_c-T_f)$

From here it is only a bit of math. We summarize the above condition:

$\kappa_s\Delta_s(T_{cs}-T_s)=-\kappa_f\Delta_f(T_{cf}-T_f)$

and simplify with $T_s = T_f=T_{wall}$ . This depends on the region you want to use the boundary condition for. For illustration we'll use the fluid side.

$\kappa_s\Delta_s(T_{cs}-T_{f})=-\kappa_f\Delta_f(T_{cf}-T_{f})$

$\Rightarrow T_{f}= T_{cs}\left(\frac{\kappa_s\Delta_s}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right)+T_{cf}\left(\frac{\kappa_f\Delta_f}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right)$

a mixed boundary condition in OpenFOAM is defined as follows:

$T_f = \text{valueFraction}\cdot\text{refValue}+(1-\text{valueFraction})(T_c+\Delta\cdot\text{refGrad})$

We can rewrite the above formula to match this:

$\Rightarrow T_{f}= T_{cs}\left(\frac{\kappa_s\Delta_s}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right)+T_{cf}\left(1-\frac{\kappa_s\Delta_s}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right)$

The actual implementation

Code:

    this->refValue() = nbrIntFld(); // This is T_s

    this->refGrad() = 0.0;

    this->valueFraction() = nbrKDelta()/(nbrKDelta() + myKDelta()); This is kappa*Delta

This is the derivation for the boundary condition.

For adding in heat fluxes due to radiation you'd slightly modify the initial set up
$Q_s +Q_{rads}= -Q_f -Q_{radf}$

I have however not referenced this to the code. And am unsure about the sign of the radiation flux definition in OpenFOAM, so this might differ. It should however be a starting point for understanding the code.

jmt · August 9, 2021, 16:06

Hi everyone, this is a very nice discussion. I am also working with this BC and am interested in a second-order formulation.

As written, the expression of flux continuity at the interface is $\kappa_{\mathrm{fluid}} \left( \frac{\partial T}{\partial \mathbf{n}} \right)_{\mathrm{fluid}} = \kappa_{\mathrm{solid}} \left( \frac{\partial T}{\partial \mathbf{n}} \right)_{\mathrm{solid}}$ .

The derivatives are expanded via first-order differencing and the formulation is re-arranged for the interface temperature.

Has anyone had success with a second-order BC for this? For example, replacing the temperature derivative with a second-order upwind type expression and re-arranging for interface temperature. This would require the two wall-adjacent cells which I suppose could be tricky to obtain.

Thanks.

piu58 · August 10, 2021, 00:01

The case you described is not a boundary. Thee are inner faces with different values of kappa at both sides. The temperature simulation does not need anything more.

jmt · August 10, 2021, 10:09

Thank you for your input piu58.

I was examining boundary coupling and compared chtMultiRegionFoam with an analytical solution for semi-infinite media. A grid convergence study revealed the convergence was first-order despite using second-order spatial discretization, which motivated me to examine the coupling.

The boundary condition is implemented and in fact hard-coded as a first-order scheme for interface temperature.

dlahaye · August 11, 2021, 09:59

Dear Julian,

I fail to understand your post as much as I would like to.

Are you sure that the scheme is only of first order only? How to you measure the order of the convergence? Are you sure that the order of convergence is not limited by first order schemes employed to discretize the convective terms in the equation?

The reason for raising this doubt is the fact that my limited understanding tells me that the first order discretization used to discretize the fluid/solid interface conditions (local discretization error) should *not* lower the order of the volumetric scheme (global discretization error) from second to first order (for details see book of Piet Wesseling among others).

Kind wishes, Domenico.

Eslam Reda · December 2, 2022, 01:14

Quote:

Originally Posted by Bloerb

.

The full BC including a heat source (G) is expressed as:

and

$\kappa_s \nabla T_s=Q_s = -Q_f - G = -\kappa_f \nabla T_f - G$

$\Delta=\frac{1}{\delta}$ .

$\nabla T = \frac{T_c-T_f}{\delta}=\Delta(T_c-T_f)$

$\kappa_s\Delta_s(T_{cs}-T_s)=-\kappa_f\Delta_f(T_{cf}-T_f) - G$

$T_s = T_f=T_{wall}$ .

$\kappa_s\Delta_s(T_{cs}-T_{f})=-\kappa_f\Delta_f(T_{cf}-T_{f}) - G$

$\Rightarrow T_{f}= T_{cs}\left(\frac{\kappa_s\Delta_s}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right)+T_{cf}\left(\frac{\kappa_f\Delta_f}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right )+ \frac{G}{\kappa_s\Delta_s+\kappa_f\Delta_f}$

$\Rightarrow T_{f}= T_{cs}\left(\frac{\kappa_s\Delta_s}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right)+T_{cf}\left(\frac{\kappa_f\Delta_f}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right )+ \frac{G * \kappa_f\Delta_f / \kappa_f\Delta_f}{\kappa_s\Delta_s+\kappa_f\Delta_f}$

$\Rightarrow T_{f}= T_{cs}\left(\frac{\kappa_s\Delta_s}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right)+\left(1 - \frac{\kappa_s\Delta_s}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right ) *\left( T_{cf} + G / \kappa_f\Delta_f \right)$

a mixed boundary condition in OpenFOAM is defined as follows (mixedFvPatchField.C):

Field<Type>

perator=

(

valueFraction_*refValue_

+ (1.0 - valueFraction_)

*(

this->patchInternalField()

+ refGrad_/this->patch().deltaCoeffs()

)

);

Hence,

= Field<Type>

perator

valueFraction_ = $\left(\frac{\kappa_s\Delta_s}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right)$

refValue_ = $T_{cs}$

refGrad_ = $\frac{G}{\kappa_f}$

this->patch().deltaCoeffs() = $\Delta_f$

chicha · August 4, 2023, 07:48

Do these formulation account for non-orthogonality of the face? If not, should they?

F42 · March 19, 2024, 06:54

Quote:

Originally Posted by chicha

Do these formulation account for non-orthogonality of the face? If not, should they?

Yes, deltaCoeffs includes a orthogonality correction. See

https://openfoamwiki.net/index.php/O...n;;deltaCoeffs

October 31, 2014, 08:31		#2
chriss85 Senior Member Join Date: Oct 2013 Posts: 397 Rep Power: 18	I've spent some more time thinking about this. Using the equations above inserted into the mixed BC formula: $T_{F_0}=\frac{\alpha}{\alpha + \frac{\kappa_1}{\Delta_1}} \frac{\frac{\kappa_1}{\Delta_1}T_{F_1}}{\alpha} + (1 - \frac{\alpha}{\alpha + \frac{\kappa_1}{\Delta_1}}) T_0 =\frac{\alpha}{\alpha + \frac{\kappa_1}{\Delta_1}} \frac{\frac{\kappa_1}{\Delta_1}T_{F_1}}{\alpha} + \frac{\frac{\kappa_0}{\Delta_0}}{\alpha + \frac{\kappa_1}{\Delta_1}} T_0 =\frac{\frac{\kappa_1}{\Delta_1}}{\alpha + \frac{\kappa_1}{\Delta_1}} T_{F_1} + \frac{\frac{\kappa_0}{\Delta_0}}{\alpha + \frac{\kappa_1}{\Delta_1}} T_0$ , here with 0: first region, 1: second region, so T0: temperature in first region cell, T_F_0: temperature in first region patch. I'm wondering if the signs at the radiation fluxes are correct. For positive Q_r (meaning outgoing radiation) an iteration procedure using this formula is bound to diverge, because the sum of the two weighting factors in front of the temperatures is not 1??? As an example: Consider radiation coming out of a gas (0) being absorbed completely by a solid (1). In this case, Qr_0 > 0, Qr_1 = 0 and the sum of both factors is larger than 1. In my understanding this should result in an increasing temperature on each iteration. I'm going to test this series one dimensionally with MATLAB to investigate the convergence properties. Does anyone have any better explanation for the reasoning behind these formulas or maybe a literature recommendation? roucho, atulkjoy, Zhiheng Wang and 3 others like this.

November 3, 2014, 11:06		#3
chriss85 Senior Member Join Date: Oct 2013 Posts: 397 Rep Power: 18	See http://www.cfd-online.com/Forums/ope...tml#post517173 for further results. atulkjoy and Zhiheng Wang like this.

November 26, 2018, 11:38		#6
Bloerb Senior Member Join Date: Sep 2013 Posts: 353 Rep Power: 20	At the interface between fluid and solid (or solid and solid) the following is true: and This means, that the heat flux exiting one domain enters the other and that both regions agree on temperature. The heat flux can be written as: $\kappa_s \nabla T_s=Q_s = -Q_f = -\kappa_f \nabla T_f$ The gradient at the wall can be expressed as the difference between the value at the cell center and wall face devided by the distance between those. OpenFOAM uses the inverse of that however: $\Delta=\frac{1}{\delta}$ . $\nabla T = \frac{T_c-T_f}{\delta}=\Delta(T_c-T_f)$ From here it is only a bit of math. We summarize the above condition: $\kappa_s\Delta_s(T_{cs}-T_s)=-\kappa_f\Delta_f(T_{cf}-T_f)$ and simplify with $T_s = T_f=T_{wall}$ . This depends on the region you want to use the boundary condition for. For illustration we'll use the fluid side. $\kappa_s\Delta_s(T_{cs}-T_{f})=-\kappa_f\Delta_f(T_{cf}-T_{f})$ $\Rightarrow T_{f}= T_{cs}\left(\frac{\kappa_s\Delta_s}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right)+T_{cf}\left(\frac{\kappa_f\Delta_f}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right)$ a mixed boundary condition in OpenFOAM is defined as follows: $T_f = \text{valueFraction}\cdot\text{refValue}+(1-\text{valueFraction})(T_c+\Delta\cdot\text{refGrad})$ We can rewrite the above formula to match this: $\Rightarrow T_{f}= T_{cs}\left(\frac{\kappa_s\Delta_s}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right)+T_{cf}\left(1-\frac{\kappa_s\Delta_s}{\kappa_s\Delta_s+\kappa_f\Delta_f}\right)$ The actual implementation Code: this->refValue() = nbrIntFld(); // This is T_s this->refGrad() = 0.0; this->valueFraction() = nbrKDelta()/(nbrKDelta() + myKDelta()); This is kappaDelta This is the derivation for the boundary condition. For adding in heat fluxes due to radiation you'd slightly modify the initial set up $Q_s +Q_{rads}= -Q_f -Q_{radf}$ I have however not referenced this to the code. And am unsure about the sign of the radiation flux definition in OpenFOAM, so this might differ. It should however be a starting point for understanding the code. snak, Eslam Reda, Daniel_Khazaei and 27 others like this. Last edited by Bloerb; November 27, 2018 at 03:05.*

August 10, 2021, 00:01		#8
piu58 Senior Member Uwe Pilz Join Date: Feb 2017 Location: Leipzig, Germany Posts: 744 Rep Power: 15	The case you described is not a boundary. Thee are inner faces with different values of kappa at both sides. The temperature simulation does not need anything more. __________________ Uwe Pilz -- Die der Hauptbewegung überlagerte Schwankungsbewegung ist in ihren Einzelheiten so hoffnungslos kompliziert, daß ihre theoretische Berechnung aussichtslos erscheint. (Hermann Schlichting, 1950)

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August 9, 2021, 16:06		#7
jmt Member Julian Join Date: Sep 2019 Posts: 32 Rep Power: 6	Hi everyone, this is a very nice discussion. I am also working with this BC and am interested in a second-order formulation. As written, the expression of flux continuity at the interface is $\kappa_{\mathrm{fluid}} \left( \frac{\partial T}{\partial \mathbf{n}} \right)_{\mathrm{fluid}} = \kappa_{\mathrm{solid}} \left( \frac{\partial T}{\partial \mathbf{n}} \right)_{\mathrm{solid}}$ . The derivatives are expanded via first-order differencing and the formulation is re-arranged for the interface temperature. Has anyone had success with a second-order BC for this? For example, replacing the temperature derivative with a second-order upwind type expression and re-arranging for interface temperature. This would require the two wall-adjacent cells which I suppose could be tricky to obtain. Thanks.

August 10, 2021, 10:09		#9
jmt Member Julian Join Date: Sep 2019 Posts: 32 Rep Power: 6	Thank you for your input piu58. I was examining boundary coupling and compared chtMultiRegionFoam with an analytical solution for semi-infinite media. A grid convergence study revealed the convergence was first-order despite using second-order spatial discretization, which motivated me to examine the coupling. The boundary condition is implemented and in fact hard-coded as a first-order scheme for interface temperature.

August 11, 2021, 09:59		#10
dlahaye Senior Member Domenico Lahaye Join Date: Dec 2013 Posts: 722 Blog Entries: 1 Rep Power: 17	Dear Julian, I fail to understand your post as much as I would like to. Are you sure that the scheme is only of first order only? How to you measure the order of the convergence? Are you sure that the order of convergence is not limited by first order schemes employed to discretize the convective terms in the equation? The reason for raising this doubt is the fact that my limited understanding tells me that the first order discretization used to discretize the fluid/solid interface conditions (local discretization error) should not lower the order of the volumetric scheme (global discretization error) from second to first order (for details see book of Piet Wesseling among others). Kind wishes, Domenico.

August 4, 2023, 07:48		#12
chicha New Member Join Date: Aug 2023 Posts: 1 Rep Power: 0	Do these formulation account for non-orthogonality of the face? If not, should they?