Contents

1. SURFACE-TO-SURFACE RADIATION
2. Surface-to-Surface Radiation: Introduction (i)
3. Surface-to-Surface Radiation: Introduction (ii)
4. Assumptions
5. Capabilities
6. Strategy
7. The energy equation
8. The heat balance equation
9. The net radiation flux
10. The mean convective heat flux (i)
11. The mean convective heat flux (ii)
12. The mean conductive heat flux
13. Determination of the surface temperature
14. Activation - Essential Q1 settings (i)
15. Activation - Essential Q1 settings (ii)
16. Activation - Essential Q1 settings (iii)
17. Activation - Essential Q1 settings (iv)
18. Activation - Essential Q1 settings (v)
19. Activation - Essential Q1 settings (vi)
20. Calculation of Radiative-exchange or View-factors
21. Radiative-exchange or View-factor File - RADI.DAT
22. Radiative-exchange or View-factor File (ii)
23. Example cases in the library (i)
24. Example cases in the library (ii)
25. Concluding Remarks
26. References

1. Surface-to-Surface Radiation

2. Surface-to-Surface Radiation: Introduction (i)

• The six-flux radiation model in PHOENICS facilitates the analysis and solution of radiation problems in which the fluid medium participates in the radiative heat transfer process.

• However, in many engineering processes the fluid medium is transparent to radiation.

• This presentation introduces a method, namely the surface-to-surface radiation model (S2SR), which permits the solution of radiation problems involving non-participating media.

3. Surface-to-Surface Radiation: Introduction (ii)

• This model is essentially a zonal method, in which the surfaces participating in the radiative-exchange process are sub-divided into a number of isothermal zones.

• The radiant interchange between the surfaces requires determination of matrices of corresponding geometric view-factors or alternatively the radiative-exchange factors.

• In the solution of the energy equation, the model allows for radiative exchange between the surfaces via the imposition of a heat balance at each thermal zone.

4. Assumptions

• The fluid medium is transparent to radiation.

• The user furnishes the radiative-exchange factors or the view-factors - the former allows for possible multiple reflections from other surfaces or transmission through them, while the latter assumes black-body radiation.

• Diffuse-grey surfaces, i.e. surfaces with radiative properties independent of direction (diffuse), and wavelength (grey).

• Temperature-independent radiative properties. If temperature dependency is required for radiative properties such as absorptivity this can be accommodated but requires recalculation of the radiative-exchange factors.

5. Capabilities

• The S2SR model allows the simulation of:
  • Steady or transient processes;
  • One-, two-, or three-dimensional processes;
  • Processes in Cartesian, cylindrical polar or BFC co-ordinates;
  • Radiative heat transfer in the presence of laminar or turbulent flows;
  • Heat transfer processes involving conjugate heat transfer;
  • Surfaces having fixed-heat-flux or fixed-temperature thermal boundary conditions;

• Surfaces having uniform radiative properties.

6. Strategy

• Zonal method - surfaces are divided into a finite number of thermal zones each characterised by a single surface temperature

• Compute these surface temperatures from a heat balance at the thermal zones.

• Heat balance accounts for convection, conduction and radiation as well as any additional heat fluxes which may be present at the surface.

• Use the computed surface temperature in calculating, as appropriate, the convective and conductive heat fluxes at the radiative zones.

• Solve an energy equation with these computed heat fluxes as the heat source at each zone, thus representing the required boundary condition.

7. The energy equation

• The energy conservation equation which is solved for the temperature solution is:

  d (r.cp.T) / dt + div (r.U.cp.T - k.grad (T)) = S

  t - time; T - the temperature; r - density of fluid or solid; cp- mean specific heat of fluid or solid; k - thermal conductivity of fluid or solid; S - any additional source terms per unit volume.

8. The heat balance equation

• The surface temperature of a thermal zone, when it is not prescribed, is computed from the heat balance given by:

  q"r,i - q"f - q"s + q" = 0 (1)

  where: q"r,i - is the net radiation flux incident on the surface i q"f - is the mean convective heat flux experienced by the adjacent fluid q"s - is the mean conductive heat flux experienced by the adjacent solid q" - is any other prescribed heat flux that may be present.

• Note: The term mean is used because normally the surface temperature mesh will be coarser than the standard PHOENICS mesh used for conduction and convection. Thus the conductive and convective heat fluxes used in the heat balance equation are area averaged heat fluxes.

9. The net radiation flux

• The net radiation flux incident on the surface i is given by:

  q"r,i = sum [ Gri,j.( Tj**4 - Ti**4 ) (2)

  The summation extends over all radiative surfaces.

  Gri,j - is the radiative-exchange factor between thermal surfaces i and j in W/m**2/K**4 Tj,Ti - are the absolute temperatures of surfaces j and i respectively in K.

10. The mean convective heat flux (i)

• The mean convective heat flux experienced by the adjacent fluid is given by:

  q"f = Gf.(Ti - Tf) (3)

  Tf - is the adjacent fluid temperature

  Gf - is the mean fluid-side heat-transfer coefficient in W/m**2/K given by: Gf = kf/df

  kf - is the thermal conductivity of the fluid

  df - is the normal distance from the surface to the first grid point in the adjacent fluid.

11. The mean convective heat flux (ii)

• For laminar flow kf is simply the laminar thermal conductivity.

• For turbulent flow kf is the turbulent thermal conductivity, so that Gf is given by

  Gf = St.Ur.cp.r

  St - is the Stanton number computed from the wall functions

  Ur - is the absolute magnitude of the relative velocity parallel to the surface.

12. The mean conductive heat flux

• The mean conductive heat flux is given by:

  q"s = Gs.(Ti - Ts) (4)

  Ts - is the adjacent solid temperature

  Gs - is the mean solid-side heat-transfer coefficient in W/m**2/K given by:

  Gs = ks/ds

  ks - is the thermal conductivity of the Solid

  ds - is the normal distance from the surface to the first grid point in the adjacent solid.

13. Determination of the surface temperature

• Substitution of equations (2), (3), and (4) into the heat balance equation (1), gives the equation for the surface temperature of the thermal zone i, viz.

  Ti**4.sum(Gri,j) + Ti.(Gf + Gs ) - sum(Gri,j.Tj )**4 - (Gf.Tf + Gs.Ts) - q" = 0 (5)

• Ti is calculated iteratively from equation (5) by application of the Newton-Raphson method.

14. Activation - Essential Q1 settings (i)

• The S2SR model is activated by setting: S2SR = T

• To activate the solution of the energy equation set: SOLVE(TEM1)

• If TEM1 is in Kelvin, then TMP1A is 0.0, if TEM1 is in Celsius, TMP1A should be set to 273.0

• The solution of the TEM1 equation requires definition of material type via a type flag assigned through the storage of the PRPS variable. To activate this store set:

  STORE(PRPS)

  and initialise the PRPS field using FIINIT and/or PATCH-INIT commands.

15. Activation - Essential Q1 settings (ii)

• PATCH and COVAL commands are required to define the location and type of the boundary condition for each radiative thermal zone.

• The PATCH command is used to locate thermal zones.

• For radiative PATCHES, the NAME must start with

  @RI###

  where @RI indicates that the patch is a thermal zone participating in internal radiative heat transfer and ### is a three-digit number representing the thermal zone number. These numbers must be consecutive starting at 1 and the patches have to be defined consecutively in the Q1 file.

• The TYPE must be an AREA i.e. NORTH, SOUTH etc

16. Activation - Essential Q1 settings (iii)

• The following restrictions apply in defining the LOCATION:

• If the zone is a surface of a solid for which conjugate heat transfer is present then,

  PATCH located on the solid side;

• If the zone is a surface on the domain edge or on a completely blocked region ( i.e. conjugate heat is not present) then,

  PATCH located on the fluid side.

17. Activation - Essential Q1 settings (iv)

• COVAL Command

• If the surface temperature is to be obtained from the heat balance, set:

  CO and VAL to GRND1,

• For fixed-heat-flux boundary condition set:

  CO to GRND1, and VAL to required heat flux,

• For fixed-temperature boundary condition set:

  CO to 0.0, and VAL to required temperature value.

18. Activation - Essential Q1 settings (v)

• Apertures require the same setting for CO and VAL as the fixed-temperature boundary condition, with VAL set to the temperature through the aperture.

• At surfaces where a fixed-temperature radiative boundary condition is applied, a conventional fixed wall temperature boundary condition must also be set to provide the correct fluid-side conductive/convective heat transfer.

• For turbulent flows wall functions need to be activated for all thermal zones via the WALL command.

19. Activation - Essential Q1 settings (vi)

• The radiative transfer link introduced between solid and fluid includes the conductive part. In a conjugate problem, the normal conductive link, which is enhanced for turbulent flow to include the convective part, has to be switched off, otherwise it will be included twice.

• This is done using PATCHes with names starting GP12DF. These multiply the diffusive link by the VALue set in the COVAL. (GP12 for Group 12, DFN for diffusion at North face). The next character of the patch name indicates the face of the cell - N, S, E, W, H or L.

• For example, on the north face of a group of cells:

  PATCH (GP12DFN1, CELL, 1,NX, NYG1,NYG1, 1,NZ, 1,1) COVAL (GP12DFN1, TEM1, 0.0, 0.0)

20. Calculation of Radiative-exchange or View-factors

• Radiative-exchange coefficient (REC) or view-factors (VF) must be supplied. RECs allow for emmissivity, reflectivity, absorptivity and the Stefan-Boltzmann constant, whilst VFs only allow for the Stefan-Boltzmann constant - they are for black body radiation.

• For 1- or 2-dimensional cartesian or BFC geometries, the VFs can be calculated by a supplied routine. For 3 dimensional or cylindrical polar geometries, the factors have to be supplied externally.

• If a file called RADI.DAT exists in the local directory, it will be read to provide RECs or VFs calculated by a third-party code. If the file does not exist locally, the on-board VF calculation will be initiated.

21. Radiative-exchange or View-factor File - RADI.DAT

• These can be supplied in a file called RADI.DAT. For a system of N thermal nodes, there must be N**2 exchange coefficients. The required format is:

                    * COMMENT LINE AS HEADER *
     N              - NUMBER OF THERMAL NODES
     GR(1,1) GR(1,2) GR(1,3) GR(1,4) GR(1,5)
     ....... ....... GR(1,J) ....... .......
     ....... GR(1,N)
     GR(2,1) GR(2,2) GR(2,3) GR(2,4) GR(2,5)
     ....... ....... GR(2,J) ....... .......
     ....... GR(2,N)
     .
     .
     GR(I,1) GR(I,2) GR(I,3) GR(I,4) GR(I,5)
     ....... ....... GR(I,J) ....... .......
     ....... GR(I,N)

22. Radiative-exchange or View-factor File (ii)

     .
     .
     GR(N,1) GR(N,2) GR(N,3) GR(N,4) GR(N,5)
     ....... ....... GR(N,J) ....... .......
     ....... GR(N,N)

The exchange coefficients should be written in the format 5(1PE13.6), using the following DO loop:

     
        DO 10 I=1,N
          WRITE (LU,20) (GR(I,J),J=1,N)
     10 CONTINUE
     20 FORMAT (5(1PE13.6))

23. Example cases in the library (i)

RADIATION MODEL EXAMPLES Case no. (Type SEELIB(Rn) or LOAD(Rn) to see or load; n = case no.)

1. INTERNAL RADIATIVE, CONVECTIVE AND CONDUCTIVE
   • CVD reactor radiation example 100
   • CVD reactor radiation example(BFC) 101
   • Laminar flow between parallel plate 102
   • Turb. flow between parallel plate 103
2. EXTERNAL LINEAR HEAT TRANSFER 1-D y-direction shell surface 105
3. EXTERNAL RADIATIVE HEAT TRANSFER
   • 1-D y-direction solid surface 107
   • 1-D y-direction shell surface 108
4. INTERNAL RADIATIVE HEAT TRANSFER WITH CONDUCTION
   • 1-D fixed surface temperature 111
   • 1-D fixed-flux fluid surface 113
   • 1-D fixed-flux solid surface 114

24. Example cases in the library (ii)

1. INTERNAL LINEAR AND RADIATIVE HEAT TRANSFER 1-D cartesian y-direction 115

2. INTERNAL RADIATIVE HEAT TRANSFER WITH CONDUCTION 1-D solid-fluid-solid set-up 116

3. THIN PLATE 1-D thin plate 118

4. TRANSIENT 1-D transient 119

25. Concluding Remarks

• An account has been given of the surface-to-surface radiation model.

• This radiation model is provided in a special ground, GXS2SR.

• The model is based on the zonal method and provides a means of simulating radiative heat transfer amongst surfaces in the presence of a transparent intervening medium.

• The main limitation of the model is its dependency on external software for the radiative-exchange factors.

• Future extensions to the model include:
• 3 dimensional view-factor calculations,
• Radiative-exchange factors for diffuse, grey surfaces or,
• Radiative-exchange factors for diffuse surfaces with spectral dependence, &
• temperature dependence of surface radiative properties i.e. absorptivity, reflectivity and transmissivity.

26. References

• C H Liu and E M Sparrow, ' Convective-Radiative Interaction in a Parallel Plate Channel - Application to Air-Operated Solar Collectors', J Heat and Mass Transfer Vol. 23 pp 1137-1146 1980.

• M Perlmutter and R Siegel,' Convective and Radiant Heat Transfer for Flow of a Transparent Gas in a Tube with a Gray wall', J Heat and Mass Transfer Vol. 5 pp 639-660 1961.