Transient Thermal Response of Sheet with Constant Temperature Boundaries

Note

The project file for this example may be viewed/run in PFC.[1] The data files used are shown at the end of this example.

A planar sheet of width \(L\) is initially at a uniform temperature of 0°C. The left side of the sheet is exposed to a constant temperature of 100°C, while the right side is kept at 0°C. The sheet eventually reaches an equilibrium state at a constant heat flux and unchanging temperature distribution. The analytical solution for temperature within the sheet as a function of distance from the left side, \(x\), and time, \(t\), is given by [Crank1975]

(1)\[\frac{T(x,t)}{T_1} = 1 - \frac{x}{L} - \frac{2}{\pi} \sum_{n=1}^{\infty} e^{- \kappa n^2 \pi^2 t / L^2} \left(\frac{\sin (n \pi x / L)}{n}\right)\]

where \(T_1\) is the temperature at the left side of the sheet (equal to 100°C), and \(\kappa\) is the thermal diffusivity given by

(2)\[\kappa = \frac{k}{\rho_t C_v}\]

where \(k\) is the thermal conductivity, \(\rho_t\) is the material density, and \(C_v\) is the specific heat at constant volume.

Numerical Model

This thermal problem is one-dimensional and can be modeled as a thin slice of material with constant temperatures applied to its left and right boundaries and adiabatic conditions on the upper and lower boundaries to represent thermal symmetry planes.

For this example, we create two different PFC2D models consisting of a cubic packing and a hexagonal packing (see Figures 1 and 2). For these two uniform packings, the porosity and the thermal resistance can be expressed analytically. The porosity is given by [Deresiewicz1958b]:

(3)\[\begin{split}n = \begin{cases} 1 - \pi / 4 & \mbox{ , (cubic pack)} \\ 1 - \pi / 2 \sqrt{3} &\mbox{ , (hexagonal pack)} \end{cases}\end{split}\]

The thermal resistance is found by applying Equation (28) of the PFC Thermal Formulation section to a control volume that surrounds one particle, and noting that the length of each thermal pipe contained within the control volume equals the particle radius, \(R\), to yield

(4)\[\begin{split}\eta = \frac{1}{kRt} \begin{cases} \frac{1}{2} & \mbox{ , (cubic pack)} \\ \frac{3}{2 \sqrt{3}} & \mbox{ , (hexagonal pack)} \end{cases}\end{split}\]

where \(t\) is the disk thickness (equal to unity in this case). The data file used to perform the simulation is “transient_sheet.dat”. It makes use of FISH functions defined in “transient_sheet-utils.p2fis”. Thermal properties and boundary conditions are specified, and then each model is run for 50 and 200 cycles, until the thermal equilibrium state is reached (using the model solve command). The normalized temperature distributions at the increasing thermal times for the cubic- and hexagonal-packed specimens are plotted along with the analytical solutions in Figures 3 and 4. There is a good match between the PFC2D responses and the analytical solutions.

Figure 1: Contour of ball temperature and thermal power vector field (PFC2D cubic pack).

Figure 2: Contour of ball temperature and thermal power vector field (hexagonal pack).

Figure 3: Modeled and analytical normalized temperature distributions for increasing thermal time (cubic pack).

Figure 4: Modeled and analytical normalized temperature distributions for increasing thermal time (hexagonal pack).

References

[Crank1975]

Crank, J. The Mathematics of Diffusion, 2nd Ed. Oxford: Oxford University Press, 1975.

[Deresiewicz1958b]

Deresiewicz, H. “Mechanics of Granular Matter,” in Advances in Applied Mechanics, Vol. 5, pp. 233-306. H. L. Dryden and Th. von Karman, eds. New York: Academic Press, 1958.

Data Files

transient_sheet.dat

; fname: transient_sheet.dat (2D)
;
; Itasca Consulting Group, Inc.
; ========================================================================
; Thermal option example :
;        TRANSIENT RESPONSE OF SEMI-INFINITE SOLID WITH APPLIED HEAT FLUX
;                           (CUBIC AND HEXAGONAL PACKINGS)
;
; ========================================================================
program log-file 'transient_sheet.log'
program log on
; ========================================================================
model new
model large-strain on
model configure thermal
model mechanical off thermal on

program echo off
program call 'transient_sheet-utils.p2fis'
program echo on
[global t1 = 'Transient Thermal Response of Sheet']
[global t2 = ' with Constant Temperature Boundaries']
model title [t1 + t2]
[init_params]

model save 'transient_sheet-ini'

[n_row = 10]
[n_col = 40]
[_pack_type = 1]
[close_pack(n_row,n_col,_pack_type)]
;
model solve thermal time-total 600.0
[num_soln]
[ana_soln]
model solve thermal time-total 1200.0
[num_soln]
[ana_soln]
model solve thermal unbalanced-maximum  1e-3
[num_soln]
[ana_soln]
;
model save 'transient_sheet-cubic'

model res 'transient_sheet-ini'
[n_row = 10]
[n_col = 40]
[_pack_type = 2]
[close_pack(n_row,n_col,_pack_type)]
model thermal timestep max 1.0e2  
model solve thermal time-total 600.0
[num_soln]
[ana_soln]
model solve thermal time-total 1200.0
[num_soln]
[ana_soln]
model solve thermal unbalanced-maximum  1e-3
[num_soln]
[ana_soln]
;
mode save 'transient_sheet-hexa'
; ========================================================================
program log off
program return
;EOF: transient_sheet.dat (2D)

transient_sheet-utils.p2fis

;fname: transient_sheet-utils.p2fis
;
; Itasca Consulting Group, Inc.
; ============================================================================
; Utility functions for thermal option verification problems                  
; ============================================================================
;                                                                             
; ----------------------------------------------------------------------------
fish define init_params
  global id_start=1
  global x0 = 0.0
  global y0 = 0.0
  global radius  = 1.0
  global epsilon = 1.0e-6
  ;
  global c_k = 1.6                      ; conductivity
  global c_cv = 0.2                     ; specific heat
  global c_dens = 1000.0                ; density
  global c_kappa = c_k / (c_dens*c_cv)
  global t1 = 100.0                     ; temperature along left side
  global tabn_start = 10
  global tabe_start = 11
  global tabn = tabn_start
  global tabe = tabe_start
  global n_row = 10
  global n_col = 50
  global _pack_type = 1
  global n_tot = 100   ; number of terms of analytical solution to use
  global final_params
  global therm_props
  global therm_bcs
end
; ----------------------------------------------------------------------------
fish define close_pack(nrow_,ncol_,type_)
  global xc   = x0
  global yc   = y0
  global rc   = radius
  global idc  = id_start
  local r2   = 2.0 * rc
  global dom_xmin = x0 - 4.0 *r2
  global dom_xmax = x0 + ncol_*r2 + 4.0 *r2
  global dom_ymin = y0 - 4.0 *r2
  global dom_ymax = y0 + nrow_*r2 + 4.0 *r2
  
  command
    model domain extent [dom_xmin] [dom_xmax] [dom_ymin] [dom_ymax]
  endcommand
  
  case_of type_
    case 1
      yinc = radius * 2.0
    case 2
      yinc = radius * math.sqrt(3.0)
  end_case
  loop local row (1,nrow_)
    loop local col (1,ncol_)
      command
        ball create id=[idc] position-x=[xc] position-y=[yc] ...
                    radius=[rc*(1 + epsilon)]
      end_command
      idc = idc + 1
      xc   = xc + r2
    end_loop
    yc = yc + yinc
    case_of type_
      case 1
        xc = x0
      case 2
        xc = x0 + radius * (row - (row//2) * 2)
    end_case
  end_loop
  
  final_params
  therm_props
  therm_bcs
end
; ----------------------------------------------------------------------------
fish define final_params
  global c_len = (n_col-1)*2.0*radius
end
; ----------------------------------------------------------------------------
fish define therm_props
;
  if _pack_type = 1 then
    poros = 1.0 - (math.pi/4.0)
    _eta = 1.0 / (2.0 * c_k * radius)
  else
    poros = 1.0 - math.pi/(2.0*math.sqrt(3.0))
    _eta = 3.0 / (2.0*math.sqrt(3.0) * c_k * radius)
  end_if
  _dens = c_dens / (1.0 - poros)
  command
    contact cmat default model Linear
    contact cmat thermal default model ThermalPipe property thres=[_eta]
    contact cmat thermal default inherit thres off
    ball attribute density=[_dens]
    ball thermal attribute specific-heat=[c_cv]
    model clean
  end_command
end
; ----------------------------------------------------------------------------
fish define therm_bcs
;
  global _xl1 = -0.01*radius
  global _xr1 = _xl1 + 1.01*radius
  global _xl2 = c_len - 0.01*radius
  global _xr2 = _xl2 + 1.01*radius
  command
    ball thermal init temp 0.0
    ball thermal init temp [t1] range position-x=([_xl1], [_xr1]) ; left side
    ball thermal fix  range position-x=([_xl1], [_xr1]) ; left side
    ball thermal fix  range position-x=([_xl2], [_xr2]) ; right side
  end_command
end
; ----------------------------------------------------------------------------
fish define num_soln
;
; ----- Numerical solution (along bottom row of balls)
  global tname = string.build("Numerical sol. - age = %1",thermal.time.total)
  local t = table.create(tabe)
  table.label(t) = tname
  tabe +=2
  
  loop foreach local btp ball.thermal.list
    local bp = ball.thermal.ball(btp)
    if ball.pos(bp,2) = 0.0 then
      local _z = ball.pos(bp,1)
      local _tz = _z / c_len
      table(t, _tz) = ball.thermal.temp(btp) / t1
    end_if
  end_loop
end
; ----------------------------------------------------------------------------
fish define ana_soln
;
; ----- Analytical solution (along bottom row of balls).
;       Do not test for convergence, just take the first [n_tot] terms.
  global tname = string.build("Analytical sol. - age = %1",thermal.time.total)
  local t = table.create(tabn)
  table.label(t) = tname
  tabn +=2
  loop foreach local btp ball.thermal.list
    local bp = ball.thermal.ball(btp) 
    if ball.pos(bp,2) = 0.0 then
      local _z = ball.pos(bp,1)
      local n = 0
      local tsum = 0.0
      loop while n < n_tot
        n = n + 1
        fn = float(n)
        local term = math.sin(math.pi*_z*fn/c_len)
        term = term * math.exp(-c_kappa*fn*fn*math.pi*math.pi* ...
                      thermal.time.total/(c_len*c_len))/fn
        tsum = tsum + term
      end_loop
      _tz = _z / c_len
      table(t, _tz) = 1.0 - _z/c_len - (2.0/math.pi)*tsum
    end_if
  end_loop
end
; ----------------------------------------------------------------------------
fish define init_params
  global id_start=1
  global x0 = 0.0
  global y0 = 0.0
  global radius  = 1.0
  global epsilon = 1.0e-6
  ;
  global c_k = 1.6                      ; conductivity
  global c_cv = 0.2                     ; specific heat
  global c_dens = 1000.0                ; density
  global c_kappa = c_k / (c_dens*c_cv)
  global t1 = 100.0                     ; temperature along left side
  global tabn_start = 10
  global tabe_start = 11
  global tabn = tabn_start
  global tabe = tabe_start
  global n_row = 10
  global n_col = 50
  global _pack_type = 1
  global n_tot = 100   ; number of terms of analytical solution to use
end
; ============================================================================
;EOF: transient_sheet-utils.p2fis

Endnotes

[1]	To view this project in PFC, use the program menu. Help ▼ Examples…   ⮡   PFC     ⮡   Verfications       ⮡           ⮡   TransientSheet.prj