Calculation of the particle relaxation time

Modification of the calculation of the particle relaxation time with respect to the chosen formulation for the drag coefficient

This function is called in a loop on the particles, so be careful to avoid too costly operations.

$\tau_c = \frac{m_p{C_p}_p}{PId_p^2h_e}$

$\tau_c$ : Thermal relaxation time (value to be computed)

: Particle mass

${C_p}_p$ : Particle specific heat

d_p : Particle diameter

h_e : Coefficient of thermal exchange

The coefficient of thermal exchange is calculated from a Nusselt number, itself evaluated by a correlation (Ranz-Marshall by default)

$\nu = \frac{h_ed_p}{\lambda} = 2 + 0.55{\Re_e}_p^{0.5}P_{rt}^{0.33}$

$\lambda$ : Thermal conductivity of the carrier field

${\Re_e}_p$ : Particle Reynolds number

P_{rt} : Prandtl number

In the next example we compute the relaxation time with two different formulations of the drag coefficient:

void
cs_user_lagr_rt(cs_lnum_t        p_id,
                cs_real_t        re_p,
                cs_real_t        uvwr,
                cs_real_t        rho_f,
                cs_real_t        rho_p,
                cs_real_t        nu_f,
                cs_real_t        *taup,
                const cs_real_t  dt[])
{
  /* Particles management */
  cs_lagr_particle_set_t  *p_set = cs_lagr_get_particle_set();
 
  cs_real_t p_diam = cs_lagr_particles_get_real(p_set, p_id, CS_LAGR_DIAMETER);
 
  /*===============================================================================
   * Relaxation time with the standard (Wen-Yu) formulation of the drag coefficient
   *===============================================================================*/
 
  /* This example gives the standard relaxation time as an indication:*/
 
  cs_real_t fdr;
 
  cs_real_t cd1  = 0.15;
  cs_real_t cd2  = 0.687;
 
  if (re_p <= 1000)
    fdr = 18.0 * nu_f * (1.0 + cd1 * pow(re_p, cd2)) / (p_diam * p_diam);
 
  else
    fdr = (0.44 * 3.0 / 4.0) * uvwr / p_diam;
 
  taup[p_id] = rho_p / rho_f / fdr;
 
  /*===============================================================================
   * Computation of the relaxation time with the drag coefficient of
   * S.A. Morsi and A.J. Alexander, J. of Fluid Mech., Vol.55, pp 193-208 (1972)
   *===============================================================================*/
 
  cs_real_t rec1 =  0.1;
  cs_real_t rec2 =  1.0;
  cs_real_t rec3 =  10.0;
  cs_real_t rec4 = 200.0;
 
  cs_real_t dd2 = p_diam * p_diam;
 
  if (re_p <= rec1)
    fdr = 18.0 * nu_f / dd2;
 
  else if (re_p <= rec2)
    fdr = 3.0/4.0 * nu_f / dd2 * (22.73 + 0.0903 / re_p + 3.69 * re_p);
 
  else if (re_p <= rec3)
    fdr = 3.0/4.0 * nu_f / dd2 * (29.1667 - 3.8889 / re_p + 1.222 * re_p);
 
  else if (re_p <=rec4)
    fdr = 18.0 * nu_f / dd2 *(1.0 + 0.15 * pow(re_p, 0.687));
 
  else
    fdr = (0.44 * 3.0 / 4.0) * uvwr / p_diam;
 
  taup[p_id] = rho_p / rho_f / fdr;
}

Computation of the thermal relaxation time of the particles

Modification of the computation of the thermal relaxation time of the particles with respect to the chosen formulation of the Nusselt number.

This function is called in a loop on the particles, so be careful to avoid too costly operations.

void
cs_user_lagr_rt_t(cs_lnum_t        p_id,
                  cs_real_t        re_p,
                  cs_real_t        uvwr,
                  cs_real_t        rho_f,
                  cs_real_t        rho_p,
                  cs_real_t        nu_f,
                  cs_real_t        cp_f,
                  cs_real_t        k_f,
                  cs_real_t        tauc[],
                  const cs_real_t  dt[])
{
  /* 1. Initializations: Particles management */
  cs_lagr_particle_set_t  *p_set = cs_lagr_get_particle_set();
 
  /* 2. Standard thermal relaxation time */
 
  /* This example gives the standard thermal relaxation time
   * as an indication.*/
 
  cs_real_t prt = nu_f / k_f;
 
  cs_real_t fnus = 2.0 + 0.55 * sqrt(re_p) * pow(prt, 1./3.);
 
  cs_real_t diam = cs_lagr_particles_get_real(p_set, p_id, CS_LAGR_DIAMETER);
  cs_real_t cp_p = cs_lagr_particles_get_real(p_set, p_id, CS_LAGR_CP);
 
  tauc[p_id]= diam * diam * rho_p * cp_p  / ( fnus * 6.0 * rho_f * cp_f * k_f);
}