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heat transfer equation of parcel

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Old   March 6, 2015, 07:12
Question heat transfer equation of parcel
  #1
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Neilson Whit
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Dear All

Could you please write mathematical formulation of heat transfer equation for a droplet?

It is interpreted as follows in src/lagrangian/intermediate/parcels/Templates/ThermoParcel.C

Code:
template<class ParcelType>
template<class TrackData>
Foam::scalar Foam::ThermoParcel<ParcelType>::calcHeatTransfer
(
    TrackData& td,
    const scalar dt,
    const label cellI,
    const scalar Re,
    const scalar Pr,
    const scalar kappa,
    const scalar NCpW,
    const scalar Sh,
    scalar& dhsTrans,
    scalar& Sph
)
{
    if (!td.cloud().heatTransfer().active())
    {
        return T_;
    }

    const scalar d = this->d();
    const scalar rho = this->rho();

    // Calc heat transfer coefficient
    scalar htc = td.cloud().heatTransfer().htc(d, Re, Pr, kappa, NCpW);

    if (mag(htc) < ROOTVSMALL && !td.cloud().radiation())
    {
        return
            max
            (
                T_ + dt*Sh/(this->volume(d)*rho*Cp_),
                td.cloud().constProps().TMin()
            );
    }

    htc = max(htc, ROOTVSMALL);
    const scalar As = this->areaS(d);

    scalar ap = Tc_ + Sh/As/htc;
    scalar bp = 6.0*(Sh/As + htc*(Tc_ - T_));
    if (td.cloud().radiation())
    {
        tetIndices tetIs = this->currentTetIndices();
        const scalar Gc = td.GInterp().interpolate(this->position(), tetIs);
        const scalar sigma = physicoChemical::sigma.value();
        const scalar epsilon = td.cloud().constProps().epsilon0();

        ap = (ap + epsilon*Gc/(4.0*htc))/(1.0 + epsilon*sigma*pow3(T_)/htc);
        bp += 6.0*(epsilon*(Gc/4.0 - sigma*pow4(T_)));
    }
    bp /= rho*d*Cp_*(ap - T_) + ROOTVSMALL;

    // Integrate to find the new parcel temperature
    IntegrationScheme<scalar>::integrationResult Tres =
        td.cloud().TIntegrator().integrate(T_, dt, ap*bp, bp);

    scalar Tnew =
        min
        (
            max
            (
                Tres.value(),
                td.cloud().constProps().TMin()
            ),
            td.cloud().constProps().TMax()
        );

    Sph = dt*htc*As;

    dhsTrans += Sph*(Tres.average() - Tc_);

    return Tnew;
}
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Old   June 23, 2016, 04:24
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Amir
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Dear Neilson,

Have you found any mathematics formulation for parcel energy balance or any reference that the model is written based on?

Regards,
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Old   November 24, 2016, 10:49
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Chris Cloney
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One note here is that Spu, Sh, Sph, and dhstrans are all passed in as zeros to the calcHeatTransfer function. That starts to get you close to something that looks like the textbook equations, but I am still working on it.
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