Radiative Flux of a Spectral Distribution at the Surface of a TPV Thermal Emitter Resulting from the Combustion of Biomass: Case of Palms Nut Shells

2.1 Problem formulation

In the field of energy, biomass is organic matter that can be used as a source of energy (bioenergy). This energy can be extracted from it by direct combustion. In this work, the palm nut shells were used as fuel.

During combustion of palm nut shells for heat power generation using boiler, the radiant heat from combustion flame is generally waste and unaccounted. Therefore, it is necessary to investigate the feasibility of a TPV system based on this waste heat radiative source by running simulation in order to evaluate the amount of heat fluxes received at the TPV absorber surface. The methodology applied here is based on the following:

-System components; -Modeling of the flame; -Modeling of radiative heat fluxes; -Calculation of view factor; -Modeling of the average temperature at the TPV absorber-filter surface-Modeling of Emittance

2.2 System components

Similar to all principles of energy conversion, TPV energy conversion is a method of converting thermal energy into electrical energy [11].

The principle is illustrated in Figure 1. The thermal sources can be: solar, combustion or nuclear, which are supplied to the absorber-emitter system. Radiation from the emitter is directed to the photovoltaic (PV) cell, where it is converted into electrical energy. To make the process efficient, the energy of the photons reaching the PV cell must be greater than the energy of the PV cell bandgap.

2.3 Modeling of the flame

In this work, the geometrical flame is modeled using a radiant conical model Figure 2 close to the experimental aspect, subdivided into 6 layers on the vertical direction, and each layer having a define temperature.

From this geometrical model of the flame, parameters that characterized combustion flame such as the flame length and the flame temperature can be modeled as shown in [12] as follow:

$\mathrm{L}_{\mathrm{f}}=-1,02 \mathrm{D}+0,0148 \mathrm{Q}^{\frac{2}{5}}$          (1)

where, $L_{f}$ is the flame length, m; D is flame diameter, m; Q is heat flux flow, w.

In Eq. (1) above, the heat flux flow is calculated as follow:

- For the flame development phase,

$Q=R H R_{f} * A_{f i} *\left(\frac{t}{t_{\alpha}}\right)^{2}$         (2)

$R H R_{f}:$ maximum heat flux flow per unit of area of flame, KW/m².

$A_{f i}:$ maximum surface flame area, m²; t: time, s; $t_{\propto}$: necessary time to attend maximum heat flux flow, s.

-Steady state phase,

$Q=R H R_{f} * A_{f i}$      (3)

- Decrease flame phase,

$Q=R H R_{f} * A_{f i} *\left(1-\frac{t-t_{r e d}}{t_{f i n}-t_{r e d}}\right)$          (4)

$t_{r e d}=t_{\alpha}+\,\,\frac{0,7 q_{f i}}{R H R_{f i}}-\,0,33 t_{\propto}$           (5)

$t_{e n d}=t_{r e d}+2 *\left[\frac{q_{f i}}{R H R_{f i}}-0,33 t_{\propto}-\left(t_{r e d}-t_{\propto}\right)\right]$          (6)

$q_{f i}=\frac{M_{i} * H_{u i}}{A_{f i}}$       (7)

$q_{f i}$: load heat density per unit area, MJ.m^-2; $M_{i}$: nut palm shells weight, Kg; $H_{u i}:$ lower calorific power, MJ.Kg^-1; $t_{\text {red }}$: time corresponding to the flame reduction phase, s; $t_{\text {end }}$: ending time of combustion, s.

The temperature profile of the combustion flame is giving by:

$T_{z}=20+0,25 Q_{c}^{\frac{2}{3}}\left(z-z_{o}\right)^{-\frac{5}{3}}$          (8)

where,

$T z$: is the flame temperature, ℃;

$Q_{c}:$ is the convective component of heat flux flow calculated as: $Q_{c}=0,8 Q$, W;

z: elevation along the flame axis, m, (Figure 3);

$z_{0}$ is the virtual origin position, m is given by:

$z_{0}=-1,02 D+0,00524 Q^{\frac{2}{5}}$         (9)

2.png

Figure 2. Radiation exchange between two surfaces

3.png

Figure 3. Flame modelling

2.4 Modeling of radiative heat fluxes (incident and net)

In general, the net radiative heat flux on an absorber-filter divided into m cells is given by [13]:

$q_{r a d, d f_{j}}=\sum_{k=1}^{m} \sigma \varepsilon^{*}\left(T_{f i, k}^{4}-T_{f i, j}^{4}\right) F_{d f_{j}-k}$         (10)

$F_{d j-k}=\int_{A_{j}} \frac{\cos \theta_{j} \cos \theta_{k}}{\pi S^{2}} d A_{j}$         (11)

$\cos \theta_{j}=\frac{\vec{s} \cdot \overrightarrow{n_{j}}}{s}$ and $\cos \theta_{k}=-\frac{\vec{s} \overrightarrow{n_{k}}}{s}$        (12)

$\frac{\cos \theta_{j} \cos \theta_{k}}{\pi S^{2}}=-\frac{\left(\vec{s} \cdot \overrightarrow{n_{j}}\right)\left(\vec{s} \cdot \overrightarrow{n_{k}}\right)}{\pi S^{4}}$       (13)

$\theta_{j}$ and $\theta_{k}:$ representing the angles between the normals to the faces and the line $\vec{S}$ connecting the two faces.

$F_{d j-k} \cong \frac{-1}{\pi} \sum_{i} \frac{\left(\vec{s} \cdot \overrightarrow{n_{j}}\right)\left(\vec{s} \cdot \overrightarrow{n_{k}}\right)}{s^{4}} \Delta A_{j}$           (14)

$\varepsilon^{*}=\frac{\varepsilon_{\text {flame }} \,\,\,\,\varepsilon_{\text {absorber }}}{\varepsilon_{\text {flame }}\,\,\,\,+\,\,\varepsilon_{\text {absorber }}\,\,\,\,-\,\,\varepsilon_{\text {flame }} \,\,\,\varepsilon_{\text {absorber }}}$         (15)

where,

$T_{f i, k}$: is the flame temperature at shell k; $k ; T_{f i, j}:$ is the absorber temperature at shell j; $k$;

$F_{d f_{j}-k}$: is view factor between shell j of absorber and shell k of flame;

$\sigma$: Constant of Stefan-Boltzmann (5.67*10^-8 W.m^-2.K^-4); $\varepsilon^{*}$: is the equivalent emissivity.

From Eq. (10) the temperature of the absorber-filter at shell j is been replaced by the average temperature T_c, hence:

$q_{r a d, d f_{j}}=\sum_{k=1}^{m} \sigma \varepsilon^{*}\left(T_{f i, k}^{4}-T_{c}^{4}\right) F_{d f_{j}-k}$         (16)

The balance heat as shown in the Figure 1 [5]:

$q=q_{\text {in }}-q_{\text {out }}$      (17)

where, $q_{\text {in }}=q_{\text {ra,flame }}$, radiant heat from the flame and $\mathrm{q}_{\text {out }}=\mathrm{q}_{\mathrm{ra}, \mathrm{air}}+\mathrm{q}_{\mathrm{conv}}$, heat lost from radiant heat of air and convection. Eq. (17) become,

$q=q_{r a, \text { flame }}-q_{\text {ra,air }}-q_{\text {conv }}$      (18)

with,

$q_{r a, \text { flame }}=\sum_{k=1}^{m} \sigma \varepsilon^{*}\left(T_{f i, k}^{4}-T_{c}^{4}\right) F_{d f_{j}-k}$           (19)

$q_{\text {ra,air }}=\sigma \varepsilon_{a}\left(T_{c}^{4}-T_{\text {air }}^{4}\right) F_{\text {compl }}$            (20)

$q_{\text {conv }}=\alpha_{c}\left(T_{c}-T_{\text {air }}\right)$        (21)

with $\propto_{\mathrm{c}}=1,78 \Delta \mathrm{T}^{0,25}$. The general equation when dividing the cylinder flame into m sections is:

2.5 Modeling of the average temperature at the TPV absorber surface

$q=\sum_{k=1}^{m} \sigma \varepsilon^{*}\left(T_{f i, k}^{4}-T_{c}^{4}\right) F_{d f_{j}-k}$

$-\sigma \varepsilon_{a b s}\left(T_{c}^{4}-T_{\text {air }}^{4}\right) F_{\text {compl }}-\alpha_{c}\left(T_{c}-T_{\text {air }}\right)$                  (22)

From the first principle of thermodynamics, we have:

$Q+\dot{W}=\dot{U}$        (23)

where,

Q: is the total heat flux exchange; $\dot{W}:$ is the total work done by the absorber; $\dot{U}$: is the change of internal energy.

Considering that the total work done by the absorber is negligible, Eq. (23) becomes:

$Q=\dot{U}$     (24)

$\dot{U}=\rho_{a b s} c_{p} V \frac{d T}{d t}$         (25)

where,

$\rho_{a b s}:$ absorber density, kg.m^-3; $C_{p}$: specific heat capacity, J.kg^-1.K^-1 calculated as in Ref. [8]; $V$: volume of the absorber, m³; $T:$ temperature, K; t: time, s.

$\rho_{a b s} c_{p} V \frac{d T}{d t}=A_{s}\left[q_{\text {ra,flame }}-q_{\text {ra,air }}-q_{\text {conv }}\right]$          (26)

$\rho_{a b s} c_{p} \frac{d T}{d t}=\frac{A_{s}}{V}\left\{\sum_{k} \sigma \varepsilon^{*}\left(T_{f i, k}^{4}-T_{c}^{4}\right) F_{d f_{j}-k}-\right.$

$\left.\sigma \varepsilon_{a b s}\left(T_{c}^{4}-T_{\text {air }}^{4}\right) F_{\text {compl }}-\alpha_{c}\left(T_{c}-T_{\text {air }}\right)\right\}$               (27)

$A_{s}:$ total surface of absorber.

By using the forward finite difference method, Eq. (27) becomes,

$\rho_{a b s} c_{p}(T) \frac{T_{c}^{t+1}-T_{c}^{t}}{\Delta t}=\frac{1}{V}\left\{A_{s} \sum_{k} \sigma \varepsilon^{*}\left(T_{f i, k}^{4}-\right.\right.$

$\left.T_{c}^{4}\right) F_{d f_{j}-k}-A_{s}\left(\sigma \varepsilon_{a b s}\left(T_{c}^{4 t}-T_{\text {air }}^{4}\right) F_{\text {compl }}\right)-$

$\left.A_{s}\left(\alpha_{c}\left(T_{c}^{t}-T_{\text {air }}\right)\right)\right\}$                          (28)

2.6 Modeling of emittance

If a body (solid or liquid) strongly absorbs radiation over a wavelength band, then it can be modeled as a black body on that band.

The spectral distribution of the radiation emitted by a black body is given by Planck's law. The emittance between the wavelengths λ and λ+dλ is: E_λ(λ, T)dλ.

Where E(λ, T) is the power spectral density, also called monochromatic exitance. Planck's law is:

$E_{\lambda}(\lambda, T)=\frac{2 \pi h c^{2}}{\lambda^{5}\left(\exp \left(\frac{h c}{\lambda k T}\right)-1\right)}$           (29)

where, h is the Planck constant: 6.62*10^-34m².kg/s, k the Boltzmann constant: 1.23*10^-23m².kg/s².k and c the speed of light in vacuum: 3*10⁸ m/s.

The maximum of the exitance as a function of the wavelength is given by Wien's displacement law:

$\lambda_{\max } T=2898 ; \mu m . K$           (30)

where, T is giving by Eq. (28).

The total power emittance by the black body is by definition:

$E(T)=\int_{0}^{\infty} E_{\lambda}(\lambda, T) d \lambda$          (31)

In this work, the waste heat energy resulting from the combustion of palm nut shells has been exploited as a radiative heat source for a TPV absorber system. For that we modeled the flame of combustion using conical approach. Simulations were performed to evaluate the incident and net heat fluxes at the surface of the absorber. The effect of convection and radiative heat lost as well as the position between the absorber and the conical flame, on the heat fluxes profile were analyzed. It comes out that the optimum position where the heat fluxes are maximum is 0.6 m. Moreover, combustion heat lost has an impact on the heat flux. The average temperature was also presented. Regarding the wavelengths, we obtained 2000 nm this value which corresponds to the IR. Finally, results obtained from the numerical simulations shows that radiative waste heat energy from the palm nut shells combustion can be used as radiative heat source for a TPV system.