Abstract
In many cases the conduction mechanism inside a particle can not be ignored (large particles, low thermal conductivity and high porosity) during turbulent gas–particle flows. However, the accurate solution might be difficult to apply. Therefore, we first develop here the ability to conduct accurate solution and then we define the criterion for which the internal conductivity might be ignored. A combination between commercial C.F.D. code and user defined programs was developed to predict numerically the gas–particle velocity and temperature profiles. The selected criterion (defined at the outlet of the pipe’s cross-section), referred to the relation between the computational desirable average temperature difference \((T_{{\rm p}_{_{\rm wall}}} - {{T_{\rm p}}_{_{\rm center}}})_{_{_{\rm AVERAGE}}}\) without ignoring internal heat conductivity and the average particles temperature \({T_{\rm p}}_{_{\rm AVERAGE}}\) by ignoring internal heat conductivity, determines whether to consider the heat conduction mechanism in numerical simulations or to ignore it. It was found that the average particles temperature for T p = f(r) is lower than the case when T p = constant. Also, it was found that the non-dimensional temperature difference criterion is a continuous function of [Bi × (d p/D)] for a specific geometry, various pipe and particle diameters, various particles’ thermal conductivities, constant heat flux and Re number. The numerical code enables to extend the classical criterion for Bi number of solids to various gas–particle systems and different operational conditions.
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Abbreviations
- a :
-
particle radius (m)
- Bi :
-
Biot number (ha/kp)
- c p :
-
gas specific heat [J/(kg K)]
- c pp :
-
particle specific heat [J/(kg K)]
- D :
-
pipe diameter (m)
- D * :
-
non-dimensional pipe diameter
- d p :
-
particle diameter (m)
- d p*:
-
non-dimensional particle diameter d p/D
- dr * :
-
non-dimensional radial distance interval
- dt * :
-
non-dimensional infinitesimal time increment
- f :
-
function computed by the right side of (7)
- Fo :
-
Fourier number \(({t^{\ast}t_{\rm scale} k_{\rm p}} \mathord{\left/ {\vphantom {{t^{\ast}t_{scale} k_{\rm p}} {\rho_{\rm p} C_{\rm p_p} a^2}}} \right.}{\rho_{\rm p} C_{\rm p_p} a^2})\)
- g :
-
acceleration gravity (m/s2)
- h :
-
heat transfer coefficient [W/(m2 K)]
- I :
-
integral equation (8)
- k :
-
thermal conductivity [W/(m K)]
- M p :
-
particle mass (kg)
- Nu :
-
Nusselt number
- Np :
-
number of particles
- q′′:
-
heat flux (W/m2)
- r :
-
particle radial direction/pipe coordinate (m)
- r p :
-
particle radial coordinate (m)
- r * :
-
non-dimensional particle radial direction (r/a)
- r * :
-
non-dimensional pipe radial coordinate (r/D)
- r * p :
-
non-dimensional particle radial coordinate (r p/a)
- Re :
-
Reynolds number based on pipe diameter gas properties and gas characteristic velocity
- T :
-
temperature (K)
- T * :
-
non-dimensional temperature (T − T ref/ΔT)
- \(\overline{T}_{\rm pi}, T_{\rm pi\_av}\) :
-
average particle temperature, \(3 \mathord{\left/ {\vphantom {3 {a^3}}} \right. }{a^3}\int_0^a {r_{\rm p}^2T_{\rm pi} \left({r_{\rm p}} \right)dr_{\rm p}} ({\rm K})\)
- \(T_{{\rm p}_{_{\rm AVERAGE}}}\) :
-
average particles temperature at the outlet of the pipe’s cross-section \(\sum_{i=1}^{N_{\rm p}} {{\overline {T_{\rm pi}}} \mathord{\left/{\vphantom {{\overline {T_{\rm pi}}} {N_{\rm p}}}} \right. } {N_{\rm p}}} ({\rm K})\)
- TPWCA:
-
the second selected criterion defined by: \((T_{{\rm p}_{_{\rm wall}}} - T_{{\rm p}_{_{\rm center}}})_{_{\rm AVERAGE}}/ T_{{\rm p}_{_{\rm AVERAGE}}}\)
- t :
-
Time (s)
- t*:
-
non-dimensional time t/(D/U)
- U :
-
characteristic gas velocity/free stream velocity (m/s)
- α:
-
loading ratio (particle mass flow rate/gas flow rate)
- Δt:
-
non-dimensional time interval
- ΔT :
-
temperature difference, \(q^{\prime\prime}_{\rm w}D/k_{f}\) (K)
- \(\Theta_{\rm p}^{\ast}\) :
-
non-dimensional particle temperature profile, \(\Theta_P^\ast = {\left({T_{\rm P} -T_\infty}\right)r}\mathord{\left/{\vphantom {{\left({T_{\rm P}-T_\infty} \right)r}{a\Delta T}}} \right. } {a\Delta T}\)
- λ n :
-
eigenvalues of \(\lambda_n \cot g \left({\lambda_{n}}\right) + \left({Bi - 1}\right) = 0\)
- ξ:
-
variable integration equation (5)
- ρ p :
-
particle density (kg/m3)
- AVER/average:
-
average value
- AV_OUTLET:
-
average value at the outlet cross- section
- Center:
-
temperature at particle’s center
- f :
-
gas
- fin:
-
initial condition for the gas
- i :
-
phase number
- i :
-
index number equation (5)
- j :
-
coordinate direction
- n :
-
index for time level
- N :
-
number of elements/element number
- ONE-PHASE:
-
heat transfer coefficient for one phase flow (gas)
- p, pp:
-
particle
- Po :
-
conditions for particles at t = 0
- r :
-
radial
- Uniform:
-
uniform temperature/uniform temperature assumption
- w, wall:
-
conditions on the pipe wall
- ∞:
-
infinity, reference
- * :
-
denote non-dimensional
- –:
-
denote average
References
Avila R. and Cervantes J. (1994). Analysis of the heat transfer coefficient turbulent particle pipe flow. Int J. Heat Mass Transf. 38: 1923–1932
Boothroyd R.G. and Haquest H. (1970). Fully developed heat transfer to a gaseous suspension of particles flowing turbulently in ducts of different size. J. Mech. Eng. Sci. 12: 191–200
Juncu G. (1998). Unsteady conjugate heat transfer for single particle and in multi-particle systems at low Reynolds numbers. Int. J. Heat Mass Transf. 41: 529–536
Wan Y.P., Prasad V., Wang G.-X., Sampath S. and Fincke J.R. (1999). Model and powder particle heating, melting, resolidification, and evaporation in plasma spraying processes. J. Heat Transf. 121: 691–699
Ramshaw J.D. and Chang C.H. (1993). Computational fluid dynamics modeling of multi-component thermal plasmas. Plasma Chem. Plasma Process. 116: 359–364
Fan L.S. and Zhu C. (1998). Principles of Gas-Solid Flow. Cambridge University Press, USA
Depew, C.A., Farbar, L.: Heat transfer to pneumatically conveyed glass particles of fixed size. Trans. ASME J. Heat Transf. 164–172 (1963)
Haim, M., Weiss, Y., Kalman, H., Ullmann, A.: The effect of the inlet conditions on the numerical solutions of particle-gas flows. Adv. Powder Techno 14, 87–110 (2003)
Haim M., Weiss Y. and Kalman H. (2003). Turbulent gas–particles flow in dilute state: a parametric study. Part. Sci. Technol. 21: 1–24
Haim M., Weiss Y. and Kalman H. (2007). Theoretical model for heat transfer in dilute turbulent gas–particle flows. Part. Sci. Technol. 25: 173–196
Ranz W.E. and Marshall W.R. (1952). Evaporation from drops. Chem. Eng. Prog. 48: 141–173
Kays W.M. and Crawford M.E. (1980). Convective Heat and Mass Transfer, 3rd edn. McGraw-Hill, Inc., New York
Gosman, A.D.: Ioannides, Aspects of computer simulation of liquid-fuelled combustors. AIAA 19th Aerospace Science. Meeting, Paper no. 81–0323, St Louis (1981)
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Haim, M., Kalman, H. The effect of internal particle heat conduction on heat transfer analysis of turbulent gas–particle flow in a dilute state. Granular Matter 10, 341–349 (2008). https://doi.org/10.1007/s10035-008-0093-3
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DOI: https://doi.org/10.1007/s10035-008-0093-3