Abstract
Heat transfer in turbulent duct flow of a suspension of fine particles is examined. The flow is fully developed but the temperature profiles due to heat transfer are not. Two coupled similar equations define the system and illustrate the limitations which arise through describing heat transfer between the phases in terms of a particle heat transfer coefficient. In this case, however, the equations may be reduced to two equations of the Sturm-Liouville type. The solutions of these have eigenfunctions which are closely related and they share a common series of eigenvalues. The latter conclusions are expected to apply more readily to a transport reactor of large bore which circulates fine noncohesive particles.
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Abbreviations
- A P :
-
surface area of particle
- C :
-
specific heat
- D :
-
pipe diameter
- d :
-
particle diameter
- f f, f s :
-
Ū f(η)/Ū avge, Ū s(η)/Ū avge
- g f :
-
total diffusivity function for gas, 1+ε H, f Pr/ν
- g s :
-
eddy diffusivity function for solids, ε H, s Pr/ν
- h p :
-
particle heat transfer coefficient
- k :
-
thermal conductivity
- m p :
-
mass of particle
- Nu s :
-
pipe wall Nusselt number for flow with solids
- Nu 0 :
-
” ” ” ” ” ” of air alone
- Pr :
-
Prandtl number, μC f/k f
- Re :
-
Reynolds number, Ū avge D/ν
- n :
-
particle concentration, \(\bar \rho _{{\text{ds}}} /\pi d^3 \rho _{\text{p}}\)
- q :
-
heat flux at wall
- R f, n , R s, n :
-
eigenfunctions of gas and solid phase
- r :
-
radial position in pipe
- r*:
-
distance from centre of particle
- T :
-
temperature
- t :
-
time
- U :
-
axial velocity down pipe
- Ū avge :
-
\(\frac{8}{{D^2 }}\int\limits_0^{\tfrac{1}{2}D} {\bar U_{\text{f}} r{\text{ d}}r}\)
- W s, W g :
-
solids and gas flow rates in transfer line
- x :
-
distance down pipe from point of heat addition
- α, β :
-
constants
- β 3 :
-
\(\frac{{3_{\rho {\text{f}}} C_{\text{f}} D^2 h_{\text{p}} }}{{_{\rho {\text{p}}} C_{\text{s}} k_{\text{f}} d}}\)
- β 4 :
-
\(\frac{{3_{\bar \rho {\text{ds}}} h_{\text{p}} D^2 }}{{_{\rho {\text{p}}} dk_{\text{f}} }}\)
- ε H :
-
eddy diffusivity of heat
- ε H, f(O):
-
fluid eddy diffusivity of heat in the absence of solids
- θ f, θ s :
-
gas and solid phase time averaged temperatures rendered suitably dimensionless
- either:
-
\(\theta = {{(\bar T - T_{\text{w}} )} \mathord{\left/ {\vphantom {{(\bar T - T_{\text{w}} )} {(T_{\text{o}} - T_{\text{w}} )}}} \right. \kern-\nulldelimiterspace} {(T_{\text{o}} - T_{\text{w}} )}}\) for uniform wall temperature conditions
- or:
-
\(\theta = {{2k_{\text{f}} (\bar T - T_{\text{o}} )} \mathord{\left/ {\vphantom {{2k_{\text{f}} (\bar T - T_{\text{o}} )} {qD}}} \right. \kern-\nulldelimiterspace} {qD}}\) for uniform heat flux conditions at the wall
- τ 2 n :
-
eigenvalue
- η :
-
2r/D
- ρ f :
-
gas density
- ρ ds :
-
dispersed solids density
- ρ *ds :
-
dispersed solids density in unit size range of particles
- ρ p :
-
density of material of particles
- ξ :
-
2x/D Re Pr
- ν :
-
kinematic viscosity
- μ :
-
viscosity
- f:
-
gaseous phase
- o:
-
at inlet to heated section
- p:
-
particle
- s:
-
solid phase
- w:
-
wall
- overscore:
-
refers to time average, e.g. \(\bar \psi = \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int\limits_0^T \psi {\text{ d}}t\)
- dash:
-
refers to a fluctuation from the mean value, e.g. \(\psi (t) = \bar \psi + \psi '(t)\)
References
Yannopoulos, J. C., N. J. Themelis, and W. H. Gauvin, Canad. J. Chem. Eng. 42 (1964) 231.
Sparrow, C. M., T. M. Hallman, and R. Siegel, Appl. Sci. Res. A 7 (1957) 37.
Sleicher, C. A. and M. Tribus, Trans. ASME 79 (1957) 789.
Soo, S. L., G. J. Trezek, R. C. Dimick and G. F. Hohenstreiter, Ind. Eng. Chem. Fund. 3 (1964) 98.
Hinze, J. O., Appl. Sci. Res. A 11 (1962) 33.
Julian, F. M. and A. E. Dukler, AIChE J. 11 (1965) 853.
Boothroyd, R. G., Trans. Inst. Chem. Eng. 44 (1966) 306.
Boothroyd, R. G., Trans. Inst. Chem. Eng. 45 (1967) 297.
Tien, C. L., ASME J. Heat Transf. 83 (1961) 183.
Depew, C. A. and L. Farbar, ASME J. Heat Transf. 85 (1963) 164.
Nosov, V. S. and N. I. Syromyatnikov, Soviet Physics Doklady 10 (1966) 672.
Babcock and Wilcox Co., rep. B. A. W. 1159, 1181, 1194, 1201, and 1207, New York 1959.
Hawes, R. I., E. Holland, G. T. Kirby, and P. R. Waller, rep. AEEW - R244/1964, Atomic Energy Establishment, Winfrith (U.K.) 1964.
Garrett, T., Heat transfer characteristics of dilute phase solids/air suspensions in turbulent pipe flow, Ph. D. thesis, University of Cambridge, Cambridge (U.K.) 1964.
Farbar, L. and M. J. Morely, Ind. Eng. Chem. 49 (1957) 1143.
Jepson, G., A. Poll, and W. Smith, Trans. Inst. Chem. Eng. 41 (1963) 207.
Quan, V. and C. L. Tien, ASME Paper 62 - HT - 15, ASME, New York 1962.
Brötz, W., J. W. Hiby, and J. W. Müller, Chemie Ing. Tech. 10 (1958) 138.
Price, P. H., Brit. J. Appl. Phys. 10 (1959) 135.
Van Zoonen, D., 3rd Europ. Cong. Chem. Eng. A54, Inst. Chem. Eng., London (U.K.) 1962.
Churchill, R. V., Modern Operational Mathematics in Engineering, McGraw-Hill, New York 1944.
Hinze, J. O., Turbulence: an Introduction to its Mechanism and Theory, McGraw-Hill, New York 1959.
Schlichting, H., Boundary Layer Theory, 4th edn., McGraw-Hill, New York 1960.
Boothroyd, R. G., J. Sci. Instr. 44 (1967) 249.
Themelis, N. J. and W. H. Gauvin, Canad. J. Chem. Eng. 40 (1962) 157.
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Boothroyd, R.G. Heat transfer in a gas borne suspension of fine particles in turbulent duct flow. Appl. sci. Res. 21, 98–112 (1969). https://doi.org/10.1007/BF00411600
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DOI: https://doi.org/10.1007/BF00411600