Unsteady Internal Forced-Convective Flow Under Dynamic Time-Dependent Boundary Temperature

Abstract

A new all-time analytical model is developed to predict transient internal forced-convection heat transfer under arbitrary time-dependent wall temperature. Slug flow condition is assumed for the velocity profile inside the tube. The solution to the time-dependent energy equation for a step wall temperature is generalized for arbitrary time variations in surface temperature using Duhamel's theorem. A harmonic boundary temperature is considered, and new compact closed-form relationships are proposed to predic: 1) fluid temperature distribution; 2) fluid bulk temperature; 3) wall heat flux; and 4) the Nusselt number. An optimum value is found for the dimensionless angular frequency of the wall temperature to maximize the heat transfer rate of the studied unsteady forced-convective process. Such dimensionless parameter depends upon the imposed-temperature angular frequency, fluid thermophysical properties, and tube geometrical parameters. A general surface temperature is considered, and the temperature field inside the medium is obtained using a superposition technique. An independent numerical simulation is performed using ANSYS® Fluent. The comparison between the obtained numerical data and the present analytical model shows good agreement: a maximum relative difference less than 4.9%. NomenclatureA = cross-sectional area, Eq. (13), m 2 a = half-width of spacing between parallel plates, or circular tube radius, m c p = heat capacity, J∕kg · K Fo = Fourier number; αt∕a 2 J = Bessel function, Eq. (6a) k = thermal conductivity, W∕m · K Nu = Nusselt number, ha∕K n = positive integer, Eq. (6b) p = 0 for parallel plate and 1 for circular tube, Eqs. (3) and (14) Pr = Prandtl number (ν∕α) q w = dimensionless wall heat flux Re = Reynolds number, 2Ua∕ν r = radial coordinate measured from circular tube centerline, m T = temperature, K U = velocity magnitude, m∕s u = velocity vector, Eq. (2) x = axial distance from the entrance of the heated section, m y = normal coordinate measured from centerline of parallel plate channel, m α = thermal diffusivity, m 2 ∕s γ = function for circular tube [Eq. (6a)] or for parallel plate [Eq. (6b)] Δφ = thermal lag (phase shift) ζ = dummy X variable η = dimensionless radial/normal coordinate for circular tube equal to r∕a or for parallel plate equal to y∕a θ = dimensionless temperature; T − T 0 ∕ΔT R λ n = eigenvalues, Eq. (6) ν = kinematic viscosity, m 2 ∕s ξ = dummy Fo variable ρ = fluid density, kg∕m 3 ψ = arbitrary function of Fo Ω = imposed-temperature angular frequency, rad∕s ω = dimensionless temperature angular frequency, Subscripts m = mean or bulk value R = reference value s = step wall (surface) temperature w = wall 0 = inlet

Integral transform method solution for heat transfer in polymer melt flow in a parallel plate single screw channel with periodic inlet temperature

“…Indeed, slug flow dampening does not change with frequency Ω. This finding casts doubts on the relevance of the slug flow assumption to represent forced convection heat transfer in ducts [2,3,6,16].…”

“…As we are chiefly concerned with the influence of the velocity profile in this particular laminar dispersion problem, we set aside the simplifying hypothesis of a slug flow as in [2,3,6,16], though slug flows will provide useful comparisons with our results. Furthermore, owing to the large thermal capacitance of the metallic barrel wall operating at a steady regulated temperature we attempt no treatment of the conjugate heat transfer problem between the flow and its wall boundary [7] and will be only considering the isothermal case here.…”

“…In the second section, the integral transform method used throughout this work is reviewed. The integral transform method is a well established analytic method in the field of forced convection heat transfer and has found numerous applications in steady state and transient heat transfer, particularly for heat exchangers [5,6,7,8,10,14,13,16,20] and recently in microfluidics [15,17].…”