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