Heat transfer: absolute temperatures and resistance accounting

Absolute temperatures are K; differences are K. Conductivity is W/(m K), convection coefficient W/(m^2 K), heat rate W, and resistance K/W. Radiation uses absolute fourth powers; passing Celsius silently changes the physics. Geometry and properties are constant, and the equations are ideal baselines.

The correlations warn with engineeringkit.ValidityWarning when an input leaves the range they were fitted to, and still return the number: dittus_boelter_nusselt (Re >= 1e4, 0.6 <= Pr <= 160), flat_plate_nusselt (0.6 <= Pr <= 60), churchill_bernstein_nusselt (Re*Pr >= 0.2), zukauskas_nusselt (0.7 <= Pr <= 500, and ST/SL >= 0.7 for aligned banks), hausen_nusselt (Re <= 2300), the laminar churchill_chu_nusselt (Ra <= 1e9) and nusselt_film_condensation (film Reynolds number <= 1800).

Conduction and convection

Full wall resistances include area: L/(k*A), ln(r2/r1)/(2*pi*k*L), and (1/r1-1/r2)/(4*pi*k). Cylinder/sphere radii require r2>r1>0. composite_wall returns each temperature drop from Q*R and includes supplied film resistances. A missing film is represented by None, not zero; zero h is rejected. At least one physical layer is required.

The legacy overall_u name returns conductance 1/sum(R) [W/K] for full resistances. It does not return area-normalized U [W/(m^2 K)]. To obtain U, divide the conductance by area. The documentation now makes this distinction explicit without changing existing numeric behavior.

fin_heat_rate_convective_tip adds the tip face, with h_tip = h unless given; h_tip = 0 is the adiabatic fin, and L = 0 the bare base area.

Lumped cooling requires Bi=h*Lc/k typically below 0.1, where Lc=V/A. The function cannot check Bi without conductivity; that assessment remains the caller’s responsibility. Semi-infinite conduction requires alpha>0 and t>0; t=0 is excluded rather than resolving the discontinuous boundary condition.

Exchangers and limits

NTU is nonnegative and Cr=Cmin/Cmax lies in [0,1]. Counterflow at Cr=1 uses NTU/(1+NTU). Stable expm1 expressions preserve tiny NTU, and log1p preserves near-equal terminal temperature differences in LMTD. Temperature crossing is excluded by requiring both terminal differences positive.

exchanger_effectiveness_crossflow takes mixed=”none” (both streams unmixed; the usual approximate formula, within 4% of the exact series for NTU <= 5 and about 1% for 2 <= NTU <= 5), “cmax” or “cmin” (exact). Every arrangement gives 1 - exp(-NTU) at Cr = 0. exchanger_effectiveness_shell_tube takes the total NTU and the number of shell passes, each with an even number of tube passes; many shells approach counterflow.

lmtd_correction_factor(P, R, shells) uses the tube-side effectiveness P = (t_o - t_i)/(T_i - t_i) and R = (T_i - T_o)/(t_o - t_i) = C_tube/C_shell, and multiplies the counterflow LMTD. It is symmetric, F(P, R) = F(PR, 1/R). A P that the arrangement cannot reach for that R (where F would fall to zero, a temperature cross) raises ValueError instead of returning NaN; add shells. The F-LMTD and effectiveness-NTU methods agree to round-off, which the tests check.

Transient conduction and numerics

plane_wall_transient takes dimensionless inputs: Fo = alpha*t/L^2 and Bi = h*L/k on the half-thickness L, and x/L measured from the midplane. The default 100 series terms are accurate for Fo above about 1e-3; near Fo = 0 the truncated series oscillates, so pass more terms. terms=1 is Heisler’s one-term chart, within 2% at the midplane for Fo > 0.2. Bi is a scalar; the eigenvalues of zeta*tan(zeta) = Bi are bracketed exactly (utils.plane_wall_eigenvalues).

conduction_1d uses evenly spaced nodes, fixed surface temperatures or adiabatic faces (None), and constant properties. The explicit scheme is stable only for r = alpha*dt/dx^2 <= 1/2; it is not refused above that so the instability can be studied, and result.mesh_fourier reports r. Crank-Nicolson is unconditionally stable but rings for a few steps after a discontinuous initial or boundary condition when r is large.

Convection correlations

flat_plate_nusselt is the isothermal flat plate in parallel flow: laminar 0.332 Re^(1/2) Pr^(1/3) locally and 0.664 on average (Pohlhausen), turbulent 0.0296 Re^(4/5) Pr^(1/3), and the mixed average with transition at Re_c = 5e5. Re_c = 0 makes it turbulent from the leading edge and Re_c = inf keeps it laminar. The Pr^(1/3) laws are meant for 0.6 < Pr < 60. churchill_bernstein_nusselt covers a cylinder in crossflow for Re Pr > 0.2, properties at the film temperature. zukauskas_nusselt is the average over a tube bank: Re is based on tube_bank_max_velocity, properties on the mean fluid temperature except Pr_s, Re must lie in [10, 2e6] (enforced), and banks of fewer than 20 rows are corrected with C2 interpolated from Zukauskas’s table. In the range 100-1000 it uses Zukauskas’s constants (0.52 and 0.71, m = 0.5) where Incropera et al. suggest the single-cylinder correlation.

laminar_tube_nusselt returns 48/11 (uniform flux) or 3.657 (uniform wall temperature, Graetz’s limit) for fully developed laminar flow in a circular tube. entry_length gives 0.05 Re Pr D below Re = 2300 (Pr = 1 for the hydrodynamic length) and 10 D above it. hausen_nusselt is the mean over a thermal entry length with a developed velocity profile and uniform wall temperature; it tends to 3.66.

chilton_colburn_mass_transfer gives h_m = h/(rho cp Le^(2/3)) [m/s], with lewis_number = alpha/D_AB. It is exact between correlations with Pr^(1/3) and Sc^(1/3) (the laminar flat plate) and approximate for turbulent flow; low mass-transfer rates are assumed.

dittus_boelter_nusselt is for fully developed turbulent tube flow, Re above about 1e4 and 0.6 < Pr < 160, with typical error 25%; it is not checked. churchill_chu_nusselt is for a vertical plate with properties at the film temperature; laminar=True selects the more accurate laminar form for Ra below about 1e9. rayleigh_number takes the magnitude of the temperature difference.

Radiation and view factors

All radiation temperatures are absolute. planck_spectral_emissive_power returns W/(m^2 m) per metre of wavelength (multiply by 1e-6 for per micrometre) and is written to underflow, not overflow, at short wavelength. radiation_heat_rate treats a small gray surface in large isothermal surroundings. The view factors assume diffuse surfaces: aligned, directly opposed rectangles and coaxial parallel disks.

radiation_enclosure solves Oppenheim’s network for n diffuse gray surfaces: T, epsilon and A have one entry per surface and F[i, j] is F_ij. The rows of F must sum to one and A_i F_ij = A_j F_ji must hold (both to 1e-6), otherwise ValueError: a missing surface (an opening) is a surface at the surroundings’ temperature with epsilon = 1. Surfaces flagged reradiating are insulated; their T is ignored and solved for. The net heat rates sum to zero. two_surface_radiation is the two-surface closed form (emissivities in (0, 1]); radiation_shield_heat_flux the large parallel plates with thin shields, given one emissivity or a pair per shield. view_factor_reciprocity returns F_ji and refuses a result above one.

Two-dimensional conduction and shape factors

conduction_2d solves steady conduction in a rectangle with the four edge temperatures fixed, constant k and no generation, on an nx by ny grid that includes the edges (corners take the mean of their two edges and enter no equation). heat_rates gives W per metre of depth entering through each edge; they sum to zero to round-off. The error falls as dx^2. The shape factors return S [m] with Q = S k dT: buried sphere and cylinder (isothermal ground surface, cylinder length much larger than D), eccentric cylinders and two parallel cylinders. Geometries that overlap raise ValueError.

Transient cylinders and spheres

cylinder_transient and sphere_transient take Fo = alpha t/r_o^2 and Bi = h r_o/k on the outer radius (not V/A) and r/r_o. Their eigenvalues (utils.cylinder_eigenvalues, utils.sphere_eigenvalues) are bracketed between known zeros. For small Bi they tend to the lumped exp(-2 Bi Fo) and exp(-3 Bi Fo); terms=1 is Heisler’s one-term chart, good for Fo > 0.2.

Phase change

neumann_stefan_front is the one-phase Stefan problem: the original phase stays at the melting point and equal densities are assumed. Ste = cp*dT/latent must be positive. nusselt_film_condensation is laminar film condensation on a vertical wall (film Reynolds number below about 1800; ripples above about 30 raise h by 10-20%); pass Rohsenow’s h_fg + 0.68*cp*(Tsat - Ts) to include film subcooling. rohsenow_boiling_heat_flux is saturated nucleate pool boiling, with C_sf = 0.013 (water on polished copper) and n = 1 (water; 1.7 for other liquids) by default. It grows as dT_excess^3 and is valid only up to the critical heat flux, which it does not check; errors of 100% in flux (about 25% in superheat) are common. zuber_critical_heat_flux uses Zuber’s C = pi/24 by default; 0.149 fits large flat heaters. Both require rho_l > rho_v.

from engineeringkit.heat_transfer import composite_wall
result = composite_wall(400, 300, [(.1, 1), (.2, 2)], A=2)
# Q=1000 W, interfaces [400, 350, 300] K.
fig, ax = result.plot()

Tests check interface energy balance, cylindrical thin-wall limit, equal- temperature radiation, cooling endpoints and exchanger limiting cases, the plane-wall eigenvalue equation and fixed-surface limit, energy conservation of insulated finite-difference conduction, Planck’s integral, view-factor limits and the Neumann front equation. The curriculum models are checked against the Blasius-Pohlhausen similarity solution, Graetz’s eigenvalue, the exact crossflow series, the effectiveness-NTU/F-LMTD identity, the cylinder and sphere lumped, fixed-surface and energy limits, the 2-D Fourier series and its second-order convergence, shape-factor identities and the closed-form two- and three-surface networks. test_reference_problems.py checks five worked examples of Incropera et al. (6th edition), from a furnace wall to a counterflow oil cooler, against their printed answers. Fin formulas assume a straight constant-section fin with an adiabatic tip.