Thermal Stress and Conduction: Method

Thermal Stress and Conduction

Four separate analyses share this page, each switched on by its checkbox: the stresses that a temperature change produces in restrained members, steady one-dimensional conduction through layered walls as a resistance network, extended surfaces (fins), and the transient response of a body suddenly exposed to a fluid. Every quantity with a dictionary unit (lengths, areas, moduli, expansion coefficients, temperature changes, forces, stresses, heat rates, densities, specific heats) converts between SI and US customary units automatically; conductivities, convection coefficients, contact resistances, absolute temperatures, times and energies are plain numbers in the units named in the group titles (W/(m·K), W/(m²·K), m²·K/W, °C, s, J in SI; Btu/(h·ft·°F), Btu/(h·ft²·°F), h·ft²·°F/Btu, °F, s, Btu in US units).

Thermal stress

A free bar heated by ΔT lengthens by δ_T = α ΔT L without stress. Stress arises only when that deformation is prevented. The total strain of a member is the sum of its thermal strain α ΔT and its mechanical strain σ/E; the restraint fixes the total strain, so the mechanical strain, and with it the stress, follows.

Bar between rigid walls. With the bar attached to both walls the total strain is zero and σ = −E α ΔT: compression on heating, tension on cooling, independent of the length and the area. With an initial gap δ_g the bar expands freely until α ΔT L = δ_g, at the temperature rise δ_g/(α L); beyond that the total strain is δ_g/L and σ = −E (α ΔT − δ_g/L). A gap never produces stress on cooling. The reaction of the walls is F = σ A.

Two segments in series. One force F passes through both segments. Compatibility requires the sum of the thermal and mechanical elongations, Σ α_i ΔT L_i + F Σ L_i/(A_i E_i), to equal the gap, so F = −(Σ α_i ΔT L_i − δ_g) / Σ L_i/(A_i E_i) and σ_i = F/A_i. Beer's Concept Application 2.6 (a stepped steel bar cooled 125 °F) is this case.

Members in parallel. Members of two kinds between rigid plates (a bolt inside a sleeve, several posts under a rigid beam) share one elongation δ. Each carries F_j = A_j E_j (δ/L − α_j ΔT), tension positive, and equilibrium of the plate gives n₁ F₁ + n₂ F₂ = P with P the external load (tension positive, zero for a bolt and sleeve). Hence δ = L (P + ΔT Σ n_j A_j E_j α_j) / Σ n_j A_j E_j. The member of larger expansion ends in compression and the other in tension when P = 0. Hibbeler's Example 4.12 (steel bolt in an aluminum sleeve, 20.3 kN) and Example 4.11 (two steel posts and one aluminum post under 90 kN) are reproduced with this case.

Bimetallic strip. Two layers bonded along their length and heated uniformly bend because the layer of higher expansion is held back by the other. Timoshenko (1925) derived the curvature 1/ρ = 6 (α₂ − α₁) ΔT (1 + m)² / { h [3 (1 + m)² + (1 + m n)(m² + 1/(m n))] } with m = t₁/t₂, n = E₁/E₂ and h = t₁ + t₂; the strip curves toward the layer of lower expansion. For equal layers of one material this is 3 Δα ΔT/(2h). The tool also solves the general two-layer section directly (plane sections, strain ε₀ + κ y, zero net force and moment), which gives the same curvature and the stress through both layers: the largest stresses are at the interface, E Δα ΔT/2 for equal layers, and the outer faces carry half that with the opposite sign. The tip deflection of a cantilever strip of length L is κ L²/2 and the midspan deflection of a simply supported strip κ L²/8 (small deflections). Width enters only the layer forces.

Steady conduction: the resistance network

For steady one-dimensional conduction without heat generation the heat rate q is the same through every layer and every surface film, so each part can be written as a thermal resistance R = ΔT/q and the parts added like electrical resistances in series. The layer resistances are L/(k A) for a plane wall, ln(r₂/r₁)/(2π k L) for a cylinder of length L and (1/r₁ − 1/r₂)/(4π k) for a spherical shell; a convective surface contributes 1/(h A) with the area of that surface; an imperfect joint between two layers contributes a contact resistance R″_c/A, which produces a temperature step at the interface. Then q = (T_∞,i − T_∞,o)/ΣR, the drop across each part is q R, and every interface temperature follows. Leaving a convection coefficient blank makes the corresponding temperature a prescribed surface temperature (no film resistance). The temperature varies linearly within a plane layer, logarithmically within a cylindrical layer and as 1/r within a spherical layer; the profile is plotted.

For a cylinder or sphere, adding insulation increases the conduction resistance but also increases the outer surface area and so lowers the convection resistance. The sum has a minimum at the critical radius r_cr = k/h (cylinder) or 2k/h (sphere), with k the outer layer's conductivity and h the outer film coefficient. Below r_cr adding insulation increases the heat loss; this matters for thin wires and small tubes with poor insulation and low h, never for pipes of ordinary size. The tool reports r_cr, says which side of it the outer radius lies, and plots the heat rate against the outer radius.

Fins

A fin of uniform section conducts heat along its length while losing it by convection from its surface. With θ = T − T_∞ the energy balance is d²θ/dx² − m² θ = 0, m² = h P/(k A_c), P the perimeter and A_c the section area; for a thin rectangular fin m² ≈ 2h/(k t) and for a pin m² = 4h/(k D). Four tip conditions are offered (Incropera Table 3.4): the infinite fin, θ = θ_b e^−mx and q_f = M = √(h P k A_c) θ_b, valid within 1 % when L ≥ 2.65/m; the adiabatic tip, θ/θ_b = cosh m(L − x)/cosh mL and q_f = M tanh mL; the exact convective tip, q_f = M [sinh mL + (h/mk) cosh mL]/[cosh mL + (h/mk) sinh mL]; and the corrected-length approximation, which applies the adiabatic result to L_c = L + t/2 (rectangular) or L + D/4 (pin), accurate when h t/(2k) ≤ 0.0625.

The annular fin of uniform thickness t between radii r₁ and r₂ obeys a modified Bessel equation; with an adiabatic tip the heat rate is q_f = 2π k r₁ t θ_b m [K₁(mr₁) I₁(mr₂) − I₁(mr₁) K₁(mr₂)] / [I₀(mr₁) K₁(mr₂) + K₀(mr₁) I₁(mr₂)], the corrected length replaces r₂ by r_2c = r₂ + t/2, and the exact convective tip is solved from the general solution c₁ I₀(mr) + c₂ K₀(mr) with −k dθ/dr = h θ at r₂. The Bessel functions are evaluated by power series and the Abramowitz and Stegun asymptotic polynomials (checked against scipy to 10⁻⁷), so the efficiency is computed rather than read from the chart of Incropera Figure 3.19; for the motorcycle-cylinder fins of Example 3.9 the computed efficiency is 0.979 against the chart reading 0.95.

The fin efficiency η_f = q_f/(h A_f θ_b) compares the fin with one entirely at the base temperature; the effectiveness ε_f = q_f/(h A_c θ_b) compares it with the bare base area, and a fin is rarely worth adding unless ε_f ≥ 2. The temperature along the fin is plotted.

Transient conduction

Lumped capacitance. If the Biot number h L_c/k, with L_c = V/A_s (the half-thickness of a wall, r_o/2 of a cylinder, r_o/3 of a sphere), is below 0.1, the temperature inside the body is nearly uniform and an energy balance gives (T − T_∞)/(T_i − T_∞) = exp(−t/τ) with τ = ρ V c/(h A_s) = ρ c L_c/h. The time to reach a temperature is τ ln[(T_i − T_∞)/(T − T_∞)] and the energy transferred up to t is ρ V c (T_i − T_∞)(1 − e^−t/τ). Incropera's thermocouple of Example 5.1 (0.706 mm bead, τ = 1 s) takes 5.2 s to come within 1 °C of a 175 °C step.

Series solution. For larger Biot numbers the temperature varies through the body. The exact solution of the one-dimensional problem with uniform initial temperature and convection at the surface is θ* = (T − T_∞)/(T_i − T_∞) = Σ C_n exp(−ζ_n² Fo) f_n(x*), Fo = α t/L² with α = k/(ρ c), where the eigenvalues ζ_n are the positive roots of ζ tan ζ = Bi (wall), ζ J₁(ζ)/J₀(ζ) = Bi (cylinder) or 1 − ζ cot ζ = Bi (sphere), Bi = h L/k with L the half-thickness or the radius, and the coefficients are C_n = 4 sin ζ_n/(2ζ_n + sin 2ζ_n), (2/ζ_n) J₁(ζ_n)/[J₀²(ζ_n) + J₁²(ζ_n)] and 4 (sin ζ_n − ζ_n cos ζ_n)/(2ζ_n − sin 2ζ_n). The position functions f_n are cos(ζ_n x*), J₀(ζ_n r*) and sin(ζ_n r*)/(ζ_n r*); the energy fraction is Q/Q_o = 1 − Σ C_n exp(−ζ_n² Fo) g_n with g_n = sin ζ_n/ζ_n, 2 J₁(ζ_n)/ζ_n and 3 (sin ζ_n − ζ_n cos ζ_n)/ζ_n³. For Fo > 0.2 the first term alone is accurate to better than 2 % (the one-term approximation of Incropera Section 5.5 and Çengel Chapter 4, whose Table 5.1 / 4-1 lists ζ₁ and C₁); the tool solves the roots by bisection for any Bi and prints the one-term values beside the N-term sum, so interpolation in the table is unnecessary. Below Fo = 0.2 read the N-term values; near Fo = 0 even many terms show the Gibbs ripple of a step initial condition.

The time at which the centre reaches a target temperature is found from the one-term expression Fo = ln(C₁ f₁(0)/θ*)/ζ₁² and refined on the N-term series; the egg of Çengel's Example 4-3 (Bi = 47.8) reaches 70 °C at its centre after 14.4 min. The surface heat flux at the time entered is h (T_∞ − T_s), and when the surface area is given the energy transferred Q = (Q/Q_o) ρ V c (T_i − T_∞) is printed with V = L_c A_s (for a wall of thickness L insulated on one face, A_s is the one exposed face). The chart shows the centre and surface histories from the series with the lumped exponential for comparison.

Conventions and limits

ΔT is the temperature change of every member (positive heating). Forces and stresses are tension positive. The heat rate of the network is positive from side i to side o. The series solutions assume constant properties, a uniform initial temperature, a sudden exposure to a fluid at constant temperature with a constant h over the whole surface, and one-dimensional geometry (a wall large in its plane, a cylinder long compared with its radius). Not covered: internal heat generation, radiation (fold it into h as a radiation coefficient), multidimensional bodies (use the product solutions of Incropera Section 5.8 with the one-dimensional results given here), semi-infinite solids, fins of varying section, and temperature-dependent properties.