Heat transfer through a multilayer wall and an insulated pipe: k, q, layer temperatures, critical diameter
Choose a plane or a cylindrical wall, give the fluid temperatures, the heat transfer coefficients and the layers (material and thickness). The calculator finds the overall heat transfer coefficient, the heat flux (the linear one for a pipe), the temperatures on the layer boundaries, draws the graph and, for a pipe, finds the critical diameter of the insulation.
The plane multilayer wall
In steady heat transfer the heat flow through all layers is the same. The flux density is q = (t₁ − t₂)/R₀, where the total resistance is R₀ = 1/α₁ + Σδᵢ/λᵢ + 1/α₂ and the overall coefficient is k = 1/R₀ (W/(m²·°C)). The temperature at a layer boundary is tᵢ = tᵢ₋₁ − q·δᵢ/λᵢ. Inside a layer the temperature is linear, and the smaller λ is, the steeper the drop, so most of the drop falls on the insulation.
The cylindrical wall
For a pipe the linear resistance is convenient: R_l = 1/(α₁·π·d₁) + Σ ln(dᵢ₊₁/dᵢ)/(2π·λᵢ) + 1/(α₂·π·dₙ). The linear heat flow q_l = (t₁ − t₂)/R_l in W/m is the same for all layers, and the flow over the length L is Q = q_l·L. Inside a layer the temperature follows the logarithm of the radius: T(r) = T₁ − q_l·ln(r/r₁)/(2π·λ).
The critical diameter of insulation
On a thin pipe, added insulation increases the area that gives heat away, and the loss first grows. The maximum is at d_cr = 2λ/α₂, where λ is the conductivity of the outer layer and α₂ the heat transfer coefficient from its surface. If the outer diameter of the insulation is smaller than d_cr, insulation does harm until enough of it is added. For mineral wool (λ about 0.05 W/(m·°C)) and α₂ about 10 W/(m²·°C) the critical diameter is about 10 mm, so ordinary pipes are not affected, while for wires, thin tubes and materials with a high λ it matters.
What to enter
Give the temperatures and α for the fluids on both sides. If the temperature of the surface itself is known, leave α empty (α = ∞). The resistance of scale, rust and contacts can be added as a separate layer with its own λ. The radiative part of the heat transfer is included in α₂; it can be estimated on the radiation page.
FAQ
How does the overall coefficient k differ from the heat transfer coefficient α?
α describes the exchange of heat between a fluid and a surface (convection and radiation), k covers the whole path from one fluid to the other through the wall: 1/k = 1/α₁ + Σδ/λ + 1/α₂. So k is always smaller than the smallest α and than λ/δ of any layer.
Why is the heat flow the same through all layers?
In a steady state no heat accumulates in any layer, so the flow through every section is the same. For a plane wall this is the flux density q (W/m²), for a pipe the linear flow q_l (W/m); the flux density then falls as the radius grows.
What is the critical diameter of insulation?
The diameter d_cr = 2λ/α₂ at which the linear loss of the pipe is greatest. If the outer diameter of the insulation is smaller than it, added insulation increases the loss. For large pipes (tens of mm and more) with ordinary insulation this is no problem, for wires and thin tubes it is.
How do I account for an air gap or scale?
Add a separate layer with its own λ and the real thickness: for scale λ is of the order of 1–3 W/(m·°C) and the thickness is in millimetres. For an air gap in a wall it is easier to set an equivalent resistance through an «own material»: δ/λ equals the required R. For reference: take the values from a handbook.
Which α should I use for water in a pipe and for air outside?
Guides (for reference): for water in a pipe in turbulent flow α is of the order of 1000–10000 W/(m²·°C), for air in a room 5–10, in wind 20–40. Find the exact value on the convective heat transfer page from the numbers Nu, Re, Pr and Gr.