Solve slab heating and cooling with robust time stepping. Compare fixed and convective boundary cases. View profiles, histories, exports, and practical calculation guidance notes.
| Parameter | Example value | Note |
|---|---|---|
| Length | 0.10 m | One-dimensional slab thickness |
| Nodes | 11 | Uniform spatial grid |
| Total time | 1200 s | Heating duration |
| Time step | 60 s | Backward Euler march |
| k, ρ, cp | 45, 7800, 460 | Representative steel-like properties |
| Left boundary | 100 °C fixed | Prescribed face temperature |
| Right boundary | 15 W/m²·K at 25 °C | Convective cooling |
| Expected trend | Center temperature rises gradually | Boundary effects move inward over time |
The calculator solves one-dimensional transient heat conduction in a slab. It uses the backward Euler implicit finite difference method.
Thermal diffusivity is α = k / (ρcp).
The internal node equation is:
(1 + 2Fo)Tin+1 - FoTi-1n+1 - FoTi+1n+1 = Tin
where Fo = αΔt / Δx².
For a fixed boundary, the node temperature is prescribed directly.
For a convective boundary, the surface control volume uses:
(1 + 2Fo + 2FoBi)Tsn+1 - 2FoTadjn+1 = Tsn + 2FoBiT∞
where Bi = hΔx / k.
The solver assembles a tridiagonal matrix at every step. It then solves the nodal temperatures with the Thomas algorithm.
Transient conduction changes with both time and position. Engineers track this change when parts heat up or cool down. The implicit method is popular because it remains stable for large time steps. That helps when simulations must cover long durations. It also supports stiff thermal problems. Thick metal parts, furnace walls, molds, and battery housings are common examples. When temperature must be tracked safely over time, an implicit scheme is often a strong choice.
This calculator models one-dimensional heat flow through a slab. The slab is divided into equally spaced nodes. Each node stores a temperature value. The program builds equations for all nodes at the new time level. That means every step uses a coupled temperature field. The result is more robust than a simple explicit update. You can study fixed-temperature faces or convective boundaries with ambient air. That makes the tool useful for heating plates, cooling panels, and basic insulation checks.
Thermal conductivity controls how fast heat spreads. Density and specific heat set thermal storage. Their ratio forms thermal diffusivity. A larger diffusivity moves thermal influence faster. Slab length and node count define the spatial resolution. Total time and time step shape the temporal resolution. Smaller steps usually improve accuracy. Larger steps can still work because the formulation is implicit. Boundary settings matter too. A fixed boundary forces the surface temperature. A convective boundary responds to fluid temperature and film coefficient.
The final nodal table shows the temperature profile along the slab. The history table shows how the center and average temperature evolve. The left and right flux values show the local conductive response near each boundary. Engineers use these outputs for thermal design checks, insulation studies, process heating, cooldown planning, and material screening. Exported tables also help with reports, validation logs, and design reviews. A good practice is to rerun the case with more nodes or a smaller time step. If key outputs barely change, the solution is likely well resolved. This simple verification step improves trust before design decisions. It also helps compare alternative materials, thicknesses, and boundary conditions. In day-to-day engineering, that saves time and reduces rework.
It means future temperatures are solved together at the next time level. The method is stable for larger time steps than a basic explicit scheme.
The Fourier number measures time-step size relative to heat diffusion. It helps you judge temporal resolution, even though the implicit method remains stable.
Yes. Use boundary temperatures or ambient temperatures above or below the initial slab temperature. The solver will predict warming or cooling accordingly.
The calculator supports fixed-temperature boundaries and convective boundaries. You can choose either type independently on the left and right slab faces.
The program slightly adjusts the step to land exactly on the requested total time. This avoids ending before or beyond the target duration.
It estimates the slab energy increase or decrease relative to the initial state. It uses area, density, heat capacity, and nodal temperature change.
Use more nodes when you expect steep gradients near boundaries. Start with 11 to 31 nodes, then refine until key temperatures change very little.
No. This page is for one-dimensional slabs only. Use a two-dimensional or three-dimensional model when lateral gradients are important.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.