Lumped Capacitance Cooling Calculator

Convection coefficient h (W/(m²·K))

Exposed surface area A (m²)

Density ρ (kg/m³)

Body volume V (m³)

Specific heat cp (J/(kg·K))

Thermal conductivity k (W/(m·K))

Initial temperature Ti (K)

Ambient temperature T∞ (K)

Elapsed time t (s)

Model: Lc = V/A; Bi = h Lc/k < 0.1; τ = ρ V cp/(h A); T(t) = T∞ + (Ti − T∞) exp(−t/τ)

The body is treated as spatially uniform in temperature; properties, exposed area, and ambient temperature remain constant. Convection is the only heat exchange: no radiation, phase change, internal generation, or changing boundary conditions. A small Biot number is necessary for this model, but does not establish an engineering guarantee.

Calculations use only your local inputs. No material properties are looked up. This educational idealized model does not guarantee engineering sizing or safety.

Enter SI values and calculate.

Calculation report

References: MIT heat-transfer equation sheet (PDF) · MIT thermal-energy notes (PDF)

A QUICK WALKTHROUGH

How to use this tool

Enter the listed SI inputs; all material properties and geometry come from your inputs.
Select Calculate and review the model assumptions and the results.
Copy the report or download it as plain text. Changing an input clears the previous result.

Formula and SI units

Lc = V/A; Bi = h Lc/k < 0.1; τ = ρ V cp/(h A); T(t) = T∞ + (Ti − T∞) exp(−t/τ). Inputs use the units shown beside each field. Temperatures, where present, are absolute kelvin.

Assumptions and limits

Local processing

Calculations use only your local inputs. No material properties are looked up. This educational idealized model does not guarantee engineering sizing or safety.

Official course references

MIT OpenCourseWare, Introduction to Heat Transfer equation sheet: https://ocw.mit.edu/courses/2-051-introduction-to-heat-transfer-fall-2015/e32f70f7e00f34953447b606cea0421f_MIT2_051F15_EqnSheet_Q2_v3.pdf ; Thermal Energy, heat-transfer notes: https://ocw.mit.edu/courses/16-050-thermal-energy-fall-2002/87d9f4544b7fd64a77201382500d057c_10_part3.pdf

GOOD TO KNOW

Common questions

Are material properties supplied automatically?

No. Choose values appropriate to your material and conditions from your own source.

Does this establish a safe engineering design?

No. It is an idealized educational calculation and does not verify the material, boundary conditions, or design.

Are inputs uploaded or saved?

No. The calculation and optional plain-text report are generated in your browser. The tool does not upload or persist the inputs.