Biot Number and Lumped Capacitance Calculator: Transient Cooling and Heating of a Solid

Cooling or heating of a solid that stays one temperature.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose the shape and enter its size, the convection coefficient and the material properties.

Enter the starting temperature, the temperature of the fluid and the time.

The calculator gives the Biot number, the time constant, the temperature at that time and the time to reach your target.

Formulas used

  • Bi = \frac{h L_c}{k}
  • \tau = \frac{\rho c_p L_c}{h}
  • T(t) = T_a + (T_i - T_a)e^{-t/\tau}

Worked example

With the default numbers in the calculator above:

  1. Characteristic length (volume divided by surface area): 3.3333 mm
  2. Biot number: 0.0027778
  3. Time constant: 249.6 s
  4. Temperature at the time you entered: 167.79 C
  5. Time to reach the target temperature: 460.72 s (7.679 minutes)

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

When can I use the lumped capacitance method?

When the Biot number is 0.1 or less. Then the inside of the solid stays almost the same temperature as the surface.

What is the characteristic length?

The volume of the body divided by the surface area exposed to the fluid. It is half the thickness of a plate, half the radius of a long cylinder and a third of the radius of a sphere.

What does the time constant mean?

The time for the temperature difference to fall to about 37 percent of its starting value. After 3 time constants about 95 percent of the change has happened.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any heat transfer text (lumped system analysis, the Biot number).

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