Biot Number Equations Formulas Calculator

Unsteady State Heat Transfer Dimensionless Value

Problem:

Solve for Biot number.

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	Biot Number
	heat transfer coefficient
	characteristic length
	thermal conductivity

References - Books:

Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.

Background

The Biot number (Bi) is a dimensionless quantity used extensively in heat transfer calculations. Named after the French physicist Jean-Baptiste Biot, this number provides significant insights into the heat conduction and convection processes. The Biot number compares the internal resistance to heat conduction within a body to the external resistance to heat convected from its surface. This comparison is crucial in determining whether the temperature within a body can be assumed uniform (lump system approach) or if temperature gradients must be considered.

Equation

The Biot number is defined using the following equation:

Bi = hL / k_s

where:

h is the heat transfer coefficient (W/m²·K)
L is the characteristic length (m)
k_s is the object's thermal conductivity (W/m·K).

Depending on the medium and materials involved, an alternative form using thermal conductivity (k_s) of the surrounding fluid or material may be used in some contexts.

How to Solve

Solving the Biot number involves:

Identify the heat transfer coefficient (h): This is typically given or can be determined based on the conditions around the object (such as fluid velocity, temperature differences, etc.).
Determine the characteristic length (L): The characteristic length is often calculated based on geometry, such as the object's volume divided by the surface area for three-dimensional bodies.
Establish the thermal conductivity (k_s): The material's thermal conductivity must be known or obtained from material property tables.
Apply the Biot equation: Insert the values into the Biot equation to obtain the dimensionless Biot number. Bi = hL / k_s

Example

Let's consider a sphere with the following parameters:

Heat transfer coefficient of h = 50 W/m²·K

Diameter of the sphere of D = 0.1 m

Thermal conductivity of k_s = 200 W/m·K

The characteristic length for a sphere (typically the radius) L = D / 2 = 0.05 m

Plugging the values into the Biot equation:

Bi = 50 W/m²·K x 0.05 m / 200 W/m·K

Bi = 2.5 / 200

Bi = 0.0125

This low Biot number implies a minimal temperature gradient within the sphere, validating the lumped capacitance model.

Fields/Degrees Where It Is Used

Mechanical Engineering: Solving heat transfer problems in engines, turbines, and machinery components.
Chemical Engineering: Heat transfer is crucial in reactor design and analysis.
Civil Engineering: In evaluating heat flow through building materials for energy efficiency.
Aerospace Engineering: Critical in analyzing thermal protection systems and materials tolerating extreme temperatures.
Environmental Engineering: Used in modeling thermal processes in environmental systems, including soil heat flux and aquatic systems' thermal behavior.

Real-Life Applications

Design of Heat Exchangers: Ensure that heat transfer effectiveness is optimized while maintaining the structural integrity of the materials.
Thermal Analysis of Electronics: Cooling mechanisms in electronic devices to prevent overheating.
Medical Device Manufacturing: Control thermal conditions in devices such as incubators and sterilizers.
Construction Industry: Insulation design for buildings to effectively manage heat loss/gain.
Automotive Industry: Enhancing engine cooling systems for better performance and efficiency.

Common Mistakes

Incorrect Characteristic Length: Using incorrect geometry leads to erroneous Biot number calculations.
Ignoring Material Anisotropy: Not accounting for directional thermal conductivity variations.
Overlooking Convection Coefficient Variations: Assuming a constant convection coefficient when conditions change significantly.
Assuming Lumped Capacitance Applicability: Misusing it in systems with significant internal temperature gradients Bi > 0.1.
Misidentifying Fluid Dynamics Influence: Neglecting the effects of fluid movement around the object affecting h.

Frequently Asked Questions

Q: What is the significance of a high Biot number?
A: A high Biot number (>>1) indicates that thermal resistance within the object is much higher than surface convection resistance, leading to significant internal temperature gradients.
Q: When can the lumped capacitance method be applied?
A: It is valid when the Biot number is less than 0.1, implying that internal conduction resistance is much lower than surface convection resistance, resulting in negligible temperature gradients within the object.
Q: How is the characteristic length determined for irregular shapes?
A: The volume ratio to the surface area or other geometrically appropriate dimensions can approximate the characteristic length for irregular shapes.
Q: What does a Biot number of 1 signify?
A: A Biot number of 1 suggests that the internal and external thermal resistances are comparable, necessitating a more detailed analysis of conduction and convection effects.
Q: Can the Biot number be negative?
A: No, the Biot number is always a non-negative value as it represents a ratio of positive thermal resistances.