Biot Number

Introduction

Named after the physicist Jean Baptiste-Biot(1774-1862), the Biot number is a unit-less/dimensionless number used in problems dealing with heat transfer. The Biot number allows for the calculation of a ratio between the heat transfer resistances within a system and the surroundings of the system, as well as its surface. The number essentially takes into consideration the interactions between the convection at a solid's surface and the conduction in the solid.[1] Transient or non-steady state problems with heat transfer in a solid require the use of the Biot number to analyze the temperature history of an object being heated or cooled over time. Problems that deal with high Biot numbers greater than 1 are more difficult to analyze since the temperature distribution is not uniform, as convection is the leading heat transfer mechanism.[1]

Biot Number Description

The definition of the Biot number is:

$\mathrm{B}\mathrm{i}=\frac{L_{C}h}{k}$ where: ${{L}}_{{C}}=\frac{V_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}}}{A_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}}$

• h= where h stands for the heat transfer coefficient or film coefficient. This is also referred to as the convective heat transfer coefficient. Typically, it will be presented in units of Watts per meter squared times kelvin [W/m^2*K] or British Thermal Units per hour times squared foot times Fahrenheit [Btu/Hr*ft^2*F]
• L_C= where the term L signifies the volume of the object per surface area of the object. It is presented typically in units of meters [m] or feet [ft].
• k_b = where k stands for the thermal conductivity of the object. Typically presented in units of Watts per Meter times Kelvin [W/m*K]

Biot Number in Negligible Internal Resistance

When the Biot number is lower than 0.1, it is assumed that the body being analyzed has negligible internal resistance and therefore it can be assumed that convection is the dominating heat transfer mechanism.[2] In a classic example, a steel ball at a temperature of T_0 at a time t=0 is immersed in tub of cold water at T=∞. When making a heat balance for this solid sphere and assuming a constant heat transfer coefficient h, the heat transfer from the fluid to the sphere should equal the change in internal energy of the object; defined by the equation: hAΔT=Cp*ρ*V dT where CpρV stands for the lumped thermal capacitance of the system. This equation tells us the temperature history of the solid sphere with respect to time. When arranging this equation and integrating between the times T=T of zero, at a time=0, and T=T at a time=t.[2]

$\frac{T-T_{\mathrm{\infty }}}{T_0-T_{\mathrm{\infty }}}=e^{\mathrm{-}\mathrm{X}}$ Where -X= (h*A/Cp*ρ*V)t

• A= stands for the surface area of the solid. typically given in units of meters squared [m^2] or foot squared [ft^2]
• ρ= where ρ stands for the density of the solid. Typically given in units of kilograms per meters cubed [Kg/m^3] or units of pounds of mass per cubic feet [Lbm/ft^3]
• V= where V stands for the volume of the sphere. Typically given in units of meters cubed [m^3] or cubic feet [ft^3]

As we make the assumption of negligible internal, we can assume a Biot number lower than 0.1

${\mathrm{N}}_{\mathrm{B}}\mathrm{i}=\frac{h\times x_1}{k}$ <0.1

• X₁ = stands for the ratio of the volume of the solid to the area of the solid
• N_Bi = stands fir the Biot Number, a dimensionless number

Utilizing the above equations, we can determine the objects temperature history. This is sometimes referred to as the Newtonian heating or cooling method or the Lumped Capacity Method.[2]