Newton's Law of Cooling is a key concept in heat transfer. It explains how objects cool down by losing heat to their surroundings. The law states that the rate of heat loss is proportional to the temperature difference between the object and its environment.

Understanding this law helps us predict cooling rates in various situations. It's used in engineering, food safety, and even forensics to estimate time of death. However, it has limitations and doesn't account for all factors affecting heat transfer in real-world scenarios.

Newton's Law of Cooling

Law and Mathematical Representation

• States the rate of heat transfer between an object and its surroundings is proportional to the temperature difference between them
• Represented mathematically by the equation: $q = hA(Ts - T∞)$
  • $q$ = heat transfer rate (W)
  • $h$ = convective heat transfer coefficient (W/m²·K)
  • $A$ = surface area (m²)
  • $Ts$ = surface temperature (K)
  • $T∞$ = ambient fluid temperature (K)
• The negative sign in the equation indicates heat is transferred from the object to the surroundings when the object's temperature is higher than the ambient temperature (heat flows from hot to cold)
• Assumes the heat transfer coefficient remains constant during the cooling process and the object's temperature is uniform throughout

Assumptions and Limitations

• Constant heat transfer coefficient throughout the cooling process
  • In reality, the heat transfer coefficient may vary with changes in temperature, fluid properties, or flow conditions
• Uniform object temperature
  • The law assumes the entire object is at a single temperature, which may not be the case for objects with significant internal temperature gradients or non-uniform heating/cooling
• Negligible thermal radiation
  • Newton's Law of Cooling only considers convective heat transfer and assumes that radiative heat transfer is negligible compared to convection
  • In situations where radiative heat transfer is significant (high temperatures or materials with high emissivity), the law may not accurately predict the cooling behavior

Factors Affecting Convective Heat Transfer

Fluid Properties

• Thermal conductivity
  • Higher thermal conductivity allows for more efficient heat transfer between the surface and the fluid (metals vs. gases)
• Density
  • Affects the fluid's ability to transport heat away from the surface (water vs. air)
• Viscosity
  • Influences the formation and thickness of the thermal boundary layer (honey vs. water)
• Specific heat capacity
  • Determines the amount of heat a fluid can absorb or release for a given temperature change (water vs. oil)

Flow Conditions

• Fluid velocity
  • Higher velocities enhance convective heat transfer by promoting mixing and reducing the thickness of the thermal boundary layer (wind speed)
• Flow regime (laminar or turbulent)
  • Turbulent flow results in higher convective heat transfer coefficients due to increased mixing and disruption of the thermal boundary layer (smoke from a cigarette vs. a chimney)
• Geometry of the surface
  • The shape and orientation of the surface affect the flow patterns and the development of the thermal boundary layer (flat plate vs. cylinder)
• Surface roughness and coatings
  • Rough surfaces or the presence of surface coatings can impact the convective heat transfer coefficient by altering the flow characteristics near the surface (smooth vs. textured surfaces)

Temperature Difference

• The temperature difference between the surface and the fluid drives the convective heat transfer process
• Larger temperature differences result in higher heat transfer rates (hot coffee vs. iced tea)

Applying Newton's Law of Cooling

Calculating Heat Transfer Rate

• Determine the convective heat transfer coefficient ($h$), surface area ($A$), surface temperature ($Ts$), and ambient fluid temperature ($T∞$)
• Substitute these values into the equation: $q = hA(Ts - T∞)$
• Example: Cooling a hot metal plate in air
  • Given: $h = 20 W/m²·K$, $A = 0.5 m²$, $Ts = 350 K$, $T∞ = 300 K$
  • Calculate: $q = 20 W/m²·K × 0.5 m² × (350 K - 300 K) = 500 W$

Calculating Temperature Change

• When the heat transfer rate ($q$) is known, the temperature change of an object can be calculated using the equation: $q = mc(dT/dt)$
  • $m$ = mass of the object (kg)
  • $c$ = specific heat capacity (J/kg·K)
  • $dT/dt$ = rate of temperature change (K/s)
• Example: Cooling a hot steel ball in water
  • Given: $q = -200 W$, $m = 1 kg$, $c = 500 J/kg·K$
  • Calculate: $dT/dt = q / (mc) = -200 W / (1 kg × 500 J/kg·K) = -0.4 K/s$

Lumped Capacitance Method

• Simplifies the analysis of convective cooling problems when the Biot number ($Bi = hL/k$) is less than 0.1
  • $L$ = characteristic length (m)
  • $k$ = thermal conductivity of the object (W/m·K)
• Assumes a uniform temperature distribution within the object
• Enables the use of a simplified version of Newton's Law of Cooling: $dT/dt = -(hA/mc)(T - T∞)$

Convective Heat Transfer Coefficient

Definition and Units

• Measure of the effectiveness of heat transfer between a surface and a fluid
• Expressed in units of W/m²·K
• Represents the rate of heat transfer per unit area per unit temperature difference between the surface and the fluid

Dependence on Fluid Properties and Flow Conditions

• Thermal conductivity
  • Higher thermal conductivity of the fluid leads to a higher convective heat transfer coefficient (water vs. air)
• Density, viscosity, and specific heat capacity
  • Affect the fluid's ability to transport and store heat, influencing the convective heat transfer coefficient
• Fluid velocity
  • Increased fluid velocity typically enhances the convective heat transfer coefficient by promoting mixing and reducing the thickness of the thermal boundary layer (natural vs. forced convection)
• Flow regime (laminar or turbulent)
  • Turbulent flow generally results in higher convective heat transfer coefficients due to increased mixing and disruption of the thermal boundary layer (smooth vs. rough pipe flow)

Empirical Correlations

• Dittus-Boelter equation for turbulent flow in pipes
  • $Nu = 0.023 Re^{0.8} Pr^n$
  • $Nu$ = Nusselt number (dimensionless)
  • $Re$ = Reynolds number (dimensionless)
  • $Pr$ = Prandtl number (dimensionless)
  • $n = 0.4$ for heating and $0.3$ for cooling
• Churchill-Bernstein equation for external flow over a flat plate
  • $Nu = 0.3 + \frac{0.62 Re^{0.5} Pr^{0.33}}{\left[1 + \left(\frac{0.4}{Pr}\right)^{0.67}\right]^{0.25}} \left[1 + \left(\frac{Re}{282,000}\right)^{0.625}\right]^{0.8}$
• These correlations estimate the convective heat transfer coefficient based on fluid properties, flow conditions, and geometry