Conducción de Calor en Estado Transitorio y Número de Biot

Analysis of Transient Heat Conduction Systems

Understanding Transient vs. Steady-State Conditions

• The analysis of heat conduction in transient states assumes that temperature changes over time, contrasting with steady-state conditions where variables remain constant.

• In transient systems, temperature is a function of both position (x, y, z in Cartesian coordinates or radius and angles in cylindrical coordinates) and time, complicating the problem significantly.

• A differential equation can involve up to four independent variables due to the multivariable nature of temperature dependence on both space and time.

Example: Temperature Simulation

• An example simulation shows how the temperature of a pot subjected to heat transfer evolves over time; initially at ambient temperature (~20°C), it rises as heat is applied.

• At 38 minutes into the simulation, temperatures range from 50°C to 60°C across different points on the pot, illustrating spatial variation in temperature.

Concentrated System Assumption

• Some systems are simplified by assuming they behave as concentrated systems where temperature does not vary with position; for instance, a cup maintains a constant temperature (T1).

• This assumption reduces complexity by eliminating spatial derivatives from the heat conduction equations since all points have the same temperature.

Heat Transfer Dynamics

• When considering thermal treatment processes like heating steel bars in an oven at constant temperatures for extended periods, these materials reach thermal equilibrium.

• Upon removal from the oven, their uniform initial temperatures will change over time as they interact with their environment.

Convection vs. Conduction

• For concentrated systems assumed to have uniform internal temperatures (e.g., T1), heat transfer primarily occurs through convection rather than conduction when interacting with surrounding media.

• The effectiveness of convective heat transfer must exceed conductive transfer for this analysis to hold true; this relationship is quantified using a dimensionless number known as Biot number.

Mathematical Modeling of Heat Transfer

• To model these systems mathematically: consider solid bodies characterized by mass (m), volume (V), density (ρ), and an initial temperature that varies over time until reaching a final state dependent on time.

Heat Transfer Analysis

Understanding Temperature Differences in Heat Transfer

• The analysis begins by considering the surrounding temperature, which is assumed to be greater than the object's temperature over time. This approach remains valid even if the surrounding temperature is lower, as it only alters the direction of heat transfer.

• An energy balance equation is introduced, stating that the heat transfer to an object at a specific moment must equal the increase in energy within that object during that same moment.

Newton's Law of Cooling and Heat Transfer Equations

• The heat transfer by convection is described using Newton's law of cooling: H cdot A cdot (T_textmedium - T_textbody) , where T changes over time. This equals the heat absorbed by conduction, represented as m cdot C_p cdot dT .

• The mass of the object can be expressed in terms of density and volume ( m = rho V ). This substitution allows for a clearer formulation involving temperature gradients.

Deriving a Differential Equation

• By substituting mass with density times volume into the previous equations, a single differential equation emerges with time and temperature as its variables.

• Rearranging this equation leads to isolating dt , allowing for separation of variables. Constants such as convective coefficient H , area, density, and specific heat capacity C_p are acknowledged.

Integration and Solving for Temperature Over Time

• The derived equation can be integrated to find a function relating temperature to time. It involves manipulating terms so that integration becomes feasible.

• The integral results in a logarithmic expression representing how temperature evolves over time relative to initial conditions.

Final Formulation and Exponential Representation

• After integrating, constants are factored out leading to an expression involving natural logarithms.

• The final form shows how changes in temperature relate through exponential functions rather than logarithmic ones for easier interpretation.

Understanding Time Constants in Thermal Systems

The Concept of Time Constant (B)

• The time constant, denoted as B, is a crucial factor in systems where variables change over time, such as gas density and volumetric expansibility in solids.

• B has units of inverse seconds (s⁻¹), which can be derived from dimensional analysis. This indicates that the product of B and time (t) must be dimensionless.

Exponential Behavior of Temperature

• The temperature change in a system can be predicted using the time constant; different values of B lead to varying rates at which temperature approaches equilibrium.

• As time progresses, the temperature approaches an asymptotic value (T∞), indicating that higher values of B result in faster thermal equilibrium.

Heat Transfer Calculations

• To calculate total heat transfer during transient states, Newton's law of cooling is applied. Heat transfer varies with time until it stabilizes at steady-state conditions.

• The rate of heat transfer depends on the convective coefficient, surface area, and temperature difference between the body and its environment.

Maximum Heat Absorption

• The maximum heat a body can absorb is limited by its ambient temperature (T∞). This relationship emphasizes that greater initial temperatures yield more significant heat absorption potential.

Applicability Criteria: Biot Number

• The validity of assumptions regarding uniform temperature within solid bodies is assessed using the Biot number—a dimensionless criterion based on characteristic length.

• Named after physicist Jean-Baptiste Biot, this number helps determine whether conduction or convection dominates heat transfer processes.

Characteristic Length Calculation

• Characteristic length (lc) is defined as volume divided by surface area. It varies for different geometries: thickness for flat plates, radius/2 for cylinders, and radius/3 for spheres.

Understanding Biot Number Dynamics

• The Biot number compares convective resistance to conductive resistance within a body. A low Biot number indicates minimal internal temperature gradients and efficient heat transfer.

• High resistances to conduction or convection hinder overall heat transfer efficiency. Thus, understanding these resistances aids in validating thermal assumptions made during analysis.

Understanding the Biot Number and Heat Transfer in Spheres

The Concept of Concentrated Bodies

• The validity of using a concentrated body assumption is discussed, particularly when the temperature gradient is negligible. This approximation holds true if the Biot number is less than or equal to 0.1.

• An example involving a copper sphere removed from an oven illustrates heat transfer principles. The conductivity of copper is noted as 400 W/m°C, emphasizing its efficiency in conducting heat.

Heat Transfer Mechanisms

• The discussion highlights that heat transfer occurs through both convection with air and conduction within the sphere itself. Determining whether the system can be treated as a concentrated body depends on calculating the Biot number.

• The characteristic length for a sphere is defined as its radius divided by 3 or its diameter divided by 6, which are crucial for further calculations.

Calculating Characteristic Length and Biot Number

• To find the characteristic length, it’s calculated using volume and surface area formulas: volume = (diameter³)/6 and area = π(diameter²). This results in a characteristic length of 0.02 meters for this specific case.

• The Biot number calculation involves multiplying the convective heat transfer coefficient (15 W/m²°C) by the characteristic length and dividing by thermal conductivity (401 W/m°C). This dimensionless number helps assess whether assumptions about heat transfer can be applied.

Validity of Assumptions

• After performing dimensional analysis, it confirms that units cancel appropriately, leading to a valid dimensionless result for the Biot number calculation.