Temperature of Body by Lumped Heat Capacity Method Calculator

Lumped Heat Capacity Method Formula:

\[ T = e^{\left( \frac{-h \cdot A_c \cdot \tau}{\rho_B \cdot c \cdot V} \right)} \cdot (T_0 - T_{\infty}) + T_{\infty} \]

Heat Transfer Coefficient (h):

W/m²·K

Surface Area for Convection (A_c):

m²

Time Constant (τ):

Density of Body (ρ_B):

kg/m³

Specific Heat Capacity (c):

J/kg·K

Volume of Object (V):

m³

Initial Temperature (T₀):

Bulk Fluid Temperature (T_∞):

Temperature at Time T (T):

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1. What is the Lumped Heat Capacity Method?

The Lumped Heat Capacity Method is a simplified approach to analyze transient heat conduction problems where the temperature gradient within the body is negligible. This method assumes that the entire body has a uniform temperature at any given time.

2. How Does the Calculator Work?

The calculator uses the Lumped Heat Capacity formula:

\[ T = e^{\left( \frac{-h \cdot A_c \cdot \tau}{\rho_B \cdot c \cdot V} \right)} \cdot (T_0 - T_{\infty}) + T_{\infty} \]

Where:

\( h \) — Heat transfer coefficient (W/m²·K)
\( A_c \) — Surface area for convection (m²)
\( \tau \) — Time constant (s)
\( \rho_B \) — Density of body (kg/m³)
\( c \) — Specific heat capacity (J/kg·K)
\( V \) — Volume of object (m³)
\( T_0 \) — Initial temperature of object (K)
\( T_{\infty} \) — Temperature of bulk fluid (K)

Explanation: The equation calculates the temperature of an object at any given time using exponential decay function, accounting for heat transfer properties and time.

3. Importance of Temperature Calculation

Details: Accurate temperature prediction is crucial for thermal analysis, cooling/heating system design, material processing, and various engineering applications where temperature control is essential.

4. Using the Calculator

Tips: Enter all parameters with appropriate units. Ensure all values are positive and within reasonable physical limits for accurate results.

5. Frequently Asked Questions (FAQ)

Q1: When is the lumped heat capacity method applicable?
A: This method is applicable when the Biot number (Bi) is less than 0.1, indicating that internal resistance to heat transfer is negligible compared to surface convection.

Q2: What are the limitations of this method?
A: The method assumes uniform temperature throughout the body, which may not be valid for large objects or materials with low thermal conductivity.

Q3: How does time constant affect temperature change?
A: A smaller time constant indicates faster temperature change, while a larger time constant means slower thermal response.

Q4: Can this method be used for cooling and heating?
A: Yes, the method works for both cooling (when T_∞ < T₀) and heating (when T_∞ > T₀) processes.

Q5: What units should be used for input parameters?
A: All parameters should be in SI units: meters, kilograms, seconds, Kelvin, Watts, and Joules for consistent results.