Energy Balance in Lumped Systems

What conditions make the assumption of a lumped system valid?

Which of the following is NOT a factor that can lead to transient heat transfer?

What is the temperature difference calculated for the uranium element based on the provided rate of heat transfer?

Which aspect is characteristic of transient heat transfer?

Which is an example of a situation leading to transient heat transfer?

What is the specific heat capacity of the material in the kettle?

What is the formula for calculating the rate of heat transfer (Q)?

If the kettle's temperature is reduced from 55oC to 43oC, what is the temperature difference (ΔT)?

What is the overall heat-transfer coefficient (U) given for the system?

What is the density ( ho) of the material in the kettle?

What does 𝑐𝑝 represent in the context of energy balance in a lumped system?

From the equation for the total amount of heat transfer, which of the following expressions accurately describes the relationship?

What happens to the total heat transfer (𝑄) as the specific heat capacity (𝑐𝑝) increases?

In the equation for maximum heat transfer 𝑄𝑚𝑎𝑥 = 𝑚𝑐𝑝 (𝑇∞ − 𝑇𝑖), what does the term 𝑇∞ represent?

What is the significance of the heat-transfer coefficient in heat transfer?

The relationship 𝑓(𝑇 𝑡 − 𝑇∞) = ℎ𝐴𝑠(𝑇 𝑡 − 𝑇∞) represents what principle in heat transfer?

What does the parameter 𝑏 in the equation 𝑇 𝑡 - 𝑇∞ = 𝑒^{-𝑏𝑡} signify?

Which factor does NOT directly influence the specific heat capacity (𝑐𝑝) of a solid?

Energy balance for a solid body considers temperature changes over time, defined by the equation involving heat transfer rate.
Mass (m) is calculated as the product of density (ρ) and volume (V).
The differential temperature change (dT) is expressed as the difference between the temperature (T) and the ambient temperature (T∞).

Integration of the energy balance from initial conditions leads to a relationship defining the temperature at any time (T_t).
The heat transfer rate is modeled by Newton's Law of Cooling, where Q̇_t represents the rate of heat transfer between the solid and its environment.

Total heat transfer: Q = m c_p (T_t - T_i) where T_i is the initial temperature.
Maximum possible heat transfer: Q_max = m c_p (T∞ - T_i).

Heat flux (q) is proportional to the temperature difference (ΔT), with U representing the overall heat-transfer coefficient: q = U ΔT and Q = U A ΔT.
Key variables include heat flux (q in W/m²), rate of heat transfer (Q in W), area (A in m²), and temperature difference (ΔT in K).

A kettle cooling from 55°C to 43°C involves parameters like specific heat capacity (c_p = 3500 J/kg), density (ρ = 1000 kg/m³), and a known overall heat transfer coefficient (U = 150 W/(m²K)).
The interfacial area through which heat flows is given as A = 6 m², with cooling water at a constant temperature of 35°C.

Heat release rate calculated: 0.25×10^6 W results in 50,000 W/m for a 5 m long system.
Temperature gradient across materials is determined using the known heat transfer parameters.

Transient heat transfer is influenced by time-varying temperature conditions due to various operational factors like batch processes, disturbances, and automatic controls.
A lumped system assumes uniform temperature change over time without spatial variation, valid in conditions where materials are highly conductive or well-mixed.