Energy Balance in Lumped Systems

Questions and Answers

What conditions make the assumption of a lumped system valid?

When the medium has high conductivity and is small or well mixed (correct)

When the temperature is affected by significant exothermic reactions

When the medium is poorly mixed and has low conductivity

When the temperature varies significantly with position

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

Disturbances from other processes

Consistent cooling water temperatures (correct)

Batch processes

Start up and shut down of processes

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

110 K

140 K

121 K (correct)

100 K

Which aspect is characteristic of transient heat transfer?

Temperature varies with time and not with position Signup and view all the answers

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

Temperature fluctuations due to changes in cooling water temperature affected by weather Signup and view all the answers

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

3500 J/kg Signup and view all the answers

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

Q = U A \delta T Signup and view all the answers

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

12 K Signup and view all the answers

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

150 W/(m2K) Signup and view all the answers

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

1000 kg/m3 Signup and view all the answers

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

Specific heat capacity of the solid Signup and view all the answers

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

𝑄 = 𝑚𝑐𝑝 (𝑇 𝑡 - 𝑇𝑖) Signup and view all the answers

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

It increases given a constant temperature difference. Signup and view all the answers

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

Ambient or surrounding temperature Signup and view all the answers

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

It quantifies the rate of heat transfer per unit area. Signup and view all the answers

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

Convection heat transfer Signup and view all the answers

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

Rate of change of temperature Signup and view all the answers

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

Surface area of the solid Signup and view all the answers

Study Notes

Energy Balance in a Lumped System

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∞).

Heat Transfer Dynamics

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.

Key Formulas

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).

Overall Heat-transfer Coefficient

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).

Example Application: Cooling a Tank

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.

Worked Example Solution Insight

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 Characteristics

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.

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Description

This quiz focuses on the energy balance equations for solid bodies in lumped systems. Participants will explore the relationships between temperature, time, and other parameters in the context of heat transfer. Understanding these concepts is crucial for applications in thermal analysis and engineering.