Heat transfer is an important concept in engineering and science, involving three main types: conduction, convection, and radiation. In steady-state heat transfer, the temperature remains constant over time. However, in unsteady or transient heat transfer, the temperature changes with time as energy moves within a system.
This type of heat transfer is common in real-life situations like cooling electronic devices or heating materials in factories. To study how heat spreads over time, we use methods like lumped parameter analysis, finite difference techniques, or numerical models like the finite volume method, depending on the system's shape and size
In this article, I will discuss :
- Lumped System Analysis : Mathematical tools required for unsteady heat transfer.
a. Biot Number
b. Nusselt Number- Characteristics Length (Lc)
a. Type of Characteristics Length Measurement- Lumped Parameter System and Time Constant (τ)
a. Case 1 : Body is surrounded by heat source
b. Case 2 : Body is surrounded by heat sink- Fourier Number
- Illustration with Solution
- Bonus Point : Misinterpreting d in Reynolds Number
1. Lumped System Analysis
A lumped system in heat transfer is a simplified analysis method that assumes the temperature of an object is uniform throughout its volume at any given time.
Lumped System Analysis is Applicable When:
- The body has an infinite thermal conductivity
- The body has zero thermal resistance to heat conduction
- The biot number (Bi) is less than or equal to 0.1
1.1 Biot Number
After reading 3rd point in condition for applying lumped system analysis, one question in your mind may arise that : " What is biot number ? What's need of introducing this term in unsteady state of heat transfer ? "
It is dimensionless parameter that compares the rate of heat conduction within an object to the rate of heat convection between the objects and its surrounding.
When the biot number is small (Bi <0.1), the temperature gradient within system is negligible, validating lumped system approach.
We can defined Biot number mathematically such as :
where:
- Lc represents the characteristic length
- k is the thermal conductivity
- h is the heat transfer coefficient
- A represents the surface area
1.2 Nusselt Number
The Biot number is defined for solids only. However, when dealing with fluids instead of solids, we can characterise heat transfer properties using the Nusselt number (Nu).
The Nusselt number is a dimensionless parameter that measures the ratio of convection heat transfer to conduction heat transfer in a fluid. It can be expressed mathematically as:
where:
- hx is the heat transfer coefficient
- Lc is the characteristic length
- kf is the thermal conductivity of the fluid
2. Characteristics Length
Characteristic length is a measure used to compare the heat transfer abilities of different materials, objects or shapes. The characteristic length (Lc) helps engineers measure and compare how heat moves through different shapes and materials.
A larger characteristic length typically means slower heat transfer, assuming the same thermal conductivity. That's why thin materials transfer heat more efficiently than thick ones.
Types of Characteristic Length Measurements
a. For External Flow (Outside the object) :
- Lc is measured along the direction the fluid flows
- For a cylinder in cross-flow, use the diameter (Lc = d)
b. For Internal Flow (Inside of the object) :
- Lc for internal flow is equal to cross-sectional area divided by perimeter.
- This represents how efficiently heat can transfer through the flow passage.
c. For Larger Wall :
- Characteristics length (Lc) for large wall is its thickness.
3. Lumped Parameter System
Case 1 : Body is surrounded by heat source
for lumped system, T = T(t) as temperature gradient for object is negligible (no change of temperature throughout its volume)
According to equation of conservation of energy,
at any time (t),
Integrating on both sides, we get
Apply boundary layer condition :
from equation (1) and (2),
writing equation(3) in term of characteristics length (Lc) :
Time Constant (τ)
The thermal time constant (τ) is a fundamental concept in unsteady state heat transfer, studied as a essential parameter for defining and understanding chemical engineering systems. This time constant characterises how quickly a system responds to temperature changes.
Time constant can be defined as the time taken by a lumped body to change temperature by 63.2% of its initial temperature differences (T∞ - Ti).
The temperature change in a lumped system follows an exponential relationship given by:
where:
- ρ = density
- Lc = characteristic length
- c = specific heat capacity
- h = heat transfer coefficient
At t = τ, the temperature difference reduces to:
θ= θi / e = 0.368 θi
This means that when time equals the thermal time constant, the temperature difference has decreased to 36.8% of its initial value, or the system has completed 63.2% of its total temperature change.
The graph plots temperature (T) against time (t), showing:
Temperature Parameters
- T∞: The ambient or final temperature (upper horizontal asymptote)
- Ti: Initial temperature
- θi: Initial temperature difference (T∞ - Ti)
The response curve shows how the temperature approaches the final value of ambient temperature asymptotically. As time pass, excess temperature (T∞ - T = θ) decrease.
The graphical representation at two distinct temperature response curves (labeled 1 and 2) that demonstrate the fundamental behaviour of systems with different thermal time constants (τ₁ and τ₂).The curves demonstrate that τ₁ < τ₂, where:
- Curve 1 (blue) reaches the target temperature more rapidly
- Curve 2 (pink) shows a slower approach to the final temperature
Conclusion : Systems with smaller time constants respond faster to temperature changes than those with larger time constants.
Case 2 : Body is surrounded by heat sink
So, according to equation of conservation of energy :
Integrating on both sides,
where:
- θ = T - T∞ (temperature difference)
- h = heat transfer coefficient
- A = surface area
- m = mass of body
- c = specific heat capacity
At t = τ, the temperature difference reduces to:
Graph plotted between temperature (T) and time (t) for body surrounded by heat sink
- The cooling curve shows an exponential decay means rate of temperature change (dT/dt) decreases over time.
- At t = τ, the temperature difference reduces to 36.8% of its initial value.
Comparative Analysis of Heat Transfer Phenomena: Source (Case 1) vs. Sink (Case 2)
Heat Source | Heat Sink | |
Process | Heating of body | Cooling of body |
Excess Temperature | θ = T∞ − T (positive) | θ = T - T∞ (positive) |
Initial Excess Temperature | θ = T∞ − Ti (positive) | θ = Ti - T∞ (positive) |
Equation of conservation of energy | For heating a body with no internal heat generation, this simplifies to: | For heating a body with no internal heat generation, this simplifies to: |
4. Fourier Number
A Fourier number is a dimensionless quantity that is used to describe the relationship between the rate of heat conduction and the rate at which heat is stored in material.
Conduction Rate:
For Heat Storage Rate:
from equation (5) and (6), Fourier number can be expressed mathematically as,
Also, Fourier number can be expressed in term of thermal diffusivity
Illustration
Misinterpreting d in Reynolds Number : Most Common Myth
When dealing with the Reynolds number (Re), an essential parameter in fluid dynamics and heat transfer, many students and professionals mistakenly assume that d, as the diameter of a pipe or cylinder in equation.
However, this is not correct! Let’s analyse what d truly represents and why this myth needs to be addressed.
What does d actually represent?
The term d is not exclusively the diameter but it represents the characteristic linear dimension of the system depending on the specific geometry and flow situation. Here are some examples to clarify:
- Flow Inside a Pipe or Tube :
For flow inside a pipe, d represents the diameter of the pipe, which is straightforward. - Flow Over a Flat Plate :
For flow over a flat plate, d represents the length of the plate along the direction of the flow. Here, the length determines how much of the plate the fluid has interacted with, which directly affects the boundary layer development and viscous forces. - Flow Past a Sphere or Cylinder :
For flow around a sphere or cylinder, d represents the diameter of the object. - Flow Through an Annular Space :
In the case of an annular space (e.g., flow between two concentric cylinders), d represents the hydraulic diameter: - Flow Over Complex Shapes :
For irregular or non-standard geometries, ddd is chosen based on the problem’s physical characteristics, like the chord length of an airfoil or edge length of a square duct.
Why Does This Misinterpretation Happen?
- Simplistic Teaching
In introductory fluid mechanics, problems often involve pipes or cylindrical objects, leading students to assume ddd always refers to diameter. The idea of "characteristic linear dimension" is not always stressed, leaving students unaware about application of Reynolds number to non-cylindrical geometries. - Formula Memorisation
Many students focus on memorising formulas without understanding their derivation or the physical meaning of each term.