Lumped System Analysis
- Interior temperatures of somebodies remain essentially uniform at all times during a heat transfer process.
- The temperature of such bodies is only a function of time, T = T(t).
- The heat transfer analysis based on this idealization is called lumped system analysis.
Consider a body of the arbitrary shape of mass m, volume V, surface area A, density ρ and specific heat Cp initially at a uniform temperature Ti.

- At time t = 0, the body is placed into a medium at temperature T∞ (T∞ >Ti) with a heat transfer coefficient of h. An energy balance of the solid for a time interval dt can be expressed as:
heat transfer into the body during dt = the increase in the energy of the body during dt
h A (T∞ – T) dt = m Cp dT - With m = ρV and change of variable dT = d(T – T∞):
- Integrating from t = 0 to T = Ti
- Using the above equation, we can determine the temperature T(t) of a body at time t, or alternatively, the time t required for the temperature to reach a specified value T(t).
Note that the temperature of a body approaches the ambient temperature T∞ exponentially.
- A large value of b indicates that the body will approach the environment temperature in a short time.
- b is proportional to the surface area but inversely proportional to the mass and the specific heat of the body.
- The total amount of heat transfer between a body and its surroundings over a time interval is: Q = m Cp [T(t) – Ti] The behavior of lumped systems can be interpreted as a thermal time constant as shown in fig. below:
- Where Rt is the resistance to convection heat transfer and Ct is the lumped thermal capacitance of the solid. Any increase in Rt or Ct will cause a solid to respond more slowly to changes in its thermal environment and will increase the time response required to reach thermal equilibrium.
The criterion for Lumped System Analysis
- Lumped system approximation provides great convenience in heat transfer analysis. We want to establish a criterion for the applicability of the lumped system analysis. A characteristic length scale is defined as:
Lc= V/A - A nondimensional parameter, the Biot number, is defined:
- The Biot number is the ratio of the internal resistance (conduction) to the external resistance to heat convection.
- Lumped system analysis assumes a uniform temperature distribution throughout the body, which implies that the conduction heat resistance is zero. Thus, the lumped system analysis is exact when Bi = 0.
- It is generally accepted that the lumped system analysis is applicable if Bi≤ 0.1
- Therefore, small bodies with high thermal conductivity are good candidates for lumped system analysis. Note that assuming h to be constant and uniform is an approximation
Fourier number:
Fourier number is given by:
- The temperature of a body in the unsteady state can be calculated at any time only when the Biot number < 0.1.
Characteristic Length: Characteristic length is denoted by lc.
Characteristic Length for different Sections
Transient Conduction in Large Plane Walls, Long Cylinders, and Spheres
- The lumped system approximation can be used for small bodies of highly conductive materials.
- But, in general, the temperature is a function of position as well as time.
- Consider a plane wall of thickness 2L, a long cylinder of radius r0, and a sphere of radius r0 initially at a uniform temperature Ti
- We also assume a constant heat transfer coefficient of h and neglect radiation. The formulation of the one‐dimensional transient temperature distribution T(x,t) results in a partial differential equation (PDE), which can be solved using advanced mathematical methods. For the plane wall, the solution involves several parameters:
T = T (x, L, k, α, h, Ti, T∞)
where α = k/ρCp.
- By using dimensional groups, we can reduce the number of parameters.
Θ=Θ(x, Bi,τ) - To find the temperature solution for the plane wall, i.e. Cartesian coordinate, we should solve Laplace’s equation with boundary and initial conditions:
So, we can write:
where,
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