Integro-partial differential equation

In summary, the integro-partial differential equation is derived using the energy balance equation and the definition of the kernel in terms of ξ. It is a governing equation for simultaneous conduction-radiation heat transfer within a cylinder, taking into account the assumptions of neglecting convection, assuming blackbody radiation, and modeling heat transfer in only one direction. The model geometry is also shown in Figure 5-7.
  • #1
mehadi06me
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Hello brothers
Anyone please help me derivation of this integro-partial differential equation (attached below).This is a governing equation for simultaneous conduction-radiation heat transfer within a cylinder.The figure is attached below.where ρ is the density, Cp is the heat capacity, κ is the thermal conductivity, σ is Stefan’s constant (the Stefan-Boltzmann constant), ε is the emissivity, and k(x, x') is the kernel corresponding to the radiation view factor. This equation arises in the physical description of 1D heat conduction and radiation along a pipe. Figure 5-7 shows the model geometry.
Before setting up the model, make the following assumptions:
• Inside the tube, neglect convection and consider only radiation and conduction.

• Assume blackbody radiation with ε = 1.

• Model heat transfer only in the x direction (assume θ symmetry).

• The pipe’s outer wall is perfectly insulated so that no heat escapes to the outside world by either radiation or conduction.

The definition of the kernel k(x, x') is given as a function of ξ (attached below)

where ξ = | x − x' |/ Di ;Help me please...
 

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  • #2
The derivation of the integro-partial differential equation can be done by applying the energy balance at each point x along the length of the cylinder. The energy balance equation is given by:ρ Cp ∂T/∂t + ∂/∂x(κ∂T/∂x) = σε (T^4-T_0^4) + ∫ k(x,x')T(x')dx' where T(x) is the temperature at any point x, T_0 is the ambient temperature, and k(x,x') is the kernel corresponding to the radiation view factor. The first term on the right-hand side of the equation represents the heat transfer due to radiation, while the second term represents the heat transfer due to conduction.To solve this equation, we must first express the kernel in terms of ξ, as given in the definition of k(x,x'). The integral of k(x,x') can then be expressed as:∫ k(x,x')T(x')dx' = ∫ k(ξ)T(x-ξDi)dξ Substituting this expression into the energy balance equation yields the required integro-partial differential equation:ρ Cp ∂T/∂t + ∂/∂x(κ∂T/∂x) = σε (T^4-T_0^4) + ∫ k(ξ)T(x-ξDi)dξ
 

1. What is an integro-partial differential equation?

An integro-partial differential equation is a type of mathematical equation that involves both partial derivatives and integrals. It is used to describe the relationship between an unknown function and its derivatives, taking into account both local and non-local effects.

2. What is the difference between an integro-partial differential equation and a regular partial differential equation?

The main difference between an integro-partial differential equation and a regular partial differential equation is the presence of integrals. In a regular PDE, the unknown function and its derivatives are related only through partial derivatives, while in an integro-PDE, integrals are also involved, making it a more complex equation to solve.

3. What are some real-world applications of integro-partial differential equations?

Integro-PDEs have a wide range of applications in various fields, including physics, engineering, finance, and biology. They are used to model complex systems such as fluid flow, heat transfer, and population dynamics, among others.

4. How are integro-partial differential equations solved?

Solving integro-PDEs is a challenging task, and there is no general method that can be applied to all types of equations. However, there are various numerical methods, such as finite difference, finite element, and spectral methods, that can be used to approximate the solution.

5. What are some open research problems in the field of integro-partial differential equations?

There are still many open research problems in the field of integro-PDEs, including the development of efficient and accurate numerical methods, the study of their long-time behavior and stability, and the analysis of their sensitivity to initial and boundary conditions. Additionally, there is ongoing research on the application of integro-PDEs to new areas, such as machine learning and data science.

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