Knowledge card

Substituted Effective Diffusion

Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory

T⁻¹Θ

Symbol

\tilde{k}_{e} f f [Δ T_{0} x / (1 - γ t)^{2}] [c / (ν_{f} (1 - γ t))] θ ″

The substituted conduction and radiative-conduction block uses the independent prescribed thermal excess and the similarity curvature factor; it carries the common scale cΔT₀x/(1−γt)³.

Main equation

Level 0 of 12

Brief — named quantities

About levels

Why the levels load one at a time

Only level 0 travels with this article; the expansion runs to level 12. The complete expansion is several megabytes, so deeper levels are fetched one at a time when you ask for them.

To go deeper still, open any quantity in the reading as its own concept card — it carries its own derivation and its own depth ladder.

((\frac{k}{ρC p})_{hn f} (\frac{Δ T _{0} x}{( 1 - γ t ) ^{2}}) (\frac{c}{ν _{f} ( 1 - γ t )}) θ^{''} S1.1 + (\frac{1}{( ρC p ) _{hn f}}) (\frac{16 σ * T \infty ^{3}}{3 k *}) (\frac{Δ T _{0} x}{( 1 - γ t ) ^{2}}) (\frac{c}{ν _{f} ( 1 - γ t )}) θ^{''})

Dimension:T⁻¹Θ (s⁻¹ K)Maths:ℝ scalar(·, ·, ·) — A single number (real numbers), depending on another quantity, another quantity, another quantityDimensionally verifiedMathematically verifiedFully reduced to base quantities

In words:plus.

Open full derivation chain: 2 stepsDerivation

1
Open Step 1: Given / definition · The source relation this block transforms: conduction and radiation gathered as one effective-d…
The source relation this block transforms: conduction and radiation gathered as one effective-diffusion block — the Rosseland-linearised radiation acts as an extra conduction.Given / definition
Level 0 · more available
Brief — named quantities
About levels
Why the levels load one at a time
Only level 0 travels with this article. The complete expansion is several megabytes, so deeper levels are fetched one at a time when you ask for them — and how deep it goes is not known until you get there.
To go deeper still, open any quantity in the reading as its own concept card — it carries its own derivation and its own depth ladder.
How many levels remain is not yet known — the depth is discovered one level at a time.
$(k \nabla T S2.1 - q_{r})$
★
Open Step 2: Substitution · Substitute ∂²T/∂y²=(T_r−T∞)[c/(ν_f(1−γt))]θ″ with T_r−T∞=ΔT₀x/(1−γt)² into both diffusive fluxe…
Substitute ∂²T/∂y²=(T_r−T∞)[c/(ν_f(1−γt))]θ″ with T_r−T∞=ΔT₀x/(1−γt)² into both diffusive fluxes.Substitution Substitution · ∂²T/∂y²=(Tr−T∞)(c/(ν_f(1−γt)))θ″
Open operand record: Substitution ∂²T/∂y²=(Tr−T∞)(c/(ν_f(1−γt)))θ″
Substitution:L0 $\partial^{2} T / \partial y^{2} = (T r - T \infty) (c / (ν_{f} (1 - γ t))) θ ″$ L1 $\frac{\partial ^{2} T}{\partial y ^{2}} ? = (T r - T \infty) (\frac{c}{ν _{f} ( 1 - γ t )}) θ^{''}$
Open term-change ledger: 2 records
Open record 1: k∇T → (k/(ρCp))_hnf(ΔT₀x/(1−γt)²)(c/(ν_f(1−γt)))θ″
Consumes:L0 $k \nabla T$ Produces:L0 $(k / (ρC p))_{h} n f (Δ T_{0} x / (1 - γ t)^{2}) (c / (ν_{f} (1 - γ t))) θ ″$ The temperature-curvature map substitutes Fourier conduction: the effective diffusivity times the wall excess times the similarity ratio times θ″.(k/(ρCp))_hnf(ΔT₀x/(1−γt)²)(c/(ν_f(1−γt)))θ″
Open record 2: q_r → (1/(ρCp)_hnf)(16σ*T∞³/3k*)(ΔT₀x/(1−γt)²)(c/(ν_f…
Consumes:L0 $q_{r}$ Produces:L0 $(1/ (ρC p)_{h} n f) (16 σ * T \infty^{3} /3 k *) (Δ T_{0} x / (1 - γ t)^{2}) (c / (ν_{f} (1 - γ t))) θ ″$ The linearised Rosseland flux divergence substitutes through the same curvature map — identical θ″ structure with the radiative conductivity 16σ*T∞³/(3k*); the linearisation folds its sign into the diffusion block.(1/(ρCp)_hnf)(16σ*T∞³/3k*)(ΔT₀x/(1−γt)²)(c/(ν_f(1−γt)))θ″
Level 0 · more available
Brief — named quantities
About levels
Why the levels load one at a time
Only level 0 travels with this article. The complete expansion is several megabytes, so deeper levels are fetched one at a time when you ask for them — and how deep it goes is not known until you get there.
To go deeper still, open any quantity in the reading as its own concept card — it carries its own derivation and its own depth ladder.
How many levels remain is not yet known — the depth is discovered one level at a time.
$((\frac{k}{ρC p})_{hn f} (\frac{Δ T _{0} x}{( 1 - γ t ) ^{2}}) (\frac{c}{ν _{f} ( 1 - γ t )}) θ^{''} S1.1 + (\frac{1}{( ρC p ) _{hn f}}) (\frac{16 σ * T \infty ^{3}}{3 k *}) (\frac{Δ T _{0} x}{( 1 - γ t ) ^{2}}) (\frac{c}{ν _{f} ( 1 - γ t )}) θ^{''})$

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Domain Analysis

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Step-by-step unit check (2 steps)

Step 1 given
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Step 2 substitution
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Reads as: k̃_eff[ΔT₀x/(1−γt)²][c/(ν_f(1−γt))]θ″

“k̃_eff[ΔT₀x/(1−γt)²][c/(ν_f(1−γt))]θ″” is analysed by expanding its full defining equation (its children and their relations), not as a bare symbol. Use Check this concept’s units above for the same result.

Mathematical Analysis

Object type, lawful operations, conditions, comparisons, and derivation-step checks.

Check one step at a time (2 steps)

Step 1 — given

Step 2 — substitution

Essence
S1The substituted effective diffusion: conduction and radiative conduction, both proportional to θ″.
S1.1essenceThe sum of (k/(ρCp))_hnf(ΔT₀x/(1−γt)²)(c/(ν_f(1−γt)))θ″ and (1/(ρCp)_hnf)(16σ*T∞³/3k*)(ΔT₀x/(1−γt)²)(c/(ν_f(1−γt)))θ″.

Dimension I

Definition

What it is — and what it is not

What it is: S2The effective thermal diffusion: conduction plus the Rosseland-linearised radiative conduction, one diffusion block.
S2.1essencek∇T reduced by q_r.

Dimension II

In practice

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Encyclopedia

Substituted Effective Diffusion

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