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How a temperature gradient develops

Explain how a temperature gradient develops over time and approaches a linear steady state in a uniform bar.

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  • University Physics Year 1
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Why this matters + quick links

This page gives the UY1 working model/result for Build up of temperature gradient. You reuse it when you turn thermal intuition into calculable models (energy balances, process reasoning, and kinetic/statistical links), with careful sign/units checks.

At a glance

  • Prerequisites: Fourier’s law, heat capacity, conservation of energy.
  • Outcomes: describe transient temperature evolution toward steady state.
  • Key result: temperature field evolves by a diffusion equation before becoming linear at steady state.
  • Common trap: assuming linear profile appears immediately after boundary temperatures are imposed.

Setup

Uniform rod of length L, area A, density ρ, specific heat c, conductivity k.

  • Initial condition: T(x,0) = T₀.
  • Boundary conditions: left end fixed at Tₕ, right end fixed at Tc (Tₕ > Tc).
  • Lateral surface insulated.

Physical interpretation:

  • The heat equation is a diffusion law: temperature “spreads” in time the way concentration spreads in diffusion.
  • Thermal diffusivity
    α = k/(ρ c)
    measures how fast a material equalizes temperature: large k helps spread heat, while large ρ c means the material stores lots of energy per kelvin (slower response).

Core derivation / explanation

Take a small slice dx. Energy balance:

ρ cA dx (∂ T)/(∂ t) = Q dot ₓ-Q dot _(x + dx).

Using Fourier law Q dot = -kA ∂ T/∂ x and Taylor expansion,

Q dot ₓ-Q dot _(x + dx) = kA((∂² T)/(∂ x²))dx.

Hence heat equation:

(∂ T)/(∂ t) = α(∂² T)/(∂ x²), α≡k/(ρ c).

At steady state, ∂ T/∂ t = 0, so

d²T/dx² = 0 ⇒ T(x) = ax + b,

which is linear between boundaries.

Units check:

[α] = (W m⁻¹K⁻¹)/(kg m⁻³ · J kg⁻¹K⁻¹) = m²s⁻¹.

Physical sanity check:

  • Larger α means faster approach to steady state.
  • If Tₕ = Tc, gradient vanishes and the rod becomes uniform in temperature.
Pitfalls (transient conduction)
  • The timescale τ∼ L²/α is an order-of-magnitude estimate. Exact solutions depend on boundary conditions and geometry. - Many real setups impose boundary temperatures through convection, not perfectly fixed end temperatures (see UY1: convection and UY1: Convective Heat Transfer).

Estimating the response time

Worked example 1

Rod response time

Problem

Estimate the diffusion response time of a uniform rod with L = 0.20 m and α = 1.0 × 10⁻⁵ m²s⁻¹.
Show full solution

  1. Use the diffusion length scale

    Method

    Estimate τ from L²/α.

    Reason

    Diffusion spreads over a distance proportional to the square root of elapsed time, so the time grows as the length squared.

    Working

    Answer

    τ∼L²/α = 0.20²/(1.0 × 10⁻⁵) = 4.0 × 10³ s.

  2. Interpret the estimate

    Working

    Answer

    τ∼4.0 × 10³ s ≈ 1.1 h. This is a characteristic diffusion timescale, not an exact time to reach steady state; the numerical factor depends on the boundaries and the chosen settling criterion.

Practice set

Check your understanding 1

For the same material and boundary conditions, how does the diffusion timescale change when the rod length doubles?

Show hint
Compare (2L)²/α with L²/α.
Show answer
It increases by a factor of four: τ ∝ L² at fixed α, so the ratio is 2² = 4.

Check your understanding 2

What mathematical condition on T(x,t) defines a steady temperature field?

Show hint
Steady means unchanged with time at each position.
Show answer
∂ T/∂ t = 0 everywhere. A spatial gradient can still exist; steady state does not mean thermal equilibrium.

Check your understanding 3

A uniform rod has insulated sides, fixed end temperatures and no internal heat source. Why can its temperature profile be curved during the transient but linear at steady state?

Show hint
Use ∂ T/∂ t = α∂²T/∂ x² in the two regimes.
Show answer
During the transient, nonzero curvature produces a local temperature change. At steady state the time derivative is zero, so d²T/dx² = 0 and T(x) is linear.

Summary + next steps

  • Temperature gradients are formed dynamically, not instantaneously.
  • Transient conduction is governed by a diffusion PDE.
  • Linear profile appears only after steady state is reached in uniform 1D conduction.

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Syllabus and review details

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