Derive the general heat conduction equation in cylindrical coordinates by applying the first law to the volume element.

For transient situations with no heat generation and azimuthal symmetry, see the derivation of Eq. 2-28 at

http://wwwme.nchu.edu.tw/Enter/html/lab/lab516/Heat%20Transfer/chapter_2.pdf

In this case, T depends upon time (t) and r only.

There is a more complicated equation where there are variations with length z and aximutual angle phi.

Why did the heat conduction equation go to therapy? Because it had some issues with its identity crisis in different coordinate systems!

Now, let's get serious (sort of). To derive the general heat conduction equation in cylindrical coordinates, we start with the first law of thermodynamics applied to a volume element.

The first law of thermodynamics states that the change in internal energy (dU) of a system is equal to the heat transfer (dQ) into the system minus the work (dW) done by the system:

dU = dQ - dW

In the case of heat conduction, there is negligible work done by the system. So, we can ignore the term (dW) and focus only on the heat transfer (dQ).

Now, let's examine a volume element in cylindrical coordinates. We assume there is a steady-state heat conduction, which means the temperature does not vary with time. This allows us to neglect the time derivative term.

In cylindrical coordinates, the volume element has dimensions dr, dz, and r dθ. The temperature at a point in the volume element can be described as T(r, θ, z). The heat flow through the sides of the volume element can be represented by qᵣ, qᵣ + Δr, qₜ, qₜ + Δθ, qz, and qz + Δz.

Now, let's apply the first law of thermodynamics to this volume element:

dU = dQ - dW

Since the heat transfer (dQ) is given by Fourier's law of heat conduction, which states that heat transfer is proportional to the temperature gradient, we can write:

dQ = -k(r, θ, z) * (∂T/∂r) * r dθ dz dr

The negative sign indicates that heat is transferred from higher to lower temperatures. k(r, θ, z) is the thermal conductivity, which can vary with position in the cylindrical coordinate system.

Now, let's integrate the heat flow terms over each side of the volume element:

qᵣ(r, θ, z) - qᵣ + Δr(r + Δr, θ, z) = -k(r, θ, z) * (∂T/∂r) * r dθ dz dr

qₜ + Δθ(r, θ + Δθ, z) - qₜ(r, θ, z) = 0 (no heat flow along the θ direction)

qz + Δz(r, θ, z + Δz) - qz(r, θ, z) = -k(r, θ, z) * (∂T/∂z) * r dθ dz dr

Now, let's apply some cunning tricks of mathematics, like taking the limit as the size of the volume element approaches zero (Δr, Δθ, and Δz go to zero).

Taking into account these limits, we can write the difference of the side heat flows as derivatives:

∂qᵣ/∂r * dr = -k(r, θ, z) * (∂T/∂r) * r dθ dz dr

∂qₜ/∂θ * r dθ = 0 (no heat flow along the θ direction)

∂qz/∂z * dz = -k(r, θ, z) * (∂T/∂z) * r dθ dz dr

Now, let's rearrange the equations and substitute the appropriate differentials:

∂qᵣ/∂r = -k(r, θ, z) * (∂T/∂r) * r

∂qₜ/∂θ = 0 (no heat flow along the θ direction)

∂qz/∂z = -k(r, θ, z) * (∂T/∂z) * r

Now, let's sum up these equations and combine them into a single equation:

∂qᵣ/∂r + ∂qₜ/∂θ + ∂qz/∂z = 0

This equation represents the general heat conduction equation in cylindrical coordinates. It states that the sum of the heat flow variations in each coordinate direction must be zero for a steady-state heat conduction.

Phew, that was quite a derivation! I hope you now have the equation stuck in your head like a catchy song.

To derive the general heat conduction equation in cylindrical coordinates, we will start by applying the first law of thermodynamics to a volume element.

Consider a cylindrical coordinate system, with coordinates r, θ, and z. Let's consider a volume element located at position (r, θ, z) with dimensions dr, dθ, and dz.

The first law of thermodynamics states that the change in internal energy (dU) of a system is equal to the heat added or removed (dQ) minus the work done by the system. In equation form, this can be written as:

dU = dQ - dW

Now, let's consider the energy transfer due to heat. The rate of heat conduction in the r-direction (q_r) can be expressed as the product of the thermal conductivity (k) and the temperature gradient in the r-direction (∂T/∂r):

q_r = -k * (∂T/∂r)

Similarly, heat conduction in the θ-direction (q_θ) and z-direction (q_z) are given by:

q_θ = -k * (1/r) * (∂T/∂θ)

q_z = -k * (∂T/∂z)

The negative signs in the equations indicate that heat conduction occurs from regions of higher temperature to regions of lower temperature.

Next, let's consider the volume element and its surrounding boundary. The heat flowing across the surfaces of the volume element in the r-direction is given by:

dq_r = q_r * 2πr * dz

Similarly, the heat flowing across the surfaces in the θ-direction is given by:

dq_θ = q_θ * r * dr * dz

The heat flowing across the surfaces in the z-direction is given by:

dq_z = q_z * 2πr * dr * dθ

Now, let's consider the change in internal energy of the volume element. This can be expressed as:

dU = ρ * cp * r * dr * dθ * dz * dT

where ρ is the density of the material, cp is the specific heat capacity, and dT is the change in temperature.

Finally, let's consider the work done by the system. In this case, we are assuming no external work, so dW is equal to zero.

Now, we can apply the first law of thermodynamics:

dU = dQ - dW

Substituting the expressions for dU, dQ, and dW, we get:

ρ * cp * r * dr * dθ * dz * dT = -(dq_r + dq_θ + dq_z) - 0

Simplifying the equation and rearranging terms, we arrive at the general heat conduction equation in cylindrical coordinates:

ρ * cp * ∂T/∂t = ∂/∂r (k * r * ∂T/∂r) + (1/r) * ∂/∂θ (k * ∂T/∂θ) + ∂/∂z (k * ∂T/∂z)

This equation represents the rate of change of temperature (∂T/∂t) in a cylindrical coordinate system, taking into account conduction in the r, θ, and z directions.

To derive the general heat conduction equation in cylindrical coordinates, we first apply the first law of thermodynamics to a small volume element within a cylindrical coordinate system.

Let's consider a cylindrical volume element with radius r, height dz, and thickness dr. The direction along the cylinder is denoted by the z-axis, and the radial direction is denoted by the r-axis.

The heat transfer across the surfaces in the r-direction (radial direction) can be expressed as follows:

Q_r = -k(dT/dr) 2πr dz

where Q_r is the heat transfer across the surfaces in the r-direction, k is the thermal conductivity of the material, T is the temperature, and (dT/dr) represents the temperature gradient along the radial direction.

Similarly, the heat transfer across the surfaces in the z-direction can be represented as:

Q_z = -k(dT/dz) πr^2 dr

where Q_z is the heat transfer across the surfaces in the z-direction.

To account for the heat being generated or absorbed within the volume element, we introduce a heat generation term, Q_g.

The net change in energy within the volume element can be given by:

dQ = ρcVdT + Q_g

where ρ is the density of the material, c is the specific heat capacity, V is the volume of the element, and dT is the change in temperature.

Now, let's apply the first law of thermodynamics to the volume element:

dQ = Q_r + Q_z + Q_g

Substituting the expressions for Q_r and Q_z, we get:

ρcVdT = -k(dT/dr) 2πr dz - k(dT/dz) πr^2 dr + Q_g

Next, we divide the equation by the volume element (V = πr^2 dz) to obtain a differential form:

ρc(dT/dt) = 1/r(d/dr)(kr(dT/dr)) + (1/r^2)(d^2T/dz^2) + Q_g

This is the general heat conduction equation in cylindrical coordinates, where (dT/dt) represents the change in temperature with respect to time t.

The equation can be further simplified depending on the specific conditions and assumptions made, but this is the general form that describes heat conduction in cylindrical coordinate systems.