Thermodynamics- Multidimensional and Transient Conduction

Question 1

What heat equation can be taken for a fluid running through a cylinder under steady state, no heat generation and axisymmetry?

Answer 1

d/dr(krdT/dr)=0

Question 2

Thermal resistance formula for conduction out of a tube radially

Answer 2

R=ln(r2/r1)/2πLk
Where L is length of pipe interested in
r is radius from centre of flowing fluid to boundary

Question 3

Formula for temperature distribution in the walls of a pipe

Answer 3

T(r)=((Ts1-Ts2)/ln(r1/r2))ln(r/r2)+Ts2
Where Ts1 is temperature of inner face of tube
Ts2 is temperature of outer face of tube
r1 and r2 follow same principle

Question 4

Solution methods to 2D conduction

Answer 4

Exact: like separation of variables
Graphical: like trapezium rule
Numerical: finite-difference/element or boundary-element method

Question 5

How does finite-difference method work?

Answer 5

It seems pointless.
Represent the physical system by a nodal network, grid or mesh. In 1D we solve along a line and break it up into points. Basically change in y over change in x for two points equidistant from the point you want. This is second order so the points are already derivatives.

Question 6

Approximate formula for finite-differences method

Answer 6

d/dx(kdT/dx)=k(T1-2T0+T-1)/Δx^2=0

Question 7

Transient conduction

Answer 7

A heat transfer process for which the temperature varies with time as well as location within a solid
dT/dt=kdT/dx
Where all ds are curly

Question 8

What does the lumped capacitance method assume about dT/dx (curly)?

Answer 8

It equals 0 so there is a spatially uniform temperature distribution

Question 9

What does the rate of change in energy stored equal in the lumped capacitance method?

Answer 9

ρVcdT/dt
Basically mCpdT/dt
This is all equal to energy in minus out plus generated which can come from radiation and convection

Question 10

Formula for Biot number

Answer 10

Bi=hLc/k
h is convective or radiative heat transfer coefficient
k is thermal conductivity
Lc is characteristic length of solid (V/As)

Question 11

What is the criterion for the applicability of the lumped capacitance method?

Answer 11

Bi«1
Means k is high so dT/dx is about 0 from:
dT/dx=-q/k

Question 12

With lumped capacitance method, what equation can be formed if negligible radiation and heat generation?

Answer 12

ρVcdT/dt=-hA(T-Tinf)

Question 13

Formula for thermal time constant

Answer 13

τt=(1/hA)ρVc
t is subscript
Basically thermal resistance times thermal capacitance

Question 14

Formula for change in thermal energy stored

Answer 14

ΔEst=-(ρVc)θ(1-e^(-t/τ))

θ is T-Tinf