Chapter 4 Transient Heat Conduction Flashcards
What does lumped system analysis represent
when the interior temperature of a body remains essentially uniform at all times during heat transfer process
What does lumped system analysis allow
Temperature to be taken as a function of Time only T(t)
What could be represented as lumped system analysis
a small copper ball
What is the energy balance for a heated solid body (words)
heat transfer into the body during dt = the increase in the energy of the body during dt
Energy balance of a solid body =
hAs(Tinf - T) dt = mcp dT
Lumped system analysis equation
(T(t) - Tinf) / Ti - Tinf = e-^bt where b = hAs / rhoVcp
Proof of lumped system analysis equation
integrat energy balance equation between T(t) and Ti t=t and t=0
What does lumped system analysis assume
T is only dependent on time not on displacement
What happens as you increase b for lumped system analysis
the lumped system will reach the ambient temperature much quicker
How does the temperature of a lumped system analysis vary with time
Temperature changes dramatically initially before slowing down
What can be assumed about resistance for valid lumped system analysis
the resistance of conduction is much smaller than the resistance of convection -> convection is limiting factor heat is quickly distributed throughout system
For lumped system what is the rate of convection heat transfer between the body and its environment at time t
Q.(t) = hAs(T(t) - Tinf)
For lumped system what is the total amount of heat transfer between the body and the surrounding medium over the time interval t=0 to t
Q = mcp (T(t) - ti)
The total maximum amount of heat transfer between the body and its surrounding is
Qmax = mcp (Tinf - Ti)
When is lumped system analysis applicable
when Bi <= 0.1
What happens when Bi <= 0.1
the temperatures within the body relative to the surroundings remain within 5 percent of each other
What is the characteristic length in the biot number defined as
Lc = V / As
How is the biot number calculated
Bi = hLc /k
What is the biot number
ratio of heat convection to heat conduction
What is the biot number equal to
convection at the surface of the body/ conduction within the body, and
conduction resistance within the body / convection resistance within the body
What are the criteria for satisfying lumped system analysis
Small Biot number there, small bodies with high thermal conductivity and low convection coefficients are best
What happens when convection coefficient h is high and k is low
large temperature differences occur between the inner and outer regions of a large solid
What is the differential equation that describes transient heat transfer
d^2 T/ dx^2 = 1/alpha * dT/dt
where alpha is thermal restivity k/rho*cp
What are the boundary conditions for a large plane wall x =0 at the centre
symmetry line at the centre, therefore dT(0,t)/dx = 0
and at x = L at surface heat transfer to via conduction = heat transfer from via convection
What is dimensionless temperature theta(x,t) =
(T(x,t) - Tinf)/(Ti - Tinf)
What is dimensionless distance X =
x/L
What is dimensionless time tau =
tau = alpha * t/L^2 = fourier number
What is the dimensionless differential equation =
d^2 theta / dX = d theta / d tau
should be partial derivatives
What is the dimensionless heat transfer coefficient
biot number = hL/k
What are the dimensionless BCs and ICs of a large plane wall equal to
BCs dtheta(0, tau)/dX = 0 and d theta(1, tau)/ dX = -Bi*theta(1, tau) ICs theta(X, 0) = 1
Why do we nondimensionalise all these quantities
reduces the number of independent variables in one dimensionalised transient conduction problems from 8 to 3
Instead of doing exact maths how do we approach transient heat transfer problems
use the graphs
What is the physical significance of the fourier number
measure of heat conducted through a body relative to heat stored
a large value means fast propagation of heat through a body
what is the ratio of the fourier number
rate at which the heat is conduced across L of a body of volume L^3, to the rate at which heat is stored in a body of volume L^3
What is a semi infinite solid
an ideaised body that has a single plane surface that extends to infinity in all directions
What ca the earth be considered as
a semi infinite medium in determine variation of temperature near its surface
When can most bodies be modeled as semi infinite solids
for short periods of times since heat does not have sufficient time to penetrate deep into the body
When can the similarity variable solution be used
when you have a constant temperature Ts on the surface of a semi infinite solid
