Chapter 4 Transient Heat Conduction

Question 1

What does lumped system analysis represent

Answer 1

when the interior temperature of a body remains essentially uniform at all times during heat transfer process

Question 2

What does lumped system analysis allow

Answer 2

Temperature to be taken as a function of Time only T(t)

Question 3

What could be represented as lumped system analysis

Answer 3

a small copper ball

Question 4

What is the energy balance for a heated solid body (words)

Answer 4

heat transfer into the body during dt = the increase in the energy of the body during dt

Question 5

Energy balance of a solid body =

Answer 5

hAs(Tinf - T) dt = mcp dT

Question 6

Lumped system analysis equation

Answer 6

(T(t) - Tinf) / Ti - Tinf = e-^bt where b = hAs / rhoVcp

Question 7

Proof of lumped system analysis equation

Answer 7

integrat energy balance equation between T(t) and Ti t=t and t=0

Question 8

What does lumped system analysis assume

Answer 8

T is only dependent on time not on displacement

Question 9

What happens as you increase b for lumped system analysis

Answer 9

the lumped system will reach the ambient temperature much quicker

Question 10

How does the temperature of a lumped system analysis vary with time

Answer 10

Temperature changes dramatically initially before slowing down

Question 11

What can be assumed about resistance for valid lumped system analysis

Answer 11

the resistance of conduction is much smaller than the resistance of convection -> convection is limiting factor heat is quickly distributed throughout system

Question 12

For lumped system what is the rate of convection heat transfer between the body and its environment at time t

Answer 12

Q.(t) = hAs(T(t) - Tinf)

Question 13

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

Answer 13

Q = mcp (T(t) - ti)

Question 14

The total maximum amount of heat transfer between the body and its surrounding is

Answer 14

Qmax = mcp (Tinf - Ti)

Question 15

When is lumped system analysis applicable

Answer 15

when Bi <= 0.1

Question 16

What happens when Bi <= 0.1

Answer 16

the temperatures within the body relative to the surroundings remain within 5 percent of each other

Question 17

What is the characteristic length in the biot number defined as

Answer 17

Lc = V / As

Question 18

How is the biot number calculated

Answer 18

Bi = hLc /k

Question 19

What is the biot number

Answer 19

ratio of heat convection to heat conduction

Question 20

What is the biot number equal to

Answer 20

convection at the surface of the body/ conduction within the body, and
conduction resistance within the body / convection resistance within the body

Question 21

What are the criteria for satisfying lumped system analysis

Answer 21

Small Biot number there, small bodies with high thermal conductivity and low convection coefficients are best

Question 22

What happens when convection coefficient h is high and k is low

Answer 22

large temperature differences occur between the inner and outer regions of a large solid

Question 23

What is the differential equation that describes transient heat transfer

Answer 23

d^2 T/ dx^2 = 1/alpha * dT/dt

where alpha is thermal restivity k/rho*cp

Question 24

What are the boundary conditions for a large plane wall x =0 at the centre

Answer 24

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

Question 25

What is dimensionless temperature theta(x,t) =

Answer 25

(T(x,t) - Tinf)/(Ti - Tinf)

Question 26

What is dimensionless distance X =

Answer 26

x/L

Question 27

What is dimensionless time tau =

Answer 27

tau = alpha * t/L^2 = fourier number

Question 28

What is the dimensionless differential equation =

Answer 28

d^2 theta / dX = d theta / d tau | should be partial derivatives

Question 29

What is the dimensionless heat transfer coefficient

Answer 29

biot number = hL/k

Question 30

What are the dimensionless BCs and ICs of a large plane wall equal to

Answer 30

``` BCs dtheta(0, tau)/dX = 0 and d theta(1, tau)/ dX = -Bi*theta(1, tau) ICs theta(X, 0) = 1 ```

Question 31

Why do we nondimensionalise all these quantities

Answer 31

reduces the number of independent variables in one dimensionalised transient conduction problems from 8 to 3

Question 32

Instead of doing exact maths how do we approach transient heat transfer problems

Answer 32

use the graphs

Question 33

What is the physical significance of the fourier number

Answer 33

measure of heat conducted through a body relative to heat stored a large value means fast propagation of heat through a body

Question 34

what is the ratio of the fourier number

Answer 34

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

Question 35

What is a semi infinite solid

Answer 35

an ideaised body that has a single plane surface that extends to infinity in all directions

Question 36

What ca the earth be considered as

Answer 36

a semi infinite medium in determine variation of temperature near its surface

Question 37

When can most bodies be modeled as semi infinite solids

Answer 37

for short periods of times since heat does not have sufficient time to penetrate deep into the body

Question 38

When can the similarity variable solution be used

Answer 38

when you have a constant temperature Ts on the surface of a semi infinite solid