Chapter 3 Steady heat conduction Flashcards
How can the wall of a house be modeled
steady and one dimensional, and thus the temperature expressed as T(x)
Heat transfer through a wall equation (words)
rate of heat transfer into the wall - rate of heat transfer out of the wall = rate of change of the energy of the wall
Heat transfer through a wall equation and how does this simplify in steady conditions
Q.in - Q.out = dEwall/dt
dEwall/dt = 0
How can fouriers law of heat conduction be applied to a wall
steady 1D Q.conwall = -kA dT/dx
under steady conditions the temperature distribution through a wall is and what does this mean
a straight line dT/dx = const, therefore T1-T2/L = dT/dx
For the thermal resistance concept what does current translate as
rate of heat transfer ie Q.
For the thermal resistance concept what does voltage translate as
temperature difference ie T1-T2
Thermal resistance for conduction
L/kA = Rcond
Thermal resistance for convection
Rconv = 1/hAs
what happens to Rconv as h increases
the thermal resistance tends towards zero, surface offers no rsistance to convection and thus it does not slow down the heat transfer process
Thermal resistance for radiation
Rrad = 1/hrad*As
What does hrad =
eff * simga *(Ts^2 +Tsurr^2)(Ts + Tsurr)
when does hcombined = hconv + h rad
as Tsurr approx Tinf
V=Ir for thermal resistance
T1-T2 = Q. * Rtotal
How does thermal resistance compare to electrical resistance
exactly the same, you can sum in series and do 1/R for parallel
what are the two assumptions in solving complex multi dimensional heat transfer problems as one dimensional using thermal resistance network
any plane normal to the x axis is isothermal (temp variation only in x)
any plane parallel to the x axis is adiabatic
Conduction resistance of a cylinder layer
Rcyl = ln(R2/R1)/2Pi()L*k
How to find the conduction resistance of a cylinder
use fouriers law -> integrate with respect to dr and dT, convert area to 2PiR*L, to find equation in terms of heat transfer, temperature difference, and geometrey of the cylinder
How does insulation work
decreases heat transfer as it increases the thermal resistance
what happens when adding insulation to a cylindrical pipe or spherical shell
additional insulation increases the conduction resistance of the insulation layer but decreases the convection resistance of the surface because of the increase in the outer surface area for convecction
Will heat transfer decrease or increase when you add insulation
it may do either depending on whether the convection resistance decreases dominates or the conduction resistance increases
How do you calculate the critical radius for a cylinder
Rcr = k/h
How do you calculate the critical radius for a sphere
Rcr = 2k/h
Draw the critcial radius heat transfer diagram
Heat transfer on y axis, radius on x axis, Q.bare and Q.max, increases from Q.bare to Q.max before decreasing again
Why do we use fins to increase heat transfer rates
by Newtons law of cooling if all variables are fixed only way to increase heat transfer is via increasing area
What is the fin equation (words)
Rate of heat conduction into the element at x = rate of heat conduction from the element at x+delta x + rate of heat convection from the element
what is the fin equation
Q.cond = Q.cond x+deltax + Q.conv
How is the area of convection determined for a fin
it is the perimeter of the fin times by delta x as looking for surface area in contact with the fluid
General equation of a fin
d^2*theta/dx^2 - m^2 * theta = 0
In the general equation for a fin what does theta stand or
T - Tinf
In the general equation for a fin what does m stand or
m^2 = hp/kAc (Ac = the cross sectional area of a fin)
In the general equation for a fin which term refers to conduction
d^2 theta / d x^2 (t
In the general equation for a fin which term refers to convection
m^2 * theta
Derive the general solution for the fin equation
see exam prep notebook
What is the general solution of the differential equation
Theta(x) = C1e^mx + C2e^-mx
What is the boundary condition at the fin base
Theta(0) = Thetab = Tb - Tinf
no heat loss has occured yet so simply temperature of the rest of the base
What is the boundary condition given by infinitely long fin
Tfin tip = Tinf
Theta(L) = T(L) - Tinf = 0 as L -> inf (relative just long enough such that temperature has reached surrounding)
Prove the Q. of a long fin
see book, should be equal to Q.longfin = sqrt(h p k Ac) *(Tb - Tinf)
How do you calculate the heat transfer from an entire fin
Use fouriers law Q. = -k Ac dT/dx, so once you have an equation in theta(x), differentiate it and take it when x =0 to get heat transfer for the entire fin
Under steady conditions what is heat transfer equal to in a fin
heat transfer from the exposed surfaces of the fin is equal to heat conduction to the fin at the base
What does negligible heat loss from the fin tip mean
Qfin tip = 0 therefore dtheta/dx at x=L will =0
What is the equation for heat transfer for an adiabatic fin tip
Q. = Q.longfin = sqrt(h p k Ac) *(Tb - Tinf) * tanh(mL)
What does the specified temperature boundary condition mean for fin
Tfintip = TL therefore theta(L) = thetaL = TL - Tinf
What is the equation for heat transfer for specified fin tip temperature
Q.longfin = sqrt(h p k Ac) *(Tb - Tinf) * (cosh(mL) - (TL-Tinf)/(Tb - Tinf)) / sinh(mL)
What is the most realistic boundary condition for fin tip
convection from the fin tip maybe including radiation, this can be used to determine heat transfer using energy balance at the fin tip
Energy balance for fin tip with convection
Fouriers at x=L = Newtons
-kAc dT/dx = hAc(T(L) - T inf)
What is the heat transfer from a fin with convection fin tip
Q.longfin = sqrt(h p k Ac) (Tb - Tinf) * (sinh(mL) + (h/mk)cosh(mL)) / (cosh(mL) + (h/mk) * sinh(mL))
What is a more practical way for accounting for heat loss through convection at a fin tip
Replace the fin length L with a corrected version where the fin tip can be counted as insulated
such that the eat transfer from a fin of length Lc with insulated tip is equal to heat transfer from the actual fin of length L with convection at the fin tip
What is the corrected fin length defined as (equation)
Lc = L + Ac/p
Lc rectangular = L + t/2 (t = thickness of in)
Lc cylindrical = L + D/4
What is the maximum heat transfer from a thin defined as and what doe sthis assume
Q.fin max = h Afin (Tb - Tinf)
zero thermal resistance or infinite thermal conductivity (Tfin = Tb) entire fin at base temp
How do you calculate the efficiency of a fin
eff = Q.fin / Q.finmax therefore Q.fin = eff *hAfin(Tb-Tinf)
What is the efficiency of a long fin =
1/mL
What is the efficiency of a adiabatic tip
tanh(mL)/mL
Triangle fins are better than rectangular fins as
they contain less material and are more efficient than rectangular profiles
Why is fin length limited
as efficiency is proportional to 1/L therefore as you increase L the less efficient the fin is
What is fin effectiveness
It compares how well a fin conducts away heat compared to just the plain surface of the heat source
What does fin effectiveness =
Q.fin / Q.no fin = Q.fin / hAb(Tb - Tinf) = heat transfer rate from the fin of base area Ab, heat transfer rate from the surface of area Ab
Relationship between fin effectiveness and efficiency
effective = (Afin/Ab) * efficiency
What is the effectiveness of a long fin =
sqrt (kp/hAc)
What should the effectiveness of a fin always be
greater than 1
What does the equation for effectiveness of a fin tell us about fin design
thermal conductivity should be high as possible (copper, aluminium)
the ratio of the perimeter of the fin to cross sectional area p/Ac should be high as possible -> use slender pin fins
Low convection heat transfer coefficient -> place fins on gas (air) side, (should be external not internal)
what is the total rate of heat transfer from a finned surface equal to
Q.total fin = Q.unfin + Q.fin
=h(Aunfin + efficiency(fin) * Afin) (Tb - Tinf)
What is the overall effectiveness for a finned surface =
Q.total fin/ Q. total no fin
What is the overall effectiveness equal to
depends on fin density, number of fins per unit length as well as the effectiveness of the individual fins
How best to judge effectiveness of a finned surface
using overal effectiveness not effectiveness of individual fins
Why does the region near the fin tip make little or no contribution to heat transfer
the gradual temperature drop along the fin means the end there is only a small temperature difference
At what value can a fin be considered infinitely long and why
when mL = 5, as makes Q.fin / Q.fin long = 1
What is the most effective value of mL
=1 as offers a good compromise between heat transfer performance and fin size
What does Q.fin/Q.finlong =
tanh(mL)
