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Heat Transfer equations
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DR
AFT
ME 470 – Heat TransferExam 1 Formulae
First Law of Thermodynamics
Est = Ein − Eout + Eg (1)
Heat Diffusion Equation
ρc∂T
∂t=
∂
∂x
[
k∂T
∂x
]
+∂
∂y
[
k∂T
∂y
]
+∂
∂z
[
k∂T
∂z
]
+ g′′′ (2)
Plane Wall without Thermal Energy Generation (0 ≤ x ≤ L)
T (x) = (Ts,2 − Ts,1)x
L+ Ts,1 and q′′x =
k
L(Ts,1 − Ts,2) (3)
Plane Wall with Thermal Energy Generation (−L ≤ x ≤ L)
T (x) =g′′′L2
2k
(
1−x2
L2
)
+ Ts and q′′x(x) = g′′′x (4)
Hollow Cylinder without Thermal Energy Generation (r1 ≤ r ≤ r2)
T (r) = (Ts,1 − Ts,2)ln(r/r2)
ln(r1/r2)+ Ts,2 and q′′r (r) =
k(Ts,1 − Ts,2)
r ln(r2/r1)(5)
Solid Cylinder with Thermal Energy Generation (0 ≤ r ≤ ro)
T (r) =g′′′r2o4k
(
1−r2
r2o
)
+ Ts and q′′r (r) =1
2g′′′r (6)
Hollow Sphere without Thermal Energy Generation (r1 ≤ r ≤ r2)
T (r) = Ts,1 − (Ts,1 − Ts,2)
[
1− (r1/r)
1− (r1/r2)
]
and q′′r (r) =k(Ts,1 − Ts,2)
r2[(1/r1)− (1/r2)](7)
Steady-State Thermal Resistances
Rwall ≡L
kARcyl ≡
ln(r2/r1)
2πLkRsph ≡
r2 − r14πr1r2k
(8)
Rconv ≡1
hARrad ≡
1
hradARfin ≡
θbqfin
(9)
Fins
εfin ≡qfin
qno fin
=qfin
hAc,bθband ηfin ≡
qfin
qmax
=qfin
hAs,finθb(10)
Convection heat loss from fin tip
θ
θb=
coshm(L− x) + (h/mk) sinhm(L− x)
coshmL+ (h/mk) sinhmLand qfin =
√
hPkAc θbsinhmL+ (h/mk) coshmL
coshmL+ (h/mk) sinhmL(11)
5
DR
AFT
Negligible heat loss from fin tip
θ
θb=
coshm(L− x)
coshmLand qfin =
√
hPkAc θb tanhmL (12)
Lc ≡ L+Ac
PLc,rectangular fin ≈ L+
t
2Lc,cylindrical fin = L+
D
4(13)
Prescribed temperature at the fin tip
θ
θb=
(θL/θb) sinhmx+ sinhm(L− x)
sinhmLand qfin =
√
hPkAc θbcoshmL− θL/θb
sinhmL(14)
Infinitely long finθ
θb= e−mx and qfin =
√
hPkAc θb (15)
where
θ ≡ T − T∞ and m ≡
√
hP
kAc
(16)
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