204 pág.


DisciplinaTransferência de Calor2.830 materiais18.511 seguidores
Pré-visualização44 páginas
s. Note, however, that the
temperature at the location x = 600 mm does not change significantly within the 150 s exposure to the
hot surroundings. Our assumption of treating the boundary at the x = 600 mm plane as adiabatic is
justified. A copper plate of 600 mm is a good approximation to a semi-infinite medium at times less
than 150 s.
Continued \u2026..
PROBLEM 5.109 (Cont.)
COMMENTS: The annotated Input screen shows the outline of the slab, the boundary conditions,
and the triangular mesh before using the Reduce-mesh option.
KNOWN: Electric heater sandwiched between two thick plates whose surfaces experience
convection. Case 2 corresponds to steady-state operation with a loss of coolant on the x = -L surface.
Suddenly, a second loss of coolant condition occurs on the x = +L surface, but the heater remains
energized for the next 15 minutes. Case 3 corresponds to the eventual steady-state condition following
the second loss of coolant event. See Problem 2.53.
FIND: Calculate and plot the temperature time histories at the plate locations x = 0, ±L during the
transient period between steady-state distributions for Case 2 and Case 3 using the finite-element
approach with FEHT and the finite-difference method of solution with IHT (\u2206x = 5 mm and \u2206t = 1 s).
ASSUMPTIONS: (1) One-dimensional conduction, (2) Constant properties, (3) Heater has negligible
thickness, and (4) Negligible thermal resistance between the heater surfaces and the plates.
PROPERTIES: Plate material (given); \u3c1 = 2500 kg/m3, c = 700 J/kg\u22c5K, k = 5 W/m\u22c5K.
ANALYSIS: The temperature distribution for Case 2 shown in the above graph represents the initial
condition for the period of time following the second loss of coolant event. The boundary conditions
at x = ±L are adiabatic, and the heater flux is maintained at oq\u2032\u2032 = 4000 W/m
 for 0 \u2264 t \u2264 15 min.
Using FEHT, the heater is represented as a plate of thickness Lh = 0.5 mm with very low thermal
capacitance (\u3c1 = 1 kg/m and c = 1 J/kg\u22c5K), very high thermal conductivity (k= 10,000 W/m\u22c5K), and a
uniform volumetric generation rate of 2 6o hq q / L 4000 W / m / 0.0005 m 8.0 10\u2032\u2032= = = ×\ufffd W/m
 for 0 \u2264 t \u2264
900 s. In the Specify | Generation box, the generation was prescribed by the lookup file (see FEHT
Help): \u2018hfvst\u2019,1,2,Time. This Notepad file is comprised of four lines, with the values on each line
separated by a single tab space:
0 8e6
900 8e6
901 0
5000 0
The temperature-time histories are shown in the graph below for the surfaces x = - L (lowest curve,
13) and x = +L (19) and the center point x = 0 (highest curve, 14). The center point experiences the
maximum temperature of 89°C at the time the heater is deactivated, t = 900 s.
Continued \u2026..
For the finite-difference method of solution, the nodal arrangement for the system is shown below.
The IHT model builder Tools | Finite-Difference Equations | One Dimensional can be used to obtain
the FDEs for the internal nodes (02-04, 07-10) and the adiabatic boundary nodes (01, 11).
For the heater-plate interface node 06, the FDE for the implicit method is derived from an energy
balance on the control volume shown in the schematic above.
in out gen stE E E E\u2032\u2032 \u2032\u2032 \u2032\u2032 \u2032\u2032\u2212 + =\ufffd \ufffd \ufffd \ufffd
a b o stq q q E\u2032\u2032 \u2032\u2032 \u2032\u2032 \u2032\u2032+ + = \ufffd
p 1 p 1 p 1 p 1 p 1 p
05 06 07 06 06 06
k k q c x
x x t
+ + + + +
\u2212 \u2212 \u2212
\u2032\u2032+ + = \u2206
\u2206 \u2206 \u2206
The IHT code representing selected nodes is shown below for the adiabatic boundary node 01, interior
node 02, and the heater-plates interface node 06. Note how the foregoing derived finite-difference
equation in implicit form is written in the IHT Workspace. Note also the use of a Lookup Table for
representing the heater flux vs. time.
Continued \u2026..
PROBLEM 5.110 (Cont.)
// Finite-difference equations from Tools, Nodes 01, 02
/* Node 01: surface node (w-orientation); transient conditions; e labeled 02. */
rho * cp * der(T01,t) = fd_1d_sur_w(T01,T02,k,qdot,deltax,Tinf01,h01,q''a01)
q''a01 = 0 // Applied heat flux, W/m^2; zero flux shown
qdot = 0 // No internal generation
Tinf01 = 20 // Arbitrary value
h01 = 1e-6 // Causes boundary to behave as adiabatic
/* Node 02: interior node; e and w labeled 03 and 01. */
rho*cp*der(T02,t) = fd_1d_int(T02,T03,T01,k,qdot,deltax)
// Finite-difference equation from energy balance on CV, Node 06
k * (T05 - T06) / deltax + k * (T07 - T06)/ deltax + q''h = rho * cp * deltax * der(T06,t)
q''h = LOOKUPVAL(qhvst,1,t,2) // Heater flux, W/m^2; specified by Lookup Table
/* See HELP (Solver, Lookup Tables). The Look-up table file name "qhvst" contains
 0 4000
 900 4000
 900.5 0
 5000 0 */
The temperature-time histories using the IHT code for the plate locations x = 0, ±L are shown in the
graphs below. We chose to show expanded presentations of the histories at early times, just after the
second loss of coolant event, t = 0, and around the time the heater is deactivated, t = 900 s.
COMMENTS: (1) The maximum temperature during the transient period is at the center point and
occurs at the instant the heater is deactivated, T(0, 900s) = 89°C. After 300 s, note that the two surface
temperatures are nearly the same, and never rise above the final steady-state temperature.
(2) Both the FEHT and IHT methods of solution give identical results. Their steady-state solutions
agree with the result of an energy balance on a time interval basis yielding Tss = 86.1°C.
0 50 100 150 200
Time, t (s) 
Surface x = -L
Center point, x = 0
Surface x = +L
800 900 1000 1100 1200
Time, t (s)
, T
Surface x = -L
Center point, x = 0
Surface x = +L
KNOWN: Plane wall of thickness 2L, initially at a uniform temperature, is suddenly subjected to
convection heat transfer.
FIND: The mid-plane, T(0,t), and surface, T(L,t), temperatures at t = 50, 100, 200 and 500 s, using
the following methods: (a) the one-term series solution; determine also the Biot number; (b) the
lumped capacitance solution; and (c) the two- and 5-node finite-difference numerical solutions.
Prepare a table summarizing the results and comment on the relative differences of the predicted
ASSUMPTIONS: (1) One-dimensional conduction in the x-direction, and (2) Constant properties.
ANALYSIS: (a) The results are tabulated below for the mid-plane and surface temperatures using the
one-term approximation to the series solution, Eq. 5.40 and 5.41. The Biot number for the heat
transfer process is
2Bi h L / k 500 W / m K 0.020 m 15 W / m K 0.67/= = \u22c5 × \u22c5 =
Since Bi >> 0.1, we expect an appreciable temperature difference between the mid-plane and surface
as the tabulated results indicate (Eq. 5.10).
(b) The results are tabulated below for the wall temperatures using the lumped capacitance method
(LCM) of solution, Eq. 5.6. The LCM neglects the internal conduction resistance and since Bi = 0.67
>> 0.1, we expect this method to predict systematically lower temperatures (faster cooling) at the
midplane compared to the one-term approximation.
Solution method/Time(s) 50 100 200 500
Mid-plane, T(0,t) (°C) 
One-term, Eqs. 5.40, 5.41 207.1 160.5 99.97 37.70
Lumped capacitance 181.7 133.9 77.69 30.97
2-node FDE 210.6 163.5 100.5 37.17
5-node FDE 207.5 160.9 100.2 37.77
Surface, T(L,t) (°C) 
One-term, Eqs. 5.40, 5.41 160.1