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Thermodynamics HW Solutions 106
Course: PHY 4803, Fall 2010School: UNF
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2 Chapter Heat Conduction Equation C1 r (T T ) dT & = k (4r 2 ) 2 = 4kC1 = 4k 2 1 Q = kA r k dx r 1 r1 2 hr2 (2.1 m)(0 25)C = 4 (30 W/m C) = 23,460 W 2.1 30 W/m C 1 2 (18 W/m 2 C)(2.1 m ) 2-29
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2 Chapter Heat Conduction Equation
C1
r (T T )
dT
&
= k (4r 2 ) 2 = 4kC1 = 4k 2 1
Q = kA
r
k
dx
r
1
r1 2 hr2
(2.1 m)(0 25)C
= 4 (30 W/m C)
= 23,460 W
2.1
30 W/m C
1
2 (18 W/m 2 C)(2.1 m )
2-29
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UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-64 A large plane wall is subjected to specified heat flux and temperature on the left surface and noconditions on the right surface. The mathematical formulation, the variation of temperature in the plate,and the rig
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-65 A large plane wall is subjected to specified heat flux and temperature on the left surface and noconditions on the right surface. The mathematical formulation, the variation of temperature in the plate,and the rig
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-66E A large plate is subjected to convection, radiation, and specified temperature on the top surface andno conditions on the bottom surface. The mathematical formulation, the variation of temperature in theplate, an
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-67E A large plate is subjected to convection and specified temperature on the top surface and noconditions on the bottom surface. The mathematical formulation, the variation of temperature in the plate,and the bottom
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-68 A compressed air pipe is subjected to uniform heat flux on the outer surface and convection on theinner surface. The mathematical formulation, the variation of temperature in the pipe, and the surfacetemperatures
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation(c) The inner and outer surface temperatures are determined by direct substitution to berInner surface (r = r1): T (r1 ) = 10 + 0.483 ln 1 + 12.61 = 10 + 0.483(0 + 12.61) = 3.91Cr1r 0.04Outer surface (r = r2): T (
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-69"GIVEN"L=6 "[m]"r_1=0.037 "[m]"r_2=0.04 "[m]"k=14 "[W/m-C]"Q_dot=300 "[W]"T_infinity=-10 "[C]"h=30 "[W/m^2-C]"f_loss=0.15"ANALYSIS"q_dot_s=(1-f_loss)*Q_dot)/AA=2*pi*r_2*LT=T_infinity+(ln(r/r_1)+k/(h*r_1)
UNF - PHY - 4803
Chapter 2 Heat Conduction EquationAssumptions 1 Heat conduction is steady and one-dimensional since there is no change with time andthere is thermal symmetry about the mid point. 2 Thermal conductivity is constant. 3 There is no heatgeneration in the c
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation&&Q = mC p T m =&Q0.450 kJ / s== 0.00134 kg / s = 4.84 kg / hC p T (4.185 kJ / kg C)(100 20) C2-38
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-71"GIVEN"r_1=0.40 "[m]"r_2=0.41 "[m]"k=1.5 "[W/m-C]"T_1=100 "[C]"Q_dot=500 "[W]"f_loss=0.10"ANALYSIS"q_dot_s=(1-f_loss)*Q_dot)/AA=4*pi*r_2^2T=T_1+(1/r_1-1/r)*(q_dot_s*r_2^2)/k "Variation of temperature""r i
UNF - PHY - 4803
Chapter 2 Heat Conduction EquationHeat Generation in Solids2-72C No. Heat generation in a solid is simply the conversion of some form of energy into sensible heatenergy. For example resistance heating in wires is conversion of electrical energy to heat
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-78 Heat is generated in a long solid cylinder with a specified surfacetemperature. The variation of temperature in the cylinder is given byT (r ) =2&gr02 r 1 + Tsk r0 80C(a) Heat conduction is steady since t
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-79"GIVEN"r_0=0.04 "[m]"k=25 "[W/m-C]"g_dot_0=35E+6 "[W/m^3]"T_s=80 "[C]""ANALYSIS"T=(g_dot_0*r_0^2)/k*(1-(r/r_0)^2)+T_s "Variation of temperature""r is the parameter to be varied"r [m]00.0044440.0088890.01
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-80E A long homogeneous resistance heater wire with specified convection conditions at the surface isused to boil water. The mathematical formulation, the variation of temperature in the wire, and thetemperature at th
UNF - PHY - 4803
Chapter 2 Heat Conduction EquationT ( 0) = T +&&g 2 gr0r0 +4k2h2= 212F +(1800 Btu/h.in 3 )(0.25 in) 2 12 in (1800 Btu/h.in 3 )(0.25 in ) 12 in + = 290.8F4 (8.6 Btu/h.ft.F)2 (820 Btu/h ft 2 F) 1 ft 1 ft Thus the centerline temperature will
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-81E"GIVEN"r_0=0.25/12 "[ft]"k=8.6 "[Btu/h-ft-F]""g_dot=1800 [Btu/h-in^3], parameter to be varied"T_infinity=212 "[F]"h=820 "[Btu/h-ft^2-F]""ANALYSIS"T_0=T_infinity+(g_dot/Convert(in^3, ft^3)/(4*k)*(r_0^2-r^2)+(
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-82 A nuclear fuel rod with a specified surface temperature is used as the fuel in a nuclear reactor. Thecenter temperature of the rod is to be determined.175CAssumptions 1 Heat transfer is steady since there is no i
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-83 Both sides of a large stainless steel plate in which heat is generated uniformly are exposed toconvection with the environment. The location and values of the highest and the lowest temperatures in theplate are to
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-85"GIVEN"L=0.05 "[m]"k=111 "[W/m-C]"g_dot=2E5 "[W/m^3]"T_infinity=25 "[C]""h=44 [W/m^2-C], parameter to be varied""ANALYSIS"T_min=T_infinity+(g_dot*L)/hT_max=T_min+(g_dot*L^2)/(2*k)h [W/m2.C]20253035404
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation550500450T m in [ C]400350300250200150100203040506070809010080901002h [W /m -C]550500450T m ax [ C]4003503002502001501002030405060702h [W /m -C]2-49
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-86 A long resistance heater wire is subjected to convection at its outer surface. The surface temperatureof the wire is to be determined using the applicable relations directly and by solving the applicabledifferenti
UNF - PHY - 4803
Chapter 2 Heat Conduction EquationAssumptions 1 Heat transfer is steady since there is no changewith time. 2 Heat transfer is one-dimensional since there isthermal symmetry about the center line and no change in theaxial direction. 3 Thermal conductiv
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-89 Heat is generated uniformly in a spherical radioactive material with specified surface temperature.The mathematical formulation, the variation of temperature in the sphere, and the center temperature are tobe dete
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-90"GIVEN"r_0=0.04 "[m]"g_dot=4E7 "[W/m^3]"T_s=80 "[C]"k=15 "[W/m-C], Parameter to be varied""ANALYSIS"T=T_s+g_dot/(6*k)*(r_0^2-r^2) "Temperature distribution as a function of r""r is the parameter to be varied"
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation800700600T [C]500400300200100000.0050.010.0150.020.0250.030.0353003500.04r [m ]12001000T 0 [ C]8006004002000050100150200250k [W /m -C]2-54400
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-91 A long homogeneous resistance heater wire with specified surface temperature is used to boil water.The temperature of the wire 2 mm from the center is to be determined in steady operation.Assumptions 1 Heat transf
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-92 Heat is generated in a large plane wall whose one side is insulated while the other side is maintainedat a specified temperature. The mathematical formulation, the variation of temperature in the wall, and thetemp
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-93"GIVEN"L=0.05 "[m]"T_s=30 "[C]"k=30 "[W/m-C]"g_dot_0=8E6 "[W/m^3]""ANALYSIS"g_dot=g_dot_0*exp(-0.5*x)/L) "Heat generation as a function of x""x is the parameter to be varied"g [W/m3]8.000E+067.610E+067.23
UNF - PHY - 4803
Chapter 2 Heat Conduction EquationVariable Thermal Conductivity2-94C During steady one-dimensional heat conduction in a plane wall, long cylinder, and sphere withconstant thermal conductivity and no heat generation, the temperature in only the plane wa
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-100 A cylindrical shell with variable conductivity is subjected to specified temperatures on both sides.The variation of temperature and the rate of heat transfer through the shell are to be determined.Assumptions 1
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-101 A spherical shell with variable conductivity is subjected to specified temperatures on both sides.The variation of temperature and the rate of heat transfer through the shell are to be determined.Assumptions 1 He
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-102 A plate with variable conductivity is subjected to specified temperatures on both sides. The rate ofheat transfer through the plate is to be determined.Assumptions 1 Heat transfer is given to be steady and one-di
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-103"GIVEN"A=1.5*0.6 "[m^2]"L=0.15 "[m]""T_1=500 [K], parameter to be varied"T_2=350 "[K]"k_0=25 "[W/m-K]"beta=8.7E-4 "[1/K]""ANALYSIS"k=k_0*(1+beta*T)T=1/2*(T_1+T_2)Q_dot=k*A*(T_1-T_2)/LT1 [W]4004254504
UNF - PHY - 4803
Chapter 2 Heat Conduction EquationSpecial Topic: Review of Differential equations2-104C We utilize appropriate simplifying assumptions when deriving differential equations to obtain anequation that we can deal with and solve.2-105C A variable is a qua
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-106C A differential equation may involve more than one dependent or independent variable. For& 2 T ( x , t ) g 1 T ( x , t )+=has one dependent (T) and 2 independent variables (xexample, the equation2ktx 2 T
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-116C A linear homogeneous differential equation of order n is expressed in the most general form asy ( n ) + f 1 ( x ) y ( n 1) + L + f n 1 ( x ) y + f n ( x ) y = 0Each term in a linear homogeneous equation contains
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-121 A long rectangular bar is initially at a uniform temperature of Ti. The surfaces of the bar at x = 0 andy = 0 are insulated while heat is lost from the other two surfaces by convection. The mathematicalformulatio
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-123E A large plane wall is subjected to a specified temperature on the left (inner) surface and solarradiation and heat loss by radiation to space on the right (outer) surface. The temperature of the rightsurface of
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-124E A large plane wall is subjected to a specified temperature on the left (inner) surface and heat lossby radiation to space on the right (outer) surface. The temperature of the right surface of the wall and therat
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-125 A steam pipe is subjected to convection on both the inner and outer surfaces. The mathematicalformulation of the problem and expressions for the variation of temperature in the pipe and on the outersurface temper
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-126 A spherical liquid nitrogen container is subjected to specified temperature on the inner surface andconvection on the outer surface. The mathematical formulation, the variation of temperature, and the rateof evap
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-127 A spherical liquid oxygen container is subjected to specified temperature on the inner surface andconvection on the outer surface. The mathematical formulation, the variation of temperature, and the rateof evapor
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-128 A large plane wall is subjected to convection, radiation, and specified temperature on the rightsurface and no conditions on the left surface. The mathematical formulation, the variation of temperaturein the wall
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-129 The base plate of an iron is subjected to specified heat flux on the left surface and convection andradiation on the right surface. The mathematical formulation, and an expression for the outer surfacetemperature
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-130 The base plate of an iron is subjected to specified heat flux on the left surface and convection andradiation on the right surface. The mathematical formulation, and an expression for the outer surfacetemperature
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-131E The concrete slab roof of a house is subjected to specified temperature at the bottom surface andconvection and radiation at the top surface. The temperature of the top surface of the roof and the rate ofheat tr
UNF - PHY - 4803
Chapter 2 Heat Conduction EquationApplying the other boundary condition at r = r1 ,&&g2g2r1 + C2 C2 = TI +r14k4kSubstituting this C2 relation into Eq. (b) and rearranging giveTI = B. C. at r = r1 :Twire (r ) = TI +&g(r12 r 2 )4 k wire(c)
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-133 A cylindrical shell with variable conductivity issubjected to specified temperatures on both sides. The rate ofheat transfer through the shell is to be determined.k(T)Assumptions 1 Heat transfer is given to be
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-135 A large plane wall is subjected to convection on the inner and outer surfaces. The mathematicalformulation, the variation of temperature, and the temperatures at the inner and outer surfaces to bedetermined for s
UNF - PHY - 4803
Chapter 2 Heat Conduction Equation2-136 A hollow pipe is subjected to specified temperatures at the inner and outer surfaces. There is alsoheat generation in the pipe. The variation of temperature in the pipe and the center surface temperature ofthe pi
UNF - PHY - 4803
Chapter 2 Heat Conduction EquationT (r ) = 37,894r 2+ 98.34 ln r + 257.2 = 257.2 473.68r 2 + 98.34 ln r4(20)The temperature at the center surface of the pipe is determined by setting radius r to be 17.5 cm, which isthe average of the inner radius an
UNF - PHY - 4803
Chapter 3 Steady Heat ConductionChapter 3STEADY HEAT CONDUCTIONSteady Heat Conduction In Plane Walls3-1C (a) If the lateral surfaces of the rod are insulated, the heat transfer surface area of the cylindrical rodis the bottom or the top surface area
UNF - PHY - 4803
Chapter 3 Steady Heat Conduction3-11C The temperature of each surface in this case can be determined from&&Q = (T1 Ts1 ) / R1 s1 Ts1 = T1 (QR1 s1 )&&Q = (Ts2 T 2 ) / Rs2 2 Ts2 = T 2 + (QRs2 2 )where Ri is the thermal resistance between the environ
UNF - PHY - 4803
Chapter 3 Steady Heat Conduction3-18 The two surfaces of a window are maintained at specified temperatures. The rate of heat loss throughthe window and the inner surface temperature are to be determined.Assumptions 1 Heat transfer through the window is
UNF - PHY - 4803
Chapter 3 Steady Heat Conduction3-19 A double-pane window consists of two 3-mm thick layers of glass separated by a 12-mm widestagnant air space. For specified indoors and outdoors temperatures, the rate of heat loss through thewindow and the inner sur
UNF - PHY - 4803
Chapter 3 Steady Heat Conduction3-20 A double-pane window consists of two 3-mm thick layers of glass separated by an evacuated space.For specified indoors and outdoors temperatures, the rate of heat loss through the window and the innersurface temperat
UNF - PHY - 4803
Chapter 3 Steady Heat Conduction3-21"GIVEN"A=1.2*2 "[m^2]"L_glass=3 "[mm]"k_glass=0.78 "[W/m-C]""L_air=12 [mm], parameter to be varied"T_infinity_1=24 "[C]"T_infinity_2=-5 "[C]"h_1=10 "[W/m^2-C]"h_2=25 "[W/m^2-C]""PROPERTIES"k_air=conductivity
UNF - PHY - 4803
Chapter 3 Steady Heat Conduction3-22E The inner and outer surfaces of the walls of an electrically heated house remain at specifiedtemperatures during a winter day. The amount of heat lost from the house that day and its its cost are to bedetermined.A
UNF - PHY - 4803
Chapter 3 Steady Heat Conduction3-24 A power transistor dissipates 0.2 W of power steadily in a specified environment. The amount of heatdissipated in 24 h, the surface heat flux, and the surface temperature of the resistor are to be determined.Assumpt
UNF - PHY - 4803
Chapter 3 Steady Heat Conduction3-26 A person is dissipating heat at a rate of 150 W by natural convection and radiation to the surroundingair and surfaces. For a given deep body temperature, the outer skin temperature is to be determined.Assumptions 1
UNF - PHY - 4803
Chapter 3 Steady Heat Conduction3-28E A wall is constructed of two layers of sheetrock with fiberglass insulation in between. The thermalresistance of the wall and its R-value of insulation are to be determined.Assumptions 1 Heat transfer through the w