PROBLEM 1.1 KNOWN: Heat rate, q, through one-dimensional wall of area A, thickness L, thermal conductivity k and inner temperature, T1. FIND: The outer temperature of the wall, T2. SCHEMATIC: ASSUMPTIONS: (1) One-dimensional conduction in the x-direction, (2) Steady-state conditions, (3) Constant properties. ANALYSIS: The rate equation for conduction through the wall is given by Fourier’s law, q q q A = -k dT dx A = kA T T L cond x x 1 2 = = ′′ ⋅ ⋅ − . Solving for T2 gives T T q L kA 2 1 cond = − . Substituting numerical values, find T C- 3000W 0.025m 0.2W / m K 10m 2 2 = × ⋅ × 415$ T C-37.5 C 2 = 415$ $ T C. 2 = 378$ < COMMENTS: Note direction of heat flow and fact that T2 must be less than T1. PROBLEM 1.2 KNOWN: Inner surface temperature and thermal conductivity of a concrete wall. FIND: Heat loss by conduction through the wall as a function of ambient air temperatures ranging from -15 to 38°C. SCHEMATIC: ASSUMPTIONS: (1) One-dimensional conduction in the x-direction, (2) Steady-state conditions, (3) Constant properties, (4) Outside wall temperature is that of the ambient air. ANALYSIS: From Fourier’s law, it is evident that the gradient, x dT dx q k ′′ = − , is a constant, and hence the temperature distribution is linear, if x q′′ and k are each constant. The heat flux must be constant under one-dimensional, steady-state conditions; and k is approximately constant if it depends only weakly on temperature. The heat flux and heat rate when the outside wall temperature is T2 = -15°C are ( ) 2 1 2 x 25 C 15 C dT T T q k k 1W m K 133.3W m dx L 0.30m − − − ′′ = − = = ⋅ = $ $ . (1) 2 2 x x q q A 133.3W m 20m 2667 W ′′ = × = × = . (2) < Combining Eqs. (1) and (2), the heat rate qx can be determined for the range of ambient temperature, -15 ≤ T2 ≤ 38°C, with different wall thermal conductivities, k. -20 -10 0 10 20 30 40 Ambient air temperature, T2 (C) -1500 -500 500 1500 2500 3500 Heat loss, qx (W) Wall thermal conductivity, k = 1.25 W/m.K k = 1 W/m.K, concrete wall k = 0.75 W/m.K For the concrete wall, k = 1 W/m⋅K, the heat loss varies linearily from +2667 W to -867 W and is zero when the inside and ambient temperatures are the same. The magnitude of the heat rate increases with increasing thermal conductivity. COMMENTS: Without steady-state conditions and constant k, the temperature distribution in a plane wall would not be linear. PROBLEM 1.3 KNOWN: Dimensions, thermal conductivity and surface temperatures of a concrete slab. Efficiency of gas furnace and cost of natural gas. FIND: Daily cost of heat loss. SCHEMATIC: ASSUMPTIONS: (1) Steady state, (2) One-dimensional conduction, (3) Constant properties. ANALYSIS: The rate of heat loss by conduction through the slab is ( ) ( ) 1 2 T T 7 C q k LW 1.4W / m K 11m 8m 4312 W t 0.20m − ° = = ⋅ × = < The daily cost of natural gas that must be combusted to compensate for the heat loss is ( ) ( ) g d 6 f qC 4312W $0.01/ MJ C t 24h / d 3600s / h $4.14/ d 0.9 10 J / MJ η × = ∆ = × = × < COMMENTS: The loss could be reduced by installing a floor covering with a layer of insulation between it and the concrete. PROBLEM 1.4 KNOWN: Heat flux and surface temperatures associated with a wood slab of prescribed thickness. FIND: Thermal conductivity, k, of the wood. SCHEMATIC: ASSUMPTIONS: (1) One-dimensional conduction in the x-direction, (2) Steady-state conditions, (3) Constant properties. ANALYSIS: Subject to the foregoing assumptions, the thermal conductivity may be determined from Fourier’s law, Eq. 1.2. Rearranging, ( ) L W 0.05m k=q 40 T T m 40-20 C x 2 1 2 ′′ = − � k = 0.10 W / m K. ⋅ < COMMENTS: Note that the °C or K temperature units may be used interchangeably when evaluating a temperature difference. PROBLEM 1.5 KNOWN: Inner and outer surface temperatures of a glass window of prescribed dimensions. FIND: Heat loss through window. SCHEMATIC: ASSUMPTIONS: (1) One-dimensional conduction in the x-direction, (2) Steady-state conditions, (3) Constant properties. ANALYSIS: Subject to the foregoing conditions the heat flux may be computed from Fourier’s law, Eq. 1.2. ( ) T T q k L 15-5 C W q 1.4 m K 0.005m q 2800 W/m . 1 2 x x 2 x − ′′ = ′′ = ⋅ ′′ = � Since the heat flux is uniform over the surface, the heat loss (rate) is q = qx A q = 2800 W / m2 3m2 ′′ × × q = 8400 W. < COMMENTS: A linear temperature distribution exists in the glass for the prescribed conditions. PROBLEM 1.6 KNOWN: Width, height, thickness and thermal conductivity of a single pane window and the air space of a double pane window. Representative winter surface temperatures of single pane and air space. FIND: Heat loss through single and double pane windows. SCHEMATIC: ASSUMPTIONS: (1) One-dimensional conduction through glass or air, (2) Steady-state conditions, (3) Enclosed air of double pane window is stagnant (negligible buoyancy induced motion). ANALYSIS: From Fourier’s law, the heat losses are Single Pane: ( ) T T 35 C 2 1 2 q k A 1.4 W/m K 2m 19,600 W g g L 0.005m − = = ⋅ = � Double Pane: ( ) T T 25 C 2 1 2 q k A 0.024 2m 120 W a a L 0.010 m − = = = � COMMENTS: Losses associated with a single pane are unacceptable and would remain excessive, even if the thickness of the glass were doubled to match that of the air space. The principal advantage of the double pane construction resides with the low thermal conductivity of air (~ 60 times smaller than that of glass). For a fixed ambient outside air temperature, use of the double pane construction would also increase the surface temperature of the glass exposed to the room (inside) air. PROBLEM 1.7 KNOWN: Dimensions of freezer compartment. Inner and outer surface temperatures. FIND: Thickness of styrofoam insulation needed to maintain heat load below prescribed value. SCHEMATIC: ASSUMPTIONS: (1) Perfectly insulated bottom, (2) One-dimensional conduction through 5 walls of area A = 4m2, (3) Steady-state conditions, (4) Constant properties. ANALYSIS: Using Fourier’s law, Eq. 1.2, the heat rate is q = q A = k T L Atotal ′′⋅ ∆ Solving for L and recognizing that Atotal = 5×W2, find L = 5 k T W q 2 ∆ ( ) ( ) 5 0.03 W/m K 35 - -10 C 4m L = 500 W 2   × ⋅  � L = 0.054m = 54mm. < COMMENTS: The corners will cause local departures from one-dimensional conduction and a slightly larger heat loss. PROBLEM 1.8 KNOWN: Dimensions and thermal conductivity of food/beverage container. Inner and outer surface temperatures. FIND: Heat flux through container wall and total heat load. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Negligible heat transfer through bottom wall, (3) Uniform surface temperatures and one-dimensional conduction through remaining walls. ANALYSIS: From Fourier’s law, Eq. 1.2, the heat flux is ( ) 0.023 W/m K 20 2 C T T 2 2 1 q k 16.6 W/m L 0.025 m ⋅ − − ′′ = = = � < Since the flux is uniform over each of the five walls through which heat is transferred, the heat load is ( ) q q A q H 2W 2W W W total 1 2 1 2 ′′ ′′� � = × = + + × � � ( ) ( ) 2 q 16.6 W/m 0.6m 1.6m 1.2m 0.8m 0.6m 35.9 W � � = + + × = � � < COMMENTS: The corners and edges of the container create local departures from one- dimensional conduction, which increase the heat load. However, for H, W1, W2 >> L, the effect is negligible. PROBLEM 1.9 KNOWN: Masonry wall of known thermal conductivity has a heat rate which is 80% of that through a composite wall of prescribed thermal conductivity and thickness. FIND: Thickness of masonry wall. SCHEMATIC: ASSUMPTIONS: (1) Both walls subjected to same surface temperatures, (2) One- dimensional conduction, (3) Steady-state conditions, (4) Constant properties. ANALYSIS: For steady-state conditions, the conduction heat flux through a one-dimensional wall follows from Fourier’s law, Eq. 1.2, ′′ q = k T L ∆ where ∆T represents the difference in surface temperatures. Since ∆T is the same for both walls, it follows that L = L k k q q 1 2 1 2 2 1 ⋅ ′′ ′′ . With the heat fluxes related as ′′ = ′′ q 0.8 q 1 2 L = 100mm 0.75 W / m K 0.25 W / m K 1 0.8 = 375mm. 1 ⋅ ⋅ × < COMMENTS: Not knowing the temperature difference across the walls, we cannot find the value of the heat rate. PROBLEM 1.10 KNOWN: Thickness, diameter and inner surface temperature of bottom of pan used to boil water. Rate of heat transfer to the pan. FIND: Outer surface temperature of pan for an aluminum and a copper bottom. SCHEMATIC: ASSUMPTIONS: (1) One-dimensional, steady-state conduction through bottom of pan. ANALYSIS: From Fourier’s law, the rate of heat transfer by conduction through the bottom of the pan is T T 1 2 q kA L − = Hence, qL T T 1 2 kA = + where ( )2 2 2 A D / 4 0.2m / 4 0.0314 m . π π = = = Aluminum: ( ) ( ) 600W 0.005 m T 110 C 110.40 C 1 2 240 W/m K 0.0314 m = + = ⋅ � � Copper: ( ) ( ) 600W 0.005 m T 110 C 110.25 C 1 2 390 W/m K 0.0314 m = + = ⋅ � � COMMENTS: Although the temperature drop across the bottom is slightly larger for aluminum (due to its smaller thermal conductivity), it is sufficiently small to be negligible for both materials. To a good approximation, the bottom may be considered isothermal at T ≈ 110 °C, which is a desirable feature of pots and pans. PROBLEM 1.11 KNOWN: Dimensions and thermal conductivity of a chip. Power dissipated on one surface. FIND: Temperature drop across the chip. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Constant properties, (3) Uniform heat dissipation, (4) Negligible heat loss from back and sides, (5) One-dimensional conduction in chip. ANALYSIS: All of the electrical power dissipated at the back surface of the chip is transferred by conduction through the chip. Hence, from Fourier’s law, P = q = kA T t ∆ or ( ) t P 0.001 m 4 W T = kW 150 W/m K 0.005 m 2 2 ⋅ × ∆ = ⋅ ∆T = 1.1 C. $ < COMMENTS: For fixed P, the temperature drop across the chip decreases with increasing k and W, as well as with decreasing t. PROBLEM 1.12 KNOWN: Heat flux gage with thin-film thermocouples on upper and lower surfaces; output voltage, calibration constant, thickness and thermal conductivity of gage. FIND: (a) Heat flux, (b) Precaution when sandwiching gage between two materials. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional heat conduction in gage, (3) Constant properties. ANALYSIS: (a) Fourier’s law applied to the gage can be written as ′′ q = k T x ∆ ∆ and the gradient can be expressed as ∆ ∆ ∆ T x = E / N SABt where N is the number of differentially connected thermocouple junctions, SAB is the Seebeck coefficient for type K thermocouples (A-chromel and B-alumel), and ∆x = t is the gage thickness. Hence, ′′ q = k E NSABt ∆ ′′ ⋅ × × × × × × q = 1.4 W / m K 350 10-6 V 5 40 10-6 V / C 0.25 10-3 m = 9800 W / m2 � . < (b) The major precaution to be taken with this type of gage is to match its thermal conductivity with that of the material on which it is installed. If the gage is bonded between laminates (see sketch above) and its thermal conductivity is significantly different from that of the laminates, one dimensional heat flow will be disturbed and the gage will read incorrectly. COMMENTS: If the thermal conductivity of the gage is lower than that of the laminates, will it indicate heat fluxes that are systematically high or low? PROBLEM 1.13 KNOWN: Hand experiencing convection heat transfer with moving air and water. FIND: Determine which condition feels colder. Contrast these results with a heat loss of 30 W/m2 under normal room conditions. SCHEMATIC: ASSUMPTIONS: (1) Temperature is uniform over the hand’s surface, (2) Convection coefficient is uniform over the hand, and (3) Negligible radiation exchange between hand and surroundings in the case of air flow. ANALYSIS: The hand will feel colder for the condition which results in the larger heat loss. The heat loss can be determined from Newton’s law of cooling, Eq. 1.3a, written as ( ) s q h T T∞ ′′ = − For the air stream: ( ) 2 2 air q 40W m K 30 5 K 1,400W m ′′ � � = ⋅ − − = � � < For the water stream: ( ) 2 2 water q 900W m K 30 10 K 18,000W m ′′ = ⋅ − = < COMMENTS: The heat loss for the hand in the water stream is an order of magnitude larger than when in the air stream for the given temperature and convection coefficient conditions. In contrast, the heat loss in a normal room environment is only 30 W/m2 which is a factor of 400 times less than the loss in the air stream. In the room environment, the hand would feel comfortable; in the air and water streams, as you probably know from experience, the hand would feel uncomfortably cold since the heat loss is excessively high. PROBLEM 1.14 KNOWN: Power required to maintain the surface temperature of a long, 25-mm diameter cylinder with an imbedded electrical heater for different air velocities. FIND: (a) Determine the convection coefficient for each of the air velocity conditions and display the results graphically, and (b) Assuming that the convection coefficient depends upon air velocity as h = CVn, determine the parameters C and n. SCHEMATIC: V(m/s) 1 2 4 8 12 ′ Pe (W/m) 450 658 983 1507 1963 h (W/m2⋅K) 22.0 32.2 48.1 73.8 96.1 ASSUMPTIONS: (1) Temperature is uniform over the cylinder surface, (2) Negligible radiation exchange between the cylinder surface and the surroundings, (3) Steady-state conditions. ANALYSIS: (a) From an overall energy balance on the cylinder, the power dissipated by the electrical heater is transferred by convection to the air stream. Using Newtons law of cooling on a per unit length basis, ( )( ) e s P h D T T π ∞ ′ = − where e P′ is the electrical power dissipated per unit length of the cylinder. For the V = 1 m/s condition, using the data from the table above, find ( ) 2 h 450W m 0.025m 300 40 C 22.0W m K π = × − = ⋅ $ < Repeating the calculations, find the convection coefficients for the remaining conditions which are tabulated above and plotted below. Note that h is not linear with respect to the air velocity. (b) To determine the (C,n) parameters, we plotted h vs. V on log-log coordinates. Choosing C = 22.12 W/m2⋅K(s/m)n, assuring a match at V = 1, we can readily find the exponent n from the slope of the h vs. V curve. From the trials with n = 0.8, 0.6 and 0.5, we recognize that n = 0.6 is a reasonable choice. Hence, C = 22.12 and n = 0.6. < 0 2 4 6 8 10 12 Air velocity, V (m/s) 20 40 60 80 100 Coefficient, h (W/m^2.K) Data, smooth curve, 5-points 1 2 4 6 8 10 Air velocity, V (m/s) 10 20 40 60 80 100 Coefficient, h (W/m^2.K) Data , smooth curve, 5 points h = C * V^n, C = 22.1, n = 0.5 n = 0.6 n = 0.8