Energy Engineering - Heat and Mass Transfer

Completed solution of the exercises book Incropera

Other

PROBLEM 1.1 KNOWN: Heat rate, q, through one-dimensional wall of area A, thickness L, thermal conductivity k and inner temperature, T 1. FIND: The outer temperature of the wall, T 2. 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, qqqA=-kdT dxA=kATT Lcondxx12 == ¢¢××- . Solving for T 2 gives TTqL kA21cond=-. Substituting numerical values, find TC-3000W 0.025m 0.2W / m K 10m2 2 =´ ״ 415 $ T C - 37.5 C 2=415 $$ TC. 2=378 $ < COMMENTS: Note direction of heat flow and fact that T 2 must be less than T 1. PROBLEM 1.2 KNOWN: Inner surface temperature and thermal conductivity of a concrete wall. FIND: Heat loss by conduction through the wall as a function of ambient air temperatures ranging from -15 to 38°C. SCHEMATIC: ASSUMPTIONS: (1) One-dimensional conduction in the x-direction, (2) Steady-state conditions, (3) Constant properties, (4) Outside wall temperature is that of the ambient air. ANALYSIS: From Fourier’s law, it is evident that the gradient, x dT dx q k¢¢ =- , is a constant, and hence the temperature distribution is linear, if x q¢¢ and k are each constant. The heat flux must be constant under one-dimensional, steady-state conditions; and k is approximately constant if it depends only weakly on temperature. The heat flux and heat rate when the outside wall temperature is T 2 = -15°C are () 2 12 x 25 C 15 C dT T T q k k 1 W m K 133.3 W m dx L 0.30 m-- - ¢¢ =- = = × = $$ . (1) 22 xx q q A 133.3 W m 20 m 2667 W¢¢ =´= ´ = . (2) < Combining Eqs. (1) and (2), the heat rate q x can be determined for the range of ambient temperature, -15 £ T 2 £ 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.Kk = 1 W/m.K, concrete wallk = 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 () () 12TT 7C q k LW 1.4 W / m K 11m 8 m 4312 W t 0.20 m -° ==״= < The daily cost of natural gas that must be combusted to compensate for the heat loss is () () g d 6 f qC 4312 W $0.01 / MJ C t 24 h / d 3600 s / h $4.14 / d 0.9 10 J / MJ h ´ =D= ´ = ´ < 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 TT m 40-20 C x 2 12′′ = −
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. () TT qk L 15-5 C W q 1.4 m K 0.005m q 2800 W/m .12 x x 2 x− ′′ = ′′ = ⋅ ′′ =
Since the heat flux is uniform over the surface, the heat loss (rate) is q = q xA 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 12 q k A 1.4 W/m K 2m 19, 600 W gg L 0.005m - ==× =$ Double Pane: () T T 25 C 2 12 q k A 0.024 2m 120 W aa 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 = 4m 2, (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 A total ′′⋅∆ Solving for L and recognizing that A total = 5×W 2, 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 TT 2 21 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 () qq A qH2W 2W W W total 1 2 1 2 ¢¢ ¢¢ éù =´ = + + ´ ëû ()( ) 2 q 16.6 W/m 0.6m 1.6m 1.2m 0.8m 0.6m 35.9 Wéù =++´= ëû < COMMENTS: The corners and edges of the container create local departures from one- dimensional conduction, which increase the heat load. However, for H, W 1, W 2 >> 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 D where DT represents the difference in surface temperatures. Since DT is the same for both walls, it follows that L = L k k q q121 22 1 ×¢¢ ¢¢. With the heat fluxes related as ¢¢= ¢¢ q 0.8 q 12 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 TT 12 qkA L - = Hence, qL TT 12 kA =+ where () 2 22 A D / 4 0.2m / 4 0.0314 m . pp== = 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 22 ⋅× ∆= ⋅ ∆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 D D and the gradient can be expressed as D DD T x = E/N S ABt where N is the number of differentially connected thermocouple junctions, S AB is the Seebeck coefficient for type K thermocouples (A-chromel and B-alumel), and Dx = t is the gage thickness. Hence, ¢¢ q= kE NS ABt D ¢¢״ ´ ´´ ´ ´ q = 1.4 W / m K 350 10-6 V 54010-6 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/m 2 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 qhTT ¥ ¢¢ =- For the air stream: () 22 air q 40 W m K 30 5 K 1, 400 W m¢¢ éù =×--= ëû < For the water stream: () 22 water q 900 W m K 30 10 K 18, 000 W 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/m 2 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 = CV n, 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/m 2×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, ()( ) esPhDTT p ¥ ¢ =- 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 450 W m 0.025 m 300 40 C 22.0 W m K p =´ -=× $ < 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/m 2×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 6080100 Coefficient, h (W/m^2.K) Data , smooth curve, 5 pointsh = C * V^n, C = 22.1, n = 0.5n = 0.6n = 0.8 PROBLEM 1.15 KNOWN: Long, 30mm-diameter cylinder with embedded electrical heater; power required to maintain a specified surface temperature for water and air flows. FIND: Convection coefficients for the water and air flow convection processes, h w and h a, respectively. SCHEMATIC: ASSUMPTIONS: (1) Flow is cross-wise over cylinder which is very long in the direction normal to flow. ANALYSIS: The convection heat rate from the cylinder per unit length of the cylinder has the form ()( ) q = h D T T s π ′ − ∞ and solving for the heat transfer convection coefficient, find () q h = . D T T s π ′ − ∞ Substituting numerical values for the water and air situations: Water () 28 10 W/m h = = 4,570 W/m K 0.030m 90-25 C 3 2 w π × ⋅ ×
< Air () 400 W/m h = 65 W/m K. 0.030m 90-25 C 2 a π =⋅ ×
< COMMENTS: Note that the air velocity is 10 times that of the water flow, yet h w ≈ 70 × h a. These values for the convection coefficient are typical for forced convection heat transfer with liquids and gases. See Table 1.1. PROBLEM 1.16 KNOWN: Dimensions of a cartridge heater. Heater power. Convection coefficients in air and water at a prescribed temperature. FIND: Heater surface temperatures in water and air. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) All of the electrical power is transferred to the fluid by convection, (3) Negligible heat transfer from ends. ANALYSIS: With P = q conv , Newton’s law of cooling yields () () P=hA T T h DL T T P TT . hDL ss s π π −= − =+ ∞∞ ∞ In water, TC+2000 W 5000 W / m K 0.02 m 0.200 ms 2= ⋅×× × 20 $ π T C + 31.8 C = 51.8 C.s=20 $$ $ < In air, TC+2000 W 50 W / m K 0.02 m 0.200 ms 2= ⋅×× × 20 $ π T C + 3183 C = 3203 C.s=20 $$ $ < COMMENTS: (1) Air is much less effective than water as a heat transfer fluid. Hence, the cartridge temperature is much higher in air, so high, in fact, that the cartridge would melt. (2) In air, the high cartridge temperature would render radiation significant. PROBLEM 1.17 KNOWN: Length, diameter and calibration of a hot wire anemometer. Temperature of air stream. Current, voltage drop and surface temperature of wire for a particular application. FIND: Air velocity SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Negligible heat transfer from the wire by natural convection or radiation. ANALYSIS: If all of the electric energy is transferred by convection to the air, the following equality must be satisfied () PEIhATT elec s== - ¥ where () 52 A DL 0.0005m 0.02m 3.14 10 m . pp - == ´ =´ Hence, () () EI 5V 0.1A 2 h 318 W/m K 52 AT T s 3.14 10 m 50 C´ == =× - - ¥ ´$ () 2 52 5 2 V 6.25 10 h 6.25 10 318 W/m K 6.3 m/s-- =´ =´ × = < COMMENTS: The convection coefficient is sufficiently large to render buoyancy (natural convection) and radiation effects negligible. PROBLEM 1.18 KNOWN: Chip width and maximum allowable temperature. Coolant conditions. FIND: Maximum allowable chip power for air and liquid coolants. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Negligible heat transfer from sides and bottom, (3) Chip is at a uniform temperature (isothermal), (4) Negligible heat transfer by radiation in air. ANALYSIS: All of the electrical power dissipated in the chip is transferred by convection to the coolant. Hence, P = q and from Newton’s law of cooling, P = hA(T - T ¥) = h W 2(T - T ¥). In air, P max = 200 W/m 2×K(0.005 m) 2(85 - 15) ° C = 0.35 W. < In the dielectric liquid P max = 3000 W/m 2×K(0.005 m) 2(85-15) ° C = 5.25 W. < COMMENTS: Relative to liquids, air is a poor heat transfer fluid. Hence, in air the chip can dissipate far less energy than in the dielectric liquid. PROBLEM 1.19 KNOWN: Length, diameter and maximum allowable surface temperature of a power transistor. Temperature and convection coefficient for air cooling. FIND: Maximum allowable power dissipation. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Negligible heat transfer through base of transistor, (3) Negligible heat transfer by radiation from surface of transistor. ANALYSIS: Subject to the foregoing assumptions, the power dissipated by the transistor is equivalent to the rate at which heat is transferred by convection to the air. Hence, () Pq hATT elec conv s== - ¥ where () () 2 242 A DL D / 4 0.012m 0.01m 0.012m / 4 4.90 10 m . pp - éù =+ = ´+ =´ êú ëû For a maximum allowable surface temperature of 85°C, the power is () () 242 P 100 W/m K 4.90 10 m 85 25 C 2.94 W elec- =״ -=$ < COMMENTS: (1) For the prescribed surface temperature and convection coefficient, radiation will be negligible relative to convection. However, conduction through the base could be significant, thereby permitting operation at a larger power. (2) The local convection coefficient varies over the surface, and hot spots could exist if there are locations at which the local value of h is substantially smaller than the prescribed average value. PROBLEM 1.20 KNOWN: Air jet impingement is an effective means of cooling logic chips. FIND: Procedure for measuring convection coefficients associated with a 10 mm ´ 10 mm chip. SCHEMATIC: ASSUMPTIONS: Steady-state conditions. ANALYSIS: One approach would be to use the actual chip-substrate system, Case (a), to perform the measurements. In this case, the electric power dissipated in the chip would be transferred from the chip by radiation and conduction (to the substrate), as well as by convection to the jet. An energy balance for the chip yields elec conv cond rad qq q q =++ . Hence, with () conv sqhATT ¥ =- , where A = 100 mm 2 is the surface area of the chip, () elec cond rad sqq q h AT T ¥ -- = - (1) While the electric power ( qelec ) and the jet ( T¥) and surface ( Ts) temperatures may be measured, losses from the chip by conduction and radiation would have to be estimated. Unless the losses are negligible (an unlikely condition), the accuracy of the procedure could be compromised by uncertainties associated with determining the conduction and radiation losses. A second approach, Case (b), could involve fabrication of a heater assembly for which the conduction and radiation losses are controlled and minimized. A 10 mm ´ 10 mm copper block (k ~ 400 W/m×K) could be inserted in a poorly conducting substrate (k < 0.1 W/m×K) and a patch heater could be applied to the back of the block and insulated from below. If conduction to both the substrate and insulation could thereby be rendered negligible, heat would be transferred almost exclusively through the block. If radiation were rendered negligible by applying a low emissivity coating (e < 0.1) to the surface of the copper block, virtually all of the heat would be transferred by convection to the jet. Hence, qcond and qrad may be neglected in equation (1), and the expression may be used to accurately determine h from the known (A) and measured ( qelec , Ts, T¥) quantities. COMMENTS: Since convection coefficients associated with gas flows are generally small, concurrent heat transfer by radiation and/or conduction must often be considered. However, jet impingement is one of the more effective means of transferring heat by convection and convection coefficients well in excess of 100 W/m 2×K may be achieved. PROBLEM 1.21 KNOWN: Upper temperature set point, T set , of a bimetallic switch and convection heat transfer coefficient between clothes dryer air and exposed surface of switch. FIND: Electrical power for heater to maintain T set when air temperature is T ∞ = 50 °C. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Electrical heater is perfectly insulated from dryer wall, (3) Heater and switch are isothermal at T set , (4) Negligible heat transfer from sides of heater or switch, (5) Switch surface, A s, loses heat only by convection. ANALYSIS: Define a control volume around the bimetallic switch which experiences heat input from the heater and convection heat transfer to the dryer air. That is, () E - E = 0 q - hA T T 0. out in sset elec −= ∞

The electrical power required is, () q = hAT T sset elec − ∞ () q = 25 W/m K 30 10 m 70 50 K=15 mW. 2-62 elec⋅× × − < COMMENTS: (1) This type of controller can achieve variable operating air temperatures with a single set-point, inexpensive, bimetallic-thermostatic switch by adjusting power levels to the heater. (2) Will the heater power requirement increase or decrease if the insulation pad is other than perfect? PROBLEM 1.22 KNOWN: Hot vertical plate suspended in cool, still air. Change in plate temperature with time at the instant when the plate temperature is 225 °C. FIND: Convection heat transfer coefficient for this condition. SCHEMATIC: ASSUMPTIONS: (1) Plate is isothermal and of uniform temperature, (2) Negligible radiation exchange with surroundings, (3) Negligible heat lost through suspension wires. ANALYSIS: As shown in the cooling curve above, the plate temperature decreases with time. The condition of interest is for time t o. For a control surface about the plate, the conservation of energy requirement is () E - E = E dT 2hA T T Mc dt out st in ss p −−= ∞


where A s is the surface area of one side of the plate. Solving for h, find () Mc dT h= 2A T T dt p ss − ∞ ()( ) 3.75 kg 2770 J/kg K h= 0.022 K/s=6.4 W/m K 2 0.3 0.3 m 225 25 K 2 2 ×⋅ ×⋅ ×× − < COMMENTS: (1) Assuming the plate is very highly polished with emissivity of 0.08, determine whether radiation exchange with the surroundings at 25 °C is negligible compared to convection. (2) We will later consider the criterion for determining whether the isothermal plate assumption is reasonable. If the thermal conductivity of the present plate were high (such as aluminum or copper), the criterion would be satisfied. PROBLEM 1.23 KNOWN: Width, input power and efficiency of a transmission. Temperature and convection coefficient associated with air flow over the casing. FIND: Surface temperature of casing. SCHEMATIC: ASSUMPTIONS: (1) Steady state, (2) Uniform convection coefficient and surface temperature, (3) Negligible radiation. ANALYSIS: From Newton’s law of cooling, () () 2 ss s qhA T T 6hW T T ¥¥ =-= - where the output power is h P i and the heat rate is () io i q P P P 1 150 hp 746 W / hp 0.07 7833 W h =- = - = ´ ´ = Hence, () s 22 2 q 7833 W T T 30 C 102.5 C 6hW 6 200 W / m K 0.3 m ¥ =+ =°+ = ° ´×´ < COMMENTS: There will, in fact, be considerable variability of the local convection coefficient over the transmission case and the prescribed value represents an average over the surface. PROBLEM 1.24 KNOWN: Air and wall temperatures of a room. Surface temperature, convection coefficient and emissivity of a person in the room. FIND: Basis for difference in comfort level between summer and winter. SCHEMATIC: ASSUMPTIONS: (1) Person may be approximated as a small object in a large enclosure. ANALYSIS: Thermal comfort is linked to heat loss from the human body, and a chilled feeling is associated with excessive heat loss. Because the temperature of the room air is fixed, the different summer and winter comfort levels can not be attributed to convection heat transfer from the body. In both cases, the heat flux is Summer and Winter: () 22 q h T T 2 W/m K 12 C 24 W/m conv s ¢¢ =-= ״ = ¥$ However, the heat flux due to radiation will differ, with values of Summer: () () 44 8 24 4 44 2 q T T 0.9 5.67 10 W/m K 305 300 K 28.3 W/m rad s sur es - ¢¢ =-=´´ × - = Winter: () () 44 8 24 4 44 2 q T T 0.9 5.67 10 W/m K 305 287 K 95.4 W/m rad s sur es - ¢¢ =-=´´ × - = There is a significant difference between winter and summer radiation fluxes, and the chilled condition is attributable to the effect of the colder walls on radiation. COMMENTS: For a representative surface area of A = 1.5 m 2, the heat losses are q conv = 36 W, q rad(summer) = 42.5 W and q rad(winter) = 143.1 W. The winter time radiation loss is significant and if maintained over a 24 h period would amount to 2,950 kcal. PROBLEM 1.25 KNOWN: Diameter and emissivity of spherical interplanetary probe. Power dissipation within probe. FIND: Probe surface temperature. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Negligible radiation incident on the probe. ANALYSIS: Conservation of energy dictates a balance between energy generation within the probe and radiation emission from the probe surface. Hence, at any instant -E + E = 0 out g

εσAT Ess4 g =
E T D 1/ 4 g s 2 επ σ  =   
() 150W T 0.8 0.5 m 5.67 10 1/ 4 s 2 824 W/m K π   =  ×  − ⋅ T K.s=254 7. < COMMENTS: Incident radiation, as, for example, from the sun, would increase the surface temperature. PROBLEM 1.26 KNOWN: Spherical shaped instrumentation package with prescribed surface emissivity within a large space-simulation chamber having walls at 77 K. FIND: Acceptable power dissipation for operating the package surface temperature in the range T s = 40 to 85°C. Show graphically the effect of emissivity variations for 0.2 and 0.3. SCHEMATIC: ASSUMPTIONS: (1) Uniform surface temperature, (2) Chamber walls are large compared to the spherical package, and (3) Steady-state conditions. ANALYSIS: From an overall energy balance on the package, the internal power dissipation P e will be transferred by radiation exchange between the package and the chamber walls. From Eq. 1.7, () 44 rad e s s sur qPATT es == - For the condition when T s = 40 °C, with A s = pD 2 the power dissipation will be () () 4 824 44 e P 0.25 0.10 m 5.67 10 W m K 40 273 77 K 4.3 W p - éù =´ ´´ ״+- = êú ëû < Repeating this calculation for the range 40 £ T s £ 85 °C, we can obtain the power dissipation as a function of surface temperature for the e = 0.25 condition. Similarly, with 0.2 or 0.3, the family of curves shown below has been obtained. 40 50 60 70 80 90 Surface temperature, Ts (C) 2 4 6 8 10 Power dissipation, Pe (W) Surface emissivity, eps = 0.3eps = 0.25eps = 0.2 COMMENTS: (1) As expected, the internal power dissipation increases with increasing emissivity and surface temperature. Because the radiation rate equation is non-linear with respect to temperature, the power dissipation will likewise not be linear with surface temperature. (2) What is the maximum power dissipation that is possible if the surface temperature is not to exceed 85 °C? What kind of a coating should be applied to the instrument package in order to approach this limiting condition? PROBLEM 1.27 KNOWN: Area, emissivity and temperature of a surface placed in a large, evacuated chamber of prescribed temperature. FIND: (a) Rate of surface radiation emission, (b) Net rate of radiation exchange between surface and chamber walls. SCHEMATIC: ASSUMPTIONS: (1) Area of the enclosed surface is much less than that of chamber walls. ANALYSIS: (a) From Eq. 1.5, the rate at which radiation is emitted by the surface is q emitE A = A T s4 =⋅ εσ () () q = 0.8 0.5 m 5.67 10 W/m K 150 + 273 K 4 2-824 emit  ×⋅  q = 726 W. emit < (b) From Eq. 1.7, the net rate at which radiation is transferred from the surface to the chamber walls is () q = A T T 44 ssur εσ − () ()() q = 0.8 0.5 m 5.67 10 W/m K 423K - 298K 44 2-824  ×⋅   q = 547 W. < COMMENTS: The foregoing result gives the net heat loss from the surface which occurs at the instant the surface is placed in the chamber. The surface would, of course, cool due to this heat loss and its temperature, as well as the heat loss, would decrease with increasing time. Steady-state conditions would eventually be achieved when the temperature of the surface reached that of the surroundings. PROBLEM 1.28 KNOWN: Length, diameter, surface temperature and emissivity of steam line. Temperature and convection coefficient associated with ambient air. Efficiency and fuel cost for gas fired furnace. FIND: (a) Rate of heat loss, (b) Annual cost of heat loss. SCHEMATIC: ASSUMPTIONS: (1) Steam line operates continuously throughout year, (2) Net radiation transfer is between small surface (steam line) and large enclosure (plant walls). ANALYSIS: (a) From Eqs. (1.3a) and (1.7), the heat loss is () () 44 qq q AhT T T T conv rad s s sur es éù =+= -+ - ¥ êú ëû where () 2 A DL 0.1m 25m 7.85m . pp== ´ = Hence, () () 22 824444 q 7.85m 10 W/m K 150 25 K 0.8 5.67 10 W/m K 423 298 K- éù =×-+´´×- êú ëû ()()22 q 7.85m 1, 250 1, 095 w/m 9813 8592 W 18, 405 W=+=+= < (b) The annual energy loss is 11 E qt 18, 405 W 3600 s/h 24h/d 365 d/y 5.80 10 J == ´ ´ ´ = ´ With a furnace energy consumption of 11 E E/ 6.45 10 J, ff h ==´ the annual cost of the loss is 5 C C E 0.01 $/MJ 6.45 10 MJ $6450 gf == ´´ = < COMMENTS: The heat loss and related costs are unacceptable and should be reduced by insulating the steam line. PROBLEM 1.29 KNOWN: Exact and approximate expressions for the linearized radiation coefficient, h r and h ra, respectively. FIND: (a) Comparison of the coefficients with e = 0.05 and 0.9 and surface temperatures which may exceed that of the surroundings (T sur = 25°C) by 10 to 100°C; also comparison with a free convection coefficient correlation, (b) Plot of the relative error (h r - r ra)/h r as a function of the furnace temperature associated with a workpiece at T s = 25°C having e = 0.05, 0.2 or 0.9. ASSUMPTIONS: (1) Furnace walls are large compared to the workpiece and (2) Steady-state conditions. ANALYSIS: (a) The linearized radiation coefficient, Eq. 1.9, follows from the radiation exchange rate equation, () () 22 r s sur s sur hTTTT es=+ + If T s » T sur, the coefficient may be approximated by the simpler expression () 3 r,a s sur h4T TTT2 es ==+ For the condition of e = 0.05, T s = T sur + 10 = 35°C = 308 K and T sur = 25°C = 298 K, find that () () 824 223 2 r h 0.05 5.67 10 W m K 308 298 308 298 K 0.32 W m K - =´´ × + + = × < ()() 3 824 3 2 r,a h 4 0.05 5.67 10 W m K 308 298 2 K 0.32 W m K - =´ ´ ´ × + = × < The free convection coefficient with T s = 35°C and T¥ = T sur = 25°C, find that () ( ) 1/ 3 1/ 3 1/ 3 2 s h 0.98 T 0.98 T T 0.98 308 298 2.1 W m K ¥ =D = - = - = × < For the range T s - T sur = 10 to 100°C with e = 0.05 and 0.9, the results for the coefficients are tabulated below. For this range of surface and surroundings temperatures, the radiation and free convection coefficients are of comparable magnitude for moderate values of the emissivity, say e > 0.2. The approximate expression for the linearized radiation coefficient is valid within 2% for these conditions. (b) The above expressions for the radiation coefficients, h r and h r,a, are used for the workpiece at T s = 25°C placed inside a furnace with walls which may vary from 100 to 1000°C. The relative error, (h r - h ra)/h r, will be independent of the surface emissivity and is plotted as a function of T sur. For T sur > 150°C, the approximate expression provides estimates which are in error more than 5%. The approximate expression should be used with caution, and only for surface and surrounding temperature differences of 50 to 100°C. Coefficients (W/m 2×K) T s (°C) e h r h r,a h 35 0.05 0.32 0.32 2.1 0.9 5.7 5.7 135 0.05 0.51 0.50 4.7 0.9 9.2 9.0 100 300 500 700 900 Surroundings temperature, Tsur (C) 0 10 20 30 Relative error, (hr-hra)/hr*100 (%) PROBLEM 1.30 KNOWN: Chip width, temperature, and heat loss by convection in air. Chip emissivity and temperature of large surroundings. FIND: Increase in chip power due to radiation. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Radiation exchange between small surface and large enclosure. ANALYSIS: Heat transfer from the chip due to net radiation exchange with the surroundings is () q = W T - T 244 sur rad εσ () () q = 0.9 0.005 m 5.67 10 W/m K 358 - 288 K 2 824444 rad ×⋅ − q rad = 0.0122 W. The percent increase in chip power is therefore ∆P Pq rad q conv W 0.350 W ×= ×= ×=100 1000 0122 100 3 5%. . . < COMMENTS: For the prescribed conditions, radiation effects are small. Relative to convection, the effect of radiation would increase with increasing chip temperature and decreasing convection coefficient. PROBLEM 1.31 KNOWN: Width, surface emissivity and maximum allowable temperature of an electronic chip. Temperature of air and surroundings. Convection coefficient. FIND: (a) Maximum power dissipation for free convection with h(W/m 2×K) = 4.2(T - T ¥)1/4 , (b) Maximum power dissipation for forced convection with h = 250 W/m 2×K. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Radiation exchange between a small surface and a large enclosure, (3) Negligible heat transfer from sides of chip or from back of chip by conduction through the substrate. ANALYSIS: Subject to the foregoing assumptions, electric power dissipation by the chip must be balanced by convection and radiation heat transfer from the chip. Hence, from Eq. (1.10), () () 44 Pq qhATTATT elec conv rad s s sur es =+= -+ - ¥ where () 2 242 A L 0.015m 2.25 10 m .- == = ´ (a) If heat transfer is by natural convection, () () () 5/4 5/4 25/4 42 q C A T T 4.2 W/m K 2.25 10 m 60K 0.158 W conv s- =- = × ´ = ¥ () () 42 8 2 4 4 4 4 q 0.60 2.25 10 m 5.67 10 W/m K 358 298 K 0.065 W rad-- =´ ´ ×- = P 0.158 W 0.065 W 0.223 W elec=+= < (b) If heat transfer is by forced convection, () () () 242 q hA T T 250 W/m K 2.25 10 m 60K 3.375 W conv s- =-= × ´ = ¥ P 3.375 W 0.065 W 3.44 W elec=+= < COMMENTS: Clearly, radiation and natural convection are inefficient mechanisms for transferring heat from the chip. For T s = 85° C and T ¥ = 25° C, the natural convection coefficient is 11.7 W/m 2×K. Even for forced convection with h = 250 W/m 2×K, the power dissipation is well below that associated with many of today’s processors. To provide acceptable cooling, it is often necessary to attach the chip to a highly conducting substrate and to thereby provide an additional heat transfer mechanism due to conduction from the back surface. PROBLEM 1.32 KNOWN: Vacuum enclosure maintained at 77 K by liquid nitrogen shroud while baseplate is maintained at 300 K by an electrical heater. FIND: (a) Electrical power required to maintain baseplate, (b) Liquid nitrogen consumption rate, (c) Effect on consumption rate if aluminum foil ( ε p = 0.09) is bonded to baseplate surface. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) No heat losses from backside of heater or sides of plate, (3) Vacuum enclosure large compared to baseplate, (4) Enclosure is evacuated with negligible convection, (5) Liquid nitrogen (LN2) is heated only by heat transfer to the shroud, and (6) Foil is intimately bonded to baseplate. PROPERTIES: Heat of vaporization of liquid nitrogen (given): 125 kJ/kg. ANALYSIS: (a) From an energy balance on the baseplate,

E - E = 0 q - q = 0inout elec rad and using Eq. 1.7 for radiative exchange between the baseplate and shroud, () p q = AT - T. 44 pp elec sh εσ Substituting numerical values, with () A = D /4, 2 pp π find () () q = 0.25 0.3 m / 4 5.67 10 W/m K 300 - 77 K 8.1 W. 2 824444 elec π ×⋅ =   − < (b) From an energy balance on the enclosure, radiative transfer heats the liquid nitrogen stream causing evaporation,


E - E = 0 q - m h = 0inout radLN2 fg where
mLN2 is the liquid nitrogen consumption rate. Hence,
/ m LN2 = q radh fg = 8.1 W / 125 kJ / kg = 6.48 10-5 kg / s = 0.23 kg / h. × < (c) If aluminum foil ( ε p = 0.09) were bonded to the upper surface of the baseplate, () () q = q / = 8.1 W 0.09/0.25 = 2.9 W p rad,foil rad fεε and the liquid nitrogen consumption rate would be reduced by (0.25 - 0.09)/0.25 = 64% to 0.083 kg/h . < PROBLEM 1.33 KNOWN: Width, input power and efficiency of a transmission. Temperature and convection coefficient for air flow over the casing. Emissivity of casing and temperature of surroundings. FIND: Surface temperature of casing. SCHEMATIC: ASSUMPTIONS: (1) Steady state, (2) Uniform convection coefficient and surface temperature, (3) Radiation exchange with large surroundings. ANALYSIS: Heat transfer from the case must balance heat dissipation in the transmission, which may be expressed as q = P i – P o = P i (1 - h ) = 150 hp ´ 746 W/hp ´ 0.07 = 7833 W. Heat transfer from the case is by convection and radiation, in which case () () 44 ss ssur qAhT T T T es ¥ éù =-+- êú ëû where A s = 6 W 2. Hence, () ( ) () 2 224444 ss 8 7833 W 6 0.30 m 200 W / m K T 303K 0.8 5.67 10 W / m K T 303 K- =×-+´´×- éù êú ëû A trial-and-error solution yields s T 373 K 100 C»=° < COMMENTS: (1) For T s » 373 K, q conv » 7,560 W and q rad » 270 W, in which case heat transfer is dominated by convection, (2) If radiation is neglected, the corresponding surface temperature is T s = 102.5° C. PROBLEM 1.34 KNOWN: Resistor connected to a battery operating at a prescribed temperature in air. FIND: (a) Considering the resistor as the system, determine corresponding values for () inEW , ()gEW , () outEW and () stEW . If a control surface is placed about the entire system, determine the values for in E , gE , outE , and stE . (b) Determine the volumetric heat generation rate within the resistor, q (W/m 3), (c) Neglecting radiation from the resistor, determine the convection coefficient. SCHEMATIC: ASSUMPTIONS: (1) Electrical power is dissipated uniformly within the resistor, (2) Temperature of the resistor is uniform, (3) Negligible electrical power dissipated in the lead wires, (4) Negligible radiation exchange between the resistor and the surroundings, (5) No heat transfer occurs from the battery, (5) Steady-state conditions. ANALYSIS: (a) Referring to Section 1.3.1, the conservation of energy requirement for a control volume at an instant of time, Eq 1.11a, is in g out st EEE E +- =   where in outE,E correspond to surface inflow and outflow processes, respectively. The energy generation term g E is associated with conversion of some other energy form (chemical, electrical, electromagnetic or nuclear) to thermal energy. The energy storage term stE is associated with changes in the internal, kinetic and/or potential energies of the matter in the control volume. g E , stE are volumetric phenomena. The electrical power delivered by the battery is P = VI = 24V´6A = 144 W. Control volume: Resistor. in outE 0 E 144 W ==  < gst E 144 W E 0 ==  The g E term is due to conversion of electrical energy to thermal energy. The term outE is due to convection from the resistor surface to the air. Continued... PROBLEM 1.34 (Cont.) Control volume: Battery-Resistor System . in outE 0 E 144 W==  < gst E 0 E 144 W==-  The stE term represents the decrease in the chemical energy within the battery. The conversion of chemical energy to electrical energy and its subsequent conversion to thermal energy are processes internal to the system which are not associated with stE or g E . The outE term is due to convection from the resistor surface to the air. (b) From the energy balance on the resistor with volume, " = (p D 2/4)L, () () 2 53 g E q 144 W q 0.06 m / 4 0.25 m q 2.04 10 W m p =" = ´ = ´    < (c) From the energy balance on the resistor and Newton's law of cooling with A s = p DL + 2(p D 2/4), () out cv s sEqhATT ¥ == -  () () 22 144 W h 0.06 m 0.25 m 2 0.06 m 4 95 25 C ppéù =´ ´ +´ - êú ëû $ [] ()2 144 W h 0.0471 0.0057 m 95 25 C =+ - $ 2 h 39.0 W m K =× < COMMENTS: (1) In using the conservation of energy requirement, Eq. 1.11a, it is important to recognize that in E and outE will always represent surface processes and g E and stE , volumetric processes. The generation term g E is associated with a conversion process from some form of energy to thermal energy. The storage term stE represents the rate of change of internal energy. (2) From Table 1.1 and the magnitude of the convection coefficient determined from part (c), we conclude that the resistor is experiencing forced, rather than free, convection. PROBLEM 1.35 KNOWN: Thickness and initial temperature of an aluminum plate whose thermal environment is changed. FIND: (a) Initial rate of temperature change, (b) Steady-state temperature of plate, (c) Effect of emissivity and absorptivity on steady-state temperature. SCHEMATIC: ASSUMPTIONS: (1) Negligible end effects, (2) Uniform plate temperature at any instant, (3) Constant properties, (4) Adiabatic bottom surface, (5) Negligible radiation from surroundings, (6) No internal heat generation. ANALYSIS: (a) Applying an energy balance, Eq. 1.11a, at an instant of time to a control volume about the plate, in out stEE E-=   , it follows for a unit surface area. ()() () ()( ) () () 22 2 2 S S conv G 1 m E 1 m q 1 m d dt McT 1 m L c dT dt a r ¢¢ -- = =´. Rearranging and substituting from Eqs. 1.3 and 1.5, we obtain () ( ) 4 SS i i dT dt 1 Lc G T h T T r aes ¥ =--- éù ëû . () 1 3 dT dt 2700 kg m 0.004 m 900 J kg K - =´´×´ () () 4 2824 2 0.8 900 W m 0.25 5.67 10 W m K 298 K 20 W m K 25 20 C - ´-´´ × -×- éù êú ëû $ dT dt 0.052 C s= $ . < (b) Under steady-state conditions,  E st = 0, and the energy balance reduces to () 4 SS GThTT aes ¥ =+- (2) () 282442 0.8 900 W m 0.25 5.67 10 W m K T 20 W m K T 293 K - ´=´´ ×´+×- The solution yields T = 321.4 K = 48.4°C. < (c) Using the IHT First Law Model for an Isothermal Plane Wall, parametric calculations yield the following results. 0 0.2 0.4 0.6 0.8 1 Coating emissivity, eps 20 30 40 50 60 70 Plate temperature, T (C) Solar absorptivity, alphaS = 1alphaS = 0.8alphaS = 0.5 COMMENTS: The surface radiative properties have a significant effect on the plate temperature, which decreases with increasing e and decreasing a S. If a low temperature is desired, the plate coating should be characterized by a large value of e/a S. The temperature also decreases with increasing h. PROBLEM 1.36 KNOWN: Surface area of electronic package and power dissipation by the electronics. Surface emissivity and absorptivity to solar radiation. Solar flux. FIND: Surface temperature without and with incident solar radiation. SCHEMATIC: ASSUMPTIONS: Steady-state conditions. ANALYSIS: Applying conservation of energy to a control surface about the compartment, at any instant


E - E +E = 0. inout g It follows that, with the solar input, Aq AE P=0 Aq A T P=0 Aq P T. Ass SS 4 sss SS 1/ 4 s SS s s α αεσ α εσ ′′ −+ ′′ −+ ′′  + =   In the shade ()q0,S′′ = 1000 W T 364 K. 1 m 1 5.67 10 W/m K 1/ 4 s 2824 ==  ×× × ⋅  − < In the sun, 0.25 1 m 750 W/m 1000 W T 380 K. 1 m 1 5.67 10 W/m K 1/ 4 22 s 2824  ×× +  ==  ×× × ⋅  − < COMMENTS: In orbit, the space station would be continuously cycling between shade and sunshine, and a steady-state condition would not exist. PROBLEM 1.37 KNOWN: Daily hot water consumption for a family of four and temperatures associated with ground water and water storage tank. Unit cost of electric power. Heat pump COP. FIND: Annual heating requirement and costs associated with using electric resistance heating or a heat pump. SCHEMATIC: ASSUMPTIONS: (1) Process may be modelled as one involving heat addition in a closed system, (2) Properties of water are constant. PROPERTIES: Table A-6, Water ( ave T = 308 K): r = 1 f v- = 993 kg/m 3, p,fc = 4.178 kJ/kg×K. ANALYSIS: From Eq. 1.11c, the daily heating requirement is daily t QUMcT =D = D () fi Vc T Tr=- . With V = 100 gal/264.17 gal/m 3 = 0.379 m 3, () ()33 daily Q 993kg / m 0.379 m 4.178kJ/kg K 40 C 62, 900 kJ =×= $ The annual heating requirement is then, () 7 annual Q 365 days 62, 900 kJ/day 2.30 10 kJ==´ , or, with 1 kWh = 1 kJ/s (3600 s) = 3600 kJ, annual Q 6380 kWh = < With electric resistance heating, annual elec QQ = and the associated cost, C, is () C 6380 kWh $0.08/kWh $510== < If a heat pump is used, () annual elecQCOPW.= Hence, () () elec annual W Q / COP 6380kWh/ 3 2130 kWh=== The corresponding cost is () C 2130 kWh $0.08/kWh $170== < COMMENTS: Although annual operating costs are significantly lower for a heat pump, corresponding capital costs are much higher. The feasibility of this approach depends on other factors such as geography and seasonal variations in COP, as well as the time value of money. PROBLEM 1.38 KNOWN: Initial temperature of water and tank volume. Power dissipation, emissivity, length and diameter of submerged heaters. Expressions for convection coefficient associated with natural convection in water and air. FIND: (a) Time to raise temperature of water to prescribed value, (b) Heater temperature shortly after activation and at conclusion of process, (c) Heater temperature if activated in air. SCHEMATIC: ASSUMPTIONS: (1) Negligible heat loss from tank to surroundings, (2) Water is well- mixed (at a uniform, but time varying temperature) during heating, (3) Negligible changes in thermal energy storage for heaters, (4) Constant properties, (5) Surroundings afforded by tank wall are large relative to heaters. ANALYSIS: (a) Application of conservation of energy to a closed system (the water) at an instant, Eq. (1.11d), yields dU dT dT Mc c q 3q 1 dt dt dt r =="== Hence, () T f t dt c/3q dT 1 0T i r =" òò () () 333 990 kg/m 10gal 3.79 10 m / gal 4180J/kg K t 335 295 K 4180 s 3 500 W- ´´ × =-= ´ < (b) From Eq. (1.3a), the heat rate by convection from each heater is ()()() 4/3 qAqAhTT DL370TT 11 s s p ¢¢ == -= - Hence, () 3/ 4 3/ 4 q 500 W 1 T T T T 24 K s 24/3 370 DL 370 W/m K 0.025 m 0.250 m p p æö æö =+ =+ = + ç÷ ç÷ èø ×´´ ´ èø With water temperatures of T i » 295 K and T f = 335 K shortly after the start of heating and at the end of heating, respectively, T s,i = 319 K T s,f = 359 K < Continued ….. PROBLEM 1.38 (Continued) (c) From Eq. (1.10), the heat rate in air is () () 4/3 44 q DL 0.70 T T T T 1s ssur pes éù =-+- ¥ êú ëû Substituting the prescribed values of q 1, D, L, T ¥ = T sur and e, an iterative solution yields T s = 830 K < COMMENTS: In part (c) it is presumed that the heater can be operated at T s = 830 K without experiencing burnout. The much larger value of T s for air is due to the smaller convection coefficient. However, with q conv and q rad equal to 59 W and 441 W, respectively, a significant portion of the heat dissipation is effected by radiation. PROBLEM 1.39 KNOWN: Power consumption, diameter, and inlet and discharge temperatures of a hair dryer. FIND: (a) Volumetric flow rate and discharge velocity of heated air, (b) Heat loss from case. SCHEMATIC: ASSUMPTIONS: (1) Steady-state, (2) Constant air properties, (3) Negligible potential and kinetic energy changes of air flow, (4) Negligible work done by fan, (5) Negligible heat transfer from casing of dryer to ambient air (Part (a)), (6) Radiation exchange between a small surface and a large enclosure (Part (b)). ANALYSIS: (a) For a control surface about the air flow passage through the dryer, conservation of energy for an open system reduces to ()() m u pv m u pv q 0 io +- + +=  where u + pv = i and q = P elec . Hence, with () ( ) mi i mc T T , io pi o-= -  () mc T T P po i elec-=  () () P 500 W elec m 0.0199 kg/s cT T po i 1007 J/kg K 25 C == = - ×  $ m 0.0199 kg/s 3 0.0181 m / s 3 1.10 kg/m r "= = =  < () 3 4 4 0.0181 m / s V 4.7 m/s o 22 A c D 0.07 m p p ""´ == = = < (b) Heat transfer from the casing is by convection and radiation, and from Eq. (1.10) () () 44 qhAT T A T T ss sssur es =-+ - ¥ Continued ….. PROBLEM 1.39 (Continued) where () 2 A DL 0.07 m 0.15 m 0.033 m . s pp== ´ = Hence, ()() ( ) 22 2824444 q 4W/m K 0.033 m 20 C 0.8 0.033 m 5.67 10 W/m K 313 293 K- =× +´ ´´ × -$ q 2.64 W 3.33 W 5.97 W=+= < The heat loss is much less than the electrical power, and the assumption of negligible heat loss is justified. COMMENTS: Although the mass flow rate is invariant, the volumetric flow rate increases as the air is heated in its passage through the dryer, causing a reduction in the density. However, for the prescribed temperature rise, the change in r, and hence the effect on ," is small. PROBLEM 1.40 KNOWN: Speed, width, thickness and initial and final temperatures of 304 stainless steel in an annealing process. Dimensions of annealing oven and temperature, emissivity and convection coefficient of surfaces exposed to ambient air and large surroundings of equivalent temperatures. Thickness of pad on which oven rests and pad surface temperatures. FIND: Oven operating power. SCHEMATIC: ASSUMPTIONS: (1) steady-state, (2) Constant properties, (3) Negligible changes in kinetic and potential energy. PROPERTIES: Table A.1, St.St.304 () () io T T T /2 775 K=+ = : r = 7900 kg/m 3, c p = 578 J/kg×K; Table A.3, Concrete, T = 300 K: kc = 1.4 W/m×K. ANALYSIS: The rate of energy addition to the oven must balance the rate of energy transfer to the steel sheet and the rate of heat loss from the oven. With in outEE -  = 0, it follows that () elec i oPmuuq0+--= where heat is transferred from the oven. With ()sss mVWt r=  , ()() io piouu cTT -= - , and ()() () 44 oo o o oo s s s sur q2HL 2HWWL hTT TT es ¥ éù =++´-+- êú ëû ()( )coosbckWL T T/t +- , it follows that ()( )( ) elec s s s p o i o o o o o oPVWtcTT2HL2HWWL r=-+++´ () () ()( ) 44 so s s sur coosbc hT T T T k WL T T /t es éù -+ - + - êú ëû ()() 3 elec P 7900 kg/m 0.01m/s 2 m 0.008 m 578J/kg K 1250 300 K =´ ´ ×- ()() 2 2 2m 25m 2 2m 2.4m 2.4m 25m [10W/m K 350 300 K +´ ´ +´ ´ + ´ × - () ()() 8244 44 0.8 5.67 10 W/m K 350 300 K ] 1.4W/m K 2.4m 25m 350 300 K/0.5m - +´ ´ × - + × ´ - Continued.…. PROBLEM 1.40 (Cont.) ()22 elec P 694, 000W 169.6m 500 313 W/m 8400W=+ + + ()694, 000 84, 800 53,100 8400 W 840kW =+++= < COMMENTS: Of the total energy input, 83% is transferred to the steel while approximately 10%, 6% and 1% are lost by convection, radiation and conduction from the oven. The convection and radiation losses can both be reduced by adding insulation to the side and top surfaces, which would reduce the corresponding value of Ts. PROBLEM 1.41 KNOWN: Hot plate-type wafer thermal processing tool based upon heat transfer modes by conduction through gas within the gap and by radiation exchange across gap. FIND: (a) Radiative and conduction heat fluxes across gap for specified hot plate and wafer temperatures and gap separation; initial time rate of change in wafer temperature for each mode, and (b) heat fluxes and initial temperature-time change for gap separations of 0.2, 0.5 and 1.0 mm for hot plate temperatures 300 < T h < 1300°C. Comment on the relative importance of the modes and the influence of the gap distance. Under what conditions could a wafer be heated to 900°C in less than 10 seconds? SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions for flux calculations, (2) Diameter of hot plate and wafer much larger than gap spacing, approximating plane, infinite planes, (3) One-dimensional conduction through gas, (4) Hot plate and wafer are blackbodies, (5) Negligible heat losses from wafer backside, and (6) Wafer temperature is uniform at the onset of heating. PROPERTIES: Wafer: r = 2700 kg/m 3, c = 875 J/kg×K; Gas in gap: k = 0.0436 W/m×K. ANALYSIS: (a) The radiative heat flux between the hot plate and wafer for T h = 600°C and T w = 20° C follows from the rate equation, () ()()() 44 44 8 24 4 2 rad h w q T T 5.67 10 W / m K 600 273 20 273 K 32.5 kW / m s - ¢¢ =- ´ × +-+ = = < The conduction heat flux through the gas in the gap with L = 0.2 mm follows from Fourier’s law, () 2 hw cond 600 20 K TT q k 0.0436 W / m K 126 kW / m L 0.0002 m- - ¢¢ == × = < The initial time rate of change of the wafer can be determined from an energy balance on the wafer at the instant of time the heating process begins, w in out st st i dT EE E E cd dt r æö ¢¢ ¢¢ ¢¢ ¢¢ -= = ç÷ èø    where outE0¢¢ =  and in rad condEqorq.¢¢ ¢¢ ¢¢ =  Substituting numerical values, find 32 wrad 3 i,rad dT q 32.5 10 W / m 17.6 K / s dt cd 2700 kg / m 875 J / kg K 0.00078 m r ¢¢ ´ ö == = ÷ ø ´×´ < w cond i,cond dT q 68.4 K / s dt cd r ¢¢ ö == ÷ ø < Continued ….. PROBLEM 1.41 (Cont.) (b) Using the foregoing equations, the heat fluxes and initial rate of temperature change for each mode can be calculated for selected gap separations L and range of hot plate temperatures T h with T w = 20°C. In the left-hand graph, the conduction heat flux increases linearly with T h and inversely with L as expected. The radiative heat flux is independent of L and highly non-linear with T h, but does not approach that for the highest conduction heat rate until T h approaches 1200°C. The general trends for the initial temperature-time change, (dT w/dt) i, follow those for the heat fluxes. To reach 900°C in 10 s requires an average temperature-time change rate of 90 K/s. Recognizing that (dT w/dt) will decrease with increasing T w, this rate could be met only with a very high T h and the smallest L. 300 500 700 900 1100 1300 Hot plate tem perature, Th (C) 0 100 200 300 400 Heat flux (kW/m ^2) q''radq''cond, L = 1.0 m mq''cond, L = 0.5 m mq''cond, L = 0.2 m m 300 500 700 900 1100 1300 Hot plate temperature, Th (C) 0 50 100 150 200 Initial rate of change, dTw/dt (K.s ^-1) q ''radq''cond, L = 1.0 m mq''cond, L = 0.5 m mq''cond, L = 0.2 m m PROBLEM 1.42 KNOWN: Silicon wafer, radiantly heated by lamps, experiencing an annealing process with known backside temperature. FIND: Whether temperature difference across the wafer thickness is less than 2°C in order to avoid damaging the wafer. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional conduction in wafer, (3) Radiation exchange between upper surface of wafer and surroundings is between a small object and a large enclosure, and (4) Vacuum condition in chamber, no convection. PROPERTIES: Wafer: k = 30 W/m× K, 0.65. ea== " ANALYSIS: Perform a surface energy balance on the upper surface of the wafer to determine w,uT . The processes include the absorbed radiant flux from the lamps, radiation exchange with the chamber walls, and conduction through the wafer. in outEE 0¢¢ ¢¢ -=  s rad cdqq q 0 a ¢¢ ¢¢ ¢¢ --= " () w,u w, 44 sw,usurTT qTTk 0 L aes - ¢¢ --- = " " () () 4 52 8244 4 w,u 0.65 3.0 10 W / m 0.65 5.67 10 W / m K T 27 273 K - ´´ - ´ ´ × - + () w,u 30W / m K T 997 273 K / 0.00078 m 0éù -× -+ = ëû w,uT 1273 K 1000 C ==° < COMMENTS: (1) The temperature difference for this steady-state operating condition, w,u w,lTT , - is larger than 2° C. Warping of the wafer and inducing slip planes in the crystal structure could occur. (2) The radiation exchange rate equation requires that temperature must be expressed in kelvin units. Why is it permissible to use kelvin or Celsius temperature units in the conduction rate equation? (3) Note how the surface energy balance, Eq. 1.12, is represented schematically. It is essential to show the control surfaces, and then identify the rate processes associated with the surfaces. Make sure the directions (in or out) of the process are consistent with the energy balance equation. PROBLEM 1.43 KNOWN: Silicon wafer positioned in furnace with top and bottom surfaces exposed to hot and cool zones, respectively. FIND: (a) Initial rate of change of the wafer temperature corresponding to the wafer temperature w,iT 300 K,= and (b) Steady-state temperature reached if the wafer remains in this position. How significant is convection for this situation? Sketch how you’d expect the wafer temperature to vary as a function of vertical distance. SCHEMATIC: ASSUMPTIONS: (1) Wafer temperature is uniform, (2) Transient conditions when wafer is initially positioned, (3) Hot and cool zones have uniform temperatures, (3) Radiation exchange is between small surface (wafer) and large enclosure (chamber, hot or cold zone), and (4) Negligible heat losses from wafer to mounting pin holder. ANALYSIS: The energy balance on the wafer illustrated in the schematic above includes convection from the upper (u) and lower (l) surfaces with the ambient gas, radiation exchange with the hot- and cool-zone (chamber) surroundings, and the rate of energy storage term for the transient condition. in out stEE E¢¢ ¢¢ ¢¢ -=   w rad,h rad,c cv,u cv,l dT qqqq cd dt r ¢¢ ¢¢ ¢¢ ¢¢ +--= () () ()() 44 44 w w sur,c w u w l w sur,h dT TT TThTThTTcd dt es es r ¥¥ -+ -- -- -= (a) For the initial condition, the time rate of temperature change of the wafer is determined using the energy balance above with ww,iTT 300K, == () () 8244 4 824444 4 0.65 5.67 10 W / m K 1500 300 K 0.65 5.67 10 W / m K 330 300 K -- ´´ × - +´´ × - () () 22 8 W / m K 300 700 K 4 W / m K 300 700 K -×-- ×-= 3 2700 kg / m 875 J / kg K ´× () w i 0.00078 m d T / dt ´ () w i d T / dt 104 K / s = < (b) For the steady-state condition, the energy storage term is zero, and the energy balance can be solved for the steady-state wafer temperature, ww,ssTT. = Continued ….. PROBLEM 1.43 (Cont.) () () 44 4 44 4 w,ss w,ss 0.65 1500 T K 0.65 330 T K ss -+ - () () 22 w,ss w,ss 8 W / m K T 700 K 4 W / m K T 700 K 0 -× -- × -= w,ssT 1251 K = < To determine the relative importance of the convection processes, re-solve the energy balance above ignoring those processes to find () ww,ss i d T / dt 101 K / s and T 1262 K.== We conclude that the radiation exchange processes control the initial time rate of temperature change and the steady-state temperature. If the wafer were elevated above the present operating position, its temperature would increase, since the lower surface would begin to experience radiant exchange with progressively more of the hot zone chamber. Conversely, by lowering the wafer, t