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Fundamentals of Heat and Mass Transfer 6th Edition Incropera, Dewitt, Bergman, Lavine - Solutions

A thin-walled, uninsulated O.3-m-diameter duct is used to route chilled air at 0.05 kg/s through the attic of a large commercial building. The attic air is at 37°C, and natural circulation provides a convection coefficient of 2 W/m2 ∙ K at the outer surface of the duct. If chilled air enters a
The problem of heat losses from a fluid moving through a buried pipeline has received considerable attention. Practical applications include the trans-Alaska pipeline as well as power plant steam and water distribution lines. Consider a steel pipe of diameter D that is used to transport oil flowing
In order to maintain pump power requirements per unit flow rate below an acceptable level, operation of the oil pipeline of the preceding problem is subject to the constraint that the oil exit temperature Tm,o exceed 110°C. For the values of Tm,j Ts, D, ti, z, L, and k; prescribed in Problem 8.63,
You are designing an operating room heat exchange device to cool blood (bypassed from a patient) from 40 to 30°C by passing the fluid through a coiled tube sitting in a vat of water-ice mixture. The volumetric flow rate (∀) is 10-4 m3 /min; the tube diameter (D) is 2.5 mm; and Tm,j and Tm,o
Pressurized water at Tm,j = 200°C is pumped at In = 2 kg/s from a power plant to a nearby industrial user through a thin-walled, round pipe of inside diameter D = 1 m. The pipe is covered with a layer of insulation of thickness t = 0.15 m and thermal conductivity k = 0.05 Wm ∙ K. The pipe, which
Water at 290 K and 0.2 kg/s flows through a Teflon tube (k = 0.35 W/m ∙ K) of inner and outer radii equal to 10 and 13 nun, respectively. A thin electrical heating tape wrapped around the outer surface of the tube delivers a uniform surface heat flux of 2000 W/m2, while a convection coefficient
The temperature of flue gases flowing through the large stack of a boiler is measured by means of a thermocouple enclosed within a cylindrical tube as shown. The tube axis is oriented normal to the gas flow, and the thermocouple senses a temperature T, corresponding to that of the tube surface. The
In a biomedical supplies manufacturing process, a requirement exists for a large platen that is to be maintained at 45 Â± 0.25Â°C. The proposed design features the attachment of heating tubes to the platen at a relative spacing S. The thick-walled, copper tubes have an inner diameter of D; = 8
Ground source heat pumps operate by using a liquid rather than ambient air, as the heat source (or sink) for winter heating (or summer cooling). The liquid flows in a closed loop through plastic tubing that is buried at a depth for which annual variations in the temperature of the soil are much
Compare Nusselt predictions based on the Colburn. Dittus-Boelter, Sieder and Tate, and Gnielinski correlations for the fully developed turbulent flow of water in a smooth circular tube at Reynolds numbers of 4000, 104, and 105 when the average mean temperature is 295 K and the surface temperature
For a sharp-edged inlet and a combined entry region the average Nusselt number may be computed from Equation 8.63, with C = 24ReD–023 and m = 0.815 – 2.08 x 10–6 ReD [20]. Determine NuD/NuD,fd at x/D = 10 and 60 for ReD = 104 and 105.
Fluid enters a thin-waned tube of 5-mm diameter and 2-m length with a flow rate of 0.04 kg/s and a temperature of Tm,i = 85°C. The tube surface is maintained at a temperature of Ts = 25°C, and for this operating condition, the outlet temperature is Tm.o = 31.1oC. What is the outlet temperature if
Air at 3 x 10-4 kg/s and 27°C enters a rectangular duct that is I m long and 4 mm by 16 mm on a side. A uniform heat flux of 600 W/m2 is imposed on the duct surface. What is the temperature of the air and of the duct surface at the outlet?
Air at 4 x 10-4 kg/s and 27°C enters a triangular duct that is 20 mm on a side and 2 m long. The duct surface is maintained at 100°C. Assuming fully developed flow throughout the duct, determine the air outlet temperature.
Air at 25Â°C flows at 30 x 10-6 kg/s within 100-mm-long channels used to cool a high thermal conductivity metal mold. Assume the flow is hydrodynamically and thermally fully developed.(a) Determine the heat transferred to the air for a circular channel (D = 10 mm) when the mold temperature is
A device that recovers heat from high-temperature combustion products involves passing the combustion gas between parallel plates, each of which is maintained at 350 K by water flow on the opposite surface. The plate separation is 40 mm, and the gas flow is fully developed. The gas may be assumed
Air at I atm and 285 K enters a 2-m-long rectangular duct with cross section 75 mm by 150 mm. The duct is maintained at a constant surface temperature of 400 K and the air mass flow rate is 0.10 kg/s. Determine the heat transfer rate from the duct to the air and the air outlet temperature.
A double-wall heat exchanger is used to transfer heat between liquids flowing through semicircular copper tubes. Each tube has a wall thickness of t = 3 mm and an inner radius of r; = 20 mm. and good contact is maintained at the plane surfaces by tightly wound straps. The tube outer surfaces are
You have been asked to perform a feasibility study on the design of a blood warmer to be used during the transfusion of blood to a patient. This exchanger is to heat blood taken from the bank at 10Â°C to 37Â°C at a flow rate of 200 ml/min. The blood passes through a rectangular cross-section
A coolant flows through a rectangular channel (gallery) within the body of a mold used to form metal injection parts. The gallery dimensions are a = 90 mm and b = 9.5 mm, and the fluid flow rate is 1.3 x 10-3 m3/s. The coolant temperature is 15Â°C, and the mold wall is at an approximately uniform
A cold plate is an active cooling device that is attached to a heat-generating system in order to dissipate the heat while maintaining the system at an acceptable temperature. It is typically fabricated from a material of high thermal conductivity, k cp ' within which channels are machined and a
The cold plate design of Problem 8.82 has not been optimized with respect to selection of the channel width, and we wish to explore conditions for which the rate of heat transfer may be enhanced. Assume that the width and height of the copper cold plate are fixed at W = 100 mm and H = 10 mm, while
An electronic circuit board dissipating 50 W is sandwiched between two ducted, forced-air-cooled heat sinks. The sinks are 150 mm in length and have 24 rectangular passages 6 mm by 25 mm. Atmospheric air at a volumetric flow rate of 0.060 m3/s and 27Â°C is drawn through the sinks by a blower.
To slow down large prime movers like locomotives a process termed dynamic electric braking is used to switch the traction motor to a generator mode in which mechanical power from the drive wheels is absorbed and used to generate electrical current. As shown in the schematic, the electric power is
To cool electronic components that are mounted to a printed circuit board and hermetically sealed from their surroundings, two boards may be joined to form an intermediate channel through which the coolant is passed. Termed a hollow-core PCB, all of the heat generated by the components may be
A printed circuit board (PCB) is cooled by laminar, fully developed air flow in adjoining, parallel-plate channels of length L and separation distance a. The channels may be assumed to be of infinite extent in the transverse direction. and the upper and lower surfaces are insulated. The temperature
Referring to Figure 8.11 consider conditions in an annulus having an outer surface that is insulated (q: = 0) and a uniform heat flux q"i at the inner surface. Fully developed, laminar flow may be assumed to exist.(a) Determine the velocity profile u(r) in the annular region.(b) Determine the
Consider the air heater of Problem 8.38, but now with air flow through the annulus and steam flow through the inner tube. For the prescribed conditions and an outer tube diameter of Do = 65 mm, determine the outlet temperature and pressure of the air, as well as the mass rate of steam condensation.
Consider a concentric tube annulus for which the inner and outer diameters are 25 and 50 mm. Water enters the annular region at 0.04 kg/s and 25°C. If the inner tube wall is heated electrically at a rate (per unit length) of q' = 4000 W/m, while the outer tube wall is insulated, how long must the
It is common practice to recover waste heat from an oil-or gas-fired furnace by using the exhaust gases to preheat the combustion air. A device commonly used for this purpose consists of a concentric pipe arrangement for which the exhaust gases are passed through the inner pipe, while the cooler
A concentric tube arrangement, for which the inner and outer diameters are 80 mm and 100 mm, respectively, is used to remove heat from a biochemical reaction occurring in a 1-m-long settling tank. Heat is generated uniformly within the tank at a rate of 105 W/m3, and water is sup plied to the
Water at m = 0.02 kg/s and Tm.i = 20°C enters an annular region formed by an inner tube of diameter Di = 25 mm and an outer tube of diameter Do = 100 mm. Saturated steam flows through the inner tube, maintaining its surface at a uniform temperature of Ts,j = 100°C, while the outer surface of the
For the conditions of Problem 8.93, how long must the annulus be if the water flow rate is 0.30 kg/s instead of 0.02 kg/s?
Consider the air cooling system and conditions of Problem 8.31, but with a prescribed pipe length of L = 15 m.(a) What is the air outlet temperature, Tm.o? What is the fan power requirement?(b) The convection coefficient associated with airflow in the pipe may be increased twofold by inserting a
Consider sterilization of the pharmaceutical product of Problem 8.27. To avoid any possibility of heating the product to an unacceptably high temperature, atmospheric steam is condensed on the exterior of the tube instead of using the resistance heater, providing a uniform surface temperature, Ts =
An electrical power transformer of diameter 230 mm and height 500 mm dissipates 1000 W. It is desired to maintain its surface temperature at 47Â°C by supplying ethylene glycol at 24Â°C through thin-walled tubing of 20-mm diameter welded to the lateral surface of the transformer. All the heat
A bayonet cooler is used to reduce the temperature of a pharmaceutical fluid. The pharmaceutical fluid flows through the cooler, which is fabricated of 10-mm-diameter, thin-walled tubing with two 250-mm-Iong straight sections and a coil with six and a half turns and a coil diameter of 75 mm. A
(a) In conventional mold design, straight cooling (heating) passages are bored through the mold in a location where the passages will not interfere with the molded part. Determine the initial heating rate and the initial cooling rate of the mold when five 5-mm-diameter 60-mm-Iong passages are bored
Consider the pharmaceutical product of Problem 8.27. Prior to finalizing the manufacturing process, test trials are performed to experimentally determine the dependence of the shelf life of the drug as a function of the sterilization temperature. Hence, the sterilization temperature must be
An extremely effective method of cooling high-power density silicon chips involves etching micro-channels, in the back (non-circuit) surface of the chip. The channels are covered with a silicon cap and cooling is maintained by passing water through the channels.Consider a chip that is 10 mm by 10
Consider the micro-channel cooling arrangement of Problem 8.101. However, instead of assuming the entire chip and cap to be at a uniform temperature, adopt a more conservative (and realistic) approach that prescribes a temperature of Ts = 350 K at the base of the channels (x = 0) and allows for a
Due to its comparatively large thermal conductivity, water is a preferred fluid for convection cooling. However, in applications involving electronic devices water must not come into contact with the devices, which would therefore have to be hermetically sealed. To circumvent related design and
A novel scheme for dissipating heat from the chips of a multichip array involves machining coolant channels in the ceramic substrate to which the chips are attached. The square chips (L c = 5 mm) are aligned above each of the channels, with longitudinal and transverse pitches of SL = ST = 20 mm.
An experiment is designed to study micro-scale forced convection. Water at Tm,i = 300 K is to be heated in a straight, circular glass tube with a 50-μm inner diameter and a wall thickness of 1 mm. Warm water at T∞ = 350 K, V = 2 m/s is in cross flow over the exterior tube surface. The experiment
An experiment is devised to measure liquid flow and convective heat transfer rates in micro-scale channels. The mass flow rate through a channel is determined by measuring the amount of liquid that has flowed through the channel and dividing by the duration of the experiment. The mean temperature
In the processing of very long plastic tubes of 2-mm inside diameter, air flows inside the tubing with a Reynolds number of 1000, the interior layer of the plastic material evaporates into the air under fully developed conditions. Both plastic and air are at 400 K, and the Schmidt number for the
Air at 300 K and a flow rate of 3 kg/h passes upward through a 30-mm tube, as shown in the sketch. A thin film of water, also at 300 K, slowly falls downward on the inner surface of the tube. Determine the convection mass transfer coefficient for this situation.
What is the convection mass transfer coefficient associated with fully developed atmospheric airflow at 27°C and 0.04 kg/s through a 50-mm-diameter tube whose surface has been coated with a thin layer of naphthalene? Determine the velocity and concentration entry lengths.
Air flowing through a tube of 75-mm diameter passes over a 150-mm-Iong roughened section that is constructed from naphthalene having the properties M = 128.16 kg/kmol and psat(300 K) = 1.31 x 10-4 bar. The air is at 1 atm and 300 K, and the Reynolds number is ReD = 35,000. In an experiment for
Dry air at 35Â°C and a velocity of 10 m/s flows over a thin-walled tube of 20-mm diameter and 200-mm length, having a fibrous coating that is water-saturatedTo maintain an approximately uniform surface temperature of 27Â°C, water at a prescribed flow rate and temperature passes through the
Consider gas flow of mass density P and rate in through a tube whose inner surface is coated with a liquid or a sublimable solid of uniform vapor density PA.s. Derive Equation 8.84 for variation of the mean vapor density, PA,m, with distance x from the tube entrance and Equation 8.81 for the total
Atmospheric air at 25°C and 3 x 10-4 kg/s flows through a 10-mm-diameter, 1-m-long circular tube whose inner surface is wetted with a water film. Determine the water vapor density at the tube outlet, assuming the inlet air to be dry. What is the rate at which vapor is added to the air?
Using the values of density for water in Table A.6, calculate the volumetric thermal expansion coefficient at 300 K from its definition. Equation 9.4 and compare your result with the tabulated value.
Consider an object of characteristic length 0.01 m and a situation for which the temperature difference is 30°C. Evaluating thermo physical properties at the prescribed conditions, determine the Rayleigh number for the following fluids: air (1 atm, 400 K), helium (1 atm, 400 K), glycerin (285 K),
To assess the efficacy of different liquids for cooling by natural convection it is convenient to introduce a figure of merit. F N, which combines the influence of all pertinent fluid properties on the convection coefficient. If the Nusselt number is governed by an expression of the form, NuL -
In many cases, we are concerned with free convection involving gases that are contained within sealed enclosures. Consider air at 27°C and pressures of 1, 10, and 100 bars. Determine the figure of merit described in Problem 9.3 for each of these three pressures. Which air pressure will provide the
The heat transfer rate due to free convection from a vertical surface, 1 m high and 0.6 m wide, to quiescent air that is 20 K colder than the surface is known. What is the ratio of the heat transfer rate for that situation to the rate corresponding to a vertical surface, 0.6 m high and 1 m wide,
Consider a large vertical plate with a uniform surface temperature of 130°C suspended in quiescent air at 25°C and atmospheric pressure.(a) Estimate the boundary layer thickness at a location 0.25 m measured from the lower edge.(b) What is the maximum velocity in the boundary layer at this
A number of thin plates are to be cooled by vertically suspending them in a water bath at a temperature of 200C. If the plates are initially at 54°C and are 0.15 m long what minimum spacing would prevent interference between their free convection boundary layers?
A square aluminum plate 5 mm thick and 200 mm on a side is heated while vertically suspended in quiescent air at 40°C. Determine the average heat transfer coefficient for the plate when its temperature is 15°C by two methods: using results from the similarity solution to the boundary layer
Consider an array of vertical rectangular fins, which is to be used to cool an electronic device mounted in quiescent, atmospheric air at T∞ = 27Â°C. Each fin has L = 20 mm and H = 150 mm and operates at an approximately uniform temperature of Ts = = 77Â°C.(a) Viewing each fin surface as a
Determine the average convection heat transfer coefficient for the 2.5-m-high vertical walls of a home having respective interior air and wall surface temperatures of(a) 20 and 10°C and(b) 27 and 37°C
Beginning with the free convection correlation of the form given by Equation 9.24, show that for air at atmospheric pressure and a film temperature of 400 K. the average heat transfer coefficient for a vertical plate can be expressed as
A solid object is to be cooled by submerging it in a quiescent fluid, and the associated free convection coefficient is given by h = C∆T1/4, where C is a constant and ∆T = T – Tx.(a) Invoking the lumped capacitance approximation, obtain an expression for the time required for the object to
A household oven door of 0.5-m height and 0.7-m width reaches an average surface temperature of 32°C during operation. Estimate the heat loss to the room with ambient air at 22°C. If the door has an emissivity of 1.0 and the surroundings are also at 22°C, comment on the heat loss by free
An aluminum alloy (2024) plate, heated to a uniform temperature of 227°C, is al10wed to cool while vertically suspended in a room where the ambient air and surroundings are at 27°C. The plate is 0.3 m square with a thickness of 15 mm and an emissivity of 0.25.(a) Develop an expression for the
The plate described in Problem 9.14 has been used in an experiment to determine the free convection heat transfer coefficient. At an instant of time when the plate temperature was 127°C, the time rate of change of this temperature was observed to be -0.0465 K/s. What is the corresponding free
The ABC Evening News Report in a news segment on hypothermia research studies at the University of Minnesota claimed that heat loss from the body is 30 times faster in 10°C water than in air at the same temperature. Is that a realistic statement?
Consider a vertical, single-pane window of equivalent width and height (W = L = 1m). The interior surface is exposed to the air and walls of a room, which are each at 18°C. Under cold ambient conditions for which a thin layer of frost has formed on the inner surface, what is the heat loss through
During a winter day, the window of a patio door with a height of 1.8 m and width of 1.0 m shows a frost line near its base. The room wall and air temperatures are 15°C.(a) Explain why the window would show a frost layer at the base rather than at the top.(b) Estimate the heat loss through the
A vertical, thin pane of window glass that is 1 m on a side separates quiescent room air at T∞,i = 20Â°C from quiescent ambient air at T∞.o = - 20Â°C. The walls of the room and the external surroundings (landscape. buildings, etc.) are also at Tsur,i = 20Â°C and Tsur,o = -20Â°C,
Consider the conditions of Problem 9.19, but now allow for a difference between the inner and outer surface temperatures, Ts,i and Ts,o of the window. For a glass thickness and thermal conductivity of t g = 10 mm and kg = 1.4 W/m ∙ K. respectively, evaluate Ts,i and Ts,o What is the heat loss
Consider the conveyor system described in Problem 7.24, but under conditions for which the conveyor is not moving and the air is quiescent. Radiation effects and interactions between boundary layers on adjoining surfaces may be neglected.(a) For the prescribed plate dimensions and initial
A thin-walled container with a hot process fluid at 50Â°C is placed in a quiescent, cold water bath at 100C. Heat transfer at the inner and outer surfaces of the container may be approximated by free convection from a vertical plate.(a) Determine the overall heat transfer coefficient between the
Consider an experiment to investigate the transition to turbulent flow in a free convection boundary layer that develops along a vertical plate suspended in a large room. The plate is constructed of a thin heater that is sandwiched between two aluminum plates and may be assumed to be isothermal.
A square plate of pure aluminum 0.5 m on a side and 16 mm thick is initially at 300°C and is suspended in a large chamber. The walls of the chamber are maintained at 27°C, as is the enclosed air. If the surface emissivity of the plate is 0.25, what is the initial cooling rate? Is it reasonable to
The vertical rear window of an automobile is of thickness L = 8 mm and height H = 0.5 m and contains fine-meshed heating wires that can induce nearly uniform volumetric heating, q (W/m3).(a) Consider steady-state conditions for which the interior surface of the window is exposed to quiescent air at
Determine the maximum allowable uniform heat flux that may be imposed at a wall heating panel 1 m high if the maximum temperature is not to exceed 37°C when the ambient air temperature is 25°C.
The components of a vertical circuit board, 150 mm on a side, dissipate 5 W. The back surface is well insulated and the front surface is exposed to quiescent air at 27Â°C.Assuming a uniform surface heat flux, what is the maximum temperature of the board? What is the temperature of the board for
Circuit boards are mounted to interior vertical surfaces of a rectangular duct of height H = 400 mm and length L = 800 mm. Although the boards are cooled by forced convection heat transfer to air flowing through the duct, not all of the heat dissipated by the electronic components is transferred to
A refrigerator door has a height and width of H = 1 m and W = 0.65 m, respectively, and is situated in a large room for which the air and walls are at T∞ = Tsur = 25Â°C. The door consists of a layer of polystyrene insulation (k = 0.03 W/m ∙ K) sandwiched between thin sheets of steel (s = 0.6)
Air at 3 atm and 100Â°C is discharged from a compressor into a vertical receiver of 2.5-m height and 0.75-m diameter. Assume that the receiver wall has negligible thermal resistance is at a uniform temperature, and that heat transfer at its inner and outer surfaces is by free convection from a
In the central receiver concept of a solar power plant many heliostats at ground level are used to direct a concentrated solar flux q to the receiver which is positioned at the top of a tower. However even with absorption of all the solar flux by the outer surface of the receiver losses due to free
An experimental apparatus is shown in the schematic for measuring the local convection coefficient and the boundary layer temperature distribution for a heated vertical plate immersed in an extensive, quiescent fluid. The plate is maintained at a uniform temperature by circulating a
Consider the transformer of Problem 8.97, whose lateral surface is being maintained at 47Â°C by a forced convection coolant line removing 1000 W. It is desired to explore cooling of the transformer by free convection and radiation, assuming the surface to have an emissivity of 0.80. (a) Determine
Airflow through a long, O.2-m-square air conditioning duct maintains the outer duct surface temperature at 10°C. If the horizontal duct is uninsulated and exposed to air at 35°C in the crawlspace beneath a home, what is the heat gain per unit length of the duct?
Consider the conditions of Example 9.3, including the effect of adding insulation of thickness t and thermal conductivity k = 0.035 W/m. K to the duct. We wish to now include the effect of radiation on the outer surface temperatures and the total heat loss per unit length of duct.(a) If Ts,i =
An electrical heater in the form of a horizontal disk of 400-mm diameter is used to heat the bottom of a tank filled with engine oil at a temperature of 5°C. Calculate the power required to maintain the heater surface temperature at 70°C.
Consider a horizontal 6-mm-thick, 100-mm-Iong straight fin fabricated from plain carbon steel (k = 57 W/m ∙ K, ε = 0.5). The base of the fin is maintained at 150°C, while the quiescent ambient air and the surroundings are at 25°C. Assume the fin tip is adiabatic.(a) Estimate the fin heat rate
The thermal conductivity and surface emissivity of a material may be determined by heating its bottom surface and exposing its top surface to quiescent air and large surroundings of equivalent temperatures, T∞ = Tsur = 25Â°C. The remaining surfaces of the sample/heater are well
Convection heat transfer coefficients for a heated horizontal surface facing upward may be determined by a gage whose specific features depend on whether the temperature of the surroundings is known. For configuration A, a copper disk, which is electrically heated from below, is encased in an
A circular grill of diameter 0.25 m and emissivity 0.9 is maintained at a constant surface temperature of 130°C. What electrical power is required when the room air and surroundings are at 24°C?
Many laptop computers are equipped with thermal management systems that involve liquid cooling of the central processing unit (CPU), transfer of the heated liquid to the back of the laptop screen assembly, and dissipation of heat from the back of the screen assembly by way of a flat, isothermal
Consider the roof of the refrigerated truck compartment described in Problem 7.20, but under conditions for which the truck is parked (V = 0), an other conditions remain unchanged. For as = B = 0.5, determine the outer surface temperature, Ts.o, and the heat load imposed on the refrigeration system.
The 4-m by 4-m horizontal roof of an uninsulated aluminum melting furnace is comprised of a 0.08-m-thick fireclay brick refractory covered by a 5-mm-thick steel (AISI 1010) plate. The refractory surface exposed to the furnace gases is maintained at 1700 K during operation, while the outer surface
A stereo receiver/amplifier is enclosed in a thin metallic case for which the top, horizontal surface is 0.5 m x 0.5 m on a side. The surface, which is not vented, has an emissivity of ε = 0.8 and is exposed to quiescent ambient air and large surroundings for which T∞ = Tsur = 25°C. For surface
At the end of its manufacturing process, a silicon wafer of diameter D = 150 mm, thickness δ = 1 mm, and emissivity ε = 0.65 is at an initial temperature of Ti = 325°C and is an owed to cool in quiescent, ambient air and large surroundings for which T∞ = Tsur = 25°C.(a) What is the
A 200-mm-square, 10-mm-thick tile has the thermo physical properties of pyrex (ε = 0.80) and emerges from a curing process at an initial temperature of Ti = 140°C. The backside of the tile is insulated while the upper surface is exposed to ambient air and surroundings at 25°C.(a) Estimate the
Integrated circuit (IC) boards are stacked within a duct and dissipate a total of 500 W. The duct has a square cross section with w = H = 150 mm and a length of 0.5 m. Air flows into the duct at 25°C and 1.2 m3/min, and the convection coefficient between the air and the inner surfaces of the duct
A highly polished aluminum plate of length 0.5 m and width 0.2 m is subjected to an airstream at a temperature of 23°C and a velocity of 10 m/s. Because of upstream conditions the flow is turbulent over the entire length of the plate. A series of segmented, independently controlled heaters is

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