Räkneövningar 4 av 5,

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1 Kemiteknik - Värme- och strömningsteknik Processteknikens grunder (PG) 05 Räkneövningar av 5,.0.05 kurs-assistent C. Haikarainen frågor; -7 utan svar; 8- med svar. KJ05- Q7-5 A freezer rejects 0 W of heat into a room at C. he freezer temperature is 3 C. An ice cube tray containing 0.5 kg of liquid water at 0 C is placed in the freezer and is completely solidified in 8 minutes. Is this freezer irreversible, reversible, or impossible? o determine this, calculate the coefficient of performance (COP) of the freezer and compare it with the best possible value for COP which is the COP for a Carnot cycle operating under these conditions. Ignore the energy required to cool the plastic tray and consider only the water. he latent heat of fusion of water is kj/kg. Use for the specific heat of the water (at ~ 0 C) c 95 J/(kg.K).. Old PG exam question ( p.) I en stor porös katalysatorplatta med tjockleken cm uppmättes en temperaturprofil i enlighet med Figuren intill, som kan beskrivas med ekvationen: ( x ) x x. x där x (m) är positionen räknad från en sida och är temperaturen (K). Materialets värmelednings-koefficient λ 0 W/(mK). Beräkna värmeströmmen Q (W/m ) från katalysatorn vid x 0 m och x 0.0 m. (glöm inte + eller tecket!) 3. Old PG exam question ( p.) En termosflaska är fylld med liter kaffe ( kg, specifik värmekapacitet c p, kj/(kg K)) vid 90 C. rycket i flaskan är konstant bar. Om den totala värmeöverföringskoefficienten (vid 0 C) är U 0,80 W/(m K), hur länge räcker det innan kaffet har svalnat till 80 C? Flaskans yta är 0,07 m i medeltal. Själva flaskans massa kan försummas. A Dewar flask (or thermos bottle) is filled with litre ( kg) coffee (specific heat cp, kj/(kg K) at 90 C. he pressure inside the flask is constant at bar. If the total heat transfer coefficient for heat transfer from the coffee to the surroundings (at 0 C) is equal to U 0,80 W/(m K), how long does it take until the coffee has cooled to 80 C? he average surface of the flask is A 0.07 m. he mass of the bottle itself may be neglected.

2 . Old PG exam question (8 p.) En plan murad vägg åtskiljer en eldstad, där temperaturen är 600 C, från ett lagerytrumme med temperaturen 30 C. Väggen består av två materialskikt: först 7 cm eldfast tegelmurning med värmeledningskoefficienten λ 0,85 W/(m K) och sedan 3 cm vanlig tegelvägg med λ 0, W/(m K). Värmeöverföringskoefficienten kan på eldstadssidan antas vara h 95 W/(m K) och på lagersidan h 30 W/(m K). Värmestrålningen kan försummas. a. Beräkna vilken värmeströmtäthet Q (W/m ) under fortfarighet ( steady state ) kommer att passera genom väggen och rita en graf som visar temperaturfördelningen i väggen. b. För att ytterligare minska värmeförlusten genom väggen tänker man på lagersidan lägga en isolering bestående av cellplastskivor med tjockleken 5 cm och λ 3 0,038 W/(m K), och smälttemperaturen 300 C. Beräkna den nya temperaturfördelningen i väggen då denna åtgärd har utförts. 5. Old PG exam question (8 p.) En het kvävgasström (ṁ,0 kg/s) med temperaturen C måste kylas ner och skickas därför genom ett långt rör med inre diametern D 0, m och konstanta väggtemperaturen w 00 C. En mätning utförs med ett termoelement vid L,0 m efter inloppet, vilket ger mätvärdet tc 60 C. Målet med mätningen är att beräkna genomsnittliga värmeöverföringskoefficienten <h w > mellan gasen och väggen (i medeltal över längden L) a. Kolla ifall gasflödet är turbulent eller laminärt och beräkna värdet för genomsnittliga värmeöverföringskoefficienten <h w > (W/(m K)) baserat på termoelementets värde. b. emperaturvärdet från termoelementet tc misstänks skilja från gasens temperatur g p.g.a. värmestrålning från termoelementet till rörets väggar. Anta att termoelementet är sfäriskt med en diameter d tc,0 mm och emissiviteten ε 0.5, beräkna kvävgasens riktiga temperatur ( C) och det riktiga värdet för värmeöverföringskoefficienten <h w > (W/(m K)). Data för kvävgasen: Värmeledningen λ 0,05 W/(m K); specifika värmen c p 000,0 J/(kg K); dynamiska viskositeten η,0 0-5 Pa s; Prandtls tal Pr 0,73. A hot stream of nitrogen gas (ṁ,0 kg/s) with temperature C must be cooled and therefore it is sent through a long tube with inner diameter D 0, m and constant wall temperature w 00 C. A measurement using a thermocouple is made at L,0 m from the inlet, which gives a reading of tc 60 C. he goal with the measurement is to calculate the average heat transfer coefficient <hw> between the gas and the wall (averaged over length L). a. Check if the gas flow is turbulent or laminar and calculate the value of the average heat transfer coefficient <hw> (W/(m K)) based on the thermocouple reading. b. he temperature reading of the thermocouple tc is suspected to be different from the temperature of the gas g as a result of heat radiation from the thermocouple to the tube wall. Assuming the thermocouple to be spherical with a diameter dtc,0 mm, with emissivity ε 0.5, calculate the real temperature of the nitrogen gas ( C), and calculate the correct value for the average heat transfer coefficient <hw> (W/(m K)) Data for the nitrogen gas: heat conductivity λ 0,05 W/(m K); specific heat cp 000,0 J/(kg K); dynamic viscosity η,0 0-5 Pa s; Prandtl number Pr 0,73.

3 6. Old PG exam question (8 p.) Ett cylindriskt ångrör med diametern D 70 mm, utan isoleringsmaterial, går genom ett rum där luftens ( ) och väggarnas (surr) temperatur är 5 C. Rörets temperatur på utsidan är s 00 C och emissiviteten för värmestrålningen är ε 0,8. Värmeöverföringskoefficienten för konvektiv värmeöverföring med omgivningen är h 5 W/m K. Se figuren intill. Beräkna: a. Intensiteterna för inkommande och utgående värmestrålningen G och E, i W/m. b. Den totala värmeströmmen som röret avger per längdenhet, i W/m. 7. Old PG exam question ( p) En lågtrycks vattenångaström ṁ S 0, kg/s från en kristallisator som fungerar vid 50 C och 0 mbar kpa måste kondenseras i en sk. shell-and-tube kondensor, mha. vatten från den närliggande ån. Om somrarna kan åns vatten vara C, emedan temperaturen vilken vattnet återlämnas från kondensorn till ån inte får överskrida 7 C. a. Vilket flöde skall flöda i rören och vilket genom skalet och varför? Och är horisontell eller vertikal layout att föredra och varför? b. Vad måste kondensorns kondensoreffekt vara (i kw) och hur mycket kylvatten ( in C) behövs för detta? c. Estimera ett värde för den totala värmeöverföringskoefficienten U (se abell 5.5 i kursmaterialet (text) eller del 5. sista powerpoint-slide) (i W/m K) och beräkna värmeöverförings arean A (m ). Data: Vatten Water ~ 5 C, 00 kpa Ånga Steam ~50 C, kpa Värme kapacitet Heat capacity c p J/(kg K) Dynamisk viskositet Dynamic viscosity η (Pa s),0 0-3,3 0-5 Densitet Density ρ (kg/m 3 ) 000 0,7 Värme konduktivitet Heat conductivity λ (W/(m K) 0,59 0,03 Kondensations/Förångnings värme Condensation / vaporization heat ΔH(kJ/kg) A low pressure steam flow ṁs 0, kg/s from a crystallizer that operates at 50 C and 0 mbar kpa must be condensed in a shell-and-tube condenser, using water from a nearby river. During the summer the temperature of the river water can be C, while the temperature at which the water is returned to the river from the condenser may not exceed 7 C. a. Which of the two streams shall flow inside the tubes and which shall flow through the shell, and why? And, is horizontal or vertical lay-out preferable and why? b. What must be the condensing effect (or duty ) of the condenser (in kw) and how much cooling water (in C) is needed for this? c. Estimate a value for the overall heat transfer coefficient U (see able 5.5 in course material text, or last powerpoint-slide section 5.) (in W/m K) and calculate the heat exchange surface area A (m ).

4 -6 An astronaut with a surface area of.8 m generates 50 W of body heat during a space walk. he exterior of the spacesuit is exposed to outer space, which is black at K. Both exterior and interior surfaces of the suit are silvered with an emissivity of 0.. It is necessary to keep the surface of the astronaut s body, which has an emissivity of 0.85, at no less than 6 o C in steady state. Should the suit be made of one layer of silvered material or two layers? he suit follows the contours of the astronaut s body closely; therefore assume radiation between the body and the suit is like radiation between two infinite parallel plates. Equate the generated heat to the net radiation from the body to the inside of the suit. Similarly, equate the net radiation on the inside to that on the outside of the suit. Solve for body temperature.. Conduction in the suit is neglected.. he suit and body may be represented as two infinite parallel planes. 3. he suit and body are gray, diffuse surfaces.. Space is black at K. 5. Reflections between locations on the suit exterior are negligible. 6. No radiation is incident on the astronaut from the sun or the space vehicle. Let the body be surface and the inside of the suit be surface. he radiation from the body to the inside of the suit is given by (see Fig. -35) A( ) Q σ + ε ε In steady state, the body heat is equal to the heat leaving the outside of the suit. If the suit has a single layer of silvered material, then the net radiation from the exterior of the suit to the environment, which is black at temperature 3, is ( Q Aεσ 3 ) Solving for Q + 3 Aεσ Substituting this into the first equation gives Q σ A 3 Aεσ Q + ε ε Solving for produces Q ( ) 50 W 8 Aσ W ( 50 W) (.8m ) ( K) ε ε ε m K Q Aσ 8 W (.8m ) m K o 95 K C his is above the limit of 6ºC. here is no need for a second layer of silvered material, which would produce a body temperature higher than ºC. - 6

5 3-7 An insulated copper wire with a length of. m carries 0 A of current. he copper is mm in diameter and the insulation (k 0.3 W/m oc) has a thickness of 0.8 cm. Air at 5 o C blows in crossflow over the wire to produce an external convective heat transfer coefficient of 9 W/m K. Assuming the copper is isothermal, find the copper temperature. ake the electrical resistivity of copper to be constant at. x 0-8 Ω m. Use the electrical resistivity of the wire to find the wire s electrical resistance. hen calculate the power dissipated by Joule heating. Finally add the conductive resistance across the insulation to the convective resistance and use the total resistance to compute the wire temperature.. he copper is a perfect conductor and hence, isothermal.. he thermal conductivity and electrical resistivity are constant. 3. Heat transfer is one-dimensional.. he heat transfer coefficient is uniform over the surface of the wire and independent of temperature. he radii of the inner and outer surfaces of the insulation are.5mm r m ( 000 mm/m) m r r+ 0.8cm m 00 cm he cross-sectional area, A, of the copper is x 7 π(0.0005) m Ax πr he electrical resistance of the wire is 8 ρl (. 0 Ω m)(.m) Re 0.03 Ω -7 Ax m he power produced by Joule heating in the wire is Q i R (0 A) (0.03 Ω ).8 W he thermal resistances are ln ( r / r) ln ( /0.0005) K R.89 π Lk W W π (.m) 0.3 m K K R 0.07 ha W s W 9 π (0.0085m)(. m) m K where A s is the surface area of the outside of the insulation. emperature is found from 3 Q R + or + Q ( R + R ) R C+(.8 W)( ) W K o 63 C 3-9

6 3-0 A freezer maintains one side of a slab of ice 3 cm thick at 0 o C. he other side exchanges heat with the ambient at 5 o C by combined natural convection and radiation. In steady state, the ice does not melt. Find the highest possible value of heat transfer coefficient on the ice surface exposed to the ambient. Use the resistance analogy to relate the surface temperature to the heat transfer coefficient.. he combined radiative/convective heat transfer coefficient does not depend significantly on temperature.. he thermal conductivity is constant. 3. he heat transfer coefficient is uniform over the surface of the ice. When the ice surface temperature,, is 0C, the heat transfer coefficient has its maximum value. If the heat transfer coefficient is higher than this value, the ice would begin to melt. Conduction through the ice occurs in series with combined convection/radiation at the outer surface, as shown in the figure. he conduction resistance is: L R ka he heat conducted through the ice is ka( ) Q R L For the combined convection and radiation, the resistance is R ha he heat conducted through the ice is equal to the heat arriving by convection and radiation; therefore, 3 Q ha( 3 ) R Eliminating Q gives ka( ) ha( 3 ) L Solving for h, and using the thermal conductivity of ice, W.88 [ 0 ( 0) ] k ( ) m K h L ( 3 ) (0.03 m) ( 5-0).8 W m o C 3-9

7 3-30 An insulated steel pipe carries hot water at 80 o C. he outer surface of the insulation loses heat to the environment by convection and radiation. For convection, assume h conv 5.8 W/m oc. he emissivity of the insulation is he surroundings are at 30 o C. Assume the inner surface of the insulation is at the water temperature. What is the surface temperature of the insulation? Use data on the figure below. Use the thermal resistance analogy to find the unknown temperature.. he thermal conductivity is constant.. Heat transfer is one-dimensional. 3. he heat transfer coefficient is uniform over the surface of the pipe and independent of temperature.. he insulation is gray and diffuse. he total heat transfer coefficient is the sum of radiative and convective heat transfer coefficients, thus htot hconv + hrad h ( )( conv + εσ s + surr s + surr ) From the resistance circuit w s s surr R R which becomes w s s surr ln r r htotπ r L π Lk It is not possible to solve this equation directly for the unknown temperature s because h tot is a function of s. By trial and error, the solution is o s.8 C Comments: Equation-solving software is very useful in problems of this type. Remember to use absolute temperauture (Kelvin) in the computations. 3-3