So Far:Conservation of Mass and EnergyPressure Drop in PipesFlow Measurement InstrumentsFlow Control (Valves) Types of Pumps and Pump Sizing

This Week:Energy BalanceHeat Transfer

Conservation of Energy

For steady flow systems

Energy = Heat (Q), Work (W), mass (h)

No Phase Change, E = m c ΔT

Phase Change, E = m hfg where hfg = enthalpy of vaporization or fusion

systemoutin dt

dEEE Systemoutin EEE

0dt

dEsystem

A 2 m3 water tank is filled with 1.25 m3 of hot water at 80C and 0.75 m3 of cold water at 10C. Assume that the specific heat of water is 4.2 kJ/kg.K.

a) Determine the temperature in the tank after it has been filled.

b) How much heat must be added to the tank to bring its temperature to 65C?

c) If a 30 kW electric heater is used, how long will the heating process take?

500 kg of grain (25C) is mixed with hot (80C) and cold (10C) water for mashing. The water to grain ratio (by weight) is 3:1 and the specific heat capacities of the water and grain are 4.2 and 1.7 kJ/kg.K, respectively.

a) If the desired “mash in” temperature is 38C, how much hot and cold water should be added?

(Continued) A three step mashing process, with 20 minute-long rests at 50, 62 and 72C, is desired. The mash should be heated quickly, but not too quickly between rests; with an optimal rate of 1C per minute. Neglect heat losses to the surroundings.

b) Plot the mash temperature vs. time.c) Determine the heating power required, in

kW.d) Determine the total heat required for the

mashing process, in kJ.

Two types of heat sources are available for mashing, electric resistance heaters and steam. The steam enters a heating jacket around the mash as dry, saturated steam at 300 kPa and it exits the system as wet, saturated steam at the same pressure (enthalpy of vaporization = 2150 kJ/kg).

(e) What is steam flow rate required, in kg/s?(f) If steam is used, what is the total mass of

steam required, in kg?

At the location of our brewery, electricity costs $0.14/kW-hr and the steam can be generated for $0.03 per kg.

(g) What is the mashing cost when electric resistance heaters are used?

(h) What is the cost with steam?

Energy Balance ExampleThe power goes out at your brewery due to an overheated transformer, shutting down your fermentation cooling mechanism. Consider a 25 m3 cylindroconical vessel that is full with a product at 10oC, specific heat of 3.8 kJ/kg.K, and density of 1025 kg/m3. Assuming that the sum of heat gains from the surroundings and conversion from fermentation is 7 kW, determine the temperature after 8 hours. How would the 7 kW load change over time?

Heat Transfer EquipmentMash Tun – External heating jacketKettle – External jackets/panels, internal coils, internal or external calandriaWort cooler – Plate heat exchangerFermenter – Internal or external coils or panelsPasteurizers – Plate heat exchangers, TunnelRefrigeration equipment – Shell and tube heat exchangers, evaporative condensersSteam and hot water equipment – Shell and tube

Heat Transfer Equipment

Mash Tun – External heating jacket

Steam in

Steam out

Wort

Heat Transfer Equipment

Mash Tun – External heating jacket

Heat Transfer Equipment

Wort kettle – Internal calandria

Steam

Heat Transfer Equipment

Wort kettle – External calandria

Steam

Heat Transfer Equipment

Wort kettle – Internal calandria

Heat Transfer Equipment

Plate Heat Exchanger

Heat Transfer Equipment

Plate Heat Exchanger

Heat Transfer Equipment

Shell and tube heat exchanger

Heat TransferTransfer of energy from a high temperature to low temperature

Conservation of EnergyEin – Eout = Esystem

Qin = m(u2 – u1) = mc(T2-T1)

WortQin

Heat TransferRate of Ein – Rate of Eout = Rate of E Accumulation

Calculate the rate of heat transfer required to cool 100 L/min of wort from 85 to 25C. The wort has a density of 975 kg/m3 and specific heat of 4.0 kJ/kg.K.

Wort

Qout

min

0)( outinout hhmQ

outinpout TTcmQ

Heat TransferRate of Ein – Rate of Eout = Rate of E Accumulation

WortH2O

0,,,, 22222 outOHinOHOHpOHOHin TTcmQ

0,,,, outwortinwortwortpwortwortout TTcmQ

0,,,,,, 2222 outOHinOHOHpOHoutwortinwortwortpwort TTcmTTcm

Heat TransferRate of Ein – Rate of Eout = Rate of E Accumulation

Wort is being cooled with chilled water in a heat exchanger. The wort enters at 85C with a flow rate of 100 L/min and it exits the heat exchanger at 25C. The chilled water enters at 5C with a flow rate of 175 L/min. The specific heat of the wort and water are 4.0 and 4.2 kJ/kg.K Determine the exit temperature of the chilled water.

WortH2O

ConductionTransfer of microscopic kinetic energy from one

molecule to another1-D Heat Transfer, Fourier Equation:

A 0.5 m2, 1.75 cm thick stainless steel plate (k = 50 W/m.K) has surface temperatures of 22.5 and 20C. Calculate the rate of heat transfer through the plate.

xTkAQ

TUAQ

1U

x1

k1

x2

k2

x3

k3

...

ConductionSame equations apply for multi-layer systems1-D Heat Transfer, Fourier Equation:

How would the rate of heat transfer change if a 2.5 cm thick layer of insulation (k = 0.05 W/m.K) were added to the “low” temperature side of the plate?

TUAQ

1U

x1

k1

x2

k2

x3

k3

...

ConvectionTransfer of heat due to a moving fluidNatural convection – buoyant forces drive flowForced convection – mechanical forces drive flow

Tem

pera

ture

Tfluid

Twall

Fluid Wall

wallfluidconvection TThAQ

Heat TransferOverall Heat Transfer Coefficient

For “thin walled” heat exchangers, Ai = Ao

TAUQ o

1Uconduction

xk

1Uconvection

1h

1Uo

1

houtsidexkw

1hinside

ConvectionA tube-in-tube heat exchanger carries hot wort at 85C in the inner tube and chilled water at 5C in the outer tube. The tube wall thickness is 4 mm and its thermal conductivity is 100 W/m.K. The wort film coefficient is 750 W/m2.K and the chilled water film coefficient is 3000 W/m2K. Determine the overall heat transfer coefficient and the rate of heat transfer per meter of heat exchanger length. The diameter of the pipe is 4.0 cm.

ConvectionCondensation

Constant temperature processOccurs when a saturated comes in contact with a surface with temperature below Tsat

for the vaporFilm coefficients: 5,000-20,000 W/m2.K

BoilingConstant temperature processSome surface roughness promotes boilingBubbles rise – significant natural convectionFraction of surface “wetted” effects QFig 9, page 114 in Kunze.

Log Mean Temperature DifferenceParallel Flow Counter Flow

Length

Tem

pera

ture

T1 T T2

Length

Tem

pera

ture T1

TT2

Log Mean Temperature Difference

A tube-in-tube, counterflow heat exchanger carries hot wort at 85C in the inner tube and chilled water at 5C in the outer tube. The tube wall thickness is 4 mm and its thermal conductivity is 100 W/m.K. The wort film coefficient is 750 W/m2.K and the chilled water film coefficient is 3000 W/m2K. Determine the overall heat transfer coefficient and the rate of heat transfer per meter of heat exchanger length.Calculate the LMTD.

2

1

21

lnTTTT

Tm

FoulingLayers of dirt, particles, biological growth, etc. effect resistance to heat transfer

We cannot predict fouling factors analyticallyAllow for fouling factors when sizing heat transfer

equipmentHistorical information from similar applicationsLittle fouling in water side, more on productTypical values for film coefficient, p. 122

ioodirtyo

RRUU

11

,

Heat Exchanger SizingBeer, dispensed at a rate of 0.03 kg/s, is chilled in an ice

bath from 18C to 8C. The beer flows through a stainless steel cooling coil with a 10 mm o.d., 9 mm i.d., and thermal conductivity of 100 W/m.K. The specific heat of the beer is 4.2 kJ/kg.K and the film heat transfer coefficients on the product and coolant sides are 5000 W/m2.K and 800 W/m2.K, respectively. The fouling factors on the product and coolant sides are 0.0008 and 0.00001 m2K/W. Assume that the heat exchanger is thin walled.

a. Determine the heat transfer rateb. Determine the LMTDc. Determine the overall heat transfer coefficientd. Determine the outside area requirede. Determine the length of tube required

RadiationVibrating atoms within substance give off photons

Emissivity of common substancesPolished aluminum: 0.04Stainless steel: 0.60Brick: 0.93Water: 0.95Snow: 1.00

Radiation between surface and surroundings:

4T RadiatedEnergy

4surr

4surf TT Q surfsurf A

RadiationSometimes, we’ll make an analogy to convection

A 3 cm diameter, 15 m long pipe carries hot wort at 85C. The pipe has 1.0 cm thick insulation, which has thermal conductivity of 0.08 W/m.K. The insulation exterior surface temperature is 35C and its emissivity is 0.85. The temperature of the surroundings is 20C. Determine the rate of heat loss by radiation.

surrsurfrad TT Q surfradAh

Heat LossesTotal Heat Loss = Convection + RadiationPreventing heat loss, insulation

Air – low thermal conductivityAir, goodWater – relatively high thermal conductivityWater, badVessels/pipes above ambient temperature – open pore structure to allow water vapor outVessels/pipes below ambient temperature - closed pore structure to avoid condensation

ConductionHollow cylinders (pipes)

r2

r1

1

2

12

ln2

rrrr

LAm TUAQ m