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## Lecture 5 Energy Balance

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ENERGY BALANCE

- Concerned with energy changes and energy flow in a chemical process.
- Conservation of energy – first law of thermodynamics i.e.

accumulation of energy in a system = energy input – energy output

Forms of energy

- Potential energy (mgh)
- Kinetic energy (1/2 mv2)
- Thermal energy – heat (Q) supplied to or removed from a process
- Work energy – e.g. work done by a pump (W) to transport fluids
- Internal energy (U) of molecules

m – mass (kg)

g – gravitational constant, 9.81 ms-2

v – velocity, ms-1

IUPAC convention

- heat transferred to a system is +ve and heat transferred from a system is –ve

- work done on a system is+ve and work done by a system is -ve

STEADY STATE/NON-STEADY STATE

- Non steady state - accumulation/depletion of energy in system

Uses

- Heat required for a process
- Rate of heat removal from a process
- Heat transfer/design of heat exchangers
- Process design to determine energy requirements of a process
- Pump power requirements (mechanical energy balance)
- Pattern of energy usage in operation
- Process control
- Process design & development
- etc

1) No mass transfer (closed or batch)

∆E = Q + W

2) No accumulation of energy, no mass transfer (m1 = m2= m)

∆E = 0 Q = -W

3. No accumulation of energy but with mass flow

Q + W = ∆ [(H + K + P)m]

Air is being compressed from 100 kPa at 255 K (H = 489 kJ/kg) to 1000kPa and 278 K (H = 509 kJ/kg) The exit velocity of the air from the compressor is 60 m/s. What is the power required (in kW) for the compressor if the load is 100 kg/hr of air?

SAMPLE PROBLEM

Water is pumped from the bottom of a well 4.6 m deep at a rate of 760 L/hour into a vented storage tank to maintain a level of water in a tank 50 m above the ground. To prevent freezing in the winter a small heater puts 31,600 kJ/hr into the water during its transfer from the well to the storage tank. Heat is lost from the whole system at a constant rate of 26,400 kJ/hr. What is the temperature of the water as it enters the storage tank assuming that the well water is at 1.6oC? A 2-hp pump is used. About 55% of the rated horse power goes into the work of pumping and the rest is dissipated as heat to atmosphere.

ENTHALPHY BALANCE

- p.e., k.e., W terms = 0
- Q = H2 – H1 or Q = ΔH

, where H2 is the total enthalpy of output streams and H1is the total enthalpy of input streams, Q is the difference in total enthalpy i.e. the enthalpy (heat) transferred to or from the system

continued

- Q –ve (H1>H2), heat removed from system
- Q +ve (H2>H1), heat supplied to system.

Example – steam boiler

Two input streams: stream 1- 120 kg/min. water, 30 deg cent., H = 125.7 kJ/kg; stream 2 – 175 kg/min, 65 deg cent, H= 272 kJ/kg

One output stream: 295 kg/min. saturated steam(17 atm., 204 deg cent.), H = 2793.4 kJ/kg

continued

Ignore k.e. and p.e. terms relative to enthalpy changes for processes involving phase changes, chemical reactions, large temperature changes etc

Q = ΔH (enthalpy balance)

Basis for calculation 1 min.

Steady state

Q = Hout – Hin

Q = [295 x 2793.4] – [(120 x 125.7) + (175 x 272)]

Q = + 7.67 x 105 kJ/min

STEAM TABLES

- Enthalpy values (H kJ/kg) at various P, T

Enthalpy changes

- Change of T at constant P
- Change of P at constant T
- Change of phase
- Solution
- Mixing
- Chemical reaction
- crystallisation

SAMPLE PROBLEM:

Calculate the ∆Hrxn for the following reaction:

4NH3(g) + 5O2(g) → 4NO(g) + 6H2O(g)

Given the following ∆Hfo/mole at 25oC.

NH3(g): -46.191 kJ/mol

NO(g): +90.374 kJ/mol

H2O(g): -214.826 kJ/mol

ENERGY BALANCE THAT ACCOUNT FOR CHEMICAL REACTIONS:

Most common:

- What is the temperature of the incoming or exit streams
- How much material must be introduced into the entering stream to provide for a specific amount of heat transfer.

SAMPLE PROBLEM:

An iron pyrite ore containing 85.0% FeS2 and 15.0% inert materials (G) is roasted with an amount of air equal to 200% excess air according to the reaction

4FeS2 + 11O2→ 2Fe2O3 + 8 SO2

in order to produce SO2. All the inert materials plus the Fe2O3 end up in the solid waste product, whick analyzes at 4.0% FeS2. Determine the heat transfer per kg of ore to keep the product stream at 25oC if the entering streams are at 25oC.

∆E = 0, W = 0, ∆E = 0, ∆PE = 0, ∆KE = 0

Therefore Q = ∆H

Latent heats (phase changes)

- Vapourisation (L to V)
- Melting (S to L)
- Sublimation (S to V)

Mechanical energy balance

- Consider mechanical energy terms only
- Application to flow of liquids

ΔP + Δ v2+g Δh +F = W

ρ 2

where W is work done on system by a pump and F is frictional energy loss in system (J/kg)

ΔP = P2 – P1; Δ v2 = v22 –v12; Δh = h2 –h1

- Bernoulli equation (F=0, W=0)

Example - Bernoulli eqtn.

Water flows between two points 1,2. The volumetric flow rate is 20 litres/min. Point 2 is 50 m higher than point 1. The pipe internal diameters are 0.5 cm at point 1 and 1 cm at point 2. The pressure at point 2 is 1 atm..

Calculate the pressure at point 2.

Paul Ashall, 2008

continued

ΔP/ρ + Δv2/2 + gΔh +F = W

ΔP = P2 – P1 (Pa)

Δv2 = v22 – v12

Δh = h2 - h1 (m)

F= frictional energy loss (mechanical energy loss to system) (J/kg)

W = work done on system by pump (J/kg)

ρ = 1000 kg/m3

continued

Volumetric flow is 20/(1000.60) m3/s

= 0.000333 m3/s

v1 = 0.000333/(π(0.0025)2) = 16.97 m/s

v2 = 0.000333/ (π(0.005)2) = 4.24 m/s

(101325 - P1)/1000 + [(4.24)2 – (16.97)2]/2 + 9.81.50 = 0

P1 = 456825 Pa (4.6 bar)

Sensible heat/enthalpy calculations

- ‘Sensible’ heat – heat/enthalpy that must be transferred to raise or lower the temperature of a substance or mixture of substances.
- Heat capacities/specific heats (solids, liquids, gases,vapours)
- Heat capacity/specific heat at constant P, Cp(T) = dH/dT or ΔH = integral Cp(T)dT between limits T2 and T1
- Use of mean heat capacities/specific heats over a temperature range
- Use of simple empirical equations to describe the variation of Cp with T

continued

e.g. Cp = a + bT + cT2 + dT3

,where a, b, c, d are coefficients

ΔH = integralCpdT between limits T2, T1

ΔH = [aT + bT2 + cT3 + dT4]

2 3 4

Calculate values for T = T2, T1 and subtract

Note: T may be in deg cent or K - check units for Cp!

Example

Calculate the enthalpy required to heat a stream of nitrogen gas flowing at 100 mole/min., through a gas heater from 20 to 100 deg. cent.

(use mean Cp value 29.1J mol-1 K-1or Cp = 29 + 0.22 x 10-2T + 0.572 x 10-5T2 – 2.87 x 10-9 T3, where T is in deg cent)

Heat capacity/specific heat data

- Felder & Rousseau pp372/373 and Table B10
- Perry’s Chemical Engineers Handbook
- The properties of gases and liquids, R. Reid et al, 4th edition, McGraw Hill, 1987
- Estimating thermochemical properties of liquids part 7- heat capacity, P. Gold & G.Ogle, Chem. Eng., 1969, p130
- Coulson & Richardson Chem. Eng., Vol. 6, 3rd edition, ch. 8, pp321-324
- ‘PhysProps’

Example – change of phase

A feed stream to a distillation unit contains an equimolar mixture of benzene and toluene at 10 deg cent.The vapour stream from the top of the column contains 68.4 mol % benzene at 50 deg cent. and the liquid stream from the bottom of the column contains 40 mol% benzene at 50 deg cent.

[Need Cp (benzene, liquid), Cp (toluene, liquid), Cp (benzene, vapour), Cp (toluene, vapour), latent heat of vapourisation benzene, latent heat of vapourisation toluene.]

Energy balances on systems involving chemical reaction

- Standard heat of formation (ΔHof) – heat of reaction when product is formed from its elements in their standard states at 298 K, 1 atm. (kJ/mol)

aA + bB cC + dD

-a -b +c +d (stoichiometric coefficients, νi)

ΔHofA, ΔHofB, ΔHofC, ΔHofD (heats of formation)

ΔHoR = c ΔHofC + d ΔHofD - a ΔHofA - bΔHofB

Paul Ashall, 2008

Heat (enthalpy) of reaction

- ΔHoR –ve (exothermic reaction)
- ΔHoR +ve (endothermic reaction)

Paul Ashall, 2008

ENTHALPHY BALANCE - REACTOR

Qp = Hproducts – Hreactants + Qr

Qp – heat transferred to or from process

Qr – reaction heat (ζ ΔHoR), where ζ is extent of reaction and is equal to [moles component,i, out – moles component i, in]/ νi

Paul Ashall, 2008

Hproducts

Hreactants

Qr

+ve

-ve

Qp

Note: enthalpy values must be calculated with reference to a

temperature of 25 deg cent

ENERGY BALANCE TECHNIQUES

- Complete mass balance/molar balance
- Calculate all enthalpy changes between process conditions and standard/reference conditions for all components at start (input) and finish (output).
- Consider any additional enthalpy changes
- Solve enthalpy balance equation

Paul Ashall, 2008

EXAMPLES

- Reactor
- Crystalliser
- Drier
- Distillation

References

- The Properties of Gases and Liquids, R. Reid
- Elementary Principles of Chemical Processes, R.M.Felder and R.W.Rousseau

Paul Ashall, 2008

SAMPLE PROBLEM:

Limestone (CaCO3) is converted into CaO in a continuous vertical kiln. Heat is supplied by combustion of natural gas (CH4) in direct contact with limestone using 50% excess air. Determine the kg of CaCO3 that can be processed per kg of natural gas. Assume that the following heat capacities apply.

Cp of CaCO3 = 234 J/mole –oC

Cp of CaO = 111 j/mole - oC

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