Mass and Energy Balances: Thermodynamics of Water and Steam, Lecture notes of Physics

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Mass and Energy Balances

Outline

-^ Terminology– Specific heat, enthalpy, heat, sensible heat, latent heat, saturationtemperature & pressure, saturated liquid & vapor•^ Phase diagram of water (liquid and vapor phases)– Sub-cooled liquid, condensate, steam (saturated steam: mix ofcondensate & vapor), vapor, superheated steam•^ Steam table•^ Steam quality•^ Superheated steam•^ Energy content (or enthalpy)– Liquid products, steam•^ Mass and energy balances– Batch & continuous processes (mixing & heating); direct & indirectcontact heating; indirect contact cooling; determination of steam quality

Terminology (contd.)

-^ Sensible heat (Q: J or J/s)–^ Energy transferred from hot to cold object•^ For a batch system: Q = m c

T^ p

Q is in J

• For a continuous system: Q = m c

T^ p

Q is in J/s or W

-^ Latent heat (Q: J or J/s)–^ Energy transferred during phase change without any temperature change•^ For a batch system: Q = m

^

Q is in J

-^ For a continuous system: Q = m

^

Q is in J/s or W

-^ Latent heat of vaporization (

: J/kg)vap^

-^ Energy reqd to convert 1 kg of a liquid to vapor phase w/o temperature change–^ It equals energy released when vapor condenses to liquid at that temperature• For heating foods, the energy of condensing steam is made use of– Latent heat of vaporization of water at 100 °C = 2257.06 kJ/kg• Latent heat of fusion (

: J/kg)fus^

-^ Energy required to convert 1 kg of a solid to liquid phase w/o temperature change• Latent heat of fusion of ice at 0 °C = 333.2 kJ/kg

Terminology (contd.)

-^ Saturation temperature and pressure– Any substance can exist in more than one phase at one time• For any pressure, there is a temperature at which a liquid can coexist withthe vapor phase• This pressure and the corresponding temperature are called the saturationpressure and saturation temperature respectively• Example: Water at 101.3 kPa (atmospheric pressure) has a saturationtemperature of 100 °C. At higher pressure, saturation temperature is higher.•^ Saturated liquid and saturated vapor– When only liquid phase exists at the saturation pressure andtemperature, the liquid is called saturated liquid or condensate (itsenthalpy is denoted by H

)c

• When only vapor exists at the saturation pressure and temperature,the vapor is called saturated vapor (its enthalpy is denoted by H

)v^

Saturated Steam Table 361.

Temperature(°C) What is the latent heat of vaporization of water at atmospheric pressure? = Hvap (at atm. Pr.)^

• Hv (at atm Pr.)^

= 2676.1 – 419.04 = 2257.06 kJ/kgc (at atm. Pr.)^

Steam Quality (x)

-^ The term “Steam” generally refers to saturated steam and notsuperheated steam•^ “Steam” is a mixture of liquid (condensate) and vapor•^ The enthalpy of steam (H

) is a weighted mean of enthalpy ofs

condensate (H

) and enthalpy of vapor (Hc

)v

-^ H^ = x Hs^

+ (1 – x) Hv^

c

-^ Rearranging,

x = (H

• H^ )/(H^ s cv^
• H^ )c -^ Note that

x = 0 when H

= H^ s c

-^ Also,^

x = 1 when H

= H^ s^ v

-^ Higher the steam quality, higher the value of H

s

-^ 0 ≤ x ≤ 1

OR

0% ≤ x ≤ 100%

Note: H^ & Hc^

at saturation temperature & pressure are determinedv^ from steam tables

Q versus T^ Q (Energy Supplied)

T (Temperature in ° C)

S to LS

L

L to V^

V

Latent

Latent

Q^1

Q^ =^ = 333.2 kJ/kg^1 fusion(ice)^ Q^ = c^ T = 419.04 kJ/kg^2 p(water)^ Q^ =^ = 2257.06 kJ/kg^3 vap(water)^ Q^3 S: Solid (ice)L: Liquid (water)V: Vapor (superheated steam)L+V: Saturated steam

Q^2

Energy Content

-^ Thermal energy content of a solid or liquid:m c(Tp^ Kelvin - 273)^

OR^ m c

(T- 0)p Celsius^

-^ This is based on the assumption that the energy content ofthe substance is zero at 0 °C (= 273 K)•^ Thermal energy content (or enthalpy) of steam is given by:m(H)^ s^ s^

Note : H= x (Hs^

) + (1 - x) Hv

c

Direct and Indirect Contact Heat Exchanger • Direct contact heat exchanger– Steam contacts product and mixes with it– Steam loses energy as it condenses; product gains energy– Rapid heat transfer; however, product is diluted• Vacuum may be used to remove added water• Indirect contact heat exchanger (eg. concentric metal pipes)– Steam is separated from product by a barrier (metal pipe)• Not as much or as rapid heating; however, product is not diluted• Energy transferred by steam to a product in a

direct contact

heat exchanger

is given by:

m

(H)s s^

-^ Energy transferred by steam (when steam condensescompletely) to a product in an

indirect contact heat

exchanger

is given by:

m(H- Hs^ s^

)c

Mass Balance

-^ Overall mass balance– For any system, the total mass going into the system must equalthe total mass coming out of the system plus any accumulation ofmass in the system• Batch system: mass (m) in kg• Continuous system: mass flow rate (m) in kg/s•^ Component mass balance– For any component in a system, the total mass of that componentgoing into the system must equal the total mass of that componentcoming out of the system plus any accumulation of that componentin the system•^ Fo^ r any system (or sub-system), the number of mass balanceequations = the number of components in the product– One mass balance equation can be written for each component

Applications of Mass and Energy Balance • Mixing and/or separation of streams of products withdifferent compositions• Condensation of steam outside a tube through whichproduct flows (indirect contact heating)• Mixing of steam with a product (direct contact heating)– Direct contact heating is faster than indirect contact heating– Side-effect of direct contact heating is dilution of product• Vacuum may be used to evaporate water that was added• Evaporation of water from a product to concentrate itNote: For batch systems, the term “mass” (m) is used and for continuoussystems, the term “mass flow rate” (m) is used

.^

m: kg^

. m: kg/s

Mixing of Two Products in a Batch System

(at the same temperature) m, x^1

m, x^2

m, x^3 Mixer The overall mass balance equation is :

m+ m= m^1

3

The total solids balance equation is

:^ mx1+ m^1

x2= mx3 2 3

The total water balance equation is

:^ m(1-x^1

) + m(1-x 1 2 ) = m(1-x 2 3

The total water balance equation is however NOT a totally new equation. It can beobtained by subtracting the total solids balance equation from the overall mass balanceequation. Thus, there are only TWO equations for the above system. Hence, given any 4variables in the system, the other 2 variables can be determined (Note that the 6variables in the above system are m

, m, m, x^ , x 1231

, x^ ). Also note that in a system such 23

as the one shown above,

x^ + x^ ≠ x^1 2

m, m, m: Total masses^1 2 3 x^ , x^ , x3: Solid fractions^1

Direct Heating with Steam (Continuous System)^ m, xf, c^ f^ p(f)

.^ , T^ f

m, x^ , c^ , Tp^ p^ p(p)

p .

Overall mass balance: m

• m= mf s^ p Solids balance: m

xf= mxpf p^ Energy balance: m

{c^ } T+ mf p(f)^ f^

(H^ ) = m{cs s^ p^

} Tp(p)p

H^ = (x) H^ s^ v^

• (1 – x) Hc with H^ and Hv^

being determined from steam tablesc^

. m, x, c^ , T: Mass flow rate, solids fraction, specific heat, temperature resp.p^ Subscripts: ‘f’ for feed; ‘p’ for productat the steam pressure of P or steam temperature of T

.^. ...^.

m, quality of xs^ Pressure, P^ OR^..

Temperature, T

. Steam Cold product

Warm diluted product Direct contact HX

Determination of Steam Quality m, c^ , Tc^ p(c)c^

m, c^ , Th^ p(h)h m, quality of x; Pressure, Ps^

OR^ Temperature, T Mass balance: m

• m= mc s^ h Energy balance: m

{c^ } T+ mc p(c)c^

(H^ ) = m{cs s^ h^

} Tp(h)h

Determine H

from above equations^ Also, H^ = (x) Hs^

• (1 – x) Hv^

c with H^ and Hv^

being determined from steam tablesc^ at the absolute (not gauge) steam pressure of P or steam temperature of TThus, x = (H

• H^ )/(H^ – Hs^ c^ v^

)c^

m, c, T: Mass, specificp heat, temperature resp.Subscripts: ‘c’ for cold;‘h’ for hot

Introduce cold water of known mass (m

) and temperature (Tc

) in a flaskc

Introduce steam into flask till water becomes hot (T

)h

Determine specific heat of water at T

Steam and Tfrom table of properties of waterc^ h^ Cold water^

Hot water

Use interpolation for:c^ , c^ , H, Hp(c)p(h)^ c^

v

Flask