f9b050d1-fc79-422b-847f-2891d8d7b2de 23 Erick Valentino 4 5 false false 0 decimal

&[DATE] &[TIME] - &[FILENAME]

Cooling Towers with SMath Studio

SECTION 1: Calculation of the heat to be removed in the process and graph of the enthalpy curve of saturated air

A flow of cooling water (WO) entering the tower @ 28 °C is required. The amount of refrigerant fluid is 100,000 kg hour of air. The inlet temperature of the fluid to the cooling tower (WIn) is 38 °C. The wet bulb temperature is 23 degrees Celsius.

Initial Data

Water Inlet Temperature

Water Outlet Temperature

Wet Bulb Temperature

Ambient Temperature

T.WIn 38 °C * : 311.15 K *

T.WO 28 °C * : 301.15 K *

T.WetB 23 °C * : 296.15 K *

T.Amb 24.85 °C * : 298 K *

Molecular Weight of Water

Molecular Weight of Air

Mass flow rate of "Water"

Operating Pressure

PM.Water 18.015268 g mol / * :

PM.Air 28.958600656 g mol / * :

Ṁ.Water 100000 kg hr / * :

P 1 atm * :

__________________________________________________________________________________________________________________

Solution

The CP of the Water is approximated by an estimation equation and corresponding constants obtained from: "webbook.nist.gov"

Molar Heat Capacity at constant pressure of the refrigerant (Water) as a function of temperature:

T Cp.Water_Molar 203.6060 - 1523.290 T K 1000 * / * + 3196.413 T K 1000 * / (2^* - 2474.455 T K 1000 * / (3^* + 3.855326 T K 1000 * / (2^/ + (J mol K * / * :

T.Amb Cp.Water_Molar 75.377 J mol K * / *

Using the Cool Props plugin to find the Molar Heat Capacity:

T Cp.Water_Molar_CoolProp CPMOLAR T T P P Water CoolProp_Props :

T.Amb Cp.Water_Molar_CoolProp 75.3286 J mol K * / *

__________________________________________________________________________________________________________________

Mass heat capacity at constant pressure of the refrigerant as a function of temperature:

T Cp.Water T Cp.Water_Molar PM.Water / :

T.Amb Cp.Water J g K * / 4.1841

Using the Cool Props plugin to find the Mass Heat Capacity:

T Cp.Water_CoolProp C T T P P Water CoolProp_Props :

T.Amb Cp.Water_CoolProp J g K * / 4.1814

Removing Heat

T.Mean T.WIn T.WO + 2 / : 306.15 K *

Q Ṁ.Water T.Amb Cp.Water * T.WIn T.WO - (* : 1.16224 106^* W *

Q Ṁ.Water T.Amb Cp.Water_CoolProp * T.WIn T.WO - (* : 1.161494 106^* W *

Using the Cool Props plugin

Note: we can observe that the values obtained from the CoolProp plugin library are not so different from those obtained through equations, so as not to lengthen too much the number of pages of this document with equations of state to find thermodynamic properties, we will use the CoolProp plugin

Vapor Pressure of refrigerant at boiling temperature (100 °C at sea level)

T P.V P Q 1 T T Water CoolProp_Props :

T.Sat 100 °C * : 373.15 K *

T.Sat P.V 1.0142 105^* Pa *

Humidity Ratio | humidity of the Air-Water mixture

X Kg.Water Kg.Dry_Air / ≡

T X T P.V P T P.V - / PM.Water PM.Air / * :

T.Sat X 678.4391 -

T.Amb X 0.0199

T.WetB X 0.0178

Enthalpy of Vaporization of Water as a function of temperature:

T H.V H T T Q 1 Water CoolProp_Props :

T.Amb H.V kJ kg / 2.5462 103^*

T.Sat H.V kJ kg / 2.6756 103^*

Mass Heat Capacity at constant pressure of Air as a function of ambient temperature

T Cp.Air C P P T T Air.mix CoolProp_Props :

T.Amb Cp.Air J g K * / 1.0066

T.WetB Cp.Air J g K * / 1.0065

"Ideal\0020\gas\0020\mass\0020\specific\0020\constant\0020\pressure\0020\specific\0020\heat" | Vapor

\[REGION[0bfb456f3a0e4645923947b3f3c214da]]\ \[REGION[af89cae477b44af4951047eeb60cb158]]\

T Cp.Videal CP0MASS T T Q 1 Water CoolProp_Props :

T.Amb Cp.Videal kJ kg K * / 1.8643

"Mass\0020\Specific\0020\Constant\0020\Pressure\0020\Specific\0020\Heat" | Vapor

\[REGION[0bfb456f3a0e4645923947b3f3c214da]]\ \[REGION[af89cae477b44af4951047eeb60cb158]]\

T Cp.V C T T Q 1 Water CoolProp_Props :

T.Amb Cp.V kJ kg K * / 1.9116

"Mass\0020\Specific\0020\Constant\0020\Pressure\0020\Specific\0020\Heat" | Air-Water

\[REGION[0bfb456f3a0e4645923947b3f3c214da]]\ \[REGION[af89cae477b44af4951047eeb60cb158]]\

T Cp.Air_Water C W T X Tdb T P P CoolProp_HAProps :

T.Amb Cp.Air_Water kJ kg K * / 1.0441

Total Enthalpy of an Air-Water mixture

\[REGION[bc10e2f4e1c84bf2bf60b634a4f76c4f]]\ \[REGION[e20cb8d5002b4869a5260507c42adf97]]\

T H.Air_Water H W T X C T Cp.Air_Water P P CoolProp_HAProps :

T.Amb H.Air_Water J kg / 75668.0775

Approximation T.WO K / T.WetB K / - (: 5

T C.H T Cp.Air T Cp.V T X * + (≡

T.R 0 °C * : 273.15 K *

App 5 K * :

T H T Cp.Air_Water T T.R - (* T H.V T X * + :

298 K * H J kg / 76631.9824

First Partial Derivative of Mass Enthalpy (dT) | Air-Water mixture

\[REGION[bc10e2f4e1c84bf2bf60b634a4f76c4f]]\ \[REGION[e20cb8d5002b4869a5260507c42adf97]]\

T dH T Cp.Air_Water :

T.Amb dH J kg K * / 1044.106

The derivative of the enthalpy with respect to temperature is equal to the specific heat. the derived function for Cool_HAProps is not available, however, as I show in the following expression, it is possible to obtain an almost similar value for the mixture of air and water

T dHA d(H)/d(T)|P T T P P Air CoolProp_Props :

T dHW d(H)/d(T)|P T T Q 1 Water CoolProp_Props :

T.Amb dHA J kg K * / * (T.Amb dHW J kg K * / * T.Amb X * (+ J kg K * / 1044.3565

Temperature at the Point of Tangency

vars

T.X 350 K * ≡

vars FindRoot

T.X dH T.X T.WO - (* T.WetB H.Air_Water + T.X H.Air_Water - 0 kJ kg / * ≡

T.X 273.15 k * >

T.X 350 k * <

T.X K #

T f T dH T T.WO - (* T.WetB H.Air_Water + T H.Air_Water - :

T.Ok 294.2 K * :

T.Ok f kJ kg / 0.0057

T.Ok 294.2 K * :

T.Ok f T.Ok diff #

this is the problem with the solvers
in nonlinear solvers plugin:
can't get the derivate because coolprop

Use a pure-numerical solver:

T f T K * dH T K * T.WO - (* T.WetB H.Air_Water + T K * H.Air_Water - 11 mat :

T.guess 300 K * :

T.guess K / T f al_nleqsolve 294.2023 11 mat

T.X 350 K * :

T T.Root T dH T T.WetB App + (- (* T.WetB H.Air_Water + T H.Air_Water - :

T T.Help T T.Root T FindRoot :

T.X T.Help #

T dH T.X T.WetB App + (- (* T.WetB H + T.X H -

T.X T.s T.X H T.X T.WetB App + (- (* T.WetB H + T diff T.X H - T.X FindRoot (:

T ρ.Water D T T P P Water CoolProp_Props :

T.Amb ρ.Water 997.086 kg m 3^/ *

Volumetric flow rate of the refrigerant fluid "Air"

G Ṁ.Water T.Amb ρ.Water / : 0.0279 m 3^s / *