Question

USE EXCEL OR MATLAB IF YOU NEED

Given the dry-bulb temperature, T_db and relative humidity, Φ, the wet-bulb, T_wb temperature may be calculated from the psychrometric equation (Jones, 1994):

P_vsΦ – P_ss = P_atmA (T_wb -T_db )                                              (1)

Use your input from the Table below.

The input T_wb and T_db are in ^oC (used in Eqs. (2) and (3) in Kelvin).   

P_ss = saturated vapor pressure at T_wb (kPa)

P_vs = saturated vapor pressure at T_db (kPa)   

P_ss = 10^[30.59051- 8.2 log₁₀ T_wb + 0.0024804 T_wb – 3142.31/ T_wb] ..........(2)

and

P_vs = 10^[30.59051- 8.2 log₁₀ T_db + 0.0024804 T_db – 3142.31/ T_db]     ........(3)

In the above two equations T is in Kelvin and P in kPa (you have to convert the temperatures from the input in ^oC to Kelvin).

P_atm= atmospheric pressure (kPa) (P_atm =101.325 kPa)

T_db = dry-bulb temperature (^oC)

T_wb = wet-bulb temperature (^oC)

A    = 6.66x10^-4 C^-1 (constant for sling psychrometer)

Φ = relative humidity (%) used in the calculations as (%/100). For example, for 80% relative humidity, input 0.8.

Solve Equation (1) for the wet-bulb temperature using Newton-Raphson method. The input is as follows: T_db= in ^oC and Φ= in % (to be converted in the program to fraction, i.e., Φ=50% means Φ=0.5). Set the tolerance to 10^-4 . In other words, the absolute difference between the results of two successive iterations should not be more than 10^-4. Check your answer using Psychrometric chart/Thermo Textbook. Also, to check if your results are correct, use f=100% and you should get T_wb=T_db. S is yourlisting number.

S (for future use)	Tdb (C)	Relative Humidity (%)
1	25	45

Answer 1

clc
clear
close
syms x %assume Twb=x
Patm=101.325;
A=6.66e-4;
Tdb=273+25;
phi=0.45;
Pss=10^(30.59051-(8.2*log10(x))+(0.0024804*x)-(3142.31/x));
Pvs=10^(30.59051-(8.2*log10(Tdb))+(0.0024804*Tdb)-(3142.31/Tdb));
f=Patm*A*(x-Tdb)+Pss-Pvs*phi;
x0=20;
[y,h]=newton_raphson(f,x0);
Tdb=y-273

function file for newton raphson

function [y,h]=newton_raphson(f,x0)
syms x;
g=diff(f); %The Derivative of the Function
n=4;
epsilon = 1e-4;
h=[];
for i=1:100
f0=vpa(subs(f,x,x0)); %Calculating the value of function at x0
f0_der=vpa(subs(g,x,x0)); %Calculating the value of function derivative at x0
y=x0-f0/f0_der; % The Formula
h=cat(1,h,y);
err=abs(y-x0);
if err<epsilon %checking the amount of error at each iteration
break
end
x0=y;
end
y = y - rem(y,10^-n); %Displaying upto required decimal places

Note: Using Psychometric chart @ Tdb=25 and RH=45 the Twd is 17.049 C. The result obtained using Matlab is 17.1806

Please do like the solution :-)