This document describes calculating the total heat required to heat water to 1000C and produce superheated steam. It involves numerically calculating: 1) the heat of vaporization at 100C using the saturation vapor pressure curve's slope; 2) integrating specific heats over temperature ranges; and 3) using finer step sizes to estimate derivative accuracy. The most accurate derivative approximation was centered finite difference with smaller step sizes providing more accurate results.

1. In this assignment, you will be calculating the heat required to heat up water from room
temperature, vaporize it and produce superheated steam at 1000C. As you might recall from
thermodynamics, we can calculate that total heat using the following equation: lbegin { equation }
|Delta H_{tot }= int_{T_0}{T_{vap}} C_{p,liq },dT+ Delta H_{vap, T}+ int_{T_{vap }}{T_{end}}
C_{p,gas },dT lend{equation } In this equation, Htot is the total required enthalpy ( J/mol), while Cp
,liq and Cp,vap are the specific heat of water in the liquid phase and vapor phase (J/molK)
respectively and Hvap,T(J/mol) is the latent heat of vaporization at a specific temperature T. In this
assignment, we will be using numerical differentiation and integration to obtain the required values
for the individual terms and find the total heat required to produce 1 mole of steam at 1000C from
water at room temperature (20C). For this assignment, you may assume that we are on earth and
that we are at a location where the boiling point of water is 100C and that the atmospheric
pressure is equal to 101,325Pa.In this assignment, you will be calculating the heat required to heat
up water from room temperature, vaporize it and produce superheated steam at 1000C. As you
might recall from thermodynamics, we can calculate that total heat using the following equation:
Ibegin{equation lend{equation In this equation, Htot is the total required enthalpy (J/mol), while Cp,
liq and Cp,vap are the specific heat of water in the liquid phase and vapor phase (J/molK)
respectively and Hvap,T(J/mol) is the latent heat of vaporization at a specific temperature T. In this
assignment, we will be using numerical differentiation and integration to obtain the required values
for the individual terms and find the total heat required to produce 1 mole of steam at 1000C from
water at room temperature (20C). For this assignment, you may assume that we are on earth and
that we are at a location where the boiling point of water is 100C and that the atmospheric
pressure is equal to 101,325Pa. Part 1 - Finding the heat of vaporization ( 40 points) In order to
calculate the total energy, we need to find the heat of vaporization at a temperature of 100C. We
can calculate this latent heat using the following equation, which is dependent on the slope of the
vapor-pressure curve as a function of temperature (dPsat/dT), the volume change upon
vaporization Vvap and the temperature T at which the phase change takes place (in Kelvin):
Ibegin{equation } |Delta H_{vap, T}=T{ Delta V{vap}frac{dP{ sat }}{dT} end{equation The volume of
a mole of water vapor at 100C can be approximated by the ideal gas law (where T is in Kelvin, R
is theuniversal gas constant and the pressure is in Pa ): lbegin { equation } frac{V}{n}=frac{RT}p}=
frac{8.314373.15}{101325}=0.03062hspace{1pt}mathrm{m3/mol} lend { equation } As the volume
of a mole of liquid water is negligible in when compared to this volume, we take this molar volume
to be equal to Vvap If we have an equation for the saturation vapor pressure as a function of
temperature, we can take the derivative at the desired temperature and find the latent heat. The
following equation holds for the saturation pressure of water (Pa) as function of temperature (in
Kelvin): lnPsat=T6096.9385+21.24096422.711193102T+1.673952105T2+2.433502ln(T) This
equation (rewritten to return Psat is defined as a function in the cell below in order to avoid typo's.
- Using this equation, calculate the saturation pressure of water at 80,90,100,110 and 120C and
check your answers with tabulated data (e.g. The Engineering Toolbox). ( 5 points) - With these 5
data points, you can find different approximations of the derivative at 100C based on forward,
centered and backward finite-divided differences. Calculate the derivative dPsat/dT based on your
found data points using these three techniques and aqcuire the highest possible accuracy for all
three. Comment on the differences between the found values and indicate which approximation is

2. the most reliable. (10 points) - Another method of estimating the derivative is taking the function
and calculate saturation pressures at different temperatures spaced closer to each other and
estimate the derivative using these values. Write functions that determine the centered finite-
divided difference estimation of the derivative, with an error of h2 and h4 respectively. These
functions Ild take the equation for the saturation pressure as input, together with the value at which
the derivative shoulc' nated, and a step size h to determine the derivative with. (10 points)- With
these two functions, calculate the derivative for stepsizes ranging from 0.1C,1C,10C and 50C.
Comment on the found differences and the desired step size. Would further decreasing the step
size be useful? (10 points) - What is the value of dPsat/dT that you consider to be trustworthy
based on your knowledge of numerical differentiation? Use this value to calculate Hvap, and look
at tabulated values of Hvap of water (e.g. The Engineering Toolbox) to see if your estimation is in
the right order of magnitude. (5 points) In [ ] : 1 import numpy as np 2 def psat(T): return np.exp (
6096.9385T(1)+21.24096422.711193e2T+1.673952e5T2+2.433502np. log(T)) 3 In [ ] : 1 # Use
this cell to write your code and do your calculations- The differences between the above methods
are a result of YOUR REASON HERE and the most accurate approximation of the derivative is
based on the FORWARD/BACKWARD/CENTERED (REMOVE WRONG ANSWERS) divided
difference method. - Using first and second order centered divided difference methods, the
following values for dPsat/dT are found: - We observe the following when comparing the two
different schemes YOUR ANSWER HERE, and decreasing the step size results in YOUR
ANSWER HERE. - Advantages and disadvantages of decreasing the step size are YOUR
ANSWER HERE. Further decreasing the step size (lower than 0.1C ), would YOUR ANSWER
HERE - We use dPsat/dT= YOUR ANSWER HERE to calculate Hvap. Our found value of Hvap=
YOUR ANSWER HERE (DONT FORGET THE UNITS)**