applications of calculus ass PDF

Preview

Summary

application of calculus assignment
first year first semester...

Description

The University of Sydney School of Mathematics and Statistics

Assignment 1 MATH1011: Applications of Calculus

Semester 1, 2021

Lecturers: Clio Cresswell

This individual assignment is due by 11:59pm Thursday 18 March 2021, via Canvas. Late assignments will receive a penalty. A single PDF copy of your answers must be uploaded in the Learning Management System (Canvas) at https://canvas. sydney.edu.au/courses/31390. Please submit only one PDF (or .docx) document (scan or convert other formats). It should include your SID, your tutorial time, day, room (or note as Zoom) and Tutor’s name. Please note: Canvas does NOT send an email digital receipt. We strongly recommend downloading your submission to check it. What you see is exactly how the marker will see your assignment. Submissions can be overwritten until the due date. To ensure compliance with our anonymous marking obligations, please do not under any circumstances include your name in any area of your assignment; only your SID should be present. The School of Mathematics and Statistics encourages some collaboration between students when working on problems, but students must write up and submit their own version of the solutions. If you have technical difficulties with your submission, see the University of Sydney Canvas Guide, available from the Help section of Canvas.

This assignment is worth 2.5% of your final assessment for this course. Your answers should be well written, neat, thoughtful, mathematically concise, and a pleasure to read. Please cite any resources used and show all working. Present your arguments clearly using words of explanation and diagrams where relevant. After all, mathematics is about communicating your ideas. This is a worthwhile skill which takes time and effort to master. The marker will give you feedback and allocate an overall letter grade and mark to your assignment using the following criteria: Mark 5

1

Grade Criterion A Outstanding and scholarly work, answering all parts correctly, with clear accurate explanations and all relevant diagrams and working. There are at most only minor or trivial errors or omissions. B Very good work, making excellent progress, but with one or two substantial errors, misunderstandings or omissions throughout the assignment. C Good work, making good progress, but making more than two distinct substantial errors, misunderstandings or omissions throughout the assignment. D A reasonable attempt, but making more than three distinct substantial errors, misunderstandings or omissions throughout the assignment. E Some attempt, with limited progress made.

0

F

4 3

2

No credit awarded.

c 2021 The University of Sydney Copyright 

1

1. This assignment concerns interpreting findings of the attached research paper: “Exponential sinusoidal model for predicting temperature inside underground wine cellars from a Spanish region”, co-authored by F.R. Mazarron and I. Canas. The paper models sinusoidal data. Begin by perusing the research paper to gain some loose familiarity with the problem being discussed. Then, attempt the questions below. It will be normal for you not to undertand everything exactly. Mathematicians are regularly asked to join projects in a diversity of areas of which they themselves are not experts. Mathematicians and science professionals need to understand how the mathematics fits and if the mathematics makes sense in the particular context. (a) After introducing the problem, the authors arrive at Equation (4) as their general temperature model for the subterranean cellars in Morcuera. (i) Equation (4) determines separate expressions for T(x,t) in terms of t only for each Morc1,  2π Morc2 and  Morc3. Write these explicitly in the form Ti (t) = d + A cos 365 (t − δ) , naming them T1 , T2 and T3 respectively. Give d, A and δ to 4 decimal places. (ii) What is the amplitude, mean value, period and horizontal shift (with direction) for each expression T1 , T2 and T3 ? Give the associated unit for each of these. The period should be given as an exact value. (iii) What do each of these expressions, T1 , T2 , T3 , predict the temperature to be on the first of January 2006 and first of April 2006? (iv) The paper includes figures describing experimental data gathered onsite during 2006 and 2007, where their T is also shown for comparison. The figures show temperatures with a precision of 2◦ C. Looking at the figures, the value for T on the first of January 2006 can be bounded as a ≤ T ≤ b, where a and b are the closest even integers to the value. Give a and b for T1 , T2 and T3 . (v) Considering again the figures describing experimental data gathered onsite during 2006 and 2007 where T is also shown, the value for T on the first of April 2006 appears to be almost which integer value? (vi) Considering part (iii), what would you expect the answers to be for parts (iv) and (v)? (b) Your results from part (a) should not be reconciling as the research paper appears to contain an error. This section investigates a possible origin. (i) Consider Equation (3). The first term Tm − k = 9.1 appears to align with the figures describing experimental data gathered onsite p p during 2006 and 2007. Taking As = 12.1 and to be precise π/365α = π/(365α), use the parameters discussed in section 3.3 to build T1 , T2 and T3 using Equation (3). Place T1 , T2 and T3 in the form requested in part (a) (i) above for comparison. The error in the paper means these will be different to what was found in part (a). (ii) What do the proposed equations for T1 , T2 and T3 of part (b) (i) suggest the temperatures of the cellars to be on the first of January 2006 and first of April 2006? (c) Since the term representing the horizontal shift in Equation (3) appeared to confuse

2

the authors, let’s consider a more simple looking version of this equation as follows:   √ 2π −x′ π/(365×α) (t − X) T(x,t) = (Tm − k) − As e cos 365 p p with Tm − k = 9.1, As = 12.1 and π/365α = π/(365α) as above. (i) Taking the temperature on the first of September 2006 for Morc1 and Morc2 as being 12.7◦ C and for Morc3 as being 12◦ C, find the corresponding X for each model. Hence construct each associated T . You may again name them T1 , T2 and T3 respectively. (ii) Considering the models of part (b) (i) and part (c) (i), what is the difference in their predicted temperatures for the cellars on the first of January 2006? (iii) Considering the match between the analytic and experimental air temperatures in the paper, are the differences found in part (c) (ii) to be rated as meaningful? A single sentence describing the degree of difference observed is all that is required here.

3

Energy and Buildings 40 (2008) 1931–1940

Contents lists available at ScienceDirect

Energy and Buildings j o ur n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n b ui l d

Exponential sinusoidal model for predicting temperature inside underground wine cellars from a Spanish region Fernando R. Mazarron 1, Ignacio Canas * Departamento de Construccio´n y Vias Rurales, Escuela Te´cnica Superior de Ingenieros Agro´nomos, ´cnica de Madrid, Avda. Complutense s/n, 28040 Madrid, Spain Universidad Polite

A R T I C L E

I N F O

Article history: Received 12 March 2008 Received in revised form 28 March 2008 Accepted 21 April 2008 Keywords: Wine cellar Thermal environment Underground building

A B S T R A C T

This article develops a mathematical model for determining the annual cycle of air temperature inside traditional underground wine cellars in the Spanish region of ‘‘Ribera del Duero’’, known because of the quality of its wines. It modifies the sinusoidal analytical model for soil temperature calculation. Results obtained when contrasting the proposed model with experimental data of three subterranean wine cellars for 2 years are satisfactory. The RMSE is below 1 8C and the index of agreement is above 0.96 for the three cellars. When using the average of experimental data corresponding to the 2 years’ time, results improve noticeably: the RMSE decreases by more than 30% and the mean d reaches 0.99. This model should be a useful tool for designing underground wine cellars making the most of soil energy advantages. ß 2008 Elsevier B.V. All rights reserved.

1. Introduction Nowadays numerous resources are used in building air conditioning for products and food preservation. Wine is a particular case, since temperature conditions during aging and maturation processes are decisive for its final quality. It is advisable that wine temperature values remain under 18 8C during the aging process and that no brusque temperature changes occur. Therefore, wine has been traditionally aged in subterranean cellars where soil thermal inertia produces thermal stability which provides wine with its particular characteristics. During the last century there has been a tendency to build aerial buildings for cellars, investing economic resources in thermal fitting-out of facilities. Given the energy crisis, the fossil fuels depletion problem and the increase in energy prices, it is necessary to design bioclimatic buildings reducing energy consumption. Recent construction of large subterranean or buried cellars in Spain is an example for the economic feasibility and for the advantages this type of buildings provide. Some authors have studied the advantages of subterranean buildings. Martı´n and Canas [1] stress the advantages of subterranean cellars compared to the aerial ones

* Corresponding author. Tel: +34 913365767; fax: +34 913365625. E-mail addresses: [email protected] (F.R. Mazarron), [email protected] (I. Canas). 1 Tel.: +34 913365767; fax: +34 913365625. 0378-7788/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2008.04.007

by calculating energy saving. Al-Temeemi and Harris [2] analyze the advantages and disadvantages of the earth-sheltered scheme. Prior research in estimating the internal temperature of subterranean buildings is scarce. Given the enormous influence that the earth has with regard to the internal conditions of subterranean buildings some authors have studied the possibility of applying mathematical models to calculate the ground temperature in order to estimate the wall temperature and air temperature of the subterranean buildings. To this effect, Labs [3] suggests a simplified one-dimensional sinusoidal equation that describes soil temperature fluctuations according to depth. It is based on the harmonic wave of the surface heat flow. This model has provided valid soil temperature results in the comparisons with experimental data from several regions around the world [3– 5]. Nevertheless, when this equation is applied to estimate the temperature field surrounding earth-sheltered buildings the results obtained show that the temperature of the outside face of the uninsulated basement wall exhibits large departures from the undisturbed temperature due to the heat exchange between the building and its enveloping soil. Other authors [6,7] propose Labs equation to predict long-term subterranean temperatures as input for computer models to estimate the heat transfer through a subsurface wall at varying depths. Martin and Canas [8] carried out a first estimation study for determining air temperature inside underground wine cellars by applying the formula proposed by Labs [3] and suggesting different hypothesis to calculate one of its variables. Nevertheless, obtained results had some limitations,

1932

F.R. Mazarron, I. Canas / Energy and Buildings 40 (2008) 1931–1940

such as a period of experimental data below 2 months per cellar or a very low average index of agreement (0.36), in the best scenario. Other authors have studied the thermal behaviour of earthcovered buildings using computer models and realising corresponding simulations during brief time periods. In this manner, Grindley and Hutchinson [9] study the thermal behaviours of an earthship in New Mexico. Using a computer model predicts the mean radiant temperatures to within 1.2 8C of the measured data during 3 days. Wang and Liu [10] studies the thermal environment of the courtyard style cave dwelling during 3 days. The model used is derived from a set of equations of heat balance amongst the courtyard air, cave rooms air and envelop surfaces. The predictions for the temperatures in the cave rooms are slightly different from the measured values, offering only the variation of the average daily air temperature. In case of underground wine cellars, inner temperature is highly conditioned by soil profile temperature, given poor ventilation and depth conditions. However, the mathematical models used to calculate the ground temperature when strictly applied produce different results from the experimental values given that they do not take into account the distortion that is caused by the building itself. In order to calculate inner temperature in underground wine cellars, we have started from the equation proposed by Labs [3], modifying it and suggesting a different calculation for some of its variables, which allows us to correct distortion caused by the cellar and the air inside it. 2. Geographical aspects and description of subterranean cellars 2.1. Location, climate and soil The cellars of study are located in Morcuera, a village in the Spanish province of Soria (Zone 30, X 482115, Y 4590728) at a height of 1060 m. This area is located in the far eastern part of the Ribera del Duero region and is typical for a mild Mediterranean climate featuring more than 2400 sunlight hours a year. Its main characteristic is continentality. Annual average temperature is 10.6 8C, ranging from 9.2 to 11.8 8C depending on the year. Average highest and lowest recorded temperatures are 16.7 and 4.6 8C, respectively. July is the warmest month with 20.0 8C on average and January the coldest with an average of 2.9 8C. Annual absolute highest and lowest temperatures are 37.6 and 14 8C, respectively. Climatology of the Ribera del Duero region is known for a low pluviometry, featuring annual average precipitation records

ranging from 450 to 650 mm (29% in winter, 28% in spring, 27% in autumn and 16% in summer). Soils have their origin in tertiary sediments constituted by lenticular layers of loamy sands or clay, featuring alternative chalky layers, marls and calcareous concretions. 2.2. General description of subterranean cellars Traditional cellars of Morcuera are normally made up of a cave or cellar that is dug under the ground level and a canyon or entrance tunnel ending in an outer fac¸ade. Sometimes a previous room dividing the canyon from the outer part can be found (see Fig. 1). The entrance fac¸ade is normally made of stone covered by materials from the excavation. The door is made of wood and features open ventilation holes facing north for a better inner air renovation. The canyon is a narrow corridor featuring less than 1m long and 2-m high. Sometimes it has niches on the sides for storing small barrels or tools. The canyon leads to a cave or cellar where the aging process of wine has traditionally taken place. They are normally placed at a depth between 1 and 6 m, existing both linear and branched distributions. Walls in cellars do not tend to have coatings, but just the excavated soil itself. Sometimes one or more ventilation chimneys called zarceras featuring a 50–100-cm diameter can be found. Their main function is CO2 ventilation during the wine fermentation process. A more detailed description of the characteristics and topologies of traditional cellars can be found in Pardo and Guerrero [11]. 2.3. Cellars of study The three cellars are located in a cellar area on the outskirts of the village, in a quite flat ground (Fig. 2):  Cellar no. 1: ‘‘Morc1’’

This cellar is located in the centre of the cellar area. It has a main cave and a secondary one and it was refurbished a little before starting the study. Canyon length amounts to almost 9 m and has a 228 slope. It is reinforced with stonework. The main room of the cave has been coated with bricks. It has an area of around 6 m2 and an average height of 1.9 m. Cellar ceiling is 3.1m deep on average with regard to the surface (Fig. 3).  Cellar no. 2: ‘‘Morc2’’

This cellar is located near Morc1. It also has a branched distribution with two rooms. The 6.5-m long canyon is reinforced

Fig. 1. Cross-section of a typical traditional underground wine cellar.

F.R. Mazarron, I. Canas / Energy and Buildings 40 (2008) 1931–1940

1933

Fig. 2. General view of the cellar area in Morcuera.

with stone rendering and has a 308 slope. It has two small rooms without coating featuring uncovered soil walls (Fig. 4).  Cellar no. 3: ‘‘Morc3’’ This cellar is located on the far north-eastern part of the cellar area. It is also north facing. It has a linear layout with a cave of around 12 m 2 and a more than 7-m long canyon featuring a 318 slope (Fig. 5).

2.4. Analysis of temperature inside cellars Two ONSET models of temperature and humidity data loggers have been used for monitoring inner temperature in cellars. (a) HOBO1 Pro Temperature/Relative Humidity data logger: ‘‘HPV1’’ It has an internal sensor of type thermistor. These are its characteristics:  Range: 30 8C to +50 8C.  Accuracy: 0.2 8C to +21 8C.  Resolution: 0.02 8C to +21 8C.  Response time: 34 min, typically with quiet air. (b) HOBO1 Pro V12 Temperature/Relative Humidity data logger: ‘‘HP-V2’’ This model is an upgrade of the previous model, although the type of sensor and characteristics are similar:

   

Range: 40 8C to +70 8C. Accuracy: 0.2 8C over 0–50 8C. Resolution: 0.02 8C to +25 8C. Response time: 15 min (90% with airflow 1 m2 /s).

The aim of the monitoring process is to know the actual evolution experienced by air temperature inside the cellar. Before starting the monitoring process, some previous temperature and relative humidity measurements were taken inside the cave of the subterranean cellars. It could be observed that hygrothermal conditions were very stable and presented small variations both in space and time. In order to determine the daily average temperature of air inside the cellars during the 2 years’ time monitoring process, dataloggers were placed in the centre of the cave of each cellar, at an intermediate height. A measurement interval of 15 min was set taking sensor’s response time and inner relative stability into consideration. An HP-V1 data-logger was installed in Morc1 until November 2006. Then, a new HP-V1 was placed beside it to check its proper operation. At the beginning of 2007 three more new HP-V2 dataloggers were installed. Two HP-V1 data-loggers were installed from the beginning in Morc2 and at the beginning of 2007 three more new HP-V2 were placed. In Morc3 there was only an HP-V1 sensor monitoring temperature until the beginning of 2007, when three new HP-V3 were installed.

Fig. 3. Elevation and section of Morc1.

1934

F.R. Mazarron, I. Canas / Energy and Buildings 40 (2008) 1931–1940

Fig. 4. Elevation and section of Morc2.

During these 2 years more than half a million air temperature data have been taken among the three cellars. When comparing records of the different sensors it can be noticed that the average difference among all of them is around 0.2 8C for each cellar. Taking sensor accuracy and uncertainty of air measurement into consideration, this value is not important. Hence the data of

the former sensor have been taken as representative for each cellar. Once the daily, weekly and monthly average values of variations recorded by the representative sensors for each cellar have been examined, some conclusions about thermal stability inside the cave can be drawn.

Fig. 5. Elevation and section of Morc3.

F.R. Mazarron, I. Canas / Energy and Buildings 40 (2008) 1931–1940 Table 1 Daily, weekl...