Measurement of the air exchange rate. Lesson in mathematics on the topic "Problems of mathematical statistics" (Grade 11) Testing building structures in the laboratory

The term "multiplicity" refers to the field of mathematics: from the point of view of this science, it means the number of times that a certain number is part of another number.

The concept of multiplicity

Simplifying the above, we can say that the multiplicity of one number in relation to another shows how many times the first number is greater than the second. Thus, the fact that one number is a multiple of another actually means that the larger of them can be divided by the smaller without a remainder. For example, a multiple of 3 is 6.

Such an understanding of the term "multiplicity" entails the derivation of several important consequences from it. The first of these is that any number can have an unlimited number of multiples of it. This is due to the fact that in order to get another number that is a multiple of some number, it is necessary to multiply the first of them by any positive integer value, which, in turn, has an infinite set. For example, multiples of the number 3 are the numbers 6, 9, 12, 15 and others obtained by multiplying the number 3 by any positive integer.

The second important property concerns the definition of the smallest integer that is a multiple of the one under consideration. Thus, the smallest multiple of any number is the number itself. This is due to the fact that the smallest whole result of dividing one number by another is one, and it is the division of the number by itself that provides this result. Accordingly, a number that is a multiple of the one under consideration cannot be less than this number itself. For example, for the number 3, the smallest multiple will be 3. In this case, it is actually impossible to determine the largest multiple of the one under consideration.

Numbers that are multiples of 10

Numbers that are multiples of 10 have all of these properties along with other multiples. So, from the listed properties it follows that the smallest number that is a multiple of 10 is the number 10 itself. At the same time, since the number 10 is two-digit, we can conclude that only numbers consisting of at least two characters can be a multiple of 10.

In order to get other numbers that are multiples of 10, you need to multiply the number 10 by any positive integer. Thus, the list of multiples of 10 will include the numbers 20, 30, 40, 50, and so on. It should be noted that all the numbers obtained must be divisible by 10 without a remainder. In this case, it is impossible to determine the largest multiple of 10, as in cases with other numbers.

Also, note that there is an easy, practical way to determine if the particular number in question is a multiple of 10. To do this, find out what its last digit is. So, if it is equal to 0, the number in question will be a multiple of 10, that is, it can be divided by 10 without a remainder. Otherwise, the number is not a multiple of 10.

Lesson 282

Lesson topic : Problems of mathematical statistics.

Lesson Objectives:

Tutorial: Teach students to solve processing tasks

statistical data using the concepts:

measurement volume, measurement range, mode

measurements, arithmetic mean, median

measurements, measurement options, multiplicity

options, and compile data in tables,

diagrams, graphs. Introduce concepts: frequency

options, frequency options (percentage).

Developing:

Develop student skills in problem solving

processing of statistical data using

data in the form of tables, charts, graphs.

Develop logical and mathematical thinking.

Nurturing:

Cultivate a culture of speech, building a plan

response, conscious discipline, culture

constructive thinking, activity in the lesson,

accuracy when writing on the board and in

notebooks, positive interest in what is being studied

subject.

Lesson type : Combined.

Type of lesson: Lesson in solving problems for the processing of statistical

data using data in the form of tables,

diagrams, graphs.

Teaching methods: Reproductive.

Material and technical equipment:

- Math Tutorial

Moscow Publishing Center "Academy" 201

- Math Tutorial General education disciplines

for professions and specialties of socio-economic

Moscow Publishing Center "Academy" 2011

- Mathematics Task book General educational disciplines

Primary and secondary vocational education

Moscow Publishing Center "Academy" 2012

- didactic handout (cards for

individual work)

During the classes

1. Organizational moment of the lesson

Submitting a report

2. Target orientation

(The teacher formulates the topic, goals and objectives of the lesson. Motivates students for learning activities. Explains the sequence of stages of the lesson leading to the achievement of the goal)

3. Checking homework.

4. Questions to consolidate the studied material.

1). List the main stages of the simplest statistical data processing.

2). What is called the volume of measurement?

3). What is the range of measurement?

4). What is the measurement mode?

5). What is the arithmetic mean?

6). What is a measurement option?

7). What is the median of a measurement?

Formation of mental skills

Solving problems at the blackboard

Task 1

In the data distribution table, some information has been lost. Restore her. If the volume is known to be 20, the range is 6 and the mode is 2.

Option

Sum

multiplicity

Solution

A-priory. In the column "Amount" should be the volume of measurement, i.e. 20. This volume is equal to the sum of all multiplicities, which means that the multiplicity of options “0” is 20 – (5+1+7+3) = 4.

The largest multiplicity is 7. This means that the measurement mode equal to 2 is located above it. Since the range is 6, and the largest variant is 3, the smallest variant is 3 - 6 = - 3. We place this variant in the last free column above the multiplicity 5.

Answer:

Option

Sum

multiplicity

Task 2

According to the given histogram of data distribution, find: quantity, measurement option, volume, range. measurement mode, the most remote from the variant mode and its multiplicity. Create a data distribution table.

Solution.

The number of options is the number of bars in the histogram, i.e. 7. The volume of measurement is equal to the sum of the multiplicities of all options, i.e. is equal to the sum of the heights of all seven columns: 3+2+7+3+5+4+1 = 25. The distribution table looks like this:

Option

Sum

multiplicity

1). The largest option is 10 and the smallest is 2.

2). The range is 8. (10 - 2) = 8.

3). The measurement mode is 5, since it occurred more often than others - 7 times.

4). At the greatest distance from the mode is option 10, its multiplicity is 1.

Definition: If the multiplicity of options is divided by the volume of measurement, then we get frequency options . This number shows what part (share) of all the data is the data equal to the selected option.

The frequency of variants can also be measured as a percentage.

Frequency options (percentage) =

Task 3

In the tenth grades of three schools in the microdistrict, a test dictation in the Russian language was held. According to their results, a histogram of the distribution of the received marks is shown.

a) Find: total number of works, frequency of fives, percentage frequency

deuces.

b) Fill in the summary table of data distribution.

c) Construct a histogram of the frequency distribution (in percent).

d) Construct a pie chart of frequency distribution (in percent).

Solution.

a) The histogram indicates that there were 40 twos, 50 threes, 75 fours, and 35 fives. There were 200 works in total. This is the volume of measurement. The frequency of fives is
, and the frequency (in percent) of twos is

b) Since all multiplicities are known, it is possible to fill in the entire distribution table:

Option

Sum

multiplicity

Frequency

0.25

0.375

0,175

Frequency,%

37,5

17,5

c) To build a histogram of the frequency distribution (in percent), we use the first and fourth lines. We get four vertical columns. The bases of which correspond to the received marks, and the heights are equal to the found frequencies (in percent).

d) divide the circle into four sectors. The central angle of the two sector is 20% of 360 0 . those. 720. The central angle of the triple sector is 25% of 360 0 , this is a right angle. The central angles of the four and five sectors are 135 0 and 63 0 respectively.

5. Questions to consolidate the studied material.

1). What is called the frequency options?

2). What formula is used to measure the frequency of options as a percentage?

6. The result of the lesson. Homework.

Task.

According to the given histogram of data distribution, find:

a) the number of options and the amount of measurement;

b) range and mode of measurement;

c) data distribution table;

d) the average of the measurement results.

Solution.

1) The number of options is the number of bars in the histogram, i.e. 9. The volume of measurement is equal to the sum of the multiplicities of all options, i.e. is equal to the sum of the heights of all nine columns: 5+6+3+7+4+11+5+4+5 = 50. The distribution table looks like this:

Option

Sum

multiplicity

2). The largest option is 10 and the smallest is 2.

The range is 8. (10 - 2) = 8.

The measurement mode is 7, since it occurred more often than others - 11 times.

3). The distribution table looks like this:

Option

Sum

multiplicity

4). The arithmetic mean is the quotient of dividing the sum of all measurement results by the measurement volume. It is convenient to calculate the average after the distribution table has been compiled. In this case, the calculations look like this:

To measure the air exchange rate

The company Construction Expert Bureau LLC provides services for measuring the air permeability of building envelopes and the air exchange rate in the room in accordance with GOST 31167-2009, SNiP 23-02-2003 and GOST 54852-2011.

The need to measure the air exchange rate

In accordance with SNiP 23-02-2003, clause 11.4, when accepting buildings for operation, it is necessary to carry out selective control of the air exchange rate in 2-3 rooms (apartments) or in a building at a pressure difference of 50 Pa in accordance with section 8 (of this SNiP) and GOST 31167-2009 and in case of non-compliance with these standards, take measures to reduce the air permeability of building envelopes throughout the building. Also, when accepting a building for operation, according to GOST 26629, thermal imaging quality control of the thermal protection of the building should be carried out in order to detect hidden defects and eliminate them.

When conducting thermal imaging quality control of thermal insulation of enclosing structures in accordance with GOST 54852-2011, when the defective area is located in the area of ​​the butt joint of wall panels or window block and panel, the air permeability of the butt joint should be checked in accordance with GOST 31167.

What is breathability and air exchange rate

Breathability- the property of enclosing structures to pass air. Volumetric air permeability is air permeability equal to the volumetric air flow per unit time per 1 m2 of the fence, and expressed in cubic meters per square meter per hour (m3/(m2×h)).

Depending on the direction of air movement through the building envelope, there are such concepts as infiltration and exfiltration.

Infiltration- due to the movement of air through the fences from the environment into the room due to wind, thermal and gravitational pressures that form the air pressure difference outside and inside the room.

Exfiltration is the opposite of infiltration.

Air exchange rate- the ratio during testing of the volumetric air flow to the internal volume per unit time, expressed in hours to the minus first power (h-1). In other words, this is the amount of air that is removed from the room in 1 hour and replaced with fresh air.

What is the purpose of air permeability and air exchange rate measurements?

Air permeability affects the temperature and humidity conditions of the premises, sanitary and hygienic standards, the durability of building structures, the heat balance of the building, and the ventilation system.

If the air permeability does not meet the standards, then this can lead to the following consequences:

• Heat losses through the building envelope increase, which in turn leads to a lack of thermal energy for space heating and, as a result, a decrease in temperature.
• During exfiltration, humid air accumulated in the room passes through the enclosing structures, which leads to waterlogging of building structures and, as a result, their deterioration in their thermal properties and their destruction.
• Violation of ventilation and air conditioning systems, with certain pressure drops, they do not cope with their duties, and sometimes do not work at all.
• With increased air permeability between the internal enclosing structures, penetration from neighboring premises (basement, underground car parking, attic, boiler room, boiler room, etc.) of harmful pollutants is possible.

The frequency of air exchange directly affects the health and safety of people's lives.

If the air exchange rate does not meet the standards, then this can lead to the following consequences:

• With an increased frequency of air exchange, the HVAC system cannot cope and, as a result, the temperature and humidity conditions in the room are disturbed and heat losses increase. In addition, the microclimate in the room is disturbed, people begin to experience discomfort from the increased speed of air movement.
• With a low air exchange rate, the concentration of harmful substances in the room increases, the concentration of oxygen in the air decreases, which leads to the release of carbon monoxide and oxygen starvation. Also, the concentration of water vapor in the room increases, the humidity rises and this can lead to the formation of mold in damp and poorly ventilated places.

That is why it is so important to control the parameters of air permeability and air exchange.

Equipment for measuring the air exchange rate

As a measuring equipment, a device called "Air Door" is used. It includes a specially designed calibrated fan with a maximum capacity of 14,000 m3/h, a frequency converter, a 2-channel digital micromanometer with software for controlling, measuring and monitoring the required parameters, a sliding frame with an airtight canvas for installing the fan in any door or window opening

This equipment is manufactured in the USA and Canada and meets all the requirements of international and Russian standards.

The fan in the system can operate in the mode of air injection (pressure difference is positive) and in the mode of air discharge (pressure difference is negative).

The system automatically takes measurements and controls the operation of the fan, so the air permeability test is performed with great accuracy (due to a large array of measurements) and with minimal time.

Air door Retrotec Q4E

Combined use of air doors and thermal imaging

The use of an air door allows you to improve the quality of the thermal imaging survey. The essence of the method lies in the fact that initially a thermal imager is taken without the use of an air door and all detected defects are recorded. Then an air door is installed and a guaranteed pressure difference is created between the indoor and outdoor air. After that, the thermal imager is again filmed, and since. temperatures differ from each other, the thermal imager easily detects defects associated with poor tightness of building structures. Also in this case, it is easier to interpret the nature of thermal defects, it can be said with certainty whether the defect is caused by poor thermal insulation, the presence of a cold bridge, or increased air permeability.

In addition, defects caused by increased air permeability can be detected at temperature differences of only 2-3 0C, which allows these measurements to be made at any time of the year. This is especially important for construction customers who want to somehow evaluate the work of a contractor who delivers a construction site in the summer.

Services for individuals

For individuals, we also provide air door measurement and combination services and thermal imaging. For apartment owners, this will help solve a number of the following problems:

• Lack of thermal energy during the heating season of the year (increased electricity bills).
• Increased air velocity inside the room.
• The formation of fungus on building envelopes.
• Destruction of building structures.
• The nature of thermal defects will be identified, which will save money on eliminating defects.
• Insufficient performance (shortage) of ventilation and air conditioning systems in the summer season (increased electricity bills).
• The entry of harmful contaminants into the premises.

For individual developers (owners of cottages), in addition to solving the above problems, the advantage of carrying out these measurements is as follows:

• During the construction of a house, it is possible to control the work on insulation and fastening of the vapor barrier before the start of finishing work.
• When building an energy-efficient house using supply and exhaust ventilation with a heat exchanger, it is very important that the air permeability is as low as possible. By taking measurements and shooting an object with a thermal imager, all defective places are identified and eliminated.
• Reduced breathability saves on electricity bills, gas bills, and more.

Tests of building structures in laboratory conditions

Having at our disposal a climatic chamber with dimensions of 5 m by 6 m and a height of 4 m, in addition to thermal testing of fragments of building structures, windows, doors, etc., we can also test these structures with the help of an air door for air permeability. And also to conduct joint heat engineering tests with imitation of wind pressure on the building structure in the cold compartment of the chamber.

Ways to measure blood pressure

Blood pressure is measured by a doctor or nurse on an outpatient basis or in a hospital (clinical blood pressure). Blood pressure can also be recorded by the patient himself or by relatives at home - blood pressure self-monitoring (SCAD). ABPM is performed by health workers on an outpatient basis or in a hospital setting. Clinical measurement of blood pressure has the largest evidence base for substantiating the classification of blood pressure, predicting risks, and evaluating the effectiveness of therapy. The accuracy of blood pressure measurement and, accordingly, the guarantee of the correct diagnosis of hypertension, determining the degree of its severity, depend on compliance with the rules for its measurement

To measure blood pressure, the following conditions are important:

1.1. The position of the patient

Sitting in a comfortable position: the arm is on the table and is at the level of the heart: the cuff is applied to the shoulder, its lower edge is 2 cm above the elbow.

1.2. Conditions for measuring hell

the use of coffee and strong tea is excluded for 1 hour before the study;

sympathomimetics are cancelled. including nasal and eye drops;

BP is measured at rest after a 5-minute rest; if the procedure for measuring blood pressure was preceded by significant physical or emotional stress, the rest period should be extended to 15-30 minutes.

1.3. Equipment

cuff size must match size arms: the rubber inflated part of the cuff should cover at least 80% of the circumference of the upper arm; for adults, a cuff 12-13 cm wide and 30-35 cm long (medium size) is used; it is necessary to have large and small cuffs for full and thin arms, respectively;

the mercury column or the arrow of the tonometer must be at zero before starting the measurement.

1.4. Multiplicity of measurement

to assess the value of blood pressure on each arm, at least two measurements should be performed with an interval of at least a minute; at a difference > 5 mmHg make one additional measurement; the final (recorded) value is the average of the last two measurements;

for the diagnosis of hypertension with a slight increase in blood pressure, a second measurement (2-3 times) is carried out after a few months;

with a pronounced increase in blood pressure and the presence of POM, high and very high risk of CVE, repeated blood pressure measurements are carried out after a few days.

1.5. Measurement technique

quickly inflate the cuff to a pressure of 20 mm Hg. exceeding SBP (by the disappearance of the pulse);

Blood pressure is measured with an accuracy of 2 mm Hg;

reduce the pressure in the cuff at a rate of approximately 2 mm Hg. per second;

the magnitude of the pressure at which 1 tone appears corresponds to SBP (1 phase of Korotkoff tones);

the amount of pressure at which the disappearance of tones occurs (5 phase of Korotkoff sounds) corresponds to DBP; in children, adolescents and young people immediately after exercise, in pregnant women and in certain pathological conditions, in adults, when it is impossible to determine the 5th phase, one should try to determine the 4th phase of Korotkoff's tones, which is characterized by a significant weakening of the tones;

if the tones are very weak, then you should raise your hand and perform several squeezing movements with the brush, then repeat the measurement, while not strongly squeezing the artery with the membrane of the phonendoscope;

during the initial examination of the patient, pressure should be measured on both hands: in the future, measurements are taken on the arm where blood pressure is higher:

in patients over 65 years of age. at availability DM and in persons receiving antihypertensive therapy (AHT), blood pressure should also be measured after 2 minutes of standing and standing;

it is also advisable to measure blood pressure in the legs, especially in patients under 30 years of age: the measurement is carried out using a wide cuff (the same as in obese persons): the phonendoscope is located in the popliteal fossa; to detect occlusive lesions of the arteries and assess the ankle-brachial index, SBP is measured using an ankle cuff and / or ultrasound method;

heart rate is calculated from the pulse at the radial artery (at least 30 seconds) after the second measurement of blood pressure in the sitting position.

Prefixes for multiple units

Multiple units- units that are an integer number of times greater than the basic unit of measurement of some physical quantity. The International System of Units (SI) recommends the following prefixes for denoting multiple units:

multiplicity	Console		Designation		Example
	Russian	international	Russian	international
10 1	soundboard	Deca	Yes	da	dal - decalitre
10 2	hecto	hecto	G	h	hPa - hectopascal
10 3	kilo	kilo	To	k	kN - kilonewton
10 6	mega	Mega	M	M	MPa - megapascal
10 9	giga	Giga	G	G	GHz - gigahertz
10 12	tera	Tera	T	T	TV - teravolt
10 15	peta	Peta	P	P	Pflop -	10 18	exa	Hexa	E	E	EB - exabyte
10 21	zetta	Zetta	W	Z	ZeV - zettaelectronvolt
10 24	yotta	Yotta	AND	Y	Yb - yottabyte

Binary understanding of prefixes

In programming and the computer-related industry, the same prefixes kilo-, mega-, giga-, tera-, etc., when applied to values ​​that are multiples of powers of two (for example, bytes), can mean a multiple of not 1000 , and 1024=2 10 . Which system is used should be clear from the context (for example, for the amount of RAM, the multiplicity of 1024 is used, and for the amount of disk memory, the multiplicity of 1000 is introduced by hard disk manufacturers).

1 kilobyte	= 1024 1	= 2 10	= 1024 bytes
1 megabyte	= 1024 2	= 2 20	= 1,048,576 bytes
1 gigabyte	= 1024 3	= 2 30	= 1,073,741,824 bytes
1 terabyte	= 1024 4	= 2 40	= 1,099,511,627,776 bytes
1 petabyte	= 1024 5	= 2 50	= 1,125,899,906,842,624 bytes
1 exabyte	= 1024 6	= 2 60	= 1,152,921,504,606,846,976 bytes
1 zettabyte	= 1024 7	= 2 70	= 1 180 591 620 717 411 303 424 bytes
1 yottabyte	= 1024 8	= 2 80	= 1 208 925 819 614 629 174 706 176 bytes

To avoid confusion, in April 1999 the International Electrotechnical Commission introduced a new standard for naming binary numbers (see Binary prefixes).

Prefixes for submultiple units

submultiple units, make up a certain proportion (part) of the established unit of measurement of a certain quantity. The International System of Units (SI) recommends the following prefixes for submultiple units:

Dolnost	Console		Designation		Example
	Russian	international	Russian	international
10 −1	deci	deci	d	d	dm - decimeter
10 −2	centi	centi	With	c	cm - centimeter
10 −3	Milli	milli	m	m	mm - millimeter
10 −6	micro	micro	mk	(u)	micron - micrometer, micron
10 −9	nano	nano	n	n	nm - nanometer
10 −12	pico	pico	P	p	pF - picofarad
10 −15	femto	femto	f	f	fs - femtosecond
10 −18	atto	atto	A	a	ac - attosecond
10 −21	zepto	zepto	h	z
10 −24	yokto	yocto	And	y

Origin of prefixes

Most prefixes are derived from Greek words. Deca comes from the word deca or deka (δέκα) - "ten", hecto - from hekaton (ἑκατόν) - "one hundred", kilo - from chiloi (χίλιοι) - "thousand", mega - from megas (μέγας), that is, " big", giga is gigantos (γίγας) - "giant", and tera is from teratos (τέρας), which means "monstrous". Peta (πέντε) and exa (ἕξ) correspond to five and six thousand digits and are translated as "five" and "six" respectively. Longitudinal micro (from micros, μικρός) and nano (from nanos, νᾶνος) are translated as "small" and "dwarf". From one word ὀκτώ (októ), meaning "eight", the prefixes yotta (1000 8) and yokto (1/1000 8) are formed.

As "thousand" the prefix milli, which goes back to the Latin mille, is also translated. Latin roots also have the prefixes santi - from centum ("one hundred") and deci - from decimus ("tenth"), zetta - from septem ("seven"). Zepto ("seven") comes from the Latin word septem or from the French sept.

The prefix atto is derived from the Danish atten ("eighteen"). Femto is derived from Danish (Norwegian) femten or Old Norse fimmtān and means "fifteen".

The prefix pico comes either from the French pico ("beak" or "small number"), or from the Italian piccolo, meaning "small".

Rules for using prefixes

• Prefixes should be written together with the name of the unit or, accordingly, with its designation.
• The use of two or more prefixes in a row (eg micromillifarad) is not permitted.
• The symbols for multiples and submultiples of the original unit raised to a power are formed by adding the corresponding exponent to the designation of a multiple or submultiple of the original unit, and the exponent means raising the multiple or submultiple to the power (together with the prefix). Example: 1 km² = (10³ m)² = 10 6 m² (not 10³ m²). The names of such units are formed by adding a prefix to the name of the original unit: square kilometer (not kilo-square meter).
• If the unit is a product or ratio of units, the prefix, or its designation, is usually attached to the name or designation of the first unit: kPa s / m (kilopascal second per meter). Attaching a prefix to the second factor of the product or to the denominator is allowed only in justified cases.

Applicability of prefixes

Due to the fact that the name of the unit of mass in SI - kilogram - contains the prefix "kilo", for the formation of multiple and submultiple units of mass, a submultiple unit of mass is used - grams (0.001 kg).

Prefixes have limited use with units of time: multiple prefixes don't go with them at all (nobody uses "kilosecond", although it's not formally forbidden), prefixes are only attached to the second (millisecond, microsecond, etc.). In accordance with GOST 8.417-2002, the name and designations of the following SI units are not allowed to be used with prefixes: minute, hour, day (time units), degree, minute, second (flat angle units), astronomical unit, diopter and atomic mass unit.

see also

• Non-SI unit prefix (English Wikipedia)
• IEEE standard for prefixes

Literature

SI prefixes
Multiples prefixes	deca-( 10 1 ) hecto- ( 10²) kilo- ( 10³) mega- ( 10 6 ) giga- ( 10 9 ) tera- ( 10 12 ) peta- ( 10 15 ) exa- ( 10 18 ) ·