A hollow shaft of diameter ratio 3/8 (internal dia to outer dia) is to transmit 375 kW power at 100 rpm. The maximum torque being 20% greater than the mean torque. The shear stress is not to exceed 60 N/mm2 and twist in a length of 4m not to exceed 2o. Calculate its external and internal diameters which would satisfy both the above conditions. (G= 0.85 X 105 N/mm2)

Answers

Answer 1

Answer:

External diameter = 158.15 mm mm

Internal diameter = 59.31 mm

Explanation:

We are given;

Diameter ratio; d_i = ⅜d_o

Where d_i is internal diameter and d_o is external diameter

Power;P = 375 KW = 375000 W

Rotational speed;N = 100 rpm

Max torque is 20% greater than mean torque; T_max = 1.2T_avg

Shear stress;τ = 60 N/mm²

Length; L = 4m = 4000 mm

Angle of twist; θ = 2° = 2π/180 radians

Modulus of rigidity;G = 0.85 X 10^(5) N/mm²

Formula for the power transmitted by the shaft is;

P = 2πNT_avg/60

Plugging in the relevant values, we have ;

375000 = 2π × 100T_avg/60

T_avg = (375000 × 60)/(2π × 100) = 35809.862 N.m = 35809862 N.mm

Since T_max = 1.20T_avg

Thus, T_max = 1.20(35809862) = 42971834.4 N.mm

Checking for strength, we'll use;

τ = Tr/J_p

Or since r = d/2

It can be written as;

τ = T(d_o)/2J_p - - - (1)

Where T is T_max

But Polar moment of inertia of hollow shaft is;

J_p = [π(d_o)⁴ - π(d_i)⁴]/32

Now, we are told that d_i = ⅜d_o

Thus;

J_p = [π(d_o)⁴ - π(⅜d_o)⁴]/32

J_p = (π/32) × d_o⁴(1 - 3⁴/8⁴)

J_p = 0.0926 d_o⁴

Plugging this for J_p in eq 1,we have;

τ = T(d_o)/2(0.0926d_o⁴)

Making d_o the subject gives;

d_o³ = T/(2 × 0.0926τ)

Plugging in the relevant values to give;

d_o³ = 42971834.4/(2 × 0.0926 × 60)

d_o³ = 3867155.7235421166

d_o = ∛3867155.7235421166

d_o = 156.96 mm

Thus, d_i = ⅜ × 156.96 = 58.86 mm

Checking for stiffness, we'll use;

T/J_p = Gθ/L

Again T is T_max

Plugging in the relevant values, we have;

42971834.4/0.0926 d_o⁴ = (0.85 × 10^(5) × 2π/180)/4000

464058686.825054/d_o⁴ = 0.7417649321

d_o⁴ = 464058686.825054/0.7417649321

d_o⁴ = 625614216.5028806

d_o = ∜625614216.5028806

d_o = 158.15 mm

d_i = ⅜ × 158.15 = 59.31 mm

So we will pick the highest values.

Thus;

d_o = 158.15 mm

d_i = 59.31 mm


Related Questions

Which battery cable is bolted to the vehicle frame to allow the metal structure of the vehicle to serve as a large conductor to carry current?

Answers

Answer:

The Negative battery cable is used to bolt the vehicle frame to allow the metal structure of the vehicle to serve as a large conductor to carry current.

Explanation:

Battery cable is large automotive cable. Like smaller types of automotive wire, it is available in PVC and cross-linked forms. Click here to read an earlier post explaining the differences between PVC and cross-linked wire insulation.

One type of PVC battery cable is SGT cable. It is rated to 80°C and therefore can be used in starters or battery grounds.  Cross-linked battery cables can also be used in starter and battery ground applications, and they are more resistant to heat, abrasion, and aging than PVC cable.

Two types of cross-linked battery cable are SGX and STX. They are rated to 125°C. Of the three types of battery cable (SGT, SGX, amd STX), STX has the thinnest wall, making it a popular choice for automotive applications with limited space.

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Which conditions contribute to engine deposits

Answers

Driving consistently will cause the oil to heat up and break down to create sludge.

And if oil gets too hot, it will cause metal on metal grinding due to the oil thinning out.

Hope this helps you. :)

What signal propagation phenomena causes the diffusion, or the reflection in multiple different directions, of a signal?

Answers

In the radio communication system, multipath is the propagation phenomenon that causes diffusion or reflection in multiple different directions of a signal.

Multipath is a propagation mechanism that impacts the propagation of signals in radio communication. Multipath results in the transmission of data to the receiving antenna by two or more paths. Diffusion and reflection are the causes that create multiple paths for the signal to be delivered.

Diffraction occurs when a signal bends around sharp corners; while reflection occurs when a signal impinges on a smooth object. When a signal is received through more than one path because of the diffraction or reflection, it creates phase shifting and interference of the signal.

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If the water surface elevation in reservoir B is 110 m, what must be the water surface elevation in reservoir A if a flow of 0.03 m3 /s is to occur in the cast iron pipe

Answers

The water surface elevation must be 110.2631 meters for a flow of 0.03m³ to occur in the cast pipe

For cast iron the chart has 0.0012  from Moody's chart

0.016 for cast iron

\(hf = \frac{flQ^{2}}{12.1d5}\)

\(h1 = h2+\frac{Q^{2} }{12.1} [\frac{0.0012*100}{(12/1000)^5} +\frac{0.0016*150}{(15/100)^5} ]\)

\(h1 = 110m+\frac{0.03^2}{12.1} [\f\frac{0.12}{0.00032} +\frac{0.24}{0.000759} ]\\\\h1 = 110+0.0000744[375+3162.06]\\\\= 110 + 0.2631m\\\\= 110.2631m\)

Therefore the water surface elevation must be 110.2631 meters for a flow of 0.03m³ to occur in the cast pipe

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.. You should
an eyewash station periodically.
A)
Remove
B) O Paint
C) Inspect
DO Move

Answers

Answer:

C.

Explanation:

You want to make sure it still works. You don't want to move it periodically though in case of an emergency.

The answer to this is C .

1. Discuss inverse square law? Corire law of illustration. 2. A lamp fitted with 120 degrees angled cone reflector illuminates circular area of 200 meters in diameter. The illumination of the disc increases uniformly from 0.5 meter-candle at the edge to 2 meter-candle at the center. Determine: i) the total light received ii) Average illumination of the disc ii) Average candle power of the source. 3. Discuss the floodlighting with suitable diagrams.

Answers

1. Inverse square law states that the intensity of light varies inversely with the square of the distance from the source. It can be represented mathematically as: I = k/d², where I is the intensity of light, d is the distance from the source and k is a constant of proportionality.

This law is illustrated by the fact that as the distance from the source increases, the intensity of light decreases proportionally to the square of the distance.2. Given, diameter of the circular area, d = 200 mRadius of the circular area, r = d/2 = 100 mLamp illuminates a circular area of 200 meters in diameter with the illumination of the disc increasing uniformly from 0.5 meter-candle at the edge to 2 meter-candle at the center. The average illumination can be calculated as follows:Average illumination of the disc, I = (0.5 + 2)/2 = 1.25 meter-candleThe total light received can be calculated as follows:Total light received = (2πr² × I) = (2 × π × 100² × 1.25) = 78,540 lumensAverage candle power of the source can be calculated as follows:Average candle power = Total light received/4π = 78,540/4π = 6250 lumens3. Floodlighting is the use of high-intensity artificial light to illuminate a large area.

The purpose of floodlighting is to provide a bright and uniform light over a large area, typically for outdoor sports fields, stadiums, and other large events. It can be achieved using various types of lighting fixtures, such as floodlights, spotlights, and high-intensity discharge lamps. Suitable diagrams for floodlighting are shown below:

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the following list of common malfunctions are for which sensor related instrument? plugged or partially plugged tubing leakage improperly sized tubing excessive vibration heat tracing too hot or too cold mechanical damage

Answers

The list of common malfunctions given in the question belongs to the 'impulse tubing' sensor-related instrument.

Impulse is referred to the sensing process of medium pressure, and tubing that directly connects this sensing process tapping to the transmitters is called impulse tubing. Impulse tubing is used to transmit the pressure signal from a process to a transmitter. Impulse tubing can be affected by several common malfunctions such as leakage, plugged or partially plugged tubing, improperly sized tubing, heat tracing too hot or too cold, excessive vibration, and mechanical damage.

"

Correct question is:

The following list of common malfunctions are for which sensor related instrument?

List of common malfunctions is as follows:

plugged or partially plugged tubingleakage improperly sized tubing excessive vibration heat tracing too hot or too cold mechanical damage

"

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1.3. If the surface tension coefficient of a fluid is 0,07 N/m and the radius
of the droplet is 2,5 mm. calculate:
1.3.1 surface tension force
(2)
1.3.2 difference in pressure of the droplet
(1)​

Answers

Answer:

A) F = 0.011 N

B) ΔP = 5.6 N/m²

Explanation:

We are given;

surface tension coefficient; S = 0.07 N/m

Radius; r = 2.5 mm = 0.025 m

A) Formula to find the surface tension force(F) is given by;

F = SL

Where L is effective length = 2πr

F = 0.07 × 2π × 0.025

F = 0.011 N

B) Formula for difference in pressure droplet is;

ΔP = 2S/r

Thus;

ΔP = (2 × 0.07)/0.025

ΔP = 5.6 N/m²

If you become an auto mechanic if there is damage in the parts and not repairable, what will you do?​

Answers

Answer:

In most cases, it will never get that far - the shop will cave and agree to properly diagnose your problem and correctly fix it. Usually, the mechanic that failed to do their job correctly will be back-flagged for the labor and in some cases they'll even be charged for the parts as well

Explanation:

A 10-V, 1.0-A dc current is run through a step-up transformer that has 10 turns on the input side and 20 turns on the output side. What is the output?(a) 10 V, 0.5 A.(b) 20 V, 0.5 A.(c) 20 V, 1 A.(d) 10 V, 1 A.(e) 0V, 0A.

Answers

The output of the step-up transformer is 20 V, 0.5 A (Option b).

A step-up transformer increases the voltage while decreasing the current on the output side compared to the input side. The turns ratio of the transformer is given as 10:20, which means that for every 10 turns on the input side, there are 20 turns on the output side.

Since the input voltage is 10 V and the turns ratio is 10:20, the output voltage can be calculated as follows:

Output Voltage = Input Voltage x (Number of Turns on Output Side / Number of Turns on Input Side)

= 10 V x (20 / 10)

= 20 V

Similarly, since the input current is 1.0 A and the turns ratio is 10:20, the output current can be calculated as follows:

Output Current = Input Current x (Number of Turns on Input Side / Number of Turns on Output Side)

= 1.0 A x (10 / 20)

= 0.5 A

Therefore, the output of the step-up transformer is 20 V, 0.5 A (Option b).

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By balancing information security and access, a completely secure information system can be created.A. TrueB. False

Answers

Answer: true

Explanation:

Which three items below should a driver be able to identify under the hood of a car?

Answers

Answer:

Engine oil level.

Brake fluid.

Power steering fluid.

1. Round off 2553 N to three significant figures.
2. Round off 58342 m to three significant figures.
3. Round off 68.534 s to three significant figures.

Answers

Answer:

(1) 2553 N = 2550 N

(2) 58342 m = 58300 m

(3) 68.534 s = 68.5 s

Explanation:

To round off a number to any significant number start from the last digit, round it off to 1 if the number is up to 5 and to 0 if the last digit is less than 5. Add this 1 or 0 to the preceding digit and continue the process until you are left with three non zero digits, if you are rounding off to three significant figures.

(1) Round off 2553 N to three significant figures.

= 2550 N

(2) Round off 58342 m to three significant figures.

= 58300 m

(3) Round off 68.534 s to three significant figures.

= 68.500 s (zero after decimal point is insignificant)

= 68.5 s

Hey guys can anyone list chemical engineering advancement that has been discovered within the past 20 years

Answers

Top 10 Emerging Technologies in Chemistry
Nanopesticides. The world population keeps growing. ...
Enantio selective organocatalysis. ...
Solid-state batteries. ...
Flow Chemistry. ...
Porous material for Water Harvesting. ...
Directed evolution of selective enzymes. ...
From plastics to monomers. ...

Consider steady one-dimensional heat transfer through a plane wall exposed to convection from both sides to environments at known temperatures T[infinity]1 and T[infinity]2 with known heat transfer coefficients h1 and h2. Once the rate of heat transfer has been evaluated, explain how you would determine the temperature of each surface.

Answers

Answer:

Now find the temperature of each surface, we have that the the temperature on the left side of the wall is T∞₁ - Q/h₁A and  the temperature on the right side of the wall is T∞₂ + Q/h₂A.

Note: kindly find an attached diagram to the complete question given below.

Sources: The diagram/image was researched and taken from Slader website.

Explanation:

Solution

Let us consider the rate of heat transfer through the plane wall which can be obtained from the relations given below:

Q = T∞₁ -T₁/1/h₁A = T₁ -T₂/L/kA =T₂ -T∞₂/1/h₂A

= T∞₁  - T∞₂/1/h₁A + L/kA + 1/h₂A

Here

The convective heat transfer coefficient on the left side of the wall is h₁, while the convective heat transfer coefficient on the right side of the wall is h₂. the thickness of the wall is L, the thermal conductivity of the wall material is k, and the heat transfer area on one side of the wall is A. Q is refereed to as  heat transfer.

Thus

Let us consider the convection heat transfer on the left side of the wall which is given below:

Q = T∞₁ -T₁/1/h₁A

T₁ = T∞₁ - Q/h₁A

Therefore the temperature on the left side of the wall is T∞₁ - Q/h₁A

Now

Let us consider the convection heat transfer  on the left side of the wall which is given below:

Q= T₂ -T∞₂/1/h₂A

T₂ = T∞₂ + Q/h₂A

Therefore the temperature on the right side of the wall is T∞₂ + Q/h₂A

Consider steady one-dimensional heat transfer through a plane wall exposed to convection from both sides

Four standard concrete cylinders of 6-inch diameter cross section were tested under uniformly increasing axial compressive loads 28 days after casting concrete. The maximum measured loads were 121,000 lbs, 116,700 lbs, 128,800 lbs, and 119,100 lbs. What is the average compressive strength of this concrete in ksi

Answers

Given,Four standard concrete cylinders of 6-inch diameter cross-section were tested under uniformly increasing axial compressive loads 28 days after casting concrete.

The maximum measured loads were 121,000 lbs, 116,700 lbs, 128,800 lbs, and 119,100 lbs.So, the average compressive strength of this concrete in ksi is as follows:The cross-sectional area of the cylinders, A= π r²where r= 6/2= 3 inches= 0.25 feetA = π × (0.25)²= 0.0491 ft²The maximum load on the concrete cylinders can be calculated by using the formula:σ = P/Awhere P = load on the cylinder,

and A = cross-sectional area of the cylinderσ = 121000/0.0491 = 2462631.4 lb/ft²σ = 2462631.4/144 = 17096.5 psi= 17096.5/1000 ksi= 17.1 ksiSimilarly,σ = 116700/0.0491 = 2373864.7 lb/ft²σ = 2373864.7/144 = 16486 psi= 16.5 ksiσ = 128800/0.0491 = 2623481.9 lb/ft²σ = 2623481.9/144 = 18209 psi= 18.2 ksiσ = 119100/0.0491 = 2425757.9 lb/ft²σ = 2425757.9/144 = 16847.7 psi= 16.8 ksiTherefore, the average compressive strength of this concrete in ksi is given by:Average compressive strength = (17.1 + 16.5 + 18.2 + 16.8)/4= 68.6/4= 17.15 ksi (approx)Hence, the average compressive strength of this concrete in ksi is 17.15.

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Can space debris take out a whole state

Answers

Yes depends how big the debris is

How does the action of wind affect a rock’s appearance?

Answers

Answer:

Forces like wind and water break down rocks through the processes of weathering and erosion. ... Forces like wind and water move the rock pieces. They mix with matter like sand to become sediment. Weathering and erosion help shape Earth's surface.

Answer:

As an agent of erosion, the wind will quickly break the rock into different rock types.

Explanation:

How do cars moves? and explain Please.

Answers

Answer:

when car vroom it go electric

Explanation:

i think then it goes to some spinny thing that spins and make the electric and the electic spins the spinny thing and the smoke comes outside the back

_____ draft is a mechanical draft created by air pulled through the boiler firebox by a blower located in the breaching after the boiler.

Answers

Answer:

Induced draft is a mechanical draft created by air pulled through the boiler firebox by a blower located in the breaching after the boiler.

Explanation:

An adult has a total of about 22.3 square feet (ft2) of skin. Use the fact that 1 m is approximately equal to 3.281 feet to convert this measurement to square meters (m2).

Answers

Explanation:

1 meter ≈ 3.281 ft.

so,

1 m² = 1m×1m ≈ 3.281 × 3.281 = 10.764961 ft²

now we have 22.3 ft² of skin.

that would be then

22.3/10.764961 = 2.071535605 ≈ 2.1 m² or even more rounded ≈ 2 m²

when a roof slope is 2/12 or less the rafters are often referred to as ___ or ___

Answers

When a roof slope is 2/12 or less, the rafters are often referred to as flat-roof rafters or low-slope roof rafters.

In traditional residential construction, low-slope roofs are often constructed using a different method than steep-slope roofs with angled rafters. Instead of using rafters, low-slope roofs typically employ a system of joists or trusses.

Here are a few common methods used for constructing low-slope roofs:

Joists: Low-slope roofs with relatively small spans can be constructed using joists. Joists are horizontal members that run parallel to each other and support the roof deck. They are typically made of wood or metal and are spaced at regular intervals. The roof deck, which can be made of plywood or oriented strand board (OSB), is then attached to the top of the joistsTrusses: For larger low-slope roofs or when longer spans are required, trusses are often used. Trusses are pre-engineered structural elements that consist of triangular-shaped frames made of wood or metal. These frames provide support and stability to the roof. The trusses are spaced at regular intervals and are placed directly on the load-bearing walls or support columns. The roof deck is then attached to the top of the trussesBuilt-up Roof (BUR): Another common method for low-slope roofs is the built-up roof system. This system involves applying multiple layers of roofing felt or fiberglass mats, which are then coated with bitumen (asphalt or coal tar) and gravel. The layers are built up to create a durable, waterproof membrane that protects the underlying structure. The built-up roof system can be installed on top of a rigid insulation board or directly on the roof deckSingle-ply Membrane: Single-ply membranes, such as EPDM (ethylene propylene diene terpolymer) or TPO (thermoplastic olefin), are also used for low-slope roofs. These membranes are manufactured in large rolls and are either mechanically fastened or adhered to the roof deck. Single-ply membranes provide a continuous, waterproof layer that is resistant to UV radiation and weathering.

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The UHRS platform is optimized for Edge/Internet Explorer only. You can still use your favorite browser, but keep in mind that you may experience technical issues when working on UHRS with a different browser than Edge or Internet Explorer.

UHRS is optimized for...

Answers

It is to be noted that all UHRS platforms are optimized for the popular kinds of internet browser applications.

What is a UHRS?

The Universal Human Relevance System (UHRS) is a crowdsourcing platform that allows for data labeling for a variety of AI application situations.

Vendor partners link people referred to as "judges" to offer data labeling at scale for us. All UHRS judges are bound by an NDA, ensuring that data is kept protected.

A browser is a software tool that allows you to see and interact with all of the knowledgeon the World Wide Web. Web sites, movies, and photos are all examples of this.

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Fast fourier transforms (FFT) are algorithms that speed up the computation of fourier coefficients compared to the traditional direct form discuss how the speed up is achieved

Answers

Answer:

Fast Fourier ( FFT ) algorithms speed up computation of Fourier coefficients by simply reducing the the computing time of a traditional direct form  Fourier series. it  achieves this by breaking complex DFTS into smaller DFTS to reduce its complexity and in turn reduce its computing time

Explanation:

Fast Fourier ( FFT ) algorithms speed up computation of Fourier coefficients by simply reducing the the computing time of a traditional direct form  Fourier series. it  achieves this by breaking complex DFTS into smaller DFTS to reduce its complexity and in turn reduce its computing time. an example of such FFT is Cooley-Tukey algorithm

Select the correct answer.

Which statement is true regarding the use of Kirchhoff’s laws?

A.
The voltage change across a resistor is negative when you traverse in the opposite direction of the current.
B.
Some current is lost at the junction due to resistance across all resistors in the circuit.
C.
Every loop should contain junctions from where current branches out in different directions.
D.
The voltage change across a resistor is zero when you traverse in the opposite direction of the current.
E.
The sum of currents flowing into a junction is more than the sum flowing out of it.

Answers

Answer:

A.

Explanation:

The voltage change across a resistor is negative when you traverse in the opposite direction of the current.

engineering economics​

engineering economics

Answers

Explanation:

mathematics education grade 7

Which two physical resources are abstracted by the hypervisor and presented to virtual machines? (Choose two.)
A. Libraries
B. CPU
C. Application
D. Desktop
E. Memory

Answers

The two physical resources that are abstracted by the hypervisor and presented to virtual machines are:

B. CPU (Central Processing Unit)

E. Memory

Explanation:

1. CPU (Central Processing Unit): The hypervisor abstracts the physical CPU(s) of the host machine and presents them to virtual machines as virtual CPUs (vCPUs). Each virtual machine is allocated a portion of the CPU resources, allowing them to run their own operating systems and execute their own processes independently. The hypervisor manages the scheduling and allocation of CPU resources among the virtual machines.

2. Memory: The hypervisor also abstracts the physical memory (RAM) of the host machine and presents it to virtual machines as virtual memory. Each virtual machine is assigned a specific amount of memory that it can utilize for running its applications and processes. The hypervisor handles the management and allocation of memory resources, ensuring that each virtual machine receives its allocated memory and preventing conflicts or overlaps.

It's important to note that the other options mentioned are not directly abstracted by the hypervisor:

A. Libraries: Libraries are software components that contain precompiled functions and routines. They are not physical resources that can be abstracted by the hypervisor.

C. Application: Applications are software programs that run on top of the operating system. The hypervisor virtualizes the underlying hardware resources for virtual machines, but it does not directly abstract or manage specific applications.

D. Desktop: The desktop refers to the user interface and graphical environment provided by the operating system. The hypervisor focuses on abstracting and managing hardware resources rather than the desktop interface itself.

In summary, the hypervisor abstracts and presents the physical CPU and memory resources of the host machine to virtual machines as virtual CPUs and virtual memory, respectively. This allows virtual machines to have their own isolated computing environments while sharing the underlying hardware resources efficiently.

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A 1.5 m x1.5 m square footing is supported by a soil deposit that contains a 16.5 m thick saturated clay layer followed by the bedrock. The clay has μs = 0.50 and Es = 5,000 kN/m2 . The footing base is at 1.5 m below the ground surface. Determine the maximum vertical central column load so that the elastic settlement of the footing will not exceed 50.0 mm. If the square footing is replaced by a 1.2 m wide wall footing with all other conditions remaining the same.

Required:
What will be the elastic settlement under the same footing pressure?

Answers

Answer:

somewhere around 34.2223 meters thick but that's what I am estimating.

discuss why power plant and distribution system engineers are concerned with the real power absorbed by a load with the real power

Answers

Real power absorbed by a load is important for engineers as it helps them to ensure that the load is supplied with the correct power and that the power system is operating efficiently.

What is Power?

Power is the ability to act or make decisions in a way that has an influence on others and their environment. It is the ability to control and command, to motivate others, and to shape the future. It is a form of strength and authority, and can be seen in many different forms from physical strength to money, status, and knowledge. Power can be used to create positive change and empower others, or it can be used to oppress and control.

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As part of a heat treatment process, cylindrical, 304 stainless steel rods of 100-mm diameter are cooled from an initial temperature of 500 C by suspending them in an oil bath at 30 C. If a convection coefficient of 500 W/m2 K is maintained by circulation of the oil, how long does it take for the centerline of a rod to reach a temperature of 50 C, at which point it is withdrawn from the bath

Answers

Answer:

Explanation:

Given that:

diameter = 100 mm

initial temperature = 500 ° C

Conventional coefficient = 500 W/m^2 K

length  = 1 m

We obtain the following data from the tables A-1;

For the stainless steel of the rod \(\overline T = 548 \ K\)

\(\rho = 7900 \ kg/m^3\)

\(K = 19.0 \ W/mk \\ \\ C_p = 545 \ J/kg.K\)

\(\alpha = 4.40 \times 10^{-6} \ m^2/s \\ \\ B_i = \dfrac{h(\rho/4)}{K} \\ \\ =0.657\)

Here, we can't apply the lumped capacitance method, since Bi > 0.1

\(\theta_o = \dfrac{T_o-T_{\infty}}{T_i -T_\infty}} \\ \\ \theta_o = \dfrac{50-30}{500 -30}} \\ \\ \theta_o = 0.0426\\\)

\(0.0426 = c_1 \ exp (- E^2_1 F_o_)\\ \\ \\ 0.0426 = 1.1382 \ exp (-10.9287)^2 \ f_o \\ \\ = f_o = \dfrac{In(0.0374)}{0.863} \\ \\ f_o = 3.81\)

\(t_f = \dfrac{f_o r^2}{\alpha} \\ \\ t_f = \dfrac{3.81 \times (0.05)^2}{4.40 \times 10^{-6}} \\ \\ t_f= 2162.5 \\ \\ t_f = 36 mins\)

However, on a single rod, the energy extracted is:

\(\theta = pcv (T_i - T_{\infty} )(1 - \dfrac{2 \theta}{c} J_1 (\zeta) ) \\ \\ = 7900 \\times 546 \times 0.007854 \times (500 -300) (1 - \dfrac{2 \times 0.0426}{1.3643}) \\ \\ \theta = 1.54 \times 10^7 \ J\)

Hence, for centerline temperature at 50 °C;

The surface temperature is:

\(T(r_o,t) = T_{\infty} +(T_1 -T_{\infty}) \theta_o \ J_o(\zeta_1) \\ \\ = 30 + (500-30) \times 0.0426 \times 0.5386 \\ \\ \mathbf{T(r_o,t) = 41.69 ^0 \ C}\)

Other Questions
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