We compute its convolution without knowing its values or the values of the system impulse response h[n].The ranges of the Summations and the limits of the signals need to be considered to ensure proper computation.
To compute the convolution of two signals, we can use the formula:
y[n] = ∑[k=-∞ to ∞] (x[k] * h[n-k])
Let's calculate the convolutions for each given pair of signals:
A. x[n] = alpha^n u[n], h[n] = beta^n u[n] (where alpha ≠ beta)
Using the convolution formula:y[n] = ∑[k=-∞ to ∞] (x[k] * h[n-k])
y[n] = ∑[k=-∞ to ∞] (alpha^k * beta^(n-k) * u[k] * u[n-k])
Since u[k] and u[n-k] are both 1 for k ≥ 0 and n-k ≥ 0, the sum becomes:
y[n] = ∑[k=0 to n] (alpha^k * beta^(n-k))
This sum can be simplified as follows:
y[n] = alpha^n * ∑[k=0 to n] (alpha^(k-n) * beta^n)
Using the sum of a geometric series formula:
y[n] = alpha^n * [(alpha^(n+1) - beta^(n+1)) / (alpha - beta)]
B. x[n] = h[n] = alpha^n u[n]
Following the same steps as above:y[n] = ∑[k=-∞ to ∞] (x[k] * h[n-k])
y[n] = ∑[k=-∞ to ∞] (alpha^k * alpha^(n-k) * u[k] * u[n-k])
Since u[k] and u[n-k] are both 1 for k ≥ 0 and n-k ≥ 0, the sum becomes:
y[n] = ∑[k=0 to n] (alpha^k * alpha^(n-k))
This sum can be simplified as follows:y[n] = ∑[k=0 to n] (alpha^n)
Since alpha is a constant, the sum becomes:y[n] = (n+1) * alpha^n
C. x[n] = (-1/2)^n u [n - 4], h[n] = 4^n u [2 - n]
Using the convolution formula:
y[n] = ∑[k=-∞ to ∞] (x[k] * h[n-k])
y[n] = ∑[k=-∞ to ∞] ((-1/2)^k * 4^(n-k) * u[k] * u[2-n+k])
Since u[k] and u[2-n+k] are both 1 for k ≥ 0 and 2-n+k ≥ 0, the sum becomes: y[n] = ∑[k=0 to min(n,2)] ((-1/2)^k * 4^(n-k))
D. The signal x[n] is not provided, so we cannot compute its convolution without knowing its values or the values of the system impulse response h[n].The ranges of the summations and the limits of the signals need to be considered to ensure proper computation.
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Which gas is released in the SMAW process causing a
shielding affect on the molten weld pool?
•nitrogen
•carbon dioxide
•argon
•hydrogen
Which of the following devices is a simple machine?
A.
an engine
B.
a pulley
C.
a motor
D.
a bicycle
E.
a crane
Answer:
A PULLY
Explanation:
HAD THIS ONE THAT IS THE CORRECT ANWSER
Answer:
The answer is B. a pulley
Explanation:
I hope I answered your question:)
Travel Time Problem: Compute the time of concentration using the Velocity, Sheet Flow Method for Non-Mountainous Orange County and SCS method at a 25 year storm evert.
Location Slope (%) Length (ft) Land Use
1 4.5 1000 Forest light underbrush with herbaceous fair cover.
2 2.5 750 Alluvial Fans (eg. Natural desert landscaping)
3 1.5 500 Open Space with short grasses and good cover
4 0.5 250 Paved Areas (1/4 acre urban lots)
Answer:
Total time taken = 0.769 hour
Explanation:
using the velocity method
for sheet flow ;
Tt = [tex]\frac{0.007(nl)^{0.8} }{(Pl)^{5}s^{0.4} }[/tex]
Tt = travel time
n = manning CaH
Pl = 25years
L = how length ( ft )
s = slope
For Location ( 1 )
s = 0.045
L = 1000 ft
n = 0.06 ( from manning's coefficient table )
Tt1 = 0.128 hour
For Location ( 2 )
s = 2.5 %
L= 750
n = 0.13
Tt2 = 0.239 hour
For Location ( 3 )
s = 1.5%
L = 500 ft
n = 0.15
Tt3 = 0.237 hour
For Location (4)
s = 0.5 %
L = 250 ft
n = 0.011
Tt4 = 0.165 hour
hence the Total time taken = Tt1 + Tt2 + Tt3 + Tt4
= 0.128 + 0.239 + 0.237 + 0.165 = 0.769 hour
A sign structure on the NJ Turnpike is to be designed to resist a wind force that produces a moment of 25 k-ft in one direction. The axial load is 30 kips. Soil conditions consist of a normally consolidated clay layer with following properties; su=800 psf, andγsat= 110 pcf. Design for a FOS of 3. Assume frost depth to be 3ft below grade
Solution :
Finding the cohesion of the soil(c) using the relation:
[tex]$c = \frac{q_u}{2}$[/tex]
Here, [tex]$q_u$[/tex] is the unconfined compression strength of the soil;
[tex]$c = \frac{800}{2}$[/tex]
= 400 psf
∴ The cohesion value is greater than 0
So the use of the angle of internal friction is 0
Referring to the table relation between bearing capacity factors and angle of internal friction.
For the angle of inter friction [tex]$0^\circ$[/tex]
[tex]$N_c = 5.14$[/tex]
[tex]$N_q = 1.0$[/tex]
[tex]$N_r = 0$[/tex]
Therefore,
[tex]$q_{ult} = (400 \times 5.14 )+(110 \times 3 \times 1.0)+(0.5 \times 100 \times 13 \times 0)$[/tex]
= 2386 psf
∴ Allowable bearing capacity [tex]$q_{a} = \frac{Q_{allow}}{A}$[/tex]
[tex]$=\frac{30}{B^2}$[/tex]
∴ [tex]$q_a = \frac{q_{ult}}{F.O.S}$[/tex]
[tex]$\frac{30}{B^2} = \frac{2386}{3}$[/tex]
∴ B = 0.2 ft
Therefore, the dimension of the square footing is 0.2 ft x 0.2 ft
[tex]$=0.04 \ ft^2$[/tex]
If the sum of the two numbers is 4 and the sum of their squares minus three times their product is 76,find the number
Answer:
-2 and 6
Explanation:
Let "x" and "y" be 2 numbers.
The sum of the two numbers is 4. The mathematical expression is:
x + y = 4
y = 4 - x [1]
The sum of their squares minus three times their product is 76. The mathematical expression is:
x² + y² - 3 x y = 76 [2]
If we substitute [1] in [2], we get:
x² + (4 - x)² - 3 x (4 - x) = 76
x² + 16 - 8 x + x² - 12 x + 3 x² = 76
5 x² - 20 x - 60 = 0
We apply the solving formula for second order equations and we get x₁ = 6 and x₂ = -2.
If we replace these x values in [1], we get:
y₁ = 4 - x₁ = 4 - 6 = -2
y₂ = 4 - x₂ = 4 - (-2) = 6
As a consequence, one of the numbers is 6 and the other is -2.
Two sites are being considered for wind power generation. On the first site, the wind blows steadily at 7 m/s for 3000 hours per year. On the second site, the wind blows steadily at 10 m/s for 2000 hours per year. The density of air on the both sites is 1.25 kg/m3 . Assuming the wind power generation is negligible during other times.Calculate the maximum power of wind on each site per unit area, in kW/m2 .
Solution :
Given :
[tex]$V_1 = 7 \ m/s$[/tex]
Operation time, [tex]$T_1$[/tex] = 3000 hours per year
[tex]$V_2 = 10 \ m/s$[/tex]
Operation time, [tex]$T_2$[/tex] = 2000 hours per year
The density, ρ = [tex]$1.25 \ kg/m^3$[/tex]
The wind blows steadily. So, the K.E. = [tex]$(0.5 \dot{m} V^2)$[/tex]
[tex]$= \dot{m} \times 0.5 V^2$[/tex]
The power generation is the time rate of the kinetic energy which can be calculated as follows:
Power = [tex]$\Delta \ \dot{K.E.} = \dot{m} \frac{V^2}{2}$[/tex]
Regarding that [tex]$\dot m \propto V$[/tex]. Then,
Power [tex]$ \propto V^3$[/tex] → Power = constant x [tex]$V^3$[/tex]
Since, [tex]$\rho_a$[/tex] is constant for both the sites and the area is the same as same winf turbine is used.
For the first site,
Power, [tex]$P_1= \text{const.} \times V_1^3$[/tex]
[tex]$P_1 = \text{const.} \times 343 \ W$[/tex]
For the second site,
Power, [tex]$P_2 = \text{const.} \times V_2^3 \ W$[/tex]
[tex]$P_2 = \text{const.} \times 1000 \ W$[/tex]
An insulated closed piston–cylinder device initially contains 0.3 m3 of carbon dioxide at 200 kPa
and 27°C. A resistance heater inside the cylinder is turned on and supplied heat to the gas. As a
result, the gas expanded by pushing the piston up, until the volume doubled. During this process,
6
the pressure changed according to = 4, in which the constant 6 has units of kPa.m
a) Find the mass of the hydrogen in the tank in kg.
b) Determine the work done by the gas in kJ.
To solve this problem, we can use the ideal gas law and the equation for polytropic process.
What is ideal gas law ?The ideal gas law is a fundamental law of physics that describes the behavior of an ideal gas. It relates the pressure, volume, temperature, and number of particles of a gas using the following equation:
PV = nRT
a) First, we need to find the mass of the carbon dioxide in the tank. The ideal gas law is:
PV = mRT
where P is the pressure, V is the volume, m is the mass, R is the universal gas constant, and T is the temperature. Rearranging for the mass, we get:
m = PV / RT
Substituting the given values, we have:
m = (200 kPa)(0.3 m3) / [(0.287 kPam3/kgK)(27°C + 273.15)] = 3.87 kg
So the mass of the carbon dioxide in the tank is 3.87 kg.
b) To determine the work done by the gas during the process, we can use the equation for polytropic process:
P1V1^n = P2V2^n
where P1 and V1 are the initial pressure and volume, P2 and V2 are the final pressure and volume, and n is the polytropic index. Substituting the given values, we have:
(200 kPa)(0.3 m3)^n = (4)(0.6 m3)^n
Dividing both sides by (0.3 m3)^n and taking the logarithm of both sides, we get:
log(200) + nlog(0.3) = log(4) + nlog(0.6)
Solving for n, we get:
n = log(4/200) / log(0.6/0.3) ≈ 1.235
Using the polytropic work equation:
W = (P2V2 - P1V1) / (1 - n)
Substituting the given values, we have:
W = [(4 kPa)(0.6 m3) - (200 kPa)(0.3 m3)] / (1 - 1.235) = 233.7 kJ
So the work done by the gas during the process is 233.7 kJ.
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Which of the following choices accurately contrasts a categorical syllogism with a conditional syllogism?
An argument constructed as a categorical syllogism uses deductive reasoning whereas an argument constructed as a conditional syllogism uses inductive reasoning.
A categorical syllogism contains two premise statements and one conclusion whereas a conditional syllogism contains one premise statement and one conclusion.
A categorical syllogism argues that A and B are both members of C whereas a conditional syllogism argues that if A is true then B is also true.
An argument constructed as a categorical syllogism is valid whereas an argument constructed as a conditional syllogism is invalid.
Answer:
The correct option is - A categorical syllogism argues that A and B are both members of C whereas a conditional syllogism argues that if A is true then B is also true.
Explanation:
As,
Categorical syllogisms follow an "If A is part of C, then B is part of C" logic.
Conditional syllogisms follow an "If A is true, then B is true" pattern of logic.
So,
The correct option is - A categorical syllogism argues that A and B are both members of C whereas a conditional syllogism argues that if A is true then B is also true.
Now, you get a turn to practice writing a short program in Scratch. Try to re-create the program that was shown that turns the sprite in a circle. After you have completed that activity, see if you can make one of the improvements suggested. For example, you can try adding a sound. If you run into problems, think about some of the creative problem-solving techniques that were discussed.
When complete, briefly comment on challenges or breakthroughs you encountered while completing the guided practice activity.
Pls help im giving 100 points for this i have this due in minutes
Answer:
u need to plan it out
Explanation:
u need to plan it out
Answer:
use the turn 1 degrees option and put a repeat loop on it
Explanation:
u can add sound in ur loop
A cylindrical bar of metal having a diameter of 15.7 mm and a length of 178 mm is deformed elastically in tension with a force of 49100 N. Given that the elastic modulus and Poisson's ratio of the metal are 67.1 GPa and 0.34 respectively, Determine the following:(a) The amount by which this specimen will elongate (in mm) in the direction of the applied stress. The entry field with incorrect answer now contains modified data.(b) The change in diameter of the specimen (in mm). Indicate an increase in diameter with a positive number and a decrease with a negative number.
Answer:
0.6727
-0.02017
Explanation:
diameter = 15.7
l = 178
E =elastic modulus = 67.1 Gpa
poisson ratio = 0.34
p = force = 49100N
first we calculate the area of the cross section
[tex]A=\frac{\pi }{4} d^{2}[/tex]
[tex]A=\frac{\pi }{4} (15.7)^{2} \\A = \frac{774.683}{4} \\[/tex]
A = 193.6mm²
1. Change in directon of the applied stress
[tex]= \frac{pl}{AE}[/tex]
= 49100*178/193.6*67.1*10³
= [tex]=\frac{8739800}{12990560}[/tex]
δl = 0.6727 mm
2. change in diameter of the specimen
equation for poisson distribution =
m = -(δd/d) / (δl/l)
0.34 = (δd/15.7) / (0.6727/178)
0.34 = (-δd * 178) / 15.7 * 0.6727
0.34 = -178δd / 10.56139
we cross multiply
10.56139*0.34 =-178δd
3.5908726 = -178δd
δd = 3.5908726/-178
δd = -0.02017 mm
the change in dimeter has a negative sign. it decreases
Tech A says that LED brake lights illuminate faster than incandescent bulbs. Tech B says that LED brake lights have
more visibility and last longer. Who is correct?
Answer:
Both
Explanation:
The pressure less than atmospheric pressure is known as:
Suction pressure
Negative gauge pressure
Vacuum pressure
All of the above
Answer:
answer is option (d) all of the above
I don’t know the answer to this question
Answer:
I dont know the answer either
Explanation:
Answer:
flux
Explanation:
a load of 12tonnes is put along a horizontal plane by a force at 30°to and above the flat. if the coefficient of sliding friction is 0.2 find the frictional force
Answer:
20368.917N
Explanation:
Frictional force (F) is the product of the Coefficient of friction and the normal reaction.
F = μN
Coefficient of friction, μ = 0.2
Normal reaction = MgCosθ
Mass, m = 12 tonnes = 12 * 1000 = 12000 kg
N = 12000 * 9.8 * cos30
N = 101844.58
F = 0.2 * 101844.58
F = 20368.917N
... is an actual sequence of interactions (i.e., an instance) describing one specific situation; a ... is a general sequence of interactions (i.e., a class) describing all possible ... associated with a situation. ... are used as examples and for clarifying details with the client. ... are used as complete descriptions to specify a user task or a set of related system features.
Answer:
ScenarioUse caseScenariosScenariosUse caseExplanation:
A scenario is an actual sequence of interactions (i.e., an instance) describing one specific situation; a use case is a general sequence of interactions (i.e., a class) describing all possible scenarios associated with a situation. Scenarios are used as examples and for clarifying details with the client. Use cases are used as complete descriptions to specify a user task or a set of related system features.
The movement of the piston and connecting rod assembly driven by the force of combustion turns the ________.
Answer:
the answer. for this is Crankshaft
Convert an acceleration of 12m/s² to km/h²
If the hypotenuse of a right triangle is 12 and an acute angle is 37 degrees find leg a and leg b lengths
scrity añao devid codicie
During a shrinkage limit test, a 19.3 cm3 saturated clay sample with a mass of 37 g was placed in a porcelain dish and dried in the oven. The oven-dried sample had a mass of 28 g with a final volume of 16 cm3 . Determine the shrinkage limit and the shrinkage ratio.
Answer:
shrinkae limit = 20.35%
shrinkage ratio = 1.45
Explanation:
1. to get the shrinkage limit we would first calculate the moisture content w.
w = (37-28)/28
= 9/28
= 0.3214
then the formula for shrinkage limit is
[tex][w-\frac{(V-Vd}{wd} ]*100[/tex]
w = 0.3214
V = 19.3
Vd = 16
Wd = 28
when we put these values into the formula:
[tex][0.3214-\frac{(19.3-16)}{28} ]*100\\[/tex]
= 20.35%
2. the shrinkage limit = Wd/V
= 28/19.3
= 1.45
Which option identifies what is missing in the categorical syllogistic argument below? Quadrupeds have four legs so horses are quadrupeds.
Conclusion
Minor premise
Inference
Major premise
Select the correct answer.
Alan is walking on a street and passes by a very busy construction site. He sees engineers hanging one end of various objects to the ceiling and the other end to heavy
weights. They observe the objects and then then take down notes. What are the engineers doing
Answer: the engineers are performing a tensile test
Explanation:
i took the test and got it right
A single crystal of a metal that has the FCC crystal structure is oriented such that a tensile stress is applied parallel to the [100] direction. If the critical resolved shear stress for this material is 2.00 MPa, calculate the magnitude of applied stress necessary to cause slip to occur on the (111) plane in the direction.
Answer:
Explanation:
From the given information:
The equation for applied stress can be expressed as:
[tex]\sigma_{app} = \dfrac{\tau_{CRSS}}{cos \phi \ cos \lambda}[/tex]
where;
[tex]\phi[/tex] = angle between the applied stress [100] and [111]
To determine the [tex]\phi[/tex] and [tex]\lambda[/tex] for the system
Using the equation:
[tex]\phi= cos^{-1}\Big [\dfrac{l_1l_2+m_1m_2+n_1n_2}{\sqrt{(l_1^2+m_1^2+n_1^2)(l_2^2+m_2^2+n_2^2)}}\Big][/tex]
for [100]
[tex]l_1 = 1, m_1 = 0, n_1 = 0[/tex]
for [111]
[tex]l_1 = 1 , m_1 = 1, n_1 = 1[/tex]
Thus;
[tex]\phi= cos^{-1}\Big [\dfrac{1*1+0*1+0*1}{\sqrt{(1^2+0^2+0^2)(1^2+1^2+1^2)}}\Big][/tex]
[tex]\phi= cos^{-1}\Big [\dfrac{1}{\sqrt{(3)}}\Big][/tex]
[tex]\phi= 54.74^0[/tex]
To determine [tex]\lambda[/tex] for [tex][1 \overline 1 0][/tex]
where;
for [100]
[tex]l_1 = 1, m_1 = 0, n_1 = 0[/tex]
for [tex][1 \overline 1 0][/tex]
[tex]l_1 = 1 , m_1 = -1, n_1 = 0[/tex]
Thus;
[tex]\lambda= cos^{-1}\Big [\dfrac{1*1+0*1+0*0}{\sqrt{(1^2+0^2+0^2)(1^2+(-1)^2+0^2)}}\Big][/tex]
[tex]\phi= cos^{-1}\Big [\dfrac{1}{\sqrt{(2)}}\Big][/tex]
[tex]\phi= 45^0[/tex]
Thus, the magnitude of the applied stress can be computed as:
[tex]\sigma_{app} = \dfrac{\tau_{CRSS}}{cos \phi \ cos \lambda }[/tex]
[tex]\sigma_{app} = \dfrac{2.00}{cos (54.74) \ cos (45) }[/tex]
[tex]\mathbf{\sigma_{app} =4.89 \ MPa}[/tex]
Suppose you are choosing between four different desktop computers: one is an Apple Mac Intosh and the other three are PC-compatible computers that use a Pentium 4, an AMD processor (using the same compiler as the Pentium 4), and a Pentium 5 (which does not yet exist in 2004 but has the same architecture as the Pentium 4 and uses the same compiler). Which of the following statements are true?
a. The fastest computer will be the one with the highest clock rate.
b. Since all PCs use the same Intel-compatible instruction set and execute the same number of instructions for a program, the fastest PC will be the one with the highest clock rate.
c. Since AMD uses different techniques than Intel to execute instructions,they may have different CPIs. But, you can still tell which of the two Pentium-based PCs is fastest by looking at the clock rate.
d. Only by looking at the results of benchmarks for tasks similar to your workload can you get an accurate picture of likely performance.
Answer:
d.
Explanation:
1. What is the productivity rate using cycle time for the following information:
I
Type of Work – Hauling
Average Cycle Time – 35 Minutes
Truck Capacity – 25 Tons
Crew - One Driver
Productivity Factor - 0.85
System Efficiency – 55 Minutes
per
Hour
Describe how to contribute to
zero/low carbon work outcomes
within the built environment.
Answer:
day if you workout without Zero billing that means you're not sweating. Sweating you're not losing anything that means you have zero outcomes
Explanation:
What current works best when the operator
encounters magnetic arc blow?
•DCEP
•ACEN
•CC
•AC
Answer:
AC
Explanation:
One situation when alternating current would work better than direct current is if the operator is encountering magnetic arc blow.
Current works best when the operator encounters magnetic arc blow is AC
Magnetic arc blow is simply defined as the arc deflection due to the warping of the magnetic field that is produced by electric arc current.
This is caused as a result of the following;
- if the material being welded has residual magnetism at an intolerable level
- When the weld root is being made, and the welding current is direct current which indicates constant direction and maintains constant polarity (either positive or negative).
Since it is caused by DC(Direct Current) which means constant polarity , it means the opposite will be better which is AC(alternating current) because it means that electricity direction will be switching to and fro and as such the polarity will also be revered in response to this back and forth switch manner.Thus, Current works best when the operator encounters magnetic arc blow is AC
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Find the derivative of x
Answer:
this is your answer. if mistake don't mind.
Where does Burj Khalifa located? and how many meters?
Answer:
Burj Khalifa is located in dubai UAE at over 828m
Explanation:
828 metres
Answer:
The Burji Khalifa, known as the Burj Dubai prior to its inauguration in 2010, is a skyscraper in Dubai, United Arab Emirates. With a total hight of 829.8 m and a roof hight of 828 m, the Burji Khalifa has been the tallest structure and building in the world since its topping out in 2009.
A storm with a duration of about 24 hours resultsin the following hydrograph at a gaging station on a river. The flow was 52 cubic meters per second (cms) before the rain began. The drainage area above the gaging station is 1,450 square kilometers. Use the observed hydrograph to develop a 24-hour rainfall duration unit hydrograph for this watershed.
Time 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84
(Hours)
Flow 52 52 55 56 97 176 349 450 442 370 328 307 280 259 238
(m3/s)
Time 90 66 102 108 114 120 126 132 138 144 155 156 162 168 176
(Hours)
(m3/s) 214 193 173 145 135 124 114 107 97 86 79 66 62 58 52
Answer:
attached below
Explanation:
Given data:
Gaged flow = 52 m^3 / sec
Depth covering drainage area = 1450 km^2
Develop a 24-hour rainfall duration unit hydrograph for the watershed using observed hydorgraph
Runoff flow = gaged flow - base flows
= 52 - 52 = 0 m^3/sec
For 18 hours time duration
Direct runoff volume ( Vr ) = ∑ ( QΔt1 )
where Δt = 6
∑Q = 3666 m^3/sec
hence Vr = Δt * ∑Q = 79185600 m^3
Next we will convert the Direct runoff volume to its equivalent depth covering the drainage area
= Vr / drainage area depth
= 79185600 / 1450000000 = 5.46 cm
Next we will find the unit hydrograph flows by applying the relation below
[tex]Q_{1.0cm} = Q_{5.46cm} (\frac{1.0cm}{5.46cm} )[/tex]
where 14m^3/sec = [tex]Q_{5.6cm}[/tex]
input value back to the above relation
[tex]Q_{1.0cm} = 2.57 m^3/sec[/tex]
Attached below is The remaining part of the solution tabulated below
and A graph of the unit hydorgraph for the given watershed
Note :
Base flow total = 1560
UH total = 671.30
An air heater may be fabricated by coiling Nichrome wire and passing air in cross flow over the wire. Consider a heater fabricated from wire of diameter D=1 mm, electrical resistivity rhoe=10−6Ω⋅m, thermal conductivity k=25W/m⋅K, and emissivity ε=0.20. The heater is designed to deliver air at a temperature of T[infinity]=50∘C under flow conditions that provide a convection coefficient of h=250W/m2⋅K for the wire. The temperature of the housing that encloses the wire and through which the air flows is Tsur=50∘C. If the maximum allowable temperature of the wire is Tmax=1200∘C, what is the maximum allowable electric current I? If the maximum available voltage is ΔE=110V, what is the corresponding length L of wire that may be used in the heater and the power rating of the heater? Hint: In your solution, assume negligible temperature variations within the wire, but after obtaining the desired results, assess the validity of this assumption.
Solution :
Assuming that the wire has an uniform temperature, the equivalent convective heat transfer coefficient is given as :
[tex]$h_T= \epsilon \sigma (T_s+T_{surr})(T_s^2 +T^2_{surr})$[/tex]
[tex]$h_T= 0.20 \times 5.67 \times 10^{-8} (1473+323)(1473^2 +323^2)$[/tex]
[tex]$h_T=46.3 \ W/m^2 .K$[/tex]
The total heat transfer coefficient will be :
[tex]$h_T=(250+46.3) \ W/m^2 .K$[/tex]
[tex]$=296.3 \ W/m^2 .K$[/tex]
Now calculating the maximum volumetric heat generation :
[tex]$\dot {q}_{max}=\frac{2h_t}{r_0}(T_s-T_{\infty})$[/tex]
[tex]$\dot {q}_{max}=\frac{2\times 296.3}{0.0005}(1200-50)$[/tex]
[tex]$= 1.362 \times 10^{9} \ W/m^3$[/tex]
The heat generation inside the wire is given as :
[tex]$\dot{q} = \frac{I^2R}{V}$[/tex]
Here, R is the resistance of the wire
V is the volume of the wire
∴ [tex]$\dot{q} = \frac{I^2\left( \rho \times \frac{L}{A} \right)}{A \times L}$[/tex]
[tex]$=\frac{I^2 \rho}{\left(\frac{\pi}{4}D^2 \right)}$[/tex]
where, ρ is the resistivity.
[tex]$I_{max}= \left(\frac{\dot{q}_{max}}{\rho} \right)^{1/2} \times \frac{\pi}{4}D^2$[/tex]
[tex]$I_{max}= \left(\frac{1.36 \times 10^9}{10^{-6}} \right)^{1/2} \times \frac{3.14}{4}(1 \times 10^{-3})^2$[/tex]
= 28.96 A
Now considering the relation for the current flow through the finite potential difference.
[tex]$E=I_{max} \times R$[/tex]
[tex]$E=I_{max} \times \rho \times \frac{L}{A}$[/tex]
[tex]$L=\frac{AE}{I_{max} \ \rho}$[/tex]
[tex]$L=\frac{\frac{\pi}{4} \times (1 \times 10^{-3})^2 \times 110}{28.96 \times 10^{-6}}$[/tex]
= 2.983 m
Now calculating the power rating of the heater:
[tex]$P= E \times I_{max}$[/tex]
[tex]$P= 110 \times 28.96}$[/tex]
= 3185.6 W
= 3.1856 kW