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3 years ago

Obtain a relation for the logarithmic mean temperature difference for use in the LMTD method?

Engineering

1 answer:

kolezko [41]3 years ago

8 0

Answer:

The log mean temperature difference is:

ΔT,lm=(ΔT1-ΔT2)/㏑(ΔT1/ΔT2)

Explanation:

To evaluate the equivalent average temperature difference between two fluids we consider a parallel-flow double-pipe heat exchanger (see attached diagram). The temperature of the hot and cold fluids is large at the inlet of the heat exchanger and decreases exponentially toward the outlet.

We can assume that the outer surface of the heat exchanger is well insulated and that heat transfer only occurs between the two fluids. We can also assume negligible kinetic and potential. The energy balance on each fluid can be written as the rate of heat loss from the hot fluid is equal to the rate of heat gained by the cold fluid in any section of the heat exchanger:

Q = -m,h×c,ph×dT,h (1)

where Q=rate of heat loss, m=mass flow rate, c,ph=heat capacity of the hot fluid, dT,h= differential temperature of the hot fluid

Q = m,c×c,pc×T.c (2)

where Q=rate of heat loss, m=mass flow rate, c,ph=heat capacity of the cold fluid, dT,h= differential temperature of the cold fluid

The temperature of the hot fluid change is negative and is added to make Q positive. Solving equations 1 and 2 in terms of dT:

dT.h = - Q/(m,h×c,ph)

dT.c = Q/(m,c×c,pc)

and taking the difference:

dT,h-dT,c= d(T,h - T,c) = -Q(1/(m,h×c,ph) + 1/(m,c×c,pc)) (3)

The heat transfer rate in the differential section of the heat exchanger can be expressed as:

Q = U(T,h-T,c)×dA,s (4)

where U=overall heat transfer coefficients, dA,s = differential sectional area. Substitute equation 4 into 3:

d(T,h - T,c)/(T,h - T,c) = -U×dA,s×(1/(m,h×c,ph) + 1/(m,c×c,pc)) (5)

Integrating equation 5:

㏑((T,h out - T,c out)/(T,h in - T,c in)) = -U×A,s×(1/(m,h×c,ph) + 1/(m,c×c,pc)) (6)

The first law of thermodynamics requires the rate of heat transfer from hot and cold fluid to be equal.

Q= m×c, pc×(T, c out-T, c in) (7)

Q= m×c, ph×(T,h out-T, h in) (8)

Solve equations 7 and 8 for m,c×c, pc and m,h×c, ph and substituting into equation 6:

Q = U×A,s×ΔT,lm

Where the log mean temperature difference is:

ΔT,lm=(ΔT1-ΔT2)/㏑(ΔT1/ΔT2)

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A cylindrical specimen of a titanium alloy having an elastic modulus of 107 GPa (15.5 × 106 psi) and an original diameter of 3.7

Keith_Richards [23]

Answer:

the maximum length of specimen before deformation is found to be 235.6 mm

Explanation:

First, we need to find the stress on the cylinder.

Stress = σ = P/A

where,

P = Load = 2000 N

A = Cross-sectional area = πd²/4 = π(0.0037 m)²/4

A = 1.0752 x 10^-5 m²

σ = 2000 N/1.0752 x 10^-5 m²

σ = 186 MPa

Now, we find the strain (∈):

Elastic Modulus = Stress / Strain

E = σ / ∈

∈ = σ / E

∈ = 186 x 10^6 Pa/107 x 10^9 Pa

∈ = 1.74 x 10^-3 mm/mm

Now, we find the original length.

∈ = Elongation/Original Length

Original Length = Elongation/∈

Original Length = 0.41 mm/1.74 x 10^-3

<u>Original Length = 235.6 mm</u>

5 0

3 years ago

2.4: Add a method called setValue(), and the description of setValue is: public int setValue(long searchKey) In this method, the

Yanka [14]

Answer:

Below is java code that must be used for the given question:

// highArray.java

// demonstrates array class with high-level interface

// to run this program: C>java HighArrayApp

////////////////////////////////////////////////////////////////

class HighArray

{

private long[] a; // ref to array a

private int nElems; // number of data items

//-----------------------------------------------------------

public HighArray(int max) // constructor

{

a = new long[max]; // create the array

nElems = 0; // no items yet

}

//-----------------------------------------------------------

public setValue find(long searchKey)

{ // find specified value

int j;

for(j=0; j<nElems; j++) // for each element,

if(a[j] == searchKey) // found item?

break; // exit loop before end

if(j == nElems) // gone to end?

return false; // yes, can't find it

else

return true; // no, found it

} // end find()

//-----------------------------------------------------------

public void insert(long value) // put element into array

{

a[nElems] = value; // insert it

nElems++; // increment size

}

//-----------------------------------------------------------

public void display() // displays array contents

{

for(int j=0; j<nElems; j++) // for each element,

System.out.print(a[j] + " "); // display it

System.out.println("");

}

//-----------------------------------------------------------

} // end class HighArray

////////////////////////////////////////////////////////////////

class HighArrayApp

{

public static void main(String[] args)

{

int maxSize = 100; // array size

HighArray arr; // reference to array

arr = new HighArray(maxSize); // create the array

arr.insert(77); // insert 10 items

arr.insert(99);

arr.insert(44);

arr.insert(55);

arr.insert(22);

arr.insert(88);

arr.insert(11);

arr.insert(00);

arr.insert(66);

arr.insert(33);

arr.display(); // display items

int searchKey = 35; // search for item

if( arr.find(searchKey) )

System.out.println("Found " + searchKey);

else

System.out.println("Can't find " + searchKey);

} // end main()

} // end class HighArrayApp

Explanation:

6 0

3 years ago

What is the maximal coefficient of performance of a refrigerator which cools down 10 kg of water (and then ice) to -6C. Upper he

inysia [295]

Given:

Temperature of water, $T_{1}$ = $-6^{\circ}C$ =273 +(-6) =267 K

Temperature surrounding refrigerator, $T_{2}$ = $21^{\circ}C$ =273 + 21 =294 K

Specific heat given for water, $C_{w}$ = 4.19 KJ/kg/K

Specific heat given for ice, $C_{ice}$ = 2.1 KJ/kg/K

Latent heat of fusion, $L_{fusion}$ = 335KJ/kg

Solution:

Coefficient of Performance (COP) for refrigerator is given by:

Max $COP_{refrigerator}$ = $\frac{T_{2}}{T_{2} - T_{1}}$

= $\frac{267}{294 - 267}$ = 9.89

Coefficient of Performance (COP) for heat pump is given by:

Max $COP_{heat pump}$ = $\frac{T_{1}}{T_{2} - T_{1}}$ $\frac{294}{294 - 267}$ = 10.89

6 0

4 years ago

Which of the following ranges depicts the 2% tolerance range to the full 9 digits provided?

Lyrx [107]

Answer:

the only one that meets the requirements is option C .

Explanation:

The tolerance of a quantity is the maximum limit of variation allowed for that quantity.

To find it we must have the value of the magnitude, its closest value is the average value, this value can be given or if it is not known it is calculated with the formula

x_average = ∑ $x_{i}$ / n

The tolerance or error is the current value over the mean value per 100

Δx₁ = x₁ / x_average

tolerance = | 100 -Δx₁ 100 |

bars indicate absolute value

let's look for these values for each case

x_average = (2.1700000+ 2.258571429) / 2

x_average = 2.2142857145

fluctuation for x₁

Δx₁ = 2.17000 / 2.2142857145

Tolerance = 100 - 97.999999991

Tolerance = 2.000000001%

fluctuation x₂

Δx₂ = 2.258571429 / 2.2142857145

Δx2 = 1.02

tolerance = 100 - 102.000000009

tolerance 2.000000001%

x_average = (2.2 + 2.29) / 2

x_average = 2,245

fluctuation x₁

Δx₁ = 2.2 / 2.245

Δx₁ = 0.9799554

tolerance = 100 - 97,999

Tolerance = 2.00446%

fluctuation x₂

Δx₂ = 2.29 / 2.245

Δx₂ = 1.0200445

Tolerance = 2.00445%

x_average = (2.211445 +2.3) / 2

x_average = 2.2557225

Δx₁ = 2.211445 / 2.2557225 = 0.9803710

tolerance = 100 - 98.0371

tolerance = 1.96%

Δx₂ = 2.3 / 2.2557225 = 1.024624

tolerance = 100 -101.962896

tolerance = 1.96%

x_average = (2.20144927 + 2.29130435) / 2

x_average = 2.24637681

Δx₁ = 2.20144927 / 2.24637681 = 0.98000043

tolerance = 100 - 98.000043

tolerance = 2.000002%

Δx₂ = 2.29130435 / 2.24637681 = 1.0200000017

tolerance = 2.0000002%

x_average = (2 +2,3) / 2

x_average = 2.15

Δx₁ = 2 / 2.15 = 0.93023

tolerance = 100 -93.023

tolerance = 6.98%

Δx₂ = 2.3 / 2.15 = 1.0698

tolerance = 6.97%

Let's analyze these results, the result E is clearly not in the requested tolerance range, the other values may be within the desired tolerance range depending on the required precision, for the high precision of this exercise the only one that meets the requirements is option C .

4 0

3 years ago

A) For Well A, provide a cross-section sketch that shows (i) ground elevation, (ii) casing height, (iii) depth to

Ad libitum [116K]

Don’t go on that file will give a virus! Sorry just looking out and I don’t know how to comment!

7 0

3 years ago