Statistics for Engineers and Scientists
Statistics for Engineers and Scientists
4th Edition
ISBN: 9780073401331
Author: William Navidi Prof.
Publisher: McGraw-Hill Education
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Question
Chapter 3, Problem 5SE

a.

To determine

Find the estimate of power loss.

Find the uncertainty in the estimate of power loss.

a.

Expert Solution
Check Mark

Answer to Problem 5SE

The estimate of power loss is P=1854,258±73188.9W_.

The uncertainty in the estimate of power loss is σP=73188.9W_.

Explanation of Solution

Given info:

The efficiency of the turbine is measured to be η=0.85±0.02, the head loss is measured to be H=3.71±0.1m and the flow rate is measured to be Q=60±1m3s. The specific gravity of water is γ=9,800Nm3 with negligible uncertainty. Furthermore it is given that the power loss is measured in Watts.

Calculation:

The form of the measurements of a process is,

Measuredvalue(μ)±Standard deviation(σ).

Here, for a random sample the measured value will be sample mean and the population standard deviation will be sample standard deviation.

The form of the measurements of efficiency of the turbine is η=0.85±0.02.

Here, the measured value of efficiency of the turbine is η=0.85 and the uncertainty in the efficiency of the turbine is ση=0.02.

The form of the measurements of head loss is H=3.71±0.1m.

Here, the measured value of head loss is H=3.71m and the uncertainty in the head loss is σH=0.1m.

The form of the measurements of flow rate is Q=60±1m3s.

Here, the measured value of flow rate is Q=60m3s and the uncertainty in the flow rate is σQ=1m3s.

Measured value of power loss:

The formula for power loss is P=ηγQH.

Here, η=0.85,H=3.71m and Q=60m3s.

The measured value of power loss is obtained as follows:

P=ηγQH=0.85×9,800×60×3.71=1854,258

Thus, the measured value of power loss is P=1854,258W_.

Uncertainty:

The uncertainty of a process is determined by the standard deviation of the measurements. In other words it can be said that, measure of variability of a process is known as uncertainty of the process.

Therefore, it can be said that uncertainty is simply (σ).

Standard deviation:

The standard deviation is based on how much each observation deviates from a central point represented by the mean. In general, the greater the distances between the individual observations and the mean, the greater the variability of the data set.

The general formula for standard deviation is,

s=xi2(xi)2nn1.

From the properties of uncertainties for functions of one measurement it is known that,

  • If X1,X2,...,Xn are independent measurements with uncertainties σX1,σX2,...,σXn and if U=U(X1,X2,..,Xn) is a function of X1,X2,...,Xn then the uncertainty in the variable U is σU=(UX1)2σX12+(UX2)2σX22+....+(UXn)2σXn2.

Here, η,QandH are not constants. The power loss is a function of η,QandH.

The uncertainty in the power loss is,

σP=ηγQH=(Pη)2ση2+(PQ)2σQ2+(PH)2σH2=((ηγQH)η)2ση2+((ηγQH)Q)2σQ2+((ηγQH)H)2σH2=(γQH)2×ση2+(ηγH)2×σQ2+(ηγQ)2×σH2=(2.18148×106)2×(0.02)2+(30,904.3)2×(1)2+(499,800)2×(0.1)2

              =731,88.9

Thus, the uncertainty in the power loss is σP=73188.9W_.

Estimate of the power loss:

The estimate of the measurement of a process is,

Measuredvalue(μ)±Standard deviation(σ).

Here, for a random sample the measured value will be sample mean and the population standard deviation will be sample standard deviation.

The estimate of power loss is,

P=Measured value of P±σP=1854,258±73188.9W

Thus, the estimate of the power loss is P=1854,258±73188.9W_.

b.

To determine

Find the relative uncertainty in the estimate of the power loss.

b.

Expert Solution
Check Mark

Answer to Problem 5SE

The relative uncertainty in the estimate of the power loss is σlnp=3.95%_.

Explanation of Solution

Calculation:

Relative uncertainty:

Relative uncertainty in U is the uncertainty as a fraction of the true value (mean of the measurement μU). Relative uncertainty is also called as coefficient of variation. Relative uncertainty is expressed in percentage without units.

The general formula to obtain relative uncertainty is,

σlnU=σUμU=σUU.

From part (a), the uncertainty in the estimate of power loss is σP=73188.9W and the measured value of power loss is P=1854,258Wr=0.91perμgDNA.

The relative uncertainty in the estimate of power loss is,

σlnP=σPμP=σPP=731,88.91854,258=0.0395

Thus, the relative uncertainty in estimate of power loss is σlnp=3.95%_.

c.

To determine

Find the uncertainty in the estimate of power loss when the uncertainty in η is reduced to 0.01.

Find the uncertainty in the estimate of power loss when the uncertainty in H is reduced to 0.05.

Find the uncertainty in the estimate of power loss when the uncertainty in Q is reduced to 0.5.

Compare the obtained two uncertainties.

c.

Expert Solution
Check Mark

Answer to Problem 5SE

The uncertainty in the estimate of power loss when the uncertainty in η is reduced to 0.01 is σP=63,000W_.

The uncertainty in the estimate of power loss when the uncertainty in H is reduced to 0.05 is σP=59,000W_.

The uncertainty in the estimate of power loss when the uncertainty in Q is reduced to 0.5 is σP=68,000W_.

The uncertainty in power loss is less when the uncertainty in H is reduced to 0.05.

Explanation of Solution

Calculation:

From part (a), the uncertainty in the efficiency of the turbine is ση=0.02, uncertainty in the head loss is σH=0.1m and the uncertainty in the flow rate is σQ=1m3s.

Uncertainty:

Here, η,QandH are not constants. The power loss is a function of η,QandH.

The uncertainty in the estimate of power loss is,

σP=ηγQH=(Pη)2ση2+(PQ)2σQ2+(PH)2σH2=((ηγQH)η)2ση2+((ηγQH)Q)2σQ2+((ηγQH)H)2σH2=(γQH)2×ση2+(ηγH)2×σQ2+(ηγQ)2×σH2=(2.18148×106)2×ση2+(30,904.3)2×σQ2+(499,800)2×σH2

Uncertainty in the estimate of power loss when the uncertainty in η is reduced to 0.01:

Here, ση=0.01, σH=0.1m and σQ=1m3s.

The uncertainty in the estimate of power loss when the uncertainty in η is reduced to 0.01 is,

σP=(2.18148×106)2×ση2+(30,904.3)2×σQ2+(499,800)2×σH2=(2.18148×106)2×(0.01)2+(30,904.3)2×(1)2+(499,800)2×(0.1)2=63,000

Thus, the uncertainty in the estimate of power loss when the uncertainty in η is reduced to 0.01 is σP=63,000W_.

Uncertainty in the estimate of power loss when the uncertainty in H is reduced to 0.05:

Here, ση=0.02, σH=0.05m and σQ=1m3s.

The uncertainty in the estimate of power loss when the uncertainty in H is reduced to 0.05 is,

σP=(2.18148×106)2×ση2+(30,904.3)2×σQ2+(499,800)2×σH2=(2.18148×106)2×(0.02)2+(30,904.3)2×(1)2+(499,800)2×(0.05)2=59,000

Thus, the uncertainty in the estimate of power loss when the uncertainty in H is reduced to 0.05 is σP=59,000W_.

Uncertainty in the estimate of power loss when the uncertainty in Q is reduced to 0.5:

Here, ση=0.02, σH=0.1m and σQ=0.5m3s.

The uncertainty in the estimate of power loss when the uncertainty in Q is reduced to 0.5 is,

σP=(2.18148×106)2×ση2+(30,904.3)2×σQ2+(499,800)2×σH2=(2.18148×106)2×(0.02)2+(30,904.3)2×(0.5)2+(499,800)2×(0.1)2=68,000

Thus, the uncertainty in the estimate of power loss when the uncertainty in Q is reduced to 0.5 is σP=63,000W_.

Comparison:

The uncertainty in the estimate of power loss when the uncertainty in η is reduced to 0.01 is σP=63,000W_.

The uncertainty in the estimate of power loss when the uncertainty in H is reduced to 0.05 is σP=59,000W_.

The uncertainty in the estimate of power loss when the uncertainty in Q is reduced to 0.5 is σP=68,000W_.

Here, 59,000<63,000<68,000.

Thus, the uncertainty in power loss is less when the uncertainty in H is reduced to 0.05.

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Chapter 3 Solutions

Statistics for Engineers and Scientists

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