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Ae104a10_solutions_2

Ae104a10_solutions_2 - Ae/APh 104a Homework Solution Set#2...

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Ae/APh 104a Homework Solution Set #2 Issued: October 12, 2010 1/3 Hot wire (i) (a) To find the relationship between the mean voltage, E w , the mean fluid temperature T a and the mean flow velocity U , substitute for h in equation (2) to get E 2 w R w = πd w l ( T w - T a )( A + BU n ) and E w = p R w πd w l ( T w - T a )( A + BU n ). Find expressions for (b) the sensitivity, S u , of the fluctuating voltage output from the wire seg- ment, e w , to a small fluctuation in velocity, Δ U The sensitivity S u relates small changes in the voltage to small changes in ve- locity, i.e. e w As given in lecture notes: S u = ∂E w ∂U = 0 . 5 q R w πd w l ( T w - T a )( A + BU n ) BnU n - 1 ( T w - T a ) S u = 0 . 5 BnU n - 1 q R w πd w l ( T w - T a ) A + BU n = 0 . 5 BnU n - 1 E w A + BU n . (c) the sensitivity, S t , of the fluctuating voltage output from the wire seg- ment, e w , to a small fluctuation in fluid temperature, Δ T a , S T = ∂E w ∂T a = - 0 . 5 q R w πd w l ( T w - T a )( A + BU n ) ( A + BU n S T = - 0 . 5 q R w πd w l ( A + BU n ) T w - T a = - 0 . 5 E w T w - T a such that e w may be written as follows: e w = S u Δ U + S t Δ T a . (1) 1
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(ii) To reduce the influence of temperature fluctuations at a given U , you need to reduce S T relative to S U , i.e. you want S T /S U 1. S T S U = - 0 . 5 E w T w - T a * 2 A + BU n BnU n - 1 E w = - 1 BnU n - 1 A + BU n T w - T a Since U is fixed and you are told that A and B are only weak functions of temperature, you can make S T /S U small by making ( T w - T a ) large, so for a given experiment (ambient temperature) you need to maximize T w and run the wire hot. You could also consider using one of the techniques discussed in class, but it is difficult to separate the effects of temperature and velocity by another means, making output correction of filtering almost impossible. High gain feedback is
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